Review Article | Open Access

# Leptogenesis in the Universe

**Academic Editor:**Gian Luigi Fogli

#### Abstract

Leptogenesis is a class of scenarios in which the cosmic baryon asymmetry originates from an initial lepton asymmetry generated in the decays of heavy sterile neutrinos in the early Universe. We explain why leptogenesis is an appealing mechanism for baryogenesis. We review its motivations and the basic ingredients and describe subclasses of effects, like those of lepton flavours, spectator processes, scatterings, finite temperature corrections, the role of the heavier sterile neutrinos, and quantum corrections. We then address leptogenesis in supersymmetric scenarios, as well as some other popular variations of the basic leptogenesis framework.

#### 1. The Baryon Asymmetry of the Universe

##### 1.1. Observations

Up to date no traces of cosmological antimatter have been observed. The presence of a small amount of antiprotons and positrons in cosmic rays can be consistently explained by their secondary origin in cosmic particles collisions or in highly energetic astrophysical processes, but no antinuclei, even as light as antideuterium or as tightly bounded as anti- particles, have ever been detected.

The absence of annihilation radiation excludes significant matter-antimatter admixtures in objects up to the size of galactic clusters ~20â€‰Mpc [1]. Observational limits on anomalous contributions to the cosmic diffuse -ray background and the absence of distortions in the cosmic microwave background (CMB) imply that little antimatter is to be found within ~1â€‰Gpc and that within our horizon an equal amount of matter and antimatter can be excluded [2]. Of course, at larger superhorizon scales the vanishing of the average asymmetry cannot be ruled out, and this would indeed be the case if the fundamental Lagrangian is C and CP symmetric and charge invariance is broken spontaneously [3].

Quantitatively, the value of baryon asymmetry of the Universe is inferred from observations in two independent ways. The first way is by confronting the abundances of the light elements, , , , and , with the predictions of Big Bang nucleosynthesis (BBN) [4â€“9]. The crucial time for primordial nucleosynthesis is when the thermal bath temperature falls below â€‰MeV. With the assumption of only three light neutrinos, these predictions depend on a single parameter, that is, the difference between the number of baryons and antibaryons normalized to the number of photons: where the subscript means â€ťat present time.â€ť By using only the abundance of deuterium, that is particularly sensitive to , [4] quotes: In this same range there is also an acceptable agreement among the various abundances, once theoretical uncertainties as well as statistical and systematic errors are accounted for [6].

The second way is from measurements of the CMB anisotropies (for pedagogical reviews, see [10, 11]). The crucial time for CMB is that of recombination, when the temperature dropped down to eV and neutral hydrogen can be formed. CMB observations measure the relative baryon contribution to the energy density of the Universe multiplied by the square of the (reduced) Hubble constant : that is related to through . The physical effect of the baryons at the onset of matter domination, which occurs quite close to the recombination epoch, is to provide extra gravity which enhances the compression into potential wells. The consequence is enhancement of the compressional phases which translates into enhancement of the odd peaks in the spectrum. Thus, a measurement of the odd/even peak disparity constrains the baryon energy density. A fit to the most recent observations (WMAP7 data only, assuming a CDM model with a scale-free power spectrum for the primordial density fluctuations) gives at 68% c.l. [12] There is a third way to express the baryon asymmetry of the Universe, that is, by normalizing the baryon asymmetry to the entropy density , where is the number of degrees of freedom in the plasma and is the temperature: The relation with the previous definitions is given by the conversion factor . is a convenient quantity in theoretical studies of the generation of the baryon asymmetry from very early times, because it is conserved throughout the thermal evolution of the Universe. In terms of the BBN results (1.2) and the CMB measurement (1.4) (at 95% c.l.) read The impressive consistency between the determinations of the baryon density of the Universe from BBN and CMB that, besides being completely independent, also refer to epochs with a six orders of magnitude difference in temperature, provides a striking confirmation of the hot Big Bang cosmology.

##### 1.2. Theory

From the theoretical point of view, the question is where the Universe baryon asymmetry comes from. The inflationary cosmological model excludes the possibility of a fine tuned initial condition, and since we do not know any other way to construct a consistent cosmology without inflation, this is a strong veto.

The alternative possibility is that the Universe baryon asymmetry is generated dynamically, a scenario that is known as* baryogenesis*. This requires that baryon number () is not conserved. More precisely, as Sakharov pointed out [13], the ingredients required for baryogenesis are three.(1) violation is required to evolve from an initial state with to a state with . (2) C and CP violation: if either C or CP was conserved, then processes involving baryons would proceed at the same rate as the C- or CP-conjugate processes involving antibaryons, with the overall effect that no baryon asymmetry is generated. (3) Out of equilibrium dynamics: equilibrium distribution functions are determined solely by the particle energy , chemical potential , and by its mass which, because of the CPT theorem, is the same for particles and antiparticles. When charges (such as ) are not conserved, the corresponding chemical potentials vanish, and thus . Although these ingredients are all present in the Standard Model (SM), so far all attempts to reproduce quantitatively the observed baryon asymmetry have failed. (1) In the SM is violated via the triangle anomaly. Although at zero temperature violating processes are too suppressed to have any observable effect [14], at high temperatures they occur with unsuppressed rates [15]. The first condition is then quantitatively realized in the early Universe.(2) SM weak interactions violate C maximally. However, the amount of CP violation from the Kobayashi-Maskawa complex phase [16], as quantified by means of the Jarlskog invariant [17], is only of order , and this renders impossible generating [18â€“20].(3) Departures from thermal equilibrium occur in the SM at the electroweak phase transition (EWPT) [21, 22]. However, the experimental lower bound on the Higgs mass implies that this transition is not sufficiently first order as required for successful baryogenesis [23]. This shows that baryogenesis requires new physics that extends the SM in at least two ways. It must introduce new sources of CP violation and it must either provide a departure from thermal equilibrium in addition to the EWPT or modify the EWPT itself. In the past thirty years or so, several new physics mechanisms for baryogenesis have been put forth. Some among the most studied are *GUT baryogenesis* [24â€“33], *electroweak baryogenesis* [21, 34, 35], the *affleck-Dine mechanism* [36, 37], and *spontaneous Baryogenesis* [38, 39]. However, soon after the discovery of neutrino masses, because of its connections with the seesaw model [40â€“44] and its deep interrelations with neutrino physics in general, the mechanism of baryogenesis via *Leptogenesis* acquired a continuously increasing popularity. Leptogenesis was first proposed by Fukugita and Yanagida in [45]. Its simplest and theoretically best motivated realization is precisely within the seesaw mechanism. To implement the seesaw, new Majorana singlet neutrinos with a large mass scale are added to the SM particle spectrum. The complex Yukawa couplings of these new particles provide new sources of CP violation, departure from thermal equilibrium can occur if their lifetime is not much shorter than the age of the Universe when , and their Majorana masses imply that lepton number is not conserved. A lepton asymmetry can then be generated dynamically, and SM sphalerons will partially convert it into a baryon asymmetry [46]. A particularly interesting possibility is â€śthermal leptogenesisâ€ť where the heavy Majorana neutrinos are produced by scatterings in the thermal bath starting from a vanishing initial abundance, so that their number density can be calculated solely in terms of the seesaw parameters and of the reheat temperature of the Universe.

This paper is organized as follows. In Section 2 the basis of leptogenesis is reviewed in the simple scenario of the one-flavour regime, while the role of flavour effects is described in Section 3. Theoretical improvements of the basic pictures, like spectator effects, scatterings and CP violation in scatterings, thermal corrections, the possible role of the heavier singlet neutrinos, and quantum effects, are reviewed in Section 4. Leptogenesis in the supersymmetric seesaw is reviewed in Section 5, while in Section 6 we mention possible leptogenesis realizations that go beyond the type-I seesaw. Finally, in Section 7 we draw the conclusions.

#### 2. Leptogenesis in the Single Flavour Regime

The aim of this section is to give a pedagogical introduction to leptogenesis [45] and establish the notations. We will consider the classic example of leptogenesis from the lightest right-handed (RH) neutrino (the so-called leptogenesis) in the type-I seesaw model [40, 41, 43, 44] in the single flavour regime. First in Section 2.1 we introduce the type-I seesaw Lagrangian and the relevant parameters. In Section 2.2, we will review the CP violation in RH neutrino decays induced at 1-loop level. Then in Section 2.3, we will write down the classical Boltzmann equations taking into account only decays and inverse decays of and give a simple but rather accurate analytical estimate of the solution. In Section 2.4 we will relate the lepton asymmetry generated to the baryon asymmetry of the Universe. Finally in Section 2.5, we will discuss the lower bound on mass and the upper bound on light neutrino mass scale from successful leptogenesis.

##### 2.1. Type-I Seesaw, Neutrino Masses, and Leptogenesis

With (neutrino oscillation data and leptogenesis both require .) singlet RH neutrinos , we can add the following Standard Model (SM) gauge invariant terms to the SM Lagrangian: where are the Majorana masses of the RH neutrinos, with and are, respectively, the left-handed (LH) lepton and Higgs doublets and with . Without loss of generality, we have chosen the basis where the Majorana mass term is diagonal. The physical mass eigenstates of the RH neutrinos are the Majorana neutrinos . Since are gauge singlets, the scale of is naturally much larger than the electroweak (EW) scale â€‰GeV. Hence after EW symmetry breaking, the light neutrino mass matrix is given by the famous seesaw relation [40, 41, 43, 44]: Assuming and â€‰eV, we have â€‰GeV not far below the GUT scale.

Besides giving a natural explanation of the light neutrino masses, there is another bonus: the *three* Sakharov's conditions [13] for leptogenesis are implicit in (2.1) with the *lepton number violation* provided by , the CP *violation* from the complexity of , and the *departure from thermal equilibrium condition* given by an additional requirement that decay rate is not very fast compared to the Hubble expansion rate of the Universe at temperature with
where â€‰GeV is the Planck mass, (=106.75 for the SM excluding RH neutrinos) is the total number of relativistic degrees of freedom contributing to the energy density of the Universe.

To quantify the departure from thermal equilibrium, we define the *decay parameter* as follows:
where is the *effective neutrino mass* defined as [47]
with â€‰eV. The regimes where , and are, respectively, known as weak, intermediate, and strong washout regimes.

##### 2.2. CP Asymmetry

The CP asymmetry in the decays of RH neutrinos can be defined as where is the thermally averaged decay rate defined as (here the Pauli-blocking and Bose-enhancement statistical factors have been ignored and we also assume Maxwell-Boltzmann distribution for the particle , that is, ; see [48, 49] for detailed studies of their effects) where is the decay amplitude. Ignoring all thermal effects [48, 49], (2.6) simplifies to where denotes the decay amplitude at zero temperature. Equation (2.8) vanishes at tree level but is induced at 1-loop level through the interference between tree and 1-loop diagrams shown in Figure 1. There are two types of contributions from the 1-loop diagrams: the self-energy or wave diagram (middle) [50] and the vertex diagram (right) [45]. At leading order, we obtain the CP asymmetry [51]: where the loop function is The first term in (2.9) comes from -violating wave and vertex diagrams, while the second term is from the -conserving wave diagram. The terms of the form in (2.9) are from the wave diagram contributions which can resonantly enhance the CP asymmetry if (resonant leptogenesis scenario, see Section 6.1). Notice that the resonant term becomes singular in the degenerate limit . This singularity can be regulated by using, e.g., an effective field-theoretical approach based on resummation [52]. Let us also note that at least two RH neutrinos are needed otherwise the CP asymmetry vanishes because the Yukawa couplings combination becomes real.

In the one flavour regime, we sum over the flavour index in (2.9) and obtain where the second term in (2.9) vanishes because the combination of the Yukawa couplings is real.

##### 2.3. Classical Boltzmann Equations

We work in the one-flavour regime and consider only the decays and inverse decays of . If leptogenesis occurs at GeV, then the charged lepton Yukawa interactions are out of equilibrium, and this defines the one-flavour regime. The assumption that only the dynamics of is relevant can be realized if, for example, the reheating temperature after inflation is such that are not produced. In order to scale out the effect of the expansion of the Universe, we will introduce the *abundances*, that is, the ratios of the particle densities to the entropy density :
The evolution of the density and the lepton asymmetry (the factor of 2 comes from the degrees of freedoms) can be described by the following classical Boltzmann equations (BE) [53]:
where and the decay and washout terms are, respectively, given by
with the th order modified Bessel function of second kind. and read (to write down a simple analytic expression for the equilibrium density of , we assume Maxwell-Boltzmann distribution. However, following [54], the normalization factor is obtained from a Fermi-Dirac distribution)

From (2.13) and (2.14), the solution for can be written down as follows: where is some initial temperature when leptogenesis begins, and we have assumed that any preexisting lepton asymmetry vanishes . Notice that ignoring thermal effects, the CP asymmetry is independent of the temperature (c.f. (2.11)).

###### 2.3.1. Weak Washout Regime

In the weak washout regime (), the initial condition on the density is important. If we assume thermal initial abundance of , that is, , we can ignore the washout when starts decaying at and we have On the other hand, if we have zero initial abundance, that is, , we have to consider the opposite sign contributions to lepton asymmetry from the inverse decays when is being populated () and from the period when starts decaying (). Taking this into account the term which survives the partial cancellations is given by [55] (this differs from the efficiency in [55] by the factor , which is due to the different normalization (2.16))

###### 2.3.2. Strong Washout Regime

In the strong washout regime () any lepton asymmetry generated during the creation phase is efficiently washed out. Here we adopt the *strong washout balance approximation* [56] which states that in the strong washout regime, the lepton asymmetry at each instant takes the value that enforces a balance between the production and the destruction rates of the asymmetry. Equating the decay and washout terms in (2.14), we have
where in the second approximation, we assume . The approximation no longer holds when freezes, and this happens when the washout decouples at , that is, . Hence, the final lepton asymmetry is given by (compare this to a more precise analytical approximation in [55])
The freeze out temperature depends mildly on . For â€“100 we have, for example, â€“ 10. We also see that independently of initial conditions, in the strong regime goes as .

##### 2.4. Baryon Asymmetry from EW Sphaleron

The final lepton asymmetry can be conveniently parametrized as follows:
where is known as the *efficiency factor*. In the weak washout regime () from (2.18) we have for thermal (zero) initial abundance. In the strong washout regime (), from (2.21), we have .

If leptogenesis ends before EW sphaleron processes become active (â€‰GeV), the asymmetry is simply given by At the later stage, the asymmetry is partially transferred to a asymmetry by the EW sphaleron processes through the relation [57] that holds if sphalerons decouple before EWPT. This relation will change if the EW sphaleron processes decouple after the EWPT [57, 58] or if threshold effects for heavy particles like the top quark and Higgs are taken into account [58, 59].

##### 2.5. Davidson-Ibarra Bound

Assuming a hierarchical spectrum of the RH neutrinos (), and that the dominant lepton asymmetry is from the decays, from (2.11) the CP asymmetry from decays can be written as Assuming three generations of RH neutrinos () and using the Casas-Ibarra parametrization [60] for the Yukawa couplings where , and any complex orthogonal matrix satisfying , (2.25) becomes Using the orthogonality condition , we then obtain the Davidson-Ibarra (DI) bound [61] where () is the heaviest (lightest) light neutrino mass. Applying the DI bound on (2.22)â€“(2.24), and requiring that , we obtain where the is the efficiency factor maximized with respect to (2.4) for a particular value of . This allows us to make a plot of region which satisfies (2.29) on the plane and hence obtain bounds on and . Many careful numerical studies have been carried out, and it was found that successful leptogenesis with a hierarchical spectrum of the RH neutrinos requires GeV [61â€“63] and eV [55, 64â€“66]. This bound implies that the RH neutrinos must be produced at temperatures which in turn implies the reheating temperature after inflation has to be in order to have sufficient RH neutrinos in the thermal bath. To conclude this section, let us note that the DI bound (2.28) holds if and only if all the following conditions apply. (1) dominates the contribution to leptogenesis. (2) The mass spectrum of RH neutrinos is hierarchical . (3) Leptogenesis occurs in the unflavoured regime GeV. As we will see in the following sections, violation of one or more of the previous conditions allows us to lower somewhat the scale of leptogenesis.

#### 3. Lepton Flavour Effects

##### 3.1. When Are Lepton Flavour Effects Relevant?

The first leptogenesis calculations were performed in the single lepton flavour regime. In short, this amounts to assuming that the leptons and antileptons which couple to the lightest RH neutrino maintain their coherence as flavour superpositions throughout the leptogenesis era, that is and . Note that at the tree level the coefficients and are simply the Yukawa couplings: and . However it should be kept in mind that since CP is violated by loops, beyond the tree level approximation the antilepton state is not the CP conjugate of the , that is, .

The single flavour regime is realized only at very high temperatures (GeV) when both and remain coherent flavour superpositions and thus are the correct states to describe the dynamics of leptogenesis. However, at lower temperatures scatterings induced by the charged lepton Yukawa couplings occur at a sufficiently fast pace to distinguish the different lepton flavours, and decohere in their flavour components, and the dynamics of leptogenesis must then be described in terms of the flavour eigenstates . Of course, there is great interest to extend the validity of quantitative leptogenesis studies also at lower scale GeV, and this requires accounting for flavour effects. The role of lepton flavour in leptogenesis was first discussed in [67]; however the authors did not highlight in what the results were significantly different from the single flavour approximation. Therefore, until the importance of flavour effects was fully clarified in [68â€“70], they had been included in leptogenesis studies only in a few cases [71â€“75]. Nowadays lepton flavour effects have been investigated in full detail [76â€“89] and are a mandatory ingredient of any reliable analysis of leptogenesis.

The specific temperature when leptogenesis becomes sensitive to lepton flavour dynamics can be estimated by requiring that the rates of processes () that are induced by the charged lepton Yukawa couplings become faster than the Universe expansion rate . An approximate relation gives [90, 91] which implies that (in supersymmetric case, since , we have ) where GeV, GeV, and GeV. Notice that to fully distinguish the three flavours it is sufficient that the and Yukawa reactions attain thermal equilibrium. It has been pointed out that besides being faster than the expansion of the Universe, the charged lepton Yukawa interactions should also be faster than the interactions [69, 83, 84]. In general whenever we also have . However, there exists parameter space where but . This scenario was studied in [83].

##### 3.2. The Effects on CP Asymmetry and Washout

The CP violation in decays can manifest itself in two ways [69]

(i) The leptons and antileptons are produced at different rates: where and .

(ii) The leptons and antileptons produced are not CP conjugate states: that is, due to loops effects they are slightly misaligned in flavour space.

We can rewrite the CP asymmetry for decays from (2.6) as follows: where terms of order and higher have been neglected. is the projector from state into flavour state and . At tree level, clearly, where the tree level flavour projector is given by From (3.5), we can identify the two types of CP violation, the first term being of type (i) equation (3.3) while the second being of type (ii) equation (3.4). Since , when summing over flavour indices , the second term vanishes . Note that the lepton-flavour-violating but -conserving terms in the second line of (2.9) are part of type (ii). In fact, they come from -conserving operators which have nothing to do with the unique â€‰-violating operator (the Weinberg operator [92]) responsible for neutrino masses. However, in some cases they can still dominate the CP asymmetries but, as we will see in Section 3.4, lepton flavour equilibration effects [93] then impose important constraints on their overall effects. Note also that due to flavour misalignment, the CP asymmetry in a particular flavour direction can be much larger and even of opposite sign from the total CP asymmetry . In fact the relevance of CP violation of type (ii) in the flavour regimes is what allows to evade the DI bound (2.28). As regards the washout of the lepton asymmetry of flavour , it is proportional to which results in a reduction of washout by a factor of compared to unflavoured case. As we will see next, the new CP-violating sources from flavour effects and the reduction in the washout could result in great enhancement of the final lepton asymmetry, and, as was first pointed out in [69], leptogenesis with a vanishing total CP asymmetry also becomes possible.

##### 3.3. Classical Flavoured Boltzmann Equations

Here again we only consider leptogenesis from the decays and inverse decays of . In this approximation, the BE for is still given by (2.13) while the BE for the lepton asymmetry in the flavour is given by (to study the transition between different flavour regimes (from one to two or from two to three flavours), a density matrix formalism has to be used [68, 84, 94]). Notice that as long as violation from sphalerons is neglected (see Section 4) the BEs for are independent of each other, and hence the solutions for the weak and strong washout regimes are given, respectively, by (2.19) and (2.21), after replacing and .

As an example let us assume that leptogenesis occurs around GeV, that is, in the two-flavour regime. Due to the fast Yukawa interactions gets projected onto and a coherent mixture of eigenstate . For illustrative purpose, here we consider a scenario in which lepton flavour effects are most prominent. We take both , so that both and are in the strong regime. From (2.21) we can write down the solution:
where in the last line we have used (3.5). If , then since , the second term approximately cancels, and (3.9) reduces to
We see that the final asymmetry is enhanced by a factor of 2 compared to the unflavoured case. If there exists some hierarchy between the flavour projectors, then the second term in (3.9) plays an important role and can further enhance the asymmetry. For example, we can have while . In this case, the second term can dominate over the first term. Finally from (3.9) we also notice that leptogenesis with , the so-called *purely flavoured leptogenesis* (PFL) (this can also refer to the case where the total CP asymmetry is negligible ), can indeed proceed [69, 95â€“98]. In this scenario some symmetry has to be imposed to realize the condition , as, for example, an approximate global lepton number . In the limit of exact the active neutrinos will be exactly massless. Instead of the seesaw mechanism, the small neutrino masses are explained by which is slightly broken by a small parameter (the â€śinverse seesawâ€ť) [99] which is technically natural since the Lagrangian exhibits an enhanced symmetry when [100]. In the next section, we will discuss another aspect of flavour effects which are in particular crucial for PFL.

##### 3.4. Lepton Flavour Equilibration

Another important effect is lepton flavour equilibration (LFE) [93]. LFE processes violate lepton flavour but conserve total lepton number, for example, , and can proceed, for example, via off-shell exchange of . In thermal equilibrium, LFE processes can quickly equilibrate the asymmetries generated in different flavours. In practice this would be equivalent to a situation where all the flavour projector equations (3.6) are equal, in which case the flavoured BE equation (3.8) can be summed up into a single BE: where in the two- (three-) flavour regime. In this case the BE is just like the unflavoured case but with a reduced washout which, in the strong washout regime, would result in enhancement of a factor of 2 in the two- (three-) flavour regime (c.f. (3.10)). Clearly, LFE can make PFL with impotent [56, 93]. Since LFE processes scale as while the Universe expansion scales as , in spite of the fact that PFL evades the DI bound, they eventually prevent the possibility of lowering too much the leptogenesis scale. A generic study in PFL scenario taking into account LFE effects concluded that successful leptogenesis still requires GeV [97]. A more accurate study in the same direction recently carried out in [98] showed that in fact the leptogenesis scale can be lowered down to GeV.

#### 4. Beyond the Basic Boltzmann Equations

Within factors of a few, the amount of baryon asymmetry that is generated via leptogenesis in decays is determined essentially by the size of the (flavoured) CP asymmetries and by the rates of the (flavoured) washout reactions. However, to obtain more precise results (say, within an uncertainty) several additional effects must be taken into account, and the formalism must be extended well beyond the basic BE discussed in the previous sections. In the following we review some of the most important sources of corrections, namely, spectator processes (Section 4.1), scatterings with top quarks and gauge bosons (Section 4.2), thermal effects (Section 4.3), contributions from heavier RH neutrinos (Section 4.4), and we also discuss the role of quantum corrections evaluated in the quantum BE approach (Section 4.5). Throughout this paper we use integrated BE; that is, we assume kinetic equilibrium for all particle species, and thus we use particles densities instead than particles distribution functions. Corrections arising from using nonintegrated BE have been studied for example in [101â€“104], and are generally subleading.

##### 4.1. Spectator Processes

Reactions that without involving violation of can still affect the final amount of baryon asymmetry are classified as *â€śspectator processesâ€ť* [105, 106]. The basic way through which they act is that of redistributing the asymmetry generated in the lepton doublets among the other particle species. Since the density asymmetries of the lepton doublets are what weights the rates of the washout processes, it can be expected that spectator processes would render the washouts less effective and increase the efficiency of leptogenesis. However, in most cases this is not true: proper inclusion of spectator processes implies accounting for all the particle asymmetries and in particular also for the density asymmetry of the Higgs [106]. This was omitted in Section 2 but in fact has to be added to the density asymmetry of the leptons in weighting, for example, washouts from inverse decays. Equation (2.14) would then become
where the factor of two in front of the washout term counts the leptons and Higgs gauge multiplicity. Clearly, in some regimes in which and are not sufficiently diluted by interacting with other particles, this can have the effect of enhancing the washout rates and suppressing the efficiency.

In the study of spectator processes it is fundamental to specify the range of temperature in which leptogenesis occurs. This is because at each specific temperature , particle reactions must be treated in a different way depending on if their characteristic time scale (given by inverse of their thermally averaged rates) is [89, 107] (1) much shorter than the age of the Universe: ; (2) much larger than the age of the Universe: ; (3) comparable with the Universe age: .

Spectator processes belong to the first type of reactions which occur very frequently during one expansion time. Their effects can be accounted for by imposing on the thermodynamic system the chemical equilibrium condition appropriate for each specific reaction, that is, , where denotes the chemical potential of an initial state particle and that of a final state particle (the relation between chemical potentials and particle density asymmetries is given in (5.3)). The numerical values of the parameters that are responsible for these reactions only determine the precise temperature when chemical equilibrium is attained but, apart from this, have no other relevance and do not appear explicitly in the formulation of the problem. Reactions of type cannot have any effect on the system, since they basically do not occur. All physical processes are blind to the corresponding parameters, that can be set to zero in the effective Lagrangian. In most cases this results in exact global symmetries corresponding to conserved charges, and these conservation laws impose constraints on the particle chemical potentials. Reactions of type in general violate some symmetry and thus spoil the corresponding conservation conditions, but are not fast enough to enforce chemical equilibrium. These are the only reactions that need to be studied by means of BE, and for which the precise value of the parameters that control their rates is of utmost importance.

A simple case to illustrate how to include spectator processes is the one-flavour regime at particularly high temperatures (say, GeV). The Universe expansion is fast implying that except for processes induced by the large Yukawa coupling of the top and for gauge interactions, all other -conserving reactions fall in class (ii). Then there are several conserved quantities as, for example, the total number density asymmetries of the RH leptons as well as those of all the quarks except the top. Since electroweak sphalerons are also out of equilibrium, is conserved too (and vanishing, if we assume that there is no preexisting asymmetry). then translates in the condition: where is the density asymmetries of one degree of freedom of the top doublet and color triplet which, being gauge interactions in equilibrium, is the same for all the six gauge components and is the density asymmetry of the singlet top. Hypercharge is always conserved, yielding Finally, in terms of density asymmetries chemical equilibrium for the top-Yukawa-related reactions translates into We have three conditions for four density asymmetries, which allows to express the Higgs density asymmetry in terms of the density asymmetry of the leptons as . Moreover, given that only the LH lepton degrees of freedom are populated, we have so that the coefficient weighting in (4.1) becomes and the washout is accordingly stronger.

With decreasing temperatures, more reactions attain chemical equilibrium, and accounting for spectator processes becomes accordingly more complicated. When the temperature drops below GeV, EW sphalerons are in equilibrium, and baryon number is no more conserved. Then the condition (4.2) is no more satisfied, and, more importantly, the BE equation (4.1) is no more valid since sphalerons violate also lepton number within equilibrium rates. However, sphalerons conserve , which is then violated only by slow reactions of type , and we should then write down a BE for this quantity. Better said, since at GeV all the third generation Yukawa reactions, including the ones of the -lepton, are in equilibrium, the dynamical regime is that of two flavours in which the relevant quasiconserved charges are and . The fact that only two charges are relevant is because there is always a direction in space which remains decoupled from . The corresponding third charge is then exactly conserved, its value can be set to zero, and the corresponding BE dropped. In this regime, the BE corresponding to (4.1) becomes To rewrite these equations in a solvable closed form, , , and must be expressed in terms of the two charge densities and . This can be done by imposing the hypercharge conservation condition (4.3) and the chemical equilibrium conditions that, in addition to (4.4), are appropriate for the temperature regime we are considering. They are [106] QCD sphaleron equilibrium; EW sphaleron equilibrium; -quark and -lepton Yukawa equilibrium. The â€śrotationâ€ť from the particle density asymmetries and to the charge densities can be expressed in terms of the matrix introduced in [67] and -vector introduced in [69]. For the present case, with the ordering they are [69] It is important to stress that in each temperature regime there are always enough constraints (conservation laws and chemical equilibrium conditions) to allow to express all the relevant particle density asymmetries in terms of the quasiconserved charges . This is because each time a conservation law has to be dropped (like conservation above), it gets replaced by a chemical equilibrium condition (like EW sphalerons equilibrium), and each time the chemical potential of a new particle species becomes relevant, it is precisely because a new reaction involving that particle attains chemical equilibrium, enforcing the corresponding condition. As regards the quantitative corrections ascribable to spectator processes, several numerical studies have confirmed that they generally remain below order one. Thus, differently from flavour effects, for order of magnitude estimates they can be neglected.

##### 4.2. Scatterings and CP Violation in Scatterings

Scattering processes are relevant for the production of the population, because decay and inverse decay rates are suppressed by a time dilation factor . The particles can be produced by scatterings with the top quark in -channel -exchange and , by -channel -exchange in , and by -channel -exchange in , ; see the diagrams (a) in Figure 2. Several scattering channels with gauge bosons also contribute to the production of ; the corresponding diagrams are (b) and (c) in the same figure.

**(a)**

**(b)**

**(c)**

When the effects of scatterings in populating the degree of freedom are included, for consistency CP violation in scatterings must also be included. In doing so some care has to be put in treating properly also all the processes of higher order in the couplings (, , where is a gauge coupling) with an on-shell intermediate state subtracted out. This can be done by following the procedure adopted in [108], and we refer to that paper for details.

In the first approximation, the CP asymmetry in scattering processes is the same as in decays and inverse decays [70, 109]. This result was first found in [75, 110, 111] for the case of resonant leptogenesis and was later derived in [70] for the case of hierarchical . A full calculation of the CP asymmetry in scatterings involving the top quark was carried out in [108], and the validity of approximating it with the CP asymmetry in decays was analyzed, finding that the approximation is generally good for sufficiently strong RH neutrino hierarchies, for examlpe, . Corrections up to several tens of percent can appear around temperatures of order and can be numerically relevant in case of milder hierarchies.

Regarding the scattering processes with gauge bosons such as or , their effects in leptogenesis were estimated in [108] under the assumption that it can also be factorized in terms of the decay CP asymmetry. However, with respect to scatterings involving the top quark, there is a significant difference that now box diagrams in which the gauge boson is attached to a lepton or Higgs in the loop of the vertex-type diagrams are also present, leading to more complicated expressions that were explicitly calculated in [112]. There it was shown that the presence of box diagrams implies that for scatterings with gauge bosons the CP asymmetry is different from the decay CP asymmetry even for hierarchical RH neutrinos. Still, this difference remains within a factor of two [112] so that related effects are in any case not very large. In general, it turns out that CP asymmetry in scatterings is more relevant at high temperatures () when the scattering rates are larger than the decay rate. Hence, it can be of some relevance to the final value of the baryon asymmetry when some of the lepton flavours are weakly washed out, and some memory of the asymmetries generated at high temperature is preserved in the final result.

##### 4.3. Thermal Corrections

At the high temperatures at which leptogenesis occurs, the light particles involved in the leptogenesis processes are in equilibrium with the hot plasma. Thermal effects give corrections to several ingredients in the analysis: (i) coupling constants, (ii) particle propagators (leptons, quarks, gauge bosons, and the Higgs), and (iii) CP-violating asymmetries, which we briefly discuss later. A detailed study of thermal corrections can be found in [49].

###### 4.3.1. Coupling Constants

Renormalization of gauge and Yukawa couplings in a thermal plasma is studied in [113]. In practice, it is a good approximation to use the zero-temperature renormalization group equations for the couplings, with a renormalization scale [49]. The value is related to the fact that the average energy of the colliding particles in the plasma is larger than the temperature.

The renormalization effects for the neutrino couplings are also well known [114, 115]. In the nonsupersymmetric case, to a good approximation these effects can be described by a simple rescaling of the low energy neutrino mass matrix , where for GeV GeV [49], and therefore can be accounted for by increasing the values of the neutrino mass parameters (e.g., ) as measured at low energy by (depending on the leptogenesis scale). In the supersymmetric case one expects a milder enhancement, but uncertainties related with the precise value of the top-Yukawa coupling can be rather large (see Figure 3 in [49]).

###### 4.3.2. Decays and Scatterings

In the thermal plasma, any particle with sizable couplings to the background acquires a thermal mass which is proportional to the plasma temperature. Consequently, decay and scattering rates get modified. Particle thermal masses have been thoroughly studied in both the SM and the supersymmetric SM [91, 116â€“120]. The singlet neutrinos have no gauge interactions, their Yukawa couplings are generally small, and, during the relevant era, their bare masses are of the order of the temperature or larger. Consequently, to a good approximation, corrections to their masses can be neglected. We thus need to account for the thermal masses of the leptons and Higgs doublets and, when scatterings are included, also of the third generation quarks and of the gauge bosons (and of their superpartners in the supersymmetric case). For a qualitative discussion, it is enough to keep in mind that, within the leptogenesis temperature range, we have . The most important effects are related to four classes of leptogenesis processes.(i)*Decays and inverse decays*: since thermal corrections to the Higgs mass are particularly large (), decays and inverse decays become kinematically forbidden in the temperature range in which . For lower temperatures, the usual processes can occur. For higher temperatures, the Higgs is heavy enough that it can decay: . A rough estimate of the kinematically forbidden region yields . The important point is that these corrections are effective only at . In the parameter region eV, that is favoured by the measurements of the neutrino mass-squared differences, the number density and its -violating reactions attain thermal equilibrium at and erase quite efficiently any memory of the specific conditions at higher temperatures. Consequently, in the strong washout regime, these corrections have practically no effect on the final value of the baryon asymmetry.(ii)*Î”L = 1 scatterings with top quark*: a comparison between the corrected and uncorrected rates of the top-quark scattering with the Higgs exchanged in the -channel and of the sum of the - and -channel scatterings shows that the only corrections appearing at low temperatures, and thus more relevant, are for (see Figure 7.1 in [109]). They reduce the scattering rates and suppress the related washouts. This peculiar situation arises from the fact that in the zero temperature limit there is a large enhancement from the quasimassless Higgs exchanged in the - and -channels, which disappears when the Higgs thermal mass is included.(iii)*Î”L = 1 scatterings with the gauge bosons*: here the inclusion of thermal masses is required to avoid IR divergences that would arise when massless (and ) states are exchanged in the - and -channels. A naive use of some cutoff for the phase space integrals to control the IR divergences can yield incorrect estimates of the gauge bosons scattering rates [49] and would be particularly problematic at low temperatures, where gauge bosons scatterings dominate over top-quark scatterings.

###### 4.3.3. CP Asymmetries

CP asymmetries arise from the interference of tree level and one-loop amplitudes when the couplings involved have complex phases and the loop diagrams have an absorptive part. This last requirement is satisfied whenever the loop diagram can be cut in such a way that the particles in the cut lines can be produced on shell. For the CP asymmetry in decay (at zero temperature) this is guaranteed by the fact that the Higgs and the lepton final states coincide with the states circulating in the loops. However, in the hot plasma in which decays occur, the Higgs and the lepton doublets are in thermal equilibrium and their interactions with the background introduce in the CP asymmetries a dependence on the temperature that arises from various effects(i) Absorption and reemission of the loop particles by the medium require the use of finite temperature propagators.(ii) Stimulation of decays into bosons and blocking of decays into fermions in the dense background require proper modification of the final states density distributions.(iii) Thermal motion of the decaying â€™s with respect to the background breaks the Lorentz symmetry and affects the evaluation of the CP asymmetries.(iv) Thermal masses should be included in the finite temperature resummed propagators, and they also modify the fermion and boson dispersion relations. Their inclusion yields the most significant modifications to the zero temperature results for the CP asymmetries.

The first three effects were investigated in [48] while the effects of thermal masses were included in [49]. In principle, at finite temperature, there are additional effects related to new cuts that involve the heavy neutrino lines. These new cuts appear because the heavy particles in the loops may absorb energy from the plasma and go onshell. However, for hierarchical spectrum, , the related effects are Boltzmann suppressed by that at is a tiny factor. For a nonhierarchical spectrum, the effect of these new cuts can however be sizable. A detailed study can be found in [121].

###### 4.3.4. Propagators and Statistical Distributions

Particle propagators at finite temperature are computed in the real-time formalism of thermal field theory [122, 123]. In this formalism, ghost fields dual to each of the physical fields have to be introduced, and consequently the thermal propagators have matrix structures. For the one-loop computations of the absorptive parts of the Feynman diagrams, the relevant propagator components are just those of the physical lepton and Higgs fields. The usual zero temperature propagators and acquire an additive term that is proportional to the particle density distribution : For the fermionic thermal propagators, there are other higher order corrections (see [49]). Unlike the case of bosons, the interactions of the fermions with the thermal bath lead to two different types of excitations with different dispersion relations, that are generally referred to as â€śparticlesâ€ť and â€śholesâ€ť [49]. The contributions of these two fermionic modes were studied in [124â€“126] where it was argued that in the strong washout regime they could give nonnegligible effects [126]. The leading effects in (i) are proportional to the factor that vanishes when the final states thermal masses are neglected, because the Bose-Einstein and Fermi-Dirac statistical distributions depend on the same argument, . As a consequence, the thermal corrections to the fermion and boson propagators ( and ) and the product of the two thermal corrections () cancel each other. This was interpreted as a complete compensation between stimulated emission and Pauli blocking. As regards the effects in (ii), they lead to overall factors that cancel between numerator and denominator in the expression for the CP asymmetry. (A similar cancellation holds also in the supersymmetric case. However, because of the presence of the superpartners both as final states and in the loops, the cancellation is more subtle and it involves a compensation between the two types of corrections (i) and (ii). We refer to [48] for details.) More recently, on the basis of a first principle derivation of the CP asymmetry within a quantum BE approach (see Section 4.5) it has been claimed that the statistical factor induced by thermal loops is instead , which does not vanish even in the massless approximation. This would result in a further enhancement in the CP asymmetry from the thermal effects [127].

###### 4.3.5. Particle Motion

Given that the decaying particle is moving with respect to the background (with velocity ) the fermionic decay products are preferentially emitted in the direction antiparallel to the plasma velocity (for which the thermal distribution is less occupied), while the bosonic ones are emitted preferentially in the forward direction (for which stimulated emission is more effective). Particle motion then induces an angular dependence in the decay distribution at order . In the total decay rate the anisotropy effect is integrated out, and only effects remain [48]. Therefore, while accounting for thermal motion does modify the zero temperature results, these corrections are numerically small [48, 49] and generally negligible.

###### 4.3.6. Thermal Masses

When the finite values of the light particle thermal masses are taken into account, the arguments of the Bose-Einstein and Fermi-Dirac statistical distributions are different. It is a good approximation [49] to use for the particle energies . Since now , the prefactor that multiplies the thermal corrections does not vanish anymore, and sizable corrections become possible. The most relevant effect is that the CP asymmetry vanishes when, as the temperature increases, the sum of the light particles thermal masses approaches [49]. This is not surprising, since the particles in the final state coincide with the particles in the loop, and therefore when the decay becomes kinematically forbidden, also the particles in the loop cannot go on the mass shell. When the temperature is large enough that , the Higgs can decay, and then there is a new source of lepton number asymmetry associated with . The CP asymmetry in Higgs decays can be up to one order of magnitude larger than the CP asymmetry in decays [49]. While this could represent a dramatic enhancement of the CP asymmetry, is nonvanishing only at temperatures , when the kinematic condition for its decays is satisfied. Therefore, in the strong washout regime, no trace of this effect survives. On the other hand, rather large couplings are required in order that Higgs decays can occur before the phase space closes: the decay rate can attain thermal equilibrium only when , and therefore, in the weak washout regime (), these decays always remain strongly out of equilibrium. This means that only a small fraction of the Higgs particles have actually time to decay, and the lepton asymmetry generated in this way is accordingly suppressed.

In summary, while the corrections to the CP asymmetries can be significant at (and quite large at for Higgs decays), in the low temperature regime, where the precise value of plays a fundamental role in determining the final value of the baryon asymmetry, there are almost no effects, and the zero temperature results still give a reliable approximation.

##### 4.4. Decays of the Heavier Right-Handed Neutrinos

In leptogenesis studies, the effects of are often neglected, which in many cases is not a good approximation. This is obvious, for example, when dynamics is irrelevant for leptogenesis: cannot provide enough CP asymmetry to account for baryogenesis, and implies that washout effects are negligible. It is then clear that any asymmetry generated in decays can survive and becomes crucial for the success of leptogenesis. Another case in which it is intuitively clear that effects can be important is when the RH neutrino spectrum is compact, which means that have values within a factor of a few from . Then and contributions to leptogenesis can be equally important and must be summed up. A model with compact RH neutrino spectrum in which dynamics is of crucial importance was recently discussed in [128].

It is less obvious that effects can also be important for a hierarchical RH spectrum and when is strongly coupled. This can happen because decoherence effects related to -interactions can project the asymmetry generated in decays onto a flavour direction that remains protected against washouts [67, 129â€“131]. Let us illustrate this with an example. Let us assume that a sizable asymmetry is generated in decays, while leptogenesis is inefficient and fails, that is: Assuming also a strong hierarchy and that leptogenesis occurs thermally guarantees that [131] Thus, the dynamics of and are decoupled: there are neither -related washout effects during leptogenesis nor -related washout effects during leptogenesis. The decays into a combination of lepton doublets that we denote by : The second condition in (4.8) implies that already at the -Yukawa interactions are sufficiently fast to quickly destroy the coherence of . Then a statistical mixture of and of the state orthogonal to builds up, and it can be described by a suitable diagonal density matrix. Let us consider the simple case where both and decay at GeV, so that flavour effects are irrelevant. A convenient choice for an orthogonal basis for the lepton doublets is where, without loss of generality, satisfies . Then the asymmetry produced in decays decomposes into two components: where and . The crucial point here is that we expect, in general, and, since , is protected against washout. Consequently, a finite part of the asymmetry from decays survives through leptogenesis. A more detailed analysis [131] finds that is not entirely washed out, resulting in the final lepton asymmetry .

For GeV GeV, flavour issues modify the quantitative details, but the qualitative picture, and in particular the survival of a finite part of , still holds. In contrast, for GeV, the full flavour basis is resolved, and thus there are no directions in flavour space where an asymmetry is protected, so that can be erased entirely. A dedicated study in which the various flavour regimes for decays are considered can be found in [132].

In conclusion leptogenesis cannot be ignored, unless one of the following conditions holds.(1) The reheat temperature is below . (2) The asymmetries and/or the washout factors vanish, and . (3)-related washout is still significant at GeV.

##### 4.5. Quantum Boltzmann Equations

So far we have analyzed the leptogenesis dynamics by adopting the classical BE of motion. An interesting question which has attracted some attention recently [85, 94, 121, 127, 133â€“139] is under which circumstances the classical BE can be safely applied to get reliable results and, conversely, when a more rigorous quantum approach is needed. Quantum BEs are obtained starting from the nonequilibrium quantum field theory based on the closed time-path (CTP) formulation [140]. Both, CP violation from wave function and vertex corrections are incorporated. Unitarity issues are resolved, and an accurate account of all quantum-statistical effects on the asymmetry is made. Moreover, the formulation in terms of Green functions bears the potential of incorporating corrections from thermal field theory within the CTP formalism.

In the CTP formalism, particle number densities are replaced by Green's functions obeying a set of equations which, under some assumptions, can be reduced to a set of kinetic equations describing the evolution of the lepton asymmetry and the RH neutrinos. These kinetic equations are nonMarkovian and present memory effects. In other words, differently from the classical approach where every scattering in the plasma is independent of the previous one, the particle abundances at a given time depend upon the history of the system. The more familiar energy-conserving delta functions are replaced by retarded time integrals of time-dependent kernels, and cosine functions whose arguments are the energy involved in the various processes. Therefore, the nonMarkovian kinetic equations include the contribution of coherent processes throughout the history of the kernels and the relaxation times are expected to be typically longer than the one dictated by the classical approach.

If the time range of the kernels is shorter than the relaxation time of the particles abundances, the solutions to the quantum and the classical BE differ only by terms of the order of the ratio of the timescale of the kernel to the relaxation timescale of the distribution. In thermal leptogenesis this is typically the case. However, there are situations where this does not happen. For instance, in the case of resonant leptogenesis, two RH (s)neutrinos and are almost degenerate in mass, and the CP asymmetry from the decay of the first RH neutrino is resonantly enhanced if the mass difference is of the order of the decay rate of the second RH neutrino . The typical timescale to build up coherently the CP asymmetry is of the order of , which can be larger than the timescale for the change of the abundance of the 's.

Since we need the time evolution of the particle asymmetries with definite initial conditions and not simply the transition amplitude of particle reactions, the ordinary equilibrium quantum field theory at finite temperature is not the appropriate tool. The most appropriate extension of the field theory to deal with nonequilibrium phenomena amounts to generalizing the time contour of integration to a closed time path. More precisely, the time integration contour is deformed to run from to and back to . The CTP formalism is a powerful Green's function formulation for describing nonequilibrium phenomena in field theory. It allows to describe phase-transition phenomena and to obtain a self-consistent set of quantum BE. The formalism yields various quantum averages of operators evaluated in the instate without specifying the out-state. On the contrary, the ordinary quantum field theory yields quantum averages of the operators evaluated with an instate at one end and an outstate at the other.

For example, because of the time-contour deformation, the partition function in the in-in formalism for a complex scalar field is defined to be where in the integral denotes that the time integration contour runs from to plus infinity and then back to again. The symbol represents the initial density matrix, and the fields are in the Heisenberg picture and defined on this closed time contour (plus and minus subscripts refer to the positive and negative directional branches of the time path, resp.). The time-ordering operator along the path is the standard one () on the positive branch, and the antitime-ordering () on the negative branch. As with the Euclidean-time formulation, scalar (fermionic) fields are still periodic (antiperiodic) in time, but with , . The temperature appears due to boundary condition, and time is now explicitly present in the integration contour.

We must now identify field variables with arguments on the positive or negative directional branches of the time path. This doubling of field variables leads to six different real-time propagators on the contour. These six propagators are not independent, but using all of them simplifies the notation. For a generic charged scalar field they are defined as where the last two Green's functions are the retarded and advanced Green's functions, respectively, and is the step function.

For a generic fermion field the six different propagators are analogously defined as From the definitions of the Green's functions, one can see that the hermiticity properties are satisfied. For interacting systems, whether in equilibrium or not, one must define and calculate self-energy functions. Again, there are six of them: , , , , , and . The same relationships exist among them as for the Green's functions in (4.13) and (4.14), such as The self-energies are incorporated into the Green's functions through the use of Dyson's equations. A useful notation may be introduced which expresses four of the six Green's functions as the elements of two-by-two matrices: where the upper signs refer to the bosonic case and the lower signs to the fermionic case. For systems either in equilibrium or in nonequilibrium, Dysonâ€™s equation is most easily expressed by using the matrix notation: where the superscript â€ś0â€ť on the Greenâ€™s functions means to use those for noninteracting system. It is useful to notice that Dysonâ€™s equation can be written in an alternative form, instead of (4.18), with on the right in the interaction terms: Equations (4.18) and (4.19) are the starting points to derive the quantum BE describing the time evolution of the CP-violating particle density asymmetries.

To proceed, one has to choose a form for the propagators. For a generic fermion (and similarly for scalars) one may adopt the real-time propagator in the form in terms of the spectral function where represents the fermion distribution function. Again, particles must be substituted by quasiparticles, dressed propagators are to be adopted, and self-energy corrections to the propagator modify the dispersion relations by introducing a finite width . For a fermion with chiral mass , one may safely choose where and is the effective thermal mass of the fermion in the plasma (not a chiral mass). Performing the integration over and picking up the poles of the spectral function (which is valid for quasiparticles in equilibrium or very close to equilibrium), one gets where and denote the distribution function of the fermion particles and antiparticles, respectively. The expressions (4.22) are valid for .

The above definitions hold for the lepton doublets (after inserting the chiral LH projector ), as well as for the Majorana RH neutrinos, for which one has to assume identical particle and antiparticle distribution functions and insert the inverse of the charge conjugation matrix in the dispersion relation.

To elucidate further the impact of the CTP approach and to see under which conditions one can obtain the classical BE from the quantum ones, one may consider the dynamics of the lightest RH neutrino . To find its quantum BE we start from (4.18) for the Green's function of the RH neutrino Adopting the corresponding form for the RH neutrino propagator and the center-of-mass coordinates one ends up with the following equation: This equations holds under the assumption that the relaxation timescale for the distribution functions is longer than the timescale of the nonlocal kernels so that they can be extracted out of the time integral. This allows to think of the distributions as functions of the center-of-mass time only. We have set to zero the damping rates of the particles in (4.22) and retained only those cosines giving rise to energy delta functions that can be satisfied. Under these assumptions, the distribution function may be taken out of the time integral, leadingâ€”at large timesâ€”to the so-called Markovian description. The kinetic equation (4.25) has an obvious interpretation in terms of gain minus loss processes, but the retarded time integral and the cosine function replace the familiar energy-conserving delta functions. In the second passage, we have also made the usual assumption that all distribution functions are smaller than unity and that those of the Higgs and lepton doublets are in equilibrium and much smaller than unity, . Elastic scatterings are typically fast enough to keep kinetic equilibrium. For any distribution function we may write , where denotes the total number density. Therefore, (4.25) can be rewritten as