ARS Leptogenesis

ARS Leptogenesis

Abstract

We review the current status of the leptogenesis scenario originally proposed by Akhmedov, Rubakov and Smirnov (ARS). It takes place in the parametric regime where the right-handed neutrinos are at the electroweak scale or below and the CP-violating effects are induced by the coherent superposition of different right-handed mass eigenstates. Two main theoretical approaches to derive quantum kinetic equations, the Hamiltonian time evolution as well as the Closed-Time-Path technique are presented, and we discuss their relations. For scenarios with two right-handed neutrinos, we chart the viable parameter space. Both, a Bayesian analysis, that determines the most likely configurations for viable leptogenesis given different variants of flat priors, and a determination of the maximally allowed mixing between the light, mostly left-handed, and heavy, mostly right-handed, neutrino states are discussed. Rephasing invariants are shown to be a useful tool to classify and to understand various distinct contributions to ARS leptogenesis that can dominate in different parametric regimes. While these analyses are carried out for the parametric regime where initial asymmetries are generated predominantly from lepton-number conserving, but flavor violating effects, we also review the contributions from lepton-number violating operators and identify the regions of parameter space where these are relevant.

ARS Leptogenesis

]M. Drewes, B. Garbrecht, P. Hernández1, M. Kekic, J. Lopez–Pavon, J. Racker, N. Rius, J. Salvado, D. Teresi

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1 Introduction

One of the most interesting implications of the extensions of the Standard Model (SM) with massive neutrinos is the possibility to explain the baryon asymmetry in the Universe via leptogenesis [1]. This mechanism has been shown to be robust and generic in seesaw models that involve a very high scale of new physics, much higher that the electroweak scale, . In the standard scenarios, leptogenesis takes place during the freeze-out of some heavy states that can decay violating charge-parity CP and lepton number . These high-scale scenarios have been studied extensively. For comprehensive reviews see [2] .

Akhmedov, Rubakov and Smirnov [3] (ARS) studied the possibility to generate a baryon asymmetry in type I seesaw models at a much lower scale, . The key observation is that the small Yukawa couplings required to explain neutrino masses in this low-scale scenario could be small enough to ensure that some of the sterile states [also referred to as right-handed (RH) neutrinos] might not reach thermal equilibrium before the electroweak phase transition, when sphaleron processes are switched off. ARS leptogenesis therefore is a freeze-in scenario. Pending on the parametric regime, lepton-number violating (LNV) processes may be negligible both in the generation of the asymmetries as well as in the washout because the Majorana mass of the RH neutrinos is small compared to the temperature. In that situation, the initial asymmetries in active leptons are purely flavored and lepton-number conserving (LNC), and eventually total asymmetries in the active sector arise and are approximatly counterbalanced by those in the sterile sector. If this situation survives until the electroweak phase transition, a net baryon asymmetry results, since the eventual equilibration later on can no longer be transmitted to the baryons in the absence of efficient sphaleron transitions. In other regions of parameter space LNV contributions may be relevant or even dominating in the source as well as washout terms and then must be accounted for. The aim of this chapter is to review the ARS mechanism of leptogenesis.

The model involves the simplest extension of the SM with heavy Majorana singlets

(1)

where are RH spinors (such that ), is a complex matrix, is a -dimensional complex symmetric matrix, and . The spectrum of this theory contains three lighter states with a mass matrix given by the famous seesaw formula

(2)

where , and heavy ones with masses of . The naive seesaw scaling (exact for one family) relating Yukawas with the light and heavy masses is therefore ), where is specified through (3) below.

In most of this work we will assume the minimal scenarios where . Different parametrizations of the Yukawa matrices have been used in the literature. For some purposes the parametrization in terms of the two unitary matrices that bi-diagonalize is useful:

(3)

For the purpose of parameter scanning however the Casas-Ibarra [4] parametrization is most appropriate. For it reads:

(4)

where is the PMNS matrix, is the diagonal matrix of the light neutrino masses (note that the lightest neutrino is massless because only two Majorana singlets are included), , where are the heavy neutrino masses and without loss of generality, we choose the mass basis for the RH neutrinos, is a matrix that depends on the neutrino ordering (NH, IH)

(5)

and finally is a generic two dimensional orthogonal complex matrix that depends on one complex angle . For , the parametrization is

(6)

where is a 3 dimensional complex orthogonal matrix that depends on three complex angles, .

2 Sakharov Conditions

As it is well known, three necessary Sakharov conditions need to be met in order to generate a baryon asymmetry from a symmetric initial condition. The model of Eq. (1) satisfies all of them:

  • Baryon number is violated by sphaleron processes

  • C and CP violation

Charge C is violated maximally, while CP violation arises from the complex nature of the flavor parameters and . After performing all the allowed field rephasings, it is easy to see that 3(6) physical CP-violating phases exist for .

  • Out-of-equilibrium condition

Assuming that the only interactions of the singlets are the Yukawa couplings in Eq. (1), it is easy to estimate when the states will reach thermal equilibrium. The scattering rate of the sterile states at temperature is roughly , with defined in Eq. (3). Comparing this to the Hubble expansion rate in a radiation dominated universe, and using the naive seesaw scaling:

(7)

where GeV is the temperature of the electroweak phase transition, , is the Planck mass, and the effective number of relativistic degrees of freedom at the temperature . This naive estimate shows that the singlet states thermalize quite close to the electroweak phase transition so it is not unnatural to have at least one state that has not reached thermal equilibrium by the time the sphaleron processes switch off.

As it is well known, CP-violating observables such as a lepton asymmetry, require the quantum interference of different amplitudes that involve different CP-violating phases as well as CP conserving ones. While in the standard scenario of leptogenesis from out-of-equilibrium decay the CP-conserving phases can be computed from the absorptive parts of the one-loop amplitudes, in the ARS case they are most suitably described by oscillations among the RH neutrinos.2 Indeed the sterile states are produced in flavor combinations that get modified in propagation, because they are a superposition of the mass eigenstates. Two scales are therefore relevant in the generation of the asymmetry the time when the oscillation rate is similar to the Hubble expansion or oscillation time

(8)

where , and the equilibration time when the scattering rate is of the order of the Hubble expansion:

(9)

Unless the mass eigenstates are extremely degenerate, there is a hierarchy of scales:

(10)

Note however that there are several flavors and therefore several oscillation and equilibration rates.

The generation of the asymmetry in the different flavors is effective at , so it takes place at temperatures much higher than the electroweak phase transition. At later times, when oscillations are very fast, quantum effects are no longer possible and the lepton asymmetries no longer grow, but the total lepton asymmetry keeps evolving because the equilibration rate of different flavors is different. After all the states reach the equilibration time , the asymmetry drops exponentially. If the EW phase transition happens before that, the subsequent evolution of the lepton asymmetries no longer affects the baryons and therefore whatever baryon asymmetry survives until remains.

3 CP invariants and naive estimates

CP-violating observables should be flavor basis invariant and be sensitive to the physical phases. In the case where we can consider all the flavor parameters as small so that we can Taylor expand in them, it is easy to find the basis independent invariants that usually require a minimum power of the flavor parameters. These invariants are useful because they allow to estimate the size of the asymmetry rather simply.

A well-known example is that of the SM with massless neutrinos. There is only one physical CP-violating phase in the quark sector and the lowest-order basis invariant combination of the Yukawas is the famous Jarlskog invariant:

(11)

where and is the CKM matrix. The invariant contains two factors: a rephasing invariant, the factor, that depends on mixing angles, which ensures that the phase becomes unphysical if any of the mixing angle vanishes, and a GIM factor that depends on the quark mass differences, which ensures that the same thing happens if any two quarks are degenerate. Any flavor-blind CP-violating observable such as the baryon asymmetry, if generated at in electroweak baryogenesis scenarios for example, is expected to be proportional to , which fails by many orders of magnitude as is found by direct computation [10, 11, 12].

In the seesaw model the situation is much more complicated because there are three/six independent CP-violating phases for . The flavor or weak basis (WB) invariants are constructed from products of the lepton Yukawa and mass matrices. Those sensitive to the CP-violating phases have been derived in [13] within the minimal type I seesaw model: all of them should vanish if CP is conserved, and conversely the non-vanishing of any of these invariants signals CP violation. They are invariant under the basis transformations:

(12)

where are the RH charged leptons of the SM. In general, there are 3 independent phases, and thus the same number of independent WB invariants. For , three CP-violating WB invariants can be constructed using just the neutrino Yukawa and mass matrices. Defining , and , they can be written as:

(13)
(14)
(15)

Since the are WB invariants, we can evaluate them in any basis. In the WB where the sterile neutrino mass matrix is real and diagonal (), one obtains:

(16)

with , and . From the above equations one can see that in the case there is only one independent CP-violating invariant, which at lowest order in the sterile neutrino masses is . Note that the quantities are related to the CP asymmetries which appear in unflavored leptogenesis via heavy Majorana neutrino decay,

(17)

where is a loop a function.

The remaining independent WB invariants that can be constructed involve also the charged lepton Yukawa couplings, . Since ARS leptogenesis occurs at , CP-violating phases associated with the Majorana character of the sterile neutrinos are suppressed by and do not play any role, if these effects are neglected. In this case, the appropriate CP-violating invariants at lowest order for are (only the first one for ):

(18)
(19)
(20)

where , and in the same WB as before,

(21)

with , , and

(22)

In the case , we find the third independent WB invariant at higher order in , .

One can also construct WB invariants involving and sensitive to Majorana phases, related to the flavored CP asymmetries in heavy neutrino decays, by just changing the matrix by in Eqs. (13)–(15[13], but they contain the same phases as the ones described here.

Note however that in ARS leptogenesis (and more generally in flavored leptogenesis), charged lepton Yukawa couplings are not small parameters: in fact, they mediate fast interactions that are resummed in the Boltzmann equations, leading to density matrices diagonal in the charged lepton mass basis. Therefore for the relevant CP-violating observables a perturbative expansion in the charged lepton Yukawas does not work, but rather a projection on the different flavors: the total lepton asymmetry is an incoherent sum of the flavor lepton asymmetries, which evolve independently. As a consequence the WB invariants do not appear in the final result, but the structure of the flavored CP-asymmetries is dictated by Eq. (22): for instance the (-conserving) CP asymmetry in flavor produced in the decay of the sterile neutrino is (see the accompanying article [14])

(23)

where is the projector on flavor ; and in ARS leptogenesis, where all sterile neutrino species contribute, one finds

(24)

We can still use the above WB invariants to estimate the size of the baryon asymmetry, in particular its dependence on the neutrino Yukawa couplings, , in processes like ARS leptogenesis which are flavor blind. From the above equations, we see that it must be at least fourth order in , however it turns out that , therefore the baryon number asymmetry, which is proportional to the sum of all the flavored asymmetries, should be . Thus the final asymmetry in ARS is expected to have the form

(25)

This reflects the fact that the CP asymmetries (and ) are lepton number conserving: a net lepton number asymmetry is generated only because each flavor lepton asymmetry evolves at a different rate. Notice that in the case of heavy Majorana neutrino decays, in general there is an additional piece of the CP-asymmetry,

(26)

which violates lepton number and does not vanish when summed over all flavors. Only in purely flavored leptogenesis, or in inverse seesaw scenarios where , a similar cancellation occurs.

However the dependence of the baryon asymmetry on may change if one of the lepton flavors is almost decoupled when the sphalerons freeze out, while the others have equilibrated; in this case the Yukawa couplings of the fast interacting species are not small parameters. Then, by lepton number conservation the asymmetry sequestered in the flavor is equal and opposite to the asymmetry in the rest of the leptonic sector, which is partially transformed into the baryon asymmetry. In this situation, the final result is .

Even if the final asymmetry is not WB invariant, in any basis there is still the freedom to make phase rotations of the lepton fields, and the physical quantities must be invariant under such rephasings. As a consequence, the CP-violating observables can be written in terms of independent rephasing invariants, as described in [15]. In order to find these, it is convenient to write explicitly the unitary matrices with the 3 phases that remain physical after all allowed field rephasing transformations. For we have

(27)
(28)

where represents the standard parameterization of a 33 mixing matrix, with 3 angles and 1 phase. Notice that the phases only enter in processes which depend on both, and , involving , and charged leptons; flavored and ARS leptogenesis are an example of such processes, while unflavored leptogenesis depends only on , and therefore on .

The Majorana phases of , are determined by two independent invariants of the form , while the Jarlskog invariant determines the phase . For , there is a single Majorana phase in , given by the rephasing invariant . When considering the parametric regime of ARS leptogenesis where the Majorana nature of the sterile neutrinos does not play a role, only the Dirac phase will be relevant and therefore we expect to find just the Jarlskog invariant of the matrix W, only in the case .

The remaining invariants involve also the matrix . They can be chosen as the Jarlskog invariant and two combinations of the form for two reference values of , which fix all the phases in the matrix (). If , there are only two more independent phases (), and therefore two independent rephasing invariants.

We can now write the CP asymmetries that appear in ARS leptogenesis in terms of the rephasing invariants as:

(29)

In turn, all these invariants can be written in terms of the 6 (3) independent ones for by using the unitarity of the mixing matrices .

Since the Majorana phases of do not contribute in the limit of small sterile neutrino Majorana masses that we are considering, we are left with 4 (2) invariants for . The result can be further simplified using that one of the sterile neutrinos is very weakly coupled, so in the approximation of neglecting we obtain:

(30)

Moreover, it can be shown that in this limit the phase of the matrix does not appear in Eq. (30), thus only three independent invariants contribute for , which we can choose as:

(31)
(32)
(33)

Although can be related to the above invariants, it is simpler to write the final baryon asymmetry for in terms of the four rephasing invariants . In the case , only the two independent invariants, , appear.

4 Quantum kinetic equations

The generation of lepton asymmetries is a purely quantum phenomenon and will therefore be missed in any classical treatment of the production of the sterile species. A related physical problem is that of oscillating neutrinos in the early Universe for which a formalism based on quantum kinetic equations was developed a long time ago [16]. Other approaches have been proposed [5, 17], and while there is not yet a fully unified formulation, different methods seem to give similar results.

There are two basic ingredients in any formulation of the problem: the dispersion relation of sterile neutrinos in a thermal plasma or refractive index and the scattering rates.

4.1 Dispersion relations of sterile neutrinos in a thermal plasma

The relativistic RH neutrinos acquire dispersive corrections from the thermal loop made up of Higgs and lepton doublets that can become of quantitative importance in certain parametric regions. This is in particular the case for strong mass degeneracy among the RH neutrinos or large active-sterile mixing, i.e. in the overdamped regime (cf. \srefsec:overdamped and \srefsec:largemix), where the damping time scale (or the scale associated with scattering rates) is of the same order or faster than the time scale of oscillations. This well-known refractive effect is calculated in the general context of fermions at finite temperature in Ref. [18]. Away from the mass-degenerate regime, the thermal mass corrections appear at next-to-leading order and are therefore subdominant. For leptogenesis in the strong washout regime, this aspect is discussed in the accompanying article [19].

4.2 Collision rates

For ultrarelativistic RH neutrinos, the phase space for reactions is suppressed at leading order and only opens up when Standard Model interactions, in particular mediated by gauge and top-quark Yukawa couplings are included. The leading logarithmic correction turns out to originate from the -channel exchange of a doublet lepton in a process with gauge radiation. Beyond the leading logarithm, also the leading order effects from hard scatterings as well as dispersive thermal effects have been computed.

The production of ultrarelativistic RH neutrinos has first been calculated using the imaginary-time formalism of thermal field theory in Refs. [20, 21]. The derivation in the Closed-Time-Path formalism can be found in Ref. [22]. While it appears not to be of immediate consequence for ARS leptogenesis in the mass-range considered here, the interpolation between the production of ultrarelativistic and non-relativistic RH neutrinos is addressed in Refs. [23, 24]. More important may be the effects on the rate from the slowly evolving Higgs field expectation value through the electroweak crossover that is computed in Ref. [25], but this has not yet been implemented in a phenomenological calculation on ARS leptogenesis. In the accompanying article [19], a detailed discussion of these matters is provided in Sec. 3.2.

4.3 Raffelt-Sigl formalism

The starting point of this formalism is to consider the time evolution of number density matrices represented by the expectation values of the number density operators for the particle states (right-helicity states) and antiparticles (left-helicity states):

(34)

It is assumed that effects are negligible. The diagonal elements of represent therefore the number density of the -th sterile particles with positive, negative helicity. By working out the time evolution of the number density operators at second order in perturbation theory in the Yukawa interaction, it can be shown[16] that the densities satisfy

(35)

where and are the annihilation and production rates of the sterile neutrinos, and

(36)

includes the refractive effects [18] of neutrino propagation in the thermal plasma (we have excluded those effects that are flavor blind, i.e. proportional to the identity in flavor which drop from the commutator).

The scattering rates can be written as

(37)

where is the Fermi-Dirac distribution and is the leptonic chemical potential normalized by the temperature. contain the contributions from all processes that produce an :

(38)

and processes: including resummed soft-gauge interactions. All these contributions have been computed for vanishing leptonic chemical potential in [26, 21, 24] and including the effect of a leptonic chemical potential to linear order in [27] .

Approximating

(39)

with , and inserting these functions in the kinetic equation we get:

where . The coefficients , which are functions of , can be found in [27] .

Finally we need the equations that describe the evolution of the leptonic chemical potentials. These are obtained from the equation that describes the evolution of the conserved charges in the absence of neutrino Yukawas, that is the numbers, where is the lepton number in the flavor . These numbers can only be changed by the same out of equilibrium processes that produce the sterile neutrinos and it is possible to relate the integrated rates of these equations to those of the sterile neutrinos and their densities:

(41)

where is the projector on flavor .

The relation between the leptonic chemical potentials and the approximately conserved charges, , is given for GeV by [28]

(42)

where we have defined by the relation:

(43)

Introducing finally the expansion of the Universe and changing variables to the scale factor and , the time derivative of the distribution functions changes to:

(44)

where is the Hubble expansion parameter. Assuming a radiation dominated universe with constant number of relativistic degrees of freedom for , then = constant that we can fix to one.

To simplify the numerical solution of these equations, it is common to consider momentum-averaged equations with the approximation , so that momentum can be averaged:

(45)

The averaged rates, can be found in [27] .

The baryon to entropy ratio is given by:

(46)

It is interesting to compare these equations to those used in the literature. In the seminal ARS paper [3], only top-quark scatterings were included, and the evolution of the leptonic chemical potentials was neglected. The latter approximation turns out to be too restrictive as an asymmetry builds up in the sterile sector only if at least three sterile species are involved. This approximation missed the important physical effect that the plasma at the relevant temperatures erases any quantum coherence in the charged leptons, which are in kinetic equilibrium also via their Yukawa couplings. The plasma is able therefore to select a charged-lepton flavor. This important point was addressed by Asaka and Shaposhnikov[29, 30]. They took into account the evolution of the leptonic chemical potentials and demonstrated that lepton asymmetries could arise also in the minimal case with . Compared to Eq. (LABEL:eq:rhon) and Eq. (41), the equations of Refs. [29, 30, 31, 32] included only top-quark scatterings, assumed Boltzmann statistics, neglected spectator effects and all non-linear terms in the equations, in particular those of . The latter approximation was relaxed in Refs. [26, 33, 34].

Finally, in Ref. [35], the Boltzmann approximation was kept but all relevant scatterings were included in the independent terms (and approximately for the dependent ones), and spectator effects were properly included by considering the evolution of the chemical potentials.

4.4 Closed-Time-Path formalism

Provided the initial state is known, there are exact equations that in principle describe the microscopic evolution of an ensemble of non-equilibrium systems. In a classical theory, these are given by the Liouville equation. In quantum field theory, one may choose between an operator formalism leading to von Neumann equations, as reviewed in \srefsec:operator:formalism, or a functional approach that we discuss in the present section. The functional approach leads in a direct manner to Schwinger-Dyson equations on the Closed-Time-Path (CTP) [36]. For practical reasons, the non-equilibrium reactions that are the essence of particle cosmology are often described by supplementing cross sections into classical Boltzmann equations, thus bypassing the first step of the program shown in \freffig:fundamental-to-fluid. In contrast to this practice, the ARS scenario is mostly treated in the operator or the CTP approach from the outset because at its core is the time evolution of quantum correlations among the different RH neutrinos. As indicated in \freffig:fundamental-to-fluid, further simplifications, ideally by applying controlled approximations, are necessary in order to make analytical or numerically fast predictions that are suitable for phenomenological studies.

Liouville/Schwinger-Dyson/von Neumann equations (microscopic kinetic equations)

Boltzmann-like kinetic equations for distribution functions

fluid equations for number densities/ bulk flows

controlled approximations

Figure 1: Path of deriving fluid equations for phenomenological calculations from first principles.

Both, operator and functional approach, start from first principles, and there is apparent benefit in verifying results using these complementary methods. Interesting features of the CTP approach that we mention before going into the technicalities are:

  • The functional approach makes no reference to the interaction picture states that are a necessary ingredient to the operator formalism. Rather, the state of the system can be encoded in two-point functions. This can be an advantage because in a realistic system, it is non-trivial to relate the interaction picture states to the physical initial conditions, which are non-Gaußian in general [37]. However, for leptogenesis as a weakly coupled model, this is hardly an issue because the non-Gaußian correlations can be reliably constructed using the straightforward perturbation expansion.

  • The functional approach directly leads to Feynman rules on the CTP, which appear to be more easily tractable than evaluating the commutators that occur in the operator approach. Moreover, the CTP techniques are closely related to the real-time formalism in equilibrium field theory, such that it is comparably simple to obtain the relevant self-energies in finite temperature or non-equilibrium backgrounds for leptogenesis.

  • Related to the previous point, all self-energies relevant for leptogenesis have been evaluated in the CTP formalism (see e.g. Ref [22] for the self-energy of relativistic RH neutrinos relevant in the ARS scenario), such that this approach is self-contained.

We emphasize however that given the the standard perturbative expansion appropriate for leptogenesis as a weakly coupled theory and provided the same approximations are made, operator-based and CTP approaches will also yield identical results. At the present state of the art, the choice of method is therefore a matter of preference, calculational transparency or convenience rather than accuracy.

After these more general remarks, we now apply the program sketched in \freffig:fundamental-to-fluid to derive the fluid equations for ARS leptogenesis in the CTP framework. The present emphasis is on the evolution of the state of the RH neutrinos, while the flavor dynamics of the active doublets from the perspective of the Schwinger-Keldysh CTP formalism is presented in detail in the accompanying article on flavored leptogenesis [14]. For now, we note that the ARS scenario is a variant of purely flavored leptogenesis because the initially produced total asymmetry vanishes while there are asymmetries in the individual flavors. Further, we are in the fully flavored regime since the temperatures are low enough such that there are no coherent correlations between the doublet flavors because these are erased immediately by the SM lepton Yukawa couplings, which mediate reactions that are fast compared to the Hubble rate. The source for the asymmetry, i.e. the time derivative of the lepton charge density is given by

(47)

where the reduced self-energy is defined through

(48)

i.e. with the Yukawa couplings stripped, and the spectral self-energy can be interpreted as the cut part of the Higgs-lepton loop at finite temperature [5]. The indices denote the lepton flavor, and we can delete the off-diagonal correlations since we are in the fully flavored regime. Since the two scenarios are closely related, the CTP derivation of the expression (47) is discussed in the accompanying review article on resonant leptogenesis [9].

Now we turn to the main focus of the present section, the computation of , which is the out-of-equilibrium component of the statistical propagator of the RH neutrinos . The present discussion constitutes a summary of the elaboration in Ref. [38], where more technical details are provided. The indices denote the RH neutrino flavors, while spinor indices are suppressed. Note that for the RH neutrinos, the off-diagonal correlations are crucial because these provide the CP-even phases that are required in order to obtain a CP-violating effect.

Formulation of the kinetic equations

The Schwinger-Dyson equations on the CTP are most straightforwardly formulated in terms of two-point functions. Let be a spinor field with mass matrix , then the Wightman functions are given by

(49)

where and are spinor indices, which we suppress in the following. Flavor indices can be introduced straightforwardly. From these basic correlators, we can construct the following Green’s functions:

(50a)
(50b)
(50c)
(50d)
(50e)

Corresponding quantities can be defined for other two-point functions, in particular for the self-energies , and we identity these by the same superscripts.

We take as the starting point for the derivation of kinetic equations Schwinger-Dyson equations on the CTP, which are exact in principle, cf. the scheme outlined in \freffig:fundamental-to-fluid. We then make the transition from position into Wigner space through a Fourier transformation with respect to the relative coordinate while keeping the average coordinate . For weakly coupled particles, the spectral function then exhibits a quasi-particle peak at the mass shell, and the Wigner space coordinate can be interpreted as the four momentum. In Wigner space, the convolution integrals take the form of a Moyal product [39, 40, 41, 42]. The gradient expansion consists (in the present spatially homogeneous case) of dropping temporal derivatives that are of order of the Hubble rate (if the associated process is close to equilibrium) or the particular relaxation rate (if the process is far from equilibrium). At leading order, the Moyal product corresponds to an ordinary multiplication. Applying this truncation, the Schwinger-Dyson equations on the CTP read

(51a)
(51b)

where all two-point functions depend on , and Eq. (51b) is known as the Kadanoff-Baym equation. We also write for the time coordinate, and derivatives with respect to , , do not occur provided spatial homogeneity holds.

Decomposing the Kadanoff-Baym equation (51b) into its Hermitian and anti-Hermitian part, we obtain the constraint and kinetic equations