SCATTERING PROCESSES IN ATOMIC PHYSICS, NUCLEAR PHYSICS, AND COSMOLOGY By Gavriil Shchedrin A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of PHYSICS - DOCTOR OF PHILOSOPHY 2013 ABSTRACT SCATTERING PROCESSES IN ATOMIC PHYSICS, NUCLEAR PHYSICS, AND COSMOLOGY By Gavriil Shchedrin The universal way to probe a physical system is to scatter a particle or radiation off the system. The results of the scattering are governed by the interaction Hamiltonian of the physical system and scattered probe. An object of the investigation can be a hydrogen atom immersed in a laser field, heavy nucleus exposed to a flux of neutrons, or space-time metric perturbed by the stress-energy tensor of neutrino flux in the early Universe. This universality of scattering process designates the Scattering Matrix, defined as the unitary matrix of the overlapping in and out collision states, as the central tool in theoretical physics. In this Thesis we present our results in atomic physics, nuclear physics, and cosmology. In these branches of theoretical physics the key element that unifies all of them is the scattering matrix. Additionally, within the scope of Thesis we present underlying ideas responsible for the unification of various physical systems. Within atomic physics problems, namely the axial anomaly contribution to parity nonconservation in atoms, and two-photon resonant transition in a hydrogen atom, it was the scattering matrix which led to the Landau-Yang theorem, playing the central role in these problems. In scattering problems of cosmology and quantum optics we developed and implemented mathematical tools that allowed us to get a new point of view on the subject. Finally, in nuclear physics we were able to take advantage of the target complexity in the process of neutron scattering which led to the formulation of a new resonance width distribution for an open quantum system. Copyright by GAVRIIL SHCHEDRIN 2013 ACKNOWLEDGMENTS In this section I would like to give a credit to many physicists and mathematicians who shared their knowledge and shaped my understanding of this beautifully designed Physical and Mathematical World whose elegance can be appreciated by mere mortals. The first in the list is my Thesis advisor, Prof. Vladimir G. Zelevinsky, whose knowledge and real understanding of theoretical physics and in particular nuclear physics is truly remarkable. My advisor expertise in atomic physics, nuclear physics and cosmology made possible the very existence of the present Thesis. Also his teaching talent is of the same caliber as his physics expertise. Here I would like to mention that MSU Physics & Astronomy Outstanding Faculty Teaching Award of 2003, 2004, and in the unprecedented period of 6 consecutive years from 2006 till 2011 was given to my advisor. So I am very much privileged and grateful to my great advisor for his willingness to work with such a troublemaker and crazy guy like me. Most importantly for me, besides physics of course, was his company where we shared a great deal of fun. I am very thankful to the Condensed Matter group at Michigan State. First of all, I am thankful to great physicist Prof. Mark Dykman for his active position in every respect and his teaching style. The very special thank you goes to Prof. Norman Birge, remarkable physicist and great experimentalist. His way of explaining physics was absolutely new experience for me. It was a refreshing point of view on the subject which gave me a practical understanding. Prof. Thomas Kaplan is the great soul of the condensed matter body at Michigan State. Prof. Chih-Wei Lai and Prof. John McGuire were very kind to share their time and ideas with me. iv I am grateful to the High Energy group at Michigan State. I am thankful to Prof. Wayne Repko for an exciting problem in cosmology and for the reference to a jewel in the modern physics literature, a masterpiece by the greatest living physicist Steven Weinberg, named Cosmology, published by the Oxford University Press in 2008. If someone wants to appreciate a truth beauty of the Physical World, this book will blow someone’s mind as it happened with me while I was writing this Thesis. I would like to thank Prof. Carl Schmidt for his courses on Quantum Field Theory and Mathematical Physics. Also I would like to thank Prof. James Linnemann, Prof. Carl Bromberg and Prof. Kirsten Tollefson, supported by the team of Tibor Nagy, Richard Hallstein and Mark Olson, who guided me thorough my teaching courses at Michigan State. On the top of that, Prof. James Linnemann has always been a great audience to discuss theoretical physics. The physics division where I finally settled down had become Nuclear Cyclotron facility at Michigan State. Prof. Scott Pratt who is the Graduate Program Director and who was a witness of my ups and downs during my four years long graduate life, has always been a great audience to discuss physics in a “five minutes” format. I would like to thank Prof. Alex Brown and Prof. Remco Zegers for their time and efforts during my nuclear graduate life. I am thankful to Prof. Michael Thoennessen whose invisible help has always been very much appreciated. Mathematics department at Michigan State has a number of great mathematicians. I am very thankful to great mathematician Prof. Alexander Volberg for his keen interest in physics that resulted in our joint paper. The chapter on quantum optics in this Thesis is based on our work. Also I am indebted to a wonderful mathematician Prof. Michael Shapiro, who shared his time and mathematical insights on the mathematical structures appearing in nuclear physics problems. v From my undergraduate life in St. Petersburg I would like to thank remarkable physicist Prof. Alexander N. Vasil’ev for his rich contribution to my undergraduate physics life which would have been completely empty otherwise. Also I am grateful to my undergraduate advisor, great physicist Prof. Leonti N. Labzowsky, who shared quite a bit of his time and knowledge with me. Most importantly, he shared a very interesting problem on axial anomaly that grew into a single chapter on atomic physics in this Thesis. The present list would never be a complete one without paying a tribute to a truly remarkable people, Anna G. Reznikova, Sergey G. Maloshevsky, and Inna D. Mints, all wonderful mathematicians, who helped me get started and supported me wholeheartedly during my rough undergraduate years in St. Petersburg. Finally I would like to thank all of my fellow physics graduate students at Michigan State, and in particular Pawin Ittisamai and Scott Bustabad who shared a great deal of humor during my graduate life. Also I am grateful to my undergraduate students who suffered at my physics classes and from whom I have learned quite a bit. Thank all of you! vi TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1 Chapter 2 Atomic Physics . . . . . . . . . . 2.1 Standard Model in low-energy physics 2.2 Furry theorem . . . . . . . . . . . . . 2.3 Axial Anomaly . . . . . . . . . . . . . 2.4 Axial Anomaly S -matrix . . . . . . . 2.5 Z -boson decay . . . . . . . . . . . . . 2.6 PNC amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . 5 . 6 . 8 . 10 . 14 . 21 Chapter 3 Quantum Optics . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . 3.2 Laser-assisted hydrogen recombination . . . 3.3 S -matrix . . . . . . . . . . . . . . . . . . . . 3.4 The Coulomb-Volkov wave function . . . . . 3.5 The partial cross section . . . . . . . . . . . 3.6 The summation procedure and cross section 3.7 The soft photon approximation . . . . . . . . 3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 32 33 34 37 38 39 42 43 Chapter 4 Nuclear Physics . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Resonance width distribution . . . . . . . . . . . . . . . . . 4.3 Effective non-Hermitian Hamiltonian and scattering matrix 4.4 From ensemble distribution to single width distribution . . 4.5 Doorway approach . . . . . . . . . . . . . . . . . . . . . . . 4.6 Photon emission channels . . . . . . . . . . . . . . . . . . . 4.7 Many-channel case . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 46 49 52 54 57 60 64 67 Chapter 5 Cosmology . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . 5.2 Fluctuations in General Relativity . . . 5.3 Boltzmann equation for neutrinos . . . 5.4 Momentum representation . . . . . . . 5.5 Perturbations to the energy-momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 71 77 82 85 vii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 5.7 5.8 5.9 5.10 5.11 5.12 Gravitational wave damping . . . . . . . . . . . . . . . Matter-radiation equality . . . . . . . . . . . . . . . . . Late-time evolution of the gravitational wave damping Early-time evolution of the gravitational wave damping General solution for the gravitational wave damping . . Evaluation of the convolution integrals . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 92 96 102 108 113 117 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . A Electron wave function behavior near the nucleus . B Secular equation in the general case . . . . . . . . . C Friedmann equations . . . . . . . . . . . . . . . . . D Energy density for fermions and bosons in the early E Convolution integral of spherical Bessel functions . F Convolution matrices in the late-time limit . . . . . G Convolution matrix in the early-time limit . . . . . H Convolution Matrices in the early-time limit . . . . BIBLIOGRAPHY . . . . . . . . . . . . . viii . . . . . . . . . . . . . . . . . . . . Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 124 125 126 129 133 137 139 140 . . . . . . . . . . . . . . . . 143 LIST OF FIGURES Figure 2.1 Figure 2.2 Proof of the Furry theorem on the example of a triangle Feynman graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The Feynman graphs that describe PNC effect in cesium. The double solid line denotes the electron in the field of the nucleus. The wavy line denotes the photon (real or virtual) and the dashed horizontal line with the short thick solid line at the end denotes the effective weak potential, i.e. the exchange by Z -boson between the atomic electron and the nucleus. Graph (a) corresponds to the basic M 1 transition amplitude, graph (b) corresponds to the E1 transition amplitude, induced by the effective weak potential. The latter violates the spatial parity and allows for the arrival of p-states in the electron propagator in graph (b), of which the contributions of 6p and 7p states dominate. The standard PNC effect arises due to the interference between graphs (a) and (b). Graph (c) corresponds to the axial anomaly. The thin solid lines represent virtual electrons and positrons. To graph (c), the Feynman diagram with interchanged external photon and Z -boson lines should be added. . . . . . . . . . . . . . . . . 9 Figure 2.3 Flow of momenta in the axial anomaly. . . . . . . . . . . . . . . . 12 Figure 2.4 Feynman diagram of Z -boson decay into two photons. . . . . . . . 15 Figure 2.5 Feynman diagram of π 0 -meson decay into two photons. Figure 4.1 The proposed resonance width distribution according to eq. (4.3) with a single neutron channel in the practically important case η Γ. The width Γ and mean level spacing D are measured in units of the mean value Γ . “For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation.” . . . . . . . . . . . . . . . 50 Figure 4.2 The proposed resonance width distribution according to eq. (4.28) with a single neutron channel and N gamma-channels in the practically important case η Γ. The neutron width Γ, radiation width γ , and mean level spacing D are measured in units of the mean value Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 ix . . . . . . 16 Figure 5.1 Graphical representation for the upper but one triangular structure of the matrix B2k,2l , eq. (70). The matrix indices l and k define a position of the matrix element in the xy -plane, while along z -axis we plot its value. . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Figure 5.2 Late-time evolution of the gravitational wave damping D(u) in the early Universe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Figure 5.3 Graphical representation for the upper triangular structure of the matrix F2k,2l , eq. (72). The matrix indices l and k define the position of the matrix element in the xy -plane, while along z -axe we plot its value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Figure 5.4 Damping of gravitational wave as a function of conformal time u in the early-time limit. Solid line corresponds to the exact solution of the gravitational wave damping in the early-time limit and dashed line is the approximate solution, A0 j0 (u), eq. (5.166). . . . . . . . 106 Figure 1 Graphical representation for the upper but one triangular structure of the matrix B2k,2l . The matrix indices l and k define a position of the matrix element in the xy -plane, while along z -axe we plot its value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Figure 2 Graphical representation for the upper but one triangular structure of the matrix B2k+1,2l+1 . The matrix indices l and k define a position of the matrix element in the xy -plane, while along z -axe we plot its value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 Figure 3 Graphical representation for the upper triangular structure of the matrix F2k,2l , eq. (72). The matrix indices l and k define the position of the matrix element in the xy -plane, while along z -axe we plot its value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Figure 4 Graphical representation for the upper triangular structure of the matrix A2k+1,2l . The matrix indices l and k define the position of the matrix element in the xy -plane, while along z -axe we plot its value.140 Figure 5 Graphical representation for the upper triangular structure of the matrix A2k,2l+1 . The matrix indices l and k define the position of the matrix element in the xy -plane, while along z -axe we plot its value.141 x Chapter 1 Introduction In this Thesis we summarize our results in atomic physics, quantum optics, nuclear physics, and cosmology unified by various aspects of quantum scattering theory. In Chapter 2 our object of investigation is the scattering matrix corresponding to the axial anomaly contribution to parity nonconservation in atoms. Here we explain physics of parity nonconservation in complex atoms and discuss the contribution of the axial anomaly to parity nonconservation in atomic cesium. The main result is the prediction of the emission of an electric photon by the magnetic dipole which has not been observed yet. The probability of this process is very small but the non-zero result is important from theoretical point of view. The main aspect of our calculation is related to the well known theorem of axial anomaly cancellation in the Standard Model. According to the Landau-Yang theorem, it is impossible for the real Z -boson of spin J = 1 to decay into two real photons in contrast to the allowed two-photon decay of the spinless π 0 -meson. However, if one of the photons that connects the triangular graph of the axial anomaly with the electron atomic transition, e.g., 6s − 7s transition in cesium, is virtual, the axial anomaly does not vanish. We have shown that one can see the impact of the axial anomaly in atomic physics through the parity violation in atoms. 1 Chapter 3 is devoted to the electron scattering process off an arbitrary central potential in the presence of strong laser field. Here we introduce a new method that allows one to obtain an analytical cross section for the laser-assisted electron-ion collision. As an example we perform a calculation for the hydrogen laser-assisted recombination. The standard S matrix formalism is used for describing the collision process. The S -matrix is constructed from the electron Coulomb-Volkov wave function in the combined Coulomb-laser field and the hydrogen perturbed state. By the aid of the Bessel generating function, the S -matrix is decomposed into an infinite series of the field harmonics. We have introduced a new step that results in an analytical expression for the cross section of the process. The theoretical novelty is in the application of the Plancherel theorem to the Bessel generating function. This allows one to perform summation of the infinite series of Bessel functions and thus obtain a closed analytical expression for the laser-assisted hydrogen photo-recombination process. In the field of nuclear physics presented in Chapter 4 we investigate the resonance width distribution for low-energy neutron scattering off heavy nuclei. Our interest was ignited by the recent experiments that claimed significant deviations from the routinely used chi square, or the Porter-Thomas, distribution. The unstable complex nucleus is an open quantum system, where the intrinsic dynamics has to be supplemented by the coupling of chaotic internal states through the continuum. We propose a new width distribution based on random matrix theory for a chaotic quantum system with a single decay channel as well as for an open quantum system with an arbitrary number of open channels. The revealed statistics of the width distribution exhibits distinctive properties that are characteristically different from the regularities shared by closed quantum systems. 2 In Chapter 5 our object of investigation is the space-time metric perturbed by the stressenergy tensor of neutrino scattering in the early Universe. In this Chapter we have developed a mathematical machinery that allows one to evaluate gravitational wave damping due to freely streaming neutrinos in the early Universe. The solution is represented by a convergent series of spherical Bessel functions derived with the help of a new compact formula for the convolution of spherical Bessel functions of integer order. These calculations can be compared to the tensor fluctuations of the Cosmic Microwave Background in order to reveal direct evidence of the presence of gravitational waves in the early Universe. The developed technique can be applied for the analysis of the scalar Cosmic Microwave Background fluctuations which provides a direct test of the standard inflationary cosmological model. Finally, Chapter 6 summarizes the results of our investigation on scattering processes in atomic physics, quantum optics, nuclear physics, and cosmology. Basic governing principles, such as conservation laws and detailed balance principle, unitarity and gauge invariance of the scattering matrix are given special attention throughout the Thesis. 3 Chapter 2 Atomic Physics In this chapter the main interest is parity violation in atomic physics. Parity operation is defined as a coordinate inversion r → −r , which tests the coordinate symmetry of the physical system, as well as intrinsic symmetries of fundamental physical objects. In quantum electrodynamics (QED) that fully describes electron-photon interaction, parity is conserved due to the invariance of the QED action under the spatial inversion. However, in the Standard Model that treats electromagnetic and weak forces as a unified electroweak interaction, which led to a prediction and subsequent discovery of the W and Z bosons, parity is violated. The W and Z bosons are the carriers of the so-called weak axial currents which are responsible for the parity violation. In addition to the high energy experiments, that aim at the verification of the Standard Model on a very high energy scale, one can test its predictions on a low energy scale by running tabletop experiments. In particular, we are interested in parity non-conservation effects in atoms, driven by the effective parity nonconserving interaction of electrons with the nucleus. This chapter is organized as follows. In the first subsection we briefly review the general scope of the problem. In the section on the Furry theorem we provide an important result for the fermion loop diagrams with an odd number of interaction vertices, e.g. one on Fig. 2.1. Next subsection stands for a general physical idea behind axial anomaly calculation in atomic physics. The actual calculation is presented in section Axial Anomaly S -matrix. In this 4 section we explicitly write the S -matrix for the decay of the Z -boson into two photons and apply it to our calculation. We show that due to the transversality conditions and on-shell constraint for real photons and the Z -boson, the axial anomaly diagram identically vanishes, in a full accordance with the Landau-Yang theorem. However, if one of the photons that connects the triangular graph of the axial anomaly with the electron atomic transition, e.g., 6s − 7s transition in cesium, is virtual, the axial anomaly does not vanish and contributes to the effect of parity nonconservation. The results of this chapter are based on our paper, GS and L. Labzowsky, Phys. Rev. A 80, 032517 (2009). 2.1 Standard Model in low-energy physics The problem of testing the Standard Model (SM) in the low-energy physical phenomena is one of the interesting topics in physics actively pursued in the last few decades. The SM in the low energy limit is tested in nuclei, where significant many-body enhancement of the parity nonconservation (PNC) effects was predicted and confirmed. Along with the PNC effects in nuclei, atomic physics experiments offer a high precision for validating the SM predictions. The most accurate of atomic experiments is the one with the neutral cesium atom, first proposed in [1] and performed with the utmost precision in [2]. The basic atomic transition employed in the cesium experiment was the strongly forbidden 6s − 7s transition of the valence electron with the absorption of a magnetic dipole photon, M 1. In the real experiment this very weak transition was opened by the external electric field but it does not matter for our further derivations. The Feynman graphs 5 illustrating the PNC effect in cesium are given below in Fig. 2.2. The atomic experiments are indirect and require very accurate calculations of the PNC effects to extract the value of the characteristic parameter of the SM, the Weinberg angle, which can be compared with the corresponding high-energy value. The main difficulty with the PNC calculations in neutral atoms is the necessity to take into account the electron correlations within the complex atom. Therefore the experiments with much simpler systems, such as few-electron highly charged ions (HCI) would be highly desirable. Several proposals on the subject were considered in [3–6]. At the same time relativistic corrections make heavy-Z atoms preferable due the enhancement factor for electronic wave functions. The radiative corrections to the PNC effect are important in cesium calculations to reach the agreement with the high-energy SM predictions. These radiative corrections include electron self-energy, vertex and vacuum polarization Feynman diagrams. They are even more important in the case of the HCI. The entire set of these corrections for the neutral cesium atom was calculated in [7–10]. The electron self-energy and vertex corrections for HCI were obtained in [11]; the vacuum polarization correction was given in [12]. The full set of radiative corrections including Z -boson loops has not been calculated, neither for a neutral atom nor for the HCI. Therefore the problem still cannot be considered as fully solved. 2.2 Furry theorem In quantum field theory Feynman diagrams with a high number of vertices are notoriously difficult for calculation. However, in the special case of a fermion loop diagram with an odd number of identical interaction vertices, its contribution, according to the Furry theorem, 6 γ γµ k1 p + k1 γ p q γλ p − k2 γ γν k2 Figure 2.1: Proof of the Furry theorem on the example of a triangle Feynman graph. is identically zero. To illustrate the proof of the Furry theorem we recognize that the Dirac matrices γ µ change sign and get transposed, γ µ , under charge conjugation C , C −1 γ µ C = −γ µ , (2.1) while the charge conjugated fermion propagator modifies according to C −1 G(p)C = C −1 pν γ ν + me −pν γ ν + me C= = G(−p). p2 − m2 p2 − m2 e e Therefore a typical triangle amplitude will change sign under charge conjugation, 7 (2.2) Aµνλ (k1 , k2 ) = (2.3) p + me ν p− k2 + me λ p+ k1 + me d4 p Tr γ µ 2 γ γ p − m2 (p − k2 )2 − m2 (p + k1 )2 − m2 e e e p + me ν p− k2 + me λ p+ k1 + me −→ (−1)3 d4 p Tr γ µ 2 γ γ ≡ 0. p − m2 (p − k2 )2 − m2 (p + k1 )2 − m2 e e e The same argument holds for a Feynman diagram with an arbitrary number of odd fermionboson vertices, and we thus prove the Furry theorem. However, if a Feynman diagram has an odd number of vertices of a different kind, for instance, two vector photon vertices, and a single π 0 pseudo-scalar vertex, or a single Z boson pseudo-vector vertex, the corresponding Feynman diagram has a non-zero value. 2.3 Axial Anomaly In the present work we consider a very special radiative correction to the PNC effect, presented by a triangular Feynman graph, or axial anomaly (AA). We understand the triangle AA as a fermion loop with at least one weak vertex [13]. Our conclusion will be that in a neutral atom the contribution of the axial anomaly is non-zero albeit relatively small. The leading contribution of the AA to the atomic PNC effect is depicted in Fig. 2.2 (c). This contribution corresponds to the Adler-Bell-Jackiw anomaly [14]. In this work we will concentrate exclusively on this term. The final answer, that looks like the emission of the electric photon by the magnetic dipole, can be easily understood before any real calculations are made. Suppose we have the 6s-7s transition in cesium. The virtual photon in this transition that connects the atomic electron line with the triangular graph of the axial anomaly must be of a magnetic dipole type M 1. This virtual photon is absorbed by the fermion current in the axial anomaly 8 7s 7s 7s E1 E1 6p, 7p M1 Z Z 6s 6s a 6s b c Figure 2.2: The Feynman graphs that describe PNC effect in cesium. The double solid line denotes the electron in the field of the nucleus. The wavy line denotes the photon (real or virtual) and the dashed horizontal line with the short thick solid line at the end denotes the effective weak potential, i.e. the exchange by Z -boson between the atomic electron and the nucleus. Graph (a) corresponds to the basic M 1 transition amplitude, graph (b) corresponds to the E1 transition amplitude, induced by the effective weak potential. The latter violates the spatial parity and allows for the arrival of p-states in the electron propagator in graph (b), of which the contributions of 6p and 7p states dominate. The standard PNC effect arises due to the interference between graphs (a) and (b). Graph (c) corresponds to the axial anomaly. The thin solid lines represent virtual electrons and positrons. To graph (c), the Feynman diagram with interchanged external photon and Z -boson lines should be added. 9 triangle, through a first vertex, and cannot change the parity of the fermion current flowing through this vertex. Now, the second vertex in the axial anomaly triangle that is responsible for exchange of the Z -boson between the fermion line and the cesium nucleus has the Dirac matrix γ 5 that changes the parity of the fermion line. As the result, the third vertex in the axial anomaly triangle that connects two fermion lines with an opposite parity must emit the electric photon type E1. 2.4 Axial Anomaly S-matrix We employ the standard expression for the effective parity nonconserving interaction of the atomic electron with the nucleus [15] in the form HW = APNC ρ0 (r)γ 5 , (2.4) with parity nonconservation vertex APNC , G Q APNC = − F√ W , 2 2 (2.5) Dirac pseudoscalar matrix γ 5 ,   0 1 i , γ 5 = iγ 0 γ 1 γ 2 γ 3 = − εµνρσ γ µ γ ν γ ρ γ σ =  4! 1 0 (2.6) and contact electron-nucleus interaction ρ0 (r), where one can neglect the finite size of an atomic nucleus, ρ0 (r) = δ(r). 10 (2.7) GF is the Fermi constant given in terms of the proton mass mp by 1 1 GF = 1.027 × 10−5 2 = 1.166 × 10−5 . mp GeV2 (2.8) QW is the weak charge of the nucleus, QW = −N + Z(1 − 4 sin2 θw ), (2.9) while Z and N are the numbers of protons and neutrons in the nucleus, and θw is the Weinberg angle. The currently accepted value for this parameter is sin2 θw 0.23. (2.10) The singular δ -function potential acting in the space of Dirac electron wave functions does not vanish only when electron and nucleon coordinates coincide. This approximation is valid if the transferred momentum q is much less than the mass of Z -boson, which is clearly valid for the atomic electron. For the electron in the loop this approximation is valid due to the fact that the momentum transfer to the nucleus, as one can see on Fig. 2.3, is (q − k) where q2 m2 , and as the consequence (q − k)2 Z m2 . Therefore we can parametrize the Z coupling of the electron-nucleon interaction by the contact interaction (2.4). 11 p1 γµ γρ q γν k p p+k q−k p+q γ λγ 5 ′ p1 Figure 2.3: Flow of momenta in the axial anomaly. We write down the S -matrix corresponding to the amplitude Fig. 2.2 (c) in the momentum representation (see Fig. 2.3): d4 p1 d4 p1 d4 p gρν S = (ie) Ψn s (p1 )γ ρ Ψns (p1 ) 2 (2.11) (2π)4 (2π)4 (2π)4 q +i p + me ν p+ q + me λ 5 p+ k + me PNC (q − k)A (k). Vλ ×Tr γ µ 2 γ γ γ µ 2 2 − m2 2 − m2 p − me (p + q) (p + k) e e 3 12 Here e and me are the electron charge and mass, Ψns (p) = Ψns (p)δ(p0 − εns ) (2.12) is the wave function of the bound atomic electron in the state |ns with εns being the energy of this state, the transferred momentum q is q = p1 − p 1 , (2.13) gρν is the pseudo-Euclidean metric tensor,   1 0 0 0   0 −1 0 0  gρν =   0 0 −1 0   0 0 0 −1     ,    (2.14) γ µ are the Dirac matrices,  γµ =  0 σµ −σ µ 0  γ0 ≡ β =  1 0 0 -1   ≡   0 σ −σ 0 ,  , (2.15) (2.16) and Aµ (k) is the wave function of the emitted photon, Aµ (x) = 2π −ikx , µe ω 13 (2.17) in the momentum representation. Here µ and k = (ω, k) are the four-vectors of the polarization and the momentum of the emitted photon, correspondingly. PNC for the parity-nonconserving In the momentum representation the potential Vλ interaction of the electron with the nucleus is PNC (q − k) = A Vλ PNC ρ0 (q − k)δλ0 , (2.18) where the nucleon density in the particular case of the point-like nucleus is ρ0 (q − k) = 2πδ(q0 − k0 ). (2.19) In this chapter we use the relativistic units with h = c = 1. ¯ 2.5 Z-boson decay The central element in the Feynman diagram of Fig. 2.3, the fermion triangle, involves two photon vertices and a single Z -boson vertex. First we consider the Z -boson (with spin J(Z) = 1) decay [16] into two photons, Fig. 2.4. The Landau theorem forbids this decay because a two-photon system cannot exist with angular momentum J = 1 [17, 18], in contrast to the allowed decay π 0 → γγ since J(π 0 ) = 0 [19], see Fig. 2.5. We shall derive this result in the Feynman diagram language for the S -matrix of Fig. 2.4, and see implications for the diagram in Fig. 2.3. 14 γ γµ k1 p + k1 Z q p γ λγ 5 p − k2 γ γν k2 Figure 2.4: Feynman diagram of Z -boson decay into two photons. The S -matrix of the Z -boson decay into two photons, Fig. 2.4, is proportional to S µνλ (k1 , k2 ) = p + me ν p− k2 + me λ 5 p+ k1 + me γ γ . d4 p Tr γ µ 2 γ p − m2 (p − k2 )2 − m2 (p + k1 )2 − m2 e e e (2.20) A simple change of variables in the Z -boson amplitude (2.20), k1 → k, k2 → −q, leads to the loop integral in the original PNC-amplitude, eq. (2.11). 15 (2.21) γ γµ k1 p + k1 π0 q p γ5 p − k2 γ γν k2 Figure 2.5: Feynman diagram of π 0 -meson decay into two photons. Here we shall note that the trace of a product of the Dirac γ 5 matrix with the four other distinct Dirac matrices leads to a non-vanishing result, Tr γ 5 γ τ γ µ γ ν γ λ = 4iετ µνλ , (2.22) where ετ µνλ is the unit antisymmetric fourth rank tensor defined as ε0123 = −1. All other combinations return zero result. Therefore the most general expression for the Z - 16 boson decay amplitude Sλµν (k1 , k2 ) is Sµνλ (k1 , k2 ) = A1 k1τ ετ µνλ + A2 k2τ ετ µνλ + A3 k1ν k1ξ k2τ εξτ µλ (2.23) +A4 k2ν k1ξ k2τ εξτ µλ + A5 k1µ k1ξ k2τ εξτ νλ + A6 k2µ k1ξ k2τ εξτ νλ . The expressions for Ai with 3 ≤ i ≤ 6 represent convergent integrals and are evaluated using the standard Feynman technique. The first step is to introduce Feynman variables ξi which simplify the loop integral (2.20), 1 1 1 δ(1 − ξ1 − ξ2 − ξ3 ) 1 dξ3 dξ2 = 2! dξ1 α1 α2 α3 (α1 ξ1 + α2 ξ2 + α3 ξ3 )3 0 0 0 ξ1 1 1 dξ2 . ≡ 2! dξ1 (α1 ξ2 + α2 (ξ1 − ξ2 ) + α3 (1 − ξ1 ))3 0 0 (2.24) In our specific case the variables αi are α1 = (p + k1 + k2 )2 − m2 , e (2.25) α2 = (p + k2 )2 − m2 , e (2.26) α3 = p2 − m2 . e (2.27) 17 Therefore the loop integral becomes 1 1 ≡ α1 α2 α3 [(p + k1 + k2 )2 − m2 ][(p + k2 )2 − m2 ][p2 − m2 ] e e e 1 ξ1 1 = 2! dξ1 dξ2 . (α1 ξ2 + α2 (ξ1 − ξ2 ) + α3 (1 − ξ1 ))3 0 0 (2.28) First we shall simplify the denominator in eq. (2.28) as follows: [(p + k1 + k2 )2 − m2 ]ξ2 + [(p + k2 )2 − m2 ](ξ1 − ξ2 ) + [p2 − m2 ](1 − ξ1 ) e e e 2 2 = p2 − 2p(−k2 ξ1 − k1 ξ2 ) + (−m2 + 2k2 k1 ξ2 + k1 ξ2 + k2 ξ1 ). (2.29) Further we notice a common integral iπ 2 d4 p , = (p2 + l − 2pk)3 2(l − k 2 ) (2.30) and the loop integral becomes 1 (2.31) [(p + k1 + k2 )2 − m2 ][(p + k2 )2 − m2 ][p2 − m2 ] e e e 2 1 ξ1 iπ 1 = 2! dξ1 dξ2 . 2 2 2 −m2 + 2k2 k1 ξ2 + k1 ξ2 + k2 ξ1 − (k2 ξ1 + k1 ξ2 )2 0 0 d4 p Finally the expression of eq. (2.31) can be simplified by the change of variable ξ1 −→ 1 − ξ1 , 18 (2.32) and the denominator in eq. (2.31) becomes 2 2 −m2 + 2k2 k1 ξ2 + k1 ξ2 + k2 (1 − ξ1 ) − (k2 (1 − ξ1 ) + k1 ξ2 )2 (2.33) 2 2 = −m2 + k1 ξ2 + k2 ξ1 − (k2 ξ1 − k1 ξ2 )2 . Finally the expressions for Ai with 3 ≤ i ≤ 6 will be expressed in terms of the convergent integrals 1 1 1 (ξ1 r ξ1 s ξ3 t )δ(1 − ξ1 − ξ2 − ξ3 ) 1 dξ1 dξ2 dξ3 . Jrst (k1 , k2 ) = 2 2 2 π 0 (ξ1 ξ2 (k1 + k2 )2 + ξ1 ξ3 k1 + ξ2 ξ3 k2 − m2 ) 0 0 (2.34) In order to obtain a convergent and gauge invariant expression for the amplitude Sµνλ we impose the Ward identities, k 1µ Sµνλ = 0, (2.35) k 2ν Sµνλ = 0. (2.36) In terms of the general expression (2.23), eqs. (2.35) and (2.36) lead to the desirable constraints on the divergent integrals 2 [−A2 + k1 A5 + (k1 k2 )A6 ]k1ξ k2τ εξτ νλ = 0, (2.37) 2 [−A1 + k2 A4 + (k1 k2 )A3 ]k1ξ k2τ εξτ µλ = 0. (2.38) 19 In this way the amplitude Sµνλ will be finite and gauge-invariant if we choose A2 (k1 , k2 ) and A1 (k1 , k2 ) in the form 2 A2 (k1 , k2 ) = k1 A5 + (k1 k2 )A6 , (2.39) 2 A1 (k1 , k2 ) = k2 A4 + (k1 k2 )A3 . (2.40) Finally we come to the Z -boson amplitude Sµνλ , Sµνλ (k1 , k2 ) = J110 (k1 , k2 )εµναβ k1α k2β (k1 + k2 )λ (2.41) 2 +J101 (k1 , k2 )(ελναβ k1α k2β k1µ + k1 ελµνα k2α ) 2 +J011 (k1 , k2 )(ελµαβ k1α k2β k2ν + k2 ελµνα k1α ), with the integrals Jrst (k1 , k2 ) given by eq. (2.34). Due to the transversality conditions for the Z -boson and on-shell photons expressed as (k1 + k2 )λ λ = 0, 1µ k1µ = 0, (2.42) 2ν k2ν = 0, and conditions for the real photons, 2 k1 = 0, 2 k2 = 0, 20 (2.43) we arrive at the result of the Landau theorem SZγγ = 0. But in our case one of the photons (e.g. with index 2) is virtual, as well as the Z -boson. Therefore the initial S -matrix (2.11) returns a nonzero result. 2.6 PNC amplitude The next step is to contract the Z -boson amplitude Sµνλ with the Z -boson δ -potential, photon polarization vector, and photon propagator, Sµνλ (k, q)δλ0 1µ a2ν , (2.44) where a2ν = αρ gρν γ 0 γ ρ gρν αν = = 2, q2 q2 q (2.45) with Dirac α-matrices  αµ = γ 0 γ µ =  0 σ σ 0  . (2.46) By noticing ε0µνα = −εµνα , 21 (2.47) we finally arrive at the axial anomaly amplitude, αν Sµνλ (k, q)δλ0 1µ 2 = J011 (k, q)[εµαβ kα qα 1µ (α, q) + εµνα 1µ kα αν ] q ( , [k × q])(α, q) = J011 (k, q) + ( , [α×k]) , (2.48) q2 and S -matrix for the PNC effect is 4π (2.49) 2ω0 d3 p1 d3 p1 + ( , [k × q])(α, q) Ψ (p1 )J011 (k, q) + ( , [α×k]) Ψ(p1 ). (2π)3 (2π)3 q2 S = (ie)3 APNC δ(Ef − Ein − ω0 ) × Next we turn our attention to non-relativistic analysis of the S -matrix. The lower component of the Dirac electron wave function χ is expressed in terms of the upper one ϕ via χ= (σp) ϕ. 2m (2.50) Dirac α-matrices mix these components,  ϕ ∗ χ∗ 1 1  0 σ   ϕ  1 , σ 0 χ1 (2.51) which reduces to ϕ∗ σ 1 (σp1 ) (σp1 ) + σ ϕ1 . 2m 2m 22 (2.52) The first term in square brackets in eq. (2.49) can be simplified to (q, [ × k])(σ, q) (q, [ × k])(σ, q) (σp1 ) + (σp1 ) , q2 q2 (2.53) while the second term reduces to (σ, [k× ])(σp1 ) + (σp1 )(σ, [k× ]). (2.54) Further we employ the well-known identity for the Pauli matrices, (σa)(σb) = (ab) + i(σ, [a × b]), (2.55) and owing to the transferred momentum is q = p1 − p1 , eq. (2.53) becomes (q, [ × k])(σ, q) (q, [ × k]) (q, [ × k])(σ, q) (σp1 )+(σp1 ) = (q, p1 +p1 ) . (2.56) q2 q2 q2 In order to simplify eq. (2.54) we use the identities (σ, [k × ])(σp1 ) = ([k × ], p1 ) + i(σ, [[k × ] × p1 ]]), (2.57) (σp1 )(σ, [k × ]) = (p1 , [k × ]) + i(σ, [p1 × [k × ]]). (2.58) 23 Therefore the sum of these terms is (σ, [k × ])(σp1 ) + (σp1 )(σ, [k × ]) (2.59) = (p1 + p1 , [k × ]) + i(σ, [q × [k × ]]) = (p1 + p1 , [k × ]) + i(σk)(q ) − i(σ )(qk), which represents the final expression for eq. (2.54). For the sake of convenience we introduce P = p1 + p1 , (2.60) and then the square brackets of the S -matrix in eq. (2.49) will be represented as (q, P ) (q, [ × k]) + (P, [k × ]) + i(σk)(q ) − i(σ )(qk). q2 (2.61) Now we have to explore how the obtained expression changes under change of integration variables p1 −→ −p1 , (2.62) p1 −→ −p1 . The expression in eq. (2.59) changes sign and therefore we shall turn our attention to the 2 integral (2.34) which simplifies in the case for a real emitted photon k1 = 0, 1 1−ξ1 ξ1 (ξ1 + ξ2 − 1) 1 dξ1 dξ2 , I(k, q) = 2 π 0 −m2 + 2ξ1 ξ2 (kq) + ξ1 (1 − ξ1 )q 2 0 24 (2.63) and in the non-relativistic limit we obtain 2ξ ξ (kq) 1 1 = 2 1 + 1 22 +O m2 − 2ξ1 ξ2 (kq) m m 1 m4 . (2.64) Finally, we have the expression which is even under the inversion (2.62), (q, P ) (q, [ × k]) (qk) + (P, [k × ])(qk) + i(σk)(q )(qk) − i(σ )(qk)2 . (2.65) 2 q After integration over p1 , p1 or similarly over (P, q), the first two terms in eq. (2.65) vanish because of the contraction with the external vectors µ and kµ . The third term in eq. (2.65) will disappear in the final expression due to the Wigner-Eckart theorem as we show later. The only nonvanishing term in eq. (2.65) that contributes to the probability of the PNC effect is −i(σ )(qk)2 , (2.66) and therefore the S -matrix in the nonrelativistic limit is S = (ie)3 APNC δ(Ef − Ein − ω0 ) × 4π 2ω0 (2.67) d3 p1 d3 p1 I Ψ+ (p1 )[−i(σ )(qk)2 ]Ψ(p1 ), 3 (2π)3 m5 (2π) where 1 1−ξ1 1 1 2 I=− 2 dξ1 dξ2 (ξ1 ξ2 )(ξ1 + ξ2 − 1) = . π 0 360π 2 0 25 (2.68) After performing the Fourier transformation, d3 p1 d3 p1 ∗ ϕ (p )[(qk)2 ]ϕ7s (p1 ) 3 (2π)3 6s 1 (2π) = m6 ϕ∗ (0)[−(k 6s (2.69) 2 2 ]ϕ (0) = −m6 ϕ∗ (0)k 2 d ϕ7s (r) ) 7s 6s 2 dr r=0 For brevity we will write d2 ϕ(r) ≡ ϕ (0), dr2 r=0 (2.70) and the S -matrix takes the form 1 (σ )k 2 ϕ∗ (0)ϕ7s (0) (2.71) 6s 2 360π √ 3/2 (GF m2 )QW me 2 2π ω0 p 3 √ (σ )ϕ∗ (0)ϕ7s (0), = −e δ(Ef − Ein − ω0 ) 6s 2 me mp 360π 2 2 S = e3 mAPNC δ(Ef − Ein − ω0 ) 4π 2ω0 and, owing to the standard correspondence between the S -matrix and amplitude of the process FPNC , S = −2πiFPNC δ(En s − Ens − ω0 ), (2.72) we obtain for the parity-nonconserving amplitude, e3 (GF m2 )QW p √ FPNC = 2πi 2 2 √ 3/2 me 2 2π ω0 (σ )ϕ∗ (0)ϕ7s (0). 6s 2 me mp 360π (2.73) For the Feynman diagrams a and c, Fig. 2.2, we have F7s→6s = FM1 + FPNC , 26 (2.74) and the corresponding expression for the probability of the process, after averaging over the initial electron spin projections and summing over the final spin projections, W7s→6s = WM1 + 1 2 2Re [FM1 FPNC ] + O FPNC , 2j0 + 1 m m 0 1 (2.75) where the matrix element for FPNC is √ 3/2 e3 (GF m2 )QW me 2 2π ω0 p √ FPNC = ϕ∗ (0)ϕ7s (0) 2 me 6s 2πi mp 360π 2 2 × n1 j1 l1 m1 s1 |(σ )|n0 j0 l0 m0 s0 . (2.76) In the specific case of the cesium transition we have n1 j1 l1 m1 s1 | = 6s1/2 |, (2.77) |n0 j0 l0 m0 s0 = |7s1/2 . Following Landau [20] we estimate the electron wavefunction (in atomic units) in the region r ∼ 1/Z (see Appendix A) as ϕ(r) ∼ Z 1/2 , ϕ (r) ∼ Z 5/2 . 27 (2.78) Therefore the expression (2.76) becomes e3 (GF m2 )QW p √ FPNC = 2πi 2 2 √ 3/2 me 2 2π ω0 Z 3 n1 j1 l1 m1 s11 |(σ )|n0 j0 l0 m0 s0 . 2 me mp 360π (2.79) The matrix element in eq. (2.79) is evaluated by means of the Wigner-Eckart theorem,   j J j  n j l ||AJL ||njl . n j l m |AJLM |njlm = (−1)j −m  −m M m (2.80) In the following we shall evaluate the product of two matrix elements, m0 m1 n1 j1 l1 m1 | q (−1)q µq [ ×k]−q |n0 j0 l0 m0 ∗ (2.81)  × n1 j1 l1 m1 | q (−1)q σq −q |n0 j0 l0 m0  , where µ = µ0 s is the magnetic moment of electron. Owing to the orthogonality relation between 3j -symbols,  (2j + 1)   j j2 j j j2 j  1  1 =δ δ jj mm , m1 m2 m m1 m2 m m1 m2 (2.82) we can perform the summation over the spin projections and obtain the mixed product i q (−1)q q [ ∗ ×k]−q = i( , [ ∗ ×k]) = i(k, [ × ∗ ]) ≡ (ksph ), (2.83) where we have introduced a photon spin variable in terms of the vector product of the photon 28 polarization vectors, sph = i[ × ∗ ]. (2.84) Now we return to eq. (2.65), where we disregarded the term (σk) which corresponds to n1 j1 l1 m1 | m0 m1 q (−1)q µq [ ×k]−q |n0 j0 l0 m0 ∗ (2.85)  × n1 j1 l1 m1 | q (−1)q σq k−q |n0 j0 l0 m0  . The orthogonality relation between 3j -symbols leads to i q (−1)q kq [ ×k]−q = i(k, [ ×k]) = 0, (2.86) and therefore this term does not contribute to the probability of the PNC process. The expression (2.81) simplifies to a product of the reduced matrix elements, (ksph ) n1 j1 l1 ||µ||n0 j0 l0 n1 j1 l1 ||σ||n0 j0 l0 . (2.87) Rewriting in terms of the spin operator s, µ = µ0 s, σ = 2s, 29 (2.88) and owing to the reduced matrix elements of the spin operator, s1 ||s||s0 = δs0 s1 s0 (s0 + 1)(2s0 + 1), (2.89) we get the final expression for eq. (2.81) for s0 = 1/2 µ0 (ksph ). (2.90) Introducing the probability of the process on Fig. 2.2 (a), 4 3 WM 1 = ω0 µ2 , 0 3 (2.91) we get the final answer in the form W7s→6s = WM 1 (1 + R(ν · sph )), (2.92) where ν = k/|k| is the unit vector in the direction of the emitted photon momentum. In our case R is equal to the ratio FPNC /FM 1 , where the amplitudes are expressed via the angular reduced matrix elements. Using the estimate ϕ(0)ϕ (0) ∼ α5 Z 3 for neutral atoms [20], we get for the anomaly contribution to the PNC-amplitudes on Fig. 2.2 (c) 1 FPNC ∼ 360π 2 me 2 3/2 α (GF m2 )QW α5 Z 3 . p mp (2.93) Using a well-known estimate for the PNC-amplitude [15], Fig. 2.2 (b), in neutral atoms 30 without the contribution from the axial anomaly 0 FPNC ∼ me 2 3/2 2 α Z (GF m2 )QW , p mp (2.94) 0 we get for the relative axial anomaly contribution (2.93) in terms of FPNC , FPNC ∼ (10)−3 α5 Z. 0 FPNC (2.95) We should admit that contribution of the axial anomaly to the PNC effects in neutral atoms is small for an observation in real experiments but the nonzero result is important from the theoretical point of view for understanding the axial anomaly mechanism. 31 Chapter 3 Quantum Optics In this chapter we introduce a new method that allows one to obtain in a closed form an analytical cross section for the laser-assisted electron-ion interaction. As an example we perform a calculation for the hydrogen laser-assisted recombination. The results of this chapter are based on our paper, GS and A. Volberg, J. Phys. A: Math. Theor. 44, 245301 (2011). 3.1 Introduction Electron scattering processes in the presence of a laser field play a significant role in the contemporary atomic physics [21]. A simple but sufficiently accurate theoretical model for laser-assisted atomic scattering would be helpful for many experimental studies. The standard theoretical approach for laser-assisted electron-ion collisions requires the construction of the S -matrix for the corresponding process. The electron wave function in a combined Coulomb-laser field is given by the well known Coulomb–Volkov solution. The dressed state of the atom is described by the time-dependent perturbation series. In our work we have developed a new method that allows one to derive in a closed form an analytical expression for the cross section of a laser-assisted atomic scattering. The new mathematical step is in using the Bessel generating function as an argument for 32 the Plancherel theorem. This allows one to perform the summation over the number of field harmonics so that the analytical expression for the cross section of the process can be explicitly written. As an example, we perform a calculation for the laser-assisted hydrogen recombination. 3.2 Laser-assisted hydrogen recombination The proposed method will be illustrated on a typical example of the laser-assisted hydrogen recombination process, p + e + L¯ ω0 −→ H + hω. h ¯ (3.1) The additional term L¯ ω0 , where L is the number of exchanged laser quanta, indicates the h presence of a laser field, ε = ε0 sin ω0 t, (3.2) and points out the conservation of quasienergy. Here ε0 is the amplitude of the field and ω0 is the field frequency. The differential cross section of the reaction for the standard field-free hydrogen recombination process is known [20] due to the detailed balance between the differential cross section of photo-recombination, dσf i /dΩf , and that of photo-ionization, dσif /dΩi , dσf i k 2 dσif = 2 , dΩf q dΩi where k and q are the momenta of the outgoing photon and electron, respectively. The differential photo-ionization cross section dσif /dΩi is given by [18, 30] 33 (3.3) 2 dσif 7 πe =2 dΩi hc ¯ h2 ¯ mZe2 2 W0 4 e−4ξarccotξ (1 − cos2 θ), −2πξ hω ¯ 1−e (3.4) where W0 is the ionization potential of the hydrogen atom from the continuum threshold to the ground state. Here ω is the frequency of the photon emitted at an angle θ relative to the incoming electron, and m and e are the mass and electric charge of the electron. The dimensionless parameter ξ is defined as ξ= Z¯ h η = , a0 q q (3.5) where Z is the nuclear charge (Z = 1), a0 is the Bohr radius, and η = Z¯ /a0 . h 3.3 S-matrix The S -matrix describing the photo-recombination process (3.1) in the presence of the laser field (3.2) is given by [18, 28, 29] S = −ie dt ΨH (r, t)|( · 0 h )e−i(k·r/¯ −ωt) |χe (r, t) , (3.6) where χe (r, t) is the Coulomb-Volkov wave function of the electron in the field of the proton and external laser field, , k and ω are the polarization vector, momentum and frequency of the emitted photon, respectively, and ΨH (r, t) is the electron wave function in the hydrogen 0 atom in 1s state in the laser field ε, eq. (3.2). Using the standard technique of transformation to the rotating frame we can obtain the 34 electron wave function in the total Coulomb-laser field [31] πξ χe (r, t) = e 2 Γ(1 − iξ)F (iξ, 1; i(qr − q · r)/¯ ) h (3.7) t t iq · r iEi t ie2 ie A2 dτ + q · A(τ )dτ − − × exp − , hmc 0 ¯ h ¯ h ¯ 2¯ mc2 0 h where q is the asymptotic value of the incoming electron momentum, q = |q|, r = |r|, F (a, b; x) is the confluent hypergeometric function, Ei is the initial kinetic energy of the incoming electron, and c is the speed of light. In the harmonic laser field (3.2) the vector-potential A is defined as A(t) = cε0 cos ω0 t ≡ A0 cos ω0 t. ω0 (3.8) t The quadratic term in the square brackets (3.7), (ie2 /2¯ mc2 ) 0 A2 dτ = O 1/c2 , h will be neglected [31]. Evaluation of the remaining integral over τ in (3.7) yields χe (r, t) = Γ(1 − iξ)F (iξ, 1; i(qr − q · r)/¯ ) h q·r E t iπξ × exp i + z sin ω0 t − i − , h ¯ h ¯ 2 (3.9) where the atomic parameter ξ was defined in eq. (3.5), and the field parameter z is given by z= e (q · ε0 ) , 2 m¯ ω0 h (3.10) The exact solution for the electron wave function in the discrete spectrum of a combined Coulomb-laser field is known. However, in our calculation we will assume that laser field introduces only a small perturbation for the ground state of electron in hydrogen. Therefore 35 we can use first-order perturbation theory for deriving the hydrogen wave function in the presence of the laser field [20, 28, 29], h ΨH (r, t) = e−iWn t/¯ n   h h eiω0 t/¯ e−iω0 t/¯ 1 + × ψn (r) −  2 ωmn + ω0 ωmn − ω0 m=n (3.11)  eA0 · p |n ψm (r) , m|  hmcω0 ¯ where ψm (r) is the wave function for the atomic electron in the field-free state |m with energy Wm , and ωmn = (Wm − Wn )/¯ . The summation in (3.11) is extended over the h full set of atomic electron states in the absence of the laser field. In the derivation of (3.11) it was assumed that none of the denominators were close to zero [20]. For the optical frequency of the laser we have ω0 ωn0 and therefore we get the following expression for ΨH (r, t), which holds true even over a broader frequency range [28], 0 h ΨH (r, t) = e−iW0 t/¯ 1 + 0 ie(ε0 · r) H cos ω0 t ψ0 (r), hω0 ¯ (3.12) where H ψ0 (r) = Z 3 −ηr/¯ h ≡ C e−ηr/¯ . h e 0 3 πa0 36 (3.13) 3.4 The Coulomb-Volkov wave function In order to perform the time integration in the S -matrix (3.6) we decompose the electron wave function χe (r, t) given by (3.9) over the Bessel functions JL (z) of integer order L and argument z given by (3.10). For this purpose we introduce the Bessel generating function L=+∞ exp [iz sin u] = JL (z) exp(iLu). (3.14) L=−∞ To simplify the S -matrix (3.6) we use the recurrence relation for the Bessel functions JL+1 (z) + JL−1 (z) = 2L J (z). z L (3.15) Performing the time integration in eq. (3.6) and applying the Gauss theorem we obtain the following expression for the S -matrix in the dipole approximation, (k · r)/¯ h 1, L=+∞ S = −2πi L=−∞ fL δ(W0 + hω − Ei + L¯ ω0 ), ¯ h (3.16) where πξ L fL = e 2 Γ(1 − iξ)C0 JL (z)ω(L) I1 + I2 z . (3.17) Here we have introduced the following notations: I1 = η¯ h I2 = ie ω0 H drψ0 (r) ( · r) e χ (r), r η (ε0 · r)( · r) H drψ0 (r) −(ε0 · ) + χe (r), h ¯ r 37 (3.18) (3.19) and hω(L) = Ei − W0 − L¯ ω0 ≡ Ei0 − L¯ ω0 . ¯ h h (3.20) Exploiting the integral involving the confluent hypergeometric function [18, 32], 2 2 −iξ i(q−p)·r/¯ −ηr/¯ F (iξ, 1, i(qr − q · r)) = 4π¯ 2 [p + (η − iq) ] h h dre h , (3.21) r [(q − p)2 + η 2 ]1−iξ we obtain the following expression for the integral I1 , eq. (3.18), ξ(1 − iξ) −2ξarccotξ e , I1 = 8πi¯ 4 ( · eq ) 2 h q (1 + ξ 2 )2 (3.22) and the corresponding expression for the integral I2 , eq. (3.19), −2(1+ξ)arccotξ 4 ξe e I2 = −8πi¯ h ω0 q 3 (1 + ξ 2 )2 (ε0 · ) − 2( · eq )(ε0 · eq ) (2 − iξ) , (3.23) (1 − iξ) where eq ≡ q/q is a unit vector along the electron momentum. 3.5 The partial cross section The partial cross section of the reaction (3.1) with fixed L is given by 3 e2 2 δ(W + hω − E + L¯ ω ) d k . dσL = 2π |f | h 0 0 ¯ i 2qω(L) L (2π)3 38 (3.24) The integration over ω yields 2 e2 C0 dσL (Ei − W0 − L¯ ω0 )3 h = 2 dΩ 8π q(1 − e−2πξ ) 2L L2 ×|JL (z)|2 |I1 |2 + Re(I1 I2 ) + |I |2 . 2 2 z z (3.25) The conservation of quasi-energy uniquely specifies the frequency of the emitted photon, eq. (3.20), in terms of the number of exchanged laser quanta L. In order to obtain the total cross section of the reaction (3.1) in the laser field (3.2) we must count all possibilities for the number of exchanged laser quanta. In other words, we have to perform the summation over all possible L, dσ = dΩ 3.6 L=+∞ L=−∞ dσL . dΩ (3.26) The summation procedure and cross section We introduce a new step that allows one to analytically sum up the infinite series (3.26). The summation over L in (3.26) is performed with the aid of the Plancherel theorem [33], n=+∞ 2π 1 |f (x)|2 dx = |cn |2 , 2π 0 n=−∞ (3.27) where cn are the Fourier coefficients of the function f (x), 2π 1 cn = f (x)eiπnx dx. 2π 0 39 (3.28) The application of the Plancherel theorem to the Bessel generating function (3.14) leads to the following equalities [34] : L=+∞ 2 JL (z) = 1, (3.29) 2 2 J 2 (z) = z , L L 2 (3.30) 2 L2n−1 JL (z) = 0, n ∈ Z+ (3.31) L=−∞ L=+∞ L=−∞ L=+∞ L=−∞ L=+∞ L=−∞ 2 2 4 J 2 (z) = z (4 + 3z ) . L L 8 (3.32) Performing the summation over the photon polarizations we obtain the closed analytical expression for the cross section of the process (3.1) 2 3 e2 C0 Ei0 dσ = 2 |I1 |2 + F , −2πξ ) dΩ 8π q(1 − e |I |2 3¯ ω0 z 2 2Ei0 h F = 2 + Re(I1 I2 ) + hω0 |I1 |2 ¯ 2 2 z 2Ei0 h2 ω0 z 2 (4 + 3z 2 ) 3Ei0 ¯ 2 2¯ ω0 h + |I2 |2 + Re(I1 I2 ) . 3 z z2 8Ei0 40 (3.33) In the zero-field limit, corresponding to z ≡ 0 and I2 ≡ 0, the correction term F in (3.33), that is responsible for the laser-modified cross section, identically vanishes. To the best of our knowledge, such a closed expression for the laser-assisted hydrogen recombination was not given in the literature. An explicit summation over the photon polarizations is performed as λ∗ λ = δ , µν µ ν (3.34) λ λ (a · λ )2 = a2 (1 − cos2 ϑ), where ϑ is the angle between the vector a, and the momentum of the outgoing photon k . The corresponding expressions for the integrals in eqs. (3.22) and (3.23) with an explicit summation over the photon polarizations are given by 2 −4ξarccotξ 6 π 2h8 ξ e (1 − cos2 θ), |I1 | = 2 ¯ 4 2 )3 q (1 + ξ 2 λ (3.35) 2 2 −4(1+ξ)arccotξ 6 π 2h8 e ξ e (3.36) |I2 | = 2 ¯ 2 ω0 q 6 (1 + ξ 2 )5 λ × ε2 (1 − cos2 φ)(1 + ξ 2 ) − 4(ε0 · eq )2 (2 + ξ 2 ) + 4(ε0 · eq )2 (4 + ξ 2 )(1 − cos2 θ) , 0 2 and for the interference term we find λ 2 −2(1+2ξ)arccotξ 6 π 2h8 eξ e Re(I1 I2 ) = 2 ¯ (ε0 · eq )(4 cos2 θ − 3), ω0 q 5 (1 + ξ 2 )4 41 (3.37) where θ is the angle between the momentum of the incoming electron q and the momentum of the outgoing photon k . The angle φ is formed by vectors ε0 and k . 3.7 The soft photon approximation The cross section given by eq. (3.33) is an exact result for the laser-assisted hydrogen recombination process under the used approximations for the electron and hydrogen lasermodified wave functions. To clarify the meaning of the obtained result we shall introduce a soft photon approximation. This allows one to reveal the meaning of each term in the expression for the cross section of the process given by (3.33). With this approximation the frequency of the emitted photon is independent of the number of the field harmonics L, hω = Ei − W0 − L¯ ω0 ¯ h Ei − W0 . (3.38) In this approximation the partial “soft photon” cross section is given by s 2 dσL e 2 C0 2L (Ei − W0 )3 |JL (z)|2 |I1 |2 + = 2 Re(I1 I2 ) + dΩ z 8π q(1 − e−2πξ ) L2 |I |2 . 2 2 z (3.39) Performing the summation (3.26) over L by means of the equalities (3.29)-(3.31) we obtain the following simple result: 2 2 e 2 C0 dσ s 3 |I |2 + |I2 | = 2 E 1 dΩ 2 8π q(1 − e−2πξ ) i0 . (3.40) Here the first term |I1 |2 corresponds to the standard laser-free recombination process whereas the second term |I2 |2 is responsible for the field-modified electron (3.9) and 42 hydrogen (3.12) states. In the zero-field limit I2 ≡ 0 and (3.40) recovers the standard field-free recombination cross section. We have to note that within the limits of the present approximation (3.38) the interference terms Re(I1 I2 ) are absent. In this and only this particular case does the square of the S -matrix (3.16) equal the sum of the squares of its terms. 3.8 Summary In this chapter we have developed a new method that allows one to obtain an analytical cross section for the laser-assisted electron-ion collision. The standard S -matrix formalism is used for describing the atomic collision process. The S -matrix is constructed from the electron Coulomb-Volkov wave function in the combined Coulomb-laser field, and the hydrogen laserperturbed state. By the aid of the Bessel generating function, the S -matrix is decomposed into an infinite series of the field harmonics. We have introduced a new step to obtain the analytical expression for the cross section of the process. The main theoretical novelty is the application of the Plancherel theorem to the Bessel generating function. This allows one to obtain analytically the cross section of the laser-assisted hydrogen photo-recombination process. This process has been chosen in order to verify the proposed method. The field-enhancement coefficient is evaluated in an analytical way and the final expression for a laser-assisted hydrogen recombination process is presented by the sum of the field-free hydrogen cross section and the laser-assisted addition. The developed method will allow one to reconsider a wide range of problems related to electron-ion collisions in an external field with the goal of obtaining analytical expressions for the cross sections of the corresponding scattering processes. The time-dependent problem 43 generated by the infinite series of the Coulomb-Volkov wave function is exactly separated from the spatial dependence and thus can be analytically solved by the proposed method. 44 Chapter 4 Nuclear Physics In this chapter we present our results for the resonance width distribution in open quantum systems. Recent measurements of resonance widths for low-energy neutron scattering off heavy nuclei claim large deviations from the routinely used chi square, or the Porter-Thomas distribution. We propose a new “standard” width distribution based on the random matrix theory for a chaotic quantum system with a single open decay channel. Two methods of derivation lead to a single analytical expression that recovers, in the limit of very weak continuum coupling, the Porter-Thomas distribution for small widths of experimental interest. The parameter defining the result is the ratio of typical widths Γ to the energy level spacing D . Compared to the Porter-Thomas distribution, the new distribution suppresses small widths and increases the probabilities of larger widths. In the case of a neutron scattering with open multiple photon channels we derive the resonance width distribution which happens to be shifted compared to the Porter-Thomas distribution by an average photon width. The experimental data are very sensitive to a shift of the distribution and therefore the obtained results might be useful in comparison random matrix theory with the nuclear experimental data. The results of this chapter are based on our paper, GS and V. Zelevinsky, Phys. Rev. C 86, 044602 (2012). 45 4.1 Introduction Random matrix theory as a statistical approach for exploring properties of complex quantum systems was pioneered by Wigner and Dyson half a century ago [37]. This theory was successfully applied to excited states of complex nuclei and other mesoscopic systems [38–41], evaluating statistical fluctuations and correlations of energy levels and corresponding wave functions supposedly of “chaotic” nature. The standard random matrix approach based on the Gaussian Orthogonal Ensemble (GOE) for systems with time-reversal invariance, and on the Gaussian Unitary Ensemble (GUE) if this invariance is violated, was formulated originally for closed systems with no coupling to the outside world. Although the practical studies of complex nuclei, atoms, disordered solids, or microwave cavities always require the use of reactions produced by external sources, the typical assumption was that such a probe at the resonance is sensitive to the specific components of the exceedingly complicated intrinsic wave function, one for each open reaction channel, and the resonance widths are measuring the weights of these components [42]. With the Gaussian distribution of independent amplitudes in a chaotic intrinsic wave function, the widths under this assumption are proportional to the squares of the amplitudes and as such can be described, for ν independent open channels, by the chisquare distribution with ν degrees of freedom. For low-energy elastic scattering of neutrons off heavy nuclei, where the interactions can be considered time-reversal invariant, one expects ν = 1 that is usually called the Porter-Thomas distribution (PTD) [43], χ2 (x) ν ν=1 = e−x/2 x(ν−2)/2 2ν/2 Γ[ν/2] 46 e−x/2 =√ , 2πx ν=1 (4.1) where Γ[z] is the gamma function. Recent measurements [44,45] claimed that the neutron width distributions in low-energy neutron resonances on certain heavy nuclei are different from the PTD. As a rule, the fraction of greater widths is increased, while the fraction of narrow resonances is reduced which, being approximately presented with the aid of the same standard class of functions χ2 , ν would require ν = 1. The literature discussing the scattering and decay processes in chaotic systems, see for example [46–49] and references therein, does not provide a detailed description of the width distribution for the region of relatively small widths observed in low-energy neutron resonances. There are various reasons for possible deviations from the simple statistical predictions [50–52]. First of all, the intrinsic dynamics, even in heavy nuclei, can be different from that in the GOE limit of many-body quantum chaos. If so, the detailed analysis of specific nuclei is required. As an example we can mention 232 Th, where for a long time a sign problem exists [53] concerning the resonances with strong enhancement of parity nonconservation in scattering of longitudinally polarized neutrons. The observed predominance of a certain sign of parity violating asymmetry contradicts to the statistical mechanism of the effect and may be related to the non-random coupling between quadrupole and octupole degrees of freedom [54]. The width distribution in the same nucleus reveals noticeable deviations from the PTD. The presence of a shell-model single-particle resonance serving as a doorway from the neutron capture to the compound nucleus can also make its footprint distorting the statistical pattern. Another (maybe related to the doorway resonance) effect can come from the changed energy dependence of the widths that is usually assumed to be proportional to E +1/2 for neutrons with orbital momentum . Finally, the situation is not strictly one-channel, since, along with elastic neutron scattering, many gamma-channels are open 47 as well. However, apart from structural effects, even in one-channel approximation, there exists a generic cause for the deviations from the PTD, since the applicability of the GOE is anyway violated by the open character of the system [55]. The appropriate modification of the GOE and PTD predictions, which should be applied before making specific conclusions, is our goal below. The resonances are not the eigenstates of a Hermitian Hamiltonian, they are poles of the scattering matrix in the complex plane. Their complex energies E = E − iΓ/2 can be rigorously described as eigenvalues of the effective non-Hermitian Hamiltonian [56]. As shown long ago, even for a single open channel, the statistical properties of the complex energies cannot be described by the GOE. The new dynamics is related to the interaction of intrinsic states through continuum. In the limit of strong coupling this leads to the overlapping resonances, Ericson fluctuations of cross sections, and sharp redistribution of widths similar to the phenomenon of super-radiance, see the review [57] and references therein. The control parameter of such restructuring is the ratio κ= πΓ 2D (4.2) of typical widths, Γ, to the mean spacing between the resonances, D . In the region of low-energy neutron resonances, κ is still small but in order to correctly separate the general statistical effects from peculiar properties of individual nuclei we need to have at our disposal a generic width distribution that differs from the PTD as a function of the degree of openness. 48 4.2 Resonance width distribution We need a practical tool that would allow one to compare an experimental output for an unstable quantum system with predictions of random matrix theory. We propose a new distribution function that is based, similar to the GOE, on the chaotic character of time-reversal invariant internal dynamics and corresponding decay amplitudes, but properly accounts for the continuum coupling through the effective non-Hermitian Hamiltonian. The numerical simulations for this Hamiltonian were described earlier [51, 58] but here we derive the analytical expression. In the typical case of nuclear applications, the introduced dimensionless parameter κ is bound from above by one. The super-radiant regime, κ ≥ 1, can be of special interest, including such systems as microwave cavities, and in the considered framework the formal symmetry exists, κ → 1/κ. At a large number of resonances and fixed number of open channels, after the super-radiant transition the broad state becomes a part of the background while the remaining “trapped” states return into the non-overlap regime. However, in heavy nuclei this transition hardly can be observed because earlier many new channels can be opened; in the modification of the PTD we see only precursors of this transition. Our arguments will follow two different routes which lead to the equivalent results. The final formula for the statistical width distribution can be presented as 1/2 πΓ (η−Γ) exp − N2 Γ(η − Γ) sinh 2D η 2σ   P (Γ) = C πΓ (η−Γ) Γ(η − Γ) 2D η  Here we consider N . (4.3) 1 intrinsic states coupled to a single decay channel, for example, s-wave elastic neutron scattering. The parameter D is a mean energy spacing between the 49 P 0.30 0.25 Sinh 2 Cnorm Χ1 2 Χ1 0.20 0.15 D Π 2D Π 2D 4 D 2 0.10 0.05 2 4 6 8 10 Figure 4.1: The proposed resonance width distribution according to eq. (4.3) with a single neutron channel in the practically important case η Γ. The width Γ and mean level spacing D are measured in units of the mean value Γ . “For interpretation of the references to color in this and all other figures, the reader is referred to the electronic version of this dissertation.” resonances, and C is a normalization constant. The quantity η is the total sum (the trace of the imaginary part of the effective non-Hermitian Hamiltonian that remains invariant in the transition to the biorthogonal set of its eigenfunctions) of all N widths; it appears as a parameter that fixes the starting ensemble distribution, see below eqs. (4.9) and (4.10). The possible values of widths are restricted from both sides, 0 < Γ < η . The above mentioned symmetry κ → 1/κ is reflected in the symmetry Γ → (η − Γ) of a factor in eq. (4.3) but, as was already stated, our region of interest is at Γ η . Another parameter, σ , determines the standard deviation of variable Γ evaluated consistently with the distribution 50 of eq. (4.3). In the practical region far away from the super-radiance we obtain P (Γ) = Cχ2 (Γ) 1 sinh κ 1/2 , κ (4.4) where κ is a new dimensionless combination, eq. (4.2). The PTD is recovered in the limiting case κ → 0 that corresponds to the approximation of an open quantum system by a closed one. The new element is the factor explicitly determined by the coupling strength κ. With growing continuum coupling the probability of larger widths increases. The distribution (4.3) for different ratios Γ /D is shown in Fig. 4.1. The origin of the square root in the new factor is the linear energy level repulsion typical for the GOE spectral statistics. Indeed, in the complex plane, E = E − iΓ/2, the distance between two poles Em and En is (Em − En )2 + (Γm − Γn )2 /4; after integration over all variables of other states we obtain a characteristic square root in the level repulsion, see below eq. (4.13) and the discussion after eq. (4.22). 51 4.3 Effective non-Hermitian Hamiltonian and scattering matrix In order to come to the result (4.3), we start with the general description of complex energies E = E − iΓ/2 in a system of N unstable states satisfying the GOE statistics inside the system and interacting with the single open channel through Gaussian random amplitudes. The general reaction theory [59] is constructed in terms of the elements of the scattering matrix in the space of open channels a, b, ..., S ba (E) = δ ba − iT ba (E). (4.5) Within the formalism of the effective non-Hermitian Hamiltonian H = H − (i/2)W , the T -matrix is defined as N T ba (E) = 1 Aa E − H mn n Ab∗ m m,n=1 (4.6) in terms of the amplitudes Aa connecting an internal basis state n with an open channel n a. Here we do not explicitly indicate the potential part of scattering that is not related to the internal dynamics of the compound nucleus. The anti-Hermitian part of the effective Hamiltonian is exactly represented by the sum over k open channels, k Aa Aa∗ , m n Wmn = a=1 52 (4.7) where the amplitudes can be considered real in the case of time-reversal invariance. It is important that the factorized structure of the effective Hamiltonian guarantees the unitarity of the scattering matrix. The amplitudes Aa are assumed to be uncorrelated Gaussian n quantities with zero mean and variance defined as Aa Ab = δ ab δnn η/N . The trace of n n the anti-Hermitian part of the effective Hamiltonian, η = TrW , i.e. the total sum of all N widths used in eq. (4.3), is a quantity invariant under orthogonal transformations of the intrinsic basis. The detailed discussion of the whole approach, numerous applications and relevant references can be found in the recent review article [57]. The simplest version of the R-matrix description uses instead of the amplitude T ba its approximate form, where the denominator contains poles on the real energy axis corresponding to the eigenvalues of the Hermitian part H of the effective Hamiltonian. Then the continuum coupling occurs only at the entrance and exit points of the process while the influence of this coupling on the intrinsic dynamics of the compound nucleus is neglected (in general, the Hermitian part of the Hamiltonian, H , should also be renormalized by the off-shell contributions from the presence of the decay channels). Contrary to that, the full amplitude T ba , eq. (4.6), accounts for this coupling during the entire process including the virtual excursions to the continuum and back from intrinsic states. The poles are the eigenvalues of the full effective Hamiltonian in the lower half of the complex energy plane. The experimental treatment corresponds to this full picture. According to the original experimental paper [44], the R-matrix code SAMMY [60] had been used in the experimental analysis where the relevant expression is given in the form Rcc = λ γλc γλc δ , Eλ − E − iΓλ /2 JJ 53 (4.8) and the treatment included a careful segregation of s- and p-resonances, J = J = 1/2 for an even target nucleus. In the notations of [60] λ represents a particular resonance, Eλ is the energy of the resonance. Here we can identify the intermediate states λ and their complex energies Eλ − iΓλ /2 with the eigenstates and complex eigenvalues of H, while the numerator includes the amplitudes transformed to this new basis (under time-reversal 2 invariance the scattering matrix is symmetric). In terms of the reduced width γλc and the 2 penetration factor Pc , the partial width is Γλc = 2Pc γλc . Assuming a single channel and universal energy dependence of penetration factors, the statistics of the total widths is the 2 same as that of γλc . 4.4 From ensemble distribution to single width distribution For a single-channel case, the joint distribution P (E; Γ) of all complex energy poles has been rigorously derived in [55] under assumptions of the GOE intrinsic dynamics in the closed system and Gaussian distributed random decay amplitudes. The result is given by (Γm −Γn )2 1 4 √ e−N F (E;Γ) , Γn (Γ +Γ )2 n (Em − En )2 + m 4 n (Em − En )2 + P (E; Γ) = CN m