Physics Reports vol.397

Physics Reports 397 (2004) 1 – 62 www.elsevier.com/locate/physrep

Development of turbulence in subsonic submerged jets Polina S. Landaa , P.V.E. McClintockb;∗ a

Department of Physics, Lomonosov Moscow State University, 119899 Moscow, Russia b Department of Physics, Lancaster University, Lancaster LA1 4YB, UK Accepted 13 March 2004 editor: I. Procaccia

Abstract The development of turbulence in subsonic submerged jets is reviewed. It is shown that the turbulence results from a strong ampli3cation of the weak input noise that is always present in the jet nozzle exit section. At a certain distance from the nozzle the ampli3cation becomes essentially nonlinear. This ampli3ed noise leads to a transition of the system to a qualitatively new state, which depends only slightly on the characteristics of the input noise, such as its power spectrum. Such a transition has much in common with nonequilibrium noise-induced phase transitions in nonlinear oscillators with multiplicative and additive noise. The Krylov–Bogolyubov method for spatially extended systems is used to trace the evolution of the power spectra, the root-mean-square amplitude of the turbulent pulsations, and the mean velocity, with increasing distance from the nozzle. It is shown that, as turbulence develops, its longitudinal and transverse scales increase. The results coincide qualitatively and also, in speci3c cases, quantitatively, with known experimental data. c 2004 Elsevier B.V. All rights reserved.
PACS: 47.27.−i; 47.20.Ft; 05.40.−a; 47.27.Wg Keywords: Turbulence; Submerged jet; Noise; Nonequilibrium phase transition

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Brief review of the evolution of views of turbulence as an oscillatory process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Some experimental results concerning turbulence development in jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. The main properties of jet ?ows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Evolution of power spectra of the pulsations of ?uid velocity and pressure with the distance from the nozzle exit section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. E-mail address: [email protected] (P.V.E. McClintock).

c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.03.004

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3.3. A jet as an ampli3er of acoustic disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Suppression and intensi3cation of turbulence in jets by a weak periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . 4. The analogy between noise-induced oscillations of a pendulum with randomly vibrated suspension axis and turbulent processes in a jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The main equations and dynamics of a plane jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The derivation of truncated equations for the amplitude of stochastic constituents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Generative solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. The 3rst approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Region I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Region II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. The second approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 18 22 29 30 48 50 52 53 57 58 58

1. Introduction There is an abundance of published works relating to hydrodynamic turbulence problems. It is interesting that the 3rst experimental work where the transition to turbulence was observed as the ?uid viscosity decreased, due to heating, was reported in 1839 by Hagen [1]. Over the years the volume of experimental works has increased to such an extent that it cannot even be listed in a review of this kind. As examples, we mention only a fundamental paper by Reynolds [2], where elegant experiments with stained liquid were described and intermittent behavior was 3rst discovered, the Compte-Bellot’s book [3], wherein a detailed comparison is made between turbulence power spectra in a plane channel and Kolmogorov’s spectra, and the book by Ginevsky et al. [4] in which experiments with jets are reviewed. A wide variety of books is devoted to the problem of hydrodynamic instability playing the major role in the transition to turbulence (see e.g. [5–10]). Among the many general texts we mention [11–17]. A number of books and a plethora of papers are devoted to numerical calculations of turbulence by both direct and indirect methods (see e.g. the books [18–21]). An important place in the literature is occupied by studies in which the general properties of so-called fully developed turbulence are derived and investigated. Thus Kolmogorov and Obukhov [22–24], for example, derived the power spectra of developed isotropic turbulence starting from simple dimensional arguments (see also [25]). DiIerent generalizations and re3nements of these results were achieved by Novikov [26], Procaccia et al. [27–36], Amati et al. [37] and many other researchers. Recent works, developing an approach to turbulence in the context of contemporary theoretical physics, including 3eld- and group-theoretic methods, can be also assigned to this class. Among these we mention [38–40]. It is known that, as distinct from ?ows in channels, jet ?ows are rarely, if ever, laminar. Over a wide range of Reynolds numbers, so-called hydrodynamic waves are excited and ampli3ed in the body of the jet. The amplitude of these waves decreases exponentially outside the jet shear layer. Undamped hydrodynamic waves can propagate only downstream with a velocity of the order of the ?ow velocity. The distinctive feature of hydrodynamic waves is their random character. Nevertheless, against the background of this randomness there are comparatively regular large-scale patterns known as coherent structures.

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It is very important to note that, when hydrodynamic waves interact with an obstacle or inhomogeneity, they do not undergo simple re?ection. Rather, they induce acoustic waves that propagate upstream. The acoustic waves coming up against an obstacle or an inhomogeneity, in their turn, induce hydrodynamic waves propagating downstream, and so on. Owing to these transformations feedback occurs in jet ?ows, and can excite self-oscillations. Just such a phenomenon arises in jets impinging upon e.g. a ?at plate, a wedge, a coaxial ring or a coaxial pipe [41–45]. In free jets inhomogeneities formed by vortices also induce acoustic waves, again resulting in feedback [46]. But this feedback is nonlinear, and it cannot cause the self-excitation of oscillations. Nonetheless, it exerts an in?uence on the development of turbulence and coherent structures. As will be shown below, the turbulent character of jet ?ows is caused by strong ampli3cation of the random disturbances which are always present at the jet nozzle exit section. 1 At a certain distance from the nozzle the ampli3cation becomes inherently nonlinear. The ampli3cation transforms the system to a qualitatively new state which depends only slightly on the power spectrum or other characteristics of the input disturbances. The system behaves much as though it had undergone a phase transition. The hypothesis that the onset of turbulence can usefully be considered as a noise-induced phase transition was 3rst oIered in [47]. It was based on the existence of profound parallels between turbulent processes in nonclosed ?uid ?ows and noise-induced oscillations in a pendulum with a randomly vibrated suspension axis, which undergoes such a phase transition [44,45,48–51]. Note that this hypothesis is in contradistinction with the widespread belief that the transition to turbulence arises through the excitation of self-oscillations, 3rst periodic and then chaotic [11,12]; but the latter idea does not explain the origin of the feedback mechanism responsible for exciting the self-oscillations. It is well known that instability in a nonclosed ?uid ?ows is of a convective character, but not absolute. Such an instability cannot excite self-oscillations because all disturbances drift downstream. 2 An extremely interesting manifestation of nonlinear eIects in jets lies in the possibility of exploiting them to control turbulence with the aid of acoustic waves applied at some appropriate frequency [4,52–57]. Similar control of noise-induced oscillations was demonstrated for the harmonically driven pendulum [58,51]. Through an approximate solution of the Navier–Stokes equations based on the Krylov–Bogolyubov asymptotic method, we will show that explicit consideration of the ampli3cation of the input noise allows us to account for many known experimental results within the initial part of a jet [59]. Moreover, it follows from our theory that the commonly accepted [60–65,21,4] explanation for the well-known shift of velocity pulsation power spectra towards the low-frequency region is in fact erroneous. According to this explanation, the shift of the power spectra occurs because of feedback via an acoustic wave nascent where vortex pairing occurs, as seen in experiments. We will show that the reason for the spectral shift lies in the jet’s divergence; and that this shift causes the increase of spatial scale with increasing distance from the nozzle, and results in the observed vortex pairing.

1

It should be noted that random sources are present at all points of a jet, even with no external disturbances—i.e. the so called natural ?uctuations [16]. But their in?uence is signi3cantly less than that of disturbances at the jet nozzle exit section and they can therefore be ignored. 2 It should be noted, however, that the instability of a jet ?ow in counter-current stream is of an absolute character and can result in self-excited oscillations.

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It should be noted that interesting phenomena similar to those for ordinary hydrodynamic turbulence are also observed in ?ows of super?uid helium [66]. 2. Brief review of the evolution of views of turbulence as an oscillatory process It is well known that ?uid ?ow in channels is laminar for small ?ow velocities and turbulent for large ?ow velocities [11,12,9]. The problem of how turbulence originates has long attracted the considerable attention of researchers. As is known from the Rytov memoirs [67], the Russian physicist Gorelik believed that : : :turbulence with its threshold of ‘self-excitation’, with typical hysteresis in its appearance or disappearance as the ?ow velocity increases or decreases, with paramount importance of nonlinearity for its developed (stationary) state—is self-oscillations. Their speci3c character lies in that they are self-oscillations in a continuous medium, i.e. in a system with very large number of degrees of freedom. Landau held implicitly the same viewpoint. According to Landau turbulence appears in the following manner: 3rst the equilibrium state corresponding to laminar ?ow becomes unstable and self-oscillations with a single frequency are excited. To describe the amplitude of these self-oscillations, based on physical considerations, Landau wrote a phenomenological equation similar to the truncated van der Pol equation for the amplitude of self-oscillations in a vacuum tube generator, commenting [68]: “With further increase of the Reynolds number new periods appear sequentially. As for the newly appeared motions, they have increasingly small scales”. As a result, multi-frequency self-oscillations with incommensurate frequencies, i.e. quasi-periodic motion, must set in. An attractor in the form of a multi-dimensional torus in the system phase space has to be associated with these self-oscillations. For a large number of frequencies such quasi-periodic self-oscillations diIer little in appearance from chaotic ones, which is why developed turbulence is perceived as a random process. In spite of the fact that Landau’s theory was phenomenological, and did not follow from hydrodynamic equations, it was accepted without question for a long time by almost all turbulence researchers. Moreover, this theory was further developed by Stuart [69–72] who proposed a technique for calculating the coePcients involved in the Landau equations, based on an approximate solution of the Navier–Stokes equations. However, the approximate solution sought by Stuart in the form of A(jt)ei(!t −kx) is, from a physical standpoint, incorrect. It describes a wave that is periodic in space, with a given wave number k and with a slowly time varying amplitude A(jt). Strictly speaking such a solution is true only for a ring ?ow of length L = 2 n=k, where n is an integer, i.e. for a ?ow with feedback. We note that a similar approach to hydrodynamic instability was used by many scientists, beginning from Heisenberg [73]. In the 1970s, after the discovery of the phenomenon of deterministic chaos and the realization that a multi-dimensional torus is unstable [74], the Landau theory became open to question, but the conception of self-oscillations was retained. The diIerence lay only in that, instead of quasi-periodic self-oscillations, they became spoken of as chaotic ones. Thus, according to these new ideas, the onset of turbulence is the sudden birth of a strange attractor in the phase space of certain dynamical variables [74,75]. We note that similar ideas were repeatedly expressed by Neimark (see [76]). Using

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the concept of turbulence as self-oscillations, Gaponov–Grekhov and co-workers published several articles on the simulation of turbulence, modelled in a chain of coupled oscillators [77,78]. However, we believe that turbulence arising in nonclosed ?uid ?ows is not a self-oscillatory process. As already mentioned above, the instability of nonclosed laminar ?ows is of a convective character but not absolute. This means that a disturbance arising at some point of the ?ow will not increase inde3nitely with time, but will drift downstream. It follows from this property of convectively unstable systems that they are not self-oscillatory, but are ampli3ers of disturbances. 3 For such a system to become self-oscillatory, global feedback must be introduced, e.g. by closing the system in a ring. 4 Disturbances are necessarily present in all real systems, both from external sources (technical ?uctuations) and as a result of the molecular structure of a substance (natural ?uctuations). The disturbances can be included as external forces in equations describing the system behavior. The calculation of the forces caused by the natural ?uctuations in hydrodynamic ?ows, based on the ?uctuation–dissipation theorem, was performed by Klimontovich [16]. In hydrodynamic ?ows the presence of ?uctuations, especially at the input, is crucial because they are precisely what lead eventually to the turbulent disturbances observed. It follows from this that an approach to turbulence within the framework of (deterministic) dynamical systems theory is not always appropriate. Naturally, the question arises as to how to treat the features of turbulence which, as pointed out by Gorelik, are seemingly precisely those that are inherent in self-oscillatory systems. First, the term “self-excitation” should be replaced by “loss of stability”. Furthermore, the hysteresis of turbulence, its “appearance or disappearance as the ?ow velocity increases or decreases” can be explained in terms of the speci3c character of the nonlinearity of the gain factor. Finally, the “paramount importance of nonlinearity for its fully developed (stationary) turbulent state” is quite possible in ampli3ers too, because nonlinearity of the ampli3er can have considerable in?uence on its output power spectrum. One piece of evidence suggesting that turbulence is not a self-oscillatory process comes from the numerical experiments of Nikitin [80,81]. He simulated ?uid ?ow in a circular pipe of a 3nite length and radius R with a given velocity at the input cross-section, and with so-called ‘soft’ boundary conditions at the output cross-section; these latter are 9 2 u 92  92  = 2 = 2 =0 ; (2.1) 9x2 9x 9x where u is the longitudinal velocity component,  and  are the radial and angular components of vorticity
= rot u, u = {u; v; w} is the ?ow velocity vector in cylindrical coordinates x, r and . Under these conditions a re?ected wave apparently does not appear, or is very weak. At the input cross-section of the pipe the longitudinal velocity component was taken to be in the form of the Poiseuille pro3le u0 (1 − r 2 =R2 ), weakly disturbed by a harmonic force at the frequency ! = 0:36u0 =R, i.e.,
 r2 u = u0 1 − 2 + A Re(u (r)e−i!t ) cos  ; R v = A Re(v (r)e−i!t ) cos ; 3 4

w = A Re(w (r)e−i!t ) sin  ;

(2.2)

This fact was 3rst mentioned by Artamonov [79]. In essence, this is exactly what occurs in the process of numerical simulation with periodic boundary conditions.

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Fig. 1. Instantaneous distributions of the longitudinal velocity component u in a steady regime for A=u0 = 0:04: (a) along the pipe axis (r=R = 0:02) and (b) near the pipe wall (r=R = 0:93). After [81].

where u (r), v (r) and w (r) are the components of the Orr–Sommerfeld vector-eigenfunction at frequency !, R is the pipe radius, and A is the disturbance amplitude. The velocity u0 and the pipe radius R were set such that the Reynolds number Re was equal to 4000. As the amplitude A exceeded a certain critical value (A ¿ Acr ), random high-frequency pulsations appeared in the ?ow after a short time interval. They occupied all the lower part of the pipe from x = x0 , where x0 depended only weakly on the distance r from the pipe axis. It turned out that the value of x0 decreased as A became larger. The appearance of turbulent pulsations was accompanied by corresponding deformation of the pro3le of the longitudinal constituent of the mean velocity: at the pipe axis the mean velocity decreased, whereas near the pipe wall it increased. We note that a similar deformation of the mean velocity pro3le with increasing turbulent pulsations occurs in jet ?ows as well. The instantaneous distributions of the longitudinal velocity component in a steady regime for A=u0 = 0:04 are shown in Fig. 1 [81]. As the amplitude A gradually decreased, the turbulent region drifted progressively downstream and disappeared at a certain value of A. It is known [82,8] that Poiseuille ?ow in a circular pipe, in contrast to that in a plane channel, possesses the property that laminar ?ow is stable with respect to small perturbations for any Reynolds number. However, in the case of suPciently large Reynolds numbers, such a ?ow is unstable with respect to 3nite perturbations. If an attractor existed corresponding to the turbulent mode, and if the role of the harmonic disturbance was to lead phase trajectories into the attractor basin, then turbulence should not disappear following cessation of the harmonic disturbance. It may be inferred from Fig. 1 that the development of turbulence for A ¿ Acr is associated with a peculiar phase transition at the point x = x0 induced by an ampli3cation of the noise that is always present in any numerical experiment owing to rounding errors. The harmonic disturbance plays

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Fig. 2. View of the turbulent velocity pulsations in a pipe (a) with periodic boundary conditions and (b) with the boundary conditions (2.1) and (2.2). After [80].

a dual role. First, it causes the appearance of instability and, secondly, it initiates the phase transition, much as occurs in a pendulum with a randomly vibrated suspension axis [58], or in jets under low-frequency acoustic forcing [4]. It is no accident that the transition to turbulence was observed by Nikitin only for low-frequency disturbances (for Strouhal numbers of order 0.1). Possible counter-arguments against the above ideas lie in the fact that numerical simulation results obtained with periodic boundary conditions are very close to those observed experimentally. But the data obtained by Nikitin in the numerical experiment described above are also close to numerical data for periodic boundary conditions [80]. The visual similarity of turbulent pulsations calculated for periodic conditions, and for the boundary conditions (2.1) and (2.2), is illustrated in Fig. 2 [45]. This similarity may be explained by the fact that many nonlinear oscillatory systems possess such pronounced intrinsic properties that they exhibit these properties independently of the means of excitation. Some examples of such (nonhydrodynamic) systems are described in [83]. Note that our discussion is not related to so called closed ?ows, e.g. to the Couette ?ow between two rotating cylinders or spheres (see [44]). In closed ?ows there is always feedback linking the output of the ampli3er to its input, so that they consequently become self-oscillatory. 3. Some experimental results concerning turbulence development in jets 3.1. The main properties of jet 6ows Issuing from a nozzle, a ?uid jet always noticeably diverges. This is associated with the fact that, owing to viscosity, neighboring ?uid layers are increasingly drawn into the motion. This phenomenon has come to be known as entrainment. The pro3le of the ?ow velocity changes essentially in the process. At the nozzle exit, it is nearly rectangular, whereas away from the nozzle it becomes bell-shaped: see Fig. 3a. The ?uid layer within which the mean velocity changes signi3cantly is called the shear layer or the mixing layer (see, for example, [84,86,87]). It can be seen from Fig. 3a that, within the initial part of the jet (x 6 xin ), the thickness of the mixing layer increases with increasing distance from the nozzle. At x = xin the thickness of the internal part of the mixing layer 1 becomes equal to the half-width of the nozzle outlet for a plane jet, or the nozzle radius for a circular jet, whereupon a continuous boundary layer is formed. In the vicinity of the

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Fig. 3. (a) Schematic diagram of a diverging free jet illustrating the change of its mean velocity pro3le and widening of the mixing layer. Curves 1 and 2 correspond to the internal and external boundaries of the mixing layer, respectively. (b) Schematic dependence of the relative mean velocity U=U0 along the jet axis on the distance x from the nozzle exit section.

1 0.9 0.8 U / U0

0.7 0.6 0.5 0.4 0.3 0.2 0

5

10

15

20

25

30

x/D

Fig. 4. Experimental dependence of the relative mean velocity U=U0 along the jet axis on the relative distance x=D from the nozzle exit section, for three intensities of the disturbance at the nozzle exit section: ju (0)=0:015, 0.093 0.209 (curves marked by open circles, 3lled circles and stars, respectively). After [50].

jet axis, the mean velocity 3rst decreases very slowly with increasing distance x from the nozzle. This part of the jet is called the initial part: see I in Fig. 3b. Further on, the decrease of the mean velocity becomes signi3cant. This part of the jet is called the main part: see III in Fig. 3b. Parts I and III are separated by the so-called transient part II. The length of the initial part decreases with increasing intensity of disturbances at the nozzle exit section. This can be seen in Fig. 4, where experimental dependences of the relative mean velocity U=U0 on the relative distance x=D from nozzle are plotted (D is the nozzle diameter). Results are shown for three values of the intensity of

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 the disturbances at the nozzle exit section ju (0) = u(0)2 − U (0)2 =U0 , where U0 and u(0) are the mean velocity and longitudinal component of total ?ow velocity at the center of nozzle exit section, respectively [4,21]. The main parts of plane and axially symmetric jets possess approximately the property of selfsimilarity, i.e. at all jet cross-sections the velocity pro3les are aPne-similar [14]. For a plane jet the property of self-similarity means that the jet velocity can be presented in the form u(x; y) = x− F(y=x ), where x and y are longitudinal and transverse coordinate respectively,  and  are certain numbers and F is a function of y=x . The processes in the jet main part are studied in considerable detail (see e.g. [85–93]). We will consider only the processes in the initial part of a jet. It is interesting that coherent structures are formed just in the mixing layer of initial part of the jet. They are vortex formations (bunches of vorticity). Their sizes are of the order of the thickness of the shear layer, and they are moderately long lived. The presence of coherent structures in a jet shear layer results in the intermittent behavior of a jet ?ow, especially in the neighborhood of the external boundary of a jet, where turbulent and laminar phases alternate [94]. 3.2. Evolution of power spectra of the pulsations of 6uid velocity and pressure with the distance from the nozzle exit section The randomness of the hydrodynamic waves excited in a jet manifests itself, in particular, as continuous power spectra of the pulsations of ?uid velocity and pressure. Within the initial part of the jet, these spectra are of a resonant character. Experiments show that the frequency fm corresponding to the maximum of the power spectrum within the initial part of the jet decreases as the distance from the nozzle exit increases [95–97,50]. Within the main part of the jet, the power spectra decrease monotonically with frequency. Fig. 5 shows examples of how the power spectra of the velocity pulsations evolve with distance from the nozzle exit, along the jet axis, and along a line oIset by R from the axis [50]. The abscissa in each case plots the frequency expressed in terms of the Strouhal numbers St = fD=U0 . As mentioned above, most studies of the diIerent processes in jets attribute such behavior of the pulsation power spectrum within the mixing layer to a pairing of vortices. When pairing takes place, the vortex repetition rate must be halved. Within the initial part of the jet, depending on the conditions of out?ow, from 3 to 4 pairings of vortices are usually observed [61]. The frequency fm at the end of the jet’s initial part should therefore decrease by factor of between 8 and 16, a conclusion that con?icts with experimental data. Experiments show that the frequency fm is not a step, but a smooth function of distance from the nozzle exit (see Fig. 6, where the experimental dependences of the Strouhal number Stm on the relative distance from the jet nozzle exit x=D are plotted [45]). In an attempt to resolve this con?ict, the researchers holding this viewpoint speculate that there is a statistical spread in the sites of pairing, but without explaining why there should be such a spread. The faster decrease of Stm within the mixing layer, compared to what happens on the jet axis, may result from the in?uence of nonlinear feedback caused by acoustic waves induced by vortices within the jet mixing layer. The presence of such waves is indirectly supported by the experimental data of Laufer [61]. According to these data high-frequency pulsations of ?uid velocity within

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Fig. 5. Evolution of the spectral density Sp (in decibels) of velocity pulsations u with increasing distance from the nozzle exit x=D along the jet axis (r = 0), and along a line oIset by R from the axis (r = 1). At the bottom, the spectral density for x=D = 0:5, r = 1 is shown on a larger scale. After [50].

a mixing layer near the nozzle exit are modulated by low-frequency pulsations with frequencies corresponding to Strouhal numbers St from 0.3 to 0.5. This fact can be also illustrated by the power spectrum of velocity pulsations on a line oIset by R from the jet axis for x=D = 0:5 (see Fig. 5, at the bottom). We see that the spectrum peaks at the main frequency corresponding to the Strouhal number St=3:2 and the two side frequencies corresponding to St1 =2:7 and St2 =3:7. This means that the modulation frequency corresponds to the Strouhal number 0.5. Owing to the nonlinear feedback, each jet cross-section can be considered as an oscillator with a natural frequency depending on the distance from that cross-section to the nozzle exit. It is evident that the strongest pulsations at the

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Fig. 6. The experimental dependence of the Strouhal number Stm on the relative distance x=D from the jet nozzle exit along the jet axis and within the mixing layer: (a) Petersen’s data for the mixing layer [96]; and (b) the data of [50]. In (b) the dependence on distance along the jet axis, and along a line oIset by R from the axis, are shown by squares and circles, respectively. The solid lines show the dependences Stm = C1 x−1=3 and Stm = C2 x−1 , where C1 ≈ 0:67 and C2 ≈ 1.

cross-section at coordinate x have to occur at a frequency fm that is related to x by the resonant relation xfm xfm =N ; + Uv a where Uv is the velocity of the vortex motion, 5 a is the sound velocity, and N is an integer. From this it follows at once that fm ∼ x−1 . This is precisely the dependence which was found experimentally by Petersen [96] (Fig. 6a). Outside the boundary layer where, within the initial part, inhomogeneities are very weak and nonlinear feedback is nearly absent, the decrease of fm with increasing x follows from the linear theory and is explained by the jet’s divergence as is shown below. 3.3. A jet as an ampli:er of acoustic disturbances Owing to its strong instability, a ?uid jet acts as an ampli3er of disturbances whose frequencies lie within a certain range. It is an ampli3er with a high spatial gain factor. A small acoustic disturbance at some frequency fa within this range near the nozzle transforms into a growing hydrodynamic wave. There is evidence for this in the experimental results of Crow and Champagne [98] and Chan [99]. It follows from the experimental data in Fig. 7 [98] that, above a certain value of the acoustic wave amplitude, the dependence of therelative root-mean-square pulsation of the longitudinal component of hydrodynamic velocity ju = u2 =U0 acquires a resonant character. Here u is the deviation of the longitudinal component of hydrodynamic velocity from its mean value at the acoustic wave frequency fa , measured in terms of Strouhal numbers. The latter authors consider that the resonance is caused by a combination of linear ampli3cation  and nonlinear saturation. The latter increases as the frequency of the disturbance rises. For jua = ua2 =U0 = 0:02, where ua is the oscillatory velocity in the acoustic wave, the dependence of ju on Sta = fa D=U0 is shown in Fig. 8. We see that ju is maximal for Sta ≈ 0:3. 5

It follows from visual observations and measurements of spatio-temporal correlations that Uv ≈ 0:5–0:7U0 .

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Fig. 7. The experimental dependence of ju = near the corresponding curves. After [98].

  u2 =U0 on jua = ua2 =U0 for x=D and values of the Strouhal number marked

Fig. 8. The dependence of ju on Sta for jua = 0:02, x=D = 4 constructed from the data given in Fig. 7. In the absence of acoustical disturbance ju ≈ 0:04. After [98].

Fig. 9a taken from [99] shows that the gain factor depends nonmonotonically on distance from the nozzle exit: it has a maximum at x=D = (0:75–1:25)=Sta . A theoretical dependence similar to that shown in Fig. 9a was found by Plaschko [100] by approximate solution of the linearized Euler equations for a slowly diverging jet. It is depicted in Fig. 9b. By doing so, Plaschko showed that the decrease of the gain factor away from the nozzle exit is caused by jet divergence, and not by nonlinear eIects as was claimed by a number of researchers. 3.4. Suppression and intensi:cation of turbulence in jets by a weak periodic forcing An interesting consequence of the nonlinear eIects in a jet is the possibility they provide for controlling the turbulence level and the length of the jet’s initial part by application of a weak acoustic wave, or by vibration of the nozzle, at an appropriate frequency [101,4]. In the case of

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Fig.  9. (a) The experimental dependence on x˜ = (x=D)Sta of the root-mean-square pulsation of hydrodynamical pressure

p˜ = p2 (in decibels), in the middle of the mixing layer, for a 3xed amplitude of acoustic disturbance and for diIerent Strouhal numbers. After [99]. (b) The theoretical dependence on the relative distance from the jet nozzle x=R of the gain factor K for axially symmetric pulsations of hydrodynamical pressure in a circular jet, for r=R = 1:05, Sta = 0:5. After [100].

high-frequency forcing, the hydrodynamic pulsations are suppressed, whereas at low frequencies, vice versa, there is intensi3cation of pulsations and turbulence. The experiments show that marked intensi3cation or suppression of turbulence within the initial part of a jet, induced by a periodic forcing, is accompanied by changes in the aerodynamic, thermal, diIusive and acoustic properties of the jet. All of these phenomena have been observed by diIerent researchers. It should be noted that the in?uence of acoustic forcing was 3rst studied by Ginevsky and Vlasov [102–106]. Let us consider their main results. In the case of a low-frequency harmonic acoustic forcing at a frequency f corresponding to a Strouhal number in the range 0.2–0.6, the vortices in the jet’s mixing layer are enlarged within the initial part. In turn, this results in an intensi3cation of the turbulent intermixing, thickening the mixing layer, shortening of the initial part and an increase in entrainment; at the same time, the longitudinal and radial velocity pulsations at the jet axis rise steeply. These eIects are observed independently of the direction of the jet irradiation, provided that the amplitudes of the longitudinal and radial components of oscillatory velocity in the sound wave at the jet axis near the nozzle lie in the range 0.05–2% of U0 . For the eIects to occur, the amplitude of the acoustic wave must exceed a certain threshold value. As the wave amplitude rises above this threshold, turbulent intermixing at 3rst intensi3es and then saturates. A further increase of the wave amplitude has little or no eIect on the jet. For high-frequency acoustic forcing of the jet at a frequency corresponding to a Strouhal numbers in the range 1.5–5.0, the vortices in the jet mixing layer become smaller. This results in an attenuation of the turbulent intermixing, a reduction in the thickness of the mixing layer, a lengthening of the initial part, and a decrease of entrainment. Correspondingly, the longitudinal and radial velocity pulsations on the jet axis decrease. In contrast to the eIect of low-frequency forcing, high-frequency forcing does not lead to saturation with increasing amplitude; moreover, an increase in the amplitude beyond a certain value causes, not suppression of the turbulence, but its intensi3cation (see Fig. 10 [65,107]). These eIects are observed universally for jets over a wide range of Reynolds numbers (Re = 102 –106 ), both for initially laminar and for turbulent boundary layers with a level of initial turbulence less than 10%. The foregoing can be illustrated by the experimental dependences of the ?ow relative mean velocity, and the relative root-mean-square pulsation of the longitudinal (ju ) and radial jv components of

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Fig. 10. The experimental dependences of the relative root-mean-square pulsation of the suppression factors (a) ju =j(0) u and (b) jv =j(0) v of the longitudinal and radial components of hydrodynamical velocity on the relative amplitude of acoustic pressure p˜ a measured in decibels, for Sta = 2:35, x=D = 8; j(0) and j(0) are relative pulsations of the longitudinal and u v radial velocity components in the absence of acoustic excitation. After [65].

hydrodynamic velocity on the distance from the nozzle exit along the jet axis for 3xed values of the Strouhal number (Fig. 11). All the dependences shown correspond to a 3xed value of the acoustic forcing intensity. We see that the mean velocity decreases essentially in the case of low-frequency forcing (0:2 ¡ Sta ¡ 1:5) and increases in the case of high-frequency forcing (Sta ¿ 1:5). It should be noted that, as the acoustic forcing intensity at low-frequency increases, the initial part of the jet decreases in length right down to the point where it disappears [108]. EIects similar to those described above are also observed for other means of periodic forcing of the jet: e.g. longitudinal or radial vibration of the nozzle, or a pulsating rate of ?uid out?ow from the nozzle [65,4]. Detailed experimental and numerical studies of turbulence suppression in jet ?ows were also performed by Hussain and collaborators [109–112]. We concentrate in particular on a single result of these works: that the suppression of turbulence by acoustic forcing of constant amplitude depends on its frequency nonmonotonically: it is maximal at a value of the forcing frequency that depends on the amplitude (see Fig. 12 taken from [112]). 6 Periodic forcing of a jet changes markedly the form of its power spectra. For low-frequency forcing, the power spectra of the velocity pulsations near the nozzle contain discrete constituents at the forcing frequency and its higher harmonics. An example of such a spectrum is given in Fig. 13. In the case of high-frequency forcing (see Fig. 14), the power spectra of the velocity pulsations within the jet mixing layer in the immediate vicinity of the nozzle exit also contain discrete components at the forcing frequency and its higher harmonics. At a short distance from the nozzle the second subharmonic appears in the spectrum. Next the fourth, eighth and successively higher subharmonics appear in the spectra. At suPciently large distances from the nozzle exit the spectra are decreasing almost monotonically. Kibens [113] obtained the dependences on distance from the nozzle of the Strouhal number corresponding to the spectral line of highest intensity, both along the jet axis and along a line oIset 6

We note that authors of [112] used, not the conventional Strouhal number St, but St =St=D, where  is the so-called boundary layer momentum thickness at the nozzle exit (see [14]).

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Fig. 11. Experimental dependences: of (top) the relative mean ?ow velocity along the jet axis U=U (0) and of the relative root-mean-square pulsation of the longitudinal (middle) (ju ) and radial (bottom) (jv ) components of hydrodynamic velocity (in %) on x=D under a longitudinal acoustic forcing at Sta = 0:25 (light circles), Sta = 2:75 (3lled circles). When the acoustic forcing is absent, the corresponding curves are marked by triangles. The amplitude of the oscillatory velocity in the acoustical wave on the jet axis near the nozzle exit constitutes 0.07% of U0 . After [65].

from the axis by R, for high-frequency acoustic forcing with a Strouhal number of 3.54 (Fig. 15 [113,45]). We see that these dependences are step-like, with distinct hysteresis phenomena. Adherents to the viewpoint that the decrease of Stm with distance from the nozzle for a free jet is caused by vortex pairing attribute the step-like character of the dependences to localization of the sites of pairing caused by the acoustic forcing [65,4]. In this explanation, the causes of the localization are ignored and the hysteresis phenomena are not discussed. The picture presented in Fig. 15 can also be interpreted as the successive occurrence of subharmonic resonances of higher and higher order as x increases. The transition from subharmonic resonance of one order to the next can clearly be accompanied by hysteresis, if within a certain range of x both of the resonances are stable. In the

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Fig. 12. The experimental dependences on the Strouhal number St = (=D)St of the suppression factor ju =j(0) u , where j(0) u is the relative intensity of the longitudinal velocity pulsations in the absence of acoustic forcing, for x= = 200. The plots are constructed for four values of the amplitude of the oscillatory velocity in the acoustic wave, namely 0.5% of U0 (circles), 2.5% (pluses), 3.5% (crosses) and 4.5% (squares). After [112].

Fig. 13. The power spectrum of the velocity pulsations in response to low-frequency acoustic forcing of a circular jet for Sta = 0:25, x=D = 0:5. After [50].

transition to a subharmonic resonance of higher order, the frequency has to be halved. This can manifest itself as vortex pairing. We can thus infer that the experimentally observed localization of the sites of vortex pairing, when an acoustic wave acts upon a jet, is a consequence, but not a cause, of the indicated behavior of the power spectra.

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Fig. 14. The evolution of power spectra of the velocity pulsations with increasing relative distance x=D from the nozzle exit under high-frequency acoustic forcing at Sta = 2:5. After [50].

Fig. 15. The dependence on distance from the nozzle of the Strouhal number corresponding to the spectral line of highest intensity, in the presence of high-frequency acoustic forcing for Strouhal number 3.54, along the jet axis (light circles) and along a line oIset by R from the axis (3lled circles). After [113].

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4. The analogy between noise-induced oscillations of a pendulum with randomly vibrated suspension axis and turbulent processes in a jet It seems at 3rst sight very surprising that there should exist any analogy between the development and control of turbulence in a jet, and the noise-induced oscillations of a pendulum with a randomly vibrated suspension axis. These latter oscillations and their control were 3rst studied in [114–117,58,51]. We believe that the analogy arises because the onset of turbulence in jets is a noise-induced phase transition, and the pendulum with a randomly vibrated suspension axis is an appropriate model for illustrating just such a transition [48]. In the simplest case, when additive noise is neglected, the equation describing the oscillations of a pendulum with a randomly vibrated suspension axis is ’U + 2(1 + ’˙ 2 )’˙ + !02 (1 + (t)) sin ’ = 0 ;

(4.1)

where ’ is the pendulum’s angular deviation from the equilibrium position, 2(1 + ’˙ 2 )’˙ is proportional to the moment of the friction force which is assumed to be nonlinear, !0 is the natural frequency of small oscillations, and (t) is a comparatively wide-band random process with nonzero power spectral density at the frequency 2!0 . When the intensity of the suspension axis vibration 7 exceeds a certain critical value proportional to the friction factor , excitation of pendulum oscillations occurs, and the variance of the pendulum’s angular deviation becomes nonzero. The evolution of such oscillations, and their power spectra with increasing noise intensity, found by the numerical simulation of Eq. (4.1), are shown in Fig. 16. It can be seen from the 3gure that, close to the excitation threshold, the pendulum oscillations possess the property of so-called on–oI-intermittency. This notion was 3rst introduced by Platt et al. [118], although a similar phenomenon was considered earlier in [119]. It was noted in [118] that intermittency of this kind is similar to the intermittency in turbulent ?ows. It is of importance that on–oI intermittency is possible, not only in dynamical systems, but in stochastic ones as well [120]. It results from ?uctuational transitions through the boundary of excitation [121,123]. External manifestations of on–oI intermittency are similar to those of ordinary intermittency (see e.g. [124]), i.e. over prolonged periods the pendulum oscillates in the immediate vicinity of its equilibrium position (‘laminar phases’); these slight oscillations alternate with short random bursts of larger amplitude (‘turbulent phases’). Away from the excitation threshold the duration of the laminar phases decreases and that of the turbulent ones increases, with the laminar phases ultimately disappearing altogether [121]. The variance of the pendulum’s angular deviation increases in the process. Comparing the evolution of the power spectra shown in Figs. 5 and 16, we can see that they have much in common. As described in [122], high-pass 3ltering of turbulent velocity pulsations reveals their intermittent behavior. We have studied this phenomenon both for experimental velocity pulsations in a jet measured by one of us [50] and also for the pendulum oscillations considered above. In each case we have observed on–oI intermittency after high-pass 3ltering. This fact can be considered as an additional argument in support of the parallels between noise-induced pendulum oscillations and turbulent processes in jets. 7

By intensity of the suspension axis vibration is meant the spectral density of (t) at frequency 2!0 (%(2!0 )).

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Fig. 16. Numerical simulations showing the pendulum oscillations (lower plot in each case) and their power spectra (upper plots) with increasing noise intensity for !0 = 1,  = 0:1,  = 100 and: (a) %(2!0 )=%cr = 1:01; (b) %(2!0 )=%cr = 1:56; (c) %(2!0 )=%cr = 2:44: and (d) %(2!0 )=%cr = 6:25. After [45].

It is important to note that the response of the pendulum to a small additional harmonic force (additional vibration of the suspension axis) is similar to the response of a jet to an acoustic force. For example, in the case when the intensity of the random suspension axis vibration is close to its threshold value, the dependence of the intensity of pendulum oscillations on the frequency of the additional harmonic forcing is of a resonant character, very much like a jet subject to an acoustic force (cf. Figs. 17 and 8). Just as in the case of turbulent jets, the noise-induced pendulum oscillations under consideration can be controlled by a small additional harmonic force. The inclusion of the additional force can be eIected by substitution into Eq. (4.1) of +a cos !a t in place of , where a and !a are, respectively, the amplitude and frequency of the additional vibration of the suspension axis. If the frequency of the additional forcing is relatively low, then this forcing intensi3es the pendulum oscillations and lowers the excitation threshold; vice versa, a relatively high-frequency forcing suppresses the pendulum

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Fig. 17. The dependence of & = ’2

1=2

on !a for !0 = 1,  = 0:1,  = 100, %(2!0 )=%cr = 1:01, a = 0:5. After [45].

Fig. 18. The dependences of & on the amplitude a of low-frequency vibration for !0 = 1,  = 0:1,  = 100 and: (a) %(2!0 )=%cr = 1:89, !a = 0:3; (b) %(2!0 )=%cr = 2:23, !a = 1:5. After [45].

oscillations and increases the excitation threshold. The intensi3cation of the pendulum oscillations by a low-frequency additional vibration is illustrated in Fig. 18 for two values of the vibration frequency. We see that the lower the forcing frequency is, the larger the variance of the oscillation becomes. Just as for jets [125], when the forcing amplitude becomes relatively large, the pendulum’s oscillation amplitude saturates. We now consider in detail the possibility of suppressing noise-induced pendulum oscillations by the addition of a high-frequency vibration. Numerical simulation of Eq. (4.1) with +a cos !a t in place of , where !a ¿ 2, shows that such suppression can occur. The results of the simulation are presented in Figs. 19 and 20. It is evident from Fig. 20 that, for small amplitudes of the high-frequency vibration, this vibration has little or no eIect on the noise-induced oscillations (see Fig. 19a). As the amplitude increases, however, the intensity of the noise-induced oscillations decreases rapidly and the duration of the ‘laminar’ phases correspondingly increases (see Figs. 19b–e). When the amplitude exceeds a certain critical value (for !a = 19:757 it is equal to 42) the oscillations are

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Fig. 19. Time series of ’(t) and ’(t) ˙ for !0 = 1,  = 0:1,  = 100, %(2!0 )=%cr = 5:6, !a = 19:757 and: (a) a = 5; (b) a = 15; (c) a = 30; (d) a = 40. After [45].

suppressed entirely. As the amplitude increases further the oscillations reappear, but now because the conditions required for parametric resonance come into play. For smaller frequencies !a , the behavior of the pendulum oscillations is diIerent. The dependences of the variance of the angle ’ on a for a number of values of the vibration frequency are shown in Fig. 20. It is evident that the variance of ’ at 3rst decreases, passes through a certain minimum value, and then increases again. It is important to note that this minimum value becomes smaller with increasing forcing frequency, but that it is attained for larger forcing amplitudes at higher frequencies. For suPciently high frequencies the oscillations can be suppressed entirely (the case illustrated in Fig. 19). The dependence shown in Fig. 20a closely resembles the corresponding dependence for a jet presented in Fig. 10. The dependences of & on !a for a number of 3xed amplitudes of the additional vibration are illustrated in Fig. 21. Again, these dependences closely resemble the corresponding ones for a jet shown in Fig. 12. The presence of a small additive noise, in addition to the multiplicative one in Eq. (4.1), does not change the behavior of the pendulum qualitatively, but there are large quantitative diIerences. The principal one is the impossibility of achieving full suppression of the pendulum oscillations. Nevertheless, a very marked attenuation of the oscillation intensity occurs. This is illustrated in Fig. 22. It should be emphasized that, in the case of turbulence, full suppression is of course also impossible.

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Fig. 20. The dependences of & on a for !0 = 1,  = 0:1,  = 100, %(2!0 )=%cr = 5:6 and: (a) !a = 3:5; (b) !a = 6; (c) !a = 11; (d) !a = 19:757. After [45].

Fig. 21. The dependence on !a of &=&0 , where &0 is the value of & in the absence of additional vibration, for %(2!0 )=%cr = 5:6, a = 2:5 (3lled circles), a = 5 (pluses), a = 10 (squares), and a = 20 (crosses). After [45].

5. The main equations and dynamics of a plane jet Let us consider a plane jet issuing from a nozzle of width 2d. Neglecting compressibility, we may describe the processes in such a jet by the two-dimensional Navier–Stokes equation for the stream

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Fig. 22. Time series ’(t) and ’(t) ˙ in the presence of additive noise in Eq. (4.1) with multiplicative noise (t) of variance 0.05 for: (a) a = 40; and (b) a = 50. The other parameters are the same as in Fig. 20. After [45].

function ((t; x; y) [11]: 9W( 9( 9W( 9( 9W( − + − )WW( = 0 ; 9t 9x 9y 9y 9x

(5.1)

where W = 92 =9x2 + 92 =9y2 is the Laplacian, ) is the kinematic viscosity, x is the coordinate along the jet axis, and y is the transverse coordinate. The stream function ((t; x; y) is related to the longitudinal (U ) and transverse (V ) components of the ?ow velocity by 9( 9( ; V (t; x; y) = − : (5.2) U (t; x; y) = 9y 9x We can conveniently rewrite Eq. (5.1) in terms of the stream function ((t; x; y) and the vorticity ˜ x; y) which is de3ned by +(t; ˜ x; y) = W((t; x; y) : +(t;

(5.3) x

y

In dimensionless coordinates = x=d, function and vorticity become

= y=d and time

t

= U0 t=d, the equations for the stream

 +˜ (t  ; x ; y ) = W ( (t  ; x ; y ) ; 

(5.4) 

9+˜ (t  ; x ; y ) 9( (t  ; x ; y ) 9+˜ (t  ; x ; y ) − 9t  9x 9y 

+

9( (t  ; x ; y ) 9+˜ (t  ; x ; y ) 2  ˜    W + (t ; x ; y ) = 0 ; −   9y 9x Re

(5.5)

where W is the Laplacian in terms of x and y , Re = 2U0 d=) is the Reynolds number, and U0 is the mean ?ow velocity in the nozzle center. From this point onwards the primes will be dropped. It should be noted that in so deciding on a dimensionless time, the circular frequencies ! = 2 f are measured in units of S = !d=U0 ≡ St, where St = 2fd=U0 is the Strouhal number. In accordance with the ideas presented above, the onset of turbulence is caused by random disturbances (noise) in the nozzle exit section. The authors of most of the works devoted to the stability of these small disturbances [126–128,132] split the solution into mean values and small random disturbances. In our opinion this procedure is inappropriate for two reasons: 3rstly, exact equations for the mean values are unknown; and, secondly, the random disturbances make a signi3cant contribution to the mean values. Therefore we split the solution of Eqs. (5.4) and (5.5) into dynamical

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and stochastic constituents. The dynamical constituents are described by stationary Navier–Stokes equations and diIer from the mean values of the corresponding variables because, owing to the quadratic nonlinearity, the stochastic constituents also contribute to the mean values. Ignoring noise sources everywhere except in the nozzle exit (for x = 0) we set U (t; 0; y) = ud (0; y) + 1 (t; y);

V (t; 0; y) = vd (0; y) + 2 (t; y) ;

(5.6)

where ud (0; y) and vd (0; y) are the dynamical constituents of the longitudinal and transverse velocity components, respectively, and 1 (t; y) and 2 (t; y) are random processes. It should be noted that ud (0; y) and vd (0; y), as well as 1 (t; y) and 2 (t; y), are not independent, but are related by the continuity equation. We consider 3rst the dynamical constituents of the velocity and vorticity. It follows from Eqs. (5.4) and (5.5) that the dynamical constituents ud (x; y), vd (x; y) and +d (x; y) are described by the equations 9ud (x; y) 9vd (x; y) +d (x; y) = − ; (5.7) 9y 9x 9ud (x; y) 9vd (x; y) + =0 ; (5.8) 9x 9y
 9+d (x; y) 9+d (x; y) 2 92 +d (x; y) 92 +d (x; y) ud (x; y) =0 : (5.9) + + vd (x; y) − 9x 9y Re 9x2 9y2 It is very diPcult, if not impossible, to solve these nonlinear equations exactly. Therefore we choose ud (x; y) in the form of a given function of y with unknown parameters depending on x. The shape of this function must depend on whether the out?ow from the nozzle is laminar or turbulent. For simplicity, we restrict our consideration to laminar nozzle ?ow. In this case we can set ud (x; y) so that, at the nozzle exit section, the boundary layer is close in form to that described by the Blasius equation (see [14,11])  F() for |y| 6 1 ; uBl (y) = (5.10) 0 for |y| ¿ 1 ; where a(1 − |y|) ; (5.11) 00 √ 00 = 1=(b0 Re) is the relative thickness of the boundary layer at the nozzle exit, b0 is determined by the conditions of out?ow from the nozzle, a is a parameter which depends on the de3nition of the boundary layer thickness, 8 F() is the derivative with respect to  of the Blasius function f(), described by the equation =

d3 f f d2 f + =0 d3 2 d2

(5.12)

with initial conditions f(0) = 0, df(0)=d = 0, d 2 f(0)=d2 ≈ 0:332 [14,11]. 8

If the boundary layer thickness is de3ned so that at its boundary the relative velocity is equal to 0.99, then a ≈ 5.

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Taking account of the entrainment of the ambient ?uid, we set the velocity pro3le close to (5.10) for x = 0 to the form 
 |y| − 1 1 1 − tanh q ud (x; y) = − r(x) ; (5.13) 1 + tanh(q=00 + r0 ) 0 (x) where 0 (x) and r(x) are unknown functions of x, and 0 (x) is the boundary layer thickness which is equal to 00 for x = 0, r0 = r(0). We note that pro3le (5.13) was 3rst suggested in [46]; it is similar to that given for the mean velocity in [126–128,132]. The thicknesses of inner and external boundary layers (1 (x) and 2 (x)) are de3ned by the relations: ud (x; 1 − 1 (x)) = ;

ud (x; 1 + 2 (x)) = 1 −  ;

where  is a number close to 1. As follows from (5.13) and (5.14) 1 (x) 2 (x) q + r(x) = arc tanh(2 − 1); q − r(x) = arc tanh(2 − 1) : 0 (x) 0 (x) Adding Eqs. (5.15) we obtain the relation between q and : q = 2 arc tanh(2 − 1) :

(5.14) (5.15)

(5.16)

For  = 0:95 we 3nd q ≈ 3. The substitution of (5.16) into (5.15) gives 1 (x) 1 r(x) 2 (x) 1 r(x) = − = + ; : (5.17) 0 (x) 2 q 0 (x) 2 q The form of the velocity pro3le determines the so-called shape-factor H [14], which is equal to the ratio between the displacement thickness  1+2 (0) 1 (1 − ud (0; y)) dy ∗ (0) = 0 (0) 0 and the thickness of momentum loss  1+2 (0) 1 ud (0; y)(1 − ud (0; y)) dy : (0) = 0 (0) 0 For a turbulent boundary layer, the shape-factor H has to lie in the range 1.4–1.6 [14], whereas for a laminar boundary layer it has to be signi3cantly more. Using the values of parameters calculated above we can calculate ∗ (0), (0) and H (0) for our velocity pro3le. As a result, we 3nd ∗ (0) ≈ 0:5081, (0) ≈ 0:1588 and H (0) ≈ 3:2. Thus, our velocity pro3le does correspond to a laminar boundary layer. To 3nd the unknown functions in expression (5.13), we use the conservation laws for the ?uxes of momentum and energy. Usually, they are derived for the mean values of these ?uxes starting from the Reynolds equations [86,21], and therefore contain the so-called turbulent viscosity. We derive them directly starting from Eqs. (5.7)–(5.9) for dynamical constituents. For this we transform Eqs. (5.7)–(5.9) in the following way. Substituting +d (x; y) from Eq. (5.7) into Eq. (5.9), and taking into account that within the jet’s initial part 92 ud =9x2 and 92 vd =9x2 are negligibly small, we obtain 92 ud (x; y) 92 ud (x; y) 2 93 ud (x; y) + vd (x; y) ud (x; y) − =0 : (5.18) 9x9y 9y2 Re 9y3

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Taking account of the continuity equation (5.8) we can rewrite Eq. (5.18) as
 2 92 ud (x; y) 9 9ud2 (x; y) 9(ud (x; y)vd (x; y)) + − =0 : 9y 9x 9y Re 9y2

(5.19)

By integrating Eq. (5.19) over y, we obtain the following approximate equation: 9ud2 (x; y) 9(ud (x; y)vd (x; y)) 2 92 ud (x; y) + − =0 : 9x 9y Re 9y2

(5.20)

The conservation law for the dynamical constituent of the momentum ?ux is found by integrating Eq. (5.20) over y from −∞ to ∞, taking into account that ud (x; ±∞) = 0 and 9ud (x; ±∞)=9y = 0. We thus obtain  ∞ 9 u2 (x; y) dy = 0 : (5.21) 9x −∞ d To derive the conservation law for the dynamical constituent of the energy ?ux, we multiply Eq. (5.20) by 2ud (x; y) and transform it to the form 9ud3 9 2 ud 9(ud2 vd ) 4 ud : + = 9x 9y Re 9y2

(5.22)

Integrating further Eq. (5.22) over y from −∞ to ∞ and taking into account the boundary conditions indicated above we 3nd   ∞  ∞
9ud (x; y) 2 4 9 3 u (x; y) dy = − dy : (5.23) 9x −∞ d Re −∞ 9y Because ud (x; y) is an even function of y, we obtain from (5.21) and (5.23) the following approximate equations:  ∞  ∞ 2 ud (x; y) dy = ud2 (0; y) dy ; (5.24) 0

 3

0

0

∞

ud2 (x; y)

4 9ud (x; y) dy = − 9x Re

 0

∞


9ud (x; y) 9y

2

dy :

(5.25)

Substituting (5.13) into Eq. (5.24) and taking into account that  ∞ ud2 (0; y) dy ≈ 1 0

we obtain a relationship between r(x) and 0 (x): 

  q q q 1 1 + tanh ln 2 cosh + r(x) − + r(x) − + r(x) = 0 : 0 (x) 0 (x) 2 0 (x) Within the jet part, where q 1 ; 0 (x)

(5.26)

(5.27)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

27

relationship (5.26) reduces to 2r(x) ≈ 1, i.e. r(x) ≈ r0 = 0:5. It follows from this and (5.19), (5.17) that 1 (x) ≈

0 (x) ; 3

2 (x) ≈

20 (x) : 3

(5.28)

Substituting (5.13) into Eq. (5.25), and taking account of (5.26), we 3nd the diIerential equation for 0 (x): 

 
q q 2q q −2 + r0 − 4 − cosh + r0 − 1 − tanh + r0 5 tanh 0 (x) 0 (x) 0 (x) 0 (x)




q q d0 (x) 1 −2 3 + tanh cosh + + r0 + r0 4 0 (x) 0 (x) dx 
 q 4q2 1 + tanh + r0 = 3 Re 0 (x) 0 (x) 
 
q q −2 : × 1 + tanh + r0 + cosh + r0 0 (x) 0 (x)

(5.29)

A solution of this equation can be found analytically only for condition (5.27). In this case Eq. (5.29) becomes 16q2 d0 (x) = : 3 Re 0 (x) dx

(5.30)

It follows from Eq. (5.30) that 0 (x) =



200 + 2kx;

k d0 (x) = ; dx 0 (x)

(5.31)

where k = 16q2 =(3 Re). We note that the dependence 0 (x) found here from the Navier–Stokes equations diIers from that found from the Reynolds equations [86,21] and containing the turbulent viscosity )t . Since, by Prandtl’s hypothesis [13], )t is proportional to the boundary layer thickness (x), the dependence (x) was found to be linear. The expressions for vd (x; y) and +d (x; y) can be found by exact solution of Eqs. (5.7), (5.8). As a result, we obtain  
q(|y| − 1) 16q sign y q(|y| − 1) vd (x; y) = − tanh − r0 3 Re 0 (x)(1 + tanh(q=00 + r0 )) 0 (x) 0 (x)
  q cosh(q(|y| − 1)=0 (x) − r0 ) q tanh + r0 − ln ; − 0 (x) 0 (x) cosh(q=0 (x) + r0 )

(5.32)

28

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62



 256q4 (|y| − 1)2 q sign y −2 q(|y| − 1) 1+ cosh +d (x; y) = − − r0 0 (x)(1 + tanh(q=00 + r0 )) 0 (x) 940 (x) Re2  2
 q q 256q2 −2 + r0 − 2 cosh 0 (x) 90 (x) Re2 20 (x)

 q(|y| − 1) q (|y| − 1) tanh − r0 − 0 (x) 0 (x) 


cosh(q(|y| − 1)=0 (x) − r0 ) q + r0 : (5.33) + ln − tanh 0 (x) cosh(q=0 (x) + r0 ) For condition (5.27), from (5.32) and (5.33) we 3nd the following approximate asymptotic expressions for vd and +d : vd (x; ±∞) ≈ ∓

16qr0 ; 30 (x) Re

+d (x; ±∞) ≈ ∓

256q3 r0 : 930 (x) Re2

(5.34)

Fig. 23 shows plots of ud (x; y), vd (x; y), +d (x; y) versus y for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000, x = 0 and 8. We see that for all values of y, except for narrow intervals near y = ±1, ud (x; y), vd (x; y) and +d (x; y) are nearly constant. The constant transverse velocity component for |y| ¿ 1 directed towards the jet axis accounts for the entrainment of ambient ?uid with the jet ?ow. It should be emphasized that the results obtained here concern only the dynamical constituents of the velocity and vorticity. Stochastic constituents greatly in?uence the thickness of the boundary layer, its dependence on the distance from the nozzle, and values of the mean velocities (see below). Substituting further U (t; x; y) = ud (x; y) +

9 (t; x; y) ; 9y

V (t; x; y) = vd (x; y) −

9 (t; x; y) ; 9x

˜ x; y) = +d (x; y) + +(t; x; y) +(t;

(5.35)

into Eqs. (5.4), (5.5) and taking into account (5.7)–(5.9), we 3nd the equations for the stochastic constituents (t; x; y) and +(t; x; y), which we write in the form +−W =0 ;

(5.36)

2 9+ 9+ 9+ 9 9 + ud (x; y) + vd (x; y) − +dy (x; y) + +d x (x; y) − W+ 9t 9x 9y 9x 9y Re =

9 9+ 9 9+ − ; 9x 9y 9y 9x

(5.37)

where +d x (x; y) =

9+d (x; y) ; 9x

+dy (x; y) =

9+d (x; y) : 9y

1

1

0.8

0.8

0.6

0.6

ud

ud

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

0.4

0.4

0.2

0.2

0

-1 -0.5

0.006 0.004 0.002 0 -0.002 -0.004 -0.006

0 0.5

1

y

0.001 0

-0.001 -1 -0.5

0 0.5

-0.002

1

y

(e)

-1 -0.5

(b)

0.002

(c)

-1 -0.5

0 0.5

1

y

(d)

10 5

Ωd

30 20 10 0 -10 -20 -30

0

1

vd

vd

0.5

y

(a)

Ωd

0

29

0 -5

-1 -0.5

0

y

0.5

-10

1

-1 -0.5

(f)

0 0.5

1

y

Fig. 23. Plots of various quantities versus y for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000: (a) ud (0; y); (b) ud (8; y); (c) vd (0; y); (d) vd (8; y); (e) +d (0; y); and (f) +d (8; y).

According to (5.6) the boundary conditions for Eqs. (5.36) and (5.37) are   9  9  = 1 (t; y); = −2 (t; y) : 9y x=0 9x x=0

(5.38)

6. The derivation of truncated equations for the amplitude of stochastic constituents To describe the development of turbulence, we can assume that the right-hand side of Eq. (5.37) is of the order of a small parameter j. In this case Eqs. (5.36) and (5.37) can be solved approximately by a method similar to the Krylov–Bogolyubov method for spatially extended systems [131]. We therefore seek a solution in the form of a series in j: +(t; x; y) = +0 (t; x; y) + jr1 (t; x; y) + j2 r2 (t; x; y) + : : : ; (t; x; y) =

0 (t; x; y)

+ js1 (t; x; y) + j2 s2 (t; x; y) + : : : ;

30

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

u(t; x; y) =

9 (t; x; y) = u0 (t; x; y) + jq1 (t; x; y) + j2 q2 (t; x; y) + · · · ; 9y

(6.1)

where +0 (t; x; y) and 0 (t; x; y) are generative solutions of Eqs. (5.36) and (5.37), u0 (t; x; y) = 9 0 (t; x; y)=9y; r1 (t; x; y); r2 (t; x; y); : : : ; s1 (t; x; y); s2 (t; x; y); : : : are unknown functions, and q1 (t; x; y)= 9s1 (t; x; y)=9y; q2 (t; x; y) = 9s2 (t; x; y)=9y; : : : : It should be emphasized that because of the quadratic nonlinearity the contribution of nonlinear terms into turbulent processes can be estimated only by using the second approximation of the Krylov–Bogolyubov method. Thus in expansion (6.1) we have to retain the terms up to the second order with respect to j. 6.1. Generative solutions Putting the right-hand side of Eq. (5.37) to zero and eliminating the stochastic constituent of vorticity, we obtain the generative equation for the stochastic constituent of the stream function: 9W 9t

0

+ ud (x; y)

− +dy (x; y)

9W 0 9W 0 + vd (x; y) 9x 9y

9 0 9 0 2 + +d x (x; y) − WW 9x 9y Re

0

=0 :

(6.2)

It should be noted that 3nding the generative solution is similar to the well-known problem of the linear instability of a jet ?ow. During the last three decades, this problem was studied primarily by Crighton and Gaster [126,132], Michalke [128] and Plaschko [127]. In these works a pro3le of the mean ?ow velocity for a circular jet was given, and the problem was solved approximately, mainly within the framework of linearized Euler equations. Because the coePcients of these equations depend on the coordinates, an exact analytic solution could not be found. Numerical calculations performed by these authors are in qualitative agreement with experimental data. Here we 3nd the generative solution for a plane jet based on the linearized Navier–Stokes equation (6.2) and using the dynamical constituents of velocity and vorticity calculated above. We emphasize that viscosity should be taken into account, because all terms in Eq. (6.2) are of the same order over the region of the boundary layer. We chose a plane jet, rather than a circular one, by virtue of the same reasoning as in [92]: its simple geometry and boundary conditions. We seek a partial solution of Eq. (6.2) in the form of a sum of running waves of frequency S with a slowly varying complex wave number Q(S; x): 
  ∞  x 1 (S) (t; x; y) = f (x; y) exp i St − Q(S; x) d x dS : (6.3) 0 2 −∞ 0 Taking into account that the jet diverges slowly, we can represent the√function f(S) (x; y) and the wave number Q(S; x) as series in a conditional small parameter 2 ∼ 1= Re: f(S) (x; y) = f0 (S; x; y) + 2f1 (S; x; y) + : : : ;

Q(S; x) = Q0 (S; x) + 2Q1 (S; x) + : : : ;

(6.4)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

31

where f0 (S; x; y); f1 (S; x; y); : : : are unknown functions vanishing, along with their derivatives, at y = ±∞. Substituting (6.3), in view of (6.4), into Eq. (6.2) and retaining only terms containing 3rst derivatives with respect to x we obtain the following equations for f0 (S; x; y) and f1 (S; x; y): L0 (Q0 )f0 = 0 ;

(6.5)

L0 (Q0 )f1 = iQ1 L1 (Q0 )f0 − L2 (Q0 )f0 ;

(6.6)

where


L0 (Q0 ) = i(S − ud (x; y)Q0 )

92 − Q02 9y2




+ vd (x; y)

93 9 − Q02 3 9y 9y




4  2 2 9 9 2 9 4 + iQ0 +dy (x; y) + +d x (x; y) − − 2Q0 2 + Q0 ; 9y Re 9y4 9y 
2 9 9 L1 (Q0 ) = ud (x; y) − 3Q02 + 2SQ0 − 2iQ0 vd (x; y) 9y2 9y
2  8iQ0 9 − Q02 ; − +dy (x; y) + Re 9y2  

 3 9Q0 9Q0 9 9 9 L2 (Q0 ) = S 2Q0 + + ud (x; y) + − ivd (x; y) − 3Q0 Q0 9x 9x 9x9y2 9x 9x 
9Q0 9 9 92 + − +dy (x; y) × 2Q0 9x9y 9x 9y 9x  
9Q0 93 9 9Q0 92 4i 2 2Q0 +3 : + − Q0 2Q0 + Re 9x9y2 9x 9y2 9x 9x

(6.7)

(6.8)

(6.9)

Eq. (6.5), with the boundary conditions for function f0 and its derivatives so as to be vanishing at y = ±∞, describes a non-self-adjoint boundary-value problem, where Q0 plays the role of an eigenvalue. Similar boundary-value problems, but on a 3nite interval, were studied by Keldysh [129]. Consistent with Fredholm’s well-known theorem [130] about linear boundary-value problems described by an inhomogeneous equation, Eq. (6.6) has a nontrivial solution only if  ∞  ∞ iQ1 3(S; X x; y)L1 (Q0 )f0 (S; x; y) dy − 3(S; X x; y)L2 (Q0 )f0 (S; x; y) dy = 0 ; (6.10) −∞

−∞

where 3(S; X x; y) is a complex conjugate eigenfunction of the adjoint boundary-value problem described by the equation:  
3
2 9 9 2 2 9 (vd (x; y)3) − Q0 [(S − ud (x; y)Q0 )3] X − − Q0 X i 9y2 9y3 9y
 2 2 94 3X X 9(+d x (x; y)3) 2 9 3X 4 − 2 + iQ0 +dy (x; y)3X − − 2Q + Q 3 X =0 : (6.11) 0 0 9y 4 9y4 9y2 Condition (6.10) allows us to 3nd the small correction Q1 (S; x) to the eigenvalue Q0 (S; x).

32

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

Over the region of |y| 6 y1 (x) (region I), where y1 (x) is the internal boundary of the boundary layer, ud (x; y) ≈ 1, vd (x; y) ≈ 0, +d x (x; y) ≈ 0 and +dy (x; y) ≈ 0. For this region the general solution of Eqs. (6.5) and (6.11) is f0 (y) = A1 sinh(B11 y) + A2 sinh(B12 y) + A3 cosh(B11 y) + A4 cosh(B12 y) ; 3(y) X = A˜ 1 sinh(B11 y) + A˜ 2 sinh(B12 y) + A˜ 3 cosh(B11 y) + A˜ 4 cosh(B12 y) ; where

B11 = Q0 ;

B12 =

Q02 +

i(S − Q0 ) Re 2

(6.12)

(6.13)

are the roots of the characteristic equation corresponding to Eqs. (6.5) and (6.11), and A1 , A2 , A3 , A4 , A˜ 1 , A˜ 2 , A˜ 3 and A˜ 4 are arbitrary constants. Any arbitrary disturbance can be represented as a linear combination of even and odd constituents. We consider the case of odd disturbances. In this case we can solve Eqs. (6.5) and (6.11) only for positive values of y, seeking a solution of these equations in the form f0 (y) = A1 f01 (y) + A2 f02 (y);

3X0 (y) = A˜ 1 3X01 (y) + A˜ 2 3X02 (y) ;

(6.14)

where, for |y| 6 y1 (x), the functions f01 (y), 3X1 (y) transform to sinh(B11 y), and f02 (y), 3X2 (y) transform to sinh(B12 y). It follows that f01 (y), 3X01 (y), f02 (y) and 3X02 (y) must satisfy the following initial conditions:   9f01  93X01  = = B11 ; f01 (0) = 3X01 (0) = 0; 9y y=0 9y y=0     92 3X01  93 f01  93 3X01  92 f01  3 = = 0; = = B11 ; 9y2 y=0 9y2 y=0 9y3 y=0 9y3 y=0   9f02  93X02  = = B12 ; f02 (0) = 3X02 (0) = 0; 9y y=0 9y y=0     92 3X02  93 f02  93 3X02  92 f02  3 = = 0; = = B12 : (6.15) 9y2 y=0 9y2 y=0 9y3 y=0 9y3 y=0 Over the region of |y| ¿ y2 (x) (region II), where y2 (x) is the external boundary of the boundary layer, ud (x; y) ≈ 0, vd (x; y) ≈ vd (x; ∞), +d x (x; y) ≈ +d x (x; ∞) and +dy (x; y) ≈ 0. For this region the solutions of Eqs. (6.5) and (6.11) must behave as partial solutions of the equations  
3
2 9 f0 9 f0 2 2 9f0 − Q0 f0 + vd (x; ∞) − Q0 iS 9y2 9y3 9y
4  2 2 9 f0 9f0 2 9 f0 4 + +d x (x; ∞) − − 2Q0 + Q0 f0 = 0 ; 9y Re 9y4 9y2

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62


iS

92 3X0 − Q02 3X0 9y2




− vd (x; ∞)

2 93X0 − +d x (x; ∞) − 9y Re




93 3X0 93X0 − Q02 9y3 9y

33



2 94 3X0 2 9 3X0 − 2Q + Q04 3X0 0 9y4 9y2

 =0 ;

(6.16)

satisfying the condition of vanishing at y = ∞. Such partial solutions can be written as f0 (y) = C21 exp[B21 (y − y2 (x))] + C22 exp[B22 (y − y2 (x))] ; 3X0 (y) = C˜ 21 exp[B˜ 21 (y − y2 (x))] + C˜ 22 exp[B˜ 22 (y − y2 (x))] ;

(6.17)

where C21 , C22 , C˜ 21 and C˜ 22 are arbitrary constants, and B21 , B22 , B˜ 21 and B˜ 22 are roots of the characteristic equations corresponding to Eqs. (6.16) with negative real parts. The characteristic equations for region II take the form B4 − a10 B3 − a20 B2 − a30 B + a40 = 0 ; B˜ 4 + a10 B˜ 3 − a20 B˜ 2 + a30 B˜ + a40 = 0 ;

(6.18)

where a10 = ±

vd (x; ∞) Re ; 2

a30 = ±

(+d x (x; ∞) − Q02 vd (x; ∞)) Re ; 2

a20 = 2Q02 +

iS Re ; 2 a40 = Q04 +

iSQ02 Re ; 2

(6.19)

the signs ‘+’ and ‘−’ correspond to y ¿ 0 and y ¡ 0, respectively. It follows from (6.18) and (6.19) that jth root of Eq. (6.18) for y ¡ 0 is equal to jth root for y ¿ 0 of opposite sign. That is why we need consider only y ¿ 0. In view of (6.19), the 3rst equation of (6.18) can be conveniently rewritten as
  vd (x; ∞) Re i S Re +d x (x; ∞) Re 2 2 B − Q0 + (B2 − Q02 ) = B : (6.20) B − 2 2 2 It can be shown that the right-hand side of Eq. (6.20) is small. Therefore the roots of Eq. (6.20) with negative real parts are approximately equal to B21 = −Q0 + WB1 ; where vd (x; ∞) Re B220 = 4 WB1 = −

B22 = B220 + WB2 ;

1+



8iS 16Q02 1+ 2 + 2 vd (x; ∞) Re vd (x; ∞) Re2

+d x (x; ∞) ; 2(iS − vd (x; ∞)Q0 )

WB2 =

(6.21)  ;

+d x (x; ∞)B220 : 2 vd (x; ∞)(B220 + Q02 ) + iSB220

(6.22)

34

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

Equating (6.14)–(6.17) at the point y = y2 (x), we 3nd the following boundary conditions: A1 f01 (y2 ) + A2 f02 (y2 ) = C21 + C22 ;

A1 f11 (y2 ) + A2 f12 (y2 ) = B21 C21 + B22 C22 ;

2 2 C21 + B22 C22 ; A1 f21 (y2 ) + A2 f22 (y2 ) = B21

A˜ 1 3X01 (y2 ) + A˜ 2 3X02 (y2 ) = C˜ 21 + C˜ 22 ;

3 3 A1 f31 (y2 ) + A2 f32 (y2 ) = B21 C21 + B22 C22 ;

A˜ 1 3X11 (y2 ) + A˜ 2 3X12 (y2 ) = B˜ 21 C˜ 21 + B˜ 22 C˜ 22 ;

A˜ 1 3X21 (y2 ) + A˜ 2 3X22 (y2 ) = B˜ 221a C˜ 21 + B˜ 222a C˜ 22 ; A˜ 1 3X31 (y2 ) + A˜ 2 3X32 (y2 ) = B˜ 321 C˜ 21 + B˜ 322 C˜ 22 ; where

(6.23)

   9f01  9f02  92 f01  ; f (y ) = ; f (y ) = ; 12 2 21 2 9y y=y2 9y y=y2 9y2 y=y2    92 f02  93 f01  93 f02  f22 (y2 ) = ; f31 (y2 ) = ; f32 (y2 ) = : 9y2 y=y2 9y3 y=y2 9y3 y=y2    93X01  93X02  92 3X01  3X11 (y2 ) = ; 3X12 (y2 ) = ; 3X21 (y2 ) = ; 9y y=y2 9y y=y2 9y2 y=y2    92 3X02  93 3X01  93 3X02  3X22 (y2 ) = ; 3X31 (y2 ) = ; 3X32 (y2 ) = : 9y2 y=y2 9y3 y=y2 9y3 y=y2

f11 (y2 ) =

The eigenvalues of Q0 for the basic boundary-value problem must satisfy the requirement that the determinant     f01 (y2 ) f02 (y2 ) 1 1      f11 (y2 ) f12 (y2 ) B21 B22    (6.24) D(Q0 ) =   2 2  f21 (y2 ) f22 (y2 ) B21 B22      3 3   f31 (y2 ) f32 (y2 ) B21 B22 be equal to zero. It can be shown that the eigenvalues for the adjoint boundary-value problem coincide with those for the basic boundary-value problem. The expression for D(Q0 ) can be written as 2 2 D(Q0 ) = q23 (y2 ) − (B21 + B22 )q13 (y2 ) + (B21 + B22 )q12 (y2 ) 2 2 + B21 B22 (q12 (y2 ) + q03 (y2 ) − (B21 + B22 )q02 (y2 )) + B21 B22 q01 (y2 ) ;

(6.25)

where q01 (y) = f01 (y)f12 (y) − f11 (y)f02 (y);

q02 (y) = f01 (y)f22 (y) − f21 (y)f02 (y) ;

q03 (y) = f01 (y)f32 (y) − f31 (y)f02 (y);

q12 (y) = f11 (y)f22 (y) − f21 (y)f12 (y) ;

q13 (y) = f11 (y)f32 (y) − f31 (y)f12 (y);

q23 (y) = f21 (y)f32 (y) − f31 (y)f22 (y) :

(6.26)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

35

1.3 10

1.2 1.1 1 0

6

v ph0

Γ0

8 0.5 1

4 3 2 10

0.9 0.8 0.7

2

0.6

5

10 5 3

0.5

0

0 0.5 21

0.4 0

1

(a)

2

3

4

5

6

St

7

0

1

2

(b)

3

4

5

St

Fig. 24. The dependences on the Strouhal number St for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000 and diIerent x of: (a) the gain factor 70 and (b) the wave phase velocity vph0 = S=K0 . The value of x is indicated near the corresponding curve in each case.

A direct numerical calculation of D(Q0 ), starting from Eqs. (6.5) and (6.25), gives random values on account of the need to subtract large numbers of the same order. Therefore, instead of Eq. (6.5), we solve the equations for qij (y) which follow from (6.5) and (6.26). They are 9q02 9q12 = q03 (y) + q12 (y); = q13 (y) ; 9y 9y 
i(S − ud (x; y)Q0 ) Re 9q03 2 = q13 (y) + 2Q0 + q02 (y) 9y 2

9q01 = q02 (y); 9y

Re [vd (x; y)[q03 (y) − Q02 q01 (y)] + +d x (x; y)q01 (y)] ; 2 
i(S − ud (x; y)Q0 ) Re 9q13 2 q12 (y) = q23 (y) + 2Q0 + 9y 2 +

+ Q04 q01 (y) +

Re [[iQ02 (S − ud (x; y)Q0 ) − iQ0 +dy (x; y)]q01 (y) + vd (x; y)q13 (y)] ; 2

Re 9q23 = Q04 q02 (y) + [[iQ02 (S − ud (x; y)Q0 ) − iQ0 +dy (x; y)]q02 (y) 9y 2 + vd (x; y)(q23 (y) + Q02 q12 (y)) − +d x (x; y)q12 (y)] :

(6.27)

Solving Eqs. (6.27) with initial conditions q01 (0) = q02 (0) = q03 (0) = q12 (0) = q23 (0) = 0;

2 2 q13 (0) = B11 B12 (B12 − B11 );

(6.28)

and substituting the solution found for y = y2 into (6.25), we calculate D(Q0 ). By varying Q0 until D(Q0 ) becomes equal to zero, we 3nd the eigenvalues of Q0 . The real part of the eigenvalue of Q0 gives the real wave number K0 , whereas its imaginary part gives the wave gain factor 70 . The dependences of the gain factor 70 and wave phase velocity vph0 = S=K0 on the Strouhal number St are given for Re = 25; 000, b0 = 0:1 and a number values of x in Figs. 24a and b.

36

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

10 8

v ph0

Γ0

1 2

6 4 3

2 0 0

1

2

3

(a)

4

5

6

7

1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4

1 2 3 0

1

(b)

St

2

3

4

5

St

Fig. 25. The dependences on the Strouhal number St of (a) the gain factor 70 and (b) the wave phase velocity vph0 = S=K0 for x = 0 and Re = 25; 000, b0 = 0:1 (curve 1), Re = 100; 000, b0 = 0:05 (curve 2) and Re = 100; 000, b0 = 0:02 (curve 3).

To estimate the in?uence of the Reynolds number and the thickness of the boundary layer at the nozzle exit, we have calculated the eigenvalues of Q0 for x = 0 in two cases: Re = 100; 000, b0 = 0:05 and Re = 100; 000, b0 = 0:02. In the 3rst case the thickness of the boundary layer at the nozzle exit is the same as in Fig. 24, and in the second case it is considerably larger. The results are shown in Fig. 25. We see that, for the same thickness of boundary layer at the nozzle exit the results depend only weakly on the Reynolds number, whereas the thickness of the boundary layer aIects the eigenvalues strongly. To 3nd the eigenfunction and the adjoint eigenfunction corresponding to the eigenvalue Q0 , we use Eq. (6.10) for the calculation of the correction Q1 to the eigenvalue Q0 , and we should in principle calculate the functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y) and solve the systems of equations (6.23), taking into account that their determinants are equal to zero. However, in the process of a direct numerical solution of Eqs. (6.5) and (6.11) over the region of boundary layer (region III) we face the problem of the strong instability of solutions corresponding to the functions f01 and 3X01 with respect to small rapidly increasing disturbances. This instability becomes more pronounced for larger S. The instability can be illustrated clearly if we pass in Eqs. (6.5) and (6.11) to new variables ˜ x; y) by 9(S; x; y) and 9(S; f0 (S; x; y) = C exp(9(S; x; y));

˜ x; y)) ; 3X0 (S; x; y) = C˜ exp(9(S;

(6.29)

where  9(S; x; y) =

0

y

B(S; x; y) dy;

˜ x; y) = 9(S;



y

0

˜ x; y) dy : B(S;

(6.30)

Substituting (6.29) into Eqs. (6.5) and (6.11) and taking account of (6.30) we obtain the following ˜ x; y): nonlinear equations for B(S; x; y) and B(S; 93 B 92 B 9B +3 + 4B + 2(3B2 − Q02 ) 3 2 9y 9y 9y




9B 9y

2

+ (B2 − Q02 )2

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

37

 

2  Re 9B 9B 9B 2 2 2 2 i(S − ud (x; y)Q0 ) − + 3B + B − Q0 + vd (x; y) + B(B − Q0 ) 2 9y 9y2 9y  + iQ0 +dy (x; y) + +d x (x; y)B = 0;

(6.31)


2 2˜ ˜ 9B B 9 9 B 93 B˜ 2 2 +3 + 4B˜ 2 + 2(3B˜ − Q0 ) + (B˜ 2 − Q02 )2 3 9y 9y 9y 9y  
Re 9B˜ 2 2 ˜ i(S − ud (x; y)Q0 ) + B − Q0 − iQ0 (2udy (x; y)B˜ + udyy (x; y)) − 2 9y
− vd (x; y)

 
92 B˜ 9B˜ 9B˜ Q02 2 2 2 ˜ ˜ ˜ ˜ + B(B − Q0 ) − 3vdy (x; y) +B − + 3B 9y2 9y 9y 3 

− 3vdyy (x; y)B˜ − vdyyy (x; y) + iQ0 +dy (x; y) − +d x (x; y)B˜ − +d xy (x; y) = 0 :

(6.32)

It is convenient to solve Eqs. (6.31) and (6.32) forward from y = y1 (x) to 1 and backward from y = y2 (x) to 1, and then to sew the solutions found for y = 1. In the 3rst case we should 3nd four partial solutions of these equations with initial conditions B1; 2 (S; x; y1 ) = B˜ 1; 2 (S; x; y1 ) = ±Q0 ;

B3; 4 (S; x; y1 ) = B˜ 3; 4 (S; x; y1 ) = ±B12 :

(6.33)

It is evident that the functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y), for y1 6 |y| 6 1, are equal to A1 (exp(91 (S; x; y)) − exp(92 (S; x; y))) ; 2 A2 f02 (S; x; y) = (exp(93 (S; x; y)) − exp(94 (S; x; y))) ; 2

f01 (S; x; y) =

3X01 (S; x; y) =

A˜ 1 (exp(9˜ 1 (S; x; y)) − exp(9˜ 2 (S; x; y))) ; 2

3X02 (S; x; y) =

A˜ 2 (exp(9˜ 3 (S; x; y)) − exp(9˜ 4 (S; x; y))) ; 2

(6.34)

(6.35)

where 991; 2 (S; x; y) = B1; 2 (S; x; y); 9y

99˜ 1; 2 (S; x; y) = B˜ 1; 2 (S; x; y) ; 9y

993; 4 (S; x; y) = B3; 4 (S; x; y); 9y

99˜ 3; 4 (S; x; y) = B˜ 3; 4 (S; x; y) 9y

(y1 6 |y| 6 1) :

38

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 1400

800

1200

600

1000

400

f 0i

f 0r

800 600

200 0

400 200

-200

0

-400

-200

-600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

(a)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

(b)

1400

8

1200

7 6

f0

5

800

4

600

arg

|f| 0

1000

3 2

400

1

200

0

0

(c)

-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 y

0 0.2 0.4 0.6 0.8

(d)

1 1.2 1.4 1.6 1.8 y

2 2.2

Fig. 26. The partial solutions of Eq. (6.31) for (a and b) y1 6 y 6 1 and (c and d) 1 6 y 6 y2 : Re = 25; 000, b0 = 0:1 m, x = 0, S = 22 (Q0 ≈ 44:357517 + 0:597408i). It is found that (a) and (b) all solutions tend to B3 , and (c) and (d) that the 3rst solution tends to the second one.

In the second case we should 3nd two partial solutions of Eqs. (6.31) and (6.32) with initial conditions B1 (S; x; y2 ) = B21 ;

B2 (S; x; y2 ) = B22 ;

B˜ 1 (S; x; y2 ) = B˜ 21 ;

B˜ 2 (S; x; y2 ) = B˜ 22 ;

(6.36)

where B21 , B22 , B˜ 21 and B˜ 22 are de3ned by (6.22). The functions f01 (S; x; y), f02 (S; x; y), 3X01 (S; x; y) and 3X02 (S; x; y), for 1 6 |y| 6 y2 , are equal to f01 (S; x; y) = C1 exp(:1 (S; x; y));

f02 (S; x; y) = C2 exp(:2 (S; x; y)) ;

(6.37)

3X01 (S; x; y) = C˜ 1 exp(:˜ 1 (S; x; y));

3X02 (S; x; y) = C˜ 2 exp(:˜ 2 (S; x; y)) ;

(6.38)

where 9:1; 2 (S; x; y) = B1; 2 (S; x; y); 9y

9:˜ 1; 2 (S; x; y) = B˜ 1; 2 (S; x; y) 9y

(1 6 |y| 6 y2 ) :

Numerical solution of Eqs. (6.31) and (6.32), both forward and backward, has shown that in the 3rst case all partial solutions for B tend to the third partial solution with initial condition B|y=y1 = B12 , whereas in the second case the 3rst partial solution tends to the second one with initial condition B|y=y2 = B22 (see, for example, Fig. 26, where all numerical partial solutions of Eqs. (6.31) and (6.32) with initial conditions (6.33) and (6.36) are shown for S = 22). This means that all partial solutions, except the third one in the 3rst case and the second one in the second case, are ˜ followunstable. That is why we have taken the two partial solutions of the equations for B and B, ing from the Euler equations, as the 3rst and second approximate partial solutions of Eqs. (6.31)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 40

40

30

20 0

~ B1i , B 1i

~ B1r , B 1r

20 10 0 -10

-40

-80

-30 0.9

0.95

(a)

1

1.05

-100 0.9

1.1

40

5

30

0

20

-5 -10

-30 1.05

-40 0.9

1.1

1.05

1.1

0

-20

1

1.1

-10

-20

0.95

(d)

y

1.05

10

-15

0.95

1

y

10

-25 0.9

0.95

(b)

y

~ B4i, B4i

~ B4r , B 4r

-20

-60

-20

(c)

39

1

y

Fig. 27. The partial solutions of Eqs. (6.31) and (6.39) for (a) and (b) y1 6 y 6 1 and (c) and (d) 1 6 y 6 y2 . The parameters are the same as in Fig. 25. It is seen that all solutions are stable.

and (6.32), and, for y1 6 y 6 1, ignored the√fourth partial solution with a large negative real part. The former is valid because  |Q0 | ∼ |B21 | Re, and the latter is valid because, for y1 6 y 6 1, exp( B3 (S; x; y) dy)exp( B4 (S; x; y) dy). The equations for B and B˜ following from the Euler equations are 
9B 2 2 i(S − ud (x; y)Q0 ) (6.39) + B − Q0 + iQ0 +dy (x; y) + +d x (x; y)B = 0 ; 9y
i(S − ud (x; y)Q0 )

9B˜ + B˜ 2 − Q02 9y



− iQ0 (2udy (x; y)B˜ + udyy (x; y))

+ iQ0 +dy (x; y) − +d x (x; y)B˜ − +d xy (x; y) = 0 :

(6.40)

An example of the partial solutions found in this way is given in Fig. 27 for S = 22. Comparing Figs. 27 with 26 we see that, as distinct from the case shown in Fig. 26, all the solutions found are stable. To 3nd the eigenfunctions and adjoint eigenfunctions we have to use expressions (6.34), (6.37) and, respectively, for the adjoint functions, (6.35), (6.38) and the sewing conditions for y = 1. Thus we 3nd the equations for A1 , A2 , C1 and C2 (and, correspondingly, in the case of the adjoint eigenfunctions, for A˜ 1 , A˜ 2 , C˜ 1 and C˜ 2 ): A1 A2 (exp(91 (S; x; 1)) − exp(92 (S; x; 1))) + exp(93 (S; x; 1)) 2 2 =C1 exp(:1 (S; x; 1)) + C2 exp(:2 (S; x; 1)) ;

40

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

A1 (l) (B (S; x; 1) exp(91 (S; x; 1)) − B2(l) (S; x; 1) exp(92 (S; x; 1))) 2 1 A2 (l) + B (S; x; 1) exp(93 (S; x; 1)) 2 3 =C1 B1(r) (S; x; 1) exp(:1 (S; x; 1)) + C2 B2 1(r) (S; x; 1) exp(:2 (S; x; 1)) ; A1 ((B1(l) (S; x; 1))2 exp(91 (S; x; 1)) − (B2(l) (S; x; 1))2 exp(92 (S; x; 1))) 2 A2 + (B3(l) (S; x; 1))2 exp(93 (S; x; 1)) 2 =C1 (B1(r) (S; x; 1))2 exp(:1 (S; x; 1)) + C2 (B2(r) (S; x; 1))2 exp(:2 (S; x; 1)) ; A1 ((B1(l) (S; x; 1))3 exp(91 (S; x; 1)) − (B2(l) (S; x; 1))3 exp(92 (S; x; 1))) 2 A2 + (B3(l) (S; x; 1))3 exp(93 (S; x; 1)) 2 =C1 (B1(r) (S; x; 1))3 exp(:1 (S; x; 1)) + C2 (B2(r) (S; x; 1))3 exp(:2 (S; x; 1)) ;

(6.41)

A˜ 1 A˜ 2 (exp(9˜ 1 (S; x; 1)) − exp(9˜ 2 (S; x; 1))) + exp(9˜ 3 (S; x; y)) 2 2 =C˜ 1 exp(:˜ 1 (S; x; 1)) + C˜ 2 exp(:˜ 2 (S; x; 1)) ; A˜ 1 ˜ (l) ˜ (B (S; x; 1) exp(9˜ 1 (S; x; 1)) − B˜ (l) 2 (S; x; 1) exp(92 (S; x; 1))) 2 1 +

A˜ 2 ˜ (l) B (S; x; 1) exp(9˜ 3 (S; x; 1)) 2 3 ˜ ˜ ˜ (r) ˜ =C˜ 1 B˜ (r) 1 (S; x; 1) exp(:1 (S; x; 1)) + C 2 B2 1 (S; x; 1) exp(:2 (S; x; 1)) ;

A˜ 1 ˜ (l) 2 ˜ ((B1 (S; x; 1))2 exp(9˜ 1 (S; x; 1)) − (B˜ (l) 2 (S; x; 1)) exp(92 (S; x; 1))) 2 +

A˜ 2 ˜ (l) (B (S; x; 1))2 exp(9˜ 3 (S; x; 1)) 2 3 2 2 ˜ ˜ ˜ (r) ˜ =C˜ 1 (B˜ (r) 1 (S; x; 1)) exp(:1 (S; x; 1)) + C 2 (B2 (S; x; 1)) exp(:2 (S; x; 1)) ;

A˜ 1 ˜ (l) 3 ˜ ((B1 (S; x; 1))3 exp(9˜ 1 (S; x; 1)) − (B˜ (l) 2 (S; x; 1)) exp(92 (S; x; 1))) 2 +

A˜ 2 ˜ (l) (B (S; x; 1))3 exp(9˜ 3 (S; x; 1)) 2 3 3 3 ˜ ˜ ˜ (r) ˜ =C˜ 1 (B˜ (r) 1 (S; x; 1)) exp(:1 (S; x; 1)) + C 2 (B2 (S; x; 1)) exp(:2 (S; x; 1)) :

(6.42)

1400 1200 1000 800 600 400 200 0 -200 -400 -600

600 400 200 0 -200 -400 -600 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(a)

y

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(b)

y 8

1200

6

f0

1400

arg χ , arg

1000

|χ | , |f| 0

41

800

χi , f 0i

χr , f 0r

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

800 600 400

4 2 0 -2

200 0

-4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

(c)

y

0 0.2 0.4 0.6 0.8

(d)

1 1.2 1.4 1.6 1.8

2 2.2

y

Fig. 28. Plots of the real and imaginary parts and modulus of the eigenfunction f0 , calculated for b0 = 0:1, q = 3, Re = 25; 000, x = 0 and (a) S = 10 (Q0 ≈ 19:952633 + 9:984723i) and (b) S = 18 (Q0 ≈ 37:974133 + 4:244753i).

Because the determinants of the systems of equations (6.41) and (6.42) are equal to zero, these equations allow us to 3nd a = A2 =A1 , c1 = C1 =A1 and c2 = C2 =A1 (and, correspondingly, a˜ = A˜ 2 = A˜ 1 , c˜1 = C˜ 1 = A˜ 1 and c˜2 = C˜ 2 = A˜ 1 ). Examples of the eigenfunctions and adjoint eigenfunctions constructed in this way are illustrated in Figs. 28 and 29 for x = 0, (a) S = 10 and (b) S = 18. We see that the range of the sharp change of the eigenfunctions and adjoint eigenfunctions becomes narrower with increasing S. It is important that the 3rst partial solution of Eq. (6.39) for y1 6 y 6 1, B1(l) (S; x; y), transforms uninterruptedly into the 3rst partial solution of the same equation for 1 6 y 6 y2 , B1(r) (S; x; y) (see Fig. 30). Thus, at a point 9 y = y∗ (S; x) ¿ 1 the real and imaginary parts of B1 (S; x; y) change sign. This change of sign provides an explanation of the formation of vortices within the boundary layer: on diIerent sides of the pivot point the stochastic constituents of the longitudinal velocity are oppositely directed. For S ¡ 3, the form of functions f02 (S; x; y) and 3X02 (S; x; y) means that over the region of the boundary layer (region III) the eigenfunctions f0 (S; x; y) and adjoint eigenfunctions 3X0 (S; x; y) depend strongly on the Reynolds number and diIer markedly from those found from the Euler equations. This is illustrated in Fig. 31, where these functions calculated by the way indicated above and from the Euler equations are compared for S = 1. For S ¿ 8 the eigenfunctions and adjoint eigenfunctions are practically independent of the Reynolds number, and hence may be calculated from the Euler equations. With a knowledge of the eigenfunctions and adjoint eigenfunctions, we can use Eq. (6.10) to calculate the corrections Q1 to the eigenvalues of Q0 . The values of 7 = 70 + 71 =4 and wave 9

This point is called the pivot point.

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

χ0i

χ0r

1e+16 0 -1e+16 -2e+16 -3e+16 -4e+16 -5e+16 -6e+16 -7e+16 -8e+16 -9e+16

1 1.2 1.4 1.6 y

3e+08 2.5e+08 2e+08 1.5e+08 1e+08 5e+07 0 -5e+07 -1e+08 -1.5e+08 -2e+08 -2.5e+08 0.4 0.6 0.8 1 1.2 1.4 1.6 y

0.4

0.8

1 1.2

1.4

1.6

8e+16 6e+16 4e+16 2e+16 0 -2e+16 -4e+16 -6e+16 0.4 0.6 0.8

7e+08

1.2e+17

6e+08

1e+17

5e+08

1 1.2 1.4 1.6 y

8e+16

4e+08

|χ 0 |

|χ 0|

0.6

y

χ 0i

1e+08 0 -1e+08 -2e+08 -3e+08 -4e+08 -5e+08 -6e+08 -7e+08 0.4 0.6 0.8

χ0r

42

3e+08

6e+16 4e+16

2e+08

2e+16

1e+08 0 0.4 0.6 0.8

(a)

1 1.2 1.4 1.6 y (b)

0 0.4 0.6 0.8

1 y

1.2 1.4 1.6

Fig. 29. Plots of the real and imaginary parts and modulus of the adjoint eigenfunction 3X0 , calculated for b0 = 0:1, q = 3, Re = 25; 000, x = 0 and (a) S = 10 (Q0 ≈ 19:952633 + 9:984723i) and (b) S = 18 (Q0 ≈ 37:974133 + 4:244753i).

30 20

B1i

B1r

10 0 -10 -20 -30 0.85

(a)

0.9

0.95

1

y

1.05

1.1

1.15

40 35 30 25 20 15 10 5 0 -5 -10 0.85

(b)

0.9

0.95

1

1.05

1.1

1.15

y

Fig. 30. The 3rst partial solution of Eq. (6.39) B1 (S; x; y) versus y over the range y1 6 y 6 y2 for x = 0, S = 13.

phase velocity vph = S=(K0 + K1 =4) as functions of St are shown for diIerent x in Figs. 32a and b. Comparing Figs. 32a and 24a we see that for x ¡ 1 the corrections to the eigenvalues of 70 are not small. This is caused by a rather large value of the derivative of 70 with respect to x for small x.

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

f 0i

f

0r

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0.8

1

1.2

y

(a)

1

1.2

y

(b)

1.3

1.8

1.2

1.7 1.6

1.1

1.5

χ 0i

1

χ0r

1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.8

43

0.9

1.4 1.3 1.2

0.8

1.1

0.7

1

0.6 0.8

1

y

(c)

0.9 0.8

1.2

1

1.2

y

(d)

Fig. 31. The eigenfunctions and adjoint eigenfunctions versus y over the range y1 6 y 6 y2 for x = 0, S = 1 (bold lines), and the same quantities calculated from the Euler equations (thin lines).

8

1.3 1.2

7

5

1

0.5

v ph

Γ

1.1

0

6 1

4 2

3

3

2

10

0.8 0.7

5

0.6

1

10 5 3 2

0.5

0

0 0.5

1

0.4 0

(a)

0.9

1

2

3

4

5

St

6

7

8

9

0

(b)

1

2

3

4

5

6

St

Fig. 32. The dependences on the Strouhal number St for b0 = 0:1, q = 3, r0 = 0:5, Re = 25; 000 and diIerent x of: (a) 7 = 70 + 71 =4 and (b) the wave phase velocity vph = S=(K0 + K1 =4). The value of x is indicated near the corresponding curve in each case.

The derivatives of 70 and vph0 for diIerent values of x are shown in Fig. 33. As x increases the derivatives of 70 and vph0 , and corrections to the eigenvalues of Q0 become progressively smaller. It can be seen that, as the distance from the nozzle increases, the gain factor decreases for large St and increases slightly for small St. For any given x the gain factor has a maximum at St = Stm , where the greater x is the smaller Stm becomes. It is easily shown that the shift of the gain factor maximum to the low-frequency region is caused by the jet’s divergence. Obviously, this shift of the gain factor maximum results in a shift of the turbulent pulsation power spectrum towards the low-frequency region as the distance from the nozzle increases (see below). It is interesting that, from x ≈ 1, the dependence of Stm on x is of an exponential character.

44

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.15 0

0.1 0.05

-4

v 0x

Γ0x

-2

-6

0 -0.05 -0.1

-8

-0.15

-10

-0.2 0

(a)

1

2

3

4

5

St

6

7

0

(b)

1

2

3

4

5

6

7

St

Fig. 33. The dependences of (a) 970 =9x ≡ 70x and (b) 9vph0 =9x ≡ v0x on the Strouhal number St for x = 0, 0.5, 1, 2, 3, 5 and 8.

Another important conclusion that can be drawn from Fig. 32 lies in the fact that the phase velocity of the hydrodynamic waves depends strongly on the Strouhal number, i.e. these waves are rather signi3cantly dispersive. Note also that the resonant character of the dependences of the gain factor on the Strouhal number that we have found indicates that each jet cross-section can be considered as an oscillator whose natural frequency decreases with increasing distance from the nozzle. This fact justi3es consideration of a jet as a chain of coupled resonant ampli3ers, which in turn allows us to understand the analogy between noise-induced pendulum oscillations and the turbulent processes in a jet. Neglecting the correction to the eigenfunction f0 (S; x; y), we can write the generative solutions (t; x; y), u0 (t; x; y) and +0 (t; x; y) as 0
  ∞  x 1 f0 (S; x; y) exp iSt − i Q(S; x) d x dS ; 0 (t; x; y) ≈ 2 −∞ 0 
 ∞  x 9f0 (S; x; y) 1 u0 (t; x; y) ≈ Q(S; x) d x dS ; exp iSt − i 2 −∞ 9y 0 
 ∞ 2 9 f0 (S; x; y) 1 9Q0 (S; x) 2 f0 (S; x; y) +0 (t; x; y) ≈ − Q (S; x) + i 2 −∞ 9y2 9x 
  x 9f0 (S; x; y) exp iSt − i + iQ(S; x) Q(S; x) d x dS : (6.43) 9x 0 It follows from (6.43) that the vorticity is moderately small outside the boundary layer. Knowing Q(S; x) and the expressions for u0 (t; x; y) and f0 (S; x; y) we can calculate the evolution of the velocity power spectra in the linear approximation. For the sake of simplicity, we do so only for region I. We can expand a random disturbance (t; y) of the longitudinal component of velocity at the nozzle exit into a series in the eigenfunctions of our boundary value problem. Over region I we can approximate cosh(Q(S; 0)y) with eigenvalues of Q as the eigenfunctions for x = 0. Hence, we can set (t; y) = 1 (t) cosh(Q(S; 0)y) + · · · :

(6.44)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

45

1

0.9

κ /κ 0

0.8

0.7

0.6

0.5

0.4

0.3 0

5

10

15

20

25

S

Fig. 34. Plot of %(S; 0; 0)=%0 described by (6.47).

The spectral density A1 (S)AX 1 (S) = A2 (S), where A(S) = |A1 |(S), is determined by the spectral density of 1 (t) which is denoted by us as %(S; 0). Because over region I u(S; 0; y) ≈ A1 (S)Q(S; x) cosh(Q(S; 0)y) ;

(6.45)

we 3nd A2 (S) =

%(S; 0) : K 2 (S; 0) + 72 (S; 0)

(6.46)

There is almost no experimental information about %(S; 0), but there is one work [133] giving power spectra of the longitudinal and transverse constituents of velocity pulsations over the range of the Strouhal numbers St 0–8 for a circular jet at diIerent initial turbulence levels. It can be seen from these data that the form of the spectra depends only slightly on the initial turbulence level, that the spectra of the longitudinal and transverse constituents of velocity pulsations are nearly identical, and that the spectral density decreases with increasing St. Since the dependence of the spectral densities presented in [133] on y is close to f(S)|cosh(Q(S; 0)y)|2 , where f(S) is a certain function of S, we can set %(S; 0) ≈ f(S). So, in accordance with data presented in [133], we approximate %(S; 0) by the formula: %0 %(S; 0) = ; (6.47) 1 + b1 S + b2 S2 + b3 S3 where %0 characterizes the level of the disturbances at the nozzle exit, b1 = 0:152, b2 = −0:005 and b3 = 0:000002. The plot of %(S; 0)=%0 described by (6.47) is shown in Fig. 34. Comparison with experimental results for power spectra of velocity pulsations and for the mean longitudinal velocity shows that %0 should be taken as very small. Hereinafter we will set %0 = 8 × 10−28 . It should be noted that, owing to the resonant character of the gain factor, the results are scarcely aIected by the shape of the dependence of %(S; 0).

46

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 36 32

log ( κ / κ 0)

log ( κ/ κ0 )

28 24 20 16 12 8 4 0 0

1

2

3

4

(a)

5

6

7

40 36 32 28 24 20 16 12 8 4 0

8

0

1

2

3

(b)

St

4

5

6

7

8

9

St

Fig. 35. Evolution of %(S; x; y)=%0 in the linear approximation for (a) y = 0 and (b) |y| = 0:7.

St m

St m

10

1

1 1

(a)

10

x

1

(b)

10

x

Fig. 36. The dependences of the Strouhal number Stm on the distance from the nozzle for (a) y = 0 and (b) |y| = 0:7 (the linear (zeroth) approximation). The dependences Stm ≈ 3:2x−0:36 (for y = 0) and Stm ≈ 6:5x−0:68 (for y = 0:7) are shown by solid lines.

It follows from (6.43), (6.45) and (6.46) that the spectral density of the longitudinal velocity pulsations in the linear approximation is 
 x K 2 (S; x) + 72 (S; x) 2   %l (S; x; y) = %(S; 0) 2 : (6.48) 27(S; x ) d x |cosh(Q(S; x)y)| exp K (S; 0) + 72 (S; 0) 0 The evolutions of %l (S; x; y) for y = 0 and |y| = 0:7 are shown in Figs. 35a and b. We see that in the two cases considered the power spectra diIer markedly, especially for small x. In particular, the diIerence shows up as a faster (for |y| = 0:7) decrease of the Strouhal number corresponding to the spectrum maximum (Stm ) with increasing distance from the nozzle. This is more easily seen in Fig. 36, where this decrease is given for both cases considered. As is evident from the 3gures, the experimental dependences can be well-approximated by the curves Stm ≈ 3:2x−0:36 (for y = 0) and Stm ≈ 6:5x−0:68 (for y = 0:7). Unfortunately, experimental data (see Figs. 5 and 6) are available only for the jet axis (corresponding to y = 0) and a line oIset by the radius from the axis (corresponding to y = 1). In the 3rst case our dependence is close to the experimental one, and in the second case it lies between the experimental ones. We emphasize that our results are obtained from the linear theory without taking account of nonlinear phenomena such as the pairing of vortices. These results reinforce our idea that the experimentally observed shift of the power spectrum is explained mainly by the divergence of the jet, not by the pairing of vortices.

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.7

0.6

0.6

0.5 0.4

0.4

εu

εu

0.5

0.3

0.3 0.2

0.2

0.1

0.1 0

0 0

(a)

47

1

2

3

4

x

5

6

7

8

0

(b)

1

2

3

4

5

x

Fig. 37. Plots of the mean-root-square values of turbulent velocity pulsations versus x for (a) y = 0 (ju (x; 0)) and (b) y = 0:7 (ju (x; 0:7)).

It should be noted that, as the distance from the nozzle increases, the width of the power spectra decreases signi3cantly. This means that the correlation time increases, i.e. the coherence level increases too. This is a cause of the formation of coherent structures. The mean-root-square value of the turbulent velocity pulsations, which is what is usually is measured experimentally, is equal to   1 ∞ jul (x; y) = %l (S; x; y) dS : (6.49) 0 The plots of jul (x; 0) and jul (x; 0:7) versus y are presented in Fig. 37. It is seen from Fig. 37a that the dependence of jul (x; 0) on x closely resembles the dependence of an order parameter on temperature for a slightly noisy second-order phase transition. This also reinforces our hypothesis that the onset of turbulence is a nonequilibrium noise-induced phase transition of the second order, similar to that for a pendulum with a randomly vibrated suspension axis. It is interesting that for y = 0:7 the root-mean-square value of turbulent velocity pulsations 3rst decreases and then increases with increasing x. This can be explained by the competition between the ampli3cation of the pulsations and the swift decrease of the spectrum width. The condition for validity of all results obtained in this paper is that jul (x; y)1. We see that along the jet axis the results are valid for almost the whole initial part (x ¡ 8), whereas for y = 0:7 they are valid only for x ¡ 5. The change of j(x; y) as x increases is correlated with the change of the mean velocity (see below), but these changes are not fully identical. Let us trace the changes of the group wave velocity at St = Stm , as well as of the wave lengths in the longitudinal and transverse directions (4lon (Stm ; x) = 2 =K(Stm ; x) and 4tr (Stm ; x) = 2 =7(Stm ; x), respectively), with increasing x. The results are presented in Fig. 38. It is seen that along the jet axis, as the distance from the nozzle increases, the group velocity vgr (Stm (x)) 3rst decreases and then increases, whereas for y =0:7 it decreases monotonically. The longitudinal wavelengths increase considerably in both cases, providing evidence for an increase in the scale of the turbulence in the longitudinal direction. Along the jet axis the transverse scale of turbulence increases too, whereas for y = 0:7 it changes nonmonotonically. The increase of the scale of turbulence in the longitudinal direction agrees with the experimental data and reveals itself in the pairing of vortices.

48

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 0.51 0.5

λ lon , λ tr

v gr

0.49 0.48 0.47 0.46 0.45 0.44 0

1

2

3

4

5

6

7

8

x

(a)

2

1

0

1

2

3

4

5

6

7

8

x

(b)

1.2

4.5 4

1.1

2

3.5

λlon , λ tr

1

v gr

2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

0.9 0.8 0.7

3 2.5 2 1.5 1

0.6

1

0.5

0.5

0 0

1

2

3

4

5

x

(c)

0

(d)

1

2

3

4

5

x

Fig. 38. The changes of (a, c) the group wave velocity at St = Stm and (b, d) the wave lengths in longitudinal (1) and transverse (2) directions (4lon (Stm ; x) = 2 =K(Stm ; x) and 4tr (Stm ; x) = 2 =7(Stm ; x), respectively), with increasing x: (a, b) for y = 0; (c, d) for y = 0:7.

6.2. The :rst approximation Putting 9A1 9A1 ∼ ∼ j2 ; 9x 9y substituting (6.1) into Eqs. (5.36), (5.37) and equating the terms of order j we obtain the following equations: 9s1 ; r1 − Ws1 = 0; q1 = 9y 9r1 9r1 9r1 9s1 + ud (x; y) + vd (x; y) − +dy (x; y) 9t 9x 9y 9x + +d x (x; y)

2 9s1 − Wr1 = R(t; x; y) ; 9y Re

(6.50)

where R(t; x; y) = and

0

9 0 9+0 9+0 9 0 − ; 9x 9y 9x 9y

and +0 are de3ned by expressions (6.43).

(6.51)

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

49

We suppose that only waves for which the sign of S is opposite can interact, an assumption that is justi3ed for dispersive waves. It should be noted that the interaction of waves with the same sign of S results in the generation of second harmonic, whereas the interaction of waves for which S is opposite in sign results in the appearance of a constant level. Because the diIerence 9

( S) 0

9x

9+0(S) 9+0(S) 9 0(S) − 9y 9x 9y

is small, the second harmonic is also small. It is therefore suPcient to consider the interaction only of waves for which S is opposite in sign. In so doing, it should be borne in mind that A1 (−S)= AX 1 (S) X and Q(−S) = −Q(S). Thus, we can set A1 (S)A1 (S ) = 2 A2 (S)(S + S), where A(S) = |A1 |(S). In this case R(t; x; y) is independent of t, and we can represent it as  ∞ 1 R(x; y) = Rs (S; x; y) dS : (6.52) 2 −∞ A solution of Eqs. (6.50) can be also presented in the form  ∞  ∞ 1 1 r1s (S; x; y) dS; s1 (x; y) = s1s (S; x; y) dS ; r1 (x; y) = 2 −∞ 2 −∞  ∞ 1 q1 (x; y) = q1s (S; x; y) dS ; 2 −∞

(6.53) (6.54)

Eqs. (6.50) can be solved analytically only over regions I and II. It follows from (6.43) and our numerical calculations that over region I
  x Q(S; x) d x ; 0 (S; x; y) ≈ A1 sinh(Q(S; x)y) exp iSt − i 0

+0 (S; x; y) ≈ −iA1 [sinh(Q(S; x)y) + 2Q(S; x)y cosh(Q(S; x)y)]
  x ×exp iSt − i Q(S; x) d x ;

9Q(S; x) 9x (6.55)

0

and in region II 0 (S; x; y) ≈ A1 [c1 (S; x) exp(B21 (S; x)(y

− y2 (x))) + c2 (S; x)
  x ×exp(B22 (S; x)(y − y2 (x)))] exp iSt − i Q(S; x) d x ; 

(6.56)

0

Q(S; x)+d x (x; ∞) c1 (S; x) exp(B21 (S; x)(y − y2 (x))) iS − Q(S; x)vd x (x; ∞) 
  x 2 + B22 (S; x)c2 (S; x) exp(B22 (S; x)(y − y2 (x))) exp iSt − i Q(S; x) d x ;

+0 (S; x; y) ≈ A1

0

(6.57)

50

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

where c1 (S; x) and c2 (S; x) are found from Eqs. (6.41), and B21 (S; x) and B22 (S; x) are determined by (6.21) and (6.22). For y − y2 (x)1=|B22 (S; x)| expressions (6.56) may be reduced to
  x Q(S; x) d x ; 0 (S; x; y) ≈ A1 c1 (S; x) exp(B21 (S; x)(y − y2 (x))) exp iSt − i 0

Q(S; x)+d x (x; ∞) c1 (S; x) exp(B21 (S; x)(y − y2 (x))) iS − Q(S; x)vd x (x; ∞) 
 x Q(S; x) d x : ×exp iSt − i

+0 (S; x; y) ≈ A1

0

(6.58)

6.2.1. Region I It follows from (6.55) that

 97(S; x) 9K(S; x) sinh(2K(S; x)y) + Rs (S; x; y) = −2(K (S; x) + 7 (S; x)) 2 9x 9x  
97(S; x) 9K(S; x) − 7(S; x) y cosh(2K(S; x)y) ×sin(27(S; x)y) + 2 K(S; x) 9x 9x
 x  2 7(S; x) d x : (6.59) ×A (S) exp 2 2

2

0

Eqs. (6.50) for r1s , q1s and s1s become r1s − Ws1s = 0;

q1s =

9s1s ; 9y

9r1s 2 − Wr1s = Rs (S; x; y) : 9x Re

(6.60)

Substituting (6.59) into Eqs. (6.60), and ignoring the term proportional to 1=Re, we obtain the following approximate expressions for r1s (S; x; y), s1s (S; x; y) and q1s (S; x; y):
 x  2 7(S; x) d x : r1s (S; x; y) = r˜1s (S; x; y)A (S) exp 2 0


 x  2 s1s (S; x; y) = s˜1s (S; x; y)A (S) exp 2 7(S; x) d x ;

(6.61)


 x  7(S; x) d x ; q1s (S; x; y) = q˜1s (S; x; y)A2 (S) exp 2

(6.62)

0

0

where K 2 (S; x) + 72 (S; x) r˜1s (S; x; y) ≈ − 7(S; x)



9K(S; x) 97(S; x) sinh(2K(S; x)y) + 9x 9x 


97(S; x) 9K(S; x) − 7(S; x) y cosh(2K(S; x)y) ; ×sin(27(S; x)y) + 2 K(S; x) 9x 9x 2

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

51


 7(S; x) 9K(S; x) 1 97(S; x) 7 + K(S; x) 27(S; x) K 2 (S; x) + 72 (S; x) 9x 9x 
9K(S; x) 97(S; x) y cosh(2K(S; x)y) ×sinh(2K(S; x)y) + K(S; x) − 7(S; x) 9x 9x  K 2 (S; x) + 72 (S; x) 97(S; x) − y cos(27(S; x)y) ; (6.63) 27(S; x) 9x 
1 1 9K(S; x) (372 (S; x) + K 2 (S; x))K(S; x) q˜1s (S; x; y) ≈ − 2 2 27(S; x) K (S; x) + 7 (S; x) 9x  97(S; x) cosh(2K(S; x)y) + (K 2 (S; x) − 72 (S; x))7(S; x) 9x
 9K(S; x) 97(S; x) + 2K(S; x)y K(S; x) − 7(S; x) sinh(2K(S; x)y) 9x 9x s˜1s (S; x; y) ≈ −

−

K 2 (S; x) + 72 (S; x) 97(S; x) [cos(27(S; x)y) 27(S; x) 9x 

− 27(S; x)y sin(27(S; x)y)]

:

(6.64)

It follows from (6.64) that along the jet axis, where sinh(2K(S; x)y) = sin(27(S; x)y) = 0 and cosh(2K(S; x)y) = cos(27(S; x)y) = 1, we have r1s (S; x; 0) = s1s (S; x; 0) = 0 and
1 1 9K(S; x) (372 (S; x) + K 2 (S; x))K(S; x) q˜1s (S; x; y) ≈ − 2 2 27(S; x) K (S; x) + 7 (S; x) 9x 

K 2 (S; x) + 72 (S; x) 97(S; x) 97(S; x) 2 2 − : + (K (S; x) − 7 (S; x))7(S; x) 9x 27(S; x) 9x (6.65) It should be noted that the functions r˜1s , s˜1s and q˜1s are slow functions of x. We emphasize that the function q1 (x; y) determines the dependence on x and y of the additional constant correction to the dynamical constituent of the longitudinal velocity, i.e. the change of the mean velocity due to nonlinear eIects. This change is caused by turbulent pulsations. It is intuitively obvious (and con3rmed by experiment) that the correction found over region I must be negative, i.e. turbulent pulsations must decrease the mean ?ow velocity in region I. Averaging (6.62), substituting (6.46) into q1s (S; x; y) and integrating over S we can 3nd q1 (x; y) for diIerent values of x and y 6 y1 (x). Fig. 39 shows some examples of the dependences of u(x; y) = ud (x; y) + q1 (x; y) on y at 3xed values of x, and on x at 3xed values of y. It is seen from Fig. 39a that the mean velocity pro3le becomes increasingly bell-shaped as x increases. Furthermore (see Fig. 39b), within the initial part (x 6 8) the velocity on the jet axis decreases

52

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 1 0.95

0.85

3

2





0.9 1

0.8 0.75 0.7 0

0.1

0.2

0.3

0.4

0.5

y

(a)

0.6

1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5

0.7

4

4.5

5

5.5

6

6.5

7

7.5

8

x

(b)

Fig. 39. The dependences of the mean velocity u(x; y) = ud (x; y) + q1 (x; y) taking account of the correction caused by the turbulent pulsations (q1 (x; y)) (a) on y at x = 6 (1), x = 7 (2) and x = 8 (3) and (b) on x at (from right to left) y = 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7.

signi3cantly only at its end, and then by no more than 5%. OI-axis, the velocity falls oI much faster. 6.2.2. Region II It follows from (6.58) and (6.21) that 4+d x (x; ∞)|c1 (S; x)|2 K(S; x)S (K 2 (S; x) + 72 (S; x))2 vd2 (x; ∞) + S(S − 27(S; x)vd (x; ∞)) 
+d x (x; ∞)[(K 2 (S; x) − 72 (S; x))vd (x; ∞) + 7(S; x)S] 2 2 × K (S; x) + 7 (S; x) − 2[K 2 (S; x)vd2 (x; ∞) + (S − 7(S; x)vd (x; ∞))2 ]  

 x ×exp 2 B21r (S; x)(y − y2 (x)) + 7(S; x) d x A2 (S) ; (6.66)

Rs (S; x; y) = −

0

where




+d x (x; ∞)vd (x; ∞) B21r (S; x) = −K(S; x) 1 − 2 2 2[K (S; x)vd (x; ∞) + (S − 7(S; x)vd (x; ∞))2 ]

 :

Eqs. (6.50) for r1s and s1s become r1s − Ws1s = 0 ; vd (x; ∞)

2 9r1s 9s1s + +dx (x; ∞) − Wr1s = Rs (x; y) ; 9y 9y Re

(6.67)

Ignoring the term proportional to 1=Re, we obtain the following approximate expressions for r1s (S; x; y), s1s (S; x; y) and q1s (S; x; y): r1s (S; x; y) ≈

2 (S; x) + 72 (S; x)) 2Rs (S; x; y)(B21r ; 2 2 B21r (S; x)[4(B21r (S; x) + 7 (S; x))vd (x; ∞) + +dx (x; ∞)]

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 1

2 1

0.6

yb

ud

0.8

0.4 0.2 0

-2 -1.5

-1 -0.5

0

0.5

1

1.5

2

y

(a)

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

53

2

1

0

1

2

3

4

5

6

7

8

9

x

(b)

Fig. 40. (a) An example of the velocity pro3le taking account of the stochastic constituents (curve 1) and the corresponding pro3le of the dynamical constituent of the velocity (curve 2) for x = 8 (the end of initial part); and (b) the internal (1) and external (2) boundaries of the mixing layer taking account of the stochastic constituents.

s1s (S; x; y) ≈ q1s (S; x; y) ≈

2 2B21r (S; x)[4(B21r (S; x) 2 4(B21r (S; x)

+

Rs (S; x; y) ; + 72 (S; x))vd (x; ∞) + +dx (x; ∞)]

Rs (S; x; y) 2 7 (S; x))vd (x; ∞)

+ +dx (x; ∞)

:

(6.68) (6.69)

Averaging (6.69), substituting (6.46) into q1s (S; x; y) and integrating over S we can 3nd q1 (x; y) for diIerent values of x and y ¿ y2 (x). Extrapolating the values of q1 (x; y) found here and for region I into the region of boundary layer we can estimate the mean longitudinal velocity pro3les width for diIerent values of x, taking account of stochastic constituents. An example of such pro3le for x = 8 (the end of the initial part) is given in Fig. 40a. For comparison, the corresponding pro3le of the dynamical constituent of the velocity is shown in the same 3gure. We see that these pro3les diIer substantially. It should be emphasized that the velocity pro3le that we have found by taking account of the stochastic constituents coincides in form with experimentally measured pro3les. Unfortunately, we cannot calculate exactly the full velocity pro3les, or the change of velocity, for all y because we are restricted to regions I and II and take no account of strong nonlinear eIects. However, we can calculate the width of the internal and external parts of the boundary layer. If we take for the internal boundary the plane where the mean velocity is equal to 0:95U0 , and for the external boundary the plane where the mean velocity is equal to 0:05U0 , then these boundaries are as shown in Fig. 40b. Our results demonstrate that the boundaries of the mixing layer are very far from being the straight lines adduced by many researchers. Up to a certain value of x, these boundaries nearly coincide with the boundaries of the mixing layer for the dynamical constituents. Only for larger x do they strongly move apart. 6.3. The second approximation To derive the equation for the amplitude A1 in the second approximation, we set in expansion (6.1)
  ∞  x 1 r2 (t; x; y) = C(S) sinh(Q(S; x)y) exp iSt − i Q(S; x) d x dS ; 2 −∞ 0

54

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

1 s2 (t; x; y) = 2



∞

−∞






D(S) sinh(Q(S; x)y) exp iSt − i

0

x

 Q(S; x) d x dS ;

(6.70)

where C(S) ∼ j2 and D(S) ∼ j2 are unknown amplitudes. Like the 3rst approximation, the second approximation can be found analytically only over regions I and II. For simplicity, we restrict our consideration to region I. Equating the terms of order j2 in Eqs. (5.36) and (5.37) for region I, neglecting the derivatives of Q(S; x) with respect to x and taking account of (6.70), we obtain the following system of approximate equations for C and D with determinant equal to zero: 
9A1 9A1 − coth(Q(S; x)y) ; C = −2Q(S; x) i 9x 9y
 9r1 (x; y) 9r1 (x; y) i(S − Q(S; x))C = −Q(S; x) coth(Q(S; x)y) + i A1 ; (6.71) 9x 9y where r1 (x; y) is de3ned by (6.53) and (6.61). From the condition of compatibility of Eqs. (6.71), we 3nd a truncated equation for the amplitude A1 over region I. It can be written as tanh(Q(S; x)y) where :(S; x; y) =

1 2

9A1 9A1 +i = :(S; x; y)A1 ; 9x 9y 

∞

−∞

(6.72)


 F(S; s; x; y)A20 (s) exp 2

x

0

 7(s; x ) d x ds ;

  1 9r˜1s (s; x; y) 27(s; x)r˜1s (s; x; y)) + i tanh(Q(S; x)y) F(S; s; x; y) = 2(Q(S; x) − S) 9y and A0 (S) = |A1 (S; 0; 0)|. Eq. (6.72) can be conveniently rewritten in terms of 
A1 (S; x; y) z(S; x; y) = ln A1 (S; 0; 0)

(6.73) (6.74)

(6.75)

as tanh(Q(S; x)y)

9z 9z +i = :(S; x; y) : 9x 9y

A solution of Eq. (6.76) can be represented as 
 x  ∞ 1   z(S; x; y) = w(S; s; x; y) exp 2 7(s; x ) d x ds : 2 −∞ 0

(6.76)

(6.77)

Substituting (6.77) into Eq. (6.76), taking into account that F(S; s; x; y) and w(S; s; x; y) are slow functions of x, and neglecting the derivative of w with respect to x we obtain the following equation

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62

for w(S; s; x; y): 9w − 2i7(s; x) tanh(Q(S; x)y)w = −iF(S; s; x; y)A20 (s) : 9y

55

(6.78)

The solution of Eq. (6.78) with initial condition w(S; s; x; 0) = 0 is 
 y cosh(Q(S; x)y) −2i7(s;x)=Q(S;x) w(S; s; x; y) = −iA20 (s) F(S; s; x; y ) dy : ) cosh(Q(S; x)y 0

(6.79)

After substituting (6.79) into (6.77) and integrating over s, we 3nd z(S; x; y). Knowledge of z(S; x; y), in view of (6.75), allows us to 3nd A2 (S; x; y) and the mean value of the phase shift. Taking account of (6.75) and using the assumption that |z(S; x; y)|1 we 3nd A2 (S; x; y) ≈ A20 (S)(1 + 2 Re[z(S; x; y)]) ; ’(S; x; y) − ’(S; 0; 0) ≈ Im[z(S; x; y)] ; where i z(S; x; y) = − 2  ×

0

y



∞

−∞

(6.80)


 x  %(s; 0)   exp 2 7(s; x ) d x K 2 (s; 0) + 72 (s; 0) 0 

F(S; s; x; y )




cosh(Q(S; x)y) cosh(Q(S; x)y )

−2i7(s;x)=Q(S;x)

dy ds ;

%(S; 0) is de3ned by (6.47). The spectral constituent of the longitudinal velocity pulsations is 
9A1 (S; x; y) sinh(Q(S; x)y) u(S; x; y) = A1 (S; x; y)Q(S; x) cosh(Q(S; x)y) + 9y 
 x Q(S; x) d x : ×exp iSt − i 0

(6.81)

(6.82)

Because of the smallness of z(S; x; y) we have A1 (S; x; y) ≈ A0 (S)(1+z(S; x; y)). It follows from here that the spectral density of the longitudinal velocity pulsations, with account taken of nonlinearity, can be represented as 
2 tanh(Q(S; x)y) 9z(S; x; y) ; (6.83) %(S; x; y) = %l (S; x; y) 1 + 2z(S; x; y) + Q(S; x) 9y where %l (S; x; y) is determined by (6.48). It follows from (6.81) that for y = 0, in the approximation under consideration, the relative nonlinear correction 2 tanh(Q(S; x)y) 9z(S; x; y) W%(S; x; y) = 2z(S; x; y) + Q(S; x) 9y to the spectral density %l (S; x; y) is absent. For y = 0, however, this correction is essential, and it increases with increasing y and x. Examples of the dependences of W% on the Strouhal number St for y=0:7 and a number values of x and for x=8 and three values of y are given in Figs. 41a and b, respectively. It is seen that for x 6 5 and y = 0:7 the nonlinear correction changes nonmonotonically

56

P.S. Landa, P.V.E. McClintock / Physics Reports 397 (2004) 1 – 62 3

3

2

2

0.5

0

2

-1

0

1

2

3

(a)

4

5

1

-1

6 7 8

-2 -3

1

4

∆κ

∆κ

1

2 3

-2 6

7

8

-3

9

1

2

3

(b)

St

4

5

6

7

8

9

St

Fig. 41. Examples of the dependences of W% on the Strouhal number St for (a) y = 0:7 and a number values of x, and (b) for x = 8 and three values of y = 0:01 (curve 1), 0.5 (2) and 0.7 (3). Because W% changes strongly as the values of x and y vary, we have plotted not W% but (W%)1=15 .

8e+29

3 7e+29

6e+29

1 5e+29

κ

4 4e+29

3e+29

2 2e+29

1e+29

0 1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

St

Fig. 42. Examples of the velocity pulsations spectral density with taking into account of the nonlinearity: x = 6, (curve 1) y = 0:5 and (curve 2) y = 0:7. For comparison in the same 3gure are given the corresponding dependences found in the linear approximation (curves 3 and 4, respectively).

with increasing x, even changing its sign. Only for x ¿ 5 the changes become monotone, and nearly for all values of St the correction is negative. The latter means that for these St the nonlinearity causes the saturation of turbulent pulsations. We note that the saturation occurs only from a certain value of the Strouhal number. For smaller Strouhal numbers the nonlinear ampli3cation occurs in place of the saturation. Two examples of the velocity pulsations spectral density (for x = 6, and y = 0:5 and 0.7) with taking into account of the nonlinearity are illustrated in Fig. 42. For comparison the corresponding spectral densities calculated in the linear approximation are shown in the same 3gure. It is seen that

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57

the nonlinearity results in the signi3cant decrease of the spectral density maximal value, and as a consequence in the decrease of the turbulent pulsations variance. It is seen from Fig. 42 that, for x = 6, the spectral density for y = 0:5 is more than for y = 0:7, even in linear approximation; whereas for small x it increases monotonically with increasing y. We note that 9’(S; x; y)=9x ≡ WK(S; x; y) gives a nonlinear correction to the wave number K(S; x). It is important to note that this correction depends on the transverse coordinate y. It follows from (6.80) and (6.81) that it is
 x   ∞ %(s; 0)   WK(S; x; y) ≈ − 7(s; x) exp 2 7(s; x ) d x 2 2 0 −∞ K (s; 0) + 7 (s; 0)   −2i7(s;x)=Q(S;x)
y cosh(Q(S; x)y) (6.84) F(S; s; x; y ) dy ds : ×Re cosh(Q(S; x)y ) 0 The value of WK(S; x; y) determines a nonlinear correction W4lon to the longitudinal wave length 4lon (S; x) = 2 =K(S; x): 2 WK(S; x; y) : (6.85) W4lon = − K 2 (S; x) Because nearly for all St the values of WK are negative, we can conclude that the nonlinearity causes the faster increase of turbulence scales with increasing x, in comparison with the results of linear consideration. 7. Conclusions The theoretical approach proposed above has enabled us to account for many experimental results, and to demonstrate that a number of widely-accepted interpretations are in fact erroneous. It has led us to a somewhat diIerent and, we believe, more realistic perspective. In particular: (1) Our studies show that the shift of velocity pulsation power spectra to the low-frequency domain is caused mainly by the jet divergence, not pairing of vortices, so that it can therefore be calculated within the linear approximation. (2) The observed phenomenon of vortices pairing can be accounted for in terms of the increase in the longitudinal and transverse turbulent scales, which is caused by jet divergence and not by resonance relations. (3) The transformation of the mean velocity pro3le can be found without the use of the concept of turbulent viscosity. (4) The in?uence of nonlinearity close to the jet symmetry plane (y = 0) is very small within the initial part of the jet, but increases signi3cantly as we approach the boundary layer. (5) The intensity of random disturbances at the nozzle exit necessary for the onset of turbulence may be very small. Our quasi-linear theory is valid only for such small intensities. For larger disturbance intensities, the development of turbulence is from the very outset an essentially nonlinear process. (6) The change of the velocity pulsation variance with distance from the nozzle closely resembles changing order parameter with increasing temperature for a second-order phase transition.

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That is why we guess that the onset of turbulence can be considered as a speci3c noise-induced phase transition similar to that for a pendulum with a randomly vibrated suspension axis. Acknowledgements It is a pleasure to acknowledge useful discussions with G. Kolmakov and helpful advices related to computation of A. Sil’chenko and I. Kaufman. The research was supported in part by the Engineering and Physical Sciences Research Council (UK), and by the Royal Society of London to whom PSL is indebted for a visiting research fellowship at Lancaster during which much of the review was completed. References U [1] G. Hagen, Uber die Bewegung des Wassers in engen zylidrischen RUohren, Pogg. Ann. 46 (1839) 423–442. [2] O. Reynolds, An experimental investigation of the circumstances which determine whether the motion of water shall be direct os sinouos, and the law of resistance in parallel channels, Philos. Trans. Roy. Soc., London 174 (1883) 935–982. Y Y [3] G. Compte-Bellot, Ecoulement Turbulent Entre deux Parois ParallYeles (EditeY e par le service de documentation scienti3que et technique de l’armement, Paris, 1965). [4] A.E. Ginevsky, Ye.V. Vlasov, R.K. Karavosov, Acoustic Control of Turbulence Jets, Fizmatlit, Moscow, 2001, to be published (in Russian) (English Translation: Springer, Heidelberg, 2004). [5] C.C. Lin, The Theory of Hydrodynamic Stability, Cambridge University Press, Cambridge, 1955. [6] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford, 1961. [7] R. Betchov, W.O. Criminale, Stability of Parallel Flows, Academic Press, New York, 1967. [8] M.A. Goldshtik, V.N. Sctern, Hydrodynamic Stability and Turbulence, Nauka, Novosibirsk, 1977 (in Russian). [9] P.D. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981. [10] Yu.S. Kachanov, V.V. Kozlov, V.Ya. Levchenko, The Onset of Turbulence in a Boundary Layer, Novosibirsk, Nauka, 1982 (in Russian). [11] L.D. Landau, E.M. Lifshitz, Hydrodynamics, Nauka, Moscow, 1986 (in Russian). [12] A.S. Monin, A.M. Yaglom, Statistical Gydromechanics, Vols. 1, 2, Gidrometeoizdat, Sankt-Petersburg, 1992 (in Russian). [13] L. Prandtl, FUuhrer durch die StrUomungslehre, F. Vieweg, Braunschweig, 1949. [14] H. Schlichting, Grenzschicht-Theorie, Verlag G. Braun, Karlsruhe, 1965. [15] M.A. Goldshtik, V.N. Shtern, N.I. Yavorsky, Viscous Flows with Paradoxical Properties, Nauka, Novosibirsk, 1989 (in Russian). [16] Yu.L. Klimontovich, Turbulent Motion and the Structures of Chaos, Kluwer Academic Publ., Dordrecht, 1991. [17] P. Holmes, J.L. Lumley, G. Berkooz, Turbulence, Coherent Structures and Symmetry, Cambridge University Press, Cambridge, 1996. [18] R.J. Roache, Computational Fluid Dynamics, Hermosa Publishers Albuquerque, 1976. [19] P. Bradshaw, T. Cebeci, J.H. Whitelaw, Engineering Calculation Methods for Turbulent Flow, Academic Press, New York, 1981. [20] D.C. Wilcox, Turbulence Modelling for CFD, DCW Industries Inc., La Canada, CA, 1998. [21] S.M. Belotserkovsky, A.S. Ginevsky, Simulation of Turbulent Jets and Wakes by Discrete Vortex Technique, Nauka, Moscow, 1995 (in Russian). [22] A.N. Kolmogorov, Local structure of turbulence in incompressible ?uid for very large Reynolds numbers, DAN SSSR 30 (1941) 299–303 (in Russian). [23] A.N. Kolmogorov, The energy dispersion in the case of locally isotropic turbulence, DAN SSSR 32 (1941) 19–21 (in Russian).

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Physics Reports 397 (2004) 63 – 154 www.elsevier.com/locate/physrep

Violations of fundamental symmetries in atoms and tests of uni%cation theories of elementary particles J.S.M. Ginges, V.V. Flambaum∗ School of Physics, University of New South Wales, Sydney 2052, Australia Accepted 9 March 2004 editor J. Eichler

Abstract High-precision measurements of violations of fundamental symmetries in atoms are a very e2ective means of testing the standard model of elementary particles and searching for new physics beyond it. Such studies complement measurements at high energies. We review the recent progress in atomic parity nonconservation and atomic electric dipole moments (time reversal symmetry violation), with a particular focus on the atomic theory required to interpret the measurements. c 2004 Elsevier B.V. All rights reserved.
PACS: 32.80.Ys; 11.30.Er; 12.15.Ji; 31.15.Ar

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Manifestations and sources of parity violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The nuclear spin-independent electron–nucleon interaction; the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . 2.2. Nuclear spin-dependent contributions to atomic parity violation; the nuclear anapole moment . . . . . . . . . . . . . 2.3. Z 3 -scaling of parity violation in atoms induced by the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Measurements and calculations of parity violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Summary of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Summary of calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Method for high-precision atomic structure calculations in heavy alkali-metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. E-mail address: [email protected] (V.V. Flambaum).

c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.03.005

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 4.2. Zeroth-order approximation: relativistic Hartree–Fock method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Correlation corrections and many-body perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. All-orders summation of dominating diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Screening of the electron–electron interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. The hole–particle interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3. Chaining of the self-energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Other low-order correlation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6. Empirical %tting of the energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7. Asymptotic form of the correlation potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8. Interaction with external %elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1. Time-dependent Hartree–Fock method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2. E1 transition amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3. Hyper%ne structure constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4. Structural radiation and normalization of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High-precision calculation of parity violation in cesium and extraction of the nuclear weak charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. High-precision calculations of parity violation in cesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1. Mixed-states calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2. Inclusion of the Breit interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3. Neutron distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Strong-%eld QED corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5. Tests of accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. The vector transition polarizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The %nal value for the Cs nuclear weak charge QW and implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Ongoing/future studies of PNC in atoms with a single valence electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atoms with several electrons in un%lled shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Parity nonconservation in thallium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. A method to exclude the error from atomic theory: isotope ratios and the neutron distribution . . . . . . . . . . . . 6.3. Ongoing/future studies of PNC in complex atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The nuclear anapole moment and measurements of P-odd nuclear forces in atomic experiments . . . . . . . . . . . . . . . 7.1. The anapole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Origin of the nuclear anapole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Parity violating e2ects in atoms dependent on the nuclear spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Measurement of nuclear spin-dependent e2ects in cesium and extraction of the nuclear anapole moment . . . 7.4.1. Atomic calculations and extraction of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Extraction of a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. The nuclear anapole moment and parity violating nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. The cesium result and comparison with other experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. Ongoing/future studies of nuclear anapole moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electric dipole moments: manifestation of time reversal violation in atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1. Electronic enhancement mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Enhancement of T -odd e2ects in polar diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Limits on neutron, atomic, and molecular EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Mechanisms that induce atomic EDMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. The P; T -violating electron–nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. The electron EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. P; T -violating nuclear moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P; T -Violating nuclear moments and the atomic EDMs they induce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Electric moments; the Schi2 moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1. The P; T -odd electric %eld distribution in nuclei created by the nuclear Schi2 moment . . . . . . . . . . . .

76 78 79 80 82 83 84 84 85 85 86 87 88 89 90 91 92 96 96 96 97 99 100 101 102 103 104 105 106 106 107 110 112 112 112 113 114 116 117 118 118 119 119 120 122 124 127 127 128 130

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 9.2. Magnetic moments; the magnetic quadrupole moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. The spin hedgehog . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. What mechanisms at the nucleon scale induce P; T -odd nuclear moments? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. The P; T -odd nucleon–nucleon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. The external nucleon EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Comparison of the size of nuclear moments induced by the nucleon–nucleon interaction and the nucleon EDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. Nuclear enhancement mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Close-level enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Collective enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Octupole deformation; collective Schi2 moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5. Calculations of atomic EDMs induced by P; T -violating nuclear moments; interpretation of the Hg measurement in terms of hadronic parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Current limits on fundamental P; T -violating parameters and prospects for improvement . . . . . . . . . . . . . . . . . . . . . . 10.1. Summary of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Ongoing/future EDM experiments in atoms, solids, and diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction The success of the standard electroweak model of elementary particles [1] is extraordinary. 1 It has been tested in physical processes covering a range in momentum transfer exceeding ten orders of magnitude. It correctly predicted the existence of new particles such as the neutral Z boson. However, the standard model fails to provide a deep explanation for the physics that it describes. For example, why are there three generations of fermions? What determines their masses and the masses of gauge bosons? What is the origin of CP violation? The Higgs boson (which gives masses to the particles in the standard model) has not yet been found. The standard model is unable to explain Big Bang baryogenesis which is believed to arise as a consequence of CP violation. It is widely believed that the standard model is a low-energy manifestation of a more complete theory (perhaps one that uni%es the four forces). Many well-motivated extensions to the standard model have been proposed, such as supersymmetric, technicolour, and left-right symmetric models, and these give predictions for physical phenomena that di2er from those of the standard model. Some searches for new physics beyond the standard model are performed at high-energy and medium-energy particle colliders where new processes or particles would be seen directly. However, a very sensitive probe can be carried out at low energies through precision studies of quantities that can be described by the standard model. The new physics is manifested indirectly through a deviation of the measured values from the standard model predictions. The atomic physics tests that are the subject of this review lie in this second category. These tests exploit the fact that low-energy phenomena are especially sensitive to new physics that is manifested in the violations of fundamental symmetries, in particular P (parity) and T (time-reversal), that occur in the weak interaction. The 1

The recent observation of neutrino oscillations calls for a minimal extension, the introduction of neutrino masses and mixing; see, e.g., the review [2].

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deviations from the standard model, or the e2ects themselves, may be very small. To this end, exquisitely precise measurements and calculations are required. More than 20 years ago atomic experiments played an important role in the veri%cation of the standard model. While the %rst evidence for neutral weak currents (existence of the neutral Z boson) was discovered in neutrino scattering [3], the fact that neutral currents violate parity was %rst established in atomic experiments [4] and only later observed in high-energy electron scattering [5]. Now atomic physics plays a major role in the search for possible physics beyond the standard model. Precision atomic and high-energy experiments have di2erent sensitivities to models of new physics and so they provide complementary tests. In fact the energies probed in atomic measurements exceed those currently accessible at high-energy facilities. For example, the most precise measurement of parity nonconservation (PNC) in the cesium atom sets a lower bound on an extra Z boson popular in many extensions of the standard model that is tighter than the bound set directly at the Tevatron (see Section 5). Also, the null measurements of electric dipole moments (EDMs) in atoms (an EDM is a P- and T -violating quantity) place severe restrictions on new sources of CP-violation which arise naturally in models beyond the standard model such as supersymmetry. (Assuming CPT invariance, CP-violation is accompanied by T -violation.) Such limits on new physics have not been set by the detection of CP-violation in the neutral K [6] and B [7] mesons (see, e.g., Ref. [8] for a review of CP violation in these systems). Let us note that while new physics would bring a relatively small correction to a very small signal in atomic parity violation, in atomic EDMs the standard model value is suppressed and is many orders of magnitude below the value expected from new theories. Therefore, detection of an EDM would be unambiguous evidence of new physics. This review is motivated by the great progress that has been made recently in both the measurements and calculations of violations of fundamental symmetries in atoms. This includes the discovery of the nuclear anapole moment (an electromagnetic multipole that violates parity) [9], the measurement of the parity violating electron–nucleon interaction in cesium to 0.35% accuracy [9], the improvement in the accuracy (to 0.5%) of the atomic theory required to interpret the cesium measurement [10], and greatly improved limits on atomic [11] and electron [12] electric dipole moments. The aim of this review is to describe the theory of parity and time-reversal violation in atoms and explain how atomic experiments are used to test the standard model of elementary particles and search for new physics beyond it. 2 We track the recent progress in the %eld. In particular, we clarify the situation in atomic parity violation in cesium: it is now %rmly established that the cesium measurement [9] is in excellent agreement with the standard model; see Section 5. The structure of the review is the following. Broadly, it is divided into two parts. The %rst part, Sections 2–7, is devoted to parity violation in atoms. The second part, Sections 8–10, is concerned with atomic electric dipole moments. In Section 2 the sources of parity violation, and the standard model predictions, are described. In Section 3 a summary of the measurements of parity violation in atoms is given, with particular emphasis on the measurements with cesium. Also the atomic calculations are summarized. In Section 4 we present a detailed description of the methods for high-precision atomic structure calculations 2

There are other tests of fundamental symmetries in atoms not dealt with in this review. These include tests of CPT and Lorentz invariance [13] and tests of the permutation-symmetry postulate and the spin-statistics connection [14].

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applicable to atoms with a single valence electron. The methods are applied to parity violation in cesium in Section 5 and the value for the weak nuclear charge is extracted and compared with the standard model prediction. A discussion of the new physics constraints is also presented. In Section 6 a brief description for the method of atomic structure calculations for atoms with more than one valence electron is given, and the thallium PNC work is discussed. A brief discussion of the prospects for measuring PNC along a chain of isotopes is also presented. Then in Section 7 work on the anapole moment is reviewed. A description of electric dipole moments in atoms is given in Section 8, with a summary of all the measurements and a discussion of the P; T -violating sources at di2erent energy scales. Then in Section 9 a review of P; T -violating nuclear moments is given. In Section 10 a summary of the best limits on P; T -violating parameters can be found. Concluding remarks are presented in Section 11. For a general introduction to atomic P-violation and P; T -violation we refer the reader to the excellent books by Khriplovich [15] and Khriplovich and Lamoreaux [16]. 2. Manifestations and sources of parity violation in atoms Parity nonconservation (PNC) in atoms arises largely due to the exchange of Z 0 -bosons between atomic electrons and the nucleus. The weak electron–nucleus interaction violating parity, but conserving time-reversal, is given by the following product of axial vector (A) and vector (V) currents 3 G
[C1N e Q  5 eNQ  N + C2N e Q  eNQ  5 N ] : (1) hˆ = − √ 2 N Here G = 1:027 × 10−5 =m2p is the Fermi weak constant, N and e are nucleon and electron %eld operators, respectively, and the sum runs over all protons p and neutrons n in the nucleus. The Dirac matrices are de%ned as       I 0 0 i 0 −I 0 ≡  = ; 5 = ; i = ; (2) −i 0 0 −I −I 0 and
= 2s are the Pauli spin matrices. The coeScients C1N and C2N give di2erent weights to the contributions of protons and neutrons to the parity violating interaction. To lowest order in the electroweak interaction, C1p = 1=2(1 − 4 sin2 W ) ≈ 0:04;

C1n = −1=2 ;

C2p = −C2n = 1=2(1 − 4 sin2 W )gA ≈ 0:05 ;

(3)

where gA ≈ 1:26. The Weinberg angle W is a free parameter; experimentally it is sin2 W ≈ 0:23. The suppression of the coeScients C1p and C2N due to the small factor (1 − 4 sin2 ) makes |C1n | about 10 times larger than C1p and |C2N |. There is a contribution to atomic parity violation arising due to Z 0 exchange between electrons. However, this e2ect is negligibly small for heavy atoms [17–19]. It is suppressed by a factor 3

We use units ˝ = c = 1 throughout unless otherwise stated.

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(1 − 4 sin2 )K=(QW R(Z)) compared to the dominant electron–nucleon parity violating interaction, where QW is the nuclear weak charge [see below, Eq. (6)], K is a numerical factor that decreases with Z and R(Z) is a relativistic factor that increases with Z [18]. For 133 Cs 6S1=2 − 7S1=2 , K ≈ 2 and R(Z) = 2:8 and so the suppression factor is ≈ 0:04% of the dominant amplitude [18]. This number was con%rmed in [19]. We will consider this interaction no further. 2.1. The nuclear spin-independent electron–nucleon interaction; the nuclear weak charge Approximating the nucleons as nonrelativistic, the time-like component of the interaction (Ae ; VN ) is given by the nuclear spin-independent Hamiltonian (see, e.g., [15]) G hˆW = − √ 5 [ZC1p !p (r) + NC1n !n (r)] ; (4) 2 Z and N are the number of protons and neutrons. This is an e2ective single-electron operator. The proton and neutron densities are normalized to unity, !n; p d 3 r = 1. Assuming that these densities coincide, !p = !n = !, this interaction reduces to G (5) hˆW = − √ QW !(r)5 : 2 2 The nuclear weak charge QW is very close to the neutron number. To lowest order in the electroweak interaction, it is QW = −N + Z(1 − 4 sin2 W ) ≈ −N :

(6)

This value for QW is modi%ed by radiative corrections. The prediction of the standard electroweak model for the value of the nuclear weak charge in cesium is [20] SM 133 (55 Cs) = −73:10 ± 0:03 : QW

(7)

The nuclear weak charge is protected from strong-interaction e2ects by conservation of the nuclear vector current. The clean extraction of the weak couplings of the quarks from atomic measurements makes this a powerful method of testing the standard model and searching for new physics beyond it. The nuclear spin-independent e2ects arising from the nuclear weak charge give the largest contribution to parity violation in heavy atoms compared to other mechanisms. However, note that the weak interaction (5) does not always “work”. This interaction can only mix states with the same electron angular momentum (it is a scalar). Nuclear spin-dependent mechanisms (see below), which produce much smaller e2ects in atoms, can change electron angular momentum and so can contribute exclusively to certain transitions in atoms and dominate parity violation in molecules. 2.2. Nuclear spin-dependent contributions to atomic parity violation; the nuclear anapole moment Using the nonrelativistic approximation for the nucleons, the nuclear spin-dependent interaction due to neutral weak currents is  

G hˆNC = − √  · C2p (8)
p + C2n
n !(r) ; 2 p n

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where #i = 0 i . This term arises from the space-like component of the (Ve ; AN ) coupling. Averaging this interaction over the nuclear state with angular momentum I in the single-particle approximation gives G K − 1=2  · I!(r) ; hˆINC = − √ 2 2 I (I + 1)

(9)

where K = (I + 1=2)(−1)I +1=2−l , l is the orbital angular momentum of the unpaired nucleon, and 2 = −C2 . There are two reasons for the suppression of this contribution to parity violating e2ects in atoms. First, unlike the spin-independent e2ects [Eq. (5)], the nucleons do not contribute coherently; in the simple nuclear shell model only the unpaired nucleon which carries nuclear spin I makes a contribution. Second, the factor C2 ˙ (1 − 4 sin2 W ) is small in the standard model. There is another contribution to nuclear spin-dependent PNC in atoms arising from neutral currents: the “usual” weak interaction due to the nuclear weak charge, hˆW , perturbed by the hyper%ne interaction [21]. In the single-particle approximation this interaction can be written as [21,22] ·I G !(r) ; hˆIQ = √ Q I 2

(10)

#N 1 = 2:5 × 10−4 A2=3 N ; Q = − QW 3 m p RN

(11)

with

RN = r0 A1=3 is the nuclear radius, r0 = 1:2 fm, 4 A = N + Z is the mass number, # = 1=137 is the %ne structure constant, and N is the magnetic moment of the nucleus in nuclear magnetons. For 133 Cs, N = 2:58 and Q = 0:017. However, the neutral currents are not the dominant source of parity violating spin-dependent e2ects in heavy atoms. It is the nuclear anapole moment a that gives the largest e2ects [23]. This moment arises due to parity violation inside the nucleus, and manifests itself in atoms through the usual electromagnetic interaction with atomic electrons. The Hamiltonian describing the interaction between the nuclear anapole moment and an electron is 5 G K  · I!(r) : hˆa = √ a 2 I (I + 1)

(12)

The anapole moment a increases with atomic number, a ˙ A2=3 . This is the reason it leads to larger parity violating e2ects in heavy atoms compared to other nuclear spin-dependent mechanisms. In heavy atoms a ∼ #A2=3 ∼ 0:1 − 1 [23,24]. (Note that the interaction (10), (11) also increases as A2=3 , however the numerical coeScient is very small.) The spin-dependent contributions [Eqs. (9), (10) and (12)] have the same form and produce the same e2ects in atoms. We will continue our discussion of the nuclear anapole moment and nuclear spin-dependent e2ects in atoms in Section 7. 4 Throughout the review we take the number (r0 = 1:1 or 1:2 fm) used in the work being cited in order to quote numerical results. 5 In fact, the distribution of the anapole magnetic vector potential is di2erent from the nuclear density. However, the corrections produced by this di2erence are small; see Section 7.

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2.3. Z 3 -scaling of parity violation in atoms induced by the nuclear weak charge In 1974 the Bouchiats showed that parity violating e2ects in atoms increase with the nuclear charge Z faster than Z 3 [17,25]. This result was the incentive for studies of parity violation in heavy atoms. Let us brie?y point out where the factor of Z 3 originates. Taking the nonrelativistic limit of the electron wave functions and considering the nucleus to be point-like, the Hamiltonian (5) reduces to G hˆW = √ (
· p%3 (r) + %3 (r)
· p)QW ; (13) 4 2m where m,
, p are the electron mass, spin, and momentum. The weak Hamiltonian hˆW mixes electron states of opposite parity and the same angular momentum (it is a scalar). It is a local operator, so we need only consider the mixing of s and p1=2 states. The matrix element p1=2 |hˆW |s , with nonrelativistic single-particle s and p1=2 electron states, is proportional to Z 2 QW . One factor of Z here comes from the probability for the valence electron to be at the nucleus, and the other from the operator p which, near the nucleus (unscreened by atomic electrons), is proportional to Z. The nuclear weak charge |QW | ≈ N ∼ Z. (See [17,25,15] for more details.) It should be remembered thatrelativistic e2ects are important, since Dirac wave functions diverge at r = 0, j ˙ r −1 ,  = (j + 1=2)2 − Z 2 #2 . Taking into account the relativistic nature of the wave functions brings in a relativistic factor R(Z) which increases with the nuclear charge Z. The factor R ≈ 10 when Z = 80. As a consequence, the parity nonconserving e2ects in atoms increase as p1=2 |hˆW |s ˙ R(Z)Z 2 QW ;

(14)

that is, faster than Z 3 . 3. Measurements and calculations of parity violation in atoms An account of the dramatic story of the search for parity violation in atoms can be found in the book [15]. Below we will brie?y discuss how parity violation is manifested in atoms, which experiments have yielded nonzero signals, what quantity is measured, and what is required to interpret the measurements. Parity violation in atoms produces a spin helix, and this helix interacts di2erently with rightand left-polarized light (see, e.g., Ref. [15]). The polarization plane of linearly polarized light will therefore be rotated in passing through an atomic vapour. The weak interaction mixes states of opposite parity (parity violation), e.g., |p + )|s , where the mixing coeScient ) is pure imaginary. Therefore, a magnetic dipole (M 1) transition 6 in atoms will have a component originating from an electric dipole (E1) transition between states of the same 6 Atomic PNC studies are not limited, of course, to M 1 transitions, although all unambiguous signals to date have been obtained with them (see below). An experiment on an E2 transition (6S1=2 − 5D3=2 ) with singly ionized barium is being considered at Seattle [26]; the Berkeley dysprosium experiment [27] involves quantum beats on an E1 transition. See Ref. [28].

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nominal parity, EPNC , e.g. p1=2 − p3=2 . The rotation angle per absorption length in such a transition is proportional to the ratio Im(EPNC )=M 1. While it may appear that it is more rewarding to study M 1 transitions that are highly forbidden, where there is a larger rotation angle, the ordinary M 1 transitions are in fact more convenient for experimental investigation since the angle per unit length ≈ ImEPNC M 1 (see, e.g., [15]). In measurements of parity violation in highly forbidden M 1 transitions, an electric %eld j is applied to open up the forbidden transition. The M 1 transition then contains a Stark-induced E1 component EStark which the parity violating amplitude interferes with. In such experiments the ratio Im(EPNC )= is measured, where  is the vector transition polarizability, EStark ∼ j. Atomic many-body theory is required to calculate the parity nonconserving E1 transition amplitude EPNC . This is expressed in terms of the fundamental P-odd parameters like the nuclear weak charge QW . Interpretation of the measurements in terms of the P-odd parameters also requires a determination of M 1 or . 3.1. Summary of measurements Zel’dovich was the %rst to propose optical rotation experiments in atoms [29]. Unfortunately, he only considered hydrogen where PNC e2ects are small. Optical rotation experiments in Tl, Pb, and Bi were proposed by Khriplovich [30], Sandars [31], and Sorede and Fortson [32]. These proposals followed those by the Bouchiats to measure PNC in highly forbidden transitions in Cs and Tl [25,17]. The %rst signal of parity violation in atoms was seen in 1978 at Novosibirsk in an optical rotation experiment with bismuth [4]. Now atomic PNC has been measured in bismuth, lead, thallium, and cesium. PNC e2ects were measured by optical rotation in the following atoms and transitions: in 209 Bi in the transition 6s2 6p3 4 S3=2 − 6s2 6p3 2 D5=2 by the Novosibirsk [4], Moscow [33], and Oxford [34,35] groups and in the transition 6s2 6p3 4 S3=2 − 6s2 6p3 2 D3=2 by the Seattle [36] and Oxford [37,38] groups; in 6s2 6p2 3 P0 − 6s2 6p2 3 P1 in 208 Pb at Seattle [39,40] and Oxford [41]; and in the transition 6s2 6p 2 P1=2 − 6s2 6p 2 P3=2 in natural Tl (70:5% 205 Tl and 29:5% 203 Tl) at Oxford [42,43] and Seattle [44]. The highest accuracy that has been reached in each case is: 9% for 209 Bi 4 S3=2 −2 D5=2 [35], 2% for 209 Bi 4 S3=2 −2 D3=2 [38], 1% for 208 Pb [40], and 1% for Tl [44]. The Stark-PNC interference method was used to measure PNC in the highly forbidden M 1 transitions: 6s 2 S1=2 −7s 2 S1=2 in 133 Cs at Paris [45–48] and Boulder [49,50,9] and 6s2 6p 2 P1=2 −6s2 7p 2 P1=2 in 203;205 Tl at Berkeley [51,52]. In the most precise Tl Stark-PNC experiment [52] an accuracy of 20% was reached. In 1997, PNC in Cs was measured with an accuracy of 0.35% [9]—an accuracy unprecedented in measurements of PNC in atoms. Results of atomic PNC measurements accurate to sub-5% are listed in Table 1. Several PNC experiments in rare-earth atoms have been prompted by the possibility of enhancement of the PNC e2ects due to the presence of anomalously close levels of opposite parity [53]. Another attractive feature of rare earth atoms is their abundance of stable isotopes. Taking ratios of measurements of PNC in di2erent isotopes of the same element removes from the interpretation the dependence on atomic theory [53]; see Section 6. Null measurements of PNC have been reported for M 1 transitions in the ground state con%guration 4f6 6s2 of samarium at Oxford [54,55] and for the 4f9 5d2 6s J = 10 − 4f10 5d6s J = 10 transition in dysprosium at Berkeley [27]. The upper limit for dysprosium was smaller than expected by theory.

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Table 1 Measurements of PNC in atoms with precision better than 5% Atom

Transition

Group

Year

Ref.

Measurement −Im(EPNC =M 1) (10−8 )

209

Bi Pb

4

208 205

Tl

6P1=2 − 6P3=2

133

Cs

6S1=2 − 7S1=2

3

S3=2 − 2 D3=2 P0 − 3 P1

Oxford Seattle Oxford Oxford Seattle Boulder Boulder

1991 1993 1996 1995 1995 1988 1997

[38] [40] [41] [43] [44] [50] [9]

10.12(20) 9.86(12) 9.80(33) 15.68(45) 14.68(17)

−Im(EPNC =) (mV/cm)

1.576(34) 1.5935(56)

Results of optical rotation experiments are given in terms of Im(EPNC =M 1); Stark-PNC experiments are given in terms of Im(EPNC =).

For a recent review of measurements of atomic PNC, we refer the reader to [28]; for a review of the early measurements, see, e.g., [57]. For comprehensive reviews, please see the book [15] and the more recent review [58]. 3.2. Summary of calculations The interpretation of the single-isotope PNC measurements is limited by atomic structure calculations. The theoretical uncertainty for thallium is at the level of 2.5–3% for the transition 6P1=2 −6P3=2 [59,60], and is worse for the transition 6P1=2 − 7P1=2 at 6% [59] and for lead (8%) [61] and bismuth (11% for the 876 nm transition 4 S3=2 − 2 D3=2 [61,62] and 15% for the 648 nm transition 4 S3=2 − 2 D5=2 [63]). Cesium is the simplest atom of interest in PNC experiments, it has one electron above compact, closed shells. The precision of the atomic calculations for Cs is 0.5% [10] (see also calculations accurate to better than 1%, [62,19,64]). For references to earlier calculations for the above atoms and transitions, see, e.g., the book [15]. In Table 2 we present the values of the most precise calculations for the PNC amplitudes corresponding to those atoms and transitions in which high-precision measurements (¡ 5% error) have been performed (Table 1). Using these calculations one may conclude that all parity violation experiments are in excellent agreement with the standard model. Note that the calculations were performed before the accurate measurements. It is interesting to note that the actual accuracy of the many-body calculations in all atoms (Cs, Tl, Pb, Bi) has been found to be better than claimed in the original theoretical papers! 3.3. Cesium Because of the extraordinary precision that has been achieved in measurements of cesium, and the clean interpretation of the measurements (compared to other heavy atoms), in this review we concentrate mainly on parity violation in cesium. The high precision of the nuclear weak charge extracted from cesium has made this system important in low-energy tests of the standard model

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Table 2 Most precise calculations of PNC amplitudes EPNC for atoms and transitions listed in Table 1. Units: −10−11 ieaB (−QW =N ) Atom

Transition

EPNC a

Ref.

209

Bi Pb 205 Tl

4

208

3

133

6S − 7S

26(3) 28(2) 27.0(8) 27.2(7) 0.904(5)

[61,62] [61] [59] [60] [10]

Cs

S3=2 − 2 D3=2 P0 − 3 P1 6P1=2 − 6P3=2

a The values for the PNC amplitudes for Cs [10] and for Tl 6P1=2 − 6P3=2 [60] include corrections beyond the other calculations. In particular, for Cs the contributions of the Breit interaction and vacuum polarization due to the strong nuclear Coulomb %eld are included. For Tl the Breit interaction is also included. The remaining corrections for Cs and Tl are discussed in detail in Sections 5, 6, respectively. These corrections would be inside the error bars for the other atoms and transitions in the table.

Table 3 Summary of experimental results for PNC in cesium 6S − 7S, −Im(EPNC )=; units: mV/cm Group

Year

Ref.

Value

Paris Boulder Boulder Boulder Paris

1982, 1984 1985 1988 1997 2003

[45–47] [49] [50] [9] [48]

1.52(18) 1.65(13) 1.576(34) 1.5935(56) 1.752(147)

and has made it one of the most sensitive probes of new physics. Measurements of parity violation in cesium have also opened up a new window from which parity violation within the nucleus (the nuclear anapole moment; see Section 7) can be studied. Below we list the measurements and calculations for cesium that have been performed over the years, culminating in a 0.35% measurement and 0.5% calculation. 3.3.1. Measurements Measurements of parity violation in the highly forbidden 6S − 7S transition in Cs were %rst suggested and considered in detail in the landmark works of the Bouchiats [17,25]. Measurements have been performed independently by the Paris group [45–48] and the Boulder group [49,50,9]. The results of the Cs PNC experiments are summarized in Table 3. The Paris result in the %rst row is the average [47] of their (revised) results for the measurements of PNC in the transitions 6SF=4 −7SF=4 [45] and 6SF=3 −7SF=4 [46]. (The nuclear angular momentum of 133 Cs I = 7=2 and the electron angular momentum J = 1=2, so the total angular momentum of the atom is F = 3; 4). The Paris group have very recently performed a new measurement of PNC in Cs (last row) using a novel approach, chiral optical gain [48]. Each of the Boulder results [49,50,9] cited in the table is an average of PNC in the hyper%ne transitions 6SF=4 − 7SF=3 and 6SF=3 − 7SF=4 . The accuracy of the latest result is 0.35%, several times more precise than the best measurements of parity violation in other atoms.

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The PNC nuclear spin-independent component, arising from the nuclear weak charge, makes the same contribution to all hyper%ne transitions. So averaging the PNC amplitudes over the hyper%ne transitions gives the contribution from the nuclear weak charge. PNC in atoms dependent on the nuclear spin was detected for the %rst (and only) time in Ref. [9] where it appeared as a di2erence in the PNC amplitude in di2erent hyper%ne transitions. The dominant mechanism for nuclear spin-dependent e2ects in atoms, the nuclear anapole moment, is the subject of Section 7. 3.3.2. Calculations Numerous calculations of the Cs 6S − 7S EPNC These calculations are summarized in Table 4. The performed more than 10 years ago represented a theory and parity violation in atoms. At the time,

amplitude have been performed over the years. many-body calculations [62,19], accurate to 1%, signi%cant step forward for atomic many-body these calculations were unmatched by the PNC

Table 4 Summary of calculations of the PNC E1 amplitude for the cesium 6S − 7S transition; units are −10−11 ieaB (−QW =N ) Authors

Year

Bouchiat, Bouchiat Loving, Sandars Neu2er, Commins Kuchiev, Sheinerman, Yahontov Das Bouchiat, Piketty, Pignon Dzuba, Flambaum, Silvestrov, Sushkov SchZafer, MZuller, Greiner, Johnson Ma[ rtensson-Pendrill Plummer, Grant SchZafer, MZuller, Greiner Johnson, Guo, Idrees, Sapirstein Johnson, Guo, Idrees, Sapirstein Bouchiat, Piketty Dzuba, Flambaum, Silvestrov, Sushkov Johnson, Blundell, Liu, Sapirstein Parpia, Perger, Das Dzuba, Flambaum, Sushkov Hartley, Sandars Hartley, Lindroth, Ma[ rtensson-Pendrill Blundell, Johnson, Sapirstein Safronova, Johnson Kozlov, Porsev, Tupitsyn Dzuba, Flambaum, Ginges

1974, 1975 1977 1981 1981 1983 1984, 1984 1985 1985 1985 1985, 1985, 1986 1987 1988 1988 1989 1990 1990 1990, 2000 2001 2002

a

1975

1985

1986 1986

1992

Ref.

Value

[17,25]a [65]a [66]a [67]a [68]b [69]a [70,71]b [72]b [73]b [74]b [75]b [76]c [76]b [77]a [59]b [78]b [79]c [62]b [80]c [81]b [19]b [82]b [64]bd [10]bd

1.33 1.15 1.00 0.75 1.06 0.97(10) 0.88(3) 0.74 0.886 0.64 0.92 0.754, 0.876, 0.856 0.890 0.935(20)(30) 0.90(2) 0.95(5) 0.879 0.908(9) 0.904(18) 0.933(37) 0.905(9) 0.909(11) 0.901(9) 0.904(5)

Semi-empirical calculations. Ab initio many-body calculations. c Combined many-body and semi-empirical calculations. d The di2erence between the values of [64,10] and previous ones is due to the inclusion of the Breit interaction in [64] and the Breit and strong %eld vacuum polarization in [10]. b

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measurements which were accurate to 2%. The method of calculation used in Ref. [62] is the subject of Sections 4 and 5. The method used in Ref. [19] is based on the popular coupled-cluster method, and we refer the interested reader to this work for details. In the last 10 years a series of new measurements have been performed for quantities used to test the accuracy of the atomic calculations [62,19], such as electric dipole transition amplitudes (see [83]). The new measurements are in agreement with the calculations, resolving a previous discrepancy between theory and experiment. This inspired Bennett and Wieman [83] to claim that the atomic theory is accurate to 0.4% rather than 1% claimed by theorists. Since then, contributions to the PNC amplitude have been found (a correction due to inclusion of the Breit interaction and more recently the strong-%eld radiative corrections) that enter above the 0.4% level but below 1% (see Section 5). A re-calculation of the work [62], with some further improvements, was performed recently, with a full analysis of the accuracy of the PNC amplitude. This work, Ref. [10], represents the most accurate (0.5%) calculation to date. It is described in detail in Section 5. The result of [10] di2ers from [62,19] by only ∼ 0:1% if Breit, vacuum polarization, and neutron distribution corrections are excluded. One may interpret this as grounds for asserting that the many-body calculations [62,19,64,10] have an accuracy of 0.5% in agreement with the conclusion of [83]. 4. Method for high-precision atomic structure calculations in heavy alkali-metals In this section we describe methods that can be used to obtain high accuracy in calculations involving many-electron atoms with a single valence electron. These are the methods that have been used to obtain the most precise calculation of parity nonconservation in Cs. They were originally developed in works [7,84,85,59] and applied to the calculation of PNC in Cs in Ref. [62]. In [62] it was claimed that the atomic theory is accurate to 1%. A complete re-calculation of PNC in Cs using this method, with a new analysis of the accuracy, indicates that the error is as small as 0.5% [10]. (We refer the reader to Section 5, where this question of accuracy is discussed in general; please also see Section 5 for an in-depth discussion of PNC in Cs.) In this section the method is applied to energies, electric dipole transition amplitudes, and hyper%ne structure. A comparison of the calculated and experimental values gives an indication of the quality of the many-body wave functions. Note that the above quantities are sensitive to the wave functions at di2erent distances from the nucleus. Hyper%ne structure, energies, and electric dipole transition amplitudes are dominated by the contribution of the wave functions at small, intermediate, and large distances from the nucleus. We concentrate on calculations for Cs relevant to the 6S − 7S PNC E1 amplitude (see Eq. (56) and Section 5.1.5). A brief overview of the method is presented in Section 4.1. For those not interested in the technical details of the atomic structure calculations, Sections 4.2–4.8 may be omitted without loss of continuity. 4.1. Overview The calculations begin in the relativistic Hartree–Fock (RHF) approximation. The N − 1 selfconsistent RHF orbitals of the core are found (N is the total number of electrons in the atom), and

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the external electron orbitals are found in the potential of the core electrons (the Vˆ N −1 potential). We get RHF wave functions, energies, and Green’s functions in this way. Correlation corrections to the external electron orbitals are included in second (lowest) order in the residual interaction (Vˆexact − Vˆ N −1 ), where Vˆexact is the exact Coulomb interaction between the atomic electrons. The correlations are included into the external electron orbitals by adding the correlation potential (the self-energy operator) to the RHF potential when solving for the external electron. Using the Feynman diagram technique, important higher-order diagrams are included into the self-energy in all orders: screening of the electron–electron interaction and the hole–particle interaction. The self-energy is then iterated using the correlation potential method. Interactions of the atomic electrons with external %elds are calculated using the time-dependent Hartree–Fock (TDHF) method; this method is equivalent to the random-phase approximation (RPA) with exchange. Using this approach we can take into account the polarization of the atomic core by external %elds to all orders. Then the major correlation corrections are included as corrections to electron orbitals (Brueckner orbitals). Small correlation corrections (structural radiation, normalization) are taken into account using many-body perturbation theory. 4.2. Zeroth-order approximation: relativistic Hartree–Fock method The full Hamiltonian we wish to solve is the many-electron Dirac equation 7 Hˆ =

N


[i · pi + ( − 1)m − Ze2 =ri ] +

i=1


i¡j

e2 : |ri − rj |

(15)

Here m and p are the electron mass and momentum,  and  are Dirac matrices, Ze is the nuclear charge and N is the number of electrons in the atom (N = 55 for cesium). This equation cannot be solved exactly, so some approximation scheme must be used. This is done by excluding the complicated Coulomb term and adding instead some averaged potential in which the electrons move. The Coulomb term, minus the averaged potential, can be added back perturbatively. It is well known that choosing the electrons to move in the self-consistent Hartree–Fock potential Vˆ N −1 , in the zeroth-order approximation, simpli%es the calculations of higher-order terms (we will come to this in the next section). The single-particle relativistic Hartree–Fock (RHF) Hamiltonian is hˆ0 =  · p + ( − 1)m − Ze2 =r + Vˆ N −1 ; Hˆ 0 =

 i

hˆ(i) 0 , where the Hartree–Fock potential

Vˆ N −1 = Vˆdir + Vˆexch ;

7

In Section 5.1.2 we discuss the inclusion of the Breit interaction into the Hamiltonian.

(16)

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

(a)

77

(b)

Fig. 1. Hartree–Fock (a) direct and (b) exchange diagrams for energies, %rst-order in the Coulomb interaction. The solid and dashed lines are the electron and Coulomb lines, respectively.

is the sum of the direct and nonlocal exchange potentials created by the (N − 1) core electrons n, N −1  †
n (r1 ) n (r1 ) 3 2 ˆ d r1 (r) (17) Vdir (r) = e |r − r1 | n=1 Vˆexch (r) = −e2

N −1
n=1



†

n (r1 )

(r1 ) 3 d r1 |r − r1 |

n (r)

:

(18)

The SchrZodinger equation hˆ0

i

= ji

i

;

(19)

where i , ji are single-particle wave functions and energies, is solved self-consistently for the N − 1 core electrons. The Hartree–Fock potential is then kept “frozen” and the RHF equation (16), (19) is solved for the states of the external electron. The Hamiltonian hˆ0 thus generates a complete orthogonal set of single-particle orbitals for the core and valence electrons [86]. Hartree–Fock diagrams for energies are presented in Fig. 1. The single-particle electron orbitals have the form   fnjl (r)3jlm 1 ; (20) njlm (r) = r −i#(
· n)gnjl (r)3jlm where # is the %ne structure constant,
= 2s is the electron spin, n = r=r, fnjl (r) and gnjl (r) are radial functions, and 3jlm is a spherical spinor (see, e.g., [15]). Because we are performing calculations for heavy atoms, and we are interested in interactions that take place in the vicinity of the nucleus (the weak and hyper%ne interactions), the %nite size of the nucleus must be taken into account. We use the standard formula for the charge distribution Z!(r) in the nucleus !0 !(r) = ; (21) 1 + exp[(r − c)=a]  where !0 is the normalization constant found from the condition !(r) d 3 r = 1, t = a(4 ln 3) is the skin-thickness, and c is the half-density radius. For 133 Cs we take t = 2:5 fm and c = 5:6710 fm (r 2 1=2 = 4:804 fm) [87]. Energy levels of cesium states relevant to the 6S − 7S E1 PNC transition are presented in Table 5. It is seen that the RHF energies agree with experiment to 10%.

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Table 5 Removal energies for Cs in units cm−1 State

RHF

5ˆ (2)

5ˆ

Experimenta

6S 7S 6P1=2 7P1=2

27954 12112 18790 9223

32415 13070 20539 9731

31492 12893 20280 9663

31407 12871 20228 9641

a

Taken from [88].

In order to obtain more realistic wave functions, we need to take into account the e2ect of correlations between the external electron and the core. We describe the techniques used to calculate these correlations in the following sections. 4.3. Correlation corrections and many-body perturbation theory The subject of this section is the inclusion of electron–electron correlations into the single-particle electron orbitals using many-body perturbation theory. We will see that high accuracy can be reached in the calculations by using the Feynman diagram technique as a means of including dominating classes of diagrams in all orders. The correlation corrections can be most accurately calculated in the case of alkali-metal atoms (for example, cesium). This is because the external electron has very little overlap with the electrons of the tightly bound core, enabling the use of perturbation theory in the calculation of the residual interaction of the external electron with the core. The exact Hamiltonian of an atom [Eq. (15)] can be divided into two parts: the %rst part is the sum of the single-particle Hamiltonians, and the second part represents the residual Coulomb interaction Hˆ =

N
i=1

Uˆ =


i¡j

hˆ0 (ri ) + Uˆ ;

(22) N


e2 − Vˆ N −1 (ri ) : |ri − rj | i=1

(23)

Correlation corrections to the single-particle orbitals are included perturbatively in the residual interaction Uˆ . By calculating the wave functions in the Hartree–Fock potential Vˆ N −1 for the zeroth-order approximation, the perturbation corrections are simpli%ed. The %rst-order corrections (in the residual Coulomb interaction Uˆ ) to the ionization energy vanish (the two terms in Eq. (23) cancel each other), since the correlation corrections %rst-order in the Coulomb interaction are nothing but the %rstorder Hartree–Fock ones (Fig. 1). The lowest-order corrections therefore correspond to those arising in second-order perturbation theory, Uˆ (2) . These corrections are determined by the four Goldstone diagrams in Fig. 2 [86]. They can be calculated by direct summation over intermediate states [86] or by the “correlation potential” method [71]. This latter method gives higher accuracy and, along with the Feynman diagram technique to be discussed in the following section, enables the inclusion

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 α α

β

α

α

79 α

α

β n

n n

γ

β γ

β

α

m

n

m α

(a)

(b)

(c)

(d)

Fig. 2. Second-order correlation corrections to energy of the valence electron. Dashed line is the Coulomb interaction; loop is polarization of the atomic core, corresponding to virtual creation of an excited electron and a hole. The state of the external electron is denoted by #; n, m are core states; and ,  are states outside the core.

of higher-order e2ects: electron–electron screening, the hole–particle interaction, and the nonlinear contributions of the correlation potential. The correlation potential method corresponds to adding a nonlocal correlation potential 5ˆ to the potential Vˆ N −1 in the RHF equation (16) and then solving for the states of the external electron. The correlation potential is de%ned such that its average value coincides with the correlation correction to the energy, ˆ # %j# =  # |5|  ˆ 5 # = 5(r1 ; r2 ; j# ) # (r1 ) d 3 r1 :

(24) (25)

It is easy to write the correlation potential explicitly. For example, a part of the operator 5(r1 ; r2 ; j# ) corresponding to Fig. 2(a) is given by † † −1 −1 †
 n (r4 )r24  (r4 )  (r2 )  (r3 )  (r1 )r13 n (r3 ) 2(a) 4 3 3 d r3 d r 4 : (26) 5 (r1 ; r2 ; j# ) = e j# + j n − j  − j  n;;

Note that 5ˆ is a single-electron and energy-dependent operator. By solving the RHF equation for the ˆ we obtain “Brueckner” orbitals and energies. 8 states of the external electron in the %eld Vˆ N −1 + 5, The largest correlation corrections are included in the Brueckner orbitals. See Table 5 for Brueckner energies of the lower states of cesium calculated in the secondorder correlation potential. 5ˆ (2) ≡ Uˆ (2) denotes the “pure” second-order correlation potential (without screening, etc.). It is seen that the inclusion of these corrections improves the energies signi%cantly, from the level of 10% deviation from experiment for the RHF approximation to the level of 1%. 4.4. All-orders summation of dominating diagrams We saw in the previous section that when we take into account second-order correlation corrections, the accuracy for energies is improved signi%cantly beyond that for energies calculated in the 8

Note that there is a slight distinction in the de%nition of these Brueckner orbitals and those de%ned in, e.g., [89].

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

RHF approximation. However, the corrections are overestimated. This overestimation is largely due to the neglect of screening in the electron–electron interaction. In this section we describe the calculations of three series of higher-order diagrams: screening of the electron–electron interaction and the hole–particle interaction, which are inserted into the ˆ and iterations of 5. ˆ With the inclusion of these diagrams the accuracy for correlation potential 5; energies is improved to the level of 0.1% (see Table 5). The screening of the electron–electron interaction is a collective phenomenon and is similar to Debye screening in a plasma; the corresponding chain of diagrams is enhanced by a factor approximately equal to the number of electrons in the external closed subshell (the 5p electrons in cesium) [84]. The importance of this e2ect can be understood by looking at a not dissimilar example in which screening e2ects are important, for instance, the screening of an external electric %eld in an atom. According to the Schi2 theorem [90], a homogeneous electric %eld is screened by atomic electrons (and at the nucleus it is zero). (See [91] where a calculation of an external electric %eld inside the atom has been performed.) The hole–particle interaction is enhanced by the large zero-multipolarity diagonal matrix elements of the Coulomb interaction [85]. The importance of this e2ect can be seen by noticing that the existence of the discrete spectrum excitations in noble gas atoms is due only to this interaction (see, e.g., [92]). The nonlinear e2ects of the correlation potential are calculated by iterating the self-energy operator. These e2ects are enhanced by the small denominator, which is the energy for the excitation of an external electron (in comparison with the excitation energy of a core electron) [85]. All other diagrams of perturbation theory are proportional to powers of the small parameter Qnd =\jint ∼ 10−2 , where Qnd is a nondiagonal Coulomb integral and \jint is a large energy denominator corresponding to the excitation of an electron from the core (due to an interaction in an internal electron line of the perturbation diagrams) [85]. 4.4.1. Screening of the electron–electron interaction The main correction to the correlation potential comes from the inclusion of the screening of the Coulomb %eld by the core electrons. Some examples of the lowest-order screening corrections are presented in Fig. 3. When screening diagrams in the lowest (third) order of perturbation theory are taken into account, a correction is obtained of opposite sign and almost the same absolute value as the corresponding second-order diagram [84]. Due to these strong cancellations there is a need to sum the whole chain of screening diagrams. However, this task causes diSculties in standard perturbation theory as the screening diagrams in the correlation correction cannot be represented by a simple geometric progression due to the overlap of the energy denominators of di2erent loops

(a)

(b)

(c)

Fig. 3. Lowest order screening corrections to the diagram in Fig. 2(a).

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

ε+ω

ε

81

ε

ω1 (a)

ω2

(b)

Fig. 4. Second-order correlation corrections to energy in the Feynman diagram technique.

ε+ω r

r

1

2

ε Fig. 5. Polarization operator.

[such an overlap indicates a large number of excited electrons in the intermediate states; see, e.g., Figs. 3(b) and (c)]. This summation problem is solved by using the Feynman diagram technique. The correlation corrections to the energy in the Feynman diagram technique are presented in Fig. 4. The Feynman Green’s function is of the form G(r1 ; r2 ; j) =




†

n (r1 ) n (r2 )

j − jn − i%

n

+




†

 (r1 )  (r2 )

j − j + i%

;

%→0 ;

(27)

where n is an occupied core electron state,  is a state outside the core. While the simplest way of calculating the Green’s function is by direct summation over the discrete and continuous spectrum, there is another method in which higher numerical accuracy can be achieved. As is known, the radial Green’s function G0 for the equation without the nonlocal exchange interaction Vexch can be expressed in terms of the solutions 70 and 7∞ of the SchrZodinger or Dirac equation that are regular at r → 0 and r → ∞, respectively: G0 (r1 ; r2 ) ˙ 70 (r¡ )7∞ (r¿ ), r¡ = min(r1 ; r2 ), r¿ = max(r1 ; r2 ). ˆ The exchange interaction is taken into account by solving the matrix equation Gˆ = Gˆ 0 + Gˆ 0 Vˆexch G. The polarization operator (Fig. 5) is given by  ∞ dj ˆ 1 ; r2 ; !) = G(r1 ; r2 ; ! + j)G(r2 ; r1 ; j) : 9(r (28) −∞ 2; This integration is carried out analytically, giving
† ˆ 1 ; r2 ; !) = i 9(r n (r1 )[G(r1 ; r2 ; jn + !) + G(r1 ; r2 ; jn − !)] n (r2 ) :

(29)

n

Using formulae (27) and (29), it is easy to perform analytical integration over ! in the calculation of the diagrams in Fig. 4. After integration, diagram 4(a) transforms to 2(a) and (c) and diagram 4(b) transforms to 2(b) and (d).

82

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 +

+

+ ....

Fig. 6. Screening diagram chain for e2ective polarization operator.

Fig. 7. Insertion of the hole–particle interaction into the second order correlation correction.

To include electron–electron screening to all orders in the Coulomb interaction in the Feynman diagram technique, the polarization operator is chained (Fig. 6) before integration over ! is carried out. The screened polarization operator is −1 ˆ ˆ ;(!) ˆ = 9(!)[1 + iQˆ 9(!)] :

(30)

The integration over ! is performed numerically. The integration contour is rotated 90◦ from the real axis to the complex ! plane parallel to the imaginary axis—this aids the numerical convergence by keeping the poles far from the integration contour. The all-order electron–electron screening reduces the second-order correlation corrections to the energies of S and P states of 133 Cs by 40%. 4.4.2. The hole–particle interaction In Fig. 7 we include the hole–particle interaction into the polarization loop of the second-order correlation correction. The hole–particle interaction accounts for the alteration of the core potential due to the excitation of the electron from the core to the virtual intermediate state. This electron now moves in the potential created by the N − 2 electrons, and no longer contributes to the Hartree–Fock potential. Denoting Vˆ 0 as the zero multipolarity direct potential of the outgoing electron, the potential which describes the excited and core states simultaneously is [85] ˆ Vˆ 0 (1 − P) ˆ ; Vˆ = Vˆ N −1 − (1 − P)

(31)

where Pˆ is the projection operator on the core orbitals, Pˆ =

N −1


|n n| :

(32)

n=1

The projection operator Pˆ is introduced into the potential to make the excited states orthogonal to the core states. It is easily seen that for the occupied orbitals Vˆ = Vˆ N −1 , while for the excited orbitals Vˆ = Vˆ N −1 − Vˆ 0 . Strictly one should also make subtractions for higher multipolarities and for the exchange interaction as well, however these contributions are relatively small and are therefore safe to ignore [85]. To obtain high accuracy, the hole–particle interaction in the polarization operator needs to be taken into account in all orders (Fig. 8). This is achieved by calculating the Green’s function in the potential (31) and then using it in the expression for the polarization operator (29). The screened

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

=

+

+

83

+ ....

Fig. 8. Hole–particle interaction in the polarization operator.

=

+

=

+

+

+ ....

Fig. 9. Renormalization of the Coulomb line due to the hole–particle interaction and screening.

Σ

=

+

Fig. 10. The electron self-energy operator with screening and hole–particle interaction included.

Σ

+

Σ

Σ

+

Σ

Σ

Σ

+

...

Fig. 11. Chaining of the self-energy operator.

polarization operator, with hole–particle interaction included, is found by using the Green’s function in Eq. (30). The Coulomb interaction, with screening and the hole–particle interaction included in all orders, is calculated from the matrix equation [85] Q˜ = Qˆ − iQˆ ;ˆQˆ :

(33)

This is depicted diagrammatically in Fig. 9. The series of diagrams representing the screening and hole–particle interaction can now be included into the correlation potential. This is done by introducing the renormalized Coulomb interaction (Fig. 9) and the polarization operator (Fig. 8) into the second-order diagrams according to Fig. 10. The screened second-order correlation corrections to the energies of S and P states of cesium are increased by 30% when the hole–particle interaction is taken into account in all orders. 4.4.3. Chaining of the self-energy The accuracy of the calculations can be further improved by taking into account the nonlinear contributions of the correlation potential 5ˆ (Fig. 11). The chaining of the correlation potential

84

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Fig. 12. Third-order diagrams of the interaction of a hole and particle from the loop with an external electron.

Fig. 13. Correlation corrections to occupied orbitals of closed shells.

(Fig. 10) to all orders is calculated by adding 5ˆ to the Hartree–Fock potential, Vˆ N −1 , and solving the equation (hˆ0 + 5ˆ − j) = 0

(34)

iteratively for the states of the external electron. The inclusion of 5ˆ into the SchrZodinger equation is what we call the “correlation potential method” and the resulting orbitals and energies “Brueckner” orbitals and “Brueckner” energies (see Section 4.3). Iterations of the correlation potential 5ˆ increase the contributions of 5ˆ (with screening and hole–particle interaction) to the energies of S and P states of cesium by about 10%. The %nal results for the energies are listed in Table 5. The inclusion of the three series of higher-order diagrams improves the accuracy of the calculations of the energies to the level of 0.1%. 4.5. Other low-order correlation diagrams Third-order diagrams for the interaction of a hole and particle in the polarization loop with an external electron are depicted in Fig. 12. These are not taken into account in the method described above. However, these diagrams are of opposite sign and cancel each other almost exactly [85]: the small and almost constant potential of a distant external electron practically does not in?uence the wave functions of the core and excited electrons in the loop; it shifts the energies of the core and excited electrons by the same amount. This cancellation was proved in the work [93] by direct calculation. Also, correlation corrections to the external electron energy arising from the inclusion of the self-energy into orbitals belonging to closed electron shells, depicted in Fig. 13, are small and can be safely omitted [71]. 4.6. Empirical
J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

85

considered as a way of including higher-order diagrams not explicitly included in the calculations. Comparison of quantities calculated with and without %tting can be used to test the stability of the wave functions and to estimate the contribution of unaccounted diagrams. 4.7. Asymptotic form of the correlation potential At large distances, the correlation potential 5ˆ approaches the local polarization potential [94], 5ˆ r →∞ ≈ −#e2 =2r 4 ;

(35)

where # is the polarizability of the core. This explains the universal behaviour of the correlation corrections to the energies of states of the external electron. 4.8. Interaction with external <elds In this section we will describe the procedures used to achieve high accuracy in the calculations of the interactions between atomic electrons and external %elds (in particular, we will present calculations for E1 transition amplitudes and hyper%ne structure (hfs) constants, arising from the interaction of the atomic electrons with the electric %eld of the photon and with the magnetic %eld of the nucleus, respectively). We need to calculate the e2ect of the external %eld on the wave functions of the core electrons (core polarization) and then take into account the e2ect of this polarization and the external %eld on the valence electron. This is achieved by using the time-dependent Hartree–Fock (TDHF) method. We describe this method in Section 4.8.1 and apply it to the calculation of E1 transition amplitudes and hfs constants in Sections 4.8.2 and 4.8.3, respectively. The dominant correlation corrections correspond to those diagrams in which the interactions occur in the external lines of the self-energy operator (“Brueckner-type corrections”). These diagrams are presented in Fig. 14; they are enhanced by the small energy denominator \jext corresponding to the excitation of the external electron in the intermediate states. The Brueckner-type corrections are calculated in a similar way to the correlation corrections to energy (see Sections 4.8.1–4.8.3). In Section 4.8.4 we describe the calculations of the remaining second-order corrections, i.e., “structural radiation” and normalization of states. Examples of structural radiation diagrams are presented in Fig. 15; in these diagrams the external %elds occur in the internal lines, and so their contributions are small due to the large energy denominators \jint corresponding to the excitation of the core electrons. The relative suppression of the contributions of structural radiation compared to Brueckner-type corrections is \jext =\jint ∼ 1=10. This section is largely based on the works [59,94].

Σ

Σ

Fig. 14. External %eld (denoted by a cross) in the external electron lines.

86

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Fig. 15. Structural radiation in diagram 2(a) of the self-energy operator.

4.8.1. Time-dependent Hartree–Fock method The polarization e2ects are taken into account by using the time-dependent Hartree–Fock method (see, e.g., [59,94] and references therein). We will consider how the RHF equations are modi%ed in the presence of a time-dependent %eld ˆ −i!t + fˆ † ei!t ) : hˆext = (fe

(36)

We can assume that the time-dependent single-particle orbitals are then given by ˜k = (

k

+ 7k e−i!t + )k ei!t ) ;

(37)

with corresponding eigenvalues (“quasi-energies”) −i!t i!t + %j(2) ; j˜k = jk + %j(1) k e k e

(38)

(2) are corrections to the RHF wave functions 7k , )k and %j(1) k , %jk ˆ induced by hext . The SchrZodinger equation 9 ˜ (hˆ0 ( ˜ ) + hˆext ) ˜ k = i k ; 9t can now be used to obtain equations for the corrections 7k and )k ,

(hˆ0 − jk − !)7k = −(fˆ + %Vˆext )

k

† (hˆ0 − jk + !))k = −(fˆ † + %Vˆext )

+ %jk k

k

and energies jk , respectively, (39)

k

+ %jk

k

;

(40)

(2) ˆ where we take into account terms up to %rst order; the energy shift %jk = %j(1) k = %jk =  k |f + N ˜ ˆ ˆ %Vˆext | k . The RHF Hamiltonian h0 ( ) corresponds to h0 with the Hartree–Fock potential Vˆ −1 calculated with the new core wave functions ˜ , and %Vˆext is the di2erence between the potential found in the external %eld and the RHF potential,

%Vˆext = Vˆ (N −1) ( ˜ ) − Vˆ (N −1) ( ) %Vˆext (r) = e

2

N −1
n=1



d 3 r1 [()† (r1 ) n (r1 ) + |r − r1 | n

− ()†n (r1 ) n (r) +

†

n (r1 )7n (r))

†

n (r1 )7n (r1 ))

(r1 )] :

(r) (41)

Eqs. (40), (41) should be solved self-consistently for the (N − 1) core electrons. The wave function of the external electron is then found in the %eld of the frozen core (the V N −1 approximation). So in the same way as in the RHF case, here we can %nd a complete set of orthonormal TDHF orbitals ˜ k with quasi-energy j˜k .

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

87

Fig. 16. Lowest-order (exchange) diagrams in TDHF; the cross denotes the external %eld.

The external electron transition amplitude M# from state # to state  induced by the %eld hˆext can be found by comparison of the amplitude obtained from Eqs. (37) and (40), 

˜

| #

=

 |7# e

−i!t

=



ˆ + %Vˆext | # e−i!t ; j# − j  + !

 |f

with conventional time-dependent perturbation theory, M# −i!t ˜# = # + ; e j# − j  + !

(42)

(43)

where only the resonant term (! ≈ j − j# ) is considered. Comparing Eqs. (43) and (42) gives [59,94] M# = 

ˆ + %Vˆext | # :

 |f

(44)

This formula corresponds to the well-known random-phase approximation (RPA) with exchange (see, e.g., [92]). Using the TDHF procedure described above, core polarization is included in all orders of perturbation theory. This is equivalent to summation of the diagram series presented in Fig. 16. The Brueckner-type correlation corrections are calculated in the same way as the corrections to energies. Brueckner, instead of RHF, orbitals are used for # and  in Eq. (44). This is equivalent to calculating the diagrams presented in Fig. 14 (with core polarization included in all orders in the external lines). However, it must be noted that by using this technique we neglect the diagrams in which the interaction takes place in the internal lines (see Fig. 15), although these diagrams give only small corrections; we will deal with these contributions in Section 4.8.4. Also in Section 4.8.4 we look at another second-order correction: the normalization of the many-body states. 4.8.2. E1 transition amplitudes The Hamiltonian of the electron interaction with the electric %eld E(t) = E0 (e−i!t + ei!t )

(45)

of an electromagnetic wave depends on the choice of gauge. In “length” form fˆ l = er · E0 and in ˆ −i!t + fˆ † ei!t . “velocity” form fˆ v = −ie( · E0 =!), where hˆE1 = fe It is known that in TDHF calculations the amplitude (44) is gauge invariant (see, e.g., [92,95,73]). Comparing results obtained from the two forms for the amplitudes (44) provides a test of the numerical calculation. In [94], the length and velocity forms were shown to give the same results for the E1 transition amplitudes when the correlation corrections (correlation potential, frequency shift in the velocity form operator, structural radiation, and normalization of states) are taken into account. Calculations using the dipole operator in length form are more stable than those obtained

88

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Table 6 Radial integrals of E1 transition amplitudes for Cs in di2erent approximations. The experimental values are listed in the last column (a.u.) Transition

RHF

TDHF

5ˆ (2) With %tting

5ˆ

5ˆ With %tting

6S − 6P1=2

6.464

6.093

5.499

5.497

5.509

7S − 6P1=2 7S − 7P1=2

5.405 13.483

5.450 13.376

5.198 12.602

5.190 12.601

5.204 12.612

a

Ref. Ref. c Ref. d Ref. e Ref. f Ref. g Ref. b

Experiment 5.5232(91)a , 5.4979(80)b , 5.5192(58)c , 5.512(2)d , 5.524(5)(1)e 5.185(27)f 12.625(18)g

[98]. [99]. [100]. Deduced from the van der Waals coeScient C6 . [101]. Deduced from photoassociation spectroscopy. [102]. Deduced from their measurement of the static dipole polarizability. [103]. [104].

in velocity form (see, e.g., [96,97,94]). For this reason we consider the calculations of E1 transition amplitudes in length form. As seen from Eq. (44), inclusion of the core polarization into the E1 transition amplitude is reduced to the addition of the operator %VˆE1 . Because the E1 operator can only mix opposite parity states, there is no energy shift, and so when calculating the wave function corrections 7k and )k from Eq. (40), %jk = 0. The results for E1 transition amplitudes calculated in length form between the lower states of cesium are presented in Table 6. Another test of the accuracy and self-consistency of the TDHF equations is the value of an external electric %eld at the nucleus. According to the Schi2 theorem [90], an external static electric %eld (! = 0 in TDHF) at the nucleus in a neutral atom is shielded completely by atomic electrons. For an atom with charge Zi the total static electric %eld Etot at the nucleus is [91] Etot (0) = E0 + Ee (0) = E0 Zi =Z ;

(46)

where E0 is the external %eld and Ee is the induced electron %eld. TDHF equations reproduce this result correctly. The oscillations of the electric %eld inside the atom are quite complex. We refer the reader to [91] for a plot of the electric %eld inside Tl+ . 9 4.8.3. Hyper
×r r3

Note that in the paper [91] the %gures and the captions have been switched.

(47) (48)

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89

is the vector potential created by the nuclear magnetic moment  and  is the Dirac matrix. When performing the calculation of the hyper%ne structure of heavy atoms, it is important to take into account the %nite size of the nucleus. Using a simple model in which the nucleus represents a uniformly magnetized ball, the hyper%ne interaction is given by  r × =rm3 ; r ¡ rm f(r) = (49) hˆhf = e · f(r); r × =r 3 ; r ¿ rm where the magnetic radius is taken to be rm = 1:1A1=3 fm (A is the mass number of the nucleus). Note that while the distribution of currents in the nucleus (produced by unpaired nucleons) is very complex, the hfs is only weakly dependent on its form (see, e.g., [105]). Corrections to the RHF wave functions induced by the hyper%ne interaction are calculated using Eq. (40). There is no time-variation in the hyper%ne interaction, and so we set ! = 0. The hfs equations are then (hˆ0 − j)% = −(hˆhf + %Vˆhf ) + %j

(50)

%j =  |hˆhf + %Vˆhf | :

(51)

The hyper%ne interaction does not alter the direct contribution to the core potential, so %Vˆhf = Vˆexch ( ˜ ) − Vˆexch ( ). We use the Vˆ N −1 approximation to calculate the complete set of orbitals. The external electron correction %j determines the atomic hyper%ne structure. Hyper%ne structure results for cesium are presented in Table 7. The correlations increase the density of the external electron in the nuclear vicinity by about 30%. An accurate inclusion of the correlations is therefore very important when considering interactions singular on the nucleus, for example the PNC weak interaction. 4.8.4. Structural radiation and normalization of states “Structural radiation” is the term we use for correlation corrections with the external %eld in the internal lines. (We will make a further distinction when we are dealing with the PNC E1 amplitude EPNC , since here there are two %elds. We call correlation corrections with the weak interaction in the internal lines “weak correlation potential”; see Section 5.1.) Examples of diagrams which Table 7 Calculations of the hyper%ne structure of Cs in di2erent approximations. In the last column the experimental values are listed (Units: MHz) State

RHF

TDHF

5ˆ (2) With %tting

5ˆ

5ˆ With %tting

Experiment

6S 7S 6P1=2 7P1=2

1425.0 391.6 160.9 57.6

1717.5 471.1 200.3 71.2

2306.9 544.4 291.5 94.3

2315.0 545.3 293.6 94.8

2300.3 543.8 290.5 94.1

2298.2a 545.90(9)b 291.89(8)c 94.35a

a

Ref. [106]. Ref. [107]. c Ref. [108]. b

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represent structural radiation are presented in Fig. 15. In second-order perturbation theory there is another contribution which arises due to the change of the normalization of the single-particle wave functions due to admixture with many-particle states. For the E1 transition amplitudes in length form, the following approximate formula can be used to calculate structural radiation [94], 95ˆ 1 95ˆ + D| # ; Mstr = −   |D (52) 2 9j 9j where D ≡ dˆ + %Vˆ is the E1 operator. The derivation can be found in Ref. [94]. We refer the interested reader to Refs. [94,59] for structural radiation in the velocity gauge. The normalization contribution for E1 transitions has the form [94,59]   1 95ˆ 95ˆ Mnorm =   |D| #   | |  +  # | | # : (53) 2 9j 9j The structural radiation and normalization contributions to the hyper%ne structure and E1 transition amplitudes of low-lying states of cesium are small. Their combined contribution, for both hfs and E1 amplitudes, usually lies in the range 0:1 − 1:0%. 5. High-precision calculation of parity violation in cesium and extraction of the nuclear weak charge In this section the method described in the preceding section is applied to the 6S − 7S parity violating amplitude in cesium and the value for the nuclear weak charge is extracted from the measurement of Wood et al. [9]. The value for this amplitude has been the source of much confusion recently. It has jumped around erratically in the last few years, and %nally it has stabilized. The origin of this instability is described below. In 1999, experimentalists Bennett and Wieman argued that the standard model is in contradiction with atomic experiments by 2.5  [83]. This conclusion was mainly based on their analysis of the accuracy of atomic structure calculations which were published in Ref. [62] in 1989 and in Ref. [19] in 1990, 1992. The point is that the new measurements of electromagnetic amplitudes in atoms have demonstrated that the accuracy of the atomic calculations is much better than it seemed to be ten years ago. Indeed, all disagreements between theory and experiment were resolved in favour of theory. Based on this, Bennett and Wieman reduced the theoretical error from the 1% claimed by theorists [62,19] to 0.4% and came to the conclusion that there may be new physics beyond the standard model. Particle physicists put forward new physics possibilities, such as an extra Z boson in the weak interaction, leptoquarks, and composite fermions; see, e.g., Refs. [109]. However, the 1% error placed on the atomic calculations [62,19] was based not only on the comparison of measured and calculated quantities such as E1 transition amplitudes. It was based also on an allowance for unaccounted contributions. It was soon after discovered that the Breit contribution is larger than expected (−0:6%) [110] (Section 5.1.2). Then it was suggested that strong-%eld radiative corrections could make a contribution as large as Breit [111]: the Uehling contribution has been found to give a contribution of 0.4% [112,113] (Section 5.1.4); and just

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91

recently it has been established that self-energy and vertex contributions are ≈ −0:8% [114,115] (Section 5.1.4). A recent comprehensive calculation of atomic PNC is accurate to 0.5% [10]. The results of this calculation, and the discussion of accuracy, are presented in the following sections. The %nal value for parity violation in cesium is in agreement with the standard model and tightly constrains possible new physics (Section 5.3). 5.1. High-precision calculations of parity violation in cesium  The weak interaction Hˆ W = Ni hˆiW [Eq. (5)] 10 mixes atomic wave functions of opposite parity and leads to small opposite-parity admixtures in atomic states >, >˜ = > + %>. This gives rise to E1 transitions between states of the same nominal parity. The parity violating 6S − 7S E1 transition amplitude in Cs is Hˆ E1 |6S = %(7S)|Hˆ E1 |6S + 7S|Hˆ E1 |%(6S) : EPNC = 7S|

(54)

Calculations of PNC E1 amplitudes can be performed using the following approaches: from a mixed-states approach, in which there is a small opposite-parity admixture in each state [Eq. (54)]; or from a sum-over-states approach, in which the amplitude [Eq. (54)] is broken down into contributions arising from opposite-parity admixtures and a direct summation over the intermediate states is performed [Eq. (55)]. In the sum-over-states approach, the Cs 6S − 7S PNC E1 transition amplitude is written in terms of a sum over intermediate, many-particle states NP1=2


7S|Hˆ E1 |NP1=2 NP1=2 |Hˆ W |6S 7S|Hˆ W |NP1=2 NP1=2 |Hˆ E1 |6S : (55) EPNC = + E6S − ENP1=2 E7S − ENP1=2 N There are three dominating contributions to this sum: EPNC =

7S|Hˆ E1 |6P1=2 6P1=2 |Hˆ W |6S 7S|Hˆ W |6P1=2 6P1=2 |Hˆ E1 |6S + E6S − E6P1=2 E7S − E6P1=2 +

7S|Hˆ E1 |7P1=2 7P1=2 |Hˆ W |6S + ··· E6S − E7P1=2

= 1:908 − 1:493 − 1:352 + · · · = −0:937 + · · · ;

(56)

the units are 10−11 ieaB (−QW =N ). The numbers are from the work [19] where the sum-over-states method was used; here we just demonstrate that these terms dominate. An advantage of the sum-overstates approach is that experimental values for the energies and E1 transition amplitudes can be explicitly included into the sum. This was the procedure for some of the early calculations of PNC in Cs (see, e.g., [77]). 10

It is seen from Eqs. (3), (4), by inserting the coeScients C1N , that the density !(r) is essentially the (poorly understood) neutron density in the nucleus. In the calculations, !(r) will be taken equal to the charge density, Eq. (21), and then in Section 5.1.3 we will consider the e2ect on the PNC E1 amplitude as a result of correcting for !(r).

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If one neglects con%guration mixing, the sum can be represented in terms of single-particle states; in this case, the sum also runs over core states (corresponding to many-particle states with a single core excitation),


7s|hˆE1 |np1=2 np1=2 |hˆW |6s 7s|hˆW |np1=2 np1=2 |hˆE1 |6s EPNC = : (57) + j6s − jnp1=2 j7s − jnp1=2 n In Refs. [62,19,64,10] PNC calculations were performed in the mixed-states approach, and in Ref. [19] a calculation was carried out in the sum-over-states approach also. Here we refer to the most precise calculations, 6 1% accuracy. 5.1.1. Mixed-states calculation In the TDHF method (Section 4.8.1, Eq. (37)), a single-electron wave function in external weak and E1 %elds is =

0

+ % + X e−i!t + Y ei!t + %X e−i!t + %Y ei!t ;

(58)

where 0 is the unperturbed state, % is the correction due to the weak interaction acting alone, X and Y are corrections due to the photon %eld acting alone, and %X and %Y are corrections due to both %elds acting simultaneously. These corrections are found by solving self-consistently the system of the TDHF equations for the core states (hˆ0 − j)% = −(hˆW + %VˆW )

;

(59)

(hˆ0 − j − !)X = −(hˆE1 + %VˆE1 )

;

(60)

† (hˆ0 − j + !)Y = −(hˆ†E1 + %VˆE1 )

;

(61)

(hˆ0 − j − !)%X = −%VˆE1 % − %VˆW X − %VˆE1W

;

(62)

† † (hˆ0 − j + !)%Y = −%VˆE1 % − %VˆW Y − %VˆE1W

;

(63)

where %VˆW and %VˆE1 are corrections to the core potential due to the weak and E1 interactions, respectively, and %VˆE1W is the correction to the core potential due to the simultaneous action of the weak %eld and the electric %eld of the photon. The TDHF contribution to EPNC between the states 6S and 7S is given by TDHF = EPNC

ˆ

7s |hE1

+ %VˆE1 |%

6s

+

ˆ + %VˆW |X6s + 

7s |hW

ˆ

7s |%VE1W | 6s

:

(64)

The corrections % 6s and X6s are found by solving Eqs. (59), (60) in the %eld of the frozen core (of course, amplitude (64) can instead be expressed in terms of corrections to 7s ). Now we need to include the correlation corrections to the PNC E1 amplitude. In the previous sections (Sections 4.3, 4.4, 4.8.4) we have discussed two types of corrections: the dominant Brueckner-type corrections, represented by diagrams in which the external %eld appears in the external electron line (see Fig. 17); and structural radiation, in which the external %eld acts on an internal electron line. In the case of PNC E1 amplitudes, in order to distinguish between structural radiation diagrams with di2erent %elds, we refer to diagrams with the weak interaction attached to

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Σ

Σ

93

Σ

Σ

Fig. 17. Brueckner-type correlation corrections to the PNC E1 transition amplitude in %rst order in the weak interaction; the crosses denote the weak interaction and the dashed lines denote the electromagnetic interaction.

δΣ

δΣ

Σ

Σ

δΣ

δΣ

(a)

(b)

(c)

Fig. 18. External %eld inside the correlation potential. In diagrams (a) the weak interaction is inside the correlation potential (%5 denotes the change in 5 due to the weak interaction); this is known as the weak correlation potential. Diagrams (b,c) represent the structural radiation (photon %eld inside the correlation potential). In diagram (b) the weak interaction occurs in the external lines; in diagram (c) the weak interaction is included in the electromagnetic vertex.

the internal electron line as “weak correlation potential” diagrams. Structural radiation and the weak correlation potential diagrams are presented in Fig. 18. We will consider %rst the dominating Brueckner-type corrections to the E1 PNC amplitude. ˆ Remember that the correlation potential is energy-dependent, 5ˆ = 5(j). This means that the 5ˆ operators for the 6s and 7s states are di2erent. We should consider the proper energy-dependence at least in %rst-order in 5ˆ (higher-order corrections are small and the proper energy-dependence is not important for them). The %rst-order in 5ˆ correction to EPNC is presented diagrammatically in Fig. 17. We can write this as 

ˆ

7s |5s (j7s )|%X6s

+ %

ˆ

7s |5p (j7s )|X6s

+ %Y7s |5ˆ s (j6s )|

6s

+ Y7s |5ˆ p (j6s )|%

6s

:

(65)

The nonlinear in 5ˆ contribution to the Brueckner-type correction is found using the correlation potential method (Section 4.3): the all-orders in 5ˆ contribution is calculated and from this the %rst-order contribution, found in the same method, is subtracted. The all-orders term is calculated

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

using external electron orbitals, and corrections to these orbitals induced by the weak interaction ˆ The PNC E1 amplitude is then calculated, and the photon %eld, found in the potential Vˆ N −1 + 5. using these new orbitals, in the same way as in the usual time-dependent Hartree–Fock method. The all-orders contribution to EPNC is ˆ − E (Vˆ N −1 ) : E all-orders = E (Vˆ N −1 + 5) (66) PNC

PNC

PNC

The %rst-order in 5ˆ contribution is found by placing a small coeScient a before the correlation ˆ When a1, the linear in 5ˆ contribution to EPNC dominates. Its extrapolation to potential, 5ˆ → a5. a = 1 gives the %rst-order in 5ˆ contribution. So the nonlinear in 5ˆ contribution to EPNC is [116] non-lin ˆ − EPNC (Vˆ N −1 )] − 1 [EPNC (Vˆ N −1 + a5) ˆ − EPNC (Vˆ N −1 )] : = [EPNC (Vˆ N −1 + 5) (67) EPNC a To complete the calculation of corrections second-order in the residual Coulomb interaction the weak correlation potential, structural radiation, and normalization contributions to the PNC amplitude must be included. The weak correlation potential is calculated by direct summation over intermediate states. See Section 4.8.4 for the approximate form for structural radiation in length form and for the form for the normalization of the many-body states. Due to parity violation there is an opposite-parity correction to the orbitals # ≡ # and  ≡  , #˜ = # + %# and ˜ =  + %, and to the correlation ˆ 5˜ = 5ˆ + %5. ˆ potential 5, Structural radiation is then given by ˜ ˜ ˜ 95 + 95 D|# ˜ str = − 1 |D ˜ : (68) M 2 9j 9j There are two contributions to structural radiation for the PNC E1 amplitude: one in which the electromagnetic vertex is parity conserving, the weak interaction included in the external lines: ˆ ˆ ˜ 95 + 95 D|# ˜ Fig: 18(b) = − 1 |D M ˜ (69) 2 9j 9j (see diagram (b) Fig. 18); and the other in which the weak interaction is included in the electromagnetic vertex (we call this structural radiation and not weak correlation potential): ˜ ˜ ˜ Fig: 18(c) = − 1 |D 95 + 95 D|# (70) M 2 9j 9j (see diagram (c) of Fig. 18). Note that in each case the amplitude %rst-order in the weak interaction is considered. The normalization contribution is   ˆ ˆ 9 5 9 5 1 ˜ # ˜ norm = |D| ˜ | | + #| |# : (71) M 2 9j 9j The results of the calculation [10] for the 6S −7S PNC amplitude are presented in Table 8. Taking into account all corrections discussed in this section, the following value is obtained for the 6S − 7S PNC amplitude in cesium EPNC = −0:9078 × 10−11 ieaB (−QW =N ) :

(72)

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95

Table 8 Contributions to the 6S − 7S EPNC amplitude for Cs in units −10−11 ieaB (−QW =N ) (5ˆ corresponds to the (un%tted) “dressed” self-energy operator) TDHF Brueckner-type correlations  7s |5ˆ s (j7s )|%X6s  % 7s |5ˆ p (j7s )|X6s  %Y7s |5ˆ s (j6s )| 6s  Y7s |5ˆ p (j6s )|% 6s  Nonlinear in 5ˆ correction Weak correlation potential Structural radiation Normalization Subtotal Breit Neutron distribution correction QED radiative corrections Vacuum polarization (Uehling) Self-energy and vertex Total

0.8898 0.0773 0.1799 −0.0810 −0.1369 −0.0214 0.0038 0.0029 −0.0066 0.9078 −0.0055 −0.0018 0.0036 −0.0072 0.8969

This corresponds to “Subtotal” of Table 8. This is in agreement with the 1989 result [62]. Notice the stability of the PNC amplitude. The time-dependent Hartree–Fock value gives a contribution to the total amplitude of about 98%. The point is that there is a strong cancellation of the correlation corrections. The mixed-states approach has also been performed in [19,64] to determine the PNC amplitude in cesium. However, in these works the screening of the electron–electron interaction was included in a simpli%ed way. In [19] empirical screening factors were placed before the second-order correlation corrections 5ˆ (2) to %t the experimental values of energies. Kozlov et al. [64] introduced screening factors based on average screening factors calculated for the Coulomb integrals between valence electron states. The results obtained by these groups (without the Breit interaction, i.e., corresponding to the Subtotal of Table 8) are 0.904 [19] and 0.905 [64]. 11 As a check, a pure second-order (i.e., using 5ˆ (2) ) calculation with energy-%tting was also performed in [10] (in the same way as [19]), and the result 0.904 was reproduced. Contributions of the Breit interaction, the neutron distribution, and radiative corrections to EPNC are considered in the following sections.

11

The numbers di2er from those presented in Table 4 due to the Breit interaction. In [19] a value for Breit of −0:2% of the PNC amplitude was included (this value was underestimated), while in [64] the magnetic (Gaunt) part of the Breit interaction was included and calculated to be −0:4%. See Section 5.1.2.

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5.1.2. Inclusion of the Breit interaction The Breit interaction is a two-particle operator 2

e Hˆ Breit = − 2


i · j + (i · nij )(j · nij ) i¡j

|ri − rj |

;

(73)

 are Dirac matrices, nij = (ri − rj )=|ri − rj |. It gives magnetic (Gaunt) and retardation corrections to the Coulomb interaction. A few years ago it was thought that the correction to EPNC arising due to inclusion of the Breit interaction in the Hamiltonian (15) is small (safely smaller than 1%). In the work [62] the Breit interaction was neglected, and in [19] it was only partially calculated. (Remember that these works claimed an accuracy of 1%.) The huge improvement in the experimental precision of the cesium PNC measurement in 1997 [9] and the claim of Bennett and Wieman in 1999 [83] that the theoretical accuracy is 0.4% prompted theorists to revisit their calculations. Naturally this also involves a consideration of previously neglected contributions which, while at the 1% level could be neglected, are signi%cant at the 0.4% level. Derevianko [110] calculated the contribution of the Breit interaction to EPNC and found that it is larger than had been expected. Its contribution to EPNC is −0:6%. This result has been con%rmed by subsequent calculations [117,64,10]. 5.1.3. Neutron distribution The weak Hamiltonian Eq. (5) was used to obtain the result Eq. (72) with !(r) taken to be the charge density, parametrized according to Eq. (21). However, as we mentioned in a footnote at the beginning of Section 5.1, the weak interaction is sensitive to the distribution of neutrons in the nucleus. Here we look at the e2ect of correcting for the neutron distribution. For the neutron density the two-parameter Fermi model (21) is used. The result of Ref. [118] was used in [10] for the di2erence \rnp = 0:13(4) fm in the root-mean-square radii of the neutrons rn2 1=2 and protons rp2 1=2 . Three cases which correspond to the same value of rn2 were considered: (i) cn = cp , an ¿ ap ; (ii) cn ¿ cp , an ¿ ap ; and (iii) cn ¿ cp , an = ap (using the relation rn2 ≈ 3 2 c + 75 ;2 a2n ). It is found that EPNC shifts from −0:18% to −0:21% when moving from the extreme 5 n cn = cp to the extreme an = ap . Therefore, EPNC changes by about −0:2% due to consideration of the neutron distribution. This is in agreement with Derevianko’s estimate, −0:19(8)% [119]. 5.1.4. Strong-<eld QED corrections It was noted in Ref. [111] that corrections to the PNC amplitude due to vacuum polarization by the strong Coulomb %eld of the nucleus could be comparable in size to the Breit correction. This has been con%rmed by calculations, the strong-%eld radiative corrections associated with the Uehling potential (vacuum polarization) increase EPNC by 0:4% [113,112,10]. In Ref. [120] it was pointed out that the self-energy correction can give a larger contribution to 133 Cs PNC with opposite sign (∼ −0:65%). The self-energy and vertex corrections were %rst calculated in [114] and found to be −0:73(20)% for 133 Cs. The relation between the PNC correction and radiative corrections to %nite nuclear size energy shifts was used in this work. This result was con%rmed in direct analytical calculations using Z# expansion [121,115,122] and by all-orders in Z# numerical calculations of the PNC matrix element of the 2S1=2 − 2P1=2 transition in hydrogenic ions performed in [123].

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Note that corrections occur at very small distances (r . 1=m) where the nuclear Coulomb %eld is not screened and the electron energy is negligible. Therefore, the relative radiative corrections to weak matrix elements in neutral atoms like Cs are approximately the same as for the 2S1=2 − 2P1=2 transition in hydrogenic ions. There is good agreement between the di2erent calculations for all values of Z; see [124] and the review [125]. For the strong-%eld self-energy and vertex contribution to PNC in 133 Cs we will quote the value −0:8%, which is the average value of [114,122] corresponding also to the value obtained in [123]. Above we discussed the radiative corrections to the weak matrix elements. However, the sum-overstates expression for the PNC amplitude contains also energy denominators and E1 electromagnetic amplitudes. It was shown in [120,10] that for Cs the corrections to the energies (−0:3%) and E1 matrix elements (+0:3%) cancel. The contributions of strong-%eld radiative corrections to EPNC of cesium are listed in Table 8. 5.1.5. Tests of accuracy There are two main methods used to estimate the accuracy of the PNC amplitude EPNC : (i) root-mean-square (rms) deviation of the calculated energy intervals, E1 amplitudes, and hyper%ne structure constants from the accurate experimental values; (ii) in?uence of %tting of energies and hyper%ne structure constants on the PNC amplitude. The PNC amplitude can be expressed as a sum over intermediate states. Notice that there are three dominating contributions to the 6S − 7S PNC amplitude in Cs; see Eq. (56). Each term in the sum is a product of E1 transition amplitudes, weak matrix elements, and energy denominators. Therefore, this amplitude is sensitive to the electron wave functions at all distances. (The weak matrix elements, energies, and E1 amplitudes are sensitive to the wave functions at small, intermediate, and large distances from the nucleus, respectively.) While mixed-states calculations of PNC amplitudes do not involve a direct summation over intermediate states, it is instructive to analyse the accuracy of the weak matrix elements, energy intervals, and E1 transition amplitudes which contribute to Eq. (56) calculated using the same method as that used to calculate EPNC . The accuracy of these quantities is determined by comparing the calculated values with experiment. Note that we cannot directly compare weak matrix elements with experiment. However, like the weak matrix elements, hyper%ne structure is determined by the electron wave functions in the vicinity of the nucleus, and this is known very accurately. In Section 4 we presented calculations of the energies, E1 transition amplitudes, and hyper%ne structure constants relevant to Cs 6S − 7S EPNC . The calculated removal energies are presented in Table 5. The Hartree–Fock values deviate from experiment by 10%. Including the second-order correlation corrections 5ˆ (2) reduces the error to ∼ 1%. When screening and the hole–particle interaction are included into 5ˆ (2) in all orders, the energies improve, 0:2 − 0:3%. The rms deviation between the calculated and experimental energy intervals E6S − E6P1=2 , E7S − E6P1=2 , and E6S − E7P1=2 is 0.3%. We mentioned at the end of Section 4.1 that the experimental values for energies can be %tted ˆ The stability of the amplitude EPNC exactly by placing a coeScient before the correlation potential 5. (as well as the E1 amplitudes and hfs constants) with %tting gives us an indication of the size of omitted contributions. Note that the accuracy for the energies is already very high and the remaining discrepancy with experiment is of the same order of magnitude as the Breit and radiative corrections.

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Therefore, generally speaking, we should not expect that %tting of the energy will always improve the results for amplitudes and hyper%ne structure. In fact, as we will see below, some values do improve while others do not. The overall accuracy, however, remains at the same level. ˆ Below we present results for EPNC obtained in three di2erent approximations: with un%tted 5, (2) 12 ˆ ˆ and with 5 and 5 %tted with coeScients to reproduce experimental removal energies. First, we analyse the E1 transition amplitudes and hfs constants calculated in these approximations. The relevant E1 transition amplitudes (radial integrals) are presented in Table 6. These are calculated with the energy-%tted “bare” correlation potential 5ˆ (2) and the (un%tted and %tted) “dressed” ˆ Structural radiation and normalization contributions are also included. The rms devipotential 5. ations of the calculated E1 amplitudes from experiment are the following: without energy %tting, the rms deviation is 0.1%; %tting the energy gives a rms deviation of 0.2% for 5ˆ (2) and 0.3% for ˆ Note, these correspond to the deviations between the calculations and the central the complete 5. points of the measurements. The errors associated with the measurements are in fact comparable to this di2erence. So it is unclear if the theory is limited to this precision or is in fact much better. Regardless, the uncertainty in the theoretical accuracy remains the same. The hyper%ne structure constants calculated in di2erent approximations are presented in Table 7. Corrections due to the Breit interaction, structural radiation, and normalization are included. The rms deviation of the calculated hfs values from experiment using the un%tted 5ˆ is 0.5%. With %tting, the rms deviation in the pure second-order approximation is 0.3%; with higher orders it is 0.4%. The point is to estimate the accuracy  of the s − p1=2 weak matrix elements. It seems reasonable then to use the square-root formula, hfs(s)hfs(p1=2 ). Notice that by using this approach the deviation is smaller. Without energy %tting, the rms deviation is 0.5%. With %tting, the rms deviation in the ˆ it is 0.3%. second-order calculation (5ˆ (2) ) is 0:2% and in the full calculation (5) From the above consideration it is seen that the rms deviation for the relevant parameters is 0:5% or better. Note that from this analysis the error for a sum-over-states calculation of EPNC would be larger than this, as the errors for the energies, hfs constants, and E1 amplitudes contribute to each of the three terms in Eq. (56). However, in the mixed-states approach, the errors do not add in this way. We now consider calculations of the PNC amplitude performed in [10] in di2erent approximations ˆ and with energy-%tted 5ˆ (2) and 5). ˆ The spread of the results can be used to estimate (with un%tted 5, the error. The results are listed in Table 9. It can be seen that the PNC amplitude is very stable. The PNC amplitude is much more stable than hyper%ne structure. This can be explained by the much smaller correlation corrections to EPNC (∼ 2% for EPNC and ∼ 30% for hfs; compare Table 8 with Table 9 Values for EPNC in di2erent approximations; units −10−11 ieaB (−QW =N )

EPNC

5ˆ (2) With %tting

5ˆ

5ˆ With %tting

0.901

0.904

0.903

Note that %tting 5ˆ (2) is an empirical method to estimate screening corrections (which were accurately calculated in ˆ Agreement between results with %tted 5ˆ (2) and ab initio 5ˆ shows that the %tting procedure is a reasonable way to 5). estimate omitted diagrams. 12

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Table 7). One can say that this small value of the correlation correction is a result of cancellation of di2erent terms in (65) but each term is not small (see Table 8). However, this cancellation has a regular behaviour. The stability of EPNC may be compared to the stability of the usual electromagnetic amplitudes where the error is very small (even without %tting). 13 In [10] the %tting of hyper%ne structure was also considered, using di2erent coeScients before ˆ The %rst-order in 5ˆ correlation correction (65) changes by about 10%. It was found that each 5. the PNC amplitude changes by about 0.4%. It is also instructive to look at the spread of EPNC obtained in di2erent schemes. The result of the work [10] (the number we present here) is in excellent agreement with the earlier result [62]. The only other calculation of EPNC in Cs that is as complete as [62,10] is that of Blundell et al. [19]. Their result in the all-orders sum-over-states approach is 0.909 (without Breit) and is very close to the value of 0.908 (corresponding to “Subtotal” of Table 8). A note on the sum-over-states procedure. The authors of Ref. [19] include single, double, and selected triple excitations into their wave functions. Note, however, that even if wave functions of 6S, 7S, and intermediate NP states are calculated exactly (i.e., with all con%guration mixing included) there are still some missed contributions in this approach. Consider, e.g., the intermediate state 6P ≡ 5p6 6p. It contains an admixture of states 5p5 ns6d: 6P = 5p6 6p + #5p5 ns6d + · · · . This mixed state is included into the sum (55). However, the sum (55) must include all many-body states of opposite 5 ns6d = 5p5 ns6d − #5p6 6p + · · · should also be included into parity. This means that the state 5p] the sum. Such contributions to EPNC have never been estimated directly within the sum-over-states approach. However, they are included into the mixed-states calculations [62,19,64,10]. It is important to note that the omitted higher-order many-body corrections are di2erent in the sum-over-states [19] and mixed-states [62,10] calculations. This may be considered as an argument that the omitted many-body corrections in both calculations are small. Of course, here it is assumed that the omitted many-body corrections to both values (which, in principle, are completely di2erent) do not “conspire” to give exactly the same magnitude. A comparison of calculations of EPNC in second-order with %tting of the energies is also useful in determining the accuracy of the calculations of EPNC . (Remember that this value is in agreement with results of similar calculations performed in [19,64]; see Section 5.1.1.) One can see that replacing the all-order 5ˆ by its very rough second-order (with %tting) approximation changes EPNC by less than 0.4% only. On the other hand, if the higher orders are included accurately, the di2erence between the two very di2erent approaches is 0.1% only. The maximum deviation obtained in the above analysis is 0.5%. This is the error claimed in the EPNC calculation [10]. 5.2. The vector transition polarizability The determination of the nuclear weak charge from the Stark–PNC interference measurements also requires knowledge of the vector transition polarizability . This can be found in a number

13

Note that di2erent methods also give di2erent signs of the errors for hfs. This is one more argument that the true value of EPNC is somewhere in the interval between the results of di2erent calculations in Table 9.

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of ways: (i) from a direct calculation of .  can be expressed as a sum over intermediate states and experimental E1 transition amplitudes and energies can be used [17] (see also [19,126]). However, this calculation is unstable due to strong cancellations of di2erent terms in the sum (see Ref. [126]). These cancellations are explained by the fact that  is proportional to the spin–orbit interaction, therefore for zero spin–orbit interaction the sum for  must be zero; (ii) from the measurement of the ratio of the o2-diagonal hyper%ne amplitude to the vector transition polarizability, Mhfs = [127].  is then extracted from the ratio using a theoretical determination of Mhfs ; (iii) from the measurement of the ratio of the scalar to vector polarizabilities, #=. # can be calculated accurately using experimental values for E1 transition amplitudes and energies in the sum-over-states approach (the calculation of # is much more stable than that of  [126]). There are currently two very precise determinations of . One was obtained from the analysis [128] (calculation of Mhfs ) of the measurement [83] of the ratio Mhfs =,  = 26:957(51)a3B , and another is from an analysis [10] (semi-empirical calculation of #; see [126] for details, where a similar calculation was performed) of the measurement [129] of the ratio #= using the most accurate experimental data for E1 transition amplitudes including the recent measurements of Ref. [130],  = 27:15(11)a3B . An average of these values gives  = 26:99(5)a3B :

(74)

5.3. The
(75)

(76)

(from the calculation [10] with the averaged value −0:8% of works [114,115] for the self-energy and vertex radiative corrections) and the averaged value for  [Eq. (74)], gives QW = −72:74(29)exp (36)theor

(77)

for the value of the nuclear weak charge for 133 Cs. The di2erence between this value and that SM 133 predicted by the standard model, QW (55 Cs) = −73:19 ± 0:13 [131], 14 is SM = 0:45(48) ; \QW ≡ QW − QW

(78)

adding the errors in quadrature. Let us brie?y consider the constraints on physics beyond the standard model set by the nuclear weak charge of cesium [Eq. (78)]. New physics corrections to the standard model are divided into 14

We use this value rather than the particle data group value, Eq. (7), since we use the new physics analysis of Ref. [131].

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two groups, those that originate through vacuum polarization corrections to gauge boson propagators and those that originate from all other mechanisms (such as new physics arising through vertex and self-energy diagrams and new tree-level physics). The former are termed “oblique” and the latter “direct”. A formalism for oblique corrections has been devised by Peskin and Takeuchi [132] in terms of weak isospin conserving and weak isospin breaking parameters S and T , respectively. Atomic parity violation (on a single isotope) is distinct from other electroweak probes of new physics in its almost exclusive dependence on the parameter S. Indeed, the dependence of the nuclear weak charge on the parameter T cancels almost exactly. For 133 Cs [131] oblique \QW = −0:800S − 0:007T :

(79)

The standard model value corresponds to S = T = 0 (no new physics) at values mt = 174:3 GeV for the top quark mass and MH = 100 GeV for the Higgs boson mass. 15 The constraint on S from PNC in 133 55 Cs [comparing Eqs. (78) and (79)] is S = −0:56(60) :

(80) new \QW

In terms of direct new physics, a positive [Eq. (78)] could be indicative of an extra Z boson in the weak interaction. A lower bound for the Z7 boson mass predicted in SO(10) theories can be obtained from the deviation of the measured weak charge from theory, according to [134] tree \QW ≈ 0:4(2N + Z)(MW =MZ7 )2 :

To one standard deviation, the lower bound on the mass MZ7 from parity violation in MZ7 ¿ 750 GeV :

(81) 133 55 Cs

is (82)

A lower bound of 595 GeV at 95% con%dence level has been obtained from a direct search at the Tevatron [135]. For a discussion of atomic physics sensitivities to new physics, see, e.g., [136,137,131] and references therein. For a recent analysis of electroweak tests of the standard model, including atomic parity violation, see, e.g., [131,138,139]; an earlier review on this topic [140] is also very informative. 5.4. Ongoing/future studies of PNC in atoms with a single valence electron A PNC measurement in cesium is continuing at Paris [48], where ∼ 1% precision is expected to be reached. The possibility of performing PNC measurements on the single trapped ions Ba+ and Ra+ is being considered at Seattle [26,141]. Preliminary atomic PNC calculations [142] indicate that the accuracy of calculations for Ba+ could compete with that for cesium. An experiment to measure PNC in (radioactive) francium was discussed at Stony Brook [143,144] and is currently being investigated at TRIUMF. Francium is a heavier analogue of cesium, and correspondingly the PNC e2ect is an order of magnitude (18 times [116]) larger. A new facility (TRIP) is under development at the Kernfysisch Versneller Instituut, Groningen. One aim is to measure parity violation in radioactive atoms and ions, attractive candidates being Ra+ and Fr; see Ref. [145]. The group at Paris is also considering the possibilities of PNC measurements with radioactive alkali atoms [146]. 15

The result is actually not very sensitive to MH which is currently MH ¿ 114:4 GeV at the 95% con%dence level [133].

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Note that the atomic theory is limited by the error associated with calculations of correlation corrections. It’s expected that the relative corrections due to correlations are the same in analogous atoms, e.g., cesium and rubidium. Thus, taking ratios of PNC measurements in analogous atoms possibly provides a way to circumvent the troublesome correlations in the determination of the ratio of the nuclear weak charges. [Another way to exclude atomic theory is through isotope ratios, see Section 6.2.] Light atoms by themselves may provide some advantage from the point of view of atomic theory. Breit, neutron skin, and strong-%eld QED corrections for rubidium are much smaller than for cesium. Therefore, atomic PNC calculations for rubidium could, in principle, reach a precision better than for cesium, however the amplitude is much smaller [76,59], about 6 times [59]. Various proposals to measure PNC in highly charged ions have been entertained for some time, however only now are high-precision measurements in these systems becoming feasible due to advances in the production and manipulation of highly charged ions [147]. 16 In hydrogen-like ions the close 2S1=2 , 2P1=2 levels are exploited, while in helium-like ions the crossing of opposite-parity levels at certain values of Z is taken advantage of. A very attractive feature of highly charged ions is that their sensitivity to stray electric %elds decreases with increase in Z; 17 in the hydrogen measurements 18 stray electric %elds strongly limited the determination of PNC. A brief review of proposals of PNC in hydrogen-like ions can be found in Ref. [149,147], including the recent proposal to study PNC in relativistic hydrogenic ions. For a review of proposals of PNC studies in helium-like ions, see, e.g., [46]; see also the more recent proposal of Ref. [150].

6. Atoms with several electrons in un(lled shells There are some advantages in measuring PNC e2ects in heavier atoms (where the PNC signal is expected to be larger) with relatively complicated electronic structure. However, the interpretation of the measurements in such cases is strongly impeded by the poor knowledge of the atomic wave functions. Even for Tl, which has a relatively simple structure, the atomic theory has an error of 2.5–3% [59,60]. The method we discussed in Sections 4 and 5 is applicable for heavy alkali-metal atoms. However, this method may not be accurate for atoms with more than one electron in un%lled shells. This is because the correlations between the electrons in the un%lled shells may be important. A very e2ective many-body method for atoms with several electrons above closed shells was developed in works [151,152] (see also [61]). It is a combination of many-body perturbation theory and con%guration interaction methods (it is called “MBPT+CI”). An e2ective Hamiltonian is constructed for the valence electrons using many-body perturbation theory for the interaction of the valence electrons with the core. In this way the correlations between the external electrons and the core are taken into account using MBPT while the correlations between the external electrons are calculated using the CI method. 16

Many years ago an upper limit on PNC in hydrogen-like argon was obtained [148]. A tabulation of the Z-dependence of various quantities in hydrogen-like ions is presented in Refs. [149,147]. 18 For a brief review of the (null) hydrogen measurements, see, e.g., Ref. [56]. 17

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103

A detailed description of the MBPT+CI method is beyond the scope of this review. We refer the interested reader to the works [151,152]. 6.1. Parity nonconservation in thallium High-precision measurements of parity violation in the 6P1=2 − 6P3=2 transition in thallium have been performed by the Oxford and Seattle groups,  (−15:68 ± 0:45) × 10−8 Oxford [43] (83) R ≡ Im{EPNC =M 1} = (−14:68 ± 0:17) × 10−8 Seattle [44] : We will use the latter, most precise measurement to extract the value for QW . A recent measurement of the ratio E2=M 1 indicates that the Oxford result should be rescaled to R = −15:34 ± 0:45 [153], reducing the discrepancy between the two measurements (83). Let us move on now to the calculations. The electronic con%guration for the ground state of thallium is 6s2 6p1=2 . This system can be treated as a one-electron or a three-electron system above closed shells. However, the 6p electron is energetically close to the 6s electrons, and correlations between 6s and 6p are signi%cant. This means that the method described in Sections 4, 5 will not be as e2ective for thallium as for cesium. This method was employed in the work [59] to the transition 6P1=2 − 6P3=2 , yielding the following result: EPNC = (−2:70 ± 0:08) × 10−10 ieaB (−QW =N ) :

(84)

This result includes core polarization of E1 and weak %elds calculated in the TDHF method and all second order correlation corrections (higher-order diagrams—electron–electron screening and the hole–particle interaction—are not included). Finally, empirical %tting of the energies is used to take into account some missed higher-order correlations. Consideration of the Breit interaction, Eq. (73), gives a correction [154] Breit \EPNC =EPNC = −0:0098 :

(85)

The size of radiative corrections to Tl PNC is the following. The Uehling correction is 0.94% [155] and we take the self-energy and vertex corrections to be −1:51% (average of −1:61% from [114] and −1:41% from [115]), rad: \EPNC =EPNC = −0:0057 :

(86)

The correction for the neutron distribution is small (with a relatively large error) [60], neutron \EPNC =EPNC = −0:003 :

(87)

With these corrections, Eq. (84) becomes EPNC = (−2:65 ± 0:08) × 10−10 ieaB (−QW =N ) :

(88)

(The errors associated with the corrections to Eq. (84) are well below 1%.) More recently the method MBPT+CI was applied in Ref. [60], with the result EPNC = (−2:72 ± 0:07) × 10−10 ieaB (−QW =N ) :

(89)

This method is well-suited for a system like thallium, where there are three electrons above a compact core, with the correlations between the three external electrons treated non-perturbatively.

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In this work the Gaunt interaction (the dominant, magnetic part of the Breit interaction, Eq. (73)) was included self-consistently at every stage of the calculation. Inclusion of radiative corrections shifts the value (89) slightly, EPNC = (−2:70 ± 0:07) × 10−10 ieaB (−QW =N ). The di2erence between this value and Eq. (88) is within the assigned errors. We need a value for the 6P1=2 − 6P3=2 magnetic dipole transition amplitude, M 1. Calculations of this quantity are very stable. In Ref. [60] the values M 1=1:692×10−3 a:u: and M 1=1:694×10−3 a:u: were calculated, the former using the MBPT+CI method and the latter in the MBPT method (treating thallium as a system with a single-electron above closed shells). The Seattle measurement (83), the PNC calculation (88), and the M 1 amplitude calculated in [60] lead to the following value for the nuclear weak charge of thallium, QW (205 Tl) = −116:3(1:3)exp (3:5)theor ;

(90)

in agreement with the standard model value [20] SM 205 ( Tl) = −116:67(5) : QW

(91)

6.2. A method to exclude the error from atomic theory: isotope ratios and the neutron distribution In Ref. [53] it was pointed out that atomic theory (and hence the large associated error) can be excluded by taking ratios of atomic PNC measurements along an isotope chain. Indeed, from a simple nonrelativistic consideration, the measured parity violating amplitude can be expressed as APNC = BQW ;

(92)

where the atomic theory is included into B. It is seen that in this approximation taking ratios of measurements of di2erent isotopes with neutron numbers N and N  APNC (N  ) QW (N  ) = QW (N ) APNC (N )

(93)

removes the dependence on B, assuming that the atomic structure does not change signi%cantly from isotope to isotope. However, a consideration of relativistic e2ects (the variation of the electron wave functions inside the nucleus) changes this simple formulation. It was shown in Ref. [156] that while the atomic structure cancels in the isotope ratios, there is an enhanced sensitivity to the neutron distribution !n (r), and the uncertainty associated with !n (r) may limit the extraction of new physics. The dependence of the PNC amplitude on nuclear structure e2ects can be incorporated through the correction Qnuc [156], nuc ) ; APNC = B(QW + QW

(94)

nuc = −N (qn − 1) + Z(1 − 4 sin2 W )(qp − 1) QW

(95)

where and

 qn =

!n (r)f(r) d 3 r;

 qp =

!p (r)f(r) d 3 r :

(96)

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The variation of the electron wave functions inside the nucleus is given by f(r), which is normalized to f(0) = 1. The isotope ratio R≡

QW (N  ) APNC (N  ) ≈ [1 + \qn ] ; APNC (N ) QW (N )

(97)

where \qn ≡ qn −qn . It is seen that R is sensitive, in particular, to the di2erence in the neutron distributions, and it is the error associated with this di2erence, %(\qn ), that could limit the interpretation of the ratio R of the PNC measurements in heavy atoms in terms of new physics. The isotope ratios are sensitive to a combination of new physics that is di2erent from that probed by measurements of parity violation in a single isotope. While measurements of atomic parity violation in a single isotope are sensitive to the weak isospin conserving parameter S and to new tree level physics, the isotope ratios are sensitive to both S and T through sin2 W as well as to new tree level physics. The sensitivity of the isotope ratios to oblique new physics has been investigated in [134,157,158], and the sensitivity to direct new physics in [137,158]. Nuclear structure corrections to the weak charge distribution have been calculated for di2erent isotopes of lead [158,159], cesium [159–161], barium [159], and ytterbium [159]. The calculations are model-dependent, and it is seen from the spread in the results that at this time our knowledge of the neutron distribution would preclude a determination of new physics competitive with other probes. The problem is that there is a very limited amount of empirical information on neutron distributions. In Ref. [162] new data on neutron distributions from experiments with antiprotonic atoms [118] was used to re-analyse the impact of the nuclear structure uncertainties on the extraction of new physics. It was found that the errors associated with nuclear structure are slightly reduced compared with those associated with nuclear calculations. It was suggested that isotope ratios of PNC measurements with atoms having Z . 50 may be more e2ective at searching for new physics, since the nuclear structure uncertainty in the extraction of (direct) new physics increases roughly as Z 8=3 and the required experimental uncertainty is less strict for lighter atoms [162]. The high sensitivity of isotope ratios of PNC measurements to variations in the neutron distribution could be used to determine the nuclear structure and test nuclear models [156]. Parity violating electron scattering measurements appear to be very promising for extracting information on the neutron distribution (see, e.g, [163]). It appears that such measurements could be used to signi%cantly reduce the nuclear structure uncertainties to the level where the new physics sensitivity of PNC isotope ratios is signi%cant. 6.3. Ongoing/future studies of PNC in complex atoms A thallium PNC measurement is continuing at Seattle [164] in which sub-1% accuracy is expected to be reached. Experimental studies of rare-earth elements dysprosium and ytterbium are in progress at Berkeley [165]. Studies with dysprosium are continuing despite the small upper limit found for the PNC amplitude [27]. Dysprosium appeared to be particularly attractive for PNC studies due to the presence of a state (4f9 5d2 6s J = 10) that is nearly degenerate with the state 4f10 5d6s J = 10 with opposite parity and the same angular momentum. However, the weak interaction does not mix the dominant electronic con%gurations of the nearly degenerate states, a nonzero weak matrix element shows up only through con%guration mixing and core polarization which makes it very small. An improvement in the statistical sensitivity of a few orders of magnitude is anticipated in the current

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search [27,166]. If the e2ect is of the same order of magnitude as the current upper limit, then dysprosium will provide an important test of the standard model and allow the determination of the nuclear anapole moment [27,166]. A measurement of PNC in ytterbium 6s2 1 S0 − 6s5d 3 D1 is in progress also at Berkeley [167,168]. In ytterbium there are seven stable isotopes in the range A = 168 − 176 with two of these isotopes having nonzero nuclear spin (possibility to observe the nuclear anapole moment). The PNC amplitude for the considered transition is about 100 times the size of the amplitude in Cs 6S1=2 − 7S1=2 [167,169,170]; the atomic theory should be more reliable than in the other rare-earth atoms, since here there is a closed 4f-shell. There are some interesting proposals for new experiments where large enhancements are expected. E.g., atomic calculations [171,172] indicate that parity violation in the transition 7s2 1 S0 − 7s6d 3 D1 in radium is enhanced due to a very small interval between the states 7s7p 3 P1 and 7s6d 3 D1 ; EPNC is calculated to be 100 times that of Cs 6S1=2 − 7S1=2 . 7. The nuclear anapole moment and measurements of P-odd nuclear forces in atomic experiments 7.1. The anapole moment The notion of the anapole moment was introduced by Zel’dovich [173] just after the discovery of parity violation. He noted that a particle may have a parity violating electromagnetic form factor, in addition to the usual electric and magnetic form factors. The %rst realistic example, the anapole moment of the nucleus, was considered in Ref. [23] and calculated in Ref. [24]. In these works it was also demonstrated that atomic and molecular experiments could detect anapole moments. Subsequently, a number of experiments were performed in Paris [46], Boulder [49,50], Oxford [43], and Seattle [44] and some limits on the magnitude of the anapole moment were established. However, it was only recently that a nuclear anapole moment was unambiguously detected—in 1997 the group at Boulder measured a nuclear anapole moment in 133 Cs (using atomic experiments) to an accuracy of 14% [9] (see also the recent review [186]). This is the %rst observation of an electromagnetic moment that violates fundamental discrete symmetries. Apart from the usual electric and magnetic monopole, dipole, quadrupole, etc., moments, there are other electromagnetic multipole moments that are not usually dealt with in multipole moment expansions since they give rise to contact, rather than long-range, potentials. The anapole is such a moment. It obeys time reversal invariance but violates parity conservation and charge conjugation invariance, i.e., T -even and P- and C-odd. The anapole moment arises out of an expansion of the vector potential as a series in R−1 , where R is the distance from the centre of the charge distribution (see, e.g., [174,15]). The part of the vector potential that is due to the anapole moment is 19 Aa (r) = a%3 (r) ; where

 a = −;

19

r 2 j(r) d 3 r ;

(98)

(99)

In the gauge ∇ · A = 0 there is a long-range term in the anapole vector-potential which may be removed by a gauge transformation.

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Fig. 19. Diagram showing the anapole moment, a, the toroidal current that produces it, j, and the magnetic %eld that the current creates, H.

j is the electromagnetic current density. Eq. (99) can be taken as the de%nition of the anapole moment. Notice the contact form of the potential—this is true for any T -even, P-odd moment; see, e.g., [23,15] for a proof. What kind of current distribution corresponds to an anapole moment? Such a current distribution is shown in Fig. 19. This will give a nonzero anapole moment because the places at which the current is pointing downwards are further from the centre than the places where the current is pointing upwards. Since the current in Eq. (99) is weighted by a factor of r 2 this current distribution will produce an anapole moment pointing perpendicular to the plane of the doughnut. A magnetic %eld is produced inside the current distribution as shown in the %gure. The expression for the anapole moment (99) contains the current vector j which changes its sign under re?ection of coordinates. The anapole moment is directed along the nuclear spin I: ˆ = −;r 2 j = aI=I . However, the spin I does not change its sign under coordinate re?ection. The a di2erent behaviour of the right- and left-hand sides of the relation r 2 j ˙ I under re?ection of coordinates means that the existence of the anapole moment violates parity (but it does not violate time reversal invariance). 7.2. Origin of the nuclear anapole moment A P-odd, T -even anapole moment of the nucleus arises due to the presence of a parity violating weak interaction between nucleons. The two-body P-odd nucleon–nucleon interaction, in the contact limit, has the form (see, e.g., [15]) G 1 Wˆ ab = √ ({(gab
a − gba
b ) · (pa − pb ); %3 (ra − rb )} 2m 2 p  +gab [
a ×
b ] · ∇%3 (ra − rb )) ;

(100)

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where { ; } is an anticommutator, G is the Fermi constant of the weak interaction, mp is the nucleon mass, and
, p, and r are the spins, momenta, and coordinates of the nucleons a and b.  give the strength of the weak interaction between nucleons. The dimensionless constants gab , gba , gab The e2ective one-body P-odd weak interaction between an unpaired nucleon and the nuclear core can then be obtained, G Wˆ = √ g[
· p!(r) + !(r)
· p] ; 2 2mp

(101)

where !(r) is the number density of core nucleons and g=

N Z gap + gan : A A

(102)

Here a = p; n denotes the unpaired nucleon. For the 133 Cs atom the unpaired nucleon is a proton and so g ≡ gp . Interaction (101) perturbs the wave function of the unpaired nucleon, resulting in the mixing of opposite parity states: = 0 + % , where 0 ≡ |0 is the unperturbed wave function and  % = n |n n|Wˆ |0 (E0 − En )−1 . An approximate analytical solution for the perturbed SchrZodinger equation (Hˆ 0 + Wˆ ) = E gives (see, e.g., [24,175]) = ei
·r

0

; √

(103)

where  = −gG!= 2. This means that the spin (s = 12
) of the unperturbed wave function will be rotated around the vector r by an angle of 2r. If, for example, the unperturbed wave function was in a spin up state, the spin at di2erent points for the perturbed wave function will be as shown in Fig. 20. Thus we have a spin helix, with a de%nite chirality, i.e., right- or left-handedness (see, e.g., [15]); the parity symmetry has been broken. Now consider the current and magnetic %eld produced by such a spin helix. The electromagnetic current of the unpaired nucleon in the nonrelativistic limit

Fig. 20. Diagram showing the spin helix that occurs due to the parity violating nucleon–nucleus interaction. The degree of spin rotation is proportional to the distance from the origin and the strength of the weak interaction.

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109

Fig. 21. A cross-section (in the x–z plane) of the current distribution due to the spin helix (produced by the parity violating nucleon–nucleus interaction). The anapole moment points along the z direction.

Fig. 22. A cross-section (in the x–y plane) of the magnetic %eld due to the spin helix (produced by the parity violating nucleon–nucleus interaction). The anapole moment points along the z direction.

has the form j=−

ie q[ 2mp

†

∇ − (∇

†

) ]+

e ∇×( 2mp

†


);

(104)

where q = 0 (1) for a neutron (proton) and  is the nucleon magnetic moment in nuclear magnetons. The %rst term comes from the orbital motion of the nucleon (convection or orbital current), while the second term is a magnetic moment current term. The current distribution and the magnetic %eld produced by the wave function of Eq. (103) have been calculated, e.g., in Ref. [176]. Cross-sections of these are shown in Figs. 21 and 22. Note their toroidal shapes; this means that they will produce an anapole moment.

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Using the expression for the electromagnetic current (104) in Eq. (99), the operator of the anapole ˆ ) can be written as moment, aˆ (a =  |a| aˆ = (;e=mp )[(r ×
) − (q=2)(pr 2 + r 2 p)] :

(105)

The dominant contribution to the nuclear anapole comes from the spin term and so we can express ˆ =
(e)=(2mp ) the anapole moment operator in terms of the magnetic dipole moment operator M ˆ as aˆ ≈ 2;(r × M). The anapole moment is usually described by a dimensionless parameter a de%ned by the following equation: 1 G KI a ; a= √ (106) e 2 I (I + 1) where K = (I + 12 )(−1)I +1=2−l

(107)

and l is the orbital angular momentum of the external nucleon. In Ref. [24] an approximate analytical result for a (in terms of the parameter gp ) was obtained by using the wave function (103) to calculate the mean value of the anapole moment operator (105). The result is 9 # 2=3 a = A gp (108) 10 mp r0 = 0:08gp

for

133

Cs ;

(109)

where r0 = 1:2 fm. It is the dependence on A2=3 that makes the anapole moment the dominant nuclear spin-dependent e2ect in heavy atoms. Single-particle calculations of a for Cs in the Woods–Saxon and harmonic oscillator potentials have been performed in Refs. [24,177,22,183] (note that in Refs. [177,22] the e2ects of con%guration mixing were also included semi-empirically) and many-body calculations in Refs. [178–181]. The results of these calculations are presented in Table 10; a compilation of these results in terms of DDH “best values” (see Section 7.5) can also be found in Ref. [182]. Single-particle calculations for the anapole moment are remarkably stable with the choice of potential [183]. We will quote the value aSP = 0:06gp

(110)

for 133 Cs, obtained from numerical calculations in the Woods–Saxon potential with spin–orbit interaction [24,183]. We will leave the discussion of many-body e2ects to Section 7.5. 7.3. Parity violating eCects in atoms dependent on the nuclear spin The nuclear anapole moment interacts with an atom’s electrons due to its magnetic %eld. The interaction is [using Eqs. (98) and (106)] G KI ·  a %3 (r) ; hˆa = e · A = e · a%3 (r) = √ (111) 2 I (I + 1)

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 Table 10 Calculations of the anapole moment a for

Singleparticle Many-body

133

111

Cs

Ref.

a × 102

a (DDH)

[24]a [24]b [183,179,180]c [177,22]d [178]e [179,180]f [180]g [181]h

8gp 6gp = (34f; − 9h0! − 2h1! + 1h2! − 5h0! − 5h1! ) × 102 4:9gp + 0:65gpn 5:5gp 23 × 102 f; 4:4gp + 0:45gn + 0:45gpn − 0:03gnp 2:9gp + 0:18gn + 0:36gpn − 0:02gnp (26:98f; − 7:01h0! − 1:72h1! + 0:16h2! − 4:48h0! − 2:16h1! ) × 102

0.36 0.27 0.26 0.25 0.11 0.23 0.15 0.21

In the last column the value for the anapole moment constant a is presented, using DDH best values of the meson– nucleon couplings (see Table 12). a Formulae (109) and (118). b Woods–Saxon potential with spin–orbit interaction, formulae (110) and (118). c Woods–Saxon potential with spin–orbit interaction; the contact and spin–orbit currents are included. This calculation includes some contributions beyond the single-particle approximation. d Harmonic oscillator potential with spin–orbit interaction. Con%guration mixing is taken into account semi-empirically. e Large-basis shell-model calculations; just the pion contribution was calculated. f Many-body e2ects taken into account in RPA. g Many-body e2ects taken into account in RPA. More complete treatment than Ref. [179]. h Large-basis shell-model calculations.

where A is the anapole vector potential and  is the relativistic velocity operator (Dirac matrices). The magnetic %eld corresponding to the anapole is localized inside the nucleus. Therefore the interaction with atomic electrons occurs only if the electron wave functions penetrate the nucleus. The anapole moment produces parity violating nuclear spin-dependent e2ects in atoms, and in heavy atoms it is the dominant mechanism producing such e2ects. Its e2ects, however, are indistinguishable from those arising from the parity violating neutral weak currents, and these need to be accounted for in the extraction of a from atomic measurements (we mentioned these e2ects brie?y in Section 2.2). One contribution to parity violating nuclear spin-dependent e2ects arises from Z 0 exchange in an electron–nucleus interaction, with an axial-vector Z 0 -nucleus coupling and a vector Z 0 -electron coupling [Eq. (9)]; see Ref. [184]. Another contribution arises from the perturbation of the nuclear spin-independent contribution, corresponding to the nuclear weak charge, by the hyper%ne interaction (Eq. (10)); see [21,22,185]. The total e2ective nuclear spin-dependent interaction can be expressed in the form 20 K G  · I!(r) ; (112) hˆIe2 = √  2 I (I + 1) where K − 1=2 I +1  = a − 2 + Q : (113) K K 20

Note that di2erent√de%nitions for , a , 2 , Q are used in di2erent works. For example, in [181,186,185] the e2ective Hamiltonian hˆIe2 = G= 2  · I!(r),  = a + 2 + Q . Also di2erent notations for the constants are used in these works.

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7.4. Measurement of nuclear spin-dependent eCects in cesium and extraction of the nuclear anapole moment The anapole moment can be detected by observing the amplitudes of transitions between atomic levels that violate parity. In the case of the Boulder experiment [9], it was an E1 transition between the 6S and 7S states of the cesium atom. The nuclear spin-dependent P-odd e2ects can be separated from the dominant e2ect (the weak interaction between the electron and the weak charge of the nucleus; see Sections 3, 5) by observing the parity violating e2ects in two di2erent hyper%ne transitions. The nuclear spin has di2erent relative orientations in the di2erent hyper%ne states. For 133 Cs, F = 3 and F = 4, where F is the total angular momentum of the atom (F = J + I, where J is the electronic angular momentum). In the Boulder experiment [9] the parity violating E1 transition amplitudes 6SF=4 − 7SF=3 and 6SF=3 − 7SF=4 were measured,  1:6349(80) mV=cm; 6SF=4 − 7SF=3 ; − Im(EPNC )= = (114) 1:5576(77) mV=cm; 7SF=3 − 7SF=4 ; and found to be signi%cantly di2erent, 0:077(11) mV=cm;  is the vector transition polarizability. This signals the presence of an anapole moment of the nucleus. 7.4.1. Atomic calculations and extraction of  Atomic many-body calculations are required to extract the e2ective constant  from experiment. Atomic calculations of nuclear spin-dependent e2ects in cesium have been performed in [184,187,188,177,19,185]. In [189] the value for  was extracted from [9] by taking the ratio of the nuclear spin dependent PNC amplitude to the main spin independent PNC amplitude, using the spin dependent calculation [188] and the spin independent calculation [59]. The calculations [188,59] were performed in the relativistic Hartree–Fock approximation, with the e2ects of core polarization included using the time-dependent Hartree–Fock method and correlations included through the use of Brueckner orbitals (see Section 4). These calculations were performed using the same method and computer codes, so the theoretical errors should cancel in the ratio. The value obtained in [189] is (133 Cs) = 0:442(63) :

(115)

In the recent work [185] the value (133 Cs) = 0:462(63) was obtained. The calculations were performed in zeroth-order in the relativistic Hartree–Fock approximation, and core polarization was included using the random phase approximation. This result was extracted directly from the spindependent component measured in [9], with  from [83]. The 4% di2erence between the two values for (133 Cs) is explained by correlation corrections: correlation corrections (Brueckner orbitals) are included in [188], while they are not considered in [185]. At the TDHF level, the results coincide. 7.4.2. Extraction of a The value of  contains three contributions (see Eq. (113)). The contributions to  from the nuclear spin-dependent neutral current 2 and from the combined neutral weak charge and hyper%ne interaction Q must be subtracted to give the anapole constant a . In the single-particle approximation, 2 = −C2p ≈ −0:05 [see Section 2, Eq. (9)]. Large-basis nuclear shell-model calculations for 133 Cs give 2 = −0:063 [181]. Atomic calculations for the combined weak charge and hyper%ne

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 Table 11 Nuclear calculations of 2 and atomic calculations of Q for 133

Ref.

2

Eq. (3)a [21]b [177,22] [181]d [185]e

−0.05 — −0.038c −0.063 —

Cs

133

Cs and

203;205

113

Tl. 203;205

Q

2

— 0.017 0.027 — 0.017

−0.05 — −0.027c −0.064 —

Tl

Q — 0.014 0.022 — —

a

Single-particle, standard model value with sin2 W = 0:23. Eq. (11). c Nuclear con%guration mixing was taken into account semi-empirically. d Large-basis nuclear shell model calculations. e Atomic many-body calculations in third-order perturbation theory. Core polarization is included in RPA. b

interaction have been performed in [21,22,185]. The recent calculation [185] is the most complete: it was performed in third-order perturbation theory, with core polarization taken into account in the random phase approximation. In the other calculations [21,22] the single-particle form for the operator [Eq. (10)] was used. The result of Johnson et al. [185] for 133 Cs is Q = 0:017. Results of calculations for 2 and Q are compiled in Table 11. With 2 from [181] and Q from [185], the anapole moment constant is a = 0:368(63) :

(116)

Interaction (111) is for a point-like nucleus. However, a real nucleus has a %nite size. In the works [187,188,177,19,185] the atomic calculations were performed by replacing %3 (r) with the nuclear density !(r). A more accurate treatment of %nite nuclear size e2ects was given in [176,189]. For Cs the correction is small—the corrected value for a [Eq. (116)] is a = 0:362(62) :

(117)

7.5. The nuclear anapole moment and parity violating nuclear forces The parity violating weak potential is usually parametrized in terms of a one-boson exchange model. The proton–nucleus and neutron–nucleus constants, gp and gn , can be expressed in terms of the following combination of meson–nucleon parity nonconserving interaction constants [24,190] (using the notation of Ref. [191]):

W; 5 0 1 2 0 1 gp = 2:0 × 10 W! 176 f; − 19:5h! − 4:7h! + 1:3h! − 11:3(h! + h! ) ; (118) W!

W; 5 0 1 2 0 1 gn = 2:0 × 10 W! −118 f; − 18:9h! + 8:4h! − 1:3h! − 12:8(h! − h! ) ; (119) W! f; ≡ h1; and the h’s are weak meson–nucleon coupling constants—the subscript denotes the type of meson involved and the superscript indicates whether it is an isoscalar, isovector, or isotensor interaction (0, 1, or 2). These are the e2ective constants obtained in the contact limit of the

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Table 12 DDH “best values” (f; ; h) in units 10−7 and e2ective coupling constants (g) at DDH values for the meson–nucleon couplings f;

h0!

h1!

h2!

h0!

h1!

gpn

gnp

gpp = gnn

gp

gn

4.6

−11.4

−0.19

−9.5

−1.9

−1.1

6.5

−2.2

1.5

4.5

0.2

nucleon–nucleon PNC interaction [Eq. (100)], with short-range nucleon–nucleon repulsion and long-range e2ects taken into account through the parameters W! and W; . Following [24,189], we take W! = 0:4 and W; = 0:16 using calculations of PNC for neutron and proton scattering on 4 He [192]. The standard reference values for weak meson–nucleon couplings are those of Desplanques et al. [191]; we list the DDH “best values” for the weak couplings and the e2ective coupling constants (Eqs. (118), (119) and (102); Refs. [24,193,183]) in terms of these values in Table 12 for easy reference. However, there are large uncertainties in the possible values, known as the “DDH reasonable ranges”. It is therefore important to determine the weak coupling constants experimentally. Information on these constants can be obtained from the measurement of the Cs anapole moment. 7.5.1. The cesium result and comparison with other experiments The measurement of the anapole moment a (117) and the single-particle calculation (110) give the following value for the weak interaction constant between an unpaired proton and the nuclear core [189], gpSP = 6 ± 1(exp) :

(120)

(Note that only the experimental error is included here; there is also a theoretical error from the nuclear calculation of aSP (110).) There is a lot of uncertainty about the value for f; . Following [189], we use DDH best values for h! and h! to obtain a value for f; from the measurement of the cesium anapole moment. Using Eqs. (120), (118) it is found that f;SP = (gpSP − 2) × 1:8 × 10−7 = [7 ± 2(exp)] × 10−7 :

(121)

(The contribution of ! and ! to gp is 2.) This result (121) agrees with QCD calculations [194,195] which give f; = 5–6 × 10−7 and is in agreement with the DDH best value f; = 4:6 × 10−7 (note that the DDH “reasonable range” for f; is 0:0–11:4 × 10−7 ). However, there is a serious discrepancy between the weak meson–nucleon couplings extracted from the Cs anapole moment and those extracted from other experiments. The following experiments are thought to give reliable information on the weak meson–nucleon couplings (that is, their interpretation is not hindered too much by nuclear structure uncertainties) [196,181]: the longitudinal analysing power for p ˜ p scattering at 13:6 MeV [197], 45 MeV [198], and 221 MeV [199] and p ˜ # scattering at 46 MeV [200], the circular polarization of -rays emitted from the 1081 keV state in 18 F [201], and the asymmetry of -rays emitted in the decay of the state 110 keV in polarized 19 F [202]. These experiments depend on di2erent combinations of the coupling constants (for instance, pp scattering does not contain f; while other experiments do). They are consistent and favour a small value of

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115

f; lying roughly between −1 × 10−7 and 1 × 10−7 (see [181,186]), in contradiction with the Cs anapole result. The value for f; extracted above for Cs was obtained in the single particle model. Let us brie?y discuss how this value changes when many-body e2ects are included. Core polarization e2ects have been calculated in Refs. [179,180] using the random phase approximation. In Ref. [180] it was found that for 133 Cs core polarization and pairing e2ects decrease the single-particle value for a by almost a factor of two. These calculations therefore increase the discrepancy of the cesium result with other experiments and with the DDH best value. Large-basis shell model calculations have been performed in Refs. [180,181]. These calculations also reduce the size of a , although not to the extent of the calculation [180]. In Table 10 results of single-particle and many-body calculations of a for 133 Cs are presented. It appears that consideration of many-body e2ects tends to exacerbate the disagreement between the meson–nucleon couplings extracted from the Cs anapole moment and those obtained from other experiments. Note that there may be other signi%cant corrections to the anapole moment calculations that may resolve the discrepancy between theory and experiment. For instance, consideration of the strong renormalization of the weak potential [177,203]. Another correction to the calculations could stem from the use of an improved set of strong meson–nucleon coupling constants. [In the one-boson exchange model the weak potential is formed from a product of weak and strong meson–nucleon couplings. The strong couplings g; = 13:45, g! = 2:79, g! = 8:37 and the isoscalar 7S = −0:12 and isovector 7V = 3:7 anomalous magnetic moments of the nucleon have been used in the calculations; see, e.g., [191,192,181,204]. These values were used to arrive at Eqs. (118) and (119).] In the recent work [204] it was shown that using strong coupling constants in the one-boson exchange model constrained by nucleon–nucleon phase shifts leads to an enhancement of parity violating e2ects of about a factor of two. The cesium nuclear anapole moment is not the only case in which parity violating e2ects appear to be enhanced. Parity violation in the MZossbauer transitions in 57 Fe and 199 Sn nuclei give a value for the circular polarization of -rays at least four orders of magnitude larger than calculations (see Ref. [193]). Statistical methods were applied in Refs. [205,206] to study parity violating e2ects in polarized neutron scattering from compound nuclei. It was found in Ref. [206] that the experiments give parity violating e2ects 1:7–3 times larger than those estimated using the DDH values for weak meson–nucleon couplings. The Cs anapole measurement is also inconsistent with the limit on the anapole moment of Tl [44], (Tl) = −0:22 ± 0:30 :

(122)

In fact, the Tl anapole moment itself produces problems. Its central point is of opposite sign to that predicted by theory. 21 Results of calculations of a for Tl are presented in Table 13; these results in terms of DDH “best values” can also be found in [182]. Using DDH best values for the weak couplings, the most complete many-body calculations of the anapole moment of Tl give 21

Note that while the Oxford result [43], (Tl) = 0:23 ± 1:20 [using the single-particle atomic calculation [207] used to obtain (122)], has a central point that is positive, its error is four times larger than the Seattle value [44]. The Oxford result has been misquoted in the literature.

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Table 13 Calculations of the anapole moment a for

Singleparticle Manybody

203;205

Tl

Ref.

a × 102

a (DDH)

[24]a [24]b [183,179,180]c [177,22]d [179,180]e [208]f [180]g [181]h

11gp 10gp = (56f; − 16h0! − 4h1! + 1h2! − 9h0! − 9h1! ) × 102 7:8gp + 0:85gpn 6:0gp 7:1gp + 0:35gn + 0:64gpn − 0:06gnp 5:3gp + 0:4gn 4:3gp + 0:1gn + 0:64gpn − 0:06gnp (13:98f; − 2:92h0! − 0:26h1! + 0:23h2! − 2:14h0! − 0:98h1! ) × 102

0.48 0.43 0.41 0.27 0.36 0.24 0.24 0.10

In the last column the value for the anapole moment constant a is presented, using DDH best values of the meson– nucleon couplings (see Table 12). a Formulae (108) and (118). b Woods–Saxon potential with spin–orbit interaction. Eq. (118) is used to express the e2ective coupling constant gp in terms of parity violating weak meson–nucleon couplings. c Woods–Saxon potential with spin–orbit interaction; the contact and spin–orbit currents are included. This calculation includes some contributions beyond the single-particle approximation. d Harmonic oscillator potential with spin–orbit interaction. Con%guration mixing was taken into account semi-empirically. e Many-body e2ects taken into account in RPA. f Large-basis shell-model calculations. Just the dominant spin-current contribution was considered. g Many-body e2ects taken into account in RPA. More complete treatment than Ref. [179]. h Large-basis shell-model calculations.

a (Tl) = 0:24 within RPA [180] and a (Tl) = 0:24 [208] and a (Tl) = 0:10 [181] in shell model calculations. Account of 2; Q (Table 11) brings no relief to this problem. The above value for (Tl), Eq. (122), was obtained from the nuclear spin-dependent component of the PNC optical rotation using a single-particle atomic calculation [207]. In a recent many-body calculation [209], in which the MBPT+CI method was implemented (see Section 6 for a brief description and references), the central point for the nuclear spin-dependent constant in Tl is interpreted as  = −0:20. 7.6. Ongoing/future studies of nuclear anapole moments It is clear that further experimental investigation is required to resolve these inconsistencies. An improved measurement of the Tl anapole moment is important, as is an anapole measurement involving a nucleus with an unpaired neutron (Cs and Tl have unpaired protons). This latter experiment would give information on gn which depends on a di2erent combination of weak meson–nucleon couplings that is roughly perpendicular to the current anapole measurements [compare Eqs. (118) and (119)] and so would provide a very important cross-check. We have already discussed in Sections 5.4 and 6.3 the PNC experiments underway. Nuclear spin-dependent e2ects in those atoms with nonzero nuclear spin will be measured. Enhancement of anapole moment e2ects in atoms can occur due to close levels of opposite parity. For example, calculations [171,172] show that the nuclear spin-dependent PNC amplitudes

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117

in the transitions 7s2 1 S0 − 7s6d 3 D1 and 7s2 1 S0 − 7s6d 3 D2 in 223 Ra and 225 Ra can be orders of magnitude larger than the nuclear spin-dependent transition amplitudes in Cs. Moreover, in the transition 7s2 1 S0 − 7s6d 3 D2 there is no background nuclear spin-independent amplitude, since the large change in angular momentum \J = 2 forbids it. Measurements with 223;225 Ra, sensitive to the neutron constant gn , would provide a sought-after cross-check on the parity violating meson– nucleon couplings. Note that there is a similar situation in Yb 6s2 1 S0 − 6s5d 3 D2 for those (stable) isotopes with nonzero nuclear spin 171 Yb (I = 1=2) and 173 Yb (I = 5=2). Work towards measurements of anapole moments in diatomic molecules is underway [210]. Diatomic molecules are attractive for anapole moment searches because the nuclear spindependent e2ects, but not the (usually dominating) nuclear spin-independent ones, are enhanced compared to atoms due to the presence of close rotational levels of opposite parity [211,212]; see also the review [213].

8. Electric dipole moments: manifestation of time reversal violation in atoms Violation of CP symmetry (combined symmetry of charge conjugation, C, and parity) was discovered in 1964 in the decays of the neutral K mesons [6]. It is incorporated into the standard electroweak model as a single complex phase in the quark mixing matrix (Kobayashi–Maskawa mechanism [214]). For a long time K mesons remained the only system in which CP-violation had been observed, until just recently, in 2001, when the collaborations BaBar and Belle detected it in the neutral B mesons [7]. The CP-violation seen there is consistent with the standard model predictions. However, a striking problem arises from cosmology. Sakharov proposed that CP-violation, present at the time of the Big Bang, is a necessary ingredient in the asymmetry of matter and antimatter [215]. It is well-known that standard model CP-violation is insuScient to generate the level of matter-antimatter asymmetry in the Universe. Understanding the origin of CP-violation and searching for possible new sources is therefore a very interesting and fundamental problem. If CPT is a good symmetry, as it is in gauge theories, then CP-violation is accompanied by T (time-reversal) violation. So far there has been no (undisputed) direct observation of T violation and its detection is of interest in its own right. 22 Also, detection of T -violation may shed light on the origin of CP-violation. The measurement of a permanent electric dipole moment (EDM) of, e.g., a neutron, atom, or molecule would be direct evidence of T -violation. 23 This can be seen very simply: an EDM, d, of a non-degenerate quantum system is directed along the total angular momentum F 22

CPLEAR [216] claim to have made the %rst direct observation of time-reversal violation through observation of a di2erence in the probabilities of KQ 0 transforming into K 0 and vice versa. There is a dispute as to whether or not this claim is true. For arguments in favour see, e.g., Ref. [217]; for arguments against see, e.g., Ref. [218]. 23 A parity and time-reversal violating EDM gives rise to a linear Stark shift. This is di2erent to the internal EDM in polar molecules and the induced EDM in hydrogen where the levels are essentially degenerate and so the Stark shift is linear in the case of a strong electric %eld when the interaction with the %eld is larger than the interval between the rotational levels in molecules or the ns − np level splitting in hydrogen. The case of a weak %eld gives the usual quadratic Stark shift in these systems. Neutral species are chosen for EDM experiments since they are not accelerated on application of the electric %eld required to detect the EDM.

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(the only vector specifying the system), and such a correlation can only occur if both T and P are violated (d and F behave di2erently on reversal of r and t). T violating electric dipole moments have an extraordinary sensitivity to new models of CP-violation. The reason is that the standard model CP-violation, appearing through just a single phase, is highly suppressed in such ?avour-conserving phenomena. In particular, the standard model prediction for the neutron EDM is dn (SM) ∼ 10−34 e cm [219], with the proton EDM the same order of magnitude, and the electron EDM is even smaller de (SM) . 10−38 e cm [220]. The EDMs appear only in high-order loops (3-loop diagrams for the neutron and 3- to 4-loop diagrams for the electron). The new sources of CP-violation appearing in new theories generate EDMs that are many orders of magnitude larger than the standard model values and the size of these EDMs are in reach of current experiments. Already the parameter space of popular theories such as supersymmetry, multi-Higgs models, and left-right symmetric models are strongly restricted by current measurements. A CP-violating phase also appears in quantum chromodynamics (QCD) and it can be constrained (or detected) from EDM measurements. It de%nes the strength of the P- and T -violating term in the QCD Lagrangian containing GJ G˜ J (this is analogous to FJ F˜ J in electrodynamics which reduces to E·B, clearly a P; T -violating correlation; see, e.g., Ref. [15]). The “unnatural” smallness of this phase (QCD . 10−10 , compare this to the Kobayashi–Maskawa angle %KM ∼ 1) is the famous “strong CP problem”. One solution to this problem involves the introduction of a new Goldstone boson known as the “axion” [221], and this particle has become a popular candidate for dark matter. However, after many experimental searches, there is no evidence of its existence; see, e.g., the review [222]. 8.1. Atomic EDMs Contributions to an atomic EDM arise from (i) the sum of intrinsic EDMs of the atomic constituents (averaged over the atomic state) and (ii) the mixing of opposite parity wave functions due to a P; T -odd interaction Hˆ PT . The atomic EDM induced in an atomic state K due to admixture with the opposite-parity wave functions M has the form datom = 2


K|D|M ˆ M |Hˆ PT |K M

EK − E M

= datom (F=F) ;

(123)

 where Dˆ = −e i ri is the electric dipole operator, Hˆ PT is the P; T -odd operator that mixes K with the set of wave functions M , and F is the total angular momentum of the atom corresponding to the state K. 8.1.1. Electronic enhancement mechanisms What are the mechanisms that lead to enhancement of parity and time invariance violating e2ects in atoms? Like the P-odd, T -even e2ects we discussed in the %rst part of this review, the e2ects of P; T -violation increase in atoms with: (i) a high nuclear charge Z. P; T -odd e2ects in atoms increase rapidly with Z [223,224]. Therefore, it is best to search for atomic EDMs in heavy atoms. (ii) close levels of opposite parity. From Eq. (123) we can see that if HPT mixes opposite parity levels with energies E1 ≈ E2 , then the atomic EDM induced will be enhanced, datom ˙ 1=(E1 − E2 ).

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119

This enhancement occurs, for example, in rare-earth atoms, where there are anomalously close levels of opposite parity. 8.2. Enhancement of T -odd eCects in polar diatomic molecules Sandars was the %rst to point out the enhancement of T -odd e2ects in polar diatomic molecules compared to atoms in his proposal to measure the proton EDM in TlF [225]. The idea is to polarize the molecules along an external electric %eld, thereby aligning the enormous intramolecular %eld, leading to an e2ective enhancement of the external %eld by several orders of magnitude. This enhancement mechanism is related to that arising from the presence of close rotational levels of opposite parity [212]. The two cases correspond to the respective cases of strong and weak interactions with the electric %eld compared to the rotational level spacing. See [213] for a review. 8.3. Limits on neutron, atomic, and molecular EDMs A typical EDM experiment is performed in parallel electric and magnetic %elds. The corresponding Hamiltonian is hˆ = − · B − d · E : (124) A linear Stark shift is measured by observing the change in frequency when the electric %eld is reversed (this is a measure of the P; T -odd correlation E · B). To date, permanent EDMs in neutrons, atoms, and molecules have escaped detection. Null measurements of EDMs have been obtained for the paramagnetic atoms 85 Rb [226,227], 133 Cs [228–230], 205 Tl [231–234,12], and 129 Xe in the metastable state 5p5 6s 3 P2 [235] and, very recently, the molecule YbF in the ground state [236]. An experiment aimed to measure the EDM of the ion Fe3+ in the solid state was carried out a long time ago [237]. Interest to carry out measurements in solids, in particular in gadolinium gallium garnet and gadolinium iron garnet, has been sparked recently [238,239]. In diamagnetic systems, EDM experiments have been performed for atomic 129 Xe [240,241] and 199 Hg [242,243,11] in the ground states and for the polar molecule TlF [244–248]. Measurements of EDMs in paramagnetic systems (that is, with nonzero total electron angular momentum) are most sensitive to leptonic sources of P; T -violation, in particular the electron EDM, while measurements of EDMs in diamagnetic systems (zero total electron angular momentum) are most sensitive to P; T -odd mechanisms in the hadronic sector. Both are sensitive to P; T -odd semi-leptonic processes (the electron–nucleon interaction), the former to those involving the electron spin, and the latter to those involving the nuclear spin. The most precise measurement of an atomic EDM in a paramagnetic system has been obtained for 205 Tl [12], d(205 Tl) = −(4:0 ± 4:3) × 10−25 e cm :

(125)

With the use of atomic calculations, described in Section 8.4.2, this measurement can be expressed in terms of a limit on the EDM of the electron, for which it currently sets the tightest constraint. The most precise measurement of an atomic EDM has been carried out with 199 Hg [11], d(199 Hg) = −(1:06 ± 0:49 ± 0:40) × 10−28 e cm :

(126)

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This measurement sets the best limits on a number of P; T -violating mechanisms in the hadronic sector. The nucleus of 199 Hg has an unpaired neutron. A limit on the neutron EDM extracted from (126) is competitive with the best direct neutron EDM measurements 24 performed at the Institut Laue-Langevin (ILL) and the Petersburg Nuclear Physics Institute (PNPI)  ILL [250] (1:9 ± 5:4) × 10−26 e cm; : (127) dn = (2:6 ± 4:0 ± 1:6) × 10−26 e cm; PNPI [251] As we will see in Section 10, at the most fundamental scale the limits on P; T -violating parameters from the measurement (126) are more strict than those from the neutron measurements (127) due to the presence of a nucleon–nucleon P; T -violating interaction that induces P; T -violating nuclear moments more eSciently than does an intrinsic EDM of a nucleon. It is interesting that although the 199 Hg nucleus has an unpaired neutron, the measurement (126) sets a constraint on the proton EDM (it contributes due to con%guration mixing) that is tighter than that from the measurement of the EDM of the TlF molecule (Tl has an unpaired proton) used in the ˆ past to set the proton EDM upper limit. In TlF the P; T -odd interaction is given by Hˆ = −h d
· , ˆ where
is the spin operator of the Tl nucleus,  is the unit vector along the internuclear axis, and h is Planck’s constant. The most precise limit on the P; T -odd coupling constant d in TlF is [248] d = −(0:13 ± 0:22) mHz :

(128)

The coupling constant d can be expressed, e.g., in terms of permanent EDMs of the proton and electron and P; T -violating interactions; such calculations for TlF have been performed in [225,252–254,174,21,255–257]; see, e.g., Ref. [248] for a general overview. The limit on the proton EDM following from the calculations of Ref. [257] is presented in Table 19. A more detailed comparison of limits on P; T -violating parameters will be postponed until Section 10. In the following sections we discuss the di2erent mechanisms that induce atomic EDMs. 8.4. Mechanisms that induce atomic EDMs An external electric %eld acting on a neutral atom consisting of nonrelativistic point-like charged particles with EDMs, interacting via electrostatic forces, is screened exactly at each particle [258,90]; see also [15]. 25 This occurs due to polarization of the atomic electrons by the external %eld. An atomic EDM cannot be induced in such a case. However, as shown by Schi2 [90], if magnetic or %nite-size e2ects are taken into account, there is incomplete shielding, and so atomic EDMs can in principle be measured. An atomic EDM can be induced from the following P; T -odd mechanisms: (i) an intrinsic EDM of an electron—the electron EDM can interact with the atomic %eld producing an atomic EDM that is many times larger than the single electron EDM [223]; 24

In line with the comments of Ref. [249], we present the most recent 1999 result of the ILL group, rather than the %nal value cited in Ref. [250] which is the average of their 1999 and 1990 results. 25 Actually, for moving particles like electrons the average value of the screened %eld is zero. An explanation is the following: a neutral atom is not accelerated by the homogeneous external %eld. If a charged particle inside the atom is not accelerated the electric %eld acting on this particle is zero. Therefore, the EDM of the particle has nothing to interact with, d · E = 0.

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154 atomic/molecular level

nuclear level

nucleon level

eeNN d (paramagnetic)

quark level

121 particle level CP models

eeqq

de

Higgs Supersymmetric

d (diamagnetic)

MQM

NNNN

S

dN

dq ~ dq ~ GGG

Left-right

Strong CP

qqqq d (neutron)

~ GG

Fig. 23. Flow diagram of CP-violation mechanisms at di2erent levels that induce neutron, atomic, and molecular EDMs.

(ii) a P; T -odd electron–nucleon interaction [259,252]; (iii) an intrinsic EDM of an external nucleon [225,260]; (iv) a P; T -odd nucleon–nucleon interaction—this interaction can induce P; T -odd nuclear moments that can greatly exceed the moments of single nucleons [261,174]. The latter two mechanisms (iii), (iv) can be grouped together at the nuclear scale since they both produce P; T -odd nuclear multipole moments. One may consider other exotic T -violating mechanisms such as dyon 26 vacuum polarization [262]. There are also complex mechanisms that can lead to an atomic EDM. For example, T -odd, P-even interactions can have P; T -odd e2ects due to P-odd radiative corrections (see, e.g., [16]). In Fig. 23 we present a ?owdiagram (slightly modi%ed from the review Ref. [263]) showing the CP-violating mechanisms at di2erent distance scales that induce neutron, atomic, and molecular EDMs. Read from left to right, it is seen clearly which measurements constrain which CP-violating parameters at smaller distances (and which popular CP-violating models). (Read in the other direction, it is seen which small-distance mechanisms induce large-distance CP-violating e2ects.) Calculations are required to relate the CP-violating parameters at di2erent scales. Solid lines indicate the parameters that are most strongly constrained (induced) by the parameters to the left (right), while dashed lines indicate a weaker constraint (inducing mechanism). In this review we focus primarily on atomic EDMs induced by P; T -odd nuclear moments originating from P; T -odd nuclear forces, since there have been several recent developments in this area. We will %rst discuss atomic EDMs induced by the P; T -violating electron–nucleon interaction in Section 8.4.1 and by the electron EDM in Section 8.4.2. For the sake of completion, we brie?y introduce nuclear P; T -violating moments in Section 8.4.3 before a comprehensive overview of these moments is given in Section 9. 26

A dyon is a particle with both electric and magnetic charges.

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8.4.1. The P; T -violating electron–nucleon interaction The P; T -violating electron–nucleus interaction has the following form (see, e.g., [16]): G
SP Q hˆ = √ [CN N N ei Q + CNT NQ i5 J N e Q J e] : Q 5 e + CNPS NQ i5 N ee 2 N

(129)

The real, dimensionless constants CNSP , CNPS , and CNT give the strength of the scalar–pseudoscalar, pseudoscalar–scalar, and tensor P; T -odd electron–nucleon interactions for the nucleon N . The constants C SP , C PS , and C T can be determined from measurements of atomic EDMs. Considering the nucleons to have in%nite mass, the electron–nucleus interaction reduces to  



G
p + CnT
n ·  !(r) : (130) hˆSP; T = hˆSP + hˆT = i √ (ZCpSP + NCnSP )0 5 + 2 CpT 2 p n In this approximation the term containing C PS vanishes. Notice the similarity between this expression (130) and Eqs. (4), (8). However, here the matrix element is real (the factor i is placed here to make the operator Hermitian; another factor of i arises due to the mixing of the upper and lower components of the electron wave function (20) due to 5 and i ). Therefore interaction (130) mixes atomic states of opposite parity and induces static electric dipole moments in atoms. Like their P-odd, T -even analogues (4) and (8), the nuclear spin-independent term in (130) receives coherent contributions from the nucleons inside the nucleus, whereas the smaller second term, dependent on the nuclear spin, arises from the unpaired nucleons. Each of the interactions Hˆ SP and Hˆ T can induce EDMs in paramagnetic atoms. However, only Hˆ T can open up the closed electron shells of diamagnetic atoms and thus induce an EDM; this term is dependent on the nuclear spin, while the scalar–pseudoscalar term is not. Therefore, the interaction Hˆ SP cannot by itself contribute to EDMs in diamagnetic atoms. However, by allowing for the hyper%ne interaction, measurements of EDMs of diamagnetic atoms can place limits on C SP which are, in fact, as competitive as those obtained from experiments with paramagnetic atoms. The atomic EDM induced by the P; T -odd scalar–pseudoscalar electron– nucleus interaction Hˆ SP along with the hyper%ne interaction Hˆ hf arises in the third order of perturbation theory [21], datom =


0|D|m m| ˆ Hˆ hf |n n|Hˆ SP |0 mn

(E0 − Em )(E0 − En )

+ permutations :

(131)

The matrix element of an e2ective operator constructed from Hˆ SP and Hˆ hf , Hˆ SP e2 =


(Hˆ SP |n n|Hˆ hf ) + (Hˆ hf |n n|Hˆ SP ) n

E0 − E n

;

(132)

SP SP has the form p1=2 |Hˆ SP e2 |s1=2 ˙ (ZCp + NCn )I · J. This matrix element can be related to that of   p1=2 |Hˆ T |s1=2 ˙ CpT p
p + CnT n
n · J, the brackets  denote averaging over the nuclear

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

123

Table 14 Calculations of atomic EDMs induced by P; T -odd electron–nucleon interaction Atom

datom =C SP (10−18 e cm)

datom =C T (10−20 e cm)

datom =C PS (10−23 e cm)

Cs

0.70a 0.71b 0.72d −5.1b −(7 ± 2)d −5:6 × 10−5e −4:4 × 10−5g

0.92b

2.2c

0.5b

1.5c

0.52f 0.41h 0.6b 2.0f 1.3b

1.2e 0.95g

Tl Xe Hg

−5:9 × 10−4e

6.0e

a

Ref. [259], semi-empirical calculation. Ref. [15,16], simple analytical calculations. c Using formula (137) and calculations for datom =C T from [16]. d Ref. [265], TDHF calculation with correlation corrections estimated from the size of the corrections for the corresponding electron EDM enhancement factors in [81]. e Using formulae (133), (137) and the calculation datom =C T from Ref. [266]. f Ref. [266], TDHF calculations. g Using formulae (133), (137) and the calculation datom =C T from Ref. [267]. h Ref. [267], TDHF calculation. b

state. The correspondence between datom (C SP ) and datom (C T ) [21,264,16],   Z SP N SP I Cp + Cn ↔ 1:9 × 103 (1 + 0:3Z 2 #2 )−1 A−2=3 −1 A A I  

× CpT
p + CnT
n ; p

(133)

n

where  is the nuclear magnetic moment in nuclear magnetons, can be used to obtain the sensitivity of datom to C SP from calculations of datom (C T ) for diamagnetic atoms. Calculations of atomic EDMs induced by Hˆ SP and Hˆ T are presented in Table 14. The current best limits for C SP have been obtained from the 205 Tl and 199 Hg measurements, Eqs. (125) and (126),  Tl (6 ± 6) × 10−8 ; (134) (0:40CpSP + 0:60CnSP ) = −8 (18 ± 8 ± 7) × 10 Hg and the best limit for C T is from Hg, CnT = −(5:3 ± 2:5 ± 2:0) × 10−9 : Here the simple shell model of the nucleus has been used, 
n = −(1=3)I=I .

(135)

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J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

Let us now consider the contribution to an atomic EDM arising from the pseudoscalar–scalar 1 component of the electron–nucleus interaction (129). The term lowest order in m− p is  

G 1 CpPS
p + CnPS
n ∇!(r)0 : (136) hˆPS = − √ 2 2mp p n   Again, the matrix element of this interaction p1=2 |Hˆ PS |s1=2 ˙ CpPS p
p + CnPS n
n · J can be related to that of Hˆ T . The correspondence is [21,15,16] A1=3 T (137) C : Z See Table 14 for calculations of the sensitivities of atomic EDMs to C PS . It is seen that this interaction induces EDMs much less eSciently than Hˆ SP and Hˆ T . It is the 199 Hg EDM measurement (126) that currently places the tightest constraint on C PS , C PS ↔ 4:6 × 103

CnPS = −(1:8 ± 0:8 ± 0:7) × 10−6 :

(138)

8.4.2. The electron EDM The best limits on the electron electric dipole moment are derived from measurements of atomic EDMs. Salpeter %rst noted the possibility of an enhancement of the electron EDM in atoms through consideration of the metastable 2s state in hydrogen [268]. As noted above, when magnetic e2ects are considered, the screening of the EDMs in an atom (Schi2 theorem) is lifted [90]. Sandars [223] pointed out that due to relativistic magnetic e2ects, the atomic EDM induced in heavy atoms can be strongly enhanced compared to the electron EDM inducing it. The value of the atomic EDM compared to the electron EDM is expressed through an enhancement factor K = datom =de :

(139) 3

The enhancement factor K increases with nuclear charge Z faster than Z . We will look brie?y at how an electron EDM induces an EDM of an atom as a whole; for a more detailed consideration we refer the reader to the works [269,224,15,16,270]. As we mentioned at the beginning of Section 8.1, there are two types of contributions to an atomic EDM arising from constituent EDMs. For the case  of electron EDMs these are the following: (i) the sum of the intrinsic EDMs of the electrons 0|de Ni=1 i0 5zi |0 ; and (ii) the admixture of opposite parity atomic  i ; see Eq. (123). Here 0 and states due to the pseudoscalar interaction Hˆ e = −de Ni=1 i0 2i · Eint 2 = 0 5  are Dirac matrices de%ned in Eq. (2), |0 is the unperturbed state of the atom (it is an eigenstate of the P; T -even Hamiltonian with no external electric %eld), and Eint is the internal atomic electric %eld. Let us consider for a moment  the Stark shift generated by the presence of the electron EDMs. It ˜ − de N i0 2i · Ei |0 , ˜ where |0 ˜ is an eigenstate of the P; T -even Hamiltonian has the form \E = 0| i=1 ˆ H which includes the external electric %eld and E = Eint + Eext is the total electric %eld. (When the external %eld is treated perturbatively and only terms %rst-order in this %eld are considered, this reduces to the linear Stark shift generated by the two contributions to the atomic EDM mentioned above.) It is convenient to break up the pseudoscalar interaction 0 2 · E = 2 · E + (0 − 1)2 · E, since the %rst term on the right-hand-side does not contribute to a linear Stark shift, 2 · E = (1=e)[2 · ∇; Hˆ ].

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

The enhancement factor for an electron in an atom can then be written as N

0| N (i − 1)2i · Ei |M M | N z i |0 i i int i=1 0 i=1 (0 − 1)5z |0 + 2e ; K(Jz =J ) = 0| E − E 0 M M i=1 where J is the electron angular momentum of the state |0 . The operator   0 0 (0 − 1)2 = 0 −2


125

(140)

(141)

mixes the lower components of the wave functions (20). So it is seen that this is a purely relativistic e2ect. In heavy atoms the %rst term in Eq. (140) is small compared to the second and so can be omitted in the calculations. The %rst analytical formulation of the enhancement factor was performed in Ref. [224]. The following expression was obtained for alkali atoms, in terms of quantities that can be determined experimentally,
4(Z#)3 r0m ˝c : (142) K= (J + 1)a2B (42 − 1)(J0 Jm )3=2 (Em − E0 ) m The sum over m is taken over the excited states of the external electron, J0 , Jm are e2ective principal  quantum numbers for the ground and excited states,  = (J + 1=2)2 − (Z#)2 , and r0m is the electric dipole radial integral. Taking into account mixing with only the nearest level, and assuming the values r0m = 5aB , J0 = Jm = 2, and Em − E0 = (1=10)Ry, we obtain: |K| ∼ 10

Z 3 #2 R ; J (J + 1=2)(J + 1)2

(143)

where R is a relativistic enhancement factor that increases with Z and is 1:2 for Rb and 2:8 for Fr in the ground states (R tends to unity when J is large). This simple formula (143) illustrates the dependence of the enhancement factor on Z 3 and on the angular momentum J and can be used to obtain order of magnitude estimates. It is seen that K is large for high Z and low J . For atoms with more complex con%gurations these formulae are not applicable and numerical calculations of the enhancement factor are required. As we mentioned earlier, experiments with paramagnetic atoms and molecules are most sensitive to the electron EDM. The current best limit on the electron EDM comes from the Tl measurement (125). The sensitivity of atomic thallium to the electron EDM is [271] d(205 Tl) = −585de

(144)

and accordingly the measurement (125) of the electron EDM is de = (6:9 ± 7:4) × 10−28 e cm :

(145)

There is some sensitivity of diamagnetic systems to the electron EDM [272], although this sensitivity is very weak. The dominant contribution appears in third-order perturbation theory due to consideration of the hyper%ne interaction,Eq. (131)  with Hˆ SP replaced with the interaction Hˆ e [21]. (The second-order “bare” contribution, 2 n 0|−de Ni=1 i0 2i |n n|Hˆ hf |0 =(E0 −En ), is signi%cantly smaller and can be neglected [21].) Another contribution [21] comes from the direct interaction of

126

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

the nuclear magnetic  %eld B (arising from the nuclear magnetic moment) with the electron EDM [268], Hˆ PT = −ide Ni=1 i · Bi . This latter interaction contributes in second-order, Eq. (123). The following relation between de and the P; T -odd tensor electron–nucleon interaction has been obtained [21,15,16],  

R 3 Gmp e CpT (146)
p + CnT
n ; de I=I ↔ √ 7 2;# (R − 1) p n R is a relativistic enhancement factor. Using the result of Ref. [266] for the calculation of datom (C T ) for 199 Hg and the above relation, the enhancement factor K(199 Hg) = −0:014 was found in Ref. [21]. A TDHF self-consistent calculation performed in [273] yielded the result d(199 Hg) = 1:16 × 10−2 de :

(147)

Due to huge polarization corrections, it is of opposite sign, and the same order of magnitude, as the lowest-order result, and its value may change with inclusion of correlation corrections [273]. While the enhancement factors are small for diamagnetic systems, the extraordinary precision that has been achieved in the Hg measurement (126) makes the corresponding measurement of the electron EDM, de =−(9:1±4:2±3:4)×10−27 e cm, comparable with those from the best paramagnetic EDM measurements. In Table 15 we list enhancement factors for both paramagnetic and diamagnetic atoms of experimental interest. Table 15 Enhancement factors of the electron EDM for atoms of interest

Paramagnetic

Diamagnetic a

Atom

Enhancement factor K = datom =de Semi-empirical

Rb Cs Fr Tl Xe 3 P2 Xe Hg

24a , 24.6b , 16.1c , 23.7c , 22.0c 119a , 131d , 138b , 138e , 80.3c , 106.0c , 100.4c 1150a −716i , −500e , −502c , −607c , −562c 130l , 120e

Ab initio 24.6c 114.9c , 114f , 115g 910(50)h −1041c , −301j , −179f , −585k −0.0008m , −0.0008g −0.014m , 0.0116g

Ref. [223]. Ref. [275]. c Ref. [76]. d Ref. [276]. e Ref. [224]. f Ref. [81]. g Ref. [273]. h Ref. [277]. i Ref. [278]. j Ref. [279]. k Ref. [271]. l Ref. [235]. m Ref. [21]. Extracted from the many-body calculation of datom (C T ) [266]; see Section 8.4.2. b

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

127

Paramagnetic polar diatomic molecules are attractive for electron EDM studies, in particular those with electron states 2 51=2 and 2 91=2 (see, e.g., [212,274] and the review [213]). In [212] an analytical estimate for the energy shift in such molecules (3 = 1=2) was made, ˆ | − de (0 − 1)2 · Etot | = d
· ;

|d | ∼

Z 3 #2 ede ; (42 − 1)a2B

(148)

where | are molecular orbitals built up from atomic orbitals mixed by the strong internal electric %eld. Molecules with electron ground state 2 51=2 include BaF, YbF, HgF, PbF. The same estimate (148) is valid for the metastable a(1)3 5 state in PbO with which an EDM experiment is in progress [280]; more re%ned calculations can be found in Refs. [281,282]. The recent measurement of the electron EDM in YbF yielded the result [236] de = (−0:2 ± 3:2) × 10−26 e cm

(149)

(calculations for the e2ective electric %elds have been performed in Refs. [283]). This is the %rst measurement of an EDM in a paramagnetic molecule, and while the limit on the electron EDM is not as impressive as that from Tl or even Hg, it is limited only by statistics. 8.4.3. P; T -violating nuclear moments Atomic EDMs can be induced if the nucleus possesses P; T -odd nuclear moments. These moments arise at the nucleon scale due to a P; T -violating interaction between nucleons or due to intrinsic nucleon EDMs. The induced nuclear moments can be electric or magnetic. For example, the following moments violate parity and time-reversal invariance: electric dipole, magnetic quadrupole, electric octupole. For the electric case, the interaction Hamiltonian that mixes opposite parity electron states, and induces an EDM of the atom, is of the form hˆPT = −e’, where ’ is the electrostatic potential of the nucleus corresponding to a P; T -odd charge distribution. In fact, due to Schi2’s theorem [90], there is an additional screening term which we will look at in Section 9. In the magnetic case Schi2’s theorem is not valid, and the interaction Hamiltonian of a relativistic electron with the vector potential A corresponding to a P; T -odd current distribution is simply hˆPT = e · A. See formula (123). The operator hˆPT has electronic and nuclear components. While the overall operator hˆPT is a scalar, the electronic and nuclear operators can be of any (equal) rank. Accordingly, the triangle rule for addition of angular momenta imposes restrictions on the angular momenta of the electron and nuclear states for nonzero matrix elements. For example, the electron interaction with the nuclear magnetic quadrupole moment cannot mix s and p1=2 electron states, since we must have |j1 − j2 | 6 2 6 j1 + j2 . Similarly, a static magnetic quadrupole moment of the nucleus cannot arise in nuclei with total angular momentum I ¡ 1. We leave a detailed consideration of P; T -odd nuclear moments for the next section. 9. P; T-Violating nuclear moments and the atomic EDMs they induce This section is devoted to a consideration of the P; T -odd nuclear moments that can induce atomic EDMs. In Sections 9.1 and 9.2 we look at the form of the P; T -violating electric and magnetic moments. In Section 9.3 we discuss how P; T -violating nuclear moments are induced by P; T -violating

128

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

mechanisms at the nucleon scale. Enhancement mechanisms for nuclear moments are reviewed in Section 9.4. Finally, in Section 9.5 we look at calculations of atomic EDMs induced by nuclear moments. 9.1. Electric moments; the SchiC moment When considering the P; T -odd electric moments of the nucleus, we must take note of an important screening phenomenon—the Schi2 theorem (we mentioned this at the beginning of Section 8.4). The electron screening is taken into account by using the following (screened) electrostatic potential of the nucleus (for a derivation, see, e.g., [284]):   e!(r) 3 !(r) 3 1 ’(R) = d r + (d · ∇) d r ; (150) |R − r| Z |R − r|  where ∇i ≡ 9=9Ri , !(r) is the nuclear charge density, !(r) d 3 r = Z, and  (151) d = er!(r) d 3 r = dI=I is the P; T -odd nuclear EDM. 27 We are interested in the contributions to ’ that are %rst order in the P; T -odd interaction. The %rst term on the right-hand side of Eq. (150) is P; T -odd if the charge density is distorted due to a P; T -odd interaction. The density in the second term can be considered spherical, since the nuclear EDM is P; T -violating (it arises due to a P; T -violating component of the density in Eq. (151)). If we consider the nucleus to be point-like, then we can perform a multipole expansion of potential (150) in terms of r=R. According to Schi2’s theorem, the nuclear electrostatic potential is screened by atomic electrons such that the dominant nuclear P; T -odd moment, the nuclear EDM, of a point-like nucleus cannot generate an atomic EDM. It is easy to see this from Eq. (150):     1 1 1 3 − e!(r) r · ∇ d r + (d · ∇) !(r) d 3 r = 0 : (152) R Z R The %rst nonzero P; T -odd term in Eq. (150) is then   1 1 1 1 (3) 3 e!(r)ri rj rk d r∇i ∇j ∇k + !(r)ri rj d 3 r : (153) ’ =− (d · ∇)∇i ∇j 6 R 2Z R The %rst term ri rj rk on the right-hand side of the equation is a reducible rank-3 tensor, while the second ri rj is a reducible rank-2 tensor. Separating the trace,   (154) ri rj rk = ri rj rk − 15 r 2 (ri %jk + rj %ik + rk %ij ) + 15 r 2 (ri %jk + rj %ik + rk %ij ) ri rj = [ri rj − 13 r 2 %ij ] + 13 r 2 %ij ;

(155)

it is seen that ’(3) is comprised of a rank-3 octupole potential ’octupole and a rank-1 “Schi2” potential ’Schi2 , ’(3) = ’octupole + ’Schi2 ; 27

(156)

The screening term appears as a result of a unitary transformation of the Hamiltonian which does not change the linear Stark shift. For exact atomic wave functions the result for the atomic EDM must be the same with and without the screening term. However, in real (approximate) calculations inclusion of the screening term is a must.

J.S.M. Ginges, V.V. Flambaum / Physics Reports 397 (2004) 63 – 154

129

where 1 1 1 1 1 Qij (d · ∇)∇i ∇j ’octupole = − Oijk ∇i ∇j ∇k + 6 R e 2Z R

(157)

’Schi2 = 4;S · ∇%3 (R) ;

(158)

and we have used ∇2 (1=R)=−4;%3 (R). S isthe P; T -odd nuclear Schi2 moment, Oijk is the P; T -odd nuclear electric octupole moment, and Qij = e!(r)(ri rj − 13 r 2 %ij ) d 3 r is the P; T -even nuclear electric quadrupole moment. The second term in Eq. (157) is small since only protons in the external shell contribute to Qij and there is a factor Z1 . The nuclear octupole and Schi2 moments are given by

 1 2 Oijk = e!(r) ri rj rk − r (ri %jk + rj %ik + rk %ij ) d 3 r ; (159) 5 

 1 5 1 2 3 2 3 e!(r)rr d r − d !(r)r d r = SI=I : S= (160) 10 3 Z Because the octupole moment Oijk carries three units of angular momentum it can only arise in nuclei with spin I ¿ 3=2, whereas the Schi2 moment can arise in nuclei with spin I ¿ 1=2 (due to the triangle rule for the addition of angular momenta). Of the atomic EDM measurements performed so far, only Cs (I = 7=2) has a nuclear spin large enough to have a static octupole moment; all other nuclei have spin I = 1=2. However, for these moments to induce an atomic EDM they must satisfy electronic angular momentum requirements. Due to the higher rank of the octupole moment it mixes electronic states of higher angular momentum than the Schi2 moment. This means that the atomic EDM induced by the octupole moment is smaller than that induced by the Schi2 moment because the wave functions of electrons with higher angular momentum penetrate the vicinity of the nucleus less due to the greater centrifugal barrier. The lowest value for the angular momentum of the electrons that can induce an atomic EDM due to mixing by the octupole moment is j = 3=2. [The conditions imposed on the allowed electronic angular momentum for mixing by the octupole moment is |j1 − j2 ] 6 3 6 j1 + j2 and that allowed by the electric dipole mixing is |j1 −j2 ] 6 1 6 j1 +j2 , so the conditions for inducing an atomic EDM are |j1 − j2 ] 6 1 and j1 + j2 ¿ 3.] This means that s states cannot contribute to the EDM produced by the electric octupole moment, so that in fact the octupole moment of the Cs nucleus cannot induce an atomic EDM in Cs in the ground state, as this state corresponds to a con%guration with a single electron in an s-state above closed shells. Static nuclear octupole moments and the atomic EDMs they induce have been considered in detail in Ref. [285]. The EDMs they induce in atoms are very small, so we will consider them no further. It is therefore obvious that the Schi2 moment is the only P; T -odd moment that induces an EDM in atoms with closed electron subshells such as Xe and Hg. (The nuclear magnetic quadrupole moment, which will be discussed in the next section, cannot induce an atomic EDM in systems with zero electron angular momentum, since there is no magnetic %eld of the electrons for the MQM to interact with. The same conclusion also follows from the triangle rule applied to Eq. (123).) In fact, all the atoms for which EDM measurements have been performed (133 Cs (I = 7=2; J = 1=2), 205 Tl (I = 1=2; J = 1=2), 129 Xe 3 P2 (I = 1=2, J = 2), 129 Xe 1 S0 (I = 1=2, J = 0), 199 Hg (I = 1=2, J = 0)) can have contributions from the nuclear Schi2 moment, however it is only Cs that can have an EDM arising due to a static magnetic quadrupole moment.

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Let us consider the form of the atomic EDM (123) induced  by the interaction of electrons with the Schi2 moment (158). The contact interaction Hˆ PT = −e i ’iSchi2 mixes s- and p-wave electron orbitals and produces EDMs in atoms. The expression (158) is consistently de%ned for nonrelativistic electrons. Using integration by parts, it is seen that the matrix element s| − e’Schi2 |p is %nite, s| − e’Schi2 |p = 4;eS · (∇

†

s

p )R=0

= constant :

(161)

However, atomic electrons near the nucleus are ultra-relativistic, the ratio of the kinetic or potential energy to mc2 in heavy atoms is about 100. For the solution of the Dirac equation, (∇ s† p )R→0 → ∞ for a point-like nucleus. Usually this problem is solved by a cut-o2 of the electron wave functions at the nuclear surface. However, even inside the nucleus ∇ s† p varies signi%cantly, ≈ Z 2 #2 , where # is the %ne-structure constant, Z is the nuclear charge. In Hg (Z = 80), Z 2 #2 = 0:34. A more accurate treatment requires the calculation of a new nuclear characteristic which has been termed the local dipole moment (LDM) [286]. This moment takes into account relativistic corrections to the nuclear Schi2 moment which originate from the electron wave functions. So in the nonrelativistic limit, Z# → 0, the LDM L = S. For 199 Hg, L ≈ S(1 − 0:8Z 2 #2 ) ≈ 0:75S. When considering the interaction of atomic electrons with the LDM it is de%ned as placed at the centre of the nucleus, that is the electrostatic potential is ’(R) = 4;L · ∇%3 (R) :

(162)

See Ref. [286] for the explicit form for L. It is more convenient to use a real electric %eld distribution produced by a P; T -odd perturbation. In [286] it was shown (by considering several nuclear models) that the natural generalization of the Schi2 moment potential for a %nite-size nucleus is 15S · R ’(R) = − n(R − RN ) ; (163) R5N where RN is the nuclear radius and n(R − RN ) is a smooth function which is 1 for R ¡ RN − % and 0 for R ¿ RN + %; n(R − RN ) can be taken as proportional to the nuclear density. This form for the electrostatic potential has no singularities and is suitable for relativistic atomic calculations. 9.1.1. The P; T -odd electric <eld distribution in nuclei created by the nuclear SchiC moment From the new form for the electrostatic potential Eq. (163) it can easily be seen that the Schi2 moment gives rise to a constant electric %eld inside the nucleus (see Fig. 24), E = −∇’. The correlation between the electric %eld and the nuclear spin, E ˙ I, is naturally P; T -odd. This electric %eld polarizes atomic electrons, producing an EDM of the atom. 9.2. Magnetic moments; the magnetic quadrupole moment In the gauge ∇ · A = 0 the vector potential produced by a steady current is  j(r) d3 r ; A(R) = |R − r|

(164)

where j is the vector current density. The lowest-order term in the multipole expansion of Eq. (164) is the P; T -even magnetic dipole moment, and we have no interest in this. The lowest-order P; T -odd

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131

E

Fig. 24. Constant electric %eld E inside the nucleus produced by the P; T -odd interaction (Schi2 moment %eld). E is directed along the nuclear spin I.

moment arises in second-order and is the rank-2 magnetic quadrupole moment (MQM); it appears alongside the P-odd, T -even anapole moment a (see Section 7) [174],  1 1 (2) jk rl ri d 3 r∇l ∇i (165) Ak = 2 R

1 1 1 (%ki al − %li ak ) − jklm Mim ∇l ∇i (166) = 4; 6 R = Aak + AMQM ; k where the anapole moment is given by Eq. (98) and the MQM is  Mim = − (ri jmnp + rm jinp )jn rp d 3 r :

(167)

(168)

The P; T -odd component of the current density j will give rise to a nonzero magnetic quadrupole moment. Its form is speci%c to the P; T -odd mechanism creating it. For instance, if we consider that it is produced by an external nucleon perturbed by P; T -odd nuclear forces, then we can use current (104), 

  e 2 Mij = 3 ri j + rj i − %ij
· r + 2q(ri lj + rj li ) !(r) d 3 r : (169) 2mp 3 The magnetic quadrupole moment can now be calculated using a P; T -odd perturbed density !(r). A general expression for the MQM can be constructed in terms of the total angular momentum of the system I,

M 3 2 Ii Ij + Ij Ii − I (I + 1)%ij : (170) Mij = 2 I (2I − 1) 3 The quantity M is conventionally referred to as the MQM and is de%ned as the maximum projection of Mij on the nuclear axis, M = Mzz . It is easily seen from a comparison of Eqs. (168), (170) that the magnetic quadrupole moment violates parity and time-reversal invariance.

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The interaction of electrons with a nuclear MQM induces an atomic EDM typically an order of magnitude larger than that induced by the nuclear Schi2 moment [260,174]. The ratio of the s − p electronic matrix elements is [174] s| · AMQM |p RM ; ∼ 102 A−2=3 s|’Schi2 |p RS

(171)

RM , RS are relativistic enhancement factors for the magnetic quadrupole and Schi2 moments, respectively, R → 1 as Z# → 0. It is seen that for atoms with light nuclei the contribution of the MQM dominates. The relativistic factor RS grows faster than RM with increase of Z. For example, at Z = 80, RS = 7[p1=2 ]; 5[p3=2 ] while RM = 1:8; explicit formulae can be found in Ref. [174]. (In the square brackets the angular momentum of the p electron state is speci%ed; for the MQM, p ≡ p3=2 .) At Z = 80; A = 200, the ratio (171) reaches ∼ 1. 9.2.1. The spin hedgehog In a spherically symmetric system the P; T -odd interaction induces a “spin-hedgehog” whereby the spin density is proportional to the radial vector,
˙ r [287,288]. The P; T -odd nucleon– nucleon interaction (leading to the perturbed wave functions (179), see below) produces the following distributions of the spins for protons and neutrons in the nucleus:
p (r) = Bp ∇!p (r) ;


n (r) = Bn ∇!n (r) ; (172)  2 the unperturbed nuclear density ! = | | . This collective spin distribution, however, produces no current (j(r) = ∇ ×
(r) ˙ ∇ × ∇!(r) = 0) and hence no magnetic %eld [288]. One may think that because the spin-hedgehog has no magnetic %eld it produces no e2ects. This is not the case. The spin-dependent part of the strong interaction is sensitive to this spin structure. It reduces the constants of the P; T -odd nucleon–nucleon interaction: for the case of an external proton interacting with the spin-hedgehog, )p → )p =1:5, while for a neutron, )n → )n =1:8 [288,203]. A distorted spin hedgehog in deformed nuclei produces a collective magnetic quadrupole %eld (see Section 9.4.2). Note that the spin-hedgehog is not speci%c to nuclei. For example, the P; T -odd electron–nucleon interaction (Section 8.4.1) mixes atomic states of opposite parity and induces a spin-hedgehog of the atom [287]; see, e.g., [15,16] for details. 9.3. What mechanisms at the nucleon scale induce P; T -odd nuclear moments? P; T -odd nuclear moments can arise due to an intrinsic EDM of an external nucleon or due to P; T -odd nuclear forces. The P; T -odd nuclear forces induce larger nuclear moments than a single nucleon EDM (Section 9.3.3). This is what makes atomic experiments so competitive compared to neutron experiments in probing CP-violation in the hadron sector. As we will see in Section 10, atomic experiments are more sensitive than neutron experiments to many underlying CP-violating mechanisms. 9.3.1. The P; T -odd nucleon–nucleon interaction The P; T -odd nucleon–nucleon interaction is the dominating nuclear mechanism inducing atomic EDMs in diamagnetic atoms and molecules.

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133

The P; T -odd nucleon–nucleon interaction, to %rst-order in the velocities p=mp , can be presented as [174] G 1 (()ab
a − )ba
b ) · ∇a %3 (ra − rb ) + )ab [
a ×
b ]{(pa − pb ); %3 (ra − rb )}) ; Wˆ ab = √ 2 2mp

(173)

where { ; } is an anticommutator and
; r, and p are the spins, coordinates, and momenta of the nucleons a and b. The dimensionless constants )ab and )ab characterize the strength of the P; T -odd nucleon–nucleon interaction and are determined from EDM measurements. If we consider the P; T -odd interaction between a single unpaired nucleon and a heavy spherical core, then we can average the two-particle interaction (173) over the core nucleons to obtain the e2ective single-particle P; T -odd interaction between the nucleon and core [174], G )a Wˆ = √
· ∇!A (r) : 2 2mp

(174)

Here it has been assumed that the proton and neutron densities are proportional to the total nuclear density !A (r); the dimensionless constant )a =

Z N )ap + )an : A A

(175)

Notice that there is only one surviving term from the P; T -odd nucleon–nucleon interaction (173); this is because all other terms contain the spin of the internal nucleons for which 
= 0. The shape of the nuclear density !A and the strong potential U are known to be similar; we therefore take !A (r) =

!A (0) U (r) : U (0)

(176)

Then Eq. (174) can be rewritten in the following form: !A (0) G = −2 × 10−21 ) cm : B=) √ U (0) 2 2mp

Wˆ = B
· ∇U ;

(177)

Now it is easy to %nd the solution of the SchrZodinger equation including the interaction Wˆ [174], (Hˆ + Wˆ ) ˜ = E ˜ ; ˜ = (1 + B
· ∇)

(178) ;

(179)

where is the unperturbed solution (Hˆ (179) is ! = ˜† ˜ =

†

+ B∇ · (

†


) :

= E ). The density arising from the wave function (180)

The second term is the P; T -odd part of the density which generates the nuclear P; T -odd moments. The electric dipole (151), Schi2 (160), and magnetic quadrupole (169), (170) nuclear moments induced by the P; T -odd nucleon–nucleon interaction, through the perturbed density (180),

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are [174]



Z d = −eB q − A eq S =− B 10 M=



 tI +

tI ; 1 I +1

(181) 

2 rex

5 − tI rq2 3

e B( − q)(2I − 1)tI ; mp

;

(182) (183)

2 are the mean-square radii of the nuclear where q = 0(1) for an external neutron (proton), rq2 and rex charge and external nucleon, respectively, and  I = l + 1=2 ;  1; (184) tI =  − I ; I = l − 1=2 ; I +1

where l is the orbital angular momentum of the external nucleon. The recoil e2ect for the electric moments (the motion of the nuclear core around the centre-of-mass; see Ref. [174]) has been taken into account (q → q − Z=A in the expression for d). In the single-particle model, the recoil e2ect for the Schi2 moment disappears due to the cancellation of its contributions to the %rst and second (screening) terms in Eq. (150). Note that for nuclei with an unpaired proton in the state s1=2 (in the simple shell model), such as 203;205 Tl, the Schi2 moment is reduced to the di2erence of two approximately equal terms, 2 − rq2 ) : S(s1=2 ) ˙ (rex

(185)

In obtaining numerical values for the Schi2 moment in an analytical calculation it is usually assumed 2 = rq2 = (3=5)R2N , where RN = r0 A1=3 , r0 = 1:1fm. Then the Schi2 moment Eq. (185) vanishes. that rex This cancellation makes calculations for 203;205 Tl unstable. The moments we have discussed so far are produced by a valence nucleon. In the work [289] it was shown that core nucleons make a contribution to P; T -odd moments that is comparable to that of a valence nucleon. The atom of particular interest to us is 199 Hg, which gives the best limit on the nuclear Schi2 moment. In the 199 Hg nucleus the unpaired nucleon is a neutron. It does not contribute to the Schi2 moment directly, see Eq. (182). The nuclear Schi2 moment arises due to the polarization of the protons of the core by the P; T -odd %eld of the external neutron. (The charge distribution must be distorted to give a P; T -odd correction to the charge density.) The strength of the P; T -odd interaction is de%ned by the parameter )np . A numerical calculation of the Schi2 moment for 199 Hg in the Woods–Saxon potential with spin–orbit interaction gives [290,289] S(199 Hg) = −1:4 × 10−8 )np e fm3 :

(186)

An analytical treatment of the electron relativistic corrections to the Schi2 moment (that is, calculation of the local dipole moment) of 199 Hg shows that these corrections are small—they reduce the Schi2 moment S only by about 25%. A many-body treatment of the 199 Hg Schi2 moment has been

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135

performed recently in the work [291]. A %nite-range P; T -odd nucleon–nucleon interaction was used and core polarization was calculated in the random-phase approximation. The result of Ref. [291] is S(199 Hg) = −0:0004 ggQ0 − 0:055 ggQ1 + 0:009 ggQ2 e fm3 ;

(187)

where g ≡ g;NN is the strong pion–nucleon coupling constant and gQ0 , gQ1 , gQ2 are the P; T -violating isoscalar, isovector, and isotensor pion–nucleon couplings, respectively; see, e.g., Refs. [261,292] for the form of the %nite-range P; T √ -violating interaction. Eqs. (186), (187) can be compared by using the relation [291] )np ∼ (Gm2; = 2)−1 g(gQ0 + gQ1 − 2gQ2 ) ∼ 7 × 106 g(gQ0 + gQ1 − 2gQ2 ), so Eq. (186) gives S(199 Hg) ∼ −0:09 g(gQ0 + gQ1 − 2gQ2 ) e fm3 . It is seen that while the contribution of the isovector channel does not change much from the value in Eq. (186), the isoscalar channel is suppressed by two orders of magnitude and the isotensor channel by one order of magnitude. These corrections are due largely to the inclusion of core polarization [291]. As we mentioned earlier, of the EDM experiments performed so far, only the 133 Cs measurement can be interpreted in terms of a magnetic quadrupole moment of the nucleus. A calculation in the Woods–Saxon potential with the spin–orbit interaction gives [174] e M (133 Cs) = 1:7 × 10−7 )p fm : (188) mp Single-particle calculations have been performed in [293] for the external nucleon contribution and in [294] for the core contribution. These calculations show that the MQM is very sensitive to the shape of the P; T -odd potential relative to the shape of the central %eld potential and to the spin– orbit potential. It was found that the core contribution arising from the interaction proportional to ) is comparable to that of the valence contribution, and that if ) ∼ ) then the core contribution is several times larger than the valence contribution. Results of calculations of nuclear P; T -odd moments of current interest are presented in Table 16. 9.3.2. The external nucleon EDM Even though the nucleon–nucleon interaction may be more e2ective in inducing nuclear P; T -odd moments and hence atomic EDMs, measurements of nucleon EDMs are interesting in their own right. Also, nucleon EDMs measured from atomic experiments may be compared to those from direct measurements. Currently, the limit on the neutron EDM from the Hg measurement (126) is competitive with those from direct neutron EDM searches (127); see Tables 19 and 20. Here we will merely quote the results of the work [260] for the nuclear Schi2 and magnetic quadrupole moments induced by a single unpaired nucleon (neutron or proton) with an intrinsic EDM dn; p ,  

1 1 5 2 2 dn; p tI + r − tI r ; (189) S= 10 I + 1 ex 3 q M=

1 dn; p (2I − 1)tI : mp

(190)

See, e.g., [260,16] for details. Notice the similarity between these expressions and those for the corresponding moments induced by P; T -odd nuclear forces (182), (183). In the approximations used, there is a simple correspondence between the P; T -violating parameters. For the Schi2 moment, −eqB ↔ dn; p , while for the MQM, −eB(q − ) ↔ dn; p . The second contribution (˙ ) to the MQM from the P; T -odd nuclear forces appears from the spin term in (104). This has no analogue in the

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Table 16 Calculations of P; T -violating moments in spherical nuclei induced by the P; T -odd nucleon–nucleon interaction Atom

S(e fm3 ) × 108

129

Xe Hg 133 Cs

1:75)np a −1:4)np a 3:0)p b

203;205

−2)p b 1:2)pp − 1:4)pn a e

199

Tl

M (e=mp fm) × 107

1:7)p b 1:6)p , 2:6)p , 1:8)p c −0:3)pp − 0:2)pn − 2:5)pp + 1:7)pn d

a

Calculated using wave functions and Green’s functions found in the Woods–Saxon potential with spin–orbit interaction, Refs. [290,289]. b Woods–Saxon potential with spin–orbit interaction, Ref. [174]. c Contributions of the valence nucleon carried out in di2erent potentials: Eq. (176), harmonic oscillator, Woods–Saxon, respectively. Ref. [293]. d Core contribution, Ref. [294]. e This value is comprised of a valence nucleon and core contribution. The valence proton gives a contribution of −0:04)pp − 1:4)pn while the core contribution is 1:23)pp , in the units in the table. The di2erence between the valence contributions of the works [290,289] and [174] is due to a di2erence in the potentials used; remember that in 203;205 Tl the Schi2 moment is very sensitive to the potential, see Eq. (185).

case of dn; p since a stationary EDM cannot induce a MQM. It is the orbital motion of a nucleon with an intrinsic EDM that induces the nuclear MQM [206]. In the simple shell model the unpaired neutron in the 199 Hg nucleus carries the nuclear spin I , and the Schi2 moment is induced by the EDM of just this neutron. In this picture, the induced Schi2 2 moment can be calculated using Eq. (189) and the simplifying assumptions rex = rq2 = (3=5)R2N , RN ≈ 1:1A1=3 fm, giving S(199 Hg) ≈ 2:2dn fm2 . Proton EDMs also contribute due to con%guration mixing. It is possible to estimate their contribution by comparing the experimental value of the magnetic moment of 199 Hg with that of the simple shell model; see Ref. [267]. In this way, it is found [295] that dn can be replaced by (dn + 0:1dp ), S(199 Hg) = (2:2 dn + 0:2 dp ) fm2 :

(191)

A numerical calculation of the nuclear Schi2 moment of 199 Hg induced by neutron and proton EDMs has recently been performed, with core polarization accounted for in the RPA approximation [296]. The result is [296] S(199 Hg) = (1:9 dn + 0:2 dp ) fm2 :

(192)

(See [296] for the discussion of uncertainty.) 9.3.3. Comparison of the size of nuclear moments induced by the nucleon–nucleon interaction and the nucleon EDM Here we consider the enhancement of the nuclear EDM induced by P; T -odd nuclear forces compared to that induced by an external valence nucleon using the simple one-boson exchange model,

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137

following [290,289,297]. The largest contribution to the constant ) is probably given by the lightest ;0 -meson, ggQ 0 G √ ) ≈ ;2 ; m; 2 g and gQ;0 are the constants of the strong and T -odd ; meson–nucleon interactions, √ 0 − † Q + 2(gpi Q 5 n + gQ;− pn)(; Q ) + ··· : (gni Q 5 n + gQ;0 pp);

(193)

(194)

A neutron EDM is induced through virtual creation of a ;− meson [298], dn =

e ggQ;− M ln : mp 4;2 m;

(195)

Here M ∼ m! ∼ 770 MeV is the scale at which the ;-meson loop converges. The values of ) and dn are expressed in terms of di2erent quantities, gQ;0 and gQ;− , respectively. However, for example, Q [298]. Taking |ggQ;0 | ∼ |ggQ;− |, in the model of T -violation with the -term, |ggQ;0 | = |ggQ;− | = 0:37|| it is found that [174,290,289,297] eB d ∼ ∼ 2;(m2; r03 |U (0)|)−1 ∼ 40 ; dn dn

(196)

that is, the nuclear EDM exceeds the nucleon EDM by one to two orders of magnitude. Similarly, P; T -odd nuclear forces generate all P; T -odd nuclear moments, such as Schi2 and MQM moments, 10–100 times larger than those generated by the presence of a nucleon EDM [297]. 9.4. Nuclear enhancement mechanisms So far we have considered P; T -odd nuclear moments in spherical nuclei. However, in nonspherical nuclei there is the possibility of enhancement due to (i) the presence of a low-lying level with opposite parity and the same angular momentum with respect to the ground state; and (ii) collective e2ects. 9.4.1. Close-level enhancement It is known that nuclei with nonspherical symmetry have close levels of opposite parity. It was pointed out in Ref. [299] that due to the existence of a close level of opposite parity with the same angular momentum as the ground state, the nuclear EDM can be enhanced; calculations of enhanced EDMs and MQMs were performed in [261], Schi2 moments in [174]. In the frozen frame, the contribution to the z-components of the electric dipole, Schi2, and magnetic quadrupole nuclear moments in the ground state 3 due to the close opposite parity state 3Q is T =2

Q 3| Q Tˆ |3 3|Hˆ PT |3  ; E3 − E3Q

(197)

where T = dz ; Sz ; Mzz . The magnetic quadrupole moment in heavy stable nuclei can be enhanced by an order of magnitude due to the “close level” mechanism, while for the electric dipole and Schi2 moments this enhancement hardly exceeds ∼ 5–10 times [174]. However, this “close-level”

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enhancement is not regular: it gives contributions to P; T -odd moments with di2erent magnitudes and signs even in the “nearest” nuclei, the results are unstable. 28 9.4.2. Collective enhancement While the “close-level” mechanism enhances the contribution of the external nucleon to the P; T -odd nuclear moments, there can also be a “collective” enhancement of the nuclear moments that occurs due to the contribution of many nucleons. In the work [288] it was shown that a collective magnetic quadrupole moment can be produced in deformed nuclei by P; T -odd nuclear forces. Unlike the close-level enhancement, this collective enhancement is regular: in deformed nuclei about A2=3 nucleons belong to open shells that contribute to the MQM. The P; T -odd nuclear forces create a spin hedgehog [Eq. (172)] as in the case of spherical nuclei (Section 9.2.1), however in the deformed case there is a nonzero magnetic %eld associated with it. The MQM of a deformed nucleus (in the rotating frame) can basically be calculated as a summation of the single-particle MQMs (183) of all nucleons in the open shells. Notice, from Eqs. (183), (184), that spin–orbit pairs I = l + 1=2 and I = l − 1=2 make contributions to the collective MQM of opposite sign. A suSciently large spin–orbit splitting is therefore required to avoid cancellation, and this is satis%ed in nuclei. This collective mechanism gives an order of magnitude enhancement of the nuclear MQM in deformed nuclei compared to spherical nuclei. 9.4.3. Octupole deformation; collective SchiC moments We will now move on to the collective P; T -odd nuclear moments produced by P; T -odd nuclear forces that arise in nuclei with static octupole deformation [300,284,285]. There is an enhancement of these collective moments, compared to single-particle nuclear moments, due to the collective nature of the intrinsic moments and the small energy separation between members of parity doublets. This enhancement can be as large as 1000 times. Static octupole deformation in the ground state has been demonstrated to exist in nuclei in the regions Ra–Th and Ba–Sm. It produces e2ects such as parity doublets, large dipole and octupole moments in the intrinsic frame of reference, and enhanced electric dipole and octupole transitions; see the review Ref. [301]. While it has been shown that the Schi2 and electric dipole and octupole moments are enhanced in nuclei with octupole deformation, we will focus our attention on the nuclear Schi2 moment (the EDM is not of direct interest, in atoms it is screened by atomic electrons; the atomic EDM induced by the electric octupole moment is small since it does not mix s and p electron orbitals.) The mechanism generating collective P; T -odd moments is the following. In the “frozen” body frame collective moments can exist without any P; T -violation. However, the nucleus rotates, and this makes the expectation value of these moments vanish in the laboratory frame if there is no P; T violation. (For example, the intrinsic Schi2 moment is directed along the nuclear axis, Sintr = Sintr n, and in the laboratory frame the only possible correlation n ˙ I violates parity and time reversal invariance.) The P; T -odd nuclear forces mix rotational states of opposite parity and create an average The reason is explained in Ref. [174]. Consider the P; T -odd interaction Hˆ PT given by Eq. (177), Wˆ = B
· ∇U ˙ [Hˆ ;
· p], where Hˆ is the single-particle Hamiltonian. It is seen that this has small matrix elements between close levels, Q ˙ E3 − E3Q . 3|[Hˆ ;
· p]|3 28

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139

orientation of the nuclear axis n along the nuclear spin I, nz = 2#

KM ; I (I + 1)

(198)

where #=



ˆ | + − |W

(199)

E+ − E −

is the mixing coeScient of the opposite parity states, K =|I ·n| is the absolute value of the projection of the nuclear spin I on the nuclear axis, M = Iz , and Wˆ is the e2ective single-particle potential (174). The Schi2 moment in the laboratory frame is Sz = Sintr nz = Sintr

2#KM : I (I + 1)

(200)

In the “frozen” body frame the surface of an axially symmetric deformed nucleus is described by the following expression:  
R() = RN 1 + l Yl0 () : (201) l=1

To keep the centre-of-mass at r = 0 we have to %x 1 [302]:  3
(l + 1)l l+1  1 = −3 : 4; (2l + 1)(2l + 3) l=2

(202)

Assuming that the distributions of the protons and neutrons are the same, the electric dipole moment er = 0 (since the centre-of-mass of the charge distribution coincides with the centre-of-mass) and hence there is no screening contribution to the Schi2 moment. We also assume constant density for R ¡ R(). The intrinsic Schi2 moment Sintr is then [300,284] 3
(l + 1)l l+1 92 3 √  Sintr = eZR3N ; (203) ≈ eZR3N 20; 20; 35 (2l + 1)(2l + 3) l=2 where the major contribution comes from 2 3 , the product of the quadrupole 2 ∼ 0:1 and octupole 3 ∼ 0:1 deformations. The estimate of the Schi2 moment in the laboratory frame gives [284] S ∼ #Sintr ∼ 0:05 e2 32 ZA2=3 )r03

eV ∼ 700 × 10−8 ) e fm3 ; |E+ − E− |

(204)

where r0 ≈ 1:2 fm is the internucleon distance, |E+ − E− | ∼ 50 keV. This estimate (204) is about 500 times larger than the Schi2 moment of a spherical nucleus like Hg (see Eq. (186)). See Table 17 for calculations [284] of Schi2 moments in nuclei assuming static octupole deformation. An attractive candidate for P; T -odd studies is radium, and recently several laboratories around the world have considered performing EDM experiments with it. As well as the possibility for a large Schi2 moment, the atomic EDM is large due to high Z. And if measurements can be performed for metastable atomic states, further enhancement can occur due to the presence of close opposite parity levels [171,172]. A Woods–Saxon calculation [284] gives the following magnitude

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Table 17 Nuclear Schi2 moments Sintr and S (in rotating and laboratory frames, respectively) calculated in the Woods–Saxon potential. Static octupole deformation is assumed 223

Sintr (e fm3 )a S() e fm3 ) × 108a a

Ra

24 400

225

Ra

24 300

223

221

Rn

15 1000

21 43

Fr

223

Fr

20 500

Numbers are from Ref. [284]. See [284] for details.

for the Schi2 moment of

225

Ra

S(225 Ra) = 300 × 10−8 )np e fm3 : Recently, a self-consistent calculation of the nuclear Schi2 moment of core polarization taken into account [303], S(225 Ra) = 5:06ggQ0 + 10:4ggQ1 − 10:1ggQ2 e fm3 :

(205) 225

Ra was performed, with (206)

This calculation was carried out in the approximation of the zero-range P; T -odd interaction. It was found that the Schi2 moment in the rotating frame is up to twice as large as the value calculated in [284] (see Table 17). However, the Schi2 moment in the laboratory frame was found to be suppressed by between 1.5 and 3 times due to suppression of the matrix element of the P; T -odd interaction. [Comparison between Eqs. (205), (206) can be made using the relation following Eq. (187); Eq. (205) then gives S(225 Ra) ∼ 20 g(gQ0 + gQ1 − 2gQ2 ) e fm3 .] Improved calculations (taking into account the %nite-range of the P; T -odd interaction) are in progress [303]. It is seen by comparison with the calculations for 199 Hg (186), (187) that the radium Schi2 moment is several hundred times larger. Note that S in Eq. (204) is proportional to the squared octupole deformation parameter 32 . In Ref. [304] it was pointed out that in nuclei with a soft octupole vibration mode 32 ∼ (0:1)2 , i.e., the Schi2 moments induced in nuclei with a soft octupole vibration mode are of the same magnitude as those induced in nuclei with static octupole deformation. In a recent work [305] Schi2 moments of nuclei with soft octupole vibrations were calculated and found to have a similar enhancement as in the static case. This means that a number of heavy nuclei can have large collective Schi2 moments. The e2ect of static octupole deformation on the size of the single-particle MQM was considered in [285]. It was found that generally there is no signi%cant enhancement due to this mechanism. There are several experiments in preparation aimed to detect EDMs of heavy atoms with deformed nuclei; see Section 10. 9.5. Calculations of atomic EDMs induced by P; T -violating nuclear moments; interpretation of the Hg measurement in terms of hadronic parameters Results of atomic calculations of EDMs (of current interest) induced by nuclear Schi2 moments are presented in Table 18. Of particular interest is the calculation for 199 Hg. A recent calculation for

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141

Table 18 Ground state EDMs of diamagnetic atoms induced by nuclear Schi2 moments. Units: 10−17 (S=(e fm3 ))e cm 129

223

Xe

0.27a , 0.38b

Rn

2.0c , 3.3b

199

Hg

−4.0d , −2.8b

225

Ra

−7.0c , −8.5b

a

Ref. [267]. Relativistic Hartree–Fock calculation. Ref. [295]. Average of two ab initio many-body calculations; core polarization and correlation corrections included. c Ref. [284]. Estimate found by scaling (with Z) calculations for lighter analogous atoms. Radon result scaled from xenon calculation [267]; radium result scaled from mercury calculation [290,289]. d Ref. [290,289]. Estimated from the calculation of datom (C T ) for mercury performed in Ref. [266]. b

the atomic EDM induced by the Schi2 moment yielded the result [295]  199  199 −17 S( Hg) d( Hg) = −2:8 × 10 e cm : e fm3

(207)

This value was obtained using the new %nite-size form for the Schi2 potential (163) and is the average of two calculations: one performed in the potential Vˆ N with core polarization taken into account using the TDHF method and the other performed in the Vˆ N −2 potential using the combined MBPT+CI method. The error of the result (207) is about 20%. The value used previously for d(199 Hg) induced by the nuclear Schi2 moment, d(199 Hg) = −4 (in the same units as Eq. (207)), was estimated in Refs. [290,289] from an atomic calculation [266] of the EDM induced by the tensor electron–nucleon interaction. From (126), (207), the best limit on the Schi2 moment follows: S(199 Hg) = (3:8 ± 1:8 ± 1:4) × 10−12 e fm3 :

(208)

As we have discussed, the Schi2 moment can be induced from a number of P; T -violating mechanisms: due to a P; T -violating nucleon–nucleon interaction or due to a static EDM of an unpaired nucleon. The limit on )np from Eqs. (208), (186) is )np = −(2:7 ± 1:3 ± 1:0) × 10−4 :

(209)

The limits on neutron and proton EDMs are [Eqs. (208), (191)] dn = (1:7 ± 0:8 ± 0:6) × 10−25 e cm ;

(210)

dp = (17 ± 8 ± 6) × 10−25 e cm :

(211)

Using instead the recent calculation (192) will not change the limits on dn and dp . The result (187) suggests that the limits on the isoscalar and isotensor P; T -odd couplings may be substantially weaker than those that would follow from Eq. (209) using Eq. (186).

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10. Current limits on fundamental P; T-violating parameters and prospects for improvement 10.1. Summary of limits EDM measurements have already excluded several models of CP-violation and the parameter space of currently popular models is strongly constrained. In this review we do not discuss the sensitivities of the various EDM measurements to di2erent CP-violating models; for such an analysis, see, e.g., the reviews [306,263,270] and the book [16]. The problem is that in new theories there are many free parameters, and this gives a whole range of possible values for CP-violating e2ects. Here we compare the sensitivities of di2erent measurements to fundamental CP-violating interactions by placing limits on phenomenological parameters. In Tables 19 and 20 we present the best limits on fundamental P; T -violating parameters extracted from EDM measurements in atoms, molecules, and neutrons. In previous sections we presented limits on hadronic and semi-leptonic CP-violating parameters at the nucleon scale [nucleon EDMs, Eqs. (127), (210) and (211); nucleon–nucleon interaction, Eq. (209); electron–nucleon interaction, Eqs. (134), (135) and (138)] and in the leptonic sector directly on the electron EDM [Eq. (145)]. These are summarized in Table 19. Here we constrain CP-violation at the quark scale from limits at the nucleon scale. Fig. 23 shows which of these parameters are related. We will begin with a discussion of limits on CP-violating parameters in the hadronic sector. Currently, the best limits on the neutron EDM come from direct neutron measurements (127), and the limit from mercury is not far behind (210). The best proton EDM limit comes from mercury (211). Constraints on nucleon EDMs constrain the following parameters at the quark level: quark EDMs, chromoelectric dipole moments (CEDMs) [CEDMs are analogous to quark EDMs, though with the external electromagnetic %eld replaced by a gluonic %eld], P; T -odd quark–quark four-fermion ˜ and another parameter de%ning interactions, the QCD phase de%ning the strength of the term G G, the strength of the term GG G˜ [in all models of CP violation considered this parameter is not the dominating mechanism inducing nucleon EDMs [312], so we will not discuss it further]. Limits on nucleon EDMs can also constrain P; T -odd lepton–quark four-fermion interactions which induce quark EDMs at the one-loop level [313]; however, the limits involving light quarks are weaker than those obtained at tree-level from atomic measurements. We use the following calculations to relate the nucleon and quark parameters, with the corresponding limits presented in Table 20. The limit on the QCD  term is arrived at using the relation obtained Q cm. To relate dn to the quark CEDMs d˜ q and quark EDMs dq we use in Ref. [311], dn =1:2×10−16 e the recent calculation [310], dn =(1±0:5)[0:55e(d˜ d +0:5d˜ u )+0:7(dd −0:25du )]. For the P; T -odd quark –quark interactions we use the phenomenological parameters [307,308,16] de%ned by the Hamiltonian Q J d) + ktc (ui Q J t a d)], hˆqq = √G2 [ks (qQ1 i5 q1 )(qQ2 q2 ) + ksc (qQ1 i5 t a q1 )(qQ2 t a q2 ) + kt (ui Q 5 J u)(d Q 5 J t a u)(d where t a are the SU (3) generators and the quarks q1 ; q2 = u; d. From the limit on the P; T -odd nucleon–nucleon interaction ) from mercury√follows the best limit on the P; T -odd pion–nucleon coupling g; Q they are related through ) ≈ ggQ 2=(Gm2; ), where Q (Ref. [298]) is used to place a conthe strong coupling g = 13:6. The relation |g| Q ≈ 0:027|| straint on the QCD  term, tighter than those obtained from direct neutron measurements. Limits on CEDMs of quarks can be obtained from the limit on g. Q We use the calculation of Ref. [309], gQ = 2(d˜ u − d˜ d )=(10−14 cm). Again, this limit is better than those from the neutron. Constraints on the phenomenological P; T -odd quark–quark interactions de%ned above are obtained from the constraints

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143

Table 19 Best limits on P; T -violating parameters at the nucleon level. Signs of the central points are omitted P; T -violating term

Value

System

Exp.

Theory

neutron EDM dn (e cm)

(17 ± 8 ± 6) × 10−26 (1:9 ± 5:4) × 10−26 (2:6 ± 4:0 ± 1:6) × 10−26

Hg n n

[11] [250] [251]

[260,16]

proton EDM dp (e cm)

(1:7 ± 0:8 ± 0:6) × 10−24 (17 ± 28) × 10−24

Hg TlF

[11] [248]

[260,16,295] [225,257]

)np = (2:7 ± 1:3 ± 1:0) × 10−4

Hg

[11]

[290,289]

gQ = (3:0 ± 1:4 ± 1:1) × 10−12

Hg

[11]

[298,16]

G √ C SP NQ N ei Q 5e 2

(0:40CpSP + 0:60CnSP ) = (18 ± 8 ± 7) × 10−8 (0:40CpSP + 0:60CnSP ) = (6 ± 6) × 10−8

Hg Tl

[11] [12]

[21,16] [265]

G √ C PS NQ i5 N ee Q 2

CnPS = (1:8 ± 0:8 ± 0:7) × 10−6

Hg

[11]

[21,16]

G √ C T NQ i5 J N e Q J e 2

CnT = (5:3 ± 2:5 ± 2:0) × 10−9

Hg

[11]

[266]

(6:9 ± 7:4) × 10−28 (91 ± 42 ± 34) × 10−28

Tl Hg

[12] [11]

[271] [21]

Hadronic

G √ 2

1 )
· ∇! 2mp

gQ Semi-leptonic

Leptonic electron EDM de (e cm)

Errors are experimental. Some relevant theoretical works are presented in the last column.

on g, Q with the exception of ksc with q1 = q2 and ktc for which the P; T -odd neutral pion–nucleon vertex is insensitive [16]. Limits on the P; T -odd electron–nucleon parameters can be broken down into limits√on phenomenological P; T -odd electron–quark parameters k1q , k2q , k3q de%ned according to hˆeq =G= 2[k1q qq Q ei Q 5 e+ k2q qi Q 5 J qe Q J e + k3q qi Q 5 qee]; Q see Ref. [16]. Until this point we have not mentioned the P; T -odd electron–electron interaction. This interaction induces atomic EDMs smaller than those induced by the electron–nucleon interaction. It does not bene%t from two enhancement factors present in the latter interaction: the relativistic factor arising from the outer electrons being in the nuclear vicinity; and the interaction with A nucleons (rather than two K-shell electrons in the former interaction); see Ref. [16]. The limit on the electron EDM can be reduced to limits on the P; T -odd electron–quark interaction [313] and on the electron–electron interaction [16] that are competitive with those obtained at tree level. However, we do not include these into our table.

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Table 20 Best limits on P; T -violating parameters at the quark level P; T -violating term

Value

System

Exp.

Theory

ks = (3 ± 1 ± 1) × 10−7 ks = (2 ± 5) × 10−5 ks = (3 ± 4 ± 2) × 10−5

Hg n n

[11] [250] [251]

[307,16] [307,308,16] [307,308,16]

q1 = q2

ksc = (1 ± 1 ± 1) × 10−6 ksc = (1 ± 2) × 10−4 ksc = (1 ± 1 ± 1) × 10−4

Hg n n

[11] [250] [251]

[307,16] [307,308,16] [307,308,16]

q1 = q2

ksc = (5 ± 2 ± 2) × 10−4 ksc = (1 ± 2) × 10−4 ksc = (1 ± 1 ± 1) × 10−4

Hg n n

[11] [250] [251]

[307,16] [307,308,16] [307,308,16]

G Q J d √ k ui Q 5 J ud 2 t

kt = (1 ± 1 ± 1) × 10−5 kt = (1 ± 2) × 10−5 kt = (1 ± 1 ± 1) × 10−5

Hg n n

[11] [250] [251]

[307,16] [307,308,16] [307,308,16]

G c Q J t a d √ k ui Q 5 J t a ud 2 t

ktc = (5 ± 3 ± 2) × 10−5 ktc = (1 ± 2) × 10−5 ktc = (1 ± 1 ± 1) × 10−5

Hg n n

[11] [250] [251]

[307,16] [307,308,16] [307,308,16]

CEDMs d˜ and EDMs d of quarks (e cm)

e(d˜d − d˜u ) = (1:5 ± 0:7 ± 0:6) × 10−26 e(d˜d + 0:5d˜u ) + 1:3dd − 0:3du =(3:5 ± 9:8) × 10−26 =(4:7 ± 7:3 ± 2:9) × 10−26

Hg

[11]

[309]

n n

[250] [251]

[310] [310]

QCD phase Q

Q = (1:1 ± 0:5 ± 0:4) × 10−10 Q = (1:6 ± 4:5) × 10−10 Q = (2:2 ± 3:3 ± 1:3) × 10−10

Hg n n

[11] [250] [251]

[298,16] [311] [311]

G √ k qq Q ei Q 5e 2 1q

k1u + k1d = (3 ± 1 ± 1) × 10−8 k1u + k1d = (1 ± 1) × 10−8

Hg Tl

[11] [11]

[16] [16]

G √ k qi Q 5 qee Q 2 3q

k3q = (2 ± 1 ± 1) × 10−8

Hg

[12]

[16]

G √ k qi Q 5 J qe Q J e 2 2q

k2q = (5 ± 3 ± 2) × 10−9

Hg

[11]

[16]

Hadronic G √ k qQ i q qQ q 2 s 1 5 1 2 2

G c √ k qQ i t a q1 qQ2 t a q2 2 s 1 5

Semi-leptonic

Signs of the central points are omitted. Errors are experimental. Some relevant theoretical works are presented in the last column.

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145

In relating the P; T -odd parameters at the nucleon and quark levels we used the most recent calculations available. For references to other calculations, please see, e.g., the review [263] and book [16]. 10.2. Ongoing/future EDM experiments in atoms, solids, and diatomic molecules An improved mercury EDM experiment is in progress at Seattle [314]. An EDM experiment with liquid 129 Xe in preparation at Princeton [315] looks very promising; the measured EDM is expected to be several orders of magnitude more precise than the current Hg result. Work towards a measurement of the EDM of Yb in laser traps is being carried out at Seattle [316] and Kyoto [317]. Possibilities for an EDM measurement in metastable states of Sm have been examined at Berkeley [318,165], a favourable state 4f6 6s5d 7 G1 having been found; a large uncertainty in the estimate of the electron’s enhancement factor K, stemming from sign ambiguities in the contributions to K, has put the EDM measurement on hold until this problem is resolved. Prospects for EDM experiments with alkalis in laser-traps are being investigated by di2erent groups, see, e.g., [319,320]. Work is underway towards a new generation of EDM experiments with alkalis trapped in solid helium [321], in paraSn-coated cells (using nonlinear optical rotation, see the review [322]) [323], and in cold bu2er gases [323]. One of the primary goals of the TRIP facility under construction at Groningen is to measure permanent EDMs of radioactive atoms and ions, in particular Ra (see, e.g., [145]). Groups at Argonne and at TRIUMF and Ann Arbor are also working towards EDM experiments with Ra and Rn, respectively. These atoms can have very large EDMs due to their high Z and the presence of nuclear (static/vibrational) octupole deformation (Section 9.4.3). Also, metastable atomic states of Ra have close levels of opposite parity, and this can be exploited to obtain an enhanced EDM e2ect [171,172]. There is a new generation of experiments in preparation aimed to measure EDMs of ions in solids (gadolinium gallium garnet and gadolinium iron garnet) [238,239]. Here it is expected that the sensitivity to T -violating e2ects (in particular, the electron EDM) will be improved by several orders of magnitude. The solid-state nature of the problem complicates the calculations required for interpretation of the measurements. A series of calculations have been performed in Ref. [324]. EDM experiments with both paramagnetic and diamagnetic diatomic molecules are underway. The %rst EDM experiment with paramagnetic molecules (YbF) was performed recently at the University of Sussex [236], and while it gave a looser bound on the electron EDM than the Tl experiment [12], a substantial improvement in the result is expected. Experiments with PbO excited to the metastable a(1) state have begun at Yale [280]. 11. Concluding remarks Exciting developments in violations of fundamental symmetries are expected in the next few years. The new generation of EDM experiments underway in atoms, diatomic molecules, and solids are expected to yield limits on electric dipole moments that are several orders of magnitude better than the current ones. Or perhaps an EDM will be unambiguously detected? Popular models such as supersymmetry will be put to the test. Improved precision tests of parity violation in atoms in

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a single isotope and in a chain of isotopes will provide crucial tests of physics beyond the standard model complementary to each other and to other electroweak tests. New measurements of the nuclear anapole moment are anticipated, and they will have important consequences for the theory of parity violating nuclear forces. Acknowledgements We are very grateful to D. Budker and M.G. Kozlov for their close reading of the review and for the many useful comments and suggestions they made. We would also like to thank V.A. Dzuba and M.Yu. Kuchiev for useful discussions. Some of this work was carried out at the Institute for Nuclear Theory, University of Washington, Seattle; we thank them for support and kind hospitality. This work was supported by the Australian Research Council. References [1] S.L. Glashow, Nucl. Phys. 22 (1961) 579; S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, in: N. Svartholm (Ed.), Elementary Particle Theory, Relativistic Groups, and Analyticity, Almqvist and Wiksells, Stockholm, 1968, p. 367. [2] M.C. Gonzalez-Garcia, Y. Nir, Rev. Mod. Phys. 75 (2003) 345. [3] F.J. Hasert, et al., Phys. Lett. B 46 (1973) 138; A. Benvenuti, et al., Phys. Rev. Lett. 32 (1974) 800; B.C. Barish, et al., Phys. Rev. Lett. 34 (1975) 538. [4] L.M. Barkov, M.S. Zolotorev, Pis’ma Zh. Eksp. Teor. Fiz. 27 (1978) 379 [Sov. Phys. JETP Lett. 27 (1978) 357]; L.M. Barkov, M.S. Zolotorev, Pis’ma Zh. Eksp. Teor. Fiz. 28 (1978) 544 [Sov. Phys. JETP Lett. 28 (1978) 503]; L.M. Barkov, M.S. Zolotorev, Phys. Lett. B 85 (1979) 308; L.M. Barkov, M.S. Zolotorev, Zh. Eksp. Teor. Fiz. 79 (1980) 713 [Sov. Phys. JETP 52 (1980) 360]. [5] C.Y. Prescott, et al., Phys. Lett. B 77 (1978) 347. [6] J.H. Christensen, J.W. Cronin, V.L. Fitch, R. Turlay, Phys. Rev. Lett. 13 (1964) 138. [7] BaBar Collaboration, B. Aubert, et al., Phys. Rev. Lett. 87 (2001) 091801; Belle Collaboration, K. Abe, et al., Phys. Rev. Lett. 87 (2001) 091802. [8] R. Fleischer, Phys. Rep. 370 (2002) 537. [9] C.S. Wood, S.C. Bennett, D. Cho, B.P. Masterson, J.L. Roberts, C.E. Tanner, C.E. Wieman, Science 275 (1997) 1759. [10] V.A. Dzuba, V.V. Flambaum, J.S.M. Ginges, Phys. Rev. D 66 (2002) 076013. [11] M.V. Romalis, W.C. GriSth, J.P. Jacobs, E.N. Fortson, Phys. Rev. Lett. 86 (2001) 2505. [12] B.C. Regan, E.D. Commins, C.J. Schmidt, D. DeMille, Phys. Rev. Lett. 88 (2002) 071805. [13] V.A. Kosteleck_y (Ed.), Proceedings of the Second Meeting on CPT and Lorentz Symmetry, World Scienti%c, Singapore, 2002. [14] R.C. Hilborn, G.M. Tino (Eds.), Proceedings of “Spin-Statistics Connection and Commutation Relations. Experimental Tests and Theoretical Implications”, AIP Conference Proceedings, Vol. 545, 2000. [15] I.B. Khriplovich, Parity Nonconservation in Atomic Phenomena, Gordon and Breach, Philadelphia, 1991. [16] I.B Khriplovich, S.K. Lamoreaux, CP Violation Without Strangeness, Springer, Berlin, 1997. [17] M.A. Bouchiat, C. Bouchiat, J. Phys. (Paris) 35 (1974) 899; M.A. Bouchiat, C. Bouchiat, J. Phys. (Paris) 36 (1975) 493. [18] O.P. Sushkov, V.V. Flambaum, Yad. Fiz. 27 (1978) 1308. [19] S.A. Blundell, W.R. Johnson, J. Sapirstein, Phys. Rev. Lett. 65 (1990) 1411; S.A. Blundell, W.R. Johnson, J. Sapirstein, Phys. Rev. D 45 (1992) 1602.

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Physics Reports 397 (2004) 155 – 256 www.elsevier.com/locate/physrep

Meson production in nucleon–nucleon collisions close to the threshold C. Hanhart Institut fur Kernphysik, Forschungszentrum Julich GmbH, D-52425 Julich, Germany Accepted 7 March 2004 editor: W. Weise

Abstract The last decade has witnessed great experimental progress that has led to measurements of near-threshold cross sections—polarized as well as unpolarized—of high accuracy for various inelastic nucleon–nucleon collision channels. These data, naturally, pose challenges to theorists to develop methods by which they can be understood and explained in commensurate detail. In this work we review the status of the present theoretical understanding of this class of reactions with special emphasis on model-independent methods. We discuss in detail not only the many observables involved in the reactions, but also the physical questions that can be addressed by studying them in the various reaction channels. The special advantages of nucleon–nucleon-induced reactions are stressed. Foremost among these is the use of the initial and 6nal states as a spin/isospin 6lter. This opens, for example, a window into the spin dependence of the hyperon–nucleon interaction and the dynamics of the light scalar mesons. c 2004 Elsevier B.V. All rights reserved.
PACS: 14.40.−n; 25.80.−e; 13.75.Gx; 13.75.Ev; 24.40.−h; 13.85.−t; 24.70.+s; 11.10.Rd; 21.30.Fe Keywords: Meson production in hadronic collisions; Meson and baryon resonances; Nucleon–nucleon interaction; Nucleon–hyperon interaction; Polarization; Model-independent methods

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Strong interactions at low and medium energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Theoretical approaches to the reaction NN → B1 B2 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Speci6c aspects of hadronic meson production close to the threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Remarks on the production operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. The role of di@erent isospin channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Selection rules for NN → NNx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-mail address: [email protected] (C. Hanhart). c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.03.007

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2. The 6nal state interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Heuristic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Dispersion theoretical approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The initial state interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Unpolarized observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Example: analysis of pp → dKG 0 K + close-to-threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Spin-dependent observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Equivalent observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. General structure of the amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1. Example I: polarization observables for a baryon pair in the 1 S0 6nal state . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Example II: amplitude analysis for pp → dKG 0 K + close-to-threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Spin cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Status of experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Symmetries and their violation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Investigation of charge symmetry breaking (CSB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The reaction NN → NN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Phenomenological approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. The NN interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. The N model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Additional short-range contributions and model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Chiral perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. On the signi6cance of o@-shell e@ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Lessons and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Production of heavier mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Remarks on the production operator for heavy meson production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. The reaction NN → NN or properties of the S11 (1535) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Associated strangeness production or the hyperon–nucleon interaction from production reactions . . . . . . . . . . 7.5. Production of scalar mesons or properties of the lightest scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Meson production on light nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. 2 → 2 reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Quasi-free production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Kinematical variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Collection of useful formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. De6nition of coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Spin traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. Partial wave expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. On the non-factorization of a strong 6nal state interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix E. Chiral counting for pedestrians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.1. Counting within TOPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E.2. Counting within the covariant scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166 167 170 177 178 179 181 183 187 189 194 195 197 198 200 201 203 203 206 206 209 210 211 212 225 226 228 228 229 230 233 236 237 238 238 239 240 241 241 243 243 243 244 246 247 248 248 250

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1. Introduction 1.1. Strong interactions at low and medium energies Quantum chromo dynamics (QCD), as the well-accepted theory of the strong interaction, unfolds its impressive predictive power in particular at high energies. However, with decreasing momentum transfer the mathematical structure of QCD becomes increasingly complicated because the perturbative expansion in the coupling constant no longer converges. The fact that particles carrying strong charge (color) have never been observed in an isolated state is just one prominent example of the non-perturbative character of QCD at low energy. Although a large amount of new data is available from measurements with electromagnetic probes (e.g., from MAMI at Mainz, ELSA at Bonn, and JLAB at Newport News), there is still much to be learned about the physics with hadronic probes at intermediate energies, comprising the investigation of production, decay, and interaction of hadrons. An important class of experiments in this context is meson production in nucleon– nucleon, nucleon–nucleus, and nucleus–nucleus collisions close to the production thresholds. Recently, two reviews on this subject [1,2] were published, however both had their main emphasis on the experimental aspects. In this report we will focus on recent theoretical developments and insights. In addition in neither of these reviews was the potential of using polarization—which will be the main emphasis of this work—discussed in detail. It is interesting to ask a priori what physics questions can be addressed with the production of various mesons near-threshold in pp, pd, and dd collisions with stored and extracted polarized beams. We therefore present here a brief list, naturally inNuenced by the personal taste of the author, that is not aimed at completeness, but more at giving the reader a Navor of the potential of meson production reaction in nucleonic collisions for gaining insight into the strong interaction physics at intermediate energies. • Final state interactions: In practice, it is diOcult to prepare secondary beams of unstable particles with suOcient accuracy and intensity that allow to study experimentally the scattering of those particles o@ nucleons. Here production reactions are an attractive alternative. One prominent example in this context is the hyperon–nucleon interaction that can be studied in the reaction pp → pYK, where Y denotes a hyperon ( , , etc.) and K denotes a kaon. From the invariant mass distributions of the YN system, information about the on-shell YN interaction can be extracted. We shall return to this point in Section 2.2. • Baryon resonances in a nuclear environment: By design, in nucleon–nucleon and nucleon–nucleus collisions we study systems with baryon number larger than 1. This allows to study particular resonances in the presence of other baryons and excited through various exchanged particles. One example is the N ∗ (1535), which is clearly visible as a bump in any production cross section on a single nucleon. A lot is known about this resonance already, however, many new questions can be answered by a detailed study of the reaction NN → NN . In the close-to-threshold regime, kinematics and the conservation of total angular momentum constrains the initial state to only a few partial waves. This, in combination with the dominance of the N ∗ (1535) in the production mechanism, allows for a detailed study of the NN → N ∗ N transition potential. This is relevant to understanding the behavior of the N ∗ in any nuclear environment as well as might reveal information on the structure of the resonance. For a detailed discussion we refer to Section 7.3.

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In hadronic reactions there is additionally a large number of excitation mechanisms for intermediate resonances possible. Besides pseudo-scalars and vector mesons, which are more cleanly studied in pion and photon-induced reactions, scalar excitations are possible. It was shown recently that, for instance, the Roper resonance is rather easily excited by a scalar source [3], and thus hadronic reactions might be the ideal place to study this controversial resonance [4]. • Charge symmetry breaking (CSB): Having available several possible initial isospin states (pp, pn, dd, etc.), which all have several possible spin states, allows experiments that make the study of symmetry breaking easily accessible. One example is the impact of charge symmetry breaking on the reaction pn → d0 : it leads to a forward–backward asymmetry in the di@erential cross section (cf. Section 5 and Refs. [5,6]). It is important to note that in case of the pion production the leading charge symmetry breaking operators are linked to the up–down mass di@erence. A forward–backward asymmetry in the reaction pn → d(0 )(s-wave) should allow extraction of the CSB f0 − a0 mixing matrix element—a quantity believed to give insights into the structure of the light scalar mesons. Once the leading symmetry breaking mechanism is identi6ed, one can use the speci6c signals caused by CSB to extract information on symmetry conserving matrix elements. For details on the idea as well as an application—extracting information on the existence of mesonic bound states —we refer the reader to Ref. [7]. • E8ective 9eld theory in large momentum transfer reactions: Pion production in nucleon–nucleon collisions is still amenable to treatment within chiral perturbation theory—if the expansion is adapted to the large momentum transfer typical of those reactions. This research is still in progress, however once completed it will allow not only to study systematically the CSB pion production mentioned above, but also to pin down the size of three-body forces relevant for a quantitative understanding of pd scattering, and to include the dispersive corrections to d scattering in a chiral perturbation theory analysis. The latter is necessary for an accurate extraction of the isoscalar N scattering length from deuteron reactions, which are otherwise diOcult to access. This issue will be discussed in Section 6.3. The wealth of information comes with the drawback that, apart from rare cases, it is diOcult to extract a particular piece of information from the data. For example, resonances and 6nal state interactions modify in a coherent superposition the invariant mass plot. Fortunately, polarization can act as a spin 6lter and di@erent contributions to the interaction can be singled out because they show up in the angular distributions of di@erent spin combinations. The various polarization observables will be discussed in detail in Section 4. One characteristic feature of meson production in NN collisions at the 6rst glance looks like a disadvantage with respect to a possible theoretical analysis: the large momentum transfer makes it diOcult to reliably construct the production operator. However, as we will discuss in detail, due to this the production operator is largely independent of the relative energy of a particular particle pair in the 6nal state 1 if we stay in the regime of small invariant masses. Because of this, dispersion relations can be used to extract low-energy scattering parameters of the 6nal state interaction and at the same time give a model-independent error estimate. In contrast, many reactions dominated by one-body currents that are characterized by small momentum transfers (cf. Fig. 1). This 1

If there are resonances near by this statement is no longer true. See the discussion in Section 4.

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159

FSI

FSI

d

(a)

(b)

Fig. 1. Comparison of two di@erent reactions that can be used to extract parameters of a strong 6nal state interaction. Shown is a typical low momentum transfer reaction (diagram (a), e.g., K − d →  n) and a large momentum transfer reaction (diagram (b), e.g., pp → pK + ).

makes a quantitatively controlled, model-independent analysis diOcult. For example, it was shown in Refs. [8,9] that value for the neutron–neutron ( N ) scattering length extracted from − d → nn (K − d →  n), where the initial state is in an atomic bound state, is sensitive to the short-range behavior of the baryon–baryon interaction used. 2 1.2. Theoretical approaches to the reaction NN → B1 B2 x In this section we will brieNy describe the theoretical developments for the reactions NN → B1 B2 x close to the threshold, where the Bi denote the two outgoing baryons (either NN or, e.g., in case of the production of a strange meson YN , where Y denotes a hyperon). The development of theoretical models especially for the reaction NN → NNx has a long history. For a review of earlier works we refer the reader to the book by Garcilazo and Mizutani [10]. Most of the models can be put into one of two classes: namely those that use the distorted wave Born approximation, where a production operator that is constructed within some perturbative scheme is convoluted with nucleonic wavefunctions, and truly non-perturbative approaches, where integral equations are solved for the full (NN ,NNx) coupled-channel problem. We start this section with a brief review of the second class. The obvious advantage of this kind of approach is that no truncation scheme is required with respect to multiple rescattering. This is a precondition to truly preserve three-body unitarity [11–13]. However, as we will see, this approach is technically rather involved and there is signi6cant progress still to be made that will allow for 2

This implies that meson exchange currents should contribute at the same order of magnitude (cf. discussion in Section 6.4).

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quantitative predictions, especially for reactions with unbound few nucleon systems in the initial and 6nal state. One obvious complication is to preserve the requirements of chiral symmetry, which are very important—at least for the reaction NN → NN close to the threshold, as will be discussed in the following sections, since in its non-linear realization it is necessary to consistently take into account one-, two- and more pion vertices as soon as loops are considered. Naturally, this can only be done within a truncation scheme that is consistent with chiral symmetry. As a possible solution, in Ref. [14] it was suggested to change to a linear representation, as given, e.g., by the linear sigma model. To our knowledge this area is not yet very well developed. All the calculations so far carried out for pion 3 production within the full (NN , NNx) coupledchannel system, where performed within time-ordered perturbation theory. Within this scheme, one way to impose unitarity is it to demand, that the number of pions in any given intermediate state should not to exceed a given number. This, in combination with that there is no particle number conservation, implies the fact, that the NN coupling constant generated by the equations at some intermediate time depends on the number of pions in Night at that time. The same is true for the renormalization of the nucleon pole. Put to the extreme: in the nuclear matter limit the dressed pion nucleon coupling vanishes [15] (see also the discussion in Ref. [16]). Another problem inherently present in time-ordered formalisms was pointed out by Jennings [17]: namely that it is impossible simultaneously to ful6ll the Pauli-principle and the unitarity condition within these integral equations. A consequence of this is a wrong sign for T20 for elastic d scattering at backward angles [18]. Both of these problems can be avoided within a covariant formulation of the integral equations as is given in Refs. [19,20]. Unfortunately, not only are those equations rather involved, they also need covariant T matrices for both subsystems. Although these exist (e.g., Ref. [21] for the N system and Ref. [22] for the NN system), they were not constructed consistently. Also the three-dimensional convolution approach of Refs. [23,16] does not su@er from the above-mentioned problems, but also within this scheme so far no results for the (NN; NNx) were published. On the other hand, there are the approaches that treat the production operator perturbatively (typically diagrams of the type shown in Fig. 2 are considered), while including the nucleon–nucleon or, more generally, the baryon–baryon interaction in the 6nal and initial state non-perturbatively. Therefore in this formalism three-body unitarity can be achieved only perturbatively, however, this formalism at least ful6lls the minimal requirement: to take fully into account the non-perturbative nature of the NN interaction. This is indeed necessary, as was con6rmed experimentally, since a proper inclusion of the NN 6nal state interaction is required to describe the energy dependence of the reaction pp → ppX [1,2]. A production operator that is constructed perturbatively allows in addition to treat the interactions between the subsystems without approximation and consistent with chiral symmetry (cf. discussion in Section 6.3). A priori, it is unclear according to what rules the production operator should be constructed and only a comparison with experiment can tell if the approach is appropriate or not. There is, however, one observation in favor of a perturbative treatment of the production operator: in the case of pion production, e@ective 6eld theory methods can be applied to show that as long as there is a meson exchange current in leading order it dominates the low-energy amplitude. Loops undergo strong cancellations and thus might well be neglected. 3

To be concrete in this section we will talk about pions only. They are, in fact, the only mesons that have so far been studied with integral equation approaches.

C. Hanhart / Physics Reports 397 (2004) 155 – 256 p

p

π

p

p

π

p

161 p

π

σ,ω,... T

p (a)

p

p (b)

p

p

p

(c)

Fig. 2. (a)–(c) Some possible contributions to the production operator for neutral pion production in pp collisions. Solid lines are nucleons, dashed lines are pions.

If, however, there is no meson exchange current at leading order (as is the case for the reaction pp → pp0 ) the situation is signi6cantly more involved and loops might well be signi6cant. 4 This issue will be discussed in detail in Section 6. In this introductory chapter we only wish to stress, that to our present knowledge the case of 0 seems to be an exception and, in addition to the reaction pp → pn+ all heavy meson production cross sections close to the threshold are indeed dominated by meson exchange currents. 5 We must add, however, that the of heavier mesons can also give a signi6cant contribution. An approach that lies somewhere between those that solve the integral equations for the whole NN → NN system and those that treat the production operator perturbatively is that of Refs. [22,24,25]. These works document the attempt to extend straightforwardly what is known about the phenomenology of nucleon–nucleon scattering (see for example [26]) to energies above the pion threshold. The three-body singularities stemming from both the energy-dependent pion exchange as well as the nucleon and Delta self-energies were taken into account. Via the optical theorem the pion production cross sections can then be calculated in a straightforward way. It turned out, however, that the models could not describe the close-to-threshold data. The reasons for this will become clear in Section 6. In Section 6.5 we will discuss how the recent progress in our understanding of pion production shows what is needed in order to improve, for example, the work of Ref. [24]. The construction of theoretical models for the production of mesons heavier than the pion started only recently, after the accelerator COSY came into operation. Most of these models are one meson exchange models. Those approaches will be described to some extend in Section 7.2. 1.3. Speci9c aspects of hadronic meson production close to the threshold In the near-threshold regime the available phase space changes very quickly. Thus, especially when comparing di@erent reactions, an appropriate measure of the energy with respect to that particular threshold is required. In pion production traditionally the variable , de6ned as the maximum pion 4 5

Note, however, even then a perturbative treatment of the production operator is still justi6ed. Here this phrase is to be understood to include resonance excitations as well.

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momentum in units of the pion mass, is used to measure the excess above threshold. For all heavier mesons the so-called excess energy Q, de6ned as √ √ Q = s − s(thres) is normally used. In Appendix A we give the explicit formulas that relate Q to . It is useful to understand the physical meaning of the two di@erent quantities: Q directly gives the energy available for the 6nal state. The interpretation of is somewhat more involved. In a non-relativistic, semiclassical picture the maximum angular momentum allowed can be estimated via the relation lmax  Rq , where q denotes the typical momentum of the corresponding particle and R is a measure of the range of forces. If we identify R with the Compton wavelength of the meson of mass mx , we 6nd lmax  q =mx  . This interpretation was given for example in Refs. [27,28]. When trying to compare the cross sections of reactions with di@erent 6nal states one has to choose carefully the variable that is used for the energy. In Ref. [29] it is shown that the relative strength of the total cross sections for pp → pp0 , pp → pp and pp → pp  is strikingly di@erent when comparing them at equal or at equal Q. Since the dominant 6nal state interaction in all of these reactions is the pp interaction, it appears most appropriate to compare the cross sections at equal Q, for then at any given excess energy the impact of the 6nal state interaction is equal for all reactions, which is not the case for equal values of . In Ref. [30] a di@erent energy variable was suggested, namely the phase-space volume. In order to produce a meson in nucleon–nucleon collisions the kinetic energy of the initial particles needs to be suOciently large to put the outgoing meson on its mass shell. To produce a meson of mass mx an initial momentum larger than the threshold value m2x (1) 4 is necessary. As long as we stay in a regime close to the production threshold the momenta of all particles in the 6nal state are small and therefore pi(thres) also sets the scale for the typical momentum transfer. In a non-relativistic picture a large momentum transfer translates to a small reaction volume, de6ned by a size parameter R ∼ 1=pi . Thus, already for pion production R is as small as R ∼ 0:5 fm and is getting even smaller for heavier mesons. Therefore, the two nucleons in the initial state have to approach each other very closely before the production of the meson can happen. It then should come as no surprise that it is important for quantitative predictions to understand the elastic as well as inelastic NN interaction in the initial state. However, since for all reactions we will be looking at the initial energy is signi6cantly larger than the excess energy Q, the initial state interaction should at most mildly inNuence the energy dependence. The energy dependence of the production operator should also be weak, for it should be controlled by the typical momentum transfer, which is signi6cantly larger than the typical outgoing momenta. On the other hand, in the near-threshold regime all particles in the 6nal state have low relative momenta and thus undergo potentially strong 6nal state interactions that can induce strong energy dependences. The di@erent parts of the matrix element are illustrated in Fig. 3. In the next subsections we will brieNy sketch some properties of meson production in nucleon– nucleon collisions in rather general terms with the emphasis on the gross features. Details, as well as selected results for various reactions, will be presented in the subsequent sections. |2 = mx MN + |˜ p(thres) i

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strongly Q-dep.

weakly Q-dep.

163

BB- FSI

FSI

A

A

ISI

ISI

Fig. 3. Sketch of the production reaction showing the initial state interaction amongst the two nucleons (ISI), the 6nal state interaction of all outgoing particles (FSI) as well as the production operator A as de6ned in the text. The left diagram indicates the complexity possible, whereas the right diagram shows the 6rst and potentially leading term for the FSI only.

r

r

r r (a)

r (b)

Fig. 4. (a)–(b) Illustration of the momentum mismatch. The picture shows why one should a priori expect meson exchange currents to be very important close to the threshold.

1.4. Remarks on the production operator As was mentioned above one of the characteristics of meson production in nucleon–nucleon collisions is the large momentum transfer. This leads to a large momentum mismatch for any one-body operator that might contribute to the production reaction. This reasoning becomes most transparent when looking at it in real space, as shown in Fig. 4. In case of the one-body operator (panel a) the evaluation of the matrix element involves the convolution of a rapidly oscillating function with a mildly oscillating function, which generally will lead to signi6cant cancellations. If there is a meson exchange present, as sketched in panel b, the cancellations will not be as eOcient. Thus one should expect that if there is a meson exchange current possible at leading order, it should dominate the production process. This picture was con6rmed in explicit model calculations (see e.g. Ref. [31]). As mentioned above one should in any case expect the production operator to be controlled by the large momentum transfer. Thus variations of the individual amplitudes with respect to the total energy should be suppressed by (p =p)2 , where p (p) is the relative momentum of an outgoing to particle pair (of the initial nucleons). Therefore it should be negligible close to the production threshold. This will be used for model-independent analyses of the production of heavy mesons and was checked within the meson exchange picture (see discussion in Ref. [32] and Section 2.2).

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1.5. The role of di8erent isospin channels One big advantage of meson production in nucleon–nucleon collisions as well as nucleon–nucleus and nucleus–nucleus collisions is, that besides the spin of the particles there is another internal degree of freedom to be manipulated: the isospin. Nucleons are isospin–( 12 ) particles in an SU (2) doublet (isospin t = 12 ) with t3 = + 12 and − 12 for the proton (p) and neutron (n), respectively. Thus a two-nucleon pair can be either in an isotriplet state (total isospin T = 1 with the three possible projections of the total isospin T3 = +1 for a pp state, T3 = 0 for a pn state and T3 = −1 for a nn state) or in an isosinglet state (T = 0 with T3 = 0 for a pn state). Let us 6rst concentrate on the isospin conserving situation. Even then various transitions are possible, depending on whether an isovector or an isoscalar particle is produced. Let us denote the allowed transition amplitudes by ATi Tf , where Ti (Tf ) denote the total isospin of the initial (6nal) NN system [27]. Then, in the case of the production of an isovector, the amplitudes A11 , A10 , and A01 are possible. They could be extracted individually from pp → ppx0 ˙ |A11 |2 , pp → pnx+ ˙ |A11 +A10 |2 , and pn → ppx− ˙ 12 |A11 + A01 |2 , where the factor of 12 in the latter case stems from the isospin √ factor of the initial state (we have |pn = 1= 2(|10 + |00 ) in which the isospin states are labeled by both the total isospin as well as their projection). On the other hand, for the production of an isoscalar there are two transitions possible, namely A00 and A11 , where pp → ppx ˙ |A11 |2 and pn → pnx ˙ 12 |A11 + A00 |2 . In this case the measurement of two di@erent reaction channels allows extraction of all the production amplitudes. The deuteron is an isoscalar and thus acts as an isospin √ 6lter. Accordingly, one 6nds for the production of any isovector particle x, A(pp → dx+ ) = 2A(pn → dx0 ), as long as isospin is conserved. The consequences of isospin breaking will be discussed in Section 5. The relative strength of the di@erent transition amplitudes proved to provide signi6cant information about the production operator. As an example let us look at the production of an isoscalar particle. If the production operator is dominated by an isovector exchange the corresponding isospin structure of the exchange current is Oˆ iv = (˜%1 ·˜%2 ), where %i denotes the isospin operator of nucleon i, which is clearly distinguished from the structure corresponding to an isoscalar exchange Oˆ is = 1. Thus one 6nds 6
1 for T = 1 ; TT3 |Oˆ iv |TT3 = 2T (T + 1) − 3 = −3 for T = 0 : On the other hand TT3 |Oˆ is |TT3 = 1 :

6

The easiest way to see this is to express Oˆ iv in terms of the Casimir operators of the underlying group: ˜%1 · ˜%2 = 4tˆ1 · tˆ2 = 2(Tˆ 2 − tˆ21 − tˆ22 )

with T 2  = T (T + 1) and ti2  = 34 .

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For two given NN states a di@erent total isospin implies that also either the total spin or the angular momentum are di@erent (cf. next section). Therefore, di@erent isospin states cannot interfere in the total cross section and one may expect
5 if isovector exchanges dominate ; &tot (pn → pnx) ∼ &tot (pp → ppx) 1 if isoscalar exchanges dominate : For x = a recent measurement at Uppsala [33] gave for this ratio a value of 6.5, clearly indicating a dominance of isovector exchanges. It should be stressed that the estimates given here only hold if e@ects from initial and 6nal state interactions, the spin dependence of the production operator, as well as all other dynamical e@ects are completely neglected. However, even in the most detailed studies, the bulk of the ratio stems from the isospin factors [34–36]. 1.6. Selection rules for NN → NNx In this section we will present the selection rules relevant for nucleon–nucleon-induced meson production. Those rules are based on the symmetries of the strong interaction that imply the conservation of parity, total angular momentum and isospin. A two-nucleon system has to obey the Pauli-principle, which implies (−)L+S+T = (−1) where L, S and T denote the angular momentum, total spin and total isospin of the two-nucleon system, respectively. In the case of a two proton system, for example, where T = 1, L + S needs to be even. Consequently, for T = 1 all even angular momenta, and therefore all even parity states, are spin singlet (S = 0) states and consequently have J = L. On the other hand, for T = 0 states it is the odd parity states that are spin singlets. In addition, for a reaction of the type NN → NNx we 6nd from parity conservation





(−)L = x (−)(L +l ) ;

(2)

where x denotes the intrinsic parity of particle x and L, L and l denote, respectively, the angular momentum of the incoming two-nucleon system, of the outgoing two-nucleon system and of particle x with respect to the outgoing two-nucleon system. We may now combine the two criteria to write


(−)(SS+ST ) = x (−)l ;

(3)

where SS (ST ) denotes the change in total (iso)spin when going from the initial to the 6nal NN system. As an example let us look at the reaction pn → d( )s-wave —a reaction that should provide valuable information on the light scalar resonances (note: the  channel is the dominant decay channel of the scalar–isovector meson a0 (980), cf. Section 7.5). Based on Eq. (9), the near-threshold regime is dominated by the lowest partial waves which in this case would be S  = 1 and T  = 0 (deuteron), together with a relative s wave between the two meson system and the deuteron (l = 0). In addition, the ( )s-wave system is a scalar–isovector system. Therefore we have to use x = +, l = 0 and ST = 1 in Eq. (3), yielding SS = 0. On the other hand, the initial state needs to be of even parity, but even parity T = 1 states have S = 0. Thus, for the production of a scalar–isovector state the s-wave deuteron with respect to the two meson system is prohibited and, as long as isospin is conserved, this reaction has to have at least one p-wave in the 6nal state. On the other hand, for

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Table 1 The lowest partial waves for the production of a pseudo-scalar particle in NN → NNx J

S

L

j

l

0

0

0

0

0

0 2

1 1

1 1

0 2

0 0

1 0 2

1 1 1

0 0 0

1 1 1

0 1 1

NN |(T =0) → NN |(T =1) + pseudo-scalar (e.g., part of pn → pp− ) Ss Not allowed 1 3 Sp S1 S0 p 1 0 1 3 D1 S0 p 1 2 3 3 Ps S1 P1 s 1 0 3 3 DJ PJ s 1 2

1 1 1 J

0 0 1 1

0 0 1 1

0 0 1 J

1 1 0 0

NN |(T =0) → NN |(T =0) + pseudo-scalar (e.g., part of pn → pn ) 3 1 Ss P1 S1 s 0 0 3 3 S1 S1 p 1 0 Sp 3 3 DJ S1 p 1 2 1 3 Ps S1 P1 s 1 0 1 3 D1 P1 s 1 2

0 1 J 1 1

1 1 1 0 0

0 0 0 1 1

1 1 1 1 1

1 1 1 0 0

LNN lx

(NN )i

(NN )f l

S

L

NN |(T =1) → NN |(T =1) + pseudo-scalar (e.g., pp → ppo ) 1 3 Ss P0 S0 s 1 1 Sp Not allowed 3 1 Ps S0 P0 s 0 0 3 1 D2 P2 s 0 2 NN |(T =1) → NN |(T =0) + pseudo-scalar (e.g., part of pp → pn+ ) 3 3 Ss P1 S1 s 1 1 3 1 Sp S0 S1 p 0 0 3 1 D2 S1 p 0 2 Ps Not allowed

the production of an isoscalar scalar particle, the s-wave 6nal state is allowed. For illustration in Tables 1 and 2 the lowest partial waves for the production of pseudo-scalar and scalar particles are given. In a straightforward way the corresponding selection rules for systems with nuclei in the initial and/or 6nal state can be easily derived along the same lines. Bose symmetry, for example, demands a dd system to be symmetric under exchange of the two deuterons, forcing L+S to be even. Therefore the lowest partial waves contributing to dd → , are 3 P0 → s and 5 D1 → p. Another example of selection rules at work in a system other than NN → NNx will be given in Section 4.3.1. 2. The nal state interaction The role of 6nal state interactions in production reactions has been discussed in the literature for a long time and at various levels of sophistication. Extensive discussions can be found in Refs. [37,38]. For more recent discussions focusing on the production of mesons in nucleon– nucleon collisions we refer to Refs. [39,40,32]. In this section we will approach the question of

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Table 2 The lowest partial waves for the production of a scalar particle in NN → NNx S

L

J

S

L

j

l

NN |(T =1) → NN |(T =1) + scalar (e.g., pp → ppa00 ) 1 1 Ss S0 S0 s 1 1 3 Sp P1 S0 p 1 3 3 Ps PJ PJ s 1 3 3 F2 P2 s 1

0 1 1 3

0 1 J 2

0 0 1 1

0 0 1 1

0 0 J 2

0 1 0 0

NN |(T =1) → NN |(T =0) + scalar (e.g., part of pp → pna+ 0) Ss Not allowed 3 3 Sp PJ S1 p 1 3 3 F2 S1 p 1 3 3 Ps PJ PJ s 1 3 3 F2 P2 s 1

1 3 1 3

J 2 J 2

1 1 1 1

0 0 1 1

1 1 J 2

1 1 0 0

NN |(T =0) → NN |(T =1) + scalar (e.g., part of pn → ppa− 0 ) Ss Not allowed 1 1 Sp P1 S0 p 0 3 1 Ps P1 P1 s 0

1 1

1 1

0 1

0 1

0 1

1 0

NN |(T =0) → NN |(T =0) + scalar (e.g., part of pn → pnf0 ) 3 3 Ss S1 S1 s 1 3 3 D1 S1 s 1 3 1 Sp P1 S1 p 0 1 1 Ps P1 P1 s 0

0 2 1 1

1 1 1 1

1 1 1 0

0 0 0 1

1 1 1 1

0 0 1 0

LNN lx

(NN )i

(NN )f l

the role of the 6nal state interactions from two approaches: 6rst we will present a heuristic approach that should make the physics clear, but has only limited quantitative applicability; then, using dispersion integrals, quantitative expressions will be derived, but these are less transparent. This investigation will show under what circumstances scattering parameters can be extracted from high momentum transfer production reactions, and will address questions about the accuracy of this extraction. 2.1. Heuristic approach As early as 1952 Watson pointed out the circumstances under which one would expect the 6nal state interaction to strongly modify the energy dependence of the total production cross section NN → NNx [41]. His rather convincing argument went as follows: assume in contrast to the production reaction the pion absorption on a free two-nucleon pair. Due to time reversal invariance the matrix elements for the two reactions are equal. Based on the observation that the absorption takes place in a small volume, it is obvious that the reaction is a lot more probable if, in a 6rst step, the two-nucleons come together rather closely due to their attractive interaction and that this signi6cantly more compact object then absorbs the pion. In this picture the inNuence of the two-nucleon system on the energy

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T =

A

M

(a)

+

M

T +

(b)

M

+

M

T

T

(c)

(d)

Fig. 5. Diagrammatic presentation of the distorted wave Born approximation.

dependence of the cross section is very natural. Based on this argument, one should expect 7  mx &NN →NNx ( ) ˙ d 3 q p d.p
|/x† (q )0NN (p )|2 ; 0

(4)

where q (p ) denotes the momentum of the outgoing meson (the relative momentum of the outgoing two-nucleon pair) and d 3 q = dq d.q
; energy conservation implies p 2 = MN (E − !q
), where !q
denotes the energy of the meson. In Eq. (4) 0(p ; p ) is the NN wavefunction at the energy, 2E(p ), of the 6nal NN subsystem, where E(p ) ≡ p 2 =2MN and /x denotes the wavefunction for the outgoing meson. If we now neglect the impact of the meson–nucleon interaction on the energy dependence, the meson wavefunction is given by a plane wave. For its component in the lth partial wave we get [42] l

/xl (q ) ˙ jl (q R)  q ;

(5)

where jl (q R) denotes the lth Bessel function. The term q l is the 6rst term in an expansion of q R, where R represents the range of forces. Therefore, Eq. (5) should appropriately describe the q dependence of the corresponding partial wave amplitude as long as (q R)
1. In addition, when the 6nal state interaction is very strong and shows a strong energy dependence (as, for example, the nucleon–nucleon interaction in the 1 S0 for small relative energies), we may replace the NN wavefunction by the on-shell T matrix, which can be written in the Lth partial wave T L (Ep
) = T0L ei3L (Ep
)

sin 3L (Ep
) ; p L+1

(6)

where 3L (Ep
) denotes the corresponding phase shift as deduced from elastic NN scattering data. In practice, this means that we neglect the term with the non-interacting outgoing two-nucleon pair (Figs. 5a and c) compared to that with the pair interacting (Figs. 5b and d). In case of the NN 6nal state interaction this is indeed justi6ed. In general, however, the plane wave piece is not negligible. Fortunately in this case, dispersion integrals may be used to 6x the relative strength of diagrams a and b of Fig. 5. This will be discussed in detail in the next section.

7

Here all 6nal state particles are treated non-relativistically for simplicity.

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At low energies the T matrix may be replaced by the 6rst term in the e@ective range expansion. 8 1 2L+1 p cot 3L (Ep
) = ; (7) aL where aL denotes the scattering length of the NN interaction in the partial wave characterized by L. Thus we can write  mx p 2L+1 (L; l)   2l+2 ( ) ˙ dq q : (8) &NN →NNx 1 + a2L p 4L+2 0 Since Eq. (5) only holds for (q R)
1, in Eq. (8) the aL -term has to be kept in the denominator only if aL R2L+1 . In case of the NN interaction, where R ∼ m this condition holds for the S-waves only. When data for the reaction pp → pp0 close-to-threshold became available [43,44] Eq. (8) indeed turned out to give the correct energy dependence of the total cross-section [45]. Eq. (8) contains two very important messages. First of all that a strong 6nal state interaction will signi6cantly change the invariant mass distribution of the corresponding two particle subsystem—the relevant scale parameter is (aL p ). This observation was used in Ref. [46] to extract information on the N interaction. We come back to this in Section 2. Secondly that due to the centrifugal barrier higher partial waves are suppressed in the close-to-threshold regime. As a rough estimate we can take non-relativistic kinematics to derive   4NN L (L; l) L l l+L+2 &NN ( ) ˙ (mx )2l+2L+4 ˙ 4NN 4(NN (9) )x Q →NNx 4(NN )x for all those partial waves where there is no strong FSI. Here we used the relation between Q and as given in Eq. (A.4) and 4NN = MN =2 and 4(NN )x = mx =(1 + mx =(2MN )) denote the reduced masses of the NN or more generally the baryon–baryon system and of the particle x with respect to the NN system, respectively. Eq. (9) contains two pieces of information: 6rst of all it shows that, for a given value of Q, heavier systems are more likely to be in a higher partial wave. The reason is that their maximum momentum allowed by energy conservation is larger than that of light particles. Secondly, it should give a reasonable estimate of the energy dependence of the partial cross sections for all those partial waves where the NN system is in a partial wave higher than the S-wave. In case of the pion production this 6nding was con6rmed recently in Ref. [47] for the reaction pp → pp0 as is demonstrated in Fig. 6, where the spin cross sections 1 &0 and 3 &1 are shown as a function of . We use the notation of Ref. [47] as 2S+1 &m , where S (m) denotes the total spin (spin projection) of the initial state. For the de6nition we refer to Section 4.2. It is easy to show that the lowest partial waves in the 6nal state that contribute to 1 &0 (3 &1 ) are Ps (Pp and Ds), where capital letters denote the relative angular momentum of the two-nucleons and the small letters that of the pion with respect to the two-nucleon system. Here we used that pion d-waves should be suppressed compared to NN D-waves (cf. Eq. (9)). Accordingly one should expect the Ps states to have an energy dependence given by 6 whereas the Pp states should follow an 8 dependence (cf. Eq. (9)). The corresponding 6ts are shown in the 6gure as a solid and a dashed line, respectively. The good agreement of these simple 8

Unfortunately, in the literature there are two di@erent conventions for the sign of the scattering length. The convention used here is that of Goldberger and Watson [37]; the usual convention for baryon–baryon interactions has a minus sign on the right-hand side.

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10

2S+1σ

m

[µb]

100

1

0.4

0.5

0.6

η

0.7

0.8

0.9

1

Fig. 6. Energy dependence of two di@erent spin cross sections for the reaction pp → pp0 . The data for 1 &0 (3 &1 ), shown as solid (opaque) dots, are compared to a 6t of the form y = A % with % = 6 (% = 8). The data are from Table V of Ref. [47] with an additional error of 10% on the total cross section.

forms for the energy dependence of the di@erent partial waves supports the conjecture that neither the initial state interaction nor the production operator and not even the FSI, for partial waves higher than S-waves, induce a signi6cant energy dependence beyond the centrifugal barrier. As we have seen, the energy dependence factorizes from the total amplitude. The situation is somewhat more complicated, however, when it comes to quantitative predictions for the cross section: in the presence of very strong 6nal state interactions it is not possible to separate model independently production operator and 6nal state interaction. The reason for this is that what sets the scale for the size of the matrix element is the convolution integral of the production operator with the 6nal state half o@-shell NN T -matrix. This convolution consists of a unitarity cut contribution in which the intermediate two-nucleons are on their mass shell, and a principal value piece that is sensitive to the o@-shell behavior of both the NN T -matrix as well as the production operator. This already shows that it is impossible to separate the production operator from the 6nal state interaction in a model independent way. Since o@-shell e@ects are not observable, a consistent construction of production operator and 6nal state interaction is required to get meaningful results for the observables. In order to keep the formulas simple so far we have not discussed at all the e@ect of the Coulomb interaction, although its e@ect especially on near-threshold meson production with two protons in the 6nal state is known to be quite strong [45]. To account for the Coulomb interaction in the formulas given above one should use the Coulomb modi6ed e@ective range expansion instead of Eq. (7). The corresponding formulas are e.g. given in Ref. [48]. 2.2. Dispersion theoretical approach The fact that meson production in nucleon–nucleon collisions is a high momentum transfer reaction allows for a rigorous treatment of strong 6nal state interactions within a dispersion theoretical

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approach. 9 As we will see, in those cases where the e@ective range of a particular 6nal state interaction is of the order of the scattering length, the modi6cation of the invariant mass spectrum induced by the corresponding 6nal state interaction is no longer proportional to the elastic scattering situation. However, the corrections that arise can be accounted for systematically. In addition to things that are well known and presented in various text books [37,38], we derive an integral representation for the scattering length in terms of an observable [32]. In addition one gets at the same time an integral representation of the uncertainty of the method. The latter can be estimated using scale arguments or by doing a model calculation. For example, in the case of the hyperon–nucleon interaction the scattering lengths as extracted from the available data on elastic scattering are of the order of a few Fermi, but with an uncertainty of a few Fermi—especially for the spin singlet scattering length (cf. discussion in Section 7.4). Using the kinematics for associated strangeness production to estimate of the theoretical uncertainty, we 6nd a value of 0:3 fm. Thus it is feasible from the theoretical point of view to improve signi6cantly our knowledge of the hyperon–nucleon scattering parameters through an analysis of the reaction pp → pK . To be de6nite, we will discuss the reaction pp → pK , however—as should be obvious—the complete discussion also applies to any subsystem with a suOciently strong interaction. Let us assume the p system is in a single partial wave (this can be easily extracted from polarized experiments as will be discussed in Section 4.3.1) and the third particle—here the kaon—is produced with a de6nite momentum transfer t = (p1 − pK )2 , where p1 denotes the beam momentum. The full production amplitude may then be written as  2  1 m˜ D(s; t; m 2 ) 1 ∞ D(s; t; m 2 ) 2 2 A(s; t; m2 ) = dm + dm ; (10)  −∞ m 2 − m2  m20 m 2 − m2 where m2 = m˜ 2 is the left-hand singularity closest to the physical region, m0 = MN + M corresponds to the 6rst right-hand cut, and 1 (11) D(s; t; m2 ) = (A(s; t; m2 + i0) − A(s; t; m2 − i0)) : 2i Note that, in contrast to the representation of the amplitude in case of one open channel only as in G → , D is not simply given by the imaginary part of A, since the initial state interaction also NN induces a phase in A (cf. discussion in the next section). Thus D has to be written explicitly as the discontinuity of A along the appropriate two-particle cut. The second integral in Eq. (10) gets contributions from the various possible 6nal state interactions, namely K, NK and N , where the latter is the strongest. We may thus neglect the former two for a moment to get, for m2 ¿ m20 , D(s; t; m2 ) = A(s; t; m2 )e−i3 sin 3 ;

(12)

where 3 is the elastic N phase shift. To see where this relation comes from we refer to Fig. 7: the appearance of the various unitarity cuts due to the baryon–baryon 6nal state interaction leads to a discontinuity in the amplitude along the corresponding branch cut. We thus may read almost directly o@ the 6gure that, indeed, the discontinuity in the amplitude is given by the production amplitude times the two-baryon phase-space density times the (complex conjugate of the) on-shell 9

Actually, the formalism applies to all large momentum transfer reactions. In addition to meson production in NN collisions, one may study e.g. d → B1 B2 + meson. This reaction is also discussed in Refs. [49–54].

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T

D(s,t,m2>s0) =

T

M

+

T

M

Fig. 7. Illustration of the sources of a discontinuity in the amplitude A from two baryon intermediate states. The dashed horizontal lines indicate the presence of a two baryon unitarity cut (diagrams with initial state interaction are not shown explicitly (cf. Fig. 5)).

N T -matrix. 10 Eq. (12) follows by simply using the de6nition of the elastic phase shift: 7T (m2 ) = ei3 sin 3, where 7 = p 4 denotes the phase-space density here expressed in terms of the reduced mass of the N system 4 = (MN M )=(MN + M ). We will discuss at the end of this section how to control the possible inNuence of the K interactions. The solution of (10) in the physical region can then be written as (see [38] and references therein) 2

A(s; t; m2 ) = eu(m +i0) /(s; t; m2 ) ; where



m˜ 2

dm 2 D(s; t; m 2 ) −u(m
2 ) e m 2 − m 2 −∞ and, in the absence of bound states,  1 ∞ 3(m 2 ) 2 dm : u(z) =  m20 m 2 − z 1 /(s; t; m ) =  2

(13)

(14)

(15)

The m2 dependence of / is dominated by the m2 dependence of the production operator. The momentum transfer in the production operator, however, is controlled by the initial momentum and therefore one should expect the m2 dependence of / to be weak as long as the corresponding relative momentum of the -nucleon pair is small. Thus, in a large momentum transfer reaction, the m2 dependence of the production amplitude A is governed by the elastic scattering phase shifts of the dominant two particle reaction in the 6nal state! The relation between the phase shifts and the m2 dependence of the amplitude as can be extracted from an invariant mass distribution is 9xed by analyticity and unitarity! 10

The appearance of the complex conjugation here follows from the direct evaluation of the discontinuity of the T -matrix by employing the unitarity of the S matrix: from S † S = 1 it follows, that 1 disc(T ) := (T (m2 + i0) − T (m2 − i0)) = −7|T (m2 )|2 : 2i

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So far we are in line with the reasoning of the previous section, however the exponential factor in Eq. (13) is in general not simply the elastic scattering amplitude. For illustration we investigate the form of A(m2 ) for a 6nal state interaction that is fully described by the 6rst two terms in the e@ective range expansion (cf. Eq. (7)), p ctg(3(m2 )) = 1=a + ( 12 )rp 2 , where p is the relative momentum of the 6nal state particles under consideration in their center-of-mass system. Then A can be given in closed form as [37] (p 2 + ,2 )r=2 /(s; t; m2 ) ; (16) 2   1=a + (r=2)p − ip  where , = (1=r)(1 + 1 + 2r=a). Note: The case of an attractive interaction without a bound state in this convention corresponds to a positive value of a and a positive value of r. In the limit a → ∞, as is almost realized in NN scattering, the energy dependence of A(m2 ) is given by 1=(1−iap ) as long as p
1=r. This, however, exactly agrees with the energy dependence for NN on-shell scattering. This prediction was experimentally con6rmed by the near-threshold measurements of the reaction pp → pp0 [43,44]. However, for interactions where the e@ective range is of the order of the scattering length, the numerator of Eq. (16) plays a role and thus the full production amplitude is no longer given by the on-shell elastic scattering times terms whose energy dependence is independent of the scattering parameters. It is often argued that since the full production amplitude is given by a term with a plane wave 6nal state as well as one with the strong interactions, that an appropriate parameterization is given by a Watson term (proportional to the elastic scattering of the outgoing particles) plus a constant term, and their relative strength should be taken as a free parameter. The considerations in the previous paragraphs show, however, that this is not the case: Eq. (13) (and thus also the special form given in Eq. (16)) describes the m2 dependence of the full invariant mass spectrum. One more comment is in order: In the previous section we stressed that, although the shape of the invariant mass spectra as well as the energy dependence of the total cross section are governed by the on-shell interactions in the subsystems, the overall normalization is not. This statement, based on e@ective 6eld theory arguments, is illustrated in Appendix D. On the other hand, Eq. (13) gives a closed expression for the invariant mass spectrum for arbitrary values of m2 . On the 6rst glance this look like a contradiction. However, it should be stressed that Eq. (13) does in general not allow one to relate the asymptotic form of the full production amplitude with information on the scattering parameters of the strongly interacting subsystems in the 6nal state to the amplitude in the close-to-threshold regime. Besides the trivial observation that Eq. (13) holds for each amplitude individually and that far away from the threshold there should be many partial waves contributing, there is no reason to believe that the function / is constant over a wide energy range. Therefore, we want to emphasize that the m2 dependence of Eq. (13) is controlled by the scattering phase shifts of the most strongly interacting subsystem only in a very limited range of invariant masses. In fact, the leading singularity that contributes to an m2 dependence of / is that of the t-channel meson exchange in the production operator. The m2 dependence that originates from this singularity can be estimated through a Taylor expansion of the momentum transfer and thus turns out to be governed by (p =p)2 , where p denotes the relative momentum of the 6nal state particle pair of interest. For p we may use that of the initial state particles at the production threshold, all to be taken in the over all center-of-mass system. As a consequence, for values of m2 that do not signi6cantly deviate A(s; t; m2 ) =

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from m20 (or equivalently p
p) the assumption of / being constant is justi6ed. However / can show a signi6cant m2 dependence over a large range of invariant masses. Let us now return to our main goal: namely to derive from (13) a formula that allows extraction of the elastic scattering phase shifts from an invariant mass spectrum. In this course we will derive a dispersion relation in terms of the function |A(m2 )|2 . Here we will use a method similar to the one used in [55,56]. However, in contrast to the formulas derived in these references, we will present a integral representation for the elastic scattering phase shifts from production data that involves a 6nite integration range only. For this we 6rst observe that Eq. (15) holds for purely elastic scattering only and thus is of very limited practical use. On the other hand, a signi6cant contribution stems from large values of m 2 which depends only weakly on m2 in the near-threshold region, and therefore can be absorbed into the function /. Let    2 3(m 2 ) 1 mmax 2 2 ˜ dm /(m2max ; m2 ) ; (17) A(m ) = exp  m20 m 2 − m2 − i0 ˜ 2max ; m2 ) = /(m2 )/m2 (m2 ), with where /(m max    ∞ 2) 3(m 1 2 dm : /m2max (m2 ) = exp  m2max m 2 − m2 − i0

(18)

The quantity m2max is to be chosen by physical arguments in such a way that both /(m2 ), and /m2max (m2 ) vary slowly on the interval (m20 ; m2max ). Obviously, it needs to be suOciently large that the structure in the amplitude we are interested in can be resolved. 11 On the other hand, it should be as small as possible, since this will keep the inNuence of the inelastic channels small. In order to estimate the minimal value of mmax we see from Eq. (16) that the values of p that enter in the integral given in Eq. (22) should be at least large enough that the scattering length term plays a  signi6cant role. Thus we require pmax ∼ 1=atyp . In case of the pn interaction (the spin triplet pn  scattering length is 5:4 fm) this would correspond to a value of pmax of only 10 MeV corresponding to a value for jmax = mmax − m0 —the maximum excess energy that should occur within the integral in Eq. (17) (as well as Eq. (22))—as small as 2 MeV. On the other hand, if we want to study the hyperon–nucleon interaction, we want to ‘measure’ scattering lengths of the order of a few fermi we choose atyp = 1 fm, leading to a value of 40 MeV for jmax . Note that 40 MeV is still signi6cantly smaller than the thresholds for the closest inelastic channels (that for the KN 6nal state is at 75 MeV, that for the  KN channel at 140 MeV). The integral in (17) contains an unphysical singularity of the type log(m2max − m2 ), which is ˜ 2max ; m2 ), but this does not a@ect the region near threshold. canceled by the one in /(m Notice next that the function  2 ˜ 2max ; m2 )} 1 1 mmax 3(m 2 ) log{A(m2 )= /(m 2   = dm (19) 2  2 − i0 2 2 2 2 2 2 2 2 2  m − m (m − m0 )(mmax − m ) (m − m0 )(mmax − m ) m0 11

One immediately observes that the integral given in Eq. (22) goes to zero for mmax → m0 . At the same time the theoretical uncertainty 3ammax , de6ned in Eq. (24), tends to the value of the scattering length.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

175

mmax

mo

mo

(a)

(b) 2

Fig. 8. The integration contours in the complex m plane to be used in the original treatment by Geshkenbein [55,56] (a) and in the derivation of Eq. (18) (b). The thick lines indicate the branch cut singularities.

has no singularities in the complex plane except the cut from m20 to m2max (cf. Fig. 8b) and its value below the cut equals the negative of the complex conjugate from above the cut. Hence,  m2max ˜ 2max ; m 2 )|2 1 log |A(m 2 )= /(m m2max − m2 3(m2 ) 2   P = − dm : (20) 2 2  2 2 2 2 mmax − m m − m0 m0 m 2 − m2 (m 2 − m2 ) 0

It is an important point to stress that  P

m2max

m20



1 m 2 − m20 (m 2 − m2 )



m2max − m2 2 dm = 0 2 2  mmax − m

(21)

as long as m2 is in the interval between m20 and m2max . Therefore, if the function / only weakly depends on m2 , as it should in large momentum transfer reactions, it can well be dropped in the above equation. In addition, up to kinematical factors, |A|2 agrees with the cross section for the production of a p pair of invariant mass m. We can therefore replace it in Eq. (20) by the cross section where, because of Eq. (21), all constant prefactors can be dropped. Thus we get

  2 mmax 1 M + Mp m2max − m2 2  P dm aS = lim M Mp m2 →m20 2 m2max − m 2 m20   2 d &S 1 1 : (22) × log p dm 2 dt m 2 − m20 (m 2 − m2 ) This is the desired formula: namely, the scattering length is expressed in terms of an observable. Note that Eq. (22) is applicable only if it is just a single YN partial wave that contributes to the cross section &S . In Section 4.3.1 we will discuss how the use of polarization in the initial state can be used to project out a particular spin state in the 6nal state.

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Up to the neglect of the kaon–baryon interactions, Eq. (20) is exact. Thus, 3a(th) —the theoretical uncertainty of the scattering length extracted using Eq. (22)—is given by the integral

 1 M + Mp (th)  3a = − lim M Mp m2 →m20 2  m2max ˜ 2max ; m 2 )|2 log |/(m m2max − m2 2  dm : (23) ×P 2 2  2 m − m 2 2 m0 max m − m2 (m − m2 ) 0

˜ 2max ; m2 )|2 = log |/(m2 )|2 + log |/m2 (m2 )|2 we may write 3a(th) = 3a(lhc) + 3ammax , Since log |/(m max where the former, determined by /(m2 ), is controlled by the left-hand cuts, and the latter, determined by /m2max (m2 ), by the large-energy behavior of the N scattering phase shifts. The closest left-hand singularity is that introduced by the production operator, which is governed by the momentum transfer. Up to an irrelevant overall constant, we may therefore estimate the variation of / ∼ 1 + 3(p =p)2 , where we assume 3 to be of the order of 1. Evaluation of integral (23) then gives  =p2 ) ∼ 0:05 fm ; 3a(lhc) ∼ 3(pmax 2 where we used pmax
= 24jmax with jmax ∼ 30 MeV and the threshold value for p ∼ 900 MeV. On the other hand, using the de6nition of /m2max (m2 ) given in Eq. (18) one easily derives   2 2  ∞ 3(y) dy  mmax 6 |3max | ; (24) |3a | =    2 (3=2)  pmax 0 (1 + y ) pmax

where y2 = (m − mmax )=jmax . Thus, in order to estimate 3ammax we need to make an assumption about the maximum value of the elastic N phase for m2 ¿ m2max . Recall that we implicitly assume the inelastic channel not to play a role. This assumption was con6rmed in Ref. [32] within a model calculation. The denominator in the integral appearing in Eq. (24) strongly suppresses large values of y. Since for none of the existing N models does 3max exceed 0:4 rad, we estimate 3ammax ∼ 0:2 fm : When using the phase shifts as given by the models directly in the integral, the value for 3ammax is signi6cantly smaller, since for all models the phase changes sign at energies above m2max . Combining the two error estimates, we conclude 3a(th) . 0:3 fm : In the considerations of the theoretical uncertainty of the parameters extracted so far we did not talk about the possible e@ect of the kaon–baryon interaction. Unfortunately, it cannot be quanti6ed a priori; however, it can be controlled experimentally. As will be discussed in Section 4, a signi6cant meson–baryon interaction, due, for example, to the presence of a resonance, will show up as a band in the Dalitz plot. If this band overlaps with the region of 6nal state interactions we are interested in, interference e@ects might heavily distort the signal [57]. By choosing a di@erent beam energy, the FSI region and the resonance band move away from each other. Thus there should be an energy regime where the FSI can be studied undistorted. It is therefore necessary for a controlled extraction of FSI parameters to do a Dalitz plot analysis at the same time. An additional possible cross-check

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177

of the inNuence of a meson–baryon interaction would be to do the full analysis at two di@erent beam energies separated in energy by at least the typical hadronic width (about 100–200 MeV). If the parameters extracted agree with each other, there was no substantial inNuence from other subsystems. The possible gain in experimental accuracy for the extraction of the N scattering lengths is illustrated and discussed in detail in Section 7.4. To our knowledge so far the corresponding formalism including the Coulomb interaction is not yet derived. 3. The initial state interaction As mentioned earlier, the collision energy of the two nucleons in the initial state is rather large, especially when we study the production of a heavy meson. However, as long as the excess energy is small, only very few partial waves contribute in the 6nal state and thus the conservation of total angular momentum as well as parity and the Pauli principle allow for only a small number of partial waves in the initial state. In addition, only small total angular momenta are relevant. In such a situation the standard methods developed by Glauber, which include the initial state interaction via an exponential suppression factor [58], cannot be applied. The role of the initial state interaction relevant to meson production reactions close to the threshold was discussed for the 6rst time in Ref. [40]. It should be clear a priori that for quantitative predictions the ISI has to play an important role: to allow a production reaction to proceed the nucleons in the initial state have to approach each other very closely—actually signi6cantly closer that the range of the NN interaction. Thus, especially for the production o@ heavy mesons, a large number of elastic and inelastic NN reactions can happen before the two-nucleon pair comes close enough together to allow for the meson production. Thus the e@ective initial current gets reduced signi6cantly. Technically, the inclusion of the initial state interaction in a calculation for a meson production process requires a convolution of the production operator with the half o@-shell T -matrix for the scattering of the incoming particles (cf. Figs. 5c and d). Contrary to the 6nal state interaction, for the high initial energies the energy dependence of the NN interaction is rather weak. One might therefore expect that the real part of the convolution integral is small and that the e@ect of the initial state interaction is dominated by the two-nucleon unitarity cut [40]. This contribution, however, can be expressed completely in terms of on-shell NN scattering parameters:  2 |:L |2 =  12 ei3L (p) ( L (p)ei3L (p) + e−i3L (p) ) = − L (p) sin2 (3L (p)) + 14 [1 + L (p)]2 6 14 [1 + L (p)]2 ;

(25)

where p denotes the relative momentum of the two nucleons in the initial state with the total energy E, and 3L ( L ) denote the phase shift (inelasticity) in the relevant partial wave L. Each partial wave amplitude should be multiplied by :L , de6ned in Eq. (25), in order to account for the dominant piece of the ISI. This method was used e.g. in Ref. [59]. Using typical values for phase shifts and inelasticities at NN energies that correspond to the thresholds of the production of heavier mesons, Eq. (25) leads to a reduction factor of the order of 3, thus clearly indicating that a consideration of the ISI is required for quantitative predictions. One should keep in mind, however, that Eq. (25) can

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

only give a rough estimate of the e@ect of the initial state interaction and should whenever possible be replaced by a full calculation. This issue is discussed in detail in Ref. [34]. Unfortunately, the applicability of Eq. (25) is limited to energies where scattering parameters for the low NN partial waves are available. Due to the intensive program of the EDDA collaboration for elastic proton–proton scattering [60] and the subsequent partial wave analysis documented in the SAID database [61], scattering parameters in the isospin—one channel are available up to energies that correspond to the ; production threshold. The situation is a lot worse in the isoscalar channel. Here pn scattering data would be required. At present, those are available only up to the production threshold. It is very fortunate that there is a proposal in preparation to measure spin observables in the pn system in this unexplored energy range at COSY [62]. 4. Observables In this section the various observables experimentally accessible are discussed. After some general remarks about the three-body kinematics in the 6nal state, in the next subsection we will discuss unpolarized observables. In the following subsection we will then focus on polarized observables. Even if x denotes just a single meson, the reaction NN → B1 B2 x is subject to a 6ve-dimensional phase: three particles in the 6nal state introduce 3×3=9 degrees of freedom, but the four momentum conservation reduces this number to 5. As we will restrict ourselves to the near-threshold regime, the 6nal state can be treated non-relativistically. The natural coordinate system is therefore given by the Jacobi–coordinates in the overall center-of-mass system (cf. Ref. [42]), where 6rst the relative momentum of one pair of particles is constructed and then the momentum of the third particle is calculated as the relative momentum of the third particle with respect to the two-body system. Obviously, there are three equivalent sets of variables possible. In the center-of-mass this choice reads ˜ j − Mj p ˜i Mi p p ˜ ij = ; Mi + M j ˜qk =

pk − Mk (˜ pi + p ˜ j) (Mi + Mj )˜ =p ˜k ; Mi + M j + M k

(26)

where we labeled the three 6nal state particles as ijk; qk = pk holds in the over all center-of-mass system only. For simplicity in the following, we will drop the subscripts when confusion is excluded. For reactions of the type NN → B1 B2 x it is common to work with the relative momentum of the two-nucleon system and to treat the particle x separately. From the theoretical point of view this choice is most convenient, for one is working already with the relative momentum of the dominant 6nal state interaction and the assignment of partial waves as used in Section 1.6 is straightforward. Note that for any given relative energy of the outgoing two-nucleon system j = p 2 =MN the modulus of the meson momentum |˜q  | is 6xed by energy conservation; we thus may characterize the phase space by the 5 tuple = = {j; .p
; .q
} ;

(27)

where .k = (cos(>k ); ;k ) denotes the angular part of vector ˜k. The coordinate system is illustrated in Fig. 9. The center-of-mass nucleon momentum in the initial state will be denoted by p ˜ . Throughout

C. Hanhart / Physics Reports 397 (2004) 155 – 256 B1 p'−MB1/(MB1+MB2) q'

p

x q'

179

B2 −p'− MB2/(MB1+MB2) q'

−p

Fig. 9. Illustration of the choice of variables in the over all center-of-mass system.

this report we choose the beam axis along the z-axis. Explicit expressions for the vectors appearing are given in Appendix B.1. Due to the high dimensionality of the phase space it is not possible to present the full complexity of the data in a single plot. At the end of Section 4.2 we will discuss a possible way of presenting the data through integrations subject to particular constraints. A di@erent choice would be simply to present highly di@erential observables as is done for bremsstrahlung, 12 or at least to use the two dimensional representation of the Dalitz plot to show some correlations. 4.1. Unpolarized observables As long as the initial state is unpolarized, the system has azimuthal rotation symmetry, reducing the number of degrees of freedom from 5 to 4. In the case of a three-particle decay (a 1 → 3 reaction) the physics does not depend on the initial direction. The same is true for reactions where the initial state does not de6ne a direction, like in the experiment series for pp G annihilation at rest carried out by the Crystal Barrel collaboration at LEAR (see Ref. [63] and references therein). Therefore, the number of degrees of freedom is further reduced from four to two. 13 Those are best displayed in the so-called Dalitz plot (see [64] and references therein) that shows a two dimensional representation in the plane of the various invariant masses m2ik = (pi + pk )2 . In reactions with two particles in the initial state and three particles in the 6nal state (2 → 3 reactions), however, the initial momentum de6nes an axis and therefore a single fully di@erential plot is no longer possible. Especially in the example given below, it will become clear how the appearance of the additional momentum changes drastically the situation. To account for the higher complexity of the 2 → 3 reactions, if enough statistics were collected in a particular experiment, one can present di@erential Dalitz plots—a new plot for each di@erent orientation of the reaction plane [65]. However, in what follows, we will call Dalitz plot the representation of a full data set/calculation in the plane of invariant masses, ignoring the initial direction. Obviously this means throwing out some correlations, for in general any integration reduces the amount of information in an observable. We will brieNy review the properties of a Dalitz plot, closely following Ref. [64]. In Fig. 10 a schematic picture of it is shown. In this plot also di@erent 12 It should be noted that in the early bremsstrahlung experiments the method of integration was not possible, because the detectors used had very limited angular acceptance. 13 This is quite obvious, since one can look at a three particle decay as the crossed channel reaction of two-body scattering which is well known to be characterized by two variables in the unpolarized case.

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m232 3

1

2

1 3

F(12) 2 1

F(31)

3

2

v1=v3 p2 max.

R

v1=v2 p3 max.

1

3

2

F(23) 1

2

3

I 1

3

v2=v3

2

p1 max.

m122 Fig. 10. Sketch of a Dalitz plot for a reaction with a three-body 6nal state (particles are labeled as 1, 2 and 3). Regions of possibly strong 6nal state interactions in the (ij) system are labeled by F(ij) ; a resonance in the (12)—system would show up like the band labeled as R. The region of a possible interference of the two is labeled as I .

regimes are speci6ed: those of potentially strong 6nal state interaction for the subsystem (ij) are labeled F(ij) . They should occur when particle i and j move along very closely. On the other hand, resonances will show up as bands in the Dalitz plot. Labeled as R in the 6gure, the e@ect of a resonance in the (12) system is shown. The total area of the Dalitz plot scales with the phase-space volume. Therefore, as the excess energy decreases, resonance and 6nal state interaction signals or the regions of di@erent 6nal state interactions might start to overlap, leading to interference phenomena (cf. the hatched area I in Fig. 10). It was demonstrated recently [57] that those can rather strongly distort resonance and 6nal state interaction signals. At least for known 6nal state properties, these patterns might help to better pin down resonance parameters [57]. On the other hand, as the excess energy increases the di@erent structures move away from each other and one should then be able to study them individually. This observation is of great relevance if one wants to extract parameters of a particular 6nal state interaction from a production reaction (cf. discussion in Section 2). In case of particle decays of spinless or unpolarized particles, the Dalitz plot not only contains the information about the occurrence of a resonance, but also its quantum numbers can be extracted by projecting the events in the resonance band (labeled as R in Fig. 10) on the appropriate axis (for the example of the 6gure this is the 23 axis). In the case of 2 → 3 reactions, however, this projection is not necessarily conclusive. To explain this statement we have to have a closer look at the angles of the system. First of all there is a set of angles, the so-called helicity angles, that can be constructed from the 6nal momenta only. One example is cos(>p
q
) =

p ˜  · ˜q  |˜ p | |˜q  |

(28)

C. Hanhart / Physics Reports 397 (2004) 155 – 256

181

were p and q where de6ned in Eq. (26). These angles that can be extracted from the Dalitz plot directly. In addition there are those angles that are related to the initial momentum—the Jackson angles. One example is cos(>p
p ) =

˜ p ˜ · p :  |˜ p | |˜ p|

(29)

It should be stressed that it is not in the distribution of the helicity angles but that of the Jackson angles that the subsystems reNect their quantum numbers [64]. 14 Therefore a pure Dalitz plot analysis is insuOcient for production reactions and the distributions for the Jackson angles have to be studied as well. This will be illustrated in an example in the following subsection. Note that in the presence of spin there are even additional axes in the problem. This will be discussed in detail in Section 4.2. 4.1.1. Example: analysis of pp → dKG 0 K + close-to-threshold Recently, a 6rst measurement of the reaction pp → dKG 0 K + close to the production threshold was reported [66] at an excess energy Q = 46 MeV. The data, as well as the corresponding theoretical analysis, based on the assumption that only the lowest partial waves contribute, will now be used to illustrate the statements of the previous section. It will become clear, especially, that the information encoded in the distributions of the Jackson angles and the helicity angles is rather di@erent. As can be seen in Table 2, a 6nal state that contains s-waves only is not allowed in this reaction. For later convenience 15 we work with the relative momentum of the kaon system (˜ pKK ˜ ; G ≡ p   cf. Eqs. (26)) and the deuteron momentum with respect to this system (˜qd ≡ ˜q ). Given our assumptions, that only the lowest partial waves contribute, the amplitudes that contribute to the production reaction are either linear in q or linear in p . In Ref. [67] the full amplitude was constructed (cf. also discussion in Section 7.5), however it should be clear that the spin averaged square of the matrix element can be written as ˜  · p) G 2 = C q q 2 + C p p 2 + C q (q˜ · p) ˆ 2 + C1p (p ˆ 2 |M| 0 0 1








˜  · q˜ ) + C3 (p ˜  · p)( ˆ q˜ · p) ˆ ; + C2 (p




(30)

˜  or linear in q˜ . Here pˆ = p since all terms in the amplitude are either linear in p ˜ =|˜ p| denotes the beam direction. Since the two protons in the initial state are identical, any observable has to be symmetric under the transformation p ˜ = −˜ p. This is why pˆ appears in even powers only. Figs. 11 and 12 show the data as well as a 6t based on Eq. (30). The parameters extracted are  ˜  · p)=p given in Table 3. The 6rst two panels of Fig. 11 contain the distributions of the angles (p ˆ    ˜  · q˜ )=(q p ). The and (q˜ · p)=q ˆ . The last panel contains the distribution of the helicity angle (p solid line corresponds to a complete 6t to the data including both p-waves in the KG 0 K + system as well as those in the d(KG 0 K + ) system. For the long dashed line the KG 0 K + p-waves were set to G s-waves) were set zero, whereas for the dotted line the d(KG 0 K + ) p-waves (corresponding to KK 14

As is stressed in Ref. [64], only under special conditions, namely for peripheral production as it occurs in high-energy experiments, the information in the helicity angles and the Jackson angles agrees. Close to threshold, however, meson production is not at all peripheral. 15 In Section 7.5 it will be argued, that the reaction pp → dKG 0 K + can be used to study scalar resonance a+ 0 (980). Thus we are especially interested in the partial waves of the kaon system, that should show a strong 6nal state interaction.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

dσ/dΩ [nb/sr]

182 5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

0

0

0.5 |cos(θpp')|

1

0

0

0.5 |cos(θpq')|

1

0

-1

0 -0.5 0.5 cos(θp'q')

1

dσ/dM [µb/GeV]

Fig. 11. Angular distributions for the reaction pp → dKG 0 K + measured at Q = 46 MeV [66]. The solid line shows the G s-waves as well as p-waves. To obtain the dashed (dotted) line, the parameters result of the overall 6t including both KK for the p-wave (s-wave) were set to zero (see text). The small error shows the statistical uncertainty only, whereas the large ones contain both the systematic as well as the statistical uncertainty added linearly.

1.5

1.5

1

1

0.5

0.5

0 0.98

1 1.02 MKK [GeV]

1.04

0

2.38

2.4 MdK [GeV]

2.42

Fig. 12. Various mass distributions for the reaction pp → dKG 0 K + (Line code as in Fig. 11). The small error bars show the statistical uncertainty only, whereas the large ones contain both the systematic as well as the statistical uncertainty (cf. Ref. [66]).

to zero. Thus, the 6rst two panels truly reNect the partial wave content of the particular subsystems individually, whereas the helicity angle (which can also be extracted from the Dalitz plot) shows a Nat distribution only if both subsystems are in a p-wave simultaneously. Therefore, the helicity angle can well be isotropic although one of the subsystems is in a high partial wave. Note that this
statement is true even if all particles were spinless. The only thing that would change is that C0q

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183

Table 3 Results for the C parameters from a 6t to the experimental data C0p




C0q

0 ± 0:1




C1p

1 ± 0:03




1:26 ± 0:08

C1q




C2 + 13 C3

−0:6 ± 0:1

−0:36 ± 0:17




The parameters are given in units of C0q .




and C0p would vanish (cf. Section 4.3.2). In Fig. 12 two Dalitz plot projections (invariant mass q
p
spectra) are shown. The 6rst one (d&=dmKK G ) is needed to disentangle C0 and C0 . The second one does not give any additional information. What is now the information contained in the two-dimensional Dalitz plot? Since it does not contain any information about the initial state, the parts of the squared amplitude that can be extracted from the Dalitz plot are easily derived from Eq. (30) by integrating over the beam direction [64], giving      1 q
 2 1 p
2 q
p
2 G d.p |M| = C0 + C1 q + C0 + C1 p 3 3   1 ˜  · q˜ ) : + C2 + C3 (p (31) 3





G s-wave strength (C0q + ( 1 )C1q ), the Therefore from the Dalitz plot one can extract the total KK 3

G p-wave strength (C0p + ( 1 )C1p ), as well as the strength of the interference of the two total KK 3 (C2 + ( 13 )C3 ). Note that in this particular example, all the coeOcients given in Eq. (31) (and even the C0 and C1 individually) can as well be extracted from the angular distributions given in Fig. 11 G invariant mass distribution (left panel of Fig. 12) directly; the Dalitz plot here does not and the KK provide any additional information. Obviously, as we move further away from the threshold, the complexity of the amplitude increases and the Dalitz plot contains information not revealed in the projections. To summarize, in order to allow for a complete analysis of the production data, in addition to the Dalitz plot the angular distributions of the 6nal momenta on the beam momentum need to be analyzed as well. The latter distributions are the ones that give most direct access to the partial wave content of the subsystems. 4.2. Spin-dependent observables Polarization observables for 2 → 2 reactions are discussed in great detail in Ref. [68]. In our case, however, we have one more particle in the 6nal state and therefore there are more degrees of freedom available. Here we will not only derive the expressions for the observables in terms of spherical tensors but also relate these to the partial wave amplitudes of the production matrix elements. In this section we closely follow Refs. [47,69].

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In terms of the so-called Cartesian polarization observables, the spin-dependent cross section can be written as   &(=; ˜P b ; ˜P t ; ˜Pf ) = &0 (=) 1 + ((Pb )i Ai0 (=) + (Pf )i D0i (=)) +

 ij

+



i

((Pb )i (Pt )j Aij (=) + (Pb )i (Pf )j Dij (=))  (Pb )i (Pt )j (Pf )k Aij; k0 (=) : : : ;

(32)

ijk

where &0 (=) is the unpolarized di@erential cross section, the labels i; j and k can be either x; y or z, and Pb , Pt and Pf denote the polarization vector of beam, target and the 6rst one of the 6nal state particles, respectively. All kinematic variables are collected in =, de6ned in Eq. (27). The observables shown explicitly in Eq. (32) include the beam analysing powers Ai0 , the corresponding quantities for the 6nal state polarization D0i , the spin correlation coeOcients Aij , and the spin transfer coeOcients Dij . In this context it is important to note that baryons that decay weakly, as the hyperons do, have a self analyzing decay. In other words, the angular pattern of the decay particles depends on the polarization of the hyperon, therefore, the hyperon polarization in the 6nal state can be measured without an additional polarimeter (see e.g. Ref. [70]). All those observables that can be de6ned by just exchanging ˜P b and ˜P t , such as the target analyzing power A0i , are not shown explicitly. From Eq. (32) it follows that for example 1 (33) &0 Aij; k0 = Tr(&k(f) M&i(b) &j(t) M† ) ; (2sb + 1)(2st + 1) where the &i(b) (&j(t) ) are the Pauli matrices acting in the spin space of beam and target, respectively. The production matrix element is denoted by M. In addition, sb (st ) denote the total spin of the beam (target) particles. It is straightforward to relate the polarization observables to the partial wave amplitudes that can be easily extracted from any model. For this purpose it is convenient to use spherical tensors de6ned through 1 (b) (t) † 1 )† (f2 )† Tr[%(f Tkk13qq13;k; k24qq24 = (34) k3 q3 %k4 q4 M%k1 q1 %k2 q2 M ] ; (2sb + 1)(2st + 1) where the %kq denote the spherical representation of the spin matrices 1 (35) %10 = &z ; %1±1 = ∓ √ (&x ± i&y ); %00 = 1 : 2 To relate the observables to the spherical tensors, the easiest method is to use the de6nitions of Eqs. (35) inside the various equations (34). For the observables for which the 6nal polarization remains undetected, the relations between the various T and the corresponding observables are shown in Table 4. In Table 5 a few of the observables that contain the 6nal state polarization are listed. Triple polarization observables are not listed explicitly, but it is straightforward to derive also the relevant expressions for these, such as √ 3 1100 Axx; x + Ayy; x − (Axy; y − Ayx; y ) = 2 Re(T111 −1 ) :

C. Hanhart / Physics Reports 397 (2004) 155 – 256

185

Table 4 Relations between spherical tensors and some observables that do not contain the 6nal state polarization following Ref. [69] Cartesian observable Di@erential cross section &0 Beam analyzing powers &0 Ax0 &0 Ay0 &0 Az0 Target analyzing powers &0 A0x &0 A0y &0 A0z Spin correlation parameters &0 Azz & 0 A & 0 AC &0 Axz &0 Azx &0 Ayz &0 Azy &0 [Axy + Ayx ] & 0 A[

Tk1 q1 k2 q2

Q i = q1 + q 2

T0000 (s; j)

0

*

√ −√2 Re(T1100 (s; j)) − 2 Im(T1100 (s; j)) T1000 (s; j)

1 1 0

*

√ −√2 Re(T0011 (s; j)) − 2 Im(T0011 (s; j)) T0010 (s; j)

1 1 0

(∗)

T1010 (s; j) −2 Re(T111−1 (s; j)) 2 Re(T √ 1111 (s; j)) −√2 Re(T1110 (s; j)) −√2 Re(T1011 (s; j)) −√2 Im(T1110 (s; j)) − 2 Re(T1011 (s; j)) 2 Im(T1111 (s; j)) 2 Im(T111−1 (s; j))

0 0 2 1 1 1 1 2 0

* * * (∗) *

*

(∗)

*

To simplify notation, the indices specifying the 6nal state polarization are dropped. The symbol * indicates a possible set of independent observables. Note: For pp-induced reactions more observables become equivalent, as described in the text. Those are marked by (∗). The linear combinations of spin correlation observables appearing in the table are de6ned in Eqs. (39).

After some algebra given explicitly in Appendix C, one 6nds 1 Q TD (p; ˆ q) ˆ = BL˜l;˜ : (q; ˆ p)A ˆ LD˜l;˜ : ; 4

(36)

˜ L˜l:

where D = {k1 q1 ; k2 q2 ; k3 q3 ; k4 q4 } and Q = q1 + q2 − q3 − q4 . All the angular dependence is contained in  1 ˜ l |:Q YL4 ˜ L ; l4 L4 BLQ˜l;˜ : (q; ˆ p) ˆ = ˆ l4 ˆ (37) ˜ L (p)Y ˜ l (q) 4 4 ;4 L

and ALD˜l;˜ : =

 ,;,G

l

CL,;˜l;˜,;G:D M , (M ,G)† :

(38)

Here , and ,G are multi-indices for all the quantum numbers necessary to characterize a particular partial wave matrix element and C denotes a coupling coeOcient that can be expressed in terms of Clebsch–Gordan coeOcients. Its explicit form is given in Eq. (C.9). From Eq. (36) one can derive the angular dependences of all observables.

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

Table 5 Relations between spherical tensors and some observables that do contain the 6nal state polarization Cartesian observable

Tkk13qq13kk24qq24

Q = q 1 + q 2 − q3 − q 4

Induced polarization &0 D0x &0 D0y &0 D0z Spin transfer coeOcients &0 Dzz & 0 D & 0 DC &0 Dxz &0 Dzx &0 Dyz &0 Dzy &0 [Dxy + Dyx ] & 0 DE

√ 1−100 (s; j)) √2 Re(T0000 1−100 2 Im(T0000 (s; j)) 1000 T0000 (s; j)

1 1 0

*

1000 T1000 (s; j) 1100 2 Re(T1100 (s; j)) 1−100 −2 Re(T (s; j)) 1100 √ 1000 − 2 Re(T (s; j)) 1100 √ 1−100 2 Re(T (s; j)) 1000 √ 1000 − 2 Im(T (s; j)) 1100 √ 1−100 2 Re(T1000 (s; j)) 1−100 −2 Im(T1100 (s; j)) 1100 2 Im(T1100 (s; j))

0 0 2 1 1 1 1 2 0

* * * * *

*

*

The symbol * indicates a possible set of independent observables. The linear combinations of spin correlation observables appearing in the table are de6ned in Eqs. (40).

In what follows, it is convenient to de6ne the following quantities: A = Axx + Ayy ;

AC = Axx − Ayy ;

and

AE = Axy − Ayx

(39)

and, analogously, D = Dxx + Dyy ;

DC = Dxx − Dyy ;

and

DE = Dxy − Dyx :

(40)

Using the conservation of parity and the explicit expression for the C coeOcient given in Eq. (C.9) CL,;˜l;˜,;G:D = (−)(k1 +k2 +k3 +k4 ) CL,;˜Gl;˜,;:D :

(41)

Since the parameter C is real, the analyzing powers are proportional to the imaginary part of M, M,∗G , whereas the di@erential cross section as well as the spin correlation parameters depend on the real part of M, M,∗G . Thus, it is either the real part or the imaginary part of B that contributes to the angular structure. As we will see in the following subsection, this observation allows for a straightforward identi6cation of the possible azimuthal dependences of each observable. Another obvious consequence of Eq. (41) is, that those observables for which ki is odd have to be small when only a single partial wave dominates. Thus, at the threshold, analyzing powers will vanish. The structure of Eq. (36) is general—no assumption regarding the number of contributing partial waves was necessary. However, if we want to make statements about the expected angular dependence of observables, the number of partial waves needs to be restricted. For example, if we allow for at most p-waves for the NN system as well as the particle x with respect to the NN system, then the largest value of L˜ and l˜ that can occur is 2, which strongly limits the possible >-dependences that can occur in the angular function B de6ned in Eq. (37).

C. Hanhart / Physics Reports 397 (2004) 155 – 256

187

If the spin of the particles is not detected there is no interference between di@erent spin states in the 6nal state. This leads to a severe selection rule in case of the reaction pp → ppX : since the 6nal state is necessarily in a T = 1 state the Pauli principle demands that di@erent spin be accompanied by di@erent angular momentum. Therefore, for the a reaction with a pp 6nal state the partial waves can be grouped into two sets, namely {Ss; Sd; Ds; Dp} and {Pp; Ps}, where only the members of the individual sets interfere with each other. 4.2.1. Equivalent observables All the angular dependence of the observables is contained in the function B de6ned in Eq. (37). In this subsection we will discuss some properties of B and relate these to properties of particular observables. The functional form of B enables one immediately to read o@ the allowed azimuthal dependences for each observable as well as to identify equivalent observables. To see this we rewrite Eq. (37) as  BLQ˜l;˜ : (q; ˆ p) ˆ = fL˜l;˜ :; Q; n (>p
; >q
) exp{i[(Q − n);p
+ n;q
]} ; (42) n

which directly translates into the following ;-dependences for the spherical tensors (cf. Eq. (36)): ˆ q) ˆ = TD (p;

N 

gD; n (>p
; >q
) exp{i[(Q − n);p
+ n;q
]} :

(43)

n=−N

Note that N is given by the highest partial waves that contribute to the reaction considered: (44) − L˜ max 6 (Q − N ) 6 L˜ max and N 6 l˜max ; where L˜ max (l˜max ) is given by twice the maximum baryon–baryon (meson) angular momentum. These limits are inferred by the Clebsch–Gordan coeOcient appearing in the de6nition of B in Eq. (37). Eq. (43) directly relates the real and the imaginary parts of the spherical tensors: Im(TD (>p
; ;p
+ =(2Q); >q
; ;q
+ =(2Q))) = Re(TD (>p
; ;p
; >q
; ;q
)) :

(45)

Thus, two observables are equivalent if they are given by the real and imaginary parts of the same spherical tensor with Q = 0. In Table 4 the relations of the various observables to the spherical tensors are given. Thus, using Eq. (45) we can identify the following set of pairwise equivalent observables: Ay0 ≡ Ax0 ;

A0y ≡ A0x ;

Axx − Ayy ≡ Axy + Ayx ;

(46)

and analogously for observables for which the 6nal state polarization is measured as well. Notice that there is no connection between Axx + Ayy and Axy − Ayx , for these have Q = 0 and therefore there is no transformation, such as the one given in Eq. (45), that relates real and imaginary parts of the spherical tensors. For identical particles in the initial state, as in pp-induced reactions, all observables should be equivalent under the exchange of beam and target. This further reduces the number of independent observables, for now the beam analyzing powers are equivalent to the target analyzing powers and Axz is equivalent to Azx . In Tables 4 and 5 a possible set of independent observables

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

is marked by a ∗. Those of these that are not independent for identical particles in the initial state are labeled as (∗). From the discussion in the previous section it follows (cf. Eq. (41)), that all those observables with an even (odd) value of k1 + k2 + k3 + k4 lead to a real (imaginary) value for gD; n , de6ned in Eq. (43). As a consequence, for all coeOcients appearing in the expansion of the observables, the ;-dependence is 6xed (cf. Table 4); for example, the terms that contribute to &0 , &0 Azz , and &0 A behave as cos(n(;q
− ;p
)). As was stressed above, the phase space for the production reactions is of high dimension. To allow for a proper presentation of the data as well as of calculations, one either needs high-dimensional plots (see discussion in the previous section) or the dimensionality needs to be reduced to one-dimensional quantities, 16 while, however, still preserving the full complexity of the data. As can be seen in Eq. (43), each polarization observable is described by 2N + 1 functions gD; n (>p
; >q
), where the number of relevant terms is given by the number of partial waves. In order to allow disentanglement of these functions, in Ref. [47] it was proposed to integrate each observable over both azimuthal angles under a particular constraint, / = m;p + n;q = c :

(47)

This integration projects on those terms that depend on / or do not show any azimuthal dependence at all [47]. To further reduce the dimensionality of the data, either the relative proton angle or the meson angle can be integrated to leave one with a large number of observables that depend on one parameter only. Those are then labeled as A/ij (>k ), where k is either p or q. In Figs. 13 and 14 some observables reported in Ref. [47] are shown for the energy with highest statistics, namely = 0:83. In the 6gures the data are compared to the model predictions of Ref. [71]. The solid lines are the results of the full model whereas for the dashed lines the contribution from the Delta isobar was switched o@. In Section 6 this model will be discussed in more detail. In the case of Ref. [47] the complete set of polarization observables for the reaction p ˜p ˜ → pp0 is given. Since the particles in the initial state are identical there are 7 independent observables, all functions of 5 independent parameters (cf. Table 4). In the analysis of the data presented in the same reference it was assumed that only partial waves up to p-waves in both the NN as well as the (NN ) system were relevant. Thus the various integrations described in the previous paragraph lead to 32 independent integrated observables that depend only on a single parameter. On the other hand, if the assumption about the maximum angular momenta holds, only 12 partial waves have to be considered in the partial wave analysis. Since the amplitudes are complex and two phases are not observable (cf. discussion at the end of Section 4.2), a total of 22 degrees of freedom needs to be 6xed from the data. Thus a complete partial wave decomposition for the reaction p ˜p ˜ → pp0 seems feasible.

16

For the experimental side this is the far more demanding procedure, for the angular dependence of eOciency as well as acceptance needs to be known very well over the full angular range for those variables that are integrated in order not to introduce false interferences.

C. Hanhart / Physics Reports 397 (2004) 155 – 256 2

2

ΑφΣq

1.5

2

Αφxzq

1.5 1

1

0.5

0.5

0.5

0

0

0

-0.5

-0.5

-0.5

-1

-1

-1

0.4

-0.5

0

0.5

1

-1

0.4

φp Αxz

-0.5

0

0.5

1

-1

0.2

0

0

0

-0.2

-0.2

-0.2

-0.4 -0.5

0

0.5

1

-0.5

0

0.5

1

2φ -φ Αy0 q p

0.2

0.1

0

0

0

-0.1

-0.1

-0.1

-0.5

0

0.5

1

-0.2

1

-0.5

0

0.5

1

-1

-0.5

2φ -φ Αxz p q

0.2

0.1

-1

0.5

Αφp

-1

0.1

-0.2

0

-0.4 -1

2φ -φ Αy0 p q

0.2

-0.5

y0

0.2

-1

-1

0.4

Αφy0q

0.2

-0.4

Αφzzq

1.5

1

189

0 cos(φq)

0.5

1

-0.2

-1

-0.5

0

0.5

1

Fig. 13. Some polarization observables reported in Ref. [47] for the reaction p ˜p ˜ → pp0 at = 0:83 as a function of the pion angle compared to predictions of the model of Ref. [71]. The solid lines show the results for the full model whereas contributions from the Delta where omitted for the dashed lines.

4.3. General structure of the amplitudes In this section we give the recipe for constructing the most general transition amplitude for reactions of the type NN → B1 B2 x, where we focus on spin 12 baryons in the 6nal state. A generalization to other reactions is straightforward. For further applications we refer to a recent review [72]. For simplicity, let us restrict ourselves to those reactions in which there is only one meson produced. The system is then characterized by three vectors, p ˜ ; ˜q  ; and p ˜ ; denoting the relative momentum of the two nucleons in the initial state, the meson momentum, and the relative momentum of the two nucleons in the 6nal state, respectively—in the over all center-of-mass system. In addition, as long as x denotes a scalar or pseudo-scalar meson, we 6nd 6 axial vectors, namely those that can be constructed from the above: i(˜ p×p ˜  );

i(˜ p × ˜q  );

and

i(˜ p × ˜q  )

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C. Hanhart / Physics Reports 397 (2004) 155 – 256 1.5

1.5

ΑΣφ q

1

1.5

φp Αzz

1

0.5

0.5

0.5

0

0

0

-0.5

-0.5

-0.5

-1

-1 0

0.1

0.2

0.4

0.6

0.8

-1

1

0 0.1

Αφ p y0

0.05

0.05

0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.6

0.8

1

Αφ q

0.6

0.8

1

0.4

0.6

0.8

1

0.4

0.6

0.8

1

-0.1 0.2

0.4

0.6

0.8

1

Αφ p

0.6

0

0.4

0.4

0.4

0.2

0.2

0.2

0

0.2 2φ −φ Αxz p q

0.6

xz

0

-0.2

-0.2

0.6

-0.05

0

xz

0

0.4

0

-0.1 0

0.2 2φq−φp Αy0

0.05

-0.05

-0.1

0 0.1

2φp−φq Αy0

0

-0.05

Α∆φ q

1

-0.2

-0.4 -0.4

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4 0.6 cos(θp)

0.8

1

-0.4

0

0.2

Fig. 14. Some polarization observables reported in Ref. [47] for the reaction p ˜p ˜ → pp0 at = 0:83 as a function of cos(>p ) compared to predictions of the model of Ref. [71]. The solid lines show the results for the full model whereas contributions from the Delta were omitted for the dashed lines.

and those that contain the 6nal or initial spin of the two-nucleon system ˜S; ˜S  ; and i(˜S × ˜S  ) ; where ˜S = H1T &y˜&H2 and ˜S  = H3†˜&&y (H4† )T . Here Hi denotes the Pauli spinors for the incoming (1,2) and outgoing (3,4) nucleons and ˜& denotes the usual Pauli spin matrices. If x is a vector particle, an additional axial vector, namely the polarization vector of the vector meson ˜j∗ occurs. In addition, if instead of a two-nucleon state in the continuum a deuteron occurs in the 6nal state, its polarization direction will be characterized by the same ˜j∗ . Since the energy available for the 6nal state is small (we focus on the close to threshold regime), we restrict ourselves to a non relativistic treatment of the outgoing particles. This largely simpli6es the formalism since a common quantization axis can be used for the whole system. In order to construct the most general transition amplitude that satis6es parity conservation, we have to combine the vectors and axial vectors given above so that the 6nal expression form a scalar

C. Hanhart / Physics Reports 397 (2004) 155 – 256

191

or pseudo-scalar for reactions where the produced meson has positive or negative intrinsic parity, respectively. The most general form of the transition matrix element may be written as ˜ · (˜SI ) + i˜A · (˜S  I) + (Si Sj )Bij ; M = H (II ) + iQ

(48)

where I = (H1T &y H2 ) and I = (H3† &y (H4T )† ). In addition, the amplitudes have to satisfy the Pauli principle as well as invariance under time reversal. This imposes constraints on the terms that are allowed to appear in the various coeOcients. The 9 amplitudes that contribute to B may be further decomposed according to the total spin to which ˜S and ˜S  may be coupled Bij = bs 3ij + bvk jijk + btij ; where the superscripts indicate if the combined spin of the initial and the 6nal state are coupled to 0 (s), 1 (v) or 2 (t), where btij is to be a symmetric, trace free tensor. Once the amplitudes are identi6ed the evaluation of the various observables is straightforward. In this case the polarization comes in through Hi Hi† = 12 (1 + ˜P i · ˜&) ; where ˜P i denotes the polarization direction of particle i. Using the formulas given in Appendix B.2 one easily derives (summation over equal indices is implied): ˜ 2 + |˜A|2 + |Bmn |2 ; 4&0 = |H |2 + |Q|

(49)

∗ 4A0i &0 = +ijijk (Qj∗ Qk + Bjl Bkl ) + 2 Im(Bil∗ Al − Qi∗ H ) ;

(50)

∗ 4D0i &0 = −ijijk (A∗j Ak + Blj Blk ) − 2 Im(Bli∗ Ql − A∗i H ) ;

(51)

4Aij &0 = 3ij (−|H |2 + |Q|2 − |A|2 + |Bmn |2 ) ∗ + 2 Re(jlij (Ql∗ H − A∗m Blm ) − Qi∗ Qj − Bim Bjm ) ;

  ∗ 4Dij &0 = 2 Re Qi∗ Aj + jilm Ql∗ Bmj + jjml Bil A∗m + 12 jilm jjnk Bln Bmk + Bij∗ H ;

(52) (53)

∗ 4Aij; k0 &0 = Im(3ij (2H ∗ Ak − 2Ql∗ Blk − j,Kk (A∗, AK − Bl, BlK )) ∗ + 2jlij (Ql∗ Ak − jnmk A∗n Blm + HBlk ) ∗ ∗ + 2(Qi∗ Bjk + Qj∗ Bik ) − jmnk (Bim Bjn + Bjm Bin )) :

(54)

For illustration we also give here the explicit expressions for A and AC de6ned in the previous section: &0 A = 12 (−|H |2 + |Qz |2 − |˜A|2 + |Bzn |2 ) ; ∗ ∗ Bxm − Bym Bym ) : &0 AC = − 12 (Qx∗ Qx − Qy∗ Qy + Bxm

As was stressed in the previous section, the method of spherical tensors is very well suited to identifying equivalent observables. This is signi6cantly more diOcult in the amplitude approach.

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

Table 6 List of a possible set of the independent amplitude structures that contribute to the reaction pp → pp + (pseudo-scalar) Amplitudes

Structures

H ˜ Q ˜A Bij

(˜ p×p ˜  )˜q  (˜ p·p ˜ ) p ˜ p ˜ jijk pk (˜ p · ˜q  ) 3ij (˜ p×p ˜  )˜q  pi pj (˜ p×p ˜  )˜q 

Lowest pw p·p ˜ ) p ˜  (˜ p ˜ (˜ p·p ˜ ) jijk pk (˜ p · ˜q  ) (˜ p×p ˜  )i qj (˜ p×p ˜  )i pj (˜ p · ˜q  )

˜q  (˜ p · ˜q  )  ˜q (˜ p · ˜q  ) jijk qk (˜ p · p ˜) (˜ p × ˜q  )i pj (˜ p × ˜q  )i pj (˜ p·p ˜ )

Ds Ss, Ds, Sd Ps, Pd Pp

In the last column shows the lowest partial waves for the 6nal state that contribute to the given amplitude (using the notation of Section 1.6). To keep the expressions simple we omitted to give the symmetric, trace free expressions for the terms listed in the last two lines.

However the amplitude method becomes extremely powerful if—due to physical arguments or appropriate kinematical cuts—one of the subsystems can be assumed in an s-wave, for then the number of available vectors is reduced signi6cantly and rather general arguments become possible (cf. Section 4.3.1). Since any amplitude can be made successively more complex by multiplying it by an arbitrary scalar, in most of the cases an ordering scheme is demanded in order to make the approach useful. In the near-threshold regime this is given by the power of 6nal momenta occurring—in analogy with the partial wave expansion. Actually, the amplitude approach presented here and the partial wave expansion presented in the previous subsection are completely equivalent. However, in the near-threshold regime the amplitude method is more transparent. As one goes away from the threshold the number of partial waves contributing as well as the number of the corresponding terms in the amplitude expansion increases rapidly. As a consequence the construction of the most general transition amplitude is rather involved. The partial wave expansion, on the other hand, can be easily extended to an arbitrary number of partial waves. As follows directly from Eq. (48), in the general case the matrix element M is described by 16 complex valued scalar functions. One can show, e.g. by explicit construction, that for general kinematics of the reaction NN → NNx all 16 amplitudes are independent. A possible choice is given explicitly in Table 6 for the reaction pp → pp + (pseudo-scalar). However, for particular reaction channels or an appropriate kinematical situation their number sometimes reduces drastically. For example in collinear kinematics (where p ˜ , ˜q  , and p ˜  are all parallel) the number of amplitudes that fully describes the reactions pp → pp + (pseudo-scalar) is equal to 3 (this special case is discussed in Ref. [76]). This can be directly read from Table 6, for under collinear conditions all cross products vanish and all structures of one group that are given by the di@erent vectors of the ˜ → ,˜ system collapse to one structure (Q p, ˜A → Kp, Bij → jijk pk ). Another interesting example is that of elastic pp scattering. The Pauli principle demands (cf. Section 1.6) that odd (even) parity states are in a spin triplet (singlet). Time reversal invariance requires the amplitude to be invariant under the interchange of the 6nal and the initial state. When parity conservation is considered in addition, one also 6nds that the total spin is conserved. Thus, only H and Bij will contribute to elastic pp scattering. In addition, from the two vectors available in the system (˜ p and p ˜  ) one can construct only 4 structures that contribute to the latter, namely

C. Hanhart / Physics Reports 397 (2004) 155 – 256

193

3ij (˜ p ·˜ p )b1 , jijk (˜ p ט p )k b2 , (pi pj +pj pi −( 23 )3ij )(˜ p ·˜ p )b3 , and (pi pj +pj pi −( 23 )˜ p2 3ij )(˜ p ·˜ p )b4 . 17 Thus, pp scattering is completely characterized by 5 scalar functions. As a further example and to illustrate how the formalism simpli6es in the vicinity of the production threshold, we will now discuss in detail the production of pions in NN collisions. Throughout this report, however, the formalism will be applied to various reactions. In our example there are three reaction channels experimentally accessible, namely pp → pp0 , pp → pn+ , and pn → pp− , that can be expressed in terms of the three independent transition amplitudes ATi Tf (as long as we assume isospin to be conserved), where Ti (Tf ) denote the total isospin of the initial (6nal) NN system [27] (cf. Section 1.5). As in Section 1.6, we will restrict ourselves to those 6nal states that contain at most one p-wave. We may then write for the amplitudes of A11 , H 11 = 0 ; ˜ 11 = a1 pˆ ; Q

  ˜A11 = a2 p ˜  )pˆ − 13 p ˜  + a3 (pˆ · p ˜ ; Bij11 = 0 ;

(55)

for the amplitudes of A10 , H 10 = 0 ; ˜ 10 = 0 ; Q

  ˜A10 = b2˜q  + b3 (˜q  · p) ˆ pˆ − 13 ˜q  ; ˜B10 ˆk ; ij = jijk b1 p

(56)

for the amplitudes of A01 , H 01 = 0 ;

  ˜ 01 = c1˜q  + c2 p( ˆ pˆ · ˜q  ) − 13 ˜q  ; Q ˜A01 = 0 ;      Bij01 = jnmk p ; ˜ k c3 3in 3jm + c4 3jm pˆ i pˆ n − 13 3in + c5 3in pˆ j pˆ m − 13 3jm

(57)

where pˆ denotes the initial NN momentum, normalized to 1, and p ˜  and ˜q  denote the 6nal nucleon and pion relative momentum, respectively. The coeOcients given are directly proportional to the corresponding partial wave amplitudes as listed in Table 1; e.g., a3 is proportional to the transition amplitude 1 D2 → 3 P2 s. When constructing amplitude structures for higher partial waves care has to be taken not to list dependent structures. In order to remove dependent structures the reduction formula, Eq. (B.7), proved useful. In addition, one should take care that the number of coeOcients appearing exactly matches the number of partial waves allowed. For example, the partial waves that contribute to Bij01 are 3 S1 → 3 P1 s, 3 D1 → 3 P1 s and 3 D2 → 3 P2 s. 17

Note, this choice of structures in not unique; we could also have used (˜ p×p ˜  )i (˜ p×p ˜  )j as a replacement of any t other structure in bij .

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

d2σ /dΩπ dMpp2 [pb/srMeV2]

1.0

Ay

0.5 0.0 -0.5 -1.0

60

40

20

0 0

45

90 135 θtot (deg)

180

0

45

90 135 θtot (deg)

180

Fig. 15. Sensitivity of the analyzing power as well as the di@erential cross section for the reaction pn → pp− to the sign of the 3 P0 → 1 S0 s amplitude a1 . The lines corresponds to the model of Ref. [73]: the solid line is the model prediction whereas for the dashed line the sign of a1 was reversed. The experimental data are from Refs. [74,75] at TLab = 353 MeV ( = 0:65).

The large number of zeros appearing in the above list of amplitude structures reNects the strong selection rules discussed in Section 1.6. As an example we will calculate the beam analyzing power and the di@erential cross section for the reaction p ˜ n → pp− . These observables were measured at TRIUMF [74,75] and later at PSI [77,78]—here, however, with a polarized neutron beam (see Fig. 15). In accordance with the TRIUMF experiment, where the relative NN energy in the 6nal state was restricted to at most 7 MeV, we assume that the outgoing NN system is in a relative S-wave. This largely simpli6es the expressions. We then 6nd    &0 = 14 |a1 |2 + 12 q Re a∗1 c1 + 23 c2 cos(>) ;      2 &0 Ay = 14 q Im(c1∗ c2 ) sin(2>) − 12 q Im a∗1 c1 − 13 c2 sin(>) cos(;) ; (58) ˆ y = −q sin(>) cos(;) and (˜q  · p) ˆ = q cos(>) where we used the de6nitions (˜q  × p) (cf. Appendix B.1). Thus, the forward–backward asymmetry in &0 as well as the shift of the zero in Ay directly measure the relative phase of the 3 P0 → 1 S0 s transition in A11 (a1 ) in the pion p-wave transitions of A01 (c1 and c2 ), as was 6rst pointed out in Ref. [79]. This issue will be discussed below (cf. Section 6). Note: As before we neglected here pion d-waves, since they are kinematically suppressed close to the threshold (cf. Eq. (9)). 4.3.1. Example I: polarization observables for a baryon pair in the 1 S0 9nal state As an example of the eOciency of the amplitude method, in this subsection we will present an analysis of the angular pattern of some polarization observables for the reaction pp → B1 B2 x under the constraint that the outgoing two baryon state (B1 B2 ) is in the 1 S0 partial wave and x is a pseudo-scalar, as is relevant for the reaction pp → pK . The dependence of the observables on the meson emission angle is largely constrained under these circumstances, as was stressed in Ref. [80]. In contrast to the discussion in Section 1.6, here we will not assume B1 and B2 to be

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identical particles. The results of this subsection will show how to disentangle in a model independent way the two spin components of the hyperon–nucleon interaction [32]. The analysis starts with identifying the tensors that are to be considered in the matrix element of Eq. (48). 18 For this we go through a chain of arguments similar to those in Section 1.6. Given ˜ can be non-zero. In addition, the that we restrict ourselves to a spin-zero 6nal state, only H and Q quantum numbers of the 6nal state are 6xed by lx , the angular momentum of the pseudo-scalar with respect to the two baryon system, since J = lx

and

tot = (−)(lx +1)

(59)

for the total angular momentum and the parity, respectively. Conservation of parity and angular momentum therefore gives (−)L = (−)lx +1 = (−)J +1 → S = 1 ;

(60)

and consequently we get H = 0. In addition, for odd values of lx we see from the former equation that L must be even. In pp systems, however, even values of L correspond to S = 0 states, not allowed in our case. Therefore lx must be even. We may thus make the following ansatz: ˜ = ,˜ Q p + K˜q  (˜q  · p ˜) ;

(61)

p·p ˜  ). All other coeOcients appearing in Eq. (48) where , and K are even functions of p, p and (˜ vanish. This has serious consequences for the angular dependences of the various observables. For example, the expression for the analyzing power collapses to i 1 A0i &0 = jijk (Qj∗ Qk ) = Im(K∗ ,)(˜q  · p ˜ )(˜q  × p ˜ )i : (62) 4 2 Therefore, independently of the partial wave of the pseudo-scalar emitted, for a two-baryon pair in the 1 S0 state the analyzing power A0y vanishes if the pseudo-scalar is emitted either in the xy-plane or in the zx-plane. In Ref. [32] this observation was used to disentangle the di@erent spin states of the N interaction. 4.3.2. Example II: amplitude analysis for pp → dKG 0 K + close-to-threshold In Section 4.1.1 we discussed in some detail the data of Ref. [66] for the reaction pp → dKG 0 K + based on rather general arguments on the cross section level. In this subsection we will present the corresponding production amplitude based on the amplitude method presented above. This study will allow us at the same time to extract information on the relative importance of the a+ 0 and the

(1405) in the reaction dynamics. G system or the deuteron with respect to the KK G As in Section 4.1.1, we assume that either the KK system is in a p-wave, whereas the other system is in an s-wave, calling for an amplitude linear in p or q , respectively. We use the same notation as in Section 4.1.1. Therefore the 6nal state has odd parity and thus also the amplitude needs to be odd in the initial momentum p ˜ . An odd parity ˜ isovector NN state has to be S = 1 and thus has to be linear in S, de6ned in Section 4.3. In addition, the deuteron in the 6nal state demands that each term is linear in the deuteron polarization vector j. 18

Note, here we could as well refer to Table 6 to come to the same conclusion as in this section, for the 1 S0 state is allowed for the pp system. However, the argument given is quite general and instructive.

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We therefore get for the full transition amplitude, in slight variation to Eq. (48) due to the presence of the deuteron in the 6nal state, M = Bij˜S i˜j∗j ;

(63)

where ˜j appears as complex conjugate, since the deuteron is in the 6nal state, and ˆ Bij = aSp pˆ i˜qj + bSp (pˆ · ˜q  )3ij + cSp˜qi pˆ j + dSp pˆ i pˆ j (˜q  · p) + aPs pˆ i p ˜ j + bPs (pˆ · p ˜  )3ij + cPs p ˜ i pˆ j + dPs pˆ i pˆ j (˜q  · p) ˆ ;

(64)

G system and small where capital letters in the amplitude label indicate the partial wave of the KK G system. letters that of the deuteron with respect to the KK Once the individual terms in the amplitude are identi6ed, it is straightforward to express the Ci de6ned in Section 4.1.1 in terms of them. We 6nd, for example,


C0q = 12 (|aSp |2 + |cSp |2 ) ;


C1q = |bSp |2 + 12 |bSp + dSp |2 + Re[a∗Sp cSp + (aSp + cSp )∗ (bSp + dSp )] ; ∗ C2 = a∗Sp aPs + cSp cPs :


A 6t to the experimental data revealed that, within the experimental uncertainty, C0p is compatible with zero. Thus, given the previous formulas, both aPs and cPs have to vanish individually. G s-waves in Ref. [66] it was argued that the reaction pp → Based on the strongly populated KK dKG 0 K + is governed by the production of the a+ 0 . In Ref. [81], however, it was argued that the strong G Kd interaction caused by the proximity of the (1405) resonance should play an important role as G s-wave. We now want to calculate the contribution of this well. This FSI should enhance the Kd G p-wave based on the amplitudes given in Table 3. This will illustrate partial wave relative to the Kd a further strength of the amplitude method, for within this scheme changing the coordinate system G system and in the Kd G system is trivial. The coordinate system suited to study resonances in the KK are illustrated in the left and right panels of Fig. 16. All we need to do now is express the vectors ˜  and ˜P  . We 6nd that appear in Eq. (30) in terms of Q ˜ ˜q  = ˜P  − ,Q

and

˜  + ˜P  ) ; p ˜  = 12 ((2 − ,)Q

where , = md =(md + mKG ). Obviously, the squared amplitude expressed in the new coordinates reveals the same structure as Eq. (30):
G = BQ
Q 2 + BP
P  2 + BQ
(Q ˜  · p) |M| ˆ 2 + B1p (P˜ · p) ˆ 2 0 0 1

˜  ) + B3 (P˜ · p)( ˜  · p) ˆ Q ˆ ; + B2 (P˜ · Q

(65)

where the coeOcients appearing can be expressed in terms of the C coeOcients of Eq. (30) so that, for example,


B0Q =


B1Q =

(2 − ,)2 q
,(2 − ,) 1
C0 + ,2 C0k − C2 ; 4 2 2 (2 − ,)2 q
,(2 − ,) 1
C1 + ,2 C1k − C3 : 4 2 2

(66)

C. Hanhart / Physics Reports 397 (2004) 155 – 256 Ko

K+

d

Ko

K+

d P'

p'

Q'

q'

p p

197

p p

(a)

p

p

(b)

Fig. 16. (a)–(b) Illustration of the coordinate system used in the analysis for the reaction pp → dKG 0 K + .

With these expressions at hand it is easy to verify, that K KG s-waves contribute to 83% to the total G s-waves contribute to 54% only [82]. Here we used the total s-wave cross section, whereas Kd



q G G (B0Q + ( 1 )B1Q ), as a measure strength for KK, (C0 + ( 13 )C1q ), and the total s-wave strength for Kd, 3 of the strength of the partial waves. G s-wave, it appears that the (1405) does As we do not see a signi6cant population of the Kd not play an essential role in the reaction dynamics of pp → dKG 0 K + close-to-threshold in contrast to the a+ 0. 4.4. Spin cross sections As early as 1963, Bilenky and Ryndin showed [83], that from the spin correlation coeOcients that can be extracted from measurements with polarized beam and target, the cross section can be separated into pieces that stem from di@erent initial spin states. Their results were recently re-derived [84] and the formalism was generalized to the di@erential level in Ref. [85]. With these so-called spin cross sections it can easily be demonstrated how the use of spin observables enable one to 6lter out particular aspects of a reaction. We begin this subsection with a derivation of the spin cross sections using the amplitude method of the previous subsection and then use the reaction p ˜p ˜ → pn+ as an illustrative example. Since we have the amplitude decomposition of the individual observables given in Eqs. (49)–(54), one easily 6nds &0 (1 − Axx − Ayy − Azz ) = |H |2 + |A|2 = : 1 &0 ;

(67)

&0 (1 + Axx + Ayy − Azz ) = |Qz |2 + |Bzn |2 = : 3 &0 ;

(68)

&0 (1 + Azz ) = 12 (|Qx |2 + |Qy |2 + |Bxn |2 + |Byn |2 )= : 3 &1 ;

(69)

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Table 7 List of the lowest partial waves in the 6nal state that contribute to the individual spin cross sections 2S+1

&m

Possible 6nal states for pp → bb x Slx

Plx

1

3

3

3

1

3

&0 &0 3 &1

S1 p S0 s, 1 S0 d, 3 S1 d 3 S1 s, 1 S0 d, 3 S1 d

Pj s, 1 P1 s Pj p, 1 P1 p 3 Pj p, 1 P1 p

Capital letters denote the baryon–baryon partial waves whereas small letters that of the meson with respect to the baryon–baryon system.

where the assignment of the various spin cross sections (2S+1) &MS , with S (MS ) the total spin (projection of the total spin on the beam axis) of the initial state can be easily con6rmed from the de6nition of the amplitudes in Eq. (48). As was shown in Section 1.6, in case of two-nucleon initial or 6nal states restrictive selection rules apply. For example, for pp 6nal states the isospin of the 6nal NN system is 1 and therefore states with even (odd) angular momentum have total spin 0 (1). From Table 7 it thus follows, that the pp S-wave in connection with a meson s-wave contributes only to 3 &0 . This example of how spin observables can be used to 6lter out particular 6nal states was used previously in Section 2 (cf. discussion to Fig. 6). For illustrative purposes we show in Fig. 17 the spectra for the various cross sections for the reaction p ˜p ˜ → pn+ , as a function of the relative energy of the nucleon pair in the 6nal state. The curves correspond to the model of Ref. [71] that very well describes the available data in the + production channel. The model is described in detail in Section 6. In the upper left panel the unpolarized cross section is shown. It is dominated by the Sp 6nal state (the dominant transition is 1 D2 → 3 S1 p), and from this spectrum alone it would be a hard task to extract information on 6nal states other than the 3 S1 NN state. This can be clearly seen by the similarity of the shapes of the dot–dashed line and the solid line. The spin cross sections, however, allow separation of the spin singlet from the spin triplet initial states. Naturally 1 &0 (lower left panel of Fig. 17) is now saturated by the Sp 6nal state, but in 3 &0 and 3 &1 other structures appear: the former is now dominated by the transition 3 P0 → 1 S0 s and the latter by 3 P1 → 3 S1 s. 4.5. Status of experiment In this presentation we will be rather brief on details about current as well as planned experiments, as this subject was already covered in recent reviews [1,2]. Here we only wish to give a brief list of observables and reactions that are measured already or are planned to be measured in nucleon–nucleon and nucleon–nucleus-induced reactions. In the case of pion production, measurements with vector- and tensor-polarized deuteron and vector-polarized proton targets and polarized proton beams have been carried out at IUCF [134]. Because of good 4 detection of photons and charged particles, CELSIUS, at least in the near future, is well equipped for studies involving mesons. For the production of heavier mesons,

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199

dσ/dε [µb/MeV]

8 σtot

6

3

0.6

σ1

0.5 0.4

4

0.3 0.2

2

0.1 0

0 0

5

10 15 ε [MeV]

20

25

0

5

10

15

20

25

ε [MeV]

7 3

1σ 0

6

0.5

5

σ0

0.4

4

0.3

3 0.2

2

0.1

1 0

0 0

5

10 15 ε [MeV]

20

25

0

5

10 15 ε [MeV]

20

25

Fig. 17. Demonstration of the selectivity of the spin cross sections. Shown are the spectra of the reaction p ˜p ˜ → pn+ as a function of j for an excess energy of 25 MeV. In each panel the solid line shows the full result for the corresponding cross section, the dot–dashed, long-dashed, dashed, and dotted lines show the Sp, Ss, Sd, and Pp contribution, respectively. The curves are from the model of Ref. [71].

ppπo

pnπ+

dπ+ ppη'

pKΛ ΝΝπ

Ν∆

ppω

ppη

ppφ ppao

Ecms

IUCF

pKΣ

ppfo

CELSIUS COSY

Fig. 18. The lowest meson production thresholds for single meson production in proton–proton collisions together with the corresponding energy ranges of the modern cooler synchrotrons.

COSY, due to its higher beam energies and intensities, but due most importantly to the possibility to use polarization, is in a position to dominate the 6eld during the years to come. The energy range of the di@erent cooler synchrotrons is illustrated in Fig. 18.

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Table 8 List of observables measured for various NN → NNx channels for excess energies up to Q = 40 MeV Channel

&tot

d&=d.

d&=dm

Aoi

Aij

pp → pp0 pp → pn+ pn → pp− pp → d+

[43,44,86–89] [91–93] [77] [95–104]

[90,88,87] [92] [77,74] [95,103,104]

[87–89] [77] —

[89,47] [92] [78,75] [105–107]

[47] [94]

pp → pp pn → pn pn → d

[109–117] [33] [120,114]

[113,112,118]

[113,112]

[119]

pp → pp 

[117,29,121]

pp → pK + pp → pK + 0

[122,123,46] [124]

pp → pp! pn → d!

[125] [126]

pp → pp;

[127]

pp → ppf0 =a0

[128]

pp → pp+ − pp → pn+ 0 pp → pp0 0

[129–131] [131] [131]

[129–131] [131] [131]

[129–131] [131] [131]

pp → ppK + K − pp → dK + KG 0

[132] [133]

[133]

[133]

[108]

—

[46]

— [127]

In Table 8 a list is given for the various NN -induced production reactions measured in recent years at SACLAY, TRIUMF, PSI, COSY, IUCF, and CELSIUS in the near-threshold regime (Q ¡ 40 MeV). The corresponding references are listed as well. In the subsections to come we will discuss some examples of the kind of physics that can be studied with the various observables in the many reaction channels. 5. Symmetries and their violation As was already mentioned in several places in this article, symmetries strongly restrict the allowed pattern for various observables. This leads to observable consequences. Naturally, it is therefore also straightforward to investigate the breaking of these symmetries by looking at a violation of these symmetry predictions. Probably, the most prominent example of an experiment for a storage ring is that proposed by the TRI collaboration to be performed at the COSY accelerator. The goal is to do a null experiment

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in order to put an upper limit on the strength of T -odd P-even interactions via measuring Ay; xz in polarized proton–deuteron scattering, which should vanish if time reversal invariance holds [135]. For details we refer to Ref. [136]. 5.1. Investigation of charge symmetry breaking (CSB) If the masses of the up and down quark were equal, the QCD Lagrangian were invariant under the exchange of the two quark Navors. In reality, these masses are not equal and also the presence of electro magnetic e@ects leads to small but measurable charge symmetry breaking e@ects. Note, the mass di@erence of a few MeV [137] is small compared to the typical hadronic scale of 1 GeV. Quantifying CSB e@ects therefore allows to extract information on the light quark mass di@erences from hadronic observables. As should be clear from the previous paragraph, CSB is closely linked to the isospin symmetry. However, does isospin symmetry demand an independence of the interaction under an arbitrary rotation in isospin space, a system is charge symmetric, if the interaction does not change under a 180◦ rotation in isospin space. Therefore isospin symmetry or charge independence is the stronger symmetry than charge symmetry. For an introduction into the subject we refer to Ref. [5]. The advantage of reactions with only nucleons or nuclei in the initial state is, that one can prepare initial states with well-de6ned isospin. This is in contrast to photon-induced reactions, since a photon has both isoscalar and isovector components. Therefore, in the case of meson photo- or electro-production, CSB signals can only be observed as deviations from some expected signal. (As an example of this reasoning see Ref. [138].) In the case of hadron-induced reactions on the other hand, experiments can be prepared that give a non-vanishing result only in the presence of CSB. This makes the unambiguous identi6cation of the e@ect signi6cantly easier. One complication that occurs if a CSB e@ect is to be extracted from the comparison of two cross sections with di@erent charges in the 6nal state is that of the proper choice of energy variable. Obviously, the quantity that changes most quickly close to the threshold is the phase space, so that it appears natural to compare two reaction channels that have the same phase-space volume. However, due to the di@erences in the particle masses, this calls for di@erent initial energies. In the case of a resonant production mechanism this might lead to e@ects of the same order as the e@ect of interest. In Ref. [139] this is discussed in detail for the reactions pp → d+ and pn → d0 . In this report we will concentrate on the implications of CSB on observables in NN and dd collisions. For details of the mechanisms of CSB we refer to Ref. [5]. In the corresponding class of experiments the deuteron as well as the alpha particle play an exceptional role since as isoscalars they can act as isospin 6lters. The most transparent example of a CSB reaction is dd → ,x ; where x is some arbitrary isovector. The reaction with x = o was recently measured at IUCF for the 6rst time close to the threshold [140]. The initial state as well as the , are pure isoscalars. Thus the 6nal state as an isovector cannot be reached as long as isospin is conserved. In addition pn reactions can be used for producing clean signals of CSB. More generally, whenever a pn pair has a well-de6ned isospin they behave as identical particles. Speci6cally, the di@erential cross section needs to be forward–backward symmetric because nothing should change if beam and

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x

a0 a0 π

(a)

x

f0 η

π

(b)

Fig. 19. Illustration of di@erent sources of charge symmetry breaking: diagram (a) shows CSB in the production operator through  − mixing and diagram (b) shows CSB in the propagation of the scalars. Thin solid lines denote nucleons, thick solid lines scalar mesons and dashed ones pseudo-scalar mesons. The X indicates the occurrence of a CSB matrix element.

target are interchanged. Any deviation from this symmetry is an unambiguous signal of CSB [5]. An experiment performed at TRIUMF recently claimed for the 6rst time a non-vanishing forward– backward asymmetry in pn → d0 [6]. Note that not every forward–backward asymmetry in pn reactions stems from isospin violation. A counterexample was given at the end of Section 4.3, where the di@erential cross section for pn → pp− is discussed in detail. There, in contrast to the previous example, T = 0 and 1 initial states interfere. In case of pion production in nucleon–nucleon collisions it is possible to de6ne a convergent e@ective 6eld theory (see Section 6). Within this theory it is possible to relate e@ects of CSB in these reactions directly to the up–down quark mass di@erence [5,141]. Preliminary studies show, that the relative importance of di@erent CSB mechanisms in the reaction pn → d0 and dd → ,0 are very di@erent [142], and thus it should be possible to extract valuable information on the leading CSB operators from a combined analysis of the two reactions. In the arguments given all that was used was the isovector character of the meson produced. Thus, the same experimental signals will be seen also in pn → d(0 ) [143] and dd → ,(0 ) [144]. In Ref. [67] also the analyzing power was identi6ed as a useful quantity for the extraction of the a0 −f0 mixing matrix element (see also discussion in Ref. [145]). The (0 )s-wave is interesting especially G threshold, since it should give insight into the nature of the light scalar mesons close to the KK a0 (980) and f0 (980) (cf. Section 7.5). Note that the  channel is the dominant decay channel of the a0 , which is an isovector–scalar particle. It should be stressed that the charge symmetry breaking signal in case of the scalar mesons is signi6cantly easier to interpret in comparison to the case of pion production. The reason is that the two scalar resonances of interest overlap and therefore the e@ect of CSB as it occurs in the propagation of the scalar mesons is enhanced compared to mixing in the production operator [82]. To make this statement more quantitative we compare the impact of f0 − a0 mixing in the propagation of the scalar mesons (Fig. 19b) to that of  − mixing in the production operator (Fig. 19a). We regard the latter as a typical CSB e@ect and thus as a reasonable order-of-magnitude estimate for CSB in the production operator. Observe that the relevant dimensionless quantity for this comparison is the mixing matrix element times a propagator

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(cf. Fig. 19). In the production operator the momentum transfer—at least close to the production threshold—is given by t = −MN mR , where mR denotes the invariant mass of the meson system produced (or equivalently the mass of the resonance) and MN denotes the nucleon mass. Thus, the appearance of the propagator introduces a factor of about 1=t into the amplitude, since tm2 . On the other hand, the resonance propagator is given by 1=(mR MR ), as long as we concentrate on invariant masses of the outgoing meson system close to the resonance position. Here MR denotes the width of the scalar resonance. Thus we 6nd using MR =50 MeV [146] that the CSB in the production operator is kinematically suppressed by a factor of more than MR =MN ∼ 1=20 as compared to CSB in the propagation of the scalars. In addition the mixing matrix element is enhanced in the case of f0 − a0 mixing (cf. Section 7.5) and therefore it should be possible to extract the f0 − a0 mixing matrix element from NN - and dd-induced reactions.

6. The reaction NN → NN The production of pions in nucleon–nucleon collisions has a rather special role. First of all, it is the lowest hadronic inelasticity for the nucleon–nucleon interaction and thus an important test of our understanding of the phenomenology of the NN interaction. Secondly, since pions are the Goldstone bosons of chiral symmetry, it is possible to study this reaction using chiral perturbation theory. This provides the opportunity to improve the phenomenological approaches via matching to the chiral expansion as well as to constrain the chiral contact terms via resonance saturation. Last but not least, a large number of (un)polarized data is available (cf. Table 8). After a brief history, we continue this chapter with a discussion of a particular phenomenological model for pion production near the threshold, followed by a presentation of recent results from chiral perturbation theory. 6.1. Some history In Section 1.2 it was argued, that for the near-threshold regime the distorted wave born approximation is appropriate, and here we will concentrate on those models that work within this scheme. 19 Pioneering work on pion production was done by Woodruf [147] as well as by Koltun and Reitan [31] in the 1960s. The diagrams included are shown in Figs. 2a and b where, in these early approaches, the N → N transition amplitudes (denoted by T in diagram 2b) were parameterized by the scattering lengths. When the 6rst data on the reaction pp → pp0 close to the threshold were published [43,44], it came as a big surprise that the model of Koltun and Reitan [31] underestimated the data by a factor of 5–10. This is in vast contrast to the reaction pp → pn+ reported in Ref. [91], where the discrepancy was less than a factor of 2. On the other hand, it was shown that the energy dependence of the total cross section can be understood from that of the NN FSI once the Coulomb interaction is properly included [45] (cf. Section 2). 19

In the next section a further argument in favor of the distorted wave born approximation will be given.

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Niskanen investigated whether the inclusion of the Delta isobar, as well as keeping the rather strong on-shell energy dependence of the N interaction, could help to improve the theoretical results for neutral pion production [148]. Although these improvements lead to some enhancement, the cross section was still missed by more than a factor of 3. The 6rst publication that reported a quantitative understanding of the pp → pp0 data was that by Lee and Riska [149] and later con6rmed by Horowitz et al. [150], where it was demonstrated that short range mechanisms as depicted in Fig. 2c, can give a sizable contribution. However, shortly after this discovery Hern\andez and Oset demonstrated, using various parameterizations for the N → N transition amplitude and qualitatively reproducing earlier work by Hachenberg and Pirner [151] that the strong o@-shell dependence of that amplitude can also be suOcient to remove the discrepancy between the Koltun and Reitan model and the data. Gedalin et al. [152] came to the same conclusion within a relativistic one boson exchange model. In Ref. [153] the N amplitude needed as input for the evaluation of diagram 2b was extracted from a microscopic model. Also there a signi6cant although smaller contribution from the pion rescattering was found. Thus, in this model still some additional short range mechanism is needed. In the succeeding years many theoretical works presented calculations for the pp → pp0 cross section. In Refs. [154,155] covariant one boson exchange models were used in combination with an approximate treatment of the nucleon–nucleon interaction. Both models turned out to be dominated by heavy meson exchanges and thus give further support to the picture proposed in Ref. [149]. However, in Ref. [156] the way that the anti nucleons were treated in Refs. [149,150,154] was heavily criticized: the authors argued that the anti nucleon contributions get signi6cantly suppressed once they are included non-perturbatively. It is interesting to note, that also in Bremsstrahlung the contribution from anti-nucleons in a non-perturbative treatment is signi6cantly reduced compared to a perturbative inclusion [157,158]. Additional short range contributions were also suggested, namely the D−! meson exchange current [159], resonance contributions [160] and loops that contain resonances [161], all those, however, turned out to be smaller compared to the heavy meson exchanges and the o@-shell pion rescattering, respectively. At that time the hope was that chiral perturbation theory might resolve the true ratio of rescattering and short range contributions. It came as a big surprise, however, that the 6rst results for the reaction pp → pp0 [162,163] found a rescattering contribution that interfered destructively with the direct contribution (diagram a in Fig. 2), making the discrepancy with the data even more severe. In addition, the same isoscalar rescattering amplitude also worsened the discrepancy in the + channel [164,165]. Some authors interpreted this 6nding as a proof for the failure of chiral perturbation theory in these large momentum transfer reactions [155,166]. Only recently was it demonstrated, that it is possible to appropriately modify the chiral expansion in order to make it capable of analysing meson production in nucleon–nucleon collisions. We will report on those studies in Section 6.3 that in the future will certainly prove useful for improving the phenomenological approaches (cf. Section 6.5). Before we close this section a few remarks on the o@-shell N amplitude are necessary. For this purpose we write the relevant piece of the N interaction T -matrix in the following form TN = −t (+)

1 † 1 † N  · N + t (−) N % · ( × )N ˙ ; 2 2m

(70)

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205

where t (+) (t (−) ) denote the isoscalar (isovector) component. Note that it is only the former that can contribute to the reaction pp → pp0 , 20 for the isospin structure of the latter changes the total isospin of the two-nucleon system. As long as we neglect the distortions due to the 6nal and initial state interactions, what is relevant for the discussion in this paragraph is the half o@-shell N amplitude. We thus may write t =t(s; k 2 ), where s denotes the invariant energy of the N system and k 2 is the square of the four momentum of the incoming pion. At the threshold for elastic N scattering (s = (m + MN )2 , k 2 = m2 ) we may write   m 4 (+) 2 1 1+ t (s0 ; m ) := (a1 + 2a3 ) = (−0:05 ± 0:01)m−  ; 3 MN   m 4 (− ) 2 1 1+ t (s0 ; m ) := (a1 − a3 ) = (1:32 ± 0:02)m− (71)  ; 3 MN where the a2I denote the scattering lengths in the corresponding isospin channels I . The corresponding values were extracted from data on − d atoms in Ref. [167]. Note that the dominance of the isovector interaction is a consequence of the chiral symmetry: the leading isoscalar rescattering is suppressed by a factor m =MN compared to the leading isovector contribution—the so-called Weinberg–Tomozawa term [168,169]. However, it is still remarkable, that also the higher order chiral corrections are small leaving a value consistent with zero for the isoscalar scattering length. A detailed study showed, that this smallness is a consequence of a very eOcient cancellation of several individually large terms that are accompanied with di@erent kinematical factors [170]. To be concrete: to order O(p2 ) one 6nds     2 gA2 (+) 2   t = 2 −2m c1 + q0 k0 c2 − + (q k)c3 ; (72) f 8MN where k and q denote the four momentum of the initial and 6nal pion, respectively. The values for the various ci are given in Table 9. For on-shell scattering at the threshold (q = k = (m ; ˜0)) 1 one gets t (+) (s0 ; m2 ) = −0:24m− using the values of Ref. [171]. 21 Please note, that the linear  combination of the ci appearing above turns out to be an order of magnitude smaller than the individual values. As a consequence, the on-shell isoscalar amplitude shows a rather strong energy dependence above threshold. It should not then come as a surprise, that the transition amplitude corresponding to Eq. (70), when evaluated in the kinematics relevant for pion production in NN collisions, within chiral perturbation theory turns out to be rather large numerically [162,163]. For the non-covariant expression given in Eq. (72) this translates into q = (m ; ˜0) and k = (m =2; ˜k), leading 1 to t (+) ((m + MN )2 ; −MN m ) = 0:5m−  . This is why the 6rst calculations using chiral perturbation theory found a big e@ect from pion rescattering—unfortunately increasing the discrepancy with the data. In Ref. [173] the tree level chiral perturbation theory calculations were repeated using a di@erent prescription for the energy of the exchange pion. 22 The authors found agreement with the data, but with a sign of the full amplitude opposite to the one of the direct term. 20

This is only true if we do not include the C isobar explicitly, as will be discussed in the next section. Note, this value is inconsistent with the empirical value given in Eq. (71). To come to a consistent value one has to go to one loop order as discussed in Ref. [172]. 22 This prescription was later criticized in Ref. [174]. 21

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One year earlier it was shown, that within phenomenological approaches the isoscalar transition amplitude evaluated in o@-shell kinematics is also signi6cantly di@erent from its on-shell value. For example, in the J^ulich meson exchange model it is the contribution from the iterated D t-channel exchange and the & exchange that are individually large but basically cancel in threshold kinematics in the isoscalar channel [153]. This cancellation gets weaker away from the threshold point. This rescattering contribution, however, turned out to interfere constructively with the direct term. Since chiral perturbation theory as the e@ective 6eld theory for low energy strong interactions is believed to be the appropriate tool to study pion reactions close-to-threshold, it seemed at this stage as if there were a severe problem with the phenomenology. However, as was shown in Section 4.3, there are observables that are sensitive to the sign of the s-wave pp → pp0 amplitude—relative to a p-wave amplitude that is believed to be under control—and the experimental results [74,75,77,78] agree with the sign as given by the phenomenological model (cf. Fig. 15) . Does this mean that chiral perturbation theory is wrong or not applicable? No. As we will discuss in the subsequent sections, it was demonstrated recently that the chiral counting scheme needs to be modi6ed in the case of large momentum transfer reactions. No complete calculation has been carried out up to now, but intermediate results look promising for a consistent picture to emerge in the years to come. The insights gained so far from the e@ective 6eld theory studies call also for a modi6cation of the phenomenological treatment. This will be discussed in detail in Section 6.5. 6.2. Phenomenological approaches As stressed in the previous section, the number of phenomenological models for pion production is large. For de6niteness in this section we will focus on one particular model, namely that presented in Refs. [71,73,153], mainly because it incorporates most of the mechanisms proposed in the literature for pion production in nucleon–nucleon collisions, its ingredients are consistent with the data on N scattering, and it is the only model so far whose results have been compared to the polarization data recently measured at IUCF [47,94,108]. Results of the model for the various total cross sections are shown in Fig. 20. The model is the 6rst attempt to treat consistently the NN as well as the N interaction for meson production reactions close to the threshold: both were taken from microscopic models (described in Refs. [175,176] for the NN and the N interaction, respectively). These were constructed from the same e@ective Lagrangians consistent with the symmetries of the strong interaction and are solutions of a Lippman–Schwinger equation based on time-ordered perturbation theory. Although not all parameters and approximations used in the two systems are the same, this model should still be viewed as a benchmark calculation for pion production in NN collisions within the distorted wave Born approximation. We will start this section with a description of the various ingredients of the model and then present some results. 6.2.1. The NN interaction A typical example of a so-called realistic model for NN scattering is the Bonn potential [26]. The model used for the NN distortions in the 6nal and initial states is based on this model, where a pseudo-potential is constructed based on the t-channel exchanges of all established mesons below one GeV in mass between both nucleons and Delta isobars.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

207

101 pp→pnπ+

σtot [mb]

100

pn→ppπ−

10-1 10-2 10-3 10-4 pp→dπ+

pp→ppπ0

10-1 10-2 10-3 10-4 0.1

1

0.1

η = qπ(max)/mπ

1

Fig. 20. Comparison of the model predictions of Ref. [73] to the data. The references for the experimental data can be found in Table 8. The solid lines show the results of the full model; the dashed line shows the results without the C contributions.

The interaction amongst the various dynamical 6elds in the model derived from the following Lagrange densities: fNN G (73) 5 4  · 94  ; LNN = m fNND G LNND = gNND G 4  · 4 + (74) &4O  · (94 O − 9O 4 ) ; 4MN LNN! = gNN! G 4 !4 LNN& = gNN& G &

;

(75)

;

LNNa0 = gNNa0 G  · a0

(76) ;

fN S G ˜ 4 T · 9  4 + h:c: ; m fN SD G ˜ · (94 O − 9O 4 ) LN SD = i 5 4 T m

(77)

LN S =

(78) O

+ h:c :

(79)

Note: The particle called 3 in the original Bonn publication [26] is nowadays called a0 . The ˜ as well as the 6elds are de6ned in Ref. [26]. Note that there is no tensor coupling for operator T the !NN vertex given for the 6t to the elastic NN scattering data did not need any such coupling.

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C. Hanhart / Physics Reports 397 (2004) 155 – 256 60

60

δ [degrees]

40

1S

20

20

0

0

-20

-20

-40

-40 3P

0

3

40

0

S1

3P

-10

0

1

-20 -20 -30 -40

-40 3P

1

2

D2

15 20 10 10

5 0

0 0

200

400

ELAB [MeV]

600

0

200

400

600

ELAB [MeV]

Fig. 21. The NN phase shifts for the model of Ref. [175] in the energy range relevant for pion production. The pion production threshold is at ELAB = 286 MeV. The experimental data are from Refs. [177] (triangles) and [61] (circles).

There is a di@erence between the nucleon–nucleon model used [175] and the Bonn potential [26]. The original Bonn model has an energy-dependent interaction, for it keeps the full meson retardation in the intermediate state. This, however, leads to technical problems, once the model is to be evaluated above the pion production threshold due to the occurrence of three-body singularities. In Ref. [24] those singularities were handled by solving the dynamical equations in the complex plane. Unfortunately, this method is not useful for the application in a distorted wave Born approximation. Instead, we used a model based on the so-called folded diagram formalism developed in Ref. [178]. This formalism, worked out to in6nite order, is fully equivalent to time-ordered perturbation theory. When truncated at low order, however, it leads to energy independent potentials that can formally be evaluated even above the pion production threshold. In addition, the model of Ref. [175] is constructed as a coupled-channel model including the NN as well as the NC and CC channels. This enables us to treat the C isobar on equal footing with the nucleons. The resulting phase shifts are shown in Fig. 21. Note that the model parameters were adjusted to the phase shifts below the pion production threshold only, which is located at ELAB = 286 MeV.

C. Hanhart / Physics Reports 397 (2004) 155 – 256 π

N

N

N (a)

π

N

∆

N

π

π

N

π

N

π

∆

π

N (c)

(b)

N

209

N

π

(d) π

N

V

Tππ

J=0,1 π

N (e)

Fig. 22. (a)–(e) Contributions to the potential of the model of Ref. [179].

Fig. 21 thus clearly illustrates that using this model for the NN interaction is indeed justi6ed and we may conclude that—at least up to laboratory energies of 600 MeV—the NN phenomenology is well understood. 6.2.2. The N model The N interaction that enters in the pion rescattering diagrams can be taken from a meson exchange model as well [179]. This allows a consistent treatment of the meson and nucleon dynamics. It should be stressed that this is the precondition for comparing the results of the phenomenological model to those of chiral perturbation theory, as we will do below. In addition, since we also want to include the rescattering diagram in partial waves higher than the s-wave, after a 6t to the N data the pole contributions (nucleon and Delta) need to be removed from the amplitudes in order to avoid double counting with the direct production. This is possible only within a microscopic model. The main features of the N model of Ref. [179] are that it is based on an e@ective Lagrangian consistent with chiral symmetry to leading order and that the t-channel exchanges in the isovector (D) and isoscalar (&) channel are constructed from dispersion integrals. 23 For details on how the t-channel exchanges are included we refer to Ref. [179]. The diagrams that enter the potential are displayed in Fig. 22. This potential is then unitarized with a relativistic Lippmann–Schwinger equation—in complete analogy to the nucleon–nucleon interaction. Within this model, at tree level the isoscalar and isovector N interaction are given by the corresponding t-channel exchanges. In the latter case the unitarization does not have a big inNuence in the near-threshold regime, and thus also the isovector scattering length is governed by the tree 23

There are ambiguities in how to extrapolate the results of the dispersion integrals to o@-shell kinematics. This issue is discussed in detail in Ref. [180].

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C. Hanhart / Physics Reports 397 (2004) 155 – 256 0.0

20.0 S11 15.0

-5.0

10.0

-10.0

5.0

-15.0

0.0

-20.0

δ [degrees]

1.0

P11

-1.0

S31

P31

-2.0 0.0 -3.0 -1.0

-4.0

-2.0

-5.0 P13

80.0

-1.0

P33

60.0 -2.0 40.0 -3.0 -4.0 1050

20.0

1100

1150

Ecms [MeV]

1200

0.0 1050

1100

1150

1200

Ecms [MeV]

Fig. 23. The N phase shifts for the model of Ref. [179] in the energy range relevant for pion production. The experimental data are from Refs. [183,184]. Note the di@erent scales of the various panels.

level D-exchange. The famous KFSR relation [181,182], that relates the couplings of the D-meson to pions and nucleons to the coupling strength of the Weinberg–Tomozawa term, is a consequence of this. On the other hand, for the isoscalar N interaction the e@ects of the unitarization are large and lead to an almost complete cancellation of the isoscalar potential with the iterated D-exchange in the near-threshold regime. As one moves away from the threshold value this cancellation gets weaker leading to the strong variation of the o@-shell isoscalar N T -matrix mentioned at the end of Section 6.1. In Fig. 23 the results for the model of Ref. [179] are compared to the data of Refs. [183,184]. As one can see the model describes the data well, especially in the most relevant partial waves: S11 ; S31 and P33 . 6.2.3. Additional short-range contributions and model parameters As was stressed above, a large class of additional mechanisms was suggested in the literature to contribute signi6cantly to pion production in nucleon–nucleon collisions. Since they are all of rather

C. Hanhart / Physics Reports 397 (2004) 155 – 256 pp→dπ+

A∑

pp→ppπo -1.5

-0.6

0.6

-1.6

-0.8

-1.7

-1

-1.8

-1.2

0.2

-1.9

-1.4

0

-2

A∆

-1

-0.5

0

0.5

1

-1.6 -1

-0.5

0

0.5

1

-1

0.8

0.4

0.4

0.6

0.2

0.3

0.4

-0.2

0

-0.2

-0.4

-0.1

0

0.5

1

-1

-0.5

0

0.5

1

-0.2

-0.8

1

0 0.5 cos (θπ)

1

-0.4

-1

-0.6

-1.2

-0.8

-1.4 -1

1

0.5

-0.5

-0.2

0 0.5 cos (θπ)

0

-0.8 -1

0

-0.6

-0.5

1

-0.5

-0.4

-1

0.5

-1

0

-0.4

0

0.1

0 -0.5

-0.5

0.2

0

0.2

-1

Azz

pp→pnπ+

0.8

0.4

211

-0.6 -0.5

0 0.5 cos (θπ)

1

Fig. 24. Comparison of the model predictions to the data taken from Ref. [47] (pp → ppo ), Ref. [108] (pp → d+ ) and Ref. [94] (pp → pn+ ).

short range and mainly inNuence the production of s-wave pions, in this work only a single diagram was included (heavy meson exchange through the !—Fig. 2c) to parameterize these various e@ects. Consequently, the strength of this contribution was adjusted to reproduce the total cross section of the reaction pp → pp0 close to the production threshold. The short-range contributions turn out to contribute about 20% to the amplitude. After this is done, all parameters of the model are 6xed. 6.2.4. Results The results of the model presented have already appeared several times in this report (Figs. 13, 14, 15, 17, 20 and 24)—mainly for illustrative purposes—and they are discussed in detail in Refs. [73,71]. Overall the model is rather successful in describing the data, given that only one parameter was adjusted to the total cross section for low-energy neutral pion production (see Section 6.2.3). One important 6nding is that the sign of the s-wave neutral pion production seems to be in accord with experiment, as is illustrated in Fig. 15 (cf. corresponding discussion in Section 4.3), in contrast to the early calculations using chiral perturbation theory. As we will see in Section 6.3, today we know that those early calculations using e@ective 6eld theory were incomplete. The most striking di@erences appear, however, for double polarization observables in the neutral pion channel, as shown in Figs. 13 and 14. As a general pattern the amplitudes seem to be of

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the right order of magnitude, but show a wrong interference pattern. To actually allow a detailed comparison of the model and data, a partial wave decomposition of both is necessary. Work in this direction is under way. It is striking that for charged pion production the pattern is very di@erent, for here almost all observables are described satisfactorily. In Fig. 24 the results of the model for charged and neutral pion production are compared to the data for a few observables. In contrast to the neutral channel, the charged pion production is completely dominated by two transitions, namely 3 P1 → 3 S1 s, which is dominated by the isovector pion rescattering, and 1 D2 → 3 S1 p, which governs the cross section especially in the regime of the Delta resonance. The prominence of the Delta resonance is a consequence of the strong transition 1 D2 (NN ) → 5 S2 (NC) that even shows up as a bump in the NN phase shifts (see Fig. 21). This e@ect was 6rst observed long ago and is well known (see, e.g., discussion in Ref. [185]). As a consequence, the NN → NC transition potential should be rather well constrained by the NN scattering data and is not the case for all the many transitions relevant in case of the neutral pion production, where the Sp 6nal state is not allowed due to selection rules (see Section 1.6). One might therefore hope to learn more about the SN interaction from the pion production data. One can also ask how well we know the production operator. Fortunately, at the pion production threshold it is still possible to analyze meson production in NN collisions within e@ective 6eld theory. This analysis will give deeper insight into the production dynamics, as will be explained in the following section. The two approaches are then compared in Section 6.5. 6.3. Chiral perturbation theory The phenomenological approaches, such as the one described in the previous section, lack a systematic expansion. Thus it is neither possible to estimate the associated model uncertainties nor to systematically improve the models. On the other hand, for various meson production reactions the phenomenological approaches proved to be quite successful. One might therefore hope that e@ective 6eld theories will give insights into why the phenomenology works, as was 6rst stressed by Weinberg [186]. A 6rst attempt to construct—model independently—the transition amplitude NN → NN was carried out almost 40 years ago [187–189]. The authors tried to relate what is known about nucleon– nucleon scattering to the production amplitude via the low-energy theorem of Adler and Dothan [190], which is a generalization of the famous Low theorem for conserved currents [191] to partially conserved currents via the PCAC relation (see, e.g., Ref. [192]). However, it soon was realized that the extrapolation from the chiral limit (m = 0) to the physical point can change the hierarchy of di@erent diagrams. The reason for the non-applicability of soft radiation theorems to meson production close to the threshold is easy to see: a necessary condition for the soft radiation theorem to be applicable is that the energy emitted is signi6cantly smaller than the typical energy scale, that characterizes variations of the nuclear wavefunction. In the near-threshold regime, however, the scale of variation is set by the inverse of the NN scattering lengths—thus a soft radiation theorem of the type of Low or that of Adler and Dothan could only be applicable to meson production if m
1=(MN a2 ). In reality, however, the pion mass exceeds the energy scale introduced by inverse NN scattering length by more than two orders of magnitude. The range of applicability of soft radiation theorems is discussed in Ref. [193] in a di@erent context.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

213

More recent analyses, however, show that in the case of pion production it is indeed possible to de6ne a convergent e@ective 6eld theory that allows a systematic study of the structure of the production operator. As we will show, most of the diagrams included in the Koltun and Reitan model [31] are indeed the leading operators in pion production. In addition this study will show • • • •

that the use of the distorted wave born approximation is justi6ed, why neutral pion production is the more problematic case, the importance of loop contributions, that there is a close connection between pion production in nucleon–nucleon collisions and the three nucleon problem.

The problem with strong interaction phenomena is their non-perturbative nature with respect to the coupling constants. To construct a controlled expansion it is necessary, to identify a small expansion parameter. In general this is only possible for a limited energy range. The conditio sine qua non-for constructing an e@ective 6eld theory for any system is the separation of scales characteristic for the system. Once the—in this context light—scales are identi6ed, one treats them dynamically, while all the dynamics that are controlled by the heavy scales are absorbed in contact interactions. As long as the relevant external momenta and energies are such that structures of the size of the inverse of the heavy scale cannot be resolved, this procedure should always work. Weinberg [194] as well as Gasser and Leutwyler [195] have shown that this general idea works even when loops need to be included. In case of low-energy pion physics it is the chiral symmetry that provides both preconditions for the construction of an e@ective 6eld theory, in that it forces not only the mass of the pion m , as the Goldstone boson of the chiral symmetry breaking, to be low, but also the interactions to be weak, for the pion needs to be free of interactions in the chiral limit for vanishing momenta. The corresponding e@ective 6eld theory is called chiral perturbation theory (HPT ) and was successfully applied to meson–meson [196] scattering. Treating baryons as heavy allows straightforward extension of the scheme to meson–baryon [197] as well as baryon–baryon [198–201] systems. In all these references the expansion parameter used was p= H ∼ p=(4f ) ∼ p=MN , where H denotes the chiral symmetry breaking scale, f the pion decay constant and MN the nucleon mass. Recently, it was shown that also the Delta isobar can be included consistently in the e@ective 6eld theory [202]. The authors treated the new scale, namely the Delta nucleon mass splitting C = MC − MN , to be of the order of p that is taken to be of the order of m . An additional new scale √ occurs for meson production in nucleon–nucleon collisions, namely the initial momentum pi ∼ m MN . Note that, although larger than the pion mass, this momentum is still smaller than the chiral symmetry breaking scale and thus the expansion should still converge, but slowly. A priori there are now two options to construct an e@ective 6eld theory for pion production. The 6rst option, called Weinberg scheme in what follows, treats all light scales to be of order of m . Thus, in this case, there is one expansion parameter, namely HW = m =MN . The other option is to expand in two scales simultaneously, namely m and pi . In this case the expansion parameter is  m ∼ 0:4 : H= MN

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This scheme was advocated in Refs. [163,203] and applied in Ref. [204]. The two additional scales, namely C and m , are identi6ed with C pi m p2 ∼ = H and ∼ i2 = H2 ; (80)

H

H

H

H where the former assignment was made due to the numerical similarity of the two numbers 24 (C = 2:1m and pi = 2:6m ). Only explicit calculations can reveal which one is the more appropriate approach. Within the Weinberg counting scheme, tree level calculations were performed for s-wave pion production in the reactions pp → pp0 [162,163,173] as well as pp → pn+ [164,165]. In addition, complete calculations to next-to-next-to-leading order (NNLO), where in the Weinberg scheme for the 6rst time loops appear, are available for pp → pp0 [205,206]. The authors found that some of the NNLO contributions exceeded signi6cantly the next-to-leading (NLO) terms leading them to the conclusion that the chiral expansion converges only slowly, if at all. This point was further stressed in Refs. [207,155], however, it was shown recently that as soon as the scale induced by the initial momentum is taken into account properly (expansion in H and not in HW ), the series indeed converges [203,204]. For illustration, in this section we compare the order assignment of the Weinberg scheme to that of the modi6ed scheme. In Appendix E the relevant counting rules of the new scheme are presented as well as justi6ed via application to a representative example. Especially, it is not clear a priori what scale to assign to the zeroth component l0 of the four dimensional integration volume d4 l as it occurs in covariant loops. After all, the typical energy of the system is given by m , but the momentum by pi . As is shown in the appendix by matching a covariant analysis to one carried out in time-ordered perturbation theory, in loops l0 ∼ pi . This assignment was also con6rmed in explicit calculations [204]. The starting point is an appropriate Lagrangian density, constructed to be consistent with the symmetries of the underlying more fundamental theory (in this case QCD) and ordered according to a particular counting scheme. Omitting terms that do not contribute to the order we will be considering here, we therefore have for the leading order Lagrangian [208,199,197]   1 1 1 1 2  2 2 2 L(0) = 94 94  − m2 2 + m ( · 9  ) − 4 2 2 2f2 4    1 †  · ( × ) ˙ N + N i90 − 4f2   gA † 1 ˜ ˜ + N  · ˜& · ∇ + ( · ∇) N 2f 2f2 hA ˜ [N † (T · ˜S · ∇)0 (81) C + h:c:] + · · · : 2f The expressions for interactions with more than two pions depend on the interpolating 6eld used. The choice made here was the so called sigma gauge—cf. Appendix A of Ref. [197], where also the corresponding vertex functions are given explicitly. 25 + 0C† [i90 − C]0C +

24 25

Note that C stays 6nite in the chiral limit, whereas both pi as well as m vanish. As usual, all observables are independent of the choice of the pion 6eld.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

215

For the next-to-leading order Lagrangian we get L(1) =

1 ˜ 2 N + 0† ∇ ˜ 2 0C ] [N † ∇ C 2mN 1 ˜ · ∇N ˜ + h:c:) (iN †  · ( × ∇) 8MN f2   1 g2 ˜ 2 − 2c1 m2 2 + 2 N † c2 + c3 − A ˙2 − c3 (∇) f 8mN    1 1 c4 + jijk jabc &k %c 9i a 9j b N − 2 4mN +

−

gA ˜ + h:c:] − hA [iN † T · ˙˜S · ∇0 ˜ C + h:c:] [iN †  · ˜ ˙& · ∇N 4mN f 2mN f

−

d2 d1 † ˜ N ( · ˜& · ∇)N N †N − jijk jabc 9i a N † &j %b N N † &k %c N + · · · ; f 2f

(82)

where f denotes the pion decay constant in the chiral limit, gA is the axial-vector coupling of the nucleon, hA is the SN coupling, and ˜S and T are the transition spin and isospin matrices, normalized such that Si Sj† = 13 (23ij − ijijk &k ) ;

(83)

Ti Tj† = 13 (23ij − ijijk %k ) :

(84)

These de6nitions are in line with the ones introduced in Section 6.2.1. The dots symbolize that what is shown are only those terms that are relevant for the calculations presented. As demanded by the heavy baryon formalism, the baryon 6elds N and 0C are the velocity-projected pieces of the relativistic 6elds appearing in the interactions discussed in Section 6.2.1; e.g. N = 12 (1 + v=) , where v4 denotes the nucleon 4-velocity. The terms in the Lagrangians given are ordered according to the conventional counting (p  m ). A reordering on the basis of the new scheme does not seem appropriate, for what order is to be assigned to the energies and momenta occurring depends on the topology of a particular diagram (see also Appendix E). The constants ci can be extracted from a 6t to elastic N scattering. This was done in a series of papers with successively improved methods [171,172,209–212]. However, here we will focus on the values extracted in Refs. [171,172] for it is those that were used in the calculations for pion production in nucleon–nucleon collisions. In the former work the ci were extracted at tree level and in the latter to one loop. The corresponding values are given in Table 9. In both papers the Delta isobar was not considered as explicit degree of freedom. In an e@ective 6eld theory the low-energy constants appearing depend on the dynamical content of the theory. Thus, in a theory without explicit Deltas, their e@ect is absorbed in the low-energy constants [172]. This is illustrated graphically in Fig. 25. Thus we need to subtract the Delta contribution from the values given in the 6rst columns

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Table 9 The various low-energy constants ci in units of GeV−1 i

citree

ciloop

citree (C=)

ciloop (C=)

1 2 3 4

−0.64 1.78 −3.90 2.25

−0.93 3.34 −5.29 3.63

−0.64 0.92 −1.20 0.9

−0.93 0.64 −2.59 2.28

The 6rst two columns give the values from the original references (left column: tree level calculation of Ref. [171]; right column: one loop calculation of Ref. [172]), whereas the last two columns give those values reduced by the Delta contribution as described in the text. It is those numbers that where used in the calculations presented here.

with ∆

without ∆

Fig. 25. Illustration of resonance saturation.

of Table 9. Analytical results for those contributions are given in Refs. [213,172]: 26 c2C = −c3C = 2c4C =

h2A = 2:7 GeV−1 : 9(MC − MN )

(85)

Thus, the only undetermined parameters in the interaction Lagrangian are d1 and d2 . Since they are the strength parameters of 4-nucleon contact interactions that do not contain any derivatives of the nucleon 6elds, they only contribute to those amplitudes that have an NN S-wave to NN S-wave plus pion p-wave transition. This automatically excludes a transition from an isospin triplet to an isospin triplet state, for this would demand to go from 1 S0 to 1 S0 accompanied by a p-wave pion, which is forbidden by conservation of total angular momentum. Thus, only two transitions are possible: T = 0 → T = 1, as it can be studied in pn → pp− , and T = 1 → T = 0, as it can be studied in pp → pn+ . Even more importantly, in both channels the di appear with the same linear combination d which is completely 6xed by the corresponding isospin factors: d=

1 3 (d1 + 2d2 ) ∼ 2 ; 3 f MN

(86)

where we have introduced the dimensionless parameter 3. In order for the counting scheme to work, 3 = O(1) has to hold. As we will show below, this order of magnitude is indeed consistent with the data. Please note, the four-nucleon–pion contact term with p-wave discussed here was to some extend also investigated in Ref. [214], however, within a di@erent expansion scheme. Thus direct comparison is not possible. 26

√ To match the results of the two references the large NC value has to be used for hA = 3gA = 2 2:7.

C. Hanhart / Physics Reports 397 (2004) 155 – 256

(a)

217

(b)

Fig. 26. (a)–(b) Illustration of the role of the 4N contact term in NN → NN and three nucleon scattering. Solid lines denote nucleons, dashed lines denote pions.

The parameter d is very interesting, for it is at the same time the leading short-range–long-range contribution 27 to the three nucleon force, as illustrated in Fig. 26. Naturally, it can also be 6xed from pd scattering data directly. This was done in Ref. [215]. We come back to this point below. The e@ect of the large scale on the vertices is best illustrated with an example. In L(0) the so-called Weinberg–Tomozawa (WT) term 1=(4f2 )N †  · ( × )N ˙ appears. The corresponding recoil ˜ · ∇N ˜ + h:c:) appears in L(1) . Therefore the vertex function correction 1=(8MN f2 )(iN †  · ( × ∇) derived from the WT term is proportional to (q0 +k0 )=f2 , where q0 (k0 ) denote the zeroth component of the 4-vectors for the outgoing (incoming) pion. If the WT term appears as the N vertex in the rescattering contribution (cf. Fig. 2b), in threshold kinematics k0 = m =2 and q0 = m . Also in threshold kinematics, where the three momentum transfer equals the initial momentum, we 6nd for the contribution from the recoil term p2 =(2MN f2 ). Obviously, since p2 = MN m , the contribution from the WT term and from its recoil term are of the same order. This changes if the WT term appears inside a loop, for then the scale for k0 is also set by p—in this case the recoil term is suppressed by one chiral order compared to the WT term itself. The assignments made are con6rmed by explicit calculation [204] as well as by a toy model study [174]. Since we know now the interactions of pions and nucleons we can investigate the relevance of sub-leading loops, where we mean loops that can be constructed from a low order diagram by inserting an additional pion line. Obviously, this procedure introduces at least one NN vertex ∼ p=f , a pion propagator ∼ 1=p2 , an integral measure p4 =(4)2 and either an additional NN vertex together with two-nucleon propagators, or one additional nucleon propagator together with a factor 1=f . To get the leading piece of the loops in the latter case, the integration over the energy variable has to pick the large scale and thus we are to count the nucleon propagator as 1=p, leading to an overall suppression factor for this additional loop of p2 =MN2 . In the former case, however, a topology is possible that contains a two-nucleon cut. This unitarity cut leads to an enhancement of that intermediate state, for it pulls a large scale (MN ) into the numerator. Based on this observation Weinberg established a counting scheme for NN scattering that strongly di@ers from that in  27

In this context, pion exchanges are called long ranged, whereas any exchange of heavier mesons—absorbed in the contact terms—are called short ranged.

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

as well as N scattering [186]. The corresponding factor introduced by this loop is p=MN (and in addition typically comes with a factor of ). As was stressed by Weinberg, this suppression is compensated by the size of the corresponding low energy constants of the two-nucleon interaction and therefore all diagrams that can be cut by crossing a two-nucleon line only are called reducible and the initial as well as 6nal state NN interaction is summed to all orders. This is what is called distorted wave Born approximation. There is one exception to this rule: namely when looking at the two-nucleon intermediate state close to the pion production vertex in diagram a of Fig. 2. Either the incoming or the outgoing nucleon needs to be o@-shell and thus this intermediate state does not allow for a two-nucleon unitarity cut. Therefore those two-nucleon intermediate states are classi6ed as irreducible. There is a special class of pion contributions not yet discussed, namely those that contain radiative pions (on-shell pions in intermediate states). We restrict ourselves to a kinematic regime close to the production threshold. As soon as an intermediate pion goes on-shell, the typical momentum in the corresponding loop automatically needs to be of order of the outgoing momenta. This leads to an e@ective suppression of radiative pions; e.g., for s-wave pions the e@ects of pion retardation become relevant at N5 LO. Note that each loop necessarily contributes at many orders simultaneously. The reason for this is that the two scales inherent to the pion production problem can be combined to a dimensionless number smaller than one: m =pi = H. However, what can always be done on very general grounds is to assign the minimal order at which a diagram can start to contribute and this is suOcient for an eOcient use of the e@ective 6eld theory. The next step is the consistent inclusion of the nuclear wavefunctions. However, the NN potentials constructed consistently with chiral perturbation theory [198–201] are not applicable at the pion production threshold. Therefore, we use the so-called hybrid approach originally introduced by Weinberg [216], where we convolute the production operator, constructed within chiral perturbation theory, with a phenomenological NN − NC wavefunction. We use the CCF model described in the previous section [175]. Let us start with a closer look at the production of p-wave pions, for those turned out easier to handle than s-wave pions. The reason for this pattern lies in the nature of pions as Goldstone bosons of the chiral symmetry: since in the chiral limit for vanishing momenta the interaction of pions with matter has to vanish, the coupling of pions naturally occurs in the company of either a derivative or an even power of the pion mass. 28 As a consequence, the leading piece of the NN vertex is of p-wave type, whereas the corresponding s-wave piece is suppressed as ! =MN , where ! denotes the pion energy. Note that also in the case of neutral pion photoproduction the s-wave amplitude is dominated by pion loops, whereas the p-wave amplitude is dominated by tree level diagrams [218,197]. The corresponding diagrams are shown in Table 10 up to N3 LO. So far in the literature calculations have been carried out only up to N2 LO [203]. An important test of the approach is to show its convergence. For that we need an observable to which s-wave pions do not contribute. Such an observable is given by the spin cross section 3 &1 recently measured at IUCF for the reaction 28

Only even powers of the pion mass are allowed to occur in the interaction, since, due to the Gell–Mann–Oakes– Renner relation [217], m2 ˙ mq , where mq is the current quark mass and in the interaction no terms non-analytic in the quark masses are allowed.

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219

Table 10 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for several nucleonic contributions for p-wave pion production p-Wave diagrams (nucleons only)

. .

. .

p ∼ m

p∼

q=m

LO

LO

qm =p2

LO

N2 LO

q=MN

NLO

N2 LO

pq=MN2

N2 LO

N3 LO

m M N

.

a)

.

√

Scale

b)

. c)

.

.

. .

.

d)

g)

f)

e)

. . . .

. . .

.

. . h)

.. . . ..

. i)

j)

Subleading vertices are marked as . Here q denotes the external pion momentum. For simplicity we assume the outgoing nucleons at rest.

pp → pp0 [47] (cf. Table 7 in Section 4.2). The parameter-free prediction of chiral perturbation theory compared to the data is shown in Fig. 27. As one can see, the total amplitude is clearly dominated by the leading order suggesting a convenient rate of convergence for the series. In addition, the prediction agrees with the data. Thus we are now prepared to extract the parameter d from data on the reaction pp → pn+ [219]. As it was argued above, only the amplitude corresponding to the transition 1 S0 → 3 S1 p, called a0 , is inNuenced by the corresponding contact interactions. The results of the chiral perturbation theory calculations are shown in Fig. 28. The 6gure shows four curves for di@erent values of the parameter 3 de6ned in Eq. (86), namely the result for 3 = 0 (dot–dashed line), for 3 = −0:2—the authors of Ref. [220] claim this value to yield an important contribution to Ay in Nd scattering at energies

220

C. Hanhart / Physics Reports 397 (2004) 155 – 256 80

40

3

σ1 [µb]

60

20

0 0.5

0.6

0.7

η

0.8

0.9

1

Fig. 27. Comparison of the predictions from e@ective 6eld theory at LO (dashed line) and NLO (with the N parameters from a NLO (dotted line) and an NNLO (solid line) analysis) with the 3 &1 cross section for pp → pp0 [47]. (Reproduced from Phys. Rev. Lett. with permission from AIP.)

1

a0 [µb1/2]

0.5

0

-0.5

-1 0.1

0.3

0.5

0.7

η

Fig. 28. a0 of pp → np+ in chiral perturbation theory. The di@erent lines correspond to values of the parameter related to the three-nucleon force: 3 = 1 (long-dashed line). 3 = 0 (dot–dashed line), 3 = −0:2 (solid line), and 3 = −1 (short-dashed line). Data are from Ref. [219].

of a few MeV 29 —as well as the results we get when 3 is varied within its natural range 3 = +1 and 3 = −1 shown as the long-dashed and the short-dashed curve, respectively. Thus we 6nd that the results for a0 are indeed rather sensitive to the strength of the contact interaction. This might be surprising at 6rst glance, since we are talking about a sub-leading operator, however it turned 29

The calculation of Ref. [220] su@ers from numerical problems.

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221

out that the leading (diagrams (a) and (b) in Table 10) and sub-leading contributions (the same diagrams with a C intermediate state) largely cancel. Thus, to draw solid conclusions the p-wave calculations should be improved by one chiral order, the corresponding diagrams of which are shown in Tables 10g–j. Please note that the rescattering contribution involving c4 that occurs at the same order as the d contribution turned out to be sensitive to the regulator used for the evaluation of the convolution integral with the nuclear wavefunction. This cuto@ dependence can be absorbed in d, which in turn is now cuto@-dependent. Therefore, in order to compare the results for d from the pion production reaction to those extracted from the three-nucleon system [215] a consistent calculation that is not possible at present has to be performed. Note that the d parameter as 6xed in Ref. [215] (there it is called E) also turned out to be sensitive to the regulator. On the long run, however, a consistent description of pion production and three-nucleon scattering should be a rather stringent test of chiral e@ective 6eld theories at low and intermediate energies. As should be clear from the discussion above, to yield values for the low-energy constant 3 that are compatible both calculations pd scattering as well as -production have to be performed using the same dynamical 6elds—at present the C-isobar is not considered as explicit degree of freedom in Ref. [215], but plays a numerically important role in the extraction of 3 from NN → NN. In Section 4.3 it was shown that the di@erential cross section as well as the analysing power for the reaction pn → pp− is sensitive to an interference term of the s-wave pion production amplitude of the A11 amplitude (3 P0 → 1 S0 s) and the p amplitudes of A01 : 3 S1 → 1 S0 p and 3 D1 → 1 S0 p. Obviously, the 4-nucleon contact interaction contributes to the former. Thus, once a proper chiral perturbation theory calculation is available for the s-wave pion production—whose status will be discussed in the subsequent paragraphs—the reaction pn → pp− close to the production threshold might well be the most sensitive reaction to extract the parameter d. Let us now turn to the s-wave contributions. The leading diagrams containing nucleons only are shown in Table 11; those that contain the C are shown in Table 12. Again, the Weinberg scheme and the new scheme are compared. The list is complete up to NNLO in both schemes. Please note, however, that one class of diagrams (k and l) that is of NLO in the Weinberg counting in the new scheme is pushed to N4 LO! Thus in the new counting scheme—contrary to the Weinberg scheme—the leading pieces of some loops appear one order lower than the tree level isoscalar rescattering amplitudes. As can also be read from the table, the corresponding order is m =MN . If we consider, in addition, that due to the odd parity of the pion the initial state has to be a p-wave, the loops themselves have to scale as √ m . Since no counter term non-analytic in m is allowed, the loops have to be 6nite or to cancel exactly. This requirement is an important consistency check of the new counting scheme. The details of the loops calculations in threshold kinematics can be found in Ref. [204]. Here we only give the results. Note, we only evaluate the leading order pieces of the integrals corresponding to the diagrams of Table 11. For example, in the integrals we drop terms of order m compared to l0 . After these simpli6cations, straightforward evaluation gives for the production amplitude from the loops with nucleons only

 ˜k  MN m  i 3 An = 3 gA ˜&1 · –n ; (87) F 128F2 |˜k|

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Table 11 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for several nucleonic contributions for s-wave pion production s-Wave diagrams (nucleons only)

.

p ∼ m

p∼

m =p

LO

LO

p=M

NLO

LO

p2 =MN2

N2 LO

NLO

pm =MN2

N2 LO

N2 LO

m2 =(pMN )

NLO

N2 LO

m3 =pMN2

N2 LO

N4 LO

m M N

. a)

. .

.

.

. b)

c)

. . . .

.

.

. .

.

g)

.

f)

e)

.

.

. . . . . .

. . . .

d)

. . .

. i)

h)

. j)

. .

√

Scale

. . . k)

. . .

. . . l)

.

Subleading vertices are marked as . Not shown explicitly are the recoil corrections for low-order diagrams. For example, recoil corrections to diagram (b) appear at order pm =MN2 .

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223

Table 12 Comparison of the corresponding chiral order in the Weinberg scheme (p ∼ m ) and the new counting scheme √ (p ∼ m MN ) for the leading and next-to-leading C contributions for s-wave pion production s-Wave diagrams (Deltas only)

. .

. .

b)

. . . .

. .

.

c)

. .

C ∼ m

C∼

pm =SMN

NLO

NLO

p3 =(SMN2 )

N2 LO

NLO

√

m  MN

.

a)

.

Scale

. .

.

.

.

. .

.

. .

.

d)

. . . .

. . . .

f)

e)

Subleading vertices are marked as . Not shown explicitly are the recoil corrections for low-order diagrams.

where n denotes the diagram (labels as in the 6gure). For the isospin functions we 6nd –d = −%c1 ;

–e = − 14 (%c1 + %c2 );

–f = 32 %c1 :

(88)

Those can be easily evaluated in the di@erent isospin channels. We 6nd for –(TT3 ) = TT3 |–d + –e + –f |11 (it is suOcient here to look at the pp initial state only) –(11) = −1 −

1 2

+

3 2

= 0;

–(00) = −1 + 0 +

3 2

= − 12 :

(89)

The 6rst observation is that the NLO contributions are of the order of magnitude expected by the power counting, since   M N m = 0:8H2 ; 128F2 where we used F =93 MeV. The power counting proposed in Refs. [163,203] thus is indeed capable of treating properly the large scale inherent to the NN → NN reaction. We checked that our results for the individual diagrams agree with the leading non-vanishing pieces from the calculations of Ref. [205]. 30 In addition, for almost all diagrams given in Table 11, 30

In this reference the same choice for the pion 6eld is made and thus a comparison of individual diagrams makes sense. Note that there is a sign error in the formula for diagram f in Ref. [205]. We are grateful to Myhrer for helping us to resolve this discrepancy.

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Table 13 Comparison of the results of the analytic calculation of Ref. [205] with the expectations based on the two counting schemes as discussed in the text Diagram

d

e

g

jR

k

l

Ref. [205] p∼√ m p ∼ m  MN

1.0 1.0 1.0

−1.0 1.0 1.0

0.1 1.0 0.4

0.4 1.0 0.4

0.03 1.0 0.06

0.02 1.0 0.06

The diagrams are labeled as in Fig. 11 (the label jR shows that here the recoil term of diagram j is calculated; it appears at NNLO in the Weinberg scheme as well as in the new counting scheme).

Ref. [205] gives explicit numbers for the amplitudes in threshold kinematics. It is intriguing to compare those to what one expects from the di@erent counting schemes. This is done in Table 13, where the 6rst line speci6es the particular diagram according to Table 11 and the second gives the result of the analytical calculation of Ref. [205]. 31 (normalized to the 6rst column). In the following two lines those numbers are compared to the expectations based on the counting schemes—6rst showing those for the Weinberg scheme and then those for the new counting scheme. As can clearly be seen, the latter does an impressive job of predicting properly the hierarchy of diagrams. Thus, at least when ISI and FSI are neglected, the counting scheme proposed is capable of dealing with these large momentum transfer reactions. It is striking that the sum of loops in the case of the neutral pion production vanishes (–(11) = 0 in Eq. (89)). In addition, for neutral pion production there is no meson exchange current at leading order and the nucleonic current (diagrams b and c in Table 11) gets suppressed by the poor overlap of the initial and 6nal state wavefunctions (see discussion in Section 1.4)—an e@ect not captured by the counting—and interferes destructively with the direct production o@ the Delta (diagrams a and b in Table 12). Thus the 6rst signi6cant contributions to the neutral pion production appear at NNLO. This is the reason why many authors found many di@erent mechanisms, all of similar importance and capable of removing the discrepancy between the Koltun and Reitan result and the data, simply because there is a large number of diagrams at NNLO. The situation is very di@erent for the charged pions. Here there is a meson exchange current at leading order and there are non-vanishing loop contributions. We therefore expect charged pion production to be signi6cantly better under control than neutral pion production and this is indeed what we found in the phenomenological model described in the previous section. Next, let us have a look at the loops that contain Delta isobars (cf. Fig. 12). In Ref. [204] it was shown that the individual loops diverge already at leading order, because the Delta–Nucleon mass di@erence introduces a new scale. Therefore, as was argued above, the sum of the diagrams has to cancel, for at NLO there is no counter term. It is an important check of the counting scheme that this does indeed happen. We take this as a strong indication that expanding in H is consistent with the chiral expansion. The observation that we have now established a counting scheme for reactions of the type NN → NN also has implications for the understanding of other reactions. One example is the analysis elastic d scattering, which is commonly used to extract the isoscalar N scattering length 31

In Ref. [205] the full result for the particular loops are given. Thus, any loop contains higher order contributions.

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225

that is diOcult to get at otherwise [167]. As mentioned above, due to chiral symmetry constraints the isoscalar scattering length does not get a contribution at leading order and thus, for an accurate extraction of this important quantity from data on d scattering or on d bound states, a controlled calculation of the few-body corrections is compulsory. One of these corrections are the so-called dispersive corrections (see Ref. [221] and references therein): loop contributions to the elastic scattering that have intermediate two-nucleon states. Obviously, the imaginary part of those loops gives the essential contribution to the imaginary part of the d scattering length, 32 however there is also a real part to these loops that needs to be calculated within a scheme consistent with that used for the calculation of the other contributions. Within chiral perturbation theory that has not been done up to now. Given the progress reported here, such a calculation is now feasible. The same technique can then also be used to calculate the corresponding corrections for 3 He scattering, recently calculated in chiral perturbation theory for the 6rst time [223]. 6.4. On the signi9cance of o8-shell e8ects A few years a ago there existed a strong program at several hadron facilities to measure bremsstrahlung in NN collisions with the goal to identify the true o@-shell behavior of the NN interaction. Indeed it was found that the predictions for several highly di@erential observables are signi6cantly di@erent when di@erent NN potentials are employed. On the other hand, a change in the o@-shell behavior of any T -matrix can be realized on the level of the e@ective interaction by a 6eld rede6nition and it is known since long that S-matrix elements do not change under those transformations (as long as one works within a well de6ned 6eld theory) [224,225]. Thus, for any given set of 6elds the o@-shell amplitudes might well have a signi6cant impact on the values of various observables, but it is an intrinsic feature of quantum 6eld theory that they cannot be separated from the short range interactions constructed within the same model space. In a very pedagogical way those results were presented later in Refs. [226,227]. How can one understand this seemingly contradictory situation: on the one hand o@-shell amplitudes enter the evaluation of matrix elements and in some cases inNuence signi6cantly the result (cf. discussion in previous sections), on the other hand they are claimed not to have any physical signi6cance? Following Ref. [226] to start the discussion, let us consider some general half-o@ shell N T -matrix. In a covariant form the list of its arguments contains the standard Mandelstam variables s, t as well as q2 —the four momentum squared of the o@-shell particle(s). For our discussion let this be the incoming pion and for simplicity omit all spin indices. We now de6ne TR (s; t; q2 ) via T (s; t; q2 ) = T˜ (s; t; q2 ) + (q2 − m2 )TR (s; t; q2 ) ; where T˜ (s; t; q2 ) is arbitrary up to the condition that it has to agree with the on-shell amplitude for q2 = m2 . Obviously, this is always possible and we can assume TR to be smooth around the on–shell point. Then, when introduced into the NN → NN transition amplitude, we get A=T

32

q2

1 1 WNN = T˜ 2 WNN + TR WNN ; 2 − m q − m2

About one third of the imaginary part was found to be related to the reaction d → NN [222].

(90)

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T

T

Fig. 29. Visualisation of Eq. (90).

where WNN denotes the NN vertex function. This example highlights two important facts: (i) a change in the o@-shell dependence of a particular amplitude can always be compensated via an appropriate additional contact term. The only quantities that are physically accessible are S-matrix elements and those are by de6nition on-shell; and (ii) the contribution stemming from the o@-shell N T -matrix is a short range contribution. Therefore the distinction between short range and o@-shell rescattering is arti6cial (see Fig. 29). 6.5. Lessons and outlook The previous section especially made clear that it is rather diOcult to construct a model that gives quantitatively satisfactory results for the reaction pp → pp0 . On the other hand, it turned out that for the reaction pp → pn+ current models as well as the e@ective 6eld theory approach do well. In the previous sections this di@erence was traced back to a suppression of meson exchange currents in the neutral pion production. What is the lesson to be learned from this? First of all, any model for meson production close to the threshold should contain the most prominent meson exchange currents; as long as these are not too strongly suppressed, one can expect them to dominate the total production cross section close to the threshold. However if there is no dominant meson exchange current, then there is a large number of sub-leading operators that compete with each other and make a quantitative understanding of the cross section diOcult. The optimistic conclusion to be drawn from these observations is that (also for heavier mesons) the leading meson exchange currents should give a reasonable description of the data, while the contributions from irreducible loops largely cancel. Obviously, in heavy meson production there is no reason anymore to consider pions only as the exchange particles. In this sense it is + production that is the typical case, whereas the 0 is exceptional due to the particular constraints from chiral symmetry in that channel. Note that also in the case of pion photoproduction close to the threshold the 0 plays a special role, in that the s-wave amplitude is dominated by loops [218]. Let us look in somewhat more detail at the production operator for neutral pion production. Within the model described above, the most prominent diagram for neutral pion production close to the production threshold is pion rescattering via the isoscalar pion–nucleon T -matrix that, for the kinematics given, is dominated by a one-sigma exchange (diagram b of Fig. 2, where the T -matrix is replaced by the isoscalar potential given by diagram e of Fig. 22). Within the e@ective 6eld theory the isoscalar potential is built up perturbatively. This is illustrated in Fig. 30. As was shown above, the leading piece of the one-sigma exchange gets canceled by other loops that cannot be interpreted as a rescattering diagram (the sum of diagrams d–f of Table 11 vanishes) and are therefore

C. Hanhart / Physics Reports 397 (2004) 155 – 256

+

+

+

227

+

...

Fig. 30. The one sigma exchange as it is perturbatively built up in the e@ective 6eld theory, starting from the left with the lowest order diagram (NLO). The chiral order increases by one power in H between each diagram from the left to the right.

not included in the phenomenological approach. This is an indication that in order to improve the phenomenological approach, at least in case of neutral pion production, pion loops should be considered as well. 33 There is one more important conclusion to be drawn from the insights reported in the two previous sections: We can now identify what was missing in the straightforward extension of the Bonn potential reported in Ref. [24]. This is especially relevant, since we will see that the failure to describe the low-energy pion production data in that approach does not point at a missing degree of freedom in the nucleon–nucleon phenomenology, but at an incomplete treatment of the cut structure. Diagrammatically, the dressed NN vertex function is given by

Γ

TNP

where the solid dot indicates the so-called bare vertex (in this picture it is to be understood as a parameterization of both the extended structure of the nucleon as well as meson dynamics not included in the non-pole N T -matrix T NP , such as D [229] or & [230] correlations) as well as the dressing due to the N interactions parameterized in T NP . The latter structure, however, contains a N cut not considered in Ref. [24]. It is straightforward to map the di@erent three-body cuts in the one pion exchange potential to those that must occur in the pion production operator in order to allow for a consistent description of scattering and production. This is indicated in Fig. 31: the additional cut due to the pion dynamics in the vertex function is related to the pion rescattering diagram in the production operator. Therefore, at least close to threshold in Ref. [24], the most relevant mechanism that leads to inelasticities was missed. As we saw, however, as we move away from the near-threshold region the contribution from the Delta isobar becomes more and more signi6cant. Consequently, the results of Ref. [24] look signi6cantly better at higher energies.

33

Note that within the e@ective 6eld theory approach the convergence of the series shown in Fig. 30 should also be checked, as pointed out in a di@erent context in Ref. [228].

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TNP

Γ Γ

Γ Γ

Fig. 31. Leading cut structure of the one pion exchange potential in nucleon–nucleon scattering.

7. Production of heavier mesons We start this section with some general remarks and then discuss some special examples. The phenomenology will be discussed only very brieNy; for details on this we refer to the original references or the recent reviews [1,2]. The list of reactions discussed in detail is by no means complete. For example, we will not discuss here  production, for this was already discussed in great detail in Ref. [2]. Neither will the production of vector mesons be discussed in detail. Here we refer to the recent contributions to the literature (Ref. [59] and references therein). As can be seen from the headlines already, special emphasis will now be put on the physics that can be extracted from the particular reactions. 7.1. Generalities As the mass of the produced meson increases, the initial energy needs to increase as well and consequently also the typical momentum transfer at threshold. This has various implications: • the NN interaction needed for a proper evaluation of the initial state interaction is less under control theoretically; for the T = 0 channel there is not even a partial wave analysis available above the -production threshold, due to a lack of data; • a larger momentum transfer makes it more diOcult to construct the production operator; at least one should expect that the exchanges of heavier mesons become even more relevant compared to the pion case;

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• it is not known, how to formulate a convergent e@ective 6eld theory; 34 • less is known about the interactions of the subsystems; • the treatment of three-body singularities due to the exchange of light particles requires more care. In the upcoming section we will not discuss in detail the role of the three-body singularities. Only recently was a method developed that in the future will allow inclusion of those singularities in calculations within the distorted wave born approximation [231]. This method was already applied in a toy model calculation [232], where its usefulness was demonstrated. In calculations these singularities can occur only if the initial state interaction is included. For the production of mesons heavier than the , however, no reliable model is available at present. Up to now no model for the production of heavy mesons takes three-body singularities into account, and thus their role is yet unclear. 7.2. Remarks on the production operator for heavy meson production As in Section 6 we will only look at analyses, that work within the distorted wave born approximation. For the two-baryon states nothing changes, other than that the nucleon–nucleon interaction needed for the initial state interaction gets less reliable from the phenomenological point of view as we go up in energy. The construction of the production operator, however, is now even more demanding, for with increasing momentum transfer heavier exchange mesons can play a signi6cant role. Therefore, in order to get anything useful out of a model calculation, as many reaction channels as possible should be studied simultaneously. Only in this way can the phenomenological model parameters can be 6xed and useful information be extracted. In this context also the simultaneous analysis of pp- and pn-induced reactions plays an important role, for di@erent isospin structures in the production operator will lead to very di@erent relative strengths of the two channels (cf. discussion in Section 1.5). Then, once a basic model is constructed that is consistent with most of the data, deviations in particular reaction channels can be studied. In Ref. [34] the model described in Section 6.2 was extended to production in nucleon–nucleon collisions, where the single channel N T -matrix used in the pion production calculations was replaced by the multi channel meson–baryon model of Ref. [233] to properly account for the exchange of heavy mesons. So far only s-waves were considered in the calculation. Results for the various total cross sections using two di@erent models for the 6nal state NN interaction are shown in Fig. 32. The studies of Ref. [34] indicate, that a complete calculation for NN → NN will put additional constraints on the relative phases of the various meson–baryon → N transition amplitudes. Unfortunately, up to now the is the heaviest meson that can be investigated using this kind of microscopic approach, for there is no reliable model for the NN interaction for energies signi6cantly above the production threshold. The strategy that therefore needs to be followed is to study many reaction channels consistently. This was done, for example, in a series of papers by Kaiser and others for , ,  [155], ! [234] as well as strangeness production [235] and by Nakayama and coworkers for ! [59,236,237], 34

In principle, one might expect an approach like that presented in Section 6.3 to work for the production of all Goldstone bosons. However, the next threshold after pion production is that for eta production and it already corresponds  to an initial momentum of pi = 770 MeV. Thus the expansion parameter would be m =MN = 0:8.

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pn→dη 10 σ [µb]

pn→pnη

pp→ppη

1

0.1 0

10

20 30 Q [MeV]

40

50

Fig. 32. Results for the total cross sections of the reactions pp → pp , pn → pn , and pn → d from Ref. [34] employing di@erent NN models for the 6nal state interaction. The solid lines represent the results with the CCF NN model [175] whereas the dashed–dotted lines were obtained for the Bonn B model [25]. Data are from Refs. [33,109–112,120].

; [237,238], [239,240] and  [241] production. Both groups construct the production operator as relativistic meson exchange currents, where the parameters are constrained by other data such as decay ratios. The striking di@erence between the two approaches is that the former studies s-waves only, does not consider e@ects of the initial state interaction and treats the 6nal state interaction in an approximate fashion. The latter group includes the ISI through the procedure of Ref. [40] (cf. Section 3), treats the FSI microscopically and includes higher partial waves as well. It is important to stress that, where compatible, the two approaches give qualitatively similar results, as stressed in Ref. [234]. Recently, it was observed that the onset of higher partial waves can strongly constrain the production operator [240]. In the same reference it was demonstrated that to unambiguously disentangle effects from FSI or higher partial waves, polarization observables are necessary. Thus one should expect that once a large amount of polarized data is available on the production of heavy mesons, the production operators and thus the relevant short-range physics for the production of a particular meson, can be largely 6xed. As an example, this particular issue is discussed in detail in the next section. 7.3. The reaction NN → NN or properties of the S11 (1535) The meson is a close relative of the pion, since it is also a member of the nonet of the lightest pseudo-scalar mesons. It is an isoscalar with a mass of 547:3 MeV—the di@erence in mass from the pion can be understood from its content of hidden strangeness through the Gell–Mann–Okubo mass formula [242,243]. As for the pion, the couples strongly to a resonance, but a resonance with di@erent quantum numbers: is it the positive parity C(1232) for the pion that in the C rest frame leads to p wave pion production, the production is dominated by the negative parity S11 (1535), 35 which leads 35

The quantum numbers are chosen in accordance with the partial wave in which the resonance would appear in N scattering. Therefore positive parity resonances appear in odd partial waves.

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to s wave production in its rest frame. As a consequence near-threshold production is completely dominated by the S11 (1535) resonance. This statement can well be reversed: studying production in various reaction channels close to the production threshold allows selective study of the properties of the S11 (1535) resonance in various environments. In this context it is interesting to note that the nature of this particular resonance is under discussion: within the so-called chiral unitary approach the resonance turns out to be dynamically generated [244,245], while on the other hand, detailed studies within the meson exchange approach [246], as described in Section 6, as well as quark models (see Ref. [247] and references therein) call for a genuine quark resonance. One expects that a molecule behaves di@erently in the presence of other baryons than a three-quark state due to the naturally enhanced aOnity to meson baryon states of the former. In this sense data on production in few-baryon systems should provide valuable information about this lowest negative parity excitation of the nucleon. The reaction NN → NN was studied intensively both theoretically [34,155,239,240,248–259] as well as experimentally (cf. Table 8) in recent years. The outstanding feature of production in NN collisions is the strong e@ect of the N FSI, most visible in the reaction pn → d , where in a range of 10 MeV the amplitude grows by about an order of magnitude [120]. This phenomenon was analyzed by means of Faddeev calculations, showing this enhancement to be consistent with a real part of the N scattering length of 0:4 fm [248]. It is interesting that the same value of the N scattering length was recently extracted from an analyses of the reaction d → pn [260] and that this value is consistent with that stemming from the microscopic model for meson–baryon scattering of Ref. [233]. First three-body calculations for the reaction with a two-nucleon pair in the continuum show only a minor impact of the N interaction on the invariant mass spectra [261]. Most model calculations for the production operator share the property that the production operator was calculated within the meson exchange picture. 36 This was then convoluted with the NN FSI, treated either microscopically or in some approximate fashion. In most analyses it was found that the dominant production mechanism is via the S11 (1535), with the exception of Ref. [249], where the reaction pp → pp turned out to be dominated by isoscalar meson exchanges, in analogy with the heavy meson exchanges in the reaction pp → pp0 . The largest qualitative di@erences between the various models is the relative importance of the exchanged mesons. In Refs. [155,252–255] the D-exchange turned out to be the dominant process, whereas it played a minor role in Refs. [34,256,249]. In Ref. [259] the D-exchange was not even considered, but exchange turned out to be signi6cant there. It remains to be seen, however, if models that are dominated by isoscalar exchanges, Refs. [259,249], are capable of accounting for the large ratio of pn- to pp-induced eta production, as discussed in Section 1.5. As stressed above, in this section we do not want to focus on the details of the dynamics of the production operator, but instead on the physical aspect of what we can learn about the S11 resonance from studying eta meson production in NN collisions. Thus, in the remaining part of this section we will restrict ourselves to a model-independent analysis of a particular set of data based on the amplitude method introduced in Section 4.3. Here we closely follow the reasoning of Ref. [240].

36

The only exception to the list given above is Ref. [257], where the ratio of pp- to pn-induced production is explained by the instanton force. The relation between this result and the hadronic approach is unclear [257].

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dσ/dmpp [arbitrary units]

0.3

0.2

0.1

0 0

5

10

15

mpp [MeV]

Fig. 33. Invariant mass spectrum for the two-proton system for the reaction pp → pp at Q = 15 MeV. The dashed line shows the Ss contribution, the dot–dashed line the Ps distribution and the solid line the incoherent sum of both. The data are taken from Ref. [113].

To be de6nite we will now concentrate on the measurement of angular distributions and invariant mass spectra for pp → pp at Q=15 MeV [113]. The experiment shows that the angular distribution of both the two-proton pair and of the are Nat, suggesting that only s-waves are present. On the other hand, the invariant mass distribution d&=dmpp deviates signi6cantly from what is predicted based on the presence of the strong pp FSI only (cf. discussion of Section 2). This is illustrated in Fig. 33, where the distribution for pp S-wave, s-wave (Ss) is shown as the dashed line. Since the N interaction is known to be strong, its presence appears as the natural explanation for the deviation of the dashed line from the data. The structure, however, is even more pronounced at Q = 42 MeV (cf. Ref. [240]) in contrast to what should be expected for the N interaction. On the other hand, the discrepancy between the data and the dashed curve can be well accounted for by a Ps con6guration (given by phase-space times a factor of p 2 , cf. Section 2.1). This is also illustrated in Fig. 33, where the pure Ps con6guration is shown as the dashed–dotted line and the total result—the incoherent sum of Ps and Ss cross section—as the solid line. How is this compatible with the angular distribution of the two-nucleon system being Nat? This question can be very easily addressed in the amplitude method described in Section 4.3. As long as we restrict ourselves to s-waves and at most P waves in the NN system, the only non-vanishing ˜ containing the amplitude for the Ss 6nal state, and ˜A, containing the two terms in Eq. (48) are Q, possible amplitudes for the Ps 6nal state. The explicit expressions for the amplitudes were given previously in Eqs. (55). In this particular case we 6nd for example (cf. Eqs. (49)–(54)) ˜ 2 + |˜A|2 ) ; &0 = 14 (|Q| A0i &0 = 0 ; ˜ 2 − |˜A|2 ) ; Axx &0 = 14 (|Q|

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where ˜ 2 = |a1 |2 ; |Q| 2 |˜A|2 = p (|a2 − (1=3)a3 |2 + x2 {|a3 |2 − (2=3) Re(a∗3 a2 )})

p · p). ˆ Here the amplitudes a1 , a2 and a3 correspond to the transitions 3 P0 → 1 S0 s, and p x = (˜ 1 3 S0 → P0 s, and 1 D2 → 3 P2 s, respectively. From these equations we directly read that • if a3 is negligible, the pp di@erential cross section d&=d x is Nat; • the observable &0 (1 − Axx ) directly measures the NN P-wave admixture; • any non-zero value for the analysing power is an indication for higher partial waves for the . It is easy to see that for a3 = 0 the di@erential cross section has to be Nat, since then the only partial wave that contributes to the NN P-waves is 1 S0 → 3 P0 s and a J = 0 initial state does not contain any information about the beam direction. Certainly, the N 6nal state interaction has to be present in this reaction channel as well and it is important to understand its role in combination with the two-nucleon continuum state. It should be clear, however, that in order to understand quantitatively the role of the N interaction, it is important to pin down the NN P-wave contribution 6rst. This is why a measurement of Axx for pp → pp is so important. What does it imply if the conjecture of a presence of NN P-waves at rather low excess energies were true? In Ref. [240] it is demonstrated that the need to populate predominantly the 1 S0 → 3 P0 s instead of the 1 D2 → 3 P2 s very strongly constrains the NN → S11 N transition potential. This indicates that a detailed study of NN → NN should reveal information about the S11 in a baryonic environment that might prove valuable in addressing the question of the nature of the lowest negative-parity nucleon resonance. It should be mentioned that in Ref. [262] an alternative explanation for the shape of the invariant mass spectrum was given, namely an energy dependence was introduced to the production operator. Based on the arguments given in the previous chapters, we do not believe that this is a natural explanation. However, as we just outlined, with Axx an observable exists that allows to unambiguously distinguish between the two possible explanations. The experiment is possible at COSY [263]. 7.4. Associated strangeness production or the hyperon–nucleon interaction from production reactions In Section 5 the small breaking of SU (2) isospin symmetry present in the strong interaction was discussed. If we include strange particles in the analysis we can also study the breaking of Navor SU (3). It is well-known that the light mesons and baryons can be arranged according to the irreducible representations of the group SU (3). The mass splittings within the multiplet can be well accounted for by the number of strange quarks in some baryon or meson. However, not much is known about the dynamics of systems that contain strangeness. Many phenomenological models for, e.g., hyperon–nucleon scattering [268–271] use the Navor SU (3) to 6x the meson baryon–meson couplings. The remaining parameters, like the cut-o@ parameters, are then 6t to the data. As we will discuss below, so far the existing data base for hyperon–nucleon scattering is insuOcient to judge, if this procedure is appropriate.

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80

d2σ/dΩ/dmΛp (µb/str/MeV)

500

σ (mb)

400

300

200

100

0

60

40

20

0 0

100

200

pLab (MeV)

300

2060

2080

2100

2120

mΛp (MeV)

Fig. 34. Comparison of the quality of available data for the reactions p elastic scattering (data are from [264–266]) and pp → K + p at TLab = 2:3 GeV [267]. In both panels the red curve corresponds to a best 6t to the data. In the left panel the dashed lines represent the spread in the energy behavior allowed by the data, according to the analysis of Ref. [266]; analogous curves in the right panel would lie almost on top of the solid line and are thus not shown explicitly.

As was stressed previously, e@ective 6eld theories provide the bridge between the hadronic world and QCD. In connection with systems that contain strangeness there are still many open questions. Up to now it is not clear if the kaon is more appropriately treated as heavy or as light particle. In addition, in order to establish the counting rules it is important to know the value of the SU (3) chiral condensate. For a review this very active 6eld of research as well as the relevant references we refer to Ref. [272]. To further improve our understanding of the dynamics of systems that contain strangeness, better data are needed. The insights to be gained are relevant not only for few-body physics, but also for the formation of hypernuclei [273], and might even be relevant to the structure of neutron stars (for a recent discussion on the role of hyperons in the evolution of neutron stars see Ref. [274]). Naturally, the hyperon–nucleon scattering lengths are the quantities of interest in this context. In the left panel of Fig. 34 we show the world data set for elastic N scattering. 37 In Ref. [266] a Likelihood analysis based on the elastic scattering data was performed in order to extract the low-energy N scattering parameters. The resulting contour levels are shown in Fig. 35, clearly demonstrating that the available elastic hyperon–nucleon scattering data do not signi6cantly constrain the scattering lengths: the data allow for values of (−1; 2:3) as well as (6,1) (all in fermi) for (as ; at ), respectively. Later models were used to extrapolate the data. However, also in this way the scattering lengths could not be pinned down accurately. For example, in Ref. [269] one can 6nd six di@erent models that equally well describe the available data but whose (S-wave) scattering lengths range from 0.7 to 2:6 fm in the singlet channel and from 1.7 to 2:15 fm in the triplet channel. 37

The data were taken inclusively (only the kaon was measured in the 6nal state), however, for the small invariant masses shown only the K channel is open.

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3

2

at (fm)

x

1

0

0

4

8 as (fm)

12

16

Fig. 35. Values allowed for the spin singlet and spin triplet scattering length by the N elastic scattering data according to Ref. [266]. The dark shaded area denotes the 1& range for the parameters and the light shaded area the 2& range. The cross shows the best 6t value (as = 1:8 fm and at = 1:6 fm).

Production reactions are therefore a promising alternative. In the literature the reactions K − d →  n [9], d → K + n (Ref. [53] and references therein) and pp → pK + [122] were suggested. Therefore the method described in Section 2.2, that applies to the latter two reactions, is an important step towards a model-independent extraction of the hyperon–nucleon scattering lengths. A natural question that arises is the quality of data needed e.g. for the reaction pp → pK + in order to signi6cantly improve our knowledge about the hyperon–nucleon scattering lengths. In Ref. [32] it is demonstrated that data of the quality of the Saclay experiment for pp → K + X [267], shown in right panel of Fig. 34 38 that had a mass resolution of 2 MeV, allows for an extraction of a scattering length with an experimental uncertainty of only 0:2 fm. Note, however, that the actual value of the scattering length extracted with Eq. (22) from those data is not meaningful, since the analysis presented can be applied only if just a single partial wave contributes to the invariant mass spectrum. The data set shown, however, represents the incoherent sum of the 3 S1 and the 1 S0 hyperon–nucleon 6nal state. The two spin states can be separated using polarization measurements. In Section 4.3.1 as well as in Ref. [32] it was shown, that for the spin singlet 6nal state the angular distributions of various polarization observables are largely constrained. This is suOcient to extract the spin dependence of the -nucleon interaction from the reaction pp → pK + . One more issue is important to stress: to make sure that the structure seen in the invariant mass spectrum truly stems from the 6nal state interaction of interest, a Dalitz plot analysis is necessary, for resonances can well distort the spectra. This is discussed in detail in Section 4.1 as well as at the end of Section 2.2.

38

Shown is only the low m2 tail of the data taken at TLab = 2:3 GeV and an angle of 10◦ .

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K+ Λ=

f0

f0

a0

a0

 K−

K0

Fig. 36. Graphical illustration of the leading contribution to the f0 − a0 mixing matrix element de6ned in Eq. (91).

7.5. Production of scalar mesons or properties of the lightest scalar The lightest scalar resonances a0 (980) and f0 (980) are two well-established states seen in various reactions [146], but their internal structure is still under discussion. Analyses can be found in the literature identifying these structures with conventional qqG states (see Ref. [275] and references therein), compact qq-qGqG states [276,277] or loosely bound K KG molecules [278,279]. In Ref. [280] a close connection between a possible molecule character of the light scalar mesons and chiral symmetry was stressed. It has even been suggested that at masses below 1:0 GeV a complete nonet of 4-quark states might exist [281]. Resolution of the nature of the light scalar resonances is one of the most pressing questions of current hadron physics. First of all we need to understand the multiplet structure of the light scalars in order to identify possible glueball candidates. In addition, it was pointed out recently that a0 (980), f0 (980) as well as the newly discovered Ds [282] might well be close relatives [283]. Thus, resolving the nature of the light scalar mesons allows one simultaneously to draw conclusions about the charmed sector and will also shed light on both the con6ning mechanism in light–light as well as light–heavy systems. Although predicted long ago to be large [284], the phenomenon of a0 − f0 mixing has not yet been established experimentally. In Ref. [284] it was demonstrated that the leading piece of the f0 − a0 mixing amplitude can be written as 39   2 √ pK 0 − pK2 + ; (91)

= f0 |T |a0 = igf0 K KG ga0 K KG s(pK 0 − pK + ) + O s where pK denotes the modulus of the relative momentum of the kaon pair and the e@ective coupling 2 constants are de6ned through MxK KG = gxK p . Obviously, this leading contribution is just that of KG K the unitarity cut of the diagrams shown in Fig. 36 and is therefore model-independent. Note that in Eq. (91) electromagnetic e@ects were neglected, because they are expected to be small [284]. The contribution shown in Eq. (91) is unusually enhanced between the K + K − and the KG 0 K 0 thresholds, a regime of only 8 MeV width. Here it scales as (this formula is for illustration only—the Coulomb interaction contributes with similar strength to the kaon mass di@erence [285])  m2K 0 − m2K + mu − m d ∼ ; 2 2 mˆ + ms mK + + m K 0 where mu , md and ms denote the current quark mass of the up, down and strange quark, respectively, 40 and m=(m ˆ which scale as (mu −md )=(m+m ˆ u +md )=2. This is in contrast to common CSB e@ects s ), 39

Here we deviate from the original notation of Achasov et al. in order to introduce dimensionless coupling constants in line with the standard Flatt\e parameterization. 40 Here we denote as common CSB e@ects those that occur at the Lagrangian level.

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1

|Λ| [arb. units]

0.8 0.6 0.4 0.2

0 970

980

990

1000 s [MeV]

1010

1020

Fig. 37. Modulus of the leading piece of the mixing amplitude de6ned in Eq. (91). The two kinks occur at the K + K − (at 987:35 MeV) and the KG 0 K 0 (995:34 MeV) threshold, respectively.

since they have to be analytic in the quark masses. It is easy to see that away from the kaon thresholds √

returns to a value of natural size. This s dependence of is depicted in Fig. 37. G scattering the mixing of a0 and f0 was studied in Within a microscopic model for  and KK Ref. [286]. Within this model both resonances are of dynamical origin. The only mixing mechanism considered was the meson mass di@erences. Within this model the predictions of Ref. [284] were con6rmed. As was demonstrated by Weinberg for the case of the deuteron [287], the e@ective couplings of resonances to the continuum states contain valuable information about the nature of the particles. In the case of the deuteron this analysis demonstrated that the e@ective coupling of the deuteron to the pn continuum, as can be derived from the scattering length and the e@ective range, shows that the deuteron is a purely composite system. Recently, it was demonstrated that, under certain conditions that apply in the case of a0 and f0 , the analysis can be extended to unstable scalar states as well [288]. Accurate data on the e@ective couplings of the scalars to kaons should therefore provide valuable information about their nature. As was argued in Section 5, the occurrence of scalar mixing dominates the CSB e@ects in production reactions. Therefore quantifying the mixing matrix element might be one of the cleanest ways to measure the e@ective decay constants of a0 and f0 . In Section 5 it was demonstrated that from studies of  production in NN and dd collisions one should be able to extract the a0 − f0 mixing amplitude. Those arguments were supported in Section 4.3.2, where G s-waves. We should it was shown, that the reaction pp → dKG 0 K + is indeed dominated by KK therefore expect a signi6cant signal for the mixing as well. In the years to come we can thus expect the experimental information about the scalar mesons to be drastically improved. 8. Meson production on light nuclei In this section we wish to make a few comments concerning meson production on light nuclei. Note, dd-induced meson production was discussed to some extent in Section 5. A more extensive discussion can be found in Ref. [2].

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3He

N

x

N

x

π

d (a)

N

N

x

N (b)

d

N

d

(c)

Fig. 38. Possible diagrams to contribute to the reaction Nd → 3Nx. In diagram (a) and (b) the three nucleons in the 6nal state from a bound state, whereas diagram (c) shows the dominant diagram in quasi-free kinematics. Here it is assumed that only a particular nucleon pair (to be identi6ed through proper choice of kinematics) interacts in the 6nal state.

8.1. Generalities Almost all general statements made for meson production in NN collisions apply equally well for meson production involving light nuclei. Naturally, now the selection rules are di@erent (see discussion at the end of Section 1.6) and the possible breakup of the nuclei introduces additional thresholds that must be considered in theoretical analyses. A comparison of meson production in two-nucleon collisions and in few-nucleon systems should improve our understanding of few-nucleon dynamics. As a result of a systematic study of, for example, both NN - and pd-induced reactions, a deeper insight into the importance of three-body forces should be gained. For example, a recent microscopic calculation using purely two-body input to calculate the reaction +3 He → ppp, found that the data [289,290] call for three-body correlations [291], con6rming earlier studies [292]. It should be stressed, however, that this 6eld is still in its infancy and a large amount theoretical e@ort is urgently called for. 8.2. 2 → 2 reactions The reaction channel studied best up to date is pd → 3 He x. Data exist mainly from SATURNE for x =  [293], x = ! [294], x = [295,296], as well as x =  and ; [297]. Most of the experiments were done using a polarized deuteron beam. In addition the reactions dd → ,0 [140] 41 and dd → , [298] were measured. The reaction pd → 3 He 0 could be understood quantitatively from diagram (a) of Fig. 38 [299]. However, it turned out that diagram (a) alone largely under predicted the data [300,301] for the reaction pd → 3 He , whereas diagram (b) contributes suOciently strongly to allow description of the data [300,302,303]. In all these approaches the individual amplitudes were taken from data directly; so far no microscopic calculation exists. The relevance of the pion exchange mechanism was explained in Ref. [304] as what the authors called a kinematic miracle: in the near-threshold 41

Note that this reaction is charge symmetry breaking and was discussed in Section 5.

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regime the kinematics for eta production almost exactly matches that for pp → d+ followed by + n → p, with the intermediate pion on-shell. The striking feature of reactions with an -nucleus 6nal state is the pronounced energy dependence that was already described in Section 7.3 for the reaction pn → d —often interpreted as a signal of an existing (quasi)bound state in the -nucleus system. Indeed, since the quark structure of the G is proportional to the number operator, one should expect the -nucleus interaction to ∼ uu G + dd get stronger with increasing number of light quark Navors present in the interaction region. For a more detailed discussion of this issue we refer to Ref. [2]. 8.3. Quasi-free production A deuteron is a loosely bound state of a proton and a neutron. For properly chosen kinematics, the deuteron can therefore be viewed as an e@ective neutron beam/target as alternative to neutron beams (see Ref. [2] and references therein). The corresponding diagram is shown in Fig. 38(c). Obviously, the existence of a three-nucleon bound states already shows that there must be a kinematic regime, where the interaction of all three nucleons in the 6nal state is signi6cant, namely, when all three have small relative momenta. On the other hand, if the spectator nucleon escapes completely una@ected, its momentum distribution should be given by just half the deuteron momentum, convoluted with the deuteron wavefunction times the phase-space factors. Thus, we should expect a momentum distribution for the spectator that shows in addition to a pronounced peak from the quasi-free production a long tail stemming from rescattering in the 6nal state. Experimental data exist for the proton-spectator momentum distribution for the reaction pd G → 3+ 2− p [305]. In Fig. 39 these data are shown together with the results of a calculation [306] that considers both the quasi-free piece as well as a rescattering piece. The data clearly show the quasi-elastic peak. Thus, when considering spectator momenta that are of about 100 MeV or less, to a good approximation, the reaction should be quasi-free. Experimentally, this conjecture can be checked by comparing data for pp induced reactions with those stemming from a pd initial state with a neutron spectator. Those comparisons were carried out at TRIUMF for pion production [74,75] as well as at CELSIUS for production [33,120], showing that the quasi-elastic assumption is a valid approximation. In Section 5 it was argued that a forward–backward asymmetry in the reaction pn → d(0 )s-wave is a good system from which to extract the f0 − a0 mixing matrix element. If investigated at COSY, this reaction can only be studied with a deuteron as e@ective neutron target. Since we are after a CSB e@ect, the expected asymmetry is of the order of a few percent. Thus we need to ensure a priori that the spectator does not introduce an asymmetry of this order through its strong interaction with, say, the deuteron. Theoretically, it would be very demanding to control such a small e@ect in a four-particle system. Fortunately, we can test experimentally to what extent the spectator introduces a forward–backward asymmetry by studying, for example, the reaction pd → + dn. If the reaction were purely quasi-elastic, the angular distribution of the + d system in its rest frame would be forward–backward symmetric, for it would be stemming from a pp initial state. However, any interaction of the spectator with either the pion or the deuteron, should immediately introduce some forward–backward asymmetry. Experimental investigations of this very sensitive test of the spectator approach are currently under way [307].

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events/10 MeV/c

500

400

300

200

100

0

0

0.2

0.4 Ps, GeV/c

0.6

Fig. 39. Momentum distribution of proton-spectators in the reaction pd G → 3+ 2− p. The data are taken from Ref. [305] while the solid line is based on a calculation that includes the quasi-free production as well as meson rescattering. The 6gure is taken from [306], where also details about the corresponding calculation can be found.

9. Summary The physics of meson production in nucleon–nucleon collisions is very rich. The various observables that are nowadays accessible experimentally are, for example, inNuenced by baryon resonances and 6nal state interactions as well as their interference. It is therefore inevitable to use polarized observables to disentangle the many di@erent physics aspects. From the authors personal point of view, the most important issues for the 6eld are: • that NN - and dd induced reactions are very well suited for studies of charge-symmetry breaking [5]. Especially, investigation of the reactions pn → d(0 )s-wave and dd → ,(0 )s-wave should shed light on the nature of the scalar mesons. For the experiments planned in this context we refer to Ref. [308]; • that suOciently strong 6nal state interactions can be extracted from production reactions with large momentum transfer, as, for example, pp → pK + . The condition for an accurate extraction is a measurement with high resolution, as should be possible with the HIRES experiment at COSY [309]; • that an e@ective 6eld theory was developed for large momentum transfer reactions such as NN → NN. Once pushed to suOciently high orders, those studies will not only provide us with a better understanding of the phenomenology of meson production in NN collisions but also allow us to identify the charge symmetry breaking operators that lead to the cross sections reported in Refs. [6,140]; • that resonances can be studied in NN -induced reactions. Those studies are complementary to photon-induced reactions, since the resonances can be excited by additional mechanisms, which

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241

can be selected using, for example, spin 0 particles as spin 6lter [4]. In addition, also the properties of resonances in the presence of another baryon can be systematically investigated. There are very exciting times to come in the near future, for not only does the improvement of experimental apparatus in recent years permit the acquisition of a great deal of high-precision data but also because or theoretical understanding has now also improved to a point that important physics questions can be addressed systematically.

Acknowledgements This article would not have been possible without the strong support, the lively discussions as well as the fruitful collaborations of the author with V. Baru, M. B^uscher, J. Durso, Ch. Elster, A. Gasparyan, J. Haidenbauer, B. Holstein, V. Kleber, O. Krehl, N. Kaiser, S. Krewald, B. Kubis, A.E. Kudryavtsev, U.-G. Mei_ner, K. Nakayama, J. Niskanen, A. Sibirtsev, and J. Speth. Thanks to all of you! I am especially grateful for the numerous editorial remarks by J. Durso.

Appendix A. Kinematical variables In this report we only deal with reactions, in which the energies of the 6nal states are so low that a non-relativistic treatment of the baryons is justi6ed. This greatly simpli6es the kinematics. Thus, in the center-of-mass system we may write for the reaction NN → B1 B2 x,  ˜ 2 + MN2 Etot = 2 p   
2
2 2 2 = M1 + p1 + M2 + p2 + m2x + q 2 p 2 q 2 + + ≈ M1 + M2 + 2412 2(M1 + M2 )



m2x + q 2 ;

(A.1)

where p denotes the momentum of the initial nucleons, p the relative momentum of the 6nal nucleons and q the center-of-mass momentum of the meson of mass mx . The reduced mass of the outgoing two baryon system is denoted by 412 = M1 M2 =(M1 + M2 ). The kinematical variable traditionally used for the total energy of a meson production reaction is , the maximum meson momentum in units of the meson mass. Then one gets from Eq. (A.1): Etot ( ) ≈ M1 + M2 + 2

 m2x + mx 1 + 2 : 2(M1 + M2 )

(A.2)

Another often used variable for the energy of a meson production reaction is Q=

√

s − (M1 + M2 + mx ) :

(A.3)

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C. Hanhart / Physics Reports 397 (2004) 155 – 256

Table 14 The values for TLab , Q and for the di@erent channels for pion production

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.10 1.20 1.30 1.40 1.50

pp → d+

pp → pp0

pp → pn+

TLab

Q

TLab

Q

TLab

Q

287.6 288.0 289.2 291.2 293.9 297.5 301.8 306.8 312.5 318.9 325.9 333.5 341.7 350.5 359.8 369.5 379.8 390.5 401.6 413.1 425.0 449.8 476.0 503.4 531.9 561.4

0.00 0.19 0.75 1.68 2.97 4.62 6.61 8.94 11.58 14.53 17.77 21.29 25.06 29.09 33.34 37.81 42.49 47.36 52.40 57.62 63.00 74.19 85.91 98.10 110.71 123.69

279.6 280.0 281.2 283.1 285.8 289.2 293.3 298.1 303.6 309.8 316.5 323.9 331.8 340.2 349.1 358.5 368.4 378.6 389.3 400.4 411.8 435.7 460.9 487.1 514.5 542.8

0.00 0.18 0.72 1.62 2.87 4.46 6.38 8.62 11.17 14.02 17.14 20.53 24.18 28.05 32.16 36.47 40.98 45.67 50.54 55.57 60.75 71.54 82.83 94.57 106.72 119.23

292.3 292.7 293.9 295.9 298.7 302.3 306.6 311.6 317.3 323.7 330.7 338.3 346.6 355.3 364.6 374.4 384.6 395.3 406.5 418.0 429.9 454.8 481.0 508.4 536.9 566.4

0.00 0.19 0.75 1.68 2.97 4.62 6.61 8.94 11.58 14.53 17.77 21.29 25.06 29.08 33.34 37.81 42.48 47.35 52.40 57.62 62.99 74.18 85.90 98.09 110.69 123.68

It is straightforward to express in terms of Q: (s − Mf2 − m2x )2 − 4(mx Mf )2 1 = 2mx s    4 1  Q 1−3  : 24Q 1 + mx 44 Mf + m x

(A.4)

where Mf = M1 + M2 and the reduced mass of the full system is given by 4 = Mf mx =(Mf + mx ). For the latter approximation we used that in the close-to-threshold regime Q
(Mf + mx ). Traditionally, is used for the pion production only, whereas Q is used for the production of all heavier mesons. To simplify the comparison of results for those di@erent reaction channels, we present in Table 14 the various values for , Q as well as TLab for the di@erent pion production channels.

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243

For completeness we also give the relation to the Laboratory variables TLab =

s − 4M 2 ; 2M

2 2 = TLab + 2MTLab : pLab

(A.5) (A.6)

Appendix B. Collection of useful formulas B.1. De9nition of coordinate system In this appendix we give the explicit expressions for the vectors relevant for the description of a general 2 → 3 reaction. We work in the coordinate system, where the z-axis is given by the beam momentum p ˜ . These formulas are particularly useful for Section 4.3. Thus we have       0 sin(>p
) cos(;p
) sin(>q
) cos(;q
)            p ˜ = p ˜  = p  0; p  sin(>p
) sin(;p
)  ; ˜q = q  sin(>q
) sin(;q
)  : 1 cos(>p
) cos(>q
) From these one easily derives   −sin(>p
) sin(;p
)  
) cos(;p
)  ; sin(> i(˜ p×p ˜  ) = ipp  p   0   cos(>q
) sin(>p
) sin(;p
) − cos(>p
) sin(>q
) sin(;q
)    i(˜ p × ˜q  ) = ip q   −cos(>q
) sin(>p
) cos(;p
) + cos(>p
) sin(>q
) cos(;q
)  ; −sin(>p
) sin(>q
) sin(;q
− ;p
)

(B.1)

as well as the analogous expression for i(˜ p × ˜q  ). B.2. Spin traces In this appendix some relations are given, that are useful to evaluate expression that arise from in the amplitude method appearing described in Section 4.3. &y Hi∗ HiT &y = 12 &y (1 + ˜P i · ˜&T )&y = 12 (1 − ˜P i · ˜&) :

(B.2)

The sum over the spins of the external particles leads to traces in spin space, such as tr(&i ) = 0 ;

(B.3)

tr(&i &j ) = 23ij ;

(B.4)

tr(&i &K &j ) = 2ijiKj ;

(B.5)

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tr(&, &i &K &j ) = 2(3i, 3jK + 3j, 3iK − 3ij 3,K ) :

(B.6)

To identify dependent structures the following reduction formula is useful: ˜a 2˜b · (˜c × ˜d) = (˜a · ˜b)˜a · (˜c × ˜d) + (˜a · ˜d)˜a · (˜b × ˜c) + (˜a · ˜c)˜a · (˜d × ˜b) :

(B.7)

To recall the sign or, better, the order of the vectors appearing, observe that on the right-hand side the vectors other than ˜a are rotated in cyclic order. To prove Eq. (B.7) we use 3ij jklm − 3im jklj = jkl (3ij 3m − 3im 3j ) = jkl ji, jjm, = jjm, (3k, 3li − 3ki 3l, ) = 3il jjmk − 3ki jjml : Appendix C. Partial wave expansion In this appendix we give the explicit relations between the partial wave amplitudes for reactions of the type NN → NNx and the spherical tensors de6ned in Eq. (34), where x is a scalar particle. The relations between the spherical tensors and the various observables is given in Tables 4 and 5. In terms of the partial wave amplitudes, we can write for two spin- 12 particles in the initial state 1       S MS ; p ˜ ;˜q |M |SMS ; p ˜ SG MG S ; p ˜ |M † |SG MG S ; p ˜  ;˜q  Tkk13qq13;k; k24qq24 = 16    1 )† (f2 )† ×SMS |%k(b) %(t) |SG MG S SG MG S |%(f k3 q3 %k4 q4 |S MS 1 q1 k 2 q2 1  (2LG + 1)(2L + 1) = 4 (2JG + 1)(2J + 1)

×S  MS ; L ML |j  Mj j  Mj ; l ml |JMJ SMS ; L0|JMJ 

G JG MG J ×S  MS ; LG MG L |jG  MG j jG  MG j ; lG  mG l |JG MG J SG MG S ; L0| 

  1 )† (f2 )† ×SMS |%k(b) %(t) |SG MG S SG MG S |%(f k3 q3 %k4 q4 |S MS 1 q1 k 2 q2

×Yl
m
l (qˆ )YL
ML
(pˆ  )YlG
mG
(qˆ )∗ YLG
MG
L (pˆ  )∗ M , (s; j)M ,G(s; j)† : l

(C.1)

In order to proceed the following identities are useful [310]:  (2l1 + 1)(2l2 + 1) ˆ l2 m2 (p) ˆ = Yl1 m1 (p)Y (2l + 1)4 lm

ˆ ×l1 m1 ; l2 m2 |lm l1 0; l2 0|l0 Ylm (p) √ % $ &|%kq |& = (−)q 2k + 1 12 &; k (−q)| 12 & :

(C.2) (C.3)

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The latter, for instance, allows evaluation of the matrix element of the spin operators: %(t) |SG MG S SMS |%k(b) 1 q1 k 2 q2  G G = SMS |%k(b) |m1 m2 m1 m2 |%(t) k2 q2 |S M S 1 q1 $ %$ % = SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) & '& ' 1 G × 1 (MS − m2 )|%k(b)q |m1 m2 |%(t) | ( M − m ) S 1 k q 2

=

1 1

2 2

2



 (−)q1 +q2 (2k1 + 1)(2k2 + 1) $ %$ % × SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) $ %$ % × 12 (MS − m2 ); k1 (−q1 )| 12 m1 12 m2 ; k2 (−q2 )| 12 (MG S − m1 ) :

(C.4)

It is convenient to couple the remaining spherical harmonics to a common angular momentum and to de6ne  1 YL˜M˜ L (p)Y ˆ l˜m˜ l (q)= ˆ : L˜ M˜ L ; l˜m˜ l |:Q BLQ˜l;˜ : (q; ˆ p) ˆ ; (C.5) 4 :

where we used the fact that the sum of the projections turns out to be equal to q1 + q2 = Q; B is then  1 ˜ l |:Q YL4 ˜ L ; l4 ˆ p) ˆ = ˆ l4 ˆ (C.6) BLQ˜l;˜ : (q; L4 ˜ L (p)Y ˜ l (q) 4 4 ;4 L

l

and normalized such that  ˆ p) ˆ = 3:0 3L0 d`p d`q BLQ˜l;˜ : (q; ˜ 3l0 ˜ 3Q0 ˜ :

(C.7)

Some properties of B are derived in the next section. After putting together the individual pieces we arrive at the 6nal result: 1 Q ˆ q) ˆ = BL˜l;˜ : (q; ˆ p)A ˆ LD˜l;˜ : ; TD (p; 4 ˜ L˜l:

where D = {k1 q1 ; k2 q2 ; k3 q3 ; k4 q4 }, and  ,; ,;G D CL˜l;˜ : M , (M ,G)† ALD˜l;˜ : = ,;,G

with CL,;˜l;˜,;G:D =

1 
(−)MS +MS 7S  MS ; L ML |j  Mj j  Mj ; l ml |JMJ SMS ; L0|JMJ 4 



G JG MG J ×SG MG S ; LG MG L |jG  MG j jG  MG j ; lG  mG l |JG MG J SG MG S ; L0| 

˜ L˜ M˜ ; l˜m|:Q ˜ ×L ML ; LG − MG L |L˜ M˜ l ml ; lG  − mG l |l˜m 

(C.8)

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%$ $ %   G ˜ ˜ ×L 0; LG 0|L0 l 0; l 0|l0 SMS | 12 (MS − m2 ); 12 m2 SG MG S | 12 m1 ; 12 (MG S − m1 ) $ %$ % × 12 (m1 + q1 ); k1 (−q1 )| 12 m1 12 m2 ; k2 (−q2 )| 12 (MG S − m1 ) $ %$ % × 12 (m1 − q3 ); k3 q3 | 12 m1 12 m2 ; k4 ; q4 | 12 (MG S − m1 ) ;

(C.9)

where the sum runs over {MS ; MS ; ML ; MG L ; m1 ; m1 } and  (2l + 1)(2lG  + 1)(2L + 1)(2LG + 1)(2LG + 1)(2L + 1)(2k1 + 1)(2k2 + 1) 7= : (2L˜ + 1)(2l˜ + 1)(2J + 1)(2JG + 1) Appendix D. On the non-factorization of a strong nal state interaction In this appendix we will demonstrate the need for a consistent treatment of both the NN scattering and production amplitudes in order to obtain quantitative predictions of meson-production reactions. Let us assume a separable NN potential V (p ; k) = ,g(p )g(k) ;

(D.1)

where , is a coupling constant and g(p) an arbitrary real function of p. With this potential the T -matrix scattering equation can be readily solved to yield T (p ; k) =

V (p ; k) 1 − R(p ) + i7(p )V (p ; p )

with 

R(p ) ≡ mP

 0

∞

d k

k  2 V (k  ; k  ) p 2 − k  2

(D.2)

(D.3)

and 7(p) = p4 denotes the phase-space density here expressed in terms of the reduced mass of the outgoing two-nucleon pair 4 = mN =2. Note that for an arbitrary function g(k), such as g(k) ≡ 1 as discussed below, R(p ) may be divergent. In this case R is to be understood as properly regularized. The principal value integral R(p ) given above is therefore a model-dependent quantity, for it depends on the regularization scheme used. The condition that the on-shell NN scattering amplitude should satisfy Eq. (6) relates this to the on-shell potential, V (p ; p ): R(p ) = 1 + 7(p ) cot(3(p ))V (p ; p ) ;

(D.4)

where it is assumed that (p ) = 1. This shows that, for a given potential, the regularization should be such that Eq. (D.4) be satis6ed in order to reproduce Eq. (6). Indeed, in conventional calculations based on meson exchange models, where one introduces form factors to regularize the principal value integral, the cuto@ parameters in these form factors are adjusted to reproduce the NN scattering phase shifts through Eq. (6). Conversely, for a given regularization scheme, the NN potential should be adjusted such as to obey Eq. (D.4). This is the procedure used in e@ective 6eld theories [311], where the coupling constants in the NN potential are dependent on the regularization.

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We also assume that the production amplitude M is given by a separable form, M (E; k) = Kg(k)h(p) ;

(D.5)

where K is a coupling constant and h(p) an arbitrary function of the relative momentum p of the two nucleons in the initial state. With this we can express the total transition amplitude as   1 i3(p
) M (E; p )   e : (D.6) sin(3(p )) A(E; p ) = − 7(p ) V (p ; p ) Eq. (D.6) is the desired formula for our discussion. It allows us to study the relationship between the NN potential and the production amplitude M (E; p ) explicitly as di@erent regularization schemes are used. For this purpose let us study the simplest case of a contact NN potential (setting the function g=1) in the limit p → 0. If we regularize the integrals by means of the power divergent subtraction (PDS) scheme [311] we get a4 R=− ; 1 − a4 where 4 denotes the regularization scale. Substituting this result into Eq. (D.4), we obtain   1 2a ,= m 1 − a4 for the NN coupling strength. Note that for 4 = 0 the PDS scheme reduces to that of minimal subtraction [311]. Since the total production amplitude A should not depend on the regularization scale we immediately read o@ Eq. (D.6) that K ˙ (1 − a4)−1 : Therefore the model clearly exhibits the point made in Section 2: Namely, the necessity of calculating both the production amplitude and the FSI consistently in order to allow for quantitative predictions. Appendix E. Chiral counting for pedestrians In this appendix we demonstrate how to estimate the size of a particular loop integral. This is a necessary step in identifying the chiral order of a diagram. It should be clear, however, that the same methods can be used to estimate the size of any integral. However, the importance of the chiral symmetry is that it ensures the existence of an ordering scheme that suppresses higher loops. The necessary input are the expressions for the vertices and propagators at any given order. For the chiral perturbation theory those can be found in Ref. [197]. In addition we need an estimate for the measure of the integral. Once each piece of a diagram is expressed in terms of the typical momenta/energies, one gets an estimate of the value of the particular diagram. The procedure works within both time-ordered perturbation theory and covariant theory. Obviously, for each irreducible diagram both methods have to give the same answer. If a diagram has a pure two-nucleon intermediate state, as is the case for the direct production, the covariant counting can only give the leading order piece of the counting within TOPT. In this appendix we study only diagrams that are three-particle irreducible; i.e., the topology of the diagram does not allow an intermediate two-nucleon state to go on-shell. The reducible diagrams

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require a di@erent treatment and are discussed in detail in the main text. There is another group of diagrams mentioned in the main text that is not covered by the counting rules presented, namely radiation pions. Those occur if a pion in an intermediate state goes on-shell. It is argued in the main text that these are suppressed, because the pion, in order to go on-shell, is only allowed to carry momenta of the order of the external momenta, and thus the momentum scale within the loop is of order of the pion mass and not of the order of the initial momentum. Therefore we do not consider radiation pions any further. E.1. Counting within TOPT As mentioned above, if we want to assign a chiral order to a diagram, all we need to do is to replace each piece in the complete expression for the evaluation of the matrix element by its value when all momenta are of their typical size. In case of meson production in nucleon–nucleon collisions this typical momentum is given by the initial momentum pi . Time-ordered perturbation theory contains only three-dimensional integrals and thus we do not need to 6x the energy scale in the integral. The counting rules are • the energy of virtual pions is interpreted as O(pi ) 42 and thus; ◦ every time slice that contains a virtual pion is interpreted as 1=pi (see Fig. 40), ◦ for each virtual pion line put an additional 1=pi (from the vertex factors), • interpret the momenta in the vertices as pi , • every time slice that contains no virtual pion is interpreted as 1=m ; most of these diagrams, however, are reducible (cf. main text); • the integral measure is taken as pi3 =(4)2 . Here we used that pi2 =MN  m
pi , in accordance with Eqs. (80), and thus nucleons can be treated as static in the propagators if there is an additional pion present. However if there is a time slice that contains two nucleons only, the corresponding propagator needs to be identi6ed with the inverse of the typical nucleon energy 1=m and the static approximation is very bad [155]. In the diagram of Fig. 40 three NN vertices, each  pi =f , appear as well as the NN Weinberg–Tomozawa vertex  pi =f2 . In addition the three time slices give a factor 1=pi3 and we also need to include a factor 1=pi2 , since there are two virtual pions. We therefore 6nd      3 pi 1 5 pi3 m pi 1 M TOPT  :  f f2 pi (4)2 f3 MN √ Here we used 4f  MN and pi  MN m . E.2. Counting within the covariant scheme Naturally, as TOPT and the covariant scheme are equivalent, the chiral order that is to be assigned to some diagram needs to be the same in both schemes. The reason why we demonstrate both is that 42

There is one exception to this rule: if a time derivative acts on a pion on a vertex, where all other particles are on-shell, then energy conservation 6xes the energy.

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Fig. 40. A typical loop that contributes to pion production in nucleon–nucleon collisions. Solid lines denote nucleons and dashed lines pions. The solid dots show the points of interactions. The horizontal dashed lines indicate equal time slices, as needed for the evaluation of the diagram in time-ordered perturbation theory.

here we are faced with a problem in which the typical energy scale m and the typical momentum scale pi are di@erent. In the covariant approach a four-dimensional integral measure enters and naturally the question arises whether m or pi is appropriate for the zeroth component of this measure p0 . For example, in Ref. [163] it was argued, that one should choose m , although this choice is by no means obvious from the structure of the integrals. However, given the experience we now have in dealing with loops in time-ordered perturbation theory, where these ambiguities do not occur, the answer is simple: we just have to assign that scale to p0 that will reproduce the same order for any diagram as in the counting within TOPT [204]. Once the choice for p0 is 6xed, the much easier to use covariant counting can be used to estimate the size of any loop integral. Thus have the following rules: • the energy of virtual pions is interpreted as O(pi ); 43 • each pion propagator is taken as O(1=pi2 ), • each nucleon propagator that cannot occur in a two-nucleon cut is taken as O(1=p0 ) (the leading contribution of a nucleon propagator that can occur in a two-nucleon cut is O(1=m ); most of these diagrams, however, are reducible (cf. main text)), • interpret the momenta in the vertices as pi , • the integral measure is taken as p0 pi3 =(4)2 (when the diagram allows for a two-nucleon cut the measure reads (m pi3 )=(4)2 ). 43

With the same exception as in the time-ordered situation (cf. corresponding footnote).

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Thus we have for the diagram of Fig. 40  2  2     3 1 1 m pi pi p0 pi3 pi 1 cov : M   3 2 2 2 f f pi p0 (4) f MN p0 Thus, we need to assign p0 ∼ pi in order to get the same result in both schemes. As a side result we also showed, that the nucleons are indeed static in leading order inside loops that do not have a two-nucleon cut, as pointed out in Ref. [206]. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

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Physics Reports 397 (2004) 257 – 358 www.elsevier.com/locate/physrep

Four sorts of meson D.V. Bugg∗ Queen Mary, University of London, London E1 4NS, UK Accepted 8 March 2004 editor: J.V. Allaby

Abstract An extensive spectrum of light non-strange qq/ states up to a mass of 2400 MeV has emerged from Crystal Barrel and PS172 data on pp / → Resonance → A + B in 17 4nal states. These data are reviewed with detailed comments on the status of each resonance. For I = 0, C = +1, the spectrum is complete and very secure. Six ‘extra’ states are identi4ed. Four of them have I = 0, C = +1 and spin-parities predicted for glueballs. Their mass ratios agree closely with predictions from lattice QCD calculations. However, branching ratios for decays are not
Contents 1. Introduction and survey of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Crystal Barrel experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Rate dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Monte Carlo and data selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. How to establish resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Star ratings of resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. Waters Edge, Honington, Warwicks CV36 5AA, UK. E-mail address: [email protected] (D.V. Bugg).

c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.03.008

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4. The partial wave analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Formulae for amplitudes and resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Illustration of spectroscopic notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. General comments on procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Review of I = 0, C = +1 resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Two-body channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Three-body data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. 2− states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Star ratings of states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. f0 (1710) and f0 (1770) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. SU (3) relations for two-body channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. I = 1, C = +1 resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Two-body data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. pp / → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. pp / → 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Evidence for 2 (1880) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. I = 1, C = −1 resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Vector polarisation of the ! and its implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. !0 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Observed resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. I = 0, C = −1 resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Comments on systematics of qq/ resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Slopes of trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Glueballs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. f0 (1500) and the qq/ nonet of f0 (1370); a0 (1450); K0 (1430) and f0 (1710)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Mixing of the glueball with qq/ states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Features of central production data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.  →  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Evidence for the 0− glueball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6. Sum rules for J= radiative decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. The light scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1. The  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. The  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3. Theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4. f0 (980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5. a0 (980) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.  →  and  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7. Ds (2317) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

269 270 271 271 273 273 276 278 283 287 291 292 294 296 300 302 304 306 307 308 313 321 321 324 325 328 329 333 333 337 338 338 342 346 348 348 348 351 353 353 354

1. Introduction and survey of results Many new resonances have appeared in Crystal Barrel and PS172 data on pp / annihilation in
D.V. Bugg / Physics Reports 397 (2004) 257 – 358

259

Table 1 Channels studied; decays  →  and  → 30 are both used, and  → 0 0 ; also ! → 0  and ! → + − 0 I = 0, C = +1

I = 1, C = +1

I = 0, C = −1

I = 1, C = −1

− + , 0 0 , ,  0 0 ,  0 0 ,  0 0

0 ,  0 30 , 0 0 0 0

! !0 0

− + , !0 !0

overlapping information. Results from 35 journal publications in collaboration with the St. Petersburg group are summarised. The main idea is to study s-channel resonances: pp / → Resonance → A + B ; by full partial wave analysis of 17 4nal states A + B listed in Table 1. Coupled channel analyses have been carried out, following the classic techniques applied to the N system since the 1960s, imposing the fundamental constraints of analyticity and unitarity. The majority of 4nal states are made up only of neutral particles. The Crystal Barrel has excellent  detection in a barrel of 1380 CsI crystals with good energy and angular resolution (see Section 1). Crystals cover 98% of the solid angle. Detection of charged particles is limited to 71% of the solid angle. Neutral 4nal states fall cleanly into four categories of isospin I and charge conjugation C, as shown in Table 1; − + contains both I = 0 and 1. The largest number of decay channels is available for I = 0, C = +1. For the − + 4nal state, d=d and polarisation data are included from earlier experiments [1,2]. The polarisation data play a vital role in identifying helicities and hence separating pp / 3 F2 states from 3 P2 . There is also a second point: d (1) ˙ Im(F1∗ F2 ) ; d where F1 and F2 are helicity combinations discussed later. DiOerential cross sections depend on moduli squared of amplitudes and real parts of interferences. The combination of diOerential cross sections and polarisations determines both magnitudes and phases of amplitudes uniquely. In any one data set, signi4cance levels of resonances are similar to those of N analyses above 1700 MeV; for I = 0, C = +1, the analysis of 8 channels with consistent parameters leads to extremely secure amplitudes and a complete set of the expected qq/ states in the mass range 1910–2410 MeV, each with a statistical signi4cance ¿ 25. The spectrum shown in Fig. 1 [3] is remarkably simple. Almost within errors, observed states lie on parallel trajectories of M 2 (where M is mass) vs. radial excitation number n. The mean slope is 1:14 ± 0:013 GeV2 per unit of n. These trajectories resemble the well-known Regge trajectories; however on Regge trajectories, states alternate in I and C (for example %1 , f2 , %3 , f4 , %5 ). Similar spectra are observed for other I and C, but are less complete and less precise. Fig. 2 shows the spectrum for I = 1, C = +1 [4,5]. Since u and d quarks are nearly massless, one expects results to be almost degenerate with I = 0, C = +1. This is indeed the case. However, the separation between 3 F2 and 3 P2 is much poorer for I = 1, C = +1 because no polarisation data are available. P

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(a)

(b)

(c)

(d)

Fig. 1. (a)–(d) Trajectories of I = 0, C = +1 states. Numbers indicate masses in MeV.

(a)

(b)

(c)

(d)

Fig. 2. (a)–(d) Trajectories of I = 1, C = +1 states.

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(b)

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Fig. 3. Trajectories of C = −1 states with (a) and (b) I = 0, (c) and (d) I = 1. In (b), the 3 D3 trajectory is moved one place right for clarity; in (d), 3 D2 is moved one place left.

For I = 1, C = −1, the observed spectrum is shown in Figs. 3(c) and (d) [6,7]. Here polarisation data for pp / → − + are available and do help enormously in separating magnitudes and phases of amplitudes. As a result, the observed spectrum is complete except for one high-lying J PC = 1−− state and the 3 G3 pp / state. The 3 D3 states are particularly well identi4ed. Figs. 3(a) and (b) show the spectrum of I = 0, C = −1 states [7]. Errors are the largest here. Statistics for the ! channel are typically a factor 6 weaker than for !0 . The second available channel is !0 0 and is diPcult to analyse because of broad 0+ structure in 0 0 . Several expected states are missing for 3 S1 . Nonetheless, those states which are observed are again almost degenerate with I = 1, C = −1. All observed resonances cluster into fairly narrow mass ranges (i) 1590–1700 MeV, (ii) 1930–2100 MeV, (iii) 2240–2340 MeV. These ‘towers’ of resonances are illustrated in Fig. 4. Sections 2–9 review qq/ states in detail. There are four ‘extra’ states with I = 0, C = +1 having masses close to lattice QCD predictions for glueballs and the same J P . Section 10 reviews these glueball candidates and the mixing of f0 (1500) with the nearby nonet made of f0 (1370), a0 (1450), K0 (1430) and f0 (1710). Section 11 reviews the light scalar mesons: (540), (750), f0 (980) and a0 (980). Section 12 take a brief look at what is needed in future. 2. The Crystal Barrel experiment This experiment ran at LEAR, the p/ storage ring at CERN, using nine p/ beam momenta of 600, 900, 1050, 1200, 1350, 1525, 1642, 1800 and 1940 MeV=c. The corresponding pp / mass range is

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Fig. 4. Clustering of resonances in mass.

Fig. 5. Overall layout of the Crystal Barrel detector, showing (1) magnet yoke, (2) magnet coils, (3) CsI barrel, (4) jet drift chamber, (5) multi-wire proportional chambers, (6) liquid hydrogen target.

from 1962 to 2409 MeV. Most of the data were taken during the last 4 months of LEAR operation, August–December 1996. Data from the PS172 experiment with a polarised target will also be used. This experiment ran during 1986 from 360 to 1550 MeV=c; the extension of the mass range down to 1910 MeV is important. The Crystal Barrel detector is illustrated in Fig. 5. Details are to be found in a full technical report [8]. A 4:4 cm liquid hydrogen target was surrounded sequentially by multiwire chambers for triggering (labelled 5 on the 4gure), a jet drift chamber (4) for detection of charged particles, a barrel (3) of 1380 CsI crystals covering 98% of 4 solid angle, and a magnet (1) producing a solenoidal 4eld of 1:5 T (though this was run at 1:0 T for all-neutral data). For running after 1994, the multiwire chambers were replaced by a silicon vertex detector. However, for detection of neutral 4nal states, the chambers and silicon vertex detector acted simply as vetos over a solid angle 98% of 4.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

263

Fig. 6. Distributions of (a) M () near the  from events 4tting , after cuts; (b) M (30 ) near the  from 30 events; (c) M () near the  from  data; (d) M (0 0 ) near the  from 0 0 events. Data are at 1800 MeV=c.

The CsI crystals measured photon showers with an angular resolution of 20 mrad in both polar and azimuthal angles. The photon detection ePciency was high down to energies below 20 MeV. The energy resolution was given by SE=E = 0:025=E 1=4 with E in GeV. In order to reject events which failed to conserve energy, the total energy deposited in the CsI crystals was derived on-line; the trigger selected events having a total energy within 200 MeV of that in pp / interactions [9]. Incident p/ were detected by a fast coincidence between a small proportional counter P and a Si counter SiC of radius 5 mm at the end of the beam-pipe and only a few cm from the entrance of the liquid hydrogen target. Four further silicon detectors located at the same point and segmented into quadrants were used to steer and focus the beam. Two veto counters were positioned 20 cm downstream of the target and inside the CsI barrel; they provided a trigger for interactions in the target. The data-taking rate was ∼ 60 events=s at a beam intensity of 2 × 105 p=s. / One gets a general impression of the cleanliness with which neutral particles were detected from Fig. 6. Panels (a) and (b) show  peaks in the  4nal state, from 2 events in (a) and from 30 events in (b) [10]. As an illustration of selection procedures, consider pp / → . Cuts were applied on con4dence levels (CL) as follows: (i) CL(0 ) ¡ 0:1% to remove 0 0 , (ii) CL(0 ) ¡ 0:01%, (iii) CL() ¿ CL( ), (iv) CL() ¿ 0:1. In the few cases where there was more than one  solution, selected events were required to have a CL at least a factor 10 better than the second. The entire process was followed through a Monte Carlo simulation discussed below. Observed backgrounds agreed closely with this simulation, using branching ratios of the background channels. The  peaks in both (a) and (b) were centred (purely from kinematics of the production reaction) within 1–2 MeV of the Particle Data Group value (PDG) [11] at all beam momenta; this checks systematics

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Fig. 7. Comparison of angular distributions for  from 4 events (black circles) and 8 (open triangles). Full curves show the result of the amplitude analysis. Dotted curves illustrate the acceptance for three momenta; small undulations are due to crystal sizes and overlap of showers between crystals, simulated by the Monte Carlo.

of the energy calibration. Fig. 7 shows angular distributions derived from 4 and 8 events, independently normalised to Monte Carlo simulations. The close agreement in both shapes and absolute magnitudes is a major cross-check. Figs. 6(c) and (d) show  peaks from 2 and 0 0 decays. Statistics are lower than for , but angular distributions and normalisations agree between 4 and 8 for pp / →   within statistics [10], and also for pp / →  0 [12]. Showers were selected from adjoining groups of nine CsI crystals. The relation between shower energy and pulse-height in crystals was obtained [8] by 4tting the 0 mass for  events, using all crystal combinations. Three variants of this calibration were tried. The primary eOects of different calibrations are removed by the overall kinematic 4t to energy-momentum conservation; this uses the beam momentum, which is known to better than 0.1%. The result is that diOerent calibrations altered 6 1% of selected events. However, the performance was better than that. Between one calibration and another, the overall change in accepted events was a factor 10 smaller; the few events which were lost were compensated by an almost equal number entering the sample. A more important consideration was the identi4cation of crystals which malfunctioned. For most running conditions, there were at most 1 or 2 crystals which failed in some way (noisy or dead). These were identi4ed and cut completely from both the data and the Monte Carlo simulation. Otherwise a faulty crystal could lead to a dip in the angular distribution over the angles covered by 9 adjoining crystals.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

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Fig. 8. Rate dependence observed at (a) 1800 MeV=c, (b) 1525 MeV=c, (c) 1350 MeV=c, and (d) 1050 MeV=c for the sum of (4 + 5 + 6 + 7 + 8) events. Lines show the 4tted rate dependence.

2.1. Rate dependence We now consider the absolute normalisation of cross sections. This is important in an amplitude analysis covering many momenta. A potentially serious rate dependence was found, but ultimately aOected normalisations by only ∼ 3–6%, varying slowly with beam momentum. It arose from pile-up in the CsI crystals. Beam intensities were 2×105 s−1 and interaction rates in the target were typically 6 × 103 s−1 . The scintillation from CsI has two components, with a risetime of 6
s, but a fall-time of 100
s. The discriminators used pole-zero cancellation which was supposed to eliminate the long component. But in practice large energy deposits in one crystal could produce some ringing in its discriminator output. This led to signi4cant pile-up between one event and the next. A detailed discussion is to be found in Ref. [10]. The normalisation was derived from beam counts P SiC , the length and density of the target, the observed number of events and the Monte Carlo simulation of detection ePciency. The general oO-line procedure was to reconstruct 45 exclusive 4nal states, each one kinematically 4tted to channels with 4–10. Rather few have 9 or 10. Fig. 8 shows as a function of beam rate the total of events 4tting with con4dence levels ¿10% to any channel with 4–8. Typically, ∼ 20% of triggers 4t such channels. There is obviously a strong rate dependence and it was necessary to extrapolate cross sections to zero beam rate. The rate dependence varied with beam steering. At several momenta, separate running conditions gave diOerent slopes, but extrapolated values were consistent within 3%.

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(a)

(b)

(c)

(d)

Fig. 9. Fractions of (a) 4, (b) 6, (c) 8 events and (d) 0 0 events, out of the total of 4nal (4 + 5 + 6 + 7 + 8) events at 1800 MeV=c vs. incident p/ rate.

Fortunately, the rate dependence is found to kill genuine events of all types in the same way. Let N (n) be the number of events 4tting above 10% con4dence level with n photons, and let N (4 − 8) be the sum of these from n = 4 to 8. Fig. 9 displays ratios N (n)=N (4 − 8) for 4, 6, 8 and for 0 0 as a function of beam rate. Any variation with rate is ¡ 3%. Neither is there any observed change with beam rate in 0 0 angular distributions or Dalitz plots of 6 events [13]. The conclusion is that angular distributions can be extracted safely at any beam rate; but the normalisation must be obtained by extrapolating to zero beam rate. A cross-check is also satis4ed between 0 0 data from Crystal Barrel and symmetrised − + diOerential cross sections from PS 172, see below. 2.2. Monte Carlo and data selection Data are 4tted kinematically to a large number of physics channels: 45 for (4–10). In order to assess branching ratios to every channel and cross-talk between them, 20–500 K Monte Carlo events are generated for every one of the 45 4tted channels, using GEANT. In the 4rst approximation, events 4tting the correct channel determine the reconstruction ePciency ji in that channel. Events 4tting the wrong channel measure the probability of cross-talk xij between channels i and j. More exactly, a set of 45 × 45 simultaneous equations is solved containing on the left-hand side the observed number of 4tted data events Di , and on the right-hand side reconstruction ePciencies ji and true numbers of created events Ni in every channel, plus terms allowing for cross-talk xij between channels:
xij Nj : (2) D i = j i Ni + j =i

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The solution is constrained so that the numbers of real events Ni in every channel are positive or zero. This procedure is carried out for several con4dence levels (1%, 5%, 10%, 20%) and using a wide variety of selection procedures. A choice is then made, optimising the ratio of signal to background. It turns out that this ratio is not very sensitive to con4dence level over the range 5–20%. Hence selection criteria are not usually critical. 3. How to establish resonances One objective of this review is to assign a star rating to all resonances. This is to some extent subjective, but some assessment is better than none, as a guide to further work. Before coming to details, it is necessary to review the general approach to identifying resonances. It is sometimes argued that magnitudes and phases of all amplitudes should be 4tted freely, so as to demonstrate that phases follow the variation expected for resonances. Though that is a 4ne ideal, it is unrealistic and leads to problems. Phases can only be determined with respect to some known amplitude. As examples, phase variation of the Z0 and the J= may be found from interferences with electroweak amplitudes. In + p scattering at low energies, the 0(1232) is established as a resonance by the fact that its cross section reaches the unitarity limit; interferences with S-waves then establish its resonant phase variation, since there is not enough cross section for S-waves to resonate too. Further examples are rather rare. The usual assumption is that isolated, well-de4ned peaks far from thresholds are resonances. The 2 (1670) and f4 (2050) are examples. The phases of other amplitudes may be determined using these and other established resonances as interferometers. This is the procedure followed in Crystal Barrel analyses. The f4 (2050) appears strongly in data for I = 0, C = +1. It then acts as an interferometer and determines phases of other partial waves. They have a similar phase variation with s and must therefore also resonate. In pp / → − + , diOerential cross sections establish real parts of interferences and polarisation data measure the imaginary parts of the same interferences. Together, these data establish quite precisely the phase variation of all partial waves with respect to f4 (2050). Further strong peaks appear as f4 (2300) and 2 (2267). They act as further interferometers, loosely coupled themselves in phase to the tail of f4 (2050). A strong %3 (1982) appears in pp / → − + . It is accurately determined by interference with f4 (2050). It then acts as an interferometer for I = 1, C = −1. The amplitude analysis grows like a rolling snowball. The phases of other partial waves at low mass are measured with respect to %3 (1982) and establish resonances in other partial waves at low mass. Variations of magnitudes are consistent with the observed phases. Peaks appear also in the mass range 2200–2300 MeV, all correlated in phase. Collectively, their phases are loosely coupled to the tails of lower mass states. For I = 1, C = +1 and I = 0, C = −1 there are no polarisation data. Phase information from diOerential cross sections arises only via real parts of interferences. When relative phases are close to 0◦ or 180◦ , errors of phases are large. Without some smoothing of the phases to follow resonant behaviour (or some other analytic form), noise in the analysis can run out of control and can lead to a forest of narrow peaks. Such a result is unphysical, since amplitudes should be analytic functions of s. Smoothing to analytic forms improves the determination of partial wave amplitudes dramatically.

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(a)

(c)

(b)

(d)

Fig. 10. Integrated cross sections for two-body scattering, compared with the partial wave 4t. In (a), black circles are data of Hasan et al. and open squares are those of Eisenhandler et al. In (b)–(d) the integration is from cos 1 = 0 to 0.875 and is over only one hemisphere; the  data are averaged over 4 and 8 data. For − + , the integration is over the full angular range. In (c), the dashed curve shows the 0+ intensity.

The technique used for N elastic scattering is to constrain the analysis using 4xed t dispersion relations. This works successfully up to 1 GeV. Above that, amplitudes are parametrised directly with analytic forms. The approach used in Crystal Barrel analysis is to use analyticity as a foundation, parametrising amplitudes from the outset in terms of simple Breit–Wigner resonances of constant width, or constants, or linear functions of s. Even then, caution is necessary. If a linear function develops a phase variation ¿ 90◦ , it is likely to represent a resonance. It is well known to be dangerous to 4t data with amplitudes having similar characteristics; orthogonal functions are needed. In reality, one can usually replace linear functions having rising phases as the tails of resonances well below the pp / threshold, e.g. (1800) or %3 (1690); their phase variations above the pp / threshold are small and under reasonable control. Linear variations with falling phases are a possibility but in practice are not observed. A valuable constraint in the analysis is that amplitudes for all partial waves except 1 S0 and 3 S1 go to zero at the pp / threshold; these two go to scattering lengths and are the most diPcult to deal with. There is a practical problem with the pp / threshold. Fig. 10 shows integrated cross sections for two-body channels. For neutral 4nal states 0 0 and , only J P = 0+ , 2+ , 4+ and 6+ are allowed by Bose statistics. There are no contributions from pp / S-waves, which have quantum numbers J PC =0−+ 1 −− 3 (pp / S1 ). One sees in Fig. 10 that neutral cross sections go to low values near (pp / S0 ) and 1

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the pp / threshold. However, the − + cross section goes to large values, following the familiar 1=v law. The pp / total cross sections follows a similar behaviour. The S-wave amplitudes near threshold need to contain this feature, and it becomes diPcult to separate the 1=v behaviour from resonances near the pp / threshold. A further problem is that singlet pp / states do not interfere with f4 (2050) (the triplet state 3 F4 in pp). / An experimental problem is that Crystal Barrel data at the lowest momenta 600 and 900 MeV=c straddle the lowest tower of resonances. These momenta correspond to masses of 1962 and 2049 MeV. In several cases, what can be said is that there are de4nite peaks, inconsistent with the tails of resonances below threshold. However, because of the limited number of available momenta, masses and widths become strongly correlated; precise masses are subject to some systematic uncertainty, which is correlated over resonances of diOerent J P via interference eOects. Each of these cases needs to be examined critically. For I = 0, C = +1 and I = 1, C = −1, the PS172 data for pp / → − + down to 360 MeV=c are very important in extending the mass range to 1912 MeV in small steps of p/ momentum. 3.1. Star ratings of resonances Resonances will be given ratings up to 4∗ . The highest class requires observation of 3 or more strong, unmistakable peaks and a good mass determination (with error 4M 6 40 MeV). Such states are equivalent to those in the summary table of the Particle Data Group—better than some cases like a0 (1450), which until 2003 was observed only in a single experiment. States are demoted to lower star ratings if the mass determination is poor, particularly if 5 ¿ 300 MeV. The 3∗ resonances are reasonably well established, usually in two strong channels with errors for masses 6 ± 40 MeV; some are established in 3 channels but with 4M in the range 40–70 MeV. The 2∗ resonances need con4rmation elsewhere. They are observed either (i) in one strong channel with 4M ¡ 40 MeV or (ii) in 2 strong channels with sizable error in the mass. Finally, 1∗ states are tentative: weak channels and poor mass determination. 4. The partial wave analysis The foundation for the analysis was laid in 1992–1994 while 4tting PS172 data on pp / → − + [1]. Early in this analysis it became clear that there is a large J =4 amplitude which can be identi4ed with the well known f4 (2050). At momenta ¿ 1 GeV=c, a prominent feature of the polarisation data, illustrated below in Fig. 13, is a polarisation close to +1 over much of the angular range and over most beam momenta, broken by some rapid
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likewise be 4tted in terms of resonances and simple backgrounds. A coupled channel analysis led to consistency between all sets of data and rather precise determinations of resonance parameters [13–15]. The 4nal analysis for these quantum numbers is in Ref. [3]. 4.1. Formulae for amplitudes and resonances It is easiest to consider one reaction as an example: pp / → a0 (980)0 → 0 0 . The amplitude for one partial wave is written √ i %pp / ge B‘ (p)BL (q) f= [F(s12 )Z12 + F(s13 )Z13 ] : (3) p M 2 − s − iM5 Here g is a real coupling constant, and  the strong-interaction phase for this partial wave. Blatt–Weisskopf centrifugal barrier factors B are included explicitly for production with orbital angular momentum ‘ from pp / and for coupling with angular momentum L to the decay channel (momentum q). The 4rst centrifugal barrier includes the correct threshold dependence on momentum p in the pp / channel. Formulae for B are given by Bugg et al. [16]. The radius of the centrifugal barrier is optimised at 0.83 fm in Ref. [3]. It is important only for the high partial waves where the barrier is strong and well determined. For L = 4 waves, the barrier radius is increased to 1.1 fm. The factor 1=p in Eq. √ (3) allows for the
(4)

where 5(s) is the sum of widths to decay channels. For each channel, 5(s) is proportional to the available phase space. This is the well-known FlattUe form [17], which will be used for special cases such as f0 (980) and a0(980); these resonances lie close to the KK threshold and the s-dependence of the KK width is important. There is a dispersive term m(s) given by TVornqvist [18]. For present purposes, these re4nements are not relevant. The analysis concerns high masses where many decay channels are open. Many of the decay channels are unknown. The approximation is therefore that 5 can be taken as a constant; m(s) is then zero. Backgrounds are usually parametrised as constants or as resonances below the pp / threshold. 0 0 For the   4nal state, there are two possible amplitudes for production of a0 (980) in ; F(s12 ) is the FlattUe formula for the a0 resonance between particles 1 and 2, and F(s13 ) is the corresponding amplitude between particles 1 and 3. The functions Z12 and Z13 describe the dependence of the production amplitude on the production angle 1 in the centre of mass and on decay angles 9; : for decay in the resonance rest frame. These functions are written in terms of Lorentz invariant tensors, following the methods described by Chung [19]. Partial waves with J P = 2+ and 4+ couple to pp / with ‘ = J ± 1. Multiple scattering through the resonance is expected to lead to approximately the same phase  for both ‘ values. The ratio of coupling constants gJ +1 =gJ −1 is therefore 4tted to a real ratio rJ ; this assumption can be checked in several cases and data are consistent with this assumption.

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A general comment concerns possible t-channel contributions. Singularities due to those exchanges are distant and therefore contribute mostly to low partial waves. The high partial waves with J ¿ 2 are consistent with no background. Secondly, t-channel exchange amplitudes acquire the phases of resonances to which they contribute. They may be considered as driving forces for those resonances; an illustration is the well-known Chew–Low theory [20], where nucleon exchange aOects the line shape of 0(1232). In the present analysis, line shapes of resonances cannot be distinguished in most cases from simple Briet–Wigner resonances with constant widths. 4.2. Illustration of spectroscopic notation As an example of spectroscopic notation, consider pp / → !. The initial pp / state may have spin s = 0 or 1. For singlet (s = 0) states, the total angular momentum is J = ‘. For triplet states (s = 1), J = ‘ or ‘ ± 1. The parity is P = (−1)‘ and C = (−1)‘+s . In the ! exit channel, the spin s is 1 for the !. As examples, J PC = 1−− may couple to initial pp / 3 S1 and 3 D1 partial waves having ‘ = 0 or 2; the exit ! channel has L = 1. In the analysis, 3 S1 and 3 D1 partial waves are 4tted with a ratio of coupling constants rJ = g‘=J +1 =g‘=J −1 . A second example is J PC = 3+− . This couples to pp / singlet states and decays to !0 with L = 2 or 4. These two partial waves are again 4tted with a ratio of coupling constants rJ = gL=J +1 =gL=J −1 . Multiple scattering through the resonance is expected to lead to approximately the same phase  for both values of ‘ or L. It is indeed found that all phase diOerences between ‘ = J ± 1 lie within the range 0 to ±15◦ . Where possible, ratios rJ are taken to be real, in order to stiOen the analysis. The !0 data are 4tted to sequential two-body processes: !a2 (1320), !a0 (980), b1 (1235) and !(1650)0 . As regards notation, consider !a2 . Spins 1 of the ! and 2 for the a2 combine to total spin s = 1, 2 or 3. The initial pp / 1 F3 states with J PC = 3+− may decay to !a2 with L = 1, 3 or 5. In practice, the lowest L value is always dominant; in every case, L values above the second may be omitted because of the strong centrifugal barrier. It is however necessary to consider all possible values of total spin s . 4.3. General comments on procedures The selection of amplitudes is kept as simple as possible, consistent with the data; introducing amplitudes whose signi4cance is marginal introduces noise. A rule-of-thumb is that one can normally remove amplitudes which improve log likelihood by ¡ 20. For I = 0, C = +1, where 8 data sets are available and the solution is strongly over-constrained, amplitudes contributing ¡ 40 are omitted; they contribute ¡ 0:3% of the total intensity in all cases. Programmes can optimise M and 5 as well as coupling constants and the barrier radius R. In practice, it is important to scan all M and 5 through at least 7 steps (freeing other M and 5) so as to check that there is a well de4ned parabolic optimum in log likelihood. Movements of other M and 5 reveal correlations. For a few poorly determined resonances near the pp / threshold, log likelihood
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It is bad practice to introduce broad isotropic backgrounds. These interfere over the whole Dalitz plot and can perturb resonances. A symptom of instability is the development of large destructive interferences between components. This is checked by printing out the integrated contributions from every amplitude and every pair of interferences. Well known resonances such as f2 (1270), a2 (1320), a0 (980), b1 (1235), etc. are 4xed at masses and widths quoted by the PDG; the D/S ratio of amplitudes for decay of b1 (1235) (and other resonances) are 4xed at PDG values. A general point is that programmes must be as fast and as simple as possible, since one needs to run hundreds of variants of the 4t with varying sets of amplitudes. A simple though laborious procedure to look for alternative solutions is to − >0 2 2 : =i = 4> i Here > is a parameter and >0 a target value. The denominator 4> is adjusted so that each questionable ingredient contributes =2 9 (i.e. 3). Those amplitudes which are really needed feel

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little in0 is set to zero. This procedure is applied for I = 1, C = +1, where some instabilities are discussed below; the penalty function contributes ∼ 650 to =2 compared with ∼ 650; 000 for log likelihood. Incidentally, the program optimises both =2 for some data sets (e.g. binned two-body angular distributions) and log likelihood (for three-body channels). There is a useful trick in checking J P of resonances. It is diPcult to exhibit the angular dependence of some amplitudes, for example high J P → a2 ; the angular dependence Z of Eq. (3) involves complicated correlations between production angle 1 and decay angles 9, : of the a2 . Putting any new resonance into the analysis always gives some improvement. Some of this improvement comes from the correct line shape and some from the correct angular dependence. It is useful to substitute diOerent (wrong) angles and examine how sensitive the analysis is to the right choice of angles and correlations. 5. Review of I = 0, C = +1 resonances Most resonance from the Crystal Barrel in-
1
2J + 1 J  PJ1 (cos 1) ; f++ 4 J J (J + 1)

(6)

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Table 2 I = 0, C = +1 resonances J PC

∗

Mass M (MeV)

Width 5 (MeV)

6++ 4++ 4++ 4−+ 3++ 3++ 2++ 2++ 2++ 2++ 2++ 2−+ 2−+ 2−+ 1++ 1++ 0++ 0++ 0++ 0−+ 0−+

— 4∗ 4∗ 2∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 3∗ 4∗ 3∗ 4∗ 2∗ 3∗ 3∗ 4∗ 3∗ 3∗ 2∗

2485 ± 40 2283 ± 17 2018 ± 6 2328 ± 38 2303 ± 15 2048 ± 8 2293 ± 13 2240 ± 15 2001 ± 10 1934 ± 20 2010 ± 25 2267 ± 14 2030 ± 16 1870 ± 16 2310 ± 60 1971 ± 15 2337 ± 14 2102 ± 13 2020 ± 38 2285 ± 20 2010+35 −60

410 ± 90 310 ± 25 182 ± 7 240 ± 90 214 ± 29 213 ± 34 216 ± 37 241 ± 30 312 ± 32 271 ± 25 495 ± 35 290 ± 50 205 ± 18 250 ± 35 255 ± 70 240 ± 45 217 ± 33 211 ± 29 405 ± 40 325 ± 30 270 ± 60

r

SS () 0 2:7 ± 0:5 0:0 ± 0:04

−2:2 ± 0:6 0:46 ± 0:09 5:0 ± 0:5 0:0 ± 0:08 1:51 ± 0:09

245 2185 1607 558 1173 1345 1557 468 168 1462 694 1349 882 1451

1342 1189

Observed channels , , f2 , VES , ,  , a2 , (f2 ), PDG a0 , (a2 ) [f2 ]L=1; 3 , f2  [f2 ]L=1; 3 , a2 , a0  , ,  , f2 , a2  , ,  , f2  , ,  , f2  , , f2 , a2 , GAMS, VES  × 2, WA102 f2  , [f2 ]L=2 , a0 , f0 (1500) [a2 ]L=0; 2 , a0  f2 , a2 , a0 , WA102 f2 , f2  ,  f2 , a2 , a0 , (f0 (980), ,  ) ,   × 2,  , Mark III, BES I , WA102(; 4) f0 (1500) × 2, (), ( ) a2 , f2 , , BES II

Errors cover systematic variations in a variety of 4ts with diOerent ingredients. The sixth column shows changes in S= log likelihood when each component is omitted in turn from the 4t to 0 0 data and remaining components are re-optimised. In the last column, channels in parentheses are weak compared with the others, or less reliable.

√ √

2J + 1f++ = 2J + 1f+− =

√

J TJ −1; J −

√

√

J + 1TJ +1; J ;

J + 1TJ −1; J +

√

J TJ +1; J :

(7) (8)

Note the diOerence in sign in the last two of these expressions for T -matrix elements TJ +1; J . DiOerential cross sections and polarisations are given by ∗ A0N d=d = Tr(F ∗ Y F) = 2Im(F++ F+ − ) ;

d=d = |F++ |2 + |F+− |2 :

(9) (10)

The polarisation is very sensitive to the diOerence between amplitudes for L = J ± 1, for example 3 P2 and 3 F2 amplitudes. It is also phase sensitive. The analysis of two-body data alone [13] leads directly to a unique and accurate solution for all 0+ , 2+ and 4+ states. Intensities of 4tted partial waves are shown in Fig. 14. A small detail is that the low-mass tail of a higher 6+ resonance is included, though the mass of that state is above the available mass range and is therefore not well determined. The f4 (2050) is observed strongly

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

275

Fig. 11. DiOerential cross sections for pp / → − + from 360 to 1300 MeV=c, compared with the 4t; panels are labelled H for data of Hasan et al. and E for those of Eisenhandler et al. Each distribution is normalised so as to integrate to 2.

in Fig. 14 in all of 0 0 ,  and  , as well as in 0 0 data discussed below. The two-body data required the presence of two 3 P2 states, two 3 F2 and two 0+ states. The strong peaking of the 3 P2 amplitude at the lowest mass is associated with f2 (1934), or f2 (1910) of the PDG. Fig. 9 of Ref. [13] shows the eOect of dropping each of these states one by one from the analysis. Removing any of them causes clear visual disagreements with the data as well as large changes in log likelihood. A cross-check on the analysis is possible between 0 0 data and − + . States with I = 1, C = −1 contribute to − + but are determined separately from ! data discussed below. If one allows for their small intensity in − + and folds the angular distribution forward–backwards about 90◦ , in order to remove interferences between I = 0 and I = 1, the resulting angular distribution agrees accurately in shape with 0 0 and their absolute magnitudes agree within 1:1 ± 2:3%. This is a valuable check on the entire process of selecting and reconstructing Crystal Barrel data, and the correction for rate dependence.

276

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 12. As Fig. 11, 1351–2230 MeV=c.

5.2. Three-body data Resonances with J P = 0− , 1+ , 2− , 3+ and 4− appear only in three-body channels. However, statistics for 0 0 are very high, typically ¿ 70 K events per momentum. There are two dominant channels, f2 (1270) and a2 (1320) which are both well determined. Mass projections are shown in Fig. 15. Small f0 (980), f0 (1500) and a0 (980) signals are also visible. The f4 (2050) acts as an interferometer at the lower momenta, and likewise f4 (2300) at higher momenta. Phases of many other partial waves show rather small mass dependence with respect to these f4 . This is illustrated in Fig. 16; there, M refers to the helicity in the initial pp / state. It is unavoidable that resonances need to be introduced into 3+ , 1+ and 0− partial waves. The behaviour of 2+ partial waves is somewhat more complex. The two-body data de4ne very precisely the existence of states which are dominantly 3 P2 at 1934 and 2240 MeV and 3 F2 at 2001 and 2293 MeV. The 4rst of these may be identi4ed with the f2 (1910) observed by GAMS [21] and

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

277

Fig. 13. Polarisation data for pp / → − + , compared with the 4t.

VES [22] collaborations. The small spacings of 2+ states leads to rather rapid variations of phases with mass, but the three-body data are entirely consistent with two-body results. There are then two lucky breaks. The 4rst is that data for the 4nal state  0 0 are dominated by a single resonance in f2 (1270) [23]. Dalitz plots are shown in Fig. 17. There is an obvious band due to f2 (1270) along the lower left-hand edge of these Dalitz plots. A very clear peak is observed in the integrated cross section for f2  at 2267 MeV, see Fig. 18. It is consistent with a resonance required near this mass in 0 0 , but gives a better de4nition of resonance parameters. In turn, this 2− state then acts as a powerful interferometer to determine other partial waves at high masses. The second lucky break comes from  data [24]. Although statistics are low, there is a strong peak at high mass due to the well identi4ed f0 (1500) S-wave 4nal state with quantum numbers 0− . This is shown by the curve labelled 0− in Fig. 19. There is a small but distinctive contribution

278

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 14. Intensities for partial waves for 0 0 ,  and  . Dashed curves show contributions from L = J − 1 and dotted curves from L = J + 1.

from the 2− state at 2267 MeV in the f0 (1500) D-wave. The analysis is very simple and leads to a cleanly identi4ed 0− resonance at 2285 MeV. Returning to  data, there remain a 4− state, two 1+ states and a lower 0− state which are needed in the full combined 4t. 5.3. 2− states It is necessary to discuss separately 2− states at 1645, 1870 and 2030 MeV. They appear in 0 0 0 data in the production reaction pp / → 0 (). This was 4rst studied [25] in data at just two momenta: 1200 and 1940 MeV=c, prior to the running of the other momenta. Two 2− I =0 states were observed. The lower one was initially observed decaying purely to a2 (1320) at 1645 MeV. This 4ts nicely as the expected partner of the well known I = 1 2 (1670). However, the same data contained a second strong signal in the f2 (1270) 4nal state at 1870 MeV. The second signal could not be explained as the high mass tail of 2 (1645) decaying to f2 . Furthermore, the relative strengths of the two signals were quite diOerent in data at 1200 and 1940 MeV=c. Two separate states were required. Both resonances were subsequently con4rmed by the WA102 collaboration. They observed the / [26]. However, the K K / 2 (1645) in decays to  and 2 (1870) in decays to both  and K K decay mode is not strong enough to make an interpretation likely as the ss/ partner of 2 (1645).

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

279

Fig. 15. Mass projections on to M () and M () for 0 0 data at four beam momenta; histograms show the result of the 4t. Data are uncorrected for (small) variations of acceptance; these are included in the maximum likelihood 4t.

A feature of pp / annihilation is that well-known ss/ states such as f2 (1525) are produced very weakly, if at all; the 2 (1870), in contrast, is produced strongly. A possible explanation is that it is the radial excitation of 2 (1645), but that would be expected to lie around 2000–2050 MeV, near f4 (2050). In later data, a much more extensive study was made of 0 0 0 data with statistics a factor 7 larger [27]. A further 2− state appears as expected at 2030 MeV, decaying dominantly to the a2 (1320) D-wave. The main features of the data are shown in Fig. 20 as scatter plots of M () vs. M () for three  mass ranges centred on 2 (1645), 2 (1870) and 2 (2030). In (c), there is a vertical band due to a2 (1320) from the decay of 2 (1645); the horizontal band at 1000 MeV is due to the low mass tail of the stronger 2 (1870) → f2 . In (b), the horizontal band from 2 (1870) is dominant

280

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 16. Relative phases of partial waves for 0 0 . The phases for 2+ and 4+ with M = 0 are relative to 4+ → a2  with L = 3, M = 0; the phases for 1+ , 3+ and 4+ with M = 1 are relative to 4+ → f2  with L = 3, M = 1; the phases for 0− and 2− are relative to 2− → f2  with L = 0.

and the vertical a2 (1320) band becomes weaker. In (a), the f2 (1270) becomes weaker and there is a strong peak at the intersection of f2 (1270) and a2 (1320); the amplitude analysis shows that the 2 (2030) decays dominantly to the a2 (1320) D-wave. The essential point of the analysis is shown by the branching ratios in Table 3. [These refer directly to data and are not corrected for other charge states or for other decay modes of the a2 and .] Branching ratios change dramatically with mass, requiring three separate states. The f2 (1870) decays dominantly to f2  and a0 ; the 2 (2030) has dominant a2 (1320) decays with L = 0 and 2, and the f2  decay is weak. Fig. 21 shows log likelihood for the three 2− states for mass and width.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

281

Fig. 17. Dalitz plots at all beam momenta for  0 0 data. Numerical values indicate beam momenta in MeV=c.

80 70 60 50 40 30 20 10 0 1.9

2

2.1

2.2

2.3

2.4

Fig. 18. The integrated cross sections for the 4nal state  0 0 , after correction for backgrounds and for all decay modes of  ,  and 0 ; the full curve shows the 4t from the amplitude analysis; the dotted curve shows the  contribution.

282

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 19. The 4t to the mean integrated cross section for pp / → 3 from 6 and 10 data, (b) (2285) → [f0 (1500)]L=0 (dashed), and 2 (2267) → [f0 (1500)]L=2 (chain curve).

(a)

(b)

(c)

Fig. 20. Scatter plots of M () vs. M () for three ranges of M (): (a) 1945–2115 MeV, centred on 2 (2030), (b) 1775–1945 MeV over 2 (1870) and (c) 1560–1750 MeV, centred on 2 (1645).

Table 3 Decay branching ratios for 2 (1645), 2 (1870) and 2 (2030) Resonance

2 (1645)

2 (1870)

2 (2030)

BR[a0 ]=BR[a2 ] BR[f2 ]=BR[a2 ]

0:21 ± 0:13 —

3:9 ± 0:6 4:5 ± 0:6

0:21 ± 0:02 0:25 ± 0:09

The 2 states couple to pp / 1 D2 . From the straight-line trajectories of Fig. 1, it appears that 2 (1870) is an ‘extra’ state. Isgur et al. [28] predicted the existence of hybrids around 1800–1900 MeV. The hybrid consists of q and q/ connected by a gluon having one unit of orbital angular momentum about the axis joining the quarks. It rotates like a skipping rope between q and q. / It is predicted that this orbital angular momentum will be transferred to one pair of quarks in the 4nal state [29,30]. The expected decay modes are therefore to one 3 P combination of quarks and one S-state combination. Favoured decay modes are f2 (1270) and a2 (1320), in agreement with 2 (1870) decays. However, the f2  decay mode is strikingly strong. One can use SU (3) to calculate the expected ratio BR[a2 ]L=0 =BR[f2 ], integrating their available phase space over a2 and f2 line-shapes.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f)

283

Fig. 21. S= –log likelihood vs. masses and widths of (a) and (c) 2 (1645), (b) and (d) 2 (1875), (e) and (f) 2 (2030).

If it is a qq/ state, the predicted ratio for 2 (2030) is 9.7 after correcting for all charged states and decay modes of a2 and ; this compares well with the observed 10:0 ± 1:8. However, for 2 (1870), the prediction is 8.5 compared with the observed 1:27 ± 0:7. The conclusion is that the decay to f2  is distinctive and far above the value expected for a qq/ state. It will be shown below that there is an I = 1 partner at 1880 MeV decaying to a2 (1320). If these make a hybrid pair, it is the 4rst pair to be identi4ed. Further hybrids containing hidden strangeness are then to be expected at ∼ 2000 MeV decaying to K2 (1430) and at ∼ 2120 MeV decaying to / f2 (1525) and/or K2 (1430)K. 5.4. Star ratings of states All states of Table 2 have a statistical signi4cance ¿ 25 standard deviations. The even spin states are still more signi4cant in two-body decays. Argand diagrams for  are shown in Fig. 22 and for  decays in Fig. 23. The data require a larger and more linear phase variation than can be produced by a single resonance. In electronics, it is well known that a linear phase variation with frequency may be obtained with a Bessel 4lter [31], which has a sequence of poles almost linearly spaced in frequency. Physically, such a system generates a time delay which is independent of frequency, since group velocity is proportional to the gradient of phase vs. frequency. It appears that a sequence of meson resonances likewise conspires to produce approximately a linear phase variation with mass squared s.

284

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Fig. 22. (a)–(p) Argand diagrams for prominent partial waves in 0 0 data; crosses mark beam momenta; the amplitudes move anti-clockwise with increasing beam momentum; in (a) and (b) dashed curves show the eOect of removing the lower 0− resonance.

Intensities of the main individual partial waves in 0 0 are displayed in Fig. 24. In each panel, titles indicate in descending order full, dashed, chain and dotted curves. The f4 (2050) is seen as a clear peak in , , and a2 (1320), Fig. 24(g) and is clearly 4∗ . It peaks at 2070 MeV because of the eOect of the centrifugal barrier for production from pp, / but the mass of the Breit–Wigner resonance is substantially lower, at 2018 MeV. The f4 state at 2283 MeV is the f4 (2300) of the PDG. It is not conspicuous as a peak in two-body data, but has been found by several earlier partial wave analysis of − + data by Hasan [1], Martin and Pennington [32], Martin and Morgan [33] and Carter [34]. It need not appear as a peak in pp / →  because of interference with f4 (2050). It does however appears clearly as a peak in f2  in Fig. 24(g) at 2320 ± 30 MeV. Recent VES data also display conspicuous 4+ peaks in !! at 2325±15 MeV with 5=235±40 MeV [35] and in  data with M = 2330 ± 10(stat) ± 20(syst) MeV with 5 = 235 ± 20 ± 40 MeV [36]. Peak positions agree well with Crystal Barrel. This is also a 4∗ resonance. The 4− state (pp / 1 G4 ) appears only in  data. Amplitudes are small, but rather stable in all 4ts. The dominant decay is to a0 (980), Fig. 24(m). This resonance is classi4ed as 2∗ . Log likelihood changes by 558 when it is omitted. General experience is that any contribution improving log likelihood by more than 500 has well determined parameters; any aOecting log likelihood by ¿ 300 is de4nitely needed.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

285

Fig. 23. Argand diagrams for + − amplitudes; crosses mark masses at 100 MeV intervals of mass beginning at 1900 MeV.

The upper 3+ state decays to f2  with both L = 1 and 3; it produces the strongest intensity at high mass in  data, Fig. 24(d). It also appears as a clear peak in  0 0 data. The Argand diagram of Fig. 22(h) shows two loops requiring two 3+ resonances. The lower one appears clearly as peaks in Fig. 24(d) and (l) in a0 (980) and f2 (1270) and more weakly in a2 (1320). Phases are well determined by interference with f4 (2050), so it really should be given 4∗ status. Its mass is well determined with an error of only ±8 MeV. The upper 2− state at 2267 MeV appears as a clear peak in the raw   data of Fig. 18. It is also observed clearly in  in decays to [f2 ]L=2 in Fig. 24(c)(chain curve) and to both a0 (980) and f0 (1500), Fig. 24(k). It is de4nitely 4∗ . The Argand diagram of Fig. 22(e) requires a lower 2− state also. It appears as a peak in Fig. 24(c) in both [f2 ]L=0 and a2 ; it is also seen in several decay modes in 0 0 0 data. So its existence is not in doubt. Because of its overlap with 2 (1870), its mass and width may have some systematic uncertainty, so it falls into the 3∗ category. The 4rst four 2+ states of Table 2 are all extremely well determined in two-body data and  and are all 4∗ . The f2 (1980) will be discussed later as f2 (1950) of the PDG. Coming to low spins, states are less well de4ned. The lower 1+ state (3 P1 ) is large in several channels in Figs. 24(b) and (j), dominantly f2 , a2  and a0 (980). Despite the fact that it lies at the bottom of the mass range, parameters are quite stable: M =1971±15 MeV, 5=241±45 MeV. It just deserves 3∗ status. However, the upper 1+ state is the least well determined state in  data. The

286

D.V. Bugg / Physics Reports 397 (2004) 257 – 358 20

10

15

7.5

10

5

5

2.5

(a)

0 1.9

40 20

30 20

10

2.17

0 2.44 1.9 (b)

10 2.17

2.44

(c)

0 1.9

2.17

2.44

0 1.9 (d)

2.17

2.44

2.17

2.44

2.17

2.44

2.17

2.44

60 20

40

10

20 0 1.9 (e)

2.17

2.44

2.17

2.44

6

10

4

5

2

0 1.9 (g)

2.17

2.44

8

60

20

20

10 2.17

(i)

0 2.44 1.9 (j)

1.5

2.17

2.44

0 1.9 (h) 8

30

40

0 1.9

0 1.9 (f)

15

6

6

4

4

2

2

0 1.9 (k)

2.17

2.44

0 1.9 (l)

1.5

6

1.5

1

1

4

1

0.5

0.5

2

0.5

0 1.9

(m)

2.17

2.44

0 1.9 (n)

2.17

2.44

0 1.9 (o)

2.17

2.44

0 1.9 (p)

Fig. 24. (a)–(p) Intensities 4tted to individual partial waves, and corrected for all  and 0 decays. The titles in each panel show in descending order full, dashed, chain and dotted curves; (1000) denotes the broad f0 (400–1200) in the  amplitude. Numerical values indicate factors by which small intensities have been scaled so as to make them visible.

reason is that it makes only small contributions to the distinctive channels a2 (1320) and f2 (1270), and there are large interferences with other amplitudes. As the resonance mass is changed, these interferences are able to absorb the variation with small changes in log likelihood. Nonetheless, it does improve log likelihood by 882, a very large amount. Since it also appears in  , it quali4es as 2∗ . The upper 0− state (pp / 1 S0 ) at 2285 MeV appears as a clear peak in the raw 3 data of Fig. 19. It appears again in the f0 (1500) channel in  with a magnitude consistent with the known branching ratio of f0 (1500) between  and . It also contributes to . It is classi4ed as 3∗ . The lower 0− state is close to the pp / threshold and its amplitude goes to a scattering length at

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

287

Fig. 25. (a) Mark III data for J= → 4, (b) the  mass spectum from E760 data on pp / → 0 at 3:0 GeV.

threshold. This makes it diPcult to separate from a threshold eOect. On the basis of Crystal Barrel data it is only 1∗ . However, new BES II data on J= → 4 identify this state clearly at 2040 MeV in 4 [37], raising its status to 2∗ . Finally we come to 0+ states. The upper 0+ state at 2337 MeV is remarkably well de4ned in all of − + , 0 0 and . This quali4es it as 3∗ . The f0 (2100) appears as an  peak in  data at 1940 MeV=c [38], see Fig. 14 of Section 5.1. It is also conspicuous in  as shown in Fig. 10 and contributes to  . If it is dropped from the two-body analysis, =2 increases by 4030, an enormous amount. Allowing for its low multiplicity (2J + 1), it is one of the most strongly produced states in pp / annihilation. It was discovered in J= → 4 [39] and also appears as a very clear peak in E760 data on pp / → 0 [40], but has no spin determination there; these data are illustrated in Fig. 25. Its mass and width are very well determined by the E760 data. Crystal Barrel data agree within a few MeV with that determination. It is classi4ed as 4∗ . An f0 (2020) with a width of 400 MeV has been reported by the WA102 collaboration [41] in decays to both 2 and 4. Adding this to the analysis of Crystal Barrel 0 0 data gives a signi4cant improvement in =2 of 370 and a visible improvement in 4ts to some d=d data. However, overlap and interference with the strong f0 (2100) make a determination of its mass and width diPcult. It also obscures f0 (2100) in  in Fig. 14. From Crystal Barrel data alone it is a 1∗ resonance, but its observation in two channels of WA102 data raises its status to 3∗ . In summary, all states of Table 2 are securely determined in mass except for the upper 1+ state at 2310 MeV and the lower 0− state at 2010 MeV; both are certainly required, but have rather large uncertainties in mass, ±60 MeV. 5.5. f0 (1710) and f0 (1770) Data from MARK III [42] and DM2 [43] on J= → K K/ establish the existence of f0 (1710) / It is also observed clearly in data on J= → !K K/ [44–46]. From these and the abdecaying to K K. sence of evidence for decays to , one can deduce a branching ratio [f0 (1710) → ]=[f0 (1710) → / ¡ 0:11 with 95% con4dence. The f0 (1710) is an obvious candidate for an ss/ state completing K K] a nonet with f0 (1370), a0 (1450) and K0 (1430). A radial excitation of f0 (1370) is expected in the same general mass range. There is ample evidence for this state, and the PDG used to call it X (1740). Today, all entries are lumped under f0 (1710). However, these need to be examined critically. In an analysis of Mark III data for

288

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 26. BES II data for J= → + − . (a) and (b) measured and 4tted Dalitz plots; (c) and (d)  and  mass projections; the upper histogram shows the 4t to data and the lower shaded histogram show background derived from a sidebin to the ; in (d) the dip at 1:5 GeV arises from a cut to remove K ∗ K/ ∗ ; (e) and (f) 4tted 0+ and 2+ contributions; the dashed histogram in (e) shows the  contribution.

J= → + − + − , Bugg et al. [39] identi4ed a clear peak in 4 at 1750 MeV as a 0+ resonance. The Mark III data are shown in Fig. 25. If it is identi4ed as f0 (1710) it would require a branching −4 / (8:5+1:2 ratio for J= → 4 of (10:9 ± 1:3) × 10−4 , larger than that to K K, −0:9 ) × 10 . That does not 4t nicely with an ss/ state, and suggests the presence of a second state. In J= →  there is a peak at 1765 MeV but no corresponding signal in J= → K K/ [46]. New data from BES con4rm and strengthen this observation [47]. These data are shown in Figs. 26 and 27. There is a clear peak in Figs. 26(c) and (e) at 1780 MeV; J P = 0+ is strongly favoured / but a small shoulder in Fig. 27(c) at over 2+ . In Fig. 27 there is no corresponding peak in K K, 1650 MeV; Mark III results are similar. This small shoulder is 4tted by f0 (1710) interfering with f0 (1500) and f2 (1525). The branching ratio =K K/ for the 1765 MeV peak in  is at least a factor 25 larger than for f0 (1710), so two separate states are de4nitely required. The natural interpretation of the peak in  is in terms of a second f0 (1770) decaying dominantly into  / This 4ts in with the second 0+ state expected in this mass and 4 but not signi4cantly into K K.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f)

289

Fig. 27. (a)–(f) As Fig. 28 for J= → K + K − .

range. There is however a long-standing puzzle why the f0 (1710), which seems to be dominantly ss, / should be produced in !KK while the f0 (1770), apparently mostly non-strange, should be produced in the 4nal state KK. In Crystal Barrel data on pp / → 0 , there is evidence for the same f0 (1770) decaying to  [48]. Fig. 28 shows the  mass projection for data at 900, 1050 and 1200 MeV=c. The histogram shows the best 4t without inclusion of f0 (1770). At all three momenta there is a small surplus of events, which may be 4tted well by including a further contribution from f0 (1770) with M =1770±12 MeV, 5 = 220 ± 40 MeV; these values are in good agreement with data discussed above. The 4rst three panels of Fig. 29 show log likelihood vs. mass at the three momenta. Full curves are for the spin 0 assignment and dashed curves for spin 2; spin 0 is obviously preferred. The last panel shows log likelihood vs. width. The f0 (1710) could decay to  via their ss/ components. However, ss/ 4nal states are conspicuous by their absence in pp / annihilation. In the same 0 data, a scan of the 2+ amplitude vs. mass shows no peaking at f2 (1525), so f0 (1710) appears an unlikely source of the  signal. Furthermore, mass and width 4tted to f0 (1770) appears signi4cantly diOerent to those of f0 (1710). A somewhat diOerent 4t to these data has been reported by Amsler et al. [49]. This 4t omits f0 (1770) and generates the 1770 MeV shoulder as the tail of an interference between f0 (1370) and

290

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 28. Data for pp / → 0 . Histograms show the 4t without f0 (1770) at (a) 900 MeV=c, (b) 1050 MeV=c and (c) 1200 MeV=c.

(a)

(c)

(b)

(d)

Fig. 29. Log likelihood vs. mass of f0 (1770) full lines J = 0, dashed lines J = 2, (a) at 900 MeV=c, (b) 1050 MeV=c and (c) 1200 MeV=c; (d) log likelihhod vs. width at 1050 MeV=c (full curve) and 1200 MeV=c (dashed).

f0 (1500); since they lie close to one another, they behave like a dipole with a long tail at high masses. The diPculty with this 4t is that the f0 (1370) signal required is considerably larger than is 4tted to f0 (1370) in 3 data at all momenta, see Section 6 below; the f0 (1370) branching ratio to  is known to be small compared to that for . The 4rst evidence for a signal in  at 1744 ± 15 MeV was reported by the GAMS collaboration [50,51]. It was observed to have an isotropic decay angular distribution, consistent with J P = 0+ .

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

291

5.6. SU (3) relations for two-body channels Amplitudes for pp / → 0 0 ,  and  throw valuable light on the quark content of √ resonances. / It is well known that  and  may be expressed as linear combinations of |(uu/ + dd)= 2 ¿ and ss. / Denoting the former by nn, / their quark content may be written in terms of the pseudoscalar mixing angle C as |nn / = cos C| + sin C|  ;

(12)

|ss / = −sin C| + cos C|  ;

(13) 0 0

where cos C 0:8 and sin C 0:6. Amplitudes for decay to   ,  and compactly as [13] |0 0  cos2 C f(nn) / = √ + √ | + cos C sinC|  ; 2 2 √ √ f(ss) / = D[sin2 C| + cos2 C| − 2 cos C sin C| ] :



may then be written (14) (15)

The second of these equations includes a factor D. This factor has been 4tted empirically by Anisovich to a large variety of data on strange and non-strange 4nal states at high energies [52]. Its value is √ D = 0:8–0.9 and the central value 0.85 is adopted in the Crystal Barrel analysis. Resonances R observed in data may be linear combinations of nn/ and ss: / R = cos E|qq / + sin E|ss / : Amplitudes for the decay of R to these three channels are given by cos E f(0 0 ) = √ ; 2

(16)

(17)

√ cos E f() = √ (cos2 C + 2D sin2 C tan E) ; 2

(18)

√ cos E f( ) = √ cos C sin C(1 − 2D tan E) : 2

(19)

Most
(20)

For an unmixed nn/ state the branching ratio BR[0 0 ]=BR[] would be 1=(0:8)4 =2:44. The f0 (2100) is one of the strongest 0+ resonances. It makes up (38 ± 5)% of the  signal. Intensities of  and  cross sections vs. mass are shown in Figs. 10(c) and (d). The f0 (2100) has the anomalous property that it is produced very strongly in pp / yet it decays dominantly to ss. / Below it will be

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Fig. 30. DiOerential cross sections for pp / → 0 0 (4rst two rows),  (rows 3 and 4) and  (bottom 2 rows), compared with the combined 4t (full curves). For  , the average of 4 and 8 data is used.

considered as a candidate for a 0+ glueball predicted close to this mass. A glueball would have a mixing angle of +35:6◦ . The departure of the
D.V. Bugg / Physics Reports 397 (2004) 257 – 358

293

well de4ned, but several are not. The well-de4ned resonances agree reasonably well with I = 0, C = +1. The remainder fall into line with I = 0 if one chooses one of two alternative solutions to the data. The fundamental diPculty is that there are two discrete solutions for 0  and 0  [12]. To get from one solution to the other, two 2+ amplitudes must be
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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 31. Angular distributions for 0  corrected for acceptance. Black circles show results from 4 and open triangles results from 8. Full and dashed curves show 4ts from solutions 1 and 2. Dotted curves illustrate the acceptance. Open squares show earlier results of Dulude et al. at neighbouring momenta, after renormalisation to obtain the optimum agreement with Crystal Barrel normalisation.

The 4nal element in the analysis is the observation of a 2 (1880) state, J P = 2− , in the production process pp / → 2 (1880), 2 →  in 0 0 data. This state overlaps partially with a further 2− state at 2005 MeV. They can actually be separated by the fact that 2 (2005) decays to a0 (980) whereas the 2 (1880) does not. However, for other decay channels there is potential ambiguity between these two close resonances. 6.1. Two-body data Angular distributions for  are shown in Fig. 31 [12]. There is again excellent agreement between 4 data (black circles) and 8 (open triangles) in both shapes and absolute normalisations. Open squares compare with earlier data of Dulude et al. [55] after scaling their normalisation to agree with Crystal Barrel data. Angular distributions for 0  are shown in Fig. 32. There are typically 1000 events per momentum. DiOerences between two alternative solutions are shown by full and dashed curves. Intensities of partial waves from these two solutions are shown in Fig. 33. The main diOerence lies in 2+ . Both solutions require the 3 F2 resonances at 2255 and 2030 MeV and some 3 P2 contribution. In the earliest analysis of Ref. [12] there was no input from 30 data and the upper 3 P2 resonance

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

295

Fig. 32. Angular distributions for 0  corrected for acceptance. Full and dashed curves show solutions 1 and 2, respectively. Dotted curves illustrate the acceptance.

Fig. 33. Intensities of 0+ , 2+ , 4+ and 6+ partial waves for pp / → 0  for the better solution (upper row) and poorer 3 solution (lower row). Dashed curves show intensities for P2 and 3 F4 , dotted curves for 3 F2 and 3 H4 , full curves their sum.

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(b)

(a)

(c)

Fig. 34. (a)–(c) Mass projections for all 0 0 combinations in 30 data after selecting events where M () lies within ±50 MeV of a0 (980) or alternatively M 2 () ¿ 0:6 GeV2 . Histograms show 4ts from the amplitude analysis.

at 2175 MeV was not included. Even in the 4nal combined analysis, this is the least signi4cant contribution. Both solutions require a 0+ state near 2025 MeV, though its mass and width have sizable errors. A lucky break is that independent evidence for this state appears in 0 0 0 data. A small but de4nite (1440) signal is observed there in  [56]. It is illustrated in Fig. 34. This channel may be 4tted to production amplitudes using simple Legendre polynomial for L = 0, 2 and 4. The result is complete dominance by L = 0, with no signi4cant evidence for any L = 2 or 4 amplitudes. This requires production from an initial I = 1, J PC = 0++ initial state. The cross section for production shows a distinct peak at 2025 MeV (Fig. 35). It may be 4tted with a resonance at 2025 MeV, 5 = 330 MeV after inclusion of the pp / L = 1 centrifugal barrier for production. However, even after this lucky break, the two-body data fail to locate the further state expected in the mass range 2300–2400 MeV as partner to f0 (2337). Note that the a0 expected between 1450 and 2025 MeV is also missing. As a result, the 0+ sector for I = 1 is poorly known. 6.2. pp / → 0 These data are discussed in detail in Ref. [5]. Dalitz plots are shown in Fig. 36 at all momenta. There are obvious vertical and horizontal bands at M 2 () = 1 and 1:7 GeV2 due to a0 (980) and a2 (1320). There is also a diagonal band at the highest four momenta due to f0 (1500).

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

297

Fig. 35. Cross sections for pp / → (1440) with L = 0. The curve shows the cross section for a resonance with M = 2025 MeV, 5 = 330 MeV, including the pp / ‘ = 1 centrifugal barrier at threshold.

Fig. 36. Dalitz plots for pp / → ; numbers indicate beam momenta in MeV=c.

Fig. 37 shows the  mass projection. The strong f0 (1500) is clearly visible. There is also a shoulder due to f0 (1770) at 1200–1642 MeV=c. There is one other important feature of Fig. 36. At the intersection of vertical and horizontal bands at M 2 () = 1 GeV2 , there is a distinct enhancement at the highest three momenta, in the bottom row of the 4gure. This enhancement cannot be 4tted adequately by constructive interference alone between two a0 (980) bands; such an enhancement is not there at lower momenta. The enhancement in fact arises from a rapidly growing signal due to f0 (2100), whose kinematic threshold is at 1525 MeV=c. This feature is important in two respects.

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 37. Projections of 0 data on to  mass; histograms show the 4t and beam momenta are indicated in MeV=c. Table 4 Resonances parameters 4tted to 0 data Name

JP

M (MeV)

5 (MeV)

Decay channel

SS



0−

2070 ± 35

310+100 −50



0−

2360 ± 25

300+100 −50

2 4 a2 a3

2− 4− 2+ 3+

1990 ± 30 2255 ± 30 2255 ± 20 (2031)

200 ± 40 185 ± 60 230 ± 15 (150)

a3

3+

2260 ± 50

250+100 −50

a0 (980) f0 (1500) a0 (980) f0 (2100) a0 (980) a0 (980) f2 (1950) a2 (1320) f0 (1500) a0 (980)

272 324 197 434 531 213 363 185 164 141

Values in parentheses are 4xed from 30 data. The 4nal column shows changes in log likelihood when each channel is dropped from the 4t and all others are re-optimised.

Firstly it shows the presence of f0 (2100) in , reported in Ref. [38]. Secondly, the f0 (2100) is so close to threshold that it appears only in the f0 (2100) S-wave, i.e. J P = 0− pp. / This observation helps identify the highest 0− resonance. Table 4 shows the conspicuous resonant contributions in 0 . The 4nal column shows changes in log likelihood when each resonance is omitted from the 4t. All are statistically at least 14 standard deviation signals, though uncertainties in smaller amplitudes introduce systematic errors which reduce the signi4cance level to 8–12 standard deviations.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

299

Fig. 38. (a)–(p) Intensities of individual f2  partial waves for the preferred solution. Squares and triangles show free 4ts at single energies. Full curves are for f2  4nal states. Dotted and chain curves in (a) show, respectively, the 4− f2  amplitude with L = 5 (triangles) and the  contribution. Dotted curves in (c) and (p) show  contributions. In (p), points and the full curve show integrated cross sections for 30 .

Singlet amplitudes are determined simply via Legendre polynomials for production amplitudes. Angular distributions shown in Ref. [5] verify directly the presence of 0− , 2− and 4− states. There is evidence also in 30 data for both 0− resonances, though they are determined best by the  data. Both should be regarded as 2∗ . The 2− state at ∼ 1990 MeV is conspicuous also in 30 data, Fig. 38(b). The combined 4t to both channels gives an optimum mass of 2005 ± 15 MeV and a width of 200 ± 40 MeV, as shown below in Table 5. The observation of this 2− state in a0 (980) is rather important. This resonance overlaps signi4cantly with 2 (1880) discussed below. However, 2 (1880) has a very weak decay to a0 (980) if any at all. The second order Legendre polynomial associated with the 2− state at 1990 in 0 data is very distinctive at 600, 900 and 1050 MeV=c and leaves no doubt of its presence. This state should be regarded as 3∗ . The 4− signal is entirely consistent with a rather strong and distinctive 4− signal in f2  in 30 data and should be regarded as 3∗ . The 3+ state at 2031 MeV is a con4rmation of a very strong f2  signal in 30 data, making this state 3∗ . The upper 3+ state is not well identi4ed in 30 data, so this should be regarded as 2∗ . The 2+ state at 2255 MeV con4rms a 3 F2 state in 30 and 0 and should be regarded as 3∗ . Its distinctive decay into f2 (1950) will be discussed in detail in Section 10.4.

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Table 5 Masses and widths of 4tted I = 1, C = +1 resonances JP

*

M (MeV)

5 (MeV)

4 a4 a4 a3 a3 a2 a2 a2 2 2 a1   a0

4− 4+ 4+ 3+ 3+ 2+ 2+ 2+ 2− 2− 1+ 0− 0− 0+

3∗ 3∗ 2∗ 2∗ 3∗ 3∗ 2∗ 2∗ 2∗ 3∗ 2∗ 2∗ 2∗ 2∗

2250 ± 15 2255 ± 40 2005+25 −45 2275 ± 35 2031 ± 12 2255 ± 20 2175 ± 40 2030 ± 20 2245 ± 60 2005 ± 15 2270+55 −40 2360 ± 25 2070 ± 35 2025 ± 30

215 ± 25 330+110 −50 180 ± 30 350+100 −50 150 ± 18 230 ± 15 310+90 −45 205 ± 30 320+100 −40 200 ± 40 305+70 −35 300+100 −50 310+100 −50 300 ± 35

a2 2 a1 a1 

2+ 2− 1+ 1+ 0−

(1950) (1880) (1930) (1640) (1801)

(180) (255) (155) (300) (210)

4∗

r

0:87 ± 0:27 0:0 ± 0:2 −2:13 ± 0:20 −0:05 ± 0:31 2:65 ± 0:56

−0:05 ± 0:30

SS (MeV)

M (I = 0)

Observed channels

11108 2455 2447 3154 18410 2289 1059 1308 2298 1633 2571 1955 1656 2134

2328 ± 38 2283 ± 17 2018 ± 6 2303 ± 15 2048 ± 8 2293 ± 13 2240 ± 15 2001 ± 10 2267 ± 14 2030 ± 15 2310 ± 60 2285 ± 20 2010+35 −60 2040 ± 38

a0 , f2 , () ,  , f2  , f2  a0 , f2  f0 (1500), a2 , f2  , f2 , f2 (1950) f2  ,  , f2  f2 , () a0 , f2  f2  a0 , f0 (2100), f2  a0 , f0 (1770), f2  ,  , (1440)

2638 7315 2609 232 2402

1934 ± 20 1860 ± 15 1971 ± 15

Values in parentheses are 4xed from other data. Entries in the lower part of the table are below the available mass range. Column 7 shows changes SS in log likelihood for 30 when each resonance is removed from the 4t and all remaining parameters are re-optimised. Column 8 shows masses for I = 0, C = +1 resonances from Table 2 for comparison.

6.3. pp / → 30 Table 5 shows resonance parameters from the 4nal combined 4t of all channels to the better of the two solutions mentioned above. This solution has log likelihood better by 1182. This diOerence is comparable to the eOect of removing one of the weaker resonances. The essential change in going to the poorer solution, shown in Table 6, is that the masses of the upper 4+ state and both of the upper two 2+ states slide downwards by ∼ 40 MeV. There is also a small increase in the mass of the lower 4+ state from 2005 to 2030 MeV. What is happening is that the two 4+ states, which have similar values of r4 , are tending to merge. Elsewhere it is observed that such merging of resonances of the same J P almost always gives small improvements in log likelihood through interference eOects. Generally, this should be regarded with suspicion. The masses of the highest 4+ and 2+ states are lower than for I = 0 by 60–70 MeV. If one takes the I = 0 solution as a guide, this solution looks physically less likely, as well as having the poorer log likelihood. Despite diOerences of detail between the two solutions, the general pattern of masses and widths for I = 0 and 1 is similar. Most I = 1 masses tend to lie 20 MeV lower than I = 0. Since the poorer solution drags the masses of the upper 2+ and 4+ states down, it is possible that the ambiguity everywhere has the eOect of lowering masses, through correlations between partial waves.

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301

Table 6 Masses and width of resonances in the alternative solution Name

JP

M (MeV)

5 (MeV)

SS

r

a4 a4 a2 a2

4+ 4+ 2+ 2+

2220 ± 20 2035 ± 20 2235 ± 35 2135 ± 45

345 ± 65 135 ± 45 200 ± 25 305+90 −45

2852 2063 3658 1685

0:30 ± 0:31 0:0 ± 0:2 −1:74 ± 0:36 −0:56 ± 0:53

The last column shows changes SS in log likelihood when each resonance is removed from the 4t and all remaining parameters are re-optimised.

One of the problems in the analysis is that angular distributions of three-body data change rapidly between the two lowest beam momenta, 600 and 900 MeV=c. In order to accommodate this feature, both the lower 3+ and 4+ states must be rather narrow. Popov [57] reports an I = 1, J P = 3+ peak in E852 data for [b1 (1235)] at ∼ 1950 MeV, but 4ts to a much lower mass of M =1873±1±18 MeV with 5 = 259 ± 18 ± 21 MeV. He also reports an a4 at 1952 ± 6 ± 14 MeV in [!%]L=2 with 5 = 231 ± 11 ± 21 MeV. Fig. 38 shows as full curves the intensities 4tted to individual f2  partial waves in the better solution. Intensities for single energy solutions are shown by black squares. This gives some idea of the stability of the 4t. Because of the ambiguity between two solutions, none of these states can be regarded as 4∗ . Both + 4 states are certainly required by  and f2  data; the upper one is required also by  data. The lower 3+ state is exceptionally strong, Fig. 38(g) and must be regarded as well established. The 2+ states are poorly separated between 3 P2 and 3 F2 , but a solution is not possible without three of them in the mass range 2000 MeV upwards. Except for the f2 (2255), which appears distinctively in , they should be regarded as 2∗ because of the ambiguity in the solution. The upper 2− state is required but poorly determined and can only be classed as 2∗ . A lower − 2 state is de4nitely required by the peak in f2  in Fig. 38(b). Recently the E852 collaboration has presented evidence for a similar resonance in f1 (1285) with M = 2003 ± 88 ± 148 MeV, 5 = 306 ± 132 ± 121 MeV [58]. For J P = 1+ , f2  data require a 1+ state around 2270 MeV and there is tentative supporting evidence also in ; it can only be regarded as 2∗ . There is a strong 1+ intensity at 600 MeV=c. The 0− states are better identi4ed by  data. As remarked above, there is a large change at the lowest momenta between the earliest analysis of f2  [53] and the 4nal analysis of Ref. [4]. The problem is an ambiguity at the lowest momentum between 3 P1 , 3 P2 and 3 F4 amplitudes arising from the rapid change of angular distributions with mass. The ambiguity between 3 P1 and 3 P2 is the principal problem, so both can only be regarded as tentative identi4cations: something to watch for in other experiments. The very strong 3 F3 state is well established independently of the others. Good information from any other source resolving the ambiguity between 3 P2 and 3 P1 would greatly stabilise the complete solution for I = 1, C = +1. One should be alert for a 2+ state decaying dominantly to %!, as partner to f2 (1910 − 1934) → !!. The strong rise in Fig. 38(i) at the lowest momentum suggests a contribution from a low mass

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(d)

(c)

(e)

Fig. 39. Full histograms show 4ts to 0 0 data at 1200 MeV=c compared with data; dotted histograms show 4ts without 2 (1880); (a) the  mass distribution, (b) the  mass distribution, (c) the  mass distribution for events with a  combination within ±65 MeV of a2 (1320), (d) and (e)  and  mass distributions for events with M () 1850–2050 MeV. Units of mass are MeV. 3

P2 state around 1950 MeV. However, this is right at the bottom of the available mass range and cannot be regarded as established at present. 6.4. Evidence for 2 (1880) Evidence was presented for 2 (1870) in Section 5.3. An I = 1 partner is to be expected and is indeed observed in pp / → 2 , 2 → a2 (1320), i.e. in the 0 0 4nal state [59]. The entire pattern of production cross sections is remarkably similar to that for 0 0 0 . The evidence for 2 (1880) is found at beam momenta 900, 1050 and 1200 MeV=c (as for 2 (1870)). Fig. 39 shows data at 1200 MeV=c. In (a), dashed histograms show the 4t without 2 (1880). There is a hint of a peak in  in the full histogram. The essential evidence comes from (c), (d) and (e). Without 2 (1880), the dashed histogram in (c) fails to 4t the observed a2 (1320) signal. Note that the normalisation of the curve cannot be scaled up arbitrarily to 4t the data, since it is governed by 4tted amplitudes. In (d), the a2 signal in the  mass range 1850 MeV (approximate

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

303

Fig. 40. Data of Daum et al. [62]. The upper 4gure shows the f2  D-wave intensity; the lower 4gure shows the phase diOerence between 2 (1670) and [f2 ]L=2 .

threshold) to 2050 MeV is clear. In (e) the  enhancement below 1350 MeV arises as a re<ection of the decay channel a2 (1320). Optimum parameters are M = 1880 ± 20 MeV, 5 = 255 ± 45 MeV. The signal cannot be explained as the high mass tail of 2 (1660): it is too strong and the 2 (1670) is too narrow to 4t observed cross sections. From Fig. 39(a), the signal clearly does not arise from  masses near 2005 MeV. The angular distribution for decay to a2 (1320) is consistent with S-wave decay, as expected so close to threshold; it is inconsistent with L = 1 decays. The VES collaboration also reports a threshold enhancement in a2 (1320) with J P = 2− in their − data [60] and also in + − 0 data where a2 (1320) → + − 0 . They observe a strong peak at the same mass in [f2 (1270)]L=2 [61]. Daum et al. [62] also observed a peak at 1850 MeV in [f2 (1270)]L=2 with a width of ∼ 240 MeV. Their result is reproduced in Fig. 40. There is a clear peak and a phase motion with respect to 2 (1670) entirely consistent with a second 2− resonance associated with the peak. However, they interpreted the data conservatively as arising from interference between 2 (1670) and a higher 2 (2100); that interpretation fails for Crystal Barrel data. Popov [57] reports a 2− peak in E852 data for [!%]L=1 for the combination of ! and % spins S = 1. He quotes M = 1890 ± 10 ± 26 MeV, 5 = 350 ± 22 ± 55 MeV. The mass agrees well with Crystal Barrel; the width is slightly higher. Eugenio [63] reports a similar 2− peak in E852 data for the − 4nal state. In view of all these observations, the 2 (1880) should be regarded as a 4∗ state. In Section 5.3, it was reported that 2 (1870) has a branching ratio to f0 (1270) much stronger than to a2 (1320) and inconsistent with SU (3) for a qq/ state. The 2 (1880) is observed strongly in decays to a2 (1320) by VES but no decay to the f2  S-wave is observed; although no actual

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

branching ratio is reported at present, it seems likely that 2 (1880) is also inconsistent with a qq/ state and SU (3) relations for decays to f2  and a2 . 7. I = 1, C = −1 resonances Data are available from − + , !0 and !0 4nal states. The latter two channels have been measured for ! → 0  at all momenta and for ! → + − 0 at 6 momenta: 600, 900, 1200, 1525, 1642 and 1940 MeV=c. The analysis of pp / → − + is reported in Ref. [13]. The I = 1 states with J PC = 1− − , 3− − and − − 5 cause forward–backward asymmetries in both d=d and P d=d via interferences with I = 0 states; these asymmetries are conspicuous and lead to good determinations of C = −1 amplitudes. A particular feature at low momenta is a very strong %3 (1982) which interferes with f4 (2050) over the whole momentum range around 750 MeV=c. Its mass and width are determined very accurately by the − + data. This resonance then serves as a useful interferometer in !0 and !0 data in the low-mass range. A second J PC = 3− − state is required also at 2260 MeV and serves as a similar interferometer at high masses. The %5 (2295) appears as a large signal in − + , see Fig. 14 of Section 5.1. Attention needs to be drawn to one point concerning the 1− − state at 2000 MeV. The best mass determination of this state comes from − + , because of the PS172 data available in small momentum steps down to a mass of 1912 MeV. As emphasised above, the combination of diOerential cross sections and data from a polarised target determines phases well for all partial waves. The Argand diagram for 3 D1 in Fig. 23 of Section 5.4 requires a phase variation approaching 180◦ between 1900 and 2100 MeV, hence a 1− − resonance. The 3 S1 amplitude shown on the same 4gure has a 1=v behaviour close to the pp / threshold. When one factors this 1=v dependence out, the 3 S1 amplitude also requires the same 1− − resonance. For !0 and !0 data discussed below, data from a polarised target are not available, so determinations of its mass and width are less reliable. Once it is established, the 1− − amplitude again serves as a useful interferometer in !0 and !0 data at low masses. Analysis of pp / → !0 , ! → 0  is reported in Ref. [64]; !0 data with ! → 0  are reported in Ref. [65]. Although this ! decay mode is identi4ed cleanly, polarisation information for the ! is carried away by the photon, whose polarisation is not measured. This was the reason for studying ! → + − 0 [6]; the combined analysis of all I = 1, C = −1 channels is reported there. The matrix element for ! → + − 0 is relativistically j9:4 p: p p4 where p are 4-momenta of decay pions. Non-relativistically, this matrix element becomes pi ∧ pj in the rest frame of the !, where pi are three-momenta of any two pions. The polarisation vector of the ! is therefore described by the normal ˜n to its decay plane. A measurement of this decay plane provides complete information on the polarisation of the !. That improves the amplitude analysis greatly. Conclusions should therefore be taken from Ref. [6]. The measurement of both decay channels also provides a valuable experimental cross-check, illustrated in Fig. 41. Angular distributions are measured mostly in the backward hemisphere for the !, because of the Lorentz boost from centre of mass to the lab. Figs. 41(e)–(j) show as points with errors the angular distributions for !0 from ! → + − 0 decays. The corridors show angular distributions from the partial wave analysis of ! → 0 . Both are corrected for acceptance. The

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

( j)

305

Fig. 41. M (+ − 0 ) from the data selection for (a) !0 data, (b) !0 at 900 MeV=c; the number of events vs. the square of the matrix element for ! decay in (c) !0 , (d) !0 ; (e)–(j): curves show the error corridor for !0 diOerential cross sections where ! decays to 0 ; points with errors show results for ! decays to + − 0 ; both are corrected for angular acceptance; the vertical dashed lines show the cut-oO which has been used.

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

observed agreement is a major cross-check on calibrations of the detector, data selection and Monte Carlo simulations. All of these are completely diOerent between all-neutral data and charged data. One aspect of this is the treatment of the vertex. For charged data, the vertex is determined by the intersection of the two charged tracks. For neutral data, the vertex is instead taken to be at the centre of the target. This assumption distorts the kinematic 4t slightly, but the correction is only 1% to d=d for all-neutral data. This procedure is safer than 4tting the vertex freely; that alternative leads to strong and dangerous variations of acceptance with cos 1! . Backgrounds are clearly visible in Figs. 41(a) and (b) under the !. The backgrounds are checked in (c) and (d). There, the number of events is plotted against the square of the matrix element for ! decay. The linear dependence checks that the ! signals are clean. The intercepts at |M |2 = 0 check the magnitudes of the backgrounds. 7.1. Vector polarisation of the ! and its implications A remarkable feature is observed for ! polarisation, with major physics implications. Both vector polarisation Py and tensor polarisations T20 , T21 and T22 are measured. For details and formulae, see Ref. [6]. Vector polarisations are shown in the upper half of Fig. 42. They are consistent with zero almost everywhere. Large tensor polarisations are observed at all momenta in the lower half of Fig. 42 (Re T21 ) and in Fig. 43 (T20 and T22 ). Vector polarisation depends on the imaginary parts of interferences between partial waves. Tensor polarisations depend on moduli squared of amplitudes and real parts of interferences. In particular, Re T21 contains real parts of precisely the same interference terms as appear in PY . Formulae are given by Foroughi [66] for pp → d+ , but the same formulae apply to pp / → !0 . The implication of zero vector polarisation is that relative phases of amplitudes are close to 0 or 180◦ ; the physics implications will now be discussed. Consider 4rst a given J P such as 1+ . This partial wave decays to ! with orbital angular momenta L = 0 or 2. It is not surprising that partial waves to these 4nal states have the same phase, since a resonance implies multiple scattering through all coupled channels. Indeed, it is found that relative phases are consistent with zero within errors for all resonances. [The 4nal analysis can then be stiOened by setting the relative phase angle to zero.] Vector polarisation Py can also arise from interference between singlet waves 1+ , 3+ and 5+ . Likewise, it can arise from interference between triplet waves 1− , 2− , 3− , 4− and 5− . It is remarkable that these further interferences also lead to nearly zero polarisation. It implies coherence between diOerent J P . An interpretation is suggested here. The incident plane wave may be expanded in the usual way in terms of Legendre functions:
eikz = (2‘ + 1)i‘ P‘ (cos 1) : (21) ‘

As a result, all partial waves of the initial state are related in phase through the factor i‘ . Fig. 44 shows Argand diagrams for all partial waves. Let us take as reference for triplet states the large 3 G4 (‘ = 3) amplitude. If one compares this by eye with diagrams for 3 D2 and 3 D3 , it is clear that there is a phase rotation of the diagram by ∼ 180◦ for 3 D2 ; 3 D3 is similar in phase to 3 G4 . The Argand loop for 3 S1 is again rotated by ∼ 180◦ with respect to 3 G4 . Singlet and triplet states do not interfere in d=d , Py , T20 , T21 and T22 . Therefore the phase of singlet states with respect to triplet

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

307

Fig. 42. Vector polarisation Py for !0 data at (a) 900, (b) 1200, (c) 1525 and (d) 1940 MeV=c compared with the partial wave 4t (histogram); (e)–(h) Re T21 at the same momenta.

are arbitrary in Fig. 44. The Argand loops for 1 P1 and 1 F3 are broadly similar, though there is a large oOset in the 1 P1 L = 0 amplitude. A similar phase coherence is reported for pp / → − + [1]. There, both I = 0 and I = 1 amplitudes are present. The amplitudes which diOer by 1 in orbital angular momentum for pp / are ∼ 90◦ out of phase, as suggested by Eq. (21). An analogy is given in Ref. [6]. Suppose a bottle is 4lled with irregular shaped objects, such as screws. If the bottle is shaken, the objects inside settle to a more compact form. Returning to resonances, they have well-de4ned masses, widths and coupling constants which make them individual. But a suggestion is that, in the interaction, phases adjust to retain as much coherence as possible with the incident plane wave. This is a natural way of inducing order between the observed resonances. It helps explain why all Argand diagrams show a smooth variation of phases with mass. 7.2. !0 data Fig. 45 shows the 0  mass projection from !0 data. There are clear peaks due to a0 (980) and a2 (1320). The strong increase in the a2 signal at the higher masses is due to the opening of the a2 !

308

D.V. Bugg / Physics Reports 397 (2004) 257 – 358 (a)

(b)

(d)

(g)

(e)

(c)

(f )

(h)

Fig. 43. (a)–(h) Re T22 and (e)–(h) T20 for !0 data at 900, 1200, 1525 and 1940 MeV=c, compared with the partial wave 4t (histograms).

threshold at 2100 MeV. This threshold is also observed through its direct eOect on the integrated cross section, shown in Fig. 46. In contrast, integrated cross sections for most other channels peak at low mass, e.g. Fig. 10(a) for − + in Section 3. A feature of the !0 data is shown in Fig. 47. There is a clear peak at 1665 MeV in !. The amplitude analysis favours J PC = 1− − strongly over J PC = 1+− , 2− − or 3− − . The 4tted mass and width M = 1664 ± 17 MeV, 5 = 260 ± 40 MeV agree well with PDG averages for !1 (1650). The 4nal state !1 (1650) contributes strongly to the 1+− (1 P1 pp) / resonance at 1960 MeV. 7.3. Observed resonances Table 7 shows resonances from the 4nal combined analysis of all channels. The complete spectrum of expected qq/ states is observed except for the 3 G3 state and the highest 3 S1 state expected near 2400 MeV. The 3 G3 state may easily be hidden close to the strong upper 3 D3 state; indeed, that resonance has a ratio r3 of 3 G3 = 3 D3 amplitudes signi4cantly diOerent from 0; this hints at the presence of either a 3 G3 state or quantum mechanical mixing with it. There is even a hint of the last 3 S1 state. A scan of the mass of the 1− state at 2265 MeV reveals two distinct optima in log likelihood,

D.V. Bugg / Physics Reports 397 (2004) 257 – 358 10

5

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10

15 10

5 5

-2

0 -4 0

2

4

6

-10

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0

5

10

-5

0

5

Fig. 44. Argand diagrams for partial waves 4tted to !0 in the combined analysis with !0 and + − . Crosses show beam momenta 600, 900, 1050, 1200, 1350, 1525, 1642, 1800 and 1940 MeV=c; all move anti-clockwise with increasing beam momentum. L is the orbital angular momentum in the ! channel.

one at 2265 MeV and a second weaker one at 2400 MeV; however there is not suPcient evidence to isolate this state so close to the top of the available mass range. Intensities of partial waves 4tted to !0 are shown in Fig. 48. A remarkable feature is a very strong 3 G4 state at 2230 MeV. In all other data, G-states are weak, presumably because of the strong ‘ = 4 centrifugal barrier in the pp / channel. The analysis has been checked in many ways to ensure that there is no cross-talk with 3 D2 , which is quite weak at high momenta. As an example, analyses have been carried out at single energies. The 4tted 3 G4 intensity follows the behaviour expected from the centrifugal barrier. It falls close to zero at low momenta where the 3 D2 intensity is large; this is illustrated in Fig. 4 of Ref. [64]. So there is no doubt that the huge 3 G4 amplitude is a real eOect. There is no obvious explanation. It must indicate that for some reason this state decays dominantly through !. The low mass of the 3 G4 state is also surprising; it is lower than the nearby F-states. This is in contrast to the well established behaviour of I = 0 states where masses of G-state resonances are higher than those of F-states. Table 7 shows in the right-hand entries the 4rmly established decay channels where each resonance is observed. The star rating depends on the number of channels in which each resonance is observed and the stability of resonance parameters. Figs. 49 and 50 show triplet and singlet partial waves for pp / → !0 . Brief comments will now be given, starting with the highest spins.

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Fig. 45. Projection of !0 data on to M (0 ); full histograms show the 4t.

The %5 (2300) is already known from GAMS data [67]. The determination from pp / → − + is − − also accurate, from the large 5 amplitude observed there. GAMS omitted any centrifugal barrier for the production process; this barrier leads to a peak position in Crystal Barrel data of 2340 MeV in agreement with the mass of 2330 ± 35 MeV quoted by GAMS. The 4− state at 2230 MeV is observed independently in ! decays with L = 3 and 5 and also in a0 (980)! in Fig. 49(g). Since it is the dominant signal at high masses in !, it is very clearly established and classi4ed as 4∗ . The upper 3− state is observed in 5 channels; the lower state at 1982 MeV is strong in − + and accurately determined. Both are therefore classi4ed as 4∗ . Indeed, the trajectory of 3− − states (pp / 3 D3 ) in Fig. 3(c), including the very well known %3 (1690), provides one of the best determinations of trajectory slopes. The upper 2− state at 2225 MeV is not given 4∗ status because of correlations with 3− and − 1 partial waves. The lower 2− state at 1940 MeV is one of the two most weakly established states. However, its Argand plot in Fig. 44 requires a rapid phase variation at low mass because the amplitude must go to zero at the pp / threshold. So a resonance is required, even if it is at the bottom of the available mass range and therefore somewhat uncertain in mass.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

311

Fig. 46. Integrated cross section for pp / → !0 compared with the 4t. It is corrected for acceptance and for branching ratios of ! → 0 ,  →  and 0 → .

Table 7 Resonance parameters from a combined 4t to !0 , !0 and − + , using both ! → 0  and ! → + − 0 decays J PC

*

Mass M (MeV)

Width 5 (MeV)

r

SS (!)

SS (!)

Observed channels

5− − 4− − 3− − 3− − 2− − 2− − 1− − 1− − 1− −

4∗ 4∗ 4∗ 4∗ 3∗ 2∗ 2∗ 3∗ 3∗

2300 ± 45 2230 ± 25 2260 ± 20 1982 ± 14 2225 ± 35 1940 ± 40 2265 ± 40 2110 ± 35 2000 ± 30

260 ± 75 210 ± 30 160 ± 25 188 ± 24 335+100 −50 155 ± 40 325 ± 80 230 ± 50 260 ± 45

(0) 0:37 ± 0:05 1:6 ± 1:0 0:006 ± 0:008 1:39 ± 0:37 1:30 ± 0:38 −0:55 ± 0:66 −0:05 ± 0:42 0:70 ± 0:23

473 1254 341 314 356 433 — — 562

133 153 578 138 502 93 313 834 133

, !, GAMS [!]L=3; 5 , a0 !, (! ) , !, a2 !, b1 , !  , !, a0 ! !, a2 !, a0 ! !, a0 ! , a2 ! , a2 !, GAMS , !, a0 !

5+− 3+− 3+− 1+− 1+−

— 3∗ 4∗ 2∗ 3∗

(2500) 2245 ± 50 2032 ± 12 2240 ± 35 1960 ± 35

∼ 370 320 ± 70 117 ± 11 320 ± 85 230 ± 50

(0) 0:78 ± 0:47 2:06 ± 0:20 1:2 ± 0:5 0:73 ± 0:18

195 264 3073 62 503

— 187 178 542 514

! !, a2 !, ! , b1  (!), a0 !, ! , (b1 ) (!), a2 !, b1  !, a0 !, ! 

Values of r are ratios of coupling constants gJ +1 =gJ −1 for coupling to pp / or !0 . Columns 6 and 7 show changes in S= log likelihood when each resonance is omitted and others are re-optimised. In the 4nal column, ! denotes !1 (1650); channels in parentheses are weak.

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Fig. 47. Projection of !0 data on to M (!); full histograms show the 4t including an !(1665) with 5 = 260 MeV.

All three 1− − states are observed in three channels. The %(2150) was found by GAMS [68,69] and now deserves 3∗ status. The mass scan for the upper one gives a distinct optimum at 2265, but some indication of a second at 2400 MeV. It is possible that there are two states in the high mass range and they cannot be fully resolved. The state at 2265 MeV is therefore classi4ed as 2∗ . Now consider singlet states. The 3+− state (1 F3 ) at 2245 MeV is observed in 4 channels; intensities of singlet partial waves for !0 are shown in Fig. 50. Its presence is required in ! by the 360◦ loop in the Argand diagram of Fig. 44. However, there is a rather large error of ±50 MeV in its mass. The lower 3+− state is the strongest contribution to ! and deserves 4∗ status. The partial wave analysis requires a rather large 1+− intensity in the !(1650) S-wave, as shown in Fig. 50(h). It is identi4ed directly by the peak in the ! mass distribution. The !0 data also require a rather strong 1 P1 intensity at low masses. It cannot be eliminated without a large change of 503 in log likelihood. The Argand loop of Fig. 44 for !0 requires a strong phase variation when one remembers that the amplitude must go to zero at the pp / threshold; this demands a low mass 1+ resonance, even though there is some uncertainty in its precise mass. A 4nal remark is that Matveev reports a new b1 at M = 1620 ± 15 MeV with 5 = 280 ± 50 MeV at the recent AschaOenburg conference [70]. This is included in Fig. 3(c).

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

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25 0

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0

2

2.2

2.4

0

2

2.2

2.4

2

2.2

2.4

2

2.2

2.4

0

Fig. 48. Intensities of partial waves 4tted to !0 data; L is the orbital angular momentum in the decay.

8. I = 0, C = −1 resonances There are three publications of Crystal Barrel data with these quantum numbers. The 4rst was for !0 0 data with ! decaying to 0  [71]. This was followed by an analysis of ! where ! again decays to 0  [72]. The 4nal combined analysis added data for both channels where ! → + − 0 [7]. The measurement of ! polarisation in these data improves the stability of the analysis considerably, so conclusions will be taken from that work. Features of the data are very similar to those for I = 1, C = −1 and need little comment. Figs. 51(e)–(j) compare angular distributions for ! in decays via 0  (shown by the corridors) and via ! → + − 0 (points with errors). There is agreement within errors, as there was for !0 . This checks the data selection and Monte Carlo simulations, but with lower accuracy than for !0 because of lower statistics. Fig. 52 shows vector and tensor polarisation Re T21 . Other tensor

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(a)

(b) 20

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(e)

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(l)

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(h)

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(g)

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(c)

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(m)

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50 2.5

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2

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(p)

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4

5

10

(n)

8

2

2.2

2.4

0

2

2.2

2.4

0

2

2.2

2.4

Fig. 49. (a)–(p) Intensities of dominant triplet partial waves in !0 data; dashed lines refer to the second title in panels.

polarisations are large, as for !0 , and are shown in Ref. [7]. Again the vector polarisation is consistent with zero, indicating the same phase coherence amongst the amplitudes as for !. Fig. 53 shows intensities of partial waves 4tted to ! and Fig. 54 shows Argand diagrams. Fig. 55 shows intensities of the large partial waves 4tted to !0 0 data. The analysis of this family of I = 0, C = −1 states is the weakest of all four families. Statistics for the ! channel are only ∼ 5000 events per momentum. In the !0 0 data, there is a strong f2 (1270) signal but it is superimposed on an ! physics background. Uncertainties in parametrising the broad  S-wave amplitude above 1 GeV introduce possible systematic errors into this physics background and, via interferences, into the determination of resonance masses in the !f2 channel. However, without the ! channel, the data cannot be 4tted adequately, so there is no choice but to include it. There is evidence in the 4nal analysis for all expected qq/ states except for two 1− . However, several of them have a 2∗ rating because of (i) low statistics for !, (ii) the diPculties caused by the ! physics background in !0 0 , (iii) the lack of an accepted interferometer to establish

D.V. Bugg / Physics Reports 397 (2004) 257 – 358 (a)

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(d)

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(h) 100

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2.2

(i)

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50

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0

2

2.2

2.4

( j) 30

4 20 2 0

10

2

2.2

2.4

0

2

2.2

2.4

Fig. 50. (a)–(j) Intensities of singlet partial waves in !0 data; dashed lines refer to the second title.

phase variations. Table 8 shows 4tted resonances and their star ratings. None can be classi4ed as 4∗ because of the limited number of decay channels. It is best to start with the low mass range and triplet states. The 3 D3 partial wave in !0 0 data of Fig. 55(d) is very strong in the !f2 4nal state down to the lowest beam momentum. It cannot be confused with a threshold eOect in 3 S1 because 3 S1 and 3 D3 give rise to quite diOerent tensor polarisations of the f2 , hence distinctive decay angular distributions. For ! → + − 0 decays, the correlation between the normal to the ! decay plane and the decay angular distribution of the f2 is very sensitive to the diOerent Clebsch–Gordan coePcients for 3 D3 , 3 D2 , 3 D1 and 3 S1 amplitudes. The conclusion is that the selection of the 3 D3 initial state is very distinctive. This can be demonstrated by running a crude initial analysis at low momenta with just one partial wave at a time. A small but signi4cant b1  D-wave amplitude provides further evidence for the 3 D3 initial state. Results for ! are similar. Again the normal to the ! decay plane in ! → + − 0 decays provides good separation between 3 D3 , 3 D2 and 3 D1 . The momentum dependence separates pp / S 2 and D-waves. The diOerential cross sections for ! look like (1 + 2 cos 1) after correction for acceptance; here 1 is the production angle. The strong cos2 1 term requires initial S or D-states (and 4nal P-states). The 1 P1 amplitude gives the isotropic term. In the earliest analysis of Ref. [72], the 1 F3 amplitude was negligible. Argand diagrams of Fig. 54 for 3 D3 , 3 D2 and 3 D1 ! 4nal states require phase variations approaching 180◦ between the pp / threshold and 2100 MeV (a beam momentum of 1050 MeV=c).

316

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

( j)

Fig. 51. M (+ − 0 ) from the data selection for (a) ! data, (b) !0 0 at 900 MeV=c; the number of events vs. the square of the matrix element for ! decay in (c) !, (d) !0 0 ; (e)–(j): curves show the error corridor for ! diOerential cross sections where ! decays to 0 ; points with errors show results for ! decays to + − 0 ; both are corrected for angular acceptance; the vertical dashed lines show the cut-oO which has been used.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

317

Fig. 52. Vector polarisation Py for ! data at (a) 900, (b) 1200, (c) 1525 and (d) 1940 MeV=c compared with the partial wave 4t (histogram); (e)–(h) Re T21 at the same momenta.

These partial waves dominate the ! diOerential cross section up to 1200 MeV=c. The rapid decrease with momentum of the ! cross section forces all of the 3 D partial waves to agree with quite narrow widths: 115 MeV for 3 D3 , 175 MeV for 3 D2 and 195 MeV for 3 D1 . It is not possible to make them wider and 4t the observed cross sections. Observed phase relations between 3 D3 , 3 D2 and 3 D1 make resonances in all three partial waves unavoidable in the same mass range. The 3 S1 partial wave is diPcult to separate from a pp / threshold eOect, but its phase falls naturally into line with that observed for 3 D1 . In view of the well established I = 13 D3 state at 1982 MeV, these results are not surprising. An isotropic component of the ! diOerential cross section is 4tted naturally by 1 P1 ; again the normal to the ! decay plane in ! → + − 0 isolates this partial wave. A diPculty here is that there is no interferometer to de4ne the phase of the 1 P1 amplitude. However, its mass dependence √ 4ts naturally to a Breit–Wigner amplitude multiplied by the factor %pp / =p of Eq. (3), Section 4.1. The charged decay mode of the ! also successfully locates a low mass 1 F3 state at 2025 MeV; this state evaded detection in the analysis of ! → 0 . However, it is clear that all these resonances are correlated in masses and widths. There is no established resonance to act as interferometer, so resonance masses and widths are derived from the simple assumption of Eq. (3) for Breit–Wigner resonances. Results for masses come out in fair

318

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30

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8

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20 0

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1

5

0.5

2 50 0

1 2

2.2

2.4

0

2

2.2

2.4

0

2

2.2

2.4

0

Fig. 53. Intensities of partial waves 4tted to ! data; L is the orbital angular momentum in the decay.

Table 8 Resonance parameters from a combined 4t to ! and !0 0 using both ! → 0  and ! → + − 0 decays J PC

*

Mass M (MeV)

Width 5 (MeV)

r

SS (!)

SS (!)

Observed channels

5− − 4− − 3− − 3− − 3− − 2− − 2− − 1− − 1− −

2∗ 1∗ 2=3∗ 2=3∗ 3∗ 2∗ 3∗ 1∗ 3∗

2250 ± 70 2250 ± 30 2255 ± 15 2285 ± 60 1945 ± 20 2195 ± 30 1975 ± 20 2205 ± 30 1960 ± 25

320 ± 95 150 ± 50 175 ± 30 230 ± 40 115 ± 22 225 ± 40 175 ± 25 350 ± 90 195 ± 60

(0) (0) ∼ 50 1:4 ± 1:0 0:0 ± 1:0 0:28 ± 0:59 0:60 ± 0:10 −3:0 ± 1:8 2:4 ± 0:45

677 70 145 98 752 160 948 180 1258

244 — 1436 1156 5793 356 1203 1410 393

!, b1 ! only !, !f2 , (!) !, !f2 , (b1 ) !, !f2 , b1 !, !f2 !, !f2 , b1 (!), ! !, b1

3+− 3+− 1+− 1+−

3∗ 2∗ 2∗ 2∗

2275 ± 25 2025 ± 20 2215 ± 40 1965 ± 45

190 ± 45 145 ± 30 325 ± 55 345 ± 75

−0:74 ± 0:60 0:60 ± 0:49 4:45 ± 1:16 −0:20 ± 0:17

253 341 272 1808

140 234 1020 700

!, !, !, !,

!f2 !f2 !f2 , (!) !f2 , b1 , (!)

Values in parentheses are 4xed. Values of r are ratios of coupling constants gJ +1 =gJ −1 for coupling to pp / or !. Columns 6 and 7 show changes in S= log likelihood when each resonance is omitted and others are re-optimised.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358 0

2

-2.5

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319

0 -5

-2

-7.5 -4 -10

-5

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-1

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4

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5 2 2.5 1 0 0 -2.5 -1 -2

-1

0

1

2

-5

-5

-2.5

0

2.5

Fig. 54. Argand diagrams for partial waves 4tted to ! in the combined analysis with !0 0 . Crosses show beam momenta 600, 900, 1050, 1200, 1350, 1525, 1642, 1800 and 1940 MeV=c; L is the orbital angular momentum in the ! channel.

agreement with I = 1, C = −1; there the %3 (1982) is derived precisely from interferences with the known f4 (2018). In view of the assumptions in the I = 0, C = −1 analysis, it is not surprising that I = 0 D-states have masses which are systematically slightly lower than I = 1, C = −1. This may not be a real eOect. The conclusion is that all of the strong 1− , 2− and 3− states at low masses make a pattern consistent with I = 1; they deserve 3∗ states for their existence, subject to the reservation that there may well be some systematic error in their mass scale. The weak 1 P1 and 1 F3 states at low mass are classi4ed as 2∗ . Next consider high mass states. The Argand loops of Fig. 54 for 3 D1 , 3 D2 and 3 D3 all move through approximately 360◦ between the pp / threshold and the highest mass, 2410 MeV. This is

320

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150

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25 2

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2

2.2

2.4

Fig. 55. (a)–(p) Intensities of the large partial waves 4tted to !0 0 data.

consistent with a group of higher 1− , 2− and 3− states. The structure in the Argand loop for 3 D3 indicates two separate states. This structure is apparent also in the 3 D3 intensity of Fig. 53. Intensities of 3 D1 and 3 D3 partial waves drop to low values at high masses in Fig. 53, but the loops in Argand diagrams are distinctive. The situation concerning the 3− amplitudes at high mass is somewhat surprising. The measured tensor polarisation in the !f2 4nal state in !0 0 data appears to require a strong peak in Fig. 55(f). The data 4t naturally to a strong 3 G3 state at 2255 MeV and a weaker 3 D3 state at 2285 MeV. However, this result should be treated with caution. The polarisation of the ! in the 4nal state is sensitive to the orbital angular momentum L in the 4nal state. However, it is much less sensitive to the angular momentum ‘ in the production process; 3 G3 and 3 D3 states diOer only through Clebsch–Gordan coePcients for coupling of pp / helicity states. To be sure of the separation 3 3 between D3 and G3 , data from a polarised target are needed and are not available at present. A balanced view of the results is that there is evidence for both 3 G3 and 3 D3 states, but they are not

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321

well separated by present data. In Table 8, they are shown as 2∗ =3∗ ; the 3∗ refers to their existence, but the 2∗ refers to the imprecise identi4cation of masses. The upper 2− − state is weak in ! and determined only through its interference with other neighbouring states and with the lower 2− state. It is therefore only 2∗ . The upper 3 D1 state at 2205 MeV appears weakly in ! and is determined mostly by ! data. Because of the uncertainties in parametrising the broad , it can be classi4ed only as 1∗ . For I = 1, there are two neighbouring states at 2110 and 2265 MeV. This suggests that the I = 0 state may be a blurred combination of two states. The 5− state at 2250 MeV appears quite strongly in both ! and b1 , Fig. 55(g), but the mass has a rather large error, so it only deserves 2∗ status. The 4− state appears only in ! and is 1∗ . Turning to singlet states, the 1 F3 resonance at 2275 MeV appears strongly in !f2 and is also required by ! with L = 4. The Argand diagram for 1 F3 with L = 4 completes a full 360◦ loop which can only be 4tted with two 3+ states. Since the mass is well determined, this state is given 3∗ status. The upper 1+ state appears weakly in ! and is required by !f2 . It deserves 2∗ status. In summary, the pattern of I = 0, C = −1 states is very similar to that of I = 1, but is less well de4ned because of (i) the lack of data from a polarised target, (ii) the uncertainties in 4tting the ! channel. Several of the states lying at the bottom of the mass range may be subject to systematic errors in mass. That situation is better for I = 1 because of the strong 3− state at 1982 MeV. This state is accurately determined by − + data and then acts as an interferometer with other partial waves. No such interferometer is available for I = 0, C = −1.

9. Comments on systematics of qq+ resonances The 4rst obvious comment is that there is no proof that the Crystal Barrel states discussed in previous sections are all qq/ states rather then hybrids. However, the regularity of the spectra suggests an origin common with accepted qq/ states at low mass. There is 4rm evidence for just one hybrid with exotic quantum numbers J PC =1−+ from the E852 collaboration at 1593 MeV [73]. There is also evidence for structure with the same quantum numbers near 1400 MeV from the E852 collaboration and from Crystal Barrel data at rest [74,75]. Hybrids are expected closer to the 1800 MeV mass range. So it seems possible that the 1400 MeV structure is connected with the opening of the 4rst S-wave inelastic thresholds with these quantum numbers: b1 (1235) and f1 (1285). Further hybrids are expected in the mass range above 1800 MeV, but they are predicted to be broad and may be diPcult to locate. 9.1. Slopes of trajectories Figs. 1–3 of Section 1 summarise the observed spectra. Straight lines are drawn through them. An obvious comment is that straight trajectories may be an approximation. It is to be expected, for example, that the opening of strong thresholds may perturb the precise masses of resonances. An obvious example is f2 (1565), which sits precisely at the !! threshold and couples strongly to that channel [76]. Therefore Figs. 1–3 should be taken as a general indication of the systematics. However, the straight lines do have predictive power. For example, the b1 (1620) was actually predicted very

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Table 9 Slopes of trajectories. JP

Slope (GeV2 )

4++ 3++ 3 F2 3 P2 2−+ 1++ 0++ 0−+ 3− −

1:139 ± 0:037 1:107 ± 0:078 1:253 ± 0:066 1:131 ± 0:024 1:217 ± 0:055 1:120 ± 0:027 1:164 ± 0:041 1:183 ± 0:028 1:099 ± 0:029

close to this mass from the known mass of b1 (1230) and was found just where expected [70]. A second similar example is the recently discovered h1 (1595). There are still some missing states. In the mass range 1600–1800 MeV, an f1 is expected around 1670 MeV and an  close to 1680 MeV. These low spin states are diPcult to identify in production processes in the presence of strong signals of higher spins. The 3 D2 states of both isospins are also missing around 1660 MeV. The 0+ sector for I = 1 is poorly represented at present. Further states are expected around 1760 and 2380 MeV; the latter is right at the top of the Crystal Barrel range. Also the whole situation concerning 3 S1 states is obscure at present. Table 9 summarises slopes of I = 0, C = +1 resonances, which are the best established; it also includes a very well determined slope from the I =1, J PC =3− − family, where masses are particularly accurate. The mean slope is 1:143 ± 0:013 GeV2 . At high masses, trajectories necessarily become parallel because spin–orbit, tensor and spin–spin splittings will diminish; they arise from short-range 1=r 2 interactions. At the lowest masses, the ,  and  have been omitted from the trajectories because of the likelihood that their masses are aOected by the instanton interaction. It is rather obvious from Fig. 1 that 3 F2 states lie systematically higher than 3 P2 . A splitting of ∼ 80 MeV is observed. This is natural. For the F-states, there is a centrifugal barrier at short range. The need to overcome this barrier by attraction from the con4ning potential pushes up the masses of F-states. According to this argument, masses of D states should lie ∼ 40 MeV below F-states. The mass of the best established D-state, %3 (1982), is indeed 36 ± 15 MeV below that of f4 (2018). The centroids of 3 F2 , 3 F3 and 3 F4 states with I = 0, weighted by their multiplicities 2J + 1, are at 2292 ± 15 and 2024 ± 8 MeV. The 1 F3 states with I = 0 are at 2275 ± 25 MeV and 2025 ± 20 MeV. Within the sizable errors, results are consistent with the absence of spin–spin splitting. The well established 2 (2267) lies 25 ± 21 MeV below the centroid of the triplet F-states; and the somewhat less well determined 2 (2005) lies 19 ± 18 MeV below the centroid of the F-states. However, in this last case there is the possibility that the 2 (2005) is moved upwards by the well known level repulsion by 2 (1880). In any case, these determinations are not as de4nitive as the diOerence measured by %3 (1982). One would then expect G-states to lie ∼ 40 MeV above F-states. The situation here is not clear because most G-states are located with rather large errors.

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323

Tensor and spin–orbit splitting may be assessed from the relations [77] SMLS = [ − 20M (3 F2 ) − 7M (3 F3 ) + 27M (3 F4 )]=54 ;

(22)

SMT = [ − 4M (3 F2 ) + 7M (3 F3 ) − 3M (3 F4 )]=14 :

(23)

For both multiplets with I = 0, C = +1, 3 F3 states lie highest in mass, requiring signi4cant tensor splitting. This splitting has the sign predicted by one-gluon exchange. Using a linear con4ning potential plus one-gluon exchange, Godfrey and Isgur [78] predict 4MT = 9 MeV for the lower multiplet from single gluon exchange. From Table 5, one 4nds 4MT = 20 ± 4:5 MeV for the lower multiplet and 7:1±7:3 MeV for the upper one; errors take into account observed correlations in 4tted masses. Because the same centrifugal barrier is used for 3 F4 , 3 F3 and 3 F2 states, there is almost no correlation of SMT or SMLS with the radius of the barrier. The observed spin–orbit splittings are 4MLS = 2:4 ± 4:4 MeV for the lower multiplet and −6:3 ± 5:1 MeV for the upper one; these average approximately to zero. Glozman [79,80] has associated the small spin splitting at high masses with Chiral Symmetry restoration and the string picture of hadrons. He argues that this leads to parity doubling high in the spectrum and cites many examples. It is well known that Regge trajectories arise naturally from the string picture of hadrons: quarks tied together by a
(25)

Then J = M 2 =2, where  is the mass density per unit length. Fig. 56 shows Regge trajectories constructed from mesons quoted in this review. On Regge trajectories, M 2 is plotted against J . Here the centrifugal barrier comes into play, so the slopes of these trajectories will be slightly larger than for those plotted against radial excitation. It is found to be 1:167 GeV2 . It is not clear how to assign 1− states to these trajectories. The %(1450) does not seem to 4t naturally as the 4rst radial excitation of the %(770). It has been suggested by two groups that the mass of the % should be lower, ∼ 1:29 GeV [82,83]; this has been a well-known controversy for some years. The lower mass would 4t much better to a trajectory through the well-de4ned %3 (1982). Then %(1700) and %(2000) 4t on to the next two trajectories. However, the %(1700) is approximately degenerate with 3 D3 and is often taken to be the 3 D1 state. The %(2000) is found experimentally to have a signi4cant 3 D1 component, but could be a mixed state. So 1− − states present a dilemma. Better data are needed. A way forward would be to study diOractive dissociation of photons to distinctive channels like b1 (1235) and f1 (1285). These decays can go via L=0 or 2 for production. It is likely that matrix elements for these two decays will be distinctively diOerent for initial states

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Fig. 56. Regge trajectories. 3

D1 and 3 S1 , because of the D-wave tied up in the 3 D1 state. At higher energies, decays to % and %f2 (1270) are likely to throw light on the D-wave content of the initial state. If the luxury of a linearly polarised photon beam is available, D-wave amplitudes will appear directly via azimuthal dependence of the cross section on cos  and cos 2 with respect to the plane of polarisation. This arises through the partial wave decomposition    6 3 3 |2; 2L |1; −1s − |2; 1L |1; 0s + |2; 0L |1; 1s : |1; 1 = 10 10 10 A linearly polarised photon is a superposition of initial states |1; 1 and |1; −1 and  dependent terms arise in the diOerential cross section from the real part of the interference between these components. 10. Glueballs The latest lattice QCD calculations of Morningstar and Peardon [84] predict a sequence of glueballs beginning with J PC = 0++ at about 1600 MeV, 2++ around 2200 MeV, then 0−+ and a second 0++ close by in mass. There are ‘extra’ states which do not 4t naturally into qq/ spectroscopy at 1500, 1980, 2190 and 2100 MeV with just these quantum numbers. All except the f2 (1980) appear prominently in J= radiative decays, the classic place to look for glueballs; and there is tentative evidence for the f2 (1980) too. Lattice QCD calculations suOer from a scale error of ∼(10–15)%, so one should be prepared for some overall shift in mass scale. Also the lattice calculations are presently done in the quenched

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325

Table 10 Glueball mass ratios predicted by Morningstar and Peardon [84] compared with experimental candidates described in the text Ratio

Prediction

Experiment

M (2++ )=M (0++ ) M (0−+ )=M (0++ ) M (0∗++ )=M (0++ ) M (0−+ )=M (2++ )

1.39(4) 1.50(4) 1.54(11) 1.081(12)

1.32(3) 1.46(3) 1.40(2) 1.043(36)

Errors are in parentheses.

approximation, where coupling to qq/ states is omitted. However, mass ratios are predicted with smaller errors. Table 10 compares mass ratios of the four candidate glueballs with lattice predictions. There is remarkable agreement, suggesting this is not an accidental coincidence. This coincidence was pointed out by Bugg et al. [85]. Here the properties of the candidate glueballs will be reviewed in detail. What is clear immediately is that the candidate states do not decay
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(a)

(b)

Fig. 57. (a) The shape of f0 (1370): in the 2 channel (chain curve), in the 4 channel (dashed); the full curve shows a Breit–Wigner resonance of constant width. (b) m(s).

Section 4.1; it arises through dispersive eOects from the s-dependence of 5(4) and is shown in Fig. 57(b). It has a large eOect and consequently all claims about the mass and width should be treated with some reserve. The f0 (1370) does not appear signi4cantly in the Cern–Munich data for 2  elastic scattering [101,102], where its intensity depends on 52 . It does appears clearly in the new BES II data on J= → , Fig. 26, Section 5.5; the mass found there is 1370 ± 50 MeV and the width 265 ± 40 MeV [37]. With the bene4t of hindsight, it is also clear that the f0 (1500) was observed by Kalogeropoulos and collaborators [103–105]. It was also observed as G(1590) by GAMS [106–110], but shifted in mass through interference eOects; they appreciated its signi4cance and claimed it as a glueball because of its  decay, regarded then as a touchstone for gluonic content. Since its discovery, the f0 (1500) has been identi4ed as a 4 peak in a re-analysis of Mark III data on J= → 4 [39]. The f0 (1770) and f0 (2100) appear as two further peaks. The data have been shown earlier in Fig. 25(a), Section 5.5, together with E760 data which show a striking resemblance. In the original Mark III and DM2 analyses [111,112], the peaks were not separated from a large, slowly varying %% component with J P = 0− . The analysis of J= → 4 has been con4rmed by BES I [113]. Fig. 58, taken from this analysis, shows the slowly varying 0− signal in (a), and 0+ and 2+ components in (b) and (c); these are determined by the spin-parity analysis and are not conspicuous in the raw data, shown as points with errors. The latest BES II data [114] have 7 times higher statistics and con4rm the analysis, improving on all individual components. From the early analysis of J= radiative decays, there was some confusion whether the fJ at 1710 MeV in K K/ has J = 0 or 2. This has now been established as J = 0 by a re-analysis of those data by Dunwoodie [115]. It has also been con4rmed by the Omega group [116] and from new BES II data on J= → K K/ [117]. The a0 (1450) has recently been con4rmed in decays to !% in Crystal Barrel data on pp / → !+ − 0 at rest [118]. Decays to !% are a factor 10:7 ± 2:7 times stronger than to , where it was originally discovered. The weak decay to  explains why it has not been observed there in E852 and VES data. Fig. 59(a) shows the percentage contribution of a0 (1450) as its mass is scanned; (b) shows the line shape in diOerent channels; the peak appears quite narrow in the !% channel because of the unavoidable eOect of the rapidly rising 5(%!) vs. mass. The percentage contribution 4tted to data is shown against width in (c), using a Breit–Wigner amplitude of constant width as a test; (d) shows the Argand diagram in the  channel.

120 100 80 60 40 20 0

1.5

Events/(35MeV)

Events/(35MeV)

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

2

120 100 80 60 40 20 0

2

m π+π-π+π- (GeV/c ) Events/(35MeV)

Events/(35MeV)

1.5

2

2 2

m π+π-π+π- (GeV/c ) 120 100 80 60 40 20 0

1.5

327

2

120 100 80 60 40 20 0

1.5

2 2

m π+π-π+π- (GeV/c )

m π+π-π+π- (GeV/c )

Fig. 58. (a)–(d) Components from the BES I analysis of J= → (4).

(a)

(b)

(c)

(d)

Fig. 59. (a) Percentage contribution of a0 (1450) vs. 4tted mass M using a FlattUe formula including 5(), 5(%!) and m(s). (b) The line shapes for the FlattUe amplitude itself (full curve), its contribution to !% (dashed) and its contribution to pp / → (!%) (dotted). (c) Percentage contribution of a0 (1450) vs. 4tted width for a Breit–Wigner amplitude of constant width, (d) the Argand diagram for the  channel; numerical values of mass are shown in GeV.

In conclusion, there is available a standard nonet made of f0 (1370), a0 (1450), K0 (1430) and f0 (1710), similar to known 3 P2 and 3 P1 nonets. There is some doubt within ∼ 70 MeV of the precise mass of f0 (1370): two-body data tend to favour masses of 1300–1335 MeV and four-body data favour higher masses close to 1400 MeV. The mass of a0 (1450) may be aOected by its strong coupling to the !% threshold. So it would be a mistake to make too much of mass diOerences amongst the lowest three members of this nonet. The f0 (1710) is rather further in mass from f0 (1370) than

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f2 (1525) is from f2 (1270), but the f0 (1370) and f0 (1710) could be pushed apart by the usual level repulsion due to an extra state f0 (1500) between them. 10.2. Mixing of the glueball with qq/ states There have been many studies of the mixing, based on both the observed masses and on branching / The much stronger coupling of f0 (1500) to  than ratios of states between , ,  and K K. to K K/ implies that it is not primarily an ss/ state. Amsler and Close [119] consider the mixing of nn/ and ss/ components with the glueball. If the composition of the mixed state |G is written √ / + !|ss] / |Go  + K[ 2|nn ; (26) |G = 2 2 1 + K (2 + ! ) then lowest order perturbation theory gives / |Go =(EGo − Enn/) ; K = dd|V

(27)

! = (EGo − Enn/)=(EGo − Ess/) ;

(28)

where E are masses of unmixed states, Go refers to the unmixed glueball and V is the interaction responsible for mixing. If EGo lies midway between the masses of unmixed nn/ and ss/ states, |G will acquire extra dd/ component and will lose ss/ component. The state G then moves towards a
(29)

|f0 (1500) = −0:69|Go  + 0:37|ss / − 0:62|nn / ;

(30)

|f0 (1370) = 0:60|Go  − 0:13|ss / − 0:79|nn / :

(31)

Strohmeier-Presicek et al. [121] arrive at somewhat diOerent numerical coePcients, but the overall picture is similar. All authors neglect mixing between f0 (1710) and the nearby f0 (1770). Amsler points out that the observed upper limit for f0 (1500) →  is inconsistent with a pure qq/ state [122]. In a series of papers, Anisovich and Sarantsev examine a somewhat diOerent scheme [123], using the K-matrix. They take the view that K-matrix poles represent ‘bare’, unmixed states. The coupling constants of these poles to  etc. then ‘dress’ these bare states and allow for the eOects of meson decays. The virtue of their approach is that they 4t a very large volume of data on  elastic scattering,  → K K/ and , plus many channels of pp / annihilation at rest. There is also the virtue that bare nn, / ss/ and |Go  states are 4tted directly to data. They arrive at 4ve K-matrix poles from 650 to 1830 MeV and identify the lowest nonet with T -matrix poles for f0 (980), a0 (980), K0 (1430) and f0 (1370), all strongly mixed with a very broad glueball with mass 1200–1600 MeV. The f0 (1500), a0 (1450) and f0 (1710) then fall into the 4rst radial excitation.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

329

Fig. 60. The mechanism of central production.

There can be no doubt that their amplitudes 4tted to data reproduce the same narrow T -channel resonances and the same branching ratios as are found in other analyses. The diOerence lies in how to interpret the broad component which appears so conspicuously in the  S-wave from threshold to 2 GeV and can also be built into other channels as a broad underlying background [16]. I have had extensive and friendly discussions with Anisovich and Sarantsev about this scheme. It has the awkward feature that the lowest K-matrix pole is largely ss; / an obvious question is whether this could be replaced by the  pole discussed in the next section. Another point is that the broad structure at high mass may arise from the opening of the strong 4 threshold and associated dispersive eOects. This requires 4tting 4 cross sections to data. It is rather important to establish whether their scheme is unique. An alternative is to 4t (i) , , f0 (980) and a0 (980) as the lowest nonet (not necessarily qq), / (ii) the conventional nonet of f0 (1370), a0 (1450), K0 (1430) and f0 (1770) mixed with f0 (1500), plus (iii) f0 (1770) as a member of the next nonet. The question is whether this rearrangement of the broad background fails to 4t the data, and if so how. In all schemes, there is agreement that the data favour the presence of a glueball in the mass range close to 1500 MeV. Given this and the lattice QCD calculations, the question is: where are the next three glueballs? This question is addressed in the following. 10.3. Features of central production data In central production, two protons scatter through a small angle with small momentum transfer. The standard hypothesis for the mechanism of the reaction is shown in Fig. 60. Two gluons, or a ladder of gluons making the Pomeron, undergo a collision close to the centre of mass of the system and produce a meson R. There are extensive data from the Omega and WA102 collaborations at CERN for many 4nal states. Close and Kirk found a mechanism for 4ltering out well-known qq/ states; they select collisions where the two exchanged gluons have a small diOerence in transverse momentum, SPT [124]. Well-known qq/ states such as f1 (1285), f2 (1270) and f0 (1370) are P-states and have a zero at the origin of their wave functions. The idea is that they will be suppressed close to zero transverse momentum in the collision which forms resonance R. It is observed experimentally that this does indeed suppress well-known qq/ states, leaving exotic states such as f0 (1500). The idea developed further by examining the angular dependence of the collision. Treating it in analogy to e+ e− → R, Close, Kirk and Schuler studied the azimuthal dependence of the production

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Fig. 61. (a)–(d) The  dependence for central production of f0 (1370), f0 (1500), f2 (1270) and f2 (1950).

process [125–127]. They de4ne  as the azimuthal angle between the transverse momenta of the two exchanged gluons (reconstructed from the kinematics of the outgoing protons). From their model, the azimuthal dependence is predicted uniquely for production of 0− , 1+ and 2− resonances R. The data agree closely with predictions, vindicating the analogy between e+ e− interactions where photons are exchanged and central production where gluons are exchanged. If R has quantum numbers 0+ and 2+ , the azimuthal dependence is not unique. It depends on two cross-sections which they denote as TT and LL, standing for transverse or longitudinally polarised gluons. For 0+ and 2+ mesons, results are shown in Fig. 61 and are truly remarkable. The well known qq/ states f0 (1370) and f2 (1270) have one distinctive form of angular dependence; f0 (1500) and a broad 2+ state at ∼ 1950 MeV have a quite contrary azimuthal dependence. Although the origin of this diOerence is not understood yet, it suggests something diOerent about f0 (1500) and f2 (1950). If f0 (1500) has a large glueball content, it suggests that perhaps f2 (1950) does also. The broad f2 (1950) in central production data of 4 was 4rst reported by the Omega collaboration in 1995 [128] with a width of 390 ± 60 MeV. It may be the same object as reported earlier in K ∗ K/ ∗ by LASS [129] with a width of 250 ± 50 MeV. The WA102 collaboration produced higher statistics in 1997 [130] and again in 2000 [131] in 40 , + − 0 0 and + − + − . The latest data are shown in Fig. 62. Decays are consistent with being purely to f2 (1270), where the  has parameters close to those of the  pole discussed in the next Section. The 4 peak has M = 1995 ± 22 MeV, 5 = 436 ± 42 MeV for data containing 4 charged pions and M = 1980 ± 22 MeV, 5 = 520 ± 50 MeV for 2 charged pions. Branching ratios between the three sets of data are accurately consistent with I = 0. Note in passing that Fig. 62(e) con4rms the 2 (1645) and 2 (1870) discussed earlier in Section 5.3. Also note that the 0++ signal in (d) is at 1500 MeV and narrow. In (c) there is a broader 0++ signal peaking at 1450 MeV and coming from interference between f0 (1370) and f0 (1500); the upper peak in (e) is evidence for the 0+ state called f0 (2020) in Section 5.

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

331

Fig. 62. WA102 data for central production of + − + − : (a) total mass spectrum, (b) 1++ , (c) 0++ %%, (d) 0+ , (e) 2−+ and (f) f2 (1270) compared with the 4ts.

Meanwhile Crystal Barrel observed independently a very similar broad 2+ peak in  with M = 1980 ± 50 MeV, 5 = 500 ± 100 MeV, values very close to WA102. The observation was made in pp / → 0 at momenta 1350–1940 MeV=c. The  decay angular distribution is shown in intervals of  mass in Fig. 63. In the lowest interval, 1.45–1:55 GeV, the process is dominated by a strong f0 (1500) signal and the decay angular distribution is almost
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D.V. Bugg / Physics Reports 397 (2004) 257 – 358 (a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

Fig. 63. (a)–(h) Decay angular distributions for Crystal Barrel data for pp / → 0 in slices of  mass.

The broad f2 (1950) makes a large improvement to =2 in the combined analysis of I = 0, C = +1 data [3] in the  channel. Without it, there are awkward interferences between the two 3 P2 and 3 F2 interferences, so as to reproduce not only the narrow states but generate also a broad background. Adding the broad background explicitly has the eOect of decoupling the narrow states to a large extent. The broad component 4ts with M = 2010 ± 25 MeV, 5 = 495 ± 35 MeV. It appears in  2 with g =g2 0 0 = 0:72 ± 0:06. This is to be compared with the prediction 0.41 for nn/ and 1 for a glueball. It leads to a
D.V. Bugg / Physics Reports 397 (2004) 257 – 358

333

Fig. 64. Etkin et al. data on : (a) acceptance corrected mass spectrum, (b) intensities of partial waves, (c) relative phases; curves show the 4t to three resonances.

10.4.  →  Etkin et al. [133] observe J PC = 2++ peaks in  →  at 2150 and 2340 MeV (Fig. 64). This process violates the OZI rule. The overall observed cross section is a factor 100 stronger than can be explained readily by OZI conserving processes. An obvious suggestion is that the peaks are due to SU (3) singlet states which mix strongly with a broad 2+ glueball. The narrow states can be produced via an nn/ component and decay via the ss/ component. An f2 (2150) is listed by the PDG, corresponding to observation by WA102 in  [134] and K K/ [115] and by GAMS in  [135]. However, remaining entries listed by the PDG were quoted as the f0 (2100) in the original publications. An ss/ 3 P2 state is expected ∼ 250 MeV above f2 (1934) and could well explain the f2 (2150). It would 4t in with the S-wave decay to  of the 2150 MeV peak. Then a 3 F2 ss/ state is expected ∼ 100 MeV higher. Etkin et al. claim two D-wave  states at 2297 ± 28 MeV (with 5 = 149 ± 41 MeV) and at 2339 ± 55 MeV (5 = 319+81 −69 MeV). An obvious question is whether there is really one D-wave state and an interfering broad 2+ background from f2 (1950). A re-analysis of the data is required to answer this question. Palano [136] has presented JETSET data taken at LEAR on pp / →  (Fig. 65). A peak is observed in the D0 wave (J = 2, S = 0, L = 2). It is 4tted with M = 2231 ± 2 MeV, 5 = 70 ± 10 MeV; however, there is noise in the signal and Palano remarks that results could be compatible with the Etkin et al. peak at 2297 MeV. This is because of interference with a broader stucture in the D2 wave, J = 2, S = 2, L = 2, in the 4rst panel. Phase motion is observed between the D0 and D2 components. 10.5. Evidence for the 0− glueball This is an entertaining story of unitarity and analyticity at work. It has appeared in two publications by Bugg et al. [137,138]. However, even in these publications the full rami4cations of the story were not fully appreciated and a further twist needs to be given here. Radiative J= decays are a classic hunting ground for glueballs. The J= has J PC = 1− − . A process which conserves C-parity is the radiation of one photon and two gluons. The two gluons

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D.V. Bugg / Physics Reports 397 (2004) 257 – 358

Fig. 65. JETSET data for pp / → .

Fig. 66. (a) Mass spectrum for J= radiative decays to (a) (+ − ) after selecting a0 (980) [139], (b) K ∗ K/ ∗ [140] and (c) K ± KS0 ∓ [141].

will couple preferentially to glueballs because 9S (M 2 ) ∼ 0:5 at these masses, so the intensity of coupling to qq/ states is suppressed by a large factor. / 4 and K ∗ (890)K(890) / All of the radiative decays to , K K, contain a dominant 0− 4nal state. This is illustrated in Fig. 66 with data from a variety of experiments. In (a), from Mark III [139], attention was focussed on the two narrow peaks at low mass. What is not displayed is a broad 0− signal under the peaks, covering the whole mass range; this broad component is shown in Fig. 58(a). Other panels of Fig. 66 show BES I data [113,140,141]. In all three cases there is a broad 0− peak. Wermes made the 4rst attempt to 4t this broad signal [142]. Bugg et al. [139] showed that it may be 4tted by a single resonance, as will be explained below. However, their paper failed to point out one important feature. In the production process J= → 0− , there is one unit of orbital angular momentum for production. This leads to an E3 factor in the

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

335

Fig. 67. The line shape of the broad 0− component after dividing out the E3 factor.

0.6

(G)

Γ (GeV/c2)

0.4

(B)

(D)

0.2

(A) (C) (E)

(F) 0

1.4

1.6

1.8

2

2.2

2.4

2

Mass (GeV/c )

Fig. 68. Widths 5i (s) for (A) , (B) %%, (C) !!, (D) K ∗ K/ ∗ , (E) , (F) K and (G) total; C is shown dashed for clarity.

intensity, where E is the photon energy. This factor in
M2

>i  ; − s − m(s) − iM i 5i (s)

(32)

where >i are coupling constant for each channel; 5i (s) are taken to be proportional to the phase space available for decay to each channel, multiplied by a form factor exp(−R2 p2 =6) with R=0:8 fm; p is the momentum in the decay. These widths are shown in Fig. 68 after normalisation to cross

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sections in Fig. 69. The function m(s) is an s-dependent dispersive correction to the mass:  ∞  M5tot (s ) ds 2 : m(s) = (s − M )   2 4m2  (s − s)(s − M )

(33)

The total width 5tot tends to zero at high mass suPciently rapidly that the dispersion integral converges well with a subtraction at s = M 2 . In Figs. 68 and 69, the magnitude of each 5i is 4tted freely. The 4t gives coupling constants squared for %%, !!, K ∗ K/ ∗ and  remarkably close to the ratio 3:1:4:1 expected for
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Table 11 Branching fractions of glueball candidates in J= radiative decays, compared with predictions of Close, Farrar and Li [143] State

Prediction

Observation

f0 (1500) (2190) f0 (2105) f2 (1950)

1:2 × 10−3 1:4 × 10−2 3:4 × 10−3 2:3 × 10−2

(1:03 ± 0:14) × 10−3 (1:9 ± 0:3) × 10−2 (6:8 ± 1:8) × 10−4 (2:6 ± 0:6) × 10−3

at 1800 MeV by the opening of the strong K ∗ K/ ∗ channel. The resulting line shape is rather diPcult to interpret. Perhaps, one should take a somewhat larger full-width than 350 MeV in order to allow for the shoulder at low mass. The 4tted mass 2190 ± 50 MeV is somewhat below the peak value of 2260 MeV because of the strong eOect of the K ∗ K/ ∗ width. 10.6. Sum rules for J= radiative decays Branching fractions for production of glueballs in J= → (gg) will now be considered. Close, Farrar and Li [143] make predictions for branching fractions for individual J P of gg; results depend on masses and widths of resonances. Using experimental values for these, Table 11 [85] makes a comparison of their predictions with experiment. The branching ratio for f0 (1500) is close to prediction for a glueball and far above that predicted / [14,28] is slightly above for qq. / The production of (2190) in %%, !!, , K ∗ K/ ∗ ,  and K K the prediction. The f0 (2105) has so far been observed in J= radiative decays only to  [39]. For this mode alone, the observed production is only 20% of the prediction; decays to  and  will also contribute, but their decay branching ratios relative to 4 are not presently known. The prediction for the branching fraction of the 2+ glueball is large if the width is taken to be the 500 MeV 4tted to f2 (1950). Observed decays to  and f2 (1270) account for (10±0:7±3:6)×10−4 of J= radiative decays and K ∗ K/ ∗ decays a further (7±1±2)×10−4 . If one assumes
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A D-state glueball would be suppressed even further. Possibly there is some other explanation of the enhancement of f0 (1500) and f2 (1950) at small SPT in central production. 11. The light scalars There has been much argument whether  and  exist; also whether f0 (980) and a0 (980) are part of the ground-state qq/ family or whether they are 4-quark states or K K/ ‘molecules’. Over the last 2 years valuable new information has appeared. The experimental situation will be reviewed 4rst, followed by a brief summary of theoretical approaches. The interpretation as simple qq/ states faces two problems: (i) why masses are so much lower than well-known families of 3 P1 , 3 P2 and 1 P1 mesons, (ii) why the  and  are so much lower in mass than a0 (980). An alternative explanation in terms of ‘molecular’ states is possible but not yet proven. This is an idea familiar from nuclear physics. Protons and neutrons are presumably made of quarks, but nuclear physics is sensitive only to the ‘residual’ long-range force between nucleons. The deuteron is a bound state of this long-range force; the light scalars may well originate from analogous residual forces between mesons. A pointer in this direction is that the anomalous states appear only in the scalar sector, not with other quantum numbers. It is possible that S-wave attraction is suPcient to produce resonances, but the centrifugal barrier in other partial waves is too strong. One of the topics discussed below will be radiative decays of (1020) to f0 (980) and a0 (980). It is illuminating to think about this process in reverse. Consider K K/ scattering near threshold. An S-wave attraction may originate either by meson exchanges or by direct gluonic interactions between quarks. Meson exchanges between 4 quarks arise through colour zero con4gurations. A variant is possible where quarks form intermediate coloured pairs. Whatever the origin of the attraction, we know that f0 (980) and a0 (980) resonances are formed close to the K K/ threshold. Suppose these resonances can be excited by an E1 photon with an energy of order 30 MeV. This con4gures the quarks so that u and u/ pairs or d and d/ pairs can annihilate, leaving the ss/ con4guration (1020) and hence revealing a short-range qq/ state. Wave functions are essential to the dynamics, hence matrix elements. We do not presently know wave functions well enough. In a similar fashion, excitation of the f0 (980) by an E2 transition could access a con4guration where all quarks annihilate leaving the 2+ glueball. Again, we presently have no understanding of glueball wave functions. Chiral Perturbation Theory is able to describe mesons scattering up to 1:2 GeV using an ‘EOective Lagrangian’ which parametrises empirically the interaction between mesons. The essential element in this approach is the Adler zero in  and K elastic scattering just below threshold. 11.1. The  The E791 collaboration has observed a low mass + − peak in D+ → + − + [145]; they 4t it +42 with a simple Breit–Wigner resonance with M = 478+24 −25 MeV, 5 = 324−40 ± 21 MeV. A similar low mass peak was observed earlier in DM2 data for J= → !+ − [146]. New BES II data [147] also contain this peak at ∼ 525 MeV, as shown in Fig. 70(c). This leaves no doubt about the existence of the . It appears in the Dalitz plot in (b) as a conspicuous band at the upper right-hand edge of the plot. The 4tted  component is shown cross-hatched in (e) and the 2+ component in (f); (e) also shows the coherent sum of  and a small f0 (980) contribution.

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Fig. 70. (a) The ! peak; the lower curve shows the quadratic 4t to the background. (b) the Dalitz plot for !+ − . (c) and (d) projections on to  and ! mass; histograms show the 4t and the shaded region the background estimated from sidebins; the dashed curve in (d) shows the 4tted b1 (1235) signal (two charge combinations). (e) and (f) show mass projections of f0 and f2 contributions to + − from the 4t; in (e), the shaded area shows the  contribution alone, and the full histogram the coherent sum of  and f0 (980).

Why does the  not appear the same way in  elastic scattering? The answer is discussed in a recent paper which refers to both  and  [148]. In  elastic scattering, there is a well-known Adler–Weinberg zero just below the elastic threshold [149] at sA m2 =2. To a 4rst approximation, it cancels the pole in the denominator of the amplitude near threshold, with the result that the phase shift 4(s) is rather featureless at low mass and shows little direct evidence for the pole. If the Adler zero is factored out of the amplitude, the pole becomes apparent.

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(a)

(b)

Fig. 71. (a) Phase shifts for  elastic scattering; dark points are phase shifts of Hyams et al. [101] above 450 MeV and from Ke4 [150] data below 400 MeV; open circles show 0 0 data [151,152]; the curve shows the 4t. (b) The trajectory in the complex s-plane where the real part of the amplitude goes through zero.

This Adler zero is absent in production processes, because the  pair is produced with large momentum. The production amplitude is > ; (34) T (J= → !) = 2 M − s − iM5(s) where 5 is s-dependent. For  elastic scattering, the amplitude must lie on the unitarity circle and the corresponding amplitude is M5(s) T ( → ) = 2 : (35) M − s − iM5(s) There is an extra factor M5(s) in the numerator. The denominator is common to all  reactions. If a Breit–Wigner amplitude describes the  elastic amplitude, numerator and denominator must be self-consistent. The rate for a transition is proportional to |F|2 , where F is the matrix element for the transition. The width of a resonance is proportional to this transition rate. It is therefore logical to include the Adler zero explicitly into the width. The idea behind this is that the matrix element for  elastic scattering goes to zero for massless pions as momentum k → 0. In production reactions such as J= → !, the pions are no longer ‘soft’ and no Adler zero is required. The parametrisation used in Ref. [148] is s − sA −(s − M 2 ) 5(s) = %(s) 2 f(s) exp : (36) M − sA A Here sA = m2 =2, the position of the Adler zero. Empirically, the best form for f(s) is b1 + b2 s. A combined 4t is made to BES II data, CERN-Munich phase shifts for − + → − + [102], the most recent Ke4 data [150], CP violation in K 0 decays [11] and data for − + → 0 0 [151,152]. The 4t to elastic data is shown by the curve in Fig. 71(a). There are some systematic discrepancies above 740 MeV between phase shifts of Hyams et al. (4lled circles) and 0 0 data (open circles) and the 4t runs midway between them. Fitted parameters are M = 0:9254, A = 1:082, b1 = 0:5843, b2 = 1:6663, all in units of GeV. For elastic scattering, there are other small contributions from higher resonances f0 (980), f0 (1370), f0 (1500), etc. They are accommodated by adding their phases to that of the ; this prescription satis4es elastic unitarity. In principle there could be a further background amplitude. The question

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Table 12 Pole positions of the  for the three 4ts of BES II data (and  elastic data in 4t A) Fit

Form

Pole position (MeV)

A B C

Eqs. (35)–(36) 5= ˙ %(s) 5 = constant

542 ± 25 − i(249 ± 25) 570 ± 30 − i(274 ± 35) 542 ± 32 − i(269 ± 32)

is whether the simple prescription of Eqs. (34)–(36) 4ts the data. In practice it does. A consistent 4t is found to both elastic and production data. As a check, the BES data alone have also been 4tted with (i) 5 = 50 %(s)=%(M 2 ), where %(s) is two-body phase space, (ii) a Breit–Wigner amplitude with constant width. All three pole positions are in close agreement, as shown in Table 12. The 4rst entry is the most reliable since it 4ts all data. The square of the modulus of the amplitude of Eq. (34) peaks at 460 MeV, slightly below the peak in Fig. 70(c), because of the rapidly increasing phase space for !. The pole is in reasonable agreement with the result of Colangelo et al.: M = (470 ± 30) − i(290 ± 20) MeV [153], determined from  elastic scattering. Their determination has the virtue of 4tting the left-hand cut consistently with the right-hand cut, using the Roy equations. However, they do not 4t data above 800 MeV, so the eOect of f0 (980) is omitted. Markushin and Locher [154] show a table of earlier determinations which cluster around the values in Table 12, with quite a large scatter. A popular misconception is that the pole needs to be close to the mass M where the phase shift passes through 90◦ on the real axis. That is true only when 5 is constant. Because 5 varies with complex s in an explicit way, the phase of the amplitude varies as one goes oO the real s axis to imaginary values. Fig. 71(b) shows the position on the complex plane where the phase goes through 90◦ ; there is a rapid variation with Im m. In BES data, the phase of the  amplitude is determined by strong interferences with b1 (1235). There are two b1 (1235) bands on the Dalitz plot of Fig. 70(b) and together they account for 41% of the intensity. The  accounts for 19% and f2 (1270) for most of the rest. An independent check is made on the phase of the  amplitude by dividing up the  mass range into bins 100 MeV wide from 300 to 1000 MeV. In each bin, the magnitude and phase of the  amplitude are 4tted freely, re-optimising all other components. Results for the phase are shown in Fig. 72. Curves show phases from the three parametrisations described above. [A detail is that zero phase is taken from 4t A at the  threshold. Strong interactions introduce an overall phase for all formulae, so the dashed and dotted curves have been adjusted vertically to agree with the full curve at 550 MeV; only the variation of the phase with mass is meaningful.] There is only small discrimination between the three forms. The agreement of phases with the curves demonstrates the correlation of magnitude and phase expected from analyticity. Magnitudes in individual bins
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Fig. 72. The phase of the  S-wave amplitude. The full curve shows 4t A, the dashed curve 4t B (Breit–Wigner amplitude of constant width) and the dotted curve 4t C (5 ˙ %(s)); points with errors show results 4tted to slices of  mass 100 MeV wide.

11.2. The  Evidence for the  comes from three sources: (i) data on K elastic scattering from LASS, (ii) E791 data for D+ → K − + + and (iii) BES data for J= → K + K − + − . The story runs parallel to that for the . The LASS phase shifts for K elastic scattering [156] are shown in Fig. 73(a). They are 4tted in Ref. [148] with the coherent sum of K0 (1430) and a  amplitude. The  is parametrised with s − sA 5(s) = %(s) 2 50 exp − (9M ) ; (37) M − sA sA = MK2 + m2 =2. As for , it is assumed that phases of the  and K0 (1430) add. For K0 (1430), a FlattUe form is used: (s − sA ) 5(s) = 2 [g1 %K (s) + g2 %K
(s)] ; (38) M − sA there is no improvement if decays to K are included. There is some freedom in 4tting the K0 (1430), because it rides on top of the  ‘background’. The pole position of the  is at M = (722 ± 60) − i(386 ± 50) MeV. Fitted parameters are M = 3:3 GeV, b1 = 24:53 GeV, 9 = 0:4 GeV2 . These parameters are highly correlated and one can obtain almost equally good 4ts over the range M = 2:1–4 GeV by re-optimising b1 and 9. The 4tted value of M for the  is very high; it is essential that it lies above the K0 (1430) since the phase shift does not reach 90◦ before that resonance. The K0 (1430) is 4tted with M = 1:535 GeV, g1 = 0:361 GeV, g1 =g1 = 1:0. Its pole position is (1433 ± 30 ± 10) − i(181 ± 10 ± 12) MeV, close to the values 4tted by Aston et al.: M = (1412 ± 6 MeV; 5 = 294 ± 23) MeV. [The full-width 5 is twice the imaginary part of the pole position.]

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Fig. 73. (a) K phase shifts. Points are from LASS [156]. The full curve shows the 4t to LASS data and the dashed curve the  contribution. (b) The 4t to the intensity of K elastic scattering from LASS data. (c) The intensity of the  amplitude in production processes, normalised to 1 at its peak; the full curve is from the 4t to LASS data and the dashed curve from the 4t to E791 data. (d) the  phase 4tted to BES II data compared with the curve 4tted to LASS data.

1 The 4tted K scattering length is 0:17m−  , very close to Weinberg’s prediction from current − 1 algebra of 0:18m . Fits to K phase shifts are shown in Fig. 73(a) and to the intensity of the elastic amplitude in Fig. 73(b). The 4tted  phase is shown by the dashed curve in Fig. 73(a); it reaches ∼ 70◦ at 1600 MeV.It is possible that the phase for real s does not actually reach 90◦ . A similar case arises for the well-known NN 1 S0 virtual state, where the phase shift rises to 60◦ and then falls again. The full curve in Fig. 73(c) shows the intensity of the  amplitude as it will appear in production processes; the curve is normalised to 1 at the peak at 700 MeV. In production reactions such as D+ → + , the observed intensity will be multiplied by the available three-body phase space. If the Adler zero is removed from the parametrisation of the K S-wave and from the K0 (1430), the  pole disappears. So the existence of the  is tied closely to the existence of the Adler zero. Zheng et al. have reported similar results [157]. The E791 collaboration has also reported an observation of the  pole in D+ → K − + + [158]. They parametrise the  width as 5=50 %(s)=%(M 2 ), where %(s) is K phase space; they 4nd M =797± 19 ± 43 MeV, 50 = 410 ± 43 ± 87 MeV. The corresponding pole position is at M = 721 − i292 MeV. The intensity calculated from their Breit–Wigner denominator is shown by the dashed curve in Fig. 73(c). Their  is distinctly narrower than from LASS data.

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Fig. 74. (a) The scatter plot of M (K + − ) against M (K − + ). (b) The projection on to K ± ∓ mass; the histogram shows the 4t. (c) The projection on to  mass. (d) The projection on to K mass. (e) The Dalitz plot for events where one K ± ∓ combination is in the mass range 892 ± 100 MeV. (f) The mass projection of the second K ∓ ± pair for the same selection as (e).

BES II data for the  in the channel J= → K + K − + − have been presented by Wu [159]. Strong K ∗ (890) and K0 (1430) bands are seen in the scatter plot of Fig. 74(a) and as peaks in (b). In (c) a peak due to %(770) is visible and in (d) a peak due to K1 (1270) and K1 (1410). The  appears as a broad low mass peak in the 4nal state K ∗ (890)K. In (f), one K ∗ (890) → K ± ∓ combination is chosen and then the mass distribution of the other K ∓ ± combination is plotted. The  appears as a broad K peak underneath a second K ∗ (890). In the analysis, it is necessary to show that the  signal does not arise from decays of K1 (1270) and K1 (1410), since all their decays give low mass K combinations. The broad low mass signal of Fig. 74(f) in fact arises partly from this source, but mostly from , K0 (1430) and their interferences. Two discrete solutions exist, both giving a  pole but with diOerent widths. Fig. 75(a) shows that a 4t without the  in the K S-wave is bad. One solution is obtained by including the Adler zero into the formulae, as in Eqs. (37) and (38). There is then a broad

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Fig. 75. (a) The poor 4t when the channel K ∗ (890) is removed; (b) individual contributions from  (full histogram) and K0 (1430) (dashed); (c) the coherent sum of  + K0 (1430) (dashed histogram) and the coherent sum  + K0 (1430) + K1 (1270) + K1 (1410) (full); (d) the phase of the  amplitude (full curve), compared with phases in individual slices of 100 MeV (points with errors).

 with pole position M = (760 ± 20 ± 40) − i(420 ± 45 ± 60) MeV; the K0 (1430) pole position is (1433 ± 32) − i(181 ± 16) MeV. In this solution, there is rather strong destructive interference between  and K0 (1430). Fig. 75(b) shows contributions from  (full curve) and K0 (1430) (dashed); further combinations are shown in (c). The phase of the  is determined accurately from interference between  and K0 (1430). Results are shown as points in Fig. 75(d). They agree closely with the full curve, which shows the 4t to LASS data. It is important to note that in this solution, the coherent sum of  and K0 (1430) produces a low-mass peak with a width of ∼ 400 MeV, as shown by the dashed histogram in Fig. 75(c). This narrower peak is similar to the solution found by E791. The

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eOect of the destructive interference is also to make the K0 (1430) peak appear narrower than it really is. One should note that the width 4tted to K0 (1430) by E791 is likewise low: 175 ± 12 ± 12 MeV. In the second type of solution, the Adler zero is omitted from the formulae for both  and K0 (1430). Simple Breit–Wigner amplitudes are 4tted with 5 ˙ %(s). In this solution, there is weak constructive interference between  and K0 (1430). The width 4tted to the  is smaller, 450–600 MeV, in agreement with the rather narrow  peak in Fig. 75(c). This solution can be made consistent with LASS data by adding a strong background amplitude which produces a negative phase shift proportional to centre of mass momentum; the sum of this repulsive contribution and the  generates the Adler zero [160]. Both approaches require a  pole. The diOerence between them reduces to one question. Should the Adler zero be included directly into the width of the ? Or should it appear as a destructive interference between a conventional Breit–Wigner resonance and a background amplitude arising from the left-hand cut? It might be possible to resolve this question by 4tting both right and left-hand cuts; but uncertainties about distant parts of the left-hand cut (how to include Regge poles) might obstruct this way of resolving the issue. A small pointer to the 4rst solution is / at rest 4nd K0 (1430) parameters M = 1415 ± 25 − i that Crystal Barrel data for pp / → K K (165 ± 25) MeV. 11.3. Theoretical approaches Several theory groups have tackled the question of 4tting Chiral Perturbation Theory (ChPT) to data. Ollet, Oset, PelUaez and collaborators have published a series of papers where they use ChPT for the second order driving force and then unitarise the theory by including fourth order terms [161–166]. Some parameters of ChPT are not known accurately. They vary these and 4t to data. A virtue of these calculations is that they also 4t repulsive phase shifts in the I = 2  channel and the I = 3=2 K system. Poles are required for , , f0 (980) and a0 (980). A representative set of pole positions is quoted and 4ts to data are shown in a recent review by PelUez [167]. Their  pole tends to come out around 440–i215 MeV, slightly narrower and lower than values given above; their  is typically 750–i226 MeV, again fairly narrow. Schechter and collaborators adopt an eOective Lagrangian based on ChPT [168–171]. Their conclusion is that there is mixing between qq/ and a 4-quark system, similar to that proposed by JaOe [172]. Beisert and Borasoy also adopt an eOective Lagrangian approach [173] 4tting experimental phase shifts but including  as an explicit degree of freedom. An approach based on meson exchanges was pioneered by the JVulich group of Speth et al. [174–176]. Zou and collaborators adopt this approach, 4tting data with K-matrix unitarisation [177,178]. Morgan, Pennington and collaborators emphasise unitarity and analyticity as a general approach to 4tting data [54,179,180]. Kaminski et al. likewise 4t data imposing constraints of analyticity and crossing symmetry [181]. The approach of TVornqvist is based on the linear  model, broken SU (3) and analyticity [182,183]. Fits by the Ishida group are reviewed in Ref. [184]. There is an important diOerence between the Ishida model and my approach. In the Ishida model, a short-range repulsive interaction is assumed between mesons. The  and  are then parametrised as conventional Breit–Wigner resonances with widths proportional to the phase space for decay. The coherent sum of the phases from these two sources generates the observed phase shifts and the Adler zero. In my approach, the short-range

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interaction is attractive. The scattering lengths for  and K scattering are small because there is a compensating long-range repulsive interaction. This repulsive interaction arises from the conventional con4ning potential which is usually assumed to rise linearly with r, because of the tension in the
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11.4. f0 (980) Recent BES II data on J= → + − and K + K − provide a major improvement in the parameters of f0 (980). It appears as a conspicous peak in Fig. 26, Section 5.5, interfering on its lower side with the . The important point is that it is also observed clearly in decays to K K/ (Fig. 27). It is 4tted with the FlattUe formula: 2 f = 1=[M 2 − s − i(g %(s) + gK2 K/ %(s)K K/ )] :

(39)

2 The branching ratio between K K/ and  determines gK2 K/ =g = 4:35 ± 0:25. In earlier work, this ratio / From BES data, M = 970 ± 8 MeV and was often unclear because of lack of information on K K. g12 = 145 ± 10 MeV. The value of g2 correlates rather strongly with M and g1 because the last term in Eq. (39) needs to be continued analytically below threshold so as to satisfy analyticity explicitly. The paper gives the correlations.

11.5. a0 (980) Crystal Barrel data at rest provide accurate parameters for a0 (980). Data on pp / → !0 [187] determine the full-width at half-maximum very precisely: 5 = 54:12 ± 0:34 ± 0:12 MeV. This value is used as a constraint in 4tting data on pp / → 0 0 , where the a0 (980) is conspicuous [88]. On the Dalitz plot, the K K/ threshold appears as a very clear ‘edge’. The intensity of the signal changes abruptly there. From the change in slope at this edge, the relative coupling to  and K K/ is determined rather accurately compared with most other experiments. The a0 (980) in 4tted with a 2 2 FlattUe formula like Eq. (39) with M = 999 ± 5 MeV, g = 0:221 ± 0:020 GeV, gK2 K/ =g = 1:16 ± 0:08. 11.6.  →  and  Data for (1020) → 0 and 0 0 from Novosibirsk [188–190] and the Kloe group at Daphne [191,192] are a topic of great interest. Achasov and Ivanchenko pointed out in 1989 that these data may throw light on a0 (980) and f0 (980) [193]. A vigorous debate has followed the publication of data; Boglione and Pennington [194] give an extensive list of references. I have myself made unpublished 4ts to these data and wish to draw upon one important conclusion, which supports work of Achasov and collaborators. The 4ts are made using parameters of f0 (980) / and a0 (980) given in the previous two subsections; amplitudes are taken from the K K-loop model of Achasov and equations of Achasov and Gubin [195]. Apart from the absolute normalisation, there are no free parameters. The results of the 4t to Kloe data for the  4nal states are shown in Figs. 76(a) and (b). The good 4t immediately veri4es the K K/ loop as the correct model for the process. This loop gives rise to a violent energy dependent factor |I (a; b)|2 shown in (d). This multiplies the factor E3 for the E1 transition, where E is photon energy. Agreement of the model with data leaves little doubt that the basic mechanism is understood. The situation is not quite so good for  → 0 0 . Fig. 77 shows the 4t. However, there are two complications which aOect the mass range below 0:6 GeV (not important for present conclusions). The 4rst is that the 4nal state %0 , % → 0  contributes there and is poorly known in normalisation. The second is that there may be a small  contribution there. The 4t to the f0 (980) peak using BES

D.V. Bugg / Physics Reports 397 (2004) 257 – 358

(a)

(b)

(c)

(d)

349

Fig. 76. Kloe data for  → 0 for (a)  → , (b)  → + − 0 ; full curves show the 4t. The dashed curve in (a) illustrates uncertainties from a possible form factor and the Adler zero; (c) the energy dependent factor appearing in d5=dm, but omitting |I (a; b)|2 , (d) |I (a; b)|2 itself.

Fig. 77. Kloe data for  → 0 0 ; the full curve shows the 4t and the dashed curve the f0 (980) intensity alone. 2 parameters looks slightly narrow, but can be optimised by increasing g by 1:5 to 160 MeV. With / those mild reservations, the 4t is again good to the f0 (980) peak and veri4es the K K-loop model. Achasov argues that these data reveal the origin of a0 (980) and f0 (980) [196]. The transition  → a0 (980) is OZI-violating, since ss/ must change into nn. / Physically, the argument is that ss/ states like  and f2 (1525) do not decay readily to non-strange mesons. His conclusion is that √ / a0 (980) must be a state such as JaOe [172] has proposed, made up of (uss / u/ − dss / d)= 2. The u and / d quarks in the K K-loop are transmitted to f0 (980) and a0 (980) 4nal states. JaOe’s proposal is that pairs of q and q/ form SU (3)
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Fig. 78. Momentum-space wave functions of f0 (980), full curve and a0 (980), dashed.

Fig. 79. The line shape of f0 (980), full curve; the K K/ fraction in the wave function, dotted curve; and 0:1 ∗ RMS radius, dashed curve.

/ It is not obvious how to A K K/ molecule contains the same quarks, but con4gured into K K. separate this possibility from JaOe’s model. The f0 (980) and a0 (980) have small binding energies. They must therefore be surrounded by long-range clouds of kaons. For√a given binding energy B, the wave function of the kaon pair is ˙ (1=r) exp(−Nr), where N ˙ MB; M is the reduced mass. Figs. 78 and 79 show results obtained by weighting | |2 with the line shape of f0 (980) or a0 (980). Wave functions of a0 (980) and f0 (980) are remarkably similar. This is illustrated in momentum space k in Fig. 78. It need not be so. The FlattUe formulae describing them have very diOerent parameters at face value. Nonetheless, the wave functions of f0 (980) and a0 (980) are within 10% of one another for all k. That suggests they have a common character.

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351

TVornqvist [18] points out that both a0 (980) and f0 (980) must contain a K K/ component. His Eq. (15) includes mesonic channels Ai Bi :  |qq / + i [ − (d=ds) Re Oi (s)]1=2 |Ai Bi   : (40) = 1 − i (d=ds) Re Oi (s)  Here Re O(s) = gK2 K/ 4m2K =s − 1 for s ¡ 4m2K . This formula is easily evaluated to 4nd the K K/ components in a0 (980) and f0 (980) as functions of s. At the K K/ threshold, these components are of course 100%. But they are ¿ 40% for all masses above 900 MeV. This is illustrated in Fig. 79 by the dotted curve. This 4gure also shows as the full curve the line shape of f0 (980) vs. mass. The RMS radius of the kaonic wave function is shown by the dashed curve in Fig. 79, scaled down by a factor 10 for display purposes. When the intensity of f0 (980) is above half-height, the RMS radius is ¿1:7 fm. A large kaonic cloud surrounding a0 (980) and f0 (980) is unavoidable. The photon in  radiative decays couples to kaons dominantly at large r, where the E1 matrix element has a large dipole moment e|r|. Note too that the kaons couple with charge 1, unlike the quarks which have charges 1/3 or −2=3. Achasov argues that K K/ ‘molecules’ cannot explain the observed large rates of  radiative decays. However, a quantitative assessment of the rates is needed, taking into account the large radii of the kaonic clouds. It is hard to believe that the matrix element for  radiative decays can be dominated by small radii. However, the matrix element for coupling of K K/ to f0 (980) or a0 (980) will be concentrated at small r. This matrix element may go via a short-range interaction at the quark level. Or it may go through t-channel meson exchanges; these will be dominated by exchange of K ∗ (890). This / It is then ‘natural’ that a0 (980) and f0 (980) lie close exchange will be responsible for  → K K. / to the K K threshold. In Ref. [148], the radial dependence of the matrix element for  →  was found from the Fourier transform of the matrix element in momentum space. Although this matrix element is uncertain above 1 GeV, as shown by the dashed curve in Fig. 80(a), F(r) is de4nitely concentrated at a radius ¡ 0:4 fm in (b) and (c). Although the binding of f0 (980) and a0 (980) could come either from interactions between quarks or from meson exchanges, these states must have large inert kaonic clouds, as TVornqvist emphasises. Kalashnikova and collaborators [197] re-examine an old idea of Weinberg to decide whether the a0 (980) and f0 (980) are composites of elementary particles or are ‘bare’ states. The choice between these possibilities rests upon whether the resonances have a single pole close to the K K/ threshold or a pair of poles on diOerent sheets. A single pole favours a composite, like the deuteron. Morgan made this point in a diOerent way in 1992 [198]. The BES data give a clear answer for f0 (980). There is a single nearby second-sheet pole at M = 0:999 − i0:014 GeV and a distant third-sheet pole at M = 0:828 − i0:345 GeV. For a0 (980), the situation is not so clear-cut. There is a second-sheet pole at 1:032 − i0:085 GeV and a third-sheet pole at M = 0:982 − i0:204 GeV. 11.7. Ds (2317) The discovery of the narrow Ds (2317) by Babar [199] and Cleo [200] has generated a
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(a)

(b)

(c)

Fig. 80. Matrix elements F for  →  (a) in momentum space, F(s), (b) F(r) and (c) r 2 F 2 (r). The full curves show the optimum 4t to elastic data and dashed curves show results with a stronger exponential cut-oO in s above 1 GeV2 .

Fig. 81. From Ref. [202]. The movement of the a0 (980) pole is shown as a function of the coupling constant for the I = 1 K K/ S-wave. Values of coupling constants are indicated. The inset shows a magni4ed view near the K K/ threshold.

There, they model the attraction in the I = 1 K K/ S-wave, ignoring coupling to . Fig. 81 shows the movement of the pole as the coupling constant to K K/ is varied. A virtual state appears 4rst at 0:76 GeV. As the coupling to K K/ increases, the pole migrates along the lower side of the real axis to the K K/ threshold. For still larger coupling, it becomes a bound state and slowly moves away from the K K/ threshold on the upper side of the cut. The pole remains in the vicinity of the K K/ threshold for quite a wide range of coupling constants. This leaves little doubt that the K K/ threshold plays a strong role in the dynamics. If the mass of the kaon were to change, the pole would move / The attraction in this with it. This does not necessarily mean that a0 (980) is a bound state of K K. channel may arise from a variety of processes generating attraction, either at the meson level or at the quark level.

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Likewise, the Ds (2317) appears as either a virtual or bound state of DK with some cs/ component mixed in. The regular cs/ state is predicted to move up in mass above the mass of simple quark models because of level repulsion. Similar phenomena near the BK threshold are predicted. However, there are many other competing ideas to explain Ds (2317) and other narrow states. The  and  appear close to  and K thresholds, but are forced away from those thresholds by the Adler zeros. It is good for us that this happens. If they were bound states below threshold, the Universe as we know it would be totally diOerent because of the long-range  exchange which binds nuclei. Van Beveren and Rupp predict [201] that there will be a D 0+ state not far above the D threshold, resembling  and . This could be the D0∗ (2290) reported by Belle with a width of 305 MeV [203]. 12. Future prospects It is possible to check experimentally whether f0 (1710) is a ground-state qq/ state or a radial excitation. This may be done by checking its production rate in Ds+ → f0 (1710). This decay is dominated by c → sW + , followed by W + turning into + . If the f0 (1710) is a radially excited ss/ state, as Anisovich and Sarantsev would have us believe, the decay rate will be suppressed by at least a factor 100 because its radial wave function is almost orthogonal to that of the Ds . E852 and VES have data which could locate the missing qq/ states in the mass range 1600–1800 MeV. The I = 0 hybrid partner of 1 (1600) is also missing. CLEO C is already taking data on  ; data on J= and  will follow during the next 5 years with better angular acceptance than BES II and excellent  detection. A high priority should be looking for the 2+ glueball in  → (4) as suggested above. Decays J= → ,  and  are important for full information on 0+ , 2+ and 0− glueballs and sorting out mixing schemes with /  and  would be simple and valuable. Radiative decays of qq. / A search for f0 (2100) → K K,   are almost unknown at present; the node in its radial wave function will throw light on wave functions of resonances produced in  decays. High quality data on = states should also appear. This looks an exciting program of physics over the next few years. The way forward with 1− − qq/ states is to study diOractive dissociation of the photon. What would complete the picture of qq/ states and make it more secure is to rerun the Crystal Barrel experiment using a frozen spin polarised target. Study of 0  and 30 would remove ambiguities and tighten the analysis for I = 1, C = +1. Study of !0 and ! channels would likewise improve and hopefully complete I = 1 and 0, C = −1 states. Such an experiment is technically straightforward. Data are needed right down to 360 MeV=c to cover the tower of resonances around 2 GeV. This is a possible program using the new p/ ring proposed for GSI. It could bring the long program of work on light non-strange mesons to a de4nitive conclusion. Acknowledgements It is a pleasure to thank the St. Petersburg group of Volodya Anisovich, Andrei Sarantsev, Viktor Nikonov, Viktor Sarantsev and Alexei Anisovich for their keen cooperation in analysing the Crystal Barrel data in
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cooperation on many topics over 9 years. Thanks go also to the Crystal Barrel collaboration for eOorts over many years in producing the data; Martin SuOert and Dieter Walter deserve special praise for their patient work perfecting the performance of the CsI crystals. The work has been funded for 16 years by the British Particle Physics and Astronomy Committee. Special thanks go to the Royal Society for supporting collaboration between Queen Mary and the BES collaboration over a six year period. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]

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Physics Reports 397 (2004) 359 – 449 www.elsevier.com/locate/physrep

Working with WKB waves far from the semiclassical limit Harald Friedricha; b;∗ , Johannes Trosta b

a Physik-Department, Technische Universitat Munchen, 85747 Garching, Germany Department of Theoretical Physics and Atomic and Molecular Physics Laboratories, RSPhysSE, Australian National University, Canberra, A.C.T. 0200, Australia

Accepted 1 April 2004 editor: J. Eichler

Abstract WKB wave functions are expected to be accurate approximations of exact quantum mechanical wave functions mainly near the semiclassical limit of the quantum mechanical Schr4odinger equation. The accuracy of WKB wave functions is, however, a local property of the Schr4odinger equation, and the failure of the WKB approximation may be restricted to a small “quantal region” of coordinate space, even under conditions which are far from the semiclassical limit and close to the anticlassical or extreme quantum limit of the Schr4odinger equation. In many physically important situations, exact or highly accurate approximate wave functions are available for the quantal region where the WKB approximation breaks down, and together with WKB wave functions in residual space provide highly accurate solutions of the full problem. WKB wave functions can thus be used to derive exact or highly accurate quantum mechanical results, even far from the semiclassical limit. We present a wide range of applications, including the derivation of properties of bound and continuum states near the threshold of a potential, which are important for understanding many results observed in experiments with cold atoms. c 2004 Elsevier B.V. All rights reserved.
PACS: 03.65.Sq; 03.75.Be; 34.20.Cf; 03.65.Xp Keywords: Modi>ed WKB theory; Semiclassical and anticlassical limits; Threshold e?ects; Quantum reAection

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Semiclassical theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 ∗

Corresponding author. Physics T30, Technical University Munich, James-Franck Str., 85747 Garching, Germany. Tel.: +49(89)28912729; fax: +49(89)28914656. E-mail addresses: [email protected] (H. Friedrich), [email protected] (J. Trost). c 2004 Elsevier B.V. All rights reserved. 0370-1573/$ - see front matter  doi:10.1016/j.physrep.2004.04.001

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2.1. The semiclassical and the anticlassical limit of the Schr4odinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The WKB approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Accuracy of WKB wave functions as a local property of the Schr4odinger equation . . . . . . . . . . . . . . . . . . . . . 2.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Beyond the semiclassical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Connection formulas at classical turning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The reAection phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Scattering by a repulsive singular potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Quantization in a potential well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Example: Circle billard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Example: Potential wells with long-ranged attractive tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Near the threshold of the potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Scattering lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Example: Sharp-step potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Example: Attractive homogeneous potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Near-threshold quantization and level densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Example: The Lennard-Jones potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Nonhomogeneous potential tails . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The transition from a >nite number to in>nitely many bound states, inverse-square tails . . . . . . . . . . . . . . . . . 4.5. Tunnelling through a centrifugal barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Quantum reAection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Analytical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Homogeneous potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantum reAection of atoms by surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Coupled channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

363 367 369 371 373 373 374 375 381 382 383 389 395 396 399 399 401 404 406 410 417 421 422 425 429 439 443 446 446

1. Introduction The relation between classical mechanics and quantum mechanics has been a central theme of physics ever since quantum mechanics was discovered at the beginning of the last century. Schr4odinger’s wave equation of quantum mechanics was published in 1926 [1], and in the same year Wentzel [2], Kramers [3] and Brillouin [4] developed the semiclassical approximation now known as the WKB approximation. It is largely equivalent to a formulation given earlier by Je?reys [5] in analogy to an approximation in wave optics which is expected to work well when the wavelengths are small compared to typical lengths in the physical system under investigation. The formulation of the WKB approximation is straightforward for conservative systems with just one degree of freedom and easily generalized [6] to systems with several degrees of freedom as long as they are integrable, meaning essentially, that their multidimensional classical dynamics can be decomposed into the dynamics of a corresponding number of conservative one-dimensional systems. It has been known since 1917 [7], that such generalizations are not straightforward for nonintegrable systems. Seemingly simple systems such as the three-body problem in celestial mechanics are nonintegrable, as was already pointed out by PoincarPe [8–10] who also observed that a possible

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consequence of nonintegrability is an extreme sensitivity of the dynamical evolution on small variations of the initial conditions, a phenomenon nowadays called “chaos”. The subtle complexities contained in the evolution of deterministic classical systems became a subject of general interest in the last quarter of the twentieth century, as continuous advances in computer capabilities enabled detailed numerical studies of nonintegrable classical systems on a large scale. This development rekindled interest in the fundamental relation between classical mechanics and quantum mechanics, in particular for nonintegrable systems with largely chaotic classical dynamics [11–13], but also for integrable [14,15] and almost integrable systems [16,17]. New semiclassical theories were developed, in which properties of a quantum spectrum such as the energy level density are expressed in terms of a sum over all periodic orbits of the corresponding classical system. An overwhelmingly large share of the immense body of research based on such periodic orbit theories has been devoted to “simple” model systems, mostly with one or two and rarely with more than two degrees of freedom. There have, however, also been more realistic applications in nuclear physics [13], solid state physics [18] and, in particular, atomic physics [19–23] which have brought forward many intriguing new insights on the interplay between classical dynamics and quantum mechanics. These developments have reinstated classical mechanics as a useful and relevant theory for atomic (and molecular) systems, even though quantum mechanics is the more general and undisputedly the valid theory in this domain. Semiclassical theories are expected to work well near the semiclassical limit of the Schr4odinger equation, which can be de>ned formally as the limit ˝ → 0. However, physically interesting situations do not always ful>ll the conditions of the semiclassical limit. Furthermore, semiclassical approximations may not be appropriate for all quantum states, even near the semiclassical limit. Consider, for example, an integrable system such as the circle billard [24–26]—a free particle moving in a two-dimensional space bounded by a circle. The bound states can be labelled by two good quantum numbers, one for the angular momentum around the centre of the circle and one for the radial motion; the semiclassical limit is the high-energy limit in this case. At high energies, there are many states for which both quantum numbers are large, but there are always some states for which one of the two quantum numbers is small. There is no energy regime where all states can be considered to be semiclassical. This is just one example showing the necessity to consider improvements of the conventional semiclassical approximations. Improvements of >rst-order semiclassical approximations are often formulated by including corrections of higher order in ˝ in the approximation of the Schr4odinger equation [27]. This works quite well in many cases, but it is not a general solution, because the resulting series is asymptotic and not convergent, and the dependence of physical quantities on the small parameter ˝ is in general nonanalytic near the semiclassical limit, ˝ → 0. “Soft” quantum e?ects due to interferences of contributions related to di?erent classical paths connecting given initial and >nal states may be well described via a series expansion in the small parameter ˝, but this approach is not appropriate for the description of “hard” quantum e?ects related to classically forbidden processes [28]. For example, probabilities for tunnelling through a smooth, analytic potential barrier are generally exponentially suppressed towards the semiclassical limit, so they do not contribute in any >nite order of ˝. Alternative methods of improving the conventional WKB approximation are proposed frequently, and we shall just mention a few recent examples: Bronzan [29] proposed a modi>ed WKB method which accurately reproduces the nodeless ground state of a system with one or more degrees of freedom; Gomez and Adhikari [30] introduced an algebraic reformulation of the WKB quantization condition which is useful in strongly con>ning potentials; a supersymmetric version of WKB

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quantization [31] is known [32,33] to yield exact energy eigenvalues for a large class of special (“shape invariant”) potentials. Further treatments of the one-dimensional Schr4odinger equation are based on the transformation of the second-order linear di?erential equation to a nonlinear >rst-order equation of the Ricatti type, which can be approximated via the quasilinearization method as described by Mandelzweig and Krivec [34,35]. The work reviewed in this article is based on a di?erent, less formal approach to improving the conventional WKB method. It exploits the fact, that the accuracy of WKB wave functions is a local property of the Schr4odinger equation. Away from the semiclassical limit, WKB wave functions may not be useful approximations in the whole of coordinate space, but there may be—and often are—large regions of coordinate space where the WKB wave functions are highly accurate, even if the conditions of the semiclassical limit are far from being ful>lled for the Schr4odinger equation as a whole. In many physically important situations, exact or highly accurate wave functions can be obtained for the “quantal regions” of coordinate space where the WKB approximation breaks down, e.g. when the potential in the Schr4odinger equation has a simple structure allowing analytical solutions in this region. By matching the exact or highly accurate wave functions from the quantal region to the WKB wave functions which are accurate in the “WKB region” outside the quantal region, we can construct wave functions which are accurate solutions of the Schr4odinger equation in the whole of coordinate space. These globally accurate wave functions can be used to derive quantum mechanical properties of the physical system which are highly accurate or asymptotically exact, even under conditions which are far from the semiclassical limit, all the way to the anticlassical or extreme quantum limit. This is demonstrated in a variety of applications in the following sections. Section 2 contains a brief review of some elements of semiclassical theory, including a discussion of the semiclassical and anticlassical limits of the Schr4odinger equation and of the (local) condition for the accuracy of WKB wave functions. In Section 3 we discuss improvements of the WKB method based on a generalization of the conventional connection formulas relating the WKB waves on either side of a classical turning point. This greatly increases the range of applicability and the performance of the WKB approximation as demonstrated in three di?erent situations: scattering by a repulsive potential, quantization in a potential well and classically forbidden tunnelling through a potential barrier. Section 4 is devoted to the narrow range of energies immediately above or below the threshold of a potential. In atomic and molecular physics, the near-threshold regime is of substantial current interest because it is relevant for the description of experiments involving ultra-cold atoms and molecules. Recent progress in the cooling of atoms to temperatures below the microkelvin regime and the successful construction of atomic degenerate quantum gases was rewarded with the Nobel Prize for physics in the years 1997 and 2001. There is intense and growing interest and activity in the >eld of ultra-cold atoms, e.g. in the preparation and understanding of quantum degenerate gases of bosonic and fermionic atoms [36,37], and also in the construction of atom-optical elements such as atom waveguides [38,39]. For potentials with tails falling o? faster than 1=r 2 the threshold represents the anticlassical or extreme quantum limit of the Schr4odinger equation where conventional WKB expansions break down. It turns out that several important near-threshold properties of the potential such as the quantization rule and the level density below threshold, and the mean scattering length and the reAectivity of the potential tail above threshold, are determined by three independent tail parameters which can be derived from the zero-energy solutions of the Schr4odinger equation. This information is of considerable practical use, not only in relation to cold atoms, but also for

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the accurate description of molecular reactions in near-threshold situations, e.g. for understanding the inAuence of quantum e?ects in the vibrational dissociation of molecules close to the dissociation threshold [40]. One example of “hard” quantum e?ects in the near-threshold region is quantum reAection by attractive potential tails as occur in the interaction of atoms and molecules with each other and with surfaces. In the quantum reAection of atoms by surfaces, the incoming atoms are reAected at distances of many hundreds or thousands of atomic units where the atom–surface interaction is accurately described by a local potential of the van der Waals type with retardation e?ects. The quantum reAected atoms do not reach the moderate distances of a few atomic units, where the atom–surface interaction becomes more intricate and leads to inelastic reactions or capture (sticking). For potentials falling o? faster than −1=r 2 , the probability for quantum reAection approaches unity at threshold, so it is always an important e?ect at suSciently low energies. The rapid progress in the >eld of cold atoms has made the quantum reAection regime increasingly accessible to laboratory experiments [41,42] and a number of investigations of this fundamental quantum process can be expected in the near future. The theory of quantum reAection is described in some detail in Section 5.

2. Semiclassical theory In this section we give a brief and necessarily sketchy review of some elements of semiclassical theory which will be needed for the subsequent sections. For more elaborate discussions, the reader is referred to textbooks and reviews [13,21,43–47]. 2.1. The semiclassical and the anticlassical limit of the Schrodinger equation Our starting point is the time-independent Schr4odinger equation for a particle of mass M moving under the inAuence of a potential V (r),



(r) +

2M [E − V (r)] (r) = 0 : ˝2

(1)

Most of the considerations of this section apply also to higher-dimensional systems, but subsequent sections are based essentially on the one-dimensional Schr4odinger equation (1). Planck’s constant ˝ de>nes a system-independent reference for measuring classical actions in the corresponding classical system. These depend on the local classical momentum
p(r) = 2M[E − V (r)] (2) and are given by  
Sc = p(r) dr = 2M[E − V (r)] dr :

(3)

In the more general case of D dimensions, r and p are replaced by D-dimensional vectors ˜r, p ˜ with components ri , pi , and the action that a particle accumulates on a trajectory C in 2D-dimensional

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phase space is  Sc =

D 

C i=1

pi dri :

(4)

When the potential is radially symmetric, V (˜r) = V (|˜r|), only the motion along the radial coordinate r = |˜r| is nontrivial; this is again described by a one-dimensional Schr4odinger equation (1), but the energy stored in the angular degrees of freedom shows up as a centrifugal potential after transformation of the equations of motion to spherical coordinates. It is generally accepted that quantum mechanical e?ects are important when typical classical actions are of the same order as or smaller than Planck’s constant ˝, and that the semiclassical limit, where quantum mechanical e?ects become less important, is the regime where typical classical actions are large compared to ˝. This leads to the formulation of a de>nition of the semiclassical limit of the Schr4odinger equation as “˝ → 0”, which is a little unfortunate, because ˝ is a well de>ned constant roughly equal to 10−34 J s. A more satisfactory de>nition of the semiclassical limit is to require Sc
˝

(5)

for typical actions Sc , and the “anticlassical” or “extreme quantum” limit is appropriately de>ned as Sc ˝ :

(6)

From Eq. (3) we see that increasing the mass M in the Schr4odinger equation leads towards the semiclassical limit, whereas for small M one can approach the anticlassical limit. In an atomic or molecular system, however, mass is generally a >xed and not a tunable parameter. Eq. (3) also suggests that the semiclassical limit might be reached for high energies, but this depends on the nature of the potential, as shown below. When the potential consists of a sum of homogeneous terms, the semiclassical and anticlassical limits can be reached for various combinations of large and/or small values of the energy and the strengths of the homogeneous terms, as discussed in Ref. [48], see also Section 5.3.4 in Ref. [49]. Here we give a brief derivation for one homogeneous term, i.e. for a potential with the property Vd (r) = d Vd (r) ;

(7)

the real number d denotes the degree of homogeneity, e.g., d = −1 for Coulomb potentials and d = +2 for the harmonic oscillator. Classical motion in homogeneous potentials has the property of mechanical similarity [50], i.e. if r(t) is a valid solution of the equations of motion at energy E, then r(1−d=2 t) is a solution at energy E
= d E. This rescaling of energy with a factor j ≡ d and of the coordinates according to s = r has the following e?ect on the classical action (3): 
√ Sc (jE) = 2M ds jE − Vd (s) 1 = j2

√

 2M

ds




E − j−1 Vd (s)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 1

= j2

√

1 1 = j2+d

 2M

ds

√

 2M




365

E − Vd (r)

dr




E − Vd (r) = j1=2+1=d Sc (E) :

(8)

This means that an increase in the absolute value |E| of the energy, j ¿ 1, results in an increase of the action Sc if and only if 1 1 + ¿0 ; 2 d

(9)

or equivalently d¿0

or

d¡ − 2 :

(10)

For homogeneous potentials of positive degree, such as all homogeneous oscillator potentials V ˙ |r|d , d ¿ 0, the semiclassical limit Sc =˝ → ∞ is reached for |E| → ∞, and the limit of vanishing energy, E → 0, corresponds to the anticlassical or extreme quantum limit. This intuitively reasonable behaviour also holds for negative degrees of homogeneity as long as d ¡ − 2. In contrast, for negative degrees of homogeneity in the range −2 ¡ d ¡ 0, the opposite and perhaps counterintuitive situation occurs: the limit of vanishing energy E → 0 de>nes the semiclassical limit, whereas |E| → ∞ is the anticlassical, extreme quantum limit. All attractive or repulsive Coulomb-type potentials, for which d = −1, fall into this category. This is consistent with the observation that the energy eigenvalues of the hydrogen atom scale the inverse square of the principal quantum number n and that the semiclassical limit corresponds to n → ∞, E → 0. For positive energies, the di?erential cross section for scattering by a 1=r potential in three spatial dimensions, the Rutherford cross section, is known to be identical in classical and quantum mechanics, but in two spatial dimensions this is not the case—here the quantum mechanical di?erential cross section approaches the classical result for E → 0, but for E → ∞ it deviates from the classical result and the prediction of the quantum mechanical Born approximation becomes more accurate [51–54]. The >ndings above may still be applicable for weakly perturbed homogeneous potentials, as long as the disturbance of the homogeneity (7) becomes negligible in the semiclassical or anticlassical limit. In some cases, the potential is a superposition of homogeneous terms with compatible semiclassical limits, e.g. a superposition of a quartic and an harmonic oscillator, V (r) = c2 r 2 + c4 r 4 . In this case, the high-energy limit E → ∞ corresponds to the semiclassical limit for both terms and for the potential as a whole. When the potential contains terms with incompatible limits, then the semiclassical limit may be reached neither for |E| → ∞ nor for E → 0. A hydrogen atom in a magnetic >eld is an example for such a situation. The electron feels the Coulomb attraction of the proton, and the magnetic >eld contributes a harmonic oscillator potential perpendicular to the >eld direction [19]. The semiclassical limit is not reached for high energies because of the Coulombic term, and it is not reached for low energies because of the harmonic contribution. For a given magnetic >eld strength, the system has no semiclassical limit. Note, however, that an appropriate simultaneous variation of energy and >eld strength can lead to the semiclassical or the anticlassical limit for this system [48].

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

In order to see how the variation of the strength of a homogeneous potential a?ects the proximity to the semiclassical or anticlassical limits, consider the potential Vd (r) = V0 Ud (r) ;

(11)

where Ud (r) is a given homogeneous potential with degree d and V0 is a variable strength parameter. If r(t) is a solution of the classical equations of motion for the potential Ud (i.e. V0 = 1) at energy E, then r(t) is a solution for the potential strength V0 = −d at the same energy. For the action (3) this implies, 
Sc (V0 ) = ds 2M[E − V0 Ud (s)]  =

dr




2M[E − Ud (r)] = (V0 )−1=d Sc (1) :

(12)

For constant energy, the semiclassical limit Sc =˝ → ∞ is reached for V0 → 0 when the degree d of homogeneity is positive, and for V0 → ∞ when d is negative. The case d = −2, i.e. potentials proportional to the inverse square of the coordinates, is special and important. Such potentials occur in the interaction of a monopole charge with a dipole and as centrifugal potential in the radial part of the Schr4odinger equation in two or more dimensions. The classical action (8) becomes independent of energy for d = −2, but there is still a dependence on the potential strength according to Eq. (12). Since d is negative, the semiclassical limit is for in>nite potential strength while the anticlassical limit corresponds to the limit of vanishing strength of the inverse-square potential. For nonhomogeneous potentials with two or more energy or length scales, the de>nition of the semiclassical limit is not always straightforward and unambiguous. Consider, for example the Woods-Saxon potential step, VWS (r) = −

V0 ; 1 + exp(r=)

(13)

which is characterized by its depth V0 = ˝2 (K0 )2 =(2M) and the di?useness parameter . A quantum particle approaching such a step with a total positive energy E = ˝2 k 2 =(2M) is partially reAected (see Section 5), and the reAection probability PR is known [43,46] to be   sinh[(q − k)] 2 PR = ; (14) sinh[(q + k)]
where q = k 2 + (K0 )2 is the asymptotic wave number on the down side of the step, r → −∞. In the Schr4odinger equation (1) we can consider the formal limit ˝ → 0 while keeping the energy E and potential strength V0 >xed. This corresponds to taking the limit k → ∞ and K0 → ∞, and the reAection probability (14) becomes, PR

k;K0 →∞

∼

exp(−4k) ;

(15)

which is now independent of q. If on the other hand, we respect the fact that ˝ is >xed and study the high-energy (k → ∞) limit of Eq. (1), then the parameter K0 de>ning the height of the step

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

remains constant and the formula for the reAection probability becomes, 2  (K0 )2 k →∞ PR ∼  exp(−4k) : k

367

(16)

We see that the high-energy limit for PR di?ers from the “formal” semiclassical limit (15) by a factor inversely proportional to the energy. This is another subtle example demonstrating that the high energy limit need not coincide with the semiclassical limit Sc =˝ → ∞. 2.2. The WKB approximation For vanishing or constant potential, the local classical momentum (2) is constant, and the Schr4odinger equation (1) has plane wave solutions,   i (r) ˙ exp ± pr : (17) ˝ When the potential is not constant, p is a function of r, and looking at Eq. (17) suggests a more general ansatz for the wave function,   i S(r) : (18) (r) = exp ˝ Inserting (18) in the Schr4odinger equation (1) gives a di?erential equation for the function S(r), S
(r)2 − i˝S

(r) = p(r)2 : In order to obtain an approximate solution for S(r), and thus for the wave function S(r) in a formal series in ˝, which is regarded as a small parameter,  2 ˝ ˝ S2 (r) + · · · : S(r) = S0 (r) + S1 (r) + i i

(19) (r), we expand (20)

In the spirit of conventional semiclassical theory, the expansion (20) assumes that all other relevant quantities of the dimension of an action are large compared to ˝, as discussed in the preceding section. Inserting Eq. (20) in Eq. (19) gives a di?erential equation for the Si (r), p2 (r) − (S0
)2 + i˝(S0

+ 2S0
S1
) + ˝2 (S1

+ 2S0
S2
+ (S1
)2 ) : : : = 0 :

(21)

Eq. (21) can only be ful>lled if all terms of O(˝n ) vanish independently. Starting with n = 0 we get 
(22) S0 = ±p(r) ⇒ S0 = ± p(r) dr : In the classically allowed regions, the local classical momentum (2) is real, and we shall always assume p(r) to refer to the positive square root of 2M[E − V (r)]; except for a possible sign, S0 is the classical action Sc , Eq. (3). The terms of >rst order in ˝ in Eq. (21) give, S1
= −

S0

p
(r) ; = − 2S0
2p(r)

(23)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

and, after integration, 1 S1 = − ln p(r) : 2

(24)

After inserting these results in the ansatz (18), we obtain the general form of the >rst-order WKB wave function,    r C1 i


(r) = p(r ) dr exp WKB ˝ r0 p(r)    C2 i r

+
; (25) exp − p(r ) dr ˝ r0 p(r) with arbitrary complex coeScients C1 and C2 . In classically allowed regions, Eq. (25) represents a superposition of a rightward travelling (>rst term) and a leftward travelling (second term) wave. The lower integration point r0 is a “point of reference” which determines the phase of each term; this point of reference must always be speci>ed when de>ning a WKB wave function. An alternative choice of WKB waves is given by the sine or cosine of the WKB integrals, e.g.     r   1 1

  : (26) p(r ) dr  − cos  WKB (r) ˙
˝ r0 2 p(r) Two linearly independent WKB wave functions can be obtained by choosing two di?erent phases  in (26), but they must not di?er by an integral multiple of 2. In classically forbidden regions, V (r) ¿ E, the local classical momentum (2) becomes purely imaginary, but WKB wave functions can still provide a feasible approximation of the exact solution of the Schr4odinger equation. Two independent WKB wave functions are now given by    1 r 1


; (27) exp ± (r) ˙ |p(r )| dr WKB ˝ r0 |p(r)| which are exponentially increasing or decreasing functions of r. The terms of higher order in ˝ in Eq. (21) allow a systematic derivation of the terms Sn in the expansion (20). It is convenient [27] to introduce the functions n (r), n = −1; 0; 1; 2; : : :, via  r dr
n−1 (r
); Sn
= n−1 : (28) Sn (r) = r0

The n are then given by −1 = S0
= ±p;

0 = S1
= −

p
; 2p

and the recursion relation   n 1 
 n+1 = − n + j n− j  2−1 j=0

(29)

for n ¿ 0 :

(30)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

In particular, 1 1 = S2
= − (
+ 02 ) = ± 2−1 0



p

3 p
2 − 4p2 8 p3

369

:

(31)

The series de>ned by Eq. (20) is asymptotic and does not converge. The inclusion of higher and higher terms eventually gives less accurate results. There are special potentials where the series terminates. For example, for potentials V (r) ˙

1 ; (ar + b)4

(32)

with real constants a and b, all n (r), n ¿ 1 vanish identically at zero energy, so the WKB wave functions (25) or (27) are exact solutions of the Schr4odinger equation for E = 0. For V (r) = (ar 2 + br +c)−2 and E =0, we >nd 2 ≡ 0, but we are left with the calculation of the remaining integrals in Eq. (28). 2.3. Accuracy of WKB wave functions as a local property of the Schrodinger equation The WKB approximation is, of course, expected to work well near the semiclassical limit ˝ → 0, see Section 2.1. However, since the expansion (20) depends on the coordinate r, the accuracy of a truncated expansion including a given number of terms must be expected to also depend on r. A frequently formulated condition for the accuracy of the WKB approximation is based on the requirement, that the second term in the left-hand side of Eq. (19) should be small compared to the >rst term, |i˝S

(r)||S
(r)2 | ; inserting S
(r) ≈ p(r) according to Eq. (22) gives  
   p (r)   1  d    1 ; ˝ 2  = (r)  p (r) 2  dr

(33)

(34)

which corresponds to the requirement that the local de Broglie wavelength, (r) = 2˝=p(r) should be slowly varying. Note, however, that the leading contribution to S

on the left-hand side of Eq. (33), namely

S0 , is already included via the >rst-order terms (23), (24) in the >rst-order WKB wave functions (25)–(27), so it does not make sense to require this term to be small as a condition for the accuracy of the wave functions. Indeed, the frequently accepted condition (34) is not necessary for the >rst-order WKB wave functions to be accurate. A striking counter-example is the potential (32) at energy E = 0. Although >rst-order WKB wave functions are exact solutions of the Schr4odinger equation, the derivative of the de Broglie wavelength goes to in>nity for r → ∞. As a criterion for the accuracy of the WKB wave functions, it makes more sense to require smallness of the >rst term of the series (20) which is not considered in the de>nition of the wave functions, i.e., the term (31). This idea is supported by considering the modi>ed Schr4odinger equation for which the WKB wave function is an exact solution. By calculating the second derivative of the

370

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

WKB wave function (25) [or (26) or (27)] it is easy to see that Schr4odinger-type equation  p(r)2 3 p
2 p



− WKB (r) − WKB (r) = 0 : WKB (r) + ˝2 4 p2 2p

WKB

is an exact solution of the (35)

It should be a good approximation to the solution of the original Schr4odinger equation if the third term on the left-hand side of (35) is small compared to the second term, i.e. the absolute value of the function   3 (p
)2 p

2 Q(r) = ˝ (36) − 4 p4 2p3 should be much smaller than unity, |Q(r)|1 :

(37)

This corresponds to requiring |1 =−1 | to be small, where 1 is given in Eq. (31); 1 is the derivative of S2 and (−˝2 )S2 is the >rst term in the series (20) not to be considered in the de>nition of the (>rst-order) WKB wave functions. The function Q(r) de>ned in Eq. (36) can be positive or negative, but it is always real, even in a classically forbidden region where the local classical momentum p(r) is purely imaginary. The condition (37) is a local condition, because Q is a function of r. This function has been called “badlands function” [55,56], because it is large in regions of coordinate space where the WKB approximation is bad. On the other hand, large (positive or negative) values of Q(r) mean that quantum e?ects are important in this region of coordinate space. So a more positive name for the function (36) is the “quantality function”. In regions of high quantality, the condition (37) is violated, the WKB approximation is poor and quantum e?ects are important. Now it is also clear that the simple condition Eq. (34) is not in general suScient for the WKB wave function to be an accurate approximation of an exact solution of the Schr4odinger equation. Consider a potential oscillating rapidly with small amplitude and a moderate total energy such that the local de Broglie wavelength 2˝=p(r) behaves as 1 + sin(qr)=q3=2 . In the limit of large q values, the simple condition (34) is ful>lled suggesting good applicability of the WKB approximation. However, √ the term involving the second derivative in Eq. (36) gives a contribution proportional to q sin qr, which results in a diverging quantality for large q. By analogous arguments it follows that small but nonvanishing values of |Q(r)| are not always suScient for the WKB approximation to be accurate; it is in principle possible that p


which contributes to the next-order correction 2 according to Eq. (30) is large even though |Q(r)| is small. Nevertheless, smallness of |Q(r)| is clearly a more reliable indication of the accuracy of the WKB approximation than the simple condition Eq. (34). Outside the quantal region, the WKB wave function is often an excellent approximation to the exact solution of the Schr4odinger equation, even if the global condition for semiclassical approximations, i.e. small ˝, Eq. (5), is not ful>lled. In such situations, it may be possible to >nd accurate solutions of the Schr4odinger equation in the quantal regions—by analytical or numerical means—and then construct globally accurate wave functions by appropriately matching the exact (or highly accurate) wave functions from the quantal region to WKB wave functions in the semiclassical, “WKB regions” where the condition (37) is well ful>lled. This simple philosophy underlies the various applications discussed in the subsequent sections of this review.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

371

2.4. Examples A simple set of examples is provided by step-wise constant potentials, Vsteps (r) = Vi ;

r ∈ Li ;

(38)

where {Li } represents a covering of coordinate space with a set of intervals, Li = (ri ; ri+1 ). At total energy E, the motion in each interval Li is classically allowed if E ¿ Vi and forbidden if E ¡ Vi . WKB wave functions are exact within each interval. In the classically allowed
intervals there are two independent solutions proportional to exp(±iki r) with wave number ki = 2M(E − Vi )=˝, in the classically forbidden intervals there
are two independent solutions proportional to exp(±%i r) with inverse penetration depth %i = 2M(Vi − E)=˝. The quantality function (36) is zero everywhere, except at the borders ri of the intervals, and a global exact solution of the Schr4odinger equation can be obtained by matching superpositions of the two independent solutions in each interval such that the wave function and its >rst derivative are continuous. This method gives exact results regardless of any consideration of the semiclassical limit. In fact, it is just how one would go about solving the Schr4odinger equation without any knowledge of or reference to WKB wave functions. At each step ri , the >rst derivative of the wave function should be continuous, but the second derivative should not, because in the Schr4odinger equation (1), the discontinuity in the potential must be compensated by a corresponding discontinuity in

. For a single discontinuity separating two regions 1 and 2 where V (r) is constant, we have a sharp-step potential. A quantum particle approaching the step on the upper level is partially reAected, and the reAection probability is √  √ 2  E − V1 − E − V2 q−k 2 √ PR = = √ (39) q+k E − V1 + E − V2 where q and k are the wave numbers on the down side and the up side of the step. Eq. (39) is a standard textbook result; it also follows from the reAection probability for the Woods-Saxon potential (13) in the limit of vanishing di?useness,  → 0. The formula (39) does not contain ˝, i.e. it is not a?ected by taking the formal semiclassical limit, ˝ → 0. One way of understanding this is to realize, that the characteristic length of the potential, de>ned as the distance over which it changes appreciably, is zero for the sharp step. This is always small compared to quantum mechanical lengths such as wavelengths or penetration depths, regardless how close we may be to the semiclassical limit. (See also footnote in Section 25 of Ref. [43]). In this context it is interesting to discuss the case that the potential V is itself continuous, but has a step-like discontinuity in one of its derivatives. Assume that V (i) is continuous for 0 6 i ¡ n and that the nth derivative V (n) (r) is discontinuous at a point r0 . The next derivative, V (n+1) (r), then has a delta-function singularity at r0 , and so do the (n + 1)st derivative of p(r) and the function
(r), cf. Eqs. (29), (31). The function S Sn+1 n+1 (r) has a step-like discontinuity at r0 , and this enters in order ˝n+1 in the expansion (20). The ansatz (18) thus contains a step-like discontinuity of order ˝n at the point r0 , which is incompatible with the requirement, that (r) be a continuous and (at least) twice di?erentiable function. The continuity of the wave function (18) is repaired by adding a second term with an amplitude of order O(˝n ). In the classically allowed regime, this leads to classically forbidden reAection with a reAection amplitude of order O(˝n ). The case n = 0

372

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

as for the sharp step potential leads to a reAection amplitude of order ˝0 , i.e., independent of ˝. For a continuous potential with a kink, i.e. with a step-like discontinuity in the >rst derivative, the amplitude for classically forbidden reAection vanishes as ˝ in the semiclassical limit. For analytical potentials, which are continuously di?erentiable to all orders, the amplitude for classically forbidden reAection generally vanishes exponentially in the semiclassical limit, e.g. as exp(−C=˝), see Section 5, Eqs. (253), (254). For homogeneous potentials (7), the semiclassical and anticlassical limits can be reached by appropriate variations of the energy and/or the potential strength, see Section 2.1. Negative degrees d corresponding to inverse-power potentials, V& (r) = ±

C& ; r&

&¿0 ;

(40)

occur in the description of various physical phenomena. For example: & = 1 for Coulomb potentials, & = 2 for centrifugal or monopole-dipole potentials, & = 3 for the van der Waals potential between a neutral polarizable particle and a surface, & = 4 for the interaction between a neutral and a charged particle, &=6 for the van der Waals potential between two neutral particles and &=7 for the retarded van der Waals potential between two neutral particles [57]. At zero energy, the local classical momentum (2) in the repulsive or attractive potential (40) is proportional to r −&=2 , and the quantality function (36) has a very simple form, &  & −2 & 1− r : (41) Q(r) ˙ 4 4 As discussed for the potential (32), Q(r) ≡ 0 for the special case & = 4. More importantly, Q(r) is seen to vanish for small r when & ¿ 2 and for large r when & ¡ 2. For Coulombic potentials and near-threshold energies, WKB wave functions become increasingly accurate for r → ∞. For >nite (positive or negative) energies, the inAuence of the potential becomes less important for r → ∞ and the local classical momentum tends to a constant. WKB wave functions become increasingly accurate for r → ∞ for any power & ¿ 0. For positive energies E = ˝2 k 2 =(2M), the error in the WKB wave function decreases asymptotically as 1=(kr)&+1 when its phase is correctly matched [58]. Towards small r values, a >nite energy in the Schr4odinger equation becomes increasingly irrelevant as r → 0, so the result (41) holds in this case as well. For & ¿ 2, WKB wave functions become increasingly accurate for r → 0. For attractive potentials, this singular behaviour cannot continue all the way to the origin, but there may be a region of small but nonvanishing r values where the WKB approximation is highly accurate. Repulsive inverse-power potentials can be meaningfully used all the way down to r = 0, and, for & ¿ 2, the WKB wave functions accurately describe the behaviour of the wave function near the origin. Expressing the strength C& of the potential (40) through a length & , C& = ˝2 (& )&−2 =(2M), we have   (&−2)=2  1 & r →0 &=4 : (42) ; '= (r) ˙ r exp −2' r &−2 For 0 ¡ & ¡ 2, the quantality function diverges for r → 0, and the WKB approximation fails near the origin. For the special case & = 2, the quantality function tends towards a constant as r → 0. For a repulsive inverse-square potential, C2 = ˝2 (=(2M), ( ¿ 0, the WKB wave function WKB and

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

the exact wave function r →0 1 +√( ; WKB (r) ˙ r 2

373

behave a little di?erently near the origin, r →0

1 √

(r) ˙ r 2 +

(+1=4

:

(43)

Inverse-square potentials will be discussed in more detail later on; see in particular Eq. (64) in Section 3.3 and the whole of Section 4.4. 3. Beyond the semiclassical limit As was shown in the preceding section the accuracy of the WKB approximation is a local property of the Schr4odinger equation in coordinate space rather than depending on the global validity of the conditions of the semiclassical limit. By bridging the gaps due to regions of high quantality, appropriate WKB wave functions may be constructed and used to derive accurate quantum results, even under conditions which are globally far from the semiclassical limit. In this section we discuss concrete applications to three di?erent physical situations: scattering by singular potentials, quantization in potential wells and tunnelling through potential barriers. 3.1. Connection formulas at classical turning points At a classical turning point, rt , of a one-dimensional system the local classical momentum (2) vanishes, p(rt ) = 0, and the quantality function Q(r) [Eq. (36)] is singular. The exact wave function may nevertheless be well represented by a superposition of exponentially increasing and decreasing WKB waves (27) on the classically forbidden side of rt , and/or by oscillating WKB waves (25), (26), on the classically allowed side. For a consistent description of both regions, we have to say how the WKB waves on either side of rt are to be connected. In the most general case, the connection formulas can be written as       r     N 2 1  1  r





↔ − (44) cos exp − p(r ) dr p(r ) dr  2  ; ˝  rt ˝  rt p(r) |p(r)|       r U   1 NU  1  r 1 




↔
(45) cos exp p(r ) dr  − p(r ) dr  :   ˝ rt 2 ˝ rt p(r) |p(r)| The two parameters  and N in Eq. (44) can be determined by comparing the exact solution corresponding to an exponentially decreasing wave on the classically forbidden side with the oscillating WKB waves on the allowed side. Since the decaying wave on the forbidden side is unique except for a constant factor, the parameters  and N are well de>ned. In contrast, the asymptotic behaviour of the exponentially increasing solution in Eq. (45) masks any admixtures of the decaying wave; since such an admixture signi>cantly a?ects phase and amplitude of the wave function on the allowed side, the two parameters U and NU are not well de>ned. In conventional semiclassical theory [43–47], the connection formulas are derived assuming that the potential can be approximated by a linear function of r in the vicinity of rt , and that this region extends “suSciently far” to either side of rt . “SuSciently far” means that the exact solutions of the Schr4odinger equation in the linear potential, which can be expressed in terms of Airy functions [59],

374

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

are valid until the asymptotic forms of the respective Airy functions can be matched to the WKB waves on both sides. For the well de>ned parameters  and N this conventional matching leads to  conv = ; N conv = 1 : (46) 2 The continuity equation relates the four parameters , N , U and NU [60]. To see this consider a superposition of Eqs. (44) and (45) with arbitrary complex coeScients, = A × (44) + B × (45), and calculate the current density, j = (˝=M)I( ∗
). For de>niteness we assume the classically allowed (forbidden) region to lie to the left (right) of rt . Inserting the left-hand sides of Eqs. (44) and (45) in the superposition gives the current density on the classically allowed side,  2  − U ∗ I(A B) sin ; (47) jallowed = M 2 whereas inserting the right-hand sides gives the current density on the classically forbidden side, 2 jforbidden = (48) I(A∗ B)N NU : M The conservation of the current density, jallowed = jforbidden , requires  U  −  : (49) N NU = sin 2 The ill-de>ned parameters U and NU are irrelevant for a totally reAecting potential. Here the wave function must vanish asymptotically in the classically forbidden region, so the coeScient of the exponentially growing solution (45) must vanish. In the more general case, the uncertainty in the de>nition of the barred quantities can be removed by introducing a second condition by convention, e.g. by assuming a >xed phase di?erence of =2 between the oscillating waves in Eqs. (44) and (45). Such a choice is used in the de>nition of the irregular continuum wave function in scattering theory [49,61]. From Eq. (49) it would follow that NU = 1=N . However, other choices are possible as long as the left-hand sides of Eqs. (44) and (45) remain linearly independent. 3.2. The re?ection phase In the case of total reAection at a classical turning point the current density is zero and it is possible to represent the quantum wave by a real function. The expressions (44) represent the exact wave function in the semiclassical regions on either side of the turning point. The amplitude parameter N can be used to calculate the particle density | |2 in the classically forbidden region [62], but the choice of N is not so important in the classically allowed region if the normalization of the wave function is either irrelevant or deducible from other considerations. The more important quantity in many cases is the phase , which inAuences the phase of the WKB wave in the whole of the classically allowed region. Rewriting the left-hand side of Eq. (44) as      r     ei=2 i  i  r ()

 −i


+ e (50) (r) = p(r ) dr exp p(r ) dr exp − WKB   ˝  ˝ p(r) rt

rt

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

375

reveals that  is the phase loss su?ered by the WKB wave due to reAection, the “reAection phase” [63,64]. With the conventional choice (46) of the reAection phase and the amplitude factor, the WKB wave function in the classically allowed region is     r   2 1  conv

 : (51) p(r ) dr  − cos WKB (r) =
˝  rt 4 p(r) For this approximation to be valid, the Airy function must match the exact wave function both in the classically allowed as well as in the classically forbidden region. The potential must be close to linear in a region extending at least a few times the wavelength of the Airy function on the classically allowed side and a few times the penetration depth on the forbidden side. This condition is sometimes hard to meet, even though the general condition (37) for accuracy of the WKB approximation may be well ful>lled on both sides of the classical turning point rt , e.g. when the kinetic energy of the particle is small and thus the wavelength large even far away from rt . Sometimes the slow variation of the potential compared to the wavelength and penetration depth coincides with the linearity requirement. However, there are important examples where these conditions are not ful>lled at the same time. The separation of the linearity requirement and the resulting >xation (46) of the reAection phase from the application of the WKB approximation greatly widens the range of applicability of WKB waves to a variety of new situations. A very simple example is the sharp potential step, V = V0 ,(r), at energies below the step, 0 ¡ E ¡ V0 . On the classically forbidden side of the step, r ¿ 0, the WKB wave function is pro
portional to exp(−%|r|) where % = 2M(V0 − E)=˝ is the inverse penetration depth, and this is an exact solution of the Schr4odinger equation. On the classically allowed side, r ¡ 0, the WKB wave √ function is proportional to cos(k|r| − =2) where k = 2ME=˝ is the wave number, and this too is an exact solution of the Schr4odinger equation. The problem is solved exactly in the whole of coordinate space, if the reAection phase  and amplitude factor N are chosen as  %k : (52)  = 2 arctan(%=k); N = 2 2 % + k2 √ Note that  assumes the semiclassical value =2 only for %=k whereas N =1 only for %=(2± 3)k. The sharp step potential is particularly easy to handle, because the quantal region reduces to a single point; WKB wave functions are exact everywhere, except at the discontinuity in the potential. For a billard system in more than one dimension, i.e. a particle moving in an area (or volume) with a sharp boundary, the generalization from an in>nitely repulsive boundary to a potential of >nite height simply means replacing the Dirichlet boundary condition appropriate for the in>nite step to the conditions (52) for the component normal to the bounding surface. This is one of the few examples where quantum e?ects beyond the semiclassical limit have been successfully included in the semiclassical description of multidimensional systems [24,25,65]. 3.3. Scattering by a repulsive singular potential Repulsive singular potentials have a large application >eld in particle scattering. They are excellent examples to demonstrate the usefulness of the concept of the reAection phase. Consider a repulsive

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homogeneous potential, C& ˝2 (& )&−2 ˝2 v& (r) : V&rep (r) = & = = r 2M r& 2M The quantality function (36) for the potential (53) at energy E = ˝2 k 2 =(2M) is  

&  1 & v& (r) 2 − 1 v Q(r) = (& + 1)k + (r) : & 4 r 2 [k 2 − v& (r)]3 4

(53)

(54)

From Eq. (54) we observe that Q(r) = O(r &−2 ) for small r. As already mentioned in Section 2.4, the WKB approximation becomes increasingly accurate for r → 0 as long as & ¿ 2. This holds in particular for small energies—including vanishing energy. In the special case & = 4 and k = 0 the quantality function vanishes for all r, Q(r) ≡ 0, and WKB wave functions are exact solutions of the Schr4odinger equation, see also Eq. (32). For the large-r limit one has to distinguish the cases E = 0 and E ¿ 0. While the Q(r) diverges as r &−2 (& ¿ 2; & = 4) for zero energy, it approaches zero for large r as r −&−2 for E ¿ 0. The classical turning point is given by rt = & (& k)−2=& :

(55) & −2

as r → 0, so the WKB For 0 ¡ & ¡ 2, the quantality function Q(r) diverges proportional to r approximation deteriorates towards the origin. Decaying WKB waves cannot be expected to be good approximations of the exact wave function on the classically forbidden side of rt for potentials which are less singular than 1=r 2 . On the allowed side of rt however, Q(r) vanishes faster than 1=r 2 for any positive value of & (and E ¿ 0), so the WKB approximation becomes increasingly accurate for r → ∞. Hence an oscillating WKB wave function such as on the left-hand side of Eq. (44),     r   1  2 ()

 ; (56) cos p(r ) dr  − WKB (r) =
˝  rt 2 p(r) is a valid representation of the exact wave function on the classically allowed side of the turning point, regardless of whether the corresponding decaying WKB wave functions are good approximations for r → 0, as is the case for & ¿ 2, or poor approximations, as is the case for 0 ¡ & ¡ 2. In any case, the reAection phase  is to be chosen such that the phase of the WKB wave (56) asymptotically matches the phase of the regular exact wave function which vanishes at r = 0. Inverse-square potentials, corresponding to & = 2, represent a special case, ˝2 ( V( (r) = : (57) 2M r 2 Potentials of this form are of considerable physical importance since they appear in the interaction of a monopole charge with a dipole and as centrifugal potential in radial Schr4odinger equation in two or more dimensions. The centrifugal potential depends on the angular momentum quantum number lD and is of the form (57) with ( = l3 (l3 + 1);

l3 = 0; 1; 2; : : : ;

for three dimensions, and 1 ( = (l2 )2 − ; l2 = 0; ±1; ±2; : : : ; 4 for two dimensions.

(58) (59)

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377

The Schr4odinger equation with an inverse-square potential (57) contains no length or energy scale. Its regular solution is essentially a simple Bessel function of the product kr,   1 r J0 (kr); where 0 = ( + : (60) (r) = 2 4 √ The classical turning point is given by krt = (, and the quantality function is (2 ( 3 5 + : (61) 2 3 2 4 ((kr) − () 2 ((kr) − ()2 Away from the classical turning point, the absolute value of Q(r) is small for large ( corresponding to the semiclassical limit, and large for small ( corresponding to the anticlassical limit, see Section 2.1. For large values of kr, the asymptotic form [59] of the Bessel function implies that the wave function (60) behaves as    1  kr →∞ 1 ; (62) (r) ˙ √ cos kr − (+ − 2 4 4 k Q(r) =

whereas for the WKB wave function (56),   √  kr →∞ 1 () : (63) (− WKB (r) ˙ √ cos kr − 2 2 k In conventional semiclassical theory, the reAection phase in the WKB wave function (63) is assumed to be =2 and the resulting discrepancy of the phases in Eqs. (63) and (62) is repaired with the help of the Langer modi>cation [66]. This consists of modifying the potential used to calculate the WKB wave functions according to 1 ˝2 1 ( → ( + ; V( (r) → V( (r) + : (64) 4 2M 4r 2 This also leads to a correct behaviour of the WKB wave function in the classically forbidden region:
0 ˙ (kr) for kr → 0, 0 = ( + 1=4, cf. Eq. (43) in Section 2.4. However, the classical turning point is shifted. Comparison of Eq. (62) with Eq. (63) suggests an alternative approach. The phases of exact and WKB waves can be made to match asymptotically by choosing the reAection phase  in the following way [63,64],   1 √ = + (+ − ( : (65) 2 4 Examination of higher order asymptotic (kr → ∞) terms shows that the error in the WKB wave function (63) with the reAection phase (65) is proportional to (kr)−3 . This is better by two orders in 1=(kr) than the conventional semiclassical treatment, which is based on a reAection phase =2 and the Langer modi>cation (64); in the conventional treatment, the error in the WKB wave function only falls o? as 1=(kr) asymptotically [63,64]. For the inverse-square potential (57), the semiclassical limit is independent of energy and is reached for ( → ∞, see Section 2.1 and Eq. (61), whereas the anticlassical limit corresponds to ( → 0. As one would expect, the reAection phase (65) approaches =2 in the semiclassical limit;

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in the anticlassical limit it approaches the value , as for a free particle reAected by a hard wall [Eq. (52) in the limit %=k → ∞]. The general repulsive potential (53) proportional to 1=r & with & = 2 is characterized by a strength parameter with the physical dimension of a length, & . The properties of the Schr4odinger equation at energy E = ˝2 k 2 =(2M) depend not on energy and potential strength independently, but only on the product k& . Proximity to the semiclassical limit may be expressed quantitatively by comparing the classical turning point (55) with the quantum mechanical length corresponding to the inverse wave number, k −1 . We call the ratio of these two quantities the reduced classical turning point, a = krt = (k& )1−2=& :

(66)

In the semiclassical limit, the classical turning point is large compared to k −1 , i.e. a → ∞, which corresponds to k → ∞ when & ¿ 2, but to k → 0 when 0 ¡ & ¡ 2, in accordance with Section 2.1. Conversely, the anticlassical limit is given by a → 0. The inverse-square potential (57) >ts into this √ scheme if we identify krt = ( as the reduced classical turning point a. In terms of the classically de>ned quantities, total energy E, particle mass M and strength parameter C& of the potential (53) or (57), the reduced classical turning point (66) is, a=

√ 1 1−1 pas rt E 2 & (C& )1=& 2M = : ˝ ˝

(67)

According to Eq. (67), the reduced classical turning point a is just the classical action obtained by multiplying the asymptotic (r → ∞) classical momentum pas = ˝k and the classical turning point rt , measured in units of Planck’s constant ˝. The reAection phase  in the WKB wave function (56) is directly related to the phase shift 1, which determines the asymptotics of the regular solution of the Schr4odinger equation [43,61], reg (r)

r →∞

˙ sin(kr + 1) :

(68)

Comparing this to the WKB wave function (56) gives an explicit expression for 1 in terms of ,   r  1  

p(r ) dr − kr − 1 = + lim : (69) 2 r →∞ ˝ rt 2 For potentials falling o? faster than 1=r asymptotically, the square bracket above remains >nite as r → ∞ and 1 is a well de>ned constant depending only on the wave number k. For & ¿ 2, the anticlassical limit of the Schr4odinger equation corresponds to the threshold, k → 0, and the phase loss of the WKB wave due to reAection by the singular potential (53) can be derived from the exact solution of the Schr4odinger equation for zero energy,    √ & 1=(2') 1 : (70) ; '= rK±' 2' reg (r) ˙ r &−2 This is the regular solution, reg (0)=0, which is unique to within a constant factor. Since the energy enters the Schr4odinger equation with a term of order E=O(k 2 ), the zero-energy solution (70) remains valid for small energies to order below O(k 2 ). For large values of r, the argument of the Bessel

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379

function K±' in (70) is small, and the leading behaviour of the wave function is, reg (r)

r →∞

˙ '2'

r 2(1 − ') − : 2(1 + ') &

(71)

The asymptotic behaviour of the regular solution of the Schr4odinger equation is given by Eq. (68) and, for potentials falling o? more rapidly than 1=r 3 , the low-energy behaviour of the phase shift is, k →0

1 ∼ n − a0 k ;

(72)

with the scattering length a0 (see Ref. [43] and Section 4.1). So for large r and small k we have, reg (r)

˙ k(r − a0 ) :

(73)

Comparing Eqs. (73) and (71) gives an explicit expression for the scattering lengths of repulsive inverse power-law potentials (53) with & ¿ 3, a0 = '2'

2(1 − ') & ; 2(1 + ')

'=

1 : &−2

(74)

The asymptotic (r → ∞) expression for the action integral in the WKB wave function (56) is,    √  1 r  2(1 − &1 ) a & −1

r →∞ : (75) p(r ) dr ∼ kr − a + O ˝ rt 2 2( 32 − &1 ) kr For & ¿ 3 the phase of the WKB wave function has the correct near-threshold behaviour, see Eqs. (72) and (74), when the reAection phase is given by [58], k →0

 ∼ −

2 √ 2(1 − &1 ) 2(1 − ') k& ;  3 1 (k& )1− & + 2'2' 2(1 + ') 2( 2 − & )

(& ¿ 3) :

(76)

The second term on the right-hand side of Eq. (76) is linear in a and cancels the corresponding contribution from the WKB action integral (75); the third term (proportional to a1+2' ) yields the contribution −ka0 with the scattering length a0 in Eq. (74). Near the semiclassical limit a → ∞, a semiclassical expansion for the phase shifts [67,68] can be used [58] to derive the leading contributions to the reAection phase for the repulsive homogeneous potentials (53),   √  (& + 1)2( &1 ) 1 a→∞  + : (77)  ∼ +O 2 a 12&2( 12 + &1 ) a3 For >nite values of the reduced classical turning point a, between the semiclassical (77) and anticlassical (76) limits, the reAection phases can be obtained by solving the Schr4odinger equation numerically and comparing the phase of the solution with the phase of the WKB wave function (56) as r goes to in>nity. The results are shown in Fig. 1 as functions of a for integer powers & from 2 to 7. In all cases we observe a monotonous decline from the value  in the anticlassical limit a = 0 towards the semiclassical expectation =2 for large a. The smooth dependence of the reAection phases in Fig. 1 on a and & suggests that a simple algebraic formula based on the exact result (65) for & = 2 might be e?ective. Indeed, if we replace

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Fig. 1. Exact reAection phases  for reAection by a repulsive homogeneous potential (53) as function of the reduced classical turning point (66). From [58].

the reduced classical turning point a ≡ aS =

a ; S(&)

√

( in Eq. (65) with a scaled reduced classical turning point,

2 & + 1 2( &1 ) ; S(&) = √ 3  & 2( 12 + &1 )

then the appropriate generalization of Eq. (65), namely  1  (aS )2 + − aS ; = + 2 4

(78)

(79)

reproduces the numerically calculated exact reAection phases with an error never exceeding 0:006 for all powers & shown in Fig. 1 and all energies. The scaling factor S(&) in Eq. (78) was chosen such that the term proportional to 1=a in the large-a expansion (77) is given correctly by the formula a→0 (79). Near the anticlassical limit, the approximate formula Eq. (79) corresponds  ∼  − a=S(&), and the next-to-leading term −a=S(&) does not agree exactly with the result (76). The accurate algebraic approximation (79) of the reAection phases yields a convenient and accurate approximation for the scattering phase shifts via Eq. (69), namely  √   2(1 − &1 )  1 1= −a (80) (aS )2 + − aS : − 4 2 2( 32 − &1 ) 2 4 This is illustrated in Fig. 2, where the phase shifts are plotted as functions of k& . The case & = 1:4 is included in order to show that the formula (79) also gives good results in the regime 1 ¡ & ¡ 2, where the semiclassical limit a → ∞ corresponds to the low-energy limit k → 0. For Coulomb potentials, & = 1, the phase shifts have to be de>ned with respect to appropriately distorted waves rather than plane waves (68), but the concept of the reAection phase and Eq. (79) are still useful [58,69].

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381

Fig. 2. Phase shifts 1 for scattering by a repulsive homogeneous potential (53) proportional to 1=r & , which are functions of k& . The thin solid lines are the exact results and the thin dashed lines are the conventional WKB results, which are obtained by inserting the value =2 for the reAection phase  in the expression (69). The thick dashed lines show the results obtained with Eq. (80). The curves for & = 8; & = 4 and & = 1:4 are shifted by ; 2 and 4 respectively. From [58].

3.4. Quantization in a potential well Consider a binding potential in one dimension with two classical turning points, rl to the left and rr to the right. We assume that there is a region between rl and rr , where WKB wave functions are accurate approximations to the exact solution of the Schr4odinger equation. Then each turning point can be assigned a reAection phase, l and r for the left and right turning point, respectively, and the bound state wave function can be written either as    r 1 l 1

(81) p(r ) dr − cos l (r) ˙
˝ rl 2 p(r) or as

   rr 1 r 1

p(r ) dr − cos r (r) ˙
˝ r 2 p(r)

(82)

at any point r in the WKB region. For the wave function to be continuous around r, the cosines in Eqs. (81) and (82) must agree, at least to within a sign, so the sum of the arguments, which does

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not depend on r, must be an integral multiple of . This yields the rather general quantization rule   l r 1 rr (E) + : (83) p(r) dr = n + ˝ rl (E) 2 2 We retrieve the conventional WKB quantization rule, 

1 rr (E) 0M  ; 0M = 2 ; p(r) dr = n + ˝ rl (E) 4

(84)

by inserting =2 for both l and r in (83) as prescribed by Eq. (46). The parameter 0M is the Maslov index which counts how often a phase loss of =2 occurs due to reAection at a classical turning point during one period of the classical motion. Allowing for reAection phases which are not integral multiples of =2 amounts to allowing nonintegral Maslov indices [64]. The conventional quantization rule (84) works well near the semiclassical limit, but it breaks down near the anticlassical limit of the Schr4odinger equation. The modi>ed quantization rule (83) represents a substantial generalization of the conventional WKB rule, in that it avoids the restrictive assumptions underlying the conventional choice of reAection phases at the turning points; it only requires the WKB approximation to be accurate in some, perhaps quite small region of coordinate space between the turning points. 3.4.1. Example: Circle billard As a simple example consider the circle billard [24–26], a particle moving freely in an area bounded by a circle of radius R. The quantum mechanical wave functions of this √ separable twodimensional problem can be written in polar coordinates as (r; ’) = eil2 ’ l2 (r)= r, and the radial wave function l2 (r) obeys the one-dimensional Schr4odinger equation with the centrifugal potential (57), (59); l2 = 0; ±1; ±2; : : : is the angular momentum quantum number. The √ exact solutions for 2 2 (r) at energy E=˝ k =(2M) are essentially Bessel functions [59], (r) ˙ rJ|l2 | (kr), and bound l2 l2 states for a given angular momentum l2 exist when the wave function vanishes at the boundary: J|l2 | (kn R) = 0. The >rst, second, third, etc. zeros of the Bessel function de>ne bound states with radial quantum number n = 0; n = 1; n = 2, etc., and angular momentum quantum number l2 . The circle billard has served as a popular model system for testing semiclassical periodic orbit theories, and in leading order, standard periodic orbit theories essentially yield the energy eigenvalues obtained via conventional WKB quantization of the separable system. Main [26] has recently used the circle-billard as an example to demonstrate the applicability of an extension of standard periodic orbit theories to include terms of higher order in ˝. In conventional WKB quantization (84) of the radial motion, the Maslov index is taken to be 0M = 3 in order to account for the hard-wall reAection with reAection phase  at r = R; furthermore, the centrifugal potential is taken to be (l2 )2 ˝2 =(2Mr 2 ) in accordance with the Langer modi>cation 2 2 (64). The resulting energy eigenvalues En;WKB |l2 | [in units of ˝ =(2MR )] are tabulated in Table 1 for angular momentum quantum numbers |l2 |=0 and 1 together with the exact eigenvalues En;exact |l2 | , which are just the squares of the zeros of the corresponding Bessel functions. For the modi>ed quantization rule (83) the reAection phase at the outer classical turning point r = R is also taken as , corresponding to hard-wall reAection, but the reAection phase at the inner classical turning point has, for l2 = 0,  the energy independent value =2 + (|l2 | −

(l2 )2 − 14 ) according to Eq. (65); also the centrifugal

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383

Table 1 Energy eigenvalues En; |l2 | [in units of ˝2 =(2MR2 )] for angular momentum quantum numbers l2 = 0 and |l2 | = 1 in the circle billard. The superscript “WKB” refers to conventional WKB quantization of the radial degree of freedom, “mqr” refers to the modi>ed quantization rule (83), “exact” labels the exact results and “(1)” the results obtained by Main [26] in higher-order semiclassical periodic orbit theory n

En;WKB 0

En;mqr 0

En;exact 0

En;(1)0

En;WKB 1

En;mqr 1

En;exact 1

En;(1)1

0 1 2 3 4 5 6 7 8

5.551652 30.22566 74.63888 138.7913 222.6829 326.3138 449.6839 592.7931 755.6416

5.798090 30.47498 74.88861 139.0412 222.9329 326.5637 449.9338 593.0431 755.8916

5.783186 30.47126 74.88701 139.0403 222.9323 326.5634 449.9335 593.0429 755.8914

5.804669 30.47647 74.88960 139.0418 222.9329 326.5656

14.39777 48.95804 103.2445 177.2678 271.0297 384.5306 517.7705 670.7494 843.4676

14.65833 49.21105 103.4959 177.5187 271.2803 384.7809 518.0207 670.9997 843.7178

14.68197 49.21846 103.4995 177.5208 271.2817 384.7819 518.0214 671.0002 843.7182

14.70160 49.22259 103.5015 177.5135 271.2822

potential (57), (59) is left intact—it is not subjected to the Langer modi>cation. A naive application of the modi>ed quantization rule (83) does not work for l2 = 0, i.e. for s-waves in two dimensions, because the centrifugal potential is actually attractive without the Langer modi>cation, and the WKB action integral diverges when taken from r = 0. This can be overcome by shifting the inner integration limit to a small positive value and adjusting the reAection phase accordingly as described in Ref. [70]; see also Section 4.4. The eigenvalues obtained with the modi>ed quantization rule (83) are listed as En;mqr |l2 | in Table 1.

Also included in Table 1 are the energies En;(1)|l2 | obtained via the extension of semiclassical periodic-orbit quantization to higher order in ˝ according to Main [26]. The performance of the various approximations is illustrated in Figs. 3 and 4 showing the errors approx |En; |l2 | − En;exact |l2 | | on a logarithmic scale. The energies obtained in conventional WKB quantization including the Langer modi>cation of the potential are, both for l2 = 0 and |l2 | = 1, too small by an error which is very close to 0:25˝2 =(2MR2 ) and virtually independent of n. Using the modi>ed quantization rule (83)—without the Langer modi>cation—reduces the error by one to three orders of magnitude and yields the same sort of accuracy as the higher-order periodic orbit quantization according to Ref. [26]. In contrast to the higher-order periodic orbit theory, however, the application of the modi>ed quantization rule is very simple indeed—just as simple as applying the conventional WKB quantization rule. For large angular momentum quantum numbers |l2 |, conventional WKB quantization becomes increasingly accurate and the improvement due to the modi>ed quantization rule is less dramatic. For small values of l2 , the modi>ed quantization rule is an extremely simple and powerful tool for obtaining highly accurate energy eigenvalues beyond the conventional WKB approximation. 3.4.2. Example: Potential wells with long-ranged attractive tails In this subsection we consider a deep potential well with a long ranged attractive tail, as occurs, e.g., in the interaction of atoms and molecules with each other and with surfaces. We focus our

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l2=0

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 3. Logarithmic plot of the error |En;approx − En;exact 0 | of various approximations of the energy eigenvalues En; |l2 | of the 0 circle billard for angular momentum number l2 = 0. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83), adapted for the weakly attractive centrifugal potential as described in Ref. [70].

attention on homogeneous tails, C& ˝2 (& )&−2 = − ; &¿2 : (85) r& 2M r& The repulsive homogeneous potentials (53) studied in the previous section can, in principle, be the whole potential in the Schr4odinger equation, but the attractive potentials (85) cannot, otherwise the energy spectrum would be unbounded from below. The full potential must necessarily deviate from the homogeneous form (85) in the vicinity of r = 0, e.g. in the form of a short-ranged repulsive term. For the moment we neglect the inAuence of such a short-ranged repulsive part in the potential and study the properties of the long-ranged attractive tail (85). For negative energies, E = −˝2 %2 =(2M), there is an outer classical turning point rout given by V&att (r) = −

rout = & (%& )−2=& ;

(86)

cf. Eq. (55). In analogy to Eqs. (66), (67) we now de>ne the reduced classical turning point as the ratio of the classical turning point rout and the quantum mechanical penetration depth %−1 , √ 1 1 1 |pas | rout ; (87) a = %rout = (%& )1−2=& = |E| 2 − & (C& )1=& 2M = ˝ ˝ where |pas | = ˝% is the absolute value of the asymptotic (r → ∞) local classical momentum (2), which is now purely imaginary. Again, a is a quantitative measure of the proximity to the semiclassical or the anticlassical limit, a → ∞ being the semiclassical and a → 0 the anticlassical limit.

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

385

|l2|=1

0

-1

conv. WKB

log10(error)

mod. quan. rule higher-order p.o. theory

-2

-3

-4 0

1

2

3

4 n

5

6

7

8

Fig. 4. Logarithmic plot of the error |En;approx − En;exact 1 | of various approximations of the energy eigenvalues En; |l2 | of the 1 circle billard for angular momentum number l2 = ±1. The >lled triangles are the errors obtained via conventional WKB quantization including the Langer modi>cation of the potential and the empty triangles are the errors of the eigenvalues obtained by Main [26] via higher-order periodic orbit theory. The >lled squares show the errors obtained via the modi>ed quantization rule (83).

We assume that the reAection phase r at the outer classical turning point is determined by the homogeneous tail (85) alone; its behaviour near the anticlassical limit can be derived [73] from the zero-energy solutions of the Schr4odinger equation in much the same way as for the repulsive potentials (53) in Section 3.3; for any power & ¿ 2 the result is,

 √ 2(1 − 1 ) 2(1 − ') 1 a→0   ∼ a + 2'2' + ' −  3 &1 & tan sin(') a1+2' ; ' = : (88) 2 & 2(1 + ') &−2 2( 2 − & ) For E = 0, the classical turning point is at +∞ and the reAection phase is  def (89) (0) =0 = + ' : 2 The zero-energy reAection phase 0 is one of three parameters characterizing the near-threshold properties of an attractive potential tail, as discussed in more detail in Section 4. Near the semiclassical limit, the leading behaviour of the reAection phase in the attractive homogeneous potential (85) is [73],

 √ (& + 1)2( &1 ) a→∞   ∼ + a : (90) tan 2 & 12&2( 12 + &1 ) The dependence of  on a in between the anticlassical (88) and semiclassical (90) limits can be derived from numerical solutions of the Schr4odinger equation and the results are shown in Fig. 5. The phases fall monotonically from the threshold value (=2) + ' towards the semiclassical expectation =2 at large a.

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Fig. 5. Exact reAection phases  for reAection by an attractive homogeneous potential (85) as function of the reduced classical turning point (87). From [73].

An algebraic approximation to the function (a) can now be obtained by generalizing the formula (79) to account for the fact that the value (0) in the anticlassical limit depends on the power & for the attractive potentials,  1  (91) (aR )2 + − aR :  = + 2' 2 4 Here aR stands for a scaled reduced classical turning point, and the scaling factor is chosen such that the formula (91) reproduces the leading contributions (90) to  in the semiclassical limit,

 1 & + 1 2( &1 ) a ; R(&) = √ : (92) aR = (& − 2) tan R(&) & 3  & 2( 12 + &1 ) As a concrete example for the e?ectivity of the modi>ed quantization rule (83) we discuss the Lennard-Jones potential, which is a model for molecular interactions, 

r 6  rmin 12 min : (93) −2 VLJ (r) = j r r The tail of the potential (93) has the homogeneous form (85) with & = 6; the strength parameter 6 is given by  1=4 4M(rmin )2 j = rmin (2BLJ )1=4 : (94) 6 = rmin ˝2 The potential (93) has its minimum value −j at r = rmin , and the energy eigenvalues, measured in units of j, depend only on the reduced strength parameter BLJ =2M(rmin )2 j=˝2 . The energy levels of the potential (93) and WKB approximations thereof were studied in considerable detail by Kirschner and Le Roy [71] and by Paulsson et al. [72]. Following Ref. [72] we choose a reduced strength

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Fig. 6. The Lennard-Jones potential (93) with reduced strength parameter BLJ = 104 . Table 2 Selected energy eigenvalues, in units of j, for the Lennard-Jones potential (93) with a strength parameter BLJ = 104 . En represents the exact energy, WEnconv is the error of the conventional WKB approximation (84), and WEnR is the error in the modi>ed WKB quantization (83) with l = =2 and the algebraic approximation (91) for r , & = 6; ' = 14 and R(6) = 2:08287. For a table containing all eigenvalues see Ref. [73] En

109 × WEnconv

109 × WEnR

0 1

−0.941046032 −0.830002083

−85841 −82492

−17508 −16684

11 12

−0.147751411 −0.115225891

−46115 −42250

−7522 −6589

22 23

−0.000198340 −0.000002697

−4493 −1021

−100 +42

n

parameter BLJ = 104 , for which the potential supports 24 bound states corresponding to quantum numbers n = 0; 1; : : : ; 23. The potential is illustrated in Fig. 6. A selection of energy eigenvalues, in units of j, is listed in Table 2; complete lists are contained in Refs. [72,73]. Next to the exact eigenvalues we also show the errors of the conventional WKB eigenvalues, which are obtained via the conventional quantization rule (84), and the last column shows the results obtained with the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6.

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5

log10(relative error)

higher order WKB 4

-3

1

2

3

-4 0

5

10

15

20

n

Fig. 7. Relative errors (95) of the energy eigenvalues in the Lennard-Jones potential (93) with strength parameter BLJ =104 . The >lled triangles show the errors of conventional WKB quantization (84) and the >lled squares show the errors of the modi>ed quantization rule (83) when the reAection phase l at the inner classical turning point is kept at =2 while the energy dependence of the outer reAection phase r is accounted for via the approximate algebraic formula (91), with ' = 14 and R(6) = 2:08287 for & = 6. The open triangles numbered 1 to 5 show the relative error for the highest bound state, n = 23, as obtained in Ref. [72] with successive higher-order approximations involving terms up to S2N +1 in the expansion (20); the state becomes unbound for N ¿ 5.

Although the results of conventional WKB quantization seem quite satisfactory at >rst sight, allowing for the energy dependence of the reAection phase at the outer classical turning point improves the accuracy by a factor ranging from >ve for the low-lying states to 45 and 25 for the highest two states. The usefulness of the modi>ed quantization rule becomes clearer when looking at the errors relative to the level spacing, which decreases by a factor of 500 from the bottom to the top of the spectrum,    WEn  rel  :  WEn =  (95) En − E n − 1  Fig. 7 shows the relative errors (95) obtained via conventional WKB quantization (>lled triangles) and via modi>ed quantization including the energy dependence of the outer reAection phase via Eq. (91) (>lled squares). The relative error of conventional WKB quantization grows by an order of magnitude as we approach the anticlassical limit at E = 0. In contrast, accounting for the energy dependence of the outer reAection phase keeps the relative error roughly constant at a comfortably low level. As shown in Ref. [72], higher-order WKB approximations involving terms up to S2N +1 in the expansion (20) substantially reduce the error for all but the last eigenstate, n = 23. The relative errors obtained in higher-order WKB approximations for the n = 23 state are shown as open triangles in Fig. 7. The relative error initially decreases, for N = 1; 2 and 3, but then increases with the order of the approximation. For N ¿ 5 the WKB series no longer yields a bound state with quantum

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389

number n = 23. This is an illustrative demonstration of the asymptotic nature of the expansion (20) which breaks down towards the anticlassical limit. The observation that conventional semiclassical approximations deteriorate towards threshold in a typical molecular potential such as (93) sometimes causes surprise [74], because one might expect such approximations to improve with increasing quantum number. It is, however, not surprising and actually well understood [72,73,75–78], that semiclassical approximations break down near the anticlassical or extreme quantum limit, which is at energy zero for potentials such as (93). A more detailed discussion of quantization near the anticlassical limit is presented in Section 4.2. 3.5. Tunnelling When two classically allowed regions of (in our case one-dimensional) coordinate space are separated by a localized classically forbidden region, a barrier, then a classical particle approaching the barrier from one side is inevitably reAected and cannot be transmitted to the other side. In quantum mechanics there is generally a >nite probability PT for transmission, and the probability PR for reAection is correspondingly less than unity, PR + PT = 1 :

(96)

Although the following discussion focusses on tunnelling through a classically forbidden region, it can also be applied to the transmission through a classically allowed region, for which the probability can be less than unity when quantum e?ects lead to a >nite probability for classically forbidden reAection. Such quantum reAection requires the existence of a region of substantial quantality and is discussed in detail in Section 5. When a particle approaches the barrier (or the quantal region of coordinate space) from the left, the wave function to the left of this region can be expressed by the WKB wave functions       r  1 i r i



(r) ∼
(97) p(r ) dr + Rl exp − p(r ) dr exp ˝ rl ˝ rl p(r) with the reAection amplitude Rl , and the transmitted wave to the right is    r 1 i


p(r ) dr exp (r) ∼ Tl ˝ rr p(r)

(98)

with the transmission amplitude Tl . The points rl and rr are points of reference where the phases of the WKB wave functions vanish. When the particle is incident from the right, Eqs. (97) and (98) are replaced by     r    i 1 i r



; (99) p(r ) dr + Rr exp p(r ) dr (r) ∼
exp − ˝ rr ˝ rr p(r)    1 i r

(100) (r) ∼ Tr
p(r ) dr ; exp − ˝ rl p(r) where Eq. (99) applies for r values to the right and Eq. (100) for r values to the left of the barrier (or quantal region). The reAection and transmission amplitudes in Eqs. (97)–(100) are connected

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through the reciprocity relations (see, e.g. Ref. [79]), T def Tr = Tl =T; Rr = −R∗l ∗ : (101) T If the potential becomes constant on one or both sides of the barrier, then the conventional ansatz with plane waves can be used to de>ne reAection and transmission amplitudes, e.g., 1 (r) ∼ √ {exp[ − ik(r − rr )] + Rr exp[ik(r − rr )]} ˝k

(102)

√ for a particle incident from the right. We have included the factor 1= ˝k to account for the velocity dependence of the particle density in accordance with the continuity equation. When the r dependence of the potential is negligible, the WKB waves in Eq. (99) are identical to the plane waves in (102) except for constant phase factors. The use of WKB waves to de>ne transmission and reAection amplitudes as in Eqs. (97)–(100) does not imply any semiclassical approximation of these amplitudes. The Schr4odinger equation should be solved exactly, and the exact wave functions matched to the incoming and reAected waves or the transmitted wave in the semiclassical regions on either side of the barrier (or quantal region). If there are no semiclassical regions, then the terms “incoming”, “reAected” and “transmitted” cannot be de>ned consistently. The ans4atze (97)–(100) involving WKB wave functions are more general than those using plane waves, because they can also be used when the potential depends strongly on r in the semiclassical region(s). The probabilities PT for transmission and PR for reAection are given by PT = |T |2 ;

PR = |R|2 ;

(103)

and they do not depend on whether the particle is incident from the left or the right, or on whether the amplitudes are de>ned via WKB or plane wave functions. The phase of the reAection amplitude depends on the direction of incidence according to Eq. (101), and also on whether WKB waves or plane waves are used as reference. For example, the reAection amplitude R(p) de>ned with reference to plane waves as in Eq. (102) is related to the r reAection amplitude R(WKB) de>ned via WKB waves according to Eq. (99) by r     1 r (p) (WKB)

R r = Rr : (104) exp lim 2i k(r − rr ) − p(r ) dr r →∞ ˝ rr The exponential on the right-hand side simply accounts for the di?erent phases accumulated by the reference waves on their way to rr and back. The reAection amplitudes do not depend on whether plane waves or WKB waves are used to represent the transmitted wave. If 2M(E − V ) approaches the constant values ˝2 k 2 and ˝2 q2 for r → +∞ and r → −∞ respectively, then the transition amplitude T (p) de>ned via incoming and transmitted plane waves is related to the transition amplitude T (WKB) based on WKB waves by     1 r− (p) (WKB) T =T exp lim i q(r− − rl ) − p(r) dr r± →±∞ ˝ rl    1 r+ p(r) dr : (105) −i k(r+ − rr ) − ˝ rr

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The phases of the reAection and transmission amplitudes also depend on the points of reference rl; r at which the reference waves have vanishing phase. In principle they could be chosen arbitrarily; in the presence of a potential barrier they are conveniently chosen as the left and right classical turning points, respectively. In the presence of a potential barrier, the wave function in the classically forbidden region is approximated by a superposition of exponentially increasing and decreasing WKB waves        1 r 1 r 1



: (106) A exp − |p(r )| dr + B exp + |p(r )| dr forb (r) =
˝ rl ˝ rl |p(r)| We assume that the classical turning points rl and rr are isolated, meaning that there is a region in the classically forbidden domain between rl and rr where the WKB representation (106) is valid. Instead of referring the WKB waves to the left classical turning point rl , we could equally have chosen rr as point of reference. For the derivation of an explicit expression for the transmission amplitude we consider the case that the particle is incident from the left and we make use of Eqs. (97), (98). In order to use the connection formulas (44) and (45) at the right turning point we rewrite Eq. (98) as     r  C 1 CU 1 r r U r

(r) = √ cos + √ cos ; (107) p dr − p dr − p ˝ rr 2 p ˝ rr 2 where 2Tl e−ir =2 C= ; e−ir − e−iUr

U

2Tl e−ir =2 CU = − : e−ir − e−iUr

(108)

The cosines of Eq. (107) can now be matched to the growing and decaying exponentials in the WKB region under the barrier, Eq. (106), according to the connection formulas (44) and (45). The exponentials containing integrals with lower limit rl can be rewritten in terms of exponentials of integrals with upper limit rr by introducing the factor  1 rr ,B (E) = exp[I (E)]; I (E) = |p(r)| dr ; (109) ˝ rl this gives the coeScients A and B of Eq. (106), U A = NU r ,B C;

B=

Nr C : 2,B

(110)

The subscript r indicates application of the connection formulas at the right turning point. By using the connection formulas once again at the left turning point rl we get an expression of the wave function in the classical allowed region to the left of rl . Decomposing the cosines into exponentials and comparison with the WKB wave incident from the left, Eq. (97), gives the coeScient of the incoming WKB wave, which should be unity, as A il =2 B iUl =2 e + e : Nl 2NU l

(111)

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Using Eqs. (110), (108), and (49) yields the general expression for the transmission amplitude   Nl Nr 1 i(Ul +Ur )=2 −1 i(l +r )=2 − e : (112) T = iNl Nr ,B e NU l NU r 4,B This is a very general formula, but it still contains eight parameters, namely the unbarred and barred phases  and amplitudes N at each of the two turning points, where the connection formulas have been applied. For a “dense” barrier, meaning that the exponentiated integral ,B as de>ned by Eq. (109) is large, we might choose to neglect the second term in the large brackets in Eq. (112) as subdominant. This leaves us with an approximate formula for the transmission amplitude T ≈ iNl Nr e−i(l +r )=2 =,B ;

(113)

which contains only the well-de>ned (unbarred) connection parameters. With the standard semiclassical choice (46) for the unbarred connection parameters, Eq. (113) leads to the standard WKB expression for the tunnelling probability, PTWKB (E) = |T |2 = (,B )−2 :

(114)

This formula fails near the top of a barrier, where the two turning points rl and rr coalesce. The expression (114) gives unity at the top of a barrier, whereas the exact result is generally smaller than unity. An improved formula, 1 PTKemble (E) = ; (115) 1 + (,B )2 is due to Kemble [80,81], and it is actually exact at all energies for an inverted quadratic potential, V (r) ˙ −r 2 ; in particular, PTKemble is exactly one half at the top of the barrier. This result is accurate for all barriers which can be approximated by an inverted parabola in a range of r values reaching from the top of the barrier into the semiclassical regions on either side. If we include the subdominant term in Eq. (112) and >x the connection parameters according to conventional WKB theory, Nl; r = NU l; r = 1, l; r = −U l; r = =2, we obtain the result   1 −1 ; (116) T = ,B + 4,B which can be found, e.g., in Ref. [45]. We now look at the behaviour of the tunnelling amplitude and probability near the base of a barrier, where the potential becomes asymptotically constant on at least one side. The limit of small excess energy, E =˝2 k 2 =(2M) → 0, corresponds to the anticlassical limit of the Schr4odinger equation if the potential approaches its asymptotic limit faster than 1=r 2 . For this case it can be shown [82], that both the amplitude N in the connection formula (44) and the amplitude NU in the connection √ formula (45) become proportional to k for k → 0. For potential tails falling o? faster than 1=r 2 the exponentiated √ integral (109) remains >nite at E = 0, so the transmission amplitude (112) is proportional to k near the threshold E → 0, if E = 0 represents the base of the barrier on just one side. The transmission probability through an asymmetric barrier with two di?erent asymptotic levels is thus proportional to the square root of the excess energy above the higher level, in the limit that this excess energy becomes small. If the potential approaches the same asymptotic limit

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393

on both sides and does so faster than 1=r 2 , then the transmission amplitude (112) is proportional to k and the transmission probability is proportional to E near the base. 2 For a potential barrier √ with tails vanishing faster than 1=r , the quantum mechanical tunnelling probability vanishes as E or as E at the base, E = 0. This behaviour is not reproduced by any of the semiclassical formulas (114), (115) or (116), which all predict a >nite tunnelling probability at the base. For a symmetric barrier, the left and right connection parameters are the same and we can drop the subscripts. Eq. (112) then simpli>es to  −1 N 2 1 iU 2 i T = iN ,B e − 2 e : (117) NU 4,B For the calculation of the transmission probability through a symmetric barrier the phases in Eq. (117) are eliminated via Eq. (49) giving  − 1 2 ,B 1 2 − +1 : (118) PT (E) = |T | = N2 4,B NU 2 If we keep only the dominant term, PT (E) ≈

N4 : (,B )2

As an example consider a symmetric rectangular barrier of length L,  2 ˝ (K0 )2 =(2M) for |r| ¡ L=2 ; V (r) = 0 for |r| ¿ L=2 :

(119)

(120)

This corresponds to a sharp-step potential at both classical turning points; the
phase  and ampli2 2 tude N are given in Eq. (52) as functions of the inverse √ penetration depth % = (K0 ) − k on the classically forbidden side and the wave number k = 2ME=˝ on the classically allowed side of the step. For the sharp-step potential, the WKB waves are exact except at the classical turning point, so the exponentially increasing wave function in the classically forbidden region can be uniquely de>ned; this allows the determination of the barred parameters in the connection formula (45), N NU = ; 2

U = − :

(121)

With Eqs. (52), and (121) for the connection parameters at both turning points and the explicit expression for the exponentiated WKB integral (109), ,B = e%L , the transmission amplitude (117) is T=

e%L (k

+

i%)2

4ik% : − e−%L (k − i%)2

The resulting transmission probability is  −1 2 %2 + k 2 sinh %L + 1 PT = ; 2%k

(122)

(123)

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a standard textbook result. The low-energy (k → 0) behaviour of the expression (123) is, PT ≈

1 16k 2 + O(k 4 ) ; 2 (K0 ) (,B − 1=,B )2

(124)

as could also be calculated via Eq. (118). Now consider symmetric barriers decaying asymptotically as an inverse power of the coordinate, V (r)

|r |→∞

∼

V&rep (|r|) =

˝2 (& )&−2 ; 2M |r|&

(125)

with & ¿ 2 and & positive. The turning points of the homogeneous potential, V& (|r|), are given by rl; r = ∓& (k& )−2=& . In order to calculate the phases and amplitudes in the low energy limit it suSces to consider a potential step which shows the same behaviour as the potential (125) for r → +∞ and stays classically forbidden for energies near zero on the left side, r → −∞. It follows that we can use the results for the reAection by singular repulsive potentials, Section 3.3. After taking into account the proportionality factors in Eqs. (68) and (70) we get the leading contribution to the amplitude factor N , k →0

N ∼

2'&'=2
k& ; 2(1 + ')

'=

1 ; &−2

(126)

which can be inserted into Eq. (119) to give for the tunnelling probability at the base of the barrier k →0

PT ∼

16'2&' (k& )2 : 2(1 + ')4 (,B )2

(127)

The formula (127) is accurate as long as the homogeneous behaviour (125) of the tails of the barrier continues far enough into the barrier, i.e., until the WKB wave functions (106) are accurate. Under these conditions, more accurate values for N going beyond the leading contribution (126) can be obtained by numerically solving the Schr4odinger equation for the homogeneous tail and matching the exact solution to the WKB waves on either side of the turning point. As an example we have calculated N for a homogeneous tail proportional to 1=r 8 and applied the formula (119) to the potential V (r) =

V0 : 1 + (r=)8

(128)

Fig. 8 shows the resulting tunnelling probabilities in a doubly logarithmic plot. The solid line is the exact numerically calculated tunnelling probability and the thick dashed line shows the prediction of the formula (119), with N calculated for the homogeneous tail proportional to 1=r 8 . The straight-line behaviour for small energies demonstrates the proportionality of the tunnelling probability to energy near the base of the barrier, see Eq. (127). The thin dashed line shows the conventional semiclassical result (114), which fails near the base of the barrier, because it remains >nite. For further examples see Refs. [60,82–84]. An accurate description of tunnelling is important for the understanding of energy levels in potentials with two or more wells, e.g. the energy splitting between two almost degenerate levels in a

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395

√ Fig. 8. Numerically calculated exact tunnelling probabilities (solid line) for the potential (128) with  MV0 =˝=5 together with the prediction (119) based on values of N (k) calculated numerically for a homogeneous potential tail proportional to 1=r 8 (thick dashed line). The thin dashed line shows the prediction of the conventional WKB formula (114). From [60].

symmetric double well is essentially determined by the tunnelling probability through the potential barrier separating the wells [43,85]. Modi>ed WKB quantization techniques utilizing the generalized connection formulas (44), (45) have been applied to the quantization in multiple-well potentials by several authors in the last few years, and substantial improvements over the predictions of the standard WKB procedure have been achieved. For details see Refs. [86–91].

4. Near the threshold of the potential In this section we consider potentials which vanish asymptotically, r → ∞, and we focus on the regime of small positive or negative energies near the threshold E = 0. For potentials falling o? faster than 1=r 2 ; E = 0 corresponds to the anticlassical or extreme quantum limit of the Schr4odinger equation; conventional semiclassical methods are not applicable in this case, but modi>ed methods involving exact wave functions in the quantal regions of coordinate space and WKB wave functions elsewhere can give reliable and accurate results. The Schr4odinger equation (1) contains the energy in order O(E), but the leading near-threshold behaviour of the wave functions is often determined by terms of lower order in E, see e.g. Eqs. (76), (88) for the near-threshold reAection phases in repulsive or attractive homogeneous potentials or Eq. (122) for the near-threshold transmission amplitude through a rectangular potential barrier. If exact solutions of the Schr4odinger equation are known at threshold, E = 0, then they also determine the leading behaviour of the solutions near threshold to order less than E, because the energy can be treated as a perturbation of (higher) order E in the Schr4odinger equation. Examples involving continuum states above threshold and discrete bound states below threshold are presented in this section.

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4.1. Scattering lengths Consider a potential V (r), which vanishes faster than 1=r 2 for r → ∞. For positive energies, E = ˝2 k 2 =(2M) ¿ 0, the regular solution reg (r) of the Schr4odinger equation, which is de>ned by the boundary condition reg (0) = 0, behaves asymptotically as reg (r)

r →∞

˙ sin(kr + 1) ;

(129)

and 1(k) is the phase shift of the exact wave function reg relative to the free wave sin(kr), as already discussed in Section 3.3. The phase shift approaches an integral multiple of  at threshold, as long as the potential falls o? faster than 1=r 2 . For a potential falling o? faster than 1=r 4 , the leading terms of the low-energy behaviour of the phase shift are given by the e?ective range expansion [61,92] 1 1 k →0 k →0 2 + k 2 re? ; 1 ∼ n − ka0 + O(k 3 ) : (130) k cot 1 ∼ − a0 2 The parameter a0 in Eq. (130) is the scattering length and re? the e?ective range of the potential V (r). For a potential behaving asymptotically as C4 ˝2 (4 )2 = − ; (131) r4 2M r 4 the leading near-threshold behaviour of the phase shift is given [93] by  k →0 1(k) ∼ n − ka0 − (k4 )2 ; (132) 3 so the e?ective range expansion (130) begins with the same constant term −1=a0 on the right-hand side, but there is a term linear in k preceding the quadratic one. For all potentials falling o? faster than 1=r 3 , the scattering length a0 dominates the low-energy properties of the scattering system. It is, e.g., a crucial parameter for determining the properties of Bose-Einstein condensates of atomic gases, see [36,94]. For a potential behaving asymptotically as r →∞

V (r) ∼

−

C3 ˝ 2 3 = − ; r3 2M r 3 the leading near-threshold behaviour of the phase shift is given [92] by r →∞

(133)

k →0

(134)

V (r) ∼ −

1(k) ∼ n − k3 ln(k3 ) ;

and it is not possible to de>ne a >nite scattering length. Scattering lengths depend sensitively on the positions of near-threshold bound states, so they are determined by the potential in the whole range of r values, not only by the potential tail. The mean value of the scattering length—averaged, e.g. over a range of well depths—is, however, essentially a property of the potential tail. If there is a WKB region of moderate r-values, where the exact solutions of the Schr4odinger equation (at near-threshold energies) are accurately approximated by WKB wave functions, explicit expressions for the scattering length can be derived [95–97] as brieAy reviewed below. The Schr4odinger equation at threshold (E = 0) has two linearly independent solutions, 0 and 1 , whose asymptotic (r → ∞) behaviour is given by 0 (r)

r →∞

∼ 1 + o(r −1 );

1 (r)

r →∞

∼ r + o(r 0 ) :

(135)

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397

In the WKB region, the exact threshold solutions 0 ; 1 can be written in WKB form,    ∞ 1 0; 1 1

; p0 (r ) dr − cos 0; 1 (r) = D0; 1
˝ r 2 p0 (r)

(136)

with well de>ned amplitudes D0 and D1 and the phases 0 and 1 . Here p0 (r) is the local classical momentum at threshold,
(137) p0 (r) = −2MV (r) : Since 0 (r) is the solution which remains bounded for r → ∞, the phase in the WKB form of this wave function in the WKB region is just the threshold value of the reAection phase at the outer classical turning point, which lies at in>nity for E = 0, i.e. the zero-energy reAection phase 0 , cf. Eq. (89) in Section 3.4. The asymptotic behaviour (129) of the regular solution of the Schr4odinger equation becomes reg (r)

k →0

˙ sin[k(r − a0 )] ∼ k(r − a0 )

√

(138)

when we insert the near-threshold behaviour (130) of the phase shift 1(k). To order k ˙ E, the regular solution of the Schr4odinger equation for small k thus corresponds to the following linear superposition of the zero-energy solutions 0 and 1 : reg (r)

˙ k[

1 (r)

− a0

0 (r)]

:

(139)

For values of r in the WKB region, 0 and 1 are WKB wave functions (136), so     ∞   ∞  k 1 1 1 0



− a0 D0 cos p0 (r ) dr − p0 (r ) dr − D1 cos reg (r) ˙
˝ r 2 ˝ r 2 p0 (r)   ∞  1 + 1 ˙
p0 (r
) dr
− cos −8 ; (140) ˝ r 4 p0 (r) where 8 is an angle de>ned by   − a0 + D1 =D0 ; tan tan 8 = a0 − D1 =D0 4

(141)

and + ; − stand for the sum and di?erence of the phases in Eq. (136), + = 0 + 1 ;

− = 0 − 1 :

Coming from the inner turning point rin , the WKB wave function    r 1 1 in (E)

cos p(r ) dr − WKB (r) =
˝ rin (E) 2 p(r)

(142)

(143)

is expected to be an accurate approximation of reg (r) for r values in the WKB region. The reAection phase in at the inner turning point will be near =2 if the conditions for conventional matching are ful>lled in the neighbourhood, see Section 3.2, but, even if this is not the case, in can be expected to be a smooth analytical function of the energy E near threshold, in (E) = in (0) + O(E) :

(144)

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The inner turning point rin also depends weakly and smoothly on E, and for r
values between rin and a value r in the WKB region, the local classical momentum p(r
) in Eq. (143) di?ers from its threshold value (137) only in order E near threshold. So to order less than E we can assume E = 0 in Eq. (143),    r 1 in (0) 1


: (145) (r) ≈ (r) ˙ p (r ) dr − cos reg WKB 0 ˝ rin (0) 2 p0 (r) Eqs. (140) and (145) are compatible if and only if the cosines agree at least to within a sign. This leads to an explicit expression for the angle 8 in terms of the threshold value  ∞ S(0) = p0 (r) dr (146) rin (0)

of the action integral, namely − S(0) in (0) + − − − n = (nth − n) + : (147) ˝ 2 4 4 Here we have introduced the threshold quantum number nth which ful>lls the modi>ed WKB quantization rule (83) at E = 0, 8=

S(0) in (0) 0 − = nth  : − (148) 2 2 ˝ nth is an upper bound to the quantum numbers n = 0; 1; 2 : : : of the negative energy bound states and is usually not an integer. An integer value of nth indicates a zero-energy bound state. The number of negative energy bound states is [nth ] + 1 where [nth ] is the largest integer below nth . Resolving Eq. (141) for a0 and using Eq. (147) gives D1 tan(nth  + − =4) + tan(− =4) D0 tan(nth  + − =4) − tan(− =4)    1 D1 − 1 : = sin + D0 2 tan(− =2) tan(nth )

a0 =

(149)

The exact zero-energy solutions 0 , 1 of the Schr4odinger equation may be known for a tail-region of the potential. The amplitudes D0; 1 and phases 0; 1 of the WKB form of the exact solutions can then be derived from these wave functions, if the tail-region, where the Schr4odinger equation is accurately solved by the known forms of 0 and 1 , overlaps with the WKB region, where these wave functions can be matched to the WKB form (136). The amplitudes and phases are thus tail parameters, which depend only on the potential in the tail region and not on its shape inside the WKB region, or at even smaller r values. The factor   D1 0 − 1 (150) b= sin D0 2 in front of the square bracket in Eq. (149) has the physical dimension of a length and is a characteristic property of the potential tail beyond the WKB region. The threshold quantum number nth , on the other hand, is related to the total number of bound states supported by the well and depends

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399

on the whole potential via the action integral S(0). It is likely that there is an inner WKB region in the well when nth is a large (but >nite) number; this is not a necessary condition however, as shown by the example of a shallow step potential, see below. For a uniform distribution of values of nth , the second term in the square bracket in Eq. (149) will be distributed evenly between positive and negative values, so the >rst term de>nes a mean scattering length aU0 , b b ; a0 = aU0 + : (151) aU0 = tan[(0 − 1 )=2] tan(nth ) 4.1.1. Example: Sharp-step potential The radial sharp-step potential is de>ned as,  2 K0 ; ˝2 (K0 )2 ˝2 Vst (r) = − ,(L − r) = − 2M 2M 0;

for 0 ¡ r 6 L ;

(152)

for r ¿ L :

The quantal region where the WKB approximation is poor is restricted to the single point r = L where the potential is discontinuous. The WKB approximation is exact for 0 ¡ r ¡ L regardless of whether the potential be deep or shallow, and the tail region of the potential can be any interval of r values that includes the discontinuity at r = L. The zero-energy solutions of the Schr4odinger equation with the asymptotic (here: r ¿ L) behaviour (135) are given, in the WKB region 0 ¡ r ¡ L, by st 0 (r)

= cos[K0 (L − r)] ;   L st1 st cos K0 (L − r) − 1 (r) = cos(st1 =2) 2



st1 with tan 2

 =−

1 : K0 L

(153)

The zero-energy reAection phase 0 at the outer turning point (here at r = L) is zero. The tail parameter b and the mean scattering length are thus given by 1 bst = ; aUst0 = L : (154) K0 The reAection phase at the inner turning point r = 0 is  corresponding to reAection at a hard wall, so the threshold quantum number de>ned by Eq. (148) is given by nstth  = K0 L − =2. Indeed, the number of bound states in the sharp-step potential (152) is the largest integer bounded by nstth + 1. With Eq. (151) and using tan(nstth ) = −1=tan(K0 L) we obtain the well known result [98] for the scattering length of the sharp-step potential, tan(K0 L) ast0 = L − : (155) K0 4.1.2. Example: Attractive homogeneous potentials More important and realistic examples are the homogeneous potential tails, C& ˝2 (& )&−2 = − ; (156) r& 2M r& which are characterized by a power & and a strength parameter & with the physical dimension of a length. The zero-energy wave functions which solve the Schr4odinger equation with the potential V&att (r) = −

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(156) and have the asymptotic behaviour (135) are essentially Bessel functions of order ±1=(& − 2) [55,73,76], 
2(1 + ') r (&) (&) ' (r) = J (z); (r) = 2(1 − ')' & rJ−' (z) ; ' 0 1 '' &  1=(2') & 1 ; z = 2' : (157) '= &−2 r For a suSciently rapid fall-o? of V&att (r), namely & ¿ 2, the WKB approximation becomes increasingly accurate for r → 0, see Section 2.4. The WKB region corresponds to small values of r=& and hence to large arguments of the Bessel functions in Eq. (157), so we can use their asymptotic expansion [59] to write 0 and 1 in the WKB form (136). This yields the phases and amplitudes,   0(&) = + '; 1(&) = − ' ; 2 2   ˝ 2(1 + ') ˝& (&) (&) 2(1 − ')'' : ; D1 = (158) D0 = ' '& ' ' The length parameter (150) is now given by b(&) = & '2'

2(1 − ') '1+2' sin(') = & ; 2(1 + ') 2(1 + ')2

and the mean scattering length is 2(1 − ') cos(') : aU0(&) = & '2' 2(1 + ')

(159)

(160)

It is interesting to observe, that the formula (160) for the mean scattering length of the attractive homogeneous potential (156) is very similar to the formula (74) for the true scattering length of the repulsive homogeneous potential (53) discussed in Section 3.3. It simply contains an additional factor cos('). The true scattering length for a potential with an attractive tail such as (156) depends on the whole potential via the threshold quantum number nth according to Eq. (151). If we take the threshold value in (0) of the inner reAection phase in Eq. (148) to be =2, then nth  = S(0)=˝ − (1 + ')=2 and the true scattering length is    S(0) ' − : (161) a0(&) = aU0(&) 1 − tan(')tan ˝ 2 The formulas (160) and (161) for homogeneous potential tails were >rst derived by Gribakin and Flambaum [95]. Together with Harabati these authors also derived an expression for the e?ective range re? in the next term of the expansion (130) for the phase shift [96]. For the discussion in this section we assumed that the potential falls o? faster than 1=r 3 asymptotically. Indeed, for a potential behaving asymptotically as (133), the low-energy behaviour of the phase shift is given by Eq. (134), so a >nite scattering length cannot be de>ned. Note, however, that the expression (159) for the tail parameter b(&) remains >nite for ' → 1 corresponding to & → 3, and also for higher integer values of ' corresponding to powers & = 2 + 1=' between 2 and 3.

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401

Table 3 Ratios of the tail parameter (159) and the mean scattering length (160) to the strength parameter & for attractive homogeneous potential tails (156) &

3

4

5

6

7

8

&→∞

b(&) =& aU(&) 0 =&

 —

1 0

0.6313422 0.3645056

0.4779888 0.4779888

0.3915136 0.5388722

0.3347971 0.5798855

=& 1

The mean scattering length (160) diverges with 2(1−') for positive integers ', but b(&) remains well de>ned and >nite. The ratio b(&) =& expressing the tail parameter b(&) of a homogeneous potential in units of the strength parameter & is tabulated in Table 3 for integer powers & from 3 to 8. For & = 4; : : : ; 8 Table 3 also shows the mean scattering length aU0(&) in units of & ; for homogeneous potentials (156), aU0(&) and b(&) are related by aU0(&) = b(&) =tan('). 4.2. Near-threshold quantization and level densities The generalized quantization rule as introduced in Section 3.4 reads  S(E) def 1 rout (E) in out = p(r) dr = n + + : ˝ ˝ rin (E) 2 2

(162)

This assumes that there is a WKB region between the inner classical turning point rin and the outer one rout , where WKB wave functions are accurate solutions of the Schr4odinger equation. The reAection phases in and out account for the phase loss of the WKB wave at the respective turning points. They are equal to =2 when the conditions for the conventional connection formulae are well ful>lled near the turning points, but they can deviate from =2 away from the semiclassical limit. For a potential V (r) with a deep well and an attractive tail, the reAection phase in at the inner turning point may—or may not—be close to =2; in any case it can be expected to be a smooth analytic function of the energy E, and there is nothing special about the threshold E = 0, cf. Eq. (144). The situation is di?erent near the outer turning point rout which, for a smoothly vanishing potential tail, moves to in>nity at threshold. When the potential is attractive at large distances and vanishes more slowly than 1=r 2 , then the action integral S(E) grows beyond all bounds as E → 0; the potential well supports an in>nite number of bound states and conventional WKB quantization, with out = =2 at the outer turning point, becomes increasingly accurate towards threshold. For a potential behaving asymptotically as V&att of Eq. (156) with 0 ¡ & ¡ 2, and energies E = −˝2 %2 =(2M) close enough to threshold, the action integral can be written as  rout (E)  S(E) (& )&−2 = C+ − %2 dr & r ˝ r0 √ F(&)  2(1=& − 12 ) %→0
; (163) ; F(&) = ∼ C + (%& )(2=&)−1 2& 2(1=& + 1)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

which leads to the near-threshold quantization rule, F(&) n→∞ n ∼ C

+ : (%& )(2=&)−1

(164)

The point r0 in Eq. (163) is to be chosen large enough for the potential to be accurately described by the leading asymptotic term proportional to 1=r & . The constants C, C
and C

in Eqs. (163) and (164) depend on the potential at shorter distances r ¡ r0 , but the energy dependent terms depend only on the potential tail beyond r0 , i.e. only on the power & and the strength parameter & determining the leading asymptotic behaviour of the potential tail. For a Coulombic potential tail, & = 1, F(1) = =2 we recover the Rydberg formula, En = −

R ˝2 %(n)2 =− ; 2M (n − C

)2

R=

˝2 ; 2M(21 )2

(165)

with Bohr radius 21 , Rydberg constant R and quantum defect C

− 1. The level density is de>ned as the (expected) number of energy levels per unit energy. If the quantum number n is known as a function of energy, then the level density is simply the energy derivative of the quantum number, dn=dE. Simple derivation of Eq. (164) with respect to E = −˝2 %2 =(2M) gives the near-threshold behaviour of the level density,   1 − 1  1 + 1 & 2 ˝2 dn E →0 F(&) 1 1 1 & 2 = : (166) − dE  & 2 2M(& )2 |E| For Coulombic tails, & = 1, this reduces to the well known form, √ dn E →0 1 R = : dE 2 |E|3=2

(167)

When the potential vanishes faster than 1=r 2 at large distances, then the action integral S(E) remains bounded at threshold. The number of bound states is >nite, and conventional WKB quantization deteriorates towards threshold, see Section 3.4. Based on the exact zero-energy solutions (135) introduced in Section 4.1, it is however possible to derive a modi>ed quantization rule which becomes exact in the limit E = −˝2 %2 =(2M) → 0. To order below O(E) = O(%2 ), the wave function b (r)

=

0 (r)

−%

1 (r)

r →∞

∼ 1 − %r

(168)

solves the Schr4odinger equation and, to order below O(%2 ), it has the correct asymptotic behaviour r →∞ required for a bound state at energy E, namely b ˙ exp(−%r). If there is a WKB region of moderate r values, where we can write the WKB expressions (136) for 0 (r) and 1 (r), then in this region the bound state wave function (168) has the form    ∞ 1 1 +

−; ; (169) cos p0 (r ) dr − b (r) ˙
˝ r 4 p0 (r) in analogy to Eq. (140). In Eq. (169), ; is the angle de>ned by      − D1 1 + %D1 =D0 − 2 = tan 1 + 2% tan ; = tan + O(% ) ; 1 − %D1 =D0 4 4 D0

(170)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

so ; =

+ % sin( 2− )D1 =D0 + O(%2 ), and    ∞ 1 0 1

2 − %b + O(% ) ; p0 (r ) dr − cos b (r) ˙
˝ r 2 p0 (r)

403

− 4

(171)

where b is the length parameter already introduced in Section 4.1, Eq. (150). Comparing this with the WKB wave function for the bound state at energy E,    rout (E) 1 1 out (E)

; (172) cos p(r ) dr − WKB (r) ˙
˝ r 2 p(r) yields an explicit expression for the reAection phase out at the outer turning point, namely out (E) = 0 + 2

S(E) − S(0) + 2%b + O(%2 ) : ˝

(173)

In deriving Eq. (173) we have exploited the fact, that the di?erence between the action integrals (162) at >nite energy E and (146) at energy zero is given, to order less than E, entirely by the tail parts of the integrals beyond the point r in the WKB region. Contributions from smaller distances than r to the action integral can be expected to depend smoothly and analytically on E near threshold, so their e?ect on the di?erence S(E) − S(0) will only be of order E ˙ %2 . Inserting the expression (173) for the outer reAection phase and (144) for the inner reAection phase into the quantization rule (162) yields b % + O(%2 ) ; (174)  where nth is the threshold quantum number already introduced in Section 4.1, Eq. (148). Note that the leading energy dependence of the outer reAection phase near threshold exactly cancels with the energy dependence of the action integral, so the near-threshold
quantization rule (174) has a universal form with a leading energy dependent term proportional to |E|. The parameter b determining the magnitude of the leading energy dependent term in the near-threshold quantization rule is just the tail parameter already de>ned in Eq. (150). The near-threshold quantization rule (174) applies for potential tails falling o? faster than 1=r 2 asymptotically. For potentials falling o? faster than 1=r 2 but not faster than 1=r 3 , the exact zero energy solutions (135) have next-to-leading asymptotic terms whose fall-o? is not a whole power of r faster than the leading term 1 or r. The wave function 0 remains well de>ned and its deviation from unity vanishes asymptotically. The wave function 1 may have asymptotic contributions corresponding to a nonnegative power of r (less than 1); a possible admixture of 0 would be of equal or lower order, so it seems that the de>nition of 1 has some ambiguity regarding possible admixtures of 0 . This does not a?ect the derivation of the expression (171), however, because 1 enters with a small coeScient % in the wave function (168) and only its leading term is relevant. The length parameter (150) is actually invariant with respect to possible ambiguities in the choice of 1 . From the universal form (174) of the near-threshold quantization rule, we immediately derive the leading behaviour of the near-threshold level density,  dn E →0 1 2Mb2 + O(E 0 ) : = (175) dE 2 ˝2 |E| %→0

n ∼ nth −

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

Eq. (175) is also quite universal, in that it holds for all potential tails falling o? faster than 1=r 2
. The leading contribution to the near-threshold level density is quite generally proportional to 1= |E|, and its magnitude is determined by the tail parameter b. Even though the number of bound states in a potential well with a short-ranged tail is >nite and there usually is a >nite interval
below threshold with no energy level at all, the level density at threshold becomes in>nite as 1=
|E| towards E → 0. The probability density for >nding a bound state near E = 0 diverges as 1= |E|, but the expected number of states in a small energy interval below threshold, which is obtained by integrating this
probability density, has a leading term proportional to |E|. It is worth mentioning that the considerations above, and in particular the universal formulas (174) for the near-threshold quantization rule and (175) for the near-threshold level density, apply for all potential wells with tails vanishing faster than 1=r 2 , irrespective of whether the leading asymptotic part of the tail is attractive or repulsive. The only condition is, that on the near side of the tail there be a WKB region in the well where the WKB approximation is good for near-threshold energies. The threshold properties are derived via the zero-energy solutions (135) which can be written as WKB wave functions (136) in this region and involve four independent parameters, two amplitudes D0; 1 and two phases 0; 1 . Due to the freedom to choose the overall normalization of the wave functions (143), (171), there remain three independent tail parameters, which determine the near-threshold properties of the potential and depend only on the potential tail beyond the semiclassical region. Three physically relevant tail parameters derived from D0; 1 and 0; 1 are: (i) the characteristic parameter b given by Eq. (150), which enters the universal near-threshold quantization rule (174) and determines the leading, singular contribution to the near-threshold level density (175); (ii) the mean scattering length aU0 given by Eq. (151); (iii) the zero-energy reAection phase 0 , which enters into the de>nition of the threshold quantum number nth , see Eq. (148), and is an important ingredient of the universal near-threshold quantization rule (174). The characteristic parameter b and the mean scattering length aU0 are related (151) via the di?erence − = 0 − 1 of the phases, aU0 = b=tan(− =2), so any two of the parameters b, aU0 and − contain essentially the same information when tan(− =2) is >nite. 4.2.1. Example: The Lennard-Jones potential To demonstrate the validity of the near-threshold quantization rule (174) we take a closer look at the threshold of the Lennard-Jones potential (93) which was studied in Ref. [73] and discussed in Section 3.4, 

r 6  rmin 12 min ; (176) −2 VLJ (r) = j r r it has its minimum value, −j, at r =rmin . The attractive tail of the potential is of the form (156) with & = 6, and the strength parameter 6 is given by 6 = rmin [4M(rmin )2 j=˝2 ]1=4 . The energy eigenvalues En , measured in units of j, depend only on the reduced strength parameter BLJ = 2M(rmin )2 j=˝2 , and for BLJ = 104 the potential well supports 24 bound states, n = 0; 1; : : : ; 23. In this speci>c case, the energy of the highest bound state, n=23, is near −2:7×10−6 , and its distance to the threshold is less than one seventieth of its separation to the second most weakly bound state at E22 =−0:198 : : :×10−3 , see Table 2 in Section 3.4. It is, however, still not so close to the anticlassical limit, E = 0, as can be seen by noting that the reduced classical turning point a = %rout = (%6 )2=3 , which vanishes in the anticlassical limit, is still as large as 1.56 at the energy E23 .

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

405

-5

log10|E23|

-6

-7

exact

-8

conv. WKB near-thr. qu. rule

-9

-10 0.0

0.5

a

1.0

1.5

Fig. 9. Binding energy |E23 | of the highest bound state in the Lennard-Jones potential (176) as function of the reduced classical turning point a = %rout = (%6 )2=3 , as the reduced strength parameter BLJ is varied in the range between 9800 (corresponding to a ≈ 0) and 104 (a = 1:56). The exact energies are shown as >lled circles, the >lled triangles show the prediction of conventional WKB quantization (84) and the open squares show the results obtained with the near-threshold quantization rule (174).

The immediate vicinity of the anticlassical limit can be studied by gradually reducing the depth of the potential well in order to push the highest bound state closer to threshold. When applying the near-threshold quantization rule (174), we assume the reAection phase at the inner classical turning point, which enters in the de>nition (148) of nth , to be =2, which is not exactly true. For the characteristic parameter b we ignore deviations of the potential tail from the homogeneous −1=r 6 form, i.e., we assume b = b(6) = 0:4779888 × 6 (see Table 3 in Section 4.1). The results are shown in Fig. 9 where the binding energy |E23 | of the highest bound state (in units of j) is plotted as a function of the reduced outer classical turning point a at the energy of this state. The largest a value in the >gure, a = 1:56, corresponds to a reduced strength parameter BLJ = 104 and a binding energy (in units of j) of 2:6969 × 10−6 . By gradually reducing BLJ , the exact binding energy of the n = 23 state, shown as >lled circles in Fig. 9, gets smaller and smaller, and it vanishes for BLJ ≈ 9800. At the same time, the reduced outer classical turning point a at the energy E23 decreases from its initial value 1:56 for BLJ = 10 000 to zero for BLJ ≈ 9800. The binding energy obtained via conventional WKB quantization (>lled triangles), which is based on reAection phases =2 at both inner and outer classical turning points, is almost 40% too large for a ≈ 1:5; it decreases much more slowly and reaches a >nite value near 2 × 10−7 when the exact binding energy vanishes. In contrast, the prediction of the universal near-threshold quantization rule (open squares) is not so accurate when a is larger than unity, but it improves rapidly as a decreases. For the three smallest values of a in Fig. 9 which lie between 0.05 and 0.2, the absolute error of the prediction of the near-threshold quantization rule (174) is less than 10−11 times the potential depth. This is smaller than the level spacing to the second highest state, n = 22, by a factor 10−7 and represents an improvement of

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

a factor of 10−4 over the performance of conventional WKB quantization. Note that applying the universal near-threshold quantization rule (174) is no more involved than applying conventional WKB quantization (84); it is in fact less so, because the action integral only has to be calculated at threshold, E = 0. The direct numerical integration of the one-dimensional Schr4odinger equation is, of course, always possible, but close to threshold it can be a nontrivial and subtle exercise, and it is de>nitely more time consuming than the direct application of a quantization rule. The universal near-threshold quantization rule (174) can thus be of considerable practical use, e.g., when a problem involves many repetitions of an eigenvalue calculation near threshold. 4.3. Nonhomogeneous potential tails The near-threshold properties of a potential tail falling o? faster than 1=r 2 are determined by three independent tail parameters, the length parameter b, the mean scattering length aU0 (which is, however, not de>ned for a potential falling o? as 1=r 3 ) and the zero-energy reAection phase 0 . This statement implies, that there is a region of moderate r values, where the WKB approximation is accurate for near-threshold energies. The tail parameters are then properties only of the potential tail beyond the WKB region; they do not depend on the potential in the WKB region or at even smaller r values. The fact that the leading asymptotic (r → ∞) behaviour of a potential is proportional to 1=r & does not necessarily mean, that the tail parameters are as given by Eqs. (158) [see also Eq. (89)], (159) and (160) for homogeneous tails (156). For these results to be valid, the homogeneous form (156) of the potential must be accurate not only in the limit of large r values, but all the way down to the WKB region. If the potential tail beyond the WKB region deviates signi>cantly from the homogeneous form (156), then the tail parameters di?er from the tail parameters of the homogeneous tails. The extent to which such nonhomogeneous contributions quantitatively a?ect the tail parameters was >rst studied by Eltschka et al. [82,97]. The tail parameters can be derived from the zero-energy solutions of the Schr4odinger equation in the tail region of the potential, which are then expressed in WKB form in the WKB region as described in Sections 4.1 and 4.2. For a given (nonhomogeneous) potential tail, the tail parameters can always be derived, at least numerically, from the known zero-energy wave functions. If the zero-energy solutions of the Schr4odinger equation are known analytically for the tail region of the potential, then the tail parameters can be derived analytically. Several examples are given in this section. Consider a potential tail consisting of two homogeneous terms,   (& )&−2 (&1 )&1 −2 C & C &1 ˝2 V&; &1 (r) = − & − &1 = − ; &1 ¿ & ¿ 2 : + (177) r r 2M r& r &1 In contrast to homogeneous potentials, potential tails consisting of two homogeneous terms contain an intrinsic, energy independent, length scale which can, e.g., be chosen as the position L, where the two contributions have equal magnitude,  L=

C &1 C&



1 &1 − &



(&1 )&1 −2 = (& )&−2



1 &1 − &

;

(178)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

407

they also have an intrinsic scale of depth, which can be chosen as the magnitude of one of the terms at r = L and expressed in terms of a depth parameter K0 , ˝2 (K0 )2 C& C&1 (& )(&−2)=2 (&1 )(&1 −2)=2 = & = &1 ; K0 = = : 2M L L L&=2 L&1 =2 The dimensionless parameter  (&−2)(&1 −2) 2(&1 −&) & < = K0 L = & 1

(179)

(180)

is a useful measure of the relative importance of the two contributions proportional to 1=r & and to 1=r &1 , respectively [82]. When the powers &, &1 in the potential (177) ful>ll the condition &1 − 2 = 2(& − 2) ;

(181)

zero-energy solutions of the Schr4odinger equation are available analytically [99], and analytical expressions for the tail parameters b, 0 and 1 are given in Refs. [82,97]. Zero-energy solutions are also available for the special case, & = 4;

&1 = 5 ;

(182)

which does not ful>ll the condition (181), and analytical expressions for the tail parameters as functions of < are given in Refs. [100,101]. A further example of a nonhomogeneous potential tail for which analytic zero-energy solutions of the Schr4odinger equation are known is,  −1 −1  3 r r4 ˝2 r 3 r4 + =− + ; (183) V1 (r) = − C3 C4 2M 3 (4 )2 which resembles a homogeneous tail for r-values either much larger or much smaller than the characteristic length L=

C4 (4 )2 = : C3 3

(184)

For rL the potential (183) resembles a −1=r 3 potential, as in the van der Waals interaction between a polarizable neutral atom and a conducting or dielectric surface; for r
L it resembles a −1=r 4 potential. The potential (183) was used by Shimizu [41] as a model for describing the e?ects of retardation in atom–surface interactions [57] and has been further studied in Refs. [56,101], see also Section 5.3. A natural de>nition of the intrinsic strength of the potential (183) is via the wave number K0 given by (3 )2 ˝2 (K0 )2 C3 C4 ; = 3 = 4 ⇒ K0 = 2M L L (4 )3

(185)

and the dimensionless parameter measuring the relative importance of the large-r and the smaller-r parts of the potential is, √ 3 2M C3 √ < = K0 L = = : (186) 4 ˝ C4

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

We thus have analytical results for three di?erent potential tails whose leading asymptotic (r → ∞) behaviour is proportional to −1=r & with & = 4: the two-term sum (177) with condition (181) ful>lled implying &1 = 6, the case (182) where it is not ful>lled, and Shimizu’s potential (183). A −1=r 4 potential occurs in several physically important situations, such as in the retarded atom– surface interaction mentioned above, and also as the leading contribution to the interaction between a charged projectile (electron or ion) and a neutral polarizable target (atom or molecule), when retardation e?ects are not taken into account. The strength of the leading −1=r 4 term in an ion-atom potential depends on the polarizability of the neutral atom [102]. For any two-term tail ful>lling (181), the tail parameters b, 0 and − are given as functions of the parameter (180) in Refs. [82,97] (where < is called (). For the special case (&; &1 ) = (4; 6)—for which < = (4 =6 )2 —the parameters b, aU0 = b=tan(− =2) and 0 are       2( 3 − 1 i<)  3 1 2( − i<)    4 4 − arg ; b(4; 6) = 26  41 41  sin  2( 4 − 4 i<)  4 2( 14 − 14 i<)       2( 3 − 1 i<)  2( 34 − 14 i<)   4 4  (4; 6) − arg aU0 = 26  1 1 ;  cos  2( 4 − 4 i<)  4 2( 14 − 14 i<)   

<  < 3 1 3 (4; 6) 1 − ln − 2 arg 2 − i< : (187) 0 =  + 4 2 4 4 4 For the two-term tail (182)—for which <=(4 =5 )3 —the analytical expressions for the tail parameters as functions of < are [100,101],  −1 3(5 )3 2 F with F(z) = |H 1(1) (z)|2 ; b(4; 5) = < (4 )2 3 3  F
( 23 <) (5 )3 5) aU(4; 1 + < = − ; 0 2(4 )2 F( 23 <)   J1 2 ( <) 4 5 5) (188) =  − < − 2 arctan  3 3  : (4; 0 6 3 Y1 2 ( <) 3 3

For Shimizu’s potential (183) we have < = 3 =4 according to Eq. (186), and the analytical expressions for the tail parameters are, (4 )2 F(2<)−1 with F(z) = |H1(1) (z)|2 ; 3   F
(2<) (4 )2 (V1 ) ; 1+< aU0 = − 23 F(2<)   J1 (2<) 3 (V1 ) 0 =  − 4< − 2 arctan : 2 Y1 (2<) b(V1 ) =

In Eqs. (188) and (189) H0(1) (z) = J0 (z) + iY0 (z) are the Hankel functions of order 0 [59].

(189)

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409

4,6

3

b / α1

4,5 2

V1

1

0 0

0.5

1

(a)

1.5



2

4,6 2

b / 4

4,5 V1 1

0 0 (b)

1

2



3

4

5

Fig. 10. Dependence of the length parameter b on the parameter < for two-term potential tails and Shimizu’s potential. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) [for which < = (4 =6 )2 ] and (&; &1 ) = (4; 5) [for which < = (4 =5 )3 ], respectively, and the dotted lines show the results for Shimizu’s potential (183) [for which < = 3 =4 ]. Part (a) shows b=&1 with &1 = 6, 5 and 3, respectively; part (b) shows b=4 .

The dependence of the parameter b on < is illustrated for the three potential tails in Fig. 10. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (&; &1 ) = (4; 5), respectively, and the dotted lines show the results for Shimizu’s potential (183). Part (a) shows b=&1 with &1 = 6; 5 and 3, respectively, and in the limit < → 0 we recover the results for the √ corresponding homogeneous potential tail as given by Eq. (159) and Table 3: b(6) =6 = 2( 34 )=[2 2 2( 54 )], b(5) =5 = 2( 23 )=[31=6 22( 43 )] and b(3) =3 = . Part (b) of Fig. 10 shows b=4 , and for < → ∞ we recover the result for a homogeneous −1=r 4 potential, namely b=4 = 1. Fig. 11 shows the dependence on < of the mean scattering lengths aU0 . Part (a) shows aU0 =&1 with &1 = 6; 5 and 3, respectively; part (b) shows aU0 =4 . For a homogeneous −1=r 4 potential tail, the mean 4 scattering length vanishes, because tan((4) − =2) is in>nite. A modi>cation of the −1=r tail involving signi>cant nonhomogeneity in the potential tail beyond the WKB region generally leads to a >nite mean scattering length, so the e?ect of such a modi>cation is appreciable. For homogeneous −1=r & tails with & ¿ 4, the mean scattering length is always positive, indicating that positive scattering lengths are more probable than negative ones [95]. Shimizu’s potential (183) is a nice example of a potential tail with a negative mean scattering length, implying that negative

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1

(4,6) (4,5)

2

-1

a0 / 4

a0 / α1

0

(4,6)

V1 1

(4,5)

-2

0 V1 0

(a)

1

2 

3

4

0 (b)

1

2 

3

4

Fig. 11. Dependence of the mean scattering length aU0 on the parameter < for three potential tails. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (&; &1 ) = (4; 5), respectively, and the dotted lines show the results for Shimizu’s potential (183). Part (a) shows aU0 =&1 with &1 =6, 5 and 3, respectively; part (b) shows aU0 =4 .

scattering lengths are more probable than positive ones. As < = 3 =4 becomes smaller and smaller, the relative importance of the −1=r 3 part of the potential increases, but scattering lengths and their mean can still be de>ned, because asymptotically (r → ∞) the potential falls o? as −1=r 4 . From Eq. (189), the small-< behaviour of the mean scattering length is <→0

aU0(V1 ) ∼ 23 (ln(<) + (E ) ;

(190)

where (E = 0:5772 : : : is Euler’s constant [59]. For a >xed value of 3 the mean scattering length tends to larger and larger negative values as < = 3 =4 decreases; in the limit < → 0 the mean scattering length approaches −∞ reAecting the fact that scattering lengths cannot be de>ned for a potential asymptotically proportional to 1=r 3 . The zero-energy reAection phases 0 are displayed in Fig. 12. Again, the solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) and (4; 5), and the dotted lines show the results for Shimizu’s potential (183). For < → 0 we recover the results (5) (3) 3 5 3 for the corresponding homogeneous tail as given by Eq. (158): (6) 0 = 4 , 0 = 6 , 0 = 2 . For (4) < → ∞ we recover the result expected for a homogeneous −1=r 4 tail, 0 = . Analytical expressions for the tail parameters b, aU0 and 0 of various potentials are compiled in Table 4. 4.4. The transition from a >nite number to in>nitely many bound states, inverse-square tails The near-threshold quantization rule (164) for an attractive potential tail vanishing as −1=r & with 0 ¡ & ¡ 2 becomes meaningless as & approaches the value 2 (from below). On the other hand, the tail parameter b entering in the near-threshold quantization rule (174) for potential tails vanishing faster than 1=r 2 diverges to in>nity for a (homogeneous) tail vanishing as −1=r & , when the power & approaches 2 from above implying ' → ∞, see Eq. (159). In order to understand the transition from potential wells with tails vanishing faster than −1=r 2 , which support at most a >nite number of bound states, to those with tails vanishing more slowly than −1=r 2 , which support in>nitely many

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411

(4,6)

1.5

(4,5) 0 / π

V1 1

0

1

2 

3

4

Fig. 12. Dependence of the zero-energy reAection phase 0 on the parameter < for three potential tails. The solid and dashed lines show the results for the two-term potential tails (177) for the cases (&; &1 ) = (4; 6) [for which < = (4 =6 )2 ] and (&; &1 ) = (4; 5) [for which < = (4 =5 )3 ], respectively, and the dotted lines show the results for Shimizu’s potential (183) [for which < = 3 =4 ].

bound states, it is necessary to look in some detail at potentials with tails asymptotically proportional to the inverse square of the distance, 2 ( def ˝ r →∞ : (191) V (r) ∼ V( (r) = 2M r 2 Inverse-square potentials of the form (191) with positive or negative values of the strength parameter ( occur in various physically relevant situations. Separating radial and angular motion in the twoor more-dimensional Schr4odinger equation leads to a centrifugal potential of the form (191) with ( as given by Eqs. (58) and (59) for three and two dimensions, respectively. Attractive and repulsive inverse-square potentials occur in the interaction of an electrically charged particle with a dipole, e.g. in the interaction of an electron with a polar molecule or with a hydrogen atom in a parity-mixed excited state [103–106]. If the inverse-square tail (191) is suSciently attractive, more precisely, if 1 1 def (192) ( = − g¡ − ; g¿ ; 4 4 then the potential supports an in>nite number of bound states, usually called a “dipole series”, and towards threshold, E = 0, the energy eigenvalues of the bound states depend exponentially on the quantum number n, [103,107]   2n  n→∞ En ∼ − E0 exp−  : (193) g − 14 The strength g of the attractive inverse-square tail determines the asymptotic value of the ratio of successive energy eigenvalues,   En n→∞ 2  ; (194) ∼ exp  En+1 1 g− 4

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Table 4 Characteristic length b, mean scattering length aU0 and zero-energy reAection phase 0 for several attractive potential tails, V (r) = −v(r) × ˝2 =(2M) v(r)

b

aU0

(K0 )2 ,(L − r)

1=K0

& '1+2' 2(1+')2

L & '2'

3

—

(& )&−2 r& 3 r3

(& )&−2 + r& (&1 )&1 −2 r &1

  ' 2(1−')  2(z+ )  &1 (2')2(1+')  2(z− )  ×  

2(z+ ) sin '2 − arg 2(z −)

2(1−') cos(') 2(1+')

  ' 2(1−')  2(z+ )  &1 (2')2(1+')  2(z− )  ×  

2(z+ ) cos '2 − arg 2(z −)

0

Comments

0 (' + 12 )

'=

1 &−2

¡1

(1 + ') 2 +

'=

1 &−2

¡1

<'[1 − ln( <'2 )]

&1 2

−2 arg(z+ )

<=

3 2



=&−1

z± = 3 r3

(4 )2 r4

+

3 [1 2

+ coth( < )] 2

 + <(1 − ln <2 )

—

<=

& &1

1='

1±'−i<' 2

3 4

−2 arg 2(1 − i <2 ) (4 )2 r4

+

(5 )3 r5

3(5 )3 (4 )2

F(z)−1

3

(5 ) − 2( (1 + < )2 4

F  (z) ) F(z)

5 6

−

4 3

2 arctan

3

[ r 3 +

r 4 −1 ] (4 )2

(K0 )2 1+exp[(r−L)=]

(K0 )2 exp(− r ) a

(4 )2 3

F(z)−1

 coth(K0 )



2

− (24 )3 (1 + <

F  (z) ) F(z)

L − 2((E + R(z))

2 ln(K0 )

<− 

J1=3 (z) Y1=3 (z)



3 2

 − 4<−   2 arctan YJ11(z) (z)  + 4K0  ln 2   2 0 ) +2 arg 2(iK 2(2iK0 ) 1 2

z = 23 <    (1) 2 F = H1=3 (z) <=

3 4

z = 2< F = |H1(1) (z)|2 z = 2 (iK0 ) a a



Here the semiclassical region of “small” r is r → −∞. (E is Euler’s constant and

< = ( 45 )3

2

the digamma function.

but not the explicit positions of the energy levels, which are >xed by the constant E0 in Eq. (193). This reAects the fact that there is no energy scale in a Schr4odinger equation with a kinetic energy and an inverse-square potential; if (r) is a solution at energy @, then (sr) is a solution at energy s2 @. For a pure −1=r 2 potential one can obtain a discrete bound state spectrum corresponding to the right-hand side of Eq. (193) by requiring orthogonality of the bound state wave functions at di?erent energies, but the resulting spectrum is unbounded from below, En → −∞ for n → −∞ [107]. In a realistic potential well with a suSciently attractive inverse-square tail, Eq. (192), the actual positions of the bound state energies are determined by the behaviour of the potential at small distances, where it must necessarily deviate from the pure −1=r 2 form. If the potential tail contains

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413

a further attractive term proportional to −1=r m , m ¿ 2, then as r decreases this term becomes dominant and the WKB approximation becomes increasingly accurate. Potential wells with two-term tails,   1 ˝2 (m )m−2 g r →∞ (195) V (r) ∼ Vg; m (r) = − + 2 ; m ¿ 2; g ¿ ; m 2M r r 4 support an in>nite dipole series of bound states and there may be a WKB region at moderate r values if the well is deep enough. In this case, near-threshold properties of the bound states can be derived [84,108] by matching the asymptotic (r → ∞) solutions of the Schr4odinger equation with the inverse-square term alone to zero-energy solutions of the tail (195), which are then expressed as WKB waves in the WKB region in a procedure similar to that used in Section 4.2. This results [108] in an explicit expression for the factor E0 , which asymptotically (n → ∞) determines the positions of the energy levels in the dipole series (193), 4

2˝2 (m − 2) m−2 E0 = exp M(m )2



    tan(S˜ 0 =˝) 2 C + D + + arctan : B 2 tanh(;=2)

The parameters B, ;, C and D appearing in Eq. (196) are,  1 2B B= g− ; ;= ; C = arg 2(iB); D = arg 2(i;) ; 4 m−2

(196)

(197)

and S˜ 0 is essentially the threshold value of the action integral from the inner classical turning point rin (0) to a point r in the WKB region,  m−2  m 2 S˜ 0 2 in (0)  1 r

p0 (r ) dr + − = − : ˝ ˝ rin (0) m−2 r 2 4

(198)

A condition for the applicability of the formula (196) is, that near the point r in the WKB region the potential must be dominated by the −1=r m term so that the inverse square contribution can be neglected; the sum of the integral and the term proportional to 1=r (m−2)=2 in (198) is then independent of the choice of r. More explicit solutions are available when the whole potential consists of an attractive inversesquare tail and a repulsive 1=r m core,   g ˝2 (m )m−2 (199) − 2 ; m¿2 : V (r) = 2M rm r The existence of a WKB region in the well is not necessary in this case, because analytical zero-energy solutions of the Schr4odinger equation are available for the whole potential. The zeroenergy solution which vanishes at r = 0 approximates >nite energy solutions to order less than O(E) in the region of small and moderate r values, and it can be matched to the solution which vanishes asymptotically in the presence of the attractive inverse-square tail. This yields the following

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

expression for the factor E0 de>ning the energies of the near-threshold bound states of the dipole series (193) [108], 4   2˝2 (m − 2) m−2 C+D ; E0 = exp 2 M(m )2 B

(200)

where B, C and D are as already de>ned in Eq. (197). Note that E0 is only de>ned to within a factor consisting of an integer power of the right-hand side of Eq. (194); multiplying E0 by an integer  

power of exp 2= g − 14 does not a?ect the energies in the dipole series (193) except for an appropriate shift in the quantum number n labelling the bound states. One interesting feature of dipole series is that, for a given strength ( ¡ − 14 of the attractive potential tail, the limit of large quantum numbers, n → ∞, does not coincide with the semiclassical limit. As discussed in Section 2.1, the semiclassical limit for an inverse-square potential can be approached for large absolute values of the potential strength, but not by varying the energy. As a consequence, the energy eigenvalues of a dipole series obtained via conventional WKB quantization (84) do not become more accurate in the limit n → ∞, not even when the Langer modi>cation (64) is used in the WKB calculation. The WKB energies obtained with the Langer modi>cation acquire a constant relative error in the limit n → ∞, whereas the error grows exponentially with n if they are calculated without the Langer modi>cation [109]. A potential with an attractive inverse-square tail (191) no longer supports an in>nite series of bound states, when the strength parameter ( is equal to (or larger than) − 14 . This can be expected from the breakdown of formulas such as (193), (196) and (200) when −( = g = 14 . It is also physically reasonable, considering that the inverse-square potential V(=−1=4 is the s-wave (l2 = 0) centrifugal potential for a particle moving in two spatial dimensions, see Eq. (59). The fact that the radial Schr4odinger equation for a particle moving in a plane with zero angular momentum actually contains an attractive centrifugal force has lead Cirone et al. [110] to investigate possibilities of a state being bound by the associated force, which the authors call “anticentrifugal”. Kowalski et al. [111] have contributed to a clari>cation of the issue by drawing attention to the topological e?ects occurring when a singular point is extracted from the plane as an origin for the de>nition of the polar coordinates. It is diScult to imagine a physical mechanism that would bind a free particle in a Aat plane, so the discontinuation of dipole series of bound states at the value − 14 of the strength parameter ( seems more than reasonable. A potential well with a weakly attractive inverse-square tail, i.e. with a strength parameter in the range −

1 6(¡0 ; 4

(201)

can support a (>nite) number of bound states if supplemented by an additional attractive potential. If the additional potential is regular at the origin, then the action integral from the origin to the outer classical turning point diverges because of the −1=r 2 singularity of the potential at r = 0, so a naive application of the generalized quantization rule (162) does not work. This can be overcome by shifting the inner classical turning point to a small positive value and adjusting the reAection phase in accordingly [70].

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415

The near-threshold quantization rule for a weakly attractive inverse-square tail has been studied in some detail by Moritz et al. [112], and analytical results have been derived for tails of the form   1 ˝2 (m )m−2 g r →∞ (weak) V (r) ∼ Vg; m (r) = − + 2 ; m ¿ 2; g 6 : (202) m 2M r r 4 For g ¡ 14 (i.e., excluding the limiting case g = 14 ), the near-threshold quantization rule is [112], n = nth − with

(%m =2)20 + O((%m )40 ) + O(%2 ) ; 2 2' sin(0)(m − 2) 0'[2(0)2(')]

(203)

 20 1 1 −g= (+ and ' = : (204) 0= 4 4 m−2 The >nite but not necessarily integer threshold quantum number nth in Eq. (203) is given by  m−2  m 2 1 r 2 in (0)  '

− −  : (205) p0 (r ) dr + − nth  = ˝ rin (0) m−2 r 2 4 2 

As in the discussion of Eqs. (196) and (198), the point r de>ning the upper limit of the action integral must lie in a region of the potential well where the WKB approximation is suSciently accurate and the potential is dominated by the −1=r m term, so the inverse-square contribution can be neglected; the sum of the integral and the term proportional to 1=r (m−2)=2 in (205) is then independent of the choice of r. When we express % in terms of the energy E = −˝2 %2 =(2M), the near-threshold quantization rule (203) becomes n = nth − B(−E)0 ;

(206)

with B=

(M(m )2 =(2˝2 ))0 : sin(0) (m − 2)2' 0'[2(0)2(')]2

(207)

The limiting case (=− 14 corresponding to 0=0 and '=0 requires special treatment; the near-threshold quantization rule in this case is [112],   ˝2 2=(m − 2) 1 n = nth + ; B = : (208) +O ln(−E=B) 2M(m )2 [ln(−E=B)]2 Again, nth is given by the expression (205); note that ' vanishes in this case. We now have a very comprehensive overview of near-threshold quantization in potential wells with attractive tails. Potentials falling o? as −1=r & with a power 0 ¡ & ¡ 2 support an in>nite number of bound states, and the limit of in>nite quantum numbers is the semiclassical limit. The near-threshold quantization rule (164) contains a leading term proportional to 1=(−E)1=&−1=2 in the expression for the quantum number n. For & = 2, the threshold E = 0 no longer represents the semiclassical limit of the Schr4odinger equation, but the potential still supports an in>nite number of bound states, if the attractive inverse-square tail is strong enough, Eq. (192); the near-threshold quantization rule now contains

g − 14 ln(−E) in the expression for the quantum number n, see Eq. (193). The attractive

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

inverse-square tail ceases to support an in>nite series of bound states at the value g = −( = 14 of the strength parameter, which corresponds to the strength of the (attractive) s-wave centrifugal potential for a particle in a plane. In the near-threshold quantization rule, the leading term in the expression for the quantum number now is a >nite number nth related to the total number of bound states,
and 1

the next-to-leading term contains the energy as 1=ln(−E) for ( = − 14 [Eq. (208)], or as (−E) (+ 4 for ( ¿ 14 , see Eq. (206). It is interesting to note, that the properties of potential wells with short-ranged tails falling o? faster than 1=r 2 >t smoothly into the picture elaborated for inverse-square tails when we take the strength of the inverse-square term to be zero. The near-threshold quantization rule (203) acquires the form (174) when ( = 0, 0 = 12 , and the coeScient of % becomes b(m) = with b(m) given by Eq. (159) when we also insert ' = 1=(m − 2). The discussion of weakly attractive inverse-square tails, de>ned by the condition (201), can be continued without modi>cation into the range of weakly repulsive inverse-square tails, de>ned by strength parameters in the range 3 0¡(¡ : (209) 4  The parameter 0 = ( + 14 determining the leading energy dependence on the right-hand sides of Eqs. (203) and (206) then lies in the range 1 ¡0¡1 ; (210) 2 and the leading energy dependence (−E)0 expressed in these equations is still dominant compared to the contributions of order O(E), which come from the analytical dependence of all short-ranged features on the energy E and were neglected in the derivation of the leading near-threshold terms. We can thus complete the comprehensive overview of near-threshold quantization by extending it to repulsive potential tails. For weakly repulsive inverse-square tails (209), the formulas (203) and (206) remain valid. The upper boundary of this range is given by  3 1 (= ; 0= (+ =1 ; (211) 4 4 which corresponds to the p-wave centrifugal potential in two spatial dimensions, l2 = ±1, see Eq. (59). At this limit, the near-threshold quantization rule has the form, n = nth − O(E) ;

(212)

and this structure prevails for more strongly repulsive inverse-square tails, ( ¿ 34 , and for repulsive potential tails falling o? more slowly than 1=r 2 . Repulsive tails falling o? more rapidly than 1=r 2 comply with the case ( = 0, i.e. of vanishing strength of the inverse-square term in the potential, and, provided there is a suSciently attractive well at moderate r values, the quantization rule has the form (174) with a tail parameter b and a threshold quantum number nth which also depends on the shorter-ranged part of the potential. Note that the condition (211) also de>nes the boundary between systems with a singular and a regular level density at threshold. For attractive potential tails and for repulsive potential tails falling o? more rapidly than 1=r 2 or as an inverse-square potential with ( ¡ 34 , the level density dn=dE

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417

Table 5 Summary of near-threshold quantization rules for attractive and repulsive potential tails. The second column gives the leading term(s) to the quantization rule in the limit of vanishing energy, E = −˝2 %2 =(2M) → 0. The third column lists equations where explicit expressions for the constants appearing in the second column can be found; these can apply quite generally, as in the >rst row, or to special models of potential tails with the asymptotic behaviour given in the >rst column V (r) for r → ∞ ˝2 − 2M (& )&−2 =r & , ˝2 2M

(=r 2 , ( ¡ −

Quantization rule for E → 0 0¡&¡2

1 4

(¿

3 4

3 4

˙ ±1=r , & ¿ 2



|(| −

n ∼ nth + A=ln(−E=B)

˙ +1=r & , 0 ¡ & ¡ 2 &

F(&)=(%& )

√

n ∼ nth − B(−E) n ∼ nth − O(E) n ∼ nth − O(E) 1 n ∼ nth − b% 

Refs. for constants

(2=&)−1

1 n ∼ − 2 ln(−E=E0 )=

( = − 14 − 14 ¡ ( ¡

n∼

1 

(+1=4

F(&): Eq. (163) 1 4

E0 : (196), (200) nth : (205) A, B: (208) nth : (205) B: (207) nth : (205)

nth : (148) b: (150), (154), (159), (187), (188), (189), Tables 3, 4

is singular at threshold, and the leading singular term is determined by the tail of the potential. For a repulsive inverse-square tail with ( ¿ 34 , and for a repulsive tail falling o? more slowly than 1=r 2 , the level density is regular at threshold, and the leading (constant) term depends also on the short-ranged part of the potential. A summary of the near-threshold quantization rules reviewed in the last three subsections is given in Table 5. 4.5. Tunnelling through a centrifugal barrier A potential with a repulsive tail at large distances and a deep attractive part at small distances forms a barrier through which a quantum mechanical particle can tunnel. If the repulsive tail falls o? more slowly than 1=r 2 , then the threshold represents the semiclassical limit (in the tail region), and the conventional WKB formula for the tunnelling probability (114),  1 rout (E) −2I (E) WKB ; I (E) = |p(r)| dr ; (213) PT (E) = e ˝ rin (E) involving the action integral between the inner classical turning point rin (E) and the outer classical turning point rout (E) is expected to work well near threshold, provided the conditions of the semiclassical limit are also well ful>lled around rin (0). As we saw in Section 3.5, the WKB formula (213) fails near the base of a barrier with a tail falling o? faster than 1=r 2 , because this corresponds to the anticlassical limit of the Schr4odinger equation; the exact tunnelling probability vanishes at the base whereas Eq. (213) produces a >nite result. Potential barriers involving the centrifugal term in

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

the radial Schr4odinger equation fall o? as 1=r 2 for large r, and hence lie on the boundary between these two regimes. When the potential on the near side (smaller r) of the barrier supports a WKB region where WKB wave functions are good approximate solutions of the Schr4odinger equation for near-threshold energies, then the amplitudes for transmission through and reAection by the barrier can be obtained in a way quite analogous to the methods for deriving the near-threshold quantization rules as described in Section 4.4, see Refs. [84,108,112]. Analytical results have been derived for a potential tail of the form   ( ˝2 (m )m−2 r →∞ V (r) ∼ V(; m (r) = ; m¿2 : (214) − 2M r 2 rm For suSciently large r values, the −1=r m term in the potential tail (214) can be neglected, and the wave function on the far side of the barrier at energy E = ˝2 k 2 =(2M) ¿ 0 can be approximated by the analytically known solution of the Schr4odinger equation with the 1=r 2 potential alone. In the barrier region, the analytically known zero-energy solutions for the potential (214) solve the Schr4odinger equation to order less than O(E), and a unique solution is obtained by matching to the asymptotic wave function just mentioned. In the WKB region on the near side of the barrier, the unique solution constructed as above can be written as a superposition of inward and outward travelling WKB waves. Comparing the amplitudes of inward and outward travelling waves on both sides of the barrier yields the transmission amplitude to order less than O(E), and the leading contribution to the transmission probability PT is, k →0

PT ∼ P(m; ()(km )20 ;

(215)

with the coeScient P(m; () =

42 =220 ; (m − 2)2' 0'[2(0)2(')]2

(216)

 0 = ( + 14 and ' = 20=(m − 2) are already de>ned in Eq. (204), but now, in Eqs. (215) and (216), ( can also assume nonnegative values. When the inverse-square tail originates from a centrifugal potential in three dimensions, its strength parameter ( is related to the angular momentum quantum number l3 by,  1 1 (217) ( = l3 (l3 + 1); 0 = ( + = l3 + ; 4 2 and the energy dependence of the transmission probability (215) is simply an expression of Wigner’s threshold law [113], according to which probabilities Pl3 which are suppressed by a centrifugal barrier of angular momentum quantum number l3 generally behave as E →0

Pl3 ˙ E 0 = E l3 +1=2 :

(218)

Note that the formulae (215) and (216) can be continued to negative strength parameters in the range of weakly attractive (201) or vanishing inverse-square tails; in fact, they hold for any ( ¿ − 14 . For − 14 ¡ ( 6 0, the potential tail (214) no longer contains a barrier, so transmission between the asymptotic (large r) region and the inner WKB region is classically allowed at all (positive) energies. The probability for this classically allowed transmission is, however, less than unity, because incoming

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419

waves can be reAected in the region of high quantality, where the WKB approximation is not good, see Section 2.3. Such “quantum reAection” will be discussed in more detail in Section 5. The probability for transmission from the outer asymptotic region to the inner WKB region actually goes to zero at threshold, even for weakly attractive inverse-square tails, − 14 ¡ ( 6 0, for which there is no barrier. Wigner’s threshold law (218) can be formally extended down to negative angular momentum quantum numbers in the range − 12 ¡ l3 6 0. We now discuss the accuracy of the WKB formula (213) for tunnelling probabilities through a centrifugal barrier at near-threshold energies. For an inverse-square tail (191) with ( ¿ 0, the WKB √ integral I (E) is dominated by the tail near outer classical turning point rout = (=k at near-threshold √ energies, and it diverges as ln( (=k) for k → 0. This means that the leading contribution to the √ √ WKB tunnelling probability (213) is proportional to k 2 ( ˙ E ( , in contradiction to the exact result (218) which obeys Wigner’s threshold law. This contradiction can be resolved by invoking the Langer modi>cation (64) when applying the WKB formula, but a residual error remains, because the WKB expression does not necessarily give the right coeScient of the E 0 term. For potential tails of the form (214), Moritz [84,108] found an upper bound for the WKB integral entering the expression (213), namely     20 m0 2−4=m ln Iapprox = − 1 + O((km ) ) ; (219) m−2 (km )1−2=m so the leading term for the corresponding tunnelling probability,  2m0=(m−2) e (km )20 ; PTWKB; approx = e−2Iapprox = 20

(220)

is a lower bound for the leading term of the WKB tunnelling probability (213). It turned out [114] that the expression (220) is not only a lower bound but becomes equal to the WKB tunnelling probability (213) in the limit E → 0. In the near-threshold limit, the WKB tunnelling probability (213) thus overestimates the exact tunnelling probability, which is given by Eqs. (215) and (216), by a factor G,   0+' e−2Iapprox e 2(0)2(') 2 def G = lim = : (221) k →0 PT 200−1=2 ''−1=2 For large values of the strength ( of the 1=r 2 term in the potential, 0 and ' are also large and we can express the gamma functions in Eq. (221) via Stirling’s formula [59]. This gives   1 m (→∞ +O ; (222) G ∼ 1+ 120 02 showing that the WKB treatment gives the correct leading behaviour of the near-threshold tunnelling probabilities in the limit of large angular momenta. The dependence of the factor G on 0 ≡ l3 + 12 is illustrated in Fig. 13 for powers m of the attractive term in the potential tail (214) ranging from m = 3 to m = 7. The WKB results become worse as 0 decreases and as m increases. For the realistic example 0 =3=2; m=6, which corresponds to angular momentum quantum number l3 =1 and an inter-atomic van der Waals attraction, the WKB formula (213) overestimates the exact tunnelling probabilities by 38% near threshold. Transmission probabilities can also be calculated for attractive inverse-square tails, if there is a WKB region of moderate r values where the WKB approximation is good. The equations (215)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

2.5 m=3 m=4 m=5 m=6 m=7

2.0 G

1.5

1.0 0

2

4

µ

6

8

10

Fig. 13. Behaviour of G [Eq. (221)] as function of 0 for m = 3; : : : ; 7. For a potential barrier (214) consisting of an attractive −1=r m potential and a centrifugal term corresponding to angular momentum quantum number l3 = 0 − 12 ; G gives the factor by which the conventional calculation of transmission probabilities via the WKB formula (213), including the Langer modi>cation (64), overestimates the exact result (215), (216) for near-threshold energies. From [108].

and (216) for the exact transmission probabilities through the potential tail (214) are also valid for weakly attractive or vanishing inverse-square terms, − 14 ¡ ( 6 0. In this range of values of (, the Langer modi>cation (64) actually produces an asymptotically repulsive potential with a barrier to tunnel through, so the conventional WKB formula (213) can be applied for near-threshold energies. Note, however, that the factor (221) by which the conventional WKB result overestimates the exact result is quite large in the range − 14 ¡ ( 6 0 corresponding to 0 ¡ 0 ¡ 12 and 0 ¡ ' ¡ 1=(m − 2), and diverges to +∞ for ( → − 14 corresponding to 0; ' → 0. For vanishing strength of the inverse-square term, ( = 0; 0 = 12 ; ' = 1=(m − 2), the near-threshold probability (215), (216) for transmission through the potential tail (214) reduces to, 4'1+2' PT = km P(m; ( = 0) = km = 4kb(m) ; (223) 2(1 + ')2 where b(m) is the length parameter (159), which determines the near-threshold quantization rule (174) and the level density just below threshold for a homogeneous −1=r m potential tail. Eq. (223) thus formulates a connection between the near-threshold properties of bound states at negative energies and those of continuum states at positive energies. This connection applies not only for the special case of vanishing strength of the inverse-square term, but to all strengths within the range of weak inverse-square tails [112],  1 0¡0 = ( + ¡1 : (224) 4 If we write the near-threshold quantization rule (203) as %→0

n ∼ nth  − C¡ (%)20

(225)

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421

for small negative energies E = −˝2 %2 =(2M), and the expression for near-threshold transmission probabilities (215), (216) at small positive energies E = +˝2 k 2 =(2M) as k →0

PT ∼ C¿ (k)20

(226)

then the constants C¡ and C¿ are related by C¿ = 4 sin(0)C¡ :

(227)

The constant  in Eqs. (225), (226) can be any (common) length; it is included so that the coef>cients C¡ and C¿ are dimensionless. For vanishing strength of the inverse-square term, i.e. for potentials falling o? faster than 1=r 2 , we have 20 = 1, and the product C¡  is simply the length parameter b of the potential tail as de>ned in Section 4.1, Eq. (150), see also the bottom block of Table 5. In this case, Eqs. (225) and (226) reduce to %→0

n ∼ nth  − %b;

k →0

PT ∼ 4kb ;

(228)

with the same length parameter b appearing in both equations. The relation (227) is independent of the power of the shorter-ranged attractive contribution to the potential tail (214). It seems reasonable to assume that it is a universal relation connecting the near-threshold states at positive and negative energies for potentials with weak inverse-square tails, and that this is also true for the special case (228) of potentials falling o? faster than 1=r 2 . The behaviour for this latter case is con>rmed in Section 5.1, see Eq. (241). 5. Quantum re ection Just as quantum mechanics can allow a particle to tunnel through a classically forbidden region, it can also lead to the reAection of a particle in a classically allowed region where there is no classical turning point. The term “quantum reAection” refers to such classically forbidden reAection. Quantum reAection can only occur in a region of appreciable quantality, i.e. where the condition (36) is violated. In regions where Eq. (36) is well ful>lled, motion is essentially (semi-) classical, and, in the absence of a classical turning point, the particle does not reverse its direction. Quantum reAection can occur above a potential step or barrier, or in the attractive long-range tails of potentials describing the interaction of atoms and molecules with each other or with surfaces. For potentials falling o? faster than 1=r 2 , the probability for quantum reAection approaches unity at threshold, so it is always an important e?ect at suSciently low energies. Quantum reAection had been observed to reduce the sticking probabilities in near-threshold atom-surface collisions more than twenty years ago [115–120], and the recent intense activity involving ultracold atoms and molecules has drawn particular attention to this phenomenon [121–125]. As described in Section 3.5, transition and reAection amplitudes, T and R, can be de>ned using WKB wave functions for the incoming, transmitted and reAected waves in the semiclassical regions, or plane waves if the potential tends to a constant asymptotically. The transmission and reAection probabilities (103), PT = |T |2 ;

PR = |R|2 ;

(229)

are independent of the choice of WKB waves or plane waves, but the phases of the transmission and reAection amplitudes depend on this choice [see Eqs. (104), (105)] and also on the choice of

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reference points, at which the reference waves have vanishing phase. Throughout this section, the reAection amplitude is always de>ned with respect to incoming plane waves incident from the right as in Eq. (102). Whenever the phase of R is important, we retain the subscript “r” to remind us of this choice. The point of reference is taken to be at r = 0 unless explicitly stated otherwise. For brevity, we shall refer to the absolute value of the reAection amplitude, i.e. to the square root of the reAection probability, as the re?ectivity. 5.1. Analytical results A fundamentally important example [46] is the Woods-Saxon step potential, V (r) = −

˝2 (K0 )2 =(2M) ; 1 + exp(r=)

(230)

which is treated in detail in the textbook of Landau and Lifshitz [43]. The reAection amplitude, de>ned with reference to plane waves (102) is,  
2(2ik) 2(−ik − iq) 2 q + k Rr = (231) ; q = (K0 )2 + k 2 : 2(−2ik) 2(ik − iq) q−k From the properties of the gamma functions of imaginary argument [59] the absolute value of the reAection amplitude (231) is sinh[(q − k)] |R| = ; (232) sinh[(q + k)] as already used in Section 2.1, Eq. (14). For small values of the di?useness,  → 0, the Woods-Saxon potential (230) approaches the sharp step potential already discussed in Sections 2.4 and 4.1, and the reAection amplitude becomes [cf. Eq. (39)] q−k Rr = − : (233) q+k The large-r tail of the Woods-Saxon potential (230) is an exponential function, ˝2 (K0 )2 exp(−r=) : (234) 2M The exponential potential (234) is an interesting example in itself, because the semiclassical approximation becomes increasingly accurate for r → −∞, although the r-dependence of the potential gets stronger and stronger. The Schr4odinger equation for the potential (234) is solved analytically [59] by any Bessel function of order ' = 2ik and argument z = 2K0  exp[ − r=(2)]. The solution which merges into a leftward travelling WKB wave for r → −∞ is the Hankel function (r) ˙ H'(1) (z), and matching this solution to a superposition of incoming and reAected plane waves yields the reAection amplitude, 2(2ik) (K0 )−4ik exp(−2k) : Rr = (235) 2(−2ik) V (r) = −

For any given value of k, the reAection amplitude (231) of the Woods-Saxon step actually becomes equal to the result (235) in the limit that the relative diBuseness [63] K0  is large, K0  → ∞.

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423

In the near-threshold region, the leading behaviour of the reAection amplitude (231) is k →0

Rr ∼ − [1 − 2k coth(K0 ) − 4ik((E + R{ where (E = 0:57721 : : : is Euler’s constant and contributions to the reAectivity are, |R| = 1 − 2kb + O((k)2 );

2

2 (−iK0 )})]

;

(236)

= 2
=2 is the digamma function [59]. The leading

b =  coth(K0 ) :

(237)

This means, that the reAectivity is unity at threshold, and its initial decrease from unity is linear in k, i.e. in the asymptotic velocity of the particle on the side where this velocity goes to zero. This threshold behaviour of the quantum reAectivity is very general and holds for all potential tails falling o? faster than 1=r 2 [55,119] as can be shown using the methods applied in Sections 4.1 and 4.2. Consider an attractive potential going to zero faster than 1=r 2 for large r with a semiclassical WKB region at moderate or small r values. Using the zero-energy solutions (135) of the Schr4odinger equation, which go unity resp. r for large r and behave as (136) for small r, we can construct a wave function (r) =

D0

1 (r)exp(i0 =2)

− D1 0 (r)exp(i1 =2) ; D0 D1 sin[(0 − 1 )=2]

(238)

which is proportional to a leftward travelling WKB wave of the form (100) in the semiclassical region r → 0. Matching the asymptotic (r → ∞) form of the wave function (238) to the superposition 1 kr →0 √ [exp(−ikr) + Rr exp(ikr)] ˙ 1 + Rr − ikr(1 − Rr ) ˝k

(239)

gives the leading near-threshold contribution to the reAection amplitude, k →0

Rr ∼ −

1 − ik exp[ − i(0 − 1 )=2]D1 =D0 : 1 + ik exp[ − i(0 − 1 )=2]D1 =D0

For the reAectivity |R|, Eq. (240) implies k →0

2

|R| ∼ 1 − 2kb = exp(−2kb) + O(k );

  D1 0 − 1 : b= sin D0 2

(240)

(241)

Eq. (241) implies that the probability for quantum reAection behaves as 1 − 4kb near threshold, and k →0 this is consistent with the result PT ∼ 4kb given for the transmission probability in Eq. (228) at the end of Section 4. The characteristic length parameter of the quantal region of the potential tail, namely b as de>ned in Eq. (150), determines not only the near-threshold quantization rule (174) and the near-threshold level density (175), but also the reAection and transmission properties of the potential tail near threshold. The near-threshold behaviour of the phase of the reAection amplitude also follows from (240) and the result is, k →0

arg(Rr ) ∼  − 2k aU0 ;

aU0 =

b : tan [(0 − 1 )=2]

(242)

The parameter aU0 determining the near-threshold behaviour of the phase of the reAection amplitude is just the mean scattering length de>ned in Eq. (151). Scattering lengths and mean scattering lengths

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

are only de>ned for potentials falling o? faster than −1=r 3 . For a potential proportional to −1=r 3 , V (r) = −

˝ 2 3 C3 = − ; r3 2M r 3

(243)

the wave function which is proportional to a leftward travelling
WKB wave of the form (100) in the semiclassical region r → 0 is (r) ˙ H1(1) (z)=z with z = 2 3 =r, and matching to the asymptotic waves (239) gives k →0

arg(Rr ) ∼  − 2k3 ln(k3 ) :

(244)

Note that the formula (241) for the near-threshold reAectivity holds for all potentials falling o? faster than −1=r 2 , even for those such as (243), where the phase of the reAection amplitude becomes divergent at threshold. The energy dependence of the phase of the reAection amplitude can be related to the time gain or delay of a wave packet during reAection [126]. If the momentum distribution of the incoming wave packet is sharply peaked around a mean momentum ˝k0 , then the shape of the reAected wave packet is essentially the same and the time shift can be calculated in the same way as in partial-wave scattering [127,128] where the reAection amplitude R is replaced by the partial-wave S-matrix. The derivative of arg[R(k)] with respect to k, taken at k0 , describes an apparent shift Wr in the point of reAection, Wr = −

1 d [arg(Rr )]k=k0 : 2 dk

(245)

The time evolution of the reAected wave packet corresponds to reAection of a free wave at the point r = Wr rather than at r = 0. For a free particle moving with the constant velocity v0 = ˝k0 =M this implies a time gain Wt =

M d d 2Wr [arg(Rr )]k=k0 = −˝ [arg(R)]E=˝2 k02 =(2M) : =− v0 ˝k0 d k dE

(246)

For a positive (negative) value of Wr the reAected wave packet thus experiences a time gain (delay) relative to a free particle (with the same asymptotic velocity v0 ) travelling to r = 0 and back. Note however, that the classical particle moving under the accelerating inAuence of the attractive potential is faster than the free particle; the quantum reAected wave packet may experience a time gain with respect to a free particle but nevertheless be delayed relative to the classical particle moving in the same potential (see Section 5.2). Eq. (242) implies that the near-threshold behaviour of the space shift (245) and of the time shift (246) is k →0

Wr 0∼ aU0 ;

k →0

Wt 0∼

2M aU0 : ˝k0

(247)

The near-threshold behaviour of the time shift due to reAection for a wave packet with a narrow momentum distribution is determined by the mean scattering length aU0 . Near threshold, the quantum reAected wave packet evolves as for a free particle reAected at r = aU0 .

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

425

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

5

10

15

20

25

kα Fig. 14. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of k& for various values of &. From [56].

5.2. Homogeneous potentials The threshold behaviour of the reAection amplitude as summarized by Eqs. (241) and (242) is determined by the two parameters b and aU0 which were obtained analytically for a variety of attractive potential tails in Section 4. They are given for homogeneous potentials, V&att (r) = −

C& ˝2 (& )&−2 = − r& 2Mr &

(248)

in Eqs. (159) and (160) and are tabulated in Table 3. For homogeneous potentials (248) the properties of the Schr4odinger equation do not depend on the energy E = ˝2 k 2 =(2M) and potential strength parameter & independently, but only on the product k& . For energies above the near-threshold region, analytical solutions of the Schr4odinger equation are not available (except for & = 4), and the reAection amplitudes have to be obtained numerically. Figs. 14 and 15 show the real and imaginary parts of ln Rr , namely ln |R| and arg Rr , as functions of k& for various values &. For attractive potential tails such as (248) or the Casimir-Polder-type potentials studied in the next section, the quantality function (36) tends to have its maximum absolute value near the position rE where the absolute value of the potential is equal to the total energy [56], |V (rE )| = E =

˝2 k 2 : 2M

(249)

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 4 3

φ(k)

2 1 0 -1 -2 0

1

2

3

4

5

kβα

Fig. 15. Phase  = arg Rr of the quantum reAection amplitude for the homogeneous potential (248). From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7. From [126].

In the corresponding repulsive potential −V (r), the point rE is the classical turning point. For the homogeneous potential (248) we have rE = & (k& )−2=& :

(250)

In the limit of large energies, we may use a semiclassical expression for the reAection amplitudes which was derived by Pokrovskii et al. [129,130]. We use the reciprocity relation (101) to adapt the formula of Refs. [129,130] to the reAection amplitude Rr de>ned via the boundary conditions (102),   rt 2i k →∞ p(r) dr  : (251) Rr (k)∗ ∼ i exp ˝ Here rt is the complex turning point with the smallest (positive) imaginary part. For a homogeneous potential (248) it can be written as rt = (−1)1=& rE = ei=& rE ;

(252)

where rE is de>ned by Eq. (250) and lies close to the maximum of |Q(r)|. Real values of the momentum p(r) only contribute to the phase of the right-hand side of Eq. (251), so the reAectivity |R| is una?ected by a shift of the lower integration point anywhere along the real axis. Integrating along the path r=rE = cos(=&) + i; sin(=&) with ; = 0 → 1 gives the result [56] k →∞

|R| ∼ exp(−B& krE ) = exp[ − B& (k& )1−2=& ] ;   

 −&

   1 R + i; sin B& = 2 sin 1 + cos d; : & & & 0

(253)

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427

Table 6 The coeScients B& , which are given by Eq. (253) and appear before (k& )1−2=& in the exponents describing the high-energy behaviour of the reAectivities of attractive homogeneous potential tails &

3

4

5

6

7

8

&→∞

B&

2.24050

1.69443

1.35149

1.12025

0.95450

0.83146

2=&

0 α=3 α=4

-2

α=5 α=6 log Rα

-4

-6

-8

2

4

6

8

(kβα) 1− 2/α

Fig. 16. Natural logarithm of the reAectivity of the homogeneous potential (248) as function of (k& )1−2=& for various values of &. The straight lines correspond to the behaviour (253) with the values of B& as listed in Table 6. From [56].

In terms of the energy E, the particle mass M and the strength parameter C& of the potential (248), the energy-dependent factor in the exponent is √ 1 1 1 pas rE ; (254) (k& )1−2=& = E 2 − & (C& )1=& 2M = ˝ ˝ where pas = ˝k is the asymptotic (r → ∞) classical momentum. The high-energy behaviour (253) of the reAectivity as function of ˝ is an exponential decrease typically expected for an analytical potential which is continuously di?erentiable to all orders, see the discussion in Section 2.4. Numerical values of the coeScients B& are listed in Table 6. Fig. 16 shows a plot of the logarithm of |R| as function of (k& )1−2=& for various values &, and the obvious convergence of the curves to the straight lines is strong evidence in favour of the high-energy behaviour (253). The phase of the right-hand side of Eq. (251) depends more sensitively on the choice of lower integration limit, which is not speci>ed in Refs. [129,130]. The k-dependence of the integral in the exponent is determined by the complex classical turning point (252), rt = rE [cos(=&) + i sin(=&)].

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

4 2

φ(k)

0 -2 -4 -6 0

1

2 (kβα)

3

4

5

1- 2/α

Fig. 17. Phase  = arg Rr of the reAection amplitude for the homogeneous potential (248) as function of (k& )1−2=& for various values of &. From top to bottom the curves show the results for & = 3, 4, 5, 6 and 7.

If we assume that the real part of the integral becomes proportional to ˝k × R(rt ) = ˝krE cos(=&) for large k, then the high-energy behaviour of the phase of the reAection amplitude is k →∞

arg Rr ∼ c − c0 krE = c − c0 (k& )1−2=&

(255)

with real constants c; c0 . This conjecture is supported by numerical calculations as demonstrated in Fig. 17. Eq. (255) implies that the space shift (245) is given for large energies by   2 k0 →∞ c0 1− rE : (256) Wr ∼ 2 & The space shifts (245) obtained from the numerical solutions of the Schr4odinger equation are plotted in Fig. 18 as functions of k0 & for & = 3, 4, 5, 6 and 7. Except for & = 3 and values of k0 3 less than about 0.15, the space shifts are always positive: according to Eq. (246) this corresponds to time gains relative to the free particle reAected at r =0. For & =3 and energies close to threshold there are signi>cant time delays. Note, however, that the classical particle accelerated under the inAuence of the attractive potential is faster than the free particle [with the same asymptotic velocity v0 =˝k0 =M], and its time gain is   ∞ 2M 1 1 dr = − B(&)rE ; (257) (Wt)cl = 2M ˝k0 p(r) ˝k0 0 where B(&) depends only on &,     1 1 1 1 + 2 1− : B(&) = √ 2 2 & &  Numerical values of B(&) are given in Table 7.

(258)

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

429

0.6 0.4

∆r/βα

0.2 0 -0.2 -0.4 -0.6 -0.8

0

1

2

3

4

5

k0βα Fig. 18. Space shift (245) for quantum reAection by the homogeneous potential (248) as function of k0 & for various values of &. From bottom to top the curves show the results for & = 3, 4, 5, 6 and 7. From [126]. Table 7 Numerical values of B(&) as de>ned in Eq. (258) &

3

4

5

6

7

8

&→∞

B(&)

0.862370

0.847213

0.852623

0.862370

0.872491

0.881900

1

The time gain (257) corresponds to the space shift v0 (Wt)cl = B(&)rE ; (259) (Wr)cl = 2 the classical particle which is accelerated in the potential and reAected at r = 0 eventually returns at the same time as a free particle reAected at (Wr)cl . The classical space shifts (259) are generally larger than the space shifts of the quantum reAected wave, as illustrated in Fig. 19 for the example & = 4. At high energies both the classical space shifts (259) and the quantum space shift (256) show the same dependence on k0 & , i.e., proportionality to rE , but the coeScient B(&) in the classical case is larger than the coeScient in the quantum case. At small energies, the classical space shift diverges as rE [Eq. (250)], whereas the quantum space shift remains bounded by a positive distance of the order of the potential strength parameter & , see Figs. 18, 19. Although the quantum reAected wave may experience a time gain relative to the free particle reAected at r = 0, it is always delayed relative to the classical particle which is accelerated in the attractive potential [126]. 5.3. Quantum re?ection of atoms by surfaces The probability for quantum reAection of atoms by a surface is directly accessible to measurement, because the (elastically) reAected atoms return with their initial kinetic energy, whereas those atoms

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1.25 1

∆r/βα

0.75 0.5 0.25 0 0

1

2

k0βα

3

4

5

Fig. 19. Space shift (245) for quantum reAection by the homogeneous potential (248) with & = 4. The solid line shows the space shift of the quantum reAected wave while the dot-dashed line shows the classical space shift (259). Similar results are obtained for other powers & ¿ 3. From [126].

which are transmitted through the quantal region of the potential tail and hence approach the surface to within a few atomic units are usually inelastically scattered or adsorbed (sticking). Quantum reAection has been observed in the scattering of thermal hydrogen atoms from liquid helium surfaces [118,120], and in the scattering of laser-cooled metastable neon atoms from smooth [41] or structured [131] silicon surfaces. Beyond the region of very small distances of a few atomic units, the interaction between a neutral atom (or molecule) and a conducting or dielectric surface is well described by a local van der Waals potential, corrected for relativistic retardation e?ects as described in the famous paper by Casimir and Polder in 1948 [57]. A compact expression has been given in Refs. [132,133]; in atomic units, the atom-surface potential is   ∞ (&fs )3 ∞ V@ (r) = − &d (i!)!3 exp(−2!rp&fs )h(p; @) dp d! ; (260) 2 0 1 where h(p; @) =

s − @p s−p + (1 − 2p2 ) ; s+p s + @p

with s =




@ − 1 + p2 ;

(261)

&fs ≡ 1=c=0:007297353 : : : is the >ne-structure constant and @ is the dielectric constant of the surface; &d is the frequency-dependent dipole polarizability of the projectile atom in its eigenstate labelled n0 with energy En0 [49],   | n0 | Zj=1 zj | n |2 2(En − En0 ) : (262) &d (i!) = (En − En0 )2 + !2 n

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

431

For a perfectly conducting surface, a simpler formula is obtained by taking @ → ∞ in Eq. (261) and integrating over p in Eq. (260),  ∞ 1 V∞ (r) = − &d (i!)[1 + 2&fs !r + 2(&fs !r)2 ]exp(−2&fs !r) d! 4r 3 0   ∞  x 1 [1 + 2x + 2x2 ]exp(−2x) d x : =− &d i (263) 4&fs r 4 0 &fs r For small r values, we can put r = 0 in the upper line of Eq. (263) and obtain the van der Waals potential between the atom and a conducting surface,  ∞ C3 (∞) 1 vdW V∞ (r) = − ; C (∞) = &d (i!) d! : (264) 3 r3 4 0 For >nite values of the dielectric constant @, the derivation of the small-r behaviour of the potential is a bit more subtle, but the result is quite simple [132,134], V@vdW (r) = −

C3 (@) ; r3

C3 (@) =

@−1 C3 (∞) : @+1

(265)

For large r values, we can assume the argument of &d in the lower line of Eq. (263) to be zero and perform the integral over x. This gives the highly retarded limit of the Casimir-Polder potential between the atom and a conducting surface, ret V∞ (r) = −

C4 (∞) ; r4

C4 (∞) =

3 &d (0) : 8 &fs

(266)

For >nite values of the dielectric constant @, we have [133,134] V@ret (r) = −

C4 (@) ; r4

C4 (@) =

@−1 (@)C4 (∞) ; @+1

(267)

∞ where (@) = 12 ((@ + 1)=(@ − 1)) 0 h(p + 1; @)(p + 1)−4 dp is a well de>ned smooth function which for @ = 1 to unity for @ → ∞. Explicit expressions for increases monotonically from the value 23 30 (@) and a table of values are given in Ref. [133]. The atom-surface potential behaves as −C3 =r 3 for “small” distances [Eqs. (264), (265)] and as −C4 =r 4 for large distances [Eqs. (266), (267)]. The ratio L=

C4 (4 )2 = C3 3

(268)

de>nes a length scale separating the regime of “small” r values, rL, from the regime of large r values, r
L. In Eq. (268) we have introduced the parameters 3 and 4 which express the potential strength in the respective limit in terms of a length, as for the homogeneous potentials (248). The expressions (260) and (263) have been evaluated for the interaction of a hydrogen atom with a conducting surface by Marinescu et al. [135] and for the interaction of metastable helium 21 S and 23 S atoms with a conducting surface (@ = ∞) and with BK-7 glass (@ = 2:295; (@) = 0:761425) and fused silica (@ = 2:123; (@) = 0:760757) surfaces by Yan and Babb [134]. A list of the potential parameters determining the “short”-range and the long-range parts of the respective potentials is given in Table 8.

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Table 8 Parameters determining the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) of the atom-surface potentials calculated by Marinescu et al. [135] for hydrogen and by Yan and Babb [134] for metastable helium. The length L is the distance (268) separating the regime of “small” distances from the regime large distances; < is the parameter (273) determining the relative importance of the “small”-distance regime and the large-distance regime for quantum reAection. All quantities are in atomic units Atom

H

He(21 S)

He(23 S)

@

∞

∞

2.295

2.123

∞

2.295

2.123

C3 C4 3 4 L <

0.25 73.61 919 520 294 1.77

2.6712 13091 38980 13820 4901 2.82

1.0498 3918 15320 7561 3732 2.03

0.9605 3582 14017 7230 3729 1.94

1.9009 5163 27740 8680 2716 3.20

0.7471 1545 10902 4748 2068 2.30

0.6836 1413 9975 4540 2067 2.20

The lengths 3 and 4 are natural length scales corresponding to typical distances, where quantum e?ects associated with the “short”- or long-range part of the potential are important. These distances are of the order of hundreds or thousands or even tens of thousands of atomic units. The words “small” or “short” refer to length scales small compared to these very large distances, a few tens of atomic units, say. In this regime, the Casimir-Polder potential (260), (263) is a good description of the atom–surface interaction and (semi-) classical approximations are well justi>ed, but it still lies beyond the regime of really small distances of a few atomic units, where more intricate details of the atom–surface interaction involving the microscopic structure of the atom and of the surface become important. The energies at which quantum reAection becomes important are given by wave numbers of the order of 1=3 and 1=4 , i.e. typically below 10−4 atomic units for metastable helium atoms. This corresponds to velocities of the order of centimetres per second, which are very small indeed, but not beyond the range of modern experiments involving ultra-cold atoms [41,42,131,136,137]. The “high”-energy behaviour of the reAection amplitude discussed in Section 5.2 in connection with Eqs. (251), (253), (255), (256) and Figs. 16, 17 refers to high energies relative to this near-threshold regime; these can still be well within the range of ultra-cold atoms. For the potential (263) between the atom and a conducting surface, we can also make some general statements about the next-to-leading terms at large and small separations. For large separations we can exploit the fact that the dipole polarizability (262) is an even function of the imaginary part of its argument, so V∞ (r) as given in the second line of Eq. (263) is an even function of 1=r and the next term in the large-distance expression (266) must fall o? at least as 1=r 6 ,   C4 1 r →∞ V∞ (r) ∼ − 4 + O 6 : (269) r r For small distances r we can calculate a correction to the expression (264) via a Taylor expansion of the integral in the >rst line of Eq. (263), and prudent use of the Thomas-Reiche-Kuhn sum rule

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433

[49] yields [138], Z&fs r ; (270) 4 where Z is the total number of the electrons in the atom. The second term on the right-hand side of Eq. (270) is the leading retardation correction to the van der Waals potential at “small” distances, but “small” means small compared to the lengths L listed in Table 8, and this can still be quite large in atomic units. It was >rst derived by Barton for one-electron atoms in 1974 [139]. An intriguing feature of the correction (270) to the van der Waals potential between an atom and a conducting surface is, that it is universal: it depends only on the number Z of electrons in the atom and not on its eigenstate n0 . The shape of the atom-surface potential in between the “short”-range behaviour (264), (265) and the long-range behaviour (266), (267) can be expressed in terms of a shape function vshape (r=L),

r

r C4 C3 = − 4 vshape ; (271) V (r) = − 3 vshape L L L L r →0

r 3 V∞ (r) ∼ − C3 +

x→0

x→∞

whose “short”- and long-range behaviour is given by vshape (x) ∼ 1=x3 and vshape (x) ∼ 1=x4 . Shimizu [41] analyzed his experimental data with the simple shape function v1 (x) = 1=(x3 + x4 ). This gives the potential V1 (r) [see Eq. (183)], whose near-threshold properties were already discussed in Section 4.3. The potential between a hydrogen atom and a conducting surface as calculated by Marinescu et al. [135] is well represented by a rational approximation, vH (x) =

1 + Jx ; x3 + 8x4 + Jx5

8 = 1; J = 0:31608 :

(272)

The coeScients 8 and J are actually determined by the expressions (269) and (270) respectively, so the formula (272) contains no adjusted parameters; it reproduces the tabulated values [135] of the exact hydrogen-surface potential to within 0.6% in the whole range of r values. 1 In numerical calculations it is generally advisable to work with smooth potentials which are continuous in all derivatives. Otherwise, e.g. when using a spline interpolation of tabulated potential values, discontinuities in higher derivatives of the potential can lead to remarkably irregular spurious contributions to the quantum reAection amplitude, see e.g. the discussion in Section 2.4. A detailed study of quantum reAection probabilities for potentials behaving as (264), (265) for “small” distances and as (266), (267) for large distances has been given in Ref. [56]. Which part of the potential dominantly determines the reAection probability depends on a crucial parameter √ 2MC3 3 <= √ = : (273) 4 ˝ C4 For small values of <, the quantum reAection probability resembles that of a homogeneous −1=r 3 potential (248) [with the appropriate value of 3 ] at all energies. For large values of <, the quantum reAection probability resembles that of a homogeneous −1=r 4 potential [with the appropriate value of 4 ] up to quite high energies. At very high energies the behaviour will eventually become that of 1

In Ref. [56] a numerical >t ignoring the boundary conditions imposed by Eqs. (269) and (270) led to 8=0:95; J=0:22, but did not give better results.

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0

log10PR

-1

-2

-3

-4

0.0000

0.0005

0.0010

0.0015

0.0020

k Fig. 20. Probability PR for quantum reAection of metastable 23 S helium atoms from a fused silica surface (@ = 2:123). The solid, dotted and dashed lines show the results obtained with the atom-surface potential of Yan and Babb [134], its highly retarded limit (267) and its nonretarded form (265) respectively. From [137].

a homogeneous −1=r 3 potential; this transition can be expected near the energy where the reAection probabilities predicted by Eq. (253) are the same for the homogeneous −1=r 3 potential and for the homogeneous −1=r 4 potential (with the appropriate values of 3 and 4 ), i.e. for k = (B3 =B4 )6 <=L ≈ 5:345<=L. For this value of k Eq. (253) predicts [56] a reAection probability near exp(−7:8<) for the homogeneous potentials, which is quite small when < ¿ 1. As a realistic example we refer to the quantum reAection of metastable 23 S helium atoms by fused silica surfaces, which are used in hollow optical >bres serving as atom-optical devices [136,137,140]. The atom-surface potential for this case was calculated by Yan and Babb [134], and the resulting probabilities for quantum reAection are shown as function of k as a solid line in Fig. 20. Here k = 0:001 atomic units corresponds to a perpendicular velocity of the incoming helium atoms near 30 cm=s; these and lower perpendicular velocities can be achieved experimentally when projectile atoms with somewhat larger velocities are incident at small, almost grazing, angles [41,137]. The dotted line in Fig. 20 shows the quantum reAection probability for a homogeneous −1=r 4 potential, corresponding to the highly retarded limit (267) of the atom-surface potential, with 4 = 4540 a.u. according to Table 8. The dashed line shows the results for the nonretarded van der Waals potential (265) with the appropriate strength parameter 3 = 9975 a.u. The reAectivity of the full potential corresponds quite closely to that given by the highly retarded limit of the potential up to momenta k ≈ 0:1 a.u., where the reAection probability has fallen to about 0.1%; in contrast, the unretarded van der Waals potential would lead to quantum reAection probabilities an order of magnitude smaller. This shows that the dominant contributions to the quantum reAection are generated in the highly retarded tail of the potential at distances much larger than L ≈ 2000 a.u. In order to understand the reAection of an atom beam from a surface, it is helpful to recall the space and time shifts discussed in Section 5.2. As long as the atom-surface potential is translationally

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

435

1.1

1

b 0.9 β4

v1 v2 vH

0.8

0.7 2

4

6

8

10

ρ Fig. 21. Characteristic parameter b (in units of 4 ) as function of the crucial parameter (273) for an atom-surface potential with di?erent shape functions determining the shape of the potential between the “short”- and long-range limits according to Eq. (271). The dotted line shows the results obtained with Shimizu’s [41] potential, v1 (x) = 1=(x3 + x4 ), the dot-dashed curve shows the results obtained with the smoother interpolating function, v2 (x) = [2=(x3 )] arctan[=(2x)], and the dashed line shows the results obtained with the shape function vH , Eq. (272), corresponding to the potential calculated by Marinescu et al. [135] for a hydrogen atom and a conducting surface. From [56].

invariant parallel to the surface, the parallel momentum component is conserved. For a free particle reAected at the position Wr de>ned in Eq. (245), the magnitude of the perpendicular component of momentum is conserved, but its direction is reversed at reAection. The atom beam thus appears to follow straight-line trajectories which are specularly reAected at a plane lying Wr in front of the surface. For the case above, corresponding closely to the reAection by a homogeneous −1=r 4 potential, the space shift describing the position of the apparent plane of reAection relative to the reAecting surface can be deduced from Fig. 19. The largest space shifts occur for wave numbers between 0:5=4 and 1=4 and amount to about 0:34 , i.e. the apparent plane of reAection is shifted by up to 1400 a.u. in front of the actual surface. For heavier alkali atoms reAected by a conducting surface, quantum reAection is also dominantly due to the −1=r 4 part of the atom-surface potential, and the values of 4 can be ten times larger [56] leading to maximal space shifts of the order of 104 a.u. An example for the inAuence of the shape of the potential can be obtained by studying the e?ect of di?erent shape functions (271) on the characteristic parameter b, which determines the near-threshold reAectivity according to Eq. (241). Fig. 21 shows the dimensionless ratio b=4 as function of the crucial parameter (273). The dotted line shows the results obtained with Shimizu’s potential [41], v1 (x) = 1=(x3 + x4 ), the dot-dashed curve shows the results obtained with the smoother interpolating function, v2 (x) = [2=(x3 )] arctan[=(2x)], and the dashed line shows the results obtained with the shape function vH , Eq. (272), corresponding to the potential calculated by Marinescu et al. for

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H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1.5

log (-logR)

1 0.5 0

v1, =1 v1, =10 vH, =1 vH, =10

–0.5 –1

-11

-10.5

-10

-9.5 -9 log (k) (a.u.)

-8.5

-8

-7.5

Fig. 22. Quantum reAectivities |R| as observed by Shimizu [41] for the scattering of metastable neon atoms by a silicon surface (>lled dots). The >gure shows ln(−ln |R|) as function of ln k (natural logarithms) with k measured in atomic units. The straight line in the top-right part of the >gure shows the “high”-energy behaviour expected for a homogeneous −1=r 4 potential according to Eq. (253) with 4 = 11400 a:u. The straight line in the bottom-left part of the >gure shows the near-threshold behaviour (241) for b = 4 = 11400 a:u. The curves were obtained by numerically solving the Schr4odinger equation with potential shapes given by the shape function v1 de>ning Shimizu’s potential (183) and with the shape function vH for the potential of Marinescu et al. [135] for the interaction of a hydrogen atom with a conducting surface, Eq. (272); the value of 3 de>ning the “short”-range van der Waals part of the potential was either 11400 a:u: (< = 1) or 114000 a:u: (< = 10). From [56].

a hydrogen atom and a conducting surface. The values of b lie within 5% of the large-< limit 4 when < ¿ 2 and approach the small-< limit 3 = 4 =< for < → 0. The most pronounced shape dependence is observed around < = 1. Recent measurements of quantum reAection were carried out by Shimizu for metastable neon atoms reAected, e.g., by a silicon surface [41]. The transition from the linear dependence of ln |R| on k near threshold to the proportionality of −ln |R| to k 1−2=& at “high” energies is nicely exposed by plotting ln(−ln |R|) as a function of ln k, see Fig. 22. At the “high”-energy end of the >gure, the data clearly approximate a straight line with gradient near 12 corresponding to & = 4. Fitting a straight line of gradient 12 through the last six to ten data points yields ln(−ln |R|) = 5:2 + 12 ln k (straight line in top-right corner of Fig. 22), and comparing this with ln(−ln |R|) = ln B4 + 12 ln k + 12 ln 4 according to Eq. (253) yields 4 ≈ 11400 a.u. The corresponding near-threshold behaviour (241), ln(−ln |R|) = ln(24 ) + ln k is shown as a straight line in the bottom-left corner of Fig. 22 and >ts the data well within their rather large scatter. Also shown in Fig. 22 are the results obtained by numerically solving the Schr4odinger equation with a potential given by two of the three shapes already introduced in connection with Fig. 21, and with the above value of 4 and two di?erent choices of the crucial parameter (273), namely <=1 implying 3 =4 and <=10 implying 3 =104 = 114000 a.u. The large-< curves clearly >t the data well con>rming that the quantum reAectivity is essentially that of the highly retarded −1=r 4 potential (267) in this case too. In fact, large values of

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449

1E-4

2

En e

0

rg y

1 Lo w

ln(-ln|R|2)

Reflection Coefficient, |R|2

1E-3

E igh

H

1

0.01

gy

ner

3

0.1

437

-1 -7

-6

-5

-4 -3 ln(kia)

-2

-1

0

1E-5 1E-6 1E-7 1E-8 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 kia

Fig. 23. Quantum reAection probabilities |R|2 observed by Druzhinina and DeKieviet in the scattering of ground state 3 He atoms by a rough quartz surface. The solid line shows the reAection probabilities calculated with Shimizu’s potential (183) with 3 = 650 a:u: and 4 = 350 a:u. The straight line in the bottom left-hand corner of the inset shows the low-energy behaviour (241), and the straight line with gradient 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the nonretarded −1=r 3 part of the potential. From [42], courtesy of M. DeKieviet.

the crucial parameter (273) are ubiquitous in realistic systems, so quantum reAection data provide a conspicuous and model-independent signature of retardation e?ects in atom-surface potentials [56]. In a more recent experiment, Druzhinina and DeKieviet [42] measured the probability for quantum reAection of (ground state) 3 He atoms by a rough quartz surface. The helium atoms which are transmitted all the way to the surface are reAected di?usely because of the surface roughness and contribute only negligibly to the yield of specularly reAected atoms; the specularly reAected atoms thus represent the quantum reAection yield. The results of Ref. [42] are reproduced in Fig. 23. In the label of the abscissa, ki is the incident wave number perpendicular to the surface and a=5 a.u. is the location of the minimum of a realistic atom-surface potential [141]. The authors analyzed their data using Shimizu’s potential (183); the strength parameter 4 was >xed via the known polarizability of the helium atoms [cf. Eq. (266)] and the dielectric constant of the quartz surface [cf. Eq. (267)] to be 4 = 350 a.u., and the strength parameter 3 was determined by >tting the calculated probabilities to the experimental data. This gave 3 = 650 a.u. corresponding to L = 190 a.u. and a crucial parameter < = 1:9. The straight line in the bottom left-hand corner of the inset in Fig. 23 is close to the near-threshold reAectivity (241) with b ≈ 4 = 350 a.u. as expected for the −1=r 4 part of the potential, and the straight line with slope 1=3 in the top right-hand corner shows the “high”-energy behaviour (253) expected for the −1=r 3 part of the potential with 3 = 650 a.u.

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0 -1

log10PR

-2 -3 -4 -5 -6 -7 -8 0.0

0.1

0.2 ka

0.3

Fig. 24. Quantum reAection probabilities for Shimizu’s atom surface potential (183) with 3 = 650 a:u: and 4 = 350 a:u: (solid line) in comparison with the predictions of the associated homogeneous potentials (248) proportional to −1=r 3 (dashed line) and to −1=r 4 (dotted line).

An important aim of the work in Ref. [42] was to measure quantum reAection probabilities so far above the threshold, that they are signi>cantly inAuenced by the nonretarded van der Waals part of the atom-surface potential. The high-energy data can actually be seen to approach the straight line in the upper right-hand corner of the inset in Fig. 23, but the quantum reAection probabilities are still substantially larger than the predictions for a pure −1=r 3 potential. This is illustrated in Fig. 24 comparing the quantum reAection probabilities predicted for the realistic (Shimizu’s) potential (solid line), which >t the data, with those obtained for the −1=r 3 potential alone (dashed line) and for the −1=r 4 potential alone (dotted line). The transition point k = (B3 =B4 )6 <=L ≈ 5:345<=L, where the predictions for the homogeneous potentials cross, lies at ka ≈ 2:7, which is near the highest energy for which data are included in Fig. 23. At these energies, the k-dependence of the probabilities obtained for the realistic potential is similar to that obtained for the −1=r 3 potential alone, but the absolute values are still an order of magnitude larger. The high-energy data in Fig. 23 are certainly inAuenced by the nonretarded van der Waals part of the potential, but they are also still far from the asymptotic limit where the reAection probabilities are expected to be quantitatively determined only by this nonretarded van der Waals potential. The theory of quantum reAection described in this section is based on the assumption that the quantal region is localized in the tail of the potential and that there is a semiclassical WKB region at smaller r values. These conditions are well ful>lled for suSciently attractive potential tails more singular than 1=r 2 for “small” r values, as typically occur in the interaction of atoms and molecules with each other and with surfaces. In this case the properties of quantum reAection depend only on the quantal region of the potential tail, and the quantum reAection probability approaches unity at threshold as described by Eq. (241). When the potential well is very shallow, and/or when the tail falls o? more slowly than 1=r 2 towards “small” r-values, as is the case with Coulombic tails,

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439

the quantal region of coordinate space may extend to very small r values, where more complex ingredients of the atom–surface interaction lead to a loss of atoms from the incoming (elastic) channel via inelastic reactions or adsorption (sticking). In this case, the quantum reAection amplitude can be expected to depend sensitively on the nature of the interaction at very small distances, and, due to loss of Aux into inelastic channels, the quantum reAection probability may remain smaller than unity even in the limit of vanishing energy. 5.4. Coupled channels Possible excitations of the projectile atoms or molecules or of the reAecting surface during the process of quantum reAection can be described by generalizing the one-dimensional Schr4odinger equation (1) via a coupled channel ansatz for a vector of channel wave functions [142],   1 (r)  .   K(r) =  (274)  ..  : N (r)

The Schr4odinger equation now has the form  V11 (r)   2 2  ˝ d . − + V K = EK; V ≡   .. 2M dr 2 VN 1 (r)

···

V1N (r)

..

.. .

.

···

   ; 

(275)

VNN (r)

with the diagonal channel potentials Vii (r) and the nondiagonal coupling potentials Vij (r); j = i. We assume that for r → ∞ the diagonal potentials approach constant values Ei and that the coupling potentials as well as the di?erences Ei − Vii (r) vanish more rapidly than 1=r 2 . The coupled equations (275) then become decoupled free-particle equations for large r, and the appropriate generalization of Eq. (102) to describe an incoming wave in the channel labelled 1 and reAected waves in all open channels, E − Ei ¿ 0, is, 1 (r)

r →∞

i (r)

r →∞

∼ √

1 [exp(−ik1 r) + R11 exp(ik1 r)] ; ˝k1

1 ∼ √ R1i exp(iki r); ˝ki

i = 1 ;

(276)


where ki is the asymptotic (r → ∞) wave number in the ith channel, ˝ki = 2M(E − Ei ). One approach towards solving the coupled-channel equations (275) is to introduce a local unitary transformation L(r) = U(r)K(r) such that the appropriately transformed potential is diagonal,   W11 (r) · · · 0   . .. ..  : (277) W = UVU−1 ≡  . .   .. 0

···

WNN (r)

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The Schr4odinger equation (275) then becomes an equation for the “adiabatic” channel wave functions 1 (r); : : : ; N (r) making up the vector L,   ˝2 d 2 ˝2 U[(U−1 )

L + 2(U−1 )
L
] = EL : − + W L − (278) 2M dr 2 2M For a meaningful discussion of the concept of quantum reAection for coupled channels, it is helpful that Eq. (278) decouple and become amenable to semiclassical approximations for small r values, r → 0 (or r → −∞). The Eq. (278) decouple if the square bracket containing the >rst and second derivatives of U vanishes. This is, e.g. the case if the coupling potentials Vij ; j = i vanish for small r, or if the diagonal potentials become dominant with respect to the coupling potentials, r →0

e.g. if Vii ˙ − 1=r & whereas the coupling potentials remain bounded. In these cases, U becomes the unit matrix at small distances. If both the diagonal potentials and the coupling terms behave as an inverse power of r for small r, then the asymptotic (r → 0) decoupling of the channels depends on the relation of the powers involved. If, e.g. in a two-channel example, V11 (r) and V22 (r)  are proportional to −1=r & for r → 0, and V12 (r) and V21 (r) proportional to −1=r & , then we obtain decoupled channels for small r, if and only if |&
− &| = 1 [143]. If the diagonal potentials and the coupling potentials have the same spatial dependence for small r, dfij =0 ; (279) Vij (r) = f(r) × fij ; dr then there is a decoupling of channels, but U = 1 so the decoupled channels will be superpositions of the diabatic channels i which are uncoupled for r → ∞. Decoupling of channels occurs in the limit r → 0 for many-body systems described in hyperspherical coordinates. In these coordinates, the hyperradius r stands for the root-mean-square average of the radial coordinates of all particles involved, and the remaining coordinates, the hyperangles, encompass not only all angular degrees of freedom, but also “mock angles” de>ned by the ratios of the individual radial coordinates [144]. Note however, that in applications to many-electron atoms, the diagonal potentials are proportional to 1=r at small values of the hyperradius r, and there need not be a semiclassical WKB region in the regime where the channels decouple (see the discussion in the last paragraph of Section 5.3). For the waves transmitted to small r values, r → 0 or r → −∞, the appropriate generalization of the one-channel boundary conditions (100) is,    r 1 i

j (r) ∼ T1j
(280) qj (r ) dr ; exp − ˝ rl qj (r)
where ˝qj (r) = 2M[E − Wjj (r)] is the local classical momentum in the adiabatic channel j. The form (280) implies, that the adiabatic channel j is open for transmission, i.e., that E − Wjj (r) ¿ 0 for small r. Some general statements can be made about the threshold behaviour of the quantum reAection amplitudes when potentials and coupling terms approach their asymptotic limits faster than 1=r 2 . If the incoming channel is energetically lowest and nondegenerate (E1 ¡ Ei for all i = 1) then the near-threshold behaviour of the elastic reAectivity |R11 | follows the pattern (241), k →0

|R11 | 1∼ 1 − 2bk1 = exp(−2bk1 ) + O((k1 )2 ) ;

(281)

with a characteristic length parameter b, depending only on the tails of the potentials and coupling terms Vij in the quantal region of coordinate space. SuSciently close to threshold only the elastic

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441

reAection channel and a certain number of transmission channels are open, and particle number conservation requires the sum of the transmission probabilities to grow proportional to 1 − |R11 |2 ∼ 4bk1 + O((k1 )2 ); this actually holds for each transmission probability individually, k1 →0 k1 →0
(PT )ij ˙ k1 ; T1j ˙ k1 : (282) This is a straightforward generalization of the second equation (228) to the coupled-channel situation. Analytical solutions for two coupled step potentials with a step-like coupling term are given in Ref. [142], and smoother Woods-Saxon steps as well as inverse-power potentials are discussed in Ref. [143]. Exponential potentials and coupling terms have been studied in considerable detail for isolated special cases by Osherov and Nakamura [145,146], while more general cases were treated in Refs. [147,148] using semiclassical methods. When two diagonal diabatic potentials, V11 (r) and V22 (r) cross at a point r0 , then coupling of the channels 1 and 2 leads to an avoided crossing of the corresponding adiabatic potentials W11 and W22 . In an adiabatic process, the incoming and reAected or transmitted waves remain on the potential energy curve Wii associated with the respective adiabatic channel, but the avoided crossing can be overcome by a nonadiabatic transition. The probabilities for such nonadiabatic transitions have been a topic of great interest for more than seventy years [149–152]. With the assumptions that the diabatic potential curves are linear at the crossing and that the coupling potential is constant one obtains the semiclassical Landau-Zener formula,   V12 (r0 )2 ; (283) (P1→1 )LZ = 1 − exp −2 ˝v0 WF where v0 is the velocity at the crossing point, Mv02 =2 = E − V11 (r0 ) and WF is the di?erence of the slopes of the two crossing curves, WF = |V11
(r0 ) − V22
(r0 )|. Eq. (283) actually describes the probability that the incoming wave in the channel labelled 1 (wave function 1 (r) for large r) remains on the adiabatic potential curve and is transmitted to the transmission channel 1
, which is the channel labelled 2 in the diabatic basis. Many improvements of the simple Landau-Zener formula (283) have been proposed over the years [152], but we shall focus on one aspect, namely the quenching of the curve crossing probability due to quantum reAection [153]. Consider a system of two Woods-Saxon step potentials, Ui + (i − 1)E0 ; i = 1; 2 ; Vii (r) = − (284) 1 + exp(ai r) with a Gaussian coupling potential, V12 (r) = V21 (r) = U12 exp[ − (a12 )2 (r − r0 )2 ] ;

(285)

as illustrated in Fig. 25. The performance of the simple Landau-Zener formula is illustrated in Fig. 26 for an example of very small coupling. The transition probability P1→1 = |T12 |2 , obtained by numerically solving the two-channel Schr4odinger equation (275) with the boundary conditions (276), (280) is plotted as function of k1 ≡ k. The solid line shows the exact result, which goes to zero at threshold according to Eq. (282). This vanishing transmission probability is not accounted for in the conventional Landau-Zener formula (283)—illustrated as dashed line in Fig. 26—nor in the numerous improvements introduced over the years [152]. It is due to the fact, that the incoming wave only reaches

442

H. Friedrich, J. Trost / Physics Reports 397 (2004) 359 – 449 5

Channel 2 Coupling Potential

Potential

0

Channel 1

Channel 2’

-5 -10 -15 -20

Channel 1’ -20

-10

0 r

10

20

Fig. 25. Woods-Saxon potentials (284) with the coupling potential (285) (dashed line). The parameters are U1 = 5:5, U2 = 26, U12 = 0:5, E0 = 3 and a1 = a2 = a12 = 1. The dotted line shows the quantality function (36) for the elastic channel just above threshold (k = 0:2). From [153].

QM LZ

0.00012

P1→1′

0.0001 8e-05 6e-05 4e-05 2e-05 0 0

0.2

0.4

0.6

0.8

1

k Fig. 26. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the simple Landau-Zener formula (283). From [153].

the curve-crossing region with a small probability, because quantum reAection in the quantal region of the incoming-channel potential becomes dominant towards threshold. This region is indicated by the dotted line in Fig. 25, which shows the quantality function (36) for the potential V11 at a near-threshold momentum, k = 0:2. For k larger than about 0.4, the e?ect of quantum reAection is negligible, and the conventional Landau-Zener formula reproduces the exact result to within a rather constant error of a few per cent. A straightforward improvement of the Landau-Zener formula (283) is to account for the e?ect of quantum reAection by multiplying the probability (283) by the probability for transmission through

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443

0.00012 0.0001

P1→1′

8e-05 6e-05

QM LZM

4e-05 2e-05 0

0

0.2

0.4

0.6

0.8

1

k Fig. 27. Transition probability P1→1 = |T12 |2 for the coupled Woods-Saxon potentials (284), (285). The parameters are U1 = 5:5, U2 = 26, U12 = 0:01, E0 = 3, a1 = a2 = 1, a12 = 0:05. The solid line shows the exact result and the dashed line is the prediction of the modi>ed Landau-Zener formula (286). From [153].

the quantal region of the potential tail, (P1→1 )LZM = (P1→1 )LZ (1 − |R|2 ) :

(286)

In the very-weak-coupling example in Fig. 26, the quantum reAectivity in the elastic channel is insensitive to the coupling, so we can take |R| to be given by the reAectivity (232) of an uncoupled Woods-Saxon potential. As shown in Fig. 27, the modi>ed Landau-Zener formula (286) does indeed account correctly for the e?ects of quantum reAection in this case and leads to much better agreement with the exact result. 6. Conclusion Although the semiclassical WKB approximation is generally expected to be most useful near the semiclassical limit, where quantum mechanical e?ects are small, semiclassical WKB wave functions can be used to advantage far from the semiclassical limit, even near the anticlassical, the extreme quantum limit of the Schr4odinger equation. This is because the conditions for the accuracy of the WKB wave functions are inherently local. Under conditions which are far from the semiclassical limit for the Schr4odinger equation as a whole, there may still be large regions of coordinate space where WKB wave functions are highly accurate approximations of the exact quantum mechanical wave functions; they may even be asymptotically exact. For example, for homogeneous potentials proportional to 1=r & with & ¿ 2, the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation, but WKB wave functions become exact for r → 0 at all energies, in particular at threshold and in the near-threshold region. The (local) condition for the accuracy of WKB wave functions such as (25) is conveniently expressed via the quantality function (36): |Q(r)|1. The regions of coordinate space, where |Q(r)| is not negligibly small are quantal regions where quantum e?ects such as classically forbidden tunnelling or reAection can be generated. In many physically important situations, exact or highly

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accurate quantum mechanical wave functions are available in the quantal regions and globally accurate wave functions can be constructed by matching these to WKB wave functions in the semiclassical regions. In this way, several highly accurate, sometimes asymptotically exact results have been derived via WKB waves far from the semiclassical limit of the Schr4odinger equation. Numerous examples have been given in the preceding sections. WKB wave functions are singular at a classical turning point and the connection formulas relating the oscillating WKB waves on the classically allowed side to the exponential waves on the forbidden side are usually formulated under restrictive assumptions for the potential (linearity), which are not related to whether or not WKB wave functions may be accurate away from the turning point. Allowing more general connection formulas (44), (45) greatly widens the range of applicability of WKB wave functions. Correct choice of the phase of the WKB wave (56) on the allowed side of a classical turning point via an appropriate de>nition of the reAection phase is a vital ingredient for the construction of accurate WKB wave functions in the classically allowed region. This leads, e.g. to a formula for the scattering phase shifts for repulsive inverse-power potentials which is highly accurate at all energies, and to an exact expression (74) for the scattering lengths, which determine the behaviour of the phase shifts in the anticlassical limit, see Section 3.3. For bound state problems, a generalization (83) of the conventional WKB quantization rule to allow an appropriate de>nition of the reAection phases at the classical turning points leads to greatly improved accuracy without complicating the procedure, see e.g. Figs. 3, 4 and 7, in Section 3.4. The generalized connection formulas (44), (45) also lead to more precise expressions for tunnelling probabilities, in particular near the base of a barrier, where conventional WKB expressions fail when the potential tail falls o? faster than 1=r 2 , see Section 3.5. For deep potential wells, where WKB wave functions are accurate in some region of small or moderate r values, the properties of bound states for E ¡ 0 and continuum states for E ¿ 0 are largely determined by the tail of the potential beyond this WKB region. For potentials falling o? faster than 1=r 2 , the threshold E = 0 represents the anticlassical limit of the Schr4odinger equation and the near-threshold properties of the wave functions are determined by three independent tail parameters, which can be derived by matching zero-energy solutions of the Schr4odinger equation to WKB waves in the WKB region. These three parameters are the characteristic length b, Eq. (150), the mean scattering length aU0 , Eq. (151), and the zero-energy reAection phase 0 , which is the phase loss of the WKB wave due to reAection at the outer classical turning point at threshold, see Eq. (136) in Section 4.1. Immediately below threshold, the quantization rule acquires a universal form (174) which becomes exact for E → 0 and contains the tail parameter b as well as the threshold quantum number nth , which depends on 0 and also on the threshold value of the action integral over the whole of the classically allowed region, see Eq. (148). The characteristic length b determines the leading singular contribution to the near-threshold level density according to Eq. (175), and also the near-threshold behaviour of the quantum reAectivity of the potential tail according to Eq. (241) in Section 5.1. The mean scattering length aU0 determines the near-threshold behaviour of the phase of the amplitude for quantum reAection according to Eq. (242), and hence also the time and space shifts involved in the quantum reAection process. With the correct tail parameters, Eqs. (174), (175), (241) and (242) are asymptotically exact relations for the near-threshold behaviour of the bound and continuum states. When the zero-energy solutions of the Schr4odinger equation for the potential tail beyond the WKB region are known analytically, analytical expressions for the tail parameters b, aU0 and 0 can

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be derived. A summary for various potential tails is given in Table 4 in Section 4.3. If analytical solutions of the Schr4odinger equation are not available, the tail parameters can be obtained from numerical zero-energy solutions of the Schr4odinger equation. Knowing the asymptotic (E → 0) behaviour of the bound and continuum states near threshold is of considerable practical value, because the direct numerical integration of the Schr4odinger equation for small >nite energies is an increasingly nontrivial exercise as the energy approaches zero. For potential tails falling o? more slowly than 1=r 2 , the threshold E = 0 represents the semiclassical limit of the Schr4odinger equation, and semiclassical methods are expected to work well in the near-threshold region, e.g. in the derivation of the near-threshold quantization rule (164) for the energies of the bound states in the limit n → ∞. Potential tails vanishing as 1=r 2 represent the boundary between long-ranged potentials, which support in>nitely many bound states, and shorter-ranged potentials which can support at most a >nite number of bound states. The behaviour of potentials with inverse-square tails (191) depends crucially on the strength parameter (, as discussed in detail in Section 4.4. For ( ¡ − 14 , i.e. for potential tails more attractive than the s-wave centrifugal potential in two dimensions (see Eq. (59) in Section 3.3), the potential supports an in>nite dipole series of bound states (193), in which the energy depends exponentially on the quantum number near threshold. Potentials with weak inverse-square tails, − 41 6 ( ¡ 34 , support at most a >nite number of bound states, but the leading energy dependence in the near-threshold quantization rule is still of order less than O(E), so the near-threshold level density is still singular at E =0. The properties of short-ranged potential tails falling o? faster than 1=r 2 appear as a special case of weak inverse-square tails with ( = 0. A summary of quantization rules is given in Table 5 in Section 4.4. Probabilities for transmission through the quantal region of an inverse-square tail with ( ¿ − 14
1

are proportional to E (+ 4 , which is a generalization of Wigner’s threshold law (218). For weak inverse-square tails, − 14 6 ( ¡ 34 , the transmission probability above threshold is related to the leading energy dependence in the near-threshold quantization rule below threshold, Eq. (227). For all potentials which are asymptotically (r → ∞) less repulsive than the p-wave centrifugal potential in two dimensions ((= 34 ), the leading near-threshold energy dependence in the quantization rule is of order less than O(E) and can be derived from the tail of the potential. For potentials with more repulsive tails, i.e. for inverse-square tails with ( ¿ 34 or for repulsive tails falling o? more slowly than 1=r 2 , the leading energy dependent terms in the near-threshold quantization rule are of order O(E) and include e?ects of the potential for smaller r values; they cannot be derived from the properties of the potential tail alone. These results have been derived using WKB waves to approximate the quantum mechanical wave functions only in regions where such an approximation is highly accurate or asymptotically exact. The results do not depend on the conditions of the semiclassical limit being ful>lled for the Schr4odinger equation as a whole. Indeed, the comprehensive results for the near-threshold region refer to the immediate vicinity of the anticlassical or extreme quantum limit in the case of potential tails falling o? faster than 1=r 2 . Many of the results summarized in this article are of direct practical importance in various >elds, e.g. in atomic and molecular physics, where the intense current interest in ultra-cold atoms and molecules has drawn attention to quantum e?ects speci>c to small velocities and low energies. The near-threshold phenomena studied in Sections 4 and 5 are explicit examples of such quantum e?ects. The quantum reAection of atoms moving as slowly as a few centimetres per second towards

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a surface occurs several hundreds or thousands of atomic units from the surface and can be observed in present-day experiments. Understanding this and similar phenomena is important for technical developments such as the construction of atom-optical devices. The theoretical considerations and practical applications discussed in this review refer mostly to the Schr4odinger equation for one degree of freedom. Matching the exact or highly accurate solutions of the Schr4odinger equation in the “quantal region” of coordinate space to WKB waves which are accurate in the “WKB region” can occur at any point where these regions overlap. An extension to systems with more than one degree of freedom does not seem straightforward, because matching between quantal and WKB regions would have to occur on a subspace of dimension one or more, and it is not clear whether this is easy to do in general. A more promising >eld for generalising the techniques reviewed in this article is that based on coupled ordinary Schr4odinger equations, such as the coupled channel equations used in the description of scattering and reactions in nuclear, atomic and molecular physics. Multicomponent WKB waves have been used in the treatment of coupled wave equations with the individual equations referring to spin components of Pauli or Dirac particles or di?erent Born-Oppenheimer energy surfaces in a molecular system [154–158]. Generalizing such theories to allow for signi>cant deviations from the semiclassical limit may greatly enhance their range of applicability. A simple example is the inAuence of quantum reAection on Landau-Zener curve crossing probabilities as described in Section 5.4. Acknowledgements The authors express their gratitude to the current and former collaboraters who have contributed to the results presented in this review, namely Kenneth G.H. Baldwin, Robin CˆotPe, Christopher Eltschka, Stephen T. Gibson, Xavier W. Halliwell, Georg Jacoby, Alexander Jurisch, Carlo G. Meister and Michael J. Moritz. Harald Freidrich also wishes to thank the members of the Department of Theoretical Physics and of the Atomic and Molecular Physics Laboratories, in particular Brian Robson, Steve Buckman, Bob McEachran and Erich Weigold, for hospitality and enlightening discussions during his stay at the Australian National University during spring and summer 2002/2003. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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CONTENTS VOLUME 397 P.S. Landa, P.V.E. McClintock. Development of turbulence in subsonic submerged jets J.S.M. Ginges, V.V. Flambaum. Violations of fundamental symmetries in atoms and tests of unification theories of elementary particles

1

63

C. Hanhart. Meson production in nucleon–nucleon collisions

155

D.V. Bugg. Four sorts of meson

257

H. Friedrich, J. Trost. Working with WKB waves far from the semiclassical limit

359

Contents of volume

451

doi:10.1016/S0370-1573(04)00246-7