Quantum leptogenesis I

Annals of Physics 326 (2011) 1998–2038

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Quantum leptogenesis I A. Anisimov a, W. Buchmüller b,⇑, M. Drewes c, S. Mendizabal d a

Fakultät für Physik, Universität Bielefeld, 33615 Bielefeld, Germany Deutsches Elektronen-Synchrotron DESY, 22603 Hamburg, Germany c Institute de Théorie des Phénomènes Physiques EPFL, 1015 Lausanne, Switzerland d Institut für Theoretische Physik, Goethe-Universität, 60438 Frankfurt am Main, Germany b

a r t i c l e

i n f o

Article history: Received 5 January 2011 Accepted 3 February 2011 Available online 21 March 2011 Keywords: Thermal field theory Nonequilibrium Leptogenesis Particle physics Cosmology

a b s t r a c t Thermal leptogenesis explains the observed matter–antimatter asymmetry of the universe in terms of neutrino masses, consistent with neutrino oscillation experiments. We present a full quantum mechanical calculation of the generated lepton asymmetry based on Kadanoff–Baym equations. Origin of the asymmetry is the departure from equilibrium of the statistical propagator of the heavy Majorana neutrino, together with CP violating couplings. The lepton asymmetry is calculated directly in terms of Green’s functions without referring to ‘‘number densities’’. Compared to Boltzmann and quantum Boltzmann equations, the crucial difference are memory effects, rapid oscillations much faster than the heavy neutrino equilibration time. These oscillations strongly suppress the generated lepton asymmetry, unless the standard model gauge interactions, which cause thermal damping, are properly taken into account. We find that these damping effects essentially compensate the enhancement due to quantum statistical factors, so that finally the conventional Boltzmann equations again provide rather accurate predictions for the lepton asymmetry. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Standard thermal leptogenesis [1] provides a simple and elegant explanation of the origin of matter in the universe. Baryogenesis via leptogenesis naturally emerges in grand unified extensions of the Standard Model, which incorporate right-handed neutrinos and the seesaw mechanism, and the predicted connection between the cosmological matter–antimatter asymmetry and neutrino properties is in remarkable agreement with the present evidence for neutrino masses [2]. ⇑ Corresponding author. E-mail address: [email protected] (W. Buchmüller). 0003-4916/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aop.2011.02.002

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Leptogenesis is an out-of-equilibrium process in the high-temperature symmetric phase of the Standard Model. It makes use of nonperturbative properties of the Standard Model, the sphaleron processes which change baryon and lepton number [3], and it requires CP violation in the lepton sector and quantum interference in the thermal bath. Almost all quantitative studies of leptogenesis to date are based on Boltzmann’s classical kinetic equations for the description of the nonequilibrium process [2]. In this article, we discuss a full quantum mechanical calculation of the generated lepton asymmetry based on Kadanoff–Baym equations [4] and the Schwinger–Keldysh formalism [5–7]. The main result has previously been reported in [8]. Here we give a detailed derivation of the result, discuss its interpretation and set the stage for future computations. Further work is still needed to obtain a ‘quantum theory of leptogenesis’ that can predict the cosmological matter–antimatter asymmetry in terms of neutrino properties without uncontrolable assumptions. Conventional leptogenesis calculations based on kinetic equations suffer from a basic conceptual problem: the Boltzmann equations are classical equations for the time evolution of phase space distribution functions; the involved collision terms, however, are obtained from zero-temperature S-matrix elements which involve quantum interferences. This is in contrast to other successful applications of the Boltzmann equations in cosmology, like primordial nucleosynthesis, decoupling of photons or freeze-out of weakly interacting dark matter particles, where the collision terms arise from tree-level S-matrix elements. In the case of leptogenesis, clearly a full quantum mechanical treatment is necessary to understand the range of validity of the Boltzmann equations and to determine the size of possible corrections [9]. In recent years, various attempts have been made to go beyond Boltzmann equations. In [9], a solution of Kadanoff–Baym equations for leptogenesis has been found to leading order in a derivative expansion in terms of distribution functions satisfying the Boltzmann equations. Various thermal corrections, in particular quantum statistical factors and thermal masses, have been included [10–13]. Quantum Boltzmann equations have been derived from Kadanoff–Baym equations for scalar and Yukawa theories [14,15] and for leptogenesis [16–19]. Except for [16], they do not contain memory effects, but they yield the correct statistical factors which go beyond the Boltzmann equations [8,16,17,20,19]. Quantum Boltzmann equations have important applications for resonant leptogenesis [16], flavoured leptogenesis [21,22] and N2-leptogenesis [23]. Similar techniques have been developed for electroweak baryogenesis [24–27] and for coherent baryogenesis [28]. The quantum treatment of leptogenesis discussed in this paper is entirely based on Green’s functions, thus avoiding all approximations needed to arrive at Boltzmann equations. Our work is based on [29], where the approach to thermal equilibrium has been discussed in terms of Green’s functions for a toy model, a scalar field coupled to a large thermal bath. In leptogenesis it is the heavy neutrino which is weakly coupled to the standard model plasma containing many degrees of freedom. The nonequilibrium propagator of the heavy neutrino is obtained by solving the Kadanoff–Baym equations. The induced quantum corrections of the lepton (and Higgs) propagators then yield the wanted lepton asymmetry. In general baryogenesis requires departure from thermal equilibrium. For the cosmological baryon asymmetry, this is provided by the Hubble expansion of the universe and, possibly, also by initial conditions. This can be seen in Fig. 1 where the time evolution of heavy neutrino abundance and lepton asymmetry, as predicted by the Boltzmann equations, are shown for two different initial conditions: thermal and zero heavy neutrino abundance. In the first case, the Hubble expansion leads to an excess of the neutrino abundance at T ’ 0.3M1; shortly afterwards, washout processes are no longer in equilibrium and the lepton asymmetry is ‘frozen in’. This is the standard out-of-equilibrium decay scenario of baryogenesis. In the second case, interactions with the thermal bath first bring the heavy neutrino into thermal equilibrium; due to the departure from thermal equilibrium during this time, an initial lepton asymmetry is generated. Around T ’ 0.3M1, this asymmetry is washed out and, as in the first case, the final lepton asymmetry is generated. Remarkably, the initial and the final asymmetry have about the same size. For the generation of the initial asymmetry the change of temperature due to the Hubble expansion is not important. This allows us to make a significant technical simplification in our analysis. Since our goal is the comparison of Boltzmann and Kadanoff–Baym equations, we concentrate on the computation of the initial asymmetry at constant temperature. We expect differences

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NN

0

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in

NN =0 1

in

NN =3/4 1

NN

eq

1

|NB-L|

10

-1

10

0

10

z=M1/T

1

Fig. 1. Evolution of heavy neutrino abundance NN1 and lepton asymmetry NBL for typical leptogenesis parameter: ~ 1 ¼ 8pC1 ðv ew =M 1 Þ2 ¼ 103 eV;  ¼ 106 ; the inverse temperature z = M1/T is the time variable. The dashed M1 ¼ 1010 GeV; m (full) lines correspond to thermal (vacuum) initial conditions for the heavy neutrino abundance; the dotted line represents the equilibrium abundance. From [30].

between the classical and the quantum approach to be of similar size in the generation of the final asymmetry. In our numerical analysis we shall consider temperatures T [ M, where the heavy neutrino production rate is not strongly affected by the effect of thermal masses of lepton and Higgs fields [11–13]. We consider an extension of the Standard Model with additional gauge singlet fermions, i.e., righthanded neutrinos, whose masses and couplings are described by the Lagrangian (sum over i, j),

L ¼ LSM þ mRi i

1 2





~  mRj þ mRj kij lLi /  Mij mc i mRj þ mRj mc : mRi þ lLi /k ij Ri R

ð1:1Þ

~ ¼ ir2 / ; SU (2) isospin indices have been TR ; C is the charge conjugation matrix and / Here mcR ¼ C m omitted. For simplicity, we consider the case of hierarchical Majorana masses, Mk>1  M1  M, and small Yukawa couplings of the lightest heavy neutrino N1  N, ki1  1, such that the decay width is much smaller than the mass. Leptogenesis is then dominated by decays and inverse decays of N, and it is convenient to integrate out the heavier neutrinos. From Eq. (1.1) one then obtains the effective Lagrangian

1 ~  N þ NT ki1 ClLi /  1 MNT CN þ 1 g lT /ClLj / þ 1 g lLi /Cl ~ L T /; ~ L ¼ LSM þ Ni N þ lLi /k i1 j 2 2 2 ij Li 2 ij

ð1:2Þ

with N ¼ mR1 þ mcR1 , and the familiar dimension-5 coupling

gij ¼

X k>1

kik

1 T k : M k kj

ð1:3Þ

Using this effective Lagrangian has the advantage that vertex- and self-energy contributions to the CP asymmetry in the heavy neutrino decay [31–33] are obtained from a single graph [9]. The paper is organised as follows. In Section 2 we present solutions of the Boltzmann equations for the heavy neutrino distribution function and the lepton asymmetry, which are useful for later comparison with the Kadanoff–Baym equations. Some results from nonequilibrium quantum field theory (QFT), in particular equilibrium correlation functions and Kadanoff–Baym equations, are recalled in

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Section 3. Section 4 contains some of the main results of this paper: analytic solutions of spectral function and statistical propagator for the heavy neutrino. These are needed for the computation of the lepton asymmetry, which is carried out in Section 5. A detailed comparison of the Boltzmann result and the Kadanoff–Baym result is given in Section 6, and numerical results for the generated lepton asymmetries are compared in Section 7. Summary and conclusions are given in Section 8, and various details, including equilibrium correlation functions, Feynman rules, a discussion of the zero-width limit and the computation of some integrals are contained in Appendices A–D. 2. Boltzmann equations The Boltzmann equations for the time evolution of the distribution functions of heavy neutrinos, lepton and Higgs doublets are well known [34]. As discussed in the previous section, we focus on the generation of the ‘initial asymmetry’ (cf. Fig. 1), which allows us to neglect Hubble expansion and washout terms and to work at constant temperature T. The distribution function of the heavy neutrinos is then determined by the first-order differential equation1

@ 2 fN ðt; xp Þ ¼  @t xp

Z

k;q

  ð2pÞ4 d4 ðk þ q  pÞ ky k 11 p  k½fN ðt; xp Þð1  fl ðkÞÞð1 þ f/ ðqÞÞ

 fl ðkÞf/ ðqÞð1  fN ðt; xp ÞÞ;

ð2:1Þ

with vacuum initial condition,

fN ð0; xp Þ ¼ 0;

ð2:2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here xp ¼ M 2 þ p2 ; k and q are the energies of N, l and / with equilibrium distribution functions fl and f/, respectively; the averaged decay matrix element is jM(N(p) ? l(k)/(q))j2 = 2(k k)11p  k (cf. [9]). For the momentum integrations we use the notation

Z

 ¼

Z

3

d p ð2pÞ3 2x

p



ð2:3Þ

In most leptogenesis calculations one directly computes the number density,

nN ðtÞ ¼

Z

3

d p ð2pÞ3

f N ðt; xp Þ;

ð2:4Þ

assuming kinetic equilibrium. The sum of decay and inverse decay widths, whose inverse is the time needed to reach thermal equilibrium [35], is given by

Cp ¼ ðky kÞ11

2

xp

Z

ð2pÞ4 d4 ðk þ q  pÞ p  k f l/ ðk; qÞ;

ð2:5Þ

k;q

where we have introduced the statistical factor (cf. [35])

fl/ ðk; qÞ ¼ fl ðkÞf/ ðqÞ þ ð1  fl ðkÞÞð1 þ f/ ðqÞÞ ¼ 1  fl ðkÞ þ f/ ðqÞ:

ð2:6Þ

Neglecting the momentum dependence of the heavy neutrino width (Cp  C), one easily obtains the solution of the Boltzmann equation (2.1) with vacuum initial condition,

fN ðt; xp Þ ¼ fNeq ðxp Þð1  eCt Þ;

ð2:7Þ

where the equilibrium distribution is

fNeq ðxp Þ ¼

1 ; ebxp þ 1

ð2:8Þ

and b = 1/T is the inverse temperature. 1

To simplify notation, we use the same symbol for the modulus of 3-momentum and 4-momentum, e.g., k = jkj and k = (jkj, k).

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To compute the lepton asymmetry, we need the Boltzmann equation for the lepton distribution function,

Z

h ð2pÞ4 d4 ðk þ q  pÞ jMðl/ ! NÞj2 fl ðkÞf/ ðqÞð1  fN ðt; xp ÞÞ q;p i jMðN ! l/Þj2 fN ðt; xp Þð1  fl ðkÞÞð1 þ f/ ðqÞÞ ;

@ 1 fl ðt; kÞ ¼  @t 2k

ð2:9Þ

where now Oðk4 Þ corrections to the matrix elements have to be kept. Using Eq. (2.7) one obtains for the lepton asymmetry

fLi ðt; kÞ ¼ fli ðt; kÞ  fli ðt; kÞ;

ð2:10Þ

with initial condition fLi(0,k) = 0,

fLi ðt; kÞ ¼ ii

1 k

Z q;p

ð2pÞ4 d4 ðk þ q  pÞ p  k f l/ ðk; qÞfNeq ðxp Þ

1

C

ð1  eCt Þ;

ð2:11Þ

where we have defined

ij ¼

n o 3 Im ki1 ðgk Þj1 M: 16p

ð2:12Þ

Summing over all lepton flavours, the generated lepton asymmetry is proportional to the familiar CP asymmetry [9],

¼

X i



 y  3 Im k gk 11 M   y  ¼ : ky k 11 16p k k 11

ii

ð2:13Þ

For later comparison with solutions of the Kadanoff–Baym equations, it is convenient to rewrite Eq. (2.11) as a 4-fold integral,

fLi ðt; kÞ ¼ ii

16p k

Z q;p;q0 ;k0

0

0

k  k ð2pÞ4 d4 ðk þ q  pÞð2pÞ4 d4 ðk þ q0  pÞfl/ ðk; qÞfNeq ðxp Þ

1

C

ð1  eCt Þ: ð2:14Þ

 2 ¼ 2k  k0 ðky kÞ =M 2 The integrand is now proportional to the averaged matrix element jMðl/ ! l/Þj 11 (cf. [9]), which involves the product of the 4-vectors k and k0 . At low temperatures, T  M, the integrand falls off like ebxp < ebM , i.e., the generated asymmetry is strongly suppressed. In standard leptogenesis calculations one considers the integrated lepton asymmetry,

nL ¼

XZ i

3

d k ð2pÞ3

f Li ðt; kÞ:

ð2:15Þ

The number densities nN (2.4) and nL correspond to the comoving number densities N N1 and jNBLj shown in Fig. 1, in the initial phase of the time evolution, i.e., for T J 0.3M. 3. Nonequilibrium QFT and Kadanoff–Baym equations In the following, we briefly introduce concepts and quantities from nonequilibrium quantum field theory that are necessary for our computation (cf. [36,37]). A thermodynamical system is represented by a statistical ensemble described by a density matrix .. The expectation value for an operator A is then given by

hAi ¼ Trð.AÞ;

ð3:1Þ

where we have adopted the usual normalisation Tr . = 1. Solving the initial value problem for . allows to compute all observables for all times. Direct computation of the time evolution of . is difficult. Generically, the von Neumann (or quantum Liouville) equation of motion for . can only be solved perturbatively for a reduced density matrix with an effective Hamiltonian. In most practical applications

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to date, a number of additional assumptions are made that lead to effective Boltzmann equations, which can take account of coherent oscillations,2 or quantum corrected Boltzmann equations (cf. Section 6).3 Instead of the time evolution of the density matrix, one can also directly study the equations of motion of the correlation functions of the theory. The infinitely many degrees of freedom of the initial density matrix are then mapped onto their infinitely many initial conditions. Though a full characterisation of the system in principle involves all n-point functions, it is often sufficient to study the oneand two-point function. This applies to the problem considered in this work. 3.1. Correlation functions for lepton and Higgs fields Leptogenesis occurs at temperatures above the electroweak scale where sphaleron processes are active and transfer the generated lepton asymmetry to a baryon asymmetry. Hence, the Standard Model is in the symmetric phase and the four real degrees of freedom of the Higgs doublet correspond to four massless real scalar fields. The spectral function and statistical propagator of a real scalar field /, D and D+, respectively, are defined as

D ðx1 ; x2 Þ ¼ ih½/ðx1 Þ; /ðx2 Þi; 1 Dþ ðx1 ; x2 Þ ¼ hf/ðx1 Þ; /ðx2 Þgi: 2

ð3:2Þ ð3:3Þ

Here only contributions from connected diagrams are to be included to compute the dressed correlation functions. These fulfil the symmetry relations

D ðx1 ; x2 Þ ¼ D ðx2 ; x1 Þ; Dþ ðx1 ; x2 Þ ¼ Dþ ðx2 ; x1 Þ;

ð3:4Þ ð3:5Þ

which follow directly from the definitions. The functions D± have an intuitive physical interpretation. The spectral function D is the Fourier transform of the spectral density,

qq ðt; xÞ ¼ i

Z

dy ixy   y y e D t þ ;t  ; 2p 2 2

ð3:6Þ

where we have used the relative and total time coordinates, y = t1  t2 and t = (t1 + t2)/2, respectively. The spectral density qq(t, x) characterises the density of quantum mechanical states in phase space. Propagating states, or resonances, appear as peaks in the spectral function. The statistical propagator contains the information about the occupation number of each state. In the following we shall also need the Wightman functions

D> ðx1 ; x2 Þ ¼ h/ðx1 Þ/ðx2 Þi; D< ðx1 ; x2 Þ ¼ h/ðx2 Þ/ðx1 Þi;

ð3:7Þ ð3:8Þ

±

which are related to D by

D ðx1 ; x2 Þ ¼ iðD> ðx1 ; x2 Þ  D< ðx1 ; x2 ÞÞ; 1 Dþ ðx1 ; x2 Þ ¼ ðD> ðx1 ; x2 Þ þ D< ðx1 ; x2 ÞÞ: 2

ð3:9Þ ð3:10Þ

Using microcausality and the condition for canonical quantization,

 _ 1 Þ; /ðx _ 2 Þ ½/ðx1 Þ; /ðx2 Þjt1 ¼t2 ¼ ½/ðx ¼ 0; t 1 ¼t 2  _ 2 Þ ¼ idðx1  x2 Þ; ½/ðx1 Þ; /ðx t ¼t 1

2

2

ð3:11Þ ð3:12Þ

See [38,39] for an application to neutrino oscillations. In [40,41] an approach based on first principles has been suggested that is applicable if the occupation numbers for the outof-equilibrium fields are small. 3

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one obtains boundary conditions in y = t1  t2 for D,

D ðx1 ; x2 Þjt1 ¼t2 ¼ 0;

ð3:13Þ

@ t1 D ðx1 ; x2 Þjt1 ¼t2 ¼ @ t2 D ðx1 ; x2 Þjt1 ¼t2 ¼ dðx1  x2 Þ;

ð3:14Þ



@ t1 @ t2 D ðx1 ; x2 Þjt1 ¼t2 ¼ 0:

ð3:15Þ

Note that these conditions do not depend on the physical initial conditions of the system encoded in the initial density matrix. These enter via the initial conditions for the statistical propagator. Analogous to D±, one can define the spectral function and statistical propagator for fermions. The fermionic fields in the Lagrangian (1.2) are massless left-handed leptons (Weyl fields lLi) and a massive neutrino (Majorana field N). For the massless leptons, the spectral function and statistical propagator are defined as

 lLia ðx1 Þ; lLjb ðx2 Þ ;

 1 lLia ðx1 Þ; lLjb ðx2 Þ ; ðSþLij Þab ðx1 ; x2 Þ ¼ 2

ðSLij Þab ðx1 ; x2 Þ ¼ i



ð3:16Þ ð3:17Þ

where a and b are spinor indices, and SU(2) indices were omitted for notational simplicity. The sub 5 ± script L denotes the projection to left-handed fields, i.e., S L ¼ P L S , where PL = (1  c )/2 and S are the propagators for Dirac fermions. As for bosons, we shall need the functions

 ðS>Lij Þab ðx1 ; x2 Þ ¼ lLia ðx1 ÞlLjb ðx2 Þ ; ðS< Þ ðx1 ; x2 Þ ¼ hl ðx2 Þl ðx1 Þi;

ð3:19Þ

Lia

Ljb

Lij ab

ð3:18Þ

which are related to spectral function and statistical propagator by

  SLij ðx1 ; x2 Þ ¼ i S>Lij ðx1 ; x2 Þ  S SþLij ðx1 ; x2 Þ ¼ SLij ðx1 ; x2 Þ þ S
ð3:20Þ ð3:21Þ

The propagators S± have the symmetry properties

h

iy

h

iy

c0 SLij ðx1 ; x2 Þ c0 ¼ SLji ðx2 ; x1 Þ;

ð3:22Þ

c0 SþLij ðx1 ; x2 Þ c0 ¼ SþLji ðx2 ; x1 Þ:

ð3:23Þ

The canonical quantization condition,

n

o y lLia ðx1 Þ; lLjb ðx2 Þ ¼ PLab dij dðx1  x2 Þ;

ð3:24Þ

implies the boundary condition for the spectral function

  SLij ðx1 ; x2 Þ

t1 ¼t2

¼ iPL dij dðx1  x2 Þ:

ð3:25Þ

Finally, the spectral function and statistical propagator for the Majorana field N read

  Gab ðx1 ; x2 Þ ¼ i Na ðx1 Þ; Nb ðx2 Þ ;

 1 Gþab ðx1 ; x2 Þ ¼ Na ðx1 Þ; Nb ðx2 Þ : 2

ð3:26Þ ð3:27Þ

They have the symmetries

G ðx1 ; x2 Þ ¼ G ðx2 ; x1 ÞT ; þ

þ

ð3:28Þ T

G ðx1 ; x2 Þ ¼ G ðx2 ; x1 Þ :

ð3:29Þ T

The canonical quantization condition, together with the Majorana property N ¼ CN ; implies the boundary condition

G ðx1 ; x2 Þjt1 ¼t2 ¼ ic0 dðx1  x2 ÞC 1 :

ð3:30Þ

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Fig. 2. Path in the complex time plane for nonequilibrium Green’s functions. The contour runs from some initial time x0 = ti + i (ti = 0) parallel to the real axis (x0 = t + i) up to some final time tf + i and returns to ti  i. To compute physical correlation functions for arbitrary times t > ti, one takes the limits tf ? 1 and  ? 0.

As for scalars, the physical initial conditions enter as boundary conditions for the statistical propagator. In the following, we will consider two types of initial conditions, thermal equilibrium and Gaussian initial correlations, for which we solve the equations of motion in the following section. Analogous to real scalars, the functions G? are defined as

 G>ab ðx1 ; x2 Þ ¼ Na ðx1 ÞNb ðx2 Þ ;

 G
ð3:31Þ ð3:32Þ

with the usual relations to spectral function and statistical propagator,

G ðx1 ; x2 Þ ¼ iðG> ðx1 ; x2 Þ  G< ðx1 ; x2 ÞÞ; 1 Gþ ðx1 ; x2 Þ ¼ ðG> ðx1 ; x2 Þ þ G< ðx1 ; x2 ÞÞ: 2

ð3:33Þ ð3:34Þ

3.2. Equations of motion In thermal leptogenesis, the deviation from thermal equilibrium that is necessary to create a matter–antimatter asymmetry is due to the heavy Majorana neutrinos which are out of equilibrium. The equations of motion for their correlation functions G± can be obtained via the Schwinger–Keldysh formalism [6]. The basic quantity is the Green’s function with time arguments defined on a contour C in the complex x0-plane, known as the Keldysh contour (cf. Fig. 2),

GC ðx1 ; x2 Þ ¼ hC ðx01 ; x02 ÞG> ðx1 ; x2 Þ þ hC ðx02 ; x01 ÞG< ðx1 ; x2 Þ:

ð3:35Þ

Here the h-functions enforce path ordering along the contour C. The necessity of considering Green’s functions with time arguments on the Keldysh contour (rather than the real axis) is a consequence of the fact that nonequilibrium processes are initial value problems. The system is prepared at initial time ti, its state at later times is unknown. Hence, the usual approach to define a S-matrix by projection onto asymptotic ‘in’ and ‘out’ states, sending initial and final time to infinity, cannot be applied. When using the Keldysh contour which starts and ends at the same time ti,4 no knowledge of the system’s state at t = ±1 is needed to define a generating functional for correlation functions. The Green’s function GC satisfies the Schwinger–Dyson equation

Cði @ 1  MÞGC ðx1 ; x2 Þ  i

Z

4

d x0 C RC ðx1 ; x0 ÞGC ðx0 ; x2 Þ ¼ idC ðx1  x2 Þ;

ð3:36Þ

C

where C RC ðx1 ; x0 Þ is the self-energy5 on the contour and the index of the function, also the self-energy can be decomposed as

RC ðx1 ; x2 Þ ¼ hC ðx01 ; x02 ÞR> ðx1 ; x2 Þ þ hC ðx02 ; x01 ÞR< ðx1 ; x2 Þ:

l

1

¼ cl @=@x1 . Like the Green’s

ð3:37Þ

In the Schwinger–Dyson equation (3.36) the time coordinates of GC and RC can lie on the upper or the lower branch of the contour. The familiar time-ordered Feynman propagator is obtained from GC ðx1 ; x2 Þ when both time arguments lie on the upper branch, and therefore denoted by G11. Correspondingly, GC ðx1 ; x2 Þ with both 4 Due to this fact this formalism is sometimes called ‘in–in’ formalism, in contrast to the ‘in–out’ formalism used to compute the S-matrix. 5 An explicit factor C is factorized for later convenience.

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time arguments on the lower part of the contour corresponds to an anti-time-ordered propagator, denoted as G22. For correlators with one time argument on the upper and one on the lower part of the contour, referred to as G12 and G21, the order of field operators is fixed by the path ordering: operators on the upper branch are always ‘earlier’ than those on the lower branch (cf. (3.35)). Altogether, one has

G12 ðx1 ; x2 Þ ¼ G< ðx1 ; x2 Þ; 21

ð3:38Þ

>

G ðx1 ; x2 Þ ¼ G ðx1 ; x2 Þ;

ð3:39Þ

i G ðx1 ; x2 Þ ¼ G ðx1 ; x2 Þ  signðx01  x02 ÞG ðx1 ; x2 Þ; 2 i G22 ðx1 ; x2 Þ ¼ Gþ ðx1 ; x2 Þ þ signðx01  x02 ÞG ðx1 ; x2 Þ; 2 11

þ

ð3:40Þ ð3:41Þ

the last two relations are easily verified by inserting the definitions of G±. In a perturbative expansion of the Schwinger–Dyson equation (3.36) in terms of Feynman diagrams, time arguments of internal vertices can lie on either branch. Hence, the number of contributing graphs doubles with each internal vertex since this can lie on the upper or the lower branch.6 Two upper vertices are connected by G11, two lower vertices by G22 and vertices of different type by G12 and G21. Each lower vertex leads to an additional factor 1. Like the Green’s function, also the self-energy RC , the sum of all one-particle irreducible graphs, can be dissected into components Rkl, with k and l being ‘contour indices’ as defined above. Analogous to (3.38) and (3.39) one then defines self-energies R? and, following (3.33) and (3.34), self-energies R± via the equations

R ðx1 ; x2 Þ ¼ iðR> ðx1 ; x2 Þ  R< ðx1 ; x2 ÞÞ;

ð3:42Þ

1 Rþ ðx1 ; x2 Þ ¼ ðR> ðx1 ; x2 Þ þ R< ðx1 ; x2 ÞÞ: 2

ð3:43Þ

Since the self-energies Rkl are directly related to the full Green’s functions Gkl, they also satisfy the relations (3.38)–(3.41). Using the above relations for Gkl and Rkl, one obtains, after a straightforward calculation, from the Schwinger–Dyson equation (3.36) a system of two coupled differential equations for G p , the Kadanoff– Baym equations. Due to spatial homogeneity, we can consider the equations for each Fourier mode separately,

Cðic0 @ t1  pc  MÞGp ðt1 ; t 2 Þ ¼  Cðic0 @ t1  pc  MÞGþp ðt1 ; t 2 Þ ¼ 

Z

t2

t1 Z t2

þ

ti

Z ti

0

dt C Rp ðt 1 ; t0 ÞGp ðt 0 ; t2 Þ;

ð3:44Þ

0

dt C Rþp ðt 1 ; t0 ÞGp ðt 0 ; t2 Þ t1

0

dt C Rp ðt 1 ; t 0 ÞGþp ðt0 ; t 2 Þ:

ð3:45Þ

For the lepton propagators S Lk one obtains the same equations, with C Rp replaced by the lepton self energies Pk and no charge conjugation matrix C multiplying the kinetic term. The Kadanoff–Baym equations (3.44) and (3.45) are exact. They contain all quantum and non-Markovian effects including the dependence on the initial time ti. Furthermore, in contrast to usual linear response techniques, they do not rely on any assumption regarding the size of the initial deviation from equilibrium. The equations in this form are valid for arbitrary nonequilibrium initial states which can be parameterized by Gaussian initial correlations. This covers the case considered in this work since the generated lepton asymmetry involves to leading order in the Yukawa coupling only the 2point functions of the heavy neutrino. When higher order initial correlations play a significant role, the Kadanoff–Baym formalism is still applicable, but the equation for the statistical propagator contains extra terms at ti [42]. In [36], thermalization has been studied for a scalar field theory using the equation of motion for the statistical propagator.

6

This fact is sometimes referred to as ‘doubling of degrees of freedom’.

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2007

In nonequilibrium quantum field theory, instead of distribution functions, quantum mechanical correlation functions G± characterise the state of the system. The interactions enter via the selfenergies R± which, via the generalised cutting rules, contain all possible processes. Encoding this information in the self-energies avoids potential problems related to the definition of asymptotic states for unstable particles as well as the substraction of real intermediate state contributions in Boltzmann equations. Note, finally, that the integro-differential equations (3.44) and (3.45) do not suffer from the late time uncertainties or secular terms that perturbative expansions of Boltzmann equations are often plagued with when applied to multiscale problems (cf. [36]). 3.3. Weak coupling to a thermal bath The Kadanoff–Baym equations provide a tool to study the dynamics of arbitrary nonequilibrium systems. Unfortunately, in most cases they can only be solved numerically. As discussed in the introduction, in this work we consider a rather simple system: one field that is out of equilibrium (N) is weakly coupled to a large thermal bath of Standard Model fields. This leads to a number of simplifications compared to the general case that allow to find analytic solutions. We have previously studied scalar field models of this type [29,43]. Here we extend the methods developed therein to the case of thermal leptogenesis. The Standard Model interactions keep the bath in thermal equilibrium. The corresponding time scale sSM 1/(g2T) at temperature T M is much shorter than the equilibration time sN 1/(k2M) of the heavy neutrino, which governs the generation of the lepton asymmetry: sSM  sN. Lepton number changing processes in the thermal bath are shown in Fig. 3. As in the case of Boltzmann equations discussed in Section 2, we focus on the CP violating interaction generating the lepton asymmetry that correspond to Fig. 3(e) and (f). To evaluate these graphs we need the correlation functions of lepton and Higgs fields in the thermal bath. A system in thermal equilibrium is described by the density matrix

.eq ¼

  exp bðH þ li Qi Þ  ; Tr exp bðH þ li Qi Þ

ð3:46Þ

where H is the Hamiltonian of the system, b is the inverse temperature, Qi are conserved charges and li are the corresponding chemical potentials. As expected for an initial state after inflation, we set all chemical potentials equal to zero. Equilibrium correlation functions of a spatially homogeneous system only depend on space–time differences, and it is convenient to consider the Fourier transforms,

D q ðxÞ ¼ S k ðxÞ ¼

Z

Z

4

d xeiðxx 4

d xeiðxx

0 qxÞ

0 kxÞ

D ðxÞ;

ð3:47Þ

S ðxÞ:

ð3:48Þ

The equilibrium density matrix (3.46) then corresponds to a shift in imaginary time. This leads to the well-known Kubo–Martin–Schwinger (KMS) relations (cf. [44])

Dq ðxÞ;

Sk ðxÞ;

ð3:49Þ

which imply

    1 i bx  Dþq ðxÞ ¼ i þ f/ ðxÞ Dq ðxÞ ¼  coth Dq ðxÞ; 2 2 2     1 i bx  Sk ðxÞ; Sþk ðxÞ ¼ i  fl ðxÞ Sk ðxÞ ¼  tanh 2 2 2

ð3:50Þ ð3:51Þ

where

f/ ðxÞ ¼

1 ; ebx  1

f l ðxÞ ¼

1 ; ebx þ 1

ð3:52Þ

are Bose–Einstein and Fermi–Dirac distribution functions, respectively. Note that the energy x is not on-shell.

2008

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Fig. 3. One- and two-loop contributions to the lepton self-energy corresponding to washout terms, (a)–(d), and CP violating terms which generate a lepton asymmetry, (e) and (f).

Equilibrium Green’s functions can be calculated in the real-time formalism using the contour Cb in the complex time plane, which is shown in Fig. 4. For the free equilibrium propagators of massless lepton and Higgs fields one obtains (q = jqj, k = jkj, cf. [44]),

1 sinðqyÞ; q   1 bq cosðqyÞ; coth Dþq ðyÞ ¼ 2q 2   kc SLk ðyÞ ¼ P L ic0 cosðkyÞ  sinðkyÞ ; k    1 bk kc þ ic0 sinðkyÞ þ cosðkyÞ : SLk ðyÞ ¼  P L tanh 2 2 k

Dq ðyÞ ¼

Fig. 4. Path Cb in the complex time plane for equilibrium correlation functions.

ð3:53Þ ð3:54Þ ð3:55Þ ð3:56Þ

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2009

All other propagators can be obtained as linear combinations using the relations described in the previous paragraph. A complete list is given in Appendix A. In the following sections we shall see that the calculation of the lepton asymmetry represents an initial value problem which can be treated based on the real time formalism together with the Keldysh contour Fig. 2. Thermal and nonthermal properties of the system are then encoded in the initial values of the various Green’s functions. 4. Nonequilibrium correlation functions The assumption of weak coupling to a large thermal bath with negligible backreaction in the framework of Kadanoff–Baym equations implies that self-energies for the heavy neutrinos N are computed from equilibrium propagators of bath fields only. This also corresponds to a leading order perturbative expansion in the coupling constant. Perturbative expansions of Boltzmann equations in multiscale problems are known to suffer from uncertainties, so-called secular terms, at late times. The Kadanoff–Baym equations (3.44) and (3.45) in full generality are free of secular terms and consistently include all memory effects. Nevertheless, the neglect of backreaction in the computation of R corresponds to a truncation in the perturbative expansion in the Yukawa couplings k, which might introduce similar uncertainties related to the multiscale nature of the problem. However, in the system of consideration contributions of higher order in k are not only suppressed by the smallness of the coupling, but also by the number of degrees of freedom in the bath that justify the neglect of backreaction. Hence, we expect potential problems due to secular terms not to be relevant. The assumption that the background medium equilibrates instantaneously on the time scale of the asymmetry generation leaves open the details of the equilibration process. In reality, there are effects related to the finite equilibration time and the finite size of the quasi-particles. As we shall see in Section 5, these quantities play a crucial role in the Kadanoff–Baym result for the lepton asymmetry. The self-energy for the heavy neutrino N to leading order in k is given by the diagram in Fig. 5. It contains time-translation invariant propagators of bath fields only, and hence it is also time-translation invariant. As shown in [29], this implies that also the spectral function is time-translation invari ant, G p ðt 1 ; t 2 Þ  Gp ðyÞ; y ¼ t 1  t 2 . In this case we can find the general solutions to the Kadanoff–Baym equations without further approximations. 4.1. Equation for the spectral function Let us now consider the equation for the spectral function of the Majorana neutrino. After an obvious change of variables, the Kadanoff–Baym equation (3.44) becomes,

Cðic0 @ y  pc  MÞGp ðyÞ 

Z

y 0

0

dy C Rp ðy  y0 ÞGp ðy0 Þ ¼ 0:

ð4:1Þ

Defining the Laplace transform

e  ðsÞ ¼ G p

Z 0

1

sy

dye

Gp ðyÞ;

e  ðsÞ ¼ R p

Z 0

1

sy

dye

Rp ðyÞ;

Fig. 5. One-loop contribution to the self-energies C R p of the Majorana neutrino N.

ð4:2Þ

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

one obtains from Eq. (4.1)



 e  ðsÞ ¼ ic0 G ð0Þ: e  ðsÞ G ic0 s  pc  M  R p p p

ð4:3Þ

Using the boundary condition (3.30),

Gp ð0Þ ¼ ic0 C 1 ;

ð4:4Þ

this leads to

 1 e  ðsÞ ¼  ic0 s  pc  M  R e  ðsÞ C 1 : G

ð4:5Þ

The inverse Laplace transform is given by

Gp ðyÞ ¼

Z CB

ds sy e  e G p ðsÞ; 2pi

ð4:6Þ

where CB is the Bromwich contour (see Fig. 6): The part parallel to the imaginary axis is chosen such that all singularities of the integrand are to its left; the second part is the semicircle at infinity which closes the contour at Re(s) < 0. e  ðsÞ is analytic on From the definition of the Laplace transform one can see that the self-energy R p the real s axis, but has a discontinuity across the imaginary axis. This gives rise to the spectral representation

e  ðsÞ ¼ i R p

Z

1

1

 dp0 Rp ðp0 Þ : 2p is  p0

ð4:7Þ

Note that the retarded and advanced self-energies are given by

e  ðix þ Þ ¼ RR ðxÞ; R p p

ð4:8Þ

e  ði p

ð4:9Þ

R

A pð

x  Þ ¼ R xÞ:

e  ðsÞ, These self-energies are determined by the discontinuity of R p

e  ðixÞ ¼ R e  ðix þ Þ  R e  ðix  Þ ¼ R ðxÞ; disc R p p p p

ð4:10Þ

with the real part given by the principal value, i.e.,

e  ðix Þ ¼ iP R p

Z

1

1

dp0 R ðp0 Þ 1  R ðxÞ: 2p x  p0 2 p

This representation of the self-energy is familiar from the theory at zero temperature.

Fig. 6. Bromwich contour.

ð4:11Þ

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2011

We are now ready to calculate the spectral function in terms of the self-energy R p ðxÞ. Its Laplace transform has singularities only on the imaginary axis. Hence the Bromwich contour can be deformed R i1þ R i1 as CB ! i1þ þ i1 (see Fig. 6), which yields for the spectral function

Z 1 Z 1 ds sy e  dx ðixþÞy e  dx ðixÞy e  G p ðix þ Þ þ G p ðix  Þ e e e G p ðsÞ ¼ 2p 1 2p 1 CB 2pi Z 1  dx ixy  e  e  ðix  Þ : ¼ e G p ðix þ Þ  G p 1 2p

Gp ðyÞ ¼

Z

ð4:12Þ

The Fourier transform of the spectral function,

qp ðxÞ ¼ i

Z

1

ixy

dye

1

Gp ðyÞ;

ð4:13Þ

is then given by

qp ðxÞ ¼

i   M  12 Rp ðxÞ

! i C 1 :  M þ 12 Rp ðxÞ

ð4:14Þ

Here we have assumed that the divergent contribution of the real part has already been absorbed into mass and wave function renormalisation, so that qp(x) represents the renormalised spectral density. The finite part of the self-energy is negligible because of the small Yukawa coupling. A straightforward calculation yields for the self-energy (cf. [35]),

Rp ðxÞ ¼ 2iðky kÞ11

Z

rðp; k; qÞ;

ð4:15Þ

k;q

where we have defined





rðp; k; qÞ ¼ fl/ ðk; qÞð2pÞ4 d4 ðp  k  qÞ þ d4 ðp þ k þ qÞ   þ f l/ ðk; qÞð2pÞ4 d4 ðp þ k  qÞ þ d4 ðp  k þ qÞ ;

ð4:16Þ

with the statistical factors

fl/ ðk; qÞ ¼ 1  fl ðkÞ þ f/ ðqÞ;

f ðk; qÞ ¼ f/ ðqÞ þ f ðkÞ: l/ l

ð4:17Þ

Note that k and q are on-shell, i.e., k = (k, k) and q = (q, q), whereas p = (x, p) is off-shell. The properties of the Dirac matrices and rotational invariance imply

Rp ðxÞ ¼ iap ðxÞc0 þ ibp ðxÞpc;

ð4:18Þ

where

  ap ðxÞ ¼ 2 ky k 11

Z

k rðp; k; qÞ; Z   1 bp ðxÞ ¼ 2 ky k 11 2 pk rðp; k; qÞ: p q;k

ð4:19Þ

q;k

ð4:20Þ

These functions satisfy the relations

ap ðxp Þ ¼ ap ðxp Þ;

bp ðxp Þ ¼ bp ðxp Þ:

ð4:21Þ

Using Eq. (4.18) and linearising the denominators in Eq. (4.14) in the small quantities ap(x) and bp(x), one obtains for the spectral density

2xCp ðxÞ 1   qp ðxÞ ¼   ð þ MÞC ; 2 x  x2p þ xCp ðxÞ 2

ð4:22Þ

where



Z



xCp ðxÞ ¼ xap ðxÞ þ p2 bp ðxÞ ¼ 2 ky k

11

q;k

pk

rðp; k; qÞ:

ð4:23Þ

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

On-shell, only the first of the d-functions in r(p;k,q) contributes, and one obtains the width appearing in the Boltzmann equations,

Z

2

Cp ðxp Þ ¼ ðky kÞ11

xp

p  k f l/ ðk; qÞð2pÞ4 d4 ðp  k  qÞ  Cp ;

q;k

ð4:24Þ

which satisfies the relations

Cp ðxp Þ ¼ Cp ðxp Þ ¼ Cp ðxp Þ:

ð4:25Þ

In the zero-width limit the spectral function (4.22) reduces to the familiar expression in vacuum,

qp ðxÞ ¼ 2psignðxÞdðp2  M2 Þð þ MÞC 1 :

ð4:26Þ

The spectral propagator is now obtained by evaluating the Fourier transform of the spectral function (4.22),

Gp ðyÞ ¼ i

Z

1

1

dx ixy e qp ðxÞ; 2p

ð4:27Þ

which yields the final result

  M  pc Gp ðyÞ ¼ ic0 cosðxp yÞ þ sinðxp yÞ eCp jyj=2 C 1 :

ð4:28Þ

xp

Compared to the free spectral function only an exponential damping factor appears. This is a feature of the narrow-width approximation, analogous to the scalar field case discussed in [29]. 4.2. Equation for the statistical propagator We now proceed to the solution of the second Kadanoff–Baym equation (3.45) which, choosing ti = 0, reads

Cðic0 @ t1  pc  MÞGþp ðt1 ; t 2 Þ 

Z

t1

0

0

dt C Rp ðt 1  t0 ÞGþp ðt 0 ; t2 Þ ¼ fp ðt 1  t 2 Þ;

ð4:29Þ

with the source term

fp ðt1  t2 Þ ¼ 

Z

t2

0

0

dt Rþp ðt1  t 0 ÞGp ðt 0  t2 Þ:

ð4:30Þ

The general solution of (4.29) takes the form

^ þ ðt1 ; t 2 Þ þ Gþ Gþp ðt1 ; t 2 Þ ¼ G p p;mem ðt 1 ; t 2 Þ;

ð4:31Þ

^ þ ðt 1 ; t 2 Þ is the general solution of the homogeneous equation where G p

^ þ ðt1 ; t 2 Þ  Cðic0 @t1  pc  MÞG p

Z 0

t1

0 ^ þ ðt 0 ; t2 Þ ¼ 0; dt C Rp ðt 1  t0 ÞG p

ð4:32Þ

and the ‘memory integral’, which contains non-Markovian effects, is given by

Gþp;mem ðt1 ; t 2 Þ ¼

Z

t1

dt

0

0

Z

t2

0

00

dt Gp ðt 1  t0 ÞRþp ðt0  t00 ÞGp ðt 00  t2 Þ:

ð4:33Þ

One easily verifies that the memory integral is a special solution of the inhomogeneous equation. In order to evaluate the memory integral we perform a Fourier transform of the self-energy (y = t1  t2),

Gþp;mem ðt1 ; t 2 Þ ¼

Z

dx 2p

Z

t1 0

 Z dy1 Gp ðy1 Þeixy1 Rþp ðxÞ

0

t2

 dy2 Gp ðy2 Þeixy2 eixy :

ð4:34Þ

Since the self-energy is computed with fields in thermal equilibrium, it satisfies the KMS condition (cf. (3.51))

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

i 2

Rþp ðxÞ ¼  tanh



2013



bx  Rp ðxÞ: 2

ð4:35Þ

Using the expressions (4.18) and (4.28) for self-energy and spectral function, respectively, which were derived in the previous section, it is now straightforward to calculate the memory integral explicitly. Neglecting terms OðCp Þ in the numerator, one finds

Z

t

ixy

dye

0

Gp ðyÞ ¼

1   

 ic0 xp sinðxp tÞ þ ix cosðxp tÞ eiðxþiCp =2Þt  ix

þ Z

ð4:36Þ

x2p  ðx þ iCp =2Þ2

t

ixy

dye

0

M  pc

xp

  

ix sinðxp tÞ  xp cosðxp tÞ eiðxþiCp =2Þt þ xp C 1 ; 1

Gp ðyÞ ¼

ð4:37Þ

x  ðx  iCp =2Þ2 2 p

  

 ic0 xp sinðxp tÞ  ix cosðxp tÞ eiðxiCp =2Þt  ix

þ

M  pc

xp

  

ix sinðxp tÞ  xp cosðxp tÞ eiðxþiCp =2Þt þ xp C 1 :

After inserting these expressions in Eq. (4.34) one can perform the x-integration using Cauchy’s theorem. The integrand has two poles7 in the upper-half plane at x = iCp/2 ± xp, and two poles in the lower-half plane x = iCp/2 ± xp. The choice of the contour depends on the sign of the time variables in the exponent. The result is a sum of the contributions from all four poles. The expressions appearing in the numerator can be simplified by means of Eqs. (4.21) and (4.24) for self-energy and equilibration width, respectively,



  M  pc ¼ 2iCp c0 þ ; xp xp xp       M  pc  M  pc M  pc ¼ 2iCp c0  : c0  Rp ðxp Þ c0 

c0 þ

 M  pc



Rp ðxp Þ c0 þ

xp

 M  pc

xp

ð4:38Þ ð4:39Þ

xp

Using these expressions one finally obtains for the memory integral, changing variables from (t1, t2) to (t, y),   

Gþp;mem ðt; yÞ ¼ 

1 bxp tanh 2 2

ic0 sinðxp yÞ 

M  pc

xp

cosðxp yÞ



 eCq jyj=2  eCq t C 1 :

ð4:40Þ

Asymptotically, for t ? 1, the memory integral becomes

Gþeq p ðt; yÞ ¼ 

   1 bxp M  pc tanh cosðxp yÞ eCq jyj=2 C 1 : ic0 sinðxp yÞ  xp 2 2 Gþeq p ðt; yÞ

One easily verifies that rier transform one obtains

Gþeq p ð xÞ ¼

Z

1

ð4:41Þ

indeed represents the equilibrium statistical propagator. For the Fou-

ixy

dye Gþeq p ðyÞ   1 bx 2xCp ðxÞ 1    ¼ tanh  ð þ MÞC 2 2 2 x  x2p þ xCp ðxÞ 2   1 bx ¼ tanh qp ðxÞ 2 2   i bx  ¼  tanh Gp ðxÞ; 2 2 1

ð4:42Þ ð4:43Þ

i.e., the KMS condition (cf. (3.51)) is indeed satisfied. 7 There are further poles at xn = ±ip(1 + 2n)/b, n integer. However, their contribution to Gþ p;mem is OðCp =MÞ and therefore negligible.

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

In order to obtain the general solution of the inhomogeneous Kadanoff–Baym equation we have to add to the memory integral the general solution of the homogeneous equation (4.32). This equation is identical to the Kadanoff–Baym equation for the spectral function (4.1) with t2 playing the role of an ^ þ ðt1 ; t2 Þ on the first argument t1 can be additional parameter. Hence, the functional dependence of G p obtained in the same way as for the spectral function. Applying the Laplace transform to (4.32) one finds

e þ ðs; t2 Þ ¼ G p

1 ^ þ ð0; t 2 Þ: ic G e  ðsÞ 0 p ic0 s  pc  M  R

ð4:44Þ

The inverse Laplace transform then gives

^ þ ðt1 ; t 2 Þ ¼ G ðt1 ÞCic G ^þ G 0 p ð0; t 2 Þ: p p

ð4:45Þ

^ þ ð0; t 2 Þ can now be determined by the symmetries (3.28) and (3.29) of G ^ ðt 1 ; t 2 Þ, which The function G p p imply

^ þ ðt 2 ; t1 Þ: ^ þ ðt1 ; t 2 ÞT ¼ G G p p

ð4:46Þ

This yields the result

^ þ ðt1 ; t 2 Þ ¼ G ðt1 ÞC c Gþ ð0; 0Þc C 1 G ðt 2 Þ; G 0 p 0 p p p

ð4:47Þ

Gþ p ð0; 0Þ

where is an antisymmetric matrix. Let us first consider the case of thermal initial condition,

Gþeq p ð0; 0Þ ¼

  M  pc bx 1 tanh C : 2xp 2

ð4:48Þ

From Eq. (4.47) one then obtains

    ^ þeq ðt 1 ; t2 Þ ¼  1 ic sinðxp yÞ  M  pc cosðxp yÞ tanh bxp eCq ðt1 þt2 Þ=2 C 1 : G 0 p xp 2 2

ð4:49Þ

^þ Adding this expression to the memory integral G p;mem one obtains the equilibrium statistical propaga^ þeq tor G which is independent of t = (t1 + t2)/2. Hence, as expected, the equilibrium statistical propagap tor is a solution of the full Kadanoff–Baym equation. We are particularly interested in the case of vacuum initial condition, which corresponds to zero initial abundance for heavy neutrinos in the Boltzmann case. The vacuum propagators are obtained from the equilibrium ones in the limit b ? 1. Hence we choose

Gþvac ð0; 0Þ ¼ p

M  pc 1 C : 2xp

ð4:50Þ

From Eqs. (4.31), (4.40) and (4.47) one then obtains the full solution for the statistical propagator, which interpolates between vacuum at t = 0 and equilibrium for t ? 1,

  M  pc Gþp ðt; yÞ ¼  ic0 sinðxp yÞ  cosðxp yÞ 

xp

   1 bxp Cp jyj=2 tanh e þ fNeq ðxp ÞeCp t C 1 :  2 2

ð4:51Þ

This result will be the basis for the calculation of the lepton asymmetry in the next section. All heavy neutrino propagators can be obtained as linear combinations of the spectral function G p ðyÞ and the statistical propagator Gþ p ðt; yÞ. A full list is given in Appendix A. Finally, let us emphasise that the solution of the Kadanoff–Baym equation for the statistical propagator is not related to the equilibrium propagator by a simple change of the distribution function from fNeq ðxÞ to some nonequilibrium function fNeq ðt; xÞ. This is in contrast to the assumption made

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2015

in the derivation of Quantum Boltzman equations [16,17,21,19]. For a system close to equilibrium this assumption leads to a valid approximation of the Kadanoff–Baym equations [9], but in general it is not justified.

5. Lepton asymmetries We are now ready to calculate the lepton asymmetry which is generated during the approach of the heavy Majorana neutrino N to thermal equilibrium. Our starting point is the flavour non-diagonal lepton current, which is obtained from the statistical propagator, l

jij ðxÞ ¼ tr½cl SþLij ðx; x0 Þx0 !x :

ð5:1Þ

0 ~ ~0 Since we consider a spatially homogeneous system, Sþ ij ðx; x Þ only depends on the difference x  x , and it is convenient to perform a Fourier transform. The zeroth component of the current, the ‘lepton number matrix’, is given by

h i Lkij ðt; t 0 Þ ¼ tr c0 SþLkij ðt; t 0 Þ :

ð5:2Þ

One easily verifies that for free fields in equilibrium

Lkii ðt; tÞ ¼ fli ðkÞ  fli ðkÞ;

ð5:3Þ

where fli and fli are the distribution functions of leptons and anti-leptons, respectively. The lepton number matrix Lkij(t, t0 ) can be directly computed from the self-energy corrections to the statistical propagator shown in Fig. 3: the external lepton couples to Majorana neutrino and Higgs boson, and also to Higgs boson and Higgs-lepton pair. Complex Yukawa couplings and quantum interference then lead to a non-vanishing lepton asymmetry. For a homogeneous system, the Kadanoff–Baym equation for the statistical propagator (cf. (3.45)) yields for each Fourier mode the equations

ðic0 @ t  kcÞSþLk ðt; t 0 Þ ¼

Z 0



t

dt 1 Pk ðt; t 1 ÞSþLk ðt 1 ; t 0 Þ

Z

t0

0

Z

dt1 Pþk ðt; t 1 ÞSLk ðt1 ; t 0 Þ;

ð5:4Þ

t0

SþLk ðt; t0 Þðic0 @ t0 kcÞ ¼  dt 1 SþLk ðt; t 1 ÞPk ðt 1 ; t0 Þ 0 Z t dt 1 SLk ðt; t 1 ÞPþk ðt 1 ; t 0 Þ: þ

ð5:5Þ

0

One then obtains for the time derivative of the lepton number matrix, dropping flavour indices (cf. [18]),8



@ t Lk ðt; tÞ ¼ itr ðic0 @ t þ ic0 @ t0 ÞSþLk ðt; t 0 Þ t¼t0   ¼ itr ðic0 @ t  kcÞSþLk ðt; t0 Þ þ SþLk ðt; t 0 Þðic0 @ t0 þkcÞ t¼t 0 "Z Z 0 t

¼ itr

0

þ

Z 0

t0

t

dt1 Pk ðt; t 1 ÞSþLk ðt1 ; t 0 Þ 

dt 1 SþLk ðt; t1 Þ

 0 k ðt 1 ; t Þ

P



Z 0

0 t

dt 1 Pþk ðt; t 1 ÞSLk ðt 1 ; t 0 Þ #

dt1 SLk ðt; t1 Þ

P

þ 0 k ðt 1 ; t Þ

: t¼t0

Using properties of the trace and the identity between integration domains 8

We thank C. Weniger for helpful discussions.

ð5:6Þ

2016

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Z

t

dt 1

0

Z

Z

t1

dt 2    þ

t

dt 2

0

0

Z

t2

dt 1    ¼

Z

0

t

dt 1

0

Z

t

dt2    ;

ð5:7Þ

0

one finds

Lk ðt; tÞ ¼ i

Z

t

dt1

Z

0

t

0



dt 2 tr Pk ðt 1 ; t 2 ÞSþLk ðt 2 ; t 1 Þ  Pþk ðt 1 ; t2 ÞSLk ðt 2 ; t1 Þ :

ð5:8Þ

Note that P k and Sk are self-energies and propagators of the full theory including gauge interactions of lepton and Higgs fields. Using the relations for propagators and self-energies

   1 > S þ SLk  Sk þ Pk  P
ð5:9Þ ð5:10Þ

one obtains from Eq. (5.8) an equivalent useful expression for the lepton number matrix,

Lk ðt; tÞ ¼ 

Z

t

dt 1

Z

0

0

t



dt 2 tr P>k ðt1 ; t 2 ÞSLk ðt 2 ; t1 Þ :

ð5:11Þ

We want to calculate the lepton asymmetry to leading order in the small Yukawa coupling k, which can be achieved in a perturbative expansion. For the heavy neutrino propagator appearing in the loop, the departure from the equilibrium propagator is important,9 which has been evaluated in the previous section,

Gp ðt1 ; t 2 Þ ¼ Geq p ðt 1  t 2 Þ þ Gp ðt 1 ; t 2 Þ:

ð5:12Þ

Lepton propagators and self-energies have large equilibrium contributions dominated by gauge interaction, with small corrections Oðk2 Þ,

SLk ðt 1 ; t2 Þ ¼ Seq Lk ðt 1  t 2 Þ þ dSLk ðt 1 ; t 2 Þ;

ð5:13Þ

Pk ðt1 ; t 2 Þ ¼ Peq k ðt 1  t 2 Þ þ dPk ðt 1 ; t 2 Þ;

ð5:14Þ

which include CP-violating source terms and washout terms. Clearly, inserting Peq and Seq k in Eq. (5.8) k eq 10 must yield Lk ðt; tÞ ¼ 0, since no asymmetry is generated in thermal equilibrium. As discussed in Section 2, we also neglecting washout terms for simplicity. One then obtains for the lepton number matrix Lk(t) to leading order in k,

Lk ðt; tÞ ¼ i

Z

t

dt1

0

Z

t

0



eq þ dt 2 tr dPk ðt 1 ; t2 ÞSeqþ Lk ðt 2  t 1 Þ  dPk ðt 1 ; t 2 ÞSLk ðt 2  t 1 Þ :

ð5:15Þ

Here dPk is given by the two-loop graphs shown in Fig. 7, which have to be evaluated with equilibrium propagators for lepton and Higgs fields and the nonequilibrium Majorana neutrino propagator. The equilibrium propagators with standard model gauge interactions remain to be evaluated. In the quasi-particle approximation one simply replaces energies k by complex quasi-particle energies Xk ¼ ðk2 þ m2th Þ1=2 þ icðkÞ. In the following we shall consider two approximations: free equilibrium propagators with zero chemical potential as given in Eqs. (A.1), (A.2) and (A.7), (A.8), Deq k ðyÞ ¼ Dk ðyÞ;

Seq Lk ðyÞ ¼ SLk ðyÞ;

ð5:16Þ

and, as a rough approximation to full thermal propagators, free equilibrium propagators modified by thermal damping rates, cU jyj Deq ; k ðyÞ ¼ Dk ðyÞe

9

cl jyj Seq : Lk ðyÞ ¼ SLk ðyÞe

ð5:17Þ

We show in Appendix D that the equilibrium part of the propagator does indeed not contribute to the asymmetry. Note that thermal equilibrium does not correspond to a Gaussian state [42]. Therefore one has to include contributions from npoint functions which are not determined by equilibrium 2-point functions. However, such terms do not contribute to leading order in the Yukawa coupling k. 10

2017

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Fig. 7. Two-loop contributions to the lepton self-energies P k , which lead to a nonzero lepton number densities.

Remarkably, thermal widths turn out to be qualitatively more important than thermal masses, as we shall explain in Section 6. The two contributions to the self-energy dPkij (cf. Fig. 7), ð1Þ

ð2Þ

dPkij ðt 1 ; t2 Þ ¼ Pkij ðt1 ; t 2 Þ þ Pkij ðt 1 ; t 2 Þ;

ð5:18Þ

factorize into a product of Yukawa couplings, which contains the flavour dependence, and a trace of thermal propagators, ð1Þ   Pð1Þ kij ðt 1 ; t 2 Þ ¼ 3iki1 ðgk Þj1 Pk ðt 1 ; t 2 Þ;

ð5:19Þ

ð2Þ  Pð2Þ kij ðt 1 ; t 2 Þ ¼ 3iðg kÞi1 kj1 Pk ðt 1 ; t 2 Þ:

ð5:20Þ

In the case of free equilibrium propagators for lepton and Higgs fields, we obtain for the self-energies ð1;2Þ< Pð1;2Þ> and Pk : k

2018

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Pkð1Þ> ðt1 ; t 2 Þ ¼

Z 0

1

dt 3

Z

3

3

d q0

d q

ð2pÞ3 ð2pÞ3

h

e > ðt1 ; t 3 ÞS110 ðt 2  t3 ÞD110 ðt 2  t 3 ÞD< ðt2  t1 Þ  G q q k p i < < < 22 e  G p ðt 1 ; t3 ÞSk0 ðt 2  t 3 ÞDq0 ðt2  t3 ÞDq ðt 2  t 1 Þ PL ;

Pkð1Þ< ðt1 ; t 2 Þ ¼

Z 0

1

dt 3

Z

3

3

d q0

d q 3

ð2pÞ ð2pÞ3

h

e 11 ðt1 ; t 3 ÞS>0 ðt2  t 3 ÞD>0 ðt 2  t3 ÞD> ðt2  t 1 Þ  G q q p k i e < ðt1 ; t 3 ÞS220 ðt2  t 3 ÞD220 ðt 2  t3 ÞD> ðt2  t 1 Þ PL ; G q q p k

Pkð2Þ> ðt1 ; t 2 Þ ¼

Z 0

1

dt 3

Z

3

Pkð2Þ< ðt1 ; t 2 Þ ¼

d q0

d q 3

ð2pÞ ð2pÞ3

h

0

1

dt 3 h

ð5:22Þ

3

e < ðt2 ; t 3 ÞS220 ðt 3  t1 ÞD220 ðt 3  t 1 ÞD< ðt2  t1 Þ  G q q p k i e 11 ðt 2 ; t3 ÞS<0 ðt 3  t 1 ÞD<0 ðt3  t1 ÞD< ðt 2  t 1 Þ PL ; G q q p k Z

ð5:21Þ

Z

3

ð5:23Þ

3

d q0

d q 3

ð2pÞ ð2pÞ3

e 22 ðt2 ; t 3 ÞS>0 ðt3  t 1 ÞD>0 ðt 3  t1 ÞD> ðt2  t 1 Þ  G q q p k i 11 11 > e > ðt2 ; t 3 ÞS 0 ðt3  t 1 ÞD 0 ðt 3  t1 ÞD ðt2  t 1 Þ PL : G q q p k

ð5:24Þ

Due to the chiral projections at the vertices, only the scalar parts of the nonequilibrium Majorana e >, G e <, G e 11 and G e 22 (cf. Eqs. (A.19)–(A.24)), propagators contribute, which are the same for G p p p p

e p ðt; t0 ÞPL ; G e p ðt; t0 Þ ¼ M cosðxp ðt  t0 ÞÞf eq ðxp ÞeCp ðtþt0 Þ=2 : PL Gp ðt; t 0 ÞCP L ¼ G N

xp

ð5:25Þ

The number of terms which contribute to the asymmetry Lk(t,t) can be significantly reduced by means of the following symmetry properties of the massless propagators:

S>k ðyÞ ¼ CS
D>q ðyÞ

¼

D
Sk ðyÞ 5 ; Dq ðyÞ;

c

c

 22 1 S11 ; k ðyÞ ¼ CSk ðyÞC

ð5:26Þ

 22 D11 q ðyÞ ¼ Dq ðyÞ;

ð5:27Þ

11 S11 k ðyÞ ¼ 5 Sk ðyÞ 5 ; 11 11 Dq ðyÞ ¼ Dq ðyÞ:

c

c

ð5:28Þ ð5:29Þ

Employing these transformation properties one can derive the following useful relations among different contributions to the integrand of Eq. (5.11):

h i h i ð1;2Þ> ð1;2Þ< tr Pk ðt 1 ; t2 ÞSk ðt 2  t 1 Þ ; h i h i ð1Þ> ð2Þ< tr Pk ðt1 ; t 2 ÞSk ðt 1  t 2 Þ :

ð5:30Þ ð5:31Þ

Using these relations one obtains from Eq. (5.11) the compact expression for the lepton asymmetry

 Lkii ðt; tÞ ¼ 12 Im ki1 ðgk Þi1

Z 0

t

dt 1

Z 0

t

 h i ð2Þ> dt2 Re tr Pk ðt1 ; t 2 ÞS
ð5:32Þ

n o Since Im ki1 ðgk Þj1 ¼ 16pij =ð3MÞ (cf. Eq. (2.12)), the leading dependence of the flavour-diagonal lepton asymmetry Lkii(t, t) on the Yukawa couplings is identical to the dependence of the difference fLi(t, k) of lepton and anti-lepton distribution functions appearing in the Boltzmann equations.

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2019

To proceed further in the evaluation of Lkii(t, t), the following relation can be used to simplify the integrand, 22 S22 k ðyÞDq ðyÞ



S
¼

"

    bq bk cosðkyÞ cosðqyÞ  tanh 2q 2 2 !    M  kc bk cosðkyÞ sinðqyÞ tanh  sinðkyÞ sinðqyÞ  i 2 k    bq sinðkyÞ cosðqyÞ : þ coth 2

HðyÞ

c0 coth

ð5:33Þ

One then obtains for the real part of the sum of products of thermal lepton and Higgs propagators (yij = ti  tj),

 h  i  < < 022 0< < Re tr S22 k0 ðy31 ÞDq ðy31 Þ  Sk0 ðy31 ÞDq ðy31 Þ Sk ðy21 Þ Dq ðy21 Þ " !   Hðy13 Þ bq cosððk þ qÞy coth Þ þ cosððk  qÞy Þ ¼ 21 21 16qq0 2 !!   bk cosððk þ qÞy21 Þ  cosððk  qÞy21 Þ þ tanh 2 !  0 bq 0 0 cosððk þ q0 Þy31 Þ þ cosððk  q0 Þy31 Þ  coth 2 !!  0 bk 0 0 þ tanh cosððk þ q0 Þy31 Þ  cosððk  q0 Þy31 Þ 2 !   0 kk bq sinððk þ qÞy21 Þ þ sinððk  qÞy21 Þ coth þ 0 2 kk !!   bk þ tanh sinððk þ qÞy21 Þ  sinððk  qÞy21 Þ 2 !  0 bq 0 0 0 0  coth sinððk þ q Þy31 Þ þ sinððk  q Þy31 Þ 2 !!#  0 bk 0 0 0 0 þ tanh : sinððk þ q Þy31 Þ  sinððk  q Þy31 Þ 2

ð5:34Þ

Defining the linear combinations of lepton and Higgs distribution functions (cf. (2.6)),

fl/ ðk; qÞ ¼ 1  fl ðkÞ þ f/ ðqÞ;

f ðk; qÞ ¼ f ðkÞ þ f ðqÞ; / l/ l

ð5:35Þ

and using the relations

    bq bk þ tanh ¼ 2f l/ ðk; qÞ; 2 2     bq bk  tanh ¼ 2f l/ ðk; qÞ; coth 2 2

coth

ð5:36Þ ð5:37Þ

one finds

Z t Z t Z t2 Z 1 eq C Lkii ðt; tÞ ¼  ii 32p dt 1 dt2 dt3 fN ðxp Þe 2 ðt1 þt3 Þ cosðxp y31 Þ 0 0 0 q;q0 xp    fl/ ðk; qÞ cosððk þ qÞy21 Þ þ f l/ ðk; qÞ cosððk  qÞy21 Þ   0 0 0 0  fl/ ðk ; q0 Þ cosððk þ q0 Þy23 Þ þ f l/ ðk ; q0 Þ cosððk  q0 Þy23 Þ 0  k  k  þ fl/ ðk; qÞ sinððk þ qÞy21 Þ þ f l/ ðk; qÞ sinððk  qÞy21 Þ 0  kk 0  0 0 0  fl/ ðk ; q0 Þ sinððk þ q0 Þy23 Þ þ f l/ ðk ; q0 Þ sinððk  q0 Þy23 Þ ;

ð5:38Þ

2020

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

where we have again used the notation

Z

Z

 ¼

3

d q ð2pÞ3 2q

q



The functions fl/ and f l/ are well known from Weldon’s analysis of discontinuities in finite-temperature field theory [35]. The sum of statistical factors

fl/ ðk; qÞ ¼ ð1  fl ðkÞÞð1 þ f/ ðqÞÞ þ fl ðkÞf/ ðqÞ

ð5:39Þ

corresponds to decays and inverse decays of the massive Majorana neutrinos whereas

f ðk; qÞ ¼ f/ ðqÞð1  f ðkÞÞ þ f ðkÞð1 þ f/ ðqÞÞ l/ l l

ð5:40Þ

accounts for their disappearance or appearance where a single quant, lepton or Higgs, is absorbed from or emitted into the thermal bath. The function fl/ contains the vacuum contribution, i.e., fl/ ? 1 as b ? 1, whereas f l/ ! 0. We now have to perform the three time integrations in Eq. (5.38). It is convenient to express the products of cosine’s and sine’s as sum of products of exponentials. Each term then becomes a sum of four exponentials, where the energies x, k ± q and k0 ± q0 appear in different linear combinations, and the four complex conjugate exponentials. As an example, consider the integral

I ðtÞ ¼

Z

t

dt1

Z

0

t

dt 2

Z

0

t2

C

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 e 2 ðt1 þt3 Þ ;

ð5:41Þ

0

with X1 = xp  k  q, X3 = xp  q0  k0 , and X2 = X1  X3 = k0 + q0  k  q. A straightforward calculation yields

  Ct C eCt þ cosðX2 tÞ  e 2 ðcosðX1 tÞ þ cosðX3 tÞÞ þ OðtÞ    I ðtÞ þ I  ðtÞ ¼ ; 2 2 X21 þ C4 X23 þ C4

ð5:42Þ

where

OðtÞ ¼

2  2X1 X3 þ C2  Ct sinðX2 tÞ  e 2 ðsinðX1 tÞ  sinðX3 tÞÞ

X2

ð5:43Þ

is of higher order in C at X1,3 = 0. Hence, this term does not contribute to the lepton asymmetry at leading order in C, i.e., in the Yukawa couplings. The two contributions in Eq. (5.38), without and with the prefactor k  k0 /(kk0 ), add up to a single term proportional to k  k0 /(kk0 ) where k  k0 denotes the product of 4-vectors. This is a consequence of Lorentz invariance of the vacuum contribution. The full result is now easily obtained from Eqs. (5.38) and (5.42) by adding the contributions with reversed sign of q and/or q0 , accompanied by the corresponding substitution flU ! f lU . Omitting the subleading terms O (cf. (5.43)), one finally obtains

Lkij ðt; tÞ ¼

4 X

Lakij ðt; tÞ;

ð5:44Þ

a¼1

where

Lakii ðt; tÞ ¼ ii 8p

Z q;q0

0 k  k eq 1 X ^a fN ðxp Þ C L 0 ðt; a; bÞ 0 2 a;b¼ k;q;q kk xp

ð5:45Þ

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2021

and 0

fl/ ðk; qÞfl/ ðk ; q0 Þ   2 2 2 xp  bðk0 þ q0 Þ þ C4 ðxp  aðk þ qÞÞ2 þ C4    0  eCt þ cos aðk þ qÞ  bðk þ q0 Þ t



 Ct  0 e 2 cos ðxp  aðk þ qÞÞt þ cos ðxp  bðk þ q0 ÞÞt ;

ð5:46Þ

f ðk; qÞf ðk0 ; q0 Þ l/ ^L2 0 ðt; a; bÞ ¼  l/  2 k;q;q 2 2 C2 ðxp  aðk  qÞÞ þ 4 xp  bðk0 þ q0 Þ þ C4 

0  eCt þ cos ðaðk  qÞ  bðk þ q0 ÞÞt

   Ct  0 e 2 cos ðxp  aðk  qÞÞt þ cos xp  bðk þ q0 Þ t ;

ð5:47Þ

0 fl/ ðk; qÞf l/ ðk ; q0 Þ ^L3 0 ðt; a; bÞ ¼    2 k;q;q 2 2 2 ðxp  aðk þ qÞÞ þ C4 xp  bðk0  q0 Þ þ C4    0  eCt þ cos aðk þ qÞ  bðk  q0 Þ t

   Ct  0 e 2 cos ðxp  aðk þ qÞÞt þ cos xp  bðk  q0 Þ t ;

ð5:48Þ

^L1 0 ðt; a; bÞ ¼  k;q;q

0

fl/ ðk; qÞfl/ ðk ; q0 Þ   2 2 2 ðxp  aðk  qÞÞ2 þ C4 xp  bðk0  q0 Þ þ C4    0  eCt þ cos aðk  qÞ  bðk  q0 Þ t

   Ct  0 e 2 cos ðxp  aðk  qÞÞt þ cos xp  bðk  q0 Þ t :

^L4 0 ðt; a; bÞ ¼  k;q;q

ð5:49Þ

This expression contains off-shell and memory effects which are not contained in Boltzmann equations. A detailed comparison will be given in the following section. So far we have neglected the thermal damping widths of lepton and Higgs fields due to gauge interactions, which are known to be much larger than the width of the heavy Majorana neutrino, cl cU g2T  k2M C, for M [ T. To estimate their effect we replace the free equilibrium propagators by cU jyj Deq ; k ðyÞ ¼ Dk ðyÞe

cl jyj Seq : k ðyÞ ¼ Sk ðyÞe

ð5:50Þ

This has a drastic effect on the calculation described above. For the dominant term in Eq. (5.45), ^L1k;q;q0 with a = b = 1, where the energy dominators can be OðC2 Þ, one now finds (c = cl + cU),

Lkii ðt; tÞ ¼  ii 16p

Z

0

0 kk cc   0 2 kk xp ððxp  k  qÞ þ c2 Þ ðxp  k0  q0 Þ2 þ c02

q;q0 0

 fl/ ðk; qÞfl/ ðk ; q0 ÞfNeq ðxp Þ

1

C

ð1  eCt Þ;

ð5:51Þ

where c = c(k, q) and c0 = c0 (k0 , q0 ). Note that now all memory effects have disappeared. 6. Boltzmann vs Kadanoff–Baym Let us now consider in detail the relation between the two results obtained for the lepton asymmetry: Eq. (2.11) from the Boltzmann equations and Eqs. (5.44)–(5.49) and (5.51) from the Kadanoff– Baym equations. Clearly, the overall CP asymmetry is identical in both cases and also the momentum integrations are very similar. Compared to the Boltzmann result the Kadanoff–Baym result has an additional statistical lepton-Higgs factor and expected off-shell energy denominators. Furthermore, there are 16 different terms corresponding to the various combinations of decay and inverse decay, appearance and disappearance. The most striking difference is the time dependence of the integrand: the Boltzmann

2022

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

result has a simple exponential behaviour whereas the Kadanoff–Baym result has terms rapidly oscillating with time with frequencies OðMÞ  C, a manifestation of memory effects. The time-dependence is contained in the integral I ðtÞ given in Eq. (5.41). Defining

i 2

i 2

X1 ¼ X1 þ C;

X3 ¼ X3 þ C;

ð6:1Þ

and using the identities t3 = t1 + (t2  t1) + (t3  t2) and X2 = X1  X3, one has (cf. (5.41)),

I ðtÞ ¼

Z

t

dt1 eCt1

Z

0

Z

tt 1

dt 21

0

dt 32 eiX1 t21 þiX3 t32 ;

ð6:2Þ

t 2

t 1

where tij = ti  tj. After performing the time-integrations, one obtains the result

I ðtÞ ¼

"

1 iX3

#     iX t 1  iX 1 t 1  iX 2 t iX1 t 1 e 1 e 1  e 1 e 1 ; X2 X1 jX1 j2

ð6:3Þ

which satisfies

I ð0Þ ¼ I 0 ð0Þ ¼ I 00 ð0Þ ¼ 0;

I 000 ð0Þ–0:

ð6:4Þ

For large times, t  1/C, there remains a term oscillating with time,

I ðtÞ

"

1

1

iX3 jX1 j

þ 2

#

1

ðeiX2 t  1Þ : 

ð6:5Þ

X2 X1

This is in contrast to the Boltzmann result whose time-dependence is given by

I B ðtÞ ¼

1  eCt

C

ð6:6Þ

;

with

I 0B ð0Þ–0;

I B ð0Þ ¼ 0;

ð6:7Þ

and I B ðtÞ 1=C ¼ const for large times t  1/C. Where is the Boltzmann result hidden in the Kadanoff–Baym result, and in which limit is it recovered? To answer this question it is instructive to consider a modified integral I ðtÞ, where thermal damping rates c c0 g2T are included, which affect the dependence on the time differences jt2  t1j and jt3  t2j (cf. Fig. 7),

I ðtÞ ¼

Z

t

dt1 eCt1

0

Z

Z

tt 1

dt 21

t 1

0

0

dt 32 eiX1 t21 cjt21 j eiX3 t32 c jt32 j :

ð6:8Þ

t 2

Compared to Eq. (5.41) the main difference is that the damping term in the t21-integration changes sign at t21 = 0. This is in contrast to the damping due to the Majorana neutrino decay width C. Carrying out the time-integrations one now obtains the result

I ðtÞ ¼

"

1

1

iX3 þ c ðiX1  cÞðiX 0

 1þ

cÞ

   eðiX1 cÞt eðiX1 þcÞt  1 

1 ðiX1 þ cÞðiX

 1

cÞ

   eðiX1 cÞt  1

    1 1 0   eðiX2 cc Þt eðiX1 þcÞt  1 þ eðiX1 cÞt  1    0 0 ðiX2  c  c ÞðiX1 þ cÞ ðiX2 þ c  c ÞðiX1  cÞ #   Ct 2c 1  e 2c ðiX3 c0 Þt : ð6:9Þ þ  1 e þ 2 X1 þ c2 C ðiX3 þ c0 ÞððiX2  c0 Þ2  c2 Þ The first four terms reduce to Eq. (6.3) for c = c0 = 0. Particularly interesting is the last line in Eq. (6.9), which is a contribution from the point t21 = t2  t1 = 0, where the damping term changes sign. This local contribution contains the only term which is enhanced by 1/C and has Boltzmann-like timedependence,

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

I ðtÞ I B ðtÞ ¼

2c

1  eCt

ðiX3 þ c0 ÞðX21 þ c2 Þ

C

:

2023

ð6:10Þ

Note that as consequence of thermal damping all oscillatory terms are exponentially suppressed for times t > 1/c, IðtÞ

" # 1 2c 1  eCt 2c 1 1  : þ þ C iX3 þ c0 X21 þ c2 ðiX3 þ c0 ÞððiX2  c0 Þ2  c2 Þ ðiX1 þ cÞðiX1  cÞ ðiX2 þ c  c0 ÞðiX1  cÞ ð6:11Þ

The Boltzmann-like term (6.10), which originates from the point t2 = t1, vanishes for c = 0. What is the order of magnitude of the lepton asymmetry (5.44) relative to the Boltzmann result in the case c = c0 = 0? The Kadanoff–Baym result depends on s = Ct, like the Boltzmann result, and in addition on the dimensionless parameter C/M  1. In Appendix C we shown that

Lk ðt; tÞ ! 0; fL ðt; kÞ

for

C M

! 0;

s ¼ Ct fixed:

ð6:12Þ

Hence, in this zero-width limit, due to rapid oscillations of the integrand, the lepton asymmetry obtained from the Kadanoff–Baym equation is at least OðC=MÞ relative to the Boltzmann lepton asymmetry. We are thus led to the conclusion that the lepton asymmetry obtained from the Kadanoff–Baym equations does not contain the Boltzmann result as limiting case as long as free equilibrium propagators are used for lepton and Higgs fields. This may not be too surprising. After all, the underlying assumption in our calculation has been that (gauge) interactions, much faster than heavy neutrino decay, establish kinetic equilibrium for leptons and Higgs particles. These interactions will unavoidably lead to thermal damping widths much larger than C. If these interactions are not taken into account in the calculation of the lepton asymmetry, one misses the main contribution and obtains a misleading result. This means that at present the best estimate for the full quantum mechanical lepton asymmetry is given by Eq. (5.51), which leads to a temperature dependent suppression compared to the Boltzmann result. Note that the proposed incorporation of thermal damping rates leads to a Boltzmann-like result, Eq. (5.51), which is valid for t J 1=C. For t < 1=C, all terms have to be kept, and one has @ t Lk ðt; tÞjt¼0 ¼ 0, which is a propery of the exact result (5.8), contrary to the Boltzmann approximation. 7. Numerical analysis Let us now quantitatively compare the Boltzmann result (2.14) for the lepton asymmetry

fLi ðt; kÞ ¼ fli ðt; kÞ  fli ðt; kÞ with the Kadanoff–Baym result for the lepton asymmetry



Lkii ðt; tÞ ¼ tr c0 SþLkii ðt; tÞ :

ð7:1Þ

For free fields in thermal equilibrium both expressions are identical. For the Kadanoff–Baym result we use Eq. (5.51) which includes the estimated effect of thermal widths for lepton and Higgs fields. As shown in Appendix C, the Boltzmann result (2.14) can be reduced to a two-dimensional momentum integral (cf. (C.15)),

fLi ðt; kÞ ¼ 

ii 4p

F B ðk; bÞ

1

C

 1  eCt ;

ð7:2Þ

where we have defined

1 F B ðk; bÞ ¼ k

Z

1

pmin ðkÞ

dp

Z

k0max ðpÞ k0min ðpÞ

0

k dk

0

1

xp

! 0 2xp k  M 2 2xp k  M 2 fl/ ðk; xp  kÞfNeq ðxp Þ; 1 0 2pk 2pk ð7:3Þ

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi here xp ¼ M 2 þ p2 , the bracket represents the product of 4-vectors divided by the corresponding energies, k k0 /(kk0 ), and the integration boundaries are 2

pmin ðkÞ ¼

jM 2  4k j ; 4k

0

kmin ¼

xp  p 2

0

kmax ðpÞ ¼

;

xp þ p 2

:

ð7:4Þ

The dependence on temperature (b = 1/T) enters through the equilibrium distribution functions of Higgs particles and leptons,

fl/ ðk; qÞ ¼ 1  fl ðkÞ þ f/ ðqÞ; q ¼ xp  k; 1 1 1 ; f / ðqÞ ¼ bq ; fNeq ðxp Þ ¼ bxp : fl ðkÞ ¼ bk e þ1 e 1 e þ1

ð7:5Þ ð7:6Þ

The Kadanoff–Baym result (5.51) for the lepton asymmetry, which includes effects of thermal damping, takes the same form as the Boltzmann result

Lkii ðt; tÞ ¼ 

ii 4p

1

F KB ðk; bÞ

C

ð1  eCt Þ:

ð7:7Þ

Since the integrand of the momentum integrations contains two delta-functions less than the expression for the Boltzmann result, the function FKB(k, b) can only be written as a four-dimensional integral (cf. (C.19)),

F KB ðk; bÞ ¼

1 1 p2 k

Z

1

dp

Z

0

k dk

k0min ðpÞ

pmin ðkÞ

 1

k0max ðpÞ

Z

qþ

dq

Z

q0þ

q0

q

dq

0

1

xp

! 2 02 p2 þ k  q2 p2 þ k  q02 0 fl/ ðk; qÞfl/ ðk ; q0 ÞfNeq ðxp Þ 0 2pk 2pk

cc0

 ;  0 ðxp  k  qÞ2 þ c2 ðxp  k  q0 Þ2 þ c02

ð7:8Þ

with the integration boundaries

q ¼ jp kj;

0

q0 ¼ jp k j:

ð7:9Þ 6g 2 8p

For the thermal widths we use the estimate c ’ c

T 0:1T (cf. [44]). Note that the damping in a non-Abelian plasma is considerably stronger than in an electromagnetic plasma at the same temperature. It is instructive to compare the Boltzmann and Kadanoff–Baym results with the prediction of quantum Boltzmann equations. As shown in [20,19], these equations lead to an additional statistical factor compared to Boltzmann equations, which implies for the lepton asymmetry 0

F QB ðk; bÞ ¼

1 k

Z

1

pmin ðkÞ

dp

Z

k0max ðpÞ

k0min ðpÞ

0

k dk

0

1

xp

! 0 2xp k  M 2 2xp k  M 2 0 0 fl/ ðk; xp  kÞfl/ ðk ; xp  k ÞfNeq ðxp Þ:  1 0 2pk 2pk

ð7:10Þ

In [20,19], this enhancement has been included in an effective, temperature-dependent CP asymmetry. In Fig. 8 Boltzmann and Kadanoff–Baym results for the lepton asymmetry are compared. At momenta k 0.2, where both distributions peak, the differences are less than 20%, at larger momenta they reach at most 50% (cf. Fig. 10). At temperatures T 0.3, where leptogenesis takes place for typical neutrino parameters [30,45], differences are essentially negligible. Boltzmann and quantum Boltzmann results for the lepton asymmetry are compared in Fig. 9. At momenta k 0.2, where both distributions are maximal, the differences can exceed 100%, and they remain large also at larger momenta (cf. Fig. 10). An enhancement Oð100%Þ at T 1 is qualitatively consistent with the enhancement found for the temperature-dependent CP asymmetries in [20,19].

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

T

1

1.0

B F k, β

0.8

KB

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6

0.8

1.0

k T

0.5

0.10

B F k, β

0.08

KB

0.06 0.04 0.02 0.00 0.0

0.2

0.4

0.6

0.8

1.0

k T

0.3

F k, β

0.014 0.012

B

0.010

KB

0.008 0.006 0.004 0.002 0.000 0.0

0.2

0.4

0.6

0.8

1.0

k Fig. 8. Comparison of the lepton asymmetry distribution functions obtained from Boltzmann equations (B, dot-dashed line) and Kadanoff–Baym equations (KB, full line) for three different temperatures; temperature and momentum are given in units of M.

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

T

1

2.5

B

2.0

F k, β

QB 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

k T

0.5

0.15

B

F k, β

QB 0.10

0.05

0.00 0.0

0.2

0.4

0.6

0.8

1.0

k T

0.3

0.015

B

F k, β

QB 0.010

0.005

0.000 0.0

0.2

0.4

0.6

0.8

1.0

k Fig. 9. Comparison of the lepton asymmetry distribution functions obtained from Boltzmann equations (B, dot-dashed line) and quantum Boltzmann equations (QB, dashed) for three different temperatures; temperature and momentum are given in units of M.

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

1.5

FKB k, β FB k, β

1.0

0.5

T

1

T

0.5

T

0.3

0.0 0.0

0.2

0.4

0.6

0.8

1.0

k 3.0

FB k, β

FQB k, β

2.5

T

1

T

0.5

T

0.3

2.0

1.5

1.0 0.0

0.2

0.4

0.6

0.8

1.0

k Fig. 10. Ratio of Kadanoff–Baym and Boltzmann lepton asymmetries (upper panel) and ratio of quantum Boltzmann and Boltzmann lepton asymmetries (lower panel) for three different temperatures; temperature and momentum are given in units of M.

The Kadanoff–Baym result strongly depends on the size of the thermal damping rates. For c, c0 ? 0, off-shell effects disappear, and the Kadanoff-Baym result approaches the quantum Boltzmann result. Numerically, already for c ’ c0 0.01T the differences are negligible. However, in a non-Abelian plasma, damping rates are large and, as a consequence, they almost compensate the enhancement due to the additional statistical factor contained in the quantum Boltzmann as well as the Kadanoff–Baym result. We conclude that, according to our estimates, the conventional Boltzmann equations provide rather accurate predictions for the lepton asymmetry. 8. Summary and conclusions The goal of leptogenesis is the prediction of the cosmological baryon asymmetry, given neutrino masses an mixings. In a ‘theory of leptogenesis’, it must be possible to quantify the theoretical error on this prediction. This requires to go beyond Boltzmann as well as quantum Boltzmann equations, such that the size of memory and off-shell effects can be systematically computed. In the present paper we have shown how to calculate the lepton asymmetry from first principles, i.e., in the framework of nonequilibrium quantum field theory. Our calculation is entirely based on Green’s functions, and it therefore avoids all assumptions which are needed to arrive at Boltzmann equations.

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Two key ingredients make the problem solvable. First, the thermal bath has a large number of degrees of freedom, all standard model particles, compared to only one particle out of equilibrium, the heavy neutrino. Hence, the backreaction of its equilibration on the temperature of the thermal bath can be neglected. Second, the heavy neutrino is only weakly coupled to the thermal bath and we can use perturbation theory in the corresponding Yukawa coupling k. The weak coupling of the heavy neutrino to the bath allowed us to obtain analytic expressions for the spectral function, which do not depend on initial conditions, and the statistical propagator. In Section 4 we have discussed two solutions of the Kadanoff–Baym equations, which correspond to thermal and vacuum initial conditions. The statistical propagator which interpolates between vacuum at t = 0 and thermal equilibrium at large times can then be used in the computation of the lepton asymmetry. Thermal leptogenesis has two vastly different scales, the width C of the heavy neutrino on one side, and its mass M, temperature T of the bath and thermal damping widths c on the other side,

C k2 M  c g 2 T < T K M: Typical leptogenesis parameters (cf. [2]) are C 107M, c 0.1T, T 0.3M, M 1010 GeV. The existence of interactions in the plasma, which are fast compared to the equilibration time sN = 1/C of the heavy neutrino, is always implicitly assumed to justify the use of Boltzmann equations for the calculation of the asymmetry, but their effects are usually not explicitly taken into account. The main result of this paper is the computation of the lepton asymmetry in Section 5, where the nonequilibrium propagators of the heavy neutrino and free equilibrium propagators for massless lepton and Higgs fields are used. Compared to Boltzmann and quantum Boltzmann equations, the crucial difference of the result (5.44)–(5.49) are the memory effects, oscillations with frequencies OðMÞ, much faster than the heavy neutrino equilibration time sN = 1/C. These oscillations strongly suppress the generated lepton asymmetry Lk(t, t) compared to the Boltzmann result fL(t, k). In fact, as shown in Appendix C, the ratio Lk(t, t)/fL(t, k) vanishes in the ‘zero-width’ limit C/M ? 0, with s = Ct fixed. This situation changes when the interactions, which in the Boltzmann approach are assumed to establish kinetic equilibrium, are explicitly included in the calculation. Lepton and Higgs fields in the thermal bath then acquire large thermal damping widths c g2T, which cut off the oscillations. As a consequence, the predicted lepton asymmetry is similar to the quantum Boltzmann result, except for off-shell effects which are now included. For small damping widths, c  T, the off-shell effects are negligible. They are large, however, in the standard model plasma. According to our calculation, using c 0.1T, the damping effects essentially compensate the enhancement due to the additional statistical factor of the quantum Boltzmann equations. We conclude that, after all corrections are taken into account, the conventional Boltzmann equations again provide rather accurate predictions for the lepton asymmetry. Note that the classical Boltzmann behaviour emerges at large times, t J 1=C > 1=c, while at early times all terms are of similar magnitude, and all quantum effects have to be kept. As already emphasised in [8], it is of crucial importance to include gauge interactions in the Kadanoff–Baym approach to make further progress towards a ‘theory of leptogenesis’. It remains to be seen whether the qualitative effects of thermal damping, as discussed in this paper, will then be confirmed or whether new surprises are encountered. Acknowledgements The authors thank D. Bödeker, O. Philipsen, M. Shaposhnikov and C. Weniger for helpful discussions, and J. Schmidt for sharing his expertise. This work was supported by the German Science Foundation (DFG) within the Collaborative Research Center 676 ‘‘Particles, Strings and the Early Universe’’ and by the Swiss National Science Foundation. Appendix A. Thermal propagators In the following we list all propagators, which are needed in the calculation described in Section 5, as functions of relative time y = t1  t2 and total time t = (t1 + t2)/2.

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2029

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Free massive scalar ðxq ¼ m2 þ q2 Þ

Dq ðyÞ ¼

1

xq

sinðxq yÞ;

ðA:1Þ

  bxq cosðxq yÞ; 2xq 2     1 bxq D11 ðyÞ ¼ coth x yÞ  i sinð x jyjÞ cosð q q q 2xq 2

Dþq ðyÞ ¼

1

coth

i ¼ Dþq ðyÞ  signðyÞDq ðyÞ; 2     1 bxq D22 ðyÞ ¼ coth x yÞ þ i sinð x jyjÞ cosð q q q 2xq 2 i ¼ Dþq ðyÞ þ signðyÞDq ðyÞ; 2     1 bxq D>q ðyÞ ¼ coth cosðxq yÞ  i sinðxq yÞ ; 2xq 2     1 bxq cosðxq yÞ þ i sinðxq yÞ : D
ðA:2Þ

ðA:3Þ

ðA:4Þ ðA:5Þ ðA:6Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

Free massive Dirac fermion ðxk ¼ m2 þ k Þ

Sk ðyÞ ¼ ic0 cosðxk yÞ þ

m  kc

xk

sinðxk yÞ;

   1 bxk m  kc tanh cosðxk yÞ ; ic0 sinðxk yÞ  xk 2 2     c0 bxk S11 ðyÞ ¼ x yÞsignðyÞ  i tanh x yÞ cosð sinð k k k 2 2     m  kc bxk þ tanh cosðxk yÞ  i sinðxk jyjÞ ; 2xk 2 Sþk ðyÞ ¼ 

i ¼ Sþk ðyÞ  signðyÞSk ðyÞ; 2     c0 bxk S22  cosðxk yÞsignðyÞ  i tanh sinðxk yÞ k ðyÞ ¼ 2 2     m  kc bxk tanh þ cosðxk yÞ þ i sinðxk jyjÞ ; 2xk 2 i ¼ Sþk ðyÞ þ signðyÞSk ðyÞ; 2     c bxk sinðxk yÞ S>k ðyÞ ¼ 0 cosðxk yÞ  i tanh 2 2     m  kc bxk cosðxk yÞ  i sinðxk yÞ ; tanh þ 2xk 2     c bxk sinðxk yÞ S
ðA:7Þ ðA:8Þ

ðA:9Þ

ðA:10Þ

ðA:11Þ

ðA:12Þ

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

The propagators for a massless left-handed fermion are obtained by the substitutions xk ! k ¼ jkj; S...k ! PL S...k , where PL = (1  c5)/2.  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Free massive Majorana fermion xp ¼ M 2 þ p2

  M  pc Gp ðyÞ ¼ ic0 cosðxp yÞ þ sinðxp yÞ C 1 ; xp    1 bxp M  pc þ cosðxp yÞ C 1 ; ic0 sinðxp yÞ  Gp ðyÞ ¼  tanh xp 2 2      c0 bxp 11 Gp ðyÞ ¼ cosðxp yÞsignðyÞ  i tanh sinðxp yÞ 2 2     M  pc bxp tanh cosðxp yÞ  i sinðxp jyjÞ C 1 ; þ 2xp 2      c0 bxp 22 Gp ðyÞ ¼  cosðxp yÞsignðyÞ  i tanh sinðxp yÞ 2 2     M  pc bxp tanh cosðxp yÞ þ i sinðxp jyjÞ C 1 ; þ 2xp 2      c bxp G>p ðyÞ ¼ 0 cosðxp yÞ  i tanh sinðxp yÞ 2 2     M  pc bxp tanh cosðxp yÞ  i sinðxp yÞ C 1 ; þ 2xp 2      c bxp G

ðA:13Þ ðA:14Þ

ðA:15Þ

ðA:16Þ

ðA:17Þ

ðA:18Þ

Nonequilibrium massive Majorana fermion (interpolation between vacuum at t = y = 0 and thermal equilibrium at t = 1, and memory integral)

  M  pc Gp ðyÞ ¼ ic0 cosðxp yÞ þ sinðxp yÞ eCp jyj=2 C 1 ; xp   M  pc cosðxp yÞ Gþp ðt; yÞ ¼  ic0 sinðxp yÞ  xp     1 bxp Cp jyj=2  tanh þ fNeq ðxp ÞeCp t C 1 ; e 2 2 i 11 þ Gp ðt; yÞ ¼ Gp ðt; yÞ  signðyÞGp ðyÞ; 2 i 22 þ Gp ðt; yÞ ¼ Gp ðt; yÞ þ signðyÞGp ðyÞ; 2 i G>p ðt; yÞ ¼ Gþp ðt; yÞ  Gp ðyÞ; 2 i G

ðA:19Þ

ðA:20Þ ðA:21Þ ðA:22Þ ðA:23Þ ðA:24Þ

ðA:25Þ

Appendix B. Feynman rules For completeness, we list in the following the Feynman rules for the Standard Model Lagrangian with right-handed neutrinos given in Eq. (1.2); a, b are spinor indices and a, b, . . . are SU(2) indices.

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Majorana neutrino

Lepton doublet

Higgs doublet

Vertices

2031

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Appendix C. Zero-width limit In this section we consider the Kadanoff–Baym result for the lepton asymmetry normalised to the Boltzmann result, Lk(t, t)/fL(t, k), in the zero-width limit as defined in Eq. (6.12), i.e.,

C M

! 0;

s ¼ Ct fixed:

To this end we have to evaluate the corresponding momentum integral (5.45) in this limit. C.1. Boltzmann equation Consider first the Boltzmann result for the lepton asymmetry given in Eq. (2.14),

fLi ðt; kÞ ¼  ii

16p k

Z

0

0

k  k ð2pÞ4 d4 ðk þ q  pÞð2pÞ4 d4 ðk þ q0  pÞ

q;p;q0 ;k0

 1 1  eCt :

 fl/ ðk; qÞfNeq ðxp Þ

ðC:1Þ

C

The integration over q and q0 can be performed using the d-functions, which leads to

fLi ðt; kÞ ¼ 

Z

ii 16p3

3

d p

Z

0

3 0

d k

 fl/ ðk; qÞfNeq ðxp Þ

1

C

kk 1 0 dðk þ q  xp Þdðk þ q0  xp Þ 0 kk xp qq0

ð1  eCt Þ;

ðC:2Þ

0 0 ^k ^ 0 Þ, depends on the angles bewhere q = jqj and q0 = jq0 j. The product of 4-vectors, k  k ¼ kk ð1  k tween the different momenta. It is convenient to define the angles with respect to the momentum 0 0 p: h = \(k,p), h0 = \(k0 ,p) and u0 ¼ \ðk? ; k? Þ; here k\ and k? are perpendicular to the vector p, i.e., 0 0 0 ^ and k ^ 0 are given by (see Fig. 11) k = kk + k\ and k ¼ kk þ k? . In terms of these angles the unit vectors k

0

cos h

1

C ^¼B k @ sin h A;

0

1

cos h0

C ^0 ¼ B k @ sin h0 cos u0 A;

ðC:3Þ

sin h0 sin u0

0

^k ^ 0 ¼ cos h cos h0 þ sin h sin h0 cos u0 . We then obtain with k

fLi ðt; kÞ ¼ 

ii 16p3

Z

3

d p

Z

1

02

k dk

0

0

Z

1

1

d cos h0

Z 2p

du0

0

1

xp qq0

 ð1  cos h cos h0  sin h sin h0 cos u0 Þ 0

 dðk þ q  xp Þdðk þ q0  xp Þfl/ ðk; qÞfNeq ðxp Þ

1

C

 1  eCt :

ðC:4Þ

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

Fig. 11. Integration angles.

Momentum conservation relates the energies q and q0 to p, k, k0 and the angles h and h0 , 2

q ¼ jp  kj ¼ ðp2 þ k  2pk cos hÞ1=2 ; 0

0

2

02

0 1=2

q ¼ jp  k j ¼ ðp þ k  2pk cos h Þ

ðC:5Þ ðC:6Þ

:

We can now make use of rotational invariance of the distribution function,

fLi ðt; kÞ ¼

1 4p

Z

dXk f Li ðt; kÞ:

ðC:7Þ

Changing variables,

dq ¼ 

0

pk d cos h; q

0

dq ¼ 

pk d cos h0 ; q0

ðC:8Þ

one arrives at

ii 1 fLi ðt; kÞ ¼  4p k 

1

xp

Z

dp

Z

1

0

0

k dk

Z

q

qþ

0 0

dq

Z

2

q0

q0þ

dq

0

02

p2 þ k  q2 p2 þ k  q02 1 0 2pk 2pk

dðk þ q  xp Þdðk þ q0  xp Þfl/ ðk; qÞfNeq ðxp Þ

1

C

 1  eCt ;

!

ðC:9Þ

where the limits of integration are given by the maximal and minimal value of q and q0 , respectively,

q ¼ jk pj;

0

q0 ¼ jk pj:

ðC:10Þ min 1

Consider now the argument of one d-function, X1 = xp  k  q, with X ¼ xp  k  qþ and Xmax ¼ xp  k  q (cf. Eq. (5.41)). Obviously, the conditions Xmin < 0 and Xmax > 0 limit the integra1 1 1 tion range in p for given momentum k,

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A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038 2

p>

jM 2  4k j  pmin ðkÞ: 4k

ðC:11Þ 0

Similarly, the constraint p > (M2  4k 2)/(4k0 ) restricts the integration range in k0 for given p, 0

k >

xp  p 2

0

0

 kmin ðpÞ; k <

xp þ p 2

0

 kmax ðpÞ:

ðC:12Þ

Changing again variables from q and q0 to X1 and X3, respectively, and using 0

@ðp; k ; X1 ; X3 Þ ¼ 1; 0 @ðp; k ; q; q0 Þ

ðC:13Þ

the integral can now be written as

fLi ðt; kÞ ¼  

ii 1 4p k 1

xp

Z

1

dp

Z

k0max ðpÞ

k0min ðpÞ

pmin ðkÞ

dk

Z

0

Xmax 1

Xmin 1 2

dðX1 ÞdðX3 Þ 1 

 fl/ ðk; qÞfNeq ðxp Þ

dX1

Z

Xmax 3

Xmin 3

dX3

02

p2 þ k  q2 p2 þ k  q02 0 2pk 2pk

!

 1 1  eCt :

ðC:14Þ

C

The limits of integration have been chosen such that they contain the points X1 = 0 and X3 = 0, which correspond to energy conservation, q = xp  k and q0 = xp  k0 , respectively. Hence, the integration on X1 and X3 can trivially be carried out, and we obtain the final result

ii 1 fLi ðt; kÞ ¼  4p k

Z

1

dp

pmin ðkÞ

Z

k0max ðpÞ

k0min ðpÞ

 fl/ ðk; xp  kÞfNeq ðxp Þ

1

C

dk

1

0

xp

0

2xp k  M2 2xp k  M2 1 0 2pk 2pk

ð1  eCt Þ:

!

ðC:15Þ

C.2. Kadanoff–Baym equation We are now ready to evaluate the leading contribution of the Kadanoff–Baym result for the lepton asymmetry. It is given by Eq. (5.46) with a = b = 1, and it can be written in the form

Z

0

1 C kk 0 2 f l/ ðk; qÞfl/ ðk ; q0 ÞfNeq ðxp Þ 0 2 2 0 2 C q;q0 kk xp ððxp  k  qÞ þ 4 Þððxp  k  q0 Þ2 þ C4 Þ h Ct   Ct    0  e 2  cosððxp  k  qÞtÞ e 2  cos ðxp  k  q0 Þt i   0 ðC:16Þ  sinððxp  k  qÞtÞ sin ðxp  k  q0 Þt :

Lkii ðt; tÞ ¼  ii 8p

We first change variables, (q, q0 ) ? (p, k0 ), with p = q + k = q0 + k0 , and use rotational invariance,

Lkii ðt; tÞ ¼

1 4p

Z

dXk Lkii ðt; tÞ:

ðC:17Þ

Choosing again angles according to Fig. 11, the integral (C.16) becomes

Z

Z

Z 2p

0

kk 0 0 Fðh; h ; . . .Þ kk 0 Z 1 Z 1 Z 2p d cos h d cos h0 du0 ð1  cos h cos h0 ¼

Lkii ðt; tÞ /

dXk

dXk0

1

du0

0

1

 sin h sin h0 cos u0 Þ Fðh; h0 ; . . .Þ Z 1 Z 1 ¼ ð2pÞ2 d cos h d cos h0 ð1  cos h cos h0 ÞFðh; h0 ; . . .Þ; 1

1

ðC:18Þ

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

2035

where we have used that the function F(h, h0 , . . .) does not depend on the angle u0 . As in the previous section, we now change the integration variables from (h, h0 ) to (q, q0 ), and using Eq. (C.8) we obtain

Lkii ðt; kÞ ¼ 

ii 8p3

Z

1 k

 1

1

dp

pmin ðkÞ

Z

k0max ðpÞ

k0min ðpÞ

0

k dk

0

Z

qþ

q

dq

Z

q0þ

dq

0

q0

1

xp

! 2 02 p2 þ k  q2 p2 þ k  q02 0 fl/ ðk; qÞfl/ ðk ; q0 ÞfNeq ðxp Þ 0 2pk 2pk 1

C

2   2 2 0 ðxp  k  qÞ2 þ C4 ðxp  k  q0 Þ2 þ C4 h  Ct  Ct   0  e 2  cosððxp  k  qÞtÞ e 2  cos ðxp  k  q0 Þt   0  sinððxp  k  qÞtÞ sin ðxp  k  q0 Þt ;

ðC:19Þ

where the limits of integration are given in Eqs. (C.10)–(C.12). We have restricted the integration over p and k to the range for which the intervals [q, q+] and ½q0 ; q0þ  contain points satisfying xp  k  q = 0 and xp  k0  q0 = 0, respectively. This finite part of the integral could then be Oð1=CÞ, which is required to match the Boltzmann result for the lepton asymmetry. The remaining part is Oð1Þ and therefore suppressed compared to the Boltzmann result. Remarkably, the integral (C.19) is a sum of terms each of which factorizes into a product where one factor depends on q but not on q0 , whereas the other factor depends on q0 but not on q. Hence one obtains

Lkii ðt; tÞ /

Z

1

dp

Z

k0max ðpÞ

k0min ðpÞ

pmin ðkÞ

0

k dk

0

X

P i ðq ; qþ ÞQi ðq0 ; q0þ Þ;

ðC:20Þ

i

where we have dropped the dependence of the factors P i and Qi on k, p and k0 for simplicity. Because of the factorization, we can now perform the integrations on q and q0 separately. Naively, one may think that in the zero-width limit C/M ? 0 the cosine terms can be set to one. But for large time t, they oscillate fast, which leads to a different result. Consider the following contribution to the integral (C.19),

Z

Pðq ; qþ Þ ¼ 

qþ

dq

q

FðqÞ 2

ðxp  k  qÞ2 þ C4

cosððxp  k  qÞtÞ;

ðC:21Þ

where F(q) has no poles. Changing the integration variable from q to z = 2X1/C, with X1 = xp  k  q, one obtains

Pðzmin ; zmax Þ ¼

i 2C

Z

zmax

zmin

    C 1 1 C C  eiz 2 t þ eiz 2 t ; dz F i xp  k  z zi zþ1 2

ðC:22Þ

max where zmin ¼ 2Xmin =C, with zmin < 0 and zmax > 0. In the limit C/M ? 0 with s = Ct 1 =C and zmax ¼ 2X1 fixed, the integration limits approach zmin ? 1 and zmax ? +1, respectively. The integral is now easily evaluated by means of the residue theorem leading to the result

 

CPðzmin ; zmax ÞjC!0 ¼ p F xp  k  i

    C 2s  s e2 þ F xp  k þ i e  2 2 C!0

C

s

¼ 2pFðxp  kÞe2 :

ðC:23Þ

In Eq. (C.19) the term P appears together with a second term,

P 0 ðq ; qþ Þ ¼

Z

qþ

q

dq

FðqÞ

s

2

ðxp  k  qÞ2 þ C4

e2 ;

which can be evaluated in the same way as P in the zero-width limit, yielding

ðC:24Þ

2036

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

 

CP 0 ðzmin ; zmax ÞjC!0 ¼ p F xp  k  i

C 2



  C  s e2 þ F xp  k þ i  2 C!0

s

¼ 2pFðxp  kÞ e2 :

ðC:25Þ

Clearly, the two terms P and P 0 add up to zero. The same result is obtained for the second factor Q after the q0 integration, as well as for the product of two sinus functions. We conclude that the integral (C.16) does not contain a contribution Oð1=CÞ. Hence, the ratio of Kadanoff–Baym result and Boltzmann result, Lk(t, t)/fL(t, k), approaches zero in the limit C/M ? 0, s = Ct fixed. Appendix D. Equilibrium contribution In Section 5 we argued that the equilibrium part of the heavy neutrino propagator does not contribute to the lepton asymmetry. In this section we verify this claim. The heavy neutrino propagator has an equilibrium and a nonequilibrium part,

e Gp ðt1 ; t 3 Þ ¼ Geq p ðt 1  t 3 Þ þ G p ðt 1 ; t 3 Þ;

ðD:1Þ

whose main difference lies in the time dependence,

e p ðt 1 ; t3 Þ / eC2ðt1 þt3 Þ : G

C

 2 jt 1 t 3 j Geq ; p ðt 1  t 3 Þ / e

ðD:2Þ

The computation of the lepton asymmetry in Section 5 was based on the nonequilibrium part, and it involved the time integral I (cf. Eq. (5.41)). Because of the different time dependence given in Eq. (D.2), the contribution of the equilibrium part to the asymmetry involves instead the integral

J ðtÞ ¼

Z

t

dt1

0

Z

t

dt 2

Z

0

t2

C

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 e2 jt1 t3 j ;

ðD:3Þ

0

which differs from I only with respect to the damping factor. X1, X2 and X3 are different linear combinations of energies, which satisfy X1 = X2 + X3. In order to evaluate the integral J , we have to split the time integration,

J ðtÞ ¼

Z

t

dt 1

Z

0

þ

Z

t1

dt 2

Z

0

t

dt2

C

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 e2 ðt1 t3 Þ

0

Z

t1

t2

t1

Z

C

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 e2 ðt1 t3 Þ þ

0

t2

C

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 e2 ðt3 t1 Þ

 :

ðD:4Þ

t1

Note the change of sign in the damping factor of the last two terms. As in Section 5, it is convenient to use the variables X1 ¼ X1  2i C and X3 ¼ X3  2i C, for which the integral simplifies to

J ðtÞ ¼

Z

t

dt 1 Z

t1

dt 2

Z

0

0

þ

Z

t2

t2

dt 3 eiX1 t1 þiX2 t2 þiX3 t3 þ

0 



dt 3 eiX1 t1 þiX2 t2 þiX3 t3

Z

t

t1



dt2

Z

t1

dt 3 eiX1 t1 þiX2 t2 þiX3 t3

0

:

ðD:5Þ

t1

Performing the t3 integral and using the relation X1 = X2 + X3, we obtain

J ðtÞ ¼

Z

t

dt 1

0

þ

Z

t1

dt 2 eiX1 t1

0

Z

t

t1

iX1 t 1

dt2 e

 1  iX 1 t 2 e  eiX2 t2

iX3

!#     1  iX3 t1 iX 2 t 2 iX1 t 1 1 iX1 t 2 iX3 t 1 iX2 t2 : e 1 e þe e e e iX3 iX3

ðD:6Þ

A. Anisimov et al. / Annals of Physics 326 (2011) 1998–2038

It is now straightforward to carry out the integrations over t1 and t2, which leads to "  2    J ðtÞ þ J ðtÞ ¼ CðX1 þ X3 Þ cosððX1  X3 ÞtÞ  1 þ ðcosðX1 tÞ 2 2 C2 C2 ðX1 þ 4 Þ X3 þ 4 ðX1  X3 Þ ! #   C2  Ct Ct sinððX1  X3 ÞtÞ  ðsinðX1 tÞ  sinðX3 tÞÞe 2 :  cosðX3 tÞÞe 2 þ 2X1 X3  2

2037

ðD:7Þ

Note that the expression has no pole at X1 = X3. As in Appendix C we now have to evaluate the momentum integral

S¼

Z

Xmax 1

Xmin 1

dX1

Z

Xmax 3

Xmin 3

dX3 ðJ þ J  Þ;

ðD:8Þ

with the integration limits given below Eq. (C.10). To perform the zero-width limit, we again introduce max the variables z1,3 = 2X1,3/C. For C ? 0, the limits of integration zmin 1;3 and z1;3 approach 1 and +1, respectively. The z3-integration can now be carried out by means of the residue theorem. The integrand of the remaining z1-integration has a double pole. The integration can again be performed using the residue theorem, and we find that CS approaches zero in the limit C/M ? 0, s = Ct fixed. Hence, the equilibrium part of the heavy neutrino propagator does not contribute at leading order in C/M. References [1] M. Fukugita, T. Yanagida, Phys. Lett. B 174 (1986) 45. [2] For a review and references see, for example, W. Buchmuller, R.D. Peccei, T. Yanagida, Ann. Rev. Nucl. Part. Sci. 55 (2005) 311. Available from: ; S. Davidson, E. Nardi, Y. Nir, Phys. Rept. 466 (2008) 105. Available from: . [3] V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. B 155 (1985) 36. [4] L.P. Kadanoff, G. Baym, Quantum Statistical Mechanics, Benjamin, New York, 1962. [5] J. Schwinger, J. Math. Phys. 2 (1961) 407. [6] L.V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 (Sov. Phys. JETP 20 (1965) 1018). [7] P.M. Bakschi, K.T. Mahanthappa, J. Math. Phys. 4 (1963) 1. [8] A. Anisimov, W. Buchmuller, M. Drewes, S. Mendizabal, Phys. Rev. Lett. 104 (2010) 121102. Available from: . [9] W. Buchmuller, S. Fredenhagen, Phys. Lett. B 483 (2000) 217. Available from: . [10] L. Covi, N. Rius, E. Roulet, F. Vissani, Phys. Rev. D 57 (1998) 93. Available from: . [11] G.F. Giudice, A. Notari, M. Raidal, A. Riotto, A. Strumia, Nucl. Phys. B 685 (2004) 89. Available from: <. [12] C.P. Kiessig, M. Plumacher, M.H. Thoma, Phys. Rev. D 82 (2010) 036007. Available from: . [13] A. Anisimov, D. Besak, D. Bodeker. Available from: . [14] M. Lindner, M.M. Muller, Phys. Rev. D 73 (2006) 125002. Available from: . [15] M. Lindner, M.M. Muller, Phys. Rev. D 77 (2008) 025027. Available from: . [16] A. De Simone, A. Riotto, JCAP 0708 (2007) 002. Available from: . [17] M. Garny, A. Hohenegger, A. Kartavtsev, M. Lindner, Phys. Rev. D 80 (2009) 125027. Available from: <0909.1559 [hep-ph]>. [18] M. Garny, A. Hohenegger, A. Kartavtsev, M. Lindner, Phys. Rev. D 81 (2010) 085027. Available from: . [19] M. Beneke, B. Garbrecht, M. Herranen, P. Schwaller, Nucl. Phys. B 838 (2010) 1. Available from: . [20] M. Garny, A. Hohenegger, A. Kartavtsev, Phys. Rev. D 81 (2010) 085028. Available from: . [21] V. Cirigliano, C. Lee, M.J. Ramsey-Musolf, S. Tulin, Phys. Rev. D 81 (2010) 103503. Available from: . [22] M. Beneke, B. Garbrecht, C. Fidler, M. Herranen, P. Schwaller, Nucl. Phys. B 843 (2011) 177. Available from: . [23] B. Garbrecht. Available from: . [24] T. Prokopec, M.G. Schmidt, S. Weinstock, Ann. Phys. 314 (2004) 208. Available from: . [25] T. Prokopec, M.G. Schmidt, S. Weinstock, Annals Phys. 314 (2004) 267. Available from: . [26] T. Konstandin, T. Prokopec, M.G. Schmidt, Nucl. Phys. B 716 (2005) 373. Available from: . [27] T. Konstandin, T. Prokopec, M.G. Schmidt, M. Seco, Nucl. Phys. B 738 (2006) 1. Available from: . [28] M. Herranen, K. Kainulainen, P.M. Rahkila. Available from: . [29] A. Anisimov, W. Buchmuller, M. Drewes, S. Mendizabal, Annals Phys. 324 (2009) 1234. Available from: . [30] W. Buchmuller, P. Di Bari, M. Plumacher, Nucl. Phys. B 643 (2002) 367 [Erratum-ibid. B 793 (2008) 362] [Available from: ]. [31] L. Covi, E. Roulet, F. Vissani, Phys. Lett. B 384 (1996) 169. Available from: . [32] M. Flanz, E.A. Paschos, U. Sarkar, Phys. Lett. B 345 (1995) 248 (Erratum-ibid. B 382 (1996) 447] [Available from: .

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