# Leptogenesis in the Universe

Authors: Chee Sheng Fong; Enrico Nardi; Antonio Riotto
Journal: Advances in High Energy Physics
 (2012)

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## 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.


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## Body

## 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-α particles, have ever been detected.The absence of annihilation radiationpp-→…π0→…2γ excludes significant matter-antimatter admixtures in objects up to the size of galactic clusters ~20 Mpc [1]. Observational limits on anomalous contributions to the cosmic diffuse γ-ray background and the absence of distortions in the cosmic microwave background (CMB) imply that little antimatter is to be found within ~1 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,D,  3He, 4He, and Li7, with the predictions of Big Bang nucleosynthesis (BBN) [4–9]. The crucial time for primordial nucleosynthesis is when the thermal bath temperature falls below  T≲1 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:
(1.1)η≡nB-nB-nγ|0,
where the subscript 0 means ”at present time.” By using only the abundance of deuterium, that is particularly sensitive to η, [4] quotes:
(1.2)1010η=5.7±0.6(95%c.l.).
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 T≲1eV 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 h≡H0/(100kmsec-1Mpc-1):
(1.3)ΩBh2≡h2ρBρcrit,
that is related to η through  1010η=274ΩBh2.  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 ΛCDM model with a scale-free power spectrum for the primordial density fluctuations) gives at 68% c.l. [12]
(1.4)102ΩBh2=2.258-0.056+0.057.
There is a third way to express the baryon asymmetry of the Universe, that is, by normalizing the baryon asymmetry to the entropy density  s=g*(2π2/45)T3, where  g* is the number of degrees of freedom in the plasma and  T  is the temperature:
(1.5)YΔB≡nB-nB-s|0.
The relation with the previous definitions is given by the conversion factor  s0/nγ0=7.04.  YΔB 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  YΔB the BBN results (1.2) and the CMB measurement (1.4) (at 95% c.l.) read
(1.6)YΔBBBN=(8.10±0.85)×10-11,YΔBCMB=(8.79±0.44)×10-11.
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 asbaryogenesis. This requires that baryon number (B) is not conserved. More precisely, as Sakharov pointed out [13], the ingredients required for baryogenesis are three.(1)
B violation is required to evolve from an initial state with  YΔB=0 to a state with  YΔB≠0.(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 functionsneq are determined solely by the particle energy  E, chemical potential μ, and by its mass which, because of the CPT theorem, is the same for particles and antiparticles. When charges (such as B) are not conserved, the corresponding chemical potentials vanish, and thus  nB=∫(d3p/(2π3))neq=nB-. 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 SMB is violated via the triangle anomaly. Although at zero temperature B 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 10-20, and this renders impossible generating  YΔB~10-10 [18–20].(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 [24–33], 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 [40–44] 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 SU(2)L singlet neutrinos with a large mass scale M 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  T~M, 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 “thermal leptogenesis” 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 Section2 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.


## 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-α particles, have ever been detected.The absence of annihilation radiationpp-→…π0→…2γ excludes significant matter-antimatter admixtures in objects up to the size of galactic clusters ~20 Mpc [1]. Observational limits on anomalous contributions to the cosmic diffuse γ-ray background and the absence of distortions in the cosmic microwave background (CMB) imply that little antimatter is to be found within ~1 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,D,  3He, 4He, and Li7, with the predictions of Big Bang nucleosynthesis (BBN) [4–9]. The crucial time for primordial nucleosynthesis is when the thermal bath temperature falls below  T≲1 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:
(1.1)η≡nB-nB-nγ|0,
where the subscript 0 means ”at present time.” By using only the abundance of deuterium, that is particularly sensitive to η, [4] quotes:
(1.2)1010η=5.7±0.6(95%c.l.).
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 T≲1eV 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 h≡H0/(100kmsec-1Mpc-1):
(1.3)ΩBh2≡h2ρBρcrit,
that is related to η through  1010η=274ΩBh2.  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 ΛCDM model with a scale-free power spectrum for the primordial density fluctuations) gives at 68% c.l. [12]
(1.4)102ΩBh2=2.258-0.056+0.057.
There is a third way to express the baryon asymmetry of the Universe, that is, by normalizing the baryon asymmetry to the entropy density  s=g*(2π2/45)T3, where  g* is the number of degrees of freedom in the plasma and  T  is the temperature:
(1.5)YΔB≡nB-nB-s|0.
The relation with the previous definitions is given by the conversion factor  s0/nγ0=7.04.  YΔB 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  YΔB the BBN results (1.2) and the CMB measurement (1.4) (at 95% c.l.) read
(1.6)YΔBBBN=(8.10±0.85)×10-11,YΔBCMB=(8.79±0.44)×10-11.
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 asbaryogenesis. This requires that baryon number (B) is not conserved. More precisely, as Sakharov pointed out [13], the ingredients required for baryogenesis are three.(1)
B violation is required to evolve from an initial state with  YΔB=0 to a state with  YΔB≠0.(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 functionsneq are determined solely by the particle energy  E, chemical potential μ, and by its mass which, because of the CPT theorem, is the same for particles and antiparticles. When charges (such as B) are not conserved, the corresponding chemical potentials vanish, and thus  nB=∫(d3p/(2π3))neq=nB-. 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 SMB is violated via the triangle anomaly. Although at zero temperature B 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 10-20, and this renders impossible generating  YΔB~10-10 [18–20].(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 [24–33], 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 [40–44] 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 SU(2)L singlet neutrinos with a large mass scale M 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  T~M, 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 “thermal leptogenesis” 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 Section2 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.N1  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 N1 (the so-called N1 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 N1 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 N1 mass and the upper bound on light neutrino mass scale from successful leptogenesis.


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

Withm(m≥2) (neutrino oscillation data and leptogenesis both require m≥2.) singlet RH neutrinos  NRi(i=1,m), we can add the following Standard Model (SM) gauge invariant terms to the SM Lagrangian:
(2.1)ℒI=ℒSM+iNRi¯∂NRi-(12MiNRic¯NRi+ϵabYαiNRi¯ℓαaHb+h.c.),
where  Mi are the Majorana masses of the RH neutrinos,  ℓα=(ναL,αL-) with  α=e,μ,τ and  H=(H+,H0) are, respectively, the left-handed (LH) lepton and Higgs SU(2)L doublets and  ϵab=-ϵba with ϵ12=1. 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  Ni=NRi+NRic. Since Ni are gauge singlets, the scale of Mi is naturally much larger than the electroweak (EW) scale Mi≫〈Φ〉≡v=174 GeV. Hence after EW symmetry breaking, the light neutrino mass matrix is given by the famous seesaw relation [40, 41, 43, 44]:
(2.2)mν≃-v2Y1MYT.
Assuming  Y~𝒪(1) and  mν≃Δmatm2≃0.05 eV, we have M~1015 GeV not far below the GUT scale.Besides giving a natural explanation of the light neutrino masses, there is another bonus: thethree Sakharov's conditions [13] for leptogenesis are implicit in (2.1) with the lepton number violation provided by Mi, the CP violation from the complexity of  Yiα, and the departure from thermal equilibrium condition given by an additional requirement that Ni decay rate ΓNi is not very fast compared to the Hubble expansion rate of the Universe  H(T) at temperature  T=Mi with
(2.3)ΓNi=(Y†Y)iiMi8π,H(T)=23g*π35T2Mpl,
where  Mpl=1.22×1019 GeV is the Planck mass,  g* (=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 thedecay parameter as follows:
(2.4)Ki≡ΓNiH(Mi)=m~im*,
where  m~i is the effective neutrino mass defined as [47]
(2.5)m~i≡(Y†Y)iiv2Mi,
with  m*≡(16π2v2/3Mpl)g*π/5≃1×10-3 eV. The regimes where Ki≪1,  Ki≈1 and  Ki≫1 are, respectively, known as weak, intermediate, and strong washout regimes.


### 2.2. CP Asymmetry

The CP asymmetry in the decays of RH neutrinosNi can be defined as
(2.6)ϵiα=γ(Ni→ℓαH)-γ(Ni→ℓα¯H*)∑αγ(Ni→ℓαH)+γ(Ni→ℓα¯H*)≡ΔγNiαγNi,
where  γ(i→f) 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 i, that is,  fi=e-Ei/T; see  [48, 49] for detailed studies of their effects)
(2.7)γ(i→f)≡∫d3pi(2π)32Eid3pf(2π)32Ef(2π)4δ(4)(pi-pf)|𝒜(i→f)|2e-Ei/T,
where 𝒜(i→f) is the decay amplitude. Ignoring all thermal effects [48, 49], (2.6) simplifies to
(2.8)ϵiα=|𝒜0(Ni→ℓαH)|2-|𝒜0(Ni→ℓα¯H*)|2∑α|𝒜0(Ni→ℓαH)|2+|𝒜0(Ni→ℓα¯H*)|2,
where 𝒜0(i→f) 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]:
(2.9)ϵiα=18π1(Y†Y)ii∑j≠iIm[(Y†Y)jiYαiYαj*]g(Mj2Mi2)+18π1(Y†Y)ii∑j≠iIm[(Y†Y)ijYαiYαj*]Mi2Mi2-Mj2,
where the loop function is
(2.10)g(x)=x[11-x+1-(1+x)ln(1+xx)].
The first term in (2.9) comes from L-violating wave and vertex diagrams, while the second term is from the L-conserving wave diagram. The terms of the form  (Mi2-Mj2)-1 in (2.9) are from the wave diagram contributions which can resonantly enhance the CP asymmetry if  Mi≈Mj (resonant leptogenesis scenario, see Section 6.1). Notice that the resonant term becomes singular in the degenerate limit  Mi=Mj. 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.Figure 1
The CP asymmetry in type-I seesaw leptogenesis results from the interference between tree and 1-loop wave and vertex diagrams. For the 1-loop wave diagram, there is an additional contribution fromL-conserving diagram to the CP asymmetry which vanishes when summing over lepton flavours.In the one flavour regime, we sum over the flavour indexα in (2.9) and obtain
(2.11)ϵi≡∑αϵiα=18π1(Y†Y)ii∑j≠iIm[(Y†Y)ji2]g(Mj2Mi2),
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 ofN1. If leptogenesis occurs at T≳1012GeV, then the charged lepton Yukawa interactions are out of equilibrium, and this defines the one-flavour regime. The assumption that only the dynamics of N1 is relevant can be realized if, for example, the reheating temperature after inflation is TRH≪M2,3 such that N2,3 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 ni=∫d3pfi to the entropy density s=(2π2/45)g*T3:
(2.12)Yi≡nis.
The evolution of the N1 density and the lepton asymmetry  YΔL=2YΔℓ≡2(Yℓ-Yℓ-)  (the factor of 2 comes from the SU(2)L degrees of freedoms) can be described by the following classical Boltzmann equations (BE) [53]:
(2.13)dYN1dz=-D1(YN1-YN1eq),(2.14)dYΔLdz=ϵ1D1(YN1-YN1eq)-W1YΔL,
where  z≡M1/T and the decay and washout terms are, respectively, given by
(2.15)D1(z)=γN1zsH(M1)=K1z𝒦1(z)𝒦2(z),W1(z)=12D1(z)YN1eq(z)Yℓeq,
with  𝒦n the nth order modified Bessel function of second kind. YN1eq and Yℓeq read (to write down a simple analytic expression for the equilibrium density of N1, we assume Maxwell-Boltzmann distribution. However, following [54], the normalization factor Yℓeq is obtained from a Fermi-Dirac distribution)
(2.16)YNeq(z)=452π4g*z2𝒦2(z),Yℓeq=154π2g*.From (2.13) and (2.14), the solution for  YΔL can be written down as follows:
(2.17)YΔL(z)=YΔL(zi)e-∫zizdz′W1(z′)-∫zizdz'ϵ1(z′)dYN1dz′e-∫z′zdz′′W1(z′′),
where zi is some initial temperature when N1 leptogenesis begins, and we have assumed that any preexisting lepton asymmetry vanishes YΔL0(zi)=0. Notice that ignoring thermal effects, the CP asymmetry is independent of the temperature ϵ1(z)=ϵ1 (c.f. (2.11)).


#### 2.3.1. Weak Washout Regime

In the weak washout regime (K1≪1), the initial condition on the N1 density YN1(zi) is important. If we assume thermal initial abundance of N1, that is,  YN1(zi)=YN1eq(0), we can ignore the washout when N1 starts decaying at z≫1 and we have
(2.18)YΔLt(∞)≃-ϵ1∫0∞dz'dYN1eqdz'=ϵ1YN1eq(0).
On the other hand, if we have zero initial N1 abundance, that is, YN1(zi)=0, we have to consider the opposite sign contributions to lepton asymmetry from the inverse decays when N1 is being populated (YN1<YN1eq) and from the period when N1 starts decaying (YN1>YN1eq). 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 12/π2, which is due to the different normalization  Yℓeq (2.16))
(2.19)YΔL0(∞)≃2716ϵ1K12YN1eq(0).


#### 2.3.2. Strong Washout Regime

In the strong washout regime (K1≫1) any lepton asymmetry generated during the  N1 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
(2.20)YΔL(z)≈-1W(z)ϵ1dYN1dz≃-1W(z)ϵ1dYN1eqdz=2zK1ϵ1Yℓeq,
where in the second approximation, we assume  YN1≃YN1eq. The approximation no longer holds when YΔL freezes, and this happens when the washout decouples at zf, that is, W(zf)<1. Hence, the final lepton asymmetry is given by (compare this to a more precise analytical approximation in  [55])
(2.21)YΔL(∞)=2zfK1ϵ1Yℓeq=π26zfK1ϵ1YN1eq(0).
The freeze out temperature zf depends mildly on K1. For  K1=10–100 we have, for example, zf=7– 10. We also see that independently of initial conditions, in the strong regime YΔL(∞) goes as K1-1.


### 2.4. Baryon Asymmetry from EW Sphaleron

The final lepton asymmetryYΔL(∞) can be conveniently parametrized as follows:
(2.22)YΔL(∞)=ϵ1η1YN1eq(0),
where η1 is known as the efficiency factor. In the weak washout regime (K1≪1) from (2.18) we have η1=1(=(27/16)K12<1) for thermal (zero) initial N1 abundance. In the strong washout regime (K1≫1), from (2.21), we have  η1=(π2/6zfK1)<1.If leptogenesis ends before EW sphaleron processes become active (T≳1012 GeV), the B-L asymmetry YΔB-L is simply given by
(2.23)YΔB-L=-YΔL.
At the later stage, the B-L asymmetry is partially transferred to a B asymmetry by the EW sphaleron processes through the relation [57]
(2.24)YΔB(∞)=2879YΔB-L(∞),
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 (M1≪M2,M3), and that the dominant lepton asymmetry is from the N1 decays, from (2.11) the CP asymmetry from N1 decays can be written as
(2.25)ϵ1=-316π1(Y†Y)11∑j≠1Im[(Y†Y)j12]M1Mj.
Assuming three generations of RH neutrinos (n=3) and using the Casas-Ibarra parametrization [60] for the Yukawa couplings
(2.26)Yαi=1v(DmNRDmνUν†)αi,
where DmN=diag(M1,M2,M3), Dmν=diag(mν1,mν2,mν3) and R any complex orthogonal matrix satisfying  RTR=RRT=1, (2.25) becomes
(2.27)ϵ1=-316πM1v2∑imνiIm(R1i2)∑imνi|R1i|2.
Using the orthogonality condition ∑iR1i2=1, we then obtain the Davidson-Ibarra (DI) bound [61]
(2.28)|ϵ1|≤ϵDI=316πM1v2(mν3-mν1)=316πM1v2Δmatm2mν1+mν3,
where mν3 (mν1) is the heaviest (lightest) light neutrino mass. Applying the DI bound on (2.22)–(2.24), and requiring that YΔB(∞)≥YBCMB≃10-10, we obtain
(2.29)M1(0.1eVmν1+mν3)η1max(M1)≳109GeV,
where the η1max(M1) is the efficiency factor maximized with respect to K1 (2.4) for a particular value of M1. This allows us to make a plot of region which satisfies (2.29) on the (M1,mν1) plane and hence obtain bounds on M1 and mν1. Many careful numerical studies have been carried out, and it was found that successful leptogenesis with a hierarchical spectrum of the RH neutrinos requires M1≳109GeV [61–63] and mν1≲0.1eV [55, 64–66]. This bound implies that the RH neutrinos must be produced at temperatures T≳109GeV which in turn implies the reheating temperature after inflation has to be TRH≳109GeV 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)
N1 dominates the contribution to leptogenesis.(2)
The mass spectrum of RH neutrinos is hierarchicalM1≪M2,M3.(3)
Leptogenesis occurs in the unflavoured regimeT≳1012GeV.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.


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

Withm(m≥2) (neutrino oscillation data and leptogenesis both require m≥2.) singlet RH neutrinos  NRi(i=1,m), we can add the following Standard Model (SM) gauge invariant terms to the SM Lagrangian:
(2.1)ℒI=ℒSM+iNRi¯∂NRi-(12MiNRic¯NRi+ϵabYαiNRi¯ℓαaHb+h.c.),
where  Mi are the Majorana masses of the RH neutrinos,  ℓα=(ναL,αL-) with  α=e,μ,τ and  H=(H+,H0) are, respectively, the left-handed (LH) lepton and Higgs SU(2)L doublets and  ϵab=-ϵba with ϵ12=1. 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  Ni=NRi+NRic. Since Ni are gauge singlets, the scale of Mi is naturally much larger than the electroweak (EW) scale Mi≫〈Φ〉≡v=174 GeV. Hence after EW symmetry breaking, the light neutrino mass matrix is given by the famous seesaw relation [40, 41, 43, 44]:
(2.2)mν≃-v2Y1MYT.
Assuming  Y~𝒪(1) and  mν≃Δmatm2≃0.05 eV, we have M~1015 GeV not far below the GUT scale.Besides giving a natural explanation of the light neutrino masses, there is another bonus: thethree Sakharov's conditions [13] for leptogenesis are implicit in (2.1) with the lepton number violation provided by Mi, the CP violation from the complexity of  Yiα, and the departure from thermal equilibrium condition given by an additional requirement that Ni decay rate ΓNi is not very fast compared to the Hubble expansion rate of the Universe  H(T) at temperature  T=Mi with
(2.3)ΓNi=(Y†Y)iiMi8π,H(T)=23g*π35T2Mpl,
where  Mpl=1.22×1019 GeV is the Planck mass,  g* (=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 thedecay parameter as follows:
(2.4)Ki≡ΓNiH(Mi)=m~im*,
where  m~i is the effective neutrino mass defined as [47]
(2.5)m~i≡(Y†Y)iiv2Mi,
with  m*≡(16π2v2/3Mpl)g*π/5≃1×10-3 eV. The regimes where Ki≪1,  Ki≈1 and  Ki≫1 are, respectively, known as weak, intermediate, and strong washout regimes.


## 2.2. CP Asymmetry

The CP asymmetry in the decays of RH neutrinosNi can be defined as
(2.6)ϵiα=γ(Ni→ℓαH)-γ(Ni→ℓα¯H*)∑αγ(Ni→ℓαH)+γ(Ni→ℓα¯H*)≡ΔγNiαγNi,
where  γ(i→f) 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 i, that is,  fi=e-Ei/T; see  [48, 49] for detailed studies of their effects)
(2.7)γ(i→f)≡∫d3pi(2π)32Eid3pf(2π)32Ef(2π)4δ(4)(pi-pf)|𝒜(i→f)|2e-Ei/T,
where 𝒜(i→f) is the decay amplitude. Ignoring all thermal effects [48, 49], (2.6) simplifies to
(2.8)ϵiα=|𝒜0(Ni→ℓαH)|2-|𝒜0(Ni→ℓα¯H*)|2∑α|𝒜0(Ni→ℓαH)|2+|𝒜0(Ni→ℓα¯H*)|2,
where 𝒜0(i→f) 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]:
(2.9)ϵiα=18π1(Y†Y)ii∑j≠iIm[(Y†Y)jiYαiYαj*]g(Mj2Mi2)+18π1(Y†Y)ii∑j≠iIm[(Y†Y)ijYαiYαj*]Mi2Mi2-Mj2,
where the loop function is
(2.10)g(x)=x[11-x+1-(1+x)ln(1+xx)].
The first term in (2.9) comes from L-violating wave and vertex diagrams, while the second term is from the L-conserving wave diagram. The terms of the form  (Mi2-Mj2)-1 in (2.9) are from the wave diagram contributions which can resonantly enhance the CP asymmetry if  Mi≈Mj (resonant leptogenesis scenario, see Section 6.1). Notice that the resonant term becomes singular in the degenerate limit  Mi=Mj. 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.Figure 1
The CP asymmetry in type-I seesaw leptogenesis results from the interference between tree and 1-loop wave and vertex diagrams. For the 1-loop wave diagram, there is an additional contribution fromL-conserving diagram to the CP asymmetry which vanishes when summing over lepton flavours.In the one flavour regime, we sum over the flavour indexα in (2.9) and obtain
(2.11)ϵi≡∑αϵiα=18π1(Y†Y)ii∑j≠iIm[(Y†Y)ji2]g(Mj2Mi2),
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 ofN1. If leptogenesis occurs at T≳1012GeV, then the charged lepton Yukawa interactions are out of equilibrium, and this defines the one-flavour regime. The assumption that only the dynamics of N1 is relevant can be realized if, for example, the reheating temperature after inflation is TRH≪M2,3 such that N2,3 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 ni=∫d3pfi to the entropy density s=(2π2/45)g*T3:
(2.12)Yi≡nis.
The evolution of the N1 density and the lepton asymmetry  YΔL=2YΔℓ≡2(Yℓ-Yℓ-)  (the factor of 2 comes from the SU(2)L degrees of freedoms) can be described by the following classical Boltzmann equations (BE) [53]:
(2.13)dYN1dz=-D1(YN1-YN1eq),(2.14)dYΔLdz=ϵ1D1(YN1-YN1eq)-W1YΔL,
where  z≡M1/T and the decay and washout terms are, respectively, given by
(2.15)D1(z)=γN1zsH(M1)=K1z𝒦1(z)𝒦2(z),W1(z)=12D1(z)YN1eq(z)Yℓeq,
with  𝒦n the nth order modified Bessel function of second kind. YN1eq and Yℓeq read (to write down a simple analytic expression for the equilibrium density of N1, we assume Maxwell-Boltzmann distribution. However, following [54], the normalization factor Yℓeq is obtained from a Fermi-Dirac distribution)
(2.16)YNeq(z)=452π4g*z2𝒦2(z),Yℓeq=154π2g*.From (2.13) and (2.14), the solution for  YΔL can be written down as follows:
(2.17)YΔL(z)=YΔL(zi)e-∫zizdz′W1(z′)-∫zizdz'ϵ1(z′)dYN1dz′e-∫z′zdz′′W1(z′′),
where zi is some initial temperature when N1 leptogenesis begins, and we have assumed that any preexisting lepton asymmetry vanishes YΔL0(zi)=0. Notice that ignoring thermal effects, the CP asymmetry is independent of the temperature ϵ1(z)=ϵ1 (c.f. (2.11)).


### 2.3.1. Weak Washout Regime

In the weak washout regime (K1≪1), the initial condition on the N1 density YN1(zi) is important. If we assume thermal initial abundance of N1, that is,  YN1(zi)=YN1eq(0), we can ignore the washout when N1 starts decaying at z≫1 and we have
(2.18)YΔLt(∞)≃-ϵ1∫0∞dz'dYN1eqdz'=ϵ1YN1eq(0).
On the other hand, if we have zero initial N1 abundance, that is, YN1(zi)=0, we have to consider the opposite sign contributions to lepton asymmetry from the inverse decays when N1 is being populated (YN1<YN1eq) and from the period when N1 starts decaying (YN1>YN1eq). 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 12/π2, which is due to the different normalization  Yℓeq (2.16))
(2.19)YΔL0(∞)≃2716ϵ1K12YN1eq(0).


### 2.3.2. Strong Washout Regime

In the strong washout regime (K1≫1) any lepton asymmetry generated during the  N1 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
(2.20)YΔL(z)≈-1W(z)ϵ1dYN1dz≃-1W(z)ϵ1dYN1eqdz=2zK1ϵ1Yℓeq,
where in the second approximation, we assume  YN1≃YN1eq. The approximation no longer holds when YΔL freezes, and this happens when the washout decouples at zf, that is, W(zf)<1. Hence, the final lepton asymmetry is given by (compare this to a more precise analytical approximation in  [55])
(2.21)YΔL(∞)=2zfK1ϵ1Yℓeq=π26zfK1ϵ1YN1eq(0).
The freeze out temperature zf depends mildly on K1. For  K1=10–100 we have, for example, zf=7– 10. We also see that independently of initial conditions, in the strong regime YΔL(∞) goes as K1-1.


## 2.3.1. Weak Washout Regime

In the weak washout regime (K1≪1), the initial condition on the N1 density YN1(zi) is important. If we assume thermal initial abundance of N1, that is,  YN1(zi)=YN1eq(0), we can ignore the washout when N1 starts decaying at z≫1 and we have
(2.18)YΔLt(∞)≃-ϵ1∫0∞dz'dYN1eqdz'=ϵ1YN1eq(0).
On the other hand, if we have zero initial N1 abundance, that is, YN1(zi)=0, we have to consider the opposite sign contributions to lepton asymmetry from the inverse decays when N1 is being populated (YN1<YN1eq) and from the period when N1 starts decaying (YN1>YN1eq). 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 12/π2, which is due to the different normalization  Yℓeq (2.16))
(2.19)YΔL0(∞)≃2716ϵ1K12YN1eq(0).


## 2.3.2. Strong Washout Regime

In the strong washout regime (K1≫1) any lepton asymmetry generated during the  N1 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
(2.20)YΔL(z)≈-1W(z)ϵ1dYN1dz≃-1W(z)ϵ1dYN1eqdz=2zK1ϵ1Yℓeq,
where in the second approximation, we assume  YN1≃YN1eq. The approximation no longer holds when YΔL freezes, and this happens when the washout decouples at zf, that is, W(zf)<1. Hence, the final lepton asymmetry is given by (compare this to a more precise analytical approximation in  [55])
(2.21)YΔL(∞)=2zfK1ϵ1Yℓeq=π26zfK1ϵ1YN1eq(0).
The freeze out temperature zf depends mildly on K1. For  K1=10–100 we have, for example, zf=7– 10. We also see that independently of initial conditions, in the strong regime YΔL(∞) goes as K1-1.


## 2.4. Baryon Asymmetry from EW Sphaleron

The final lepton asymmetryYΔL(∞) can be conveniently parametrized as follows:
(2.22)YΔL(∞)=ϵ1η1YN1eq(0),
where η1 is known as the efficiency factor. In the weak washout regime (K1≪1) from (2.18) we have η1=1(=(27/16)K12<1) for thermal (zero) initial N1 abundance. In the strong washout regime (K1≫1), from (2.21), we have  η1=(π2/6zfK1)<1.If leptogenesis ends before EW sphaleron processes become active (T≳1012 GeV), the B-L asymmetry YΔB-L is simply given by
(2.23)YΔB-L=-YΔL.
At the later stage, the B-L asymmetry is partially transferred to a B asymmetry by the EW sphaleron processes through the relation [57]
(2.24)YΔB(∞)=2879YΔB-L(∞),
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 (M1≪M2,M3), and that the dominant lepton asymmetry is from the N1 decays, from (2.11) the CP asymmetry from N1 decays can be written as
(2.25)ϵ1=-316π1(Y†Y)11∑j≠1Im[(Y†Y)j12]M1Mj.
Assuming three generations of RH neutrinos (n=3) and using the Casas-Ibarra parametrization [60] for the Yukawa couplings
(2.26)Yαi=1v(DmNRDmνUν†)αi,
where DmN=diag(M1,M2,M3), Dmν=diag(mν1,mν2,mν3) and R any complex orthogonal matrix satisfying  RTR=RRT=1, (2.25) becomes
(2.27)ϵ1=-316πM1v2∑imνiIm(R1i2)∑imνi|R1i|2.
Using the orthogonality condition ∑iR1i2=1, we then obtain the Davidson-Ibarra (DI) bound [61]
(2.28)|ϵ1|≤ϵDI=316πM1v2(mν3-mν1)=316πM1v2Δmatm2mν1+mν3,
where mν3 (mν1) is the heaviest (lightest) light neutrino mass. Applying the DI bound on (2.22)–(2.24), and requiring that YΔB(∞)≥YBCMB≃10-10, we obtain
(2.29)M1(0.1eVmν1+mν3)η1max(M1)≳109GeV,
where the η1max(M1) is the efficiency factor maximized with respect to K1 (2.4) for a particular value of M1. This allows us to make a plot of region which satisfies (2.29) on the (M1,mν1) plane and hence obtain bounds on M1 and mν1. Many careful numerical studies have been carried out, and it was found that successful leptogenesis with a hierarchical spectrum of the RH neutrinos requires M1≳109GeV [61–63] and mν1≲0.1eV [55, 64–66]. This bound implies that the RH neutrinos must be produced at temperatures T≳109GeV which in turn implies the reheating temperature after inflation has to be TRH≳109GeV 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)
N1 dominates the contribution to leptogenesis.(2)
The mass spectrum of RH neutrinos is hierarchicalM1≪M2,M3.(3)
Leptogenesis occurs in the unflavoured regimeT≳1012GeV.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 neutrinoN1 maintain their coherence as flavour superpositions throughout the leptogenesis era, that is ℓ1=∑αcα1ℓα and ℓ1′¯=∑αcα1'*ℓ¯α. Note that at the tree level the coefficients c and c'* are simply the Yukawa couplings: cα1=Yα1 and cα1'*=Yα1*. However it should be kept in mind that since CP is violated by loops, beyond the tree level approximation the antilepton state ℓ1′¯ is not the CP conjugate of the ℓ1, that is, cα1′≠cα1.The single flavour regime is realized only at very high temperatures (T≳1012GeV) when both ℓ1 and ℓ1′ 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, ℓ1 and ℓ1′ decohere in their flavour components, and the dynamics of leptogenesis must then be described in terms of the flavour eigenstates ℓα. Of course, there is great interest to extend the validity of quantitative leptogenesis studies also at lower scale T≲1012GeV, 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 [68–70], they had been included in leptogenesis studies only in a few cases [71–75]. Nowadays lepton flavour effects have been investigated in full detail [76–89] 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 processesΓα (α=e,μ,τ) that are induced by the charged lepton Yukawa couplings hα become faster than the Universe expansion rate H(T). An approximate relation gives [90, 91]
(3.1)Γα(T)≃10-2hα2T,
which implies that  (in supersymmetric case, since hα=mα/(vucosβ), we have T≲Tα(1+tan2β))
(3.2)Γα(T)>H(T)whenT≲Tα,
where Te≃4×104GeV, Tμ≃2×109GeV, and Tτ≃5×1011GeV. Notice that to fully distinguish the three flavours it is sufficient that the τ and μ 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 N1 interactions [69, 83, 84]. In general whenever Γτ(M1)>H(M1) we also have Γτ(M1)>ΓN1(M1). However, there exists parameter space where Γτ(M1)>H(M1) but Γτ(M1)<ΓN1(M1). This scenario was studied in [83].


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

The CP violation inNi decays can manifest itself in two ways [69](i) The leptons and antileptons are produced at different rates:(3.3)γi≠γ-i,
where γi≡γ(Ni→ℓiH) and  γ-i≡γ(Ni→ℓi′¯H*).(ii) The leptons and antileptons produced are not CP conjugate states:(3.4)CP(ℓi′¯)=ℓi′≠ℓi,
that is, due to loops effects they are slightly misaligned in flavour space.We can rewrite the CP asymmetry forNi decays from (2.6) as follows:
(3.5)ϵiα=Piαγi-P-iαγ-iγi+γ-i=Piα+P-iα2ϵi+Piα-P-iα2≃Piα0ϵi+ΔPiα2,
where terms of order 𝒪(ϵiΔPiα) and higher have been neglected. Piα is the projector from state ℓi into flavour state ℓα and ΔPiα=Piα-P-iα. At tree level, clearly, Piα=P-iα≡Piα0 where the tree level flavour projector is given by
(3.6)Piα0=YαiYαi*(Y†Y)ii.
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 ∑αPiα=∑αP-iα=1, when summing over flavour indices α, the second term vanishes  ∑αΔPiα=0. Note that the lepton-flavour-violating but L-conserving terms in the second line of (2.9) are part of type (ii). In fact, they come from d=6L-conserving operators which have nothing to do with the unique d=5 L-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 ϵiα can be much larger and even of opposite sign from the total CP asymmetry ϵi. 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 α, it is proportional to
(3.7)Wiα∝Piαγi+P-iαγ-i≃Piα0Wi,
which results in a reduction of washout by a factor of Piα0≤1 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 ϵi=0 also becomes possible.


### 3.3. Classical Flavoured Boltzmann Equations

Here again we only consider leptogenesis from the decays and inverse decays ofN1. In this approximation, the BE for YN1 is still given by (2.13) while the BE for YΔLα the lepton asymmetry in the flavour α 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]).
(3.8)dYΔLαdz=ϵ1αD1(YN1-YN1eq)-P1α0W1YΔLα.
Notice that as long as L violation from sphalerons is neglected (see Section 4) the BEs for YΔLα 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 ϵ1→ϵ1α and K1→K1α≡P1α0K1.As an example let us assume that leptogenesis occurs aroundT~1010GeV, that is, in the two-flavour regime. Due to the fast τ Yukawa interactions ℓ1(ℓ1′) gets projected onto ℓτ(ℓτ′) and a coherent mixture of e+μ eigenstate ℓe+μ(ℓe+μ′). For illustrative purpose, here we consider a scenario in which lepton flavour effects are most prominent. We take both K1τ,K1e+μ≫1, so that both YΔLτ and YΔLe+μ are in the strong regime. From (2.21) we can write down the solution:
(3.9)YΔL(∞)=YΔLτ(∞)+YΔLe+μ(∞)=π26zfK1YN1eq(0)(ϵ1τP1τ0+ϵ1e+μP1e+μ0)≃π23zfK1ϵ1YN1eq(0)+π212zfK1YN1eq(0)(ΔP1τP1τ0+ΔP1e+μP1e+μ0),
where in the last line we have used (3.5). If P1τ0≃P1e+μ0, then since ΔP1τ+ΔP1e+μ=0, the second term approximately cancels, and (3.9) reduces to
(3.10)YΔL(∞)≃π23zfK1ϵ1YN1eq(0).
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 P1τ0>P1e+μ0 while ΔP1τ≪ΔP1e+μ. In this case, the second term can dominate over the first term. Finally from (3.9) we also notice that leptogenesis with ϵ1=0, the so-called purely flavoured leptogenesis (PFL) (this can also refer to the case where the total CP asymmetry is negligible ϵ1≈0), can indeed proceed [69, 95–98]. In this scenario some symmetry has to be imposed to realize the condition ϵ1=0, as, for example, an approximate global lepton number U(1)L. In the limit of exact U(1)L the active neutrinos will be exactly massless. Instead of the seesaw mechanism, the small neutrino masses are explained by U(1)L which is slightly broken by a small parameter μ (the “inverse seesaw”) [99] which is technically natural since the Lagrangian exhibits an enhanced symmetry when μ→0 [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, ℓαH→ℓβH, and can proceed, for example, via off-shell exchange of N2,3. 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:
(3.11)dYΔLdz=ϵ1D1(YN1-YN1eq)-P1α0W1YΔL,
where P1α0=1/2(1/3) 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 (3) in the two- (three-) flavour regime (c.f. (3.10)). Clearly, LFE can make PFL with ϵ1=0 impotent [56, 93]. Since LFE N2,3 processes scale as T3 while the Universe expansion scales as T2, 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 M1≳108GeV [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 M1~106GeV.


## 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 neutrinoN1 maintain their coherence as flavour superpositions throughout the leptogenesis era, that is ℓ1=∑αcα1ℓα and ℓ1′¯=∑αcα1'*ℓ¯α. Note that at the tree level the coefficients c and c'* are simply the Yukawa couplings: cα1=Yα1 and cα1'*=Yα1*. However it should be kept in mind that since CP is violated by loops, beyond the tree level approximation the antilepton state ℓ1′¯ is not the CP conjugate of the ℓ1, that is, cα1′≠cα1.The single flavour regime is realized only at very high temperatures (T≳1012GeV) when both ℓ1 and ℓ1′ 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, ℓ1 and ℓ1′ decohere in their flavour components, and the dynamics of leptogenesis must then be described in terms of the flavour eigenstates ℓα. Of course, there is great interest to extend the validity of quantitative leptogenesis studies also at lower scale T≲1012GeV, 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 [68–70], they had been included in leptogenesis studies only in a few cases [71–75]. Nowadays lepton flavour effects have been investigated in full detail [76–89] 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 processesΓα (α=e,μ,τ) that are induced by the charged lepton Yukawa couplings hα become faster than the Universe expansion rate H(T). An approximate relation gives [90, 91]
(3.1)Γα(T)≃10-2hα2T,
which implies that  (in supersymmetric case, since hα=mα/(vucosβ), we have T≲Tα(1+tan2β))
(3.2)Γα(T)>H(T)whenT≲Tα,
where Te≃4×104GeV, Tμ≃2×109GeV, and Tτ≃5×1011GeV. Notice that to fully distinguish the three flavours it is sufficient that the τ and μ 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 N1 interactions [69, 83, 84]. In general whenever Γτ(M1)>H(M1) we also have Γτ(M1)>ΓN1(M1). However, there exists parameter space where Γτ(M1)>H(M1) but Γτ(M1)<ΓN1(M1). This scenario was studied in [83].


## 3.2. The Effects on CP Asymmetry and Washout

The CP violation inNi decays can manifest itself in two ways [69](i) The leptons and antileptons are produced at different rates:(3.3)γi≠γ-i,
where γi≡γ(Ni→ℓiH) and  γ-i≡γ(Ni→ℓi′¯H*).(ii) The leptons and antileptons produced are not CP conjugate states:(3.4)CP(ℓi′¯)=ℓi′≠ℓi,
that is, due to loops effects they are slightly misaligned in flavour space.We can rewrite the CP asymmetry forNi decays from (2.6) as follows:
(3.5)ϵiα=Piαγi-P-iαγ-iγi+γ-i=Piα+P-iα2ϵi+Piα-P-iα2≃Piα0ϵi+ΔPiα2,
where terms of order 𝒪(ϵiΔPiα) and higher have been neglected. Piα is the projector from state ℓi into flavour state ℓα and ΔPiα=Piα-P-iα. At tree level, clearly, Piα=P-iα≡Piα0 where the tree level flavour projector is given by
(3.6)Piα0=YαiYαi*(Y†Y)ii.
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 ∑αPiα=∑αP-iα=1, when summing over flavour indices α, the second term vanishes  ∑αΔPiα=0. Note that the lepton-flavour-violating but L-conserving terms in the second line of (2.9) are part of type (ii). In fact, they come from d=6L-conserving operators which have nothing to do with the unique d=5 L-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 ϵiα can be much larger and even of opposite sign from the total CP asymmetry ϵi. 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 α, it is proportional to
(3.7)Wiα∝Piαγi+P-iαγ-i≃Piα0Wi,
which results in a reduction of washout by a factor of Piα0≤1 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 ϵi=0 also becomes possible.


## 3.3. Classical Flavoured Boltzmann Equations

Here again we only consider leptogenesis from the decays and inverse decays ofN1. In this approximation, the BE for YN1 is still given by (2.13) while the BE for YΔLα the lepton asymmetry in the flavour α 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]).
(3.8)dYΔLαdz=ϵ1αD1(YN1-YN1eq)-P1α0W1YΔLα.
Notice that as long as L violation from sphalerons is neglected (see Section 4) the BEs for YΔLα 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 ϵ1→ϵ1α and K1→K1α≡P1α0K1.As an example let us assume that leptogenesis occurs aroundT~1010GeV, that is, in the two-flavour regime. Due to the fast τ Yukawa interactions ℓ1(ℓ1′) gets projected onto ℓτ(ℓτ′) and a coherent mixture of e+μ eigenstate ℓe+μ(ℓe+μ′). For illustrative purpose, here we consider a scenario in which lepton flavour effects are most prominent. We take both K1τ,K1e+μ≫1, so that both YΔLτ and YΔLe+μ are in the strong regime. From (2.21) we can write down the solution:
(3.9)YΔL(∞)=YΔLτ(∞)+YΔLe+μ(∞)=π26zfK1YN1eq(0)(ϵ1τP1τ0+ϵ1e+μP1e+μ0)≃π23zfK1ϵ1YN1eq(0)+π212zfK1YN1eq(0)(ΔP1τP1τ0+ΔP1e+μP1e+μ0),
where in the last line we have used (3.5). If P1τ0≃P1e+μ0, then since ΔP1τ+ΔP1e+μ=0, the second term approximately cancels, and (3.9) reduces to
(3.10)YΔL(∞)≃π23zfK1ϵ1YN1eq(0).
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 P1τ0>P1e+μ0 while ΔP1τ≪ΔP1e+μ. In this case, the second term can dominate over the first term. Finally from (3.9) we also notice that leptogenesis with ϵ1=0, the so-called purely flavoured leptogenesis (PFL) (this can also refer to the case where the total CP asymmetry is negligible ϵ1≈0), can indeed proceed [69, 95–98]. In this scenario some symmetry has to be imposed to realize the condition ϵ1=0, as, for example, an approximate global lepton number U(1)L. In the limit of exact U(1)L the active neutrinos will be exactly massless. Instead of the seesaw mechanism, the small neutrino masses are explained by U(1)L which is slightly broken by a small parameter μ (the “inverse seesaw”) [99] which is technically natural since the Lagrangian exhibits an enhanced symmetry when μ→0 [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, ℓαH→ℓβH, and can proceed, for example, via off-shell exchange of N2,3. 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:
(3.11)dYΔLdz=ϵ1D1(YN1-YN1eq)-P1α0W1YΔL,
where P1α0=1/2(1/3) 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 (3) in the two- (three-) flavour regime (c.f. (3.10)). Clearly, LFE can make PFL with ϵ1=0 impotent [56, 93]. Since LFE N2,3 processes scale as T3 while the Universe expansion scales as T2, 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 M1≳108GeV [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 M1~106GeV.


## 4. Beyond the Basic Boltzmann Equations

Within factors of a few, the amount of baryon asymmetry that is generated via leptogenesis inN1 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 𝒪(1) 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 [101–104], and are generally subleading.


### 4.1. Spectator Processes

Reactions that without involving violation ofB-L can still affect the final amount of baryon asymmetry are classified as “spectator processes” [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 YΔH [106]. This was omitted in Section 2 but in fact has to be added to the density asymmetry of the leptons YΔℓ in weighting, for example, washouts from inverse decays. Equation (2.14) would then become
(4.1)dYΔLdz=ϵ1D1(YN1-YN1eq)-2(YΔℓ+YΔH)W1,
where the factor of two in front of the washout term counts the leptons and Higgs gauge multiplicity. Clearly, in some regimes in which YΔℓ and YΔH 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 temperatureT, particle reactions must be treated in a different way depending on if their characteristic time scale τ (given by inverse of their thermally averaged rates) is [89, 107] (1)
much shorter than the age of the Universe:τ≪tU(T);(2)
much larger than the age of the Universe:τ≫tU(T);(3)
comparable with the Universe age:τ~tU(T).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,∑IμI=∑FμF, where μI denotes the chemical potential of an initial state particle and μF 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 T 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 (2) 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 (3) 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,T≳1013GeV). 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 B-L-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, B is conserved too (and vanishing, if we assume that there is no preexisting asymmetry). B=0 then translates in the condition:
(4.2)2YΔQ3+YΔt=0,
where YΔQ3 is the density asymmetries of one degree of freedom of the top SU(2)L doublet and color triplet which, being gauge interactions in equilibrium, is the same for all the six gauge components and YΔt is the density asymmetry of the SU(2)L singlet top. Hypercharge is always conserved, yielding
(4.3)YΔQ3+2YΔt-YΔℓ+YΔH=0.
Finally, in terms of density asymmetries chemical equilibrium for the top-Yukawa-related reactions μQ3+μH=μt translates into
(4.4)YΔQ3+12YΔH=YΔt.
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 YΔH=(2/3)YΔℓ. Moreover, given that only the LH lepton degrees of freedom are populated, we have YΔL=2YΔℓ so that the coefficient weighting W1 in (4.1) becomes 2(YΔℓ+YΔH)=(5/3)YΔL 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 belowT~1012GeV, 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 B-L, which is then violated only by slow reactions of type (3), and we should then write down a BE for this quantity. Better said, since at T≲1012GeV all the third generation Yukawa reactions, including the ones of the τ-lepton, are in equilibrium, the dynamical regime is that of two flavours in which the relevant quasiconserved charges are Δτ=B/3-Lτ and Δeμ=B/3-Leμ. The fact that only two charges are relevant is because there is always a direction in e-μ  space which remains decoupled from N1. The corresponding third charge Δeμ′ 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
(4.5)-dYΔαdz=ϵαD1(YN1-YN1eq)-2(YΔℓα+YΔH)W1(α=τ,eμ).
To rewrite these equations in a solvable closed form, YΔℓτ, YΔℓeμ, and YΔH must be expressed in terms of the two charge densities YΔτ and YΔeμ. 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] (1) QCD sphaleron equilibrium; (2) EW sphaleron equilibrium; (3)b-quark and τ-lepton Yukawa equilibrium. The “rotation” from the particle density asymmetries YΔℓα  and  YΔH to the charge densities YΔα can be expressed in terms of the A matrix introduced in [67] YΔℓα=AαβℓYΔβ(α,β=τ,eμ) and C-vector YΔH=CαHYΔα introduced in [69]. For the present case, with the ordering (eμ,τ) they are [69]
(4.6)Aℓ=1460(196-24-9156),CH=1230(41,56).
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 YΔα. This is because each time a conservation law has to be dropped (like B 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 theN1 population, because decay and inverse decay rates are suppressed by a time dilation factor ∝M1/T. The N1=N-1 particles can be produced by scatterings with the top quark in s-channel H-exchange qtc→Nℓα and q-t-c→Nℓ-α, by t-channel H-exchange in qℓ-α→Nt-c, q-ℓα→Ntc and by u-channel H-exchange in ℓαt-c→Nq, tcℓ-α→Nq-; see the diagrams (a) in Figure 2. Several scattering channels with gauge bosons also contribute to the production of N1; the corresponding diagrams are (b) and (c) in the same figure.Diagrams for various2↔2 scattering processes: (a) scatterings with the top quarks; (b) and (c) scatterings with the gauge bosons (A=B, Wi  with i=1,2,3).
(a)
(b)
(c)When the effects of scatterings in populating theN1 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 (ht2λ4, g2λ4, where g is a gauge coupling) with an on-shell intermediate state N1 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 Nj. 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, M2/M1≫10. Corrections up to several tens of percent can appear around temperatures of order T~M2/10 and can be numerically relevant in case of milder hierarchies.Regarding the scattering processes with gauge bosons such asNℓα→AH-NH→Aℓ-α or NA→ℓαH, 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 (T>M1) 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 Λ~2πT [49]. The value Λ>T 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 m(μ)=r·m, where 1.2≲r≲1.3 for 108GeV ≲μ≲1016GeV [49], and therefore can be accounted for by increasing the values of the neutrino mass parameters (e.g., m~) as measured at low energy by ≈20%-30% (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, 116–120]. 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 mH(T)≳mq3,t(T)≫mℓ(T). 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 (mH(T)≈0.4T), decays and inverse decays become kinematically forbidden in the temperature range in which mH(T)-mℓ(T)<MN1<mH(T)+mℓ(T). For lower temperatures, the usual processes N1↔ℓH can occur. For higher temperatures, the Higgs is heavy enough that it can decay: H↔ℓN1. A rough estimate of the kinematically forbidden region yields 2≲T/M1≲5. The important point is that these corrections are effective only at T>M1. In the parameter region m~>10-3eV, that is favoured by the measurements of the neutrino mass-squared differences, the N1 number density and its L-violating reactions attain thermal equilibrium at T≈M1 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)
ΔL = 1 scatterings with top quark: a comparison between the corrected and uncorrected rates of the top-quark scattering γHstop≡γ(q3t-↔ℓN1) with the Higgs exchanged in the s-channel and of the sum of the t- and u-channel scatterings γHt+utop≡γ(q3N1↔ℓt)+γ(t-N1↔ℓq-3) shows that the only corrections appearing at low temperatures, and thus more relevant, are for γHt+utop (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 ~ln(MN1/mH) from the quasimassless Higgs exchanged in the  t- and u-channels, which disappears when the Higgs thermal mass mH(T)~T~MN1 is included.(iii)
ΔL = 1 scatterings with the gauge bosons: here the inclusion of thermal masses is required to avoid IR divergences that would arise when massless ℓ (and H) states are exchanged in the t- and u-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 whichN1 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 ϵ→ϵ(T) 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 decayingN’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 N2,3 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, M2,3≫M1, the related effects are Boltzmann suppressed by exp(-M2,3/T) that at T~M1 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 2×2 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 -iSℓ0(p,mℓ)=(p-mℓ+i0+)-1 and -iDH0(p,mH)=(p2-mH2+i0+)-1 acquire an additive term that is proportional to the particle density distribution nℓ,H=[exp(Eℓ,H/T)±1]-1:
(4.7)δSℓ(T)=-2πnℓ(p-mℓ)δ(p2-mℓ2),δDH(T)=+2πnHδ(p2-mH2).
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 “particles” and “holes” [49]. The contributions of these two fermionic modes were studied in [124–126] 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 -nℓ+nH-2nℓnH that vanishes when the final states thermal masses are neglected, because the Bose-Einstein and Fermi-Dirac statistical distributions depend on the same argument, Eℓ=EH=M1/2. As a consequence, the thermal corrections to the fermion and boson propagators (nℓ and nH) and the product of the two thermal corrections (nℓnH) 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 ℓ~,H~ 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 -nℓ+nH, 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 particleN1 is moving with respect to the background (with velocity β→) 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 𝒪(β). In the total decay rate the 𝒪(β) anisotropy effect is integrated out, and only 𝒪(β2) 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 Eℓ,H=M1/2∓(mH2-mℓ2)/2M1. Since now Eℓ≠EH, the prefactor -nℓ+nH-2nℓnH 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 M1 [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 mH(T)>mℓ(T)+M1, the Higgs can decay, and then there is a new source of lepton number asymmetry associated with H→ℓN1. The CP asymmetry in Higgs decays ϵH can be up to one order of magnitude larger than the CP asymmetry in N1 decays [49]. While this could represent a dramatic enhancement of the CP asymmetry, ϵH is nonvanishing only at temperatures T≳TH~5M1, 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 m~≳(TH/M1)2m*≫m*, and therefore, in the weak washout regime (m~≲m*), 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 atT≳M1 (and quite large at T≫M1 for Higgs decays), in the low temperature regime, where the precise value of ϵ 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 ofN2,3 are often neglected, which in many cases is not a good approximation. This is obvious, for example, when N1 dynamics is irrelevant for leptogenesis: ϵ1≪10-6 cannot provide enough CP asymmetry to account for baryogenesis, and m~1≪m* implies that N1 washout effects are negligible. It is then clear that any asymmetry generated in N2,3 decays can survive and becomes crucial for the success of leptogenesis. Another case in which it is intuitively clear that N2,3 effects can be important is when the RH neutrino spectrum is compact, which means that M2,3 have values within a factor of a few from M1. Then N1 and N2,3 contributions to leptogenesis can be equally important and must be summed up. A model with compact RH neutrino spectrum in which N2,3 dynamics is of crucial importance was recently discussed in [128].It is less obvious thatN2,3 effects can also be important for a hierarchical RH spectrum and when N1 is strongly coupled. This can happen because decoherence effects related to N1-interactions can project the asymmetry generated in N2,3 decays onto a flavour direction that remains protected against N1 washouts [67, 129–131]. Let us illustrate this with an example. Let us assume that a sizable asymmetry is generated in N2 decays, while N1 leptogenesis is inefficient and fails, that is:
(4.8)m~2≫m*,m~1≫m*.
Assuming also a strong hierarchy and that leptogenesis occurs thermally guarantees that [131]
(4.9)nN1(T~M2)≈0,nN2(T~M1)≈0.
Thus, the dynamics of N2 and N1 are decoupled: there are neither N1-related washout effects during N2 leptogenesis nor N2-related washout effects during N1 leptogenesis. The N2 decays into a combination of lepton doublets that we denote by ℓ2:
(4.10)|ℓ2〉=(Y†Y)22-1/2∑αYα2|ℓα〉.
The second condition in (4.8) implies that already at T≳M1 the N1-Yukawa interactions are sufficiently fast to quickly destroy the coherence of ℓ2. Then a statistical mixture of ℓ1 and of the state orthogonal to ℓ1 builds up, and it can be described by a suitable diagonal density matrix. Let us consider the simple case where both N2 and N1 decay at T≳1012 GeV, so that flavour effects are irrelevant. A convenient choice for an orthogonal basis for the lepton doublets is (ℓ1,ℓ0,ℓ0′) where, without loss of generality, ℓ0′ satisfies 〈ℓ0′∣ℓ2〉=0. Then the asymmetry ΔYℓ2 produced in N2 decays decomposes into two components:
(4.11)ΔYℓ0=c2ΔYℓ2,ΔYℓ1=s2ΔYℓ2,
where c2≡|〈ℓ0∣ℓ2〉|2 and s2=1-c2. The crucial point here is that we expect, in general, c2≠0 and, since 〈ℓ0∣ℓ1〉=0, ΔYℓ0 is protected against N1 washout. Consequently, a finite part of the asymmetry ΔYℓ2 from N2 decays survives through N1 leptogenesis. A more detailed analysis [131] finds that ΔYℓ1 is not entirely washed out, resulting in the final lepton asymmetry YΔL=(3/2)ΔYℓ0=(3/2)c2ΔYℓ2.For109GeV ≲M1≲1012GeV, flavour issues modify the quantitative details, but the qualitative picture, and in particular the survival of a finite part of ΔYℓ2, still holds. In contrast, for M1≲109GeV, the full flavour basis (ℓe,ℓμ,ℓτ) is resolved, and thus there are no directions in flavour space where an asymmetry is protected, so that Yℓ2 can be erased entirely. A dedicated study in which the various flavour regimes for N1,2,3 decays are considered can be found in [132].In conclusionN2,3 leptogenesis cannot be ignored, unless one of the following conditions holds.(1)
The reheat temperature is belowM2.(2)
The asymmetries and/or the washout factors vanish,ϵN2η2≈0 and ϵN3η3≈0.(3)
N1-related washout is still significant at T≲109GeV.


### 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, 133–139] 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)neutrinosN1 and N2 are almost degenerate in mass, and the CP asymmetry from the decay of the first RH neutrino N1 is resonantly enhanced if the mass difference ΔM=(M2-M1) is of the order of the decay rate of the second RH neutrino ΓN2. The typical timescale to build up coherently the CP asymmetry is of the order of 1/ΔM, which can be larger than the timescale ~1/ΓN1 for the change of the abundance of the N1'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 fromt0 to +∞ and back to t0. 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(4.12)Z[J,J†]=Tr[𝒯(exp[i∫C(J(x)ϕ(x)+J†(x)ϕ†(x))d4x])ρ]=Tr[𝒯+(exp[i∫(J+(x)ϕ+(x)+J+†(x)ϕ+†(x))d4x])×𝒯-(exp[-i∫(J-(x)ϕ-(x)+J-†(x)ϕ-†(x))d4x])ρ],
where C in the integral denotes that the time integration contour runs from t0 to plus infinity and then back to t0 again. The symbol ρ 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 (𝒯+) on the positive branch, and the antitime-ordering (𝒯-) on the negative branch. As with the Euclidean-time formulation, scalar (fermionic) fields ϕ are still periodic (antiperiodic) in time, but with ϕ(t,x→)=ϕ(t-iβ,x→), β=1/T. The temperature T 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 fieldϕ they are defined as
(4.13)Gϕ>(x,y)=-Gϕ-+(x,y)=-i〈ϕ(x)ϕ†(y)〉,Gϕ<(x,y)=-Gϕ+-(x,y)=-i〈ϕ†(y)ϕ(x)〉,Gϕt(x,y)=Gϕ++(x,y)=θ(x,y)Gϕ>(x,y)+θ(y,x)Gϕ<(x,y),Gϕt¯(x,y)=Gϕ--(x,y)=θ(y,x)Gϕ>(x,y)+θ(x,y)Gϕ<(x,y),Gϕr(x,y)=Gϕt-Gϕ<=Gϕ>-Gϕt¯,Gϕa(x,y)=Gϕt-Gϕ>=Gϕ<-Gϕt¯,
where the last two Green's functions are the retarded and advanced Green's functions, respectively, and θ(x,y)≡θ(tx-ty) is the step function.For a generic fermion fieldψ the six different propagators are analogously defined as
(4.14)Gψ>(x,y)=-Gψ-+(x,y)=-i〈ψ(x)ψ¯(y)〉,Gψ<(x,y)=-Gψ+-(x,y)=+i〈ψ¯(y)ψ(x)〉,Gψt(x,y)=Gψ++(x,y)=θ(x,y)Gψ>(x,y)+θ(y,x)Gψ<(x,y),Gψt¯(x,y)=Gψ--(x,y)=θ(y,x)Gψ>(x,y)+θ(x,y)Gψ<(x,y),Gψr(x,y)=Gψt-Gψ<=Gψ>-Gψt¯,Gψa(x,y)=Gψt-Gψ>=Gψ<-Gψt¯.
From the definitions of the Green's functions, one can see that the hermiticity properties
(4.15)(iγ0Gψ(x,y))†=iγ0Gψ(y,x),(iGϕ(x,y))†=iGϕ(y,x)
are satisfied. For interacting systems, whether in equilibrium or not, one must define and calculate self-energy functions. Again, there are six of them: Σt,  Σt¯,  Σ<,  Σ>,  Σr,  and Σa. The same relationships exist among them as for the Green's functions in (4.13) and (4.14), such as
(4.16)Σr=Σt-Σ<=Σ>-Σt¯,Σa=Σt-Σ>=Σ<-Σt¯.
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:
(4.17)G~=(Gt±G<G>-Gt¯),Σ~=(Σt±Σ<Σ>-Σt¯),
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, Dyson’s equation is most easily expressed by using the matrix notation:
(4.18)G~(x,y)=G~0(x,y)+∫d4z1∫d4z2G~0(x,z1)Σ~(z1,z2)G~(z2,y),
where the superscript “0” on the Green’s functions means to use those for noninteracting system. It is useful to notice that Dyson’s equation can be written in an alternative form, instead of (4.18), with G~0 on the right in the interaction terms:
(4.19)G~(x,y)=G~0(x,y)+∫d4z3∫d4z4G~(x,z3)Σ~(z3,z4)G~0(z4,y).
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 fermionψ (and similarly for scalars) one may adopt the real-time propagator in the form  Gψt(k,tx-ty) in terms of the spectral function ρψ(k,k0)(4.20)Gψt(k,tx-ty)=∫-∞+∞dk02πe-ik0(tx-ty)ρψ(k,k0)×{[1-fψ(k0)]θ(tx-ty)-fψ(k0)θ(ty-tx)},
where fψ(k0) 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 Γψ(k). For a fermion with chiral mass mψ, one may safely choose
(4.21)ρψ(k,k0)=i(k+mψ)[1(k0+iϵ+iΓψ)2-ωψ2(k)-1(k0-iϵ-iΓψ)2-ωψ2(k)],
where ωψ2(k)=k2+Mψ2(T) and Mψ(T) is the effective thermal mass of the fermion in the plasma (not a chiral mass). Performing the integration over k0 and picking up the poles of the spectral function (which is valid for quasiparticles in equilibrium or very close to equilibrium), one gets
(4.22)Gψ>(k,tx-ty)=-i2ωψ{(k+mψ)[1-fψ(ωψ-iΓψ)]e-i(ωψ-iΓψ)(tx-ty)+γ0(k-mψ)γ0f¯ψ(ωψ+iΓψ)ei(ωψ+iΓψ)(tx-ty)},Gψ<(k,ty-tx)=i2ωψ{(k+mψ)fψ(ωψ+iΓψ)e-i(ωψ-iΓψ)(tx-ty)+γ0(k-mψ)γ0[1-f¯ψ(ωψ-iΓψ)]ei(ωψ+iΓψ)(tx-ty)},
where k0=ωψ  and  fψ,f¯ψ denote the distribution function of the fermion particles and antiparticles, respectively. The expressions (4.22) are valid for tx-ty>0.The above definitions hold for the lepton doublets (after inserting the chiral LH projectorPL), 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 C 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 neutrinoN1. To find its quantum BE we start from (4.18) for the Green's function  GN1< of the RH neutrino N1:(4.23)(i∂→x-M1)GN1<(x,y)=-∫d4z[-ΣN1t(x,z)GN1<(z,y)+ΣN1<(x,z)GN1t¯(z,y)]=∫d3z∫0tdtz[ΣN1>(x,z)GN1<(z,y)-ΣN1<(x,z)GN1>(z,y)].
Adopting the corresponding form for the RH neutrino propagator and the center-of-mass coordinates
(4.24)X≡(t,X→)≡12(x+y),(τ,r→)≡x-y,
one ends up with the following equation:
(4.25)∂fN1(k,t)∂t=-2∫0tdtz∫d3p(2π)312ωℓ(p)12ωH(k-p)1ωN1(k)|ℳ(N1→ℓH)|2×[fN1(k,t)(1-fℓ(p,t))(1+fH(k-p,t))-fℓ(p,t)fH(k-p,t)(1-fN1(k,t))]×cos[(ωN1(k)-ωℓ(p)-ωH(k-p))(t-tz)]≃-2∫0tdtz∫d3p(2π)312ωℓ(p)12ωH(k-p)1ωN1(k)|ℳ(N1→ℓH)|2×(fN1(k,t)-fℓeq(p)fHeq(k-p))×cos[(ωN1(k)-ωℓ(p)-ωH(k-p))(t-tz)].
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, leading—at large times—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, fℓfH≃fℓeqfHeq. Elastic scatterings are typically fast enough to keep kinetic equilibrium. For any distribution function we may write f=(n/neq)feq, where n denotes the total number density. Therefore, (4.25) can be rewritten as
(4.26)∂nN1∂t=-〈ΓN1(t)〉nN1+〈Γ~N1(t)〉nN1eq,〈ΓN1(t)〉=∫0tdtz∫d3k(2π)3fN1eqnN1eqΓN1(t),ΓN1(t)=2∫d3p(2π)3|ℳ(N1→ℓH)|22ωℓ2ωHωN1cos[(ωN1-ωℓ-ωH)(t-tz)],〈Γ~N1(t)〉=∫0tdtz∫d3k(2π)3fN1eqnN1eqΓ~N1(t),Γ~N1(t)=2∫d3p(2π)3fℓeqfHeqfN1eq|ℳ(N1→ℓH)|22ωℓ2ωHωN1cos[(ωN1-ωℓ-ωH)(t-tz)],
where 〈ΓN1(t)〉 is the time-dependent thermal average of the Lorentz-dilated decay width. Integrating over large times, t→∞, thereby replacing the cosines by energy-conserving delta functions:
(4.27)∫0∞dtzcos[(ωN1-ωℓ-ωH)(t-tz)]=πδ(ωN1-ωℓ-ωH),
we find that the two averaged rates 〈ΓN1〉 and 〈Γ~N1〉 coincide and we recover the usual classical BE for the RH distribution function
(4.28)∂nN1∂t=-〈ΓN1〉(nN1-nN1eq),〈ΓN1〉=∫d3k(2π)3fN1eqnN1eq∫d3p(2π)3|ℳ(N1→ℓH)|22ωℓ2ωHωN1(2π)δ(ωN1-ωℓ-ωH).
Taking the time interval to infinity, namely, implementing Fermi's golden rule, results in neglecting memory effects, which in turn results only in on-shell processes contributing to the rate equation. The main difference between the classical and the quantum BE can be traced to memory effects and to the fact that the time evolution of the distribution function is nonMarkovian. The memory of the past time evolution translates into off-shell processes.Similarly, one can show that the equation obeyed by the asymmetry reads(4.29)∂nΔLα(X)∂t=-∫d3z∫0tdtzTr[Σℓα>(X,z)Gℓα<(z,X)-Gℓα>(X,z)Σℓα<(z,X)+Gℓα<(X,z)Σℓα>(z,X)-Σℓα<(X,z)Gℓα>(z,X)].
Proceeding as for the RH neutrino equation one finds (including the moment only for the 1-loop wave contribution to the CP asymmetry ϵwα)
(4.30)∂nΔLα∂t=ϵwα(t)〈ΓN1〉(nN1-nN1eq),ϵwα(t)=-4〈ΓN1〉∑β=13Im(Y1αY1βYβ2†Yα2†)×∫0tdtz∫0tzdt2∫0t2dt1e-ΓN2(tz-t2)e-(Γℓβ+ΓH)(t2-t1)∫d3k(2π)3fN1eqnN1eq×∫d3p(2π)31-fℓβeq(p)+fHeq(k-p)2ωℓβ(p)2ωH(k-p)ωN1(k)∫d3q(2π)31-fℓαeq(q)+fHeq(k-q)2ω¯ℓα(q)2ω¯H(k-q)ωN2(k)×sin(ωN1(t-t1)+(ωℓβ+ωH)(t1-t2)+ωN2(t2-tz)+(ω¯ℓα+ω¯H)(tz-t))×Tr(M1PLpM2q),
where, to avoid double counting, we have not inserted the decay rates in the propagators of the initial and final states and, for simplicity, we have assumed that the damping rates of the lepton doublets and the Higgs field are constant in time. This should be a good approximation as the damping rates are to be computed for momenta of order of the mass of the RH neutrinos. As expected from first principles, we find that the CP asymmetry is a function of time and its value at a given instant depends upon the previous history of the system.Performing the time integrals and retaining only those pieces which eventually give rise to energy-conserving delta functions in the Markovian limit, we obtain(4.31)ϵwα(t)=-4〈ΓN1〉∑β=13Im(Y1αY1βYβ2†Yα2†)×∫0tdtzcos[(ωN1-ω¯ℓα-ω¯H)(t-tz)](ΓN22+(ωN2-ωN1)2)((Γℓβ+ΓH)2+(ωN1-ωℓβ-ωH)2)×∫d3k(2π)3fN1eqnN1eq(Γℓβ+ΓH)×(2(ωN2-ωN1)sin2[(ωN2-ωN1)tz2]-ΓN2sin[(ωN2-ωN1)tz])×∫d3p(2π)31-fℓβeq(p)+fHeq(k-p)2ωℓβ(p)2ωH(k-p)ωN1(k)×∫d3q(2π)31-fℓαeq(q)+fHeq(k-q)2ω¯ℓα(q)2ω¯H(k-q)ωN2(k)Tr(M1PLpM2q).
From this expression it is already manifest that the typical timescale for the building up of the coherent CP asymmetry depends crucially on the difference in energy of the two RH neutrinos.If we now let the upper limit of the time integral take large values, we neglect the memory effects; the CP asymmetry picks contribution only from the on-shell processes. Taking the damping rates of the lepton doublets equal for all the flavours and the RH neutrinos nearly at rest with respect to the thermal bath, the CP asymmetry reads (now summing over all flavour indices)(4.32)ϵw(t)≃-Im(YY†)122(YY†)11(YY†)22M1M2ΓN21(ΔM)2+ΓN22×(2ΔMsin2[ΔMt2]-ΓN2sin[ΔMt]),
where ΔM=(M2-M1). The CP asymmetry (4.32) is resonantly enhanced when ΔM≃ΓN2, and at the resonance point it is given by
(4.33)ϵw(t)≃-12Im(YY†)122(YY†)11(YY†)22(1-sin[ΔMt]-cos[ΔMt]).
The timescale for the building up of the CP asymmetry is ~1/ΔM. The CP asymmetry grows starting from a vanishing value, and, for t≫(ΔM)-1, it averages to the constant standard value. This is true if the timescale for the other processes relevant for leptogenesis is larger than ~1/ΔM. In other words, one may define an “average” CP asymmetry
(4.34)〈ϵw〉=1τp∫t-τptdt′ϵN1W(t′),
where τp represents the typical timescale of the other processes relevant for leptogenesis, for example, the ΔL=1 scatterings. If τp≫1/ΔM~ΓN2-1, the oscillating functions in (4.33) are averaged to zero and the average CP asymmetry is given by the value used in the literature. However, the expression (4.32) should be used when τp≲1/ΔM~ΓN2-1.The fact that the CP asymmetry is a function of time is particularly relevant in the case in which the asymmetry is generated by the decays of two heavy states which are nearly degenerate in mass and oscillate into one another with a timescale, given by the inverse of the mass difference and has been studied in [86, 141]. From (4.32) it is manifest that the CP asymmetry itself oscillates with the very same timescale and such a dependence may or may not be neglected depending upon the rates of the other processes in the plasma. If ΓN1≳ΓN2, the time dependence of the CP asymmetry may not be neglected. The expression (4.32) can also be used, once it is divided by a factor 2 (because in the wave diagram also the charged states of Higgs and lepton doublets may propagate) and the limit M2≫M1 is taken, for the CP asymmetry contribution from the vertex diagram:
(4.35)ϵv(t)≃-Im(YY†)12216π(YY†)11M1M2(2sin2[M2t2]-ΓN2M2sin[M2t]).
In this case, the timescale for this CP asymmetry is ~M2 and much larger than any other timescale in the dynamics. Therefore, one can safely average over many oscillations, getting the expression present in the literature.What has been discussed here provides only one example for which a quantum Boltzmann approach is needed. In general, the lesson is that quantum BEs are relevant when the typical timescale in a quantum physical process, such as the timescale for the unflavour-to-flavour transition or the timescale to build up coherently the CP asymmetry (of the order of1/ΔM), is larger than the timescale for the change of the abundances.


## 4.1. Spectator Processes

Reactions that without involving violation ofB-L can still affect the final amount of baryon asymmetry are classified as “spectator processes” [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 YΔH [106]. This was omitted in Section 2 but in fact has to be added to the density asymmetry of the leptons YΔℓ in weighting, for example, washouts from inverse decays. Equation (2.14) would then become
(4.1)dYΔLdz=ϵ1D1(YN1-YN1eq)-2(YΔℓ+YΔH)W1,
where the factor of two in front of the washout term counts the leptons and Higgs gauge multiplicity. Clearly, in some regimes in which YΔℓ and YΔH 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 temperatureT, particle reactions must be treated in a different way depending on if their characteristic time scale τ (given by inverse of their thermally averaged rates) is [89, 107] (1)
much shorter than the age of the Universe:τ≪tU(T);(2)
much larger than the age of the Universe:τ≫tU(T);(3)
comparable with the Universe age:τ~tU(T).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,∑IμI=∑FμF, where μI denotes the chemical potential of an initial state particle and μF 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 T 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 (2) 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 (3) 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,T≳1013GeV). 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 B-L-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, B is conserved too (and vanishing, if we assume that there is no preexisting asymmetry). B=0 then translates in the condition:
(4.2)2YΔQ3+YΔt=0,
where YΔQ3 is the density asymmetries of one degree of freedom of the top SU(2)L doublet and color triplet which, being gauge interactions in equilibrium, is the same for all the six gauge components and YΔt is the density asymmetry of the SU(2)L singlet top. Hypercharge is always conserved, yielding
(4.3)YΔQ3+2YΔt-YΔℓ+YΔH=0.
Finally, in terms of density asymmetries chemical equilibrium for the top-Yukawa-related reactions μQ3+μH=μt translates into
(4.4)YΔQ3+12YΔH=YΔt.
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 YΔH=(2/3)YΔℓ. Moreover, given that only the LH lepton degrees of freedom are populated, we have YΔL=2YΔℓ so that the coefficient weighting W1 in (4.1) becomes 2(YΔℓ+YΔH)=(5/3)YΔL 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 belowT~1012GeV, 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 B-L, which is then violated only by slow reactions of type (3), and we should then write down a BE for this quantity. Better said, since at T≲1012GeV all the third generation Yukawa reactions, including the ones of the τ-lepton, are in equilibrium, the dynamical regime is that of two flavours in which the relevant quasiconserved charges are Δτ=B/3-Lτ and Δeμ=B/3-Leμ. The fact that only two charges are relevant is because there is always a direction in e-μ  space which remains decoupled from N1. The corresponding third charge Δeμ′ 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
(4.5)-dYΔαdz=ϵαD1(YN1-YN1eq)-2(YΔℓα+YΔH)W1(α=τ,eμ).
To rewrite these equations in a solvable closed form, YΔℓτ, YΔℓeμ, and YΔH must be expressed in terms of the two charge densities YΔτ and YΔeμ. 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] (1) QCD sphaleron equilibrium; (2) EW sphaleron equilibrium; (3)b-quark and τ-lepton Yukawa equilibrium. The “rotation” from the particle density asymmetries YΔℓα  and  YΔH to the charge densities YΔα can be expressed in terms of the A matrix introduced in [67] YΔℓα=AαβℓYΔβ(α,β=τ,eμ) and C-vector YΔH=CαHYΔα introduced in [69]. For the present case, with the ordering (eμ,τ) they are [69]
(4.6)Aℓ=1460(196-24-9156),CH=1230(41,56).
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 YΔα. This is because each time a conservation law has to be dropped (like B 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 theN1 population, because decay and inverse decay rates are suppressed by a time dilation factor ∝M1/T. The N1=N-1 particles can be produced by scatterings with the top quark in s-channel H-exchange qtc→Nℓα and q-t-c→Nℓ-α, by t-channel H-exchange in qℓ-α→Nt-c, q-ℓα→Ntc and by u-channel H-exchange in ℓαt-c→Nq, tcℓ-α→Nq-; see the diagrams (a) in Figure 2. Several scattering channels with gauge bosons also contribute to the production of N1; the corresponding diagrams are (b) and (c) in the same figure.Diagrams for various2↔2 scattering processes: (a) scatterings with the top quarks; (b) and (c) scatterings with the gauge bosons (A=B, Wi  with i=1,2,3).
(a)
(b)
(c)When the effects of scatterings in populating theN1 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 (ht2λ4, g2λ4, where g is a gauge coupling) with an on-shell intermediate state N1 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 Nj. 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, M2/M1≫10. Corrections up to several tens of percent can appear around temperatures of order T~M2/10 and can be numerically relevant in case of milder hierarchies.Regarding the scattering processes with gauge bosons such asNℓα→AH-NH→Aℓ-α or NA→ℓαH, 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 (T>M1) 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 Λ~2πT [49]. The value Λ>T 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 m(μ)=r·m, where 1.2≲r≲1.3 for 108GeV ≲μ≲1016GeV [49], and therefore can be accounted for by increasing the values of the neutrino mass parameters (e.g., m~) as measured at low energy by ≈20%-30% (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, 116–120]. 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 mH(T)≳mq3,t(T)≫mℓ(T). 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 (mH(T)≈0.4T), decays and inverse decays become kinematically forbidden in the temperature range in which mH(T)-mℓ(T)<MN1<mH(T)+mℓ(T). For lower temperatures, the usual processes N1↔ℓH can occur. For higher temperatures, the Higgs is heavy enough that it can decay: H↔ℓN1. A rough estimate of the kinematically forbidden region yields 2≲T/M1≲5. The important point is that these corrections are effective only at T>M1. In the parameter region m~>10-3eV, that is favoured by the measurements of the neutrino mass-squared differences, the N1 number density and its L-violating reactions attain thermal equilibrium at T≈M1 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)
ΔL = 1 scatterings with top quark: a comparison between the corrected and uncorrected rates of the top-quark scattering γHstop≡γ(q3t-↔ℓN1) with the Higgs exchanged in the s-channel and of the sum of the t- and u-channel scatterings γHt+utop≡γ(q3N1↔ℓt)+γ(t-N1↔ℓq-3) shows that the only corrections appearing at low temperatures, and thus more relevant, are for γHt+utop (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 ~ln(MN1/mH) from the quasimassless Higgs exchanged in the  t- and u-channels, which disappears when the Higgs thermal mass mH(T)~T~MN1 is included.(iii)
ΔL = 1 scatterings with the gauge bosons: here the inclusion of thermal masses is required to avoid IR divergences that would arise when massless ℓ (and H) states are exchanged in the t- and u-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 whichN1 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 ϵ→ϵ(T) 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 decayingN’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 N2,3 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, M2,3≫M1, the related effects are Boltzmann suppressed by exp(-M2,3/T) that at T~M1 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 2×2 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 -iSℓ0(p,mℓ)=(p-mℓ+i0+)-1 and -iDH0(p,mH)=(p2-mH2+i0+)-1 acquire an additive term that is proportional to the particle density distribution nℓ,H=[exp(Eℓ,H/T)±1]-1:
(4.7)δSℓ(T)=-2πnℓ(p-mℓ)δ(p2-mℓ2),δDH(T)=+2πnHδ(p2-mH2).
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 “particles” and “holes” [49]. The contributions of these two fermionic modes were studied in [124–126] 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 -nℓ+nH-2nℓnH that vanishes when the final states thermal masses are neglected, because the Bose-Einstein and Fermi-Dirac statistical distributions depend on the same argument, Eℓ=EH=M1/2. As a consequence, the thermal corrections to the fermion and boson propagators (nℓ and nH) and the product of the two thermal corrections (nℓnH) 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 ℓ~,H~ 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 -nℓ+nH, 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 particleN1 is moving with respect to the background (with velocity β→) 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 𝒪(β). In the total decay rate the 𝒪(β) anisotropy effect is integrated out, and only 𝒪(β2) 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 Eℓ,H=M1/2∓(mH2-mℓ2)/2M1. Since now Eℓ≠EH, the prefactor -nℓ+nH-2nℓnH 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 M1 [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 mH(T)>mℓ(T)+M1, the Higgs can decay, and then there is a new source of lepton number asymmetry associated with H→ℓN1. The CP asymmetry in Higgs decays ϵH can be up to one order of magnitude larger than the CP asymmetry in N1 decays [49]. While this could represent a dramatic enhancement of the CP asymmetry, ϵH is nonvanishing only at temperatures T≳TH~5M1, 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 m~≳(TH/M1)2m*≫m*, and therefore, in the weak washout regime (m~≲m*), 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 atT≳M1 (and quite large at T≫M1 for Higgs decays), in the low temperature regime, where the precise value of ϵ 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.


## 4.3.1. 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 Λ~2πT [49]. The value Λ>T 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 m(μ)=r·m, where 1.2≲r≲1.3 for 108GeV ≲μ≲1016GeV [49], and therefore can be accounted for by increasing the values of the neutrino mass parameters (e.g., m~) as measured at low energy by ≈20%-30% (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, 116–120]. 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 mH(T)≳mq3,t(T)≫mℓ(T). 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 (mH(T)≈0.4T), decays and inverse decays become kinematically forbidden in the temperature range in which mH(T)-mℓ(T)<MN1<mH(T)+mℓ(T). For lower temperatures, the usual processes N1↔ℓH can occur. For higher temperatures, the Higgs is heavy enough that it can decay: H↔ℓN1. A rough estimate of the kinematically forbidden region yields 2≲T/M1≲5. The important point is that these corrections are effective only at T>M1. In the parameter region m~>10-3eV, that is favoured by the measurements of the neutrino mass-squared differences, the N1 number density and its L-violating reactions attain thermal equilibrium at T≈M1 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)
ΔL = 1 scatterings with top quark: a comparison between the corrected and uncorrected rates of the top-quark scattering γHstop≡γ(q3t-↔ℓN1) with the Higgs exchanged in the s-channel and of the sum of the t- and u-channel scatterings γHt+utop≡γ(q3N1↔ℓt)+γ(t-N1↔ℓq-3) shows that the only corrections appearing at low temperatures, and thus more relevant, are for γHt+utop (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 ~ln(MN1/mH) from the quasimassless Higgs exchanged in the  t- and u-channels, which disappears when the Higgs thermal mass mH(T)~T~MN1 is included.(iii)
ΔL = 1 scatterings with the gauge bosons: here the inclusion of thermal masses is required to avoid IR divergences that would arise when massless ℓ (and H) states are exchanged in the t- and u-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 whichN1 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 ϵ→ϵ(T) 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 decayingN’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 N2,3 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, M2,3≫M1, the related effects are Boltzmann suppressed by exp(-M2,3/T) that at T~M1 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 2×2 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 -iSℓ0(p,mℓ)=(p-mℓ+i0+)-1 and -iDH0(p,mH)=(p2-mH2+i0+)-1 acquire an additive term that is proportional to the particle density distribution nℓ,H=[exp(Eℓ,H/T)±1]-1:
(4.7)δSℓ(T)=-2πnℓ(p-mℓ)δ(p2-mℓ2),δDH(T)=+2πnHδ(p2-mH2).
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 “particles” and “holes” [49]. The contributions of these two fermionic modes were studied in [124–126] 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 -nℓ+nH-2nℓnH that vanishes when the final states thermal masses are neglected, because the Bose-Einstein and Fermi-Dirac statistical distributions depend on the same argument, Eℓ=EH=M1/2. As a consequence, the thermal corrections to the fermion and boson propagators (nℓ and nH) and the product of the two thermal corrections (nℓnH) 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 ℓ~,H~ 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 -nℓ+nH, 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 particleN1 is moving with respect to the background (with velocity β→) 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 𝒪(β). In the total decay rate the 𝒪(β) anisotropy effect is integrated out, and only 𝒪(β2) 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 Eℓ,H=M1/2∓(mH2-mℓ2)/2M1. Since now Eℓ≠EH, the prefactor -nℓ+nH-2nℓnH 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 M1 [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 mH(T)>mℓ(T)+M1, the Higgs can decay, and then there is a new source of lepton number asymmetry associated with H→ℓN1. The CP asymmetry in Higgs decays ϵH can be up to one order of magnitude larger than the CP asymmetry in N1 decays [49]. While this could represent a dramatic enhancement of the CP asymmetry, ϵH is nonvanishing only at temperatures T≳TH~5M1, 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 m~≳(TH/M1)2m*≫m*, and therefore, in the weak washout regime (m~≲m*), 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 atT≳M1 (and quite large at T≫M1 for Higgs decays), in the low temperature regime, where the precise value of ϵ 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 ofN2,3 are often neglected, which in many cases is not a good approximation. This is obvious, for example, when N1 dynamics is irrelevant for leptogenesis: ϵ1≪10-6 cannot provide enough CP asymmetry to account for baryogenesis, and m~1≪m* implies that N1 washout effects are negligible. It is then clear that any asymmetry generated in N2,3 decays can survive and becomes crucial for the success of leptogenesis. Another case in which it is intuitively clear that N2,3 effects can be important is when the RH neutrino spectrum is compact, which means that M2,3 have values within a factor of a few from M1. Then N1 and N2,3 contributions to leptogenesis can be equally important and must be summed up. A model with compact RH neutrino spectrum in which N2,3 dynamics is of crucial importance was recently discussed in [128].It is less obvious thatN2,3 effects can also be important for a hierarchical RH spectrum and when N1 is strongly coupled. This can happen because decoherence effects related to N1-interactions can project the asymmetry generated in N2,3 decays onto a flavour direction that remains protected against N1 washouts [67, 129–131]. Let us illustrate this with an example. Let us assume that a sizable asymmetry is generated in N2 decays, while N1 leptogenesis is inefficient and fails, that is:
(4.8)m~2≫m*,m~1≫m*.
Assuming also a strong hierarchy and that leptogenesis occurs thermally guarantees that [131]
(4.9)nN1(T~M2)≈0,nN2(T~M1)≈0.
Thus, the dynamics of N2 and N1 are decoupled: there are neither N1-related washout effects during N2 leptogenesis nor N2-related washout effects during N1 leptogenesis. The N2 decays into a combination of lepton doublets that we denote by ℓ2:
(4.10)|ℓ2〉=(Y†Y)22-1/2∑αYα2|ℓα〉.
The second condition in (4.8) implies that already at T≳M1 the N1-Yukawa interactions are sufficiently fast to quickly destroy the coherence of ℓ2. Then a statistical mixture of ℓ1 and of the state orthogonal to ℓ1 builds up, and it can be described by a suitable diagonal density matrix. Let us consider the simple case where both N2 and N1 decay at T≳1012 GeV, so that flavour effects are irrelevant. A convenient choice for an orthogonal basis for the lepton doublets is (ℓ1,ℓ0,ℓ0′) where, without loss of generality, ℓ0′ satisfies 〈ℓ0′∣ℓ2〉=0. Then the asymmetry ΔYℓ2 produced in N2 decays decomposes into two components:
(4.11)ΔYℓ0=c2ΔYℓ2,ΔYℓ1=s2ΔYℓ2,
where c2≡|〈ℓ0∣ℓ2〉|2 and s2=1-c2. The crucial point here is that we expect, in general, c2≠0 and, since 〈ℓ0∣ℓ1〉=0, ΔYℓ0 is protected against N1 washout. Consequently, a finite part of the asymmetry ΔYℓ2 from N2 decays survives through N1 leptogenesis. A more detailed analysis [131] finds that ΔYℓ1 is not entirely washed out, resulting in the final lepton asymmetry YΔL=(3/2)ΔYℓ0=(3/2)c2ΔYℓ2.For109GeV ≲M1≲1012GeV, flavour issues modify the quantitative details, but the qualitative picture, and in particular the survival of a finite part of ΔYℓ2, still holds. In contrast, for M1≲109GeV, the full flavour basis (ℓe,ℓμ,ℓτ) is resolved, and thus there are no directions in flavour space where an asymmetry is protected, so that Yℓ2 can be erased entirely. A dedicated study in which the various flavour regimes for N1,2,3 decays are considered can be found in [132].In conclusionN2,3 leptogenesis cannot be ignored, unless one of the following conditions holds.(1)
The reheat temperature is belowM2.(2)
The asymmetries and/or the washout factors vanish,ϵN2η2≈0 and ϵN3η3≈0.(3)
N1-related washout is still significant at T≲109GeV.


## 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, 133–139] 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)neutrinosN1 and N2 are almost degenerate in mass, and the CP asymmetry from the decay of the first RH neutrino N1 is resonantly enhanced if the mass difference ΔM=(M2-M1) is of the order of the decay rate of the second RH neutrino ΓN2. The typical timescale to build up coherently the CP asymmetry is of the order of 1/ΔM, which can be larger than the timescale ~1/ΓN1 for the change of the abundance of the N1'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 fromt0 to +∞ and back to t0. 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(4.12)Z[J,J†]=Tr[𝒯(exp[i∫C(J(x)ϕ(x)+J†(x)ϕ†(x))d4x])ρ]=Tr[𝒯+(exp[i∫(J+(x)ϕ+(x)+J+†(x)ϕ+†(x))d4x])×𝒯-(exp[-i∫(J-(x)ϕ-(x)+J-†(x)ϕ-†(x))d4x])ρ],
where C in the integral denotes that the time integration contour runs from t0 to plus infinity and then back to t0 again. The symbol ρ 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 (𝒯+) on the positive branch, and the antitime-ordering (𝒯-) on the negative branch. As with the Euclidean-time formulation, scalar (fermionic) fields ϕ are still periodic (antiperiodic) in time, but with ϕ(t,x→)=ϕ(t-iβ,x→), β=1/T. The temperature T 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 fieldϕ they are defined as
(4.13)Gϕ>(x,y)=-Gϕ-+(x,y)=-i〈ϕ(x)ϕ†(y)〉,Gϕ<(x,y)=-Gϕ+-(x,y)=-i〈ϕ†(y)ϕ(x)〉,Gϕt(x,y)=Gϕ++(x,y)=θ(x,y)Gϕ>(x,y)+θ(y,x)Gϕ<(x,y),Gϕt¯(x,y)=Gϕ--(x,y)=θ(y,x)Gϕ>(x,y)+θ(x,y)Gϕ<(x,y),Gϕr(x,y)=Gϕt-Gϕ<=Gϕ>-Gϕt¯,Gϕa(x,y)=Gϕt-Gϕ>=Gϕ<-Gϕt¯,
where the last two Green's functions are the retarded and advanced Green's functions, respectively, and θ(x,y)≡θ(tx-ty) is the step function.For a generic fermion fieldψ the six different propagators are analogously defined as
(4.14)Gψ>(x,y)=-Gψ-+(x,y)=-i〈ψ(x)ψ¯(y)〉,Gψ<(x,y)=-Gψ+-(x,y)=+i〈ψ¯(y)ψ(x)〉,Gψt(x,y)=Gψ++(x,y)=θ(x,y)Gψ>(x,y)+θ(y,x)Gψ<(x,y),Gψt¯(x,y)=Gψ--(x,y)=θ(y,x)Gψ>(x,y)+θ(x,y)Gψ<(x,y),Gψr(x,y)=Gψt-Gψ<=Gψ>-Gψt¯,Gψa(x,y)=Gψt-Gψ>=Gψ<-Gψt¯.
From the definitions of the Green's functions, one can see that the hermiticity properties
(4.15)(iγ0Gψ(x,y))†=iγ0Gψ(y,x),(iGϕ(x,y))†=iGϕ(y,x)
are satisfied. For interacting systems, whether in equilibrium or not, one must define and calculate self-energy functions. Again, there are six of them: Σt,  Σt¯,  Σ<,  Σ>,  Σr,  and Σa. The same relationships exist among them as for the Green's functions in (4.13) and (4.14), such as
(4.16)Σr=Σt-Σ<=Σ>-Σt¯,Σa=Σt-Σ>=Σ<-Σt¯.
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:
(4.17)G~=(Gt±G<G>-Gt¯),Σ~=(Σt±Σ<Σ>-Σt¯),
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, Dyson’s equation is most easily expressed by using the matrix notation:
(4.18)G~(x,y)=G~0(x,y)+∫d4z1∫d4z2G~0(x,z1)Σ~(z1,z2)G~(z2,y),
where the superscript “0” on the Green’s functions means to use those for noninteracting system. It is useful to notice that Dyson’s equation can be written in an alternative form, instead of (4.18), with G~0 on the right in the interaction terms:
(4.19)G~(x,y)=G~0(x,y)+∫d4z3∫d4z4G~(x,z3)Σ~(z3,z4)G~0(z4,y).
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 fermionψ (and similarly for scalars) one may adopt the real-time propagator in the form  Gψt(k,tx-ty) in terms of the spectral function ρψ(k,k0)(4.20)Gψt(k,tx-ty)=∫-∞+∞dk02πe-ik0(tx-ty)ρψ(k,k0)×{[1-fψ(k0)]θ(tx-ty)-fψ(k0)θ(ty-tx)},
where fψ(k0) 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 Γψ(k). For a fermion with chiral mass mψ, one may safely choose
(4.21)ρψ(k,k0)=i(k+mψ)[1(k0+iϵ+iΓψ)2-ωψ2(k)-1(k0-iϵ-iΓψ)2-ωψ2(k)],
where ωψ2(k)=k2+Mψ2(T) and Mψ(T) is the effective thermal mass of the fermion in the plasma (not a chiral mass). Performing the integration over k0 and picking up the poles of the spectral function (which is valid for quasiparticles in equilibrium or very close to equilibrium), one gets
(4.22)Gψ>(k,tx-ty)=-i2ωψ{(k+mψ)[1-fψ(ωψ-iΓψ)]e-i(ωψ-iΓψ)(tx-ty)+γ0(k-mψ)γ0f¯ψ(ωψ+iΓψ)ei(ωψ+iΓψ)(tx-ty)},Gψ<(k,ty-tx)=i2ωψ{(k+mψ)fψ(ωψ+iΓψ)e-i(ωψ-iΓψ)(tx-ty)+γ0(k-mψ)γ0[1-f¯ψ(ωψ-iΓψ)]ei(ωψ+iΓψ)(tx-ty)},
where k0=ωψ  and  fψ,f¯ψ denote the distribution function of the fermion particles and antiparticles, respectively. The expressions (4.22) are valid for tx-ty>0.The above definitions hold for the lepton doublets (after inserting the chiral LH projectorPL), 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 C 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 neutrinoN1. To find its quantum BE we start from (4.18) for the Green's function  GN1< of the RH neutrino N1:(4.23)(i∂→x-M1)GN1<(x,y)=-∫d4z[-ΣN1t(x,z)GN1<(z,y)+ΣN1<(x,z)GN1t¯(z,y)]=∫d3z∫0tdtz[ΣN1>(x,z)GN1<(z,y)-ΣN1<(x,z)GN1>(z,y)].
Adopting the corresponding form for the RH neutrino propagator and the center-of-mass coordinates
(4.24)X≡(t,X→)≡12(x+y),(τ,r→)≡x-y,
one ends up with the following equation:
(4.25)∂fN1(k,t)∂t=-2∫0tdtz∫d3p(2π)312ωℓ(p)12ωH(k-p)1ωN1(k)|ℳ(N1→ℓH)|2×[fN1(k,t)(1-fℓ(p,t))(1+fH(k-p,t))-fℓ(p,t)fH(k-p,t)(1-fN1(k,t))]×cos[(ωN1(k)-ωℓ(p)-ωH(k-p))(t-tz)]≃-2∫0tdtz∫d3p(2π)312ωℓ(p)12ωH(k-p)1ωN1(k)|ℳ(N1→ℓH)|2×(fN1(k,t)-fℓeq(p)fHeq(k-p))×cos[(ωN1(k)-ωℓ(p)-ωH(k-p))(t-tz)].
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, leading—at large times—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, fℓfH≃fℓeqfHeq. Elastic scatterings are typically fast enough to keep kinetic equilibrium. For any distribution function we may write f=(n/neq)feq, where n denotes the total number density. Therefore, (4.25) can be rewritten as
(4.26)∂nN1∂t=-〈ΓN1(t)〉nN1+〈Γ~N1(t)〉nN1eq,〈ΓN1(t)〉=∫0tdtz∫d3k(2π)3fN1eqnN1eqΓN1(t),ΓN1(t)=2∫d3p(2π)3|ℳ(N1→ℓH)|22ωℓ2ωHωN1cos[(ωN1-ωℓ-ωH)(t-tz)],〈Γ~N1(t)〉=∫0tdtz∫d3k(2π)3fN1eqnN1eqΓ~N1(t),Γ~N1(t)=2∫d3p(2π)3fℓeqfHeqfN1eq|ℳ(N1→ℓH)|22ωℓ2ωHωN1cos[(ωN1-ωℓ-ωH)(t-tz)],
where 〈ΓN1(t)〉 is the time-dependent thermal average of the Lorentz-dilated decay width. Integrating over large times, t→∞, thereby replacing the cosines by energy-conserving delta functions:
(4.27)∫0∞dtzcos[(ωN1-ωℓ-ωH)(t-tz)]=πδ(ωN1-ωℓ-ωH),
we find that the two averaged rates 〈ΓN1〉 and 〈Γ~N1〉 coincide and we recover the usual classical BE for the RH distribution function
(4.28)∂nN1∂t=-〈ΓN1〉(nN1-nN1eq),〈ΓN1〉=∫d3k(2π)3fN1eqnN1eq∫d3p(2π)3|ℳ(N1→ℓH)|22ωℓ2ωHωN1(2π)δ(ωN1-ωℓ-ωH).
Taking the time interval to infinity, namely, implementing Fermi's golden rule, results in neglecting memory effects, which in turn results only in on-shell processes contributing to the rate equation. The main difference between the classical and the quantum BE can be traced to memory effects and to the fact that the time evolution of the distribution function is nonMarkovian. The memory of the past time evolution translates into off-shell processes.Similarly, one can show that the equation obeyed by the asymmetry reads(4.29)∂nΔLα(X)∂t=-∫d3z∫0tdtzTr[Σℓα>(X,z)Gℓα<(z,X)-Gℓα>(X,z)Σℓα<(z,X)+Gℓα<(X,z)Σℓα>(z,X)-Σℓα<(X,z)Gℓα>(z,X)].
Proceeding as for the RH neutrino equation one finds (including the moment only for the 1-loop wave contribution to the CP asymmetry ϵwα)
(4.30)∂nΔLα∂t=ϵwα(t)〈ΓN1〉(nN1-nN1eq),ϵwα(t)=-4〈ΓN1〉∑β=13Im(Y1αY1βYβ2†Yα2†)×∫0tdtz∫0tzdt2∫0t2dt1e-ΓN2(tz-t2)e-(Γℓβ+ΓH)(t2-t1)∫d3k(2π)3fN1eqnN1eq×∫d3p(2π)31-fℓβeq(p)+fHeq(k-p)2ωℓβ(p)2ωH(k-p)ωN1(k)∫d3q(2π)31-fℓαeq(q)+fHeq(k-q)2ω¯ℓα(q)2ω¯H(k-q)ωN2(k)×sin(ωN1(t-t1)+(ωℓβ+ωH)(t1-t2)+ωN2(t2-tz)+(ω¯ℓα+ω¯H)(tz-t))×Tr(M1PLpM2q),
where, to avoid double counting, we have not inserted the decay rates in the propagators of the initial and final states and, for simplicity, we have assumed that the damping rates of the lepton doublets and the Higgs field are constant in time. This should be a good approximation as the damping rates are to be computed for momenta of order of the mass of the RH neutrinos. As expected from first principles, we find that the CP asymmetry is a function of time and its value at a given instant depends upon the previous history of the system.Performing the time integrals and retaining only those pieces which eventually give rise to energy-conserving delta functions in the Markovian limit, we obtain(4.31)ϵwα(t)=-4〈ΓN1〉∑β=13Im(Y1αY1βYβ2†Yα2†)×∫0tdtzcos[(ωN1-ω¯ℓα-ω¯H)(t-tz)](ΓN22+(ωN2-ωN1)2)((Γℓβ+ΓH)2+(ωN1-ωℓβ-ωH)2)×∫d3k(2π)3fN1eqnN1eq(Γℓβ+ΓH)×(2(ωN2-ωN1)sin2[(ωN2-ωN1)tz2]-ΓN2sin[(ωN2-ωN1)tz])×∫d3p(2π)31-fℓβeq(p)+fHeq(k-p)2ωℓβ(p)2ωH(k-p)ωN1(k)×∫d3q(2π)31-fℓαeq(q)+fHeq(k-q)2ω¯ℓα(q)2ω¯H(k-q)ωN2(k)Tr(M1PLpM2q).
From this expression it is already manifest that the typical timescale for the building up of the coherent CP asymmetry depends crucially on the difference in energy of the two RH neutrinos.If we now let the upper limit of the time integral take large values, we neglect the memory effects; the CP asymmetry picks contribution only from the on-shell processes. Taking the damping rates of the lepton doublets equal for all the flavours and the RH neutrinos nearly at rest with respect to the thermal bath, the CP asymmetry reads (now summing over all flavour indices)(4.32)ϵw(t)≃-Im(YY†)122(YY†)11(YY†)22M1M2ΓN21(ΔM)2+ΓN22×(2ΔMsin2[ΔMt2]-ΓN2sin[ΔMt]),
where ΔM=(M2-M1). The CP asymmetry (4.32) is resonantly enhanced when ΔM≃ΓN2, and at the resonance point it is given by
(4.33)ϵw(t)≃-12Im(YY†)122(YY†)11(YY†)22(1-sin[ΔMt]-cos[ΔMt]).
The timescale for the building up of the CP asymmetry is ~1/ΔM. The CP asymmetry grows starting from a vanishing value, and, for t≫(ΔM)-1, it averages to the constant standard value. This is true if the timescale for the other processes relevant for leptogenesis is larger than ~1/ΔM. In other words, one may define an “average” CP asymmetry
(4.34)〈ϵw〉=1τp∫t-τptdt′ϵN1W(t′),
where τp represents the typical timescale of the other processes relevant for leptogenesis, for example, the ΔL=1 scatterings. If τp≫1/ΔM~ΓN2-1, the oscillating functions in (4.33) are averaged to zero and the average CP asymmetry is given by the value used in the literature. However, the expression (4.32) should be used when τp≲1/ΔM~ΓN2-1.The fact that the CP asymmetry is a function of time is particularly relevant in the case in which the asymmetry is generated by the decays of two heavy states which are nearly degenerate in mass and oscillate into one another with a timescale, given by the inverse of the mass difference and has been studied in [86, 141]. From (4.32) it is manifest that the CP asymmetry itself oscillates with the very same timescale and such a dependence may or may not be neglected depending upon the rates of the other processes in the plasma. If ΓN1≳ΓN2, the time dependence of the CP asymmetry may not be neglected. The expression (4.32) can also be used, once it is divided by a factor 2 (because in the wave diagram also the charged states of Higgs and lepton doublets may propagate) and the limit M2≫M1 is taken, for the CP asymmetry contribution from the vertex diagram:
(4.35)ϵv(t)≃-Im(YY†)12216π(YY†)11M1M2(2sin2[M2t2]-ΓN2M2sin[M2t]).
In this case, the timescale for this CP asymmetry is ~M2 and much larger than any other timescale in the dynamics. Therefore, one can safely average over many oscillations, getting the expression present in the literature.What has been discussed here provides only one example for which a quantum Boltzmann approach is needed. In general, the lesson is that quantum BEs are relevant when the typical timescale in a quantum physical process, such as the timescale for the unflavour-to-flavour transition or the timescale to build up coherently the CP asymmetry (of the order of1/ΔM), is larger than the timescale for the change of the abundances.


## 5. Supersymmetric Leptogenesis

### 5.1. What’s New?

Supersymmetric leptogenesis constitutes a theoretically appealing generalization of leptogenesis for the following reason: while the SM equipped with the seesaw provides the simplest way to realize leptogenesis, such a framework is plagued by an unpleasant fine-tuning problem. For a nondegenerate spectrum of heavy Majorana neutrinos, successful leptogenesis requires generically a scale for the singlet neutrino masses that is much larger than the EW scale [61] but at the quantum level the gap between these two scales becomes unstable. Low-energy supersymmetry (SUSY) can naturally stabilize the required hierarchy, and this provides a sound motivation for studying leptogenesis in the framework of the supersymmetrized version of the seesaw mechanism. Supersymmetric leptogenesis, however, introduces a certain conflict between the gravitino bound on the reheat temperature and the thermal production of the heavy singlets neutrinos [142–145]. In this section, we will not be concerned with the gravitino problem, nor with its possible ways out but focus on the new features SUSY brings in for leptogenesis.The supersymmetric type-I seesaw model is described by the superpotential of the minimal supersymmetric SM (MSSM) with the additional terms:(5.1)W=12MpqNpcNqc+λαpNpcℓαHu,
where p,q=1,2,… label the heavy singlet states in order of increasing mass and α=e,μ,τ labels the lepton flavour. In (5.1) ℓ, Hu, and Nc are, respectively, the chiral superfields for the lepton and the up-type Higgs SU(2)L doublets and for the heavy SU(2)L singlet neutrinos defined according to usual conventions in terms of their LH Weyl spinor components (e.g., the Nc supermultiplet has scalar component N~* and fermion component NLc). Finally the SU(2)L index contraction is defined as ℓαHu=ϵρσℓαρHuσ with ϵ12=+1.Originally, the issue of MSSM leptogenesis was approached in conjunction with SM leptogenesis [49, 51] as well as in dedicated studies [146, 147]. However in this first analysis, several features that are specific of SUSY in the high temperature regime relevant for leptogenesis, in which soft SUSY breaking parameters can be effectively set to zero, had been overlooked. In that case, in spite of the large amount of new reactions, the differences between SM and MSSM leptogenesis can be resumed by means of simple counting of a few numerical factors [109, 130, 148], like, for example, the number of relativistic degrees of freedom in the thermal bath, the number of loop diagrams contributing to the CP asymmetries, and the multiplicities of the final states in the decays of the heavy neutrinos and sneutrinos, and one can estimate [109]
(5.2)YΔB(∞)MSSMYΔB(∞)SM≈{2(strongwashout),22(weakwashout).
Recently it was pointed out in [89] that in fact MSSM leptogenesis is rich in new and nontrivial features, and genuinely different from the simpler realization within the SM. The two important effects are as follows.(a)
If the SUSY breaking scale does not exceed by much 1 TeV, aboveT~5×107GeV the particle and superparticle density asymmetries do not equilibrate [149], and it is mandatory to account in the BE for the differences in the number density asymmetries of the boson and fermion degrees of freedom. This can be given in terms of a nonvanishing gaugino chemical potential.(b)
When soft SUSY breaking parameters are neglected, additional anomalous global symmetries that involve bothSU(2)L and SU(3)C fermion representations emerge [150]. As a consequence, the EW and QCD sphaleron equilibrium conditions are modified with respect to the SM, and this yields a different pattern of sphaleron-induced lepton-flavour mixing [67, 69, 70]. In addition, a new anomaly-free exactly conserved ℛ-symmetry provides an additional constraint that is not present in the SM, and a careful counting reveals that four independent quantities, rather than the three of the SM case, are required to give a complete description of the various particle asymmetries in the thermal bath, with the additional quantity corresponding to the number density asymmetry of the heavy scalar neutrinos.Although the modifications above are interesting from the theoretical point of view, a quantitative comparison with the results obtained when the new effects are ignored shows that the corrections remain below𝒪(1) [89]. (This modification would be important for certain supersymmetric leptogenesis scenarios which contain new sources of CP violation e.g. soft leptogenesis  (see Section 6.2).)  Finally, it should be pointed out that in the supersymmetric case, the temperature in which the lepton flavour effects (see Section 3) come into play is enhanced by a factor of (1+tan2β) since the charged Yukawa couplings are given by  hα=mα/(vucosβ).The purpose of the following sections is twofold. We describe the chemical equilibrium conditions and conservation laws for MSSM in conjunction to SM. In Section5.2 we list the constraints that hold independently of assuming a regime in which particle and sparticle chemical potentials equilibrate (superequilibration (SE) regime) or do not equilibrate (nonsuperequilibration (NSE) regime). In Section 5.3 we list the constraints that hold only in the SE regime and in Section 5.4 the ones that hold in NSE regime. The question of NSE is irrelevant in the SM since there are no superparticles. Hence the parts relevant for the SM are Sections 5.2 and 5.3, with the chemical potential of the gaugino set to zero, the chemical potential of the down-type higgsino replaced by the minus of the uptype higgsino (see (5.11) and (5.12)), and all the quantities related to superparticles replaced by the ones for particles.


### 5.2. General Constraints

We first list in items(1), (2), and (3) the conditions that hold in the whole temperature range MW=T≲1014GeV. Conversely, some of the Yukawa coupling conditions given in items (4) and (5) will have to be dropped as the temperature is increased and the corresponding reactions go out of equilibrium. First we will relate the number density asymmetry of a particle Δn≡n-n- for which a chemical potential can be defined to its chemical potential. For both bosons (b) and fermions (f) this relation acquires a particularly simple form in the relativistic limit mb,f≪T, and at first order in the chemical potential  μb,f/T≪1:
(5.3)Δnb=gb3T2μb,Δnf=gf6T2μf.
For simplicity of notations, in the following we denote the chemical potentials with the same notation that labels the corresponding field: ϕ≡μϕ.(1)
At scales much higher thanMW, gauge fields have vanishing chemical potential W=B=g=0 [57]. This also implies that all the particles belonging to the same SU(2)L or SU(3)C multiplets have the same chemical potential. For example ϕ(I3=+(1/2))=ϕ(I3=-(1/2)) for a field ϕ that is a doublet of weak isospin I→, and similarly for color.(2)
Denoting byW~R, B~R, and g~R the RH winos, binos, and gluinos chemical potentials and by ℓ,Q (ℓ~,Q~) the chemical potentials of the (s)lepton and (s)quarks LH doublets, the following reactions: Q~+g~R→Q, Q~+W~R→Q, ℓ~+W~R→ℓ, ℓ~+B~R→ℓ, imply that all gauginos have the same chemical potential: -g~=Q-Q~=-W~=ℓ-ℓ~=-B~, where W~, B~, and g~ denote the chemical potentials of LH gauginos. It follows that the chemical potentials of the SM particles are related to those of their respective superpartners as
(5.4)Q~,ℓ~=Q,ℓ+g~,Hu,d=H~u,d+g~,u~,d~,e~=u,d,e-g~.
The last relation, in which  u,d,e≡uR,dR,eR denote the RH SU(2)L singlets, follows, for example, from u~Lc=uLc+g~ for the corresponding LH fields, together with  uLc=-uR, and from the analogous relation for the SU(2)L singlet squarks.(3)
Before EW symmetry breaking hypercharge is an exactly conserved quantity, and we can assume a vanishing total hypercharge for the Universe:(5.5)∑i(YΔQi+2YΔui-YΔdi)-∑α(YΔℓα+YΔeα)+YΔH~u-YΔH~d=0.(4)
When the reactions mediated by the lepton Yukawa couplings are faster than the Universe expansion rate (see Section3.1 for the temperature regime when lepton Yukawa interactions are in thermal equilibrium), the following chemical equilibrium conditions are enforced:
(5.6)ℓα-eα+H~d+g~=0,(α=e,μ,τ).
If the temperature is not too low lepton flavour equilibration (see Section 3.4) induced by off-diagonal slepton soft masses will not occur. We assume that this is the case, and thus we take the three ℓα to be independent quantities.(5)
Reactions mediated by the quark Yukawa couplings enforce the following six chemical equilibrium conditions:(5.7)Qi-ui+H~u+g~=0,(ui=u,c,t),(5.8)Qi-di+H~d+g~=0,(di=d,s,b).The upquark Yukawa coupling maintains chemical equilibrium between the LH and RH uptype quarks up to T~2·106GeV. Note that when the Yukawa reactions of at least two families of quarks are in equilibrium, the mass basis is fixed for all the quarks and squarks. Intergeneration mixing then implies that family-changing charged-current transitions are also in equilibrium: bL→cL and tL→sL imply Q2=Q3; sL→uL and cL→dL imply Q1=Q2. Thus, up to temperatures T≲1011GeV, that are of the order of the equilibration temperature for the charm Yukawa coupling, the three quark doublets have the same chemical potential:
(5.9)Q≡Q3=Q2=Q1.
At higher temperatures, when only the third family is in equilibrium, we have instead Q≡Q3=Q2≠Q1. Above T~1013 when (for moderate values of tanβ) also b-quark (as well as the τ-lepton) SU(2)L singlets decouple from their Yukawa reactions, all intergeneration mixing becomes negligible and Q3≠Q2≠Q1.


### 5.3. Superequilibration Regime

At relatively low temperatures, additional conditions from reactions in chemical equilibrium hold. Since the constraints below apply only in the SE regime, we number them including this label.6SE:
Equilibration of the particle-sparticle chemical potentialsμϕ=μϕ~ [149] is ensured when reactions like ℓ~ℓ~→ℓℓ are faster than the Universe expansion rate. These reactions are induced by gaugino interactions, but since they require a gaugino chirality flip, they turn out to be proportional to its mass mg~ and can be neglected in the limit mg~→0.Furthermore, since theμ parameter of the HuHd superpotential term is expected to be of the order of the soft gaugino masses, it is reasonable to consider in the same temperature range also the effect of the higgsino mixing term, which implies that the sum of the up- and downhiggsino chemical potentials vanishes. The rates of the corresponding reactions, given approximately by Γg~~mg~2/T and Γμ~μ2/T, are faster than the Universe expansion rate up to temperatures
(5.10)T≲5·107(mg~,μ500GeV)2/3GeV.The corresponding chemical equilibrium relations enforce the conditions:(5.11)g~=0,(5.12)H~u+H~d=0.
7
SE
:
Up to temperatures given by (5.10) the MSSM has the same global anomalies than the SM, that are the EW SU(2)L-U(1)B+L mixed anomaly and the QCD chiral anomaly. They generate the effective operators OEW=Πα(QQQℓα) [151] and OQCD=Πi(QQuLicdLic) [151–153]. Above the EW phase transition reactions induced by these operators are in thermal equilibrium, and the corresponding conditions read
(5.13)9Q+∑αℓα=0,(5.14)6Q-∑i(ui+di)=0,
where we have used the same chemical potential for the three quark doublets (5.9), which is always appropriate in the SE regime below the limit (5.10).Equations (5.6), (5.7)–(5.9), together with the SE conditions (5.11)-(5.12), the two anomaly conditions (5.13)-(5.14), and the hypercharge neutrality condition equation (5.5), give 11+2+2+1=16 constraints for the 18 chemical potentials. Note however that there is one redundant constraint, that we take to be the QCD sphaleron condition, since by summing up (5.7) and (5.8) and taking into account (5.9), (5.11), and (5.12), we obtain precisely (5.14). Therefore, like in the SM, we have three independent chemical potentials. We can define three linear combinations of the chemical potentials corresponding to the SU(2)L anomaly free flavour charges Δα≡B/3-Lα that being anomaly-free and perturbatively conserved by the low energy MSSM Lagrangian evolve slowly because the corresponding symmetries are violated only by the heavy Majorana neutrino dynamics. Their evolution needs to be computed by means of three independent BEs. In terms of the abundances (2.12) the density of the Δα charges normalized to the entropy density can be written as
(5.15)YΔα=3[13∑i(2YΔQi+YΔui+YΔdi)-(2YΔℓα+YΔeα)-23YΔg~].
The expression above is completely general and holds in all temperature regimes, including the NSE regime (see Section 5.4).The density asymmetries of the doublet leptons and higgsinos, that weight the washout terms in the BE, can now be expressed in terms of the anomaly-free charges by means of theA matrix and C vectors introduced, respectively, in [67, 69] that are defined as:
(5.16)YΔℓα=AαβℓYΔβ,YΔH~u,d=CαH~u,dYΔα.
Here and in the following we will give results for the A and C matrices for the fermion states. We recall that in the SE regime the density asymmetry of a scalar boson that is in chemical equilibrium with its fermionic partner is given simply by YΔb=2YΔf with the factor of 2 from statistics.


#### 5.3.1. First Generation Yukawa Reactions out of Equilibrium (SE Regime)

As an example let us now consider the temperaturesT≳4·106(1+tan2β)GeV, when the d-quark Yukawa coupling can be set to zero (in order to remain within the SE regime we assume tanβ~1). In this case the equilibrium dynamics is symmetric under the exchange u↔d (both chemical potentials enter only the QCD sphaleron condition equation (5.14) with equal weights) and so must be any physical solution of the set of constraints. Thus, the first condition in (5.8) can be replaced by the condition d=u, and again three independent quantities suffice to determine all the particle density asymmetries. The corresponding result is
(5.17)Aℓ=13×2148(-90612012075-688287528-688),CH~u=-CH~d=-12148(37,52,52).
Note that since in this regime the chemical potentials for the scalar and fermion degrees of freedom of each chiral multiplet equilibrate, the analogous results for YΔℓα+YΔℓ~α can be obtained by simply multiplying the A matrix in (5.17) by a factor of 3. This gives the same A matrix obtained in the nonsupersymmetric case in the same regime (see, e.g., (4.12) in [69]). The C matrix (multiplied by the same factor of 3) differs from the nonsupersymmetric result by a factor 1/2. This is because after substituting H~d=-H~u (see (5.12)) all the chemical potential conditions are formally the same than in the SM with H~u identified with the chemical potential of the scalar Higgs, but since C expresses the result for number densities, in the SM, a factor of 2 from boson statistics appears for the SM Higgs. This agrees with the analysis in [58] and is a general result that holds for SUSY within the SE regime.


### 5.4. Nonsuperequilibration Regime

At temperatures above the limit given in (5.10) the Universe expansion is fast enough that reactions induced by mg~ and μ do not occur. Setting to zero in the high temperature effective theory these two parameters has the following consequences  (i)
Condition equation (5.11) has to be dropped, and gauginos acquire a nonvanishing chemical potential g~≠0 (corresponding to the difference between the number of LH and RH helicity states). The chemical potentials of the members of the same matter supermultiplets are no more equal (nonsuperequilibration) but related as in (5.4).(ii)
Condition (5.12) also has to be dropped, and the chemical potentials of the up- and downtype Higgs and higgsinos do not necessarily sum up to zero.(iii)
The MSSM gains two new global symmetries:mg~→0 yields a global R-symmetry, while μ→0 corresponds to a global symmetry of the Peccei-Quinn (PQ) type.


#### 5.4.1. Anomalous and Nonanomalous Symmetries

Two linear combinationsR2 and R3 of R and PQ, having, respectively, only SU(2)L and SU(3)C mixed anomalies, have been identified in [150]:
(5.18)R2=R-2PQ,R3=R-3PQ.
The authors of [150] have also constructed the effective multifermions operators generated by the mixed anomalies:
(5.19)O~EW=Πα(QQQℓα)H~uH~dW~4,(5.20)O~QCD=Πi(QQucdc)ig~6.
Given that three global symmetries B, L, and R2 have mixed SU(2)L anomalies (but are free of SU(3)C anomalies) we can construct two anomaly-free combinations, the first one being B-L which is only violated perturbatively by NcℓHu and the second anomaly free combination which is also an exact symmetry of the MSSM+seesaw in the NSE regime [89]:
(5.21)ℛ=53B-L+R2,
and is exactly conserved. In the SU(3)C sector, besides the chiral anomaly we now have also R3 mixed anomalies. Thus also in this case anomaly-free combinations can be constructed, and in particular we can define one combination for each quark superfield. Assigning to the LH supermultiplets chiral charge χ=-1 these combinations have the form [89]:
(5.22)χqL+κqLR3,
where, for example, κuLc=κdLc=1/3 and κQL=2/3. Note that since R3 is perturbatively conserved by the complete MSSM+seesaw Lagrangian, when the Yukawa coupling of one quark is set to zero, the corresponding charge (5.22) will be exactly conserved.


#### 5.4.2. Constraints in the Nonsuperequilibration Regime

In the NSE regime, the conditions listed in items6SE and 7SE of the previous section have to be dropped, but new conditions arise.6NSE:
The conservation law for theℛ charge yields the following global neutrality condition:
(5.23)ℛtot=∑fΔnfℛf+∑bΔnbℛb+ΔnN~1ℛN~1=T26(∑i(2Qi-5ui+4di)+2∑α(ℓα+eα)+5H~d-H~u+31g~)-ΔnN~1=0.The last terms in both lines of (5.23) correspond to the contribution to ℛ-neutrality from the lightest sneutrino asymmetry ΔnN~1=nN~1-nN~1* with charge ℛN~1=-ℛNc=-1. Note that since in general N~1 is not in chemical equilibrium, no chemical potential can be associated to it, and hence this constraint needs to be formulated in terms of its number density asymmetry that has to be evaluated by solving a BE for YΔN~≡YN~1-YN~1* (see Section 5.5).7NSE:
The operators in (5.19) induce transitions that in the NSE regime are in chemical equilibrium. This enforces the generalized EW and QCD sphaleron equilibrium conditions [150]:
(5.24)3∑iQi+∑αℓα+H~u+H~d+4g~=0,2∑iQi-∑i(ui+di)+6g~=0,that replace (5.13) and (5.14).8NSE:
The chiral-R3 charges in (5.22) are anomaly-free, but clearly they are not conserved by the quark Yukawa interactions. However, when a quark supermultiplet decouples from its Yukawa interactions, an exact conservation law arises. (Note that hu,d→0 implies u and d decoupling, but Q1 decoupling is ensured only if also hc,s→0.)  The conservation laws corresponding to these symmetries read
(5.25)T26[3qR+6(qR-g~)]+13R3tot=0,(5.26)T262[3QL+6(QL+g~)]-23R3tot=0and hold forqR=ui,di and QL=Qi in the regimes when the appropriate Yukawa reactions are negligible. Note the factor of 2 for the QL chiral charge in front of the first square bracket in (5.26) that is due to SU(2)L gauge multiplicity. In terms of chemical potentials and ΔnN~1, the total R3 charge in (5.25) and (5.26) reads
(5.27)R3tot=T26(82g~-3∑i(2Qi+11ui-4di)+∑α(16ℓα+13eα)+16H~d-14H~u)+ΔnN~1R3N~1,whereR3N~1=-1. As regards the leptons, since they do not couple to the QCD anomaly, by setting he→0, a symmetry under chiral supermultiplet rotations is directly gained for the RH leptons implying Δne+Δn~e=0 and giving the condition:
(5.28)e-23g~=0.No analogous condition arises for the lepton doublets relevant for leptogenesis, since by assumption they remain coupled via Yukawa couplings to the heavyN’s.In the NSE regime there are different flavour mixing matrices for the scalar and fermion components of the leptons and Higgs supermultiplets. To express more concisely all the results, it is convenient to introduce a newC vector to describe the gaugino number density asymmetry per degree of freedom in terms of the relevant charges:
(5.29)YΔg~=Cag~YΔa,withΔa=(Δα,ΔN~).


#### 5.4.3. First Generation Yukawa Reactions out of Equilibrium (NSE Regime)

As an example, in the temperature range between108 and 1011GeV, and for moderate values of tanβ, all the first generation Yukawa couplings can be set to zero. Using for u,d conditions equation (5.25) and for e condition equation (5.28) as implied by hu,d,he→0 we obtain
(5.30)Aℓ=19×162332(-1981173398733987-825326634-14757114761-80552663414761-147571-8055),Cg~=-11162332(163,165,165,-255),CH~u=-1162332(3918,4713,4713,95),CH~d=13×162332(5413,9712,9712,-252),
where the rows correspond to (YΔe,YΔμ,YΔτ,YΔN~). For completeness, in (5.30) we have also given the results for CH~d even if only the uptype Higgs density asymmetry is relevant for the leptogenesis processes. Note that neglecting the contribution of ΔnN~1 to the global charges ℛtot in (5.23) and R3tot in (5.27) corresponds precisely to setting to zero the fourth column in all the previous matrices. Then, analogously with the SE and SM cases, within this “3-column approximation” all particle density asymmetries can be expressed just in terms of the three YΔα charge densities.


### 5.5. Supersymmetric Boltzmann Equations

In order to illustrate how the new effects described above modify the structure of the BE, here we write down a simpler expressions in which only decays and inverse decays are included: (the complete set of BE including decays, inverse decays, and scatterings with top-quark is given in the Appendix of  [89].)
(5.31)sHzdYN1dz=-(YN1YN1eq-1)γN1,(5.32)sHzdYN~+dz=-(YN~+YN~1eq-2)γN~1,(5.33)sHzdYΔN~dz=-YΔN~YN~1eqγN~1-32γN~1∑aCag~YΔaYℓeq+⋯,(5.34)sHzdYΔαdz=-ϵα[(YN1YN1eq-1)γN1+(YN~+YN~1eq-2)γN~1]+P1α0(γN1+12γN~1)∑a(Aαaℓ+CaH~u+Cag~)YΔaYℓeq,
where γN~1 is the corresponding thermally averaged decay rate for RH sneutrino N~1. In (5.32) we have introduced the overall sneutrino abundance YN~+=YN~1+YN~1*, while YΔN~≡YN~1-YN~1* in (5.33) is the sneutrino density asymmetry that was already introduced in Section 5.4. In the washout terms we have normalized the charge densities YΔa=(YΔα,YΔN~) to the equilibrium density of a fermion with one degree of freedom Yℓeq. In (5.31) and (5.34) we have also neglected for simplicity all finite temperature effects. Taking these effects into account would imply, for example, that the CP asymmetry for N~ decays into fermions is different from the one for decays into scalars [49], while we describe both CP asymmetries with ϵα. A few remarks regarding (5.33) are in order. In the SE regime g~=0, and thus it would seem that the sneutrino density asymmetry YΔN~ vanishes. However, this only happens for decays and inverse decays, and it is no more true when additional terms related to scattering processes, that are represented in the equation by the dots, are also included (see [147] and the Appendix of [89]). Therefore, also in the SE regime YN~1 and YN~1* in general differ. However, in this case recasting their equations in terms of two equations for YN~+ and YΔN~ is just a convenient parametrization. On the contrary, in the NSE regime this is mandatory, because the sneutrinos carry a globally conserved ℛ-charge and YΔN~ is required to formulate properly the corresponding conservation law. As we have seen, this eventually results in YΔN~ contributing to the expressions of the lepton flavour density asymmetries in terms of slowly varying quantities.In [89] a complete numerical analysis was carried out, and it was shown that numerical corrections with respect to the case when NSE effects are neglected remain at the 𝒪(1) level. This is because only spectator processes get affected, while the overall amount of CP violation driving leptogenesis remains the same than in previous treatments.


## 5.1. What’s New?

Supersymmetric leptogenesis constitutes a theoretically appealing generalization of leptogenesis for the following reason: while the SM equipped with the seesaw provides the simplest way to realize leptogenesis, such a framework is plagued by an unpleasant fine-tuning problem. For a nondegenerate spectrum of heavy Majorana neutrinos, successful leptogenesis requires generically a scale for the singlet neutrino masses that is much larger than the EW scale [61] but at the quantum level the gap between these two scales becomes unstable. Low-energy supersymmetry (SUSY) can naturally stabilize the required hierarchy, and this provides a sound motivation for studying leptogenesis in the framework of the supersymmetrized version of the seesaw mechanism. Supersymmetric leptogenesis, however, introduces a certain conflict between the gravitino bound on the reheat temperature and the thermal production of the heavy singlets neutrinos [142–145]. In this section, we will not be concerned with the gravitino problem, nor with its possible ways out but focus on the new features SUSY brings in for leptogenesis.The supersymmetric type-I seesaw model is described by the superpotential of the minimal supersymmetric SM (MSSM) with the additional terms:(5.1)W=12MpqNpcNqc+λαpNpcℓαHu,
where p,q=1,2,… label the heavy singlet states in order of increasing mass and α=e,μ,τ labels the lepton flavour. In (5.1) ℓ, Hu, and Nc are, respectively, the chiral superfields for the lepton and the up-type Higgs SU(2)L doublets and for the heavy SU(2)L singlet neutrinos defined according to usual conventions in terms of their LH Weyl spinor components (e.g., the Nc supermultiplet has scalar component N~* and fermion component NLc). Finally the SU(2)L index contraction is defined as ℓαHu=ϵρσℓαρHuσ with ϵ12=+1.Originally, the issue of MSSM leptogenesis was approached in conjunction with SM leptogenesis [49, 51] as well as in dedicated studies [146, 147]. However in this first analysis, several features that are specific of SUSY in the high temperature regime relevant for leptogenesis, in which soft SUSY breaking parameters can be effectively set to zero, had been overlooked. In that case, in spite of the large amount of new reactions, the differences between SM and MSSM leptogenesis can be resumed by means of simple counting of a few numerical factors [109, 130, 148], like, for example, the number of relativistic degrees of freedom in the thermal bath, the number of loop diagrams contributing to the CP asymmetries, and the multiplicities of the final states in the decays of the heavy neutrinos and sneutrinos, and one can estimate [109]
(5.2)YΔB(∞)MSSMYΔB(∞)SM≈{2(strongwashout),22(weakwashout).
Recently it was pointed out in [89] that in fact MSSM leptogenesis is rich in new and nontrivial features, and genuinely different from the simpler realization within the SM. The two important effects are as follows.(a)
If the SUSY breaking scale does not exceed by much 1 TeV, aboveT~5×107GeV the particle and superparticle density asymmetries do not equilibrate [149], and it is mandatory to account in the BE for the differences in the number density asymmetries of the boson and fermion degrees of freedom. This can be given in terms of a nonvanishing gaugino chemical potential.(b)
When soft SUSY breaking parameters are neglected, additional anomalous global symmetries that involve bothSU(2)L and SU(3)C fermion representations emerge [150]. As a consequence, the EW and QCD sphaleron equilibrium conditions are modified with respect to the SM, and this yields a different pattern of sphaleron-induced lepton-flavour mixing [67, 69, 70]. In addition, a new anomaly-free exactly conserved ℛ-symmetry provides an additional constraint that is not present in the SM, and a careful counting reveals that four independent quantities, rather than the three of the SM case, are required to give a complete description of the various particle asymmetries in the thermal bath, with the additional quantity corresponding to the number density asymmetry of the heavy scalar neutrinos.Although the modifications above are interesting from the theoretical point of view, a quantitative comparison with the results obtained when the new effects are ignored shows that the corrections remain below𝒪(1) [89]. (This modification would be important for certain supersymmetric leptogenesis scenarios which contain new sources of CP violation e.g. soft leptogenesis  (see Section 6.2).)  Finally, it should be pointed out that in the supersymmetric case, the temperature in which the lepton flavour effects (see Section 3) come into play is enhanced by a factor of (1+tan2β) since the charged Yukawa couplings are given by  hα=mα/(vucosβ).The purpose of the following sections is twofold. We describe the chemical equilibrium conditions and conservation laws for MSSM in conjunction to SM. In Section5.2 we list the constraints that hold independently of assuming a regime in which particle and sparticle chemical potentials equilibrate (superequilibration (SE) regime) or do not equilibrate (nonsuperequilibration (NSE) regime). In Section 5.3 we list the constraints that hold only in the SE regime and in Section 5.4 the ones that hold in NSE regime. The question of NSE is irrelevant in the SM since there are no superparticles. Hence the parts relevant for the SM are Sections 5.2 and 5.3, with the chemical potential of the gaugino set to zero, the chemical potential of the down-type higgsino replaced by the minus of the uptype higgsino (see (5.11) and (5.12)), and all the quantities related to superparticles replaced by the ones for particles.


## 5.2. General Constraints

We first list in items(1), (2), and (3) the conditions that hold in the whole temperature range MW=T≲1014GeV. Conversely, some of the Yukawa coupling conditions given in items (4) and (5) will have to be dropped as the temperature is increased and the corresponding reactions go out of equilibrium. First we will relate the number density asymmetry of a particle Δn≡n-n- for which a chemical potential can be defined to its chemical potential. For both bosons (b) and fermions (f) this relation acquires a particularly simple form in the relativistic limit mb,f≪T, and at first order in the chemical potential  μb,f/T≪1:
(5.3)Δnb=gb3T2μb,Δnf=gf6T2μf.
For simplicity of notations, in the following we denote the chemical potentials with the same notation that labels the corresponding field: ϕ≡μϕ.(1)
At scales much higher thanMW, gauge fields have vanishing chemical potential W=B=g=0 [57]. This also implies that all the particles belonging to the same SU(2)L or SU(3)C multiplets have the same chemical potential. For example ϕ(I3=+(1/2))=ϕ(I3=-(1/2)) for a field ϕ that is a doublet of weak isospin I→, and similarly for color.(2)
Denoting byW~R, B~R, and g~R the RH winos, binos, and gluinos chemical potentials and by ℓ,Q (ℓ~,Q~) the chemical potentials of the (s)lepton and (s)quarks LH doublets, the following reactions: Q~+g~R→Q, Q~+W~R→Q, ℓ~+W~R→ℓ, ℓ~+B~R→ℓ, imply that all gauginos have the same chemical potential: -g~=Q-Q~=-W~=ℓ-ℓ~=-B~, where W~, B~, and g~ denote the chemical potentials of LH gauginos. It follows that the chemical potentials of the SM particles are related to those of their respective superpartners as
(5.4)Q~,ℓ~=Q,ℓ+g~,Hu,d=H~u,d+g~,u~,d~,e~=u,d,e-g~.
The last relation, in which  u,d,e≡uR,dR,eR denote the RH SU(2)L singlets, follows, for example, from u~Lc=uLc+g~ for the corresponding LH fields, together with  uLc=-uR, and from the analogous relation for the SU(2)L singlet squarks.(3)
Before EW symmetry breaking hypercharge is an exactly conserved quantity, and we can assume a vanishing total hypercharge for the Universe:(5.5)∑i(YΔQi+2YΔui-YΔdi)-∑α(YΔℓα+YΔeα)+YΔH~u-YΔH~d=0.(4)
When the reactions mediated by the lepton Yukawa couplings are faster than the Universe expansion rate (see Section3.1 for the temperature regime when lepton Yukawa interactions are in thermal equilibrium), the following chemical equilibrium conditions are enforced:
(5.6)ℓα-eα+H~d+g~=0,(α=e,μ,τ).
If the temperature is not too low lepton flavour equilibration (see Section 3.4) induced by off-diagonal slepton soft masses will not occur. We assume that this is the case, and thus we take the three ℓα to be independent quantities.(5)
Reactions mediated by the quark Yukawa couplings enforce the following six chemical equilibrium conditions:(5.7)Qi-ui+H~u+g~=0,(ui=u,c,t),(5.8)Qi-di+H~d+g~=0,(di=d,s,b).The upquark Yukawa coupling maintains chemical equilibrium between the LH and RH uptype quarks up to T~2·106GeV. Note that when the Yukawa reactions of at least two families of quarks are in equilibrium, the mass basis is fixed for all the quarks and squarks. Intergeneration mixing then implies that family-changing charged-current transitions are also in equilibrium: bL→cL and tL→sL imply Q2=Q3; sL→uL and cL→dL imply Q1=Q2. Thus, up to temperatures T≲1011GeV, that are of the order of the equilibration temperature for the charm Yukawa coupling, the three quark doublets have the same chemical potential:
(5.9)Q≡Q3=Q2=Q1.
At higher temperatures, when only the third family is in equilibrium, we have instead Q≡Q3=Q2≠Q1. Above T~1013 when (for moderate values of tanβ) also b-quark (as well as the τ-lepton) SU(2)L singlets decouple from their Yukawa reactions, all intergeneration mixing becomes negligible and Q3≠Q2≠Q1.


## 5.3. Superequilibration Regime

At relatively low temperatures, additional conditions from reactions in chemical equilibrium hold. Since the constraints below apply only in the SE regime, we number them including this label.6SE:
Equilibration of the particle-sparticle chemical potentialsμϕ=μϕ~ [149] is ensured when reactions like ℓ~ℓ~→ℓℓ are faster than the Universe expansion rate. These reactions are induced by gaugino interactions, but since they require a gaugino chirality flip, they turn out to be proportional to its mass mg~ and can be neglected in the limit mg~→0.Furthermore, since theμ parameter of the HuHd superpotential term is expected to be of the order of the soft gaugino masses, it is reasonable to consider in the same temperature range also the effect of the higgsino mixing term, which implies that the sum of the up- and downhiggsino chemical potentials vanishes. The rates of the corresponding reactions, given approximately by Γg~~mg~2/T and Γμ~μ2/T, are faster than the Universe expansion rate up to temperatures
(5.10)T≲5·107(mg~,μ500GeV)2/3GeV.The corresponding chemical equilibrium relations enforce the conditions:(5.11)g~=0,(5.12)H~u+H~d=0.
7
SE
:
Up to temperatures given by (5.10) the MSSM has the same global anomalies than the SM, that are the EW SU(2)L-U(1)B+L mixed anomaly and the QCD chiral anomaly. They generate the effective operators OEW=Πα(QQQℓα) [151] and OQCD=Πi(QQuLicdLic) [151–153]. Above the EW phase transition reactions induced by these operators are in thermal equilibrium, and the corresponding conditions read
(5.13)9Q+∑αℓα=0,(5.14)6Q-∑i(ui+di)=0,
where we have used the same chemical potential for the three quark doublets (5.9), which is always appropriate in the SE regime below the limit (5.10).Equations (5.6), (5.7)–(5.9), together with the SE conditions (5.11)-(5.12), the two anomaly conditions (5.13)-(5.14), and the hypercharge neutrality condition equation (5.5), give 11+2+2+1=16 constraints for the 18 chemical potentials. Note however that there is one redundant constraint, that we take to be the QCD sphaleron condition, since by summing up (5.7) and (5.8) and taking into account (5.9), (5.11), and (5.12), we obtain precisely (5.14). Therefore, like in the SM, we have three independent chemical potentials. We can define three linear combinations of the chemical potentials corresponding to the SU(2)L anomaly free flavour charges Δα≡B/3-Lα that being anomaly-free and perturbatively conserved by the low energy MSSM Lagrangian evolve slowly because the corresponding symmetries are violated only by the heavy Majorana neutrino dynamics. Their evolution needs to be computed by means of three independent BEs. In terms of the abundances (2.12) the density of the Δα charges normalized to the entropy density can be written as
(5.15)YΔα=3[13∑i(2YΔQi+YΔui+YΔdi)-(2YΔℓα+YΔeα)-23YΔg~].
The expression above is completely general and holds in all temperature regimes, including the NSE regime (see Section 5.4).The density asymmetries of the doublet leptons and higgsinos, that weight the washout terms in the BE, can now be expressed in terms of the anomaly-free charges by means of theA matrix and C vectors introduced, respectively, in [67, 69] that are defined as:
(5.16)YΔℓα=AαβℓYΔβ,YΔH~u,d=CαH~u,dYΔα.
Here and in the following we will give results for the A and C matrices for the fermion states. We recall that in the SE regime the density asymmetry of a scalar boson that is in chemical equilibrium with its fermionic partner is given simply by YΔb=2YΔf with the factor of 2 from statistics.


### 5.3.1. First Generation Yukawa Reactions out of Equilibrium (SE Regime)

As an example let us now consider the temperaturesT≳4·106(1+tan2β)GeV, when the d-quark Yukawa coupling can be set to zero (in order to remain within the SE regime we assume tanβ~1). In this case the equilibrium dynamics is symmetric under the exchange u↔d (both chemical potentials enter only the QCD sphaleron condition equation (5.14) with equal weights) and so must be any physical solution of the set of constraints. Thus, the first condition in (5.8) can be replaced by the condition d=u, and again three independent quantities suffice to determine all the particle density asymmetries. The corresponding result is
(5.17)Aℓ=13×2148(-90612012075-688287528-688),CH~u=-CH~d=-12148(37,52,52).
Note that since in this regime the chemical potentials for the scalar and fermion degrees of freedom of each chiral multiplet equilibrate, the analogous results for YΔℓα+YΔℓ~α can be obtained by simply multiplying the A matrix in (5.17) by a factor of 3. This gives the same A matrix obtained in the nonsupersymmetric case in the same regime (see, e.g., (4.12) in [69]). The C matrix (multiplied by the same factor of 3) differs from the nonsupersymmetric result by a factor 1/2. This is because after substituting H~d=-H~u (see (5.12)) all the chemical potential conditions are formally the same than in the SM with H~u identified with the chemical potential of the scalar Higgs, but since C expresses the result for number densities, in the SM, a factor of 2 from boson statistics appears for the SM Higgs. This agrees with the analysis in [58] and is a general result that holds for SUSY within the SE regime.


## 5.3.1. First Generation Yukawa Reactions out of Equilibrium (SE Regime)

As an example let us now consider the temperaturesT≳4·106(1+tan2β)GeV, when the d-quark Yukawa coupling can be set to zero (in order to remain within the SE regime we assume tanβ~1). In this case the equilibrium dynamics is symmetric under the exchange u↔d (both chemical potentials enter only the QCD sphaleron condition equation (5.14) with equal weights) and so must be any physical solution of the set of constraints. Thus, the first condition in (5.8) can be replaced by the condition d=u, and again three independent quantities suffice to determine all the particle density asymmetries. The corresponding result is
(5.17)Aℓ=13×2148(-90612012075-688287528-688),CH~u=-CH~d=-12148(37,52,52).
Note that since in this regime the chemical potentials for the scalar and fermion degrees of freedom of each chiral multiplet equilibrate, the analogous results for YΔℓα+YΔℓ~α can be obtained by simply multiplying the A matrix in (5.17) by a factor of 3. This gives the same A matrix obtained in the nonsupersymmetric case in the same regime (see, e.g., (4.12) in [69]). The C matrix (multiplied by the same factor of 3) differs from the nonsupersymmetric result by a factor 1/2. This is because after substituting H~d=-H~u (see (5.12)) all the chemical potential conditions are formally the same than in the SM with H~u identified with the chemical potential of the scalar Higgs, but since C expresses the result for number densities, in the SM, a factor of 2 from boson statistics appears for the SM Higgs. This agrees with the analysis in [58] and is a general result that holds for SUSY within the SE regime.


## 5.4. Nonsuperequilibration Regime

At temperatures above the limit given in (5.10) the Universe expansion is fast enough that reactions induced by mg~ and μ do not occur. Setting to zero in the high temperature effective theory these two parameters has the following consequences  (i)
Condition equation (5.11) has to be dropped, and gauginos acquire a nonvanishing chemical potential g~≠0 (corresponding to the difference between the number of LH and RH helicity states). The chemical potentials of the members of the same matter supermultiplets are no more equal (nonsuperequilibration) but related as in (5.4).(ii)
Condition (5.12) also has to be dropped, and the chemical potentials of the up- and downtype Higgs and higgsinos do not necessarily sum up to zero.(iii)
The MSSM gains two new global symmetries:mg~→0 yields a global R-symmetry, while μ→0 corresponds to a global symmetry of the Peccei-Quinn (PQ) type.


### 5.4.1. Anomalous and Nonanomalous Symmetries

Two linear combinationsR2 and R3 of R and PQ, having, respectively, only SU(2)L and SU(3)C mixed anomalies, have been identified in [150]:
(5.18)R2=R-2PQ,R3=R-3PQ.
The authors of [150] have also constructed the effective multifermions operators generated by the mixed anomalies:
(5.19)O~EW=Πα(QQQℓα)H~uH~dW~4,(5.20)O~QCD=Πi(QQucdc)ig~6.
Given that three global symmetries B, L, and R2 have mixed SU(2)L anomalies (but are free of SU(3)C anomalies) we can construct two anomaly-free combinations, the first one being B-L which is only violated perturbatively by NcℓHu and the second anomaly free combination which is also an exact symmetry of the MSSM+seesaw in the NSE regime [89]:
(5.21)ℛ=53B-L+R2,
and is exactly conserved. In the SU(3)C sector, besides the chiral anomaly we now have also R3 mixed anomalies. Thus also in this case anomaly-free combinations can be constructed, and in particular we can define one combination for each quark superfield. Assigning to the LH supermultiplets chiral charge χ=-1 these combinations have the form [89]:
(5.22)χqL+κqLR3,
where, for example, κuLc=κdLc=1/3 and κQL=2/3. Note that since R3 is perturbatively conserved by the complete MSSM+seesaw Lagrangian, when the Yukawa coupling of one quark is set to zero, the corresponding charge (5.22) will be exactly conserved.


### 5.4.2. Constraints in the Nonsuperequilibration Regime

In the NSE regime, the conditions listed in items6SE and 7SE of the previous section have to be dropped, but new conditions arise.6NSE:
The conservation law for theℛ charge yields the following global neutrality condition:
(5.23)ℛtot=∑fΔnfℛf+∑bΔnbℛb+ΔnN~1ℛN~1=T26(∑i(2Qi-5ui+4di)+2∑α(ℓα+eα)+5H~d-H~u+31g~)-ΔnN~1=0.The last terms in both lines of (5.23) correspond to the contribution to ℛ-neutrality from the lightest sneutrino asymmetry ΔnN~1=nN~1-nN~1* with charge ℛN~1=-ℛNc=-1. Note that since in general N~1 is not in chemical equilibrium, no chemical potential can be associated to it, and hence this constraint needs to be formulated in terms of its number density asymmetry that has to be evaluated by solving a BE for YΔN~≡YN~1-YN~1* (see Section 5.5).7NSE:
The operators in (5.19) induce transitions that in the NSE regime are in chemical equilibrium. This enforces the generalized EW and QCD sphaleron equilibrium conditions [150]:
(5.24)3∑iQi+∑αℓα+H~u+H~d+4g~=0,2∑iQi-∑i(ui+di)+6g~=0,that replace (5.13) and (5.14).8NSE:
The chiral-R3 charges in (5.22) are anomaly-free, but clearly they are not conserved by the quark Yukawa interactions. However, when a quark supermultiplet decouples from its Yukawa interactions, an exact conservation law arises. (Note that hu,d→0 implies u and d decoupling, but Q1 decoupling is ensured only if also hc,s→0.)  The conservation laws corresponding to these symmetries read
(5.25)T26[3qR+6(qR-g~)]+13R3tot=0,(5.26)T262[3QL+6(QL+g~)]-23R3tot=0and hold forqR=ui,di and QL=Qi in the regimes when the appropriate Yukawa reactions are negligible. Note the factor of 2 for the QL chiral charge in front of the first square bracket in (5.26) that is due to SU(2)L gauge multiplicity. In terms of chemical potentials and ΔnN~1, the total R3 charge in (5.25) and (5.26) reads
(5.27)R3tot=T26(82g~-3∑i(2Qi+11ui-4di)+∑α(16ℓα+13eα)+16H~d-14H~u)+ΔnN~1R3N~1,whereR3N~1=-1. As regards the leptons, since they do not couple to the QCD anomaly, by setting he→0, a symmetry under chiral supermultiplet rotations is directly gained for the RH leptons implying Δne+Δn~e=0 and giving the condition:
(5.28)e-23g~=0.No analogous condition arises for the lepton doublets relevant for leptogenesis, since by assumption they remain coupled via Yukawa couplings to the heavyN’s.In the NSE regime there are different flavour mixing matrices for the scalar and fermion components of the leptons and Higgs supermultiplets. To express more concisely all the results, it is convenient to introduce a newC vector to describe the gaugino number density asymmetry per degree of freedom in terms of the relevant charges:
(5.29)YΔg~=Cag~YΔa,withΔa=(Δα,ΔN~).


### 5.4.3. First Generation Yukawa Reactions out of Equilibrium (NSE Regime)

As an example, in the temperature range between108 and 1011GeV, and for moderate values of tanβ, all the first generation Yukawa couplings can be set to zero. Using for u,d conditions equation (5.25) and for e condition equation (5.28) as implied by hu,d,he→0 we obtain
(5.30)Aℓ=19×162332(-1981173398733987-825326634-14757114761-80552663414761-147571-8055),Cg~=-11162332(163,165,165,-255),CH~u=-1162332(3918,4713,4713,95),CH~d=13×162332(5413,9712,9712,-252),
where the rows correspond to (YΔe,YΔμ,YΔτ,YΔN~). For completeness, in (5.30) we have also given the results for CH~d even if only the uptype Higgs density asymmetry is relevant for the leptogenesis processes. Note that neglecting the contribution of ΔnN~1 to the global charges ℛtot in (5.23) and R3tot in (5.27) corresponds precisely to setting to zero the fourth column in all the previous matrices. Then, analogously with the SE and SM cases, within this “3-column approximation” all particle density asymmetries can be expressed just in terms of the three YΔα charge densities.


## 5.4.1. Anomalous and Nonanomalous Symmetries

Two linear combinationsR2 and R3 of R and PQ, having, respectively, only SU(2)L and SU(3)C mixed anomalies, have been identified in [150]:
(5.18)R2=R-2PQ,R3=R-3PQ.
The authors of [150] have also constructed the effective multifermions operators generated by the mixed anomalies:
(5.19)O~EW=Πα(QQQℓα)H~uH~dW~4,(5.20)O~QCD=Πi(QQucdc)ig~6.
Given that three global symmetries B, L, and R2 have mixed SU(2)L anomalies (but are free of SU(3)C anomalies) we can construct two anomaly-free combinations, the first one being B-L which is only violated perturbatively by NcℓHu and the second anomaly free combination which is also an exact symmetry of the MSSM+seesaw in the NSE regime [89]:
(5.21)ℛ=53B-L+R2,
and is exactly conserved. In the SU(3)C sector, besides the chiral anomaly we now have also R3 mixed anomalies. Thus also in this case anomaly-free combinations can be constructed, and in particular we can define one combination for each quark superfield. Assigning to the LH supermultiplets chiral charge χ=-1 these combinations have the form [89]:
(5.22)χqL+κqLR3,
where, for example, κuLc=κdLc=1/3 and κQL=2/3. Note that since R3 is perturbatively conserved by the complete MSSM+seesaw Lagrangian, when the Yukawa coupling of one quark is set to zero, the corresponding charge (5.22) will be exactly conserved.


## 5.4.2. Constraints in the Nonsuperequilibration Regime

In the NSE regime, the conditions listed in items6SE and 7SE of the previous section have to be dropped, but new conditions arise.6NSE:
The conservation law for theℛ charge yields the following global neutrality condition:
(5.23)ℛtot=∑fΔnfℛf+∑bΔnbℛb+ΔnN~1ℛN~1=T26(∑i(2Qi-5ui+4di)+2∑α(ℓα+eα)+5H~d-H~u+31g~)-ΔnN~1=0.The last terms in both lines of (5.23) correspond to the contribution to ℛ-neutrality from the lightest sneutrino asymmetry ΔnN~1=nN~1-nN~1* with charge ℛN~1=-ℛNc=-1. Note that since in general N~1 is not in chemical equilibrium, no chemical potential can be associated to it, and hence this constraint needs to be formulated in terms of its number density asymmetry that has to be evaluated by solving a BE for YΔN~≡YN~1-YN~1* (see Section 5.5).7NSE:
The operators in (5.19) induce transitions that in the NSE regime are in chemical equilibrium. This enforces the generalized EW and QCD sphaleron equilibrium conditions [150]:
(5.24)3∑iQi+∑αℓα+H~u+H~d+4g~=0,2∑iQi-∑i(ui+di)+6g~=0,that replace (5.13) and (5.14).8NSE:
The chiral-R3 charges in (5.22) are anomaly-free, but clearly they are not conserved by the quark Yukawa interactions. However, when a quark supermultiplet decouples from its Yukawa interactions, an exact conservation law arises. (Note that hu,d→0 implies u and d decoupling, but Q1 decoupling is ensured only if also hc,s→0.)  The conservation laws corresponding to these symmetries read
(5.25)T26[3qR+6(qR-g~)]+13R3tot=0,(5.26)T262[3QL+6(QL+g~)]-23R3tot=0and hold forqR=ui,di and QL=Qi in the regimes when the appropriate Yukawa reactions are negligible. Note the factor of 2 for the QL chiral charge in front of the first square bracket in (5.26) that is due to SU(2)L gauge multiplicity. In terms of chemical potentials and ΔnN~1, the total R3 charge in (5.25) and (5.26) reads
(5.27)R3tot=T26(82g~-3∑i(2Qi+11ui-4di)+∑α(16ℓα+13eα)+16H~d-14H~u)+ΔnN~1R3N~1,whereR3N~1=-1. As regards the leptons, since they do not couple to the QCD anomaly, by setting he→0, a symmetry under chiral supermultiplet rotations is directly gained for the RH leptons implying Δne+Δn~e=0 and giving the condition:
(5.28)e-23g~=0.No analogous condition arises for the lepton doublets relevant for leptogenesis, since by assumption they remain coupled via Yukawa couplings to the heavyN’s.In the NSE regime there are different flavour mixing matrices for the scalar and fermion components of the leptons and Higgs supermultiplets. To express more concisely all the results, it is convenient to introduce a newC vector to describe the gaugino number density asymmetry per degree of freedom in terms of the relevant charges:
(5.29)YΔg~=Cag~YΔa,withΔa=(Δα,ΔN~).


## 5.4.3. First Generation Yukawa Reactions out of Equilibrium (NSE Regime)

As an example, in the temperature range between108 and 1011GeV, and for moderate values of tanβ, all the first generation Yukawa couplings can be set to zero. Using for u,d conditions equation (5.25) and for e condition equation (5.28) as implied by hu,d,he→0 we obtain
(5.30)Aℓ=19×162332(-1981173398733987-825326634-14757114761-80552663414761-147571-8055),Cg~=-11162332(163,165,165,-255),CH~u=-1162332(3918,4713,4713,95),CH~d=13×162332(5413,9712,9712,-252),
where the rows correspond to (YΔe,YΔμ,YΔτ,YΔN~). For completeness, in (5.30) we have also given the results for CH~d even if only the uptype Higgs density asymmetry is relevant for the leptogenesis processes. Note that neglecting the contribution of ΔnN~1 to the global charges ℛtot in (5.23) and R3tot in (5.27) corresponds precisely to setting to zero the fourth column in all the previous matrices. Then, analogously with the SE and SM cases, within this “3-column approximation” all particle density asymmetries can be expressed just in terms of the three YΔα charge densities.


## 5.5. Supersymmetric Boltzmann Equations

In order to illustrate how the new effects described above modify the structure of the BE, here we write down a simpler expressions in which only decays and inverse decays are included: (the complete set of BE including decays, inverse decays, and scatterings with top-quark is given in the Appendix of  [89].)
(5.31)sHzdYN1dz=-(YN1YN1eq-1)γN1,(5.32)sHzdYN~+dz=-(YN~+YN~1eq-2)γN~1,(5.33)sHzdYΔN~dz=-YΔN~YN~1eqγN~1-32γN~1∑aCag~YΔaYℓeq+⋯,(5.34)sHzdYΔαdz=-ϵα[(YN1YN1eq-1)γN1+(YN~+YN~1eq-2)γN~1]+P1α0(γN1+12γN~1)∑a(Aαaℓ+CaH~u+Cag~)YΔaYℓeq,
where γN~1 is the corresponding thermally averaged decay rate for RH sneutrino N~1. In (5.32) we have introduced the overall sneutrino abundance YN~+=YN~1+YN~1*, while YΔN~≡YN~1-YN~1* in (5.33) is the sneutrino density asymmetry that was already introduced in Section 5.4. In the washout terms we have normalized the charge densities YΔa=(YΔα,YΔN~) to the equilibrium density of a fermion with one degree of freedom Yℓeq. In (5.31) and (5.34) we have also neglected for simplicity all finite temperature effects. Taking these effects into account would imply, for example, that the CP asymmetry for N~ decays into fermions is different from the one for decays into scalars [49], while we describe both CP asymmetries with ϵα. A few remarks regarding (5.33) are in order. In the SE regime g~=0, and thus it would seem that the sneutrino density asymmetry YΔN~ vanishes. However, this only happens for decays and inverse decays, and it is no more true when additional terms related to scattering processes, that are represented in the equation by the dots, are also included (see [147] and the Appendix of [89]). Therefore, also in the SE regime YN~1 and YN~1* in general differ. However, in this case recasting their equations in terms of two equations for YN~+ and YΔN~ is just a convenient parametrization. On the contrary, in the NSE regime this is mandatory, because the sneutrinos carry a globally conserved ℛ-charge and YΔN~ is required to formulate properly the corresponding conservation law. As we have seen, this eventually results in YΔN~ contributing to the expressions of the lepton flavour density asymmetries in terms of slowly varying quantities.In [89] a complete numerical analysis was carried out, and it was shown that numerical corrections with respect to the case when NSE effects are neglected remain at the 𝒪(1) level. This is because only spectator processes get affected, while the overall amount of CP violation driving leptogenesis remains the same than in previous treatments.


## 6. Beyond Type-I Seesaw and Beyond the Seesaw

There exist many variants of leptogenesis models beyond the standard type-I seesaw. In this section, we try to classify them into appropriate groups. Unavoidably there would be some overlap; that is, a hybrid model of leptogenesis which can belong to more than one group, for example, soft leptogenesis (Section6.2) from resonantly enhanced CP asymmetry could rightly fall under resonant leptogenesis (Section 6.1). However we try our best to categorize them according to the main features of the model, and, when appropriate, they will be quoted in more than one place. Clearly, the number of beyond type-I seesaw leptogenesis models is quite large. We have not attempted in any way to be exhaustive, and we apologize in advance for the unavoidable several omissions.


### 6.1. Resonant Leptogenesis

A resonant enhancement of the CP asymmetry inN1 decay occurs when the mass difference between N1 and N2 is of the order of the decay widths. Such a scenario has been termed “resonant leptogenesis,” and has benefited from many studies in different formalisms (see  [154]  for a comparison of the different calculations)  [50, 52, 75, 85–87, 110, 111, 155–168] (see [169] for a recent review). The resonant effect is related to the self-energy contribution to the CP asymmetry. Consider, for simplicity, the case where only N2 is quasidegenerate with N1. Then, the self-energy contribution involving the intermediate N2 to the total CP asymmetry (we neglect important flavour effects [75]) is given by
(6.1)ϵ1(self-energy)=-M1M2ΓN2M2M22(M22-M12)(M22-M12)2+M12ΓN22Im[(λ†λ)122](λ†λ)11(λ†λ)22.
The resonance condition reads
(6.2)|M2-M1|=ΓN22.
In this case
(6.3)|ϵ1(resonance)|≃12|Im[(λ†λ)122]|(λ†λ)11(λ†λ)22.Thus, in the resonant case, the asymmetry is suppressed by neither the smallness of the light neutrino masses, nor the smallness of their mass splitting, nor small ratios between the singlet neutrino masses. Actually, the CP asymmetry could be of order one (more accurately,|ϵ1|≤1/2).With resonant leptogenesis, the BEs are different. The densities ofN1 and N2 are followed, since both contribute to the asymmetry, and the relevant timescales are different. For instance, the typical timescale to build up coherently the CP asymmetry is unusually long, of order 1/ΔM. In particular, it can be larger than the timescale for the change of the abundance of the sterile neutrinos. This situation implies that for resonant leptogenesis quantum effects in the BE can be significant [85, 86, 141, 170] (see Section 4.5).The fact that the asymmetry could be large, independently of the singlet neutrino masses, opens up the possibility of low scale resonant leptogenesis. Models along these lines have been constructed in [75, 156, 161, 171]. It is a theoretical challenge to construct models where a mass splitting as small as the decay width is naturally achieved. For attempts that utilize approximate flavour symmetries see, for example, [157, 158, 160, 164, 166, 172], while studies of this issue in the framework of minimal flavour violation can be found in [87, 162, 163]. The possibility of observing resonant CP violation due to heavy RH neutrinos at the LHC was studied in [173, 174].


### 6.2. Soft Leptogenesis

The modifications to standard type-I leptogenesis due to SUSY have been discussed in Section5. The important parameters are the Yukawa couplings and the singlet neutrino parameters, which appear in the superpotential equation (5.1). SUSY must, however, be broken. In the framework of the MSSM extended to include singlet neutrinos (MSSM+N), there are, in addition to the soft SUSY breaking terms of the MSSM, terms that involve the singlet sneutrinos N~i, in particular bilinear (B) and trilinear (A) scalar couplings. These terms provide additional sources of lepton number violation and of CP violation. Scenarios where these terms play a dominant role in leptogenesis have been termed “soft leptogenesis” [102, 107, 175–196] (see also [54] for a recent review).Soft leptogenesis can take place even with a single RH neutrino because the presence of theB term implies that N~ and N~† states mix to form two mass eigenstates with mass splitting proportional to B itself. Furthermore when B~ΓN~, the CP asymmetry is resonantly enhanced realizing the resonant leptogenesis scenario (see Section 6.1). In the following we will consider a single generation MSSM+N. The relevant soft SUSY terms involving N~, the SU(2)L gauginos  λ~2±,0, the U(1)Y gauginos λ~1, and the three sleptons ℓ~α in the basis in which charged lepton Yukawa couplings are diagonal are given by
(6.4)-ℒsoft=M~2N~*N~+(AYαϵabN~ℓ~αaHub+12BMN~N~+h.c.)+12(m2λ~2±,0¯PLλ~2±,0+m1λ~1¯PLλ~1+h.c.),
where for simplicity, proportionality of the bilinear and trilinear soft breaking terms to the corresponding SUSY invariant couplings has been assumed: BM=BM and Aα=AYα. The Lagrangian derived from (5.1) and (6.4) is characterized by only three independent physical phases: ϕA≡arg(AB*), ϕg2≡(1/2)arg(Bm2*) and ϕgY≡(1/2)arg(Bm1*), which can be assigned to A and to the gaugino coupling operators g2,gY, respectively. As mentioned earlier, a crucial role in soft leptogenesis is played by the N~-N~† mixing to form the mass eigenstates
(6.5)N~+=12(eiΦ/2N~+e-iΦ/2N~*),N~-=-i2(eiΦ/2N~-e-iΦ/2N~*),
where Φ≡arg(BM) and the corresponding mass eigenvalues are M±2=M2+M~2±|BM|. Without loss of generality, we can set Φ=0, which is equivalent to assigning the phases only to A and Yα.It has been pointed out that the CP asymmetries for the decays ofN~± into scalars and fermions have opposite sign and cancel each other at the leading order [176, 177, 192], resulting in a strongly suppressed total CP asymmetry ~𝒪(msoft3/M3) where msoft is the scale of soft SUSY breaking terms. There are two possibilities that can rescue leptogenesis. Firstly, thermal effects which break SUSY can spoil this cancellation [176, 177, 192]. Secondly, nonsuperequilibration effects (see Section 5.4) which imply that lepton and slepton asymmetries differ can also spoil this cancellation [107]. The CP asymmetries for the decays of N~± into scalars and fermions are, respectively, given by
(6.6)ϵαs(T)=ϵ-αΔs(T),ϵαf(T)=-ϵ-αΔf(T),
where ϵ-α is the temperature independent term of ~𝒪(msoft/M) which contains contributions from the self-energy correction, vertex correction, and the interference between the two. In the limit T→0, we have Δs(T),Δf(T)→1/2, and thus the inclusion of thermal effects and/or nonsuperequilibration is mandatory to avoid the cancellation between the asymmetries into scalars and fermions.We can make a rough estimate of the scale relevant for soft leptogenesis by requiring|ϵ-|~msoft/M≳10-6 which gives M≲109 GeV for msoft~1 TeV. Hence soft leptogenesis always happens in the temperature regime where lepton flavour effects are relevant [190]. In general, the CP asymmetry from self-energy contribution requires B≪msoft to be resonantly enhanced. However it was shown in [195] that flavour effects can greatly enhance the efficiency, and eventually B~msoft is allowed. The nice feature of soft leptogenesis is that the tension with the gravitino problem gets generically relaxed and, in the lower temperature window, is completely avoided.


### 6.3. Dirac Leptogenesis

The extension of the SM with singlet neutrinos allows for two different ways for generating tiny neutrino masses. The first one is the seesaw mechanism which has at least three attractive features(i)
No extra symmetries (and, in particular, no global symmetries) have to be imposed.(ii)
The extreme lightness of neutrino masses is linked to the existence of a high scale of new physics, which is well motivated for various other reasons (e.g., gauge unification).(iii)
Lepton number is violated, which opens the way to leptogenesis.The second way is to impose lepton number and give to the neutrinos Dirac masses. A priori, one might think that all three attractive features of the seesaw mechanism are lost. Indeed, one must usually impose additional symmetries. But one can still construct natural models where the tiny Yukawa couplings that are necessary for small Dirac masses are related to a small breaking of a symmetry. What is perhaps most surprising is the fact that leptogenesis could proceed successfully even if neutrinos are Dirac particles, and lepton number is not (perturbatively) broken [197, 198]. Such scenarios have been termed “Dirac leptogenesis” [198–207].An implementation of the idea is the following. A CP-violating decay of a heavy particle can result in a nonzero lepton number for LH particles and an equal and opposite nonzero lepton number for RH particles, so that the total lepton number is zero. For the charged fermions of the SM, the Yukawa interactions are fast enough that they quickly equilibrate the LH and the RH particles, and the lepton number stored in each chirality goes to zero. This is not true, however, for Dirac neutrinos. The size of their Yukawa couplings isλ≲10-11, which means that equilibrium between the lepton numbers stored in LH and RH neutrinos will not be reached until the temperature falls well below the electroweak breaking scale. To see this, note that the rate of the Yukawa interactions, given by Γλ~λ2T, becomes significant when it equals the expansion rate of the Universe,  H~T2/mpl. Thus, the temperature of equilibration between LH and RH neutrinos is T~λ2mpl~(λ/10-11)2 MeV, that is, well below the temperature when sphalerons, after having converted part of the LH lepton asymmetry into a net baryon asymmetry, are switched off.A specific example of a supersymmetric model where Dirac neutrinos arise naturally is presented in [199]. The Majorana masses of the N-superfields are forbidden by U(1)L. The neutrino Yukawa couplings are forbidden by a U(1)N symmetry where, among all the MSSM+N fields, only the N superfields are charged. The symmetry is spontaneously broken by the vacuum expectation value of a scalar field χ that can naturally be at the weak scale, 〈χ〉~vu. This breaking is communicated to the MSSM+N via extra, vector-like lepton doublet fields, ϕ+ϕ-, that have masses Mϕ much larger than vu. Consequently, the neutrino Yukawa couplings are suppressed by the small ratio 〈χ〉/Mϕ. The CP violation arises in the decays of the vector-like leptons, whereby Γ(ϕ→NHuc)≠Γ(ϕ-→NcHu) and Γ(ϕ→Lχ)≠Γ(ϕ-→Lcχc). The resulting asymmetries in N and in L are equal in magnitude and opposite in sign. Finally note that the main phenomenological implication of Dirac leptogenesis is the absence of any signal in neutrinoless double beta decays.


### 6.4. Triplet Scalar (Type-II) Leptogenesis

One can generate seesaw masses for the light neutrinos by tree-level exchange ofSU(2)L-triplet scalars T [44, 208–211]. The relevant new terms in the Lagrangian are
(6.7)ℒT=-MT2|T|2+12([λL]αβℓαℓβT+MTλϕϕϕT*+h.c.).
Here, MT is a real mass parameter, λL is a symmetric 3×3 matrix of dimensionless, complex Yukawa couplings, and λϕ is a dimensionless complex coupling. Since this mechanism necessarily involves lepton number violation and allows for new CP-violating phases, it is interesting to examine it in the light of leptogenesis [81, 180, 185, 212–233]. One obvious problem in this scenario is that, unlike singlet fermions, the triplet scalars have gauge interactions that keep them close to thermal equilibrium at temperatures T≲1015 GeV. It turns out, however, that successful leptogenesis is possible even at a much lower temperature. This subsection is based in large part on [220] where further details and, in particular, an explicit presentation of the relevant BE can be found.The CP asymmetry that is induced by the triplet scalar decays is defined as follows:(6.8)ϵT≡2Γ(T-→ℓℓ)-Γ(T→ℓ-ℓ-)ΓT+ΓT-,
where the overall factor of 2 comes because the triplet scalar decay produces two (anti)leptons.To calculateϵT, one should use the Lagrangian in (6.7). While a single triplet is enough to produce three light massive neutrinos, there is a problem in leptogenesis if indeed this is the only source of neutrinos masses. The asymmetry is generated only at higher loops and is unacceptably small. It is still possible to produce the required lepton asymmetry from a single triplet scalar decays if there are additional sources for neutrino masses, such as type I, type III, or type II contributions from additional triplet scalars. Define mII (mI) as the part of the light neutrino mass matrix that comes (does not come) from the contributions of the triplet scalar responsible for ϵT:
(6.9)m=mII+mI.
Then, assuming that the particles exchanged to produce mI are all heavier than T, we obtain the CP asymmetry
(6.10)ϵT=14πMTvu2BLBHIm[Tr(mII†mI)]Tr(mII†mII),
where BL(BH) is the tree-level branching ratio to leptons (Higgs doublets). If these are the only decay modes, that is, BL+BH=1, then BL/BH=Tr(λLλL†)/(λHλH†), and there is an upper bound on the asymmetry
(6.11)|ϵT|≤14πMTvu2BLBH∑imνi2.
Note that, unlike the singlet fermion case, |ϵT| increases with larger mνi.As concerns the efficiency factor, it can be close to maximal,η~1, in spite of the fact that the gauge interactions tend to maintain the triplet abundance very close to thermal equilibrium. There are two necessary conditions that have to be fulfilled by the decay rates T→ℓ-ℓ- and T→ϕϕ in order that this will happen [220](1)
One of the two decay rates is faster than theTT- annihilation rate.(2)
The other decay mode is slower than the expansion rate of the Universe.The first condition guarantees that gauge scatterings are ineffective: the triplets decay before annihilating. The second condition guarantees that the fast decays do not wash out strongly the lepton asymmetry: lepton number is violated only by the simultaneous presence ofT→ℓ-ℓ- and T→ϕϕ.Combining a calculation ofη with the upper bound on the CP asymmetry (6.11), successful leptogenesis implies a lower bound on the triplet mass MT varying between 109 GeV and 1012 GeV, depending on the relative weight of mII and mI in the light neutrino mass.Interestingly, in the supersymmetric framework, “soft leptogenesis” (see Section6.2) can be successful even with the minimal set of extra fields—a single T+T-—that generates both neutrino masses and the lepton asymmetry [180, 185].


### 6.5. Triplet Fermion (Type-III) Leptogenesis

One can generate neutrino masses by the tree-level exchange ofSU(2)L-triplet fermions Tia [234–236] (i denotes a heavy mass eigenstate while a is an SU(2)L index) with the Lagrangian
(6.12)ℒTa=[λT]αkτρσaℓαρϕσTka-12MiTiaTia+h.c.
Here τa are the Pauli matrices, Mi are real mass parameters, and λT is a 3×3 matrix of complex Yukawa couplings.This mechanism necessarily involves lepton number violation and allows for new CP-violating phases so we should examine it as a possible source of leptogenesis [228, 237–243]. This subsection is based in large part on [238] where further details and the relevant BE can be found.As concerns neutrino masses, all the qualitative features are very similar to the singlet fermion case. As concerns leptogenesis there are, however, qualitative and quantitative differences. With regard to the CP asymmetry from the lightest triplet fermion decay, the relative sign between the vertex and self-energy loop contributions is opposite to that of the singlet fermion case. Consequently, in the limit of strong hierarchy in the heavy fermion masses, the asymmetry in triplet decays is three times smaller than in the decays of the singlets. On the other hand, since the triplet has three components, the ratio between the final baryon asymmetry andϵη is three times bigger. The decay rate of the heavy fermion is the same in both cases. This, however, means that the thermally averaged decay rate is three times bigger for the triplet, as is the on-shell part of the ΔL=2 scattering rate.A significant qualitative difference arises from the fact that the triplet has gauge interactions. The effect on the washout factorη is particularly significant for m~≪10-3eV, the so-called weak washout regime (note that this name is inappropriate for triplet fermions). The gauge interactions still drive the triplet abundance close to thermal equilibrium. A relic fraction of the triplet fermions survives. The decays of these relic triplets produce a baryon asymmetry, with
(6.13)η≈M1/1013GeV(form~≪10-3eV).
The strong dependence on M1 results from the fact that the expansion rate of the Universe is slower at lower temperatures. On the other hand, for m~≫10-3 eV, the Yukawa interactions keep the heavy fermion abundance close to thermal equilibrium, so the difference in η between the singlet and triplet case is only 𝒪(1). Ignoring flavour effects, and assuming strong hierarchy between the heavy fermions, [216] obtained the lower bound
(6.14)M1≳1.5×1010GeV.
When the triplet fermion scenario is incorporated in a supersymmetric framework, and the soft breaking terms do not play a significant role, the modifications to the previous analysis are by factors of 𝒪(1).


## 6.1. Resonant Leptogenesis

A resonant enhancement of the CP asymmetry inN1 decay occurs when the mass difference between N1 and N2 is of the order of the decay widths. Such a scenario has been termed “resonant leptogenesis,” and has benefited from many studies in different formalisms (see  [154]  for a comparison of the different calculations)  [50, 52, 75, 85–87, 110, 111, 155–168] (see [169] for a recent review). The resonant effect is related to the self-energy contribution to the CP asymmetry. Consider, for simplicity, the case where only N2 is quasidegenerate with N1. Then, the self-energy contribution involving the intermediate N2 to the total CP asymmetry (we neglect important flavour effects [75]) is given by
(6.1)ϵ1(self-energy)=-M1M2ΓN2M2M22(M22-M12)(M22-M12)2+M12ΓN22Im[(λ†λ)122](λ†λ)11(λ†λ)22.
The resonance condition reads
(6.2)|M2-M1|=ΓN22.
In this case
(6.3)|ϵ1(resonance)|≃12|Im[(λ†λ)122]|(λ†λ)11(λ†λ)22.Thus, in the resonant case, the asymmetry is suppressed by neither the smallness of the light neutrino masses, nor the smallness of their mass splitting, nor small ratios between the singlet neutrino masses. Actually, the CP asymmetry could be of order one (more accurately,|ϵ1|≤1/2).With resonant leptogenesis, the BEs are different. The densities ofN1 and N2 are followed, since both contribute to the asymmetry, and the relevant timescales are different. For instance, the typical timescale to build up coherently the CP asymmetry is unusually long, of order 1/ΔM. In particular, it can be larger than the timescale for the change of the abundance of the sterile neutrinos. This situation implies that for resonant leptogenesis quantum effects in the BE can be significant [85, 86, 141, 170] (see Section 4.5).The fact that the asymmetry could be large, independently of the singlet neutrino masses, opens up the possibility of low scale resonant leptogenesis. Models along these lines have been constructed in [75, 156, 161, 171]. It is a theoretical challenge to construct models where a mass splitting as small as the decay width is naturally achieved. For attempts that utilize approximate flavour symmetries see, for example, [157, 158, 160, 164, 166, 172], while studies of this issue in the framework of minimal flavour violation can be found in [87, 162, 163]. The possibility of observing resonant CP violation due to heavy RH neutrinos at the LHC was studied in [173, 174].


## 6.2. Soft Leptogenesis

The modifications to standard type-I leptogenesis due to SUSY have been discussed in Section5. The important parameters are the Yukawa couplings and the singlet neutrino parameters, which appear in the superpotential equation (5.1). SUSY must, however, be broken. In the framework of the MSSM extended to include singlet neutrinos (MSSM+N), there are, in addition to the soft SUSY breaking terms of the MSSM, terms that involve the singlet sneutrinos N~i, in particular bilinear (B) and trilinear (A) scalar couplings. These terms provide additional sources of lepton number violation and of CP violation. Scenarios where these terms play a dominant role in leptogenesis have been termed “soft leptogenesis” [102, 107, 175–196] (see also [54] for a recent review).Soft leptogenesis can take place even with a single RH neutrino because the presence of theB term implies that N~ and N~† states mix to form two mass eigenstates with mass splitting proportional to B itself. Furthermore when B~ΓN~, the CP asymmetry is resonantly enhanced realizing the resonant leptogenesis scenario (see Section 6.1). In the following we will consider a single generation MSSM+N. The relevant soft SUSY terms involving N~, the SU(2)L gauginos  λ~2±,0, the U(1)Y gauginos λ~1, and the three sleptons ℓ~α in the basis in which charged lepton Yukawa couplings are diagonal are given by
(6.4)-ℒsoft=M~2N~*N~+(AYαϵabN~ℓ~αaHub+12BMN~N~+h.c.)+12(m2λ~2±,0¯PLλ~2±,0+m1λ~1¯PLλ~1+h.c.),
where for simplicity, proportionality of the bilinear and trilinear soft breaking terms to the corresponding SUSY invariant couplings has been assumed: BM=BM and Aα=AYα. The Lagrangian derived from (5.1) and (6.4) is characterized by only three independent physical phases: ϕA≡arg(AB*), ϕg2≡(1/2)arg(Bm2*) and ϕgY≡(1/2)arg(Bm1*), which can be assigned to A and to the gaugino coupling operators g2,gY, respectively. As mentioned earlier, a crucial role in soft leptogenesis is played by the N~-N~† mixing to form the mass eigenstates
(6.5)N~+=12(eiΦ/2N~+e-iΦ/2N~*),N~-=-i2(eiΦ/2N~-e-iΦ/2N~*),
where Φ≡arg(BM) and the corresponding mass eigenvalues are M±2=M2+M~2±|BM|. Without loss of generality, we can set Φ=0, which is equivalent to assigning the phases only to A and Yα.It has been pointed out that the CP asymmetries for the decays ofN~± into scalars and fermions have opposite sign and cancel each other at the leading order [176, 177, 192], resulting in a strongly suppressed total CP asymmetry ~𝒪(msoft3/M3) where msoft is the scale of soft SUSY breaking terms. There are two possibilities that can rescue leptogenesis. Firstly, thermal effects which break SUSY can spoil this cancellation [176, 177, 192]. Secondly, nonsuperequilibration effects (see Section 5.4) which imply that lepton and slepton asymmetries differ can also spoil this cancellation [107]. The CP asymmetries for the decays of N~± into scalars and fermions are, respectively, given by
(6.6)ϵαs(T)=ϵ-αΔs(T),ϵαf(T)=-ϵ-αΔf(T),
where ϵ-α is the temperature independent term of ~𝒪(msoft/M) which contains contributions from the self-energy correction, vertex correction, and the interference between the two. In the limit T→0, we have Δs(T),Δf(T)→1/2, and thus the inclusion of thermal effects and/or nonsuperequilibration is mandatory to avoid the cancellation between the asymmetries into scalars and fermions.We can make a rough estimate of the scale relevant for soft leptogenesis by requiring|ϵ-|~msoft/M≳10-6 which gives M≲109 GeV for msoft~1 TeV. Hence soft leptogenesis always happens in the temperature regime where lepton flavour effects are relevant [190]. In general, the CP asymmetry from self-energy contribution requires B≪msoft to be resonantly enhanced. However it was shown in [195] that flavour effects can greatly enhance the efficiency, and eventually B~msoft is allowed. The nice feature of soft leptogenesis is that the tension with the gravitino problem gets generically relaxed and, in the lower temperature window, is completely avoided.


## 6.3. Dirac Leptogenesis

The extension of the SM with singlet neutrinos allows for two different ways for generating tiny neutrino masses. The first one is the seesaw mechanism which has at least three attractive features(i)
No extra symmetries (and, in particular, no global symmetries) have to be imposed.(ii)
The extreme lightness of neutrino masses is linked to the existence of a high scale of new physics, which is well motivated for various other reasons (e.g., gauge unification).(iii)
Lepton number is violated, which opens the way to leptogenesis.The second way is to impose lepton number and give to the neutrinos Dirac masses. A priori, one might think that all three attractive features of the seesaw mechanism are lost. Indeed, one must usually impose additional symmetries. But one can still construct natural models where the tiny Yukawa couplings that are necessary for small Dirac masses are related to a small breaking of a symmetry. What is perhaps most surprising is the fact that leptogenesis could proceed successfully even if neutrinos are Dirac particles, and lepton number is not (perturbatively) broken [197, 198]. Such scenarios have been termed “Dirac leptogenesis” [198–207].An implementation of the idea is the following. A CP-violating decay of a heavy particle can result in a nonzero lepton number for LH particles and an equal and opposite nonzero lepton number for RH particles, so that the total lepton number is zero. For the charged fermions of the SM, the Yukawa interactions are fast enough that they quickly equilibrate the LH and the RH particles, and the lepton number stored in each chirality goes to zero. This is not true, however, for Dirac neutrinos. The size of their Yukawa couplings isλ≲10-11, which means that equilibrium between the lepton numbers stored in LH and RH neutrinos will not be reached until the temperature falls well below the electroweak breaking scale. To see this, note that the rate of the Yukawa interactions, given by Γλ~λ2T, becomes significant when it equals the expansion rate of the Universe,  H~T2/mpl. Thus, the temperature of equilibration between LH and RH neutrinos is T~λ2mpl~(λ/10-11)2 MeV, that is, well below the temperature when sphalerons, after having converted part of the LH lepton asymmetry into a net baryon asymmetry, are switched off.A specific example of a supersymmetric model where Dirac neutrinos arise naturally is presented in [199]. The Majorana masses of the N-superfields are forbidden by U(1)L. The neutrino Yukawa couplings are forbidden by a U(1)N symmetry where, among all the MSSM+N fields, only the N superfields are charged. The symmetry is spontaneously broken by the vacuum expectation value of a scalar field χ that can naturally be at the weak scale, 〈χ〉~vu. This breaking is communicated to the MSSM+N via extra, vector-like lepton doublet fields, ϕ+ϕ-, that have masses Mϕ much larger than vu. Consequently, the neutrino Yukawa couplings are suppressed by the small ratio 〈χ〉/Mϕ. The CP violation arises in the decays of the vector-like leptons, whereby Γ(ϕ→NHuc)≠Γ(ϕ-→NcHu) and Γ(ϕ→Lχ)≠Γ(ϕ-→Lcχc). The resulting asymmetries in N and in L are equal in magnitude and opposite in sign. Finally note that the main phenomenological implication of Dirac leptogenesis is the absence of any signal in neutrinoless double beta decays.


## 6.4. Triplet Scalar (Type-II) Leptogenesis

One can generate seesaw masses for the light neutrinos by tree-level exchange ofSU(2)L-triplet scalars T [44, 208–211]. The relevant new terms in the Lagrangian are
(6.7)ℒT=-MT2|T|2+12([λL]αβℓαℓβT+MTλϕϕϕT*+h.c.).
Here, MT is a real mass parameter, λL is a symmetric 3×3 matrix of dimensionless, complex Yukawa couplings, and λϕ is a dimensionless complex coupling. Since this mechanism necessarily involves lepton number violation and allows for new CP-violating phases, it is interesting to examine it in the light of leptogenesis [81, 180, 185, 212–233]. One obvious problem in this scenario is that, unlike singlet fermions, the triplet scalars have gauge interactions that keep them close to thermal equilibrium at temperatures T≲1015 GeV. It turns out, however, that successful leptogenesis is possible even at a much lower temperature. This subsection is based in large part on [220] where further details and, in particular, an explicit presentation of the relevant BE can be found.The CP asymmetry that is induced by the triplet scalar decays is defined as follows:(6.8)ϵT≡2Γ(T-→ℓℓ)-Γ(T→ℓ-ℓ-)ΓT+ΓT-,
where the overall factor of 2 comes because the triplet scalar decay produces two (anti)leptons.To calculateϵT, one should use the Lagrangian in (6.7). While a single triplet is enough to produce three light massive neutrinos, there is a problem in leptogenesis if indeed this is the only source of neutrinos masses. The asymmetry is generated only at higher loops and is unacceptably small. It is still possible to produce the required lepton asymmetry from a single triplet scalar decays if there are additional sources for neutrino masses, such as type I, type III, or type II contributions from additional triplet scalars. Define mII (mI) as the part of the light neutrino mass matrix that comes (does not come) from the contributions of the triplet scalar responsible for ϵT:
(6.9)m=mII+mI.
Then, assuming that the particles exchanged to produce mI are all heavier than T, we obtain the CP asymmetry
(6.10)ϵT=14πMTvu2BLBHIm[Tr(mII†mI)]Tr(mII†mII),
where BL(BH) is the tree-level branching ratio to leptons (Higgs doublets). If these are the only decay modes, that is, BL+BH=1, then BL/BH=Tr(λLλL†)/(λHλH†), and there is an upper bound on the asymmetry
(6.11)|ϵT|≤14πMTvu2BLBH∑imνi2.
Note that, unlike the singlet fermion case, |ϵT| increases with larger mνi.As concerns the efficiency factor, it can be close to maximal,η~1, in spite of the fact that the gauge interactions tend to maintain the triplet abundance very close to thermal equilibrium. There are two necessary conditions that have to be fulfilled by the decay rates T→ℓ-ℓ- and T→ϕϕ in order that this will happen [220](1)
One of the two decay rates is faster than theTT- annihilation rate.(2)
The other decay mode is slower than the expansion rate of the Universe.The first condition guarantees that gauge scatterings are ineffective: the triplets decay before annihilating. The second condition guarantees that the fast decays do not wash out strongly the lepton asymmetry: lepton number is violated only by the simultaneous presence ofT→ℓ-ℓ- and T→ϕϕ.Combining a calculation ofη with the upper bound on the CP asymmetry (6.11), successful leptogenesis implies a lower bound on the triplet mass MT varying between 109 GeV and 1012 GeV, depending on the relative weight of mII and mI in the light neutrino mass.Interestingly, in the supersymmetric framework, “soft leptogenesis” (see Section6.2) can be successful even with the minimal set of extra fields—a single T+T-—that generates both neutrino masses and the lepton asymmetry [180, 185].


## 6.5. Triplet Fermion (Type-III) Leptogenesis

One can generate neutrino masses by the tree-level exchange ofSU(2)L-triplet fermions Tia [234–236] (i denotes a heavy mass eigenstate while a is an SU(2)L index) with the Lagrangian
(6.12)ℒTa=[λT]αkτρσaℓαρϕσTka-12MiTiaTia+h.c.
Here τa are the Pauli matrices, Mi are real mass parameters, and λT is a 3×3 matrix of complex Yukawa couplings.This mechanism necessarily involves lepton number violation and allows for new CP-violating phases so we should examine it as a possible source of leptogenesis [228, 237–243]. This subsection is based in large part on [238] where further details and the relevant BE can be found.As concerns neutrino masses, all the qualitative features are very similar to the singlet fermion case. As concerns leptogenesis there are, however, qualitative and quantitative differences. With regard to the CP asymmetry from the lightest triplet fermion decay, the relative sign between the vertex and self-energy loop contributions is opposite to that of the singlet fermion case. Consequently, in the limit of strong hierarchy in the heavy fermion masses, the asymmetry in triplet decays is three times smaller than in the decays of the singlets. On the other hand, since the triplet has three components, the ratio between the final baryon asymmetry andϵη is three times bigger. The decay rate of the heavy fermion is the same in both cases. This, however, means that the thermally averaged decay rate is three times bigger for the triplet, as is the on-shell part of the ΔL=2 scattering rate.A significant qualitative difference arises from the fact that the triplet has gauge interactions. The effect on the washout factorη is particularly significant for m~≪10-3eV, the so-called weak washout regime (note that this name is inappropriate for triplet fermions). The gauge interactions still drive the triplet abundance close to thermal equilibrium. A relic fraction of the triplet fermions survives. The decays of these relic triplets produce a baryon asymmetry, with
(6.13)η≈M1/1013GeV(form~≪10-3eV).
The strong dependence on M1 results from the fact that the expansion rate of the Universe is slower at lower temperatures. On the other hand, for m~≫10-3 eV, the Yukawa interactions keep the heavy fermion abundance close to thermal equilibrium, so the difference in η between the singlet and triplet case is only 𝒪(1). Ignoring flavour effects, and assuming strong hierarchy between the heavy fermions, [216] obtained the lower bound
(6.14)M1≳1.5×1010GeV.
When the triplet fermion scenario is incorporated in a supersymmetric framework, and the soft breaking terms do not play a significant role, the modifications to the previous analysis are by factors of 𝒪(1).


## 7. Conclusions

During the last few decades, a large set of experiments involving solar, atmospheric, reactor, and accelerator neutrinos have converged to establish that the neutrinos are massive. The seesaw mechanism extends the Standard Model in a way that allows neutrino masses, and it provides a nice explanation of the suppression of the neutrino masses with respect to the electroweak breaking scale. Furthermore, without any addition or modification, it can also account for the observed baryon asymmetry of the Universe. The possibility of giving an explanation of two apparently unrelated experimental facts—neutrino masses and the baryon asymmetry—within a single framework that is a natural extension of the Standard Model, together with the remarkable “coincidence” that the same neutrino mass scale suggested by neutrino oscillation data is also optimal for leptogenesis, makes the idea that baryogenesis occurs through leptogenesis a very attractive one.Leptogenesis can be quantitatively successful without any fine tuning of the seesaw parameters. Yet, in the nonsupersymmetric seesaw framework, a fine-tuning problem arises due to the large corrections to the mass-squared parameter of the Higgs potential that are proportional to the heavy Majorana neutrino masses. Supersymmetry can cure this problem, avoiding the necessity of fine tuning; however, it brings in the gravitino problem [144] that requires a low reheat temperature after inflation, in conflict with generic leptogenesis models. Thus, constructing a fully satisfactory theoretical framework that implements leptogenesis within the seesaw framework is not a straightforward task.From the experimental side, the obvious question to ask is if it is possible to test whether the baryon asymmetry has been really produced through leptogenesis. Unfortunately it seems impossible that any direct test can be performed. To establish leptogenesis experimentally, we need to produce the heavy Majorana neutrinos and measure the CP asymmetry in their decays. However, in the most natural seesaw scenarios, these states are simply too heavy to be produced, while if they are light, then their Yukawa couplings must be very tiny, again preventing any chance of direct measurements.Lacking the possibility of a direct proof, experiments can still provide circumstantial evidence in support of leptogenesis by establishing that (some of) the Sakharov conditions for leptogenesis are realized in nature. Planned neutrinoless double beta decay (0νββ) experiments (GERDA [244], MAJORANA [245], and CUORE [246]) aim at a sensitivity to the effective 0νββ neutrino mass in the few ×10 meV range. If they succeed in establishing the Majorana nature of the light neutrinos, this will strengthen our confidence that the seesaw mechanism is at the origin of the neutrino masses and, most importantly, will establish that the first Sakharov condition for the dynamical generation of a lepton asymmetry (L violation) is realized. Proposed SuperBeam facilities [247, 248] and second generation off-axis SuperBeam experiments (T2HK [249] and NOνA [250]) can discover CP violation in the leptonic sector. These experiments can only probe the Dirac phase of the neutrino mixing matrix. They cannot probe the Majorana low energy or the high energy phases, but the important point is that they can establish that the second Sakharov condition for the dynamical generation of a lepton asymmetry is satisfied. As regards the third condition, that is, that the heavy neutrino decays occurred out of thermal equilibrium, it might seem the most difficult one to test experimentally. In reality the opposite is true, and in fact we already know that an absolute neutrino mass scale of the order of the solar or atmospheric mass differences is perfectly compatible with sufficiently out of equilibrium heavy neutrinos decays.Given that we do not know how to prove that leptogenesis is the correct theory, we might ask if there is any chance to falsify it. Indeed, future neutrino experiments could weaken the case for leptogenesis, or even falsify it, mainly by establishing that the seesaw mechanism is not responsible for the observed neutrino masses. By itself, failure in revealing signals of0νββ decays will not disprove leptogenesis. Indeed, with normal neutrino mass hierarchy one expects that the rates of lepton-number-violating processes are below experimental sensitivity. However, if neutrinos masses are quasidegenerate or inversely hierarchical, and future measurements of the oscillation parameters will not fluctuate too much away from the present best fit values, the most sensitive 0νββ decay experiments scheduled for the near future should be able to detect a signal [251]. If instead the limit on |mββ| is pushed below ~10 meV (a quite challenging task), this would suggest that either the mass hierarchy is normal or neutrinos are not Majorana particles. The latter possibility would disprove the seesaw model and standard leptogenesis. Thus, determining the order of the neutrino mass spectrum is extremely important to shed light on the connection between 0νββ decay experiments and leptogenesis. In summary, if it is established that the neutrino mass hierarchy is inverted and at the same time no signal of 0νββ decays is detected at a level |mee|≲10meV, one could conclude that the seesaw is not at the origin of the neutrino masses and that (standard) leptogenesis is not the correct explanation of the baryon asymmetry. As concerns CP violation, a failure in detecting leptonic CP violation will not weaken the case for leptogenesis in a significant way. Instead, it would mean that the Dirac CP phase is small enough to render L-conserving CP-violating effects unobservable.Finally, the CERN LHC has the capability of providing information that is relevant to leptogenesis, since it can play a fundamental role in establishing that the origin of the neutrino masses is not due to the seesaw mechanism, thus leaving no strong motivation for leptogenesis. This may happen in several different ways. For example (assuming that the related new physics is discovered), the LHC will be able to test if the detailed phenomenology of any of the following models is compatible with an explanation of the observed pattern of neutrino masses and mixing angles: supersymmetricR-parity violating couplings and/or L-violating bilinear terms [252, 253]; leptoquarks [254, 255]; triplet Higgs [256, 257]; new scalar particles of the type predicted in the Zee-Babu [258, 259] types of models [260–262]. It is conceivable that such discoveries can eventually exclude the seesaw mechanism and rule out leptogenesis.



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