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Instructions to authors Aims and scope Physics Reports keeps the active physicist up-to-date on developments in a wide range of topics by publishing timely reviews which are more extensive than just literature surveys but normally less than a full monograph. Each Report deals with one speciﬁc subject. These reviews are specialist in nature but contain enough introductory material to make the main points intelligible to a non-specialist. The reader will not only be able to distinguish important developments and trends but will also ﬁnd a suﬃcient number of references to the original literature. Submission In principle, papers are written and submitted on the invitation of one of the Editors, although the Editors would be glad to receive suggestions. Proposals for review articles (approximately 500–1000 words) should be sent by the authors to one of the Editors listed below. The Editor will evaluate proposals on the basis of timeliness and relevance and inform the authors as soon as possible. All submitted papers are subject to a refereeing process. Editors J.V. ALLABY (Experimental high-energy physics), EP Division, CERN, CH-1211 Geneva 23, Switzerland. E-mail: [email protected] D.D. AWSCHALOM (Experimental condensed matter physics), Department of Physics, University of California, Santa Barbara, CA 93106, USA. E-mail: [email protected] J.A. BAGGER (High-energy physics), Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore MD 21218, USA. E-mail: [email protected] C.W.J. BEENAKKER (Mesoscopic physics), Instituut–Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands. E-mail: [email protected] G.E. BROWN (Nuclear physics), Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, NY 11974, USA. E-mail: [email protected] D.K. CAMPBELL (Non-linear dynamics), Dean, College of Engineering, Boston University, 44 Cummington Street, Boston, MA 02215, USA. E-mail: [email protected] G. COMSA (Surfaces and thin ﬁlms), Institut fur . Physikalische und Theoretische Chemie, Universit.at Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] J. EICHLER (Atomic and molecular physics), Hahn-Meitner-Institut Berlin, Abteilung Theoretische Physik, Glienicker Strasse 100, 14109 Berlin, Germany. E-mail: [email protected] M.P. KAMIONKOWSKI (Astrophysics), Theoretical Astrophysics 130-33, California Institute of Technology, 1200 East California Blvd., Pasadena, CA 91125, USA. E-mail: [email protected] M.L. KLEIN (Soft condensed matter physics), Department of Chemistry, University of Pennsylvania, Philadelphia, PA 19104-6323, USA. E-mail: [email protected]

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A.A. MARADUDIN (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] D.L. MILLS (Condensed matter physics), Department of Physics and Astronomy, University of California, Irvine, CA 92697-4575, USA. E-mail: [email protected] H. ORLAND (Statistical physics and ﬁeld theory), Service de Physique Theorique, CE-Saclay, CEA, 91191 Gif-sur-Yvette Cedex, France. E-mail: [email protected] R. PETRONZIO (High-energy physics), Dipartimento di Fisica, Universita" di Roma – Tor Vergata, Via della Ricerca Scientiﬁca, 1, I-00133 Rome, Italy. E-mail: [email protected] S. PEYERIMHOFF (Molecular physics), Institute of Physical and Theoretical Chemistry, Wegelerstrasse 12, D-53115 Bonn, Germany. E-mail: [email protected] I. PROCACCIA (Statistical mechanics), Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] E. SACKMANN (Biological physics), Physik-Department E22 (Biophysics Lab.), Technische Universit.at Munchen, . D-85747 Garching, Germany. E-mail: [email protected] A. SCHWIMMER (High-energy physics), Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel. E-mail: [email protected] R.N. SUDAN (Plasma physics), Laboratory of Plasma Studies, Cornell University, 369 Upson Hall, Ithaca, NY 14853-7501, USA. E-mail: [email protected] W. WEISE (Physics of hadrons and nuclei), Institut fur . Theoretische Physik, Physik Department, Technische Universit.at Munchen, . James Franck Strae, D-85748 Garching, Germany. E-mail: [email protected] Manuscript style guidelines Papers should be written in correct English. Authors with insuﬃcient command of the English language should seek linguistic advice. Manuscripts should be typed on one side of the paper, with double line spacing and a wide margin. The character size should be suﬃciently large that all subscripts and superscripts in mathematical expressions are clearly legible. Please note that manuscripts should be accompanied by separate sheets containing: the title, authors’ names and addresses, abstract, PACS codes and keywords, a table of contents, and a list of ﬁgure captions and tables. – Address: The name, complete postal address, e-mail address, telephone and fax number of the corresponding author should be indicated on the manuscript. – Abstract: A short informative abstract not exceeding approximately 150 words is required. – PACS codes/keywords: Please supply one or more PACS-1999 classiﬁcation codes and up to 4 keywords of your own choice for indexing purposes. PACS is available online from our homepage (http://www.elsevier.com/locate/physrep). References. The list of references may be organized according to the number system or the nameyear (Harvard) system. Number system: [1] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The inverse scattering transform – Fourier analysis for nonlinear problems, Studies in Applied Mathematics 53 (1974) 249–315. [2] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

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[3] B. Ziegler, in: New Vistas in Electro-nuclear Physics, eds E.L. Tomusiak, H.S. Kaplan and E.T. Dressler (Plenum, New York, 1986) p. 293. A reference should not contain more than one article. Harvard system:

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Physics Reports 375 (2003) 1 – 103 www.elsevier.com/locate/physrep

Magnetic light scattering in low-dimensional quantum spin systems P. Lemmensa;∗ , G. G'untherodtb , C. Grosc a

Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany b 2. Physikalisches Institut, RWTH Aachen, D-56056 Aachen, Germany c Institut f*ur Theoretische Physik, Univ. des Saarlands, D-66041 Saarbr*ucken, Germany Received 1 July 2002 editor: D.L. Mills

Abstract An overview of one- and two-dimensional quantum spin systems based on transition-metal oxides and halides of current interest is given, such as spin-Peierls, spin-dimer, geometrically frustrated and ladder systems. The most signi1cant and outstanding contributions of magnetic light scattering to the understanding of these materials are discussed and compared to results of other spectroscopies and thermodynamic measurements. c 2002 Elsevier Science B.V. All rights reserved. PACS: 75.10.Jm; 78.30.−j Keywords: Transition metal compounds; Low-dimensional quantum spin; Magnetic light scattering

Contents 0. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Excitations in low-dimensional spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Important inorganic quantum spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Structural considerations and important parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Quasi-zero-dimensional compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Dimer compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. The Cu-tetrahedra systems Cu2 Te2 O5 X2 , X=Br,Cl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Quasi-two-dimensional compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. The 1/5-depleted system CaV4 O9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. The Shastry–Sutherland system SrCu2 (BO3 )2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3. High temperature superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. Tel.: +49-711-689-1631; fax: +49-711-689-1632. E-mail address: [email protected] (P. Lemmens).

c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 3 2 1 - 6

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2.3.4. The diluted square lattice system K2 V3 O8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Low-dimensional cuprates: new compounds related to high-temperature superconductors . . . . . . . . . . . . . . . . . 2.4.1. The two-leg ladder SrCu2 O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. The chain/ladder system (Sr,Ca)14 Cu24 O41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Low-dimensional vanadates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. The AV2 O5 family of compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2. The alternating chain system (VO)2 P2 O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Low-dimensional halides and pnictides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. The chain system KCuF3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. The ACuCl3 family of compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3. The Haldane system CsNiCl3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4. Yb4 As3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7. Inorganic systems with spin-Peierls and related instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1. CuGeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2. NaV2 O5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. Magnetic parameters of selected spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Magnetic light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Two-magnon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Light scattering in high-temperature superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Light scattering in quasi-one-dimensional spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. The limit of large dimerization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Non-zero temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Defect-induced light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Spinon light scattering in CuGeO3 : experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Raman scattering from spin-1/2 ladders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Magnetic bounds states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Bound states in CuGeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Three-magnon scattering in CuGeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Defect-induced bound states in CuGeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Bound states in NaV2 O5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Experimental observations in NaV2 O5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. De1ciency and substitutions on the Na site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3. Theoretical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Bound states in (VO)2 P2 O7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Bound states in SrCu2 (BO3 )2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. EKect of substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2. Polarization dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Quasielastic scattering in low-dimensional spin systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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0. Preface This article reviews recent progress in magnetic light scattering in one- and two-dimensional quantum spin systems. These systems received considerable attention during the past decade both from a theoretical and an experimental point of view. Following the investigations of the two-dimensional superconducting cuprates and the search for related transition-metal oxides a fascinating 1eld of copper oxide compounds, vanadates, manganites and nickelates opened up. These compounds show

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eKects of strong electronic correlations and magnetism in low dimensions, in particular interplay between spin, charge and orbital degrees of freedom. The theory of magnetism in one dimension, on the other hand, has a history reaching back to the origin of quantum mechanics. This is due to the fact that a spin chain allows more easily analytical or numerical solutions. It was found that the suppression of “trivial” long-range magnetic order sets the stage for an enormous complexity of possible ground states, exotic quasiparticles and many-body states. Understanding these eKects is the most intriguing challenge at present. A central concept in describing these low-dimensional quantum spin systems is that of a spin liquid. This ground state is dominated by strong quantum Ouctuations, pronounced spin–spin correlations and a suppression of long-range magnetic order. The Heisenberg chain with isotropic antiferro-magnetically coupled (s = 1=2) spins represents such a state in the sense that the spin–spin correlations decay algebraically. It is therefore often denoted as a critical spin liquid. An interesting situation occurs when competing interactions lead to a sudden change of the excitation spectrum, e.g., the opening of an excitation gap or the formation of long-range magnetic order. These quantum phase transitions are driven or controlled by the exchange coupling parameters, the exchange topology or by spin vacancies. The excitation gap may be realized with or without a spontaneously broken translational symmetry. The spin-Peierls transition and the related charge ordering instability discovered in the inorganic compounds CuGeO3 and NaV2 O5 , respectively, represent the case of broken translational symmetry. These compounds allow to investigate the excitation spectrum going from a homogeneous gapless to a dimerized state just as a function of temperature. In the two-leg spin ladder system SrCu 2 O3 and the chain/ladder system Sr 14 Cu24 O41 an excitation gap is realized without breaking translational symmetry. These compounds are discussed as model systems for an electronic mechanism of high-temperature superconductivity. The steady improvement of understanding also leads to surprising reinterpretations of compounds that have been investigated for years. The formerly canonical example of a spin ladder, the vanadium compound (VO)2 P2 O7 , is now recognized as a spin chain with strongly alternating coupling constants. This result has profound consequences for the interpretation of its low-energy excitations. Very important compounds that bridge one and two dimensions and still do not show long-range magnetic order are the spin frustrated system SrCu 2 (BO3 )2 and the 1/5-depleted square lattice system CaV4 O9 . Light scattering experiments or other spectroscopic methods like inelastic neutron scattering have been used to investigate both the above cited and many more compounds. One of the most signi1cant aspects of light scattering experiments is the observation of magnetic singlet bound states. These states originate from strong triplet–triplet interaction and characterize the excitation spectrum of the spin system. Recent theoretical progress has enabled a more detailed understanding of these eKects. Parameters like dimerization, frustration, interchain coupling, and spin–phonon coupling have an important impact on the ground state and the excitations of a quantum spin system. This review is organized as follows: After a brief description of the excitations and the phase diagram of quantum spin systems given in Section 1, important low-dimensional model-type spin systems, recent experimental results and their interpretations are discussed in Section 2. Up to now no comprehensive review on this rapidly growing 1eld exists that also considers materials aspects. Therefore, we try to balance between well established results and very recent developments. In Section 3 magnetic light scattering in low-dimensional spin systems is reviewed. The following

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Sections 4 and 5 discuss magnetic bound states and quasielastic scattering, respectively. Section 6 1nally summarizes aspects of the present knowledge in this 1eld and gives an outlook to future developments. This review bene1ted from discussions and collaborations with many colleagues, too numerous to mention here completely. We would like to thank especially E.Ya. Sherman and R. ValentQR for carefully reading the manuscript. P.L. and G.G. would like to express their gratitude to the recent and previous members of the light scattering group in Aachen, e.g., K.Y. Choi, J. Pommer, A. Ionescu, M. Fischer, M. Grove, G. Els, and P.H.M. van Loosdrecht. Further valuable discussions with F. Mila, W. Brenig, K. Ueda, M. Sigrist, W. Metzner, B. L'uthi, G.S. Uhrig, C. Pinettes, M. Udagawa, V. Gnezdilov and Yu.G. Pashkevich are gratefully acknowledged. We are obliged to C. Geibel, F. Steglich, H. Kageyama, M. Isobe, Yu. Ueda, J. Akimitsu, H. Tanaka, M. Johnsson, P. Millet, A. Revcolevschi, B.C. Sales, W. Assmus, S. Barilo and their collaborators for providing numerous samples. Furthermore, P.L. acknowledges gratefully a stipend of the Venture Business Laboratory for a visit to the Institute of Integrated Arts and Sciences (IIAS), Hiroshima University. 1. Excitations in low-dimensional spin systems In strongly correlated electron systems with integer number of electrons per site the low-energy excitations are usually given by the spin degrees of freedom. This situation is properly described by the Heisenberg exchange spin Hamiltonian. If, in addition, the exchange is restricted to low dimensions, then spin chains, spin ladders, and respective systems with a more complex exchange geometry are realized. These systems exhibit a number of unusual properties which are related to strong quantum Ouctuations. These properties will be addressed in the following. One-dimensional s = 1=2 spin systems (spin chains) with uniform nearest neighbor exchange coupling show according to the Lieb–Schultz–Mattis theorem a degeneracy of the singlet ground state with triplet excitations [1]. Assuming negligible spin anisotropies even for T = 0 the ground state is gapless and not magnetically ordered [2,3]. It is described by the Bethe Ansatz [4,5]. The spin– spin correlations are algebraically decaying, typical for a quantum critical state. Triplet excitations in such a system are not described as magnons (bosons) but as massless domain wall-like s = 1=2 spinons (fermions). These spinons are created as pairs, e.g., by an exchange process. Their dynamical structure factor is therefore given by a gapless two-particle continuum restricted by a lower and an upper dispersing boundary. In Fig. 1.1 a sketch of the spinon creation and the spinon continuum is given. The spectral weight of the continuum is dominant close to its lower boundary. A quantum phase transition from a gapless critical state into a gapped state (disordered spin liquid) is induced by dimerization, i.e. an alternation of the coupling constants to nearest neighbors ± = (1 ± )J Jnn nn along the chain or by a suUcient frustration = Jnnn =Jnn due to next nearest neighbor antiferromagnetic exchange Jnnn [6]. With dimerization the spinons are con1ned into massive triplet excitations. This con1nement of spinon and antispinon composite objects (triplets) is discussed similar to the quark con1nement in particle physics [7,8]. The resulting quantum disordered ground state is characterized by short-ranged exponentially decaying spin–spin correlations. In many cases the system is allowed to be described as an arrangement of spin dimers. The resulting lifted degeneracy

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

(a)

5

E

(b) ∆ 01 0

k

π

Fig. 1.1. Spin excitations on a homogeneous chain. On the left-hand side following (a) – (c) the generation of two spinons (vertical bars) by a spin Oip and consecutive exchange processes is given. On the right-hand side the corresponding two-spinon continuum is shown. The spectral weight of the continuum is maximal on the dispersing lower branch. Fig. 1.2. Sketch of spin excitations on a dimerized chain. Breaking a dimer (a) – (b) corresponds to the singlet/triplet gap 01 with the respective triplet dispersion shown on the right-hand side. The continuum of “free” triplets is reached for energies E ¿ 201 . For small k there exists a 1nite slope of the dispersion relation.

Fig. 1.3. Phase diagram (T = 0) of spin chains in dependence of dimerization and spin frustration . For = 0 and 6 c = 0:2412 (dark grey bar) a gapless quantum critical ground state exists. The remaining dashed region denotes the phase space of the gapped quantum disordered state.

of triplet and singlet excitations leads to an energy gain of the system. Fig. 1.2 shows a sketch of the excitation processes in a dimerized chain with the respective energy dispersion. The dimerization or alternation of the coupling constants connects k = 0 and k = , therefore a small part of the continuum spectral weight (˙ 2 ) is also expected for k 6 =2 [6]. In Fig. 1.2 this contribution is neglected. A (T = 0) phase diagram of dimerized and frustrated spin chains is given in Fig. 1.3. The points ( = 0; = 0) and ( = 0; = 0:5) correspond to the Bethe Ansatz and the Majumdar–Ghosh point, respectively. For = 0 and c ¡ ¡ 0:3 the gap remains numerically small [9–12]. The simplest representative of the quantum disordered state, however, is the two-leg spin ladder with an approximately equal or larger exchange coupling along the rungs with respect to the coupling along the legs of the ladder [13]. The singlet ground state is composed of spin dimers on the rungs. Here, the term spin liquid is even more appropriate as it is not based on a broken translational symmetry. An excitation in this picture of strong dimerization corresponds to the breaking of one dimer. The energy related to this process is the singlet–triplet gap 01 , see Fig. 1.2. A coupling of more than two chains to three-, four- or 1ve-leg ladders leads to the experimentally proven

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Fig. 1.4. Dispersion of magnetic excitations of a homogeneous two-leg ladder with J⊥ = 2J . The triplet branch as well as the lower boundary of the two-particle continuum are given by full lines. The dashed lines mark singlet (S = 0) and triplet (S = 1) bound states [21].

conjecture that ladders with an even number of legs have a spin gap while odd-leg ladders are gapless [14,15]. 1 In the limit of a large number of coupled chains a two-dimensional Heisenberg system is obtained and the magnitude of the respective spin gap is going to zero. This limit may also be used to understand the two-dimensional high-temperature superconductors (HTSC). Weakly doped twoand three-leg ladders have been theoretically investigated in this context [19,22–24]. However, also in two dimensions spin dimer ground states with a gapped excitation spectrum are realized. This happens either if the exchange topology is modi1ed to favor a dimer ground state, e.g., in removing 1/5 of the spins from a square lattice [25–27], or due to strong frustration (next nearest neighbor interaction) [28]. The triplet–triplet interactions that are responsible for the opening of the gap also lead to magnetic bound states, i.e. triplet excitations that are bound to singlet, triplet or quintuplet states [6,21,29–31]. The former two states are characterized by a well-de1ned excitation with an energy reduced with respect to the energy of the two-particle continuum of “free” triplets. If interchain or magnetoelastic interactions are dominant, bound states consist of soliton–antisoliton pairs [32–36]. Neglecting these eKects, the binding energy of a bound state in a dimerized spin chain originates from frustration. The maximum number of bound states of a spin chain is restricted to one singlet and one triplet state. In spin ladders with an additional diagonal frustration the number of bound states and their binding energy is less limited and increasing with frustration. In Fig. 1.4 the excitation spectrum of a homogeneous two-leg s = 1=2 ladder is shown with a singlet and a triplet bound state at the lower 1 It should be mentioned that the combined eKect of dimerization and interchain interaction may also lead to a vanishing spin gap in a quantum spin system. In a certain parameter space of a two-leg ladder [16] with additional frustration, 4-spin cyclic exchange or interchain interactions, quantum phase transitions to gapless phases have been observed [17,18]. On the other hand, there is theoretical evidence for a spin gap in doped three-leg ladders for a certain set of exchange coupling constants [19,20].

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boundary of the two-particle continuum [21]. A quantum phase transition at a critical frustration into a gapless phase is understood as a condensation process of many-particle bound states together with a general softening of the excitation spectrum [30]. More generally, magnetic bound states may therefore be used to study the triplet–triplet interaction, determine the coupling parameters and the phase diagram of the system. If defects, either as localized non-magnetic vacancies or as mobile carriers, are introduced into a quantum spin system its excitation spectrum may change drastically [37–41]. In a 2D square lattice doping with mobile carriers destroys long-range NQeel order and leads to the opening of a pseudo-gap for spin and charge excitations. The eKect of localized spin vacancies in dimerized spin chains or in even-leg spin ladders is diKerent. Here, a transition from a gapped into an ordered and gapless NQeel-type state is induced by a seemingly negligible amount of vacancies. This eKect is based on the doping of only weakly bound spinons by every vacancy. Thereby additional excitations are introduced in the gap corresponding to staggered moments for sites far from the vacancy and increasing spin– spin correlations [38,42,43]. Interchain interaction leads to the occurrence of magnetic order at 1nite temperature. The coexistence of true long-range magnetic order and dimerization is possible if spatial variations of the competing order parameters are taken into account [44]. This means that the excitation spectrum of such a system has two features, the gapped triplet mode due to dimerization and the gapless “spin wave mode” [45]. The emergence of antiferromagnetism keeps the structure and the energy scales of these modes essentially unchanged as the transfer of spectral weight from the gapped to the spin wave mode is realized with only a small reduction of the gap and an increase of the spin wave velocity. The latter mode is damped with a broadening proportional to the square of the wave vector [46]. Similar arguments have recently been used to describe the interplay or competition of disorder-induced antiferromagnetism and superconductivity in Heavy Fermion compounds [45]. Comprehensive experimental studies concerning the eKect of spin vacancies exist for spin ladder and dimerized spin chain systems. Some results including light scattering data will be presented in the next section. The eKect of mobile carriers on the gapped excitation spectrum of a spin liquid is directly related to the problem of an electronic mechanism for HTSC and not yet understood completely [47–49]. In the 2D CuO2 square lattice the doped holes are believed to form self-organized slowly Ouctuating arrays of metallic stripes in which the motion of holes shows a locally quasi-one-dimensional character [50– 52]. A spin gap or pseudo-gap is then the result of the spatially con1ned Mott-insulating regions of the material in the proximity of the metallic stripes. This eKect has been described by the term “topological doping” [48]. Corresponding theoretical studies of weakly doped two- and three-leg ladders con1rm these ideas in the sense that a tendency toward a binding eKect of holes either into a superconducting condensate or charge ordered ground state exists [22–24]. The excitation spectrum of the latter system is of special interest as it is separated into a gapless Luttinger-liquid (odd channel) and an insulating gapped spin liquid phase (even channel) [20,24]. In some sense this spectrum represents or mimics the scenario of spin and charge separation discussed for HTSC. A quantum phase transition into a superconducting state with d-wave character has been predicted for the three-leg ladder at higher doping levels [24]. In a very simpli1ed picture the two channels of the three-leg ladder may be understood as a “plain” ladder coupled to a chain. Holes hop back and forth from the chain to the ladder system. In the ladder they prefer to form pairs minimizing the number of broken dimers. Hopping back

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into the chain system this correlation is “partially transferred” into the conducting channel [20,24]. The experimental part of this problem is far from being completely settled and therefore under progress. Although only few spin ladder systems are available, unusual experimental results exist, pointing to k-dependent relaxation rates diKerent from the undoped material [53]. In the following section a thorough review of the presently known inorganic low-dimensional compounds will be given focusing on the questions discussed above. 2. Important inorganic quantum spin systems A general advantage of the use of inorganic compounds like transition-metal halides and oxides is the availability of large single crystals of considerable quality. This advantage is based on their high thermal stability. Recent developments in crystal growing techniques together with the wider availability of optically heated travelling solvent Ooating zone (TSFZ) furnaces [54] enabled systematic substitution studies for several inorganic compounds. These investigations themselves provided further motivation due to surprising results in the sense of rich phase diagrams and a modi1cation of the low-energy excitations [36,37,40,42,55–58]. Unfortunately only few compounds have been realized where doping of mobile carriers is achieved. Here the application of high pressure either during crystal growth or later during the determination of phase diagrams proved to be important and is still growing in acceptance. In contrast, the use of soft chemical routes, such as hydrothermal growth, is still limited, partly due to problems in the controllability of the growth parameters [59]. The sample quality of compounds that have a singlet ground state may be judged by measurements of the magnetic susceptibility in dependence of temperature. Every spin defect induces a local moment that is observed as a “Curie contribution” to the magnetic susceptibility. 2.1. Structural considerations and important parameters Three building principles exist that reduce the superexchange of a 3d ion–oxygen con1guration to less than three dimensions. These are on the one hand an enlarged distance or missing bridging oxygen between two 3d ion sites or on the other hand a superexchange path with an angle different from 180◦ . Due to the Kanamori–Goodenough–Anderson rule (vanishing AF superexchange via perpendicular oxygen O2p-orbitals) a non-collinear exchange path leads to a weakening of the magnetic exchange of, e.g., neighboring CuO chains [61,62]. Finally, so-called lone pair cations, such as As3+ , Se4+ ; Te4+ , Pb2+ or Bi3+ in combination with halides can be used to form crystal structures with unique magnetic topologies that show a reduced connectivity [63–65]. In this way compounds representing chains, zig-zag double chains or ladders with diKerent number of legs are realized [66,67]. Fig. 2.1 shows a comparison of several possible 3d ion–oxygen con1gurations. In Table 2.2 at the end of Section 2 some representative low-dimensional spin systems and their magnetic properties are listed. Experimental parameters to expand the phase diagram are pressure, that may control the exchange interaction, and frustration that is strongly related to triplet–triplet interaction and the existence of magnetic bound states. Furthermore, the application of an external magnetic 1eld might lead to interesting behavior. For instance in a fermionic picture (strictly applicable for a 1D system) the magnetic 1eld controls some 1lling of the quantum spin system. In several systems plateaus in the

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

(a)

(b)

(c)

9

(d)

Fig. 2.1. Overview of 3d ion–oxygen con1gurations realized in low-dimensional transition metal compounds: (a) a linear 3d ion–O-chain, (b) a non-collinear 3d ion–O2 -chain with reduced exchange, (c) a frustrated double chain (zig-zag chain), and (d) two ladders with a frustrated weak interladder coupling. The thick (thin) lines mark dominant (reduced) exchange coupling paths. The small circles denote the positions of the transition metal ions, e.g., Cu2+ or V4+ with s = 1=2. The large circles denote O2− . The magnetic properties of these networks depend in addition on the relative strength of the exchange coupling constants [60].

magnetization as a function of the 1eld exist that are divided by step-like transitions (see Sections 2.3 and 2.6.2). In some systems, transitions into incommensurate magnetic phases are observed [68–70]. So far, the available literature on light scattering experiments mainly discusses s = 1=2 compounds. The reported experiments on s = 1 Haldane systems [71,72] are limited in number [73–76]. 2.2. Quasi-zero-dimensional compounds A series of compounds made out of weakly coupled units, mostly spin dimers or spin-tetrahedra, can be viewed in 1rst approximation as zero-dimensional. Depending on the strength of the couplings in between the units, one might view these compounds also as quasi-one-dimensional or quasi-two-dimensional. An example is the spin-1/2 ladder compound CaV2 O5 , in which the inter-rung coupling is so weak that it might be viewed in 1rst approximation as made up of dimers with only very weak interdimer couplings [77,78]. 2.2.1. Dimer compounds A spin dimer system with still noticeable interdimer interaction is realized in the copper ditellurate CuTe2 O5 . The structure is shown in the inset of Fig. 2.2. The lone pair cation Te4+ yields a magnetic insulation of pairs of strongly distorted and edge-sharing CuO6 octahedra with respect to each other W within the pair of octahedra and 5:28 A W to the next [81]. The resulting Cu–Cu distances are 3:18 A pair [79]. The magnetic susceptibility of CuTe2 O5 shows a maximum at Tmax = 56:5 K and a strong decrease for smaller temperatures. It can be described with the dimer model ˙ [3 + exp( · J )]−1 , = 1=kB T [82] and an intradimer coupling J = 90 K. The high temperature susceptibility corresponds to a Curie–Weiss temperature of = 41 K. Concerning the eKect of dimerization, also the spin-Peierls system CuGeO3 and the related charge ordering system NaV2 O5 are very attractive. They allow to investigate both the homogeneous and dimerized state of a spin chain system as function of temperature. With respect to larger degrees of dimerization and a coupling of two chain systems we refer to (VO)2 P2 O7 as discussed in Section 2.5.2.

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Fig. 2.2. Magnetic susceptibility (dots) and lattice structure of CuTe2 O5 with edge-sharing CuO6 octahedra forming Cu dimers. The line corresponds to a dimer 1t [82] with J = 90 K. The open (full) spheres in the inset represent oxygen (tellurium) ions [79].

t1 Chain B

t2 t3

Chain A

t5 Chain B

b

c Fig. 2.3. Topology of the intra- and inter-dimer hopping matrix elements of one bc-layer of CsV2 O5 . Note the close resemblance to the compound (VO)2 P2 O7 , see Fig. 2.30. The 1lled triangles denote projections of V5+ O4 -tetrahedra, the 1lled circles the spin-1/2 V4+ -ions. The magnitude of the interchain hopping matrix-elements t2 and t5 are small [80].

The weakly coupled spin-1/2 dimer compounds (VO)2 P2 O7 (see Section 2.5.2 and the crystal structure in Fig. 2.30) and CsV2 O5 [83] are structurally very similar. Both systems contain planes of weakly coupled dimers with a topology illustrated in Fig. 2.3. CsV2 O5 contains one of these planes per unit-cell and (VO)2 P2 O7 contains two inequivalent planes. A density functional theory (DFT) and susceptibility analysis has found that CsV2 O5 might be viewed alternatively as a strongly dimerized chain compound with a dimerization parameter of = 0:8 [80]. The dimer compound [84,85] CaCuGe2 O6 is very unusual. Crystallographically [84,86] the dominant topological structure are chains of edge-sharing CuO6 octahedra running along the c-axis. The spin-1/2 Cu2+ -ions form nevertheless spin-dimers in between pairs of third-nearest-neighbors

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J3 J1

Fig. 2.4. Topology of CaCuGe2 O6 . The full and dashed lines denote the 1rst- and third nearest-neighbor exchange couplings J1 and J3 , respectively, with J1 =J3 ≈ −0:2, J3 = 67 K, and J1 parallel to the crystallographic c-axis.

a

2

1 x

2

1

d

z

y

c

z y

x

x

Fig. 2.5. Topology of Cu2 Te2 O5 X2 , with X = Br; Cl. The ovals and the squares denote the CuO3 X units which are linked by intra-tetrahedra couplings J1 and J2 . The in-plane (ab) couplings are Jx ; Ja and Jd . The in-chain (c) couplings are Jc and Jz .

located on adjacent chains. This very unusual physics is possibly due to the near-to-90◦ Cu–O–Cu superexchange angle along the c-axis [86]. This leads to a strong reduction of the exchange coupling which, due the Kanamori–Goodenough–Anderson rule [87–89], actually becomes slightly negative (ferromagnetic) [86]. (Fig. 2.4.) 2.2.2. The Cu-tetrahedra systems Cu2 Te2 O5 X2 , X=Br,Cl In Fig. 2.5, a projection of the topology of Cu2 Te2 O5 X2 with X = Br; Cl is presented. It contains tetrahedra of Cu2+ ions which are weakly coupled in the (ab)-plane and along the c-axis. The magnetic susceptibility as function of temperature is shown in Fig. 2.6. The susceptibility at higher temperature contains a similar Curie–Weiss-like term for both compounds. Towards lower temperatures a maximum at 30 K (Br) and 23 K (Cl) is observed in (T ) [64]. This susceptibility can be 1tted in 1rst approximation [63] by non-interacting tetrahedra with intra-tetrahedra

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.

.

.

Temperature (K) Fig. 2.6. Magnetic susceptibility of Cu2 Te2 O5 X2 , with X = Br; Cl for diKerent magnetic 1elds. The inset shows the derivative 9=9T at B = 0:1 and 5 T with the transitions assigned by arrows [64]. Fig. 2.7. Speci1c heat of Cu2 Te2 O5 X2 , with X = Br; Cl close to the transition in magnetic 1elds up to 13:5 T. Note the two diKerent temperature scales for the respective compounds. The inset gives the 1eld dependence of the transition temperatures [64].

exchange couplings J1 ≈ J2 ≈ 43 and 38:5 K, respectively, for the bromide and the chloride compound. The two compounds undergo an ordering transition [64], presumably of antiferromagnetic nature, at T0(Br) = 11:4 K and TN(Cl) = 18:2 K. The smaller transition temperature of the bromide is attributed to the approximately 6% larger unit cell volume of the bromide related to a weaker inter-tetrahedra coupling. The ordering in Cu2 Te2 O5 Br 2 is anomalous concerning thermodynamic and spectroscopic properties. As a 1rst example note the opposite behavior of the bromide compared to the chloride concerning the unusual shape of the derivative 9=9T (T = T0 ) shown in the inset of Fig. 2.6. Thermodynamic properties. The speci1c heat Cp (T ) of Cu2 Te2 O5 Cl2 (see upper curve with upper temperature scale in Fig. 2.7) shows a mean 1eld like transition with a sizeable anomaly at TN and a weak dependence on magnetic 1elds. These data point to a 3D-antiferromagnetic ordering of a system with only weak spin anisotropies and appreciable inter-tetrahedra coupling. The related energy scales are characterized by the ratio of the transition temperature to the maximum temperature in (T ); T0 =Tmax = 0:78 for Cu2 Te2 O5 Cl2 . For Cu2 Te2 O5 Br 2 the ratio T0 =Tmax = 0:27 is much smaller implying the freezing out of a large part of the magnetic triplet excitations already for T ¿ T0 . The derivative 9(T )=9T reveals a small but well discernible step which has the opposite sign compared to Cu2 Te2 O5 Cl2 (Fig. 2.6), i.e. the slope for T ¡ TN is smaller than for T ¿ TN [64]. The speci1c heat of Cu2 Te2 O5 Br 2 at B = 0 (see lower curves with lower temperature scale in Fig. 2.7) shows a broadened anomaly at T = T0 (0) = 11:4 K. This proves that this transition also occurs in the absence of an external 1eld. Applying a magnetic 1eld leads to a very strong increase of the size of the anomaly, by more than a factor of 3 at B=13 T, and to a pronounced shift of T0 to higher temperatures, e.g. to T0 (B = 13 T) = 12:4 K. This shift corresponds to a larger magnetization of the ordered state compared to the disordered state. The 1eld dependence follows a power law

Intensity (arb. units)

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

T = 3K

13

Cu 2 Te 2 O5 Br2

450

5K 7K 300

8K

3K 9K

150

0

20

40

10K 14K 22K 60 80

Raman shift (cm-1)

Fig. 2.8. Raman spectra of Cu2 Te2 O5 Br 2 as function of temperature in (cc) polarization parallel to the crystallographic c-axis [64].

with T0 (B) = T0 (0) + a · Bn and n = 1:41 ± 0:05. The magnetic entropy at T0 involved with the instability has been roughly estimated to be only Sm (T0 ) 1:8 J=K mol = 0:16 · Rln2=spin, whereas for Cu2 Te2 O5 Cl2 , the entropy at TN is much larger, Sm (TN ) 4:1 J=K mol = 0:36 · Rln2=spin. One might view these compounds, especially Cu2 Te2 O5 Br 2 , as weakly coupled tetrahedra compounds close to a quantum critical transition between an antiferromagnetic state and a spin liquid [92,93]. On the other hand, some of the presented results provide evidence for a sizable magnetic anisotropy and an additional Dzyaloshinskii–Moriya (DM) interaction in the spin system. This interaction is allowed due to the low symmetry exchange path lacking the inversion center. High-1eld magnetization experiments performed on powder samples in 1elds up to 50 T point into the same direction [94]. A 1nite but small slope is observed in Cu2 Te2 O5 Br 2 for “small 1elds” up to 15 T followed by a smooth change of slope for higher 1elds. This behavior is attributed to a mixing of triplets to the singlet ground state. The pseudo-spin gap of this compound is therefore attributed to an unique interplay of quantum Ouctuations with anisotropies. Magnetic light scattering. Of special interest is the possible occurrence of low-lying intra-tetrahedra singlet-excitations in magnetic Raman scattering [64,91]. Indeed Raman scattering has been the 1rst tool to demonstrate the instability. In Fig. 2.8 Raman spectra of Cu2 Te2 O5 Br 2 are shown in c-axis polarization as a function of temperature. The high-temperature spectra are dominated by a triangle-shaped scattering continuum centered at 61 cm−1 = 88 K. This position matches the energy scale 2J = 86 K previously determined from a 1t of the magnetic susceptibility [63]. The continuum has been discussed as a two-magnon-like scattering process of a tetrahedra chain supported by the “in-chain” selection rule [95,96]. The total linewidth and low energy onset at 40 cm−1 both point to an appreciable inter-tetrahedra coupling. For temperatures T ¡ 9 K ¡ T0(Br) , a sharp maximum develops with a soft mode-like behavior. This mode reaches its maximum energy of 18 cm−1 at the lowest temperatures. The change of the low-energy excitation spectrum is accompanied by an increase of intensity on the right-hand shoulder of the scattering continuum leading to a small shift of the triangle to higher energies. No other transition related anomalies have been found in optical spectroscopy, especially no phonon anomalies were observed at higher energy. The two magnetic signals are only observed with both electric 1eld vectors parallel to the crystallographic c-axis corresponding to the chains of tetrahedra. This symmetry selection rule and the

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Fig. 2.9. Raman spectra of Cu2 Te2 O5 Br 2 at T =2:3 K in (cc) polarization in magnetic 1elds parallel to the crystallographic c-axis. The inset shows low frequency data with higher spectral resolution [90].

temperature dependence of the low-energy mode are similar to the properties of singlet bound states observed, e.g. in the dimerized phase (T ¡ TSP = 14:5 K) of CuGeO3 . In the latter spin chain system frustration leads to a binding eKect of two elementary triplets to a well-de1ned mode at the energy 1:78 [97]. The physics and phenomenology of such states will be further described in Section 4. An important diKerence between this pure singlet mode and the low-energy excitation in Cu2 Te2 O5 Br 2 is given by its quite large magnetic 1eld dependence. In Fig. 2.9 Raman spectra are shown with magnetic 1elds parallel to the c-axis of the compound. The maximum frequency shifts non-linearly with the magnetic 1eld leading to an increase by 4:5 cm−1 in 6 T. Also this shift might tentatively be attributed to a mixing of triplet states into the ground state due to DM interaction. At higher 1elds (B ¿ 4 T) a second peak is induced at higher energies (≈ 22 cm−1 ). This peak does not shift with further increase of 1eld. This complex behavior is not yet understood in detail. A comparison of the Raman response of Cu2 Te2 O5 Br 2 with the related Cu2 Te2 O5 Cl2 is given in Fig. 2.10. It is obvious that the low-energy spectral response of the two compounds is completely diKerent. In the chloride the magnetic scattering is less pronounced and more spread out. This corresponds to a larger inter-tetrahedra coupling. An interesting aspect is that substitution studies allow a continuous tuning of the composition and to study the crossover between the two systems. In samples with mixed composition a continuous shift of the phonon frequencies is observed as given in Fig. 2.11 for some selected phonon modes. Generally the spectrum of optical phonons is softer for the bromide compared to the chloride. This is related to the approximately 6% larger volume of the unit cell of the bromide. Preliminary investigations of the magnetic excitation spectrum show with increasing Cl concentration a shift of the soft mode to higher frequency and splitting into at least three peaks that 1nally overlap and form the continuum shown in Fig. 2.10 for X = Cl. 2.3. Quasi-two-dimensional compounds Only very few 2D spin systems with a quantum disordered ground state are discussed in the literature in which the spin gap is comparably large and not due to spin anisotropies. In 2D this spin liquid ground state should follow from exotic exchange topologies as both the square and the triangular lattice have a NQeel-like ground state [98]. Examples of 2D spin gap systems are the KagomQe lattice, the 1/5-depleted square lattice [25–27,99–101], and the Shastry–Sutherland

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Fig. 2.10. Comparison of the Raman response of Cu2 Te2 O5 Br 2 and Cu2 Te2 O5 Cl2 at T = 3 K in (cc) polarization. The curve for X = Br is shifted vertically for clarity [91]. Fig. 2.11. Phonon Raman spectra of Cu2 Te2 O5 (Br 1−x ; Clx )2 as function of x in (cc) polarization [90].

Fig. 2.12. (a) Topology of V4+ in CaV4 O9 forming a 1/5-depleted square lattice with three sublattices (A, B, C). (b) Topology of Cu2+ in SrCu2 (BO3 )2 in the ab plane. The intradimer (J1 ) and interdimer (J2 ) exchange couplings are given by solid and dashed lines, respectively.

lattice [28] shown in Fig. 2.12. The latter two exchange topologies are realized in CaV4 O9 and SrCu2 (BO3 )2 , respectively. 2.3.1. The 1/5-depleted system CaV4 O9 The compound CaV4 O9 with a singlet–triplet gap 01 =135 K [25,102,103] is a 2D arrangement of plaquette singlet units of four V4+ atoms (s=1=2) within four corner-linked VO5 pyramids. These are coupled by an exchange between the V spins within edge-linked VO5 pyramids [103]. A simplifying sketch of the corresponding 1/5-depleted 2D spin structure is displayed in Fig. 2.12(a). This structure

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is built up by three plaquette systems, denoted by A, B and C. The exchange coupling constants are therefore divided into intra-plaquette coupling JA , acting between spins on a link in an A plaquette, and inter-plaquette coupling JBC , acting on links shared by the B and C plaquettes. Theoretically several possible ground states have been identi1ed, e.g., a “plaquette RVB” state or, in contrast, a spin-Peierls state with a spontaneously broken lattice symmetry [26,27]. Their existence is controlled by the ratio of the coupling constants. More general, the experimentally observed gap formation has been discussed either due to the coupling of two meta-plaquette systems (i.e. plaquettes formed by n.n.n. links) or due to a considerable eKect of frustration by intra-plaquette next nearest neighbor exchange interaction [104]. Theoretical and experimental results on this interesting compound are reviewed in Refs. [105,106]. Recently the structurally and electronically related SrV4 O9 has been prepared using hydrothermal synthesis and found to have a spin gap of similar magnitude as in CaV4 O9 [59]. 2.3.2. The Shastry–Sutherland system SrCu2 (BO3 )2 The layered compound SrCu2 (BO3 )2 consists of buckled CuBO3 planes and Sr layers. A projection of this structure on the ab plane realizes the frustrated 2D exchange topology shown in Fig. 2.12(b). Its magnetic ground state is a direct product of singlet dimers with a spin gap 01 =34 K [107,110,111]. Despite the problematic viscosity of borate melts, large single crystals can be prepared using an optically heated travelling solvent Ooating zone (TSFZ) furnace [112]. A review of experimental properties of this system is given in Ref. [113]. SrCu2 (BO3 )2 is the only realization of the Shastry–Sutherland lattice studied theoretically nearly 20 years ago [11]. Its exactly solvable dimer ground state is a 2D analogue of the Majumdar–Ghosh point [10] of a 1D spin chain with a strong frustrating next nearest neighbor exchange coupling, = J2 =J1 = 0:5, see Fig. 1.3. The triplet excitations in this type of frustrated lattice are almost localized leading to very Oat triplet branches compared to the energy scale of the exchange coupling constants [113,114]. The corresponding “spin wave” velocities should therefore be very small leading to strong triplet–triplet interaction especially at the high symmetry points of the magnetic Brillouin zone. On the other hand, with increasing interdimer exchange coupling a quantum phase transition of the spin system from an exact dimer ground state to an antiferromagnetically ordered gapless state is expected. Then the exchange topology is equivalent to a 2D Heisenberg square lattice. The quantum critical point separating these phases is therefore controlled by the ratio x = J2 =J1 of the next nearest interdimer coupling J2 to the nearest neighbor intradimer coupling J1 . A critical ratio of xc = 0:691 has been determined theoretically [28,115]. A modi1cation of this ratio by pressure or substitutions would allow to study the crossover between these two competing states in a unique way. While substitution experiments in SrCu2 (BO3 )2 did not prove to be successful, a still preliminary investigation of magnetic susceptibility under pressure has indeed shown a decrease of the spin gap as expected for a closer proximity to the antiferromagnetic state [116]. Thermodynamic properties. The magnetic susceptibility of SrCu2 (BO3 )2 at ambient conditions shows a maximum (Tm = 15 K) and a rapid decrease toward lower temperatures with an exponential tail for T → 0 [107], see Fig. 2.13. A similar behavior is observed in the speci1c heat of the spin C system with a maximum (Tm p = 8 K) [108] given in Fig. 2.14. It is noticeable that the maxima in both thermodynamic quantities are observed at temperatures much smaller than the singlet–triplet gap and nearly one order of magnitude smaller than the

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Fig. 2.13. Magnetic susceptibility of SrCu2 (BO3 )2 . The inset shows the low temperature evolution with a 1t using a dimer model as a dashed and dotted line [107]. Fig. 2.14. Low-temperature speci1c heat c=T of SrCu2 (BO3 )2 in magnetic 1elds up to 12 T. The dashed line corresponds to the estimated phonon contribution to the speci1c heat [108].

estimated exchange coupling constants J1 = 100 K and J2 ≈ 68 K [28]. The resulting ratio x = 0:68 is just below the above given value of xc , placing this material in the dimerized phase, close to the boundary of the NQeel state. A more recent analysis of the bandwidth of the triplet branch favors smaller but still comparable values of x ≈ 0:65 [117]. A further theoretical investigation gives evidence for x ≈ 0:6 and an unexpected instability in the triplet channel at a critical ratio xt = 0:63 [118]. The localized nature of the triplet excitations in SrCu2 (BO3 )2 also shows up as 1ve consecutive transitions from gapped to gapless phases observed in magnetic 1elds up to 50 T. The gapped phases are identi1ed as plateaus in the normalized magnetization at m=0, 1/3, 1/4, and 1/8. The corresponding magnetic 1elds are H ¡ 20 T; H ≈ 30 T and H ≈ 40 T [107,119]. The plateau at m = 1=3 is attributed to a stripe phase of triplets [120]. These experiments are the 1rst observation of a magnetic 1eld-induced “triplet–singlet-lattice” in a 2D spin system. Until recently magnetic phase diagrams of similar complexity have only been discussed for 1D spin systems, e.g., for the frustrated double chain system NH4 CuCl3 . A thorough discussion of these topology-induced quantization phenomena is given in Section 2.6.2. Investigations of ESR [121], IR [122] and neutron scattering [114] on SrCu2 (BO3 )2 observed triplet excitations with a spin gap of = 34 K and a very small dispersion of only 2 K pointing to their extremely localized character. In addition, a second triplet branch 01 = 55 K with an assumed larger dispersion of 17 K was detected. This branch can be interpreted as a triplet bound state of two elementary triplets (see discussion below). Frustration due to the interdimer coupling J2 can

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lead to the reduction of the ratio 01 =01 = 1:62 below 2 (corresponding to non-interacting magnons) [114]. In recent neutron scattering experiments the dispersion of the second triplet branch could not be con1rmed. The k dependence of 01 is comparable with a set of disrupted sheets with diKerent energies in diKerent regions of k space [123]. These lead to an enlarged spectral weight at 01a =58 K and 01b =75 K, instead of the earlier single dispersing branch. The origin of these exciton-like levels may be related to DM interaction, strong spin–phonon interaction or their interplay. It should be noted that these experimental results question the earlier determination of the magnetic coupling constants described above. The complexity of these results on a seemingly simple model system with an exact singlet ground state proves one more time the large potential of quantum spin systems for unconventional collective excitations. For these states inelastic light scattering is a valuable complementary experimental tool. Magnetic light scattering. In Raman scattering experiments on SrCu2 (BO3 )2 a rich excitation spectrum with an additional drastic evolution with temperature has been observed [124]. At high temperatures (T ¿ 01 ) the light scattering spectra show strong quasielastic scattering with a Lorentzian spectral function. For intermediate temperatures this signal is gradually replaced by a broad maximum with a center of spectral weight at 50 cm−1 , a frequency comparable with the doubled spin gap. The temperature dependence of the scattering intensities of the respective signals has been mapped on the magnetic susceptibility and the speci1c heat of the spin system [124]. This kind of mapping has also successfully been used in the case of CuGeO3 and (VO)2 P2 O7 and will further be discussed in Section 5. The spectrum changes drastically for very small temperatures compared with the spin gap (T 01 ) where the above-discussed scattering intensities are completely suppressed. Four well-de1ned modes with energies i00 = 30; 46; 56 and 70 cm−1 appear [124]. It is interesting to note that the splitting of these modes in Raman scattering is comparable to the splitting of the triplet bound state branch observed in neutron scattering. Nevertheless, the modes observed in Raman scattering experiments do not show an appreciable magnetic 1eld dependence in contrast to the triplet states. These experiments will further be discussed in Section 4.4. Examples of the respective Raman data are given in Fig. 4.17. The above-described bound state scattering is only observable with the electric 1eld vectors parallel to the ab plane of the compound. There exist no inter-layer singlet bound states. There is, however, a phonon mode with peculiar properties. This pure inter-layer mode observed in spectra of (cc) polarization is strongly anharmonic. As its energy is comparable to the energy scale of the magnetic system it has been investigated in detail. Fig. 2.15 shows the strong softening and broadening of linewidth for temperatures T ¿ 70 K with a frequency of only 35 cm−1 at T = 300 K. In the inset the frequency of this mode is shown as function of temperature. The small decrease observed in the low temperature limit (T ¡ = 34 K) is larger than the experimental error and must be related to an appreciable spin–phonon coupling. Furthermore related second and higher harmonics are displayed in Fig. 2.16. The very small frequency shift of the second harmonic mode points to a negligible dispersion and a large corresponding density of states of the underlying phonon mode. These eKects motivated a X-ray study that indeed observed a structural phase transition of SrCu2 (BO3 )2 at Tc = 395 K [125]. For T ¿ Tc the compound has an inversion center and completely Oat CuBO3 planes. Atomic displacements involved in the transition are directed perpendicular to the ab-plane leading to a buckling of the planes for T ¡ Tc . In the magnetic susceptibility a small step-like decrease is induced due to a change of the inter-layer couplings that are also frustrated [126,118]. The intra-layer coupling constants and their ratio are not aKected.

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Fig. 2.15. Softening of a phonon mode in SrCu2 (BO3 )2 for (cc) polarization as function of temperature. The inset shows the frequency of the mode. The line is a guide to the eye [109].

Fig. 2.16. Anharmonicity of the interlayer phonon mode at 62 cm−1 in SrCu2 (BO3 )2 and the related second harmonic mode as function of temperature [109].

The phonon spectrum of the alternating chain system (VO)2 P2 O7 has some similarities to the scenario found in SrCu2 (BO3 )2 as discussed in Section 2.5.2 and displayed, e.g., in Fig. 2.32. In (VO)2 P2 O7 the frequency of the phonon at 123 cm−1 softens by 10%. Comparing the behavior of (VO)2 P2 O7 with SrCu2 (BO3 )2 a relation to the high symmetry crystallographic phase of (VO)2 P2 O7 might be appropriate. This phase is stabilized under high pressure and has a higher symmetry and a smaller number of independent alternating spin chains within one unit cell [127,128]. Also strong

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spin–phonon coupling has been discussed as the primary origin for the observed anharmonicity of optical phonons in (VO)2 P2 O7 [129]. 2.3.3. High temperature superconductors In contrast to the 2D systems with strongly frustrated or unconventional exchange topology, the “pseudo-gap” observed in underdoped high HTSC for T ¿ Tc cannot be attributed to the exchange topology. Here the CuO2 planes form a “conventional” square lattice. The opening of this gap as a partial and anisotropic suppression of the excitation spectrum is observed together with crossover-like eKects [130]. DiKerent experimental techniques have been successfully used to investigate this phenomenon, such as resistivity measurements, Hall eKect, Raman [131,132] and IR spectroscopy, angle-resolved photoemission [7], but also NMR spectroscopy [133,134]. Therefore, these eKects also involve spin degrees of freedom. The gap formation has often been associated with short-range ordered spin singlets (Zhang–Rice singlets) [46,135–138] and with Ouctuating charge ordered stripes (see also the discussion in Section 1) [48]. The model of the “Nearly Antiferromagnetic Fermi Liquid” (NAFL) highlights the role of strong antiferromagnetic spin Ouctuations at the so-called “hot spots” of the Brillouin zone (k = ( ; 0) and (0; )). These points are connected by the AF wave vector kAF that remains important due to the close lying NQeel phase in the temperature-doping phase diagram of HTSC. At these hot spots precursors of a damped spin density wave are expected. These may be detectable as a pseudo or spin gap with an energy scale of approximately ≈ 2 J. The NAFL model leads to the same symmetry of the pseudo-gap as observed in angle-resolved photoemission [139,140]. However, it is not clear whether it describes the energy scale of the pseudo-gap correctly. In several experiments pseudo-gaps with ≈ 300 K have been observed. This energy scale is much smaller than the exchange coupling constants of HTSC. A strong criticism of the NAFL scenario can also be found in Ref. [141]. 2.3.4. The diluted square lattice system K2 V3 O8 Recently, also vanadates have been searched for new two-dimensional compounds [67]. The inherent Oexibility in coordination and electronic con1guration of the vanadium ion allows to realize exotic exchange topologies that have never before been investigated concerning their magnetic properties. An interesting candidate of this family of compounds is K2 V3 O8 with an interconnection of magnetic (s = 1=2) V4+ O4 pyramids and non-magnetic (s = 0) V5+ O4 tetrahedra. The topology shown in Fig. 2.17 leads to a diluted square lattice with reduced exchange couplings (J = 12:6 K). Due to a next neighbor exchange path with missing inversion center in between two V4+ sites, this system has an appreciable DM interaction and an additional c-axis anisotropy. The resulting weak ferromagnetism and 1eld-induced spin reorientation leads to a pronounced and sharp increase of the magnetization for T 6 TN = 4 K with 1elds parallel to the plane [142]. The ordered moment at low temperatures is 0:72!B , i.e. reduced from the expected 1!B . Raman data of this system in Fig. 2.18 show scattering continua at low energies that might tentatively be attributed to two-magnon scattering [143]. In 2D antiferromagnets two-magnon scattering gives a dominant contribution in the B1g scattering geometry. The underlying local exchange process between sublattices is also allowed at elevated temperatures above the NQeel transition. Therefore this scattering intensity is not limited to low temperatures.

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Fig. 2.17. Crystal structure of K2 V3 O8 showing a plane of interconnected V4+ O4 pyramids and V5+ O4 tetrahedra (darker and brighter shaded, respectively) [142]. Dark spheres correspond to K − ions, while the vanadium sites are omitted.

Fig. 2.18. Low-energy Raman spectra of K2 V3 O8 in B1g and A1g symmetry. The maxima are marked by dashed lines. In B2g symmetry no low-energy scattering has been observed [143].

In K2 V3 O8 the scattering intensity is also observed in the symmetry-forbidden A1g geometry. The presence of a symmetry-forbidden signal may be ascribed to the breakdown of the selection rules due to a non-secular term like the DM interaction. Unlike the cuprates, however, no evidence for the existence of a two-magnon signal in B2g symmetry has been found. The double maximum in the B1g scattering geometry remains also unexplained in this scenario. Neutron scattering experiments do not give evidence for a splitting of zone-boundary magnon branches of similar magnitude [145]. Compared to this neutron data the peak position of the continuum at 27 cm−1 corresponds to 3 J, a relation that matches a classical estimate for an exchange process in 2D. For a quantum, s = 1=2 spin system a factor 2.7 is expected especially if the reduced ordered moment is taken into account. With increasing temperature the peak frequency of the spectrum shifts to lower frequency while the linewidth increases for both symmetries. The magnetic scattering persists at least up to ∼ 10TN which is unprecedented for two-magnon scattering in similar quantum spin systems. In Fig. 2.19 the

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. . . . . .

Fig. 2.19. Temperature dependence of linewidth and frequency of the scattering maxima in K2 V3 O8 (full squares) in comparison with the 2D K2 NiF4 (open squares) and the 3D antiferromagnet KNiF3 (full triangles) [143,144].

temperature dependence of linewidth and frequency of the two-magnon scattering is compared with the earlier data from the typical two-dimensional and three-dimensional antiferromagnets K2 NiF4 and KNiF3 , respectively [143,144]. The seemingly ineKective thermal Ouctuations point to an involved energy scale which is diKerent from that of the spin system. As a candidate orbital degrees of freedom should be discussed. The square pyramidal local symmetry of the V4+ is compatible with an orbital degeneracy and low energy spin–orbital coupled modes [146,147]. Phonon–orbital coupled modes have, e.g., been discussed to explain Raman scattering results in the undoped manganites LaMnO3 [148]. 2.4. Low-dimensional cuprates: new compounds related to high-temperature superconductors While searching for new cuprates that are related to HTSC in their structural and electronic properties a very fascinating class of materials has been discovered. These compounds might show a similar tendency toward superconductivity [14,15,24]. Lightly doped two-leg and three-leg ladders should in their spin and charge degrees of freedom represent a minimal model for underdoped HTSC. Following the prediction of Rice and Sigrist [22,23], superconductivity was discovered in the Sr cuprate Sr 0:4 Ca13:6 Cu24 O41 at Tc = 12 K and with a pressure P = 3:5 GPa [150]. Fig. 2.20 shows the crystal structure of this compound containing CuO2 chains and Cu2 O3 ladders. It is important not to overemphasize the similarity between the quasi-2D HTSC and the 1D Sr cuprates discussed here [151]. The HTSC parent compounds are antiferromagnetic insulators, the parent compound of the superconducting ladder material instead is a non-magnetic Mott insulator with a well-de1ned spin gap. This diKerence is partly related to the crystallographic structure. The HTSC are dominated by 2D CuO2 square planes, the ladder compounds are built, on the other hand, by layers of low-dimensional Cu–O networks with a variety of diKerent coordinations as shown in Fig. 2.1 [66]. Additional Sr or (La,Ca) layers separate these networks and provide a layered character of the crystal structure. The magnetic properties of these systems are only similar to HTSC in the sense that the Cu–O–Cu superexchange path is of comparable strength. The Cu–O-networks lead

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Cu O Sr ,Ca

Fig. 2.20. Lattice structure of (Sr; Ca)14 Cu24 O41 with CuO2 chains and Cu2 O3 ladders [149].

to a quasi-one-dimensional exchange that suppresses magnetic order [2]. To soften the above given statement it should be remarked that the compound Sr 14−x Cax Cu24 O41 which is the only example of a superconducting low-dimensional Sr cuprate also shows NQeel ordering. However, the transition temperature is small and the ordered moments are observed on the Cu sites in the chain sublattice. Further details will be discussed in Section 2.4.2. Other important examples of Sr cuprates are the two-leg ladder SrCu2 O3 , the three-leg ladder Sr 2 Cu3 O5 [152], the zig-zag chain SrCuO2 (corner-sharing double chain) and the linear chain system Sr 2 CuO3 [153,154]. In angle-resolved photoemission experiments on the zig-zag chain SrCuO2 [155] and later on the linear chain system Sr 2 CuO3 [156] spin–charge separation into spinon and holon branches have been observed. In the linear chain system Sr 2 CuO3 a picosecond switching of the optical absorption has been observed and related to the genuine electronic properties of the 1D Mott insulator [157]. It has not been possible to induce free carriers in bulk samples of these compounds. On the other hand, they play an important role as model systems [151,158]. In contrast, using a 1eld-eKect device with an insulating Al2 O3 oxide barrier on a thin 1lm, doping and superconductivity has been achieved near the interface with a CuO2 in1nite-layer [159] and a Cu2 O3 ladder system [160]. In the case of the in1nite-layer system, the electric-1eld induced doping enabled hole and electron conductivity with a respective maximum superconducting transition temperatures of Tch−max =89 K and Tce−max = 34 K just by tuning the gate voltage of the device [159]. In the ladder system [CaCu2 O3 ]4 1eld-induced hole doping led to a transition temperature Tch = 14 K, similar to the pressure-induced superconductivity in the spin ladder system (Sr; Ca)14 Cu24 O41 . The properties of this compound will be discussed in detail in Section 2.4.2. 2.4.1. The two-leg ladder SrCu2 O3 The two-leg ladder system SrCu2 O3 is an extensively studied cuprate. For this exchange topology a singlet ground state and a large (≈ 0:5 J) gap to triplet excitations is expected [13,15]. Unfortunately,

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this compound is only available as powder samples. Therefore, most system parameters show a quite large variation in the published literature. The exchange coupling constants parallel and perpendicular to the ladder direction are J = 850–2000 K and J⊥ = 750–1000 K [154]. The existence of a spin gap with 01 = 280–400 K has been veri1ed (380 K in neutron scattering) [154,161,162]. In following experimental [43,163,164] as well as theoretical studies [37,38,40–42] the inOuence of substitutions on the gap and the spin–spin correlations has been investigated. Antiferromagnetic 3D long-range order with a maximum TN =8 K for x = 0:04 has been observed in Sr(Cu1−x ; Znx )2 O3 . Even for very small Zn and Ni substitution levels, e.g., for x = 0:01, this order is still well-de1ned [164,165]. Neutron scattering investigations on the singlet–triplet gap have shown that its size is independent of the Zn substitution x. The spectral weight of the singlet–triplet transitions decreases monotonically with x [163]. This observation is compatible with the existence of a 1nite in-gap density of states which increases with x and also corresponds to the observed linear contribution to the speci1c heat [165]. For larger substitution levels (x = 0:04) almost all singlets are destroyed. Therefore, the spin gap closes. A simple estimation using the minimum k value of a 1nite chain and the des Cloizeaux– Pearson dispersion of triplets [5] leads to the critical concentration xcr = 20 = 2 J as function of the unrenormalized spin gap and the exchange coupling J [166]. NMR investigations have shown that an impurity induces a localized magnetic moment at the impurity site and small staggered moments in the entire region near the impurity [43]. These experimental observations con1rm the expected changes of the excitations scheme of a spin ladder in the presence of spin vacancies. 2.4.2. The chain/ladder system (Sr,Ca)14 Cu24 O41 The compound Sr 14−x Cax Cu24 O41 is a complicated system with both CuO2 chains and Cu2 O3 ladders. A substitution of Sr by the isovalent Ca as well as applied pressure lead to a transfer of holes from the chains to the ladders and their delocalization. Superconductivity under pressure is observed with a maximum transition temperature of Tc = 12 K in the doped system [150,168]. Structural considerations and Ca substitution. The crystal structure of this compound is shown in Fig. 2.20. This almost commensurate system is built up by sublattices of chains and ladders in the ac-plane. The sublattices are given by the dashed rectangles in Fig. 2.21. The c-axis lattice parameter W With isovalent Ca substitution corresponds approximately to clattice ≈ 10 · cchain ≈ 7 · cladder 27 A. and with applied pressure the b- and c-axis parameters decrease leading to strong changes of the electronic properties, e.g., a reduction of the anisotropy in the resistivity [169]. In Fig. 2.22 the resistivity of a single crystal with x = 11:5 is shown as a function of temperature and pressure. Increasing pressure the anisotropy of the resistivity #a =#c at T = 50 K decreases from 80 (P = 0) to 10 (P = 4:5 GPa) [169]. Magnetic properties. The static magnetic properties of Sr 14 Cu24 O41 for T ¡ 300 K are dominated by the chain system as the second gap originating from the ladder sublattice is much larger. Using magnetic susceptibility a singlet–triplet gap chain =140 K has been determined [170]. The gap in the ladder has been investigated using NMR, neutron and Raman scattering. The respective gap values are ladder = 550 K in NMR or ladder = 375 K in neutron scattering [154,171]. The Raman scattering data support the neutron scattering results [60]. A similar disparate variation of the gap values from NMR and neutron as well as Raman scattering is found for the dependence of the ladder gap on Ca substitution. While in NMR experiments a strong

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Fig. 2.21. Structure of (a) the chain and (b) the ladder sublattice in (Sr; Ca)14 Cu24 O41 in the ac-plane and with the c-axis lattice parameters cchain and cladder indicated. (c) and (d) depict two models for charge ordering in the CuO2 chains as described in the text [167]. Squares in (c) denote dimers due to Zhang–Rice singlets and double squares in (d) denote extended singlets. Fig. 2.22. EKect of pressure on the resistivity of Sr 14−x Cax Cu24 O41 with x = 11:5 (a) along the ladder direction (#c ) and (b) perpendicular to the ladder (#a ). The inset shows the pressure dependence of Tc [169].

decrease of the gap on substitution from ladder = 550 to 270 K (x = 11:5) has been observed [154], the corresponding neutron experiments have not indicated any change [171]. A similar behavior is found in Raman scattering [60]. Fits to the triplet dispersions measured by neutron scattering have been used to determine the exchange coupling constants in the ladder. They are J = 1500 K, J⊥ = 830 K (J = 1040 K, J⊥ = 750 K) for x = 0 (x = 11:5) [171,172]. Optical spectroscopy. In measurements of the optical conductivity the substitution of Ca for Sr leads to a rapid transfer of the spectral weight from the high to the low energy region. This behavior is similar to observations in HTSC with increasing hole concentration [173]. In Sr 14−x Cax Cu24 O41 it is the consequence of a charge redistribution from the chains to the ladders as the substitution is isovalent. The transferred spectral weight has been used to estimate the hole density in each structural element. This is given in the inset of Figs. 2.23 and 2.24. A later study using polarization-dependent X-ray absorption spectroscopy found a less pronounced charge transfer with substitution and a signi1cant contribution from orbitals on the rungs of the ladders [176].

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Fig. 2.23. The eKective electron number per Cu (left-hand scale) of Sr 14−x Ax Cu24 O41 determined from an optical conductivity sum rule is plotted as a function of energy at T = 300 K. NeK per ladder Cu is indicated on the right-hand scale. The valencies of Cu on both chain and ladder sites are given in the inset [173]. Fig. 2.24. The a-axis optical conductivity of %a (!) Sr 14−x Cax Cu24 O41 is shown at various temperatures for x = 8 (upper panel) and x = 11 (lower panel). In the inset the low-frequency spectra are shown [174].

In recent experiments the observation of a “pseudo-gap” and a collective mode in the optical conductivity of Sr 14−x Cax Cu24 O41 with x = 8–11 has been claimed [174]. This is based on the diminishing of spectral weight for frequencies below 600 cm−1 (x = 11) and 1100 cm−1 (x = 8) with decreasing temperature for polarization parallel to the a-axis. The term “pseudo-gap” has been used for this phenomenon as it resembles the optical conductivity of underdoped HTSC [130]. However, there are also important diKerences. The energy of the pseudo-gap in HTSC does not strongly depend on doping. Furthermore, the derived gap energies for the chain/ladder compound do neither correspond in their magnitude nor in their dependence on Ca substitution to the neutron scattering results. Therefore, and as a gap of primary spin character is observed here in the charge degrees of freedom, conclusions based on IR absorption data are not unambiguous. In addition, optical conductivity in c-axis polarization shows a doping-dependent peak at 50 cm−1 (x=11) and 100 cm−1 (x=8) [174]. FIR investigations on samples with x=0 identi1ed a pronounced frequency dependence in the optical conductivity forming a peak at 2 cm−1 for temperatures up to 150 K [177]. This peak may be connected with slowly Ouctuating charge dynamics or a charge density wave instability.

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27

Fig. 2.25. Raman light scattering spectra of Sr 14 Cu24 O41 in intraladder (cc) polarization at 10 K (thin line) and 350 K (thick line). The gap features, low temperature modes, and the frequency shift of the phonon at 312 cm−1 that is sensitive to charge ordering are marked by arrows. The spectra are corrected for the Bose factor and show the imaginary part of the susceptibility [60,175].

In Raman scattering experiments on non-substituted Sr 14 Cu24 O41 , shown in Fig. 2.25, a decrease of the scattering intensity below 200 cm−1 and an enhancement of a shoulder near 490 cm−1 have been observed upon cooling [60,175]. This renormalization of scattering intensity for frequencies smaller than 200 cm−1 (290 K) and 490 cm−1 (700 K) corresponds to 2chain and 2ladder . These values are in good agreement with the above-discussed neutron scattering and susceptibility data. Furthermore, charge ordering in the chain system is evident as a shift of a phonon mode from 306 to 312 cm−1 and the appearance of additional side bands at lower energy for T = 10 K. These observations will be discussed further below. In Fig. 2.26 it is shown that with Ca substitution the gap feature of the ladder system near 490 cm−1 is smeared out but does not vanish or decrease in energy. In contrast, the smaller gap in the chain sublattice near 200 cm−1 is destroyed for x ¿ 5. This supports the negligible substitution dependence of the spin gap in the ladder system investigated in neutron scattering. The anomalies of the phonon mode at 312 cm−1 related to charge ordering in the chains are completely suppressed on substitution. In addition, a very low-frequency mode is observed near 14 cm−1 at T =10 K that disappears in the substituted samples. Instead, quasielastic scattering is observed. The origin of this signal is unclear and it may be related to an additional modulation of the charge density. Additional medium energy modes are induced for lower temperatures in the frequency range 350 –400 cm−1 corresponding to 1:5ladder . They are marked by arrows in Fig. 2.25 [60]. The intensity of these modes vanishes with increasing temperature with a power law and faster than the intensity of the gap-related shoulder at 2ladder [175]. No magnetic 1eld dependence of the above described features has been observed in 1elds up to 6 T. A room temperature investigation of the scattering intensity at higher frequencies

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Sr14-x Ca xCu 24 O41

Intensity (arb. un its)

(cc)

T = 10 K

x=0 x=2 x=5 x = 12

100

200

300

400

500

600

Raman shift (cm-1)

Fig. 2.26. Raman light scattering spectra of Ca-substituted Sr 14−x Cax Cu24 O41 with x =0–12 in intraladder (cc) polarization. The dashed lines are guides to the eye. The background of the scattering intensity is indicated by a dotted line [60].

and in dependence of La and Y substitution may be found in Ref. [178]. In these experiments additional modes have been proposed to originate from IR-active phonons. The high-energy regime of the Raman spectra in Sr 14−x−y Cax Yy Cu24 O41 has intensively been investigated and analyzed by several groups as function of substitutions and in a resonance study. Important parts of the corresponding results will be discussed in Section 3.5. Early studies found a very sharp peak in Sr 14 Cu24 O41 in (cc) polarization at 3003 cm−1 with a high-energy shoulder forming two broader maxima at 3700 and 4300 cm−1 [179,180]. A similar maximum in (aa) polarization is weaker in intensity and the high-energy peaks are missing. From these peak positions exchange coupling constants J = 1000 cm−1 and J⊥ = 950 cm−1 of the ladder system have been estimated. These experimental data have been theoretically reanalyzed by Freitas et al. [181,182] and Schmidt et al. [77] concerning coupling constants and higher order Raman scattering eKects. In Fig. 3.7 some of the results for the undoped La6 Ca8 Cu24 O41 are compiled together with model calculations [77]. It has very early been highlighted that the two-magnon signal shows an anomalously small linewidth at low temperatures, i.e. only about 1/10 of the corresponding linewidth of roughly 1000 cm−1 in 2D systems [179,180]. Comparing these results with the undoped compound La6 Ca8 Cu24 O41 it is clear that the small lineshape must somehow be related to the doping of the system. This observation is in some objection to the localization of the holes on the spin chain sub-system at ambient pressure and the given substitution levels. Recently the two-magnon scattering has been reinvestigated and con1rmed, showing that the intensity and lineshape of the two-magnon peak is strongly dependent on the energy of the incident light [183]. These resonance eKects are proposed to result from the proximity of the charge-transfer gap to the incident energy. A 1nal understanding of the small linewidth and related phenomena is still missing.

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

29

Properties of the chain system. While even in Sr 14 Cu24 O41 the Cu valency is +2.25 the charge carriers are localized in the chain sublattice. The compound is an insulator [185]. This localization is connected with a dimerization and charge ordering in the chains. Corresponding superstructure peaks are observed in X-ray scattering. Their intensities increase for temperatures below 50 K and completely vanish only for temperatures up to room temperature [167]. The corresponding dimers are formed in the chains between the copper spins that are separated by 2 times the nearest neighbor Cu–Cu distance between the nearest neighbor copper ions, while the interdimers separation is 4 times that distance. The dimerization is discussed to originate from the formation of ordered Zhang–Rice singlets (see Fig. 2.21(c) or (d)) [167]. Here, the squares and double squares denote a Zhang–Rice singlet (Cu3+ ) and an extended singlet, respectively. In NMR experiments the existence of both Cu2+ and Cu3+ in the chains has been veri1ed [53]. The signal of the Cu3+ state splits into two peaks for T ¡ 200 K. In neutron scattering investigations the observed triplet dispersion along the c-axis is described by an antiferromagnetic intradimer coupling Jnnn = 134 K between next nearest neighbor Cu ions and a ferromagnetic interdimer coupling J2 = −13:2 K [172]. A second neutron study found two triplet branches of the chains system, an acoustic and an optic branch. This leads to an intradimer coupling as given above, an interdimer coupling along the c-axis Jc = 9 K, and an interdimer coupling along the a-axis Ja = 8:6 K. The dimer con1guration indicates a quasi-2D hole ordering of the states with s = 1=2 and s = 0. The latter s = 0 spin states are due to Zhang–Rice singlets [186]. The charge ordering-induced phonon anomalies in Sr 14 Cu24 O41 can easily be correlated with the gap formation in the chain subcell as both quantities are determined simultaneously by a Raman scattering experiment. In Figs. 2.25 and 2.26 it is evident that in the energy regime of 250 –350 cm−1 modes shift with temperature and form additional satellite peaks. This temperature dependence can precisely be mapped on the temperature dependence of superstructure peak intensities from X-ray scattering experiments [167]. The phonon frequencies are therefore related to the order parameter of charge ordering in the chain system. Fig. 2.27 shows the evolution of the A1 mode at 304 cm−1 that shifts to 312 cm−1 at lower temperatures with the formation of additional split-oK modes at 294 and 303 cm−1 . Noteworthy is the non-monotonous change of frequency with the largest slope at approximately T ≈ 150 K for Sr 14 Cu24 O41 . This mode has recently be assigned to an O-vibration of the chain along the diagonal of the ab plane [187]. Comparing these shifts with respective results W at T = 150 K can be on substituted samples a coherence length of charge ordering of only 27 A estimated using the averaged distance between the substitution sites in the chain. Following a simple model developed for similar phonon anomalies in NaV2 O5 due to the Coulomb energy contribution to the lattice force constant [188], see Section 2.52, the shift of the phonon frequency is given by a Coulomb contribution cou leading to * = (+ + cou )=M . Additional modes may be activated due to a back-folding of phonon branches by the 5-time increase of the chain unit cell [186]. On the other hand, supposing that the mode at 304 cm−1 is an in-phase vibration, then in the case of charge asymmetry, the normal coordinates are linear combinations of the in-phase and out-of-phase vibrations, and therefore new “in phase” modes appear in the Raman spectra. If this mixing is strong enough, each of the new modes can be interpreted as a vibration of one Cu ion. In NaV2 O5 such a mixing of in-phase and out-of-phase modes has been identi1ed to explain the observed phonon anomalies due to charge ordering and charge transfer [189,188]. The gradual increase and smooth crossover of the phonon frequency displayed in Fig. 2.27 points towards short range Ouctuations typical for charge ordering in quasi-1D systems. It has not only been

30

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Fig. 2.27. Enlarged view on the phonon anomaly at 312 cm−1 in Sr 14 Cu24 O41 in interchain (aa) polarization (left panel). The right panel shows the phonon frequency as function of temperature for a pure (x = y = 0) sample as well as for Ca (x = 0) and Zn (y = 0) substituted samples. Crossover temperatures (T ∗ ) are indicated by dashed lines [184].

observed in the present compound and in NaV2 O5 but also in the stripe ordering of charge and spin in La2 NiO4+ [188–190]. The phonon frequency in Sr 14 Cu24 O41 shows a sudden increase of slope at a certain temperature (T ≈ 200 K). This eKect must be related to an increase of the coherence length and induced by an additional interchain or magnetoelastic coupling. These details have not been resolved in the X-ray experiments as incoherent scattering is lost. Therefore, the phonon-probe is more sensitive to local charge ordering [167]. The eKect of diKerent substitutions on the phonon mode at 312 cm−1 is presented in Fig. 2.28. While Ca substitution suppresses the appearance of the split-oK modes quite fast, i.e. already at a Ca content of x = 2 these modes are strongly reduced in intensity, Zn and Ni substitution on the Cu site do not suppress the split-oK modes at all. This is based on the charge transfer from the chains to the ladders induced by Ca substitution. This charge transfer suppresses the charge ordering and the dimerization of the spins in the chain subsystem. Thereby also the evolution of the spin gap in the chain system is suspended. However, even at high Ca substitutions x ¿ 5 still a frequency shift of the phonon is observable while the spin gap is not detectable anymore (right panels of Fig. 2.28). This is related to the diKerent coherence lengths of the spin and the lattice system. The non-linear dependence of the spin gap in the chain system on the charge ordering/transfer estimated by the integrated and normalized phonon shift demonstrates clearly that this gap formation needs a well-established coherence of charge ordering. EAect of pressure and the phase diagram. Under applied pressure holes are transferred from the chain to the ladder system [154,170]. The carriers are delocalized with further pressure. The

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

31

Fig. 2.28. EKect of Ca, Zn and Ni substitution on the phonon anomaly at 312 cm−1 in Sr 14 Cu24 O41 (left panel). In the right panels the temperature-induced integrated frequency shift of the phonon at 312 cm−1 is taken as a measure of the charge ordering and related to the spin gap of the chain system [184].

eKects observed in resistivity shown in Fig. 2.22 are discussed as a dimensional crossover to a two-dimensional system [169]. Superconductivity appears in Sr 0:4 Ca13:6 Cu24 O41 at Tc = 12 K and an applied pressure of P =3:5 GPa [150]. NMR experiments on a sample with x =12 indicate a collapse of the gap to a “pseudo-spin gap” under pressure [191] as shown in Fig. 2.29. This is meant to describe a reduced but not completely diminished density of states. In contrast, preliminary neutron scattering experiments under pressure do not support this result [192]. The diKerent behavior of the ladder spin gap observed in neutron scattering and NMR may be related to the fact that the former method probes a coherent singlet–triplet transition while the latter samples low-energy relaxations via partly localized states in the gap. The experimental results therefore indicate a more complex low-energy spectrum than resolved by neutron scattering. This spectrum may contain magnetic bound states [60] and collective modes [174] as evidenced in optical experiments. However, a limited amount of sample inhomogenities that would not show up in neutron scattering experiments may also explain the inconsistencies. Recently an additional AF order (10−1 –10−2 !B ) of Cu moments on the chain and ladder site has been found in Ca substituted (Sr; Ca)14 Cu24 O41 . Long range ordering is also observed in (Ca; La)14 Cu24 O41 that shows no superconductivity under pressure. The NQeel temperature is TN = 2 and 10 K for (Sr; Ca)14 Cu24 O41 and (Ca; La)14 Cu24 O41 , respectively. In the former compound TN increases strongly for further Ca substitution x ¿ 12:5. Therefore, in Sr 14−x Cax Cu24 O41 the two phases of superconductivity and antiferromagnetism are separated only by pressure [193,194]. Recent neutron scattering and !SR experiments on samples with x = 11:5 and ambient pressure observe a coexistence of the singlet state and antiferromagnetism with an inhomogeneous distribution of the internal 1elds [195].

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Relaxation rate (10 3 s-1 )

10

0

Sr14-x Ca xCu 24 O41

x=12

30 kbar 10

-1

αΤ 1 bar 10 -2

∆ =350 Κ 10

100

Temperature (K)

Fig. 2.29. EKect of pressure on the NMR relaxation rate and the spin gap in Sr 14−x Cax Cu24 O41 with x = 12. The measurements at ambient pressure (open circles) show a 1t (full line) expected for a spin gap of the ladder with ladder = 350 K. For a pressure of 30 kbar (full circles) a weaker temperature dependence is observed (a power law is shown as a dashed curve) [191].

The above discussed investigations are very important as they try to clarify whether superconductivity in Sr 14−x Cax Cu24 O41 is induced in a low-dimensional gapped spin liquid state or whether applying pressure “simply” increases the dimensionality. In the latter case this compound would better be characterized as a quasi-two-dimensional system. This approach is somewhat supported by estimates of the upper critical 1eld Hc2 that show only a weak anisotropy in the ladder plane [196]. From the present experimental status, however, the scenario of a doped spin liquid should be favored. Recently a further spin ladder compound, -Cu0:66 V2 O5 , showing evidence of superconductivity under pressure has been found [197]. This compound and related low-dimensional vanadates will be discussed in the following paragraph. 2.5. Low-dimensional vanadates There exist an even larger number of low-dimensional vanadium–oxygen compounds than of cuprates as more diverse coordinations and a broader composition range are accessible [67]. Many systems show eKects of strong electronic correlations, e.g., charge and/or orbital ordering, Mott metal–insulator transitions and spin density wave ground states. Important parameters of the phase diagrams are temperature, magnetic 1eld, composition and pressure [198]. Recently vanadium compounds have been systematically searched for interesting low-dimensional spin systems [199]. Vanadates allow diKerent coordinations with oxygen going from VO4 -tetrahedra with V5+ to VO5 -pyramids, where both V4+ and V5+ valence states are observed, to VO6 -octahedra with distinct V3+ ions. Especially for the pyramidal coordination small, seemingly negligible changes of the local coordination may lead to drastic changes of the electronic state of vanadium, i.e. a valence change from a non-magnetic V5+ to the s = 1=2 state V4+ . This opens up interesting aspects of correlated charge ordering and lattice instabilities as, e.g., realized in the quarter-1lled spin ladder compound

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

33

-NaV2 O5 . The above given geometrical elements can be arranged to form chains, ladders or more complicated 2D arrangements, leading to a multitude of possible vanadium oxides [67]. 2.5.1. The AV2 O5 family of compounds The system AV2 O5 , with A = Li, Na, Cs, Mg and Ca is a family of V compounds that covers a broad range of low-dimensional systems such as spin chains and ladders. The compound CaV2 O5 is described best as a two-leg spin ladder [99,200]. The resulting singlet ground state has a spin gap of 01 =500 K. The compound MgV2 O5 represents a two-leg spin ladder with an additional strongly frustrated coupling between the ladders [201,202]. The respective gap 01 = 15 K is much smaller. The compound CsV2 O5 has static dimers each formed by two V4+ O5 edge-sharing square pyramids and a gap 01 = 160 K [83,199]. In contrast, LiV2 O5 is a quasi-1D anisotropic-chain system [203] without a spin gap [200,83,204]. The quarter-1lled two-leg spin ladder -NaV2 O5 attracted considerable interest from both experimental as well as theoretical side [205–208]. This compound is one of 1ve existing Nax V2 O5 phases at ambient conditions. In the following it is denoted as NaV2 O5 as its earlier determined room temperature non-centrosymmetric structure Pmn21 proved to be wrong [205]. On the contrary, the centrosymmetric structure Pmnm is presently used [209]. NaV2 O5 shows a singlet formation with 01 = 85 K for T ¡ TSP = 34 K accompanied by a lattice instability. It is therefore often assigned to spin–Peierls materials. It will be discussed in more detail in Section 2.7.2. The correlation between structural parameters, the exchange coupling constants and the exchange topology of the AV2 O5 family is described in Refs. [199,202]. The related beta-bronzes -A0:33 V2 O5 with A=Na, Ag, Li as well as with A=Sr and Cu that have been investigated in the 1980s in the context of bipolaron formation have recently found a renewed interest [210–212]. The crystal structure of these compounds contains three diKerent V sites (two zig-zag chain sites and one ladder site). In addition, there exist “channels” containing the mono- or divalent cations. For the series with monovalent cations A = Na, Ag and Li charge ordering (TCO = 136, 90 and 180 K) of V4+ : s=1=2 on one chain site is observed [210] at low temperatures. This charge ordering is extremely sensitive to the cation stoichiometry. All other V-sites remain non-magnetic (V5+ ). For even lower temperatures NQeel ordering takes place (TN = 24, 27 and 7 K). Transport experiments at room temperature show a quasi-1D metallic behavior. The charge ordering transitions are observed as kinks and an abrupt increase of the resistivity. Evidence for an additional structural instability exists for A = Na at approximately Tc = 250 K. In this latter system superconductivity under hydrostatic pressure of P = 7 GPa has been observed with a maximum transition temperature of Tc = 9 K [197]. The beta-bronzes with divalent cations are even more spectacular since for A = Sr, Ca and Cu charge ordering of V4+ : s = 1=2 is expected on the ladder sites and experimentally evidenced by NMR experiments [213,210]. Therefore in these compounds a doped spin ladder is realized. In -Cu0:66 V2 O5 transport experiments under pressure (P=4:5 GPa) show evidence of superconductivity with Tc =5 K [197]. In this compound Cu1+ , that partly occupies several distinct sites in the channels of the crystal structure, plays the role of a non-magnetic dopant of the system. The structural ambiguity of the Cu site complicates the determination of the electronic structure and the eKect of charge transfer from the channel to the ladder sites. Further experimental and theoretical studies on these compounds are underway.

34

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 (a)

(b)

Fig. 2.30. Projected view on the crystal structure of (VO)2 P2 O7 in c-axis and a-axis direction showing (a) the ladder and (b) the alternating chain, respectively. V ions are shown in grey, P in light gray and O in black [214].

2.5.2. The alternating chain system (VO)2 P2 O7 In Fig. 2.30 the crystal structure of (VO)2 P2 O7 (vanadyl pyrophosphate) is shown. This compound consists of chains along the b-axis direction with a double V–O–P–O–V link through phosphate groups alternating with a V–O–V link between edge-sharing VO5 square pyramids. In a-axis direction a ladder with double V–O–V legs is formed. The magnetic susceptibility of this compound shows a maximum at 74 K and a rapid decrease at lower temperatures. Together with neutron scattering experiments on powder samples low-dimensional spin Ouctuations, a singlet ground state and a gap in the excitation spectrum have been deduced [215]. However, since it was not possible to resolve the dominant exchange path direction the structural element along the a-axis lead to the erroneous interpretation of (VO)2 P2 O7 as a spin ladder compound [15,216]. Recently, neutron scattering experiments on a batch of aligned single crystals proved the b-axis direction to be the dominant exchange path [217–219]. The triplet dispersion is strongest in this direction. This leads to the assignment of (VO)2 P2 O7 as a chain system with a strong static alternation of the exchange coupling. Perpendicular to the chains a negligible dispersion with k along

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

(0,-2,ξ)

(η,-2,0)

15

35

(0,ζ-2,0)

Energy (meV)

(VO)2 P2 O7 10

5

0 1

1/2 η

0

0

1/2 ξ

1

0

1/2 ζ

1

Fig. 2.31. Triplet dispersion in (VO)2 P2 O7 determined by neutron scattering experiments. The 1rst and the second triplet branch with 01 = 37 K and 01 = 69 K are shown with 1lled and open symbols, respectively [217].

the c-axis and a weak ferromagnetic dispersion along the a-axis, i.e. parallel to the ladders, are observed. A spin wave 1t to the triplet dispersion results in exchange coupling constants of J1 = 120 K and J2 = 100 K along the chains (b-axis direction) and a singlet–triplet excitation gap of 01 = 37 K. The zone-boundary energy of this branch is EZB = 180 K [220]. The larger coupling (J1 ) is realized via the phosphate groups. The static alternation = (J1 − J2 )=(J1 + J2 ) = 0:09 is much larger compared to spin-Peierls compounds where values of only = 0:015– 0.05 are reported for T ¡ TSP . ESR investigations on (VO)2 P2 O7 later supported by NMR point to even larger values of =0:2 [220,221]. Therefore, the understanding of the properties of (VO)2 P2 O7 in comparison with spin-Peierls systems is important. Theories of dimerized spin systems using numerical approaches are often more practical for larger alternations [29]. For these calculations (VO)2 P2 O7 may be used as a model system. The main peculiarity of the excitation spectrum of (VO)2 P2 O7 is the observation of a second triplet branch 01 = 69 K in neutron scattering experiments with a comparable dispersion as the 1rst lower-energy triplet branch. This dispersion is shown in Fig. 2.31. The second triplet branch has been assigned tentatively to a triplet bound state of a strongly alternating chain [217]. This would have been the 1rst experimental observation of such a signal reported in literature. Later experimental investigations, however, found evidence for two chain systems with diKerent alternations within one unit cell of this compound. This will be discussed further below.

36

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 700

(VO)2 P2 O7

600

c(bb )-c 500

10 K 70 K 100 K 150 K 200 K 250 K

Im χ

400 300 200 100 0 90

100

110

120

130

-1

Raman shift (cm ) Fig. 2.32. Phonon anomalies of (VO)2 P2 O7 in intrachain (bb) light scattering polarization [214].

The second branch has considerable intensity over the whole Brillouin zone as shown by neutron scattering. Therefore, the assignment to a triplet bound state has been criticized. Calculations show that the spectral weight of such a bound state should be dominant at the zone boundary and not at the / point of the Brillouin zone. This inconsistency may be solved by an additional interaction, i.e. a more complicated exchange topology [222]. Indeed, an additional frustrating interchain exchange term across the phosphate group can increase the spectral weight of the triplet bound state [31,223]. In a detailed Raman scattering study this problem has been investigated experimentally [214]. Here, no sign for strong frustration has been detected as it should manifest itself in a large binding energy and spectral weight of a singlet bound state. Respective data are discussed in Section 4.3. However, there is evidence for strong spin–phonon coupling showing up as Ouctuation-induced quasielastic scattering and phonon anomalies [214]. A further Raman and IR scattering study concentrating on phonons and the respective symmetry analysis can be found in Ref. [224]. In Fig. 2.32 low-energy Raman spectra are shown. A strong anharmonicity of a phonon of (VO)2 P2 O7 at 123 cm−1 is observed. This mode most probably corresponds to the phosphate group that mediates the exchange along the alternating chain. Its energy coincides with the energy 01 (k) at the zone-boundary EZB = 180 K ≡ 125 cm−1 . The frequency shift of this and a second anomalous phonon mode at 70 cm−1 is shown in Fig. 2.33. The observed softening of the phonon frequency corresponds to a decrease of the corresponding force constant by 20%. In interchain (aa) scattering con1guration the eKects of spin–phonon coupling are similar drastic, leading to a complete vanishing of the phonon intensity of the 123 cm−1 -mode for temperatures T ¿ 80 K (see Fig. 2.34). This temperature is comparable to 201 = 72 K. The strong spin–phonon interaction has been taken as an alternative reason for a large spectral weight of the tentative triplet bound state 01 in the center and at the boundary of the Brillouin zone [129,214].

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

37

Frequency (T) /Frequency (10K)

1.02 1.00

(VO)2 P2 O7 0.98 0.96 70 cm

-1

(cc)

123 cm

-1

(cc)

123 cm

-1

(bb)

0.94 0.92 0.90 0

50

100

150

200

250

300

Temperature (K) Fig. 2.33. Anharmonic temperature dependence of the frequency of the phonons in (VO)2 P2 O7 at 70 and 123 cm−1 in intrachain (bb) and interchain (cc) polarization on a normalized frequency scale [214].

80

123 cm

-1

c(aa)-c

1.0

0.8

Im χ

60

0.6 100 K

50 0.4 40 K

40

norm. int. Intensity

(VO)2 P2 O7

70

0.2 5K 30 115

120

125

130

Raman shift (cm-1)

0

20

40

60

80

100

0.0

Temperature (K)

Fig. 2.34. Phonon anomalies in Raman spectra of (VO)2 P2 O7 in interchain (aa) polarization in dependence of temperature (left panel). The intensity of the mode at 123 cm−1 is shown in the right panel as a function of temperature. The dashed line is a guide to the eye [214].

Strong spin–phonon coupling is also observed as Ouctuations of the energy density of the spin system (see Section 5). These are detected as strong quasielastic light scattering and have been used to determine the magnetic contribution to the speci1c heat. A comparison of these results for VOPO with a DMRG calculation [225] leads to a quantitative agreement supporting strongly the alternating chain model with strong spin–phonon coupling for (VO)2 P2 O7 .

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Table 2.1 Spin gaps and intrachain exchange interactions of two spin subsystems A and B in (VO)2 P2 O7 Spin subsystem

(K)

J1 (K)

J2 (K)

Chain A Chain B

68 35

136 124

92 103

0.19 0.09

The data are derived from 31 P and 51 V NMR experiments. The data of the subsystem B correspond to the parameters derived from neutron scattering experiments as explained in the text [220].

As discussed in Section 2.3.2 the existence of a high symmetry phase at elevated pressure and/or temperature may also play an important role in inducing anharmonicity and enhancing spin–phonon coupling. Phonon anomalies in SrCu2 (BO3 )2 have been undoubtedly attributed to a structural phase transition into a centrosymmetric phase for T ¿ Tc = 395 K. In (VO)2 P2 O7 a transition into a high pressure phase is observed at a pressure of P = 3 GPa and temperatures between 400 and 500 K [127,128]. This transition is certainly related to the observed anomalies. In this scenario it is understandable that the high-pressure phase of (VO)2 P2 O7 does not show evidence for phonon anomalies at low temperatures [226]. The triplet bound state tentatively observed in neutron scattering motivated a detailed NMR study [220] that raised doubts concerning this interpretation. The temperature dependence of 31 P and the 51 V NMR shows two independent components leading to two diKerent spin gaps. The gaps are determined to be A = 68 K and B = 35 K. These data are in excellent agreement with the branches observed in neutron scattering. See Table 2.1 for important parameters that have been derived from these measurements. The NMR data are understandable taking two independent magnetic subsystems with distinct propagating triplet branches into account. The doubling of the gap in the second spin system with respect to the 1rst due to a diKerent chain alternation is therefore proposed to be accidental. The spin subsystems are associated with two crystallographically inequivalent V chains along the b-axis. The inequivalence is proposed to be based on a very small diKerence in the V–O–V distances W [220]. The two branches observed in neutron scattering are therefore the result of a (Yd = 0:02 A) large unit cell (with eight formula units and four chains) with two chain systems, A and B, diKering in the alternation of their exchange coupling (see Table 2.1). It is important to notice that the weak ferromagnetic coupling in a-axis direction acts between the same kind of chains. Therefore both chain systems stay decoupled. Furthermore, the larger alternation ( = 0:19) is identical to the value reported in ESR experiments [221]. 2.6. Low-dimensional halides and pnictides In contrast to oxides, many low-dimensional halides show more or less distinct spin–orbit coupling that enforces the description of their magnetic subsystem as Ising or XXZ-models with either easy-axis or easy-plane anisotropies. We will discuss here only compounds with negligible spin anisotropies. Examples are KCuF3 , the ACuCl3 family of compounds, the Haldane system CsNiCl3 and a rare earth compound, Yb4 As3 .

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

(a)

39

(b)

Fig. 2.35. Crystal structure and spin topology of ACuCl3 , with A = Tl, K: (a) projection of the crystal structure along the a-axis, (b) exchange coupling constants of the three-dimensional dimer topology [232].

2.6.1. The chain system KCuF3 The compound KCuF3 represents a one-dimensional Heisenberg antiferromagnet in the sense of small interchain exchange interactions and negligible spin anisotropies. In spite of the large intrachain exchange coupling J = 196 K, 3D NQeel ordering is observed only at TN = 39 K. The speci1c heat of the spin system shows a maximum at T = 180 K due to one-dimensional spin Ouctuations [227]. This compound played an important role in the 1rst spectroscopic investigations of the spinon continuum using neutron scattering [228,229]. Experiments were successfully compared to the M'uller ansatz, an analytic solution for the two-spinon contribution of the Bethe solution [230]. If no additional interaction, e.g. next nearest neighbor coupling, exists the 1D Heisenberg Hamiltonian commutes with the Raman exchange light scattering operator. Therefore, in light scattering experiments, in contrast to, e.g. CuGeO3 , no spinon continuum has been observed. Instead, intrachain polarized quasielastic light scattering has been detected with an intensity I (T ) which strongly increases with temperature. This contribution can be described by a correlation function of the magnetic energy density and its Ouctuations. (See Section 5 for a detailed discussion.) The renormalized intensity I (T )=T 2 has successfully been compared with the magnetic contribution to the speci1c heat determined by birefringence measurements [227]. In more recent neutron scattering experiments evidence for a longitudinal magnon has been found [231]. This mode is observed in the antiferromagnetically ordered phase of the spin-1/2 quasi-one-dimensional system and signals a crossover from one-dimensional to three-dimensional behavior. Furthermore it indicates a reduction in the ordered spin moment of the spin-1/2 antiferromagnet that has successfully been compared to quantum 1eld theory calculations. 2.6.2. The ACuCl3 family of compounds The copper halides KCuCl3 ; TlCuCl3 and NH4 CuCl3 share a monoclinic structure with the space group P21 =c. The exchange topology consists of double chains of edge-sharing CuCl6 octahedra running parallel to the crystallographic a-axis, which are separated by the cations as shown in Fig. 2.35. The magnetic system has earlier been modelled using 3 coupling constants, describing the intrachain

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

and a frustrating interchain coupling of the system [234]. This chain has been mapped on the exchange topology of a strongly dimerized spin chain with next nearest neighbor exchange interaction [235,236]. Recently, the former two compounds are discussed within a 3D coupled spin dimer model using 6 exchange coupling constants [237–240,232]. While KCuCl3 and TlCuCl3 de1nitely have a singlet ground state and a spin gap as expected for a dimer system, the appropriate exchange topology and the low-energy model Hamiltonian for NH4 CuCl3 are still under debate. In KCuCl3 the triplet dispersion for k parallel and perpendicular to the chains is comparable to the smallest gap 01 (k) = 32 K of the triplet branch at k = (1; 0; 1) [237,238,241,242]. The intradimer interaction has been determined to be J ≈ 48 K. The dimers themselves are weakly 3D coupled with the largest interdimer coupling given by J⊥ ≈ 5:2 K. The magnetic susceptibility shows a maximum at Tmax = 30 K and a pronounced drop toward lower temperature. This dependence has been 1tted √ successfully for T ¡ 15 K to (T ) ≈ (1= T ) · exp(−=kB T ), with = 35 K [234]. This expression has earlier been derived for antiferromagnetically coupled spin ladders [243]. From recent high 1eld magnetization data coupling constant that are a factor 0.8– 0.9 smaller compared to the above given values have been determined for TlCuCl3 [232]. With rising temperature neutron scattering experiments show a Oattening and decrease of bandwidth of the triplet branches while keeping the spin gap approximately constant. In addition, a thermally activated broadening of the triplet branch (lifetime / ≈ exp(−01 =kB T )) and a loss of its intensity is observed for temperatures up to 30 K. These eKects are interpreted as an eKective diminishing of the interdimer coupling as function of temperature due to thermally induced triplet–triplet interaction or due to spin–phonon interaction [244]. In !SR experiments free electron spins are observed for temperatures down to 6 K, followed by a spin-glass-like slowing down of the dynamics at lower temperatures [245,246]. In TlCuCl3 the maximum in the magnetic susceptibility is observed at Tmax = 38 K a value similar to KCuCl3 . However, neutron scattering experiments resolve a smallest gap of only 01 = 7 K at k = (1; 0; 1). An analysis of the triplet dispersion leads to an intradimer interaction J ≈ 60 K while the interdimer exchange coupling constant J⊥ ≈ 18 K is a factor 3 larger compared to the respective coupling constant in KCuCl3 [247,239,240]. The much larger bandwidth of the triplet branches compared with the gap is therefore attributed to a breakdown of the weakly coupled dimer model for TlCuCl3 [239]. Treating the interdimer interactions in mean 1eld, the magnetic susceptibility can be modelled by the expression (T ) ≈ (=(3 + exp(J + J )) for eKective independent dimers, with = 1=(kB T ) and J the sum of the interdimer interactions. Using this expression the ratio J=(kB Tmax ) = 1:6 leads to J + J = 60 K in good agreement with the analysis of the neutron scattering experiments [239]. In light scattering experiments on KCuCl3 and TlCuCl3 magnetic continua with energies comparable to the exchange coupling constants are observed. In addition multiphonon scattering intensities with energies above the single phonon frequency spectrum show up [233]. In Fig. 2.36 respective spectra in TlCuCl3 are shown at 4 and 293 K in (aa) and (cc) polarization. The anomalous scattering intensities are only observed in the interdimer (aa) polarization. With increasing temperature the maximum at 46 cm−1 in TlCuCl3 shifts to higher energy and is strongly reduced in intensity. In KCuCl3 in addition to the maximum at 56 cm−1 a second maximum is observed at elevated temperatures and an energy of 34 cm−1 . For KCuCl3 and TlCuCl3 transitions into gapless 3D ordered magnetic phases are induced by magnetic 1elds as well-de1ned cusp-like minima in the magnetization at Hc = 22:3 and 5:7 T

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

41

Fig. 2.36. Raman spectra of the spin dimer system TlCuCl3 in (aa) and (cc) polarization. A scattering continuum and several multi-phonon modes are observed in (aa) polarization and at low temperatures (full line) [233].

[247,248,232], respectively. For H ¿ Hc the slope of the magnetization m(H ), i.e. the magnetic susceptibility, is 1nite. These transitions do not depend on the orientation of the magnetic 1eld with respect to the crystallographic axes. Furthermore, the phase lines show an upward curvature towards higher 1elds for higher temperature. For both systems the 1eld-induced transition has been described as a Bose–Einstein condensation of dilute magnons [249]. This is based on the scaling of the critical temperature with magnetic 1eld, Tc (h) ˙ |h − hc |1=4 , with 4 = 2:3 and 2.1, respectively. The description of an isotropic Bose–Einstein system with a Hartree–Fock model would lead to 4 ≈ 3=2. In the mixed system (Tl1−x Kx )CuCl3 with exchange bond randomness, the 1eld-induced magnetic ordering is suppressed [240]. Recent NMR and ultrasonic experiments point to an appreciable lattice distortion and 1rst-order-like character of the transition [250]. The sharp peak observed in the ultrasonic attenuation as function of magnetic 1eld or temperature shows a hysteresis [251]. Finally, NH4 CuCl3 is believed to have a magnetic (triplet) ground state in contrast to the singlet ground state of the other two copper halides. This distinction is tentatively attributed to interchain interaction [252]. However, as also supported by the magnetic phase diagram, the triplet ground was proposed to result from the combined eKect of interchain interaction and dimerization [16]. In measurements of speci1c heat evidence for a three-dimensional magnetic ordering with TN = 1:3 K has been found that is suggested to involve only 1=4 of the spins. The remaining spins are proposed to remain in a singlet state. The magnetic entropy at this transition is very small as the magnetic correlations are well-developed at this low temperature [253]. The magnetic phase diagram of NH4 CuCl3 is of special interest as it contains gapped phases induced by a magnetic 1eld. These phases originate from quantum eKects and not from magnetic anisotropies. The magnetization shows a sequence of 1ve well-de1ned transitions (Hc = 5; 12:8; 17:9; 24:7; 29:1 T) and plateaus at m = 1=4; 3=4 and 1, with m per copper site in units of g!B [16,236,252]. For small 1elds and in between the plateaus the magnetization M (H ) increases roughly linear.

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

In contrast, the magnetization of the spin chain system copper benzoate shows a continuously 1eldinduced gap as a transition-like increase of the magnetization. This gap is related to a staggered magnetization that is again attributed to an alternating g-tensor and a DM interaction [254,255]. In this phase magnetic bound states with energies E 6 3 cm−1 are theoretically predicted. Theoretically, the plateaus observed in NH4 CuCl3 and in some other 1D s = 1=2 spin systems correspond to a topologically induced quantization (similar to the quantum hall eKect) denoted as a “1eld-induced Haldane gap” or “triplet–singlet-lattice” phase. They should ful1ll the Oshikawa– Yamanaka–AZeck condition n(s − m) = integer, with n the spatial period of the spin ground state (number of spins per unit cell), s the magnitude of the spin and m the magnetization [256]. Recent theoretical studies have shown that in dependence of the coupling parameters plateaus may be observed for m = 0 and 1=3 in three-leg ladders, for m = 0 and 1=2 in two-leg ladders and in frustrated alternating Heisenberg chains [256–259]. Following these arguments the plateau at m = 1=4 in NH4 CuCl3 cannot correspond to a diluted dimer ordering of a frustrated alternating Heisenberg chain as suggested in Ref. [258]. Here, the proposed division of the spin system into a triplet and a singlet portion as evidenced form speci1c heat data might explain the phase diagram [253]. Recently two other arguments have been used in connection with the complicated behavior of NH4 CuCl3 in a magnetic 1eld: The up to now ignored orientational transition of the NH4 -tetrahedra at Torient = 70 K [260] may induce an additional modulation of the exchange coupling constants or an enlargement of the unit cell. Furthermore, it has been suggested that a weak coupling of two diKerent ladder topologies may be helpful to suppress a plateau at m = 1=2 [261]. 2.6.3. The Haldane system CsNiCl3 The Haldane gap of an integer-spin Heisenberg chain is besides dimerization, frustration and ladder topology another route to a gapped excitation spectrum [71,72]. A Haldane gap has 1rst been observed in CsNiCl3 , which contains s = 1 Ni2+ chains [73,262,263]. This compound has exchange coupling constants of J = 16:6 K and J⊥ = 0:29 K for intra- and inter-chain exchange respectively. These values and a small easy axis anisotropy of D = −0:62 K have been determined for temperatures below TN = 4:4 K using the dispersion of magnon branches in the 3D NQeel ordered phase. The observed singlet–triplet gap 01 = 15:4 K and the 2 -periodicity of the triplet branch in the disordered phase (T ¿ TN : no symmetry of the chain is broken) are solely due to the “hidden order” described by Haldane. 2.6.4. Yb4 As3 In the mixed-valency and semi-metallic rare earth compound Yb4 As3 , charge ordering (TCO = 290 K) leads to s = 1=2 Yb3+ chains that are only weakly coupled. Charge ordering in a quantum spin system is also realized in NaV2 O5 and plays a role in Sr 14 Cu24 O41 . In Yb4 As3 a very large linear speci1c heat contribution has been observed at low temperatures [264]. This contribution is related to fermionic spinon excitations on the chains as the carrier concentration is very small for T ¡ TCO [265]. Indeed, in neutron scattering experiments a gapless spinon dispersion has been observed as expected for a s = 1=2 Heisenberg chain system with J = 40 K [266–268]. It has therefore been emphasized that this compound realizes a “novel route” to Heavy Fermion systems [269].

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The physical picture of Yb4 As3 got another twist when inelastic neutron scattering experiments [270] found a gap to (all) magnetic excitations. This gap opens in the presence of an external magnetic 1eld, con1rming a prediction [271] by Schmidt et al. based on an interpretation of previous speci1c heat data [264]. Several proposals have been made in order to explain this unusual behavior [272]. The 1rst model [273] is based on the observation that a staggered DM interaction, which generates an eKective staggered g-tensor, is allowed [274,275] in this 4f-compound. It is known that a staggered g-tensor leads to a gap in an external 1eld [254,255]. The second model [272], based on a mean-1eld analysis of the anisotropic spin-chain, proposes that a gap opens in the presence of a uniform transversal magnetic 1eld. Raman light scattering experiments on Yb4 As3 show quasielastic scattering for intrachain polarization while in crossed polarization a gap-like decrease of the scattering is observed for Y! ¡ 100 cm−1 [184]. In addition, multiphonon excitation due to spin–phonon coupling exist. These latter results are comparable to observations in the zig-zag chain compound SrCuO2 and in the chain system Sr 2 CuO3 [276]. 2.7. Inorganic systems with spin-Peierls and related instabilities The Peierls instability of one-dimensional metals was introduced by Sir R. Peierls in his textbook “Quantum Theory of Solids” [277]. It is based on the divergence of the electronic susceptibility at k = 2kF and the instability to a periodic distortion of the chain. The resulting opening of an energy gap and insulating ground state overcompensates the energy loss due to the lattice distortion [3]. The spin-Peierls transition of one-dimensional insulating spin systems is the magnetic analogue of the Peierls transition. The gain of energy results from a dimerization of the spin degrees of freedom. The coupling of the spin to the phonon system leads to the formation of a non-magnetic singlet ground state in a dimerized system. The observation of a singlet–triplet gap in the magnetic excitation spectrum as well as a lattice distortion is a critical proof of this ground state. This transition was 1rst investigated in the 1970s in the organic compounds (MEM)(TCNQ)2 [278] or (TTF)(Cu-BDT) [279,280] and later in charge transfer salts [281]. For an introduction to the phenomenology of the spin-Peierls transition and respective compounds we refer to the review article of Bray [282]. The discovery of a spin-Peierls transition at TSP = 14:3 K in CuGeO3 by Hase et al. [283] and of a spin-Peierls like transition (combined with charge ordering) at TSP = 34 K in NaV2 O5 by Isobe et al. [207] opened up a much broader range of experimental methods and investigations. This led to a fruitful extension of this 1eld from both experimental as well as theoretical point of view [284]. The compound CuGeO3 represents a system with considerable relative strength of competing interactions [285,286]. This is due to AF next nearest neighbor exchange interaction along the spin chain (frustration). The later found NaV2 O5 shows a quarter-1lled ladder structure [287] with a dimerization instability that is more complicated due to the interplay of hopping and intersite Coulomb interaction. Both compounds are interesting and important to study. Their properties and magnetic parameters will be discussed in the following. 2.7.1. CuGeO3 The compound CuGeO3 crystallizes in an orthorhombic structure (Pbmm) with the lattice parameW b = 8:43 A, W and c = 2:95 A W [288]. A classi1cation of the k = 0 vibrational modes can ters a = 4:81 A;

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Fig. 2.37. Lattice structure of CuGeO3 . The CuO2 chains are formed by O2 and Cu. The Ge site is hidden within the GeO4 tetrahedron.

be found in Refs. [289,290]. The compound may be described by ribbons of CuO2 and GeO4 chains sharing one oxygen atom (O1) in the bc-plane, as shown in Fig. 2.37. The CuO2 chains (including O2) that are aligned along the c-axis provide a superexchange path via a Cu–O–Cu bonding angle very close to 90◦ [61,291]. This special exchange path leads to a weakening of the nearest neighbor c exchange Jnn compared to, e.g., a 180◦ -exchange in Sr-cuprates and a considerable strength of the c next nearest neighbor exchange Jnnn along the c-axis. This antiferromagnetic exchange to the next nearest neighbor leads to a frustration of the spin system with the frustration parameter = Jnnn =Jnn . The relevance of frustration is supported by magnetic susceptibility measurements that show a much too broad maximum at Tm = 60 K [283] in comparison with the Bonner–Fisher prediction c [292] for a Heisenberg spin chain. Considering a 1nite Jnnn a much better agreement with experiment is found [293,294]. Using the above mentioned susceptibility data and neutron scattering results c c [295,296] exchange constants of Jnn = 150 K; Jnnn = 30 K with = 0:2 have been found [293,294]. Frustration-induced Raman scattering [97] discussed in Section 3.4 leads to a rough estimate of c Jnn = 150 K and = 0:24 [97] while a careful analysis of susceptibility and magnetostriction data c points to a larger frustration and corresponding exchange parameters, = 0:35– 0.37 and Jnn = 160 K, respectively [297,298]. A further origin of frustration in CuGeO3 is the coupling of the spin system to so-called c “hot phonons” that have a frequency close to Jnn [299]. This non-adiabatic coupling adds an eKective frustration term to the Hamiltonian of the magnetic subsystem. Singlet formation and transition-induced modes. A drop in the magnetic susceptibility [283] and the observation of superstructure reOections in X-ray [300,301] or neutron scattering experiments [302] mark the spin-Peierls transition at TSP = 14:3 K. The accompanied lattice distortion has a wave vector kSP = (1=2; 0; 1=2). This means a doubling of the lattice cell parameters in a- and c-axis direction. While this vector describes the new periodicity of the phonon system kAF =(0; 1; 1=2) gives the periodicity of the magnetic system. In Fig. 2.38(a) a sketch of the two vectors in the Brillouin

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 (a)

E

(b)

a*

45

½ kSP =[½,0,½]

[0,0,0] b*

1

[0,0]

½ kAF =[0,1,½]

1 c*

½

b*

c*

} 01

kAF=[0,1,½]

Fig. 2.38. (a) Sketch of the relevant k-vectors and (b) the neutron dispersion of the acoustic triplet branch in the dimerized phase of CuGeO3 .

zone and in Fig. 2.38(b) the periodicity of the triplet dispersions observed in neutron scattering are shown [303,304]. In CuGeO3 the dimer positions alternate perpendicular to the c-axis direction, i.e. are staggered along the b-axis of the unit cell. This has been concluded from a careful analysis of the intensity of the superstructure reOections [305] and has important consequences for the observation of an incommensurate magnetic phase in an applied magnetic 1eld [16]. The lattice displacements of the atomic positions connected with the spin-Peierls transition may be described as due to an alternating rotation of the GeO4 tetrahedrons around the Ge atoms or around the O1–O1 axis, which induces displacements of Cu atoms along the c-axis and a twisting of the CuO2 ribbons around the c-axis. All atomic displacements are smaller than 2×10−3 [284,301]. Therefore, the superstructure reOections are very weak in intensity. There are four constituting phonon modes of the CuO2 chain dimerization, as shown in Fig. 2.39. A calculation of their microscopic spin–phonon coupling constants is given in Ref. [306]. The modes ;1 and ;2 have the largest values as they eUciently modulate the Cu–O–Cu bonding angle of the CuO2 chains. Three of these modes are observable in Raman scattering for temperatures below TSP as they are zone-folded from the high symmetry Brillouin zone boundary (1=2; 0; 1=2) to the zone center at 107, 370 and 821 cm−1 [308]. The fourth mode nearly coincides in frequency with a high-energy cut-oK of a density of states at 224 cm−1 observed in Raman scattering (see Fig. 3.3). This high-energy cut-oK may originate from a two-magnon density of states [309] as it is close to the maximum of the c-axis dispersion of magnon-like triplets observed in neutron scattering [295,296] for temperatures below TSP at E(1=2; 0; 1=4) ≈ 170 K = 118 cm−1 . Due to strong spin–phonon coupling the density of states observed in Raman scattering may also be of mixed spin–phonon origin as it is equally well described by a Fano-like lineshape [310] of a single phonon coupled to a continuum of magnetic excitations. The soft mode problem. In spite of the enormous experimental eKort linked to the spin-Peierls transition in CuGeO3 no anomalous phonon behavior could be detected. It is therefore now well established that especially no phonon soft mode exists close to TSP [291,302,307,311,312]. This peculiarity is not easy to understand as a lattice instability may be based on a macroscopic population of a phonon branch ;qSP . This is easy to achieve with ;qSP → 0 for T = TSP .

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

Fig. 2.39. Displacement patterns of the T2+ eigenmodes in CuGeO3 involved in the spin-Peierls distortion for T ¡ TSP . The shaded areas are the CuO4 plaquettes which form the Cu chains. The Cu atoms are in the center of each plaquette, the corners are formed by oxygen (O2). The oxygen (O1) are represented by the open circles with the Ge ions in between them. The given frequencies correspond to zone center optical phonons and have been determined by a force constant model to the phonon dispersions [306,307].

The spin-Peierls-induced Raman mode of CuGeO3 at 30 cm−1 has been proposed to be the soft mode of this system as its energy as function of temperature scales with the square root of the superstructure reOection intensities observed in elastic neutron and X-ray scattering [313]. However, this mode goes soft approaching the spin-Peierls temperature only from low temperatures. For T ¿ TSP it does not exist. In general, and taking phonons in the adiabatic limit into account, the spin-Peierls transitions of the considered systems have been predicted for arbitrary small spin–phonon interaction [314]. On the other hand, optical phonons in an anti-adiabatic (high frequency) limit lead to a transition only above a critical coupling. It has been suggested that in this limit the mechanism of the transition is diKerent and no soft modes should be observable [299,315]. It has also been proposed that in CuGeO3 the four phonon modes (;1 –;4 ) with T2+ symmetry (see Fig. 2.39) involved in the spin-Peierls distortion prevent the observation of a soft mode [307]. In a paper of Gros and Werner following an extended Cross–Fischer approach [314] these problems have been reconciled. Only phonons involved in the lattice instability and with a bare frequency less than a critical value completely soften. Phonons with frequencies ;qSP ¿ 01 do not soften but harden slightly in frequency. This eKect has indeed been observed experimentally in CuGeO3 for T ≈ TSP [316]. The earlier investigated organic spin-Peierls compounds have generally a much softer phonon spectrum and are therefore always in the limit ;qSP ¡ 01 with pronounced phonon soft modes [299,317]. The magnetic excitation spectrum. For T ¡ TSP well-de1ned magnon-like triplet branches develop out of a spinon continuum observed above TSP [318]. Fig. 2.38(b) shows a sketch of the neutron scattering results. They exhibit a 3D dispersion with a k-dependent singlet–triplet gap 01 and a splitting in an applied magnetic 1eld of the expected magnitude (g ≈ 2) [295,296]. The dispersion

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of 01 along the b-axis points to a non-negligible interchain interaction with an exchange coupling J b ≈ 0:1J c . The observation of these well-de1ned triplet branches on the other hand proves that the spinon excitations de1nitely change their character with the dimerization of the spin system [29]. The magnon-like modes should now be understood as triplets that consist of bound spinons. A further binding eKect below TSP is observed in the triplet spectrum as a gap of magnitude comparable to 01 now separating the triplet branch from a continuum of free unbound spinons. The spectral weight of the latter, however, is reduced for T 6 TSP [319]. The binding eKect on the spinon excitations to form triplets mainly depends on dimerization. However, magnetic and magnetoelastic interchain interaction [32] and frustration [29] are also important factors, as will be discussed in Section 4. Taking the available neutron scattering data [295,319,320] and the k-vectors k/ = (0; 0; 0); kSP = (1=2; 0; 1=2), and kAF =(0; 1; 1=2) illustrated in Fig. 2.38, the singlet–triplet gap of CuGeO3 at the high symmetry points has the following values: /01 = 24–30 K = k01AF and k01SP = 65 K. ESR-experiments [321] on thermally excited transitions in the Zeeman-split triplet branch (H = 0) observe a gap with a size similar to k01AF , while infrared absorption experiments [322] observe a gap comparable to the size of k01SP . Theoretical 1D models neglect the magnon-dispersion perpendicular to the chain along c, the spin-Peierls gap of theoretical 1D models need therefore to be identi1ed with the average gap (/01 + k01SP )=2 = 4 meV = 46:4 K. The reduced gap ratio 2/01 =kB TSP = 3:4 is close to the weak coupling “BCS”-value. Evidence for Dzyaloshinskii–Moriya interaction. This inconsistency is based on an anisotropy of the spin interaction that manifests itself most directly in a second optical triplet branch observed only recently in neutron scattering experiments [323]. The second branch has a considerable spectral weight in the whole Brillouin zone and may be constructed by folding the so-far known acoustic triplet branch along the b- and c-axis about the middle of the Brillouin zone. It may originate from an additional and so far not considered antisymmetric part of the exchange Hamiltonian, e.g., a DM interaction [324,325]. The DM interaction is compatible with the absence of an inversion center in between two copper sites along the b-axis direction of CuGeO3 . Deviations from the assumed centrosymmetric Pbmm structure have been observed in Raman scattering [326] and X-ray scattering experiments, and have been discussed in this context [324]. Also ESR experiments point into the same direction [327]. In X-ray scattering experiments unexpected superstructure reOections compatible with the point group P21 21 2 have been found in carefully annealed single crystals. This proposal corresponds to a superstructure with a non-centrosymmetric copper site in the unit cell 2ap · bp · 4cp , with ap · bp · cp the pseudo-unit cell of the Pbmm structure. This superstructure is based on a coherent arrangement of diKerently distorted CuO6 octahedra along the c-axis. It can only be veri1ed experimentally if the corresponding structural coherence length is large enough [325]. The singlet bound state. In inelastic light scattering experiments gap excitations are visible due to frustration-induced exchange light scattering [97]. Similar to the usual two-magnon scattering in 3D antiferromagnets combining two magnons with opposite k, in CuGeO3 a singlet bound state of two strongly interacting triplet states is formed. This bound state shows in contrast to spin–orbit coupling-induced one-magnon scattering no splitting in an applied magnetic 1eld [313] and de1nes the singlet–singlet gap as 00 = 30 cm−1 = 43:2 K. The energy of this singlet corresponds to 2/01 renormalized to lower energy by a binding energy of the triplets [328]. Following calculations of Bouzerar et al. [29] the energy and spectral weight of this singlet bound state strongly support an

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CuGeO3

U D

gH / 2T

SP

(0) (T/K)

I 1.0

I 0.5

0.94

D 0.90 0.2

0.0 0.0

0.4

0.6

0.5

0.8

1.0

T/TSP(0)

Fig. 2.40. Magnetic-1eld vs. temperature phase diagram of CuGeO3 on a reduced scale showing the uniform (U), the dimerized spin-Peierls (D) and the magnetic 1eld-induced incommensurate phase (I). The inset shows the hysteresis at the D=I phase boundary observed in magnetostriction experiments [329]. Fig. 2.41. Schematic phase diagram of substituted CuGeO3 with the uniform (U), dimerized spin-Peierls (D), and 3D antiferromagnetic phase (AF). The solid and the dashed lines give the phase boundaries for Zn and Si substituted samples, respectively.

c exchange constant of Jnn = 150 K and a frustration parameter = 0:24. A further discussion of the properties of magnetic bound states is given in Section 4. The incommensurate phase in a magnetic Eeld. Applying a magnetic 1eld at low temperatures a 1rst-order transition to an incommensurate (I) magnetic phase is observed. The tricritical point is at Tc = 11 K and Hc = 12:25 T [330]. The phase diagram including the uniform (U), the dimerized (D) spin-Peierls, and the magnetic phase is shown in Fig. 2.40 [329]. The lattice distortion in the I phase is described by a soliton-like lattice close to the D=I phase boundary with a continuous change to a sinusoidal distortion at higher 1elds. Further aspects of the I phase are discussed in Refs. [68–70,322,329]. Substitutions and long-range AF order. A very fascinating aspect of this compound is the possibility of substitutions. Several extensive studies of substitution eKects of transition metals on the copper site (Ni [331–333], Co [334], Mg [335], Zn [328,336–341]) and Si on the Ge site [57,328,338,340,342–344] have been performed and summarized in Refs. [58,333,345,346]. With a few percent of substitutions the spin-Peierls transition is suppressed drastically and 1nally disappears. On the contrary, in the magnetic susceptibility a Curie–Weiss contribution proportional to the defect concentration is observed. At lower temperature and even for very small substitution levels long-range AF ordering is induced (TN 6 5 K) with the easy axis along the crystallographic c-axis. For small x this order is in coexistence with the dimerization [339]. For higher concentration levels

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the superstructure peaks corresponding to the spin-Peierls phase exhibit increasing disorder. Finally the magnetic ordering disappears again as shown in Fig. 2.41. Applying hydrostatic pressure the decrease of the singlet–triplet gap and TSP can partially be undone. This is caused by an increasing frustration with pressure that localizes the triplet excitations and restricts the suppression of the order parameter to a smaller region near the impurity [306,348]. There exists no critical concentration for the observation of long-range order with Zn substitution. In carefully prepared single crystals of Cu1x Znx GeO3 with x 6 5 × 10−3 a minimum NQeel temperature of 0:0285 K has been observed. This indicates that the dimerization sustains the coherence of the antiferromagnetic phase of the spin polarization in impurity-doped systems and is consistent with the theory of the impurity-doped spin-Peierls systems [347]. In contrast for Mg and Ni substitution a 1rst-order transition from a dimerized AF to a pure homogeneous AF is observed with a critical concentration of xc =0:02 [335,332]. At this concentration TSP is zero and TN jumps to a higher value. The occurrence of 3D long-range magnetic order in a quasi-one-dimensional system may be explained in the simple picture of cut chain segments that weight the interchain interactions stronger than in the in1nite chain. However, this argument would not explain why this ordering disappears again at higher concentration levels. On the other hand magnetic ordering due to disorder is a phenomenon observed in geometrically frustrated spin systems [285,349]. A proper theoretical description of this behavior has to include that even a very small substitution level in a spin chain drastically changes its excitation spectrum. In a dimerized, non-perturbed system a gapped state with exponentially decaying spin–spin correlations is realized. A local substitution or disorder induces algebraically decaying spin–spin correlations as a transition to a quantum critical state. This transition is described as a quantum critical point. Strongly simplifying, the observed NQeel ordering results from a shift of this quantum critical point from T = 0 to 1nite temperature due to interchain interaction [39,42]. Theoretical studies of non-magnetic impurities in spin chains and ladders found general interest as a step toward understanding the eKect of doping mobile carriers in quantum spin systems [37–42]. Earlier Monte Carlo studies showed an enhanced four-spin susceptibility and therefore an enhancement of the spin-Peierls instability in random or disordered spin chains [350,351]. Non-magnetic impurities in strongly dimerized spin chains create loose s = 1=2 spins which randomly introduce states within the magnetic excitation gap. These states should be observable as low-energy excitations (doped spinons) that form a weakly dispersing impurity branch inside the gap. For smaller dimerization many-body s = 1=2 states are spread over several lattice spacings enhancing the spin– spin correlation at short distances [39,42]. The above mentioned scenario has been discussed to explain substitution-induced NQeel ordering in the spin ladder Sr(Cu; Zn)2 O3 (Section 2.4.1) and in CuGeO3 . Enhanced short-range dimer Ouctuations [37] have also been studied in Zn as well as Si substituted CuGeO3 using Raman scattering of a dimerization-induced phonon mode at 370 cm−1 and neutron scattering of the superstructure reOections [328,339,341,352]. The intensity of the phonon mode at 370 cm−1 as function of (decreasing) temperature shows an increase for T / TSP . This has been investigated in a Zn-substitution study (x=0) depicted in Fig. 2.42 [328]. With 1nite substitution level the maximum intensity of this mode is suppressed. In samples that do not show a thermodynamic phase transition (x = 0:06), however, the intensity stays 1nite. Its onset shifts to higher temperature with increasing substitution levels. As for x = 0:06 an onset of the dimerization-induced mode at

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

integrated Intensity (a.u.)

50

4000

(Cu1-xZnx)GeO3

3000

x=0

x=0.018

2000

x=0.035

1000

x=0.06

0 0

5

10

15

20

25

30

35

40

Temperature (K)

Fig. 2.42. EKect of Zn substitution in CuGeO3 on the intensity of a dimerization-induced phonon mode at 370 cm−1 [328].

Tonset ≈ 25 K is observed, i.e. at a temperature above the spin-Peierls temperature of the pure samples, this shift is understood as a 1ngerprint of locally enhanced dimer Ouctuations. Similar but more pronounced eKects have been observed in Si-substituted samples. In consistency with X-ray studies Si substitution on the germanium site induces lattice strains and enhances drastically the amount of disorder in the lattice cell [328,344]. A further eKect of substitutions is the introduction of defect-induced bound states in the singlet– triplet gap of the excitation spectrum. This has been observed in light scattering spectra of CuGeO3 as a well-de1ned mode at about half the energy of the singlet bound state of the pure samples as described in Section 4.1.2 [32,36,353]. Concluding the discussion of the magnetic parameters and properties, it should be obvious that CuGeO3 is not an ideal representative of a spin-Peierls system. Especially the deviations of the spin system from the Heisenberg Hamiltonian, i.e. a non-negligible interchain interaction and intrachain frustration, evidence for a DM interaction and the complicated spin–phonon coupling prevent its closed-up and unambiguous description within a set of a few parameters. 2.7.2. NaV2 O5 The compound NaV2 O5 shows a phase transition at TSP = 35 K with combined spin-Peierls and charge-ordering characteristics [207,208]. It is formed by edge-sharing VO5 square pyramids in the ab plane separated by chains of sodium ions located between the ab layers. If the superexchange via the oxygen square pyramids along the b-axis is taken as the dominant exchange path, the compound may be understood as an arrangement of weakly coupled (90◦ exchange) ladders. As only one electron (spin) per two vanadium sites is available the ladders are quarter-1lled [287]. A complete charge ordering on a linear chain of V4+ (s = 1=2) and a chain of V5+ (s = 0) with distinguishable vanadium sites leads to the earlier reported non-centrosymmetric Pmn21 structure [205]. However, this is in disagreement with structural investigations at room temperature using synchrotron X-ray diKraction, neutron scattering [209], and also with recent NMR experiments [354,355]. Presently a centrosymmetric structure Pmnm is used for the high-temperature phase [209], shown in Fig. 2.43. In NMR experiments for temperatures above TSP only one vanadium site is observed whereas below TSP two sites are resolved. The structure of the low-temperature phase is under discussion.

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c b a Fig. 2.43. Lattice structure of NaV2 O5 . A ladder along the b-axis is formed by two rows with mixed V4:5+ O5 square pyramids that share corners. This ladder is only weakly coupled to a neighboring ladder as corresponding V-ions are shifted by half a unit cell. These pyramids share edges. The site of vanadium (omitted here) is in the basal plane center of each pyramid and slightly shifted towards the respective apical oxygen. The rows of black spheres denote sodium chains.

The acentric Fmm2 point group or one of its subgroups are proposed [355,356]. The Fmm2 point group is built by an alternation of non-distorted and distorted, charge ordered chains. A corresponding V4:5 -site, however, is not observed in NMR [354,355,357]. A similar structure has been investigated theoretically in the more general context of a coexisting charge-density wave and spin-Peierls order [358]. Van Smaalen et al. have proposed [359] that stacking disorder in the low-temperature phase of -NaV2 O5 of plane with only zig-zag charge ordering, as illustrated in Fig. 2.49, may lead to an average space group Fmm2, as observed by X-ray. This route would then reconcile the various experimental results discussed above, in particular the transverse magnon splitting observed by neutron scattering [360]. Symmetry analysis and room temperature Raman spectra. At room temperature the k = 0 vibrational modes of NaV2 O5 with the space group Pmnm (D13 2h ) can be classi1ed [189]: /vib = 8Ag ⊕ 3B1g ⊕ 8B2g ⊕ 5B3g 3Au ⊕ 7B1u ⊕ 4B2u ⊕ 7B3u ; where the even modes are Raman active and the odd B modes are infrared active. For the space group Pmn21 (C72v ) with two distinguishable crystallographic vanadium sites the result is [361]: /vib = 15A1 ⊕ 8A2 ⊕ 7B1 ⊕ 15B2 : Room temperature Raman spectra of NaV2 O5 for interladder (cc) polarization and all light scattering polarizations accessible in the ab plane are shown in Fig. 2.44 [189,362]. The experiments in the temperature range from room temperature to TSP are clearly incompatible with the space group Pmn21 and con1rm the proposed centrosymmetric space group Pmnm, which exhibits one vanadium site. However, in these spectra some other unusual features exist. There is quasielastic scattered light

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..

Fig. 2.44. Room temperature Raman light scattering spectra of NaV2 O5 for interladder (cc) polarization and all light scattering polarizations accessible in the ab plane. The broadened central peak in (cc) polarization is due to the experimental setup [189,362]. Fig. 2.45. Triplet dispersion in NaV2 O5 along the b-axis determined by neutron scattering. The points mark the results of (3; k; 0)-scan pro1les with 1xed energy transfer. The solid lines are resolution convoluted 1ts. The corresponding dispersion is shown in the inset [373].

in intraladder (bb) scattering con1guration, i.e. with the polarization of the incident and scattered light along the eKective chain direction. This scattering is not observed in other polarizations. It vanishes upon cooling the sample below T = 100 K. In other quantum spin systems a comparable scattering contribution has been observed and attributed to Ouctuations of the magnetic energy density. This will be discussed further in Section 5. The broad maximum observed in (aa) polarization at 640 cm−1 has been discussed in the context of light scattering on charge kinks [363]. A second transition-induced quasielastic light scattering contribution in crossed (ab) polarization will be described below. Magnetic properties and the singlet ground state. In magnetic susceptibility measurements a broad maximum in (T ) is observed at Tm ≈ 300 K. This temperature dependence is in agreement with the Bonner–Fisher prediction for a dominant 1D exchange path (Jnn = 560 K) [207]. Band structure calculations for a quarter-1lled ladder show that the spin degrees of freedom involve eKective orbital spins of two neighboring V sites on the ladder. These orbital spins indeed form an eKective spin chain with respect to the low-energy excitations as the corresponding charge transfer gap is estimated to

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53

be CT ≈ 0:7 eV [287]. The interchain coupling in this eKective spin model is very weak. Therefore, no eKect of frustration on the magnetic susceptibility is observed [364]. A detailed discussion of the electronic structure of NaV2 O5 will be given below. The singlet formation at TSP =34 K is detected by an exponential drop in the magnetic susceptibility comparable to CuGeO3 . The observation of superstructure peaks in X-ray scattering and zone-folded phonon modes in Raman scattering prove a lattice dimerization with a propagation vector kSP = (1=2; 1=2; 1=4), i.e. an increase of the volume of the unit cell by a factor of 16 [208,365–367]. The spin-Peierls nature of the dimerization transition, however, is questionable due to several arguments and experimental 1ndings discussed below, e.g., the very small magnetic 1eld dependence of the transition temperature and the substantial entropy release at TSP [368]. The singlet–triplet gap is estimated using speci1c heat: 01 = 67 and 84 K [369,370], magnetic susceptibility: 85 K [208], ESR: 93:5 K [371], and NMR: 98 and 108 K [357,354]. In recent ESR experiments a magnetic 1eld-induced peak splitting (g = 1:97) has been observed, con1rming the triplet nature of the excited state [372]. Neutron scattering results and the triplet dispersion. In neutron scattering experiments on powder samples a broad maximum at the energy 115 K is observed as well as hints for dynamic 2D spin correlations [365]. More recent investigations on single crystals con1rm k01AF = 115 K, with kAF = (h; 1=2; 0) and h = integer. In addition, a steep dispersion of the triplet branch in b-axis direction and a negligible dispersion along the c-axis is detected [373,374]. This determines unambiguously the b-axis as the dominant exchange path direction. Fig. 2.45 shows corresponding scans in k space at various 1xed energies identifying the gap and the steep dispersion [373]. In a-axis direction the dispersion is distinctively smaller. This branch is degenerate at kAF and shows an unexpected splitting in the middle of the Brillouin zone. This leads to two singlet–triplet gaps, ah; 1=2; 0 = 89 K and bh; 1=2; 0 = 130 K, with h = half -integer. In addition, the form factor of this branch has an unusual intensity modulation with h. It changes with the periodicity of 3 reciprocal lattice units [373,374]. A comparable splitting has not been observed in CuGeO3 . Therefore, a simple explanation due to a cell doubling-induced folding of the unit cell in a-axis direction may be excluded. As discussed below, it is compatible [360] with a zig-zag charge ordering of the 2V 4:5+ → V 4+ + V 5+ . Fig. 2.46 shows the corresponding neutron scattering results in dependence on k [374]. Nature of the transition. There are further experimental observations in NaV2 O5 that diKer from the usual spin-Peierls picture and point toward a charge ordering preceding or accompanying the lattice distortion [375]. The intensity of the superstructure reOections [365] shows a poor agreement with the Cross–Fisher relation [314] 01 ˙ 2=3 . Furthermore, the entropy involved with the transition at TSP = 34 K and the value of the reduced gap 201 =kB TSP = 4:8–6:6 K are much larger compared to CuGeO3 . A conventional spin-Peierls mechanism can also be excluded in view of the very small magnetic 1eld dependence of TSP (H ) [369]. There exists even evidence for an additional 1rst-order transition slightly below TSP from thermal expansion experiments [370]. This additional transition at T = 33 K has been interpreted as a charge ordering transition since in NMR experiments two V sites are observed for T ¡ TSP [354]. Later however, this assignment has been criticized and the anomaly has been attributed to a possible phase separation [375]. Ohwada et al. have studied the nature of the transition under pressure of up to 1.3 GPa and found a “devil’s staircase” of crystallographic transitions [376], which could be modelled accurately by the ANNI-model [377]. The occurrence of the devil’s staircase, see Fig. 2.47, has important implications

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P

C

C

C

C C C C

C

C C C C C C

Fig. 2.46. Triplet dispersion in NaV2 O5 along the a-axis determined by neutron scattering (full dots). The solid and broken line indicate the calculated dispersions based on a zig-zag charge ordered ground state for k = 0:5 and k = 0 r.l.u., respectively. The solid and broken arrows correspond to possible transitions in neutron scattering and ESR experiments, respectively [373]. Fig. 2.47. Devil’s staircase of crystallographic transitions for NaV2 O5 with diKerent ordering pattern Cp=q along the c-direction [376].

for the microscopic description of the underlying crystallographic transitions [376]. Assuming that these crystallographic transitions are driven by diKerent charge-ordering patterns along c, the analysis via the ANNI-model implies in-plane and inter-plane eKective coupling constants of the order of 30 –40 K. Since the screened V–V Coulomb matrix-elements are of order 0.3–0:4 eV [378], it follows that the eKective couplings driving the transition must contain magnetic or elastic components and the assumption [375,379,380] that the Coulomb interaction drives the transition at TSP = 34 K is questionable because TSP is much smaller than the V–V Coulomb matrix-elements. Microscopic properties of the room-temperature state. The electronic state at room temperature has been studied intensively [287,364,381] and gives a clue towards possible scenarios for the dimerization transition in NaV2 O5 . The contributions to the electronic Hamiltonian are illustrated in Fig. 2.48 [287,381]. It contains a rung-hopping ta ≈ −0:35 eV and two inter-rang hoppings tb and td along the ladder. In the 1rst DFT-analysis td was neglected and tb was found to be ≈ −0:175 eV. EKectively only the combined hopping amplitude tb + td = −0:175 eV can be determined from the bands at the Fermi-edge, the anti-bonding bands above the Fermi-edge can be described better [381] assuming a substantial td = −0:12 eV. The occurrence of a large diagonal hopping td opens the possibility of diagonal dimer formation in the low-T phase. The interladder couplings t1 and t2 are small and do not inOuence the character of the charge excitations [382].

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t2

B tb

td

A t1

C

E D

ta

Fig. 2.48. Tight-binding parameters for NaV2 O5 in the room-temperature state. Note that there is a certain freedom in dividing the eKective hopping along the ladder, tb + td = −0:175 eV into the leg-hopping tb ≈ −0:055 eV and the diagonal part td ≈ −0:12 eV. The other parameters are ta ≈ −0:375 eV with t1 and t2 being small. One arrow on a V–V rung denotes a spin-1/2 moment, the rungs with two/zero spin illustrate a possible charge-transfer excitation. Another common notation is t⊥ = ta and t = tb [287,381]. Fig. 2.49. Diagonal zig-zag charge ordering in NaV2 O5 showing the inequivalence of neighboring pairs of V sites (AB and BC) due to the diKerence in the V4+ occupation (marked as circles) in neighboring ladders. The lines denote ta and tb in the ladder while the dashed line marks t1 (compare Fig. 2.48). Note that the hopping matrix elements ED and DC are in general diKerent [360].

The DFT-estimate [287] for the onsite Hubbard-U of U ≈ 2:8 eV has been con1rmed by clusteranalysis of resonant inelastic X-ray scattering results [383]. The screened V–V Coulomb matrixelements are of the same order of magnitude [378,384] 0.3–0:8 eV as the hopping-matrix elements, but the√eKective Coulomb rung–rung interaction has been shown [378] to be reduced by a factor 1 − 1= 2 ≈ 0:3. It is therefore to be expected, like in LiV2 O5 (see discussion further below), that the degree of charge ordering in the low-T phase is less than complete [203]. Please note that in this model the transition temperature TSP = 34 K = 2:93 meV is small in comparison with the above given parameters. It is therefore still under discussion whether interladder frustration [375] is responsible for the reduction of the transition temperature. The eKectively one-dimensional spin structure of NaV2 O5 is caused by the occupation of a bonding state of a rung by one electron with the energy −ta . Considering the strong on-site Coulomb repulsion the cost for an electron transfer to the next rung is at least CT = 2ta (see Fig. 2.48 for an illustration). Nature of the charge ordering in the low-temperature state. Charge ordering phenomena and other instabilities like spin density wave transitions or transitions to commensurate or incommensurate antiferromagnetic ground states are a general property of quarter-1lled quasi-one-dimensional systems [358,379,385]. Especially charge disproportionation as observed in many organic conductors (charge transfer salts with metallic countercations) is driven by intersite Coulomb interaction [386].

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The energetically preferred ground state con1guration is a zig-zag charge ordering with dimers diagonal to the ladder. This con1guration minimizes the intraladder Coulomb energy and takes contributions due to lattice distortions in the ab plane of NaV2 O5 into account [375,379,380]. Fig. 2.49 shows a sketch of zig-zag charge ordering. This type of charge ordering has been predicted [360] from an analysis of the splitting of the neutron triplet branch in a-axis direction and has been con1rmed by subsequent X-ray analysis studies [359]. Possible consequences resulting from the zig-zag charge ordering shown in Fig. 2.49 are [375]: • The charge transfer gap may be increased to CT ≈ U1 , with U1 the next nearest neighbor (intersite) Coulomb repulsion inside ladders [360,375]. An electronically driven transition has been predicted to occur [387] for U1 ¿ 2t⊥ but not found in direct diagonalization-studies [388] for U1 ≈ 2t⊥ . • The eKective spin-exchange coupling Jb along the ladder is reduced. This eKect is experimentally observed comparing the coupling constants from 1ts to magnetic susceptibility for T ¿ TSP (560 K) with T ¡ TSP (440 K) [360,381]. • An alternation of this coupling along the ladder and thereby the opening of the singlet–triplet gap may be induced by the inequivalence of the sites (see Fig. 2.49 [375]). The formation of diagonal singlets or dimers either as interladder or intraladder singlets is discussed controversially [375,379]. • The larger V4+ pushes out neighboring oxygen and thereby stabilizes an occupation of the neighboring plaquettes with the smaller V4+ . These in-plane lattice distortions considerably stabilize the charge ordering [360,375]. • kink-like excitations coupled to the lattice can appear in Raman scattering [363] Pretransitional Fuctuations. The existence of an apical oxygen in NaV2 O5 is additionally noteworthy as it leads to an enhancement of electronic intersite correlations [188]. The asymmetry of the elementary cell with the apical oxygen of the VO5 -pyramid either above or below the V site and a W lead to a large Coulomb 1eld at the relatively small distance of this oxygen to the V (da ≈ 1:61 A) V sites. As a result a charge redistribution between the V ions is followed by a lattice displacement (see Fig. 2.50). Realized in an out-of-phase vibration of the V ions within one rung this displacement couples the bonding and antibonding states discussed above as an origin of the eKective chain (their energy separation is 2t⊥ ). In the charge ordered state the Coulomb force contribution of this displacement feeds back to the charge disproportionation, thereby stabilizing it. In other words, the 1eld of the apical oxygen enhances the asymmetry of the V ions arising due to correlations and modulates the eKective exchange coupling. Estimates show that strong phonon-driven charge and spin Ouctuations are expected as well as the appearance of a new Raman-active mode [188]. In CuGeO3 only few hints exist [389,390] for dimer or structural Ouctuations far above the spin-Peierls transition. This is related to the fact that its transition is governed by long-range magnetoelastic interactions leading to a mean-1eld like behavior. However, in NaV2 O5 the case is diKerent as in quarter-1lled systems the inOuence of intersite Coulomb interaction is stronger. Therefore, the experimental evidence for pretransitional Ouctuations or charge ordering, that will be discussed now, should be taken very seriously. This evidence is found both in thermodynamic as well as in spectroscopic data: A longitudinal mode of the sound velocity shows a transition-related anomaly that persists in a temperature range 0:8TSP 6 T 6 2TSP [391,392]. In IR absorption a transition-induced

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(b)

(a)

da z -z Fig. 2.50. Sketch of the charge disproportionation in NaV2 O5 due to an V ion displacement (±z) along the c-axis in the 1eld of the apical oxygen. The distance of the apical oxygen to the vanadium is given by da . The dashed circle (full circle) denotes the oxygen (vanadium) site. In (a) the situation in the homogeneous high temperature and (b) in the charge ordered phase is shown. The square pyramid oxygen sites are omitted [188].

phonon at 722 cm−1 is observed for temperatures up to 2TSP [393]. In Raman scattering experimental spectra in crossed (ab) polarization show unexpected phonons of A1g symmetry for temperatures TSP ¡ T 6 80 K. This observation is compatible with a breaking of the inversion symmetry at this temperature. In X-ray diKraction the superstructure peaks allow to determine correlation lengths that show a crossover from 3D to 2D lattice Ouctuations at about 50 K [394]. Fluctuations are also evident in NaV2 O5 as a quasielastic light scattering intensity that shows a pronounced temperature dependence far above but also below TSP . This broad feature, as shown in Fig. 2.51, is observed in crossed (ab) polarization [189]. Its intensity increases (with decreasing temperatures) below TCO ≈ 80 K, with TCO de1ned as a Ouctuation or crossover temperature of the electronic system. The intensity then drops again sharply with decreasing temperature below TSP . As discussed below, in this temperature range the intensity of magnetic bound states start to rise. The quasielastic scattering contribution is therefore assigned to magnetic light scattering as it competes with the scattering intensity of the bound states [189,362]. A comparable eKect is observed in (bb) polarization as a broad scattering continuum in the frequency range 50 cm−1 6 Y! 6 160 cm−1 . It has a similar intensity as function of temperature, however, in this polarization the Ouctuations close to the dimerization transition have a gapped spectrum, i.e. there is no scattering contribution for frequencies Y! ¡ 50 cm−1 [189]. The phonon system shows more direct evidence for the onset of charge ordering at T ¿ TSP . A distinct phonon anomaly exists in A1g symmetry at 422 cm−1 . This excitation displayed in Fig. 2.52 is very likely the phonon mode discussed above which is coupled to intersite charge Ouctuations [189,188]. It shows a hardening below the transition, remarkable changes of the phonon lineshape and 1nally the evolution of a distinct second mode at 394 cm−1 for the lowest temperatures. The onset of the frequency dependence with lowering temperature is again at TCO = 80 K. The temperature dependence of the phonon energy in Fig. 2.52 shows a continuous, gradual change at TCO instead of a transition. This behavior is attributed to the low dimensionality of the system. A similar “soft crossover” identi1ed as a charge ordering has been observed in La2 NiO4:133 . For this 2D compound with a Ni–O square plane, stripe order into 1D domains of charge and spin is discussed. The intensity of the charge-order superstructure peak sets in for T ¡ TCO = 200 K triggering the magnetic order at Tm = 110 K [51]. In addition to the very similar temperature dependencies of

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Intensity (arb. units)

24K 60K

(ab)

NaV2 O 5

300

Intensity (arb. units)

30K

150

5K

100

50

0

50

100

150

200

NaV2O5 100 K

250

(bb)

200

40 K

150

5K 100 -1

-1

429cm

394cm

250

-1

Raman shift (cm )

380

400

420

440

460

-1

Raman shift (cm ) phonon energy (cm )

0.8

430

TSP= 34 K

-1

Norm. Intensity

1.0

0.6 0.4 0.2 0.0

TCO = 80 K

428 426 424 422 420

0

20

40

60

Temperature (K)

80

100

0

20

40

60

80

100

120

140

Temperature (K)

Fig. 2.51. Quasielastic light scattering in NaV2 O5 in crossed (ab) polarization and its normalized intensity as function of temperature [189]. Fig. 2.52. Raman light scattering spectra of NaV2 O5 (upper panel) and excitation energy of a A1g phonon at 422 cm−1 in intraladder (bb) polarization (lower panel). The onset of the phonon hardening (TCO ) and the transition temperature (TSP ) are marked by arrows [189].

the superstructure intensities of charge ordering in both compounds also the reduced energy scales TCO =Tm = 1:81 and TCO =TSP = 2:4 are comparable. Fluctuations in the spin and phonon system of NaV2 O5 support a scenario for the singlet ground state that is motivated by the observation of three magnetic bound states in Raman scattering experiments [189,362,366]. These states poses mixed polarization selection rules. A dynamic superposition of several energetically nearly degenerate dimer con1gurations is compatible with this observation. The con1gurations should also include diagonal dimers on the ladder. This scenario resembles the dynamic pseudo-Jahn–Teller eKect or an RVB model, emphasizing either the importance of phonons or electronic correlations [362]. The observation of gap-related excitations with energies that depend on the time or frequency scale of the respective experiment would be a critical proof. The large variation of the gap values determined by diKerent measurement techniques at least suggests such non-adiabatic eKects in NaV2 O5 . For a complete discussion of the bound state properties motivating this approach see Section 4.

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Fig. 2.53. Crystal structure of >-LiV2 O5 . Shown are the two equivalent xy-planes. There are two kinds of VO5 square-pyramids, containing the V(1) and V(2)-ions, respectively. Note the substantial corrugation of the xy-plane, compare with the lattice structure of NaV2 O5 presented in Fig. 2.43.

Comparison with LiV2 O5 . Since NaV2 O5 is an insulator, it was initially thought [83,206] to contain two inequivalent V-ions, with electronic con1guration (3d)0 and (3d)1 and with the V-(3d)1 ions forming zig-zag spin-1/2 chains. Re1ned crystallographic data analysis at room temperature [209,287,395] showed the existence of only one equivalent V-ion in a mixed-valence (3d)0:5 con1guration. It was then shown by DFT-calculation [287,364] that the insulating nature of NaV2 O5 arises from a bonding–antibonding splitting of V–O(1) –V orbitals. The moment-carrying bonding orbital is there delocalized, giving rise to the possibility of further charge ordering phenomena at lower temperatures, as discussed in the previous section. LiV2 O5 is, on the other hand, charge ordered at all temperatures [396,397] and has a very similar lattice structure as NaV2 O5 (see Fig. 2.53). A re1ned DFT-analysis has shown that the degree of charge ordering is not complete, but roughly 70:30 [203]. The degree of charge ordering determines the nature of the underlying physical model: A 50:50 charge distribution (no ordering like in the high-temperature phase of NaV2 O5 ) implies a quarter-1lled ladder compound. Complete charge ordering 0:100 would imply, on the other hand, a zig-zag chain compound. The DFT-result for LiV2 O5 of a 70:30 charge-ordering implies, on the other hand, an asymmetricladder model [203] with Coulomb coupling in between adjacent ladders [388]. These 1ndings suggest that the degree of charge-ordering in the low-temperature phase of NaV2 O5 , which is to date not known [398,399], would have substantial inOuence on the underlying physical model. 2.8. Magnetic parameters of selected spin systems In Table 2.2 magnetic parameters of selected spin systems are summarized. Due to frustration, exchange topology or spin–phonon coupling many compounds prefer other ground states than long-range magnetic order.

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Table 2.2 Selected inorganic one-dimensional spin systems and their magnetic parameters Compound

J

J⊥

01

TN

TSP

System

YBa2 Cu3 O6 SrCu2 (BO3 )2 KCuF3 Sr 2 CuO3 SrCuO2 SrCu2 O3 Sr 14−x Cax Cu24 O41 (VO)2 P2 O7 CuGeO3 NaV2 O5

1400 70 187–203 1300 2100 850 –2000 1050 –1500 100 –120 120 –160 440 –560

1400 — — — — 750 –1000 750 –830 — 15 15

— 34 — — — 280 – 420 120 –133;550 –270 37, 69 24 –30 84 –110

— 0.6 — — — — — — 0.24 – 0.37 —

— — — — — — — 0.09, 0.19 0.012– 0.05 0.03– 0.09

≈ 500 — 39 11 — — 2– 4 — — —

— — — — — — – — 14.3 34

2D square plane 2D frustrated Chain Chain Zig-zag chain Ladder Ladder+chain Alternating chains Spin-Peierls 1/4 –1lled ladder

All units are in [K]. J denotes the exchange coupling constants along the chains and ladders, J⊥ the exchange coupling perpendicular to the chain or ladder, and are parameters of frustration and dimerization. For CuGeO3 the magnitude of the dimerization is still under discussion [306]. The transition temperatures for NQeel and spin-Peierls ordering (TN ; TSP ) are given. In the case of Sr 14−x Cax Cu24 O41 the singlet–triplet gaps for the CuO2 chain and the Cu2 O3 ladder sublattice are given. For this compound we include the variation of the parameters due to Ca substitution or diKerent experimental techniques. For SrCu2 (BO3 )2 the values given under J and correspond to the larger intradimer exchange coupling (J1 ) and the ratio of the coupling constants (J2 =J1 ) with J2 the interdimer coupling constant. References for the data are given in the respective sections to be found above.

3. Magnetic light scattering Light scattering in magnetic compounds is a well established method. The study of spin waves or magnons in ferromagnetic and antiferromagnetic compounds has been used extensively to determine coupling constants, anisotropies and magnon–magnon interaction. A review of the principal eKects and basic theory can be found in the textbook “Light Scattering in Magnetic Solids” by Cottam and Lockwood [400]. The coupling of light to elementary excitations of a solid is described by a diKerential scattering cross section. Its calculation reduces to the determination of the correlation function (k )∗ k!* and is therefore based on the determination of the general susceptibility tensor (r) [400]. This leads to an eKective Raman Hamiltonian R = EI (r)ES ; (3.1) r

;

with the electric 1eld vector EI and ES of the incident and scattered light, respectively. Expanding the susceptibility (r) in powers of spin operators [401] we obtain: (r) = 0 (r) + K! (r)Sr! + G!* (r)Sr! Sr* +

T

!;*

!

!;*

* M!* (r; )Sr! Sr+ + higher order terms :

(3.2)

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Fig. 3.1. Magnetic Raman scattering of the 3D antiferromagnet FeF2 . (a) Both one- and two-magnon scattering are observed at 50 and 155 cm−1 , respectively (T = 15 K). (b) Measurements in an applied magnetic 1eld of 5.2 T show a splitting only of the one-magnon signal (T = 20 K).

The 1rst term in Eq. (3.2) corresponds to Rayleigh scattering, the second and third term are linear and quadratic in the spin operators and induce spin Oips at a single ionic site r. The complex tensors K and G determine the strength of the coupling of light to the spin system associated with spin– orbit coupling. They may be evaluated using Faraday rotation and magnetic linear birefringence. In inelastic light scattering the resulting one magnon signal has oK-diagonal and diagonal elements of the susceptibility tensor. 3.1. Two-magnon scattering Two-magnon scattering which is observed with appreciable intensity in antiferromagnets is induced by the fourth term in Eq. (3.2) that couples spin operators at diAerent sites r and r + . A pair of excitations of exchange coupled magnetic sites on opposite sublattices with YS z = 0 and wave vectors of similar amplitude but diKerent sign are induced. The eKective Hamiltonian reads: * R = EI ES M!* (r; )Sr! Sr+ : (3.3) r; !;*

In Fig. 3.1 a Raman spectrum of FeF2 is shown [400]. This Ising-type antiferromagnet (TN = 78:4 K) is one of the rare cases where both one- and two-magnon scattering are observed with comparable intensities. More usual is either the observation of one-magnon scattering in systems with strong spin–orbit coupling or two-magnon scattering in systems that are close to an isotropic exchange. The typical width of the 2-magnon response in 3-dimensional, ordered antiferromagnets is much smaller than the characteristic energy, as evidenced for FeF2 in Fig. 3.1. This is in contrast to the 2-magnon intensity for quasi one-dimensional systems, which we will discuss further below. The symmetry of Eq. (3.3) with respect to the spin operators and the electric 1eld vectors leads to special selection rules in the case that the magnetic system has a dominant exchange path. For one-dimensional spin systems two-magnon scattering is observed only in the A1g scattering component, with EI and ES both parallel to the crystallographic direction of the exchange path. To observe this type of scattering the Raman operator is required not to commute with the exchange

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Hamiltonian. Therefore, any additional perturbation of the exchange Hamiltonian, such as a dimerization or frustration of the spin system is helpful. The Raman light scattering intensity I scales with I ˙ J 2 . For 2D antiferromagnets, such as the parent compounds of the high-temperature superconductors, magnetic scattering is allowed only in the B1g scattering component, i.e. with perpendicular in-plane electric 1eld vectors. As the symmetry selection rules described by Eq. (3.3) only apply to non-resonant scattering they may be broken if the energy of the intermediate state matches the energy of some electronic excitation levels. 3.2. Light scattering in high-temperature superconductors The most prominent and complete examples of light scattering studies in 2D antiferromagnets have been performed on the parent compounds of the HTSC, e.g., YBa2 Cu3 O6+ [402,403], La2 CuO4 [404] or Bi2 Sr 2 (Ca1−x Yx )Cu2 O8 [131,132]. They have been studied to understand the temperature-doping phase diagram, pseudo-gap phenomena and related eKects [405,406]. Two-magnon light scattering [144,407] in the NQeel phase of the undoped HTSC shows some peculiarities that strongly deviate from the behavior of other antiferromagnets [408–410]. The two-magnon density of states with a maximum at an energy of approximately 2:7 J is not only observed in the B1g light scattering component but also in the B2g and A1g components. Additionally, the maxima of scattering intensity are strongly asymmetric and broadened to higher energy. There are pronounced resonance eKects by tuning the energy of the incident light. Although a coupling of the magnons to phonons and strong magnon–magnon interaction might describe this behavior qualitatively, the resonance eKects point to an electronic origin. A triple resonant scattering in the virtual intermediate state of the Raman process on the charge transfer gap of these compounds has successfully been used to explain the breakdown of the symmetry selection rules and anomalous lineshapes [408,409]. Low-energy correlations in weakly doped HTSC are discussed in the context of stripe-like dynamic charge and spin correlations [50]. These aspects have been studied using light scattering in the isostructural nickelates [411–414]. Commensurate and static charge and spin ordering to stripes is observed in La1:66 Sr 0:33 NiO4 at T ¡ TCO = 240 K and T ¡ Tm = 190 K, with TCO and Tm the charge and spin ordering temperatures, respectively. The HTSC and nickelates both share the square plane structural element. However, the linear Cu–O–Cu superexchange in HTSC with J ≈ 800–1700 K is stronger compared with the Ni–O–Ni superexchange with J ≈ 350 K. 3.3. Light scattering in quasi-one-dimensional spin systems For light scattering in quasi-one-dimensional spin systems other selection rules apply compared to 2D antiferromagnets, see Eq. (3.3). Due to the dominant exchange path along one crystallographic direction only magnetic scattering with the polarization of EI and ES both parallel to this direction is allowed. Furthermore, the Hamiltonians of magnetic exchange scattering and the magnetic spin system should not commute. This will be explained further in discussing the Hamiltonian of a static spin-Peierls system including frustration. It has been successfully used to model the Raman scattering results on the spin-Peierls compound CuGeO3 for temperatures above and below TSP = 14:3 K [97]. The Hamiltonian for a spin-Peierls chain is given by [(1 + (−1)i )Si · Si+1 + Si · Si+2 ] : (3.4) HSP = Jnn i

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

ωI

63

ωS

Fig. 3.2. Sketch of an exchange light scattering process on two singlet states on a spin chain. !I and !S denote the frequency of the incident and scattered photon, respectively.

In Eq. (3.4) the exchange coupling to nearest neighbors Jnn Si · Si+1 is modulated for T 6 TSP by the dimerization = 0. A competing exchange coupling to the next nearest neighbor given by Si · Si+2 , with the frustration parameter = Jnnn =Jnn , changes the excitation spectrum and ground state properties considerably as described in Section 1. For an alternating spin chain with dimerization > the Raman operator corresponding to Eq. (3.1) has the form: R= (1 + >(−1)i )Si · Si+1 : (3.5) i

We compare the Hamiltonian in Eq. (3.4) with the Raman operator in Eq. (3.5) taking into account a dimerization >. For = > = 0 there is no exchange scattering without the frustration term as [HSP ( = 0); R] = 0. With = 0, the model (Eq. (3.4)) leads to magnetic Raman scattering which has been observed experimentally in the spin-Peierls compound CuGeO3 [97,308,309,415]. 3.3.1. The limit of large dimerization In the limit of large dimerization the exchange light scattering process in 1D spin systems may be illustrated on a small system of four spins S1 ; S2 , S3 and S4 [416]. The singlet ground state |s; s is formed by a product of two singlets s12 and s34 . The Raman operator (3.5) for this system simpli1es to R23 =S2 ·S3 . Using the usual spin operators z + Si ; Si = Six + iSiy , and Si− = Six − iSiy leads to R23 = S2z S3z + (1=2)(S2+ S3− + S2− S3+ ) :

(3.6)

The exchange Raman process is then given by R23 |s; s = (1=4)(−|t0 ; t0 + |t1 ; t−1 + |t−1 ; t1 ) ;

(3.7)

where t0; ±1 denotes the triplet state with the corresponding S z component. The total spin before and after the application of R23 is zero. Two elementary magnons are created in the spin system. This is sketched in Fig. 3.2 [410]. Note, that the two magnons are created (in the limit of large dimerization) on adjacent dimers. 3.3.2. Non-zero temperature For T = 0 the lowest triplet branch may be thermally populated. Achieving this situation in a spin-Peierls systems means that the reduced gap ratio 2SP =TSP should not be too large, e.g.,

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not larger than the mean 1eld value 3.5. This requirement is ful1lled in the case of CuGeO3 and (VO)2 P2 O7 but not in the case of NaV2 O5 . We study a situation where the Raman process uses |t0 ; s as an initial state [416]. This leads to R23 |t0 ; s = (1=4)(−|s; t0 + |t1 ; t−1 − |t−1 ; t1 ) :

(3.8)

Here, the total spin before and after the application of R23 is unity as requested by the exchange light scattering process. The 1rst term in (3.8) does not change the number of triplets in the system. The other terms induce a transition from one to two triplets. Therefore, this process is called three-magnon scattering since it involves a magnon in the initial state and two magnons after the scattering process [416]. 3.3.3. Defect-induced light scattering An interesting aspect of dimerized quantum spin systems is the substitution of a magnetic site by a non-3d ion creating a spin vacancy. In principle every substituted site breaks up one dimer and frees one spinon [37,417]. This is noticeable as a Curie–Weiss contribution to the magnetic susceptibility. The question now arises whether this substitution may lead to similar well-de1ned bound states as in the pure dimerized phase. Observations using NMR and ESR would support such eKects since states in the spin-Peierls gap are induced by the substitution [37]. Furthermore, also a well-de1ned binding energy may be realized due to a binding eKect of the spinon to the defect site [36,353]. Taking a spin system with singlets at the sites (i − 2; i − 1), a substitution at site i − 2 leads to a spinon at site i − 1. The spinon moves freely without changing the number of singlets but changing their position as long as the position of the singlets does not change the energy of the system. In contrast, taking additional interactions into account this motion of the spinon away from the defect site costs energy. In 1rst approximation, i.e. for a suUciently large frustration, this binding energy is linear in the distance between the spinon and the defect site. In this sense the eKect of next nearest neighbor intrachain interaction and interchain or magnetoelastic interaction is expected to be similar [32,353]. From the theoretical side a light scattering process on such a spinon or dopant bound state is possible [416,353]. Starting from the Raman exchange light scattering Hamiltonian in Eq. (3.5) and using the limit of large dimerization the spin system now includes a spinon at site i −1. We consider the singlet at site (i; i+1), the relevant part of R is R=(1−>)Si−1 ·Si . Applying R to the | ↑-spinon at i − 1 and the adjacent singlet at (i; i + 1) yields √ (3.9) R| ↑; s = (1 − >)[(1=4)| ↑; t0 − 1=(2 2)| ↓; t1 ] =

1−> 1−> |s; ↑ + √ [| ↑; ↑; ↓ − | ↓; ↑; ↑] : 4 4 2

(3.10)

The arrows indicate the Sz component of the spinon and s; t0 or t±1 stand for a singlet and a triplet with Sz = 0 or Sz = ±1 for two spins. The eKect of the Raman operator may be summarized in shifting the spinon by one singlet spin pair to the right by two sites (the 1rst term in Eq. (3.10)) and the creation of a n.n.n. singlet (the second term in Eq. (3.10)) in between sites i − 1 and i + 1. The total spin of the system remains unchanged. Thus the Raman operator is able to induce transitions in between diKerent so-called dopant-bound spinons (DBS), e.g. in between the ground-state and the 1rst excited DBS states. This exchange

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65

400

Intensity (a.u.)

CuGeO3 300

T=40K

12K

200

100

5K 0

100

200

300

400

500

600

Raman shift (cm-1)

Fig. 3.3. Intrachain (cc) polarized Raman light scattering spectra of CuGeO3 for temperatures above and below TSP =14:3 K (the curves have been given an oKset for clarity).

scattering mechanism should show no magnetic 1eld dependence since initial and 1nal states are both triplets. Furthermore, the limit of strong dimerization used to facilitate the discussion is not mandatory as a comparable description is also valid in the limit of weak dimerization [353]. Experimental results that are consistent with this scenario have been obtained in Zn-substituted CuGeO3 and will be discussed in Section 4.1.2. 3.4. Spinon light scattering in CuGeO3 : experimental results Intrachain (cc) polarized Raman spectra of CuGeO3 are shown in Fig. 3.3 for temperatures above and below TSP =14:3 K [328]. Above TSP (40 K) a broad continuum extending from 150 to 500 cm−1 has attributed to spinon scattering. Lowering the temperature the intensity of this signal is suppressed (see Fig. 4.4 (open squares)). On the other hand, transition-induced modes appear that will be discussed in Section 4.1. The continuum is observed only for both incident and scattered light polarizations parallel to the CuO2 chain direction. These polarization selection rules are consistent with the dominant exchange path of a 1D spin system. The magnetic Raman light scattering intensity given by Eqs. (3.4) and (3.5) has been modelled using diKerent theoretical approaches [97,418–420]. In Fig. 3.4 typical experimental results of the spinon continuum observed in CuGeO3 in the homogeneous phase T ¿ TSP (T =20 K) are compared to theoretical model calculations [97]. Phonon lines have been subtracted from the measurement for clarity. The calculation has used a solitonic mean-1eld approach to evaluate the Raman intensity at 1nite temperatures [97,419]. Comparable results have also been obtained by other groups [418,420]. The observed broad continuum in the frequency range 150 –550 cm−1 is a consequence of the frustration term proportional to in Eq. (3.4) since the nearest neighbor term in (3.4) conserves the number of spinons. Therefore, this scattering contribution has been named “frustration-induced scattering” [97]. The derived coupling constants that give a qualitative good description of the experimental data (J = 150 K; = 0:24) are in agreement with results from 1ts to the magnetic susceptibility [294]; however, they are smaller than results derived from thermal expansion data [297]. As the model used in Eqs. (3.4) and (3.5) neglects phonon degrees of freedom as well as

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Fig. 3.4. Comparison of experimental Raman spectra (squares) with calculations (lines) for CuGeO3 in the homogeneous phase (T = 20 K). Phonon lines have been subtracted for clarity. Theoretical results are determined with = > = 0 for (a) Jnn = 150 K; = 0:24 (dashed dotted line) and (b) Jnn = 170 K; = 0:2 (dotted line).

interchain interaction the agreement between theory and experiment is promising enough to identify the continuum in Fig. 3.4 as of spinon scattering origin. For temperatures below TSP the Raman response of the spin-Peierls system changes considerably due to the dimerization , as shown in Fig. 3.3 [60,328]. As a reasonable theoretical approach a direct diagonalization of a system of 28 spins has been performed to determine the four-spin correlation function. The calculated excitation spectrum has been approximated analytically. A following study of the dependence on > in Eq. (3.5) and a comparison with experiment have led to > = 0:12. As shown in Fig. 3.5, the decrease of the intensity of the continuum and the existence of the sharp mode at 30 cm−1 is well described. The strong eKect of the >-term on the calculated spectra has been explained by matrix-element eKects of the Raman operator (Eq. (3.5)) [97]. An alternative description of the mode at 30 cm−1 as a magnetic singlet bound state will be discussed in Section 4. 3.5. Raman scattering from spin-1/2 ladders The very broad Raman response of spin-1/2 chain compounds like CuGeO3 , as evidenced in Fig. 3.4, is due to the 2-spinon-continuum illustrated in Fig. 1.1. The magnetic excitation spectrum of 2-leg ladders is, on the other hand, dominated by magnons (spin-1 excitations). This is illustrated in Fig. 1.4. The magnetic Raman response for a two-leg ladder is therefore quite narrow, especially for small ratios of J =J⊥ . In Fig. 3.6 the Raman response of CaV2 O5 from Ref. [421] is compared with a 1t using J =J⊥ = 0:09; J⊥ = 431 cm−1 (solid line) assuming an experimental resolution of / = 3 cm−1 . The long dashed line gives a 1t with J =J⊥ = 0:125; J⊥ = 447 cm−1 ; / = 0, the thin horizontal line corresponds to the oKset of the 1ts to account for a scattering background [77].

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500.0

CuGeO3

T=5K

Intensity (a.u.)

400.0

300.0

200.0

100.0

0.0 0.0

100.0

200.0

300.0

400.0

500.0

-1

Raman shift (cm )

Fig. 3.5. Comparison of experimental Raman spectra (squares) with calculations (bold line) for CuGeO3 in the dimerized spin-Peierls phase (T = 5 K). Phonon lines have been subtracted from the spectra with the exception of the spin-Peierls-induced phonon mode at 107 cm−1 .

Fig. 3.6. Experimental Raman response of CaV2 O5 [421] compared with diKerent model calculations as described in the text [77]. Fig. 3.7. Raman response of La6 Ca8 Cu24 O41 . Noisy thin lines: experimental data [179] in (aa) (dashed) and (cc) (solid) polarization scaled to the same constant value between 2000 and 2500 cm−1 . The thick solid line represents the theoretical result for J =J⊥ =1100 cm−1 and a resolution / =3 cm−1 [77]. The inset gives the 1-triplet dispersions !(k) for J =J⊥ and the cyclic 4-spin term parameter xcyc = 0; 0:05; 0:10; 0:15 (solid, dashed, long dashed, dotted) [77].

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An interesting aspect of magnetic Raman scattering from two-leg ladders is the possible occurrence of two maxima in the response for larger values of the ratio J =J⊥ due to the minimum in the magnon dispersion relation near k = 0 (see Fig. 1.4). The theoretically predicted second high-frequency maximum [77] has not yet been observed in Raman experiments. As an example we show in Fig. 3.7 the Raman response of La6 Ca8 Cu24 O41 , which can be modelled by J⊥ = J . Here, the thick solid line is the theory for J = J⊥ = 1100 cm−1 and a resolution / = 3 cm−1 [77]. The thin horizontal line corresponds to an oKset of the 1t to account for background. The non-observance of the second magnetic Raman peak in the experimental data by Sugai et al. (1999) [179] may be due to an additional cyclic 4-spin term in the Hamiltonian, discussed in the context of spin-1/2 ladders [77,422] or due to overlapping contributions from the two-particle continuum (see Fig. 1.4).

4. Magnetic bounds states Magnetic bound states of elementary triplets are a general feature of all quantum spin systems with a singlet ground state and a gapped excitation spectrum. The bound state energies are reduced below the energy of the two-triplet continuum. Important parameters concerning this binding eKect are spin frustration as given by a competing next nearest neighbor exchange interaction as well as interchain exchange or magnetoelastic interaction. The properties of magnetic bound states such as energy, spectral weight and multiplicity characterize the spin–spin correlations or instabilities of a quantum spin system. One-dimensional exchange topologies analyzed theoretically concern the dimerized and frustrated spin chain [6,29,423–425], the one-dimensional Hubbard model [426], spin chains with interchain [32] or spin–phonon interaction [33–36], spin ladders [21], and frustrated spin ladders [17,30]. Additional bound states are also induced by defects in spin chain systems with frustration or interchain coupling [36,353]. With respect to 2D systems the strongly frustrated Shastry–Sutherland lattice [118,427,428], the frustrated [429] or 1/5-depleted square lattice [27,28] and the KagomQe lattice [430] have been investigated. The latter and related systems with strong frustration and a degenerate ground state are recently in the center of interest as they exhibit a gapped elementary triplet spectrum coexisting with and 1lled up with collective singlet (S = 0) excitations. These non-magnetic excitations may have energies smaller than the gap, depending on a delicate balance of the coupling constants of the systems [431,432,63]. In a dimerized spin chain (purely magnetic 1D model) the magnetic excitation spectrum contains a triplet branch 01 and a corresponding two-particle continuum of triplet excitations for energies above 201 (k = 0). If a frustration of the spin system due to next nearest neighbor intrachain exchange or an interchain interaction is taken into account well-de1ned magnetic bound states develop [6,29,32]. These states consist of strongly interacting triplet excitations. The high-energy cutoK of these bound states is again given by 201 (k). Above this energy the bound states dissolve in the two-particle continuum. For a critically √ frustrated spin chain (J2 =J1 =c =0:24) one singlet bound state is predicted with an energy 00 = 301 . A further triplet bound state with a considerable spectral weight exists only for larger values of J2 [6,29].

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2.2

2.0

1.8

CL

R

1.6

1.4

1.2

1.0 0.00

δ=0.4 δ=0.2 δ=0.1 δ=0.05 δ=0.02

0.10

x

x x

ED

xx x x xx

DMRG

0.20

α

αc

0.30

0.40

0.50

Fig. 4.1. Excitation diagram !(k) of a dimerized ( = 0:2) and frustrated ( = 0:2) antiferromagnetic Heisenberg chain [29]. Starting from the singlet ground state with increasing energy a triplet state (full dots), a singlet bound state (open squares) and a second triplet state (closed triangles) with a minimum k = kmin have been found. Fig. 4.2. EKect of the frustration on the ratio R = 00 =01 of the singlet vs. the triplet state energy (for k = 0). DiKerent dimerization levels are investigated using exact diagonalization √ (ED) and density matrix renormalization group theory (DMRG). For = c = 0:2412 the “classical limit” with R = 3 is recovered.

In Fig. 4.1 the calculated excitation diagram of a frustrated alternating Heisenberg chain is given for moderate frustration [29]. It is evident that the spectral weight of the second triplet bound state is restricted to a smaller phase space with an energy close to the continuum (201 ). The ratio R = 00 =01 of the singlet vs. the triplet state energy characterizes the binding eKect and thereby the triplet interactions. In calculations using exact diagonalization and DMRG (Fig. 4.2) frustration has been found to be the main parameter controlling the binding energy of the singlet bound state vs. the triplet. In these purely one-dimensional systems the dimerization is of less importance for the ratio R. This eKect may also be characterized by the “spinon velocity” that is reduced with frustration leading to a larger density of states for k → 0 [423]. The excitation spectrum of a two-leg Heisenberg ladder is not so well established. In 1rst approximation it may be compared with a single frustrated chain. At least one singlet and one triplet bound state should exist. The strong coupling singlet ground state consists of a dimer on every rung of the ladder [21]. This situation changes drastically with intraladder frustration [17]. In the limit of dominant frustration the system is in a gapped “Haldane” phase, which can be mapped on a s = 1 chain. The dimer phase, realized for smaller frustration, and the “Haldane” phase are separated by a 1rst-order transition line (quantum phase transition). Especially for coupling parameters close to the transition a complicated behavior of low-lying singlet and triplet bound states has been discussed [30]. An increasing number of bound states with enlarged binding energy should exist. These states soften below the one-particle triplets [30]. Finally, at the transition, many-particle states form which intermix strongly with each other. This general softening of both singlet and triplet modes at k = 0 and

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the resulting degeneracy of in1nite-particle states mark the quantum phase transition. As the size of the bound states in real space as well as their spectral weight increase dramatically this scenario should also have a profound eKect at 1nite temperature, i.e. an increasing number of bound states should be observable [30]. Further important aspects of bound state formation especially in real systems are interchain and magnetoelastic interaction [32]. The corresponding states are often denoted as soliton/antisoliton bound states. In the following we will use the limit of strong dimerization to elucidate this eKect. Interchain or magnetoelastic interactions lead to an energetically preferred dimer pattern. Breaking a dimer forms two seemingly independent solitons in the surrounding of the well ordered dimers. A consequent displacement of one soliton with respect to the other distorts the dimer pattern and increases the energy of the system. In this way a binding potential which is approximately linear with distance is realized. The state with lowest energy in this soliton/antisoliton binding potential is a triplet. A number of further excitation levels is realized, the soliton bound states. As the potential is steeper with increasing interaction, a smaller number of bound states is expected in systems that are closer to two dimensions [32]. For larger distances of the solitons, corresponding to energies above 2, new soliton/antisoliton states are created. In the limit of vanishing interaction a soliton continuum of quasi-free particles is formed. From the experimental point of view the importance of interchain interaction on the magnetic excitation spectrum of quasi-one dimensional compounds is not clear. This is caused by the fact that most of the so-far investigated systems exhibit both a non-negligible spin–phonon and interchain interaction. Regardless of this complexity, the observed phenomena were discussed and compared in most cases with purely magnetic model calculations. Experimental evidence for singlet bound states, excited triplet states (three-magnon process) and defect-induced states have been found in Raman scattering. In neutron scattering experiments triplet modes interpreted as a bound states have been observed. A complete discussion of the respective compounds and corresponding references will be given below. Frustration is a very important aspect for 2D spin systems as it leads both to a gap in the triplet spectrum and an appreciable binding of the singlet states. A model system within this respect is the strongly frustrated SrCu2 (BO3 )2 [111]. In the underlying 2D orthogonal dimer lattice triplets are strongly localized [11,28] and form bound states [118,427,428]. For this exchange topology a quantum critical point dividing a spin liquid from a NQeel state has been postulated. The phase boundary is controlled by the ratio of the interdimer to the frustrating intradimer exchange coupling. A comparison of calculations of the magnetic susceptibility with measurements suggests that the compound is indeed close to the critical point [28,115,117]. Furthermore, a large number of low-energy singlet modes are observed in Raman scattering experiments that are attributed to two- and three-particle bound states [124]. For further discussion see Section 4.4. To assign experimentally the observed excitations to magnetic bound states and distinguish them from alternative explanations a number of stringent criteria have been developed. The polarization selection rules and preferred energies in the interval [01 ; 201 ] have already been discussed. In addition, the application of a magnetic 1eld should identify one-magnon scattering as a splitting or shift of the signal. The strength of the obligatory spin–orbit coupling may be judged by the g-value of the compound. Furthermore, the intensity or spectral weight of the additional modes as function of temperature or defect concentration is a useful characteristic. The singlet bound state as a composite state at the /-point is very sensitive to any Ouctuation. It should therefore be

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1.0

(cc)

T (K)

(c)

T (K)

(I - I min) / ∆I

Intensity (a.u.)

CuGeO 3

71

CuGeO 3

0.5

0.0 5

Brillouin shift (cm -1 )

10

15

20

25

Temperature (K)

Fig. 4.3. Brillouin light scattering spectra of CuGeO3 in intrachain (cc) polarization for several temperatures showing: (a) the singlet bound state response (the curves have been given an oKset for clarity); (b) three-magnon scattering for T ¡ TSP ; and (c) the disappearance of the scattering intensity for T ¿ TSP [416]. Fig. 4.4. Renormalized scattering intensities of the spin-Peierls-induced modes in CuGeO3 . The integrated intensity of the spinon continuum is marked by open squares. The modes at 369, 224, and 30 cm−1 are given by closed squares, closed triangles, and open circles, respectively [328].

destroyed with rising temperature or due to defects. In contrast, two-magnon scattering (from the zone boundary) persists as paramagnon scattering for T TN due to short-range correlations [400]. To distinguish magnetic bound states from phonon-related bound states several arguments based on the frequency and lineshape of the observed modes may be used. They will be described in Section 4.2. 4.1. Bound states in CuGeO3 The thoroughly studied inorganic spin-Peierls system CuGeO3 has been used as a model system concerning bound states in dimerized spin chains [6,21,29,32,33,35,353]. In neutron scattering experiments a triplet state, 01 = 2:1–2:4 meV, with a second, similar gap to a continuum is observed [295,296,320,319]. In light scattering experiments a sharp asymmetric mode is observed at 00 = 30 cm−1 ≈ 1:7801 . The appearance of this mode in the dimerized phase of CuGeO3 is shown in Fig. 4.3(a) [416]. These experiments have been performed using a Sandercock-type tandem Fabry– Perot interferometer in 90◦ scattering geometry. The mode at 30 cm−1 has been discussed controversially as a “1D two-magnon density of states”, a soft mode connected with the spin-Peierls transition or as a continuum of bound states [97,308,309, 313,326,415,433]. It is now understood as a singlet bound state 00 of triplet excitations [328,416]. This mode is only observable in light scattering experiments with intrachain polarization, i.e. incident and scattered light are polarized along the chain direction. This polarization selection rule is consistent with the spin conserving nature (no spin Oip, i.e. YS =0) of the exchange light scattering mechanism, as described in Section 3. In addition this low-energy mode shows no shift or splitting in an applied magnetic 1eld [313].

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Raman shift (cm -1)

(a)

CuGeO3

Temperature (K)

Fig. 4.5. Temperature dependence of the energy of the singlet bound state (open symbols) and the three-magnon scattering at 18 cm−1 (1lled symbols). Solid lines are guides to the eye. (b) Temperature dependence of the intensity of the 18-cm−1 shoulder. The solid line corresponds to an activated T -dependence of the intensity of the three-magnon signal [416].

The temperature dependence of the integrated light scattering intensity of several spin-Peierlsinduced modes is shown in Fig. 4.4 [328]. For elevated temperatures but still below the spin-Peierls transition T 6 TSP the intensity of the singlet bound state at 30 cm−1 decreases faster compared to the intensity of the dimerization-induced phonon modes at 369 and 224 cm−1 . This approximately linear decrease of integrated intensity with rising temperature shown in Fig. 4.4 by the open circles is a “1ngerprint” of bound states as they are very sensitive to thermal Ouctuations. The same eKect leads to an enormous broadening comparable with the energy of this mode for temperatures still far below T = TSP . The strong decrease of the scattering intensity of the bound state and its broadening limits the temperature range in which the frequency of the bound state can be determined. This temperature dependence is composed of a product of 01 (T ) and the temperature dependence of the binding energy (a weak function of the dimerization). As seen in Fig. 4.5(a) (open squares), this temperature dependence is non-mean 1eld-like. It is not-identical to 01 (T ) determined by neutron scattering experiments [295]. 4.1.1. Three-magnon scattering in CuGeO3 With rising temperature the lowest triplet branch in CuGeO3 is thermally populated and a new scattering channel opens up at lower frequency. This three-magnon scattering corresponds to transitions between thermally excited states, i.e. from the triplet branch at 01 to the continuum of two-triplet excitations with its low-energy onset at 201 . Therefore, 01 is also expected to be the low-energy boundary for this process. A theoretical description of this process is given in Section 3.3.2. In Fig. 4.3(a) Brillouin spectra of CuGeO3 at several temperatures below TSP are shown. It is evident which at elevated temperatures the bound state at 30 cm−1 is not the only signal. At temperatures above 7 K a shoulder at frequencies of about 18 cm−1 develops that at a temperature of 13 K is comparable in intensity with the singlet bound state. At temperatures above 14 K this signal disappears again. As shown in Fig. 4.5(a), the energy of this mode has a similar temperature dependence as the singlet at 30 K [416]. It must therefore be closely related to the singlet–triplet gap 01 . The ratio of both energies corresponds to the binding energy of the singlet bound state. The intensity

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of the 18 cm−1 mode, however, shows just the opposite temperature dependence compared with the intensity of the singlet bound state (compare open circles in Fig. 4.4 with Fig. 4.5(b). The 1t in Fig. 4.5(b) corresponds to the temperature dependence that would be expected for a thermally activated process. The light scattering intensity at 18 cm−1 is totally intrachain polarized and observable neither in scattering in crossed nor in circular light polarization as all magnetic scattering in CuGeO3 . No magnetic 1eld dependence of these signals could be found up to 6 T [304,416]. These experimental results rule out one-magnon scattering which should have antisymmetric Raman tensor elements and should display a Zeeman splitting. Therefore, all experimental observations support the interpretation of the mode at 18 cm−1 as a three-magnon scattering process between excited triplet states. 4.1.2. Defect-induced bound states in CuGeO3 The eKect of substitutions either on the Cu site or on the Ge site of CuGeO3 may be summarized as a strong change of the low-energy excitations in the dimerized phase as well as the creation of long-range NQeel order. For a summary of substitution eKects we refer to Sections 2.7.1 and 3.3.3. Every spin vacancy breaks up one dimer and frees one spinon [37,417]. As there is a binding eKect of the spinon to the defect site [32,36] and the Raman operator allows the excitation of the ground state of this spinon [353] it may lead to well-de1ned modes in Raman scattering experiments, the so-called dopant-bound spinons [416]. Indeed, in early Raman scattering experiments in CuGeO3 a mode with an energy smaller than the singlet bound state 00 has been observed. Some of these and later results have been tentatively attributed to one-magnon scattering [326,352,434]. Only recently more detailed experiments together with appropriate theoretical approaches have become available [36,353] (see also Section 3.3.3). In describing these experimental results we will concentrate on the region with low and intermediate substitution level to minimize possible defect interactions. Furthermore, a substitution with a negligible eKect on the global lattice properties and a very local disruption of the intrachain exchange path is desirable. All this is realized for Zn substitution of the Cu site of CuGeO3 [328,345]. There is no strong change of the lattice constants or the phonon frequencies with Zn doping and samples with a small and homogeneous Zn distribution are easily prepared [338]. In contrast, substitutions of Ge by Si lead to strong eKects of lattice pressure due to the diKerent ionic radii and possible changes of the interchain interaction [328,344]. In addition to the observed dopant-bound spinon in CuGeO3 , the eKect of substitutions are twofold. On the one hand the singlet bound state (of the pure system) is broadened and suppressed in spectral weight as a coherent con1ned state of two triplets [328,434]. On the other hand short-range dimer Ouctuations are enhanced [37]. The latter eKect has been studied in Zn as well as Si-substituted samples using Raman scattering of the dimerization-induced phonon mode at 370 cm−1 and X-ray scattering in Mg-substituted samples [328,341,352,435]. The eKect of substitutions on the singlet bound state is demonstrated in Fig. 4.6 presenting Brillouin spectra of Cu1−x Znx GeO3 with x =0–3.3% [304,353]. The singlet bound state (SBS) is well resolved as a sharp and asymmetric peak at 30 cm−1 . With Zn substitution the intensity of this mode drops drastically. Already for x=2:0% no de1nite mode is observed anymore. In the left panels of this 1gure the loss of integrated intensity and the frequency shift of the singlet bound state with substitution are given.

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 3.5

Cu1-xZnxGeO3

1.2

Cu1-xZnxGeO3

*1/6

2.5

1.0

0%

DBS

1.5 0.5

*1/5

1.0

DBS

2.2 K

0.8

0.6

DBS

0.2%

SBS

32

0.66% -1

Brillouin shift ( cm )

28

1.4%

DBS 2.0% 3.3%

10 0

1

2

8K

30

SBS 30

15

6K

0.4

32

Brillouin shift (cm-1)

SBS

DBS

3

4

Zn concentration (%)

5

10 15 20 25 30 35 40 45

Brillouin shift (cm -1)

Intensity (a.u.)

0.5

x=0.66%

SBS

1.0

SBS 2.0

Intensity ( a.u.)

Intensity (a.u.)

3.0

28

SBS 26

10 K 12 K

14

DBS

14 K

12 2

4

6

8

10

Temperature (K)

5

10

15

20

25

30

35

40

45

Brillouin shift ( cm-1)

Fig. 4.6. Brillouin spectra of Cu1−x Znx GeO3 ; x = 0–3.3% in intrachain (cc) polarization are shown on the right panel with the singlet bound state (SBS) and the defect-induced dopant-bound spinon (DBS) (T =2:2 K). The intensity corresponding to x = 0 is reduced by 1/6 and the SBS at x = 0:2% by 1/5. The upper and the lower left panel show the integrated intensity and the frequency of the modes as function of x [353].

Fig. 4.7. Temperature dependence of the singlet bound state (SBS) and the dopant-bound spinon (DBS) in intrachain polarized Brillouin spectra of Cu1−x Znx GeO3 , with x = 0:66% (right panel). The temperature dependence of the integrated intensity and the peak position of the modes are given in the upper and lower left panel [353].

A new well-de1ned excitation, denoted as a dopant-bound spinon (DBS), is observed at nearly half the frequency of the singlet bound state [353]. The integrated intensity of this mode increases roughly linear with increasing x for x 6 1% (see upper left panel of Fig. 4.6). This linear increase corresponds to the desirable case of non- or weakly interacting defects. For larger x the intensity of this mode decreases again. In this concentration region the lineshapes of the two modes and the dependence of the frequencies of the modes on x are very similar. In addition, both modes are only observed in the presented intrachain (cc) scattering geometry. The temperature dependence (2.2–14 K) of the Brillouin spectra for 1xed x = 0:66% in Fig. 4.7 indicates further similarities of the two modes [353]. The integrated intensity, the energy as well as the linewidth vary in a comparable way with temperature. The left-hand panels of this 1gure show a detailed analysis of the data. Therefore, both modes must be related to the singlet–triplet gap. A further important piece of experimental evidence is the application of a magnetic 1eld as it should discriminate possible one-magnon scattering contributions from magnetic bound states. The former scattering intensity should show a splitting or a shift of the observed signals. In Fig. 4.8 a comparison of Brillouin spectra with B = 0 and 4 T of Cu1−x Znx GeO3 with x = 0:66% is shown

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

75

0.5

Cu1-xZnxGeO3 *2

x=0.66% (cc) Intensity (a.u.)

0.4

B=4T 0.3

0.2

B=0T 0

10

20

30

40

50

Brillouin shift (cm -1)

Fig. 4.8. Brillouin spectra of Cu1−x Znx GeO3 with x = 0:66% in intrachain (cc) polarization with an applied magnetic 1eld of B = 0 and 4 T (T = 2:2 K) [304].

[304]. Neither the singlet bound state nor the dopant-bound spinon show an eKect of the applied magnetic 1eld. In concluding, the presented data undoubtedly prove the existence of bound states of a substitutioninduced spinon at the defect site. Neither the above discussed three-magnon light scattering process observed at elevated temperature nor one-magnon scattering can explain the observed experimental results. 4.2. Bound states in NaV2 O5 The compound NaV2 O5 diKers in several aspects from CuGeO3 . This is summarized in its description as a quarter-1lled ladder that on one hand adds electronic degrees of freedom to the system and on the other hand, as a ladder, has a diKerent exchange topology. The dimerization-induced modes are altered from CuGeO3 with respect to their multiplicity and polarization selection rules. Similar to CuGeO3 , the investigation of the temperature dependence of the intensity, frequency and half-width of these additional modes lead to a clear distinction between modes at lowenergy that are candidates for magnetic bound states and simple zone-folded phonon modes due to the crystallographic distortion. From phonon and low-energy quasielastic scattering strong Ouctuations are evident with an onset at TCO ≈ 80 K. This has been discussed in Section 2.7.2. These Ouctuations may also be important for the understanding of the magnetic bound states in NaV2 O5 . 4.2.1. Experimental observations in NaV2 O5 A comparison of intraladder (bb) spectra at T = 100 K and T = 5 K (Fig. 4.9) indicates several new modes in the low-temperature phase of NaV2 O5 [362]. In addition, a step-like decrease of the background scattering intensity is observed for frequencies Y! ¡ 120 cm−1 = 83:5 K. This marks

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 200

Intensity (arb.units)

(bb)

NaV2 O 5

67

180 160

246

107

948

396

140

134

120

652

100

202

80

T=5K

60

T=100K

40 0

200

400

600

800

1000

Raman shift (cm -1)

Fig. 4.9. Raman light scattering spectra of NaV2 O5 at T = 100 and 5 K in intraladder (bb) polarization. The additional modes in the low-temperature phase are marked [362].

the onset of a gap with 01 = 86 K. This value is smaller than k01AF = 115 K for kAF = (h; 1=2; 0) and h; 1=2; 0 = 89 K with h = half -integer observed in neutron scattering h = integer, but is comparable to 01 [373,374]. In Fig. 4.10 the intensity and linewidth of selected modes visible in the low-temperature phase are displayed as function of temperature [362]. Three low-energy modes at 67, 107, and 134 cm−1 have a particular temperature dependence of the intensity comparable with dimerization-induced phonon modes. The intensity increases more gradually upon cooling, with a smaller slope. No saturation in intensity is observed toward lowest temperature. Additionally, these three modes strongly broaden and soften slightly toward lower frequencies approaching TSP from below. In contrast, the modes at 202, 246, and 948 cm−1 follow the scheme of zone-folded phonons. They neither show any broadening nor any shift in frequency as they become Raman active due to the lowering of the symmetry for T 6 TSP . Selection rules and magnetic Eeld dependence. An analysis of the selection rules of the Raman modes, given in Fig. 4.11, shows, however, that the bound state candidates with the exception of the mode at 107 cm−1 are of mixed symmetry, i.e. the 67 and 134 cm−1 -modes appear in interchain (aa) as well as intrachain (bb) and crossed (ab) polarizations [362]. The 107 cm−1 -mode is observed only in intrachain (bb) polarization. The latter would be the preferable polarization selection rule of a magnetic bound state in a model spin chain. With 01 = 85 K ≡ 60 cm−1 from susceptibility measurements these three low-energy modes are situated below or near 201 = 120 cm−1 . This gap value is also con1rmed in the Raman light scattering experiments as a drop in the background intensity. As the number and selection rules of the singlet bound states diKer from what has been observed in CuGeO3 we will carefully check this assignment. In view of the very close energy of the 67 cm−1 -mode to the singlet–triplet energy gap (01 = 60 cm−1 ), this mode might be one-magnon scattering allowed due to spin–orbit coupling [436]. However, the g-value in NaV2 O5 is even closer to 2 than in CuGeO3 [207,437], pointing to

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 (a)

1.0

200

67 cm-1

(bb)

NaV2 O5

107 cm-1

0.6

T=60K

134 cm-1

100

NaV2 O 5

0.4

T=5K 67 cm-1

0.2

107 cm-1 134 cm-1

0.0

(b) 202 cm -1

1.0

246 cm -1

0.8

67 cm-1

948 cm -1 0.6

Intensity (arb. units)

Norm. intensity

0.8

FWHM (normal.)

77

0 200

(aa) 100

T=5K

T=60K 0 200

(ab)

0.4

107 cm-1

T=5K

100

0.2 0.0

T=60K 5

10

15

20

25

Temperature (K)

30

35

0 0

50

100

150

200

250

-1

Raman shift (cm )

Fig. 4.10. (a) Integrated intensity of selected transition-induced modes observed in NaV2 O5 for temperatures below TSP and (b) normalized half-width (FWHM) of the additional modes as function of temperature [362]. Fig. 4.11. Low-energy Raman light scattering spectra of NaV2 O5 for three light scattering polarizations in the ab plane of the single crystal [362].

a negligible orbital momentum in this compound. Recently, however, a combination of the electronic structure and the DM interaction, allowed by symmetry in this material, has been discussed to describe a observable Raman cross-section of one- and two-magnon scattering [438]. Applying a magnetic 1eld may lift the three times degeneracy of a triplet state resulting in a splitting or shift of the 67 cm−1 -mode. As shown in Fig. 4.12 neither a shift, a splitting, nor even a broadening of this mode in a 1eld up to 7 T is observed [189]. Therefore, a simple one-magnon interpretation has been ruled out. In a similar approach the mode at 134 cm−1 has been identi1ed as the corresponding two-magnon scattering [439]. This, however, would be surprising. Already in two-dimensional spin systems two-magnon scattering is strongly renormalized to lower frequencies due to magnon–magnon interaction. Coherent magnetic oscillations and time-resolved probes. New excitations related to the magnetic system of NaV2 O5 have recently been observed in pump–probe femtosecond time-domain spectroscopy. A mode at 127 cm−1 , corresponding to an energy nearly twice the spin gap, is attributed to a bound state. The selection rule of the coherent oscillation following from the circular polarized

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

0.08

(ba)

N aV2 O5

T=5 K

(bb)

NaxV2O5

T=1.5 K

x=0.97

0.07

0.06

x=0.98

Intensity (a.u.)

Intensity (arb.units)

B = 7T 0.05

0.04

x=0.99

4T 0.03

phonon

0.02

x=1.0

0T 0.01 50

55

60

65

70

75

Raman shift ( cm ) -1

80

60

80

100

120

140

-1

Raman shift (cm )

Fig. 4.12. Magnetic 1eld dependence of the Raman mode at 67 cm−1 of NaV2 O5 in crossed (ba) light scattering polarization. No splitting or broadening in a 1eld up to 7 T is observed [189]. Fig. 4.13. EKect of Na de1ciency in NaV2 O5 , with x = 0:97–1 on Raman light scattering spectra in intrachain (bb) polarization [445,446].

light used for the pump or probe beam points to its origin in a spin-Oip excitation from the singlet ground state [440]. Other excitation at 90.6, 181, 230, 303, 420, 450 and 531 agree within 3 cm−1 with phonons observed in Raman scattering. In the pump–probe experiments they show diKerent temperature dependencies and selection rules as the bound state at 127 cm−1 . The application of this and other time-resolved techniques is at present rather limited. Exciting results exits for the 1D spin chain system Sr 2 CuO3 , were an instantaneous photoinduced absorption and picosecond recovery due to very large dipole coupling between nearly degenerate one and two-photon states is observed. These eKects are based on the electronic properties of the 1D Mott insulator [157]. Further investigations on other strongly correlated and low-dimensional systems are under way. Phonon-related bound states. Also phonon-related bound states may be considered. Indeed, such phenomena were observed for example in YbS [441–443]. These modes were interpreted as excitons interacting with a LO phonon at !0 giving rise to exciton–phonon bound states with frequencies ! = n · !0 , with integer n. The polarization selection rules are identical with the LO phonon. Finally,

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

79

the linewidth of these phonon bound states is a linear function of n. Integer multiples of excitation energies can be found in NaV2 O5 by assuming an !0 of about 30 cm−1 . However, no such mode could be detected in light scattering experiments. Besides, the light scattering intensity of the bound states in NaV2 O5 appears not only in the fully symmetric scattering components (aa); (bb), but also in the crossed (ab) component. A similar disagreement is found for the linewidth that does not show a systematic broadening. Also the scenario of folded phonons with an additional coupling to the spin and charge degrees of freedom has been discussed. Recent infrared absorption studies on NaV2 O5 showing numerous modes with partly weak intensity in close proximity to the Raman excitations follow a similar route [444]. These approaches are to the present understanding not compatible with results from temperature and substitution studies. Summarizing, the experimental results in NaV2 O5 are not compatible with an approach based on phonon bound states. 4.2.2. DeEciency and substitutions on the Na site One of the most subtle problems preparing Nax V2 O5 samples is to control the Na content x [199,447]. Deviations from the nominal stoichiometry result in a shift of the ratio between V4+ and V5+ toward the non-magnetic (s = 0) V5+ . This leads to an eKective substitution. Further possible substitutions in NaV2 O5 are Li and Ca substitution on the Na site [199]. Systematic studies of Na de1ciencies, as reported in Refs. [200,367], indeed show that the drop in the magnetic susceptibility is suppressed and TSP is slightly shifted toward lower temperatures. Therefore, this Na-de1ciency is also a promising tool to discuss the bound states. A direct comparison of spectra with x = 0:97–1 in intrachain (bb) scattering con1guration is displayed in Fig. 4.13 [445,446]. Solely, the three bound states show a considerable shift in frequency and a strong broadening. The zone-folded phonon modes do not show this shift toward lower frequencies. The frequency shift of the bound states is quite naturally explained by assuming that the Na de1ciency leads to a reduced singlet–triplet gap and hence to a smaller energy of the magnetic bound states. For Li substitution on the Na site a similar but somewhat smaller shift of the bound states is observed. In Fig. 4.14 spectra of Na1−x Lix V2 O5 are shown [445]. For larger Li concentration (7%) only a broad maximum remains of the bound state scattering contribution. It should be noted, that all magnetic bound states react in a similar way to the substitution while the eKect on the phonon modes is weaker. These results are comparable to the eKect of Zn substitution on the magnetic bound state in CuGeO3 [328]. 4.2.3. Theoretical considerations Concluding the experimental data it is stated that the modes at 67, 107 and 134 cm−1 are described best as singlet bound states. They, however, are not consistent in number as well as in selection rules with the excitation spectrum of a frustrated and dimerized 1D Heisenberg chain, which exhibits one singlet bound state. The intensity of this singlet bound state should only be observed in intrachain scattering con1guration, i.e. with light scattering polarization parallel to the dominant exchange path. In the high-temperature phase (T ¿ TSP ) NaV2 O5 is described as a quarter-1lled ladder with charge degrees of freedom. However, since the charge transfer gap CT ≈ 1 eV is large, the low-energy excitations are dominated by spin Ouctuations of one spin per rung along the ladder direction. This leads to a mapping of the system on a spin 1/2 Heisenberg chain [287]. Actually, the mode at

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

(bb)

Na1-xLixV2O 5

400

x=0.07

350

Intensity (arb. units)

T= 5K

300

x=0.05

250 200

x=0.024 150 100

x= 0

50 0 0

100

200

300

400

500

600

Raman shift (cm-1)

Fig. 4.14. EKect of Li substitution in Na1−x Lix V2 O5 , with x = 0– 0.07, on Raman light scattering spectra in intrachain (bb) polarization [445].

107 cm−1 = 1:78 × 01 is consistent both in selection rule and energy with the singlet bound state in CuGeO3 (30 cm−1 = 1:78 × 01 ). Therefore, we identify this mode as the singlet bound state of an eKective chain. To describe the two unpolarized bound states, a quasi-2D system or a cluster model [398,448] in the ab plane of the compound may be relevant. This means taking into consideration dimers parallel, perpendicular as well as diagonal to the ladder. The observed bound states have a small energy separation compared to CT ≈ 1 eV. Actually, their energy scale is given by the singlet–triplet gap 01 [373,374]. Therefore, the character of the excitations is predominantly magnetic allowing a description of the bound states in the context of a pure spin model. The analogy of this situation to the dynamical Jahn–Teller eKect motivated the proposal of a dynamic dimer formation as a ground state of the system [362,188]. The superposition of diKerent energetically nearly degenerate dimer con1gurations may show up in the triplet channel of the excitation spectrum. This phenomenological approach, however, has later not been supported by synchrotron X-ray scattering experiments as the low-temperature phase showed only evidence for a static stacking disorder of zig-zag charge ordering patterns in NaV2 O5 [359]. 4.3. Bound states in (VO)2 P2 O7 The compound (VO)2 P2 O7 was for many years a canonical example for a spin ladder compound (see Section 2.5.2). Only recently neutron scattering experiments on single crystals have determined that the dominant exchange path is along the b-axis direction. In addition, two strongly dispersing triplet modes (01 = 25 cm−1 and 01 = 46 cm−1 ) have been observed. The second mode has been interpreted as the triplet bound state of an alternating spin chain [217]. The experimental spectral weight of both modes has been found to be comparable. However, no triplet continuum has been

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103 20

(bb)

(aa)

81

(ab)

10 K 4K

Im χ

15

(VO)2 P2 O7 20 K

10 20 K 10 K 75 K 5

20 K

40 K 70 K

0

20

30

40

50

60

20

30

40

50

60

20

30

40

50

60

-1

Raman shift (cm ) Fig. 4.15. Low-energy Raman scattering in (VO)2 P2 O7 with light scattering polarizations in the ab plane. The spectra are corrected by the Bose factor. A shoulder with an onset at 45 cm−1 is observed in intrachain (bb) as well as interchain (aa) polarization [214,449].

observed like in CuGeO3 [319]. To clarify which model applies, such as a purely 1D magnetic chain, frustration or a more complicated scenario including spin–phonon coupling, other spectroscopic information together with theoretical modelling is needed. In calculations using a “simple” alternating spin chain the spectral weight of the triplet bound states has been found to be restricted to a limited part of the Brillouin zone. This result is in contrast to the experimental neutron scattering observations [222] and motivated two re1ned theoretical studies taking an additional diagonal coupling (frustration) via the phosphate group into account [31,223]. With this approach a much better description of the neutron scattering experiments has −1 has been postubeen achieved. In addition, a singlet bound states with k=0 00 = 1:6801 = 42 cm lated [223]. Concluding, a frustrated quasi-2D spin model might describe the excitations observed in neutron scattering experiments with respect to their energy and spectral weight across the whole Brillouin zone. In Raman light scattering experiments a shoulder of the scattering intensity is observed at low temperatures with an onset at 45 cm−1 , an energy only slightly renormalized with respect to 201 = 50 cm−1 determined by neutron scattering experiments (see Fig. 4.15) [214,217,449]. This intensity is masked at higher energy (Y! ¿ 60 cm−1 ) by phonon scattering and shows up with light polarizations both parallel and perpendicular to the dominant exchange direction. It is not observed in crossed light polarization. This excludes two-magnon scattering of an usual 2D spin system. As no splitting or shift of the shoulder is evident in an applied magnetic 1eld, one-magnon scattering is excluded [214,449]. A later Raman study on (VO)2 P2 O7 showed evidence for a peak at 90 cm−1 and attributed this mode to the second gap [224]. The shoulder at 45 cm−1 is observed for T ¡ 50 K in interchain (aa) polarization and for T ¡ 75 K ≈ 201 in intrachain (bb) polarization. This temperature is comparable to the size of the gap itself (see Fig. 4.16). The intensity of this shoulder shows a linear increase with decreasing temperature

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P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

norm. Intensity at 47 cm

-1

1.2

1.0

(aa) (bb)

(VO)2 P2 O7 0.8

0.6

0.4

0.2

0.0

0

10

20

30

40

50

60

70

80

Temperature (K)

Fig. 4.16. Normalized intensity of the Raman scattering at 47 cm−1 with respect to the intensity at lower frequency in (VO)2 P2 O7 as function of temperature. Open circles: (aa) light polarization; black squares: (bb) light polarization [214].

comparable to the bound states in NaV2 O5 or CuGeO3 . All these observations allow an assignment of this shoulder to a singlet bound state with a negligible binding energy. As this binding is weaker than predicted for the frustrated quasi-2D spin system, a purely magnetic model for the excitation spectrum in (VO)2 P2 O7 may not be appropriate. The observed anomalies of phonons, with energies close to the magnetic excitations, point to this alternative direction (see Section 2.5.2). Theoretically, phonon- or magnetoelasticity-induced bound state phenomena are expected if optical phonon modes have an energy close to the magnetic spectrum of the compound [33–35]. This scenario might be relevant for the case of (VO)2 P2 O7 . 4.4. Bound states in SrCu2 (BO3 )2 Low-energy light scattering in SrCu2 (BO3 )2 exhibits a drastic evolution with decreasing temperature [124]. Quasielastic scattering observed at high temperatures (T ¿ 01 ) is replaced by a broad two-magnon-like continuum centered near 50 cm−1 . The selection rules of these intensities yield B2 symmetry, i.e. the light polarization vectors along (ab) connect next nearest dimers within the ab plane of the compound [124]. In this polarization no Raman scattering process from the dimer ground state is allowed [118,428]. The observed scattering intensity is therefore due to scattering on thermally induced and strongly localized triplet states of the Shastry–Sutherland lattice. Consequently for T → 0 all scattering is suppressed in this geometry due to the large spin gap (see Fig. 4.17). The temperature dependencies can be mapped on thermodynamic data, such as the magnetic susceptibility and the speci1c heat of the spin system as shown in Fig. 4.18. The temperature dependence of the intensity of the 30-cm−1 bound state is shown in Fig. 4.18(a). The solid line is a 1t based on the strong damping of this state due to scattering on thermally excited triplet states. The decrease of the scattering intensity IB with temperature is governed by the density of excitations and follows the equation IB (T ) ˙ (1 − Ae−=kB T ), where A is a constant. A reasonable agreement is found with only one free parameter (A = 215:7) and 1xing = 34 K from experimental data. In Fig. 4.18(b) the temperature dependence of the continuum between 20 and 80 cm−1 is well described by the temperature-dependent susceptibility [107]. In (c) the quasielastic

P. Lemmens et al. / Physics Reports 375 (2003) 1 – 103

83

1.2

200

(a)

(a)

J1 0.8

J2

(ab) 50K

0.4

295K

IB (T) bound state intensity

100 0.0

Intensity (arb. units)

Intensity (arb. units)

7K

4K 1.5K

0 200

(a' b'+c) T = 2.1 K

(b)

(a’b’)

0T 6T

0.8

(b)

0.4

χ(T) continuum intensity

0.0

(c)

1.5K 0.8

100

50K

20 40 60 80 Raman shift (cm-1)

4K

0.4

cm(T) 2 quasielastic intensity I / T

295K 0

20

40

60

80 -1

Raman shift (cm )

100

0.0

2

4

6

8

10

12

14

16

Temperature (K)

Fig. 4.17. Raman spectra of SrCu2 (BO3 )2 in two scattering geometries: (a) light polarization (ab) and (b) rotated by 45◦ , i.e. (a b ). The upper inset shows Cu-dimers of 1/4 of the unit cell. The respective polarization (arrows) of incident and scattered light is given with respect to the crystallographic axes. The lower inset shows spectra for B = 0 (full line) and 6 T (dotted line) in (a b + c) polarization. An additional Ag -phonon is marked by an arrow [124]. Fig. 4.18. Mapping of the temperature dependence of normalized Raman scattering intensities (open symbols) of the bound states, the intensity of the continuum and of the quasielastic scattering on thermodynamic quantities (lines) [124].

magnetic scattering intensity divided by T 2 is well described by the speci1c heat according to Eq. (5.1) in Section 5 [108]. It is noteworthy that a mapping of the intensity of the Raman scattering continuum on the magnetic susceptibility also works for the frustrated spin chain system CuGeO3 and corresponds to replacing a four-spin correlation function by a two-spin function [124]. The theoretical implications of this experimental evidence that has not been investigated so far. For very low temperatures compared with the spin gap (T 01 ) four well-de1ned modes with energies i00 =30, 46, 56 and 70 cm−1 appear. A re1ned analysis discovered a further broadened mode centered around 100 cm−1 [109]. These excitations are primarily observed in the (a b ) scattering con1guration corresponding to B1 symmetry selection rules and polarization vectors that connect nearest dimers. Only this scattering geometry allows exchange Raman scattering on the Shastry– Sutherland lattice if no excited triplet states are present. The modes neither split nor shift in an applied magnetic 1eld up to 6 T as shown in the inset of Fig. 4.17. In (ab) polarization two weaker

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shoulders of intensity appear near 25 and 40 cm−1 for T 01 . The frequencies of these modes are identical to the triplet excitations observed in neutron scattering and ESR (24.5 and 40 cm−1 ). Applying a magnetic 1eld, at least for the former mode a de1nite shift corresponding to the triplet nature of the underlying excitation has been detected. If the above mentioned modes are normalized to the energy of the singlet–triplet gap 01 = 24:5 cm−1 they correspond to ratios of 1:2501 –2:901 . The sequence of modes is astonishingly similar to the reduced energy of low-temperature modes found in CuGeO3 (1:7801 ) and NaV2 O5 (1:101 ; 1:7801 and 2:201 ). Therefore, and as the triplet excitations in SrCu2 (BO3 )2 are even more localized and strongly interacting compared with frustrated spin chains, all above given modes are assigned to magnetic bound states [124]. Several theoretical studies of the magnetic excitation spectrum support this interpretation [118,427,428]. 4.4.1. EAect of substitutions Comparing the behavior of SrCu2 (BO3 )2 with other quantum spin system it is obvious that here the triplet excitations are even stronger spatially localized and the spin–spin correlation length is extremely small. As shown in the previous sections substitution experiments are a sensitive probe to this degree of localization. Furthermore and related to this question, the phase diagram of this compound and the assumed proximity of SrCu2 (BO3 )2 to a NQeel phase opens the up the chance to study quantum criticality. Here Zn substitutions inducing spin vacancies in the system might help at least to shift the compound into the cross over regime. Several substitution studies have been performed by H. Kageyama and Yu. Ueda, ISSP Tokyo using Zn substitutions on the Cu site to induce spin vacancies. Furthermore the eKect of Ba and Ca substitution on the Sr site has been studied. The latter, in a certain solubility window, decreases or increases the crystallographic c-axis and the distance of the CuBO3 planes, respectively. The results of these studies are easy to summarize: No decrease of the spin gap or emergence of an antiferromagnetic instability has been observed. Instead a defect-induced low-temperature Curie-tail is induced together with a Oattening of the plateaus in high-1eld magnetization [450]. In Fig. 4.19 the eKect of Zn substitution on the low-energy Raman spectra of SrCu2 (BO3 )2 are shown at three diKerent temperatures, T = 1:5, 4 and 5 K, respectively [109]. Obviously the eKect of the substitution on the excitation spectra is limited, only the intensities of the modes are reduced. In this sense this trivial eKect of substitutions may be approximated by an eKective increase of temperature. The non-existent energy shift is compatible with a constant and substitution-independent global magnitude of the spin gap. Only in a very limited concentration range an additional broad intensity at 16 –18−1 is observed which might be related to a breaking of the translational invariance. All these experimental facts are well understood considering thermally and substitution-induced triplets as quasilocalized on each Cu dimer on the Shastry–Sutherland lattice. Therefore, substitutions locally switch oK the spin gap without leading to a global eKect on the singlet state. 4.4.2. Polarization dependence The magnetic bound states in SrCu2 (BO3 )2 are purely in-plane polarized. As shown in Fig. 4.20 scattering intensity appears only in (aa) and (a b ) polarization. Furthermore in the investigated frequency range no additional interlayer bound states show up in (cc) polarization. The only scattering intensity observed in this spectral range is a phonon with a large anharmonicity. This interlayer

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Fig. 4.19. Low-energy Raman scattering of Zn-substituted Sr(Cu1−x ; Znx )2 (BO3 )2 in (a b ) polarization at 1.5, 4 and 5 K [109]. The measurements with x = 0 (full line), x = 0:01 (dashed) and 0.02 (dotted) are vertically displaced for clarity.

Fig. 4.20. Low-energy Raman scattering for SrCu2 (BO3 )2 in interlayer and intralayer polarizations at T = 1:6 K. The measurements in intralayer polarizations are vertically displaced upwards for clarity. The (aa), (cc) and (ac) polarizations show an additional phonon mode marked by an arrow. No evidence for additional interlayer modes is found [109].

phonon mode with a frequency of 58 cm−1 at T = 4 K is discussed further in Section 2.3.2 (see Fig. 2.15). It is related to a structural phase transition at T = 395 K [125]. Its anharmonicity and its energy comparable to the spin system might induce an enhancement of the spin–phonon interaction. Further investigations of both experimental and theoretical nature are underway.

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5. Quasielastic scattering in low-dimensional spin systems Raman spectra of several magnetic systems show quasielastic light scattering as a strongly broadened central peak with Lorentzian or Gaussian lineshape and with speci1c selection rules. A canonical example for 3D AF spin systems is the cubic, quasi-isotropic compound KNiF3 in which magnetic energy Ouctuations for temperatures close to the NQeel ordering (TN =248:5 K) give rise to quasielastic scattering [452]. In low-dimensional spin systems quasielastic scattering is a very general feature, too. The temperature dependence of this intensity can be used to characterize the low-energy excitations of the system. It is especially sensitive to the opening of a spin gap and can be very useful to determine the magnetic part of the speci1c heat. In direct measurements of the speci1c heat very often the lattice degrees of freedom dominate the magnetic part. Spin systems investigated so far are the “pure” spin chain KCuF3 [227], the charge ordered rare earth spin chain Yb4 As3 [175], the spin-Peierls compound CuGeO3 [451], the alternating spin chain system (VO)2 P2 O7 [214], the quarter-1lled spin ladder NaV2 O5 , [189] and very recently the strongly frustrated 2D spin dimer compound SrCu 2 (BO3 )2 [124,453]. In compounds with a dimerization transition (TSP ) the coupling to optical phonons leads to Ouctuation contributions that are observed for temperatures slightly above TSP . This quasielastic scattering is considered to redistribute its spectral weight to soliton bound states (T ¡ TSP ) [315] and to be connected to the non-observance of soft modes at the transition [316]. In the above-mentioned 1D spin systems quasielastic scattering should only be observed in intrachain scattering con1guration, i.e. with the light polarization parallel to the eKective chain direction. A comparison of diKerent scattering geometries including right angle scattering in CuGeO3 is shown in Fig. 5.1. Quasielastic scattering is indeed only observed in (cc), i.e. intrachain polarization. A similar investigation for NaV2 O5 is shown in Fig. 2.44. Here quasielastic scattering is also observed only in intraladder (bb) polarization [189,362]. In this sense (VO)2 P2 O7 seems to be an exception since quasielastic scattering is primarily observed in interchain (aa) polarization. In a-axis direction neutron scattering data give evidence only for a Oat ferromagnetic dispersion compared to the steep antiferromagnetic dispersion along the b-axis. Uhrig and Normand [31] proposed a quasi-2D model with an additional diagonal AF exchange for (VO)2 P2 O7 . The 2D spin dimer compound SrCu2 (BO3 )2 shows this scattering contribution in (ab) con1guration, i.e. in polarization yielding B2 symmetry [124]. These results are described and analyzed in Section 4.4 (see Figs. 4.17(a) and 4.18) together with other magnetic scattering contributions. The experimentally observed spectral weight of the quasielastic scattering contribution shows generally a strong increase with increasing temperature. In the same temperature range the magnetic contribution to the speci1c heat has a maximum. The quasielastic scattering has been explained within two diKerent scenarios: Spin diKusion [454, 455] that is forbidden in perfect 1D spin systems and would lead to a Gaussian lineshape of the central line [456] or Ouctuations of the energy density of the spin system [457]. The latter mechanism leads to a Lorentzian spectral function and is important for systems with non-negligible spin–phonon coupling. The energy of a magnetic system is not constant, but Ouctuates about its mean value [458]. Already a weak spin–phonon coupling increases the spectral weight of these Ouctuations as it decreases their time scale.

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Fig. 5.1. Symmetry analysis of quasielastic light scattering in CuGeO3 . (a) for the frequency range −100 to 100 cm−1 and (b) for the frequency range −100 to 500 cm−1 in right angle scattering geometry [451]. Fig. 5.2. Quasielastic light scattering in CuGeO3 for intrachain (cc) polarization and diKerent temperatures [451].

In Fig. 5.2 a Raman experiment of CuGeO3 displays quasielastic scattering as function of temperature [451]. In this compound very strong spin–phonon coupling is evident for two phonon branches [306]. Therefore, the Lorentzian spectral function leads to a very good description of the data. For smaller temperatures the intensity of the scattering contribution is diminished as the Ouctuations are suppressed. From this quasielastic light scattering intensity one can easily deduce the magnetic contribution to the speci1c heat, Cm . This analysis has been performed for KCuF3 [227], CuGeO3 [451] and (VO)2 P2 O7 [214]. Therefore, we brieOy summarize the theory of Reiter and Halley [457,458]. The scattering intensity can be calculated using the Fourier components of a correlation function of the magnetic energy density. For high temperatures and hydrodynamic conditions this function can be simpli1ed leading to an expression that connects the scattering intensity with the magnetic speci1c heat Cm : I (!) ˙

Cm T 2 DT k 2 ; !2 + (DT k 2 )2

(5.1)

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Fig. 5.3. Analysis of the quasielastic light scattering in CuGeO3 in terms of the magnetic speci1c heat Cm [451]. The data (black dots) are smoothed (line) and then compared with calculations using the given coupling constants. The dotted and dashed lines correspond to J = 150 K and = 0:24 [294] or J = 160 K and = 0:36 [293].

with DT = K=Cm the thermal diKusion coeUcient, K the magnetic thermal conductivity, and ! ∼ DT k 2 !0 leading to k 2k0 sin A=2, with A the scattering angle, and !0 ; k0 characterized by the incident light. Analyzing experiments using Eq. (5.1) the scattering background is assumed to be constant. Especially contributions from Rayleigh scattering should be reduced using a suitable experimental setup and scattering geometry. For transparent samples right angle scattering geometry is most favorable. In Fig. 5.3 results for CuGeO3 in 90◦ scattering geometry are shown. The resulting Cm has been compared with theoretical models considering frustrated spin chains [451]. A qualitative agreement with the exchange coupling and the frustration parameter from thermodynamic experiments [297,298] has been obtained. In Raman experiments on (VO)2 P2 O7 strong phonon anomalies have been observed for modes with energies comparable to the magnetic excitation spectrum. Therefore, assuming strong spin–phonon coupling, the existence of Ouctuations of the energy density seems reasonable in this compound. For a comprehensive discussion we refer to Section 2.5. Indeed, quasielastic scattering is observed in interchain (aa) scattering con1guration. The respective data are shown in Fig. 5.4. This intensity strongly decreases with lowering temperature. Calculations using a Lorentzian and a Gaussian lineshape are given in this 1gure as a solid and a dashed line, respectively. The maxima observed near 60 and 70 cm−1 are due to phonon scattering and should be ignored here.

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(aa)

(VO)2 P2 O7

400

89

Lorentzian Gaussian

150

Intensity (arb. units)

200 K 300

200 K 100

75 K 200

40

55 K

50 60 70 80 -1 Raman shift (cm )

90

40 K 100

20

40

60

80

Raman shift (cm-1)

Fig. 5.4. Quasielastic scattering intensity in (VO)2 P2 O7 determined in interchain (aa) polarization for diKerent temperatures. The dashed and solid line show a calculation using a Gaussian and Lorentzian lineshape, respectively [214]. The inset is an enlarged view of the data for T = 200 K.

1.0

(VO)2 P2O 7

0.6

0.4

2

2

K / T or Cm (arb. units)

0.8

DMRG

0.2

0.0

0

50

100

150

200

250

300

Temperature (K)

Fig. 5.5. Speci1c heat of (VO)2 P2 O7 (black squares) determined from quasielastic light scattering shown in Fig. 5.4. The dashed curve is a guide to the eye. The full curve is a DMRG calculation of the speci1c heat of an alternating Heisenberg chain [214,215].

Following the arguments given above the magnetic speci1c heat was determined from the quasielastic scattering using Eq. (5.1). The result is shown in Fig. 5.5 [214]. A direct measurement of this quantity is not yet available. However, DMRG calculations [225] of the magnetic speci1c heat with the parameters J = 110 K and = 0:09 K are shown for comparison in the same 1gure. The maximum of the magnetic speci1c heat determined by light scattering experiments is at Tmax = 55 K. The corresponding maximum in the magnetic susceptibility is at 74 K [221]. The DMRG calculation for a strongly alternating Heisenberg chain agrees well with the experimental data.

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This supports the scenario of (VO)2 P2 O7 as an alternating chain system with considerable spin– phonon coupling. 6. Conclusions and outlook Over the past decade the understanding of low-dimensional strongly correlated electron systems improved enormously. From the experimental side this is related to the signi1cant progress made in materials science of complex transition-metal oxides. Indeed, quantum spin systems with chain, ladder, depleted-planar, and other exchange geometries have been designed and investigated. One motivation for these studies has been the possible connection between high-temperature superconductivity and low dimensionality. In the undoped and insulating parent compounds of the high-temperature superconductors long-range NQeel order is observed. To get more insight in the suppression of antiferromagnetic long-range order upon doping, spin liquids, such as, e.g., one-dimensional half-1lled spin systems with the above given topologies have been studied. In these model-type systems long-range order is suppressed without doping. The compound Sr 14−x Cax Cu24 O41 shows indeed superconductivity under pressure. The spin gaps of the ladders and the dimerized chains agree well with the behavior expected for low-dimensional spin systems. A suppression of the Raman scattering intensity has been found for frequencies below 200 and 490 cm−1 . These frequencies correspond to the doubled singlet–triplet gaps of the chain and the ladder sublattice, respectively. Upon Ca substitution the electronic properties of Sr 14−x Cax Cu24 O41 cross over into a 2D regime. Still, Raman and neutron scattering experiments undoubtedly prove that the magnitude of the spin gap in the ladders is preserved. Charge ordering in the dimerized chains is observed in Raman scattering as a shift of a phonon mode and the appearance of additional side bands. Upon substitution these phonon anomalies and the gap feature are suppressed in the Raman spectra. Taking into account a charge transfer from the chains to the ladders with increasing Ca substitution makes this eKect understandable as it suspends the dimerization of the chains. Much attention has been devoted to NMR on Sr 14−x Cax Cu24 O41 under applied pressure. These investigations claim a change of the ladder gap from a distinct gap to a “pseudo-spin gap”. However, NMR data at ambient pressure as a function of Ca substitution are already in conOict with Raman and neutron scattering experiments. This may be due to a higher sensitivity of NMR to incoherent low-energy states. These states may exist either due to structural defects or mobile carriers. To clarify this problem, further spectroscopic experiments under pressure are needed. Summarizing the present status, most experimental results including Raman scattering are in favor of a preserved spin gap and low dimensionality of the ladder sublattice in the doped system. A situation of similar complexity that is also linked to the problem of a doped quantum spin system is realized in NaV2 O5 . Raman scattering experiments in this compound point to charge ordering with an onset at higher temperatures (TCO = 80 K) than the subsequent dimerization transition (TSP = 34 K). The observed phonon anomalies point to a temperature scale (TCO =TSP ) of this eKect that is comparable to the 1D stripe formation in the nickelate La2 NiO4 . The polarization selection rules of the magnetic bound states in NaV2 O5 support the proposed zigzag charge ordering. A further interesting realization of a spin liquid is given in the two-dimensional compounds CaV4 O9 and SrCu2 (BO3 )2 . In these systems spin gaps exist due to their exotic exchange topologies. Theoretical studies on 1/5-depleted square lattices and the orthogonal dimer (Shastry–Sutherland)

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lattice indeed proposed a manifold of possible singlet ground states and a rich excitation spectrum. Experimentally the gap formation and triplet bound states were observed in ESR, NMR and neutron scattering. In SrCu2 (BO3 )2 a multitude of singlet bound states was detected in light scattering experiments. The observed polarization selection rules of these modes and quasielastic scattering (at higher temperatures) reOect the two-dimensional nature of the underlying spin–spin correlations. This reOects the two-dimensional nature of the underlying spin–spin correlations. In the one-dimensional spin systems discussed here the maximum energy of the bound states is restricted to the doubled singlet– triplet gap. This is not the case for SrCu2 (BO3 )2 and is attributed to a strong frustration-induced localization of the triplets. Light scattering experiments proved to be a versatile tool to investigate compounds that represent low-dimensional quantum spin systems. This is based on the high sensitivity and energy resolution on comparably small samples, and, furthermore on the speci1c symmetry information that may be gained from an analysis of the Raman scattering tensor. A further promising aspect of these experiments is the easier feasibility in which high hydrostatic pressure may be applied. This parameter may be used to increase the interchain interaction or the frustration of a spin system in a controlled way. Experimental investigations on pure and substituted CuGeO3 found an increase of the frustration with pressure that leads to a “revival” of the spin-Peierls transition in the substituted compounds. Pressure studies will surely gain increased signi1cance in the future. The observation of singlet bound states in light scattering experiments has opened up a new way of characterizing triplet–triplet interactions. These states can be regarded as the magnetic analogue to excitonic states in semiconductors. Magnetic light scattering as a spin conserving exchange process is complementary to magnetic neutron scattering in which these singlet bound states cannot be observed. Singlet bound states are now believed to be a general phenomenon in systems with a gapped excitation spectrum. From the theoretical side three approaches have been undertaken to understand the binding eKect of triplets. BrieOy, they may be characterized as taking into account frustration in dimerized magnetic systems, interchain interaction, or the important role of spin–phonon coupling. In the experimental data presented here the spectral weight and binding energy of the bound states were found to be dominated by frustration, e.g., due to next nearest neighbor exchange interaction in CuGeO3 , or by a complicated exchange topology in SrCu2 (BO3 )2 and NaV2 O5 . The experimental identi1cation of magnetic bound states is based on the investigation of the spectral weight and lineshape of the respective scattering intensity as function of dimerization, magnetic 1eld and polarization selection rules. In all cases the spectral weight of the singlet bound states showed a linear increase with decreasing temperature. This characteristics is linked to their nature as bound two-particle states in a system with strong Ouctuations and short-ranged spin–spin correlations. The spin-Peierls compound CuGeO3 has been used as a model system in this respect. The singlet bound state observed here corresponds in energy, polarization selection rules and other properties exactly to the expected behavior of a frustrated alternating chain system. The importance of frustration is also highlighted by the observation of spinon scattering for T ¿ TSP . This scattering intensity is only allowed due to frustration. In the “pure” non-frustrated spin chain system KCuF3 no corresponding signal was observed. In substituted CuGeO3 the defect-induced dopant-bound spinon is observed in light scattering as a well-de1ned mode. This mode demonstrates distinctly the binding eKect of spinons to a defect site. Finally, for the 1rst time a three-magnon process was observed and successfully modelled as scattering from a thermally populated triplet branch to a higher lying one. It is noteworthy that this scattering process is lost in any bosonic description where only quadratic terms in the expansion

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of the Raman tensor in terms of the spin are kept. This process is a generic feature of gapfull low-dimensional spin liquids. In summary, we provided an overview of selected aspects in the 1eld of low-dimensional quantum spin systems. Most of the major issues discussed here were investigated using inelastic light scattering. This spectroscopy together with appropriate theoretical models undoubtedly gives important insight into interactions and will therefore also play an active role in the future. The physics of low-dimensional spin systems is an integral part of the 1eld of strongly correlated electrons. It developed into a fascinating area of solid state physics and quantum magnetism in particular. Also in the more technologically relevant area of magnetism in reduced dimensions the basic understanding of electronic correlations and quantum spin systems achieved so far will prove to be important. Magnetic nanostructures scaled down to the quantum dot or molecular level will be crucial entities for quantum information processing. Future developments that are diUcult to foresee will surely lie in the materials science of new complex compounds. In this 1eld the emerging use of preparation concepts as the “lone pair approach” and crystallographic database searching strategies are noticeable. These compounds might have similarly extraordinary properties as the high-temperature superconductors. The large number of very recently discovered or reinvestigated materials with exotic properties and unique magnetic exchange topologies point into this promising direction. References [1] A. Auerbach, Interacting Electrons and Quantum Magnetism, Graduate Texts in Contemporary Physics, Springer, New York, 1994; P. Fazekas, Lecture Notes on Electron Correlations and Magnetism, World Scienti1c, Singapore, 1999. [2] N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [3] K. Yosida, Theory of Magnetism, Solid-State Sciences, Vol. 122, Springer, New York, 1996. [4] H.A. Bethe, Z. Phys. 71 (1931) 205. [5] J. des Cloizeaux, J.J. Pearson, Phys. Rev. 128 (1962) 2131. [6] G.S. Uhrig, H.J. Schulz, Phys. Rev. B 54 (1996) R9624. [7] R.B. Laughlin, Phys. Rev. Lett. 79 (1997) 1726. [8] R.B. Laughlin, in: Y.M. Cho, J.B. Hong, C.N. Yang (Eds.), Proceedings of the Inaugural Conference of the Asia-Paci1c Center for Theoretical Physics, World Scienti1c, Singapore, 1998. [9] F.D.M. Haldane, Phys. Rev. B 25 (1982) 4925. [10] C.K. Majumdar, J. Phys. C 3 (1970) 911. [11] B.S. Shastry, B. Sutherland, Phys. Rev. Lett. 47 (1981) 964. [12] R. Chitra, S. Pati, H.R. Krishnamurthy, D. Sen, S. Ramasesha, Phys. Rev. B 52 (1995) 6581. [13] E. Dagotto, J. Riera, D.J. Scalapino, Phys. Rev. B 45 (1992) 5744. [14] B.G. Levi, Phys. Today 49 (1996) 17. [15] E. Dagotto, T.M. Rice, Science 271 (1996) 618. [16] D.C. Cabra, M.D. Grynberg, Phys. Rev. Lett. 82 (1999) 1768. [17] Z. Weihong, V. Kotov, J. Oitmaa, Phys. Rev. B 57 (1998) 11439. [18] Y. Honda, T. Horiguchi, cond-mat/0106426 (2001), unpublished. [19] D.C. Cabra, A. Honecker, P. Pujol, Phys. Rev. B 58 (1998) 6241. [20] S.R. White, D.J. Scalapino, Phys. Rev. B 57 (1998) 3031. [21] O.P. Sushkov, V.N. Kotov, Phys. Rev. Lett. 81 (1998) 1941. [22] T.M. Rice, S. Gopalan, M. Sigrist, Europhys. Lett. 23 (1993) 445. [23] M. Sigrist, T.M. Rice, F.C. Zhang, Phys. Rev. B 49 (1994) 12058. [24] T.M. Rice, S. Haas, M. Sigrist, F.C. Zhang, Phys. Rev. B 56 (1997) 14655.

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Physics Reports 375 (2003) 105 – 218 www.elsevier.com/locate/physrep

Oscillations of neutrinos and mesons in quantum $eld theory Mikael Beuthe∗ Institut de Physique Theorique, Universite Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Received 1 September 2002 editor: R. Petronzio

Abstract This report deals with the quantum $eld theory of particle oscillations in vacuum. We $rst review the various controversies regarding quantum-mechanical derivations of the oscillation formula, as well as the di1erent $eld-theoretical approaches proposed to settle them. We then clear up the contradictions between the existing $eld-theoretical treatments by a thorough study of the external wave packet model. In particular, we show that the latter includes stationary models as a subcase. In addition, we explicitly compute decoherence terms, which destroy interferences, in order to prove that the coherence length can be increased without bound by more accurate energy measurements. We show that decoherence originates not only in the width and in the separation of wave packets, but also in their spreading through space–time. In this review, we neither assume the relativistic limit nor the stability of oscillating particles, so that the oscillation formula derived with $eld-theoretical methods can be applied not only to neutrinos but also to neutral K and B mesons. Finally, we discuss oscillations of correlated particles in the same framework. c 2002 Elsevier Science B.V. All rights reserved. PACS: 14.60.Pq; 14.40.−n; 03.65.Bz; 12.15.Ff Keywords: Oscillation; Neutrino; Meson; Quantum $eld theory

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Oscillations in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Mixing in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Oscillations with plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Time to space conversion of the oscillating phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

107 110 111 114 116

Corresponding author. Present address: Observatoire Royal de Belgique, Avenue Circulaire 3, B-1180 Bruxelles, Belgium. E-mail address: [email protected] (M. Beuthe). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 5 3 8 - 0

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2.3.1. Standard oscillation phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Nonstandard oscillation phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Problems with the plane wave treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Intermediate wave packets and other improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillations in quantum $eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Review of the literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Five questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The external wave packet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. The Jacob–Sachs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. The process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3. The propagation amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Mixed propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. The Bavor-mixing amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. The simplest consistent model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. The overlap function for Gaussian wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1. Gaussian wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2. Gaussian overlap function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Propagation amplitude and dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Pole integration and wave packet correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Three propagation regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Laplace’s and stationary phase methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. No-dispersion regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Transversal-dispersion regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. Longitudinal-dispersion regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Estimates of the dispersion thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1. No-dispersion condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2. Dispersion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oscillations of stable particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Oscillations without dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2. Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Oscillations with transversal dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1. Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Oscillations with longitudinal dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1. Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2. Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. A shortcut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis of the probability in the stable case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Oscillations at large distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1. Oscillation formula at large distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2. Oscillation phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3. Coherence length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4. Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.5. Energy–momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Oscillations at short distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Answers at last . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Existence of a stationary limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Unbounded coherence length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 118 120 121 126 126 128 129 130 130 131 133 134 136 137 138 139 141 143 143 143 145 145 146 149 152 153 153 153 154 155 155 155 157 157 158 161 161 161 164 166 167 167 168 169 171 173 173 174 174 175

M. Beuthe / Physics Reports 375 (2003) 105 – 218 7.3.3. No mass prefactors at large distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4. No plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Oscillations of unstable particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1. Nonexponential corrections due to mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Oscillation formula for unstable particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. Wigner–Weisskopf e1ective mass matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Oscillations in the B0 BH 0 system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Selected topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1. Normalization of the transition probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Coherence, incoherence and decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. Incoherent e1ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.2. Coherent e1ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Unstable source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Correlated oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Energy–momentum correlation at the source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. No recoil oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Oscillations of correlated mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. The Jacob--Sachs theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Unstable particle, with b ¡ M1 ¡ m ¡ M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Unstable particle, with M1 ¡ b ¡ m ¡ M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Stable particle, with M1 ¡ m ¡ M2 ¡ b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Stable particle, with M1 ¡ m ¡ b ¡ M2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107 175 176 177 177 178 181 183 185 187 187 189 190 191 193 194 194 195 196 199 201 202 205 205 206 208 210 210 210

1. Introduction Nearly $fty years ago, the puzzle led Gell-Mann and Pais [134] to discover that interaction (or Bavor) eigenstates need not coincide with particles having well-de$ned masses and lifetimes. The concept of Bavor-mixing was born. Pais and Piccioni [219] went one step further and suggested that the propagation of a Bavor eigenstate, as a coherent superposition of mass eigenstates, leads to the partial conversion of the initial Bavor into other Bavors. The Bavor-mixing transition probability oscillates in space with a wavelength depending on the mass di1erences. The theory of Bavor-mixing and particle oscillations was soon vindicated in a series of experiments on the kaon system [190,114,148,171]. Similar oscillations were observed in 1987 with B mesons [13]. By analogy, Pontecorvo proposed in 1957 the hypothesis of neutrino–antineutrino oscillations [229,230]. Neutrino Bavor-mixing, strictly speaking, was suggested somewhat later [203], and the two-Bavor oscillation case, as well as its application to solar neutrinos, was examined at the end of the 1960s by Pontecorvo and Gribov [231,149]. Neutrino oscillations are, however, much more diKcult to observe than meson oscillations, because of the small masses and the weak interactions of the neutrinos. They remained for 30 years a theoretical subject (see for example the reviews [66,65]), with the noteworthy exception of the solar neutrino experiment in the Homestake Mine [98,97,93,34].

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In the 1990s, the persistent de$cit in solar neutrinos (in comparison with the predictions of solar models [34,40,95]) observed at Homestake was con$rmed by the experiments Kamiokande-II [161,122], SAGE [1,2], GALLEX [24,157], GNO [16] and Super-Kamiokande [123,126]. Moreover the Sudbury Neutrino Observatory recently announced the detection of a nonelectronic component in the solar neutrino Bux [9,10], the total Bux being compatible with the predictions of solar models. All these observations can be explained by neutrino mixing and oscillations [47,43,38,127,211]. On another length scale, the observation of an angular dependence of the atmospheric muon–neutrino Bux by the Super-Kamiokande Collaboration [124,125,175,170] gave a huge impetus to the neutrino oscillation hypothesis (note that a global muon–neutrino depletion had already been observed in other atmospheric neutrino experiments [160,121,51,18,15]). Preliminary results from the long-baseline experiment K2K [11,164] con$rm these results by showing a depletion in an arti$cially produced muon–neutrino beam. Finally, controversial evidence for neutrino oscillations has appeared in the laboratory experiment LSND [28,29] but has not been con$rmed (nor totally excluded) by the KARMEN experiment [26,27,92]. Particle oscillations are an interference phenomenon which allows the measurement of extremely small quantities, such as masses too small to be measurable by other means (mass di1erences between conjugate mesons [114,148,13] or neutrinos [113,180]), or CP violation (measured nearly forty years ago in neutral meson systems [91,77] and yet to be measured in the leptonic sector [103]). Thus, it is a privileged tool in the discovery of new physics. Nowadays, B meson and neutrino oscillations have a huge importance in the determination of the unknown parameters of the Standard Model and its minimal extension, in particular the mixing matrix elements (including the CP violation phase) and the neutrino masses. Besides serving as a delicate probe in particle physics, particle oscillations are a good testing ground for quantum mechanics. Einstein–Podolsky–Rosen correlations [162,100,196] have been studied theoretically [57,58,116,228] as well as experimentally [25] in the neutral K and B systems. Bell inequalities for neutral kaons are the subject of active research [22,59,135,76]. Furthermore, fundamental issues of quantum mechanics are raised by oscillation experiments. The determination of the conditions under which the coherence between the di1erent mass eigenstates is maintained is crucial for the phenomenology of neutrino oscillations [182,204]. The question of the correlation between neutral mesons, in processes where B0 BH 0 is included in the $nal state, is important for an estimate of the oscillation frequency [153]. Considering the practical importance of the oscillations, the theoretical framework describing them should be rock-solid. Surprisingly, the validity of the standard treatment of oscillations, resorting to plane waves in a simple quantum-mechanical model, was not seriously questioned until neutrino oscillations $nally gained credibility in the 1990s. Two pioneering articles should be noted. First, a quantum $eld model of kaon oscillations was proposed by Sachs in 1963 [238], but was soon forgotten, until the use of Sachs’ reciprocal basis was recently revived for a rigorous quantum-mechanical treatment of meson oscillations [77,17]. Second, Kayser showed in 1981 that neutrino oscillations are in contradiction with the plane wave assumption of the standard treatment of oscillations, and he proposed to modelize the oscillating particles with wave packets [176]. His model, however, was not widely accepted and numerous articles have continued to appear to this day, discussing the ambiguities of the plane wave standard treatment, such as the proper choice of the frame [201] or the equality of the energies or momenta of the di1erent mass eigenstates [136]. Although Kayser’s wave packet model is a serious improvement on the standard plane wave treatment, it also su1ers from serious defects: Bavor states are ill-de$ned [142]; the nonrelativistic

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limit is problematic if the mass eigenstates are not nearly degenerate [142]; the question about the equality (or not) of the energy of the di1erent mass eigenstates is not answered [155,248]; the possibility remains of obtaining nonstandard oscillation formulas [245]; the relation between the coherence length, beyond which oscillations vanish, and the conditions of production and detection is vague (problem of the determination of the wave packet size [182]); $nally, it is not adequate for the oscillations of unstable particles. In order to solve these problems, a quantum $eld treatment of neutrino oscillations, similar to Sachs’s model, has been proposed by Giunti et al. [145]. The model consists in treating the oscillating particle as an internal line of a Feynman diagram, propagating between a source and a detector, which are represented by ingoing and outgoing external wave packets. This approach has been followed by other authors, but no agreement has yet been reached on the correct assumptions and calculation methods: 1. There is disagreement on whether the interfering mass eigenstates have the same energy [150] or whether their energy is spread; the existence of propagating wave packets, associated to the mass eigenstates, is also at stake [151]. 2. Whereas oscillations vanish if the 3-momentum is measured too accurately, it is not clear whether the same thing happens when the energy is perfectly measured [183,144,139]. 3. Nonstandard formulas for neutrino oscillations have been derived in speci$c cases, either showing a strong dependence of the Bavor-mixing transition probability on the neutrino mass ratios [242], or having a ‘plane wave’ behavior [163], or leading to oscillations of charged leptons in pion decay [247]. Moreover, this formalism cannot be applied, as it is, to meson oscillations or to unstable neutrinos. The existence of neutrino mass makes indeed not only neutrino mixing possible but also neutrino decay [208,75,184,192]. Although it cannot alone explain the data, neutrino decay has a long history as a possible explanation for the solar neutrino anomaly [36,222,39,55,54,5,6,202,90,41,169,42,50]. Moreover this hypothesis was compatible until recently with atmospheric neutrino data [46,195,117, 45,89,202], although it now seems ruled out [180]. Neutrino decay has also been discussed with respect to supernovae [119,193]. In this report, we intend to reconcile the di1erent $eld-theoretical treatments in order to solve the above problems. Our $rst aim is to argue in favor of a $eld theory approach of oscillations. We want to show that it provides, on the one hand, the best available derivation of the oscillation formula and gives, on the other hand, a good explanation of the characteristics of the oscillation process (origin of the oscillations, observability conditions, boundary conditions). Our second aim is to strengthen the phenomenological basis of the neutrino oscillation formula through its uni$cation with the mesonic oscillation formula. We thus extend the formalism to the unstable case, so that the same formula can be applied to K and B mesons, to neutrinos and to unstable neutrinos. In Section 2, we discuss the quantum-mechanical approaches, so as to understand the problems addressed in most of the articles and the di1erent solutions adopted in the literature. The existing $eld-theoretical treatments are reviewed in Section 3, and their contradictions are summarized in $ve questions, which will serve as a guideline in the calculations of Sections 4–6; these questions will be answered in Sections 7.3 and 10. In Sections 4 and 5, we explain all the assumptions behind the external wave packet model. The relativistic case and the nonrelativistic (but nearly mass degenerate) cases are treated in a uni$ed way, and the oscillating particle is allowed to be unstable.

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Oscillations of stable particles are analyzed in Section 6. With the aim of reconciling the di1erent points of view in the literature, we discuss di1erent methods of computation, which allow to piece together a complete physical interpretation in Section 7. We shall see that this formalism lends itself to a wave packet interpretation of the oscillating particle and can thus take up the physical insights of Kayser’s intermediate wave packet model. The dispersion (or spreading in space–time of the amplitude) is taken into account, as it is important for mesons and might be relevant for solar and supernova neutrinos. We can then show that the existence of the coherence length is due not only to the separation of wave packets, but also to their dispersion. We compute the explicit dependence of the coherence length on the conditions of production and detection, to check that it increases without bound with the accuracy of energy measurements. We also show in which sense the oscillation formula can be seen as a superposition of mass eigenstates with the same energy, in order to clear up the misunderstanding of stationary boundary conditions. In addition, we analyze in Section 7 the nonstandard oscillation formulas derived within $eld-theoretical models and we disprove them. Oscillations of unstable particles are examined in Section 8. Nonexponential corrections to mixing are computed for the $rst time in quantum $eld theory, and they are shown to be negligible. We clearly state the correspondence between the $eld-theoretical treatment and the Wigner–Weisskopf and reciprocal basis formalisms. Some more topics are discussed in Section 9: the normalization of the transition probability, the size of the external wave packets and the instability of the source. Finally, we examine correlated oscillations (DANE, KEKB and BABAR experiments) in Section 10, including an analysis of the possible energy–momentum correlation at the source. In the same framework, oscillations of recoil particles are shown to be nonexistent. Let us mention four restrictions. First, we shall only consider free propagation in vacuum. Interactions with background matter can have a dramatic e1ect on mixing, as demonstrated by the Mikheyev–Smirnov–Wolfenstein (or MSW) e1ect [256,206,207,189]. In practice, the MSW oscillations are washed out by averaging over the energy spectrum [227,204], so that the question of intrinsic decoherence discussed in the present report is not very important in that case. It is true that a $eld-theoretical formalism for oscillations with matter interactions should be developed to deal with dense astrophysical environments [234,232,86,106], but this goes far beyond the limits of this review. Second, we shall not explain the nonperturbative quantum $eld theory of particle oscillations, as proposed by Blasone, Vitiello and co-workers [73,71]. This approach is indeed of a completely di1erent nature than the perturbative $eld-theoretical treatments discussed in this report. We shall however comment on their results in Section 3.1. Third, we shall not examine exotic mechanisms, such as oscillations of massless neutrinos due to a violation of the Equivalence Principle [235], to resonant spin-Bavor precession [12,44], to Bavor-changing (or nonuniversal) interactions [118] or to dissipation [199,52] (comparative reviews can be found in [195,220,132]). Finally, we consider neither the gravitational e1ects on oscillations [257,194] nor the possibility of extra dimensions [109,99]. 2. Oscillations in quantum mechanics We $rst discuss the concept of particle mixing as it is a requirement for oscillations. We then present the standard derivation of the oscillation formula, stressing the problematic choice of basis

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for unstable particles, as well as the ambiguities arising in the conversion of an oscillation in time into an oscillation in space. This section ends with a review of the attempts to solve these problems with more sophisticated quantum-mechanical models. 2.1. Mixing in quantum mechanics The possibility of particle mixing arises from the fundamental principles of quantum theory. The measurement of an observable causes the initial state to jump into an eigenstate of the operator associated to this observable. Thus the act of measurement determines a basis of physical states, made up of all the eigenstates of the corresponding operator. The di1erent bases associated to the di1erent operators are linked by linear transformations; a state expanded in a given basis can be seen as a mixing of the eigenstates of that basis. While one gets used to the quantum superposition principle as long as wave functions correspond to rather abstract properties of a system, it is always a bit shocking to apply it to particles, which are associated in our mind with classical concepts. The two-slit experiment with particles is the best known example, but particle mixing, appearing when an interacting eigenstate is a superposition of di1erent mass eigenstates, is hardly less unsettling. The textbook case is given by the neutral kaon system: the decay pattern of a K 0 into pions is well explained by the superposition of two mass eigenstates, a CP-even state decaying quickly into two pions, and a CP-odd state decaying more slowly into three pions. Thus a K 0 can be seen as a mixing of a KS (K short) and a KL (K long), having well-de$ned masses and widths (the so-called mass eigenstate basis). Whereas pionic decays do not allow to identify the $nal strangeness (K 0 or KH 0 ), it is possible to do so in the decays of kaons into e, with the result that one observes a nonnegligible probability to detect a transition from the initial K 0 into a KH 0 . This is the phenomenon referred to as particle oscillation. The Bavor-mixing transition probability oscillates indeed with the propagation distance, unless the oscillation is averaged by some mechanism, in which case the transition probability simply follows the exponential decay law for each mass eigenstate. Particle oscillations can be compared to a 2-slit interference experiment, which requires $rstly that several paths must be available to a particle, and secondly that these paths should not be distinguishable by the experimental conditions. In particle oscillations, the di1erent mass eigenstates are the analog of the di1erent paths, since they are eigenstates of the space–time evolution operators Pˆ and Hˆ . Of course, their other quantum numbers should be equal. The equivalent of the second requirement is that the masses should not be distinguishable by the experimental apparatus. Under these conditions, the probability to observe the propagating particle in a given state, other than a mass eigenstate, oscillates in space (and in time) with a wavelength inversely proportional to the mass di1erence, because of the interference between the mass eigenstate amplitudes. Thus particle oscillations occur if the initial and $nal states are not mass eigenstates and if it is impossible to ascertain which mass eigenstate has contributed as an intermediate state. This last assumption implies that mass di1erences should be much smaller than the typical energies involved in the processes, with the result that the oscillation wavelength is macroscopically large and the particle oscillates over macroscopic distances. Of course, oscillations are only observable in processes where the propagation distance is of the order of, or much larger than, the oscillation wavelength. Otherwise it is perfectly adequate to work with interaction (or Bavor) eigenstates, as it has been done until now with neutrinos in accelerator experiments. Note that one often talks about particle oscillations even if the oscillation term is

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averaged by some mechanism, such as an integration over the energy spectrum, though this situation could be more appropriately called incoherent mixing. Let us be more precise about the basis pertaining to the oscillation phenomenon. The Hamiltonian should be such that it can be split into a propagation Hamiltonian Hpropag , describing the free propagation of the particle, and an interaction Hamiltonian Hint , describing the interactions producing the propagating particle. These two subsets of the Hamiltonian can be distinguished if there exists a quantum number, called 6avor, conserved by Hint , but violated by Hpropag . The appropriate basis for the production of the particle is made up of eigenstates of both Hint and the Bavor operator. It is called the 6avor basis. The basis relevant to the propagation of the particle is the one in which Hpropag is diagonal and is called the propagation basis, or mass basis. This basis does not coincide with the Bavor basis, since the Bavor operator does not commute with Hpropag . The simplest example is given by stable neutrinos, in which case the Hamiltonian Hpropag includes the mass matrix, whereas the Hamiltonian Hint includes the weak interactions of the neutrinos with the bosons Z and W . The Bavor is here the leptonic number, which can be electronic, muonic or tauic and maybe sterile. This number is conserved by the weak interactions, but is violated by a nondegenerate mass matrix which mixes neutrinos of di1erent Bavors. As regards unstable particles, the Wigner–Weisskopf method [191,172,209] is used to build a non-Hermitian e1ective Hamiltonian Hpropag , with the aim of describing both propagation and decay. Hint includes the interactions involved in the production of the oscillating particles. In the case of neutral K (or B) mesons, Hpropag includes, on the one hand, the degenerate e1ective mass matrix (due to Yukawa, strong and electromagnetic interactions) and, on the other hand, the weak interactions causing the decay and the Bavor-mixing transitions K 0 ↔ KH 0 (or B0 ↔ BH 0 ). Hint is an e1ective Hamiltonian describing the strong interactions of the mesons; it can be built with chiral perturbation theory. The Bavor is here the strangeness (or bottomness) with values ±1, which is conserved by strong interactions but violated by weak interactions. The non-Hermiticity of Hpropag means that the neutral meson system does not evolve in isolation, because of its possible decay. The relationship between the propagation and Bavor basis can be written as Uj |j (0) ; (1) | (0) = j

where U is the matrix diagonalizing Hpropag , if this Hamiltonian has been expressed from the start in the Bavor basis. The states |j (0) are the propagation (or mass) eigenstates, at time t = 0, with well-de$ned masses and widths, belonging to the propagation basis, whereas the states | (0) are the Bavor eigenstates. The properties of the matrix U depend on the Hamiltonian Hpropag . In the case of a stable oscillating particle, the matrix U arises from the diagonalization of the mass matrix and is unitary. In the case of an unstable oscillating particle, Hpropag is necessarily non-Hermitian, as explained above, so that the time evolution is nonunitary, i.e. the oscillating particle is allowed to decay. The matrix U diagonalizing that Hamiltonian is generally not unitary, although it can be unitary if some symmetry makes the Hamiltonian normal 1 [81].

1

H is normal if [H; H † ]=0.

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The computation of an amplitude requires the knowledge of the scalar product values of the eigenstates. In the Bavor basis, the following orthogonality properties hold: (0) | (0) = ;

(2)

since Hint is Hermitian and commutes with the Bavor operator. In the propagation basis, Hpropag is Hermitian only if the oscillating particle is stable. Thus the scalar product of the Bavor eigenstates must be de$ned by transforming the Bavor scalar products (2): Uj−1 Ui−1† : (3) i (0) | j (0) =

These scalar products become orthogonal if the matrix U is unitary, which is true if the particle is stable or, more generally, if Hpropag is normal. In the example of neutral kaons, the approximation of CP symmetry makes Hpropag normal and U unitary. Since the violation of this symmetry is of the order of 10−3 , the right-hand side of Eq. (3) is of the same order in the neutral kaon system. This lack of orthogonality implies that the states |i (0) can be normalized in di1erent ways. The normalization factors do not matter as long as they do not appear in the $nal result. A normalization problem arises however in the case of unstable particles if we try to compute an amplitude involving a mass eigenstate in the initial or $nal state. For example, the amplitude |Htotal |KL depends on the normalization chosen for the state |KL . In that case, the normalization ambiguity is of the order of the direct CP violation parameter , so that it cannot be neglected [61]. This problem does not arise in the $eld-theoretical approach, where the kaons are considered as intermediate states, which are not directly observed. It must be stressed that the orthogonality properties in the Bavor basis are valid only if all mass eigenstates are kinematically allowed [63,220,96]. For example, let us suppose that there are four stable neutrinos such that mi = 0 for i = 1; 2; 3 and m4 1 GeV. Under 1 GeV, the Bavor scalar products become ∗ : (0) | (0) = − U4 U4

(4)

This diKculty is a $rst indication of the problems arising in the de$nition of a Bavor basis. Is it really possible to interpret a Bavor eigenstate as an observable particle? In quantum mechanics, each stable particle can be associated with an irreducible representation of the PoincarUe group, with a given mass and a given spin. In that framework, the Bavor eigenstates do not correspond to particles, unless they are degenerate in mass. The problem is not easier to solve in quantum $eld theory. Whereas creation and annihilation operators of mass eigenstates are well-de$ned, transformation (1) does not yield creation and annihilation operators of Bavor states that satisfy canonical commutation relations [142]. One could thus wonder whether Bavor eigenstates really exist or not. A phenomenological argument in favor of their existence is that, for obvious practical reasons, no one has yet devised an experiment involving only one mass eigenstate of a system of oscillating particles. 2 On the contrary, oscillating particles are produced and detected in a superposition of mass eigenstates formally equivalent to a Bavor eigenstate. In that sense, Bavor eigenstates are observed, whereas mass eigenstates are not. 2

This might have been possible for neutral kaons, had the CP symmetry been exact in that system, making the propagation basis coincide with the CP basis. Decay channels respecting CP (K → 2 ; 3 ) would have then allowed a nonambiguous determination of the propagating eigenstate.

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The case of an unstable particle di1ers from the stable case in two respects. First, the decay widths of the propagating eigenstates can vary greatly, allowing the isolation of a nearly pure beam of the longest-lived eigenstate (so the phenomenologist is tempted to declare this state ‘observable’). Second, there is no state corresponding to an unstable particle in the Hilbert space of physical states. These particles are best described by S-matrix poles in quantum $eld theory (so the theoretician is tempted to declare these unstable states ‘not observable’). 2.2. Oscillations with plane waves Many articles and books give derivations of the oscillation formula with plane wave states (see for example Refs. [66,208]). In this section and in the next, we examine this method in its $ne details, emphasizing subtle points and doubtful assumptions. Following the rules of quantum mechanics, the transition probability between an initial state of Bavor at time t = 0 and position x = 0 and a $nal state of Bavor at time t and position x is given by A→ (t; x) = (0) |exp(−i Hˆ propag t + iPˆ · x)| (0) ; where Hˆ propag is the Hamiltonian operator and Pˆ is the generator of translations. The states |; (0) are taken to be energy–momentum eigenstates, so that a plane wave treatment will follow. This amplitude can be computed with the help of the change of basis (1): A→ (t; x) = (0)| Uj e−ij −j |j (0) j

−1 Uj e−ij −j Uj | (0) ; = (0)|

(5)

j;

where the phase is given by j = Ej t − p j · x ; and the decay term is de$ned by m j j t : j = 2Ej

(6) (7)

The symbols Ej , pj , mj and j stand for the energy, momentum, mass and width of the mass eigenstate |j (0). The factor pj · x is often dropped, either because the mass eigenstate is assumed to be in its rest frame (the time t is then the proper time), or because of an assumption of equal momenta for all mass eigenstates (the factor pj · x then does not appear in the probability). These assumptions cannot be justi$ed and neither will be assumed in the following derivation. If the particle is unstable, Hpropag is an e1ective Hamiltonian, computed with the Wigner–Weisskopf method in the rest frame of the decaying particle. Hence, the phase depends on the proper time, j =mj , so that a boost is necessary to obtain expressions (6) and (7) valid in the laboratory frame. However the concept of a rest frame has no meaning in the case of a superposition of di1erent mass eigenstates, which may have di1erent velocities. Thus the choice of the boost is ambiguous because it is not clear whether a common boost should be used for the di1erent mass eigenstates, or whether each mass eigenstate is boosted by a di1erent velocity. This diKculty is another indication of the

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problems arising in the correspondence between a Bavor state and a particle. It will be seen in Section 2.3 that the choice of the boost has a big impact on the value of the oscillation length. With the help of the orthogonality property (2), amplitude (5) can be rewritten as −1 Uj e−ij −j Uj : (8) A→ (t; x) = j

In the case of K and B mesons, the oscillation formula (8) is sometimes written in the following way [210]: Uj e−ij −j Ui† i (0) | j (0) : (9) A→ (t; x) = i; j

The equivalence of this expression with Eq. (8) can be checked with the scalar products (3). Another way to obtain the transition amplitude (8) for neutral mesons is to de$ne a reciprocal basis [238,110,17,77,243]. This method consists in de$ning two mass bases: the kets belong to the in basis and are the right-eigenvectors of the Hamiltonian, whereas the bras belong to the out basis (or reciprocal basis) and are the left-eigenvectors of the Hamiltonian. Their Hermitian conjugate vectors are not used to write amplitudes. The Hamiltonian can then be expressed in a diagonal form: |j (0)in j out j (0) | ; (10) Hˆ propag = j

where the j = mj − ij =2 are the complex eigenvalues of the Hamiltonian in the rest frame of the oscillating particle (in the mass-degenerate limit). The new bases are related to the Bavor basis by Uj | j (0)in ; | (0) = j

(0)| =

−1 out j (0) | Uj

:

(11)

j

With these notations, a kind of orthogonality property is restored: out i (0) | j (0)in

= ij :

The oscillation formula obtained with these new bases is identical to Eq. (8), since it is simply another way to decompose Hˆ propag . The physical interpretation of the reciprocal basis is not clear and its use is not really necessary, since the same result can be obtained without it. Actually, the reciprocal basis method is a technical trick which is best understood from a $eld-theoretical viewpoint. After all, the new bases were $rst introduced [238] in a quantum $eld theory approach as left- and right-eigenvectors of the propagator representing the oscillating particle. As emphasized by Enz and Lewis [110], they should always appear through Eq. (10), underlining their unphysical and intermediate character. For example, the quantity | |Htotal | KL |2 is not an observable, as mentioned in Section 2.1. In the special case of a Hermitian Hpropag , U is unitary and j = 0. The amplitude then reads U j e−iEj t+ipj ·x U∗j : A→ (t; x) = j

This is the expression commonly used for neutrinos, except that the term pj · x is often dropped for the reasons mentioned after Eq. (7).

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In the general case of a non-Hermitian Hamiltonian Hpropag , the transition probability is given by the squared modulus of the oscillation amplitude (8): P→ (t; x) = Ui Ui−1 Uj∗ (U −1 )∗j e−i(i −j )−i −j : (12) i; j

The particle oscillation phenomenon is manifest in the oscillatory behavior of the interference terms in the transition probability (12), with a phase i − j depending on the point (t; x) of detection. Regarding antiparticle oscillations, the CPT theorem applied to the amplitude (8) gives the relationship AH→H (t; x) = A→ (t; x); so that the oscillation formula for antiparticles is obtained from Eq. (12) by the exchange of and . Note that CP violation arises from terms in Eq. (12) breaking the ↔ symmetry, since it appears through a di1erence between |AH→H (t; x)|2 and |A→ (t; x)|2 . 2.3. Time to space conversion of the oscillating phase 2.3.1. Standard oscillation phase There has been some controversy about the conversion of the oscillation phase i − j , appearing in Eq. (12), into a measurable quantity. Since the propagation time is not measured in oscillation experiments, 3 a prescription is needed to get rid of the time dependence in the phase di1erence. The numerous prescriptions proposed in the literature are somewhat confusing. They can be classi$ed by expanding the phase around an average energy or momentum. Since the oscillating particle is on-shell, the energy Ej can be expressed in function of the momentum pj . The phase j can then be expanded around an average momentum p, not very di1erent from pj or pk , and an average mass m. Although the mass m can be very di1erent from mj or mk in the ultra-relativistic case, the expansion is correct as long as the mass di1erence m2j = m2j − m2 is small with respect to the energy. The momentum di1erence pj = pj − p is expected to be of the same order than m2j . In one spatial dimension, the expansion of the phase reads, to $rst order in m2j and pj , m2j (13) j = pj2 + m2j tj − pj x = Et − px + t + (vt − x)pj + Etj ; 2E with the average energy and velocity de$ned by E = p2 + m2 and v=p=E, respectively. A di1erent time tj has been allowed for each mass eigenstate, and expanded around an average time t, with tj = tj − t. Of course, the following arguments will only be correct to $rst order in m2j =2E, but it is useless to argue about further orders in a Bawed approach such as the plane wave treatment. All prescriptions leading to the standard oscillation formula set tj = 0, i.e. they impose that interference only takes place for equal propagation times (and lengths) for the di1erent mass eigenstates. This equal time prescription has been explicitly stated [181,177,178,201], in reaction against articles proposing di1erent detection times [244,245]. It has also been legitimated by an equal velocity prescription [245,101], which is seen to be equivalent to the previous prescription with the help of 3

However time measurements are important according to Okun [217] in an experiment performed at IHEP (Serpukhov) [23].

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the classical relation tj = x=vj . However the equal velocity condition leads to Ej =Ei = mj =mi , which is very unlikely for neutrinos [218]. Note that imposing equal times ti = tj , in the laboratory frame, also means imposing equal proper

times i = j if the classical relation j = tj2 − x2 is used. The last relation implies that a boost of velocity vj = x=tj is used to go from the rest frame of the mass eigenstate mj to the laboratory frame. Thus the question of the choice of the correct boost boils down to the question of the equality of propagation times in the laboratory frame. A second prescription is needed in order to obtain the standard oscillation formula. It could be called the classical propagation condition: vt − x = 0. It imposes that the term (vt − x)pj , appearing in Eq. (13), is negligible in comparison with m2j t=2E. This condition can be weakened to |vt −x|t. Since plane waves are delocalized in space–time, this condition cannot be justi$ed without a more sophisticated treatment, for example with wave packets. These two prescriptions are suKcient to derive the following formula: m2ij |x| |x| i − j ∼ = 2 osc ; (14) = Lij 2|p| obtained from a three-dimensional generalization of Eq. (13). The oscillation length Losc ij is de$ned by 4 |p| Losc ; (15) ij = m2ij where m2ij = m2i − m2j is assumed to be positive. The classical propagation condition also justi$es the substitution t → E|x|=|p| in i; j (see Eq. (7)). The transition probability (12) together with phase (14) form the standard oscillation formula used to $t the experimental data (see [154] for an application to two-Bavor neutrino oscillations). Of course, probability (12) should $rst be averaged over the energy spectrum and over the region of production and detection (see Section 9.2.1). If the oscillation is completely washed out by these averaging procedures, the transition probability (12) can be simpli$ed by the substitution e−i(i −j ) → ij . Although the equal time prescription and the classical propagation condition are suKcient to obtain the standard oscillation phase (14), additional prescriptions leading to the same result are commonly found in the literature: 1. The equal momentum prescription is the most common: pi = pj = 0. As seen above, this assumption is not necessary. Moreover, it is impossible to impose experimental conditions such that the momentum uncertainty is zero, since the oscillations are destroyed by a momentum measurement more accurate than the mass di1erence. Thus, this prescription is groundless. 2. The equal energy prescription has been recently advocated by Lipkin: Ei = Ej = 0 [197]. It has the advantage of avoiding the classical propagation condition since it leads to pj = −m2j =2p, so that the time dependence completely drops from the phase di1erence j − k (at least if the equal time prescription is assumed), and the standard oscillation phase (14) is directly obtained. In principle, oscillation experiments are feasible with a zero energy uncertainty, since spatial oscillations are not expected to vanish in that case (note that a quantum $eld treatment is necessary to prove it). In practice, the energy uncertainty is far from being negligible and is often of the

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same order of magnitude as the momentum uncertainty. Thus the equal energy prescription is only justi$ed if an extremely small uncertainty on the energy is imposed by the physical properties of the process itself. Do we have any theoretical reason to expect that this uncertainty is smaller than the mass di1erence? No convincing arguments supporting that assumption have been given until now. For example, 4 Lipkin computes the energy–momentum uncertainties pK , EK of the kaon in the process p− → &K 0 [198]. If the proton is at rest in a lattice, its momentum uncertainty, due to the Debye temperature of the crystal, can be estimated at pp ∼ 103 eV. Lipkin estimates pK to be of the same order of magnitude, whereas he neglects EK ∼ (pp )2 =mp . However EK is still much larger than the mass di1erence mK ∼ 10−6 eV, which is the most important mass scale in the experiment. Moreover the pion momentum uncertainty gives a $rst order contribution to EK and should not be neglected. Even if one has shown that Ep, there is a long way to go to show that Em. Another argument against the equality of energies is that it holds only in one particular frame. For example, if the energies of the di1erent mass eigenstates are equal in the decay of a pion at rest ( → '), the energy di1erence becomes approximately equal to the momentum di1erence if the pion is relativistic [136]. Thus the equal energy prescription should be shown to be true in the laboratory frame for any experimental conditions. However there is a much more reasonable way of looking at the equal energy prescription. It consists in seeing it as the result of a time average washing out the interference between wave packet components having di1erent energies. There is then no need to prove that the energy uncertainty is zero. This argument will be considered in Section 2.5. 3. Energy–momentum conservation at the production: $rst proposed by Winter [255], this recurring prescription [75,146,201,244,245,140,136,252] allows to compute explicitly pj . It has often been used to show that neither the momenta nor the energies of the di1erent mass eigenstates are equal. In the example of the pion decay at rest ( → '), the energies and momenta of the muon and neutrino can be computed exactly if the energy–momentum of the pion is perfectly known. However this knowledge is usually not available: when the energy–momentum spread of the source is much larger than the mass di1erence m2ij , it is meaningless to compute the exact values of the energies and momenta to order m2ij . A more detailed examination of this question requires wave packets instead of plane waves, or even better, quantum $eld theory. 2.3.2. Nonstandard oscillation phase Controversial prescriptions leading to nonstandard oscillations formulas involve di:erent propagation times ti = tj = 0 or, equivalently, di1erent proper times i = j . Let us parametrize pj by pj = p + ( − 1)

4

m2j ; 2p

(16)

Lipkin gives at least three reasons for the equality of energies: the strict energy conservation discussed here [197,198], the Bavor-energy factorization [155,198] and the stationarity resulting from a time average [197,198]. The last two arguments are discussed in Section 2.5.

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where is a dimensionless number of order unity. The corresponding energy and velocity can be written as Ej = E +

m2j ; 2E

vj = v + ((1 − v2 ) − 1)

m2j : 2pE

(17)

The momenta are equal if = 1 whereas the energies are equal if = 0. As explained above, it has also been proposed to determine through energy–momentum conservation at production, leading to a value of of order unity. In any case, the time di1erence is computed with the help of the classical relation tj = x=vj and reads tj = (E 2 − m2 )

m2j x : 2p3 E

Inserting this value in Eq. (13) and using t = x=v, we obtain 1 − m2j x j = Et − px + 1 + + 2 : v 2p

(18)

With the equal momentum prescription ( = 1), the corresponding oscillation length will be smaller by a factor 2 than the standard value given in Eq. (15) [197]. With the equal energy prescription ( = 0), the oscillation length will be smaller by a factor 1 + v−2 than in Eq. (15). Thus the equal energy prescription may also lead to a nonstandard oscillation length, contrary to what was claimed in [197]. With the energy–momentum conservation prescription, the oscillation length will be smaller than in Eq. (15) by a factor depending on the value of , which depends on the energy [244]. These formulas can also be applied to the case of correlated oscillations, such as (1020) → K 0 KH 0 or ((4s) → B0 BH 0 . In the center-of-mass frame of the resonance, the di1erent time prescription leads to an oscillation length smaller by a factor 2 than the standard value, since the equality |pi | = |pj | valid in that frame leads to = 1 [245]. Another disturbing consequence of the di1erent time prescription is the oscillation of recoil particles, for example & in − p → &K 0 [244], or the muon in → ' [246,247]. This is easily seen by applying Eq. (13) to the recoil particle. Although m2j = 0 (the recoil particle has only one mass eigenstate), i = j because ti = tj . The oscillation of particles having only one mass eigenstate is unacceptable since it leads to nonconservation of the detection probability of this particle. This is not the case when there are several mass eigenstates, as the sum of the detection probabilities of the di1erent mass eigenstates is always equal to 1 for a given propagation distance. The treatment of neutrino oscillations in → ' proposed by Field [112] also resorts to the di1erent time prescription. This author claims that the di1erent neutrino mass eigenstates are detected at the same space–time point but are produced at di1erent space–time points. Following Field, the oscillation phase would not only receive a contribution from the neutrino path di1erence but also from the path di1erence of the source which decays at di1erent times. Field computes the $rst contribution with the energy–momentum conservation prescription, which gives for a source at rest an oscillation phase larger than the standard result by a factor 2 (see Eq. (18) with v=1). The second contribution, which only appears in Field’s article, is computed in the same way. The oscillation

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phase obtained by Field therefore di1ers from the standard result (14). Field’s method also leads to the prediction of muon oscillations because of the use of the di1erent time prescription. 5 Another example of the di1erent time prescription can be found in Malyshkin and Kulsrud’s analysis [204] of the time variations of the solar neutrino Bux, which leads to a result di1erent from Parke’s formula [224]. As we do not consider here oscillations in matter, this new oscillation formula will not be discussed further, although our comments on the di1erent time prescription also apply to that case. The di1erent time prescription has been strongly criticized by several authors within the plane wave framework [201,181,177,178,197,198]. Their argument, 6 which can be stated as ‘interference only occurs between states taken at the same space–time point’, does not hold when examined in a wave packet or $eld-theoretical model. Kiers et al. [182,183] have indeed shown that the coherent character of the detection process allows wave functions at di1erent space–time points to interfere. In particular, a long coherent measurement in time may be used to revive oscillations, even after the mass eigenstate wave packets have completely separated spatially (see Section 2.5). The question of the correct time prescription is thus subtler than it seems at $rst sight. The crux of the matter is to take into account the production and detection processes. In this way, each increment in the phase associated to the propagating particle, due to a slightly di1erent production (respectively detection) point, is cancelled by a decrement in the phase of the wave packet of the source (respectively detector) [140]. For example, this cancellation can be implemented in the intermediate wave packet model (see Section 2.5) by computing the total amplitude as an overlap of the propagating wave packet with the source and detector wave packets [137–139]. Note however that this mechanism is much more natural in a quantum $eld model such as the external wave packet model (see Section 4), where the amplitude is integrated over all possible microscopic production and detection points, with the result that the phase depends only on average (i.e. macroscopic) production and detection points. In conclusion, neither the equal time prescription nor the di1erent time prescription can be justi$ed in the plane wave approach, although the choice of the prescription has an important e1ect on value of the oscillation length. Moreover other prescriptions, such as the classical propagation condition and the equal energy prescription, cannot be understood within the plane waves formalism. A wave packet or quantum $eld treatment is thus inescapable. Let us also insist on the dubiousness of the arguments using energy–momentum conservation. In most cases, they are invalidated by the energy–momentum spectrum of the source: di1erent energy–momentum components of the source can contribute to di1erent mass eigenstates. 2.4. Problems with the plane wave treatment In Section 2.3, it has been shown that the plane wave treatment of particle oscillations cannot deal in a satisfactory way with the time dependence of the oscillating phase. Besides, this approach implies a perfectly well-known energy–momentum and an in$nite uncertainty on the space–time localization 5

Field’s criticism of wave packets models will be examined in Section 2.5. A more convincing (though indirect) argument has been given by Giunti [137]: the use of the di1erent time prescription in the double-slit experiment leads to a wrong interference pattern. 6

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of the oscillating particle. Oscillations are destroyed under these assumptions [176]. On the one hand, the perfect knowledge of the energy–momentum allows to determine which mass eigenstate propagates. On the other hand, the spatial delocalization makes impossible the measurement of the oscillation length. A correct oscillation formula should include observability conditions in such a way that the oscillation term vanishes if either 1. the energy–momentum uncertainty is smaller than the mass di1erence between the interfering mass eigenstates, or 2. the oscillation length is of the same order, or smaller, than the uncertainty on the position of the source or of the detection point of the oscillating particle. Another kind of problem is not speci$c to the plane wave treatment, but a1ects all approaches where the oscillating particle is considered to be directly observable. On the one hand, Bavor eigenstates are ill-de$ned for stable particles, because we do not know how to de$ne creator and annihilation operators of Bavor states satisfying canonical (anti)commutation relations [142]. On the other hand, mass eigenstates are ill-de$ned for unstable particles, since they are in general not orthogonal (except if the Hamiltonian is normal, see Section 2.1). The solution to these problems simply consists in considering the oscillating particle as an intermediate state. Actually this stand reBects well the experimental situation, where the oscillating particle is not directly observed. One rather detects the particles in interaction with the oscillating state, both at the source and at the detector. The Bavor transition probability should thus be computed with observable particles as initial and $nal states [187,142,145,237]. Finally, we should mention two other problems regarding unstable particles. These diKculties arise because of the nonrelativistic Wigner–Weisskopf method used to compute the e1ective Hamiltonian. First, interference between di1erent mass eigenstates is forbidden in nonrelativistic quantum mechanics (Bargmann superselection rule [48,173,133]). The argument is the following. The invariance of the SchrWodinger equation under Galilean transformations determines the transformation law of a quantum state: it is multiplied by a phase factor depending on the mass and space–time position of the state. Thus di1erent mass eigenstates transform di1erently, so that the relative phase in a superposition of such eigenstates is not conserved under Galilean transformations. Therefore a coherent superposition of di1erent mass eigenstates is forbidden. Second, unstable particles cannot be consistently described in nonrelativistic quantum mechanics for the same reason at the origin of the Bargmann superselection rule: transitions between di1erent mass eigenstates are forbidden [173]. Unstable states cannot be considered as asymptotic states. Thus they do not appear in the Hilbert space of physical states and must be treated in quantum $eld theory where they appear as complex poles of the full propagator [253]. 2.5. Intermediate wave packets and other improvements Some of the problems of the plane wave treatment are solved by the intermediate wave packet model, in which a wave packet is associated with each propagating mass eigenstate. Note that this model is usually discussed with respect to neutrinos, i.e. in the relativistic limit. Nussinov was the $rst to put forward the existence of wave packets as the cause of a coherence length beyond which oscillations vanish [216]. Oscillations with wave packets were then studied in detail by Kayser [176].

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Fig. 1. Decoherence due to a wave packet width *x larger than the oscillation length.

Fig. 2. Decoherence due to the nonoverlapping of the di1erent mass eigenstates wave packets.

The oscillation formula was later explicitly computed with Gaussian wave packets by Giunti et al. [141,139,137]. The intermediate wave packet model shows that oscillations vanish if *x & Losc ij , i.e. if the uncertainty over the position is larger than the oscillation length (Fig. 1). For minimal uncertainty wave packets, this condition can be rewritten as *p . m2ij =2p, i.e. oscillations are forbidden if the momentum spread of the wave packets is smaller than the mass di1erence between the interfering eigenstates. Thus oscillations are destroyed by energy–momentum measurements aiming to determine which mass eigenstate propagates. The other main result of the model is the existence of a coherence length beyond which oscillations vanish [216,176,141]. Its usual explanation is that wave packets associated to di1erent mass eigenstates have di1erent group velocities. Hence, wave packets progressively separate, and interference disappears when they do not overlap anymore (Fig. 2). We shall see in Section 7.1.3 that dispersion is also at the origin of the coherence length. In the intermediate wave packet model, the classical propagation condition, |vt − x|t, is automatically implemented by the space–time localization of the wave packet. However the question of the equal or di1erent time prescription is not solved, since the times of production and detection are not speci$ed by the formalism. As regards the question of the equality of energies or momenta, it is in principle no longer relevant, since the energy–momentum is spread out. In fact, the oscillation length depends only on the zeroth order values (in a m2ij -expansion) of the average energies and momenta, because of the cancellation between the time and space parts of the phase (see Eqs. (13) and (16) with pj equal to the average momentum). The time dependence of the probability is dealt with by averaging over time [141]. It is thus not necessary to worry about the equality of energies or momenta. Nevertheless there is still much argument about the equality of the energy spectra, or the equality of the average energies, momenta or velocities. For example, Grossman and Lipkin [155] imposed a same energy spectrum for all mass eigenstates, so that the probability of $nding a particle with

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the wrong Bavor vanishes for all times at the position of the source. It can be objected that this boundary condition is automatically satis$ed without imposing an identical energy spectrum, since the probability to detect the oscillating particle at the source becomes negligible once the wave packet has left it. Equality of average velocities has been proposed by Takeuchi et al. [250,251], but is very unlikely for neutrinos, as noted by Okun and Tsukerman [218], because the ratio of the average energies Ei =Ej should then be equal to the mass ratio mi =mj . Actually there is another way of looking at this question. Starting with arbitrary wave packets, it is simple to check that interference occurs only between wave packet components with the same energy, at least if the oscillation process is strictly stationary. This line of thought has been advocated by Sudarsky et al. [249] (within a $eld-theoretical model), by Lipkin [197,198] (with wave packets) and by Stodolsky [248] (within a density matrix formalism). Let us explain it with the intermediate wave packet model. In one dimension, the wave function corresponding to the mass eigenstate mi can be written as dE Ai (E) e−iE(t −t0 )+iki (x−x0 ) ; i (x; t) = where ki = E 2 − m2i and (t0 ; x0 ) are the coordinates of the source. The Bavor-mixing transition probability is a linear superposition of terms i (x; t) j∗ (x; t). If the time of emission is unknown, the probability is averaged over t0 , so that it becomes a superposition of terms like ∗ dE Ai (E)A∗j (E) ei(ki −kj )(x−x0 ) : (19) dt0 i (x; t) j (x; t) = 2 This equation makes clear that interference occurs only between wave packet components with the same energy. The time-averaged transition probability in the wave packet model is equivalent to an incoherent sum over energy eigenstates (‘incoherent’ means a sum in the probability, not in the amplitude). Eq. (19) can thus be seen as a sum over plane waves of energy E and momentum ki , weighed by the factor Ai (E)A∗j (E). The equivalence between a wave packet and a plane wave decomposition of a stationary beam was already observed by Dicke and Wittke [104] in connection with electron-interference experiments, and by Comsa [94] in connection with neutron-interferometry. The same issue was recently raised by Kiers et al. [182] and by Stodolsky [248], this time in connection with neutrinos. Of course the time integral in Eq. (19) only yields a delta function if the time interval is in$nite, i.e. if the process is strictly stationary. In practice, there is always some available information on the time of emission. If the time integral is limited to the interval XT , the delta function becomes a narrow peak of width 1=XT , which can be neglected as long as it is small in comparison with the mass di1erence m2ij =2E. This condition can be written as XT Tijosc , where Tijosc is the oscillation time. A few more results have been obtained with wave packets. Kiers et al. have used the equivalence between wave packets and plane waves, explained above, to show that it is always possible to increase the coherence length by more accurate energy measurements at the detector [182]. After Giunti and Kim [139] showed how to take into account the detector’s momentum uncertainty in the intermediate wave packet model, Giunti [137] did the same with the detector’s energy uncertainty, so as to reproduce Kiers et al.’s prediction. Wave packets have also been applied by Nauenberg to study correlated oscillations of neutrinos or mesons [212], and to the propagation of neutrinos in matter [143,225]. Finally, it is worth mentioning that the wave packet approach is explained in great

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detail in Kim and Pevsner’s book [185], is discussed by Fukugita and Yanagida [128] and has been reviewed by Zra lek [258]. The intermediate wave packet model has been recently discussed by De Leo et al. [102], who recover the standard oscillation formula under two conditions. The $rst condition, which they call ‘minimal slippage’, is equivalent to the existence of the coherence length discussed above, i.e. mass eigenstate wave packets should overlap at detection otherwise there is decoherence. The second condition, which they call ‘noninstantaneous creation’ or ‘pure Bavor creation event-wise’, means that interference occurs between wave packet components corresponding to di1erent initial wave packet points, so that the standard oscillation phase (14) is obtained. This mechanism, which was already explained by Giunti and Kim [140] (see end of Section 2.3.2), is automatically included in quantum $eld treatments where the oscillating particle is coupled to a source and detector (see Section 4). In the same article, De Leo et al. argue that the oscillation formula might depend on the wave packet shape, for example if the wave packet is a superposition of two well-separated Gaussian peaks, each having its own phase factor. In that case, there would be a succession of coherence/decoherence/coherence/decoherence regions as the mass eigenstates wave packets shift one with respect to the other. Moreover, the constant phase di1erence between the two Gaussian peaks crops up in the oscillation phase in the second coherence regime. While surprising phenomena indeed arise for such special wave packet shapes, it seems diKcult to set up an experiment in which a speci$c wave packet shape is maintained for all the particles within the produced Bux. In practice we should average over all wave packet shapes, with the result that only the typical width of the wave packet will have an e1ect on the $nal oscillation formula. All other e1ects due to the di1erent shapes of the wave packets cancel each other out. Besides, the analysis of oscillations in a faraway region (such as the second coherence regime mentioned above) should take into account the fact that dispersion wipes out oscillations at large distances (see Section 7.1.3). The representation of the oscillating particle by a minimal uncertainty wave packet has been criticized by Field [112], who claims that oscillating particles follow classical space–time trajectories and that the oscillation phase is due to the propagation time di1erence between di1erent mass eigenstates (see Section 2.3). One of his main arguments is that the neutrino momentum spread in → ' is of the order of decay width ' of the muon, so that neutrino oscillations should be completed suppressed by decoherence in wave packet models. Moreover, the Heisenberg relation is violated if one admits that the neutrino spatial uncertainty is of the order of the cm. These arguments do not hold since the neutrino momentum spread reBects the momentum uncertainties of the pion source and of the emitted muon, which are much larger than ' . Finally, Field criticizes models where the neutrino wave packet is coupled to a wave packet source, by claiming that the initial quantum state is not the same for all amplitudes, so that interference is forbidden. This criticism is not justi$ed, since it makes no sense to cut the source wave packet into disjointed parts which are then interpreted as di1erent initial states. Although they solve some of the problems of the plane wave approach, intermediate wave packets are not satisfying for many reasons: 1. The question of the existence of a Fock space for the weak eigenstates remains. 2. Oscillating particles are not, and cannot be, directly observed. It would be more convincing to write a transition probability between observable particles, involved in the production and detection of the oscillating particle.

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3. The wave packet shapes, sizes and normalizations are arbitrary. In particular, the amplitudes of production and detection are not taken into account. This is not a problem in the relativistic limit or for nearly degenerate masses, in which cases these amplitudes can be factorized from the sum over the mass eigenstates. However this factorization is not possible for nonrelativistic particles (unless they are nearly degenerate in mass). 4. The coherence length depends on the di1erence between the group velocities of the wave packets, the exact values of which are unknown in the nonrelativistic case. 7 There is no reason to believe that the group velocities could be determined, to order m2ij =2p, by energy–momentum conservation at the source, contrary to what is claimed in Ref. [141]. 5. The coherence length increases with the precision of the energy–momentum measurements at the source and at the detector. Whereas the momentum precision is limited by the condition *p & m2ij =2p, there is, a priori, no such limitation on the energy accuracy and, therefore, no bound on the coherence length. Although it has been claimed [139] that such a bound exists, the intermediate wave packet model cannot settle that question, because the energy uncertainty has to be inserted by hand into the model. 6. Experiments measure a particle Bux, not a particle density. The transition probability must thus be converted to a Bux density, involving inverse velocity factors. These enhance nonrelativistic mass eigenstates and strongly skew the probability [141]. This e1ect is due to the neglect of the amplitudes of production and detection and to an analysis restricted to one dimension. 7. Although it has not been controversial in the wave packet model, why should the times appearing in the di1erent wave packets be identical? A justi$cation of this equal time prescription is necessary to rule out the nonstandard oscillation formulas discussed in Section 2.3. 8. It is not possible to analyze in the wave packet model the corrections to the oscillation probability due to the possible instability of the propagating particle. 9. The inBuence of the source instability on the observability of the oscillations cannot be studied in that framework. A quantum-mechanical model of neutrino oscillations solving the $rst three problems has been put forward by Rich [237]. This author considers the oscillating particle as an intermediate state and computes the probability transition between initial and $nal observable states with second-order time-dependent perturbation theory of standard quantum mechanics. Since this model takes into account the production and detection processes, it has the other advantage of not requiring the equal time prescription. The spirit of Rich’s model is similar to the stationary boundary condition models resorting to $eld-theoretical methods (see Section 3.1), but it has the disadvantage of being nonrelativistic. Ancochea et al. have tried to solve the diKculty of converting the probability density for neutral kaons into a Bux [21]. In their nonrelativistic wave packet model, they construct Bavor probability currents associated to a SchrWodinger equation. A problem arising in this approach is that the Bavor currents are not conserved [258]. Moreover, no one knows how to extend this method to the relativistic case. 7

Eq. (17) can be interpreted as giving the group velocity of a mass eigenstate wave packet. In the relativistic limit (v → 1), the -dependence drops from this equation, so that the velocity can be determined up to order m2ij =2p without needing to know .

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Sassaroli has proposed an hybrid model for neutrino oscillations, going half-way to quantum $eld theory, in which a coupled system of two Dirac equations is quantized [239]. However boundary conditions cannot be applied consistently, unless lepton Bavor wave functions are considered as observable and the relativistic limit is taken. This diKculty was already noted by Giunti et al. [142]. The review of intermediate wave packet models would not be complete without mentioning its latest and most sophisticated version, as proposed by Giunti [138]. Instead of representing the oscillating particle by a superposition of arbitrary wave packets, Giunti computes the form of the intermediate wave packet from basic principles. More precisely, it consists in creating the intermediate wave packet from quantum $eld interactions between the external wave packets involved in the production process of the oscillating particle. The wave packet then evolves in space–time, before interacting with quantum $eld interactions at the detector. This interacting wave packet model is very close in method, spirit and results to the external wave packet model at the core of the present report. It will be further discussed in Section 5.2. 3. Oscillations in quantum eld theory 3.1. Review of the literature Few authors deny that the most rigorous treatment of oscillations is done in the quantum $eld theory framework. However, although the quantum $eld computations in the literature all reproduce in some limit the naive quantum-mechanical formula given by Eqs. (12) and (14), there is not yet an agreement in which respect they di1er from the naive formula. There are two reasons for the lack of agreement between the existing quantum $eld derivations of the oscillation formula: $rst, the di1erent authors use di1erent physical assumptions, and second, they use di1erent approximation schemes to compute the transition probability. The $eld-theoretical approach to particle oscillations is quite old. Already in 1963, Sachs [238] applied S-matrix methods to neutral kaon interferences, 8 using a model developed earlier with Jacob [166] for unstable particles. Let us call this model, as applied to nonoscillating particles, the Jacob–Sachs model. Its application to systems of mixed particles will be called the external wave packet model. In this model, the particle to be studied is represented by its propagator; it propagates between a source and a detector, where wave packets representing the external particles are in interaction. Much later, Sudarsky et al. [249] studied the inBuence of a spatially varying potential on the neutral kaon system. They resort to a one-dimensional model similar to Sachs’ but do not specify the contour of integration, so that the $niteness of their $nal expressions is not guaranteed. Neither Sachs nor Sudarsky et al. studied the observability conditions of oscillations. Another simpli$ed model was proposed by Beuthe et al. [61], with the aim of modelizing experiments at CPLEAR and DANE. Their model is not satisfying since they use external wave packets localized in time but not in space, which does not correspond to actual experiments. 8

Quoting Sachs, from [238]: ‘The question of whether it is the momenta or the energies of these particles that are to be taken equal in the treatment of interference phenomena has often been raised. [Our] method provides a clear answer to this question since it is based on an analysis of the phenomena in terms of wave packets’. Nearly 40 years after, new articles continue to appear, wondering about the equality of energies or momenta.

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As regards neutrinos, Kobzarev et al. analyzed neutrino oscillations with a bare-bones quantum $eld model [187]: the source and detector are in$nitely heavy nuclei, so that the propagation distance is perfectly known, whereas the propagation time is left undetermined. No constraints on oscillations are discussed in that article. Next came an important article by Giunti et al. [142], showing that it is impossible to build a Fock space for Bavor states, because the mixing of the ladder operators for mass eigenstates does not yield Bavor ladder operators satisfying canonical (anti)commutation relations. This observation strikes a blow to the quantum-mechanical wave packet approach, which should not be mistaken for the quantum $eld model with external wave packets, as it is sometimes the case in the literature. Nonrelativistic corrections to the neutrino propagation are explicitly computed in Ref. [142] in a few examples. In a pioneering article, Giunti et al. [145] studied neutrino oscillations within a Gaussian external wave packet model. They derive a localization condition (no oscillations if Losc . *x ) and a coherence length beyond which oscillations vanish. These conditions agree with those obtained in the quantum-mechanical picture [216,176,141]. A later paper by Giunti et al. [144] contains essentially identical results as in Ref. [145], but with more generality. Cardall used the same model [85], paying greater attention to the normalization of the event rate and to the spin structure. A model close in spirit to the external wave packet model was proposed by Kiers and Weiss [183]. These authors couple the oscillating neutrino with localized source and detector, which are idealized by oscillators. They show how the coherence length increases with the energy precision at the detector. Note that they had already predicted this phenomenon using elementary quantum mechanics [182]. Another kind of model arises from the use of stationary boundary conditions, leading to a unique value for the energy of the oscillating particle. The simplest example is the Kobzarev et al. model mentioned above [187]. Going a bit further, Grimus and Stockinger proposed a model with external particles represented either by bound states or by plane waves. They obtain a localization condition but no coherence length. In two other papers, Grimus et al. [151,152] studied the inBuence of an unstable source on the oscillations, with the model of the previous paper modi$ed by a Wigner–Weisskopf approximation for the unstable source. Ioannisian and Pilaftsis analyzed neutrino oscillations [163] within a scalar version of the Grimus–Stockinger model. They claim to have found a novel form of neutrino oscillations at short distance, which they call ‘plane wave oscillations’. Stationary boundary conditions were also used by Cardall and Chung [87] to study the MSW e1ect in quantum $eld theory. Note that a quantum $eld derivation of the MSW e1ect has been proposed by Mannheim [205], who represents the interaction with matter with an e1ective potential, whereas this potential has been derived with $nite temperature $eld theory by NWotzold and Ra1elt [214] and by Pal and Pham [223]. Campagne [84] studied a neutrino source decaying in Bight, such as a relativistic pion, with $eld-theoretical methods. He sidesteps external wave packets by limiting arbitrarily the interactions regions with the help of step functions, but it is only a trick to replace complicated interactions at the source and detector by simple stationary boundary conditions. Decay in Bight has also been studied by Dolgov [105], within a simpli$ed external wave packet model modi$ed by a Wigner–Weisskopf approximation. Another approach (source–propagator models) consists in using the propagator in con$guration space coupled to a source but not to a detector. Srivastava et al. chose this method to modelize correlated oscillations of two kaons [245] and neutrino oscillations [247]. Since external wave packets are absent, time–space conversion problems cannot be avoided and lead to nonstandard oscillation

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lengths or recoil oscillations (see Section 2.3). Shtanov also used a source–propagator model, and claims to have found a strong dependence of the oscillation formula on the neutrino masses if the source and detector are very well localized in space–time [242]. In a completely di1erent line of thought, Blasone, Vitiello and other researchers have attempted to de$ne a Fock space of weak eigenstates and to derive a nonperturbative oscillation formula. The main results of these studies are summarized in [72]. Note that a previous formalism developed in [73,14,67,158] lead to an oscillation formula not invariant under reparametrization [129,74,130,69] and was replaced by a more satisfying theory in [71,68], which was further developed in [72,167, 168,70]. This new theory aims at de$ning Bavor creation and annihilation operators, satisfying canonical (anti)commutation relations, by means of Bogoliubov transformations. As a result, new oscillation formulas are obtained for fermions and bosons, with the oscillation frequency depending surprisingly not only on the di1erence but also on the sum of the energies of the di1erent mass eigenstates. Apart from the speculative nature of the enterprise, the drawbacks of the approach are the dependence on time, not on space, of the oscillation formula (Lorentz covariance is broken), as well as the lack of observability conditions. Although these studies are very interesting from a fundamental point of view, it is not obvious whether the new features of the Blasone–Vitiello oscillation formulas are observable in practice. Since these new oscillation formulas tends to the standard oscillation formula (12) in the relativistic limit or if the mass eigenstates are nearly degenerate, we can focus on the case of a nonrelativistic oscillating particle having very distinct mass eigenstates. In that case, p ∼ m2 =2E, so that either *p . m2 =2E or p . *p . Under these conditions, the quantum theory of measurement says that interference between the di1erent mass eigenstates vanishes. Once the oscillation terms have been averaged to zero, the Blasone–Vitiello formulas do not di1er anymore from the standard oscillation formula (12). Therefore, the Blasone–Vitiello formalism does not seem to be relevant to the phenomenology of oscillations on macroscopic distances. This observation does not detract from the theoretical worth of that approach. All the above models, whether for mesons or for neutrinos, can be grouped in four categories: external wave packet models, stationary boundary conditions models, source–propagator models and Blasone–Vitiello models. In the following sections, the connection between external wave packet models and stationary boundary condition models will be studied and the Ioannisian–Pilaftsis and Shtanov models will be analyzed in detail. 9 Finally let us mention again the interacting wave packet model recently proposed by Giunti [138] which was described at the end of Section 2.5, since it bridges the gap between the intermediate wave packet model and quantum $eld treatments. Moreover, its results are equivalent to those obtained in the external wave packet model. 3.2. Five questions The contradictions between the existing quantum $eld derivations of the oscillation formula can be summarized into $ve questions: 1. Whereas the external wave packet model allows to associate intermediate ‘wave packets’ to the oscillating particle, it is not possible to do so in models using stationary boundary conditions. 9

Some of our results have been published in Ref. [60].

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2.

3.

4. 5.

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Moreover, the coherence length is $nite in the former case, but in$nite in the latter. Is it possible to see the oscillation formula derived in the models using stationary boundary conditions [187,150– 152,163,87], as a particular case of the oscillation formula derived in the models using external wave packets [145,144,85]? Kiers et al. [182] have shown in a quantum-mechanical model that the value of the coherence length depends on the accuracy of the energy–momentum measurements at the detector. This e1ect has been con$rmed by quantum $eld theory calculations [144,183], but it is not clear whether oscillations survive a perfect measurement of the energy [144,139]. Does a perfect knowledge of the energy lead to an in$nite coherence length or is there decoherence anyway? For a source strongly localized in space and time, Shtanov has derived an oscillation formula where each oscillating exponential exp(−ij ) is multiplied by a prefactor depending on the mass mj . Unless the masses mj are nearly degenerate, these prefactors strongly modify the standard oscillation formula given by Eqs. (12)–(14). Is this result correct? Ioannisian and Pilaftsis [163] claim to have found a novel form of neutrino oscillations (‘plane wave oscillations’), if the spatial spread of the source and detector is of macroscopic size. Does such behavior exist? As in the quantum-mechanical treatment of Section 2.3, the correlation between an oscillating particle and other particles (or recoil particles) at the source has been said to modify the oscillation length with respect to Eq. (14) and to bring about oscillations of the recoil particles [245,247]. Is this assertion true?

In order to answer these questions, we shall use a model such that all speci$c models used in the articles cited above can be recovered in some limit. Moreover, to extend the usual treatment of a stable relativistic oscillating particle, our model will also allow the oscillating particle to be nonrelativistic and/or unstable and will take into account the dispersion. Answers to questions 1– 4 can be found in Section 7.3 while question 5 is treated in Section 10.

4. The external wave packet model The numerous problems arising in the plane wave approach of oscillations (see Section 2.4) and in the intermediate wave packet method (see Section 2.5) show that the oscillating particle cannot be treated in isolation. The oscillation process must be considered globally: the oscillating states become intermediate states, not directly observed, which propagate between a source and a detector. This idea is easily implemented in quantum $eld theory, where intermediate states are represented by internal lines of Feynman diagrams. Quantum $eld theory has the advantage of providing a relativistic treatment from the start, which is required to study the mixings of relativistic and nonrelativistic particles. It also allows to describe unstable particles in a consistent way. The oscillating particle is represented by a relativistic propagator, which determines the space–time evolution and the possible decay of the particle. Boosts and the consideration of speci$c rest frames become pointless. Since interactions are included in the amplitude, equal or di1erent time prescriptions are not needed anymore. Particles interacting with the oscillating particles at the source and at the detector are described by wave packets. This external wave packet model was $rst proposed by Sachs [238] for kaons and by Giunti et al. [145] for neutrinos.

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As appears clearly from the review of Section 3.1, there is an ongoing controversy in the literature about whether the boundary conditions (i.e. the source and the detector) should be taken as stationary or not. In other words, can the time independence of most oscillation experiments be translated into the assumption that the energy of the process is perfectly de$ned? Although it is true that most sources are stationary from a macroscopic point of view (for example, the Bux of solar neutrinos is steady), there is no reason to think that it should be the case from a microscopic point of view. Whereas it can be reasonable to make a stationary approximation in a quantum-mechanical model where we do not have any information on the microscopic processes (for example a density matrix model as in [248]), the same approximation is very dubious in quantum $eld theory, which describes the interactions of individual particles. After all, perturbative quantum $eld theory applies to one-particle propagation processes, in which a stationary source and detector are the exception, rather than the rule. As emphasized by Cardall [85], the Sun is certainly not stationary at the atomic scale, and neither is a detector composed of bound state particles. For example, a water Cerenkov detector sees charged lepton wave packets with $nite energy and time spread. The $nite character of the spread is partly due to the limited coherence time of the bound state particle that has interacted with the incoming neutrino. Boundary conditions can be considered as stationary at the microscopic level when the energy uncertainty at the source or detector is smaller than the inverse propagation distance, i.e. *eP; D . 1=L. This constraint is extremely stringent. In the example of atmospheric neutrinos, the process can be considered as stationary if *eP; D . 10−19 MeV, which is not satis$ed in current oscillation experiments. Although it seems diKcult to argue that the energy uncertainty is smaller than the mass di1erence, it is possible to take up another stand regarding stationary boundary conditions. In the same way as in Section 2.5, it consists in arguing that interference occurs between wave packet components with the same energy, because of the time average on the transition probability. As it is unrealistic to consider in isolation an interference for a given energy, we should take care to integrate the probability (computed with stationary boundary conditions) over the wave packet energy width. At that point a question arises: is this energy width determined by the source or by the detector? There is thus information to be gained by working with nonstationary boundary conditions. Moreover, we shall show that stationary boundary conditions can be imposed in the external wave packet model by assigning zero velocities to some states, whereas other states are represented by plane waves. In Section 4.1, we describe the Jacob–Sachs model which is the prototype for the external wave packet model. We then examine the diagonalization of mixed propagators in Section 4.2. Next, we compute the transition amplitude of the process associated to particle oscillations in Section 4.3. In Section 4.4, we discuss the simplest model in which oscillations can be consistently described. Finally, we present in Section 4.5 the external wave packet model in its Gaussian version. 4.1. The Jacob–Sachs model The $rst version of the external wave packet model was developed by Jacob and Sachs [166] to describe the propagation of a nonoscillating unstable particle. 4.1.1. The process The propagating process of a particle between a source and a detector (extended in space–time and indicated by dotted circles) is symbolized by Fig. 3.

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Fig. 3. Propagation of a particle between a source and a detector, centered in xP and xD .

On Fig. 3, momenta Bow from left to right. PI represents the set of incoming particles, of total momentum q, arriving in the production region (or source), which is centered around the point (tP ; xP ). PF represents the set of outgoing particles, of total momentum k, coming from the production region, with the exception of the intermediate particle whose propagation is studied. DI , DF and (tD ; xD ) are de$ned similarly, but apply to the detection process. If the intermediate particle decays, DI (q ) can be seen as an outgoing state. The interaction points at production and detection are noted x and x , respectively. The external particles are assumed to be stable, whereas the intermediate particle is allowed to be unstable. Although it is possible to describe external unstable particles by considering a more global process where all unstable particles are considered as intermediate states, the technical diKculties involved in computing the corresponding amplitudes are enormous. All computations will be carried out for a scalar propagating particle since it can be argued (see Section 4.3) that the spin has no impact on the characteristics of oscillations. With the aim of localizing the production region around (tP ; xP ), the in- and outgoing particles at point x are represented by wave packets which overlap (in con$guration space) around (tP ; xP ). As the energy–momenta are often measured, the wave packets are supposed to be well-localized in momentum space, around their average momentum. The detection region is localized in the same way. Note that there is no diKculty in replacing certain wave packets by bound states or by plane waves, if needed. For example, it is practical to use plane waves if an emitted particle is not observed [150]. If the dotted circles are removed, the above picture can be interpreted as a Feynman diagram. The internal line represents either a particle or an antiparticle. The experimental conditions are chosen such that a quasi-real particle propagates on a macroscopic distance, transferring positive energy from x to x . The energy–momenta of the initial and $nal states are such that the S-matrix element, corresponding to the above process, is evaluated at the singularity of the propagator of the intermediate particle. If x 0 − x0 is a macroscopic time, it will be seen that the intermediate particle (positive energy component) contributes to the propagation, but not the intermediate antiparticle (negative energy component). 4.1.2. Wave packets A wave packet representing a given state | of mass m can be expressed in momentum space [226] as | = [dk] (k) | k ;

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where (k) is the wave function in momentum space taken at time t = 0, and |k belongs to the basis of one-particle states of momentum k, also taken at time t = 0. In the interaction picture, the time-dependence of the basis and of the wave function cancel each other so that the state | is independent of time. That is why we may take (k) and |k at time t = 0. The following notation is used: dk 1 ; (2 )3 2E(k) √ where E(k) = k2 + m2 . The normalization of free states is chosen as follows [226]: [dk] =

k | p = 2E(k) (2 )3 (3) (p − k) ; so that

| =1

if

dk | (k)|2 = 1 : (2 )3

In con$guration space, the wave function reads dk (x; t) = (k) e−iE(k)t+ik·x : (2 )3

(20)

(21)

It is here necessary to use the time-dependent wave function, i.e. (k) exp(−iE(k)t), since the wave function is considered alone. If the wave packet represents a particle with an approximately known momentum K, the wave function (k) is sharply peaked at k = K. The con$guration space wave function ˜ (x; t = 0) has a maximum at the point x = 0 if (K + k ) = (K − k ). On this assumption, the wave function will be noted (k; K). Wave packets centered in x0 at time t0 are built with the help of the space–time translation operator exp(iPˆ · x0 ), where x0 = (t0 ; x0 ). If a wave packet in momentum space is given by 1(k; K; x0 ; t0 ) = (k; K) eiE(k)t0 −ik·x0 ;

(22)

the corresponding wave packet in con$guration space will be peaked at the point x0 at time t0 : dk ˜ t; K; x0 ; t0 ) = (k; K) e−iE(k)(t −t0 )+ik·(x−x0 ) : 1(x; (2 )3 Without loss of generality, let us choose to work with only one particle in PI (q), in PF (k), in DI (q ) and in DF (k ). The extension to a larger number of initial and $nal particles is straightforward and would only complicate the notation. The wave packets are built such that those involved in the production are centered at xP at time tP , whereas those involved in the detection are centered at xD at time tD . They are noted [145,144] |PI = [dq]1PI (q; Q; xP ; tP )|PI (q) ; |PF =

[dk]1PF (k; K; xP ; tP )|PF (k) ;

M. Beuthe / Physics Reports 375 (2003) 105 – 218

|DI = |DF =

133

[dq ]1DI (q ; Q ; xD ; tD )|DI (q ) ; [dk ]1DF (k ; K ; xD ; tD )|DF (k ) :

Gaussian wave packets are discussed in Section 4.5.1. 4.1.3. The propagation amplitude The general formula of the connected amplitude corresponding to Fig. 3 is given by 4 ˆ − 1|PI ; DI ; A = PF ; DF | T exp −i d x HI where HI is the interaction Lagrangian for the intermediate particle and Tˆ is the time ordering operator. Let g be the coupling constant in HI . Expanding the amplitude to order g2 , and inserting the wave packets expressions, we obtain [dk]1P∗F [dq ]1DI [dk ]1D∗ F Aplanewave (q; k; q ; k ) (23) A= [dq]1PI with

Aplanewave (q; k; q ; k ) =

4

d x MP (q; k)e

−i(q−k)·x

d 4 x MD (q ; k )e−i(q −k )·x G(x − x) ;

(24)

where MP (q; k) and MD (q ; k ) are the interaction amplitudes at production and detection. The propagator G(x − x) has been obtained by $eld contraction and reads d 4 p −ip·(x −x) e G(p2 ) ; (25) G(x − x) = (2 )4 where G(p2 ) = i(p2 − M02 + i)−1 is the free propagator in momentum space and M0 is the bare mass of the propagating particle. It is assumed that renormalization has been carried out so that M0 can be replaced by the physical mass m. The external particles are on shell: q0 = EPI (q) = q2 + m2PI ; and so on. We have supposed that the particle (p0 ¿ 0) propagates from x to x and the antiparticle (p0 ¡ 0) from x to x. If interactions at the source and at the detector are such that the particle propagates from x to x and the antiparticle from x to x , the contraction of the $elds would have yielded the propagator d 4 p ip·(x −x) H e G(p2 ) : G(x − x) = (2 )4 In that case, the di1erent phase sign would select the antiparticle pole in the subsequent contour integral. If the propagating particle is unstable, the complete propagator should replace the free propagator in the amplitude. For this purpose, the amplitude is expanded to all orders and one sums over all diagrams corresponding to one-particle-irreducible insertions (1PI self-energy) in the propagator

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[78,226]. This sum is a geometrical series in the 1PI self-energy −i6(p2 ) and yields the same formula as above, except that G(p2 ) represents now the complete propagator: i G(p2 ) = 2 : (26) 2 p − M0 − 6(p2 ) + i It is convenient to proceed to a change of variables in Eq. (23): x → x + xP

and

x → x + xD ;

where xP = (tP ; xP ) and xD = (tD ; xD ). Amplitude (23) becomes d4 p (p0 ; p)G(p2 ) e−ip·(xD −xP ) ; A= (2 )4

(27)

where the overlap function (p0 ; p) represents the overlap of the incoming and outgoing wave packets, both at the source and at the detector. It is de$ned by 0 4 ipx 4 −ipx −iqx (p ; p) = d x e d x e [dq] PI (q; Q) e [dk] P∗F (k; K) eikx ×

[dq ]

DI (q

−iq x

;Q )e

[dk ]

∗

DF (k

; K ) eik x MP (q; k)MD (q ; k ) :

(28)

All external particles are on-shell. Note that the overlap function is independent of xP and xD . The integrals over x and x in Eq. (28) yield delta functions, which impose energy–momentum conservation at the source and the detector. Remark also that the overlap function depends on the directions of incoming and outgoing momenta. Most computations in $eld theory apply to microscopic processes, where in- and outgoing states can be approximated by plane waves. In that case, the wave functions PI , PF , DI and DF become delta functions, so that the overlap function (p0 ; p) is proportional to (p0 ; p) ∼ (4) (K + K − Q − Q )(4) (p − Q + K)MP (Q; K)MD (Q ; K ) ; where Q = ( Q2 + m2PI ; Q) and so on. The momentum integral in Eq. (27) is then easy to perform and yields A ∼ (4) (K + K − Q − Q )G((Q − K)2 ) e−i(Q−K)·(xD −xP ) : This expression cannot lead to oscillations in a superposition of several amplitudes, since the phase of the exponential does not depend on the mass of the intermediate state. Thus, oscillations vanish in the plane wave limit. In the case of an amplitude corresponding to a macroscopic process, such as the propagation on a macroscopic distance, the overlap function is not a delta function so that the momentum integral in Eq. (27) is not trivial (see Section 4.4 for a simple exact model). The main contribution comes from the pole associated to the propagating particle, so that the phase of the exponential will depend on the mass. 4.2. Mixed propagators In Section 2.1, mixing was de$ned in quantum mechanics as the result of the noncoincidence of the Bavor basis and the mass basis. The de$nition of mixing is similar in $eld theory, except that

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135

it applies to $elds, not to physical states. This di1erence allows to bypass the problems arising in the de$nition of Bavor and mass bases. The complete Lagrangian is split in a propagation Lagrangian Lpropag and an interaction Lagrangian Lint . These two subsets of the Lagrangian can be distinguished if there is a 6avor transformation which is a symmetry of Lint but not of Lpropag . Particle mixing occurs if the propagator built from Lpropag , and representing the creation of a particle of Bavor at point x and the annihilation of a particle of Bavor at point x , is not diagonal, i.e. not zero for = . The Lagrangian Lpropag always includes the kinetic and the mass terms. If the particle is unstable, Lpropag also contains the interaction causing the decay. In the simplest case, illustrated by the light neutrinos e , ' and , the propagating particle is stable and the form of its fundamental Lagrangian is known (although the mass matrix might have its origin in an e1ective Lagrangian). Lpropag contains the mass matrix, generated by Yukawa interactions, whereas Lint includes the weak interactions. In the Bavor basis (called the gauge eigenstates basis for neutrinos), the mixed propagator is nondiagonal. It is well-known [208,64] that Lpropag can be diagonalized by a unitary transformation on the $elds: 10 † = Vj j ; (29) j

where Greek and Latin indices refer to the Bavor and to the mass basis, respectively. It could however turn out that only a subset of V is used to describe the mixing in an oscillation experiment. One reason could be that some of the mixed states are too heavy to be produced (see Eq. (4)). In this case, the mixing matrix for the light states can be nonunitary [63,220,96]. Another reason could be that some of the mixed states are sterile, i.e. they only interact through mass mixing. In that case, the mixing matrix for the nonsterile states is rectangular and does not satisfy unitarity [240]. The propagator is de$ned by the time-ordered two-point function: G (x − x) = 0|T ( (x )∗ (x))|0 : Since Wick’s theorem applies to $elds in the mass basis, we must substitute Eq. (29) in this equation. In the scalar case, $eld contraction yields † G (x − x) = Vj GD; jj (x − x)Vj ; (30) j

where GD; jj (x − x) is the free propagator (with D for ‘diagonal’) of a scalar particle of mass mj : i d 4 p −ip(x −x) e GD; jj (x − x) = : 4 2 (2 ) p − m2j + i The derivation of the mixed propagator for a spin one-half particle is similar. The treatment of unstable particles is more involved. The complete Lagrangian is again split in Lint and Lpropag , but the interactions causing the decay should be included in Lpropag . The free propagator is then replaced by the complete propagator, obtained by an in$nite sum on the self-energy −i6(p2 ) [49]: iG −1 (p2 ) = p2 1 − M02 − 6(p2 ) ; 10

The matrix V corresponds to the matrix U t of Section 2.1.

(31)

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M. Beuthe / Physics Reports 375 (2003) 105 – 218

in an obvious matrix notation. The propagation eigenstates are characterized by the complex poles of the complete propagator zj = m2j − imj j or, equivalently, by the zeros of the inverse propagator. It is always possible to diagonalize the propagator if its eigenvalues are distinct: G(p2 ) = V −1 (p2 )GD (p2 ) V (p2 ) ;

(32)

2

where GD (p ) is the diagonal matrix the elements of which are given by i GD; jj (p2 ) = 2 : 2 p − mj + imj j

(33)

The matrix V is not necessarily unitary, since the self-energy matrix for unstable particles is usually not normal (see Section 2.1). Moreover, it depends on the energy. If the mass eigenstates are nearly degenerate, the self-energy can be approximated by its value at p2 = m2 and the matrix V becomes constant. In Section 8.1, we show that the energy dependence of the self-energy generates corrections to the amplitude in inverse powers of the propagation distance L. These corrections are due to production thresholds of multi-particles states [241,166,78]. 4.3. The 6avor-mixing amplitude In order to derive the Bavor-mixing amplitude, the Jacob–Sachs model of Section 4.1 has to be modi$ed to take into account the di1erent Bavors. The process corresponding to the production at the source of a particle of Bavor , and the detection of a particle of Bavor at a detector placed at a macroscopic distance from the source, can still be symbolized by the Fig. 3 of Section 4.1.1, except that the intermediate particle should be considered as a superposition of di1erent mass eigenstates. The initial Bavor is tagged, for example, by the outgoing state PF (k), whereas the $nal Bavor can be tagged by the outgoing state DF (k ). If it is impossible to identify the Bavor at the detector (ex: K 0 ; KH 0 → + − ), one should sum over the di1erent Bavors. If the energy dependence of the matrix V diagonalizing the propagator can be neglected (see Eq. (32)), the amplitude corresponding to the global process can be expressed as a linear combination of amplitudes Aj corresponding to the propagation of di1erent mass eigenstates: A( → ; T; L) = Vj−1 Aj Vj ; (34) j

where the average propagation time T is de$ned by T =xD0 −xP0 and the average propagation distance by L = xD − xP . The partial amplitude Aj has the same form as the propagation amplitude of an isolated particle, given in Eq. (27): d4 p 0 Aj = (p0 ; p)GD; jj (p2 ) e−ip T +ip·L : (35) (2 )4 The overlap function (p0 ; p) is de$ned by Eq. (28) and the propagator for the jth mass eigenstate by Eq. (33). As noted in Section 4.1.1, it will be enough for our purpose to work with a scalar oscillating particle. When the spin is taken into account, three cases must be distinguished: 1. If the particle is relativistic and the interactions at the source and detector are chiral, only one helicity eigenstate contributes to the propagation. The spin structure can then be factorized and the

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137

computation can proceed with a scalar propagator. The two mentioned requirements are satis$ed for light neutrino oscillations, since the energy threshold of the detectors is much higher than the light neutrino mass scale, and because of the chirality of the Standard Model Bavor-changing interactions. 2. If the particle is nonrelativistic, the spin structure is approximately equal for the di1erent helicities provided that the mass eigenstates are nearly degenerate: mi ∼ = mj ≡ m. The helicity factors can then be factorized from the sum over the mass eigenstates: spin 2 |H (s; m)|2 |Ascalar P→ (T; L) ∼ → (T; L)| ; s

where H (s; m) includes all helicity dependent factors (with the index s referring to the spins of

the external particles). However the source and detector contributions to |H (s; m)|2 cannot s be disentangled if the particle is not relativistic, so that the whole process cannot be factorized into a product of source/propagation/detection probabilities (this point was emphasized in Refs. [145,87]). As the mass eigenstates are nearly degenerate, it seems natural to de$ne the oscillation probability as the modi$cation introduced by the mass di1erence: spin

P→ (T; L) =

P→ (T; L)

spin

P→ (T; L)|mi =mj

;

so that the calculation of the oscillation can be done as if the oscillating particle were scalar. The discussion is very similar for a relativistic particle with nonchiral interactions. 3. If the particle is nonrelativistic and the mass eigenstates have very di1erent masses, each speci$c process has to be computed separately, with the inBuences of the amplitudes of production and detection carefully taken into account for each mass eigenstate. This situation is not really relevant to oscillations, but rather to incoherent mixing, since oscillations are averaged to zero in such experimental conditions. In other words, the contributions of the Feynman diagrams corresponding to the di1erent mass eigenstates are summed in the probability, not in the amplitude. In this report, we always assume the condition m2ij E 2 , under which oscillations are observable, so that the third case is not considered. Some explicit examples of the third case have been given by Giunti et al. [142], and by Kiers and Weiss [183]. The conditions of factorization of the transition probability have been discussed by Cardall and Chung [87]. 4.4. The simplest consistent model A simple model, in which the Bavor-changing amplitude (Eqs. (34)–(35)) can be computed exactly, has been proposed by Kobzarev et al. [187]. In this model, a charged lepton, represented by a plane wave, collides with an in$nitely heavy nucleus, situated in xP . The neutrino produced at that point has a de$nite energy equal to the energy of the incident lepton. At point xD , the neutrino collides with a second in$nitely heavy nucleus and, as a result of this collision, a charged lepton is emitted with an energy equal to the neutrino energy. Let us see what these assumptions mean in the wave packet notation of Section 4.1.2. There are two ingoing states at the source, the lepton and the nucleus. The lepton is represented by a plane

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wave, so that PI; lept (ql ; Ql )

∼ (3) (ql − Ql ) :

The uncertainty on the momentum of the nucleus is in$nite, so that the ingoing and outgoing momentum wave functions are constant: PI; nucl (qn ; Qn )

∼ const

and

PF; nucl (kn ; Kn )

∼ const :

The situation is similar at the detector, except that there are one ingoing and two outgoing states. The overlap function (28) is easily computed: (p0 ; p) ∼ (p0 − Ein )(p0 − Eout ) ;

(36)

where Ein is the energy of the incoming lepton at the source and Eout is the energy of the outgoing lepton at the detector. The partial amplitude (35) becomes Aj ∼ (Ein − Eout ) d 3 p GD; jj (Ein ; p)eip·L ; where L = xD − xP . The Fourier transform of the propagator with respect to its momentum can be computed by contour integration and yields 1 Aj ∼ (Ein − Eout )eipj L ; (37) L 2 − m2j and L = |L|. where pj = Ein The transition probability between an initial state of Bavor and a $nal state of Bavor is given by the squared modulus of the amplitude (34) with Aj given by Eq. (37). Oscillations between mass eigenstates mi and mj arise from interference terms Ai A∗j ∼ ei(pi −pj )L . Thus, the ij-interference term oscillates with a frequency equal to m2ij L ; (38) (pi − pj )L ∼ = 2pm 2 2 2 2 − m2 , with m the mass in the degenerate limit. Phase (38) where mij ≡ mi − mj and pm = Ein is equal to the standard oscillation phase (14) calculated in the quantum-mechanical plane wave treatment. This very simple model has the advantage of being consistent, unlike the plane wave approach in quantum mechanics. It could be recommended as a pedagogical tool. It is however not suKcient for a thorough study of the oscillation phenomenon. First, it cannot describe unstable particles, since they cannot decay into an in$nitely heavy state. Moreover, the approximation of plane waves and stationary states is too strong to allow the study of the observability conditions of oscillations. Another drawback of this model is that the amplitude is independent of the direction of L. Last, a spatial localization of the source or the detector more precise than the Compton wavelength of the lightest external particle is unphysical [165]. 4.5. The overlap function for Gaussian wave packets In most cases, it is not possible to compute exactly the Bavor-changing amplitude (34). The technical diKculty lies in the energy–momentum integration in Eq. (35). The two exceptions are,

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139

on the one hand, the Kobzarev et al. model of Section 4.4 and, and the other hand, the limit in which all the external wave packets are plane waves (in which case oscillations vanish as shown in Section 4.1.3). To proceed further, we have to be more speci$c about the shape of the overlap function (28). In order to answer the questions on the coherence length, it is useful to work with an overlap function which depends explicitly on the energy and 3-momentum uncertainties at the source and at the detector. This can be done by approximating the in- and outgoing particles with Gaussian wave packets, as $rst proposed by Giunti et al. [145,144]. The general case of arbitrary wave packets can then be analyzed as a superposition of Gaussian wave packets. Note that the Gaussian approximation is used by most authors since it allows analytical integrations. 4.5.1. Gaussian wave packets General wave packets were de$ned in Section 4.1.2. The wave function of a Gaussian wave packet can be written at time t = 0 in momentum space as (p − p> )2 2 3=4 : (39) exp − > (p; p> ) = 2 2 *p> 4*p> We chose to set the initial phase to zero. This wave packet is thus centered in x = 0 at time t = 0 (see Eq. (22)). Recall that the normalization is given by Eq. (20). The width *p> is the momentum uncertainty of the wave packet, as can be checked by computing the variance of the operator pˆ a : 2 ; (pa )2 > = >|(pˆ a − pH a> )2 |> = *p>

with the average momentum given by pH a> = >|pˆ a |> = p>a : It is also useful to de$ne a width *x> in con$guration space by *p> *x> = 1=2. The wave function is given in con$guration space by Eq. (21). This integral is Gaussian energy is expanded to second order around the average momentum p> : 1 p2 + m2> ∼ ((p − p> )2 − (v> · (p − p> ))2 ) ; = E> + v> · (p − p> ) + 2E> where E> = p>2 + m2> and v> = p> =E> . The wave packet is thus given in con$guration space (2 *2 )−3=4 (x − v> t)@−1 (x − v> t) ˜ > (x; t) = √ x> ; exp −E> t + ip> · x − 2 4*x> det @

if the (40) by (41)

where the matrix notation is implicit and with the matrix @ de$ned by @ij = ij + (ij − v>i v>j )

2it*p2 : E>

(42)

The spatial uncertainty of the wave packet can be computed in con$guration space with Eq. (41): 2 [(Re @−1 )−1 ]aa ; (xa )2 > = >|(xˆa − xHa> )2 |> = *x>

where the average position is given by xHa> = >|xˆa |> = v>a t :

(43)

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If the z axis is chosen along v> , we obtain −1 −1 xx

−1 −1 yy

[(Re @ ) ] = [(Re @ ) ] [(Re @−1 )−1 ]zz = 1 +

4 2 4m4> *p> t : 6 E>

4 2 4*p> t =1+ ; E>2

(44) (45)

These equations show that the wave packet begins to spread in directions transverse to v> at t ∼ = 2 2 E> =2*p> , whereas the spreading along v> begins at the later time t ∼ . In this report, we = E>3 =2m2> *p> refer to the time-dependence of Eq. (43) as the dispersion of the wave packet. Although dispersion will be neglected in the external wave packets, we shall see that the propagation amplitude associated to the oscillating particle spreads in the same way. In order to study the stationary limit v> → 0, it is of great interest to compute the energy uncertainty of the wave packet (as proposed in Ref. [138]). The average energy EH > and the average squared energy should both be computed to order *p2 . In momentum space, the average energy is given by ˆ = E> + (3 − v>2 ) EH > = >|E|>

*p2 : 2E>

(46)

The second term comes from the second order term in the expansion of the energy operator around 2 the average momentum. Remarkably the average energy EH > is di1erent from E> = p> + m2> . The average squared energy can be computed exactly: >|Eˆ 2 |> = E>2 + 3*p2 ;

(47)

so that the squared energy uncertainty is given to order *p2 by (E)2 > = >|(Eˆ − EH > )2 |> = v>2 *p2 :

(48)

In con$guration space, the same results are obtained only if the dispersion (i.e. time spread) of the wave packet is taken into account. We compute to order *p2 (additional terms appear at higher order) the average energy and squared energy: 9@ i ˆ ; (49) >|E|> = E> − Tr 4 9t iE> 9@ Tr : (50) >|Eˆ 2 |> = E>2 + v>2 *p2 − 2 9t With the help of Eq. (42), we can check that Eqs. (49) and (50) are equal to Eqs. (46) and (47), respectively (but only if dispersion is not neglected). Thus the squared energy uncertainty is the same in both representations as expected (similar computations for the momentum uncertainty show that its value is the same in momentum and con$guration space even if dispersion is neglected). Note that the energy uncertainty coincides at $rst order with the naive expectation: ∼ E = (p> + p)2 + m2 − p>2 + m2 ∼ (51) = v> · p : It could be objected that the squared energy uncertainty does not vanish to order *p4 =E>2 in the limit v> → 0. However external wave packet models are always built with nonspreading wave packets,

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141

i.e. one sets @ij = ij . If this approximation is taken seriously, we can forget about the original expression in momentum space. In that case, the average energy and squared energy computed in con$guration space are shifted to E> and E>2 + v>2 *p2 , respectively, with no correction of higher order in *p2 (see Eqs. (49) and (50)), so that the energy uncertainty is still given by Eq. (48). Thus the energy uncertainty of nonspreading wave packets vanishes in the stationary limit v> → 0 to all orders in *p2 . This result is of great interest to study the stationary limit of the external wave packet model. For example, the bound state wave function used by Grimus and Stockinger [150] coincides with the wave function of a nonspreading wave packet at rest, i.e. Eq. (41) with v> = 0 and @ = 1. Besides, plane waves can be obtained by taking *x> → ∞ in Eq. (41). 4.5.2. Gaussian overlap function We now compute the overlap function (28) with nonspreading external wave packets as done in Ref. [144]. The factors MP (q; k) and MD (q ; k ) multiplying the exponential vary slowly over the width of the wave packet and can be approximated by their value at the average momentum. They can be factorized outside the sum over the mass eigenstates since it is assumed that the neutrinos are either relativistic or nearly degenerate in mass. The wave packets PI; F and DI; F have the form given in Eq. (39). Expanding the energy to $rst order around the average momenta, the momentum √ integrations in Eq. (28) give results similar to Eq. (41) but with @ = 1 and additional factors 2E coming from the integration measure. For example we obtain (x − vPI t)2 −iqx [dq] PI (q; Q)e ; (52) = NPI exp −iEPI (Q)t + iQ · x − 2 4*xP I 2 −3=4 where NPI = (2 *xP ) (2EPI (Q))−1=2 is a normalization constant. I Doing the same for the other wave packets, we can write the overlap function as

(p0 ; p) = N with P (p

0

; p) =

P (p

0

; p)

∗

D (p

0

; p) ;

x2 − 2vP · xt + @P t 2 d x exp i(p − EP )t − i(p − pP ) · x − 2 4*xP 4

0

(53) ;

where EP = EPI − EPF , pP = Q − K. The function D (p0 ; p) is de$ned in the same way, with the index P replaced by D, except for the energy–momentum which is de$ned so as to be positive: ED = EDF − EDI and pD = K − Q . The constant N includes the normalization constants as well as the factors MP; D evaluated at the maxima of the wave packets. A new width *pP has been de$ned by *pP *xP = 1=2, with 1 1 1 = 2 + 2 : 2 *xP *xPI *xPF *pP can be interpreted as the momentum uncertainty at the source. The width *xP is mainly determined by the external particle with the smallest space width. This is expected since the production region depends on the overlap in space–time of the external wave packets. The symbol vP is de$ned by vPI vP F 2 vP = *xP : + 2 2 *xP *xPF I

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It can be interpreted as the velocity of the production region, approximately equal to the velocity of the particle with the smallest spatial spread (unless the velocities of the di1erent in- and outgoing particles are very di1erent). The symbol @P , satisfying 0 6 @P 6 1, is de$ned by 2 vPI vP2 F 2 : @P = *xP + 2 2 *xP *xPF I Recall in all above de$nitions that there might be more than one in- and outgoing state in the production and detection process. Integrating over x and t, we obtain P (p

0

−3 −1 ; p) = 2 *pP *eP exp(−fP (p0 ; p)) ;

(54)

with fP (p0 ; p) =

(p − pP )2 (p0 − EP − (p − pP ) · vP )2 + ; 2 2 4*pP 4*eP

(55)

where 2 2 2 = *pP (@P − vP2 ) 6 *pP : *eP

(56)

The quantity *eP can be interpreted as the energy uncertainty at the source, or also as the inverse of the time of overlap of wave packets during the production process. Indeed, we can show that *2 xP 2 = (v − v )2 ; *eP 2 *2 4*x x ¡

where the sum is over all wave packets involved in the production process. This sum is dominated by the term including the two smallest wave packets in con$guration space (unless their velocities are nearly equal). If *x1 is the smallest width and *x2 the second smallest, we obtain |v1 − v2 | 1 ∼ overlap : (57) *eP ∼ *x2 TP overlap

where TP is de$ned as the duration of the production process. Thus, *eP can be interpreted as the energy uncertainty at the source, since it is proportional to the inverse of the time of overlap of overlap the external wave packets at the source. The quantities *xD , *pD , vD , @D , *eD , TD have similar de$nitions and properties. Note that stationary boundary conditions are recovered by setting vP; D = 0 and *eP; D = 0, with *pP; D di1erent from zero. Besides, we shall need to know the stationary limit of |vP; D |=*eP; D . Since −1 vP is the velocity of the production region and *eP is the duration of the production process, the ratio |vP |=*eP is bounded by the size SP of the macroscopic region of production. The argument is similar for the ratio |vD |=*eD . Thus we assume the following constraints: |vP; D | . SP; D : (58) *eP; D These bounds are very conservative, since we shall see that stationary models such as those found in Refs. [150,163] are recovered by setting |vP; D |=*eP; D = 0. In the example of the Grimus–Stockinger

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model [150], an initial stationary neutron (vPI;n = 0) decays into a stationary proton (vPF;pr = 0), a ‘plane-wave’ electron (*xPF;el = ∞) and the intermediate antineutrino. At detection, the antineutrino collides with a stationary electron (vDI =0) and the outgoing antineutrino and electron are represented as plane waves (*xDF; = *xDF;el = ∞). As argued at the end of Section 4.5.1, all these stationary states are limiting cases of wave packets. As the propagation distance is macroscopic, only processes satisfying global conservation of energy–momentum have a nonnegligible probability of occurring. Since our aim is not to prove this well-known fact, we impose that pP = pD ≡ p0

and

EP = ED ≡ E0 ;

(59)

but we still have vP = vD . These approximations allow to do expansions around p0 and E0 . An associated velocity can be de$ned by v0 = p0 =E0 . 5. Propagation amplitude and dispersion 5.1. Introduction In Section 4, we described the external wave packet model and we applied it to Bavor-mixing transitions. All the information relevant to the source and detector was included in what we called the overlap function (Eq. (28)). We computed this overlap function, on the one hand, in the pedagogical case of in$nitely heavy source and detector (Eq. (36)) and, on the other hand, in the more general case of Gaussian external wave packets (Eqs. (53)–(55)). It is now possible to evaluate the transition amplitude. Although the Bavor-mixing amplitude, as de$ned in Eqs. (34) and (35), is the common starting point of most $eld-theoretical models found in the literature, the obtained oscillation formulas do not always agree. In Section 3, several conBicting results were mentioned, namely stationary states versus nonstationary states, the possible existence of plane wave oscillations and of mass prefactors. Actually, it will be seen in Sections 6 and 7 that these apparent contradictions vanish if we proceed to a careful evaluation of the integrals present in Eq. (35). In the present section, the possible methods of evaluating amplitude (35) are compared. We shall see that the choice of the integration method depends on the propagation distance, so that the distance range can be divided in three regimes. We then evaluate the amplitude in the three scenarios. 5.2. Pole integration and wave packet correspondence Since the experimental conditions are such that the propagating particle is on-shell, the main contribution to the transition amplitude (35) comes from the pole of the propagator. Although integrals on p0 and p can, in principle, be done in any order, it is practical to integrate $rst on the energy, i.e. to do the pole integration on the energy, in order to make possible an interpretation of the amplitude in terms of propagating wave packets. In Section 6.4, we shall discuss another method of integration (for stable oscillating particles) that has the advantages of being shorter and of clearing up the meaning of the stationary limit. However, this last method neither lends itself to a wave packet picture nor can be easily extended to unstable oscillating particles.

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The crucial step in the pole integration is the choice of an appropriate contour in the complex plane. The contour should be carefully chosen as the analytic continuation of most overlap functions diverges at in$nity in the complex plane. The integration on the energy can be performed with the help of the Jacob–Sachs theorem [166]. This theorem is based on the assumption that the energy spectrum of all incident particles is limited to a $nite range. Thus the overlap function (E; p) is distinct from zero only for p2 =E 2 −p2 within certain bounds (with E ¿ 0). On this interval, (E; p) is taken to be in$nitely di1erentiable. In that case, the Jacob–Sachs theorem says that the asymptotic value (T → ∞) of the energy integral in Eq. (35) is given by its residue at the pole below the real axis. Thus, the evaluation of the partial amplitude Aj with the Jacob–Sachs theorem yields d3 p (Ej (p); p) e−ij (p)−j (p) ; Aj ∼ (60) = (2 )4 Ej (p) where Ej (p) =

m2j + p2

j (p) = Ej (p)T − p · L j (p) =

m j j T : 2Ej (p)

(61) (62) (63)

For Gaussian external wave packets, the overlap function is given by Eqs. (53)–(55). In principle, this function should be cut o1 outside the energy range determined by experimental conditions so as to satisfy the conditions of the Jacob–Sachs theorem. However, these corrections are very small and will be neglected in the computations. The reader is referred to Appendix A for more details on the validity conditions of this theorem. It is interesting to note that amplitude (60) is mathematically equivalent to the amplitude obtained in the intermediate wave packet model [176], in which the mass eigenstates are directly represented by wave packets. The overlap function (Ej (p); p) corresponds to the wave function of the jth mass eigenstate. Thus it makes sense, in an external wave packet model, to talk about mass eigenstate ‘wave packets’ associated with the propagating particle. This wave packet picture brings up a problem, as the overlap function takes into account not only the properties of the source, but also of the detector. This is unusual for a wave packet interpretation and not satisfying for causality. This point was recently clari$ed in an interesting article by Giunti [138] which was already mentioned in Section 2.5. In this paper, Giunti proposes a sophisticated version of the intermediate wave packet model, in which the wave packet of the oscillating particle is explicitly computed with $eld-theoretical methods in terms of external wave packets. The intermediate wave packet depends only on the function P (p0 ; p) (see Eq. (54)). The translation of this wave packet to the detection point yields a factor exp[ − ij (p)]. At the detector, its interaction with external wave packets gives rise to the function D (p0 ; p). Remarkably, the total amplitude is exactly the same as Eq. (60). However, the propagating wave packet now depends only on the properties of the source through the function P (p0 ; p). The only drawbacks of this model are that it needs the equal time prescription and that it is not applicable to unstable oscillating particles.

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145

Fig. 4. Laplace’s method is preferable if the phase varies slowly (upper drawing), whereas the method of stationary phase is preferable if the phase varies rapidly (lower drawing). The overlap function (resp. the cosine of the phase) is indicated on both drawings by a continuous (resp. dotted) line.

5.3. Three propagation regimes 5.3.1. Laplace’s and stationary phase methods Unfortunately, the integration over the 3-momentum in Eq. (60) cannot be done analytically. Resorting to the explicit form (53)–(55) of the overlap function valid for Gaussian external wave packets, we see that integral (60) can be approximated by means of an asymptotic expansion for −2 −2 which two kinds of large parameters can be used. On the one hand, *pP; D and *eP; D are large parameters appearing in the overlap function (53)–(55). They suggest a second order expansion of the integrand around the maximum pj of the overlap function, followed by a Gaussian integration: this is called Laplace’s method [111,53]. On the other hand, T and L are large parameters appearing in the phase. They suggest a second order expansion of the integrand around the stationary point pcl; j of the phase, followed by a Gaussian integration: this is called the method of stationary phase [111,53]. The competition between these two asymptotic behaviors implies a detailed study of the oscillation of the phase around the average momentum pj (Fig. 4). The expansion of the phase in Eq. (60) should be compared with the expansion of the overlap function. Although both methods are expected to lead roughly to the same answer in the case of the propagation of a single particle, it should be checked whether the delicate compensation mechanism resulting in the oscillation phase is independent of the method chosen. In Section 5.3.2, it will be shown that Laplace’s method is preferable if the dispersion (i.e. spreading in time) of the amplitude is negligible. This is certainly the case suKciently near the source but is not true anymore beyond some threshold. From the discussion of Section 4.5.1, it is expected that the spreading of the amplitude $rst begins in directions transversal to the propagation axis, before becoming signi$cant along this axis. It is thus appropriate to divide the propagation range into three

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regimes: the no-dispersion regime, the transversal-dispersion regime and the longitudinal-dispersion regime. Laplace’s method will be used in the $rst regime in all directions and in the second regime along the propagation axis, whereas the method of stationary phase will be used in the second regime along transversal directions and in the third regime in all directions. As the phase of an amplitude corresponding to a propagating particle oscillates in pL, the interference between two amplitudes corresponding to two mass eigenstates provides a phase of the form pL, where = m2ij =2p02 , with m2ij = m2i − m2j . When the conditions of production and detection of the oscillating particle are studied with the help of Gaussian wave packets, a new parameter is introduced, namely *p2 . Therefore, the parameter can also appear with other dimensionless coeKcients like (p=*p )2 and (*p L)2 . These new coeKcients are smaller than the original coeKcient pL if the distance L is macroscopic. The new contributions will be neglected in the phase, since they are much smaller than the oscillating phase pL. On the other hand, they should not be neglected in the modulus of the amplitude, where they are the leading terms in the -expansion and yield observability conditions of oscillations. We shall thus calculate the transition probability to O(2 ) in the real part of the argument of the exponential and to O() in the phase. Gaussian integrations will be consistent with the -expansion if the extremum is computed to O(), the real part of the argument of the exponential to O(2 ), the phase to O(), the $rst derivatives to O() and the second derivatives to O(0 ). 5.3.2. No-dispersion regime As discussed above, the integration over the 3-momentum in Eq. (60) is done by Laplace’s method if the phase varies slowly over the width of the overlap function, i.e. if the particle is suKciently near the source. The aim of this section is to determine the extension of this no-dispersion regime and to compute the amplitude Aj in this range. The $rst step consists in analyzing the form of the overlap function (53)–(55). The inBuence of the decay term j (p) (Eq. (63)) will be neglected in this section since we are near the source. The study of the overlap function amounts to study the argument of the exponential, i.e. of the function fj (p) de$ned by fj (p) = fP (Ej (p); p) + fD (Ej (p); p) ;

(64)

where fP; D are de$ned by Eq. (55) (with the constraint (59)) and Ej (p) by Eq. (61). It is convenient to expand all results in small mass di1erences around (E0 , p0 , m0 , v0 ), where m20 = E02 − p02 and v0 = p0 =E0 . The expansion parameter is noted and refers collectively to all m2j =2E02 , where m2j = m2j − m20 . The value pj minimizing fj (p) is given to O() by m2j ; (65) pj = p0 + (uP + uD ) 2E0 where v0 − vP; D uP; D = : 2*eP; D The associated energy Ej = pj2 + m2j can be expanded to O() as m2j ; (66) Ej = E0 + ˜ 2E0

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147

where =1+v ˜ 0 ·uP +v0 ·uD . The values of the dimensionless coeKcients and can be computed but their explicit expressions will not be needed. It is suKcient to know that ˜ → 0 in the stationary limit. A velocity vj = pj =Ej is also de$ned for future use. We are now going to approximate the overlap function as a Gaussian and compute its three characteristic widths. At the extremum pj , the Hessian matrix of fj (p) reads to O(0 ) @ab ≡

1 92 fj ab (p ) = + uPa uPb + uDa uDb ; j 2 9pa 9pb 4*p2

where the momentum width *p is de$ned by 1 1 1 = 2 + 2 ; *p2 *pP *pD

(67)

and is approximately equal to the smallest width among the production and detection momentum widths. The associated width *x in con$guration space is de$ned by *p *x = 1=2. The matrix @ab determines the range of p values for which the overlap function (Ej (p); p) is not negligible. As @ab is symmetric, it can be diagonalized by an orthogonal coordinate transformation. The eigenvalues of @ab are *x2 = *x2±

1 ; 4*p2

1 1 1 = 2 + (uP2 + uD2 ) ± 4*p 2 2

(uP2 + uD2 )2 − 4(uP × uD )2 :

(68)

The eigenvector associated with *x2 is in the direction of uP ×uD , whereas the eigenvectors associated with *x2± belong to the plane de$ned by uP and uD . In the limit |uP ||uD | (resp. |uP ||uD |), the 2 eigenvalues *x2 and *x2− become degenerate and the eigenvector associated with *x+ becomes aligned with uP (resp. uD ). This is also the case in the limit of parallel uP and uD . These limits are relevant to the case of stationary boundary conditions which are examined below. Let us choose coordinate axes (ex ; ey ; ez ) coinciding with the normalized eigenvectors associated 2 2 with (*x2 ; *x2− ; *x+ ), respectively. The quantities (*p2 ; *p2 − ; *p+ ) (with *p± *x± =1=2) can be interpreted as the momentum widths of the overlap function, since they give constraints on the range of p values for which the overlap function is nonnegligible: |px − pjx | . *p ; |py − pjy | . *p− ; |pz − pjz | . *p+ :

(69)

The case of the stationary limit is of special interest since this assumption is used by several authors [187,150,151,163,87]. Recall that stationary boundary conditions are obtained in the external wave packet model by taking vP; D → 0 and *eP; D → 0 with |vP; D | . *eP; D SP; D (see Eq. (58)). In this limit, the velocity v0 becomes aligned with the axis ez , x; y v0x; y ∼ vP; D → 0 ;

(70)

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and two eigenvalues become degenerate whereas the third diverges: *x2− → *x2 ; 2 *x+ →

1 + uP2 + uD2 → ∞ : 4*p2

(71)

In other words, the transversal widths (i.e. in directions orthogonal to p0 ) are given by *p in the stationary limit, whereas the longitudinal width (i.e. in the direction of p0 ) is given by *p+ → 0. The second step consists in comparing the expansion of the overlap function with the expansion of the phase j (p) around pj , which reads T (pa − pja )Rab (pb − pjb ) ; (72) j (p) ∼ = j (pj ) + (vj T − L)(p − pj ) + 2E0 where Rab = ab − v0a v0b . The second derivatives have been evaluated to O(0 ). Laplace’s method will be preferred to the method of stationary phase if the phase j (p) varies slowly over the ‘bump’ of the overlap function. As *p ¿ *p− ¿ *p+ , the variation of the phase will become important in the $rst place along the direction px , then in the direction py and $nally in the direction pz . The criterion for the use of Laplace’s method in all three directions px; y; z will thus be determined by considering the largest momentum width *p . The insertion of constraints (69) into phase (72) yields $rst order conditions for a slowly varying phase, |vjx T − Lx |*p . 1 ; |vjy T − Ly |*p− . 1 ; |vjz T − Lz |*p+ . 1 ; as well as a second order condition, T 2 * .1 ; E0 p

(73)

(74)

where we have used the property qa Rab qb 6 q2 and the fact that *p is the largest width. For a given T satisfying Eq. (74), it is always possible to $nd a range of L values so that conditions (73) are satis$ed. For other L values, the amplitude is negligible as will be checked on the result (see Eq. (79)). Thus the criterion allowing to choose the integration method is determined by Eq. (74): the integration on p will be done by Laplace’s method if T . E0 =*p2 or, equivalently (with L = |L| and p0 = |p0 |), if p0 (75) L. 2 ; *p since conditions (73) impose the relation L ∼ = v0 T as long as *x+ L. For *x+ & L (stationary limit), we now show that condition (75) is directly obtained without going through condition (74). The overlap function imposes in that limit that |p| = E02 − m2j , so that we are left with an angular integration with the angular part of the integrand given by p · p0 exp + ip · L : (76) 2*p2

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Condition (75) shows that the angular variation of the phase in (76) is slow with respect to the angular variation of the overlap function, in which case Laplace’s method will give good results. Therefore, condition (75) constitutes a good criterion for the use of Laplace’s method whether the stationary limit is taken or not. Condition (75) is usually not veri$ed in oscillation experiments, because L=*x p0 =*p in most cases. This condition is the same than the one under which Ioannisian and Pilaftsis [163] obtain ‘plane wave’ oscillations. If condition (75) is satis$ed, the evaluation of integral (60) can be done by Laplace’s method and yields Aj = N*p *p− *p+ exp(−iEj T + ipj · L − fj (pj ) − Fj (T ) − j (pj )) ; where N absorbs numerical constants. The function Fj (T ) is de$ned by −1 T 1 Fj (T ) = (vj T − L)t @ + i R (vj T − L) ; 4 2E0

(77)

(78)

where @ab and Rab are considered as matrices. In the framework of the wave packet interpretation developed after Eq. (60), the function exp(−Fj (T )) plays the part of the space–time envelope of the wave packet associated with the jth mass eigenstate. The elements of the matrix Re(@+i(T=2E0 )R)−1 constrain the extent of the wave packet envelope in space–time. As T increases, the wave packet spreads because of the i(T=E0 )R term. Thus the dispersion of the wave packet is due to the second order term in the expansion of the phase j (pj ). Therefore, condition (74) or, equivalently, condition (75) means that dispersion has not yet begun in any direction, transversal or longitudinal. For that reason, the range of L values de$ned by L . p0 =*p2 is called the no-dispersion regime. Note that this threshold was already met in the analysis of Gaussian wave packets in Section 4.5.1 (see Eq. (44)). Of course, the above interpretation is not valid for *x+ & L, in which case the propagation time T becomes indeterminate and dispersion loses its meaning. Now that the origin of dispersion has been clari$ed, the term in Eq. (78) including R can be neglected with respect to @. Moreover, we choose to approximate vj by v0 in Fj (T ). In comparison with the oscillation formulas that will be derived in the transversal- and longitudinal-dispersion regimes, this approximation will lead to the absence of the coherence-length term, since this term exclusively arises, when the dispersion is neglected, from the velocity di1erence vi − vj . Dropping the index j, the wave packet envelope in Eq. (77) can then be written in the coordinate system diagonalizing @ as F(T ) =

(v0x T − Lx )2 (v0y T − Ly )2 (v0z T − Lz )2 + + ; 2 4*x2 4*x2− 4*x+

(79)

which shows that conditions (73) assumed for Laplace’s method are required to obtain a nonnegligible amplitude Aj . In other words, p0 and L should be nearly parallel in order to have an amplitude signi$cantly di1erent from zero. We shall see that this constraint explicitly appears in the $nal oscillation formula (Eq. (111)). 5.3.3. Transversal-dispersion regime Let us now assume that condition (75) is not satis$ed, i.e. L & p0 =*p2 . In that case, Laplace’s method cannot be used to integrate on all three components px; y; z in amplitude (60) since dispersion

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becomes signi$cant. However, the spreading of the amplitude is not identical in all directions. More speci$cally, the onset of dispersion in the direction p0 can be delayed by two factors. First, the matrix element Rab present in Eq. (72) leads to a relativistic contraction (of 1 − v02 ) in the direction p0 of the dispersion of the amplitude (see Eq. (78)). Second, the momentum width along p0 is given for *eP; D *x (i.e. in the stationary limit (71)) by a vanishing *p+ . Thus Laplace’s method is valid for a longer time T in the direction p0 than in directions transverse to this vector. For this reason, the choice of the integration method in the direction p0 will be postponed for a short while, whereas the method of stationary phase will be preferred for momentum integrations in directions transverse to p0 . Let the z axis be along L, i.e. L = Lez . As in Section 5.3.2, the examination of amplitude (60) shows that the quick variation of the phase averages the amplitude to zero unless p0 and L are nearly parallel. The method of stationary phase can thus be applied in directions px; y , the stationary points of which are given by px = py = 0. The result of the method of stationary phase for the transverse momenta in integral (60) can be written as follows: Ng(l) dp exp(−ij (p) − fj (p) − j (p)) ; (80) Aj = T − i' where p ≡ pz and j (p) = p2 + m2j T − pL ;

(81)

fj (p) = fjP (p) + fjD (p) ; with

fjP (p) =

2

(p − p0 ) + 2 4*pP

(82)

p2 + m2j − E0 − (p − p0 )vP 2 4*eP

2 ;

(83)

where vP = vPz and p0 = p0z . The average energy E0 has been rede$ned so as to absorb a factor p0x vPx + p0y vPy . The de$nition of fjD (p) is similar. Finally, j (p) expresses the possible decay of the oscillating particle: m j j T j (p) = : (84) 2 p2 + m2j The function g(l) comes from the transversal part of the overlap function. It expresses the geometrical constraint between the direction of observation l = L=L and the momentum p0 : (p0 × l)2 ; (85) g(l) = exp − 4*p2 where *p is de$ned by Eq. (67). The function g(l) restricts the propagation to a cone of axis p0 and angle arcsin(*p =p0 ) (see Fig. 6 in Section 7.1). −3 −1 −3 −1 Numerical constants have been included in N , as well as 4-volume factors *pP *eP and *pD *eD , 2 and an energy factor E0 . The constant ' = E0 =2*p comes from the overlap function and acts as a cut-o1 for small T . As T is macroscopic, the prefactor 1=(T − i') can be approximated by 1=T so as to give a prefactor 1=L2 in the transition probability. This expected geometrical decrease is seen

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to originate in the transverse dispersion of the wave packet corresponding to the oscillating particle. As in Section 5.3.2, it is convenient to de$ne a reference mass m0 and a velocity v0 by m20 = E02 − p02 ;

(86)

v0 = p0 =E0 :

(87)

As before, the choice of the method to perform the longitudinal momentum integration (80) is done by comparing the expansions of the phase and of the overlap function around the value pj for which fj (p) + j (p) is extremal (the decay term is not neglected in this section). The explicit value of pj will be computed in Section 6.3 for the stable case and in Section 8.2 for the unstable case. The expansions of fj (p) and j (p) are given by (p − pj )2 fj (p) ∼ ; = fj (pj ) + 4*p2 e1 j (p) ∼ = j (pj ) + (vj T − L)(p − pj ) +

m2j T (p − pj )2 ; 2E03

(88)

where Ej = pj2 + m2j and vj = pj =Ej . The e:ective widths *p e1 and *x e1 are de$ned by 1 *p2 e1

=

1 1 (v0 − vP )2 (v0 − vD )2 + 2 + + ; 2 2 2 *pP *pD *eP *eD

1 : (89) 2 Note that *p e1 has the same form as the asymptotic value of *p+ (see Eq. (71)). With the help of the wave packet correspondence discussed at the end of Section 5.2, the e1ective width *pe1 can be interpreted as the energy–momentum width of the oscillation process, since it is the width of the overlap function. It is dominated by the smallest among the energy uncertainties (recall that *eP; D 6 *pP; D ). The e1ective width *x e1 is then approximately equal either to the production or to the detection time uncertainty, depending on which one is the largest. Laplace’s method is preferable if phase (88) varies slowly over the width *p e1 , i.e. if the two following conditions are satis$ed: *p e1 *x e1 =

|vj T − L|2*p e1 . 1 ;

(90)

m2j T 2 4* .1 : 2E03 p e1

(91)

As in Section 5.3.2, the $rst order constraint (90) will be included in the result of Laplace’s method. Thus the criterion allowing to choose between Laplace and stationary phase methods is given by Eq. (91). In other words, it is better to use Laplace’s method if T is smaller than a dispersion time disp Tj de$ned by disp

Tj

=

E03 : 2m2j *p2 e1

(92)

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The term ‘dispersion time’ is justi$ed by the fact that it is the time at which the longitudinal dispersion of the amplitude becomes important, more precisely twice the initial size. A dispersion disp disp disp disp length Lj can be de$ned by Lj = v0 Tj . The distance range p0 =*p2 . L . Lj will be called disp

the transversal-dispersion regime. For L & Lj , the stationary phase method is more accurate: this distance range will be called the longitudinal-dispersion regime. Note that this threshold was already met in the analysis of Gaussian wave packets in Section 4.5.1 (see Eq. (45)). Various estimates of the dispersion length are discussed in Section 5.4.2, showing that the concept of dispersion length is relevant to nonrelativistic particles such as K and B mesons, as well as to supernova neutrinos, and possibly to solar neutrinos. In the transversal-dispersion regime, the evaluation of amplitude (80) as a Gaussian integral around pj gives Ng(l)*p e1 (vj T − L)2 1 ; exp −iEj T + ipj L − fj (pj ) − j (pj ) − Aj = disp disp 4*x2 e1 1 + iT=Tj T 1 + iT=T j

(93) where N absorbs numerical constants and pj is the value for which fj (p) + j (p) is extremal. Amplitude (93) behaves as a wave packet of group velocity vj and space–time extent (1 + disp disp (T=Tj )2 )1=2 *x e1 . If the longitudinal dispersion is neglected (Tj = ∞), amplitude (93) is similar to Eq. (18) of Ref. [144]. 5.3.4. Longitudinal-dispersion regime At suKciently large distance, dispersion becomes signi$cant and all neutrinos propagating freely disp enter into the longitudinal-dispersion regime. In this regime (L & Lj ), it has been argued that integral (80) should be evaluated with the method of stationary phase. The stationary point of the phase j (p) is given by vcl

pcl; j = mj

2 1 − vcl

;

(94)

where vcl = L=T . It can be interpreted as the classical momentum of a particle of mass mj , travelling at the classical velocity vcl . Of course, a stationary point exists only for T ¿ L. Otherwise Laplace’s method must be used, but the amplitude is nearly zero in that case anyway. The evaluation of amplitude (80) as a Gaussian integral around pcl; j gives (p ))2 (f Ng(l)*p e1 cl ; j j ; exp −imj T 2 − L2 − fj (pcl; j ) − j (pcl; j ) + *p2 e1 Aj = disp disp 1 + iT=Tj T 1 + iT=Tj (95) where fj (p) is de$ned by Eq. (82), fj (p) refers to its derivative and N absorbs numerical constants. The wave packet interpretation of amplitude (95) is not obvious but the shape of the associated wave packet can be studied by an expansion around the maximum of the amplitude.

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153

Fig. 5. The propagation range can be divided into three regimes according to the dispersion.

5.4. Estimates of the dispersion thresholds In Section 5.3, it was shown that the propagation range could be divided into three regimes separated by two thresholds (Fig. 5). The $rst threshold is determined by the no-dispersion condition (75) and separates the no-dispersion regime from the transversal-dispersion regime at Lnodisp =p0 =*p2 . disp The second threshold (see Eq. (92)) is given by the dispersion length Lj = p0 E02 =2m2j *p2 e1 and separates the transversal-dispersion regime from the longitudinal-dispersion regime. It is not straightforward to estimate the value of these thresholds, mainly because of our ignorance of the external wave packet sizes. 5.4.1. No-dispersion condition The $rst threshold is rarely of the order of a macroscopic distance. A far-fetched example where the propagation distance could be smaller than the no-dispersion threshold Lnodisp = p0 =*p2 is given by Ioannisian and Pilaftsis [163]: atmospheric neutrinos are detected in the no-dispersion regime if *x & 10−2 cm (with L ∼ 1000 km and p ∼ 1 GeV). Although such a macroscopic size of the neutrino ‘wave packet’ is not totally excluded (see Section 5.4.2), it is unlikely to be veri$ed. However, this case will be examined in detail in Section 6.1 both for reason of completeness and because it was argued in Ref. [163] that the corresponding oscillation formula would di1er signi$cantly from the standard result. 5.4.2. Dispersion length This threshold is a priori more interesting than the $rst, as it is of the order of a macroscopic distance. As said above, the main diKculty lies in the lack of knowledge of the external wave packet sizes. In the case of neutrinos, another uncertainty arises because of the dependence of the dispersion length on the absolute mass scale, instead of a dependence on the mass di1erences. Some examples are given below, partly based on the wave packet size discussion of Section 9.2.2. We should always remember that *p e1 is determined by the smallest width among the production and detection energy or momentum widths. First, the neutrinos in the experiment LSND [28,29] have a momentum around 30 MeV. The neutrino mass can be taken to be 1 eV, and the production and detection widths should be around 0:01 MeV [152]. The corresponding dispersion length, around 107 m, is much longer than the 30 m length scale of the experiment. Second, atmospheric neutrinos have a momentum around 1 GeV and let us take a mass of 1 eV as a bound. An estimate of the source width can be obtained from the inverse pion and muon lifetimes and is given by *pP ∼ 10−14 MeV in the case of a pion source. The detector width can be estimated at *xD ∼ 10−10 m [237], i.e. *pD ∼ 10−3 MeV, and is dominated by

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the source width. The corresponding dispersion length, larger than 1033 km, is completely irrelevant to the experiment. It is however doubtful that the source width is truly estimated by the inverse source lifetime (see the discussion of Section 9.2.2). Third, solar neutrinos have a momentum around 1 MeV and let us take a mass of 1 eV as a bound. Various estimates of the source width have been given, between 10−7 and 10−3 MeV [216,188,200,185,182,204]. The detector width can be estimated as above at *pD ∼ 10−3 MeV. The dispersion length, varying between 102 and 1010 km for a 1 eV neutrino mass (depending on the chosen source width), could be relevant to the experiment, but is much larger for smaller neutrino masses. As regards mesons, kaons in the CPLEAR experiment have a momentum around 550 MeV [120]. If the width is guessed to be 0:01 MeV, the dispersion length, around 10−6 m, is much smaller than the length scale of the experiment. More generally, the ratio between the dispersion length and the decay length (de$ned by Ldecay = p=m) is given by E2 Ldisp = ; Ldecay *p2 e1 2m so that dispersion occurs for all quasi-stable particles (m), and in particular for B mesons. In conclusion, if the estimates above are truthful, dispersion cannot be ignored for K and B mesons, and might be important for solar neutrinos (and certainly for supernova neutrinos). It is then necessary to develop a formalism taking into account the dispersion, in which case the use of the stationary phase method cannot be avoided.

6. Oscillations of stable particles In Section 5, the choice of the method of evaluation of amplitude (35) was seen to depend on the distance range within which the oscillating particle detected. Three distance ranges were distinguished, depending on whether the amplitude had begun to spread in the transverse and longitudinal directions (with respect to the direction of propagation). The amplitude was then evaluated in the three possible cases, yielding formulas (77), (93) and (95) corresponding to the no-dispersion, transversal- and longitudinal-dispersion regimes respectively. In this section, 11 we compute the transition probability for stable oscillating particles. The transition probability is then obtained by squaring the amplitude and averaging over the macroscopic propagation time T , which is not observed in experiments: − 1 ∗ − 1∗ P→ (L) ∼ dT Ai A∗j : Vi Vi Vj Vj (96) i; j

We shall come back to the proportionality factor in Sections 7 and 9.1. The oscillation formula is derived in the three di1erent propagation regimes in Sections 6.1–6.3. A shorter method of integration, valid for stable oscillating particles in the second and third regimes, is then discussed in Section 6.4. 11

The reader not interested in all computational details can skip this section. Its results are summarized in Section 7.

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155

6.1. Oscillations without dispersion 6.1.1. Amplitude For L . p0 =*p2 , the momentum integration in Eq. (60) is done by Laplace’s method, yielding for stable oscillating particles (see Eq. (77) with j (pj ) = 0) the following amplitude:   2 2 mj Aj = N*p *p− *p+ exp−iEj T + ipj · L − − F(T ) ; (97) 4*˜ m E0 where pj is given by Eq. (65), and F(T ) =

(v0x T − Lx )2 (v0y T − Ly )2 (v0z T − Lz )2 + + ; 2 4*x2 4*x2− 4*x+

(98)

with *x , *x± de$ned by Eqs. (68). The parameter *˜ m , which has the dimension of a width, is related to fj (pj ) (see Eq. (64)) by 2 m2j fj (pj ) = : 4*˜ m E0 6.1.2. Probability The last step towards the oscillation formula consists in computing the time average of the transition probability, which is a Gaussian integral on T : (m2i )2 + (m2j )2 (Ei − Ej )2 ∗ − 2F(T0 ) ; dT Ai Aj = Ng˜ exp −iij (T0 ) − − (99) 4F (T0 ) 16*˜ 2m E02 where T0 is the solution of F (T0 ) = 0, while the phase ij (T0 ) is given by ij (T0 ) = (Ei − Ej )T0 − (pi − pj ) · L : Nonexponential factors are included in Ng˜. The function F(T ) can be rewritten as ˜ 2 (˜v0 T − L) F(T ) = ; 4*x2 where

(100)

(101)

√ v˜ 0 = *x @−1 v0 ; √ L˜ = *x @−1 L ;

−2 ). It is then easy to compute with @−1 = diag(*x−2 ; *x−−2 ; *x+ v˜ 0 · L˜ T0 = 2 ; v˜ 0 ˜ 2 (˜v0 × L) ; F(T0 ) = 4*x2 v˜ 20 v˜ 2 F (T0 ) = 02 : 2*x

(102) (103) (104)

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Actually, the time interval of integration XT is $nite. The approximation XT → ∞ is good if XT is larger than the width of the overlap function given by Eq. (104): |v0 |XT & *˜ x e1 ;

(105)

where *˜ x e1 =

|v0 | *x : |˜v0 |

(106)

The insertion of Eq. (103) in Eq. (99) shows that F(T0 ) plays the role of a directional constraint since dT Ai A∗j is nonnegligible if L˜ is nearly parallel to v˜ 0 . In order to express this constraint in terms of nontilded quantities, two cases should be distinguished: *x+ ∼ *x and *x+ *x . For *x+ ∼ *x , dT Ai A∗j is nonnegligible if L is nearly parallel to v0 : v0 (107) L= L + O(*x+ ) : |v0 | With substitution (107), the value of T0 given by Eq. (102) becomes T0 = L=|v0 | + O(*x+ ), so that phase (100) becomes m2ij ij (T0 ) = (L + O(*x+ )) ; 2p0

(108)

which is equal to the standard oscillation phase (14) if *x+ L. If *x+ *x , v˜ 0 and L˜ z tend to zero so that F(T0 ) should directly be studied as a function of v0 and L. General conclusions for arbitrary *x± can be drawn from the study of the quadratic form in (Lx ; Ly ; Lz ) associated with F(T0 ) = 1. This analysis shows that there is an eigenvalue s3 = 0 corresponding to an eigenvector along v0 . The two other eigenvalues s1; 2 are positive (with s2 6 s1 ), so that the surface F(T0 ) = 1 in (Lx ; Ly ; Lz )-space is a cylinder of elliptical section with an axis along v0 . This geometrical picture can be interpreted as imposing that the components of L orthogonal to v0 should be smaller than 1=s2 , whereas there is no constraint at all on the component of L along v0 . In other words, the probability is maximal within a cylinder of axis v0 . In the stationary limit (*x+ → ∞), the lengthy expressions of the nonzero eigenvalues become s1 →

1 ; 4*x2

s2 →

*x2 (v0z )2 1 : 2 2 4*x2 *x+ (v0x )2 + *x+ (v0y )2 + *x2 (v0z )2

(109)

The properties of the stationary limit, namely Eqs. (58), (70) and (71), lead to the bound 1=s2 . SP; D . In the stationary limit, the components Lx; y (which are then orthogonal to v0 ) should thus be smaller than SP; D . The time-averaged probability is therefore always negligible in directions other than along the average momentum p0 . Thus the evaluation of the phase ij (T0 ) yields ij (T0 ) =

m2ij (L + O(SP; D )) ; 2p0

which is equal to the standard oscillation phase (14) if SP; D L.

(110)

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157

With results (99), (103), (104) and (108) (or (110)), the Bavor-mixing transition probability (96) can be written for a propagation distance satisfying SP; D L . p0 =*p2 as ˜ 2 (˜v0 × L) P→ (L) ∼ Ng˜ exp − Vi Vi−1 Vj∗ Vj−1∗ 2*x2 v˜ 20 i; j 

(m2i )2 + (m2j )2 L  ×exp −2 i osc − − 2 2 2 2 Lij 16*˜ m E0

2  ˜*˜ x e1  ; Losc ij

(111)

where ˜ and *˜ x e1 are de$ned by Eqs. (66) and (106), respectively. Ng˜ includes all prefactors independent of L as well as normalization factors. The oscillation length Losc ij is de$ned by 4 |p0 | Losc : (112) ij = m2ij As discussed above, the probability is maximal within a cylinder of axis v0 and radius smaller than SP; D . There is no decrease in 1=L2 since the spatial spreading of the ‘wave packet’ has not yet begun. 6.2. Oscillations with transversal dispersion 6.2.1. Amplitude disp For p0 =*p2 . L . Lj , the momentum integration in Eq. (60) is done by the method of stationary phase in the transverse directions and by Laplace’s method in the longitudinal direction, yielding in the stable case (see Eq. (93) with j (pj ) = 0) Ng(l)*p e1 (vj T − L)2 1 ; (113) exp −iEj T + ipj L − fj (pj ) − Aj = disp disp 4*x2 e1 1 + iT=Tj T 1 + iT=Tj with fj (p) and *p e1 de$ned by Eqs. (82) and (89), respectively. We had not yet determined the value pj for which fj (p) is extremal. The computation of pj to O() yields m2j ; (114) pj = p0 + ( − 1) 2p0 where the dimensionless number is de$ned by 1 vP (v0 − vP ) vD (v0 − vD ) ; (115) − − = *p2 e1 2 2 *p2 *eP *eD with *p de$ned by Eq. (67). The associated energy Ej = pj2 + m2j and velocity vj = pj =Ej are given to O() by m2j E j = E0 + ; (116) 2E0 m2j : (117) vj = v0 + ((1 − v02 ) − 1) 2p0 E0 The parameter has been de$ned so as to be in correspondence with the notation of Ref. [144]. The symbol ! appearing in that article is related to our notation by ! = *p2 =*p2 e1 . Note that the authors

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of Ref. [144] do not compute explicitly and also take the relativistic limit v0 = 1. The explicit value of is very interesting to know, since = 0 in the case of stationary boundary conditions, in which case all mass eigenstates have the same energy E0 . The value of fj (p) at order 2 is given by 2 m2j 1 + O(3 ) ; (118) fj (pj ) = 2 4*m 2E0 where *m is de$ned by 1 1 1 (vP − vD )2 1 2 + : = *p e1 + 2 2 2 2 *m2 *p2 *eP *eD *eP *eD

(119)

*m will be called the mass width, as it imposes a constraint between the masses mi; j and the ingoing energy–momentum (see Section 7.1.5). 6.2.2. Probability As in Section 6.1.2, the last step towards the oscillation formula consists in computing the time average of the transition probability. Actually, the computation can be simpli$ed by considering separately two cases. The examination of Eq. (118) shows indeed that the amplitude Aj (Eq. (113)) is negligible unless |m2j =2E0 | . *m . Together with the same condition on Ai , it puts a constraint on the mass di1erence: 2 m2ij . 2*m2 ; where m2ij = m2i − m2j : 2E0 There are two possible cases: 1. the masses are nearly degenerate; 2. if not, the masses must be very small in comparison with the energy E0 , i.e. the oscillating particles are relativistic. We $rst consider the case of nearly degenerate masses. If |mi − mj |mi ; mj , it is possible to disp disp and to work with only one dispersion time de$ned by T disp = make the approximation Ti ∼ = Tj 3 2 2 E0 =2m˜ 0 *p e1 , with m˜ 0 the mass in the degenerate limit. The integrand in (96) can be written as 2 2 2 2 2 2 2 g (l)* N (m ) + (m ) e1 i j p exp − − iij (T; L) − fij (T; L) ; Ai A∗j = 16*m2 E02 T 2 1 + (T=T disp )2 with the phase given by ij (T; L) = (Ei − Ej )T − (pi − pj )L −

T T disp

(vi T − L)2 − (vj T − L)2 1 ; 1 + (T=T disp )2 4*x2 e1

and the function fij (T; L) given by (vi T − L)2 + (vj T − L)2 1 : fij (T; L) = 1 + (T=T disp )2 4*x2 e1 The prefactor N 2 absorbs numerical constants.

(120)

(121)

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159

The time integral in Eq. (96) can be evaluated with Laplace’s method. We want to compute the transition probability to order 2 in the real part of argument of the exponential, and to order in the phase. This again implies computing the minimum Tij of fij (T; L) to order , fij (Tij ; L) to order 2 , its second derivative to order 0 , and the $rst derivative of the phase to order . The minimum of fij (T; L) is reached to order 0 for T = L=v0 , and to order for vi + vj − 2v0 L Tij = 1− + O(2 ) ; (122) 2v0 v0 If expansion (117) is used, the minimum of fij (T; L) reads fij (Tij ; L) =

L2 (vi − vj )2 1 + ‘2 8v02 *x2 e1 E04

‘2 = 2 ((1 − v˜20 ) − 1)2 8*p e1 m˜ 40 1 + ‘2

m2ij 2p0

2 + O(3 ) ;

(123)

where ‘ = L=(v0 T disp ) and 1 − v˜20 = m˜ 20 =E02 . The velocity v0 has been replaced by v˜0 in vi; j because |(v02 − v˜20 )|1. The value of the second derivative of fij (T; L) reads 2v02 *p2 e1 1 d 2 fij (Tij ; L) = + O() : 2 dT 2 1 + ‘2 The value of the phase at T = Tij is ij (Tij ; L) =

m2ij L + O(2 ) : 2p0

(124)

(125)

The derivative of the phase is given to order by dij 2‘ (1 − v˜20 ) + ‘2 E0 m2ij 2 (Tij ; L) = Ei − Ej − * (vi − vj )L = + O(2 ) ; dT 1 + ‘2 p e1 1 + ‘2 2m˜ 20

(126)

where expansions (116) and (117) have been used. As above, the velocity v0 has been replaced by v˜0 in vi; j because |(v02 − v˜20 )|1. The second derivative of the phase is of order and thus does not contribute to the transition probability to order 2 , at least in the argument of the exponential. The approximation of the time average integration by Laplace’s method can now be done and yields g2 (l) dT Ai A∗j = v0 N 2 *p e1 2 L  2 2  (m2i )2 + (m2j )2 * L L x e1  ; (127) ×exp−2 i osc − − 2 2 − Lij Losc 16*m2 E02 Lcoh ij ij where N 2 absorbs numerical constants. The oscillation length Losc for the masses mi and mj is ij given by 4 p0 Losc : (128) ij = m2ij

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Without loss of generality, m2ij is taken to be positive. The coherence length Lcoh is de$ned by ij p0 osc 1 Lcoh Lij : ij = √ * 2 p e1

(129)

It is assumed that the time interval XT used to average is larger than the width of the overlap function: v0 XT & *x e1

(‘1) :

(130)

Moreover, the time separation T sep between the wave packet peaks is supposed to be smaller than the time interval XT . This assumption is true as long as the distance is not hugely larger than the oscillation length, since Eq. (122) shows that T sep is of the order of T sep = |T ii − T jj | ∼

1 L ; osc Lij v0 p0

(131)

where 1=v0 p0 is of the order of a microscopic time. The violation of the condition XT & T sep gives rise to the interesting possibility of detecting separate pulses, corresponding to the di1erent mass eigenstates [176,236]. For example, a time-dependence of the neutrino burst from the supernova SN1987A has been searched for (see [34,234] for reviews), but only upper mass limits have been derived. Let us now consider relativistic particles with very di:erent masses. Without loss of generality, disp disp we suppose that mi mj . As Ti Tj , the approximation of taking only one dispersion time for disp

the two mass eigenstates is not valid anymore. Therefore, we shall suppose that L . Li and show disp that the decoherence sets in before the dispersion length Li is reached. The calculation of the time average proceeds as in the nearly mass degenerate case, with the approximations ‘i2 1 (with disp ‘i =L=Li ) and 20 (the particle is relativistic), so that the same result is obtained (see Eq. (127)). disp Now, the coherence length Lcoh is shorter than the dispersion length Li if m2ij =m2i & *p e1 =E0 . ij This condition is always true for very di1erent masses, since in that case m2ij =m2i ∼ = 1. Therefore, the interference becomes negligible before the dispersion of the heaviest mass eigenstate begins, so that result (127) is also valid for very di1erent masses. In conclusion, the Bavor-mixing transition probability (96) for a propagation distance L = Ll disp disp satisfying p0 =*p2 . L . min(Li ; Lj ) has the same form whatever the mass values and reads P→ (L) ∼ v0 N 2 *p e1 

g2 (l) Vi Vi−1 Vj∗ Vj−1∗ L2 i; j

(m2i )2 + (m2j )2 L ×exp−2 i osc − − 2 2 Lij 16*m2 E02

*x e1 Losc ij

2

−

2   : coh L

Lij

(132)

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6.3. Oscillations with longitudinal dispersion 6.3.1. Amplitude disp For L & Lj , the momentum integration in Eq. (60) is done by the method of stationary phase in all directions, yielding in the stable case (see Eq. (95) with j (pcl; j ) = 0): (p ))2 (f Ng(l)*p e1 cl ; j j ; (133) Aj = exp −imj T 2 − L2 − fj (pcl; j ) + *p2 e1 disp disp 1 + iT=Tj T 1 + iT=Tj where fj (p) and *p e1 are de$ned by Eqs. (82) and (89), respectively. The stationary point is given by Eq. (94): mj vcl mj L pcl; j = =√ : 2 T 2 − L2 1 − vcl The corresponding energy is given by mj T 2 2 √ Ecl; j = pcl : ; j + mj = T 2 − L2

(134)

(135)

The value of fj (pcl; j ) is given by fj (pcl; j ) = fjP (pcl; j ) + fjD (pcl; j ) with fjP; D (pcl; j ) =

(pcl; j − p0 )2 (Ecl; j − E0 − (pcl; j − p0 )vP; D )2 + ; 2 2 4*pP; 4*eP; D D

(136)

(p ) + f (p ), with while the value of fj (pcl; j ) is given by fj (pcl; j ) = fjP cl; j cl; j jD fjP; D (pcl; j ) =

pcl; j − p0 Ecl; j − E0 − (pcl; j − p0 )vP; D + (vcl − vP; D ) : 2 2 2*pP; 2*eP; D D

6.3.2. Probability As in Sections 6.1.2 and 6.2.2, the last step towards the oscillation formula consists in computing the time average of the transition probability. Examination of the term fj (pcl; j ) in Eq. (133) shows that the amplitude Aj is nearly zero unless |pcl; j − p0 | . *pP; D (see Eq. (136)). This condition means that the interference term Ai A∗j will be negligible unless |pcl; i − pcl; j | . *pP; D , that is |mij |=m . *pP; D =p (with m referring to mi or mj ). In other words, the interference term is negligible if the masses are not nearly degenerate. This result is in agreement with the conclusion of Section 6.2.2: if the masses mi and mj are very di1erent, decoherence sets in before the dispersion length is reached. disp ∼ disp It is thus possible to work with only one dispersion time T disp = Ti = Tj . It is understood disp ∗ disp is taken to be Tj . The that for noninterference terms, Aj Aj , the dispersion time T i.e. dT integral to be computed is dT Ai A∗j , with Ai A∗j =

N 2 g2 (l)*p2 e1 exp(−i˜ ij (T; L) − f˜ij (T; L)) ; T 2 1 + (T=T disp )2

(137)

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with the phase given by ˜ ij (T; L) = mij T 2 − L2 + *p2 e1

T T disp

(fi (pcl; i ))2 − (fj (pcl; j ))2 ; 1 + (T=T disp )2

(138)

and the function f˜ij (T; L) de$ned by f˜ij (T; L) = fi (pcl; i ) + fj (pcl; j ) − *p2 e1

(fi (pcl; i ))2 + (fj (pcl; j ))2 : 1 + (T=T disp )2

We would like to evaluate dT Ai A∗j by Laplace’s method, in the same way as in Section 6.2.2. This involves expanding the argument of exponential (137) in powers of small mass di1erences. However, we should take care not to expand mj around m0 , as there is no guarantee, in the relativistic case, that m √ j = mj − m0 is much smaller than m0 . This is linked to the extreme sensitivity on T of the factor T 2 − L2 in the relativistic case. Thus the masses mi and mj should be expanded around m˜ 0 = (mi + mj )=2. A new expansion parameter is de$ned by m˜ j = mj − m˜ 0 . The parameters m˜ j are said to be of order . Unfortunately, the value T = T˜ 0 minimizing f˜ij (T; L) to order 0 cannot be computed exactly. An approximate solution would be any T satisfying approximately pcl; j ∼ = p0 and the exact solution can ˜ 0 L=p0 is chosen as the approximate solution, with E˜ 0 = be computed by perturbation around it. If T = E p02 + m˜ 20 (this solution satis$es pcl; j =p0 to order 0 ), the value of T˜ 0 , to order m˜ 20 = m˜ 20 −m20 , reads E˜ 0 L m˜ 20 m˜ 20 + (1 − )L ; T˜ 0 = p0 2p03 E0 where is de$ned by Eq. (115). The expansions in m˜ 20 around m0 and in m˜ j around m˜ 0 will be performed to the same order. It can be checked that the value of T minimizing f˜ij (T; L) to order is still given by T = T˜ 0 (the reason is that m˜ 0 is the average of mi and mj ). The value of f˜ij (T; L) at its minimum reads 2 2 2 2 2 4 2 (m ) + ( m ˜ ) m m ˜ m ˜ E ‘ ij 0 ij 0 0 0 2 2 + ((1 − v˜0 ) − 1) + O(3 ) ; f˜ij (T˜ 0 ; L) = p0 8*m2 E02 8m˜ 40 *p2 e1 1 + ‘2 (139) where ‘ = L=(v0 T disp ) as before, *m is de$ned by Eq. (119) and v˜0 = p0 = E˜ 0 . The value of the second derivative of f˜ij (T; L) with respect to T reads 2v02 *p2 e1 1 d 2 f˜ij ˜ ( T ; L) = + O() : (140) ij 2 dT 2 1 + ‘2 This equation shows that the spatial width (linked to the time width by a factor v0 ) of the wave packet associated to the oscillating particle increases linearly with L for ‘1 (i.e. at a distance much larger than the dispersion length): *p e1 m2j spatial width = *x e1 1 + ‘2 ∼ L: (141) p0 E02 This expression agrees with the quantum-mechanical result [176,185], derived by observing that the relation t = (m2 =p2 E)Lp, obtained from t = L=v (with L kept $xed), leads to L = (m2 =E 2 ) (*p =p)L.

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The value of the phase at T = T˜ 0 is m2ij m˜ 0 mij ˜ ij (T˜ 0 ; L) = L= L + O(2 ) ; p0 2p0

(142)

where as before m2ij = m2i − m2j . The derivative of the phase is given by d ˜ ij ˜ (1 − v˜20 ) + ‘2 E0 mij + O(2 ) : (T 0 ; L) = dT 1 + ‘2 m˜ 0

(143)

The second derivative of the phase is of order (m˜ j )2 and can thus be neglected. Eqs. (139), (140), (142) and (143) can be compared with Eqs. (123), (124), (125) and (126). The relation mij = m2ij =2m˜ 0 , valid for nearly degenerate masses, may be used. From that comparison, it is clear that the $nal result will be the same as Eq. (132). In conclusion, the Bavor-mixing transition probability (96) for a propagation distance L = Ll satisfying L & Ldisp reads g2 (l) P→ (L) ∼ v0 N 2 *p e1 2 Vi Vi−1 Vj∗ Vj−1∗ L i; j 

(m2i )2 + (m2j )2 L ×exp−2 i osc − − 2 2 Lij 16*m2 E02

*x e1 Losc ij

2

−

2   ; coh L

Lij

(144)

where the relation 2m˜ 20 (mij )2 + 2(m˜ 20 )2 = (m2i )2 + (m2j )2 , valid for nearly degenerate masses, has coh de$ned respectively by Eqs. (128) and (129). Recall that we can set been used, and with Losc ij , Lij disp ∼ disp Li = Lj = Ldisp in this regime, since decoherence occurs in the transversal-dispersion regime if the mass eigenstates are not nearly degenerate. As in Section 6.2.2, the time interval XT used to average is assumed to be large enough. At $rst sight, we should have v0 XT & ‘*x e1 in the limit ‘1 but this condition is always violated at some large distance. Actually, it is suKcient to suppose that XT & T osc =2 ( = E0 = m˜ 0 is the usual Lorentz factor) so that the phase can freely oscillate around its average value (see Eq. (143) in the limit ‘1). This last condition is easily satis$ed. Putting together this condition with the conditions on XT valid for the no-dispersion and transversal-dispersion regime (Eqs. (105) and (130)), we obtain (with *˜ x e1 ∼ *x e1 ): Tijosc * : (145) XT & max x e1 ; 2 v0 The relative strength of the constraints present in Eq. (145) is linked to the relative values of the dispersion and coherence lengths: Tijosc *x e1 disp ? 2 ↔ Lij ? Lcoh : ij v0 It is striking that two di1erent methods of approximation, a priori valid in di1erent regimes, give the same oscillation formula, Eqs. (132) and (144). The dispersion length Ldisp does not play any special role in the $nal result. Each method is thus accurate enough to be extended to the whole range of distances. However, it will be seen in Section 7 that the physical interpretation depends on disp the relative values of L and Li; j .

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6.4. A shortcut We now explain another method to derive the oscillation formula (in the transversal- and longitudinal dispersion regime) which does not require the notion of a dispersion length. Although this method is shorter, it is not obvious how to apply it to unstable oscillating particles. Moreover, it does not make the wave packet picture clearly apparent. Returning to Eq. (35), we can choose to integrate on the 3-momentum before doing the energy integral. In particular, the integration on the 3-momentum can be done with the help of the Grimus– Stockinger theorem [150]. Let (p) be a 3 times continuously di1erentiable function on R3 such that itself and all its $rst and second derivatives decrease at least like 1=p2 for |p| → ∞. Then, for a real number A ¿ 0, √ (p)eip·L L→∞ 2 2 √ i AL ( d3 p → − Al) e + O(L−3=2 ) ; A − p2 + i L where L = |L| and l = L=L. For A ¡ 0, the integral decreases like L−2 . The remaining energy integral in amplitude (35) can be done by a saddle-point approximation [144]. However, it is quicker to perform $rst the time average in probability (96), which yields a delta function, and makes one of the energy integrations trivial: N2 ∗ dT Ai Aj = 2 dE (E; qi l) ∗ (E; qj l) ei(qi −qj )L ; (146) L where (E; p) is the overlap function de$ned by Eqs. (53)–(55) and qj = E 2 − m2j . N 2 absorbs numerical constants. Actually the time interval of integration XT is $nite, so that the delta function is only an approximation certainly valid 12 for XT & Tijosc . Eq. (146) shows that the transition probability can be interpreted as an incoherent sum (i.e. occurring in the probability) over energy eigenstates: interference occurs only between the components of (E; p) having the same energy [249]. In this way, the correspondence between models with and without stationary boundary conditions is obvious: the time-integrated nonstationary probability is equivalent to the energy-integrated stationary probability. For example, the oscillation formula obtained by Grimus and Stockinger with stationary boundary conditions [150] has the form of the integrand in the right-hand side of Eq. (146). This equivalence con$rms that the stationary case can be obtained from the more general nonstationary case in the limit of a vanishing energy width. Note however that the stationary limit cannot be realized in experiments and that the oscillation formula is always averaged over the energy spectrum. In Section 5.2, we have shown how a wave packet can be associated with the amplitude Ai , so that oscillations can be seen, like in the quantum-mechanical treatment, as the result of an interference between propagating wave packets. Thus equivalence (146) shows that this physical picture still holds in the case of stationary boundary conditions, provided that an incoherent sum over the energy is performed, contrary to what was claimed in Ref. [151]. Therefore, a wave packet picture can always be associated with the oscillation formula (once the incoherent energy average has been done), though this physical picture is well hidden in the formalism using stationary boundary conditions. 12

As seen from Eq. (145), this condition can be weakened to XT & max(v0−1 *x e1 ; −2 Tijosc ).

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This equivalence between the time-integrated nonstationary probability and the energy-integrated stationary probability is similar to the equivalence we have met, in quantum-mechanical models, between intermediate wave packet models and stationary approaches (see Eq. (19)). However, the question of the equality of propagation times does not crop up in the quantum $eld theory formalism. Recall that integrations over microscopic space–time variables are included in the overlap function, with the result that the phase depends only on the average time T and L. Note that the time average on the probability in Sections 6.2 and 6.3 could also have been done before the longitudinal-momentum integration in the amplitude, apparently yielding a delta function which makes one of the momentum integrals trivial. However this method is spoilt in 3 dimensions by the prefactor |T − i'|−2 . Instead of a delta function, one obtains a delta function look-alike of width '−1 ∼ *p2 =E0 , introducing an additional momentum uncertainty which 2 is larger than the mass di1erence m2 =E since Losc ij & p0 =*p . For this reason, it was preferable to avoid this shortcut in Sections 6.2 and 6.3 (though it yields the same $nal answer). Moreover, it was interesting for the physical interpretation to postpone the time average, so as to obtain the explicit dependence of the amplitude on time and distance as shown in Eqs. (113) and (133). If the coordinate system is chosen so that L is oriented along a coordinate axis, it is easy to rewrite integral (146) as N 2 g2 (l) ∗ dT Ai Aj = dE ei(qi −qj )L−fi (E)−fj (E) ; (147) L2 with the de$nitions fj (E) = fjP (E) + fjD (E) and 2 2 E − E0 − E 2 − m2j − p0 E 2 − m2j − p0 vP + ; fjP (E) = 2 2 4*pP 4*eP

(148)

where vP and p0 are the components of vP and p0 along L, while E0 has been rede$ned so as to absorb the transversal part of p0 · vP . The de$nition of fjD (E) is similar. The geometrical constraint g(l) is de$ned by Eq. (85). Since the phase in Eq. (147) has no stationary point, there is no problem in using Laplace’s method to integrate over the energy E. The integrand is maximal for Eij = E0 +

m2i + m2j + O(2 ) ; 4E0

where the dimensionless number is de$ned by Eq. (115), and ∼ m2i ∼ m2j . The value of fi (E) + fj (E) at the extremum reads 2 (m2i )2 + (m2j )2 *x e1 2 + 2 + O(3 ) ; fi (Eij ) + fj (Eij ) = Losc 16*m2 E02 ij where *x e1 , *m and Losc ij are de$ned by Eqs. (89), (119) and (128), respectively. The value of the second derivative of fi (E) + fj (E) at the extremum reads 1 d 2 (fi + fj ) 1 (Eij ) = 2 2 + O() : 2 dE 2 2v0 *p e1

(149)

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The expansion of the phase around the extremum reads ij (E) ∼ =−

m2ij m2ij L+ L(E − Eij ) : 2p0 2p02 v0

(150)

The second derivative of the phase is of order and can be neglected with respect to the second derivative of fi +fj . The approximation of integral (147) by Laplace’s method is now straightforward and yields g2 (l) dT Ai A∗j = v0 N 2 *p e1 2 L  2 2  (m2i )2 + (m2j )2 * L L x e1  : (151) ×exp−2 i osc − − 2 2 − Lij Losc 16*m2 E02 Lcoh ij ij The linear superposition of the di1erent partial transition probabilities dT Ai A∗j gives the same result as those obtained in the transverse- and longitudinal-dispersion regimes (Eqs. (132) and (144)). Though the computation is shorter, the origin of the decoherence and localization terms is not as clear (see Section 7), because ‘wave packets’ in con$guration space do not appear explicitly at any stage of the calculation. Moreover, the oscillation formula (111) valid in the no-dispersion regime cannot be exactly reproduced.

7. Analysis of the probability in the stable case In Section 6, the oscillation probability was computed in the three distance ranges studied in Section 5, yielding the formulas (111), (132) and (144). In this section, we analyze the di1erent terms appearing in these formulas and we explain their origin. This analysis will allow us to answer most of the questions raised in Section 3. Before analyzing the oscillation formulas, let us sum up the assumptions used in its derivation. The oscillating particle is stable, and propagates in vacuum over a macroscopic distance L, i.e. L1=p0 , where p0 is the average momentum of the particle. The velocity of the oscillating particle is arbitrary, but the condition m2ij E02 is assumed to be satis$ed. This means that nonrelativistic particles are supposed to have nearly degenerate masses (if it were not the case, oscillations would vanish anyway). The oscillation formula has been derived for a scalar particle, but this assumption is not very restrictive, since the spin structure factorizes from the sum on the mass eigenstates as long as m2ij E02 . In order to obtain a time-independent formula, the transition probability has been averaged over a time interval XT satisfying XT & max(v0−1 *x e1 ; −2 Tijosc ). We have also supposed that the time separation T sep between the wave packets, given by Eq. (131), is smaller than the time interval XT . The dispersion has been taken into account. The oscillation formula valid at large distance (i.e. in the second and third regimes) will be studied $rst since it is the generic experimental case.

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Fig. 6. The probability of detection is maximal within a cone at large distance and within a cylinder at short distance.

7.1. Oscillations at large distance 7.1.1. Oscillation formula at large distance At large distance (L & p0 =*p2 ), the Bavor-mixing transition probability for a stable particle of arbitrary velocity propagating over a distance L = Ll, with dispersion taken into account, is given in a very good approximation by Eqs. (132), (144) or (151): Ng g2 (l) P→ (L) = Vi Vi−1 Vj∗ Vj−1∗ L2 i; j 

×exp−2 i

L − Losc ij

(m2i )2

+ (m2j )2 16*m2 E02

− 2 2

*x e1 Losc ij

2

−

2  L  ; coh

Lij

(152)

where g(l) is the geometrical factor de$ned by Eq. (85): (p0 × l)2 ; g(l) = exp − 4*p2 with *p de$ned by Eq. (67). Recall that the function g(l) restricts the propagation to a cone of axis p0 and angle arcsin(*p =p0 ) (see Fig. 6). Ng is a normalization constant determined by the conservation of the probability: L2 dM P→ (L) = 1 :

−1 = V † for stable From Eq. (152), we can see that P→ (L) is independent of L, since V particles (unless the indices ; are restricted to active Bavors in the presence of sterile Bavors). Note that the unitarity relation is only veri$ed to order because of the approximations made in the computations. Thus the normalization constant Ng is $xed by (153) Ng dM g2 (l) = 1 : The question of the normalization of the probability will be addressed in more detail in Section 9.1. The oscillation and coherence lengths are de$ned by Eqs. (128) and (129), respectively: 4 p0 1 p0 osc and Lcoh Lij : (154) Losc ij = √ ij = * m2ij 2 p e1

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The e1ective width *x e1 is de$ned by *p e1 *x e1 = 1=2, with *p e1 given by Eq. (89): 1 *p2 e1

=

1 1 (v0 − vP )2 (v0 − vD )2 + + + ; 2 2 2 2 *pP *pD *eP *eD

The dimensionless parameter is de$ned by Eq. (115): 1 1 vP (v0 − vP ) vD (v0 − vD ) 2 : + 2 − − = *p e1 2 2 2 *pP *pD *eP *eD The mass width *m is de$ned by Eq. (119): 1 1 (vP − vD )2 1 1 2 + : = *p e1 + 2 2 2 2 *m2 *p2 *eP *eD *eP *eD

(155)

(156)

(157)

As will be made clear shortly the oscillation formula (152) reduces to the standard formulas (12) and (14) (with the additional property of 1=L2 geometrical decrease) if the observability conditions |p0 × l| . *p , LLcoh and Losc ij ij SP; D are satis$ed. 7.1.2. Oscillation phase Each term of the sum appearing in Eq. (152) oscillates with a phase given by 2 L=Losc ij . This phase is independent of the details of production and detection up to order (except the dependence on the momentum p0 , of course). Its origin can be better understood by going back to the expression of the probability before the time average is done. 1. In the transversal-dispersion regime, the phase is given by Eq. (120). In the limit T T disp , the third term of (120) can be neglected and the phase reads ij (T; L) ∼ = (Ei − Ej )T − (pi − pj )L : Using expansions (114) and (116), we can write E j T − pj L ∼ = E0 T − p0 L +

m2j (L + (v0 T − L)) : 2p0

(158)

The explicit expression for is not useful, as we shall see below. The oscillating phase ij (T; L) can now be written ij (T; L) ∼ = 2

L v0 T − L + 2 : Losc Losc ij ij

(159)

Since |v0 T − L| . *x e1 (see Eq. (113)), the second term of the right-hand side of Eq. (159) is negligible if * e1 1 : (160) || xosc Lij Such a constraint is indeed present in the third exponential term of probability (152), so that either the second term of the right-hand side of Eq. (159) does not contribute to the phase, or the corresponding interference term in the probability is negligible.

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Since the leading term of the oscillating phase does not depend on , the phase is independent to order of the exact values of Ei; j and pi; j . In other words the phase is independent of the conditions of production and detection. The two fundamental reasons for this independence are clearly seen in Eq. (158): (a) the particle is on-shell: Ej = pj2 + m2j ; (b) the particle is well localized in space–time: |v0 T − L| . *x e1 . 2. In the longitudinal-dispersion regime, the phase is given by Eq. (138). In the limit T T disp , the second term of the right-hand side of Eq. (138) can be neglected. The phase now reads ˜ ij (T; L) ∼ = mij T 2 − L2 : Using expansions (142) and (143) around the average propagation time T˜ 0 , the phase can be written in the limit ‘1 as E0 mij L (T − T˜ 0 ) : (161) ˜ ij (T; L) ∼ = 2 osc + Lij m˜ 0 The second term of this equation is small if the spatial spread of the wave packet is smaller than 2 Losc ij = , where is the usual Lorentz factor. Since the possible time range is constrained by the width of the overlap function, i.e. |T − T˜ 0 | . ‘*x e1 =v0 (see Eq. (140)), the second term of the right-hand side of Eq. (161) is negligible if *p e1 L 1 ; (162) p0 Losc ij i.e. if LLcoh ij . Such a constraint is indeed present in the last exponential term of probability (152). If it is satis$ed, the oscillating phase is equal to 2 L=Losc ij , as usual. If not, the ij interference term vanishes and decoherence occurs. 7.1.3. Coherence length The last exponential term in probability (152) shows that the ij-interference term vanishes at a distance larger than the coherence length Lcoh ij . This decoherence, predicted by Nussinov [216], has two possible origins, since both the overlap function and the derivative of the phase contribute to 2 exp(−(L=Lcoh ij ) ). The physical explanation depends on whether the coherence length is larger than the dispersion length or not. 13 disp

disp

1. If the coherence length is smaller than the dispersion length (here Ldisp = min(Li ; Lj )), decoherence takes place in the transversal-dispersion regime. In that case, the decoherence term comes mainly from the function fij (Tij ; L), i.e. from the overlap function. This is clear from Eq. (123), with the dispersion neglected for simplicity: ‘1. This decoherence arises from the progressive separation of the wave packets, due to the di1erent group velocities vi and vj : if T is large enough, both terms |vi T − L| and |vj T − L| cannot remain small. disp disp 2. If the coherence length is larger than the dispersion length (here Ldisp ∼ = Li ∼ = Lj ), decoherence takes place in the longitudinal-dispersion regime. In that case, the decoherence term does not 13

As noted in Section 6.2.2 the condition Lcoh & Ldisp can be written as m2 =m2 . v02 *p =p0 , which is equivalent to say that the masses are nearly degenerate.

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Fig. 7. In the transversal-dispersion regime (II), the coherence length is due to the nonoverlap of wave packets, whereas it is due in the longitudinal-dispersion regime (III) to their dispersion (the coherence length was not computed in the no-dispersion regime).

come anymore from the separation of the wave packets since the packets spread out beyond the dispersion length as quickly as they separate (see Eq. (141)). Still, decoherence arises from the variation of the phase over the width of the overlap function (see Eq. (161)). The interference term is averaged to zero by the time integral when the spatial spread of the ‘wave packet’ 2 becomes larger than Losc ij = . A similar mechanism has been observed in connection with neutron interferometry [186]. All in all, the coherence length arises not only from the separation of wave packets, as it is usually explained in the literature: in the case of nearly degenerate masses, it can also originate in a too large dispersion of the wave packet in comparison with the oscillation length (Fig. 7). Remark that the phenomena of separation of wave packets and dispersion contribute to the coherence length in both regimes: if approximations such as ‘1 (or ‘1) are not made, terms proportional to L2 come from fij (Tij ; L) (or f˜ij (T˜ 0 ; L)) and from the squared derivative of the phase, divided by the second derivative of fij (or of f˜ij ). Furthermore, the transition between the two regimes is not visible in the oscillation formula, because the two decoherence mechanisms smoothly blend. Besides the above explanations in con$guration space, the coherence length can be also understood in energy–momentum space, where it is due to the variation of the phase over the ‘wave packet’ width *p e1 (see Eq. (150)). The explanation in momentum space is thus simpler, but the two mechanisms at the origin of the coherence length in con$guration space are more intuitive. As dispersion is usually neglected in the literature, it is interesting to derive the oscillation formula disp with this approximation (Tj = ∞). Such a computation in the transversal-dispersion regime yields the same result as when dispersion is taken into account (Eq. (152)), except that the following substitution has to be made: Lcoh ij →

Lcoh ij (FALSE) : |(1 − v02 ) − 1|

(163) disp

The incorrect multiplying factor has its origin in Eq. (117): in the limit Tj = ∞, the coherence length term arises only from the di1erence between the group velocities vi and vj . However the

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factor |(1 − v02 ) − 1| tends to 1 in the relativistic limit, so that substitution (163) becomes trivial in that limit. This observation explains why our result (152) coincides with Eq. (26) of Ref. [144], as the authors of this article, while neglecting dispersion, consider only relativistic neutrinos. Note however that even relativistic neutrinos spread at large distances so that a calculation neglecting disp dispersion such as in Ref. [144] is only valid for L . Lj . It is interesting to observe that the coherence length increases when a long coherent measurement in time is performed at the detector, even if the oscillating ‘wave packets’ have separated spatially [182,183]. In that case, the energy uncertainty at detection goes to zero, *eD → 0, so that the e1ective width also goes to zero, *p e1 , and the coherence length becomes in$nite, Lcoh ij → ∞. After all these theoretical considerations, it must be said that the decoherence at the level of the wave packet is irrelevant in most experiments, since it is usually dominated by decoherence e1ects originating in the energy spread of the beam. Di1erent situations are discussed in Section 9.2.2. Finally, let us remark that the exponential decrease in L2 of this decoherence term is model-dependent. It results here from the Gaussian approximation. However, the de$nition of the coherence length is model-independent, apart from a multiplying constant. 7.1.4. Localization A third kind of term appearing in the Bavor-mixing transition probability (152) are localization terms, that is, observability constraints imposing that the oscillation length should be larger than the 2 space–time uncertainty: Losc ij & *x e1 . This condition can be rewritten as mij =p0 . *p e1 , stating that oscillations vanish if the energy–momentum measurements allow to distinguish between the di1erent mass eigenstates. Two localization terms appear in probability (152), while one more is implicitly assumed when applying the Jacob–Sachs theorem: 1. The term containing *m can be rewritten (m2ij )2 (m2i + m2j )2 (m2i )2 + (m2j )2 = + : 32*m2 E02 32*m2 E02 16*m2 E02

(164)

The $rst term of the right-hand side of Eq. (164) can be written as a localization term: Losc ij & *x ;

(165)

as *m ∼ v0 *p whether the stationary limit is taken or not. The second term of the right-hand side of Eq. (164) is not a localization term and will be discussed in Section 7.1.5. 2. The term containing is also a localization term and imposes that Losc ij & ||*x e1 :

(166)

It is not obvious whether this constraint is stronger than the previous one, i.e. whether it is possible that ||*x e1 *x . This situation might arise from an energy uncertainty much smaller than the momentum uncertainty. In that case, de$nition (155) of the e1ective width shows that *x e1 *x . Say, for example, that the energy uncertainty at the detection goes to zero, i.e. *eD → 0.

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Fig. 8. In the no-dispersion and transversal-dispersion regimes (I and II), the localization condition is due to the initial spread of the wave packet, whereas it is due in the longitudinal-dispersion regime (III) to their nonoverlap.

With the help of Eqs. (155), (156) and (58), we see that * →0 |vD | . SD ; (167) ||*x e1 eD→ *eD where SD is the size of the macroscopic detection region. Thus the localization term does not give a stronger constraint than Losc ij & SD . This constraint is always satis$ed, as it is equivalent to the constraint obtained by averaging the transition probability over the production region (see Section 9.2.1). Therefore, the coherence length can be increased without bound by more accurate energy measurements, contrary to what was claimed in Refs. [144,139]. Note that this is not true if the accuracy of the 3-momentum measurements is increased, as the localization makes the oscillations vanish when the corresponding spatial uncertainty becomes larger than the oscillation length. Of course the opposite conclusions would be reached if experiments measured time, not distance. 3. The contour integral used in the Jacob–Sachs theorem yields a third localization constraint. The poles corresponding to the mass eigenstates mi and mj cannot be both included in the same contour integration unless |m2ij |=p0 . *p . This third constraint is not stronger than the previous ones and can be ignored. As was the case for the coherence length, there are two origins for the localization term containing (Fig. 8). 1. In the limit ‘1 (transversal-dispersion regime), this term comes from the variation of the phase over the width of the wave packet (see Eq. (159)). If the oscillation length is smaller than the width of the wave packet, the interference term is averaged to zero by the time integration. 2. In the limit ‘1 (longitudinal-dispersion regime), this term arises from the overlap function (see Eq. (139)). Thus it comes from the separation of the wave packets, which remains constant in the longitudinal-dispersion regime. Recall that the coherence length was shown to arise, in the transversal-dispersion regime, from the wave packet separation and, in the longitudinal-dispersion regime, from the variation of the phase. Therefore, the coherence-length and localization conditions in the transversal-dispersion regime transform respectively into localization and coherence-length conditions in the longitudinaldispersion regime. The situation is summarized in Table 1.

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Table 1 Origins of the coherence length and localization terms

Coherence length Localization

Transversal-dispersion regime

Longitudinal-dispersion regime

Decreasing overlap of wave packets Initial spread of each wave packet

Increasing dispersion of each wave packet Constant overlap of wave packets

7.1.5. Energy–momentum conservation The last term of the oscillation probability (152) to be discussed is the second term in the decomposition of Eq. (164). This term could be a matter of concern since it does not vanish in the limit mi = mj . Note that this kind of term is not speci$c to our computation. For example, it would appear in the oscillation formula (26) of Ref. [144] if the terms Sa (Ea ) present in Eq. (22) of that article had been expanded beyond zeroth order in m2a =Ea2 . The second term in Eq. (164) imposes that |m2i + m2j − 2m20 | . *m : E0

(168)

As m0 is related to the average in- and outgoing momentum by m20 = E02 − p02 , condition (168) means that the mass eigenstates have to be on-shell with respect to (E0 ; p0 ) within the uncertainty *m . For example, this constraint is impossible to satisfy in the mixing of relativistic and nonrelativistic neutrinos if the available energy–momentum is such that only the lightest neutrino can be produced. However, it has no e1ect on the oscillations in the two cases considered in the present report, namely relativistic particles or nearly mass degenerate particles. Condition (168) should simply be considered as expressing the conservation of energy–momentum. Such kinematical constraints are usually not included in the oscillation formula, though they rightly belong to it. A complete computation of the transition probability should not only include this energy–momentum constraint, but also similar terms arising from the interaction amplitudes MP; D , from the prefactor resulting from the Gaussian integration and from the spin structure of the propagator. Actually the masses of the oscillating particle should be expected to appear not only through mass di1erences but also through their absolute values. 7.2. Oscillations at short distance At short distance (L . p0 =*p2 ), the oscillation formula is given by Eq. (111): ˜ 2 (˜v0 × L) P→ (L) = Ng˜ exp − Vi Vi−1 Vj∗ Vj−1∗ 2*x2 v˜ 20 i; j 

(m2i )2 + (m2j )2 L ×exp−2 i osc − − 2 2 Lij 16*˜ 2m E02

2  ˜*˜ x e1  ; Losc ij

(169)

where *˜ x e1 and Losc ij are de$ned by Eq. (106) and Eq. (112), respectively. Ng˜ includes all prefactors independent of L as well as normalization factors.

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The normalization of the probability can be done in the same way as in Eq. (153): 2 ˜ (˜ v × L) 0 Ng˜ L2 dM exp − =1 : 2*x2 v˜ 20

(170)

As in the case of oscillations at large distance, it can be argued that the two last exponential terms in Eq. (169) yield an observability condition not stronger than Losc ij & SP; D (see Section 7.1.4). It could also be interesting to understand the origin of the oscillation phase by going back to the amplitude, before doing the time average. This discussion can be found in Ref. [60]. Actually, the oscillation formula (169) is very similar to the formula valid at large distance (Eq. (152)). There are three main di1erences. First there is no coherence-length term in Eq. (169), which is due to the neglect of the terms beyond O(0 ) in the evaluation of F(T0 ) (see Eq. (79)). Second, the geometrical decrease in 1=L2 is lacking in Eq. (169), which is explained by the fact that the dispersion of the oscillating particle ‘wave packet’ is not yet signi$cant. Finally, the directional constraint present in Eq. (169) con$nes the propagation to a cylinder of radius r (with *x . r . SP; D ), whereas the directional constraint present in Eq. (152) restricts the oscillating particle propagation to a cone of angle arcsin(*p =p0 ) (Fig. 6 in Section 7.1.1). This di1erent behavior is also a result of the absence of dispersion for L . p0 =*p2 . In conclusion, the oscillation formula (169) reduces to the standard formulas (12) and (14) if the observability conditions |v0 × L| . |v0 |*x and Losc ij SP; D are satis$ed. 7.3. Answers at last In Section 3, our review of $eld-theoretical treatments brought to the fore several questions: • Is there a limit in which the oscillation formula, derived with stationary boundary conditions, can be obtained from the oscillation formula derived with external wave packets? • Does an in$nitely precise measurement of the energy lead to an in$nite coherence length, or is there decoherence anyway? This question is inseparable from the $rst one. • Does a strong localization of the source lead to a dependence of oscillation formula on mass prefactors? • Do ‘plane waves oscillations’ exist? The external wave packet model used in this report is of suKcient generality to allows us to clarify and answer these questions. 7.3.1. Existence of a stationary limit Two important results concerning stationary boundary conditions have been demonstrated. First, there is no contradiction between models with stationary boundary conditions and those with external wave packets. The former type of model can be obtained from the latter in a smooth limit. Let us state again the gist of the problem. Stationary boundary conditions, given by vP; D = 0 and *eP; D = 0, lead to an in$nite e1ective width *x e1 and thus to an in$nite coherence length Lcoh (see Eqs. (154) ij and (155)). Hence, the localization term depending on *x e1 =Losc seems to diverge in Eq. (152), ij with the result that the ij-interference term seems to vanish in the stationary limit *x e1 → ∞. If it were true, it would be impossible to increase without limit the coherence length by performing long

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coherent measurements. This would be in contradiction with stationary boundary condition models, which have a zero energy uncertainty but an in$nite coherence length. However the product *x e1 remains $nite, as shown by Eq. (167), so that the interference term does not vanish. For example, the Grimus–Stockinger model [150] is obtained in the limit = 0, *x e1 → ∞ with *x e1 = 0. The latter condition means that this model can be recovered from the external wave packet model if the stationary limit has the property vP; D =*eP; D = 0 (see discussion after Eq. (58)). The only localization condition comes from the momentum spreads of the source and detector (see Eq. (165)). Note also that the amplitude Aj (see Eq. (60)) takes in the stationary limit the form of a plane wave, with a well-de$ned energy and momentum (see Eqs. (114)–(116) with → 0). The second important result is equivalence (146) between the time-integrated nonstationary probability and the energy-integrated stationary probability. Since oscillations can be described in the former model by interfering ‘wave packets’, this interpretation is also valid in the latter, contrary to what was claimed in [151]. Thus the incoherent superposition of stationary amplitudes, which have a plane wave form, reproduces the characteristics of wave packet propagation. Of course, this is not true if the probability is not integrated over the energy spectrum, but we have already argued that a source and a detector with zero energy uncertainties do not constitute realistic boundary conditions. 7.3.2. Unbounded coherence length As shown by Kiers et al. [182], a more precise measurement of the energy at detection increases the coherence length. This feature is reproduced within the external wave packet model: Lcoh ∼ ij *x e1 ∼ 1=*eD (see Eqs. (154) and (155)). In the limit *eD → 0, the external wave packet model becomes equivalent to a stationary model (even if the source is nonstationary), as the detector picks up wave packet’s components having exactly the same energy. Contrary to what was asserted in Refs. [144,139], oscillations do not vanish in that limit since we argued above that *x e1 remains $nite. 7.3.3. No mass prefactors at large distance Contrary to what was claimed by Shtanov [242], mass prefactors never appear in front of the oscillating exponentials present in the transition probability (152). Let us $rst explain what these ‘mass prefactors’ mean. Shtanov derives oscillation formulas for neutrinos within a source–propagator model in con$guration space. Shtanov computes the convolution of the neutrino propagator with a source, keeping only the phase in the oscillating exponentials but neglecting the contributions depending on the width *x of the source. With this last approximation, the space–time localization (i.e. the ‘wave packet envelope’) of the amplitude is lost. Let us consider his model which will be considered in a scalar version for an easier comparison withour results. The propagator for a scalar particle propagating from x to x is asymptotically (mj (x − x)2 1) given in con$guration space by √ √ mj −imj (x −x)2 e ; (171) Aj ∼ ((x − x)2 )3=4 where (x − x)2 is the Lorentz interval. The convolution of Eq. (171) with a monochromatic source yields without problem the standard oscillation formula (12). However, convolution with a strongly localized source (*x; t . 1=E) leads to an amplitude that keeps its dependence on the mass prefactor

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√

mj . In that case, the transition probability is not equivalent to the oscillation formula (12), unless the masses mi; j are nearly degenerate. We now proceed to show that mass prefactors only appear at an intermediate stage of the calculation, but not in the $nal result. Note $rst that amplitude (171) is in correspondence with our amplitude (95), computed with the stationary phase method, since the prefactor in Eq. (133) can be disp rewritten, for T Tj , as √ mj *p e1 ∼ 2 ; disp (T − L2 )3=4 T 1 + iT=Tj where de$nition (92) of the dispersion time has been used. This prefactor coincides with the prefactor in Eq. (171). However, the subsequent time average of the transition probability completely cancels this dependence on the mass, yielding Eq. (144). This can be seen by expanding the argument of the exponential in Eq. (133) around the average propagation time. The width with respect to T of the amplitude Aj is found to be equal to *p e1 T=mj v0 3cl (where cl is the Lorentz factor associated with the velocity vcl = L=T ), thereby providing a mj -dependent factor that cancels the mj prefactor in the Gaussian integration. Shtanov does not obtain such a result, since he does not perform any time average on the probability. Besides, mass prefactors would remain even if a time average were performed on the probability: no ‘wave packet envelope’ appears indeed in Shtanov’s amplitude, so that the cancellation mechanism explained in the previous paragraph is not at work. For that reason, Shtanov wrongly concludes that the mass prefactor remains if the source is strongly localized. oscillation Another non-standard formula, derived by Shtanov for mass eigenstates satisfying m2 (x − x)2 m1 (x − x)2 , reduces to the standard formula when decoherence mechanisms are taken into account. In conclusion, Shtanov’s computations in con$guration space do not lead to new oscillation formulas. 7.3.4. No plane waves Ioannisian and Pilaftsis [163] derive a formula for neutrino oscillations which exhibits a plane wave behavior if the condition L . p0 =*p2 is satis$ed (with *x Losc ij , as usual). Their term ‘plane wave oscillations’ means that the oscillation amplitude Aj does not decrease as the inverse of the distance, that its phase depends on the direction L as pj · L, and that there is no geometrical constraint linking the average 3-momentum and the direction of observation. If it were true, the oscillation length measured by a speci$c detector would depend on the direction of the total momentum of the initial particles. The plane wave condition is satis$ed, for example, by taking *x ∼ 1 cm, L ∼ 1000 km and p ∼ 1 GeV. These conditions might be veri$ed for atmospheric neutrinos. In Section 6.1.2, we have proved that the detection probability of an oscillating particle, derived under the condition L . p0 =*p2 , is negligible in directions other than the average momentum whether the stationary limit is taken or not. Therefore, the oscillation phase has the standard form given in Eq. (14) and no ‘plane wave oscillations’ can be observed, contrary to the Ioannisian and Pilaftsis’ claim. Also, the absence of the 1=T (or 1=L) factor in Eq. (77), as noted by the same authors, is easily understood by noting that the absence of dispersion entails that the amplitude does not decrease with the distance (the probability is maximal within a cylinder as pictured on Fig. 6 of Section 7.1.1).

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It is interesting to understand why a directional constraint is missing for L . p0 =*p2 in Ioannisian and Pilaftsis’ result, as this fact explains their ‘plane wave oscillations’ prediction. At the end of their computations, Ioannisian√and Pilaftsis obtain an amplitude Aj whose dominant term depends on exp(ipj |L |), where |L | = L 2 is the complex ‘norm’ of a complex vector L = L − 2i*x2 p0 (their results are translated into our notation through the correspondence qj → pj , ˜k → p0 , l2 → 4*x2 , ˜L → L , ˜l → L). For Lp0 =*p2 , the quantity |L | can be expanded and the argument of the exponential reads to second order 2 p p · L · L) (p j 0 0 2 2 2 L − : − 2 ipj |L − 2i*x p0 | ∼ = 2*x p0 pj + ipj p0 4*x p0 p02 The two last terms were neglected in Ref. [163], though L*x . They lead to the following directional constraint: (v0 × L)2 ; (172) exp − 4*x2 v02 where the factor 4 becomes a factor 2 when the amplitude is squared. Therefore, Ioannisian and Pilaftsis’ result includes a directional constraint forbidding plane wave oscillations. Note that the stationary limit assumption |vP; D |=*eP; D = 0 leads to s1 = s2 (see Eq. (109)), so that constraint (172) becomes equal to the one present in Eq. (111). Thus the condition |vP; D |=*eP; D = 0 seems generic for models with stationary boundary conditions, since it was also applied in the case of the Grimus–Stockinger model (see discussion after Eq. (58) and at the end of Section 4.5.2). In conclusion, the analysis of the transition probability (111) derived in the external wave packet model under condition (75) has shown that ‘plane wave oscillations’ do not exist. This negative result was con$rmed by a reexamination of the formula derived in Ref. [163]. 7.4. Conclusion Through the discussions of this section the external wave packet model has shown its power and generality. This model can either reproduce in some limit the $eld-theoretical oscillation formulas found in the literature, or allows to understand why they are wrong. Thus the three categories of models described in Section 3.1 (external wave packet models, stationary boundary condition models, source–propagator models) really make one. Furthermore, the external wave packet model has the advantage of associating a clear physical picture to the di1erent stages of the calculations. For example, the role played by the dispersion in the observability conditions (coherence length and localization) could be cleared up in Section 7.1, while the threshold L = p0 =*p2 identi$ed by Ioannisian and Pilaftsis could be interpreted as indicating the onset of dispersion. 8. Oscillations of unstable particles First, we study nonexponential corrections speci$c to mixed unstable particles, and then examine in which respect the oscillation formula obtained in the stable case is modi$ed by the decay term. Next, we establish the relation between our formalism and the Wigner–Weisskopf–Lee–Oehme–Yang model. Finally, we apply the formulas derived in this section to the B0 BH 0 system.

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8.1. Nonexponential corrections due to mixing If the oscillating particle is unstable, the Bavor-mixing amplitude (34) is a superposition of mass-eigenstate amplitudes Aj given by Eq. (35). However, these formulas were derived on the assumption that the diagonalization matrices V appearing in Eq. (34) are constant. This hypothesis is only true in the limit of a negligible renormalized self-energy. This approximation cannot be made for unstable particles, since their decay arises from the imaginary part of this self-energy. Actually, nonexponential corrections are always important at large time for unstable particles (though until now not observable), so it is worth checking whether the energy–momentum dependence of the diagonalization matrix has a measurable e1ect on the transition probability. Nonexponential corrections to the propagation of an unmixed particle of well-de$ned mass are studied in Appendix A, in the context of the derivation of the Jacob–Sachs theorem. These corrections have two origins. First, they are due to the bounded character of the energy–momentum spectrum of the particle. Second, they are due to multi-particle thresholds, if these are included in the energy–momentum spectrum. The $rst type of correction is easily calculated for mixed propagators, since it has nothing to do with the diagonalization matrices. The corrections can be evaluated separately for each partial amplitude Ai , and are negligible (see Appendix A). In order to compute the second type of corrections, the energy-dependent diagonalization matrices are kept inside the energy–momentum integral (35). Let us consider the case of P 0 PH 0 mixing in the limit of no CP violation (P is a K or a B meson). Starting from Eq. (31), the inverse propagator for the neutral meson system can be parametrized by [238] 2 2 2 2 − m − f (p ) −f p H(p ) 00 0 0 ; (173) iG −1 (p2 ) = 2 −f00 p2 − m2 − f00 (p2 ) H (p ) where m is the renormalized mass in the degenerate limit and −if (p2 ) are the renormalized complex self-energies. The diagonal elements are equal because of CPT symmetry. In the limit of CP 2 CP symmetry, the o1-diagonal can be taken to be equal: f00H(p2 ) = f00 H (p ). This propagator can be diagonalized as follows: 2 2 2 −1 − m − f (p )) 0 (p 1 1 V (p2 ) : −iG(p2 ) = V −1 (p2 ) 0 (p2 − m22 − f2 (p2 ))−1 Thresholds of multi-particle states appear in the renormalized self-energies fj (p2 ) and in the diagonalization matrix V (p2 ). Let us de$ne the function I (T ) by I (T ) = dE (E; p)G (p2 ) e−iET ; where G (p2 ) is the nondiagonal propagator in Bavor space and (E; p) is the overlap function de$ned by Eq. (28). The Fourier transform of this function with respect to the 3-momentum yields the amplitude A( → ; T; L). Since we are interested by nonexponential corrections to mixing, let us study the e1ect of a threshold p2 =b2 on the matrix element I00H(T ). The corresponding propagator

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matrix element is G00H(p2 ) =

−1 −1 (p2 )V10H(p2 ) (p2 )V20H(p2 ) iV01 iV02 + : p2 − m21 − f1 (p2 ) p2 − m22 − f2 (p2 )

(174)

The function I (T ) is computed with the method explained in Appendix A. The integration on the energy is done by a contour integration including the two poles z1; 2 appearing in the right-hand side of Eq. (174). The function I (T ) is equal to the sum of the poles’ contribution J , the contributions J1; 2 of the energy–momentum thresholds of the overlap function (E; p), as well as the contribution Jb of the multi-particle threshold: I00H(T ) = J + J1 + J2 + Jb : J is given by the residues of the two poles zj = m2j − imj j : (z1 ; p) −i√z1 +p2 T (z2 ; p) −i√z2 +p2 T −1 −1 (z1 )V10H(z1 ) e + V02 (z1 )V20H(z2 ) e : J = V01 z1 + p 2 z2 + p 2 Since CP violation is neglected, the propagation eigenstates are the CP eigenstates K1; 2 ∼ K 0 ± KH 0 , so that the matrix V is given at the pole by 1 1 1 V (z1 ) ∼ : (175) = V (z2 ) ∼ =√ 2 1 −1 We shall take into account, in the evaluation of Jb , that these expressions are not valid far from the poles. An estimate of J is given by 1 J ∼ (176) (m2 ; p)e−2 T=2 |e−iXmT −XT=2 − 1| ; m where m is the mass in the degenerate limit, Xm=m1 −m2 , X =1 −2 and we have approximated m2 + p 2 ∼ = m. The contributions J1 and J2 , due to the bounded spectrum of the overlap function, can be computed separately on each term of the right-hand side of Eq. (174), exactly as in the one-particle case. The result is likewise negligible. Let us examine in detail the contribution of Jb . Eq. (216) of Appendix A shows that Jb depends on the di1erence between the complete propagator G (z) and its analytic continuation G; II (z): √2 2 ∞ d! (z(!); p)(G00;H II (z) − G00H(z))e−!T : (177) Jb = −ie−i b +p T 0

The di1erence GII (z) − G(z) can be computed with parametrization (173). To $rst order in f , we have (178) G00;H II (z) − G00H(z) = −i f00;H II (z) − f00H(z) G00H(z)G00;H II (z) : For large T , the dominant contribution to Jb comes from the ! values near zero, that is z = b2 , because of the decreasing exponential. It is thus enough to know how the integrand tends to zero near the threshold z = b2 . Just under the real axis (z = x − i), the spectral representation of the self-energy given by Eq. (213) entails that f00;H II (z) − f00H(z) = f00H(x + i) − f00H(x − i) = 2i Im f00H(x + i) :

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The value at the pole of the function f00H(x + i) can be related to experimentally known quantities with the help of Eqs. (173) and (175): mX Im f00H(m2 + i) ∼ : =− 2 As the main decay channels of K 0 and KH 0 are two-pion decays, the nondiagonal self-energy Im f00H(x + i) has a two-particle threshold behavior: mX x − b2 : Im f00H(x + i) ∼ =− 2 m2 − b 2 Eq. (178) can then be rewritten as G00;H II (z) − G00H(z) ∼ = −mX

z − b2 G H(z)G00;H II (z) : m2 − b 2 0 0

Substituting y = !T in Eq. (177) and expanding the integrand in 1=T with the help of the z(!) parametrization given by Eq. (217), we obtain y 2 z = b2 − 2i b + p2 + O(T −2 ) : T It follows that √ m3=2 X 2 −i b2 +p2 T (b ; p) e ; Jb ∼ T −3=2 2 (m − b2 )5=2 where we have approximated b2 + p2 ∼ = m2 and used the fact that the self-energy is negligible at the threshold. Thus Jb is of the order of X 1 (b2 ; p) : (179) Jb ∼ (QT )−3=2 Q m It is now possible to compare the estimates of J and Jb , given by Eqs. (176) and (179), respectively. At small times: Jb ∼ J if XmT |Xm|T −3=2 |X| ∼ ∼ sin : (QT ) Q 2 2 It is known that |X| . |Xm| (from the experimental data in the case of the K mesons [156] and from theoretical predictions in the case of B mesons [115]). Thus, Jb ∼ J if T ∼ Q−1 . Since Q is of the order of 0.2–1 GeV, for the K and B mesons, nonexponential corrections are dominant at small times for T . 10−24 s. This time range is not observable. At large times: J ∼ Jb if |X| |X|L3=2 ∼ (L T )−3=2 ; Q Q5=2 where L = min(1 ; 2 ), that is if |X| Q − 2 ln : L T − 3 ln (L T ) ∼ 5 ln L L For kaons, Q ∼ 220 MeV so that L T ∼ 190. The amplitude is negligible at such large times so that nonexponential corrections are not observable. For B mesons, Q ∼ 1 GeV and X=L can be e−L T=2 ∼ (QT )−3=2

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estimated at 4 × 10−3 for the Bd0 and at 10−1 for the Bs0 [115], so that L T ∼ 168 in the former case and L T ∼ 162 in the latter. Thus, nonexponential corrections are not observable in either case. Estimate (179) of nonexponential corrections to the propagation of mixed neutral mesons is in agreement with the theoretical formulas obtained by Chiu and Sudarshan (Eq. (3.30) of [88]), and by Wang and Sanda (Eq. (59) of [254]). The authors of these two papers compute nonexponential corrections in a quantum-mechanical framework and extend the Wigner–Weisskopf–Lee–Oehme– Yang formalism [191,172,209] (see also [215]). In conclusion, nonexponential corrections can be neglected in the propagation of quasi-stable mixed particles. Therefore, the Bavor-mixing amplitude (34) with constant matrices V can be used not only in the stable case, but also for an unstable oscillating particle. 8.2. Oscillation formula for unstable particles The approximation of the amplitude Aj and of the time average with Laplace’s or the stationary phase methods can be transposed for an unstable oscillating particle. The decay term j (p) introduces a new small parameter, namely mj j =p2 . This parameter is of order = m2ij =2p2 , or smaller, since decay = p=mj j should be of the order, or larger, than the oscillation length the mean decay length Lj 2 Losc = 4 p=m , or else the oscillations would not be observable. In parallel with the discussion ij ij of Section 5.3.1, the inBuence of this new parameter on the transition probability shows up in the argument of the exponential, through combinations with large dimensionless coeKcients. Let us inspect the possible combinations of with the available large parameters, where refers either to (pLosc )−1 or to (pLdecay )−1 . First, if the conditions of production and detection are not considered, only one such coeKcient is available, i.e. pL. This combination gives the oscillation term exp(−2 iL=Losc ) and the usual exponential decrease of the probability in exp(−L=Ldecay ). Next, the analysis of the propagation with external wave packets adds a new parameter, i.e. *p e1 . The combinations (p=*p e1 )2 and (*p e1 L)2 give the following contributions to the argument of the exponential (with minus signs omitted): 1. Decay-independent terms, like the decoherence term, (L=Lcoh )2 , and the localization term, (*x e1 =Losc )2 , which were already obtained in the stable case. 2. Mixing-independent terms, like (*p e1 L=p Ldecay )2 and (*x e1 =Ldecay )2 . The $rst is completely negligible in comparison with the exponential decrease at order . The second imposes that Ldecay *x e1 , which is true for quasi-stable particles. 3. Mixing-decay cross terms, like *x2 e1 =Losc Ldecay and *p2 e1 L2 =p2 Losc Ldecay . With the help of the condition O(Ldecay ) & O(Losc ), it can be seen that the new contributions, with respect to the stable case, do not impose new constraints for the observability of the oscillations. They are thus neglected in the transition probability. We have yet to check whether the oscillation phase is modi$ed or not by the widths j . Let us consider $rst the transversal-dispersion regime (the no-dispersion regime can be treated similarly). The position pj of the maximum is shifted by the decay term j (p). The new maximum is the solution of fj (p) + j (p) = 0 ;

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where fj (p) and j (p) are de$ned by Eqs. (82) and (84), respectively. It can be computed as before to $rst order in ( referring to m2j =2E02 or to mj j =2E02 ) and is equal to pj = pj(0) + pj() ;

(180)

where pj(0) is the solution for j = 0, given by Eq. (114), whereas pj() is the contribution from the decay term, given by pj()

. p0

*p2 e1 E02

T decay

Tj

:

(181) decay

= E0 =mj j . Although the expansion parameter mj j =2E02 is The decay time is de$ned by Tj multiplied in the exponential by the large parameter T , the shift pj() is small and expansion (180) decay

is valid as long as the propagation time T is not much larger than the decay time Tj corresponding energy Ej = pj2 + m2j and velocity vj = pj =Ej are given by

. The

Ej = Ej(0) + Ej() ;

(182)

vj = vj(0) + vj() ;

(183)

where Ej(0) and vj(0) are the solutions for j =0, given by Eqs. (116) and (117), respectively, whereas the contributions from the decay term are equal to Ej() = v0 pj() and vj() = m20 pj() =E03 . Apart from the modi$ed values of pj , Ej and vj , the longitudinal momentum integration with Laplace’s method yields the same results as in Section 6.2.1. The phase of the interference term Ai A∗j is thus given, before the time average, by Eq. (120), but with the modi$ed values (180), () (182) and (183). It can be split in a -independent part (0) ij (T; L), and a -dependent part ij (T; L). The latter is equal to ‘2 () () () ; (184) ij (T; L) = (v0 T − L)(pi − pj ) 1 − 1 + ‘2 where ‘ = T=T disp and de$nition (92) of T disp has been used. Using the condition |v0 T − L| . *x e1 (see Eq. (113)) and the values of pi;() j given by Eqs. (180) and (181), we have T * T p e1 |ij() (T; L)| . v0 decay − decay ; E0 T T i

j

which is much smaller than 1 in the observable domain O(T ) . O(T decay ). Therefore, the oscillation phase is not modi$ed by the width: |ij() (T; L)||(0) ij (T; L)| : Let us now turn our attention to the longitudinal-dispersion regime. Of course, the stationary point pcl; j is not shifted by the decay term j (p), but the phase receives a contribution from the derivative of the decay term. More precisely, phase (138) becomes *p2 e1 ‘ ((fi (pcl; i ) + i (pcl; i ))2 − (fj (pcl; j ) + j (pcl; j ))2 ) (185) ˜ ij (T; L) = mij T 2 − L2 + 1 + ‘2

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where ‘ = T=T disp as above. The derivative of the decay term is equal to j (pcl; j ) = −

L p0 : 2Ecl2 ; j Ldecay j

The terms depending on i; j are proportional to *p e1 j . They will be small in the observable domain O(T ) . O(T decay ), so that they can be neglected in phase (185). Thus the oscillation phase is not modi$ed by the decay term at any distance. Result (152), obtained decay for a stable oscillating particle, is only modi$ed by the usual exponential decrease exp(−L=Lij ), with decay

Lij

=

2p0 : mi i + m j j

(186)

In conclusion, the Bavor-mixing transition probability for a long-lived unstable particle (i.e. satdecay isfying *x e1 Lij ), having a relativistic or a nonrelativistic velocity (but always with m2ij E02 ), and with dispersion taken into account, is given for L & p0 =*p2 by Ng g2 (l) L − 1 ∗ − 1∗ P→ (L) = Vi Vi Vj Vj exp − decay L2 Lij i; j 

×exp−2 i

L − Losc ij

(m2i )2

+ (m2j )2 16*m2 E02

− 2 2

*x e1 Losc ij

2

−

2  L  : coh

Lij

(187)

The de$nitions of the characteristic lengths and parameters appearing in this formula are gathered below Eq. (152). For L . p0 =*p2 , the oscillation formula for stable oscillating particles is similarly modi$ed by the insertion of a decay term into Eq. (111). 8.3. Wigner–Weisskopf e:ective mass matrix Oscillating neutral mesons are usually described in quantum mechanics with the Wigner–Weisskopf method as extended by Lee et al. [191,172,209]. In this framework, the evolution of a two-meson state satis$es a SchrWodinger-like equation with a complex (and non-Hermitian) e1ective mass matrix H , which can be split into a Hermitian mass matrix M and a Hermitian decay matrix , H =M −i. It would be interesting to establish a correspondence between this method and the $eld-theoretical formula (187). First, let us simplify the oscillation formula (187). Note that the coherence length can be neglected for particles such as the neutral K or B mesons, since the oscillation length is comparable with the decay length: Lcoh ij decay

Lij

∼

p0 Losc ij 1 : *p e1 Ldecay ij

Next, the quasi-degeneracy of the mass eigenstates makes it impossible to set up experimental conditions which destroy oscillations. Thus the localization terms (containing the parameters *m

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and ) drop from Eq. (187). These approximations lead to the following simpli$ed formula: Ng g2 (l) L L : (188) Vi Vi−1 Vj∗ Vj−1∗ exp − decay − 2 i osc P→ (L) = L2 Lij Lij i; j The comparison of Eq. (188) with the Wigner–Weisskopf–Lee–Oehme–Yang theory is more easily done at the level of the amplitude. Whereas the true amplitude depends on both T and L, probability (188) can, equivalently, be generated from the following e:ective amplitude Ng g(l) −1 j m0 L Vj ; Vj exp −i mj − i (189) Ae1 ( → ; L) = L 2 p0 j where the masses are assumed to be nearly degenerate (mi ∼ = mj ∼ = m0 ). If M is a diagonal matrix the diagonal terms of which are given by mj − ij =2, the e1ective amplitude can be written as Ng g(l) m0 L −1 V exp −iM V Ae1 ( → ; L) = L p0 Ng g(l) m0 L exp −iMBavor = ; L p0 where MBavor = V −1 MV . The mass matrix MBavor corresponds to the e1ective Hamiltonian in the Lee–Oehme–Yang theory. We recognize the factor m0 L=p0 as being the classical proper time of propagation, common to the two mass eigenstates. However, let us again emphasize that this observation does not justify the equal time prescription, which was explained in Section 2.3. It must not be forgotten that T and L are the average propagation time and length, and that interference takes place in a time and space range de$ned by the width of the e1ective ‘wave packet’. The e1ective oscillation amplitude con$rms the soundness of the reciprocal basis treatment in quantum mechanics explained in Section 2.2. Recalling that the diagonalization matrix V is related to the matrix U diagonalizing the states through V = U t , Hamiltonian (10) can be written as Hˆ propag = | Vj−1 e−i j t Vj | ; (190) j

where j = mj − ij =2. If the time t is taken to be m0 L=p0 , the matrix element |Hˆ propag | is in correspondence with the amplitude Ae1 ( → ; L) given by Eq. (189). The $eld-theoretical method will thus give the same results as the reciprocal basis formalism in the case of nearly degenerate unstable states. Both formalisms are easier to use that the cumbersome nonorthogonal basis (3), since any transition amplitude can be simply computed as a matrix product. Since the reciprocal basis formalism has already been applied to the study of CP violation (see [238,110,17,77,243]), we shall only give one example in Section 8.4 of the application of formula (189) to the computation of a CP asymmetry. Let us end this section by a remark on the observability of intermediate unstable particles. It is tempting to associate a well-de$ned mass eigenstate with the exponential behavior in e−i j t in Eqs. (189) or (190), but normalized mass eigenstates cannot be de$ned if the mass matrix MBavor is not normal (see Section 2.1). The reason is that a $nal state can be produced by all mass channels. In practice, it is sometimes possible to maximize the probability of a channel (since the lifetimes j

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can be very di1erent). In that case, one talks about decays into a mass eigenstate represented by its reciprocal basis vector out j (0)| [77]. 8.4. Oscillations in the B0 BH 0 system B0 BH 0 oscillations were $rst measured in 1987 [13], but the discovery of CP violation in this system is very recent [3,30]. We compute here as an example the following CP asymmetry [213]: (Bd0 (L) → f) − (BH 0d (L) → f) ACP (Bd0 → f; L) = ; (191) (Bd0 (L) → f) + (BH 0d (L) → f) where the index f refers to a CP eigenstate. The propagation eigenstates have nearly equal decay widths [115], and must be distinguished by their di1erent masses. Let us de$ne Xmd ≡ mH − mL , where the indices refer to BL , for Light, and BH , for Heavy. In the limit of a constant self-energy matrix f , the diagonalizing matrix of the mixed propagator (173) can be parametrized by * 1 1 1 v v −1 −1 V=√ and V = √ ; (192) 2* * −1 2* * −* where *2 = f00 H =f00H. Note that the phase of this parameter depends on the phase convention chosen for Bavor states. The physics should be invariant [77] under |B0 → ei |B0

and

|BH 0 → eiH|BH 0 :

(193)

Under the ket rephasing (193), * transforms as * → ei(−)H *. This parameter is related to the notations of [77] through * = −qB =pB . The deviation of |*| from 1 parametrizes the amount of CP violation in mixing, called ‘indirect CP violation’. The constant v in Eq. (192) is an arbitrary normalization, underlining the fact that the meson cannot be considered as an asymptotic state. Its decay amplitude should always be included in the full amplitude. Theoretical calculations ([77] and references therein) show that |*| − 1 ∼ O(10−4 ), which is much smaller than the present experimental uncertainty, so that only a small CP violation occurs in the mixing. On the other hand, a much larger CP violation is expected in the decays (‘direct CP violation’), since three quark generations are involved in processes such as Bd → J= KS . Thus, CP violation in mixing is usually neglected for B mesons (i.e. |*| = 1), which is the opposite of what occurs with neutral kaons. The computation of asymmetry (191) requires the knowledge of the amplitude Tf (L), corresponding to the propagation of an initial B0 . Starting from the e1ective oscillation amplitude (189), we can write Tf (L) as C L m 0 0 0 − 1 Tf (L) ∼ (M(B → f); M(BH → f))V exp −iM V ; p0 0 where the e1ective mass matrix M is equal to diag(mL − iL =2; mH − iH =2). It will be useful to de$ne the parameter 'f by M(BH 0 → f) : (194) 'f = M(B0 → f)

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Under rephasing (193) of the kets, 'f → e−i(−)H 'f . If |'f | = 1, ‘direct CP violation’ occurs in decay amplitudes to the state f. Whereas the quantities * and 'f are not invariant under rephasing (193), the product *'f is invariant. This quantity is related to the notations of [77] through *'f = − f . If the $nal state is a CP eigenstate and the transition dominated by a single CKM amplitude (for example Bd0 → J= KS ), *'f can be expressed in terms of the elements of the quark mixing matrix, so that its measurement is very important for the determination the CKM matrix elements. The amplitude Tf (L) can be written as Tf (L) ∼

CM(B0 → f) ((1 + *'f )e−imL (m0 =p0 )L−(m0 =2p0 )L L 2 + (1 − *'f )e−imH (m0 =p0 )L−(m0 =2p0 )H L ) ;

(195)

where p0 is the modulus of the total momentum of the $nal pions. We also need to compute the amplitude Tf (L), corresponding to the propagation of an initial BH 0 . The same method yields Tf (L) ∼

C∗ M(B0 → f) ((1 + *'f )e−imL (m0 =p0 )L−(m0 =2p0 )L L 2* − (1 − *'f )e−imH (m0 =p0 )L−(m0 =2p0 )H L ) :

(196)

With the insertion of Eqs. (195) and (196) and the approximations |*| = 1 and H = L , the asymmetry (191) becomes m0 L m0 L 0 dir 0 interf 0 ACP (Bd → f; L) = ACP (Bd → f) cos Xmd + ACP (Bd → f)sin Xmd ; p0 p0 where direct CP violation, de$ned by 0 Adir CP (Bd → f) =

1−| 1+|

2 f| 2 f|

=

1 − |'f |2 ; 1 + |'f |2

has been separated from CP violation coming from the interference between the mixing and the decay: 0 Ainterf CP (Bd → f) =

2 Im f 2 Im (*'f ) =− : 2 1 + | f| 1 + |'f |2

Three di1erent kinds of CP violation have been met in this section: indirect, direct and interference CP violations, signaled by |*| = 1, |'f | = 1 and Im(*'f ) = 0, respectively (see [77, p. 78]). If the $nal state is J= KS , a theoretical study (see for example [115]) shows that the CP violation parameter is in a very good approximation equal to J= KS = e−2i , where is one of the angles of the unitary triangle. The CP violation in the neutral kaon system that is included in the $nal state can be neglected. The direct and interference asymmetries become 0 Adirect CP (Bd → J= KS ) = 0

and

0 Ainterf CP (Bd → J= KS ) = −sin 2 :

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The quantities sin 2 and | | were recently measured by the BABAR [30,31] and Belle [3,4] collaborations: sin 2BABAR = 0:741 ± 0:067(stat) ± 0:033(syst) ; sin 2Belle = 0:719 ± 0:074(stat) ± 0:035(syst) ; |

BABAR |

= 0:948 ± 0:051(stat) ± 0:017(syst) :

The value of the interference asymmetry is an indication of CP violation in the B system, whereas the value of | | is consistent with no direct CP violation. 9. Selected topics In Section 9.1, we examine the question of the normalization of the transition probability. We then discuss in Section 9.2 the relation between coherent and incoherent e1ects, as well as the di1erent estimates of the wave packet size. Finally, the case of an unstable source is reviewed in Section 9.3. 9.1. Normalization of the transition probability Until now, we have not explained why the time-averaged squared modulus of the amplitude (see Eq. (96)) can be interpreted as the Bavor-mixing transition probability. Actually, the interpretation of expression (152) as a transition probability was a bit of a guess. As a matter of fact, the steps

between the amplitude A( → ) = j Vj−1 Aj Vj and a normalized event rate, containing an expression which can be interpreted as a Bavor-mixing transition probability, are not straightforward. In the case of the large-distance oscillation formula, the constant prefactors v0 N 2 *p e1 appearing in Eq. (132) (or in Eqs. (144) and (151)) have been shoved into a constant Ng , which was normalized in Eq. (153) in order to obtain a unitary evolution in the stable case. The normalization of the short-distance oscillation formula (111) proceeded in the same way. Whereas these prefactors can be factorized from the sum on the mass eigenstates if m2ij E02 , this is not possible for a mixing of a relativistic mi and a nonrelativistic mj mass eigenstate. Cardall has made an attempt [85] to go all the way to a normalized event rate. Although he claims to resort to arbitrary external wave packets, his model is equivalent to the Gaussian external wave packet model, since he uses Gaussian approximations to perform the integrals. We shall explain how Cardall’s argument can be applied to the external wave packet model developed in Sections 4–6. Only the large-distance case will be considered (L & p0 =*p2 ); the short-distance case can be treated similarly. The expression A( → ; T; L)A∗ ( → ; T; L) (see Eq. (34)) is proportional to a transition probability between one-particle states. In order to obtain a formula applicable to experiments, the external wave packets should be interpreted as densities of particles. Cardall proposes three rules of correspondence: 1. Let us $rst examine the external wave packets normalization constants NPI , NPF , NDI and NDF . They come from the overlap function and are included in the factor N 2 appearing in the transition probability (132). Eq. (52) shows that NPI is proportional to the modulus of the initial wave

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function at production. The constant NDI is interpreted in the same way. The constants NP2I and ND2I are then interpreted as the initial state particle densities at production and detection, respectively: NP2I →

dQ f(Q; xP ) (2 )3 2EPI

and

ND2I →

dQ f(Q ; xD ) ; (2 )3 2EDI

where f is the phase space density. The normalization constants NP2F and ND2F are interpreted as particle densities for the $nal state particles at production and detection: NP2F →

dK (2 )3 2EPF

and

ND2F →

dK : (2 )3 2EDF

Note that there may be more than one particle in the initial and $nal states. 2. Let us now consider the production and detection widths. Recall that the factor N 2 in Eq. (132) −3 −1 −3 −1 2 contains a volume factor V 2 = (2−4 4 *pP *eP *pD *eD ) coming from the overlap function (see Eq. (54)). One factor V is interpreted as an integration on macroscopic space–time variables: V → dxP d xP0 dxD d xD0 : Since the production time is not measured, it is integrated over in the event rate. This integration is equivalent to the time average over the propagation time T done in the present report. 3. Finally, the e1ective width *p e1 , appearing in front of the exponential in Eq. (132), is interpreted as the result of an integration over the energy spectrum of the oscillating particle. Furthermore, the second volume factor V , the geometrical factor g2 (l) and the term including *m are interpreted, in the limit mi = mj ≡ m˜ 0 , as a product of delta functions constraining the direction of L and the energy of the oscillating particle. More precisely, 2 dE p0 − (p0 · l)2 (m˜ 20 − m20 )2 8 √ (4) (p − pP )(4) (p − pD ) ; → − v0 *p e1 V exp − 2 2 2 2*p 8*m E0 2 where p = (E; E 2 − m˜ 20 l), and with the approximations pP = pD = p0 , vP = vD = 0, so that the energy and momentum decouple. The relationship is exact in the limit *pP = *pD = 0 (with the approximations for the momenta and velocities just mentioned). These three rules lead to the macroscopic event rate at the detector at time xD0 : dQ dQ 0 f(Q; x ) f(Q ; xD ) d(Q; Q ; xP ; xD ) ; d(xD ) = dxP dxD P (2 )3 (2 )3 with the constraint xP0 = xD0 − L=v0 . The single particle event rate is given by d(Q; Q ; xP ; xD ) = dE 6ux × Pmix × d* : The 6ux is the Bux of oscillating particles of energy E, produced at xP and detected at xD . It includes a phase space factor for $nal state particles at production, dK, as well as the delta function (4) (p − pP ), the interaction vertex |MP (Q; K)|2 , the geometrical factor 1=L2 , and a velocity factor |v0 − vD |. The factor d* is the cross section for the interaction of particle of mass m˜ 0 in the detector. It includes a phase space factor for $nal state particles at production, dK (not integrated over if

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189

these momenta are measured), as well as the delta function (4) (p − pD ), the interaction vertex |MD (Q ; K )|2 and the MHller factor |v0 − vD |−1 . Finally, Pmix is identi$able as the Bavor mixing transition probability:  2  2 2 2 2 (m ) v0 L L ij  : − Vi Vi−1 Vj∗ Vj−1∗ exp−2 i osc − + 2 Pmix = 2 2 coh L * 32p * L m ij 0 ij e1 p i; j In comparison with P→ (L), given by Eq. (152), the geometrical decrease and the normalization factor, i.e. Ng g2 (l)=L2 , are included in the Bux and in d*. The approximation m0 = (mi + mj )=2 has also been made in each interference term. In the case of a mixing of relativistic and nonrelativistic particles, the Bavor-mixing probability does not factorize from the amplitudes of production and detection. The interference terms are however negligible because occurs. The dependence on the mass mj should be kept in decoherence 2 the noninterference terms dT |Aj | , with the result that the prefactor v0 in (132) should be replaced by vj . Velocity-dependent prefactors were already derived in the intermediate wave packet model of Giunti et al. [141], except that the one-dimensional treatment in that article leads to a peculiar dependence in vj−1 (in our case, the conversion of the 1=T 2 prefactor into a vj2 =L2 prefactor yields an additional v2 , leading to the expected linear dependence in vj ). Cardall’s correspondence rules are still valid, although the Bux and the cross-section now depend on the mass eigenstate. For example, m˜ 0 and v0 are replaced respectively by mj and vj in the third rule. The single particle event rate is replaced by d(Q; Q ; xP ; xD ) = |Vj |2 |Vj−1 |2 dE 6uxj × d*j : j

Mixings of relativistic and nonrelativistic neutrinos have been studied by Ahluwalia and Goldman [8], who identify the third mass eigenstate with the 33:9 MeV particle suggested by the KARMEN experiment [26]. Note that the amplitude of production (and also of detection) should be computed separately for the relativistic and nonrelativistic neutrinos. 9.2. Coherence, incoherence and decoherence Generally speaking, decoherence is said to occur in particle oscillations if the interference terms in the transition probability are averaged to zero by some mechanism. In that case, the transition probability becomes independent of the distance. We have seen in Section 7 that this phenomenon appears in the large-distance Bavor-mixing transition probability (152) through the localization term and through the coherence length. Localization conditions, such as Losc ij & *x , determine whether there is decoherence from the start. If decoherence only occurs beyond a certain distance, the threshold is called the coherence length. These two phenomena are closely connected, since the coherence-length and localization conditions in the transversal-dispersion regime transform into the localization and coherence-length conditions in the longitudinal-dispersion regime, respectively. In the end, both mechanisms of decoherence originate in the wave packets widths of the external particles. Other e1ects lead to similar constraints on the oscillations. They can be classi$ed in coherent or incoherent e1ects. In quantum $eld theory, a coherent e1ect has to be taken into account in the amplitude, whereas an incoherent e1ect is incorporated into the computation only at the level of the

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probability [151]. Actually, this distinction has to be made only because approximations made in the computations often make the intrinsic decoherence e1ects disappear. For example, there are endless discussions about the energy and momentum coherence in the plane wave treatment of oscillations, since this approximation destroys all natural decoherence mechanisms. This should be contrasted with the fact that coherence-length and localization conditions appear explicitly in the oscillation formula (152) obtained in the external wave packet model. Moreover, decoherence between di1erent energy components is automatic in the external wave packet model (see Section 6.4). Note that one should be careful not to confuse the term ‘(in)coherent’, referring here to a constraint applied to the amplitude or to the probability, with the term ‘(de)coherence’, referring to the existence or disappearance of oscillations. 9.2.1. Incoherent e:ects First, the energy–momentum spread of the beam has to be taken into account [37,66,34]. For example, the average of the oscillation term over a Gaussian momentum distribution of width Xp and mean value pH gives  2  2 L L Xp L (p − p) H  ; dp exp −2 i osc − ∼ exp−2 i osc − 2 2 H Lij 2(Xp)2 pH LHosc Lij ij 2 where LHosc H ij = 4 p=m ij . A new coherence length can be de$ned by pH Hosc 1 Lij : LHcoh ij = √ 2 Xp

This new coherence length can be obtained from the coherence length (154), derived in the external wave packet model, by substituting the beam spread Xp for the e1ective width *p e1 . This result could be expected from the discussion of Section 6.4, by extending the incoherent sum over the energy from a *p e1 range to a XE = Xp=v0 range. Second, the macroscopic propagation distance L is not perfectly known [149,66]. A Gaussian average over the macroscopic region of production gives  2  H 2 L L XL (L − L)  ; dL exp −2 i osc − ∼ exp−2 i osc − 2 2 Lij 2(XL)2 Lij Lijosc where LH is the average propagation distance and XL is the size of the source. Thus a new localization condition has to be satis$ed: Losc ij & XL : This condition is similar to the localization condition (166) which comes from a coherent e1ect. The only thing to do is to substitute the size of the source XL for the e1ective width *x e1 . As noted several times ([104,94,237,182,248], and references therein), it is impossible, in stationary cases, to distinguish wave packets from an incoherent plane wave superposition with the same energy–momentum spectrum. This observation leads directly to the generalization of the quantum $eld computation of the transition probability already done with external Gaussian wave packets. Arbitrary wave packets, with

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191

space width and momentum widths not minimizing the uncertainty, lead to an oscillation probability similar to Eq. (152) (or Eq. (187) if the particle is unstable), except that the relation *x e1 *p e1 = 1=2 does not hold anymore. This result can be understood as follows. If the arbitrary external wave packets are decomposed in Gaussian wave packets, the amplitude Ai becomes a superposition of Gaussian amplitudes. These do not interfere with each other if their phases are very di1erent. Thus each Gaussian amplitude will mostly interfere with itself, from which an incoherent superposition follows. In practice, the size of the region of production is usually much larger than the size of the wave packets. Similarly, the energy–momentum spread of the beam is usually much larger than the energy–momentum spread of the wave packet. Incoherent e1ects are thus very often dominant. At worst, the sizes of the regions of production and detection and of the energy–momentum spectrum are determined by the characteristics of the wave packet. Similar decoherence mechanisms were found by Gabor [131] in connection with electron-interference experiments. This author obtains three decoherence factors, coming from an average over the size of the source, from an average over the energy spectrum, and from the path di1erence between the interfering beams. The two $rst e1ects have been explained in this section and the last e1ect can be related to the wave packet separation in particle oscillations. 9.2.2. Coherent e:ects Recall that coherent e1ects are constraints on the oscillation process, which should be applied at the level of the amplitude. Most coherent e1ects can be expressed through their inBuence on the sizes of the external wave packets associated to the production and detection of the oscillating particle. Unfortunately, the estimate of a wave packet size is not an easy matter and no consensus exists on whichever evaluation method is the best. Let us consider $rst a solar neutrinos. Only the cases of line spectra (pep or 7 Be) deserve careful thought, since the energy average always dominates coherent e1ects for continuum spectra (except if a detector with a extremely high energy resolution is invented). The most commonly discussed constraint on the wave packet size comes from the pressure broadening, that is, the interruption of coherent emission due to collisions of the emitting atoms. Nussinov [216], Loeb [200] and Kim and Pevsner [185] estimate the wave packet size of the parent nuclei at *x ∼ 10−6 cm, whereas Krauss and Wilczek [188] propose *x ∼ 10−4 cm. Kiers et al. [182] claim that the small wave packets of captured electrons give a stronger constraint *x ∼ 6 × 10−8 cm. However, none of these estimates gives a momentum width *p as large as the 1 keV energy spread of the solar neutrino line spectra [188,221,35]. The main contribution to this energy spread comes from the thermal energy spread of the captured electron [188], as well as from the Doppler shift due to the thermal motion of the emitting nucleus [221]. These e1ects cannot be modelized at the level of the amplitude and belong thus to incoherent e1ects. Finally, Malyshkin and Kulsrud [204] compute the e1ect of Coulomb collisions on the solar neutrino Bux. They obtain a quantity *a , equivalent to the wave packet width, and consistent with Nussinov and Loeb’s estimates. While Malyshkin and Kulsrud agree that pressure broadening can be neglected with respect to the line width, they claim that it could be relevant to the decoherence in the case of the continuous solar spectra, for a detector of very high resolution and a very long oscillation length. This conclusion is in contradiction with the increase of the coherence length with the resolution of the detector: the coherence length is mainly determined by the energy

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resolution of the detector, if it is smaller than the wave packet size at the source. All these results show that coherent e1ects seem to be irrelevant to solar neutrinos. As regards neutrinos from supernovae, the source wave packet width has been estimated by Anada and Nishimura at *x ∼ 10−14 cm for neutrinos from the supernova core [19], and at *x ∼ 10−9 cm for neutrinos from the neutrino sphere [20]. Since p ∼ 10 MeV, core neutrinos decohere before oscillating (p=*p ∼ 0:1), whereas sphere neutrinos might oscillate if the oscillation length is not too short (p=*p ∼ 104 ). However, the incoherent momentum spread (p=Xp ∼ 1–10) destroys any interference e1ect left, unless the oscillation length is, by chance, comparable to the supernova–Earth distance [236]. Another type of neutrino source is a radioactive nucleus in an atomic lattice, which is the case for reactor neutrinos. The wave packet size can be estimated by Rich [237] and Grimus and Stockinger [150] at *x ∼ 10−10 m. The wave packet size of the emitted electrons is larger and can be neglected [185]. With the neutrino energy around 1 MeV, decoherence occurs beyond p=*p ∼ 103 oscillation lengths, so that there are no coherent e1ects relevant for laboratory experiments. Of course, oscillations may vanish because of the incoherent energy average. Coherent e1ects may also appear because of the $nite lifetime of the source decay , which interrupts the classical emission of the wave train and limits the size of the wave packet to *x ∼ cdecay [182]. This e1ect could be relevant to atmospheric neutrinos and in accelerators. For quasi-stable sources, the constraint Losc ij & *x can be very stringent. For example, in the case of → ', the length of the wave packet is bounded by cdecay ∼ 7:8 m and could be macroscopically large. However this argument is only valid as long as the decay point of the source is not observed at all. As emphasized by Kayser [176], the detection of a $nal state, such as the muon in → ', can localize the decay point to a precision much better than either cdecay or the macroscopic size of the production region. An extreme example is given by the neutrinos from the decay of a nucleus with decay ∼ 1 s. The observation of the particle allows to pin down the decay point to a precision much better than either cdecay ∼ 105 km or the production region, for example a nuclear reactor. Thus the relation between the decay time of the source and the wave packet size of the oscillating particle is not direct. The decay time only puts an upper bound to the wave packet length. Some papers dealing with the $nite lifetime of the source are reviewed in Section 9.3. Since the coherence length depends not only on the characteristics of the source, but also on those of the detector, the minimal wave packet size in the detection process must also be checked. It can be roughly estimated at *x ∼ 10−10 m, that is, *p ∼ 10−3 MeV [151]. In all cases, this momentum width is not larger than the energy spread and can be neglected. As regards K and B mesons, particle decay takes place before decoherence occurs: Lcoh ij decay

Lij

∼

p0 i + j p0 1 ; ∼ *p e1 2|mi − mj | *p e1 decay

and Lij are given by Eqs. (154) and (186), respectively. where Lcoh ij To sum up, coherent e1ects are most likely irrelevant with respect to incoherent e1ects. This discussion also shows that, in practice, we are far from being able to increase the coherence length by more accurate energy measurements.

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9.3. Unstable source In principle, the instability of the source can be taken into account in a $eld-theoretical model by considering the source as another internal line of the global Feynman diagram describing the process. The diKculty, of technical nature, consists in integrating on both propagators, with a constraint on the decay point which can be either nonexistent, or very stringent. The case of an unstable source decaying in Bight has been considered by Campagne in a $eldtheoretical model [84]. Besides the usual condition Losc ij & *xP; D , this author obtains a new localizadecay decay osc is either the decay length p=m of the unstable source, tion condition Lij & LP , where LP or the length of the decay tunnel, if the latter is shorter. However, this treatment is not a real improvement on the one proposed by Rich, who had derived earlier the same results in the framework of time perturbation theory in quantum mechanics [237]. It is not satisfactory that macroscopic quantities, such as the lengths of the source, of the target and of the pion decay tunnel, are treated in Ref. [84] on the same footing as microscopic quantities. The case of an unstable source at rest has been considered in detail by Grimus et al. [151,152]. They use their previous $eld-theoretical model [150], modi$ed by a quantum-mechanical Wigner– Weisskopf approximation, in order to take into account the $nite lifetime of the source. Besides the already known condition Losc ij & *xP; D , they obtain a new localization condition: *P 1 osc L ; . mP P 4 ij

(197)

where *P , mP and P are respectively the wave packet momentum width, the mass, and the decay width of the unstable parent particle. In brief, the unstable source should not move on distances larger than the oscillation length during its lifetime, or else the oscillations vanish. They $nd that this condition is most likely satis$ed in the experiments LSND [28,29] and KARMEN [26,27]. 2 2 The same authors have also found a new coherence length, Lcoh ij = 4E =mij P , which is however completely irrelevant, since it is much larger than the coherence lengths discussed previously (P *P for all weakly unstable particles). The widths *xP; D can be estimated at 10−2 MeV for LSND and KARMEN [152], so that the condition Losc ij & *xP; D is also satis$ed. A slightly di1erent model for → ' is proposed by Dolgov [105]. This author computes the transition amplitude by coupling the neutrino propagator to external wave packets at the source, but not at the detector (source–propagator model). The $nite lifetime of the source is taken into account with a Wigner–Weisskopf approximation, like in the articles discussed above [151,152], but without the restriction of a source at rest. Dolgov considers $rst the case where the decay point of the muon is not registered (but its energy–momentum is perfectly known) and computes an amplitude of spatial width = p =m . The oscillating phase is given by L L − v0 T v · v0 with = 2 2 ; ij (T; L) = 2 osc + Lij Losc v0 − v · v0 ij where v and v0 are the velocities of the pion and neutrino, respectively. With the help of the constraint 0 . v0 T − L . , the second term of the phase is seen to be negligible if Losc ij . For a source at rest, = 0 and this localization condition vanishes. When the muon decay point is registered, Dolgov obtains an oscillation phase equal to the standard result as long as the localization condition * Losc is satis$ed (where * is the size of the wave packet of the initial pion). ij

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It would be interesting to study intermediate situations where the muon is registered with a space– time uncertainty and to compute the time-independent oscillation probability. This brief account of the treatment of oscillations from an unstable source shows that there is still work to do, especially regarding decays in Bight.

10. Correlated oscillations 10.1. Introduction Experiments where two correlated mesons oscillate together are very interesting for the study of CP violation [82,83,77] and can provide tests of the Einstein–Podolsky–Rosen (EPR) e1ect [25,57,58,116,228]. At the present time, the process (1020) → K 0 K 0 is studied by the experiment DANE at Frascati [7]. The process ((4s) → B0 B0 is studied at B factories such as KEKB at Tsukuba [4] or BABAR at Stanford [31,33], where asymmetric collisions allow the measurement of the B pathlengths. In principle, correlated oscillations could occur with particles not conjugated to each other, for example the lepton could oscillate between e and ' and the neutrino between e and ' in the process → l. However, the charged lepton masses are too di1erent for such oscillations to be observable (but note that if the masses were close enough to allow oscillations, it would be impossible to identify the Bavor). In this section, we develop the formalism appropriate for correlated oscillations and we apply it to oscillations of correlated mesons. We then disprove a claim of a nonstandard oscillation length. The question of the oscillation of recoil particles, like & in p → &K or ' in → ', can be examined in the same framework, allowing us to answer the $fth question posed in Section 3.2. The treatment of one-particle oscillations with the external wave packet model of Section 4 can be easily extended by associating a propagator with each oscillating particle. Consider the two successive processes: (e+ e− )(pP ) → RX ;

(198)

R → P1 P2 → f1 (pD1 )f2 (pD2 ) ;

(199)

where pP and pD1; 2 are the average momenta of the source R and of the $nal states, respectively. We would like to apply the external wave packet model to the second process. In principle this model should be modi$ed to include the decay width of the source R, since one external particle at an interaction point is not suKcient to localize it. However the decay widths of the (1020) and ((4s) are large, so that their decay point will be close to their production point. The latter can be localized with the e± external wave packets. In the case of a small decay width, it will be seen below that the correlated oscillations become independent of the production point. Note that the initial Bavors of the oscillating particles cannot be observed at the source and must be summed over. For example, B0 BH 0 and BH 0 B0 are both produced in the ((4s) decay. In contradistinction to the one-particle oscillation case, this sum does not destroy the oscillations, because of the Bavor correlation between the oscillating particles. Indeed, two mesons of the same Bavor cannot be observed in the above process at the same time in the center of mass frame (EPR e1ect).

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Let us $rst write the amplitude corresponding to process (199). The masses of the $rst particle are noted m1a , m1b , etc., while the masses of the second particle are noted m2i , m2j , etc. The two sets coincide if P1 and P2 are conjugated mesons, but it is not necessarily the case. The average production and detection points are noted xP and yD1 , yD2 , respectively. The partial amplitude, corresponding to the propagation of the eigenstates of mass m1a , from xP to yD1 , and of mass m2i , from xP to yD2 , can be written as 4 Aa; i = d p1 d 4 p2 (p1 ; p2 )G1a (p12 )G2i (p22 )e−ip1 ·(yD1 −xP )−ip2 ·(yD2 −xP ) ; with the overlap function given by (p1 ; p2 ) = N

0 P (p1

+ p20 ; p1 + p2 )

0 0 D1 (p1 ; p1 ) D2 (p2 ; p2 )

:

The production overlap function P is de$ned by Eq. (54). The detection overlap function D1 is de$ned by the same equation, where pD1 , ED1 and vD1 have been substituted to pP , EP and vP , respectively. The de$nition of D2 is similar. The integration on the energies p1;0 2 can be done with the help of the Jacob–Sachs theorem (211), yielding √ 2 √ 2 Aa; i ∼ d 3 p1 d 3 p2 (z1 ; z2 ; p1 ; p2 )e−i z1 +p1 T1 +ip1 ·L1 e−i z2 +p2 T2 +ip2 ·L2 ; (200) where T1; 2 = tD1; 2 − tP , L1; 2 = yD1; 2 − xP are the average propagation time and distance of Pi , and zi is the pole of the propagator Gi (p2 ). Since our aim is not to prove the conservation of energy–momentum between the initial state R and the $nal states f1; 2 , we set pD 1 + p D 2 = p P

and

ED 1 + E D 2 = E P :

As before, it will be useful to de$ne reference masses m1 and m2 through m21; 2 = ED2 1; 2 − pD2 1; 2 . The velocities v1; 2 are de$ned by v1; 2 = pD1; 2 =ED1; 2 (they should not be confused with vD1; 2 , which appear in the overlap function and refer to the velocities of the detection regions, as explained in Section 4.5.2). 10.2. Factorization The momentum integrations in amplitude (200) can be easily evaluated, either with Laplace’s method or with the stationary phase method, provided that the energy–momentum correlation is not too stringent at the source. More precisely, the integrations on p1 and p2 can be done independently if the energy uncertainty at the source is larger than the energy uncertainties at the detection points (recall that the momentum width is always larger than the energy width). For example, the production widths for the (1020) decay and the ((4s) decay can be estimated by their average decay widths, which are 4.26 and 14 MeV, respectively [156]. Both widths are larger than typical detector uncertainties. In that case, the energy–momentum width at the source can be neglected and the overlap function factorizes in one-particle overlap functions: (z1 ; z2 ; p1 ; p2 ) ∼ =

1 (z1 ; p1 ) 2 (z2 ; p2 )

:

(201)

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Thus the resulting amplitude factorizes in one-particle oscillation amplitudes. The rest of the computation proceeds as in the one-particle case, except when the detection times are measured: the time average is then done only once, on the production time. If the energy–momentum uncertainty is smaller at the production point than at the detection points, energy–momentum correlations are introduced at the source so that the amplitude cannot be factorized. However, the oscillation formula is not expected to be modi$ed, as long as the energy– momentum uncertainty at the source is larger than the mass di1erence between the interfering mass eigenstates: *pP & m2ij =pD1; 2 . This condition resembles the localization condition derived in the one-particle oscillation case, which stated that oscillations vanish if the uncertainty on the position of the source is larger than the oscillation length. This similarity is misleading: it will be seen correlated oscillations do not vanish if *pP . m2ij =pD1; 2 , since the knowledge of the source energy– momentum is not suKcient to ascertain which mass eigenstates are produced. This is because the energy–momentum of the source is shared between two intermediate particles, instead of one as in the one-particle oscillation model presented in Section 4. Moreover, the position and time of the production process can be reconstructed, in principle, from the $nal states characteristics, so that the oscillation pattern is not washed out by an ill-de$ned production point. The discussion of the above constraint is somewhat academic, as it is satis$ed for the experimentally studied processes, → K 0 K 0 and ((4s) → B0 B0 . Nevertheless, the question of whether the violation of this constraint changes the oscillation formula will be examined, partly as a matter of principle, and partly because it will be useful for the discussion of the recoil oscillation conundrum. 10.3. Energy–momentum correlation at the source There are two reasons to study more carefully energy–momentum correlations at the source. First, this analysis yields a correlated localization condition, arising from the source. In particular, it allows to check explicitly that a zero momentum width at the source does not wash out the oscillations. Second, it will be useful for the examination of the so-called recoil oscillations. Unfortunately the treatment of the full 3-dimensional case is involved, because the correlation at the source might link a variation in the longitudinal momentum of one particle, with a variation in the transversal momentum of the other. If the transversal dispersion is neglected, the integrals can in principle be evaluated with Laplace’s method in three dimensions, but the results are lengthy. For that reason, only the collinear case will be presented. Momenta, velocities and lengths can take positive or negative values, with the sign indicating the direction. Since the stationary phase and Laplace’s method give the same results (see Section 6), we choose the latter for the longitudinal momenta integrations. Besides, the dispersion will be neglected. Recall that the masses of the $rst particle are noted m1a , m1b , etc., while the masses of the second particle are noted m2i , m2j , etc. Suppose that the overlap function is maximal for p1 = p1a and p2 = p2i . As before, these momenta can be computed by expanding the argument of the overlap function in small mass di1erences. The expansion parameters are given by m21a = m21a − m21 and m22i = m22i − m22 . At $rst order in mass di1erences, the average momenta of the particles are p1a = pD1 +

*x2e1 3 c2 − *x2e1 2 c1 ; 4S

(202)

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p2i = pD2 +

*x2e1 3 c1 − *x2e1 1 c2 ; 4S

197

(203)

where S = *x2e1 1 *x2e1 2 − *x4e1 3 ¿ 0. In momentum space, the e1ective widths read 1 *p2 e1 1; 2 1 *p2 e1 3

=

=

(v1; 2 − vD1; 2 )2 1 (v1; 2 − vP )2 1 + + + ; 2 2 2 2 *pP *eP *pD *eD 1; 2 1; 2

1 (v1 − vP )(v2 − vP ) + : 2 2 *pP *eP

(204)

They are related to the e1ective widths in con$guration space by *pe1 1; 2;3 *xe1 1; 2;3 = 1=2. Their name is justi$ed below by their appearance as second order coeKcients in the expansion of the overlap function. These expressions can be compared to the e1ective width (89) in the one-particle oscillation case. The mass di1erences are included in the constants ck : vk − v D k vk − vP ck = k + (1 + 2 ) ; 2 2 *eDk *eP with 1 = m21a =2ED1 and 2 = m22i =2ED2 . If the overlap function is noted = N exp(−f(p1 ; p2 )), the expansion of the argument of the exponential around its extremum can be written as f(p1 ; p2 ) = f(p1a ; p2i ) +

(p1 − p1a )2 (p2 − p2i )2 (p1 − p1a )(p2 − p2i ) + + : 4*p2 e1 1 4*p2 e1 2 2*p2 e1 3

The evaluation in one dimension of integral (200) by Laplace’s method yields m1 1 T1 m 2 2 T2 − iE2i T1 + ip2i L1 − Aa; i ∼ exp(−f(p1a ; p2i )) exp −iE1a T1 + ip1a L1 − 2E1a 2E2i ×exp

−1 2 (v1a T1 − L1 )2 + *x2e1 1 (v2i T2 − L2 )2 (* 4S xe1 2

−2*x2e1 3 (v1a T1 − L1 )(v2i T2 − L2 )) ; (205) 2 where v1a = p1a =E1a , with E1a = p1a + m2a . The velocity v2i and the energy E2i are similarly de$ned. Once more ‘wave packets’ can be associated with the oscillating particles 1 and 2, but they are correlated by the $nite width *xe1 3 . The phase can be expanded around pD1; 2 and m1; 2 . For example, m21a m21a ∼ + L1 ; (206) 1a = E1a T1 − p1a L1 = ED1 T1 − pD1 L1 + (v1 T1 − L1 ) p1a + 2p1 2p1 where p1a = p1a − p1 . The phase di1erence between two amplitudes vanishes but for the oscillation term already encountered in the previous section, provided the wave packet is suKciently localized, so that the term proportional to v1 T1 − L1 in (206) is negligible. At the extremum, the argument of the overlap function reads f(p1a ; p2i ) =

21 + 22 + 21 2 ; 64S

(207)

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and yields a localization condition, that is, the mass di1erences cannot be too large compared with the energy–momentum widths. The coeKcients and read 1 (v2 − vD1 )2 (v2 − vD2 )2 1 + + = 2 2 2 2 *pP *eP *D2 *eD *eD 1 2 1 1 (v2 − vD2 )2 1 (vP − vD1 )2 1 + 2 + + 2 2 + ; + 2 2 2 2 2 2 2 *pD2 *eD2 *pP *eD1 *eP *pD1 *eP *eD1 *pD1 *eD1 *p2 e1 2 =

1 1 (v1 − vD1 )(v2 − vD2 ) (v1 − vD1 )(v2 − vD1 ) (v2 − vD2 )(v1 − vD2 ) + + + 2 2 2 2 2 2 *pP *eP *D2 *eD * *pP *eD *2 eD2 1 1 eD2 1 1 1 (v1 − vD1 )(vP − vD1 ) (v2 − vD2 )(vP − vD2 ) ; + 2 + + 2 2 2 2 *eP *pD *eD *pD *eD 1 1 2 2

−2 −2 −2 where *pD = *pD + *pD . The coeKcient is obtained from by the exchange of the indices 1 ↔ 2. 1 2 What happens when the energy–momentum uncertainty at the source goes to zero? Three features of amplitude (205) must be examined: the localization condition (207) given by the function f(p1a ; p2i ), the value of the phase and the ‘wave packet’ e1ect on the time average. In the limit *pP → 0 (so that *eP → 0 too), the argument of the overlap function becomes at the extremum

lim f(p1a ; p2i ) =

*pP →0

((v2 − vD1 )1 + (v1 − vD1 )2 )2 ((v2 − vD2 )1 + (v1 − vD2 )2 )2 + 2 2 4(v1 − v2 )2 *eD 4(v1 − v2 )2 *eD 1 2 +

(1 + 2 )2 : 2 4(v1 − v2 )2 *pD

Thus a small, or even zero energy–momentum uncertainty at the source, does not destroy the oscillations. This result con$rms our expectations, since a zero width at the source does not give information on which mass eigenstates propagate. On the other hand, the function f(p1a ; p2i ) di2 2 verges when either *pD or *pD goes to zero, yielding the expected localization conditions at the 1 2 detectors. In the limit of a zero energy–momentum width at the source, wave packets associated to oscillating particles are of in$nite extent. This phenomenon was expected, since the time of production TP , included in T1 and T2 , becomes ill-de$ned. However, the average on the time of production does not destroy the oscillations, at least if the detection times are measured. The explanation follows. In the limit *pP → 0, Eqs. (202) and (203) show that the quantities p1a =p1a −pD1 and p2i =p2i −pD2 become lim p1a = − lim p2i = −

*pP →0

*pP →0

1 + 2 : v1 − v 2

Note that these values can be directly obtained, in the one-dimensional case, from energy–momentum conservation at the source. The phase of the amplitude (see Eq. (206)) can then be written as lim (1a + 2i ) = 1

*pP →0

L1 − L2 − v2 (T1 − T2 ) L1 − L2 − v1 (T1 − T2 ) + 2 ; v1 − v 2 v1 − v 2

(208)

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199

where the term of order zero in the mass di1erences has been dropped. Since T1 − T2 = tD1 − tD2 and L1 − L2 = yD1 − yD2 , the phase is independent of the production point. However, the production point is implicit in phase (208) when the classical limit is taken. The substitution of the classical relations L1; 2 = v1; 2 T1; 2 shows indeed that the expressions multiplying 1 and 2 are the propagation times T1 and T2 , respectively. In other words the space–time coordinates of the production point can be reconstructed from the coordinates of detection and the velocities of the oscillating particles, with the exception of the v1 = v2 case (recall that velocities of the same sign have the same direction). While complete computations in three dimensions are complicated, it can be seen that the phase di1erence becomes independent of the production coordinates, in the limit of zero energy–momentum uncertainty at production: lim (1a + 2i − 1b − 2j ) = E1ab (T1 − T2 ) − p1ab · (L1 − L2 ) :

*pP →0

(209)

10.4. No recoil oscillations It has been claimed that particles produced together with mixed states also oscillate because of the energy–momentum recoil. For example, the & baryon, in the process − p → &K 0 , is supposed to be in a superposition of two energy–momentum eigenstates, so that its detection probability should oscillate in space [244]. Furthermore the oscillation frequency of the kaon is modi$ed by the momentum recoil against the &. In another example, muons produced in → ' are supposed to oscillate in space, because of the momentum recoil against the mixed state of the neutrino [246,247]. As explained in Section 2.3, recoil oscillations arise in quantum-mechanical models if di1erent propagation times are associated to the di1erent mass eigenstates. Recoil oscillations are unacceptable from basic principles, since the oscillation of the detection probability of a nonmixed state means that probability is not conserved at all distances. On the contrary, when several mass eigenstates are mixed, the sum of the detection probabilities of the di1erent mass eigenstates is always equal to 1 for a given propagation distance. This assertion has been questioned in several papers in the framework of quantum mechanics. These refutations involve either the consideration of the di1erent proper times associated with the oscillating particle and the recoil particle [201,258,80], or the use of classical trajectories [107], or the use of energy–momentum conservation for the average energy–momenta of the propagating wave packets [258,212]. The source–propagator approach of Shtanov [242] is not a real improvement on the arguments of Ref. [107], since Shtanov treats con$guration space variables, which have a microscopic role, as classical macroscopic variables. Quantum $eld theory allows to understand which quantum-mechanical explanations are correct and important. The correlated oscillation model of the previous section is easily adapted to the treatment of recoil oscillations. Suppose that the $rst particle is in a superposition of several mass eigenstates m1a , whereas the second particle has only one mass eigenstate m2 (it is the so-called recoil particle). Since the phase of amplitude (205) depends only on the average propagation times T1; 2 and distances L1; 2 , which are common to the di1erent mass eigenstates, no recoil oscillations will be observed as long as these quantities are well-de$ned. This can be checked explicitly if *pP & *pD1; 2 , as the overlap function factorizes in that case. Thus the only case requiring a careful examination is when the uncertainty on the position of the source is larger than the oscillation length, i.e. when

200

M. Beuthe / Physics Reports 375 (2003) 105 – 218

*pP . m21ab =2p1 . This condition is not satis$ed in the process − p → &K 0 . The momentum uncertainty of the proton, in this $xed-target experiment, is around 3 keV, i.e. much larger than the kaon mass di1erence of 3 × 10−9 keV [198]. Therefore, the problem of recoil oscillations does not arise in the &K 0 complex. In contradistinction to the case of the − p collision, it is possible to consider a decay with a sharp momentum. Its theoretical minimal uncertainty is given by its 2:5×10−8 eV decay width, which is smaller than the neutrino mass di1erences. This case has been treated, with essentially correct quantum-mechanical arguments, by Dolgov et al. [107] and within a model coupling the ' and propagators with a source wave packet by Dolgov [105]. On the one hand, if only the neutrino is observed, the detailed study of the one-particle oscillation case in Section 6 has shown that neutrino oscillations occur, provided that the momentum width associated with the muon is larger than the inverse oscillation length. On the other hand, if only the muon is observed, no oscillations in the probability of muon detection occur. The reason is that the sum over the neutrino Bavors makes the transition probability (152) independent of the distance (apart from the geometrical decrease in L−2 ): L2 dM P→ (L) = 1 :

Furthermore oscillations also vanish after integration over the unknown source decay coordinates. This point was discussed in Section 9.3. Finally, the only case left is the detection of both muon and neutrino, coming from the decay of a pion endowed with a sharp momentum. As explained above, this process can be treated like a correlated oscillation, but with only one mass eigenstate m2 for the muon. Formula (209) shows that the oscillation of the probability is independent of the production point, so that the average over the production point does not have any e1ect on the phase. Moreover, the oscillation term can be expressed as depending only on the detection point of the neutrino, and on the reconstructed decay point (see Eq. (208) with 2 = 0): lim (1a + 2 ) = 1

*pP →0

L1 − L2 − v2 (T1 − T2 ) : v1 − v 2

The substitution of the classical relations L1; 2 = v1; 2 T1; 2 in the above equation shows indeed that the phase di1erence depends only on the classical propagation time of the neutrino: lim (1a + 2 ) = 1 T1 =

*pP →0

m21a L1 : 2pD1

In short the probability to detect both muon and neutrino can be interpreted as the result of a neutrino oscillation alone. Recall that the above relation can be obtained in a simple way in one dimension by energy–momentum conservation at the source. As regards the experimental data, oscillations of recoil particles have not been probed (in the case of the → ', it would only be possible for a neutrino oscillation length shorter than the muon decay length). However, the neutral kaon mass di1erence obtained from strangeness oscillation experiments is consistent with the one obtained from regeneration experiments [201].

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201

10.5. Oscillations of correlated mesons We compute here the amplitude associated to the correlated oscillations of a pair of neutral mesons produced in e+ e− annihilation. The phenomenology of correlated K and B mesons is discussed in [108,79,159,77] and in [82,83,62,115,77], respectively. Each meson oscillates between its components PL –PH , before decaying into $nal states f1 (k1 ) and f2 (k2 ) at spacetime points y1 and y2 : e+ e− → R(q) → P 0 PH 0 → f1 (k1 )f2 (k2 ) ; where q, k1 and k2 are the corresponding energy–momenta. Since each $nal state can be produced by either P 0 or PH 0 , the two amplitudes arising from the exchange of P 0 and PH 0 as intermediate states must be coherently added. The resonance quantum numbers of and ((4s), J PC = 1− − , are conserved by the strong interactions at the source, so that the pair P 0 PH 0 is in an antisymmetric state under P and under C. Thus the relative sign of the two contributions to R → f1 f2 is negative [196]. The total amplitude before antisymmetrization is given by Eq. (200). In Section 10.2, it was shown that the energy–momentum correlation at the source can be neglected for the resonances and ((4s), so that the overlap function factorizes as in Eq. (201). The amplitude for correlated oscillations can thus be approximated by the product of two amplitudes, each corresponding to a single oscillation. Furthermore, it was shown in Section 8.3 that the oscillation amplitude, in the case of nearly degenerate unstable particles, can be replaced by the e1ective amplitude (189) depending only on the distance L. In the end the antisymmetrized amplitude to detect a f1 at distance L1 and a f2 at distance L2 can be written as 1 0 −1 −iM m0 L1 =p1 −1 −iM m0 L2 =p2 Tf1 f2 ∼ MP (M01 M01 e V e V (M02 M02 H )V H )V 0 1 1 0 −same expression with ↔ ; 0 1 where MP = M(R → P 0 PH 0 ). The matrix M is de$ned by M = diag(mL − iL =2; mH − iH =2), m0 is the mass of the kaon in the degenerate limit and pj are the norms of the 3-momenta of the $nal states fj . The diagonalization matrix V is given by Eq. (192). The amplitudes M0j and M0jH stand for M(P 0 → fj ) and M(PH 0 → fj ), respectively. The amplitude can be written as (1 + *'f1 )(1 + *'f2 ) Tf1 f2 ∼ MP M01 M02 2* ×(−Tf2 e−i(mL −(i=2)L )m0 L1 =p1 −i(mH −(i=2)H )m0 L2 =p2 + Tf1 e−i(mH −(i=2)H )(m0 L1 =p1 )−i(mL −(i=2)L )m0 L2 =p2 ) ; where 'f is de$ned by Eq. (194) and Tf = (1 − *'f )=(1 + *'f ). Note that the amplitude vanishes for identical $nal states (f1 = f2 and p1 = p2 ) and identical propagation distances (L1 = L2 ), as expected [100]. In the center-of-mass frame, the equality of momenta p1 = p2 ≡ p implies that the

202

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interference term oscillates like m0 (mH − mL ) (L1 − L2 ) : cos p

(210)

This oscillation formula coincides with the quantum-mechanical result obtained with the equal time prescription, whereas the di1erent time prescription leads to an oscillation length shorter by a factor 2 [245]. As already explained in Section 2.3, the quantum $eld treatment shows that neither prescription is meaningful, since the interference takes place over a space and time range, determined by the e1ective ‘wave packet’ width. We can only say that, in the end, the di1erent mass eigenstates have the same average propagation time or distance, so that the oscillation formula will agree with the one obtained with the identical time prescription. As regards the experimental data, Kayser [178] has shown, in the case of the process ( → B0 B0 , that the mass di1erence obtained from the oscillation formula (210) is in agreement with the mass di1erence extracted from single B oscillations. 11. Summary and outlook Although the plane wave derivation of the vacuum oscillation formula has often been criticized in the literature, it is still used in most articles and textbooks. There are two reasons for this. First, the oscillation formula obtained in this way is believed to be correct by most physicists, in spite of the numerous inconsistencies present in its derivation. Second, the other approaches are not felt to be completely satisfying, with the consequence that more sophisticated treatments are not considered worth the e1ort. Let us $rst point out that we do not claim that the oscillation formula obtained with the plane wave approach should be revised. However, we have argued that this approach becomes unacceptable when all its inconsistencies are added up: the perfect knowledge of the momentum precludes spatial oscillations, observability conditions (such as *x . Losc or L . Lcoh ) are not taken into account, Bavor states are ill-de$ned, unstable oscillating particles cannot be consistently described in that model. Furthermore, the plane wave derivation requires two prescriptions: the classical propagation condition (|vt − x|t) and the equal time prescription (ti = tj ). Whereas the $rst prescription can be justi$ed in a quantum-mechanical wave packet approach, the second prescription can only be proved right in a $eld-theoretical treatment. This is an important argument in favor of the latter approach, since the di1erent time prescription (ti = tj ) leads to oscillation formulas di1ering signi$cantly from the standard result. We have also shown that an energy–momentum prescription (such as Ei = Ej or pi = pj ) is not necessary to derive the oscillation formula. Once the necessity of a more sophisticated approach is understood, we are faced with the choice between two main methods: on the one hand, a quantum-mechanical treatment associating wave packets with the propagating mass eigenstates and, on the other hand, a $eld-theoretical treatment where the oscillating particle is considered as an internal line of a Feynman diagram. The $rst possibility is rich in physical insights (such as the existence of a coherence length) but not satisfactory in many respects: Bavor states are still ill-de$ned, the equal time prescription is still needed, the nonrelativistic limit is problematic if the mass eigenstates are not nearly degenerate, the size of the wave packet is hard to estimate, the coherence length is not well-de$ned in the nonrelativistic limit, the case of a vanishing energy uncertainty is not included, and $nally the treatment is inadequate for

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203

unstable particles. Whereas most of these problems are solved in the interacting wave packet model of Giunti [138], this last model requires quantum $eld theory to compute the interactions and is as complex as the external wave packet model. Derivations of the oscillation formula resorting to $eld-theoretical methods are not very popular, although they seem the only way out. The $rst reason is that they are thought to be very complicated. The second reason is that the existing quantum $eld computations of the oscillation formula do not agree in all respects. The aim of our report was to counter both objections. The $rst of these is easily refuted by noting that the oscillation formula for a stable particle can be derived in a very simple $eld-theoretical model, the Kobzarev et al. model [187], in which the source and the detector are approximated by in$nitely heavy nuclei. This is the simplest model of all in which the oscillation formula can be consistently derived. The second objection could only be countered by a detailed and complete computation of the oscillation formula with all approximations carefully considered. In this report, we have shown that all existing $eld-theoretical treatments (with the noteworthy exception of the Blasone–Vitiello approach [72]) can be included in the so-called external wave packet model, where the oscillating particle is described as an internal line of a Feynman diagram and propagates between a source and a detector localized with in- and outgoing wave packets. In particular, we have paid attention to the case of a vanishing energy uncertainty (stationary limit) and its wave packet interpretation, so that we could prove that the Grimus–Stockinger model [150] is a subcase of the external wave packet model. Two other limits have been considered. First, we have proved that oscillations near the source can be described by the standard oscillation formula, contrary to what was claimed by Ioannisian and Pilaftsis [163]. Second, we have also shown that oscillations far from a well-localized source are also described by the standard oscillation formula, contrary to Shtanov’s claim [242]. In order to include all these cases in our formalism, it has been necessary to evaluate the amplitude with two approximation schemes: Laplace’s and stationary phase methods. As a result, the propagation range was divided into three regimes, distinguished by the dispersion (or spreading in space–time) of the amplitude. In the $rst regime, near the source, the dispersion is negligible and Laplace’s method is suKcient. This case corresponds to Ioannisian and Pilaftsis’ limit. In the second regime, the transversal dispersion becomes important, so that both approximation methods need to be used. In the third regime, far from the source, the longitudinal dispersion becomes important and the stationary phase method is suKcient. This case corresponds to Shtanov’s limit. We have also discussed another method of computation (in energy–momentum space) valid for stable oscillating particles, as it clears up in which sense interference occurs between same energy states. The computation of the intrinsic decoherence at the wave packet level has yielded two well-known observability conditions for oscillations. However, our physical explanation di1ers from the standard one by taking into account the dispersion of the amplitude, so that the origin of decoherence depends on the distance at which it occurs. The $rst condition of observability of oscillations, Losc ij *x , is either due to the initial spread of the associated ‘wave packets’, or to their constant overlap at large distance. Similarly the existence of a coherence length (yielding the second observability condition LLcoh ij ) is either due to the separation of ‘wave packets’, or to their dispersion. Our 3-dimensional treatment has also yielded a third observability condition, |p0 ×l| . *p (modi$ed into |v0 ×L| . v0 *x at short distance) which has the obvious geometrical interpretation of constraining the propagation within a cone (respectively a cylinder at short distance). It is now clear that the coherence length tends to in$nity in the stationary limit, as noted by Kiers et al. [182].

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The oscillation of unstable particles, whether mesons or neutrinos, has been analyzed in the same framework, with the result that the oscillation formula is only modi$ed, as expected, by the well-known exponential decay term. Nonexponential corrections to the oscillation formula have been estimated for the $rst time in quantum $eld theory. Finally we have given a $eld-theoretical treatment of correlated oscillations, with the aim of disproving the existence of recoil oscillations. Our work con$rms that the standard vacuum oscillation formula given by Eqs. (12) and (14) is correct if the observability conditions mentioned above are satis$ed. This fact is rather remarkable, in the light of the numerous inconsistencies which were present in the plane wave derivation of this formula. It can be explained by the following observations: • the energy and momentum present in the phase are linked by the on-shell condition, and the average propagation time and length are linked by a constraint equivalent to a wave packet localization in space–time. For these reasons, the Ej T and pj L components of the phase cancel but for the usual oscillating factor m2 L=2p; • only same energy components interfere because of the time average, so that the time-averaged transition probability computed in the external wave packet model is equivalent to the energy-integrated transition probability computed with plane waves; • from the previous argument, decoherence e1ects arising from external wave packets cannot be distinguished from decoherence e1ects arising from averages over the production (and detection) region and over the energy spectrum; since the former e1ects are dominated by the latter, they have no inBuence on the oscillation formula. For 10 years, new experimental results in neutrino physics have stimulated the research on the theory of particle oscillations, leading to the development of numerous (and sometimes conBicting) $eld-theoretical approaches. In this report, we have tried to unify these treatments and to extend them to oscillations of K and B mesons. We indeed believe that the neutrino oscillation formula is much strengthened by considering it in the same framework as the mesonic oscillation formula, whose parameters have been con$rmed by other methods such as regeneration in matter. We hope to have convinced the reader that $eld-theoretical models provide, on the one hand, the most secure foundation to the oscillation formula and give, on the other hand, a good physical understanding of oscillation phenomena, whatever the particles involved. One case needing further investigation is the decay in Bight of the source of the oscillating particle, which is important for atmospheric neutrinos and some laboratory neutrino experiments. It would also be interesting to apply a $eld-theoretical treatment to cascade decays, such as B → J= K → J= fK , where double-Bavor oscillations occur. These processes are useful for the determination of the signs of the BH BL mass and lifetime di1erences [32,179,77,233]. The $eld-theoretical approach faces its biggest challenge in the description of neutrino oscillations in matter. A $rst task consists in the rederivation of the MSW oscillation formula (see [205,214,223,87] for a few attempts). Another diKcult task is to describe the nonstationary evolution of neutrinos in supernovae or in the early universe, where interaction rates are in competition with the Bavor oscillation period [234,86]. The turn of the century marks a new golden age for CP violation experiments, thanks to the B factories, as well as the golden era of neutrino oscillation experiments. These favorable auspices should be a great stimulation for further research on the theoretical foundations of oscillations.

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Acknowledgements I am deeply indebted to Jeanne De Jaegher for a thorough reading of the manuscript, which led to lengthy discussions on the quantum mechanics of oscillations, as well as the rewriting of some sections. Discussions with Jean Pestieau, Andreas Veithen, Jacques Weyers, Carlo Giunti and Christian Cardall are gratefully acknowledged. I thank Giuseppe Vitiello and Massimo Blasone for having organized in Vietri a workshop dedicated to the theoretical foundations of particle oscillations. It is a pleasure to thank Michel and Brita Beuthe for their comments as well as for travel support. I must also express my gratitude to Karel and CUecile De Jaegher and to Attilio Rivoldini for their computer-related assistance. Finally, I thank the UniversitUe Catholique de Louvain and its Institut de Physique ThUeorique for $nancial support. Appendix A. The Jacob--Sachs theorem Let

(E; p) be a function which is distinct from zero only within certain bounds: (E; p) = 0

for 0 ¡ M12 ¡ p2 ¡ M22 and E ¿ 0 ;

with p2 = E 2 − p2 . On this interval (E; p) is taken to be in$nitely di1erentiable. Let the function I (T ) be de$ned by I (T ) = dE (E; p)G(p2 )e−iET ; where the function G(p2 ) is the complete scalar propagator in momentum space. We are going to prove that the function I (T ) has the following asymptotic behavior: √ 2 Z T →∞ ( z0 + p2 ; p)e−i z0 +p T ; I (T ) → z0 + p 2

(211)

where z0 is the pole of the integrand and Z is the residue. This section follows Ref. [166]. It will be easier to work with an overlap function depending explicitly on p2 , so we write (p2 ; p). We choose to implement the assumption of compact domain of (p2 ; p) by (p2 ; p) = (p2 − M12 )n (p2 − M22 )n M(p2 ; p) (p2 ; p) = 0

for 0 ¡ M12 ¡ p2 ¡ M22 (n ¿ 0) ;

otherwise :

The symmetrical behavior at M12 and M22 is chosen only to simplify the algebra. The function M is taken to be analytic over the domain of interest. Under the change of variable z = p2 , the function I (T ) becomes 2 √ 2 1 M2 d z(z + p2 )−1=2 (z; p)G(z)e−i z+p T : I (T ) = 2 M12 All functions in the integrand are analytically continued in the complex plane. For example, the real propagator is considered as the limit of an analytic function of the complex variable z: G(p2 ) = lim G(z) : z →p2 +i

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The full propagator associated to a scalar $eld can be written in the Schwinger’s spectral representation [241] as G(z) =

z−

M02

i ; − 6(z)

where M0 is the bare mass of the scalar $eld and 6(z) is an analytic function de$ned by ∞ *(s) ; 6(z) = z ds z −s 2 b

(212)

(213)

which can be interpreted as the self-energy of the scalar $eld. The positive real function *(x) satis$es *(x) = 0 for x 6 b2 . b2 is the invariant mass of the lightest multi-particle state in interaction with the scalar $eld corresponding to the propagator. This spectral representation has the advantage of being in direct correspondence with the full propagator (Eq. (26)), obtained in perturbation theory by an in$nite sum over the self-energy insertions. If x is on the real axis, it can be checked that 6(x + i) − 6(x − i) = −2 ix*(x) : The self-energy can be written as 6(x ± i) = M 2 + u(x) ∓ iv(x) ;

(214)

where v(x) = x*(x) and M 2 is an in$nite constant which is absorbed in the renormalized mass M 2 : M 2 = M02 + M 2 : Note that M is not the physical mass since it is not the pole of the propagator. The propagator G(z) has branch points at the thresholds corresponding to the multi-particle states. The $rst branch point is at z = b2 . We assume that the other branch points are above M22 . The analytic continuation of G(z) into the second Riemann sheet [78] is such that the analytically continued function GII (z) just below the real axis is equal to the original function just above the real axis. The analytically continued function 6II (z) of the function 6(z) is de$ned in the same way, 6II (x − i) = 6(x + i), so that GII (z) =

z−

M02

i : − 6II (z)

(215)

GII (z) has a pole at z0 = m2 − im. Di1erent cases have to be considered, according to the stability or instability of the particle, and according to the positions of the pole z0 and threshold b2 . The pole is assumed to be inside the contour (otherwise the particle cannot be emitted). A.1. Unstable particle, with b ¡ M1 ¡ m ¡ M2 The path of integration is shown in Fig. 9. The analytic expressions of the paths Pj read Pj : z = (−i! + Mj2 + p2 )2 − p2 (j = 1; 2) ;

M. Beuthe / Physics Reports 375 (2003) 105 – 218

207

Fig. 9. Integration contour if b ¡ M1 ¡ m ¡ M2 .

where ! ranges from zero to !∞ on P1 and from !∞ to zero on P2 . The analytic expression of the path P reads P : z = (−i!∞ + M 2 + p2 )2 − p2 ; where M ranges from M1 to M2 , with the limit !∞ → ∞ understood. The integral I (T ) is equal to I (T ) = J + J1 + J2 + J : J is the contribution of the pole z0 , whereas J1 , J2 and J are the contributions of P1 , P2 and P , respectively. Their analytic expressions read 1. Contribution of the pole:

√ 2 J = Z (z0 + p2 )−1=2 (z0 ; p) e−i z0 +p T : Setting XM ∼ = |m − M1; 2 |, a rough estimate of J is given by J ∼ m2n−1 (XM )2n e−T=2 . 2. Contribution of the sickle-shaped paths: ∞ √ j −i Mj2 +p2 T d! (z(!); p)GII (z(!))e−!T : Jj = i(−1) e 0

At large T , the dominant contribution to the integral comes from the ! values near zero, because of the decreasing exponential. Since the integrand tends to zero with ! ( (Mj2 ; p) = 0)), the asymptotic behavior of the integral depends on the way the integrand tends to zero. Setting y = !T and expanding in 1=T , we obtain: √ 2 2 Jj ∼ (M22 − M12 )n (Mj2 + p2 )n=2 M(Mj2 ; p)GII (Mj2 )T −(n+1) e−i Mj +p T : The corrections to this formula are of order (XMT )−(n+2) , where XM ≈ Mj − m, that is, they are of the order of the uncertainty on the particle mass. It is assumed that the function M(z; p) diverges more slowly than exp(−!T ) on the paths Pj as ! tends to in$nity. This assumption is true for a wide class of functions, in particular for Gaussians, whereas their path integral diverges on half-circles at in$nity. A rough estimate of Jj is given by Jj ∼ m2n−1 (XM )n−1 T −(n+1) .

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3. Contribution from the path at in$nity: M2 √ 2 2 2 M − !∞ T J =e dM (z(M ))GII (z(M ))e−i M +p T : M 2 + p2 M12 If M satis$es the same conditions at in$nity as above, J ∼ exp(−!∞ T ) and tends to zero as !∞ → ∞. In conclusion, the contribution of the pole is a decreasing exponential in T , whereas the contributions due to the bounded character of the energy spectrum decrease in inverse powers of T . At small T , Jj =J ∼ (XMT )−(n+1) , which is nonnegligible for XMT . 1. Below that value, the asymptotic evaluation of Jj is not valid anymore, because terms in (XMT )−(n+2) have been neglected. First, let us consider weakly decaying particles. In the case of the KS0 , the mass is measured with a precision of XM ≈ 10−2 MeV. Thus nonexponential corrections will be important for T . 10−19 s, which is not observable since the KS0 lifetime is 0:89 × 10−10 s. In the case of the B0 , XM ≈ 2 MeV, so that nonexponential corrections will be important for T . 10−22 s, which is not observable since the B0 lifetime is 1:29 × 10−12 s. Next, let us next consider resonances. In the example of S(1232), XM ≈ 2 MeV, so that nonexponential corrections are important for T . 10−22 s, which is large compared to the inverse width equal to 5 × 10−24 s. Thus the propagation of resonances can never be modelized by the contribution of the pole alone. In the case of the weakly decaying particles, power law corrections are also important at large times and dominate if T & 2(n + 1) ln(XMT ) = 2(n + 1)(ln(T ) + ln(XM=)) : For KS0 , the two-particle threshold is characterized by n = 1=2 and XM= ∼ O(1010 ), so that the threshold of the nonexponential behavior is given by T & 69, at which time the amplitude will be much too small to be observable. For B0 , taking n = 1=2 and XM= ∼ O(109 ), we obtain T & 62, with the same conclusion as above. To sum up the analysis of the propagation of a weakly decaying particle (with no multi-particle thresholds included in the spectrum), the function I (T ) is very well approximated in the observable time domain by Eq. (211). A.2. Unstable particle, with M1 ¡ b ¡ m ¡ M2 In this case, the multi-particle threshold is included in the energy spectrum. It generates new power law corrections to the amplitude. The path of integration is shown in Fig. 10. The contributions of the paths Pj are the same as in the previous case, except that G(z) replaces GII (z) on P1 , since this path lies on the $rst Riemann sheet. The asymptotic value of Jj as T → ∞ does not change. The contributions of the paths Pbj do not cancel each other out, since they do not lie on the same Riemann sheet. Their sum reads √ 2 1 Jb = d z(z + p2 )−1=2 (z; p)(GII (z) − G(z)) e−i z+p T 2 Pb ∞ √ −i b2 +p2 T = −ie d! (z(!); p)(GII (z) − G(z)) e−!T : (216) 0

M. Beuthe / Physics Reports 375 (2003) 105 – 218

209

Fig. 10. Integration contour if M1 ¡ b ¡ m ¡ M2 .

The analytic expression of the paths Pb reads Pb : z = (−i! + b2 + p2 )2 − p2 ;

(217)

where ! ranges from zero to !∞ , with the limit !∞ → ∞ understood. The function I (T ) is equal to I (T ) = J + J1 + J2 + Jb + J : The asymptotic behavior of Jb is studied in the same way as for Jj . As before, the dominant contribution to Jb , for large T , comes from ! values near zero. Again, we set y = !T and expand the integrand in 1=T . Recall that GII (z) is de$ned by the analytic continuation of G(x + i) below the cut, GII (x − i) = G(x + i). Its discontinuity through the cut can be computed with Eqs. (212), (214) and (215): 2v(x) GII (x − i) − G(x − i) = : 2 (x − M − u(x))2 + v2 (x) In the example of K 0 , the self-energy is given at $rst order in g2 by the pion bubble diagram (where g is the coupling constant between a kaon and two pions), yielding g2 1 − b2 =z ; v(z) = 4 where b = 2m is the two-pion threshold. The function u(z) is also of order g2 . The evaluation of Jb with the same asymptotic method as used for Jj gives √ (b2 + p2 )1=4 g2 2 −i b2 +p2 T (b ; p)e : Jb = −i(−2i )1=2 T −3=2 4 b (b2 − M 2 − u(b2 ))2 The corrections to this formula are of order (QT )−5=2 , where Q ≡ M − b is the energy release on decay of the unstable particle. Noting that cutting rules give v(m2 ) = m, the coupling constant g2 can be replaced by its expression in function of , m and Q. A rough estimate of Jb is then given by 1 (b2 ; p) : Jb ∼ (QT )−3=2 Q m

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At small T , the ratio Jb =J is of the order of Jb ∼ (QT )−3=2 ; J Q which is much smaller than 1 if QT 1 and =Q1. This is the case for weakly decaying particles. For example, =Q ≈ 10−14 for KS0 . It is not true for resonances. For example, =Q ≈ 0:8 for S(1232). Below the value QT ≈ 1, the asymptotic computation of Jb is not valid anymore since corrections in (QT )−5=2 have been neglected. At large T , the power law contribution Jb dominates the pole contribution if T − 3 ln(T ) & 5 ln(Q=) ; so that the nonexponential time thresholds are given by S T & 165 for KS0 , L T & 202 for KL0 0 and L; H T & 157 for BL; H (in the last case, we have taken Q = 1 MeV). Thus nonexponential e1ects are not observable at large times for weakly decaying particles. In the case of resonances, the contribution of Jb is always of the same order or larger than the pole contribution. Resonances do not propagate macroscopically, since their width is of the same order as the typical energy of the processes. To sum up the analysis of the propagation of a weakly decaying particle, including the multiparticle thresholds, the function I (T ) is very well approximated in the observable time domain by Eq. (211). A.3. Stable particle, with M1 ¡ m ¡ M2 ¡ b This case can be examined like the $rst one, except that the pole is real, so that the result is given by Eq. (211), with = 0. There is no exponential decrease of the amplitude. Power law corrections are negligible at large T . A.4. Stable particle, with M1 ¡ m ¡ b ¡ M2 This case can be examined like the second one, except that the pole is real. The result is again given by Eq. (211), with = 0. The ratio Jb =J is of the order of Jb =J ∼ (mT )−3=2 g2 =Q2 , where g is the coupling constant with the particles produced at the threshold. The contribution of Jb is too small to be observed at small T in the current experiments, and is negligible at large T . References [1] A.I. Abazov, et al., Search for neutrinos from Sun using the reaction 71 Ga(e ; e− )71 Ge, Phys. Rev. Lett. 67 (1991) 3332. [2] [SAGE Collaboration], J.N. Abdurashitov, et al., Measurement of the solar neutrino capture rate by SAGE and implications for neutrino oscillations in vacuum, Phys. Rev. Lett. 83 (1999) 4686. [3] [Belle Collaboration], K. Abe, et al., Observation of large CP violation in the neutral B meson system, Phys. Rev. Lett. 87 (2001) 091802. [4] [Belle Collaboration], K. Abe, et al., An improved measurement of mixing-induced CP violation in the neutral B meson system, Phys. Rev. D 66 (2002) 071102.

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Physics Reports 375 (2003) 219 – 326 www.elsevier.com/locate/physrep

Electroweak radiative corrections in high energy processes Michael Melles Paul Scherrer Institut (PSI), CH-5232 Villigen, Switzerland Received 10 October 2002 editor: R. Petronzio

Abstract Experiments at future colliders will attempt to unveil the origin of electroweak symmetry breaking in the TeV range. At these energies the Standard Model (SM) predictions have to be known precisely in order to disentangle various viable scenarios√such as supersymmetry and its manifestations. In particular, large logarithmic corrections of the scale ratio s=M , where M denotes the gauge boson masses, contribute signi2cantly up to and including the two loop level. In this paper we review recent progress in the theoretical understanding of the electroweak Sudakov corrections at high energies up to subleading accuracy in the SM and the minimal supersymmetric SM (MSSM). We discuss the symmetric part of the SM Lagrangian at high energies yielding the e5ective theory employed in the framework of the infrared evolution equation (IREE) method. Applications are presented for important SM and MSSM processes relevant for the physics program of future linear colliders including higher order purely electroweak angular dependent corrections. The size of the higher order subleading electroweak corrections is found to change cross sections in the several percent regime at TeV energies and their inclusion is thus mandatory for predictions of high energy processes at future colliders. c 2002 Elsevier Science B.V. All rights reserved. PACS: 12.15.−y

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Unbroken gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Sudakov double logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Gribov’s factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Infrared evolution equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E-mail address: [email protected] (M. Melles). c 2002 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 5 5 0 - 1

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2.4. Subleading corrections from splitting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Anomalous scaling violations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Renormalization group corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Virtual corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2. Real gluon emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Broken gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Fermions and transverse degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. The equivalence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Longitudinal degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Semi-inclusive cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Electroweak RG corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Angular dependent corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Sudakov logarithms in softly broken supersymmetric models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Fully inclusive cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Physical 2elds at 2xed order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1. One loop results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2. Two loop results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Gauge boson production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Higgs production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Charged MSSM Higgs production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Heavy quark production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Operators and quantum numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1. Symmetric and physical gauge 2elds and gauge couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2. Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.3. Explicit values for Y , Q, T 3 , C, (I A )2 , (I Z )2 , (I W )2 , C ew , and I ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Electroweak Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

236 240 242 243 246 251 252 261 263 268 269 273 278 282 283 284 287 292 293 296 301 304 309 309 310 310 310 311 312 314 324

1. Introduction The Standard Model (SM) of particle physics [1–5] has enjoyed unprecedented success over the last decades. The discovery of the top quark at the Tevatron [6,7] leaves the Higgs particle [8–11] as the last undiscovered ingredient to complete the SM. While it is possible that the SM remains valid up to energies far beyond experimental reaches, most theorists view the SM as an e5ective theory which is embedded in a larger theory usually containing uni2cation of the gauge interactions at a high scale MGUT . This expectation seems well motivated due to the presence of light neutrino masses established at Super-Kamiokande [12] in connection with a seesaw mechanism involving MGUT . Also coupling uni2cation in the minimal supersymmetric SM (MSSM) points to the existence of a higher scale in nature where the forces unify. If, however, the SM is the e5ective low energy theory of a more complete and uni2ed theory at MGUT , the hierarchy problem must be taken seriously. Supersymmetry is able to stabilize the quadratic

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221

divergences in the scalar sector by canceling these terms with the corresponding superpartner loop divergences if, and only if, the superpartner mass splittings are not much larger than the weak scale. Another possibility currently discussed is that there are large extra dimensions at the TeV scale [13,14], however, such a scenario only trades one problem (the existence of a large scale MGUT ) for another (the existence of large extra dimensions of the “right” size). In any case, while the SM works extremely well, it does not explain electroweak symmetry breaking (EWSB). A negative mass squared is introduced by hand in the SM, but in the larger theory the reason for EWSB is expected to be dynamical such as in typical SUGRA models [15]. While many possible extensions of the SM exist, only experiments at future colliders will shed light on the origin of EWSB expected to lie in the TeV regime. At this point a few general remarks about the usage of the expression EWSB are appropriate in order to not be misleading. It has been known for some time now that the Higgs mechanism does not lead to a breaking of the local gauge invariance on the lattice [16]. In general, all vacuum expectation values (v.e.v.’s) of gauge dependent operators (such as 0| (x)|0) can be shown to vanish. As was pointed out in Ref. [17], the crucial point about the continuum version in the conventional perturbative formulation of the Higgs mechanism is not as much the existence of a v.e.v. v, but rather the existence of a non-trivial orbit minimizing the Higgs-potential. The apparent breaking of the original symmetry by v is due to it being a gauge choice (which always breaks the gauge symmetry). In other words, if we were to reformulate the full theory in terms of only gauge invariant operators, then no symmetry breaking would be visible (but of course new operators would occur describing, for instance, the di5erent masses of the electroweak gauge bosons). Since it was also shown in Ref. [17] that the di5erence between the manifestly gauge invariant picture and the conventional perturbative formulation vanishes for observables in the small coupling limit, we prefer to use the standard terminology and operators in which the original symmetry is hidden. It is in this sense, that we use the expression “broken gauge theory” below. The high precision measurements of SLC/LEP have limited the room for extensions of the SM considerably and in general, they cannot deviate from the SM to a large extent without evoking so-called conspiracy e5ects. It would therefore be very desirable to have a leptonic collider at hand in the future in order to answer questions posed by discoveries made at the LHC and possibly the Tevatron. In particular, if only a light Higgs is discovered, say at 115 GeV, then it is mandatory to investigate all its properties in detail to experimentally establish the Higgs mechanism including a possible reconstruction of the potential and of course of the Yukawa couplings. In addition one would have to look for additional heavy Higgs-bosons which could easily escape detection at the hadronic machines, but can be discovered at the -option at TESLA [18–20] up to masses reaching 80% of the c.m. energy. If any supersymmetric particle would be found in addition, it is necessary to clarify and/or test the relations between couplings and properties of all new particles in as much detail as possible in a complementary way to what would already be known by that time. The overall importance of leptonic colliders would thus be to clarify the physics responsible for the EWSB which in turn means it must be a high precision machine. On the theory side this means that e5ects at the 1% level should be under control in both the SM as well as all extensions that are viable at that point. The purpose of the present work is to summarize the recent activities and results relevant on this level of precision from electroweak radiative corrections at energies much larger than the gauge boson masses and to apply these corrections to processes relevant to the linear collider program. This does not mean that the corrections are

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negligible for hadronic machines, however, for the high precision illustrations we focus here on e+ e− machines in the TeV range. At the expected level of precision required to disentangle new physics e5ects from the SM in the O(6 1%) regime, higher order electroweak radiative corrections cannot be ignored at energies in the TeV range. As a consequence, there has been a lot of interest recently in the high energy behavior of the SM [21–29]. The largest contribution is contained in electroweak double logarithms (DL) of the Sudakov type and a comprehensive treatment of those corrections is given in Ref. [28] to all orders. The e5ects of the mass-gap between the photon and Z-boson has been considered in recent publications [30,31] since spontaneously broken gauge theories lead to the exchange of massive gauge bosons. In general one expects the SM to be in the unbroken phase at high energies. There are, however, some important di5erences of the electroweak theory with respect to an unbroken gauge theory. Since the physical cuto5 of the massive gauge bosons is the weak scale M ≡ MW ∼ MZ ∼ MH , pure virtual corrections lead to physical cross sections depending on the infrared “cuto5”. Only the photon needs to be treated in a semi-inclusive way. Additional complications arise due to the mixing involved to make the mass eigenstates and the fact that at high energies, the longitudinal degrees of freedom are not suppressed. Furthermore, since the asymptotic states are not group singlets, it is expected that fully inclusive cross sections contain Bloch–Nordsieck violating electroweak corrections [33]. It has by now been established that the exponentiation of the electroweak Sudakov DL calculated in Ref. [28] via the infrared evolution equation method (IREE) [34,35] with the 2elds of the unbroken phase is indeed reproduced by explicit two loop calculations with the physical SM 2elds [30,31,36]. One also understands now the origin of previous disagreements. The results of Ref. [26], based on fully inclusive cross sections in the photon, is not gauge invariant as already pointed out in Ref. [28]. The factorization used in Ref. [25] is based on QCD and e5ectively only takes into account contributions from ladder diagrams. In the electroweak theory, the three boson vertices, however, do not simply cancel the corresponding group factors of the crossed ladder diagrams (as is the case in QCD) and thus, infrared singular terms survive for left handed fermions (right handed ones are e5ectively Abelian) in the calculation of Ref. [25]. The IREE method does not encounter any such problems since all contributing diagrams are automatically taken into account by determining the kernel of the equation in the e5ective regime above and below the weak scale M . It is then possible to calculate corrections in the e5ective high energy theory in each case yielding the same result as calculations in the physical basis. Thus, the mass gap between the Z-boson and the photon can be included in a natural way with proper matching conditions at the scale M . For longitudinally polarized gauge bosons it was shown in Ref. [37,38] that the leading and subleading (SL) kernel can be obtained from the Goldstone boson equivalence theorem. We specify in the next section how the high energy e5ective theory is obtained from the SM and illustrate the approach followed in the main part of this work. 1.1. The standard model The complete classical Lagrangian Lclass of the electroweak SM (EWSM) reads in terms of the physical 2elds, i.e. the mass and charge eigenstates A ; Z ; W± ; H; l; ; u, and d, the would-be Goldstone 2elds ± and , and the physical parameters e; MW ; MZ ; MH ; Mf , and V, as

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follows [39]: Lclass =

f=l;;u;d

+

i

i

e [I 3 fQ L fiL − sw2 Qf fQi fi ]Z sw cw W; f i

√

e [uQLi Vij dLj W+ + dQ Li Vij† ujL W− ] 2sw

√

e [QLi lLi W+ + lQLi Li W− ] 2sw

i; j

+

[fQi (iE − mf )fi − eQf fQi fi A ]

i

f=l;;u;d

+

223

− 14 |9 A − 9 A − ie(W− W+ − W− W+ )|2 2 1 cw − + − + − 9 Z − 9 Z + ie (W W − W W ) 4 sw 2 1 cw + + + + + + − 9 W − 9 W − ie(W A − W A ) + ie (W Z − W Z ) 2 sw 2 1 e e − + + 9 (H + i) − i W + iMZ Z + i Z (H + i) 2 sw 2cw sw 2 2 2 c e − s w w + + + + + + 9 + ieA − ie Z − iMW W − i W (H + i) 2cw sw 2sw 1 MH2 1 − MH2 H 2 − e H − + + |H + i|2 2 2sw MW 2 2 1 MH2 2 − + 2 + |H + i| −e 2 2 8sw2 MW mf; i 3 Q − e (fQi fi H − 2IW; f ifi 5 fi ) 2s M w W i f=l;;u;d

+

i; j

−

i

e √ 2sw √

md; j L mu; i R (uQ i Vij dLj + + dQ Li Vij† ujR − ) − (uQ i Vij dRj + + dQ Ri Vij† ujL − ) MW MW

e ml; j L R + QR L − (Qi li + li i ) : 2sw MW

(1)

The quantization of the EWSM requires the introduction of a gauge-0xing term and of Faddeev–Popov 0elds. We introduce a gauge-2xing term of the form 1 1 1 + − (C A )2 − (C Z )2 − C C (2) L2x = − 2A 2Z W

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with linear gauge-2xing operators C ± = 9 W± ∓ iMW W ± ; C Z = 9 Z − MZ Z ; C A = 9 A :

(3)

This general linear gauge contains 2ve independent gauge parameters a ; a=A; Z; ±, and a ; a=Z; ±, ( ) . where (±) ≡ W For W = W and Z = Z the terms involving the would-be Goldstone 2elds in (3) cancel the mixing terms V 9 in the classical Lagrangian (1) up to irrelevant total derivatives. This gauge is called ’t Hooft–Feynman gauge and is used in the following if not stated otherwise. The corresponding Faddeev–Popov ghost-2eld Lagrangian reads cw + 2 + + Z u + Lghost = −uQ (9 9 + W MW )u + ie(9 uQ ) A − sw cw Z + + A − ie(9 uQ )W u − u sw 1 c2 − sw2 Z − eMW W uQ + (H + i)u+ − + uA − w u 2sw 2cw sw − (u+ → u− ; W + → W − ; + → − ; i → −i) cw Z (9 uQ )(W+ u− − W− u+ ) sw 1 1 Z Z + − − + Hu − ( u + u ) − eMZ Z uQ 2cw sw 2sw

− uQ Z (9 9 + Z MZ2 )uZ − ie

− uQ A 9 9 uA + ie(9 uQ A )(W+ u− − W− u+ ) :

(4)

Adding up all terms (1), (2) and (4) we obtain the complete Lagrangian of the EWSM suitable for higher-order calculations, LGSW = Lclass + L2x + Lghost

(5)

The Feynman rules which can be derived from the Lagrangian in Eq. (5) are given in Appendix A.2 in the ’t Hooft–Feynman gauge for the physical 2elds of the broken gauge theory. At high energies and for processes that are not mass suppressed or dominated by resonances we can neglect particle masses and terms connected to the vacuum expectation value (v.e.v.) of the broken gauge theory to the level of SL accuracy [40]. Thus, instead of the Lagrangian in Eq. (5) we use a high energy approximation of Lsymm which is based on the 2elds of the unbroken phase in the symmetric basis and neglect all terms with a mass dimension. It is composed of a Yang–Mills part, a Higgs and a fermion part which are given by [41] LYM = − 14 (9 Wa − 9 Wa + g#abc Wb Wc )2 − 14 (9 B − 9 B )2 ;

(6)

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225

where #abc is the totally antisymmetric tensor of SU(2). The Higgs part consists of a single complex scalar SU(2) doublet 2eld with hypercharge Y = 1: +

(x) &(x) = (7) 0 (x) with 0 (x)= √12 (H (x)+i(x)) and where the v.e.v. is neglected. + ; − and denote the would-be Goldstone bosons and H the physical Higgs 2eld. They couple to the gauge 2elds via LH = (D &)† (D &) ;

(8)

where we omit the self coupling part (the potential) and the covariant derivative is given by Y (9) D = 9 − igT a Wa + ig B : 2 The left handed fermions transform as doublets and the right handed ones as singlets under the gauge group. The fermionic part of the symmetric Lagrangian is then given by LF = (LQj i D Lj ) + (RQ j i D Rj ) − (LQj Gjl Rl & + h:c:) : (10) j

j

j;l

The covariant derivative acting on right handed 2elds contains no term proportional to g. The Yukawa coupling matrices are denoted by Gjl noting that for up-quarks, the charge conjugated Higgs 2eld must be used. The high energy e5ective symmetric part of the Lagrangian is then given by Lsymm = LYM + LH + LF + L2x + LFP ;

(11)

where the corresponding ghost and gauge 2xing terms are given by L2x = − 12 [(F W )2 + (F B )2 ]

(12)

1 F W = √ 9 Wa ; W

(13)

with 1 F B = √ 9 B B

and LFP = uQ - (x)

.F - 0 u (x) ; ./ 0 (x)

(14)

where .F - =./ 0 (x) is the variation of the gauge 2xing operators F - under the in2nitesimal gauge transformations characterized by / 0 (x). The Faddeev–Popov ghosts are denoted by u- (x). The corresponding Feynman rules are thus analogous to a theory with an unbroken SUL (2) × UY (1) and fermions or scalars in the fundamental representation, respectively. The new ingredient in LF is the Yukawa term. In addition we have in the gauge boson sector the coupling of the gauge bosons to scalars through the covariant derivative in LH . This e5ective regime corresponds to region (I) in Fig. 1 where the wavy line separates the transverse sector (analogous to an unbroken gauge theory) and the scalar sector, where for the would-be Goldstone bosons the equivalence theorem (E.T.) must be used. Note that all gauge bosons contained in Lsymm are massless and an infrared cuto5 will treat Wa and B 2elds in the same way.

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s

f,γ, ZT,WT

|k⊥|

H,φ (E.T.)

SUL(2)xUY(1) I (m=0)

M |k⊥|

QED II (m)

λ √ Fig. 1. The schematic depiction of the e5ective high energy regimes ( sM 3) in the framework of the infrared evolution equation method. In region (I), the high energy corrections are obtained e5ectively in the unbroken SUL (2) × UY (1) theory described by Lsymm in Eq. (11) where all terms connected to the v.e.v. can be neglected to SL accuracy. For external fermions and transverse gauge bosons this picture contains at the subleading level Yukawa interactions and contributions from Higgs doublets to the anomalous scaling violations. For external longitudinal gauge bosons ( ={ + ; − ; }) the equivalence theorem (E.T.) is employed yielding e5ectively a scalar theory charged under the unbroken gauge group. Again this scenario contains at the subleading level Yukawa terms introduced by the spontaneous symmetry breaking. For all charged particles, the soft photon e5ects, regulated here by a 2ctitious photon mass 3, are included by integrating in region (II) which incorporates pure QED e5ects including mass terms. In the calculation 3 is replaced by a cuto5 on the exchanged |k⊥ |. The matching condition is given by the requirement that the high energy solution in region (I) is obtained if the infrared cuto5 is chosen to be the gauge boson mass M .

In the following we always use the ’t Hooft–Feynman gauge. For the high energy regime, the Lagrangian in Eq. (11) is convenient since it allows for an approach via the IREE method [34,35] described in Section 3 and thus, for a consistent treatment of higher order SL corrections at high energies. In Ref. [40] it was proven that Eq. (11) at one loop to SL accuracy gives the same results as calculations based on the physical Lagrangian in Eq. (5). The approach in Ref. [40] uses collinear Ward identities to show that SL contributions from the v.e.v. part of the Lagrangian (5) do not contribute additional terms not already contained in Lsymm . In particular this means that for longitudinal degrees of freedom at high energy we can employ the Goldstone boson equivalence theorem and to SL accuracy, we treat the would-be Goldstone bosons + ; − and as physical degrees of freedom in the ultrarelativistic limit. At higher orders, all terms related to the renormalization of the Goldstone bosons are sub-subleading (SSL). √ Fig. 1 also indicates that this approach is only valid in the high energy regime sM and that the QED corrections from below the weak scale must be included by appropriate matching conditions at M . Thus the overall approach consists of identifying the relevant degrees of freedom in regions (I) and (II), integrating out the contributions to SL accuracy and by matching the solution found in II in such a way that at the weak scale M the solution in region (I) is reproduced. 1.2. Organization of the paper The paper is organized as follows. In Section 2 we summarize the various ingredients needed to calculate SL virtual corrections in unbroken gauge theories. While these corrections do not lead to

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227

physical observables in those theories, the IREE approach allows for an application of the results of Section 2 to broken gauge theories in the high energy limit in Section 3. The QED e5ects from the region below the weak scale are implemented with the appropriate matching conditions. As mentioned above, we use the term “broken gauge theories” in the sense that the local symmetry is hidden due to the degeneracy of the vacuum ground state and thus not evident in the physical states. The associated local BRST relations, however, still hold [16,17]. In Section 4 the results summarized in Section 3 are applied to speci2c processes relevant to a future linear collider program. In particular the importance of the higher than one loop corrections is emphasized. We present our concluding remarks in Section 5 and discuss lines of future work needed for precision predictions at future TeV colliders. 2. Unbroken gauge theories In this section we summarize the results obtained for virtual corrections in unbroken gauge theories at high energies. These contributions will be crucial for the high energy regime of the SM in Section 3. 2.1. Sudakov double logarithms The high energy asymptotics of electromagnetic processes was calculated many years ago within the framework of QED [42]. In particular the amplitude for e+ e− elastic scattering at a 2xed angle (s ∼ |t| ∼ |u|m2 32 , where m is the electron and 3 a 2ctitious 1 photon mass) in the DL approximation has the form s m2 2 M = MBorn 5 ; (15) ; m2 3 2 where MBorn is the Born amplitude for e+ e− scattering and 5 is the Sudakov form factor. The DL approximation applies in the energy regime e2 e2 s m2 2 s log ∼ log log ∼1 ; 462 m2 462 m2 32

(16) √ where the QED coupling e2 =461. Thus each charged external particle e5ectively contributes 5 to the total amplitude. The Sudakov form factor appears in the elastic scattering of an electron o5 an external 2eld [42]. It is of the form s m2 e2 s m2 5 = exp − : (17) ; R ; m2 3 2 862 m2 3 2 To specify R it is convenient to use the Sudakov parametrization of the momentum of the exchanged virtual photon: k = vp1 + up2 + k⊥ 1

(18)

3 plays the role of the infrared cut-o5. In physical cross sections the divergence in 3 of the elastic amplitude is canceled with the analogous divergences in processes with soft photon emissions.

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for massless fermions and m2 m2 p 2 + u p2 − p1 + k⊥ k = v p1 − s s

(19)

for massive fermions. p1 and p2 are the initial and 2nal momenta of the scattered electron and in the following we denote the Euclidean component 2 2 k⊥ = −k⊥ ¿0 ;

(20)

R(s=m2 ; m2 =32 ) can then be written as the integral over u and v after rewriting the measure as d 4 k = d 2 k⊥ d 2 k with 2 2 d 2 k⊥ = |k⊥ |d|k⊥ |d = 12 dk⊥ d = 6dk⊥ ;

(21)

s du dv ; (22) 2 where we turn the coordinate system such that the p1 ; p2 plane corresponds to 0; x and the y; z coordinates to the k⊥ direction so that it is purely spacelike (see Eq. (20)). The last equation follows from pi2 = 0, i.e. pi2x ≈ pi20 and d 2 k = |9(k 0 ; k x )=9(u; v)|du dv ≈

(p10 p2x − p20 p1x )2 ≈ (p10 p20 − p2x p1x )2 = (p1 p2 )2 = (s=2)2 : Integrating according to the DL phase space of Fig. 2 (where plays the role of 3): 1 1 1 1 s m2 /(suv − 32 ) = ; du dv R 2 v=s 2 u=s m2 3 2 u + m v + m 0 0 1 1 du dv /(u − m2 v=s)/(v − m2 u=s)/(suv − 32 ) ; ≈ u 0 v 0

(23)

(24)

where s ∼ |t| ∼ 2p1 p2 . The 2rst two factors in the integrand correspond to the propagators of the virtual fermions which occur in the one-loop triangle Sudakov diagram. The /-function appears as a result of the integration of the propagator of the photon over its transverse momentum k⊥ : k2

i i i 2 = + 6.(suv − 32 − k⊥ ) →P 2 2 2 2 − 3 + i# suv − 3 − k⊥ + i# suv − 32 − k⊥

(25)

writing it in form of the real and imaginary parts (the principle value is indicated by P). The latter does not contribute to the DL asymptotics and at higher orders gives subsubleading contributions. We note that the main contribution comes from the region near the photon mass shell: 2 suv = 32 + k⊥ :

To DL accuracy Eq. (24) gives for 3m: s m2 1 s m2 2 s R log = ; + log log ; m2 3 2 2 m2 m2 32

(26)

(27)

where the result comes equally from two di5erent kinematical regions, vu and uv as is evident from Fig. 2. Therefore one can write R = 2r.

M. Melles / Physics Reports 375 (2003) 219 – 326

229

Virtual Sudakov DL-Phase Space 1

v

µ m

µm s 2 µ s 0

2

µ m

µ µm s s

1

u Fig. 2. The virtual Sudakov DL-phase space in massive QED for the function R in the {u; v} representation. The cuto5 plays the role of 3 for m. The shaded area is the region of integration and is symmetric with respect to u and v. For ¿ m the relevant phase space is mass independent.

We can obtain physical insight by presenting the two equal contributions separately. In the 2rst region, with vu, the virtual photon is emitted along p1 and the parameter v is given by the ratio of energies of the photon and the initial electron. Here instead of u, it is convenient to use Eq. (26) to replace it by the square of the transverse momentum component of the photon. Then integrating 2 over v and k⊥ according to the DL phase space in Fig. 3 gives 1 2 s 2 min(|k⊥ |=m;1) 2 2 dk⊥ dk⊥ s m dv sv dv (28) r ; 2 = √ 2 2 √ 2 2 2 m 3 v k⊥ + m v 3= s v 3 2 k⊥ 32 | k⊥ | = s in the DL approximation, which may be evaluated to give half of R. The quantity r is proportional to the probability √ wi of the emission of a soft and almost collinear photon from an external particle with energy s and mass mi , i.e. e2 s m2i 2 wi (s; 3 ) = 2 r : (29) ; 46 m2i 32 Q f, Q then analogous If several charged particles participate in a process, for example e+ e− → fff contributions appear for each external line, provided all external invariants are large and of the same order. This leads to the general result

n 1 2 M = MBorn exp − wi (s; 3 ) ; (30) 2 i=1 where n is the number of external lines corresponding to charged particles. In summary the soft emissions described by the Sudakov form factor is a quasi-classical e5ect which does not depend on

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M. Melles / Physics Reports 375 (2003) 219 – 326 Virtual Sudakov DL-Phase Space

v

1

µ

2 2

m

2

m s

2

µ s 0

µ

2

2

m

k 2⊥

s

2 Fig. 3. The virtual Sudakov DL-phase space in massive QED for the function R in the {k⊥ ; v} representation. The shaded area is the region of integration. For m the cuto5 plays the role of 3 in the text. For ¿ m the relevant phase space is mass independent as in the {u; v} representation of Fig. 2.

the hard dynamics of the process. In particular there are no quantum mechanical interference e5ects in the DL Sudakov corrections for large scattering angles. 2.2. Gribov’s factorization theorem In this section we discuss a factorization theorem due to Gribov [43–46]. It was originally derived in bremsstrahlung o5 hadrons at high energies in the context of QED but can appropriately be extended to non-Abelian gauge theories. We follow the original derivation for real emission processes noting that form of factorization of virtual corrections must be analogous due to the KLN theorem [47,48]. Consider bremsstrahlung o5 a fermion with mass m in the laboratory system. We denote the invariants according to the notation depicted in Fig. 4 as follows: s = (p1 + p2 )2 ;

t2 = (p2 − p2 )2 :

(31)

The usual eikonal argument is that for 2p1 km2 , only the diagram on the l.h.s. is large, yielding (neglecting m in M ): # ∗ p1 M (s ; t2 ) : M=− (32) p1 k At large energies we have 2p1 k = 2|p1 | |k|(1 − cos /) +

|k| 2 m : |p1 |

(33)

M. Melles / Physics Reports 375 (2003) 219 – 326

p1

k

p1’

p1

231

p1’ k

p2

p2’

p2

p2’

Fig. 4. Bremsstrahlung in a process involving charged fermions. At high energies only the external legs contribute to DL accuracy.

Thus for M to be large we need in any case: |k|=|p1 |1. It follows that the condition |p1 | |k|/ 2 m2 should be ful2lled for small emission angles /. Gribov observed, however, that M is large in broader region! In the region 2p1 k ¿ m2 ; |k|=|p1 |1 we have with p1 = (p10 ; 0; 0; |p1 |); k = (k 0 ; k 0 sin /; 0; k 0 cos /) and #∗ = (0; cos /; ±i; −sin /): M=−

2 M (s ; t2 ) |k|/

(34)

i.e. a suScient condition is: |k⊥ | ≈ |k|/m. The proof proceeds as follows: M is taken on the mass shell in order to ensure gauge invariance! In covariant form the conditions read: 2 2p1 k 4p1 kp2 k 2p2 k 2 = 1; 1; k⊥ m2 : s s s We can then write the amplitude in a gauge invariant form ∗ # p2 # ∗ p 1 − M (s; t) M= p2 k p1 k

(35)

(36)

and write it in the following way: M = p1 M1 + p2 M2 + q M3 ;

q = p2 − p2 :

(37)

Gauge invariance yields: p1 kM1 + p2 kM2 + qkM3 = 0 :

(38)

At high energies t2 − t1 = 2qk = (2kp1 =s)2qp2 + (2kp2 =s)2qp1 + 2|k⊥ | |q⊥ | ≈ 2|k⊥ | |q⊥ |. Thus, 2qk is small, so that there is no need to distinguish between s; s and between t1 ; t2 in the inner on-shell amplitude M of Eq. (36). Eq. (38) then reads p1 kM1 = −p2 kM2 : 2

We consider here only the case of initial state radiation in analogy to Refs. [43,44].

(39)

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k

p1

p1’

p1

p2’

p2

k

p1’

κ1 p2

p2’

Fig. 5. Higher order splittings determining the imaginary part of the scattering amplitude. At high energies, Gribov proofed that the pole terms in the variables 2pi k do not dominate the amplitude and that the large terms factorize with respect to 1=|k⊥ |.

Thus M (s; t) − p2 kT (s; t; p1 k; p2 k) ; p1 k M (s; t) − p1 kT (s; t; p1 k; p2 k) ; M2 = p2 k M1 = −

(40) (41)

where the only pole of M1 is at p1 k, that of M2 at p2 k. M3 and T have no singularity at p1 k or p2 k, and p2 p1 M = (42) − [M (s; t) + p1 kp2 kT ] + q M3 ; p2 k p1 k 1 2 2 ∗ M (s; t) + k⊥ sT + #∗ qM3 : M = # M = − (43) |k⊥ | 4 Since M3 and T are functions of p1 k; p2 k they could be of order of the 2rst term. To show that this is not the case, consider the imaginary part (discontinuity) of M in p1 k; p2 k: Im M = − 12 |k⊥ |s Im T + #∗ q Im M3 :

(44)

It is determined by all possible splittings (not in s; t1 ; t2 ) like the ones depicted in Fig. 5. The simplest two particle intermediate state contains the amplitude ∗ # @1 # ∗ p1 2 − Ms ((p1 + k)2 ; q ) ; Ms = (45) @1 k p1 k where @1 is the momentum of the charged particle. As p1 is large and (p1 + k)2 s; |1 | is also large and along p1 : q 1 p1 ≈ + ⊥ ; @1 k p1 k @1 k

(46)

is the component of perpendicular to p . Thus where q⊥ 1 1

Ms =

# ∗ q⊥ 2 Ms ((p1 + k)2 ; q ) : @1 k

(47)

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233

We therefore observe that the large terms cancel! For higher splittings the cancellation proceeds analogously since in the intermediate states all particles formed at high energies are parallel to the original particle momentum. Thus, the large contributions to the original bremsstrahlung amplitude are given by M=−

2 M (s; t) |k⊥ |

(48)

2 for 2p1 ks; 2p2 ks and k⊥ 6 m2 . The cuto5 is introduced for later convenience. An anal2 2 ogous factorization in 1=k⊥ then holds for virtual corrections with k⊥ ¿ 2 since the sum of real and virtual corrections must be independent of the infrared cuto5. In order to treat non-Abelian gauge theories we need to introduce a gauge invariant cuto5 on all virtual particles with momentum @i :

2i⊥ ¿ 2 ;

@i = vi p1 + ui p2 + @i⊥

(49)

and it is understood that AQCD in order to remain in the perturbative regime. The crucial point now is that determines both the positions of the thresholds in the variables 2p1 k; 2p2 k and the minimum momentum transfers [49]: (p1 − k)2 = (@1 + @2 )2 u2 u1 2 2 2 + 2⊥ 1 + − k⊥ = 1⊥ 1 + ¿ 42 u1 u2

(50)

2 for k⊥ 2 . Thus the cut starts from 42 and

|(k − q1 )2 | = (q1⊥ − k⊥ )2 − s(u1 − u)(v1 − v) ¿ 2 :

(51)

Now the same dispersive arguments are applicable to QCD as they were in QED. Thus we can consider again the simplest situation, when the additional soft gauge boson is emitted in the process with all invariants large. Of course, for the emission of a boson almost collinear to the particle the direction of the particle with momentum pi , the invariant 2kpi is small in comparison with s. In the case of non-Abelian gauge theories the corresponding amplitude for the emission of 2 a soft gauge boson with small k⊥ 2 has, according to the Gribov theorem as derived above, the following form in non-Abelian theories: Ma (p1 ; : : : ; pn ; k; 2 ) =

n j=1

gs

# ∗ pj a T (j)M(p1 ; : : : ; pn ; 2 ) : kpj

(52)

where gs denotes the QCD (or SU (N )) coupling. The possible corrections to this factorized expres2 sion are of the order of k⊥ =2 . However, to DL accuracy, we can substitute 2 in the arguments of 2 the scattering amplitudes by its boundary value k⊥ . Notice that the amplitude on the r.h.s. of (52) is taken on-the-mass shell, which guarantees its gauge invariance. The result (52) is highly non-trivial in the Feynman diagram approach. It means, that the region of applicability of the classical formulas for the Bremsstrahlung amplitudes is signi2cantly enlarged at high energies. The form of the virtual factorization and the subsequent resummation is the topic of the following section.

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pj

p1

k

p2

pl

Fig. 6. Feynman diagrams contributing to the infrared evolution equation (54) for a process with n external legs. In a general covariant gauge the virtual gluon with the smallest value of k⊥ is attached to di5erent external lines. The inner scattering amplitude is assumed to be on the mass shell.

2.3. Infrared evolution equations Sudakov e5ects have been widely discussed for non-Abelian gauge theories, such as SU (N ) and can be calculated in various ways (see, for instance, [50–58]). We consider here the scattering amplitude in the simplest kinematics when all its invariants slj = 2pl pj are large and of the same order slj ∼ s. A general method of 2nding the DL asymptotics (not only of the Sudakov type) is based on the infrared evolution equations describing the dependence of the amplitudes on the infrared cuto5 of the virtual particle transverse momenta [34,35]. This cuto5 plays the same role as 3 in QED, but, unlike 3, it is not necessary that it vanishes and it may take an arbitrary value. It can be introduced in a gauge invariant way by working, for instance, in a 2nite phase space volume in the transverse direction with linear size l ∼ 1=. Instead of calculating asymptotics of particular Feynman diagrams and summing these asymptotics for a process with n external lines it is convenient to extract the virtual particle with the smallest value of |k⊥ | (k⊥ ⊥ | of the other virtual particles are much pj ; pl ) in such a way, that the transverse momenta |k⊥ bigger 2 2 : k 2⊥ k⊥

(53)

2 plays the role of the initial infrared cut-o5 2 . For the other particles k⊥ In particular, the Sudakov DL corrections are related to the exchange of soft gauge √ bosons, see Fig. 6. For this case the integral over the momentum k of the soft (i.e. |k 0 | s) virtual boson with the smallest k⊥ can be factored o5, which leads to the following infrared evolution equation: n p j pl d4 k i gs2 2 M(p1 ; : : : ; pn ; ) = MBorn (p1 ; : : : ; pn ) − 4 2 2 2 (26) s k⊥ 2 k + i (kpj )(kpl ) j;l=1; j =l

2 ×T a (j)T a (l)M(p1 ; : : : ; pn ; k⊥ );

(54)

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235

2 where the amplitude M(p1 ; : : : ; pn ; k⊥ ) on the right hand side is to be taken on the mass shell, but 2 2 with the substituted infrared cuto5: → k⊥ . From Eq. (18) and the on-shell condition (26) it is 2 clear that pj pl =(kpj )(kpl ) = 2=k⊥ and that Eq. (54) has the required factorized form for the virtual corrections according to the discussion in Section 2.2. The generator T a (l) (a = 1; : : : ; N ) acts on the color indices of the particle with momentum pl . The non-Abelian gauge coupling is g. In Eq. (54), and below, k⊥ denotes the component of the gauge boson momentum k transverse to the particle emitting this boson. Note that in Sudakov DL corrections there are no interference e5ects, so that we can talk about the emission (and absorption) of a gauge boson by a de2nite (external) particle, namely by a particle with momentum almost 2 collinear to k. It can be expressed in invariant form as k⊥ ≡ min(2(kpl )(kpj )=(pl pj )) for all j = l. The above factorization is directly related to the non-Abelian generalization of the Gribov theorem in Eq. (52). The form in which we present Eq. (54) corresponds to a covariant gauge for the gluon with momentum k. In this region for j = l we have pj pl =kpj El =!, where El is the energy of the particle with momentum pl and ! the frequency of the emitted gauge boson. Using the conservation of the total non-Abelian group charge: n

2 T a (j)M(p1 ; : : : ; pj ; : : : ; pn ; k⊥ )=0 ;

(55)

j=1

we can reduce the double sum over the gauge boson insertions in Eq. (54) to a single sum over external legs. In addition it is convenient to use the Sudakov parametrization analogously to Eq. (18) 2 and to replace the variable u by k⊥ according to Eq. (26). The infrared evolution equation then takes on the form n 2 min(|k⊥ |=ml ;1) 2gs2 s dk⊥ dv M(p1 ; : : : ; pn ; 2 ) = MBorn (p1 ; : : : ; pn ) − 2 √ (46)2 v k 2 | k⊥ | = s ⊥ l=1

2

×Cl M(p1 ; : : : ; pn ; k⊥ ) ;

(56)

where Cl is the eigenvalue of the Casimir operator T a (l)T a (l) (Cl = CA for gauge bosons in the adjoint representation of the gauge group SU (N ) and Cl = CF for fermions in the fundamental representation). The di5erential form of the infrared evolution equation follows immediately from (56): 9M(p1 ; : : : ; pn ; 2 ) = K(2 )M(p1 ; : : : ; pn ; 2 ) ; 9 log(2 )

(57)

where K(2 ) ≡ −

n

1 9Wl (s; 2 ) 2 9 log(2 )

(58)

l=1

with Wl (s; 2 ) =

gs2 Cl r 462

s m2l ; m2l 2

:

(59)

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As in the Abelian case, Wl is the probability to emit a soft and almost collinear gauge boson from the particle l with mass ml , subject to the infrared cut-o5 on the transverse momentum. Note again that the cut-o5 is not taken to zero. The function r is determined by (28) for arbitrary values of the ratio ml =. To logarithmic accuracy, we obtain from (59): s 9Wl (s; 2 ) gs2 Cl log : = − 2 2 9 log( ) 86 max(2 ; m2l )

(60)

The infrared evolution equation (57) should be solved with an appropriate initial condition. In the case of large scattering angles, if we choose the cut-o5 to be the large scale s then clearly there are no Sudakov corrections. The initial condition is therefore M(p1 ; : : : ; pn ; s) = MBorn (p1 ; : : : ; pn ) ;

(61)

and the solution of (57) is thus given by the product of the Born amplitude and the Sudakov form factors (Fig. 7):

n 1 2 2 Wl (s; ) : (62) M(p1 ; : : : ; pn ; ) = MBorn (p1 ; : : : ; pn ) exp − 2 l=1

Therefore we obtain an exactly analogous Sudakov exponentiation for the gauge group SU (N ) to that for the Abelian case, see (30). Theories with semi-simple gauge groups can be considered in a similar way. 2.4. Subleading corrections from splitting functions At high energies, where particle masses can be neglected, the form of soft and collinear divergences is universal. In this regime it is then appropriate to employ the formalism of the Altarelli–Parisi approach [59] and to calculate the corresponding splitting functions. We will do so below only for the virtual case. An important observation in this connection is that at high energies, all subleading terms are either of the collinear√or the RG type. This can be seen as follows: The types of soft, i.e. |k 0 | s, divergences in loop corrections with massless particles, unlike the collinear logarithms, can be obtained by setting all k dependent terms in the numerator of tensor integrals to zero (since the terms left are of the order of the hard scale s). Thus it is clear that the tensor structure which emerges is that of the inner scattering amplitude in Fig. 8 taken on the mass-shell, times a scalar function of the given loop correction. In the Feynman gauge, for instance, we 2nd for the well known vertex corrections the familiar three-point function C0 and for higher point functions we note that in the considered case all infrared divergent scalar integrals reduce to C0 multiplied by factors of 1=s etc. The only infrared divergent three point function is given by d4 k 1 : (63) C0 (s=2 ) ≡ 4 2 2 2 2 k⊥ ¿2 (26) (k + i#)(k + 2pj k + i#)(k − 2pl k + i#) 2 The function C0 (s=2 ) is fastly converging for large k⊥ and we are interested here in the re2 gion s in order to obtain large logarithms. Then logarithmic corrections come from the region 2 k⊥ s|u|; s|v|s (the strong inequalities give DL, the simple inequalities single ones) and we can

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write to logarithmic accuracy: ∞ ∞ ∞ s6 2 2 du dv dk⊥ C0 (s= ) = 2(26)4 −∞ 2 −∞ 1 2 (suv − k⊥ + i#)(suv − k⊥ + su + i#)(suv − k⊥ − sv + i#) 1 du 1 dv ∞ 2 si62 2 dk /(k2 − 2 ).(suv − k⊥ ) ≈ 2(26)4 −1 su −1 sv −∞ ⊥ ⊥ 1 i du 1 dv /(suv − 2 ) ≈ 2(46)2 s −1 u −1 v 1 i du 1 dv /(suv − 2 ) = (46)2 s 0 u 0 v 1 i i du 1 dv s = = log2 2 : (46)2 s 2 u 2 v 2(46)2 s ×

2

s

2

(64)

su

Thus, no single soft logarithmic corrections are present in C0 (s=2 ). In order to see that this result is not just a consequence of our regulator, we repeat the calculation for a 2ctitious gluon mass. 3 In this case we have d4 k 1 2 : (65) C0 (s=3 ) ≡ (26)4 (k 2 − 32 + i#)(k 2 + 2pj k + i#)(k 2 − 2pl k + i#) It is clear that C0 (s=32 ) contains soft and cuto5 3, which plays the role of in this 2 1 i 2 2 3 − i# log + C0 (s=3 ) = (46)2 s 2 −s

collinear divergences (kpj; l ) and is regulated with the case. Integrating over Feynman parameters we 2nd 62 : (66) 3

We are only interested here in the real part of loop corrections of scattering amplitudes since they are multiplied by the Born amplitude and the imaginary pieces contribute to cross sections at the next to next to leading level as mentioned above. In fact, the minus sign inside the double logarithm corresponds precisely to the omitted principle value contribution of Eq. (25) in the previous calculation. Thus, no single soft logarithmic correction is present in the case when particle masses can be neglected. This feature prevails to higher orders as well since it has been shown that also in non-Abelian gauge theories the one-loop Sudakov form factor exponentiates [50–58]. In case we would keep mass-terms, even two point functions, which in our scheme can only yield collinear logarithms, would contain a soft logarithm due to the mass-renormalization which introduces a derivative contribution (see for instance Ref. [60]). In conclusion, all leading soft corrections are contained in double logarithms (soft and collinear), and subleading logarithmic corrections in 3

Note that this regulator spoils gauge invariance and leads to possible inconsistencies at higher orders. Great care must be taken for instance when a three gluon vertex is regulated inside a loop integral.

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pj

p1

p2

pl k

Fig. 7. In an axial gauge, all collinear logarithms come from corrections to a particular external line (depending on the choice of the four vector n satisfying n Aa = 0) as illustrated in the 2gure. In a covariant gauge, the sum over all possible insertions is reduced to a sum over all n-external legs due to Ward identities. Overall, these corrections factorize with respect to the Born amplitude.

a massless theory, with all invariants large (sj; l = 2pj pl ∼ O(s)) compared to the infrared cuto5, are of the collinear type or renormalization group logarithms. The universal nature of collinear type logarithmic corrections can then easily be seen in an axial gauge where collinear logarithms are related to corrections on a particular external leg depending on the choice of the four vector n [61–64]. A typical diagram is depicted in Fig. 7. In a general covariant gauge this corresponds (using Ward identities) to a sum over insertions in all n external legs [28]. We can therefore adopt the strategy to extract the gauge invariant contribution from the external line corrections on the invariant matrix element at the subleading level. The results of the above discussion are thus important in that they allow the use of the Altarelli–Parisi approach to calculate the subleading contribution to the evolution kernel of Eq. (57). We are here only concerned with virtual corrections and use the universality of the splitting functions to calculate the subleading terms. For brevity we discuss both scalar as well as conventional QCD simultaneously. In each case one only needs to switch o5 the other type of 2eld in the fundamental representation to obtain the case of interest. The 0-function in both cases di5ers in the non-glue part but since this di5erence is of no consequence in our later discussion we do not distinguish between the two. For the purpose of calculating SL virtual corrections we use the virtual quark, scalar quark and gluon contributions to the V V splitting functions Pqq (z); PssV (z) and Pgg (z) describing the probability to emit a soft and/or collinear virtual particle with energy fraction z o5 the original external line four momentum. The in2nite momentum frame corresponds to the Sudakov parametrization with lightlike vectors. In general, the splitting functions PBA describe the probability of 2nding a particle B inside a particle A with fraction z of the longitudinal momentum of A with probability PBA to 2rst order [59]: -s PBA dt ; dPBA (z) = (67) 26 where the variable t = log s=2 for our purposes. It then follows [59] that -s z(1 − z) Q |VA→B+C |2 2 d log k⊥ ; dPBA (z) = 2 26 2 k⊥ spins

(68)

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239

where VA→B+C denotes the elementary vertices and PBA (z) =

z(1 − z) Q |VA→B+C |2 : 2 2 k⊥

(69)

spins

2 in Eq. (68) is s and it is thus directly related to dt. The upper bound on the integral over dk⊥ Regulating the virtual infrared divergences with the transverse momentum cuto5 as described above, we 2nd the virtual contributions to the splitting functions for external quark, scalar quarks and gluon lines: s V Pqq (z) = CF −2 log 2 + 3 .(1 − z) ; (70) s PssV (z) = CF −2 log 2 + 4 .(1 − z) ; (71) s 4 QCD V Pgg (z) = CA −2 log 2 + .(1 − z) : (72) 0 CA 0

The functions can be calculated directly from loop corrections to the elementary processes [65–68] and the logarithmic term corresponds to the leading kernel of Section 2.3. We introduce virtual distribution functions which include only the e5ects of loop computations. These ful2ll the Altarelli–Parisi equations 4 -s 1 dy 9q(z; t) V = q(z=y; t)Pqq (y) ; (73) 9t 26 z y -s 1 dy 9s(z; t) = s(z=y; t)PssV (y) ; (74) 9t 26 z y 9g(z; t) -s 1 dy V g(z=y; t)Pgg (y) : (75) = 9t 26 z y R V + PBA , where R denotes the contribution from real The splitting functions are related by PBA = PBA 5 gauge boson emission. PBA is free of logarithmic corrections and positive de2nite. The subleading term in Eq. (72) indicates that the only subleading corrections in the pure glue sector are related to a shift in the scale of the coupling. These corrections enter with a di5erent sign compared to the conventional running coupling e5ects. For fermion and scalar external lines there is an additional subleading correction from collinear terms which is not related to a change in the scale of the coupling.

4

Note that the o5 diagonal splitting functions Pqg and Pgq , etc. do not contribute to the virtual probabilities to the order we are working here. In fact, for virtual corrections there is no need to introduce o5-diagonal terms as the corrections factorize with respect to the Born amplitude. The normalization of the Eqs. (70)–(72) corresponds to calculations in two to two processes on the cross section level with the gluon symmetry factor 12 included. The results, properly normalized, are process independent. 5 Pqq was 2rst calculated by V.N. Gribov and L.N. Lipatov in the context of QED [68,69].

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Inserting the virtual probabilities of Eqs. (70)–(72) into Eqs. (73)–(75) we 2nd -s CF s 2 s log 2 − 3 log 2 q(1; t) = q0 exp − ; 26 -s CF s 2 s log 2 − 4 log 2 s(1; t) = s0 exp − ; 26 -s CA 4 QCD s 2 s log 2 − g(1; t) = g0 exp − ; 0 log 2 26 CA 0

(76) (77) (78)

C − 13 TF nf with CA = 3; CF = 4=3 and TF = 12 . These functions describe the total where 00QCD = 11 12 A contribution for the emission of virtual particles (i.e. z = 1), with all invariants large compared to the cuto5 , to the densities q(z; t); s(z; t) and g(z; t). The normalization is on the level of the cross section. For the invariant matrix element we thus 2nd at the subleading level for processes with n external lines: M(p1 ; : : : ; pn ; gs ; ) = M(p1 ; : : : ; pn ; gs )   nq ng ns 1 1 1 ×exp− Wqj (s; 2 ) − Wsi (s; 2 ) − Wgl (s; 2 ) 2 j=1 2 i=1 2

(79)

l=1

with nq + ng + ns = n, and - s CF s 2 2 s log 2 − 3 log 2 ; Wq (s; ) = 46 - s CF s 2 2 s log 2 − 4 log 2 ; Ws (s; ) = 46 - s CA 4 QCD s s log2 2 − Wg (s; 2 ) = 00 log 2 : 46 CA

(80) (81) (82)

The functions Wq ; Ws and Wg correspond to the probability of emitting a virtual soft and/or collinear gauge boson from the particle q; s; g subject to the infrared cuto5 . Typical diagrams contributing to Eq. (79) in a covariant gauge are depicted in Fig. 8. In massless QCD there is no need for the label Wqj ; Wsi or Wgl , however, we write it for later convenience. The universality of the splitting functions is crucial in obtaining the above result. 2.5. Anomalous scaling violations The solution presented in Eq. (79) determines the evolution of the virtual scattering amplitude M(p1 ; : : : ; pn ; gs ; ) for large energies at 2xed angles and subject to the infrared regulator . In the massless case there is a one to one correspondence between the high energy limit and the infrared limit as only the ratio s=2 enters as a dimensionless variable [70,71]. Thus, we can generalize the Altarelli–Parisi equations (73)–(75) to the invariant matrix element in the language of the renormalization group. For this purpose, we de2ne the infrared singular (logarithmic) anomalous

M. Melles / Physics Reports 375 (2003) 219 – 326

p1

pj

pl

pj

p1

k

p2

241

k

p2

pl

Fig. 8. Feynman diagrams contributing to the infrared evolution equation (84) for a process with n external legs. In a general covariant gauge the virtual gluon with the smallest value of k⊥ is attached to di5erent external lines. The inner scattering amplitude is assumed to be on the mass shell.

dimensions CF -s CF -s CA -s t; 5s (t) ≡ t; 5g (t) ≡ t : (83) 46 46 46 Infrared divergent anomalous dimensions have been derived in the context of renormalization properties of gauge invariant Wilson loop functionals [72]. In this context they are related to undifferentiable cusps of the path integration and the cusp angle pj pl =2 gives rise to the logarithmic nature of the anomalous dimension. In case we use o5-shell amplitudes, one also has contributions from end points of the integration [72]. The leading terms in the equation below have also been discussed in Refs. [73–75] in the context of QCD. With these notations we 2nd that Eq. (79) satis2es 9 1 1 QCD 9 +0 + ng 5g (t) + g + nq 5q (t) + q 9t 9gs 2 2 1 M(p1 ; : : : ; pn ; gs ; ) = 0 ; (84) + ns 5s (t) + s 2 5q (t) ≡

to the order we are working here and where M(p1 ; : : : ; pn ; gs ; ) is taken on the mass-shell. The di5erence in the sign of the derivative term compared to Eq. (57) is due to the fact that instead of di5erentiating with respect to log 2 we use log s=2 . The anomalous dimensions are given by g = −-s =6 00QCD = −-s =6( 11 C − nf =3 TF ); s = −CF -s =6 and q = −CF 43 -=6. As mentioned above 12 A in pure scalar QCD the 0-function di5ers in the non-glue part from 0QCD . The quark–antiquark operator anomalous dimension q or in scalar QCD s enter even for massless theories as the quark antiquark operator leads to scaling violations through loop e5ects since the quark masslessness is not protected by gauge invariance and a dimensionful infrared cuto5 needs to be introduced. Thus, although the Lagrangian contains no m Q or m2 ∗ term, quantum corrections lead to the anomalous scaling violations in the form of q or s . The factor 12 occurs since we write Eq. (84) in terms of each external line separately. 6 For the gluon, the scaling violations due to the infrared cuto5 are 6

In case of a massive theory, we could, for instance avoid the anomalous dimension term q by adopting the pole mass de2nition. In this case, however, we would obtain terms in the wave function renormalization, and in any case, the one to one correspondence between UV and IR scaling, crucial for the validity of Eq. (84), is violated.

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δ q q:

δ 2:

Fig. 9. The two counterterms contributing to the quark anomalous dimension q = 9=9 log Q2 (−.qq Q + .2 ) or s = 9=9 log Q2 (−.s∗ s + .2 ). Here Q denotes the MS dimensional regularization mass parameter. Due to divergences in loop corrections there are scaling violations also in the massless theory.

manifest in terms of an anomalous dimension proportional to the 0-function since the gluon mass is protected by gauge invariance from loop corrections. Thus, in the bosonic sector the subleading terms correspond e5ectively to a scale change of the coupling. Fig. 9 illustrates the corrections to the external quark–antiquark lines from loop e5ects. Except for the infrared singular anomalous dimension (Eq. (83)), all other terms in Eq. (84) are the standard contributions to the renormalization group equation for S-matrix elements [76]. In QCD, observables with infrared singular anomalous dimensions, regulated with a 2ctitious gluon mass, are ill de2ned due to the masslessness of gluons. In the electroweak theory, however, we can legitimately investigate only virtual corrections since the gauge bosons will require a mass. Eq. (84) will thus be very useful in Section 3. 2.6. Renormalization group corrections In this section we review the case of higher order RG-corrections in unbroken gauge theories like QCD following Ref. [77]. Explicit comparisons with higher order calculations for the on-shell Sudakov form factor revealed that the relevant RG scale in the respective diagrams is indeed the perpendicular Sudakov component [78–82]. It should be noted, however, that in particular for the massive cases, the discussion below is not valid close to any of the thresholds. In such cases it is useful to consider “physical renormalization schemes” such as discussed in Refs. [83–85] which display a gauge invariant, continuous and smooth Uavor threshold behavior with automatic decoupling of heavy particles. For our purposes here, we give correction factors for each external line below. The universal nature of the higher order SL-RG corrections can be seen as follows. Consider the gauge invariant fermionic part (∼ nf ) as indicative of the full 00QCD term (replacing nf =3=TF ( 11 C −00QCD )). 12 A 2n−1 n 2 s= ), this loop correction must be folded In order to lead to subleading contributions, i.e. O(-s log with the exchange of a gauge boson between two external lines (producing a DL type contribution) like the one depicted in Fig. 10. Using the conservation of the total non-Abelian group charge, i.e. Eq. (55), the double sum over all external insertions j and l is reduced to a single sum over all n external legs. Thus these types of corrections can be identi2ed with external lines at higher orders. The same conclusion is reproduced by the explicit pole structure of MS renormalized scattering amplitudes at the two loop level in QCD [86]. The results presented in Ref. [86] have been con2rmed recently by explicit massless two loop QCD calculations [98–103]. In addition, from the expression in Ref. [86] it can be seen that the SL-RG corrections are independent of the spin, i.e. for both quarks and gluons the same running coupling argument is to be used. This is a consequence of the

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243

Fig. 10. A QED diagram at the two loop level yielding a SL-RG correction. The explicit result obtained in Ref. [87] for 1 4 1 QED 4 the case of equal masses relative to the Born amplitude was − 36 e =1664 log3 s=m2 = 12 00 e =1664 log3 s=m2 . This result is reproduced exactly by including a running coupling into the one loop vertex correction diagram. The argument of the coupling must depend on the component of the loop momentum (going into the fermion loop) which is perpendicular to the external fermion momenta. In QCD, although more diagrams contribute, the net e5ect is just to replace 00QED → 00QCD in the above expression.

fact that these corrections appear only in loops which can yield DL corrections on the lower order level and as such, the available DL phase space is identical up to group theory factors. We begin with the virtual case. 2.6.1. Virtual corrections The case of virtual SL-RG corrections for both massless and massive partons has been discussed in Ref. [88] with a di5erent Sudakov parametrization. Below we show the identity of both approaches. The form of the corrections is given in terms of the probabilities WiV (s; 2 ). To logarithmic accuracy, they correspond to the probability to emit a soft and/or collinear virtual parton from particle i at high energies subject to an infrared cuto5 . At the amplitude level all expressions below are universal for each external line and exponentiate according to

n 1 Wi (s; 2 ) ; (85) M(p1 ; : : : ; pn ; gs ; ) = MBorn (p1 ; : : : ; pn ; gs ) exp − 2 i=1 V where n denotes the number of external lines. We begin with the massless case. Massless QCD. In the following we denote the running QCD-coupling by 2 -s (k⊥ )=

-s (2 )

1+

(-s (2 )=6)00QCD

2 log (k⊥ =2 )

≡

-s (2 ) : 2 1 + c log (k⊥ =2 )

(86)

Up to two loops the massless 0-function is independent of the chosen renormalization scheme and is gauge invariant in minimally subtracted schemes to all orders [89]. These features will also hold for the derived renormalization group correction factors below in the high energy regime. The scale denotes the infrared cuto5 on the exchanged k⊥ between the external momenta pj ; pl , where the Sudakov decomposition is given by k = vpl + upj + k⊥ , such that pj k⊥ = pl k⊥ = 0. The cuto5 2 2 serves as a lower limit on the exchanged Euclidean component k⊥ = −k⊥ ¿ 0 as in the previous

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sections. In order to avoid the Landau pole we must choose ¿ AQCD . Thus, the expressions given in this section correspond for quarks to the case where m. For arbitrary external lines we then have

- s Ci 2 W˜ DL iV (s; ) = 26

s

2

2 dk⊥ 2 k⊥

1 2

k⊥ =s

dv -s Ci s = log2 2 : v 46

(87)

The RG correction is then described by including the e5ect of the running coupling from the scale 2 to s according to [78–82] (see also discussions in Refs. [88,90]):

2 1 dk⊥ -s (2 ) dv 2 2 2 2 2 k⊥ k⊥ =s v 1 + c log(k⊥ = ) 1 -s (2 )Ci 1 -s (2 ) -s (2 ) s − 1 + 2 log ; = log 2 log 26 c -s (s) c -s (s)

Ci 2 W˜ RG iV (s; ) = 26

s

(88)

where Ci = CA for gluons and Ci = CF for quarks. For completeness we also give the subleading terms of the external line correction which is of course also important for phenomenological applications. The terms depend on the external line and the complete result to logarithmic accuracy is given by (s; 2 ) WgRG V

-s (2 )CA = 26

(s; 2 ) WqRG V

-s (2 )CF = 26

1 s log 2 c 1 s log 2 c

-s (2 ) -s (2 ) s 1 2 QCD log (89) 0 log 2 − 1 + 2 log − -s (s) c -s (s) CA 0

-s (2 ) -s (2 ) 3 1 s log − 1 + 2 log − log 2 : -s (s) c -s (s) 2

(90)

It should be noted that the subleading term in Eq. (89) proportional to 00QCD is not a conventional renormalization group corrections but rather an anomalous scaling dimension, and enters with the opposite sign [37] compared to the conventional RG contribution (see Section 2.5). Massive QCD. Here we give results for the case when the infrared cuto5 m, where m denotes the external quark mass. We begin with the case of equal external and internal line masses: 2 Equal masses. Following Ref. [88], we use the gluon on-shell condition suv = k⊥ to calculate the integrals. We begin with the correction factor for each external massive quark line. Following the diagram in Fig. 2 we 2nd CF 2 W˜ RG qV (s; ) = 26 ×

0

1

du u

0

1

m2 dv m2 /(suv − 2 )/ u − v / v− u v s s

-s (m2 ) 1 + c log (suv=m2 )

M. Melles / Physics Reports 375 (2003) 219 – 326

CF = 26

2 =s

− =

=m

m=s

2 =s

du u

du u

1

2 =su

dv + v

1

=m

du u

1

m2 =su

dv v

m2 =s -s (m2 ) dv du 1 dv − 1 + c log (suv=m2 ) 2 =su v m=s u (s=m2 )u v 1 s 1 -s (2 ) -s (m2 ) log 2 log − 1 + 2 log : c m -s (s) c -s (s)

-s (m2 )CF 26

245

1

(91)

The -dependent terms cancel out of any physical cross section (as they must) when real soft Bremsstrahlung contributions are added and c=-s (m2 )00QCD =6 for massive quarks. In order to demonstrate that the result in Eq. (91) exponentiates, we calculated in Ref. [88] the explicit two loop renormalization group improved massive virtual Sudakov corrections, containing a di5erent “running 2 scale” in each loop. It is of course also possible to use the scale k⊥ directly. In this case we have according to the diagram in Fig. 3: s 2 =m2 2 k⊥ 2 1 dk⊥ dk⊥ -s (m2 ) dv dv + 2 2 2 2 2 v k⊥ 1 + c log (k⊥ =m2 ) m 2 k⊥ k⊥ 2 k⊥ =s =s v -s (2 ) -s (m2 ) -s (m2 )CF 1 s 1 = log 2 log − 1 + 2 log 26 c m -s (s) c -s (s)

2 W˜ RG qV (s; ) =

CF 26

m2

(92)

which is the identical result as in Eq. (91). For completeness we also give the subleading terms of the pure one loop form factor which is again important for phenomenological applications. The complete result to logarithmic accuracy is thus given by WqRG (s; 2 ) = V

-s (m2 )CF 26

1 s log 2 c m

log

1 s -s (2 ) -s (m2 ) 3 m2 − 1 + 2 log − log 2 − log 2 : -s (s) c -s (s) 2 m (93)

For m = Eq. (93) agrees with Eq. (90) in the previous section for massless quarks. Unequal masses. In this section we denote the external mass as before by m and the internal mass by mi and thus, the constant c = -s (m2i )00QCD =6. We consider only the case at high energies taking the 2rst two families of quarks as massless. The running of all light Uavors is implicit in the nf term of the 00QCD function. The result is then given by s 2 =m2 2 k⊥ 2 1 dk⊥ dk⊥ -s (m2i ) dv dv + 2 2 2 2 2 v k⊥ 1 + c log (k⊥ =m2i ) m 2 k⊥ k⊥ 2 k⊥ =s =s v s 1 -s (m2 ) 1 m2 -s (2 ) -s (m2i )CF 1 log 2 log − 1 + log + log 2 : = 26 c m -s (s) c -s (s) c mi

CF 2 W˜ RG qV (s; ) = 26

m2

(94)

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It is evident that the e5ect of unequal masses is large only for a large mass splitting. In QCD, we always assume scales larger than AQCD and with our assumptions we have only the ratio of mt =mb leading to signi2cant corrections. The full subleading expression is accordingly given by s -s (2 ) -s (m2i )CF 1 RG 2 log 2 log −1 WqV (s; ) = 26 c m -s (s) 1 -s (m2 ) 1 m2 s 3 m2 + log + log 2 − log 2 − log 2 c -s (s) c 2 m mi s -s (2 ) -s (m2i )CF 1 log 2 log −1 = 26 c m -s (s) 1 -s (m2i ) m2 -s (m2 ) 3 s + 2 : (95) − log log − log c -s (m2 ) -s (s) 2 m2 2 For m = mi Eq. (95) agrees with Eq. (93) in the previous section for equal mass quarks. If we want to apply the above result for the case of QED corrections later, then there is no Landau pole (at low energies) and we can have large corrections of the form mb =me , etc. In this case the running coupling term is given by 2 )= e2 (k⊥

1 − 13 e2 =462

e2 nf j=1

2 Qj2 NCj log k⊥ =m2j

(96)

and instead of Eq. (95) we have  2  ef 1 e2 (2 ) s RG 2 WfV (s; ) = 2 log 2 log 2 −1 86  c m e (s)    nf 1 e2 2 j e2 (m2 )  m2  3 m2  s 1 1− Q N log 2 − log 2 − log 2 + 2 log 2 c e (s) 3 462 j=1 j C 2 m  mj and where c = − 13 e2 =462

nf

j=1

(97)

Qj2 NCj .

2.6.2. Real gluon emission We discuss the massless and massive case separately since the structure of the divergences is di5erent in each case. For massive quarks we discuss two types of restrictions on the experimental requirements, one in analogy to the soft gluon approximation. The expressions below exponentiate on the level of the cross section, i.e. for observable scattering cross sections they are of the form dH(p1 ; : : : ; pn ; gs ; expt ) = dHBorn (p1 ; : : : ; pn ; gs ) n 2 [Wi; R (s; 2 ; expt ) − Wi; V (s; 2 )] ; ×exp i=1

(98)

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247

where the sum in the exponential is independent of and only depends on the cuto5 expt de2ning the experimental cross section. We begin with the massless case. Emission from massless partons. In this section we consider the emission of real gluons with a cuto5 k⊥ 6 expt , related to the experimental requirements. For massless partons we have at the DL level: √ expt 2 s 2 dk C d! -s Ci s i DL 2 2 2 s 2 s ⊥ = log ; (99) − log W˜ iR (s; ; expt ) = 2 2 6 2 46 2 k⊥ expt | k⊥ | ! and thus for the RG-improved correction: √ expt 2 s 2 dk -s (2 ) d! C i 2 2 ⊥ W˜ RG iR (s; ; expt ) = 2 2 2 6 2 k⊥ |k⊥ | ! 1 + c log k⊥ = 2 expt -s (2 ) Ci -s (2 ) 1 -s (2 ) 1 s 1 : = − 1 − log log 2 log + 2 log 2 2 26 c c s c -s (expt ) -s (expt ) (100) This expression depends on as it must in order to cancel the infrared divergent virtual corrections. In fact the sum of real plus virtual corrections on the level of the cross section is given by 2 WiRG (s; 2 ; expt ) − WiRG (s; 2 ) R V 2 expt 1 Ci s -s (s) -s (s) + log = QCD log 2 log − log 2 2 s c -s (expt ) -s (expt ) 200

2 expt Ci 6 -s (s) = QCD log − log 2 20 s -s (expt ) -(s)00QCD

and thus independent of . The full expressions to subleading accuracy are thus 2 expt s -s (2 ) CA -s (2 ) 1 1 RG 2 2 log 2 log log WgR (s; ; expt ) = − − 1 2 26 c c s -s (expt ) 1 -s (2 ) s 2 QCD + 2 log ; 00 log 2 − 2 c -s (expt ) CA 2 WqRG (s; 2 ; expt )= R

1 s log 2 c

(102)

2 expt -s (2 ) 1 log log − 1 − 2 c s -s (expt ) s 1 -s (2 ) 3 + 2 log : − log 2 2 c -s (expt ) 2 CF -s (2 ) 26

(101)

(103)

All divergent (-dependent) terms cancel when the full virtual corrections are added. Emission from massive quarks. In the case of a massive quark, i.e. m, the overall infrared divergence is not as severe. This means we can discuss di5erent requirements which all have the correct divergent pole structure canceling the corresponding terms from the virtual contributions. We divide the discussion in two parts as above.

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Equal masses. The constant c = -s (m2 )00QCD =6 below. We have the following expression without a running coupling: WqR (s;

≈

2

 -s C F     26

 -s C F    26

- s CF = 6

2 ; expt )

2 expt

2

dk⊥

2

√

s

| k⊥ |

2 k⊥ d! 2 ! (k⊥ + m2 =s!2 )2

1 s m2 m2 1 s s log2 2 + log 2 log 2 − log 2 − log2 2 2 m m 2 expt

;

s s m2 2 s s log 2 + log 2 log 2 + log 2 − log 2 log 2 m m m expt expt 2

mexpt ; (104)

;

expt m :

If we want to employ a restriction analogously to the soft gluon approximation, we 2nd independently of the quark mass [37,30]: 2 WqR (s; 2 ; expt )

- s CF = 6 -s CF ≈ 26

2 expt

2

2

dk⊥

√

expt

| k⊥ |

2 k⊥ d! 2 ! (k⊥ + m2 =s!2 )2

1 s s m2 m2 s s s log2 2 + log 2 log 2 − log 2 + log 2 − log 2 log 2 2 m m m expt expt

:

(105)

In all cases above we have not taken into account all subleading collinear logarithms related to real gluon emission. In order to now proceed with the inclusion of the running coupling terms it is convenient to 2rst consider only the DL phase space in each case. Thus we 2nd 2

expt 2 m 2 2 2 dk dk 1 (m )C s s F s RG 2 2 ⊥ ⊥ W˜ qR (s; ; expt ) = log 2 + log 2 2 2 26 m k⊥ k⊥ k⊥ k2 2 m2 1 + c log ⊥2 m 2 2 s s -s ( ) -s (m )CF 1 1 log 2 log ≈ − 1 + log 2 2 26 c m c -s (expt ) expt 1 -s (m2 ) ; mexpt + 2 log (106) 2 c -s (expt ) and 2

-s (m )CF 2 2 W˜ RG qR (s; ; expt ) = 26 ≈

2 expt

2

2 dk⊥ s 1 log 2 2 2 m 1 + c log k⊥ =m2 k⊥

s -s (m2 )CF 1 -s (2 ) log 2 log ; 2 26 c m -s (expt )

expt m :

(107)

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249

The full subleading expressions are thus given by s s -s (2 ) -s (m2 )CF 1 1 RG 2 2 log 2 log WqR (s; ; expt ) ≈ − 1 + log 2 2 26 c m c -s (expt ) expt 1 -s (m2 ) m2 ; mexpt + 2 log − log 2 c 2 -s (expt )

(108)

and 2 WqRG (s; 2 ; expt )≈ R

-s (m2 )CF 26

1 s -s (2 ) 2 log 2 log ; expt m : + log 2 2 c m -s (expt ) expt

(109)

In case we also impose a cut on the integration over ! we have independently of the relation between m and expt assuming only m2 s: 2 2

expt 2 2 (m expt =s) 2 2 2 dk dk (m )C s expt s F 2 2 ⊥ ⊥ log 2 + m2 expt log 2 W˜ RG 2 qR (s; ; expt ) = 2 2 26 m k⊥ k⊥ k⊥ 2 s

1 2 1 + c log k⊥ =m2 s -s (2 ) -s (m2 )CF 1 log 2 log ≈ −1 2 26 c m -s (expt ) ×

2 2 -s (expt -s (expt ) m2 =s) s 1 1 log + + log 2 log 2 2 c expt -s (expt m2 =s) c2 -s (expt )

:

(110)

This expression agrees with the result obtained in Ref. [88] where the gluon on-shell condition 2 k⊥ = suv was used and one integral over one Sudakov parameter was done numerically. In Ref. [88] it was also shown that the RG-improved virtual plus soft form factor also exponentiates by explicitly calculating the two loop RG correction with each loop containing a running coupling of the 2 corresponding k⊥ . The full subleading expression for the RG-improved soft gluon emission correction is thus given by 2 -s (expt ) s s -s (2 ) -s (m2 )CF 1 1 RG 2 2 log 2 log log log WqR (s; ; expt ) ≈ − 1 + 2 2 2 2 26 c m c -s (expt m =s) -s (expt ) expt 2 -s (expt m2 =s) 1 s m2 (111) + 2 log − log 2 + log 2 2 c -s (expt expt ) for the equal mass case. The case of di5erent external and internal masses is again important for applications in QED and will be discussed next.

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Unequal masses. While the gluonic part of the 0-function remains unchanged we integrate again only from the scale of the massive fermion which is assumed to be in the perturbative regime. For applications to QED, however, we need the full expressions below. Here we discuss only the case analogous to the soft gluon approximation. Considering again only the high energy scenario we have for the case of an external mass m and a fermion loop mass mi :   2 m2 expt 2 2 2 2 expt expt  s dk⊥ dk⊥ -s (m2i )CF  s 2 2 log 2 + m2 expt log 2  W˜ RG 2  qR (s; ; expt ) = 2 2 26 m k⊥ k⊥ k⊥ 2 s

1 2 1 + c log k⊥ =m2i s -s (2 ) -s (m2i )CF 1 log 2 log ≈ −1 2 26 c m -s (expt m2 =s) ×

2 2 -s (expt -s (expt ) m2 =s) m2i 1 1 + log + log 2 log 2 2 c expt -s (expt m2 =s) c2 -s (expt )

:

(112)

This expression agrees with the result obtained in Eq. (110) for the case mi = m. The full subleading expression for the RG-improved soft gluon emission correction is thus given by s -s (2 ) -s (m2i )CF 1 RG 2 2 log 2 log WqR (s; ; expt ) ≈ −1 2 26 c m -s (expt m2 =s) 2 2 -s (expt -s (expt ) m2 =s) m2i s 1 m2 1 + log − log 2 + log 2 + log 2 log 2 2 c expt -s (expt m2 =s) c2 -s (expt ) expt -s (2 ) -s (m2i )CF 1 s −1 = log 2 log 2 26 c m -s (expt m2 =s) 2 -s (expt m2 =s) s m2 1 -s (m2i ) : (113) log − log 2 + log 2 + 2 2 2 c -s (expt ) -s (expt ) expt As mentioned above, this expression is more useful for applications in QED or if the mass ratios are very large. In QED we have again the running coupling of the form given in Eq. (96), and Eq. (113) becomes  2 2 e e2 (expt m2 =s) s e2 (2 ) 1 f 1 RG 2 2 log 2 log 2 2 WfR (s; ; expt ) ≈ 2 log − 1 + 2 86 c m c2 e (expt m2 =s) e2 (expt ) 

  nf 2 2 2 e 1 m s expt × 1 − Q2 N j log 2  − log 2 + log 2  ; 3 462 j=1 j C mj expt

(114)

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251

nf where again c = − 13 e2 =462 j=1 Qj2 NCj . This concludes the discussion of SL-RG e5ects in QCD. As a side remark we mention that for scalar quarks, the same function appears as for fermions since the DL-phase space for both cases is identical. Only 00 di5ers in each case.

3. Broken gauge theories In the following we will apply the results obtained in the previous sections to the case of spontaneously broken gauge theories. It will be necessary to distinguish between transverse and longitudinal degrees of freedom. The physical motivation in this approach is that for very large energies, sMW2 ≡ M 2 , the electroweak theory is in the unbroken phase, with e5ectively an SU (2) × U (1) gauge symmetry as described by the high energy symmetric part of the Lagrangian in Eq. (11). We will calculate the corrections to this theory and use the high energy solution as a matching condition for the regime for values of ¡ M . We begin by considering some simple kinematic arguments for massive vector bosons. A vector boson at rest has momentum k = (M; 0; 0; 0) and a polarization vector that is a linear combination of the three orthogonal unit vectors e1 ≡ (0; 1; 0; 0);

e1 ≡ (0; 0; 1; 0);

e3 ≡ (0; 0; 0; 1) :

(115)

After boosting this particle along the 3-axis, its momentum will be k = (Ek ; 0; 0; k). The three possible polarization vectors are now still satisfying: k #j = 0;

#j2 = −1 :

(116)

Two of these vectors correspond to e1 and e2 and describe the transverse polarizations. The third vector satisfying (116) is the longitudinal polarization vector #L (k) = (k=M; 0; 0; Ek =M ) ;

(117)

i.e. #L (k) = k =M + O(M=Ek ) for large energies. These considerations illustrate that the transversely polarized degrees of freedom at high energies are related to the massless theory, while the longitudinal degrees of freedom need to be considered separately. Another manifestation of the di5erent high energy nature of the two polarization states is contained in the Goldstone boson equivalence theorem. It states that the unphysical Goldstone boson that is “eaten up” by a massive gauge boson still controls its high energy asymptotics. A more precise formulation is given below in Section 3.3. Thus we can legitimately use the results obtained in the massless non-Abelian theory for transverse degrees of freedom at high energies and for longitudinal gauge bosons by employing the Goldstone boson equivalence theorem. Another di5erence to the situation in an unbroken non-Abelian theory is the mixing of the physical 2elds with the 2elds in the unbroken phase. These complications are especially relevant for the Z-boson and the photon.

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3.1. Fermions and transverse degrees of freedom The results we obtain in this section are generally valid for spontaneously broken gauge theories, however, for de2niteness we discuss only the electroweak Standard Model. The physical gauge bosons are thus a massless photon (described by the 2eld A ) and massive W ± and Z bosons (described correspondingly by 2elds W± and Z ): 1 W± = √ (W1 ± iW2 ) ; 2

(118)

Z = cos /w W3 + sin /w B ;

(119)

A = −sin /w W3 + cos /w B :

(120)

Thus, amplitudes containing physical 2elds will correspond to a linear combination of the massless 2elds in the unbroken phase. The situation is illustrated schematically for a single gauge boson external leg in Fig. 11. In case of the W ± bosons, the corrections factorize with respect to the physical amplitude. In the general case let us denote physical particles (2elds) by f and particles (2elds) of the unbroken theory by u. Let the connection between them be denoted by f = u C fu u, where the sum is performed over appropriate particles (2elds) of the unbroken theory as in Eqs. (118)–(120). Note that, in general, physical particles, having de2nite masses, do not belong to irreducible representations of the symmetry group of the unbroken theory (for example, the photon and Z bosons have no de2nite isospin). On the other hand, particles of the unbroken theory, belonging to irreducible representations of the gauge group, have no de2nite masses. Then for the amplitude Mf1 ; :::; fn (p1 ; : : : ; pn ; 2 ) with n physical particles fi with momenta pi and infrared cut-o5 2 , the general case for virtual corrections is given by M

f1 ; :::; fn

2

(p1 ; : : : ; pn ; ) =

n "

C fi ui Mu1 ; :::; un (p1 ; : : : ; pn ; 2 ) :

(121)

u1 ;:::;un i=1

On the one loop level and to subleading accuracy, Eq. (121) must also include the correct counterterms for the commonly chosen on-shell scheme. In this scheme the on-shell photon, for instance, does not mix with the Z-boson, thus including all mixing e5ects into the massive neutral Z-boson sector. More details are given below. In the following we give results only for the amplitudes Mu1 ; :::; un (p1 ; : : : ; pn ; 2 ) keeping in mind that in general the physical amplitudes must be obtained via Eq. (121). For fermions, transverse W ± , longitudinal gauge bosons or Higgs bosons, no linear combination arises, i.e. the universal corrections below factorize automatically with respect to the physical Born process. Only photons and Z-bosons are a5ected by this complication for the obvious reasons discussed above. To logarithmic accuracy, all masses can be set equal: MZ ∼ MW ∼ MHiggs ∼ M

√ and the energy is considered to be much larger, sM . The left and right handed fermions are correspondingly doublets (T = 1=2) and singlets (T = 0) of the SU (2) weak isospin group and have

M. Melles / Physics Reports 375 (2003) 219 – 326

γ

W3

= -sw δ W

= cw δ W

W =2

+cw δB

+/-

-0.5

δW

B

+sw δ B

W +/-i 2

1

-0.5

W =

B

W3

Z

253

W

2

δW

+/-

δW

Fig. 11. The schematic corrections to external gauge boson emissions in terms of the 2elds in the unbroken phase of the electroweak theory. There are no mixing terms between the W3 and B 2elds for massless fermions. We denote cos /w by cw and sin /w by sw . For W ± 2nal states, the corrections factorize with respect to the physical amplitude. In general, one has to sum over all 2elds of the unbroken theory with each amplitude being multiplied by the respective mixing coeScient as given in Eq. (121). At one loop, also the renormalization conditions must be included.

hypercharge Y related to the electric charge Q, measured in units of the proton charge, by the Gell–Mann–Nishijima formula Q = T 3 + Y=2. The value for the infrared cuto5 can be chosen in two di5erent regimes (see Fig. 1): √ I sM and II M . The second case is universal in the sense that it does not depend on the details of the electroweak theory and will be discussed below. In the 2rst region we can neglect spontaneous symmetry breaking e5ects (in particular terms connected to the v.e.v.) and consider the theory with 2elds B and Wa as given by Lsymm in Eq. (11). One could of course also calculate everything in terms of the physical 2elds, however, we emphasize again that in this case we need to consider the photon also in region (I). The omission of the photon would lead to the violation of gauge invariance since the photon contains a mixture of the B and W3 2elds.

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M. Melles / Physics Reports 375 (2003) 219 – 326

In region (I), the renormalization group equation (or generalized infrared evolution equation) (84) in the case of all mi ¡ M reads 7

nf ng 9 9 9 1 1 1 +0 + 0 + 00 − nB 00 + 5fk (t) + kf 5gi (t) − nW 9t 9g 9g 2 6 2 6 2 i=1 k=1

×M⊥ (p1 ; : : : ; pn ; g; g ; ) = 0 ;

(122)

where the index ⊥ indicates that we consider only ng transversely polarized external gauge bosons with nW + nB = ng and nf denotes the number of external fermion lines. The two 0-functions are given by 0(g(Q 2 )) =

g3 (Q 2 ) 9g(Q 2 ) ≈ −0 ; 0 9 log Q 2 862

(123)

3 2 9g (Q 2 ) g (Q ) ≈ −00 ; 0 (g (Q )) = 9 log Q 2 862

2

(124)

with the one-loop terms given by 00 =

11 C 12 A

− 13 ngen −

1 n ; 24 h

00 = − 59 ngen −

1 n 24 h

;

(125)

where ngen denotes the number of fermion generations [91,92] and nh the number of Higgs doublets. Eq. (122) describes the one loop RG corrections correctly. At higher orders the subleading RG terms must be included according to the discussion in Section 3.5. The infrared singular anomalous dimensions read 2

Yi i 5f; Ti (Ti + 1) + t ; (126) g (t) = 46 46 2 where Ti and Yi are the total weak isospin and hypercharge, respectively, of the particle emitting the soft and collinear gauge boson. Analogously,

- 1 + .i; R m2i m2i - Yi 2 i f = −3 Ti (Ti + 1) + + ; (127) + .i; L 46 46 2 46 4 M2 4M 2 where the last two terms only contribute for quarks of the third generation. i denotes the isospin partner of i. The presence of Yukawa terms and also the Higgs contribution to the 0-functions in Eq. (125) are remnants of the spontaneous broken symmetry which leads to di5erences even in the transverse sector compared to unbroken gauge theories as is obvious from the form of Lsymm in Eq. (11). In terms of the corresponding logarithmic probabilities we thus have the following 7

Note, that the amplitude on the right hand side is in general a linear combination of 2elds in the unbroken phase according to Eq. (121). In addition, in the electroweak theory matching will be required at the scale M and often on-shell renormalization of the couplings e and sin /w is used. In this case one has additional complications in the running coupling terms due to the di5erent mass scales involved below M . Details are presented in Section 3.5.

M. Melles / Physics Reports 375 (2003) 219 – 326

expression for fermions from the virtual splitting function approach: Yi2 s 2 2 2 s Ti (Ti + 1) + tan /w log 2 − 3 log 2 Wfi (s; ) = 46 4 1 + .i; R m2i s m2i + log 2 : + . i; L 2 2 4 M 4M For external transversely polarized gauge bosons:

s - - Yi 2 2 2 s Wgi (s; ) = log 2 − .i; W 00 + .i; B 00 log 2 : Ti (Ti + 1) + 46 46 2 6 6

255

(128)

(129)

The initial condition for Eq. (122) is given by the requirement that for the infrared cuto5 2 = s we obtain the Born amplitude. The solution of (122) is thus given by ⊥ M⊥ (p1 ; : : : ; pn ; g; g ; ) = MBorn (p1 ; : : : ; pn ; g; g )

ng - Yi 2 s 1 Ti (Ti + 1) + log2 2 : ×exp − 2 i=1 46 46 2 s - + nW 00 + nB 00 log 2 26 26 nf 1 - Yk 2 s 2 s Tk (Tk + 1) + log 2 − 3 log 2 − 2 46 46 2 k=1 s m2k - 1 + .k; R m2k ; log 2 + .k; L + 46 4 M2 4M 2

(130)

where we neglect RG corrections for now. These will be discussed thoroughly in Section 3.5. nW and nB denote the number of external W and B 2elds, respectively. The SU (2) × U (1) group factors in the exponential can be written in terms of the parameters of the broken theory as follows: 2 g2 2 2 Yi = ei2 + g2 (Ti (Ti + 1) − (Ti3 )2 ) + (T 3 − sin2 /w Qi )2 ; g Ti (Ti + 1) + g 2 cos2 /w i where the three terms on the r.h.s. correspond to the contributions of the soft photon (interacting with the electric charge ei = Qi g sin /w ), the W ± and the Z bosons, respectively. Although we may rewrite solution (130) in terms of the parameters of the broken theory in the form of a product of three exponents corresponding to the exchanges of photons, W ± and Z bosons, it would be wrong to identify the contributions of the diagrams without virtual photons with this expression for the particular case ei2 = 0. This becomes evident when we note that if we were to omit photon lines then the result would depend on the choice of gauge, and therefore be unphysical. Only for /w = 0, where the photon coincides with the B gauge boson, would the identi2cation of the ei2 term with the contribution of the diagrams with photons be correct.

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M. Melles / Physics Reports 375 (2003) 219 – 326

We now need to discuss the solution in the general case. In region (I) we calculated the scattering amplitude for the theory in the unbroken phase in the massless limit. Choosing the cuto5 in region (II), M , we have to only consider the QED contribution. In this region we cannot necessarily neglect all mass terms, so we need to discuss the subleading terms for QED with mass e5ects. If mi , the results from massless QCD can be used directly by using the Abelian limit CF = 1. In case mi we must use the well known next to leading order QED results, e.g. [93], and the virtual probabilities take the following form for fermions:  2 ei s  2 s  mi ;   (46)2 log 2 − 3 log 2 ; 2 wfi (s; ) = (131)  s m2i s ei2  2 s  log 2 − 1 2 log 2 ; +log 2 − 3 log 2 ; mi :  (46)2 mi mi mi Note, that in the last equation the full subleading collinear logarithmic term [60] is used in distinction to Ref. [93]. In the explicit two loop calculation presented in Ref. [94] it can be seen that the full collinear term also exponentiates at the subleading level in massive QED. For W ± bosons we have analogously: # $ s ei2 M2 2 2 s wwi (s; ) = log 2 − 1 2 log 2 + log : (132) (46)2 M M2 In addition we have collinear terms for external on-shell photon lines from fermions with mass mj and electromagnetic charge ej up to scale M :  nf ej2 j M2   N log ; mj ;   3 462 C  2 2 2 wi (M ; ) = (133) nf  1 ej2 j M2   N log 2 ; mj :  3 462 C mj j=1 Note that automatically, wi (M 2 ; M 2 ) = 0. At one loop order, this contribution cancels against terms from the renormalization of the QED coupling up to scale M . For external Z-bosons, however, there are no such collinear terms since the mass is large compared to the mi . Thus, the corresponding RG-logarithms up to scale M remain uncancelled. The appropriate initial condition is given by Eq. (130) evaluated at the matching point = M . Thus we 2nd for the general solution in region (II): ⊥ M⊥ (p1 ; : : : ; pn ; g; g ; ) = MBorn (p1 ; : : : ; pn ; g; g )

ng s - Yi 2 - 1 2 s Ti (Ti + 1) + 00 + nB 00 log 2 log + nW ×exp − 2 2 i=1 46 46 2 M 26 26 M

nf - Yk 2 % 2 s s & 1 Tk (Tk + 1) + log − 3 log − 2 46 46 2 M2 M2 k=1

M. Melles / Physics Reports 375 (2003) 219 – 326

+ 46

n

−

1 + .k; R m2k m2k + . k; L 4 M2 4M 2

s log 2 mt n

257

nf

1 exp − (wfi (s; 2 ) − wfi (s; M 2 )) 2 i=1

w 1 1 (wwi (s; 2 ) − wwi (s; M 2 )) − w (M 2 ; 2 ) 2 i=1 2 i=1 i

⊥ (p1 ; : : : ; pn ; g; g ) =MBorn

ng s - Yi 2 1 Ti (Ti + 1) + log2 2 ×exp − 2 i=1 46 46 2 M - s + nW 00 + nB 00 log 2 26 26 M

nf 1 - Yk 2 % 2 s s & log − − 3 log Tk (Tk + 1) + 2 46 46 2 M2 M2 k=1 s m2k - 1 + .k; R m2k log 2 + .k; L + 46 4 M2 4M 2 mt n M2 ei2 s s m2i m2i 1 log 2 log + 2 log log + 3 log ×exp − 2 i=1 (46)2 mi M 2 M2 m2i m2i n 1 1 ei2 i m2i M2 − −2 log 2 N log 2 : 2 i=1 3 462 C mi

(134)

The last equality holds for mi M and we have replaced the matching scale M by mt in the Yukawa enhanced subleading terms since the coeScients are unambiguously determined and the argument in the corresponding logarithm must be mt [38,29]. It is important to note again that, unlike the situation in QCD, in the electroweak theory we have in general di5erent mass scales determining the running of the couplings of the physical on-shell renormalization scheme quantities. We have written the above result in such a way that it holds for arbitrary chiral fermions and transversely polarized gauge bosons. In order to include physical external photon states in the on-shell scheme, the renormalization condition is given by the requirement that the physical photon does not mix with the Z-boson. This leads to the condition that the Weinberg rotations in Fig. 11 at one loop receive no RG-corrections. Thus, above the scale M the subleading collinear and RG-corrections cancel for physical photon and Z-boson states. Since the Yukawa enhanced terms are novel features in broken gauge theories as compared to the situation in QCD we use the non-Abelian generalization of the Gribov theorem in the following to prove the correctness of our splitting function approach for speci2c processes. Since we are interested here in corrections to order O(-n log2n−1 s=M 2 ), each additional loop correction to the universal subleading terms in the previous section must yield two logarithms, i.e. we are considering DL-corrections to the basic process like the inner fermion loop in Fig. 13. It is of particular

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M. Melles / Physics Reports 375 (2003) 219 – 326

k1

k

k1

k

l l k2

k1

k

k2

k1

k

l l k2

k2

Fig. 12. Two loop Feynman diagrams yielding Yukawa enhanced logarithmic corrections to the third generation of fermions in the 2nal state. The Ward identity in Eq. (135) assures that in the Feynman gauge, the sum of all vertex and self energy diagrams does not lead to additional SL logarithms at the two loop level. Only corrections to the original one loop vertex need to be considered and lead to the exponentiation of Yukawa terms in the fermionic SM sector to SL accuracy.

importance that all additional gauge bosons must couple to external legs, since otherwise only a subleading term of order O(-n log2n−2 MQ 2 ) would be generated. All subleading corrections generated by the exchange of gauge bosons coupling both to external Goldstone bosons and inner fermion lines cancel analogously to a mechanism found in Ref. [95] for terms in heavy quark production in -collisions in a Jz = 0 state. Formally this can be understood by noting that such terms contain an infrared divergent correction. The sum of those terms, however, is given by the Sudakov form factor. Thus any additional terms encountered in intermediate steps of the calculation cancel. This point can be understood also from the principle of gauge invariance. At the two loop level for instance we have to consider diagrams of the type depicted in Fig. 12. They involve Vertex corrections 5 (k12 ; l2 ; (k1 −l)2 ) and self energy terms K((k1 −l)2 ) with the same overall Yukawa-term structure. Writing the gauge coupling in the symmetric basis for clarity since we are considering a regime where s = (k1 − k2 )2 M 2 , where M is the gauge boson mass. In any case, local gauge invariance is not violated in the SM and for heavy particles in the high energy limit, we can perform the calculation in a basis which is more convenient. For our purposes we need to investigate terms containing three large logarithms in those diagrams. Since the would-be Goldstone boson loops at one loop only yield a single logarithm it is clear that the gauge boson loop momentum l must be soft. Thus we need to show that the UV logarithm originating from the k integration is identical (up to the sign) in both diagrams. We can therefore neglect the loop momentum l inside the fermion

M. Melles / Physics Reports 375 (2003) 219 – 326

e

f, k1

+

f , k1

e

bt

e

+

259

...

-

...

-

e f , k2

f , k2

Fig. 13. Feynman diagrams yielding Yukawa enhanced logarithmic corrections to the third generation of fermions in the 2nal sate. The inner scattering amplitude is taken on the mass shell. No DL-corrections originate from the inner loop. At higher orders, the subleading corrections are given in factorized form according to the non-Abelian generalization of Gribov’s theorem as described in the text. Corrections from gauge bosons inside the Goldstone-boson loop give only sub-subleading contributions. DL-corrections at two and higher loop order are given by gauge bosons coupling to (in principle all) external legs as schematically indicated.

loop. It is then straightforward to see that 9 K(k12 ) = 5 (k12 ; 0; k12 ) ; 9k1

(135)

where the full sum of all contributing self energy and vertex diagrams must be taken. Thus, we have established a Ward identity for arbitrary Yukawa couplings of scalars to fermions and thus, the identity of the UV singular contributions. The relative sign is such that the generated SL logarithms of the diagrams in Fig. 12 cancel each other. The existence of such an identity is not surprising since it expresses the fact that also the Yukawa sector is gauge invariant. We are thus left with gauge boson corrections to the original vertices in the on-shell renormalization scheme such as depicted in Fig. 13. At high energies we can therefore employ the non-Abelian version of Gribov’s bremsstrahlung theorem. The soft photon corrections are included via matching as discussed above. For the one loop process in Fig. 13, for instance, we include only corrections with top and bottom quarks and assume on-shell renormalization. Thus the corrections at higher orders factorize with respect to the one loop fermion amplitude and M“Born” (p1 ; : : : ; pn ) = M1loop (p1 ; : : : ; pn ). Note that the latter is also independent of the cuto5 since the fermion mass serves as a natural regulator. In principle we can choose the top-quark mass to be much larger than for instance. In our case we have for the electroweak DL corrections at the weak scale = M : 2 2 Y 2 s g g Wlew (s; M 2 ) = log2 2 : Ti (Ti + 1) + (136) 2 2 166 166 4 M We now want to consider speci2c processes relevant at future e+ e− colliders and demonstrate how to apply the non-Abelian version of Gribov’s factorization theorem for the higher order corrections. The subleading corrections are then compared to the general splitting function result in Eqs. (128) and (130). Below we use the physical 2elds for the respective contributions. From the arguments of Section 6.1 it is now straightforward to include also top-Yukawa terms for chiral quark 2nal states. These terms occur for left handed bottom as well as top quark external lines. The situation for a typical Drell–Yan process is depicted in Fig. 13 where for the inner scattering

260

M. Melles / Physics Reports 375 (2003) 219 – 326

amplitude we have two contributions. We neglect all terms of order O(m2f =s; M 2 =s). Using on-shell renormalization we 2nd for the inner amplitude on the left in Fig. 13 for a right handed electron in the initial and a left handed bottom quark in the 2nal state from the ± loop for the sum of the and Z contributions according to the Feynman rules in Appendix A.2: a

e4 m2t e+ | |eR− 4sM 2 sw2 cw2 L fL |l=(2l − k1 − k2 ) |fR dn l × + a .DY ct 2 n 2 (26) (l − mf + i#)((l − k1 )2 − M 2 + i#)((l − k2 )2 − M 2 + i#)

ADY 1loop = −

=−

i e4 m2t M e+ | |eR− fL | |fR (B23 − B23 ): 3262 4sM 2 sw2 cw2 L

(137)

M = −log s=M 2 . For the diagram on the right The scalar functions at high energy evaluate to B23 − B23 ± in Fig. 13 we have for the bottom again only the contribution. Here we 2nd for the sum of the and Z contributions: b

e4 m2t Qt e+ | |eR− 2sM 2 sw2 cw2 L fL |l= l=|fR dn l × + b .DY ct (26)n (l2 − M 2 + i#)((l − k1 )2 − m2t + i#)((l − k2 )2 − m2t + i#)

ADY 1loop = −

=

e4 m2t Qt i M e+ | |eR− fL | |fR (B23 − B23 ): 3262 2sM 2 sw2 cw2 L

(138)

In all cases we renormalize on-shell, i.e. by requiring that the vertex vanishes when the momentum transfer equals the masses of the external on-shell lines. All on-shell self energy contributions do not contribute in this scheme. For external left handed top quarks, the ± loop is mass suppressed and we only have to consider the and H corrections. They are given by replacing Qt → 2Qt (Tt3 )2 and Qt → 12 Qt in Eq. (138). It turns out that the ZH contributions equal the corrections from the ± and Z ± in the case of the bottom calculation. The Born amplitude is given by e2 (Qf − Tf3 )eL+ | |eR− fL | |fR : scw2  e2  − +  i   6sc2 eL | |eR fL | |fR ; fL = tL ; bL w = 2  2 e    i 2 eL+ | |eR− fR | |fL ; fR = tR scw 3

DY =i MBorn

2

(139)

for top and bottom quarks. In all cases, log Mm2 terms can be savely neglected to the accuracy we t are working. Thus we 2nd for left handed quarks of the third generation: g2 1 m2t s DYL DY (140) .f; tL =bL log 2 M1loop (p1 ; : : : ; p4 ) = MBorn (p1 ; : : : ; p4 ) 1 − 1662 4 M 2 M

M. Melles / Physics Reports 375 (2003) 219 – 326

261

For right handed external top quarks we have ± , and H corrections. In that case we observe that the ZH , ± and Z ± loops have an opposite sign relative to the left handed case. For the corrections corresponding to the topology shown on the right in Fig. 13 we must replace Qt in Eq. (138) by Qf − Tf3 = 16 for the ± graph. The same contribution is obtained by adding the H and loops and we 2nd: g2 1 m2t s DYR DY (141) .f; tR log 2 M1loop (p1 ; : : : ; p4 ) = MBorn (p1 ; : : : ; p4 ) 1 − 1662 2 M 2 M At higher orders we note that the exchange of gauge bosons inside the one loop process is subsubleading and we arrive at the factorized form analogous to the Yukawa corrections for external Goldstone bosons. Since these corrections are of universal nature we can drop the speci2c reference to the Drell–Yan process and the application of the generalized Gribov-theorem for external fermion lines to all orders yields:

nf 1 M(p1 ; : : : ; pn ; 2 ) = M1loop (p1 ; : : : ; pn ) exp − Wlew (s; M 2 ) (142) 2 l=1

Wlew (s; M 2 )

is given in Eq. (136) and the quantum numbers are those of the external fermion where lines. Since at high energies all fermions can be considered massless we can again absorb the chiral top-Yukawa corrections into universal splitting functions as in Ref. [37]. Thus in the electroweak theory we 2nd to next to leading order the corresponding probability for the emission of gauge bosons from chiral fermions subject to the cuto5 are given by Eqs. (128) and (130). The corrections from below the scale M need to be included via matching as described above. For physical observables, soft real photon emission must be taken into account in an inclusive (or semi inclusive) way and the parameter 2 in (134) will be replaced by parameters depending on the experimental requirements. This will be brieUy discussed in Section 3.4. Next we turn to longitudinal degrees of freedom after 2rst reviewing the Goldstone boson equivalence theorem. 3.2. The equivalence theorem At high energies, the longitudinal polarization states can be described with the polarization vector is given in Eq. (117). The connection between S-matrix elements and Goldstone bosons is provided by the equivalence theorem [105–107]. It states that at tree level for S-matrix elements for longitudinal bosons at the high energy limit M 2 =s → 0 can be expressed through matrix elements involving their associated would-be Goldstone bosons. We write schematically in case of a single gauge boson (Fig. 14): Mw ± ± M(WL ; phys ) = M( ; phys ) + O √ ; (143) s Mz M(ZL ; phys ) = iM(; phys ) + O √ : (144) s The problem with this statement of the equivalence theorem is that it holds only at tree level [108,109]. For calculations at higher orders, additional terms enter which change Eqs. (143) and (144).

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M. Melles / Physics Reports 375 (2003) 219 – 326 +

+

e

WL

e

+

φ

+

-

φ

-

→

M /s → 0 2

-

-

e

WL

e

Fig. 14. The pictorial Goldstone boson equivalence theorem for W -pair production in e+ e− collisions. The correct DL-asymptotics for longitudinally polarized bosons are obtained by using the quantum numbers of the charged would-be Goldstone scalars at high energies.

Because of the gauge invariance of the physical theory and the associated BRST invariance, a modi2ed version of Eqs. (143) and (144) can be derived [108] which reads Mw ± ± k M(W (k); phys ) = Cw Mw M( (k); phys ) + O √ ; (145) s Mz ; (146) k M(Z (k); phys ) = iCz Mz M((k); phys ) + O √ s where the multiplicative factors Cw and Cz depend only on wave function renormalization constants and mass counterterms. Thus, using the form of the longitudinal polarization vector of Eq. (117) we can write Mw ± ± M(WL (k); phys ) = Cw M( (k); phys ) + O √ ; (147) s Mz : (148) M(ZL (k); phys ) = iCz M((k); phys ) + O √ s We see that in principle, there are logarithmic loop corrections to the tree level equivalence theorem. The important point in our approach, however, is that the correction coeScients are not functions of the energy variable s: Cw = Cw (; Q M; g; g );

Cz = Cz (; Q M; g; g ) :

(149)

The pictorial form of the Goldstone boson equivalence theorem is depicted in Fig. 14 for longitudinal W -boson production at a linear e+ e− collider. In the following we denote the logarithmic variable t ≡ log s=2 , where is a cuto5 on the transverse part of the exchanged virtual momenta k of all involved particles, i.e. 2 2 6 k⊥ ≡ min(2(kpl )(kpj )=(pl pj ))

(150)

for all j = l. The non-renormalization group part of the evolution equation at high energies is given on the invariant matrix element level by Eq. (84): 9 M(L(k); 9t

phys )

= K(t)M(L(k);

phys )

(151)

M. Melles / Physics Reports 375 (2003) 219 – 326

263

and thus, after inserting Eqs. (147), (148) we 2nd that the same evolution equation also holds for M( (k); phys ). The notation here is L = {WL± ; ZL } and = { ± ; }, respectively. Thus, the log s=2 dependence in our approach is unrelated to the corrections to the equivalence theorem, and in general, is unrelated to two point functions in a covariant gauge at high energies where masses can be neglected. This is a consequence of the physical on-shell renormalization scheme where the MS renormalization scale parameter Q ∼ M . Physically, this result can be understood by interpreting the correction terms Cw and Cz as corrections required by the gauge invariance of the theory in order to obtain the correct renormalization group asymptotics of the physical Standard Model 2elds. Thus, their origin is not related to Sudakov corrections. In other words, the results from the discussion of scalar QCD in Section 2 should be applicable to the subleading scalar sector in the electroweak theory regarding a non-Abelian scalar gauge theory as the e5ective description in this range according to Lsymm in Eq. (11). The only additional complication is the presence of subleading Yukawa enhanced logarithmic corrections which will be discussed below. It is also worth noticing, that at one loop, the authors of Ref. [29] obtain the same result for the contributions from the terms of Eq. (149). √ In their approach, where all mass-singular terms are identi2ed and the renormalization scale Q = s, these terms are canceled by additional corrections from mass and wave function counterterms. At higher orders it is then clear that corrections from two point functions are subsubleading in a covariant gauge. 3.3. Longitudinal degrees of freedom According to the discussion of the previous section we can use Goldstone bosons in the high energy regime as the relevant degrees of freedom for longitudinal gauge boson production. Thus, the Higgs boson and the would-be Goldstone bosons actually receive the same corrections in high energy processes (up to purely electromagnetic terms). Regulating the virtual infrared divergences with the transverse momentum cuto5 as described above, we 2nd the virtual contributions to the splitting functions for external Goldstone and Higgs bosons: V P V± ± (z) = P (z)V = PHH (z)

=

Ti (Ti + 1) + tan2 /w

Yi 2

2

s 3 m2t −2 log 2 + 4 − .(1 − z) : 2 M2

(152)

The functions can be calculated directly from loop corrections to the elementary processes in analogy to QCD [65–67] and the logarithmic term corresponds to the leading kernel of Ref. [37]. We introduce virtual distribution functions which include only the e5ects of loop computations. These ful2ll the Altarelli–Parisi equations in analogy to Eq. (74): 1 9 (z; t) g2 dy V = 2 (z=y; t)P (y) : (153) 9t 86 z y R V + P , where R denotes the contribution from real The splitting functions are related by P = P boson emission. P is free of logarithmic corrections and positive de2nite.

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Inserting the virtual probability of Eq. (152) into Eq. (153) we 2nd: 2 Yi s g2 s 2 (1; t) = 0 exp − 2 Ti (Ti + 1) + tan /w log2 2 − 4 log 2 86 2 s 3 m2t : log 2 + 2 M2

(154)

These functions describe the total contribution for the emission of virtual particles (i.e. z=1), with all invariants large compared to the cuto5 , to the densities (z; t) ( = { ± ; ; H}). The normalization is not per line but on the level of the cross section. For the invariant matrix element involving n external scalar particles we thus 2nd at the subleading level: n 1 W (s; 2 ) ; (155) M(p1 ; : : : ; pn ; g; 2 ) = MBorn (p1 ; : : : ; pn ; g)exp − 2 i=1 i where Wi (s; 2 )

g2 = 1662

3 m2t s s Yi2 2 2 s log 2 − 4 log 2 + Ti (Ti + 1) + tan /w log 2 : (156) 4 2 M2

The functions Wi correspond to the probability of emitting a virtual soft and/or collinear gauge boson from the particle subject to the infrared cuto5 . Typical diagrams contributing to Eq. (156) in a covariant gauge are depicted in Fig. 8. The universality of the splitting functions is crucial in obtaining the above result. Again, since the Yukawa enhanced terms are novel features in broken gauge theories as compared to the situation in QCD we use the non-Abelian generalization of the Gribov theorem in the following to prove the correctness of our splitting function approach for speci2c processes using on-shell renormalization of the external Goldstone bosons. Since the three fermion loop is more complicated than the situation in the fermionic sector above, we provide some more details in deriving the respective Ward identity. At the two loop level, we need to consider the diagrams displayed in Fig. 15. The corresponding two loop amplitudes read (neglecting l outside the fermion loop): dn l d n k (k1 − k2 ) Tr[(Gr !r + Gl !l )(k − k1 )2k2 (k − k1 + l=)(Gr !r + Gl !l )k] ; (157) (46)n (46)n (l2 − 32 )(k2 + l)2 (k1 − l)2 k 2 (k − k1 )2 (k − k1 + l)2

dn l (46)n

d n k (k1 − k2 ) Tr[(Gr !r + Gl !l )(k − k1 + l=)(Gr !r + Gl !l )k]4k1 k2 ; (46)n (l2 − 32 )(k2 + l)2 (k1 − l)2 k 2 (k − k1 + l)2 (k1 − l)2

(158)

where we omit common factors and the scalar masses taking M ∼ 3 for clarity. The soft photon corrections must also be included via matching. The Gr; l denote the chiral Yukawa couplings and !r; l = 12 (1 ± 5 ). The gauge coupling is again written in the symmetric basis. For our purposes we need to investigate terms containing three large logarithms in those diagrams. Since the fermion

M. Melles / Physics Reports 375 (2003) 219 – 326

k1

k

265

k1

k

l

l k2

k2

Fig. 15. Two loop corrections involving Yukawa couplings of scalars to fermions. The Ward identity in Eq. (135) assures that in the Feynman gauge, the sum of both diagrams does not lead to additional SL logarithms at the two loop level. Only corrections to the original one loop vertex need to be considered and lead to the exponentiation of Yukawa terms in the SM to SL accuracy.

loops at one loop only yield a single logarithm it is again clear that the gauge boson loop momentum l must be soft. Thus we need to show that the UV logarithm originating from the k integration is identical (up to the sign) in both diagrams. We can therefore neglect the loop momentum l inside the fermion loop. We 2nd for the fermion loop vertex 5 (k12 ; 0; k12 ) belonging to Eq. (157): Tr[(Gr !r + Gl !l )(k − k1 ) (k − k1 )(Gr !r + Gl !l )k] k 2 (k − k1 )2 (k − k1 )2 =

4Gr Gl (2k1 (k 2 − k1 k) + k (k12 − k 2 )) : k 2 (k − k1 )4

(159)

This we need to compare with the self energy loop K(k12 ) from Eq. (158): 9 Tr[(Gr !r + Gl !l )(k − k1 )(Gr !r + Gl !l )k] 9k1 k 2 (k − k1 )2 =

2k1 (k 2 − k1 k) + k (k12 − k 2 ) 9 4Gr Gl (k1 k − k 2 ) = 4G G : r l 9k1 k 2 (k − k1 )2 k 2 (k − k1 )4

(160)

In short we have established the analogous Ward identity in Eq. (135) in the longitudinal would-be Goldstone boson sector. The relative sign is again such that the generated SL logarithms of the diagrams in Fig. 15 cancel each other. The existence of such an identity expresses the fact that also the longitudinal Yukawa sector is gauge invariant. At higher orders this Ward identity ensures the corresponding cancellations to SL accuracy. Also in an axial gauge the corrections can be seen to factorize accordingly since in this gauge DL terms originate only from on-shell two point functions. We are thus left with gauge boson corrections to the original vertices in the on-shell renormalization scheme such as depicted in Fig. 16. At high energies we can therefore employ the non-Abelian version of Gribov’s bremsstrahlung theorem.

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φ , k1 +

+

e

t

...

-

e

φ , k2 -

Fig. 16. A Feynman diagram yielding Yukawa enhanced logarithmic corrections in the on-shell scheme. At higher orders, the subleading corrections are given in factorized form according to the non-Abelian generalization of Gribov’s theorem as described in the text. Corrections from gauge bosons inside the top-loop give only sub-subleading contributions. DL-corrections at two and higher loop order are given by gauge bosons coupling to (in principle all) external legs as schematically indicated.

Thus the corrections at higher orders factorize again with respect to the one loop amplitude and M“Born” (p1 ; : : : ; pn )=M1loop (p1 ; : : : ; pn ). Note that the latter is also independent of the cuto5 since the fermion mass serves as a natural regulator. The subleading corrections are then compared to the general splitting function approach yielding Eq. (155). In the case of the amplitude of Fig. 16 we must use the quantum numbers of the associated Goldstone bosons and we have the following Born amplitude MBorn (p1 ; : : : ; p4 ) = i

e2 e− | |eL+ (k1 − k2 ) 2scw2 R

(161)

and at one loop we have two fermion loops contributing (ttb and bbt). The renormalization condition is provided by the requirement that the corrections vanish at the weak scale, i.e. for s = M 2 , which amounts to subtracting the vertex for that case. The 2rst diagram of the two is, according to the Feynman rules of Appendix A.2 using the physical 2elds, given by e 4 m2 t e eR− | |eL+ c+ Attb 1loop (p1 ; : : : ; p4 ) = 3 2 s2 2sM w ;Z

× =

t t Tr{!− l=!+ (l= − k2 ) (c+ !+ + c − !− )(l= + k1 )} dn l + .ttb ct 2 2 (26)n (l2 − mb + i#)((l + k1 )2 − mt + i#)((l − k2 )2 − m2t + i#)

e4 m2t 3iQt M e− | |eL+ (B23 − B23 )(k1 − k2 ) ; 1662 cw2 2sM 2 sw2 R

(162)

f f where !± = 12 (1±5 ) and the chiral couplings are given by c± =Qf for the photon and c+ =sw =cw Qf f 2 3 ttb and c− = (sw Qf − Tf )=sw cw for Z-bosons, respectively. The counterterm .ct is chosen such that the logarithmic corrections vanish for s = M 2 . Thus, the sum of the scalar functions is to logarithmic

M. Melles / Physics Reports 375 (2003) 219 – 326

267

M accuracy B23 − B23 = −log s=M 2 . Analogously, we have for the bbt quark loop: e 4 m2 t e Abbt eR− | |eL+ c+ 1loop (p1 ; : : : ; p4 ) = 3 2 s2 2sM w ;Z

× =−

b b !+ + c − !− )(−l= + k2 )} dn l Tr{!+ (−l=)!− (−l= − k1 ) (c+ + .bbt ct 2 2 n 2 2 (26) (l − mt + i#)((l + k1 ) − mb + i#)((l − k2 )2 − m2b + i#)

3i(Qb − Tb3 ) e4 m2t M e− | |eL+ (B23 − B23 )(k1 − k2 ) : 1662 cw2 2sM 2 sw2 R

Adding both results (162) and (163) we 2nd g2 3 m2t s M1loop (p1 ; : : : ; p4 ) = MBorn (p1 ; : : : ; p4 ) 1 − log 2 1662 2 M 2 M and the all orders result to subleading accuracy is given by

n 1 M(p1 ; : : : ; pn ; 2 ) = M1loop (p1 ; : : : ; pn ) exp − Wlew (s; M 2 ) 2

(163)

(164)

(165)

l=1

where Wlew (s; M 2 ) is given in Eq. (136). The subleading Yukawa corrections from the Altarelli-Parisi in Eq. (155) agree with the corresponding results from the application of the Gribov-theorem in Eq. (165). For longitudinal Z-boson and Higgs production, we note that there is only one non-mass suppressed elementary vertex with two neutral scalars, namely the ZH vertex. As mentioned above, universal terms are related to the massless limit. For the “Born amplitude” of the Higgs–strahlung vertex we have e ZH MBorn = (k − k2 ) : (166) 2sw cw 1 The universal Yukawa corrections to both external and H states from an o5 shell Z line are then given by the corrections depicted in the inner fermion loop of Fig. 17. Here we 2nd e3 m2t 4M 2 sw2 n t t Tr{5 (l=)(l= − k2 ) (c+ !+ + c − !− )(l= + k1 )} d l × + .ZH ct 2 2 n 2 2 (26) (l − mt + i#)((l + k1 ) − mt + i#)((l − k2 )2 − m2t + i#)

AZH 1loop (p1 ; : : : ; p3 ) = 3

=

6Tt3 e3 m2t M (B23 − B23 )(k1 − k2 ) 1662 sw cw 4M 2 sw2

=

ZH MBorn

and thus ZH M1loop (p1 ; : : : ; p3 )

e2 m2t 3 s 1− log 2 2 2 2 2 166 sw M M

(167)

:

(168)

From the same line of reasoning as for the charged Goldstone bosons we 2nd that the all orders result is given by Eq. (165). At the subleading level, this is equivalent to the corresponding corrections obtained in Eq. (155).

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M. Melles / Physics Reports 375 (2003) 219 – 326

χ , k1

Z, q t

...

q2=s

H, k 2 Fig. 17. A Feynman diagram yielding Yukawa enhanced logarithmic corrections to external longitudinal Z-bosons and Higgs lines in the on-shell scheme. At higher orders, the subleading corrections are given in factorized form according to the non-Abelian generalization of Gribov’s theorem as described in the text. Corrections from gauge bosons inside the top-loop give only sub-subleading contributions.

In addition to the Sudakov corrections in Eq. (156) we also have to include terms corresponding to the renormalization of the mass terms in the Yukawa coupling of the Born amplitude (∼ m2t =M 2 ; m2H =M 2 ) at the one loop level [29]. At higher orders, mass renormalization terms are connected to two point functions and thus subsubleading. The relevant higher order SL-RG terms, however, will be discussed Section 3.5. 3.4. Semi-inclusive cross sections In order to make predictions for observable cross sections, the unphysical infrared cuto5 2 has to 2 2 be replaced with a cuto5 expt , related to the lower bound of k⊥ of the other virtual particles of those gauge bosons emitted in the process which are not included in the cross section. We assume that 2 2 expt ¡ M 2 , so that the non-Abelian component of the photon is not essential. The case expt ¿ M2 is much more complicated and is discussed in Ref. [28] through two loops at the DL level. We can write the expression for the semi-inclusive cross section in a compact way as follows: dH(p1 ; : : : ; pn ; g; g ; expt ) = dHelastic (p1 ; : : : ; pn ; g; g ; ) exp(wexpt (s; mi ; ; expt )) :

(169)

The dependence in this expression cancels and the semi-inclusive cross section depends only on the parameters of the experimental requirements. In terms of all the virtual corrections discussed above the full result is given by: dH(p1 ; : : : ; pn ; g; g ; expt )

n nf ng = dHBorn (p1 ; : : : ; pn ; g; g ) exp − Wgi (s; M 2 ) − Wfi (s; M 2 ) − W i (s; M 2 )

i=1

i=1

i=1

M. Melles / Physics Reports 375 (2003) 219 – 326

269

nf nw 2 2 ×exp − (wfi (s; ) − wfi (s; M )) − (wwi (s; 2 ) − wwi (s; M 2 )) i=1

−

n

i=1

wi (M 2 ; m2j ) × exp(wexpt (s; mi ; ; expt )) ;

(170)

i=1

where ng denotes the number of transversely polarized gauge bosons and nf the number of external fermions. This expression omits all RG corrections, even at the one loop level. The functions W and w correspond to the logarithmic probability to emit a soft and/or collinear particle per line, where the capital letters denote the probability in the high energy e5ective theory and the lower case letter the corresponding one from pure QED corrections below the weak scale. The matching condition is implemented such that for = M only the high energy solution remains. For the contribution from scalar 2elds = { ± ; ; H } above the scale M we have W i (s; M 2 ) in Eq. (156) with - = g2 =46 and tan2 /w = - =-. The last term is written as a logarithm containing the top quark mass mt rather than the weak scale M since these terms always contain mt as the heaviest mass in the loop correction [38]. Wfi (s; M 2 ) and Wgi (s; M 2 ) are given in Eqs. (128) and (129) respectively. Again we note that for external photon and Z-boson states we must include the mixing appropriately as discussed in Section 3.1. For the terms entering from contributions below the weak scale we have wfi (s; 2 ) given in Eq. (131), wwi (s; 2 ) in Eq. (132) and wi (s; 2 ) in Eq. (133) for the virtual corrections. For real photon emission we have in the soft photon approximation: wexpt (s; mi ; ; expt )  n ei2 s  2 s 2 s  −log ; + log − 3 log  2  2 2 2  (46) expt i=1 = n   s s m2i s ei2 s  2  log 2 − 1 2 log 2 + log 2 2 log 2 log 2 − 1 ;  (46)2 mi mi expt mi i=1

mi (171) mi

where n is the number of external lines and the upper case applies only to fermions since for W ± we have ¡ M . Note that in all contributions from the regime ¡ M we have kept mass terms inside the logarithms. This approach is valid in the entire Standard Model up to terms of order O(log mt =M ). Note, however, that in the Yukawa enhanced terms the replacement M → mt is legitimate. The overall -dependence in the semi-inclusive cross section cancels and we only have a dependence on the parameter expt related to the experimental energy resolution. All universal electroweak Sudakov corrections at DL and SL level exponentiate. 3.5. Electroweak RG corrections The way to implement the SL-RG corrections is clear from the discussion in Section 2.6. At high energies, the DL phase space is essentially described by an unbroken SU (2) × U (1) theory in which we can calculate the high energy contributions. In this regime, all particle masses can be neglected and we have to consider the following virtual electroweak DL phase space integral with running

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couplings in each gauge group:

       T (T + 1)-(2 ) s 1 2 2 (2 )  dk dv =4)1 (Y i i i RG 2 ⊥ W˜ iV (s; ) = + 2 2 2 26 2 k⊥ k2  k⊥ =s v   1 + c log k⊥ log ⊥    1 + c 2 2 s 1 -(2 )Ti (Ti + 1) 1 -(2 ) -(2 ) log 2 log − 1 + 2 log = 26 c -(s) c -(s) - (2 )Yi2 1 1 - (2 ) s - (2 ) + − 1 + 2 log ; log 2 log 86 c - (s) - (s) c

(172)

where -(2 )=g2 (2 )=46, - (2 )=g 2 (2 )=46, c =-(2 )00 =6 and analogously, c =- (2 )00 =6. In each case, the correct non-Abelian or Abelian limit is reproduced by letting the corresponding couplings of the other gauge group approach zero. In this way it is easy to see that the argument of the running couplings can only be what appears in Eq. (172). The form of Eq. (172) is valid for fermions, transversely and longitudinally polarized external lines but (omitted) subleading terms as well as the quantum numbers of the weak isospin Ti and the weak hypercharge Yi di5er. In order to implement the missing soft photon contribution, we choose the analogous form of solution in Eq. (114) and have to implement it in such a way that for = M Eq. (172) is obtained. The full result for the respective semi-inclusive cross sections is then given by dHRG (p1 ; : : : ; pn ; g; g ; expt ) = dHBorn (p1 ; : : : ; pn ; g(s); g (s)) ng n nf RG 2 RG 2 RG 2 Wgi (s; M ) − Wfi (s; M ) − W i (s; M ) × exp −

i=1

i=1

i=1

nf

× exp − (wfRG (s; 2 ) − wfRG (s; M 2 )) i i i=1

n nw wi (M 2 ; m2j ) exp(wRG (s; mi ; ; expt )) ; − (wwRGi (s; 2 ) − wwRGi (s; M 2 )) − expt i=1

(173)

i=1

where nf denotes here again the number of external fermions. The argument of the gauge couplings in the Born cross section indicate the one loop renormalization of the couplings which is not included in the exponential expressions but which at one loop is genuinely subleading: s -(M 2 ) 2 log 2 ; -(s) = -(M ) 1 − 00 (174) 6 M 2 s 2 - (M ) log 2 ; (175) - (s) = - (M ) 1 − 00 6 M

M. Melles / Physics Reports 375 (2003) 219 – 326

where -(M 2 ) = e2 (M 2 )=46sw2 and - (M 2 ) = e2 (M 2 )=46cw2 with   nf 2 2 1 e M Q2 N j log 2  e2 (M 2 ) = e2 1 + 3 462 j=1 j C mj

271

(176)

and e2 =46 = 1=137. If there are non-suppressed mass ratios in the Born term, also these terms need to be renormalized at one loop (see Ref. [29]). Higher order mass renormalization terms would then be sub-subleading. The function W RG (s; M 2 ) is given by i s 1 -(M 2 ) -(M 2 )Ti (Ti + 1) 1 -(M 2 ) 2 log − 1 + log (s; M ) = log W RG i 26 c M2 -(s) c2 -(s) - (M 2 )Yi2 1 1 - (M 2 ) s - (M 2 ) + − 1 + 2 log log 2 log 86 c M - (s) - (s) c -(M 2 ) s - (M 2 ) Yi2 3 -(M 2 ) m2t s − Ti (Ti + 1) + 4 log 2 − log 2 ; 46 46 4 M 2 46 M 2 mt (177) where we again have mt in the argument of the Yukawa enhanced correction [38]. Analogously for fermions we have -(M 2 ) -(M 2 )Ti (Ti + 1) 1 -(M 2 ) s 1 RG 2 Wfi (s; M ) = log 2 log − 1 + 2 log 26 c M -(s) c -(s) - (M 2 )Yi2 1 - (M 2 ) s - (M 2 ) 1 + log 2 log − 1 + 2 log 86 c M - (s) - (s) c -(M 2 ) s - (M 2 ) Yi2 − 3 log 2 Ti (Ti + 1) + 46 46 4 M s m2i -(M 2 ) 1 + .i; R m2i log : (178) − + . i; L 46 4 M2 4M 2 m2t The last term contributes only for left handed bottom and for top quarks as mentioned above and f denotes the corresponding isospin partner for left handed fermions. s 1 -(M 2 ) -(M 2 )Ti (Ti + 1) 1 -(M 2 ) RG 2 log 2 log − 1 + 2 log Wgi (s; M ) = 26 c M -(s) c -(s) - (M 2 )Yi2 1 - (M 2 ) s - (M 2 ) 1 + log 2 log − 1 + 2 log 86 c M - (s) - (s) c s -(M 2 ) - (M 2 ) − .i; W 00 + .i; B 00 log 2 : (179) 6 6 M

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Again we note that for external photon and Z-boson states we must include the mixing appropriately as discussed in Ref. [37]. For the terms entering from contributions below the weak scale we have for fermions:  2 s 1 s 1 e2 (2 ) e2 (2 ) 3 ei   log 2 log 2 − 1 + 2 log 2 − log 2 ; mi ;   862 c e (s) c e (s) 2         ei2  1 s 3 s e2 (2 ) m2  log − 1 − log log − log ; RG 2 (180) wfi (s; ) = 862  c m2 e2 (s) 2 m2 2        nf  2 2 2 2   1 e e (m )  m  1 2 j   1− ; mi ; Q N log + 2 log 2  j C  c e (s) 3 462 m2j  j=1

nf 2 j where c = − 13 e2 =462 j=1 Q N . Analogously, for external W-bosons and photons we 2nd:  j C s M2 e2 (2 ) ei2  1 RG 2 log 2 log 2 − 1 − log 2 wwi (s; ) = 2 86  c M e (s)  nf 2 1 e 1 e (M )  M  2 j 1− ; + 2 log 2 Q N log j C c e (s) 3 462 j=1 m2j   nf  ej2 j 1 M2   N log ; mj ;   2  3 j=1 462 C wi (M 2 ; 2 ) = nf 2  2   1 ej N j log M ; m :   j 3 462 C m2j j=1 2

2



2

(181,182)

(183)

Note that the function wi (M 2 ; 2 ) does not receive any RG corrections to the order we are working since it contains only SL terms. For the virtual corrections and for real photon emission we have in the soft photon approximation:  n 2 e2 expt s 1 e2 (2 ) 1  i   log log log − 1 −  2  862 c 2 c s e2 (expt )   i=1      s e2 (2 ) 3 1   log ; mi ; log − +  2 2 2  c 2 e (expt ) 2      n  s ei2 1 e2 (2 ) RG log 2 log 2 2 −1 (184) wexpt (s; mi ; ; expt ) = 862 c m e (expt m2 =s)   i=1       nf 2 2 2 2  2  e ( m =s) 1 e 1 expt expt   1 − Qj2 NCj log 2  + 2 log   2 2 (2 )  c 3 46 e m expt j  j=1      m2 s    − log 2 + log 2 ; mi ; expt

M. Melles / Physics Reports 375 (2003) 219 – 326

273

where n is the number of external lines and nf fermions propagating in the loops folded with the DL integrals. The upper case applies only to fermions since for W ± we have ¡ M . Note that in all contributions from the regime ¡ M we have kept mass terms inside the logarithms. For the running above the weak scale M we use only the massless 00 , 00 terms with ngen = 3. This approach is valid in the entire Standard Model up to terms of order O(log mt =M ). In order to brieUy discuss the size of the SL-RG corrections obtained here, we present a numerical comparison. For this purpose we will only √ compare the terms which are new in the RG analysis, i.e. the running from the weak scale M to s. We are thus interested in e5ects starting at the two loop level and want to compare the relative size of the RG-improved form factors to the pure Sudakov RG RG terms. It is therefore of interest to compare the ratios (e{−Wi } − e{−Wi } )=e{−Wi √} for the various particle labels i. Since the physical scales in the problem are given by M and s, the lower and upper limits of the couplings are given accordingly by these scales for the functions Wi . Fig. 18 compares the respective ratios for various SM particles. For de2niteness we take M = 80 GeV, mt = 174 GeV, sw2 = 0:23, -(M 2 ) = 1=128=sw2 , - (M 2 ) = 1=128=cw2 , 00 = 19=24 and 00 = −41=24. The di5erence between the curves using M 2 and those using s as the scales in the conventional Sudakov form factors is a measure of the inherent scale uncertainty which is removed by the RG-improved Sudakov form factors WiRG . The largest e5ect is obtained in the gauge boson sector [77]. For external { + ; − ; ; H } particles we have at 1 TeV a di5erence between the curves of about 0:35% per line on the level of the cross section, growing to 0:65% at 2 TeV. The situation is very similar for transversely polarized W + ; W − particles where it reaches about 0.4% at 1 TeV and 0.8% at 2 TeV per line on the cross section level. For left handed quarks of the third generation the size of the corrections is about 0.15% at 1 TeV per line on the level of the cross section and 0.33% at 2 TeV. These corrections are thus considerably smaller and only needed if precisions below the one percent level are necessary from the theory side. For right handed top quarks the e5ect is even smaller since only the running of - enters and it is thus negligible for most applications. The form of the two curves in case of right handed tops di5ers markedly from the other three cases because at the energies displayed, the dominant e5ect is actually due to subleading Yukawa enhanced corrections 2 2 (∼ -) since the DL terms are proportional √ to - and since the ratio mt =M is of the size of an additional logarithm for these values of s. In general it can be seen that—where the DL terms dominate—the renormalization group improved results are indeed in-between the upper and lower bounds given by the respective scale choices in the conventional Sudakov form factors. Indeed also for right handed top quarks this pattern is observed if only DL corrections are taken into account. It should be emphasized again that also the QED-RG corrections can be sizable since large mass ratios with light particles occur. These should of course also be implemented in a full SM prediction at TeV energies keeping in mind that one must always be far above any particle thresholds. 3.6. Angular dependent corrections In this section we discuss the important contribution from angular dependent logarithmic corrections. In massless four fermion processes these were 2rst given by Ref. [27] at the SL level and in Ref. [110] at the SSL level. Since the size of the observed corrections is large, these terms are important for future collider phenomenology. In Ref. [111] a general way of treating SL angular terms for arbitrary processes including mass terms was presented. The regime of validity assumes

274

M. Melles / Physics Reports 375 (2003) 219 – 326 RG

RG

(exp{-Wi } - exp{-Wi}) / exp{-Wi } 0.004

0.01

0.002

0.0075 0.005

0

+

0.0025

-

i={ φ ,φ ,χ,H}

-0.002

+

-

i={WT,WT}

0

-0.004

-0.0025

-0.006

-0.005

-0.008

-0.0075

-0.01

-0.01

-0.012

-0.0125 1000

2000

3000

4000

1000

5000

2000

√s

3000

4000

5000

4000

5000

√s x 10

-2

0.004 0

0.003 0.002

-0.05

0.001 -0.1

i={tL,bL}

0 -0.001

i={tR}

-0.15

-0.002 -0.2

-0.003 -0.004

-0.25

-0.005 1000

2000

3000 2

4000 ,

2

: (M ), (M )

5000

1000

2000

3000 ,

: (s), (s)

Fig. 18. This 2gure compares the renormalization group improved probabilities WiRG with the conventional Sudakov exponentials Wi for various external particle lines. The comparison is made with the indicated scale choices for the √ functions Wi and takes into account only the RG corrections from the scale M to s. Taking the di5erence between the two curves is a measure of the uncertainty removed by the RG e5ects. The variations in the scale of the coupling in the Wi functions is largest in the scalar (Goldstone and Higgs boson) sector and for transverse W ± where the e5ect is about 0:8% at 2 TeV per line on the level of the cross section. In general, the RG improved form factors di5er by fractions of percent/line and need to be taken into account at future colliders if the experimental accuracy is in the percentile regime.

that all |2pi pj |M 2 . In order to better understand the origin of angular dependent corrections in the electroweak theory, consider the case of massive QED and the right handed SM depicted in Fig. 19. We will show that the mass terms do not lead to new e5ects compared to those in the massless case. The two scalar integrals of Fig. 19, regularized with gauge boson mass terms 31 and 32 , are given in massive QED or in the case of right handed (massive) fermions by d 4 l2 1 d 4 l1 a II/ = 4st 2 2 4 4 2 (26) (26) (l1 − 31 )((p1 − l1 ) − m21 )((p1 − l1 − l2 )2 − m21 ) 1 × ; (185) ((p2 + l1 )2 − m22 )(l22 − 322 )((p3 + l2 )2 − m23 )

M. Melles / Physics Reports 375 (2003) 219 – 326 p1 l2 IIθ =

p1

l1

IIaθ

275

l1

l2

IIbθ

p2 +

p2

p3

p3

p1

p1

l1 ∼

p2 x

Is

It

p3

l2

p2

p3

Fig. 19. Angular dependent two loop on-shell diagrams. The sum of IIa/ and IIb/ factorizes in massive QED and the right handed SM into the product of the two one loop corrections (each with a di5erent invariant and mass terms) in leading order.

IIb/

d 4 l1 (26)4

d 4 l2 (26)4

1 − − l2 − m21 )((p1 − l1 − l2 )2 − m21 ) 1 ; × ((p2 + l1 )2 − m22 )(l22 − 322 )((p3 + l2 )2 − m23 )

= 4st

(l21

312 )((p1

)2

(186)

denoting s = 2p1 p2 , t = 2p1 p3 and where the mi are the masses of the external charged particles on their mass shell. Thus, it is straightforward to see that the sum of the two diagrams factorizes to leading order: d 4 l2 l21 + l22 − 2p1 (l1 + l2 ) d 4 l1 a b II/ + II/ = 4st (26)4 (26)4 (l21 − 312 )((p1 − l1 )2 − m21 )((p1 − l1 − l2 )2 − m21 ) 1 × ((p2 + l1 )2 − m22 )(l22 − 322 )((p1 − l2 )2 − m21 )((p3 + l2 )2 − m23 ) d 4 l1 2s ≈ 2 2 4 2 (26) (l1 − 31 )((p1 − l1 ) − m21 )((p2 + l1 )2 − m22 ) 2t d 4 l2 × : (187) 2 2 4 2 (26) (l2 − 32 )((p1 − l2 ) − m21 )((p3 + l2 )2 − m23 ) The omitted cross term 2l1 l2 leads only to corrections containing three logarithms at the two loop level. It is thus on the same level as the approximation in the beginning of our discussion which only considers scalar integrals and can therefore be neglected. To DL accuracy we can employ the

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Sudakov technique, parametrizing the loop momenta along the external four momenta as m2 m2 l1 ≡ v1 p1 − 1 p2 + u1 p2 − 2 p1 + l1 ⊥ ; s s 2 m1 m23 p 3 + u2 p 3 − p 1 + l2 ⊥ : l2 ≡ v2 p1 − t t

(188) (189)

Thus, after rewriting the measure and integrating over the perpendicular components we 2nd for the case of two photons with mass 3 (omitting the principle value parts): 1 dv1 1 du1 1 m21 m22 a b 2 /(su1 v1 − 3 )/ u1 − II/ + II/ ∼ 2 v1 / v1 − u1 86 v1 0 u 1 s s 0 1 m21 m23 dv2 1 du2 1 2 v 2 / v2 − u2 /(tu2 v2 − 3 )/ u2 − × 2 86 v 2 0 u2 t t 0 3m2 =s m2 =s 1 2 dv1 1 du1 dv1 1 du1 dv1 1 du1 1 − − = 2 86 v1 32 =sv1 u1 32 =s v1 32 =sv1 u1 32 =s 3m2 =s v1 sv1 =m22 u1 3m1 =s m2 =s 1 du1 1 dv1 du1 1 dv1 − − u1 32 =su1 v1 32 =s 3m1 =s u1 su1 =m21 v1 3m3 =t m2 =t 1 3 dv2 1 du2 dv2 1 du2 dv2 1 du2 1 − − × 2 86 v2 32 =tv2 u2 32 =t v2 32 =tv2 u2 32 =t 3m3 =t v2 tv2 =m23 u2 3m1 =t m2 =t 1 du2 1 dv2 du2 1 dv2 − − u2 32 =tu2 v2 32 =t 3m1 =t u2 tu2 =m21 v2 s s 1 1 1 s 1 s 1 2 s 2 s + log 2 log 2 + log 2 log 2 = 2 − log 2 − log 86 4 m2 2 3 2 3 m1 4 m1 m2 1 1 1 1 t 1 t t t 2 t 2 t × 2 − log 2 − log (190) + log 2 log 2 + log 2 log 2 : 86 4 m3 2 3 2 3 m1 4 m1 m3 The important point about the result in Eq. (190) is not only the factorized form in terms of the two massive one loop form factors but also the fact that the fermion mass terms correspond to each external on shell line in the amplitude. Thus, by rewriting the term in the bracket of the last line in Eq. (190) as 1 t t 1 1 t 1 t t t − log2 2 − log2 + log 2 log 2 + log 2 log 2 4 m3 2 3 2 3 m1 4 m1 m3 t m2 m2 1 1 log2 21 + log2 23 = log2 2 − 2 3 4 3 3 2 2 1 1 s t 2 s 2 m1 2 m3 log 2 + log 2 ≈ log 2 + log 2 log − ; (191) 2 3 3 s 4 3 3

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277

we see that the SL angular terms are indeed independent of the fermion mass terms. For the corrections involving the invariant u ≡ 2p2 p3 the situation is analogous. In the case of the SM with right handed massive fermions we need to consider in addition the exchange of Z-bosons. The results read s 1 1 IIa/ + IIb/ ∼ 2 log2 2 86 2 M 2 2 1 1 s t 1 2 s 2 m1 2 m3 log 2 + log 2 × 2 (192) log 2 + log 2 log − 86 2 3 3 s 4 3 3 for the case of a photon and a Z-boson with mass M , and for the case of two Z’s we have 1 1 s t s s 1 1 log2 2 × 2 log2 2 + log 2 log (193) IIa/ + IIb/ ∼ 2 86 2 M 86 2 M M s Again we see the independence of the SL angular terms on the fermion mass terms and in addition, the fact that the gauge boson mass gap does not spoil the type of factorization in the right handed SM. This type of factorization can be generalized on theoretical grounds to the situation in the general SM. It should be noted that all fermion mass singularities in the SM only arise through photon radiation or coupling renormalization. The latter is not important in our discussion here and is anyhow sub-subleading at higher orders. The exchange of the heavy gauge bosons does not lead to fermion mass singular terms assuming that all mi 6 M . This case is analogous to QCD where angular terms factorize in matrix form [86]. Thus, only corrections where one heavy gauge boson and one photon are involved are novel features in the SM. In this case, however, the type of factorization analogous to Eq. (190) for fermion mass and soft terms follows from the factorized form of real emission corrections. They are of factorized form and the KLN theorem then leads to the analogous situation for the sum of all virtual corrections. The soft terms must also factorize since we can always de2ne an observable by only allowing soft photon radiation, i.e. chosing XE 6 M . From the arguments presented in Ref. [111] it then follows that ui ; :::; uin

MSL1

({pk }; {ml }; M; 3) nf ng 1 RG 1 RG 2 W (s; M ) − W (s; M 2 ) = exp − 2 i=1 gi 2 i=1 fi n n n 1 ˜Va ˜VQ a s 2pl pk 1 RG I i ;ik I i ;il log 2 log W i (s; M 2 ) + 2 − k l 2 i=1 86 M s V =B;W a k=1 l¡k

nf

a

n

w 1 RG 1 2 (wfi (s; 32 ) − wfRG (s; M )) − (wRG (s; 32 ) − wwRGi (s; M 2 )) i 2 i=1 2 i=1 wi n n n 1 / / w (M 2 ; m2j ) + (wkl (s; 32 ) − wkl (s; M 2 )) − 2 i=1 i

× exp −

k=1 l¡k

ui1 ; :::; ui ; :::; ui ; :::; uin

×MBorn

k

l

({pk }; {ml }) ;

(194)

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M. Melles / Physics Reports 375 (2003) 219 – 326

where ng denotes the number of external gauge bosons (in the symmetric basis), nf the number of external fermions and n the number of external scalars (including Higgs particles). The notation used in the regime below the scale M is analogous; note that we use a photon mass regulator in this case. The 2elds u have a well de2ned isospin, but for angular dependent terms involving CKM mixing e5ects, one has to include the extended isospin mixing appropriately in the corresponding couplings I˜Via;ik of the symmetric basis. The expressions for the virtual probabilities W RG are given k in Section 2.6. In addition we denote e2 s 2pl pk (195) Qk Ql log 2 log 862 3 s for the angular dependent corrections from the soft QED regime, where the Qj denote the electrical charges of the external lines. These terms are not manifestly positive and thus not related to emission probabilities and also process dependent. The diagonal terms do not involve (CKM-extended) isospin rotated Born matrix elements. These occur only from the angular terms above the scale M . Thus, the way the electroweak angular terms factorize is also in an exponentiated operator form as in QCD. Eq. (194) is valid for arbitrary external lines. / (s; 32 ) = wkl

3.7. Sudakov logarithms in softly broken supersymmetric models In the introduction we already discussed the motivations for considering supersymmetric extensions of the SM. Since supersymmetry must be broken in nature (if it is at all relevant to physics at the electroweak scale), the original simplicity is lost and in the case of the MSSM over one-hundred parameters are needed to describe the model. In order for supersymmetry to stabilize the hierarchy problem, though, two conditions need to be ful2lled. Firstly, the masses of the superpartner particles must not be much larger than the electroweak scale and secondly, it must be a “softly broken” symmetry, i.e. broken by mass terms and couplings with a positive mass dimension. Thus, at energies in the TeV regime, these two conditions (assumed to be ful2lled in the MSSM or the NMSSM for instance) lead to the following consequences for radiative corrections of electroweak origin. If we assume that the mass scale of the superpartner particles are not much larger than the weak scale, say less than 500 GeV, and energies in the TeV regime, the results for the DL corrections outlined above for the SM can be applied straightforwardly to the MSSM. The reason is that the gauge couplings are preserved under supersymmetry and no additional spin 1 particles are exchanged. The appropriate quantum numbers in the eigenvalues of the casimir operators are the same as those of the SM partners. In case the superpartner masses are larger than 500 GeV, additional double logarithms need to be taken into account in a way outlined in Ref. [28]. In the following we assume that we can neglect such terms, i.e. that all particles in the MSSM have a mass below 500 GeV. At the subleading level, the situation in general is less clear at higher orders. For SL angular dependent terms, the same reasoning as above goes through since they originate only from the exchange of spin 1 gauge bosons and can thus be resummed as in the SM (see Section 3.6). Box-type diagrams exchanging supersymmetric particles in the s-channel do not contribute to SL angular terms. The same holds for all universal SL corrections which involve the exchange of SM particles since they are properties of the external particles only.

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279

New types of SL Sudakov corrections are, however, involved in the exchange of supersymmetric particles as discussed in Refs. [112–114] at the one loop level in the on-shell production of superpartner particles in e+ e− collisions. In the following we discuss the results obtained in those works. We begin with the corrections contributing in particular Yukawa terms from the 2nal state corrections. The 2nal result of calculating all terms contributing to the Yukawa sector includes terms depending on tan 0, which is the ratio of the two v.e.v.’s from the two Higgs doublet sector of the MSSM. Since tan 0 could be as large as 60, the bottom Yukawa terms are crucial in supersymmetric models. Also higher order terms need to be considered as in the SM and the exponentiation of the MSSM Yukawa terms follows from the same arguments as in the SM. In particular the Ward identity Eq. (135) holds [114]. Again, here we assume that the susy masses are close to the electroweak scale. These corrections apply to both the SM as well as the superpartner production. In particular the process of charged Higgs production seems to be well suited for an indirect measurement of tan 0 ¿ 14 of better than O(25%), and a few percent for tan 0 ¿ 25 [114]. The important point to note here is not only the precision but in particular the fact that this determination of tan 0 is independent of soft breaking terms to SL accuracy and thus model independent. This is due to the fact that the soft breaking contributions are constants and can be eliminated via subtraction if a series of precise measurements is performed at various energy scales. In addition this approach is scheme and gauge invariant. Also the SL-gauge terms get modi2ed by loops containing novel superpartner contributions. If we neglect the mass di5erence between the SM fermionic particle and its scalar partner we observe an exact supersymmetry relation 8 described by (for 3 = M ): =dHeBorn dHeSL + − + − e →fQ0 f0 e →fQ0 f0 - -

- -

= dHSL + −

=dHBorn e-+ e-− →f˜Q0 f˜0 e- e- →f˜Q0 f˜0

g2 (m2s )Te-− (Te-− + 1) 1 ˜ g2 (m2s ) 2 s 3 s log = exp − − 00 log 2 862 M2 3 462 ms 2 (m2 ) s g s 1 s − log2 2 − 0˜ 0 log3 2 3262 M 3 462 ms

2 s g2 (m2s ) g 2 (m2s ) Ye-− 2 log 2 + Te-− (Te-− + 1) + 2 2 86 86 4 M g 2 (m2s )Ye2−

−

8

g2 (m2s )Tf˜0 (Tf˜0 + 1) 862

1 ˜ g2 (m2s ) 2 s 3 s log − 00 log 2 M2 3 462 ms

Here we take a common mass also for the two chiral superpartners. Since in general the mass eigenstates are di5erent from the Uavor eigenstates if their masses di5er when supersymmetry breaking is included, mixing e5ects must then also be considered.

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−

g 2 (m2s )Yf2˜

0

3262

1 ˜ g 2 (m2s ) s s log − log3 2 0 M 2 3 0 462 ms

2

2

s g2 (m2s ) g 2 (m2s ) Yf˜0 2 log 2 + Tf˜0 (Tf˜0 + 1) + 2 2 86 86 4 M

2 mˆ 2f˜ s g2 (m2s ) 1 + .0; R mˆ f˜ log 2 − + .0; L 2 2 2 86 2 M 2M ms

g2 (m2s ) s % t − log 2 (tan2 /w Ye-− Yf˜0 + 4Ie3− If3˜0 ) log 862 M u .-; L .0; L −t −u ; + .d; f˜ log − .u; f˜ log s s tan2 /w Ye-− Yf˜0 =4 + Ie3− If3˜ -

(196)

0

where Tj denotes the total weak isospin of the particle j, Yj its weak hypercharge and at high s the invariants are given by t = −s=2(1 − cos /) and u = −s=2(1 + cos /). The helicities are those of the fermions (f) whose superpartner is produced. In addition we denote mˆ f˜ = mt =sin 0 if f˜ = t˜ ˜ f˜ denotes the corresponding isopartner of f. ˜ For particles other than and mˆ f˜ = mb =cos 0 if f˜ = b. those belonging to the third family of quarks/squarks, the Yukawa terms are negligible. Eq. (196) depends on the important parameter tan 0 = vu =vd , the ratio of the two vacuum expectation values, and displays an exact supersymmetry in the sense that the same corrections are obtained for the fermionic sector in the regime above the electroweak scale M . Here we assume that the asymptotic MSSM 0-functions can be used from the scale ms ∼ mf˜ ∼ M with ng nh 3 − ; 0˜ 0 = CA − 4 2 8 g2 (s) =

5 nh 0˜ 0 = − ng − ; 6 8

g2 (m2s )

1 + 0˜ 0 (g2 (m2s )=462 ) log (s=m2s )

;

(197) 2

g (s) =

g 2 (m2s )

1 + 0˜ 0 (g 2 (m2s )=462 ) log (s=m2s )

;

(198)

where CA = 2, ng = 3 and nh = 2. In practice, one has to use the relevant numbers of active particles in the loops. These terms correspond to the SL-RG corrections just as in the case of the SM as discussed in Ref. [77] but now with the MSSM particle spectrum contributing. They originate only from RG terms within loops which without the RG contribution would give a DL correction. It should be noted that the one-loop RG corrections do not exponentiate and are omitted in the above expressions. They are, however, completely determined by the renormalization group in softly broken supersymmetric theories such as the MSSM and sub-subleading at higher than one loop order. Relation (196) is expected since in unbroken supersymmetry both chiral fermions and the superpartner sfermions are part of the same supermultiplet. In the diagrammatic evaluation, however, the identity expressed in (196) is a highly non-trivial check on the overall correctness of the calculation as di5erent particles and loops contribute in each case. Since in the real world mf˜ = mf , the corresponding matching terms containing light fermion mass need to be included as well.

M. Melles / Physics Reports 375 (2003) 219 – 326

281

For charged Higgs production we have analogously dHeSL + − e →H + H − - -

g2 (m2s ) 1 ˜ g2 (m2s ) 2 s 3 s exp − Te-− (Te-− + 1) log − 00 log 2 862 M2 3 462 ms g 2 (m2s )Ye2− s 1 ˜ g 2 (m2s ) 2 s 00 − log − log3 2 2 2 2 326 M 3 46 ms

2 s g2 (m2s ) g 2 (m2s ) Ye-− 2 log 2 + Te-− (Te-− + 1) + 2 2 86 86 4 M g2 (m2s ) 1 ˜ g2 (m2s ) 2 s 3 s − TH (TH + 1) log − 00 log 2 862 M2 3 462 ms s 1 ˜ g 2 (m2s ) g 2 (m2s )YH2 2 s log − 0 log3 2 − 3262 M 2 3 0 462 ms

s g 2 (m2s ) YH2 g2 (m2s ) 2 log 2 TH (TH + 1) + + 2 2 86 86 4 ms g2 (m2s ) m2t m2b s 2 2 −3 cot 0 + 2 tan 0 log 2 2 2 166 M M ms 1 g2 (m2s ) s t −t t 2 2 + .-; R tan /w log : − log 2 .-; L log + 2cw log M 2cw2 u s u 462

= dHeBorn + − + − - e- →H H

(199)

It should be noted here that the Yukawa terms proportional to tan 0 are quite large due to the additional factor of 3 = NC from the quark loops [114]. Overall, both Yukawa contributions in Eq. (196) and (199) reinforce the Sudakov suppression factor of the leading double logarithmic terms. The higher order exponentiation of the universal terms uses the Ward identity (135) for the respective corrections. The angular terms are of SM origin only as described above and are thus treated as in Section 3.6. In addition we have the matching terms stemming form mass terms of both the gauge bosons and particles involved in the process. In any case it would also be helpful for collider experiments to have a full one loop calculation in the full supersymmetric theory in order to have a better understanding of the size of constant terms omitted in our high energy approximation. These terms, however, are diScult to discuss at this point since they depend strongly on how supersymmetry breaking is realized in nature. Neutral Higgs production in the MSSM depends also on the parameter -, the mixing angle between the CP even neutral Higgs particles. For these terms the corresponding dependence on tan 0 is thus not as evident as in the charged Higgs case without independent knowledge on - but results can be found in Ref. [114]. For transverse gauge boson production it should be clear that the 0 functions in Eqs. (179) should be replaced by the corresponding MSSM expression given in Eq. (197).

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Longitudinal gauge boson production is analogous to Higgs production after the application of the Goldstone equivalence theorem. 3.8. Fully inclusive cross sections Physical observables in the electroweak theory depend on the infrared cuto5, the gauge boson mass M , as was discussed in Section 3.4. This feature is closely related to electroweak symmetry breaking itself since it indicates that in the unbroken limit M → 0 cross sections become unobservable. Thus, we would expect in this limit con2ning e5ects limiting the types of initial or 2nal state particles to be in an isospin singlet state. In the broken physical theory, this is evidently not the case and physical particles carry a non-Abelian group charge. It is for this reason that even fully inclusive cross sections are expected to depend on log2 s=M 2 . Also in QCD the Bloch–Nordsieck violating terms are present in fully inclusive cross sections and only the initial state color averaging eliminates the infrared problem. Thus we see that masslessness of the gauge bosons in a non-Abelian theory is intimately connected with con2nement and therefore color neutral initial and 2nal states. The realization of this phenomenon in the electroweak theory was emphasized in Ref. [33] where these “Bloch–Nordsieck violating” terms were calculated by means of the coherent state operator formalism from QCD. The basic idea is provided by the observation that in collider experiments one does not take the average over, say, the e− e+ and e+ initial states since the asymptotic states carry weak isospin quantum numbers. The authors of Ref. [115] conclude that these log2 s=M 2 terms in fully inclusive electroweak processes are due only from initial state W ± corrections. There are, however, possibly additional terms from the Abelian sector due to the fact that the hypercharge is broken in the physical 2elds. These terms, however, can be neglected for light fermions. By then considering the leading coherent state operators for the soft and hard (Born) parts of the scattering amplitude, the authors of Ref. [115] 2nd that the Bloch–Nordsieck violating terms exponentiate in the form of the Sudakov form factor in the adjoint representation. For initial state fermions with two isospin components the results for the fully inclusive cross sections are then given by Born Born Born Born H11 = H22 = 12 (H11 + H12 + [H11 − H12 ] exp(−2FS )) ;

(200)

Born Born Born Born + H12 − [H11 − H12 ] exp(−2FS )) ; H12 = H21 = 12 (H11

(201)

where g2 s log2 2 (202) 2 166 M is the Sudakov form factor and the factor of 2 in the exponential argument corresponds to the Casimir eigenvalue in the adjoint representation, i.e. T (T + 1) with T = 1. A generalization to transversely polarized initial states is given in Ref. [116]. Denoting the triplet representation as +; 3; − the authors obtain for the case of fermion boson scattering: FS =

Born Born Born H1+ = H2− = 12 (H1+ + H1Born − + [H1+ − H1− ] exp(−2FS )) ;

(203)

Born Born Born + H1Born H1− = H2+ = 12 (H1+ − − [H1+ − H1− ] exp(−2FS )) ;

(204)

Born + H1Born H13 = H23 = 12 (H1+ + H1− ) = 12 (H1+ − ) :

(205)

M. Melles / Physics Reports 375 (2003) 219 – 326

283

In the case of transverse boson boson scattering we have Born 1 Born 1 1 H++ = H– = H++ ( 3 + 12 exp(−2FS ) + 16 exp(−6FS )) + H− + ( 3 − 2 exp(−2FS ) Born 1 ( 3 − 13 exp(−6FS )) ; + 16 exp(−6FS )) + H3+

(206)

Born 1 Born 1 1 ( 3 − 12 exp(−2FS ) + 16 exp(−6FS )) + H− H−+ = H+− = H++ + ( 3 + 2 exp(−2FS ) Born 1 ( 3 − 13 exp(−6FS )) ; + 16 exp(−6FS )) + H3+ Born Born Born + H− H3+ = H3− = 13 {[H++ + ](1 − exp(−6FS )) + H3+ (1 + 2 exp(−6FS ))} :

(207) (208)

In Ref. [117] results for the longitudinal and Higgs sector are presented employing the equivalence theorem. For a light Higgs the inclusive cross sections are related as follows: H − − = H + + ; H + h = H − h = H + 3 ;

(209)

∗ ∗ exp(−(2 + tan2 /w )FS ) ; Hhh = H33 = 12 (H + + + H + − ) + Re H Born 0 0 → 0 0

(210)

∗ ∗ exp(−(2 + tan2 /w )FS ) : H3h = 12 (H + + + H + − ) − Re H Born 0 0 → 0 0

(211)

The individual results can be obtained from Born Born H + + + H − + + 2H3 + = H Born + + + H − + + 2H3 + ;

(212)

Born Born H + + + H − + − 2H3 + = (H Born + + + H − + − 2H3 + )exp(−2FS ) ;

(213)

Born Born H + + − H − + + 2 Im H3 + →h + = H Born + + − H − + + 2 Im H3 + →h + ;

(214)

Born Born H + + − H − + − 2 Im H3 + →h + = (H Born + + − H − + − 2 Im H3 + →h + )exp(−2FS ) :

(215)

In general, the e5ects are sizable. In particular for initial state leptons and for gauge bosons O(10%) e5ects can be expected in the TeV range. In Ref. [118] an initial step towards electroweak splitting functions is taken for a spontaneously broken SU (2) gauge group. In this model there is no gauge boson mass gap, however, the important di5erence to the QCD case is that the initial state isospin averaging is omitted. A rigorous extension to the full electroweak theory would be important since the size of SL terms in non-negligible. 3.9. Physical 0elds at 0xed order In this section we present results obtained in real 2xed order perturbative calculations performed with the physical SM 2elds in the on-shell scheme. It is pivotal to compare these results with the corrections obtained with the IREE method expounded on above where the calculation is performed in the high energy e5ective theory including appropriate matching conditions at the weak scale in order to include pure QED e5ects. Fortunately, a lot of e5ort has been spent in perturbative electroweak calculations at high energies in order to allow for a cross check for all degrees of freedom at one loop. At the two loop level calculations at the DL level are available in the regime we are interested in. We begin, however, at the lowest order. It should be mentioned that in order to

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M. Melles / Physics Reports 375 (2003) 219 – 326

obtain an accuracy in the 1% regime, it is not suScient to include only logarithmic corrections since for processes involving unsuppressed mass-ratios in the Born cross section (such as + − → + − ) constants can be much larger. Also 62 terms and Yukawa enhanced constants can be of the order of several % and thus, for the precision objectives of the linear collider a full one loop calculation is necessary in most cases. At higher orders, however, these constant terms can be neglected. 3.9.1. One loop results There have been various calculations at the one loop level, however, for our purposes it is convenient to discuss the results of Ref. [29] in more detail where general formulas are derived for one loop logarithmic corrections at high energies. For all SL Sudakov logarithms we 2nd agreement with the results presented in Ref. [29]. The universality of the corrections in the physical theory, however, is more diScult to see since the Ward identities are more involved in the broken gauge theory. We summarize results presented in Ref. [29] for DL and one loop angular terms containing log s=M 2 log u=t type contributions. In addition there are also the on-shell parameter renormalization of the couplings, masses and mixing angles. All results are in the ’t Hooft–Feynman gauge and the techniques discussed in this section are valid only at the one loop level. In order to obtain double logarithmic corrections one can drop the dependence of the numerator on the loop momentum. In this approximation the one-loop corrections give Q i :::i :::i :::i n −4ie2 pk pl IiV iak (k)IiV ial (l)M01 k l n d4 q k l .Mi1 :::in = (216) 4 (q2 − M 2 )[(p + q)2 − m2 ][(p − q)2 − m2 ] (26) k l Va k l ± k=1 l¡k Va =A;Z;W

and in leading order, using the high-energy expansion of the scalar three-point function [96], one obtains n 1 Q i :::i :::i :::i IiV iak (k)IiV ial (l)M01 k l n [L(|rkl |; MV2a ) − .Va A L(m2k ; 32 )] (217) .Mi1 :::in = k l 2 ± k=1 l =k Va =A;Z;W

where the short hand notations L(|rkl |; M 2 ) :=

rkl e2 log2 2 2 166 M

2 ) L(s) := L(s; MW

l(rkl ; M 2 ) :=

2 l(s) := l(s; MW )

e2 rkl log 2 2 166 M

(218) (219)

are used with 2 rkl = (pk + pl )2 ∼ 2pk pl MW

(220)

The DL term containing the invariant rkl depends on the angle between the momenta pk and pl . Writing |rkl | + L(|rkl |; s) (221) L(|rkl |; M 2 ) = L(s; M 2 ) + 2l(s; M 2 )log s the angular-dependent part is isolated in logarithms of rkl =s, and gives a subleading soft–collinear (SSC) contribution of order l(s) log(|rkl |=s), whereas terms L(|rkl |; s) can be neglected in LA. The remaining part, together with the additional contributions from photon loops in (217), gives the leading soft-collinear (LSC) contribution and is angular-independent. This contribution corresponds

M. Melles / Physics Reports 375 (2003) 219 – 326

285

to the universal part discussed in the previous sections in the symmetric basis. The eikonal approximation (216) applies to chiral fermions, Higgs bosons, and transverse gauge bosons, and depends on their gauge couplings I Va (k) which can be found in Appendix A.1. Since the longitudinal polarization vectors (117) grow with energy, matrix elements involving longitudinal gauge bosons have to be treated with the equivalence theorem, i.e., they have to be expressed by matrix elements involving the corresponding Goldstone bosons. As discussed in Section 3.2, the equivalence theorem for Born matrix elements receives no DL one-loop corrections. Therefore, the soft-collinear corrections for external longitudinal gauge bosons can be obtained using the simple relations ±

.DL M:::WL ::: = .DL M:::

±

:::

;

.DL M:::ZL ::: = i.DL M::::::

(222)

from corrections (217) for external Goldstone bosons. The invariance of the S matrix with respect to global SU(2) × U(1) transformations implies (up to mass terms): 0 = .Va Mi1 :::in = ie IiV iak (k)Mi1 :::ik :::in (223) k

k

which is the analogue of Eq. (55) for the electroweak theory. For external Goldstone 2elds extra contributions proportional to the Higgs vacuum expectation value appear, which are, however, irrelevant in the high-energy limit. Using (223), the LSC logarithms in (217) can be written as a single sum over external legs, .LSC Mi1 :::in =

n k=1

i :::ik :::in

1 .LSC ik ik (k)M0

:

After evaluating the sum over A, Z, and W, in (217), the correction factors read 1 MZ2 LSC ew Z 2 2 em 2 2 .ik ik (k) = − Cik ik (k)(s) − 2(I (k))ik ik log 2 l(s) + deik ik Qk L (s; 3 ; mk ) : 2 MW

(224)

(225)

The 2rst term represents the DL symmetric-electroweak part and is proportional to the electroweak Casimir operator C ew de2ned in (A.10). This is always diagonal in the SU(2) indices, except for external transverse neutral gauge bosons in the physical basis (A.14), where it gives rise to mixing between amplitudes involving photons and Z bosons. The second term originates from Z-boson loops, owing to the di5erence between MW and MZ , and 2 MW em 2 2 2 L (s; 3 ; mk ) := 2l(s)log + L(MW ; 32 ) − L(m2k ; 32 ) (226) 32 contains all logarithms of pure electromagnetic origin. The LSC corrections for external longitudinal gauge bosons are directly obtained from (225) by using the quantum numbers of the corresponding Goldstone bosons.

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The contribution of the second term of (217)–(221) remains a sum over pairs of external legs, .SSC Mi1 :::in =

n

k=1 l¡k Va

=A;Z;W ±

i :::ik :::il :::in

1 .iVaik; SSC i il (k; l)M0 k

l

(227)

with angular-dependent terms. The exchange of soft, neutral gauge bosons contributes with SSC 2 2 .A; i ik i il (k; l) = 2[l(s) + l(MW ; 3 )]log k

l

SSC .Z; i ik i il (k; l) = 2l(s) log k

l

|rkl | A I (k)IiAl il (l); s ik i k

|rkl | Z I (k)IiZl il (l) s ik i k

(228)

and, except for I Z in the neutral scalar sector H; (see Appendix A.1), the couplings I N are diagonal matrices. The exchange of charged gauge bosons yields ±

.iW ik i;SSC il (k; l) = 2l(s) log k

l

|rkl | ± I (k)Ii∓ (l) l il s ik ik

(229)

and owing to the non-diagonal matrices I ± (k) [cf. (A.17), (A.22) and (A.26)], contributions of SU(2)-transformed Born matrix elements appear on the left-hand side of (227). In general, these transformed Born matrix elements are not related to the original Born matrix element and have to be evaluated explicitly. The SSC corrections for external longitudinal gauge bosons are obtained from (227) with the equivalence theorem (222), i.e. the couplings and the Born matrix elements for Goldstone bosons 9 have to be used on the right-hand side of (227). The application of the above formulas is illustrated in Section 4 for the case of 4-particle processes, where owing to r12 = r34 , r13 = r24 and r14 = r23 , (227) reduces to 2 2[l(s) + l(MW ; MV2a )] .SSC Mi1 i2 i3 i4 = Va =A;Z;W ±

& |r12 | % Va Q Q i i i i i i i i Ii i1 (1)IiV ia2 (2)M01 2 3 4 + IiV ia3 (3)IiV ia4 (4)M01 2 3 4 × log 1 2 3 4 s & |r13 | % Va Q Q i i i i i i i i Ii i1 (1)IiV ia3 (3)M01 2 3 4 + IiV ia2 (2)IiV ia4 (4)M01 2 3 4 + log 1 3 2 4 s % & |r14 | Va i1 i2 i3 i4 i1 i2 i3 i4 VQ a VQ a Va Ii i1 (1)Ii i4 (4)M0 + Ii i2 (2)Ii i3 (3)M0 +log 1 4 2 3 s

(230)

and the logarithm with rkl = s vanishes. Note that this formula applies to 4 → 0 processes, where all particles or antiparticles and their momenta are incoming. Predictions for 2 → 2 processes are obtained by substituting outgoing particles (antiparticles) by the corresponding incoming antiparticles (particles). 9

Note that for Goldstone bosons , the equivalence theorem as well as the couplings (A.23) and (A.21) contain the imaginary constant i.

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287

In addition to the angular terms in Eq. (230) at one loop there are also SL terms from logarithms connected to parameter renormalization. These include the RG corrections to the mixing angles, electromagnetic charge, Yukawa couplings and scalar self couplings in the on-shell scheme. The corresponding terms are given in Ref. [29]. In Refs. [37,38] we have shown that the splitting function approach employed for the electroweak radiative correction in the high energy regime described by Lsymm is indeed justi2ed at the one loop level. For this cross check it is crucial that the DL and the non-Yukawa SL terms factorize with respect to the same group factor. As can be seen by the explicit results in Eqs. (128) and (156), these terms are given by g2 g 2 Yi2 # 2 s s $ log (231) T (T + 1) + − 3 log i i 1662 1662 4 M2 M2 for fermions and g2 g 2 Yi2 # 2 s s $ log T (T + 1) + − 4 log i i 1662 1662 4 M2 M2

(232)

for scalars. The correct reproduction of the explicit high energy approximation based on the physical 2elds in Ref. [97] constitutes a very strong check on the overall approach. Since also the corrections from region (II) in Fig. 1 are found to agree with Ref. [97] (including soft photon bremsstrahlung), the matching condition at the weak scale is con2rmed at the subleading level for both transverse as well as longitudinal degrees of freedom. The latter includes in particular the treatment of the would-be Goldstone bosons in the on-shell scheme to SL accuracy at high energies according to the e5ective Lagrangian Lsymm in Eq. (11). This concludes our discussion of the one loop case. More de2nitions and representations in the physical basis are presented in Appendix A.1. 3.9.2. Two loop results Also at the two loop level there have been checks performed with 2xed order calculations in terms of the physical SM 2elds. While the level of agreement demonstrated at the one loop level is impressive, it is very important for the desired accuracy of future linear colliders that there is agreement with real two loop DL calculations and the IREE method in order to trust predictions at the SL level. We follow Refs. [30,36] to demonstrate the method used and to show how non-exponentiating terms cancel in the electroweak theory in the ’t Hooft–Feynman gauge. Analogous results have been obtained in the Coulomb gauge in Refs. [31,32] for external fermion lines and recently also for longitudinal and transverse gauge bosons. In this gauge, the DL corrections are related to two point functions and thus directly to external legs. The intermediate steps, however, are more cumbersome and therefore we only discuss the ’t Hooft–Feynman gauge. In this gauge, a complete set of Feynman rules is listed in Appendix A.2. We begin with the case for right handed fermions coupled to a gluon, or in general a gauge singlet. At the one loop level, the electroweak corrections are depicted in Fig. 20. The fermion masses are neglected for simplicity. They can, however, be added without changing the nature of the higher order corrections. For right handed fermions we only need to consider the neutral electroweak gauge

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Fig. 20. The one loop electroweak SM Feynman diagrams leading to DL corrections in the ’t Hooft–Feynman gauge for g → fR fQL . Only the vertex corrections from the neutral Z-boson (zigzag-lines) and the photon propagators contribute. At higher orders only corrections to these two diagrams need to be considered in the DL approximation. The photonic corrections are regulated by a 2ctitious mass terms 3. In physical cross sections, the 3-dependence is canceled by the e5ect of the emission of soft and collinear bremsstrahlung photons.

bosons, i.e. we are concerned with an U (1)R × U (1)Y gauge theory which is spontaneously broken to yield the Z-boson and photon 2elds. The DL-contribution of a particular Feynman diagram is thus given by Mk = MBorn Fk

(233)

where the Fk are given by integrals over the remaining Sudakov parameters at the n-loop level:

n n " 1 1 dui dvi ef2 Fk = Lk : (234) 862 i=1 0 0 ui vi The Lk describe the regions of integration which lead to DL corrections. At one loop, the diagrams of Fig. 20 lead to

1 ef2 du 1 dv sw2 R (1) R 2 2 /(suv − 3 ) + 2 /(suv − M ) MDL = MBorn 1 − 2 86 0 u 0 v cw

ef2 sw2 R 2 s 2 s log 2 + 2 log (235) = MBorn 1 − 1662 3 cw M2 which is the well known result from QED plus the same term with a rescaled coupling (see the Feynman rules in Appendix A.2) and infrared cuto5. The restriction to right handed fermions allows us to focus solely on the mass gap of the neutral electroweak gauge bosons. The W ± only couples to left handed doublets. At the two loop level we have to consider more diagrams than in the QED case. The relevant Feynman graphs that give DL corrections in the ’t Hooft–Feynman gauge are depicted in Fig. 21. Only these corrections can yield four logarithms at the two loop level in the ’t Hooft–Feynman gauge. Otherwise one cannot obtain the required pole terms (as is well known in QED [95]). They contain diagrams where the exchanged gauge bosons enter with di5ering on-shell regions, i.e. di5ering integration regions which give large DL corrections. It is instructive to revisit the case of pure QED corrections, since the topology of the graphs yielding DL contributions in the ’t Hooft–Feynman gauge is unchanged. In QED at the two loop level, the scalar integrals

M. Melles / Physics Reports 375 (2003) 219 – 326

289

Fig. 21. The two loop electroweak SM Feynman diagrams leading to DL corrections in the ’t Hooft–Feynman gauge for g → fR fQL . The neutral Z-boson (zigzag-lines) and the photon propagators possess di5erent on-shell regions due to the mass gap.

corresponding to the 2rst row of Fig. 21 are given by: 1 du1 1 dv1 1 du2 1 dv2 QED /(su1 v1 − 32 )/(su2 v2 − 32 )/(u1 − u2 )/(v1 − v2 ) ; S1 ≡ u v u v 1 1 2 2 0 0 0 0 (236) 1 du1 1 dv1 1 du2 1 dv2 QED S2 ≡ /(su1 v1 − 32 )/(su2 v2 − 32 )/(u1 − u2 )/(v2 − v1 ) : u 1 0 v 1 0 u 2 0 v2 0 (237) Thus, in QED we 2nd the familiar result 1 du1 1 dv1 1 du2 1 dv2 S1QED + S2QED = /(su1 v1 − 32 )/(su2 v2 − 32 )/(u1 − u2 ) u v u v 1 1 2 2 0 0 0 0 2 1 1 du1 dv1 1 1 1 2 su1 2 s log 2 = ; log 2 = 3 2 2 3 32 =s u1 32 =su1 v1 2

(238)

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which yields the second term of the exponentiated one loop result in Eq. (235) for sw → 0. In the electroweak theory, we also need to consider the remaining diagrams of Fig. 21. The only di5erences occur because of the rescaled coupling according to the Feynman rules in Appendix A.2 and the fact that the propagators have a di5erent mass. Thus the second row of Fig. 21 leads to 1 sw2 du1 1 dv1 1 du2 1 dv2 M; 3 M; 3 S1 + S 2 = 2 /(su1 v1 − M 2 )/(su2 v2 − 32 )/(u1 − u2 ) c w 0 u 1 0 v1 0 u 2 0 v2 1 sw2 du1 1 dv1 1 su1 log2 2 = 2 cw M 2 =s u1 M 2 =su1 v1 2 3 2 sw2 1 1 M2 1 4 s 3 s 4 M (239) = 2 log 2 − log 2 log 2 + log 2 ; cw 8 3 6 3 3 24 3 where we indicate the gauge boson masses of the two propagators in the scalar functions. Analogously we 2nd for the remaining two rows 1 du1 1 dv1 1 du2 1 dv2 sw2 3; M 3; M S1 + S2 = 2 /(su1 v1 − 32 )/(su2 v2 − M 2 )/(u1 − u2 ) c w 0 u 1 0 v1 0 u 2 0 v 2 1 sw2 du2 1 dv2 1 # 2 s su2 $ log 2 − log2 2 = 2 cw M 2 =s u2 M 2 =su2 v2 2 3 3 s2 1 s s c2 log2 2 log 2 − 2w (S1M; 3 + S2M; 3 ) = w2 (240) cw 4 M 3 sw and S1M; M

+

S2M; M

1 sw4 = 2 cw4

s 1 log2 2 2 M

2

:

(241)

Thus, we 2nd for the full two loop electroweak DL-corrections (2)

(2) R R MDL ≡ MBorn (1 + .(1) R + .R ) :

with .(1) R

ef2 =− 1662

and

.(2) R

= =

ef2 862 ef2 862

2

s2 s s log 2 + w2 log2 2 3 cw M

%

2

(242) (243)

S1QED + S2QED + S1M; 3 + S2M; 3 + S13; M + S23; M + S1M; M + S2M; M

2

1 4 s s s s s2 s4 log 2 + w2 log2 2 log2 2 + w4 log4 2 8 3 4cw M 3 8cw M

2 2 ef2 s s s 1 − log2 2 + w2 log2 2 = 2 1662 3 cw M

&

(244)

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291

which is precisely the second term of the exponentiated one loop result in the process of g → fR fQL . In Ref. [30] we showed that by using the appropriate quantum numbers listed in Appendix A.1, that the result of the IREE method of Section 3.1 gives indeed the same result. For left handed fermions the calculation was performed in the ’t Hooft–Feynman gauge in Ref. [36]. It is instructive to list the intermediate steps of the calculation in order to identify terms not present in the case of unbroken gauge theories. We write the form of the correction as (2)

(2) L L MDL ≡ MBorn (1 + .(1) L + .L ) ;

(245)

where the one loop result is given by

ef2 ef2 g2 g 2 Y 2 s (1) 2 s log2 2 : log 2 − Tf (Tf + 1) + − .L = − 2 2 2 2 166 3 166 166 4 166 M

(246)

The two loop contribution can be written as a sum of the straight ladder diagrams, the crossed ladder and the three boson diagrams, which also enter due to the charged gauge boson exchange. Respectively we denote these terms as .(2) L = sl + cl + tb :

(247)

The reason for this is that in QCD, the non-Abelian contribution of cl (∼ CA ) is cancelled by the tb terms. The explicit results are 

2 2 e Y 1  ef4 s s s f 2 f g2 Tf (Tf + 1) + g sl = log4 2 + − ef2 log2 2 log2 2 4 646  24 3 4 4 3 M

2 Y 1 s 2 f log4 2 + ef2 g2 Tf (Tf + 1) + g − ef2 + 24 M 4   s s 1 s 1 log4 2 − log3 2 log 2 ; (248) × 6 M 3 M 3    

2 4 2  ef s 1 2 Yf log4 2 +  g2 Tf (Tf + 1) + g − ef2 − g4 Tf (Tf + 1) cl = 4 646  12 3 4

2 2 Yf − ef2 g2 Tf (Tf + 1) + g 4

2

Yf2 − ef2 g Tf (Tf + 1) + g 4  % & s  3 1 4 s 3 s 2 2 log − log log 2 ; + 2g e Qf Tf 6 M2 M2 3 

s 1 × log4 2 + ef2 12 M

tb =

1 6464

g4 Tf (Tf + 1)

2

2

1 s s 1 s − log4 2 + log3 2 log 2 6 M 3 M 3

s 1% 4 s 1 s& 3 s log4 2 − 2g2 e2 Qf Tf3 log : − log log 12 M 6 M2 M2 32

(249) (250)

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Thus we see that we have 

2 2  e4 e Y 1 s s s f f f 2 log4 2 + − ef2 log2 2 log2 2 g2 Tf (Tf + 1) + g .(2) L = 4 646  8 3 4 4 3 M

2 2 Yf − ef2 g2 Tf (Tf + 1) + g 4

+ 1 = 2

ef2 s − log2 2 − 2 166 3

2

  1 s log4 2 8 M 

ef2 g2 g 2 Y 2 − T (T + 1) + f f 1662 1662 4 1662

s log2 2 M

2 (251)

which is precisely half the square of the 2rst order correction. The novel feature here is that the way the cancellation happens is more complicated in the electroweak theory as already the straight ladder diagrams possess non-exponentiating terms. Only the full sum displays the familiar exponentiation. In the framework of the IREE method, however, these complications are naturally accounted for by carefully examining the kernel in both the high energy regime according to Lsymm in Eq. (11) and the region where only QED e5ects enter. At the scale M the appropriate matching conditions have to be employed. On the other hand, this complication is the reason for the wrong result in Ref. [25] which is based on QCD factorization properties. 4. Applications In this section we apply the results presented in Section 3 to speci2c processes at a future e+ e− collider. As mentioned in the introduction, such a machine must be a precision tool to disentangle and clarify the physics discovered at the LHC or the Tevatron. For our purposes we de2ne precision as e5ects that change cross section at the 1% level. We would like to emphasize that at that level it is not suScient to calculate only logarithmic corrections to all orders. At one loop, constant terms such as m2t =M 2 can lead to non-negligible e5ects and thus, a full one loop analysis is needed in these cases. In this work we want to emphasize the e5ect of the higher order corrections where constant terms are negligible. For all SL terms we can use the results summarized in Eq. (173) and the angular terms as described in Section 3.6. In the following we will display only the “pure” electroweak corrections originating above the weak scale M relative to the Born cross section. The importance of the angular dependent corrections is also discussed in Refs. [27,110,111] at higher orders and in [104] at the one loop level. While we focus below on the linear collider case, also for hadronic machines the electroweak corrections in the TeV range are important and must be included. In Ref. [119] the e5ect of one-loop logarithmic electroweak radiative corrections on WZ and W production processes at the LHC was studied for instance. Using the leading-pole approximation these corrections were implemented into Monte Carlo programs for pp → ll l lQ ; ll . The authors 2nd that electroweak corrections lower the predictions by 5 –20% in the physically interesting region of large transverse momentum and small rapidity separation of the gauge bosons. We begin in Section 4.1 with transverse and longitudinal gauge boson production, followed by Higgs boson production in Section 4.2 and heavy quark production in Section 4.3. For the latter

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293

two cases we also include supersymmetric Sudakov e5ects, in particular the dependence on tan0. For all two to two processes we denote s s t = − (1 − cos /); u = − (1 + cos /) (252) 2 2 which is valid at high energies and where / is the scattering angle between the initial and 2nal state particles. For de2niteness, we use the following parameters: mt = 174 GeV;

M = 80:35 GeV;

sw2 = 0:22356;

1 e2 (M 2 ) = 46 128

(253)

in all numerical results displayed. Furthermore we use g = e=sw and g = e=cw below. The angular terms at one loop are obtained from Eq. (230) which was derived in Ref. [29] except for the case of Higgs–strahlung. All energy units in the 2gures are in GeV. 4.1. Gauge boson production In this section we discuss the e5ect of the higher order pure electroweak corrections to transverse and longitudinal gauge bosons production at the cross section level. These processes, in particular the longitudinal ones, are important at high energies for the unitarity of the theory. If no Higgs boson should be found, the W ± sector has to become strongly interacting in the TeV range in order to preserve unitarity. It would therefore be important to know when and how the new dynamics would deviate from the perturbative SM predictions. Fig. 22 displays the energy dependence of the respective cross section in W ± production for the various polarizations. All universal terms are given by Eqs. (177)–(179). In addition we have angular terms which read relative to the Born amplitude in the high energy e5ective regime described by Lsymm in Eq. (11): −t t # g2 (M 2 ) s t$ / log ; (254) .e+ e− →W + W − = − log 2 log + 1 − T R L T 862 M u u s B;W a 1 g2 (M 2 ) s t −t / 2 ; (255) .e+ e− → + − = − log 2 log + 2cw log 2 2 R L 86 M 2c u s w a B;W B;W a

./e+ e− → + − = − L R

g 2 (M 2 ) s t log 2 log : 2 86 M u

In terms of their numerical coeScients relative to the Born cross section we have t$ −t e2 (M 2 ) t # s eL ; W T log dH/eL ; WT = dHBorn −8:95 log + 1 − ; log u u s 862 M2 −t e2 (M 2 ) t s eL ; W L eL ; W L dH/ = dHBorn − 5:76 log + 13:9 log log 2 ; u s 862 M t e2 (M 2 ) s eR ; W L dH/eR ; WL = dHBorn −2:58 log log 2 2 u 86 M

(256)

(257) (258) (259)

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EW

- dσB) / dσB

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

: eRφ exp () : eRφ 1loop

-0.7

-0.8

: eLφ exp () : eLφ 1loop

-0.9

: eLWT exp () : eLWT 1loop

1000

1500

2000

2500

3000

3500

4000

4500

5000

√s Fig. 22. The purely electroweak virtual corrections relative to the Born cross section in transverse and longitudinal W ± production in e+ e− collisions at 90◦ scattering angle as a function of the c.m. energy. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. It is clearly visible that the di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

and RG corrections which at one loop are given by s g2 (M 2 ) eL ; W T eL ; W T −200 ; = dHBorn log dHRG 462 M2 41 − 82cw2 + 22cw4 g2 (M 2 ) s eL ; W L eL ; W L log 2 ; dHRG = dHBorn 12cw2 462 M 41 g 2 (M 2 ) s eR ; W L eR ; W L : = dHBorn log 2 dHRG 2 12 46 M

(260) (261) (262) (263)

The corrections are written relative to the Born cross section and Fig. 22 shows the one loop and all orders results in each case to SL accuracy. The largest e5ect can be seen in the transverse sector

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

295

EW

(dσSL - dσDL ) / dσB 0.04

0.02

0

-0.02

-0.04

-0.06

-0.08

: eR φ

-0.1

: eLφ

-0.12

: eLWT -0.14

1000

1500

2000

2500

3000

3500

4000

4500

5000

√s Fig. 23. The purely electroweak virtual corrections relative to the Born cross section production in e+ e− collisions at 90◦ scattering angle as a function of the c.m. energy. 2gure for each symbol. Given are the full one loop and the resummed corrections to It is clearly visible that the di5erence between the two approaches, originating from non-negligible at TeV energies and necessitates the inclusion of the higher order SL below the weak scale are omitted.

in transverse and longitudinal W ± The polarization is indicated in the DL and SL accuracy in each case. SL terms at the two loop level, is terms. Pure QED corrections from

for left handed electrons where the resummed terms are of order 30% at 1 TeV, 55% at 3 TeV and 65% at 5 TeV. The respective cross section for right handed electrons is mass suppressed [97]. Also for longitudinal W ± (∼ ± via the equivalence theorem) we have signi2cant e5ects. For left handed electrons the resummed terms are of order 20% at 1 TeV, 38% at 3 TeV and 46% at 5 TeV. For right handed electrons the resummed corrections are of order 11% at 1 TeV, 22% at 3 TeV and 26% at 5 TeV. In each case it can be seen that the one loop contributions alone are insuScient in the TeV regime and di5er from the resummed results in the percentile range at 1 TeV, of order 10% at 3 TeV and even more beyond. It is, however, not only important to take higher order DL corrections into account but also the higher order SL terms. This can be seen in Fig. 23 where the one loop terms agree (to SL accuracy) and at higher orders in one case only the DL and in the other also the SL terms are kept. Again the largest e5ect occurs in the transverse W ± sector. The di5erence between the higher order SL and

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DL corrections is of order 4% at 1 TeV, 10.5% at 3 TeV and 14.5% at 5 TeV. Also for longitudinal W ± production the e5ects are signi2cant. For left handed electrons the di5erence between the higher order SL and DL terms changes cross section by 1.8% at 1 TeV, 3% at 3 TeV and 3.5% at 5 TeV. For right handed electrons we have di5erences of 0.5% at 1 TeV, 2% at 3 TeV and 3% at 5 TeV. Thus, at these energies the SL terms can be as large as the leading terms and must not be omitted. For the longitudinal gauge bosons, there is a partial cancellation between the subleading Sudakov and Yukawa enhanced terms. Therefore, the overall e5ect is larger in the transverse sector but still large in the longitudinal one. Also the angular terms are signi2cant as can be seen in Fig. 24. We treat the angular terms at higher orders as described in Section 3.6 and the one loop terms from Eqs. (257)–(259). The 2gure displays the e5ect for 2xed c.m. energy of 1 and 3 TeV and we have written all angular terms in such a way that they are proportional to the Born cross section. This is always possible but involves factor of t=u, etc. It can be seen that the angular corrections are large and vary for the resummed contributions by almost 20% at 1 TeV and 17% at 3 TeV for transverse W ± production for scattering angles between 50◦ and 130◦ . Also for longitudinal W ± production the corrections are large. For left handed electrons the cross section changes by about 19% at 1 TeV and 20% at 3 TeV in the same angular range. For the same range of scattering angles, the right handed electrons the resummed angular terms change cross section by about 4.5% at 1 TeV and 5% at 3 TeV. The one loop corrections are even larger and lead to signi2cantly di5erent results. Thus, the higher order terms are very important and it is mandatory to investigate if the two loop angular corrections in the full electroweak theory are indeed given by the product of the one loop terms and the Sudakov form factor. In e+ e− collisions, the only non-mass suppressed longitudinal Z process is the Higgs–strahlung process. As such we discuss it in the next section together with other Higgs production processes. 4.2. Higgs production In this section we discuss light SM Higgs production processes relevant to the linear collider program. We begin with the Higgs–strahlung process e+ e− → H. The cross section for this process is smaller than the W -fusion process at TeV energies, however, it would still be utilized to collect more Higgs events. The electroweak corrections, however, will reduce the cross section. All universal corrections are given by Eqs. (178) and (177). In addition we have angular terms which read relative to the Born amplitude in the high energy e5ective regime described by Lsymm in Eq. (11): −u −t g2 (M 2 ) 4cw2 s / + log : (264) .e+ e− →H = − log 2 log R L 862 cw2 − sw2 M s s B;W a Numerically we have for relative to the Born cross section: −t s −u e2 (M 2 ) eL ; H eL ; H dH/ = dHBorn −50:25 log log 2 + log s s 862 M

(265)

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297

- dσB) / dσ B

EW

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

: eR φ exp () : eR φ1loop

-0.35

: eL φ exp () : eL φ 1loop

-0.4

: eLWT exp () : eLWT 1loop

-0.45

-0.5 50

60

70

80

90

100

110

120

130

110

120

130

θ (√s=1TeV) (dσ

- dσB) / dσB

EW

-0.2

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: eR φ exp () : eR φ1loop

-0.7

: eL φ exp () : eL φ1loop

-0.8

: eLWT exp () : eLWT 1loop

-0.9

50

60

70

80

90

100

θ (√s=3TeV)

Fig. 24. The purely electroweak virtual corrections relative to the Born cross section in transverse and longitudinal W ± production in e+ e− collisions at 1 and 3 TeV c.m. energy as a function of the scattering angle. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. It is clearly visible that the di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

and RG corrections which at one loop are given by 41 − 82cw2 + 60cw4 g2 (M 2 ) s eL ; H eL ; H dHRG = dHBorn log 2 ; 12cw2 (sw2 − cw2 ) 462 M

(266)

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- dσB) / dσB

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: eR H χ1loop : eL Hχ exp () : eL H χ1loop

-0.4

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4500

5000

√s Fig. 25. The purely electroweak virtual corrections relative to the Born cross section in Higgs–strahlung in e+ e− collisions at 90◦ scattering angle as a function of the c.m. energy. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

eR ; H eR ; H dHRG = dHBorn

41 g 2 (M 2 ) s log 2 2 12 46 M

:

(267) (268)

Fig. 25 depicts the changes of the cross section as a function of c.m. energy for the two electron polarizations at 90◦ scattering angle. Only Z-exchange contributes. The resummed corrections reduce the cross section by about 11% at 1 TeV, 23% at 3 TeV and about 28% at 5 TeV for left handed electrons. For right handed electrons, the e5ect is actually an increase in the cross section at energies up to about 2 TeV due to the large positive angular terms. The reduction at 3 TeV is about 5% and 14% at 5 TeV for the resummed cross section. Again we can see that the one loop predictions di5er in the percentile regime at 1 TeV and by about 7% at 3 TeV. At higher energies the di5erence grows even more rapidly. Thus, for the linear collider in the TeV range, higher order contributions are necessary.

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

(dσ

299

- dσB) / dσB

0.3

0.25

: eLHχ exp () (1 TeV) : eLHχ 1loop (1 TeV)

0.2

: eLHχ exp () (3 TeV) : eLHχ 1loop (3 TeV)

0.15

0.1

0.05

0

-0.05

-0.1

-0.15 50

60

70

80

90

100

110

120

130

θ Fig. 26. The purely electroweak virtual corrections relative to the Born cross section in Higgs–strahlung in e+ e− collisions at 1 and 3 TeV c.m. energy as a function of the scattering angle. Only cross sections involving left handed electrons receive SL angular corrections. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

In Fig. 26 the angular dependence for the Higgs–strahlung process is displayed. Only the cross section involving left handed electrons possess angular dependent terms. The corrections are symmetric with respect to the central scattering angle (i.e. symmetric in u ↔ t). The resummed cross section are consistently lower due to the large positive angular one loop result in Eq. (265). At 1 TeV, the corrections change by about 14% and at 3 TeV by about 18% over the displayed angular range. Again we see how important those terms are for future collider experiments in the TeV range. We now want to brieUy discuss two other important Higgs production processes at e+ e− colliders, namely the W fusion process and Higgs–strahlung o5 top quarks. Both of these processes involve three 2nal state particles and given the multiplicity of 2nal state scattering states, we restrict ourselves here to a discussion only of the energy-dependence at DL accuracy. All universal DL corrections are given by the DL terms in Eqs. (128) and (156). We begin with the W -fusion process e+ e− → e Qe H in Fig. 27. Only left handed electrons contribute. This process yields the largest cross section at high energies for Higgs production.

300

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

(dσ DL - dσB) / dσB -0.2

-0.3

-0.4

-0.5

-0.6

: eL νL H exp () : eL νL H 1loop

-0.7

-0.8

1000

1500

2000

2500

3000

3500

4000

4500

5000

√s Fig. 27. The purely electroweak virtual corrections relative to the Born cross section in Higgs radiation o5 W ± (fusion) in e+ e− collisions as a function of the c.m. energy. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to DL accuracy in each case. It is clearly visible that the di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted. The SL terms are omitted for simplicity, however, are needed for a full treatment.

The resummed DL corrections reduce the cross section by about 25% at 1 TeV, 46% at 3 TeV and 53% at 5 TeV. The one loop DL predictions are o5 by about 5% at 1 TeV, 15% at 3 TeV and 20% at 5 TeV relative to the Born cross section! While the SL terms can change this picture signi2cantly, the typical size of the corrections should be correctly depicted in Fig. 27. In Fig. 28 we display the electroweak DL corrections to Higgs productions o5 top quarks for the various polarizations. This process is crucial in order to determine the top-Higgs Yukawa coupling [120]. The DL terms reduce the cross section for eL tL by about 24% at 1 TeV, 43% at 3 TeV and 52% at 5 TeV. For eR tL by about 18% at 1 TeV, 35% at 3 TeV and 42% at 5 TeV. For eL tR by about 17% at 1 TeV, 33% at 3 TeV and 40% at 5 TeV and 2nally for eR tR by about 11% at 1 TeV, 21% at 3 TeV and 27% at 5 TeV. Again the one loop DL corrections di5er signi2cantly, especially for left handed polarizations.

M. Melles / Physics Reports 375 (2003) 219 – 326

301

EW

(dσDL - dσB) / dσB -0.1

-0.2

-0.3

-0.4

: eRtLH exp () : eRtLH 1loop

-0.5

: eLtLH exp () : eLtLH 1loop

-0.6

: eLtRH exp () : eLtRH 1loop

-0.7

: eRtRH exp () : eRtRH 1loop -0.8 1000

1500

2000

2500

3000

3500

4000

4500

5000

√s

Fig. 28. The purely electroweak virtual corrections relative to the Born cross section in Higgs radiation o5 2nal state top-quarks in e+ e− collisions as a function of the c.m. energy. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to DL accuracy in each case. It is clearly visible that the di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted. The SL terms are omitted for simplicity, however, are needed for a full treatment.

In all Higgs production processes discussed in this section, the electroweak radiative corrections are important and can reduce the cross section considerably. Even for the top-Yukawa measurement at 800 GeV at TESLA or the NLC, corrections are of O (20%) and the di5erence between one loop and resummed and be a few percent. Therefore a full higher order SL analysis is warranted for this process in addition to the QCD corrections. 4.2.1. Charged MSSM Higgs production In this section we discuss the e5ect of MSSM Sudakov e5ects in charged Higgs production. The relevant radiative corrections are given in Eq. (199) for the universal and angular dependent Sudakov terms above the susy scale set by ms = mH under the assumptions stated in Section 3.7. The angular dependent corrections can also be obtained from Eqs. (258) and (259) via the replacement M → mH , which for our purposes, however, is of SSL accuracy in our “light susy” mass assumption.

302

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

(dσ

- dσB) / dσB

0.05

0

-0.05

-0.1

mH = 300 GeV

-0.15

: eRH+ (tan = 10) : eLH+ (tan = 10)

-0.2

: eRH+ (tan = 40) : eLH+ (tan = 40) -0.25 1000

1500

2000

2500

3000

3500

4000

4500

5000

√s Fig. 29. The purely electroweak MSSM virtual corrections to charged Higgs production above the susy scale set by mH = 300 GeV to SL accuracy. The dependence is shown for two characteristic choices of tan 0 for both right and left handed electron polarizations.

In addition we have RG contributions at one loop which read s -2 (m2H )00 + - 2 (m2H )00 eL ; H + eL ; H + log 2 ; dH/ = dHBorn −2 2 2 6(-(mH ) + - (mH )) mH dH/eR ; H+

=

eR ; H+ dHBorn

- (m2H ) ˜ s 00 log 2 ; −2 6 mH

(269)

(270)

where 0˜ 0 and 0˜ 0 are given in Eq. (198). In Fig. 29 the energy dependence for the process e+ e− → H + H − is depicted for two typical values of tan 0 = 10 and tan 0 = 40. The scattering angle is held 2xed at / = 90◦ since the angular dependence is analogous to the case of longitudinal W production in Fig. 24. The helicity of the electrons is indicated and only the resummed all orders SL results are presented. The value of the heavy charged Higgs particles is mH = 300 GeV. For a di5erent heavy Higgs mass, the displayed results shift accordingly. Fig. 29 shows that the e5ect of the MSSM Sudakov corrections is somewhat reduced compared to the SM case. This is almost entirely due to the di5erent mass scale used. The left handed cross section are more suppressed than the right handed ones due to the larger DL group factors. In addition,

M. Melles / Physics Reports 375 (2003) 219 – 326 (dσ

EW

303

- dσB) / dσB

0.05

0.025

0

-0.025

-0.05

-0.075

mH = 300 GeV -0.1

-0.125

: eRH+ (1 TeV) : eLH+ (1 TeV)

-0.15

: eRH+ (3 TeV) : eLH+ (3 TeV) 5

10

15

20

25

30

35

40

tan β

Fig. 30. The purely electroweak MSSM virtual corrections for charged Higgs production above the susy scale set by √ mH = 300 GeV to SL accuracy as a function of tan 0. The dependence is shown for two characteristic choices of s for both right and left handed electron polarizations.

larger values for tan 0 lead to an enhanced suppression since the overall sign of the Yukawa terms √ is negative. The dependence on tan 0 is explicitly shown in Fig. 30 for c.m. energies of s = 1 TeV √ and s = 3 TeV. At 1 TeV the right and left handed cross section the same overall dependence on tan 0, di5ering by about 7% in the range displayed. At 3 TeV, the left handed cross section is more suppressed but the overall tan 0 dependence is similar. The cross section can di5er by about 13% in the range between tan 0 = 6 to tan 0 = 40. This strong dependence, especially for larger values on tan 0 originates from both, the size of the Yukawa terms as well as from the number of colors in the quark loops. In Ref. [114] this was utilized to suggest a measurement of tan 0 at CLIC with a relative precision better than 25% (a few percent for large values tan 0 ¿ 25). More importantly, this determination of tan 0 to SL accuracy does not depend on the soft breaking terms (which are constants) and is scheme and gauge invariant. While we focussed here only on the case of heavily charged Higgs production, also the neutral Higgs, fermion and sfermion processes have important information on MSSM parameters contained in the coeScients of large Sudakov logarithms and should be fully exploited at such a collider.

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M. Melles / Physics Reports 375 (2003) 219 – 326

4.3. Heavy quark production In this section we discuss heavy quark production at the linear collider. These processes can be used to measure -s above the production threshold and should be fully understood. A general result for the angular terms relative to the Born amplitude is given by 2 2 (M ) g s t ./e+ e− →fQ0 f0 = − log 2 [tan2 /w Ye-− Yf0 + 4Te3− Tf30 ] log 2 - 166 M u B;W a .-; L .0; L + 2 tan /w Ye-− Yf0 =4 + Te3− Tf30 -

−t −u − .u; f log .d; f log s s

;

(271)

where the last line only contributes for left handed fermions and the d; u symbols denote the corresponding isospin quantum number of f. We begin with top production. All universal terms are given by Eqs. (178) and (177). The angular terms and RG corrections which at one loop are given by t s −u e2 (M 2 ) eL ; tL eL ; t L dH/ = dHBorn 4:9 log − 16:3 log (272) log 2 ; u s 862 M t e2 (M 2 ) s eR ; tL 0:86 log (273) log 2 ; dH/eR ; tL = dHBorn 2 u 86 M t e2 (M 2 ) s eL ; tR eL ; tR (274) log 2 ; dH/ = dHBorn 1:72 log u 862 M t e2 (M 2 ) s eR ; tR 3:43 log (275) log 2 dH/eR ; tR = dHBorn 2 u 86 M and

e2 (M 2 ) s −12:2 = log 2 ; 862 M 2 2 e (M ) s eR ; tL eR ; tL log 2 ; dHRG = dHBorn 8:8 862 M 2 2 e (M ) s eL ; tR eL ; tR log 2 ; dHRG = dHBorn 8:8 862 M 2 2 e (M ) s eR ; tR eR ; tR log 2 : dHRG = dHBorn 8:8 862 M eL ; tL dHRG

eL ; t L dHBorn

(276) (277) (278) (279)

Fig. 31 displays the energy dependence of the corrections for central scattering angles for the various polarizations. The di5erence between the resummed and the one loop contributions is in the several percent range only above 1 TeV. The largest corrections are again obtained in the case where both fermions are left handed. The overall corrections are large and non-negligible. Also the angular terms, depicted in Fig. 32, are signi2cant, especially for eL tL where they change cross section by about 20% for 1 and 3 TeV over the angular range between 50◦ and 130◦ . This is another indication that the higher order angular terms need to be understood in the full electroweak theory.

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

(dσ

305

- dσB) / dσB

0

-0.05

-0.1

-0.15

-0.2

-0.25

: eR tL exp () : eR tL 1loop

-0.3

: eLtL exp () : eLtL 1loop

-0.35

: eLtR exp () : eLtR 1loop

-0.4

: eR tR exp () : eR tR 1loop

-0.45 1000

1500

2000

2500

3000

3500

4000

4500

5000

√s

Fig. 31. The purely electroweak virtual corrections relative to the Born cross section in top quark production in e+ e− collisions at 90◦ scattering angle as a function of the c.m. energy. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

For bottom quark production bR does not contain SL-Yukawa terms. All universal corrections are given by Eqs. (178) and (177). In addition we have angular terms and RG corrections which at one loop are given by −t e2 (M 2 ) t s eL ; b L eL ; b L dH/ (280) = dHBorn − 4:04 log + 19:8 log log 2 ; u s 862 M t e2 (M 2 ) s eR ; b L eR ; b L (281) = dHBorn 0:86 log log 2 ; dH/ u 862 M t e2 (M 2 ) s eL ; b R −0:86 log ; (282) log dH/eL ; bR = dHBorn u 862 M2 t e2 (M 2 ) s eR ; b R eR ; b R (283) = dHBorn −1:72 log log 2 dH/ u 862 M

306

M. Melles / Physics Reports 375 (2003) 219 – 326 EW

- dσB) / dσB

(dσ 0.05

0

-0.05

-0.1

: eR tL exp () : eR tL 1loop : eLtL exp () : eLtL 1loop

-0.15

: eLtR exp () : eLtR 1loop

-0.2

: eR tR exp () : eR tR 1loop -0.25 50

60

70

80

90

100

110

120

130

θ (√s=1TeV) EW

(dσ

- dσB) / dσB

0

-0.05

-0.1

-0.15

-0.2

-0.25

-0.3

: eR tL exp () : eR tL 1loop

-0.35

: eLtL exp () : eLtL 1loop

-0.4

: eL tR exp () : eL tR 1loop

-0.45

: eR tR exp () : eR tR 1loop 50

60

70

80

90

100

110

120

130

θ (√s=3TeV)

Fig. 32. The purely electroweak virtual corrections relative to the Born cross section in top quark production in e+ e− collisions at 1 and 3 TeV c.m. energy as a function of the scattering angle. The polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

and eL ; b L dHRG

=

eL ; b L dHBorn

e2 (M 2 ) s −16:6 log 2 2 86 M

;

(284)

M. Melles / Physics Reports 375 (2003) 219 – 326 (dσ

EW

307

- dσ B) / dσB

0

-0.1

-0.2

: eRbR exp () : eRbR 1loop : eLbR exp () : eLbR 1loop

-0.3

: eRbL exp () : eRbL 1loop : eLbL exp () : eLbL 1loop

-0.4

1000

1500

2000

2500

3000

3500

4000

4500

5000

√s Fig. 33. The purely electroweak virtual corrections relative to the Born cross section in bottom quark production in e+ e− collisions at 90◦ scattering angle as a function of the c.m. energy. The electron polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

eR ; b L dHRG

=

eR ; b L dHBorn

e2 (M 2 ) s 8:8 log 2 2 86 M

;

e2 (M 2 ) s 8:8 = log 2 ; 862 M e2 (M 2 ) s eR ; b R eR ; b R 8:8 : = dHBorn log dHRG 862 M2 eL ; b R dHRG

eL ; b R dHBorn

(285)

(286) (287)

Fig. 33 demonstrates that in the energy range displayed, the electroweak corrections are actually positive for eR bR . This is mainly due to the RG corrections for the right handed coupling (g ) which is Abelian and therefore increases with energy. Secondly it is due to the fact that the Yukawa terms are absent and the DL terms are partially o5set by the SL terms. The remaining corrections are similar to the top production discussion.

308

M. Melles / Physics Reports 375 (2003) 219 – 326 - dσB) / dσB

EW

(dσ 0.05

0

-0.05

: eRbR exp () : eRbR 1loop -0.1

: eLbR exp () : eLbR 1loop : eR bL exp ()

-0.15

: eR bL 1loop : eLbL exp ()

-0.2

: eLbL 1loop

-0.25 50

60

70

80

90

100

110

120

130

θ (√s=1TeV) EW

(dσ

- dσB) / dσB

0

-0.1

: eRbR exp ()

-0.2

: eRbR 1loop : eLbR exp () : eLbR 1loop

-0.3

: eRbL exp () : eRbL 1loop -0.4

: eLbL exp () : eLbL 1loop -0.5 50

60

70

80

90

100

110

120

130

θ (√s=3TeV)

Fig. 34. The purely electroweak virtual corrections relative to the Born cross section in bottom quark production in e+ e− collisions at 1 and 3 TeV c.m. energy as a function of the scattering angle. The electron polarization is indicated in the 2gure for each symbol. Given are the one loop and the resummed corrections to SL accuracy in each case. The di5erence between the two approaches is non-negligible at TeV energies and necessitates the inclusion of the higher order terms. Pure QED corrections from below the weak scale are omitted.

The angular terms, depicted in Fig. 34, show a similar behavior to the case of top quark production in Fig. 32 in that the largest contribution is again for the purely left handed case. The overall size of the corrections di5ers and the eR bR contributions stay positive for most of the displayed angular range.

M. Melles / Physics Reports 375 (2003) 219 – 326

309

5. Outlook Electroweak radiative corrections at high energies have received much attention recently due to their importance at experiments in the TeV regime. It is not only the phenomenological importance, however, that has led to a surge in interest into the high energy behavior of the SM but also the fact that conceptually new e5ects enter due to EWSB. The main di5erences to the case of unbroken gauge theories originate from the longitudinal sector and the fact that three of the gauge bosons acquire masses, while the photon stays massless. In addition, the asymptotic states carry a non-Abelian group charge, the weak isospin and are superpositions of the 2elds of the unbroken phase. As a consequence, fully and semi-inclusive cross section show double and single logarithmic dependencies on the ratio of the energy and the gauge boson mass, and longitudinal degrees of freedom are not mass suppressed. Thus, observables in the SM depend on the infrared cuto5, the weak scale, which in this case, however, is a physical parameter. At one loop, general methods exist which allow to calculate corrections relevant to precision measurements at future colliders. Also at higher orders, a general approach to SL accuracy is available, based on the high energy approximation of the symmetric part of the SM Lagrangian and is phenomenologically necessary. Focusing on techniques to calculate the higher order corrections, we have summarized the present status of virtual electroweak radiative corrections to SM and MSSM high energy processes. In the framework of the IREE method, the high energy e5ective theory is based on the high energy limit of the SM Lagrangian in the symmetric limit where all terms connected to the v.e.v. can be neglected to SL accuracy. The QED corrections below the weak scale are incorporated with the appropriate matching conditions. This approach is so far the only one able to allow for a two loop calculation of the DL and SL terms which are relevant for TeV experiments at future colliders. We have shown that the one loop terms are insuScient when the c.m. energy is larger than 1 TeV and that both, DL and SL terms at the two loop level are necessary at the several percent level. Also angular dependent corrections cannot be neglected and it should be investigated if their calculation at the SSL level is needed for some observables. It should also be investigated if large Yukawa constants can be treated in a systematic manner to SSL accuracy. In this context it is also important to consider the emission of real gauge bosons above the weak scale even at the SL level. In summary, there exists a way to calculate all higher order virtual SL electroweak radiative corrections to high energy processes. The approach is in agreement with all available one loop calculations in terms of the physical SM or MSSM 2elds and at the two loop level to DL accuracy. These terms are crucial for the experiments at future colliders in the TeV regime since the e5ects of new physics expected in this range can be rather small.

Acknowledgements I would like to thank A. Denner, S. Pozzorini, E. Accomando for valuable discussions and D. Wyler for his support of this work.

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Appendix A. In this appendix we list the relevant quantum numbers of the physical SM 2elds and the corresponding Feynman rules of the full SM in the ’t Hooft–Feynman gauge. A.1. Operators and quantum numbers In this section we present the generators of the physical gauge group and various group-theoretical matrices used in Section 3.9. We follow the discussion of Ref. [29]. The notation for the components of such matrices is M’i ’i (’) ;

(A.1)

where the argument ’ represents a multiplet and 2xes the representation for the matrix M , whereas ’i are the components of the multiplet. Explicit representations for left- and right-handed fermions (’ = fL ; fR ; fL ; fR ), for gauge bosons (’ = V ) and for the scalar doublet (’ = &) are given below. Where the representation is implicit, the argument ’ is omitted. For the eigenvalues of diagonal matrices one has M ’ i ’i = . ’ i ’ i M’i

(A.2)

A.1.1. Symmetric and physical gauge 0elds and gauge couplings For physical gauge bosons one needs to take special care of the e5ect of Weinberg rotation (mixing). The symmetric basis Va = B; W 1 ; W 2 ; W 3 , is formed by the U(1) and SU(2) gauge bosons, which transform as a singlet and a triplet, respectively, and quantities in this basis are denoted by a tilde. The physical basis is given by the charge and mass eigenstates Va = A; Z; W + ; W − . The physical charged gauge bosons, W± =

W 1 ∓ iW 2 √ ; 2

(A.3)

are pure SU(2) states, whereas in the neutral sector the SU (2) and U (1) components mix, and the physical 2elds N = A; Z are related to the symmetric 2elds N˜ = B; W 3 by the Weinberg rotation,

cw −sw (A.4) N = UN N˜ (/w )N˜ ; U (/w ) = cw sw with cw = cos /w and sw = sin /w . In the on shell renormalization scheme the Weinberg angle is 2xed by cw =

MW MZ

(A.5)

The gauge couplings are given by the generators of global gauge transformations. In the symmetric basis, they read IB = −

1 Y ; cw 2

1 a a I˜W = T ; sw

a = 1; 2; 3 ;

(A.6)

M. Melles / Physics Reports 375 (2003) 219 – 326

311

where Y is the weak hypercharge and T a are the components of the weak isospin. In the physical basis one has I A = −Q;

IZ =

T 3 − sw2 Q ; sw cw

I± =

1 ± 1 T 1 ± iT 2 √ T = sw sw 2

(A.7)

with Q = T 3 + Y=2. A.1.2. Casimir operators The SU(2) Casimir operator is de2ned by C=

3

(T a )2

(A.8)

a=1

Loops involving charged gauge bosons are often associated with the product of the non-Abelian charges C − (T 3 )2 (I W )2 := (A.9) [I H I −H ] = 2 s w H=± and if one includes the contributions of neutral gauge bosons, one obtains the e5ective electroweak Casimir operator 1 Y 2 1 Va VQ a cw C := I I = 2 + 2 C : (A.10) cw 2 sw ± Va =A;Z;W

For irreducible representations (fermions and scalars) with isospin T’ , the SU(2) Casimir operator is proportional to the identity and reads C’i ’i (’) = .’i ’i C’ ;

C’ = T’ [T’ + 1] :

(A.11)

Physical gauge bosons have a reducible representation as already discussed in Section 3. In the ˜ ) is a diagonal 4 × 4 matrix symmetric basis C(V C˜ V˜ a V˜ b = .ab C˜ V˜ a

(A.12)

with U(1) and SU(2) eigenvalues C˜ B = 0;

C˜ W a = 2 :

(A.13)

The transformation of a matrix like (A.12) to the physical basis, yields a 4 × 4 matrix with diagonal 2 × 2 block structure, i.e. without mixing between the charged sector (W ± ) and the neutral sector (N = A; Z). In the neutral sector C(V ) becomes non-diagonal owing to mixing of the U (1) and SU (2) eigenvalues,

2 −s c s w w w ˜ −1 (/w )]NN = 2 CNN = [U (/w )CU ; (A.14) −sw cw cw2

312

M. Melles / Physics Reports 375 (2003) 219 – 326

whereas in the charged sector it remains diagonal, CW H W H = 2.HH :

(A.15)

A.1.3. Explicit values for Y , Q, T 3 , C, (I A )2 , (I Z )2 , (I W )2 , C ew , and I ± Here a list of the eigenvalues (or components) of the operators Y; Q; T 3 ; C; (I A )2 ; (I Z )2 ; (I W )2 ; ew C , and I ± is given following Ref. [29], that have to be inserted in the general results. For incoming particles or outgoing antiparticles the values for the particles have to be used, for incoming antiparticles or outgoing particles the values of the antiparticles. @ @ T Fermions. The fermionic doublets f@ = (f+ ; f− ) transform according to the fundamental or trivial representations, depending on the chirality @ = L; R. Except for I ± , the above operators are diagonal. For lepton and quark doublets, L@ = (@ ; l@ )T and Q@ = (u@ ; d@ )T , their eigenvalues are Y=2

Q

T3

C

(I A )2

(I Z )2

(I W )2

C ew

L ; QL ∓ 12

0

± 12

3 4

0

1 2 4sw cw2

1 2sw2

1 + 2cw2 4sw2 cw2

lL ; lQL ∓ 12

∓1

∓ 12

3 4

1

(cw2 − sw2 )2 4sw2 cw2

1 2sw2

1 + 2cw2 4sw2 cw2

lR ; lQR ∓1

∓1

0

0

1

sw2 cw2

0

1 cw2

uL ; uQ L ± 16

± 23

± 12

3 4

4 9

(3cw2 − sw2 )2 36sw2 cw2

1 2sw2

sw2 + 27cw2 36cw2 sw2

dL ; dQ L ± 16

∓ 13

∓ 12

3 4

1 9

(3cw2 + sw2 )2 36sw2 cw2

1 2sw2

sw2 + 27cw2 36cw2 sw2

uR ; uQ R ± 23

± 23

0

0

4 9

4 sw2 9 cw2

0

4 9cw2

dR ; dQ R ∓ 13

∓ 13

0

0

1 9

1 sw2 9 cw2

0

1 9cw2

(A.16)

For left-handed fermions, I ± (fL ) have the non-vanishing components IfHH f−H (fL ) = −IfHQ

Q −H fH

.HH (fQ L ) = √ 2sw

(A.17)

whereas for right-handed fermions I ± (fR ) = 0. Scalar 0elds. The symmetric scalar doublet, &=( + ; 0 )T ; &∗ =( − ; ∗0 )T , transforms according to the fundamental representation, and its quantum numbers correspond to those of left-handed leptons (A.16) with + ↔ lL ;

0 ↔ L ;

− ↔ lL ;

∗0 ↔ L :

(A.18)

M. Melles / Physics Reports 375 (2003) 219 – 326

313

After symmetry breaking the neutral scalar 2elds are parametrized by the mass eigenstates 1 0 = √ (v + H + i) : 2

(A.19)

With respect to this basis S = (H; ) the operators Q; C; (I N )2 , and C ew remain unchanged, while T 3 and Y become non-diagonal in the neutral components

Y 1 0 −i 3 =− TSS = − ; (A.20) 2 SS 2 i 0 and Z Z = −IH = IH

i

(A.21)

2sw cw

The W± couplings read H H H IS −H = −I H S = .HH IS

(A.22)

with IHH := −

H ; 2sw

IH := −

i 2sw

(A.23)

Gauge 0elds. For transversely polarized external gauge bosons one has to use the adjoint representation. In the symmetric basis the diagonal operators have eigenvalues Y=2

Q

T3

C

(I A )2

(I Z )2

(I W )2

C cw

W±

0

±1

±1

2

1

cw2 sw2

1 sw2

2 sw2

W3

0

0

0

2

0

0

2 sw2

2 sw2

B

0

0

0

0

0

0

0

0

(A.24)

In the neutral sector, owing to the Weinberg rotation, the non-trivial operators C cw ; C and (I W )2 become non-diagonal in the physical basis N = A; Z, with components 2

−s c s w w 1 2 w cw (A.25) CNN = (I W )2NN = 2 CNN = sw2 sw −sw cw cw2 whereas the trivial operators Y=2 = Q = T 3 = (I A )2 = (I Z )2 = 0 remain unchanged. In the physical basis the non-vanishing components of the I ± couplings are H H H INW −H = −IW H N = .HH IN

with IAH = −H;

IZH = H

cw sw

(A.26) (A.27a)

314

M. Melles / Physics Reports 375 (2003) 219 – 326

A.2. Electroweak Feynman rules In this appendix we list the Feynman rules of the SM in the ’t Hooft–Feynman gauge including the counterterms in a way appropriate for the concept of generic diagrams [41]. I.e. we write down generic Feynman rules obtained from the classic Lagrangian in Eq. (5) and give the possible actual insertions. We omit any 2eld renormalization constants for the unphysical 2elds. For brevity we introduce the shorthand notation c = cw ;

s = sw :

(A.27b)

In the vertices all momenta are considered as incoming. Propagators: for gauge bosons V = ; Z; W in the ’t Hooft–Feynman gauge (i = 1)

for Faddeev–Popov ghosts G = u ; uZ ; uW

for scalar 2elds S = H; ;

and for fermion 2elds F = fi

In the ’t Hooft–Feynman gauge we have the following relations: Mu = 0;

Mu Z = M = M Z ;

Tadpole:

VV counterterm:

Mu± = M = MW :

(A.28)

M. Melles / Physics Reports 375 (2003) 219 – 326

315

with the actual values of V1 ; V2 and C1 ; C2 , without renormalization of the gauge-2xing term W+ W− : C1 = C3 = .Z W ; ZZ : C1 = C3 = .Z ZZ ;

2 2 C2 = MW .Z W + .MW ;

C2 = MZ2 Z ZZ + .MZ2 ;

AZ : C1 = C3 = 12 .Z AZ + 12 .Z ZA ; AA : C1 = C3 = .Z AA ;

C2 = MZ2 12 .Z ZA ;

C2 = 0

(A.29a)

with renormalization of the (’t Hooft–Feynman) gauge-2xing term C3 = 0;

C1 ; C2 as above

(A.29b)

VS counterterm:

with the actual values of V; S and C, without renormalization of the gauge-2xing term 2 .MW ± ∓ W 1 ; W : C = ±MW 2 .Z + 2 MW .MZ2 ZZ 1 Z : C = iMZ 2 .Z + ; MZ2 A : C = iMZ 21 .Z ZA

(A.30a)

with renormalization of the (’t Hooft–Feynman) gauge-2xing term C =0 :

(A.30b)

SS-counterterm:

with the actual values of S1 ; S2 and C1 ; C2 , without renormalization of the gauge-2xing term HH : C1 = .ZH ; : C1 = 0;

C2 = MH2 .Z H + .MH2 ; e .t ; 2s MW e .t C2 = − : 2s MW

C2 = −

+ − : C1 = 0;

(A.31a)

316

M. Melles / Physics Reports 375 (2003) 219 – 326

with renormalization of the (’t Hooft–Feynman) gauge-2xing term HH : C1 = .ZH ; : C1 = 0;

C2 = MH2 .Z H + .MH2 ;

C2 = −

+ − : C1 = 0;

e .t + .MZ2 ; 2s MW

C2 = −

e .t 2 + .MW : 2s MW

(A.31b)

FF-counterterm:

with the actual values of F1 ; FQ 2 and CL ; CR ; CS− ; CS+   CL = 12 (.Zijf; L + .Zijf; L† ); CR = 12 (.Zijf; R + .Zijf; R† ) ;    fj fQi : CS− = mf; i 21 .Zijf; L + mf; j 21 .Zijf; R† + .ij .mf; i ;     C + = m 1 .Z f; R + m 1 .Z f; L† + . .m : f; i 2 f; j 2 ij f; i ij ij S

(A.32)

VVVV coupling:

with the actual values of V1 ; V2 ; V3 ; V4 and C, 1 .s + + − − W ; W W W W : C = 2 1 + 2.Ze − 2 + 2.Z s s c c2 1 .s + − W ZZ + .Z AZ ; W W ZZ : C = − 2 1 + 2.Ze − 2 2 + .Z + .Z s c s s  c 1 .s 1 ZZ 1 AA W    C = s 1 + 2.Ze − c2 s + .Z + 2 .Z + 2 .Z + − W W AZ : 2    − 1 .Z AZ − 1 c .Z ZA ; 2 2 s2 c W+ W− AA : C = −[1 + 2.Ze + .Z W + .Z AA ] + .Z ZA : s

(A.33)

M. Melles / Physics Reports 375 (2003) 219 – 326

317

VVV coupling:

with the actual values of V1 ; V2 ; V3 and C, 1 1 c ZA .Z ; AW+ W− : C = 1 + .Ze + .Z W + .Z AA − 2 2 s 1 c 1 .s 1 1 + .Ze − 2 + .Z W + .Z ZZ + .Z AZ : ZW+ W− : C = − s c s 2 2

(A.34)

SSSS coupling:

with the actual values of S1 ; S2 ; S3 ; S4 and C, 2 .t 3 MH2 .s .MH2 e .MW H HHHH : C = − 2 2 1 + 2.Ze − 2 + ; + − + 2.Z 2 4s MW s 2s MW MH2 MH2 MW 2 HH .t .s .MH2 e .MW 1 MH2 H ; + − + .Z : C = − 2 2 1 + 2.Ze − 2 + 2 4s MW s 2s MW MH2 MH2 MW HH 2 .t 3 MH2 .s .MH2 e .MW ; : C = − 2 2 1 + 2.Ze − 2 + + − 2 4s MW s 2s MW MH2 MH2 MW 2 .t .s .MH2 e .MW 1 M2 ; + − : C = − 2 H2 1 + 2.Ze − 2 + 2 4s MW s 2s MW MH2 MH2 MW 2 .t .s .MH2 e .MW 1 MH2 : + − : C = − 2 2 1 + 2.Ze − 2 + 2 2s MW s 2s MW MH2 MH2 MW SSS coupling:

(A.35)

318

M. Melles / Physics Reports 375 (2003) 219 – 326

with the actual values of S1 ; S2 ; S3 and C, 2 .t 3 MH2 .s .MH2 e 1 .MW + HHH : C = − 1 + .Ze − + − + 2 2 2 2s MW s 2s MW MH 2 MW MH 2 H .s .MH2 e .t 1 .MW 1 MH2 + 1 + .Ze − + − :C =− 2 2 2 2s MW s 2s MW MH 2 MW MH H

3 H .Z 2

;

1 + .Z H 2

VVSS coupling:

with the actual values of V1 ; V2 ; S1 ; S2 and C, 1 .s + − W H W W HH : C = 2 1 + 2.Ze − 2 ; + .Z + .Z 2s s W+ W+ 1 .s W : C = 1 + 2.Z ; − 2 + .Z e 2s2 s W + W − + − s2 − c2 AZ .s (s2 − c2 )2 2 ZZ + .Z .Z ; ZZ : C = 1 + 2.Z + + e 2s2 c2 (s2 − c2 )c2 s sc  s2 − c 2 1 .s 1 ZZ 1 AA  C = 1 + 2.Ze + 2 + 2 .Z + 2 .Z  sc (s − c2 )c2 s ZA + − :   (s2 − c2 )2 1 ZA  .Z + .Z AZ ; + 2s2 c2 2 +

−

AA + − : C = 2[1 + 2.Ze + .Z AA ] +

s2 − c2 ZA .Z ; sc

s2 − c2 .s 1 ZZ H + .Z + .Z ; ZZHH : C = 2 2 1 + 2.Ze + 2 2s c c2 s s2 − c2 .s 1 ZZ + .Z ; ZZ : C = 2 2 1 + 2.Ze + 2 2s c c2 s ZAHH 1 1 : C = 2 2 .Z ZA ; 2s c 2 ZA

:

(A.36)

M. Melles / Physics Reports 375 (2003) 219 – 326

1 1 AZ 1 .c 1 W 1 H 1 ZZ 1 + 2.Ze − − W Z H : C = − + .Z + .Z + .Z .Z ; 2c c 2 2 2 2s 2 1 1 ZA 1 .s 1 W 1 H 1 AA ± ∓ 1 + 2.Ze − + .Z + .Z + .Z .Z ; − W A H : C = − 2s s 2 2 2 2c 2 i 1 AZ i .c 1 W 1 ZZ 1 + 2.Ze − + .Z + .Z .Z ; ∓ W± Z ∓ : C = ∓ 2c c 2 2 2s 2 i 1 ZA i .s 1 W 1 AA ± ∓ 1 + 2.Ze − + .Z + .Z .Z : ∓ W A : C = ∓ 2s s 2 2 2c 2 ±

319

∓

(A.37)

VSS coupling:

with the actual values of V; S1 ; S2 and C, i 1 ZA .Z ; 2cs 2 s2 − c2 .s 1 H 1 ZZ i 1 + .Ze + + .Z + .Z ; ZH : C = − 2cs c2 s 2 2 1 AA s2 − c2 1 ZA + − ; A : C = − 1 + .Ze + .Z + .Z 2 2sc 2 1 .s 1 ZZ s2 − c 2 1 1 + .Ze + 2 + .Z − .Z AZ ; Z + − : C = − 2sc (s − c2 )c2 s 2 2 1 .s 1 W 1 H 1 + .Ze − + .Z + .Z ; W ± ∓ H : C = ∓ 2s s 2 2 i .s 1 W ± ∓ 1 + .Ze − + .Z : W :C =− 2s s 2

AH : C = −

SVV coupling:

(A.38)

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M. Melles / Physics Reports 375 (2003) 219 – 326

with the actual values of S; V1 ; V2 and C, 2 .s 1 .MW 1 H 1 W + .Z ; + + .Z HW+ W− : C = MW 1 + .Ze − 2 s s 2 MW 2 2 1 2s2 − c2 .s 1 .MW 1 H ZZ HZZ : C = MW 2 1 + .Ze + ; + .Z + .Z + 2 sc c2 s 2 MW 2 1 1 ZA .Z ; sc2 2 2 s 1 1 .s 1 .MW 1 W 1 ZZ ± ∓ + W Z : C = −MW 1 + .Ze + 2 − MW .Z AZ ; + .Z + .Z 2 c c s 2 MW 2 2 2 2 s 1 ZA 1 .MW 1 W 1 AA ± ∓ W A : C = −MW 1 + .Ze + − MW + .Z + .Z (A.39) .Z : 2 2 MW 2 2 c2 HZA : C = MW

VFF coupling:

with the actual values of V; FQ 1 ; F2 and C + ; C − ,  % & f; R f; R †   C + = −Qf .ij (1 + .Ze + 12 .Z AA ) + 12 (.Zij + .Zij ) + .ij gf+ 21 .Z ZA ; fQi fj : % &   C − = −Qf .ij (1 + .Ze + 1 .Z AA ) + 1 (.Zijf; L + .Zijf; L† ) + .ij g− 1 .Z ZA ; f 2 2 2 

 .gf+ 1 ZZ 1 1  f; R f; R † + +  C = gf .ij 1 + + + .Z + (.Zij + .Zij ) − .ij Qf .Z AZ ;    2 2 2 gf ZfQi fj :

 .gf− 1 ZZ   − −  . 1 + + 12 (.Zijf; L + .Zijf; L† ) − .ij Qf 21 .Z AZ ; = g + 2 .Z C  ij f  g− f

W+ uQ i dj :

    C + = 0;         

1 .s 1 W + .Z ) + .Vij Vij (1 + .Ze − C− = √ s 2 2s 1 + (.Ziku; L† Vkj + Vik .Zkjd; L ) ; 2 k

M. Melles / Physics Reports 375 (2003) 219 – 326

W− dQ j ui :

    C + = 0;         

W− Qi lj : C + = 0; W− lQj i : C + = 0; where

1 .s 1 W + 2 .Z ) + .Vji† Vji† (1 + .Ze − C− = √ s 2s 1 d; L† † † u; L + (.Zjk Vki + Vjk .Zki ) ; 2 k 1 .s 1 W 1 ; L† l; L − √ + .Z + (.Zii + .Zii ) ; .ij 1 + .Ze − C = s 2 2 2s 1 .s 1 W 1 l; L† ; L − + .Z + (.Zii + .Zii ) ; C = √ .ij 1 + .Ze − s 2 2 2s

s 1 .s ; = − Qf .Ze + 2 c c s 3 2 3 IW; IW; s2 − c2 .s f − s Qf f − − ; .gf = .Ze + + .gf+ : gf = sc sc c2 s

gf+

321

s = − Qf ; c

(A.40)

.gf+

(A.41)

The vector and axial vector couplings of the Z-boson are given by 3 2 IW; 1 − f − 2s Qf + vf = (gf + gf ) = ; 2 2sc SFF coupling:

3 IW; 1 − f + af = (gf − gf ) = : 2 2sc

with the actual values of S; FQ 1 ; F2 and C + ; C − ,   1 1 .s .mf; i .MW 1 H +  C =− + .ij mf; i 1 + .Ze − − + 2 .Z   2s MW s mf; i MW         f; R f; L †   + 12 (mf; i .Zij + .Zij mf; j ) ;    HfQi fj :   1 1 .s .mf; i .MW 1 H  −  + C =− .ij mf; i 1 + .Ze − − + .Z    mf; i 2s MW s MW 2          + 12 (mf; i .Zijf; L + .Zijf; R† mf; j ) ;  

(A.42)

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  1 3 .s .mf; i .MW 1 +   + C = i 2IW; f .ij mf; i 1 + .Ze − −   2s MW s mf; i MW         f; R f; L †  1  + 2 (mf; i .Zij +ij mf; j ) ;    fQi fj :   .m 1 .s .M 1 f; i W  − 3  C = −i 2IW; f + .ij mf; i 1 + .Ze − −    2s MW s mf; i MW         f; L f; R † 1  + 2 (mf; i .Zij + .Zij mf; j ) ;     1 1 .s .md; j .MW +   √ + C V 1 + .Z + .Vij md; j = − m − − ij d; j e   s md; j MW  2s MW        1 u; L † d; R   + (.Zik Vkj md; j + Vik md; k .Zkj ) ;   2  k + uQ i dj :   1 1 .mu; i .MW  − .s  √ C m 1 + .Z + mu; i .Vij = V − + −  u; i ij e s   MW mu; i MW 2s        1  u; R † d; L  + (.Zik mu; k Vkj + mu; i Vik .Zkj ) ;   2 k   1 1 .s .mu; i .MW † +   √ C = + .Vji† mu; i : Vji mu; i 1 + .Ze − − +   M s m M  2s W u; i W      $   1 # d; L† † † u; R   + V m + V m .Z .Z ;  u; i u; k jk ki jk ki  2  k − dQ j ui :   1 1 .s .md; j .MW  † −  √ + C =− md; j Vji 1 + .Ze − + md; j .Vji† −    M s m M 2s W d; j W       $  1 # d; R†   + .Zjk md; k Vki† + md; j Vjk† .Zkiu; L ;   2 k    C + = − √1 ml; i .ij 1 + .Ze − .s + .ml; i − .MW + 1 (.Z ; L† + .Z l; R ) ; ii ii s ml; i MW 2 2s MW + Qi lj :   − C =0 ;  +  C =0 ; −Q $ (A.43) lj i : 1 ml; i .s .ml; i .MW 1 # l; R† ; L −  + .Zii + .Zii .ij 1 + .Ze − − +  C = −√ s ml; i MW 2 2s MW

M. Melles / Physics Reports 375 (2003) 219 – 326

323

VGG coupling:

with the actual values of V; GQ 1 ; G2 and C, c 1 ZA 1 AA ± ± ∓ AuQ u : C = ± 1 + .Ze + .Z .Z ; 2 s 2 1 c 1 .s 1 ZZ ± ± + .Z ± .Z AZ ; ZuQ u : C = ∓ 1 + .Ze − 2 s c s 2 2 c 1 .s 1 W ; W± uQ ± uZ : C = ± 1 + .Ze − 2 + .Z s c s 2 c 1 .s 1 W ± Z ∓ + .Z ; W uQ u : C = ∓ 1 + .Ze − 2 s c s 2 ( ' W± uQ ± u : C = ∓ 1 + .Ze + 12 .Z W ; ( ' W± uQ u∓ : C = ± 1 + .Ze + 12 .Z W SGG coupling:

with the actual values of S; GQ 1 ; G2 and C, 2 1 2s2 − c2 .s 1 .MW 1 H Z Z + ; + .Z HuQ u : C = − 2 MW 1 + .Ze + 2 2sc c2 s 2 MW 2 2 1 .s 1 .MW 1 H ± ± + ; + .Z HuQ u : C = − MW 1 + .Ze − 2 2s s 2 MW 2 2 1 .s 1 .MW ± ± + ; uQ u : C = ∓i MW 1 + .Ze − 2 2s s 2 MW 2 1 s2 − c2 .s 1 .MW MW 1 + .Ze + + ; ± uQ Z u∓ : C = 2 2sc c2 s 2 MW

(A.44)

324

M. Melles / Physics Reports 375 (2003) 219 – 326

2 .s 1 .MW s2 − c 2 1 ; uQ u : C = MW 1 + .Ze + 2 + 2 2sc (s − c2 )c2 s 2 MW 2 1 .MW ± uQ ± u : C = MW 1 + .Ze + : 2 2 MW ±

± Z

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(A.45)

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Physics Reports 375 (2003) 327 – 410 www.elsevier.com/locate/physrep

Method of multiple scales in quantum optics Maciej Janowicz∗ Institute of Physics, Polish Academy of Sciences, Al. Lotnikow 32/46, 02-668 Warsaw, Poland Accepted 1 October 2002 editor: J: Eichler

Abstract Applications of the method of multiple scales to the quantum-optical problems are reviewed. After a preliminary study of applications to the classical and quantum anharmonic oscillator, several examples in the spontaneous emission, resonance /uorescence, and cavity quantum electrodynamics are analyzed. A preliminary account of application of the method to a model of Bose–Einstein condensates is given. A common thread throughout the main body of the review is the stabilization of the wave function and population of a level due to the phase modulation by an external agent. c 2002 Published by Elsevier Science B.V. PACS: 42.50−p; 02.30Mv; 42.50Hz; 42.65Sf Keywords: Quantum optics; Method of multiple scales; Anharmonic oscillator; Spontaneous emission; Resonance /uorescence; Cavity quantum electrodynamics

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Anharmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Heisenberg equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Time evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Comparison with numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Classical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Quantum-mechanical case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Resonance /uorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Corresponding author. Carl-von-Ossietzky University, Fachbereich 8-Physik, 26111 Tel.: +49-441-798-3996; fax: +49-441-798-3201. E-mail addresses: [email protected], [email protected] (M. Janowicz). c 2002 Published by Elsevier Science B.V. 0370-1573/03/$ - see front matter PII: S 0 3 7 0 - 1 5 7 3 ( 0 2 ) 0 0 5 5 1 - 3

Oldenburg,

328 332 332 336 341 341 343 348 365

Germany.

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4.1. General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Modulation of atomic frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Population trapping in cascade systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Bichromatic excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Atom–Deld interactions in cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Two-level externally excited atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Two-mode cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. The four-level “Lambda” system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Description of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3. Application of multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4. SchrIodinger-cat states and population dynamics in a restricted two-mode two-photon interaction . . . 6. Fields and atoms in cavities with oscillating mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Electromagnetic Deld in the oscillating cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Atom–Deld interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Multiple scales and cold boson gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

366 368 371 373 379 380 382 385 385 386 388 393 395 396 399 401 407 407 407

1. Introduction This introductory section serves the following purposes: Drstly, it explains the motivation to study the method of multiple scales (MMS) in Quantum Theory. Then, a most elementary example is shown to give an idea why the method can be successful. Thirdly, very brief historical and bibliographical remarks are given to place the topic in a suitable context in the development of non-linear physics. Undoubtedly, the computer methods play an ever-growing role in the recent development of scientiDc research, especially in physics. This is true in quantum optics and electrodynamics as well. Even very complicated master equations, typical for this Deld of research, can be handled by modern high-speed computers. In the 1990s, an additional very powerful tool for computational analysis of quantum optical systems was developed, namely the Monte-Carlo simulation of the stochastic SchrIodinger equations. In this connection, one might rightly ask whether the approximate methods of applied mathematics can still be of any use, and, in particular, whether they are still useful in such a subject as quantum optics. This work deals with these methods, especially with perturbation theory, and it tries to justify their usefulness. The most important fact about them is that they can strengthen and stimulate our intuition, leading to better understanding of the problem considered. Sometimes, the role of the approximate methods is still greater: they give us the language to speak about the behavior of physical systems. Probably the most striking example are the Feynman diagrams which give graphical representation of perturbation expansions. In this work we concentrate on applications of a speciDc method, called the method of multiple scales, developed in classical mechanics to deal with non-linear equations containing a small parameter. It will be shown that this method is by no means restricted to the classical domain but can also be applied to the Heisenberg equations of motion as well as to the SchrIodinger equation. Whenever possible and meaningful, we shall try to apply this method directly to the SchrIodinger

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equation for the time-evolution operator. This is because the time-evolution operator—together with an initial density matrix—provides the most general description of a physical system. There is a simple reason why the method of multiple scales can be eLcient in the quantum-optical context. In quantum optics, we very often have to do with systems of few degrees of freedom described by some Hamiltonian operators. These operators have the dimension of energy and usually consist of several terms some of which contain numerical factors of the dimension of energy or frequency much smaller than analogous factors standing at other terms. But a Hamiltonian is a generator of the time evolution of the quantum system. Therefore, in the time evolution of the system one can often very naturally distinguish several diMerent time scales deDned by diMerent terms in the Hamiltonian. Let us now brie/y explain why an ordinary—or regular—perturbation expansion is very often insuLcient to obtain reasonable approximation in both classical and quantum dynamics. Indeed, as is well known in non-linear mechanics, the regular perturbation expansion applied to systems with the so-called internal resonance brings about secular terms in the approximate solutions. The term “secular” has been ascribed in astronomy to expressions which grow with time t as powers of t. Such expressions cannot be usually accepted even in solutions to problems in classical mechanics because they like to appear even in the systems which are known to be periodic. But the situation in quantum case is much worse. Indeed, if an approximate expression for the time evolution operator was to contain terms growing (say—linearly) in time, the canonical commutation relations would be broken and the probability would not be conserved. This is in striking contradiction to the most essential postulates of Quantum Theory. Hence the need for a “good” quantum perturbation expansion method, which does not allow such pathological situations. In the following chapters we will show such a method—the method of multiple scales—in action. To illustrate the diLculties encountered when using regular perturbation expansions let us consider a trivial example. Let a(t) be a quantity depending on time and let it satisfy the following Drst-order diMerential equation: a˙ = −i!a − a ;

(1)

with the initial condition a(0) = b where “dot” denotes diMerentiation over time, and !. Let us pretend for a moment that we are not able to solve this equation exactly and must resort to a perturbation expansion; that is, we can solve it if either ! or is present in the right-hand side, but not for both of them multiplying a. To proceed, let us introduce a dimensionless time variable = !t so that d=dt = !d=d. Let, in addition, j = =!1. Then we have da = −ia − ja : (2) d According to the prescription of the regular perturbation theory we write a = a0 + ja1 + j2 a2 + · · · ; substitute this expansion into Eq. (2), and compare terms on both sides standing at the same power of j assuming the initial conditions to be a0 (0) = b, ai (0) = 0; i = 1; 2; : : : : Then we obtain in the zeroth order d a0 = −ia0 ; d and in the Drst order d a1 = −ia1 − a0 : d

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And it is in the Drst order that the diLculty appears. When we solve for a0 , a0 (t) = be−i , substitute to the equation for a1 , and try to solve the resulting inhomogeneous equation, we Dnd a1 (t) = −be−i which means that our perturbation expansion can be valid only for extremely short times. In the higher orders the situation becomes even worse, the powers of t which enter the approximate solution grow with the order of perturbation expansion. To Dnd a remedy, let us make use of the fact that in the above example ! and deDne two diMerent time scales. Thus, we can try to improve the situation by assuming that all ai (i = 0; 1; : : :) depend separately on two “times”, say, T0 and T1 , such that T0 = and T1 = j. Then we have d=d = 9=9T0 + j9=9T1 . The perturbation expansion is now slightly diMerent: 9a0 = −ia0 ; 9T0 9a1 9a0 + = −ia1 − a0 : 9T0 9T1 The zeroth-order solution is now a0 = b(T1 )e−iT0 . On substituting this solution into the equation for a1 we can avoid the troubles connected with the inhomogeneous term on the right-hand side if we “absorb” it to the equation satisDed by 9a0 =9T1 . This means that we should write 9b(T1 ) −iT0 9a0 = e = −a0 = −b(T1 )e−iT0 9T1 9T1 so that b(T1 ) = be−T1 while a1 becomes zero in view of the initial conditions. This way we have actually obtained the exact solution (after expressing T1 and T0 in terms of t, ! and ). Needless to say, in less trivial examples getting exact solutions by the method of multiple scales happens very rarely. The rest of our work is devoted to an analysis of some of those less trivial problems which appear in quantum optics. The great eLciency of MMS has Drst been recognized in the Deld of non-linear diMerential equations and classical non-linear mechanics. The major developments have included non-linear oscillations, boundary layer problems, /uid dynamics and aerodynamics, and the theory of spacecraft motion. In this connection we should mention that there already exist several excellent books dealing with MMS. Among the monographs written from the point of view of applied mathematics one should mention the books by Kevorkian and Cole [1,2], by Nayfeh [3,4] and by Nayfeh and Mook [5]. An outstanding exposition of approximate and asymptotic methods of applied mathematics including MMS as their crown is contained in the book by Bender and Orszag [6]. A classical exposition of perturbation methods of /uid mechanics including MMS is given in [7]. In quantum mechanics, one of the early attempts to apply the methods of non-linear oscillations to the Heisenberg equations of motion appeared in the work by Ackerhalt and RzTaz˙ ewski [8], which also contained famous Ackerhalt’s exact solutions to the Heisenberg equations in the Jaynes–Cummings model [9,10] (we shall use the abbreviation “JCM” for the Jaynes–Cummings model). The approach used in [8] is similar to the Lindstedt–Poincare method of non-linear mechanics which may be considered to Dt in the general framework of the multiple scales techniques. But still before [8] there had been several ideas to use multiple scales to describe the spontaneous emission in the SchrIodinger picture [11,12]. Those ideas were later applied to the case of the radiation (including superradiance eMects) from thin

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Dlms [13–15] as well as to the spontaneous decay processes in the presence of radiationless transitions [16]. Another quantum-optical Deld of principal interest in which MMS found its application in 1970s was the interaction of a single- and many-atom systems with external laser radiation: problems of this type were considered in [17–19]. The years 1980 –1995 brought about the interest in the asymptotic methods in laser physics. Both the method of multiple scales [20,21,24] and other methods, Drst of all the Krylov–Bogolyubov–Mitropolskii techniques [22,23,95], were employed, mostly, however, on the level of semi-classical equations. Those interesting applications in the laser theory are already covered in the review papers by Erneux [25,26] and will not be discussed here. In the 1990s, MMS was also applied to provide a general link between the internal (quantum) and external (semi-classical) motion of atomic and molecular systems within the framework of the Wigner-function approach [27]. The coupling between external and internal degrees of freedom of an atom in the laser Deld was considered—using the method of multiple scales—in [28]. Further applications of MMS, mostly in the cavity quantum electrodynamics, are described in [29–31]. The remaining part of our review is organized as follows. In Section 2 we discuss the problem of the dynamics of an anharmonic oscillator from the point of view of MMS. Using the results of [36,37] as well as our own, we show that the anharmonic oscillator admits high-quality perturbative solutions for both the Heisenberg and the SchrIodinger equations. In Section 3 some applications to the problem of spontaneous emission are studied. Section 4 is devoted to the interaction of few-level atoms with external laser Delds. The excitations of model atoms with non-monochromatic radiation, e.g., having modulated amplitude, modulated phase, or coming from two independent lasers with diMerent frequencies, are especially interesting for us. Section 5 deals with the interaction of atoms with one or two modes of quantized electromagnetic Deld; such model interactions are very often employed in the cavity quantum electrodynamics. In Section 6 some aspects of the Deld quantization in cavities with moving mirrors and atomic interaction with quantized Delds in such cavities are discussed in adiabatic approximations. Section 7 contains attempts to apply the multiple-scales methodology to the problems of tunneling of the cold gases in the double-well potential. In Section 8 we provide several remarks, both optimistic and skeptical, about the possible further applications of the method of multiple scales in the domain of quantum optics and, more generally, quantum mechanics of simple systems. It is to be noted that the present work has been written on the so-called “physical” level of rigor. No attempt has been made for mathematical justiDcation in terms of proofs; thus, all the calculations are purely “formal” from the mathematical point of view. In fact, the series which we will have to deal with, are, in generic cases, asymptotic and divergent. A heuristic rule has been even oMered by Van Dyke [7] which states that an asymptotic solution to a problem is usually divergent if it depends on two independent scales (length scales in his case or time scales in ours). This statement does not, of course, depreciate the usefulness of asymptotic multiple-scales expansions as will be seen in the following sections on explicit examples. On the other hand, except for the following section, we shall try to concentrate as much as possible on the physical contents of solutions provided by the method we discuss. Our strategy in Section 2 which deals with anharmonic oscillator is to provide numerical evidence for eLciency (as well as limitations) of MMS; in the following sections we shall rather try to obtain qualitative and semi-quantitative predictions from MMS. We Dnish this introduction with two remarks about notation. All the operators in our formulae will be denoted by letters with carets. Dots over symbols denote diMerentiation over the time t.

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2. Anharmonic oscillator This section is devoted to the anharmonic oscillator, one of the best investigated model systems in quantum mechanics. It has been studied in much detail and from many points of view, in particular, by Bender and Wu [33], Hioe et al. [34], and by Stevenson [35]. It has acquired such a popularity because of the following important characteristics: • Its classical counterpart, associated with the so-called DuLng equation, is a very useful model to study non-linear oscillations (in particular, it forms an approximation to the equation of pendulum and shares some of its features), and to test methods of non-linear mechanics. • It is still relatively simple and allows for detailed study with almost arbitrary precision, but at the same time it is amazingly non-trivial, cf. [33]. • It is considered to be the simplest model of the quantum Deld theory corresponding to selfinteracting boson theory with local quartic self-interaction; therefore, the theory of quantum anharmonic oscillations is sometimes called a “zero-dimensional” Deld theory . 2.1. Heisenberg equations of motion Quite recently, Bender and Bettencourt [36,37] have tested the method of multiple scales for possible usefulness in the quantum Deld theory by applying it to the Heisenberg equations of motion in the anharmonic oscillator case, thus oMering the Drst “pure” example of using this tool of non-linear mechanics in quantum dynamics. They have started with a brief analysis of the classical DuLng equation with a small parameter multiplying the non-linear term and have shown how the ordinary (regular) perturbative expansion leads to secular terms proportional to time due to resonant coupling in successive order. The DuLng equation is, in fact, the most standard example used in all books about perturbation theory (cf. [1–3,6]), but the authors of [36,37] have discussed it as classical counterpart of their quantum results. To make our review reasonably self-contained, we shall brie/y summarize the application of MMS to the DuLng equation (in this we shall follow [36]), referring the reader to the above-mentioned books for details. The DuLng equation reads d2 x + x + 4jx3 = 0 ; (3) dt 2 where x is a classical variable—the position of a non-linearly oscillating particle, and j is a small parameter. We shall solve it perturbatively with the initial conditions x(0) = 1 and x (0) = 0. Let us Drst try to use just an ordinary expansion into a series of j, ∞ x(t) = jn x n (t) ; n=0

with the initial conditions x n (0) = n0 and xn (0) = 0. In the zeroth and Drst order in j we get x0 + x0 = 0

(4)

x1 + x1 = −4x03 :

(5)

and

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333

The solution to Eq. (4) that satisDes the initial conditions is x0 (t)=cos t, so that the right-hand side of Eq. (5) becomes equal to −cos(3t) − 3 cos t. But then Eq. (5) becomes an equation of the forced harmonic oscillator with the force being in resonance with the free oscillations. This inevitably leads to a solution with the envelope growing linearly in time. In our case it is x1 = 18 cos(3t) − 18 cos t − 32 t sin t :

(6)

This solution contains the secular term ∼ t sin t, which has, of course, nothing to do with the correct solution which remains bounded for all times. This is perhaps even more self-evident physically than mathematically; on one hand, the potential grows very fast to inDnity with x → ±∞ so that the motion must be bounded; on the other hand, the energy of any closed physical system cannot grow indeDnitely without being added from the environment. One can improve the ordinary expansion by reordering and resummation of the most secular terms [36,37] provided that one is able to recognize them. But the less secular terms would still remain, and, in order to get rid of them, one would rather like to employ a less cumbersome approach from the very beginning. MMS enables one to do this. Indeed, let us assume that the variable x depends separately on two time variables, t and = jt: x(t) = X0 (t; ) + jX1 (t; ) + O(j2 )

(7)

Then, in the zeroth and the Drst order with respect to j we Dnd 92 X0 + X 0 = 0 9t 2

(8)

92 92 3 X + X = −4X − 2 X0 : 1 1 0 9t 2 9t9

(9)

and

The general solution to Eq. (8): X (t) = A()cos t + B()sin t is then substituted into (9), and the amplitudes A() and B() are found from the requirement of absence of the term proportional to time. This can be achieved by the requirement that the coeLcients of sin t and cos t on the right-hand side vanish. As a result we get dB 3 3 = − A3 − AB2 ; d 2 2

(10)

dA 3 3 3 2 = B + AB : d 2 2

(11)

By multiplication of (10) by B() and (11) by A() one obtains a “conservation law”: d C() = 0 ; d where C() =

1 ([A()]2 + [B()]2 ) ; 2

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so that C() plays the role of an adiabatic invariant: it changes only in the higher orders of perturbation expansion. Using the above two formulae, we rewrite (10) and(11) in terms of C: dB = −3C(0)A ; (12) d dA = 3C(0)B ; (13) d which, on using the initial conditions, provides us with the secularity-free O(1) approximation to x: 3 X0 (t; ) = cos 1 + j t : (14) 2 which contains the Drst-order corrections to the frequency. In the following, Bender and Bettencourt have proceeded with the quantum version of the DuLng equation: d2 qˆ + qˆ + 4jqˆ3 = 0 (15) dt 2 (where now qˆ is the position operator), which can be derived from the following Hamiltonian: Hˆ = 12 pˆ2 + 12 qˆ2 + jqˆ4

(16)

by writing down the Heisenberg equations of motion and eliminating the momentum operator. The initial conditions q(0) ˆ = qˆ0 and p(0) ˆ = pˆ0 are such that the canonical commutation relations [qˆ0 ; pˆ0 ] = i˝ are fulDlled. To apply MMS, Bender and Bettencourt write ˆ ) = Qˆ 0 (t; ) + jQˆ 1 (t; ) + O(j2 ) q(t) ˆ = Q(t;

(17)

so that the operator of position of the oscillator now depends on two independent time variables. Substitution of Eq. (17) into (15) and collecting coeLcients of consecutive powers of j gives: 92 ˆ Q0 + Qˆ 0 = 0 ; 9t 2

(18)

and 2 92 ˆ ˆ 1 = −4Qˆ 30 − 2 9 Qˆ 0 : + Q Q 1 9t 2 9t9 The general solution to (18) is given by

ˆ ˆ t + B()sin t ; Qˆ 0 (t) = A()cos

(19)

(20)

from which it follows that in the zeroth order p(t) ˆ = Bˆ cos t − Aˆ sin t + O(j) ; where Aˆ and Bˆ are operators which depend on but not on t. To fulDll the canonical commutation relations these operators have to satisfy ˆ ˆ [A(); B()] = i˝ ; ˆ = qˆ0 and B(0) ˆ while the initial conditions require that A(0) = pˆ0 .

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Now the right-hand side of Eq. (19) is evaluated. In order to prevent secularity, the coeLcients of cos t and sin t are set to zero to obtain d Bˆ 1 ˆ = (−3Aˆ3 − Bˆ AˆBˆ − Bˆ Bˆ Aˆ − AˆBˆ B) (21) d 2 and d Aˆ 1 ˆ 3 ˆ : (22) = (3B + AˆBˆ Aˆ + AˆAˆBˆ + Bˆ AˆA) d 2 The above equations imply the “conservation” law for the operator Hˆ 1 = 1=2(Aˆ2 + Bˆ 2 ): d Hˆ 1 =0 ; d from which it follows that Hˆ 1 = 12 (pˆ20 + qˆ20 ) : Canonical commutation relations allow to rewrite (22) and (21) as 3 d Bˆ = − (Hˆ 1 Aˆ + AˆHˆ 1 ) (23) d 2 and d Aˆ 3 ˆ ˆ = (H 1 B + Bˆ Hˆ 1 ) : (24) d 2 These (still apparently complicated) operator diMerential equations admit an elegant solution in terms of a generalized Weyl ordering of operators ˆ = W [qˆ0 cos(3Hˆ 1 ) + pˆ0 sin(3Hˆ 1 )] ; A()

(25)

ˆ = W [pˆ0 cos(3Hˆ 1 ) − qˆ0 sin(3Hˆ 1 )] B()

(26)

and (cf. [36,37] for the deDnition of the generalized Weyl product), which can be rewritten in a compact form to give the complete O(1) solution qˆ0 cos(t + 3Hˆ 1 jt) + cos(t + 3Hˆ 1 jt)qˆ0 Qˆ 0 (t; ) = 2 cos(3jt˝=2) pˆ0 sin(t + 3Hˆ 1 jt) + sin(t + 3Hˆ 1 jt)pˆ0 : (27) 2 cos(3jt˝=2) Taking into account that the coeLcient which multiplies t in the arguments of trigonometric function is an operator, Bender and Bettencourt have inferred that MMS allows to perform a kind of an operator mass renormalization. Taking the matrix elements of Qˆ 0 between the states n − 1| and |n of the unperturbed, linear oscillator they have concluded that the energy level diMerences are, to the Drst order in j, equal to 1 + 3n˝j in full accord with the standard Drst-order Rayleigh–SchrIodinger perturbation calculations. In the following development Bender and Bettencourt have solved an old problem of discrepancy between the Rayleigh–SchrIodinger perturbation theory and the WKB method regarding the form +

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of asymptotic wave functions. By suitable reorganization and summation of perturbation series they have shown that both approaches actually give the same results. This is of interest from the point of view of MMS since the WKB approach can be viewed as a variant of the multiple scales technique [6]. The asymptotic solutions obtained in [36,37] can be used to obtain other characteristics of the anharmonic oscillator. A simple example will be discussed in Section 2.3. 2.2. Time evolution operator The following considerations are parallel to those of [36,37] but use the method of multiple scales applied directly to the SchrIodinger equation for the time-evolution operator. Our model of a (scalar, quartic) anharmonic oscillator is deDned by the following Hamiltonian operator: 1 2 1 pˆ + m!2 xˆ2 + gxˆ4 ; Hˆ = (28) 2m 2 where xˆ and pˆ are position and momentum operators, m is the mass of the oscillator, ! is the frequency of linear (unperturbed) oscillations, and g is the “coupling constant” measuring anharmonicity. For a moment we shall keep all the constants (like the Planck constant, frequency, and the mass of the oscillator) in our formulae, because we want to deDne a dimensionless small parameter in terms of them. It is convenient—and very much in the spirit of quantum optics—to introduce the creation and annihilation operators a and a† deDned as i m! xˆ + pˆ ; aˆ = 2˝ m! i m! † xˆ − pˆ : aˆ = 2˝ m! The inverse formulae are ˝ (aˆ + aˆ† ) xˆ = 2m! and

pˆ = −i

˝m! (aˆ − aˆ† ) : 2

The harmonic terms in Hamiltonian (28) obviously give ˝!(aˆ† a+1=2). ˆ Expanding the operator xˆ4 in † terms of the operators aˆ and aˆ gives, after a little of algebra, the following form of the anharmonic term: ˝2 g 1 4 2 †4 †3 †2 ˆ ˆ + [aˆ + 4aˆ aˆ + 6aˆ + h:c:] ; 6 N +N + (29) gxˆ = 4m2 !2 2 ˆ and “h.c.” means “Hermitian-conjugate where Nˆ denotes the excitation number operator, N = aˆ† a, terms”. On the right-hand side we can now recognize the Rayleigh–SchrIodinger Drst-order correction to the energy of the nth level, except that we still have the operator Nˆ instead of the number n.

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Let us now rewrite the Hamiltonian as Hˆ = HC + Hˆ 0 + Hˆ 1 ; where the c-number quantity HC is given by 3 ˝2 g 1 HC = ˝! + ; 2 4 m2 ! 2 the new zeroth-order Hamiltonian Hˆ 0 reads Hˆ 0 = ˝!1 aˆ† aˆ ; where 3 ˝g 2 m2 ! 2 and, Dnally, the perturbing term is !1 = ! +

Hˆ 1 = ˝(6Nˆ 2 + [aˆ†4 + 4aˆ†3 aˆ + 6aˆ†2 + h:c]) ; where g˝ : 4m2 !2 The SchrIodinger equation for the complete time-evolution operator reads d Uˆ i˝ = (HC + Hˆ 0 + Hˆ 1 )Uˆ ; dt but we immediately factor out the c-number phase and write −i Uˆ = exp HC t Uˆ ˝ =

so that Uˆ satisDes the equation d Uˆ 1 i = (Hˆ 0 + Hˆ 1 )Uˆ : dt ˝ Let us now introduce a dimensionless time variable

(30)

(31)

= !1 t ; so that d d = !1 : dt d

The SchrIodinger equation for Uˆ now reads d Uˆ = (hˆ0 + jhˆ1 )Uˆ ; d where hˆ0 = aˆ† aˆ = Nˆ ;

hˆ1 = 6Nˆ 2 + [aˆ†4 + 4aˆ†3 aˆ + 6aˆ†2 + h:c:] ;

(32)

(33)

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and we have Dnally deDned the parameter which in the following will be considered small, i.e., much smaller than 1: ˝g = j= : (34) !1 4m2 !3 + 6˝g The above derivation of a simple basic equation may be thought too long, but we believe the expression of a small dimensionless expansion parameter in terms of physical constants and parameters is a matter of principal importance. We are now ready to apply MMS and obtain approximate solutions. Under the assumption that j is small, we may look for the perturbative solution in the terms of the expansion: Uˆ () = Uˆ 0 (T0 ; T1 ; : : :) + jUˆ 1 (T0 ; T1 ; : : :) + j2 Uˆ 2 (T0 ; T1 ; : : :) + · · · ;

(35)

where all the operators Uˆ i are supposed to depend separately on all the “times” T0 ; T1 ; T2 ; : : : ; and are subject to the initial conditions: U0 (0; 0; : : :) = 1;

U1 (0; 0; : : :) = U2 (0; 0; : : :) = · · · = 0 :

(36)

The dimensionless times Ti correspond to various scales of dynamics of the system and are deDned as Tm = jm : In the following sections we shall encounter situations in which such simple deDnitions of time scales turn out to be insuLcient. On substituting expansion (35) into (32) and collecting terms with equal powers of j, we obtain in the zeroth order 9Uˆ 0 = hˆ0 Uˆ 0 ; (37) i 9T0 in the Drst order 9Uˆ 0 9Uˆ 1 +i = hˆ0 Uˆ 1 + hˆ1 Uˆ 0 ; i 9T0 9T1 in the second order 9Uˆ 1 9Uˆ 0 9Uˆ 2 +i +i = hˆ0 Uˆ 2 + hˆ1 Uˆ 1 ; i 9T0 9T1 9T2

(38)

(39)

and so on. We can try to solve them systematically order by order. In the zeroth order we obtain ˆ Uˆ 0 = e−ih0 T0 Vˆ 0 (T1 ; T2 ; : : :) = exp(−iNˆ T0 )Vˆ 0 (T1 ; T2 ; : : :)

while in the Drst order, on writing ˆ Uˆ 1 (T0 ; T1 ; T2 ; : : :) = e−iN T0 UWˆ 1 (T0 ; T1 ; T2 ; : : :) ; we obtain 9Vˆ 0 9UWˆ 1 ˆ ˆ ˆ 0 )Vˆ 0 ; +i = (eiN T0 hˆ1 e−iN T0 )Vˆ 0 = 6Nˆ 2 Vˆ 0 + A(T i 9T0 9T1

(40) (41)

(42)

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339

where ˆ 0 ) = aˆ†4 e4iT0 + 4aˆ†3 ae A(T ˆ 2iT0 + 6aˆ†2 e2iT0 + h:c : It is clear that on the right-hand side of Eq. (42) there exists one term—6Nˆ 2 —which, after integration over T0 , would give a result linear in T0 , that is, a secular term of the kind which we attempt to avoid. Therefore, we absorb this dangerous term in the derivative of V0 over T1 by writing i

9Vˆ 0 = 6Nˆ 2 Vˆ 0 ; 9T1

(43)

with the result ˆ2 Vˆ 0 (T1 ; T2 ; : : :) = e−6iN T1 Wˆ 0 (T2 ; : : :) :

(44)

All other terms on the right-hand side of Eq. (42) can be integrated to give ˆ 0 )Vˆ 0 ; UWˆ 1 (T0 ; T1 ; T2 ) = Xˆ 1 (T1 ; T2 ; : : :) − B(T

(45)

with ˆ 0 ) = 1 aˆ†4 e4iT0 + 2aˆ†3 ae B(T ˆ 2iT0 + 3aˆ†2 e2iT0 − h:c : 4 The above equations Dnish our calculations up to the Drst order. In the second order the formalism becomes somewhat algebraically involved, but computations are still fairly straightforward. We have i

9Uˆ 2 9Xˆ 1 (T1 ; : : :) 9Wˆ 0 ˆ ˆ ˆ2 + ie−iN T0 + ie−iN T0 e−6iN T1 9T0 9T1 9T2 ˆ ˆ 0 )e−6iNˆ 2 T0 Wˆ 0 ] : =Nˆ Uˆ 2 + hˆ1 e−iN T0 [Xˆ 1 (T1 ; : : :) − B(T

(46)

As before, we factor out the operator exp(−iNˆ T0 ) ˆ Uˆ 2 = e−iN T0 UWˆ 2 ;

and obtain i

9UWˆ 2 9Wˆ 0 9Xˆ 1 (T1 ; : : :) ˆ2 +i + ie−6iN T1 9T0 9T1 9T2 ˆ

ˆ

ˆ 2 T1

ˆ 0 )e−6iN =(eiN T0 hˆ1 e−iN T0 ) [Xˆ 1 (T1 ; : : :) − B(T

Wˆ 0 (T2 ; : : :)]

ˆ 2 T1

ˆ 0 )] [Xˆ 1 (T1 ; : : :) − B(T ˆ 0 )e−6iN =[6Nˆ 2 + A(T

Wˆ 0 (T2 ; : : :)] :

(47)

ˆ 0 )Xˆ 1 (T1 ; : : :) Now, we have to identify possible sources of secular terms. It is clear that both A(T 2 ˆ ˆ ˆ and 6N B(T0 )V 0 (T1 ; T2 ; : : :) which appear on the right-hand side of Eq. (47) can be integrated over T0 without any problem. On the other hand, integration over T0 of the term 6Nˆ 2 Xˆ 1 (T1 ; : : :) would

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give a term linear in T0 . We still cannot require, however, that i

9Xˆ 1 (T1 ; : : :) = 6Nˆ 2 Xˆ 1 (T1 ; : : :) 9T1

with the obvious solution: ˆ2 Xˆ 1 (T1 ; : : :) = e−6iN T1 Yˆ 1 (T2 ; : : :) ;

(48)

because the terms ˆ 2 T0

−e6iN

ˆ 0 )B(T ˆ 0 )e−6iNˆ 2 T0 = Cˆ A(T

Drst need to be checked for the presence and character of operators which would possibly cause secular terms to appear. A somewhat boring algebra, characterized by an extensive application of the commutation relations for the creation and destruction operators leads to the following result: Cˆ = −2(2Nˆ + 1)[17Nˆ (Nˆ + 1) + 21]Wˆ 0 (T2 ; : : :) + Cˆ 1 (T0 ; T1 )Wˆ 0 (T2 ; : : :) ; where Cˆ 1 (T0 ; T1 ) can already be integrated over T0 without giving any dangerous terms. Thus, we must require Wˆ 0 (T2 ) = exp(2i(2Nˆ + 1)[17Nˆ (Nˆ + 1) + 21]T2 )Zˆ ;

(49)

where the operator Zˆ can depend on higher-order times Tm , m ¿ 3. We can now set Xˆ 1 as in Eq. (48) for Cˆ does not contain any operators which would have to be absorbed in 9X1 =9T1 . Integration over T0 in Eq. (47) (under the condition that we have already eliminated “bad” terms by “absorbing” them in Xˆ 1 and Wˆ 0 as shown just above) completes the second-order solution. We shall not provide the whole complicated formula. The important second-order contributions are contained in Xˆ 1 and in W0 . One can, in principle, continue the computation to obtain (more and more complicated) higher-order ˆ 0 ) and B(T ˆ 0 ) contains four-linear terms in terms. It is interesting that, although the product of A(T the creation and annihilation operators, the exponential multiplying Zˆ contains Nˆ in a polynomial of the third, and not the fourth order. On the other hand, if we stop computation at the present stage, the only remaining exercise is to impose the initial conditions. In particular, Zˆ must be the ˆ unit operator and Yˆ 1 (0; : : :) has to be equal to B(0). The above procedure and results for asymptotic forms of the time-evolution operator well demonstrate the power of MMS. Although the fact that in the lowest order the eMective Hamiltonian should be proportional to the diagonal part of the “intensity” xˆ2 could be guessed just by looking at Hamiltonian (28), MMS not only provides a precise numerical coeLcient of such eMective energy operator, but also gives corrections not contained in the exponential terms as well as allows for eLcient (if long) computations of higher-order approximations containing numerical coeLcients which are rather diLcult to guess, cf. Eq. (49). As our Hamiltonian is time-independent, the approximate evolution operator can be used to obtain both the approximate time dependence of the operators in the Heisenberg picture and the evolution of the wave function. The second part of the following section is devoted to an analysis of the dynamics of the expectation value of coordinate xˆ of the anharmonic oscillator, computed using our asymptotic results as well as “exact” (numerical) results.

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Comparison of exact and approximate solutions to the Duffing equation 3 exact solution purely harmonic ordinary perturbative solution multiple-scales solution

2

x

1

0

-1

ε = 0.1, x0 = 1

-2

-3 0

2

4

6

8

10

12

14

t Fig. 1. Comparison of exact and approximate solutions to the DuLng equation for the time-dependence of the dimensionless coordinate x of the anharmonic oscillator. The initial conditions are: p(0) = 0, x(0) = 1:0, and the parameter j = 0:1.

2.3. Comparison with numerical results 2.3.1. Classical case To begin with, we will show Drst how good (or bad) the method of multiple scales can be when applied to the anharmonic oscillator in the classical case. Extensive comparisons of this type already exist, cf., e.g., [6], so we shall be very concise. In Figs. 1–6 we have compared four types of solutions: (i) exact solutions (the simple anharmonic oscillator admits, of course, an exact analytical solution in terms of elliptic integrals; we have not, however, plotted these analytic solution, but simply integrated numerically the DuLng equation and displayed obtained numbers); (ii) solutions for purely harmonic oscillations without any non-linearity; (iii) ordinary (regular) perturbative solutions containing secular terms; and Dnally, (iv) classical solutions obtained in [36,37] with the help of MMS. In all cases, the initial value of momentum has been taken equal to zero. Three families of initial conditions of position x0 and parameters j have been considered: (i) x0 = 1:0, j = 0:1; (ii) x0 = 5:0, j = 0:005; (iii) x0 = 0:5, j = 1. Figs. 1–3 display the position of the oscillator x as a function of time t, while Figs. 4–6 show the corresponding pattern in the phase plane (x; p).

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M. Janowicz / Physics Reports 375 (2003) 327 – 410 Comparison of exact and approximate solutions to the Duffing equation 10 exact solution purely harmonic multiple-scales solution

x

5

0

-5

ε = 0.005, x0 = 5 -10 0

2

4

6

8

10

12

14

t

Fig. 2. The same as in Fig. 1, but for diMerent initial conditions: p(0) = 0, x(0) = 5:0, and for j = 0:005.

It is clear that in all three cases (i) – (iii) the MMS solutions are much superior to the simple harmonic approximation and regular perturbative solution. For the parameters of the family (i) (Figs. 1 and 4) we observe that the approximate solution becomes out of phase with the exact one at time of the order of 10, but the qualitative picture in the phase space is excellent for all times. It is quite interesting that MMS works well even for large amplitudes (Fig. 2), provided that j is small enough—there is only a little “dephasing” even for t = 15, the picture in the phase space (Fig. 5) is quite satisfactory as well (the regular perturbative approximation is not shown since it blows up very fast). It is also interesting that even though the “small” parameter j is actually equal to 1, MMS again provides reliable results for relatively large times on condition that the initial amplitude is O(1). Let us notice, however, that the time for which one might still expect MMS to be a good approximation evidently depends on the initial conditions, and for very large initial amplitudes one should expect large deviations from the true solution. Let us repeat the last statement in a somewhat diMerent context. Let us imagine we try to describe a non-linear classical system with a small parameter in the “SchrIodinger picture”, i.e., using the classical distribution function in the phase space (classical density matrix), #, which satisDes the Liouville equation. We can write down the

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Comparison of exact and approximate solutions to the Duffing equation 1.5

ε = 1, x0 = 0.5

1

0.5

x

0

-0.5

-1

exact solution purely harmonic ordinary perturbative solution multiple-scales solution

-1.5

-2 0

2

4

6

8

10

t

Fig. 3. The same as in Fig. 1, but for the initial conditions: p(0) = 0, x(0) = 0:5, and for j = 1:0.

solution to the Liouville equation if we know the solutions to the Hamilton equations for q(t); p(t). Now, if # is initially spread out in a large region of the phase space, there will be some trajectories for which MMS will work well, but there is a non-zero probability that the system will evolve from such a point in the phase space that our approximation will become completely unrealistic. This means that we have to be very careful when large /uctuations are possible in the system; they impose additional constraints on the usefulness of MMS, beside the necessity of the presence of a small parameter. We have indicated this problem—somewhat trivial, perhaps, in the classical setting—anticipating more subtle and diLcult situation in quantum mechanics. On the other hand, let us stress that in our examples we have used only the O(1) MMS solution which has only contained corrections to the phase, but no amplitude corrections. This shows that even the minimum eMort to obtain such absolutely lowest-order approximation can bring impressive success. 2.3.2. Quantum-mechanical case Here we shall compare exact numerical solutions and their MMS approximations assuming that the system is initially in a coherent state. We shall discuss the dynamics of expectation values x ˆ of the position operator xˆ of the oscillator as functions of time. Coherent states have been chosen here as initial states since, on the one hand, they are usually assumed to be, in a sense, very

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Comparison of exact and approximate solutions to the Duffing equation - phase space picture

ε = 0.1, x0 = 1

1

p

0.5

0

-0.5

-1

exact solution purely harmonic ordinary perturbative solution multiple-scales solution

-1

-0.5

0

0.5

1

x Fig. 4. Comparison of exact and approximate solutions to the DuLng equation in the phase space (x; p). The initial conditions are: p(0) = 0, x(0) = 1:0, and the parameter j = 0:1.

close to classical states, while, on the other hand, quantum-mechanical /uctuations are already very pronounced for coherent states. They are conveniently expressed in terms of the number states of harmonic oscillator, which are eigenstates |n of the operator Nˆ : Nˆ |n = n|n ;

(50)

where n is a non-negative integer. In the coordinate representation number states are products of Hermite polynomials and a Gaussian function. Coherent states are eigenstates of the annihilation operators a|$ ˆ = $|$ ;

(51)

where $ is a complex number. Any coherent state can be written as the following superposition of the number states: |$ =

∞ $n 2 √ |n e−(1=2)|$| : n! n=0

(52)

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Comparison of exact and approximate solutions to the Duffing equation - phase space picture 8 exact solution purely harmonic multiple-scales solution

6

4

p

2

0

-2

-4

-6 ε = 0.005, x0 = 5 -8 -8

-6

-4

-2

0

2

4

6

8

x Fig. 5. The same as in Fig. 4, but for diMerent initial conditions: p(0) = 0, x(0) = 5:0, and for j = 0:005.

To compute asymptotic dynamics of the system, we have used the O(1) approximation to the time-evolution operator, which already contains Drst-order “phase” corrections, and is given by ˆ ˆ2 Uˆ 0 = e−iN e−6ijN :

(53)

Let us consider the dynamics of expectation values of x, ˆ which are displayed in Figs. 7–12 (in units of ˝=(2m!)) and form a counterpart of the previous classical comparison. Let us Drst notice that, as is clear from Eqs. (53) (as well as from intuitive representations), for our asymptotic approximation to be meaningful, it is not enough that j itself is small. We should, in fact, have at least j Nˆ (0) = jn1. W Figs. 7–9 deal with the case where this product is equal to 3 × 10−3 , while in Figs. 10–12 we have jnW = 3 × 10−2 . In Fig. 7 we show the short-time behavior of x ˆ for j = 10−4 and nW = 30:0 as given by exact numerical solution to the SchrIodinger equation as well as by application of U0 . To plot this Dgure we have deliberately chosen quite large ’s in order to be able to see the diMerence between the exact and approximate results. For shorter times the agreement is, naturally, still better, for, in fact, the exact and the MMS solutions are almost identical. Obviously, for jnW still smaller, we obtain even better agreement.

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Comparison of exact and approximate solutions to the Duffing equation - phase space picture

1

ε = 1, x0 = 0.5

p

0.5

0

-0.5

-1

exact solution purely harmonic ordinary perturbative solution multiple-scales solution

-1

-0.5

0 x

0.5

1

Fig. 6. The same as in Fig. 4 but for the initial conditions p(0) = 0, x(0) = 0:5, and for j = 1.

In Fig. 8, keeping jnW = 3 × 10−3 we have changed j to 10−3 and nW to 3, and displayed our solutions for longer time. Now, the anharmonic oscillator is an example of systems in which the “grain-like” nature of the Quantum Theory manifest itself by the presence of the so-called collapses and revivals of oscillations. Fig. 8 shows the dynamics approaching the Drst collapse; it is clear that MMS solutions give good qualitative approximation for the location of collapse on time axis (the same is true about the Drst revival, not shown in the Dgure). The quantitative agreement of exact and approximate solution is satisfactory, to say the least, for times smaller than the time of the Drst collapse. The situation changes for worse when j becomes equal to 10−2 . In Fig. 9 we can see that for dimensionless times of about 40 the approximate solution, while qualitatively still sound, becomes strongly out of phase from the exact solution. This may be somewhat surprising in view of our expectations associated with the classical theory when MMS usually gives satisfactory results for the dimensionless time ∼ 1=j. But in the quantum case with j ∼ 10−2 the /uctuations start to spoil the results even for nW as small as ∼ 0:1. With growing nW this eMect becomes more and more serious, and Dnally devastating. Nevertheless, for the parameters of Fig. 9, MMS provides reasonably good qualitative picture of oscillations and good approximation of their amplitudes.

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Quantum anharmonic oscillator: comparison of exact and approximate results 6 exact solution O(1) first-order MMS

ε = 0.0001, = 30.0 4

<x(t)>

2

0

-2

-4

-6 250

255

260

265 τ

270

275

280

Fig. 7. Comparison of exact and approximate solutions to the problem of an anharmonic oscillator: time-dependence of the expectation value x(t) in units of ˝=(2m!). Initially the system is in the coherent state, and the important parameters are j = 10−4 ; nW = 30:0.

In Figs. 10–12 the product jnW is one order of magnitude larger than in Figs. 7–9 which leads to large inaccuracies of MMS for not very short times. In Figs. 10 and 11 we display the long-time dynamics of x ˆ for j = 0:001 and nW = 30. It is clear that the MMS completely misses the true revivals (both small and large ones) of the oscillations, and gives a large-scale false revivals. In addition, when we have attempted to plot the O(j) MMS approximation (i.e. that obtained with the help U0 + jU1 ) we have realized that, for jnW ∼ 10−2 , the norm of the wave function is rather poorly conserved. It oscillates with the amplitude of up to 0:2. This is because the time-evolution operator is unitary only up to the order j, and the factor multiplying j2 can become fairly large as it contains fourth-order polynomials of the creation and annihilation operators. Thus, in the case of still quite small jnW we would have to use the second-order correction to keep unitarity. Let us be quick to mention, however, that even with this imperfect unitarity, MMS is much superior to the ordinary regular perturbation expansion of the time-evolution operator (the results of the latter are not even shown since they blow up very fast). The short-time agreement (up to ∼ 10) between MMS and exact results is also displayed in Fig. 12, where j = 0:01 and nW = 3:0. The MMS approximation predicts collapses and revivals of the

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Quantum anharmonic oscillator: comparison of exact and approximate results exact solution O(1) first-order MMS

0.3

ε = 0.001, = 3.0

0.2

<x(t)>

0.1

0

-0.1

-0.2

-0.3 120

130

140

150

160 τ

170

180

190

200

Fig. 8. The same as in Fig. 7, but for diMerent parameters j and n: W j = 10−3 , nW = 3.

oscillations, but, starting with the Drst revival, its predictions about times of collapses and revivals are unreliable. Again, the Drst-order approximation gave here rather large (up to 20%) non-conservation of probability. As a general conclusion from these quantum-mechanical comparison, we notice that MMS usually gives good results for shorter times and smaller j than might be expected in view of the spectacular success of its classical version. But this only means that “quantum” MMS—like every method of applied mathematics—has its own region of validity which depends on a particular problem. Another conclusion which we would like to draw in the end of this section is that the Drst-order approximation may become insuLcient, and that both phase- and amplitude corrections of higher order may be necessary if the results are to be reliable. 3. Spontaneous emission By “spontaneous emission” we mean the emission of light by an excited atom in the absence of initial excitations of electromagnetic modes. As such, this process belongs to the most important

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349

Quantum anharmonic oscillator: comparison of exact and approximate results 1.5 exact solution O(1) first-order MMS

ε = 0.01, = 0.3 1

<x(t)>

0.5

0

-0.5

-1

-1.5 0

10

20

30

40

50

60

τ Fig. 9. The same as in Fig. 7, but for diMerent parameters j and n: W j = 10−2 , nW = 0:3.

ones in nature. In addition, correct theory of spontaneous emission, Drst developed by Dirac within his time-dependent quantum perturbation formalism, opened, together with fundamental works of Heisenberg, Pauli and Jordan, the era of quantum Deld theory. Hence the problem of spontaneous emission (including the problem of widths of spectral lines of spontaneously emitted light) under various conditions remains one of the most interesting ones in quantum theory and is still intensively studied. As attempts of Fermi [38] as well as Crisp and Jaynes [39] (cf. [40] as well) to build a “neoclassical” theory of spontaneous emission seem to have shown, the details of the process of spontaneous emission of light require the Deld quantization for their description. Hence our discussion of spontaneous emission from the point of view of MMS must start with the concept of quantized electromagnetic Deld. Usually, quantum opticians Dnd it reasonable and convenient to quantize the electromagnetic Deld in a large-volume Dctitious box or cavity by imposing the periodic boundary conditions on the walls of the cavity (cf., e.g., [41]). In all practical calculations we Dnally take the limit of inDnite size of the cavity and replace summation over all cavity modes by integration over the wave vector.

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M. Janowicz / Physics Reports 375 (2003) 327 – 410 Quantum anharmonic oscillator: exact results 15 exact solution

ε = 0.001, = 30.0

10

<x(t)>

5

0

-5

-10

-15 0

200

400

600

800

1000

τ

Fig. 10. Long-time behavior of x(t) for j = 10−3 , nW = 30—exact numerical results.

The electromagnetic vector potential in the cavity is written in the form 2 ˝ ˆ t) = {el; aˆl; eikl ·r + el;? aˆ†l; e−ikl ·r } ; A(r; 2!l j0 V l

(54)

=1

where V is the cavity volume, the subscript l is the index of cavity modes (it in fact consists of three indices, l1 ; l2 ; l3 , is the polarization index, el; , el;? are the polarization vectors, kl is the wave vector, !l = kl c is the frequency of the (l; ) mode, while aˆl and aˆ†l are the annihilation and creation operators which satisfy the canonical commutation relations: [aˆl; ; aˆ†l ; ] = ll ; ;

(55)

whereas all other commutators made of these operators vanish. In the following we shall restrict ourselves to the description in terms of linear polarizations in free space, so that the polarization vectors el; will be assumed to be real. The wave vector k can be expressed by the

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351

Quantum anharmonic oscillator: approximate results 15 O(1) first-order MMS ε = 0.001, = 30.0 10

<x(t)>

5

0

-5

-10

-15 0

200

400

600

800

1000

τ

Fig. 11. Long-time behavior of x(t) for j = 10−3 , nW = 30—MMS results.

multi-index l as k=

2( (l1 ; l2 ; l3 ) ; L

and the summation over l in (55) is the summation over all three indices l1 ; l2 ; l3 . It is to be noted that the sums over l are very often written as a sum over k, and then the creation and annihilation operators bear the index k as well. In order to Dnd the decomposition of the electric Deld in terms of the annihilation and creation operators, it is necessary to choose the gauge. The most natural though not explicitly Lorentz-invariant choice is the Coulomb (radiation) gauge in which we have ˆ t) = 0 ; ∇ · A(r; and the total electric Deld is given by ˆ ˆ t) = − 9A(r; t) − ∇*(r; ˆ t) E(r; 9t

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Quantum anharmonic oscillator: comparison of exact and approximate results 4 exact solution O(1) first-order MMS ε = 0.01, = 3.0

3

2

<x(t)>

1

0

-1

-2

-3

-4 0

20

40

60

80

100

τ Fig. 12. The same as in Fig. 7, but for diMerent parameters j and n: W j = 10−2 , nW = 3:0.

where *ˆ is the static potential to be expressed by the operator variables characterizing matter Delds. In the regions of space free of charges we can put *ˆ = 0, and, by noticing that for non-interacting Delds aˆl; t = aˆl e−i!l t ; ? i !l t aˆ? ; l; t = aˆl e

(56)

we obtain immediately: ˆ t) = i E(r;

2 l

=1

˝!l el; {aˆl; exp(ikl · r) − aˆ†l; exp(−ikl · r)} ; 2j0 V

while the magnetic induction decomposition reads 2 ˝ ˆ t) = −i (el; × kl ){aˆl; exp(ikl · r) − aˆ†l; exp(−ikl · r)} : B(r; 2!l j0 V l

=1

(57)

(58)

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353

Relation (57) remains true in the regions where the matter Delds are non-zero provided, however, that Eˆ is replaced by Eˆ T which is the transverse part of the electric Deld operator. Now we are ready to express the radiation Deld Hamiltonian in terms of modes 1 1 ˆ2 3 2 ˆ ˆ d r j0 ET (r) + H= ˝!l aˆ†l aˆl ; (59) B (r) = 2 ,0 l;

where we have omitted the inDnite vacuum energy term (1=2) (˝!l ). The advantage of the box quantization is that one has to do with operators a; ˆ aˆ† enumerated by discrete indices, and not with the operator-valued distributions, and one can easily make the limit of atomic coupling with only one mode, as is standard in Cavity Quantum Electrodynamics. The passage from the discrete mode—decomposition of quantized Delds given above, to the decomposition in terms of operator-valued distributions in k space is quite simple; one should only write 1=(2()3=2 √ instead of 1= V and replace summation by integration over k. The major and very widespread assumption about the atomic degrees of freedom which is to be employed here is that the atom has only Dnite number of discrete levels. In fact, for the purposes of this section it is enough to consider the model consisting of just two or three levels. There is a huge amount of experiments in which such a two-level model was perfectly suLcient and justiDed, cf., e.g., [42–44]. If the energy of the Drst (|1) atomic level is E1 and of the second (|2) E2 , then we may write the atomic energy operator as Hˆ A = E1 |1 1| + E2 |2 2| ;

(60)

or, using the convenient “.” notation n ˆ Ei .ˆii ; HA =

(61)

i=1

where .ˆij = |i j| and n = 2 for two-level atoms. Many authors have found it convenient to use symbols .ˆ+ to denote .ˆ21 , and .ˆ− to denote .ˆ12 . The operators Rˆ ± = (1=2).ˆ± have also been extensively used. As always in non-relativistic physics, we can choose the zero of energy according to our convenience. If we choose, for instance, E1 = 0, the two-level-atom Hamiltonian can be written as Hˆ A = ˝!.ˆ22 ;

(62)

where ! = E2 =˝; or, we can choose such zero of energy that the upper level has the energy +(1=2)(E2 − E1 ), and the lower one the energy −(1=2)(E2 − E1 ). Then the atomic Hamiltonian reads 1 (63) Hˆ A = ˝!(.ˆ22 − .ˆ11 ) : 2 The diMerence .ˆ22 − .ˆ11 is often denoted by .ˆz , since in the SchrIodinger picture it acts in the Hilbert space C2 as one of the ordinary Pauli matrices, the latter being deDned as 0 1 0 −i 1 0 .ˆx = ; .ˆy = ; .ˆz = : 1 0 i 0 0 −1

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M. Janowicz / Physics Reports 375 (2003) 327 – 410

The problem of the quantum-optical interaction Hamiltonian in the non-relativistic setting is surprisingly non-trivial. There exists an elegant and authoritative derivation of the most correct nonrelativistic Hamiltonian from the coupled operator Maxwell–Dirac equations, provided in [50]. However, further approximations are usually made to render it in a maximally simpliDed form. There are two basic forms of such simpliDed interaction Hamiltonian Hˆ AF which are extensively used in quantum optics (which, after all, may be considered as a part of non-relativistic quantum electrodynamics). The Drst one stems from the “minimal-coupling” Hamiltonian and can be written as e e2 ˆ 2 ˆ ˆ H AF = |i i| − pˆ · A(r) + (64) A (r) |j j| ; m 2m ij ˆ is the electromagnetic potential. In the context where m is the electron mass, e is its charge, and A of spontaneous emission one usually makes two further approximations. Firstly, one assumes that, due to the smallness of atomic size, the electromagnetic potential changes only very little within the atom. Thus, it should be a very good approximation to calculate the potential not at the position of the electron, but at the center of mass of the atom. In addition, to obtain the overall picture of spontaneous emission one may neglect the A2 term present in (64). Let us notice, however, that the latter approximation could be very dangerous if one is interested in the details of atomic dynamics, (cf. [45,46]), or the exact non-relativistic values of radiative energy-level shifts. If we disregard this objection, we obtain from Eq. (64) e ˆ Hˆ AF = − A(r i|p|j. ˆ (65) 0) ij ; m ij where r0 is the position of the atomic center of mass. The second form of the interaction Hamiltonian: the “−d · E” coupling stems from the coupling of the dipole to the electric Deld already known from the elementary electrostatics. The interaction energy of a dipole in an external electric Deld is equal to minus scalar product of the dipole moment and the Deld itself. Hence we write Hˆ AF = −dˆ · Eˆ T (r0 ) ; where dˆ is the dipole moment operator of the atom, and Eˆ T is the operator of the transverse electric Deld at the center of mass of the atom, the mode decomposition of which is given by (57). This form of the interaction Hamiltonian already assumes the electric dipole approximation. It can be obtained from the minimal coupling form by a unitary transformation called the Power–Zienau transformation provided, however, that one neglects an additional term containing the square of the (transverse) atomic polarization, [47–49]. If we now take into account that atoms do not have permanent dipole moments (unless they are in an external static Deld), so that the diagonal matrix elements of d vanish, we obtain ˆ .ˆij · E(r ˆ 0) : Hˆ AF = − i|d|j (66) i=j

In the two-level model of an atom the above summation only involves two terms. With the energy operators given above, we are ready to consider the spontaneous emission from the point of view of MMS. The method of multiple scales has been Drst applied to the problem of

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355

spontaneous emission in the papers by Yung-Chang Lee and coworkers [11–13] in the context of superradiance [51,52]. Superradiance is a term given to the spontaneous emission of a collection of N atoms characterized by the intensity proportional to N 2 under the absence of any external agent (excitation); such an eMect clearly involves high degree of coherence of emission in spite of the fact that the spontaneous emission is usually (and rightly) considered as a powerful coherence-destroyer. For the superradiance to be present, all atoms of the sample must “feel” the presence of all remaining atoms (hence the sample should not have large spatial size), and each atom should experience a signiDcant energy loss, cf. [51]. In their Drst paper on the subject of spontaneous emission, Lee et al. studied one- and two-atom system coupled to the electromagnetic Deld. They showed that the usual exponential ansatz used rather ad hoc by Weisskopf and Wigner [53] in their theory of natural line shape follows “naturally” from the multiple-scales analysis. They started from the following Hamiltonian governing a system containing a collection of atoms (in the two-level approximation) and the radiation Deld, written in the interaction picture: Hˆ =

i

j[d? k exp(ik · xi ) exp(i!0k t)

k

·Cˆ k Rˆ + (i) + dk exp(−ik · xi ) exp(−i!0k t)Cˆ †k Rˆ − (i)] ;

(67)

where Cˆ k and Cˆ †k denote the photon creation and annihilation operators of the mode labeled by k; Rˆ + and Rˆ − are the atomic raising and lowering operators for the ith atom, j is a parameter explained √ below, while dk and d? k are the atom-Deld coupling constants proportional to ek ·d12 = !k with ek are the polarization vectors. Finally, !0 denotes the energy diMerence (divided by ˝) between the atomic excited and ground states (assumed identical for all atoms), !k is the frequency associated with the wave vector k, and !0k = !0 − !k . In [11] the authors have clearly assumed that summation over k also involves the summation over polarizations, and the anti-resonant terms leading to emission together with excitation as well as spontaneous absorption have been omitted. The possibility of application of MMS in the present context relies on the fact that, physically, there are at least two natural and very diMerent time scales in the system. The Drst one is associated with the atomic transition frequency !0 (relevant electromagnetic mode frequencies are close to it), the second one is the inverse of the spontaneous decay rate (typically ∼ 10−7 !0 ), which, however, does not appear explicitly in the interaction Hamiltonian. For this reason Lee et al. have introduced an artiDcial “small” parameter j which in the end will be set equal to 1. The true, natural small parameter appears at a later stage of the MMS procedure. Following the Wigner–Weisskopf SchrIodinger-picture approach [53,54], Lee et al. have written the wave function of the system in the form |6(t) = a(t)|2; 0k +

bk (t)|1; 1k ;

(68)

k

where |2; 0k denotes the state with the atom excited and zero photons in the Deld, while |1; 1k describes the atom in the ground state and one photon present in the mode k. Then, amplitudes a(t)

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M. Janowicz / Physics Reports 375 (2003) 327 – 410

and bk (t) satisfy the following system of linear diMerential equations: d i!0k t jd? bk ; i a(t) = ke dt k

d bk = jdk e−i!0k a(t) : dt The above system of equations, to be solved with the initial conditions i

(69)

bk (0) = 0 ;

a(0) = 1;

although linear, is still fairly complicated, Drst of all because it describes a physical system with inDnitely many degrees of freedom. The authors of [11] proceed by deDning the new time variables Tn = jn t and using MMS with respect to the perturbation parameter j. Thus, the SchrIodinger amplitudes are looked for in the form ∞ a(t) = jn a(n) (T0 ; T1 ; T2 ; : : :) ; n=0

bk (t) =

∞

jn bk(n) (T0 ; T1 ; T2 ; : : :) ;

n=0

with the initial conditions a(T0 ; T1 ; T2 ; : : :) = 1 ; bk (T0 ; T1 ; T2 ; : : :) = 0 : Substituting the above expansions into (69) and collecting terms standing at equal powers of j one obtains i

9a(0) =0 ; 9T0

9b(0) i k =0 ; 9T0 (1) 9a(0) 9a i!0k T0 (0) = + d? bk ; i ke 9T0 9T1 k 9b(0) 9b(1) k = dk e−i!0k T0 a(0) ; + k i 9T0 9T1 (2) 9a(1) 9a(0) 9a i!0k T0 (1) = + + d? bk ; i ke 9T0 9T1 9T2 k (1) (0) 9b 9b 9b(2) k = dk e−i!0k T0 a(1) : + k + k i 9T0 9T1 9T2

(70)

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357

The requirement of absence of secular terms leads to the conclusion that both a(0) and b(0) are k independent not only of T0 , as is clear from Eqs. (70), but also of T1 . The same statement is true about a(1) and b(1) k . But in the second order in j one gets s T0 (0) 9a a(2) = −T0 − ds |dk |2 ds1 ei!0k s1 a(0) : (71) 9T2 0 0 k

In addition to MMS, Lee et al. had to employ the assumption that T0 is large, that is, much larger than 1=!0 . From this assumption it follows that T0 P i!0k s ds e = i8(!0k ) = i − i((!0k ) ; (72) !0k 0 where P denotes the Cauchy principal value, and T0 s T0 1 − ei!0k T0 i!0k s1 ∼ iT0 8(!0k ) : ds ds1 e =i −i 2 !0k !0k 0 0 The latter equation is only valid for large T0 , as well. Using (73) in (72) we obtain Dnally 9a(0) +i |dk |2 8(!0k )a(0) = 0 ; 9T2

(73)

(74)

k

which can also be written as a(0) (T2 ) = exp(−i!s T2 − s T2 ) ;

(75)

where we have introduced the symbols 1 !s = |dk |2 P ; !0k k

s = (

|dk |2 (!0k ) :

k

If Eq. (75) is fulDlled, the right-hand side in (71) is equal to zero for large T0 , and no secular terms arise in the second order on time scales much larger that 1=!0 . The authors of [11] have also shown that a(2n+1) = 0 for n ¿ 1, and that, for large T0 , a(2n) = 0 for n ¿ 2. Thus, when t becomes large, we Dnally obtain a(t) ≈ e−i!s t −s t ;

(76)

where j has been replaced by 1. The amplitudes bk can be obtained from (76) by simple integration which gives b k = dk

e−i!0k t e−i(!s −is )t − 1 : !0k + !s − i

(77)

The last formula shows that the spectrum of emitted light is Lorentzian. It is to be noted that the results contained in Eqs. (76) and (77) are obviously well known. Also, MMS alone cannot actually provide the correct spectral line width . The additional relation (72),

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M. Janowicz / Physics Reports 375 (2003) 327 – 410

which is valid, strictly speaking, only for inDnite T0 , has had to be used; and one can comment that it is also used in standard approaches to the problem of spontaneous emission. In fact, the need to use Eq. (72) (or an equivalent relation) is related to time-reversal invariance of the Hamiltonian and SchrIodinger equation: to get any non-zero one has to break this invariance “by hand” regardless of which method one uses (this is not less true about the von Neumann equation for the density matrix). Nevertheless, it is to be stressed that no additional exponential or other ansatz has had to be employed in [11] to obtain the solution. In [11], the problem of spontaneous emission from two atoms located at the points x1 and x2 has been addressed as well, and the method of multiple scales has been applied to solve it. The SchrIodinger-picture wave function has been assumed to have the form |6(t) = b1 (t)|1; 0; 0k + b2 (t)|0; 1; 0k + bk (t)|1; 1; 1k ; (78) k

which means that the subspace of the Hilbert space allowed by the model consists of (a) the state with the Drst atom in the excited state, and the second one in the ground state with no photon in the Deld; (b) the state with the Drst atom in the ground state, the second atom excited, and no excitation of the electromagnetic Deld; (c) the state with both atoms in their ground states and one quantum of Deld in the mode k. The initial conditions are such that only the Drst atom is excited, b1 (0) = 1, b2 (0) = bk (0) = 0. Now, MMS works almost algorithmically (up to the necessity of using (72) once again) and gives in the second order 9b(0) (0) 1 = −f11 b(0) 1 − f12 b2 ; 9T2 9b(0) (0) 2 = −f21 b(0) 1 − f22 b2 ; 9T2

(79)

where fij = i!ij = ij ; !ij =

ij = (

|dk |2 P

k

1 ikxij e ; !0k

|dk |2 eikxij (!0k ) ;

and xij = xi − xj . After solving the above pair of equations, Lee et al. could obtain results for all three SchrIodinger amplitudes. In the next paper, Lee and Lee [12] generalized the procedure to the case of N atoms. Instead of system (79), using MMS they obtained the following equations for the amplitudes of having precisely one, jth atom in the excited state: N dbi =− fij bj (t) ; dt j=1

where fij = i

k

|dk |2 eikxij 8(!0 − !k ) :

(80)

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359

Using these formulae, Lee and Lee discussed several geometrical arrangements of atoms in lattices and obtained intensities and dispersion relations by numerically solving system (80). It is clear that MMS served there only as a starting point for the analysis while the main physical results of excitation energy trapping and single-peaked intensity spectrum [13] were not related to the method. MMS provided here a system of equations to be solved numerically, which is much more convenient than the initial system of diMerential equations for the SchrIodinger amplitudes. Investigations of many-atom spontaneous emission found their continuation in the paper [14] and, in the context of exciton superradiance, in [15]. System (69) is an inDnite system of, in fact, uncountably many (the sum is to be understood as an integral if we get rid of the quantization box) ordinary diMerential equations. It is clear that, if we know the amplitude a(t), all the amplitudes bk can be obtained just by integration. Therefore, an interesting question arises: can we replace all the modes bk by just one or a few time-dependent variables, so that the dynamics of a(t) is unchanged, and all bk can be computed on the Dnal stage of calculations by integrating the second equation of system (69) with already known a(t). In some cases, this question has been answered aLrmatively (cf. e.g., [55] as an important example) leading to the concept of “faked vacuum” [56] or “pseudomodes”, [57–59]. The concept of pseudomodes turns out to be especially convenient in the theoretical analyses of radiating atoms in cavities, in photonic crystals, waveguides, or any other systems in which the coupling constant is to a good approximation not a /at, but rather a well-localized function of frequency. As an example, let us consider such a coupling constant in a Lorentzian form: 1 √ dk = g! = √ ; (81) g (! − !c ) − i ( where !c is a central frequency of “reservoir” which may consist, for instance, of the electromagnetic modes of a cavity. The parameter is the rate of damping of the excitations of reservoir (the leakage rate of, say, the cavity excitations), while g measures the eMective strength of the interactions of the atom and the reservoir. The total Hamiltonian of the system reads Hˆ = Hˆ A + Hˆ F + Hˆ AF ;

(82)

where the Hamiltonian of a two-level atom Hˆ A is given by Hˆ A = ˝!0 .ˆ22

(83)

and ˝!0 is the energy gap between two atomic levels. The Deld Hamiltonian is given by ∞ d! aˆ†! aˆ! ; Hˆ F = ˝

(84)

while the interaction Hamiltonian is written in the form ∞ ˆ H FA = ˝ d!(g! aˆ†! .ˆ12 + g!? .ˆ21 aˆ! ) ;

(85)

0

0

where .ˆij = |i j| are the atomic raising and lowering operators, while aˆ! and aˆ†! are annihilation and creation operators of the Deld. We have clearly ignored the dependence of the above operators on the k vector; this is justiDed by the mode structure of the Delds in cavity, as well as by the fact that we are not interested here in any spatial relations.

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Dividing the total Hamiltonian into two parts, Hˆ = Hˆ 0 + Hˆ 1 with ∞ † ˆ d! aˆ! aˆ! ; H 0 = ˝!0 .ˆ22 + 0

(86)

we write the total time evolution operator as the product: Uˆ = exp((−i=˝)Hˆ 0 t)Uˆ I ;

(87)

where the interaction-picture time-development operator satisDes the equation i˝

d Uˆ I = Hˆ I Uˆ I : dt

(88)

The interaction-picture Hamiltonian HI is given by ∞ ∞ † † ? d!
0

(89)

where
where |2; 0 is the state with the excited atom and no excitation of the Deld, and |1; 1! is a state with the atom deexcited and one quantum present in the Deld. Writing the SchrIodinger equation and eliminating the amplitudes =! from the resulting system, we obtain the simple integro-diMerential equation for $: t $˙ = − G(t; t )$(t ) dt (91) 0

with the kernel G(t; t ) given by ∞ d!|g! |2 exp[ − i
(92)

The integral can obviously be performed exactly, but to simplify its form one can observe that the lower limit of integration can be set approximately equal to −∞ because the two frequencies !0 and !c are assumed to be approximately equal and very large, that is, much larger than both and g! . From now on we shall assume the perfect resonance between the atom and the cavity Deld, !0 = !c . Then the result of integration in (92) is G(t; t ) = g exp(−|t − t |) : Now, it is a very simple matter to Dnd out the appropriate pseudomode. We immediately verify that the system √ $˙ = −i gB ; √ B˙ = −B − i g$ (93)

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gives, upon the elimination of B, the same integro-diMerential equation for $ as the system containing all amplitudes b! . The variable B is thus the required pseudomode. The solution of Eq. (93) is evidently trivial. But let us now suppose that the atomic energy is not constant. That is, the atom is in/uenced by some external low-frequency laser Deld which serves to modulate !0 . A practical setup to perform such modulation is described in [60]. We can take into account the eMect of modulation by the replacement of !0 by !0 (1 − j sin ?t) in the atomic Hamiltonian. In the following we shall use the parameter f = j!0 (this is the modulation depth which is much smaller than !0 but may or may not be much smaller—or much larger—than ?). In this case of a modulated atomic frequency, we can work again in the (slightly modiDed) interaction picture to obtain the following diMerential equations for the SchrIodinger amplitude $ and pseudomode B √ f $˙ = −i g exp i cos(?t) B (94) ? and

f B˙ = −B − i g exp −i cos(?t) $ : ? √

(95)

This system is somewhat more complicated due to the time-dependent coeLcients and has been analyzed numerically in [61]. Quite naturally, a Floquet-type analysis could be performed, but we can apply here a simple version of MMS to strengthen intuitive reasoning. In the following we shall need the simple equality: exp[ ± ia cos(?t)] = J0 (a) + 2

∞

(±i)n Jn (a)cos(n?t) ;

(96)

n=1

where a = f=? and Jn (x) are the Bessel functions of an integer order n and argument x. √ We shall assume that g is much smaller than either ? or d. We deDne new dimensionless time √ variable , = ?t, and the small parameter = g=?. Then Eqs. (94) and (95) take the form d$ f = −i exp i cos() B ; d ? f dB = − B − i exp −i cos() $ : (97) d g ? There is a family of various non-trivial relations between parameters which can easily be explored. We shall restrict ourselves, however, to two cases only. Let us Drst take the limit of the weak coupling, g. More speciDcally, let us assume that =(1=2 )g21 and the (dimensionless) parameter 1 is of the order of 1. Then we have d$ f = −i exp i cos() B ; dt ? f dB = −1 B − i exp −i cos() $ : d ?

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Although the example is almost

too simple to apply MMS, let us still do this. We write Tn = n ,

$ = n=0 $n (T0 ; T1 ; : : :), and B = n=0 Bn (T0 ; T1 ; : : :). From the zeroth-order equations in we obtain $0 = $0 (T1 ; T2 ; : : :) ; B0 = e−1 T0 B00 (T1 ; T2 ; : : :) : In the Drst order the secular terms cannot appear, so it turns out that $00 and B00 do not depend on T1 , while $1 and B1 are given by the following expressions:

inT0 ∞ J0 (a) n+1 e e−inT0 e−1 T0 B00 (T2 ; : : :) ; − i Jn (a) + $1 = $10 (T1 ; T2 ; : : :) + i 1 in − −in − 1 1 n=1 and J0 (a) $00 (T2 ; : : :) 1 inT0 ∞ e −inT0 n+1 + $00 (T2 ; : : :) : (−i) Jn (a) + in + 1 −in + 1 n=1

B1 = B10 (T1 ; T2 ; : : :)e−1 T0 − i

In the second order, however, we obtain 9$2 9$1 9$00 J0 (a) + + = −ieia cos T0 B10 (T1 ; T2 ; : : :)e−1 T0 − i $00 (T2 ; : : :) ; 9T0 9T1 9T2 1 and this means that the secular terms will appear unless we require that ∞ −J02 (a) Jn2 (a) 9$00 $00 ; = − 9T2 1 n − i 1 n=1 which means that we have got an eMective decay constant ∞ J 2 (a) nJn2 (a) A2 = 0 + ; 2 + 2 1 n 1 n=1 so the probability that the excited state is still populated at time t is approximately equal to Pe (t) = |$00 |2 ≈ e−2A2 T2 = e−2(gA2 =?)t : By choosing an appropriate a = f=? we can to some extent manipulate the rate of the decay of the excited state. However, we cannot “turn oM” the spontaneous emission this way since ? and are of the same order and driving Deld cannot detune the atom from the reservoir. As the second case we shall consider the medium-strength-coupling with =g21 , where, as before, 1 is of the order of 1, so that g and are of the same order. Then, we have the following diMerential equations for $ and B: d $ = −ieia cos B ; d dB = −1 B − ie−ia cos $ : d

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363

Now, using MMS again, we Dnd that in the zeroth order $0 and B0 do not depend on T0 , while in the Drst order the secular terms vanish provided that $0 and B0 satisfy 9$0 = −iJ0 (a)B0 ; 9T1 9B0 = −1 B0 − iJ0 (a)$0 : 9T1 From the above formulae it is clear that the quantity J0 (a) plays the role of an eMective coupling constant between the atom and the pseudomode, and that it is possible to change the dynamics of the spontaneous emission in quite a spectacular way by changing the modulation depth f. In particular, we may choose f and ? in such a way that J0 (a) = 0. This way, by driving the atom √ in and out of resonance with the reservoir (large ?, ? g) as well as imposing the resonance condition J0 (a) = 0 we can suppress the spontaneous emission. Obviously, such a suppression of spontaneous decay may happen only in the case of a reservoir with a well-localized spectrum,

Suppression of spontaneous emission by modulation |α|, for a = 2.40483 |α|, for a = 0.0 |B|, for a = 2.40483 |B|, for a = 0.0

1.2

1

|α|, |B|

0.8

0.6

0.4

0.2

0 0

5

10

15

20

25 τ

30

35

40

45

50

Fig. 13. Suppression of spontaneous emission of an atom coupled to a Lorentzian reservoir. Time dependence of the moduli of amplitudes |$()| and |B()| is shown for a = 2:40483 and 0. The other parameters are 1 = 2:0, = 0:1.

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Suppression of population exchange by modulation |α1|, for a = 2.40483 |α2|, for a = 2.40483 |α1|, for a = 0.0 |α2|, for a = 0.0

1.2

1

|α1|, |α2|

0.8

0.6

0.4

0.2

0 0

5

10

15

20

25 τ

30

35

40

45

50

Fig. 14. Suppression of energy exchange between two atoms coupled to a Lorentzian reservoir. Time dependence of |$1 ()| and |$2 ()| is shown for a = 2:40483 and 0. The other parameters are 1 = 2:0, = 0:1.

√ otherwise the condition g? is not fulDlled. The above conclusions remain true in the case of the strong-coupling limit g. The suppression of spontaneous emission in a system with modulation is illustrated in Fig. 13, where we have plotted the dependence of the square root of population of the excited state on time as well as the time dependence of the pseudomode amplitude B for diMerent values of a and for = 0:1, 1 = 2: From Fig. 13 it is clear that the suppression of the decay of |$| is very strong even for quite large values of the small parameter . Let us also address the problem of interaction of two spontaneously decaying atoms coupled to the Lorentzian reservoir and such that they are driven by an external Deld in the same way as above. This means that the interaction Hamiltonian contains the term .ˆ(1) ˆ(2) ˆij with the ij + . ij instead of . obvious meaning of the superscripts (we still omit any spatial dependencies), and the wave function is given by ∞ |6(t) = $1 (t)|2; 1; 0 + $2 (t)|1; 2; 0 + =! (t)|1; 1; 1! 0

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365

which means that our Hilbert space consists of the states with either the Drst or the second atom excited and no Deld excitation, and the states with both atoms in the ground state and one photon in the mode ! present. The equations of motion for the amplitudes $1 and $2 read √ $˙1 = −i geia cos ?t B ; √ $˙2 = −i geia cos ?t B ; √ B˙ = −B − i ge−ia cos ?t ($1 + $2 ) ; a rather simple generalization of Eqs. (94) and (95). In the case of medium-strength and strong coupling we immediately infer from the Drst-order of MMS that not only the emission can be √ suppressed if ? g and J0 (a) = 0, but also the energy exchange between the two atoms will be very slow, because they are coupled via the pseudomode only. Indeed, simple numerical calculations conDrm this result, as is seen in Fig. 14, where |$1 | and |$2 | are plotted versus for a = 0 and for a = 2:40483. The initial values are $1 (0) = 1, $2 (0) = 0:001, B = 0. This interesting example of resonance suppression of interaction Dnishes our section devoted to the spontaneous emission. 4. Resonance uorescence In this section we consider our Drst examples from the circle of problems associated with coherent excitation and dynamics. In fact we shall only touch several important topics connected with the interaction of atomic matter with the electromagnetic Deld under the assumption that atoms can be approximated by just few levels while the electromagnetic Deld is classical. What is more, the in/uence of atoms on the Deld itself is assumed to be negligible; the latter assumption allows the Deld to be identical with “applied” or “external” one. The only dynamical variables of the system are atomic variables. On the other hand, the variation of Delds in time will be considered as prescribed. In addition, in all but one example it is assumed that the frequency or frequencies of the applied Deld are close to resonance with one or more of the characteristic frequencies of atomic transitions. At Drst the damping will be ignored, but it will appear in latter stages of computations to make the models more realistic. The major source of references containing a huge amount of information about methods and results in coherent atomic excitations is the monumental treatise by Shore [62]. Among other valuable references one should mention [51,63]. After some very brief description of Drst important papers on the application of MMS to the dynamics of coherently excited atoms, we present basic ideas associated with the correlation-function approach. Then we derive an eMective Hamiltonian describing a two-level atom under the in/uence of a strongly non-resonant and weak external Deld. It is shown that such a Deld produces eMective modulation of atomic frequency. This way we provide a justiDcation for one of the eMective Hamiltonians used in the previous section. Then we consider the problem of the atomic population trapping in a three-level cascade system. The last subsection is devoted to the application of MMS in the case of bichromatic excitation.

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4.1. General considerations The method of multiple scales has been Drst applied to the problem of coherent atomic excitations by Wong et al. [17–19]. In the Drst paper of the series, Wong et al. have developed their general theory and applied it to two test cases: the Bloch–Siegert shift and the two-photon resonance excitation of a molecule. An interesting and important feature of the approach of these authors is that they have developed a technique to avoid the problem of small denominators (cf., e.g., [64]) in all required orders, and not just to elimination of secular terms. They have worked in the SchrIodinger picture, but a realization of their technique in terms of the Heisenberg equations is possible as well. We shall not attempt to show somewhat involved details of their computations, but to apply techniques similar to theirs in some speciDc examples. In their second work, Garrison et al. have presented their multiple scales analysis of the Doppler-free population inversion induced by a two-photon resonance. Both these works contain comparisons of MMS predictions with numerical results. The paper [19] is devoted to the problem of a two-photon resonance coupled to a one photon resonance through a common energy level. In one of the remaining parts of this section we shall use MMS to study multichromatic excitations of two- and three-level atoms using MMS in the setting close to that oMered in [17]. We shall, however, use the Heisenberg equations of motion in order to include losses. The Heisenberg equations of motion lead naturally to study the evolution of correlation functions in time. Our exposition of the basic equations for the correlation-function dynamics will be patterned after that of [65]. To derive the Heisenberg equation of motion, we write the two-level atomic Hamiltonian in one of the forms shown in the previous section: Hˆ A = 12 ˝!0 .ˆz :

(98)

The Hamiltonian describing interactions with the external electromagnetic Deld is chosen to be Hˆ AF = −dˆ · E(t) ;

(99)

where, as in the previous section, E(t) denotes the time-dependent (classical) electric Deld, and dˆ is the atomic dipole moment operator. For two-level atoms the latter operator has the following representation: dˆ =

2

dij .ˆij ;

(100)

i; j=1

ˆ where dij = i|d|j, and for the atoms not exposed to static electric or magnetic Delds the diagonal elements are equal to zero. One has to notice, however, that in order to account for scattering and damping phenomena, the atom has to be coupled to a reservoir. This can be modeled in many diMerent ways. For instance, one can start with the full Hamiltonian of the form Hˆ T = Hˆ A + Hˆ AF + Hˆ R + Hˆ AR ;

(101)

where Hˆ R denotes the Hamiltonian of the reservoir and Hˆ AR the coupling between the atom and the reservoir. The Heisenberg equations of motion are then solved partially by “adiabatic elimination” of all the degrees of freedom of the reservoir (in principle this can be done with the help of MMS as well), as shown, e.g., in [41]. The remaining equations of motion contain non-Hamiltonian

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367

terms—damping constants T and L , and quantum-mechanical noise operators Bˆ+ (t), Bˆ− (t), and ˆ Bz (t), about which we shall assume that they are correlated to the delta function and that their mean values vanish. Taking into account the form of the Hamiltonian and the above simple considerations concerning damping, we obtain C1 (t) .ˆz + Bˆ− (t) ; .ˆ˙− = −(T + i!).ˆ− − i 2 C1 (t) .ˆz + Bˆ+ (t) ; .ˆ˙+ = −(T − i!).ˆ+ + i 2 .ˆ˙z = −L .ˆz + iC1 (t)(.ˆ+ − .ˆ− ) − L + Bˆz (t) ;

(102)

where C1 (t) = 2d12 · E(t)=˝, and we have written .ˆ− instead of .ˆ12 as well as .ˆ+ instead of .ˆ21 . We shall now perform a transformation to a rotating frame, rotating with a frequency !L close to one of the characteristic frequencies of the Deld E(t). The transformation is deDned by the unitary operator −i ˆ U (t) = exp (!L t − DL ).ˆz ; 2 so that the Hamiltonian of the system atom plus Deld reads Hˆ A + Hˆ F = 12 \.ˆz − dˆ · E(t)[ei(!L t −DL ) .ˆ+ + e−i(!L t −DL ) .ˆ− ] ;

(103)

where < = !0 − !L . As a result, the raising and lowering operators transform as .ˆ− → .ˆ− e−i(!L t −DL ) ; .ˆ+ → .ˆ+ ei(!L t −DL ) ; and Eq. (102) takes the form C(t) .ˆz + Bˆ− (t) ; .ˆ˙− = −(T + i<).ˆ− − i 2 C? (t) .ˆz + Bˆ+ (t) ; .ˆ˙+ = −(T − i<).ˆ+ + i 2 .ˆ˙z = −L .ˆz + i(C(t).ˆ+ − C? (t).ˆ− ) − L + Bˆz (t) ;

(104)

where C(t) = C1 (t) exp[i(!L t − DL )]. From this system of equations we obtain two other systems for the expectation values and for the correlation functions which will play the key role in the following subsections. By taking the expectation values of Eqs. (104) we get .ˆ˙− = −(T + i<) .ˆ− − i

C(t) .ˆz ; 2

.ˆ˙+ = −(T − i<) .ˆ+ + i

C(t) .ˆz ; 2

.ˆ˙z = −L .ˆz + iC(t)( .ˆ+ − .ˆ− ) − L :

(105)

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On the other hand, deDning /uctuations of the operators .ˆ± and .ˆz by the formulae .ˆ± = .ˆ± − .ˆ± ; .ˆz = .ˆz − .ˆz ; and three correlation functions c+ , c− , and cz : c+ (t; t ) = .ˆ+ (t).ˆ− (t ) ; c− (t; t ) = .ˆ− (t).ˆ− (t ) ; cz (t; t ) = .ˆz (t).ˆ− (t ) ;

(106)

we obtain the following system for the latter: d C(t) c− (t; t ) = −(T + i<)c− (t; t ) − i cz (t; t ); dt 2 C? (t) d c+ (t; t ) = −(T − i<)c+ (t; t ) + i cz (t; t ); dt 2 d cz (t; t ) = −L cz (t; t ) − iC? (t)c− (t; t ) + iC(t)c+ (t; t ) ; dt

(107)

where we have used the assumption about Bˆ operators speciDed above, as well as the assumption that .ˆ±; z (t ) and Bˆ±; z (t) are uncorrelated for t 6 t. This assumption leads to results equivalent to those obtained using the so-called Quantum Regression Hypothesis. Eqs. (107) are subject to the following initial conditions which follow from the Pauli matrix algebra: c+ (t ; t ) = 12 [1 + .z (t )] − | .+ (t )|2 ; c− (t ; t ) = − .− (t )2 ; cz (t ; t ) = −[1 + .z (t )] .− (t ) :

(108)

In the case of excitation by just one external resonant Deld, the above equations (as well as their density-matrix-elements counterparts) have led to the theoretical predictions [66–69] of the three-peaked spectra of resonance /uorescence (called Mollow triplets), the triumphant experimental vindication [43] of which has been one of the major achievements of the heroic period of the physics of laser-atom interactions. 4.2. Modulation of atomic frequency We begin with an MMS description of the dynamics of a two-level atom coupled to a Deld with the frequency much smaller than the characteristic frequency of atomic transitions. The losses will not be taken into account and we shall concentrate on the eMective time evolution operator. We will show that a simple eMective Hamiltonian emerges in a natural way in this system.

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369

We start with the energy operator Hˆ = 12 ˝!0 .ˆz − 2dˆ · E0 cos(!t) ;

(109)

and assume that both ! and ?, the latter deDned by ?=

2d12 · E0 ; ˝

(110)

are much smaller than !0 . We shall deDne our small parameter as j = ?=!0 and write ! = jE!0 , where E is a dimensionless parameter and E=O(1). Introduction of a new dimensionless time variable = !0 t leads to the following SchrIodinger equation for the time-evolution operator: 1 d Uˆ = .ˆz − j cos(jE).ˆx Uˆ ; (111) i d 2 where .ˆx = .ˆ+ + .ˆ− . We now introduce new time variables deDned as T0 = ;

T1 = j;

T2 = j2 (j; ) ;

for it will shortly become evident that the simplest choice of Tn =jn does not work in this example. From the above equation it follows that 9 9 (j; ) 9 9 9 = +j + j2 : 9 9T0 9T1 9 9T2 The expansion of the time-evolution operator is standard: U = U0 + jU1 + j2 U2 + · · · : In the zeroth-order approximation we Dnd immediately −i ˆ .z T0 Vˆ (T1 ; T2 ; : : :) ; U 0 = exp 2

(112)

which naturally can be written as a linear combination of .ˆz and .ˆ0 = 1. In the Drst order we get i where

9Vˆ 9UWˆ 1 +i = −cos(ET1 )Mˆ (T0 )Vˆ (T1 ; : : :) ; 9T0 9T1

i .ˆz T0 U1 UWˆ 1 = exp 2

and the matrix Mˆ (T0 ), equal to 0 eiT0 ; Mˆ (T0 ) = 0 e−iT0

(113)

(114)

(115)

can be written as a linear combination of the matrices .ˆx and .ˆy . We can see from Eq. (113) that no secular terms can appear in the Drst order, hence (9Vˆ )=(9T1 ) = 0, and we may simply integrate

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over T0 to obtain UWˆ 1 = cos(ET1 )Mˆ 1 (T0 ) ; where

Mˆ 1 (T0 ) =

0

ei T 0

−e−iT0

0

(116) Vˆ (T2 ; : : :) + Xˆ (T1 ; T2 ; : : :) :

(117)

Writing Uˆ 2 = e−(i=2).ˆz T0 UWˆ 2 ; we get in the second order i

9UWˆ 1 9 9Vˆ 9UWˆ 2 +i +i = −cos2 (ET1 )Mˆ (T0 )Mˆ 1 (T0 )Vˆ − cos(ET1 )Mˆ (T0 )Xˆ : 9T0 9T1 9 9T2

(118)

Since, however, Mˆ (T0 ) · Mˆ 1 (T0 ) = −.ˆz ; the Drst term on the right-hand side of (118) is equal to cos2 (ET1 ).ˆz Vˆ = 12 (1 + cos(2ET1 ))Vˆ ; and, in order to avoid secular terms, we must require that Vˆ fulDlls the condition i

1 1 9 9Vˆ = [1 + cos(2ET1 )].ˆz Vˆ = [1 + cos(2jE)].ˆz Vˆ : 9 9T2 2 2

(119)

On the other hand, Xˆ is independent of T1 . We would be unable to integrate (119) without the presence of on the left-hand side. But as it is present, we simply write 9 1 = [1 + cos(2jE)] ; 9 2 so that 1 1 + sin(jE) ; (j; ) = 2 2jE

(120)

while Vˆ (T2 ; : : :) = e−i.ˆz T2 : It follows that, up to the second order, Uˆ 0 can be written as 1 j 2 [1 + j ] + sin(2jE) : U0 = exp −i.ˆz 2 2E

(121)

(122)

By recalling the deDnition of , we can recast Eq. (122) into the form Uˆ eM ≡ Uˆ 0 = exp[ − iHˆ eM (t)=˝] ;

(123)

M. Janowicz / Physics Reports 375 (2003) 327 – 410

where the eMective Hamiltonian HeM reads 1 ?2 ?2 !0 + + sin(2!t) .ˆz : HeM = ˝ 2 !0 2!0

371

(124)

This means that slowly varying and very strongly detuned external Delds mainly give rise to the modulation of the atomic frequency, which is a very intuitive result, for such a Deld should not induce any transitions—that is, no changes in the population inversion .z should take place. 4.3. Population trapping in cascade systems In this section we shall use MMS and results of the previous subsection to obtain population trapping in the two-photon system analogous to that considered by Agarwal and Harshawardhan [77]. It consists of a three-level atom with the levels |1, |2, |3, coupled to two external Delds. The Drst one couples only the levels |1 and |2, and, having very small frequency, leads just to the modulation of the frequency of the atomic transition |1 → |2. On the other hand, the second Deld is in a two-photon resonance with the levels |1 and |3, that is, the Deld frequency is equal to the energy separation of the two latter levels (divided by ˝). However, it is assumed that the ˆ ˆ ˆ matrix elements of the dipole moment of the form 1|d|3 vanish, while 1|d|2 as well as 2|d|3 are diMerent from zero. Such a ladder system with the energies E3 ¿ E2 ¿ E1 is sometimes called a I system. In the following we shall choose zero of the energy such that E1 = 0 and write E2 = ˝!21 , E3 = ˝!31 . The Hamiltonian which we begin with is thus Hˆ = ˝!21 .ˆ22 + ˝j!ˆ 21 (.ˆ22 − .ˆ11 ) · sin(!m t) + ˝!31 .ˆ33

−2d12 · E0 cos(!L t)(.ˆ12 + .ˆ21 ) − 2d12 · E0 cos(!L t)(.ˆ23 + .ˆ32 ) ;

(125)

where !m is the modulation frequency, corresponding to ! from the previous subsection. To get rid of unwanted terms in the above Hamiltonian, we perform the unitary transformation generated by Hˆ 0 = ˝(!L .ˆ22 + 2!L .ˆ33 + j(.ˆ22 − .ˆ11 )sin !m t) : The transformed Hamiltonian reads Hˆ = ˝\.ˆ22 − 2d12 · E0 cos !L t (.ˆ12 e−i!L t+2ir(t) + .ˆ21 ei!L t −2ir(t) )

−2d23 · E0 cos !L t (.ˆ23 ei!L t −ir(t) + .ˆ32 e−i!L t+2ir(t) ) ;

(126)

where < = !21 − !L and r(t) =

j!21 (cos !m t − 1) : !m

We shall now perform the rotating wave approximation, disregarding all terms in (126) which oscillate with the frequency ∼ 2!L . These terms provide small corrections of the Bloch–Siegert-shift type, analyzed from the point of view of MMS in [17]. On neglecting these fast oscillating terms, we obtain the following simpler Hamiltonian which is already suitable for a simple multiple scales

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analysis: Hˆ = ˝\.ˆ22 − 12 ˝?12 (.ˆ12 e2ir(t) + .ˆ21 e−2ir(t) ) − 12 ˝?23 (.ˆ23 e−ir(t) + .ˆ32 eir(t) ) ;

(127)

where ?ij = 2dij · E0 =˝. We assume that ?ij <, !m <. On using the new symbols Jij and Jm such that ?ij = j\Jij and !m = j\Jm we may write the following SchrIodinger equation for the time-evolution operator (the dynamics of which is generated by Hˆ ) in time = \t: d Uˆ = hˆ0 + jhˆ1 ; d

(128)

where hˆ0 = .ˆ22 and h1 = − 12 (J12 [e2ir() .ˆ12 + h:c:] + J23 [e−ir() .ˆ23 + h:c:]) : We deDne the diMerent “times” in the standard way, Tn = jn , n = 0; 1; 2; : : :, and obtain in the zeroth order Uˆ 0 = exp[ − i.ˆ22 T0 ]Vˆ (T1 ; T2 ; : : :) :

(129)

In the Drst order, upon using a deDnition which has already become standard Uˆ 1 = exp(−i.ˆ22 T0 )UWˆ 1 (T0 ; T1 ; : : :) ; we Dnd that Vˆ does not depend on T1 and that UWˆ 1 is given by ˆ 0 ; T1 ; : : :) + Xˆ (T1 ; T2 ; : : :) ; UWˆ 1 = B(T where ˆ 0 ; T1 ; : : :) = 1 [J12 (.ˆ21 e−2ir(T1 ) eiT0 − h:c:) + J23 (.ˆ23 e−ir(T1 ) e−iT0 − h:c:)]Vˆ (T2 ) : B(T 2 In the derivation of the last equation we have used the equalities ei.ˆ22 T0 .ˆ12 e−i.ˆ22 T0 = .ˆ12 e−iT0 ; ei.ˆ22 T0 .ˆ23 e−i.ˆ22 T0 = .ˆ23 eiT0 ; as well as analogous formulae for .ˆ21 and .ˆ32 . The variable r() has been written as r(T1 ) since we have !m t = j\Jm =< = jJm = Jm T1 . ˆ 0 ; T1 ; : : :) such that To proceed, let us also introduce a new operator A(T ˆ 0 ; T1 ; : : :) = ei.ˆ22 T0 hˆ1 e−i.ˆ22 T0 : A(T Then in the second order of the perturbation expansion we Dnd i

9UWˆ 2 9UWˆ 1 9Vˆ ˆ 0 ; T1 ; : : :)(Xˆ (T1 ; : : :) + B(T ˆ 0 ; T1 ; : : :)Vˆ ) : +i +i = A(T 9T0 9T1 9T2

(130)

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373

Using simple operator algebra of the .ˆ matrices we realize that the terms on the right-hand side of (130) that can exhibit secular behavior after integration over T0 are −1 2 (131) {J12 (.ˆ11 − .ˆ22 ) + J223 (.ˆ33 − .ˆ22 ) + J12 J23 (.ˆ13 eir(T1 ) + .ˆ31 e−ir(T1 ) )}Vˆ (T2 ; : : :) : 4 The physical meaning of these terms is quite clear. The diagonal ones in .ˆij represent the energy shift of the three involved levels proportional to the intensity of the applied Deld. More importantly, there are terms representing the eMective two-photon coupling between the levels |1 and |3. This coupling is, however, modulated. Using Eq. (96) we may extract the terms in Eq. (131) which do not contain T1 explicitly, and write 9Vˆ !21 −1 2 2 {J (.ˆ11 − .ˆ22 ) + J23 (.ˆ33 − .ˆ22 )} + J0 i J12 J23 (.ˆ13 + .ˆ31 )Vˆ ; = (132) 9T2 4 12 \Jm while the part of (131) which does depend explicitly on T1 can be absorbed into (9Xˆ )=(9T1 ). The expression multiplying Vˆ on the right-hand side of Eq. (132) can be considered as an eMective Hamiltonian generating dynamics of a two-photon transition in the cascade system we consider. If it happens, however, that !21 =0 ; J0 \Jm the coupling between levels |1 and |3 will eMectively be equal to zero and the initial population of these levels will be stabilized. Of course, one cannot turn the coupling completely oM, since the transition operators will be present in Xˆ (T1 ). Nevertheless, such a resonant dynamical suppression of the two-photon oscillations should be observable—both the absorption and resonance /uorescence lines should be very weak. Suppression of the Rabi oscillations caused by this type of modulation has been Drst predicted for another system in [77,78]. Several other interesting interference eMects which can be found in the phase-modulated systems have been analyzed in [79–81]. 4.4. Bichromatic excitation In the previous example we have considered, in fact, excitations of a three-level system by two applied Delds with very diMerent frequencies. Now, we want to consider the bichromatic excitations in a stricter sense: the two applied laser Delds will be assumed to have almost equal frequencies close to the one-photon resonance. Thus, the two-level atom model will suLce for our purposes. Such a model of bichromatic excitations was studied extensively in the 1980s and 1990s, see, e.g., [70–76]. Here we shall concentrate on the case of one Deld being much stronger that the other, while the frequency diMerence between them will be of the same order as the Rabi frequency of the weak (probe) Deld. Such a problem has been analyzed in [75]. Our starting point are systems (105) and (107). The external Deld C(t) is given by C(t) = 2(?1 cos(!1 t) + ?2 cos(!2 t))ei!L t :

(133)

We can still choose an appropriate !L . For our purposes it is convenient to use !L = !1 , so that the strong Deld labeled “1” is in the exact “resonance” with the frequency of the rotating system. Let us denote the diMerence of frequencies of the two excitation Delds by B, B = !1 − !2 . On neglecting

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the oscillations with very large frequencies ∼ 2!L , we obtain C(t) = ?1 + ?2 e−iBt :

(134)

We shall assume that ?1 is much larger than all the other parameters of the dimension of frequency. Naturally, many diMerent relations between the other parameters are possible (and it is possible to analyze the other cases using MMS). We shall, however, consider only the case ?1 ?2 ∼ B ∼ < ∼ ; where we have assumed for simplicity that L = T . To begin with, we shall address the problem of the dynamics of the expectation values. As in all considerations before, we introduce the dimensionless time variable such that = ?1 t. Let us also denote the vector made of the expectation values of the atomic operators .ˆ± , .ˆz by S, that is     s− .ˆ−        (135) S=  .ˆ+  =  s+  : sz .ˆz We shall look for the solution for S in the form of the perturbation series: S = S0 + jS1 + j2 S2 ; where, say, j = max(<=?1 ; B=?1 ; ?2 =?1 ; =?1 ). System (105) takes the form d S = (M0 + jM1 )S + jA ; d where the matrices Mˆ 0 , Mˆ 1 are given by   0 0 − 2i   i ; Mˆ 0 =   0 0 2  −i i 0   −(E + i) 0 − 2i J2 e−ij8   ij8  i Mˆ 1 =  0 −(E − i) J e 2 2  ; −iJ2 eij8 iJ2 eij8 −E

(136) (137)

and new parameters are deDned by the scalings < = j?1 ; ?2 = j?1 J2 ; B = j?1 8 ; = j?1 E :

(138)

The vector A is equal to (0; 0; −E)T . Clearly, our system is such that MMS can be useful, so we deDne the new time scales Tn = jn , n = 0; 1; 2; : : :, and obtain in the zeroth order 9S0 = Mˆ 0 S0 : (139) 9T0

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375

The matrix Mˆ 0 has the eigenvalues i, −i and 0. Let us denote the system of right eigenvectors of this matrix by vj , while uj will form the system of its left eigenvectors. The index j = 1 will correspond to the eigenvalue i, j = 2—to eigenvalue −i, and j = 3—to eigenvalue 0. We have the biorthogonality relation uj · vk = jk :

(140)

The solution to (139) is given by S0 =

3

Aj (T1 ; T2 ; : : :)ej T0 vj :

(141)

j=1

The explicit form of the normalized eigenvectors is given by v1 = 12 (−1; 1; 2)T ; v2 = 12 (1; −1; 2)T ; v3 =

(1; 1; 0)T ;

√1

2

and u1 = 12 (−1; 1; 1) ; u2 = 12 (1; −1; 1) ; u3 =

√1

2

(1; 1; 0) :

In the Drst order we have 9S1 9S0 + = Mˆ 0 S1 + Mˆ 1 S0 − A ; 9T0 9T1

(142)

and it seems quite natural to represent S1 in terms of the eigenvectors of M0 : S1 =

3

Bj (T0 ; T1 ; T2 ; : : :)ej T0 vj :

(143)

j=1

Substitution of (143) into (142) gives 3 9Bj 9Aj + = e− j T0 Ak (uj Mˆ 1 vk )ek T0 − uj · A ; 9T0 9T1

(144)

k=1

where we have used the fact that vk are eigenvectors of M0 . Computation of all products uj Mˆ 1 vk is straightforward and gives the following equations for Bj : A3 E 9B1 9A1 −iT0 i T0 ; (145) A1 (iJ2 cos(8T1 ) − E)e + √ (i + J2 sin(8T1 )) − + =e 9T0 9T1 2 2

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A3 E 9B2 9A2 i T0 i T0 ; A2 (−iJ2 cos(8T1 ) − E)e + √ (−i + J2 sin(8T1 )) − + =e 9T0 9T1 2 2

(146)

9B3 9A3 A1 A2 + = √ (i − J2 sin(8T1 ))eiT0 + √ (−i − J2 sin(8T1 ))e−iT0 − EA3 : 9T0 9T1 2 2

(147)

We infer that the following equations for Aj should hold: 9A1 = (iJ2 cos(8T1 ) − E)A1 ; 9T1 9A2 = (−iJ2 cos(8T1 ) − E)A2 ; 9T1 9A3 = −EA3 ; 9T1

(148)

from which we obtain the Drst-order solutions for Aj : A1 (T1 ; : : :) = $1 (T2 ; : : :) exp[i(J2 =8) sin(8T1 ) − ET1 ] ; A2 (T1 ; : : :) = $2 (T2 ; : : :) exp[ − i(J2 =8) sin(8T1 ) − ET1 ] ; A3 (T1 ; : : :) = $3 (T3 ; : : :) exp[ − ET1 ] :

(149)

From Eqs. (145)–(147) and (149) it follows that Bj are given by iE −iT0 A3 √ (− + iJ2 sin(8T1 )) − ; B1 = C1 (T1 ; : : :) + e 2 2 iE iT0 A3 √ (− − iJ2 sin(8T1 )) + B2 = C2 (T1 ; : : :) + e ; 2 2 1 B3 = C3 (T1 ; : : :) + √ [eiT0 A1 ( + iJ2 sin(8T1 )) + e−iT0 A2 ( − iJ2 sin(8T1 ))] : 2

(150) (151)

These results Dnish the Drst-order computations. In the second order we have 9S2 9S1 9S0 + + = Mˆ 0 S2 + Mˆ 1 S1 : 9T0 9T1 9T2

(152)

Again, we represent the solution for S2 as an expansion in terms of the right eigenvectors of the matrix Mˆ 0 : S2 =

3

Dk (T0 ; : : :)vk ek T0 ;

k=1

to obtain 3

9Dj (T0 ; : : :) 9Bj (T0 ; : : :) 9Aj (T1 ; : : :) + + = Bk (T0 ; : : :)uj Mˆ 1 · vk e(k −j )T0 : 9T1 9T2 9T0 k=1

(153)

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Performing the multiplication of the matrix Nˆ , Nˆ jk = uj Mˆ 1 vk by the vector with entries Bk we convince ourselves that in order to avoid secular terms in the second order the following equations must hold: 9C1 9A1 1 + = (iJ2 cos(8T1 ) − E)C1 + ( + iJ2 sin(8T1 ))(i + J2 sin(8T1 ))A1 ; 9T1 9T2 2 9C2 9A2 1 + = (−iJ2 cos(8T1 ) − E)C1 + ( − iJ2 sin(8T1 ))(−i + J2 sin(8T1 ))A2 ; 9T1 9T2 2 9C3 9A3 E + = −EC3 + √ : 9T1 9T2 2

(154) (155) (156)

Writing C1; 2 = exp[ ± i(J2 =8) sin(8T1 ) − ET1 ]E1; 2 ; where the plus sign refers to the index 1, and minus sign to the index 2, we Dnally get 2 J22 + T2 ; $1 = $10 exp i 2 4 2 J22 + T2 ; $2 = $20 exp −i 2 4 $3 = $30 ;

(157)

and E1 = F1 (T2 ; : : :) −

i $1 sin(28T1 ) ; 8

E2 = F2 (T2 ; : : :) +

i $2 sin(28T1 ) ; 8

E3 = F3 (T2 ; : : :)e−ET1 + √ ; 2

(158)

where $i0 , i = 1; 2; 3 can depend on T3 and higher times. The formal MMS analysis has thus been accomplished up to the second order except that we do not provide long and not very impressive solutions for Dj . There is only one qualitatively important feature contained in the Dj , namely, D1 and D2 have terms which guarantee that sz will eventually reach the value −1, and not 0; this is necessary to make the model self-consistent. Before addressing the problem of the time-dependent /uorescence spectra let us just observe that, if the atom is initially in, say, the ground state, the approximate expression for sz is given by J2 1 2 −ET1 2 (159) sz = −e cos T0 + sin(8T1 ) + (2 + J2 )T2 ; 8 4 which can also be rewritten by recalling the deDnition of Tn and as 2 < ?2 ?22 1 −t 2 t : + sz (t) = −e cos ?1 t + sin(Bt) + B 4 ? 1 ?1

(160)

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Besides a small correction to the frequency of the oscillations of inversion sz , we also have the modulation of frequency due to the sinusoidal term under the cosine function. This means that the atomic inversion actually oscillates with all the frequencies ?1 ± nB with natural n, where B is the diMerence between the carrier frequencies of the excitation Delds, and ?1 = ?1 + (1=4)(2<2 =?1 + ?22 =?1 ). It is interesting that the roles of B and < are completely diMerent, the latter providing just a shift of the frequencies. It might seem at Drst sight that we do not observe any particular resonance condition involving B and ?2 . But the resonance is in fact present: if we use the identities ∞ cos(z sin s) = J0 (z) + 2 J2n (z) cos(2ns) n=1

and sin(z sin s) = 2

∞

J2n+1 (z) sin((2n + 1)s) ;

n=0

we realize that once again there is a kind of resonance if the Rabi frequency of the weak Deld and the diMerence of the carrier frequencies are such that J0 (?2 =B) = 0. In this case the oscillations with the frequency ?1 are turned out and the inversion oscillates just with the frequencies ?1 ± nB with non-zero n. Under the above initial conditions, we have the following simple zeroth-order solutions for s± : ?2 −t s± = ∓ie sin ?t + sin(Bt)t ; B where ? = ?1 + <2 =(2?1 ) + ?12 =(4?1 ). From the expression for s+ we can immediately obtain the coherent part of the time-dependent spectrum, as it is given by [82] 2 1 T Scoh (!; T ) = s+ (t) exp[ − i(! − !L )t] T 0 (cf. [83] for extensive discussion of the time-dependent spectrum). Using the above deDnition we obtain 2 ∞ 1 f1 (!; T ) − f2 (!; T ) Jm (a) 2 (161) Scoh (!; T ) = ; 2T m=−∞ + (? + mB − (! − !L ))2 where f1 (!; T ) = g(1 − e−T cos(gT )) ; f2 (?; T ) = sin(gT ) ; and g = ? + mB − (! − !L ) : It can be seen that, quite naturally, the spectra contain lines centered at ! − !L = ±(? + mB), m being an integer (positive or negative), and the factor which is responsible for the intensity of each line contains the Bessel functions of the argument a = ?2 =B. Thus, by choosing a particular value of

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379

a we may get rid of one or more lines, or at least strongly diminish its intensity. This is also true about the principal sidebands of the Mollow spectrum which can be “turned oM” by choosing such a that J0 (a) = 0. This oMers an interesting possibility of an experiment with slow (adiabatic) change of either the Rabi frequency of the weak component, or the detuning B, or both, and monitoring the spectrum—we should observe appearance and disappearance of certain lines when a crosses zeros and local maxima of consecutive Bessel functions. Let us Dnally notice that the spectrum itself oscillates in time, being modulated by harmonics of the detuning. The expression for incoherent part of the spectrum is somewhat complicated (and will not be provided here), because of the long asymptotic formula for the correlation function necessary to compute it, i.e. c+ (t; t ). This correlation function is obviously obtained in the same manner as s+ , except that the initial conditions are not that simple. But recalling the fact that c+ (t; t ) must satisfy almost the same equation as s+ , the initial conditions, and the deDnition of the incoherent part of the spectrum: T t 2 Sinc (!; T ) = Re c+ (t; t ) exp[ − i(! − !L )(t − t )] dt dt ; T 0 0 we can easily realize that the incoherent part of the spectrum has lines centered at ! − !L at the distance equal to either zero, or ±(? + mB), or ±(2? + mB) for integer m. These results, the analogs of which can also be found, e.g., in the case of amplitude- or phasemodulated excitation Delds, Dnish our discussion of applications of MMS in the Deld of coherent atomic excitations. 5. Atom–#eld interactions in cavities In this section we shall discuss some examples of application of MMS to the problem of interaction of few-level atoms with the quantized electromagnetic Deld in microwave cavities. In such systems the mean number of photons is usually not large enough to neglect the in/uence of the atom on the Deld. Therefore, one should treat the whole atom–Deld system self-consistently. Nevertheless, it turns out that striking similarities exist between the atom–Deld systems in cavities, and atomic system driven by external laser Delds. Let us provide some numbers related to the considered interactions. In the microwave cavities, the characteristic frequencies of electromagnetic modes are of the order of 109 –1010 Hz. The interaction strengths, on the other hand, are of the order of 104 –105 Hz. The cavity losses (due to the interactions of electromagnetic Deld with anything other than the atom under consideration) can be as small as 100 Hz, but obviously larger losses are also possible. It is to be noted that careful analysis of systems, examples of which are discussed here, led, in the 1980s, to the discovery of collapses and revivals of the oscillations in the atomic population inversion, one of the most important achievements of the quantum theory in that decade, since it provided one of the most direct and convincing proofs of the quantum, “grain-like” nature of light [84–88,10,44]. We shall discuss three examples in this section. The Drst one is a two-level atom interacting with both one mode of the cavity Deld and with a strong microwave external Deld. This system will be used to show how MMS leads to eMective multi-photon coupling. Then we shall address

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the problem how the interactions of a two-level atom with one cavity mode are in/uenced by the presence of a second detuned mode of electromagnetic Deld. The third example involves a four-level atom interacting with two cavity modes in the “M” conDguration. Again, building up eMective two-photon Hamiltonians is shown, but in a very diMerent context: one of the possible realizations of the SchrIodinger-cat states and a system with fully predictable collapses and revivals are discussed. 5.1. Two-level externally excited atom Let us consider a two-level atom with the energy gap between its levels equal to ˝!0 , which is coupled to both the quantized cavity Deld and strong external Deld which can be considered classical. It will be assumed that the coupling of the atom to all cavity modes except one is inessential (though it can be studied using MMS). The Hamiltonian of the system is given by Hˆ = ˝!0 .ˆ22 + ˝!aˆ† a − 2d12 · E0 cos(!L t)(.ˆ12 + .ˆ21 ) − ˝g(.ˆ12 + .ˆ21 )(aˆ + aˆ† ) ;

(162)

where a; ˆ aˆ† are the cavity Deld destruction and creation operators, g represents the coupling of the Deld with the atom, E0 is the amplitude of the external Deld, and d12 is the dipole moment matrix element assumed to be real. We perform a unitary transformation of this Hamiltonian using the following unitary operator: ˆ Uˆ = exp(−i[!L (.ˆ22 + aˆ† a)])

and neglect all the terms that oscillate with frequencies ±2!L in the resulting expression. This way we obtain the Hamiltonian 1 Hˆ = ˝\.ˆ22 + ˝<0 aˆ† aˆ − ˝?(.ˆ12 + .ˆ21 ) + ˝g(.ˆ21 aˆ + aˆ† .ˆ12 ) ; 2 where < = ! − !L , <0 = !0 − !L , and ? = 2d12 · E0 =˝. The problem is exactly solvable if < = <0 . Our goal here is to obtain a family of eMective Hamiltonians if the coupling between the atom and the cavity Deld is much smaller than the Rabi frequency of the external Deld, that is, if g?. As usual, we start with changing the time variable form t to = ?t, and deDning the small parameter j = g=?. The SchrIodinger equation for the time evolution operator becomes 1 d Uˆ † † = − (.ˆ12 + .ˆ21 ) + j[.ˆ22 + 0 aˆ aˆ + (.ˆ21 aˆ + aˆ .ˆ12 )] Uˆ ; (163) d 2 where we have assumed for a moment that < and <0 are of the same order of smallness as g and we substituted = <=(j?) as well as 0 = <0 =(j?). In the simplest case, the external Deld is just in exact resonance with the atom, = 0. For simplicity we shall concentrate on this case, but the case of = 0 presents actually no diLculties. Eq. (163) can be rewritten as i

d Uˆ = (hˆ0 + jhˆ1 )Uˆ ; d

(164)

with the obvious meanings of the operators hˆ0 and hˆ1 . We expand the time-evolution operator Uˆ as Uˆ = Uˆ 0 + jUˆ 1 + j2 Uˆ 2 + · · ·, and the time scales are deDned as Tn = jn , n = 0; 1; 2; : : : . In the

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381

zeroth order we get i ˆ U 0 = exp (.ˆ12 + .ˆ21 )T0 Vˆ (T1 ; T2 ; : : :) ; 2 while in the Drst order we must consider 9UWˆ 1 9Vˆ i +i = [0 aˆ† a + exp(ihˆ0 T0 )g(.ˆ21 aˆ + aˆ† .ˆ12 ) exp(−ihˆ0 T0 )]Vˆ ; 9T0 9T1 with already standard meaning of UWˆ 1 . The right-hand side can be computed to obtain g [0 a† a + (.12 + .21 )(a + a† ) 2 ig g cos T0 (.ˆ21 − .ˆ21 )(aˆ − aˆ† ) + sin T0 (.ˆ22 − .ˆ11 )(aˆ − aˆ† )]Vˆ : 2 2 To avoid any secular terms we must require that 9Vˆ g i = 0 aˆ† aˆ + (.ˆ12 + .ˆ21 )(aˆ + aˆ† ) Vˆ : 9T1 2 The solution for Vˆ (T1 ) is clearly given by g Vˆ (T1 ; T2 ; : : :) = exp[ − i(0 aˆ† aˆ + (.ˆ12 + .ˆ21 )(aˆ + aˆ† ))T1 ]Wˆ (T2 ; : : :) ; 2 which we shall also write as ˆ 1 ]Wˆ Vˆ = exp[ − iKT +

(165)

ˆ while for UWˆ 1 we get (this equality deDnes the operator K), ig g † † ˆ sin T0 (.ˆ21 − .ˆ12 )(aˆ − aˆ ) − cos T0 (.ˆ22 − .ˆ11 )(aˆ − aˆ ) Vˆ : UW 1 = Xˆ (T1 ; : : :) − (166) 2 2 It is interesting to note that the Drst-order eMective Hamiltonian which is, up to a constant, equal ˆ contains anti-resonant terms of the form of .ˆ12 aˆ as well as of .ˆ21 aˆ† , as if the rotating wave to K, approximation was not performed. This is absolutely correct because all the time the atom is excited by the external Deld, and emitting a photon from the ground state means essentially that the atom transfers photons from the external Deld to the cavity mode. Let us check how the Deld operators evolve in slow time T1 if we neglect Uˆ 1 . From Eq. (165) we Dnd immediately g g a(T ˆ 1 ) = a(0) ˆ + (.ˆ12 + .ˆ21 ) e−i0 T1 − (.ˆ12 + .ˆ21 ) ; 20 20 g g † † aˆ (T1 ) = aˆ (0) + (.ˆ12 + .ˆ21 ) ei0 T1 − (.ˆ12 + .ˆ21 ) ; (167) 20 20 from which it follows that the photon number operator behaves according to the formula g2 Nˆ (T1 ) = aˆ† (0)a(0) ˆ + 2 (1 − cos(0 T1 )) 20 +

g (.ˆ12 + .ˆ21 )[a(0)(1 ˆ − e−i0 T1 ) + aˆ† (0)(1 − ei0 T1 )] ; 20

(168)

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which means that, in accordance with our expectations, that the photon number operator oscillates even if initially there were no photons in the cavity and the atom enters it in the ground state; the frequency of oscillations is g<0 =?. In the second order of our perturbation expansion we Dnd that, to avoid secular terms, W (T2 ) must satisfy the following equation: 9Wˆ g2 g2 g † † (.ˆ12 + .ˆ21 ) 2aˆ aˆ + 2 + 1 − (aˆ + aˆ ) Wˆ : i =− 9T2 4 0 0 We can see that in the second order the cavity Deld starts to provide additional corrections to the Rabi frequency “felt” by the atom, which are proportional to the cavity photon number, but there are also constant factors multiplying .ˆ12 + .ˆ21 . This subsection has provided a short introduction to the techniques used to extract eMective Hamiltonians in Cavity QED. A similar approach will be used in the following considerations where there are no external Delds, but both the atomic and cavity subsystems are more complicated. 5.2. Two-mode cavity In this subsection we are interested in the in/uence of the second highly detuned cavity mode on the standard interaction of the Drst cavity mode with the two-level atom. It is quite evident that in every cavity there are plenty of out-of-resonance modes and we shall try to provide some insight into the problem of their signiDcance. We start with the Hamiltonian of the entire system: Hˆ = ˝[!0 .ˆ22 + !1 aˆ†1 aˆ1 + !2 aˆ†2 aˆ2 + 1 (.ˆ12 + .ˆ21 )(aˆ1 + aˆ†1 ) + 2 (.ˆ12 + .ˆ21 )(aˆ2 + aˆ†2 )] ;

(169)

where the subscripts at aˆ and aˆ† are introduced to distinguish the two modes. For reasons which will become clear later we shall assume that the expectation value of the operator Nˆ = .ˆ22 + aˆ†1 a1 does not vanish. We write the time-evolution operator Uˆ = exp[ − (i=˝)Hˆ t] as Uˆ = Uˆ p Uˆ , where Uˆ p = exp{−i[!0 (.ˆ22 + a†1 a1 + a†2 a2 )]} ; and assume that the atom and the Drst mode are in the exact resonance, !0 = !1 . The dynamics of the time-evolution operator Uˆ are generated by the Hamiltonian Hˆ = ˝[\aˆ†2 aˆ + 1 (aˆ†1 .ˆ12 + .ˆ21 a) ˆ + 2 (aˆ†2 .ˆ12 + .ˆ21 a)] ˆ ;

(170)

where the terms oscillating like exp(±2!0 t) have been neglected, and < = !2 − !0 . As always, we introduce a new dimensionless time equal to the product of the largest frequency—< in our case—and the physical time t, = \t. Also, we rescale the coupling constants i : i = jgi <, i = 1; 2. This way we obtain the SchrIodinger equation: i

d Uˆ = (hˆ0 + jhˆ1 )Uˆ ; d

where hˆ0 = aˆ†2 aˆ2 and hˆ1 = g1 (aˆ†1 .ˆ12 + .ˆ21 aˆ1 ) + g2 (aˆ†2 .ˆ12 + .ˆ21 a) ˆ :

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383

Again, we introduce the time variables: Tn =jn ; n=0; 1; 2; : : :, and in the zeroth-order approximation we obtain Uˆ 0 = exp(−iaˆ†2 aˆ2 T0 )V (T1 ; T2 ; : : :) : As in all our examples with the time-evolution operator, we shall simplify the Drst-order equations by writing Uˆ 1 = exp(−ihˆ0 T0 )UWˆ 1 so that the Drst-order equation is given by 9UWˆ 1 9Vˆ i +i = kˆ1 Vˆ ; (171) 9T0 9T1 where ˆ ˆ kˆ1 = eih0 T0 hˆ1 e−ih0 T0 = g1 (a† .ˆ12 + .ˆ21 a) ˆ + g2 (aˆ† .ˆ12 eiT0 + .ˆ21 ae ˆ −iT0 ) : 1

2

It is therefore clear that we have to solve the following two equations: 9Vˆ = kˆ2 Vˆ = g1 (aˆ†1 .ˆ12 + .ˆ21 a) ˆ Vˆ (172) i 9T1 and 9UWˆ 1 i = g2 (.ˆ21 aˆ2 e−iT0 + aˆ†2 .ˆ12 eiT0 ) : (173) 9T0 The solution for Vˆ is easy to obtain since we essentially just have to write down the time-evolution operator for the Jaynes–Cummings model with exact resonance. An elegant way to obtain suchan operator has been shownin [89]. We deDne the operators Sˆij such that Sˆii = .ˆii and Sˆ12 = aˆ†1 .ˆ12 = Nˆ ˆ Nˆ , and easily check that Sˆij have the same algebraic properties as .ˆij . as well as Sˆ21 = .ˆ21 a= Therefore, our operator Vˆ can be represented as Vˆ (T1 ; T2 ; : : :) = exp[ − ig1 Nˆ (S12 + S21 )T1 ]Wˆ (T2 ) = [cos(g1 Nˆ T1 ) + i sin(g1 Nˆ T1 )(Sˆ12 + Sˆ21 )]Wˆ (T2 ) ; where we have taken advantage of the fact that Nˆ commutes with all Sˆ . For UWˆ we get UWˆ 1 = g2 (.ˆ21 aˆ2 e−iT0 − aˆ†2 .ˆ12 eiT0 )Vˆ + Xˆ (T1 ; : : :) :

ij

1

In the second order we have 9UWˆ 2 9UWˆ 1 9Vˆ i +i +i = k1 UWˆ 1 ; 9T0 9T1 9T2 where UWˆ 2 is deDned by Uˆ 2 = exp[ − ihˆ0 T0 ]UWˆ 2 . A simple algebra leads to the following form of the right-hand side of the above equation: g1 (aˆ†1 .ˆ12 + .ˆ21 a) ˆ Xˆ + g22 (.ˆ11 aˆ†2 aˆ2 − .ˆ22 aˆ2 aˆ†2 )Vˆ + g1 g2 (.ˆ11 aˆ†1 aˆ2 e−iT0 − .ˆ22 aˆ1 aˆ†2 eiT0 )Vˆ : Taking into account that the only term in UWˆ 1 depending on T1 is Xˆ , we obtain that the following conditions must be fulDlled if we require absence of the secular terms: 9Yˆ 9Wˆ ˆ ˆ i (174) +i = eik 2 T1 g22 (.ˆ11 aˆ†2 aˆ2 − .ˆ22 aˆ2 aˆ†2 )e−ik 2 T1 Wˆ ; 9T1 9T2

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where Yˆ (T1 ; T2 ; : : :) is deDned by the relation Xˆ = exp(−ikˆ2 T1 )Yˆ : Simple computation leads to the following expression for the right-hand side of Eq. (174): g2 R:H:S:(174) = 2 [1 − (2aˆ†2 aˆ2 + 1){cos(2g1 Nˆ T1 )(Sˆ22 − Sˆ11 ) 2 − i sin(2g1 Nˆ T1 )(Sˆ21 − Sˆ12 )}]Wˆ : It follows that Wˆ must contain just a trivial c-number term: g2 Wˆ = exp −i 2 T2 ; 2

(175)

while the expression for Yˆ is somewhat more complicated:

2 g 1 † 2 sin(2g1 Nˆ T1 )(Sˆ22 − Sˆ11 ) Yˆ (T1 ; : : :) = i (2aˆ2 aˆ2 + 1) 2 2g2 Nˆ i cos(2g1 Nˆ T1 )(Sˆ12 − Sˆ21 ) Wˆ + Yˆ 1 (T2 ; : : :) : + (176) 2g1 Nˆ We shall stop our perturbative analysis at this point. Let us summarize what we know about the time evolution operator. We have Uˆ () = Uˆ 0 () + jUˆ 1 () + j2 Uˆ 2 (), and 2 ˆ ˆ Uˆ 0 = e−ih0 T0 e−ik 2 T1 e−i=2g2 T2 ;

2 ˆ Uˆ 1 = e−ih0 T0 e−i=2g2 T2

+e

+

−ikˆ2 T1

2g1

i

Nˆ

(177) ˆ

g2 (.ˆ21 aˆ2 e−iT0 − aˆ†2 .ˆ12 )e−ik 2 T1

g2 i 2 (2aˆ†2 aˆ2 + 1) 2

cos(2g1

1

2g1

sin(2g1

Nˆ

Nˆ T1 )(S21 − S12 )

Nˆ T1 )(Sˆ22 − Sˆ11 )

;

(178)

and 2 ˆ ˆ Uˆ 2 = g1 g2 e−ih0 T0 e−i=2g2 T2 (.ˆ11 aˆ†1 aˆ2 e−iT0 + .ˆ22 aˆ1 aˆ†2 eiT0 e−ik 2 T1 ) :

(179)

Although the above expressions are a bit long, we can draw from them several qualitative conclusions. First of all, it is interesting to note that Uˆ 0 does not contain any coupling between the atom and the second mode. Such a coupling we would obtain in Uˆ 0 only in the third-order analysis, and it would express itself by the purely dispersive terms, proportional to the photon number in the second mode and to the inversion, but it would not contain raising and lowering atomic operators. This means that the in/uence of the second mode on the atomic dynamics is contained only in Uˆ 1 and higher-order Uˆ n . And, as we can see from (178), this in/uence is either just through the intensity of radiation in the second mode, with no change in the number of quanta in that mode, or,

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although the second mode does couple to the atomic dipole moment, this coupling it accompanied by the fast-varying factors exp(±iT0 ). In Uˆ 2 we Dnd the term responsible for the coupling of the Drst and the second mode of radiation Deld without any change of the population of atomic levels. However, in this low order the coupling is again accompanied by phase factors which vary fast in time. We might thus reasonably expect that, for times of the order of ∼ 1=j2 (which means t ∼ <=g2 where g ∼ min(g1 ; g2 )) the in/uence of the detuned mode can be visible through very fast and very low-amplitude oscillations of inversion or the resonant-mode photon number, without causing transitions. Let us note that the amplitude of the oscillations depends linearly on the number of photons in the detuned mode, so that certain caution seems to be necessary (taking into account conclusions of the section about an anharmonic oscillator) in the case of quasi-mesoscopic cavity Delds when the mean photon number in detuned modes could be of the order of 102 . But even in this case the product of j and the second-mode photon number is of the order of 10−3 for realistic detunings and coupling constants. 5.3. The four-level “Lambda” system 5.3.1. Introductory remarks In this section, devoted to an analysis of the dynamics of a four-level atom with two cavity modes, we want to provide an insight into how eMective Hamiltonians of quite a complicated quantum system can be obtained using MMS. Our considerations turn out to be much simpliDed by the fact that the system under consideration is actually exactly solvable, that is, the time-evolution operator can be written down exactly. This is done using an algebraic technique developed for the Jaynes–Cummings-like models by Yu et al. [89]. Application of MMS is justiDed, nevertheless, because the exact solution is too diLcult to be of extensive use; simpler eMective Hamiltonians and eMective time-evolution operators are to be obtained. The following considerations are based on the paper [29]. Our system is a four-level generalization of the three-level non-resonant “M-system” considered, e.g., in [91,92]. Let us note that there has been some discrepancy between the authors of [91] and [92] regarding the structure of the eMective Hamiltonian which describes the coupling between two lower levels of their model. We believe that the method of multiple scales, since it is universal and eLcient, brings resolution to this discrepancy. It also provides several other interesting eMective Hamiltonians not considered in [91] or [92]. We shall also provide a two-mode system which can exhibit predictable collapses and revivals—this is a special type of the four-level “M-system” with two degenerate upper levels. Unlike in the similar one-mode case, however, the dynamics of inversion, though described by a closed-form expression, can also be only quasi-periodic and thus quite erratic. In addition, we propose a Stern–Gerlach type of experiment to produce a SchrIodinger-cat-type state in the cavity. Again, the best candidate for such an experiment is the degenerate (in the two upper levels) system if the coupling constants are such that the Stark shifts are identical. The rest of this section is organized as follows. First, we present the model and attempt to solve it exactly. In the following subsection we use MMS to extract a family of eMective Hamiltonians for our generalized “M-system”. Then we analyze the production of the SchrIodinger-cat state of the cavity modes and the two-mode regular collapses-and-revivals pattern.

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5.3.2. Description of the model We consider a model system consisting of a four-level atom interacting with two modes of the cavity electromagnetic Deld. We shall assume a very speciDc coupling between the atom and the modes which is a direct generalization of the “M-system” considered, e.g., in [91]. More precisely, the system is deDned by the following Hamiltonian: 4 ˆ H= Ei .ˆii + ˝ !i aˆ†i aˆi + ˝ (g1j aˆ1 .ˆj1 + g2j aˆ2 .ˆj2 + h:c:) : (180) i=1

i=1;2

j=3;4

Thus, the Drst (lowest) level is coupled to the third and fourth only via the Drst mode of the cavity Deld, while the second level is coupled to the upper ones only via the second mode. So far, we have not assumed any relationship between the atomic energies and the mode frequencies except for the rotating wave approximation; we will try to solve the problem exactly. By “exact solution” we mean an explicit expression for the time-evolution operator. We start by writing the Hamiltonian as a linear combination of operators with simple algebraic properties. This task is simpliDed by the observation that for some given m photons in the Drst mode and n photons in the second one, the following states of the system can be populated: |4; m; n, |3; m; n, |1; m + 1; n and |2; m; n + 1. This suggests that the problem can be reduced to the four-level system in an external time-independent (within the rotating wave approximation) Deld if we Dnd appropriate constants of motion. Let us further observe that, beside the Hamiltonian itself, the system indeed possesses at least two further constants of motion (the excitation number operators) which turn out to be very useful: Nˆ 1 = aˆ†1 aˆ1 + .ˆ22 + .ˆ33 + .ˆ44 (181) and (182) Nˆ 2 = aˆ†2 aˆ2 + .ˆ11 + .ˆ33 + .ˆ44 : The operators Nˆ 1 and Nˆ 2 commute not only with the Hamiltonian, but also with the free part of the Hamiltonian and with all products of the type aˆi .ˆji ; i = 1; 2; j = 3; 4, as well as aˆ†i aˆj .ˆij ; i; j = 1; 2. Let us deDne the new operators as 1 1 Sˆ31 = aˆ1 .ˆ31 ; Sˆ41 = aˆ1 .ˆ41 ; Nˆ 1 Nˆ 1 1 Sˆ32 = aˆ2 .ˆ32 ; Nˆ 2

1 Sˆ42 = aˆ2 .ˆ42 ; Nˆ 2

Sˆii = .ˆii ; i = 1; 2; 3; 4;

Sˆ34 = .ˆ34 :

(183)

These operators, together with their Hermitian conjugates and the operator Sˆ12 = Sˆ13 Sˆ32 = Sˆ14 Sˆ42 (and its conjugate) span the Lie algebra su(4) with the usual commutator as the Lie product. Actually, it is of even more importance that they form an associative matrix algebra, isomorphic to that spanned by the operators .ˆij . Let us for instance calculate the product 1 1 1 Sˆ12 Sˆ24 = Sˆ13 Sˆ32 Sˆ24 = aˆ†1 .ˆ13 aˆ2 .ˆ32 aˆ†2 .ˆ24 ˆ ˆ N1 N2 Nˆ 2 1 1 = aˆ†1 .ˆ13 (aˆ†2 aˆ2 + 1).ˆ34 : ˆ ˆ N2 N1

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But 1 † 1 ˆ 1 † (aˆ2 aˆ2 + 1).ˆ34 − (aˆ2 aˆ2 + 1 − Nˆ 2 ).ˆ34 N 2 .ˆ34 = ˆ ˆ ˆ N2 N2 N2 =

1 † (aˆ2 aˆ2 + 1 − aˆ†2 aˆ2 − .ˆ11 − .ˆ33 − .ˆ44 ).ˆ34 = 0 ; Nˆ 2

hence Sˆ12 Sˆ24 = (1= Nˆ 1 )a†1 .ˆ14 = Sˆ14 . One can now write the Hamiltonian in terms of the operators Sˆij : Hˆ =

4

Ei Sˆii + ˝

i=1

!i aˆ†i aˆi + ˝

i=1;2

(gij Sˆji + h:c:) :

(184)

i=1;2 j=3;4

By applying the unitary transformation: Wˆ = exp[(−i=˝)(!1 Nˆ 1 + !2 Nˆ 2 )t] ; we get the following “interaction” Hamiltonian which only contains the operators Sˆij and Nˆ i without the explicit presence of the free Deld terms: Hˆ = (E1 − ˝!2 )Sˆ11 + (E2 − ˝!1 )Sˆ22 + (E3 − ˝!1 − ˝!2 )Sˆ33 + (E4 − ˝!1 − ˝!2 )Sˆ44 + ˝ Nˆ i (gij Sˆji + gij? Sˆij ) : i=1;2 j=3;4

If we choose the zero of energy in such a way that E4 = ˝(!1 + !2 ), we obtain 3 ˆ ˆ H = −˝
(185)

i=1;2 j=3;4

where <1 = (E4 − E1 )=˝ − !1 , <2 = (E4 − E2 )=˝ − !2 , <3 = (E4 − E3 )=˝. The matrix representation of the Hamiltonian is given by   −<1 0 ˆ†13 ˆ†14    0 −<2 ˆ†23 ˆ†24    (186)  ;  ˆ  ˆ −< 0 23 3  13  ˆ24 0 0 ˆ14 ? where ˆij = gij Nˆ i and ˆ? = g Nˆ i . All the ˆ operators commute. ij ij Thus, the problem of the exact solution for our model has been reduced to the problem of the diagonalization of a quadratic matrix of the fourth degree. This can always be performed, but the necessary intermediate step—the computation of the eigenvalues and eigenvectors—introduces a terrible algebraic mess which obscures both the formal structure of the Hamiltonian and the underlying physics. Therefore, despite the fact that the system admits an exact solution, in the next section we will apply an eLcient perturbation scheme—the method of multiple scales—to extract

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the physical information contained in the Hamiltonian Hˆ , provided that a natural small parameter exists in it.

5.3.3. Application of multiple scales To apply MMS, it will be√ assumed that the absolute values of at least two of
(187)

i=1;2 j=3;4

that is, we have i

d uˆ ˆ = huˆ : d

There are two natural time scales of the evolution: one connected with the part of the Hamiltonian containing detunings, the other connected with the coupling constants. Let us therefore introduce the new standard time variables: T0 = ;

T1 = j;

T2 = j2 ;

so that we have again 9 d 9 9 = +j + j2 + ··· : d 9T0 9T1 9T2

(188)

Before proceeding any further, let us introduce the following notation that will be used throughout ˆ like F(T ˆ 0 ), denote this section. Namely, in all the formulae below, the symbols that start with F, operators which contain exponential functions of T0 such that the factors multiplying T0 in the exponents are O(1). Such operators can be safely integrated without any danger of obtaining small (or zero) denominators. This notation has been introduced here because the system considered in this section is the most complicated of those analyzed in this paper. To obtain an asymptotic expression for uˆ we expand it as uˆ = uˆ 0 + juˆ 1 + j2 uˆ 2 + · · · :

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Taking into account Eqs. (187) and (188) we get i

9uˆ 0 = hˆ0 uˆ 0 ; 9T0

(189)

i

9uˆ 1 9uˆ 0 + = hˆ0 uˆ 1 + hˆ1 uˆ 0 ; 9T0 9T1

(190)

i

9uˆ 2 9uˆ 1 9uˆ 0 + + = hˆ0 uˆ 2 + hˆ1 uˆ 1 + hˆ2 uˆ 0 : 9T0 9T1 9T2

(191)

The Hamiltonian in (187) does not contain terms of the second-order in j so that Eqs. (189)–(191) become a bit simpler in this case since one has hˆ2 = 0. However, in the following considerations we shall meet the case of hˆ2 = 0 as well. Even in the case of purely dispersive interactions with no transitions between levels we can have internal resonances, the appearance of which completely changes the physical situation. In order to analyze them, we must consider several sub-cases. A. Case 1: i − j = O(1) Here we have to consider the case in which the diMerences between the ’s are of the same order as the ’s themselves. Eq. (189) can easily be solved: uˆ 0 = exp[ − i hˆ0 T0 ]v(T ˆ 1 ; T2 ) = exp(i(1 Sˆ11 + i2 Sˆ22 + i3 Sˆ33 )T0 )v(T ˆ 1 ; T2 ) :

(192)

To solve Eq. (190) we write uˆ 1 = exp[ − ihˆ0 T0 ]uWˆ 1 (T0 ; T1 ; T2 ) ; and Eq. (190) takes the form i

9uW 1 9v ˆ ˆ +i = eih0 T0 hˆ1 e−ih0 T0 v(T ˆ 1 ; T2 ) : 9T0 9T1

(193)

The Hamiltonian hˆ1 in the “interaction representation” with respect to hˆ0 is ˆ ˆ eih0 T0 hˆ1 e−ih0 T0 = (mˆ †13 Sˆ13 e−i13 T0 + mˆ 13 Sˆ31 ei13 T0 ) + (mˆ †14 Sˆ14 e−i1 T0 + mˆ 14 Sˆ41 ei1 T0 )

+ (mˆ †23 Sˆ23 e−i23 T0 + mˆ 23 Sˆ32 ei23 T0 ) + (mˆ †24 Sˆ24 e−i2 T0 + mˆ 24 Sˆ42 ei2 T0 ) ;

(194)

where ij = i − j . We can see that, in the Drst order, all the terms on the right-hand side of Eq. (193) are of the ˆ 0 ) and hence they can be integrated without the appearance of secular terms. Therefore, we type F(T put 9v=9T ˆ 1 = 0, and integrate Eq. (193). Let x(T ˆ 1 ; T2 ) denote a solution to the homogeneous equation i 9uWˆ 1 =9T0 = 0. To solve Eq. (191) we write ˆ

uˆ 2 = e−ih0 T0 uWˆ 2 (T0 ; T1 ; T2 ) :

(195)

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Then Eq. (191) becomes i

9uWˆ 1 9vˆ 9uWˆ 2 ˆ ˆ +i +i = eih0 T0 hˆ1 e−ih0 T0 uWˆ 1 : 9T0 9T1 9T2

(196)

ˆ 1 since vˆ does not depend on T1 . Taking into account (194) we Dnd the Here 9uWˆ 1 =9T1 = 9x=9T right-hand side of (196): e

ihˆ0 T0

|mˆ 13 |2 ˆ |mˆ 14 |2 ˆ −ihˆ0 T0 ˆ ˆ (S 33 − Sˆ11 ) + (S 44 − Sˆ11 ) uW 1 = h1 e 13 1

|mˆ 23 |2 ˆ |mˆ 24 |2 ˆ ˆ ˆ ˆ 0) : + (S 33 − S 22 ) + (S 44 − S 22 ) v + F(T 23 2

(197)

By |mˆ ij |2 we mean |gij |2 Nˆ i =J2 . Now, the Drst four terms on the right-hand side of Eq. (197) do not depend on T0 . Thus, if we try to integrate over T0 , this will immediately lead to unphysical terms proportional to T0 . We can avoid their appearance very easily by equating i 9v=9T ˆ 2 to the Drst four terms in Eq. (197). The solution to the equation produced in this way is obvious: |mˆ 13 |2 ˆ |mˆ 14 |2 ˆ (S 33 − Sˆ11 ) + (S 44 − Sˆ11 ) v(T ˆ 2 ) = exp −i 13 1 |mˆ 23 |2 ˆ |mˆ 24 |2 ˆ ˆ ˆ + (S 33 − S 22 ) + (S 44 − S 22 ) T2 : 23 2

(198)

If we combine this result with Eq. (192), and then go back to time t and deDne the eMective Hamiltonian as Hˆ eM = i˝Uˆ †0 d Uˆ 0 =dt, Uˆ 0 = Wˆ uˆ 0 , we obtain 3 2 ˆ 2 ˆ | | N N |g |g Hˆ eM 14 1 24 2 = !1 Nˆ 1 + !1 Nˆ 2 − Sˆ44
(199)

Our eMective Hamiltonian in Eq. (199) is thus diagonal which is a characteristic feature of purely dispersive interactions. It is to be observed that the shifts of all atomic levels depend on the mean excitation numbers. We shall see later that very diMerent forms of eMective Hamiltonians arise if an internal resonance is present in the system.

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B. Case 2: 1 − 2 = O(j) Let us now assume that, although all the <’s are much larger than the eMective coupling constants, one of their diMerences, <1 − <2 is a small quantity, comparable with the eMective couplings. In √ this case we write <2 = <1 − (<1 − <2 ) and deDne , as min(|<1 |; |<3 |), and J as max(|<12 |; gij ni ) where
= hˆ0 + jhˆ1 ;

(200)

where lower case and mˆ ij ’s are deDned as before and 12 = <12 =J. However, this is not enough to substantially change the eMective Hamiltonian as given by Eq. (199). Indeed, proceeding as before, we obtain almost exactly the same eMective Hamiltonian, with the only diMerence being that all the <2 ’s are replaced by <1 ’s and all the <23 ’s by <13 ’s in all those (and only those) terms which contain |gij |2 . Substantial changes are introduced to the @rst-order correction only; the operator xˆ is ˆ 2 ), where y(T ˆ 2 ) can be expressed in terms no longer T1 -independent: x(T1 ; T2 ) = exp[ − i12 Sˆ22 T1 ]y(T of the new v. ˆ C. Case 3: 1 − 2 = O(j2 ) Now we assume that we have almost exact internal resonance:E1 + ˝!1 is almost equal to E2 + ˝!2 , so their diMerence is small even when compared with |gij | Nˆ i . While , and J are deDned as before, we have 12 = <12 ,=(J2 ). Again, j = J=,. The Hamiltonian (for the evolution in time ) is now given by † hˆ = −1 (Sˆ11 + Sˆ22 ) − 3 Sˆ33 + j (mˆ ij Sˆij + mˆ ij Sˆji ) + j2 12 Sˆ22 = hˆ0 + jhˆ1 + j2 hˆ2 : (201) i; j

In the Hamiltonian above, the index i takes the values of 1; 2 while the index j takes the values 3; 4. The zeroth-order approximation is given by uˆ 0 = exp[i(1 (Sˆ11 + Sˆ22 ) + 3 Sˆ33 )T0 ]v(T ˆ 1 ; T2 ) :

(202)

To obtain the Drst-order approximation we compute the Hamiltonian hˆ1 in the “interaction representation” with respect to hˆ0 : ˆ ˆ eih0 T0 hˆ1 e−ih0 T0 = (mˆ †13 Sˆ13 e−i13 T0 + mˆ 13 Sˆ31 ei13 T0 )

+ (mˆ †14 Sˆ14 e−i1 T0 + mˆ 14 Sˆ41 ei1 T0 ) + (mˆ †23 Sˆ23 e−i13 T0 + mˆ 23 Sˆ32 ei13 T0 ) + (mˆ †24 Sˆ24 e−i1 T0 + mˆ 24 Sˆ42 ei1 T0 ) :

(203)

ˆ 0 ), we have 9v=9T Because the right-hand-side of Eq. (203) is of the type F(T ˆ 1 = 0 and hence ˆ ˆ 9uW 1 =9T1 = 9x=9T ˆ 1 , where uW 1 , xˆ and uW 2 are deDned as for Case 1.

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We have to solve to the second order an expression similar in Eq. (197): i

9uWˆ 2 9xˆ 9vˆ ˆ ˆ ˆ +i +i = eih0 T0 hˆ1 e−ih0 T0 uWˆ 1 + eih0 T0 hˆ2 e−ih0 T0 vˆ 9T0 9T1 9T2 |mˆ 14 |2 ˆ |mˆ 23 |2 ˆ |mˆ 24 |2 ˆ |mˆ 13 |2 ˆ ˆ ˆ ˆ ˆ (S 33 − S 11 ) + (S 44 − S 11 ) + (S 33 − S 22 ) + (S 44 − S 22 ) vˆ = 13 1 13 1

mˆ †13 mˆ 23 mˆ †14 mˆ 24 ˆ − S 12 + h:c: v + 12 Sˆ22 + Fˆ 1 (T0 ) : + (204) 13 1

Note the presence of non-diagonal terms in the above expression. We may put i 9x=9T ˆ 1 = 0 and ˆ ˆ i 9uW 2 =9T0 = F 1 (T0 ). Then the solution for vˆ is obvious and we immediately Dnd the eMective Hamiltonian: 3 Hˆ eM |g13 |2 Nˆ 1 ˆ ˆ ˆ = !1 N 1 + ! 2 N 2 −
|g14 |2 Nˆ 1 ˆ |g23 |2 Nˆ 2 ˆ (S 44 − Sˆ11 ) + (S 33 − Sˆ22 ) <1 <13

? ? g23 Nˆ 1 Nˆ 2 g14 g24 Nˆ 1 Nˆ 2 ˆ g13 S 12 + h:c: : + − <13 <1 +

(205)

The above eMective Hamiltonian again contains intensity-dependent Stark shifts of all levels as well as the coupling between the Drst and the second level. Its form may justify the assumption of Gerry and Eberly that to study an eMective interaction between the states |1 and |2 it is enough to restrict oneself to near resonant cases, i.e. two detunings, <1 and <2 , which are equal or almost equal. If this is not the case, one can use a diagonal eMective Hamiltonian and investigate the eMective coupling between the two lower levels only as a small correction. Needless to say, having linearized both the total Hamiltonian and all eMective Hamiltonians, we could very easily obtain an explicit expression for all eMective time-evolution operators. D. Case 4: 1 − 2 = O(j2 ); 3 = O(j2 ) Let us now consider the case of a very √ small energy gap between the levels |3 and |4. We deDne: , = min(|<1 |; |<2 |), J = max(|gij | Ni ), 12 = <12 ,=(J2 ), 3 = <3 ,=(J2 ), j = J=,. This case is quite interesting because we obtain an eMective coupling not only between the two lower levels, but also between the two upper ones. In fact, calculations of exactly the same type as before lead to the eMective Hamiltonian 1 Hˆ eM = −
(206)

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As it has been pointed out in [91], the eMective interaction between the two lower levels is zero-photon: there is neither gain nor loss of photons but any transition from one level to another is connected with the exchange of photons between modes. The Hamiltonian above shows that between the two upper levels we have a zero-photon coupling in an even stricter sense: there are Rabi oscillations in the sub-system consisting of states |3 and |4, but there is no exchange of photons between the two modes. E. Case 5: 1 − 2 = O(1); 3 = O(j2 ) Now , = min(|<1 |; |<2 |), and the other variables are deDned as before. In this case we obtain an eMective Hamiltonian which contains just the eMective interaction between the two upper levels, without any coupling between the lower ones: 3 Hˆ eM 1 =−
1 2 ˆ N 2 (Sˆ44 − Sˆ22 )) (|g23 |2 Nˆ 2 (Sˆ33 − Sˆ22 ) + g24 <2

? ˆ ? ˆ N 1 g23 g24 N2 ˆ g13 g14 S 34 + h:c: : + + <1 <2

+

(207)

This Hamiltonian provides a two-mode generalization of that proposed by [96]. Unlike in the Case C, this time only the upper levels are eMectively coupled, and the operator coupling constant signiDcantly diMers from the one present in Eq. (205). We will use the Hamiltonian given by (207) in the next subsection to “produce” a kind of SchrIodinger-cat state in the cavity. In this section we apply a variant of the method of multiple scales to extract eMective interactions in a “M-type” four-level system with weak coupling. It is to be stressed that the notation has been greatly simpliDed by the previous linearization of our system, but MMS would work equally well even if we had no idea about how to linearize the initial Hamiltonian. 5.3.4. SchrAodinger-cat states and population dynamics in a restricted two-mode two-photon interaction We shall now consider the possible appearance of some special states of the cavity Delds from a superposition of the products of simple coherent states with large mean number of photons. By a turn of phrase, such states are called “SchrIodinger-cat” states. They are quite important from a fundamental point of view as their existence is a reply to Einstein’s and SchrIodinger’s objections against quantum theory, based on the fact that it is diLcult to Dnd any coherent superposition of quantum states on a macroscopic level. In fact, on the one hand, we can now produce such coherent superpositions in high-Q cavities (see, e.g., [90]) and in optical traps, but on the other, there is a satisfactory explanation of the diLculty in observing such superpositions in terms of the decoherence introduced by the coupling with the reservoir which is always present. One of the most important non-classical features of models of the Jaynes–Cummings type is the presence of the so-called collapses and revivals in the dynamics of the populations of atomic states. Usually, however, one cannot Dnd a closed formula to express these dynamics. Asymptotic and numerical methods (e.g. see [84,86]) show that when time increases, collapses and revivals usually appear less and less regularly and then Dnally in an erratic way, due to the strong overlap between

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neighboring revivals. Phoenix and Knight [96] have found a system which exhibits perfectly regular dynamics of collapses and revivals. The system described by our eMective Hamiltonian in Case 5 is another example which may have this feature. In this section we will assume—without the loss of generality—that the gij ’s are real. Let us begin with SchrIodinger-cat type states. We will assume— just in order to obtain a closed-form expression—that g13 =g14 ; g23 =g24 and <3 =0. This means that the upper levels are perfectly degenerate: not only are their energies (without interactions) equal; the Stark shifts are also the same. Let us rewrite the Hamiltonian containing the eMective coupling between the levels |3 and |4 as Hˆ eM = E11 Sˆ11 + E22 Sˆ22 + ˝(!1 + !2 )(Sˆ33 + Sˆ44 ) + ˝!1 nˆ1 + ˝!2 nˆ2 + Hˆ I ;

(208)

where nˆi = aˆ†i aˆi , i = 1; 2, and Hˆ I is given by 2 2 g13 ˆ 2g13 ˆ ˆ 2g2 g2 N 1 S 11 + 23 Nˆ 2 Sˆ22 + ˝ N 1 + 23 Nˆ 2 (Sˆ33 + Sˆ44 + Sˆ34 + Sˆ43 ) : (209) Hˆ I = −˝ <1 <2 <1 <2 Products of the excitation number operators and Sˆjk ’s can be written in terms of the photon number 2 =
(210)

Let us denote the time of /ight of the atom through the cavity by t0 (we will identify this time with the time of the atom–cavity interaction). After a time t0 + t1 , we perform a measurement to project the wave functions onto one of the states 3| or 4|; because the atomic energies in these two states are the same, we propose to use a Stern–Gerlach type of experiment if the upper levels diMer in their mJ value—atoms in state |3 /y to a diMerent spatial region than those in state |4 and we can then measure the atomic energy in an ionization chamber to project on states |3 or |4. Let us suppose that we have found the atom in state |4. Then the Deld in the cavity will be in the state |6Deld = 12 exp(−i(!1 + !2 )(t0 + t1 )) exp(−i(!1 nˆ1 + !2 nˆ2 )(t0 + t1 )) ×[(c4 − c3 )|$|= + (c4 + c3 )e−2i(1 +2 )t0 |$e−2i1 t0 |=e−2i2 t0 ] :

(211)

If we choose i and t0 appropriately, the expression in the square bracket above can be equal, e.g., to (c4 − c3 )|$|= + (c4 + c3 )| − $| − =, taking a shape characteristic to cavity SchrIodinger-cats. Let us note that, unlike in the usual dispersive interaction of the Jaynes–Cummings atom with a cavity mode, it is not actually necessary to prepare the atom in a superposition of two states. It can be prepared in state |3 or in state |4, or in any superposition of these states (provided that c3 = c4 and c3 = −c4 ), and we can perform projection onto either of states 3| or 4|—in any case, a SchrIodinger-cat state will arise, without any additional (=2 pulses applied to the atom before measurement. On the other hand, if there is no degeneracy in the two upper levels and the interactions are purely dispersive (the eMective Hamiltonian is as in Case 1 of the previous section), we can prepare the atom in a superposition of, say, the two upper levels before it enters the cavity and then apply an additional pulse after it leaves the cavity. The subsequent measurement of atomic energy will produce two-mode SchrIodinger cats.

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Let us now consider the population dynamics for the case of the exact degeneracy of levels |3 and |4; that is, the assumptions about gij are the same as above in this section. Let the system be initially prepared in the state |6(0) = |4|$|=. Then by straightforward calculation we Dnd that Sˆ44 (t) = 12 [1 + Re(exp[2i(1 + 2 )t + |$|2 (e2i1 t − 1) + |=|2 (e2i2 t − 1)])] :

(212)

Thus the population dynamics in our system can be expressed by a closed-form formula. The overlap between neighboring revivals does not appear as a consequence of the fact that in our eMective Hamiltonian the coupling contains the actual Deld intensities rather than their square roots. Nevertheless the population dynamics are, in general, only quasi-periodic in time, since in the time dependence of Sˆ44 we get all frequencies of the type k1 + l2 , where k; l are arbitrary integers. Contributions of various frequencies are weighted by the products of Bessel and modiDed Bessel functions. Thus, the behavior of Sˆ44 (t) can be both quite regular and fairly bizarre, depending on the relation between the ’s and mean number of photons, as well as on the scale of time of the observation. The above results Dnish our considerations about the four-level two-mode “M” system. Analogous approach can obviously be developed for other interesting conDgurations of levels and modes in Cavity QED. 6. Fields and atoms in cavities with oscillating mirror There has been considerable interest in studies of the behavior of the electromagnetic Deld in a cavity with oscillating mirror due to interesting eMects which may be found in such system; these eMect include: quantum vacuum radiation [98–103], non-classical statistics of these photons [104–108], as well as dynamical modiDcation of the Casimir forces cf., e.g., [109–111]. The problem of the interaction dynamics of an atom and radiation in a cavity with oscillating mirrors has certain peculiarities. First, it is to be noted that the problem is analytically very diLcult, almost intractable, if the cavity walls are not assumed to be perfectly conducting, that is, perfectly re/ecting. But then it turns out that very fundamental troubles appear since, in general, the quantum Deld theory of systems with boundary conditions depending on time is by no means well understood [112]. More speciDcally, in perfectly conducting cavities with oscillating mirrors the probability is not conserved, in other words, the SchrIodinger picture does not exist. This has been proved by showing the non-existence of the time-evolution operator [98], the very object in which we are most interested in here. The situation can be improved in several ways. On the one hand, one can still try to get rid of the assumption of inDnitely conducting walls and try to use a relatively simple but less idealized model of the wall [109,110]. Another solution consists in including the oscillating wall as a part of the dynamical system [113]; a Hamiltonian which includes the position and momentum of the wall is well deDned. Finally, motivated by the present-day experimental possibilities, one can consider only adiabatic (slow) motion of the walls [114,100,30]; for such motion one can Dnd an approximate eMective Hamiltonian of the Deld in the cavity, and the atom–Deld interaction Hamiltonian is always well-deDned. It is precisely the third approach we shall examine in this section. In the following subsection we shall consider the electromagnetic Deld in a one-dimensional cavity with one oscillating mirror to get an eMective free-Deld Hamiltonian. Then we shall couple the Deld

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with a two-level atom to obtain an interesting eMect of stabilization of the excited-state population. In the whole discussion of Section 6 we shall consider any cavity losses to be negligibly small. 6.1. Electromagnetic @eld in the oscillating cavity We shall consider here idealized one-dimensional cavities only. In such cavities, the electromagnetic Deld is speciDed completely if the electromagnetic vector potential A is known. Due to the one-dimensional character of the problem, the polarizations of the Deld completely decouple and the vectorial character of A is irrelevant. For the Deld of TE polarization, one can prove [98] that the electric Deld should vanish on the walls (even if they are moving), and the same is true for the component of A perpendicular to the walls. Let this component be called u. It depends on time and one spatial coordinate, say, z. Let us also assume that one mirror is Dxed at z = 0, while the other one oscillates periodically so that it is located at z = q(t) = L + Af(?t) = Lg(?t) where L is the size of the cavity at rest. The variable A is the amplitude of the oscillations, f is a periodic function of time, and g(?t) is deDned as 1+(A=L)f(?t). Let us introduce the dimensionless variables =(c=L)t, x = z=q(t), and J = (L=c)?. Then, from the wave equation satisDed by u: 1 92 u 92 u = 2 (213) c2 9t 2 9x we obtain 2 2 g˙ 2 9u 1 92 u 92 u gI 9u g˙ 92 u 2 g˙ 9 u + 2x = ; (214) − x − 2x + x 92 g2 9x g 9x g 9x9 g2 9x2 g2 9x2 where g˙ = dg=d. The boundary conditions to be fulDlled by u in terms of the new variables are very simple: u has to vanish for x = 0 and 1. We look for the solution for u in terms of the series ∞ 2 u= An () sin(n(x) (215) L n=1 so that the boundary conditions are fulDlled automatically. Substitution of (215) into (214) gives g˙ 2 gI g˙ AI m + 2 2 wmn An − wmn An − 2 wmn A˙ n g n g n g n −

g˙ 2 1 vmn An + 2 (m()2 Am = 0 ; g2 n g

where

wmn = 2n(

and vmn = 2(n()

1

x sin(m(x) cos(n(x) d x

0

2

(216)

0

1

x2 sin(m(x) sin(n(x) d x :

We shall make the following calculations somewhat easier by assuming that g is a purely sinusoidal function: A (217) g() = 1 + sin(J) : L

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We shall now introduce the following adiabatic assumptions: (i) the amplitude of the oscillations of the mirror is much smaller than the size of the cavity, AL; (ii) the frequency of the oscillations of the moving mirror is much smaller than the fundamental frequency of the cavity, !1 = (c=L, that is, ?!1 ; (ii) the two ratios just deDned are of the same order, that is, if we deDne the small parameter j as j = A=L, then J can be written as J = j, with , = O(1). Eq. (216) can be conveniently rewritten including only terms up to the order j2 : wmn A˙ n + (m()2 [1 − 2j sin(j,) + 3j2 sin2 (j,)]Am = 0 : (218) AI m − 2j2 , cos(j,) n

We now apply the method of multiple scales to obtain the approximate solution to (218). The amplitudes Am are written as (1) 2 (2) Am = A(0) m + jAm + j Am + · · · ;

and we introduce the following “time” variables: T0 = ;

T1 = j;

T2 = jD(T1 ) ;

so that 9 9 9 d = +j + j2 D (T1 ) + ··· d 9T0 9T1 9T2 and 92 92 92 = + 2j + j2 92 9T02 9T0 9T1

92 92 + D (T ) 1 9T0 9T2 9T12

;

where the superscript “prime” standing at D denotes the derivative of the function D over its argument T1 . In the zeroth-order of the perturbation expansion we obtain 2 9 2 + (m() A(0) (219) m =0 9T02 with the solution −im(T0 im(T0 + a? : A(0) m = am (T1 ; T2 ; : : :)e m (T1 ; T2 ; : : :)e

(220)

In the Drst order we get 2 9 9am −im(T0 9a? m im(T0 2 (1) A + (m() = 2 im( e − im( e m 9T1 9T1 9T02 im(T0 ): + 2(m()2 sin (,T1 )(am e−im(T0 + a? me

(221)

To avoid secular terms we must require that the following conditions are fulDlled: 9am = im( sin(,T1 )am ; 9T1 9a? m = −im( sin(,T1 )a? m : 9T1

(222)

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Their solutions are given by m( [cos(,T1 ) − 1] ; am = $(T2 ) exp −i , m( ? [cos(,T a? = $(T ) exp i ) − 1] : 2 1 m ,

(223)

We may stop at this point, considering $’s as constants independent of T2 and writing the solution to Eq. (221) A(1) m = 0, which can be done since the inhomogeneities have been absorbed in the amplitudes am and a? m . In this case we realize that the combined solutions (220) and (223) can be obtained if we write equations for the amplitudes am , a? m as Hamilton’s equations, where the Hamiltonian is given by Hˆ eM = !m (1 − j sin(?t))a? where !m = m(c=L ; (224) m am m

with the canonical Poisson brackets: {am ; ia? n } = mn : Then we can try to perform a quite naive quantization just by replacing the Poisson brackets by the corresponding commutators and raise the complex amplitudes am , a? m to the status of destruction and creation operators while the form of the Hamiltonian remains the same (Eq. (224)). We can also continue the multiple-scales procedure to obtain Am in the second order. We write −im(T0 im(T0 A(1) + b? m = bm (T1 ; T2 )e m (T1 ; T2 )e

and obtain the following equation for the second-order correction A(2) m : 2 9 92 2 (2) A + (m() = −2 + 2(m()2 sin(,T1 ) A(1) m m 2 9T 9T 9T0 0 1 2 9A(0) 9 92 n (0) A − + 2D (T ) + 2, cos(,T ) w − 3(m()2 sin2 (,T1 )A(0) 1 1 mn m m : 9T0 9T2 9T 9T12 0 n

(225)

(226)

To avoid any terms proportional to powers of T0 in the solution, the following relations must hold: 9bm = im( sin(,T1 )bm ; 9T1 9b? m = −im( sin(,T1 )b? m ; 9T1 ? so that the dependence of bm , b? m on T1 is the same as that of am , am . In addition, we must require that the relation

iD (T1 )

9$m 1 = [i, cos(,T1 )$m + 2i, cos(,T1 )wmm $m + 2m( sin2 (,T1 )$m ] 9T2 2

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holds (together with its complex conjugate). This leads to the solution for D: i m( m( T1 − sin(2,T1 ) ; D(T1 ) = sin(,T1 )(1 + 2wmm ) + 2 2 4, and for $m , $m = $Wm e−iT2 = $Wm e−ijD(T1 ) . It is to be noted that the solution obtained in the second order does not allow us to write down an eMective Hamiltonian of the form Hˆ eM = !m (t)$m? $m : m

If, however, j is suLciently small, we can restrict ourselves to the Drst-order approximation and use the quantum version of Hamiltonian (224). This is what we shall do in our description of the atom–Deld interaction. Realistic values of j will be discussed at the end of this subsection together with conditions of experimental veriDcation of our results. 6.2. Atom–@eld interactions We shall use here the two-level model of an atom and the approximate Deld Hamiltonian given by Eq. (224), except that we shall disregard coupling of the atom to any modes except of the “lower” ones (coupling with the other modes produces very fast-varying terms). Thus, we start with the energy operator Hˆ = ˝!1 (1 − j sin?t)aˆ† aˆ + ˝!22 .ˆ22 + ˝G(.ˆ21 aˆ + aˆ† .ˆ12 ) ;

(227)

aˆ† ,

aˆ and neglected anti-resonant terms. In the above equation where we have omitted the index 1 at G denotes the atom–Deld coupling constant. To proceed, let us Drst notice that the excitation-number operator Nˆ = .ˆ22 + aˆ† aˆ is an operator constant of motion despite the modulation. We may simplify the form of the Hamiltonian by writing the total time-evolution operator Uˆ as the product Uˆ = Wˆ Uˆ 1 , where j!1 Wˆ = exp −i!1 Nˆ t − i (228) [cos(?t) − 1]aˆ† aˆ : ? The operator Uˆ 1 satisDes the SchrIodinger equation i where

d ˆ U 1 = Hˆ 1 (t)Uˆ 1 ; dt

Hˆ 1 (t) = \Sˆ22 + G

(229) Nˆ (Sˆ21 e−ir(t) + Sˆ12 eir(t) ) ;

and we have introduced the following notation: < = ! − !1 ; Sˆii = .ˆii ;

for i = 1; 2 ;

Sˆ12 = Nˆ −1=2 aˆ† .ˆ12 ; Sˆ21 = Nˆ −1=2 .ˆ21 aˆ ;

(230)

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and r(t) =

j!1 [cos(?t) − 1] : ?

The excitation-number operator commutes with all operators Sˆij , and the latter form a realization of the same algebra (both in the sense of Lie algebra and associative algebra) as the atomic operators .ˆij . Using once more expansion (96) we get

√ j!1 ˆ ij!1 =? ˆ −ij!1 =? Nˆ d Uˆ 1 = G N √ J0 S 21 e + S 12 e i dt ? N ∞ √ j!1 Nˆ ˆ i[j!1 =?−k((=2)] ˆ −i[j!1 =?−k((=2)] cos k?t Uˆ 1 ; + 2G N √ (S 21 e + S 12 e ) Jk ? N k=1

(231)

where N = Nˆ and we have assumed that < = 0, that is, under the absence of the oscillations of the wall the cavity Deld is in exact resonance with the atom. It is now important to realize that the arguments of the Bessel functions are actually O(1) if A=L and ?=!1 are of the same order, which is what we assume. But this means that we can quite easily Dnd such ? and A that J0 (j!1 =?) will actually be equal to zero. Using again the method of multiple scales with the new small parameter = GN=? we Dnd the approximate solution for Uˆ 1 in the Drst order:

√ ∞ N 1 ˆ G Uˆ 1 = 1 − 2i (S 21 )ei[j!1 =?−k((=2)] ? k k=1 j! 1 −i [ j ! =? − k((=2)] 1 sin(k?t) Vˆ 1 ; + Sˆ12 e Jk (232) ? where

j!1 j!1 i(j!1 =?) ˆ −i(j!1 =?) ˆ ˆ ˆ G N t − i(e G Nˆ t : S 21 + e S 12 ) sin J0 V 1 = cos J0 ? ?

(233)

From Eqs. (232) and (233) it follows that the Rabi oscillations in the cavity will be strongly inhibited if the zeroth Bessel function vanishes since in this case Vˆ 1 ≈ 1 and the population inversion Sˆ22 would oscillate only due to very small-amplitude and high-frequency terms containing higher Bessel functions in (232). Let us notice that for microwave cavities we have !1 ∼ 1010 Hz, L ∼ 1 cm, and G ∼ 104 –105 Hz. This means that if we make one wall of the cavity from a piezoelement, all assumptions made during our MMS analysis will be fulDlled if the piezoelement oscillates with the amplitude A ∼ 1 m and frequency ? ∼ 106 Hz, and the initial number of photons in the cavity is of the order of 1. To the author’s knowledge, preparation of such cavities and piezoelements is entirely within the possibilities of the current experimental technique. Very similar suppression of the Rabi oscillations have been predicted for a cavity partially Dlled with the dielectric with a time-varying dielectric constants. The quantitative aspect of this prediction also heavily relies on the

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401

application of MMS [31]. The above remarks Dnish our examples of the application of MMS in the Cavity QED. 7. Multiple scales and cold boson gases In this section we shall only touch the extremely fashionable topics of the physics of Bose– Einstein condensates (cf., e.g., [115–118]). The scope of this review as well as (lack of) the author’s competence do not allow for even a very short presentation of this somewhat cross-disciplinary branch of quantum optics and statistical mechanics. We shall only consider the interesting behavior of a system of cold bosons conDned to a double-well potential in the two-mode approximation without and with external modulation of the well. Such a system has been considered in the context of Bose–Einstein condensates e.g. in [119–123]. Our strategy in this section is as follows: after the formulation of the problem in terms of the many-body Hamiltonian, we shall make the two-mode approximation and consider the semi-classical problem. We shall then Dnd approximate solutions of the latter problem using MMS. In our semi-classical treatment we shall take into account losses due to the coupling with the remaining modes as well as due to the in/uence of the environment. We utilize here the simplest way to introduce losses, just by adding the damping constants to the semi-classical and mean-Deld equations. That such an approach is suLcient in the context of BEC is by no means evident, and we employ it for the sake of simplicity. A more reDned approach has been outlined in [124] and applied in [125]. Let us start with writing down the Hamiltonian for the system of many bosons in the conDning potential U (r; t): 2 ˝ V 0 3 † † † † ˆ ˆ ˆˆ ; Hˆ = d r ∇ ˆ · ∇ ˆ + ˆ U (r; t) ˆ + (234) 2m 2 where ˆ and ˆ † are the boson Deld operators, m is the mass of a single atom, V0 =4(˝2 a=m measures the strength of the two-body interactions (assumed to be modeled by the Dirac-delta-function potential in the Drst-quantization picture), and a is the s-wave scattering length. For instance, the trap potential can be taken to be a symmetric double well in the x direction and harmonic in the y, z directions [119,123]: U (r) = b(x2 − q02 )2 + 12 m!t2 (y2 + z 2 ) :

(235)

In the above equation !t is the trap frequency in the y–z plane and b measures the strength of the conDnement in the x direction. The linearized motion in each well is harmonic with the frequency !0 = q0 (8b=m)1=2 , and we shall take !0 = !t following [123]. The energy eigenstates of the double well may be approximated as symmetric and antisymmetric combinations of the localized states with energy eigenvalues given in the Drst-order perturbation theory by E± = E0 ± ˝?=2, where E0 is the ground-state energy of the almost harmonic potential near the bottom of each well, while the energy-level splitting ? is given by 3 q2 !0 20 exp[ − q02 (2r02 )] ; 8 r0 where r0 = ˝=2m!0 . ?=

(236)

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The two-mode approximation consists in the following almost trivial mode decomposition of the Deld operators: ˆ (r) =

2

cˆn un (r) ;

n=1

ˆ † (r) =

2

cˆ†n un? (r) ;

(237)

n=1

where un (r) = u0 (r − rn ), n = 1; 2, and u0 (r) is the normalized single-particle ground state of the harmonic potential near each of the two stable minima of the double-well potential, located at rn . The operators cˆn , cˆ†n are the annihilation and creation operators in the local modes in the nth well. In this two-mode approximation the Hamiltonian takes the form ˝? † (cˆ cˆ1 + cˆ†1 cˆ2 ) + ˝R[cˆ†12 cˆ21 + cˆ†22 cˆ22 ] ; Hˆ 1 = ˝!0 (cˆ†1 cˆ1 + cˆ†2 cˆ2 ) + 2 2 where 2(a˝ R= m

(238)

d 3 r|u0 (r)|4

is a measure of strength of the interparticle interaction. The creation and annihilation operators satisfy the obvious commutation relations [cˆ1 ; cˆ†1 ] = [cˆ2 ; cˆ†2 ] = 1 ; and all other commutators vanish. From Hamiltonian (238) we derive the following semi-classical equations of motion supported by the damping: i

db1 ? = b2 + 2Rn1 b1 − 1 b1 ; dt 2

i

db2 ? = b1 + 2Rn2 b2 − 2 b2 ; dt 2

(239)

plus corresponding equations for b? i , where we have replaced the operators cˆi with complex variables bi exp(−i!0 t), and where ni = b? b i i . The parameters 1 , 2 are damping constants which in principle need not be equal. These semi-classical counterparts of the Heisenberg equations contain the damping terms due to the coupling to other modes as well as small coupling to the environment. In addition, let us now rescale the amplitudes: √ = → N= ; √ =? → N =?

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(where N is the total number of particles) to obtain the equations for which the name “mean-Deld” is more appropriate than “semi-classical”: d=1 ? i = =2 + 2RN |=1 |2 =1 − 1 =1 ; dt 2 d=2 ? = =1 + 2RN |=2 |2 =2 − 2 =2 : dt 2 Let us now deDne the following new variables: 1 Sx = (|=2 |2 − |=1 |2 ) ; 2 i

(240)

−i ? (b b2 − c:c:) ; 2 1 1 b2 + c:c:) : (241) Sz = (b? 2 1 The closed system of equations for Sx , Sy , Sz can be obtained only in the case of 1 = 2 , since we have dSx 1 1 = −?Sy − (1 + 2 )Sx + (2 − 1 )(|=1 |2 + |=2 |2 ) ; dt 2 2 Sy =

dSy = ?Sx − 4RNSx Sy − (1 + 2 )Sy ; dt dSz = 4RNSx Sy − (1 + 2 )Sz : (242) dt Let us concentrate on this case, and let = 1 = 2 . The sign of ? is irrelevant and we shall change it, ? → −?. Let us also introduce a new time variable = 2t. Finally, since we have to do with the semi-classical theory, we may perform an arbitrary scaling of the dependent variables. Let us choose it in the following way: Sx = Rx ; 2RN Sy =

2 Ry ; RN?

2 Rz : RN? With these substitutions, the mean-Deld equations become simply dRx = Ry − Rx ; d Sz =

dRy = #Rx − Ry − Rx Rz ; d dRz = R x Ry − R z ; d

(243)

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where # = −?2 =(42 ). This is evidently the system of the Lorenz equations (confer e.g. with [126,127]) with the “Prandtl number” . = 1, the “aspect ratio” equal to 1, but with the negative “Rayleigh’s number”. It is somewhat surprising that the only parameter in the above equations that is not simply a constant does not contain the particle number N , and depends exclusively on the ratio ?=. Naturally, this is true only if = 0 and 1 =2 . Unfortunately, despite the interesting relation to the Lorenz model which is so rich in spectacular properties, our system with negative # is in fact very trivial. Indeed, the origin (Rx = Ry = Rz = 0) is the only Dxed point of the system, it is asymptotically stable, and it actually is a strong attractor. The system usually reaches the origin quite fast for not very large values of −#. Physically, it means that the system performs rather trivial, though non-linear, oscillations, moving from one well to the other with the exponentially decaying amplitude, and no bifurcations, let alone more complicated eMects, can ever appear. And if we allow for 1 = 2 , the situation does not improve: the dynamics of the variable S0 = (1=2)(|=1 |2 + |=2 |2 ) do not depend on the other variables of the system, and, as a result, the origin is again the only Dxed point. In the paper [121] it has been shown that system (240) without damping has two types of stationary solutions (except the zero solution): one with populations equally distributed in both wells, and the other with an unbalanced distribution of atoms over the wells. By repeating the procedure of [121] we realize that even the smallest damping present in the system completely destroys both types of non-zero stationary solutions, radically trivializing the mean Deld dynamics; this corresponds to the above Dndings about the negative “Rayleigh’s number” in the associated Lorenz system. So far our considerations have rather had negative character. Let us, however, following Holthaus and Stenholm [121,122], introduce the modulation of the double well with the frequency !, so that instead of (240) we have d=1 ? i = =2 + 2RN |=1 |2 =1 + (, sin(!t) − )=1 ; dt 2 d=2 ? = =1 + 2RN |=2 |2 =2 − (, sin(!t) + )=2 ; (244) dt 2 where we have again assumed for simplicity that 1 = 2 = . As a matter of fact, the two-mode approximation is meaningful only for small condensates with N ∼ 103 at most (cf. [119,121] for more precise estimations). We shall now assume that the number of condensed atoms is just about 102 and we will try to perform perturbation expansion assuming that 2RN ? and i ? for i = 1; 2. Our goal is to check to what extent modulation and losses can aMect the tunneling. To make our computations easier, we introduce the following complex variables: i

S+ = 12 (Sz + iSy );

S− = 12 (Sz − iSy ) ;

(245)

to obtain the following system of equations for the dynamics of S− , S+ , Sx in the “time” = ?t: 2 i 4iRN , Sx S− + 2i sin(jJ) − S− ; S˙− = − Sx + i 2 ? ? ? 2 i 4iRN , Sx S+ − 2i sin(jJ) − S+ ; S˙+ = Sx − i 2 ? ? ? 2 Sx ; S˙x = i(S+ − S− ) − ?

(246)

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405

where the dot over the symbol denotes diMerentiation over and jJ = !=?—we have thus assumed that the modulation frequency is much smaller than ?. One might ask what happens if ! and ? are of the same order, in particular, what happens when they are equal. Unfortunately, in this case MMS leads to only very slight simpliDcation, and equations we have to solve in the Drst order are only a little bit easier than the system we have started with. Below, we obtain the Drst-order approximate solutions under the above assumption about !. In addition, we require that 4RN=?=j1, 2,=?=jp, p = O(1), and 2=? = jE, E = O(1). We form the vector S having the components (S− ; S+ ; Sx ), and write S = S0 + jS1 + · · ·

(247)

n

as well as Tn = j , and we observe that the zeroth-order problem has already been solved in the section about the bichromatic excitation of the two-level atom, since in the lowest order we have 9S0 = Mˆ 0 S0 ; (248) 9T0 where  i  0 0 −  2    ˆ i  (249) M0 =  : 0 0  2  i

−i

0

The solution is given by S0 =

3

Aj (T1 ; : : :)vj ej T0 ;

(250)

j=1

where vj , j = 1; 2; 3, are the eigenvectors of the matrix Mˆ 0 to the eigenvalues j , and 1 = i, 2 = −i, 3 = 0. Thus, our problem is again singular, since the zeroth-order matrix is non-invertible. Nevertheless, MMS allows to obtain an approximate solution in this non-linear problem as well. Since all the Si , i = x; y; z must be real, we have A2 = A? 1. In the Drst order we have 9S1 9S0 + = Mˆ 0 S1 + N(S) ; (251) 9T0 9T1 where the vector N contains non-linear terms, modulation and losses:   iSx0 S−0 + ip sin(JT1 )S+0 − ES−0    (252) N=  −iSx0 S+0 − ip sin(JT1 )S−0 − ES+0  : −ESx0 As in the case of the analysis of bichromatic excitation of an atom, we look for the Drst-order solution in the form S1 =

3 j=1

Bj (T0 ; T1 ; : : :)vj ej T0 ;

(253)

406

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to obtain, after somewhat boring though simple calculations: −i 9B1 9A1 + = √ (A1 A3 + A2 A3 e−2iT0 + pA3 sin(JT1 )e−iT0 ) − EA1 : 9T0 9T1 2

(254)

The equation for B2 and A2 is obtained by taking complex conjugate of (253), while for B3 and A3 we have i 9B3 9A3 + = √ (−A21 e2iT0 + A22 e−2iT0 + p sin(JT1 )(A1 eiT0 − A2 e−iT0 )) − EA3 : 9T0 9T1 2

(255)

From the above equations it follows that, to avoid singular terms in the Drst order, we must require that the following conditions determining Aj be fulDlled: i 9A1 = − √ A1 A3 − EA1 ; 9T1 2 9A2 i = √ A2 A3 − EA2 ; 9T1 2 9A3 = −EA3 : 9T1

(256)

Let us observe that the modulation does not change the phases of the zeroth-order solutions, the latter given by i −ET1 ; A1 = $1 exp −ET1 + √ $3 e E 2 i −ET1 A2 = $2 exp −ET1 − √ $3 e ; E 2 A3 = $3 e−ET1 :

(257)

Since we restrict ourselves to the Drst-order approximation, the variables $i , i = 1; 2; 3, which in principle depend on all higher times Tn , n ¿ 2, have been set constant. Integrating over T0 we immediately Dnd three Bj : 1 1 B1 (T0 ; T1 ) = C1 + √ A2 (T1 )A3 (T1 )e−2iT0 + √ pA3 (T1 ) sin(JT1 )e−iT0 ; 2 2 2 1 1 B2 (T0 ; T1 ) = C2 + √ A1 (T1 )A3 (T1 )e2iT0 + √ pA3 (T1 ) sin(JT1 )eiT0 ; 2 2 2 1 p B3 (T0 ; T1 ) = C3 − √ (A21 e2iT0 + A22 e−2iT0 ) + √ sin(JT1 )(A1 eiT0 + A2 e−iT0 ) ; 2 2 2

(258) (259) (260)

where Cj , j = 1; 2; 3 are constants of integration which no more depend on T1 ; T2 ; : : : and are to be found from the initial conditions. We can see √ that in the zeroth order the dynamics of Sz = S+ + S− is almost trivial, for in this order Sz ∼ 2A3 so the dominant feature of the evolution of Sz in time are damped oscillations with the frequency 1. On the other hand, the zeroth-order dynamics of Sy and Sx contain damped

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407

modulation of the frequency, with the initial modulation depth depending on the initial conditions for Sz . The latter fact is obviously a consequence of the non-linearity of the system. 8. Conclusions In this paper we have reviewed several examples of quantum-optical applications of a peculiar method of changing the variables called the method of multiple scales, with the restriction that only time scales are multiple. Evidently, there exist systems described by partial diMerential equations in which we have to introduce many spatial variables as well; the foremost example is the system of Maxwell–Bloch equations (cf., e.g., [128]) for which the application of MMS leads to the famous reduced system integrable by the Inverse Scattering Transform. The multiple-scales analyses oMered in this review have usually been dealing with only few degrees of freedom. For non-linear systems with many degrees of freedom, MMS on its own does not, in general, oMer very much help. For instance, in the case of spontaneous emission, the application of the method could be made sound only via the introduction of the concept of pseudomodes, thus eMectively reducing the number of degrees of freedom to three. Usually, we cannot expect that MMS provides us with substantial simpliDcation of a problem with many degrees of freedom without additional insight speciDc to this problem. This is particularly true about systems with uncountably many resonant and quasi-resonant modes. It is to be noted, however, that the analysis of partial diMerential equations can also involve MMS if we may expect, on physical grounds, that one or very few fundamental frequencies are distinguished. This is illustrated, for instance, in a book by Nayfeh [3] where the non-linear relativistic SchrIodinger equation serves as an example. On the other hand, one can also apply MMS in some systems where there no natural small parameter can be found. For instance, in the case of parametric ampliDers and lasers one can study the dynamics of the systems close to the Hopf bifurcations, where the small parameter appears due to the expansion of a physical control parameter near its critical value at which the bifurcation takes place [22,23]. It is the author’s hope that the examples provided above constitute a convincing argument for further application of MMS in the domain of quantum optics, non-relativistic quantum electrodynamics, and quantum statistical mechanics, sometimes just conDrming our intuition in a semi-quantitative way, and sometimes developing it. Acknowledgements On various stages of development of the manuscript, the author has discussed the method of multiple scales with Professors Luiz Davidovich and Stig Stenholm. It is a great pleasure and honor to thank them for their time and advice. References [1] J. Kevorkian, J.D. Cole, Perturbation Methods in Applied Mathematics, Springer, New York, 1981. [2] J. Kevorkian, J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, New York, 1996.

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Physics Reports 375 (2003) 411 – 443 www.elsevier.com/locate/physrep

A general parametric model for the gain of gas avalanche counters with particular attention to non-cylindrical geometries J.E. Bateman∗ Instrumentation Department, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, UK Accepted 1 November 2002 editor: J.V. Allaby

Abstract A simple, general parametric model of the avalanche process in gas counters is described. Applicable to all the common forms of gas avalanche detector—wire, microstrip, point anode and parallel gap, the model describes the gain process in terms of two pseudo-physical constants which are e.ectively invariant over the working range of any given detector con0guration. For counter operation over a wide range of conditions (e.g. very di.erent gas pressures) the model is extended so that four parameters are required to model the gain. Applications of the model to the characterisation, operation and design of a variety of counter types are given. c 2003 Elsevier Science B.V. All rights reserved. PACS: 07.85.Fv Keywords: Gas avalanche counter; Proportional counter; Gain model

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The gain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The single wire proportional counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The GMSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. The point anode detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The parallel gap detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Discussion of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The extended model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The ballistic fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗

Tel.: +44-1235-821-900; fax: +44-1235-445-753. E-mail address: [email protected] (J.E. Bateman).

c 2003 Elsevier Science B.V. All rights reserved. 0370-1573/03/$ - see front matter doi:10.1016/S0370-1573(02)00599-9

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6. Applications of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Gain as a function of anode dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Counter spark stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Design criteria for micromegas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. Gain stabilisation of Cowing-gas counters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Operating gas counters away from ambient conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Gain variation with gas composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

428 429 429 432 434 436 440 441 442 442

1. Introduction Since their invention in the early twentieth century gas counters have proved extremely adaptable as detectors of ionising radiation, not only on account of the ease with which the detecting medium may be changed (e.g. 3 He for slow neutrons or xenon for hard X-rays), but also due to the geometric Cexibility of the sensitive volume and the avalanche region. Thus planar (parallel gap) [1], linear (wire) [2] or point (needle) [3] detectors have all been successfully used in a wide range of applications, some with imaging capability and some without. Recent technological advances associated with the electronics industry such as micro-lithography and precision printed circuit manufacture have revolutionised one’s ability to fabricate the small electrode structures required in many designs. The three geometries as implemented by this new technology are exhibited in the compteur a trous (CAT) [4], the gas microstrip detector (GMSD) [5] and the microdot detector (MDD) [6]. In the absence of a general model of the avalanche process, design and operation of the devices is largely empirical. There is an extensive literature with models for the gain of the standard single wire proportional counter [7–16], which however, in general exhibit little practical utility for the design and operation of gas counters. This report elaborates a simple model of the avalanche gain process which is applicable to all detector geometries and which allows the behaviour of the gas gain to be explored as a function of the geometry and the thermodynamic gas variables pressure (P) and temperature (T ). This provides useful assistance in the design, characterisation and operation of the new families of gas avalanche detectors. 2. The gain model In an electron avalanche process the controlling parameter is the speci0c ionisation (), the number of ion pairs being generated per unit length at any given point in the high 0eld region in which the ampli0cation is occurring. In gases is generally known as the Townsend CoeJcient. The total multiplication or gas gain (M ) achieved by an electron in the transit from cathode to anode is given by: rc ln M = − dr ; (1) ra

where the integral is performed along the electron path from the cathode to the anode.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

413

An electron drifting in a high electric 0eld can acquire enough energy in the free path between atomic interactions to (at lower 0eld values) excite the gas atoms and (as the 0eld increases) ionise them, so contributing to the speci0c ionisation . The excitation processes absorb electron kinetic energy and so keep low (although in binary gas mixtures excited atoms can generate ionisation by secondary interactions such as Penning processes and photo-ionisation by Cuorescent photons). The simplest model for neglects these e.ects and considers only the collision ionisation process in which an energetic electron directly ionises a gas atom. This is characterised by the parameters— W the threshold energy for ionisation (the outer-most atomic electron binding energy) and the cross-section for the ionisation process. Any electron of the population (which is assumed to have a Maxwellian energy distribution) with a kinetic energy above this value is assumed to cause an ionisation event. This leads to the following expression for [17]: 1 −W = exp ; (2) E where is the electron mean free path and E is the local electric 0eld strength. If is the e.ective cross-section for the ionisation process we can write =

1 ; NL

(3)

where NL is Loschmidt’s number, the number of atoms (molecules) per unit volume. In turn the gas laws yield the relation NL =

NA P ; RT

(4)

where NA is Avogadro’s number and R is the gas constant. Since the ambient variables P and T always appear as a ratio it is convenient to de0ne q=

P : T

Combining Eqs. (2)–(5) yields WNA NA = exp − q R R(E=q)

(5)

(6)

in which we see that =q is a simple function of E=q, including both the electric 0eld and the ambient dependencies. E=q is known as the reduced electric 4eld. Comparing Eq. (6) with Eq. (2) shows that the quantity q = R=NA and Eq. (6) can be written as −W 1 exp ; (6a) = q q E=q q where the parameters W and q represent invariant physical parameters of the gas 0lling. In Eq. (6) the electric 0eld is in general a function of position, i.e. E = E(r). Substituting the appropriate function of E(r) for any given geometry of gas detector and integrating Eq. (1) with the functional dependence of given in Eq. (6) gives the equation describing the gas gain as a function of Vac (the anode–cathode potential di.erence) and the ambient variable q.

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2.1. The single wire proportional counter The single wire proportional counter is generally taken as the canonical gas avalanche detector and it is the subject of most modelling studies. It consists of a 0ne wire of radius ra stretched down the axis of a cylindrical tube of radius rc which is 0lled with the active gas. A potential di.erence Vac is applied between the electrodes with the wire positive with respect to the cathode. The electric 0eld is radially symmetric and is described by E(r) =

Vac ; ln(rc =ra )r

(7)

where r is the radial distance from the counter axis. Substituting this expression for E and integrating Eq. (1) from the cathode (rc ) to the anode surface (ra ) gives the gain formula. The rapid dependence of on E(r) and the steep decline of E with r makes the integral insensitive to the cathode limit which is set to in0nity in order to simplify the 0nal expression Vac AB ; (8) ln(M ) = exp − A Vac where A = W ln(rc =ra ) (V )

(9)

and B=

ra NA q ra = : R

(10)

In this (the simplest) version of the model the gain is entirely determined by the two parameters A and B with the gas thermodymamic dependence contained in B. Values of A and B obtained by 0tting an experimental gain curve give directly the basic physical parameters W and by inversion of Eqs. (9) and (10). The gain of a single wire proportional counter consisting of a 20 m diameter gold-plated tungsten wire mounted axially in a brass tube of 20 mm inner diameter and 0lled with a Cowing argon +7:5% methane gas mixture was studied. Fig. 1 shows the measured data for the gain (plotted as ln M ) as a function of the anode–cathode bias potential (Vac ) at constant q. Here it can be seen that Eq. (8) provides an excellent 0t over a wide range of gain (70 ¡ M ¡ 17000) with 0tted values: A=75:367 V and B=13:788. Using the counter geometry and Eqs. (9) and (10) we 0nd W =10:9 eV and = 0:725 m. These are plausible values for an argon-based gas mixture at ambient conditions. Eq. (10) predicts that B should be a linear function of q. Fig. 2 shows a plot of ln M as a function of q at Vac = 1320 V, taken over the range of q values accessible in a Cow counter due to changes in the ambient pressure and temperature (940 mb ¡ P ¡ 1030 mb; 18C ¡ T ¡ 24◦ C). The data 0ts satisfactorily to a straight line function for the parameter AB=Vac . However, the presence of a signi0cant o.set in the 0tting function shows that the relationship predicted by Eq. (10) is not ful0lled exactly, and the 0t is simply the local tangent to a curve which is gently concave down. This is an indication of the limitations of the model which will be discussed in detail below.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

415

ln(M) (M = Avalanche Gain)

11

9

7

5

MSGD (Ps=500µm, As=5µm, Cs=250µm, Ar + 25%iC H ) 4 10 y = x/A*exp(-A*B/x), A=56.166, B=3.8048,r2=0.99981 Single Wire Counter (ra=10µm, rc =10mm, Ar+7.5%CH ) 4 y = x/A*exp(-A*B/x), A=75.367, B=13.788,r2=0.99998

3 300

900

600

1500

1200

Anode Potential Vac (V)

Fig. 1. The gas gain curves (ln M versus Vac ) for a single wire cylindrical proportional counter and a gas microstrip detector (GMSD) with the corresponding 0ts of the simple gain model.

ln(M) @ Vac = 1320.31V

8.2 Data from Wire Counter of Figure 1 y = a*exp(-(b + c*x)), r2 = 0.98904, a = 17.641, b = 0.22262, c = 0.16564

8.1

8.0

7.9 3.25

3.30

3.35

3.40

3.45

3.50

q (=P/T) (mb/K)

Fig. 2. A plot of the logarithm of the gas gain of the wire counter as a function of the gas ambient variable q (=P=T ) over the range experienced by a Cow counter due to ambient changes in P (pressure) and T (absolute temperature).

2.2. The GMSD The GMSD consists of a pattern of interleaved, parallel metallic strips laid down on a semiconducting glass surface [5]. A 0ne anode strip (typically 10 m wide(As )) is Canked by cathode strips which are typically ≈ 100 m wide (Cs ), in a pattern which repeats every few hundred m (i.e. pitch = Ps ). In spite of the planar structure, the basic geometry of the avalanche region is similar to that of the single wire counter so it is not surprising that the gain of the GMSD 0ts well to Eq. (8). Fig. 1 shows such a 0t to experimental data from a GMSD.

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443

Experiment shows that the electric 0eld in the region in which the avalanche occurs near the GMSD anode strip is close to that of a cylindrical counter with ra = As =2 and rc = (Ps − Cs )=2. Thus Eqs. (9) and (10) can be used for A and B. If this is done with the 0t in Fig. 1, W is found to be 12:2 eV and ; 0:657 m, plausible values for operation at ambient conditions and close to those found above for the single wire counter. The GMSD is usually operated as a three terminal device with a drift electrode placed parallel to the glass plate at a suitable (≈ 10 mm) distance to de0ne the active detector volume. As described in detail elsewhere [18], the drift electrode potential (Vd ) contributes to the 0eld in the avalanche region above the anode strip so that the correct bias for equation (8) is V = Vac + Vd , where ≈ 0:01. Eq. (8) generally achieves an excellent 0t to the experimentally measured gain whether Vac or V is used (though, of course the 0tted values of W and di.er slightly). It is noted in Fig. 1 that the bias potential required for GMSD operation is approximately one half that required for the single wire counter. This is, of course, simply a reCection of the di.erence of a factor of ≈ 50 in the anode–cathode gap of the two devices. 2.3. The point anode detector The point anode detector takes many detailed forms [3]; however, the basic element is a quasispherical anode structure near which the avalanches occur. For the purposes of modelling, the electric 0eld near the anode is assumed to be that between two concentric spheres: the anode surface of radius ra and the cathode surface of radius rc . Thus E(r) =

Vac : ((1=ra ) − (1=rc ))r 2

(11)

The rapid dependence of E on r and the similarly rapid dependence of on E (Eq. (2)) mean that the positioning and shape of the cathode generally have a weak e.ect on the gas gain and that the assumption of a spherically symmetric 0eld near the anode is valid. The gas gain is evaluated by substituting Eqs. (11) and (2) in Eq. (1) and integrating. As in the wire counter case (and by the same justi0cation), the integral is simpli0ed by setting rc to in0nity. The result is √ Vac B ln M = ; (12) erfc 2 A Vac where A = W

1 1 − ra rc

(V)

(13)

and Wra2 B=

1 1 − ra rc

(V) :

(14)

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

417

11

10

ln(M)

9

10 x 10 Pin Connector, Ar + 25%iC4H10 y = 0.886*sqrt(x/A)*exp(-0.9062*B/x - 0.8751*sqrt(B/x)), std-err:0.0404 A=0.11246, B=1943.1, r2=0.99948

8

Parallel Gap (1mm) Ar + 7.5%CH4 y = A*exp(-A*B/x), std-err:0.00956 A=233.42, B=23.961, r2=0.99993

7

2-D Pin Detector, Ar + 20%CH4 y = 0.886*sqrt(x/A)*exp(-0.9062*B/x - 0.8751*sqrt(B/x)), std-err:10.7 A=0.085400, B=9068.0, r2=0.99966

6 1000

2000

3000

4000

5000

Anode Potential Vac (V)

Fig. 3. A plot of the gain curves (ln M versus Vac ) for a parallel gap counter, the 10 × 10 pin array [20] and the imaging pin detector [19] with the corresponding simple model 0ts.

Rearranging Eqs. (13) and (14) gives √ AB (eV) ; W= ra ((1=ra ) − (1=rc )) ra B : = A

(15) (16)

erfc(x) is the complementary error function which requires a numerical integration for each evaluation. In order to simplify the 0tting process using Eq. (12), erfc(x) is approximated by the expression erfc(x) ≈ exp(−0:9062x2 − 0:8751x)

(17)

which gives a precision of a few % throughout the range 0 ¡ x ¡ 4, in which the value of erfc(x) is signi0cant. Pin detectors use electronic connector pins to provide the quasi-spherical anodes of the counters. Fig. 3 shows the 0ts of Eq. (12) to two pin detector designs. The right hand data set (squares) shows the gain curve of the imaging pin detector [19] which consists of a 2 mm diameter ball connector mounted in a gas enclosure with the cathode readout structures at a distance of ≈ 30 mm from the

418

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

anode. It is operated in a gas mixture of argon +20% methane. A potential of +Vac is applied to the pin anode with the readout cathodes at earth potential. Applying Eqs. (15) and (16) to the 0t parameters of the imaging pin detector data in Fig. 3 yields W = 28:8 eV and = 3:07 m. The array pin detector [20] is fabricated from an electronic connector array of 100 pins with hemispherical ends which are spaced at 2:54 mm intervals on a grid. The pin tips are brought up close to a cathode plane with holes (aligned on the pins) through which the ionisation released in an active gas space are drawn into the avalanche region above each pin tip. The pins are held at earth potential and −Vac is applied to the cathode plane. The pin diameter is 0:48 mm and the average distance to the cathode plane is ≈ 0:7 mm (rc = 1:02 mm). The gain curve for a pin in this detector is shown as the open circles in Fig. 3 along with the 0t to Eq. (12). Applying Eqs. (15) and (16) to the 0t parameters of the array pin detector data in Fig. 3 yields W = 19:33 eV and = 1:83 m. The explicit dependence of the gain on the ambient variable q can be obtained by using Eqs. (3) and (4) to substitute in Eq. (12). 2.4. The parallel gap detector In the parallel gap detector the avalanche takes place in the high electric 0eld between two accurately parallel planar electrodes, the 0rst of which is a form of mesh through which the X-ray generated ionisation may pass from the active volume of the detector into the high 0eld region. In this case the electric 0eld is uniform and given by E=

Vac ; d

(18)

where d is the separation between the anode and cathode planes. Substitution of this in Eqs. (2) and (1) followed by integration leads to the relation: −AB ; (19) ln M = A exp Vac where d ;

(20)

B=W :

(21)

A=

The parallel gap detector used to produce the gain curve in Fig. 3 (0lled circles) consisted of a 10 mm deep conversion gap separated from a 1 mm deep avalanche gap by a stainless steel mesh plane. The mesh was used as the readout electrode (at earth potential) and +Vac was applied to the rear foil electrode. Applying Eqs. (20) and (21) to the 0t parameters of the parallel gap detector data in Fig. 3 yields W = 23:96 eV and = 4:28 m.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

419

3. Discussion of the model While the model clearly is capable of accurately 0tting the gain curves of all the gas counters studied, its limitations are clearly seen in the systematic variation seen in the values of the two parameters W and , which should be invariant for a given gas. (Admittedly a variety of argon-based mixtures are used above, but the systematic variation clearly bears little relation to the changes in the gas mixture.) What is dramatically di.erent in the various detectors is the value of the electric 0eld near the anode (where the avalanche occurs). Table 1 shows and W tabulated against the anode 0eld for all the devices tested. A clear picture emerges with smoothly increasing from 0.657 (GMSD) to 4:28 m (parallel gap) as the reduced anode electric 0eld decreases from 140 × 103 VK=(cm-mb) to 4:52 × 103 VK=(cm-mb). W shows a more erratic tendency to increase from ≈ 11 to ≈ 25 eV over the same range. This behaviour is clari0ed by comparison of the functional dependence for given by Eq. (6a) with the experimental data for available for argon [21]. (Fig. 4.) In this preliminary analysis the model 0ts are applied directly to the experimental data, as will generally be the case in practical application. As will be discussed in detail in Section 5 below, the electronic shaping times of the pulse ampli0er used in each case (and acting on the variable charge delivery pro0le of the counter) modify the situation somewhat (especially when avalanching in a wide range of electric 0elds is considered). The model is quite tolerant of these e.ects and, while the precise values of W and change slightly, the general trend with electric 0eld strength is not a.ected. (This is a useful property of the model since the exact pulse shape properties are not usually known.) The e.ect of varying charge delivery pro0les is seen in the somewhat erratic values of the W parameter observed in Table 1 when compared with the more systematic values observed in Table 3 after the pulse shape corrections of Section 5 are applied. Fig. 4 shows that in the range of reduced 0eld typical of wire and GMSD detectors (¿ 5 × 104 VK=(cm-mb)) =q is well approximated by Eq. (6a) with W = 15:26 eV and q = 3:094 × 10−4 , i.e. = 3:094=3:41 = 0:91 m, values similar to those observed with these detectors. In the region of reduced 0eld typical of the pin and the parallel gap detectors (103 –104 VK/(cm-mb)) the 0t to the Table 1 The values of W and obtained from the simple model 0ts for all the detector types (Figs. 1 and 3) shown with the corresponding anode reduced electric 0eld

Reduced anode electric 0eld Xa =103 (VK/(cm-mb)) Mean free path (m) Ionisation threshold W (eV)

Gas microstrip detector

Cylindrical wire detector

Pin array detector

Imaging pin detector

140.0

56.4

26.0

10.9

0.657 12.2

0.725 10.9

1.83 19.33

3.07 28.8

Parallel gap detector 4.52

4.28 23.96

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 Reduced Townsend Coefficient α/q (Ion Pairs K / (cm-mb))

420

10000 1000

Argon Data from: S.C.Brown, Basic data of Plasma Physics, MIT Press, Cambridge Mass,1959

100

α/q = 1/(λq)exp[-W/(λq)/(E/q)]

10 Argon Data y=1/3.094e-4*exp(-15.26/3.094e-4/x) y=1/0.001599*exp(-27.83/.001599/x)

1 0.1 103

104

105

Reduced Electric Field X(=E/q) (V K / (cm-mb))

Fig. 4. A comparison of the reduced (0rst) Townsend coeJcient of argon with the simple model with two sets of 0xed values of W and q corresponding to the typical reduced electric 0eld X (=E=q) in the wire counter and the parallel gap counter.

experimental data for =q gives W =27:83 eV and =0:001599=3:41=4:69 m. (Note: q=3:41mb=K at 20◦ C and 1000 mb). Again these values tend to match to those obtained with the pin and parallel gap detectors (Table 1). These results inspire the search for a suitable functional form for W and q as a function of the reduced 0eld (X =E=q) so that the changes in the avalanche mechanisms with X can be satisfactorily captured by the simple model for . Fig. 5 shows the argon data modelled with the parameterisations: W = a + b=X ;

(22)

q = c + d=X :

(23)

The resulting 0t is excellent and the resulting functions W (X ) and q(X ) are shown in Fig. 6. Fig. 7 shows the same data plotted as W (X ) and (X ) (at ambient conditions: q = 3:41 mb=K). Bearing in mind that practical detectors always operate with variable fractions of various quenchers in the argon (which modify the situation a little), the message of Fig. 7 is that for most types of gas detectors which avalanche in reduced 0eld values ¿ 104 VK=(cm-mb), the variation in W and is quite restricted. Given that the total avalanche process in a wire counter (for instance) occurs over at most a factor of three in X , the assumption of constant W and is not unreasonable and the success of the model is understandable. In certain detectors it is possible to force the 0eld con0guration beyond that describable by the simple model. Fig. 8 shows the gain curve of a GMSD design capable of giving a very wide range of gain (As = 5 m; Cs = 250 m; Ps = 500 m) [22] with the simple model not succeeding in 0tting the data over the whole range. The parameterisation of W and derived above indicates

Reduced Townsend Coefficient α/q (ion pairs K / (cm-mb))

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

421

Data from : S.C.Brown, Basic data of Plasma Physics, MIT Press, Cambridge Mass,1959

1000

10 Argon Data y=1/(2.9444e-4+3.5835/x)*exp(-(15.00+2.5537e4/x)/x/(2.9444e-4+3.5835/x))

0.1

0.001 103

104

105

Reduced Electric Field X (=E/q)

V K / (cm-mb)

Fig. 5. A 0t of the argon Townsend coeJcient over the whole range of X with the parameterisations of W (X ) and q(X ) used in the extended gain model (Eqs. (22), (23)).

40

λq cm-mb / K

0.003 λq = 2.9444e-4+3.5835/x W = 15+2.5537e4/x 30

0.002

0.001 20 0 103

104 Reduced Electricfield X (=E/q)

Ionisation Threshold Energy W (eV)

0.004

105 V K / (cm-mb)

Fig. 6. A plot of the 0ts of W (X ) and q(X ) to the argon data.

an obvious generalisation of the basic gain formulae in Eqs. (8)–(10). We now permit the gain constants A and B to be functions of Vac of the form: A = a + b=Vac and B = 1=(c + d=Vac ). As Fig. 8 shows, this simple generalisation produces an excellent 0t over the whole operating range of the detector.

422

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

40

8 λ at P=1000mb, T= 293.15K W = 15+2.557e4/x

6

30 4

2

20

Ionisation Threshold Energy W (eV)

Effective Electron Mean Free Path λ (µm)

10

0 103

104

105

Reduced Electric Field X (=E/q)

V K / (cm-mb)

Fig. 7. A plot of the 0ts of W (X ) and (X ) to the argon data in ambient conditions.

Avalanche Gain

105 104 103 GMSD Data: Ps=500µm, Cs=250µm, As=5µm, Argon + 25%iC4H10 [22] y = exp(x/(a+b/x)*exp(-(a+b/x)/x/(c+d/x))), Std-err = 3.46% a=28.779,b=35.919, c=-20.390, d=527.72 y = exp(x/48.172*exp(-48.172*4.2354/x)), Std-err = 6.64%

102 101 300

400

500

600

700

800

Anode Potential Vac (V)

Fig. 8. The gain versus bias (Vac ) data for a GMSD operating over a very wide gain range [22] with a simple model 0t (dashed line) and a simple parameterisation of A and B in terms of Vac suggested by the W (X ); (X ) functions.

4. The extended model The success of the parameterisations for W and q (speci0ed in Eqs. (22) and (23)) in representing the argon experimental data for =q as a function of X (Fig. 5) points the way to an extended model for the avalanche gain process. The exact solution would involve substituting relations (22) and (23) in Eq. (6a) which would be integrated according to Eq. (1). Such an integration could only be performed numerically and thus is of little use as a general tool. A simpler approximation to this process is indicated by the success of the 0t to the GMSD gain data in Fig. 8 of the parameterisation of W and q in terms of Vac .

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 106

423

Simple Model: W=10.33eV λq=2.21e-4mbcm/K

105

Extended Model: W = 9.99+1.927e4q/E(eV)

Gas Gain

λq = 1.32e-4 + 4.985q/E(mbcm/K)

104

Cylindrical PC - Argon + 7.5%Methane Vac = 1320V rc = 1cm, q=3.41mb/K

103 Simple model: y=exp(1320/10.33/ln(1/x)*exp(-10.33*ln(1/x)*x*3.41/2.214e-4/1320))

2

Extended Model: y = exp(1320/(9.99+1.927e4*3.41*x*ln(1/x)/1320)/ln(1/x) *exp(-(9.99+1.927e4*3.41*x*ln(1/x)/1320)*ln(1/x)*x*3.41 /1320/(1.32e-4+4.985*3.41*x*ln(1/x)/1320)))

101 0.00001

0.0001

0.001

0.01

Anode (Wire) Radius (cm)

Fig. 9. A comparison of the gain versus anode radius plots derived from the simple model and the extended model for a wire counter at constant Vac .

Noting that the avalanche takes place close to the anode and that the range of reduced 0eld (X ) in the avalanche region is not too great, we accept the gain equation calculated on the basis of constant W and q and parameterise them in the gain formula with the functions de0ned in Eqs. (22) and (23) with the X value set to that at the anode of the detector (Xa ). Thus for the single wire proportional counter we can write Xa =

Vac : ra ln(rc =ra )q

From Eqs. (8)–(10) we can write Vac W ln(rc =ra )ra q ln(M ) = : exp − W ln(rc =ra ) qVac

(24)

(25)

Substituting relations (22) and (23) with X = Xa we get Vac {a + (bqra ln(rc =ra )=Vac }ln(rc =ra )ra q ln(M ) = : (26) exp − {a + (bqra ln(rc =ra )=Vac }ln(rc =ra) {c + (dqra ln(rc =ra )=Vac }Vac This equation gives the explicit dependence of the gas gain on all the relevant variables in terms of four parameters: a; b; c; d which (in turn) de0ne the behaviour of W and q over a wide range of reduced 0eld. While this formulation is still an approximation, it is reasonable to expect that it will be useful over a wider range of operation than the simple model. The parameters a; b; c; d can be obtained by 0tting them to a plot of ln M versus q using Eq. (26). Using the data of Fig. 2 for this purpose yields: W = 9:99 + 1:927 × 104 =X (eV) and q = 1:32 × 10−4 + 4:985=X (mbcm=K). Fig. 9 compares the predictions of the extended model (Eq. (26)) and the simple model (Eqs. (8)–(10) with constant W = 10:33 eV and q = 2:21 × 10−4 mbcm=K) for the gas gain of

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443

a single wire counter as a function of the wire radius (Vac = 1320 V). Both models produce a similar curve with good agreement in the region of high X (small anode radii) in which the calibration was made. However, for larger anode radii the two predictions diverge quite rapidly. This result simply con0rms the expectation that the 0xed W and q of the simple model can only give accurate results when the range of X is restricted to the calibration region, though it gives the general behaviour over a wider range. The parameterisations of W and q in terms of the reduced anode 0eld can be generated for the point anode and parallel gap counters in the same way as done above for the single wire cylindrical counter. Substitution in the gain formulae of the simple model then leads to the equivalent formulae to Eq. (26) for these cases. 5. The ballistic fraction Gas avalanche counters (virtually without exception) are operated in pulse mode. In order to achieve this, some form of pulse shaping is applied to the charge waveform delivered by the detector; typically resistor–capacitor di.erentiation/integration (CR-RC). In this case the counter waveform is e.ectively sampled at a time (=RC) after the avalanche generated by the incident particle (e.g. X-ray). The formation of the charge waveform is governed by the movement (in the electric 0eld of the detector) of the electrons and ions released in the avalanche. In devices which may be viewed as simple two-terminal capacitors, the charge induced on the electrodes may be deduced simply by considering the energy withdrawn by the drifting charges from the external power supply. The situation can be further simpli0ed by the fact that the avalanche usually occurs very close to the anode so that the electrons have a very short distance to travel and they also have velocities which are (on average) hundreds of times greater than those of the ions. The problem thus reduces to consideration of the movement of the avalanche ion cloud from the anode to the cathode. This analysis is given in detail in Ref. [23]. The whole of the charge signal liberated in the avalanche is not available to the detecting circuits until the ions reach the cathode. This period is called the positive ion transit time (I ) and can range in value from s to ms depending on the counter geometry. For practical purposes the ampli0er shaping time is kept to ≈ 1 s or less and this generally means that only a fraction of the true avalanche gain is recorded. This is known as the ballistic fraction. For the simpler counter geometries the energy analysis permits the evaluation of the charge waveform from a fast event (such as an X-ray) Q(t) to be calculated. (The waveform generated by a charged particle (which can leave an extended trail of ionisation in the detector) is spread out by the drift of the primary electrons to the avalanche zone and is slower and more complex.) The results for three simplest cases are as follows: Parallel gap: Q(t) t = ; (27) Q I where I =

d2 ; !Vac

(28)

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

425

where Q is the total charge in the avalanche event, d is the width of the avalanche gap, ! the positive ion mobility and Vac the potential di.erence across the gap. Cylindrical wire: Q(t) 1 t = ln 1 + ; (29) Q 2 ln(rc =ra ) R where R =

ra2 ln(rc =ra ) 2!Vac

(30)

and R is the parameter which determines the fast part of the waveform. Expression (29) attains unity at the ion collection time I =

ra2 ln(rc =ra ){rc2 − ra2 } : 2!Vac

(31)

In general rc ra , it follows that r 2 I R ∼ = a2 : rc Spherical electrodes: 1 Q(t) = Q (1 − ra =rc )

(32)

1 1− (1 + t=R )1=3

;

(33)

where R =

ra3 (1=ra − 1=rc ) 3!Vac

(34)

I =

(rc3 − ra3 )(1=ra − 1=rc ) : 3!Vac

(35)

and

If rc ra , then r 3 I R ∼ = a3 : rc

(36)

These formulae can be used to estimate the charge waveforms of X-ray pulses in the di.erent detectors (of which the gain curves are presented in Figs. 1 and 3) with varying degrees of precision as Fig. 10 shows. Within the limitation of the assumption of the ion mobility of argon ions for all the di.erent ions present (≈ 1:7 cm2 =(Vs)), the representation of the cylindrical wire counter, the parallel gap counter and the imaging pin counter are essentially accurate on the time scale presented in Fig. 10. (Vac is chosen at a typical operating point of each detector.) In the pin array detector the ions are collected by both the cathode and the drift electrode. However, since a low drift 0eld is generally used, most go to the cathode and the curve in Fig. 10 will be reliable in the 0rst microsecond or so which concerns us.

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443 1.0

Q(t) / Q0

0.8 0.6 GMSD - Bellazini and Spezziga Calculation [24]

0.4

Pin Array 1/3 y = 1/(1-.24/1.02)*(1 - 1/(1+x/0.0664) ) Cylindrical Single Wire counter y = 1/2/6.91*ln(1+x/1.45e-3)

0.2

Imaging Pin Detector 1/3 y = 1.034*(1- 1/(1+x/0.412) ) Parallel Gap y = x/3.92

0 0

1

2

3

4

5

Time (µs)

Fig. 10. A comparison of the charge delivery waveforms Q(t)=Q0 of the various types of gas avalanche detectors, each at their typical operating conditions.

In the GMSD, drift 0elds are generally high with a large proportion (¿ 50%) of the ions travelling to the drift electrode. The structure of the 0eld pattern near the anodes approximates to that of a cylindrical detector of cathode radius (Ps − Cs )=2 (see Section 2.2 above) with a typical I of a few hundred ns, and that in the drift section to a parallel gap with a typical I of a few hundred s. Fortunately the electrode pattern on the GMSD plate (with much more cathode area than anode area) functions as a Frisch grid and ensures that the very slow component of Q(t) is very small indeed (a few percent). The complex 0eld structure of the GMSD means that a full numerical solution (using Greens theorem) is required to calculate Q(t) and no simple formulae are possible. The numerical solution of Bellazzini and Spezziga [24] for a typical GMSD structure is shown in Fig. 10. It shows that there is a sharp rise to about 95% in about 200 ns followed by a very slowly rising plateau which is estimated to terminate at 100 s. In fact the front edge can be 0tted quite accurately with the form of Eq. (29), but only up to t ≈ 200 ns. Table 2 summarises the numerical values of R and I for all the detectors. Here we see an important characteristic of gas avalanche counters, namely the very large ratio which generally exists between the fast part of the pulse and the 0nal ion collection time. The build up of positive ion space charge in the long drift is one of the limiting rate factors on the counter as it eventually distorts the counter electric 0eld and depresses the gain. With ampli0er shaping time constants () of 1 s, the ballistic fraction (FB ) for each counter can be read o. each plot as the value of Q(t)=Q at t = 1 s. In Fig. 10 FB varies from 25% for the parallel gap counter to 96% for the GMSD, showing what a signi0cant e.ect that it has on the gain calibration. Further, R is a linear function of Vac so that FB can change signi0cantly over the operational bias range. In order to produce corrected gain curves, the data of Figs. 1 and 3 were divided by the function Q(1)=Q as a function of Vac (of the corresponding counter) and the new gain curves 0tted as described in Section 2 above to generate new values of W and . Table 3 shows the resulting values which demonstrate a systematic increase in both parameters for all detectors, the smaller FB , the larger the change. Only the GMSD which has FB close to unity shows negligible

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

427

Table 2 The charge pulse waveform parameters R and I for the various types of gas detectors (at their appropriate operating biases and ambient gas conditions) Gas microstrip detector

Cylindrical wire detector

Pin array detector

Charge pulse fast component (R )

0:103 ns

1:45 ns

66:4 ns

Positive ion collection time (I )

100 sa

1:45 ms

2:94 msa

Imaging pin detector 0:412 s 11:12 ms

Parallel gap detector — 3:92 s

a Indicates the ion transit time to the drift electrode; otherwise it is to the nearest cathode. The small proportion of positive ions which penetrate the grid of the parallel gap counter (≈ 5%) will experience a transit time of ≈ 5 ms.

Table 3 The values of W and obtained from the simple model 0ts for all the detector types (Figs. 1 and 3) after correction had been made to the gain data for the ballistic fraction of each detector

Reduced anode electric 0eld Xa =103 (VK/(cm-mb)) Mean free path (m) Ionisation threshold W (eV)

Gas microstrip detector

Cylindrical wire detector

Pin array detector

Imaging pin detector

140.0

56.4

26.0

10.9

0.672 12.22

1.06

2.06

12.06

20.45

3.07 32.5

Parallel gap detector 4.52

4.28 30.9

Also shown is the corresponding anode reduced electric 0eld.

change. (Because no formulae are available for Q(t)=Q in this case and the correction is very small, a constant value of FB was used throughout the range of Vac .) Fig. 11 shows these corrected values of W and plotted against the anode reduced 0eld for all the detectors. The values 0t reasonably well to the expected function (Eq. (23), but the W values fall into two groups depending on the quencher concentration. Thus the two points taken with 7.5% methane can be seen to be consistent with a di.erent curve (Eq. (22)) to that for the curves with 20 –25% of quencher. The increase of W with the quencher concentration is a reCection of the large inelastic scattering cross-section of the quencher compared with argon. The dependence of R on the ion mobility (!) makes FB sensitive to changes in the pressure (P) and (absolute) temperature (T ) of the gas. Elementary kinetic theory shows that this dependence is !˙

T 1=2 : P

(37)

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 10

40 W (eV) y = a + b/x, a=10.419, b=92.576 y = a + b/x, a=10.815, b=238.27 λ (µm) y = a + b/x,std-err:0.557 a=0.78171, b=27.906

8

30 6

4 20 2

0

10 1

10

100

Xa /1000 (Anode reduced field - VK/cm-mb)

Effective Ionisation Threshold W (eV)

Effective Electron Mean Free path λ (µm)

428

Fig. 11. Here the values of W and derived from the data of Figs. 1 and 3 (after they have been corrected for the ballistic fraction) are plotted against the reduced anode electric 0eld in each detector. W is seen to be more sensitive to the gas composition than so that the points measured in argon +7:5% methane clearly lie on a di.erent curve (dashed) from those with 20 –25% quencher.

If the ampli0er time constant is below the knees of the Q(t) curves (Fig. 10) (or I in the case of the parallel gap) then the apparent gain of a gas counter will decrease with increasing gas pressure rather faster than indicated by the basic gain model. Fig. 10 also shows the Cexibility of point and line anode designs, in that there is always a relatively fast (sub s) rise to the charge pulse over a wide range of anode dimensions so that a reasonable fraction of the charge signal is available (≈ 50%). With parallel gap counters the gap dimension is critical and must be sub-millimetre if microsecond pulses are to be used at ambient pressure.

6. Applications of the model It is clear from the above discussion that the pulse gain available from any gas avalanche detector is accurately described by the simple models developed in Section 2 only if the ballistic fraction FB remains roughly constant and the reduced anode electric 0eld Xa does not vary by a large factor over the range of study. The 0rst condition can be met by ensuring that the ampli0er shaping time constant is long enough that the peak sampling time is above the knee of the Q(t)=Q0 curve throughout the study range. If the range of Xa is large (decades) then the extended model must be used and the parameters a; b; c; d evaluated for accurate modelling. However, as Fig. 9 shows, the simple model is capable of representing the general behaviour of the gain over 3 decades of Xa with the main caveat being that it systematically underestimates it at the lowest reduced 0eld values (largest radii in Fig. 9).

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

429

Wire Radius (mm) 0

0.002

0.004 0.006 0.008 0.010

Avalanche Gain

12000

Pin Detector (sperical anode), Argon+20% Methane, Cathode Radius (rc) = ∞, Vac =1125V Cylindrical wire Counter, Argon+7.5%Methane, Cathode Radius(rc) =1cm, Vac=1050V

9000

Parallel Gap counter, Argon+7.5%Methane, Gap (d) =1mm, Vac=590V

6000 3000 0 0

0.1

0.2

0.3

0.4

Pin Radius and Parallel Gap Width (mm)

Fig. 12. A comparison of the behaviour of the gain of the wire, pin and parallel gap counters as a function of the anode dimension at constant Vac , as predicted by the simple model in ambient conditions.

6.1. Gain as a function of anode dimension Accepting the limitations noted above, the simple model can be used to compare the gain of the three types of avalanche counters as a function of their anode dimension. Using the 0t parameters W and derived from the experimental data of Figs. 1 and 3, the gain formulae (8), (12) and (19) are used to plot the gain of the wire counter, pin detector and parallel gap as a function of the wire radius, pin radius and gap width—Fig. 12. All the detectors display the same characteristic behaviour with the gain peaking at a wire radius of 0:9 m, a pin radius of 36 m and a gap width of 111 m. The bias potential has been chosen so as to give a gain of ≈ 10 000 in each case. The plots of Fig. 12 imply that there is an optimal design for each type of detector, if for no other reason than the fact that at the peak of the gain curve each detector is maximally insensitive to dimensional irregularities in the gain-determining structure. Since it is easy to generate GMSD anodes of various widths using the lithographic process, the gain of GMSD anodes of width 2,3,4,5 and 10 m was measured. Fig. 13 shows the result with the gain peaking at an anode width of 3 m (radius = 1:5 m) and the gain curve 0tting reasonably well to the approximate wire formula given in Section 2.2. The optimal radius from the GMSD data (1:5 m) is larger than the prediction of Fig. 12 (0:9 m); however, as noted above, precise agreement is not to be expected and the order of magnitude is satisfactory. 6.2. Counter spark stability The high gain of gas counters means that sparking is inevitable if any positive feedback processes arise. Typically these are electron photo-emission, 0eld emission and electron emission from neutralising ions at the cathode. These processes are sensitive to the electric 0eld strength (Ec ) at the cathode and it is generally found that this parameter is a good diagnostic of the maximum gain from a detector (when other things such as the gas mixture are equal). In Fig. 14 the simple model for

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443 2500

Avalanche Gain of GMSD

2000

Argon + 25% Isobutane Pitch = 300µm, Cathodewidth = 90µm Substrate = S8900

1500

1000 Maximum Stable Gain

500

Gain at VC = -600V, VD = -2000V y = exp(600/a/ln(c/x)*exp(-a*ln(c/x)*b*x/600)), r2 = 0.98391, a=15.744,b=1.3766, c=110.87

0 0

4

8

12

Nominal Anode Width (µm)

Fig. 13. Experimental data showing the gain maximum in a GMSD counter as a function of the anode width [18]. The wire counter 0t uses the approximations developed for the GMSD dimensions in Section 2.2.

Cathode Electric Field (Ec) for M=5000 (V/cm)

190 Single Wire PC (ra=0.001cm, rc=1cm), Argon + 7.5%methane at STP 170

150

Ec = Vac / rc ln(rc /ra)

130

110

90 0

2

4

6

8

10

Wire Anode Radius (µm)

Fig. 14. This plot shows the electric 0eld strength at the cathode of a wire counter (rc = 10 mm) as the anode radius varies as predicted by the simple model when the gain is held constant at M = 5000.

the wire counter is used to calculate Vac and thence Ec using the relation Ec =

Vac rc ln(rc =ra )

(38)

Gap Electric Field (Eg) (V/cm) @ M=5000

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

431

2.0x105 M = 5000,Argon+7.5% methane, STP 1.5x105 ⇐ Maximum Gain Gap Width 1.0x105

0.5x105

0 0.005

0.01

0.02

0.05

0.1

Width of Parallel Gap (cm)

Fig. 15. The electric 0eld strength in a parallel gap counter as the gap width varies as predicted by the simple model when the gain is held constant at M = 5000.

in the case that a constant gain of M = 5000 is demanded. (Since formula (8) cannot be inverted simply, this is done numerically.) Fig. 14 shows that the cathode 0eld decreases monotonically with the anode radius and, at the maximum of the gain curve in Fig. 12, Ec is below 100 V=cm. This behaviour illustrates the well-known design criterion that high gain wire counters should have very small anode wire diameters. In the GMSD the e.ective rc decreases to fractions of a millimetre and the stability correspondingly decreases. The di.erent geometry means that Ec is determined as much by the width of the cathode strip as the anode–cathode gap. This behaviour is demonstrated in the results presented in Ref. [22]. However, as the data in Fig. 13 shows, the maximum gain rises steadily as the anode width decreases. As an aside we note that the parameters determining the ballistic fraction R and I (Eqs. (30) and (31)) both vary as ra2 ln(rc =ra ) and so FB also increases as the anode radius decreases. The same argument holds a fortiori for the pin detector since the electric 0eld in the anode-cathode space falls o. as 1=r 2 , making Ec very low. The pin array reduces this stability somewhat by bringing the intermediate cathode to within ≈ 1 mm of the pin tips. The situation is very di.erent in the parallel gap detector in which the full anode 0eld strength required for the avalanche is present at the cathode electrode. Eq. (19) may be simply inverted to get Vac as a function of M and hence the gap 0eld Eg (=Vac =d). Explicitly this is Eg =

1 Wq : q {ln(qd= q) − ln(ln M )}

(39)

Fig. 15 shows a plot of Eg as a function of the gap width d at a constant gain of M = 5000. In this case the gap 0eld increases monotonically as the gap width decreases and at the maximum of the gain curve in Fig. 12, has a value of 6 × 104 V=cm. This graph shows why it is, in general, diJcult

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443

to get stable, high gains from parallel gap detectors. The high cathode 0eld strengths impose very limiting conditions on the quality of the electrodes and the quenching capacity of the gas. 6.3. Design criteria for micromegas Micromegas is a parallel gap detector with an avalanche gap of dimensions less than 1 mm [25]. The simple model is useful for evaluating some of the important operating parameters of the device. Fig. 12 shows that using the parameters derived from the detector with a gap of 1 mm, there is a maximum in the gain versus gap curve at d = 111 m (for a gain of 9000) and it is interesting to look at the properties of the device around the gain maximum. From Eq. (19) we can write explicitly: qd Wqd ln M = exp − ; (40) q qV where the explicit product q is retained as an approximate invariant, and V is the potential di.erence across the avalanche gap. Di.erentiating with respect to d we get 1 Wq 1 9M = ln M − : (41) M 9d d qV The gain maximum in Fig. 12 is now identi0ed by setting 1=M 9M=9d = 0, giving the position of the maximum: dmax =

qV ; Wq

(42)

i.e. the gap required for the gain maximum varies linearly with the operating potential. Thus in order to sit on the maximum, a unique potential and thus gain is determined. It also follows from Eq. (42) that the electric 0eld in the gap is the same whenever a gap is operated on the peak: i.e. Eg =Wq= q. Using the experimental parameters from the 1 mm gap this evaluates to Eg = 5:33 × 104 V=cm. By eliminating V from Eq. (41) by substituting from Eq. (40) we can quantify the gain stability as a function of the gap width at a given operating gain qd 1 9M ln M 1 + ln(ln M ) − ln : (43) = M 9d d q This function is plotted in Fig. 16 with M = 5000; q = 3:41 mb=K and q = 1:529 × 10−3 cmmb=K. As expected, the di.erential is zero at the gain maximum. However, the very asymmetric behaviour either side of the peak means that if a variable gain is desired from the detector it is clearly preferable to choose the gap so that the operating region lies (e.ectively) on the upper side of the gain maximum on this plot. In the gap range 100 m to 1 mm the relative di.erential is moderately uniform with a maximum of 300 cm−1 . This translates into a gain variation of 3% per m of change in d. Since X-ray pulse height spectra in gas counters generally have a spread of ≈ 10% FWHM or greater, this means that Cuctuations of a few m are tolerable in the gap so operation exactly on the gain maximum is not necessary if this tolerance can be achieved.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

433

1000

1/M ∂M/ ∂d (cm-1)

500

0

-500

-1000 0.001

M = 5000,Argon+7.5%methane,STP

0.01

0.1

Parallel Gap Width (d) (cm)

Fig. 16. The relative sensitivity of the gain of a parallel gap counter operated at constant gain (M = 5000) to changes in the gap width (d) as a function of the gap width—as predicted by the simple model.

It is often desirable to operate gas counters with Cowing gas. In this case the gain becomes sensitive to the ambient variable q. In order to quantify this we di.erentiate Eq. (40) with respect to q: 1 9M ln M Wqd = 1− : (44) M 9q q qV It is immediately obvious that substituting dmax from Eq. (42) into Eq. (44) makes the di.erential zero, so that by operating at the gain maximum, one can obtain complete immunity from q shifts. Eliminating V from Eq. (44) in favour of ln M gives the behaviour of this di.erential as a function of the gap width: 1 9M ln M qd = 1 + ln(ln M ) − ln : (45) M 9q q q Fig. 17 shows the plot of this function with the gain 0xed at M = 5000 and the same experimental parameters as before. In this case the obvious operating point is as close to the gain maximum as possible. Allowing for a reasonable operating range in gain and for the sensitivity of the gain maximum to q, it should be possible to keep the relative di.erential to the order of 1 K=mb which is the level of stability achieved by a wire counter. This gives a gain shift of ≈ 0:3% per K and 0:1% per mb. The maximum of the gain curve in Fig. 12 represents a triple point characterised by the variable values (Vmax ; dmax ; Mmax ) which are all uniquely determined by the gas and ambient variables (W; q; q). Combining Eqs. (40) and (42) gives the complete speci0cation Vmax = exp(1)W ln(Mmax ) ;

(46)

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443

M = 5000, Argon+7.5% methane 1/M ∂M/∂q ( K/mb)

5

0

-5

0.001

0.01

0.1

Parallel Gap Width (d) (cm)

Fig. 17. The relative sensitivity of the gain of a parallel gap counter operated at constant gain (M = 5000) to changes in the gas ambient variable (q) as a function of the gap width—as predicted by the simple model.

and dmax = exp(1)

q ln(Mmax ) : q

(47)

Since the practical design situation requires a speci0ed gas gain, the gain is used as the independent parameter. If Eq. (47) is inverted, it is obvious that Mmax is an exponential function of dmax . Thus reducing the gap width rapidly reduces the gain available at the desirable triple point operating conditions. 6.4. Gain stabilisation of
(48)

where q is again retained as an approximate invariant. Eq. (48) shows that M is a function of Vac and q so that an excursion in M is given by dM =

9M 9M dq : dVac + 9Vac 9q

(49)

Since the term on the left of Eq. (48) is ln M , it is more appropriate to write (49) as dM 1 9M 1 9M dVac + = dq : M M 9Vac M 9q

(50)

The servo condition for using Vac to stabilise the changes in q is dM =0 ; M

(51)

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

435

or 1 9M 1 9M dq dVac = − M 9Vac M 9q

(52)

giving us the di.erential equation (1=M )(9M=9q) dVac =− : dq (1=M )(9M=9Vac )

(53)

For convenience we write S =−

(1=M ) (9M=9q) ; (1=M ) (9M=9Vac )

(54)

and so integrating Eq. (53) we get the servo relation Vac = V0 + S(q − q0 ) ;

(55)

where V0 and q0 are operating bias and ambient conditions to which we wish to servo the gain. This is a particularly simple (linear) correction function provided the function S is e.ectively constant over the range of Vac and q met with in practice. The following analysis shows that this is in fact the case and experiment con0rms it also. Di.erentiating Eq. (48) with respect to q gives W ln(rc =ra )ra 1 9M =− ln M M 9q qVac and with respect to Vac gives 1 W ln(rc =ra )ra q 1 9M = ln M + M 9Vac Vac qVac2

(56)

(57)

Substituting relations (56) and (57) into (54) yields S=

W ln(rc =ra )ra : q(1 + (W ln(rc =ra )ra q= qVac ))

(58)

In order to evaluate S an actual case must be considered. In the Wide Angle X-ray Scattering (WAXS) detector [26] operating with 17% dimethylether (DME) in argon at a gain of M = 1080; Vac = 500 V and 0tting to Eq. (48) gives (W ln(rc =ra )ra q)= q = 174 V. Since the second term in the denominator of Eq. (58) is ¡ 1 (0:348), the e.ect of the small changes in q and Vac involved in the process are attenuated and, further, the sense of change of Vac in the servo process follows that of q so minimising any change in S throughout the parameter range required by the servo process. Thus, the simple linear relation of Eq. (55) can be used to perform the stabilisation of the gain against the e.ect of changes in ambient conditions, using only a single constant S, which can be evaluated by the following simple calibration:

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J.E. Bateman / Physics Reports 375 (2003) 411 – 443

Vc = 490V (Unstabilised) y = a*exp(-b*x),std-err:5.1 a=9916.6, b=0.75557

Gas Gain

800

Vc = 490 +40.58(q-3.411)V (Stabilised) y = 754.564

760

720

mean: 754.564 sdev: 4.6385 (0.62%)

3.30

3.35

3.40

3.45

3.50

q (mbar/K)

Fig. 18. The gas gain of a GMSD detector with Cowing gas measured over a period of 3 weeks, plotted as a function of q. The squares represent the gain at constant Vac and the circles the gain when Vac is adjusted in accord with the calibrated servo relation (55).

In order to evaluate the key servo parameter S it is necessary to evaluate the two parameters (1=M )(9M=9q) and (1=M )(9M=9Vac ) near the desired operating point q0 . The simplest way to do this is to note that if a 0t of the form M = a exp(bx) is made to any gain function M (x) then (1=M )(9M=9x) is simply the parameter b in the exponential 0t. In view of the approximate constancy of S, it is not necessary to be excessively precise about the reference values of the second variable in each data set. It is also clear that in measuring b, the units of the ordinate are immaterial and there is no need to calibrate the gain. However, when working directly in PHA channels it is important to ensure that any digital o.set has been measured and removed from the peak channel data. The b parameters of the 0ts obtained from the WAXS detector data [27] allow us to evaluate S = 0:767=0:0189 = 40:58 VK=mB and a servo function of Vc = 490 + 40:58(q − 3:411)V ;

(59)

where q0 is set arbitrarily to P = 1000 mb and T = 20◦ C. Fig. 18 shows how the counter gain (circles) is controlled against changes in the ambient variable (q) when this condition is applied. The RMS error on the controlled mean gas gain of 754.56 is 4.64 giving a fractional error of 0.62%. This is achieved over a range of q corresponding to 986 ¡ P ¡ 1017 mbar and 16:4 ¡ T ¡ 26:4◦ C. The unstabilised gain (squares) at Vc = 490 V is also shown for comparison. 6.5. Operating gas counters away from ambient conditions One of the outstanding advantages of gas counters is their ability to function with a wide range of gas 0llings at ambient, hyperbaric (above ambient) or hypobaric (below ambient) pressures. Hyperbaric operation is often used to enhance the detection eJciency for high energy X-rays and

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

437

9

ln(Gain )

8

7

p = 1.2, 2, 3, 4, 5, 6 bar 6

5 400

eqn: y = x/a*exp(-b/x), R:0.0494, max dev:0.04450 a=89.22, b=295.9 eqn: y = x/a*exp(-b/x), R:0.0321, max dev:0.03065 a=73.76, b=259.3 eqn: y = x/a*exp(-b/x), R:0.0397, max dev:0.02920 a=66.78, b=121.7 eqn: y = x/a*exp(-b/x), R:0.0773, max dev:0.06715 a=80.20, b=291.1 eqn: y = x/a*exp(-b/x), R:0.0205, max dev:0.02449 a=69.51, b=197.1 eqn: y = x/a*exp(-b/x), R:0.0641, max dev:0.05456 a=63.44, b=87.53

550

700

850

1000

1150

Anode Voltage Vac (V)

Fig. 19. The gain data curves from a GMSD operated with a xenon +20% CO2 mixture at a series of hyperbaric gas pressures. (From Ref. [28].) Simple model 0ts are also shown.

neutrons, while hypobaric operation is often used for the detection of ultra-violet light. Potential uses of the gain model are the analysis of gain data over a wide range of q and prediction of the gain outside the calibration range when required. Suitable data is available for analysis from the work of Shekhtman and his collaborators [28] who tested a GMSD with a hyperbaric xenon 0lling. Fig. 19 shows the avalanche gain curves presented in Ref. [28] for a GMSD with a pattern pitch (Ps ) of 200 m, an anode width (As ) of (3 m) and a cathode width of 60 m, operated with a 0lling of xenon +20%CO2 at pressures of 1.2, 2, 3, 4, 5, 6 bars. The analysis uses the simple version of the model to provide smooth 0ts to the sparse data (Fig. 19) so that interpolated gain values can be obtained at a 0xed Vac (630.5V) to provide a plot of ln M versus q over the operating range. This plot is shown in Fig. 20. This data can now be used to 0t ln M to q using Eq. (26) (the extended model) and so obtain the constants a,b,c,d. (The GMSD approximations for ra and rc of Section 2.2 are used.) Before attempting the 0t, Eq. (26) is simpli0ed as follows ( + q=Vac )q Vac exp − ; (60) ln M = ( + q=Vac ) Vac ($ + %q=Vac ) where = a ln(rc =ra ); = bra (ln(rc =ra ))2 ; $ = c=ra ; % = d ln(rc =ra ). For practical purposes, there is no need to unscramble the variables, a; b; c; d unless explicit representations are required for W and q. The 0t of Eq. (60) is shown in Fig. 20. The 0tted values of ; ; $; % can now be used to predict the gain curves as a function of Vac for the di.erent pressures. Fig. 21 shows the predictions superimposed on the experimental data. As Fig. 21 indicates, the agreement of the model with the data is good up to a pressure of 3bar, but consistently overestimates the gain at the higher pressures. One reason for this is the fact that the 0tting process (ln M versus q) was only carried out at a single value of Vac at the low end of

438

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 9 y=630.5/(a+b*x/630.5)*exp(-(a+b*x/630.5)*x/630.5/(c+d*x/630.5)), std-err:44.7,r2=0.99805, a=59.995, b=576.56, c=2.9012,d=40.294

8 Interpolated Data from Figure 19, V =630.5V

ln(M)

7

6

5

4 0

5

10

15

20

25

q (mb/K)

Fig. 20. A plot of ln M versus q at Vac = 630:5 V extracted from the data of Fig. 19 using the simple model 0ts to interpolate. A 0t to the extended model is shown from which its parameters are deduced. x = 630.5V y=x/(59.995+576.56*3.41*6.0/x)*exp(-(59.995+576.56*3.41*6.0/x)*3.41*6.0/x/(2.9012+40.294*3.41*6.0/x)) P = 6bar y=x/(59.995+576.56*3.41*5.0/x)*exp(-(59.995+576.56*3.41*5.0/x)*3.41*5.0/x/(2.9012+40.294*3.41*5.0/x)) P = 5bar y=x/(59.995+576.56*3.41*4.0/x)*exp(-(59.995+576.56*3.41*4.0/x)*3.41*4.0/x/(2.9012+40.294*3.41*4.0/x)) P = 4bar y=x/(59.995+576.56*3.41*3.0/x)*exp(-(59.995+576.56*3.41*3.0/x)*3.41*3.0/x/(2.9012+40.294*3.41*3.0/x)) P = 3bar y=x/(59.995+576.56*3.41*2.0/x)*exp(-(59.995+576.56*3.41*2.0/x)*3.41*2.0/x/(2.9012+40.294*3.41*2.0/x)) P = 2bar y=x/(59.995+576.56*3.41*1.2/x)*exp(-(59.995+576.56*3.41*1.2/x)*3.41*1.2/x/(2.9012+40.294*3.41*1.2/x)) P = 1.2bar

15 q Data fitted only at this bias potential →

ln(M)

10

5

0 400

550

700

850

1000

1150

Anode Potential Vac (V)

Fig. 21. A comparison of the experimental gain data of Fig. 19 with the predictions of the extended model using the parameters obtained from Fig. 20.

the data set so there is every reason to expect the extrapolation to drift away at the other extreme of the set. Ideally the 0t of ln M to q would be carried out by optimising over both Vac and q simultaneously over the full range of both variables.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

106

439

P = 0.333bar

Gas Gain

P = 1bar

104

y =exp(1320/(9.99+1.927e4*0.333*3.41*x*ln(1/x)/1320)/ln(1/x) *exp(-(9.99+1.927e4*0.333*3.41*x*ln(1/x)/1320)*ln(1/x)*x*0.333*3.41 /1320/(1.32e-4+4.985*0.333*3.41*x*ln(1/x)/1320))) y =exp(1320/(9.99+1.927e4*3.41*x*ln(1/x)/1320)/ln(1/x) *exp(-(9.99+1.927e4*3.41*x*ln(1/x)/1320)*ln(1/x)*x*3.41 /1320/(1.32e-4+4.985*3.41*x*ln(1/x)/1320))) y =exp(1320/(9.99+1.927e4*3*3.41*x*ln(1/x)/1320)/ln(1/x) *exp(-(9.99+1.927e4*3*3.41*x*ln(1/x)/1320)*ln(1/x)*x*3*3.41 /1320/(1.32e-4+4.985*3*3.41*x*ln(1/x)/1320))) y =exp(1320/(9.99+1.927e4*10*3.41*x*ln(1/x)/1320)/ln(1/x) *exp(-(9.99+1.927e4*10*3.41*x*ln(1/x)/1320)*ln(1/x)*x*10*3.41 /1320/(1.32e-4+4.985*10*3.41*x*ln(1/x)/1320)))

P = 3bar P = 10bar

102

Cylindrical PC-Argon + 7.5%Methane Vac = 1320V rc=1cm,q(1bar,20C)=3.41mb/K W = 9.99 + 1.927e4q/E (eV) λq = 1.32e-4 + 4.985q/E (mbcm/K)

100 0.00001

0.0001

0.001

0.01

Anode Radius (cm)

Fig. 22. A plot showing the predicted variation of the optimum wire radius in the wire counter as the gas pressure is varied over a large range. The parameters of the extended 0t are derived from a 0t to the data of Fig. 2.

A further consideration is the e.ect of the gas pressure on the ion mobility and hence on the fast component of the charge pulse r (Eq. (30)). The fast component r is proportional to P=Vac and inspecting Fig. 21 we see that at the maximum gain used r increases by a factor of just over three between P = 1:2 bar and P = 6 bar. The ampli0er shaping time constant is not known, but as the GMSD is designed for fast counting, it may be assumed to be short and in this case could lead to a signi0cant decrease in the ballistic fraction FB at the higher gas pressures. This would reproduce the type of disagreement seen in Fig. 21 at the higher pressures. The necessity of the extended model when covering a large range of q is seen if one considers the prediction of Eq. (59) for Vac in the WAXS GMSD for operation at a pressure of 6 bars (remember this relation is only calibrated for use close to ambient pressure). It predicts a rise of a factor of 2.4 in Vac . As we see from Fig. 21, the increase in Vac observed in the xenon GMSD is only a factor of 1.6. This di.erence is caused by the dependence of W and q on the reduced anode electric 0eld (Eqs. (22) and (23)). As seen in Fig. 12, all detector types have an optimum anode dimension at which the gain maximises at a 0xed Vac , and the various operating parameters tend to optimise. The simpler mathematics of the parallel gap case shows that this condition is in fact a triple point at which the three parameters Vmax ; dmax and Mmax are locked together. Eq. (47) shows that dmax is inversely proportional to q. Although similar simple explicit relations cannot be produced for the wire (GMSD) and spherical anode devices, their behaviour is inevitably parallel. Thus, in general we expect that the optimum anode structure will decrease in size as q increases. In Fig. 22 the parameter 0ts to Eq. (26) from the data of Fig. 2 are used to predict the behaviour of the gas gain of a wire counter with rc = 1 cm operated at Vac = 1320 V in argon + 7:5% methane at a range of gas pressures as the anode radius is varied. Typically, the optimum anode radius decreases to sub-micron values as the pressure rises above ambient. Conversely, in hypobaric operation, the anode dimension should be increased. The GMSD is a technology which can realise such narrow anode widths. However, as the gas pressure rises, the destructive potential of sparks increases and the low mass of metal in the track makes it extremely vulnerable to spark damage. Using the same modelling data as used in Fig. 14 to predict the cathode electric 0eld we 0nd that for a 0xed gain, Vac minimises at the gain

440

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

maximum. The stored energy available to the spark is 1=2CVac2 , where C is the total stray capacity of the anode structure. Thus, provided the value of C is kept to a minimum operation at the triple point is favourable. A further protective mechanism is available in the choice of the metal of the anode strip; a resistive metal such as chromium can be used to limit the spark currents. The optimal operating conditions of a parallel gap counter under hyperbaric operation are predicted by the simple model in Eqs. (46) and (47). The simple model predicts a linear decrease in dmax as the pressure (i.e. q) is increased. The e.ect of the increase in q with q (as predicted by the extended model) can be seen to cancel out some of this decrease, so that the e.ect of pressure change is not as extreme as this. However, for typical avalanche gains (M ≈ 5000) the parallel gap will decrease to much less than 100 m for signi0cant over-pressures. As noted in Section 6.3, the gap electric 0eld Eg = Wq= q and so increases approximately linearly with q (W and q vary in the same sense with q—see Fig. 6). Thus values of Eg ¿ 105 V=cm are easily exceeded and the stored energy per unit area of gap (1=2&0 Eg2 dmax ) rises proportionately. Thus, while the grids of the parallel gap structure may be more robust than the anode strips of the GMSD, the immunity to spark damage may not be any better and the stability of the counter against sparking is liable to be considerably poorer in hyperbaric operation. 6.6. Gain variation with gas composition It was noted in Section 3 that W and q are pseudo-physical parameters, and that, in combination they represent the behaviour of the competing inelastic electron scattering processes in the gas. The presence of a quencher component in the counter gas is essential for stable avalanche operation at useful gas gains. However, the quencher, by its very nature, introduces a series of inelastic scattering processes, which result in the gain of the counter being sensitive to the fraction of quencher in the gas mixture. Fig. 23 shows a series of gain curves obtained from a GMSD as the fraction of isobutane quencher in argon is increased from 5% to 100%. Using the simple model, gain 0ts to all the data sets are made and the values of W and q extracted using relations (8)–(10) with the GMSD approximations of Section 2.2. Fig. 24 plots the derived values of W and as a function of the quencher fraction. In Fig. 24 we note that as the quencher fraction tends to zero W tends to ≈ 14 ev which is close to the ionisation potential of argon. However, in pure isobutane, W = 24:5 eV which is many times the ionisation potential of isobutane. This behaviour reCects the large number of dissipative inelastic interactions possible in isobutane (e.g. vibrational and rotational modes). The results shown in Figs. 23 and 24 have obvious practical implications for counter operation: The highest gain data point in each curve represents the highest stable gain obtainable with that gas mixture. Inspection shows that the maximum gain is available at an isobutane fraction of 75%. Since the UV absorption of the quencher is proportional to its density, it follows that for hyperbaric operation the fraction can be reduced in proportion to the pressure (i.e. the partial pressure of the quencher is kept constant). Thus if operation is required at 5 bar, the fraction of isobutane need only be 5% instead of the usual 25%. As Fig. 23 shows, this adjusts W and q in such a way as to achieve a gain of 1000 at a bias potential of 476 V instead of 630 V, reducing the stored energy by almost a factor of two. The reduction in the maximum gain available is usually not a problem as the neutrons or high energy X-rays being detected give large pulse heights.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 4000

Avalanche Gain

3000

y = exp(x/a*exp(-b/x)), std-err:7.47, a=74.385, b=574.99 y = exp(x/a*exp(-b/x)), std-err:3.7, a=66.428, b=522.28 y = exp(x/a*exp(-b/x)), std-err:2.17, a=59.18, b=460.31 y = exp(x/a*exp(-b/x)), std-err:2.34, a=49.888, b= 224.44 y = exp(x/a*exp(-b/x)), std-err:1.67, a=55.473, b=314.35 y = exp(x/a*exp(-b/x)), std-err:3.81, a=43.852, b=215.15

441

75%

100%

2000

Isobutane Fraction in Argon

50% 25%

10% 5%

1000

0 300

450

600

750

900

1050

V' (=Vac+βVd) (V)

24.5

1.2 W (eV) λ (µm)

21.0

0.8

17.5

0.4 GMSD Approximations: rc = (Ps-Cs)/2 = 105µm ra = As /2 = 5µm 0 0

20

40

60

80

Effective Ionisation Threshold W (eV)

Effective Electron Mean Free Path λ (µm)

Fig. 23. The sequence of gain curves obtained with a GMSD as the fraction of isobutane quencher in argon is varied from 5% to 100% [18] with simple model 0ts.

14.0 100

Fraction of IB in Argon (%)

Fig. 24. The values of W and for the GMSD gas, as a function of the quencher fraction as derived from the simple model 0ts to the data of Fig. 23.

7. Conclusions Mathematical models of the avalanche gain process have been developed for all the typical gas counter geometries. Based on the pseudo-physical parameters W (the notional ionisation threshold energy) and q (proportional to the inelastic electron scattering cross-section) the simple models have been shown to work well for typical counter operation around ambient conditions when W

442

J.E. Bateman / Physics Reports 375 (2003) 411 – 443

and q may be assumed to be constant. Comparison of the Townsend coeJcient of argon with the form used in the model shows that W and q can be represented as simple functions of the reduced electric 0eld X , (Eqs. (22), (23)). Since the avalanche takes place in the electric 0eld close to the cathode the corrected values of W and q can simply be represented in terms of the anode potential, Vac ; q and four new constants. Incorporating these formulations of W and q into the simple models leads to the extended models, of which Eq. (26) is an example. The extended models can now be used to characterise the gain performance of the various counter types over the widest range of operating conditions, examples of which have been given. Careful consideration has been given to the e.ect of the 0nite charge collection times (ballistic fraction) on the evaluated parameters (W and q). For reliable use of the gain models it is necessary to ensure that the pulse ampli0er shaping time is long enough to ensure that the sampling time is above the knee of the charge delivery time curve Q(t) throughout the operational range (Fig. 10). Application of the models to the design of gas counters shows that (for 0xed gas conditions) all possess an optimal anode dimension (Fig. 12). This is a triple point with the gain, operating potential and anode dimension locked together. In ambient conditions, the typical optimal anode dimensions are ≈ 1 m radius for a wire (microstrip) detector, ≈ 40 m radius for a point anode detector and ≈ 100 m for a parallel gap detector. The optimal dimension varies inversely with the gas pressure (i.e. q) (Eq. (47)). At the triple point the di.erential sensitivity to dimensional and ambient changes tend to be minimised, and, in the case of wire, strip and point detectors, the electric 0elds minimised. Throughout the analysis the gas ambient variable q(=P=T ) has been used. This ensures that the formulations may be applied equally to sealed counters (when the density, and hence are temperature invariant) and Cow counters when the temperature must also be taken into account. Acknowledgements The equipments and results on which the above model is based were the results of the e.orts of many colleagues at Rutherford Appleton Laboratory (and elsewhere). In particular I would like to thank my colleagues (present and former) J.F. Connolly, P. Mackay, R. Stephenson, D.M. Duxbury, and J.A. Mir for their support. References [1] G. Charpak, W. Dominik, J.C. Santiard, F. Sauli, N. Solomey, Nucl. Instrum. and Methods A 274 (1989) 275–290. [2] S.C. Curran, J.D. Craggs, Counting Tubes, Butterworth, London, 1949. [3] G. Comby, et al., CEN-Saclay report DphE/STIPE 79/01/02/36. G. Comby, P. Mangeot, IEEE Trans. Nucl. Sci. NS-27 (1980) 106. [4] F. Bartol, et al., J. Phys. III France 6 (1996) 337. [5] A. Oed, Nucl. Instrum. and Methods A 263 (1988) 351–359. [6] S.F. Biagi, T.J. Jones, Nucl. Instrum. and Methods A 361 (1995) 72–76. [7] M.E. Rose, S.A. Kor., Phys. Rev. 59 (1941) 850. [8] W. Diethorn, NYO-6628 (1956). [9] A. Williams, R.I. Sara, Int. J. Appl. Rad. Isotopes 13 (1962) 229. [10] A. Zastawny, J. Sci. Instr. 43 (1966) 179. [11] M.W. Charles, J. Phys. E 5 (1972) 95.

J.E. Bateman / Physics Reports 375 (2003) 411 – 443 [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

443

W. Bambynek, Nucl. Instrum. and Methods 112 (1973) 103. R.S. Wol., Nucl. Instrum. and Methods 115 (1974) 461. J. Planinic, Nucl. Instrum. and Methods 136 (1976) 165. H. Miyahara, M. Watanabe, T. Watanabe, Nucl. Instrum. and Methods A 241 (1985) 186–190. Y. Uozumi, T. Sakae, M. Matoba, H. Ijiri, N. Koori, Nucl. Instrum. and Methods A 324 (1993) 558–564. J.A. Crowther, Ions, Electrons and Ionising Radiations, Edward Arnold, London 1959. J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, J. Lipp, J.A. Mir, R. Stephenson, J.E. Simmons, E.J. Spill, IEE Trans. Nuc. Sci NS-49 (4) (2002) 1644–1650. J.E. Bateman, Nucl. Instrum. and Methods A 240 (1985) 177–187. J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, A.S. Marsh, J.E. Simmons, R. Stephenson, Nucl. Instrum. and Methods A 482 (2002) 707–714. S.C. Brown, Basic Data of Plasma Physics, MIT Press, Cambridge, MA, 1959. J.E. Bateman, R. Barlow, G.E. Derbyshire, J.A. Mir, R. Stephenson, IEE Trans. Nuc. Sci. NS-49 (3) (2002) 894–898. F. Sauli, Principles of operation of multiwire proportional and drift chambers, CERN 77-09. ∗ R. Bellazzini, M.A. Spezziga, Revista del Nuovo Cimento 17 (1994) 1–91. Y. Giomataris, Ph. Rebourgeard, J.P. Robert, G. Charpak, Nucl. Instrum. and Methods A 376 (1996) 29. J.E. Bateman, J.F. Connolly, G.E. Derbyshire, D.M. Duxbury, J. Lipp, J.A. Mir, J.E. Simmons, E.J. Spill, R. Stephenson, R.C. Farrow, W.I. Helsby, B.R. Dobson, R. Mutikainen, I. Suni, Nucl. Instrum. and Methods A 477 (2002) 29–36. J.E. Bateman, D.M. Duxbury, Calibration of the servo equation for a gas microstrip detector, Rutherford Appleton Laboratory Report RAL-TR-2001-038, (http://www-dienst.rl.ac.uk/library/2000/tr/raltr-2001038.pdf). G.D. Minakov, Yu.N. Pestov, V.S. Prokopenko, L.I. Shekhtman, Nucl. Instrum. and Methods A 326 (1993) 566–569.

445

CONTENTS VOLUME 375 P. Lemmens, G. Gu¨ntherodt, C. Gros. Magnetic light scattering in low-dimensional quantum spin systems

1

M. Beuthe. Oscillations of neutrinos and mesons in quantum ﬁeld theory

105

M. Melles. Electroweak radiative corrections in high energy processes

219

M. Janowicz. Method of multiple scales in quantum optics

327

J.E. Bateman, A general parametric model for the gain of gas avalanche counters with particular attention to non-cylindrical geometries

411

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445

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