H. Primack, U. Smilansky / Physics Reports 327 (2000) 1}107
THE QUANTUM THREE-DIMENSIONAL SINAI BILLIARD } A SEMICLASSICAL ANALYSIS
Harel PRIMACK!, Uzy SMILANSKY" !Fakulta( t fu( r Physik, Albert-Ludwigs Universita( t Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany "Department of Physics of Complex Systems, The Weizmann Institute, Rehovot 76100, Israel
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Physics Reports 327 (2000) 1}107
The quantum three-dimensional Sinai billiard } a semiclassical analysis Harel Primack!,*, Uzy Smilansky" !Fakulta( t fu(r Physik, Albert-Ludwigs Universita( t Freiburg, Hermann-Herder-Str. 3, D-79104 Freiburg, Germany "Department of Physics of Complex Systems, The Weizmann Institute, Rehovot 76100, Israel Received June 1999; editor: I. Procaccia Contents 1. Introduction 2. Quantization of the 3D Sinai billiard 2.1. The KKR determinant 2.2. Symmetry considerations 2.3. Numerical aspects 2.4. Veri"cations of low-lying eigenvalues 2.5. Comparing the exact counting function with Weyl's law 3. Quantal spectral statistics 3.1. The integrable R"0 case 3.2. Nearest-neighbour spacing distribution 3.3. Two-point correlations 3.4. Auto-correlations of spectral determinants 4. Classical periodic orbits 4.1. Periodic orbits of the 3D Sinai torus 4.2. Periodic orbits of the 3D Sinai billiard } classical desymmetrization 4.3. The properties and statistics of the set of periodic orbits 4.4. Periodic orbit correlations 5. Semiclassical analysis 5.1. Semiclassical desymmetrization 5.2. Length spectrum 5.3. A semiclassical test of the quantal spectrum
4 10 10 12 15 17 18 19 19 23 25 28 28 29 32 35 41 47 48 51 53
5.4. Filtering the bouncing-balls I: Dirichlet}Neumann di!erence 5.5. Filtering the bouncing-balls II: mixed boundary conditons 6. The accuracy of the semiclassical energy spectrum 6.1. Measures of the semiclassical error 6.2. Numerical results 7. Semiclassical theory of spectral statistics 8. Summary Acknowledgements Appendix A. E$cient quantization of billiards: BIM vs. full diagonalization Appendix B. Symmetry reduction of the numerical e!ort in the quantization of billiards Appendix C. Resummation of D using the LM Ewald summation technique Appendix D. `Physicala Ewald summation of GT(q) 0 Appendix E. Calculating D(3) 00 Appendix F. The `cubic harmonicsa >(c) LJK F.1. Calculation of the transformation coe$cients a(L) cJK,M F.2. Counting the >(c) 's LJ
* Corresponding author. Tel.: #49-761-203-7622; fax: #49-761-203-7629. E-mail addresses:
[email protected] (H. Primack),
[email protected] (U. Smilansky) 0370-1573/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 0 9 3 - 9
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H. Primack, U. Smilansky / Physics Reports 327 (2000) 1}107 Appendix G. Evaluation of l(q ) p G.1. Proof of Eq. (10) G.2. Calculating l(q ) p Appendix H. Number-theoretical degeneracy of the cubic lattice H.1. First moment H.2. Second moment
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Appendix I. Weyl's law Appendix J. Calculation of the monodromy matrix J.1. The 3D Sinai torus case J.2. The 3D Sinai billiard case Note added in proof References
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Abstract We present a comprehensive semiclassical investigation of the three-dimensional Sinai billiard, addressing a few outstanding problems in `quantum chaosa. We were mainly concerned with the accuracy of the semiclassical trace formula in two and higher dimensions and its ability to explain the universal spectral statistics observed in quantized chaotic systems. For this purpose we developed an e$cient KKR algorithm to compute an extensive and accurate set of quantal eigenvalues. We also constructed a systematic method to compute millions of periodic orbits in a reasonable time. Introducing a proper measure for the semiclassical error and using the quantum and the classical databases for the Sinai billiards in two and three dimensions, we concluded that the semiclassical error (measured in units of the mean level spacing) is independent of the dimensionality, and diverges at most as log +. This is in contrast with previous estimates. The classical spectrum of lengths of periodic orbits was studied and shown to be correlated in a way which induces the expected (random matrix) correlations in the quantal spectrum, corroborating previous results obtained in systems in two dimensions. These and other subjects discussed in the report open the way to extending the semiclassical study to chaotic systems with more than two freedoms. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 05.45.#b; 03.65.Sq Keywords: Quantum chaos; Billiards; Semiclassical approximation; Gutzwiller trace formula
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1. Introduction The main goal of `quantum chaosa is to unravel the special features which characterize the quantum description of classically chaotic systems [1,2]. The simplest time-independent systems which display classical chaos are two-dimensional, and therefore most of the research in the "eld focused on systems in 2D. However, there are very good and fundamental reasons for extending the research to higher number of dimensions. The present paper reports on our study of a paradigmatic three-dimensional system: The 3D Sinai billiard. It is the "rst analysis of a system in 3D which was carried out in depth and detail comparable to the previous work on systems in 2D. The most compelling motivation for the study of systems in 3D is the lurking suspicion that the semiclassical trace formula [2] } the main tool for the theoretical investigations of quantum chaos } fails for d'2, where d is the number of freedoms. The grounds for this suspicion are the following [2]. The semiclassical approximation for the propagator does not exactly satisfy the time-dependent SchroK dinger equation, and the error is of order +2 independently of the dimensionality. The semiclassical energy spectrum, which is derived from the semiclassical propagator by a Fourier transform, is therefore expected to deviate by O(+2) from the exact spectrum. On the other hand, the mean spacing between adjacent energy levels is proportional to +d [3] for systems in d dimensions. Hence, the "gure of merit of the semiclassical approximation, which is the expected error expressed in units of the mean spacing, is O(+2~d), which diverges in the semiclassical limit +P0 when d'2! If this argument were true, it would have negated our ability to generalize the large corpus of results obtained semiclassically, and checked for systems in 2D, to systems of higher dimensions. Amongst the primary victims would be the semiclassical theory of spectral statistics, which attempts to explain the universal features of spectral statistics in chaotic systems and its relation to random matrix theory (RMT) [4,5]. RMT predicts spectral correlations on the range of a single spacing, and it is not likely that a semiclassical theory which provides the spectrum with an uncertainty which exceeds this range, can be applicable or relevant. The available term by term generic corrections to the semiclassical trace formula [6}8] are not su$cient to provide a better estimate of the error in the semiclassically calculated energy spectrum. To assess the error, one should substitute the term by term corrections in the trace formula or the spectral f function which do not converge in the absolute sense on the real energy axis. Therefore, to this date, this approach did not provide an analytic estimate of the accuracy of the semiclassical spectrum. Under these circumstances, we initiated the present work which addressed the problem of the semiclassical accuracy using the approach to be described in the sequel. Our main result is that in contrast with the estimate given above, the semiclassical error (measured in units of the mean spacing) is independent of the dimensionality. Moreover, a conservative estimate of the upper bound for its possible divergence in the semiclassical limit is O(Dlog +D). This is a very important conclusion. It allows one to extend many of the results obtained in the study of quantum chaos in 2D to higher dimensions, and justi"es the use of the semiclassical approximation to investigate special features which appear only in higher dimensions. We list a few examples of such e!ects: f The dual correspondence between the spectrum of quantum energies and the spectrum of actions of periodic orbits [9}11] was never checked for systems in more than 2D. However, if the universality of the quantum spectral correlations is independent of the number of freedoms, the corresponding range of correlations in the spectrum of classical actions is expected to depend on
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the dimensionality. Testing the validity of this prediction, which is derived by using the trace formula, is of great importance and interest. It will be discussed at length in this work. f The full range of types of stabilities of classical periodic orbits that includes also the loxodromic stability [2] can be manifest only for d'2. f Arnold's di!usion in the KAM regime is possible only for d'2 (even though we do not encounter it in this work). Having stated the motivations and background for the present study, we shall describe the strategy we chose to address the problem, and the logic behind the way we present the results in this report. The method we pursued in this "rst exploration of quantum chaos in 3D, was to perform a comprehensive semiclassical analysis of a particular yet typical system in 3D, which has a wellstudied counterpart in 2D. By comparing the exact quantum results with the semiclassical theory, we tried to identify possible deviations which could be attributed to particular failures of the semiclassical approximation in 3D. The observed deviations, and their dependence on + and on the dimensionality, were used to assess the semiclassical error and its dependence on +. Such an approach requires the assembly of an accurate and complete databases for the quantum energies and for the classical periodic orbits. This is a very demanding task for chaotic systems in 3D, and it is the main reason why such studies were not performed before. When we searched for a convenient system for our study, we turned immediately to billiards. They are natural paradigms in the study of classical and quantum chaos. The classical mechanics of billiards is simpler than for systems with potentials: The energy dependence can be scaled out, and the system can be characterized in terms of purely geometric data. The dynamics of billiards reduces to a mapping through the natural PoincareH section which is the billiard's boundary. Much is known about classical billiards in the mathematical literature (e.g. [12]), and this information is crucial for the semiclassical application. Billiards are also very convenient from the quantal point of view. There are specialized methods to quantize them which are considerably simpler than those for potential systems [13]. Some of them are based on the boundary integral method (BIM) [14], the KKR method [15], the scattering approach [16,17] and various improvements thereof [18}20]. The classical scaling property is manifest also quantum mechanically. While for potential systems the energy levels depend in a complicated way on + and the classical actions are non-trivial functions of E, in billiards, both the quantum energies and the classical actions scale trivially in + and JE, respectively, which simpli"es the analysis considerably. The particular billiard we studied is the 3D Sinai billiard. It consists of the free space between a 3-torus of side S and an inscribed sphere of radius R, where 2R(S. It is the natural extension of the familiar 2D Sinai billiard, and it is shown in Fig. 1 using three complementary representations. The classical dynamics consists of specular re#ections from the sphere. If the billiard is desymmetrized, specular re#ections from the symmetry planes exist as well. The 3D Sinai billiard has several advantages. It is one of the very few systems in 3D which are rigorously known to be ergodic and mixing [21}23]. Moreover, since its introduction by Sinai and his proof of its ergodicity [21], the 2D Sinai billiard was subject to thorough classical, quantal and semiclassical investigations [15,17,21,24}27]. Therefore, much is known about the 2D Sinai billiard and this serves us as an excellent background for the study of the 3D counterpart. The symmetries of the 3D Sinai billiard greatly facilitate the quantal treatment of the billiard. Due to the spherical symmetry of the inscribed obstacle and the cubic-lattice symmetry of the billiard (see Fig. 1(c)) we are able to use the
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Fig. 1. Three representations of the 3D Sinai billiard: (a) original, (b) 48-fold desymmetrized (maximal desymmetrization) into the fundamental domain, (c) unfolded to R3.
KKR method [15,28}30] to numerically compute the energy levels. This method is superior to the standard methods of computing generic billiard's levels. In fact, had we used the standard methods with our present computing resources, it would have been possible to obtain only a limited number of energy levels with the required precision. The KKR method enabled us to compute many thousands of energy levels of the 3D Sinai billiard. The fact that the billiard is symmetric means that the Hamiltonian is block-diagonalized with respect to the irreducible representations of the symmetry group [31]. Each block is an independent Hamiltonian which corresponds to the desymmetrized billiard (see Fig. 1(b)) for which the boundary conditions are determined by the irreducible representations. Hence, with minor changes one is able to compute a few independent spectra that correspond to the same 3D desymmetrized Sinai billiard but with di!erent boundary conditions } thus one can easily accumulate data for spectral statistics. On the classical level, the 3D Sinai billiard has the great advantage of having a symbolic dynamics. Using
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Fig. 2. Some bouncing-ball families in the 3D Sinai billiard. Upper "gure: Three families parallel to the x, y and z axis. Lower "gure: top view of two families.
the centers of spheres which are positioned on the in"nite Z3 lattice as the building blocks of this symbolic dynamics, it is possible to uniquely encode the periodic orbits of the billiard [27,32]. This construction, together with the property that periodic orbits are the single minima of the length (action) function [27,32], enables us to systematically "nd all of the periodic orbits of the billiard, which is crucial for the application of the semiclassical periodic orbit theory. We emphasize that performing a systematic search of periodic orbits of a given billiard is far from being trivial (e.g. [2,33}36]) and there is no general method of doing so. The existence of such a method for the 3D Sinai billiard was a major factor in favour of this system. The advantages of the 3D Sinai billiard listed above are gained at the expense of some problematic features which emerge from the cubic symmetry of the billiard. In the billiard there exist families of periodic, neutrally stable orbits, the so called `bouncing-balla families that are illustrated in Fig. 2. The bouncing-ball families are well-known from studies of, e.g., the 2D Sinai and the stadium billiards [15,17,37,38]. These periodic manifolds have zero measure in phase space (both in 2D and in 3D), but nevertheless strongly in#uence the dynamics. They are
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responsible for the long (power-law) tails of some classical distributions [39,40]. They are also responsible for non-generic e!ects in the quantum spectral statistics, e.g., large saturation values of the number variance in the 2D Sinai and stadium billiards [37]. The most dramatic visualization of the e!ect of the bouncing-ball families appears in the function D(l),+ cos(k l) } the `quantal n n length spectruma. The lengths l that correspond to the bouncing-ball families are characterized by large peaks that overwhelm the generic contributions of unstable periodic orbits [38] (as is exempli"ed by Fig. 28). In the 3D Sinai billiard the undesirable e!ects are even more apparent than for the 2D billiard. This is because, in general, the bouncing balls occupy 3D volumes rather than 2D areas in con"guration space and consequently their amplitudes grow as k1 (to be contrasted with k0 for unstable periodic orbits). Moreover, for R(S/2 there is always an in"nite number of families present in the 3D Sinai billiard compared to the "nite number which exists in the 2D Sinai and the stadium billiards. The bouncing balls are thoroughly discussed in the present work, and a large e!ort was invested in devising methods by which their e!ects could be "ltered out. After introducing the system to be studied, we shall explain now the way by which we present the results. The semiclassical analysis is based on the exact quantum spectrum, and on the classical periodic orbits. Hence, the "rst sections are dedicated to the discussion of the exact quantum and classical dynamics in the 3D Sinai billiard, and the semiclassical analysis is deferred to the last sections. The sections are grouped as follows: f Quantum mechanics and spectral statistics (Sections 2 and 3). f Classical periodic orbits (Section 4). f Semiclassical analysis (Sections 5}7). In Section 2 we describe the KKR method which was used to numerically compute the quantum spectrum. Even though it is a rather technical section, it gives a clear idea of the di$culties encountered in the quantization of this system, and how we used symmetry considerations and number-theoretical arguments to reduce the numerical e!ort considerably. The desymmetrization of the billiard according to the symmetry group is worked out in detail. This section ends with a short explanation of the methods used to ensure the completeness and the accuracy of the spectrum. The study of spectral statistics, Section 3, starts with the analysis of the integrable billiard (R"0) case. This spectrum is completely determined by the underlying classical bouncing-ball manifolds which are classi"ed according to their dimensionality. The two-point form factor in this case is not Poissonian, even though the system is integrable. Rather, it re#ects the number-theoretical degeneracies of the Z3 lattice resulting in non-generic correlations. Turning to the chaotic (R'0) cases, we investigate some standard statistics (nearest-neighbour, number variance) as well as the auto-correlations of the spectral determinant, and compare them to the predictions of RMT. The main conclusion of this section is that the spectral #uctuations in the 3D Sinai billiard belong to the same universality class as in the 2D analogue. Section 4 is devoted to the systematic search of the periodic orbits of the 3D Sinai billiard. We rely heavily on a theorem that guarantees the uniqueness of the coding and the variational minimality of the periodic orbit lengths. The necessary generalizations for the desymmetrized billiard are also explained. Once the algorithm for the computation of periodic orbits is outlined, we turn to the de"nition of the spectrum of lengths of periodic orbits and to the study of its statistics. The number of periodic orbits with lengths smaller than ¸ is shown to proliferate
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exponentially. We check also classical sum rules which originate from ergodic coverage, and observe appreciable corrections to the leading term due to the in"nite horizon of the Sinai billiard. Turning our attention to the two-point statistics of the classical spectrum, we show that it is not Poissonian. Rather, there exist correlations which appear on a scale larger than the nearest spacing. This has very important consequences for the semiclassical analysis of the spectral statistics. We study these correlations and o!er a dynamical explanation for their origin. The semiclassical analysis of the billiard is the subject of Section 5. As a prelude, we propose and use a new method to verify the completeness and accuracy of the quantal spectrum, which is based on a `universala feature of the classical length spectrum of the 3D Sinai billiard. The main purpose of this section is to compare the quantal computations to the semiclassical predictions according to the Gutzwiller trace formula, as a "rst step in our study of its accuracy. Since we are interested in the generic unstable periodic orbits rather than the non-generic bouncing balls, special e!ort is made to eliminate the e!ects of the latter. This is accomplished using a method that consists of taking the derivative with respect to a continuous parameterization of the boundary conditions on the sphere. In Section 6 we embark on the task of estimating the semiclassical error of energy levels. We "rst de"ne the measures with which we quantify the semiclassical error, and demonstrate some useful statistical connections between them. We then show how these measures can be evaluated for a given system using its quantal and semiclassical length spectra. We use the databases of the 2D and 3D Sinai billiards to derive the estimate of the semiclassical error which was already quoted above: The semiclassical error (measured in units of the mean spacing) is independent of the dimensionality, and a conservative estimate of the upper bound for its possible divergence in the semiclassical limit is O(Dlog +D). Once we are reassured of the reliability of the trace formula in 3D, we return in Section 7 to the spectral statistics of the quantized billiard. The semiclassical trace formula is interpreted as an expression of the duality between the quantum spectrum and the classical spectrum of lengths. We show how the length correlations in the classical spectrum induce correlations in the quantum spectrum, which reproduce rather well the RMT predictions. The work is summarized in Section 8. To end the introductory notes, a review of the existing literature is in order. Only very few systems in 3D were studied in the past. We should "rst mention the measurements of 3D acoustic cavities [41}45] and electromagnetic (microwaves) cavities [46}49]. The measured frequency spectra were analysed and for irregular shapes (notably the 3D Sinai billiard) the level statistics conformed with the predictions of RMT. Moreover, the length spectra showed peaks at the lengths of periodic manifolds, but no further quantitative comparison with the semiclassical theory was attempted. However, none of the experiments is directly relevant to the quantal (scalar) problem since the acoustic and electromagnetic vector equations cannot be reduced to a scalar equation in the con"gurations chosen. Therefore, these experiments do not constitute a direct analogue of quantum chaos in 3D. This is in contrast with #at and thin microwave cavities which are equivalent (up to some maximal frequency) to 2D quantal billiards. A few 3D billiards were discussed theoretically in the context of quantum chaos. Polyhedral billiards in the 3D hyperbolic space with constant negative curvature were investigated by Aurich and Marklof [50]. The trace formula in this case is exact rather than semiclassical, and thus the issue of the semiclassical accuracy is not relevant. Moreover, the tetrahedral that was treated had
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exponentially growing multiplicities of lengths of classical periodic orbits, and hence cannot be considered as generic. Prosen considered a 3D billiard with smooth boundaries and 48-fold symmetry [19,20] whose classical motion was almost completely (but not fully) chaotic. He computed many levels and found that level statistics reproduce the RMT predictions with some deviations. He also found agreement with Weyl's law (smooth density of states) and identi"ed peaks of the length spectrum with lengths of periodic orbits. The majority of high-lying eigenstates were found to be uniformly extended over the energy shell, with notable exceptions that were `scarreda either on a classical periodic orbit or on a symmetry plane. Henseler et al. treated the N-sphere scattering systems in 3D [51] in which the quantum mechanical resonances were compared to the predictions of the Gutzwiller trace formula. A good agreement was observed for the uppermost band of resonances and no agreement for other bands which are dominated by di!raction e!ects. Unfortunately, conclusive results were given only for non-generic con"gurations of two and three spheres for which all the periodic orbits are planar. In addition, it is not clear whether one can infer from the accuracy of complex scattering resonances to the accuracy of real energy levels in bound systems. Recently, Sieber [52] calculated the 4]4 stability (monodromy) matrices and the Maslov indices for general 3D billiards and gave a practical method to compute them, which extended our previous results for the 3D Sinai billiard [53,54]. (See also Note added in proof.) 2. Quantization of the 3D Sinai billiard In the present section we describe the KKR determinant method [28}30,55] to compute the energy spectrum of the 3D Sinai billiard, and the results of the numerical computations. The KKR method, which was used by Berry for the 2D Sinai billiard case [15], is most suitable for our purpose since it allows to exploit the symmetries of the billiard to reduce the numerical e!ort considerably. The essence of the method is to convert the SchroK dinger equation and the boundary conditions into a single integral equation. The spectrum is then the set of real wavenumbers k where the corresponding secular determinant vanishes. As a matter of fact, we believe that only n with the KKR method could we obtain a su$ciently accurate and extended spectrum for the quantum 3D Sinai billiard. We present in this section also some numerical aspects and verify the accuracy and completeness of the computed levels. We go into the technical details of the quantal computation because we wish to show the high reduction factor which is gained by the KKR method. Without this signi"cant reduction the numerical computation would have resulted in only a very limited number of levels [46,48]. The reader who is not interested in these technical details should proceed to Section 2.4. To avoid ambiguities, we strictly adhere to the conventions in [56]. 2.1. The KKR determinant We "rst consider the 3D `Sinai torusa, which is the free space outside of a sphere of radius R embedded in a 3-torus of side length S (see Fig. 1). The SchroK dinger equation of an electron of mass m and energy E is reduced to the Helmholtz equation: +2t#k2t"0,
k,J2mE/+ .
(1)
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The boundary conditions on the sphere are taken to be the general linear (self-adjoint) conditions: (2) i cos a ) t#sin a ) R ( t"0 , n where n( is the normal pointing outside the billiard, i is a parameter with dimensions of k, and a3[0, p/2] is an angle that interpolates between Dirichlet (a"0) and Neumann (a"p/2) conditions. These `mixeda boundary conditions will be needed in Section 5 when dealing with the semiclassical analysis. Applying the KKR method, we obtain the following quantization condition (see [54] for a derivation and for details): det[A (k)#kP (kR; i, a)d d ]"0, l, l@"0, 1, 2,2, !l4m4l, !l@4m@4l@ , lm,l{m{ l ll{ mm{ (3) where k is the wavenumber under consideration and A (k),4pil~l{ + i~LC D (k), ¸"0, 1, 2,2, M"!¸,2, ¸ , (4) lm,l{m{ LM,lm,l{m{ LM LM 1 D (k),(!ik) + h`(kSo)>H (Xq )# d , (5) LM L LM J4p L0 q Z3 M0N | @ p 2p C , dh d/> (h, /)>H (h, /)> (h, /) , (6) LM,lm,l{m{ LM lm l{m{ 0 0 iR cos a ) n (kR)!kR sin a ) n@ (kR) l l (7) P (kR; i, a), l iR cos a ) j (kR)!kR sin a ) j@ (kR) l l "cot[g (kR; i, a)] . (8) l In the above j , n , h` are the spherical Bessel, Neumann and Hankel functions, respectively [56], l l l > are the spherical harmonics [56] with argument Xq in the direction of q, and g are the lm l scattering phase shifts from the sphere, subject to the boundary conditions (2). The physical input to the KKR determinant is distributed in a systematic way: The terms A (k) contain information only about the structure of the underlying Z3 lattice, and are lm,l{m{ independent of the radius R of the inscribed sphere. Hence they are called the `structure functionsa [28,30]. Moreover, they depend on a smaller number of `building blocka functions D (k) which LM contain the in"nite lattice summations. The diagonal term kP (kR)d d contains the information l ll{ mm{ about the inscribed sphere, and is expressed in terms of the scattering phase shifts from the sphere. This elegant structure of the KKR determinant (3) prevails in more general situations and remains intact even if the Z3 lattice is replaced by a more general one, or if the `harda sphere is replaced by a `softa spherical potential with a "nite range (`mu$n-tina potential) [28}30]. This renders the KKR a powerful quantization method. In all these cases the structure functions A depend lm,l{m{ only on the underlying lattice, and the relation (8) holds with the appropriate scattering matrix. Thus, in principle, the structure functions (or rather D ) can be tabulated once for a given lattice LM (e.g. cubic) as functions of k, and only P need to be re-calculated for every realization of the l potential (e.g. changing R). This makes the KKR method very attractive also for a large class of generalizations of the 3D Sinai billiard. The determinant (3) is not yet suitable for numerical computations. This is because the lattice summations in D are only conditionally convergent and have to be resummed in order to give LM
C
P P
D
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absolutely and rapidly convergent sums. This is done using the Ewald summation technique, which is described in Appendices C}E. The further symmetry reductions of the KKR determinant, which are one of the most important advantages of this method, are discussed in the following. 2.2. Symmetry considerations As can be seen from Eqs. (4)}(8) and from Appendix C, the main computational e!ort involved in computing the KKR determinant is consumed in the lattice sums D (k) which need to be LM evaluated separately for every k. Therefore, it is imperative to use every possible means to economize the computational e!ort invested in calculating these functions. For this purpose, we shall exploit the cubic symmetry of the 3D Sinai billiard as well as other relations that drastically reduce the computational e!ort. 2.2.1. Group-theoretical resummations For the practical (rapidly convergent) computation, the functions D are decomposed into LM three terms which are given in Appendix C (see also Appendix D). Eqs. (C.16)}(C.19) express D(2) as a sum over the direct cubic lattice, whereas, D(1) is a sum over the reciprocal cubic lattice, LM LM which is also a cubic lattice. Thus, both sums can be represented as D(j) (k)" + f (j)(o; k)>H (Xq ), j"1, 2 . LM LM q Z3 | We show in Appendix G that lattice sums of this kind can be rewritten as
(9)
f (j)(o ; k) p + >H (X ( qp ) , (10) LM g l(q ) ( p g|Oh p where O is the cubic symmetry group [31], and q ,(i , i , i ) resides in the fundamental section h p 1 2 3 04i 4i 4i . The terms l(q ) are integers which are explicitly given in Appendix G. The inner 1 2 3 p sums are independent of k, and can thus be tabulated once for all. Hence the computation of the k dependent part becomes 48 times more e$cient (for large, "nite lattices) when compared to (9) due to the restriction of q to the fundamental section. p A further reduction can be achieved by a unitary transformation from the M> N basis to the LM more natural basis of the irreducible representations (irreps) of O : h D(j) (k)"+ LM q
>(c) (X),+ a(L)H > (X) , (11) LJK cJK,M LM M where c3[1,2,10] denotes the irrep under consideration, J counts the number of the inequivalent irreps c contained in ¸, and K"1,2, dim(c) is the row index within the irrep. The functions >(c) are known as the `cubic harmonicsa [57]. Combining (10) and (11), and using the unitarity of LJK the transformation as well as the `great orthonormality theorema of group theory [31] we arrive at D(j) (k)"+ a(L)H D(j) (k) , sJ,M LJ LM J f (j)(o ; k) p >(s)H(Xq ) . D(j) (k)"48+ p LJ LJ l(q ) q p p
(12) (13)
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The superscript (s) denotes the totally symmetric irrep and the subscript k was omitted since s is one-dimensional. The constant coe$cients a(L)H can be taken into the (constant) coe$cients sJ,M C resulting in LM,lm,l{m{ A (k)"4pil~l{+ i~LD (k)C , lm,l{m{ LJ LJ,lm,l{m{ LJ D (k)"D(1)(k)#D(2)(k)#D(3) (k)d , LJ LJ LJ 00 L0
(14) (15)
C "+ a(L)H C . (16) LJ,lm,l{m{ sJ,M LM,lm,l{m{ M We show in Appendix F that for large ¸ the number of D (k)'s is smaller by a factor +1/48 LJ than the number of D (k)'s. This means that the entries of the KKR determinant are now LM computed using a substantially smaller number of building blocks for which lattice summations are required. Thus, in total, we save a saving factor of 482"2304 over the more naive scheme (4)}(6). 2.2.2. Number-theoretical resummations In the above we grouped together lattice vectors with the same magnitude, using the geometrical symmetries of the cubic lattice. One can gain yet another reduction factor in the computational e!ort by taking advantage of a phenomenon which is particular to the cubic lattice and stems from number theory. The lengths of lattice vectors in the fundamental sector show an appreciable degeneracy, which is not connected with the O symmetry. For example, the lattice vectors (5, 6, 7) h and (1, 3, 10) have the same magnitude, J110, and are not geometrically conjugate by O . This h number-theoretical degeneracy is both frequent and signi"cant, and we use it in the following way. Since the square of the magnitude is an integer we can write
C
D
= 48 D(j) (k)" + f (j)(o "Jn; k) + >(s)H(Xqp ) . (17) LJ p LJ l(q ) 2 n/1 op /n p The inner sums incorporate the number-theoretical degeneracies. They are k independent, and therfore can be tabulated once for all. To show the e$ciency of (17) let us restrict our lattice summation to o 4o (which we always p .!9 do in practice). For large o the number of lattice vectors in the fundamental domain is po3 /36, .!9 .!9 and the number of summands in (17) is at most o2 . Thus, the saving factor is at least po /36. In .!9 .!9 fact, as shown in Appendix H, there are only (asymptotically) (5/6)o2 terms in (17), which sets the .!9 saving factor due to number-theoretical degeneracy to be po /30. In practice, o "O(100) and .!9 .!9 this results in a reduction factor of about 10, which is signi"cant. 2.2.3. Desymmetrization The symmetry of the 3D Sinai torus implies that the wavefunctions can be classi"ed according to the irreps of O [31]. Geometrically, each such irrep corresponds to speci"c boundary conditions h on the symmetry planes that de"ne the desymmetrized 3D Sinai billiard (see Fig. 1). This allows us to `desymmetrizea the billiard, that is to restrict ourselves to the fundamental domain with speci"c boundary conditions instead of considering the whole 3-torus. We recall that the boundary conditions on the sphere are determined by P (k) and are independent of the irrep under l
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consideration. For simplicity, we shall restrict ourselves to the two simplest irreps which are both one-dimensional: c"a: This is the totally antisymmetric irrep, which corresponds to Dirichlet boundary conditions on the planes. c"s: This is the totally symmetric irrep, which corresponds to Neumann boundary conditions on the planes. The implementation of this desymmetrization is straightforward (see [54] for details) and results in a new secular equation: det[A(c) (k)#kP (kR)d d ]"0 , lj,l{j{ l ll{ jj{
(18)
where c is the chosen irrep and A(c) (k)"4pil~l{+ i~LD (k)C(c) , lj,l{j{ LJ LJ,lj,l{j{ LJ
(19)
C(c) " + a(l) a(l{)H C LJ,lj,l{j{ cj,m cj{,m{ LJ,lm,l{m{ mm{
(20)
. (21) " + a(L)H a(l) a(l{)H C sJ,M cj,m cj{,m{ LM,lm,l{m{ Mmm{ The desymmetrization of the problem has a few advantages: Computational ezciency: In Appendix F we show that for large ¸'s the number of cubic harmonics >(c) that belong to a one-dimensional irrep is 1/48 of the number of the spherical LJK harmonics > . Correspondingly, if we truncate our secular determinant such that ¸4¸ , then LM .!9 the dimension of the new determinant (18) is only 1/48 of the original one (3) for the fully symmetric billiard. Indeed, the desymmetrized billiard has only 1/48 of the volume of the symmetric one, and hence the density of states is reduced by 48 (for large k). However, due to the high cost of computing a determinant (or performing a singular-value decomposition) [58] the reduction in the density of states is over-compensated by the reduction of the matrix size, resulting in a saving factor of 48. This is proven in Appendix B, where it is shown in general that levels of desymmetrized billiards are computationally cheaper than those of billiards which possess symmetries. Applied to our case, the computational e!ort to compute a given number N of energy levels of the desymmetrized billiard is 48 times cheaper than computing N levels of the fully symmetric billiard. Statistical independence of spectra: The spectra of di!erent irreps are statistically independent since they correspond to di!erent boundary conditions. Thus, if the fully symmetric billiard is quantized, the resulting spectrum is the union of 10 independent spectra (there are 10 irreps of O h [31]), and signi"cant features such as level rigidity will be severely blurred [59]. To observe generic statistical properties and to compare with the results of RMT, one should consider each spectrum separately, which is equivalent to desymmetrizing the billiard. Rigidity: The statistical independence has important practical consequences. Spectral rigidity implies that it is unlikely to "nd levels in close vicinity of each other. Moreover, the #uctuations in the spectral counting functions are bounded. Both features of rigidity are used in the numerical algorithm which computes the spectrum, as is described in more detail in Section 2.3.
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To summarize this subsection, we have demonstrated that the high symmetry features of the 3D Sinai billiard are naturally incorporated in the KKR method. This renders the computation of its spectrum much more e$cient than in the case of other, less symmetric 3D billiards. Thus, we expect to get many more levels than the few tens that can be typically obtained for generic billiards [46,48]. In fact, this feature is the key element which brought this project to a successful conclusion. We note that other specialized computation methods, which were applied to highly symmetric 3D billiards, also resulted in many levels [19,20]. This completes the theoretical framework established for the e$cient numerical computation of the energy levels. In the following we discuss the outcome of the actual computations.
2.3. Numerical aspects We computed various energy spectra, de"ned by di!erent combinations of the physically important parameters: 1. The radius R of the inscribed sphere (the side S was always taken to be 1). 2. The boundary conditions on the sphere: Dirichlet/Neumann/mixed: 04a4p/2. 3. The boundary conditions on the symmetry planes of the cube: Dirichlet/Neumann. These boundary conditions correspond to the antisymmetric/symmetric irrep of O , respectively. Due h to the lattice periodicity, Dirichlet (Neumann) boundary conditions on the symmetry planes induce Dirichlet (Neumann) also on the planes between neighbouring cells. The largest spectral stretch that was obtained numerically corresponded to R"0.2 and Dirichlet boundary condition everywhere. It consisted of 6697 levels in the interval 0(k4281.078. We denote this spectrum in the following as the `longest spectruma. The practical application of (18) brings about many potential sources of divergence: The KKR matrix is in"nite-dimensional in principle, and each of the elements is given as an in"nite sum over the cubic lattice. To regulate the in"nite dimension of the matrix we use a physical guideline, namely, the fact that for l'kR the phase shifts decrease very rapidly toward zero, and the matrix becomes essentially diagonal. Therefore, a natural cuto! is l "kR, which is commonly used (e.g. .!9 [17]). In practice, one has to go slightly beyond this limit, and to allow a few evanescent modes: l "kR#l . To "nd a suitable value of l we used the parameters of the longest spectrum .!9 %7!/ %7!/ and computed the 17 eigenvalues in the interval 199.5(k(200 with l "0, 2, 4, 6, 8, 10 (l has %7!/ .!9 to be odd). We show in Fig. 3 the successive deviations of the computed eigenvalues between consecutive values of l . The results clearly indicate a 10-fold increase in accuracy with each %7!/ increase of l by 2. A moderately high accuracy of O(10~4) relative to level spacing requires %7!/ l "8 which was the value we used in our computations. %7!/ To regulate the in"nite lattice summations in D we used successively larger subsets of the LJ lattice. The increase was such that at least twice as many lattice points were used. Our criterion of convergence was that the maximal absolute value of the di!erence between successive computations of D was smaller than a prescribed threshold: LJ max DDi !Di`1D(e . LJ LJ LJ
(22)
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Fig. 3. Accuracy of eigenvalues as a function of the number of evanescent modes l . The case considered was R"0.2 %7!/ and Dirichlet boundary conditions everywhere. The "gure shows the absolute di!erences of the eigenvalues between two successive values of l , multiplied by the smooth level density. That is, `0}2a means dM (k )Dk (l "2)!k (l "0)D %7!/ n n %7!/ n %7!/ ,D*N D. We show 17 eigenvalues in the interval 199.5(k(200. n
The threshold e"10~6 was found to be satisfactory, and we needed to use a sub-lattice with maximal radius of 161. The KKR program is essentially a loop over k which sweeps the k-axis in a given interval. At each step the KKR matrix M(k) is computed, and then its determinant is evaluated. In principle, eigenvalues are obtained whenever the determinant vanishes. In practice, however, the direct evaluation of the determinant su!ers from a few drawbacks: f The numerical algorithms that are used to compute det M(k) are frequently unstable. Hence, it is impossible to use them beyond some critical k which is not very large. f For moderately large k's, the absolute values of det M(k) are very small numbers that result in computer under#ows (in double precision mode), even for k-values which are not eigenvalues. f Due to "nite precision and rounding errors, det M(k) never really vanishes for eigenvalues. A superior alternative to the direct calculation of the determinant is to use the singular-valuedecomposition (SVD) algorithm [58], which is stable under any circumstances. In our case, M is real and symmetric, and the output are the `singular valuesa p which are the absolute values of the i eigenvalues of M. The product of all of the singular values is equal to Ddet MD, which solves the stability problem. To cure the other two problems consider the following `conditioning measurea: $*. M(k) r(k), + ln p (k) . (23) i i/1 The use of the logarithm circumvents the under#ow problem. Moreover, we always expect some of the smallest singular values to re#ect the numerical noise, and the larger ones to be physically relevant. Near an eigenvalue, however, one of the `relevanta singular values must approach zero, resulting in a `dipa in the graph of r(k). Hence, by tracking r as a function of k, we locate its dips and take as the eigenvalues the k values for which the local minima of r are obtained. Frequently,
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one encounters very shallow dips (typically ;1) which are due to numerical noise and should be discarded. To ensure the location of all of the eigenvalues in a certain k interval, the k-axis has to be sampled densely. However, oversampling should be avoided to save computer resources. In order to choose the sampling interval *k in a reasonable way, we suggest the following. If the system is known to be classically chaotic, then we expect the quantal nearest-neighbour distribution to follow the prediction of random matrix theory (RMT) [2]. In particular, for systems with time-reversal symmetry: p P(s)+ s, s;1, s,(k !k )dM ((k #k )/2) , n`1 n n n`1 2
(24)
where dM (k) is the smooth density of states. The chance of "nding a pair of energy levels in the interval [s, s#ds] is P(s) ds. The cumulative probability of "nding a pair in [0, s] is therefore crudely given by
P
I(s)+
s
p P(s@) ds@+ s2, 4
s;1 . (25) 0 A more re"ned calculation, taking into account all the possible relative con"gurations of the pair in the interval [0, s] gives p Q(s)+ s2, s;1 . 6
(26)
If we trace the k-axis with steps *k and "nd an eigenvalue, then the chance that there is another one in the same interval *k is Q(*kdM (k)). If we prescribe our tolerance Q to lose eigenvalues, then we should choose s(Q) 1 *k" + dM (k) dM (k)
S
6Q . p
(27)
In the above, we assumed that the dips in r(k) are wide enough, such that they can be detected over a range of several *k's. If this is not the case and the dips are very sharp, we must re"ne *k. In our case dips were quite sharp, and in practice we needed to take Q of the order 10~5}10~6. 2.4. Verixcations of low-lying eigenvalues After describing some numerical aspects of the computation, we turn to various tests of the integrity and completeness of the computed spectra. In this subsection we compare the computed low-lying eigenvalues for R'0 with those of the R"0 case. In the next one we compare the computed stair-case function to Weyl's law. The theoretical background for the comparison between low-lying eigenvalues to those of the R"0 case is as follows. The lowest l value, for which there exist antisymmetric cubic harmonics, is l"9 [57]. Consequently, for cases with Dirichlet conditions on the symmetry planes, the lowest l-values in the KKR matrix is l"9. Thus, for kR(9 the terms P (kR) in equation (18) are very l small, and the matrix approximately equals the matrix obtained for an empty tetrahedron. The
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Fig. 4. The unfolded di!erences *N for the low-lying levels of the 3D Sinai billiard with R"0.2 and Dirichlet n everywhere. We indicated by the vertical line k"45 the theoretical expectation for transition from small to large *N. The line *N"0 was slightly shifted upwards for clarity. Fig. 5. N (k) for the longest spectrum of the 3D billiard. The data are smoothed over 50 level intervals. 04#
`empty tetrahedrona eigenvalues can be calculated analytically: 2p kR/0" Jl2#m2#n2, n S
0(l(m(n .
(28)
We hence expect k +kR/0 for k [9/R . (29) n n n Similar considerations were used by Berry [15] for the 2D Sinai billiard, where he also calculated the corrections to the low-lying eigenvalues. In Fig. 4 we plot the unfolded di!erence *N ,dM (k )Dk !kR/0D for the longest spectrum (R"0.2, Dirichlet everywhere). One clearly n n n n observes that indeed the di!erences are very small up to k"9/0.2"45, and they become of order 1 afterwards, as expected. This con"rms the accuracy and completeness of the low-lying levels. Moreover, it veri"es the correctness of the rather complicated computations of the terms A which are due to the cubic lattice. lj,l{j{ 2.5. Comparing the exact counting function with Weyl's law It is by now a standard practice (see e.g. [17]) to verify the completeness of a spectrum by comparing the resulting stair-case function N(k),dMk 4kN to its smooth approximation NM (k), n known as `Weyl's lawa. In Appendix I we derive Weyl's law for the 3D Sinai billiard (Eq. (I.13)), and now consider the di!erence N (k),N(k)!NM (k). Any jump of N by $1 indicates 04# 04# a redundant or missing eigenvalue. In fact, this tool is of great help in locating missing eigenvalues. In Fig. 5 we plot N for the longest spectrum. It is evident that the curve #uctuates around 0 with 04# no systematic increase/decrease trends, which veri"es the completeness of the spectrum. The
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average of N over the available k-interval is (!4)]10~4 which is remarkably smaller than any 04# single contribution to NM (note that we had no parameters to "t). This is a very convincing veri"cation for both the completeness of the spectrum as well as the accuracy of the Weyl's law (I.13). We also note that the typical #uctuations grow quite strongly with k. This is due to the e!ects of the bouncing-ball families (see Section 1) and will be discussed further in Section 3.3. 3. Quantal spectral statistics Weyl's law provides the smooth behaviour of the quantal density of states. There is a wealth of information also in the #uctuations, and their investigation is usually referred to as `spectral statisticsa. Results of spectral statistics that comply with the predictions of random matrix theory (RMT) are generally considered as a hallmark of the underlying classical chaos [2,17,24,59,60]. In the case of the Sinai billiard we are plagued with the existence of the non-generic bouncingball manifolds. They in#uence the spectral statistics of the 3D Sinai billiard. It is therefore desirable to study the bouncing balls in some detail. This is done in the "rst subsection, where we discuss the integrable case (R"0) that contains only bouncing-ball manifolds. For the chaotic cases R'0 we consider the two simplest spectral statistics, namely, the nearest-neighbour distribution and two-point correlations. We compute these statistics for the levels of the 3D Sinai billiard, and compare them to RMT predictions. In addition, we discuss the two-point statistics of spectral determinants that was recently suggested by Kettemann, Klakow and Smilansky [61] as a characterization of quantum chaos. 3.1. The integrable R"0 case If the radius of the inscribed sphere is set to 0, we obtain an integrable billiard which is the irreducible domain whose volume is 1/48 of the cube. It is plotted in Fig. 6. This tetrahedron billiard is a convenient starting point for analysing the bouncing-ball families, since it contains no unstable periodic orbits but only bouncing balls. Quantum mechanically, the eigenvalues of the tetrahedron are given explicitly as: k
2p " Jn2#m2#l2, 0(n(m(l3N . (nml) S
(30)
The spectral density d (k)"+= d(k!k ) can be Poisson resummed to get R/0 0:n:m:l (nml) S3k2 + sinc(kSJp2#q2#r2) d (k)" R/0 96p2 Z pqr| S2k S2k q2 ! + J (kSJp2#q2)! + J kS p2# 32p Z 0 2 16J2p pq|Z 0 pq| 3S S S # + cos(kSp)# + cos k p 16p Z 8J2p J2 Z p| p| S S 5 # + cos k p ! d(k!0) . 16 6J3p p|Z J3
A
B
A
B
A S
B
(31)
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In the above sinc(x),sin(x)/x, sinc(0),1, and J is the zeroth-order Bessel function. Let us 0 analyse this expression in some detail. Terms which have all summation indices equal to 0 give the smooth part of the density, and all the remaining terms constitute the oscillatory part. Collecting the smooth terms together we get dM
R/0
S3k2 S2k S 5 (k)" ! (1#J2)# (27#9J2#8J3)! d(k!0) . 96p2 32p 144p 16
(32)
This is Weyl's law for the tetrahedron, which exactly corresponds to (I.13) with R"0 (except the last term for which the limit RP0 is di!erent). As for the oscillatory terms, it is "rst useful to replace J (x) by its asymptotic approximation [62] 0 which is justi"ed in the semiclassical limit kPR:
S A B
p 2 cos x! , xPR . J (x)+ 0 4 px
(33)
Using this approximation we observe that all of the oscillatory terms have phases which are of the form (k]length#phase). This is the standard form of a semiclassical expression for the density of states of a billiard. To go a step further we notice that the leading-order terms, which are proportional to k1 ("rst line of (31)), have lengths SJp2#q2#r2 which are the lengths of the periodic orbits of the 3-torus, and therefore of its desymmetrization into the tetrahedron. This conforms with the expressions derived by Berry and Tabor [63,64] for integrable systems. The other, sub-leading, oscillatory contributions to (31) correspond to `impropera periodic manifolds, in the sense that their dynamics involves non-trivial limits. Some of these periodic orbits are restricted to symmetry planes or go along the edges. Of special interest are the periodic orbits that are shown in Fig. 6. They are isolated, but are neutrally stable and hence are non-generic. Their contributions are contained in the last two cosine terms of (31), and the one with length S/J3 is the shortest neutral periodic orbit. Other sub-leading oscillatory contributions are discussed in [54]. We therefore established an interpretation in terms of (proper or improper) classical periodic orbits of the various terms of (31). 3.1.1. Two-point statistics of the integrable case We continue by investigating the two-point statistics of the tetrahedron, which will be shown to provide some non-trivial and interesting results. Since we are interested in the limiting statistics as kPR, we shall consider only the leading term of (31), which is the "rst term. Up to a factor of 48, this is exactly the density of states d 3 of the cubic 3-torus, and thus for simplicity we shall dwell on T the 3-torus rather than on the tetrahedron:
A
B
2p S3k2 + sinc(kSo) . (34) d 3 (k)" + d k! o " T S 2p2 q Z3 q Z3 | | We observe that both the quantal spectrum and the classical spectrum (the set of lengths of periodic orbits) are supported on the cubic lattice Z3, and this strong duality will be used below. The object of our study is the spectral form factor, which is the Fourier transform of the two-point correlation function of the energy levels [59]. For billiards it is more convenient to work with the eigenwavenumbers k rather than with the eigenenergies E . Here the form factor is n n
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Fig. 6. Upper: geometry of the tetrahedron (R"0) billiard. Lower: neutral periodic orbits in the desymmetrized 3D Sinai. The billiard is indicated by boldface edges. Dot-dash line: The shortest neutral periodic orbit of length S/J3. Double dot-dash line: Neutral periodic orbit of length S/J2.
given by
K
K
1 n2 2 K(q; k)" + exp[2pidM (k)k q] . n N n/n1
(35)
In the above N,n !n #1, and k are the eigenvalues in the interval [k 1 , k 2 ] centred around 2 1 n n n k"(k 1 #k 2 )/2. It is understood that the interval contains many levels but is small enough such n n that the average density is almost a constant and is well approximated by dM (k).
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In the limit qPR the phases in (35) become random in the generic case, and therefore K(q)P1. However, if the levels are degenerate, more care should be exercised, and one obtains 1 1 K(q; k)" + g (k )" +@ g2(k ), qPR , (36) k n k i N N n i where g (k ) is the degeneracy of k and the primed sum is only over distinct values of k . Since k n n i N"+@ g (k ) we obtain i k i +@ g2(k ) Sg2(k)T K(q; k)" i k i " k , qPR , (37) +@ g (k ) Sgk (k)T i k i where S ) T denotes an averaging over k 's near k. In the case of a constant g the above expression i reduces to K(qPR)"g, but it is important to note that K(qPR)OSgT for non-constant degeneracies. Using the relation o"kS/(2p) (see Eq. (34)) and Eqs. (H.6), (H.8) in Appendix H we get Sg2(kS/(2p))T bS K 3 (q; k)" o " k, qPR , (38) T Sg (kS/(2p))T 2p o where b+9.8264 is a constant. That is, contrary to the generic case, the saturation value of the form factor grows linearly with k due to number-theoretical degeneracies. Turning to the form factor in the limit qP0, we "rst rewrite (34) as d 3 (k)"dM (k)#+ A sin(k¸ ). T j j j Then, using the diagonal approximation as suggested by Berry [4,65], and taking into account the degeneracies gl (¸ ) of the lengths we have j 1 + @g2l (¸ )DA D2d(q!¸ /¸ ), q;1 . (39) K(q; k)" j j j H 4dM 2(k) j In the above the prime denotes summation only over distinct classical lengths, and ¸ ,2pdM (k) is H called the Heisenberg length. The coe$cients A are functions of ¸ and therefore can be replaced j j by the function A(q). For q large enough such that the periodic manifolds have a well-de"ned classical density dM (l), the summation over delta functions can be replaced by multiplication with #¸ dM (l)/Sgl (l)T with l"¸ q such that H H #pDA2(q)DdM (l) Sg2l (l)T #, q;1 . (40) K(q; k)" 2dM (k) Sgl (l)T
A
B
A straightforward calculation shows that the term in brackets is simply 1, which is the generic situation for the integrable case (Poisson statistics) [4,66]. Hence, we obtain Sg2l (l)T K(q; k)" , qP0 . Sgl (l)T
(41)
Since, as we noted above, the lengths of the classical periodic orbits are supported on the Z3 lattice, we can write using l"So: Sg2(l/S)T bk2S2 " q, qP0 , K(q; k)" o p Sg (l/S)T o
(42)
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Fig. 7. The scaled quantal form factor of the tetrahedron for various k-values compared with GUE and Poisson. Note the log}log scales.
where we used again Eqs. (H.6), (H.8). This is a very surprising result, since it implies that contrary to the generic integrable systems, which display Poisson level statistics with K"1, here KJq which is typical to chaotic systems! This peculiarity is manifestly due to the number-theoretical degeneracies of Z3. If we now combine the two limiting behaviours of the form factor in the simplest way, we can express it as a scaled RMT-GUE form factor: K 3 (q; k)+K ) K (cq) (43) T = GUE where K "Sbk/(2p) and c"2Sk. For the tetrahedron we have the same result with K PK /48 = = = and cPc/48. This prediction is checked and veri"ed numerically in Fig. 7 where we computed the quantal form factor of the tetrahedron around various k-values. The agreement of the two asymptotes to the theoretical prediction (43) is evident and the di!erence from Poisson is well beyond the numerical #uctuations. 3.2. Nearest-neighbour spacing distribution We now turn to the chaotic case R'0. One of the most common statistical measures of a quantum spectrum is the nearest-neighbour distribution P(s). If fact, it is the simplest statistics to compute from the numerical data. We need only to consider the distribution of the scaled (unfolded) spacings between neighbouring levels: s ,NM (k )!NM (k )+dM (k )(k !k ) . (44) n n`1 n n n`1 n It is customary to plot a histogram of P(s), but it requires an arbitrary choice of the bin size. To avoid this arbitrariness, we consider the cumulant distribution:
P
I(s),
s
0
ds@P(s@)
(45)
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Fig. 8. Di!erences of integrated nearest-neighbour distribution for R"0.2 (up) and R"0.3 (down). Set d1, 2, 3 refer to the division of the spectrum into 3 domains. Data are slightly smoothed for clarity.
for which no bins are needed. Usually, the numerical data are compared not to the exact P (s) RMT but to Wigner's surmise [2], which provides an accurate approximation to the exact P (s) in RMT a simple closed form. In our case, since we found a general agreement between the numerical data and Wigner's surmise, we choose to present the di!erences from the exact expression for I (s) GOE taken from Dietz and Haake [67]. In Fig. 8 we show these di!erences for R"0.2, 0.3 and Dirichlet boundary conditions (6697 and 1994 levels, respectively). The overall result is an agreement between the numerical data and RMT to better than 4%. This is consistent with the general wisdom for classically chaotic systems in lower dimensions, and thus shows the robustness of the RMT conjecture [27] for higher-dimensional systems (3D in our case). Beyond this general good agreement it is interesting to notice that the di!erences between the data and the exact GOE for R"0.2 seem to indicate a systematic modulation rather than a statistical #uctuation about the value zero. The same qualitative result is obtained for other boundary conditions with R"0.2, substantiating the conjecture that the deviations are systematic and not random. For R"0.3 the di!erences look random and show no particular pattern. However, for the upper third of the spectrum one observes structures which are similar to the R"0.2 case (see Fig. 8, lower part).
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Currently, we have no theoretical explanation of the above-mentioned systematic deviations. They might be due to the non-generic bouncing balls. To assess this conjecture we computed P(s) for R"0.2, 0.3 with Dirichlet boundary conditions in the spectral interval 150(k(200. The results (not shown) indicate that the deviations are smaller for the larger radius. This is consistent with the expected weakening of the bouncing-ball contributions as the radius grows, due to larger shadowing and smaller volumes occupied by the bouncing-ball families. Hence, we can conclude that the bouncing balls are indeed prime candidates for causing the systematic deviations of P(s). It is worth mentioning that a detailed analysis of the P(s) of spectra of quantum graphs show similar deviations from P (s) [68]. RMT 3.3. Two-point correlations Two-point statistics also play a major role in quantum chaos. This is mainly due to their analytical accessibility through the Gutzwiller trace formula as demonstrated by Berry [4,65]. There is a variety of two-point statistical measures which are all related to the pair-correlation function [59]. We chose to focus on R2(l) which is the local variance of the number of levels in an energy interval that has the size of l mean spacings. The general expectation for generic systems, according to the theory of Berry [4,65], is that R2 should comply with the predictions of RMT for small values of l (universal regime) and saturate to a non-universal value for large l's due to the semiclassical contributions of short periodic orbits. The saturation value in the case of generic billiards is purely classical (k-independent). The e!ect of the non-generic bouncing-ball manifolds on two-point spectral statistics was discussed in the context of 2D billiards by Sieber et al. [37] (for the case of the stadium billiard). They found that R2 can be decomposed into two parts: A generic contribution due to unstable periodic orbits and a non-generic contribution due to bouncing balls: R2(l)+R2 (l)#R2 (l) . (46) UPO "" The term R2 has the structure: "" R2 (l)"kF (l/dM (k)) , (47) "" 45!$*6. where F is a function which is determined by the bouncing balls of the stadium billiard, and is 45!$*6. given explicitly in [37]. In particular, for large values of l the term R2 #uctuated around an "" asymptotic value: R2 (l)+kF (R), lPR . (48) "" 45!$*6. One can apply the arguments of Sieber et al. [37] to the case of the 3D Sinai billiard and obtain for the leading-order bouncing balls (see (34)): R2 (l)+k2F (l/dM (k)) , (49) "" 3D4" with F characteristic to the 3D Sinai billiard. Asymptotically, we expect 3D4" R2 (l)+k2F (R), lPR . (50) "" 3D4" The function F can be written down, albeit it contains the areas of the cross-sections of the 3D4" various bouncing-ball manifolds, for which we have no explicit expressions. Therefore, we shall investigate the scaling features of R2 without insisting on its explicit form. ""
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Fig. 9. The number variance R2(l) for the longest spectrum. Upper plot: full l-range, lower plot: a magni"cation of small l range. Fig. 10. Rescaled number variance (51) for the longest spectrum.
The numerical computations of R2 for the longest spectrum (R"0.2, Dirichlet everywhere) are shown in Fig. 9. We divided the spectrum into four intervals such that dM did not vary much within each interval. This is a pre-requisite for a meaningful semiclassical analysis. It is evident from the "gure that for small values of l (up to +1) there is an agreement with GOE. Moreover, the agreement with GOE is much better than with either GUE or Poisson, as expected. This is in agreement with the common knowledge in quantum chaos [59], and again, substantiates the RMT conjecture also for chaotic systems in 3D. For larger l values there are marked deviations which saturate into oscillations around k-dependent asymptotic values. It is clearly seen that the saturation values grow faster than k, which is consistent with (50). To test (49) quantitatively, we plotted in Fig. 10 the rescaled function: 1 S2 (q; k), [R2(qdM (k))!R2 (qdM (k))] "" GOE k2
(51)
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Fig. 11. Comparison between the number variances for two di!erent radii R"0.2, 0.3 of the inscribed sphere of the 3D Sinai billiard. In both cases we considered the spectral interval 120(k(160 and used Dirichlet boundary conditions.
which according to (49) is the k-independent function F (q). Indeed, there is a clear data collapse 3D4" for q[5, and the saturation values of S2 are of the same magnitude for all values of k. This veri"es "" (49) and demonstrates the important part which is played by the bouncing balls in the two-point (long-range) statistics. For generic systems the agreement between R2 and RMT should prevail up to lH, where ¸ (k) 2pdM (k) lH" H " . ¸ ¸ .*/ .*/
(52)
In the above ¸ is Heisenberg length and ¸ is the length of the shortest periodic orbit. For the H .*/ cases shown in Fig. 9 the value of lH is of the order of 100. Nevertheless, the deviations from the universal predictions start much earlier. This is again a clear sign of the strong e!ect of the bouncing-balls. To substantiate this claim, we compare in Fig. 11 the number variances for R"0.2 and 0.3 in the same k interval and with the same boundary conditions (Dirichlet). The in#uence of the bouncing balls is expected to be less dominant in the R"0.3 case, since there are fewer of them
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with smaller cross sections. This is indeed veri"ed in the "gure: The agreement with GOE predictions lasts much longer (up to l+6) in the R"0.3 case, and the saturation value is smaller, as expected. 3.4. Auto-correlations of spectral determinants The two-point correlations discussed above are based on the quantal spectral densities. Kettemann et al. [61] introduced the auto-correlations of quantal spectral determinants as a tool for the characterization of quantum chaos. Spectral determinants are de"ned as Z(E)"0 Q E"E , (53) n that is, they are 0 i! E is an eigenenergy. The (unnormalized) correlation function of a spectral determinant is de"ned as
P
A
B A
B
1 E`*E@2 u u C(u; E), ZH E@! , u;*E . dE@ Z E@# *E 2dM 2dM E~*E@2 There are various motivations to study the function C(u) [61]:
(54)
1. There is a marked di!erence in the behaviour of C(u) for rigid and non-rigid spectra. For completely rigid spectra the function C(u) is oscillatory, while for Poissonian spectra it rapidly decays as a Gaussian. For the RMT ensembles it shows damped oscillations which are due to rigidity. 2. The function C(u) contains information about all n-point correlations of the spectral densities. Thus, it is qualitatively distinct from the two-point correlations of spectral densities and contains new information. 3. The Fourier transform of C(u) exhibits in an explicit and simple way symmetry properties which are due to the reality of the energy levels. 4. In contrast to spectral densities, the semiclassical expressions for spectral determinants can be regularized using the method of Berry and Keating [69]. Regularized semiclassical spectral determinants contain a "nite number of terms, and are manifestly real for real energies. 5. The semiclassical expression for C(u) is closely related to the classical Ruelle zeta function. To study C(u) numerically, regularizations are needed. For the 3D Sinai billiard the longest spectrum was divided into an ensemble of 167 intervals of N"40 levels, and each interval was unfolded to have mean spacing 1 and was centered around E"0. For each unfolded interval I the j function C (u) was computed using equation (69) of [61], with *E"JN. The ensemble average j function C(u) was normalized such that C(0)"1. The results of the computation are shown in Fig. 12. The agreement with RMT is quite good up to u+3, that is for short energy scales for which we indeed expect universality to hold. 4. Classical periodic orbits In this section we present a comprehensive study of the periodic orbits of the 3D Sinai billiard. By `periodic orbitsa we mean throughout this section generic, isolated and unstable periodic orbits
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Fig. 12. The two-point correlation function of spectral determinants C(u) for the 3D Sinai billiard (longest spectrum). The spectrum was divided into 167 intervals of 40 levels each and the average correlation function is shown. The continuous line is the RMT-GOE theoretical curve, and the dashed line is the numerical correlation. The correlation function is normalized to 1 for u"0. With kind permission from the authors of [61].
which involve at least one bounce from the sphere. Thus, bouncing-ball orbits are not treated here. The classical periodic orbits are the building blocks for the semiclassical Gutzwiller trace formula, and are therefore needed for the semiclassical analysis to be presented in the next sections. 4.1. Periodic orbits of the 3D Sinai torus We found it necessary and convenient to "rst identify the periodic orbits of the symmetric 3D Sinai billiard on the torus, and to compute their lengths and stabilities. The periodic orbits of the desymmetrized 3D Sinai billiard could then be derived by an appropriate classical desymmetrization procedure. The basic problem is how to "nd in a systematic (and e$cient) way all the periodic orbits of the 3D Sinai billiard up to a given length ¸ . In dealing with periodic orbits of the Sinai billiard it is .!9 very helpful to consider its unfolded representation that tessellates R3 } as is shown in Fig. 1. We start by considering the periodic orbits of the fully symmetric 3D Sinai billiard on the torus (ST). This case is simpler than the desymmetrized billiard, since it contains no boundaries and the tiling of the R3 space is achieved by simple translations along the cubic lattice Z3. In the unfolded representation every orbit is described by a collection of straight segments which connect spheres. At a sphere, the incident segment re#ects specularly. A periodic orbit of period n is not necessarily periodic in the unfolded representation, but rather, it obeys the restriction that the segments repeat themselves after n steps modulo a translation by a lattice vector (see Fig. 13). If we "x an origin for the lattice, we can assign to every orbit (not necessarily periodic) a `code worda by concatenating
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Fig. 13. Representation of a periodic orbit of the Sinai 2-torus. Left: one cell representation, Right: unfolded representation.
the `addressesa (locations of the centers on the Z3 lattice) of the spheres from which it re#ects. The code word can consist of either the absolute addresses of the spheres or alternatively, the address of the sphere relative to the previous one. We shall adopt the latter convention and use the relative addresses as the `lettersa from which the code word is composed. This relative coding has the advantage that a periodic orbit is represented by a periodic code word. The number of possible letters (`alphabeta) is obviously in"nite and the letter (0, 0, 0) is excluded. A periodic orbit can be represented by any cyclic permutation of its code. To lift this ambiguity, we choose a convenient (but otherwise arbitrary) lexical ordering of the letters and use the code word which is lexically maximal as the unique representative of the periodic orbit: (periodic orbit of ST) C ="(w , w ,2, w ), 1 2 n ="maxM=, PK =, PK 2=,2, PK n~1=N ,
w 3Z3C(0, 0, 0) , i
(55)
where PK ="(w , w ,2, w , w ) is the operation of a cyclic permutation of the code word. 2 3 n 1 Let us consider the code word = with n letters: ="(w , w ,2, w ), w "(w , w , w ) . (56) 1 2 n i ix iy iz It relates to the n#1 spheres centred at c "(0, 0, 0), c "w , c " w #w ,2, c " 1 2 1 3 1 2 n`1 w #2#w . Let us choose arbitrary points on each of the spheres, and connect them by 1 n straight segments. We get a piecewise straight line which leads from the "rst to the last sphere.
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Fig. 14. A shadowed (classically forbidden) periodic orbit of the Sinai 3-torus.
In general, this line is not a classical orbit because the specular re#ection conditions are not satis"ed. To "nd a periodic orbit, we specify the positions of the points on each sphere by two angles h , u . The length of the line is a function of M(h ,u )D N. Periodic orbits on the ST must i i i i i/1,2,n have identical coordinates for the "rst and the last points (modulo a lattice translation), hence h "h , u "u and we have only 2n independent variables to completely specify a periodic n`1 1 n`1 1 set of segments, with length: n ¸ (h ,2, h , u ,2, u )" + ¸ (h , h , u , u ) , W 1 n 1 n i i i`1 i i`1 i/1
(57)
where ¸ are the lengths of the segments that correspond to the letter w . To satisfy the condition of i i specular re#ection we require that the length ¸ is extremal with respect to any variation of its W variables. The following theorem guarantees two essential properties of the coding and of the periodic orbits which are identi"ed as the extrema of (57) [27,32]: Theorem. To each code word W of the 3D ST there corresponds at most one periodic orbit which is the only minimum of ¸ . W The theorem contains two statements: First, that periodic orbits are necessarily minima of the length, and not saddles or maxima. Second, that there are no local minima besides the global one. The phrase `at mosta in the theorem above needs clari"cation: For each code word = the length function ¸ is a continuous function in all of its variables over the compact domain which is W the union of the spheres. Therefore ¸ must have a global minimum within this domain. This W minimum can be, however, classically forbidden, meaning that at least one of its segments cuts through one or more spheres in the lattice (that might or might not be a part of the code) rather than re#ecting from the outside. This obstruction by an intervening sphere will be called `shadowinga. An example is shown in Fig. 14. The forbidden periodic orbits are excluded from the set of classical periodic orbits. (They also do not contribute to the leading order of the trace formula [15,70] and therefore are of no interest in our semiclassical analysis.) If all the segments are classically allowed, then we have a valid classical periodic orbit. Finally, we would like to mention that the minimality property was already implied in the work of Sieber [60], and the explicit versions of the theorem were proved simultaneously by Bunimovich [27] (general formulation, applies in particular to the 3D case) and Schanz [32] (restricted to the 2D Sinai billiard). The number of letters in the codes of periodic orbits of length less than ¸ can be bounded .!9 from above by the following argument. To each letter w there corresponds a minimal segment
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length ¸ (w)'0 which is the minimum distance between the spheres centred at (0, 0, 0) and at .*/ w"(w , w , w ): x y z ¸ (w)"SJw2#w2#w2!2R . .*/ x y z
(58)
In the above, S is the lattice constant (torus's side) and R is the radius of the sphere. The smallest possible ¸ (w) is obtained for w"(1, 0, 0) and equals S!2R,¸ . We readily conclude that .*/ .*/ the code word cannot contain more letters than the integer part of ¸ /¸ . .!9 .*/ We are now in a position to formulate an algorithm for a systematic search of all the periodic orbits of length up to ¸ of the 3D Sinai torus: .!9 1. Collect all of the admissible letters into an alphabet. An admissible letter w satis"es (a) wO(0, 0, 0). (b) w is not trivially impossible due to complete shadowing, e.g., like (2, 0, 0)"2](1, 0, 0). (c) ¸ (w)4¸ . .*/ .!9 2. De"ne an arbitrary lexical order of the letters. 3. From the admissible alphabet construct the set of admissible code words ="(w ,2, w ), such 1 n that (a) ¸ (=),+n ¸ (w )4¸ . .*/ i/1 .*/ i .!9 (b) w Ow } no a priori complete shadowing. i i`1 (c) = is lexically maximal with respect to cyclic permutations: ="maxMPK i=, i"0,2, n!1N. 4. For each candidate code word = minimize numerically the function ¸ . According to the W theorem, there should be exactly one minimum, which is the global one. 5. Check whether the resulting periodic orbit is shaded. Accept only periodic orbits which are not shaded. Once the periodic orbit is identi"ed, its monodromy (stability) matrix is computed according to the recipe given in Appendix J. 4.2. Periodic orbits of the 3D Sinai billiard } classical desymmetrization If we desymmetrize the ST into the Sinai billiard (SB), we still "nd that the SB tessellates the R3 space. Hence, each periodic orbit of the ST is necessarily also a periodic orbit of the SB. The converse is not true, i.e., periodic orbits of the SB are not necessarily periodic in ST. However, it is easy to be convinced that if a periodic orbit of SB is repeated su$ciently many times, it becomes also periodic in ST. An example is shown in Fig. 15. From a more abstract point of view, this is because the cubic group O is "nite. Thus in principle one could use the algorithm given above to h systematically "nd all the periodic orbits of the SB. This is, however, highly ine$cient because by analysing the group O we "nd that in order to "nd all the periodic orbits of the SB up to ¸ we h .!9 must "nd all of the periodic orbits of ST up to 6¸ . Due to the exponential proliferation .!9 of periodic orbits this would be a colossal waste of resources which would diminish our ability to compute periodic orbits almost completely. To circumvent this di$culty, without losing the useful uniqueness and minimality properties which apply to the ST, we make use of the property that
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Fig. 15. Desymmetrization of orbits from the Sinai torus to the Sinai billiard. For clarity we show an example in 2D. Left: a primitive periodic orbit in the ST. Right: the corresponding periodic orbit in the SB. We observe that the latter is 4 times shorter than the former.
periodicity in the SB is synonymous to periodicity in ST modulo an element g( 3O . This simple h geometrical observation is a manifestation of the fact that the tiling of R3 by the SB is generated by the group O ?Z3. Thus, we can represent the periodic orbits of the SB by using their unfolded h representation, augmented by the symmetry element g( according to which the periodic orbits closes: Periodic orbit of SB C= K ,(=; g( )"(w , w ,2, w ; g( ) . 1 2 n
(59)
The coding is not yet well-de"ned since a given periodic orbit can in general be represented by several codes. Similarly to the case of the ST, there is a degeneracy with respect to the starting point. However, in the case of the SB this is not simply related to cyclic permutations. Rather, if a periodic orbit is described by (w , w ,2, w ; g( ) then it is also described by 1 2 n (w , w ,2, w , g( w ; g( ), (w , w ,2, g( w , g( w ; g( ),2 , 2 3 n 1 3 4 1 2 (g( w , g( w ,2, g( w ; g( ), (g( w , g( w ,2, g( 2w ; g( ),2 , 1 2 n 2 3 1 F
(60)
(g( ((g( )~1w , g( ((g( )~1w ,2, g( ((g( )~1w ; g( ),2, (g( ((g( )~1w , w , w ,2, w ; g( ) . 1 2 n n 1 2 n~1 In the above /(g( ) is the period of g( , which is de"ned as the smallest natural number for which g( ((g( )"e( , where e( is the identity operation. For O in particular /(g( )3M1, 2, 3, 4, 6N. The above h generalized cyclic permutation invariance is due to the periodicity modulo g( of the periodic orbits of the SB in the unfolded representation. In addition to the generalized cyclic invariance there is also a geometrical invariance of orbits of the SB in the unfolded representation. Indeed, if we operate on an orbit in the unfolded representation with any hK 3O we obtain the same orbit in h the SB. This symmetry is carried over also to the codes. If a periodic orbit is described by (w , w ,2, w ; g( ) then it is also described by 1 2 n (hK w , hK w ,2, hK w ; hK g( hK ~1) ∀hK 3O . 1 2 n h
(61)
To summarize, a periodic orbit of the SB can be encoded into a code word up to degeneracies due to generalized cyclic permutations and geometrical operations. The set of operations which relate
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the various codes for a given periodic orbit is a group to which we refer as the invariance group. In order to lift this degeneracy and to obtain a unique mapping of periodic orbits of the SB to code words we need to specify a criterion for choosing exactly one representative. There are many ways of doing this, but we found it convenient to apply the natural mapping of periodic orbits of the SB to those of the ST, and there, to choose the maximal code. More speci"cally: 1. Select the alphabet according to the rules prescribed in the preceding subsection, and de"ne ordering of letters. 2. Extend the word = K into = I : = I ,(w , w ,2, w , g( w , g( w ,2, g( w , g( 2w ,2, g( ((g( )~1w ,2, g( ((g( )~1w ) . 1 2 n 1 2 n 1 1 n
(62)
The code = I describes the periodic orbit of the SB which is continued /(g( ) times to become periodic in the ST. Applying a generalized cyclic permutation on = K is equivalent to applying the standard cyclic permutation on = I . Applying a geometrical operation hK on = K is equivalent to operating letter by letter with hK on = I . The invariance group corresponding to = I is H"C?O , where C is the group of cyclic permutations of order n ) /(g( ). The simple decompoh sition of H is due to the commutativity of C and O , and it greatly facilitates the computations. h 3. If = I is maximal with respect to the invariance group H, then the corresponding = K is the representative of the periodic orbit. A comment on the uniqueness of this selection process is appropriate at this point. For any = K we can uniquely construct the corresponding = I and the invariance group and check the maximality of = I . Hence, we are able to uniquely decide whether = K is a valid representative code or not. However, there are cases in which more than one = K correspond to the same maximal = I . It is straightforward to show that in these cases the basic code word = is symmetric under some operation(s): ="kK =, kK 3O . To such symmetric codes must correspond symmetric periodic h orbits, which is necessitated by the uniqueness theorem for the ST. But for the SB the symmetry of the orbit means that it is wholly contained in a symmetry plane, and therefore is not a proper classical orbit. Such orbits are nevertheless required for the semiclassical analysis and will be treated in the next section when dealing with semiclassical desymmetrization. In summary, we have shown so far that the mapping of a given proper periodic orbit to a code is well-de"ned and unique. In order for the coding to be useful and powerful, we need to establish uniqueness in the opposite direction, that is to show that for a given (unsymmetrical) = K there corresponds at most one (proper) classical periodic orbit. The mapping = K C= I is very useful in that respect. Indeed, if there were two distinct periodic orbits of the SB with the same coding = K , then we could repeat them /(g( ) times to get two distinct periodic orbits of the ST with the same code = I , which is in contradiction with the theorem above. This proves the uniqueness of the relation between codes and periodic orbits. To facilitate the actual computation of periodic orbits of the SB, we have to establish their minimality property, similarly to the ST case. We need to prove that the length of a periodic orbit is a minimum, and that it is the only minimum. The minimality of a periodic orbit of the SB is proven by using again the unfolding to periodic orbits of ST, and noting that a minimum of ¸ I is W
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necessarily also a minimum of ¸ K , since the latter is a constrained version of the former. Thus, W periodic orbits of the SB are minima of ¸ K . We "nally have to show that there exists only a single W minimum of ¸ K . The complication here is that, in principle, a minimum of ¸ K does not necessarily W W correspond to a minimum of ¸ I , since there are, in general, more variables in the latter. We resolve W this di$culty by using arguments from the proof of Schanz [32] as follows. A necessary condition for minimality is that orbits are either externally re#ected from the scatterers or cut through them in straight segments. Internal re#ections are not allowed for a minimum. Thus, if we extend a minimum of SB to ST, we necessarily get an orbit with no internal re#ections. According to Schanz [32], there is exactly one such orbit, which is the minimum in ST. This proves the uniqueness of the (global) minimum of ¸ K in SB. W These results allow us to use essentially the same algorithm as for the ST for the systematic search of periodic orbits of the SB. We need to extend the codes and the length functions to include a group element g( , and to modify the rules according to which we choose an admissible and lexically maximal code word = K . One also has to modify the computations of the monodromy matrix, as described in Appendix J. 4.3. The properties and statistics of the set of periodic orbits The algorithm described above is capable of "nding all of the periodic orbits up to any desired length. Before discussing the properties of this set, we "nd it appropriate to display a few typical periodic orbits, which were computed for the desymmetrized billiard with R"0.2 (and S"1). The orbits are represented in an unfolded way in Figs. 16}19. In this subsection we shall study in detail the spectrum of lengths of periodic orbits, a small interval thereof is shown in Fig. 20. Each horizontal strip provides the lower end of the length spectrum of Sinai billiards with 0.024R40.36. The spectrum corresponding to the lowest value of R shows clustering of the lengths near the typical distances of points of the Z3 lattice (1, J2, J3, 2,2). Once R is increased, some of the periodic orbits which were allowed for the smaller R are decimated because of the increased e!ect of shadowing. However, their lengths become shorter, resulting in the proliferation of the periodic orbits with their length. This is best seen in the spectrum which corresponds to the largest value of R } the graphics is already not su$ciently "ne to resolve the individual lengths. After these introductory comments, we now study the length spectrum in detail, and compare the theoretical expectations with the numerical results. The exponential proliferation of the periodic orbits puts a severe limit on the length range which we could access with our "nite computer resources. However, we were able to compute the periodic orbits for a few values of the radius R, and concentrated on the R"0.2 case in order to be able to perform a semiclassical analysis of the longest quantal spectrum (see next section). For this radius we found all the 586,965 periodic orbits up to length 5. This number of periodic orbits includes repetitions and time-reversed conjugates. We also computed for this radius all the 12,928,628 periodic orbits up to length 10 which have no more than 3 re#ections. This comprises the database on which we based our further numerical studies and illustrations. The systematic algorithm which was used to produce this data set, together with a few tests which will be described here and in the next section, lead us to believe that the data set is both accurate and complete.
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Fig. 16. A sample of periodic orbits of the desymmetrized 3D Sinai billiard with S"1, R"0.2 with a single re#ection. The periodic orbits are shown in the unfolded representation. The `fulla spheres are those from which the periodic orbit re#ects. The `fainta dotted spheres are those from which there is no re#ection. Fig. 17. A sample of periodic orbits of the 3D SB with 2 re#ections.
Periodic orbits are expected to proliferate exponentially (e.g., [2]). That is, the number N (l) of -%/ periodic orbits of length less than l should approach asymptotically [2]: exp(jl) , lPR, N (l)+ -%/ jl
(63)
where j is the topological entropy (per unit length). To examine the validity of the above formula in our case we use the numerical data to compute
A
1 (l), ln l
B
+ ¸ , (64) j L%3' yLj yl where ¸ is a length below which we do not expect universality (i.e. the law (63)) to hold. The %3' exponential proliferation implies: j
j
/6.
1 ln j (l)+ lnDejl!ejL%3' D! Pj, lPR. /6. l l
(65)
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Fig. 18. A sample of periodic orbits of the 3D SB with 4 re#ections. Fig. 19. A sample of periodic orbits of the 3D SB with 7 re#ections. The bottom periodic orbit undergoes 8 re#ections.
Therefore, we expect j (l) to approach a constant value j when l is su$ciently larger than ¸ . In /6. %3' Fig. 21 we show the results of the numerical computation of j for the R"0.2 database and for /6. ¸ "2.5. The "gure clearly indicates a good agreement between the data and the theory (65) for %3' j"3.2. One of the hallmarks of classically ergodic systems is the balance between the proliferation of periodic orbits and their stability weights due to ergodic coverage of phase space. This is a manifestation of the uniform coverage of phase space and is frequently referred to as the `Hannay}Ozorio de Almeida sum rulea [71]. It states that ¸ p p(l),+ d(l!¸ )P1, lPR, j Ddet(I!M )D j PO
(66)
where ¸ is the primitive length and M is the stability (monodromy) matrix [2] (see Appendix J for p j explicit expressions). The above relation is meaningful only after appropriate smoothing. For generic billiards the only classical length scale is the typical length traversed between re#ections, and we expect (66) to approximately hold after a few re#ections. In the Sinai billiard we are faced
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Fig. 20. Length spectra of periodic orbits for Sinai billiards with R values between 0.02 and 0.36 in steps of *R"0.02. The vertical bars indicate the lengths of periodic orbits. Fig. 21. The quantity j (cf. RHS of Eq. (64)) computed from the periodic orbit database of R"0.2. We used /6. ¸ "2.5. The theoretical "t is according to Eq. (65). %3'
with the problem of an `in"nite horizona, that is, that the length of free #ight between consecutive re#ections is unbounded. This is just another manifestation of the existence of the bouncing-ball families. According to [39,40] this e!ect is responsible for a non-generic power-law tail in p(l): a(R) p(l)+1! , l
(67)
where a(R) is a parameter that depends on the radius R. When R increases the in#uence (measure in con"guration space) of the bouncing balls is reduced, and we expect a(R) to decrease. To check (67) we computed numerically the cumulant:
P
l
¸ p dl@p(l@)+ + , DI!M D L%3' j %3' j L yL yl which should be compared to the theoretical expectation: P(l)"
P(l)"(l!¸
%3'
A B
)!a(R)ln
l
¸
.
(68)
(69)
%3'
The results are shown in Fig. 22. We considered R"0.2 and 0.3 and included periodic orbits up to ¸ "10 with number of re#ections n43. The restriction on n facilitates the computation and is .!9 justi"ed for moderate values of l since the contributions from higher n's are small. The observed deviation between the theoretical and numerical curves for R"0.3 at lZ8 is due to the fact that periodic orbits with n"4 become signi"cant in this region. The above numerical tests con"rm the validity of (67), with a(R) which is a decreasing function of R. In particular, for the length interval considered here, there is a signi"cant deviation from the fully ergodic behaviour (66).
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Fig. 22. The function P(l) (cf. RHS of Eq. (68)) computed for R"0.2 and 0.3 and "tted according to Eq. (69). We also show the asymptotic prediction (66).
The sum-rule (66) which formed the basis of the previous analysis is an expression of the ergodic nature of the billiards dynamics. In the next subsection we shall make use of similar sum-rules which manifest the ergodicity of the PoincareH map obtained from the billiard #ow by, e.g., taking the surface of the sphere and the tangent velocity vector as the PoincareH section. The resulting return-map excludes the bouncing-ball manifolds since they do not intersect the section. However, their e!ect is noticed because between successive collisions with the sphere the trajectory may re#ect o! the planar faces of the billiard an arbitrary number of times. Thus, the number of periodic orbits which bounce n times o! the sphere (n-periodic orbits of the map) is unlimited, and the topological entropy is not well de"ned. Moreover, the length spectrum of n-periodic orbits is not bounded. These peculiarities, together with the fact that the symbolic code of the map consists of an in"nite number of symbols, are the manifestations of the in"nite horizon of the unfolded Sinai billiard. The return map itself is discontinuous but it remains area preserving, so the formulas which we use below, and which apply to generic maps, can be used here as well. The classical return probability is de"ned as the trace of the n-step classical evolution operator (see, e.g., [72] and references therein). It is given by n p,j ;(n), + , (70) P Ddet(I!M )D j j| n where n is the number of times the periodic orbit re#ects from the sphere, P is the set of all n n-periodic orbits, n is the period of the primitive periodic orbit of which j is a repeated traversal. p,j As a consequence of the ergodic nature of the map ;(n)P1 in the limit nPR. However, due to the e!ect of the in"nite horizon, the number of periodic orbits in P is in"nite, and in any numerical n simulation it is important to check to what degree the available data set satis"es the sum rule. For this purpose we de"ne the function n p,j ;(l; n), + H(l!¸ ) , j P Ddet(I!M )D j j in n
(71)
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Fig. 23. Upper plot: the function ;(l; n) (cf. Eq. (71)) for the cases R"0.4, n"1, 2, 3. Lower plot: the function R;(l; n)/Rl for the same cases. Both plots indicate the saturation of the classical return probability in spite of the in"nitely many periodic orbits in P . n
which takes into account only n-periodic orbits with ¸ 4l. In Fig. 23 we plot ;(l; n) for R"0.4 j and n"1, 2, 3. The results clearly indicate that for the present data saturation is reached, and once n52 the asymptotic value is very close to 1. Even at n"1 one gets ;(n"1)+0.8 which is surprisingly close to 1, bearing in mind that we are dealing with the "xed points of the map! It should be noted that to reach saturation in the case R"0.4, n"3 one needs 536,379 periodic orbits up to l"12, whose computation consumes already an appreciable amount of time. Thus, we are practically restricted to the few lowest n's in our computations. As can be seen in Fig. 23 the function R;(l; n)/Rl is mostly supported on a "nite interval of ¸ values. Its width will be denoted by *¸(n).
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4.4. Periodic orbit correlations In the previous subsection we discussed various aspects of the one-point statistics of the classical periodic orbits, and demonstrated their consistency with the standard results of ergodic theory. Here, we shall probe the length spectrum further, and show that this spectrum is not Poissonian. Rather, there exist correlations between periodic orbits which have far-reaching e!ects on the semiclassical theory of spectral statistics of the quantum billiard. The semiclassical theory will be dealt with in Section 7, and here we restrict ourselves to purely classical investigations. Above we introduced the PoincareH return map of the sphere, and have shown that the ergodicity of this map implies a sum rule for the set of n-periodic orbits of the map. We de"ne the weighted density of lengths of n-periodic orbits as follows: d (l; n), + AI d(l!¸ ) , (72) #j j j|Pn where AI are given by j n (!1)bj p,j AI " , (73) j Ddet(I!M )D1@2 j and b is the number of times the trajectory re#ects from the planar boundaries. The amplitudes j AI are related to the standard semiclassical amplitudes A de"ned in (99) by AI "pnp A /¸ . j j j j j j The density (72) is di!erent from the density p(l) de"ned previously (66) since: (a) it relates to the subset of the n-periodic orbits of the return map of the sphere, (b) it assigns a signed weight to each of the d-functions located at a particular length, and (c) the absolute value of the weights in (72) are the square roots of the weights in (66). Densities with signed weights are not encountered frequently in spectral theory, but they emerge naturally in the present context. At this point the de"nition of d (l; n) might look unfamiliar and strange, but the reason for this particular choice will become #clear in the sequel. To examine the possible existence of correlations in the length spectrum, we study the corresponding auto-correlation function
P
R (dl; n), #-
=
dl d (l#dl/2; n) d (l!dl/2; n) . ##-
(74)
0 The two-point form factor is the Fourier transform of R (dl; n), and it reads explicitly as #`= 2 K (k; n)" e*kxR (x; n) dx" + AI exp(ik¸ ) . ##j j ~= j|Pn The form factor has the following properties:
P
K
K
(75)
f K (k; n) is a Fourier transform of a distribution and therefore it displays #uctuations, which #become stronger as the number of contributing orbits increases. Therefore, any discussion of this function requires some smoothing or averaging. We shall specify the smoothing we apply in the sequel.
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f At k"0,
K
K
2 . (76) K (0; n)" + AI j #j|Pn Because of the large number of periodic orbits, the sum of the signed amplitudes is e!ectively reduced due to mutual cancellations. Its value can be estimated by assuming that the signs are random. Hence, K (0; n)+ + DAI D2 , j #j|Pn which will be shown below to be bounded. f At large values of k,
(77)
(78) K (k; n)+ + g DAI D2 for kPR, j j #j|Pn where g is the number of isometric periodic orbits of length ¸ . Since large #uctuations are j j endemic to the form factor, this relation is meaningful when k-averaging is applied. Comparing the last sum with (70) we can write K (k; n)+Sn g T;(n) for kPR. (79) #p,j j In our case of the 3D SB, n "n for the large majority of the periodic orbits in P , which is the p,j n generic situation for chaotic systems. Also, g "2 for almost all the periodic orbits with n53. j Thus, one can safely replace Sn g T with 2n for large n. Moreover, as we saw above, ;(n)P1 p,j j for large n, hence K P2n for large k and n. #f If the length spectrum as de"ned above were constructed by a random sequence of lengths with the same smooth counting function ;(l; n), or if the phases were picked at random, one would obtain the Poisson behaviour of the form factor, namely, a constant 2p . K (k; n)+Sn g T;(n) for k' #p p *¸(n)
(80)
Here, *¸(n) is the e!ective width of the length distribution de"ned above. Thus, we could identify two-point correlations in the classical length spectrum by computing K (k; n) and observing deviations from the k-independent expression (80). #4.4.1. Numerical tests We used the periodic orbit database at our disposal to compute the form factors for several values of n and R. In each case presented we made sure that the function ;(l; n) is numerically saturated. This guarantees that the (in"nitely many) neglected periodic orbits have very small weight, and are thus insigni"cant. In Fig. 24 we present the numerical results, where we plotted the function
P
1 k C (k; n), dk@ K (k@; n) , ##k!k .*/ .*/ k
(81)
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Fig. 24. The averaged classical form factor C (k; n) (cf. (81)) of the 3D SB for several values of n and R. We also plot the #averaged form factors with signs of the amplitudes scrambled, without cross-family terms, and with amplitudes averaged over family (length-correlation only). See text for details.
designed to smooth the #uctuations in K (k; n) [11]. We started the integration at k '0 to #.*/ avoid the large peak near k"0, which otherwise overwhelms the results. In any case, the neglected small-k region is irrelevant for the semiclassical theory of quantal spectral correlations. Analysing the results, we note that the asymptotic form factors (denoted in Fig. 24 as `full classical form factora) approach constant values, which are indeed close to 2n, as predicted. More signi"cant are the deviations from the constant (Poissonian) result at low k, which demonstrate unambiguously the existence of correlations in the classical spectra. The structure of the form factor indicates that the classical spectrum is rigid on the scale of a correlation length j(n;R), which can be de"ned as the inverse of the k value at which the form factor makes its approach to the asymptotic value [11]. In the following we shall describe a few tests which prove that the observed correlations are real, and not a numerical artifact or a trivial consequence of the way in which the length spectral density is de"ned. The spectral density d (l; n) has an e!ective "nite width *¸(n) which was de"ned above. The fact #that the lengths are constrained to this interval induces trivial correlations which appear on the
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scale *¸(n), and we should check that this scale is su$ciently remote from the correlation scale j(n;R). To this end, and to show that the observed classical correlations are numerically signi"cant, we scrambled the signs of the weights AI by multiplying each of them with a randomly chosen sign. j We maintained, however, the time-reversal symmetry by multiplying conjugates by the same sign. The resulting form factors (denoted as `Scrambled signsa in Fig. 24), are consistent with the Poissonian value 2n for essentially all k values, and the di!erence between the scrambled and unscrambled data is large enough to add con"dence to the existence of the classical correlations. This indicates also that the correlations are not due to the e!ective width of d (l; n), since both the #scrambled and unscrambled data have the same e!ective width. On the other extreme, one might suspect that the classical correlations are due to rigidity on the scale of one mean spacing between lengths of periodic orbits. This is certainly not the case, since the typical mean length spacing for the length spectra shown in Fig. 24 is 10~3}10~4, which implies a transition to the asymptotic value for much larger k-values than observed. We therefore conclude that the correlation length j(n; R) is much larger than the mean spacing between neighbouring lengths. This is the reason why various studies of the length-spectrum statistics [60,73] claimed that it is Poissonian. Indeed it is Poissonian on the scale of the mean spacing where these studies were conducted. The correlations become apparent on a very di!erent (and much larger) scale, and there is no contradiction. The coexistence of a Poissonian behaviour on the short length scales, and apparent rigidity on a larger scale was discussed and explained in [11]. It was suggested there that a possible way to construct such a spectrum is to form it as a union of N<1 statistically independent spectra, all having the same mean spacing DM , and which show spectral rigidity on the scale of a single spacing. The combined spectrum with a mean spacing DM /N will be Poissonian when tested on this scale, since the spectra are independent. However, the correlations on the scale DM will persist in the combined spectrum. A simple example will illustrate this construction. Take a random (Poissonian) spectrum with a mean spacing 1. Generate a shifted spectrum by adding jM <1 to each spectral point and combine the original and the shifted spectra to a single spectrum. On the scale 1 the combined spectrum is Poissonian. However, the fact that each spectral point is (rigidly) accompanied by another one, a distance jM apart, is a correlation which will be apparent at the scale jM only. We use this picture in our attempt to propose a dynamical origin of the length correlations. 4.4.2. The dynamical origin of the correlations As was already mentioned, the idea that periodic orbit correlations exist originates from the quantum theory of spectral statistics which is based on trace formulas. The classical correlations are shown to be a manifestation of a fundamental duality between the quantum and the classical descriptions [9,11]. However, the e!ect is purely classical, and hence should be explained in classical terms, without any reference to the quantum mechanical analogue. The essential point is to "nd the classical origin of the partition of the periodic orbits to independent and uncorrelated families, as was explained in the previous section. So far, all the attempts to "nd the classical roots of these correlations failed, and till now there is no universal theory which provides the classical foundations for the e!ect. For the Sinai billiard in 3D there seems to exist a physical}geometrical explanation, which is consistent with our data, and which is supported by further numerical tests. Consider the Sinai billiard with a sphere with a vanishingly small radius. In this case, all the periodic orbits which are encoded by words = built of the same letters w are isometric, i
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independently of the ordering of the letters or the attached symmetry element g( . This phenomenon can be clearly seen in the spectrum of lengths corresponding to R"0.02 in Fig. 20. In this case, it is clear that the spectrum of lengths is a union of `familiesa of periodic orbits, each family is characterized by a unique set of building blocks w , which are common to the family members. i When the radius R increases and becomes comparable to the linear dimension of the billiard, the approximate isometry and the resulting correlations breaks down, and one should use a more re"ned and restrictive de"nition of a family. The aim is to "nd a partition to families which will restrict the membership in a family to the smallest set, without losing any of the correlation features. The most restrictive de"nition of a family in the present context will be to include all the periodic orbits which share the same ="(w , w ,2, w ) part of the code and have di!erent 1 2 n admissible g( symmetry elements. Words which are built of the same letters but in a di!erent order de"ne di!erent families. Since there are 48 possible g( 's, each family consists of at most 48 members and will be denoted by X(=). It should also be noted that the signs of the weights AI within a family j do not change with R since they re#ect the parity of g( . The partition of the set of periodic orbits in families is not particular to just a few orbits, but rather, is valid for the entire set. This partition is the proposed source of the correlations that were observed in the form factor. This concept is illustrated in Fig. 25, and graphic representations of two families are displayed in Fig. 26. The most outstanding feature which emerges from Fig. 26 is that the orbits occupy a very narrow volume of phase-space throughout most of their length, and they fan out appreciably only at a single sphere. The above arguments suggest that the main source of correlations are the similarities of orbits within each family X(=). To test this argument we performed a numerical experiment, in which we excluded the inter-family terms of the form factor, leaving only the intra-family terms. This excludes family}family correlations and maintains only correlations within the families. The results are shown in Fig. 24 (denoted by `Neglecting cross-family contributionsa). The obvious observation is that the form factors were only slightly a!ected, proving that periodic-orbit correlations do not cross family lines! Thus, the main source of correlations is within the families X(=). We mention that very similar results are obtained if inter-family sign randomization is applied instead of the exclusion of cross-terms. We note, that in most cases a periodic orbit and its time-reversal conjugate do not belong to the same family. Thus, neglecting the cross-family terms leads to partial breaking of time-reversal, which we compensated for by rectifying the intra-family form factor such that it will have the same asymptotic value as the full one. It is interesting to check whether the correlations are due to the lengths or due to the size of the amplitudes. To examine that, we not only neglected the cross-family terms, but also replaced the amplitudes AI within each family by constants multiplied by the original signs, such that the overall j asymptotic contribution of the family does not change. The results are also plotted in Fig. 24 (denoted as `Length correlations onlya). The resulting (recti"ed) form factors display slightly diminished correlations. However there is no doubt that almost all correlations still persist. This proves that the correlations between the magnitudes of the weights play here a relatively minor role, and the correlations are primarily due to the lengths. There are a few points in order. First, the numerical results presented here concerning the classical correlations are similar to those of Ref. [11]. However, here we considered the classical mapping rather than the #ow, and this reduces the numerical #uctuations signi"cantly. Using the mapping also enables the quantitative comparison to the semiclassical theory, which will be discussed in Section 7. Second, it is interesting to enquire whether the average number of family
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Fig. 25. Two periodic orbits which are members of the same family of the quarter 2D SB. The two periodic orbits have the same =, hence they re#ect o! the same discs. But they correspond to two diwerent symmetry elements, and hence are di!erent. For simplicity, the illustration is made for the 2D SB, but the same principle applies also to the 3D Sinai billiard. Left: unfolded representation, right: standard representation. Fig. 26. Two families of periodic orbits of the 3D SB, represented in the unfolded representation of the SB. The faint spheres do not participate in the code.
members N (n; R) increases or decreases with n. Since, if it decreases, our explanation of the origin &!. of correlations becomes invalid for large n. The numerical results clearly indicate that N (n; R), &!. computed as a weighted average with the classical weights, increases with n, which is encouraging. For example, for the case R"0.4 we obtained N "9.64, 18.31, 21.09, 28.31 for n"1, 2, 3, 4, &!. respectively. Thus, we were able to identify the grouping of orbits into `familiesa with the same code word = but with di!erent symmetry g( as the prominent source of the classical correlations in the 3D
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Sinai billiard. In each family the common geometric part of the code = sets the mean length and the di!erent group elements g( introduce the modulations. This pattern repeats for all the families, but the lengths of di!erent families are not correlated. This "nding conforms very well with the general scheme which was proposed to explain the typical correlations in the classical spectrum [9,74]. However, a classical derivation of a quantitative expression for the correlations length j is yet to be done. 4.4.3. Length correlations in the 3-Torus The ideas developed above about the correlations between periodic orbits in the fully chaotic billiard, have an analogue in the spectrum of lengths of periodic tori in the integrable case of the 3-torus. In Section 3 we studied the quantum 3-torus of size S and showed in Section 3.1.1 that due to number theoretical degeneracies, the quantum form factor is not Poissonian. The form factor displays a negative (repulsive) correlation which levels o! at qH"1/c"1/(2Sk). This can be transcribed into an expression for the correlation length of the classical spectrum in the following way. Expressing qH in units of length we obtain ¸H"2pdM (k)qH"S2k/2p .
(82)
Consequently, kH(¸)"2p¸/S2 ,
(83)
from which we read o! j(¸)"S2/¸ .
(84)
Since the lengths of the periodic orbits are of the form ¸ "S]Jinteger, the minimal spacing j between periodic orbits near length ¸ is D (¸)"S2/2¸ , .*/ and therefore
(85)
j(¸)"2D (¸) . (86) .*/ In other words, the classical correlation length of the 3-torus coincides (up to a factor 2) with the minimal spacing between the periodic orbits. Therefore, j(¸) indeed signi"es the correlation length scale between periodic orbits, which is imposed by their number-theoretical structure.
5. Semiclassical analysis In the previous sections we accumulated information about the quantum spectrum and about the periodic orbits of the 3D Sinai billiard. The stage is now set for a semiclassical analysis of the billiard. We shall focus on the analysis of the semiclassical Gutzwiller trace formula [2] that reads in the case of the Sinai billiard: = d(k), + d(k!k )+dM (k)#d (k)#+ A cos(k¸ ) . n "" j j n/1 PO
(87)
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The quantum spectral density on the RHS is expressed as the sum of three terms. The term dM is the smooth density of states (see Appendix I). The term d consists of the contributions of the "" non-generic bouncing-ball manifolds. It contains terms of the form (31) with di!erent prefactors (which are possibly 0) due to partial or complete shadowing of the bouncing-ball family by the sphere. The last term is the contribution of the set of generic and unstable periodic orbits, where ¸ denote their lengths and A are semiclassical amplitudes. One of the main objective of the j j present work was to study the accuracy of (87) by a direct numerical computation of the di!erence between its two sides. This cannot be done by a straightforward substitution, since three obstacles must be removed: f The spectrum of wavenumbers k was computed for the fully desymmetrized Sinai billiard. To n write the corresponding trace formula, we must remember that the folding of the Sinai torus into the Sinai billiard introduces new types of periodic orbits due to the presence of symmetry planes, edges and corners. Strictly speaking, the classical dynamics of these orbits is singular, and becomes meaningful only if proper limits are taken. As examples we mention periodic orbits that bounce o! a corner, or that are wholly con"ned to the symmetry planes. These periodic orbits are isolated and unstable, and should not be confused with the bouncing-ball families which are present both in the ST and in the SB. For periodic orbits that re#ect from a corner but are not con"ned to symmetry planes, the di$culty is resolved by unfolding the dynamics from the SB to the ST as was described in the previous section. Periodic orbits which are con"ned to symmetry planes are more troublesome since there is more than one code word = K which correspond to the same periodic orbit. We denote the latter as `impropera. The 3D Sinai billiard is abundant with improper periodic orbits, and we cannot a!ord treating them individually as was done, e.g. by Sieber [60] for the 2D hyperbola billiard. Rather, we have to "nd a general and systematic method to identify them and to calculate their semiclassical contributions. This will be done in the next subsection. (The semiclassical contributions of the improper periodic manifolds for the integrable case R"0 were discussed in Section 3.1.) f As it stands, Eq. (87) is a relation between distributions rather than between functions, and hence must be regulated when dealing with actual computations. Moreover, even though our quantum and classical databases are rather extensive, the sums on the two sides of the equation can never be exhausted. We overcome these problems by studying the weighted `length spectruma obtained from the trace formula by a proper smoothing and Fourier transformation. It is de"ned in Section 5.2. f Finally, we must "nd ways to rid ourselves from the large, yet non-generic contributions of the bouncing-ball families. This was achieved using rather elegant tricks which are described in Sections 5.4 and 5.5 below. 5.1. Semiclassical desymmetrization To derive the spectral density of the desymmetrized Sinai billiard we make use of its expression in terms of the (imaginary part of the) trace of the SB Green function. This Green function satis"es the prescribed boundary conditions on all the boundaries of the fundamental domain, and the trace is taken over its volume. In the following we shall show how to transform this object into a trace over the volume of the entire ST, for which all periodic orbits are proper (no symmetry planes). This
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will eliminate the di$culty of treating the improper orbits. To achieve this goal we shall use group-theoretical arguments [31,60,75}77]. The "nal result is essentially contained in [78]. When desymmetrizing the ST into SB, we have to choose one of the irreps of O to which the h eigenfunctions of the SB belong (see Section 2.2.3). We denote this irrep by c. We are interested in the trace of the Green function of the SB over the volume of the SB which is essentially the density of states: ¹,Tr G(c) (r, r@) . (88) SB SB One can apply the projection operation [31] and express G(c) using the Green function of the ST: SB 1 G(c) (r, r@)" + s(c)H(g( )G (r, g( r@) , (89) SB ST l ( c g|Oh where s(c)(g( ) is the character of g( in the irrep c and l is the dimension of c. It can be veri"ed that the c above G satis"es the inhomogeneous Helmholtz equation with the correct normalization, and it SB is composed only of eigenfunction that transform according to c. Thus, 1 ¹" + s(c)H(g( )Tr G (r, g( r@) . (90) SB ST l ( c g|Oh To relate Tr with Tr we use the relation SB ST G (r, r@)"G (hK r, hK r@) ∀hK 3O (91) ST ST h which can be proven by e.g. using the spectral representation of G . In particular, we can write ST 1 (92) G (r, r@)" + G (hK r, hK r@) . ST ST 48 K h|Oh Combining (92) with (90) we get 1 + s(c)H(g( )Tr G (hK r, hK g( r@) ¹" SB ST 48l ( K c g,h|Oh 1 " + s(c)H(hK g( hK ~1)Tr G (hK r, (hK g( hK ~1)hK r@) SB ST 48l ( K c g, h|Oh 1 " + s(c)H(kK )Tr G (hK r, kK hK r@) . (93) SB ST 48l K K c h,k|Oh To obtain the second line from the "rst one, we recall that the character is the trace of the irrep matrix, and we have in general Tr(ABC)"Tr(CAB), therefore s(g( )"s(hK g( hK ~1). The third line is obtained from the second one by "xing hK and summing over g( . Since hK g( hK ~1"hK g( hK ~1 Q g( "g( 1 2 1 2 the summation over g( is a rearrangement of the group. We now apply the geometrical identity
P
P
+ d3r f (hK r)" d3r f (r) ST h|Oh SB K
(94)
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to cast (93) into the desired form: 1 + s(c)H(g( )Tr G (r, g( r@) , ¹" ST ST 48l ( c g|Oh
(95)
where we relabelled kK as g( for convenience. The result (95) is the desired one, since ¹ is now expressed using traces over ST which involve no symmetry planes. Semiclassically, formula (95) means that we should consider all the periodic orbits of the ST modulo a symmetry element g( to get the density of states of the SB. Therefore, the di$culty of handling improper orbits is eliminated, since in the ST all of the isolated periodic orbits are proper. Let us elaborate further on (95) and consider the various contributions to it. A proper periodic orbit of the SB with code (=; g( ) has 48 realizations in the ST which are geometrically distinct. They are obtained from each other by applying the operations of O . These conjugate periodic orbits are h all related to the same g( and thus have the same lengths and monodromies. Consequently they all have the same semiclassical contributions. Hence, their semiclassical contribution to ¹ is the same as we would get from naively applying the Gutzwiller trace formula to the SB, considering only proper periodic orbits. This result is consistent with our "ndings about classical desymmetrization (Section 4.2 above). For the improper periodic orbits there is a di!erence, however. There are genuine semiclassical e!ects due to desymmetrization for unstable periodic orbits that are con"ned to planes or to edges, notably large reduction in the contributions for Dirichlet conditions on the symmetry planes. To demonstrate this point, let us consider in some detail an example of the periodic orbit that traverses along the 8-fold edge AE in Fig. 1. For the ST (no desymmetrization) its semiclassical contribution is A "R/2p . 1
(96)
For the SB there are 8 code words that correspond to the periodic orbit(s) which traverses along this 8-fold edge. A calculation yields for the semiclassical contribution:
C
A BD
R 2!b A " 2$2J1!2b$b 8 8p 1!b
,
(97)
where b,R/S. The upper sign is for the case of the totally symmetric irrep, and the lower one for the totally antisymmetric irrep. In the antisymmetric case we get for b;1: A /A +(b/2)4 , 8 1
(98)
which means that the desymmetrization greatly reduces the contribution of this periodic orbit in case of Dirichlet boundary conditions on the planes. For the case of our longest spectrum (R"0.2, S"1) this reduction factor is approximately 2]10~4 which makes the detection of this periodic orbit practically impossible. For Neumann boundary conditions the contribution is comparable to the ST case and is appreciable. Formula (95) together with the algorithm described above are the basis for our computations of the semiclassical contributions of the periodic orbits of the SB. Speci"cally, the contribution of
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a code = K is given by ¸10K K K s(c)H(g( )p K W , (99) AK " W W W pl r Ddet(I!M K )D1@2 c W where ¸10K is the length of the periodic orbit, K K " (number of distinct realizations of = K W under O W)/48 and r is the repetition index. The term p K is due to the re#ections from the spheres h W and is determined by the boundary conditions on them. For Neumann boundary conditions p K "1, for Dirichlet boundary conditions p K "(!1)n, where n is the number of bounces. W W 5.2. Length spectrum Having derived the explicit expression for the semiclassical amplitudes for the SB (99), we are in position to transform the trace formula (87) to a form which can be used for numerical computations which test its validity. We de"ne the length spectrum as the Fourier transform of the density of states:
P
1 1 `= D(l ), d(k)e*kl dk" + e*kn l . J2p ~= J2p n
(100)
For convenience, we de"ne d(!k),d(k)Nk "!k and the sum is carried out for all ~n n n3ZCM0N. Using the trace formula (87) we obtain semiclassically
S
p A [d(l!¸ )#d(l#¸ )] . D (l)"DM (l)#D (l)#+ j j 4# "" 2 j (101) PO In the above DM (l) is a singularity at l"0 which is due to the smooth density of states. The length spectrum is sharply peaked near lengths of periodic orbits hence its name. To regularize (101) such that it can be applied to "nite samples of the quantum spectrum, we use a weight function and construct the weighted length spectrum [79]:
P
1 1 `= w(k!k@)d(k@)e*k{l dk@" D(w)(l; k), + w(k!k )e*kn l , n J2p n J2p ~=
(102)
where w is a weight function (with an e!ective "nite support) that is concentrated at the origin. The corresponding semiclassical expression is: A (103) D(w)(l; k)"DM (w)(l )#D(w)(l )#+ j [w( (l!¸ )e*k(l~Lj )#w( (l#¸ )e*k(l`Lj )] , j j 4# "" 2 PO where w( (l),(1/J2p):`= w(k)e*kl dk is the Fourier transform of w(k). ~= In principle, d(k) and D(l) contain the same information and are therefore equivalent. However, for our purposes, it is advantageous to use the length spectrum D(l ) (and in practice D(w)(l; k)) rather than the spectral density (87) for the following reasons: f The regularized semiclassical length spectrum, D(w), is absolutely convergent for suitably chosen 4# weight functions [79] (e.g. Gaussians). This is in contrast with the original trace formula (87).
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Fig. 27. Absolute value of the quantal length spectra DD(w)D with a Gaussian window, p"30, compared to the theoretical prediction (105). The location of the shortest bouncing ball is indicated by the vertical line.
f There is an exact mathematical result [80] that states that for billiards the singular supports of D(l) and of D (l) are the same, if the in"nite spectra are considered. This exact quantum}classical 4# result speci"cally relates to the length spectra. It is therefore useful to identify and treat transient e!ects (e.g. di!raction contributions) for "nite spectra using the length representation. f The trace formula (87) can be considered as a means to quantize a chaotic system, since it expresses the quantal density of states in terms of the classical length spectrum. However, in practice, this is not convenient because the semiclassical amplitudes are only leading terms in asymptotic series in k (equivalently in +). For "nite values of k there can be large deviations due to sub-leading corrections [6,7] and also due to signi"cant di!raction corrections [8,81,38]. Treating the trace formula the other way (`inverse quantum chaologya) is advantageous because the quantal amplitudes have all equal weights 1. f The appearance of peaks in both d(k) and D(l) comes as a result of the constructive interference of many oscillatory contributions. Any missing or spurious contribution can blur the peaks (see Fig. 27 for an example with a single energy level missing). For the energy levels we have a good control on the completeness of the spectrum due to Weyl's law (see Appendix I). As discussed above, this is not the case for periodic orbits where we do not have an independent veri"cation of their completeness. Hence it is advantageous to use the energy levels which are known to be complete in order to reproduce peaks that correspond to the periodic orbits. f For the Sinai billiard the low-lying domain of the spectrum is peculiar due to e!ects of desymmetrization (see Section 2.4). For Dirichlet boundary conditions on the planes, the levels k R(9 are very similar to those of the integrable case (R"0). The `chaotica levels for which n the semiclassical approximation is valid (k R'9) thus start higher up, which makes the n semiclassical reproduction of them very di$cult in practice even with the use of Berry}Keating resummation techniques [69]. On the other hand, using the quantum levels we can reproduce a few isolated length peaks, as will be seen in the sequel. In the following we shall demonstrate a stringent test of the completeness and of the accuracy of the quantal spectrum using the length spectrum. Then we shall investigate the agreement between
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the quantal and the semiclassical length spectra. We shall employ a technique to "lter the e!ects of the bouncing balls, such that only generic contributions remain. 5.3. A semiclassical test of the quantal spectrum In the following we use the length spectrum in order to develop a stringent test of the completeness and integrity of the quantal spectrum. This supplements the integrity and completeness analysis of the quantal spectrum done in Sections 2.4 and 2.5. The idea is to focus on an isolated contribution to the length spectrum that can be compared to an analytical result. In Section 3.1 we discussed the integrable billiard (R"0) and observed that there are contributions to the density of states due to isolated but neutral periodic orbits. The shortest periodic orbit of this kind has length S/J3+0.577S and was shown in Fig. 6. Its contribution must prevail for R'0 until it is shadowed by the inscribed sphere, which occurs at R"S/J6+0.41S. Being the shortest bouncing ball, it is isolated from the lengths of other bouncing balls. The only generic periodic orbit, for R"0.2, which comes near this length, is the shortest unstable periodic orbit of length 0.6S. However, for Dirichlet boundary conditions on the planes the latter is practically eliminated due to symmetry e!ects as was discussed in Section 5.1. Since other periodic orbits are fairly distant, this shortest bouncing ball is an ideal test-ground of the length spectrum. Using (31) and a Gaussian window:
C
D
1 (k!k@)2 w(k!k@)" exp ! , 2p2 J2pp2
(104)
one obtains the contribution of the shortest bouncing ball to the length spectrum: e*k(l~S@J3) D(w) (l; k)" exp[!(l!S/J3)2p2/2] . 4#,4)035%45v"" (6p)3@2
(105)
Due to its isolation, one expects that the shortest bouncing ball gives the dominant contribution to the length spectrum near its length. Thus, for l+S/J3, one has DD(8)D+DD(w) D. The latter is 4# 4#,4)035%45v"" independent of k. To test the above relation, we computed the quantal length spectrum D(w) for R"0 and R"0.2 for two di!erent values of k, and compared with (105). The results are shown in Fig. 27, and the agreement is very satisfactory. To show how sensitive and stringent this test is, we removed from the R"0.2 quantal spectrum a single level, k "175.1182, and studied the e!ect on the length spectrum. As is seen in the 1500 "gure, this is enough to severely damage the agreement between the quantum data and the theoretical expectation. Therefore, we conclude that our spectrum is complete and also accurate to a high degree. 5.4. Filtering the bouncing-balls I: Dirichlet}Neumann diwerence The ultimate goal of our semiclassical analysis is to test the predictions due to Gutzwiller's trace formula. Since the 3D Sinai is meant to be a paradigm for 3D systems, we must remove the in#uence of the non-generic bouncing-ball families and "nd a way to focus on the contributions of the generic and unstable periodic orbit. This is imperative, because in the 3D Sinai billiard the
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Fig. 28. Quantal length spectra for R"0 and 0.2 compared to semiclassical length spectrum for R"0.2 that contains only generic, unstable periodic orbits. In all cases k"160, p"30. The locations of the bouncing balls are indicated: daggers for 2-parameter bouncing balls that occupy 3D volume in con"guration space, stars for 2D bouncing balls and crosses for 1D bouncing balls.
bouncing balls have contributions which are much larger than those of the generic periodic orbits. Inspecting Eqs. (31) and (99), we "nd that the contributions of the leading-order bouncing balls are stronger by a factor of k than those of the generic periodic orbits. This is worse than in the 2D case, where the factor is Jk. To show how overwhelming is the e!ect of the bouncing balls, we plot in Fig. 28 the quantal lengths spectra DD(w)D for R"0 and 0.2 (Dirichlet everywhere) together with DD(w)D which contains contributions only from generic and unstable periodic orbits. One observes 4# that all the peaks in the quantal length spectra are near lengths of the bouncing balls. Contributions of generic periodic orbits are completely overwhelmed by those of the bouncing balls and cannot be traced in the quantal length spectrum of R"0.2. Also, we see that for R"0.2 the peaks are in general lower than for R"0. This is because of the (partial or complete) shadowing e!ect of the inscribed sphere that reduces the prefactors of the bouncing balls as R increases. In the case of the 2D Sinai billiard it was possible to analytically "lter the e!ect of the bouncing balls from the semiclassical density of states [37,17]. In three dimensions this is much more di$cult. The functional forms of the contributions to the density of states of the bouncing balls are given in (31), but it is a di$cult geometric problem to calculate the prefactors which are proportional to the cross sections of the bouncing-ball manifolds in con"guration space. The desymmetrization makes this di$culty even greater and the calculations become very intricate. In addition, there is always an inxnite number of bouncing-ball manifolds in the 3D Sinai. This is in contrast with the 2D Sinai, in which a "nite (and usually quite small for moderate radii) number of bouncing-ball families exist. All this means that an analytical subtraction of the bouncing-ball contributions is very intricate and vulnerable to errors which are di$cult to detect and can have a devastating e!ect on the quantal-semiclassical agreement. In order to circumvent these di$culties, we present in the following an e$cient and simple method to get rid of the bouncing balls. The idea is simple: the bouncing balls are exactly those periodic orbits that do not re#ect from the sphere. Therefore, changing the boundary conditions on the sphere does not a!ect the bouncing-ball contributions. Thus, the semiclassical density of states
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Fig. 29. Dirichlet}Neumann di!erence length spectra for R"0.2, with k"100, p"30. The semiclassical length spectrum is computed according to (108). The daggers, stars and crosses indicate the positions of the bouncing balls (refer to Fig. 28) and the vertical bars indicate the positions of the generic, unstable periodic orbits.
for Dirichlet/Neumann boundary conditions on the sphere is d "dM #d #d(04#) . (106) D@N D@N "" D@N The di!erence d !d is hence independent (in leading approximation in k) of d and has the D N "" standard form of a trace formula: ,d (k)!d (k)"[dM (k)!dM (k)]#+ (A(D)!A(N)) cos(k¸ ) . (107) D N D N j j j PO Here A(D), A(N) are the coe$cients that correspond to Dirichlet and Neumann cases, respectively. In j j fact, for Dirichlet, each re#ection with the sphere causes a sign change, while for Neumann there are no sign changes. Therefore, d
D~N
G
2A(D) odd number of reflections , j A(D~N),A(D)!A(N)" j j j 0 even number of reflections ,
(108)
and we expect to observe in the length spectrum of d contributions only due to generic periodic D~N orbits with an odd number of re#ections. The results of the numerical computations are presented in Fig. 29 where we compare the quantal (exact) vs. semiclassical (theoretical) length spectra. We observe on the outset that in contrast to Fig. 28 the quantal and semiclassical length spectra are of similar magnitudes and the bouncing balls no longer dominate. The peaks near lengths that correspond to the bouncing balls are greatly diminished, and in fact we see that the peak corresponding to the shortest bouncing ball (l+0.577) is completely absent, as predicted by the theory. Even more important is the remarkable agreement between the quantal and the semiclassical length spectra which one observes near various peaks (e.g. near l"0.75, 1.25, 2). Since the semiclassical length spectrum contains only generic contributions from unstable periodic orbits, this means that we demonstrated the existence and the correctness of these Gutzwiller contributions in the quantal levels. Therefore, one can say that at least as far as length spectra are concerned, the semiclassical trace formula is partially successful. There are, however, a few locations for which there is no agreement between the quantal and the semiclassical length spectra.
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The places where this discrepancy takes place are notably located near 3D bouncing-ball lengths. This suggests that there are `remnantsa of the bouncing-ball contributions that are not "ltered by the Dirichlet}Neumann di!erence procedure. It is natural to expect that these remnants are most prominent for the strongest (3D) bouncing balls. The origin of these remnants are the periodic orbits that are exactly tangent to the sphere. As an example, consider the 3D bouncing-ball families that are shown in Fig. 2 (upper part). The corresponding tangent orbits constitute a 1-parameter family that surrounds the sphere like a `coronaa. For a single tangent traversal their contributions acquire opposite signs for Dirichlet and Neumann boundary conditions on the sphere. Hence, the Dirichlet}Neumann di!erence procedure still include these contributions which is apparent in the large discrepancy near l"1. For two tangent traversals the Dirichlet and Neumann contributions have the same sign and hence cancel each other. This is indeed con"rmed in Fig. 29 where we observe that near l"2 there is no discrepancy between the quantal and the semiclassical length spectra. The above-mentioned tangent orbits belong to the set of points in phase space in which the classical mapping is discontinuous. Semiclassically they give rise to di!raction e!ects. Tangent orbits were treated for the 2D case in our work [38,81]. To eliminate their e!ects we need to sharpen our tools and to "nd a better "ltering method than the Dirichlet}Neumann di!erence procedure. This is performed in the following using mixed boundary conditions. 5.5. Filtering the bouncing-balls II: mixed boundary conditions The idea behind the Dirichlet}Neumann di!erence method was to subtract two spectra which di!er only by their boundary conditions on the sphere. This can be generalized, if one replaces the discrete `parametera of Dirichlet or Neumann conditions by a continuous parameter a, and studies the di!erences of the corresponding densities of states d(k; a )!d(k; a ). In Section 2 we discussed 1 2 the mixed boundary conditions regarding the exact quantization of the 3D SB and gave the a-dependent expressions for the quantal phase shifts. Mixed boundary conditions were extensively discussed in [82,83]. To include the mixed boundary conditions in the semiclassical trace formula we generalize the results of Berry [15]. There, he derived the trace formula for the 2D Sinai billiard from an expansion of the KKR determinant in terms of traces. If one uses the 3D KKR matrix with (8) and performs a similar expansion, the result is a modi"cation of the Gutzwiller terms as follows: A cos(k¸ )PA cos(k¸ #n p#/ ) , (109) j j j j j j nj i cot a / "(!2) + arctan . (110) j k cos h(j) i i/1 Here A are the semiclassical coe$cients for the Dirichlet conditions on the sphere (cf. Eq. (99)) and j n counts the number of re#ections from the sphere. The angles h(j) are the re#ection angles from the j i sphere measured from the normal of the jth periodic orbit. It is instructive to note that the phases (110) above are exactly the same as those obtained by a plane wave that re#ects from an in"nite plane with mixed boundary conditions (2). This is consistent with the local nature of the semiclassical approximation. A prominent feature of the mixed boundary conditions which is manifest in
A
B
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(109) is that they do not a!ect the geometrical properties (length, stability) of the periodic orbits. Rather, they only cause a change of a phase which depends on the geometry of the periodic orbit. This is due to the fact that the mixing parameter a has no classical analogue. The invariance of periodic orbits with respect to a renders the mixed boundary conditions an attractive parameter for, e.g. investigations of parametric statistics. This was discussed and demonstrated in detail in [83]. We are now in a position to apply the mixed boundary conditions to get an e$cient "ltering of the bouncing-ball contributions. We "rst note that if we "x i, then the levels are functions of a: k "k (a). Let us consider the derivative of the quantal counting function at a"0: n n RN(k; a) R dI (k), "+ H[k!k (a)] n Ra Ra a/0 a/0 n dk (a) d(k!k ) , (111) "+ ! n n da a/0 n where k "k (0) are the Dirichlet eigenvalues. Hence, the quantity dI is a weighted density of states n n with delta-peaks located on the Dirichlet eigenvalues. The semiclassical expression for dI does not contain the leading contribution of the bouncing balls, since this contribution is independent of a. The semiclassical contributions of the isolated periodic orbits to dI are of the form A B cos(k¸ ), where j j j 2k nj B " + cos h(j) . (112) j i i i/1 This is easily derived from (109) and (110). Since the re#ection angles h(j) are in the range [0, p/2], i the coe$cient B vanishes if and only if h(j)"p/2 for all i"1,2, n , which is an exact tangency. j i j Therefore, exactly tangent periodic orbits are also eliminated by the derivative method. This is the desired e!ect of the mixed boundary conditions method that serves to further clean the spectrum from sub-leading contributions of the bouncing balls. We summarize Eqs. (111) and (112):
A
K
B
K
A
B
Rk . (113) dI (k)"+ v d(k!k )+(4.005))#+ A B cos(k¸ ), v , ! n n n 5%3. j j j n Ra a/0 n PO To check the utility of dI and to verify (113) we computed both sides of (113) for R"0.2 and i"100. The quantal spectrum was computed for a"0.003 and the derivatives v were obtained n by the "nite di!erences from the a"0 (Dirichlet) spectrum. The coe$cients B were extracted from j the geometry of the periodic orbits. In Figs. 30 and 31 the length spectra are compared. The agreement between the quantal and the semiclassical data is impressive, especially for the lower l-values. There are no signi"cant remnants of peaks near the bouncing-ball locations, and the peaks correspond to the generic and unstable periodic orbits. This demonstrates the utility of using dI as an e$cient means for "ltering the spectrum from the non-generic e!ects. The quantal-semiclassical agreement of the length spectra is not perfect, however, and it is instructive to list possible causes of this disagreement. We "rst recall that the semiclassical amplitudes A are the leading terms in an asymptotic series, hence we expect corrections of order j 1/k to the weights of periodic orbits. They are denoted as + corrections and were treated in detail by Gaspard and Alonso [6] and by Alonso and Gaspard [7]. In our case, however, 1/k+1/100 and
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Fig. 30. Length spectra for the mixed boundary conditions derivative method (113). Data are for R"0.2, k"150, p"30, i"100. The dashed line represents DD(w)(l)!D(w)(l)D. The daggers, stars and crosses indicate the positions of the 4# bouncing balls (refer to Fig. 28) and the vertical bars indicate the positions of the generic, unstable periodic orbits. Fig. 31. Continuation of Fig. 30 to 2.54l45. We did not indicate the locations of unstable periodic orbits due to their enormous density.
these corrections are not expected to be dominant. More important are di!raction corrections which are also "nite k e!ects that stem from the existence of a concave component (the sphere) in the billiard. Several kinds of di!raction corrections to the trace formula were analysed for 2D billiards. Vattay et al. [8] considered creeping orbits, and we considered in [38,81] penumbra corrections. (The penumbra is the region in phase space which is close to tangency: Dl!kRD+(kR)1@3, where +l is the angular momentum.) We list the various di!raction corrections in the following: Creeping orbits: These are orbits which are classically forbidden. They `creepa over concave parts of the billiard, and their semiclassical contribution is exponentially small in k1@3, which should be negligible for the k values that we consider. Exactly tangent orbits: These were already mentioned above, and we showed that their contributions are eliminated to a large extent by the mixed boundary conditions procedure. For 2D systems we found, however, that this is true in leading order only, and there are small remnants of the tangent orbits in the weighted density dI [38]. The magnitude of the remnants in 2D is O(1/Jk), which is smaller than O(k0) of a generic unstable periodic orbit. In 3D, a similar analysis shows that the remnants of each family of tangent orbits is O(k0) which is the same magnitude as for unstable periodic orbits. Reviewing Figs. 30 and 31, we can observe some of the peaks of the quantalsemiclassical di!erence near lengths that correspond to exactly tangent orbits. Unstable and isolated periodic orbits that traverse the penumbra: We have shown in [81,38] that for periodic orbits which just miss tangency with a concave component of the billiard boundary, there is a correction to the semiclassical amplitude A which is of the same magnitude as A itself. j j These O(1) di!raction corrections are the most important corrections to the trace formula for generic billiards. For periodic orbits which re#ect at an extreme forward direction from a concave component, the amplitude A is very small due to the extreme classical instability. If we include j
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di!raction corrections, the semiclassical contributions of these orbits get much larger. Therefore, the semiclassical contributions of periodic orbit that traverse the penumbra must be radically corrected. Moreover, we found that if one considers all the periodic orbits up to the Heisenberg length ¸ ,2pdM (k) (necessary to obtain a resolution of one mean level spacing), then almost all of H the periodic orbits are vulnerable to penumbra di!raction corrections. Classically forbidden periodic orbits that traverse the shaded part of the penumbra: Penumbra di!raction e!ects lead to semiclassical contributions from periodic orbits that slightly traverse through a concave component. Since they do not relate to classically allowed orbits, they represent new contributions to the trace formula rather than corrections of existing ones. Their magnitudes are comparable to those of generic unstable periodic orbits. The above list of corrections, which was compiled according to studies of 2D billiards, suggests that there is a wealth of e!ects that must be considered if one wishes to go beyond the Gutzwiller trace formula. It is very di$cult to implement these corrections systematically even for 2D billiards, and it goes beyond the scope of the present work to study them further for the 3D Sinai billiard. We mention in passing that except exact tangency, the penumbra e!ects are transient and depend on k. According to the mathematical theorem [80] mentioned above, the quantal and the semiclassical length spectra are asymptotically the same. The signi"cance of our "ndings in this section is that we have shown that the quantal-semiclassical agreement is achieved already for "nite and moderate values of k and that the corrections are not very large (for the l-range we looked at). This is very encouraging, and justi"es an optimistic attitude to the validity of the semiclassical approximation in 3D systems. However, obtaining accurate energy levels from the trace formula involves many contributions from a large number of periodic orbits. Therefore, one cannot directly infer at this stage from the accuracy of the peaks of the length spectrum to the accuracy of energy levels. There is a need to quantify the semiclassical error and to express it in a way which makes use of the above semiclassical analysis. This is done in Section 6.
6. The accuracy of the semiclassical energy spectrum One of the most important applications of the trace formula is to explain the spectral statistics and their relation to the universal predictions of Random Matrix Theory (RMT) [4,5]. However, a prerequisite for the use of the semiclassical approximation to compute short-range statistics is that it is able to reproduce the exact spectrum within an error comparable to or less than the mean level spacing! This is a demanding requirement, and quite often it is doubted whether the semiclassical approximation is able to reproduce precise levels for high-dimensional systems on the following grounds. The mean level spacing depends on the dimensionality (number of freedoms) of the system, and it is O(+d) [3]. Gutzwiller [2] quotes an argument by Pauli [87] to show that in general the error margin for the semiclassical approximation scales as O(+2) independently of the dimensionality. Applied to the trace formula, the expected error in units of the mean spacing, which is the "gure of merit in the present context, is therefore expected to be O(+2~d). We shall refer to this as the `traditional estimatea. It sets d"2 as a critical dimension for the applicability of the semiclassical trace formula and hence for the validity of the conclusions which are drawn from it.
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The few systems in d'2 dimensions which were numerically investigated display spectral statistics which adhere to the predictions of RMT as accurately as their counterparts in d"2 [53,54,19]. Thus, the traditional estimate cannot be correct in the present context, and we shall explain the reasons why it is inadequate. In this section we shall develop measures for the accuracy of the semiclassical energy levels. We shall then derive formulas to evaluate these measures. Using our quantal and classical (periodic orbits) databases for the 2D and 3D Sinai billiards, we shall apply the formulas and get numerical bounds for the semiclassical errors. The problem of the accuracy of the energy spectrum derived from the semiclassical trace formula was hardly discussed in the literature. Gutzwiller quotes the traditional estimate of O(+2~d) [2,75]. Gaspard and Alonso [6], Alonso and Gaspard [7] and Vattay et al. [8] derived explicit and generic + corrections for the trace formula, but do not address directly the issue of semiclassical accuracy of energy levels. Boasman [88] estimates the accuracy of the BIM [14] for 2D billiards in the case that the exact kernel is replaced by its asymptotic approximation. He "nds that the resulting error is of the same magnitude as the mean spacing, in agreement with the traditional estimate. However, the dependence of the semiclassical error on the dimensionality is not established. We also mention a recent work by Dahlqvist [89] in which the semiclassical error due to penumbra (di!raction) e!ects is analytically estimated for the 2D Sinai billiard. The results are compatible with the ones reported here. 6.1. Measures of the semiclassical error In order to de"ne a proper error measure for the semiclassical approximation of the energy spectrum one has to clarify a few issues. In contrast with the EBK quantization which gives an explicit formula for the spectrum, the semiclassical spectrum for chaotic systems is implicit in the trace formula, or in the semiclassical expression for the spectral determinant. To extract the semiclassical spectrum we recall that the exact spectrum, ME N, can be obtained from the exact n counting function: = N(E), + H(E!E ) n n/1 by solving the equation
(114)
N(E )"n!1, n"1, 2,2 . (115) n 2 In the last equation, an arbitrarily small amount of smoothing must be applied to the Heavyside function. In analogy, one obtains the semiclassical spectrum ME4#N as [50] n N (E4#)"n!1, n"1, 2,2 , (116) 4# n 2 where N is a semiclassical approximation of N. Note that N with which we start is not 4# 4# necessarily a sharp counting function. However, once ME4#N is known, we can `rectifya the smooth n N into the sharp counting function Nj [5]: 4# 4# = Nj(E), + H(E!E4#) . (117) 4# n n/1
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The most obvious choice for N is the Gutzwiller trace formula [2] truncated at the Heisenberg 4# time, which is what we shall use. Alternatively, one can start from the regularized Berry}Keating Zeta function f (E) [69], and de"ne N "NM !(1/p) Im log f (E#i0), which yields N "Nj. 4# 4# 4# 4# 4# Next, in order to de"ne a quantitative measure of the semiclassical error, one should establish a one-to-one correspondence between the quantal and the semiclassical levels, namely, one should identify the semiclassical counterparts of the exact quantum levels. In classically chaotic systems, for which the Gutzwiller trace formula is applicable, the only constant of the motion is the energy. This is translated into a single `gooda quantum number in the quantum spectrum, which is the ordinal number of the levels when ordered by their magnitude. Thus, the only correspondence which can be established between the exact spectrum ME N and its semiclassical approximation, n ME4#N, is n E % E4# . (118) n n This is to be contrasted with integrable systems, where it is appropriate to compare the exact and approximate levels which have the same quantum numbers. The natural scale on which the accuracy of semiclassical energy levels should be measured is the mean level spacing (dM (E))~1. We shall be interested here in the mean semiclassical error, and possible measures are the mean absolute di!erence e(1)(E),SdM (E )DE !E4#DT n n n E or the variance
(119)
e(2)(E),S(dM (E )(E !E4#))2T , (120) n n n E where S ) T denotes averaging over a spectral interval *E centred at E. The interval *E is large enough so that the mean number of levels *E ) dM (E)<1. Yet, *E is small enough on the classical scale, such that dM (E)+constant over the interval considered. We shall now compare two di!erent estimates for the semiclassical error. The "rst one is the traditional estimate:
G
e53!$*5*0/!-"O(+2~d)P
const, d"2 R,
d53
as +P0
(121)
(cf. Section 1). It claims that the semiclassical approximation is (marginally) accurate in two dimensions, but it fails to predict accurate energy levels for three dimensions or more. We emphasize that the traditional estimate is a qualitative error measure, emerging from global error estimate of the time propagator. Hence, it cannot be directly connected to either e(1) or e(2). We mention it here since it is the one usually quoted in the literature. One may get a di!erent estimate of the semiclassical error, if the Gutzwiller trace formula (GTF) is used as a starting point. Suppose that we have calculated N to a certain degree of precision, and 4# we compute from it the semiclassical energies E4# using (116). Denote by *N the higher order n 4# terms which were neglected in the calculation of N . The expected error in E4# can be estimated by 4# n including *N and calculating the energy di!erences d . That is, we consider 4# n N (E4##d )#*N (E4##d )"n!1 . (122) 4# n n 4# n n 2
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Combining (116) and (122) we get (to "rst order in d ) n !*N (E4#) !*N (E4#) 4# n . 4# n + (123) d + n RN (E4#)/RE dM (E4#) n 4# n In the above we assumed that the #uctuations of N around its average are not very large. Thus, 4# e(1),GTF+dM (E4#)Dd D+*N (E4#) . (124) n n 4# n Let us apply the above formula and consider the case in which we take for N its mean part NM , 4# and that we include in NM terms of order up to (and including) +~m, m4d. For *N we use both the 4# leading correction to NM and the leading order periodic orbit sum which is formally (termwise) of order +0. Hence, e(1),GTF "O(+~m`1)#O(+0)"O(+.*/(~m`1, 0)) . (125) NM We conclude that approximating the energies only by the mean counting function NM up to (and not including) the constant term, is already su$cient to obtain semiclassical energies which are accurate to O(+0)"O(1) with respect to the mean density of states. Note again that no periodic orbit contributions were included in N . Including less terms in NM will lead to a diverging 4# semiclassical error, while more terms will be masked by the periodic orbit (oscillatory) term. One can do even better if one includes in N the smooth terms up to and including the constant term 4# (O(+0)) together with the leading-order periodic orbit sum which is formally also O(+0). The semiclassical error is then e(1),GTF"O(+1) . (126) 10 That is, the semiclassical energies measured in units of the mean level spacing are asymptotically accurate independently of the dimension! This estimate grossly contradicts the traditional estimate (121) and calls for an explanation. The "rst point that should be noted is that the order of magnitude (power of +) of the periodic orbit sum, which we considered above to be O(+0), is only a formal one. Indeed, each term which is due to a single periodic orbit is of order O(+0). However, the periodic orbit sum absolutely diverges, and at best it is only conditionally convergent. To give it a numerical meaning, the periodic orbit sum must therefore be regularized. This is e!ectively achieved by truncating the trace formula or the corresponding spectral f function [26,69,84,90]. However, the cuto! itself depends on +. One can conclude that the simple-minded estimate (126) given above is at best a lower bound, and the error introduced by the periodic orbit sum must be re-evaluated with more care. This point will be dealt with in great detail in the sequel, and we shall eventually develop a meaningful framework for evaluating the magnitude of the periodic orbit sum. The disparity between the traditional estimate of the semiclassical error and the one based on the trace formula can be further illustrated by the following argument. The periodic orbit formula is derived from the semiclassical propagator K using further approximations [2]. Therefore one 4# wonders, how can it be that further approximations of K actually reduce the semiclassical error 4# from (121) to (126)? The puzzle is resolved if we recall that in order to obtain e(1),GTF above we 10 separated the density of states into a smooth part and an oscillating part, and we required that the smooth part is accurate enough. To achieve this, we have to go beyond the leading Weyl's term and to use specialized methods to calculate the smooth density of states beyond the leading order.
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Fig. 32. Illustration of DN(E)!Nj(E)D for small deviations between quantum and semiclassical energies: e(1);dM ~1,DM . 4# The quantum staircase N(E) is denoted by the full line and the semiclassical staircase Nj(E) is denoted by the dashed line. 4# The di!erence is shaded.
These methods are mostly developed for billiards [82,91,92]. In any case, to obtain e(1),GTF we have 10 added additional information which goes beyond the leading semiclassical approximation. A direct check of the accuracy of the semiclassical spectrum using the error measures e(1), e(2) is exceedingly di$cult due to the exponentially large number of periodic orbits needed. The few cases where such tests were carried out involve 2D systems and it was possible to check only the lowest (less than a hundred) levels (e.g. [60,73]). The good agreement between the exact and the semiclassical values con"rmed the expectation that in 2D the semiclassical error is small. In 3D, the topological entropy is typically much larger [50,54], and the direct test of the semiclassical spectrum becomes prohibitive. Facing with this grim reality, we have to introduce alternative error measures which yield the desired information, but which are more appropriate for a practical calculation. We construct the measure d(2)(E),SDN(E)!Nj(E)D2T . (127) 4# E As before, the triangular brackets indicate averaging over an energy interval *E about E. We shall now show that d(2) faithfully re#ects the deviations between the spectra, and is closely related to e(1) and e(2). Note that the following arguments are purely statistical and apply to every pair of staircase functions. Suppose "rst that all the di!erences E4#!E are smaller than the mean spacing. Then, DN!NjD n n 4# is either 0 or 1 in most of the cases (see Fig. 32). Hence, DN!NjD"DN!NjD2 along most of the 4# 4# E-axis. Consequently, d(2)(E)+SDN(E)!Nj(E)DT for small deviations . (128) 4# E The right-hand side of the above equation (the fraction of non}zero contributions) equals e(1). Thus, d(2)+e(1) for small deviations .
(129)
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If, on the other hand, deviations are much larger than one mean spacing, the typical horizontal distance dM DE!E D should be comparable to the vertical distance DN!NjD, and hence, in this limit n 4# d(2)+e(2) for large deviations . (130) Therefore, we expect d(2) to interpolate between e(1) and e(2) throughout the entire range of deviations. This behaviour was indeed observed in a numerical tests which were performed to check the above expectations [93]. Moreover, it was shown in [93] that d(2) is completely equivalent to e(2) when the spectral counting functions are replaced with their smooth counterparts, provided that the smoothing width is of the order of one mean level spacing and the same smoothing is applied to both counting functions. That is, d(2) +e(2) for all deviations . (131) 4.005) In testing the semiclassical accuracy, this kind of smoothing is essential and will be introduced by truncating the trace formula at the Heisenberg time t ,hdM . These properties of the measure d(2), H and its complete equivalence to e(2) for smooth counting functions, renders it a most appropriate measure of the semiclassical error. We now turn to the practical evaluation of d(2). To perform the energy averaging, we choose a positive window function w(E@!E) which has a width *E near E and is normalized by :`= dE@ w2(E@)"1. It falls o! su$ciently rapidly so that all the expressions which follow are well ~= behaved. We consider the following counting functions that have an e!ective support on an interval of size *E about E: NK (E@; E),w(E@!E)N(E@) ,
(132)
NK j(E@; E),w(E@!E)Nj(E@) . (133) 4# 4# The functions NK and NK j are sharp staircases, since the multiplication with w preserves the 4# sharpness of the stairs (it is not a convolution!). We now explicitly construct d(2)(E) as
P P
`= dE@DNK (E@; E)!NK j(E@; E)D2 4# ~= `= dE@DN(E@)!Nj(E@)D2w2(E@!E) . (134) " 4# ~= To obtain d(2) we need to smooth N and Nj over a scale of order of one mean spacing. One can, 4.005) 4# e.g., replace the sharp stairs by error functions. As for Nj, we prefer to simply replace it with the 4# original N , which we assume to be smooth over one mean spacing. That is, we suppose that 4# N contains periodic orbits up to Heisenberg time. Hence, 4# `= d(2) (E)"
[email protected])(E@)!N (E@)D2 w2(E@!E) . (135) 4.005) 4# ~= A comment is in order here. Strictly speaking, to satisfy (131) we need to apply the same smoothing to N and to Nj, and in general Nj,4.005)ON , but there are di!erences of order 1 between the 4# 4# 4# two functions. However, since our goal is to determine whether the semiclassical error remains "nite or diverges in the semiclassical limit +P0, we disregard such inaccuracies of order 1. d(2)(E)"
P
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If a more accurate error measure is needed, then more care should be practised in this and in the following steps. Applying Parseval's theorem to (135) we get
P
1 `= d(2) (E)" dtDDK (t; E)!DK (t; E)D2 , 4.005) 4# + ~= where
P
1 `= dE@ NK 4.005)(E@; E) exp(iE@t/+) , DK (t; E), J2p ~=
P
1 `= DK (t; E), dE@ NK (E@; E) exp(iE@t/+) . 4# 4# J2p ~=
(136)
(137)
(138)
We shall refer to DK , DK as the (regularized) quantal and semiclassical time spectra, respectively. 4# These functions are the analogs of the length spectra D(l; k) used in Section 5 for the billiard problem. The analogy becomes clear by invoking the Gutzwiller trace formula and expressing the semiclassical counting function as a mean part plus a sum over periodic orbits. We have +A (E) N (E)"NM (E)#+ j sin[S (E)/+!l p/2] , j j 4# ¹ (E) 10 j
(139)
where A "¹ /(p+r JDdet(I!M )D) is the semiclassical amplitude of the jth periodic orbit, and j j j j ¹ , S ,l , M , r are its period, action, Maslov index, monodromy matrix and repetition index, j j j j j respectively. Then, the corresponding time spectrum reads 1 +A (E) DK (t; E)+DM (t; E)# + j 4# 2i ¹ (E) 10 j + ]Me(*@ )*Et`Sj (E)+w( ([t#¹ (E)]/+)!e(*@+)*Et~Sj (E)+w( ([t!¹ (E)]/+)N . (140) j j In the above, the Fourier transform of w is denoted by w( . It is a localized function of t whose width is *t++/*E. The sum over the periodic orbits in D therefore produces sharp peaks centred 4# at times that correspond to the periods ¹ . The term DM corresponds to the smooth part j and is sharply peaked near t"0. To obtain (140) we expanded the actions near E to "rst order: S (E@)+S (E)#(E@!E)¹ (E). We note in passing that this approximate expansion of S can be j j j j avoided altogether if one performs the Fourier transform over +~1 rather than over the energy. This way, an action spectrum will emerge, but also here the action resolution will be "nite, because the range of +~1 should be limited to the range where dM (E; +) is approximately constant. It turns out, therefore, that the two approaches are essentially equivalent, and for billiards they are identical. The manipulations performed thus far were purely formal, and did not manifestly circumvent the di$cult task of evaluating d(2) . However, the introduction of the time spectra and formula (136) 4.005) put us in a better position than the original expression (134). The advantages of using the time spectra in the present context are the following:
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f The semiclassical time spectrum DK (t; E) is absolutely convergent for all times (as long as the 4# window function w is well behaved, e.g. it is a Gaussian). This statement is correct even if the sum (140) extends over the entire set of periodic orbits! This is in contrast with the trace formula expression for N (and therefore NK ) which is absolutely divergent if all of the periodic orbits 4# 4# are included. f Time scale separation: As we noted above, the time spectrum is peaked at times that correspond to periods of the classical periodic orbits. This allows us to distinguish between various qualitatively di!erent types of contributions to d(2) . 4.005) We shall now pursue the separation of the time scales in detail. We "rst note that due to NK , NK 4# being real, there is a t % (!t) symmetry in (136), and therefore the time integration can be restricted to the limits 0 to #R: d(2)"(1/+):`= 2"(2/+):=2 . We now divide the time axis ~= 0 into four intervals: 04t4*t:
*t4t4t
%3'
:
t 4t4t : %3' H
t 4t(R: H
The shortest time scale in our problem is *t"+/*E. The contributions to this time interval are due to the di!erences between the exact and the semiclassical mean densities of states. This is an important observation, since it allows us to distinguish between the two sources of semiclassical error } the error that emerges from the mean densities and the error that originates from the #uctuating part (periodic orbits). Since we are interested only in the semiclassical error that results from the #uctuating part of the spectral density, we shall ignore this regime in the following. This is the non-universal regime [65], in which periodic orbits are still sparse, and cannot be characterized statistically. The `ergodica time scale t is purely %3' classical and is independent of +. In this time regime periodic orbits are already in the universal regime and are dense enough to justify a statistical approach to their proliferation and stability. The upper limit of this interval is the Heisenberg time t "hdM (E), which is the H time that is needed to resolve the quantum (discrete) nature of a wavepacket with energy concentrated near E. The Heisenberg time is `quantala in the sense that it is dependent of +: t "O(+1~d). H This is the regime of `longa orbits which is e!ectively truncated from the integration as a result of the introduction of a smoothing of the quantal and semiclassical counting functions, with a smoothing scale of the order of a mean level spacing.
Dividing the integral (136) according to the above time intervals, we can rewrite d(2) : 4.005) t%3' tH = 2dt d(2) (E)" DDK (t; E)!DK (t; E)D2 # # 4.005) 4# + tH *t t%3' ,d(2) #d(2)#d(2) . (141) 4)035 . -0/' As explained above, d(2) can be ignored due to smoothing on the scale of a mean level spacing. The -0/' integral d(2) is to be neglected for the following reason. The integral extends over a time interval 4)035 which is "nite and independent of +, and therefore it contains a "xed number of periodic orbits contributions. The semiclassical approximation provides, for each individual contribution, the
AP P P B
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leading order in +, and therefore [80] we should expect d(2) P0 as +P0 . (142) 4)035 Our purpose is to check whether the semiclassical error is "nite or divergent as +P0, and to study whether the rate of divergence depends on dimensionality. Eq. (142) implies that d(2) cannot a!ect 4)035 d(2) in the semiclassical limit and we shall neglect it in the following. We remain with d(2) +d(2) , (143) 4.005) . which will be our object of interest from now on. The fact that t is extremely large on the classical scale renders the calculation of all the periodic H orbits with periods less than t an impossible task. However, sums over periodic orbits when H the period is longer than t tend to meaningful limits, and hence, we would like to recast the %3' expression for d(2) in the following way. Write d(2) as . . 2 tH d(2)" dtSDDK (t)!DK (t)D2T (144) . 4# t + %3' t 2 tH SDDK (t)!DK (t)D2T 4# t dtSDDK (t)D2T ] (145) " t + %3' SDDK (t)D2T t t 2 tH tH , dtSDDK (t)D2T ]C(t)" envelope]correlation , (146) t + %3' %3' t t where the parametric dependence on E was omitted for brevity. The smoothing over t is explicitly indicated to emphasize that one may use a statistical interpretation for the terms of the integrand. This is so because in this domain, the density of periodic orbits is so large that within a time interval of width +/*E there are exponentially many orbits whose contributions are averaged due to the "nite resolution. We note now that we can use the following relation between the time spectrum and the spectral form factor K(q):
P
P P
C
SDDK (t)D2T K(q) t dt" dq + 4p2q2
P
D
(147)
where q,t/t is the scaled time. The above form factor is smoothed according to the window H function w. Hence,
P
1 1 K(q)C(q) d(2) + dq . (148) 4.005) 2p2 q2 q%3' For generic chaotic systems we expect that K(q) agrees with the results of RMT in the universal regime q'q [4,24,65]. Therefore %3' q4K(q)4gq for q (q41 , (149) %3' where g"1 for systems which violate time reversal symmetry, and g"2 if time reversal symmetry is respected. This implies that the evaluation of d(2) reduces to 4.005) g 1 C(q) d(2) + dq , (150) 4.005) 2p2 q %3' q
P
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where we took the upper bound gq for K(q). The dependence on + in this expression comes from the lower integration limit which is proportional to +d~1 as well as from the implicit dependence of the function C on +. Formula (150) is our main theoretical result. However, we do not know how to evaluate the correlation function C(q) from "rst principles. The knowledge of the + corrections to each of the terms in the semiclassical time spectrum is not su$cient since the resulting series which ought to be summed is not absolutely convergent. Therefore we have to recourse to a numerical analysis, which will be described in the next section. The numerical approach requires one further approximation, which is imposed by the fact that the number of periodic orbits with t(t is prohibitively H large. We had to limit the database of periodic orbits to the domain t(t with t ;t ;t . #16 %3' #16 H The time t has no physical origin, it represents only the limits of our computational resources. #16 Using the available numerical data we were able to compute C(t) numerically for all t (t(t %3' #16 and we then extrapolated it to the entire domain of interest. We consider this extrapolation procedure to be the main source of uncertainty. However, since the extrapolation is carried out in the universal regime, it should be valid if there are no other time scales between t and t . %3' H 6.2. Numerical results We used the formalism and de"nitions presented above to check the accuracy of the semiclassical spectra of the 2D and 3D Sinai billiards. The most important ingredient in this numerical study is that we could apply the same analysis to the two systems, and by comparing them to give a reliable answer to the main question posed in this section, namely, how does the semiclassical accuracy depend on dimensionality. The classical dynamics in billiards depends on the energy (velocity) trivially, and therefore the relevant parameter is the length rather than the period of the periodic orbits. Likewise, the quantum wavenumbers k are the relevant variables in the quantum description. From now on n we shall use the variables (l, k) instead of (t, E), and use `length spectraa rather than `time spectraa. The semiclassical limit is obtained for kPR and O(+) is equivalent to O(k~1). Note also that for a billiard NM (k)+Akd where A is proportional to the billiard's volume. We start with the 2D Sinai billiard, which is the free space between a square of edge S and an inscribed disc of radius R, with 2R(S. Speci"cally, we use S"1 and R"0.25 and consider the quarter desymmetrized billiard with Dirichlet boundary conditions for the quantum calculations. The quantal database consists of the lowest 27,645 eigenvalues in the range 0(k(1320, with eigenstates which are either symmetric or antisymmetric with respect to re#ection on the main diagonal. The classical database consists of the shortest 20,273 periodic orbits (including time reversal, re#ection symmetries and repetitions) in the length range 0(l(5. For each orbit, the length, the stability determinant and the re#ection phase were recorded. The numerical work is based on the quantum spectra and on the classical periodic orbits which were computed by Schanz and Smilansky [17,94] for the 2D billiard. We begin the numerical analysis by demonstrating numerically the correctness of Eq. (142). That is, that for each individual contribution of a periodic orbit, the semiclassical error indeed vanishes in the semiclassical limit. In Fig. 33 we plot DD!D D for l"0.5 as a function of k. This length 4# corresponds to the shortest periodic orbit, that is, the one that runs along the edge that connects the circle with the outer square. For D we used the Gutzwiller trace formula. As is clearly seen 4#
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Fig. 33. The absolute di!erence between the quantal and the semiclassical (Gutzwiller) length spectra for the 2D Sinai billiard at l"0.5. This length corresponds to the shortest unstable periodic orbit. The average log}log slope is about !1.1, indicating approximately k~1 decay. The data were averaged with a Gaussian window. Fig. 34. The functions C(l; k) for quarter 2D Sinai billiard S"1, R"0.25 with Dirichlet boundary conditions. The window w(k@!k) was taken to be a Gaussian with standard deviation p"60. We averaged C(l; k) over l-intervals of +0.2 in accordance with (145) to avoid sharp peaks due to small denominators. The averaging, however, is "ne enough not to wash out all of the features of C(l; k). The vertical bars indicate the locations of primitive periodic orbits, and the daggers indicate the locations of the bouncing-ball families.
from the "gure, the quantal}semiclassical di!erence indeed vanishes (approximately as k~1), in accordance with (142). We emphasize again that this behaviour does not imply that d(2) vanishes in the semiclassical limit, since the number of periodic orbits included depends on k. It implies only that d(2) vanishes in the limit, since it consists of a "xed and "nite number of periodic orbit 4)035 contributions. We should also comment that penumbra corrections to individual grazing orbits introduce errors which are of order k~c with 0(c(1 [38,81]. However, since the de"nition of `grazinga is in itself k dependent, one can safely neglect penumbra corrections in estimating the large k behaviour of d(2) . 4)035 We now turn to the main body of the analysis, which is the evaluation of d(2) for the 2D Sinai . billiard. Based on the available data sets, we plot in Fig. 34 the function C(l; k) in the interval 2.5(l(5 for various values of k. One can observe, that as a function of l the functions C(l; k) #uctuate in the interval for which numerical data were available, without exhibiting any systematic mean trend to increase or to decrease. We therefore approximate C(l; k) by C(l; k)+const ) f (k),C (k) . (151) !7' As mentioned above, we extrapolate this formula in l up to the Heisenberg length ¸ "2pdM (k) and H using (150) we obtain C (k) d(2),2D " !7' ln(¸ /¸ )"C (k)O(ln k) . 4.005) H %3' !7' 2p2
(152)
The last equality is due to ¸ "O(kd~1). To evaluate C (k) we averaged C(l; k) over the interval H !7' ¸ "3.5(l(5"¸ and the results are shown in Fig. 35. We choose ¸ "3.5 because the %3' #16 %3'
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Fig. 35. Averaging in l of C(l; k) for 2D Sinai billiard as a function of k.
density of periodic orbits is large enough for this length (see Fig. 34) to expect universal behaviour of the periodic orbits. (For the Sinai billiard described by #ow the approach to the invariant measure is algebraic rather than exponential [39,40], and thus one cannot have a well-de"ned ¸ . %3' At any rate, the speci"c choice of ¸ did not a!ect the results in any appreciable way.) Inspecting %3' C (k), it is di$cult to arrive at "rm conclusions, since it seems to #uctuate around a constant !7' value up to k+900 and then to decline. If we approximate C (k) by a constant, we get !7' a `pessimistica value of d(2): d(2),2D (k)"O(ln k)"O(ln +) `pessimistica 4.005) while if we assume that C (k) decays as a power law, C (k)"k~b, b'0, then !7' !7' d(2),2D (k)"O(k~b ln k)P0 `optimistica . 4.005) Collecting the two bounds we get
(153)
(154)
O(k~b ln k)4d(2),2D (k)4O(ln k) . (155) 4.005) Our estimates for the 2D Sinai billiard can be summarized by stating that the semiclassical error diverges no worse than logarithmically (meaning, very mildly). It may well be true that the semiclassical error is constant or even vanishes in the semiclassical limit. To reach a conclusive answer one should invest exponentially larger amount of numerical work. There are a few comments in order here. Firstly, the quarter desymmetrization of the 2D Sinai billiard does not exhaust its symmetry group, and in fact, a re#ection symmetry around the diagonal of the square remains. This means, that the spectrum of the quarter 2D Sinai billiard is composed of two independent spectra, which di!er by their parity with respect to the diagonal. If we assume that the semiclassical deviations of the two spectra are not correlated, the above measure is the sum of the two independent measures. It is plausible to assume also that both spectra have roughly the same semiclassical deviation, and thus d(2),2D is twice the semi4.005) classical deviation of each of the spectra. Secondly, we recall that the 2D Sinai billiard contains `bouncing-balla families of neutrally stable periodic orbits [15,17,37]. We have subtracted their leading-order contribution from DK such that it includes (to leading order) only contributions from
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Fig. 36. Veri"cation of Eq. (156) for the quarter 2D Sinai billiard. We plot I(m),:2dm@ K(m@)/m@ 2 and compare the m quantum data with RMT. The minimal m corresponds to ¸ "3.5. The integration is done for smoothing, and we "x %3' the upper limit to avoid biases due to non-universal regime. Note the logarithmic scale. Fig. 37. The numerical values of d(2) for the quarter 2D Sinai billiard. We included also the contribution d(2) of the 4.005) 4)035 non-universal regime. The contributions from the time interval t 4t4t are contained in d(2) , and d(2) is %3' #16 .,#16 .,%95 the extrapolated value for t 4t4t (refer to Eq. (141) and to the end of Section 6.1). #16 H
generic, isolated and unstable periodic orbits. This is done since we would like to deduce from the 2D Sinai billiard on the 2D generic case in which the bouncing-balls are not present. (In the Sinai billiard, which is concave, there are also di!raction e!ects [38,81], but we did not treat them here.) Thirdly, the analogue of (147) for billiards reads K(m) SDDK (l)D2T dl" dm l 4p2m2
(156)
when m,l/¸ . In Fig. 36 we demonstrate the compliance of the form factor with RMT GOE H using the integrated version of the above relation, and taking into account the presence of two independent spectra. Finally, it is interesting to know the actual numerical values of d(2),2D (k) for 4.005) the k values that we considered. We carried out the computation, and the results are presented in Fig. 37. One observes that for the entire range we have d(2),2D (k)+0.1;1, which is very 4.005) encouraging from an `engineeringa point of view. We now turn to the analysis of the 3D Sinai billiard. We use the longest quantal spectrum (R"0.2, Dirichlet) and the classical periodic orbits with length 0(l(5. To treat the 3D Sinai billiard we have to somewhat modify the formalism which was presented above. This is due to the fact that in the 3D case the contributions of the various non-generic bouncing-ball manifolds overwhelm the spectrum [53,54], and unlike the 2D case, it is di$cult to explicitly eliminate their (leading-order) contributions (cf. the discussion in Section 5.4). Since our goal is to give an indication of the semiclassical error in generic systems, it is imperative to avoid this dominant and non-generic e!ect. We shall use the mixed boundary conditions, which were discussed in Section 5.5 and were shown to largely "lter the bouncing-ball e!ects. Speci"cally, we consider dI (cf. (113)) for our
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purposes. Let us construct the weighted counting function
P
k
dk@ dI (k@)"+ v H(k!k ) . (157) n n 0 n The function NI is a staircase with stairs of variable height v . As explained above, its advantage n over N is that it is semiclassically free of the bouncing balls (to leading order) and corresponds only to the generic periodic orbits [83]. Similarly, we construct from dI the function NI . Having 4# 4# de"ned NI , NI , we proceed in analogy to the Dirichlet case. We form from NI , NI the functions 4# 4# NK , NK , respectively, by multiplication with a window function w(k@!k) and then construct the 4# measure d(2) as in (134). The only di!erence is that the normalization of w must be modi"ed to account for the `velocitiesa v such as n NI (k),
dM ~1(k)+ v2Dw(k !k)D2"1 . (158) n n n The above considerations are meaningful provided the `velocitiesa v are narrowly distributed n around a well-de"ned mean v(k), and we consider a small enough k-interval, such that v(k) does not change appreciably within this interval. Otherwise, d(2) is greatly a!ected by the #uctuations of v n (which is undesired) and the meaning of the normalization is questionable. We shall check this point numerically. To demonstrate the utility of the above construction using the mixed boundary conditions, we return to the 2D case. We set i"100p, and note that the spectrum at our disposal for the mixed case was con"ned to the interval 0(k(600. First, we want to examine the width of the distribution of the v 's. In Fig. 38 we plot the ratio of the standard deviation of v to the mean, n n averaged over the k-axis using a Gaussian window. We use the same window also in the calculations below. The observation is that the distribution of v is moderately narrow and the n width decreases algebraically as k increases. This justi"es the use of the mixed boundary conditions as was discussed above. One also needs to check the validity of (156), and indeed we found compliance with GOE also for the mixed case (results not shown). We next compare the functions C(l; k) for both the Dirichlet and the mixed boundary conditions. It turns out that also in the mixed case the functions C(l; k) #uctuate in l with no special tendency (not shown). The averages C (k) !7' for the Dirichlet and mixed cases are compared in Fig. 39. The values in the mixed case are systematically smaller than in the Dirichlet case which is explained by the e$cient "ltering of tangent and close to tangent orbits that are vulnerable to large di!raction corrections [81,38]. However, from k"250 on the two graphs show the same trends, and the values of C in both !7' cases are of the same magnitude. Thus, the qualitative behaviour of d(2) is shown to be 4.005) equivalent in the Dirichlet and mixed cases, which gives us con"dence in using d(2) together with 4.005) the mixed boundary conditions procedure. We "nally applied the mixed boundary conditions procedure to compute d(2) for the 4.005) desymmetrized 3D Sinai with S"1, R"0.2 and set i"100. We "rst veri"ed that also in the 3D case the velocities v have a narrow distribution } see Fig. 38. Next, we examined Eq. (156) using n quantal data, and discovered that there are deviations form GOE (Fig. 40). We have yet no satisfactory explanation of these deviations, but we suspect that they are caused because the ergodic limit is not yet reached for the length regime under consideration due to the e!ects of the
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Fig. 38. Calculation of Q,JSv2T!Sv T2/DSv TD for quarter 2D Sinai billiard (up) and for the desymmetrized 3D Sinai n n n billiard (down).
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Fig. 39. Comparison of C (k) for Dirichlet and mixed boundary conditions for the quarter 2D Sinai billiard. We used !7' a Gaussian window with p"40. Fig. 40. Check of Eq. (156) for the desymmetrized 3D Sinai billiard. The minimal m corresponds to ¸ "2.5. The %3' function I(m) is de"ned as in Fig. 36. Note the logarithmic scale.
in"nite horizon which are more acute in 3D. Nevertheless, from observing the "gure as well as suggested by semiclassical arguments, it is plausible to assume that K(m)Jm for small m. Hence, this deviation should not have any qualitative e!ect on d(2) according to (150). Similarly to the 2D case, the function C(l; k) #uctuates in l, with no special tendency (Fig. 41). If we average C(l; k) over the universal interval ¸ "2.54l4¸ "5, we obtain C (k) which is shown in Fig. 42. The %3' #16 !7' averages C (k) are #uctuating with a mild decrease in k, and therefore we can again conclude that !7' O(k~b ln k)4d(2),3D 4O(ln k) , 4.005)
(159)
where the `optimistica measure (leftmost term) corresponds to C (k)"O(k~b), b'0, and the !7' `pessimistica one (rightmost term) is due to C (k)"const. In other words, the error estimates !7' (155), (159) for the 2D and the 3D cases, respectively, are the same, and in sharp contrast to the traditional error estimate which predicts that the errors should be di!erent by a factor O(+~1). On the basis of our numerical data, and in spite of the uncertainties which were clearly delineated, we can safely rule out the traditional error estimate. Our main "nding is that the upper bound on the semiclassical error is a logarithmic divergence, both for a generic 2D and 3D systems (Eqs. (155), (159)). In this respect, there are a few points which deserve discussion. To begin, we shall try to evaluate d(2) using the explicit expressions for the leading corrections 4.005) to the semiclassical counting function of a 2D generic billiard system, as derived by Alonso and Gaspard [7]:
C
D
A Q N(k)"NM (k)#+ j sin k¸ # j #O(1/k2) , j ¸ k j j
(160)
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Fig. 41. The functions C(l; k) for desymmetrized 3D Sinai billiard S"1, R"0.2 with mixed boundary conditions. We took a Gaussian window with p"20, and smoothed over l-intervals of +0.3. The upper vertical bars indicate the locations of primitive periodic orbits. Fig. 42. Averaging in l of C(l; k) for 3D Sinai billiard as a function of k. The averaging was performed in the interval ¸ "2.5(l(5"¸ . %3' #16
where A are the standard semiclassical amplitudes, ¸ are the lengths of periodic orbits and Q are j j j the k-independent amplitudes of the 1/k corrections. The Q 's are explicitly given in [7]. j We ignored in the above equation the case of odd Maslov indices. If we calculate from N(k) the corresponding length spectrum DK (l; k) using a (normalized) Gaussian window w(k@!k)" (1/Jpp2) 4 exp[!(k@!k)2/(2p2)], we obtain iJp A + j [e*k(l~Lj )~* Qj @ke~(l~Lj )2p2@2!e*k(l`Lj )`* Qj @ke~(l`Lj )2p2@2] . DK (l; k)+ 2Jp 4 j ¸j
(161)
In the above we regarded the phase e*Qj @k as slowly varying. The results of Alonso and Gaspard [7] suggest that the Q are approximately proportional to the length of the corresponding periodic j orbits: Q +Q¸ . j j We can therefore well approximate DK as
(162)
iJp A DK (l; k)+ e~*Ql@k + j [2]"e~*Ql@kDK , 4#vGTF ¸ 2Jp 4 j j
(163)
where DK is the length spectrum which corresponds to the semiclassical Gutzwiller trace 4#vGTF formula for the counting function (without 1/k corrections). We are now in a position to evaluate the semiclassical error, indeed: d(2) (k)"2 4.005)
P
LH
L
.*/
P
dlDDK (l; k)!DK (l; k)D2"8 4#vGTF
LH
L
.*/
A B
dl sin2
Ql DDK (l; k)D2 . 2k
(164)
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If we use Eq. (156) and K(l)+gl/¸ (which is valid for l(¸ for chaotic systems), we get H H 2g QLH @(2k) sin2(t) 2g LH dl Ql dt sin2 " . (165) d(2) (k)+ 4.005) t p2 p2 .*/ l 2k .*/ QL @(2k) L For kPR we have that
P
A B P
P
P
QL.*/ @(2k) sin2(t) QL.*/ @(2k) dt dt ) t"O(1/k2) + t 0 0 which is negligible, hence we can replace the lower limit in (165) with 0:
(166)
P
2g QLH @2k sin2(t) dt d(2) (k)+ . (167) 4.005) t p2 0 This is the desired expression. The dimensionality enter in d(2) (k) only through the power of k 4.005) in ¸ . H Let us apply Eq. (167) to the 2D and the 3D cases. For 2D we have to leading order that ¸ "Ak, where A is the billiard's area, thus, H 2g QA@2 sin2(t) dt "const"O(k0) (168) d(2),2D (k)+ !/!-:5*#!t p2 0 which means that the semiclassical error in 2D billiards is of the order of the mean spacing, and therefore the semiclassical trace formula is (marginally) accurate and meaningful. This is compatible with our numerical "ndings. For 3D, the coe$cients Q were not obtained explicitly, but we shall assume that they are still j proportional to ¸ (Eq. (162)) and therefore that (167) holds. For 3D billiards ¸ "(
P
d(2),3D (k)"O(ln k) . (169) !/!-:5*#!That is, in contrast to the 2D case, the semiclassical error diverges logarithmically and the semiclassical trace formula becomes meaningless as far as the prediction of individual levels is concerned. This statement is compatible with our numerical results within the numerical dispersion. However, it relies heavily on the assumption that Q +Q¸ , for which we can o!er no j j */ 3D justi"cation. We note in passing, that the logarithmic divergence persists also for d'3. Another interesting point relates to integrable systems. It can happen that for an integrable system it is either di$cult or impossible to express the Hamiltonian as an explicit function of the action variables. In that case, we cannot assign to the levels other quantum numbers than their ordinal number, and the semiclassical error can be estimated using d(2). However, since for integrable systems K(q)"1, we get that
P
1 1 C(q) d(2),*/5 + dq . 4.005) 2p2 q2 %3' q
(170)
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Therefore, for deviations which are comparable to the chaotic cases, C(q)"O(1), we get d(2),*/5 "O(+1~d) which is much larger than for the chaotic case and diverges for d51. 4.005) Formula (150) for the semiclassical error contains semiclassical information in two respects. Obviously, C(q), which describes the di!erence between the quantal and the semiclassical length spectra, contains semiclassical information. But also the fact that the lower limit of the integral in (150) is "nite is a consequence of semiclassical analysis. If this lower limit is replaced by 0, the integral diverges for "nite values of +. Therefore, the fact that the integral has a lower cuto!, or rather, that D is exactly 0 below the shortest period, is a crucial semiclassical ingredient in our analysis. Finally, we consider the case in which the semiclassical error is estimated with no periodic orbits taken into account. That is, we want to calculate SDN(E)!NM (E)D2T which is the number variance E R2(x) for the large argument x"*EdM (E)<1. This implies C(q)"1, and using (150) we get that d(2) "g/(2p2) ln(t /t ), which in the semiclassical limit becomes g/(2p2) ln(t )"O(ln +). This 4.005) H %3' H result is fully consistent and compatible with previous results for the asymptotic (saturation) value of the number variance R2 (see for instance [65,95,96]). It implies also that the pessimistic error bound (153) is of the same magnitude as if periodic orbits were not taken into account at all. (Periodic orbits improve, however, quantitatively, since in all cases we obtained C (1.) Thus, if !7' we assume that periodic orbit contributions do not make N worse than NM , then the pessimistic 4# error bound O(ln +) is the maximal one in any dimension d. This excludes, in particular, algebraic semiclassical errors, and thus refutes the traditional estimate O(+2~d).
7. Semiclassical theory of spectral statistics In Section 3 we studied several quantal spectral statistics of the Sinai billiard and have shown that they can be reproduced to a rather high accuracy by the predictions of Random Matrix Theory (RMT). In the present section we would like to study the spectral two-point correlation function in the semiclassical approximation, and to show how the classical sum rules and correlations of periodic orbits, which were de"ned in Section 4, can be used to reconstruct, within the semiclassical approximation, the predictions of RMT. The starting point of the present discussion is the observation that the semiclassical spectrum can be derived from a secular equation of the form [26,84] Z (k), det(I!S(k))"0 , 4#
(171)
where S(k) is a (semiclassically) unitary matrix which depends parametrically on the wavenumber k. In the semiclassical approximation, the unitary operator S(k) can be considered as the quantum analogue of a classical PoincareH mapping, which for billiard systems in d dimensions, is the classical billiard bounce map. The dimension N(k) of the Hilbert space on which S(k) acts, can be expressed within the semiclassical approximation, in terms of the phase-space volume of the PoincareH section M as follows: M N(k)"[N(k)], N(k)" , (2p+)d~1
(172)
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where [ ) ] stands for the integer value. For a billiard in two dimensions N(k)"Lk/p, where L is the circumference of the billiard. In the case of the fully desymmetrized 3D Sinai billiard, for which we consider the sphere return map, N(k)"k2R2/48. The reason why we de"ned the smooth function N(k) will become clear in the sequel. The eigenvalues of S(k) are on the unit circle: Mexp(ih (k))NN(k) . If for a certain k, one of the l l/1 eigenphases is an integer multiple of 2p, then Eq. (171) is satis"ed, and this value of k belongs to the spectrum. Because of this connection between the billiard spectrum on the k-axis and the eigenphase spectrum on the unit circle, the statistics of k-intervals can be read o! the corresponding statistics of the eigenphase intervals averaged over an appropriate k-interval where N(k) is constant [26,84]. For this reason, it is enough to study the eigenphase statistics, and if they can be reproduced by the predictions of RMT for the relevant circular ensemble, the wavenumber spectral statistics will conform with the prediction of RMT for the corresponding Gaussian ensemble. The spectral density of the matrix S(k) can be written as N(k) N(k) 1 = (h; k), + d(h!h (k))" # + (e~*nhtr Sn#e*nhtr (Ss)n) . l 2p 2p l/1 n/1 The corresponding two-point correlation function is derived by computing d
2.
TP
A
B A
(173)
BU
g g 2p dh d h# ; k d h! ; k , (174) 2. 2. 2 2 2p 0 where S ) T denotes an average over a k-interval where N(k) takes the constant value N. The two-point spectral form factor is de"ned as the Fourier coe$cients of C (g), and by substituting 2 (173) in (174), one "nds that they are equal to (1/N)SDtr Sn(k)D2T. RMT provides an explicit expression 2p C (g)" 2 N
AB
1 n SDtr Sn(k)D2T "K RMT b N N
,
(175)
where b is the standard ensemble label [59]. The most important fact to be noticed is that n, the `topological timea, is scaled by N, which plays here the ro( le of the Heisenberg time. For a Poisson ensemble, 1 SDtr Sn(k)D2T "1 . P0*440/ N
(176)
From now on we shall be concerned with the circular orthogonal ensemble (COE: b"1). The function K (q) is a monotonically increasing function which starts as 2q near the origin, and COE bends towards its asymptotic value 1 in the vicinity of q"1. For an explicit expression consult, e.g. [72]. Our aim is to show that the semiclassical expression for (1/N)SDtr Sn(k)D2T reproduces this behaviour when the correlations of periodic orbits are properly taken into account. Recalling that the unitary matrix S(k) is the quantum analogue of the PoincareH map, one can express tr Sn(k) in terms of the n-periodic orbits of the mapping. If the semiclassical mapping is hyperbolic, and the billiard bounce map is considered, one gets [72] n p,j tr Sn(k)+ + e*kLj (!1)bj . P Ddet(I!M )D1@2 j j| n
(177)
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Here P is the set of all n-periodic orbits of the bounce map, n is the period of the primitive orbit n p,j of which the n-periodic orbit is a multiple. The monodromy matrix is denoted M , ¸ is the length, j j and b is the number of bounces from the boundaries (for a Dirichlet boundary condition). Note j that when the PoincareH section consists of a part of the boundary (as is the case for the sphere return map in the 3D Sinai billiard), b can be di!erent from n. Recalling the de"nition of j the classical density d (l; n) (72) in Section 4.4, and realizing that the pre-exponential factors are just #the AI coe$cients (73), we deduce that within the semiclassical approximation, j 2p = e*nh d (h; k) dh"tr Sn(k)+ e*kl d (l; n) dl . (178) 2. #0 0 This equation is of fundamental importance, because it expresses the duality between the quantum mechanical spectral density and the classical length density via their Fourier transforms [11]. Hence, the spectral form factors of the classical and the quantum spectral distributions are also related by
P
P
TK
KU
2 + AI e*kLj . (179) j j|Pn We have shown already in Section 4.4 that the length spectrum as de"ned by the classical density (72) contains non-trivial correlations. They appear on a scale j(n; R) which is inversely proportional to the value of k where the classical correlation function approaches it asymptotic value gn. What remains to be seen now is the extent by which the semiclassical expression (179) reproduces the expected universal scaling and the detailed functional dependence on the scaled topological time q"n/N as predicted by RMT. The large k limit of K (k; n) was written explicitly in (79) and veri"ed numerically: #K (kPR; n)+Sn g T ) ;(n)+2n . (180) #p p This limit corresponds to the limit n/N(k)P0 so that 1 1 1 SDtr Sn(k)D2T" SK (k; n)T" #N N N
(1/N)SDtr Sn(k)D2T+2n/N ,
(181)
which is identical to the behaviour of K (q) in the small q limit [59]. Therefore, the classical COE uniform coverage of phase space guarantees the adherence to RMT in the limit qP0. This result was derived originally by Berry in his seminal paper [4]. It is the `diagonal approximationa which can be used as long as the range of k values is larger than j(n; R)~1. In other words, this approximation is valid on the scale on which the classical length spectrum looks uncorrelated. This observation shows that the domain of validity of the diagonal approximation has nothing to do with the `Ehrenfest timea, sometimes also called the `log + timea. Rather, it depends on the correlation length in the classical spectrum j(n; R), as displayed by the classical form factor. Given the classical correlation function, K (k; n), it cannot be meaningfully compared to the #COE result at all values of the parameters. This is because once N(k)"1, one cannot talk about quantum two-point correlations, since the spectrum consists of a single point on the unit circle. In other words, this is the extreme quantum limit, where the Hilbert space consists of a single state. Therefore, the k-values to be used must exceed in the case of the 3D Sinai billiard k "J96/R, .*/ which corresponds to N(k)"2. Hence, the values of q"n/N which are accessible are restricted to the range 04q4n/2.
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Fig. 43. Comparing the classical form factor with the universal RMT predictions for various cases of the 3D SB.
In Fig. 43 we summarize our numerical results by comparing the form factor obtained from periodic orbit theory K with the theoretical RMT prediction K . What we actually show is the #COE running average,
P
1 q q@ dq@ K (q@) , C(q), (182) n #q 0 where K (q),K (k(q; n); n). The corresponding COE curve (cf. Eqs. (175) and (179)) is given by ##1 q dq@ K (q@) . (183) C(q), COE q 0 The `diagonal approximationa curve is obtained by replacing K (q) by 2q, namely, classical COE correlations are ignored. The data sets which were chosen are those for which su$ciently many periodic orbits were computed so that the sum rule ;(n; l)+1 was satis"ed. We did not include the n"1 data because they are non-generic. As clearly seen from the "gure, the data are consistent with the RMT expression and they deviate appreciably from the diagonal approximation. This is entirely due to the presence of classical correlations, and it shows that the classical correlations are indeed responsible for the quantitative agreement. Note also that the data represents four di!erent combinations of n and R, which shows that the classical scaling is indeed consistent with the universal scaling implied by RMT. In Fig. 44 we present essentially the same data, but integrated and plotted using the variable k, similarly to Section 4. The integration started at k for .*/ a meaningful comparison with RMT. Again, we observe the quantitative agreement, which is especially good for the higher n values (n"3, 4). In Section 4.4 we showed that the classical correlations originate to a large extent from the X(=) families of periodic orbits. Moreover, the form factor which was calculated by neglecting crossfamily contributions was much smoother than the original one. It is therefore appealing to take advantage of this smoothness and compare the numerical and theoretical form factors themselves instead of their running averages. We de"ne
P
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Fig. 44. The classical form factor compared with the universal RMT predictions for various cases of the 3D SB in the variable k.
P
1 k(N`1) dk@ K (k@; n) K (N; n),SK (k; n)T " ###N k(N#1)!k(N) k(N)
(184)
which is the semiclassical ensemble average of the form factor. In Fig. 45 we compare K (N; n) with #N ) K (n/N). The classical form factor included intra-family contribution only, and we multiplied COE it by a factor such that asymptotically it will match the theoretical value 2n. This factor compensates for the partial breaking of time-reversal symmetry and for the fact that the classical saturation is to values slightly below 2n for the n's under consideration. One observes that the agreement is quite good, and in any case the classical form factor is sharply di!erent from the diagonal approximation, meaning that classical correlations are important. In Fig. 46 we present the same results with q"n/N as the variable. It again shows that the classical form factor agrees with the COE expression beyond the validity range of the diagonal approximation. The range of q where a good agreement is observed increases with n as expected, but the estimated domain of valid comparison q(2n seems to be too optimistic.
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Fig. 45. The classical form factor K (N; n) compared to RMT COE. The variable is N. #-
In summary, we can say that the present results show that the semiclassical theory based on the Gutzwiller trace formula is capable to reproduce the COE form factor beyond the `diagonal approximationa. To do this, one has to include the classical correlations in the way which was done here, and once this is done, there is no need to augment the theory by uncontrolled `higher ordera or `di!ractivea corrections as was done in [85,86] and by others. The results obtained in the present section are corroborated by a recent analysis of periodic orbit lengths correlations in billiards constructed from octagonal modular domains in the hyperbolic plain [74]. The same quality of agreement was obtained between the classical form factor and the corresponding RMT result. These billiards are in two dimensions, and therefore the scaling laws depend di!erently on k, and the fact that the resulting scaled quantities agree with the expressions derived from RMT gives further support to the line of thinking developed here. We have grounds to believe that the classical correlations are universal in hyperbolic systems, and have to do with the self-similar organization of the set of periodic orbits. The previous numerical studies which were conducted also on di!erent systems support this conjecture [9,11].
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Fig. 46. The classical form factor K (N; n) compared to RMT COE. The control variable is q"n/N. #-
8. Summary In the present paper we tried to provide a complete description of a paradigmatic threedimensional quantum system which is chaotic in the classical limit } the three-dimensional Sinai billiard. This study is called for especially because most of the detailed investigations in the "eld were carried out for systems in two dimensions. Our main purpose in this study was to emphasize and clarify issues which are genuinely related to the three dimensional character of the system. The question which concerned us most was whether the semiclassical approximation } the main theoretical tool in the "eld } is su$ciently accurate for the spectral analysis of systems in three dimensions. We were able to obtain accurate and extensive databases for the quantum energy levels and for the classical periodic orbits. These allowed us to check various properties of the quantum spectrum, and in particular to study the applicability of the semiclassical approximation. The main conclusion from our work is that contrary to various expectations, the semiclassical accuracy,
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measured in units of the mean spacing, does not diverge as a +~(d~2). Our numerical tests and analytical arguments indicate an error margin which at worst diverges weakly (logarithmically) with +. One of the main problems which we had to overcome was how to separate the generic features which are common to all chaotic systems, from the system speci"c attributes, which in the present case are the `bouncing balla manifolds of periodic orbits. We should emphasize that in d dimensions the bouncing-ball manifolds contribute terms of order k(d~1)@2, which are much larger than the order 1 contributions due to generic periodic orbits. Hence it is clear that as the dimension increases, the extracting of generic features becomes more di$cult, and one has also to control higher + corrections, such as, e.g., di!raction corrections to the bouncing ball contributions. We developed a method to circumvent some of these di$culties which was su$cient for the 3D Sinai billiard case, namely, we focused on the derivative of the spectrum with respect to the boundary condition. This method is a powerful means which can also be used in other instances, where non-generic e!ects should be excluded. One of the issues which are essential to the understanding of trace formulae and their application, was "rst mentioned by Gutzwiller in his book, under the title of the `third entropya [2]. Gutzwiller noticed that in order that the series over periodic orbits can be summed up (in some sense) to a spectral density composed of d functions, the phases of the contributing terms should have very special relations. The more quantitative study of this problem started when Argaman et al. [9] de"ned the concept of periodic orbit correlations. The dual nature of the quantum spectrum of energies and the classical spectrum of periodic orbit was further developed in [11]. It follows that the universality of the quantum spectral #uctuations implies that the correlation length in the spectrum of the classical actions depends on the dimensionality in a speci"c way. This was tested here for the "rst time, and the mechanism which induces classical correlations was discussed. Our work on the Sinai billiard in three dimensions proved beyond reasonable doubt that the methods developed for two dimensional chaotic systems can be extended to higher dimensions. Of utmost importance and interest is the study of classical chaos and its quantum implications in many body systems. This is probably the direction to which the research in `quantum chaosa will be advancing.
Acknowledgements The research performed here was supported by the Minerva Center for Non-Linear Physics, and by the Israel Science Foundation. Many colleagues helped us during various stages of the work. We are indebted in particular to Michael Berry, Eugene Bogomolny, Leonid Bunimovich, Doron Cohen, Barbara Dietz, Eyal Doron, Shmuel Fishman, Klaus Hornberger, Jon Keating, Diettrich Klakow, Daniel Miller, Zeev Rudnick, Holger Schanz, Martin Sieber and Iddo Ussishkin for numerous discussions, for suggestions and for allowing us to use some results prior to their publication. In particular we thank Klaus Hornberger and Martin Sieber for their critical reading of the manuscript and for their remarks and suggestions. HP is grateful for a MINERVA postdoctoral fellowship, and wishes to thank Reinhold BluK mel and John Briggs for their hospitality in Freiburg. The Humboldt foundation is acknowledged for supporting US stay in Marburg, Germany during the summer of 1998 when much of the work on this manuscript was carried out.
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Appendix A. E7cient quantization of billiards: BIM vs. full diagonalization In this appendix we wish to compare two possible quantization schemes for billiards: direct diagonalization (DD) of the Hamiltonian matrix vs. the boundary integral method (BIM) (see e.g. [14,88]). The diagonalization is a generic method to solve the time-independent SchroK dinger equation, while the BIM is specialized for billiards. To compare the two methods, we estimate the complexity of computing all of the eigenvalues up to a given wavenumber k. To "nd the matrix elements of the Hamiltonian we treat the billiard boundaries as very high potential walls. The linear dimension M(k) of the Hamiltonian matrix that is needed for "nding eigenvalues around k is M
DD
(k)"O
AA B B
S d "O((kS)d) , j
(A.1)
where S is the typical linear dimension of the billiard, j"2p/k is the wavelength and d is the dimensionality of the billiard. The above estimate is obtained by enclosing the billiard in a hypercube with edge S and counting the modes up to wavenumber k. The numerical e!ort to "nd eigenvalues of a matrix is of order of its linear dimension to the power 3. Thus, the numerical e!ort to "nd all the eigenvalues of the billiard up to k using DD is estimated as C (k)"O(M3 (k))"O((kS)3d) . (A.2) DD DD The expected number of eigenvalues up to k is given to a good approximation by Weyl's law, which for billiards reads N(k)"O((kS)d) .
(A.3)
Thus, the numerical e!ort to calculate the "rst (lowest) N eigenvalues of a billiard in d dimension in the direct Hamiltonian diagonalization is C (N)"O(N3) (A.4) DD which is independent of the dimension. As for the BIM, one traces the k-axis and searches for eigenvalues rather than obtaining them by one diagonalization. This is done by discretizing a kernel function on the boundary of the billiard and looking for zeroes of the resulting determinant. The linear dimension of the BIM matrix is M
BIM
AA B B
"O
S d~1 "O((kS)d~1)"O(N1~1@d) . j
(A.5)
This estimate is obtained from discretizing the boundary of the billiards which is of dimension d!1 by hypercubes of edge j. The numerical e!ort of calculating the determinant once is c (k)"O(M3 (k))"O((kS)3(d~1)) . (A.6) BIM BIM (In practice, one often uses the SVD algorithm [58], which is much more stable than a direct computation of the determinant and has the same complexity.) Using the relation (A.3) we "nd that
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the numerical e!ort to "nd an eigenvalue near the Nth one is estimated by c (N)"O(N3~3@d) . (A.7) BIM To get the above result we assumed that a "xed number of iterations (evaluations of the determinant) is needed to detect each eigenvalue, which is justi"ed at least for the case where level repulsion is expected. Thus, the complexity to calculate all the eigenvalues up to the Nth is C (N)"O(N)c (N)"O(N4~3@d) . BIM BIM In particular,
(A.8)
G
O(N5@2) for d"2 , C (N)" BIM O(N3) for d"3 . We conclude that the BIM is more e$cient than DD for 2 dimensions, and for 3 dimensions they are of the same level of complexity. In practice, however, it seems that the BIM is better also in 3 dimensions, since the DD matrices can be prohibitly large, and manipulating them (if possible) can be very expensive due to memory limitations. Also one has to take into account that due to evanescent modes, the numerical proportionality factor in (A.5) is actually close to 1, while for (A.1) the factor can be large if high accuracy is desired. This is due to the fact that the o!-diagonal matrix elements of the Hamiltonian decay only like a power law due to the sharp potential and hence very large matrices are needed in order to obtain accurate eigenvalues.
Appendix B. Symmetry reduction of the numerical e4ort in the quantization of billiards Consider a d-dimensional billiard which is invariant under a group G of geometrical symmetry operations. We want to compare the numerical e!ort that is needed to compute the lowest N eigenvalues of the fully symmetric billiard with that of computing the lowest N eigenvalues of the desymmetrized billiard. By `desymmetrizeda we mean the following: if X is the full billiard domain, then the desymmetrized billiard u is such that 6 ( G g( u"X. If one uses the direct diagonalization g| (DD) of the Hamiltonian matrix, then there is no advantage to desymmetrization, because the prefactor in (A.4) should not depend on the shape of the billiard if its aspect ratio is close to 1. Therefore, the numerical e!ort of computing the lowest N levels of either the fully symmetric or the desymmetrized billiard is more or less the same using DD. On the other hand, as we show in the sequel, desymmetrization is very advantageous within the framework of the BIM. We "rst note that considering a particular irreducible representation c of G is equivalent to desymmetrization of the billiard together with imposing boundary conditions that are prescribed by c. The dimension of c is denoted as d and the order of G is denoted as NG . Given a complete c basis of functions in which the functions are classi"ed according to the irreps of G, then the fraction of the basis functions that belong to the irrep c is d2/NG ,F . This is also the fraction of c c eigenvalues that belong to c out of the total number of levels, when we consider a large number of levels. Using the notations of Appendix A, we thus have M(c) (k)"F M (k) , BIM c BIM N(c)(k)"F N(k) , c
(B.1)
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where the quantities with superscript c correspond to the desymmetrized billiard, and the others to the fully symmetric one. Using (A.5) and repeating arguments from Appendix A results in C(c) (N)"F3@dC (N) . (B.2) BIM c BIM In the equation above we replaced N(c)PN. Thus, the decrease in the density of states is more than compensated by the reduction in the size of the secular matrix and the overall numerical e!ort is diminished by a factor of F3@d. For example, in the case of the 3D Sinai billiard and for c a one-dimensional irrep, the saving factor is
A B
12 3@3 1 " F3@d" c 48 48
(B.3)
which is a very signi"cant one.
Appendix C. Resummation of D using the Ewald summation technique LM In general, the Ewald summation technique is used to calculate (conditionally convergent) summations over lattices MqN: S"+ f (q) .
(C.1)
q
One splits the sum S into two sums S , S which depend on a parameter g: 1 2 S"S #S "+ f (q; g)#+ f (q; g) . 1 2 1 2 q q
(C.2)
This splitting is usually performed by representing f (x) as an integral, and splitting the integral at g. The idea is to resum S on the reciprocal lattice Mu N using the Poisson summation formula: 1
P
S "+ ddo exp(2piqu) f (q; g),+ fK (u; g) , 1 1 1 u u
(C.3)
and to choose g such that both S and S will rapidly converge. 1 2 We need to apply the Ewald summation technique to D (k), given explicitly in Eq. (5), which LM constitute the main computational load. This is because they include summations over the Z3 lattice which need to be computed afresh for each new value of k. It is possible to apply the Ewald technique directly to each D , but it is much simpler to take an indirect route: we shall LM Ewald resum the free Green function on the 3-torus, and then read o! the D 's as expansion LM coe$cients. We start with the free outgoing Green function on the three-dimensional torus: 1 exp(ikDq!qD) GT(q)"! + , 0 4p q Z3 Dq!qD |
(C.4)
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Fig. 47. Contours for the integral evaluation of GT. In the above h,arctan[Im(k)/Re(k)]/2!p/4. 0
where we took the side of the torus to be 1 for simplicity and de"ned q,r!r@. To split the sum we use an integral representation of the summands [55,30]:
P
C
D
2 = k2 exp(ikDq!qD) exp !(q!q)2m2# " dm , 4m2 Dq!qD Jp 0(C)
(C.5)
where the integration contour C is shown in Fig. 47. It is assumed that k has an in"nitesimal positive imaginary part, which is taken to 0 at the end of the calculation. We now deform the contour into C@ (see Fig. 47), such that it runs along the real axis for m'Jg/2, and split the integral at this point as follows: GT(q)"GT(q)#GT(q) , 0 1 2 1 GT(q)"! + 1 2pJp q
P
J
(C.6)
C
D
(C.7)
C
D
(C.8)
k2 g@2 exp !(q!q)2m2# dm , 4m2 0(C{)
P
1 k2 = exp !(q!q)2m2# GT(q)"! + dm . 2 4m2 2pJp q J g@2
The summation in GT is rapidly convergent, due to the fact that we integrate over the tails of 2 a rapidly decaying function in m (faster than a Gaussian), and we start further on the tail when o grows. In order to make GT also rapidly convergent, we need to Poisson resum it. We use the 1 identity
C
D
pJp (2pg)2 + exp[!(q!q)2m2]" + exp #i(2pu)q , 4m2 m3 u q
(C.9)
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which is obtained by explicitly performing the integrals of the Poisson summation. Thus,
P
"+ u
J
g@2 dm expM[k2!(2pg)2]/4m2N m3 0(C{)
1 GT(q)"! + exp(2piuq) 1 2u
(C.10)
exp(2piuq) expM[k2!(2pg)2]/gN . k2!(2pg)2
(C.11)
The second line was obtained from the "rst one by performing the integrals explicitly. The expression obtained for GT is also rapidly convergent, and is suitable for computations. We thus 1 succeeded in rewriting GT as two rapidly converging sums (C.11), (C.8). We note that the results 0 (C.11), (C.8) are valid for general lattices, the cubic lattice being a special case [30]. The heart of the above resummation of GT was the integral representation (C.5) which is 0 non-trivial. In Appendix D we present an alternative derivation of the above results using more intuitive, physical arguments. It remains to extract the D 's from the resummed GT. The basic relation is [30] LM 0
C
k n (kq) 0 GT(q)" + j (kq)> (Xq ) D (k)# d 0 L LM LM j (kq) L0 J4p 0 LM
D
.
(C.12)
Using expansion theorems [56] applied to (C.8) and (C.11) one can rewrite GT as 0
G
e~(2pg)2@g GT(q)" + > (Xq ) + 4piLek2@g>H (Xu )j (2pgq) LM L 0 LM k2!(2pg)2 u LM !+ q
2iL
P
>H (Xq ) Jp LM
=
J
g@2
C
k2 dm j (!2ioqm2) exp !(o2#q2)m2# L 4m2
DH
.
(C.13)
Comparing Eqs. (C.12) and (C.13), and using the orthogonality of the spherical harmonics > (Xq ), LM one obtains
C
D
1 k D (k)" + 2#+ 2! n (kq)d . LM L0 j (kq) u J4p 0 q L
(C.14)
This is the Ewald-resummed expression of D (k). It has the interesting feature that even though LM each of the terms explicitly depends on q, the total expression is independent of q. The same applies also to g. This freedom can be used to simplify the expression (C.14), since for qP0 the spherical Bessel functions simplify to powers [30] (aq)L j (aq)P , L (2¸#1)!!
(C.15)
which are computationally less demanding. Taking the limit is straightforward for ¸O0, while for D there is a complication due to the singularity of n (kq). As shown in Appendix E this 00 0
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singularity is exactly cancelled by the q"0 term, resulting in a "nite expression also for D . The 00 "nal result is D "D(1) #D(2) #D(3) d , LM LM LM 00 L0
(C.16) 2
e~(2pg) @g D(1) "4piLk~Lek2@g+ (2pg)L>H (Xu ) , LM LM k2!(2pg)2 u
P
C
(C.17)
D
2L`1k~L k2 = D(2) " dm m2L exp !o2m2# + oL>H (Xq ) , (C.18) LM LM 4m2 Jp qE0 J g@2 Jg = (k2/g)n D(3) "! , (C.19) + 00 2p n!(2n!1) n/0 with the convention gLD "1. This completes the task of Ewald-resumming the building g/0,L/0 blocks D (k) into rapidly convergent series. LM Appendix D. `Physicala Ewald summation of GT(q) 0 In this appendix we present a derivation of the results (C.11), (C.8) by a method that is di!erent than the one used in Appendix C. The present method is physically appealing and does not require the use of complicated integral representations. It is inspired by Appendix B of Kittel's book [97] which deals with the problem of calculating Madelung constants (electrostatic potentials) of ion crystals. In the sequel we use q,r!r@ and adopt the following notational convention: For any quantity X(q) we add a superscript ¹ to denote its lattice sum: XT(q),+ X(q!q) .
(D.1)
q
We start from the Helmholtz equation for G : 0 (D.2) (+2r #k2)G (q)"d(q) . 0 Due to linearity, the function GT satis"es 0 (+2r #k2)GT(q)"dT(q) . (D.3) 0 The RHS of (D.3) can be interpreted as a `charge distributiona which is composed of point charges on a lattice. Each such point charge d(q!q) induces a `potentiala G (q!q)" 0 !exp(ikDq!qD)/(4pDq!qD) which is long-ranged due to the sharpness of the charge. (This is in analogy to the electrostatic case.) Hence, the lattice sum of potentials GT is conditionally conver0 gent. To overcome this di$culty we introduce an arbitrary charge distribution j(q) and rewrite (D.3) as GT(q)"GT(q)#GT(q) , 0 1 2 (+2#k2)GT(q)"jT(q) , 1 (+2#k2)GT(q)"dT(q)!jT(q) . 2
(D.4) (D.5) (D.6)
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We want j(q!q) to e!ectively screen the d(q!q) charges, making G short-ranged. This will 2 result in rapid convergence of GT. (Note, that Eqs. (D.4)}(D.6) hold also for the quantities without 2 the T superscript due to linearity.) On the other hand, j(q) must be smooth enough, such that GT will rapidly converge when Poisson resumed. It is hence plausible to choose a (spherically 1 symmetric) Gaussian charge distribution for j(q): j(q)"A exp(!aq2) ,
(D.7)
where A and a are yet arbitrary parameters. We calculate "rst G (q) by rewriting the inducing charge as an integral over d charges, and using 2 the fact that each d charge contributes G to the potential: 0
P
(+2#k2)G (q)"d(q)!j(q)"d(q)! d3Qj(Q)d(q!Q) . 2 Hence,
(D.8)
P
G (q)"G (q)! d3Qj(Q)G (q!Q) 2 0 0
C AB
D
P
A = p 3 2 dt e~a(t`q)2cos(kt) . (D.9) e~k @4a # "G (q) 1!A 0 2aq a 0 The "rst term is long-ranged due to G , and the second term is short-ranged due to the integral that 0 is rapidly decreasing as a function of q. To make G short ranged, we thus have to set the coe$cient 2 of G to 0, which is satis"ed if we choose 0 a 3 k2 . (D.10) exp A"A(k, a)" p 4a
AB A B
Therefore, we get for GT a rapidly convergent sum: 2
P
Ja ek2@4a 1 = dt exp[!a(t#Dq!qD)2] cos(kt) . + GT(q)"! 2 2pJp q Dq!qD 0
(D.11)
This can be re-expressed in a more compact form using complement error functions with complex arguments:
C
A
1 ik 1 GT(q)"! + Re exp(!ikDq!qD) erfc JaDq!qD! 2 2p q Dq!qD 2Ja
BD
,
(D.12)
where
P
1 = erfc(z), e~u2 du . Jp z
(D.13)
To calculate GT we can directly Poisson resum (D.11). Alternatively, we can use again the 1 Helmholtz equation for GT (D.5) to simplify the calculations. We expand GT in the reciprocal 1 1
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lattice:
P
GT(q)"+ d3o exp(2piqu)G (q!q) 1 1 u
P
"+ exp(2piqu) d3o exp(!2piqu)G (q),+ exp(2piqu)G u , 1 1 u u
(D.14)
where the second line was obtained from the "rst one by shifting the origin of the integration. Similarly for jT(q): j(q)"+ exp(2piqu)ju .
(D.15)
u
Inserting (D.14) and (D.15) into (D.5) and using the orthogonality of the Fourier components, we get the simple relation between G u and ju : 1 ju G u" . (D.16) 1 k2!(2pg)2 When inserted back into (D.14) we "nally get for GT: 1 exp(2piuq)exp[k2!(2pg)2/4a] . GT(q)"+ 1 k2!(2pg)2 u
(D.17)
This expression is identical with (C.11) if we set 4a"g. It can be shown [55] that also the expressions for GT, (C.8) and (D.12) are identical. However, Eq. (D.12) is more convenient if one 2 needs to compute GT(q), since it involves well-tabulated computer-library functions [58] and saves 0 the burden of numerical integrations. On the other hand, the expression (C.8) is more convenient as a starting point for calculating D (k). LM To summarize, we re-derived the Ewald-resummed form of GT(q) using the underlying 0 Helmholtz equation. We used a physically intuitive argument of screening potentials that was shown to be equivalent to the more abstract integral representation of G (q), Eq. (C.5). 0 Appendix E. Calculating D(3) 00 We need to calculate (refer to Eq. (C.14) and its subsequent paragraph):
G
C
P
A
BDH
k2 1 1 cos(kq) 1 = D(3) ,lim dm exp !q2m2# ! , (E.1) p J 4m2 j (kq) J4p q 00 q?0 0 g@2 where we used the explicit expression n (x)"!cos(x)/x. Taking the limit of the denominator is 0 trivial, since j (kq)P1. For qP0 we can write 0 1 cos(kq) 1 " #O(q) , (E.2) J4p q J4pq
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which contains 1/q singularity. As for the term with the integral, we expand exp(k2/4m2) in a Taylor series, and transforming to the variable t"qm one gets:
P
A
B
P
1 = k2 1 = (kq)2n = ! dm exp !q2m2# dt t~2n e~t2 . "! + p J 4nn! J 4m2 pq n/0 q g@2 g@2 For n"0:
P
=
P
=
AP P B
Jp 1 = qJg@2 ! dt e~t2" ! Jgq#O(q2) . 2 2 J 0 0 q g@2 For n'0 we integrate by parts:
J
dt e~t2"
A
B
1 ~2n`1 1 e~gq2@4#O(q~2n`3) . dt t~2n e~t2" Jgq 2n!1 2
(E.3)
(E.4)
(E.5)
q g@2 Collecting everything together back to (E.1), the 1/q singularities cancel, and we remain with the "nite expression: Jg = (k2/g)n D(3) "! . + 00 2p n!(2n!1) n/0
(E.6)
Appendix F. The `cubic harmonicsa >(c) LJK F.1. Calculation of the transformation coezcients a(L) cJK,M We want to calculate the linear combinations of spherical harmonics that transform according to the irreducible representations of the cubic group O . This problem was addressed by von der Lage h and Bethe [57] which coined the term `cubic harmonicsa for these combinations. They gave an intuitive scheme that was used to calculate the "rst few cubic harmonics, but their arguments are di$cult to extend for large ¸'s. Moreover, their method is recursive, because one has to orthogonalize with respect to all lower lying combinations. This is cumbersome to implement numerically and might result in instabilities for large ¸'s. The only other work on the subject that we are aware of [98] specializes in the symmetric representation and gives only part of the combinations. It also expresses the results not in terms of spherical harmonics, but rather as polynomials that are di$cult to translate to > 's. LM We describe in the following a simple and general numerical method to calculate the cubic harmonics in a non-recursive way. This is based on a general theorem that states that a function f (c) transforms according to the irrep c i! it satis"es [31] PK (c)f (c)"f (c)
(F.1)
where PK (c) is the projection operator onto the subspace that belongs to c: l PK (c)" c + s(c)H(g( )g( . N ( G g|G
(F.2)
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We denoted by l the dimension of c, N is the number of elements in the group G, and s(c)(g( ) are c G the characters. The realization of PK (c) as a matrix in an arbitrary basis results in general in an in"nite matrix. However, in the case of the cubic harmonics, we know that O LO(3), thus the h operations of g( 3O do not mix di!erent ¸'s. Hence, working in the > basis, we can write the h LM cubic harmonics as the "nite combinations: `L H (F.3) >(c) (X)" + a(L) > (X) , LJ cJ,M LM M/~L where J enumerates the irreps c in ¸. For simplicity we consider 1-dimensional irreps. Applying (F.1), (F.2) to (F.3) and using the Wigner matrices D(L)(g( ) to express the operations of g( on > [56], we get the following (2¸#1)](2¸#1) linear system: LM + [P(c,L) !d ]a(L) "0 , MM{ cJ,M{ MM{ M{ where H
(F.4)
1 (F.5) P(c,L) " + s(c)H(g( )D(L) (g( ) . MM{ MM{ 48 ( g|G The above equations are best solved using SVD algorithm [58], and the (orthonormalized) eigenvectors that belong to the zero singular values are the required coe$cients a(L)H . For cJ,M{ multi-dimensional irreps one needs to classify the cubic harmonics also with respect to the row K inside the irrep. This can be done by simple modi"cation of the above procedure, using the appropriate projectors [31]. The above general procedure can be simpli"ed for speci"c irreps. In the following we shall concentrate on the completely antisymmetric irrep c"a and further reduce the linear system (F.4). We "rst note that the antisymmetric cubic harmonics must satisfy per de"nition: g( >(a) (X)"s(a)(g( )>(a) (X)"(!1)(1!3*5: 0& g( )>(a) (X) ∀g( 3O . LJ LJ LJ h We then choose a few particular g( 's for which the operations on > (X) are simple: LM r( (xyz),(!xyz): r( > (h, /)"> (h,!/)"(!1)M> (h, /) , x x LM LM L~M r( (xyz),(x!yz): r( > (h, /)"> (h, p!/)"> (h, /) , y y LM LM L~M r( (xyz),(xy!z): r( > (h, /)"> (p!h, /)"(!1)L`M> (h, /) , z z LM LM LM (h, /) . p( (xyz),(yxz): p( > (h, /)"> (h, p !/)"(!i)M> L~M xy xy LM LM 2 Applying (F.6) and (F.7) to (F.3) results in the following `selection rulesa:
(F.6)
(F.7)
a(L)H "0, ¸O2p#1, MO4q, p, q3N , aJ,M (F.8) H H a(L) "!a(L) aJ,~M aJ,M which reduces the number of independent coe$cients to be computed by a factor of 16. The form of the projector matrix P(aL) can also be greatly reduced, if we observe that the group O can be h
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written as the following direct multiplication: O "G ?G , h 3 16 G "Me( , c( , c( 2N e( "identity, c( (xyz)"(yzx) , 3 G "Me( , p( N?Me( , r( N?Me( , r( N?Me( , r( N 16 xy x y z and consequently, the projector can be written as
(F.9) (F.10) (F.11)
PK (a)"PK PK , (F.12) 3 16 PK "e( #c( #c( 2 , (F.13) 3 PK "(e( !p( )(e( !r( )(e( !r( )(e( !r( ) . (F.14) 16 xy x y z The operator PK acts as the identity on the subspace de"ned by (F.8) and hence we need to 16 consider only the operation of PK . Simple manipulations give the following set of equations: 3 p H + 2d(L) !d a(L) "0, q'0, ¸"2p#1 . (F.15) 4q,4q{ aJ,4q{ 4q,4q{ 2 q{;0 The matrices d(L) are the `reduceda Wigner matrices, which are real [56], thus the resulting 4q,4q{ coe$cients are also real. The above is a square linear system, which is 8 times smaller than the general one (F.4).
C
AB
D
F.2. Counting the >(c) 's LJ The number of the irreps c of O that are contained in the irrep ¸ of O(3) is given by the formula h [31]: 1 (F.16) N " + s(c)H(g( )s (g( ) L cL 48 ( g|Oh where s (g( ) are the characters of the irrep ¸. An explicit calculation shows that the main L contributions for large ¸'s come from the identity and from the inversion operations, thus l (2¸#1) . N +[1$(!1)L] c cL 48
(F.17)
where the $ corresponds to the parity of c. Since for l -dim irrep we have l basis functions, and c c there are 2¸#1 basis functions in the irrep ¸, the fraction of cubic harmonics that belong to c is F + 1 l2 c 48 c in accordance with the general relation
(F.18)
+ l2"48 . c c Consequently, the fraction of cubic harmonics that belong to the Kth block of c is
(F.19)
F +1l . cK 48 c
(F.20)
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Appendix G. Evaluation of l(q ) p G.1. Proof of Eq. (10) We need to prove the relation + > (X ( q )"l(q ) + > (Xq ) , (G.1) LM { LM g pq q g|Oh {|S( p ) where q ,(i, j, k) is the unique vector in the set O q which resides in the fundamental domain p h i5j5k50, S(q ) is the collection of all distinct vectors obtained by the operations g( q , g( 3O , p p h and l(q ) is an integer. p (
Proof. Let H be the set of all g( 3O under which q is invariant: h p g( q "q Qg( 3H . p p The set H is a subgroup since
(G.2)
1. The identity e( 3H. 2. The set H is closed under multiplication, since if g( , g( 3H then g( (g( q )"g( q "q . 1 2 1 2 p 1 p p 3. The set H is closed under inversion: g( ~1q "g( ~1(g( q )"q . p p p The order of (number of terms in) H is denoted as NH . We assume that H is the maximal invariance subgroup, and construct the right cosets g( H"Mg( hK ,2N. According to [31] there are 1 N "48/NH mutually exclusive such cosets C ,2, C c . (The number 48 is the order of O .) Their c 1 N h union is O . For each coset C we can de"ne h i q ,C q (G.3) i i p which is meaningful due to the invariance of q under H. p We want to prove the following Lemma. q Oq iw iOj. i j Proof. Assume the opposite, then in particular g( q "g( q i p j p Q (g( ~1g( )q "q j i p p Q g( ~1g( "h3H j i Q g( "g( h i j Q C "C i j in contradiction to the assumption. The last line was obtained using the rearrangement theorem [31] applied to the group H. h
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We now set Nc S(q )" Z q i p i/1 l(q )"NH "integer p and since + ( h "+Nc + ( i we proved (G.1). h g|O i/1 g|C
(G.4) (G.5)
G.2. Calculating l(q ) p We give an explicit expression of l(q ). Consider q "(i, j, k) such that i5j5k50 without loss p p of generality. Then l(q )"l (q )l (q ) , p p p s p 1, iOjOkOi ,
(G.6)
l (q )" 2, i"jOk or iOj"k or i"kOj , p p 6, i"j"k ,
(G.7)
G
l (q )"2(j;%30 */$*#%4) . (G.8) s p We prove this formula in the following. First we observe that O can be decomposed as h O "P ?S , (G.9) h 3 3 P "group of permutation of 3 numbers , (G.10) 3 S "M$$$N"3 sign changes . (G.11) 3 Let H , H be the subgroups of P , S , respectively, under which q is invariant. P S 3 3 p Lemma. H"H ?H . P S Proof. Let g( "p( s( , where g( 3O , p( 3P and s( 3S . This representation of g( is always possible h 3 3 according to (G.9). If s( N H then s( q Oq , thus necessarily there is at least one sign change in s( q S p p p with respect to q . Consequently, g( q Oq , because permutations only change the order of indices p p p and cannot restore the di!erent sign(s). We conclude that g( N H. Thus, g( 3HNs( 3H . For every S g( 3H we must have therefore g( q "p( s( q "p( q "q which proves that also p( 3H . h p p p p P We conclude that NH "order(H ) ) order(H ). This is manifest in Eqs. (G.6)}(G.8). P S Appendix H. Number-theoretical degeneracy of the cubic lattice H.1. First moment The following arguments are due to Keating [99]. We "rst need to estimate the fraction of integers that can be expressed as a sum of 3 squares. The key theorem is due to Gauss and Legendre
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and states that q"i2#j2#k2, i, j, k3N Q qO4m(8l#7), m, l3N .
(H.1)
From this we can estimate that the fraction of integers which cannot be expressed as a sum of 3 squares of integers is
A
B
1 1 1 1 1# # #2 " . 4 42 6 8
(H.2)
In the above we used the fact (which is easily proven) that if q"4m(8l#7) then m, l are uniquely determined. Therefore, asymptotically only 5/6 of the integers are expressible as a sum of three squares. Our object of interest is the degeneracy factor g (o) de"ned as o g (o),d(j3Z3 D i"o) . (H.3) o The number of Z3-lattice points whose distance from the origin is between o and o#*o is estimated by considering the volume of the corresponding spherical shell: N +4po2*o . o Since o2 is an integer, the number of integers in the same interval is
(H.4)
n +2o*o . o Taking into account that only 5/6 of the integers are accessible, we obtain
(H.5)
N 12p o " Sg (o)T" o. o (5/6)n 5 o
(H.6)
H.2. Second moment Here we use a result due to Bleher and Dyson [100], brought to our attention by Rudnick: N 16p2 f(2) + g2(Jk)"cN2#error, c" +30.8706 . o 7 f(3) k/1 Di!erentiating by N and considering only integers for which g O0 one obtains o Sg2(o)T+12co2+74.0894o2 . o 5 Therefore, Sg2(o)T/Sg (o)T+bo, b"c/p+9.8264 . o o
(H.7)
(H.8) (H.9)
Appendix I. Weyl's law A very important tool in the investigation of eigenvalues is the smooth counting function, known as Weyl's law. For billiards it was thoroughly discussed e.g. by Balian and Bloch [82] and by Baltes and Hilf [91]. We construct in the following the expression for the 3D Sinai billiard. In general, it
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99
takes on the form NM (k)"N k3#N k2#N k#N , (I.1) 3 2 1 0 where we included terms up to and including the constant term. In fact, for the nearest-neighbour and two-point spectral statistics the constant term N is unimportant, since it shifts the unfolded 0 spectrum x ,NM (k ) uniformly. Nevertheless, for completeness we shall calculate this term. We n n enumerate the contributions in the case of Dirichlet boundary conditions one by one and then write down the full expression. Fig. 6 should be consulted for the geometry of the billiard. N : There is only one contribution due to the volume of the billiard: 3
A
B
volume 1 4 N " " S3! pR3 . 3 6p2 288p2 3
(I.2)
N : The contribution is due to the surface area of the planes#sphere: 2 surface 1 N "! "! [6(1#J2)S2!7pR2] . 2 16p 384p
(I.3)
N : Here we have contributions due to the curvature of the sphere and due to 2-surface edges: 1 Curvature:
P
C
D
1 R 1 # "! , 72p R (s) R (s) 2 1 463&!#% where R , R are the principal local radii of curvature. 1 2 1 N#637!563%" 1 12p2
ds
(I.4)
Edges: We have 6 plane}plane edges and 3 plane}sphere edges. Their contributions are given by
A
B
A
B
1 a S R 9p 95 p N%$'%4" + ! j ¸" (27#9J2#8J3)# ! , 1 24p p j 144p 24p 8 12 a %$'%4 j
(I.5)
where ¸ are the lengths of the edges, and a are the corresponding angles. j j N : There are three terms here due to square of the curvatures, 3-surface corners and curvature 0 of the edges: Curvature2: 1 N#637!563%2" 0 512p
P
463&!#%
ds
C
D
1 1 2 ! "0 . R (s) R (s) 2 1
(I.6)
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3-surface corners: In the 3D Sinai billiard we have 6 corners due to intersection of 3 surfaces; 3 of them are due to intersection of 3 symmetry planes and the other 3 are due to intersection of 2 symmetry planes and the sphere. The corners are divided into 4 types as follows: 1]a,(453, 54.743, 36.263) , 3]b,(453, 903, 903) , 1]c,(603, 903, 903) , 1]d,(903, 903, 903) . As for the corners b, c, d which are of the type (/, 903, 903) there is a known expression for their contribution [91]:
A
B
(I.7)
c "! 5 , c "! 1 , c "! 1 . c 36 d 64 b 128
(I.8)
1 p / c "! ! . ( 96 / p Therefore,
As for the corner a, we calculate its contribution from the R"0 integrable case (`the pyramida). The constant term in the case of the pyramid is !5/16 [53] and originates only from 3-plane contributions (there are no curved surfaces or curved edges in the pyramid). The pyramid has 4 corners: 2 of type a and 2 of type b. Using c above we can therefore eliminate c : b a 2c #2(! 5 )"! 5 Nc "! 15 . a 128 16 a 128
(I.9)
Hence, the overall contribution due corners in the 3D Sinai is N3v463&!#%"1(! 15 )#3(! 5 )#1(! 1 )#1(! 1 )"! 5 . 0 128 128 36 64 18
(I.10)
Curvature of edges: We have 3 edges which are curved. They are 903 edges that are due to plane-sphere intersections. Baltes and Hilf [91] quote the constant term (!1/12)#(1/256)(H/R) for the circular cylinder, where H is the height and R is the radius of the cylinder. We conclude from this that the H-independent term !1/12 is due to the curvature of the 903 edges between the 2 bases and the tube. Assuming locality, it is then plausible to conjecture that the contribution due to the curvature of a 903 edge is 1 ! 48p
P
dl , R(l)
(I.11) %$'% where R(l) is the local curvature radius of the edge. When applied to our case (R(l)"!R), we get N#637. %$'%" 1 . 0 64
(I.12)
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101
Putting everything together we get
A
B
1 4 1 NM (k)" S3! pR3 k3! [6(1#J2)S2!7pR2]k2 288p2 3 384p
C
#
A
S 3 11 (27#9J2#8J3)#R ! 144p 64 32p
BD
151 k! . 576
(I.13)
Appendix J. Calculation of the monodromy matrix The monodromy matrix measures the linear response to in"nitesimal displacements of the initial conditions of a classical orbit. Its eigenvalues determine the stability of the orbit. Due to the symplectic form of the equations of motion, if j is an eigenvalue of the monodromy matrix then also jH, 1/j and 1/jH [2]. Therefore, generically the eigenvalues come in groups of four: j"exp($u$iv), u, v3R .
(J.1)
In d dimensions the monodromy has 2(d!1) eigenvalues. Therefore, only for d53 the generic situation (291) can take place. In two dimensions there are only two eigenvalues and consequently one obtains the following three possible situations (which are special cases of (J.1) with either u or v set to 0): f Elliptic: j "exp($iv), stable orbit. 1,2 f Parabolic: j "1 or j "(!1), neutrally stable orbit. 1,2 1,2 f Hyperbolic: j "exp($u) or j "!exp($u), unstable orbit. 1,2 1,2 The parabolic case with the `#a sign is denoted as `direct parabolica and with `!a sign it is denoted as `inverse parabolica. Similar terminology applies to the hyperbolic case. The generic case (J.1) is designated as `loxodromic stabilitya [2]. J.1. The 3D Sinai torus case We wish to calculate explicitly the 4]4 monodromy matrices in the case of the 3D Sinai torus. There are (at least) two possible ways to tackle this problem. One possibility is to describe the classical motion by a discrete (Hamiltonian, area-preserving) mapping between consecutive re#ections from the spheres. The mapping is generated by the straight segment that connects the two re#ection points, and the monodromy can be explicitly calculated from the second derivatives of the generating function. This straightforward calculation was performed for the 2D case (for general billiards) e.g. in [72] and it becomes very cumbersome for three dimensions. Rather, we take the alternative view of describing the classical motion as a continuous #ow in time, as was done e.g. by Sieber [60] for the case of the 2D hyperbola billiard. We separate the motion into the sections of free propagation between spheres and re#ections o! the spheres, and the monodromy
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matrix takes the general form M"Mn`10nMn 2M302M2 M201M1 , (J.2) 1301 3%& 1301 3%& 1301 3%& where Mi`10i describes the free propagation from sphere i to sphere i#1 and Mi describes the 1301 3%& re#ection from the sphere i. To explicitly calculate the matrices one has to choose a well-de"ned (and convenient) coordinate system, which is a non-trivial task in three dimensions. If we denote the direction along the orbit by `1a, then we have two more directions, denoted henceforth `2a and `3a. Hence there is a rotation freedom in choosing the directions 2 and 3. For convenience of calculation of M we choose the following local convention for coordinates: Near sphere i there exists the plane 3%& P which is uniquely de"ned (except for normal incidence) by the incoming segment, the outgoing i segment and the normal to the sphere i at the re#ection point. Direction 1 is obviously in P . We i uniquely de"ne direction 3 to be perpendicular to P along the direction of the cross product of the i outgoing direction with the normal. Direction 2 is then uniquely de"ned as e( ,e( ]e( such that 2 3 1 a right-handed system is formed. Obviously e( is contained in P . The uniqueness of the local 2 i coordinate system guarantees that the neighbourhoods of the initial and the "nal points of the periodic orbits are correctly related to each other. To account for the local coordinate systems we need to apply a rotation between every two re#ections that aligns the `olda system to the `newa one. Hence, M"Mn`10nMn`10nMn 2M302M302M2 M201M201M1 . (J.3) 1301 305 3%& 1301 305 3%& 1301 305 3%& We should also "x the convention of the rows and columns of M in order to be able to write explicit expressions. It is chosen to be
AB AB A A A
dq dq 2 2 dp dp 2 "M 2 . dq dq 3 3 dp dp 3 &*/!3 */*5*!A detailed calculations gives the explicit expressions for M , M and M : 1301 3%& 305 1 ¸ /p 0 0 i`10i 0 1 0 0 , Mi`10i" 1301 0 0 1 ¸ /p i`10i 0 0 0 1 !1 ! 2p i R #04 b Mi " 3%& 0 0
B
0
0
0
!1
0
0
0
1 2p#04 bi R 0
0
0
cos a i`10i 0
Mi`10i" 305 sin a i`10i 0
cos a
i`10i 0
sin a i`10i
B
,
(J.4)
(J.5)
(J.6)
1
!sin a i`10i 0 cos a i`10i 0
0
B
!sin a i`10i . 0 cos a i`10i
(J.7)
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103
In the above p is the absolute value of the momentum which is a constant, ¸ is the length of i`10i the orbit's segment between spheres i and i#1, b is the re#ection angle with respect to the normal i of the sphere i and a is the angle that is needed to re-align the coordinate system from sphere i`10i i to i#1. Even though the entries of M are dimensional, the eigenvalues of M are dimensionless. Hence, the eigenvalues cannot depend on p, which is the only variable with dimensions of a momentum. (All other variables have either dimension of length or are dimensionless.) Therefore, one can set p"1 for the sake of the calculations of the eigenvalues of M. The formulas above for the monodromy were veri"ed numerically for a few cases against a direct integration of the equations of motion near a periodic orbit of the Sinai torus. We mention the work of Sieber [52] which extends the calculation of the monodromy matrix to an arbitrary billiard in three dimensions. J.2. The 3D Sinai billiard case We next deal with the calculation of the monodromy matrix for the periodic orbits of the desymmetrized 3D Sinai billiard. In principle, one can follow the same procedure as above, and calculate the monodromy as for the Sinai torus case, this time taking into account the presence of the symmetry planes. A re#ection with a symmetry plane is described by
A
M1-!/%" 3%&
0
B
0
!1
0 0
!1 0 0
0
0
1 0
0
0
0 1
,
(J.8)
which is simply M with RPR. This method, however, is computationally very cumbersome 3%& because of the need to fold the orbit into the desymmetrized Sinai billiard. Instead, we can use the monodromy matrix that is calculated for the unfolded periodic orbit, because the initial and "nal (phase space) neighbourhoods are the same modulo g( . A calculation shows, that in order to align the axes correctly, one needs to reverse direction 3 if g( is not a pure rotation:
A
1 0
0
0 1 0 MK " W 0 0 p(g( ) 0 0
0
0 0 0
G
(J.9)
p(g( )
where p(g( ) is the parity of g( : p(g( )"
B
M6/&0-$%$ , WK
#1, g( is a rotation ,
!1, g( is an improper rotation (rotation#inversion) .
(J.10)
The above formulas were veri"ed numerically for a few cases by comparing the result (J.9) to a direct integration of the classical dynamics in the desymmetrized Sinai billiard.
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[81] H. Primack, H. Schanz, U. Smilansky, I. Ussishkin, Penumbra di!raction in the quantization of dispersing billiards, Phys. Rev. Lett. 76 (1996) 1615}1618. [82] R. Balian, B. Bloch, Distribution of eigenfrequencies for the wave equation in a "nite domain I. Three-dimensional problem with smooth boundary surface, Ann. Phys. 60 (1970) 401}447. * [83] M. Sieber, H. Primack, U. Smilansky, I. Ussishkin, H. Schanz, Semiclassical quantization of billiards with mixed boundary conditions, J. Phys. A 28 (1995) 5041}5078. [84] E.B. Bogomolny, Semiclassical quantization of multidimensional systems, Nonlinearity 5 (1992) 805. [85] I.L. Aleiner, A.I. Larkin, Divergence of the classical trajectories and weak localization, Phys. Rev. B 54 (1996) 14 423. [86] R.S. Whitney, I.V. Lerner, R.A. Smith, Can the trace formula describe weak localization? Waves Random Media 9 (1999) 179}200; cond-mat/9902328. [87] W. Pauli, AusgewaK hlte Kapital aus der Feldquantisierung, in: C. Enz (Ed.), Lecture Notes, ZuK rich, 1951. [88] P.A. Boasman, Semiclassical accuracy for billiards, Nonlinearity 7 (1994) 485}533. * [89] Per Dahlqvist, Error of semiclassical eigenvalues in the semiclassical limit: an asymptotic analysis of the Sinai billiard, J. Phys. A 32 (1999) 7317}7344; chao-dyn/9812017. [90] B. Georgeot, R.E. Prange, Exact and quasiclassical Fredholm solutions of quantum billiards, Phys. Rev. Lett. 74 (1995) 2851}2854. [91] H.P. Baltes, E.R. Hilf, Spectra of Finite Systems, Bibliographisches Institut, Mannheim, 1976. [92] M.V. Berry, C.J. Howls, High orders of the Weyl expansion for quantum billiards: resurgence of the Weyl series, and the Stokes phenomenon, Proc. Roy. Soc. London A 447 (1994) 527}555. [93] H. Primack, U. Smilansky, On the accuracy of the semiclassical trace formula, J. Phys. A 31 (1998) 6253}6277. [94] H. Schanz, Investigation of two quantum chaotic systems, Ph.D. Thesis, Humboldt University, Berlin, 1997, LOGOS Verlag, Berlin, 1997. [95] E. Bogomolny, C. Schmit, Semiclassical computations of energy levels, Nonlinearity 6 (1993) 523}547. [96] R. Aurich, J. Bolte, F. Steiner, Universal signatures of quantum chaos, Phys. Rev. Lett. 73 (1994) 1356}1359. [97] C. Kittel, Introduction to Soild State Physics, Wiley, New York, 1953. [98] S. Golden, T.R. Tuttle Jr., Coordinate-permutable cubic harmonics and their determination, Phys. Rev. B 42 (1990) 6916}6920. [99] Jon Keating, private communication. [100] P.M. Bleher, F.J. Dyson, Mean square limit for lattice points in spheres, Acta Arithmetica 68 (1994) 383}393. [101] T. Papenbrock, T. Prosen, Quantization of a billiard model for interacting particles, Phys. Rev. Lett. 84 (2000) 262; chao-dyn/9905008. [102] T. Hesse, Semiklassische Untersuchung Zwei- und dreidimensionaler Billiardsysteme, Ph.D. thesis, UniversitaK t Ulm (1997).
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ORIGIN AND PROPAGATION OF EXTREMELY HIGH-ENERGY COSMIC RAYS
Pijushpani BHATTACHARJEE!, GuK nter SIGL" !Indian Institute of Astrophysics, Bangalore 560 034, India "Astronomy & Astrophysics Center, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Origin and propagation of extremely high-energy cosmic rays Pijushpani Bhattacharjee!,*, GuK nter Sigl" !Indian Institute of Astrophysics, Bangalore 560 034, India "Astronomy&Astrophysics Center, Enrico Fermi Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA Received August 1999; editor: M. Kamionkowski
Contents 1. Introduction and scope of this review 2. The observed cosmic rays 2.1. Detection methods at di!erent energies 2.2. The measured energy spectrum 2.3. Events above 1020 eV 2.4. Composition 2.5. Anisotropy 2.6. Next-generation experiments on ultrahighenergy cosmic ray, c-ray, and neutrino astrophysics 3. Origin of bulk of the cosmic rays: general considerations 3.1. Energetics 3.2. Galactic versus extragalactic origin of the bulk of the CR 3.3. Acceleration mechanisms and possible sources 4. Propagation and interactions of ultra-highenergy radiation 4.1. Nucleons, nuclei, and the Greisen} Zatsepin}Kuzmin cuto! 4.2. UHE photons and electromagnetic cascades 4.3. Propagation and interactions of neutrinos and `exotica particles
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4.4. Signatures of galactic and extragalactic magnetic "elds in UHECR spectra and images 4.5. Constraints on EHECR source locations 4.6. Source search for EHECR events 4.7. Detailed calculations of ultra-high-energy cosmic-ray propagation 4.8. Anomalous kinematics, quantum gravity e!ects, Lorentz symmetry violations 5. Origin of UHECR: acceleration mechanisms and sources 5.1. Maximum achievable energy within di!usive shock acceleration mechanism 5.2. Source candidates for UHECR 5.3. A possible link between gamma-ray bursts and sources of E'1020 eV events? 6. Non-acceleration origin of cosmic rays above 1020 eV 6.1. The basic idea 6.2. From X particles to observable particles: hadron spectra in quarkPhadron fragmentation 6.3. Cosmic topological defects as sources of X particles: general considerations 6.4. X particle production from cosmic strings
* Corresponding author. E-mail addresses:
[email protected] (P. Bhattacharjee),
[email protected] (G. Sigl) 0370-1573/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 1 - 5
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P. Bhattacharjee, G. Sigl / Physics Reports 327 (2000) 109}247 6.5. X particles from superconducting cosmic strings 6.6. X particles from decaying vortons 6.7. X particles from monopoles 6.8. X particles from cosmic necklaces 6.9. A general parametrization of production rate of X particles from topological defects 6.10. TDs, EHECR, and the baryon asymmetry of the Universe 6.11. TeV-scale Higgs X particles from topological defects in supersymmetric theories 6.12. TDs themselves as EHECR particles 6.13. EHECR from decays of metastable superheavy relic particles
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6.14. Cosmic rays from evaporation of primordial black holes 7. Constraints on the topological defect scenario 7.1. Low-energy di!use c-ray background: role of extragalactic magnetic "eld and cosmic infrared background 7.2. Constraints from primordial nucleosynthesis 7.3. Constraints from distortions of the cosmic microwave background 7.4. Constraints on neutrino #uxes 8. Summary and conclusions Acknowledgements References
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Abstract Cosmic-ray particles with energies in excess of 1020 eV have been detected. The sources as well as the physical mechanism(s) responsible for endowing cosmic-ray particles with such enormous energies are unknown. This report gives a review of the physics and astrophysics associated with the questions of origin and propagation of these extremely high-energy (EHE) cosmic-rays in the Universe. After a brief review of the observed cosmic rays in general and their possible sources and acceleration mechanisms, a detailed discussion is given of possible `top-downa (non-acceleration) scenarios of origin of EHE cosmic rays through decay of su$ciently massive particles originating from processes in the early Universe. The massive particles can come from collapse and/or annihilation of cosmic topological defects (such as monopoles, cosmic strings, etc.) associated with Grand Uni"ed Theories or they could be some long-lived metastable supermassive relic particles that were created in the early Universe and are decaying in the current epoch. The highest energy end of the cosmic-ray spectrum can thus be used as a probe of new fundamental physics beyond Standard Model. We discuss the role of existing and proposed cosmic-ray, gamma-ray and neutrino experiments in this context. We also discuss how observations with next generation experiments of images and spectra of EHE cosmic-ray sources can be used to obtain new information on Galactic and extragalactic magnetic "elds and possibly their origin. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 98.70.Sa; 98.70.Vc; 95.30.Cq Keywords: Ultrahigh energy cosmic rays; Topological defects; Acceleration mechanism; Cosmic ray interaction
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1. Introduction and scope of this review The cosmic rays (CR) of extremely high-energy (EHE) } those with energy Z1020 eV [1}3,6}8] } pose a serious challenge for conventional theories of origin of CR based on acceleration of charged particles in powerful astrophysical objects. The question of origin of these extremely high-energy cosmic rays (EHECR)1 is, therefore, currently a subject of much intense debate and discussions [4,5,9], for a recent brief review, see Ref. [10]. The current theories of origin of EHECR can be broadly categorized into two distinct `scenariosa: the `bottom-upa acceleration scenario, and the `top-downa decay scenario, with various di!erent models within each scenario. As the names suggest, the two scenarios are in a sense exact opposite of each other. In the bottom-up scenario, charged particles are accelerated from lower energies to the requisite high energies in certain special astrophysical environments. Examples are acceleration in shocks associated with supernova remnants, active galactic nuclei (AGNs), powerful radio galaxies, and so on, or acceleration in the strong electric "elds generated by rotating neutron stars with high surface magnetic "elds, for example. In the top-down scenario, on the other hand, the energetic particles arise simply from decay of certain su$ciently massive particles originating from physical processes in the early Universe, and no acceleration mechanism is needed. The problems encountered in trying to explain the EHECR in terms of acceleration mechanisms have been well documented in a number of studies; see, e.g., Refs. [11}14]. Even if it is possible, in principle, to accelerate particles to EHECR energies of order 100 EeV in some astrophysical sources, it is generally extremely di$cult in most cases to get the particles come out of the dense regions in and/or around the sources without losing much energy. Currently, the most favorable sources in this regard are perhaps a class of powerful radio galaxies (see, e.g., Refs. [15}20] for recent reviews and references to literature), although the values of the relevant parameters required for acceleration to energies Z100 EeV are somewhat on the side of extreme [13]. However, even if the requirements of energetics are met, the main problem with radio galaxies as sources of EHECR is that most of them seem to lie at large cosmological distances, <100 Mpc, from Earth. This is a major problem if EHECR particles are conventional particles such as nucleons or heavy nuclei. The reason is that nucleons above K70 EeV lose energy drastically during their propagation from the source to Earth due to Greisen}Zatsepin}Kuzmin (GZK) e!ect [21,22], namely, photoproduction of pions when the nucleons collide with photons of the cosmic microwave background (CMB), the mean-free path for which is & few Mpc [23]. This process limits the possible distance of any source of EHE nucleons to [100 Mpc. If the particles were heavy nuclei, they would be photo-disintegrated [24,25] in the CMB and infrared (IR) background within similar distances (see Section 4 for details). Thus, nucleons or heavy nuclei originating in distant radio galaxies are unlikely to survive with EHECR energies at Earth with any signi"cant #ux, even if they were accelerated to energies of order 100 EeV at source. In addition, since EHECR are hardly de#ected at least by the large-scale intergalactic and/or Galactic magnetic "elds, their arrival directions
1 We shall use the abbreviation EHE to speci"cally denote energies EZ1020 eV, while the abbreviation UHE for `Ultra-High Energya will sometimes be used to denote EZ 1 EeV, where 1 EeV"1018 eV. Clearly, UHE includes EHE but not vice versa.
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should point back to their sources in the sky (see Section 4 for details). Thus, EHECR o!er us the unique opportunity of doing charged particle astronomy. Yet, for the observed EHECR events so far, no powerful sources along the arrival directions of individual events are found within about 100 Mpc [26,12].2 There are, of course, ways to avoid the distance restriction imposed by the GZK e!ect, provided the problem of energetics is somehow solved separately and provided one allows new physics beyond the Standard Model of particle physics; we shall discuss those suggestions later in this review. In the top-down scenario, on the other hand, the problem of energetics is trivially solved from the beginning. Here, the EHECR particles owe their origin to decay of some supermassive `Xa particles of mass m <1020 eV, so that their decay products, envisaged as the EHECR particles, X can have energies all the way up to &m . Thus, no acceleration mechanism is needed. The sources X of the massive X particles could be topological defects such as cosmic strings or magnetic monopoles that could be produced in the early Universe during symmetry-breaking phase transitions envisaged in grand uni"ed theories (GUTs). In an in#ationary early Universe, the relevant topological defects could be formed at a phase transition at the end of in#ation. Alternatively, the X particles could be certain supermassive metastable relic particles of lifetime comparable to or larger than the age of the Universe, which could be produced in the early Universe through, for example, particle production processes associated with in#ation. Absence of nearby powerful astrophysical objects such as AGNs or radio galaxies is not a problem in the top-down scenario because the X particles or their sources need not necessarily be associated with any speci"c active astrophysical objects. In certain models, the X particles themselves or their sources may be clustered in galactic halos, in which case the dominant contribution to the EHECR observed at Earth would come from the X particles clustered within our Galactic Halo, for which the GZK restriction on source distance would be of no concern. In this report we review our current understanding of some of the major theoretical issues concerning the origin and propagation of EHECR with special emphasis on the top-down scenario of EHECR origin. The principal reason for focusing primarily on the top-down scenario is that there already exists a large number of excellent reviews which discuss the question of origin of ultra-high-energy cosmic rays (UHECR) in general and of EHECR in particular within the general bottom-up acceleration scenario in details; see, e.g., Refs. [11,28,16}20]. However, for completeness, we shall brie#y discuss the salient features of the standard acceleration mechanisms and the predicted maximum energy achievable for various proposed sources of UHECR. We would like to emphasize here that this is primarily a theoretical review; we do not discuss the experimental issues (for the obvious reason of lack of expertise), although, again, for completeness, we shall mention the major experimental techniques and brie#y review the experimental results concerning the EHECR. For an excellent historical account of the early experimental
2 Very recently, it has been suggested by Boldt and Ghosh [27] that particles may be accelerated to energies &1021 eV near the event horizons of spinning supermassive black holes associated with presently inactive quasar remnants whose numbers within the local cosmological universe (i.e., within a GZK distance of order 50 Mpc) may be su$cient to explain the observed EHECR #ux. This would solve the problem of absence of suitable currently active sources associated with EHECR. A detailed model incorporating this suggestion, however, remains to be worked out.
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developments and the "rst claim of detection of an EHECR event, see Ref. [29]. For reviews on UHECR experiments in general and various kinds of experimental techniques used in detecting UHECR, see Refs. [30}32]. For a review of the current experimental situation concerning EHECR, see the recent review by Yoshida and Dai [33]. Overviews of the various currently operating, up-coming, as well as proposed future EHECR experiments can be found, e.g., in Refs. [5,9]. By focusing primarily on the top-down scenario, we do not wish to give the wrong impression that the top-down scenario explains all aspects of EHECR. In fact, as we shall see later, essentially each of the speci"c top-down models that have been studied so far has its own peculiar set of problems. Indeed, the main problem of the top-down scenario, in general, is that it is highly model dependent and invariably involves as-yet untested physics beyond the Standard Model of particle physics. On the other hand, it is precisely because of this reason that the scenario is also attractive } it brings in ideas of new physics beyond the Standard Model of particle physics (such as Grand Uni"cation) as well as ideas of early-Universe cosmology (such as topological defects and/or massive particle production in in#ation) into the realms of EHECR where these ideas have the potential to be tested by future EHECR experiments. The physics and astrophysics of UHECR are intimately linked with the emerging "eld of neutrino astronomy as well as with the already established "eld of c-ray astronomy which in turn are important subdisciplines of particle astrophysics (for a review see, e.g., Ref. [34]). Indeed, as we shall see, all scenarios of UHECR origin, including the top-down models, are severely constrained by neutrino and c-ray observations and limits. We shall also discuss how EHECR observations have the potential to yield important information on Galactic and extragalactic magnetic "elds. The plan of this review is summarized in the Table of Contents. Unless otherwise stated or obvious from the context, we use natural units with +"c"k "1 B throughout.
2. The observed cosmic rays In this section we give a brief overview of CR observations in general. Since this is a very rich topic with a tradition of almost 90 years, only the most important facts can be summarized. For more details the reader is referred to recent monographs on CR [35,36] and to rapporteur papers presented at the biennial International Cosmic Ray Conference (ICRC) (see, e.g., Refs. [37}39]) for updates on the data situation. The relatively young "eld of c-ray astrophysics which has now become an important sub"eld of CR astrophysics, can only be skimmed even more super"cially and for more information the reader is referred to the proceedings of the Compton c-ray symposia and the ICRC and to Refs. [40}42], for example. We will only mention those c-ray issues that are relevant for UHECR physics. Similarly, the emerging "eld of neutrino astrophysics [43] will be discussed only in the context of ultra-high energies for which a possible neutrino component and its potential detection will be discussed later in this review. 2.1. Detection methods at diwerent energies The CR primaries are shielded by the Earth's atmosphere and near the ground reveal their existence only by indirect e!ects such as ionization. Indeed, it was the height dependence of this
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Fig. 1. The CR all particle spectrum [509]. Approximate integral #uxes are also shown.
latter e!ect which lead to the discovery of CR by Hess in 1912. Direct observation of CR primaries is only possible from space by #ying detectors with balloons or spacecraft. Naturally, such detectors are very limited in size and because the di!erential CR spectrum is a steeply falling function of energy, roughly in accord with a power law with index !2.7 up to an energy of K2]1016 eV (see Fig. 1), direct observations run out of statistics typically around a few 100 TeV ("1014 eV) [44]. For the neutral component, i.e. c-rays, whose #ux at a given energy is lower than the charged CR #ux by several orders of magnitude, this statistical limit occurs at even lower energies, for example, around 100 GeV for the instruments on board the Compton Gamma Ray Observatory (CGRO) [45]. The space-based detectors of charged CR traditionally use nuclear emulsion stacks such as in the JACEE experiment [46]; now-a-days, spectrometric techniques are also used which are advantageous for measuring the chemical composition. For c-rays, for
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example, the Energetic Gamma Ray Experiment Telescope (EGRET) on board the CGRO uses spark chambers combined with a NaI calorimeter. Above roughly 100 TeV, the showers of secondary particles created in the interactions of the primary CR with the atmosphere are extensive enough to be detectable from the ground. In the most traditional technique, charged hadronic particles, as well as electrons and muons in these extensive air showers (EAS) are recorded on the ground [47] with standard instruments such as water Cherenkov detectors used in the old Volcano Ranch [1] and Haverah Park [3] experiments, and scintillation detectors which are used nowadays. Currently operating ground arrays for UHECR EAS are the Yakutsk experiment in Russia [6] and the Akeno Giant Air Shower Array (AGASA) near Tokyo, Japan, which is the largest one, covering an area of roughly 100 km2 with about 100 detectors mutually separated by about 1 km [8]. The Sydney University Giant Air Shower Recorder (SUGAR) [2] operated until 1979 and was the largest array in the Southern hemisphere. The ground array technique allows one to measure a lateral cross section of the shower pro"le. The energy of the shower-initiating primary particle is estimated by appropriately parametrizing it in terms of a measurable parameter; traditionally this parameter is taken to be the particle density at 600 m from the shower core, which is found to be quite insensitive to the primary composition and the interaction model used to simulate air showers [48]. The detection of secondary photons from EAS represents a complementary technique. The experimentally most important light sources are the #uorescence of air nitrogen excited by the charged particles in the EAS and the Cherenkov radiation from the charged particles that travel faster than the speed of light in the atmosphere. The "rst source is practically isotropic whereas the second one produces light strongly concentrated on the surface of a cone around the propagation direction of the charged source. The #uorescence technique can be used equally well for both charged and neutral primaries and was "rst used by the Fly's Eye detector [7] and will be part of several future projects on UHECR (see Section 2.6). The primary energy can be estimated from the total #uorescence yield. Information on the primary composition is contained in the column depth X (measured in g cm~2) at which the shower reaches maximal particle density. The average .!9 is related to the primary energy E by of X .!9 SX T"X@ ln(E/E ) . (1) .!9 0 0 Here, X@ is called the elongation rate and E is a characteristic energy that depends on the primary 0 0 composition. Therefore, if X and X@ are determined from the longitudinal shower pro"le .!9 0 measured by the #uorescence detector, then E and thus the composition, can be extracted after 0 determining the E from the total #uorescence yield. Comparison of CR spectra measured with the ground array and the #uorescence technique indicate systematic errors in energy calibration that are generally smaller than &40%. For a more detailed discussion of experimental EAS analysis with the ground array and the #uorescence technique see, e.g., the recent review by Yoshida and Dai [33] and Refs. [30}32]. In contrast to the #uorescence light, for a given primary energy, the output in Cherenkov light is much larger for c-ray primaries than for charged CR primaries. In combination with the so-called imaging technique } in which the Cherenkov light image of an electromagnetic cascade in the upper atmosphere (and thus also the primary arrival direction) is reconstructed [49] } the Cherenkov technique is one of the best tools available to discriminate c-rays from point sources against the strong background of charged CR. This technique is used, for example, by the High Energy
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Gamma Ray Astronomy (HEGRA) experiment (now 5 telescopes of 8.5 m2 mirror area) [50] and by the 10 m Whipple telescope [51] with threshold energies of K500 and 200 GeV, respectively. In the Southern hemisphere, the Collaboration of Australia and Nippon (Japan) for a GAmma Ray Observatory in the Outback (CANGAROO) experiment [52] currently consists of two 7 m imaging atmospheric Cherenkov telescopes at Woomera, Australia, with an energy threshold of 200 GeV. Another new experiment which is at the completion stages of construction and testing is the Multi-Institution Los Alamos Gamma Ray Observatory (MILAGRO) [53] which is a water (rather than atmospheric)-Cherenkov detector that detects electrons, photons, hadrons and muons in EAS, has a 24-h duty cycle, `all-skya coverage, and good angular resolution (40.43 at 10 TeV), and is sensitive to c-rays in the energy range from &200 GeV to &100 TeV. For c-rays, therefore, an as yet unexplored window between a few tens of GeV and K200 GeV remains which may soon be closed by large-area atmospheric Cherenkov detectors [54]. For a detailed review of this "eld of very high-energy c-ray astronomy see, e.g., Refs. [40}42]. Finally, muons of a few hundred GeV and above have penetration depths of the order of a kilometer even in rock and can thus be detected underground. The Monopole Astrophysics and Cosmic Ray Observatory (MACRO) experiment [55], for example, located in the Gran Sasso laboratory near Rome, Italy, has a rock overburden of about 1.5 km, consists of K600 tons of liquid scintillator and acts as a giant time-of-#ight counter. Operated in coincidence with the Cherenkov telescope array EAS-TOP [56] located above it, it can for instance be used to study the primary CR composition around the `kneea region [57] (see Fig. 1). A similar combination is represented by the Antarctic Muon And Neutrino Detector Array (AMANDA) detector and the South Pole Air ShowEr array (SPASE) of scintillation detectors. AMANDA consists of strings of photomultiplier tubes of a few hundred meters in length deployed in the antarctic ice at depths of up to 2 km, and reconstructs tracks of muons of energies in the TeV range [58]. 2.2. The measured energy spectrum Fig. 1 shows a compilation of the CR all-particle spectrum over the whole range of energies observed through di!erent experimental strategies as discussed in Section 2.1. The spectrum exhibits power-law behavior over a wide range of energies, but comparison with a "t to a single power law (dashed line in Fig. 1) shows signi"cant breaks at the `kneea at K4]1015 eV and, to a somewhat lesser extent, at the `anklea at K5]1018 eV. The sharpness of the knee feature is a not yet resolved experimental issue, particularly because it occurs in the transition region between the energy range where direct measurements are available and the energy range where the data come from indirect detection by the ground array techniques whose energy resolution is typically 20% or worse [44]. For example, the EAS-TOP array observed a sharp spectral break at the knee within their experimental resolution [59], whereas the AGASA [60] and CASA-MIA [61] data support a softer transition. Figs. 2}6 show the CR data above 1017 eV measured by di!erent experiments. The ankle feature was "rst discussed in detail by the Fly's Eye experiment [7]. The slope between the knee and up to K4]1017 eV is very close to 3.0 (Fig. 1); then it seems to steepen to about 3.2 up to the dip at K3]1018 eV, after which it #attens to about 2.7 above the dip. As will be discussed in Section 2.4,
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Fig. 2. Energy spectrum of UHECR measured by the AGASA experiment. The dashed curve represents the spectrum expected for extragalactic sources distributed uniformly in the Universe. The numbers attached to the data points are the number of events observed in the corresponding energy bins. (From M. Takeda et al. [8].) Fig. 3. Fly's Eye monocular energy spectrum. Dots: data. Lines: predicted spectra for source energy cuto! at di!erent energies. Solid line: cuto! at 1019.6 eV. Dashed line: cuto! at 1020 eV. Chain line: cuto! at 1021 eV. (From Yoshida and Dai [33].)
Fig. 4. Fly's Eye stereo energy spectrum. Dots: data. Dotted line: best "t in each region. Dashed lines: a two-component "t. (From Yoshida and Dai [33].) Fig. 5. The Haverah Park energy spectrum. (From Yoshida and Dai [33].)
the Fly's Eye also found evidence for a change in composition to a lighter component above the ankle, that is correlated with the change in spectral slope. The situation at the high end of the CR spectrum is as yet inconclusive and represents the main subject of the recent strong increase of theoretical and experimental activities in UHECR physics
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Fig. 6. The Yakutsk energy spectrum. (From Yoshida and Dai [33].)
which also motivated the present review. The present data (see Figs. 2}6) seem to reveal a steepening just below 1020 eV, but above that energy signi"cantly more events have been seen than expected from an extrapolation of the GZK `cuto! a at K1020 eV. This is perhaps the most puzzling and hence interesting aspect of UHECR because a cuto! is expected at least for extragalactic nucleon primaries irrespective of the production mechanism (see Section 4.1). Even for conventional local sources, the maximal energy to which charged primaries can be accelerated is expected to be limited (see Section 5) and it is generally hard to achieve energies beyond the cuto! energy. 2.3. Events above 1020 eV The "rst published event above 1020 eV was observed by the Volcano Ranch experiment [1]. The Haverah Park experiment reported 8 events around 1020 eV [3], and the Yakutsk array saw one event above this energy [6]. The SUGAR array in Australia reported 8 events above 1020 eV [2], the highest one at 2]1020 eV. The world record holder is still a 3.2]1020 eV event which was the only event above 1020 eV observed by the Fly's Eye experiment [7], on 15 October 1991. Probably the second highest event at K2.1]1020 eV in the world data set was seen by the AGASA experiment [8] which meanwhile detected a total of 6 events above 1020 eV (see Fig. 2). The Fly's Eye and the AGASA events have been documented in detail in the literature and it seems unlikely that their energy has been overestimated by more than 30%. For more detailed experimental information see, e.g., the review [33]. Theoretical and astrophysical implications of these events are a particular focus of the present review. For an overview of speci"c source searches for these events see Section 4.6. 2.4. Composition We will discuss the question of composition here only for CR detected by ground-based EAS detectors, i.e., for CR above &100 TeV, only. Information on the chemical composition is mainly
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provided by the muon content in case of ground arrays and by the depth of shower maximum for optical observation of the EAS. Just to indicate the qualitative trend we mention that, for a given primary energy, a heavier nucleus produces EAS with a higher muon content and a shower maximum higher up in the atmosphere on average compared to those for a proton shower. The latter property can be understood by viewing a nucleus as a collection of independent nucleons whose interaction probabilities add, leading to a faster development of the shower on average. The higher muon content in a heavy nucleus shower is due to the fact that, because the shower develops relatively higher up in the atmosphere where the atmosphere is less dense, it is relatively easier for the charged pions in a heavy nucleus shower to decay to muons before interacting with the medium. The spectral and compositional behavior around the knee at K4]1015 eV may play a crucial role in attempts to understand the origin and nature of CR in this energy range, as will be discussed in little more detail in Section 3. Indeed, there are indications that the chemical composition becomes heavier with increasing energies below the knee [44]. Around the knee the situation becomes less clear and most of the experimental results, such as from the SOUDAN-2 [62], the HEGRA [63], and the KArlsruhe Shower Core and Array DEtector (KASCADE) [64] experiments, seem to indicate a substantial proton component and no signi"cant increase in primary mass. Recent results, for example, from the Dual Imaging Cherenkov Experiment (DICE) seem to indicate a lighter composition above the knee which may hint to a transition to a di!erent component [65], but evidence for an increasingly heavy composition above the knee has also been reported by the KASCADE collaboration [66] and by HEGRA [67]. Based on the analysis discussed above, Eq. (1), the Fly's Eye collaboration reported a composition change from a heavy component below the ankle to a light component above, that is correlated with the spectral changes around the ankle [7]. However, this was not con"rmed by the AGASA experiment [8,33]. In addition, there have been suggestions that the observed energy dependence of SX T could be caused by air shower physics rather than an actual composition .!9 change [68]. One signature of a heavy nucleus primary would be the almost simultaneous arrival of a pair of EASs at the Earth. Such pairs would be produced by photodisintegration of nuclei by solar photons and could be used to measure their mass, as was pointed out quite early on [69]. This e!ect has been reconsidered recently in light of existing and proposed UHECR detectors [70,71]. At the highest energies, observed EAS seem to be consistent with nucleon primaries, but due to poor statistics and large #uctuations from shower to shower, the issue is not settled yet. Some scenarios of EHECR origin, such as the top-down scenario discussed in Sections 6 and 7, predict the EHECR primaries to be dominated by photons and neutrinos rather than nucleons. Distinguishing between photon and nucleon-induced showers is, however, extremely di$cult at UHE and EHE regions } the standard muon-poorness criterion of photon induced showers relative to nucleon-induced showers, applicable at lower (1014}1016 eV) energies, does not apply to the UHE region. It has been claimed that the highest energy Fly's Eye event is inconsistent with a c-ray primary [72]. It should be noted, however, that at least for electromagnetic showers, EAS simulation at EHE is complicated by the Landau}Pomeranchuk}Migdal (LPM) e!ect and by the in#uence of the geomagnetic "eld [73]. Furthermore, in the simulations, EAS development depends to some extent on the hadronic interaction event generator which complicates data interpretation [74]. De"nite conclusions on the composition of the EHECR, therefore, have to
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await data from next generation experiments. Together with certain characteristic features of the photon-induced EHECR showers due to geomagnetic e!ects [73], the large event statistics expected from the next generation experiments will hopefully allow to distinguish between photon and nucleon EHECR primaries. In turn, accelerator data together with EAS data can be used to constrain, for example, the cross section of protons with air nuclei at center-of-mass energies of 30 TeV [75]. The hypothesis of neutrinos or new neutral particles as EHECR primaries will be discussed in Sections 4.3.1 and 4.3.2, respectively. 2.5. Anisotropy For a recent compilation and discussion of anisotropy measurements see Ref. [76]. Fig. 7 shows the summary "gure from that reference. Implications of these anisotropy measurements will be
Fig. 7. A compilation of anisotropy measurements ("rst harmonic Fourier amplitude and phase). Northern and southern hemisphere results are denoted by upward-pointing and downward-pointing triangles, respectively. (From Clay and Smith [76].)
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discussed brie#y in the next section where we discuss the origin of CR in general. For discussions of subtleties involved in the measurements and interpretation of anisotropy data, choice of coordinate systems used in presenting anisotropy results, etc., see e.g., Ref. [30]. The anisotropy amplitude is de"ned as DdD"(I
!I )/(I #I ) , (2) .!9 .*/ .!9 .*/ where I and I are the minimum and maximum CR intensity as a function of arrival direction. .*/ .!9 Very recently, results have been presented on the anisotropy of the CR #ux above K1017 eV from the Fly's Eye [77] and the AGASA [78] experiments. Both experiments report a small but statistically signi"cant anisotropy of the order of 4% in terms of Eq. (2) toward the Galactic plane at energies around 1018 eV. These analyses did not reveal a signi"cant correlation with the Supergalactic Plane, whereas earlier work seemed to indicate some enhancement of the #ux from this plane [79}81]. In addition, the newest data seem to indicate that also the events above 1020 eV are consistent with an isotropic distribution on large scales [82], as far as that is possible to tell from about 15 events in the world data set. At the same time, there seems to be signi"cant small scale clustering [83]. 2.6. Next-generation experiments on ultrahigh-energy cosmic ray, c-ray, and neutrino astrophysics As an upscaled version of the old Fly's Eye Cosmic Ray experiment, the High Resolution Fly's Eye detector is currently under construction at Utah, USA [84]. Taking into account a duty cycle of about 10% (a #uorescence detector requires clear, moonless nights), the e!ective aperture of this instrument will be K600 km2 sr, about 10 times the AGASA aperture, with a threshold around 1017 eV. Another project utilizing the #uorescence technique is the Japanese Telescope Array [85] which is currently in the proposal stage. Its e!ective aperture will be about 15}20 times that of AGASA above 1017 eV, and it can also be used as a Cherenkov detector for TeV c-ray astrophysics. Probably, the largest up-coming project is the international Pierre Auger Giant Array Observatories [86] which will be a combination of a ground array of about 1700 particle detectors mutually separated from each other by about 1.5 km and covering about 3000 km2, and one or more #uorescence Fly's Eye type detectors. The ground array component will have a duty cycle of nearly 100%, leading to an e!ective aperture about 200 times as large as the AGASA array, and an event rate of 50}100 events per year above 1020 eV. About 10% of the events will be detected by both the ground array and the #uorescence component and can be used for cross calibration and detailed EAS studies. The energy threshold will be around 1019 eV. For maximal sky coverage it is furthermore planned to construct one site in each hemisphere. The southern site will be in Argentina, and the northern site probably in Utah, USA. Recently, NASA initiated a concept study for detecting EAS from space [87] by observing their #uorescence light from an Orbiting Wide-angle Light-collector (OWL). This would provide an increase by another factor &50 in aperture compared to the Pierre Auger Project, corresponding to an event rate of up to a few thousand events per year above 1020 eV. Similar concepts such as the AIRWATCH [88] and Maximum-energy air-Shower Satellite (MASS) [89] missions are also being
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discussed. The energy threshold of such instruments would be between 1019 and 1020 eV. This technique would be especially suitable for detection of very small event rates such as those caused by UHE neutrinos which would produce horizontal air showers (see Section 7.4). For more details on these recent experimental considerations see Ref. [9]. New experiments are also planned in c-ray astrophysics. The Gamma ray Large Area Space Telescope (GLAST) [90] detector is planned by NASA as an advanced version of the EGRET experiment, with an about 100 fold increase in sensitivity at energies between 10 MeV and 200 GeV. For new ground-based c-ray experiments we mention the Very Energetic Radiation Imaging Telescope Array System (VERITAS) project [91,92] which consists of eight 10 m optical re#ectors which will be about two orders of magnitude more sensitive between 50 GeV and 50 TeV than WHIPPLE. A similar next generation atmospheric imaging Cherenkov system with up to 16 planned telescopes is the High Energy Stereoscopic System (HESS) project [93]. Furthermore, the Mayor Atmospheric Gamma-ray Imaging Cherenkov Telescope (MAGIC) project [94] aims to build a very large atmospheric imaging Cherenkov telescope with 220 m2 mirror area for detection of c-rays between 10 and 300 GeV, i.e. within the as yet unexplored window of c-ray astrophysics. The CANGAROO experiment in Australia plans to upgrade to four 10 m telescopes and lower the threshold to 100 GeV. Finally, another strategy to explore this window utilizes existing solar heliostat arrays, and is represented by the Solar Tower Atmospheric Cherenkov E!ect Experiment (STACEE) [95] in the USA, the Cherenkov Low Energy Sampling & Timing Experiment (CELESTE) in France [96], and the German-Spanish Gamma Ray Astrophysics at ALmeria (GRAAL) experiment [97]. High-energy neutrino astronomy is aiming towards a kilometer scale neutrino observatory. The major technique is the optical detection of Cherenkov light emitted by muons created in charged current reactions of neutrinos with nucleons either in water or in ice. The largest pilot experiments representing these two detector media are the now defunct Deep Undersea Muon and Neutrino Detection (DUMAND) experiment [98] in the deep sea near Hawai and the AMANDA experiment [58] in the South Pole ice. Another water-based experiment is situated at Lake Baikal [99]. Next generation deep sea projects include the French Astronomy with a Neutrino Telescope and Abyss environmental RESearch (ANTARES) [100,101] and the underwater Neutrino Experiment SouthwesT Of GReece (NESTOR) project in the Mediterranean [102], whereas ICECUBE [103] represents the planned kilometer scale version of the AMANDA detector. Also under consideration are neutrino detectors utilizing techniques to detect the radio pulse from the electromagnetic showers created by neutrino interactions in ice [104]. This technique could possibly be scaled up to an e!ective area of 104 km2 and a prototype is represented by the Radio Ice Cherenkov Experiment (RICE) at the South Pole [105]. The radio technique might also have some sensitivity to the #avor of the primary neutrino [106]. Neutrinos can also initiate horizontal EAS which can be detected by giant ground arrays such as the Pierre Auger Project [107,108]. Furthermore, as mentioned above, horizontal EAS could be detected from space by instruments such as the proposed OWL detector [87]. Finally, the search for pulsed radio emission from cascades induced by neutrinos or cosmic rays above &1019 eV in the lunar regolith could also lead to interesting limits [109]. More details on neutrino astronomy detectors are contained in Refs. [110,43,111], and some recent overviews on neutrino astronomy can be found in Refs. [112,113].
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3. Origin of bulk of the cosmic rays: general considerations The question of origin of cosmic rays continues to be regarded as an `unsolved problema even after almost 90 years of research since the announcement of their discovery in 1912. Although the general aspects of the question of CR origin are regarded as fairly well-understood now, major gaps and uncertainties remain, the level of uncertainty being in general a function that increases with energy of the cosmic rays. The total CR energy density measured above the atmosphere is dominated by particles with energies between about 1 and 10 GeV. At energies below K1 GeV the intensities are temporally correlated with the solar activity which is a direct evidence for an origin at the Sun. At higher energies, however, the #ux observed at Earth exhibits a temporal anticorrelation with solar activity and a screening whose e$ciency increases with the strength of the solar wind, indicating an origin outside the solar system. Several arguments involving energetics, composition, and secondary c-ray production suggest that the bulk of the CR between 1 GeV and at least up to the knee region (see Fig. 1) is con"ned to the Galaxy and is probably produced in supernova remnants (SNRs). Between the knee and the ankle the situation becomes less clear, although the ankle is sometimes interpreted as a cross over from a Galactic to an extragalactic component. Finally, beyond K10 EeV, CR are generally expected to have an extragalactic origin due to their apparent isotropy, but ways around this reasoning have also been suggested. In the following, we give a somewhat more detailed account of these general considerations, separating the discussion into issues related to energetics, Galactic versus extragalactic origin, and acceleration mechanisms and the possible sources of CR. We reserve a more comprehensive discussion of the origin of UHECR above &1017 eV (which make only a small part of the total CR energy density, but are the main focus of this review) for Sections 5 and 6. 3.1. Energetics As mentioned above, the bulk of the CR observed at the Earth is of extrasolar origin. The average energy density of CR is thus expected to be uniform at least throughout most of the Galaxy. If CR are universal, their density should be constant throughout the whole Universe. As a curiosity we note in this context that the mean energy density of CR, u , is comparable to the CR energy density of the CMB. It is not clear, however, what physical process could lead to such an `equilibrationa, which is thus most likely just a coincidence. We will see in the following, that indeed a universal origin of bulk of the CR is nowadays not regarded as a likely possibility. If the CR accelerators are Galactic, they must replenish for the escape of CR from the Galaxy in order to sustain the observed Galactic CR di!erential intensity j(E). Their total luminosity in CR must therefore satisfy ¸ "(4p/c):dE d
(3)
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where o is the mean density of interstellar gas and Z is the mean charge number of the CR ' particles. The mean energy density of CR and the total mass of gas in the Milky Way that have been inferred from the di!use Galactic c-ray, X-ray and radio emissions are u "(4p/c):dE Ej(E)K1 eV cm~3 and M &o <&4.8]109M , respectively. Hence, simple CR ' ' _ integration yields
P
Ej(E) &1.5]1041 erg s~1 . ¸ &M dE CR ' X(E)
(4)
This is about 10% of the estimated total power output in the form of kinetic energy of the ejected material in Galactic supernovae which, from the energetics point of view, could therefore account for most of the CR. We note that the energy release from other Galactic sources, e.g. ordinary stars [35] or isolated neutron stars [115] is expected to be too small, even for UHECR.3 Together with other considerations (see Section 3.2) this leads to the widely held notion that CR at least up to the knee predominantly originate from "rst-order Fermi acceleration (see below) in SNRs. Another interesting observation is that the energy density in the form of CR is comparable both to the energy density in the Galactic magnetic "eld (&10~6 G) as well as that in the turbulent motion of the gas, u &B2/8p&1 o v2 , (5) CR 2 ' 5 where o and v are the density and turbulent velocity of the gas, respectively. This can be expected ' 5 from a pressure equilibrium between the (relativistic) CR, the magnetic "eld, and the gas #ow. If Eq. (5) roughly holds not only in the Galaxy but also throughout extragalactic space, then we would expect the extragalactic CR energy density to be considerably smaller than the Galactic one which is another argument in favor of a mostly Galactic origin of the CR observed near Earth (see Section 3.2). We note, however, that, in order for Eq. (5) to hold, typical CR di!usion time scale over the size of the system under consideration must be smaller than its age. This is not the case, for example, in clusters of galaxies if the bulk of CR are produced in the member galaxies or in cluster accretion shocks [117]. 3.2. Galactic versus extragalactic origin of the bulk of the CR The energetical considerations mentioned above already provide some arguments in favor of a Galactic origin of the bulk of the CR. Another argument involves the production of secondary c-rays from the decays of neutral pions produced in interactions of CR with the baryonic gas throughout the Universe: For given densities of the CR and the gas, the resulting c-ray #ux can be calculated quite reliably [118] and the predictions can be compared with observations. This has been done, for example, for the Small Magellanic Cloud (SMC). The observed upper limits [119] turn out to be a factor of a few below the predictions assuming a universal CR density. The CR 3 This conclusion may, however, change somewhat with the recent detection of certain soft-gamma repeaters [116] which seem to indicate the existence of a subclass of pulsars with dipole magnetic "elds as large as a few times 1014 G. This may increase the available magnetic energy budget from pulsars by two to three orders of magnitude.
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density at the SMC should, therefore, be at least a factor of a few smaller than the local Galactic density. As a second test we mention the search for a CR gradient (e.g. [120}122]: For a Galactic CR origin one expects a decrease of CR intensity with increasing distances from the Galactic center which should be encoded in the secondary c-ray emission that can be measured by space based instruments such as EGRET. The observational situation is, however, not completely settled yet [123]. Whereas the spatial variation of the c-ray #ux "ts rather well, the observed spectrum appears to be too #at compared to the one expected from the average CR spectrum. Since the average CR spectrum throughout the Galaxy is generally steepened (compared to the spectrum at the source) by di!usion in Galactic magnetic "elds, the observed relatively #at secondary c-ray spectrum may be interpreted as if the secondary c-rays are produced by CR interactions mainly at the (Galactic) sources rather than in the interstellar medium. This interpretation, however, requires that the escape time of CR from their sources be energy-independent. Some information on CR origin is in principle also contained in the distribution of their arrival directions which has been discussed in Section 2.5 (see Fig. 7). Below &1014 eV, the amplitude of the observed anisotropy, &10~3, is statistically signi"cant and roughly energy independent. Above 1014 eV, observed anisotropy amplitudes are generally statistically insigni"cant with possible exceptions between K1015 eV and 1016 eV [76] and again close to 1018 eV, the latter correlated with the Galactic plane [77,78]. A possible clustering towards the Supergalactic Plane for energies above a few tens of EeV was claimed [79}81], but has not been con"rmed by more recent studies [77,78]. Since charged CR at these energies are hardly de#ected by the Galactic magnetic "eld, the apparent lack of any signi"cant anisotropy associated with the Galactic plane implies that the high-energy end of the CR spectrum is most likely to have an extragalactic origin (see Section 4.6). For Galactic sources, detailed models of CR di!usion and c-ray production in the Galaxy have been developed (see, e.g., Refs. [124,125]). These models are generally based on the energy loss } di!usion equation that will be discussed below in the context of UHECR propagation [see Eq. (36)], with the di!usion constant generalized to a di!usion tensor. This tensor and other parameters in these models can be obtained from "ts to the observed abundances of nuclear isotopes. It is often su$cient to consider a simpli"ed model, the so called `leaky boxa model (see Refs. [35,36] for detailed discussions) in which the di!usion term is approximated by a loss term involving a CR containment time t . Fits to the data lead to t &107 yr below CR CR &1016 eV with only a weak energy dependence. This is in turn consistent with observed anisotropies which, below &1014 eV, can be interpreted by the Compton}Getting e!ect [126] which describes the e!ect of the motion of the observer relative to an isotropic distribution of CR. In this case, the relative motion is a combination of the motion of the solar system within the Galaxy and the drift motion of the charged CR di!using and/or convecting in the interstellar medium. The magnitude and the weak energy dependence of the anisotropy in this energy range can be interpreted as arising out of di!usion of CR predominantly along the tangled incoherent component of the Galactic magnetic "eld. In summary, CR composition and anisotropy data provide further evidence for a Galactic origin for energies at least up to the knee region of the spectrum. In this context, the knee itself is often interpreted as a magnetic decon"nement e!ect such that CR above the knee leave the Galaxy relatively faster, leading to steepening of the spectrum
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above the knee. In addition, the maximum energy achieved in shock acceleration is proportional to the primary charge and could also lead to a spectral steepening (see Section 3.3). Alternatively, the knee has also been interpreted as being caused by the #ux contribution from a strong single source [127]. Finally, since the range of electrons above &1011 eV becomes smaller than t due to synchroCR tron and inverse Compton losses, the electronic CR component at such energies, which is about 1% of the hadronic #ux, is undoubtedly of Galactic origin. This can also be explained by acceleration in SNRs. 3.3. Acceleration mechanisms and possible sources There are basically two kinds of acceleration mechanisms considered in connection with CR acceleration: (1) direct acceleration of charged particles by an electric "eld, and (2) statistical acceleration (Fermi acceleration) in a magnetized plasma. In the direct acceleration mechanism, the electric "eld in question can be due, for example, to a rotating magnetic neutron star (pulsar) or, a (rotating) accretion disk threaded by magnetic "elds, etc. The details of the actual acceleration process and the maximum energy to which a particle can be accelerated depend on the particular physical situation under consideration. For a variety of reasons, the direct acceleration mechanisms are, however, not widely favored these days as the CR acceleration mechanism. Apart from disagreements among authors about the crucial details of the various models, a major disadvantage of the mechanism, in general, is that it is di$cult to obtain the characteristic power-law spectrum of the observed CR in any natural way. However, as pointed out by Colgate [128], a power law spectrum does not necessarily point to Fermi acceleration, but results whenever a fractional gain in energy of a few particles is accompanied by a signi"cantly larger fractional loss in the number of remaining particles. We will not discuss the direct acceleration mechanism in any more details, and refer the reader to reviews, e.g., in Refs. [35,11,15,128]. The basic idea of the statistical acceleration mechanism originates from a paper by Fermi [129] in 1949: Even though the average electric "eld may vanish, there can still be a net transfer of macroscopic kinetic energy of moving magnetized plasma to individual charged particles (`test particlesa) in the medium due to repeated collisionless scatterings (`encountersa) of the particles either with randomly moving inhomogeneities of the turbulent magnetic "eld or with shocks in the medium. Fermi's original paper [129] considered the former case, i.e., scattering with randomly moving magnetized `cloudsa in the interstellar medium. In this case, although in each individual encounter the particle may either gain or lose energy, there is on average a net gain of energy after many encounters. The original Fermi mechanism is nowadays called `second-ordera Fermi mechanism, because the average fractional energy gain in this case is proportional to (u/c)2, where u is the relative velocity of the cloud with respect to the frame in which the CR ensemble is isotropic, and c is the velocity of light. Because of the dependence on the square of the cloud velocity (u/c(1), the second-order Fermi mechanism is not a very e$cient acceleration process. Indeed, for typical interstellar clouds in the Galaxy, the acceleration time scale turns out to be much larger than the typical escape time (&107 yr) of CR in the Galaxy deduced from observed isotopic ratios of CR. In addition, although the resulting spectrum of particles happens to be a power law in energy, the power-law index depends on the cloud velocity, and so the superposed
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spectrum due to many di!erent sources with widely di!erent cloud velocities would not in general have a power-law form. A more e$cient version of Fermi mechanism is realized when one considers encounters of particles with plane shock fronts. In this case, the average fractional energy gain of a particle per encounter (de"ned as a cycle of one crossing and then a re-crossing of the shock after the particle is turned back by the magnetic "eld) is of "rst order in the relative velocity between the shock front and the isotropic-CR frame. Currently, the `standarda theory of CR acceleration } the so-called `Di!usive Shock Acceleration Mechanisma (DSAM) is, therefore, based on this "rst-order Fermi acceleration mechanism at shocks. For reviews and references to original literature on DSAM, see, e.g., Refs. [130}132,15,19,20]. An important feature of DSAM is that particles emerge out of the acceleration site with a characteristic power-law spectrum with a power-law index that depends only on the shock compression ratio, and not on the shock velocity. Shocks are ubiquitous in astrophysical situations: in the interplanetary space, in supernovae in interstellar medium, and even in cosmological situations as in radio-galaxies. The basic ideas of the DSAM have received impressive con"rmation from in situ observations in the solar system, in particular, from observations of high-energy particles accelerated at the Earth's bow shock generated by collision of the solar wind with the Earth's magnetosphere; see, again the reviews in Refs. [130}132,15] for references. We will discuss the DSAM again in connection with UHECR in Section 5. Here we only note that for a given acceleration site, there is a maximum energy achievable, E , which is limited .!9 either by the size of the shock (which has to be larger than the gyroradius of the particles being accelerated) or by the time scale of acceleration up to this energy (which has to be smaller than the lifetime of the shock and also smaller than the shortest time-scale of energy losses). From a theoretical point of view, SNRs are not only attractive (and maybe the only serious) candidate of Galactic CR origin in terms of power (see Section 3.1) but also in terms of the maximum achievable CR energy, which is estimated to lie somewhere between 1012 and 1017 eV. In addition, the observed constant beryllium-to-iron abundance ratio in the atmospheres of stars of di!erent metallicity is another indicator that at least the carbon, nitrogen and oxygen CR, that produce beryllium by spallation with interstellar hydrogen and helium (this being the main production channel for beryllium), have to be accelerated in SNRs [133]. For recent discussions of the relevance of composition for the origin of Galactic CR, see Ref. [134] for lithium, beryllium, and boron in particular, and Ref. [135] in general. The DSAM theory of CR acceleration in SNRs has been worked out in considerable details; see, e.g., Refs. [130,131,136,137]. Support to the shock-acceleration scenario for hadronic CR is given by experimental indications that while the composition below the knee region becomes heavier with energy (see Section 2.4), the composition is relatively less dependent on rigidity (,pc/Ze, where p is the momentum and Ze is the charge, and c is the speed of light). This is expected for shock acceleration for which the maximum rigidity should be equal for all nuclei. Furthermore, the observed X-ray emission from SNRs seems to be caused by synchrotron radiation of electrons with energies up to &100 TeV. Assuming that nuclei are accelerated as well, this implies #uxes consistent with Galactic CR acceleration in SNR shocks [138]. As another e!ect, the interactions with the surrounding matter of protons accelerated in SNRs produce neutral pions, and the resulting #ux of secondary c-rays from SNRs has been predicted as well; see, e.g., Refs. [139}144]. Nowadays, given the existence of space and ground based c-ray detecting systems (see Section 2.1), the SNR acceleration paradigm for Galactic CR origin can also
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be tested by searching for these secondary c-rays. As of now, the situation is still somewhat inconclusive since no "rm detection of such c-rays has been reported (see, e.g., Refs. [145,146]). Furthermore, the SNR scenario almost certainly does not explain UHECR which consequently would constitute a separate component. Pulsars and neutron stars in close binary systems have also been discussed as alternative Galactic CR sources for which the maximum energy in principle may even reach the UHECR energy range. However, an origin of the bulk of the cosmic rays in X-ray binary systems is contradicted by the complete absence of detectable TeV radiation from Cygnus X-3 and Hercules X-1, as reported by the Chicago Air Shower Array-MIchigan Anti (CASA-MIA) experiment [147]. A comprehensive scenario for the origin of CR based exclusively on "rst-order Fermi acceleration has been proposed by Biermann [16]. In this scenario, the sources are (a) supernovae exploding into the interstellar medium, for energies up to &1015 eV, (b) supernovae exploding into a predecessor stellar wind, for energies up to &1017 eV, and (c) the hot spots of powerful radio-galaxies for the highest energies. It is claimed that this scenario meets every observational test to date. A criticism of shock acceleration as the origin of CR has been given by Colgate [128]. Instead, acceleration in the electric "elds produced by reconnection of twisted magnetic "elds has been suggested as a mechanism that could operate in a much larger fraction in space than shock acceleration and up to the highest observed CR energies. This is due to the wide-spread presence of helical magnetic "elds carrying excess angular momentum from mass condensations in the Universe. Apart from proposed laboratory experiments [128], it is, however, presently not clear how to observationally discriminate this scenario of CR origin from the shock acceleration scenario. Also, the power law index of the predicted spectra does not fall out of this scenario naturally and may strongly depend on the speci"c environment. Plaga [148] has presented a scenario where all extrasolar hadronic CR are extragalactic in origin and accumulate in the Galaxy due to `magnetic #ux trappinga. It was claimed that the c-ray #ux levels from the Magellanic clouds is not a suitable test of this scenario and that the ankle in the energy spectrum appears as a natural consequence of this scenario. The opposite possibility that all CR nuclei above a few GeV and up to the highest energies observed, and all electrons and c-rays above a few MeV are of Galactic origin has also been put forward by Dar and collaborators [149]. In this scenario the acceleration sources have been suggested to be the hot spots in the highly relativistic jets from merger and accretion induced collapse of compact stellar objects, the so called microblazars, within our own Galaxy and its halo. The same objects in external galaxies could also give rise to cosmological c-ray bursts. Finally, to close this short summary with a very speculative possibility of CR origin, we note that it is known that charged and/or polarizable particles interacting with the electromagnetic zeropoint #uctuations are accelerated stochastically [150,151]. The discussion of this e!ect goes back to Einstein and Hopf [152] who investigated classical atoms interacting with classical thermal radiation. The acceleration rate X,dE/dt for a proton is given by [150] XK(3/5p)C2u5 /m [1013 eV s~1 , (6) 0 N where C"2e2/3m is the radiation damping constant, m the nucleon mass, and u is a frequency N N 0 that is smaller than the Compton frequencies of the quarks. In an energy range where energy losses are negligible, the resulting acceleration spectrum must have the form j(E)JE~1 due to the
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Lorentz invariance of the spectrum of the vacuum #uctuations. The latter is also the reason that a net acceleration results because it implies the absence of a drag force. The spectrum typically cuts o! exponentially at energies where the acceleration time ¹ KE/X becomes larger than the !## proton attenuation time at that same energy due to loss processes. It seems, however, unlikely that this acceleration process plays a signi"cant role in CR production because, for given typical baryon densities, the predicted hard spectrum tends to overproduce CR #uxes at high energies.
4. Propagation and interactions of ultra-high-energy radiation Since implications and predictions of the spectrum of UHECR depend on their composition which is uncertain, we will in this chapter review the propagation of all types of particles that could play the role of UHECR. We start with the hadronic component, continue with discussion on electromagnetic cascades initiated by UHE photons in extragalactic space, and then comment on more exotic options such as UHE neutrinos and new neutral particles predicted in certain supersymmetric models of particle physics. We then discuss how propagation can be in#uenced by cosmic magnetic "elds and what constraints on the location of UHECR sources are implied. The role played by these constraints in the search for sources of EHECR beyond 1020 eV is discussed. Finally, the formal description of CR propagation by transport equations is brie#y reviewed, with an account of the literature on analytical and numerical approaches to their solution. Before proceeding, we set up some general notation. The interaction length l(E) of a CR of energy E and mass m propagating through a background of particles of mass m is given by " `1 1!kbb " p(s) , l(E)~1" de n (e) dk (7) " 2 ~1 where n (e) is the number density of the background particles per unit energy at energy e, " b "(1!m2 /e2)1@2 and b"(1!m2/E2)1@2 are the velocities of the background particle and the " " CR, respectively, k is the cosine of the angle between the incoming momenta, and p(s) is the total cross section of the relevant process for the squared center of mass (CM) energy
P
P
s"m2#m2#2eE(1!kbb ) . (8) " " The most important background particles turn out to be photons with energies in the infrared and optical (IR/O) range or below, so that we will usually have m "0, b "1. A review of the universal " " photon background has been given in Ref. [153]. It proves convenient to also introduce an energy attenuation length l (E) that is obtained from E Eq. (7) by multiplying the integrand with the inelasticity, i.e. the fraction of the energy transferred from the incoming CR to the recoiling "nal state particle of interest. The inelasticity g(s) is given by
P
dp 1 dE@ E@ (E@, s) , g(s),1! dE@ p(s)
(9)
where E@ is the energy of the recoiling particle considered in units of the incoming CR energy E. Here by recoiling particle we usually mean the `leadinga particle, i.e. the one which carries most of the energy.
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If one is mostly interested in this leading particle, the detailed transport equations (see Section 4.7) for the local density of particles per unit energy, n(E), are often approximated by the simple `di!usion equationa R n(E)"!R [b(E)n(E)]#U(E) (10) 5 E in terms of the energy loss rate b(E)"E/l (E) and the local injection spectrum U(E). Eq. (10) applies E to a particle which loses energy at a rate dE/dt"b(E), and is often referred to as the continuous energy loss (CEL) approximation. The CEL approximation is in general good if the non-leading particle is of a di!erent nature than the leading particle, and if the inelasticity is small, g(s);1. For an isotropic source distribution U(E, z) in the matter-dominated regime for a #at Universe (X "1), 0 Eq. (10) yields a di!erential #ux today at energy E, j(E), as
P
3 zi,.!9 dE (E, z ) i U(E, z ) , j(E)" t (11) dz (1#z )~11@2 i 0 i i i 8p dE 0 where t is the age of the Universe, E (E, z ) is the energy at injection redshift z in the CEL 0 i i i approximation, i.e. the solution of dE/dt"b(E) (with b(E) including loss due to redshifting), E (E, 0)"E with t"t /(1#z)3@2. The maximum redshift z corresponds either to an absolute i 0 i,.!9 cuto! of the source spectrum at E "E (E, z ) or to the earliest epoch when the source became .!9 i i,.!9 active, whichever is smaller. For a homogeneous production spectrum U(E), this simpli"es to j(E)K(1/4p)l (E)U(E) , (12) E if l (E) is much smaller than the horizon size such that redshift and evolution e!ects can be ignored. E Eqs. (11) and (12) are often used in the literature for approximate #ux calculations. 4.1. Nucleons, nuclei, and the Greisen}Zatsepin}Kuzmin cutow Shortly after its discovery, it was pointed out by Greisen [21] and by Zatsepin and Kuzmin [22] that the cosmic microwave background (CMB) radiation "eld has profound consequences for UHECR: With respect to the rest frame of a nucleon that has a su$ciently high energy in the cosmic rest frame (CRF, de"ned as the frame in which the CMB is isotropic), a substantial fraction of the CMB photons will appear as c-rays above the threshold energy for photo-pion production, E-!",5)3,m #m2/(2m )K160 MeV. The total cross section for this process as a function of the c n n N c-ray energy in the nucleon rest frame, E-!", is shown in Fig. 8. Near the threshold the cross section c exhibits a pronounced resonance associated with single pion production, whereas in the limit of high energies it increases logarithmically with s"m2 #2m E-!" [154]. The long tail beyond the N N c "rst resonance is essentially dominated by multiple pion production, Nc PN(np), n'1 (c stands " " for the background photon). For a background photon of energy e in the CRF, the threshold energy E-!",5)3 translates into a corresponding threshold for the nucleon energy, c E "[m (m #m /2)]/eK6.8]1016(e/eV)~1 eV . (13) 5) p N p Typical CMB photon energies are e&10~3 eV, leading to the so-called Greisen}Zatsepin}Kuzmin (GZK) `cuto! a at a few tens of EeV where the nucleon interaction length drops to about 6 Mpc as can be seen in Fig. 9. Detailed investigations of di!erential cross sections, extending into the multiple pion production regime, have been performed in the literature, mainly for the purpose of
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Fig. 8. The total photo-pion production cross section for protons (solid line) and neutrons (dashed line) as a function of the photon energy in the nucleon rest frame, E . -!"
Fig. 9. The nucleon interaction length (dashed line) and attenuation length (solid line) for photo-pion production and the proton attenuation length for pair production (thin solid line) in the combined CMB and the estimated total extragalactic radio background intensity shown in Fig. 10 below.
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calculating secondary c-ray and neutrino production; for recent discussions and references to earlier literature see, e.g., Refs. [155}157]. Below this energy range, the dominant loss mechanism for protons is production of electron} positron pairs on the CMB, pc Ppe`e~, down to the corresponding threshold " E "[m (m #m )]/eK4.8]1014(e/eV)~1 eV . 5) e N e
(14)
Therefore, pair production by protons (PPP) in the CMB ensues at a proton energy E&5]1017 eV. The "rst detailed discussion of PPP in astrophysics was given by Blumenthal [158]. PPP is very similar to triplet pair production by electrons, ec Pee`e~ (see Section 4.2.), " where `electrona, e, means either an electron or a positron in the following. Away from the threshold the total cross section for a nucleus of charge Z is well approximated by the one for triplet pair production, multiplied by Z2. Parametric "ts to the total cross section and the inelasticity for PPP over the whole energy range were given in Ref. [159]. The resulting proton attenuation length is shown in Fig. 9. The next important loss mechanism which starts to dominate near and below the PPP threshold is redshifting due to the cosmic expansion. Indeed, all other loss processes are negligible, except possibly in very dense central regions of galaxies: The interaction length due to hadronic processes which have total cross sections of the order of 0.1 barn in the energy range of interest, for example, is lK3]105(X h2)~1 MpcZ107 Mpc, where 0.009[X h2[0.02 [160], with X the average cosmic " " " baryon density in units of the critical density, and h the Hubble constant H in units of 0 100 km s~1 Mpc~1. For neutrons, b-decay (nPpe~l6 ) is the dominant loss process for E[1020 eV. The neutron e decay rate C "m /(q E), with q K888.6$3.5 s the laboratory lifetime, implies a neutron range n N n n of propagation R "q E/m K0.9(E/1020 eV) Mpc . n n N
(15)
The dominant loss process for nuclei of energy EZ1019 eV is photodisintegration [24,25,161,162] in the CMB and the IR background (IRB) due to the giant dipole resonance. Early calculations [25] suggested a loss length of a few Mpc. Recent observations of multi-TeV c-rays from the BL Lac objects Mrk 421 and Mrk 501 suggest [163,164], however, an IRB roughly a factor 10 lower than previously assumed, which is also consistent with recent independent calculation [165] of the intensity and spectral energy distribution of the IRB based on empirical data primarily from IRAS galaxies. This tends to increase the loss length for nuclei [166]. Recent detailed Monte Carlo simulations [167}169] indicate that, with the reduced IR background, the CMB becomes the dominant photon background responsible for photodisintegration and, for example, leads to a loss length of K10 Mpc at 2]1020 eV. This loss length plays an important role for scenarios in which the highest-energy events observed are heavy nuclei that have been accelerated to UHE (see, e.g., Ref. [26]): The accelerators cannot be much further away than a few tens of Mpc. Speci"c #ux calculations for the source NGC 253 have been performed in Ref. [170]. Apart from photodisintegration, nuclei are subject to the same loss processes as nucleons, where the respective thresholds are given by substituting m by the mass of the nucleus in Eqs. (13) N and (14).
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4.2. UHE photons and electromagnetic cascades As in the case of UHE nucleons and nuclei, the propagation of UHE photons (and electrons/ positrons) is also governed by their interaction with the cosmic photon background. The dominant interaction processes are the attenuation (absorption) of UHE photons due to pair-production (PP) on the background photons c : cc Pe`e~ [171], and inverse Compton scattering (ICS) of " " the electrons (positrons) on the background photons. Early studies of the e!ect of PP attenuation on the cosmological UHE c-ray #ux can be found, e.g., in Ref. [172]. The c-ray threshold energy for PP on a background photon of energy e is E "m2/eK2.6]1011(e/eV)~1 eV , (16) 5) e whereas ICS has no threshold. In the high-energy limit, the total cross sections for PP and ICS are p K2p K3 p (m2/s) ln s/2m2 (s<m2) . (17) PP ICS 2 T e e e For s;m2, p approaches the Thomson cross section p ,8pa2/3m2 (a is the "ne structure e ICS T e constant), whereas p peaks near the threshold Eq. (16). Therefore, the most e$cient targets for PP electrons and c-rays of energy E are background photons of energy eKm2/E. For UHE this e corresponds to e[10~6 eVK100 MHz. Thus, radio background photons play an important role in UHE c-ray propagation through extragalactic space. Unfortunately, the universal radio background (URB) is not very well known mostly because it is di$cult to disentangle the Galactic and extragalactic components. Observational estimates have been given in Ref. [173], and an early theoretical estimate was given in Ref. [174]. Recently, an attempt has been made to calculate the contribution to the URB from radio-galaxies and AGNs [175], and also from clusters of galaxies [176] which tends to give higher estimates. The issue does not seem to be settled, however. At frequencies somewhere below 1 MHz the URB is expected to cut o! exponentially due to free}free absorption. The exact location of the cut-o! depends on the abundance and clustering of electrons in the intergalactic medium and/or the radio source and is uncertain between about 0.1 and 2 MHz. Fig. 10 compares results from Ref. [175] with Ref. [174] and the observational estimate from Ref. [173]. In the extreme Klein}Nishina limit, s<m2, either the electron or the positron produced in the e process cc Pe`e~ carries most of the energy of the initial UHE photon. This leading electron can " then undergo ICS whose inelasticity (relative to the electron) is close to 1 in the Klein}Nishina limit. As a consequence, the upscattered photon which is now the leading particle after this two-step cycle still carries most of the energy of the original c-ray, and can initiate a fresh cycle of PP and ICS interactions. This leads to the development of an electromagnetic (EM) cascade which plays an important role in the resulting observable c-ray spectra. An important consequence of the EM cascade development is that the e!ective penetration depth of the EM cascade, which can be characterized by the energy attenuation length of the leading particle (photon or electron/positron), is considerably greater than just the interaction lengths [177]; see Figs. 11 and 12). As a result, the predicted #ux of UHE photons can be considerably larger than that calculated by considering only the absorption of UHE photons due to PP. EM cascades play an important role particularly in some exotic models of UHECR origin such as collapse or annihilation of topological defects (see Section 6) in which the EHECR injection spectrum is predicted to be dominated by c-rays [178]. Even if only UHE nucleons and nuclei are
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Fig. 10. Contributions of normal galaxies (dotted curves), radio galaxies (long dashed curve), and the cosmic microwave background (short dashed curve) to the extragalactic radio background intensity (thick solid curves) with pure luminosity evolution for all sources (upper curves), and for radio galaxies only (lower curves), from Ref. [175]. Dotted band gives an observational estimate of the total extragalactic radio background intensity [173] and the dot-dash curve gives an earlier theoretical estimate [174] (From Protheroe and Biermann [175].)
Fig. 11. Interaction lengths (dashed lines) and energy attenuation lengths (solid lines) of c-rays in the CMB (thin lines) and in the total low-energy photon background spectrum shown in Fig. 10 with the observational URB estimate from Ref. [173] (thick lines), respectively. The interactions taken into account are single and double pair production.
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Fig. 12. Energy attenuation lengths of electrons for various processes: solid lines are for triplet pair production, and dashed lines for inverse Compton scattering in the CMB (thin lines) and in the total low-energy photon background spectrum shown in Fig. 10 with the observational URB estimate from Ref. [173] (thick lines). The dotted lines are for synchrotron emission losses in a large-scale extragalactic magnetic "eld of r.m.s. strength of 10~11 G (upper curve) and 10~10 G (lower curve), respectively.
produced in the "rst place, for example, via conventional shock acceleration (see Section 5), EM cascades can be produced by the secondaries coming from the decay of pions which are created in interactions of UHE nucleons with the low energy photon background. The EM cascading process and the resulting di!use c-ray #uxes in the conventional acceleration scenarios of UHECR origin were calculated in the 1970s; see, e.g., Refs. [179}181]. The EM cascades initiated by `primarya c-rays and their e!ects on the di!use UHE c-ray #ux in the topological defect scenario of UHECR were "rst considered in Ref. [178]. All these calculations were performed within the CEL approximation which, as described above, deals with only the leading particle. However, the contribution of non-leading particles to the #ux can be substantial for cascades that are not fully developed. A reliable calculation of the #ux at energies much smaller than the maximal injection energy should therefore go beyond the CEL approximation, i.e., one should solve the relevant Boltzmann equations for propagation; this is discussed in Section 4.7. Cascade development accelerates at lower energies due to the decreasing interaction lengths (see Figs. 11 and 12) until most of the c-rays fall below the PP threshold on the low-energy photon background at which point they pile up with a characteristic E~1.5 spectrum below this threshold [35,182}184]. The source of these c-rays are predominantly the ICS photons of average energy
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SE T"E (1!4SsT/3m2) arising from interactions of electrons of energy E with the background c e e e at average squared CM energy SsT in the Thomson regime. The relevant background for cosmological propagation is constituted by the universal IR/O background, corresponding to e[1 eV in Eq. (16), or E K1011 eV. Therefore, most of the energy of fully developed EM cascades 5) ends up below K100 GeV where it is constrained by measurements of the di!use c-ray #ux by EGRET on board the CGRO [185] and other e!ects (see Section 7). Flux predictions involving EM cascades are therefore an important source of constraints of UHE energy injection on cosmological scales. This is further discussed in Sections 6 and 7. It should be mentioned here that the development of EM cascades depends sensitively on the strength of the extragalactic magnetic "elds (EGMFs) which is rather uncertain. The EGMF typically inhibits cascade development because of the synchrotron cooling of the e`e~ pairs produced in the PP process. For a su$ciently strong EGMF the synchrotron cooling time scale of the leading electron (positron) may be small compared to the time scale of ICS interaction, in which case, the electron (positron) synchrotron cools before it can undergo ICS, and thus cascade development stops. In this case, the UHE c-ray #ux is determined mainly by the `directa c-rays, i.e., the ones that originate at distances less than the absorption length due to PP process. The energy lost through synchrotron cooling does not, however, disappear; rather, it reappears at lower energies and can even initiate fresh EM cascades there depending on the remaining path length and the strength of the relevant background photons. Thus, the overall e!ect of a relatively strong EGMF is to deplete the UHE c-ray #ux above some energy and increase the #ux below a corresponding energy in the `lowa (typically few tens to hundreds of GeV) energy region. These issues are further discussed in Section 4.4.1. The lowest-order cross sections, Eq. (17), fall o! as ln s/s for s<m2. Therefore, at EHE, e higher-order processes with more than two "nal state particles start to become important because the mass scales of these particles can enter into the corresponding cross section which typically is asymptotically constant or proportional to powers of ln s. Double pair production (DPP), cc Pe`e~e`e~, is a higher-order QED process that a!ects " UHE photons. The DPP total cross section is a sharply rising function of s near the threshold that is given by Eq. (16) with m P2m , and quickly approaches its asymptotic value [186] e e p K172a4/36pm2K6.45 lbarn (s<m2) . (18) DPP e e DPP begins to dominate over PP above &1021}1023 eV, where the higher values apply for stronger URB (see Fig. 11). For electrons, the relevant higher-order process is triplet pair production (TPP), ec Pee`e~. " This process has been discussed in some detail in Ref. [187] and its asymptotic high-energy cross section is p K(3a/8p) p ((28/9)ln s/m2!218/27) (s<m2) TPP T e e with an inelasticity of
(19)
gK1.768(s/m2)~3@4 (s<m2) . (20) e e Thus, although the total cross section for TPP on CMB photons becomes comparable to the ICS cross section already around 1017 eV, the energy attenuation is not important up to &1022 eV because g[10~3 (see Fig. 12). The main e!ect of TPP between these energies is to create
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a considerable number of electrons and channel them to energies below the UHE range. However, TPP is dominated over by synchrotron cooling (see Section 4.4), and therefore negligible, if the electrons propagate in a magnetic "eld of r.m.s. strength Z10~12 G, as can be seen from Fig. 12. Various possible processes other than those discussed above } e.g., those involving the production of one or more muon, tau, or pion pairs, double Compton scattering (ec Pecc), c}c scattering " (cc Pcc), Bethe}Heitler pair production (cXPXe`e~, where X stands for an atom, an ion, or " a free electron), the process cc Pe`e~c, and photon interactions with magnetic "elds such as pair " production (cBPe`e~) } are in general negligible in EM cascade development. The total cross section for the production of a single muon pair (cc Pk`k~), for example, is smaller than that for " electron pair production by about a factor 10. Energy loss rate contributions for TPP involving pairs of heavier particles of mass m are suppressed by a factor K(m /m)1@2 for s<m2. Similarly, e DPP involving heavier pairs is also negligible [186]. The cross section for double Compton scattering is of order a3 and must be treated together with the radiative corrections to ordinary Compton scattering of the same order. Corrections to the lowest-order ICS cross section from processes involving m additional photons in the "nal state, ec Pe#(m #1)c, m 51, turn out c " c c to be smaller than 10% in the UHE range [188]. A similar remark applies to corrections to the lowest-order PP cross section from the processes cc Pe`e~#m c, m 51. Photon}photon " c c scattering can only play a role at redshifts beyond K100 and at energies below the redshiftdependent pair production threshold given by Eq. (16) [189}191]. A similar remark applies to Bethe}Heitler pair production [190]. Photon interactions with magnetic "elds of typical galactic strength, &10~6 G, are only relevant for EZ1024 eV [192]. For extragalactic magnetic "elds (EGMFs) the critical energy for such interactions is even higher. 4.3. Propagation and interactions of neutrinos and `exotica particles 4.3.1. Neutrinos Neutrino propagation: The propagation of UHE neutrinos is governed mainly by their interaction with the relic neutrino background (RNB). The average squared CM energy for interaction of an UHE neutrino of energy E with a relic neutrino of energy e is given by SsTK(45 GeV)2(e/10~3 eV)(E/1015 GeV) .
(21)
If the relic neutrino is relativistic, then eK3¹ (1#g /4) in Eq. (21), where ¹ K1.9(1#z) K" l " l 1.6]10~4(1#z) eV is the temperature at redshift z and g [ 50 is the dimensionless chemical " potential of relativistic relic neutrinos. For non-relativistic relic neutrinos of mass m [ 20 eV, l eKmax[3¹ , m ]. Note that Eq. (21) implies interaction energies that are typically smaller than l l electroweak energies even for UHE neutrinos, except for energies near the Grand Uni"cation scale, EZ1015 GeV, or if m Z1 eV. In this energy range, the cross sections are given by the Standard l Model of electroweak interactions which are well con"rmed experimentally. Physics beyond the Standard Model is, therefore, not expected to play a signi"cant role in UHE neutrino interactions with the low-energy relic backgrounds. The dominant interaction mode of UHE neutrinos with the RNB is the exchange of a =B boson in the t-channel (l #l6 Pl #lM ), or of a Z0 boson in either the s-channel (l #l6 P+M ) or the i j i j i i t-channel (l #l6 Pl #l6 ) [193}196]. Here, i, j stands for either the electron, muon, or tau #avor, i j i j
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where iOj for the "rst reaction, l denotes a charged lepton, and f any charged fermion. If the latter is a quark, it will, of course, subsequently fragment into hadrons. As an example, the di!erential cross section for s-channel production of Z0 is given by dp i 6 j 0 M /dk"(G2 s/4p) [M2 /((s!M2 )2#M2 C2 )] [g2 (1#kH)2#g2 (1!kH)2] , (22) l `l ?Z ?ff F Z Z Z Z L R where G is the Fermi constant, M and C are mass and lifetime of the Z0, g and g are the usual F Z Z L R dimensionless left- and right-handed coupling constants for f, and kH is the cosine of the scattering angle in the CM system. The t-channel processes have cross sections that rise linearly with s up to sKM2 , with M the W W =B mass, above which they are roughly constant with a value p (sZM )&G2 M2 &10~34 cm2. t W F W Using Eq. (21) this yields the rough estimate p (E, e)&min[10~34, 10~44(s/MeV2)] cm2 t &min[10~34, 3]10~39(e/10~3 eV)(E/1020 eV)] cm2 .
(23)
In contrast, within the Standard Model the neutrino-nucleon cross section roughly behaves as p (E)&10~31(E/1020 eV)0.4 cm2 (24) lN for E Z 1015 eV (see discussion below at end of Section 4.3.1). Interactions of UHE neutrinos with nucleons are, however, still negligible compared to interactions with the RNB because the RNB particle density is about 10 orders of magnitude larger than the baryon density. The only exception could occur near Grand Uni"cation scale energies and at high redshifts and/or if contributions to the neutrino-nucleon cross section from physics beyond the Standard Model dominate at these energies (see below at end of Section 4.3.1). It has recently been pointed out [197] that above the threshold for =B production the process l#cPl=` becomes comparable to the ll processes discussed above. Fig. 13 compares the cross sections relevant for neutrino propagation at CM energies around the electroweak scale. Again, for UHE neutrino interactions with the RNB the relevant CM energies can only be reached if (a) the
Fig. 13. Various cross sections relevant for neutrino propagation as a function of s [194,197]. The sum + f fM does not j j j include f "l , l , t, =, or Z. (From D. Seckel [197].) j i i
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UHE neutrino energy is close to the Grand Uni"cation scale, or (b) the RNB neutrinos have masses in the eV regime, or (c) at redshifts z Z 103. Even then the lc process never dominates over the ll process. At lower energies there is an additional lc interaction that was recently discussed as potentially important besides the ll processes: Using an e!ective Lagrangian derived from the Standard Model, Ref. [198] obtained the result p (s)K9]10~56 (s/MeV2)5 cm2, supposed to be c`l?c`c`l valid at least up to s [ 10 MeV2. Above the electron pair production threshold the cross section has not been calculated because of its complexity but is likely to level o! and eventually decrease. Nevertheless, if the s5 behavior holds up to sK a few hundred MeV2, comparison with Eq. (23) shows that the process c#lPc#c#l would start to dominate and in#uence neutrino propagation around E&3]1017(e/10~3 eV) eV, as was pointed out in Ref. [199]. For a given source distribution, the contribution of the `directa neutrinos to the #ux can be computed by integrating Eq. (11) up to the interaction redshift z(E), i.e. the average redshift from which a neutrino of present day energy E could have propagated without interacting. This approximation neglects the secondary neutrinos and the decay products of the leptons created in the neutral current and charged current reactions of UHE neutrinos with the RNB discussed above. Similarly to the EM case, these secondary particles can lead to neutrino cascades developing over cosmological redshifts [195]. Approximate expressions for the interaction redshift for the processes discussed above have been given in Refs. [35,200] for CM energies below the electroweak scale, assuming relativistic, nondegenerate relic neutrinos, m [ ¹ , and g ;1. Approaching the electroweak scale, a resonl l " ance occurs in the interaction cross section for s-channel Z0 exchange at the Z0 mass, s"M2 K(91 GeV)2, see Eq. (22). The absorption redshift for the corresponding neutrino energy, Z EK1015 GeV(e/10~3 eV)~1 drops to a few (or less for a degenerate, relativistic RNB) and asymptotically approaches constant values of a few tens at higher energies. An interesting situation arises if the RNB consists of massive neutrinos with m &1 eV: Such l neutrinos would constitute hot dark matter which is expected to cluster [201], for example, in galaxy clusters. This would potentially increase the interaction probability for any neutrino of energy within the width of the Z0 resonance at E"M2 /2m "4]1021(eV/m ) eV. Recently, it Z l l has been suggested that the stable end products of the `Z-burstsa that would thus be induced at close-by distances ([50 Mpc) from Earth may explain the highest energy cosmic rays [202,203] and may also provide indirect evidence for neutrino hot dark matter. These end products would be mostly nucleons and c-rays with average energies a factor of K5 and K40 lower, respectively, than the original UHE neutrino. As a consequence, if the UHE neutrino was produced as a secondary of an accelerated proton, the energy of the latter would have to be at least a few 1022 eV [202], making Z-bursts above GZK energies more likely to play a role in the context of non-acceleration scenarios (see Sections 6 and 7). Moreover, it has subsequently been pointed out [204] that Z production is dominated by annihilation on the non-clustered massive RNB compared to annihilation with neutrinos clustering in the Galactic halo or in nearby galaxy clusters. As a consequence, for a signi"cant contribution of neutrino annihilation to the observed EHECR #ux, a new class of neutrino sources, unrelated to UHECR sources, seems necessary. This has been con"rmed by more detailed numerical simulations [205] where it has, however, also been demonstrated that the most signi"cant contribution could come from annihilation on neutrino dark matter clustering in the Local Supercluster by amounts consistent with expectations. In the
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absence of any assumptions on the neutrino sources, the minimal constraint comes from the unavoidable production of secondary c-rays contributing to the di!use #ux around 10 GeV measured by EGRET: If the Z-burst decay products are to explain EHECR, the massive neutrino overdensity f over a length scale l has to satisfy f Z 20 (l /5 Mpc)~1, provided that only neutrinos l l l l leave the source, a situation that may arise in top-down models if the X particles decay exclusively into neutrinos (see Ref. [206] for a model involving topological defects and Ref. [207] for a scenario involving decaying superheavy relic particles). If, instead, the total photon source luminosity is comparable to the total neutrino luminosity, as in most models, the EGRET constraint translates into the more stringent requirement f Z 103(l /5 Mpc)~1. This bound can only be relaxed if most l l of the EM energy is radiated in the TeV range where the Universe is more transparent [205]. Furthermore, the Z-burst scenario requires sources that are optically thick for accelerated protons with respect to photo-pion production because otherwise the observable proton #ux below the GZK cuto! would be comparable to the neutrino #ux [204]. A systematic parameter study of the required overdensity, based on analytical #ux estimates, has been performed in Ref. [208]. Recently, it has been noted that a degenerate relic neutrino background would increase the interaction probability and thereby make the Z-burst scenario more promising [209]. A neutrino asymmetry of order unity is not excluded phenomenologically [210] and can be created in the early Universe, for example, through the A%eck}Dine baryogenesis mechanism [211] or due to neutrino oscillations. The authors of Ref. [209] pointed out that for a neutrino mass m K0.07 eV, a value l suggested by the Super}Kamiokande experiment [212], and for sources at redshifts of a few, the #ux of secondary Z-decay products is maximal for a RNB density parameter X K0.01. Such l neutrino masses, however, require the sources to produce neutrinos at least up to 1022 eV. UHE neutrinos from the decay of pions, that are produced by interactions of accelerated protons in astrophysical sources, must have originated within redshifts of a few. Moreover, in most conventional models their #ux is expected to fall o! rapidly above 1020 eV. Examples are production in active galactic nuclei within hadronic models [213}218], and `cosmogenica neutrinos from interactions of UHECR nucleons (near or above the GZK cuto!) with the CMB (see, e.g., Refs. [219,220]). The latter source is the only one that is guaranteed to exist due to existence of UHECR near the GZK cuto!, but the #uxes are generally quite small. Therefore, interaction of these UHE neutrinos with the RNB, that could reveal the latter's existence, can, if at all, be important only if the relic neutrinos have a mass m Z 1 eV [193]. Due to the continuous release of l UHE neutrinos up to much higher redshifts, most top-down scenarios would imply substantially higher #uxes that also extend to much higher energies [200]. Certain features in the UHE neutrino spectrum predicted within such top-down scenarios, such as a change of slope for massless neutrinos [195] or a dip structure for relic neutrino masses of order 1 eV [196,203], have therefore been proposed as possibly the only way to detect the RNB. However, some of the scenarios at the high end of neutrino #ux predictions have recently been ruled out based on constraints on the accompanying energy release into the EM channel (see Section 7). Since in virtually all models UHE neutrinos are created as secondaries from pion decay, i.e. as electron or muon neutrinos, q-neutrinos can only be produced by a #avor changing =B t-channel interaction with the RNB. The #ux of UHE q-neutrinos is therefore usually expected to be substantially smaller than the one of electron and muon neutrinos, if no neutrino oscillations take place at these energies. However, the recent evidence from the Superkamiokande experiment for nearly maximal mixing between muon and q-neutrinos with D*m2D"Dm2k !m2q DK5]10~3 eV2 l l
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[212] would imply an oscillation length of ¸ "2E/D*m2D"2.6]10~6(E/PeV)(D*m2D/5] 04# 10~3 eV2)~1 pc and, therefore, a rough equilibration between muon and q-neutrino #uxes from any source at a distance larger than ¸ [221]. Turning this around, one sees that a source at distance 04# d emitting neutrinos of energy E is sensitive to neutrino mixing with D*m2D"2E/dK1.3] 10~16 (E/PeV)(d/100 Mpc)~1 eV2 [222,223]. Under certain circumstances, resonant conversion in the potential provided by the RNB clustering in galactic halos may also in#uence the #avor composition of UHE neutrinos from extraterrestrial sources [224]. In addition, such huge cosmological baselines can be sensitive probes of neutrino decay [225]. Neutrino detection: We now turn to a discussion of UHE neutrino interactions with matter relevant for neutrino detection. UHE neutrinos can be detected by detecting the muons produced in ordinary matter via charged-current reactions with nucleons; see, e.g., Refs. [226,227,230] for recent discussions. Corresponding cross sections are calculated by folding the fundamental standard model quark-neutrino cross section with the distribution function of the partons in the nucleon. These cross sections are most sensitive to the abundance of partons of fractional momentum xKM2 /2m E, where E is the neutrino energy. For the relevant squared momentum W N transfer, Q2&M2 , these parton distribution functions have been measured down to xK0.02 W [228]. (It has been suggested that observation of the atmospheric neutrino #ux with future neutrino telescopes may probe parton distribution functions at much smaller x currently inaccessible to colliders [229]). Currently, therefore, neutrino-nucleon cross sections for E Z 1014 eV can be obtained only by extrapolating the parton distribution functions to lower x. Above 1019 eV, the resulting uncertainty has been estimated to be a factor 2 [227], whereas within the dynamical radiative parton model it has been claimed to be at most 20% [230]. An intermediate estimate using the CTEQ4-DIS distributions can roughly be parameterized by [227] p (E)K2.36]10~32(E/1019 eV)0.363 cm2 (1016 eV [ E [ 1021 eV) . (25) lN Improved calculations including non-leading logarithmic contributions in 1/x have recently been performed in Ref. [231]. The results for the neutrino-nucleon cross section di!er by less than a factor 1.5 with Refs. [227,230] even at 1021 eV. Neutral-current neutrino-nucleon cross sections are expected to be a factor 2}3 smaller than charged-current cross sections at UHE and interactions with electrons only play a role at the Glashow resonance, l6 eP=, at E"6.3]1015 eV. e Furthermore, cross sections of neutrinos and anti-neutrinos are basically identical at UHE. Radiative corrections in#uence the total cross section negligibly compared to the parton distribution uncertainties, but may lead to an increase of the average inelasticity in the outgoing lepton from K0.19 to K0.24 at E&1020 eV [232], although this would probably hardly in#uence the shower character. Neutrinos propagating through the Earth start to be attenuated above K100 TeV due to the increasing Standard Model cross section as indicated by Eq. (25). Detailed integrations of the relevant transport equations for muon neutrinos above a TeV have been presented in Ref. [231], and, for a general cold medium, in Ref. [233]. In contrast, q-neutrinos with energy up to K100 PeV can penetrate the Earth due to their regeneration from q decays [223]. As a result, a primary UHE q-neutrino beam propagating through the Earth would cascade down below K100 TeV and in a neutrino telescope could give rise to a higher total rate of upgoing events as compared to downgoing events for the same beam arriving from above the horizon. As mentioned above, a primary q-neutrino beam could arise even in scenarios based on pion decay, if l !l mixing k q
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occurs with the parameters suggested by the Super-Kamiokande results [221]. In the PeV range, q-neutrinos can produce characteristic `double-banga events where the "rst bang would be due to the charged-current production by the q-neutrino of a q whose decay at a typical distance K100 m would produce the second bang [222]. These e!ects have also been suggested as an independent astrophysical test of the neutrino oscillation hypothesis. In addition, isotropic neutrino #uxes in the energy range between 10 TeV and 10 PeV have been suggested as probes of the Earth's density pro"le, whereby neutrino telescopes could be used for neutrino absorption tomography [234]. New interactions: It has been suggested that the neutrino-nucleon cross section could be enhanced by new physics beyond the electroweak scale in the CM, or above about a PeV in the nucleon rest frame; see Eq. (21). For the lowest partial wave contribution to the cross section of a point-like particle this would violate unitarity [235]. However, two major possibilities have been discussed in the literature for which unitarity bounds seem not to be violated. In the "rst, a broken SU(3) gauge symmetry dual to the unbroken SU(3) color gauge group of strong interaction is introduced as the `generation symmetrya such that the three generations of leptons and quarks represent the quantum numbers of this generation symmetry. In this scheme, neutrinos can have e!ectively strong interaction with quarks and, in addition, neutrinos can interact coherently with all partons in the nucleon, resulting in an e!ective cross section comparable to the geometrical nucleon cross section [236]. However, the massive neutral gauge bosons of the broken generation symmetry would also mediate #avor changing neutral current (FCNC) processes, and experimental bounds on these processes indicate that the scale of any such new interaction must be above &100 TeV. The second possibility is that there may be a large increase in the number of degrees of freedom above the electroweak scale [237]. A speci"c implementation of this idea is given in theories with n additional large compact dimensions and a quantum gravity scale M &TeV 4`n that has recently received much attention in the literature [238] because it provides an alternative solution (i.e., without supersymmetry) to the hierarchy problem in Grand Uni"cations of gauge interactions. In such scenarios, the exchange of bulk gravitons (Kaluza}Klein modes) leads to an extra contribution to any two-particle cross section given by [239] p K4ps/M4 K10~27(M /TeV)~4(E/1020 eV) cm2 , (26) ' 4`n 4`n where the last expression applies to a neutrino of energy E hitting a nucleon at rest. Note that a neutrino would typically start to interact in the atmosphere for p Z 10~27 cm2, i.e. for lN E Z 1020 eV, assuming M K1 TeV. The neutrino therefore becomes a primary candidate for 4`n the observed EHECR events. A speci"c signature of this scenario in neutrino telescopes based on ice or water as detector medium would be the absence of events above the energy E where c p grows beyond K10~27 cm2. The corresponding signature in atmospheric detectors such as the ' Pierre Auger detectors would be a hardening of the spectrum above the energy E . Furthermore, c according to Eq. (26), the average atmospheric column depth of the "rst interaction point of neutrino-induced EAS in this scenario is predicted to depend linearly on energy. This should be easy to distinguish from the logarithmic scaling, Eq. (1), expected for nucleons, nuclei, and c-rays. There are, however, astrophysical constraints on M which result from limiting the emission 4`n of bulk gravitons into the extra dimensions. The strongest constraints in this regard come from nucleon}nucleon bremsstrahlung in type II supernovae [240]. These contraints read
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M Z 50 TeV, M Z 4 TeV, and M Z 1 TeV, for n"2, 3, 4, respectively, and, therefore, n54 is 6 7 8 required if neutrino primaries are to serve as a primary candidate for the EHECR events observed above 1020 eV. This assumes that all extra dimensions have the same size given by r KM~1 (M /M )2@nK2]10~17(TeV/M )(M /M )2@n cm , (27) n 4`n P- 4`n 4`n P- 4`n where M denotes the Planck mass. The above lower bounds on M thus translate into the P4`n corresponding upper bounds r [ 3]10~4 mm, r [ 4]10~7 mm, and r [ 2]10~8 mm, respecn n n tively. The neutrino primary hypothesis of EHECR together with other astrophysical and cosmological constraints thus provides an interesting testing ground for theories involving large compact extra dimensions representing one possible kind of physics beyond the Standard Model. In this context, we mention that in theories with large compact extra dimensions mentioned above, Newton's law of gravity is expected to be modi"ed at distances smaller than the length scale given by Eq. (27). Indeed, there are laboratory experiments measuring gravitational interaction at small distances (for a recent review of such experiments see Ref. [241]), which also probe these theories. Thus, future EHECR experiments and gravitational experiments in the laboratory together have the potential of providing rather strong tests of these theories. These tests would be complementary to constraints from collider experiments [242]. In the context of conventional astrophysical sources, the relevant UHE neutrino primaries could, of course, only be produced as secondaries in interactions of accelerated protons of energies at least 1021 eV with matter or with low-energy photons. This implies strong requirements on the possible sources (see Section 5). 4.3.2. Supersymmetric particles Certain supersymmetric particles have been suggested as candidates for the EHECR events. For example, if the gluino is light and has a lifetime long compared to the strong interaction time scale, because it carries color charge, it will bind with quarks, anti-quarks and/or gluons to form color-singlet hadrons, so-called R-hadrons. This can occur in supersymmetric theories involving gauge-mediated supersymmetry (SUSY) breaking [243] where the resulting gluino mass arises dominantly from radiative corrections and can vary between &1 and &100 GeV. In these scenarios, the gluino can be the lightest supersymmetric particle (LSP). There are also arguments against a light quasi-stable gluino [244], mainly based on constraints on the abundance of anomalous heavy isotopes of hydrogen and oxygen which could be formed as bound states of these nuclei and the gluino. However, the case of a light quasi-stable gluino does not seem to be settled. In the context of such scenarios a speci"c case has been suggested in which the gluino mass lies between 0.1 and 1 GeV [245]. The lightest gluino-containing baryon, udsg8 , denoted S0, could then be long-lived or stable, and the kinematical threshold for c } S0 `GZKa interaction would be " higher than for nucleons, at an energy given by substituting the S0 mass M 0 for the nucleon S mass in Eq. (13) [246]. Furthermore, the cross section for c } S0 interaction peaks at an energy " higher by a facor (m 0 /m )(mH !m 0 )/(mD !m ) where the ratio of the mass splittings between S N S N the primary and the lowest lying resonance of the S0 (of mass mH ) and the nucleon satis"es (mH !m 0 )/(mD !m ) Z 2. As a result of this and a somewhat smaller interaction cross section of S N S0 with photons, the e!ective GZK threshold is higher by factors of a few and sources of events above 1019.5 eV could be 15}30 times further away than for the case of nucleons. The existence of
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such events was, therefore, proposed as a signal of supersymmetry [246]. In fact, Farrar and Biermann reported a possible correlation between the arrival direction of the "ve highest energy CR events and compact radio quasars at redshifts between 0.3 and 2.2 [247], as might be expected if these quasars were sources of massive neutral particles. Undoubtedly, with the present amount of data the interpretation of such evidence for a correlation remains somewhat subjective, as is demonstrated by the criticism of the statistical analysis in Ref. [247] by Ho!man [248] and the reply by Farrar and Biermann [249]). Still, it can be expected that only a few more events could con"rm or rule out the quasar hypothesis. Meanwhile, however, accelerator constraints have become more stringent [250,251] and seem to be inconsistent with the scenario from Ref. [245]. However, the scenario with a `tunablea gluino mass [243] still seems possible and suggests either the gluino-gluon bound state gg8 , called glueballino R , or the isotriplet g8 !(uu6 !ddM ) , called o8, as the lightest quasi-stable R-hadron. 0 8 For a summary of scenarios with light gluinos consistent with accelerator constraints see Ref. [252]. Similar to the neutrino primary hypothesis in the context of acceleration sources (see Section 4.3.1), a speci"c di$culty of this scenario is the fact that, of course, the neutral R-hadron cannot be accelerated, but rather has to be produced as a secondary of an accelerated proton interacting with the ambient matter. As a consequence, protons must be accelerated to at least 1021 eV at the source in order for the secondary S0 particles to explain the EHECR events. Furthermore, secondary production would also include neutrinos and especially c-rays, leading to #uxes from powerful discrete acceleration sources that may be detectable in the GeV range by spaceborne c-ray instruments such as EGRET and GLAST, and in the TeV range by ground based c-ray detectors such as HEGRA and WHIPPLE and the planned VERITAS, HESS, and MAGIC projects. At least the latter three ground based instruments should have energy thresholds low enough to detect c-rays from the postulated sources at redshift z&1. Such observations in turn imply constraints on the required branching ratio of proton interactions into the R-hadron which, very roughly, should be larger than &0.01. These constraints, however, will have to be investigated in more detail for speci"c sources. It was also suggested to search for heavy neutral baryons in the data from Cherenkov instruments in the TeV range in this context [253]. A further constraint on new, massive strongly interacting particles in general comes from the character of the air showers created by them: The observed EHECR air showers are consistent with nucleon primaries and limits the possible primary rest mass to less than K50 GeV [254]. With the statistics expected from upcoming experiments such as the Pierre Auger Project, this upper limit is likely to be lowered down to K10 GeV. It is interesting to note in this context that in case of a con"rmation of the existence of new neutral particles in UHECR, a combination of accelerator, air shower, and astrophysics data would be highly restrictive in terms of the underlying physics: In the above scenario, for example, the gluino would have to be in a narrow mass range, 1}10 GeV, and the newest accelerator constraints on the Higgs mass, m Z 90 GeV, would require the presence of a D term of an ) anomalous ;(1) gauge symmetry, in addition to a gauge-mediated contribution to SUSY X breaking at the messenger scale [243]. Finally, SUSY could also play a role in top-down scenarios where it would modify the spectra of particles resulting from the decay of the X particles (see Section 6.2.1).
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4.3.3. Other particles Recently, it was suggested that QCD instanton-induced interactions between quarks can lead to a stable, strong bound state of two K"uds particles, a so-called uuddss H-dibaryon state with a mass M K1700 MeV [255]. This particle would have properties similar to the sypersymmetric H S0 particle discussed in the previous section, i.e. it is neutral and its spin is zero. Its e!ective GZK cuto! would, therefore, also be considerably higher than for nucleons, at approximately 7.3]1020 eV, according to Ref. [255]. It would thus also be a primary candidate for the observed EHECR events that could be produced at high redshift sources. 4.4. Signatures of galactic and extragalactic magnetic xelds in UHECR spectra and images Cosmic magnetic "elds can have several implications for UHECR propagation that may leave signatures in the observable spectra which could in turn be used to constrain or even measure the magnetic "elds in the halo of our Galaxy and/or the extragalactic magnetic "eld (EGMF). 4.4.1. Synchrotron radiation and electromagnetic cascades As already mentioned in Section 4.2, the development of EM cascades strongly depends on presence and strength of magnetic "elds via the synchrotron loss of its electronic component: for a particle of mass m and charge qe (e is the electron charge) the energy loss rate in a "eld of squared r.m.s. strength B2 is dE/dt"!4p (B2/8p) (qm /m)4(E/m )2 . (28) 3 T e e For UHE protons this is negligible, whereas for UHE electrons the synchrotron losses eventually dominate over their attenuation (due to interaction with the background photons) above some critical energy E &1020(B/10~10 G)~1 eV that depends somewhat on the URB (see Fig. 12). 53 Cascade development above that energy is essentially blocked because the electrons lose their energy through synchrotron radiation almost instantaneously once they are produced. In this energy range, c-ray propagation is therefore governed basically by absorption due to PP or DPP, and the observable #ux is dominated by the `directa or `"rst generationa c-rays, and their #ux can be calculated by integrating Eq. (11) up to the absorption length (or redshift). Since this length is much smaller than the Hubble radius, for a homogeneous source distribution this reduces to Eq. (12), with l (E) replaced by the interaction length l(E). E Thus, for a given injection spectrum of c-rays and electrons for a source beyond a few Mpc, the observable cascade spectrum depends on the EGMF. As mentioned in Section 4.2, the hadronic part of UHECR is a continuous source of secondary photons whose spectrum may therefore contain information on the large-scale magnetic "elds [256]. This spectrum should be measurable down to K1019 eV if c-rays can be discriminated from nucleons at the &1% level. In more speculative models of UHECR origin such as the topological defect scenario that predict domination of c-rays above &1020 eV, EGMFs can have even more direct consequences for UHECR #uxes and constraints on such scenarios (see Section 7.1). The photons coming from the synchrotron radiation of electrons of energy E have a typical energy given by E K6.8]1013(E/1021 eV)2(B/10~9 G) eV , 4:/
(29)
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which is valid in the classical limit, E ;E. Constraints can arise when this energy falls in a range 4:/ where there exist measurements of the di!use c-ray #ux, such as from EGRET around 1 GeV [185], or upper limits on it, such as at 50}100 TeV from HEGRA [257], and between K6]1014 eV and K6]1016 eV from CASA-MIA [258]. For example, certain strong discrete sources of UHE c-rays such as massive topological defects with an almost monoenergetic injection spectrum in a 10~9 G EGMF would predict c-ray #uxes that are larger than the charged cosmic-ray #ux for some energies above K1016 eV and can therefore be ruled out [259]. 4.4.2. Deyection and delay of charged hadrons Whereas for electrons synchrotron loss is more important than de#ection in the EGMF, for charged hadrons the opposite is the case. A relativistic particle of charge qe and energy E has a gyroradius r KE/(qeB ) where B is the "eld component perpendicular to the particle ' M M momentum. If this "eld is constant over a distance d, this leads to a de#ection angle (30) h(E, d)Kd/r K0.523q(E/1020 eV)~1(d/1 Mpc)(B /10~9 G) . ' M Magnetic "elds beyond the Galactic disk are poorly known and include a possible extended "eld in the halo of our Galaxy and a large scale EGMF. In both cases, the magnetic "eld is often characterized by an r.m.s. strength B and a correlation length l , i.e. it is assumed that its power # spectrum has a cut-o! in wave number space at k"2p/l and in real space it is smooth on scales # below l . If we neglect energy loss processes for the moment, then the r.m.s. de#ection angle over # a distance d in such a "eld is (2dl /9)1@2 # K0.83 q(E/1020 eV)~1(d/10 Mpc)1@2(l /1 Mpc)1@2(B/10~9 G) (31) h(E, d)K # r ' for d Z l , where the numerical prefactors were calculated from the analytical treatment in # Ref. [260]. There it was also pointed out that there are two di!erent limits to distinguish: For dh(E, d);l , particles of all energies `seea the same magnetic "eld realization during their # propagation from a discrete source to the observer. In this case, Eq. (31) gives the typical coherent de#ection from the line-of-sight source direction, and the spread in arrival directions of particles of di!erent energies is much smaller. In contrast, for dh(E, d)
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particles would have passed the observer already, whereas lower-energy particles would not have arrived yet. Similarly to the behavior of de#ection angles, the width of the spectrum around E would be much smaller than E if both d is smaller than the interaction length for stochastic 0 0 energy loss and dh(E, d);l . In all other cases the width would be comparable to E . # 0 Constraints on magnetic "elds from de#ection and time delay cannot be studied separately from the characteristics of the `probesa, namely the UHECR sources, at least as long as their nature is unknown. An approach to the general case is discussed in Section 4.7. 4.5. Constraints on EHECR source locations Nucleons, nuclei, and c-rays above a few 1019 eV cannot have originated much further away than K50 Mpc. For nucleons this follows from the GZK e!ect (see Fig. 9, the range of nuclei is limited mainly by photodisintegration on the CMB (see Section 4.1), whereas photons are restricted by PP and DPP on the CMB and URB (see Fig. 11). Together with Eq. (31) this implies that above a few 1019 eV the arrival direction of such particles should in general point back to their source within a few degrees [12]. This argument is often made in the literature and follows from the Faraday rotation bound on the EGMF and a possible extended "eld in the halo of our Galaxy, which in its original form reads Bl1@2[10~9 G Mpc1@2 [262,263], as well as from the known # strength and scale height of the "eld in the disk of our Galaxy, B K3]10~6 G, l [ 1 kpc. ' ' Furthermore, the de#ection in the disk of our Galaxy can be corrected for in order to reconstruct the extragalactic arrival direction: Maps of such corrections as a function of arrival direction have been calculated in Refs. [264,265] for plausible models of the Galactic magnetic "eld. The de#ection of UHECR trajectories in the Galactic magnetic "eld may, however, also give rise to several other important e!ects [266] such as (de)magni"cation of the UHECR #uxes due to the magnetic lensing e!ect mentioned in the previous section (which can modify the UHECR spectrum from individual sources), formation of multiple images of a source, and apparent `blindnessa of the Earth towards certain regions of the sky with regard to UHECR. These e!ects may in turn have important implications for UHECR source locations. However, important modi"cations of the Faraday rotation bound on the EGMF have recently been discussed in the literature: The average electron density which enters estimates of the EGMF from rotation measures, can now be more reliably estimated from the baryon density X h2K0.02, " whereas in the original bound the closure density was used. Assuming an unstructured Universe and X "1 results in the much weaker bound [267] 0 B [ 3]10~7(X h2/0.02)~1(h/0.65)(l /Mpc)~1@2 G , (33) " # which suggests much stronger de#ection. However, taking into account the large-scale structure of the Universe in the form of voids, sheets, "laments, etc., and assuming #ux freezing of the magnetic "elds whose strength then approximately scales with the 2 power of the local density, leads to more 3 stringent bounds: Using the Lyman a forest to model the density distribution yields [267] B [ 10~9}10~8 G
(34)
for the large-scale EGMF for coherence scales between the Hubble scale and 1 Mpc. This estimate is closer to the original Faraday rotation limit. However, in this scenario the maximal "elds in the sheets and voids can be as high as a lG [267}269].
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Therefore, according to Eqs. (31) and (34), de#ection of UHECR nucleons is still expected to be on the degree scale if the local large-scale structure around the Earth is not strongly magnetized. However, rather strong de#ection can occur if the Supergalactic Plane is strongly magnetized, for particles originating in nearby galaxy clusters where magnetic "elds can be as high as 10~6 G [262,263,270] (see Section 4.6) and/or for heavy nuclei such as iron [26]. In this case, magnetic lensing in the EGMF can also play an important role in determining UHECR source locations [311,316]. 4.6. Source search for EHECR events The identi"cation of sources of EHECR has been attempted in it least two di!erent ways: First, it has been tried to associate some of the EHE events with discrete sources. For the 300 EeV Fly's Eye event, potential extragalactic sources have been discussed in Ref. [26]. Prominent objects that are within the range of nuclei and nucleons typically require strong magnetic bending, such as Cen A at K3 Mpc and K1363 from the arrival direction, Virgo A (13}26 Mpc, K873), and M82 (3.5 Mpc, K373). The Seyfert galaxy MCG 8}11}11 at 62}124 Mpc and the radio galaxy 3C134 of Fanaro!-Riley (FR) class II are within about 103 of the arrival direction. Due to Galactic obscuration, the redshift (and thus the distance) of the latter is, however, not known with certainty, and estimates range between 30 and 500 Mpc [271]. A powerful quasar, 3C147, within the Fly's Eye event error box at redshift zK0.5 has been suggested as a neutrino source [72]. A potential problem of this option is that standard neutrino-nucleon cross sections predict an interaction probability of neutrinos near 1020 eV of &10~5 in the atmosphere. As long as de#ection in the EGMF is not too strong [see Eq. (31)], the required large neutrino #ux would most likely imply a comparable nucleon #ux below the GZK cuto! that is not observed [272]. We note, however, that, in contrast to interactions with the RNB, the CM energy of a neutrino-nucleon collision at that energy is a few hundred TeV where new physics beyond the electroweak scale could enhance the neutrino-nucleon cross section (see discussion in Section 4.3.1). For the highest energy AGASA event, a potential source for the neutrino option is the FR-II galaxy 3C33 at K300 Mpc distance, whereas the FR-I galaxy NGC 315 at K100 Mpc is a candidate in case of a nucleon primary. A Galactic origin for both the highest energy Fly's Eye and AGASA event seems only possible in case of iron primaries and an extended Galactic halo magnetic "eld [273]. Second, identi"cation of UHECR sources with classes of astrophysical objects has been attempted by testing statistical correlations between arrival directions and the locations of such objects. The Haverah park data set and some data from the AGASA, the Volcano Ranch, and the Yakutsk experiments were tested for correlation with the Galactic and Supergalactic plane, and positive result at a level of almost 3p was found for the latter case for events above 4]1019 eV [79]. An analysis of the SUGAR data from the southern hemisphere, however, did not give signi"cant correlations [80]. More recently, a possible correlation of a subset of about 20% of the events above 4]1019 eV among each other and with the Supergalactic Plane was reported by the AGASA experiment, whereas the rest of the events seemed consistent with an isotropic distribution [81,83]. Results from a similar analysis combining data from the Volcano Ranch, the Haverah Park, the Yakutsk, and the Akeno surface arrays in the northern hemisphere [274], as well as from these and the Fly's Eye experiment [275] were found consistent with that, although no "nal conclusions can
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be drawn presently yet. These "ndings give support to the hypothesis that at least part of the EHECR are accelerated in objects associated with the Supergalactic Plane. However, it was subsequently pointed out [276] that the Supergalactic Plane correlation at least of the Haverah Park data seems to be too strong for an origin of these particles in objects associated with the large-scale galaxy structure because, within the range of the corresponding nucleon primaries, galaxies beyond the Local Supercluster become relevant as well. As a possible resolution it was suggested [277,271] that the possible existence of strong magnetic "elds with strengths up to lG and coherence lengths in the Mpc range, aligned along the large-scale structure [268], could produce a focusing e!ect of UHECR along the sheets and "laments of galaxies. A recent study claims, however, consistency of the arrival directions of UHECR with the distribution of galaxies within 50 Mpc from the Cfa Redshift Catalog [278]. The case of UHECR correlations with the large-scale structure of galaxies, therefore, does not seem to be settled yet. Correlations between arrival directions of UHECR above 4]1019 eV and c-ray burst (GRB) locations have also been investigated. Although the arrival directions of the two highest energy events are within the error boxes of two strong GRBs detected by BATSE [279], no signi"cant positive result was found for the larger UHECR sample [280]. This may be evidence against an association of UHECR with GRBs if their distance scale is Galactic, but not if they have an extragalactic origin because of the implied large time delays of UHECR relative to GRB photons (see Section 5.3). Furthermore, whereas no enhancement of the TeV c-ray #ux has been found in the direction of the Fly's Eye event in Ref. [281], a weak excess was recently reported in Ref. [282]. Finally, a statistically signi"cant correlation between the arrival directions of UHECR events in the energy range (0.8}4)]1019 eV and directions of pulsars along the Galactic magnetic "eld lines has been claimed for the Yakutsk EAS data in Ref. [283]. It would be interesting to look for similar correlations for the data sets from other UHECR experiments. 4.7. Detailed calculations of ultra-high-energy cosmic-ray propagation In order to obtain accurate predictions of observable CR spectra for given production scenarios, one has to solve the equations of motion for the total and di!erential cross sections for the loss processes discussed in Sections 4.1}4.4. If deviations from rectilinear propagation are unimportant, for example, if one is only interested in time-averaged #uxes, one typically solves the coupled Boltzmann equations for CR transport in one spatial dimension either directly or by Monte Carlo simulation. In contrast, if it is important to follow three-dimensional trajectories, for example, to compute images of discrete UHECR sources in terms of energy and time and direction of arrival in the presence of magnetic "elds, the only feasible approach for most purposes is a Monte Carlo simulation. We describe both cases brie#y in the following. 4.7.1. Average yuxes and transport equations in one dimension Computation of time-averaged #uxes from transport equations or one-dimensional Monte Carlo simulation is most relevant for di!use #uxes from many sources and for spectra from discrete sources that emit constantly over long time periods. This is applicable at su$ciently high energies such that de#ection angles in potential magnetic "elds are much smaller than unity. Formally, the Boltzmann equations for the evolution of a set of species with local densities per energy n (E) i
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are given by
P P P
P P
1!b b " i + p [s"eE(1!b b )] i?j " i 2 ~1 j dp `1 1!b b@ " j n (E@) j?i [s"eE@(1!b b ), E]#U dl + (35) # dE@ de n (e) " j " j i dE 2 ~1 j for an isotropic background distribution (here assumed to be only one species), with our notation (see Eqs. (7) and (8)) extended to several species. We brie#y summarize work on solving these equations for the propagation of nucleons, nuclei, c-rays, electrons, and neutrinos in turn. Nucleons and nuclei: Motivated by conventional acceleration models (see Section 5), many studies on propagation of nucleons and nuclei have been published in the literature. Approximate analytical solutions of the transport equations can only be found for very speci"c situations, for example, for the propagation of nucleons near the GZK cuto! (e.g. [284}287]) and/or under certain simplifying assumptions such as the CEL approximation Eqs. (10)}(12) for nucleons (e.g. [288,289]) and c-rays (e.g. [178]). The CEL approximation is excellent for PPP because of its small inelasticity. For pion production, due to its stochastic nature implied by its large inelasticity, the CEL approximation tends to produce a sharper pile-up right below the GZK cuto! compared to exact solutions [290]. It still works reasonably well as long as many pion production events take place on average, i.e. for continuous source distributions and for distant discrete sources. Numerical solutions for nucleons solve the transport equations either directly [291,290,156] or through Monte Carlo simulation [292,26,293,294]. Monte Carlo studies of the photodisintegration histories of nuclei have "rst been performed in Ref. [25] and subsequently in Refs. [26,167,169]. Electromagnetic cascades: Numerical calculations of average c-ray #uxes from EM cascades beyond the analytical CEL approximation are more demanding due to the exponential growth of the number of electrons and photons and are usually not feasible within a pure Monte Carlo approach. Such simulations have been performed mainly in the context of topological defect models of UHECR origin (see Section 6). Calculations of the photon #ux between K100 MeV and K1016 GeV (the Grand Uni"cation Scale) have been presented in Refs. [259,295] where a hybrid Monte Carlo matrix doubling method [296] was used, and in Refs. [156,205,206] where the transport equations are solved by an implicit numerical method. Such calculations play an important role in deriving constraints on top-down models from a comparison of the predicted and observed photon #ux down to energies of K100 MeV (see Section 7). EM cascade simulations are also relevant for the secondary c-ray #ux produced from interactions of primary hadrons [290] and its dependence on cosmic magnetic "elds [256]. Under certain circumstances, this secondary #ux can become comparable to the primary #ux [184]. Analytical calculations have been performed for saturated EM cascades [182]. These calculations show that the cascade spectrum below the pair production threshold has a generic shape. This has also been used to derive constraints on energy injection based on direct observation of this cascade #ux or on a comparison of its side e!ects, for example, on light element abundances, with observations. We will discuss these issues in Sections 7.1 and 7.2. As a "rst application of numerical transport calculations we present the e!ective penetration depth of EM cascades, which we de"ne as the coe$cient l (E) in Eq. (12), where j(E) is the c-ray #ux E R n (E)"!n (E) de n (e) " t i i
`1
dl
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Fig. 14. E!ective penetration depth of EM cascades, as de"ned in the text, for the strongest theoretical URB estimate (solid lines), and the observational URB estimate from Ref. [173] (dashed lines), as shown in Fig. 10, and for an EGMF ;10~11 G (thick lines), and 10~9 G (thin lines), respectively.
resulting after propagating a homogeneous injection #ux U(E). Fig. 14 shows results computed for the new estimates of the IR background from Ref. [163], and for some combinations of the URB and the EGMF. Neutrino yuxes: Accurate predictions for the UHE neutrino #ux have become more relevant recently due to several proposals for a km3 scale neutrino observatory [43]. Fluxes of secondary neutrinos from photo-pion production by UHECR have been calculated numerically, e.g., in Refs. [290,196,156], by solving the full transport equations for nucleons. Because of the small redshifts involved, the neutrinos can be treated as interaction-free, and the main uncertainties come from the poorly known injection history of the primary nucleons (see Fig. 31). In top-down scenarios, neutrinos are continuously produced up to very high redshifts and secondaries produced in neutrino interactions can enhance the UHE neutrino #uxes compared to the simple absorption approximation used in Refs. [200,297]. By solving the full Boltzmann equations for the neutrino cascade, unnormalized spectral shapes of neutrino #uxes from topological defects have been calculated in Ref. [195], and absolute #uxes in Ref. [196]. Semianalytical calculations of c-ray, nucleon, and neutrino #uxes for a speci"c class of cosmic string models predicting an absolute normalization of the UHECR injection rate (see Section 6.4.6) have been performed in Ref. [298]. Recently, an integrated code has been developed which solves the coupled full transport equations for all species, i.e., nucleons, c-rays, electrons, and neutrinos concurrently [205,206]. This allows, for example, to make detailed predictions for the spectra of the nucleons and c-rays produced by resonant Z0 production of UHE neutrinos on a massive RNB which could serve as a signature of hot dark matter [202,203] (see Section 4.3.1).
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4.7.2. Angle-time-energy images of ultra-high-energy cosmic-ray sources In Section 4.4 we gave simple analytical estimates for the average de#ection and time delay of nucleons propagating in a cosmic magnetic "eld. Here, we review approaches that have been taken in the literature to compute e!ects of magnetic "elds on both spectra and angular images (and their time dependence) of sources of UHE nucleons. Strong deyection: An exact analytical expression for the distribution of time delays that applies in the limit dh(E, d)
(37)
As mentioned in Section 3.2, Eq. (36) and its generalization to an anisotropic di!usion tensor plays a prominent role also in models of Galactic CR propagation. We stress here that while this equation provides a good description of the propagation of Galactic CR for energies up to the knee, it has rather limited applicability in studying UHECR propagation which often takes place in the transition regime between di!usion and rectilinear propagation (see below). Small deyection: For small de#ection angles and if photo-pion production is important, one has to resort to numerical Monte Carlo simulations in three dimensions. Such simulations have been performed in Ref. [303] for the case dh(E, d)
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In Refs. [304,293,294] the Monte Carlo simulations were performed in the following way: The magnetic "eld was represented as Gaussian random "eld with zero mean and a power spectrum with SB2(k)TJknH for k(k and SB2(k)T"0 otherwise, where k "2p/l characterizes the # # # numerical cut-o! scale and the r.m.s. strength is B2":= dk k2SB2(k)T. The "eld is then calculated 0 on a grid in real space via Fourier transformation. For a given magnetic "eld realization and source, nucleons with a uniform logarithmic distribution of injection energies are propagated between two given points (source and observer) on the grid. This is done by solving the equations of motion in the magnetic "eld interpolated between the grid points, and subjecting nucleons to stochastic production of pions and (in case of protons) continuous loss of energy due to PP. Upon arrival, injection and detection energy, and time and direction of arrival are recorded. From many (typically 40 000) propagated particles, a histogram of average number of particles detected as a function of time and energy of arrival is constructed for any given injection spectrum by weighting the injection energies correspondingly. This histogram can be scaled to any desired total #uence at the detector and, by convolution in time, can be constructed for arbitrary emission time scales of the source. An example for the distribution of arrival times and energies of UHECR from a bursting source is given in Fig. 15.
Fig. 15. Contour plot of the UHECR image of a bursting source at d"30 Mpc, projected onto the time}energy plane, with B"2]10~10 G, l "1 Mpc, from Ref. [304]. The contours decrease in steps of 0.2 in the logarithm to base 10. The # dotted line indicates the energy-time delay correlation q(E, d)JE~2 as would be obtained in the absence of pion production losses. Clearly, dh(E, d);l in this example, since for E(4]1019 eV, the width of the energy distribution at # any given time is much smaller than the average (see Section 4.4). The dashed lines, which are not resolved here, indicate the location (arbitrarily chosen) of the observational window, of length ¹ "5 yr. 0"4
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We adopt the following notation for the parameters: q denotes the time delay due to magnetic 100 de#ection at E"100 EeV and is given by Eq. (32) in terms of the magnetic "eld parameters; ¹ denotes the emission time scale of the source; ¹ ;1 yr corresponds to a burst, and ¹ <1 yr S S S (roughly speaking) to a continuous source; c is the di!erential index of the injection energy spectrum; N denotes the #uence of the source with respect to the detector, i.e., the total number of 0 particles that the detector would detect from the source on an in"nite time scale, "nally; L is the likelihood function of the above parameters. By putting windows of width equal to the time scale of observation over these histograms one obtains expected distributions of events in energy and time and direction of arrival for a given magnetic "eld realization, source distance and position, emission time scale, total #uence, and injection spectrum. Examples of the resulting energy spectrum are shown in Fig. 16. By dialing Poisson statistics on such distributions, one can simulate corresponding observable event clusters. Conversely, for any given real or simulated event cluster, one can construct a likelihood of the observation as a function of the time delay, the emission time scale, the di!erential injection spectrum index, the #uence, and the distance. In order to do so, and to obtain the maximum of the likelihood, one constructs histograms for many di!erent parameter combinations as described above, randomly puts observing time windows over the histograms, calculates the likelihood function from the part of the histogram within the window and the cluster events, and averages over di!erent window locations and magnetic "eld realizations.
Fig. 16. Energy spectra for a continuous source (solid line), and for a burst (dashed line), from Ref. [304]. Both spectra are normalized to a total of 50 particles detected. The parameters corresponding to the continuous source case are: ¹ "104 yr, q "1.3]103 yr, and the time of observation is t"9]103 yr, relative to rectilinear propagation with S 100 the speed of light. A low energy cuto! results at the energy E "4]1019 eV where q S "t. The dotted line shows how the S E spectrum would continue if ¹ ;104 yr. The case of a bursting source corresponds to a slice of the image in the q }E S E plane, as indicated in Fig. 15 by dashed lines. For both spectra, d"30 Mpc, and c"2.
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In Ref. [293] this approach has been applied to and discussed in detail for the three pairs observed by the AGASA experiment [81], under the assumption that all events within a pair were produced by the same discrete source. Although the inferred angle between the momenta of the paired events acquired in the EGMF is several degrees [305], this is not necessarily evidence against a common source, given the uncertainties in the Galactic "eld and the angular resolution of AGASA which is K2.53. As a result of the likelihood analysis, these pairs do not seem to follow a common characteristic; one of them seems to favor a burst, another one seems to be more consistent with a continuously emitting source. The current data, therefore, does not allow one to rule out any of the models of UHECR sources. Furthermore, two of the three pairs are insensitive to the time delay. However, the pair which contains the 200 EeV event seems to signi"cantly favor a comparatively small average time delay, q [10 yr, as can be seen from the likelihood function 100 marginalized over ¹ and N (see Fig. 17). According to Eq. (32) this translates into a tentative S 0 bound for the r.m.s. magnetic "eld, namely, B [ 2]10~11(l /1 Mpc)~1@2(d/30 Mpc)~1 G , (38) # which also applies to magnetic "elds in the halo of our Galaxy if d is replaced by the lesser of the source distance and the linear halo extent. If con"rmed by future data, this bound would be at least two orders of magnitude more restrictive than the best existing bounds which come from Faraday rotation measurements [see Eq. (34)] and, for a homogeneous EGMF, from CMB anisotropies [306]. UHECR are therefore at least as sensitive a probe of cosmic magnetic "elds as other measures in the range near existing limits such as the polarization [307] and the small-scale anisotropy [308] of the CMB.
Fig. 17. The likelihood, L, marginalized over ¹ and N as a function of the average time delay at 1020 eV, q , S 0 100 assuming a source distance d"30 Mpc. The panels are for pair d 3 through d 1, from top to bottom, of the AGASA pairs [81] (see Section 4.7.2). Solid lines are for c"1.5, dotted lines for c"2.0, and dashed lines for c"2.5.
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More generally, con"rmation of a clustering of EHECR would provide signi"cant information on both the nature of the sources and on large-scale magnetic "elds [309]. This has been shown quantitatively [294] by applying the hybrid Monte Carlo likelihood analysis discussed above to simulated clusters of a few tens of events as they would be expected from next generation experiments [5] such as the High Resolution Fly's Eye [84], the Telescope Array [85], and most notably, the Pierre Auger Project [86] (see Section 2.6), provided the clustering recently suggested by the AGASA experiment [81,83] is real. The proposed satellite observatory concept for an Orbiting Wide-angle Light collector (OWL) [87] might even allow one to detect clusters of hundreds of such events. Five generic situations of the time-energy images of UHECR were discussed in Ref. [294], classi"ed according to the values of the time delay q induced by the magnetic "eld, the emission E time scale of the source ¹ , as compared to the lifetime of the experiment. The likelihood calculated S for the simulated clusters in these cases presents di!erent degeneracies between di!erent parameters, which complicates the analysis. As an example, the likelihood is degenerate in the ratios N /¹ , or N /*q , where N is the total #uence, and *q is the spread in arrival time; these 0 S 0 100 0 100 ratios represent rates of detection. Another example is given by the degeneracy between the distance d and the injection energy spectrum index c. Yet another is the ratio (dq )1@2/l , that E # controls the size of the scatter around the mean of the q !E correlation. Therefore, in most E general cases, values for the di!erent parameters cannot be pinned down, and generally, only domains of validity are found. In the following, the reconstruction quality of the main parameters considered is summarized. The distance to the source can be obtained from the pion production signature, above the GZK cut-o!, when the emission timescale of the source dominates over the time delay. Since the time delay decreases with increasing energy, the lower the energy E , de"ned by q C K¹ , the higher the C E S accuracy on the distance d. The error on d is, in the best case, typically a factor 2, for one cluster of K40 events. In this case, where the emission timescale dominates over the time delay at all observable energies, information on the magnetic "eld is only contained in the angular image, which was not systematically included in the likelihood analysis of Ref. [294] due to computational limits. Qualitatively, the size of the angular image is proportional to B(dl )1@2/E, whereas the # structure of the image, i.e., the number of separate images, is controlled by the ratio d3@2B/l1@2/E. # Finally, in the case when the time delay dominates over the emission timescale, with a time delay shorter than the lifetime of the experiment, one can also estimate the distance with reasonable accuracy. Some sensitivity to the injection spectrum index c exists whenever events are recorded over a su$ciently broad energy range. At least if the distance d is known, it is in general comparatively easy to rule out a hard injection spectrum if the actual c Z 2.0, but much harder to distinguish between c"2.0 and 2.5. If the lifetime of the experiment is the largest time scale involved, the strength of the magnetic "eld can only be obtained from the time-energy image because the angular image will not be resolvable. When the time delay dominates over the emission time scale, and is, at the same time, larger than the lifetime of the experiment, only a lower limit corresponding to this latter time scale can be placed on the time delay and hence on the strength of the magnetic "eld. When combined with the Faraday rotation upper limit Eq. (34), this would nonetheless allow one to bracket the r.m.s. magnetic "eld strength within a few orders of magnitude. In this case also, signi"cant
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information is contained in the angular image. If the emission time scale is larger then the delay time, the angular image is obviously the only source of information on the magnetic "eld strength. The coherence length l enters in the ratio (dq )1@2/l that controls the scatter around the mean of # E # the q !E correlation in the time-energy image. It can therefore be estimated from the width of this E image, provided the emission time scale is much less than q (otherwise the correlation would not E be seen) and some prior information on d and q is available. E An emission time scale much larger than the experimental lifetime may be estimated if a lower cut-o! in the spectrum is observable at an energy E , indicating that ¹ Kq C . The latter may, in C S E turn, be estimated from the angular image size via Eq. (32), where the distance can be estimated from the spectrum visible above the GZK cut-o!, as discussed above. An example of this scenario is shown in Fig. 18. For angular resolutions *h, timescales in the range 3]103 (*h/13)2(d/10 Mpc) yr[¹ Kq [1042107 (E/100 EeV)~2 yr (39) S E could be probed. The lower limit follows from the requirement that it should be possible to estimate q from h , using Eq. (32), otherwise only an upper limit on ¹ , corresponding to this same E E S number, would apply. The upper bound in Eq. (39) comes from constraints on maximal time delays in cosmic magnetic "elds, such as the Faraday rotation limit in the case of cosmological large-scale "eld (smaller number) and knowledge on stronger "elds associated with the large-scale galaxy structure (larger number). Eq. (39) constitutes an interesting range of emission timescales for many conceivable scenarios of ultra-high energy cosmic rays. For example, the hot spots in certain powerful radio galaxies that have been suggested as ultra-high-energy cosmic-ray sources [286], have a size of only several kpc and could have an episodic activity on time scales of &106 yr. A detailed comparison of analytical estimates for the distributions of time delays, energies, and de#ection angles of nucleons in weak random magnetic "elds with the results of Monte Carlo simulations has been presented in Ref. [310]. In this work, de#ection was simulated by solving a stochastic di!erential equation and observational consequences for the two major classes of source scenarios, namely continuous and impulsive UHECR production, were discussed. In agreement with earlier work [261] it was pointed out that at least in the impulsive production scenario and for an EGMF in the range 0.1}1]10~9 G, as required for cosmological GRB sources (see Section 5.3 below), there is a typical energy scale E &1020.5}1021.5 eV below which the #ux is " quasi-steady due to the spread in arrival times, whereas above which the #ux is intermittent with only a few sources contributing. General case: Unfortunately, neither the di!usive limit nor the limit of nearly rectilinear propagation is likely to be applicable to the propagation of UHECR around 1020 eV in general. This is because in magnetic "elds in the range of a few 10~8 G, values that are realistic for the Supergalactic Plane [277,271], the gyro radii of charged particles is of the order of a few Mpc which is comparable to the distance to the sources. An accurate, reliable treatment in this regime can only be achieved by numerical simulation. To this end, the Monte Carlo simulation approach of individual trajectories developed in Refs. [293,294] has recently been generalized to arbitrary de#ections [311]. The Supergalactic Plane was modeled as a sheet with a thickness of a few Mpc and a Gaussian density pro"le. The same statistical description for the magnetic "eld was adopted as in Refs. [293,294], but with a "eld power-law index n "!11, representing a turbulent Kolmogorov-type spectrum, and weighted H 3 with the sheet density pro"le. It should be mentioned, however, that other spectra, such as the
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Fig. 18. (a) Arrival time-energy histogram for c"2.0, q "50 yr, ¹ "200 yr, l K1 Mpc, d"50 Mpc, corresponding 100 S # to BK3]10~11 G. Contours are in steps of a factor 100.4"2.51; (b) Example of a cluster in the arrival time}energy plane resulting from the cut indicated in (a) by the dashed line at qK100 yr; (c) The likelihood function, marginalized over N and c, for d"50 Mpc, l K Mpc, for the cluster shown in (b), in the ¹ !q plane. The contours shown go 0 # S 100 from the maximum down to about 0.01 of the maximum in steps of a factor 100.2"1.58. Note that the likelihood clearly favors ¹ K4q . For q large enough to be estimated from the angular image size, ¹ <¹ can, therefore, be S 100 100 S 0"4 estimated as well.
Kraichnan spectrum [312], corresponding to n "!7, are also possible. The largest mode H 2 with non-zero power was taken to be the largest turbulent eddy whose size is roughly the sheet thickness. In addition, a coherent "eld component B is allowed that is parallel to the sheet and # varies proportional to the density pro"le.
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When CR backreaction on the weakly turbulent magnetic "eld is neglected, the di!usion coe$cient of CR of energy E is determined by the magnetic "eld power on wavelengths comparable to the particle Larmor radius, and can be approximated by [313] 1 B D(E)K r (E) . (40) 3 ' := dk k2SB2(k)T 1@r' (E) As a consequence, for the Kolmogorov spectrum, in the di!usive regime, where q Z d, the E di!usion coe$cient should scale with energy as D(E)JE1@3 for r [ ¸/(2p), and as D(E)JE in ' the so-called Bohm di!usion regime, r Z ¸/(2p). This should be re#ected in the dependence of the ' time delay q on energy E: From the rectilinear regime, q [ d, hence at the largest energies, where E E q JE~2, this should switch to q JE~1 in the regime of Bohm di!usion, and eventually to E E q JE~1@3 at the smallest energies, or largest time delays. Indeed, all three regimes can be seen in E Fig. 19 which shows an example of the distribution of arrival times and energies of UHECR from a bursting source. In a steady-state situation, di!usion leads to a modi"cation of the injection spectrum by roughly a factor q , at least in the absence of signi"cant energy loss and for a homogeneous, in"nitely E extended medium that can be described by a spatially constant di!usion coe$cient. Since in the non-di!usive regime the observed spectrum repeats the shape of the injection spectrum, a change to a #atter observed spectrum at high energies is expected in the transition region [299]. From the spectral point of view this suggests the possibility of explaining the observed UHECR #ux above K10 EeV including the highest energy events with only one discrete source [301].
Fig. 19. The distribution of time delays q and energies E for a burst with spectral index c"2.4 at a distance d"10 Mpc, E similar to Fig. 15, but for the Supergalactic Plane scenario discussed in the text. The turbulent magnetic "eld component in the sheet center is B"3]10~7 G. Furthermore, a vanishing coherent "eld component is assumed. The inter-contour interval is 0.25 in the logarithm to base 10 of the distribution per logarithmic energy and time interval. The three regimes discussed in the text, q JE~2 in the rectilinear regime EZ200 EeV, q JE~1 in the Bohm di!usion regime E E 60 EeV[E[200 EeV, and q JE~1@3 for E[60 EeV are clearly visible. E
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The more detailed Monte Carlo simulations reveal the following re"nements of this qualitative picture: The presence of a non-trivial geometry where the magnetic "eld falls o! at large distances, such as with a sheet, tends to deplete the #ux in the di!usive regime as compared to the case of a homogeneous medium. This is the dominant e!ect as long as particles above the GZK cuto! do not di!use, this being the case, for example, for an r.m.s. "eld strength of B [ 5]10~8 G, dK10 Mpc. The simple explanation is that the "xed total amount of particles injected over a certain time scale is distributed over a larger volume in case of a non-trivial geometry due to faster di!usion near the boundary of the strong "eld region. With increasing "eld strengths the di!usive regime will extend to energies beyond the GZK cuto! and the increased pion production losses start to compensate for the low-energy suppression from the non-trivial geometry. For very strong "elds, for example, for B Z 10~7 G, dK10 Mpc, the pion production e!ect will overcompensate the geometry e!ect and reverse the situation: In this case, the #ux above the GZK cuto! is strongly suppressed due to the di!usively enhanced pion production losses and the #ux at lower energies is enhanced. Therefore, there turns out to be an optimal "eld strength that depends on the source distance and provides an optimal "t to the data above 10 EeV. The optimal case for d"10 Mpc, with a maximal r.m.s. "eld strength of B "10~7 G in the plane center is shown .!9 in Fig. 20. Furthermore, the numerical results indicate an e!ective gyroradius that is roughly a factor 10 higher than the analytical estimate, with a correspondingly larger di!usion coe$cient compared to Eq. (40). In addition, the #uctuations of the resulting spectra between di!erent magnetic "eld realizations can be substantial, as can be seen in Fig. 20. This is a result of the fact that most of the
Fig. 20. The average (solid histogram) and standard deviation (dashed lines) with respect to 15 simulated magnetic "eld realizations of the best "t spectrum to the data above 1019 eV for the scenario of a single source in a magnetized Supergalactic Plane. This best "t corresponds to a maximal magnetic "eld in the plane center, B "10~7 G, with all .!9 other parameters as in Fig. 19. 1 sigma error bars are the combined data from the Haverah Park [3], the Fly's Eye [7], and the AGASA [8] experiments above 1019 eV.
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magnetic "eld power is on the largest scales where there are the fewest modes. These considerations mean that the applicability of analytical #ux estimates of discrete sources in speci"c magnetic "eld con"gurations is rather limited. Angular images of discrete sources in a magnetized Supercluster, in principle, contain information on the magnetic "eld structure. For the recently suggested "eld strengths between &10~8 G and K1 lG the angular images are large enough to exploit that information with instruments of angular resolution in the degree range. An example where a transition from several images at low energies to one image at high energies allows one to estimate the magnetic "eld coherence scale is shown in Fig. 21.
Fig. 21. Angular image of a point-like source in a magnetized Supergalactic Plane, corresponding to one particular magnetic "eld realization with a maximal magnetic "eld in the plane center, B "5]10~8 G, all other parameters .!9 being the same as in Fig. 20. The image is shown in di!erent energy ranges, as indicated, as seen by a detector of K13 angular resolution. A transition from several images at lower energies to only one image at the highest energies occurs where the linear de#ection becomes comparable to the e!ective "eld coherence length. The di!erence between neighboring shade levels is 0.1 in the logarithm to base 10 of the integral #ux per solid angle.
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The newest AGASA data [83], however, indicate an isotropic distribution of EHECR. To explain this with only one discrete source would require the magnetic "elds to be so strong that the #ux beyond 1020 eV would most likely be too strongly suppressed by pion production, as discussed above. This suggests a more continuous source distribution which may also still reproduce the observed UHECR #ux above K10 EeV with only one spectral component [314]. A more systematic parameter study of sky maps and spectra in UHECR in di!erent scenarios is therefore now being pursued [315,316]. Intriguingly, scenarios in which a di!use source distribution follows the density in the Supergalactic Plane within a certain radius, can accomodate both the large-scale isotropy and the small-scale clustering revealed by AGASA if a magnetic "eld of strength BZ0.05 lG permeates the Supercluster [316]. Fig. 22 shows the angular distribution in Galactic coordinates in such a scenario for di!erent "eld strengths and source distribution radii. The integral angular distributions with respect to the Supergalactic Plane for two such cases are shown in Fig. 23. Detailed Monte Carlo simulations performed on these distributions reveal that the anisotropy decreases with increasing magnetic "eld strength due to di!usion and that small-scale clustering increases with coherence and strength of the magnetic "eld due to magnetic lensing. Both anisotropy and clustering also increase with the (unknown) source distribution radius. Furthermore, the discriminatory power between models with respect to anisotropy and clustering strongly increases with exposure [316]. Finally, the corresponding solid angle integrated spectra show negligible cosmic variance for di!use sources and "t the data well both for B "0.05 lG and for B "0.5 lG, as shown in .!9 .!9 Fig. 24. As a result, a di!use source distribution associated with the Supergalactic Plane can explain most of the currently observed features of ultra-high-energy cosmic rays at least for "eld strengths close to 0.5 lG. The large-scale anisotropy and the clustering predicted by this scenario will allow strong discrimination against other models with next generation experiments such as the Pierre Auger Project. 4.8. Anomalous kinematics, quantum gravity ewects, Lorentz symmetry violations The existence of UHECR beyond the GZK cuto! has prompted several suggestions of possible new physics beyond the Standard Model. We have already discussed some of these suggestions in Section 4.3 in the context of propagation of UHECR in the extragalactic space. Further, in Section 6 we will discuss suggestions regarding possible new sources of EHECR that also involve postulating new physics beyond the Standard Model. In the present section, to end our discussions on the propagation and interactions of UHE radiation, we brie#y discuss some examples of possible small violations or modi"cations of certain fundamental tenets of physics (and constraints on the magnitude of those violations/modi"cations) that have also been discussed in the literature in the context of propagation of UHECR. For example, as an interesting consequence of the very existence of UHECR, constraints on possible violations of Lorentz invariance (VLI) have been pointed out [317]. These constraints rival precision measurements in the laboratory: If events observed around 1020 eV are indeed protons, then the di!erence between the maximum attainable proton velocity and the speed of light
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Fig. 22. Angular distribution in Galactic coordinates in scenarios where the UHECR sources with spectral index c"2.4 are distributed according to the matter density and r.m.s. magnetic "eld strength in the Local Supercluster, following a pancake pro"le with scale height of 5 Mpc and scale length and maximal "eld strength B in the plane center as .!9 indicated. The observer is within 2 Mpc of the Supergalactic Plane whose location is indicated by the solid line and at a distance d"20 Mpc from the plane center. The absence of sources within 2 Mpc from the observer was assumed. The color scale shows the intensity per solid angle, and the distributions are averaged over 4 magnetic "eld realizations with 20 000 particles each.
has to be less than about 1]10~23, otherwise the proton would lose its energy by Cherenkov radiation within a few hundred centimeters. Possible tests of other modes of VLI with UHECR have been discussed in Ref. [318] and Ref. [319] discusses constraints on VLI in the context of horizontal air-showers generated by cosmic rays in general. Gonzalez}Mestres [318], Coleman and Glashow [320], and earlier, Sato and Tati [321] and Kirzhnits and Chechin [322] have also suggested that due to modi"ed kinematical constraints the GZK cuto! could even be evaded by
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Fig. 23. The distribution of events above 6]1019 eV in Supergalactic latitude for two scenarios with a di!use source distribution in a magnetized Supergalactic Plane, for B "0.5 lG (thick histogram), and for B "0.05 lG (thin .!9 .!9 histogram), assuming 1.63 angular resolution. The full angular distributions for these cases were shown in the lower and upper panel of Fig. 22, respectively. The dash}dotted curve represents a completely isotropic distribution.
Fig. 24. Best "t to the data above 1019 eV of the spectra predicted by two scenarios with di!use sources in a magnetized Supergalactic Plane whose predicted angular distributions were shown in the top and bottom panel of Fig. 22 and in Fig. 23. The thick histogram is for B "0.5 lG, with the observer 2 Mpc above the Supergalactic Plane, and the thin .!9 histogram is for B "0.05 lG, with the observer in the plane center. The cosmic variance between di!erent realizations .!9 is negligible.
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allowing a tiny VLI too small to have been detected otherwise. Similar consequences apply to other energy loss processes such as pair production by photons above a TeV with the low-energy photon background [323]. It seems to be possible to accommodate such e!ects within theories involving generalized Lorentz transformations [324] or deformed relativistic kinematics [325]. Furthermore, it has been pointed out [326] that violations of the principle of equivalence (VPE), while not dynamically equivalent, also produce the same kinematical e!ects as VLI for particle processes in a constant gravitational potential, and so the constraints on VLI from UHECR physics can be translated into constraints on VPE such that the di!erence between the couplings of protons and photons to gravity must be less than about 1]10~19. Again, this constraint is more stringent by several orders of magnitude than the currently available laboratory constraint from EoK tvoK s experiments. As a speci"c example of VLI, we consider an energy-dependent photon group velocity RE/Rk" c[1!sE/E #O(E2/E2 )] where c is the speed of light in the low-energy limit, s"$1, and 0 0 E denotes the energy scale where this modi"cation becomes of order unity. This corresponds to 0 a dispersion relation c2k2KE2#s E3/E , (41) 0 which, for example, can occur in quantum gravity and string theory [327]. The kinematics of electron}positron pair production in a head-on collision of a high-energy photon of energy E with a low-energy background photon of energy e then leads to the constraint eK(E/4)(m2/E E #h h )#s E2/4E , (42) e 1 2 1 2 0 where E and h &O(m/E ) are, respectively, the energy and outgoing momentum angle (with i i i respect to the original photon momentum) of the electron and positron (i"1, 2). For the case considered by Coleman and Glashow [317] in which the maximum attainable speed c of the i matter particle is di!erent from the photon speed c, the kinematics can be obtained by substituting c2!c2 for sE/E in Eq. (42). i 0 Let us de"ne a critical energy E "(m2E )1@3K15(E /m )1@3 TeV in the case of the energy# e 0 0 Pdependent photon group velocity, and E "m /Dc2!c2D1@2 in the case considered by Coleman and # e i Glashow. If s(0, or c (c, then e becomes negative for EZE . This signals that the photon can i # spontaneously decay into an electron}positron pair and propagation of photons across extragalactic distances will in general be inhibited. The observation of extragalactic photons up to K20 TeV [328,329] therefore puts the limits E ZM or c2!c2Z!2]10~17. In contrast, if s'0, or 0 Pi c 'c, e will grow with energy for EZE until there is no signi"cant number of target photon i # density available and the Universe becomes transparent to UHE photons. A clear test of this possibility would be the observation of Z100 TeV photons from distances Z100 Mps [330]. In addition, the dispersion relation (41) implies that a photon signal at energy E will be spread out by *tK(d/c)(E/E )K1(d/100 Mpc)(E/TeV)(E /M )~1 s. The observation of c-rays at energies 0 0 PEZ2 TeV within K300 s from the AGN Markarian 421 therefore puts a limit (independent of s) of E Z4]1016 GeV, whereas the possible observation of c-rays at EZ200 TeV within K200 s from 0 a GRB by HEGRA might be sensitive to E KM [331]. For a recent detailed discussion of these 0 Plimits see Ref. [332]. A related proposal originally due to KosteleckyH in the context of CR suggests the electron neutrino to be a tachyon [333]. This would allow the proton in a nucleus of mass m(A, Z) for mass
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number A and charge Z to decay via pPn#e`#l above the energy threshold E "m(A, Z)] e 5) [m(A, Z$1)#m !m(A, Z)]/Dm e D which, for a free proton, is E K1.7]1015/(Dm e D/eV) eV. e l 5) l Ehrlich [334] claims that by choosing m2e K!(0.5 eV)2 it is possible to explain the knee and l several other features of the observed CR spectrum, including the high-energy end, if certain assumptions about the source distribution are made. The experimental best-"t values of m2e from l tritium beta decay experiments are indeed negative [335], although this is most likely due to unresolved experimental issues. In addition, the values of Dm2e D from tritium beta decay experiments l are typically larger than the value required to "t the knee of the CR spectrum. This scenario also predicts a neutron line around the knee energy [336].
5. Origin of UHECR: acceleration mechanisms and sources As mentioned in Section 3.3, the "rst-order Fermi acceleration in the form of DSAM when applied to shocks in supernova remnants can accelerate particles to energies perhaps up to &1017 eV (see, e.g., Ref. [16]), but probably not much beyond. Thus, SNRs are unlikely to be the sources of the UHECR above &1017 eV. Instead, for UHECR, one has to invoke shocks on larger scales, namely extragalactic shocks. Several papers have, therefore, focussed on extragalactic objects such as AGNs and radio-galaxies as possible sites of UHECR acceleration. Reviews on this topic can be found, e.g., in Refs. [12}20]. Below, we brie#y discuss the issue of maximum achievable energy within DSAM and then discuss the viability or otherwise of the extragalactic sources that have been proposed as acceleration sites for CR up to the highest energies of Z1020 eV. We also brie#y mention acceleration of UHECR in pulsars. 5.1. Maximum achievable energy within diwusive shock acceleration mechanism There is a large body of literature on the subject of DSAM. We urge the reader to consult the reviews in Refs. [130}132,15,19,20] for details and original references. In the simplest version of DSAM, one adopts a so-called test-particle approximation in which the shock structure is given a priori and is not a!ected by the particles being accelerated. One also assumes a non-relativistically moving plane-parallel shock front. The magnetic "eld is assumed parallel to the shock normal. The inhomogeneities of the magnetic "eld are assumed to scatter particles e$ciently so as to result in a nearly isotropic distribution of the particles. Under these circumstances, one gets (see, for example, Ref. [36], for a text-book derivation) a universal power-law energy spectrum of the accelerated particles, N(E)JE~q, with index q"(r#2)/(r!1), where r"u /u is the shock 1 2 compression ratio, u and u being the upstream- and downstream velocities of the #uid in the 1 2 rest-frame of the shock. The shock compression ratio r is related to the adiabatic index of the #uid. For typical astrophysical situations, one gets r(4 and hence q'2. There are several issues that complicate this simple picture. Among these are issues associated with (a) the e!ects of a more realistic shock geometry, (b) back-reaction of the accelerated particles on the shock structure and its e!ect on the resulting particle spectrum, (c) ultra-relativistic shocks (which may be relevant for acceleration of particles in GRBs, for example), (d) the question of generation of the magnetic #uctuations which are necessary for scattering of particles and which determine the mean-free path of particles and hence the relevant di!usion coe$cients which, in
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turn, e!ect the spectral shape of the accelerated particles, and so on. For example, it has been recently claimed [337] that for strong shocks inclusion of back-reaction e!ects can result in a signi"cantly harder spectrum with q"1.5 compared to the canonical spectrum with q'2. On the other hand, for ultrarelativistic shocks in the limit CPR (C being the Lorentz factor of the shock), the spectrum becomes softer (qK2.2) than the canonical q"2 spectrum [338]. Furthermore, it has been claimed recently that Fermi-type shock acceleration by relativistic blast waves leads to an energy gain of a factor KC2 in the "rst shock crossing cycle, but only by a factor K2 in following cycles because particles do not have time to re-isotropize upstream before the next cycle [339]. For a recent review of particle acceleration at relativistic shocks see Ref. [20]. The above issues are, however, subjects of considerable on-going debates and discussions. Here we will not go into the subtleties associated with these issues. Instead, we focus directly on the question of the maximum achievable energy, following the analysis of Ref. [12]. In relativistic shocks the cuto! energy E for the source spectrum of accelerated cosmic rays is, # in the test particle approximation, generally given by Ze BR, the product of the charge Ze of the cosmic-ray particle, the magnetic "eld B and the size R of the shock, multiplied by some factor of order unity [11,36,17,15,340,341]. However, for the highest energies the mean free path of the particle becomes comparable to the shock size R, which has to be properly taken into account in calculating E . # The acceleration of a particle of energy E in an astrophysical shock is governed by the equation dE/dt"E/¹ , (43) !## where ¹ is the energy-dependent acceleration time. For DSAM, the slope q of the resulting !## power-law energy spectrum, dN/dEJE~q, of the particles is related to ¹ and ¹ , the mean !## %4# (in general also energy dependent) escape time by [36,340] q"1#(¹
/¹ ) . (44) !## %4# For "rst-order Fermi acceleration at nonrelativistic shocks, ¹ is usually given by !## ¹ "[3/(u !u )](D /u #D /u ) , (45) !## 1 2 1 1 2 2 where u , u are the up- and downstream velocities of material #owing through the shock in its rest 1 2 frame, and D and D are the corresponding di!usion coe$cients, respectively. Di!usion is 1 2 dominated by magnetic pitch angle scattering caused by inhomogeneities in the magnetic "eld [131]. Therefore, the mean free path j is bounded from below by the gyroradius r "E/(Ze B) ' multiplied by some factor g, and so D and D (for ultra-relativistic particles) can be estimated as 1 2 D , D &j/3 Z gE/3Ze B . (46) 1 2 For non-relativistic shocks, g is usually set equal to 1 [36,340]. However, as we deal with the UHECR, we have to consider relativistic shocks because they provide the most powerful accelerators. Monte Carlo simulations of such relativistic shocks show that g can be as large as K40 [342], leading to an e!ective slowing-down of acceleration compared to a naive extrapolation of Eq. (45) (with gK1) to the relativistic shock case. This is, however, partially compensated for by an additional factor which reaches a value of about 10 in highly inclined, and by about a factor of 13.5 in parallel [342], relativistic shocks, respectively, in the limit u P1. There is, however, some 1 disagreement on this issue [343].
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Putting everything together and minimizing ¹ from Eq. (45) as a function of u and u in the !## 1 2 interval [0,1] we arrive at ¹ Z(g/2.25)(E/Ze B) . (47) !## On the other hand, as long as the di!usion approximation is valid, i.e., as long as j(R corresponding to E(E ,Ze BR/g, the escape time is given by ¹ "R2/j, whereas for $*&& %4# E5E the particles are, to a good approximation, freely streaming out of the shock region and $*&& ¹ "R. Using Eqs. (44) and (47), we thus get %4# q(E'E )&1#E/2.25E . (48) $*&& $*&& De"ning the cuto! energy E as the energy where the source spectral index becomes 3 (recall that # the slope of the CR spectrum observed at the earth is around 2.7 in the UHE region) yields E ,E &1017 eV Z(R/kpc)(B/kG) . (49) # q/3 We have assumed here that the magnetic "eld is parallel to the shock normal. If that is not the case there will be an electric "eld E"u ]B in the shock rest frame. Together with di!usion e!ects this 1 causes a drift acceleration of charged particles along the shock front [344] to a maximal energy E which, for magnetic "eld B substantially inclined to the shock normal, is given by .!9 E "Ze u BR&1018 eV Zu (R/kpc)(B/kG) . (50) .!9 1 1 This is about one order of magnitude larger than Eq. (49) if u approaches the speed of light. 1 However, the electric "eld E is expected to be much smaller in general due to plasma e!ects so that rather special conditions have to be ful"lled in order that such high energies can be approached. We shall, therefore, take the estimate in Eq. (49) as a `benchmarka estimate for the maximum achievable energy within DSAM. 5.2. Source candidates for UHECR Irrespective of the precise acceleration mechanism, there is a simple dimensional argument, given by Hillas [11], which allows one to restrict attention to only a few classes of astrophysical objects as possible sources capable of accelerating particles to a given energy. In any statistical acceleration mechanism, there must be a magnetic "eld (B) to keep the particles con"ned within the acceleration site. Thus, the size R of the acceleration region must be larger than the diameter of the orbit of the particle &2r . Including the e!ect of the characteristic velocity bc of the magnetic scattering ' centers one gets the general condition [11] (B/kG)(R/kpc)'2(E/1018 eV)1/Zb .
(51)
As argued by Hillas [11], the above condition also applies to direct acceleration scenarios (as may operate for example in the pulsar magnetosphere), in which the electric "eld arises due to a moving magnetic "eld. The dimensional argument expressed by Eq. (51) is often presented in the form of the famous `Hillas diagrama shown in Fig. 25, which shows that to achieve a given maximum energy, one must have acceleration sites that have either a large magnetic "eld or a large size of the acceleration region. Thus, for example, only a few astrophysical sources } among them, AGNs, radio-galaxies, and pulsars } satisfy the conditions necessary (but may or may not be
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Fig. 25. The Hillas diagram showing size and magnetic "eld strengths of possible sites of particle acceleration. Objects below the corresponding diagonal lines cannot accelerate protons (iron nuclei) to 1020 eV. bc is the charateristic velocity of the magnetic scattering centers. (This version courtesy Murat Boratav.)
su$cient) for acceleration up to &1020 eV. Currently, therefore, most discussions of astrophysical acceleration mechanisms for EHECR have focussed on these objects. Below we brie#y summarize the status of these objects as possible acceleration sites for CR up to EHECR energies. 5.2.1. AGNs and radio-galaxies AGNs and radio-galaxies could be the main contributors to extragalactic CR. Several arguments support this possibility: "rst, at least two BL Lacertae objects, a certain class of AGNs, have been observed in c-rays above K10 TeV, namely Markarian 421 [328] and Markarian 501 [329]. Photons of such high energies may be produced by the decay of pions produced in interactions of the accelerated protons with the ambient matter in these sources (see, e.g., Ref. [345]) rather than by inverse Compton scattering of low-energy photons by accelerated electrons (e.g., [346]) in these
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sources. Second, it has been pointed out, that the energy content in the di!use c-ray background measured by EGRET is comparable to the one required for an extragalactic proton injection spectrum proportional to E~2 up to 1020 eV if it is to explain the observed UHECR spectrum above the ankle at 1018.5 eV [164]. This is expected if the di!use photons again result from the decay of pions produced by the accelerated protons and the subsequent propagation (cascading) of those photons. On the other hand, the fast variability of #ares recently observed may favor the acceleration of electrons as an explanation of the highest energy photons observed by ground Cherenkov telescopes [347], which has triggered reconsideration of theoretical #ux predictions in these models [348,349]. More generally, both proton and electron acceleration could provide energy-dependent contributions to the c-ray #ux. To settle this question will require more data from the optical up to the TeV range, whose current status is reviewed in Ref. [350]. Recently, a #are with hour-scale variability was observed simultaneously in X-rays and very high-energy c-rays from Markarian 421 [351]; the implications of this observation for the emission mechanism(s) of the radiation in the di!erent wavebands are, however, not clear at this stage. The physics of AGNs and radio-galaxies in the context of the possibility of these objects being the sources of UHECR have been reviewed extensively in recent literature; see, e.g., Refs. [13}17] for original references. Estimates of the typical values of R and B for the central regions of AGNs give [215] R&0.02 pc and B&5 G. These values when substituted into Eq. (49) above yield E &1019 eV for protons. This number can be uncertain perhaps by a factor of few. So, although # AGNs are unlikely to be the sources of the EHECR above 1020 eV, a good part of the UHECR below &few]1019 eV could in principle originate from AGNs. However, the major problem here is that the accelerated protons are severely degraded due to photo-pion production on the intense radiation "eld in and around the central engine of the AGN. In addition, there are energy losses due to synchrotron and Compton processes. Taking into account the energy losses simultaneously with the energy gain due to acceleration, Norman et al. [13] conclude that neither protons nor heavy nuclei are likely to escape from the central regions of AGNs with energies much above &1016 eV. There is also a suggestion (see, e.g., Szabo and Protheroe in Ref. [213]) that the photo-pion production process could convert protons into neutrons, which could then escape from the central region of the AGN, and the neutron could later decay to protons through b-decay. However, neutrons themselves are also subject to photo-pion production in the dense radiation "eld, and so neutrons above &1016 eV also cannot escape from the central regions of AGNs [13]. One can thus conclude that the central regions of AGNs are unlikely to be the sources of the observed UHECR.4 Currently, perhaps the most promising acceleration sites for UHECR are the so-called hot-spots of Fanaro!}Riley type II radio-galaxies; for reviews and references see Refs. [15,17,13]. The issue of maximum energy achievable in this case has also been reviewed by Norman et al. [13]. The energy
4 Although UHE nucleons cannot escape from AGN cores, the associated UHE neutrinos from the decay of the photo-produced pions can. The integrated contribution from all AGNs may then produce a di!use high-energy neutrino background that may be detectable [213]. Clearly, from the discussion above, this potential contribution to the UHE neutrino background would, however, be unrelated to the sources of the observed UHECR, and would also not be subject to the Waxman}Bahcall bound which does not apply to `hiddena sources [218,352] (see below).
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loss due to photo-pion production at the source is not signi"cant at hot spots because the density of the ambient soft photons at the hot-spots is thought to be not high enough. Depending on the magnetic "eld strength at the hot-spots (which is crucial but happens to be the most uncertain parameter in this consideration), a maximum energy of even up to &1021 eV seems to be possible. So these radio-galaxies could, in principle, be sources of UHECR including the EHECR above &1020 eV. However, the main problem with radio-galaxies as sources of the EHECR is their locations: the radio-galaxies that lie along the arrival directions of individual EHECR events are situated at large cosmological distances (Z100 Mpc) from Earth [26] (see also Section 4.6), in which case, because of the GZK e!ect discussed earlier, the particles do not survive with EHECR energies even if they are produced with such energies at the source. Thus, at the present time, although it seems hot-spots of radio-galaxies may well be the sources of UHECR above &1017 eV, it seems di$cult to invoke them as sources of the observed EHECR events above 1020 eV.5 The ultimate test for the case of AGNs and radio-galaxies as proton accelerators and for the origin of CR at least up to the GZK cuto! is expected to come from neutrino astronomy: practically no neutrinos are produced in the non-hadronic AGN models with electron acceleration. In contrast, if jets (as opposed to cores) of AGNs and radio-galaxies are the main sources of extragalactic CR, the secondary c-rays and neutrinos are created by pion production and the energy content in the di!use neutrino #ux can be normalized to the c-ray #ux produced by these AGNs [214,215]. It has recently been pointed out by Waxman and Bahcall [218] that a comparison with the UHECR #ux around 1019 eV leads to another bound on the di!use neutrino #ux that is more stringent by about two orders of magnitude, as long as accelerated protons are not absorbed in the source. This upper bound has become known as the Waxman}Bahcall upper bound. A subsequent more detailed numerical study by Mannheim, Protheroe, and Rachen pointed out possible loopholes to this bound and claim that it only applies to neutrino energies between &1016 and &1018 eV [353]. However, according to Bahcall and Waxman [352], the attempts presented in Ref. [353] to evade the bound on di!use neutrino #uxes from optically thin sources are in con#ict with observational evidence and the Waxman}Bahcall bound is robust. We recall, however, that the Waxman}Bahcall bound does not apply to sources that are optically thick for protons, such as could be the case for AGN cores (see above). Also, this bound does not directly apply to top-down scenarios because there neutrinos are produced as primaries, not secondaries, with #uxes that are considerably higher than the nucleon #uxes. As will be discussed in Section 7.4, in top-down scenarios the di!use neutrino #uxes are still bounded by the observed di!use GeV c-ray background. Recently, Boldt and Ghosh [27] have advanced the interesting suggestion that EHECR particles may be accelerated near the event horizons of spinning supermassive black holes associated with presently inactive quasar remnants. The required emf is generated by the black-hole-induced rotation of externally supplied magnetic "eld lines threading the horizon. This suggestion avoids
5 The distance problem with radio-galaxies may, however, be avoidable if the EHECR are a possible new kind of supersymmetric particle (S0) [245}247] which could be produced by accelerated protons through the photo-production process in the dense regions of some compact radio-galaxies. These S0 particles being electrically neutral would be able to escape from the source, and their speci"c particle physics properties may allow them to avoid the drastic loss process associated with the GZK e!ect; see Section 4.3.2 for a discussion.
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the problem of the dearth of currently active galactic nuclei, quasars and/or radio galaxies within acceptable distances ([50 Mpc) to serve as possible sources of EHECR events. Boldt and Ghosh estimate the number of supermassive black holes of required mass Z109M associated with _ `deada (no jet) quasars within a volume of radius &50 Mpc to be su$cient to explain the observed EHECR #ux. The exact process by which the rotational energy of the spinning supermassive black hole goes into accelerating protons to EHECR energies and the process by which the required magnetic "eld is generated and sustained remain to be spelled out, however. If the expectations of Ref. [27] are borne out by more detailed modeling, acceleration of protons to a maximum energy of &1021 eV would be possible, and in that case, dead quasars in our local cosmological neighborhood would indeed be one of the most promising sources of EHECR. 5.2.2. Pulsars As seen from the Hillas diagram, Fig. 25, pulsars are potential acceleration sites for UHECR. Most of the acceleration scenarios involving pulsars rely upon direct acceleration of particles in the strong electrostatic potential drop induced at the surface of the neutron star by unipolar induction due to the axially symmetric rotating magnetic "eld con"guration of the rotating neutron star. The maximum potential drop for typical pulsars can in principle be as high as &1021 eV (see, e.g., Ref. [115]). The component of a particle's momentum perpendicular to the local magnetic "eld line is damped out due to synchrotron radiation, and so the particles are forced to move along and are accelerated by the electric "eld component along the magnetic "eld lines. However, in any realistic model, the large potential drop along the magnetic "eld lines is signi"cantly short circuited by electrons and positrons moving in the opposite directions along the "eld lines } the source of the electron}positron pairs being the pair-cascade initiated by strong curvature radiation from seed electrons accelerated along the curved magnetic "eld lines. Acceleration models with pulsars have been reviewed, for example, in Refs. [35,15] and more recently in Ref. [115]. The general conclusion seems to be that, for isolated neutron stars (without companion), acceleration of particles to energies beyond &1015 eV is di$cult. Another class of acceleration models [354] utilize large electric "elds produced by unipolar induction in accretion disks around rotating neutron stars or black holes. An accretion disk threaded by a large-scale poloidal magnetic "eld produces a radial component of an electric "eld in the disk, which can be utilized for particle acceleration. However, energy loss through interaction of the accelerated particles in the ambient photon "eld around the central compact object prevents the maximum achievable energy from exceeding beyond about &1015 eV (see, e.g., the review by Takahara [15]). Several other accretion disk-based models are reviewed, for example, in Refs. [35,15]. None of these models is, however, capable of accelerating particles to UHECR energy regions. While the pulsar acceleration models mentioned above all deal with direct acceleration, there exists another class of models which attempt to utilize the statistical shock acceleration mechanism in accretion shocks around compact objects such as neutron stars or black holes; for a review, see, e.g., Ref. [15]. Again, it is di$cult to go past &1015 eV when energy loss processes are taken into account. Finally, we mention that recent discovery [116] of a `magnetara } a pulsar with a very high magnetic "eld } associated with the Soft Gamma Repeater SGR 1900#14 indicates the existence of a class of pulsars with dipole magnetic "elds approaching &1015 G. Obviously, for pulsars with
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such high magnetic "elds, the energy budget available for particle acceleration can be 2}3 orders of magnitude larger than the canonical estimates based on pulsar magnetic "eld of &1012 G, although it is not clear if the energy loss processes } a generic problem with acceleration around compact objects } can be gotten around. Recently, it has been suggested [355] that iron ions from the surfaces of newly formed strongly magnetized pulsars may be accelerated through relativistic MHD winds. It is claimed that pulsars whose initial spin periods are shorter than &4(B /1013 G)1@2 ms, where B is the surface magnetic 4 4 "eld, can accelerate iron ions to greater than &1020 eV. These ions can pass through the remnant of the supernova explosion that produced the pulsar without su!ering signi"cant spallation reactions. Clearly, in this scenario, the composition of EHECR is predicted to be dominantly iron nuclei (the relativistic wind may also accelerate some lighter nuclei though). These predictions will be testable in the up-coming experiments. 5.2.3. Other candidate sources A variety of other candidate UHECR acceleration sites have been studied in literature. Among these are acceleration in Galactic wind termination shocks [356], in shocks created by colliding galaxies [357], in large-scale shocks resulting from structure formation in the Universe [13], such as shocks associated with acceretion #ow onto galaxy clusters and cluster mergers [358,359], and so on. While for some of these sites E can reach the UHE region (depending on the magnetic .!9 "eld strength, which is highly uncertain), it is generally di$cult to stretch E beyond 1020 eV. .!9 5.3. A possible link between gamma-ray bursts and sources of E'1020 eV events? Cosmological GRBs most likely contribute a negligible fraction to the low-energy CR #ux around 100 GeV [360], as compared to SNRs, the favorite CR source below the knee. In contrast, a possible common origin of UHECR and cosmological GRBs was pointed out in Refs. [361,362], mainly based on the observation that the average rate of energy emission required to explain the observed UHECR #ux is comparable to the average rate of energy emitted by GRBs in c-rays. In addition, the predicted spectrum seems to be consistent with the observed spectrum above K1019 eV for proton injection spectra JE~2.3B0.5 [288], typical for the Fermi acceleration mechanism which is supposed to operate in dissipative wind models of GRBs. Because the rate of cosmological GRBs within the cone observed by the experiments out to the maximal range of EHECR beyond the GZK cuto! (K50 Mpc, see Section 4.1) is only about one per century, the likelihood of observing such UHECR from GRBs within the few years over which these UHECR experiments have been active is very small, unless cosmic magnetic "elds lead to time delays of at least 100 years. The cosmological GRB scenario for UHECR therefore necessarily implies a lower limit on magnetic "elds which in case of a large scale "eld characterized by an r.m.s. strength B and a coherence scale l [see Section 4.4, Eq. (32)] reads # BZ10~10(E/1020 eV)(d/30 Mpc)~1(l /1 Mpc)~1@2 G . (52) # This is an important observational test of this scenario. In particular, the observation of N arrival directions that are di!erent within the typical de#ection angle given by Eq. (31), strengthens the bound in Eq. (52) by a factor N1@2. The recently observed isotropy of arrival directions up to the highest energies [8] may in that respect already pose a problem to this scenario if one takes into
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account observational limits on the large-scale EGMF [363]. In addition, the energetic requirements have also become more severe due to recent GRB observations that indicate a larger distance scale than assumed at the time when the UHECR-GRB connection was proposed [363]. In the dissipative wind model of GRBs a plasma of photons, electrons, positrons, with a small load of baryons, is accelerated to ultrarelativistic Lorentz factors c<1, and at some dissipation radius r , a substantial part of the kinetic energy is converted to internal energy by internal shocks $ [364]. At this point the plasma is optically thin and part of the internal energy is radiated in the form of the c-rays that give rise to the GRB. In addition, the highly relativistic random motion in the wind rest frame resulting from dissipation is expected to build up magnetic "elds close to equipartition with the plasma, which in turn give rise to second order Fermi acceleration of charged particles. In the following, we brie#y review the conditions derived in Ref. [361] on the wind parameters that are required to accelerate protons to energies beyond 100 EeV. The most crucial constraint arising from acceleration itself comes from the requirement that the acceleration time in the wind rest frame, t Kr , should be smaller than the comoving expansion ! ' time t Kr /c. In terms of the comoving magnetic "eld strength B, this condition reads $ $ BZE/er K3]104(E/1020 eV)(r /1013 cm)~1 G , (53) ' $ where E is the proton energy in the observer frame, and we have used r KE/(ceB). ' The time scale for pion production losses on the c-rays in the plasma is given by l K10/(n p ), E,c c p where p K10~28 cm2 is the asymptotic high-energy pion production cross section (see Section p 4.1), the factor 10 takes into account the inelasticity of K0.1, and the comoving c-ray density n can be expressed in terms of the c-ray luminosity ¸ and typical energy e in the observer frame, c c c n K¸ /(4pr2ce ). The condition l 't then leads to an additional lower limit on B, c c $ c E,c ! B Z 20(¸ /1051 erg s~1)(r /1013 cm)~2(c/300)~2 G . (54) c $ Finally, the condition that the synchrotron loss length l ,E/(dE/dt) be larger than t , E,4:/ 4:/ ! where (dE/dt) is given by Eq. (28), leads to an upper limit on B, 4:/ B [ 3]105(c/300)2(E/1020 eV)~2 . (55) The three conditions (53)}(55) can be satis"ed simultaneously, provided r Z1012(c/300)~2(E/1020 eV)3 cm . (56) $ These values have to be set in relation to the time scale t of a GRB, via r [c2t . Therefore, GRB $ GRB eventually all conditions can be satis"ed, provided cZ40(E/1020 eV)3@4(t /s)~1@4, which GRB are reasonable values within most of the dissipative wind models. By rewriting Eq. (53) in terms of the equipartition "eld B in the comoving frame, (B/B )2Z0.15(c/300)2(E/1020 eV)2] %1 %1 (¸/1051 erg s~1)~1, where ¸ is the total luminosity, it is obvious that for reasonable wind luminosities and Lorentz factors, the magnetic "eld is not far from equipartition. The main proton energy losses in the GRB scenario summarized above are synchrotron radiation and pion production. Both processes give rise to secondaries, photons in the "rst, and photons and neutrinos from pion decay in the second case, and the resulting #uxes of these secondaries have subsequently been estimated in the literature [365}369]. Refs. [365,370] computed the neutrino #ux around 1014 eV correlated with GRBs in the dissipative wind model and showed that several tens of events should be observed with a km scale
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neutrino observatory, with a large #uctuation of the number of detected events from burst to burst [371]. This neutrino #ux has been proposed to be used as an extremely long neutrino baseline to test neutrino properties such as #avor mixing (see also discussion in Section 4.3.1) and the limits of validity of the relativity principles (see also discussion in Section 4.8). Ref. [366] investigated the neutrino #ux from the same process at energies above 1019 eV and found it to be detectable by AIRWATCH-type experiments [88] such as the MASS [89]. A correlation of a fraction of all UHE neutrinos with GRBs would thus be a strong test of the GRB scenario of UHECR origin. An experimental upper limit of 0.87]10~9 cm~2 upward going neutrino-induced muons per average GRB has been set by the MACRO detector [372]. The question of the maximal possible neutrino energies from GRBs and also blazars was reconsidered recently, resulting in typical numbers of &1019 eV: Ref. [373] gives a detailed account of loss processes and Ref. [374] focuses on neutrino production associated with external shocks in GRB "reball models in the afterglow phase. In addition, there would be a background of UHE neutrinos from the interaction of UHECR with the CMB outside of the GRB which was found to be detectable as well [156,366]. The distribution of this background was argued to be an indicator of the distribution of the source population of UHECR which could be used to distinguish between the major theoretical scenarios. Similarly to the case of AGN and radio-galaxy sources, the neutrino #ux level from GRBs is limited by the di!use GeV c-ray background (see Section 5.2.1). The synchrotron emission associated with proton acceleration to UHE in the cosmological GRB scenario has been found to carry away a fraction of about a percent of the total burst energy. At energies around 10 MeV this signal should be detectable with the proposed GLAST satellite experiment, while above a few hundred GeV, ground-based air Cherenkov telescopes should be sensitive enough to detect this #ux [367,368]. Ref. [369] even claims that synchrotron emission should give rise to afterglows that extend into the TeV range. These c-ray #uxes above &10 MeV would thus provide another strong signature of proton acceleration up to UHE in GRBs that should be testable in the near future. This signature may already have been observed in the 10}20 TeV range [375] (see also Ref. [376]), and the resulting cascade c-ray #ux in the GeV range has been pointed out as a possible explanation of the di!use c-ray #ux observed at these energies [377,378]. This would imply the phenomenal total liberated energy of &1056 erg per GRB (assuming isotropic radiation) and the lower limit cZ500 on the Lorentz factor, which is consistent with the "reball model outlined above. Recently, it was claimed that an explanation of the observed UHECR spectrum in the context of acceleration in GRBs requires speci"c GRB progenitors such as binary pulsars [339]. Emission of high-energy c-rays and neutrinos in GRBs associated with supernova explosions in massive binary systems, whose existence was recently suggested by observations, has been discussed in Ref. [379].
6. Non-acceleration origin of cosmic rays above 1020 eV 6.1. The basic idea As discussed in the Section 5, the shock acceleration mechanism is a self-limiting process: for any given set of values of dimension of the acceleration region ("xed by, say, the radius R of the shock) and the magnetic "eld strength (B), simple criterion of Larmor containment of a particle of charge
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Ze within the acceleration region implies that there is a maximum energy E &ZeBR up to .!9 which the particle can be accelerated before it escapes from the acceleration region, thus preventing further acceleration. The observed EHECR events above 1011 GeV, therefore, pose a serious challenge for any acceleration mechanism because a value of E Z1011 GeV can barely be .!9 achieved in even the most powerful astrophysical objects for reasonable values of R and B associated with these objects [11}13]. The problem becomes more acute when one recognizes that the actual energy at the source has to be signi"cantly larger than the observed energy of the particles because of energy loss during propagation as well as in the immediate vicinity of the source. In addition, there is the problem of absence of any obviously identi"able sources for the observed EHECR events, as discussed in Section 4.6. Because of these di$culties, there is currently much interest in the possibility that the EHECR events may represent a fundamentally di!erent component of cosmic rays in the sense that these particles may not be produced by any acceleration mechanisms at all; instead, these particles may simply be the result of decay of certain massive particles (generically, `Xa particles) with mass m '1011 GeV originating from high-energy processes in the early Universe. As we shall discuss X below, such non-acceleration or `top-downa decay mechanism (as opposed to conventional `bottom-upa acceleration mechanism) of production of extremely energetic particles in the Universe today are possible and may indeed be naturally realized within the context of uni"ed theories of elementary particle interactions in the early Universe. The basic idea of a top-down origin of cosmic rays can be traced back to Georges Lemam( tre [380] and his theory of `Primeval Atoma, the precursor to the Big Bang model of the expanding Universe. The entire material content of the Universe and its expansion, according to Lemam( tre, originated from the `super-radioactive disintegrationa of a single atom of extremely large atomic weight, the Primeval Atom, which successively decayed to `atomsa of smaller and smaller atomic weights. The cosmic rays were envisaged as the energetic particles produced in intermediate stages of decay of the Primeval Atom } they were thus `glimpses of the primeval "reworksa [380]. Indeed, Lemam( tre regarded cosmic rays as the main evidential relics of the Primeval Atom in the present Universe. Of course, we now know that the bulk of the observed cosmic rays are of recent (post-galactic) origin, and not cosmological. In particular, as far as the EHECR are concerned, we now know that the existence of CMBR, which was unknown in Lemam( tre's time, precludes the origin of the observed EHECR particles in very early cosmological epoch (or equivalently at very large cosmological distances) because of the GZK e!ect discussed in Section 4. Nevertheless, it is interesting that one of the earliest scenarios considered for the origin of cosmic rays was a cosmological top-down, non-acceleration scenario, rather than a bottom-up acceleration scenario. In the modern version of the top-down scenario of cosmic-ray origin, the X particles (the possible sources of which we shall discuss below) typically decay to quarks and leptons. The quarks hadronize, i.e., produce jets of hadrons containing mainly light mesons (pions) with a small percentage of baryons (mainly nucleons). The pions decay to photons, neutrinos (and antineutrinos) and electrons (and positrons). Thus, energetic photons, neutrinos and charged leptons, together with a small fraction of nucleons, are produced directly with energies up to &m without X any acceleration mechanism. In order for the decay products of the X particles to be observed as EHECR particles today, three basic conditions must be satis"ed: (a) The X particles must decay in recent cosmological
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epoch, or equivalently at non-cosmological distances ([100 Mpc) from Earth } otherwise the decay products of the X particles lose all energy by interacting with the background radiation, and hence do not survive as EHECR particles. A possible exception is the case where neutrinos of su$ciently high energy originating from X particle decay at large cosmological distances <100 Mpc give rise to EHE nucleons and/or photons within 100 Mpc from Earth through decay of Z bosons resonantly produced through interaction of the original EHE neutrino with the thermal relic background (anti)neutrinos, if neutrinos have a small mass of order &eV; see Section 4.3.1, (b) the X particles must be su$ciently massive with mass m <1011 GeV, and (c) X the number density and rate of decay of the X particles must be large enough to produce detectable #ux of EHECR particles. In the present section, we "rst discuss the nature of the expected production spectra of observable particles (nucleons, photons, neutrinos) resulting from the decay of X particles in general. We then discuss in detail a particular realization of the top-down scenario in which the X particles are the supermassive gauge bosons, Higgs bosons and/or superheavy fermions produced from cosmic topological defects (TDs) like cosmic strings, magnetic monopoles, superconducting cosmic strings etc., which could be formed in symmetry-breaking phase transitions associated with Grand Uni"ed Theories (GUTs) in the early Universe. We then discuss the possibility that the X particles could be some long-lived metastable supermassive relic particles of non-thermal origin produced, for example, through vacuum #uctuations during a possible in#ationary phase in the early Universe. It has been suggested that such metastable supermassive long-lived particles could constitute (a part of) the dark matter in the Universe, and a fraction of these particles decaying in the recent epochs may give rise to the EHECR. 6.2. From X particles to observable particles: hadron spectra in quarkPhadron fragmentation The precise decay modes of the X particles would depend on the speci"c particle physics model under consideration. In the discussions below we shall assume that X particles decay typically into quarks and leptons, irrespective of the sources of the X particles.6 By far the largest number of `observablea particles (nucleons, photons, neutrinos) resulting from the decay of the X particles are expected to come through the hadronic channel, i.e., through production of jets of hadrons by the quarks. The process of `fragmentationa of the quarks into hadrons is described by QCD. The spectra of various particles at production are, therefore, essentially determined by QCD, and not by any astrophysical processes. The actual decay mechanism of the X particles into quarks and leptons, and the multiplicities and the spectra of these quarks and leptons may depend upon the origin and nature of the X particles themselves. Nevertheless, the spectra of hadrons in the jets created by individual quarks should be reasonably independent of the origin of the quarks themselves. We, therefore, "rst discuss the expected spectra of hadrons (and the resulting spectra of nucleons, photons and neutrino } the latter two from decay of pions) in individual jets created by individual quarks. 6 In some supersymmetric models, the decay products of the X particles may also include squarks and/or sleptons. However, for m much above the typical supersymmetry breaking scale of order TeV, the supersymmetric and X non-supersymmetric particles would behave essentially similarly.
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The hadron spectra under discussion should be very similar to those measured for e`e~Pq6 qPhadrons process in colliders. The actual process by which a single high-energy quark gives rise to a jet of hadrons is not understood fully yet } it involves the well-known (and unsolved) `con"nementa problem of QCD. However, various semi-phenomenological approaches have been developed which describe the fragmentation spectra that are in good agreement with the currently available experimental data on inclusive hadron spectra in quark/gluon jets in a variety of high-energy processes. In these approaches, the process of production of a jet containing a large number of hadrons by a single high-energy quark (or gluon) is `factorizeda into three stages. The "rst stage involves `harda processes involving large momentum transfers, whereby the initial high-energy quark emits `bremsstrahlunga gluons which in turn create more quarks and gluons through various QCD processes (q6 q pair production by gluons, gluon bremsstrahlung by the produced quarks, gluon emission by gluons, and so on). These hard processes are well described by perturbative QCD. Thus a single high-energy quark gives rise to a `parton cascadea } a shower of quarks and gluons } which, due to QCD coherence e!ects, is con"ned in a narrow cone or jet whose axis lies along the direction of propagation of the original quark. In the semi-phenomenological approaches to the jet fragmentation process, the "rst stage of the process, i.e., the parton cascade development, described by perturbative QCD, is terminated at a cut-o! value, Sk2 T1@2 &1 GeV, of the typical transverse momentum. Thereafter, the second M #65v0&& stage involving the non-perturbative con"nement process is allowed to take over, binding the quarks and gluons into `primarya color neutral objects. In the third stage, the unstable primary `hadronsa decay into known hadrons. The second and the third stages are usually described by the available phenomenological hadronization models such as the LUND string fragmentation model [381] or the cluster fragmentation model [382]. Detailed Monte Carlo numerical codes now exist [382}384] which incorporate the three-stage process outlined above. These codes provide a reasonably good description of a variety of relevant experimental data. 6.2.1. Local Parton}Hadron Duality There is an alternative approach that is essentially analytical and has proved very fruitful in terms of its ability to describe the gross features of hadronic jet systems, such as the inclusive spectra of particles, the particle multiplicities and their correlations, etc., reasonably well. This approach is based on the concept of `Local Parton}Hadron Dualitya (LPHD) [385]. Basically, in this approach, the second stage involving the non-perturbative hadronization process mentioned above is ignored, and the primary hadron spectrum is taken to be the same, up to an overall normalization constant, as the spectrum of partons (i.e., quarks and gluons) in the parton cascade after evolving the latter all the way down to a cut-o! transverse momentum Sk2 T1@2 &R~1& M #65v0&& few hundred MeV, where R is a typical hadronic size. A rigorous `proof a of LPHD at a fundamental theoretical level is not yet available. However, the fact that LPHD gives a remarkably good description of the experimental data including recent experimental results from LEP, HERA and TEVATRON [386] gives strong indications of the general correctness of the LPHD hypothesis in some average sense. The fundamental basis of LPHD is that the actual hadronization process, i.e., the conversion of the quarks and gluons in the parton cascade into color neutral hadrons, occurs at a low virtuality scale of order of a typical hadron mass independently of the energy of the cascade initiating
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primary quark, and involves only low momentum transfers and local color re-arrangement which do not drastically alter the form of the momentum spectrum of the particles in the parton cascade already determined by the `harda (i.e., large momentum transfer) perturbative QCD processes. Thus, the non-perturbative hadronization e!ects are lumped together in an `unimportanta overall normalization constant which can be determined phenomenologically. A good quantitative description of the perturbative QCD stage of the parton cascade evolution is provided by the so-called Modi"ed Leading Logarithmic Approximation (MLLA) [387] of QCD, which allows the parton energy spectrum (which is a solution of the so-called DGLAP evolution equations) to be expressed analytically in terms of functions depending on two free parameters, namely, the e!ective QCD scale K (which "xes the e!ective running QCD coupling %&& strength a%&&) and the transverse momentum cut-o! QI . For the case QI "K , the analytical result 4 0 0 %&& simpli"es considerably, and one gets what is referred to as the `limiting spectruma [385,386] for the energy distribution of the partons in the cascade, which has the following form:
P
C
D D B
dN 4C n@2 dl cosh a#(2m/>!1)sinh a B@2 x 1!35 " F C(B) e~Ba dx b p (4N /b)>(a/sinh a) # ~n@2 a 16N 1@2 #> ]I [cosh a#(2(m/>)!1)sinh a] . B sinh a b
AC
(57)
Here dN is the number of partons carrying a fraction between x and x#dx of the energy 1!35 E "E of the original jet-initiating quark q, m"ln(1/x), >"ln(E /K ), +%5 q +%5 %&& a"[tanh~1(1!2m/>)#il], I is the modi"ed Bessel function of order B, where B"a/b with B a"[11N /3#2n /(3N2)] and b"[(11N !2n )/3], n being the number of #avors of quarks and # & # # & & N "3 the number of colors, and C "(N2!1)/2N "4. # F # # 3 Eq. (57) gives us the spectrum of the partons in the jet. By LPHD hypothesis, the hadronic fragmentation function (FF), i.e., the hadron spectrum, dN /dx (with x"E /E , E being the ) ) +%5 ) energy of a hadron in the jet), due to hadronization of a quark q, is given by the same form as in Eq. (57), except for an overall normalization constant K(>) that takes account of the e!ect of conversion of partons into hadrons: x dN /dx"K(>)x dN /dx (58) ) 1!35 with now x"E /E on both sides of the equation. ) +%5 Phenomenologically, for given values of K and E , the normalization constant can be %&& +%5 determined simply from overall energy conservation, i.e., from the condition :1 x[dN (>, x)/ ) 0 dx] dx"1. The value of K is not known a priori, but a "t to the inclusive charged particle %&& spectrum in e`e~ collisions at center-of-mass energy E "2E &90 GeV (Z-resonance) gives #. +%5 K#) &250 MeV, while the value of K is found to be typically &1.3 at LEP energies. %&& An important characteristic of the MLLA spectrum (57) treated as a function of the variable m is the existence of a maximum at m given by .!9 m
">[1#JC/>!C/>] (59) .!9 2 with C"a2/(16bN ). The existence of this maximum is directly related to the suppression of soft # gluon multiplication in the cascade due to QCD color coherence e!ect. Recent analysis [388] of
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relevant LEP data have provided experimental con"rmation of the energy evolution of m as .!9 predicted by Eq. (59). For asymptotically high energies of interest, i.e., for E
3/(36N )]1@2. The full MLLA spectrum (57) can be approximated well by # a `distorted gaussiana [386] in terms of calculable higher moments of m. Note that, within the LPHD picture, there is no way of distinguishing between various di!erent species of hadrons. Phenomenologically, the experimental data can be "tted by using di!erent values of K for di!erent species of particles depending on their masses. For our consideration of %&& particles at EHECR energies, all particles under consideration will be extremely relativistic, and since, in our case, E &m /few
(62)
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At EHECR energies, for which x&E/m ;1 (since m can be as large as &1016 GeV and we are X X interested in EHECR with energies E&few]1011 GeV), spectrum (61) can be well approximated [397] by a power law, dN /dEJE~a with a&1.3. Spectrum (62) is also approximated by a power ) law with a&1.5. The MLLA spectrum (57) is less well approximated by a single power law, but can be approximated by two or more segmented power laws. It should be emphasized here that, in using the MLLA#LPHD hadron spectrum in the calculation of particle spectra in the top-down scenario, one should keep it in mind that there is a great deal of uncertainty involved in extrapolating the QCD (MLLA#LPHD) spectra } which have been tested so far only at relatively `lowa energies of &100 GeV } to the extremely high energies of our interest, namely, Z1014 GeV. For example, there could be thresholds associated with new physics beyond the standard model which may alter the spectra as well as content of the particles in the jets. One example of possible new physics is supersymmetry (SUSY). If SUSY `turns ona at an energy scale of say M &1 TeV, then the QCD cascade development process is expected to involve not SUSY only the usual quarks and gluons, but also their supersymmetric partners (squarks, gluinos) with equal probability as long as QI 2'M2 , where QI 2 is the `virtualitya (i.e., the 4-momentum SUSY transfer squared) involved in various sub-processes contributing to the cascade. The virtuality of the cascade particles steadily decreases as the cascade process progresses. Once QI 2 falls below M2 , the SUSY particles in the cascade would decouple from the cascade process and eventually SUSY decay into the stable lightest supersymmetric particles (LSPs). Thus, as pointed out by Berezinsky and Kachelrie{ [399], some fraction of the particles in the QCD cascade resulting from the decay of X particles may be in the form of high or even ultra-high-energy LSPs. Indeed, Berezinsky and Kachelrie{ [399] have claimed that LSPs may take away a signi"cant fraction (&40%) of the total energy of the jet, which must be properly taken into account in normalizing the spectra and calculating the yield of the `observablea hadrons (pions and nucleons). Inclusion of SUSY in the parton cascading process also changes the shape of the fragmentation spectrum [400]. The limiting QCD MLLA spectrum is still given by Eq. (57) with constant a replaced by a "11N /3 and SUSY # "9!n , and the maximum of the spectrum is given by Eq. (59) with b replaced by b SUSY & C replaced by C "a2 /(16b N ). Thus the maximum of the SUSY-QCD spectrum is SUSY SUSY SUSY # shifted to higher m (lower energy) relative to the non-SUSY MLLA QCD spectrum. The properties of LSPs are model dependent, but due to a variety of phenomenological reasons [399,401] the UHE LSPs themselves are unlikely to be the candidates for the observed EHECR events except possibly for the case of a neutral bound state of light gluino and uds hadron [245,246]. Supersymmetry is by no means the only possible kind of new physics beyond the standard model. Nevertheless, from the discussions above, we see that EHECR may, in fact, provide an interesting probing ground for search for new physics beyond the standard model. For some recent discussions of the e!ects of the SUSY versus non-SUSY QCD spectra on the "nal evolved particle spectra, see Ref. [206]. The various hadronic fragmentation spectra discussed above are displayed in Fig. 26 for comparison. More recently, attempts have been made [391] to calculate the injection spectra of nucleons, photons and neutrinos resulting from X particle decay by directly using numerical Monte Carlo event generators as incorporated in the HERWIG [382] and JETSET [383] programs. Results of
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Fig. 26. The fragmentation function for E "5]1015 GeV in MLLA approximation with SUSY (thick solid line +%5 peaking at 1012 GeV) and without SUSY (thin solid line) [see Eq. (57)] in comparison to the older expressions Eq. (61) (dashed line) and Eq. (62) (dotted line).
Ref. [391] indicate that although for m &103 GeV the Monte Carlo results agree with the X MLLA#LPHD predictions (which is not surprising since both MLLA#LPHD as well as Monte Carlo event generators are suitably parametrized to "t the existing collider data),7 signi"cant di!erences with the spectra predicted from MLLA#LPHD appear for m <103 GeV. In X particular, the spectra of photons and neutrinos seem to di!er signi"cantly from the nucleon spectrum at high x values (whereas in the MLLA#LPHD picture they are assumed to be roughly similar; see below). More importantly, nucleons seem to be almost as abundant as photons and neutrinos in certain ranges of x values (speci"cally, in the range 0.2 [ x [ 0.4), contrary to the general expectation that baryon production should be suppressed relative to meson (and hence to photon and neutrino) production at all x, independently of m .8 Unfortunately, due to the very X nature of these Monte Carlo calculations, it is di$cult to understand the precise physical reason for
7 The Monte Carlo calculations of Ref. [391] are done with standard, non-SUSY QCD. 8 In the string fragmentation scheme of hadronization as implemented in the LUND Monte Carlo program JETSET [383], for example, the yield of mesons (and hence the photons and neutrinos resulting from their decay) is always expected to dominate over baryons. This is because, meson formation involves breaking of a color #ux tube through nucleation of a quark}antiquark pair whereas baryon formation involves formation of a diquark}antidiquark pair, the probability for which is considerably suppressed compared to that for quark}antiquark pair formation. In the HERWIG program too, in which the hadronization scheme involves initial formation of clusters of partons which subsequently break up into color-neutral 3-quark states (baryons) and quark}antiquark states (mesons), the predicted baryon/meson ratio is generally always considerably less than unity, at least so at currently accessible accelerator energies (see, e.g., Ref. [402] for a review of baryon production in e`e~ annihilations).
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the unexpectedly high relative baryon yield for certain values of x in the case of large m . Clearly, X more work needs to be done on this important issue. An important point to note about the particle spectra displayed in Fig. 26 is that all these spectra are generally harder than the production spectra predicted in conventional shock acceleration theories, which, as discussed in the previous section, by and large predict power-law di!erential production spectra with a52. This fact has important consequences; it leads to the prediction of a pronounced `recoverya [200] of the evolved nucleon spectrum after the GZK `cut-o! a and the consequent #attening of the spectrum above &1011 GeV in the top-down scenario. Under certain circumstances, a relatively hard production spectrum may also naturally give rise to a `gapa in the measured EHECR spectrum [403]. The importance of a relatively hard production spectrum of EHECR from possible `fundamentala processes was "rst emphasized by Schramm and Hill [404]. For a power-law di!erential spectrum JE~a, the index a"2 is a natural dividing line between what can be characterized as `softa and `harda spectra: For soft spectra (a'2), the total particle multiplicity (JE1~a) as well as the total energy (JE2~a) are both dominated by the lower limits of the relevant integrals, which means that most of the energy is carried by the large number of low-energy particles. Such a spectrum is ine$cient in producing a signi"cant #ux of extremely high-energy particles. For hard spectra (1(a(2), on the other hand, although the total particle multiplicity is still dominated by very low-energy particles, the energy is mainly carried o! by a few extremely energetic particles. Thus, hard spectra such as the ones generically predicted within the top-down scenario involving QCD cascade mechanism discussed above, seem to be more `naturala from the point of view of producing EHECR than soft spectra generally predicted in shock acceleration scenarios. 6.2.2. Nucleon, photon and neutrino injection spectra With a given hadronic fragmentation function, dN /dx, we can obtain the nucleon, photon and ) neutrino injection spectra due to decay of all X particles at any time t as described below. We shall i assume that nucleons and pions are produced with the same spectrum; however, see Ref. [391] and the discussions above. Let n5 (t) denote the rate of decay of X particles per unit volume at any time t. Let us assume that X each X particle, on average, undergoes NI -body decay to N quarks (including antiquarks) and q Nl leptons (neutrinos and/or charged leptons), so that NI "N #Nl , and that the available energy q m is shared roughly equally by the NI primary decay products of the X. Then, the nucleon X injection spectrum, U (E , t ), from the decay of all X particles at any time t can be written as N i i i U (E , t )"n5 (t )N f (NI /m )dN /dx , (63) N * * X * q N X ) where E denotes the energy at injection, f is the nucleon fraction in the hadronic jet produced by * N a single quark, and x"NI E /m . * X The photon injection spectrum from the decay of the neutral pions (p0P2c) in the jets is given by (see, for example, Ref. [405])
P
mX @NI dE U 0 (E, t ) , n * E E* where U 0 (E, t )K1[(1!f )/f ]U (E, t ) is the neutral pion spectrum in the jet. n * 3 N N N * U (E , t )K2 c * *
(64)
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Similarly, the neutrino (l #l6 ) injection spectrum resulting from the charged pion decay k k [pBPkBl (l6 )] can be written, for E <m , as [219,200] k k * n mX @NI dE k (E , t )K2.34 Un?kl U ` ~ (E, t ) , (65) (lk `l6 k ) * * * E (n `n ) * 2.34E where U ` ~ K2U 0 . n (n `n ) The decay of each muon (from the decay of a charged pion) produces two more neutrinos and an electron (or positron): kBPeBl (l6 ) l6 (l ). Thus each charged pion eventually gives rise to three e e k k neutrinos: one l , one l6 and one l (or l6 ), all of roughly the same energy. So the total l #l6 k k e e k k injection spectrum will be roughly twice the spectrum given in Eq. (65), while the total l #l6 e e spectrum will be roughly same as that in Eq. (65). The full l and l spectra resulting from decay of k e pions and the subsequent decay of muons can be calculated in details following the procedure described in the book by Gaisser [36]. Note that, if the hadron spectrum in the jet is generally approximated by a power law in energy, then nucleon, photon and neutrino injection spectra will also have the same power-law form all with the same power-law index [405,219]. As mentioned earlier, it is generally expected that mesons (pions) should be the most numerous particles in the hadronic jets created by quarks coming from the decay of X particles. Thus, U 0 /U K1((1!f )/f )K10 and U ` ~ /U &20 for f &3% This means that, in terms of n N 3 N N (n `n ) N N number of particles at production, the decay products of the pions, i.e., photons and neutrinos, dominate over nucleons at least by factors of order 10. Since neutrinos su!er little attenuation and can come to us unattenuated from large cosmological distances (except for absorption due to fermion pair production through interaction with the cosmic thermal neutrino background, the path length for which is <100 Mpc; see Section 4.3), their #uxes are expected to be the largest among all particles at the highest energies. However, their detection probability is much lower compared to those for protons and photons.9 Photons also far outnumber nucleons at production. However, the propagation of extragalactic EHE photons is in#uenced by a number of uncertain factors such as the level of the URB and the strength of the EGMF (see Section 4.2). Depending on the level of the URB and EGMF and the distance of the X particle source, the photon #ux may dominate over the nucleon #ux and thus dominate the `observablea di!use particle #ux, at EHECR energies. Indeed, the prediction [178] of a possible large photon/nucleon ratio ('1) at su$ciently high EHECR energies is a distinguishing feature of the top-down scenario of origin of EHECR, and can be used as a signature for testing the scenario in forthcoming experiments. This has been discussed recently in more details in Refs. [406,206]. In this context, note that photons and neutrinos in the top-down scenario are primary particles in the sense that they are produced directly from the decay of the pions in the hadronic jets. In contrast, photons and neutrinos in conventional acceleration scenarios can be produced only through secondary processes } they are mainly produced by the decay of photo-produced pions resulting from the GZK interactions of primary EHECR nucleons with CMBR photons. Of course,
P
9 The EHE neutrinos of TD origin would, however, be potentially detectable by the proposed space-based detectors like OWL and AIRWATCH, and ground-based detectors like Auger, Telescope Array, and so on; see discussions in Section 7.4.
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these secondary neutrinos and photons would also be there in the top-down scenario, but their #uxes are sub-dominant to the primary ones. Once the injection spectra for nucleons, photons, and neutrinos are speci"ed, the evolution of these spectra and the "nal predicted #uxes of various particles can be obtained by considering various propagation e!ects discussed in detail in Section 4. The predicted particle #uxes in the top-down scenario are discussed later in Section 7, where we also discuss various signatures of the scenario in general and constraints from various observations. Obviously, while the shapes of the "nal particle spectra are determined by the injection spectra (which are "xed by QCD as explained above) and various propagation e!ects, the absolute magnitudes of the #uxes will be "xed by the source function n5 , the production and/or decay rate of X the X particles, in di!erent realizations of the non-acceleration scenario.10 Unfortunately, n5 is X highly model dependent and depends on free parameters of the particular top-down model under consideration. Because of this reason, it has not been possible to predict the absolute #ux levels in the top-down models with certainty; only certain plausible models have been identi"ed. Below, we shall discuss the expected values of n5 in some speci"c top-down models and examine their e$cacy X with regard to EHECR. But, "rst, in order to have some idea of the kind of numbers involved, we perform a simple (albeit crude) benchmark calculation of n5 required to obtain a signi"cant X contribution to the measured EHECR #ux. 6.2.3. X particle production/decay rate required to explain the observed EHECR yux: A benchmark calculation Since in top-down models photons are expected to dominate the `observablea EHECR #ux, let us assume for simplicity that the highest-energy events are due to photons. To be speci"c, let us assume a typical 2-body decay mode of the X into a quark and a lepton: XPql. The quark will produce a hadronic jet. The photons from the decay of neutral pions in the jet carry a total energy E K(1]0.9]1)m ( f /0.9)"0.15m ( f /0.9), where f is the fraction of the total energy of the c,T05!3 2 X n X n n jet carried by pions,11 and we have assumed that the quark and the lepton share the energy m equally. Assuming a power-law photon injection spectrum with index a, dN /dE JE~a, with X c c c 0(a(2, and normalizing with the total photon energy E , we get the photon injection c,T05!spectrum from the decay of a single X particle as dN /dE "(0.6/m )(2!a)( f /0.9)(2E /m )~a . (66) c c X n c X We can neglect cosmological evolution e!ects and take the present epoch values of the relevant quantities involved, since photons of EHECR energies have a cosmologically negligible path length of only few tens of Mpc for absorption through pair production on the universal radio background.
10 In the topological defect models discussed below, the X particles are generally assumed to have extremely short lifetimes, so they decay essentially instantaneously as soon as they are released from the defects. Therefore, in the topological defect models, n5 , for all practical purposes, refers to the production rate of X particles from the defects. In X contrast, in models in which X particles are metastable, long-lived particles (with lifetime Z age of the Universe) possibly produced during an in#ationary epoch in the early Universe, n5 refers to the decay rate of the X particles. X 11 We assume that the jet consists of only pions and nucleons, all with the same spectrum, and that all particles are ultrarelativistic. Thus, f is also the total pion fraction in terms of number of particles in the jet. n
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With these assumptions, and assuming that the (sources of the) X particles are distributed uniformly in the Universe, the photon #ux j (E ) at the observed energy E is simply given by c c c j (E )K(1/4p)l(E )n5 dN /dE , (67) c c c X c c where again l(E ) is the pair production absorption path length of a photon of energy E . c c Normalizing the above #ux to the measured EHECR #ux, we get
A
B A B A BA B
BA
B A B A BA B
BA
l(E ) ~1 E2j(E) c (n5 ) K1.2]10~46 X,0 EHECR 10 Mpc 1 eV cm~2 s~1 sr~1
A
m 1~a 0.5 X 1016 GeV 2!a
B
2E a~1.5 1016 GeV
0.9 cm~3 s~1 . (68) f n The subscript 0 stands for the present epoch. In terms of energy injection (Q), de"ned by n5 "Q/m , the above requirement reads X X l(E ) ~1 E2j(E) 2E a~1.5 c (Q ) K1.2]10~21 0 EHECR 10 Mpc 1 eV cm~2 s~1 sr~1 1016 GeV ]
A
A
m 2~a 0.5 X 1016 GeV 2!a
B
0.9 eV cm~3 s~1 . (69) f n For a "ducial value of the observed EHECR #ux given by E2j(E)K1 eV cm~2 s~1 sr~1 at the "ducial energy E"1011 GeV, and for m "1016 GeV, a"1.5, and f "0.9, the above estimates X n indicate that in order for a generic top-down mechanism to explain the measured EHECR #ux, the X particles must (be produced and/or) decay in the present epoch at a rate of &1]1035 Mpc~3 yr~1, or in more `down-to-eartha units, about &13AU~3 yr~1, i.e., about 10 X particles within every solar system-size volume per year over a volume of radius 10 Mpc. The above numbers are uncertain (most likely overestimate [206]) by perhaps as much as an order of magnitude or so, depending on the decay mode of the X particle, the fraction of energy m that goes into nucleons versus pions, the form of the hadronic fragmentation function that X determines the injection spectra of various particles, the absorption path length of EHECR photons, electromagnetic cascading e!ect (which is discussed in Section 4 and which we have neglected here), and so on. Nevertheless, we think the numbers derived above do serve as crude benchmark numbers. The above rough estimate assumes that the (sources of the) X particles are distributed uniformly in the Universe, so the #ux above refers to the di!use #ux. In principle, EHECR events could also be produced by isolated nearby bursting sources of X particles, in which case, depending on the distance of the source, the above energy and number density requirements would be di!erent. In addition, in the case of long-lived X particles of primordial origin, as also in the case of certain kinds of topological defects such as monopolonium (see below), the (sources of) X particles could be clustered in our Galactic halo [389,406,391] as well as in clusters of galaxies, in which cases the required values of n5 and Q will depend on the clustering factor and clustering X,0 0 length-scale. With the above rough estimates in mind, we now proceed to discuss possible sources of the X particles. ]
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6.3. Cosmic topological defects as sources of X particles: general considerations Cosmic topological defects (TDs) [407}409] } magnetic monopoles, cosmic strings, domain walls, superconducting cosmic strings, etc., as well as various hybrid systems consisting of these TDs } are predicted to form in the early Universe as a result of symmetry-breaking phase transitions envisaged in GUTs. Topological defects associated with symmetry-breaking phase transitions are well known in condensed matter systems; examples there include vortex lines in super#uid helium, magnetic #ux tubes in type-II superconductors, disclination lines and `hedgehogsa in nematic liquid crystals, and so on. Recent laboratory experiments (see [409] for references) on vortex-"lament formation in the super#uid transition of 3He (which occurs at a temperature of a few millikelvin) have provided striking con"rmation of the basic Kibble-Zurek [407}409] picture of TD formation in general that was initially developed within the context of TD formation in the early Universe. It is sometimes thought that the existence of TDs in the Universe today is inconsistent with the idea of an in#ationary early Universe } after all, one of the motivations behind the development of the in#ationary paradigm (for reviews, see, e.g., Refs. [410,411]) was to get rid of the unwanted TDs like superheavy magnetic monopoles and domain walls by diluting their abundance through the exponential expansion of the Universe characteristic of an in#ationary phase. However, it has been recently realized that TDs could be produced in non-thermal phase transitions occurring during the preheating stage after in#ation [412,413], and models can be constructed in which interesting abundances of `harmlessa TDs so formed can exist in the Universe today. This mechanism of TD production involves explosive particle production due to stimulated decay of the in#aton oscillations through parametric resonance e!ect, which leads to large "eld variances for certain "elds coupled to the in#aton "eld. These large "eld variances in turn lead to symmetry restoration for those "elds even if the actual reheat temperature after in#ation is small. Subsequent reduction of the "eld variances due to the continuing expansion of the Universe would again cause a symmetry breaking phase transition at which TDs could be formed. Thus, TDs can exist even if there was an early in#ationary phase of the Universe. In general, a TD has a `corea of size &g~1, g being the vacuum expectation value of the Higgs "eld in the broken symmetry phase. The Higgs "eld is zero and the symmetry is unbroken at the center of this core } the center being a line for a cosmic string, a point for a monopole and a two-dimensional surface for a domain wall } while far outside the core the symmetry is broken and the gauge and the Higgs "elds are in their true ground states. It is in this sense that the object is referred to as a `defecta } it is a region of unbroken symmetry (`false vacuuma) surrounded by broken symmetry regions (`true vacuuma). The energy densities associated with the gauge and the Higgs "elds are higher within the defect core than outside. TDs are topologically stable due to non-trivial `windinga of the Higgs "eld around the defect cores. This topological stability ensures the `trappinga of the excess energy density associated with the gauge and the Higgs "elds inside the defect cores, which is what makes TDs massive objects. The mass scale of a defect is "xed by the energy (or temperature) scale of the symmetry breaking phase transition at which the defect is formed. Thus, if we denote by ¹ the critical temperature of # the defect-forming phase transition in the early Universe, then the mass of a monopole formed at that phase transition is roughly of order ¹ , the mass per unit length of a cosmic string is of order #
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¹2, and the mass per unit area of a domain wall is of order ¹3. For generic symmetry breaking # # potentials of the Higgs "eld, ¹ &g. # The TDs can be thought of as `constituteda of the `trappeda quanta of massive gauge- and Higgs "elds of the underlying spontaneously broken gauge theory.12 Under certain circumstances, quanta of some massive fermion "elds could also be trapped inside the defects due to their coupling with the defect-forming gauge and Higgs "elds. We shall generically denote the massive particles trapped inside TDs as `Xa particles which can be `supermassivea with mass m that can be as large X as &1016 GeV if the TDs under consideration are associated with breaking of a GUT symmetry. Due to their topological stability, once formed in the early Universe, the TDs can survive forever with X particles trapped inside them. However, from time to time, some TDs, through collapse, annihilation or other processes, can release the trapped X particles [414}418,390,392}399]. Decays of these X particles can give rise to extremely energetic nucleons, neutrinos and photons with energies up to &m [200,178]. Depending on the parameters involved, some of these processes X can give a signi"cant contribution to, and possibly explain, the measured EHECR #ux above &1020 eV [12,403,295,297,406,206]. There is a large body of literature on the subjects of nature and classi"cation of topological defects of various kinds and their formation and evolution in the early Universe. We will not attempt to review these topics here; instead we refer the reader to Refs. [408,409] for a comprehensive review and list of references. Historically, much of the early considerations of cosmic topological defects had to do with their gravitational e!ects, namely, the possible role of cosmic strings in providing the seeds for formation of galaxies and large scale structure in the Universe, the possibility of magnetic monopoles being a candidate for the dark matter in the Universe, the recognition of the disastrous role of massive domain walls in cosmology, and so on. The particle aspect of TDs } that TDs harbor massive quanta of gauge, Higgs, and possibly other "elds inside them, and that these massive particles could, under certain circumstances, be released from TDs with possibly important consequences } did not receive much attention early on. In 1982, two independent works [414,415] pointed out that collapsing closed loops of cosmic strings [414], and monopole}antimonopole annihilations [415] could be sources of massive X particles of GUT scale mass &1016 GeV in the Universe, and that baryon number violating decays of these X particles could be responsible for the observed baryon asymmetry of the Universe; however, the possible connection of these massive X particles with EHECR was not explored then.13 In 1983 Hill [392] pointed out that decay of supermassive X particles released from monopole}antimonopole annihilation through formation and eventual collapse of metastable monopolonium (monopole}antimonopole bound states) could give rise to very energetic particles, and soon it was also pointed out [404] that these energetic particles could be observable as extremely high-energy cosmic rays. After the discovery of superconducting cosmic
12 We shall restrict ourselves to considerations of `locala TDs, that is, the TDs arising from breaking of local symmetries only. `Globala topological defects are possible in theories with spontaneously broken global symmetries. There are no massive gauge bosons for global defects; a large portion of the energy density of a global defect resides in the form of massless Goldstone bosons, and massive scalar particles play only a subdominant role. 13 Recently, however, it has been pointed out [419] that both EHECR and the baryon asymmetry of the Universe may arise from the decay of X particles released from TDs. For a brief discussion of this possibility, see Section 6.10.
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string solutions by Witten [420] in 1985, Ostriker et al. [421] pointed out that massive charge carriers whose mass can be as large as the GUT scale, &1016 GeV, could be spontaneously emitted from superconducting cosmic string loops which attained a certain maximum current (beyond which the string would lose its superconductivity). Following this, Hill et al. [393] calculated the expected ultra-high-energy nucleon and neutrino spectra resulting from the decay of the massive charge carriers (X particles) released from superconducting cosmic string loops. Soon thereafter, ultra-high-energy cosmic-ray spectrum resulting from decay of X particles released from ordinary (i.e., non-current-carrying) cosmic strings due to cusp evaporation process [394,395], as well as due to collapse or multiple self-intersections of closed cosmic string loops [397,396] were considered. A general formulation of calculating the #ux of ultra-high-energy particles from decay of X particles released from TDs in general was given in Ref. [200] in which a convenient parametrization of the production rate of X particles from any kind of TDs was proposed. This facilitated calculation of expected UHE particle spectra due to decay of X particles released from TDs in general without considering speci"c TD models. The importance of the expected high #ux of photons relative to nucleons from the decay of X particles at the highest energies as a signature of the TD scenario in general was pointed out in Ref. [178]. The recent detections of the cosmic-ray events above 1020 eV by the Fly's Eye as well as the AGASA experiments and the realization of the di$culties faced by conventional acceleration scenarios in explaining these events [12,26,403,13] have led to a renewed interest in the TD scenario of origin of EHECR. In what follows, we discuss various X particle production processes involving various di!erent kinds of TDs that have been studied so far, and discuss their e$cacies with regard to EHECR keeping in mind the rough estimates of the X particle production rate required to explain EHECR discussed above. Among the various kinds of TDs, cosmic strings are perhaps the most well studied, analytically as well as numerically, in terms of their properties pertaining to their formation and evolution in the Universe. It is, therefore, possible to make some quantitative estimates of X particle production rates due to various cosmic string processes with relatively less number of free parameters as compared to processes involving other TDs. Because of this, and for illustrative reasons, we "rst discuss the cosmic string processes in some details, and then discuss other defects brie#y. 6.4. X particle production from cosmic strings The release of X particles from cosmic strings requires removal of local topological stability of (segments of ) cosmic strings. The topological stability of cosmic strings is due to non-trivial winding of the phase of the relevant (complex) Higgs "eld around the string. This local stability is removed whenever the string is in such a con"guration that con#icting demand is placed on the phase of the Higgs "eld, leading to `unwindinga of the phase. This happens, for example, (a) at the point of intersection of two string segments, (b) if a closed loop of string shrinks (due to energy loss through, e.g., gravitational radiation) down to a radius of the order of the width of the string, and (c) in the `cusp evaporationa process. Brief descriptions of these processes are given below. Intersection and intercommuting of string segments. The cosmic string has a small but "nite width w&g~1, where g is the vacuum expectation value (VEV) of the relevant Higgs "eld. The energy per
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unit length of the string is k&g2. Because of the "nite width of the string, two intersecting string segments have to overlap on top of each other over a length scale of order w&g~1 near the point of intersection. The Higgs "eld phase becomes unde"ned in the overlapping region, and a topology removal event takes place, whereby the energy contained in the overlapping region is released in the form of massive X particles. The remaining string segments then `exchange partnersa and reconnect so as to maintain the continuity of the phase of the Higgs "eld. This process of `intercommutinga of strings has been veri"ed by numerical simulations of cosmic string interactions. This is also the fundamental process that leads to formation of closed loops of string from self-intersections of long strands of string and to splitting of any closed loop into smaller daughter loops when the parent loop self-intersects. The energy released in the form of X particles at each intercommuting event is &kw&g, and since m &g, we see that, modulo factors of order unity, roughly of order one X particle is released at X each intercommuting event. The contribution of this process to the cosmic-ray #ux will depend on the rate of occurrence of these intercommuting events which, as we shall see below, is negligibly small. Final stage of loop shrinkage. Similarly, each closed loop of string after shrinking down to a radius of order the width of the string will produce roughly of order one X particles, most of the energy of the original loop having been radiated away in the form of gravitational radiation. Cusp evaporation. The cusp evaporation is another process by which X particles can be released from cosmic strings. Cusps are points on a string at which the string at an instant of time moves with the speed of light. The radius of curvature of the string at a cusp point becomes very small. Existence of cusps is a generic feature [422] of equations of motion of closed loops of string described by the Nambu}Goto action [408]. Strictly speaking, the Nambu}Goto action is valid only for a mathematically in"nitesimally thin string with no width, and in fact, in this case, the radius of curvature of the string at the cusp is mathematically unde"ned. For a realistic string with a small but "nite width, the Nambu}Goto action provides a good description of the string motion as long as radius of curvature of the string is much larger than the width of the string. Clearly, the Nambu}Goto action breaks down at the cusp points; nevertheless, the Nambu}Goto description of a string with initially no cusp shows that `almosta cusp points tend to develop, at which, due to "nite width of the string, overlapping of two string segments takes place, leading to `evaporationa [423] of the overlapped regions of the string in the form of X particles. The existence of cusps was initially shown for closed loops [422], but cusps can also occur on long strands of strings due to the presence of small-scale structures such as `kinksa (where the tangent vector to the string changes discontinuously). Cusps can be formed when kinks propagating on the string in opposite directions collide with each other [424]. The length of string involved in a cusp is [423,424] l &f2@3w1@3, where f is a characteristic length of the small-scale structure on # strings (basically the interkink distance); for a loop of length ¸ with no or few kinks, the cusp length l is given by the same expression as above with f replaced by ¸. The energy released in the form of # X particles due to a single cusp event is &kl , and hence the number of X particles released in # a single cusp evaporation event on a long string is &(gf)2@3, while that for a closed loop of length ¸ is &(g¸)2@3.
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Very recently, it has been pointed out in Ref. [425] that the value of cusp length [423] l &¸2@3w1@3 mentioned above is actually valid for only a special class of cusps. For a more generic # cusp, including the e!ects of the Lorentz contraction of the string core, Ref. [425] "nds l &(¸w)1@2, which is smaller than the previous value by a factor of (w/¸)1@6. Since ¸ will typically # be a cosmological length scale whereas w is a microscopic length scale, the factor (w/¸)1@6 will be an extremely small factor which will drastically reduce the e!ectiveness of the cusp evaporation process in producing observable cosmic ray #ux. We shall continue to use the `olda estimate [423] of l in the calculations below as a sort of `upper limita on l . We shall see that even this upper # # limit will be too low to yield observable cosmic ray #ux. For a given l , the contribution of the cusp evaporation process to the cosmic-ray #ux will # depend upon the number of cusp evaporation events on loops and long strings occurring per unit volume per unit time, which can be determined only if we know the number densities of closed loops and long strings and the rate of occurrence of cusps on each loop. In order to estimate the number densities of closed loops and long strings, let us brie#y recall the salient features of evolution of cosmic strings in the Universe [408,409]. 6.4.1. Evolution of cosmic strings Immediately after their formation, the strings would be in a random tangled con"guration. One can de"ne a characteristic length scale, m , of the string con"guration in terms of the overall 4 mass-energy density, o , of strings through the relation 4 o "k/m2 , (70) 4 4 where k denotes the string mass (energy) per unit length. Initially, the strings "nd themselves in a dense medium, so they move under a strong frictional damping force. The damping remains signi"cant [408] until the temperature falls to ¹[(Gk)1@2g, where G,1/M2 is Newton's constant Pand g is the symmetry-breaking scale at which strings were formed. [Recall, for GUT scale cosmic strings, for example, g&1016 GeV, k&g2&(1016 GeV)2, and so Gk&10~6.] In the friction dominated epoch, a curved string segment of radius of curvature r quickly achieves a terminal velocity J1/r. The small-scale irregularities on the strings are, therefore, quickly smoothed out. As a result, the strings are straightened out and their total length shortened. This means that the characteristic length scale m describing the string con"guration increases and consequently the 4 energy density in strings decreases with time as the Universe expands. Eventually m becomes 4 comparable to the causal horizon distance &t. At about this time, the ambient density of the Universe also becomes dilute enough that damping becomes unimportant so that the strings start moving relativistically. Beyond this point, there are two possibilities. Causality prevents the length scale m from growing 4 faster than the horizon length. So, either (a) m keeps up with the horizon length, i.e., m /t becomes 4 4 a constant, or (b) m increases less rapidly than t. In the latter case, the string density falls less 4 rapidly than t~2. On the other hand, we know that the radiation density in the radiationdominated epoch as well as matter density in the matter-dominated epoch both scale as t~2. Clearly, therefore, in case (b) the strings would come to dominate the density of the Universe at some point of time. It can be shown that this would happen quite early in the history of the Universe unless the strings are very light, much lighter than the GUT scale strings. A string
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dominated early Universe would be unacceptably inhomogeneous con#icting with the observed Universe.14 The other possibility, case (a) above, which goes by the name of `scalinga hypothesis, seems to be more probable, as suggested by detailed numerical as well as analytical studies [408,409]. The numerical simulations generally "nd that the string density does reach the scaling regime given by o J1/t2, and then continues to be in this regime. It is, however, clear that in order for this to 4,4#!-*/' happen, strings must lose energy at a certain rate. This is because, in absence of any energy loss, the string con"guration would only be conformally stretched by the expansion of the Universe on scales larger than the horizon so that m would only scale as the scale factor Jt1@2 in the radiation 4 dominated Universe, and Jt2@3 in the matter dominated Universe. In both cases, this would fail to keep the string density in the scaling regime, leading back to string domination. In order for the string density to be maintained in the scaling regime, energy must be lost by the string con"guration per unit proper volume at a rate o5 satisfying the equation 4,-044 o5 "!2(RQ /R) o #o5 , (71) 4,505!4 4,-044 where the "rst term on the right-hand side is due to expansion of the Universe, R being the scale factor of the expanding Universe. In the scaling regime o5 "!2o /t, which gives 4,505!4 o5 "!o /t in the radiation dominated Universe, and o5 "!(2/3)o /t in the matter 4,-044 4 4,-044 4 dominated Universe. The important question is, in what form does the string con"guration lose its energy so as to maintain itself in the scaling regime? One possible mechanism of energy loss from strings is formation of closed loops. Occasionally, a segment of string may self-intersect by curling up on itself. The intersecting segments may intercommute, leading to formation of a closed loop which pinches o! the string. The closed loop would then oscillate and lose energy by emitting gravitational radiation and eventually disappear. It can be shown that this is indeed an e$cient mechanism of extracting energy from strings and transferring it to other forms. The string energy loss rate estimated above indicates that scaling could be maintained by roughly of order one closed loop of horizon size (&t) formed in a horizon size volume (&t3) in one hubble expansion time (&t) at any time t. In principle, as far as energetics is concerned, one can have the same e!ect if, instead of one or few large loops, a large number of smaller loops are formed. Which one actually happens depends on the detailed dynamics of string evolution, and can only be decided by means of numerical simulations. Early numerical simulations seemed to support the large (i.e., & horizon size) loop formation picture. Subsequent simulations with improved resolution, however, found a lot of small-scale structure on strings, the latter presumably being due to kinks left on the strings after each crossing and intercommuting of string segments. Consequently, loops formed were found to be much smaller in size than horizon size and correspondingly larger in number. Further simulations showed that the loops tended to be formed predominantly on the scale of the cut-o! length imposed for reasonable resolution of the smallest size loops allowed by the given resolution scale of 14 However, a string dominated recent Universe } dominated by `lighta strings formed at a phase transition at about the electroweak symmetry breaking scale } is possible. Such a string dominated recent Universe may even have some desirable cosmological properties [426]. Such light strings are, however, not of interest to us in this discussion.
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the simulation. It is, however, generally thought that the small-scale structure cannot continue to build up inde"nitely, because the back-reaction of the gravitational radiation from the kinky string itself would eventually stabilize the small-scale structure at a scale f&CGkt, where C&100 is a geometrical factor. The loops would, therefore, be expected to be formed predominantly of size &f, at any time t. Although much smaller than the horizon size, these loops would still be of `macroscopica size, much larger than the microscopic string width scale (&g~1&k~1@2). These loops would, therefore, also oscillate and eventually disappear by emitting gravitational radiation. Thus, according to above picture, the dominant mechanism of energy loss from strings responsible for maintaining the string density in the scaling regime would be formation of macroscopic-size (
G
dn (¸, t)" d¸
2 K#1 1 , ¸4KCGkt , 3x2 K t2(¸#CGkt)2 0,
¸'KCGkt .
15 Long strings are de"ned as string segments with radius of curvature Z the horizon length &t.
(76)
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It is often convenient to de"ne n (t)"¸(dn/d¸)(¸, t) as the number density of loops in a length L interval *¸&¸ around ¸. Using Eq. (76) we see that n (t)Jt~4¸ for ¸;CGkt and L n (t)Jt~2¸~1 for ¸
(79)
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where f is a numerical factor accounting for the e$ciency of the cusp evaporation process, and we # have assumed that two cusps appear every oscillation period of the loop [422]. Folding this with the loop length distribution function of Eq. (76), we get the number density of X particles produced per unit time per unit volume due to all loops at any time t:
P
dn n5 #641"4f m~1kw1@3 d¸ ¸~1@3 (¸, t) . X # X d¸
(80)
Using Eq. (76) we see that the dominant contribution to the integral in the above equation comes from loops of length ¸&CGkt, giving n5 #641K4f m~1kw1@3 (2/3x2)[(K#1)/K] (CGk)~4@3t~10@3 X # X "4f (2/3x2) [(K#1)/K]C~4@3(Gk)~1M2@3t~10@3 . (81) # PTaking f "1, x"0.3, K"1, C"100, and with t K2.06]1017(X h2)~1@2 sec for the age of the # 0 0 Universe, we get the rate of X particle production in the present epoch due to cusp evaporation from all cosmic string loops as n5 #641K6.4]10~56(Gk/10~6)~1 cm~3 s~1 . (82) X,0 Comparing this rate with that in Eq. (68), we see that the X particle production rate due to cusp evaporation from cosmic string loops is too small (by about ten orders of magnitude) to give any signi"cant contribution to cosmic-ray #ux. Note that n5 #641 generally increases with decreasing value of k. This is due to the fact that for lower X values of k (lighter strings) the gravitational radiation rate is reduced, so loops survive longer and consequently the number density of loops present at any time is larger, giving a larger contribution to X particle production rate. This at "rst suggests [394] that for su$ciently light strings (i.e., for su$ciently small values of Gk) the X particle production rate due to cusp evaporation may even exceed the rate required for su$cient cosmic ray production, thereby giving a lower limit on Gk, i.e., a lower limit on the energy scale of any string-forming phase transition. However, this turns out not to be the case [395]: For too small values of Gk, the energy loss of the string loops through gravitational radiation becomes so small that the cusp evaporation process itself becomes the dominant energy loss mechanism which then determines the number density of loops. Detailed calculations [395] show that when the loop length distribution function, Eq. (76), is modi"ed (for small values of Gk) to include the energy loss of the loop due to cusp evaporation itself, then there is no lower limit on Gk. Indeed, n5 #641 increases with decreasing values of Gk reaching a peak at X around Gk&10~15, and n5 #641 then decreases with further decrease in the value of Gk. The peak X value of n5 #641, however, still remains about four orders of magnitude below the value required for X producing su$cient cosmic ray #ux. Let us now consider cusp evaporation from long strings. Cusps can be formed on long strings due to collisions of kinks traveling on long strings [424]. The total length of string in a given volume < is ¸&
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of X particles due to cusp evaporation from the long string network in the scaling solution (m "xt) 4 can be written as n5 #641,LSKs(gf)2@3(fm )~2"sx~2C~4@3(Gk)~1M2@3t~10@3 . (83) X 4 PExcept for an overall factor of order unity, this is same as the X particle production rate in the case of cusp evaporation from loops. Thus, X particle production from cusps on cosmic strings gives negligible contribution to cosmic-ray #ux. As already mentioned earlier, the cusps assumed above are special ones. Taking into account the fact that for a generic cusp the energy released is actually much smaller [425], one can conclude that the cusp evaporation process leads to utterly negligible cosmic-ray #ux. 6.4.5. Collapse or repeated self-intersections of closed loops It is clear from the above discussions that in order to produce X particles with a large enough rate so as to be relevant for cosmic rays, macroscopically large lengths of strings are required to be involved in the X particle production process. One such process is complete collapse or repeated self-intersections of closed loops [396]. It is known [427] that any initially static non-circular loop, or any loop con"guration that can be described by single-frequency Fourier modes, collapses into a double-line con"guration at a time ¸/4 after its birth, ¸ being the length of the loop. (Recall, period of oscillation of a loop of length ¸ is ¸/2.) In such overlapped con"gurations, the entire string would annihilate into X particles [414]. Such completely collapsing con"gurations are, however, likely to be very rare. Nevertheless, this kind of collapsing loops serve as an example of a general class of situations in which macroscopically large fraction of the energy of cosmic string loops is dissipated in the form of X particles on a time scale much shorter than the time scale q &(CGk)~1¸ of decay of the loops due to energy loss through gravitational radiation. For ' example, one can think of a situation in which a large loop self-intersects and splits into two smaller loops, and each daughter loop self-intersects and splits into two further smaller loops, and so on. Under such a circumstance, it can be seen [429] that a single initially large loop of length ¸ can break up into a debris of tiny loops (of size &g~1, thereby turning into X particles) on a time-scale q &¸. Since, as discussed earlier, loops are expected to be born at any time t with typical size $%"3*4 ¸&KCGkt;t, we see that the above time-scale of break-up of a large loop into X particles is much less than the Hubble time, and very much less than the gravitational decay time scale q . ' Following Ref. [396] let us suppose that a fraction f of the total energy in all newly born loops X at any time t goes into non-relativistic X particles of mass m on a time scale much shorter than the X gravitational decay time scale q . Using Eqs. (72) and (73) we then get ' n5 (t)"f (k/m ) (2/3x2) t~3 , (84) X X X where xK0.3. Eq. (68) then implies that in order to explain EHECR we require f [email protected]]10~5 , (85) X 16 where we have taken l (E "300 EeV)"50 Mpc, m &g&k1@2, and de"ned g "(g/1016 GeV). c X 16 One should keep in mind that Eq. (85) is valid provided the cosmic string loops producing the X particles are distributed in a spatially homogeneous manner. For a given value of f , there is an independent constraint on g which comes from the fact that X 16 any electromagnetic radiation injected at the EHECR energies would initiate an electromagnetic
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cascade (see Section 4.2) whereby a part of the injected energy would show up at lower energies (in the 10 MeV}100 GeV region) where existing measurements of the di!use gamma-ray background constrain any such energy injection. The energy density that would go into the cascade radiation is approximately given by [406] (see Section 7.1) u K1 m n5 t , (86) #!4 2 X X 0 and the measured gamma-ray background in the 10 MeV}100 GeV region [185] imposes the constraint [406] u 42]10~6 eV cm~3 . #!4 Eqs. (84), (86), and (87) together imply the condition
(87)
f g2 49.6]10~6 . (88) X 16 The requirement of Eq. (85) can be satis"ed (so that we are able to explain the EHECR) without violating the cascade constraint (88) only for g lying in the range 9.2]1012 GeV [ g [ 1.2]1015 GeV with f in the corresponding range 6.7]10~4 [ f [ 1 satisfying f K2.8] X X X 10~5g~3@2. 16 The above discussions indicate that a cosmic string scenario of EHECR with m much above X &1015 GeV may be di$cult to reconcile with the low-energy di!use gamma-ray constraint. On the other hand, for m &1015 GeV, cosmic strings can be responsible for EHECR without X violating the gamma-ray background constraint provided that a fraction few]10~4 of the newly born loops at any time t goes into X particles on a time scale much smaller than the Hubble time &t. The above crude analytical estimates indicate that the measured `lowa-energy di!use gammaray background provides an important constraint on the mass (or energy) of the decaying particle if the number densities of these particles are normalized so as to explain the EHECR. This important fact was "rst pointed out in Ref. [430] and has subsequently been emphasized in many studies [295,156,297,418,406,206]. As discussed earlier below Eq. (69), the energy injection rate needed to explain the EHECR is uncertain, and the estimate of Eq. (69) is probably an overestimate in which case the upper limit on m derived above (for the viability of cosmic string scenario of EHECR) may be pushed up to X &1016 GeV, a typical GUT scale. Indeed, recent detailed numerical calculations [206] show that, for a large range of other parameters, TD scenarios of EHECR in general are consistent with all observational data for m up to &1016 GeV, but not much above this value. X Coming now to the question of f , it is not known what fraction of loops may be born in X collapsing and/or repeatedly self-intersecting con"gurations such that essentially all their energy eventually turns into X particles. In principle, numerical simulations of loop self-intersections should be able to answer this question, but in practice the simulations have so far lacked the necessary resolution. On the theoretical side, Siemens and Kibble [431] have shown that selfintersection probability of a loop increases exponentially with the number of harmonics needed to describe the loop con"guration. In particular, since kinks are high harmonic con"gurations, loops having kinks have high probability of self-intersection. Since the loops formed from selfintersection of long strings (or from splitting of existing loops) invariably have kinks on them, it is not inconceivable that an interesting fraction (&few]10~4) of loops at any time may indeed
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undergo repeated self-intersections and rapidly deteriorate into tiny loops which decay into X particles. Note that if loops always self-intersect and thus quickly turn into X particles, i.e., if f &1, then the the constraint (85) gives g49.2]1012 GeV, which would rule out the existence of X GUT-scale cosmic strings [396] because of excessive X particle production and the resulting overproduction of EHECR. At the same time, for f K1, cosmic strings with g&1013 GeV can X explain the EHECR #ux without violating the low energy cascade c-ray constraint. Note that in the process of repeated self-intersection and splitting o! of loops, a fraction of the total energy of a parent loop is likely to go into kinetic energy of the daughter loops at each splitting [429]. Depending on this fraction the daughter loops may get substantial kicks at their birth. The smallest loops which eventually turn into X particles may, therefore, be relativistic [429] and hence well dispersed in space. Thus, although at any time there might be relatively few initially large loops within our Hubble volume so that the distribution of those initially large loops might be highly inhomogeneous and anisotropic, the X particles themselves resulting from repeated selfintersection and splitting of those loops and the resulting cosmic rays may be more isotropically and uniformly distributed in the sky. However, there may be a problem if the X particles are too relativistic, and the authors of Ref. [406], in particular, have argued within the context of a speci"c loop fragmentation model that in the relativistic X particle case it is hard to obtain su$cient number of X particles to explain the EHECR #ux without violating the cascade constraint (88) for any reasonable values of g. However, this conclusion seems to be speci"c to the particular loop fragmentation model considered in Ref. [406], and can be evaded in other loop fragmentation scenarios. For example, in the Siemens}Kibble scenario [431] mentioned above in which all cosmic string loops quickly break up into X particles, thus giving f &1, it can be shown [432] that the EHECR #ux can be explained X without violating the c-ray cascade constraint, provided the strings are su$ciently light, g[3.1]1013 GeV, and f , the fraction of energy of any parent loop (in its rest frame) that goes KE into the kinetic energy of daughter loops, is not too large, f [few percent. KE It may be mentioned here that an EHECR scenario involving lighter (i.e., lighter than GUT scale) cosmic strings with, e.g., g&few]1013 GeV, has an advantage over one with heavier strings because the number density of loops of such light strings would be larger than that for heavier strings. Recall (from the discussions following Eq. (76)) that the number density of loops at any time is proportional to (Gk)~1 and the average separation between the loops is proportional to (Gk)1@3, while the typical length of a loop is proportional to Gk. Thus, while for GUT-scale strings with g&1016 GeV (i.e., Gk&10~6), there are only about 2.4(X h2)3@2 loops within a typical `GZK 0 volumea of radius &50 Mpc, the number would be larger by a factor of 106 for strings with g&1013 GeV (Gk&10~12). Thus the problem of lack of enough cosmic string loops [433,428,416] encountered in the GUT-scale cosmic string scenario of origin of EHECR may be solved with su$ciently light string loops undergoing repeated splittings and thereby producing su$ciently energetic X particles relevant for EHECR. 6.4.6. Direct emission of X particles from cosmic strings Finally, we consider the recent suggestion [417] that X particles may be directly radiated from cosmic strings. From the results of their new numerical simulations of evolution of cosmic strings, authors of Ref. [417] have claimed that if loop production is not arti"cially restricted by imposing a cuto! length for loop size in the simulation, then loops tend to be produced predominantly on the
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smallest allowed length scale in the problem, namely, on the scale of the width of the string. Such small loops promptly collapse into X particles. In other words, according to Ref. [417], there is essentially no loop production at all } the string energy density is maintained in the scaling regime by energy loss from strings predominantly in the form of direct X particle emission, rather than by formation of large loops and their subsequent gravitational radiation. It should be mentioned here that this result, which implies a radical departure from the results of earlier numerical simulations of evolution of cosmic strings [408,409], has been questioned recently [434]. However, the basic issues involved here are quite complex and currently rather ill-understood, and as such the results of Ref. [417] cannot be ruled out at this time. If the results of Ref. [417] are correct, then X particles are directly produced by cosmic strings at a rate given by Eq. (84) with f "1. Comparing this rate X with that in Eq. (68), we see that in order for cosmic rays from cosmic strings not to exceed the observed cosmic-ray #ux, the string-forming symmetry breaking scale g is constrained [396,417,418,298] as g[1013 GeV. Thus, in this case, GUT scale cosmic strings with g&1016 GeV will be ruled out [396,417,418,298] } because they would necessarily overproduce EHECR } while at the same time cosmic strings formed at a phase transition with g&1013}1014 GeV would be a `naturala source of EHECR. Note, however, that the typical radius of curvature of long strings today, and hence the typical distance between neighboring long strings is of the order of the Hubble distance &t . Therefore, 0 the observed EHECR can be due to direct emission of X particles from long strings only in the case of accidental proximity of a long-string segment lying within say &50 Mpc from us [406]. In this case, however, the observed arrival directions of the EHECR events is predicted to be highly anisotropic. In particular, one should expect the sources of individual EHECR events to trace out a linear or "lamentary region of sky [396] corresponding to the long-string con"guration. This prediction should be testable with the upcoming large-area EHECR detectors. Before closing this discussion on cosmic strings as possible sources of EHECR, it is worthwhile to mention that cosmic string formation at a phase transition with desired g&1013}1014 GeV rather than at the GUT scale transition with g&1016 GeV is not hard to envisage. For example, the symmetry breaking SO(10)PSU(3)]SU(2)]U(1) ];(1) can take place at the GUT uni"caY tion scale M &1016 GeV; with no U(1) subgroup broken, this phase transition produces no GUT strings. However, the second U(1) can be subsequently broken with a phase transition at a scale &1014 GeV or lower to yield the cosmic strings relevant for EHECR. Note that these strings would be too light to be relevant for structure formation in the Universe and their signature on the CMBR sky would also be too weak to be detectable. Instead, the extremely high energy end of the cosmic ray spectrum may o!er a probing ground for signatures of these `lighta cosmic strings. We add that cosmic strings of mass scales somewhat below the GUT scale could also be produced in non-thermal phase transitions associated with the preheating stage after in#ation, as mentioned in Section 6.3. 6.5. X particles from superconducting cosmic strings Superconducting cosmic strings (SCSs) [420] are cosmic strings carrying persistent electric currents. The current can be carried either by a charged Higgs "eld having a non-zero vacuum expectation value inside the string (thus breaking the electromagnetic gauge invariance inside the string and thereby making the string superconducting), or by a charged fermion "eld living as
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a `zero modea inside the string due to coupling of the fermion to the string-forming Higgs "eld. (Here zero mode refers to the fact that the relevant fermions are massless inside the string whereas they have a "nite mass outside the string.) In the case of non-abelian cosmic strings the superconductivity of the string can also be due to a charged vector "eld condensate inside the string. Superconducting strings would also in general carry electric charges due to charges trapped at the formation of the string by the Kibble mechanism and/or due to inter-commuting of string segments with di!erent currents. A review of basic properties of SCSs can be found in Ref. [408]. Superconducting strings cannot sustain currents beyond a certain critical current J . In the case # of fermionic superconductivity, this happens because the density of the charge carriers at the critical current becomes degenerate enough that the Fermi momentum inside the string exceeds the mass of the fermion in the vacuum outside the string, at which point the fermions above the fermi sea cease to be trapped on the string and begin to be ejected into the vacuum outside the string. Similarly, in the bosonic case, the energy density in the charged scalar "eld condensate inside the string at the maximum current becomes high enough to cause restoration of the broken electromagnetic symmetry inside the string, as a result of which the string loses its superconductivity. The magnitude of J is model dependent, but the string forming symmetry-breaking scale # g provides an upper bound on J , namely, J 4J Keg, both for bosonic as well as fermionic # # .!9 superconductivity. (Here e is the elementary electronic charge.) If an SCS achieves the critical current, the charge carriers will be expelled from the string. Outside the string, the charge carriers are massive with a mass that } depending on the particle physics model } can be as large as the GUT-scale &1016 GeV. These massive charge carriers would then be the X particles whose decay may give rise to extremely energetic cosmic-ray particles. There can be a variety of mechanisms of setting up the initial current on the string. Apart from small-scale current and charge #uctuations induced on the string at the time of the superconducting phase transition, large-scale coherent (dc) currents can be induced on macroscopically large string segments (of horizon scale &t) as the string moved through a possible primordial magnetic "eld in the Universe. Or a string can pick up a current due to its motion through the Galactic magnetic "eld, for example. In addition, any closed loop of oscillating SCS in an external magnetic "eld would have an ac current, and there would also be short-wavelength ac contribution to the current on the scale of the small-scale wiggles on long strings. The evolution of current-carrying SCSs is considerably more complicated than that of `ordinarya non-current-carrying cosmic strings, and is rather poorly understood at the present time. It is therefore di$cult to make concrete predictions about contributions of SCSs to EHECR. One possible model of X particle production from SCSs with fermionic superconductivity and the resulting EHECR #ux was "rst studied by Hill et al. (HSW) [393]. Their model was based on the scenario of evolution of current-carrying SCS closed loops suggested by Ostriker et al. (OTW) [421]. In the OTW scenario, initial currents on closed SCS loops are induced due to the changing magnetic #ux of a primordial magnetic "eld linked by the loops } the #ux change being due to the expansion of the Universe. Once an initial current is set up, a current-carrying, oscillating closed loop of SCS loses energy through electromagnetic as well as gravitational radiation, and as a result the loop shrinks in size. This, in turn, leads to an increase of the dc component of the current on the loop (JJ¸~1, ¸ being the instantaneous length of the loop) due to conservation of the initial
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magnetic #ux linked by the loop. Eventually, the current on the loop would reach the saturation value J "J , at which point charge carriers } the massive X particles } would be emitted from 4 # the loop. The X particle emission rate from a current saturated SCS loop in the case of fermionic superconductivity can be estimated as follows [393]: The number of fermions of a given chirality plus antifermions of the opposite chirality per unit length of the string is n "p /p, where p is F F F the Fermi momentum. The current on the string is related to the Fermi momentum through the relation J"en "ep /p. The string is saturated when p "m , where m is the mass of the F F F F F fermion (to be identi"ed with the X particle in this case). Thus J "em /p. The mass of the fermion 4 F arises from its Yukawa coupling with the symmetry-breaking Higgs "eld responsible for the formation of the string. Thus m "gg, where g (assumed [ 1) is the Yukawa coupling constant F and g is the VEV of the string-forming Higgs "eld. The total number of fermions plus antifermions inside a SCS loop of length ¸ in the saturated regime is simply N (t)"(m /p)¸(t). After it is F F saturated, the loop continues to radiate energy and shrink. As the saturated loop shrinks, the Fermi momentum remains constant at p "m , the current remains constant at J , and so the shrinkage F F 4 of the saturated SCS loop is accompanied by fermion emission at a rate given by NQ "(m /p)¸Q . F F The loop shrinkage rate ¸Q in the saturated regime is in general determined by the combined rate of electromagnetic (e.m.) plus gravitational energy radiation from the loop. However, depending on the values of the Yukawa coupling g and the symmetry-breaking scale g, either the e.m. or the gravitational radiation may dominate. The e.m. radiation power from a SCS loop (without cusp16) with current J is given by [421] P "c J2, where c K100. The gravitational energy loss rate, P , is given by Eq. (74). In %. %. %. '3!7 order to have EHECR particles, we shall require that m "gg51012 GeV, and since one generally F expects g[1, we shall require gZ1012 GeV, i.e., GkZ10~14 and 10~7(Gk)~1@2[g41. From these conditions one can see that, for a saturated SCS loop with 10~14[Gk[10~8, e.m. radiation dominates over gravitational radiation for all allowed values of g that satisfy the requirement mentioned above. On the other hand, for a saturated loop with Gk'10~8, e.m. radiation dominates if 10(Gk)1@2[g41, and gravitational radiation dominates if 10~7(Gk)~1@2[g(10(Gk)1@2. If ¸Q is determined by the e.m. radiation, then the fermion emission rate from a saturated SCS loop is given by NQ "(4/p2) a c g3g , (89) F %. %. where a " 1 is the e.m. "ne-structure constant, and we have used kKg2 and m "gg. If %. 137 F gravitational radiation dominates (which requires g to be su$ciently small), then NQ "(g/p) (CGk)g . (90) F Eq. (89) or (90) gives the fermion emission rate from a single saturated loop. To "nd the total number density of fermions (X particles) produced by all saturated SCS loops per unit time at any time t, we need to know the number density of saturated SCS loops in the Universe as a function of cosmic time t. It is here that things become rather complicated and model-dependent. The
16 Loops with cusps may have signi"cantly higher radiated electromagnetic power [435].
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evolution of the length distribution function for current-carrying SCS loops is not known. HSW [393] assumed that the loop formation rate for SCSs is same as that for ordinary cosmic strings and their evolution in the pre-saturation regime is governed by gravitational radiation loss as discussed in Section 6.4.1.17 One expects that at any given cosmic time t, all existing loops of length below a certain `saturation lengtha ¸ will have achieved the saturation current, and each of these 4 loops would be emitting fermions at a rate given by Eq. (89) or (90), as the case may be. The saturation length ¸ depends on the details of the manner in which the initial current is induced on 4 loops, the magnitude of the initial current on a loop (which depends on the strength of the ambient magnetic "eld), the subsequent magnetic "eld history experienced by the loop, and so on. In the OTW scenario, in which the initial currents on loops are induced by the removal (due to expansion of the Universe) of a primordial magnetic "eld whose energy density scales as that of the universal radiation background, ¸ is roughly constant in time [393]. In this case, with the loop 4 length distribution function given by that for ordinary cosmic strings [Eq. (76)], one sees that the X particle production rate, n5 (t)Jt~4. HSW [393] showed that for certain ranges of parameter X values, this scenario can produce EHECR #ux comparable with observed #ux. In a more general situation, depending on the magnetic "eld history, the saturation length ¸ can 4 increase or decrease with time, and consequently the time dependence of n5 would be di!erent. X According to Ref. [393], in scenarios where ¸ grows with time } such a scenario may obtain, 4 for example, if the intergalactic magnetic "elds are increased by dynamo e!ects } the value of saturation length in the present epoch ¸ (t ) would be smaller than its value in the OTW scenario, 4 0 and the resulting absolute value of n5 in the present epoch would be insu$cient to explain the X observed EHECR #ux. On the other hand, as we shall discuss in Section 7, there is a general problem in situations where ¸ decreases with time because then the energy injection in the early 4 epochs would turn out to be unacceptably large from considerations of distortion of the CMBR and primordial nucleosynthesis [437] if the absolute value of n5 in the present epoch were such as X to explain the EHECR #ux. Thus, in general it seems di$cult to invoke SCS loops (at least in the case of fermionic superconductivity) as possible sources of EHECR. We should stress that the above conclusion hinges upon the assumption that evolution of SCSs in the unsaturated regime is similar to that of ordinary cosmic strings. This is highly uncertain, and as of now, no detailed numerical simulations comparable to those available for ordinary cosmic strings have been done for the study of evolution of a network of superconducting strings. Even the dynamics of a single current-carrying SCS loop is uncertain. The assumption of homologous shrinkage of saturated SCS loops assumed above is probably too simplistic. Indeed, the loop can fold onto itself in complicated shapes with one or more self-intersections, leading to enhanced emission of the charge carriers. A variety of other instabilities can appear (see, e.g., Ref. [438] and references therein for a recent discussion of these issues). In addition, the e!ects of the ac current modes as well as the e!ects of the plasma in which the strings move are highly uncertain. It is also possible that SCS loops may, in fact, never achieve the saturation current at all; instead they may form stable `vortonsa } charge- and current-carrying SCS loops stabilized against shrinkage by angular momentum [439] } which themselves may be relevant for EHECR (see below).
17 The evolution of the current in SCS loops under combined e.m. plus gravitational radiation is discussed in a di!erent context in Ref. [436].
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Apart from the uncertainties inherent to the physics of SCSs in general, there are several `astrophysicala uncertainties associated with the proposal of SCSs as possible sources of EHECR. First, as already mentioned above, the mechanism of induction of initial current on SCSs is uncertain due to uncertainties in our knowledge of the magnetic "eld history of the Universe. Second, it has been pointed out [440] that even if SCS loops achieve saturation current and emit the massive charge carriers (X particles), the energetic particles resulting from the decay of the X particles are likely to be quickly degraded in energy due to synchrotron losses and other processes occurring in the high magnetic "eld region around the string. Within the framework of the Standard Model it has been claimed that most of the energy is radiated as thermal neutrinos with a temperature of roughly 10 MeV that may be observable by underground detectors and be comparable to the atmospheric neutrino #ux at these energies, whereas the emitted c-rays may give rise to coincident GRBs [441]. The problem of degradation of energetic particles may, however, be avoided [393] if the charge carriers have a lifetime su$ciently long that they may be able to drift into the weak-"eld region far away from the string before decaying. Another possibility arises if the string has mainly ac current: In this case, there can be sections of the string with large electric charge but small current, and high-energy particles can escape through those regions [406]. To summarize this discussion on SCSs, then, the range of possibilities here are so large and our current state of knowledge of evolution of SCSs is so uncertain that a de"nite conclusion regarding the viability or otherwise of SCSs as sources of EHECR cannot be made at this stage. The simplest models that have been studied so far generally fail to produce su$cient EHECR #ux. However, more work will be needed in this regard. 6.6. X particles from decaying vortons A SCS loop possessing both a net charge as well as a current can, under certain circumstances, be stabilized against collapse by the angular momentum of the charge carriers. Such stable SCS loops of microscopic dimension, called `vortonsa [439], do not radiate classically and essentially behave like particles with quantized charge and angular momentum. A vorton can be characterized by essentially two integer `quantuma numbers: (a) N, the total winding number of the phase of the charge carrier scalar "eld condensate along the length of the loop, which is responsible for the conserved current on the loop,18 and (b) Z, which is related to the total charge Q"Ze on the loop. A vorton generally tends to evolve towards a `chirala state19 in which DZDKDND, and angular momentum LKZNKN2. The characteristic vorton radius R obtained by minimizing the total energy of a SCS loop is given by R K(2p)~1@2DNZD1@2g~1, 7 7 4 where g is the string-forming symmetry breaking scale. Note that g may, in general, be di!erent 4 4 from (larger than) the symmetry breaking scale g associated with the appearance of superconducp tivity in the string. However, the most favorable conditions for vorton formation occur when g and 4 g are not too widely di!erent. For GUT-scale vortons with g &g &1016 GeV, a rough estimate p 4 p 18 We consider here the case of bosonic superconductivity of the string. However, similar arguments apply for fermionic case also because of formal equivalence of bosons and fermions in (1#1)-dimensional "eld theory on the string world-sheet; for more details, see Ref. [408]. 19 The name chiral refers to the fact that in this case the rotation velocity of the vorton approaches the speed of light.
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[442] gives NKZ&100, and so R &10~28 cm, but these estimates can be o! by several orders of 7 magnitude depending on the detailed dynamics of the vorton formation process. (Meta)stable vortons with lifetime greater than the age of the Universe can be a dark matter candidate. However, in some cases, their predicted abundance in the early Universe is so large as would overclose the Universe at early times, in which case the particle physics models under consideration (which predict vorton formation) have to be ruled out. These considerations strongly constrain the vorton formation energy scale in the early Universe. For more details on these issues and for detailed discussions and references on properties, formation and evolution of vortons, see, e.g., Refs. [442}444]. Vortons can be relevant for EHECR in two ways: Although classically stable, a vorton can decay by quantum mechanical tunnelling process. Such metastable vortons decaying in the present epoch can release the massive charge carrier particles with mass m [g which can act as the X particles p p of the top-down scenario of EHECR if m Z1012 GeV and if vortons exist today with a su$cient p abundance. This possibility has been studied in Ref. [445]. Alternatively, vortons, being highly charged particles, could be accelerated to extremely high energies in some astrophysical sites, and thus vortons themselves could act as the EHECR particles [446]. Here we brie#y discuss the "rst possibility (decaying vortons); the second possibility will be discussed brie#y in Section 6.12.2. Authors of Ref. [445] have studied an approximate semiclassical model of vorton decay through quantum tunnelling originally suggested by Davis [447]. This involves calculating the tunnelling probability of a chiral vorton con"guration with NKZ units of topological winding number to change to a con"guration with N!1 units (with the accompanying emission of one quantum of the charge carrier "eld of mass m &g ), through a barrier of height *E in energy and spatial width p p *R, where R is the radius of the vorton. The tunnelling rate, or the inverse of the lifetime of the vorton, is generally given by q~1&m exp!(*E *R). From simple consideration of energy 7 7 conservation, one can show that [445] *E *RKN. Thus vortons with larger initial N have longer lifetime. In order to be present and decaying in the present epoch, the vortons must have N larger than a certain minimum value N &ln(t g ) (we have assumed g &g ). On the other hand, from .*/ 0 p 4 p the point of view of obtaining su$cient EHECR #ux, the vorton lifetime (and hence N) should not be too large, for a given vorton abundance in the Universe. The vorton abundance and the typical values of N (which as explained above determines the vorton lifetime) depend on the detailed dynamics of the vorton formation process and are rather poorly understood at present. But, in general, it turns out that the joint requirements on the vorton lifetime and abundance (in order to obtain su$cient EHECR #ux) place con#icting demands on the vorton formation energy scale. Thus at the present time it seems rather di$cult to explain EHECR with decaying vortons. According to a recent study [444], the vorton density is most sensitive to the order of the string forming phase transition and relatively insensitive to the details of the subsequent superconducting phase transition. For a second-order string-forming phase transition, vorton production is cosmologically disastrous (because they overclose the Universe) and hence unacceptable for g (&g ) 4 p in the range 105 GeV[g [1014 GeV. For a "rst-order string-forming phase transition, the exclu4 sion range is somewhat narrower: 109 GeV[g [1012 GeV. For g <1014 GeV, no vortons are 4 4 expected to form. On the other hand, vortons formed at g [105 GeV (109 GeV for a "rst-order 4 phase transition) can provide a (part) of the dark matter and are, therefore, cosmologically interesting. However, these low mass-scale vortons are unlikely to be relevant for EHECR because the typical mass of the emitted charge carriers (X particles) are then too low to produce EHECR
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particles. Note, however, that analysis of Ref. [444] still leaves open a window of potentially interesting vorton density for g Kg &1012}1014 GeV, which may then be relevant for EHECR if 4 p the appropriate requirement on the vorton lifetime can be met. We mention here that, as far as EHECR are concerned, vortons as possible EHECR sources would behave very much like the possible long-lived, superheavy metastable relic X particles which are discussed in a general way in Section 6.13. Thus, like standard cold dark matter, vortons would be expected to cluster in the Galactic halo and so their density in the Galactic halo would be signi"cantly enhanced over their average cosmological density. The dominant contribution of vortons to the EHECR #ux would then come from this clustered component with the concomitant advantages and disadvantages that are discussed later in Section 6.13. We add, however, that vortons are highly charged particles and as such they should be subject to a variety of cosmological constraints applicable to massive highly charged particles. The vortons also have a circulating current and hence behave essentially as point magnetic dipoles. These attributes may or may not have dramatic e!ects on their clustering properties, but remain to be studied in detail. 6.7. X particles from monopoles Compared to vortons and superconducting strings, magnetic monopoles as topological defects are perhaps somewhat more well studied in terms of their formation and evolution in the Universe. Formation of magnetic monopoles is essentially inevitable in most realistic GUT models. They lead to the well-known monopole overabundance problem, which historically played a major role in the development of the idea of in#ationary cosmology. For a review of monopoles and their cosmological implications, see e.g., Ref. [411]. The relevance of monopoles as possible sources of X particles in a top-down scenario of EHECR arose from the works of Hill [392] and Schramm and Hill [404]. If monopoles were formed at a phase transition in the early Universe, then, as Hill [392] suggested in 1983, formation of metastable monopole}antimonopole bound states } `monopoloniuma } is possible. At any temperature ¹, monopolonia would be formed with binding energy E Z¹. The initial radius r of " * a monopolonium would be r &g2 /(2E ), where g is the magnetic charge (which is related to the * . " . electronic charge e through the Dirac quantization condition eg "N/2, N being the Higgs "eld . winding number characterizing the monopole as a TD). Classically, of course, the monopolonium is unstable. Quantum mechanically, the monopolonium can exist only in certain `stationarya states characterized by the principal quantum number n given by r"n2aB , where n is a positive integer, . r is the instantaneous radius, and aB "8a /m is the `magnetica Bohr radius of the mono. % M polonium. Here a "1/137 is the `electrica "ne-structure constant, and m is the mass of % M a monopole. Since the Bohr radius of a monopolonium is much less than the Compton wavelength (size) of a monopole, i.e., aB ;m~1, the monopolonium does not exist in the ground (n"1) state, because . M then the monopole and the antimonopole would be overlapping, and so would annihilate each other. However, a monopolonium would initially be formed with n<1. It would then undergo a series of transitions through a series of tighter and tighter bound states by emitting initially photons and subsequently gluons, Z bosons, and "nally the GUT X bosons. Eventually, the cores of the monopole and the antimonopole would overlap, at which point the monopolonium would annihilate into X particles. Hill showed that the life time of a monopolonium is proportional to the
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cube of its initial radius. Depending on the epoch of formation, some of the monopolonia formed in the early Universe could be surviving in the Universe today, and some would have collapsed in recent epochs including the present epoch. The resulting X particles could then be a source of EHECR. The monopolonia collapsing in the present epoch would have been formed in the early Universe at around the epoch of primordial nucleosynthesis [392,398]. At that epoch, the monopole-plasma energy exchange time scale would still be smaller than the expansion time scale of the Universe [398], so the relevant monopolonium abundance at formation can be reasonably well described in terms of the classical Saha ionization formalism. On the other hand, the e`e~ annihilations at a temperature of &0.3 MeV (i.e., shortly after the nucleosynthesis epoch) signi"cantly reduces the e!ectiveness of monopole-plasma scatterings in maintaining thermal equilibrium of the monopoles. Thus although the relevant monopolonia are formed when the monopoles are still in thermal equilibrium, their subsequent `spiraling ina and collapse mostly occurs in a situation when the monopoles are e!ectively decoupled from the background plasma. Thus, the lifetime of the relevant monopolonia can be calculated to a good approximation by using the standard `vacuuma dipole radiation formula given by Hill [392]. The X particle production from collapsing monopolonia and the resulting EHECR #ux was studied in details in Ref. [398]. As in the case of collapse and/or successive self-intersections of cosmic string loops, the X particle production rate n5 (t) due to monopole-antimonopole annihilaX tions through monopolonia formation turns out to be proportional to t~3. The e$cacy of the process with regard to EHECR, however, depends on two parameters, namely, (a) the monopolonium-to-monopole fraction at formation (m ) and (b) the monopole abundance. The latter is & unknown. However, for a given monopole abundance, m is in principle calculable by using the & classical Saha ionization formalism. Phenomenologically, since a monopole mass is typically m &40m (so that each monoM X polonium collapse can release &80 X particles), we see from Eq. (68) that one requires roughly (only!) a few monopolonium collapse per decade within roughly every Solar system-size volume over a volume of radius &few tens of Mpc centered at Earth. Whether or not this can happen depends, as already mentioned, on m as well as on the monopole abundance, the condition [398] & being (X h2)hm K1.7]10~8(m /1016 GeV)1@2 [10 Mpc/l(E "300 EeV)], where X is the mass M & X c M density contributed by monopoles in units of closure density of the Universe. Thus, as expected, larger the monopole abundance, smaller is the monopolonium fraction m required to explain the & EHECR #ux. Note that, since m must be less than unity, the above requirements can be satis"ed as long & as (X h2)h'1.7]10~8(m /1016 GeV)1@2. Recall, in this context, that the most stringent bound M X on the monopole abundance is given by the Parker bound (see Ref. [411]), (X h2) [ M P!3,%3 4]10~3(m /1016 GeV)2. The estimate of m obtained by using the Saha ionization formalism M & [392,398] shows that the resulting requirement on the monopole abundance (in order to explain the EHECR #ux) is well within the Parker bound mentioned above. The monopolonium collapse, therefore, is an attractive scenario in this regard. A detailed study of the kinetics of monopolonium formation is needed to determine the monopolonium fraction at formation. The above scenario assumes, of course, that the well-known monopole overabundance problem (see, e.g., Ref. [411]) is `solveda by some mechanism, e.g., in#ation, but at the same time the scenario also assumes that a small but interesting relic abundance of monopoles was somehow left
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behind. Such a relic abundance could have been produced, for example, thermally during the reheating stage after the in#ationary phase. Also, various out-of-equilibrium processes, such as a phase transition to a transient superconducting phase [448,449] giving rise to transient magnetic #ux tubes connecting monopole}antimonopole pairs, could well enhance monopolonium abundances beyond the equilibrium estimates mentioned above and be more easily compatible with the required numbers derived above for explaining the EHECR #ux. These possibilities remain to be studied in detail. Recently, the kinetics of monopolonium formation process has been studied in Ref. [450] by solving the relevant Boltzmann equation. The authors of Ref. [450] claim that the resulting monopolonium abundance is too low to be able to explain the EHECR #ux. This is based on the observation that the typical energy loss time scale of monopolonium with the plasma due to friction is smaller than the Hubble time by a factor K10m /M <1 before recombination such M Pthat bound states can be formed only after recombination. Instead, Ref. [450] suggests a di!erent (non-thermal) mechanism of monopolonium formation in which essentially all monopoles and antimonopoles are connected by strings formed at a relatively low energy phase transition (at &100 GeV). The monopole `magnetica #ux is assumed to be completely con"ned inside the strings } the monopoles are also assumed to have no other uncon"ned charges } so that the monopolonia decay mainly through emission of gravitational radiation (rather than electromagnetic radiation), with lifetimes comparable to the age of the Universe. It is claimed that in this scenario the relic abundance of monopolonia can be su$cient to explain the EHECR #ux. This mechanism, however, remains to be studied in detail. An interesting possibility is that monopolonia, unlike monopoles, may be clustered in the Galactic halo. Monopoles may be accelerated by the Galactic magnetic "eld causing them to escape (`evaporatea [411]) from the halo even if they were initially clustered there. However, monopolonia, being magnetically neutral, should be immune to the Galactic magnetic "eld: it is easy to check that typical Galactic magnetic "eld strength of &few lG is too weak to `ionizea a monopolonium. In this respect, the clustering properties of monopolonia in the Galactic halo should be very similar to the standard Cold Dark Matter. Thus, as mentioned in the case of vortons above, the density of monopolonia in the Galactic halo may be signi"cantly enhanced over their average cosmological density in the Universe [406]. This means that the actual universal monopolonium abundance required for explaining the EHECR could be even lower than the estimates obtained above assuming unclustered distribution of monopolonia in the Universe. The signatures of clustered monopolonia as sources of EHECR will in many respects be similar to those of metastable massive relic particles discussed in Section 6.13. 6.8. X particles from cosmic necklaces A cosmic necklace is a possible hybrid topological defect consisting of a closed loop of cosmic string with monopole `beadsa on it. Such a hybrid defect was "rst considered by Hindmarsh and Kibble [451]. Such hybrid defects could be formed in a two-stage symmetry-breaking scheme such as GPH];(1)PH]Z . In such a symmetry breaking, monopoles are formed at the "rst step of 2 the symmetry breaking if the group is semisimple. In the second step, `Z a strings are formed, and 2 then each monopole gets attached to two strings, with monopole magnetic #ux channeled along
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the string. Possible production of massive X particles from necklaces has been pointed out in Ref. [390]. The evolution of the necklace system is not well understood. The crucial quantity is the dimensionless ratio r,m /(kd), where m denotes the monopole mass, k is the string energy per M M unit length, and d is the average separation between a monopole and its neighboring antimonopole along the string. For r;1, the monopoles play a subdominant role, and the evolution of the system is similar to that of ordinary strings. For r<1, the monopoles determine the behavior of the system. Authors of Ref. [390] assume that the system evolves to a con"guration with r<1. This is a crucial assumption, which remains to be veri"ed by numerical simulations. If this assumption holds, then one may expect that the monopoles sitting on the strings would tend to make the motion of the closed necklaces aperiodic, leading to frequent self-intersections of these necklaces and to eventual rapid annihilation of the monopoles and antimonopoles trapped on necklaces. This would lead to X particle production. The X particle production rate from necklaces is given by [390] n5 &r2k/m t3 . (91) X X Except for numerical factors, this equation has the same form as Eq. (84) for cosmic string loops with k replaced by r2k. For suitable choices of values of r2k, necklaces can explain the observed EHECR. One advantage of the necklace scenario is that, for su$ciently large values of r, the distance between necklaces can be small enough that su$cient number of necklaces may be expected within a typical `GZKa radius of few tens of Mpc. For su$ciently large r, necklaces may also cluster within the Local Supercluster, and may even cluster on galactic scales [406]. Again, necklaces clustered in the Galactic halo could be an attractive source of EHECR. A somewhat related monopole-string system, namely, a network of monopoles connected by strings [452] as possible sources of EHECR was studied in Ref. [453]. This system is obtained by replacing the factor Z in the symmetry-breaking scheme mentioned above by Z with N'2. 2 N In this case, after the second stage of symmetry breaking, each monopole gets attached to N'2 strings. Each monopole is pulled in di!erent directions because of the tension in the strings attached to it. The net acceleration su!ered by a monopole due to these pulls causes it to radiate gauge boson quanta, mostly photons and gluons (monopoles carry both ordinary magnetic charge as well as color-magnetic charge). However, the predicted #ux at EHECR energies turns out to be too low to explain the observed #ux for all reasonable values of the parameters of the system [453,406]. 6.9. A general parametrization of production rate of X particles from topological defects From the above discussions on various di!erent kinds of topological defects as possible sources of EHECR, it is clear that di!erent kinds of defects in general produce X particles at di!erent rates. It was suggested in Ref. [200] that X-particle production rate for any general TD process may, on dimensional grounds, be expressible in terms of the two fundamental parameters entering in the problem, namely, the mass-scale m (which, in turn, is related to the symmetry-breaking scale X at which the relevant TDs were formed) and the Hubble time t in the form (in natural units with +"c"1) n5 (t)"imp t~4`p , X X
(92)
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where i and p are dimensionless constants whose values depend on the speci"c process involving speci"c TDs under consideration, or alternatively, in the form n5 (t)"(Q /m )(t/t )~4`p , (93) X 0 X 0 where Q is the rate of energy injected in the form of X particles of mass (energy) m per unit 0 X volume in the present epoch, and t denotes the present age of the Universe. The quantity 0 Q depends on the speci"c TD process under consideration. 0 The above forms for n5 are expected to be valid for any TD systems for which there is no intrinsic X time and energy scales involved other than the Hubble time t and mass scale m . This is the case in X situations in which the TDs under consideration evolve in a scale-independent way. As discussed above, this `scalinga is indeed a property of evolution of cosmic strings. The same is true for X particle production from monopolonia and necklaces.20 For a given TD process, the quantity Q is in principle calculable. However, in practice, for 0 essentially all kinds of TDs as discussed above, the evolutionary properties of the TD systems are not known well enough to allow us to calculate the values of Q a priori in a parameter-free 0 manner. Nevertheless, for a given value of the parameter p, the above parametrization of the X particle production rate allows us to study the TD scenario of EHECR in a general way (i.e., without referring to any speci"c TD process) by suitably normalizing the value of Q so as to 0 explain the observed EHECR data and then checking to see if the value of Q so obtained is 0 consistent or not with other relevant data (such as the di!use gamma ray background in the 10 MeV}100 GeV region; see Section 7). Except for the cusp evaporation process, other relevant X particle production processes involving cosmic strings studied so far are characterized by Eq. (93) with p"1, as are the processes involving monopolonia and necklaces. The decaying vorton scenario is characterized by p"2. On the other hand, superconducting cosmic string scenarios studied so far correspond to p(1. As we shall discuss in Section 7, TD processes with p(1 generally lead to unacceptably high rate of energy injection in the early cosmological epochs, which would cause excessive 4He photodisintegration and CMBR distortion [437], and are, therefore, currently unfavored in the context of EHECR. 6.10. TDs, EHECR, and the baryon asymmetry of the Universe In the TD scenario of EHECR origin, the X particles typically belong to some Grand Uni"ed Theory. The decay of the X particles may, therefore, involve baryon number violation. Based on this observation, it has been suggested [419] that there may be a close connection between the EHECR and the observed baryon asymmetry of the Universe (BAU). Indeed it may be the case that both arise from the decay of X particles released from TDs. Production of X particles from TDs is an irreversible process, so the standard out-of-thermal-equilibrium condition necessary for the creation of BAU is (and hence the famous Sakharov conditions are) automatically satis"ed. 20 In the case of hybrid defects such as necklace, there are more than one mass scales involved. However, the time dependence of n5 is still expressible in the form of Eq. (92) or (93); cf. Eq. (91). X
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In this scenario, the X particles released from TDs in the early epochs produce the BAU, whereas the EHECR are due to X particle decay in the recent epoch. As indicated by Eq. (93), the rate of X particle production by TDs in the early epochs was higher than it is now. Normalizing the present-day rate of X particle production from the requirement to explain the EHECR #ux, the total integrated baryon asymmetry produced by decays of all X particles released from TDs at all epochs in the past can be calculated. As pointed out in Ref. [419], depending on the amount of baryon number violation in each X particle decay, which unfortunately is unknown and is model dependent, the net baryon asymmetry produced can account for or at least be a signi"cant fraction of the observed BAU. Thus, if this scenario is correct, then not only the extremely high-energy cosmic rays, but the entire low-energy baryonic content of the Universe today may at some stage or another have arisen from decay of massive particles from TDs, and the EHECR observed today would then represent the baryon creation process `in actiona in the Universe today. The baryon asymmetry should, in principle, be re#ected in the observed EHECR, but it will be extremely di$cult, if not impossible, to detect this in EHECR. Realistic calculations including all relevant baryon number violating processes within speci"c GUT models will be needed to explore this idea further. 6.11. TeV-scale Higgs X particles from topological defects in supersymmetric theories We have so far dealt with X particles of mass <1011 GeV produced by topological defects. Recently, however, it has been realized (see Ref. [418] for details and other references) that in a wide class of supersymmetric gauge theories, the relevant Higgs boson can be `lighta, of mass m &TeV H (the `softa supersymmetry breaking scale), whereas the gauge boson can be much heavier with mass m [1016 GeV, the GUT scale. In these theories, therefore, topological defects can simultaneously V be sources of the Tev mass-scale Higgs bosons as well as the GUT mass-scale gauge bosons. It has been suggested [418] that while the superheavy gauge bosons may act as the X particles generating the EHECR, there is now an additional direct source of energy injection at the TeV scale due to decays of the Higgs bosons, which may contribute signi"cantly to the extragalactic di!use c-ray background above &10 GeV which also seems to be di$cult to explain in terms of conventional sources. Some implications of this are discussed further in Section 7. 6.12. TDs themselves as EHECR particles Strictly speaking, the subject of this section belongs to Section 5 because the basic ideas discussed below involve acceleration mechanisms rather than any top-down decay mechanism. Nevertheless, since the objects which are accelerated are topological defects themselves, we discuss them here. Two situations have been discussed in literature, involving monopoles and vortons: we discuss them in turn. 6.12.1. Monopoles as EHECR particles It has been suggested by Kephart and Weiler [454}456], following an earlier suggestion by Porter [457], that magnetic monopoles of mass m &109}1010 GeV may themselves act as the M EHECR particles. This is an attractive suggestion because, from the point of view of energetics, monopoles can indeed be easily accelerated to the requisite EHECR energies by the Galactic
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magnetic "eld. A monopole of minimum Dirac magnetic charge q "e/2a (where a"e2/4pK M 1/137 is the "ne-structure constant) will typically acquire a kinetic energy E &q B¸ JNK5.7]1020(B/(3]10~6 G))(¸ /300 pc)1@2(R/30 kpc)1@2 eV (94) K M # # in traversing through the Galactic magnetic "eld region of size R&30 kpc containing a coherent magnetic "eld B&3]10~6 G with a coherence length ¸ &300 pc, where N&R/¸ is the average # # number of coherent magnetic domains encountered and the JN factor takes account of the random di!erence in the magnetic "eld orientations within di!erent coherent domains. In order to ensure that air-showers induced by monopoles contain relativistic particles, the monopoles themselves must be su$ciently relativistic which requires that the monopole mass be m [1010 GeV. Such relatively low mass monopoles must be formed at a symmetry-breaking scale M [109 GeV. These monopoles would also be interesting because they would be free of the usual monopole over-abundance problem associated with GUT-scale monopoles of mass &1017 GeV formed at the GUT symmetry-breaking phase transition at a scale of &1016 GeV. In fact, it is a curious coincidence [454,455] that the observed EHECR #ux lies just three to four orders of magnitude below the `Parker limita (see, e.g., [411]) on the Galactic monopole #ux for monopoles of the required mass &1010 GeV. This interesting coincidence has prompted Kephart and Weiler to speculate that this possible connection between EHECR and monopoles may be a hint towards some dynamical reason that forces the monopole #ux to saturate the Parker bound.21 There are, however, several uncertainties in this monopole scenario of EHECR. The precise mechanism behind, and the nature of, monopole-induced air showers are largely unknown. A monopole is expected to have an intrinsic strongly interacting hadronic `clouda around it, of typical dimension &K~1 & few fm. Thus monopoles, like protons, are expected to have a typical QCD strong interaction cross section for interaction with air nuclei. In addition, a variety of other monopole}nucleus interactions are possible, such as enhanced monopole-catalyzed baryon number violating processes with a strong cross section of &10~27 cm2 [459], possible binding of nuclei to monopoles [460] (in which case the monopole}air interaction would resemble a relativistic nucleus}nucleus collision), and so on. Bound states of charged particles and monopoles as EHECR primaries have also been considered recently in Ref. [461] where it has been suggested that the EAS spectrum created by such primaries should exhibit a line spectrum component. This speci"c prediction should be easy to test with next generation UHECR experiments. Several other possible strong as well as electromagnetic interactions of monopoles with nuclei are mentioned by Weiler [456]. One of the major problems, however, is that although in many cases the relevant cross sections can be large, the required large inelasticities (i.e., large energy transfers) are generally di$cult to realize for a massive particle like the monopole [401]. Another problem arises from considerations of distributions of arrival directions of individual EHECR events [462]. The arrival directions of monopole primaries are expected to show preference for the local Galactic magnetic "eld directions. However, the arrival directions of the observed EHECR events seem to show no such preference [462]. Moreover, Monte Carlo calculations [462] indicate that the expected spectrum of monopoles accelerated in the Galactic 21 A scenario in which monopoles `naturallya occur with an abundance at the level of the Parker saturation limit was discussed earlier in connection with certain phase transitions in some superstring theories [458].
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magnetic "eld is very di!erent from that of the observed UHECR. However, the spectrum beyond 1020 eV is not yet well measured, and only future measurements of the EHECR spectrum by the up-coming large-area detectors will hopefully be able to settle this issue. Thus, it is not yet possible to completely rule out the monopole scenario of EHECR. However, it seems rather unlikely [401] at the present time. Clearly, more theoretical work especially on the air-showering aspects of monopoles are needed. 6.12.2. Vortons as EHECR particles Vortons were already discussed in Section 6.6. Vortons are highly charged particles and can, therefore, be accelerated in powerful astrophysical objects such as AGNs, radio galaxy hot-spots, and so on, if vortons are present in those objects with su$cient abundance } a possibility if vortons are (at least a part of) the ubiquitous dark matter. Authors of Ref. [446] have proposed vortons of mass m &Zm with Z&100 and m&g &g &109 GeV as the EHECR particles. Like the case of V 4 p monopoles discussed above, accelerating vortons to the requisite energies seems to be no problem. The main problem, however, is that the interaction properties of vortons with ordinary matter and the kind of atmospheric air-showers they are likely to generate are highly uncertain, and so nothing much de"nite can be said about this possibility. For details on one particular model studied so far, see Ref. [446], which suggests that vortons should produce a line spectrum component of EAS, similar to the case of charged particle}monopole bound state primaries [461]. 6.13. EHECR from decays of metastable superheavy relic particles 6.13.1. General considerations It has been suggested recently by Kuzmin and Rubakov [463] and by Berezinsky et al. [389] (see also Ref. [464]) that EHECR may be produced from decay of some metastable superheavy relic particles (MSRPs) of mass m Z1012 GeV and lifetime larger than or comparable to the age of the X Universe.22 The long but "nite lifetime of MSRPs could be due to slow decay of MSRPs through non-perturbative instanton e!ects [463] or through quantum gravity (wormhole) e!ects [389] which induce small violation of some otherwise conserved quantum number associated with the MSRPs. The (almost) conserved quantum number could be due to some global discrete symmetry, for example. In other words, but for the instanton and/or quantum gravity e!ects, the MSRPs would have been absolutely stable, and their long but "nite lifetime is then due to small violation of some protector symmetry. Possible candidates for MSRPs and their possible decay mechanisms giving them long lifetime have been discussed in the context of speci"c particle physics/superstring theory models in Refs. [464,466,391,467,468]. Several non-thermal mechanisms of production of MSRPs in the postin#ationary epoch in the early Universe have been studied. They include gravitational production through the e!ect of the expansion of the background metric on the vacuum quantum #uctuations of the MSRP "eld [469,472], or creation during reheating at the end of in#ation if the MSRP "eld couples to the in#aton "eld [463,470]. The latter case can be divided into three subcases, namely 22 This possibility, with a speci"c MSRP candidate from superstring theory called `cryptona [465] in mind, was noted several years ago in a footnote in Ref. [397]. However, it was not explored there further.
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`incoherenta production with an abundance proportional to the MSRP annihilation cross section, non-adiabatic production in broad parametric resonances with the in#aton "eld during preheating, and creation in bubble wall collisons if in#ation is completed by a "rst order phase transition [471]. A recent review of MSRP production mechanisms associated with in#ation is given in Ref. [413]. Under certain circumstances MSRPs can even exist in the present-day Universe with su$cient abundance so as to act as non-thermal superheavy dark matter. The MSRPs of mass Z1012 GeV, if they are to act as superheavy dark matter, would have to have been created in non-thermal processes, because the unitarity limit on the self-annihilation cross section imposes [473] an upper bound of &500 TeV on the mass of any relic dark matter particle species that was once in thermal equilibrium in the early Universe. Another possibility is the thermal production of such MSRPs with a subsequent substantial entropy production, for example, by thermal in#ation [474]. Various phenomenological aspects of the MSRP decay scenario of EHECR origin has recently been studied further in Refs. [391,406,475}478]. In order to explain the observed EHECR #ux, a speci"c relationship between the abundance of MSRPs and their lifetime must be satis"ed. We can see this from the discussions already presented in Section 6.2.3. Let us assume for simplicity that each MSRP decay produces roughly a similar number of quarks and/or leptons as in the case of X particles from topological defects (however, this need not [463] be the case } see below). Also let us "rst assume that the MSRPs (we shall call them X particles throughout this section) are distributed uniformly across the Universe, and as in Section 6.2.3 we consider decay of the X particles in the present epoch only. The decay rate is simply n5 "n /q , where q is the lifetime X X X X as de"ned through the usual exponential decay law. The quarks from the decay of these X particles will hadronize in the same way as discussed in Section 6.2, producing a photon and neutrino dominated injection spectrum. Then using Eq. (68) with a"1.5 we see that in order to produce the observed #ux of EHECR (assuming they are photons), we require q K2.8]1021(X h2)(l /100 Mpc)(1012 GeV/m )1@2 yr , (95) X X E X where X is the cosmic average mass density contributed by the MSRPs in units of the critical X density o K1.05]10~4h2 GeV cm~3, h is the present value of Hubble constant in units of c 100 km s~1 Mpc~1, and l is the e!ective penetration depth of UHE radiation. E Thus, for X h2&1 (maximal value allowed), one requires q &3]1021 yr for m "1012 GeV X X X and l "100 Mpc. Recall that in order to produce the EHECR particles (assuming they E are nucleons and/or photons), the relic X particles must decay in the present epoch. Thus q X cannot be much smaller than &1010 yr because then most of the MSRPs would have already decayed in earlier epochs.23 With q &t &1010 yr, one needs only a tiny MSRP abundance X 0 X h2&3]10~12 (for m "1012 GeV) to explain the EHECR #ux. X X Currently, neither the abundance X nor the lifetime q of the proposed MSRPs is known with X X any degree of con"dence. Clearly, in order to produce (but not overproduce) the EHECR #ux, some "ne tunning between X and q , which should approximately satisfy the condition given by X X Eq. (95), is needed. Note that since m has to be Z1012 GeV (to explain the EHECR), the MSRP X lifetime cannot exceed &1022 yr. 23 On the other hand, q
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It is expected that the MSRPs would behave like the standard cold dark matter (CDM) particles in the Universe. If X h2&1, then MSRPs would be the CDM or at least a signi"cant part of it. The X important point noted in Ref. [389] is that irrespective of the value of the universal MSRP density X , the ratio m "o /o of mass density contribution from the X particles to that from CDM X X X CDM particles should be roughly same everywhere in the Universe, since both X particles and CDM particles would respond to gravity in the same way. Thus, the value of m within individual galactic X halos should be same as that in the extragalactic space. Thus, since CDM particles (by de"nition) are clustered on galactic halo scales, so will be the X particles, and that by the same factor. In particular, MSRPs are expected to be clustered in our own Galactic Halo (GH). Using the solar neighborhood value of the DM density in the GH, oH K0.3 GeV cm~3 (see, e.g., Ref. [411]) as CDM,_ a reference value for the average CDM density oH in the GH, one can write the average number CDM density of X particles in the GH as nH K3]10~13(X /X )(1012 GeV/m )(oH /0.3 GeV cm~3) cm~3 , (96) X X CDM X CDM where X "o /o gives the cosmic average mass density of CDM. Noting that the cosmic CDM CDM c average number density of X particles is n#04K10~16(X h2)(1012 GeV/m ) cm~3, we see that the X X X GH contribution to the predicted EHECR #ux from MSRP decay exceeds the extragalactic (EG) MSRP decay contribution by a factor f given by f,j /j "f(nH /n#04)(R /l )K15f(0.2/X h2)(R /100 kpc)(100 Mpc/l ) , (97) H %9 X X H E CDM H E where R is the radius of the GH, and f is a geometric factor of order unity determined H by the spatial distribution of the X particles in the Halo. If the X particles constitute a dominant fraction of the CDM in the GH, then their density must fall o! as 1/r2 at large Galactocentric distances in order to explain the #at rotation curve of the Galaxy. In the situation in which the X particles do not constitute the dominant component of the CDM in the Galaxy, the spatial distribution of X particles in the GH may, depending on the nature of their interaction with other matter, be di!erent from that of CDM. However, if the X particles are only very weakly interacting with other particles } interacting mainly through gravity } then they will be expected to be distributed like the CDM irrespective of their density contribution. Thus, for an isothermal density pro"le [479], for example [406,475}477], fK6.7. For another model [480], fK2. From the above discussion, it is clear that in this scenario the contribution of X particles to the EHECR #ux will be dominated by the GH component over the EG component by a factor of order 10 or more except probably for the case of neutrino #ux for which one should replace l in the E above equation by the neutrino absorption length, <100 Mpc. For neutrinos the EG contribution would be comparable to or greater than the GH contribution. For UHECR protons with energy E;E &5]1019 eV, the e!ective energy attenuation length scale becomes large (<100 Mpc), GZK so the EG proton component can also be comparable to the GH proton component at these energies. The immediate conclusion [389,406] that follows from the above discussion is that in this GH dominated X particle decay (GHXPD) scenario, the spectrum of UHECR, which is predicted to be photon dominated, should show (almost) complete absence of the GZK suppression. The required energy injection rate in the form of X particles is also not constrained by the EGRET measured c-ray background because there is practically no EHE c-ray cascading } R is too small compared H
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to EHE photon interaction length of Z10 Mpc (see Fig. 11). The apparent absence of the GZK suppression in the recent UHECR data from the AGASA experiment [8] may be taken as indicative [82] of support to the GHXPD scenario, although the AGASA EHECR data can also be consistent with a general EG X particle decay scenario (involving topological defects, for example) with p"1 of Eq. (93) (see Ref. [206] for the most recent calculations). Note in this context that a general top-down scenario in which X particles are MSRPs can be e!ectively described by Eq. (93) with p"2. Also, as mentioned earlier, some kinds of topological defect sources of (unstable) X particles such as monopolonium, vortons, and possibly necklaces can also be clustered in the GH, and will produce EHECR spectrum having roughly the same characteristics as that in the MSRP related GHXPD scenario being discussed here. An example of the #uxes predicted in the GHXPD scenario is shown in Fig. 30. Note that with the GH contribution dominating over the EG component, the condition expressed by Eq. (95) should be modi"ed } the r.h.s. of Eq. (95) will now get multiplied by the factor f of Eq. (97). Thus, for a given q the required universal abundance of MSRPs, X , will be a factor X X f lower than what is indicated by Eq. (95). Finally, we note that the number of quarks and/or leptons resulting from the decay of the MSRP X particle has been assumed to be around 2 or 3 in practically all calculations done so far. This is reasonable in a perturbative decay picture. On the other hand, as pointed out by Kuzmin and Rubakov [463], instanton mediated decay processes (which may be needed for making the MSRPs su$ciently long-lived) typically lead to multiparticle "nal states. Thus instanton mediated X particle decays will typically give a relatively large number (&10) of quarks with a fairly #at distribution in energy, rather than typically 1 or 2 quarks expected in the perturbative X decay scenario implicitly assumed, for example, in the topological defect scenario. In other words, the numbers NI and N used in the discussion of Section 6.2.2 may be quite di!erent from what q are usually assumed in the top-down scenario. This may leave a distinguishing signature in the predicted spectrum which may help in distinguishing the topological defect scenario from the MSRP scenario [463]. 6.13.2. Anisotropy An important signal of the GHXPD scenario of EHECR, pointed out by Dubovsky and Tinyakov [475], is the expected anisotropy of the EHECR #ux. Because of the o!-center location of the Solar system with respect to the (assumed spherical) GH, some amount of anisotropy of the EHECR #ux measured at Earth is expected. This is important because an experimental con"rmation of the absence of the predicted anisotropy will be a de"nite evidence against the GHXPD model. The expected anisotropy has been calculated in Refs. [406,475}477] in several di!erent GH models. The predicted anisotropy varies from about 10}40% depending on the parameters of the GH model. The strongest measure of the anisotropy in this model is associated with the large ratio of the predicted UHECR #ux from the direction of the galactic center (GC) to that from the direction of the galactic anticenter (GA) [475,406]. Unfortunately, except for one southernhemisphere EAS array, namely, the now-defunct SUGAR array in Sydney, Australia, none of the major currently operating as well as closed EAS arrays (AGASA, Fly's Eye, Haverah Park, Yakutsk } all located in the northern hemisphere) can `seea the GC. So this strongest signal of the predicted anisotropy will have to wait to be tested until the southern hemisphere detector of the Pierre Auger project is built and made operational. In the meantime, large northern hemisphere
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EAS arrays should see a dip in the measured EHECR #ux from the direction of the GA relative to the direction perpendicular to the Galactic plane [406]. Another strong signal of the GHXPD model are the expected `dipsa in the GH #ux enhancement factor f of Eq. (97), due to direct #ux from relatively close-by EG dark matter clumps represented by objects such as the the Andromeda galaxy or the Virgo cluster [389,406,475,476]. According to Ref. [477], in the case in which the GH component becomes dominant over the EG component at EZ1019 eV, the latest AGASA data already rule out the GHXPD model. On the other hand, in the case in which the GH component makes the dominant contribution to the #ux measured by AGASA only at E Z 1020 eV, the model cannot yet be ruled out even for the largest predicted anisotropy in the model: the number of events (&6) is too low to allow any statistically signi"cant conclusion to be drawn. On the other hand, according to the analysis of Ref. [476] (BSW) which includes the data from the now defunct SUGAR array in the analysis, the GHXPD model is ruled out, because these authors claim that the predicted anisotropies are much higher than those observed and that the predicted #ux from the Andromeda galaxy is not seen. On the basis of their analysis, BSW claim that the GHXPD scenario cannot contribute more than &10% to the observed UHECR #ux. However, Berezinsky and Mikhailov [477] have pointed out that in the light of the latest AGASA data [8] the GHXPD contributes dominantly to the observed #ux only at energies E'1020 eV where the SUGAR array most probably observed no events, so the inclusion of the SUGAR array data could bias the analysis signi"cantly. Clearly, this important question of anisotropy remains unresolved. In addition, the recent evidence for small-scale clustering found by the AGASA experiment [83] may require a signi"cant clustering of the dark matter component consisting of the decaying X particles on scales of the Galactic halo. As already mentioned, the up-coming large arrays, especially the southern hemisphere array of the Pierre Auger project will have the capability to unambiguously resolve the issue. For a recent more detailed study of the question of anisotropy in the GHXPD model, see, Ref. [481]. In scenarios where neutrinos have a mass in the eV range and UHE neutrinos from MSRP decays at large cosmological distances <100 Mpc give rise to EHECR within 100 Mpc from Earth through decay of Z bosons resonantly produced by interactions of the primary neutrino with the relic neutrinos (see Section 4.3.1), the expected EHECR angular distribution is not expected to correlate with the GH. Rather, the EHECR anisotropy is then expected to correlate with the large scale structure at a certain redshift range which is determined by the resonance condition for the annihilating primary neutrinos [207]. 6.14. Cosmic rays from evaporation of primordial black holes A recent proposal that does not involve the production of X particles and therefore, in the strict sense does not belong to the top-down scenarios, concerns the production of UHECR during evaporation of primordial black holes [482]. It was claimed that the required black hole abundance is consistent with observational constraints on the local rate of black hole evaporation. This mechanism may, however, not contribute to the UHECR #ux at all due to the formation of a photosphere around the black hole whose typical temperature is of the order of the QCD scale [483]. The photosphere strongly reduces the possibility of observing individual black holes with temperatures greater than the QCD scale, since the high-energy particles emitted from the black hole are recycled to lower energies in the photosphere. However, primordial black holes may
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provide a signi"cant contribution to the Galactic #ux of protons, electrons, their antiparticles, and c-rays in the 100 MeV range, a fact that has extensively been used to in turn constrain the number of black holes formed in the early Universe [484].
7. Constraints on the topological defect scenario Scenarios of UHECR production that are related to new physics near the Grand Uni"cation scale exhibit a striking di!erence to conventional acceleration models: whereas acceleration is tied, in one form or another, to astrophysical objects and magnetized shocks associated with them and took place at redshifts not greater than a few, energy release associated with Grand Uni"cation scale physics takes place not only today, but already in the early Universe up to temperatures corresponding to the Grand Uni"cation scale. Consequently, this energy release } that can roughly be estimated by normalizing its present day rate to the value required to explain the observed UHECR #ux } can imply other e!ects that are constrained by observations. Such e!ects are a di!use c-ray background below &100 GeV (see Section 7.1) and its in#uence on light element abundances through photo-disintegration of light nuclei (Section 7.2), and on distortions of the CMB (Section 7.3), as well as comparatively high di!use neutrino #uxes above &1018 eV (Section 7.4). 7.1. Low-energy diwuse c-ray background: role of extragalactic magnetic xeld and cosmic infrared background In Section 4.2 the development of EM cascades and the resulting di!use c-ray background were discussed qualitatively, and Section 4.4.1 addressed how magnetic "elds can modify the picture. Here, we demonstrate concrete implications of these general aspects for the low-energy di!use c-ray background predicted by TD scenarios. A simple analytical estimate of the saturated cascade c-ray spectrum can be given in the case when the CMB is the only relevant low-energy photon background in which the cascading takes place (E is now the photon energy) [35,485]:
G
(E/E )~1.5 for E(E , x x u #!4 j#!4(E)K for E (E(E , (98) ] (E/E )~2 c x x # E2(2#ln(E /E )) x # x 0 for E 'E . # Here, u is the total release of EM energy per unit volume above E , and E and E are #!4 # # x characteristic redshift dependent energies given by [486] E Km2/22¹ K(4.9]104 GeV)/(1#z) , # e CMB (99) E K0.04E K(2]103 GeV)/(1#z) , x # where ¹ K2.735(1#z) K is the CMB temperature at the epoch characterized by redshift z. Ref. CMB [182] comes to a very similar result for the cascade spectrum for all photon background spectra that fall o! steeper than e~2. It was, furthermore, pointed out that near E the spectrum steepens to # E~5 when photon}photon scattering becomes important [189]. This e!ect, however, becomes
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important only at redshifts larger than about 100 and therefore does not play a role in viable TD scenarios where most of the cascade #ux is created at lower redshifts. The in#uence of photon backgrounds at energies above the CMB energies such as the IR/O backgrounds can approximately be taken into account by multiplying the right-hand side of Eq. (98) with exp[!q(E)], where q(E) is the optical depth for pair creation on these backgrounds at energy E over the distance to the source. Due to the short interaction lengths for PP and ICS above E , the cascade spectrum forms # essentially instantaneously on cosmological time scales. For E[E , PP is strongly suppressed and # ordinary pair production, i.e. the Bethe}Heitler (BH) process, and Compton scattering (CS) become the dominant processes. The photon energy attenuation length is then given by l~1(E)"n [(1!>)pBH(E)#(>/4)pBH(E)#(1!>/2)g (E)p (E)] , (100) E B H H% CS CS where n is the baryon number density, > is the mass fraction of 4He, pBH and pBH are the BH cross B H H% sections on hydrogen and 4He, and p and g are cross section and inelasticity for CS. For CS CS z[103, l (E)ZH~1(1#z)~3@2, and the Universe is therefore transparent for the cascade photons E 0 which can be directly observed as di!use c-ray background. In Fig. 27 the maximal instantaneous release of energy u that results from comparing Eq. (98) with the observed c-ray background at #!4 200 MeV is shown as a function of redshift.
Fig. 27. Maximal energy release in units of the CMB energy density allowed by the constraints from the observed c-ray background [185] at 100 MeV (dotted curve limiting left most range) and 5 GeV (dotted curve limiting next to left most range), CMB distortions (dashed curve, from Ref. [500]), and 4He photo-disintegration as a function of redshift z. These bounds apply for instantaneous energy release at the speci"ed redshift epoch.
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The measurement of the di!use c-ray background between 30 MeV and 100 GeV [185] imposes constraints on the total amount of EM energy density u K4.5]10~6 eV cm~3 injected and #!4 recycled by cascading to lower energies. These constraints and their role for TD models have "rst been pointed out in Ref. [430]. There, cascading in the CMB and the IR/O backgrounds was considered for EM injection at comparatively low redshifts. An analytical estimate based on Eq. (98) with u determined from the EHE c-ray #uxes in CEL approximation, Eq. (12), #!4 neglecting any EGMF, was given in Ref. [437]. Since the di!use c-ray spectrum observed between K100 MeV and K100 GeV is roughly JE~2.1 [185], the most stringent limit is obtained at the highest energy. As a result, scenarios with p"0 such as the simplest TD models involving superconducting cosmic strings are ruled out altogether [437]. Further, in case of a power-law fragmentation function, scenarios with p"1 such as models involving ordinary cosmic strings or annihilation of magnetic monopoles and antimonopoles require the power-law index to satisfy qZ2!3/(3#log (m /1023 eV)) , (101) 2 10 X e.g., qZ1.7 for a GUT scale mass m "1016 GeV. X More accurate numerical calculations take into account the IR/O background as well as any EGMF and the development of unsaturated cascades at UHE and its impact on the normalization of the energy release. For the following the new estimates of the IR background [163] are assumed. In addition, to be conservative in terms of scenarios obeying all constraints, the strongest URB version from Ref. [175] (see Fig. 10) is assumed. Fig. 28 shows results from Ref. [206] for the time averaged c-ray and nucleon #uxes in a typical TD scenario, assuming no EGMF, along with current observational constraints on the c-ray #ux. The spectrum was optimally normalized to allow for an explanation of the observed EHECR events, assuming their consistency with a nucleon or c-ray primary. The #ux below [2]1019 eV is presumably due to conventional acceleration in astrophysical sources and was not "t. Similar spectral shapes have been obtained in Ref. [295], where the normalization was chosen to match the observed di!erential #ux at 3]1020 eV. This normalization, however, leads to an overproduction of the integral #ux at higher energies, whereas above 1020 eV, the "ts shown in Figs. 28 and 29 have likelihood signi"cances above 50% (see Ref. [403] for details) and are consistent with the integral #ux above 3]1020 eV estimated in Refs. [7,8]. The PP process on the CMB depletes the photon #ux above 100 TeV, and the same process on the IR/O background causes depletion of the photon #ux in the range 100 GeV}100 TeV, recycling the absorbed energies to energies below 100 GeV through EM cascading (see Fig. 28). The predicted background is not very sensitive to the speci"c IR/O background model, however [487]. The scenario in Fig. 28 obviously obeys all current constraints within the normalization ambiguities and is therefore quite viable. We mention at this point, however, that TD scenarios are constrained by the true extragalactic contribution to the di!use c-ray background which might be signi"cantly smaller than limits on or measurements of the di!use c-ray #ux. This was indeed suggested recently [488] for the EGRET measurements near 1 GeV which could be dominated by contributions from the Galactic halo. In addition, the bulk of the extragalactic c-ray background may be caused by unresolved blazars [489], although there is some disagreement on this issue [490]; for example, a recent analysis indicates that only K25% of the di!use extragalactic emission can be attributed to unresolved c-ray blazars [491]. Other sources such as secondary c-rays from the interactions of CR con"ned in galaxy clusters also contribute to the extragalactic c-ray background [117]. In any of these cases
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Fig. 28. Predictions for the di!erential #uxes of c-rays (solid line) and protons and neutrons (dotted line) in a TD model characterized by p"1, m "1016 GeV, and the decay mode XPq#q, assuming the supersymmetric modi"cation of X the fragmentation function, Eq. (57), with a fraction of about 10% nucleons. The calculation used the code described in Ref. [206] and assumed the strongest URB version shown in Fig. 10 and an EGMF ;10~11 G. 1 sigma error bars are the combined data from the Haverah Park [3], the Fly's Eye [7], and the AGASA [8] experiments above 1019 eV. Also shown are piecewise power law "ts to the observed charged CR #ux (thick solid line) and the EGRET measurement of the di!use c-ray #ux between 30 MeV and 100 GeV [185] (solid line on left margin). Points with arrows represent upper limits on the c-ray #ux from the HEGRA [257], the Utah-Michigan [510], the EAS-TOP [511], and the CASA-MIA [258] experiments, as indicated.
Fig. 29. Same as Fig. 28, but for an EGMF of 10~9 G.
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the constraints on energy injection in TD models discussed here would consequently become more stringent, typically by a factor 2}3. However, above K10 GeV, another component might be necessary and a cascade spectrum induced by decay of heavy particles of energy beyond K100 GeV seems to "t the observed spectrum between 10 and 100 GeV [418], as can also be seen in Fig. 28. Furthermore, the c-ray background constraint can be circumvented by assuming that TDs or the decaying long lived X particles do not have a uniform density throughout the Universe but cluster within galaxies [389]. An example for this case will be discussed further below. Fig. 29 shows results for the same TD scenario as in Fig. 28, but for a high EGMF &10~9 G (somewhat below the current upper limit [262]). In this case, rapid synchrotron cooling of the initial cascade pairs quickly transfers energy out of the UHE range. The UHE c-ray #ux then depends mainly on the absorption length due to pair production and is typically much lower [178,256]. (Note, though, that for m Z1025 eV, the synchrotron radiation from these pairs can be X above 1020 eV, and the UHE #ux is then not as low as one might expect.) We note, however, that the constraints from the EGRET measurements do not change signi"cantly with the EGMF strength as long as the nucleon #ux is comparable to the c-ray #ux at the highest energies, as is the case in Figs. 28 and 29. A more detailed discussion of viable TD models depending on uncertainties in the fragmentation function, the mass and dominant decay channel of the X particle, the URB, and the EGMF, has been given in Ref. [206]. The results of Ref. [206] di!er from those of Ref. [295] which obtained more stringent constraints on TD models because of the use of the older fragmentation function Eq. (62), and a stronger dependence on the EGMF because of the use of a weaker EGMF which lead to a dominance of c-rays above K1020 eV. Dubovsky and Tinyakov [492] have recently pointed out that there could be an extra component of c-rays that would dominate the #ux shown in Figs. 28 and 29 around the trough at K1015 eV due to synchrotron emission from the electron component of the extragalactic cascade hitting the Galactic magnetic "eld. For TD models, such a component is predicted to be close to the observational upper limit from the CASA-MIA experiment [258] (see Figs. 28 and 29). The detection of such a component would, however, not necessarily be a signature of the top-down origin of EHECR because a similar component would for the same reason be expected as an extension of the di!use c-ray background around &100 GeV if this background is produced by sources whose spectrum extends to Z100 TeV. Observations [328,329] suggest that to be the case for blazars which are also expected to signi"cantly contribute to the di!use c-ray #ux observed by EGRET [489,490]. The energy loss and absorption lengths for UHE nucleons and photons are short ([100 Mpc). Thus, their predicted UHE #uxes are independent of cosmological evolution. The c-ray #ux below K1011 eV, however, scales as the total X particle energy release integrated over all redshifts and increases with decreasing p [437]. For m "2]1016 GeV, scenarios with p(1 are therefore ruled X out (as can be inferred from Figs. 28 and 29), whereas constant comoving injection models (p"2) are well within the limits. Since the EM #ux above K1022 eV is e$ciently recycled to lower energies, the constraint on p is in general less sensitive to m then expected from earlier CEL-based X analytical estimates [430,437]. A speci"c p"2 scenario is realized in the case where the supermassive X particles have a lifetime longer than the age of the Universe and contribute to the cold dark matter and cluster with the large-scale structure, as discussed in Section 6.13. An example of the expected #uxes in this scenario is shown in Fig. 30. In this context, we mention that a possible additional contribution to the c-ray
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Fig. 30. Same as Fig. 28, but for the case p"2, where the decaying X particles are long lived and contribute to cold dark matter, assuming an overdensity of 104 on the scale of the Galactic halo, K100 kpc. Points with arrows represent approximate upper limits on the di!use neutrino #ux from the Frejus [501], the EAS-TOP [502], and the Fly's Eye [503] experiments, as indicated. The projected sensitivity for the Pierre Auger project is using the acceptance estimated in Ref. [108], and the one for the OWL concept study is based on Ref. [87], both assuming observations over a few years period.
#ux from the synchrotron emission (in the Galactic magnetic "eld) of the electrons/positrons produced in the decays of the MSRPs clustered in the Galactic halo has recently been calculated in Ref. [493]. This contribution, however, seems to be smaller than the contribution from cascading of the extragalactic c-ray component shown in Fig. 30. Moreover, at least below K1015 eV, both components are unlikely to be detectable even with next generation experiments because of the dominant well established #uxes that are higher by at least a factor of 1000 at the relevant energies and are presumably due to conventional sources such as blazars. We now turn to signatures of TD models at UHE. The full cascade calculations predict c-ray #uxes below 100 EeV that are a factor K3 and K10 higher than those obtained using the CEL or absorption approximation often used in the literature [406], in the case of strong and weak URB, respectively. Again, this shows the importance of non-leading particles in the development of unsaturated EM cascades at energies below &1022 eV. Our numerical simulations give a c/CR #ux ratio at 1019 eV of K0.1. The experimental exposure required to detect a c-ray #ux at that level is K4]1019 cm2 s sr, about a factor 10 smaller than the current total experimental exposure. These exposures are well within reach of the Pierre Auger Cosmic Ray Observatories [86], which may be able to detect a neutral CR component down to a level of 1% of the total #ux. In contrast, if the EGMF exceeds &10~11 G, then UHE cascading is inhibited, resulting in a lower UHE c-ray spectrum. In the 10~9 G scenario of Fig. 29, the c/CR #ux ratio at 1019 eV is 0.02, signi"cantly lower than for no EGMF. It is clear from the above discussions that the predicted particle #uxes in the TD scenario are currently uncertain to a large extent due to particle physics uncertainties (e.g., mass and decay
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Table 1 Some viable p"1 TD scenarios explaining EHECR at least above 100 EeV m ! X
Fig.
URB
EGMF"
FF
f N
Mode
Q0 #
[GZK$
ZGZK$
1013
f l Z400 Mpc for high ll High Any [ Med [10~11 High [10~11 [ Med [10~11 Any [10~11
URB, No EGMF% No-SUSY 10% No-SUSY [10% No-SUSY 10% Any [10% * *
ll qq qq ql ql ll, ll
[31 1.4 1.4 0.88 0.93 1.3
c N N N c c
c N c c c c
1014
f l Z150 Mpc for high ll High Any [Med [10~10 Any [10~11 Any [10~11
URB, No EGMF% No-SUSY 10% No-SUSY [10% Any [10% * *
ll qq qq, ql ql ll, ll
[19 1.3 1.3 0.97 1.4
c N c#N N c
c c#N, N& c c c
1015
f l Z500 Mpc for high ll Any Any [Med [10~11 Any [10~11
URB, No EGMF% Any 10% Any [10% * *
ll qq, ql, ql qq, ql, ql ll, ll
[25 1.3 1.3 1.3
c N
c
c
c
ll qq qq qq, ql, ql ll, ll
[2.0 1.6 1.3 1.9 1.6
c N c, N&
c c#N, N& c, c#N'
c
c
1016 28, 29
f l Z3000 Mpc for high ll High any High [10~9 Any [10~11 [Med [10~11
URB, No EGMF% SUSY 10% No-SUSY 10% Any [10% * *
!in GeV. "in Gauss. #maximal total energy injection rate at zero redshift in units of 10~23 h eV cm~3 s~1. $dominant component of `visiblea TD #ux below and above GZK cuto! at K70 EeV; no entry means di!erent composition is possible, depending on parameters. %viable for eV mass neutrinos if their overdensity f over a scale l obeys the speci"ed condition, for the case of high URB l l and vanishing EGMF; for weaker URB/stronger EGMF, the condition on f l relaxes/becomes more stringent, ll respectively. &for EGMF Z10~10 G. 'for EGMF Z10~9 G.
modes of the X particles, the quark fragmentation function, the nucleon fraction f , and so on) as N well as astrophysical uncertainties (e.g., strengths of the radio and infrared backgrounds, extragalactic magnetic "elds, etc.). More details on the dependence of the predicted UHE particle spectra and composition on these particle physics and astrophysical uncertainties are contained in Ref. [206]. We summarize in Table 1 (adapted from Ref. [206]), some of the scenarios that are capable of explaining the observed EHECR #ux at least above 100 EeV, without violating any observational constraints. The predicted compositions of the particles below and above the GZK cuto! are also indicated. We stress here that there are viable TD scenarios which predict nucleon #uxes that are comparable to or even higher than the c-ray #ux at all energies, even though c-rays
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dominate at production. This occurs, e.g., in the case of high URB and/or for a strong EGMF, and a nucleon fragmentation fraction of K10%; see, for example, Fig. 29. Some of these TD scenarios would therefore remain viable even if EHECR induced EASs should be proven inconsistent with photon primaries (see, e.g., Ref. [72]). The normalization procedure to the EHECR #ux described above imposes the constraint Q0 [10~22 eV cm~3 s~1 within a factor of a few [295,206,297] for the total energy release rate EHECR Q from TDs at the current epoch (see also the benchmark calculation in Section 6.2.3). In most TD 0 models, because of the unknown values of the parameters involved, it is currently not possible to calculate the exact value of Q from "rst principle, although it has been shown that the required 0 values of Q (in order to explain the EHECR #ux) mentioned above are quite possible for certain 0 kinds of TDs (see Section 6 for details). Some cosmic string simulations suggest that strings may lose most of their energy in the form of X particles and estimates of this rate have been given [417]. If that is the case, the constraint on Q0 translates into a limit on the symmetry breaking scale EHECR g and hence on the mass m of the X particle: g&m [1013 GeV [418,298]. Independently of X X whether or not this scenario explains EHECR, the EGRET measurement of u leads to a similar #!4 bound, Q0 [2.2]10~23 h(3p!1) eV cm~3 s~1, which leaves the bound on g and m practically EM X unchanged. We may mention here that in most supersymmetric (SUSY) GUT models, X particle (gauge and Higgs boson) masses smaller than about 1016 GeV are disfavored from proton decay and other considerations; see, e.g., Refs. [494,495]. However, if the relevant topological defects are formed not at the GUT phase transition, but at a subsequent phase transition after the GUT transition (see the "nal paragraph of Section 6.4.6), then depending on the model, the associated X particles may not mediate baryon number violating processes, and so the proton decay constraints may not apply to them, and then X particles of mass m (1016 GeV are not X ruled out. 7.2. Constraints from primordial nucleosynthesis For zZ103, the Universe becomes opaque to the cascade photons discussed in the previous section [485]. These photons typically survive for a time Kl (E) during which they have a certain E chance to photo-disintegrate a 4He nucleus, with corresponding e!ective cross sections p%&&3 (E) D@ H% for production of D or 3He. This is possible as long as cascade cut o! energy E (z) is larger than the # 4He photo-disintegration threshold, E4H%"19.8 MeV, or, from Eq. (99), z[3]106. Instantaneous 5) generation of a cascade spectrum j#!4(E) then leads to creation of D and 3He with number densities c n 3 K(>/4)n :dE l (E)4pj#!4(E)p%&&3 (E), or, using Eq. (100), D@ H% B E c D@ H% 4pj (E)>p 3 (E) c D@ H% . (102) n 3 K dE D@ H% 4(1!>)pBH(E)#>pBH(E)#(4!2>)g (E)p (E) H H% CS CS The measured photo-disintegration cross sections immediately imply
P
(3He/D) Z8 , (103) 1)050 i.e., `cascading nucleosynthesisa predicts much more 3He than D. On the other hand, data imply (3He/D) [1.13 [496] for the abundances at the time of solar system formation and chemical _ evolution suggests (3He/D) [(3He/D) for the primordial abundances. Therefore, 4He photop _ disintegration cannot be the predominant production mechanism of primordial D and 3He, and
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altogether one has [437] ((3He#D)/H) [5]10~5 , (104) 1)050 which translates into an upper bound of EM energy release at redshifts 3]106ZzZ103 by either applying Eq. (102) or more detailed Monte Carlo simulations [485]. Fig. 27 shows the resulting maximal allowed instantaneous energy release as a function of redshift. We see that the resulting constraints from primordial nucleosynthesis on the TD models discussed in Section 6 are comparable to but independent of the constraints from the di!use c-ray background discussed in Section 7.1 [437]. 7.3. Constraints from distortions of the cosmic microwave background Early non-thermal electromagnetic energy injection can also lead to a distortion of the cosmic microwave background. We focus here on energy injection during the epoch prior to recombination. A comprehensive discussion of this subject is given in Ref. [497]. Regarding the character of the resulting spectral CMB distortions there are basically two periods to distinguish: "rst, in the range 3]106Kz 'z'z K105 between the thermalization redshift z and the Comptoniz5) : 5) ation redshift z , a fractional energy release *u/u leads to a pseudo-equilibrium Bose}Einstein : spectrum with a chemical potential given by kK0.71*u/u. This relation is valid for negligible changes in photon number which is a good approximation for the Klein}Nishina cascades produced by the GUT particle decays we are interested in [497]. Second, in the range z 'z'z K103 between z and the recombination redshift z the resulting spectral distortion : 3%# : 3%# is of the Sunyaev}Zel'dovich type [498] with a Compton y parameter given by 4y"*u/u. The most recent limits on both k and y were given in Ref. [499]. The resulting bounds on *u/u for instantaneous energy release as a function of injection redshift [500] are shown as the dashed curve in Fig. 27. The resulting bounds for the TD models are less stringent than the bounds from cascading nucleosynthesis and the observable c-ray background, but they still allow one to rule out the simplest model for superconducting cosmic strings corresponding to p"0. 7.4. Constraints on neutrino yuxes As discussed in Section 6, in TD scenarios most of the energy is released in the form of EM particles and neutrinos. If the X particles decay into a quark and a lepton, the quark hadronizes mostly into pions and the ratio of energy release into the neutrino versus EM channel is rK0.3. The resulting di!use UHE neutrino #uxes have been calculated in the literature for various situations: Ref. [195] contains a discussion of the (unnormalized) predicted spectral shape and Ref. [394] computes the absolute #ux predicted by speci"c processes such as cusp evaporation on ordinary cosmic strings. Ref. [200] calculated absolute #uxes for the scenarios discussed in Section 6 by using a simple estimate of the neutrino interaction redshift as mentioned in Section 4.3.1, and Ref. [196] improved on this by taking into account neutrino cascading in the RNB. None of these works, however, took into account cosmological constraints on TD models such as the ones discussed in Sections 7.1}7.3.
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Fig. 31. Predictions for the summed di!erential #uxes of all neutrino #avors (solid lines) from the atmospheric background for di!erent zenith angles [512] (hatched region marked `atmospherica), from proton blazars that are photon optically thick to nucleons but contribute to the di!use c-ray #ux [216] (`proton blazara), from UHECR interactions with the CMB [259] (`cosmogenica), for the TD model from Ref. [200] with p"0 (`BHS0a) and p"1 (`BHS1a), and for the TD model from Fig. 28, assuming an EGMF of [10~12 G (`SLBY98a, from Ref. [206]). Also shown are the #uxes of c-rays (dotted line), and nucleons (dotted lines) for this latter TD model. The data shown for the CR #ux and the di!use c-ray #ux from EGRET are as in Figs. 28 and 29. Points with arrows represent approximate upper limits on the di!use neutrino #ux from the Frejus [501], the EAS-TOP [502], and the Fly's Eye [503] experiments, as indicated. The projected sensitivity for the Pierre Auger project is using the acceptance estimated in Ref. [108], and the one for the OWL concept study is based on Ref. [87], both assuming observations over a few years period.
Fig. 31 shows predictions of the total neutrino #ux for the same TD model on which Fig. 28 is based, as well as some of the older estimates from Ref. [200]. In the absence of neutrino oscillations the electron neutrino and antineutrino #uxes that are not shown are about a factor of 2 smaller than the muon neutrino and antineutrino #uxes, whereas the q-neutrino #ux is in general negligible. In contrast, if the interpretation of the atmospheric neutrino de"cit in terms of nearly maximal mixing of muon and q-neutrinos proves correct, the muon neutrino #uxes shown in Fig. 31 would be maximally mixed with the q-neutrino #uxes. To put the TD component of the neutrino #ux in perspective with contributions from other sources, Fig. 31 also shows the atmospheric neutrino #ux, a typical prediction for the di!use #ux from photon optically thick proton blazars [216] that are not subject to the Waxman}Bahcall bound and were normalized to recent estimates of the blazar contribution to the di!use c-ray background [491], and the #ux range expected for `cosmogenica neutrinos created as secondaries from the decay of charged pions produced by UHE nucleons [259]. The TD #ux component clearly dominates above &1019 eV. In order to translate neutrino #uxes into event rates, one has to fold in the interaction cross sections with matter that were discussed in Section 4.3.1. At UHE these cross sections are not directly accessible to laboratory measurements. Resulting uncertainties therefore translate directly to bounds on neutrino #uxes derived from, for example, the non-detection of UHE muons
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produced in charged-current interactions. In the following, we will assume the estimate Eq. (25) based on the Standard Model for the charged-current muon-neutrino-nucleon cross section p if lN not indicated otherwise. For an (energy-dependent) ice or water equivalent acceptance A(E) (in units of volume times solid angle), one can obtain an approximate expected rate of UHE muons produced by neutrinos with energy 'E, R(E), by multiplying A(E)p (E)n 2 (where n 2 is the nucleon density in water) lN HO HO with the integral muon neutrino #ux KEj k . This can be used to derive upper limits on di!use l neutrino #uxes from a non-detection of muon induced events. Fig. 31 shows bounds obtained from several experiments: The Frejus experiment derived upper bounds for EZ1012 eV from their non-detection of almost horizontal muons with an energy loss inside the detector of more than 140 MeV per radiation length [501]. The EAS-TOP collaboration published two limits from horizontal showers, one in the regime 1014}1015 eV, where non-resonant neutrino-nucleon processes dominate, and one at the Glashow resonance which actually only applies to l6 [502]. The e Fly's Eye experiment derived upper bounds for the energy range between &1017 eV and &1020 eV [503] from the non-observation of deeply penetrating particles. The AKENO group has published an upper bound on the rate of near-horizontal, muon-poor air showers [504]. Horizontal air showers created by electrons or muons that are in turn produced by charged-current reactions of electron and muon neutrinos within the atmosphere have recently been pointed out as an important method to constrain or measure UHE neutrino #uxes [107,108] with next generation detectors. Furthermore, the search for pulsed radio emission from cascades induced by neutrinos or cosmic rays above &1019 eV in the lunar regolith with the NASA Goldstone antenna has lead to an upper limit comparable to the constraint from the Fly's Eye experiment [109]. The p"0 TD model BHS0 from the early work of Ref. [200] is not only ruled out by the constraints from Sections 7.1}7.3, but also by some of the experimental limits on the UHE neutrino #ux, as can be seen in Fig. 31. Further, although both the BHS1 and the SLBY98 models correspond to p"1, the UHE neutrino #ux above K1020 eV in the latter is almost two orders of magnitude smaller than in the former. The main reason for this are the di!erent #ux normalizations adopted in the two papers: First, the BHS1 model was obtained by normalizing the predicted proton #ux to the observed UHECR #ux at K4]1019 eV, whereas in the SLBY98 model the actually `visiblea sum of the nucleon and c-ray #uxes was normalized in a optimal way. Second, the BHS1 assumed a nucleon fraction about a factor 3 smaller [200]. Third, the BHS1 scenario used the older fragmentation function Eq. (61) which has more power at larger energies (see Fig. 26). Clearly, the SLBY98 model is not only consistent with the constraints from Sections 7.1}7.3, but also with all existing neutrino #ux limits within 2}3 orders of magnitude. What, then, are the prospects of detecting UHE neutrino #uxes predicted by TD models? In a 1 km3 2p sr size detector, the SLBY98 scenario from Fig. 31, for example, predicts a muonneutrino event rate of K0.15 yr~1, and an electron neutrino event rate of K0.089 yr~1 above 1019 eV, where `backgroundsa from conventional sources should be negligible. Further, the muon-neutrino event rate above 1 PeV should be K1.2 yr~1, which could be interesting if conventional sources produce neutrinos at a much smaller #ux level. Of course, above K100 TeV, instruments using ice or water as detector medium, have to look at downward going muon and electron events due to neutrino absorption in the Earth. However, q-neutrinos obliterate this Earth shadowing e!ect due to their regeneration from q decays [223]. The presence of q-neutrinos, for example, due to mixing with muon neutrinos, as suggested by recent experimental results from
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Super-Kamiokande, can therefore lead to an increased upward going event rate [221]; see discussion in Section 4.3.1. For recent compilations of UHE neutrino #ux predictions from astrophysical and TD sources see Refs. [217,505] and references therein. For detectors based on the #uorescence technique such as the HiRes [84] and the Telescope Array [85] (see Section 2.6), the sensitivity to UHE neutrinos is often expressed in terms of an e!ective aperture a(E) which is related to A(E) by a(E)"A(E)p (E)n 2 . For the cross section of lN HO Eq. (25), the apertures given in Ref. [84] for the HiRes correspond to A(E)K3 km3]2p sr for EZ1019 eV for muon neutrinos. The expected acceptance of the ground array component of the Pierre Auger project for horizontal UHE neutrino-induced events is A(1019 eV)K20 km3 sr and A(1023 eV)K200 km3 sr [108], with a duty cycle close to 100%. We conclude that detection of neutrino #uxes predicted by scenarios such as the SLBY98 scenario shown in Fig. 31 requires running a detector of acceptance Z10 km3]2p sr over a period of a few years. Apart from optical detection in air, water, or ice, other methods such as acoustical and radio detection [43] (see, e.g., Ref. [104] and the RICE project [105] for the latter) or even detection from space [87] appear to be interesting possibilities for detection concepts operating at such scales (see Section 2.6). For example, the OWL satellite concept, which aims to detect EAS from space, would have an aperture of K3]106 km2 sr in the atmosphere, corresponding to A(E)K6]104 km3 sr for EZ1020 eV, with a duty cycle of K0.08 [87]. The backgrounds seem to be in general negligible [196,506]. As indicated by the numbers above and by the projected sensitivities shown in Fig. 31, the Pierre Auger Project and especially the OWL project should be capable of detecting typical TD neutrino #uxes. This applies to any detector of acceptance Z100 km3 sr. Furthermore, a 100 d search with a radio telescope of the NASA Goldstone type for pulsed radio emission from cascades induced by neutrinos or cosmic rays in the lunar regolith could reach a sensitivity comparable or better than the Pierre Auger sensitivity above &1019 eV [109]. A more model-independent estimate [297] for the average event rate R(E) can be made if the underlying scenario is consistent with observational nucleon and c-ray #uxes and the bulk of the energy is released above the PP threshold on the CMB. Let us assume that the ratio of energy injected into the neutrino versus EM channel is a constant r. As discussed in Section 7.1, cascading e!ectively reprocesses most of the injected EM energy into low-energy photons whose spectrum peaks at K10 GeV [487]. Since the ratio r remains roughly unchanged during propagation, the height of the corresponding peak in the neutrino spectrum should roughly be r times the height of the low-energy c-ray peak, i.e., we have the condition max [E2j k (E)]Kr max [E2j (E)]. Imposing E l E c the observational upper limit on the di!use c-ray #ux around 10 GeV shown in Fig. 31, max [E2j k (E)][2]103r eVcm~2s~1 sr~1, then bounds the average di!use neutrino rate above E l PP threshold on the CMB, giving R(E)[0.34 r[A(E)/(1 km3]2p sr)](E/1019 eV)~0.6 yr~1 (EZ1015 eV) .
(105)
For r[20(E/1019 eV)0.1 this bound is consistent with the #ux bounds shown in Fig. 31 that are dominated by the Fly's Eye constraint at UHE. We stress again that TD models are not subject to the Waxman}Bahcall bound because the nucleons produced are considerably less abundant than and are not the primaries of produced c-rays and neutrinos. In typical TD models such as the one discussed above where primary neutrinos are produced by pion decay, rK0.3. However, in TD scenarios with r<1 neutrino #uxes are only limited by the condition that the secondary c-ray #ux produced by neutrino interactions with the RNB be below
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the experimental limits. An example for such a scenario is given by X particles exclusively decaying into neutrinos [206] (although this is not very likely in most particle physics models, but see Ref. [206] and Fig. 32 for a scenario involving topological defects and Ref. [207] for a scenario involving decaying superheavy relic particles, both of which explain the observed EHECR events as secondaries of neutrinos interacting with the RNB). Another possibility is the existence of hidden sector (mirror) topological defects radiating hidden (mirror) matter which interacts only gravitationally or via superheavy particles, whereas the mirror neutrinos can maximally oscillate into ordinary neutrinos [507]. Such scenarios could induce appreciable event rates above &1019 eV in a km3 scale detector. A detection would thus open the exciting possibility to establish an experimental lower limit on r. Being based solely on energy conservation, Eq. (105) holds regardless of whether or not the underlying TD mechanism explains the observed EHECR events. The transient event rate could be much higher than Eq. (105) in the direction to discrete sources which emit particles in bursts. Corresponding pulses in the EHE nucleon and c-ray #uxes would only occur for sources nearer than K100 Mpc and, in case of protons, would be delayed and dispersed by de#ection in Galactic and extragalactic magnetic "elds [260,304]. The recent observation of a possible correlation of CR above K4]1019 eV by the AGASA experiment [81] might suggest sources which burst on a time scale t ;1 yr. A burst of Kr[A(E)/1 km3] " 2p sr](E/1019 eV)~0.6 neutrino induced events within a time t could then be expected. Associated " pulses could also be observable in the GeV}TeV c-ray #ux if the EGMF is smaller than K10~15 G in a signi"cant fraction of extragalactic space [508]. In contrast, the neutrino #ux is comparable to (not signi"cantly larger than) the UHE photon plus nucleon #uxes in the models involving metastable superheavy relic particles discussed in
Fig. 32. Flux predictions for a TD model characterized by p"1, m "1014 GeV, with X particles exclusively decaying X into neutrino-antineutrino pairs of all #avors (with equal branching ratio), assuming a neutrino mass m "1 eV. For l neutrino clustering, the lower limit from Table 1 was assumed, corresponding to an overdensity of K30 over a scale of l K5 Mpc. The calculation assumed the strongest URB version shown in Fig. 10 and an EGMF;10~11 G. The line l key is as in Figs. 28 and 30.
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Section 6.13; see Fig. 30. This can be understood because the neutrino #ux is dominated by the extragalactic contribution which scales with the extragalactic nucleon and c-ray contribution in exactly the same way as in the unclustered case, whereas the extragalactic `visiblea contribution is much smaller in the clustered case. Such neutrino #uxes would be hardly detectable even with next generation experiments. 8. Summary and conclusions It is clear from the discussions in the previous sections that the problem of origin of EHECR continues to remain as a major unsolved problem. The EHECR present a unique puzzle: recall that for lower-energy cosmic rays (below about 1016 eV) there is a strong belief that these are produced in supernova remnants (SNRs) in the Galaxy. However, because of the twists and turns the trajectories of these particles su!er in propagating through the Galactic magnetic "eld, it is not possible to point back to the sources of these particles in the sky, and one has to take recourse to indirect arguments, such as those involving c-ray production or other secondary particle abundances, to establish the `evidencea that indeed these `lowa-energy cosmic rays are produced in SNRs. The EHECR particles, on the other hand, are hardly a!ected by any intergalactic and/or Galactic magnetic "elds. Also, because of energy losses they su!er during their propagation through the intergalactic space, the EHECR particles } if they are `standarda particles such as nucleons, heavy nuclei and/or photons } cannot originate at distances much further away than about 60 Mpc from Earth. So knowing (observationally) the arrival directions of the individual EHECR events, we should be able to do `particle astronomya, i.e., trace back to the sources of these EHECR particles in the sky. Yet, all attempts toward this have generally failed to identify possible speci"c powerful astronomical sources (such as active galactic nuclei, radio galaxies, quasars, etc., that could in principle produce these particles). In some cases arrival directions of some EHECR events point to some quasars or radio galaxies, but those are generally situated at distances well beyond 60 Mpc. Apart from the problem of source identi"cation, the other basic question is that of energetics. What processes are responsible for endowing the EHECR particles with the enormous energies beyond 1011 GeV? As we have discussed in this review, conventional `bottom-upa acceleration mechanisms are barely able to produce particles of such energies when energy loss processes at the source as well as during propagation are taken into account. The conventional bottom-up models thus face a two-fold problem: energetics as well as source location. It seems to be becoming increasingly evident } though this is far from established yet } that to solve both problems together, some kind of new physics beyond the Standard Model of particle physics may be necessary. In this report we have discussed several such proposals that are currently under study. We summarize them here. Currently, two types of distinct `scenariosa are being studied: In one of these scenarios, one still works within the framework of the bottom-up scenario and assumes that there exist sources where the requirements of energetics are met somehow; the source distance problem is then solved by postulating any of the following: f The EHECR are a new kind of particle, for example, a supersymmetric particle whose interactions with the CMB is such that they su!er little or no signi"cant energy loss during
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propagation, so that they can reach Earth from cosmological distances. However, in this proposal, the new particles must be produced as secondary particles at the source or during propagation through interactions of a known accelerated particle such as proton with the medium. This has other implications such as associated secondary c-ray production which will have to confront various constraints from observation. The EHECR are known particles but they are endowed with new interaction properties. One possibility along this line is the suggestion that the neutrino interaction cross section at the relevant energies receives dominant contributions from physics beyond the Standard Model which causes neutrinos to interact in the atmosphere. In order not to violate unitarity bounds, this cross section cannot be a point cross section and/or high partial waves have to contribute signi"cantly. EHECR are `exotica particles such as magnetic monopoles which are fairly easily accelerated to the requisite energies by the Galactic magnetic "eld. This is an attractive proposal, because neither energetics nor source distance is much of a problem. However, the problem with this proposal seems to be that magnetic monopoles, because of their heavy mass, are unlikely to induce relativistic air-showers. EHECR are nucleons and/or c-rays that are produced within the GZK distance limit of &60 Mpc from Earth by interactions of su$ciently high-energy neutrinos with the thermal relic neutrino background. This mechanism is most e!ective if some species of neutrinos have a small mass of &eV, in which case the neutrinos play the role of hot dark matter in the Universe. This mechanism, if it leaves some characteristic signature in the spectrum and/or composition of EHECR, has the potential to o!er an indirect method of `detectinga the relic massive neutrino dark matter background, but is also strongly constrained by associated c-ray production. EHECR are nucleons, but one allows small modi"cations to the fundamental laws of physics. One speci"c proposal along this line is to postulate a tiny violation of Lorentz invariance, too small to have been detected so far, which allows avoidance of the GZK cuto! e!ect for nucleons so that nucleons can arrive to Earth from cosmological distances.
Clearly, all the above proposals involve new physics beyond the Standard Model. In the other scenario, generally called `top-downa scenario, there is no acceleration mechanism involved: The EHECR particles arise simply from the decay of some su$ciently massive `Xa particles presumably originating from processes in the early Universe. There is thus no problem of energetics. One clear prediction of the top-down scenario is that EHECR should consist of `elementarya particles such as nucleons, photons, and even possibly neutrinos, but no heavy nuclei. Two classes of possible sources of these massive X particles have been discussed in literature. These are: f X particles produced by collapse, annihilation or other processes involving systems of cosmic topological defects such as cosmic strings, magnetic monopoles, superconducting cosmic strings and so on, which could be produced in the early Universe during symmetry-breaking phase transitions envisaged in Grand Uni"ed Theories. In an in#ationary early Universe, the relevant topological defects could be formed at a phase transition at the end of in#ation. The X particle mass can be as large as a typical GUT scale &1016 GeV. f X particles are some long-living metastable superheavy relic particles (MSRP) of mass Z1012 GeV with lifetime comparable to or much larger than the age of the Universe. These
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X particles could be produced in the early Universe in particle production processes associated with in#ation, and they could exist in the present day Universe with su$cient abundance as to be a good candidate for (at least a part of) the cold dark matter in the Universe. These MSRP X particles are expected to cluster in our Galactic Halo. Decay of these X particles clustered in the Galactic Halo (GH) can easily explain the apparent absence of the GZK cuto! in the recent EHECR data. In both of these top-down mechanisms, the decay of the X particles predominantly produce photons and neutrinos, and their spectra at production is "xed by particle physics (QCD). The topological defect (TD) model, as we have discussed in details in this review, is severely constrained by existing measurements and limits on the di!use gamma ray background at energies below 100 GeV that would receive a contribution (through electromagnetic cascading process) from decay of TD-produced X particles at large cosmological distances. The MSRP X particles clustered in the GH are, however, not constrained by the di!use gamma ray measurements. Instead, the MSRP scenario is (or rather will be) constrained by measurements of the anisotropy of UHECR, because a rather strong anisotropy is expected in this model because of the o!-center location of the Sun with respect to the Galactic center. In the MSRP decay model, the observed spectrum of EHECR should essentially be the unmodi"ed injected spectrum resulting from the decay of the MSRPs. The precise measurements of the EHECR spectrum can then be a probe of QCD at energies much beyond that currently available in accelerators. The same would apply in the case of processes involving topological defects that can be clustered in the GH (as in the case of monopolonium). In addition to o!ering a variety of di!erent kinds of probes of new physics, the measurement of the EHECR spectrum by the next generation large detectors will be expected to yield signi"cant new information about fundamental physical processes in the Universe today. For example, observations of images and spectra of EHECR by the next generation experiments will be able to yield new information on Galactic and especially the poorly known extragalactic magnetic "elds [262,263] and possibly their origin. This could lead to the discovery of a large-scale primordial magnetic "eld which potentially could open still another new window into processes occurring in the early Universe.
Acknowledgements We are most grateful to the late David Schramm whose insights, encouragements and constant support had been crucial to us in our e!orts in this exciting area of research over the past several years. Indeed it was he who "rst urged us to undertake the project of writing this Report. We also wish to thank Felix Aharonian, Peter Biermann, Paolo Coppi, Veniamin Berezinsky, Chris Hill, Karsten Jedamzik, Sangjin Lee, Martin Lemoine, Angela Olinto, (the late) Narayan Rana, Qaisar Sha", Floyd Stecker, and Shigeru Yoshida for collaborations at various stages. We wish to thank Shigeru Yoshida for providing us with the "gures containing the results of various UHECR experiments. In this context we also thank Peter Biermann, Murat Boratav, Roger Clay, Raymond Protheroe, Andrew Smith, David Seckel and Masahiro Teshima for allowing us to use various "gures from their papers in this Review. We acknowledge stimulating discussions with colleagues
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and collaborators mentioned above, as well as with Ivone Albuquerque, John Bahcall, Pasquale Blasi, Silvano Bonazola, Brandon Carter, Daniel Chung, Jim Cronin, Arnon Dar, Luke Drury, Torsten Ensslin, Glennys Farrar, Raj Gandhi, Haim Goldberg, Yuval Grossman, Francis Halzen, Mark Hindmarsh, Michael Kachelrie{, Karl-Heinz Kampert, Edward Kolb, Thomas Kutter, Vadim Kuzmin, Eugene Loh, Norbert Magnussen, Alfred Mann, Karl Mannheim, Hinrich Meyer, Motohiko Nagano, Patrick Peter, Rainer Plaga, Clem Pryke, JoK rg Rachen, Georg Ra!elt, Esteban Roulet, Wolfgang Rhode, Ina Sarcevic, Subir Sarkar, Paul Sommers, Parameshwaran Sreekumar, Leo Stodolsky, Tanmay Vachaspati, Heinz VoK lk, Eli Waxman, Tom Weiler, Gaurang Yodh, Enrice Zas, Arnulfo Zepeda, and Igor Zheleznyk. PB wishes to thank Ramanath Cowsik and Kumar Chitre for their interest and encouragement, and Arnold Wolfendale for communications. PB was supported at NASA/Goddard by a NAS/NRC Resident Senior Research Associateship award during a major part of this work, and he thanks John Ormes and Floyd Stecker for the hospitality and for providing a friendly and stimulating work environment at LHEA (Goddard). PB also acknowledges support under the NSF US-India collaborative research grant (No. INT-9605235) at the University of Chicago. GS was supported by DOE, NSF and NASA at the University of Chicago, by the Centre National de la Recherche Scienti"que at the Observatoire de Paris-Meudon, and by the Max-Planck-Institut fuK r Physik in Munich, Germany.
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[469] D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. D 59 (1999) 023501. [470] D.J.H. Chung, E.W. Kolb, A. Riotto, Phys. Rev. Lett. 81 (1998) 4048. [471] E.W. Kolb, D.J.H. Chung, A. Riotto, e-print hep-ph/9810361, in: H.V Klapdor-Kleingrothaus, L. Baudis (Eds.), Proceedings of the Second International Conference on Dark Matter in Astro and Particle Physics, Heidelberg, Germany, 20}25 July 1998. [472] V. Kuzmin, I. Tkachev, Phys. Rev. D 59 (1999) 123006; JETP. Lett. 68 (1998) 271. [473] K. Greist, M. Kamionkowski, Phys. Rev. Lett. 64 (1990) 615. [474] see, e.g., T. Asaka, M. Kawasaki, T. Yanagida, Phys. Rev. D 60 (1999) 103518. [475] S.L. Dubovsky, P.G. Tinyakov, Pisma Zh. Eksp. Teor. Fiz. 68 (1998) 99; JETP. Lett. 68 (1998) 107. [476] A. Benson, A. Smialkowski, A.W. Wolfendale, Astropart. Phys. 10 (1999) 313. [477] V. Berezinsky, A.A. Mikhailov, Phys. Lett. B 449 (1999) 237. [478] Y. Chikashige, J. Kamoshita, e-print astro-ph/9812483. [479] J. Caldwell, J. Ostriker, Astrophys. J. 251 (1981) 61; J. Bahcall et al., Astrophys. J. 265 (1983) 730. [480] J.F. Navarro, C.S. Frenk, S.D.M. White, Astrophys. J. 462 (1996) 563. [481] G.A. Medina Tanco, A.A. Watson, e-print astro-ph/9903182, Astropart. Phys. 12 (1999) 25; astro-ph/9905240, in: Proceedings of 26th International Cosmic Ray Conference, Utah, 1999, to appear. [482] A. Barrau, e-print astro-ph/9907347, Astropart. Phys., to appear. [483] A.F. Heckler, Phys. Rev. D 55 (1997) 480; Phys. Rev. Lett. 78 (1997) 3430. [484] for a review see B.J. Carr, J.H. MacGibbon, Phys. Rep. 307 (1998) 141. [485] R.J. Protheroe, T. Stanev, V.S. Berezinsky, Phys. Rev. D 51 (1995) 4134; M. Kawasaki, T. Moroi, Astrophys. J. 452 (1995) 506. [486] J. Ellis, G.B. Gelmini, J.L. Lopez, D.V. Nanopoulos, S. Sarkar, Nucl. Phys. B 373 (1992) 399. [487] P.S. Coppi, F.A. Aharonian, Astrophys. J. 487 (1997) L9. [488] A. Smialkowski, A.W. Wolfendale, L. Zhang, Astropart. Phys. 7 (1997) 21. [489] See, e.g., F.W. Stecker, M.H. Salamon, Astrophys. J. 464 (1996) 600. [490] A. MuK cke, M. Pohl, e-print astro-ph/9807297, in Proceedings of BL Lac Phenomena, June 1998, Turku/Finland, Publ. Astron. Soc. of the Paci"c Conference Series, to appear. [491] R. Mukherjee, J. Chiang, in: M. Catanese, J. Quinn, T. Weekes (Eds.), Proceedings of VERITAS Workshop on TeV Astrophysics of Extragalactic Sources, Astropart. Phys. 11 (1999), Astropart. Phys. 11 (1999) 213. [492] S.L. Dubovsky, P.G. Tinyakov, e-print astro-ph/9906092. [493] P. Blasi, Phys. Rev. D 60 (1999) 023514. [494] R. Arnowitt, P. Nath, Phys. Rev. Lett. 69 (1992) 725; Phys. Rev. D 49 (1994) 1479. [495] V. Lukas, S. Raby, Phys. Rev. D 55 (1997) 6986. [496] J. Geiss, in: N. Prantzos, E. Vangioni-Flam, M. Casse (Eds.), Origin and Evolution of the Elements, Cambridge University Press, Cambridge, 1993, p. 107. [497] W. Hu, J.S. Silk, Phys. Rev. D 48 (1993) 485; see also references therein. [498] Ya.B. Zel'dovich, R. A. Sunyaev, Astrophys. Space Sci. 4 (1969) 301. [499] J.C. Mather et al., Astrophys. J. 420 (1994) 439. [500] E.L. Wright et al., Astrophys. J. 420 (1994) 450. [501] W. Rhode et al., Astropart. Phys. 4 (1996) 217. [502] M. Aglietta et al. (EAS-TOP collaboration), in: Proceedings of 24th International Cosmic Ray Conference, Istituto Nazionale Fisica Nucleare, Rome, Italy, 1995, Vol. 1, 638. [503] R.M. Baltrusaitis et al., Astrophys. J. 281 (1984) L9; Phys. Rev. D 31 (1985) 2192. [504] M. Nagano et al., J. Phys. G 12 (1986) 69. [505] M. Roy, H.J. Crawford, e-print astro-ph/9808170, Astropart. Phys., submitted for publication. [506] P.B. Price, Astropart. Phys. 5 (1996) 43. [507] V.S. Berezinsky, A. Vilenkin, e-print hep-ph/9908257. [508] E. Waxman, P.S. Coppi, Astrophys. J. 464 (1996) L75. [509] S. Swordy, private communication. The data represent published results of the LEAP, Proton, Akeno, AGASA, Fly's Eye, Haverah Park, and Yakutsk experiments. [510] J. Matthews et al., Astrophys. J. 375 (1991) 202. [511] M. Aglietta et al. (EAS-TOP collaboration), Astropart. Phys. 6 (1996) 71. [512] P. Lipari, Astropart. Phys. 1 (1993) 195.
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BOSON INTERFEROMETRY IN HIGH-ENERGY PHYSICS
R.M. WEINER!," ! Physics Department, University of Marburg, Marburg, Germany " Laboratoire de Physique TheH orique, Univ. Paris-Sud, Orsay, France
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Boson interferometry in high-energy physics R.M. Weiner!,",* !Physics Department, University of Marburg, Marburg, Germany "Laboratoire de Physique The& orique, Univ. Paris-Sud, Orsay, France Received February 1999; editor: J. Bagger
Contents 1. Introduction 2. The GGLP e!ect 2.1. The wave-function approach 2.2. Quantum optical methods in BEC 3. Final state interactions of hadronic bosons 3.1. Electromagnetic "nal state interactions 3.2. Strong "nal state interactions 4. Currents 4.1. Classical versus quantum currents 4.2. Classical currents 4.3. Primordial correlator, correlation length and space}time distribution of the source 4.4. Production of an isospin multiplet 4.5. Photon interferometry. Upper bounds of BEC 4.6. Coherence and lower bounds of Bose}Einstein correlations 4.7. Quantum currents 4.8. Space}time form of sources in the classical current formalism
252 253 253 258 267 267 271 275 275 276 278 280 285 289 292 293
4.9. The Wigner function approach 4.10. Dynamical models of multiparticle production and event generators 4.11. Experimental problems 5. Applications to ultrarelativistic nucleus}nucleus collisions 5.1. BEC, hydrodynamics and the search for quark}gluon plasma 5.2. Pion condensates 6. Correlations and multiplicity distributions 6.1. From correlations to multiplicity distributions 6.2. Multiplicity dependence of Bose}Einstein correlations 6.3. The invariant Q variable in the space}time approach: higher-order correlations; `intermittencya in BEC? 7. Critical discussion and outlook Acknowledgements References
299 304 308 309 309 322 324 324 331
335 337 340 340
Abstract Intensity interferometry and in particular that based on Bose}Einstein correlations (BEC) constitutes at present the only direct experimental method for the determination of sizes and lifetimes of sources in particle
* Corresponding address: 112 Avenue Felix Faure, 75015 Paris, France. E-mail address: [email protected] (R.M. Weiner) 0370-1573/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 1 4 - 3
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and nuclear physics. The measurement of these is essential for an understanding of the dynamics of strong interactions which are responsible for the existence and properties of atomic nuclei. Moreover, a new state of matter, quark matter, in which the ultimate constituents of matter move freely, is within the reach of present accelerators or those under construction. The con"rmation of the existence of this new state is intimately linked with the determination of its space}time properties. Furthermore, BEC provides information about quantum coherence which lies at the basis of the phenomenon of Bose}Einstein condensation seen in many chapters of physics. Coherence and the associated classical "elds are essential ingredients in modern theories of particle physics including the standard model. Last but not least besides this `applicativea aspect of BEC, this e!ect has implications for the foundations of quantum mechanics including the understanding of the concept of `identical particlesa. Recent theoretical developments in BEC are reviewed and their application in high-energy particle and heavy-ion reactions is analysed. The treated topics include: (a) a comparison between the wave-function approach and the space}time approach based on classical currents, which predicts `surprisinga particle}anti-particle BEC, (b) the study of "nal state interactions, (c) the use of hydrodynamics, and (d) the relation between correlations and multiplicity distributions. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 13.85.!t; 25.75.Gz Keywords: Bose}Einstein correlation; Quark}gluon plasma
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1. Introduction The method of photon intensity interferometry was invented in the mid-1950s by Hanbury} Brown and Twiss for the measurement of stellar dimensions and is sometimes called the HBT method. In 1959}1960 G. Goldhaber, S. Goldhaber, W. Lee and A. Pais discovered that identical charged pions produced in p6 }p annihilation are correlated (the GGLP e!ect). Both the HBT and the GGLP e!ects are based on Bose}Einstein correlations (BEC). Subsequently Fermi}Dirac correlations for nucleons have also been observed. Loosely speaking, both these correlation e!ects can be viewed as a consequence of the symmetry (antisymmetry) properties of the wave function with respect to permutation of two identical particles with integer (half-integer) spin and are thus intrinsic quantum phenomena. At a higher level, these symmetry properties of identical particles are expressed by the commutation relations of the creation and annihilation operators of particles in the second quantisation (quantum "eld theory). The quantum "eld approach is the more general approach as it contains the possibility to deal with creation and annihilation of particles and certain phenomena like the correlation between particles and antiparticles can be properly described only within this formalism. Furthermore, at high energies, because of the large number of particles produced, not all particles can be detected in a given reaction and therefore one measures usually only inclusive cross sections. For these reactions the wave-function formalism is impractical. Related to this is the fact that the second quantisation provides through the density matrix a transparent link between correlations and multiplicity distributions. This last topic has been in the centre of interest of multiparticle dynamics for the last 20 years (we refer among other things to Koba}Nielsen}Olesen (KNO) scaling and `intermittencya). Furthermore, one of the most important properties of systems made of identical bosons which is responsible for the phenomenon of lasing is quantum statistical coherence. This feature is also not accessible to a theoretical treatment except in "eld theory. The present review is restricted to Bose}Einstein correlations which constitute by far the majority of correlations papers both of theoretical and experimental nature. This is due to the fact that BEC present important heuristic and methodological advantages over Fermi}Dirac correlations. Among the "rst we mention the fact that quantum coherence appears only in BEC. Among the second, one should recall that pions are the most abundantly produced secondaries in highenergy reactions. In the last few years there has been a considerable surge of interest in boson interferometry. This can be judged by the fact that at present there is no meeting on multiparticle production where numerous contributions to this subject are presented. Moreover, since 1990 [1] meetings dedicated exclusively to this topic are organised; this re#ects the realisation that BEC on their own are an important subject. This development is in part due to the fact that at present intensity interferometry constitutes the only direct experimental method for the determination of sizes and lifetimes of sources in particle and nuclear physics. Since soft strong interactions which are responsible for multiparticle production processes cannot be treated by perturbative QCD, phenomenological approaches have to be used in this domain and space}time concepts are essential elements in these approaches. That is why intensity interferometry has become an indispensable tool in the investigation of the dynamics of high-energy reactions. However, this alone could not explain the explosion of interest in BEC if one did not take into account the search for quark}gluon plasma which has mobilised the attention of most of the
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nuclear physics and of an appreciable part of the particle physics community. For several reasons the space}time properties of `"reballsa produced in heavy ion reactions are key to the process of understanding whether quark matter has been formed. The main emphasis of the present review will be on theoretical developments which took place after 1989}1990. Experimental results will be mentioned only insofar as they illustrate the theoretical concepts. For a review of older references see. e.g. the paper by Boal et al. [2]. In the 1990s the single most important theoretical event was in our view the space}time generalisation of the classical current formalism. For a detailed presentation of this generalisation and its applications up to 1993 the reader is referred to Ref. [3]. For a review of experimental results in e`}e~ reactions see [4]. Finally a more pedagogical and more complete treatment of the theory of BEC can be found in [5]. There are two categories of papers not mentioned: (i) Those which the reviewer was unaware of; he apologises to the authors of these papers for this. (ii) Those which he considered as irrelevant or as repetitions of previous work. The large number of papers quoted despite these restrictions shows that an exhaustive listing of references on BEC is not trivial.
2. The GGLP e4ect In the period 1954}1960 Hanbury}Brown and Twiss developed the method of optical intensity interferometry for the determination of (angular sizes) of stars (see e.g. [6]). The particle physics equivalent of the Hanbury}Brown}Twiss (HBT) e!ect in optics is the Goldhaber, Goldhaber, Lee and Pais (GGLP) e!ect [7,8] which we shall describe schematically below. However, when going over from optics to particle physics the following point has to be considered: in particle physics one does not measure distances r in order to deduce (di!erences of) momenta k and thus angular sizes, but one measures rather momenta in order to deduce distances. This explains why GGLP were not aware of the HBT method.1 Up to a certain point there are two ways of approaching the theory of intensity interferometry: the wave-function approach and the "eld theoretical approach. Although the "rst one is only a particular case of the second one it is still useful because it allows sometimes a more intuitive understanding of certain concepts and in particular that of the distinction between boson and fermion correlations. We shall start therefore with the wave-function approach. 2.1. The wave-function approach Let us consider for the beginning a source which consists of a number of discrete emission points i each of which is characterised by a probability amplitude F (r)"F d(r!r ) . i i i
1 For a comparison of optical and particle physics intensity interferometry see e.g. [5].
(2.1)
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Let tk (r) be the wave function of an emitted particle. The total probability P(k) to observe the emission of one particle with momentum k from this source is obtained by summing the contributions of all points i. If this summation is done incoherently the single particle probability reads P(k)"+ DF t(r )D2 . (2.2) i i i Instead of discrete emission points consider now a source the emission points of which are continuously distributed in space and assume for simplicity that the wave functions are plane waves tk (r)& exp(ikr). The sum over x will now be replaced by an integral and we have i
P
P(k)" DF(r)D2 d3r .
(2.3)
Similarly, the probability to observe two particles with momenta k , k is 1 2
P
P(k k )" Dt D2f (r ) f (r ) d3r d3r . 1,2 1 2 1 2 1 2
(2.4)
Here t "t (k , k ; r , r ) is the two-particle wave function and we have introduced the 1,2 1,2 1 2 1 2 density distribution f"DFD2. Suppose now that the two particles are identical. Then the two-particle wave function has to be symmetrised or antisymmetrised depending on whether we deal with identical bosons or fermions. Assuming plane waves we have t "(1/J2)[e*(k1 r1 `k2 r2 )$e*(k1 r2 `k2 r1 )] 1,2
(2.5)
with the plus sign for bosons and the minus sign for fermions. With this wave function one obtains P(k , k )"D fI (0)D2$D fI (k !k )D2 . 1 2 I I 1 2
(2.6)
The incoherent summation corresponds to random #uctuations of the amplitudes F . i Following the GGLP experiment [8] consider the space points x and x within a source, 1 2 so that each point emits two identical particles (equally charged pions in the case of GGLP) with momenta k and k . These particles are detected in the registration points r and r so that in 1 2 1 2 r only particles of momentum k and in r only particles of momentum k are registered (see 1 1 2 2 Fig. 1). Because of the identity of particles one cannot decide which particle pair originates in x and which in x . 1 2 Assuming that the individual emission points of the source act incoherently GGLP derived Eq. (2.6) which for bosons leads to the second-order correlation function C (k , k ),P (k , k )/P (k )P (k )"1#D fI (q)D2 , 2 1 2 2 1 2 1 1 1 2
(2.7)
where fI is the Fourier transform of f and q the momentum di!erence k !k . 1 2 Eq. (2.7) shows quite clearly how in particle physics momentum (correlation) measurements can yield information about the space}time structure of the source.
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Fig. 1. The GGLP experiment schematically.
In the case of coherent summation one gets instead C "1 (see e.g. [2]). We deduce from this 2 result (see also below) that coherence reduces the correlation and that a purely coherent source has a correlation function which does not depend on its geometry.2 2.1.1. Newer correlation measurements in p6 }p annihilation The original GGLP experiment [7] measured the correlation function in terms of the opening angle3 of a pion pair. The GGLP experiment has been repeated in the last few years at LEAR (see e.g. [12] where also older references are quoted4). In these newer experiments the three momenta of particles were measured, although one continued to use as a variable for the correlation function the invariant momentum di!erence Q, as suggested already in [8]. One of the remarkable observations made in all annihilation reactions is that the intercept of the second-order correlation function C (k, k) appears consistently to exceed the canonical value of 2 2 reaching values up to 4. (This e!ect was possibly not seen in the original experiment [7] because of the averaging over the magnitudes of the momenta.) In [14] this e!ect was attributed to resonances while subsequently [15,12] a (nonchaotic) Skyrmion-type superposition of coherent states was proposed as an alternative explanation. Another possible explanation of this intriguing observation may be that in annihilation processes squeezed states (see Section 2.2) are produced, while this is not the case in other processes. Indeed, it is known that squeezed states can lead to overbunching e!ects (see below). Furthermore, as shown in Ref. [16], squeezed states may be preferentially produced in rapid reactions. Or, according to some authors [15], annihilation is a more rapid process than other reactions
2 See also [9,10] for an attempt to approach the issue of coherence and chaos by using wave packets. 3 The use of the opening angle as a kinematical variable in BEC studies was readopted in Ref. [11] where it was recommended as a tool for the investigation of "nal state interactions. 4 For a reanalysis of the results of an older annihilation experiment see [13].
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occurring at higher energies. Given the importance of squeezed states, further experimental and theoretical studies of this issue are highly desirable.5 2.1.2. Resonances, apparent coherence and other experimental problems: the j factor It is known that in multiparticle production processes, an appreciable fraction of pions originates from resonance decays. Resonances act in opposite ways on the correlation function of pion pairs. On the one hand, the interference between pions originating from short-lived resonances and `directlya produced pions leads to a narrow peak in the second-order correlation function at small values of q [17,18]. On the other hand, long-lived resonances give rise to pions which are beyond the range of detectors and this leads to an apparent decrease of the intercept of the second-order correlation function C (k, k). These modi"cations of the intercept are important among other 2 things, because their understanding is essential in the search for coherence through the intercept criterion. As mentioned already one of the most immediate consequences of coherence is the decrease of the correlation function at small q. Because of the large number of di!erent resonances produced in high-energy reactions a quantitative estimate of their e!ects is possible only via numerical techniques. In Sections 4.9.1 and 5.1.3 we present a more detailed discussion of the in#uence of resonances on BEC within the Wigner function formalism. For older references on this topic see also [2,19,20]. Another experimental di$culty is that, because of limited statistics or certain technical problems, sometimes not all degrees of freedom can be measured in a given experiment. This leads to an e!ective averaging over the non-measured degrees of freedom and hence also to an e!ective reduction of the correlation function. As a matter of fact BEC experiments have shown from the very beginning that the extrapolation of the correlation function to q"0 usually never led to the maximum value C (0)"2 permitted by Eq. (2.7). 2 To take into account empirically this e!ect experimentalists introduced into the correlation function a correction factor j. Thus Eq. (2.7), e.g. was modi"ed into C (k , k )"1#jD fI (q)D2 . (2.8) 2 1 2 Because one initially thought that formally this generalisation o!ers also the possibility to describe partially coherent sources, the corresponding parameter j was postulated to be limited by (0,1) and called the `incoherencea factor; indeed j"0 leads to a totally coherent source and j"1 to a totally chaotic one. Unfortunately, this nomenclature is not quite correct, as explained below. It should also be mentioned that there exists a strong correlation between the empirical values of j and those of the `radiusa which enter fI (see e.g. [21] for a special study of this issue). 2.1.3. The limitations of the wave-function formalism The wave-function formalism presented in the previous subsection has severe shortcomings: (a) The correlation function (2.7) depends only on the di!erence of momenta q and not also on the sum k #k , in contradiction with experimental data. Although we will see in Section 4.3 that 1 2 this limitation disappears automatically in the quantum statistical ("eld theoretical) approach, it 5 In Ref. [16] the e!ect of chaotic superpositions of squeezed states in BEC was also studied. It is shown that such a superposition always leads to overbunching, while pure squeezed states can lead also to antibunching.
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can also be remedied within the "rst quantisation formalism by using the Wigner function approach. In this approach the source function is from the beginning a function both of coordinates and momenta and therefore the correlation function depends on k and k separately. There 1 2 is, of course, a price to pay for this procedure as it involves a semiclassical approximation.6 (b) The wave-function formalism may be useful when exclusive reactions are considered as was the case, e.g. in the GGLP paper. Indeed t in Eq. (2.5) is just the wave function of the two-boson 1,2 system, i.e. an assumption that two and only two bosons are produced is made. At low energies, i.e. low average multiplicities, this condition can be satis"ed. However at high energies the identi"cation of all particles is very di$cult and up to now has not been done. Therefore, one measures practically always inclusive cross sections. This means that instead of (2.5) one would have to use a wave function which describes the two bosons in the presence of all other produced particles. To obtain such a wave function one would have to solve the SchroK dinger equation of the many body, strongly interacting system, which is not a very practical proposal. Related to this is the di$culty to treat higher-order correlations within the wave-function formalism. (c) The correlation function C in the wave-function formalism is independent of isospin and is 2 thus the same for charged and neutral particles. We shall see in Section 4.4 that in a more correct quantum "eld theoretical approach, this is not the case. This will a!ect among other things the bounds of the correlations and will lead to quantum statistical particle}antiparticle correlations which are not expected in the wave-function formalism. (d) Coherence cannot be treated adequately (see below). The correlation functions derived in this subsection refer in general to incoherent sources and attempts to introduce coherence within the wave-function formalism are rather ad hoc parametrisations. However, coherence is the most characteristic and important property of Bose}Einstein correlations among other things because it is the basis of the phenomenon of Bose}Einstein condensates found in many chapters of physics, like superconductors, super#uids, lasers, and the recently discovered atomic condensates [24]. It would be very surprising if coherence would not be found also in particle physics given the fact that the wavelengths of the emitted particles are of the same order as that of the sources. Furthermore, as pointed out in this connection in [25] modern particle physics is based on spontaneously broken symmetries. The associated "elds are coherent. That is why one of the main motivations of BEC research should be the measurement of the amount of coherence in strong interactions. For this purpose the formalism of BEC has to be generalised to include the presence of (partial) coherence and this again can be done correctly only within quantum statistics, i.e. quantum "eld theory. We conclude this subsection with the observation that the wave function or wave packet approach may, nevertheless, be useful in BEC for the investigation of "nal state interactions (see below) or for the construction of event generators, where phases or quantum amplitudes are ignored anyway. Also, for correlations between fermions where coherence is absent the wavefunction formalism may be an adequate substitute, although here too a "eld theoretical approach is possible.
6 String models [22,23] also use a Wigner-type formalism. Here, it is postulated that there exists a `formationa time q and therefore the particle production points are distributed around t2!x2"q2. This implies among other things a correlation between particle production points and momenta.
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2.2. Quantum optical methods in BEC In high-energy processes in which the pion multiplicity is large, we may in general expect the methods of quantum statistics (QS)7 to be useful. For BEC in particular they turn out to be indispensable. These methods have been applied with great success particularly in quantum optics (QO), super#uidity, superconductivity, etc. What distinguishes optical phenomena from those in particle physics are conservation laws and "nal state interactions which are present in hadron physics. At high energies and high multiplicities the "rst are not important. Neglecting for the moment the "nal state interactions also, QS reduces then to QO and we may take over the formalism of QO to interpret the data on multipion production at high energies, provided we consider identical pions. Given the general validity of QS (or QO), it is then clear that any model of multiparticle production must satisfy the laws of quantum statistics and this has far reaching consequences, independent of the particular dynamical mechanism which governs the production process. The main tools in the QO formalism are de"ned below.8 Coherent states and squeezed coherent states: Coherent states DaT are eigenstates of the (oneparticle) annihilation operator a aDaT"aDaT . Squeezed coherent states are eigenstates DbT of the two-particle annihilation operator 4 b"ka#las
(2.9)
(2.10)
with DkD2!DlD2"1 ,
(2.11)
so that bDbT "bDbT . (2.12) 4 4 One of the remarkable properties of these states which explains also their name is that for them the uncertainty in one variable can be squeezed at the expense of the other so that (2.13) (*p)25u/2 (*q)241/2u, 4 4 or vice versa. The importance of this remarkable property lies among other things in the possibility to reduce quantum #uctuations and this explains the great expectations associated with them in communication and measurement technology as well as their interest from a heuristic point of view. It has been found recently in [16] that squeezed states appear naturally when one deals with rapid phase transitions (explosions). Indeed consider the transition from a system a to a system b and assume that it proceeds rapidly enough for the relation between the creation and annihilation
7 By QS we understand in the following the density matrix formalism within second quantisation. 8 A review of the applications of quantum optical methods to multiparticle production up to 1988 can be found in [26].
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operators and the corresponding "elds in the two `phasesa to remain unchanged. Mathematically, this process will be described by postulating at the moment of this transition the following relations between the generalised coordinate q and the generalised momentum p of the "eld: q"(1/J2E )(bs#b)"(1/J2E )(as#a) , b a (2.14) p"iJ(E /2)(bs!b)"iJ(E /2)(as!a) , a b as, a are the free "eld creation and annihilation operators in the `phase aa and bs, b the corresponding operators in the `phase ba. Eq. (2.14) holds for each mode p. Then we get immediately a connection between the a and b operators, a"b cosh r#bs sinh r , as"b sinh r#bs cosh r
(2.15)
with r"r(p)"1 log(E /E ) . a b 2 Transformation (2.15) is just the squeezing transformation (2.10) with k"cosh r,
l"sinh r
(2.16)
(2.17)
which proves the statement made above. The observation of squeezed states in BEC may thus serve as a signal for such rapid transitions. Furthermore, the existence of isospin induces in hadronic BEC certain e!ects which are speci"c for squeezed states. This topic will be discussed in Section 4.4. From the point of view of BEC what distinguishes ordinary coherent states from squeezed states is the following: for coherent states the intercept C (k, k)"1 while for squeezed states it can take 2 arbitrary values. In Fig. 2 one can see such an example. Expansions in terms of coherent states. Coherent states form an (over)complete set so that an arbitrary state D f T can be expanded in a unique way in terms of these states. Of particular use is the expansion of the density matrix o in terms of coherent states. For a pure coherent state the density operator reads o"DaTSaD .
(2.18)
For an arbitrary density matrix case we have
P
o" P(a)DaTSaD d2a .
(2.19)
Here P is a weight function which usually, but not always, has the meaning of a probability. The normalisation condition for the density operator translates in terms of the P representation as
P
tr o" P(a) d2a"1 . Eq. (2.19) is called also the Glauber}Sudarshan representation.
(2.20)
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Fig. 2. Second-order correlation function C (k, k),g(2) as a function of the squeezing parameter r for pure squeezed 2 states (from Ref. [27]).
The knowledge of P(a) is (almost) equivalent to the knowledge of the density matrix. However in most cases the exact form of P(a) is not accessible and one has to be content with certain approximations of it. Among these approximations the Gaussian form is privileged because: (i) one can prove that P(a) is of Gaussian form for a certain physical situation which is frequently met in many-body physics. (ii) its use introduces an enormous mathematical simpli"cation. Proposition (i) is the subject of the central limit theorem which states that if 1. the number of sources becomes large; 2. they are stationary in the sense that their weight function P(a) depends only on the absolute value DaD; 3. they act independently, then P(a) is Gaussian. These conditions are known to be ful"lled in most cases of optics and presumably also in high-energy physics. Chaotic "elds and in particular systems in thermal equilibrium are described by a Gaussian density matrix. One of the reasons why the Gaussian form for P plays such an important part in correlation studies is the fact that for a Gaussian P(a) all higher-order correlations can be expressed in terms of the "rst two correlation functions (see e.g. Refs. [3,5]). On the other hand, the coherent state representation is particularly important for correlation studies because in this representation all correlation functions can be expressed in terms of the creation and annihilation operators as and a of the "elds (particles). This follows from the Fourier expansion of an arbitrary "eld in second quantisation n(x)"+ [a e~*kx#as e*kx] . (2.21) k k k This property will be used extensively in Section 4.2 within the classical current formalism.
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Correlation functions. The "rst-order correlation function reads G(1)(x, x@),¹r[ons(x)n(x@)] .
(2.22)
Higher (nth)-order correlation functions are de"ned analogously by G(n)(x 2x , x x ),¹r[ons(x )2ns(x )n(x ) n(x )] . (2.23) 1 n n`1 2 2n 1 n n`1 2 2n In quantum "eld theory because of the mathematical complexity of the problem exact solutions of the "eld equations are available only in special cases. One such case will be discussed later on. However for strong interactions9 even for this case one has to use phenomenological parametrisations of the correlation functions and determine the parameters (which have a de"nite physical meaning) by comparing with experiment. In optics for stationary chaotic "elds two particular parametrisations are used: 1. Lorentzian spectrum: G(1)(x , x )"Sn T e~@x1 ~x2 @@m . 1 2 #) 2. Gaussian spectrum:
(2.24)
G(1)(x , x )"Sn T e~@x1 ~x2 @2@m2 , (2.25) 1 2 #) m is the coherence length in x-space and Sn T is the mean number of particles associated with the #) chaotic "elds. In [25] it has been proposed to use the analogy between time and rapidity in applying the methods of quantum optics to particle physics. Indeed in optics processes are usually stationary in time while in particle physics the corresponding stationary variable (in the rapidity plateau region) is rapidity.10 Pure coherent or pure chaotic "elds are just extreme cases. In general one expects partial coherence, i.e. a superposition of coherent and chaotic "elds n"n #n . (2.26) #0)%3%/5 #)!05*# This leads for the Lorentzian case, e.g. to a second-order correlation function of the form C (x, x@)"1#2p(1!p) e~@x~x{@@m#p2 e~2@x~x{@@m , (2.27) 2 where p is the chaoticity, which varies between 0 (for purely coherent sources) and 1 (for totally chaotic sources). Eq. (2.8) is seen to be a particular form of the above equation for j"p"1 and it is clear herefrom that j does not describe (partial) coherence as its name would imply. The presence of coherence introduces a new term into C . However, it is remarkable that the number of free 2 parameters in Eq. (2.27) is the same as in Eq. (2.8). Formally, it appears as if there would act two
9 By strong interactions we mean here interactions for which perturbative methods are inapplicable. They are present not only in hadronic physics but also in quantum optics. 10 This property does not hold for other variables.
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sources rather than one, but the `weightsa and the space}time characteristics of these two sources are in a well-de"ned relationship. This circumstance had been forgotten up to 1989 [28] both by experimentalists and theorists. The reason for this is the fact that during the 1980s the wave-function formalism was dominating the BEC literature, especially the experimental one. From the foregoing discussion it should be clear that there are various reasons besides coherence why the bunching e!ect in BEC is reduced. However, it should also be clear that the empirical description of this state of a!airs through the j factor is possible only for totally chaotic sources. Since in an experiment this is never known a priori, this implies that the "tting of data with a formula of this type is misleading and should be avoided, the more so because the correct formula (2.27) does not contain more free parameters than Eq. (2.8). In the example presented above these free parameters are p and m for Eq. (2.27) and j and the e!ective radius (which enters in fI ) for Eq. (2.8), respectively. Using the rapidity}time analogy of Ref. [25] for a partially chaotic "eld the second-order correlation function in rapidity is given by Eq. (2.27), with x being replaced by rapidity y. The fact that the last two terms in Eq. (2.27) are in a well-de"ned relationship and depend in a characteristic way on the two parameters p and m is a consequence of the superposition of the two xelds (coherent and chaotic) and distinguishes a partially coherent source from a source which is a superposition of two independent chaotic intensities. Because of this the form (2.27) was proposed in [28] to be used as a signal for detection of coherence in BEC.11 As a matter of fact, an attempt in this direction was made in an experimental study by Kulka and LoK rstad [29]. In this analysis BEC data from pp and p6 p reactions at Js"53 GeV were used to compare various forms of correlation functions. Among other things one considered formulae of QO type for rapidity C "1#2p(1!p) e~@y1 ~y2 @@m#p2 e~2@y1 ~y2 @@m 2 (corresponding to a Lorentzian spectrum) and
(2.28)
(2.29) C "1#2p(1!p) e~@y1 ~y2 @2@m2#p2 e~2@y1 ~y2 @2@m2 2 (corresponding to a Gaussian spectrum) as well as arbitrary superpositions of two chaotic sources of exponential or Gaussian form, respectively. C "1#j e~@y1 ~y2 @@m#j e~2@y1 ~y2 @@m , (2.30) 2 1 2 (2.31) C "1#j e~@y1 ~y2 @2@m2#j e~2@y1 ~y2 @2@m2 . 2 2 1 Here j and j represent arbitrary weights of the two chaotic sources. 1 2 Because of the limited statistics no conclusion could be drawn as to the preference of the QO form versus the two-source form. Similar inconclusive results were obtained when one replaced y !y in the above equations by the invariant momentum di!erence Q2"(k !k )2. 1 2 1 2
11 Eq. (2.27) is a special case of superposition of coherent and chaotic "elds; it can be considered as corresponding to point-like coherent and chaotic sources and a momentum independent chaoticity; superpositions of more general "nite-size sources are considered in Section 4.4 (Eq. (4.40)) and Section 4.8.
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2.2.1. Higher-order correlations We have mentioned above that a characteristic property of the Gaussian form of density matrix (not to be confused with the Gaussian form of the correlator or the Gaussian form of the space}time distribution) is the fact that all higher-order correlation functions are determined just by the "rst two correlation functions. Since all BEC studies in particle physics performed so far assume a Gaussian density matrix, the reader may wonder why it is necessary to measure higher-order correlation functions. There are at least three reasons for this: (i) The conditions of the applicability of the above theorem and in particular the postulate that the number of sources is in"nite and that they act independently can never be ful"lled exactly. (ii) In the absence of a theory which determines from "rst principles the "rst two correlation functions, models for these quantities are used, which are only approximations. The errors introduced by these phenomenological parametrisations manifest themselves di!erently in each order and thus violate the above theorem even if (i) would not apply. Moreover, for certain parametrisations of the correlation functions the phases of the chaotic and coherent amplitudes disappear from the second-order correlation function (see Section 4.8) and are present only in higher-order correlation functions. (iii) In experiments, because of limited statistics and sometimes also because of theoretical biases not all physical observables are determined, but rather averages over certain variables are performed, which again introduce errors which propagate (and are ampli"ed) from lower-to-higher correlations. Conversely, by comparing correlation functions of di!erent order, one can test the applicability of the theorem quoted above and pin down more precisely the parameters which determine the "rst two correlation functions (e.g. the chaoticity p and the correlation length m in Eqs. (2.28) and (2.29)), which is essentially the purpose of particle interferometry. The phenomenological application of these considerations will be discussed in the following as well as in Section 6.3 for the particular case of the invariant Q variable, but arguments (i)}(iii) have general validity. It would be a worthwhile research project to compare the deviations introduced in the relation between lower and higher correlation functions, due to (i) with those introduced by (ii) and (iii). The simpli"cation brought by the variable Q can be enhanced by a further approximation proposed by Biyajima et al. [30]. With the notation Q "k !k the analogue of Eq. (2.29) can be ij i j written C "1#2p(1!p) exp(!R2Q2 )#p2 exp(!2Q2 R2) . 2 12 12
(2.32)
For the third-order correlation function one obtains C "1#2p(1!p)[expM!R2Q2 N#expM!R2Q2 N#expM!R2Q2 N] 3 12 13 23 #p2[expM!2R2Q2 N#expM!2R2Q2 N#expM!2R2Q2 N] 12 13 23 #2p2(1!p)[expM!R2(Q2 #Q2 )N#expM!R2(Q2 #Q2 )N 12 23 13 23 #expM!R2(Q2 #Q2 )N]#2p3[expM!R2(Q2 #Q2 #Q2 )N] . 12 13 12 13 23
(2.33)
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Ref. [30] proposed to use symmetrical con"gurations for all two-particle momentum di!erences, i.e. to consider Q independent of (i, j). For C this assumption does not of course introduce any ij 2 modi"cations. However for higher orders the simpli"cation is important. Thus, e.g. for the third-order correlation with Q "Q "Q and with the de"nition Q2 "Q2 #Q2 #Q2 12 13 23 5)3%% 12 13 23 Eq. (2.33) becomes C "1#6p(1!p) expM!1R2Q2 N#3p2(3!2p) expM!2R2Q2 N 3 3 5)3%% 3 5)3%% #2p3 expM!R2Q2 N . (2.34) 5)3%% In Ref. [30] similar expressions for C and C , again for a Gaussian correlator, were given. 4 5 These relations for higher-order BEC were subjected to an experimental test in Ref. [31], using the UA1 data for p6 p reactions at Js"630 and 900 GeV. For reasons which will become clear immediately, we discuss here this topic in some detail. The procedure used in [31] for this test consisted in determining R and p separately for each order q of the correlation and comparing these values for di!erent q. It was found that a Gaussian correlator did not "t the data. Next in [31], one tried to replace the Gaussian correlator by an exponential (see Eq. (2.28)). To do this one substituted simply in the expressions for the correlation functions of Ref. [30] the factor exp(!R2Q2) with exp(!RQ). Such a procedure was at hand given the fact that for C the QO formulae both for an exponential correlator and a Gaussian 2 correlator were known [28] and their comparison suggested just this substitution, as seen from Eq. (2.29). In [31] one used then for the exponential correlator the relations (2.35) C%.1*3*#!-"1#2p(1!p) exp(!RQ )#p2 exp(!2Q R) , 2 12 12 C%.1*3*#!-"1#6p(1!p) exp(!1RQ )#3p2(3!2p) exp(!2RQ ) 3 3 5)3%% 3 5)3%% #2p3 exp(!RQ ). (2.36) 5)3%% With these modi"ed formulae one still could not "nd in [31] a unique set of values p and R for all orders of correlation functions. However now a clearer picture of the `disagreementa between the QO formalism and the data emerged. It seemed that while the parameter p was more or less independent of q, the radius R increased with the order q in a way which could be approximated by the relation R "RJ1q(q!1) . (2.37) q 2 However in [32] it was shown that the "ndings of [31] and in particular Eq. (2.37) not only did not contradict QS but on the contrary constituted a con"rmation of it. While Eq. (2.35) for the second-order correlation function coincides with that derived in quantum optics for an exponential spectrum, this is not the case with the expressions for higher-order correlations C%.1*3*#!q (Eq. (2.36)). The formulae for C , C and C corresponding in QS to an exponential correlator and 3 4 5 derived in [32] di!er from the empirical ones used in [31]. As an example we quote C "1#6p(1!p) exp(!(1/J3)RQ )#3p2(3!2p) exp(!(2/J3)RQ ) 3 5)3%% 5)3%% #2p3 exp(!J3RQ ). 5)3%%
(2.38)
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As observed in [32] and as one easily can check by comparing Eq. (2.36) with Eq. (2.38), one can make the empirical formulae for C used in [31] coincide with the correct ones, by replacing the q parameter R with a scaled parameter R and the relation between R and R , is nothing else but 4 4 R "R , where R is given by Eq. (2.37). The fact that this happens for three di!erent orders, i.e. for 4 q q C , C and C makes a coincidence quite improbable. 3 4 5 By empirically modifying the formulae of higher-order BEC for the exponential case, paper [31] had explicitly violated QS and the `phenomenologicala relationship (2.37) between R and q just compensated this violation. The fact that this compensation and the "nal agreement between theory and experiment was not perfect is not surprising and is discussed in [32]. Besides reasons (i)}(iii) mentioned above, one has to take into account the fact that the QO formalism in which Eqs. (2.35) and (2.33) are based assumes stationarity in Q, i.e. assumes that the correlator depends only on the di!erence of momenta k !k and not also on their sum. As mentioned already this condition is in general 1 2 not ful"lled in BEC. Furthermore, the parameter p, if it is related to chaoticity, is in general momentum dependent (see Section 4.8). Also, the symmetry assumption, Q independent of i, j, may be too strong. Besides these theoretical caveats, i,j there are also experimental problems, related to the fact that the UA1 experiment is not a dedicated BEC experiment and thus su!ers from speci"c diseases, which are common to almost all particle physics BEC experiments performed so far. Among other things, there is no identi"cation of particles (only 85% of the tracks recorded are pions), and the normalisation of correlation functions is the `conventionala one, i.e. not based on the single inclusive cross sections as the de"nition of correlation functions demands (see (2.7) and Section 4.11), but rather uses an empirically determined `backgrounda ensemble.
In the mean time further theoretical and experimental developments took place. On the theoretical side a new space}time approach to BEC was developed [3,33] which is more appropriate to particle physics and which contains as a special case the QO formalism. In particular, the two exponential features of the correlation function is recovered. On the experimental side a new technique for the study of higher-order correlations was developed, the method of correlation integrals which was applied [34] to a subset of the UA1 data in order to test the above-quoted QO formalism. The "ts were restricted to second- and third-order cumulants only. Again it was found that by extracting the parameters p and R from the secondorder data, the `predicteda third-order correlation, this time by using a correct QO formula, di!ered signi"cantly from the measured one. If con"rmed, such a result could indicate that the QO formalism provides only a rough description of the data and that higher precision data demand also more realistic theoretical tools. Such tools are the QS space}time approach to BEC presented in Section 4.3. A further, but more remote possibility would be to look for deviations from the Gaussian form of the density matrix. However, it seems premature to speculate along these lines given the fact that the procedure used to test the relation between the second- and third-order correlation functions has to be quali"ed. Indeed in [34] one did not perform a simultaneous "t of second- and third-order data to check the QO formalism. Such a simultaneous "t appears necessary before drawing conclusions, because as mentioned above (see (ii) and (iii)), the errors involved in `guessinga the form of the correlator, and the fact that the variable Q does not characterise completely the two-particle correlation, limit the applicability of the theorem which reduces higher-order correlations to "rst and second ones. As a matter of fact, it was found [35] (see also Section 6.3) in a comparison of the QS space}time approach with higher-order correlation data, that the second-order correlation data is quite insensitive to the values of the parameters which enter the correlator, while once higher-order data are used in a simultaneous "t, a strong delimitation of the acceptable parameter values results.
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Thus there are several possible solutions if one restricts the "t to the second-order correlation and the correct one among these can be found only by "tting simultaneously all correlations. If by accident one chooses in a lower correlation the wrong parameter set, then the higher correlations cannot be "tted anymore.12 Before ending these phenomenological considerations in which the variable Q played a major part, a few remarks about its use may be in order. The invariant momentum diwerence Q. BEC studies in particle physics use often a privileged variable namely the squared momentum di!erence: (2.39) Q2"(k !k )2"(k !k )2!(E !E )2 . 1 2 1 2 1 2 It owes its special role to the fact that it is a relativistic invariant and it has already been used in the pioneering paper by GGLP [8]. It also has the advantage that it involves all four components of the momenta k simultaneously so that the intercept of the correlation function C (k, k) coincides 2 with C (Q"0). Thus by measuring C as a function of one single scalar quantity Q one gets 2 2 automatically the intercept. This is not the case with other single scalar quantities used in BEC like y !y or k !k which characterise the intercept only partially. 1 2 M,1 M,2 On the other hand, Q su!ers from certain serious diseases which make its use for practical interferometrical purposes questionable. The "rst and most important de"ciency of Q is the fact that it mixes time and space coordinates: the associated quantity R in the conventional parametrisation of the correlation function C "1#j exp(!R2Q2) is neither a radius nor a lifetime, but a combination of these, which 2 cannot be easily disentangled. Another de"ciency of Q, which is common to all single scalar quantities is the circumstance that it does not fully characterise the correlation function. Indeed, the second-order correlation function C is in general a function of six independent quantities which 2 cannot be replaced by a single variable. An improvement on Q was proposed by Cramer [36] with the introduction of coalescence variables which constitute a set of three boost invariant variables to replace for, a pair of identical particles, with the single variable Q. They are related to Q by Q2"2m2(C #C2 #C2 ), where L T R C , C , C denote longitudinal, transverse and radial coalescence respectively, and m the mass of L T R the particle. They have the properties that C "0 when y "y , C "0 when / "/ and L 1 2 T 1 2 C "0 when either m "m or k "k . Here m is the transverse mass and / the azimuthal angle R 1 2 1 2 i in the transverse plane. It is shown in [36] that with these new variables a Lorentz invariant separation of the space- and time-like characteristics of the source is possible, within the kinematical assumptions involved by the particular choice of the coalescence variables. This separation is however rather involved. In [36] the coalescence variables are used for the introduction of Coulomb corrections into second- and higher-order correlation functions. Another way to compensate in part for the fact that one single variable does not characterise completely the two-particle system, but which maintains the use of Q is, as explained above, to consider higher-order correlations. 12 The fact that in [34] the correlations were normalised by mixing events rather than by comparing with the single inclusive cross sections in the same event, as prescribed by the de"nition of the correlation functions may also in#uence the applicability of the central limit theorem.
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3. Final state interactions of hadronic bosons One of the most important di!erences between the HBT e!ect in optics and the corresponding GGLP e!ect in particle physics is that in the "rst case we deal with photons while in the second case with hadrons. While photons in a "rst approximation do not interact, hadrons do interact. This interaction has two e!ects: (i) it in#uences the correlation between identical hadrons and (ii) it leads to correlations also between non-identical hadrons. This review deals only with correlations due to the identity of particles and in particular with Bose}Einstein correlations. Therefore only e!ect (i) will be discussed.13 E!ect (i) is usually described in terms of "nal state interactions. In some theoretical studies (see e.g. [38] and references quoted there) emission of particles at di!erent times is also treated as an e!ective "nal state interaction. From the BEC point of view the "nal states interaction constitutes in general an unwanted background, which has to be subtracted in order to obtain the `truea quantum statistical e!ect on which interferometry measurements are based. That this is not always a trivial task will be shown in the following. There are two types of "nal state interactions in hadronic interferometry: electromagnetic, traded under the generic name of Coulomb interactions, and strong. Furthermore, one distinguishes between one-body "nal state interactions and many-body "nal state interactions. 3.1. Electromagnetic xnal state interactions The plane wave two-body function used in the considerations above (see Eq. (2.5)) applies of course only for non-interacting particles. As a "rst step towards a more general treatment consider charged particle interferometry. As a matter of fact the vast majority of BEC studies, both of experimental and theoretical nature, refer to charged pions. For two-particle correlations, we will have to consider the interaction of each member of a pair with the charge of the source and the Coulomb interaction between the two particles constituting the pair. The "rst e!ect will a!ect primarily the single-particle probabilities and is not expected to depend on the momentum di!erence q. Attention has been paid so far mostly to the second e!ect, i.e. the modi"cation of the twoparticle wave function due to the Coulomb interaction between the two particles. While initially, having in mind the Gamow formula, it was assumed that this e!ect is (for small q values) quite important, at present serious doubts about these estimates have arisen. The model dependence of corrections for this e!ect makes it almost imperative that experimental data should be presented also without Coulomb corrections, so that it should be left to the reader the possibility of introducing (or not introducing) corrections according to her/his own prejudice.
13 The reader interested in correlations between non-identical particles is referred to [2] for the period up to 1990; for more recent literature see [37], where correlations in low energy heavy ion reactions are reviewed.
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3.1.1. Coulomb correction and `overcorrectiona As usual one separates the centre-of-mass motion from the relative motion. For the last one the scattering wave function reads u(r)P e*kz#r~1g(h, /)e*kr , (3.40) r?= where the relative position vector r has polar coordinates r, h, /. The form of the function g depends on the scattering potential. In the Coulomb case the corresponding SchroK dinger equation can be solved exactly and the correction to the two-particle wave function and the correlation function can be calculated. So far at least three sophisticated procedures have been used for this purpose. In [39] the value of the square modulus of the wave function u in the origin r"0 was proposed as a correction term G to the correlation function C . 2 Up to non-interesting factors this is the Gamow factor which reads Du(0)D2"2pg/(exp(2pg)!1)"G(g) .
(3.41)
Here g"am /q (3.42) n and q"Dk !k D. m is the pion mass. 1 2 n However as pointed out by Bowler [40] and subsequently also by others, in BEC this approximation may be questionable. Indeed in a typical e`}e~ reaction, e.g. the source which gives rise to BEC has a size of the order of 1 fm which is a large number compared with a typical `Coulomb lengtha r de"ned as the classical turning point where the kinetic energy balances the 5 potential Coulomb energy: q8 2/2m "e2/r , (3.43) 3%$ 5 where q8 "q/2 and m is the reduced mass of the pion pair. For a typical BEC momentum 3%$ di!erence of q"100 MeV, one gets from (3.43) r "0.08 fm. This suggests that by taking the value 5 of the wave function at r"0 one overestimates the Coulomb correction. More recently, Biyajima and collaborators [41] (see also [42]) have considered a further correction to the correction proposed by Bowler, which decreases even more the Coulomb e!ect and which is also of heuristic interest. In [41] it is pointed out that the wave function (3.40) used by Bowler has not taken into account the symmetry of the two-particle system. It has to be supplemented by an exchange term so that the r.h.s. of (3.40) becomes (e*qz#e~*qz)#[ f (h, /)#f (p!h, /#p)]r~1e*qr .
(3.44)
The corrections due to this new e!ect are of the same order as those found in [40] and go in the same direction. Another approach to the Coulomb correction in BEC has been suggested by Baym and Braun-Munzinger [43]. Starting from the observation of [40] about the classical turning point these authors propose the use for heavy ion reactions of a classical Coulomb correction factor arising from the assumption that the Coulomb e!ect of the pair is negligible for separations less than an initial radius r . This model is tested by comparing its results with experimental data on 0
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n`n~, n~p, n`p correlations in heavy ion collisions14 (Au}Au at AGS energies). The assumption behind this comparison is that the observed correlations are solely due to the Coulomb e!ect.15 Indeed qualitatively this seems to be the case: thus the data for the n`n~ and n~p correlations show a bunching e!ect characteristic of an attractive interaction while the data for the n`p correlation show an antibunching e!ect, characteristic of repulsion. After this test the authors compare their correction with the Gamow correction and "nd that the last one is much stronger. Wherefrom they also conclude that the Gamow factor overestimates quite appreciably the Coulomb e!ect.16 An even stronger conclusion is reached by Merlitz and Pelte [47] from the solution of the time-dependent SchroK dinger equation for two identical charged scalar bosons in terms of wave packets. These authors "nd that the `expecteda Coulomb e!ect in the correlation function is obliterated by the dispersion of the localised states and is thus unobservable. This makes the interpretation of experimentally observed n`n~ correlations in terms of Coulomb e!ects even more doubtful. The theoretical studies of the Coulomb e!ect in BEC quoted so far are based on the solution of the SchroK dinger equation and apply in fact only for the non-relativistic case. While one might argue that the relative motion of two mesons in BEC is for small q non-relativistic, this is not true for the single-particle distributions (see below). Therefore, in principle, one should replace the solution of the SchroK dinger equation in the Coulomb "eld used above by the corresponding solution of the Klein}Gordon equation. This apparently has not yet been done, with the exception of of a calculation by Barz [48] who investigated the in#uence of the Coulomb correction on the measured values of radii. He found an important change of these radii due to the Coulomb "eld only for momenta 4200 MeV. Finally, one must mention that the corrections of the wave function described above do not take into account the fact that the charge distribution of a meson is not point like, but has a "nite extension of the order of 1 fm. This means that in principle the SchroK dinger (or the Klein}Gordon) equation has to be solved with a Coulomb potential modi"ed by this "nite size e!ect. Given the great sensitivity of BEC on small corrections in the wave function, this might be a worthwhile enterprise for future research. As a matter of fact it is known from atomic physics (isotopic and isomeric shifts and hyper"ne structure) that these "nite size e!ects lead to observable consequences.
Besides the wave function e!ect which in#uences the BEC due to directly produced particles or those originating from short-lived resonances, one has to consider [40] the Coulomb overcorrection applied to pairs of which one particle is a daughter of a long-lived state. This e!ect may bias the correction by up to 20%. Coulomb correction for higher-order correlations. The Coulomb corrections discussed above were limited to single- and two-body interactions. In present high-energy heavy ion reactions we have
14 The measured n`n~ correlations were also used in two recent experimental papers [44,45] to estimate the Coulomb correction. Why such a procedure is questionable is explained below. 15 This assumption has to be quali"ed among other things because the "nal state strong interactions e!ects due to resonances can also in#uence these correlations. Furthermore there also exists a quantum statistical correlation for the n`n~ system (see Sections 4.4, 4.7 and 4.8) which, however may be weak. 16 See however also Ref. [46] where rescattering is added to the classical Coulomb e!ect and where somewhat di!erent results are obtained. It is unclear whether the strong position-momentum correlations implicit in this rescattering model do not violate quantum mechanics.
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already events with hundreds of particles and in the very near future at the relativistic heavy ion collider at Brookhaven (RHIC) this number will increase by an order of magnitude. Then many-body "nal state interactions may become important. Unfortunately, the theory of manybody interactions even for such a `simplea potential as the Coulomb one is apparently still unmanageable. The long-range nature of the electromagnetic interaction does not make this task simpler. Bowler [40] sketched a scenario for Coulomb screening based on the string model. In such a model particles are ordered in space}time, so that, e.g. at least one n` must be situated between the members of a n~n~ pair. While the net in#uence of this e!ect on n`n~ is expected to be small, for a n~n~ pair the situation is di!erent, because instead of repulsion one obtains attraction. In the case of long-range interactions such an e!ect may become important if the n` propagates together 1 with the n~n~ pair. This happens if Q &Q &Q . To take care of this e!ect Bowler suggests 2 3 12 13 23 the replacement C(Q )PC(Q )SC~1(Q )C~1(Q )T 23 23 12 13 k3
(3.45)
for the Coulomb n~n~ correction and C(Q )PC(Q )SC~1(Q )C(Q )T 12 12 23 13 k3
(3.46)
for the Coulomb n`n~ correction. Here S2T 3 symbolises averaging with respect to the mok mentum of particle 3. While on the average SC~1(Q )C(Q )T 23 13 k3
(3.47)
is unity, at small Q the function C(Q) oscillates rapidly and therefore the factor SC~1(Q )C~1(Q )T 12 13 k3
(3.48)
is sensitive to the distribution of Q , Q associated with the local source. According to [40] this 12 13 last factor for n`n~ pairs does not exceed 0.5% but for like-sign pairs no estimate is provided. Coulomb and resonance ewect in single inclusive cross sections. At a "rst look one might be tempted to believe that for single inclusive cross sections in heavy ion reactions the estimate of Coulomb e!ects is straightforward. Unfortunately, this is not the case and so far there is no reliable theoretical estimate of this e!ect. This is so because the produced charged secondaries do not move simply in the electromagnetic "eld of the colliding nuclei but at the same time interact with all the other secondaries. In view of this situation, recently an attempt has been made to put in evidence experimentally the Coulomb e!ect in the single inclusive cross section of pions in heavy ion reactions [49]. In this experiment an excess of negative pions over positive pions in Pb}Pb reactions at 158 AGeV was observed which the authors of [49] attributed to the Coulomb interaction of produced pions with the nuclear "reball. However, this interpretation has been challenged in [50] where, in a detailed hydrodynamical simulation it was shown that a similar excess in the n~/n` ratio is expected as a consequence of resonance (especially hyperon) decays. This quali"cation goes in the same direction as that mentioned above with respect to the exaggeration of the e!ects of Coulomb interaction in BEC. It also illustrates the complexity of the many-body problem of heavy ion reactions even for weak interactions like the electromagnetic one, which in principle are well known.
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3.2. Strong xnal state interactions This is a very complex problem because we are dealing with non-perturbative aspects of quantum chromodynamics. That there is no fully satisfactory solution to this problem can be seen from the very fact that we have at least three di!erent approaches to it. As will become clear from the following, these di!erent approaches must not be used simultaneously, as this would constitute double (or triple) counting which is also why a fourth solution proposed here which is of heuristic nature will appear in many cases more appealing and more e$cient. 3.2.1. Final state interactions through resonances The majority of secondaries produced in high-energy collisions are pions out of which a large fraction (between 40% and 80%) arises from resonances.17 Since the resonances have "nite lifetimes and momenta, their decay products are created in general outside the production region of the `directa pions (i.e., pions produced directly from the source) and that of the resonances. As a consequence, the two-particle correlation function of pions re#ects not only the geometry of the (primary) source but also the momentum spectra and lifetimes of resonances [2]. Kaons are much less a!ected by this circumstance [52]; however, correlation experiments with kaons are much more di$cult because of the low statistics. For a more detailed discussion of kaon BEC see Section 5.1.3. The (known) resonances have been taken into account explicitly within the wave-function formalism (see e.g. [17,53,18]), the string model [22,23] or within other variants of the Wigner function formalism (see e.g. [19,54}56]). The drawback of this explicit approach is that it is rather complicated, it is usually applicable only at small momenta di!erences q and it presupposes a detailed knowledge of resonance characteristics, including their weights, which, with few exceptions, cannot be measured directly and have to be obtained from event generators18 or other models.19 With these essential caveats in mind one "nds that the distortion of the two-particle correlation function due to resonance decay leads to two obvious e!ects: (a) the e!ective radius of the source increases, i.e., the width of the correlation function decreases, and (b) due to the "nite experimental resolution in the momentum di!erence the presence of very long-lived resonances leads to an apparent decrease of the intercept of the correlation function. E!ect (b) is particularly important if one wants to draw conclusions from the intercept about a possible contribution of a coherent component in multiparticle production. In hydrodynamical studies of multiparticle production processes one considers resonances within the Wigner function approach (see Section 5.1.3). 3.2.2. Density matrix approach In Ref. [59] one describes the e!ect of strong "nal state interactions by constructing a density matrix based on an e!ective Lagrangian of Landau}Ginzburg form as used in statistical physics 17 For experimental estimates, see Ref. [51]. 18 That event generators are not a reliable source of information for this purpose was demonstrated in the case of e`}e~ reactions in [57,58]. 19 In Ref. [56] the weights were determined from thermodynamical considerations.
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and quantum optics (see also [60]). This is a more theoretical approach as it allows to study the e!ect of the strength of the interaction g on BEC, albeit in an e!ective Lagrangian description. One writes the density matrix
P
o"Z~1 dnDnTe~F(n)SnD,
P
Z" dne~F(n) ,
(3.49)
where F is the analogue of the Landau}Ginzburg free energy and the integrals are functional integrals over the "eld n. This "eld is written as a superposition of coherent n and chaotic "elds # n : #) n"n (y)#n (y) . (3.50) # #) The variable y refers in particular to rapidity. The total mean multiplicity SnT is related to the "eld n by SnT"SDn2DT .
(3.51)
Similar relations hold for the coherent and chaotic parts of n. One assumes stationarity in y, i.e. the "eld correlator G(y, y@)"Sn(y)n(y@)T depends only on the di!erence y!y@,*y. One writes the Landau}Ginzburg form for F as
P C
F(n)"
y
K K
dy an(y)#b
D
Rn(y) 2 #gDn(y)D4 , Ry
(3.52) 0 where a, b and g are constants. The strong interaction coupling is represented by g. The constants a, b can be expressed in terms of Sn T and the `coherence lengtha m which is de"ned through the #) correlator G (see Eq. (2.24)). The main result of these rather involved calculations is that while the interaction does not play any signi"cant role in the value of the intercept C (0) it plays an important part in (C (*yO0)). 2 2 This situation is illustrated in Figs. 3 and 4. Thus it is seen in Fig. 3 that all C curves for various 2 g coincide in the origin. The e!ect of the interaction in this approach is similar to that of (short-lived) resonances, i.e. it leads to a decrease of the width of the correlation function. The sensitivity of C on g suggests that the shape of the correlation function can, in principle, be used 2 for the experimental determination of g. One "nds furthermore that there is no g-dependence for purely chaotic or purely coherent sources. This observation suggests that for a strongly coherent or chaotic "eld the "nal state interaction does not manage to disturb the correlation. Note that the curves in Fig. 4 intersect at some *y which depends on the chaoticity. This is characteristic for correlations treated by quantum statistics, when one has a superposition of coherent and chaotic "elds, and is a manifestation of the appearance of two (di!erent) functions in the correlation function. We recall (see Section 2.2) that for the particular case of a Lorentzian spectrum one gets, in the absence of xnal state interactions, C (y, y@)"1#2p(1!p)e~@y~y{@@m#p2e~2@y~y{@@m 2 instead of the empirical relation C (y, y@)"1#j exp(!Dy!y@D/m) . 2
(3.53)
(3.54)
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Fig. 3. Second-order correlation function as given by the quantum statistical formalism of Ref. [59] for various values of the coupling constant g for i"0.5 (from [59]). The parameter i is related to the chaoticity p via the relation i"Sn T/Sn T"1/p!1 where p"Sn T/SnT and n"n #n is the total multiplicity. > is the maximum rapidity. # #) #) # #) Fig. 4. The same as in Fig. 3 for various values of i at g"0 (from [59]).
As shown in Fig. 5 the parametrisation (3.54) leads to parallel curves for various values of j, while the more correct parametrisation of Ref. [59] leads to intersecting curves. This is due to the fact that the relation for C derived in [59] contains as a particular case (for g"0) Eq. (3.53) and 2 retains the essential feature of Eq. (3.53), which consists in the superposition of two exponentials. 3.2.3. Phase shifts For charged pions the strong "nal state interactions can also be described by phase shifts. It is known that for an isospin I"2 state (this is the isospin of a system of two identically charged pions) the corresponding strong interaction is repulsive. However it has been suggested [61] that the range of strong interactions is smaller than the size of the hadronic source and therefore the correlation should be essentially una!ected by this e!ect.20 Even for particle reactions like hadron}hadron or e`}e~ the size of the source is of the order of 1 fm while the range of interaction is only 0.2 fm. On the other hand, the e!ective size quoted above arises because of the joint contribution to BEC of direct pairs and resonances. So it is interesting to analyse these two contributions separately.
20 The separation between resonances and phase shifts is of course not rigorous because phase shifts re#ect also the e!ect of resonances; however as long as phase shifts constitute a small e!ect, this should not matter.
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Fig. 5. Second-order correlation function for various values of the `incoherencea parameter j as given by the phenomenological equation (3.54).
Two identically charged pions are practically always produced together with a third pion of opposite charge. Then according to Bowler [61] one has also to consider the I"0 attractive interaction between the oppositely charged pions and this compensates largely for the I"2 state interaction. Thus it appears that also in particle reactions only resonances play an important role in "nal state strong interactions. The considerations about "nal state interactions made in this subsection treat separately Coulomb and strong interactions. This is permitted as long as we deal with small e!ects or when the ranges of the two types of interactions do not overlap. For very small distances this is not anymore the case. Furthermore, the Gamow and the phase shift corrections are based on the wave-function formalism which ignores the possibility of creating particles. However, when entering the non-classical region the well-known di$culties of the wave-function formalism become visible (Klein paradox). To consider this e!ect, in Ref. [62] the joint contribution of the strong interaction potential and the Coulomb potential are analysed in a version of the Bethe}Salpeter equation for spinless particles. It is found that as expected also from the considerations presented above the strong interaction diminishes appreciably the Gamow correction. 3.2.4. Ewective currents As will be explained below the most satisfactory approach to BEC is at present the classical current approach, based on quantum "eld theory. Here three types of source characteristics appear: the chaoticity, the correlation lengths/times and the space}time dimensions. It is obvious that these quantities already contain information about the nature of the interaction and therefore it is quite natural to consider them as e!ective parameters which describe all the e!ects of strong "nal state
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interactions. This suggests that rather than starting with `barea non-interacting currents and introducing afterwards "nal state interactions, it might be preferable to assume from the beginning that the currents as de"ned by the "eld theoretical formalism are `e!ectivea and therefore already contain all "nal state interactions. This approach to strong "nal state interactions is probably the most recommendable one at the present stage for BEC in reactions induced by hadrons or leptons because it: (1) is simpler; (2) avoids double counting; (3) avoids the use of poorly known resonance characteristics; (4) avoids the use of the ill-de"ned concept of "nal state interactions for strong interactions. For heavy ion reactions, when hydrodynamical methods are used the explicit consideration of resonances can be practised up to a certain point without major di$culties and then the strong "nal state interactions can be taken into account through these resonances (see Section 5.1.3). To conclude this discussion of "nal state interactions in BEC, it is interesting to note that in boson condensates the "nal state interactions might be di!erent than in normal hadronic sources. In a condensate the Bose "eld becomes long range in con"guration space. This can be understood as a consequence of the fact that in a condensate the e!ective mass of the "eld carrier vanishes. Indeed a calculation [63] based on the chiral sigma model shows that the e!ective range of the pion "eld can increase several times due to this e!ect.
4. Currents 4.1. Classical versus quantum currents In Section 2.2 we were concerned mainly with the properties of "elds and did not ask the question where these "elds come from. In the present section we shall pose this question and try to answer it. We start by recalling the de"nition of correlation functions within quantum "eld theory. Let as(k) and a (k) be the creation and annihilation operator of a particle of momentum k, where i i the index i labels internal degrees of freedom such as spin, isospin, strangeness, etc. The n-particle inclusive distribution is 1 dnpi1 2in n "(2p)3n < 2E ¹r(o as1 (k )2asn (k )a n (k )2a 1 (k )) , j & i 1 i n i n i 1 p du 2du 1 n j/1 where du "d3k /(2p)32E * * * is the invariant volume element in momentum space. With the notation dnpi1 2in Gi1 2in (k ,2, k ),(1/p) , n 1 n du 2du 1 n the general n-particle correlation function is de"ned as Ci1 2in (k ,2, k )"Gi1 2in (k ,2, k )/Gi1 (k ) ) 2 ) Gin (k ) . n 1 n n 1 n 1 1 n n
(4.1)
(4.2)
(4.3)
(4.4)
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The particles which the operators as and a are associated with are the quanta of the "eld (which we denote in general by /); these particles as well as the density matrix o refer to the "nal state where measurements take place. On the other hand, we usually know (or guess) the density matrix only in the initial state. Therefore, we have to transform the above expression so that eventually the density matrix in the "nal state o is replaced by the density matrix in the initial state o , while the & * "elds will continue to refer to the "nal state. To emphasize this we wrote in Eq. (4.1) o . We have & o "So Ss , (4.5) & * so that G (k)"(2p)(2E ) ¹rMo Ssas(k)a(k)SN , (4.6) 1 1 * G (k , k )"(2p)6(2E )(2E ) ¹rMo Ssas(k )as(k )a(k )a(k )SN . (4.7) 2 1 2 1 2 * 2 1 1 2 Thus, if the initial conditions i.e. o are given, in principle the knowledge of the S matrix su$ces * to calculate the physical quantities of interest. In one case the S matrix can even be derived without approximations. This happens when the currents are classical and we shall discuss this case in some detail in this and the following section. Before doing this, we shall consider brie#y the more general case when the currents are not necessarily classical. The S matrix is given by the relation
GP
H
S"T exp i d4x¸ (x) , */5
(4.8)
where the interaction lagrangian ¸ is a functional of the "elds /. T is the chronological */5 time-ordering operator; we shall use below also the antichronological time-operator T I . Consider for simplicity a scalar "eld produced by a current J. Then ¸ (x),J(x)/(x) . (4.9) */5 Eqs. (4.8) and (4.9) allow us now to calculate the correlations we are interested in terms of the currents after eliminating the "elds. One obtains thus P (k)"¹rMo Js (k)J (k)N , (4.10) 1 i H H P (k , k )"¹rMo T I [Js (k )Js (k )]T[J (k )J (k )]N , (4.11) 2 1 2 i H 1 H 2 H 1 H 2 where the label H stands for the Heisenberg representation. Now the cross sections depend only on the currents and the density matrix in the initial state. The appearance of the time ordering operators T and T I in Eqs. (4.10) and (4.11) is a reminder of the fact that the current J is here an operator. 4.2. Classical currents Besides the fact that in this case an exact, analytical solution of the "eld equations is available and that the limits of this approximation are quite clear, the classical current has the important
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advantage that in it the space}time characteristics of the source are clearly exhibited and thus contact with approaches like the Wigner approach and hydrodynamics are made possible. The assumption that the currents are classical implies that J is a c number and then the order in Eq. (4.11) does not matter. This approximation can be used in particle physics when the currents are produced by heavy particles (e.g. nucleons) and/or when the momentum transfer q is small compared with the momentum K of the emitting particles.21 In Section 4.7 a new criterion for the applicability of the classical current assumption in terms of particle}antiparticle correlations will be presented. The classical current formalism was introduced to the "eld of Bose}Einstein correlations in [64,65,39]. In this approach, particle sources are treated as external classical currents J(x), the #uctuations of which are described by a probability distribution PMJN. From many points of view like, e.g. understanding the space}time properties of the sources or the isotopic spin dependence of BEC this approach is superior to any other approach. This has become clear only in the last years [33,3] when a systematic investigation of the independent physical quantities which enter the dynamics of correlation functions has been made (see below). The classical current formalism in momentum space is mathematically identical with the coherent state formalism used in quantum statistics and in particular in quantum optics (see Section 2.2), the classical currents in k-space J(k) being proportional to the eigenvalues of the coherent states DaT. This explains the importance of the coherent state formalism for applications in particle and nuclear physics. The density matrix is
P
o" DJPMJNDJTSJD ,
(4.12)
where the symbol DJ denotes an integration over the space of functions J(x), and the statistical weight PMJN is normalised to unity,
P
DJPMJN"1 .
(4.13)
(The reader will recognise in Eq. (4.12) the P-representation introduced in Section 2.2.) Expectation values of "eld operators can then be expressed as averages over the corresponding functionals of the currents, e.g., ¹r(oas(k )2as(k )a(k ) a(k )) 1 n n`1 2 n`m n (!i) n`m i "< SJH(k )2JH(k )J(k ) J(k )T . (4.14) < 1 n n`1 2 n`m j/1 J(2p)32Ej j/m`1 J(2p)32Ej In the following, we shall discuss the special case where the #uctuations of the currents J(x) are described by a Gaussian distribution PMJN. The reasons for this choice are given in Section 2.2.
21 For multiparticle production processes this also implies a constraint on the multiplicity and/or the momenta k of the produced particles.
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As in quantum optics we write the current J(x) as the sum of a chaotic and a coherent component J(x)"J (x)#J (x) #)!05*# #0)%3%/5
(4.15)
with J (x)"SJ(x)T , (4.16) #0)%3%/5 J (x)"J(x)!SJ(x)T . (4.17) #)!05*# By de"nition, SJ (x)T"0. The case SJ(x)TO0 corresponds to single-particle coherence. #)!05*# The Gaussian current distribution is completely determined by specifying its "rst two moments: the "rst moment coincides, because of Eq. (4.17), with the coherent component, I(x),SJ(x)T
(4.18)
and the second moment is given by the 2-current correlator D(x, x@),SJ(x)J(x@)T!SJ(x)TSJ(x@)T"SJ
#)!05*#
(x)J (x@)T . #)!05*#
(4.19)
4.3. Primordial correlator, correlation length and space}time distribution of the source We come now to a more recent development [33,3] of the current formalism which has shed new light on both fundamental and applicative aspects of BEC. There are two in principle independent aspects of physics which come together in the phenomenon of BEC in particle and nuclear physics. One refers to the geometry of the source and goes back to the original Hanbury}Brown and Twiss interference experiment in astronomy. The `geometrya is characterised by the size of the source, e.g. the longitudinal and transverse radius R and R , , M respectively, and the lifetime of the source R . 0 The second aspect is related to the dynamics of the source and is expressed through correlation lengths. In the following, we will use two correlation lengths ¸ , ¸ and a correlation time ¸ .22 As , M 0 a consequence of the "nite space}time size of sources in particle physics one cannot in general separate the geometry from the dynamics in the second- (and higher-)order correlation function. This separation is possible (see below) only by using simultaneously also the single inclusive cross Section [33].23 Assuming Gaussian currents, one may take the point of view that the purpose of measuring n-particle distributions is to obtain information about the space}time form of the coherent component, I(x), and of the correlator of the chaotic components of the current, D(x, x@). In practice,
22 We refer here to short range correlations. See Section 6.1.3 for a distinction between short- and long-range correlations. 23 For systems in local equilibrium Makhlin and Sinyukov [66] introduced a length scale (called in [67] `length of homogeneitya) which characterises the hydrodynamical expansion of the source and can be di!erent from the size of the system. Further references on this topic can be found, e.g. in [68,69].
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because of limited statistics it is necessary to consider simple parametrisations of these quantities, and to use the information extracted from experimental data to determine the free parameters (such as radii, correlation lengths, etc.). The approach considered in [3] di!ers in some fundamental aspects from those applications of the density matrix approach in particle physics which are performed in momentum (rapidity) space and are limited usually to one dimension (see however [70] where rapidity and transverse momentum were considered). The approach of Ref. [3] is a space}time approach in which the parameters refer to the space}time characteristics of the source. This approach has important heuristic advantages compared with the momentum (rapidity) space approach as explained below. Among other things, in the quantum statistical (QS) space}time approach the parameters of the source as de"ned above can be considered as e!ective parameters which contain already the entire information which one is interested in and which one could obtain from experiment, and thus distinguishing between directly produced particles and resonance decay products could amount to double counting. The apparent proliferation of parameters brought about by the QS approach is compensated by this heuristic and practical simpli"cation. Furthermore, a new and essential feature of the approach of Ref. [3] as compared with previous applications of the current formalism [65,39] which assume ¸"0, is the xnite correlation length (time) ¸. This fact has important theoretical and practical consequences. It leads among other things to an e!ective correlation between momenta and coordinates, so that, e.g. the second-order correlation functions depend not only on the di!erence of momenta q"k !k but also on the sum k #k . This non-stationarity 1 2 1 2 property, which is observed in experiment, is usually associated with expanding sources and treated within the Wigner function formalism. However from the considerations presented above it follows that expansion is in general not a necessary condition for non-stationarity in q. It will be shown how expanding sources can be treated without the Wigner formalism, which restricts unnecessarily the applicability of the results to small q values (see Section 4.8). The distinction between correlation lengths and radii is possible only in the current formalism; the Wigner formalism provides just a length of homogeneity. The results which follow from the space}time approach [3] include: (i) The existence of at least 10 independent parameters that enter into the correlation function; (ii) new insights into the problem of partial coherence; (iii) isospin e!ects: n0n0 correlations are di!erent from nBnB correlations; there exists a quantum statistical (anti)correlation between particles and antiparticles (n`n~ in this case). These e!ects are associated with the presence of squeezed states in the density matrix, which in itself is a surprising and unexpected feature in conventional strong interaction phenomenology. It turns out that soft pions play an essential role in the experimental investigation of BEC, both with respect to the e!ect of particle}antiparticle correlations as well as in the investigation of the coherence of the source. Depending on the relative magnitude of the parameters of the coherent and chaotic component, soft particles can either enhance or suppress the coherence e!ect. Consider "rst the case of an in"nitely extended source. The correlation of currents at two space}time points x and y is described by a primordial correlator SJ(x)J(y)T "C(x!y) . 0
(4.20)
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Note that C depends only on the di!erence x!y. The correlator C(x!y) contains some characteristic length (time) scales ¸, the so-called correlation lengths (times).24 In the current formalism used in [39] ¸"0. C(x!y) is a real and even function of its argument. In the rest frame of the source, it is usually parametrised by an exponential, C(x!y)"C exp[!Dx !y D/¸ !Dx!yD/¸] 0 0 0 0 or by a Gaussian,
(4.21)
C(x!y)"C exp[!(x !y )2/2¸2 !(x!y)2/2¸2] . (4.22) 0 0 0 0 However, it should be clear that in principle any well-behaved decreasing function of (x!y) is a priori acceptable, and in practice it is usually up to the experimenter to decide which particular form is more appropriate. Ansatze (4.21) and (4.22) need to be modi"ed for the case of an expanding source (see Sections 4.8 and 4.9) where each source element is characterised not only by a correlation length ¸k but also by a four-velocity uk. As a matter of fact, the form of the function C is irrelevant as long as one is interested in the general statements of the theoretical quantum statistical (current) formalism. In practical applications, of course, in order to obtain concrete information about the source and the medium (i.e., about ¸) the form of C has to be speci"ed. In principle, a full dynamical theory is expected to determine the functional form of the correlation function; however at present this `fundamentalista approach is not applicable.25 One uses instead a phenomenological approach like that re#ected in Eqs. (4.21) and (4.22). E!ects of the geometry of the source are taken into account by introducing the space}time distributions of the chaotic and of the coherent component, f (x) and f (x), respectively. The #) # expectation values of the currents, I(x) and D(x, x@), take non-zero values only in space}time regions where f and f are non-zero. Thus, one may write # #) I(x)"f (x) , (4.23) # D(x, x@)"f (x)C(x!x@) f (x@) . (4.24) #) #) We turn now to an important new aspect of the current approach. 4.4. Production of an isospin multiplet Following [3] we generalise the previous results to the case of an isospin multiplet and derive explicit expressions for the single inclusive distributions and correlation functions of particles that form an isotriplet (such as the n`, n~ and n0-mesons). For the sake of de"niteness, we will refer to pions in the discussion below, but it should be understood that the formalism is applicable to an arbitrary isomultiplet.
24 These correspond to the `coherence lengthsa used in the quantum optical literature. 25 Indeed one may hope that for strongly interacting systems lattice QCD may provide in future the correlation length ¸.
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Consider the production of charged and neutral pions, n`, n~ and n0, by random currents. The initial interaction Lagrangian is written L "J (x)n~(x)#J (x)n`(x)#J (x)n0(x) . (4.25) */5 ` ~ 0 The current distribution is completely characterised by its "rst two moments. They read for the case of an isotriplet I (x),SJ (x)T, i, i@"#,!, 0 , i i D (x, x@),SJ (x)J (x@)T!SJ (x)TSJ (x@)T . ii{ i i{ i i{ Invariance of the chaotic 2-current correlator under rotation in isospin space implies D (x, x@)"D (x, x@),D(x, x@) , 00 `~ D (x, x@)"D (x, x@)"D (x, x@)"D (x, x@)"0 . `` ~~ `0 0~ The corresponding current distribution is PMJ N"(1/N) exp[!AMJ N] i i where in coordinate representation
(4.27)
(4.28)
PPP
DJ DJ DJ exp[!AMJ N] , ` ~ 0 i
N"
(4.26)
(4.29)
and
PP
AMJ N" i
C
d4x d4y (J (x)!I (x))M(x, y)(J (y)!I (y)) ` ` ~ ~
D
1 # (J (x)!I (x))M(x, y)(J (y)!I (y)) 0 0 0 2 0
(4.30)
with M(x, x@)"D~1(x, x@) .
(4.31)
For the general case of a partially coherent source, the single inclusive distributions of pions of charge i (i"#,!, 0) can be expressed as the sum of a chaotic component and a coherent component, (1/p)dpi/du"(1/p)dpi/duD #(1/p)dpi/duD #)!05*# #0)%3%/5
(4.32)
(1/p)dpi/duD "D(k, k) , #)!05*#
(4.33)
(1/p)dpi/duD "DI(k)D2 . #0)%3%/5
(4.34)
with
and
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In general, the chaoticity parameter p will be momentum dependent, p(k)"D(k, k)/(D(k, k)#DI(k)D2) .
(4.35)
To write down the correlation functions in a concise form, one introduces the normalised current correlators d "D(k , k )/[D(k , k ) ) D(k , k )]1@2, rs r 4 r r 4 4
dI "D(k ,!k )/[D(k , k ) ) D(k , k )]1@2 , rs r 4 r r 4 4
(4.36)
where the indices r, s label the particles. Note that dI is the same function as d but for the change of sign of one of its variables. One may express the correlation functions in terms of the magnitudes and the phases of the correlators: ¹ ,¹(k , k )"Dd(k , k )D, ¹I ,¹I (k , k )"DdI (k , k )D , rs r 4 r 4 rs r 4 r 4 /#),/#)(k , k )"Arg d(k , k ), /I #),/I #)(k , k )"Arg dI (k , k ) rs r 4 r 4 rs r 4 r 4
(4.37)
and the phase of the coherent component /#,/#(k )"Arg I(k ) . r r r
(4.38)
The same notation will be used for the chaoticity parameter p ,p(k ) . r r
(4.39)
The two-particle correlation function is C``(k , k )"1#2Jp (1!p ) ) p (1!p )¹ cos(/#) !/# #/# )#p p ¹2 . 2 1 2 1 1 2 2 12 12 1 2 1 2 12
(4.40)
In [3] higher-order correlation functions up to and including order 5 are given. In the absence of single-particle coherence the two-particle correlation functions for di!erent pairs of n`, n~, n0 mesons read C``(k , k )"1#Dd D2, C`~(k , k )"1#DdI D2 , 2 1 2 12 2 1 2 12 C`0(k , k )"1, C00(k , k )"1#Dd D2#DdI D2 2 1 2 2 1 2 12 12
(4.41)
These results [71] were surprising in that they disagreed with some of the preconceived notions on Bose}Einstein correlations. For instance, it was commonly assumed that without taking into account "nal state interactions and in the absence of coherence, the maximum of the two-particle correlation of identical pions is 2 (for k "k ). It was also assumed that there 1 2 are no correlation e!ects among di!erent kinds of pions because these particles are not identical. (This last assumption is even sometimes used in normalising the experimental data on CBB with 2 respect to C`~.) 2 Results (4.41) show that these assertions are not necessarily true. In particular, looking at the two pion correlations one can see that in addition to the familiar correlations of identical particles (the terms Dd D2) there are particle}antiparticle } in this case, n`n~ } correlations (the terms DdI D2). 12 12 The n0 has both terms, as it is identical with its antiparticle. Essentially, this last fact is the
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explanation for the appearance of the &&surprising'' e!ects. Obviously it is a speci"c quantum "eld e!ect. It will be shown in Section 4.8, that for soft pions and for small lifetimes of the source the terms DdI D2 can in principle become comparable with the conventional terms Dd D2. This implies 12 12 that the distribution C00(k , k ) of two neutral pions can be as large as 3, and the maximum value 2 1 2 of C`~ is 2 (instead of 1). The corresponding limit of C000 is 15 (instead of 6), and that of C``~ is 2 3 3 6 (instead of 2). However, it should be noted that these are merely upper limits, which for massive particles are not reached except for sources of in"nitesimally small lifetimes. For soft photons however the situation is di!erent (see below). The `newa terms, proportional to Sal (k )al (k )T are due to the non-stationarity (in k space) of 1 2 the source. While in quantum optics time stationarity is the rule, in particle physics this is not the case because of the "nite lifetime and "nite radius of the sources. The existence of a non-vanishing expectation value of the products a(k )a(k ) is what one would expect (see Section 2.2) from 1 2 two-particle coherence (squeezing), just as Sa(k)TO0 follows from ordinary (one-particle) coherence (note that the latter has not been assumed here). The fact that squeezing which, as mentioned above, is quite an exceptional situation in optics, discovered only recently, is a natural consequence of the formalism for present particle physics, is possibly one of the most startling results obtained recently in BEC. A characteristic feature of isospin squeezed states is that they are two-mode states and that for static sources they lead to anti-correlations. Finally, it should be pointed out again that the above results } in particular, the fact that C00 in general di!ers from C~~ } are consistent with 2 2 isospin symmetry. In closing this section one should note that the existence of particle}antiparticle correlations is not restricted to pions but applies also to other systems, e.g. neutral kaons. In principle, thus there exist also K KM quantum statistical correlations. However, since K particles cannot be observed 0 0 0 except in linear combinations with KM in the form of K and K , the QS particle}antiparticle 0 4 correlation e!ect has to be disentangled from the K K or K K Bose}Einstein correlation (K and 4 4 - 4 K are of course bosons and thus subject to BEC) which exist also in the wave-function formalism which ignores the intrinsic `newa K KM correlation. As a matter of fact K K correlations have 0 0 4 4 been observed experimentally; however, no attempt has been made so far to extract from them the surprising e!ects.26 (For more recent experiments see [72] and for a theoretical analysis of the `olda e!ects and their possible application in CP violation phenomena see [73].)27
26 Because of their larger mass as compared with that of pions, these e!ects may be even more quenched than in the case of pions, except for sources of very short lifetime (see Section 4.8). 27 The analysis in the preceding subsections referred to the production of an isotriplet assuming just the symmetry between the isospin components (see Eq. (4.25)) of the current. In principle, for strong interactions the conservation of isospin I must also be considered. While the chaotic part is not a!ected by this condition, the coherent component is in#uenced by conservation of isospin [74]. In particular, this can lead to an additive positive term in the correlation function and thus to an increase of the bounds of BEC for pions. It remains to be seen whether this e!ect can be distinguished from the e!ect of long-range correlations (see Section 6.1.3). Moreover in hadronic reactions and in particular those involving nuclei such an e!ect would be suppressed because of the following circumstance. The initial state has to be averaged over all components of isospin I which is a "rst `dilutinga factor. Furthermore, the e!ect is appreciable only for low total isospin. This total isospin has to be shared by the chaotic component I and the coherent #) component I : I"I #I . The "rst one arises mostly from resonances with di!erent isospin values, so that even if the # # #) total isospin takes its minimum value (I"0), I and therefore I can take larger values. #) #
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4.4.1. An illustrative model of uncorrelated point-like random sources To clarify the origin of di!erent terms in the functions Cab(k , k ), let us consider, a source 2 1 2 consisting of N point-like random sources N J (x)" + j (x )d(t!t )d3(x!x ), a"#,!, 0 (4.42) a a i i i i/1 and assume that the currents j (x ) at di!erent points x are mutually independent and have the a i i same statistical properties, i.e. S jH(x )j (x )T"d S jHj T . a i b j ij a b
(4.43)
We also assume in this section that S j T"0 , a
(4.44)
i.e. ignore a possible coherent component S j T to make the presentation more transparent. a Now the one-particle distribution is SJH(k)J (k)T"N ) S jHj T a a a a
(4.45)
and the two-particle distribution takes the form SJH(k ) J (k ) JH(k ) J (k )T a 1 a 1 b 2 b 2 N N " + S jH(x )j (x )jH(x )j (x )T# + S jH(x )j (x )TS jH(x )j (x )T a i a i b i b i a i a i b j b j i/1 iEj N # + S jH(x )j (x )TS jH(x )j (x )Te*(k1 ~k2 )(xi ~xj ) b i a i a j b j iEj N (4.46) # + S jH(x )jH(x )TS j (x )j (x )Te*(k1 `k2 )(xi ~xj ) . a j b j a i b i iEj Let us consider separately the four di!erent terms on the right-hand side of Eq. (4.46). The "rst term corresponds to two particles being emitted from a single point (Fig. 6a); it is proportional to the number of emitting points. The second term describes an independent emission of two particles from di!erent points (Fig. 6b). The third term, being non-zero for a"b, describes an interference e!ect of direct and exchange diagrams, characteristic of identical particles, emitted from di!erent points (Fig. 6c). This is the usual BE-correlation e!ect. The fourth term describes an interference of two-particle emissions from di!erent points (Fig. 6d) (`two-particle sourcesa). It is non-zero for real currents with a"b (n0n0) and for complex currents with JH"J (n`n~), that is for particle}antiparticle associative emission. In this simple model, it a b represents the `surprisinga e!ects discussed above.
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Fig. 6. Diagrams contributing to di!erent terms of the two-particle correlator in the point-like random source model (from Ref. [3]).
This diagrammatic illustration of the `surprisinga BE-correlations is due to Bowler [75], who found that these e!ects derived for the "rst time in Ref. [71] can be understood in terms of the qualitative considerations mentioned above. Bowler derived the `newa e!ects from the string model. 4.5. Photon interferometry. Upper bounds of BEC The advantage of photon BEC resides in the fact that photons are not in#uenced by "nal state interactions. Photons present also an interesting subject of theoretical research from the general BEC point of view, since they are spin-one bosons while pions and kaons used in hadronic BEC are scalar particles. We shall see below that this supplementary degree of freedom has speci"c implications for BEC. Last but not least, photon correlations are for various reasons, discussed below, of particular interest in the search of quark matter. We present some of the results of [76], which contain as a special case those of [77] and where these topics are discussed. Consider a heavy ion reaction where photons are produced through bremsstrahlung from protons in independent proton}neutron collisions.28 The corresponding elementary dipole currents are jj(k)"(ie/mk0)p ) e (k) , j 28 Photon emission from proton}proton collisions is suppressed because it is of quadrupole form.
(4.47)
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where p"p !p is the di!erence between the initial and the "nal momenta of the proton, e is the * & j vector of linear polarization and k the photon momentum; e and m are the charge and mass of the proton, respectively. The total current is written as N Jj(k)" + e*kxn jj(k) . (4.48) n n/1 For simplicity, we will discuss in the following only the case of pure chaotic currents SJj(k)T"0 and refer for coherence e!ects to the original literature [76,77]. In analogy to the considerations of the previous subsection the index n labels the independent nucleon collisions which take place at di!erent space}time points x . These points are assumed to be randomly distributed in the n space}time volume of the source with a distribution function f (x) for each elementary collision. The current correlator is proportional to products of the form
A
B
N (4.49) + Spi pj T ej 2 (k ) . j 2 n n n/1 For central collisions due to the axial symmetry around the beam direction one has for the momenta the tensor decomposition SJj1 (k )Jj2 (!k )T"ei 1 (k ) 1 2 j 1
Spi pj T"1p dij#d lilj , (4.50) n n 3 n n where l is the unit vector in the beam direction and p , d are real positive constants. In [77] an n n isotropic distribution of the momenta was assumed. This corresponds to the particular case d "0. n The generalisation to the form (4.50) is due to [76]. The summation over polarisation indexes is performed using the relations S(ei ) p )(ej ) p )T"1(ei ) ej)d , ll{ l{ 3 2 + ei (k)ej (k)"dij!ninj , j j j/1 where n"k/DkD. We write below the results for the second-order correlation function
(4.51)
C (k , k )"o (k , k )/o (k )o (k ) 2 1 2 2 1 2 1 1 1 2 for two extreme cases: (1) Uncorrelated elementary currents (isotropy) (p
(4.53)
C (k , k ; pO0, d"0)"1#1[1#(n ) n )2][D fI (k !k )D2#D fI (k #k )D2] , 1 2 1 2 1 2 2 1 2 4 leading to an intercept
(4.54)
C (k, k)"3#1 D fI (2k)D2 2 2 2 limited by values (3, 2). 2 (2) Strong anisotropy (p;d)
(4.55)
C (k , k ; p"0, dO0)"1#D fI (k !k )D2#D fI (k #k )D2 2 1 2 1 2 1 2
(4.52)
(4.56)
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Fig. 7. The linear polarisation vectors e (k) for two photons with momenta k and k (from Ref. [82]). j 1 2
with an intercept C (k, k)"2#D fI (2k)D2 (4.57) 2 limited this time by values (2,3). Note that due to the form of the photon}current interaction (4.47) in this (strong anisotropy) case the photons emerge practically completely polarised so that the summation over polarisations does not a!ect the correlation. These results are remarkable among other things because they illustrate the speci"c e!ects of photon spin on BEC. Thus while for (pseudo)scalar pions the intercept is a constant (2 for charged pions and 3 for neutral ones) even for unpolarised photons the intercept is a function of k. For a graphical illustration and explanation of this fact see Fig. 7. It is seen that to perform the summation over polarisation implied by Eq. (4.51) only one direction of the linear polarisation can be chosen to be equal for both photons, while the other polarisation direction di!ers by the angle h between the momenta k , k . 1 2 One thus "nds that, while for a system of charged pions (i.e. a mixture of 50% positive and 50% negative) the maximum value of this intercept Max C (k, k) is 1.5, for photons Max C (k, k) exceeds 2 2 this value and this excess re#ects the space}time properties (represented by fI (k)), the degree of (an)isotropy of the source represented by the quantities p and d, and the supplementary degree of freedom represented by the photon spin. As a consequence of the fact that fI is a decreasing function of its argument, in Eqs. (4.54) and (4.56) the terms with fI (k #k ) are in general smaller than the terms with fI (k !k ), except for 1 2 1 2 small momenta k. The fact that the di!erences between charged pions and photons are enhanced for soft photons reminds us of a similar e!ect found with neutral pions (see Section 4.4). Neutral pions are in general more bunched than identically charged ones and this di!erence is more pronounced for soft pions. This similarity is not accidental, because photons as well as n0 particles are neutral and this circumstance has quantum "eld theoretical implications which are also mentioned below. We see thus that photon BEC can provide information both about the space}time form of the source represented by f and the dynamics which are represented by d.29,30 29 See [78,79] where photon correlation experiments in low energy (100 MeV/nucleon) heavy ion reactions are reported. For a theoretical discussion of these experiments see [80]. 30 The relation between photon interferometry and the formation length of photons is discussed in [81].
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The results on photon correlations presented above refer to the case where the sources are `statica i.e. not expanding. Expanding sources were considered in [82] within a covariant formalism. Some of the results above, in particular Eqs. (4.54) and (4.55), which had been initially derived by Neuhauser [77], were challenged by Slotta and Heinz [83]. Among other things, these authors claim that for photon correlations due to a chaotic source `the only change relative to 2-pion interferometry is a statistical factor 1 for the overall strength of the 2 correlation which results from the experimental averaging over the photon spina. In [83] an intercept 3 is derived which is 2 in contradiction with the results presented above and in particular with Eq. (4.55) where besides the factor 3 there appears 2 also the k dependent function 1D fI (2k)D2. Similar statements can be found in previous papers [84}87] where more detailed 2 applications concerning heavy ion reactions based on this assertion of [83] are presented. Some of the papers quoted above were criticised immediately after their publication in [88,89,82] and the paper [83] was written with the intention to settle this `controversya. It should be pointed out here that the reason for the di!erence between the results of [77,76] on the one hand and those of Ref. [83] on the other is mainly that in [83] a formalism was used which is less general than that used in [77,76] and which is inadequate for the present problem. This implies among other things that unpolarised photons cannot be treated in the way proposed in [83] and that the results of [77,76] are correct. In [83] the following formula for the second-order correlation function is used: g8 (q, K)g8 lk(!q, K) C(k , k )"1# kl . 1 2 g8 k (0, k )g8 k (0, k ) k 1 k 2
(4.58)
Here g8 is the Fourier transform of a source function, q"k !k and K"1(k #k ). This formula is a particular case of 1 2 2 1 2 a more general formula for the second-order correlation function derived by Shuryak [64] using a model of uncorrelated sources, when emission of particles from the same space}time point is negligible (see Section 4.4.1). Since this equation is sometimes used in the recent literature without giving the reader the possibility of evaluating the approximations used in its derivation, we will sketch this derivation in the following. In [64] one starts with the current correlator SJH(x )J (x )T"d J (x, *x) , i 1 j 2 ij i
(4.59)
where J (x) is the current emitted by point x and i x"(x #x )/2, 1 2
*x"x !x . 1 2
(4.60)
Eq. (4.59) assumes that the individual currents (iOj) are uncorrelated. With the notation II (q, K) for the Fourier i transform of I (x, *x) and II (q, K),+ II (q, K) the inclusive single-particle distribution reads i i
TK P
KU
=(k)" + e*kxJ (x) d4x "+ II (0, k)"II (0, k) j i j i
(4.61)
and the two-particle distribution is given by
TK P
=(k k )" + (e*k1 x1 `*k2 x2 #e*k1 x2 `*k2 x1 )J (x )J (x ) dx dx i 1 j 2 1 2 1 2 i,j
KU 2
(4.62)
or "nally =(k , k )"II (0, k )II (0, k )#DII (q, K)D2#+ [SJH(k )JH(k )J (k )J (k )T 1 2 1 2 i 1 i 2 i 1 i 2 i ! II (0, k )II (0, k )!DII (q, K)D2] . i 1 i 2 i
(4.63)
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The "rst term on the r.h.s. of Eq. (4.63) is the product of one-particle distributions and the second term is the conventional interference term, corresponding to Fig. 6c. By going over to the Wigner source function (see also Section 4.9)
P
g (x, K)" d4ye~*KySJH(x#1y)J (x!1y)T , kl k 2 l 2
(4.64)
these "rst two terms result in Eq. (4.58) of [83]. However as is clear from Eq. (4.63) there exists also a third term, neglected in Eq. (4.58) and which corresponds to the simultaneous emission of two particles from a single point (x ) as indicated in i Fig. 6d. While for massive particles this term is in general suppressed, this is not true for massless particles and in particular for soft photons. In [77,76] this additional term had not been neglected as it was done in [83] and therefore it is not surprising that Ref. [83] could not recover the results of Refs. [77,76]. The neglect of the term corresponding to emission of two particles from the same space}time point is not permitted in the present case. As mentioned in Section 4.4.1, in a model of uncorrelated point-like random sources like the present one, emission of particles from the same space}time point corresponds in a "rst approximation to particle}antiparticle correlations and this type of e!ect leads also to the di!erence between BEC for identically charged pions and the BEC for neutral pions. This is so because neutral particles coincide with the corresponding antiparticles. (As a consequence of this, e.g. while for charged pions the maximum of the intercept is 2, for neutral pions it is 3 (see Section 4.4).) Photons being neutral particles, similar e!ects like those observed for n0}s are expected and indeed found (see above). This inconsequent application of the current formalism invalidates the conclusions of Ref. [83] and con"rms and strengthens the criticism expressed in [88,89,82] of the papers [84}87]. The fact that for unpolarised photons Max C (k, k) is 2, can be understood by realising that a system of unpolarised photons consists on the average of 50% 2 photons with the same helicities and 50% photons with opposite helicities. The "rst ones contribute to the maximum intercept (of the unpolarized system) with a factor of 3 and the last ones with a factor of 1 (corresponding to unidentical particles). For the sake of clari"cation it must be mentioned that Ref. [83] contains also other incorrect statements. Thus the claim in [83] that the approach by Neuhauser `does not correctly take into account the constraints from current conservationa is unfounded as can be seen from Eq. (4.51) which is an obvious consequence of current conservation (see e.g. Eq. (7.61) in [90]). Last but not least the statement that because the tensor structure in Eq. (20) of Ref. [82] is parametrised in terms of k and k separately `instead of only in terms of K, leading to spurious terms in the tensor 1 2 structure which eventually result in their spurious momentum-dependent prefactora has also to be quali"ed. As mentioned above, Eq. (4.58) to which this observation about the K dependence of [83] refers is not general enough for the problem of photon interferometry.
4.6. Coherence and lower bounds of Bose}Einstein correlations We mentioned in the previous subsections that the intercepts of the second- and higher-order correlation functions can deviate from the canonical values derived within the wave function formalism. This e!ect is important for at least three reasons: (i) It illustrates the limitations of the wave-function approach. (ii) It can in principle (provided other e!ects like "nal state interactions are taken into account) be used for the determination of the degree of coherence. (iii) It can serve as a test of models of BEC, since the value of the intercept follows from very general quantum statistical considerations, in particular the Gaussian nature of the density matrix. In most BEC models the intercept is identical to the maximum of the correlation function and therefore it can be studied by limiting the discussion to chaotic sources as was done in the previous subsection where the upper bounds of correlation functions were investigated. On the other hand, the minimum of the correlation functions is determined both by the form of the density matrix and the amount of coherence (see Ref. [88]) because coherence leads to a decrease of the correlation
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function. This will be illustrated below by discussing the lower bounds of this function. We will show among other things (see [3]) that in the quite general case of a Gaussian density matrix, for a purely chaotic system the two-particle correlation function must always be greater than one. On the other hand, in the presence of a coherent component the correlation function may take values below unity. Some implications for experimental and theoretical results found in the literature will be discussed here as well as in Section 5.1.6. We have seen in Section 4.4 that for identically charged bosons (e.g., n`) the two-particle correlation function reads C``(k , k )"1#2Jp (1!p ) ) p (1!p )¹ cos(/#) !/# #/# )#p p ¹2 . 2 1 2 1 1 2 2 12 12 1 2 1 2 12
(4.65)
For neutral bosons like photons, or n0's, the terms dI (k , k ) also appear (see Eq. (4.36)) in the BEC r 4 function: C00(k , k )"1#2Jp (1!p ) ) p (1!p )¹ cos(/#) !/# #/# )#p p ¹2 2 1 2 1 1 2 2 12 12 1 2 1 2 12 #2Jp (1!p ) ) p (1!p )¹I cos(/I #) !/# !/# )#p p ¹I 2 . 1 1 2 2 12 12 1 2 1 2 12
(4.66)
Let us "rst consider the case of a purely chaotic source. Insertion of p(k),1 in Eqs. (4.65) and (4.66) immediately yields C (k , k )51. In the case of partial coherence, the terms containing 2 1 2 cosines come into play and consequently C may take values below unity. Eqs. (4.65) and (4.66) 2 imply that C~~(k , k )52/3 and C00(k , k )51/3. Because of the cosine functions in (4.65) and 2 1 2 2 1 2 (4.66) one would expect C as a function of the momentum di!erence q to oscillate between values 2 above and below 1. Such a behaviour of the Bose}Einstein correlation function has been observed in high-energy e`}e~ collision experiments (see e.g., Ref. [91]), but apparently not in hadronic reactions. This observation was interpreted as a consequence of "nal state interactions in Ref. [75]. If "nal state interactions determine this e!ect, it is unclear why the e!ect is not seen in hadronic reactions. On the other hand, if coherence is responsible for it, this would be easier to understand. Indeed multiplicity distributions of secondaries in e`e~ reactions are much narrower (almost Poisson-like) than in pp reactions, which is consistent with the statement that hadronic reactions are more chaotic than e`}e~ reactions [92].31
So far, two methods have been proposed for the detection of coherence in BEC: the intercept criterion [93] (C (k, k)(2) and the two-exponent structure of C [28]. Both these methods have 2 2 their di$culties because of statistics problems or other e!ects. The observation of C (k , k )(1 2 1 2 could constitute a third criterion for coherence. In [84,85] the two-particle correlation function has been calculated for photons emitted from a longitudinally expanding system of hot and dense hadronic matter created in ultrarelativistic nuclear collisions. For such a system, the particles are emitted from a large number of independent source elements (#uid elements), and consequently one would expect the multiparticle "nal state to be described by a Gaussian density matrix. However, although the system is assumed to be purely chaotic the correlation function calculated in [84] is found to take values signi"cantly below unity.
31 This statement is not necessarily in contradiction with the empirical observation that the j factor in e`}e~ reactions appears in general to be larger than in p}p reactions, given the fact that j is not a true measure of coherence.
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Clearly, this is in contradiction with the general result derived above from quantum statistics (C 51 for a chaotic system). 2 The reason for this violation of the general bounds derived for a purely chaotic source is in this concrete case the use of an inadequate approximation in the evaluation of the space}time integrals. However as pointed out in [89] the expression for the two-particle inclusive distribution used in Ref. [84] (Eq. (3) of that paper), which in our notation takes the form
P P
P (k , k )" d4x d4x g(x , k )g(x , k )[1#cos((k !k )(x !x ))] , 2 1 2 1 2 1 1 2 2 1 2 1 2
(4.67)
is also unsatisfactory32 because for certain physical situations it can lead to values below unity for the two-particle correlation function even if the integrations are performed exactly. To see this, consider, e.g., the simple ansatz g(x, k)"const. exp[!a(x!bk)2]d(t!t ) , (4.68) 0 where a and b are free parameters. The expression for P (k , k ) used in Ref. [84] then yields 2 1 2 C (k , k )"1#exp[!q2/2a] cos[bq2] . (4.69) 2 1 2 Clearly, if b exceeds a~1 the above expression will oscillate and take values below unity. On the other hand, in the current formalism (see below) one obtains with the same ansatz for g C (k , k )"1#exp[!q2/2a]51 . (4.70) 2 1 2 Thus, Eq. (4.67) can lead to values C (1 if there is a strong correlation between the momentum 2 of a particle and the space}time coordinate of the source element from which it is emitted. Such correlations between x and k can occur in the case of an expanding source. The reason for this pathological behaviour is that the simultaneous speci"cation of coordinates and momentum as implied by Eq. (4.67) is constrained in quantum mechanics by the Heisenberg uncertainty relation and any violation of this constraint leads necessarily to a violation of quantum mechanics. This violation manifests itself sometimes, as in the present case, through a violation of the conservation of probability. This phenomenon is also met when using the Wigner function, which for this reason cannot always be associated with a bona"de probability amplitude. We will discuss this problem also in Section 5.1.6. The above considerations concerning bounds for the BEC functions refer to the case of a Gaussian density matrix. In general, a di!erent form of the density matrix may yield correlation functions that are not constrained by the bounds derived here. For instance, we have seen in Section 2.2 that for squeezed states C can take arbitrary positive values. Moreover, for particles 2 produced in high-energy hadronic or nuclear collisions, the #uctuations of quantities such as impact parameter or inelasticity may introduce additional correlations which may also a!ect the bounds of the BEC functions.
32 This formula appears apparently for the "rst time in [94] and was criticised (for other reasons) already in [95]. It is nevertheless used in certain event generators for heavy ion reactions (see Section 4.10).
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4.7. Quantum currents The results derived in the previous section, in particular the isospin dependence of BEC, were obtained in the assumption that the currents were classical. The question arises up to what point these conclusions survive in a fully quantum treatment of the problem. It would also be important to get a more precise criterion for the phenomenological applicability of the classical assumption, besides the no-recoil prescription. This question was discussed in [96] where it was found that the `surprisinga e!ects not only persist when the currents are quantum, but that they can serve as an experimental estimate of the size of the quantum corrections. We shall sketch brie#y in the following the results of Ref. [96]. As in the classical current case one starts with the interaction Lagrangian ¸ (x),J (x)n(~)(x)#J (x)n(`)(x)#J (x)n0(x) (4.71) */5 (`) (~) 0 The currents J(`), J(~), J0 are operators which we assume again for simplicity as not depending on the n(B) and n0 "elds ([J, n]"0). Taking into account the di!erent isospin components in Eqs. (4.10) and (4.11) we "nd that as in the classical current case the single and double inclusive cross sections depend on these components, e.g.33 G(~)(k)"¹rMo J(`)(!k)J(~)(k)N , (4.72) 1 i H H G(~`)(k , k )"¹rMo T I [J(`)(!k )J(~)(!k )]T[J(~)(k )J(`)(k )]N . (4.73) 2 1 2 i H 1 H 2 H 1 H 2 From now on we shall omit the label H and assume that all operators are written in the Heisenberg representation. Assuming a Gaussian density matrix one gets G(0)(k)"Fn(k, k) , (4.74) 1 G(~)(k)"G(`)(k)"F#)(k, k) , (4.75) 1 1 (4.76) G(~~)(k , k )"G(~)(k )G(~)(k )#DF#)(k , k )D2 , 2 1 2 1 1 1 2 1 2 G(00)(k , k )"G(0)(k )G(0)(k )#DFn(k , k )D2#DU/(k , k )D2 , (4.77) 2 1 2 1 1 1 2 1 2 1 2 G(~`)(k , k )"G(~)(k )G(`)(k )#DU#)(!k , k )D2 , (4.78) 2 1 2 1 1 1 2 1 2 where functions F and U are de"ned for charged particles (upper index ch) and for neutral ones (upper index n) as follows: F#)(k , k ),|T MJ(`)(!k )J(~) (k )N}"|J(`)(!k )J(~)(k )} , 1 2 # ^ 1 > 2 1 2 U#)(k , k ),|T MJ(`)(!k )J(~)(k )N}"|T I MJ(`)(!k )J(~)(k )N} , 1 2 # ^ 1 ^ 2 1 2 (k )N}"|J(0)(!k )J(0)(k )} , F/(k , k ),|T MJ(0)(!k )J(0) 1 2 1 2 # ^ 1 > 2 U/(k , k ),|T MJ(0)(!k )J(0)(k )N}"|T I MJ(0)(!k )J(0)(k )N} . 1 2 # ^ 1 ^ 2 1 2
33 For reasons of notational simplicity we have replaced Js(k) by J(!k).
(4.79) (4.80) (4.81) (4.82)
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In contrast to the classical current approach [3] which deals with only one type of two-current correlator we have here two di!erent kinds of two-current correlators depending on their ordering prescriptions. Moreover and most remarkably, the di!erence between these two correlators is re#ected by the di!erence between !! and #! correlations. It thus follows from [96] that the `surprisinga e!ects found in [71], i.e. the presence of particle}antiparticle Bose}Einstein type correlations and a new term in the Bose}Einstein correlation function for neutral particles are reobtained, but under a more general form which contains also the quantum corrections. These equations also prove that the above e!ects are not an artefact of the classical current formalism but have general validity. Moreover and most remarkably, from the above equations follows that the diwerence between the ewects of the classical and quantum currents resides in just these `new ewectsa and in particular in the diwerence between 00 and !! correlations, i.e. in the #! correlations. This result can serve as an estimate of the importance of quantum corrections to the classical current formalism of BEC. Since #! correlations are in general small, it follows that the classical current approach is a good approximation, except for very short-lived sources, where the #! correlations become comparable to the !! correlations. It also follows that the experimental measurement of #! correlations is a highly rewarding task, since they are a rather unique tool for the investigation of two very interesting e!ects in BEC, namely squeezed states and quantum corrections. 4.8. Space}time form of sources in the classical current formalism In [3] two types of sources were considered, a `statica one which corresponds to a source in rest and an expanding one. We will present below some of the results, as they exemplify certain important features of the space}time approach within the classical current formalism. A static source: The space}time distributions of static sources, as well as the primordial correlator, are parametrised as Gaussians: (4.83) f (x)"exp(!x2 /R2 !x2 /R2 !x2 /R2 ) , #) 0 #),0 , #),, M #),M f (x)"exp(!x2 /R2 !x2 /R2 !x2 /R2 ) , (4.84) # 0 #,0 , #,, M #,M C(x!y)"exp[!(x !y )2/2¸2 !(x !y )2/2¸2 !(x !y )2/2¸2 ] . (4.85) 0 0 0 , , , M M M Note that the term static here does not imply time independence but rather a speci"c time dependence de"ned by Eqs. (4.83) and (4.84) corresponding to source elements being at rest. This is to be contrasted to the expanding source, discussed in the next section, which explicitly contains velocities of source elements. The main justi"cation for this particular form of parametrisation is mathematical convenience, because, as will be shown below, for this case the correlation functions in momentum space can be calculated analytically and the physical implications can be read immediately. In Eqs. (4.83)}(4.85), R and R (a"0, o, E) are the lifetimes, transverse radii and longitudinal #),a #,a radii of the chaotic source and of the coherent source, respectively, and ¸ (a"0, o, E) are the a correlation time and the corresponding correlation lengths in transverse and in longitudinal direction. The relative contributions of the chaotic and the coherent component are determined by "xing the value of the (momentum dependent) chaoticity parameter p at some arbitrary scale
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(in this case, at k"0): p ,p(k"0) . (4.86) 0 The model contains 10 independent parameters: the radii and lifetimes of the chaotic and of the coherent source, the correlation lengths in space and time, and the chaoticity p . In [3] it is 0 assumed that ¸ "¸ ,¸, i.e., that the medium is isotropic, which leaves us with nine indepen, M dent parameters. With the de"nitions R2 "R2 ¸2/(R2 #¸2) (a"0, o, E) aL #),a a #),a a one may write the single inclusive distribution in the form
(4.87)
E(1/p)d3p/d3k"E(1/p)d3p/d3kD (p s (k)#(1!p )s (k)) , k/0 0 #) 0 # where
(4.88)
s (k)"exp[!E2R2 /2!k2 R2 /2!k2 R2 /2] , (4.89) #) 0L , ,L M ML s (k)"exp[!E2R2 /2!k2 R2 /2!k2 R2 /2] . (4.90) # #,0 , #,, M #,M The scales which determine the mean energy}momentum of the coherently produced particles are given by the inverse lifetime and radii, R~1, of the coherent source. For the chaotically produced #,a particles, these scales are given by the inverse of a combination of correlation lengths and dimensions of the chaotic source, R~1. Eq. (4.87) implies that R 4R . The radius of the chaotic aL aL #),a source enters the single inclusive distribution only in combination with the correlation length ¸. This feature which occurs also for higher-order correlations leads to the important consequence that experimental measurements of BEC do not provide separately information about radii (lifetimes) of sources, nor about correlation lengths (-times), but rather about the combination of these quantities as given by Eq. (4.87). On the other hand, by measuring both the single and the double inclusive distribution one can disentangle radii from correlation lengths. It follows from Eqs. (4.88)}(4.90) that in the presence of partial coherence in general (i.e., unless R "R ) the single inclusive distribution is a superposition of two Gaussians of di!erent widths. aL #,a If the geometry of the coherent source is the same as that of the chaotic source, one has R "R 'R , which would imply that coherently produced particles can be observed #,a #),a La predominantly in the soft regime. However, if the coherent radii are small compared to the chaotic ones, this situation is reversed. As a next step, consider the correlation functions. The correlation function of two negatively charged pions is C~~(k , k )"1#2Jp (1!p ) ) p (1!p )¹ cos(/#) !/# #/# )#p p ¹2 . (4.91) 2 1 2 1 1 2 2 12 12 1 2 1 2 12 For the Gaussian parametrisations all phases in the second-order correlation function disappear,34 /#) "/I #) "/#"/I #"0 , 12 12 j j
(4.92)
34 This is not the case anymore for an expanding source, e.g. (see below) or in general for higher-order correlations.
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and
C
(E !E )2(R2 !R2 ) (k !k )2(R2 !R2 ) 2 #),0 0L ! 1,, 2,, #),, ,L ¹ "exp ! 1 12 8 8
D
(k !k )2(R2 !R2 ) 2M #),M ML , ! 1M 8
C
(4.93)
(E #E )2(R2 !R2 ) (k #k )2(R2 !R2 ) 2 #),0 0L ! 1,, 2,, #),, ,L ¹I "exp ! 1 12 8 8
D
(k #k )2(R2 !R2 ) 2M #),M ML . ! 1M 8
(4.94)
The two-particle correlation function C~~ is the sum of a purely chaotic term (J¹2 ) and 2 12 a mixed term (J¹ ). The momentum dependence of the chaoticity parameter, p"p(k), implies 12 a momentum dependence of the contribution of the mixed term relative to that of the purely chaotic term. To see how this a!ects the interplay between the two terms (i.e., the interplay between the two Gaussians), it is useful to explicitly insert the momentum dependence of the chaoticity parameter by writing p "p(k )"p /A r r 0 r
(r"1, 2)
(4.95)
with A ,A(k )"p #(1!p )S r r 0 0 rr
(r"1, 2) ,
(4.96)
and
C
(E2#E2)(R2 !R2 ) (k2 #k2 )(R2 !R2 ) s c,0 0L ! r, s, c,, ,L S "exp ! r rs 4 4 (k2 #k2 )(R2 !R2 ) sM c,M ML ! rM 4
D
.
(4.97)
With this, C~~ takes the form 2 C~~(k , k )"1#[(2p (1!p )S /A A )]¹ #(p2 /A A )¹2 . (4.98) 2 1 2 0 0 12 1 2 12 0 1 2 12 The momentum dependence of the relative contributions of the purely chaotic and of the mixed term is re#ected in the factor S . Depending on the sign of the combinations R2 !R2 , S may 12 #,a aL 12 act either as a suppression factor or as an enhancement factor of the mixed term relative to the chaotic term. This is a consequence of the fact that, in contrast to the case of the correlation function C derived within the wave-function formalism, where C depends only on the di!erence 2 2 of momenta k !k now the correlation function depends also on k #k .35 1 2 1 2 35 Till recently this desirable physical property, which is observed in most experimental data on BEC, was considered to be a consequence of the expansion of the source and used to be derived within the Wigner function formalism, which is also a particular case of the classical current formalism. As shown in the example treated above (see [33]), it can be considered also a consequence of the (partial) coherence of a non-expanding source.
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It is instructive to discuss the tilde terms that give rise to the particle}antiparticle correlations for parametrisation (4.83)}(4.85) of a static source. For the sake of transparency, consider only the purely chaotic case, p "0. The correlation functions of like and unlike charged pions then take 0 the form C~~(k , k )"1#¹2 , (4.99) 2 1 2 12 C`~(k , k )"1#¹I 2 . (4.100) 2 1 2 12 From (4.93) and (4.94) it can be seen that the `newa ¹I terms that appear in the par12 ticle}antiparticle correlations are in general small compared to the `ordinarya ¹ terms that 12 determine the particle}particle correlations. The term ¹I gives rise to an anticorrelation e!ect due 12 to the factor in Eq. (4.94) containing the sum k #k , if the "rst factor, containing E #E , is not 1 2 1 2 too small. The latter is possible, if the time duration of the pion emission process and/or the pion energies are su$ciently small. We thus expect an enhanced contribution of the `newa terms for soft pions. The appearance of anticorrelations36 is, as mentioned above, a general property of squeezed states, which are present in the space}time formalism of [3]. The tilde terms arise as a consequence of the non-stationarity of the source. In the limit of a stationary source, R PR, and ¹I P0. An 0 12 upper limit is given by C`~"1#D¹I D241#exp[!(R2 !R2 )m2] . (4.101) 2 12 0 0L n In the limit ¸
36 Some authors have recently called them `back-to-backa correlations. 37 This particular possibility was suggested in [97]. Andreev [98] suggested a time evolution scenario for the medium e!ect. 38 Anticorrelations in disoriented chiral condensates are considered in [99].
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dependent. While many of the studies of Bose}Einstein correlations for expanding sources have followed, with slight variations, a Wigner function type of approach, the use of the Wigner approach is in general too restrictive and is recommendable only in the case where a full-#edged hydrodynamical description of the system is performed. In the present section, following [3], we shall therefore start with a more general discussion of the expanding source which is based on the space}time current correlator and the space}time form of the coherent component and which is not a!ected by the semi-classical and small q approximations inherent in the Wigner function approach. We introduce the variables q, g and x , with , q"Jx2 !x2 , g"1ln [(x #x )/(x !x )] . (4.102) 0 , 2 0 , 0 , Here q is the proper time, x the coordinate in the longitudinal direction (e.g. the collision axis in , p}p reactions or the jet axis in e`}e~ reactions) and g the space}time rapidity. An ansatz which is invariant under boosts of the coordinate frame in longitudinal direction will be considered (Fig. 8). Physically, this ansatz is motivated by the prejudice that the single inclusive distribution in rapidity is #at. The space}time distributions of the chaotic and of the coherent source and the correlator are then parametrised as f (x)& exp[!(q!q )2/(dq )2] exp(!x2 /R2 ) . (4.103) #) 0,#) #) M #) f (x)&exp[!(q!q )2/(dq )2] exp(!x2 /R2) (4.104) # 0,# # M # C(q !q , g !g , x !x ) 1 2 1 2 M,1 M,2 "exp[!(q !q )2/2¸2!(2q q /¸2)sinh2((g !g )/2)!(x !x )2/2¸2 ] . (4.105) 1 2 q 1 2 g 1 2 M,1 M,2 M The model contains again 10 independent parameters: the proper time coordinates of the chaotic and the coherent source, q , q , their widths in proper time, dq and dq , the transverse radii, 0,#) 0,# #) # R and R , the correlation lengths ¸ , ¸ and ¸ , and the chaoticity parameter p . #) # q M g 0
Fig. 8. Geometry of the boost-invariant source (from Ref. [3]).
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In order to be able to obtain explicit expressions for the single inclusive distribution and the correlation functions, a further simplifying assumption is made, namely, that dq "dq "0. #) # Eqs. (4.103) and (4.104) then take the form f (x)&d(q!q ) exp(!x2 /R2 ) , (4.106) #) 0,#) M #) f (x)&d(q!q ) exp(!x2 /R2) . (4.107) # 0,# M # Now the results no longer depend on the correlation length ¸ , and one is left with seven q independent parameters: q , q , R , R , ¸ , ¸ and p . 0,#) 0,# #) # M g 0 The Fourier integrations necessary to obtain D(k , k ) and I(k) can be performed by doing 1 2 a saddle point expansion; this should provide a good approximation if a ,m q /2<1, i iM 0,#) With the de"nitions:
b,q2 /2¸2<1 . 0,#) g
(4.108)
R2 ,R2 ¸2 /(R2 #¸2 ) , (4.109) L #) M #) M c ,q (m !m )/¸2m m , c8 ,q (m #m )/¸2m m , (4.110) 12 0,#) 1M 2M g 1M 2M 12 0,#) 1M 2M g 1M 2M the single inclusive distribution can be written as the sum of a chaotic and a coherent term E(1/p)d3p/d3k"(p s (k)#(1!p )s (k))E(1/p)d3p/d3kD 0 #) 0 # k/0
(4.111)
with s (k)"(m /m ) exp[!k2 R2 /2] , (4.112) #) p M M L s (k)"(m /m ) exp[!k2 R2/2] , (4.113) # p M M # where m is the transverse mass of the pions emitted. The momentum dependence of the chaoticity M parameter takes the form p "p(k )"p /A r r 0 r
(r"1, 2)
(4.114)
with A ,A(k )"p #(1!p )S r r 0 0 rr
(r"1, 2) ,
(4.115)
and S "exp[!(k2 #k2 )(R2!R2 )/4] . (4.116) rs rM sM # L Unless R "R , the transverse momentum distribution is a superposition of two Gaussians of # L di!erent widths. The rapidity distribution is uniform, dN/dy"const., as a result of boost invariance. In opposition to what is assumed usually in simpli"ed quasi-hydrodynamical treatments, the transverse radius of the chaotic source, R , cannot be determined independently by measuring #) only the single inclusive distribution, as the quantity R which sets the scale for the mean L transverse momentum of the chaotically produced particles is a combination of R and the #) correlation length ¸ . M
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We recall that the second-order correlation functions C``(k , k )"1#2Jp (1!p ) ) p (1!p )¹ cos(/#) !/# #/# )#p p ¹2 (4.117) 2 1 2 1 1 2 2 12 12 1 2 1 2 12 are de"ned in terms of the magnitudes and phases of d and dI , ¹ , ¹I , /#) and /I #) of the rs re rs rs rs rs chaotic source as well as of the phases of the coherent component, /#. The expressions for these r quantities read for an expanding source: ¹ "(1#c2 )~1@4 exp[2!(k !k )2(R2 !R2 )/8] , 12 12 12 1 2 1M 2M #) L ¹I "(1#c82 )~1@4 exp[2!(k !k )2(R2 !R2 )/8] , (4.118) 12 12 12 1 2 1M 2M #) L /#) "[bc /(1#c2 )](y !y )2!q (m !m )!1arctan c , (4.119) 12 12 12 1 2 0,#) 1M 2M 2 12 /I #) "[bc8 /(1#c82 )](y !y )2!q (m #m )!1arctan c8 , (4.120) 12 12 12 1 2 0,#) 1M 2M 2 12 /#"!q m . (4.121) j 0,# jM One thus "nds again that the correlation functions do not depend separately on the geometrical radii R or on the correlation lengths ¸ but rather on the combination R de"ned in (4.109). This L expression reduces in the limit R <¸ to ¸ and in the limit. R ;¸ to R. The model considered #) #) in [39] is thus a particular case of the space}time approach [3] for ¸"0. As in the static case the tilde terms give rise to the particle}antiparticle correlations. For a purely chaotic system the intercept of the p`p~ correlation function is C`~(k, k)"1#(1#4(b/a)2)~1@2"1#(1#4(q /m ¸2)2)~1@2 . (4.122) 2 0,#) M g We conclude this section with the observation that in [3] a correspondence between the correlation length ¸ in the primordial correlator C(x!y) and the temperature ¹ for a pion source that exhibits thermal equilibrium was established. In the limit of large volume <JR3 and lifetime R of the system, it reads 0 ¸&¹~1 . (4.123) 4.9. The Wigner function approach As mentioned previously, the experimental observation of the fact that the two particle correlation function depends not only on the di!erence of momenta q"k !k but also on the sum 1 2 k #k led to the introduction and the use [100] of a `sourcea function within the well-known 1 2 Wigner function formalism of quantum mechanics.39 While it turned out later that this property of the correlation function can be derived within the current formalism without the semiclassical approximations involved by the Wigner formalism, this formalism is still useful when applied within a hydrodynamical context.
39 An attempt to consider the correlation between coordinates and momentum was also performed earlier within the ordinary wave-function formalism by Yano and Koonin [94] who proposed a formula for the second-order correlation function of form (4.67). However, this form turned out subsequently to have pathological features as it leads in some cases to a violation of the lower bounds of the correlation function (see Section 5.1.6). The reason for this misbehaviour is mentioned in Section 4.6 and will also be discussed in the following.
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The Wigner function approach for BEC was proposed in a non-relativistic form in Ref. [100] and subsequently generalised in [101,3] (see also [64,95]). The Wigner function called also source function, g(x, k), may be regarded as the quantum analogue of the density of particles of momentum k at space}time point x in classical statistical physics. It is de"ned within the wave-function formalism as
P P
A A
BA BA
B B
1 1 g(x, k, t)" d3x@tH x# x@, t t x! x@, t e*kx{ 2 2 1 1 " d3k@tH k# k@, t t k! k@, t e~*k{x 2 2
(4.124)
and is related to the coordinate and momentum densities by the relations
P P
n(x, t)" d3k g(x, k, t) ,
(4.125)
n(k, t)" d3x g(x, k, t) ,
(4.126)
respectively. Due to its quantum nature the function g(x, k) takes real but not necessarily positive values. Although Eq. (4.124) is nothing but a de"nition which does not imply any approximation, its form suggests that it might be useful when simultaneous information about coordinates and momenta are desirable, provided of course that the limits imposed by uncertainty relations are not violated. As a matter of fact as will be shown below, the Wigner function is useful for BEC only if a more stringent condition is ful"lled, namely that the di!erence of momenta q of the pair is small, as compared with the individual momenta of the produced particles. It is thus clear that its applicability is more restricted than that of the classical current approach, where only the `no recoila condition, i.e. small total momentum of produced particles, as compared with the momentum of incident particles, must be respected. This circumstance is often overlooked when comparing theoretical predictions based on the Wigner approach with experimental data. In particular, it also follows that the application of the Wigner formalism to data has necessarily to take into account from the beginning resonances which dominate the small q region. It turns out that the use of the Wigner function for BEC is heuristically justi"ed only in special cases as, e.g. when a coherent hydrodynamical study is performed, i.e. when the observables are related to an equation of state and when simultaneously single- and higher-order inclusive distributions are investigated. Unfortunately only very few papers, where the Wigner function formalism is used, are bona"de hydrodynamical studies. The majority of papers in this context are `quasi-hydrodynamicala (see Sections 5.1.5 and 5.1.6) in the sense that the form of the source function is expressed in terms of ewective physical variables like temperature or velocity, which are not related by an equation of state. In this case the application of the Wigner approach is a `luxurya which is not justi"ed. This is a fortiori true since, as will be shown in the following, the Wigner approach is mathematically not simpler than the classical current approach, of which it is a particular case. Thus the space}time model [3] presented above (see Section 4.8) is more general than the Wigner approach, albeit it is not more complicated and does not have more independent parameters.
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In the second quantisation g(x, k) is de"ned in terms of the correlator Sas(k )a(k )T by the relation i j
P
Sas(k )a(k )T" d4x exp[!ix (kk!kk)] ) g[x, 1(k #k )] . i j k i j 2 i j
(4.127)
This is a natural generalisation of (4.126) to which it reduces in the limit k "k . i j Accordingly, for the second-order correlation function one writes
P P
P (k , k )" d4x d4x [g(x , k )g(x , k )#g(x , K)g(x , K) exp[iq (xk !xk )]] , 1 2 1 1 2 2 1 2 k 1 2 2 1 2
(4.128)
where Kk"(kk #kk )/2 and qk"kk !kk are the mean momentum and momentum di!erence of 1 2 1 2 the pair.40 The relation between this Wigner approach and the classical current approach is established by expressing the r.h.s. of Eq. (4.127) in terms of the currents. One has
P T A B A BU
1 z z g(x, k)" d4z J x# J x! exp[!ikkz ] . (4.129) k 2 2 2JE E (2p)3 i j The derivation of the Wigner formalism from the classical current formalism has the important advantage that it avoids violations of quantum mechanical bounds as those mentioned previously. Note that in the r.h.s. of Eq. (4.128) enters the o!-mass shell average momentum 1(k #k ) 2 2 1 which is not equal to the on-mass shell average K"1JE2!m2 #m2 where E is the total energy 2 1 2 of pair (1, 2). This means among other things that in this approach it is not enough to postulate the source function g in order to determine the second (and higher-order) correlation function C , but 2 further assumptions are necessary. Usually one neglects the o!-mass shellness, i.e. one approximates E by the sum E #E where E are the on-shell energies of particles (1, 2), which means that 1 2 i one neglects quantum corrections41 which is permitted as long as k !k "q is small.42 1 2 As mentioned already, the use of the Wigner formalism is worthwhile within a true hydrodynamical approach when the relation with the equation of state is exploited. In this case the probability to produce a particle of momentum k from the space}time point x depends on the #uid velocity, uk(x), and the temperature, ¹(x), at this point, and one has 1 JE E Sas(k )a(k )T" i j i j (2p)3
P
(1/2)(kk#kk) dp (x ) i j k k ) exp[!ix (kk!kk)] k i j R exp[(1/2)(kk#kk)u (x )/¹ (x )]!1 i j k k & k (4.130)
40 For neutral particles, there are additional contributions to P (k , k ) which play a role for soft particles and which 2 1 2 will be neglected here. 41 That these corrections can be important has also been shown in [102]. 42 It is sometimes argued that the relevant q range in BEC is given by D~1 where D is a typical length scale of the source and therefore for heavy ion reactions this should be allowed. This is not quite correct, because the shape of the correlation function from which one determines the physical parameters of the source is not given just by the values of the correlation function near the origin, but depends also on its values at large q.
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Here, dpk is the volume element on the freeze-out hypersurface R where the "nal state particles are produced. We will discuss applications of this approach in Section 5. 4.9.1. Resonances in the Wigner formalism For a purely chaotic source, the formalism in order to take into account the e!ects of resonance decays on the Bose}Einstein correlation function can be found, e.g. in [18,54]. An extension of this approach is due to [56,101] which allows to consider also the e!ect of coherence and provides rather detailed and subsequently, apparently, con"rmed predictions for heavy ion reactions. It is based on the Wigner function formalism. The correlation function of two identical particles of momenta k and k can be written as 1 2 (4.131) C (k , k )"1#(A A /A A ) , 2 1 2 12 21 11 22 where the matrix elements A are given in terms of source functions g(x, k) as follows: ij
P
A "JE E Sas(k )a(k )T" d4xg(x , kk)e*qkxk . i j k ij i j
(4.132)
A typical source function reads g (x , pk) (4.133) g(x , pk)"gdir(x , pk)# + res?p k k p k 2 res/o,u,g, where the labels dir and resPp refer to direct pions and to pions which are produced through the decay of resonances (such as o, u, g, etc.), respectively. The contribution from a particular resonance decay is estimated in [56,101] using kinematical and phase space considerations as well as the source function of that resonance. The source distribution for the direct production of pions and resonances is calculated assuming local thermodynamical and chemical equilibrium as is appropriate for a hydrodynamical treatment.
P
pk dp (x@ )d4(x !x@ ) k k k k . (4.134) R exp[[pku (x@ )!B k (x@ )!S k (x@ )]/¹ (x@ )]!1 k k a B k a S k & k Here a denotes the particular resonance and dpk is the di!erential volume element and the integration is performed over the freeze-out hypersurface R. uk(x) and ¹ are the four-velocity of & the #uid element at point x and the freeze-out temperature, respectively. B and S are the baryon number and the strangeness of the particle species labelled a, respectively, and k and k are the B S corresponding chemical potentials. J is the spin of the particle. This approach is then extended [56] to include also a coherent component resulting in a secondorder correlation function of the form 2J#1 gdir(x , pk)" a k (2p)3
C (k , k )"1#2p (1!p )Re d #p2 Dd D2 , 2 1 2 %&& %&& 12 %&& 12 where p is an e!ective chaoticity related to the true chaoticity p 43 via %&& dir p "p (1!f res)#f res . %&& dir 43 It is assumed that only directly produced particles have a coherent component.
(4.135)
(4.136)
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f res is the fraction of particles arising from resonances. The form of this equation is the same as that derived previously for a partially coherent source within the current formalism and manifests the characteristic two-component structure. The sensitivity of the correlation function on the chaoticity parameter p can be estimated, dir e.g. from the intercept (see Eq. (4.135)) I "C (k, k)"1#2p !p2 . o 2 %&& %&&
(4.137)
The fractions of pions produced directly (chaotically and coherently) and from resonances are f dir"p Ap /A , f dir"(1!p )Ap /A , f res"R Ares/A #) dir ii ii #0 dir ii ii res/o,u,g,2 ii ii
(4.138)
with f dir#f dir#f res"1 . #) #0 In Fig. 9 the intercept of the correlation function is shown as a function of p and f res. In order dir to read o! the fraction of direct chaotically produced particles, p , from the intercept of dir the correlation function, one has to extract the e!ective chaoticity p according to Eq. (4.137) %&& and then correct for the fraction of pions from resonance decays. Note that p (p . In parti%&& dir cular, if a large fraction of pions arises from resonance decays, p P1 and one needs very %&&
Fig. 9. Intercept of two-particle correlation function in the presence of coherence and resonances (from Ref. [56]).
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precise measurements of the two-particle correlation function at small q to determine the true chaoticity, p . dir A further complication arises if a fraction of particles is the decay products of long-lived resonances.44 This topic as well as the problem of misidenti"cation are discussed in [56]. 4.10. Dynamical models of multiparticle production and event generators Due to the lack of a full-#edged theory of multiparticle production in strong interactions di!erent models of multiparticle dynamics were proposed. Bose}Einstein correlations measurements have been used either to test a particular model or/and to determine some of its parameters. Among other things these models were used to predict the dependence of the chaoticity on the type of reaction. In the following, we will sketch the main theoretical ideas on which these models are based and mention brie#y their relation to data. One of the "rst models of particle production from which de"nite predictions on BEC can be derived is the Schwinger model [104] for e`}e~ reactions. It visualises the source as an onedimensional string in a coherent state and thus predicts the absence of any bunching e!ect. A similar prediction follows from the bremsstrahlung model [105]. Recoilless bremsstrahlung can be described by a classical current which also corresponds to a coherent state. Given the fact that in all hadron production processes BEC, i.e. a bunching e!ect has been seen, it follows that the above two models are ruled out by experiment. More complex predictions follow from a dual topological model due to Giovannini and Veneziano [106] which associates the processes e`}e~Phadrons to a unitarity cut in one plane, reactions induced by Pomeron exchange to a cut in two planes, and annihilation reactions p6 }p to a cut in three planes. This model predicts then among other things that for p~p~ BEC the intercepts C (k, k) of the second-order correlation functions for the above reactions should satisfy 2 the following relation: [Ce`e~(k, k)!2]/[Cpp(k, k)!2]/[Cann(k, k)!2]"1/1/1 . (4.139) 2 2 2 23 (A similar, but quantitatively di!erent relationship is predicted for p`p~ correlations.) Despite the fact that since the publication of this paper in 1977 many experimental BEC studies of these reactions have been performed, the above predictions could not be tested quantitatively in a convincing manner. This is due among other things to experimental di$culties (see Section 4.11) and illustrates the unsatisfactory status of experimental BEC investigations. A qualitative remark can however be made: the expectation that the annihilation reaction leads to more bunching than other reactions is apparently con"rmed (see e.g. Ref. [12] and Section 2.1.1). As to the di!erence between e`}e~ reactions and hadronic reactions the experimental situation is rather confused (see also below).
44 In some papers [103] pions originating from long-lived resonances are associated with a `haloa while those coming form short-lived resonances or directly produced are related to a `corea. Then it is claimed among other things that the `corea parameters of the source (like radius and j factor) can be obtained from the data just by eliminating the small Q points and "tting only the remaining points. Even if such a separation would be clear cut (there are doubts about this because of the u resonance), it would be of course dependent on the resolution of the detector.
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A somewhat related dynamical model based on Reggeon theory, has been already proposed in [107]. A straightforward extension of this formalism to heavy ion reactions does not work as it predicts that the longitudinal radius is of `hadronica size [108]. A di!erent approach to BEC based on the classical current formalism is proposed in [109]. The currents are associated with the chains of the dual parton model, and contrary to what is assumed in other applications of the classical current formalism, all the phases of these elementary currents are "xed, so that the source is essentially coherent. This is a special case of the classical current approach presented in Sections 4.2, 4.3 and 4.8 where allowance is made both for a chaotic and coherent component. The model is intended to work for p}p reactions where the authors state that resonances do not play an important role. It explains, according to the authors, the dependence of the j parameter in the empirical formula for the second-order correlation function C "1#j exp(!R2q2) (4.140) 2 on the multiplicity and energy. Unfortunately, the claim that in p}p reactions pions are only directly produced is unfounded. Furthermore, there are other factors which in#uence the multiplicity dependence of j (see Section 6.2) which are not considered in [109] and which are of more general nature. An orthogonal point of view for the interpretation of the same j factor (also for directly produced pions, only,) is due to [110]. In this approach the source is made of totally chaotic elementary emitting cells which are occupied by identical particles subject to Bose}Einstein statistics. Di!erent cells are independent so that correlations between particles in di!erent cells lead to j"0, while correlations between particles in the same cell are characterised by j"1. From the interplay of these two types of correlations, one obtains with an appropriate weighting, large j values in e`}e~ reactions and small j values in p}p reactions, as in [109], but within a completely di!erent approach. We conclude the discussion of these two approaches by the following remarks. Besides the reservations about the role of directly produced pions in BEC expressed above and which presents the two approaches in a rather academic light, it is unclear whether the j factor in e`}e~ reactions is larger than in p}p reactions as assumed in [110]. This issue awaits a critical analysis of the speci"c experimental set-ups. The fact that quite di!erent approaches lead to similar conclusions about the j factor con"rms that the parametrisation of the second-order correlation function in the form (4.140) is (see also Section 2.2) inadequate. We discuss now other two, closely related, approaches, which make more detailed predictions about the form of the correlation function in e`}e~ reactions: Refs. [22,75] on the one hand, and Refs. [23,111,121] on the other. Both approaches are based on a variant of the string model (for a more extended review of this topic see e.g. [113]). Such a string represents a coloured "eld formed between a quark q and an antiquark q6 , which tend to separate. Because of con"nement the break-up of the string can be materialised only through creation of new qq6 pairs, which are the mesons produced in the reaction. The di!erence between the Schwinger model of con"nement on the one hand and the models of Bowler and Andersson}Hofmann}RingneH r on the other is that in the former the "eld couples directly and locally to a meson, while in the latter ones the quarks, which constitute the meson, are created at di!erent points. This feature destroys the coherence inherent in the Schwinger model and makes the Bose}Einstein bunching e!ect possible. For massless quarks the second-order
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correlation function can be approximated by the relation [23] C "1#Scos(i*A)/cosh(b*A/2)T , 2
(4.141)
where *A denotes the di!erence between the space}time areas of coloured "elds spanned by the two particles, i is the string tension and b a parameter characterising the decay probability of the string. For massive quarks the formulae become more involved and were approximated analytically in [22] or calculated numerically in [23,111,112]. In this model the correlation function depends both on the di!erence of momenta k !k as well as on their sum k #k , re#ecting the 1 2 1 2 correlation between the momentum of the particle and the coordinate of its production point. This is a consequence of the fact that string models use a Wigner-function-type approach. From the above equations it follows that there are two length scales in the problem, one associated with i and the other with b. Phenomenologically these correspond to q and q . Both these lengths are , M correlation lengths rather than geometrical radii. (As a matter of fact there is no geometrical radius in the string model.) Their magnitudes are quite di!erent. In both string approaches one obtains a di!erence between BEC for identically charged and neutral pions as found in [71]. However while in [22] there is room for coherence, this is apparently not the case for [23,111,112], which predict a totally chaotic source. Furthermore in [22] an energy dependence of the BEC is predicted (the correlation function is expected to shrink with increasing energy), while in [23,111,112] the correlation function does not depend on energy.45 A rather discordant note in this string concerto [22,23] is represented by the paper by Scholten and Wu [114]. These authors, using a di!erent hadronisation mechanism conclude that dynamical correlations, at least in e`}e~ reactions dominate over BEC correlations so that BEC cannot be used to infer information about the size and lifetime of the source.46 This point of view seems too extreme, as it is contradicted by some simple empirical observations: in e`}e~ reactions, as well as in all other reactions, correlations between identical particle are observed which are much stronger than those of non-identical ones, the correlation functions are (in general) monotonically decreasing functions of the momentum di!erence q and in nuclear reactions the `radiia obtained from identical particle correlations increase with the mass number of the participating nuclei. All these observations are in agreement with what one would expect from BEC, which suggests that dynamical correlations cannot distort this picture too much. However a thorough comparison of BEC in di!erent reactions, using the same experimental techniques, appears highly desirable. Event generators. The model [23] was implemented by Sjostrand [115] into JETSET under the name LUBOEI by modifying a posteriori the momenta of produced pions so that identical pairs of pions are bunched according to [23]. This manipulation `by handa violates energy}momentum
45 To make the model more realistic in Ref. [23] resonances were included according to the variant of the Lund model (JETSET) in use at that time (1986) and agreement with e`}e~ data was found. Subsequently however it was pointed out in [58] that some resonance weights used in [23] were incorrect, so and the agreement mentioned above was probably accidental. 46 What concerns e`}e~ reactions similar scepticism was expressed by Haywood [4].
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conservation which was imposed at the beginning in JETSET. To compensate for this, the momenta are rescaled so that energy}momentum conservation is restored. However this rescaling introduces spurious long-range correlations, which bias the BEC. Nevertheless, in general this program leads to a reasonable description of the bunching e!ect in the second-order correlation function.47 More re"ned features of BEC which re#ect the quantum mechanical essence of the e!ect, cannot be obtained of course. One reason for this, of rather technical nature, is due to the fact that the ad hoc modi"cation of two-particle correlations does not yet include many-body correlations, re#ected in the symmetrisation (or antisymmetrisation) of the entire wave function. Another reason of fundamental character is that event generators like any Monte Carlo algorithm deal in general with probabilities48 and therefore cannot account for quantum e!ects, which are based on phases of amplitudes.49 The event generator JETSET was further developed by LoK nnblad and SjoK strand [118,119] and used to estimate the in#uence of BEC on the determination of the mass of = in e`}e~ reactions, a subject of high current interest for the standard model and in particular for the search of the Higgs particle. This e!ect was also studied using di!erent event generators in [120,121]. The argument of LoK nnblad and SjoK strand is the following. Consider the reaction e`}e~P=`=~Pq q6 q q6 when both ='s decay into hadrons. Then according to 1 2 3 4 [122] the typical space}time separation of the decay vertices of the =` and the =~ is less than 0.1 fm (at LEP 2 energies) and thus much smaller than a typical hadronic radius (&0.5 fm). There will thus be a Bose}Einstein interference between a pion from the =` and a pion (with the same charge) from the =~ and one cannot establish unambiguously the `parenthooda of these pions. This prevents then in this model a precise determination of the invariant mass of the ='s. In [118] algorithms for the inclusion of this e!ect into the determination of the mass of the = are proposed and for certain scenarios mass corrections of the order of 100 MeV at 170 GeV c.m. energy are obtained. However, as emphasised in [118] other scenarios with less or no e!ect of BEC on the mass determination of the = are possible. Thus in [120,121] e!ects of the order of only 20 MeV are found. For more details we refer the reader to the original literature. This aspect of BEC is interesting in itself as it illustrates the possible applications of this e!ect in electroweak interactions, a domain which is beyond the usual application domain of BEC, i.e. that of strong interactions. The Lund model was applied also to heavy ion reactions and then extended to include BEC (e.g. the SPACER version [55] of the Lund model). The topic of event generators for heavy ion reactions is of current interest because of the ongoing search for quark}gluon plasma. Padula et al. [95] suggested to use for this purpose the Wigner function formalism in order to take into account explicitly the correlation between momenta and coordinates, as implied by the inside}outside cascade approach. This is evidently another way of expressing the non-stationarity of the correlation function mentioned above. (An explicit introduction of momentum}coordinate correlations in
47 See, however, Ref. [116]. 48 For heavy ion reactions the `quantum molecular dynamicsa (QMD) model [117] attempts to surpass this de"ciency by using wave functions rather than probabilities as input. However, this model also neglects (anti)symmetrisation e!ects and cannot be used for interferometry studies. 49 It is interesting to mention that for one string Andersson and Hofmann [23] proposed a formulation of the BEC e!ect in terms of amplitudes. However this procedure cannot be used to generate events.
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particle physics event generators like JETSET/LUBOEI is not necessary, because the nonstationarity is delivered `free housea by the string model used in the LUND generator.) On the other hand, the Wigner formalism may present also another advantage as emphasised more recently by Bialas and Krzywicki [123]. This has to do with the important di$culty mentioned above and which is inherent in all event generators, namely the probabilistic nature of Monte Carlo methods. The Wigner function has in certain limits the meaning of a wave function and thus provides quantum amplitudes. The proposal of Bialas and Krzywicki consists then in starting from the single-particle distribution X (k) constructed from non-symmetrised particle 0 wave functions as produced by conventional event generators and writing the Wigner function g(k; x)"X (k)w(k; x) , (4.142) 0 where w(k; x) is the conditional probability that given that the particles with momenta k , k ,2, k 1 2 n are present in the "nal state, they are produced at the points x ,2, x . Then the art of the model 1 n builder consists in guessing the probability w(k; x). This may be easier than guessing from the beginning the exact Wigner function. For example, a simple ansatz would be to assume that the likelihood to produce a particle from a given space point is statistically independent of what happens to other particles. This means that w(k; x) can be factorised in terms of the individual particles. Implementations of this scheme were discussed in [124,125]. When using the Wigner formalism or any model (like those used in event generators) which speci"es momenta and coordinates simultaneously, one must of course watch that the correlations between momenta and coordinates do not become too strong. This apparently has not always been done.50 That such a procedure is dangerous since it can lead to unphysical antibunching e!ects, i.e. to the violation of unitarity was already mentioned in [89] (see Section 4.6). This point has been reiterated recently, e.g. in [127}129]. Concluding this section one should emphasise that event generators are just an experimental tool, sometimes useful in the design of detectors or for getting rudimentary information about experimentally inaccessible phase. Often they are however abused, e.g. to search for `newa phenomena: if agreement between data and event generators is found, one states that no `newa physics was found. Such a procedure is unjusti"ed, because agreement with a model or an event generator is often accidental. Furthermore, for the reasons mentioned above event generators cannot be used to obtain the `truea correlation function, i.e. they are no substitute for a bona"de HBT experiment (see also [130] for a critical analysis of transport models from the point of view of interferometry). 4.11. Experimental problems The confrontation of model predictions with experimental BEC data has been hampered by two major facts: (i) most models are idealisations, i.e. they use assumptions which are too strong. Examples of such assumptions are: neglect of "nal state interactions, boost invariance, particular
50 E.g. it is unclear to us whether the `purea multiple scattering approach of [126] satis"es the above constraint.
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analytical forms of the correlation functions; (ii) for various reasons in almost no BEC experiment so far a `truea or `completea correlation function was measured, i.e. a correlation function as de"ned by C (k , k )"P (k , k )/P (k )P (k ) . (4.143) 2 1 2 2 1 2 1 1 1 2 Here P and P are the double and single inclusive cross sections, respectively. What is measured 2 1 usually is instead a function which di!ers from Eq. (4.143) in several respects. The normalisation of C is not done in terms the product of single inclusive cross sections P but 2 1 in terms of a `backgrounda double inclusive cross section, which is obtained either by considering pairs of (identically charged) particles which come from di!erent events, or by considering oppositely charged particles, or by simulating P with an event generator which does not contain 2 BEC. One does not (yet) measure the full correlation function C in terms of its six independent 2 variables, but rather projections of it in terms of single variables like the momentum di!erence q, rapidity di!erence y !y , etc. 1 2 Last but not least, the intercept of the correlation function, which contains important information about the amount of coherence, cannot be really measured at present because (a) one does not yet control su$ciently well the "nal state interactions which contribute to the intercept; (b) its experimental determination implies an extrapolation to q"0. Such an extrapolation can be performed only if the analytical form of the correlation function at q50 is known, which is not the case. For these reasons at present it is di$cult to test quantitatively a given model, except when its predictions are very clear cut. This circumstance limits certainly the usefulness of BEC as a tool in determining the exact dynamics of a reaction.
5. Applications to ultrarelativistic nucleus}nucleus collisions 5.1. BEC, hydrodynamics and the search for quark}gluon plasma The use of BEC in the search for quark}gluon plasma is in most cases based on hydrodynamics. This is so because the space}time evolution of the system can be assumed to be given by the equations of hydrodynamics the solutions of which are di!erent depending whether a QGP is formed or not. In this way, hydrodynamics also provides information about the equation of state (EOS). QGP being a (new) phase it is described by a speci"c EOS which is di!erent from that of ordinary hadronic matter. The proof that this phase has been seen must include information about its EOS and thus the combination of hydrodynamics with BEC constitutes the only consistent way through which the formation of QGP can be tested. QCD predicts that the phase transition from hadronic matter to QGP takes place only when a critical energy density is exceeded. To measure this density we need to know the initial volume of the system. While via photon interferometry (see Sections 4.5 and 4.6) one can in principle measure the dimensions (and thus the energy density) of the initial state, hadron interferometry yields information only about the "nal freeze-out stage when hadrons are created. To obtain information about the initial state with hadronic probes, hydrodynamical models have to be used in order to
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extrapolate backwards from the "nal freeze-out stage where hadrons are created, to the interesting initial stage. The lifetime of the system as given by BEC is also an important piece of information for QGP search. Indeed, in order to decide whether we have seen the new phase, we have to measure the lifetime of the system. Only lifetimes exceeding signi"cantly typical hadronic lifetimes (10~23 s) could prove the establishment of QGP (see below). 5.1.1. General remarks about the hydrodynamical approach Besides the main advantages of hydrodynamics related to the information about initial conditions, freeze-out and equation of state, for the study of BEC in particular hydrodynamics is very useful because it provides the single inclusive distributions which are intimately connected with higher-order distributions as well as the weights and the space}time and momentum distributions of resonances, which strongly in#uence the correlations. The phenomenological applications of the hydrodynamical approach to data are however hampered by two circumstances. (i) While the ultimate goal of BEC is the extraction of the minimum set of parameters which include radii and coherence lengths both for the chaotic and coherent components of the source, in practice, mainly because of limited statistics (but also because of an inadequate analysis of the data) one has to limit oneself to the determination of a reduced number of parameters, which we call in the following `e!ective radiia R and `e!ective chaoticitya p . In reality, R is a combination of %&& %&& %&& correlation lengths51 ¸ and geometrical lengths R as introduced in Sections 4.3 and 4.8. Only in the particular case where one length scale is much smaller than the other, can one assume that one measures a `purea radius or a `purea correlation length. For simplicity, in the following, we shall assume that this is the case and in particular we assume that R<¸ so that, R reduces to ¸. This %&& limit might perhaps correspond to what is seen in experiment, if one considers the expansion of the system in the hadronic phase. (In the high-temperature limit ¸+¹~1, see Section 4.8). (ii) The presentation of the data is still biased by theoretical prejudices. Instead of a consistent hydrodynamical analysis, much simpli"ed models are used (see Section 5.1.5 where these models are presented under the generic name of quasi-hydrodynamics) for this presentation and therefore to obtain the real physical quantities, one would have to solve a complicated mathematical `inversea problem, i.e. one would have to reconstruct the raw data from those presented in the experimental papers and then apply the correct theoretical analysis to these. This has not been done so far and even if the statistics is su$cient for this purpose, the outcome is questionable because of the di$culties implied by the numerics. (That is why it would be desirable that experimentalists and theorists perform a joint analysis of the data or at least that the data should be presented also in `rawa form.) The nearest approximation to the solution of the `inversea problem found in the literature, is that of [101,150,131] based on the application of the HYLANDER code by the Marburg group: It consists in "tting the results of the hydrodynamical calculation to the Gaussian form used by experimentalists:
A
1 C (k , k )"1#j exp[!1q2 R2 !1q2 R2 !1q2 R2 ],1#j exp ! + (qR)2 2 1 2 2 , , 2 065 065 2 4*$% 4*$% 2 51 In the following the concept of length refers to space}time.
B
(5.1)
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and comparing with the inverse width of the correlation function as presented in the experimental papers.52 Here qk,pk !pk , Kk,1(pk #pk ) and q and K denote the components of q and K in 1 2 2 1 2 , , beam direction, and q and K the components transverse to that direction; q is the projection of M M 065 the transverse momentum di!erence, q on the transverse momentum of the pair, 2K , and q the M M 4*$% component perpendicular to K . (For a source with cylindrical symmetry, the two-particle M correlation function can be expressed in terms of the "ve quantities K , K , q , q and q .) , M , 4*$% 065 R , R , R are e!ective parameters, associated via Eq. (5.1) to the corresponding q components. , 4*$% 065 Eq. (5.1) is equivalent to an expansion of the correlation function C(q, K) for small q. The use of Eq. (5.1) for the representation of correlations data implies then that one does not measure the geometrical radius of the system but the length of homogeneity, which means that energy density determinations based on BEC are an overestimate.53 To take into account the fact that the correlation function depends in general not only on the momentum di!erence q"k !k but also 1 2 on the sum K"1(k #k ), the parameters R and j are assumed to be functions of K and rapidity 2 1 2 1(y #y ). 2 1 2 Hadron BEC refer to the freeze-out stage. This stage is usually described by the Cooper}Frye formula [136]: dN g E " i dk (2p)3
P
p dpk k , (5.3) exp((p uk!k !k )/¹ )!1 p k s b & which describes the distribution of particles with degeneracy factor g and four-momentum i pk emitted from a hypersurface element dpk with four-velocity uk.54 After the cascading of the resonances we obtain the "nal observable spectra. 5.1.2. Transverse and longitudinal expansion The equations of hydrodynamics are non-linear and therefore good for surprises. An illustration of this situation is represented by the realisation, described in more detail below, that the naive
52 In [132] it was recommended that experimentalists should use the more complete formula C (k , k )"1#j exp[!1q2 R2 !1q2 R2 !1q2 R2 !2q q R2 ] , (5.2) 065 , 065, 2 1 2 2 , , 2 065 065 2 4*$% 4*$% where 2q q R2 is called `crossa term. For a more detailed discussion of its meaning and dependence on the 065 , 065, coordinate system, see. Ref. [133]. Of course, in view of the de"ciencies of the entire phenomenological procedure outlined above and discussed in greater length in Section 5.1.5, these details are of limited importance. In particular they do not a!ect the conclusions discussed here. 53 In [134,135] a distinction is made between the `locala length of homogeneity ¸ (x, k) and the `hydrodynamicala ) length, % (x) which is the ensemble average of the former. ) 54 In most applications particles produced with momenta p pointing into the interior of the emitting isotherm k (p dpk(0) were assumed to be absorbed and therefore their contribution to the total particle number was neglected. In k Ref. [137] this e!ect was indeed estimated to be negligible and recent attempts to reconsider it could not change this conclusion. Another e!ect is the interaction of the freeze-out system with the rest of the #uid. This e!ect can be estimated by comparing the evolution of the #uid with and without the frozen-out part. This is done by equating the frozen-out part with that corresponding in the equation of hydrodynamics to the case p "0. The #uid parameters are modi"ed by this k procedure at a level not exceeding 10% [138]. The in#uence of the freeze-out mechanism on the determination of radii via BEC has been discussed recently in several papers; see. e.g. [139] and references quoted therein.
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Fig. 10. Bose}Einstein correlation functions in longitudinal and transverse direction, for the three-dimensional (solid lines) and the one-dimensional calculations (dashed lines) (from Ref. [101]).
intuition about the role of transverse expansion in the determination of the transverse and longitudinal radius may be completely misleading. Only a systematic analysis based on (3#1)-D hydrodynamics clari"ed this issue. In the present section we will discuss Bose}Einstein correlations of pions and kaons produced in nuclear collisions at SPS energies in the framework of relativistic hydrodynamics. Concrete applications were done for the symmetric reactions S#S and Pb#Pb at 200 AGeV. Many of the theoretical results were predictions at the time they were obtained. These predictions were subsequently con"rmed in experiment. In [140] it was found that the transverse radius extracted from data on Bose}Einstein correlations (BEC) for O#Au at 200 AGev reached in the central rapidity region a value of about 8 fm. It was then natural to conjecture that this could be an indication of transverse #ow [141}143]. In the meantime the experimental observation in itself has been quali"ed [144] and it now appears that the transverse radius obtained from the BEC data does not exceed a value of 4}5 fm (see however Ref. [143]). Motivated by this situation in Ref. [101] an investigation55 of the role of three-dimensional hydrodynamical expansion on the space}time extension of the source was performed and compared with a (1#1)-D calculation. Contrary to what one might have expected it was found that transverse yow does not increase the transverse radius. On the other hand, a strong dependence of the longitudinal radius on the transverse expansion was established. Fig. 10 shows two typical examples of the Bose}Einstein correlations as functions of q and , q for the one- and the three-dimensional hydrodynamical solution. The dependence of C (q ) on M 2 , transverse expansion agrees qualitatively with what one would expect. For a purely longitudinal expansion, the e!ective longitudinal radius of the source is larger than in the case of threedimensional expansion, which is re#ected in a decrease of the width of the correlation function (see also in Fig. 11 below). On the other hand, the results for C (q ) were at a "rst glance rather surprising. Naively one 2 M might have expected that the transverse #ow would lead to an increase of the transverse radius, i.e., to a narrower correlation function C (q ). However, in Fig. 10 the curves that describe the 2 M 55 In Ref. [145] the dependence of the e!ective transverse radius on the transverse velocity "eld was investigated for a "xed freeze-out hypersurface. The e!ects of transverse expansion on the shape and position of the hypersurface are not considered there.
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Fig. 11. Dependence of the longitudinal and transverse radii extracted from Bose}Einstein correlation functions on the rapidity y and average momentum K of the pair. As before, solid lines correspond to the three-dimensional and dashed K M lines to the one-dimensional results. The open circles indicate values of R obtained from Eq. (5.4) (from Ref. [101]). ,
one- and the three-dimensional results are almost identical. If anything, one would conclude that the e!ective transverse radius is smaller in the presence of transverse expansion. This e!ect can however be explained if one takes a closer look at the details of the hydrodynamic expansion process as investigated in Ref. [101]. Due to the correlation between the space}time point where a particle is emitted, and its energy}momentum, the e!ective radii obtained from Bose}Einstein correlation data present a characteristic dependence on the average momentum of the pair, Kk. Fig. 11 shows the dependence of the e!ective radii R , R and R on the rapidity , 4*$% 065 y and the mean transverse momentum of the pair K , both for the one- and for K M the three-dimensional calculation. The longitudinal radius R becomes considerably smaller , (by a factor of 2}3) if transverse expansion is taken into account. For the one-dimensional case, an approximate analytic expression has been derived for the y - and the K -dependence of the K M longitudinal `radiia in Refs. [146,147]56 (see also [95]): R "J(2¹ /m )q /cosh(y ) , , & M o K
(5.4)
56 In this reference Eq. (5.4) is used to disprove the applicability of hydrodynamics to p}p reactions in the ISR energy range (Js"53 GeV). Such a conclusion seems dangerous given the approximations involved both in the derivation of this formula as well as in the interpretation of the BEC measurements in the above reactions.
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where m "(m #m )/2 is the average transverse mass of the two particles, ¹ is the freeze-out M M1 M2 & temperature and q "(Ru /Rx)~1 is the inverse gradient of the longitudinal component of the o , four-velocity in the centre (at x"0).57 In [101] one "nds that this approximate expression describes R (K , y ) for S#S reactions quite well, both for the one- and the three-dimensional , M K case (see Fig. 11). However for Pb#Pb reactions the same formula fails to account for the data [148]. This is not surprising, because Eq. (5.4) is based on the assumption of boost invariance, i.e. no stopping. This assumption is not justi"ed at SPS energies where there is considerable stopping. The inelasticity increases with atomic number and this may explain the breakdown of the above formula. This exempli"es the limitations of the boost-invariance assumption, an assumption which must not be taken for granted but in special circumstances. In [149] it was proposed to use the information obtained from "tting the single inclusive distribution to constrain the parameters that enter into the hydrodynamic description, and then to calculate the transverse radius directly. Indeed, let R denote the hypersurface in Minkowski space on which hadrons are produced. Then one can de"ne, e.g. a transverse radius :R R pkdp /(exp[(pku !k)/¹]!1) k k R " , (5.5) M :R pkdp /(exp[(pku !k)/¹]!1) k k where u , ¹ and k denote the four-velocity, temperature, and chemical potential on the hypersurk face R, respectively as in Eq. (5.3). It is interesting to note that this method for the determination of transverse radii based on the single inclusive cross sections provides a geometrical radius while the use of the second-order correlation function provides a coherence length (length of homogeneity). Comparing the e!ective transverse radius R extracted from the Bose}Einstein correlation function to the mean M transverse radius as calculated directly in [149] according to Eq. (5.3), one "nds in [101] that the two results agree to an accuracy of about 10%. This conclusion is con"rmed and strengthened in a more recent study by Schlei [131] for kaon correlations.
Of course, this approach can be used only if a solution of the equations of hydrodynamics is available; with quasi-hydrodynamical methods this is not possible. 5.1.3. Role of resonances and coherence in the hydrodynamical approach to BEC This problem was investigated using an exact (3#1)-D numerical solution of hydrodynamics in [150]. The source distribution g(x, k) was determined from a three-dimensional solution of the relativistic hydrodynamic equations. Fig. 12 illustrates the e!ect of successively adding the contributions from o, u, D and g decays to the BEC correlation functions of directly produced (thermal) n~ (dotted lines), in longitudinal and in transverse direction. The width of the correlation progressively decreases as the decays of resonances with longer lifetimes are taken into account, and the correlation loses its Gaussian shape. The long-lived g leads to a decrease of the intercept. Pion versus kaon interferometry. Ideally, a comparison of pion and kaon interferometry should lead to conclusions concerning possible di!erences in the space}time regions where these particle 57 Expression (5.4) for R denotes in fact the length of homogeneity ¸ mentioned above. It refers to the region within , ) which the variation of Wigner function is small. By de"nition ¸ 4R where R is the geometrical radius. )
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Fig. 12. Bose}Einstein correlation functions of negatively charged pions, in longitudinal and transverse direction. The separate contributions from resonances are successively added to the correlation function of direct (thermal) n~ (dotted line). The solid line describes the correlation function of all n~ (from Ref. [156]).
decouple from the hot and dense matter. It was proposed that kaons may decouple (freeze out) at earlier times and higher temperatures than pions [151]. Indeed, preliminary results had indicated that the e!ective longitudinal and transverse source radii extracted from nn correlations were signi"cantly larger than those obtained from KK correlations [152]. However, seen in Fig. 12, the BEC of pions are strongly distorted by the contributions from resonance decay. It was pointed out in Ref. [52] in a study based on the Lund string model that such distortions are not present for the BEC of kaons, and that consequently for the e!ective transverse radii one expects R (KB)(R (nB), even in the absence of any di!erence in the freeze-out geometry of directly M M produced pions and kaons. These conclusions were con"rmed in [156] within the hydrodynamical approach and one found furthermore that this e!ect is even more pronounced if one considers longitudinal rather than transverse radii. Furthermore, the interplay between coherence and resonance production which was not considered in [52] was studied in [150]. There are also some striking di!erences between [52,150] in the resonance production cross sections used. In Fig. 13, BEC functions of n~ (solid lines) and of K~ (dashed lines) are compared, at k "0 M and k "1 GeV/c, respectively. The dotted lines correspond to the BEC function of thermally M produced n~. It can be seen that the distortions due to the decay contributions from long-lived resonances disappear only at large k . Fig. 14 shows the e!ective radii R , R and R as M , 4*$% 065 functions of rapidity and transverse momentum of the pair, both for n~n~ (solid lines) and for K~K~ pairs (dashed lines). For comparison, the curves for thermally produced pions (dotted lines)
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Fig. 13. Correlation functions of all n~ (solid lines), thermal n~ (dotted lines) and all K~ (dashed lines), for k "0 and for M k "1 GeV/c (from Ref. [56]). M
Fig. 14. Dependence of the longitudinal and transverse radii extracted from Bose}Einstein correlation functions on the rapidity y and average momentum k of the pair, for all n~ (solid lines), thermal n~ (dotted lines) and all K~ (dashed k M lines). The full circles were obtained by substituting the value St (z"0, r )T"2 fm/c for the average lifetime of the & M system (calculated directly from hydrodynamics by averaging over the hypersurface) into Eq. (5.4), with ¹ "0.139 GeV & (from Ref. [56]).
are also included. The e!ective longitudinal radii extracted from n~n~ correlations are considerably larger than those obtained from K~K~ correlations. In the central region the two values for R di!er by a factor of &2. For the transverse radii, the factor is &1.3. A comparison between ,
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Fig. 15. n~n~ Bose}Einstein correlation functions in the presence of partial coherence (from Ref. [56]).
results for K~ and thermal n~ shows that part of this e!ect can be accounted for by kinematics (the pion mass being smaller than the kaon mass; see also Eq. (5.4)). Nevertheless, the large di!erence between the widths of pion and kaon correlation functions is mainly due to the fact that pion correlations are strongly a!ected by resonance decays, which is not the case for the kaon correlations. In the hydrodynamic scenario of Ref. [56], about 50% of the pions in the central rapidity region are the decay products of resonances [149], while less than 10% of the kaons are created in resonance decays (KHPKn dominates, contributing with about 5%). In Ref. [56] the problem of coherence within the hydrodynamical approach to BEC was also investigated. Fig. 15 shows the n~n~ correlation functions in the presence of partial coherence. In order to extract e!ective radii from Bose}Einstein correlation functions in the presence of partial coherence, Eq. (5.1) must be replaced by the more general form
C
D
C
D
1 1 C (k , k )"1#j ) p2 exp ! + (qR)2 #Jj ) 2p (1!p )exp ! + (qR)2 . 2 1 2 %&& %&& %&& 2 4
(5.6)
5.1.4. Comparison with experimental data Some of the predictions made in [56] for S#S reactions could be checked experimentally in Refs. [153,154], in particular the rapidity and transverse momentum dependence of radii and remarkable agreement was found. In [155] the hydrodynamical calculations were extended to Pb#Pb reactions and compared with S#S reactions and, where data were available, with experiment. The calculation of Bose}Einstein correlations (BEC) was performed using the
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Fig. 16. E!ective radii extracted from Bose}Einstein correlation functions as a function of the rapidity y of the pair and k the transverse average momentum K of the pair for all pions (from Ref. [155]). M
formalism outlined in Ref. [150] including the decay of resonances. The hadron source was assumed to be fully chaotic. Figs. 16 and 17 show the calculations for the e!ective radii R , R and R as functions of , 4*$% 065 rapidity y and transverse momentum K of the pion pair compared to the corresponding NA35 K M and preliminary NA49 data [157,158], respectively. All these calculations, which in the case of S#S had been true predictions, agree surprisingly well with the data.58 This suggests that our understanding of BEC in heavy ion reactions has made progress and con"rms the usefulness of the Wigner approach when coupled with full-#edged hydrodynamics. An important issue in comparing data with theory is the detector acceptance of a given experiment. This is also discussed in detail in [155]. Another application of hydrodynamics to the QGP search in heavy ion reactions is due to Rischke and Gyulassy [159] who investigate the ratio r"R /R . Based on considerations due to Pratt [100], this quantity had been proposed by 065 4*
%$58 The EOS used in the hydrodynamical studies quoted above included a phase transition from QGP to hadronic matter. How critical this assumption is for the agreement with data is yet unclear and deserves a more detailed investigation. On the other hand, the very use of hydrodynamics is based on the assumption of local equilibrium, and this equilibrium is favoured by the large member of degrees of freedom due to a QGP.
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Fig. 17. E!ective radii extracted from Bose}Einstein correlation functions as a function of the transverse average momentum K of the pair for all pions compared with data (from Ref. [155]). M
Bertsch and collaborators [160] as a signal of QGP. Under certain circumstances one could expect that for a long-lived QGP phase r should exceed unity, while for a hadronic system, due to "nal state interactions, the out and side sizes should be comparable. The authors of [159] performed a quantitative hydrodynamical study of r in order to check whether this signal survives a more realistic investigation, albeit they did not take into account resonances. For directly produced pions it is found that r indeed re#ects the lifetime of an intermediate (QGP) phase. However we have seen from [56,155], that for pion BEC, when resonances are considered, hydrodynamics with an EOS containing a long-lived QGP phase, leads (in agreement with experiment) to values of r of order unity. To avoid this complication, in [161] it was proposed to consider kaons at large k . However even this proposal may M have to be quali"ed, besides the fact that it will be very di$cult to do kaon BEC at large k . Firstly, one has to recall that M the entire formalism on which the r signal is based and in particular the parametrisation (5.1) is questionable. Secondly it remains to be proved that this signal survives if one imposes simultaneously the essential constraint due to the single inclusive distribution.59 Furthermore, it is unclear up to what values of k the Wigner formalism, (which is a particular M case of the classical current formalism) on which the theory is based, is applicable. For these reasons the determination of the lifetime of the system via `purea hydrodynamical considerations is certainly an alternative which deserves to be considered seriously, despite its own di$culties.
59 I am indebted to B.R. Schlei for this remark.
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5.1.5. Bose}Einstein correlations and quasi-hydrodynamics As mentioned already, the initial motivation for proposing the Wigner function formalism for BEC was to explain why the experimentally observed second-order correlation functions C were 2 depending not only on the momentum di!erence q"k !k but also on their sum K"k #k . 1 2 1 2 However in the mean time it was shown [33,3] that this feature follows from the proper application of the space}time approach in the current formalism even without assuming expansion. Furthermore, the Wigner formalism is useful only at small q and cannot be applied in the case of strong correlations between positions and momenta while the current formalism is not limited by these constraints. As explained above the use of the Wigner formalism can be defended from the point of view of economy of thought if combined with bona"de hydrodynamics and an equation of state. This notwithstanding, besides a few real, albeit numerical, hydrodynamical calculations, most phenomenological papers on BEC in heavy ion reactions (see e.g. [67,162,163,102,164}168, 68,69,169}178] have used the Wigner formalism without a proper hydrodynamical treatment, i.e. without solving the equations of hydrodynamics; hydrodynamical concepts like velocity and temperature were used just to parametrise the Wigner source function. While such a procedure may be acceptable as a theoretical exercise, it is certainly no substitute for a professional analysis of heavy ion reactions. This is a fortiori true when real data have to be interpreted.60 As exempli"ed in previous sections such a procedure is unsatisfactory, among other things because it can lead to wrong results. The use of this `quasi-hydrodynamicala approach is even more surprising if one realises the fact that the Wigner formalism not only is not simpler than the more general current61 formalism but it is also less economical. The number of independent parameters necessary to characterise the BEC within the Wigner formalism of Ref. [166] is62 10, i.e. it is as large as that in the current formalism. However the 10 parameters of [166] describe a very particular source,63 as compared with that of the current formalism: besides the fact that the correlation function source is assumed to be Gaussian, it is completely chaotic and it can provide only the length of homogeneity ¸ [67]. For ) the search of quark}gluon plasma, however, the geometrical radius R is relevant, because the energy density is de"ned in terms of R and the use of ¸ instead of R leads to an overestimate of the ) 60 A recent experimental paper [45] where such a procedure is used is a good illustration of the limits of quasihydrodynamical models. The analysis here concentrates on the resolution of the ambiguity between temperature and transverse expansion velocity of the source. It is clear that such an ambiguity is speci"c to quasi-hydrodynamics and is from the beginning absent in a correct hydrodynamical treatment. Moreover even this result may have to be quali"ed given some assumptions which underlay this analysis. To quote just two: (i) The assumption of boost invariance made in [45] decouples the longitudinal expansion from the transverse one. This not only a!ects the conclusions drawn in this analysis but prevents the (simultaneous) interpretation of the experimental rapidity distribution. (ii) The neglect of long-lived resonances which strongly in#uences the j factor and thus also the extracted radii. Of course, despite the richness of the data, no attempt to relate the observations to an equation of state can be made within this approach. 61 We remind that the classical current formalism, in opposition to the Wigner function formalism, does not apply only to small q and to semiclassical situations. Furthermore it allows for a correct treatment of new phenomena like particle}antiparticle correlations. 62 For each value of K a Gaussian ellipsoid is described by three spatial extensions, one temporal extension, three components of the velocity in the local rest frame and the three Euler angles of orientation. 63 To consider such an approach as `model independenta [165,179] is misleading.
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energy density.64 Furthermore, the physical signi"cance of the parameters of the Wigner source is unclear if the Gaussian assumption does not hold.65 Not only is there no a priori reason for a Gaussian form, but on the contrary, both in nuclear and particle physics as well as in quantum optics, there exists experimental evidence that in many cases an exponential function in DqD is at small q a better approximation for C than a Gaussian. Furthermore, in the presence of coherence, 2 no single simple analytical function, and in particular no single Gaussian is expected to describe C . This is a straightforward consequence of quantum statistics. 2 Given the fact that good experimental BEC data are expensive both in terms of accelerator running time and man power, the use of inappropriate theoretical tools, when more appropriate ones are available, is a waste which has to be avoided. For the reasons quoted above we will not discuss in more detail the numerous and sometimes unnecessarily long papers which use quasihydrodynamical methods. Concluding remarks. In a consistent treatment of single and double inclusive cross sections for identical pions via a realistic hydrodynamical model, resonances play a major role leading to an increase of e!ective radii of sources. E!ective longitudinal radii are more sensitive to the presence of resonances than transverse ones. From the hydrodynamical treatment we learn that the hadron source (the real "reball) is represented by a very complex freeze-out hypersurface (see Ref. [155]). The longitudinal and transverse extensions of the "reball change dynamically as a function of time, rather than show up in static e!ective radii. Thus, the interpretation of BEC measurements is also complicated. When no quantitative comparison with data is intended, besides the current formalism, analytical approximations of the equations of hydrodynamics can be useful, because they allow a better qualitative understanding of hydrodynamical expansion. However, when a quantitative interpretation of experiments is intended and in particular a connection with the equation of state is looked for, the only recommendable method is full-#edged hydrodynamics. 5.1.6. Photon correlations and quasi-hydrodynamics Photon correlations have been investigated within the context of quark}gluon plasma search, since they present certain methodological advantages as compared with hadrons. While experimentally genuine photon BEC in high-energy heavy ion reactions have not yet been unambiguously identi"ed, because of the strong n0 background and the small cross sections for photon production, there are several theoretical studies devoted to this topic. The advantage of photon BEC resides in the fact that, while correlations between hadrons are in#uenced by "nal state interactions, photon correlations are `cleana from this point of view. For high-energy physics, photons present another important advantage that they can provide direct information about the early stages of the interaction when quarks and gluons dominate and hadrons have not yet been created. In particular photon BEC contain information about the lifetime of the quark}gluon plasma [84}87,180,89]. Among other things it was argued, e.g. in [180] and con"rmed in [89]
64 One may argue that the `length of homogeneitya [67] ¸ de"ned in terms of the Wigner function is a particular case ) of the correlation length ¸ de"ned in terms of the correlator (see Section 4.3). While ¸ is always limited from above by R, ) ¸ can be either smaller or larger than R. 65 Even if a Gaussian form would hold for directly produced pions, resonances would spoil it [56,155,131].
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using a more correct formalism (see below), that the correlation function C in the transverse 2 direction can serve as a signal for QGP as it is sensitive to the existence of a mixed phase. Unfortunately, some of these studies (Refs. [84}87,180]) besides being based on quasi-hydrodynamics, use an input formula for the second-order BEC, which is essentially incorrect (see Section 4.6). Besides this, some approximations made in [84], e.g. are inadequate. This question was analysed in more detail in [89] where it was found that only some of the results of Ref. [84] survive a more critical analysis. The formula for the two-particle inclusive probability used in [84}87,180] reads
P P
P (k , k )" d4x d4x g(x , k )g(x , k )[1#cos((k !k )(x !x ))] , 2 1 2 1 2 1 1 2 2 1 2 1 2
(5.7)
while Ref. [89] uses the more correct formula
P P
A
BA
B
k #k k #k 2 g x , 1 2 [1#cos((k !k )(x !x ))] . P (k , k )" d4x d4x g x , 1 1 2 1 2 1 2 1 2 2 1 2 2 2 (5.8)
(From Eq. (5.8) one gets for the second-order correlation function Eq. (4.58).) It is found in [89] that two rather surprising properties of the two-photon correlation function presented in [84] are artefacts of inappropriate approximations in the evaluation of space}time integrals. In [84], it has been claimed that the BEC function in the longitudinal direction (a) oscillates and (b) takes values below unity. As property (b) is inconsistent with general quantum statistical bounds, it was important to clarify the origin of this discrepancy. On the other hand, it has been con"rmed in [89] that the correlation function in the transverse direction does exhibit oscillatory behaviour in the out component of the momentum di!erence. Furthermore, in this reference a change of the BEC function in *y from Gaussian to a two-component shape with decreasing transverse photon momentum was found which may serve as evidence for the presence of a mixed phase and, hence, as a QGP signature. However even after correcting the wrong input BEC formula it is questionable whether the other approximations made in Refs. [84}87,180,89] may not invalidate the above result. Besides the use of a simpli"ed hydrodynamical solution one has to recall that (i) the Wigner formalism like the more general classical current formalism, is limited to small momenta k of produced particles (no recoil approximation), (ii) besides this general limitation to small k the Wigner formalism is useful only for small di!erences of momenta q and for weak correlations between coordinates and momenta, and (iii) although Eq. (4.58) does not su!er from the violation of unitarity disease mentioned in Section 4.6 of Chapter 3, it is based on an approximation which makes it sometimes inapplicable for photons (see Section 4.5). From the above discussion one may conclude that the experimental problems of photon BEC are matched by theoretical problems yet to be solved. 5.2. Pion condensates One of the most interesting phenomena related to Bose}Einstein correlations is the e!ect of Bose condensates. The remarkable thing about this e!ect is that it is not speci"c for particle or nuclear
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physics, but occurs in various other chapters of physics like superconductivity and super#uidity. Moreover, recently not only the condensation of a gas of atoms has been experimentally achieved [185] but the quantum statistical coherence of these systems has been experimentally proven through Bose}Einstein correlations [24]. The proposal to use BEC for the detection of condensates was made a long time ago [59]. The more recent developments in heavy ion reactions have made this subject of current interest. Already in experiments at the SPS (e.g., Pb#Pb at E "160 AGeV) secondary particles are formed at high number densities in rapidity space "%!. [181] and in future experiments at RHIC and the LHC one expects to obtain even higher multiplicities of the order of a few thousand particles per unit rapidity. If local thermal (but not chemical) equilibrium is established and the number densities are su$ciently large, the pions may accumulate in their ground state and a Bose condensate may be formed.66 A speci"c scenario for the formation of a Bose condensate, namely, the decay of short-lived resonances, was discussed in Ref. [183] where conditions necessary for the formation of a Bose condensate in a heavy ion collision were investigated. In Ref. [183] it was found that if a pionic Bose condensate is formed at any stage of the collision, it can be expected to survive until pions decouple from the dense matter, and thus it can a!ect the spectra and correlations of "nal state pions. In Ref. [184] one investigated the in#uence of such a condensate on the single inclusive cross section and on the second-order correlation function of identically charged pions (Bose}Einstein correlations BEC) in hadronic reactions for expanding sources. A hydrodynamical approach was used based on the HYLANDER routine. The Bose}condensate a!ects the single inclusive momentum distributions EdN/d3k, the momentum-dependent chaoticities p and the Bose}Einstein correlation functions C only over a limited momentum range. This is due to the fact that in 2 a condensate there exists a maximum velocity (which implies also a maximum momentum di!erence q ) and leads to a very characteristic structure in single and double inclusive spectra. .!9 In Fig. 18 the results of the numerical evaluations of the single inclusive momentum distributions E(dN/d3k), the momentum-dependent chaoticities p and the Bose}Einstein correlation functions C are shown for a spherically and for a longitudinally expanding source. The presence of 2 a Bose}condensate of only 1% results in a decrease of the intercept by about 15%. Furthermore due to a limited value of q a part of the tail of the two-particle correlation functions is not .!9 a!ected by the pionic Bose-condensate and a peak appears. To what extent such peaks can be observed in experimental data depends among other things on the size of the source, the details of the freeze-out, the width of the momentum distribution in the bosonic ground state, and detector acceptance. Plasma droplets? If the phase transition from hadronic matter to QGP and vice versa is of "rst order then one could expect the formation of a mixed phase, in which QGP and hadronic matter coexist. Such a mixed phase manifests itself in the hydrodynamical evolution of the system [186] and it in#uences among other things the transverse momentum distribution of photons as seen in Section 5.1.6. It was suggested by Seibert [187] that the mixed phase could also lead to a granular structure which might be seen in the #uctuations of the velocity distributions of secondaries
66 Recently a di!erent type of pion condensate, the disordered chiral condensate, has received much attention in the literature. It has been argued [182] that such a condensate would lead to the creation of squeezed states.
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Fig. 18. Single inclusive spectra, momentum dependent chaoticities and two-particle BE correlation functions for a spherically and a longitudinally expanding source. The di!erent line styles correspond to di!erent condensate densities n compared to the thermal number densities n . k is the pionic chemical potential (from Ref. [184]). #0 5) n
produced in the hadronisation stage. Pratt et al. [188] (see also [189] proposed subsequently that in Bose}Einstein correlations, too, one might see a signature of this granularity.
6. Correlations and multiplicity distributions 6.1. From correlations to multiplicity distributions An important physical observable in multiparticle production is the multiplicity distribution67 P(n), i.e., the probability to produce in a given event n particles. The link between the multiplicity
67 The multiplicity distribution will be denoted sometimes in the following also by MD.
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distribution P(n) and BEC is represented by the density matrix o, since it is the same o which appears in the de"nition of P(n): P(n),SnDoDnT
(6.1)
and the de"nition of correlation functions (see e.g. Eq. (4.4)). Usually, one expresses o in terms of the P(a) representation (see Section 2.2) and by using for P(a) simple analytical expressions one is able to derive the most characteristic forms of P(n) in an analytical form, like the Poisson or the negative binomial representation (see e.g. [5]). There exist however physically interesting cases where no analytical expression for the multiplicity distribution P(n) exists, but instead the moments of P are given. From the phenomenological point of view the approach to MD via moments presents sometimes important advantages because it allows the construction of an ewective density matrix from the knowledge of a few physical quantities like the correlation lengths and mean multiplicity, which in turn can be obtained from experiment (see e.g. [190]). In the following we will address this aspect of the problem, especially since in this way the link between correlations and MD becomes clearer. We start by recalling some de"nitions. Besides the normal moments of the MD given by SnqT,+ P(n)nq , n one uses frequently the factorial moments
(6.2)
U ,+ P(n)(n(n!1) ) 2 ) (n!q#1)) . (6.3) q n These can be expressed in terms of the inclusive correlation functions o through the relation q n! U " du 2 du o (k ,2, k )" . (6.4) q 1 q q 1 q (n!q)! X X
P
P
T
U
Eqs. (6.3) and (6.4) illustrate the fundamental fact that the inclusive cross sections o and thus the q correlation functions determine the moments of the MD, which are nothing but the integrals of o . q Although this relation between moments of MD and correlation functions is a straightforward aspect of multiparticle dynamics, the connection between MD and correlations has often been overlooked. This is in part due to the fact that measurements of correlations are, for reasons of statistics and other technical considerations, frequently performed in di!erent (narrower) regions of phase space than measurements of MD.68 However the importance of the use of the relationship between MD and correlations can hardly be overemphasised, just because of the di!erent experimental methods used in the investigation of these two observables. In the absence of a theory of
68 This is, e.g. the case where MD are measured with no proper identi"cation of the particles, while BEC refer of course to identical particles. Thus the UA5 experiment [191], which discovered the violation of KNO scaling in MDs, measured only charged particles, without distinguishing between positive and negative charges. However, the rapidity region accessible in this experiment was much broader than the corresponding region in the UA-1 experiment [192] where correlation measurements were performed and where a distinction between positive and negative charges could be made.
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multiparticle production, the form of the correlators and the amount of chaoticity are unknown and have therefore to be parametrised and then determined experimentally in correlation experiments. Both the parametrisation and the measurements are a!ected by errors. Similar considerations apply for MD, but because of the di!erent experimental conditions under which correlation measurements and MD measurements are made, the corresponding errors are di!erent. (See also the discussion of the importance of higher-order correlation, Section 2.2.1.) Therefore from a phenomenological and practical point of view, MD and correlations are rather complimentary and have to be interpreted together. In the following we will exemplify the usefulness of this point of view. 6.1.1. Rapidity dependence of MD in the stationary case The dependence of moments of multiplicity distributions P(n) on the width of the bins in momentum or rapidity space has been in the centre of multiparticle production studies for the last 15 years. It got much attention after: (1) the experimental observation [191] by the UA5 collaboration that the normalised moments of P(n) in the rapidity plateau region increase with the width of the rapidity window *y; (2) the proposal by Bialas and Peschanski [193] that this behaviour, which at a "rst look was power like may re#ect `intermittencya, i.e. the absence of a "xed scale in the problem, which could imply that self-similar phenomena play a role in multiparticle production. However soon after this proposal was published it was pointed out in [194] that the quantum statistical approach, presented in Section 2.2 and which implies a xxed scale,69 predicts a similar functional relationship between the moments of MD and *y. In Fig. 19 from [194] some examples of this behaviour are plotted and compared with experimental data. For small !dy the (semilogarithmic) plot can be approximated by a power function as indicated by the data. Recalling that the QS formalism applies to identical particles, it follows that BEC could be at the origin of the so-called intermittency e!ect. This point of view was corroborated subsequently also in [195,196] and was con"rmed experimentally by the observation that the `intermittencya e!ect is strongly enhanced when identical particles are considered70 and/or when studied in more than one dimension.71 For a more recent very clear con"rmation of this point of view refer to the studies by Tannenbaum [197]. Further developments related to `intermittencya will be discussed in Section 6.3. 6.1.2. Rapidity dependence of MD in the non-stationary case The assumption of translational invariance in rapidity permitted to apply the quantum optical formalism, in which time has the analogous property, to MD and led to a simple interpretation of the observed broadening of the MD with the decrease of the width of the rapidity window in high-energy reactions. However stationarity in rapidity is expected to hold only in the central region (and only at high energies). Indeed experimental data on proton(antiproton)}proton
69 This scale is the correlation length m of Eqs. (2.24) or (2.25). 70 The UA5 data refer to a mixture of equal numbers of positive and negative particles; this dilutes the BEC e!ect. 71 This last observation also supports the idea that BEC is the determining factor in intermittency because the integration over transverse momentum implied by a one-dimensional y investigation diminishes the BEC.
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Fig. 19. Normalised factorial moments U of order q in "nite (pseudo)rapidity windows of width dy around y "0, q CMS plotted against dg (from Ref. [194]).
collisions in the energy range 52(Js(540 GeV show that if one considers shifted rapidity bins along the rapidity axis the MD in these bins depend on the position of the bin: in the central region the MD is broader and can be described by a negative binomial distribution while in the fragmentation region it is narrower and can be described by a Poisson MD [198]. The mean multiplicity in the central region increases faster (approximately like s1@4) with the energy than in the fragmentation region. This was interpreted in [198] as possible evidence for the existence of two sources, one of chaotic nature localised in the central region and another coherent in the fragmentation region. This interpretation is in line with the folklore that gluons which interact stronger (than quarks) form a central blob which may be equilibrated, while the fragmentation region is populated by throughgoing quarks associated with the leading particles.72 A few years later the NA35-collaboration [200] measured BEC in 16O}Au reactions at 200 GeV/nucleon in a relatively broad y region and found evidence for a larger and more chaotic source in the central rapidity region, and a smaller and more coherent source in the fragmentation region. It was then natural to correlate (see Ref. [28]) the two observations, i.e. that referring to MD
72 For a microscopic interpretation of this e!ect in terms of a partonic stochastic model, see Ref. [199].
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and that referring to BEC. Taken together the credibility of the conjecture made in Ref. [198] is strongly enhanced, just because we face here di!erent physical observables and di!erent experiments, each with its own speci"c corrections and biases. Moreover it was pointed out in [28] that experiment [200] did not necessarily imply that the two sources were independent, but could also be interpreted as due to a single partially coherent source. Indeed consider in a simpli"ed approach as used, e.g. in quantum optics a superposition n of two "elds one coherent denoted by n and another chaotic denoted by n so that # #) n"n (k(1))#n (k(2)) and Q "k(1)!k(2). Assuming boost invariance the correlator depends # M #) M M M M only on k and we have for the second-order correlation function M C "1#2p(1!p)e~Q2M R2M @2#p2e~Q2M R2M , (6.5) 2 where the transverse `radiusa R plays the role of the correlation length and M p"SDn TD2T/(Dn D2#SDn TD2T) . #) # #) Assume now that the chaoticity p is rapidity dependent so that in one rapidity region, denoted by (A), p(A)+1. In that region then the third term in Eq. (6.5) dominates, i.e. C +1#p2(A)e~Q2M R2M (A) . (6.6) 2 In the parametrisation used in [200], according to which we have two independent sources, this suggests that the e!ective radius of source A is R (A). Conversely, for the more coherent region M denoted by B, p(B) is small and C reads 2 C +1#2p(B)(1!p(B))e~Q2M R2M (B)@2 (6.7) 2 with an e!ective radius R (B)/J2(R (A). M M This corresponds qualitatively to the observations made in Ref. [200]. Unfortunately, these observations have not yet been con"rmed by another, independent experiment so that the reader should view these considerations with prudence.73 In any case they prove the usefulness of a global analysis which incorporates both BEC and MD. On the other hand, a dedicated simultaneous investigation of the rapidity dependence of these two observables appears very desirable. 6.1.3. Energy dependence of MD and its implications for BEC; long-range yuctuations in BEC and MD This subject has been discussed recently in [202]. Besides the rapidity dependence, the dependence of MD on the centre of mass energy of the collision, Js, constitutes an important topic in the study of high-energy multiparticle production processes. This energy dependence is usually discussed in terms of the violation [191] of KNO scaling [203]. KNO scaling implies that the normalised moments SnmT/SnTm are constant as a function of s (for high energies, i.e. large SnT, these moments coincide with the normalised factorial moments). For charged particles it turned out that while KNO scaling is approximately satis"ed over the range of ISR energies (20 GeV4Js460 GeV), it is violated if one goes to SPS-Collider energies (200 GeV4 Js4900 GeV), i.e. one "nds a considerable increase of multiplicity #uctuations with increasing
73 For various caveats concerning the analysis of BEC and MD data see Refs. [200,191,198,28,60,201].
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energy. In the following, we will show how BEC can be used to understand the origin of this s-dependence. To do this, one needs to distinguish between long-range (dynamical) correlations (LRC) and short-range correlations (SRC).74 For identical bosons, one important type of SRC are BEC which re#ect quantum statistical interference. In addition, there exist dynamical SRC like "nal state interactions, which however are quite di$cult to be separated from BEC. Up to 1994 one usually assumed that LRC do not play an important part in BEC measurements (see however [195,196]) in the sense that for Q'1 GeV the two-particle correlation functions do not signi"cantly exceed unity. However in Ref. [202] evidence, based on observational data, was presented showing that this is not the case and that new and important information about LRC is contained in the BEC data obtained by the UA1-Minimum-Bias Collaboration [192]. In principle, the observed increase of multiplicity #uctuations with Js could be due to a change of the SRC as seen in BEC, i.e. of the chaoticity and radii/lifetimes. This possibility was discussed in [207] but could not be tested because of the lack of identical particle data for multiplicity distributions at Collider energies. At that time only the UA5 data [191] for multiplicity distributions of charged particles were available. Furthermore, up to this point, the e!ect of LRC had only been studied in terms of two-particle correlations as a function of rapidity di!erence, i.e. in one dimension. With the advent of the newly analysed UA1 data [192] for identical particles in three dimensions (essentially in Q2"!(k !k )2) this situation changed. The analysis of [202] led to the conclusion of the 1 2 existence in BEC data of long-range #uctuations in the momentum space density of secondaries and to the realisation that the increase with energy of multiplicity #uctuations is to a great extent due to an increase of the asymptotic values of the m-particle correlation functions C!4:.15., i.e. their m values in the limit of large momentum di!erences where BEC do not play a role.75 In Ref. [192], the UA1 collaboration presents the two-particle correlation of negatively charged secondaries as a function of the invariant momentum di!erence squared Q2"(k !k )2! 1 2 (E !E )2. The data (see Fig. 20) have two unusual features: (I) at large Q2 the correlation function 1 2 saturates above unity, and (II) at small Q2 it takes on values above 2. The higher-order correlation functions also exhibit property (I) [192].76 By comparing the asymptotic values of the correlation functions at large momentum di!erences C!4:.15. (m"2,2, 5), with the normalised factorial m moments, / ,Sn(n!1) ) 2 ) (n!m#1)T/SnTm in the momentum space region DyD43, k ' m M 0.15 GeV one "nds that the contribution of the BE interference peak to the moments is negligible for such large rapidity windows. Herefrom one concludes in [202] that (I) indicates the presence of LRC in the momentum space density of secondary particles and that it is quite plausible (see below) that (II) has to a great extent the same explanation. We sketch here the arguments of Ref. [202]. In general, LRC may be related to #uctuations in impact parameter or inelasticity, or #uctuations in the number of sources. In what follows, let us label these #uctuations by a parameter a.
74 The importance of making this distinction was pointed out, among other things, in [204,205]. In [206] rapidity correlations were measured for events at "xed multiplicity in order to get rid of the e!ect of LRC. 75 The e!ect of LRC on BEC was discussed in Ref. [208], where several models speci"c for nucleus}nucleus collisions were considered, but at that time no evidence for this e!ect could be found. 76 For higher-order correlations the equivalent of property (II) is C 'm! (see also below). The values of C (k,2, k) m m for m'2 are apparently not yet available.
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Fig. 20. Second-order correlation function for negative particles at Js"630 GeV (from Ref. [202]).
The m-particle Bose}Einstein correlation function at a "xed value of a is given by C (k ,2, k Da)"o (k ,2, k Da)/o (k Da) ) 2 ) o (k Da) , m 1 m m 1 m 1 1 1 m where ol (k ,2, kl Da) are the l-particle inclusive distributions. 1 The #uctuations in a are described by a probability distribution h(a) with
P
da h(a)"1 .
(6.8)
(6.9)
If the experiment does not select events at "xed a, the measured inclusive distributions are o (k ,2, k )"So (k ,2, k Da)T , m 1 m m 1 m where the symbol S2T denotes an average over the #uctuating parameter a, i.e.
P
SX(a)T, da h(a)X(a)
(6.10)
(6.11)
The m-particle correlation function at the intercept reads C (k,2, k)"m! SamT/SaTm , (6.12) m where the symbols S T refer to averaging with respect to h(a). At large momentum di!erences one has C (k ,2, k )PSamT/SaTm"C!4:.15. for Dk !k DPR (iOj, i, j"1,2, m) , (6.13) m 1 m m i j i.e., the m-particle correlation functions can have intercepts above m! and saturate at values above unity for large momentum di!erences.
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The most obvious candidate for the function h(a) is the inelasticity distribution which describes the event-to-event #uctuations of the inelasticity K. With the identi"cation a,Sn(K)T, where Sn(K)T is the mean multiplicity at inelasticity K, one obtains from the above considerations a "rst `experimentala information about this important physical quantity at Collider energies. Previous experimental information about this distribution was derived in [209] from the data of Ref. [210] at JsK17 GeV. The conclusions obtained in [202] about LRC are based among other things on the di!erent normalisations used in di!erent experiments. Some tests related to these conclusions are proposed: f Analysis of BEC at lower energies (NA22 range) as well as at Js"1800 GeV with the same normalisation as that used by the UA1 Collaboration. The values of C at Q2'1 GeV2 and 2 possibly also at very small Q2 obtained in this way should exceed those obtained with the "xed multiplicity normalisation in the same experiments. The following inequalities for C (Q2'1 (GeV)2), should be observed if this normalisation is used: 2 C (NA22)(C (UA1)(C (Tevatron) . 2 2 2 f Analysis of BEC at UA1 energies with the same normalisation as that used so far by the NA22 and Tevatron groups ("xed multiplicity). The enhancement of C at large Q (and possibly also at 2 small Q) observed so far, should disappear to a great extent. 6.2. Multiplicity dependence of Bose}Einstein correlations The operators for the "eld (intensity) and number of particles do not commute. This means that measurements of `ideala BEC can be performed only when no restriction on the multiplicity n, which #uctuates from event to event, is made. In practice, however, very often such restrictions are imposed, either because of technical reasons or because of theoretical prejudices. To the last category belong considerations imposed by the search for QGP in high-energy heavy ion reactions. Thus one expects that by selecting events with n5n , where n is in general an energy.*/ .*/ dependent quantity, one gets information about the interesting `centrala collisions. Another reason why multiplicity constraints are of practical importance for QGP experiments is the need to compare various QGP signals in a given event and at the same time determine for that event the radius, lifetime and chaoticity of the source, among other things in order to be able to estimate the energy density achieved in that event. This means that for QGP search it is interesting to perform interferometry measurements for single events, which of course have a given multiplicity. For these reasons the investigation of the multiplicity dependence of BEC constitutes an important enterprise which we shall address in the following section. 6.2.1. The quantum statistical formalism Correlation functions de"ned in quantum statistics and used in quantum optics refer to ensemble averages of intensities77 I of "elds n, where I(k , y)"Dn(k , y)D2 . M M 77 Herefrom the name `intensity interferometrya.
(6.14)
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The total multiplicity n of identical particles over a given phase space region is given by
P P
n" dk2 dyDn(k , y)D2 . M M
(6.15)
Both the "eld n(k , y) and the intensity I(k , y) are stochastic variables. Averaging over an M M appropriate ensemble, we get the mean total multiplicity
P P
SnT" dk2 dySDn(k , y)D2T . M M
(6.16)
In [211] the dependence of BEC within the QS formalism on the total multiplicity n and on n was investigated and it was found that the size of this e!ect is (especially at low SnT) .*/ surprisingly large and must not be ignored, as had been done before. Both the n constraint and .*/ the SnT constraint lead to a decrease of the correlation function C at "xed y "y !y , i.e. to an 2 g 1 2 antibunching e!ect. The last e!ect can be approximated, except for very large n and small y by g a simple analytical formula C(n) (y )+C (y )(n!1)/nf , (6.17) (2) g (2) g 2 where Cn denotes the correlation function at "xed n and f is the reduced factorial moment. 2 2 These results show that BEC parameters like radii, lifetimes, and chaoticity do depend on the particular experimental conditions under which the measurements are performed. It is worth mentioning that a multiplicity dependence of BEC was observed experimentally for p}p and a}a reactions at E "53 and 31 GeV, respectively already in [212]. There it was found that the transverse radius increases with the cm multiplicity of charged particles n . This e!ect was interpreted by Barshay [213] to be a consequence of the impact #)!3'%$ parameter dependence. The same e!ect was seen in heavy ion reactions [200] and got a similar interpretation in [176].78 The interpretation in terms of impact parameter dependence could be checked directly in heavy ion reactions since here the impact parameter can be determined on an event-by-event basis. Another mechanism for the increase of radius with multiplicity was proposed by Ryskin [214]. He pointed out that in high multiplicity events, which many authors associate with large transverse momenta of partons and thus with a regime where perturbative QCD applies, one expects that the size of the hadronization region should increase with the multiplicity like Jn.
A related topic is the dependence of BEC on the rapidity density d"*n/*y, which has also been observed experimentally in p6 }p reactions at the CERN SPS Collider [215] and at the Fermilab tevatron [216]. Using a parametrisation C "1#j exp(!R2Q2) , (6.18) 2 where Q is the invariant momentum transfer, it was found that R increased with d while j decreased with d. This last observation is compatible with the results from [211]; a more quantitative comparison would be possible only if, among other things, the data were parametrised in a way more consistent with quantum statistics. Thus the four momentum di!erence Q is not an ideal
78 The approach of [176] combines simpli"ed (1-D) hydrodynamics with a multiple scattering model, which also exploits the impact parameter dependence.
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variable for BEC (see Section 2.1.5) and the coherence e!ect has to be taken into account as outlined in Section 2.2 and not by the simple empirical j factor. 6.2.2. The wave-function formalism; `pasersa? The dependence on total multiplicity of BEC was investigated also within the wave-function formalism. In Section 2.1 where the GGLP theory was presented it was pointed out that the wave-function formalism may be useful for exclusive processes or for event generators. Indeed, in a "rst approximation, the wave function t of a system of n identical bosons, e.g. can be obtained n from the product of n single-particle wave functions t by symmetrisation. Then the calculation of 1 C(n) is in principle straightforward and follows the lines of GGLP. (2) However, when the multiplicities n become large (say n'20) the explicit symmetrisation of the wave-function formalism becomes di$cult. This leads Zajc [217] to use numerical Monte Carlo techniques for estimating n particle symmetrised probabilities, which he then applied to calculate two-particle BEC. He was thus able to study the question of the dependence of BEC parameters on the multiplicity. For an application of this approach to Bevalac heavy ion reactions, see [218]. Using as input a second-order BEC function parametrised in the form (6.18) in Ref. [217] it was found (and we have seen above that this was con"rmed in [211],) that the `incoherencea parameter j decreased with increasing n.79 This of course does not mean that events with higher pion multiplicities are denser and more coherent. On the contrary, Ref. [217] concluded that the above results have to be used in order to eliminate the bias introduced by this e!ect into experimental observations.80 The authors of [219,220] however did not share this opinion. Ref. [219] went even so far as to deduce the possible existence of pionic lasers from considerations of this type. Ref. [219] starts by proposing an algorithm for symmetrising the wave functions which presents the advantages that it reduces the computing time very much when using numerical techniques, which is applicable also for Wigner-type source functions and not only plane wave functions, and which for Gaussian sources provides even analytical results. Subsequently in Ref. [221] wave packets were symmetrised and in special cases the matrix density at "xed and arbitrary n was derived in analytical form. This algorithm was then applied to calculate the in#uence of symmetrisation on BEC and multiplicity distributions. As in [217] it is found in [219] that the symmetrisation produces an e!ective decrease of the radius of the source, a broadening of the multiplicity distribution P(n) and an increase of the mean multiplicity as compared to the non-symmetrised case. What is new in [219] is (besides the algorithm) mainly the meaning the author attributes to these results. In a concrete example Ref. [219] considers a non-relativistic source distribution S in the absence of symmetrisation e!ects: S(k, x)"[1/(2pR2m¹)3@2]exp(!(k /¹)!(x2/2R2))d(x ) 0 0
(6.19)
79 In [217] the clumping in phase space due to Bose symmetry was also illustrated. 80 The same interpretation of the multiplicity dependence of BEC was given in [211]. In this reference the nature of the `fakea coherence induced by "xing the multiplicity is even clearer, as one studies there explicitly coherence in a consistent quantum statistical formalism.
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where k /¹"k2/2D2 . (6.20) 0 Here ¹ is an e!ective temperature, R an e!ective radius, m the pion mass and D a constant with dimensions of momentum. Let g and g be the number densities before and after symmetrisation, respectively. In terms of 0 S(k, x) we have
P
g " S(k, x) d4k d4x 0
(6.21)
and a corresponding expression for g with S replaced by the source function after symmetrisation. Then one "nds [219] that g increases with g and above a certain crtitical density g#3*5, g diverges. 0 0 This is interpreted in Ref. [219] as `passinga. The reader may be rightly puzzled by the fact that while g has a clear physical signi"cance the number density g and a fortiori its critical value have no physical signi"cance, because in nature 0 there does not exist a system of bosons the wave function of which is not symmetrised. Thus contrary to what is alluded to in Ref. [219], this paper does not address really the question how a condensate is reached. Indeed the physical factors which induce condensation are, for systems in (local) thermal and chemical equilibrium,81 pressure and temperature and the symmetrisation is contained automatically in the distribution function f"1/[exp[(E!k)/¹]!1]
(6.22)
in the term !1 in the denominator; E is the energy and k the chemical potential. To realise what is going on it is useful to observe that the increase of g can be achieved by 0 decreasing R and/or ¹. Thus g can be substituted by one or both of these two physical quantities. 0 Then the blow-up of the number density g can be thought of as occurring due to a decrease of ¹ and/or R. However this is nothing but the well-known Bose}Einstein condensation phenomenon. While from a purely mathematical point of view the condensation e!ect can be achieved also by starting with a non-symmetrised wave function and symmetrising it afterwards `by handa, the causal, i.e. physical relationship is di!erent: one starts with a bosonic, i.e. symmetrised system and obtains condensation by decreasing the temperature or by increasing the density of this bosonic system. Another confusing interpretation in [219] relates to the observation made also in [217] that the symmetrisation produces a broadening of the multiplicity distribution (MD). In particular, starting with a Poisson MD for the non-symmetrised wf one ends up after symmetrisation with a negative binomial. This is a simple consequence of Bose statistics, and must not be associated with the so-called pasing e!ect. In fact for true lasers the opposite e!ect takes place. Before `condensinga, i.e. below threshold their MD is in general broad and of negative binomial form corresponding to a chaotic (thermal) distribution while above threshold the laser condensate is produced and as such corresponds to a coherent state and therefore is characterised by a Poisson MD. Last but not least
81 For lasers the determining dynamical factor is among other things the inversion.
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the fact that this broadening increases with n is not, as suggested in [219,220], due to the approach to `lasing criticalitya, but simply to the fact that the larger the n, the larger the number of independent emitters is and the better the central limit theorem applies. This theorem states (see Section 2.2) that the "eld produced by a large number of independent sources is chaotic. Finally a terminological remark appears necessary here. We believe that names like `pasera or pionic laser used in the papers quoted above are unjusti"ed and misleading. The only characteristic which the systems considered in these papers possibly share with lasers is the condensate property, i.e. the bunching of particles in a given (momentum) state. However lasers are much more than just condensates; one of the main properties of lasers which distinguishes them from other condensates is the directionality, a problem which is not even mentioned in the `pasera literature.82,83 6.3. The invariant Q variable in the space}time approach: higher-order correlations; `intermittencya in BEC ? The issue of apparent power-like rapidity dependence of moments of MD was discussed in Section 6.1.1 where it was pointed out that this dependence could in principle be understood within the QS formalism without invoking the idea of intermittency. However this was not the end of the story because: (1) it was observed [192] that the power-like behaviour extends also to BEC data (in the invariant variable Q). This was surprising because up to that moment BEC data could usually be "tted by a Gaussian or exponential function, albeit these data did not extend to such small Q values as those measured in [192]; 2) Bialas [227] (see also [228,229]) proposed that the source itself has no "xed size, but is #uctuating from event to event with a power distribution of sizes. Since the measurements made in [192] were in the invariant variable Q rather than *y, one had to understand whether the QS approach which implies "xed scales does not lead to a similar behaviour in Q. Refs. [230,231] addressed this question and proved that, indeed, by starting from a space}time correlator with a "xed correlation length and a source distribution with a "xed radius, one gets after integrations over the unobserved variables a correlation function which is power like in a limited Q range. In the following, we shall sketch how this happens. The two-particle Bose}Einstein correlation function is de"ned as C (k , k )"o (k , k )/[o (k ) ) o (k )] 2 1 2 2 1 2 1 1 1 2
(6.23)
where o (k) and o (k , k ) are the one- and two-particle particle inclusive spectra, respectively. 1 2 1 2 The two-particle correlation function projected on Q2 is C (Q2)"1#(I (Q2)/I (Q2)) 2 2 11
(6.24)
82 For a model of directional coherence, not necessarily related to pion condensates, see [222]; experimental hints of this e!ect have possibly been seen in [223]. 83 For another investigation of the e!ect of symmetrization on the single inclusive cross section cf. [224,225]. In [225] second-order correlation functions are also considered (see also Ref. [226] for this topic).
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Fig. 21. C (Q2) for the static source model (dashed line) and for the expanding source model (solid line) compared to the 2 UA1 data. The dotted line shows a power-law "t (from Ref. [230]).
with the integrals
P P P P
I (Q2)" du du d[Q2#(kk !kk )2]o (k )o (k ) , 11 1 2 1 2 1 1 1 2
(6.25)
I (Q2)" du du d[Q2#(kk !kk )2](o (k , k )!o (k )o (k )) , 2 1 2 1 2 2 1 2 1 1 1 2 where du,d3k/(2n)32E is the invariant phase space volume element. Two types of sources, a static one and an expanding one, were considered (cf. Section 4.8). The parameters employed are physically meaningful quantities in the sense that they give the lifetime, radii and correlation lengths of the source. This is not the case for the ad hoc parametrisation of C (Q2), e.g., with 2 a Gaussian C (Q2)"1#j e~R2Q Q2 . (6.26) 2 Q To illustrate the behaviour of the correlation function as a function of Q2, one applied the formalism to describe UA1 data in the phase space region DyD43.0 and k 5150 MeV. In M [230,231] C (Q2) was calculated for the static and for the expanding source by Monte Carlo 2 integration (for the static source, approximate analytical results could be obtained only for DyD'1.5, where they agree with the numerical results [231]). Fig. 21 shows the results of "ts to the UA1 data84 for the static source (dashed line) and for the expanding source (solid line), which were 84 The C (Q2) data of [192] were normalised so that at large Q2, C (Q2)+1. An explanation for the experimental 2 2 observation of [192] that C (Q2) exceeds by a multiplicative factor of &1.3 both the upper and lower `conventionala 2 limits of 2 and 1, respectively, is discussed in Ref. [202].
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obtained under the assumption of a purely chaotic source, p "1 (the data are consistent with an 0 amount of 410% coherence,85 but the sensitivity to p was not su$cient to further constrain the 0 degree of coherence within these limits). For comparison, the result of a power law "t (dotted line) as suggested in Ref. [192] C (Q2)"a#b ) (Q2/1 GeV2)~( (6.27) 2 is also plotted. One "nds that the data can be well described both with the static and the expanding source model with reasonable values for the radii, lifetime and correlation length. The results of above show that the power-like behaviour of C (the same holds for higher order 2 correlations) can be reproduced by assuming a conventional space}time source with "xed parameters, i.e., without invoking `intermittencya.86 This conclusion of [230] is strengthened by an explicit consideration of resonances in [231]. The advantages of the QS formalism as compared with the wave-function formalism emerge clearly also in the problem of higher-order correlations. In the QS formalism higher-order correlations are treated on the same footing as lower-order ones and emerge just as consequences of the form of the density matrix. Therefore questions found sometimes in the literature like `what is the in#uence of higher-order correlations on the lower onea do not even arise in QS and in fact do not make sense. We emphasised in Section 2.2.1 the importance of higher-order correlations for the phenomenological determination of the form of the correlation function and this applies in particular when a single variable like Q is used instead of the six independent degrees of freedom inherent in the correlator. For this reason the space}time integration at "xed Q has been used recently [35] also in the study of higher-order correlations and applied to the NA22 data [233]. It was found among other things that an expanding source with "xed parameters as de"ned in Section 4 can account for the data up to and including the fourth order, con"rming in a "rst approximation the Gaussian form of the density matrix. The fact that previous attempts in this direction like [233,34] met with di$culties may be due to the fact that in these two experimental studies the QO formalism in momentum space was used and possibly also because no simultaneous "t of all orders of correlations was performed, as was the case in [35].
7. Critical discussion and outlook For historical reasons related to the fact that the GGLP e!ect was observed for the "rst time in annihilation at rest when (almost) exclusive reactions were studied, the theory of BEC was initially based on the wave-function formalism. This formalism is not appropriate for inclusive reactions in high-energy physics, among other things because it yields a correlation function which depends
85 The older UA-1 data [31] were limited to larger Q2 values. Furthermore, the quantum statistical interpretation [32] of these data is di!erent from that in [230], which is based on space}time concepts. This explains the di!erent values of chaoticity obtained in [230,231], on the one hand and in Refs. [31,32] on the other hand. 86 A similar point of view is expressed in [232] for the particular case of bremsstrahlung photons.
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only on the momentum di!erence q and not also on the sum of momenta, it does not take into account isospin, and cannot treat adequately coherence. This last property is essential as it leads to one of the most important applications of BEC, i.e. to condensates. Furthermore, this property strongly a!ects another essential application of BEC, namely the determination of sizes, lifetimes, correlation lengths and correlation times of sources. The ad hoc parametrisation of the correlation function under the form (2.8) where j is supposed to take into account (in)coherence is unsatisfactory. An improvement based on an analogy with quantum optics leads to an additional term (with the same number of parameters). A more complete and more correct treatment of BEC is provided by the space}time formalism of classical currents. Almost all studies of BEC assume a Gaussian density matrix. Possible deviations from this form could and should be looked for by studying higher-order correlations. The classical current approach is based on an exact solution of the equations of quantum "eld theory and it can be considered at present the most advanced and complete description of BEC. It introduces a primordial correlator of currents which is characterised by "nite correlation lengths and correlation times. The geometry of the source is an independent property of the system and it is here that the traditional radii and lifetimes enter. The form of the geometry and of the correlator is not given by the theory and it is up to experiment to determine these. For a Gaussian density matrix the space}time approach within the classical current formalism leads to a minimum of 10 independent parameters; these include geometrical and dynamical scales as well as the chaoticity. Phenomenologically, these scales can be separated only by considering simultaneously single and double inclusive cross sections. Experimentally, this separation as well as the determination of all parameters has not yet been performed and constitutes an important task for the future. This should be done not only for particle reactions but also for heavy ion reactions, as an alternative to the quasi-hydrodynamical approach which cannot provide this separation. It would also be desirable to extend the parametrisation for an expanding source by dropping the assumption of boost invariance. Besides the conventional ## or !! pion correlations there exist also #! correlations, which become important for sources of small lifetimes. These `surprisinga "eld theoretical e!ects represent squeezed states, which unlike what happens in optics, appear in particle physics `for freea. These e!ects can be used to investigate the di!erence between classical and quantum currents. Their detection constitutes one of the most important challenges for future experiments. BEC are in#uenced by "nal state interactions. Coulomb "nal state interactions do not play a major part except at very small q. These small values have apparently not yet been reached in experiment and it is questionable whether they will be reached in the foreseeable future. In heavy ion reactions this is due to the large values of radii which multiply q and even for typical scales of 1 fm the present resolution of detectors is not su$cient to make this e!ect very important. Nevertheless, since Coulomb corrections have been studied so far only within the wave-function formalism and usually by applying the SchroK dinger equation, it would be desirable to extend this study by considering the Klein}Gordon equation which is more appropriate for mesons. Even more interesting would be to study Coulomb corrections within the classical current formalism. Resonances play an important part in "nal state interactions and progress has been achieved in their understanding, in particular in heavy ion reactions, where their in#uence has been investigated using solutions of the equations of hydrodynamics within the Wigner function formalism.
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BEC have been investigated in e`}e~, hadron}hadron and heavy ion reactions; however a systematic comparison of results using the same parametrisation and normalisation is yet to be done.87 Correlations are intimately related to multiplicity distributions which can serve as complementary tools in the determination of the parameters of sources. Therefore a systematic investigation of BEC and multiplicity distributions in the same phase space region is desirable. How useful this can be has been shown by proving in this way the in#uence of long-range correlations on BEC. BEC can be useful in heavy ion reactions and for the search of quark}gluon plasma if in particular one of the two conditions is satis"ed: (A) the investigation is based on full-#edged hydrodynamics, implying the solution of the equations of hydrodynamics with explicit consideration of the equation of state. (B) the investigation is based on the classical current space}time approach. Case (A) has the advantage that the dependence of the equation of state on the phase transition may be re#ected also in single inclusive cross sections and in the BEC. (So far, however, the sensitivity of BEC on the equation of state has not yet been proven with present data [234].) It has however the disadvantage that its applicability is restricted because it is based on the Wigner function, which is a particular case of the classical current formalism, and which is useful only for small q and not too strong correlations between momenta k and coordinates x. Case (B) has the disadvantage that there is no contact with the equation of state. However it has the advantage it is not restricted to small q and weak correlations between k and x. Unfortunately, many of the theoretical papers on BEC in heavy ion reactions do not satisfy either condition (A) or condition (B). This is the case with most of the quasi-hydrodynamical papers which use a parametrisation of the source function based on qualitative hydrodynamical considerations without the use of an equation of state and without solving the equations of hydrodynamics. This quasi-hydrodynamical approach has the disadvantages of (A) and (B) but none of their advantages.88 There are many yet unsolved problems in the investigation of BEC, some of them of theoretical, but most of them of experimental nature (see also Ref. [235]). While an analysis of the experimental BEC deserves a special review, some of the obvious reasons for this unsatisfactory experimental situation are: (1) Most of the BEC experiments performed so far use inadequate detectors, because they are not dedicated experiments but rather by-products of experiments planned for other purposes. What is needed among other things is track-by-track detection and improved identi"cation of particles. (2) Insu$cient statistics. An improvement of statistics especially at small q by at least one order of magnitude is necessary to address some of the problems enumerated above. 87 At present even for the same type of reaction di!erent normalisations are frequently used and this can lead to apparently di!erent results, as exempli"ed in the case of NA22 and UA1 data. Tests for a better understanding and elimination of these discrepancies have been proposed (see Section 6.1.3). 88 Quasi-hydrodynamics as compared with hydrodynamics has the supplementary disadvantage that it does not allow a separation between geometrical radii and correlation lengths. Moreover it does not have even the excuse of simplicity, since the number of free parameters in the quasi-hydrodynamical approach is as large as that in the classical current space}time approach.
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(3) Incorrect or incomplete parametrisations of the correlation functions. Very often and in part because of (2) not all six independent variables of C are measured, but projection of these. Very 2 popular among these projections is the relativistically invariant variable Q. This is not a good variable for BEC studies, because among other things it mixes the space and time variables in an uncontrollable way. Furthermore, in most parametrisations, coherence is not (or inadequately) considered. (4) Inadequate normalisations. Practically, all BEC experiments use a normalisation procedure of the correlation function which does not correspond to its de"nition. This de"nition relates the double (or higher order) inclusive cross section to the product of single inclusive cross sections and not to an `uncorrelated backgrounda. The solution of the problems mentioned above will make of boson interferometry what it is supposed to be: a reliable method for the determination of sizes, lifetimes, correlation lengths, and coherence of sources in subatomic physics. A more pedagogical presentation of the theory of Bose}Einstein correlations, which discusses also its quantum optical context, including a comparison between the HBT and the GGLP e!ects and between photon and hadron intensity interferometry can be found in the book by the author [5]. Some of the most representative theoretical and experimental papers on BEC and which are frequently quoted within the present review have been reprinted in a single volume in Ref. [236].
Acknowledgements I am indebted to D. Strottman for a careful reading of the manuscript and for many helpful comments.
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A. AchuH carro, T. Vachaspati / Physics Reports 327 (2000) 347}426
SEMILOCAL AND ELECTROWEAK STRINGS
Ana ACHUD CARRO!,", Tanmay VACHASPATI# !Department of Theoretical Physics, UPV-EHU, 48080 Bilbao, Spain "Institute for Theoretical Physics, University of Groningen, The Netherlands #Physics Department, Case Western Reserve University, Cleveland, OH 44106, USA
AMSTERDAM } LAUSANNE } NEW YORK } OXFORD } SHANNON } TOKYO
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Semilocal and electroweak strings Ana AchuH carro!,", Tanmay Vachaspati#,* !Department of Theoretical Physics, UPV-EHU, 48080 Bilbao, Spain "Institute for Theoretical Physics, University of Groningen, The Netherlands #Physics Department, Case Western Reserve University, Cleveland, OH 44106, USA Received August 1999; editor: J. Bagger Contents 1. Introduction 1.1. The Glashow}Salam}Weinberg model 2. Review of Nielsen}Olesen topological strings 2.1. The Abelian Higgs model 2.2. Nielsen}Olesen vortices 2.3. Stability of Nielsen}Olesen vortices 3. Semilocal strings 3.1. The model 3.2. Stability 3.3. Semilocal string interactions 3.4. Dynamics of string ends 3.5. Numerical simulations of semilocal string networks 3.6. Generalisations and "nal comments 4. Electroweak strings 4.1. The Z string 5. The zoo of electroweak defects 5.1. Electroweak monopoles 5.2. Electroweak dyons 5.3. Embedded defects and W-strings 6. Electroweak strings in extensions of the GSW model 6.1. Two Higgs model 6.2. Adjoint Higgs model
350 353 356 357 358 361 363 363 366 372 374 374 379 381 381 383 383 385 387 389 389 390
7. Stability of electroweak strings 7.1. Heuristic stability analysis 7.2. Detailed stability analysis 7.3. Z-string stability continued 7.4. Semiclassical stability 8. Superconductivity of electroweak strings 8.1. Fermion zero modes on the Z-string 8.2. Stability of Z-string with fermion zero modes 8.3. Scattering of fermions o! electroweak strings 9. Electroweak strings and baryon number 9.1. Chern}Simons or topological charge 9.2. Baryonic charge in fermions 9.3. Dumbells 9.4. Possible cosmological applications 10. Electroweak strings and the sphaleron 10.1. Content of the sphaleron 10.2. From Z-strings to the sphaleron 11. The 3He analogy 11.1. Lightning review of 3He 11.2. Z-string analog in 3He 12. Concluding remarks and open problems Acknowledgements References
* Corresponding author. E-mail address: [email protected] (T. Vachaspati) 0370-1573/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 1 5 7 3 ( 9 9 ) 0 0 1 0 3 - 9
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Abstract We review a class of non-topological defects in the standard electroweak model, and their implications. Starting with the semilocal string, which provides a counterexample to many well-known properties of topological vortices, we discuss electroweak strings and their stability with and without external in#uences such as magnetic "elds. Other known properties of electroweak strings and monopoles are described in some detail and their potential relevance to future particle accelerator experiments and to baryon number violating processes is considered. We also review recent progress on the cosmology of electroweak defects and the connection with super#uid helium, where some of the e!ects discussed here could possibly be tested. ( 2000 Elsevier Science B.V. All rights reserved. PACS: 11.10.!z; 11.27.#d Keywords: Strings; Electroweak; Semilocal; Sphaleron
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1. Introduction In a classic paper from 1977 [102], a decade after the S;(2) ];(1) model of electroweak L Y interactions had been proposed [52], Nambu made the observation that, while the Glashow}Salam}Weinberg (GSW) model does not admit isolated, regular magnetic monopoles, there could be monopole}antimonopole pairs joined by short segments of a vortex carrying Z-magnetic "eld (a Z-string). The monopole and antimonopole would tend to annihilate but, he argued, longitudinal collapse could be stopped by rotation. He dubbed these con"gurations dumbells1 and estimated their mass at a few TeV. A number of papers advocating other, related, soliton-type solutions2 in the same energy range followed [41], but the lack of topological stability led to the idea "nally being abandoned during the 1980s. Several years later, and completely independently, it was observed that the coexistence of global and gauge symmetries can lead to stable non-topological strings called `semilocal stringsa [127] in the sin2 h "1 limit of the GSW model that Nambu had considered. Shortly afterwards it was 8 proved that Z-strings were stable near this limit [123], and the whole subject made a comeback. This report is a review of the current status of research on electroweak strings. Apart from the possibility that electroweak strings may be the "rst solitons to be observed in the standard model, there are two interesting consequences of the study of electroweak and semilocal strings. One is the unexpected connection with baryon number and sphalerons. The other is a deeper understanding of the connection between the topology of the vacuum manifold (the set of ground states of a classical "eld theory) and the existence of stable non-dissipative con"gurations, in particular when global and local symmetries are involved simultaneously. In these pages we assume a level of familiarity with the general theory and basic properties of topological defects, in particular with the homotopy classi"cation. There are some excellent reviews on this subject in the literature to which we refer the reader [53,32,116]. On the other hand, electroweak and semilocal strings are non-topological defects, and this forces us to take a slightly di!erent point of view from most of the existing literature. Emphasis on stability properties is mandatory, since one cannot be sure from the start whether these defects will actually form. With very few exceptions, this requires an analysis on a case by case basis. Following the discussion in [33], one should begin with the de"nition of dissipative con"gurations. Consider a classical "eld theory with energy density ¹ 50 such that ¹ "0 everywhere 00 00 for the ground states (or `vacuaa) of the theory. A solution of a classical "eld theory is said to be dissipative if lim max ¹ (x, t)"0 . 00 t?= x
(1)
We will consider theories with spontaneous symmetry breaking from a Lie group G (which we assume to be "nite-dimensional and compact) to a subgroup H; the space V of ground states of the 1 Or monopolia, after analogous con"gurations in super#uid helium [95]. 2 One example, outside the scope of the present review, are so-called vorticons, proposed by Huang and Tipton, which are closed loops of string with one quantum of Z boson trapped inside.
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theory is usually called the vacuum manifold and, in the absence of accidental degeneracy, is given by V"G/H. The classi"cation of topological defects is based on the homotopy properties of the vacuum manifold. If the vacuum manifold contains non-contractible n-spheres then "eld con"gurations in n#1 spatial dimensions whose asymptotic values as rPR `wrap arounda those spheres are necessarily non-dissipative, since continuity of the scalar "eld guarantees that, at all times, at least in one point in space the scalar potential (and thus the energy) will be non-zero. The region in space where energy is localized is referred to as a topological defect. Field con"gurations whose asymptotic values are in the same homotopy class are said to be in the same topological sector or to have the same winding number. In three spatial dimensions, it is customary to use the names monopole, string3 and domain wall to refer to defects that are point-like, one- or two-dimensional, respectively. Thus, one can have topological domain walls only if n (V)O1, topological strings only if n (V)O1 and topological 0 1 monopoles only if n (V)O1. Besides, defects in di!erent topological sectors cannot be deformed 2 into each other without introducing singularities or supplying an in"nite amount of energy. This is the origin of the homotopy classi"cation of topological defects. We should point out that the topological classi"cation of textures based on n (V) has a very di!erent character, and will not 3 concern us here; in particular, con"gurations from di!erent topological sectors can be continuously deformed into each other with a "nite cost in energy. In general, textures unwind until they reach the vacuum sector and therefore they are dissipative. It is well known, although not always su$ciently stressed, that the precise relationship between the topology of the vacuum and the existence of stable defects is subtle. First of all, note that a trivial topology of the vacuum manifold does not imply the non-existence of stable defects. Secondly, we have said that a non-trivial homotopy of the vacuum manifold can result in non-dissipative solutions but, in general, these solutions need not be time independent nor stable to small perturbations. One exception is the "eld theory of a single scalar "eld in 1#1 dimensions, where a disconnected vacuum manifold (i.e. one with n (V)O1) is su$cient to prove the existence 0 of time independent, classically stable `kinka solutions [55,33]. But this is not the norm. The O(3) model, for instance, has topological global monopoles [16] which are time independent, but they are unstable to angular collapse even in the lowest non-trivial winding sector [54]. It turns out that the situation is particularly subtle in theories where there are global and gauge symmetries involved simultaneously. The prototype example is the semilocal string, described in Section 3. In the semilocal string model, the classical dynamics is governed by a single parameter b"m2/m2 that measures the square of the ratio of the scalar mass, m , to the vector mass, m (this is 4 7 4 7 the same parameter that distinguishes type I and type II superconductors). It turns out that: f When b'1 the semilocal model provides a counterexample to the widespread belief that quantization of magnetic #ux is tantamount to its localization, i.e., con"nement. The vector boson is massive and we expect this to result in con"nement of magnetic #ux to regions of width
3 The names cosmic string and vortex are also common. Usually, `vortexa refers to the con"guration in two spatial dimensions, and `stringa to the corresponding con"guration in three spatial dimensions; the adjective `cosmica helps to distinguish them from the so-called fundamental strings or superstrings.
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given by the inverse vector mass. However, this is not the case! As pointed out by Hindmarsh [59] and Preskill [109], this is a system where magnetic #ux is topologically conserved and quantized, and there is a "nite energy gap between the non-zero #ux sectors and the vacuum, and yet there are no stable vortices. f When b(1 strings are stable4 even though the vacuum manifold is simply connected, p (V)"1. Semilocal vortices with b(1 are a remarkable example of a non-topological defect 1 which is stable both perturbatively and to semiclassical tunnelling into the vacuum [110]. As a result, when the global symmetries of a semilocal model are gauged, dynamically stable non-topological solutions can still exist for certain ranges of parameters very close to stable semilocal limits. In the case of the standard electroweak model, for instance, strings are (classically) stable only when sin2 h +1 and the mass of the Higgs is smaller than the mass of the Z boson. 8 We begin with a description of the Glashow}Salam}Weinberg model, in order to set our notation and conventions, and a brief discussion of topological vortices (cosmic strings). It will be su$cient for our purposes to review cosmic strings in the Abelian Higgs model, with a special emphasis on those aspects that will be relevant to electroweak and semilocal strings. We should point out that these vortices were "rst considered in condensed matter by Abrikosov [2] in the non-relativistic case, in connection with type II superconductors. Nielsen and Olesen were the "rst to consider them in the context of relativistic "eld theory, so we will follow a standard convention in high energy physics and refer to them as Nielsen}Olesen strings [103]. Sections 3}5 are dedicated to semilocal and electroweak strings, and other embedded defects in the standard GSW model. Electroweak strings in extensions of the GSW model are discussed in Section 6. In Section 7 the stability of straight, in"nitely long electroweak strings is analysed in detail (in the absence of fermions). Sections 8 and 10 investigate fermionic superconductivity on the string, the e!ect of fermions on the string stability, and the scattering of fermions o! electroweak strings. The surprising connection between strings and baryon number, and their relation to sphalerons, is described in Sections 9 and 10. Here we also discuss the possibility of string formation in particle accelerators (in the form of dumbells, as was suggested by Nambu in the 1970s) and in the early universe. Finally, Section 11 describes a condensed matter analog of electroweak strings in super#uid helium which may be used to test our ideas on vortex formation, fermion scattering and baryogenesis. A few comments are in order: f Unless otherwise stated we take space time to be #at, (3#1)-dimensional Minkowski space; the gravitational properties of embedded strings are expected to be similar to those of Nielsen}Olesen strings [51] and will not be considered here. A limited discussion of possible cosmological implications can be found in Sections 3.5 and 9.4. f We concentrate on regular defects in the standard model of electroweak interactions. Certain extensions of the Glashow}Salam}Weinberg model are brie#y considered in Section 6 but 4 We want to stress that, contrary to what is often stated in the literature, the semilocal string with b(1 is absolutely stable, and not just metastable.
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f f
f f
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otherwise they are outside the scope of this review; the same is true of singular solutions. In particular, we do not discuss isolated monopoles in the GSW model [51,31], which are necessarily singular. No family mixing e!ects are discussed in this review and we also ignore S;(3) colour c interactions, even though their physical e!ects are expected to be very interesting, in particular in connection with baryon production by strings (see Section 9). Our conventions are the following: space time has signature (#,!,!,!). Planck's constant and the speed of light are set to one, +"c"1. The notation (x) is shorthand for all space-time coordinates (x0, xi), i"1, 2, 3; whenever the x-coordinate is meant, it will be stated explicitly. We also use the notation (t, x). Complex conjugation and hermitian conjugation are both indicated with the same symbol, (s), but it should be clear from the context which one is meant. For fermions, tM "tsc0, as usual. Transposition is indicated with the symbol ( T). One "nal word of caution: a gauge "eld is a Lie Algebra valued one-form A"A dxk" k Aa ¹adxk, but it is also customary to write it as a vector. In cylindrical coordinates (t, o, u, z), k A"A dt#A do#A du#A dz is often written A"A tK #A o( #(A /o)u( #A z( , In spherit o r z t o r z cal coordinates, (t, r, h, u), A"A dt#A dr#A dh#A du is also written A"A tK #A r( # t r h r t r (A /r)hK #(A /r sin h)u( . We use both notations throughout. h r
1.1. The Glashow}Salam}Weinberg model In this section we set out our conventions, which mostly follow those of [30]. The standard (GSW) model of electroweak interactions is described by the Lagrangian ¸"¸ # + ¸ #¸ . (2) b f fm &!.*-*%4 The "rst term describes the bosonic sector, comprising a neutral scalar /0, a charged scalar /`, a massless photon A , and three massive vector bosons, two of them charged (=B) and the k k neutral Z . k The last two terms describe the dynamics of the fermionic sector, which consists of the three families of quarks and leptons
ABABAB l e e
l k k
l q q
u
c
t
d
s
b
.
(3)
1.1.1. The bosonic sector The bosonic sector describes an S;(2) ];(1) invariant theory with a scalar "eld U in the L Y fundamental representation of S;(2) . It is described by the Lagrangian L ¸ "¸ #¸ #¸U !<(U) (4) b W Y
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with ¸ "!1=a =kla, a"1, 2, 3 , W 4 kl (5) ¸ "!1> >kl , Y 4 kl where =a "R =a !R =a #geabc=b =c and > "R > !R > are the "eld strengths for the kl k l l l k l kl k l l k S;(2) and ;(1) gauge "elds, respectively. Summation over repeated S;(2) indices is underL Y L stood, and there is no need to distinguish between upper and lower ones: e123"1. Also,
KA
BK
2 ig ig@ ¸U "DD UD2, R ! qa=a ! > U , j j j j 2 2
(6)
<(U)"j(UsU!g2/2)2 ,
(7)
where qa are the Pauli matrices,
A B
q1"
0 1
,
1 0
A
q2"
0 !i i
0
B
,
A
q3"
1
0
B
0 !1
,
(8)
from which one constructs the weak isospin generators ¹a"1qa satisfying [¹a, ¹b]"ieabc¹c. 2 The classical "eld equations of motion for the bosonic sector of the standard model of the electroweak interactions are (ignoring fermions)
A
B
g2 DkD U#2j UsU! U"0 , k 2
(9)
i D =kla"jka " g[UsqaDkU!(DkU)sqaU] , l W 2
(10)
i R >kl"jk " g@[UsDkU!(DkU)sU] , l Y 2
(11)
where D =kla"R =kla#geabc=b =klc. l l l When the Higgs "eld U acquires a vacuum expectation value (VEV), the symmetry breaks from S;(2) ];(1) to ;(1) . In particle physics it is standard practice to work in unitary gauge L Y %. and take the VEV of the Higgs to be SUTT"g(0, 1)/J2. In that case the unbroken ;(1) subgroup, which describes electromagnetism, is generated by the charge operator
A B
1 0 > Q,¹3# " 2 0 0
(12)
and the two components of the Higgs doublet are charge eigenstates
A B
U"
/` /0
.
(13)
> is the hypercharge operator, which acts on the Higgs like the 2]2 identity matrix. Its eigenvalue on the various matter "elds can be read-o! from the covariant derivatives D "R !ig=a ¹a! k k k ig@> (>/2) which are listed explicitly in Eqs. (6) and (24)}(28). k
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In unitary gauge, the Z and A "elds are de"ned as Z ,cos h =3 !sin h > , A ,sin h =3#cos h > , (14) k 8 k 8 k k 8 k 8 k and =B,(=1Gi=2)/J2 are the = bosons. The weak mixing angle h is given by tan h ,g@/g; k k k 8 8 electric charge is e"g sin h cos h with g ,(g2#g@2)1@2. z 8 8 z However, unitary gauge is not the most convenient choice in the presence of topological defects, where it is often singular. Here we shall need a more general de"nition in terms of an arbitrary Higgs con"guration U(x): Z ,cos h na(x)=a !sin h > , A ,sin h na(x)=a #cos h > , k 8 k 8 k k 8 k 8 k where Us(x)qaU(x) na(x),! Us(x)U(x)
(15)
(16)
is a unit vector by virtue of the Fierz identity + (UsqaU)2"(UsU)2. In what follows, we omit a writing the x-dependence of na explicitly. Note that na is ill-de"ned when U"0, so in particular at the defect cores. The generators associated with the photon and the Z-boson are, respectively, Q"na¹a#>/2,
> ¹ "cos2 h na¹a!sin2 h "na¹a!sin2 h Q , 8 Z 8 82
(17)
while the generators associated with the (charged) = bosons are determined, up to a phase, by the conditions [¹`, ¹~]"na¹a"¹ #sin2 h Q, (¹`)s"¹~ . (18) Z 8 (note that if na"(0, 0, 1), as is the case in unitary gauge, one would take ¹B"(¹1$i¹2)/J2.) There are several di!erent choices for de"ning the electromagnetic "eld strength but, following Nambu, we choose [Q, ¹B]"$¹B,
A "sin h na=a #cos h > , (19) kl 8 kl 8 kl where =a and > are "eld strengths. The di!erent choices for the de"nition of the "eld strength kl kl agree in the region where D U"0 where D is the covariant derivative operator; in particular k k this is di!erent from the well known 't Hooft de"nition which is standard for monopoles [65]. (For a recent discussion of the various choices see, e.g. [63,62,121].) And the combination of S;(2) and ;(1) "eld strengths orthogonal to A is de"ned to be the Z "eld strength: kl Z "cos h na=a !sin h > . (20) kl 8 kl 8 kl 1.1.2. The fermionic sector The fermionic Lagrangian is given by a sum over families plus family mixing terms (¸ ). Family &. mixing e!ects are outside the scope of this review, and we will not consider them any further. Each family includes lepton and quark sectors ¸ "¸ #¸ f l q
(21)
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which for, say, the "rst family are
AB
l (22) ¸ "!iWM ckD W!ie6 ckD e #h(e6 UsW#WM Ue ) where W" e l k R k R R R e L u !iu6 ckD u !idM ckD d ¸ "!i(u6 , dM ) ckD R k R R k R q L k d L /` u !G (u6 , dM ) d #dM (/~, /H) d L /0 R R d L !/H u !G (u6 , dM ) u #u6 (!/0, /`) (23) u L /~ R R d L where /H and /~ are the complex conjugates of /0 and /` respectively. h, G and G are Yukawa d u couplings. The indices L and R refer to left- and right-handed components and, rather than list their charges under the various transformations, we give here all covariant derivatives explicitly:
AB C A B C A B
ABD ABD
AB A
BA B
l ig ig@ D W"D " R ! qa=a # > k k e k k 2 2 k L D e "(R #ig@> )e , k R k k R u u ig ig@ D " R ! qa=a ! > , k d k k 2 6 k d L L i2g@ > u , D u " R ! k k R 3 k R
AB A A A
BA B
B B
ig@ D d " R # > d . k R k 3 k R
l
e L
,
(24) (25) (26) (27) (28)
One xnal comment: Electroweak strings are non-topological and their stability turns out to depend on the values of the parameters in the model. In this paper we will consider the electric charge e, Yukawa couplings and the VEV of the Higgs, g/J2, to be given by their measured values, but the results of the stability analysis will be given as a function of the parameters sin2 h and 8 b"(m /m )2 (the ratio of the Higgs mass to the Z mass squared); we remind the reader H Z that sin2h +0.23, m ,g g/2"91.2 GeV, m ,gg/2"80.41 GeV and current bounds on the 8 Z z W Higgs mass m ,J2jg are m '77.5 GeV, and an unpublished bound m '90 GeV. H H H 2. Review of Nielsen}Olesen topological strings We begin by reviewing Nielsen}Olesen (NO) vortices in the Abelian Higgs model, with emphasis on those aspects that are relevant to the study of electroweak strings. More detailed information can be found in existing reviews [53].
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2.1. The Abelian Higgs model The theory contains a complex scalar "eld U and a ;(1) gauge "eld which becomes massive through the Higgs mechanism. By analogy with the GSW model, we will call this "eld > . The k action is
P C
A
B
D
g2 2 1 S" d4x DD UD2!j UsU! ! > >kl , k 2 4 kl
(29)
where D "R !iq> is the ;(1)-covariant derivative, and > "R > !R > is the ;(1) "eld k k k kl k l l k strength. The theory is invariant under ;(1) gauge transformations: U(x)Pe*qs(x)U(x)"UK (x),
> (x)P> (x)#R s(x)">K (x) k k k k which give D U(x)PDK UK (x)"e*qs(x)D U. k k k The equations of motion derived from this Lagrangian are
A
B
(30)
g2 D DkU#2j DUD2! U"0 , k 2 (31)
Rk> "!iqUsDa U . kl l
Before we proceed any further, we should point out that, up to an overall scale, the classical dynamics of the Abelian Higgs model is governed by a single parameter, b"2j/q2, the (square of the) ratio of the scalar mass to the vector mass.5 The action (29) contains three parameters (j, g, q), which combine into the scalar mass J2jg"m ,l~1, the vector mass qg"m ,l~1, and 4 4 7 7 an overall energy scale given by the vacuum expectation value of the Higgs, g/J2. The rescaling J2 g UK (x), x" x( , U(x)" qg J2
g >" >K k J2 k
(32)
changes the action to
P
1 d4x [DD UD2!1b(UsU!1)2!1> >kl] , S" k 2 4 kl q2
(33)
where now D "R !i> and we have omitted hats throughout for simplicity. In physical terms k k k this corresponds to taking l as the unit of length (up to a factor of J2) and absorbing the ;(1) 7 charge q into the de"nition of the gauge "eld, thus UPU/SDUDT,
xPx/J2l , 7
e> P> J2l , k k 7
EPE/SDUD2T .
5 b is also the parameter that distinguishes superconductors or type I (b(1) from type II (b'1).
(34)
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The energy associated with (29) is
P C
A
B
D
g2 2 E" d3x DD UD2#DD UD2#j UsU! #1E2#1B2 , 0 i 2 2 2
(35)
where the electric and magnetic "elds are given by F "E and F "e Bk, respectively 0i i ij ijk (i, j, k"1, 2, 3). Modulo gauge transformations, the ground states are given by > "0, U"ge*C/ k J2, where C is constant. Thus, the vacuum manifold is the circle V"MU3C D UsU!1g2"0N+S1 . 2
(36)
A necessary condition for a con"guration to have "nite energy is that the asymptotic scalar "eld con"guration as rPR must lie entirely in the vacuum manifold. Also, D U must tend to zero, k and this condition means that scalar "elds at neighbouring points must be related by an in"nitesimal gauge transformation. Finally, the gauge "eld strengths must also vanish asymptotically. Note that, in the Abelian Higgs model, the last condition follows from the second, since 0"[D , D ]U"!iq> U implies > "0. But this need not be the case when the Abelian Higgs k l kl kl model is embedded in a larger model. Vanishing of the covariant derivative term implies that, at large r, the asymptotic con"guration U(x) must lie on a gauge orbit; U(x)"g(x)U 0
where g(x)3G and U 3V . 0
(37)
where U is a reference point in V. Note that, since all symmetries are gauge symmetries, the set 0 of points that can be reached from U through a gauge transformation (the gauge orbit of U ) 0 0 spans the entire vacuum manifold. Thus, V"G/H"G /H , where G indicates the group -0#!- -0#!-0#!of gauge, i.e. local, symmetries. On the other hand, the spaces V and G /H need not coincide -0#!- -0#!in models with both local and global symmetries, and this fact will be particularly relevant in the discussion of semilocal strings. 2.2. Nielsen}Olesen vortices In what follows we use cylindrical coordinates (t, o, u, z). We are interested in a static, cylindrically symmetric con"guration corresponding to an in"nite, straight string along the z-axis. The ansatz of Nielsen and Olesen [103] for a string with winding number n is U"(g/J2) f (o)e*nr, q> "nv(o), > "> "> "0 r o t z
(38)
(that is, >"v(o) du or Y"u( v(o)/o), with boundary conditions f (0)"v(0)"0, f (o)P1, v(o)P1 as oPR .
(39)
Note that, since > "> "0, and all other "elds are independent of t and z, the electric "eld is z t zero, and the only surviving component of the magnetic "eld B is in the z direction.
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Fig. 1. The functions f , v for a string with winding number n"1 (top panel) and n"50 (bottom panel), for NO NO b,2j/q2"0.5. The radial coordinate has been rescaled as in Eq. (32), o( "qgo/J2.
Substituting this ansatz into the equations of motion we obtain the equations that the functions f and v must satisfy f @(o) n2f (o) ! [1!v(o)]2#jg2(1!f (o)2) f (o)"0 , f A(o)# o2 o v@(o) vA(o)! #q2g2f 2(o)[1!v(o)]"0 . o
(40)
In what follows, we will denote the solutions to the system (40), (39) by f and v ; they are not NO NO known analytically, but have been determined numerically; for n"1, b"0.5, they have the pro"le in Fig. 1. At small o, the functions f and v behave as on and o2 respectively; as oPR, they approach their asymptotic values exponentially with a width given by the inverse scalar mass, m , and the inverse 4 vector mass, m , respectively, if b(4. For b'4 the fall-o! of both the scalar and the vector is 7 controlled by the vector mass [105]. One case in which it is possible to "nd analytic expressions for the functions f and v is in the NO NO limit nPR [6]. Inside the core of a large n vortex, the functions f and v are f (o)"((q/4n)m m o2)n@2e~qm4 m7 o2@8, v(o)"(1/4n)m m o2 (41) 4 7 4 7 to leading order in 1/n, and the transition to their vacuum values is controlled by a "rst integral W( f, f @, v, v@)"const. Large n vortices behave like a conglomerate of `solida n"1 vortices. The area scales as n, so the radius goes like Jn¸ , where ¸ "2(Jm m )~1. The transition region between 0 0 4 7 the core and asymptotic values of the "elds is of the same width as for n"1 vortices Fig. 1 shows
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the functions f , v for n"50, b"0.5 (note that for b'1 these multiply winding solutions are NO NO unstable to separation into n"1 vortices which repel one another [26,66]). 2.2.1. Energy considerations The energy per unit length of such con"gurations (static and z-independent) is therefore
P C
A
BD
1 g2 2 E" d2x DD UD2# B2#j UsU! , m 2 2
(42)
where m, n"1, 2 and B"R > !R > is the z-component of the magnetic "eld. m n n m In order to have solutions with "nite energy per unit length we must demand that, as oPR, D U, DUD2!g2/2 and > all go to zero faster than 1/o. k mn The vacuum manifold (36) is a circle and strings form when the asymptotic "eld con"guration of the scalar "eld winds around this circle. The important point here is that there is no way to extend a winding con"guration inwards from o"R to the entire xy plane continuously while remaining in the vacuum manifold. Continuity of the scalar "eld implies that it must have a zero somewhere in the xy plane. This happens even if the xy plane is deformed, and at all times, and in three dimensions one "nds a continuous line of zeroes which signal the position of the string (a sheet in space time). Note that the string can have no ends; it is either in"nitely long or a closed loop. The zeroes of the scalar "eld are forced by the non-zero topological degree of the map S1PV ,
(43)
uPU(o"R, u) , usually called the winding number of the vortex; the resulting vortices are called topological because they are labelled by non-trivial elements of the "rst homotopy group of the vacuum manifold (where non-trivial means `other than the identity elementa). Thus, n (V)"p (S1)O1, is a neces1 1 sary condition for the existence of topological vortices. Vortices whose asymptotic scalar "eld con"gurations are associated with the identity element of p (V) are called non-topological. In 1 particular, if V is simply connected, i.e. p (V)"1, one can only have non-topological vortices. 1 A few comments are needed at this point. 2.2.2. Quantization of magnetic yux Recall that B is the z-component of the magnetic "eld. The magnetic #ux F through the Y xy-plane is therefore
P
P
F , d2x B" Y
P
2p
2pn R s du" r q
(44) 0 o/= and is quantized in units of 2p/q. This is due to the fact that U(o"R, u)"ge*qs(r)/J2, D U"g/J2[iqR s!iq> ]"0 and U must be single-valued, thus q[s(2p)!s(0)]"2pn. The r r r integer n is, again, the winding number of the vortex. Y ) dl" =
2.2.3. Magnetic pressure In an Abelian theory, the condition $ ) B"0 implies that parallel magnetic "eld lines repel. A two-dimensional scale transformation xPj x where the magnetic "eld is reduced accordingly to
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keep the magnetic #ux constant, BK "K~2B(x/K), reduces the magnetic energy :d2x B2/2 by K2. What this means is that a tube of magnetic lines of area S can lower its energy by a factor of K2 by 0 spreading over an area K2S . 0 Note that later we will consider non-Abelian gauge symmetries, for which $ ) BO0 and the energy can also be lowered in a di!erent way. In this case, one can think of the gauge "elds as carrying a magnetic moment which couples to the `magnetica "eld and, in the presence of a su$ciently intense magnetic "eld, the energy can be lowered by the spontaneous creation of gauge bosons. In the context of the electroweak model, this process is known as =-condensation [11] and its relevance for electroweak strings is explained in Section 7. 2.2.4. Meissner ewect and symmetry restoration In the Abelian Higgs model, as in a superconductor, it is energetically costly for magnetic "elds to coexist with scalar "elds in the broken symmetry phase. Superconductors exhibit the Meissner e!ect (the expulsion of external magnetic "elds), but as the sample gets larger or the magnetic "eld more intense, symmetry restoration becomes energetically favourable. An example is the generation of Abrikosov lattices of vortices in type II superconductors, when the external magnetic "eld reaches a critical value. The same phenomenon occurs in the Abelian Higgs model. In a region where there is a concentration of magnetic #ux, the coupling term q2A2U2 in the energy will tend to force the value of the scalar "eld towards zero (its value in the symmetric phase). This will be important to understand the formation of semilocal (and possibly electroweak) strings, where there is no topological protection for the vortices, during a phase transition (see Section 3.5). The back reaction of the gauge "elds on the scalars depends on the strength of the coupling constant q. When q is large (in a manner that will be made precise in Section 3.5) semilocal strings tend to form regardless of the topology of the vacuum manifold. 2.3. Stability of Nielsen}Olesen vortices Given a solution to the classical equations of motion, there are typically two approaches to the question of stability. One is to consider the stability with respect to in"nitesimal perturbations of the solution. If one can establish that no perturbation can lower the energy, then the solution is called classically stable. Small perturbations that do not alter the energy are called zero modes, and signal the existence of a family of con"gurations with the same energy as the solution whose stability we are investigating (e.g. because of an underlying symmetry). If one can guess an instability mode, this approach is very e$cient in showing that a solution is unstable (by "nding the instability mode explicitly) but it is usually much more cumbersome to prove stability; mathematically the problem reduces to an eigenvalue problem and one often has to resort to numerical methods. A stability analysis of this type for Nielsen}Olesen vortices has only been carried out recently by Goodband and Hindmarsh [56]. An analysis of the stability of semilocal and electroweak strings can be found in later sections. A second approach, due to Bogomolnyi, consists in "nding a lower bound for the energy in each topological sector and proving that the solution under consideration saturates this bound. This immediately implies that the solution is stable, although it does not preclude the existence of zero
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modes or even of other con"gurations with the same energy to which the solution could tunnel semiclassically. We will now turn to Bogomolnyi's method in the case of Nielsen}Olesen vortices. 2.3.1. Bogomolnyi limit and bounds Consider the scalar gradients (D U)sD U#(D U)sD U"[(D #iD )U]s(D #iD )U!i[(D U)sD U!(D U)sD U] 1 1 2 2 1 2 1 2 1 2 2 1 "D(D #iD )UD2!i[R (UsD U)!R (UsD U)]#iUs[D , D ]U . (45) 1 2 1 2 2 1 1 2 Note that the second term on the RHS of (45) is the curl of the current J "!iUsD U, and that i i { J ) dl tends to zero as oPR for con"gurations with "nite energy per unit length (because D U i must vanish faster than 1/o). Now use the identity [D , D ]U"!iqF U"!iqBU to rewrite 1 2 12 the energy per unit length as follows:
P C P C
A
BD
g2 2 E" d2x D(D $iD )UD2#1B2$qBUsU#j UsU! 1 2 2 2
G A
1 g2 " d2x D(D $iD )UD2# B$q UsU! 1 2 2 2
BH
A
BD
g2 2 2 #(j!1q2) UsU! 2 2
P
g2 $q B d2x . 2
(46)
The last integral is the total magnetic #ux, and we saw earlier that it has to be an integral multiple of 2p/q, so we can write, introducing b"2j/q2, E"2p($n)
PC A
g2 # 2
C A
1 g2 D(D $iD )UD2# B$q UsU! 1 2 2 2
BD
g2 2 #1q2(b!1) UsU! , 2 2
BD
2
(47)
where the plus or minus signs are chosen so that the "rst term is positive, depending on the sign of the magnetic #ux. Note that, if b51 the energy is bounded below by E5SUsUTqF , Y where F is the magnetic #ux.6 Y
(48)
6 When b"1, the masses of the scalar and the vector are equal, and the Abelian Higgs model can be made supersymmetric. In general, bounds of the form (energy)5(constant)](#ux) are called Bogomolnyi bounds, and their origin can be traced back to supersymmetry.
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If b"1, there are con"gurations that saturate this bound: those that satisfy the "rst-order Bogomolnyi equations (D $iD )U"0, B$q(UsU!1g2)"0 1 2 2 or, in terms of f (o) and v(o), f @(o)#($n)((v(o)!1)/o) f (o)"0,
(49)
($n)v@(o)#(q2g2/2)o( f 2(o)!1)"0 .
(50)
However, when b'1 there does not exist a static solution with E"pDnDg2 since requiring, e.g., B#q(UsU!g2/2)"0 and (UsU!g2/2)"0 simultaneously would imply B"0, which is inconsistent with the condition on the total magnetic #ux, :B d2x"2pn/q. This has an e!ect on the stability of higher winding vortices when b'1: if n'1 the solution breaks into n vortices each with a unit of magnetic #ux [26], which repel one another. If n"1 there are stable static solutions, but with an energy higher than the Bogomolnyi bound. This is because the topology of the vacuum manifold forces a zero of the Higgs "eld, and then competition between magnetic and potential energy "xes the radius of the solution. The same argument shows that n"1 strings are stable for every value of b. One still has to worry about angular instabilities, but a careful analysis by [56] shows there are none. The dynamics of multivortex solutions is governed by the fact that when b(1 vortices attract, but with b'1 they repel [66]. This can be understood heuristically from the competition between magnetic pressure and the desire to minimize potential energy by having symmetry restoration in as small an area as possible. The width of the scalar vortex depends on the inverse mass of the Higgs, l , that of the magnetic #ux tube depends on the inverse vector boson mass, l . If b(1, have 4 7 m 'm so l (l (the radii of the scalar and vector tubes). The scalar tubes see each other "rst 7 4 7 4 } they attract. Whereas if b'1, the vector tubes see each other "rst } they repel. For b"1 there is no net force between vortices, and there are static multivortex solutions for any n. In the Abelian Higgs case they were explicitly constructed by Taubes [69] and their scattering at low kinetic energies has been investigated using the geodesic approximation of Manton [96] by Ruback [114] and, more recently, Samols [117]. For b(1, Goodband and Hindmarsh [56] have found bound states of two n"1 vortices oscillating about their centre of mass. 3. Semilocal strings The semilocal model is obtained when we replace the complex scalar "eld in the Abelian Higgs model by an N-component multiplet, while keeping only the overall phase gauged. In this section we will concentrate on N"2 because of its relationship to electroweak strings, but the generalization to higher N is straightforward, and is discussed below. 3.1. The model Consider a direct generalization of the Abelian Higgs model where the complex scalar "eld is replaced by an S;(2) doublet UT"(/ , / ). The action is 1 2 1 g2 2 S" d4x D(R !iq> )UD2! > >kl!j UsU! , (51) k k 4 kl 2
P C
A
BD
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where > is the ;(1) gauge potential and > "R > !R > its "eld strength. Note that this is just k kl k l l k the scalar sector of the GSW model for g"0, g@"g "2q, i.e. for sin2 h "1, and =a "0. z 8 k Let us take a close look at the symmetries. The action is invariant under G"S;(2) ] '-0"!;(1) , with transformations -0#!e*qc(x) 0 / 1 , > P> #R c(x) , UPe*qc(x)U" (52) k k k 0 e*qc(x) / 2 under ;(1) , and -0#!cos(a )#in sin(a ) i(n !in )sin(a ) / 2 3 2 1 2 2 1 , > P> UPe*aa qaU" (53) k k i(n #in )sin(a ) cos(a )!in sin(a ) / 1 2 2 2 2 3 2 under S;(2) , where a"Ja2 #a2 #a2 3[0, 4p) is a positive constant and n "a /a is a a '-0"!1 2 3 a constant unit vector. Note that a shift of the function c(x) by 2p/q leaves the transformations una!ected. The model actually has symmetry G"[S;(2) ];(1) ]/Z ; the Z identi"cation '-0"!-0#!2 2 comes because the transformation with (a, c) is identi"ed with that with (a#2p, c#p/q). Once U acquires a vacuum expectation value, the symmetry breaks down to H";(1) exactly as in the GSW model, except for the fact that the unbroken ;(1) subgroup is now global (for instance, if the VEV of the Higgs is SUTT"g(0, 1)/J2, the unbroken global ;(1) is the subgroup with n "n "0, n "1, qc"a/2). Thus, the symmetry breaking is [S;(2) ];(1) ]/ 1 2 3 '-0"!-0#!Z P;(1) . 2 '-0"!Note also that, for any xxed U a global phase change can be achieved with either a global 0 ;(1) transformation or a S;(2) transformation. The signi"cance of this fact will become -0#!'-0"!apparent in a moment. Like in the GSW model, the vacuum manifold is the three sphere
A
A
BA B
BA B
V"MU3C2 D UsU"1g2N+S3 , (54) 2 which is simply connected, so there are no topological string solutions. On the other hand, if we only look at the gauged part of the symmetry, the breaking looks like ;(1)P1, identical to that of the Abelian Higgs model, and this suggests that we should have local strings. After symmetry breaking, the particle content is two Goldstone bosons, one scalar of mass m "J2jg and a massive vector boson of mass m "qg. In this section it will be convenient to use 4 7 rescaled units throughout; after the rescaling (32), and dropping hats, we "nd
P C
D
1 b q2S" d4x D(R !i> )UD2! > >kl! (UsU!1)2 , k k 4 kl 2
(55)
and, as in the Abelian Higgs case, b"m2/m2"2j/q2 is the only free parameter in the model. The 4 7 equations of motion D DkU#b(DUD2!1)U"0, k
Rk> "!iUsDa U . kl l
(56)
are exactly the same as in the Abelian Higgs model but replacing the scalar "eld by the S;(2) doublet, and complex conjugation by Hermitian conjugation of U. Therefore, any solution UK (x), >K (x) of (31) (in rescaled units) extends trivially to a solution U (x), (> ) (x) of the semilocal k 4k 4-
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model if we take U (x)"UK (x)U , (> ) (x)">K (x) (57) 40 k 4k with U a constant S;(2) doublet of unit norm, Us U "1. In particular, the Nielsen}Olesen string 0 0 0 can be embedded in the semilocal model in this way. The con"guration U"f (o)e*nrU , >"nv (o) du (58) NO 0 NO remains a solution of the semilocal model with winding number n provided f and v are NO NO the solutions to the Nielsen}Olesen equations (40). In this context, the constant doublet U is 0 sometimes called the &colour' of the string (do not confuse with S;(3) colour!). One important di!erence with the Abelian Higgs model is that a scalar perturbation can remove the zero of U at the centre of the string, thereby reducing the potential energy stored in the core. Consider the energy per unit length, in these units, of a static, cylindrically symmetric con"guration along the z-axis:
P C
D
E b 1 " d2x (R > !R > )2#D(R !i> )UD2# (UsU!1)2 . m n n m m m (g2/2) 2 4
(59)
Note, "rst of all, that any "nite energy con"guration must satisfy (R !i> )/ P0, (R !i> )/ P0, /M / #/M / P1 as oPR . m m 1 m m 2 1 1 2 2 (As before, m, n"1, 2 and (o, u) are polar coordinates on the plane orthogonal to the string.) This leaves the phases of / and / undetermined at in"nity and there can be solutions where both 1 2 phases change by integer multiples of 2p as we go around the string; however, there is only one ;(1) gauge "eld available to compensate the gradients of / and / , and this introduces a correlation 1 2 between the winding in both components: the condition of "nite energy requires that the phases of / and / di!er by, at most, a constant, as oPR. Therefore, a "nite energy string must tend 1 2 asymptotically to a maximal circle on S3:
A
UPe*nr
ae*C
B
,e*nrU 0 J1!a2
A
B
n >Pn du or YP u( , o
(60)
where 04a41 and C are real constants, and determine the &colour' of the string. A few comments are needed at this point. f Note that the choice of U is arbitrary for an isolated string (any value of U can be rotated into 0 0 any other without any cost in energy) but the relative &colour' between two or more strings is "xed. That is, the relative value of U is signi"cant whereas the absolute value is not. 0 f The number n is the winding number of the string and, although it is not a topological invariant in the usual sense (the vacuum manifold, S3, is simply connected), it is topologically conserved. The reason is that, even though any maximal circle can be continuously contracted to a point on S3, all the intermediate con"gurations have in"nite energy. The space that labels "nite energy con"gurations is not the vacuum manifold but, rather, the gauge orbit from any reference point U 3V, and this space (G /H ) is not simply connected: p (G /H )"p (;(1)/1)"Z. 0 -0#!- -0#!1 -0#!- -0#!1
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Thus, con"gurations with di!erent winding numbers are separated by in"nite energy barriers, but this information is not contained in p (V).7 1 f On the other hand, because p (V)"1, the existence of a topologically conserved winding 1 number does not guarantee that winding con"gurations are non-dissipative either. In contrast with the Abelian Higgs model, a "eld con"guration with non-trivial winding number at o"R can be extended inwards for all o without ever leaving the vacuum manifold. Thus, the fact that p (G /H )O1 only means that "nite energy "eld con"gurations fall into inequivalent 1 -0#!- -0#!sectors, but it says nothing about the existence of stable solutions within these sectors. f Thus, we have a situation where p (V)"p (G/H)"p (S3)"1 but p (G /H )"p (S1)"Z , (61) 1 1 1 1 -0#!- -0#!1 and the e!ect of the global symmetry is to eliminate the topological reason for the existence of the strings. Notice that this subtlety does not usually arise because these two spaces are the same in theories where all symmetries are gauged (like GSW, Abelian Higgs, etc.). We will now show that, in the semilocal model, the stability of the string depends on the dynamics and is controlled by the value of the parameter b"2j/q2. Heuristically we expect large b to mimic the situation with only global symmetries (where the strings would be unstable), whereas small b resembles the situation with only gauge symmetries (where we expect stable strings). 3.2. Stability Let us "rst prove that there are classically stable strings in this model. We can show this analytically for b"1 [127]. Recall the expression of the energy per unit length (59). The analysis in the previous section goes through when the complex "eld is replaced by the S;(2) doublet, and we can rewrite
P
E "2pDnD# d2x[DD U$iD UD2#1(B$(UsU!1))2#1(b!1)(UsU!1)2] , 1 2 2 2 (g2/2)
(62)
choosing the upper or lower signs depending on the sign of n. Since n is "xed for "nite energy con"gurations this shows that, at least for b"1, a con"guration satisfying the Bogomolnyi equations (D $iD )U"O, B$(UsU!1)"0 , (63) 1 2 is a local minimum of the energy and, therefore, automatically stable to in"nitesimal perturbations. But these are the same equations as in the Abelian Higgs model, therefore the semilocal string (58) automatically saturates the Bogomolnyi bound (for any &colour' U ). Thus, it is classically stable for 0 b"1. This argument does not preclude zero modes or other con"gurations degenerate in energy. Hindmarsh [59] showed that, for b"1 there are indeed such zero modes, described below in Section 3.2.3. 7 The fact that the gauge orbits sit inside V"G/H without giving rise to non-contractible loops can be traced back to the previous remark that every point in the gauge orbit of U can also be reached from U with a global transformation. 0 0
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We have just proved that, for b"1, semilocal strings are stable. This is surprising because the vacuum manifold is simply connected and a "eld con"guration that winds at in"nity may unwind without any cost in potential energy.8 The catch is that, because p (G /H )"p (;(1))"Z is 1 -0#!- -0#!1 non-trivial, leaving the ;(1) gauge orbit is still expensive in terms of gradient energy. As we come in from in"nity, the "eld has to choose between unwinding or forming a semilocal string, that is, between acquiring mostly gradient or mostly potential energy. The choice depends on the relative strength of these terms in the action, which is governed by the value of b, and we expect the "eld to unwind for large b, when the reduction in potential energy for going o! the vacuum manifold is high compared to the cost in gradient energy for going o! the ;(1) orbits, and vice versa. Indeed, we will now show that, for b'1, the n"1 vortex is unstable to perturbations in the direction orthogonal to U [59] while, for b(1, it is stable. For b"1, some of the perturbed 0 con"gurations become degenerate in energy with the semilocal vortex and this gives a (complex) one-parameter family of solutions with the same energy and varying core radius [59]. 3.2.1. The stability of strings with b'1 Hindmarsh [59] has shown that for b'1 the semilocal string con"guration with unit winding is unstable to perturbations orthogonal to U , which make the magnetic #ux spread to in"nity. As 0 pointed out by Preskill [109], this is remarkable because the total amount of #ux measured at in"nity remains quantized, but the #ux is not con"ned to a core of "nite size (which we would have expected to be of the order of the inverse vector mass). The semilocal string solution with n"1 is, in rescaled units, U "f (o)e*rU , > "v (o) du . (64) 4NO 0 4NO However, as pointed out in [59], this is not the most general static one-vortex ansatz compatible with cylindrical symmetry. Consider the ansatz U"f (o)e*rU #g(o)e*mrU , >"v(o) du , (65) 0 M with DU D"DU D"1 and UM U "0. The orthogonality of U and U ensures that the e!ect of 0 M 0 M 0 M a rotation can be removed from U by a suitable S;(2)];(1) transformation, therefore the con"guration is cylindrically symmetric. For the con"guration to have "nite energy we require the boundary conditions f (0)"g@(0)"v(0)"0 and fP1, gP0, vP1 as oPR. We know that if g"0 the energy is minimized by the semilocal string con"guration f"f , v"v , because the problem is then identical to the Abelian Higgs case. The question is NO NO whether a non-zero g can lower the energy even further, in which case the semilocal string would be unstable. The standard way to "nd out is to consider a small perturbation of (64) of the form g"/(o)e*ut and look for solutions of the equations of motion where g grows exponentially, that is, where u2(0. The problem reduces to "nding the negative eigenvalue solutions to the 8 In the Nielsen}Olesen case a con"guration with a non-trivial winding number must go through zero somewhere for the "eld to be continuous. But here, a con"guration like UT(o"R)"g(0, e*r)/J2 can gradually change to UT(o"0)"g(1, 0)J2 as we move towards the centre of the `stringa without ever leaving the vacuum manifold. This is usually called &unwinding' or &escaping in the third dimension' by analogy with condensed matter systems like nematic liquid crystals.
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SchroK dinger}type equation
C
A B
D
1 d d (v(o)!m)2 ! o # #b( f (o)2!1) t(o)"u2t(o) . o do do o2
(66)
First of all, it turns out that it is su$cient to examine the m"0 case only. Note that, since 04v(o)41, for m'1 the second term is everywhere larger than for m"1, so if one can show that all eigenvalues are positive for m"1 then so are the eigenvalues for m'1. But for m"1 the problem is identical to the analogous one for instabilities in f in the Abelian Higgs model, and we know there are no instabilities in that case. Therefore it is su$cient to check the stability of the solution to perturbations with m"0 (negative values of m also give higher eigenvalues than m"0). If m"0, the above ansatz yields
P C
D
= 1 (1!v)2 v2 1 E "2p o ( f @)2#(g@)2# (v@)2# f 2# g2# b( f 2#g2!1)2 do (67) 2o2 o2 o2 2 (g2/2) 0 for the (rescaled) energy functional (59). Notice that a non-zero g at o"0 (where fO1) reduces the potential energy but increases the gradient energy for small values of o. If b is large, this can be energetically favourable (conversely, for very small b, the cost in gradient energy due to a non-zero g could outweigh any reduction in potential energy). Indeed, Hindmarsh showed that there are no minimum-energy vortices of "nite core radius when b'1 by constructing a one-parameter family of con"gurations whose energy tends to the Bogomolnyi bound as the parameter o is increased: 0 o o2 ~1@2 o2 ~1@2 o2 ~1 o2 f (o)" 1# 1# , g(o)" 1# , v(o)" . (68) o o2 o2 o2 o2 0 0 0 0 0 The energy per unit length of these con"gurations is E"pg2(1#1/3o2 ) which, as o PR, tends 0 0 to the Bogomolnyi bound. This shows that any stable solution must saturate the Bogomolnyi bound, but this is impossible because, when b'1, saturation would require B"0 everywhere, which is incompatible with the total magnetic #ux being 2p/q (see the comment after Eq. (50)). While this does not preclude the possibility of a metastable solution, numerical studies have found no evidence for it [59,7]. All indications are that, for b'1, the semilocal string is unstable towards developing a condensate in its core which then spreads to in"nity. Thus, the semilocal model with b'1 is a system (see Fig. 2) where magnetic #ux is quantized, the vector boson is massive and yet there is no con"nement of magnetic #ux.9
C
D
C
D
C
D
3.2.2. The stability of strings with b(1 Semilocal strings with b(1 are stable to small perturbations (see Fig. 3). Numerical analysis of the eigenvalue equations [59,60] shows no negative eigenvalues, and numerical simulations of the solutions themselves indicate that they are stable to z-independent perturbations [7,4], including those with angular dependence. Note that the stability to z-dependent perturbations is automatic,
9 Preskill has emphasized that the `mixinga of global and local generators is a necessary condition for this behaviour, that is, there must be a generator of H which is a non-trivial linear combination of generators of G and G [109]. '-0"!-0#!-
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Fig. 2. A two-dimensional simulation of the evolution of a perturbed isolated semilocal string with b'1, from [7]. The plot shows the (rescaled) energy density per unit length in the plane perpendicular to the string. b"1.1. The initial conditions include a large destabilizing perturbation in the core, UT(t"0)"(1, f (o)e*r), which is seen to destroy the NO string.
as they necessarily have higher energy. These results are con"rmed by studies of electroweak string stability [57,6] taken in the limit h Pp/2. 8 3.2.3. b"1 zero modes and skyrmions Substituting the ansatz (65) into the (rescaled) Bogomolnyi equations for n"1 gives v(o)!1 f @(o)# f (o)"0 , o v(o) g(o)"0 , g@(o)# o
(69)
v@(o)#o( f 2(o)#g2(o)!1)"0 . When b"1 we showed earlier that the semilocal string f"f , g"0, v"v saturates the NO NO Bogomolnyi bound, so it is necessarily stable (since it is a minimum of the energy). There may exist, however, other solutions satisfying the same boundary conditions and with the same energy. Hindmarsh showed that this is indeed the case by noticing that the eigenvalue equation has a zero-eigenvalue solution [59]
C P
o
D
v(o( ) , o(
t "const , (70) 0 0 which signals a degeneracy in the solutions to the Bogomolnyi equations. (Note that the &colour' at in"nity, U , is "xed, so this is not a zero mode associated with the global S;(2) transformations; its 0 dynamics have been studied in [85].) t"t exp ! 0
do(
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Fig. 3. The evolution of a string with b(1. The initial con"guration is the same as in Fig. 2 but now, after a few oscillations, the con"guration relaxes into a semilocal string, UT"(0, f (o)e*r): b"0.9. NO
It can be shown that the zero mode exists for any value of g, not just g"0; the Bogomolnyi equations (69) are not independent since g(o)"q f (o)/o (71) 0 is a solution of the second equation for any (complex) constant q . Solving the other two equations 0 leads to the most general solution with winding number one and centred at o"0. It is labelled by the complex parameter q , which "xes the size and orientation of the vortex: 0 / q 1 1 " 0 expM1u(o; Dq D)N , (72) 2 0 / oe*r Jo2#Dq D2 2 0
A B
A B
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where u"ln DUD2 is the solution to +2u#2(1!eu)"+2 ln(o2#Dq D2), uP0 as oPR . (73) 0 If q O0, the asymptotic behaviour of these solutions is very di!erent from that of the Nielsen} 0 Olesen vortex; the Higgs "eld is non-zero at o"0 and approaches its asymptotic values like O(o~2). Moreover, the magnetic "eld tends to zero as B&2Dq D2o~4, so the width of the #ux tube 0 is not as well-de"ned as in the q "0 case when B falls o! exponentially. These q O0 solutions 0 0 have been dubbed &skyrmions'. In the limit Dq DP0, one recovers the semilocal string solution (64), 0 with u"ln( f 2 ), the Higgs vanishing at o"0 and approaching the vacuum exponentially fast. NO On the other hand, when Dq D<1, u+0 the scalar "eld is in vacuum everywhere and the 0 solution approximates a CP1 lump [59,86]. Thus, in some sense, the &skyrmions' interpolate between vortices and CP1 lumps. 3.2.4. Skyrmion dynamics We have just seen that, for b"1 the semilocal vortex con"guration is degenerate in energy with a whole family of con"gurations where the magnetic #ux is spread over an arbitrarily large area. It is interesting to consider the dynamics of these `skyrmionsa when bO1 [60,19]: large skyrmions tend to contract if b(1 and to expand if b'1. The timescale for the collapse of a large skyrmion increases quadratically with its size [60]. Thus large skyrmions collapse very slowly. Benson and Bucher [19] derived the energy spectrum of delocalized &skyrmion' con"gurations in 2#1 dimensions as a function of their size. More precisely, they de"ned an &antisize' s"E /E as the ratio of the magnetic energy :d2x B2/2 to the total energy (59). Note that .!'/%5*# 505!when the #ux lines are concentrated, magnetic energy is high compared to the other contributions, and vice versa. Thus, sP0 corresponds to the limit in which the magnetic #ux lines are spread over an in"nitely large area, which explains the name &antisize'. For large skyrmions } those with s4b/(1#b) } they concluded that the minimum energy con"guration among all delocalized con"gurations with antisize s satis"es E(b, s)"2p
g2 b 2 b!s(b!1)
(74)
(if s'b/(1#b) the analysis does not apply). Therefore, energy decreases monotonically with decreasing s for b'1 and increases monotonically for b(1, con"rming that delocalized con"gurations tend to grow in size if b'1 and shrink if b(1. This behaviour is observed in numerical simulations [3]. Benson and Bucher [19] have pointed out that in a cosmological setting the expansion of the Universe could drag the large skyrmions along with it and stop their collapse. The simulations in #at space are at least consistent with this, in that they show that delocalized con"gurations tend to live longer when arti"cial viscosity is increased, but a full numerical simulation of the evolution of semilocal string networks has not yet been performed and is possibly the only way to answer these questions reliably. Finally, we stress that the magnetic #ux of a skyrmion does not change when it expands or contracts (the winding number is conserved) but this does not say anything about how localized the #ux is. In contrast with the Abelian Higgs case, the size of a skyrmion can be made arbitrarily large with a "nite amount of energy.
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3.3. Semilocal string interactions 3.3.1. Multivortex solutions, b"1, same colour Multi-vortex solutions in 2#1 dimensions corresponding to parallel semilocal strings with the same colour have been constructed by Gibbons et al. [51] for the critical case b"1. Their analysis closely follows that of [69] in the case of the Abelian Higgs model, and starts by showing that, as in that case, the full set of solutions to the (second-order) equations of motion can be obtained by analysing the solutions to the ("rst-order) Bogomolnyi equations. In the Abelian Higgs model, solutions with winding number n are labelled by n unordered points on the plane (those where the scalar "eld vanishes) which, for large separations, are identi"ed with the positions of the vortices. In the semilocal model, the solutions have other degrees of freedom, besides position, describing their size and orientation. Assuming without loss of generality that the winding number n is positive, and working in temporal gauge > "0, any solution with winding number n is speci"ed (up to symmetry 0 transformations) by two holomorphic polynomials n P (z)" < (z!z ) n r r/1 ,zn#p zn~1#2#p z#p n~1 1 0 and Q (z),q zn~1#2#q z#q , (75) n n~1 1 0 where z"x#iy is a complex coordinate on the xy plane. The solution for the Higgs "elds is, up to gauge transformations,
A B
A B
6 / Q 1 " e(1@2)u(z,z) n , (76) / P JDP D2#DQ D2 2 n n n where the function u(z, z6 )"ln(D/ D2#D/ D2) must satisfy 1 2 +2u#2(1!eu)"+2 ln(DP D2#DQ D2) , (77) n n and tend to 0 as DzDPR. Although its form is not known explicitly, Ref. [51] proved the existence of a unique solution to this equation for every choice of P and Q (if P and Q have a common n n n n root then exp[u/2] has a zero there, so the expression for the Higgs "eld is everywhere well-de"ned). The gauge "eld can then be read o! from the Bogomolny equations (63). This generalizes (72) to arbitrary n. The coe$cients of P (z), Q (z) parametrize the moduli space, C2n. n n The Nielsen}Olesen vortex has Q "0. If P O0, then in regions where DQ D;DP D one "nds n n n n 1Q 2 Q Q 2 n , D/ D& n , v&1! n D/ D&1! (78) 1 2 2P P P n n n indicating that the scalar "elds fall o! as a power law, as opposed to the usual exponential fall o! found in NO vortices. The same is true of the magnetic "eld.
K K
K K
K K
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The low-energy scattering of semilocal vortices and skyrmions with b"1 was studied in [86] in the geodesic approximation of [96]. The behaviour of these solitons was found to be analogous to that of CP1 lumps but without the singularities, which are smoothed out in the core. 3.3.2. Interaction of parallel strings, b(1, diwerent colours Ref. [7] carried out a numerical study in two dimensions of the interaction between stable (b(1) strings with di!erent `coloura with non-overlapping cores. It was found that the strings tend to radiate away their colour di!erence in the form of Goldstone bosons, and there is little or no interaction observed. The position of the strings remains the same during the whole evolution while the "elds tend to minimize the initial relative S;(2) phase (see Fig. 4).
Fig. 4. A numerical simulation of the interaction between two parallel semilocal strings with di!erent &colour', from Ref. [7]. The initial con"guration has one string with UT "(0, f (o )e*r1 ) and the other with UT "(i f (o )e*r2 , 0), where 1 1 2 2 (o , u ) are polar coordinates centred at the cores of each string. The energy density of the string pair is plotted in the i i plane perpendicular to the strings. The colour di!erence is radiated away in the form of Goldstone bosons, and the strings cores remain at their initial positions: b"0.5.
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Thus, we expect interactions between in"nitely long semilocal strings with di!erent colours to be essentially the same as for Nielsen}Olesen strings. This expectation is con"rmed by numerical simulations of two- and three-dimensional semilocal string networks [3,4], discussed in Section 3.5. 3.4. Dynamics of string ends Note that, in contrast with Nielsen}Olesen strings, there is no topological reason that forces a semilocal string to continue inde"nitely or form a closed loop. Semilocal strings can end in a `clouda of energy, which behaves like a global monopole [59]. Indeed, consider the following asymptotic con"guration for the Higgs "eld:
A
B
cos 1h g 2 U" , (79) J2 sin 12h e*r which is ill-de"ned at h"p and at r"0. We can make the con"guration regular by introducing pro"le functions such that the Higgs "eld vanishes at those points:
A
B
h (r, h) cos 1h g 1 2 U" , (80) J2 h2 (r, h) sin 12h e*r where h and h vanish at r"0 and h (r, p)"0. This con"guration describes a string in the z(0 1 2 2 axis ending in a monopole at z"0. At large distances, r<1, the Higgs "eld is everywhere in vacuum (except at h+p) and we "nd UssU&x, just like for a Hedgehog in O(3) models. On the other hand, the con"guration for the gauge "elds resembles that of a semi-in"nite solenoid; the string supplies ;(1) #ux which spreads out from z"0. This is the h Pp/2 limit of a con"guration "rst discussed by Nambu [102] in the context of the 8 GSW model, see Section 5, but here the energy of the monopole is linearly divergent because there are not enough gauge "elds to cancel the angular gradients of the scalar "eld. Angular gradients provide an important clue to understand the dynamics of string ends. If b(1, numerical simulations show that string segments grow to join nearby segments or to form loops (see Figs. 5 and 6) [4]. This con"rms analytical estimates in Refs. [51,60]. In other cases the string segment collapses under its own tension, with the monopole and antimonopole at the ends annihilating each other. 3.5. Numerical simulations of semilocal string networks As the early Universe expanded and cooled to become what we know today it is very likely that it went through a number of phase transitions where topological (and possibly non-topological) defects are expected to have formed according to the Kibble mechanism [76,140,53]. Although the cosmological evidence for the existence of such defects remains unclear [9], there is plenty of experimental evidence from condensed matter systems that networks of defects do form in symmetry-breaking phase transitions [104], the most recent con"rmation coming from the Lancaster}Grenoble}Helsinki experiments in vortex formation in super#uid helium [17]. An important question is whether semilocal (and electroweak) strings are stable enough to form in a phase transition.
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Fig. 5. Loop formation from semilocal string segments. The "gure shows two snapshots, at t"70 and 80, of a 643 numerical simulation of a network of semilocal strings with b"0.05 from Ref. [4], where the ends of an open segment of string join up to form a closed loop (see Section 3.5 for a discussion of the simulations). Subsequently, the loops seem to behave like those of topological cosmic string, contracting and disappearing. Fig. 6. The growth of string segments to form longer strings. The "gure shows two snapshots, at time t"60 and 70 of a large 2563 numerical simulation of a network of semilocal strings with b"0.05 from Ref. [4]. Note several joinings of string segments, e.g. two separate joinings on the long central string, and the disappearance of some loops. The di!erent apparent thickness of strings is entirely an e!ect of perspective. The simulation was performed on the Cray T3E at the National Energy Research Scienti"c Computing Center (NERSC). See Section 3.5 for a discussion.
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We defer discussion of the electroweak case to Section 9.4. Here we want to review recent numerical simulations of the formation and evolution of a network of b(1 semilocal strings [3}5] which show that such strings should indeed form in appreciable numbers in a phase transition. The results suggest that, even if no vortices are formed immediately after U has acquired a non-zero vacuum expectation value, the interaction between the gauge "elds and the scalar "elds is such that vortex formation does eventually occur simply because it is energetically favourable for the random distribution of magnetic "elds present after the phase transition to become concentrated in regions where the Higgs "eld has a value close to that of the symmetric phase. Even though they do not account for the expansion of the Universe, these simulations represent a "rst step towards understanding semilocal string formation in cosmological phase transitions and they have already provided very interesting insights into the dynamical evolution of such a network. 3.5.1. Description of the simulations From a technical point of view, the numerical simulation of a network of semilocal strings has additional complications over that of ;(1) topological strings. Because there are not enough gauge degrees of freedom to cancel all of the scalar "eld gradients, the existence of string cores depends crucially on the way the "elds (scalar and gauge) interact. Another problem, generic to all non-topological strings, is that the winding number is not well de"ned for con"gurations where the scalar is away from a maximal circle in the vacuum manifold, and this makes the identi"cation of strings much more di$cult than in the case of topological strings. The strategy proposed in [3] to circumvent these problems is to follow the evolution of the gauge "eld strength in numerical simulations, since the "eld strength provides a gauge invariant indicator for the presence of vortices. The initial conditions are obtained by an extension of the Vachaspati}Vilenkin algorithm [130] appropriate to non-topological defects, plus a short period of dynamical evolution including a dissipation term (numerical viscosity) to aid the relaxation of con"gurations in the &basin of attraction' of the semilocal string. As with any new algorithm, it is essential to check that it reproduces previously known results accurately, and this has been done in [3]. Note that setting / "0 in the semilocal model obtains 2 the Abelian Higgs model, thus comparison with topological strings is straightforward, and it is used repeatedly as a test case, both to check the simulation techniques and to minimise systematic errors when quoting formation rates. In particular, the proposed technique is tested in a twodimensional toy model (representing parallel strings) in three di!erent ways: (a) restriction to the Abelian Higgs model gives good agreement with analytic and numerical estimates for cosmic strings in [130]; (b) the results are robust under varying initial conditions and numerical viscosities (see Fig. 8), and (c) they are in good agreement with previous analytic and numerical estimates for semilocal string formation in [7,60]. The results are summarized in Fig. 9. We refer the reader to Refs. [3}5] for details; however, a few comments are needed to understand those "gures. f The study takes place in #at space time. Temporal gauge and rescaled units (32) are chosen. Gauss' law, which here is a constraint derived from the gauge choice > "0, is used to test the 0 stability of the code.
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Fig. 7. The #ux tube structure in a two-dimensional semilocal string simulation with b"0.05, from Ref. [3]. The upper panel (t"0) shows the initial condition after the process described in the text. The lower panel shows the con"guration resolved into "ve #ux tubes by a short period of dynamical evolution (t"100). These #ux tubes are semilocal vortices.
f Space is discretized into a lattice with periodic boundary conditions. The equations of motion (56) are solved numerically using a standard staggered leapfrog method; however, to reduce its relaxation time an ad hoc dissipation term was added to each equation (gUQ and g>Q , respecti ively). A range of strengths of dissipation was tested, and it did not signi"cantly a!ect the number densities obtained. The simulations displayed in this section all have g"0.5. f The number density of defects is estimated by an extension of the Vachaspati}Vilenkin algorithm [130] by "rst generating a random initial con"guration for the scalar "elds drawn from the vacuum manifold, which is not discretized, and then "nding the gauge "eld con"guration that minimizes the energy associated with (covariant) gradients.10 If space is a grid of dimension N3, the correlation length is chosen to be some number p of grid points (p"16 in [3,4]; the size of the lattice is either N"64 or N"256.) To obtain a reasonably smooth con"guration for the
10 In fact, it turns out that the energy-minimization condition is redundant, since the early stages of dynamical evolution carry out this role anyway.
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Fig. 8. A test of the sensitivity of the results to the choice of initial conditions in a two-dimensional simulation with the algorithm proposed in Section 3.5. The plot shows the number of semilocal strings formed per initial two-dimensional correlation volume. Each point is an average over 10 simulations. Squares indicate that the vacuum initial conditions described in the text were used, while open circles indicate that non-vacuum (thermal) initial conditions were used. Both sets of initial conditions are seen to give comparable results. Statistical results are derived from a large suite of simulations (700 in all) carried out on a 643 grid (from Ref. [3]). Fig. 9. The ratio of lengths of semilocal and cosmic strings as a function of the stability parametr b, from [4].
scalar "elds, one throws down random vacuum values on a (N/p)3 subgrid; the scalar "eld is then interpolated onto the full grid by bisection. Strings are always identi"ed with the location of magnetic #ux tubes. For cosmic strings, the two-dimensional toy model accurately reproduces the formation rates of [130]. For semilocal strings, on the other hand, the initial con"gurations generated in this way have a complicated #ux structure with extrema of di!erent values (top panel of Fig. 7), and it is far from clear which of these, if any, might evolve to form semilocal vortices; in order to resolve this ambiguity, the initial con"gurations are evolved forward in time. As anticipated, in the unstable regime b'1 the #ux quickly dissipates leaving no strings. By contrast, in the stable regime b(1 string-like features emerge when con"gurations in the `basin of attractiona of the semilocal string relax unambiguously into vortices (bottom panel of Fig. 7). Since the initial conditions are somewhat arti"cial, the results were checked against various other choices of initial conditions, in particular di!erent initial conditions for the gauge "eld and also `thermala initial conditions for the scalar "eld (see Fig. 8 and Ref. [3] for a precise description of the initial conditions). All the initial conditions in [3,4] had zero initial velocities for the "elds. Initial conditions with non-zero "eld momenta have not yet been investigated. 3.5.2. Results and discussion These simulations give very important information on the dynamics and evolution of a network of semilocal strings. In particular, they con"rm our discussion in the previous subsection of the behaviour of the ends of string segments, and of strings with di!erent colours. String segments are seen to grow in order to join nearby ones or form closed loops, and very short segments are also
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observed to collapse and disappear. The colour degrees of freedom do not seem to introduce any new forces between strings. Because the strings tend to grow or form closed loops, time evolution makes the network resemble more and more a network of topological strings (NO vortices) but with lower number densities.11 Note that the correlation length in the simulations is constrained to be larger than the size of the vortex cores, to avoid overlaps. This results in a minimal value of the parameter b of around 0.05 (if b is lowered further, the scalar string cores become too wide to "t into a correlation volume, in contradiction with the vacuum values assumed in a Vachaspati}Vilenkin algorithm). Fig. 9 shows the results for seven di!erent values of b by taking several initial con"gurations on a 643 grid smoothed over every 16 grid-points. As expected, for b(1 the formation rate depends on b, tending to zero as b tends to 1. The ratio of semilocal string density to cosmic string density in an Abelian Higgs model for the same value of b is less than but of order one. For the lowest value of b simulated (b"0.05), the semilocal string density is about one-third of that of cosmic strings. One "nal word of caution about the possible cosmological implications of these simulations. We mentioned above that numerical viscosity was introduced to aid the relaxation of con"gurations close to the semilocal string. In an expanding Universe the expansion rate would provide some viscosity, though g would typically not be constant. This may have an important e!ect on the production of strings. Indeed, note the di!erent numbers of upward and downward pointing #ux tubes in Fig. 7, despite the zero net #ux boundary condition. The missing #ux resides in the smaller &nodules', made long-lived by the numerical viscosity; these are none other than the &skyrmions' described in Section 3. As was explained there, the natural tendency of skyrmions when b(1 is to collapse into strings, but the timescale for collapse increases quadratically with their size and Benson and Bucher [19] have argued that the e!ect of the expansion could stop the collapse of large skyrmions almost completely. On the other hand, one expects skyrmions to be formed with all possible sizes, so the e!ect of the expansion on the number density of strings remains an open question. Another important issue that has not yet been addressed is whether these semilocal networks show scaling behaviour, and whether reconnections are as rare as the above simulations suggest. Both would have important implications for cosmology. However, the answer to these and other questions may have to wait until full numerical simulations are available. 3.6. Generalisations and xnal comments (i) Charged semilocal vortices. The semilocal string solution described earlier in this section is strongly static and z-independent, by which we mean that D (U)"D (U)"0. It is possible to relax t z these conditions while still keeping the Lagrangian and the energy independent of z. The idea is that, as we move along the z-direction, the "elds move along the orbit of the global symmetries; in other words, Goldstone bosons are excited. Abraham has shown that it is possible to construct semilocal vortices with "nite energy per unit length carrying a global charge [1] in the Bogomolnyi limit b"1.12 They satisfy 11 However, one important point is that no intersection events were observed in the semilocal string simulations, so the rate of reconnection has not been determined. 12 By contrast, charged solutions with D (U)O0 in the Abelian Higgs model have in"nite energy per unit length [72]. 0
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a Bogomolnyi-type bound and are therefore stable. Perivolaropoulos [106] has constructed spinning vortices (however these have in"nite energy per unit length). (ii) Semilocal models with S;(N) ];(1) symmetry. The generalization of semilocal strings '-0"!-0#!to so-called extended Abelian Higgs models with an N-component multiplet of scalars whose overall phase is gauged is straightforward [127,59], and has been analysed in detail in [60,51]. The strings are stable (unstable) for b(1 (b'1) and for b"1 they are degenerate in energy with skyrmionic con"gurations labelled by an N!1 complex vector. For winding n, and widely separated vortices, the Nn complex parameters that characterize the con"gurations can be thought of as the n positions in R2&C and the (N!1)n &orientations'. (iii) Semilocal monopoles and generalized semilocality. We have seen that semilocal strings have very special properties arising from the fact that n (G/H)"0 but n (G /H )O0. 1 1 -0#!- -0#!An immediate question is whether it is possible to construct other non-topological defects such that n (G/H)"0 but n (G /H )O0 . k k -0#!- -0#!-
(81)
This possibility would be particularly interesting in the case of monopoles, k"2, since they might retain some of the features of global monopoles, in particular a higher annihilation rate in the early Universe. Surprisingly, the answer seems to be negative. Within a very natural set of assumptions, it was shown in [127] that the condition (81) can only be satis"ed if the gauge group G is Abelian, -0#!and therefore one cannot have semilocal monopoles (nor any other defects satisfying conditions (81) with k'1). However, Preskill has remarked that it is possible to de"ne a wider concept of semilocality [109] by considering the larger approximate symmetry G which is obtained in the limit where !11309 gauge couplings are set to zero. The symmetry G is partially broken to the exact symmetry !11309 G&G ]G (modulo discrete transformations) when the gauge couplings are turned on. It is -0#!'-0"!then possible to have generalized semilocal monopoles associated with non-contractible spheres in G /H which are contractible in the approximate vacuum manifold G /H even -0#!- -0#!!11309 !11309 though they are still non-contractible in the exact vacuum manifold G/H. Another obvious possibility is to have topological monopoles with `coloura, by which we mean extra global degrees of freedom, if the symmetry G&G ]G is such that the gauge orbits are '-0"!-0#!non-contractible two spheres, n (G /H )O1. Given that there are no semilocal monopoles 2 -0#!- -0#![127], these monopoles must have n (G/H)O1, so they are topologically stable, and they have 2 additional global degrees of freedom. (iv) Semilocal defects and Hopf xbrations. In the semilocal model, the action of the gauge group "bres the vacuum manifold S3 as a non-trivial bundle over S2&CP1, the Hopf bundle. The fact that this bundle is non-trivial is at the root of conditions (61), and is ultimately the reason why the topological criterion for the existence of strings fails. In view of this, Hindmarsh [60] has proposed an alternative de"nition of a semilocal defect: it is a defect in a theory whose vacuum manifold is a non-trivial bundle with "bre G /H . -0#!- -0#!Extended Abelian Higgs models [60] are similarly related to the "brations of the odd-dimensional spheres S2N~1 with "bre S1 and base space CPN~1. A natural question to ask is if the remaining Hopf "brations of spheres can also be realized in a "eld theoretic model. This question S3 S4 "bration in a quaternionic model. Other was answered a$rmatively in [61] for the S7P
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non-trivial bundles were also implemented in this paper, but to date the "eld theory realization of S7 S8 Hopf bundle remains an open problem. the S15P (v) Monopoles and textures in the semilocal model. Since the gauge "eld is Abelian, div B"0, and isolated magnetic monopoles are necessarily singular in semilocal models. The only way to make the singularity disappear is by embedding the theory in a larger non-Abelian theory which provides a regular core, or by putting the singularity behind an event horizon [51]. One important question that has not yet been addressed is if the scalar gradients in these spherical monopoles make them unstable to angular collapse into a #ux tube. A related system where this happens is in O(3) global monopoles where the spherically symmetric con"guration is unstable. In the semilocal case, it is possible that the pressure from the magnetic "eld might prevent the instability towards angular collapse. Finally, note that, because n (S3)"Z, there is also the possibility of textures in the semilocal 3 model (51). In contrast with purely scalar O(4) models, their collapse seems to be stopped by the pressure from the magnetic "eld [60]. Of course, they can still unwind by tunnelling. (vi) We should point out that systems related to the semilocal model have been studied in condensed matter. In [28], the system was an unconventional superconductor where the role of the global SU(2) group was played by the spin rotation group. In [135] the hypothetical case of an `electrically chargeda A-phase of 3He, i.e. a superconductor with the properties of 3He-A, was considered (see Section 11.1 for a brief discussion of the A and B phases of 3He). In this case the global group was SO(3), the group of orbital rotations. Both papers discussed continuous vortices in such superconductors, which correspond to the `skyrmionsa discussed here.
4. Electroweak strings In this section we introduce electroweak strings. There are two kinds: one, more precisely known as the Z-string, carries Z-magnetic #ux, and is the type that was discussed by Nambu and that becomes stable as it approaches the semilocal limit. It is associated with the subgroup generated by ¹ "na¹a!sin2 h Q . Z 8 There are other strings in the GSW model that carry S;(2) magnetic #ux, called =-strings. There is a one-parameter family of = strings which are all gauge equivalent to one another, and they are all unstable. They are generated by a linear combination of the S;(2) generators ¹` and ¹~. These will be discussed in more detail in the next section. 4.1. The Z string Modulo gauge transformations, the con"guration describing a straight, in"nitely long Z-string along the z-axis is [123]
AB
0 g U" f (o)e*r , NO 1 J2
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2 2 v (o) NO u( , Z"! v (o) du or, in vector notation, Z"! (82) NO g g o z z A "=B"0 , k k where f and v are the Nielsen}Olesen pro"les that solve Eqs. (40). It is straightforward to show that this is a solution of the bosonic equations of motion (alternatively, one can show that it is an extremum of the energy [123]). Eqs. (82) describe a string with unit winding. The solutions with higher winding number can be constructed in an analogous way. Note that the winding number is not a topological invariant; the unstable string can decay by unwinding until it reaches the vacuum sector. The solution (82) reduces to the semilocal string in the limit sin2 h "1, and therefore it is 8 classically stable for b(1 and unstable for b'1 (see Section 7), where b is now the ratio between the Higgs mass, J2jg and the Z-boson mass g g/2, thus z b"8j/g2 . (83) z The Z-string con"guration is axially symmetric, as it is invariant under the action of the generalised angular momentum operator M "¸ #S #I , (84) z z z z where ¸ , S and I are the orbital, spin and isospin parts, respectively, de"ned in Section 9.2. z z z The Z-string carries a Z-magnetic #ux F "4p/g (85) Z z thus particles whose Z charge is not an integer multiple of g /2 will have Aharonov}Bohm z interactions with the string (see Section 8.3). The Z-string can terminate on magnetic monopoles (such con"gurations are discussed in Section 5). When a string terminates, the discrete Aharonov}Bohm interaction can be smoothly deformed to the trivial interaction. The smoothness is provided by the presence of the magnetic #ux of the monopole. Note that, in the background given by (82), the covariant derivative becomes g d ,D D "R #i z [!2(¹3!Q sin2 h )]Z 8 k k k Zv453*/' k 2
(86)
in particular, left and right fermion "elds couple to Z with di!erent strengths, since the e!ective k Z-charge q"!2(¹3!Q sin2 h ) (87) 8 has di!erent values, q "q $1. (Note that q is proportional to the string generator ¹ , de"ned in R L z Eq. (17); the proportionality factor has been introduced for later convenience.) This will be important when discussing scattering. Note also that, for the Higgs "eld, q"diag(!cos 2h , 1) . (88) 8 Ambj+rn and Olesen [10] and, more recently, Bimonte and Lozano [22] have derived Bogomolnyi-type bounds for periodic con"gurations in the GSW model. They consider static con"gurations such that physical observables are periodic in the xy-plane and cylindrically symmetric in each cell.
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If A is the area of the basic cell, they "nd that the energy (per unit length) satis"es E5(1/2e2)m2 (2g@F !m2 A) W Y W
if m 5m , H Z
(89)
E5(1/2e2)(m m /m )2(2g@F !m2 A) if m (m , W H Z Y W H Z where F is the magnetic #ux of the hypercharge "eld through the cell. Note that the top line of (89) Y reduces to the familiar E5SUsUTqF for the Abelian Higgs and semilocal case in the gP0 limit Y (with q"g@/2). In the non-Abelian case the bound involves an area term and therefore does not admit a topological interpretation. In the Bogomolnyi limit, m "m , the bound is saturated for con"gurations satisfying the "rst H Z order Bogomolnyi equations D #iD U"0 , 1 2
A
B
g@ g2 > # UsU! "0 , 12 2 2 sin2 h 8
(90)
Wa #1gUsqaU"0 . 12 2 A solution to these equations describing a lattice of Z-strings was constructed in [22]. Other periodic con"gurations with symmetry restoration had been previously found in the presence of an external magnetic "eld in [10].
5. The zoo of electroweak defects The electroweak Z-string is one member in the zoo of electroweak defects. Other members include the electroweak monopole, dyon and the =-string. The latter fall in the class of `embedded defectsa and this viewpoint provides a simple way to characterize them. The electroweak sphaleron is also related to the electroweak defects. 5.1. Electroweak monopoles To understand the existence of magnetic monopoles in the GSW model, recall the following sequence of facts: f The Z-string does not have a topological origin and hence it is possible for it to terminate. f As the hypercharge component of the Z-"eld in the string is divergenceless it cannot terminate. Therefore it must continue from within the string to beyond the terminus. f However, beyond the terminus, the Higgs is in its vacuum and the hypercharge magnetic "eld is massive. Then, if the massive hypercharge #ux was to continue beyond the string, it would cost an in"nite amount of energy and this is not possible. f The only means by which the hypercharge "eld can continue beyond the terminus is in combination with the S;(2) "elds such that it forms the massless electromagnetic magnetic "eld.
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Fig. 10. The outgoing hypercharge #ux of the monopole passing through the surface R!S should equal the incoming hypercharge #ux through the Z-string.
So the terminus of the Z-string is the location of a source of electromagnetic magnetic "eld, that is, a magnetic monopole [102]. We now make this argument more quantitative. Assume that we have a semi-in"nite Z-string along the !z-axis with terminus at the origin (see Fig. 10). Let us denote the A- and Z-magnetic #uxes through a spatial surface by F and F . A Z These are given in terms of the =- and >-#uxes by taking surface integrals of the "eld strengths (see Eqs. (19) and (20)). Therefore F "cos h F !sin h F , F "sin h F #cos h F , (91) Z 8 n 8 Y A 8 n 8 Y where we have denoted the S;(2) #ux (parallel to na in group space) by F and the hypercharge n #ux by F . Y Now consider a large sphere R centered on the string terminus. The "eld con"guration is such that there is only A-#ux through R except near the South pole (S) of R, where there is only a Z magnetic #ux. Hence, F DR "0, F D "0 . AS Z ~S Together with (91) this gives,
(92)
F DR "tan h F DR , F D "!cot h F D . 8 Y ~S nS 8 YS n ~S The hypercharge #ux must be conserved as it is divergenceless. So
(93)
F DR "!F D ,F , Y ~S YS Y and, inserting this and (93) in (91) yields
(94)
F DR "F /cos h , F D "F /sin h . A ~S Y 8 ZS Y 8
(95)
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Now the #ux in the Z-string along the !z-axis is quantized in units of 4p/g (recall z g "e/cos h sin h gives the coupling of the Z boson to the Higgs "eld). Therefore, for the unit z 8 8 winding string, F D "4p/g . ZS z Then (95) yields
(96)
(97) F "(4p/g )sin h , F DR "(4p/g )tan h "(4p/e)sin2 h . z 8 8 Y z 8 A ~S Hence the terminus of the string has net A-#ux emanating from it and hence it is a magnetic monopole. The electromagnetic #ux of the electroweak monopole appears to violate the Dirac quantization condition. However, this is not true since one must also take the Z-string into account when deriving the quantization condition relevant to the electroweak monopole. This becomes clearer when we work out the magnetic #ux for the S;(2) "elds. Using (93) with (97), the net non-Abelian #ux is F "F D #F DR "4p/g (98) n nS n ~S just as we would expect for a 't Hooft-Polyakov monopole [65]. That is, the Dirac quantization condition works perfectly well for the S;(2) "eld and the monopole charge is quantized in units of 4p/g. Another way of looking at (98) is to say that the electroweak monopole is a genuine S;(2) monopole in which there is a net emanating ;(1) LS;(2) #ux. The structure of the theory, n however, only permits a linear combination of this #ux and hypercharge #ux to be long range and so there is a string attached to the monopole. But this string is made of Z "eld which is orthogonal to the electromagnetic "eld and so the string does not surreptitiously return the monopole electromagnetic #ux. Also, the magnetic charge on the monopole is conserved and electroweak monopoles can only disappear by annihilating with antimonopoles. It is useful to have an explicit expression describing the asymptotic "eld of the electroweak monopole and string. Nambu's monopole-string con"guration, denoted by (UM , = M a , >M ), is k k cos(h/2) g UM " , (99) J2 sin(h/2)e*r
A
B
where h and u are spherical coordinates centred on the monopole, and the gauge "eld con"guration is g= M a "!eabcnbR nc#i cos2 h na(UM sR UM !R UM sUM ) , (100) k k 8 k k g@>M "!i sin2 h (UM sR UM !R UM sUM ) , (101) k 8 k k where na is given in Eq. (16). Note that there is no regular electroweak con"guration that represents a magnetic monopole surrounded by vacuum in the GSW model. 5.2. Electroweak dyons Given that the electroweak monopole exists, it is natural to ask if dyonic con"gurations exist as well. We now write down dyonic con"gurations that solve the asymptotic "eld equations [126].
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The existence of such con"gurations is implicit in Nambu's original paper in the guise of what he called `externala potentials [102]. Essentially, the dyon solution is an electroweak monopole together with a particular external potential. The ansatz that describes an electroweak dyon connected by a semi-in"nite Z string is U"UM ,
(102)
=a"= M a!dt nafQ /cos h , (103) 8 >">M !dt fQ /sin h , (104) 8 where f"f(t, x), overdots denote partial time derivatives and barred "elds have been de"ned in the previous subsection. We now need to insert this ansatz into the "eld equations and to "nd the equation satis"ed by f. Some algebra leads to RiR fQ "0, R RifQ "0 , i t which can be solved by separating variables, f"m(t) f (x) .
(105) (106)
This leads to mK "0, +2f"0 .
(107)
The particular solution that we will be interested in is the solution that gives a dyon. Hence, we take q sin h cos h 1 8 8 , m"m t, f (r)"! (108) 0 4pm r 0 where m and q are constants. Now, using (108), together with (103), (104) and (106), we get the dyon 0 electric "eld E "(q/4p)r/r3 . (109) A For a long segment of string, the monopole and the antimonopole at the ends are well separated and we can repeat the above analysis for both of them independently. Therefore, the electric charge on the antimonopole at one end of a Z-string segment is uncorrelated with the charge on the monopole at the other end of the string. This means that we can have dyons of arbitrary electric charge at either end of the string. The situation will change with the inclusion of fermions since these can carry currents along the string and transport charge from monopole to antimonopole. This completes our construction of the dyon-string system in the GSW model. As of now, the charge q on the dyon is arbitrary. Quantum mechanics implies that the electric charge must be quantized. If we include a h term in the electroweak action (but no fermions):
P
g2h S " d4 x=a = I kla , h 32p2 kl
(110)
where = I kla"1ekljp=a , 2 kl
(111)
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then the charge quantization condition becomes q"(n#(h/2p))e .
(112)
This agrees with the standard result for dyons [139]. In the GSW model with fermions, it is known that the h term can be eliminated by a rotation of the fermionic "elds. This argument can be turned around to argue that the CP violation in the mass matrix of the fermions will lead to an e!ective h term and so the electroweak monopoles will indeed have a fractional charge with h being related to the CP violation in the mass matrix. The precise value of the fractional electric charge on electroweak monopoles has not yet been calculated and remains an open problem. It should be mentioned that, even though the electric charge on an electroweak dyon can be fractional as in (112), the total electric charge on the dyon-string system is always integral because the CP violating fractional charge on the monopole is equal and opposite to that on the antimonopole. 5.3. Embedded defects and =-strings A very simple way of understanding the existence of electroweak string solutions is in terms of embedded defects. While this method does not shed any light on the stability of the electroweak string, it does provide a scheme for "nding other solutions. The idea is that the electroweak symmetry group contains several ;(1) subgroups which break completely when the electroweak symmetry breaks. Corresponding to each such breaking, one might have a string solution. A more complete analysis tells us when such a solution can exist [128,15,35,87]. Consider the general symmetry breaking GPH .
(113)
Suppose G is a subgroup of G which, in this process, breaks down to G WH. Then we ask the %." %." question: when are topological defects in the symmetry breaking G PG WH (114) %." %." also solutions in the full theory? An answer to this question requires separating the gauge "elds into those that transform within the G subgroup and those that do not. Similarly, the Higgs "eld %." components are separated into those that lie in the embedded vector space of scalar "elds and those that do not. Then, it is possible to write down general conditions under which solutions can be embedded [15,35]. Here we shall not describe these conditions but remark that the Z-string is due to the embedded symmetry breaking ;(1) P1 , (115) Z where the ;(1) is generated by ¹ , de"ned in Eq. (17). Now, there are other ;(1)'s that can be Z Z embedded in the GSW model which lie entirely in the S;(2) factor. For example, we can choose ;(1) which is generated by ¹1 (one of the o!-diagonal generators of S;(2)). Since we have 1 ;(1) P1 (116) 1
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when the electroweak symmetry breaks, there is the possibility of another string solution in the GSW model. Indeed, it is easily checked that this string can be embedded in the GSW model and the solution is called a =-string. By considering a one-parameter family of ;(1) subgroups generated by ¹ " cos(f)¹1#sin(f)¹2 , f we can generate a one-parameter family of =-strings
A
B
cos u g U" f (o) , ie~*f sin u J2 NO
(117)
(118)
=1"!(2/g)cos f v
(o) du, =2"!(2/g)sin f v (o) du , (119) NO NO and all other "elds vanish. Although the string solutions are gauge equivalent for di!erent values of f, the parameter does take on physical meaning when considering multi-string con"gurations in which the value of f is di!erent for di!erent strings [15]. Note that the generator (117) can be obtained from ¹1 by the action of the unbroken (electromagnetic) group, ¹ "e*fQ¹1e~*fQ . (120) f With this in mind, Lepora et al. [87,88] have classi"ed embedded vortices. The idea is that, for a general symmetry breaking GPH, the Lie algebra of G, G, decomposes naturally into a direct sum of the space H of generators of the unbroken subgroup H (the ones associated with massless gauge bosons) and the space M of generators associated with massive gauge bosons: G"H#M. The action of H on the subspace M further decomposes M into irreducible subspaces. The classi"cation of embedded vortices is based on this decomposition, as we now explain. Recall (Eq. (37)) that "nite energy vortices are associated with gauge orbits on the vacuum manifold.13 Choosing a base point U in the vacuum manifold, each embedded vortex can be 0 associated to a Lie algebra generator which is tangent to the gauge orbit describing the asymptotic scalar "eld con"guration of the vortex. The unbroken subgroup H at U `rotatesa the various 0 gauge orbits among themselves as in Eq. (120). Thus, the action of H splits the space of gauge orbits into irreducible subspaces. Except for critical values of the coupling constants (which could lead to so-called combination vortices), it can be shown [15,87] that embedded vortices have to lie entirely in one of these irreducible subspaces. If the subspaces have dimension greater than one, then there may be a family of gauge-equivalent vortices. In the GSW model, for instance, the Lie Algebra decomposes into H#M #M where H is 1 2 spanned by the charge Q, M is a one-dimensional subspace spanned by ¹ (corresponding to the 1 Z Z-string) and M is a two-dimensional subspace comprising all =-string generators ¹ . 2 f
13 The gauge orbits are geodesics of a squashed metric on the vacuum manifold which is di!erent from the isotropic metric relevant to the scalar sector [89].
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Both the =- and the Z-string are embedded string solutions in the GSW model. What makes the Z-string more interesting is its unexpected stability properties. It can be shown [87] that only those vortices lying in one-dimensional subspaces can have a stable semilocal limit. Thus, embedded vortices belonging to a family are always unstable. Another important di!erence is that the Z-string is known to terminate on magnetic monopoles but this is not true of the =-string. The =-string can terminate without any emanating electromagnetic "elds since it is entirely within the S;(2) sector of the GSW model. It is straightforward to embed domain walls in the GSW model. There are no embedded monopoles in the GSW model since there is no S;(2) subgroup that is broken to ;(1).
6. Electroweak strings in extensions of the GSW model Electroweak strings have been discussed in various extensions of the GSW model. We describe some of this work below. We do not, however, discuss extensions in which topological strings are produced at the electroweak scale [38,23]. 6.1. Two Higgs model As discussed in Section 5.3, the Z-string is an embedded string in the GSW model. The general conditions that enable the embedding are valid even with a more complicated Higgs structure. Here we will consider the two Higgs doublet model which is inspired by supersymmetric extensions of the GSW model. In a two Higgs doublet model, the Higgs structure of the GSW model is doubled so that we have scalars U and U and the scalar potential is [73] 1 2 l2 2 l2 2 l2 l2 2 <(U , U )"j Us U ! 1 #j Us U ! 2 #j Us U ! 1 # Us U ! 2 1 2 1 1 1 2 1 1 2 2 2 2 2 2 2 2 3
A
B
A
D
C
C
B
CA
B A D
2 l l #j [(Us U )(Us U )!(Us U )(Us U )]#j Re(Us U )! 1 2 cos m 4 1 1 2 2 1 2 2 1 5 1 2 2
C
DC
l l l l 2 #j Im(Us U )! 1 2 sin m #j Re(Us U )! 1 2 cos m 6 1 2 1 2 7 2 2
BD
D
l l Im(Us U )! 1 2 sin m . (121) 1 2 2
Here l and l are the respective VEVs of the two doublets, j are coupling constants and the 1 2 i parameter m is a phase. In polar coordinates, the solution for the two Higgs Z-string is
AB AB
0 , U "l f (o)e*r 1 1 1 1
(122)
0 U "l f (o)e*r , 2 2 2 1
(123)
2 v(o) Z"! u( g o z
(124)
390
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with the pro"le functions satisfying di!erential equations similar to the Abelian}Higgs case. These have been studied in Ref. [39] where the stability has also been analyzed (also see [120]). Other cylindrically symmetric solutions have been considered in [13]. 6.2. Adjoint Higgs model The GSW model with an additional S;(2) "eld in the adjoint representation, v, is what we shall refer to as the `adjoint Higgs modela. The impact of the adjoint "eld on electroweak defects was considered in Ref. [75]. The bosonic sector of the adjoint Higgs model is ¸"¹ #D(R #igea=a )vD2!<(U, v)#¸ , (125) %8 k k f where ¹ is the gradient part of the bosonic sector of the electroweak Lagrangian, ¸ is the %8 f fermionic part of the Lagrangian, ea "e (a, i, j"1, 2, 3) and ij aij <(U, v)"!k2 UsU!k2 v2#j (UsU)2#j v4#av2UsU#bv ) UssU . (126) 2 3 2 3 If we impose an additional Z symmetry on the Lagrangian under UP#U, vP!v, the 2 symmetry is ([S;(2) ];(1) ]/Z )]Z and we must set b"0. In what follows, we shall only L Y 2 2 consider this case and henceforth ignore the last (cubic) term in the potential. In this case, an additional simpli"cation is that the leptons and quarks do not couple to v and so ¸ is identical to f the fermionic Lagrangian of the GSW model. (If the Z symmetry is absent, the cubic term in the 2 potential is allowed but is constrained to be small by experiment.) In a cosmological context, as the universe cools down from high temperatures, if the parameters lie in a certain range [75] there will "rst be a phase transition in which the adjoint "eld gets a VEV. The VEV of the adjoint will break the S;(2) factor of the high-temperature symmetry group to ;(1). If the VEV of v is along the (0, 0, 1) direction, the generator of this ;(1) will be ¹3 and we will denote the unbroken subgroup as ;(1) . So the symmetry-breaking pattern at this stage is 3 ([S;(2)];(1) ]/Z )]Z P([;(1) ];(1) ]/Z )]Z (127) Y 2 2 3 Y 2 2 and topological magnetic monopoles will be produced with pure ;(1) #ux. 3 At a lower temperature, the doublet "eld will also get a VEV with the e!ect ([;(1) ];(1) ]/Z )]Z P;(1) 3 Y 2 2 %. where, as usual, the electromagnetic charge operator is
(128)
Q"¹3#1> . (129) 2 The electromagnetic component (A) from the monopoles is massless but the orthogonal part (Z) of the #ux is massive and gets con"ned to a string. This is the Z-string. In addition, the breaking of the Z factor gives domain walls. 2 In the second stage of symmetry breaking, the Z-string is topological and hence is stable. The presence of magnetic monopoles from the earlier symmetry breaking means that the Z-strings can break by terminating on monopoles. But, as the monopoles form at a higher energy scale, their mass is much larger than the energy scale at which strings form and which sets the scale for the tension in the string. So the string can only break by instanton processes.
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At a yet lower temperature, the VEV of the adjoint turns o!. This makes no di!erence to the symmetry structure of the model (apart from restoring the Z symmetry and eliminating 2 the domain walls) and hence no signi"cant di!erence to the monopoles connected by strings. However, it does a!ect the stability of the strings since the monopoles are no longer topological.
7. Stability of electroweak strings 7.1. Heuristic stability analysis As described in [123], the Z-string goes over into the semilocal string in the limit h Pp/2 and 8 hence the stability of the Z-string should match on continuously to that of the semilocal string. Therefore we expect that Z-strings should be stable if h is close to p/2 and m 4m . 8 H Z The stability analysis to certain subsets of perturbations can be carried out much more easily than to the completely general perturbations. The subset includes perturbations in the Higgs "eld and =-"elds separately. Such analyses may be found in [123,124,15] and [107]. (i) Higgs xeld perturbations: Perturbations in the Higgs "eld alone have maximum destabilizing e!ect for h "p/4 [15] and, in this case, it is easy to see that the Z-string is unstable. Consider the 8 one-parameter family of "eld con"gurations U(x; m)" cos m U (cos m x)#sin m U , 0 M
(130)
Z (x; m)" cos m Z (cos m x) , j (0)j
(131)
where the string solution is denoted by the 0 subscript, m3[0, p/2] and
AB
g 1 U " . M J2 0
(132)
For m"0 the "eld con"guration is the unperturbed Z-string while for m"p/2 it is the vacuum. The energy per unit length of this "eld con"guration can be evaluated and is found to be E(m)"cos2 m E(m"0) .
(133)
Hence the energy per unit length of the string is a monotonically decreasing function of m and so the string is unstable to decay into the vacuum. (ii) Incontractible two spheres: James [70], and Klinkhamer and Olesen [81] have constructed the Z- and =-string solutions by considering incontractible two spheres in the space of electroweak "eld con"gurations in two spatial dimensions. The idea was introduced by Taubes [119] and was used by Manton to construct the sphaleron [97,80]. The procedure (known as the `minimaxa procedure) is to construct a set of "eld con"gurations that are labelled by some parameters k . If i this set is incontractible in the space of "eld con"gurations, then there exist (subject to certain assumptions [97]) values of the parameters for which the "eld con"guration extremizes the energy functional. For example, Klinkhamer and Olesen [81] give the following construction for the
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Z-string in terms of a two-parameter (k, l) family of "eld con"gurations p/24[kl]4p: ="0, >"0 U"(1!M1!h(o)Nsin[kl])
g
AB 0
J2 1
,
="!f (o)Ga¹a, >"f (o)sin2 h F3, 8 0 g X; , U"h(o) 1 J2
(134)
04[kl]4p/2:
AB
(135)
where = and > are Lie algebra-valued 1-forms (e.g. ="=a ¹a dxk), [kl],max(DkD, DlD), k Fa¹a"2i;~1 d; , (136) Ga¹a"X;[F1¹1#F2¹2# cos2 h F3¹3];~1X~1 , 8 ;(k, l, u)"!i sin kq !i cos k sin lq !i cos k cos l sin uq # cos k cos l cos u1 , 1 2 3 X";(k, l, u"0)~1 ,
(137) (138) (139)
and the functions f (o) and h(o) satisfy the boundary conditions f (0)"0"h(0), f (R)"1"h(R) .
(140)
This set of "eld con"gurations labelled by the parameters k, l3[!p, p] de"nes an incontractible two sphere in the space of "eld con"gurations. This is seen by considering the "elds as if they were de"ned on the three sphere on which the coordinates are u, k and l and then showing that the "eld con"gurations de"ne a topologically non-trivial mapping from this S3 to the vacuum manifold which is also an S3. Then the minimax procedure says that there is an extremum of the energy at some value of the parameters. By inserting the "eld con"gurations into the energy functional, it can be checked that the extremum occurs at k"0"l, when the con"guration coincides with that of the Z-string. Furthermore, for h 4p/4, the extremum is a maximum and hence the Z-string is 8 unstable. A very similar analysis has been done [70,81] for the =-string con"rming the result [15] that it is always unstable. (iii) W-condensation: There is also a well-known [11] instability to perturbations in the =-"elds alone called `=-condensationa. Application of this instability to the Z-string may be found in [107,124,125,6]. A heuristic argument goes as follows. The energy of a mass m, charge e and spin s particle in a uniform magnetic "eld B along the z-axis is given by E2"p2#m2#(2n#1)eB!2eB ) s , (141) z where n"0, 1, 2,2 labels the Landau levels and p is the momentum along the z-axis. Now, if z s"1, the right-hand side can be negative for p "0, n"0 provided z B'm2/e . (142)
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This signals an instability towards the spontaneous creation of spin one particles in su$ciently strong magnetic "elds [11]. In our case, the magnetic "eld is a Z-magnetic "eld and this couples to the spin one =-particles. If the string thickness is larger than the Compton wavelength of the =-particles, the Z-magnetic "eld may be considered uniform. Also, the relevant charge in this case is the Z-charge of the =-bosons and is g cos2h . The constraint that the string be thick so that the Z-magnetic "eld Z 8 appears uniform and that the charge not be too small means that h should be small. Hence the 8 instability towards =-condensation applies for small h . This analysis can be performed more 8 quantitatively [107] with the result that there is a relatively hard bound sin2 h '0.8 for the string 8 to be stable to =-condensation. 7.2. Detailed stability analysis To analyse the stability of electroweak strings, we perturb the string solution, extract the quadratic dependence of the energy on the perturbations and then determine if the energy can be lowered by the perturbations by solving a SchroK dinger equation. The analysis is quite tedious [71,6,57,93,94] and here we will only outline the main steps. We use the vector notation in this section for simplicity. The general perturbations of the Z-string are (/ , / , dZ, W a6 , A) , M ,
(143)
where a6 "1, 2, / and / are scalar "eld #uctuations de"ned by M ,
A
U"
/
/
M #/ NO ,
B
,
(144)
dZ is de"ned by Z"Z
NO
#dZ .
(145)
(The subscript NO means that the "eld is the unperturbed Nielsen}Olesen solution for the string as described in Section 2.) The "elds W a6 , A are perturbations since the unperturbed values of these "elds vanish in the Z-string. The perturbations can depend on the z-coordinate and the z-components of the vector "elds can also be non-zero. However, since the vortex solution has translational invariance along the z-direction, it is easy to see that it is su$cient to consider z independent perturbations and to ignore the z-components of the gauge "elds. This follows from the expression for the energy resulting from the Lagrangian in Eq. (4) where the relevant z-dependent terms in the integrand are 1Ga Ga #1F F #(D U)s(D U) 2 i3 i3 4 Bi3 Bi3 3 3
(146)
and explicitly provide a positive contribution to the energy. Hence, we drop all reference to the z-coordinate with the understanding that the energy is actually the energy per unit length of the string.
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Now we calculate the energy of the perturbed con"guration, discarding terms of cubic and higher order in the in"nitesimal perturbations. We "nd E"(E #dE )#E #E #E , (147) NO NO M # W where E is the energy of the Nielsen}Olesen string and dE is the energy variation due to the NO NO perturbations / and dZ. The term E is due to the perturbation / in the upper component of the , M M Higgs "eld, E is the cross-term between perturbations in the Higgs and gauge "elds, while E is # W the contribution from perturbing the gauge "elds alone:
P
E " d2x[DdM / D2#jg2( f 2!1)D/ D2] , j M M M
P
g E "i z cos h d2x[Us¹a6 d U!(d U)s¹a6 U]=a6 , j 8 j j # 2
(148) (149)
with d de"ned in (86), j
P
E " d2x[cW 1]W 2 ) $]Z#1D$]W 1#cW 2]ZD2 W 2 #1D$]W 2#cZ]W 1D2#1g2f 2(W a6 )2#1($]A)2] , (150) 2 4 2 where c,g cos h , 8 dM ,R !i(g /2) cos(2h )Z (151) j j z 8 j and the f and Z "elds in the above equations are the unperturbed "elds of the string. The two instabilities discussed in the previous subsection can also be seen in Eq. (147). First consider perturbations in the Higgs "eld alone. Then only E is relevant. For h "p/4, dM "R , M 8 j j and E is the energy of a particle described by the wavefunction / in a purely negative potential M M in two dimensions since f 241 everywhere. It is known that a purely negative potential in two dimensions always has a bound state.14 Hence, the energy can be lowered by at least one perturbation mode and so the string is unstable when h "p/4. The instability towards =8 condensation can be seen in E . The term with c can be negative and its strength is largest for small W h . Hence =-condensation is most relevant for small h . 8 8 Returning to the full stability analysis, we "rst note that the perturbations of the "elds that make up the string do not couple to the other available perturbations, i.e. the perturbations in the "elds f and v only occur inside the variation dE . Now, since we know that the Nielsen}Olesen string NO with unit winding number is stable to perturbations for any values of the parameters then necessarily, dE 50 and the perturbations / and dZ cannot destabilize the vortex. Then, we are NO , justi"ed in ignoring these perturbations and setting dE "0. Also we note that the A boson only NO appears in the last term of Eq. (150) and this is manifestly positive. So we can set A to zero. 14 For some potentials though, the wavefunction of the bound state may have singular (though integrable) behaviour at the origin and such bound states would be inadmissible for us since we require that the perturbations be small. This turns out not to be the case for the potential in Eq. (148).
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The remaining perturbations can be expanded in modes: / "s(o)e*mr M for the mth mode where m is any integer. For the gauge "elds we have
C C
(152)
D D
1 W1" M fM (o) cos(nu)#f (o) sin(nu)Ne( # M!hM (o) sin(nu)#h (o) cos(nu)Ne( , 1 1 o o 1 1 r
(153)
1 W2" M!fM (o) sin(nu)#f (o) cos(nu)Ne( # MhM (o) cos(nu)#h (o) sin(nu)Ne( 2 2 o o 2 2 r
(154)
for the nth mode where n is a non-negative integer. The most unstable mode is the one with m"0 and n"1. This is because these have the lowest gradient energy and are the only perturbations that can be non-vanishing at o"0. Further analysis shows that the string is most unstable to the h #h mode. Hence, we can ignore f , 1 2 i h !h and the barred variables. A considerable amount of algebra then yields 1 2 s (155) dE[s, m ]"2p do o(s, m )O ` ` m ` where O is a 2]2 matrix di!erential operator and
P
A B
m "(h #h )/2 . (156) ` 1 2 Before proceeding further, note that a gauge transformation on the "elds does not change the energy. However, we have not "xed the gauge in the preceding analysis and hence it is possible that some of the remaining perturbations, (s, m ), might correspond to gauge degrees of freedom and ` may not a!ect the energy. So we now identify the combination of perturbations s and m that are ` pure gauge transformations of the string con"guration. The S;(2) gauge transformation, exp(igt), of an electroweak "eld con"guration leads to "rst-order changes in the "elds of the form dU"igtU , d= "!iD(0)t , (157) 0 i i where = "=a¹a, t"ta¹a, and the 0 index denotes the unperturbed "eld and covariant i i derivative. In our analysis above, we have "xed the form of the unperturbed string and so we should restrict ourselves to only those gauge transformations that leave the Z-string con"guration unchanged. (For example, dU should only contain an upper component and no lower component.) This constrains t to take the form
A
t"s(o)
B
0
ie~*r
!ie*r
0
,
(158)
where s(o) is any smooth function. This means that perturbations given by
A B s(o)
m (o) `
A
"s(o)
!gg f (o)/J2
B
2(1!2 cos2 h v(o)) 8
(159)
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are pure gauge perturbations that do not a!ect the string con"guration. Therefore, such perturbations cannot contribute to the energy variation and must be annihilated by O. Then, in the two-dimensional space of (s, m ) perturbations, we can choose a basis in which one direction is ` pure gauge and is given by (159) and the other orthogonal direction is the direction of physical perturbations. The physical mode can now be written as f(o)"(1!2 cos2 h v(o))s(o)#(gg f (o)/2J2)m (o) . ` 8 So now the energy functional reduces to one depending only on f(o):
(160)
P
dE[f]"2p do o fOM f ,
(161)
where OM is the di!erential operator
A
B
1 d o d #;(o) OM "! o do P do `
(162)
and
A B
f @2 4S 1 d of @ ` # ;(o)" # , P f 2 g2g2o2f 2 o do P f ` ` where
(163)
g2g2o2f 2 P "(1!2 cos2 h v)2# ` 8 4
(164)
C
D
d v@ (1!2 cos2 h v) g2g2f 2 4 cos4 h v@2 8 #o 8 ! 2 cos2 h . (165) S (o)" 8 ` P (o) do o P (o) 4 ` ` The question of Z-string stability reduces to asking if there are negative eigenvalues u of the SchroK dinger equation OM f"uf .
(166)
The eigenfunction f(o) must also satisfy the boundary conditions f(o"0)"1 and f@(0)"0 where prime denotes di!erentiation with respect to o. In this way the stability analysis reduces to a single SchroK dinger equation which can be solved numerically. The results of the stability analysis are shown in Fig. 11 as a plot in parameter space (m /m , sin2 h ), demarcating regions where the Z-string is unstable (that is, where negative H Z 8 u exist) and stable (negative u do not exist). It is evident that the experimentally constrained values: sin2 h "0.23 and m /m '0.9 lie entirely inside the unstable sector. Hence the Z-string in the 8 H Z GSW model is unstable. The stability analysis of the Z-string described above leaves open the possibility that the string might be stable in some special circumstances such as, the presence of extra scalar "elds, or a magnetic "eld background, or fermions. We now describe some circumstances in which the Z-string stability has been analysed.
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Fig. 11. The Z-string is stable in the triangular shaded region of parameter space. At sin2 h "0.5, the string has 8 a scaling instability. The experimentally allowed parameters are also shown.
7.3. Z-string stability continued The stability of Z-strings has been studied in various other circumstances: (i) Thermal ewects: In [64] the authors examined thermal e!ects on Z-string stability using the high-temperature e!ective potential and found slight modi"cations to the stability. The conclusion is that Z-strings in the GSW model are unstable at high temperatures as well. In the same paper, left}right symmetric models were studied and it was found that these could contain stable strings that are similar to the Z-string. (ii) Extra scalar xelds: It is natural to wonder if the presence of extra scalar "elds in the model can help provide stability. In [39] the stability was examined in the physically motivated two Higgs doublet model with little advantage. In [131] it was shown that an extra (globally) charged scalar "eld could enhance stability. The extra complex scalar "eld, t, is coupled to the electroweak Higgs by a term DtD2UsU and hence the charges have lower energy on the string where UsU&0 than outside the string where U has a non-zero VEV. This is exactly as in the case of non-topological solitons or Q-balls [113,46,33]. However, scalar global charges attract and this can cause an instability of the charge distribution along the string [34,131]. For realistic parameters, stable Z-strings do not seem likely even in the presence of extra scalar "elds. (iii) Adjoint scalar xeld: A possible variant of the above scheme is that an S;(2) adjoint can be included in the GSW model as described in Section 6.2. Now, since the Z-string is topological within the second symmetry breaking stage in Eq. (128), it is stable. However, to be consistent with current experimental data the VEV of the S;(2) adjoint must vanish at a lower energy scale. At this stage the Z-string becomes unstable. Hence, in this scheme, there could be an epoch in the early universe where Z-strings would be stable. (iv) External magnetic xeld: An interesting possibility was studied by Garriga and Montes [49] when they considered the stability of the Z-string placed in an external electromagnetic magnetic "eld of "eld strength B parallel to the string. First, note that B should be less than B "m2 /e, # W
398
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Fig. 12. The triangular regions depict the parameter range for which the electroweak vacuum and the Z-string are both stable in the presence of a uniform external magnetic "eld whose strength is proportional to K. For a range of magnetic "eld (K&0.85), stable strings are possible even with the experimentally constrained parameter values.
otherwise the vacuum outside the string is unstable to =-condensation [11]. Then they found that the Z-string could be stable if B'JbB , where b"m2 /m2 should be less than 1 for stability of the # H Z ambient vacuum. The region of stability for a few values of the magnetic "eld (given by K"g B/2m2 ) is sketched in Fig. 12. For a certain range of K&0.85, stable Z-strings in the GSW z Z model are still just possible. A way to understand the enhanced stability of the Z-string in a magnetic "eld is to realize that the =-condensation instability is due to the interaction of =3 "sin h A #cos h Z and =B. k 8 k 8 k k The Z-string itself has a Z magnetic #ux. Then the external electromagnetic #ux can serve to lower the net =3 #ux. This reduces the e$ciency of =-condensation and makes the string more stable. Another viewpoint can be arrived at if we picture the Z-string instability to be one in which the string breaks due to the production of a monopole-antimonopole pair on the string. If the external magnetic "eld is oriented in a direction that prevents the nucleated magnetic monopoles from accelerating away from each other, it will suppress the monopole pair production process, leading to a stabilization of the string for su$ciently strong magnetic "elds. (v) Fermions: The e!ect of fermions on the stability of the Z-string has been considered in Refs. [40,100,83,91]. Naculich [100] found that fermions actually make the Z-string unstable. In [91] it was argued that this e!ect of fermions is quite general and also applies to situations where the strings form at a low energy scale due to topological reasons but can terminate on very massive monopoles formed at a very high energy scale. This most likely indicates that the Z-string solution itself should be di!erent from the Nielsen}Olesen solution when fermions are included. We shall describe these results in greater detail in Section 8 after discussing fermion zero modes on strings. Z-strings have also been considered in the presence of a cold bath of fermions [24]. The e!ect of the fermions is to induce an e!ective Chern}Simons term in the action which then leads to a long-range magnetic "eld around the string.
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7.4. Semiclassical stability Preskill and Vilenkin [110] have calculated the decay rate of electroweak strings in the region of parameter space where they are classically stable. The instability is due to quantum tunneling and is calculated by "nding the semiclassical rate of nucleation of monopole-antimonopole pairs on electroweak strings. The bounce action is found to be S&(4p2/g2)a /a (167) = 4 where the strings are classically stable if the ratio of parameters a /a is larger than 1. (a /a is the = 4 = 4 ratio of energy in the magnetic #ux when it is spread over an in"nite area to that if it is con"ned within the string.) The semiclassical decay probability of the string per unit length per unit time is proportional to exp[!S]. The decay rate gets suppressed as we approach the semilocal string (gP0) thus the semilocal string is also stable semiclassically.
8. Superconductivity of electroweak strings 8.1. Fermion zero modes on the Z-string Here we shall consider the fermionic sector of the GSW model in the "xed background of the unit winding Z-string for which the solution is given in Eq. (82). The Dirac equations for a single family of leptons and quarks are obtained from the Lagrangian in Section 1.1.2. These have been solved in the background of a straight Z string in [40,50,99]. The analysis is similar to that for ;(1) strings [133,68] since the Z-string is an embedded ;(1) string in the GSW model (see Section 5.3). A discussion of the fermion zero modes in connection with index theorems can be found in [82,79]. In polar coordinates with the Z-string along the z-axis, a convenient representation for the c matrices is
A
co"
A
ct"
B A
0
e~*r
0
0
!e*r
0
0
0
0
0
0
!e~*r
0
0
e*r
0
B
q3
0
0
!q3
A
, cz"
0
, cr"
B
1
!1 0
B
0
!ie~*r
0
0
!ie*r
0
0
0
0
0
0
ie~*r
0
0
ie*r
0
A B
, c5"
0 1 1 0
.
,
(168)
(169)
(Note that the derivative ckR is given by ctR #coR #c(R /o#czR .) Then the electron has k t o ( z a zero-mode solution
AB 1
AB 0
0 1 e " t (o) , e " it (o) , L 1 R !1 0 4 0
1
(170)
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where t@ #(qv/o)t "!h(g/J2) ft , 4 1 1 (q!1)v g t@ ! t "!h ft . 4 4 o J2 1
(171) (172)
In these equations q is the eigenvalue of the operator q de"ned in Eq. (87) and denotes the Z-charge of the various left-handed fermions. (For the electron, q"cos(2h ).) The boundary conditions are 8 that t and t should vanish asymptotically. This means that there is only one arbitrary constant 1 4 of integration in the solution to Eqs. (170)}(172). This may be taken to be a normalization of t 1 and t . 4 For the d quark, the solution is the same as in Eqs. (170)}(172) except that q"1!(2/3) sin2 h . 8 For the u quark the solution is
AB
AB
0
u " L
1 0
1
0 t (o) , u " it (o) , 2 R 3 1 0
!1
where
(173)
t@ !(qv/o)t "!G (g/J2) f t , 3 2 2 u g (q#1)v t@ # t "!G ft , 3 3 u o J2 2
(174) (175)
with q"!1#(4/3) sin2 h . Note that (171), (172) are related to (174), (175) by qP!q. 8 The right-hand sides of the neutrino Dirac equations (corresponding to Eqs. (171) and (172)) vanish since the neutrino is massless. The solutions can be found explicitly in terms of the string pro"le equations in the case when the Higgs boson mass (m "J2jg) equals the Z boson mass H (m "g g/2) [50]. Recall that the string equations in the m "m case are [26] Z z H Z f @"( f/o)(1!v) , (176) v@"(m2 /2)o(1!f 2) Z yielding the useful relation
P
(177)
A B
v m o do "ln Z , o f
(178)
where we have included a factor of m to make the argument of the logarithm dimensionless. Now Z the zero-mode pro"le functions for the massless fermions are
A B
t "c m3@2 1 1 Z
A B
m o ~q m o q~1 Z , t "c m3@2 Z , 4 4 Z f f
(179)
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where c and c are independent constants that can be chosen to normalize the left- and 1 4 right-handed fermion states and the spinors are given in (170). The boundary condition that the left-handed fermion wavefunction should vanish at in"nity is only satis"ed if q'0. Hence, (179) can only give a valid solution for q'0 for the left-handed fermion. If we also require normalizability, we need q'1. (Note that there is no singularity at o"0 because fJo when o&0.) If we have a left-handed fermion with q4!1, the correct equations to use are the equations corresponding to the up-quark equations given in (174) and (175) and these are solved by letting qP!q in (179). In this case, the spinors are given in (173). For the electroweak neutrino, the right-handed component is absent and q"!1. This means that the neutrino has the same spinor structure as the left-handed up quark and the solution is that in (179) with q replaced by #1. Therefore, the wavefunction falls o! as 1/o and the state is strictly not normalizable } the normalization integral diverges logarithmically. However, depending on the physical situation, one could be justi"ed in imposing a cut-o!. For example, when considering closed loops of string, the cuto! is given by the radius of the loop. Next we give the explicit solutions to the Dirac equations in (171) and (172) in the case when the fermion mass (m "hg/J2) is equal to the scalar mass which is also equal to the vector mass. This f so-called `super-Bogomolnyia limit is not realized in the GSW model but may be of interest in other situations (for example, in supersymmetric models). Then, if the charge on the left-handed fermion vanishes (q"0), the solution can be veri"ed to be t (o)"Nm3@2(1!f (o)2) , (180) 1 Z t (o)"2Nm1@2( f (o)/o) (1!v(o)) , (181) 4 Z where N is a dimensionless normalization factor. For the same set of parameters, the solution for the up-quark equations can be written by using the transformation qP!q in the above solutions. Further, this solution can also be derived using supersymmetry arguments [132,37]. The left-handed fermion wavefunctions found above can be multiplied by a phase factor exp[i(E t!pz)] and the resulting wavefunction will still solve the Dirac equations provided p E "e p , (182) p i where i labels the fermions, and e "#1"e , e "!1"e . (183) l u e d In other words, l and u travel parallel to the string #ux while e and d travel anti-parallel to the L string #ux. We should mention that the picture of quarks travelling along the Z-string (see Fig. 13) may be inaccurate since QCD e!ects have been totally ignored. At the present time it is not known if the strong forces of QCD will con"ne the quarks on the string into mesons and baryons (for example,
Fig. 13. The direction of propagation of quark and lepton zero modes on the Z-string.
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pions and protons). Further, the electromagnetic interactions of the particles on the string might lead to bound states of electrons and protons on the string. This would imply a picture where hydrogen (and other) atoms are the fundamental entities that live on the string. 8.2. Stability of Z-string with fermion zero modes In Fig. 14 we show the e!ect that perturbations of order e in the Z-string "elds have on the fermion (u and d quarks) zero modes. The zero momentum modes acquire an O(e) mass while the non-zero momentum modes get an O(e2) mass. For the perturbation analysis to make sense, we require that the u and d quark zero momentum modes are either both "lled or both empty. In that case, the O(e) terms in the variation in the energy will cancel and we will be left with something that is O(e2). In fact, e2 N 1 *E"! Dm D2¸ + , (184) 1 2 k k/1 where m is a matrix element having to do with the interactions of the u and d quarks, ¸PR is the 1 length of the string on which periodic boundary conditions have been imposed, and NPR is a cut-o! on the energy levels which are labeled by k. The crucial piece of this formula is the minus sign which shows that the energy of the string is lowered due to perturbations [100]. In Ref. [91] it was argued that an identical calculation could be done for any classically stable string that could terminate on (supermassive) magnetic monopoles. However, in the low-energy theory, the strings are e!ectively topological and hence, it seems unlikely that fermions can lead to an instability. This suggests that the bosonic string con"guration gets modi"ed by the fermions and the stability analysis around the Nielsen}Olesen solution may be inappropriate. So far, the stability analysis with fermions presented here only considered the zero modes and ignored the in"nitely many massive fermion modes. Very recently, Groves and Perkins [58] have
Fig. 14. The e!ect of perturbations of the Z-string on fermion zero modes.
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analysed the full spectrum of massless and massive fermionic modes in the background of the electroweak string. They then calculate the e!ect of the Dirac sea on the stability of electroweak strings by calculating the renormalised energy shift of the Dirac sea when a Z-string is perturbed by introducing a non-zero upper component to the Higgs doublet. This energy shift is negative and so destabilises the string, but it is small, leading them to conclude that if positive energy fermionic states are populated, it is conceivable that the total fermionic contribution could be to stabilise the string. This work is still in progress. In the meantime, the stability of Z-strings remains an open question. 8.3. Scattering of fermions ow electroweak strings The elastic scattering of fermions o! semilocal and electroweak strings has been considered in [48,36,92]. As mentioned earlier, we expect Aharonov}Bohm scattering. The main feature of the cross-section is that the scattering violates helicity [48]. It is straightforward to show that the helicity operator R ) P, where Ri"eijkcicj is the spin operator and Pi are the canonical momenta, does not commute with the hamiltonian. If UT"(/`, /0), the commutator is proportional to (D/0) terms. Consider for a moment the usual representation of Dirac matrices,
A B 0 1
c0"
1 0
A
, ci"
B
0
!qi
qi
0
A
B
1 0 , c " . 5 0 !1
(185)
Then, for an incoming electron, one "nds
A
[H, R ) P]"ih
B
0
qj(Dj/0)s
qjDj/0
0
,
(186)
where h is the Yukawa coupling and (D /0) is given in Eq. (86). Therefore helicity-violating j processes can take place in the core of the string. A preliminary calculation by Ganoulis in Ref. [48] showed that, for an incoming plane wave, the dominant mode of scattering gives identical cross sections for positive and negative helicity scattered states. More precisely, for an incoming electron plane wave of momentum k, energy u and positive helicity it was found that, to leading order, dp dk
K
A
B
1 u!k 2 & sin2(pq ) , R k 2u
(187)
B where u2"k2#m2, q is the Z-charge of the right fermion "eld, given in Eq. (87) (recall that right % R and left fermion "elds have di!erent Z-charges, q "q $1). R L A more detailed calculation was done by Davis et al. [36], and later extended by Lo [92], using a &top hat' model for the string
G
f (r)"
0,
r(R ,
G
v(r)"
0,
r(R ,
(188) 2/g , r'R , z which is expected to be a reasonable approximation since the scattering cross section in the case of (topological) cosmic strings has been shown to be insensitive to the core model [108]. Note that there is a discontinuous jump in the fermion mass and string #ux; however the wavefunctions are g/J2, r'R ,
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matched so that they are continuous at r"R. In this approximation, the left and right "elds decouple in the core of the string, so helicity violating processes are concentrated at r"R. The authors of [36,92] con"rmed that, in the massive case, there are helicity-conserving and helicity-#ip scattering cross sections of equal magnitude. The latter goes to zero in the massless limit (in that case, the left and right "elds decouple, and no helicity violation is possible), suggesting that helicity violation may be stronger at low energies. For `fractional string #uxa (i.e. for fractional q) the cross section is of a modi"ed Aharonov}Bohm form, and independent of string radius. For integer q it is of Everett form [43] (the strong interaction cross section is suppressed by a logarithmic term). Another interesting feature has to do with the ampli"cation of the fermionic wavefunction in the core of the string. Lo [92] has remarked that there is a regime in which the scattering cross section for electroweak strings is much less sensitive to the fermion charge (that is, to sin2 h ) than for 8 cosmic strings. In contrast with, e.g., baryon number violating processes, which show maximal enhancement only for discrete values of the fractional #ux, the helicity violating cross section for electroweak strings in the regime k&m, kR;1 shows a plateau for 0(sin2 h (1/2 where 8 ampli"cation is maximal and the cross-section becomes of order m~1. This can be traced back to f the asymmetry between left and right "elds; while the wavefunction ampli"cation is a universal feature, di!erent components of the fermionic wavefunction acquire di!erent ampli"cation factors in such a way that the total enhancement of the cross section is approximately independent of the fermionic charge, q (or, equivalently, of sin2 h ). 8 Elastic scattering is independent of the string radius for both electroweak and semilocal strings (for integral #ux there is only a mild dependence on the radius coming from the logarithmic suppression factor in the Everett cross-section). Since the cross-section is like that of ;(1) strings, we would expect electroweak and semilocal strings to interact with the surrounding plasma in a way that is analogous to topological strings.
9. Electroweak strings and baryon number As "rst shown by Adler [8], and Bell and Jackiw [18], currents that are conserved in a classical "eld theory may not be conserved on quantization of the theory. In the GSW model, one such current is the baryon number current and the anomalous current conservation equation is N R jk " F [!g2=a = I akl#g@2> >I kl] , k B 32p2 kl kl
(189)
where jk is the expectation value of the baryon number current operator + b : tM ckt: where the B s s sum is over all the species of fermions labeled by s, t is the fermion spinor and b is the baryon s number for species s and the operator product is normal ordered. Also, N denotes the number of F families, and tilde the dual of the "eld strengths. The anomaly equation can be integrated over all space leading to
P
N *Q " F dt d3x[!g2=a = I akl#g@2> >I kl]"*Q B 32p2 kl kl CS
(190)
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with,
P
CA
B
D
g N Q " F d3xe g2 =aij=ak! e =ai=bj=ck !g@2>ij>k . ijk CS 32p2 3 abc
(191)
Here, *( ) ) denotes the di!erence of the quantities evaluated at two di!erent times, Q is the B baryonic charge and the surface currents and integrals at in"nity are assumed to vanish. Q is CS called the Chern}Simons, or topological, charge and can be evaluated if we know the gauge "elds. The left-hand side of Eq. (190) evaluates the baryon number by counting the fermions directly. We describe the evaluation of both the right- and left-hand side for fermions on certain con"gurations of Z-strings in the following subsections. Finally, in Section 9.4 we brie#y comment on possible applications to cosmology. 9.1. Chern}Simons or topological charge We will be interested in the Chern}Simons charge contained in con"gurations of Z-strings. Then, we set all the gauge "elds but for the Z-"eld to zero in the expression for the Chern-Simons charge, yielding
P
g2 z cos(2h ) d3x Z ) B , Q "N CS F 32p2 8 Z
(192)
where B denotes the magnetic "eld in the Z gauge "eld: Bi "eijkR Z . Z Z j k The terms on the right-hand side have a simple interpretation in terms of a concept called `helicitya in #uid dynamics [20]. Essentially, if a #uid #ows with velocity * and vorticity x"$]*, then the helicity is de"ned as
P
h" d3x * ) x .
(193)
Since the helicity measures the velocity #ow along the direction of vorticity, it measures the corkscrew motion (or twisting) of the #uid #ow. A direct analog is de"ned for magnetic "elds:
P
h " d3x A ) B B
(194)
which is of the same form as the terms appearing in (192). Hence, the Chern}Simons charge measures the twisting of the magnetic lines of force. The helicity associated with the Z "eld alone is given by
P
H " d3x Z ) B . Z Z
(195)
If we think in terms of #ux tubes of Z magnetic "eld, H measures the sum of the link and twist Z number of these tubes: H "¸ #¹ . Z Z Z
(196)
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Fig. 15. A pair of linked loops. Fig. 16. A circular Z-string loop of radius a threaded by n Z-strings.
For a pair of unit winding Z #ux tubes that are linked once as shown in Fig. 15 the helicity is H "2F2 , (197) Z Z where F is the magnetic #ux in each of the two tubes. Note that the helicity is positive for the Z strings shown in Fig. 15. If we reversed the direction of the #ux in one of the loops, the magnitude of H would be the same but the sign would change. For the Z-string, we also know that Z F "4p/g (198) Z z and so Eq. (192) yields [129] Q "N cos(2h ) . CS F 8
(199)
9.2. Baryonic charge in fermions The baryon number associated with linked loops of Z-string has been evaluated in Ref. [50] by studying the fermionic zero modes on such loops. This corresponds to evaluating the left-hand side of Eq. (190) directly in terms of the fermions that carry baryon number. The calculation involves adding the baryonic charges of the in"nite Dirac sea of fermions living on the string together with zeta function regularization. To understand why the linking of loops leads to non-trivial e!ects, note that the quarks and leptons have a non-trivial Aharanov}Bohm interaction with the Z-string. So the Dirac sea of fermions on a loop in Fig. 15 is a!ected by the Z-#ux in the second loop. This shifts the level of the Dirac sea in the ground state leading to non-trivial baryonic and other charges. Instead of considering the linked loops as shown in Fig. 15 it is simpler to consider a large circular loop of radius aPR in the xy-plane threaded by n straight in"nite strings along the z-axis
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(Fig. 16). Then the fermionic wavefunctions take the form (200) t "e~*(Ep t~pp)t(0)(r), t "e~*(Ep t~(p~n@a)p)t(0)(r) , L R R L where the functions with superscript (0) are the zero-mode pro"le functions described in Section 8.1 and p is a coordinate along the length of the circular loop. From these wavefunctions, the dispersion relation for a zero-mode fermion on the circular loop is u "e (k!qZ) , (201) k i where q is the Z-charge of the fermion, e is de"ned in Eq. (183), u is related to the energy E by i u,aE, and k to the momentum p by k,ap3Z. Z is the component of the gauge "eld along the circular loop multiplied by a and is given by Z,2n/g . (202) z The crucial property of the dispersion relation is that, if there is an Aharanov}Bohm interaction between the Z-string and the fermion, u cannot be zero for any value of k since k is an integer but k qZ is not. The Z- , A- and baryon number (B) charges of the leptons and quarks are shown in Table 1. Note that we use 2q /g to denote the Z-charge and this is identical to the eigenvalue of the operator Z z q de"ned in Eq. (87) and also to q used in the previous section. The energy of the fermions is found by summing over the negative frequencies } that is, the Dirac sea } and so the energy E due to a single fermion species is 1 ~ei = 1 (203) E" +u "e + (k!qZ) , k i a a k/kF where k denotes the Fermi level } the value of k for the highest "lled state. Therefore we need to F sum a series of the type = = S" + (k!qZ)" + (k#k !qZ) . F k/kF k/0 The sum is found using zeta function regularization
(204)
S"f(!1, k !qZ)"! 1 !1(k !qZ)(k !qZ!1) . F F 12 2 F
(205)
Table 1 Summary of Z-, electric and baryonic charges for the leptons and quarks. The charges q are for the left-handed fermions Z and s2,sin2h 8
2q /g Z z q /e A q B
l L
e
d
u
!1 0 0
1!2s2 !1 0
1!2s2/3 !1/3 1/3
!1#4s2/3 2/3 1/3
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With this result, the energy contribution from the ith species of fermions takes the form
C
D
1 e 2 1 1 1 # k(i)!q Z# i ,! # K2 . E "! i i 2 24a 2a F 24a 2a i
(206)
Adding the contributions due to di!erent members of a single fermion family, we get 2aE"K2#K2#3K2#3K2 . (207) l e d u Next we can calculate the angular momentum of the fermions in the circular loop background. The system has rotational symmetry about the z-axis and this enables us to de"ne the generalized angular momentum operator as the operator that annihilates the background "eld con"guration [71]: M "¸ #S #nI , z z z z where
(208)
¸ "!i1 R/Ru , (209) z S is the spin operator, and, the isospin operator is given in terms of the ;(1) (hypercharge) and z S;(2) charges } q and q respectively } of the "eld in question: 1 2 1 2q 2q 2 ¹3! 1 1 . I " (210) z 2 g g@
CA B A B D
The isopin operator acts via a commutator bracket on the gauge "elds and by ordinary matrix multiplication on the Higgs "eld and fermion doublets. We are interested in the angular momentum of the chiral fermions on the circular loop which lies entirely in the xy-plane. The fermions in the zero modes therefore have S "0. (The spin of the z fermions is oriented along their momenta which lies in the xy-plane.) The action of ¸ is found by z acting on the fermion wavefunctions such as in Eq. (200) (remembering to let nP!n for the neutrino and up quark). The action of I is found by using the charges of the fermions given in the z GSW model de"ned in Section 1.1.2. We then "nd
AB A
B
C
D C
AB A
B
l (k(l)#n)l u (k(u)#n/3)u L " L , M L " L , (211) z e z k(e)e d (k(d)!2n/3)d L L L L M e "k(e)e , M u "(k(u)#n/3)u , M d "(k(d)!2n/3)d , (212) z R R z R R z R R where the k(i) are de"ned above in Eq. (201). Now summing over states, as in the case of the energy, we "nd the total generalized angular momentum of the fermions on the circular loop: M
D C
D C
D
1 2 1 1 2 3 2n 1 2 3 n 1 2 1 ! k(e)! ! k(d)! ! # k(u)# # . M" k(l)#n# 2 F 2 F 2 F 2 2 3 2 3 2 2 F
(213)
Note that though the gauge "elds do not enter explicitly in the generalized angular momentum, they do play a role in determining the angular momentum of the ground state through the values of the Fermi levels. The calculation of the electromagnetic and baryonic charges and currents on the linked loops is similar but has a subtlety. To "nd the total charge, a sum over the charges in all "lled states must be
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done. This leads to a series of the kind = S " + 1. q k/kF To regularize the divergence of the series, it is written as
(214)
= S "lim + (k!qZ)j . (215) q j?0 k/kF The subtlety is that the gauge invariant combination k!qZ is used as a summand rather than k or some other gauge non-invariant expression [98]. Once again zeta function regularization is used to get
C
D
= 1 S " + (k#k !qZ)0"f(0, k !qZ)"! k !qZ! . q F F F 2 k/0 With this result, the contribution to the charge due to fermion i is
C
(216)
D
e Q "e q6 k(i)!q Z# i "e q6 K , i i i F i i i i 2
(217)
where q6 is the charge carried by the ith fermion of the kind that we wish to calculate. i (Note that q6 can represent any charge } electric, baryonic etc. } and is, in general, di!erent from the i Z-charge q .) i The currents along the string are given by tM czt where cz is given in Eq. (169). This gives J "e Q /2pa . (218) i i i By adding the contributions due to each variety of fermion, expressions for the energy, angular momentum, charges and currents for one loop threaded by n have been found in [50]. These results are reproduced in Table 2. It is reassuring to note that in the ground state, the baryon number of the single loop is given by nN cos 2h in agreement with the calculation of the Chern}Simons F 8 number. The energy of the fermionic ground state shows a complicated dependence on x as is demonstrated in Fig. 17. Note that E(x) does not have a monotonic dependence on x and the energy of strings that are linked n times bears no simple relation to those linked m times. In particular, the Table 2 Expressions for the energy, generalized angular momentum, charges and currents in terms of x"2n sin2 h /3. We have 8 omitted the multiplicative factor N in all the expressions for convenience F
aE M Q /e A B 2paJ /e A 2paJ B
x3(0, 1/3)
(1/3, 1/2)
(1/2, 2/3)
(2/3, 1)
12x2!6x#1 0 0 !3x#1 !8x#2 !x
12x2!9x#2 n!1 !1 !3x#1 !8x#3 !x
12x2!15x#5 2!n #1 !3x#2 !8x#5 1!x
12x2!18x#7 0 0 !3x#2 !8x#6 1!x
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Fig. 17. The energy of the ground state of linked loops versus x"2n sin2 h /3. 8
energy does not continue to decrease as we consider strings that have higher linkage. The lowest energy possible, however, is when x"1/4 and for n"1, this corresponds to sin2 h "3/8, which is 8 also the value set by Grand Uni"ed models. It is not clear if this is simply a coincidence or if there is some deeper underlying reason [74]. 9.3. Dumbells In his 1977 paper, Nambu discussed the possible occurrence of electroweak monopoles and strings in particle accelerators. There are two issues in this discussion: the "rst is the production cross-section of solitonic states in particle collisions, and the second is the signatures of such states if they are indeed produced in an accelerator. The answer to the "rst question is not known though it is widely believed that the process is suppressed not only by the large amount of energy required but also due to the coherence of the solitonic state. The second question was addressed by Nambu [102] and he estimated the energy and lifetime of electroweak strings that may be possible to detect in accelerators. To "nd the energy of a Z-string segment, Nambu treated the monopoles at the ends as hollow spheres of radius R inside which all "elds vanish. A straightforward variational calculation in units of g+246 GeV then gives the monopole mass 4p M" sin5@2 h 8 3e
S
m H m W
(219)
and radius
S
sin h 8 . m m (220) H W The string segment is approximated by a cylindrical tube with uniform Z magnetic #ux with all other "elds vanishing. This gives R"
A B
m 2 , q"p H o" m Jm m Z H Z for the core radius and string tension, respectively.
(221)
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Now, if the monopoles are a distance l apart, the total energy of the system is E"2M!(Q2/4pl)#ql ,
(222)
which is clearly minimized by l"0, i.e. the string can minimize its energy by collapsing. The tendency to collapse can be countered by a centrifugal barrier if the string segment (`dumbella) is rotating fast enough about a perpendicular axis. The energy and angular momentum of a relativistic dumbell has been estimated by Nambu to be E&1plq, 2 where
¸&1pl2q , 8
lq/2M"v2/(1!v2)
(223)
(224)
with v&1 being the velocity of the poles. The expressions for E and ¸ imply the existence of asymptotic Regge trajectories, ¸&a@ E2 0 with slope
(225)
a@ "1/2pq&(m /m ) TeV~2 . (226) 0 Z H which, if found, would be a signature of dumbells. The orbiting poles at the ends of the rotating dumbell will radiate electromagnetically and this energy loss provides an upper bound to the lifetime of the con"guration. An estimate of the radiated power from the analysis of synchrotron radiation in classical electrodynamics (see e.g. [67]) gives P&8p]137(q/M)2 sin4 h . 3 8 Therefore the decay width C"P/E is given by CKE/¸
(227)
(228)
and for large angular momentum, can lead to signi"cant lifetimes (compared to E~1). To obtain numerical estimates, note that the above estimates are valid only if the dumbell length is much greater than the width of the Z-string. This imposes a lower bound on the angular momentum: ¸
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due to the angular momentum of the dumbell.15 A careful analysis of these factors has not yet been performed and is a vital open problem that may become experimentally relevant with the next generation of accelerators. 9.4. Possible cosmological applications The role of electroweak strings in cosmology depends on their abundance during and after the electroweak phase transition. If this abundance is negligible, electroweak strings may at best only be relevant in future accelerator experiments (see Section 9.3). If, however, there is a cosmological epoch during which segments and loops of electroweak strings were present, they could impact on two observational consequences: the "rst is the presence of a primordial magnetic "eld, and the second is the generation of a cosmological baryon number. What is perhaps most remarkable is that the two consequences might be related } the baryonic density of the universe would be related to the helicity of the primordial magnetic "eld [125,112]. (i) Primordial magnetic xelds: A gas of electroweak string segments is necessarily accompanied by a gas of electroweak monopoles. The eventual collapse and disappearance of electroweak strings removes all the electroweak monopoles but the long-range magnetic "eld emanating from the monopoles is expected to remain trapped in the cosmological plasma since that is a very good electrical conductor. This will then lead to a residual primordial magnetic "eld in the present universe. A quantitative estimate of the resulting primordial magnetic "eld cannot be made with con"dence but a dimensional estimate is possible. An estimate for the average #ux through an area ¸2"N2/¹2, where N is a dimensionless number that relates the length scale of interest, ¸, to the cosmological thermal correlation length ¹~1, was obtained in [122,125], and then translated into the average magnetic "eld through that area. The result is BD &¹2/N . (230) !3%! (Magneto-hydrodynamical considerations provide a lower bound &1012 cm s on ¸ at the present epoch.) It is important to remember that the above is an areal (i.e. #ux) average, de"ned by [42]
TA P B U
BD , !3%!
1 2 1@2 , dS ) B A
(231)
where the surface integral is over an area A and S ) T denotes ensemble averaging. (ii) Baryon number: A gas of electroweak string segments and loops would, in general, contain some helicity density of the Z-"eld. When the electroweak strings eventually annihilate, it is possible that the helicity gets converted into baryon number [129,125]. However, in Refs. [44,45] it is argued that fractional quantum numbers of a soliton are unrelated to the number of particles produced when the soliton decays. Instead, only the change in the winding of the Higgs "eld in a process that starts out in the vacuum and ends up in the vacuum can be related to the particle 15 In the stability analysis for a "nite piece of string of length ¸, the eigenvalues of the stability equation are shifted by a contribution of order p2/¸2 with respect to the in"nitely long case, thus for su$ciently short segments the radial decay mode could become stable. Longitudinal collapse might then be stabilized by rotation, as explained above.
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number. This would imply that we would have to consider the formation of electroweak strings together with their decay before we can "nd the resulting baryon number. Such a calculation has not yet been attempted. An interesting question is to consider what happens to the helicity in the Z-"eld after the strings disappear. One possibility is that the helicity gets transferred to a frozen-in residual magnetic "eld after the strings have decayed. To see this, consider a linked pair of loops as in Fig. 15. The strings can break by nucleating monopole}antimonopole pairs, and then the string segments can shrink, "nally leading to monopole annihilation. If this process happens in the early universe, the loops will be surrounded by the ambient plasma which will freeze-in the magnetic "eld lines. Hence, after the strings have disappeared, we will be left with a linked pair of magnetic "eld lines. In other words, the original helicity in the Z-"eld has been transferred to helicity in the A-"eld. This argument relies on the freezing-in of the magnetic "eld emanating from the monopoles and in the real setting the physics can be much more complicated. However, a connection between the baryon abundance of the universe and the properties of a primordial magnetic "eld seems tantalizing. Stable strings at the electroweak scale: If in more exotic models, strings at the electroweak scale were stable and had the superconducting properties discussed above, they could be responsible for baryogenesis [14] and the presence of primary antiprotons in cosmic rays [118]. The production of antiprotons follows on realizing that any strings tangled in the galactic plasma would be moving across the galactic magnetic "eld. In the rest frame of the string, the changing magnetic "eld causes an electric "eld along the string according to Faraday's law. The electric "eld along the string raises the levels of the u- and d-quark Dirac seas (see Fig. 18), as well as the electron Dirac sea (not shown in the "gure). This means that the electric "eld produces quarks and leptons on the string. The electric charges of the particles are in the ratio e : u : d ::!1 :#2/3 :!1/3 and the rate of production of these particles due to the applied electric "eld is proportional to the charges. Furthermore, the quarks come in three colours and so for every electron that is produced, 3]2/3"2 u-quarks and 3]1/3"1 d-quark are also produced. As a result, the net electric charge produced is 1](!1)#2](2/3)#1](!1/3)"0. However, net baryon number 2](1/3)#1](1/3)"1 is produced because the quarks carry baryonic charge 1/3 while the baryonic charges of the leptons vanish. Depending on the orientation of the string, either baryons or antibaryons will be produced. Some of these would then be emitted from the string and would arrive on earth as cosmic rays. Formation of strings in the electroweak phase transition: Early attempts to understand the formation rates of electroweak strings were made in [125] based on the statistical mechanics
Fig. 18. The dispersion relations for the u and d quark zero modes are shown. The "lled states are denoted by solid circles while dashes denote un"lled states. For convenience, periodic boundary conditions are assumed along the string and so the momentum takes on discrete values.
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of strings. The estimates indicate that a density of strings will be formed immediately after the phase transition. However, the application of string statistical mechanics to electroweak strings may not be justi"ed and so other avenues of investigation are needed. An alternative approach to study electroweak string formation was taken by Nagasawa and Yokoyama [101]. They assumed a thermal distribution of scalar "eld values and gradients, and estimated the probability of obtaining a string-like scalar "eld con"guration. The conclusion was that electroweak vortex formation in a thermal system is totally negligible. One possible caveat is that the technique used in [101] ignores the e!ect of gauge "elds, which we know are signi"cant in the formation of related objects such as semilocal strings. In [115], Sa$n and Copeland have evolved the classical equations of motion to study the formation of electroweak strings, and they found the presence of the gauge "elds led to larger string densities than one would have inferred from the scalar "elds alone, at least when sin2 h "0. However, this study does not directly address the question of 8 string formation in a phase transition because no measure has been placed on the choice of initial conditions and their choice may be too restrictive. Most recently, a promising development has taken place [29] } calculations in lattice gauge theory have been done to study the electroweak phase transition and there is evidence that electroweak strings will form. Further studies along these lines will provide important and quantitative insight into the formation of electroweak strings. Using the results on the formation of semilocal strings, we can gain some intuition about the formation of electroweak strings in the region of parameter space close to the semilocal limit (the region of stability in Fig. 11). We have seen that semilocal strings with b(1 have a non-zero formation rate, increasing as bP0. Initially, short segments of string are seen to grow and join nearby ones because this reduces the gradient energy at the ends of the strings. The ends of electroweak strings are proper magnetic monopoles, and therefore the scalar gradients are cancelled much more e$ciently by the gauge "elds, but as sin h P0 the cores of the monopoles get larger 8 and larger, and they could begin to overlap with nearby monopoles, so it is possible that short segments of electroweak string will also grow into longer ones.
10. Electroweak strings and the sphaleron The sphaleron is a classical solution in the GSW model that carries baryon number N /2, where F N is the number of fermion families [97,80]. For h "0, the asymptotic form of the sphaleron F 8 Higgs "eld is
A
B
cos h U " , 41) sin h e*r
(232)
while the gauge "elds continue to be given by Eqs. (100) and (101) in which UM should be replaced by U . (Note that the hypercharge gauge "eld vanishes for h "0.) Inside the sphaleron, the Higgs 41) 8 "eld vanishes at one point. The sphaleron also has a magnetic dipole moment that has been evaluated for small values of h . The reason that the sphaleron is important for particle physics is 8 that its energy de"nes the minimum energy required for the classical violation of baryon number in the GSW model.
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As has already been described in Section 9, non-trivial baryon number can be associated with linked and twisted segments of electroweak string. Further, for speci"c values of the link and twist, the baryon number of a con"guration of Z-strings can also be N /2. This raises the question: are F sphalerons related to Z-string segments? An early paper to draw a connection between the various solutions in the GSW model is Ref. [47]. In [129,63,125,62], however, a direct correspondence between the "eld con"guration of the Z-string and the sphaleron was made. 10.1. Content of the sphaleron In [63] Hindmarsh and James evaluated the magnetic charge density and current density within the sphaleron. A subtlety in this calculation is that there is no unique de"nition of the electromagnetic "eld when the Higgs "eld is not everywhere in the vacuum. The choice adopted in [63] (and also the choice in this review) is F%."sin h =a na#cos h > . (233) ij 8 ij 8 ij The evaluation of the magnetic charge density (which is proportional to the divergence of the magnetic "eld strength) clearly shows that the sphaleron contains a region with positive magnetic charge density and a region with negative magnetic charge density. Furthermore, the total charge in, say, the positive charge region agrees with the magnetic charge of a monopole. In addition, there is a #ux of Z magnetic "eld connecting the two hemispheres. This would seem to con"rm that the sphaleron consists of a Z-string segment. However, this is not the full picture. In addition to the string segment, Hindmarsh and James "nd that the electric current is non-zero in the equatorial region and is in the azimuthal (e( ) direction. r 10.2. From Z-strings to the sphaleron The scalar "eld con"guration for a "nite segment of Z-string was given in Section 9.3:
A
B
cos(H/2) U 6" , .. sin(H/2) e*r
(234)
where cos H,cos h !cos h 6 #1 (235) . . and the angles h and h 6 are measured from the monopole and antimonopole, respectively, as . . shown in Fig. 19. It is straightforward to check that (234) yields the monopole "eld con"guration close to the monopole (h 6 P0) and the antimonopole con"guration close to the antimonopole (h Pp). It also . . yields a string singularity along the straight line joining the monopole and antimonopole (h "p, h 6 "0). However, there are other Higgs "eld con"gurations that also describe monopoles . . and antimonopoles:
A
B
A
B
cos(h /2) sin(h 6 /2) . . U "e*c , U 6 "e*c . . . sin(h /2)e*r cos(h 6 /2)e*r . .
(236)
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Fig. 19. De"nition of the coordinate angles h and h 6 . The azimuthal angle, u, is not shown. . .
Next consider the Higgs "eld con"guration:
A
B
sin(h /2) sin(h 6 /2)e*c#cos(h /2) cos(h 6 /2) . . . . U 6 (c)" .. sin(h /2) cos(h 6 /2)e*r!cos(h /2) sin(h 6 /2)e*(r~c) . . . .
(237)
together with the gauge "elds given by Eqs. (100) and (101) with UM replaced by U 6 (c). When we .. take the limit h 6 P0 we "nd the monopole con"guration (with c"0) and when we take h Pp . . the con"guration is that of an antimonopole (with arbitrary c) provided we perform the spatial rotation uPu#c. Note that the asymptotic gauge "elds agree since these are determined by the Higgs "eld. The monopole and antimonopole in (237) also have the usual string singularity joining them. This means that the con"guration in Eq. (237) describes a monopole and antimonopole pair that are joined by a Z-string segment that is twisted by an angle c. The Chern}Simons number of one such segment can be calculated [129] and is Q "N cos 2h c/2p . CS F 8
(238)
If c"p/cos(2h ) then the Chern}Simons number of the twisted segment of string is N /2 and is 8 F precisely that of the sphaleron. Given that the segment with twist p/cos(2h ) has Chern}Simons number equal to that of the 8 sphaleron, it is natural to ask if some deformation of it will yield the sphaleron. This deformation is not hard to guess for the h "0 case. In this case, if we let the segment size shrink to zero, we 8
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Fig. 20. The thick solid line is the location of the Z-string for a dumbell con"guration and the dashed curves lie in the equatorial plane and are drawn to guide the eye. The dotted lines depict lines of magnetic #ux. The arrows show the orientation of the vector n( J!UssU. Fig. 21. The "eld con"guration for a stretched sphaleron as in Fig. 20. Only one magnetic "eld line is shown.
have h "h 6 "h and the Higgs "eld con"guration of Eq. (237) gives . . cos h . U 6 (c"p)" .. sin h e*r
A
B
(239)
This is exactly the scalar "eld con"guration of the sphaleron for h "0 (Eq. (232)). Note that the 8 asymptotic gauge "elds continue to be given by Eqs. (100) and (101) and satisfy the requirement that the covariant derivatives of the Higgs "eld vanish. Encouraged by this successful connection in the h "0 case, it was conjectured in [129,125] that 8 the sphaleron can also be obtained by collapsing a twisted segment of Z-string with Chern}Simons number N /2 for any h . If true, this would mean that the asymptotic Higgs "eld con"guration, U , F 8 S for the sphaleron for arbitrary h is given by 8 sin2(h/2)e*cS #cos2(h/2) U " (240) S sin(h/2) cos(h/2)e*r(1!e~*cS )
A
B
where c "p/cos(2h ). S 8 The twisting of the magnetic "eld lines in the sphaleron con"guration has been further clari"ed in [62]. The direction of magnetic "eld lines is shown for a dumbell in Fig. 20 and for a `stretcheda sphaleron in Fig. 21. (The asymptotic "elds for the stretched sphaleron are identical to those for the sphaleron and the twisted Z-string.) In the stretched sphaleron case, the magnetic "eld line twists around the vertical string segment by an angle p (for h P0) as one goes from monopole to 8 antimonopole. This twist provides non-trivial Chern}Simons number to the con"guration [129]. On physical grounds it seems reasonable that there should be a critical value of twist at which one can get a static solution for a Z-string segment. This is because the segment likes to shrink under its own tension but the twist prevents the shrinkage and is equivalent to a repulsive force between the monopole and antimonopole. (This idea owes its origin to Taubes [119], who
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discovered a solution containing a monopole and an antimonopole in an O(3) model in which the Coulomb attraction is balanced by the relative misorientation of the magnetic poles.) Then, if the string is su$ciently twisted, the attractive force due to the tension and the repulsive force due to the twist will balance and a static solution can exist. So far we have been assuming that the only dynamics of the segment is towards collapsing or expanding of the string segment. However, since we are dealing with twisted segments, we should also include the rotational dynamics associated with twisting and untwisting. So, while any twist greater than a certain critical twist might successfully prevent the segment from collapsing, only a special value of the twist can give a static solution to the rotational dynamics. Furthermore, we expect that this solution will be unstable towards rotations that twist and untwist the string segment. This would be the unstable mode of the sphaleron. Similar connections between the =-string and the sphaleron have also been constructed in [12]. A connection to multisphalerons [77] has not been established but seems plausible. The relation between strings and other axially symmetric solutions [78,27], in particular bisphalerons [84], is an interesting open problem.
11. The 3He analogy The symmetry structure of 3He closely resembles the electroweak symmetry group and hence we expect the analog of electroweak strings to exist in 3He [134,136,138]. Indeed, this analog is called the n"2 vortex. We now explain this correspondence in greater detail. 11.1. Lightning review of 3He 3He nuclei have spin 1/2 and two such nuclei form a Cooper pair which is the order parameter for the system. Unlike 4He, the pairing is a spin triplet (S"1) as well as an orbital angular momentum triplet (¸"1). As a result there are 3]3 components of the wavefunction of the Cooper pair } that is, the order parameter has 9 complex components. Hence, the order parameter is written as a 3 by 3 complex-valued matrix: A with a (spin index) and i (spatial index) ranging ai from 1 to 3. At temperatures higher than a few milli-Kelvin the system is invariant under spatial rotations (SO(3) ) as well as rotations of the spin degree of freedom of the Cooper pair (SO(3) ). Another L S symmetry is under overall phase rotations of the wavefunction (;(1) ) and the corresponding N conserved charge is particle number (N). Hence the symmetry group is G"SO(3) ]SO(3) ];(1) . L S N
(241)
There are several possible phases of 3He corresponding to di!erent expectation values of the order parameter. In the A-phase, the orbital angular momenta of the Cooper pairs are all aligned and so are the spin directions. This corresponds to A "D dK t , ai 0 a i
(242)
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Fig. 22. Depiction of the A- and B-phases of 3He. In the A-phase, the spin orientations of all the Cooper pairs are parallel and so are the orbital orientations. In the B-phase, the relative orientation of the spin and orbital orientations are "xed in all the Cooper pairs but neither the spin nor the orbital orientations of the various Cooper pairs are aligned.
where D &10~7 eV is the temperature-dependent gap amplitude, the real unit vector dK is the spin 0 a part of the order parameter, and (243) t "(m( #in( )/J2 i i i with m( and n( being orthogonal unit vectors, is the orbital part of the order parameter. This expectation value of the order parameter leads to the symmetry breaking: ]Z . (244) GP;(1) 3 ];(1) 3 L ~N@2 2 S The reason why a ;(1) subgroup of SO(3) ];(1) survives the symmetry breaking can be derived L N from the expectation value in Eq. (242). A spatial rotation of the order parameter is equivalent to a phase rotation of t and this phase can be absorbed by a corresponding ;(1) rotation of the i N order parameter. Hence, just as in the electroweak case, a diagonal ;(1) subgroup remains unbroken. The ;(1) 3 survives since rotations about the dK -axis leave the order parameter invariant. S The non-trivial element of the residual discrete Z symmetry corresponds to a sign inversion of 2 both t and dK . A depiction of the A- and B- phases is shown in Fig. 22 (after [90]). i In the B-phase, neither the orbital angular momenta nor the spin directions of the di!erent Cooper pairs are aligned. But the angle between the direction of the angular momenta and the spin direction is "xed throughout the sample. Hence in the B-phase, independent rotations of the orbital angular momenta and of spin are no longer symmetries. However, a simultaneous rotation of both orbital angular momenta and spin remains an unbroken symmetry. In other words, a diagonal subgroup of SO(3) ]SO(3) remains unbroken. Therefore, in the B-phase the order parameter is S L written as A "3~1@2e*(R (n( , h) , ai ai
(245)
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where / is a phase and the 3]3 matrix R describes relative rotations of the spin and orbital ai degrees of freedom about an axis n( and by angle h. The symmetry-breaking pattern is GPSO(3) 3 3 . (246) L `S This symmetry breaking resembles the chiral symmetry breaking transition studied in QCD (with two #avors of quarks) and may be useful for experimentally investigating phenomenon such as the formation of `disoriented chiral condensatesa [25]. The B-phase does not resemble the electroweak model and hence we will not discuss it any further. We shall also not discuss the various other phases of 3He (for example, the A phase) which are known to occur. (For a useful 1 chart of the phases, see Section 6.2 of Ref. [134].) In addition to the continuous symmetries, there are a number of discrete symmetries that arise in the phases of 3He. These are important for the classi"cation of topological defects in 3He. A description may be found in [116]. 11.2. Z-string analog in 3He Clearly the A-phase closely resembles the electroweak symmetry breaking because of the mixing of the generator of the non-Abelian group (SO(3) ) and the Abelian group (;(1) ). The orbital part L N of the order parameter is responsible for this pattern of symmetry breaking and hence t plays the i role of the electroweak Higgs "eld U. The connection, however, is indirect since t is a complex i 3 vector while U is a complex doublet. The idea is that the 3He-A real vector lK
H%A
w]ws "i "m( ]n( wsw
(247)
is analogous to the electroweak real vector lK
"!UssU/UsU . (248) %8 The electroweak Z-string is a non-topological solution for which the Higgs "eld con"guration is
AB
0 g U" f (r)e*r . 1 J2
(249)
For this con"guration lK "z( . %8 The vacuum manifold M of 3He-A has A p (M )"Z (250) 1 A 4 and hence there are topological Z vortices in 3He-A. The vortices occur in classes labeled by 4 n"$1/2, 1. The vortices with n equal to an even integer are topologically equivalent to the vacuum. The non-trivial topological vortices (labeled by n"$1/2, 1) cannot be the equivalent of the non-topological Z-string. However, the topologically trivial n"!2 vortex is also seen in 3He-A. The order parameter for this vortex is A (o, u)"D z( [e*nrf (o)(x( #iy( )#e*(n`2)rf (o)(x( !iy( )] , aj 0 a 1 j j 2 j j
(251)
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where f (o) and f (o) are two pro"le functions with f (R)"1, f (R)"0, f (0)"0 1 2 1 2 1 and f (0) depending on n. In correspondence with the electroweak Z-string, the n"2 vortex 2 has lK "z( . However, the order parameter need not vanish at the centre of the vortex H%A for certain members of the n"2 class of vortices. For example, with n"!2, we may have f (0)O0. 2 The n"2 vortex is not topological and can be continuously deformed into the vacuum manifold. The con"guration at the terminus of the n"2 vortex is called the hedgehog or monopole lK "r( (the radial unit vector). This texture is the direct analog of the electroweak magnetic H%A monopole (lK "r( ) at the terminus of a Z-string. %8 The n"2 discontinuous vortex is unstable but even so has been observed in 3He. In the laboratory, the rotation of the sample stabilizes the n"2 vortex. This seems to be closely analogous to the result of Garriga and Montes [49] who "nd that electroweak strings can be stabilized by external magnetic "elds (Section 7.3). Before proceeding further, it is prudent to remind ourselves of some important di!erences between the (bosonic sector of the) GSW model and 3He. The symmetries in 3He are all global whereas the symmetries in the GSW model are all local. So the n"2 discontinuous vortex is like a global analog of the Z-string. Another important di!erence is in the discrete symmetries in the two systems. The symmetry structure of the GSW model is really [S;(2)];(1)]/Z since 2 the Z elements 1 and !1 which form the center of S;(2) also occur in ;(1). On the contrary, the 2 symmetry group of 3He-A has a multiplicative Z factor which gives rise to the non-trivial 2 topology of the vacuum manifold. It is important to note that we cannot expect 3He to provide an exact replica of the GSW model. However, the similar structures of the two systems means that certain issues can be experimentally addressed in the 3He context while they are far beyond the reach of current particle physics experiments. An issue of this kind is the baryon number anomaly in the GSW model and the anomalous generation of momentum in 3He. As described in Section 8, there are fermionic zero modes on the Z-string and an electric "eld applied along the Z-string leads to the anomalous production of baryon number. What is the corresponding analog in 3He? At "rst sight, 3He does not have the non-Abelian gauge "elds that the electroweak string has and so it seems that the analogy is doomed. But this is not true. The point is that the physics of fermionic zero modes has to do with the dynamics of fermions on the xxed background of the Z-string. Likewise, in 3He we can be interested in the dynamics of quasiparticles in the "xed background of the n"2 vortex. As far as the interaction of quasiparticles with the order parameter background is concerned, one can think of the 3He-A vortex as being due to a ("ctitious) gauge "eld Z@k. Then the interaction of quasiparticles with the order parameter is of the form j Z@k which is exactly analogous to the interaction of quarks and leptons with the k Z-boson. Just as in the electroweak case, the 3He quasiparticles have zero modes on the vortex. In close analogy with the scenario where the motion of a superconducting string through an external magnetic "eld leads to currents along the string (Section 9.4), the velocity of the 3He vortex through the super#uid leads to an anomalous #ow of quasiparticles but this time in the direction perpendicular to the vortex. This #ow causes an extra force on the vortex as it moves through the super#uid that can be monitored experimentally. Such a force was measured in the Manchester experiment [21,137] and is in excellent agreement with theoretical predictions. Hence the Manchester experiment veri"es the anomalous production of quasiparticle momentum on moving
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vortices and the corresponding production of baryon number on electroweak strings moving through a magnetic "eld.
12. Concluding remarks and open problems Quantum "eld theory has been very successful in describing particle physics. Yet the successes have mostly been relegated to perturbative phenomena. A more spectacular level of success will be achieved when our "eld theoretic description of particle physics is con"rmed at the non-perturbative level. The "rst non-perturbative objects that are likely to be encountered in this quest are topological defects and their close cousins that we have described in this review. The search for topological defects can be conducted in accelerator experiments or in the cosmological realm via astronomical surveys. These searches are complementary } only supermassive topological defects can be evident in astronomical surveys, while only the lightest defects can potentially be produced in accelerators. Foreseeable accelerator experiments give us access only to topological defects at the electroweak symmetry breaking scale. So it is very important to understand the defects present in the standard electroweak model and all its viable extensions. One may hope that the structure of defects will yield important clues about the underlying symmetry of the standard model. With this hope, we have described wide classes of defects present in "eld theories. These defects are not all topological and this is relevant to the standard electroweak model which also lacks the non-trivial topology needed to contain topological defects. The absence of topology in the model means that the defect solutions cannot be enumerated in topological terms and neither can their stability be guaranteed. We have described, however, how the existence of defect solutions may still be derived by examining the topological defects occurring in subspaces of the model. The electroweak defects can be thought of as being topological defects that are embedded in the electroweak model. The issue of stability of the defect solution is yet more involved and has not yet been fully resolved in the presence of fermions. That the electroweak Z-string is stable for large h (within the 8 bosonic sector) was inspired by the discovery of semilocal strings and their stability properties. The explicit stability analysis of the electroweak string marks out the region of parameter space in which the Z-string is stable. Then it is clear that the Z-string is unstable for the parameters of the standard model. In certain viable extensions of the standard model and under some external conditions (such as an external magnetic "eld), the standard electroweak Z-string can still be stable. Even if the Z-string is unstable, it is possible that the lifetime of segments of string is long enough so that they can be observed in accelerators. This possibility was discussed in the "rst paper on the subject by Nambu [102]. The discovery of Z-string segments would truly be historic since it would con"rm the existence of magnetic monopoles in particle physics. However, the rate of formation of Z-string segments and their lifetime has not yet been studied in detail. Some of the di$culties in this problem lie quite deep since they involve the connection of perturbative particle physics to the non-perturbative solitonic features. Additionally, the in#uence of fermions on electroweak strings needs further investigation. Electroweak strings may play a cosmological role in the genesis of matter over antimatter as is evident since con"gurations of electroweak string have properties that are similar to the electroweak sphaleron. The challenge here is to determine the number density of electroweak strings
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formed during the electroweak phase transition and their decay rate. Note that the formation of topological strings has been under constant examination over the last two decades and only now, with some experimental input, are we beginning to understand their formation. The cosmological formation of electroweak defects has not been addressed with as much vigour. Recently though, there have been spurts of activity in this area, with lattice calculations beginning to shed interesting insight [29]. It is very likely that further lattice results will be able to give quantitative information about the formation of electroweak strings at the electroweak phase transition. While particle physics experiments to detect electroweak strings are quite distant, experiments in condensed matter systems to study topological defects are becoming more feasible and can be used to test theoretical ideas that are relevant to both particle physics and condensed matter physics. Already there are experiments that test theories of the formation of topological vortices. We can also expect that condensed matter experiments might some day test the formation of defects that are not topological. The experiments on He3 are most relevant in this regard since it contains close analogs of electroweak strings. Furthermore, ideas relating to the behaviour of fermions in the background of electroweak strings can also be tested in the realm of He3. This makes for exciting physics in the years to come which will stimulate the growth of particle physics, cosmology and both, theoretical and experimental, condensed matter physics.
Acknowledgements We are grateful to G. Volovik for very useful comments on Section 11, to M. Groves and W. Perkins for an early draft of their paper, and to M. Hindmarsh for the sphaleron "gures in Section 10. AA thanks K. Kuijken and L. Perivolaropoulos for help with some of the "gures in Sections 2 and 3, and J. Urrestilla for pointing out several typos in an earlier draft. This work was supported by a NATO Collaborative Research Grant CRG 951301, and our travel was also partially supported by NSF grants PHY-9309364 (AA), as well as by UPV grant UPV 063.310-EB187/98 and CICYT grant AEN-93-0435. TV was partially supported by the Department of Energy, USA. TV thanks the University of the Basque Country, and AA thanks Case Western Reserve University, Pierre Van Baal and Leiden University for their hospitality. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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CONTENTS VOLUME 327 H. Primack, U. Smilansky. The quantum three-dimensional Sinai billiard } a semiclassical analysis
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P. Bhattacharjee, G. Sigl. Origin and propagation of extremely high-energy cosmic rays
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R.M. Weiner. Boson interferometry in high-energy physics
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A. AchuH carro, T. Vachaspati. Semilocal and electroweak strings
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