Leptogenesis in the Universe

Abstract

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

1. The Baryon Asymmetry of the Universe

1.1. Observations

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

The absence of annihilation radiation 𝑝𝑝→…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 ⁷Li, 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: 𝑛𝜂≡^𝐵−𝑛_𝐵𝑛_𝛾0, (1.1) where the subscript 0 means ”at present time.” By using only the abundance of deuterium, that is particularly sensitive to 𝜂, [4] quotes: 10¹⁰𝜂=5.7±0.6(95%c.l.). (1.2) 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): Ω_𝐵ℎ²≡ℎ²𝜌_𝐵𝜌_crit, (1.3) that is related to 𝜂 through 10¹⁰𝜂=274Ω_𝐵ℎ². 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] 10²Ω_𝐵ℎ²=2.258^+0.057_−0.056. (1.4) 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: 𝑌_Δ𝐵≡𝑛_𝐵−𝑛_𝐵𝑠₀. (1.5) The relation with the previous definitions is given by the conversion factor 𝑠₀/𝑛_𝛾0=7.04. 𝑌_Δ𝐵 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 𝑌_Δ𝐵 the BBN results (1.2) and the CMB measurement (1.4) (at 95% c.l.) read 𝑌^BBN_Δ𝐵=(8.10±0.85)×10⁻¹¹,𝑌^CMB_Δ𝐵=(8.79±0.44)×10⁻¹¹. (1.6) The impressive consistency between the determinations of the baryon density of the Universe from BBN and CMB that, besides being completely independent, also refer to epochs with a six orders of magnitude difference in temperature, provides a striking confirmation of the hot Big Bang cosmology.

1.2. Theory

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

The alternative possibility is that the Universe baryon asymmetry is generated dynamically, a scenario that is known as baryogenesis. This requires that baryon number (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 𝑌_Δ𝐵=0 to a state with 𝑌_Δ𝐵≠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 functions 𝑛_eq 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 𝑛_𝐵=∫(𝑑³𝑝/(2𝜋³))𝑛_eq=𝑛_𝐵. Although these ingredients are all present in the Standard Model (SM), so far all attempts to reproduce quantitatively the observed baryon asymmetry have failed. (1) In the SM 𝐵 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 𝑌_Δ𝐵∼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 Section 2 the basis of leptogenesis is reviewed in the simple scenario of the one-flavour regime, while the role of flavour effects is described in Section 3. Theoretical improvements of the basic pictures, like spectator effects, scatterings and CP violation in scatterings, thermal corrections, the possible role of the heavier singlet neutrinos, and quantum effects, are reviewed in Section 4. Leptogenesis in the supersymmetric seesaw is reviewed in Section 5, while in Section 6 we mention possible leptogenesis realizations that go beyond the type-I seesaw. Finally, in Section 7 we draw the conclusions.

2. 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

With 𝑚(𝑚≥2) (neutrino oscillation data and leptogenesis both require 𝑚≥2.) singlet RH neutrinos 𝑁_𝑅𝑖(𝑖=1,𝑚), we can add the following Standard Model (SM) gauge invariant terms to the SM Lagrangian: ^ℒ_𝐼^=ℒ_SM^+𝑖𝑁_𝑅𝑖/^𝜕𝑁_𝑅𝑖⁻¹₂^𝑀_𝑖^𝑁𝑐_𝑅𝑖^𝑁_𝑅𝑖^+𝜖_𝑎𝑏^𝑌_𝛼𝑖^𝑁_𝑅𝑖^ℓ𝑎_𝛼^{𝐻𝑏,+h.c.} (2.1) where 𝑀_𝑖 are the Majorana masses of the RH neutrinos, ℓ_𝛼=(𝜈_𝛼𝐿,𝛼⁻_𝐿) with 𝛼=𝑒,𝜇,𝜏 and 𝐻=(𝐻⁺,𝐻⁰) are, respectively, the left-handed (LH) lepton and Higgs SU(2)_𝐿 doublets and 𝜖_𝑎𝑏=−𝜖_𝑏𝑎 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 𝑁_𝑖=𝑁_𝑅𝑖+𝑁^𝑐_𝑅𝑖. Since 𝑁_𝑖 are gauge singlets, the scale of 𝑀_𝑖 is naturally much larger than the electroweak (EW) scale 𝑀_𝑖≫⟨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]: 𝑚_𝜈≃−𝑣²𝑌1𝑀𝑌^𝑇. (2.2) Assuming 𝑌∼𝒪(1) and 𝑚_𝜈≃Δ𝑚²_atm≃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: the three Sakharov's conditions [13] for leptogenesis are implicit in (2.1) with the lepton number violation provided by 𝑀_𝑖, the CP violation from the complexity of 𝑌_𝑖𝛼, and the departure from thermal equilibrium condition given by an additional requirement that 𝑁_𝑖 decay rate Γ_𝑁𝑖 is not very fast compared to the Hubble expansion rate of the Universe 𝐻(𝑇) at temperature 𝑇=𝑀_𝑖 with Γ_𝑁𝑖=𝑌^†𝑌_𝑖𝑖𝑀_𝑖(𝑇)=28𝜋,𝐻3𝑔_∗𝜋³5𝑇²𝑀_pl, (2.3) where 𝑀_pl=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 the decay parameter as follows: 𝐾_𝑖≡Γ_𝑁𝑖𝐻𝑀_𝑖=𝑚_𝑖𝑚_∗, (2.4) where 𝑚_𝑖 is the effective neutrino mass defined as [47] 𝑚_𝑖≡𝑌^†𝑌_𝑖𝑖𝑣²𝑀_𝑖, (2.5) with 𝑚_∗≡(16𝜋²𝑣²/3𝑀_pl)𝑔_∗𝜋/5≃1×10⁻³ eV. The regimes where 𝐾_𝑖≪1, 𝐾_𝑖≈1 and 𝐾_𝑖≫1 are, respectively, known as weak, intermediate, and strong washout regimes.

2.2. CP Asymmetry

The CP asymmetry in the decays of RH neutrinos 𝑁_𝑖 can be defined as 𝜖_𝑖𝛼=𝛾𝑁_𝑖→ℓ_𝛼𝐻−𝛾𝑁_𝑖→ℓ_𝛼𝐻^∗∑_𝛼𝛾𝑁_𝑖→ℓ_𝛼𝐻+𝛾𝑁_𝑖→ℓ_𝛼𝐻^∗≡Δ𝛾^𝛼_𝑁𝑖𝛾_𝑁𝑖, (2.6) 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, 𝑓_𝑖=𝑒^{−𝐸𝑖/𝑇}; see [48, 49] for detailed studies of their effects) 𝛾𝑑𝑖→𝑓≡³𝑝_𝑖(2𝜋)³2𝐸_𝑖𝑑³𝑝_𝑓(2𝜋)³2𝐸_𝑓(2𝜋)⁴𝛿⁽⁴⁾𝑝_𝑖−𝑝_𝑓𝒜𝑖→𝑓²𝑒^{−𝐸𝑖/𝑇}, (2.7) where 𝒜(𝑖→𝑓) is the decay amplitude. Ignoring all thermal effects [48, 49], (2.6) simplifies to 𝜖_𝑖𝛼=𝒜₀𝑁_𝑖→ℓ_𝛼𝐻²−𝒜₀𝑁_𝑖→ℓ_𝛼𝐻^∗2∑_𝛼𝒜₀𝑁_𝑖→ℓ_𝛼𝐻²+𝒜₀𝑁_𝑖→ℓ_𝛼𝐻^∗2, (2.8) 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]: 𝜖_𝑖𝛼=118𝜋𝑌^†𝑌_{𝑖𝑖𝑗≠𝑖}Im𝑌^†𝑌_𝑗𝑖𝑌_𝛼𝑖𝑌^∗_𝛼𝑗𝑔𝑀²_𝑗𝑀²_𝑖+118𝜋𝑌^†𝑌_{𝑖𝑖𝑗≠𝑖}Im𝑌^†𝑌_𝑖𝑗𝑌_𝛼𝑖𝑌^∗_𝛼𝑗𝑀²_𝑖𝑀²_𝑖−𝑀²_𝑗, (2.9) where the loop function is 𝑔(𝑥)=√𝑥1()1−𝑥+1−1+𝑥ln1+𝑥𝑥. (2.10) 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 (𝑀²_𝑖−𝑀²_𝑗)⁻¹ in (2.9) are from the wave diagram contributions which can resonantly enhance the CP asymmetry if 𝑀_𝑖≈𝑀_𝑗 (resonant leptogenesis scenario, see Section 6.1). Notice that the resonant term becomes singular in the degenerate limit 𝑀_𝑖=𝑀_𝑗. 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 from 𝐿-conserving diagram to the CP asymmetry which vanishes when summing over lepton flavours.

In the one flavour regime, we sum over the flavour index 𝛼 in (2.9) and obtain 𝜖_𝑖≡_𝛼𝜖_𝑖𝛼=118𝜋𝑌^†𝑌_{𝑖𝑖𝑗≠𝑖}Im𝑌^†𝑌²_𝑗𝑖𝑔𝑀²_𝑗𝑀²_𝑖, (2.11) where the second term in (2.9) vanishes because the combination of the Yukawa couplings is real.

2.3. Classical Boltzmann Equations

We work in the one-flavour regime and consider only the decays and inverse decays of 𝑁₁. 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 𝑇_RH≪𝑀_2,3 such that 𝑁_2,3 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 𝑛_𝑖=∫𝑑³𝑝𝑓_𝑖 to the entropy density 𝑠=(2𝜋²/45)𝑔_∗𝑇³: 𝑌_𝑖≡𝑛_𝑖𝑠. (2.12) The evolution of the 𝑁₁ density and the lepton asymmetry 𝑌_Δ𝐿=2𝑌_Δℓ≡2(𝑌_ℓ−𝑌_ℓ) (the factor of 2 comes from the SU(2)_𝐿 degrees of freedoms) can be described by the following classical Boltzmann equations (BE) [53]: 𝑑𝑌_𝑁1𝑑𝑧=−𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1,𝑑𝑌_Δ𝐿𝑑𝑧=𝜖₁𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1−𝑊₁𝑌_Δ𝐿, (2.13)(2.14) where 𝑧≡𝑀₁/𝑇 and the decay and washout terms are, respectively, given by 𝐷₁(𝑧)=𝛾_𝑁1𝑧𝑠𝐻𝑀₁=𝐾₁𝑧𝒦₁(𝑧)𝒦₂(𝑧),𝑊₁(𝑧)=12𝐷₁(𝑧)𝑌^eq_𝑁1(𝑧)𝑌^eq_ℓ, (2.15) with 𝒦_𝑛 the 𝑛th order modified Bessel function of second kind. 𝑌^eq_𝑁1 and 𝑌^eq_ℓ 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 𝑌^eq_ℓ is obtained from a Fermi-Dirac distribution) 𝑌^eq_𝑁(𝑧)=452𝜋⁴𝑔_∗𝑧²𝒦₂(𝑧),𝑌^eq_ℓ=154𝜋²𝑔_∗. (2.16)

From (2.13) and (2.14), the solution for 𝑌_Δ𝐿 can be written down as follows: 𝑌_Δ𝐿(𝑧)=𝑌_Δ𝐿𝑧_𝑖𝑒^{−∫𝑧𝑧𝑖𝑑𝑧′𝑊1(𝑧′)}−^𝑧_𝑧𝑖𝑑𝑧^′𝜖1𝑧^′𝑑𝑌_𝑁1𝑑𝑧^′𝑒^{−∫𝑧𝑧′𝑑𝑧′′𝑊1(𝑧′′)}, (2.17) where 𝑧_𝑖 is some initial temperature when 𝑁₁ leptogenesis begins, and we have assumed that any preexisting lepton asymmetry vanishes 𝑌⁰_Δ𝐿(𝑧_𝑖)=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 𝑌_𝑁1(𝑧_𝑖) is important. If we assume thermal initial abundance of 𝑁₁, that is, 𝑌_𝑁1(𝑧_𝑖)=𝑌^eq_𝑁1(0), we can ignore the washout when 𝑁₁ starts decaying at 𝑧≫1 and we have 𝑌^𝑡_Δ𝐿(∞)≃−𝜖₁^∞₀𝑑𝑧^′𝑑𝑌^eq_𝑁1𝑑𝑧^′=𝜖₁𝑌^eq_𝑁1(0). (2.18) On the other hand, if we have zero initial 𝑁₁ abundance, that is, 𝑌_𝑁1(𝑧_𝑖)=0, we have to consider the opposite sign contributions to lepton asymmetry from the inverse decays when 𝑁₁ is being populated (𝑌_𝑁1<𝑌^eq_𝑁1) and from the period when 𝑁₁ starts decaying (𝑌_𝑁1>𝑌^eq_𝑁1). 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 𝑌^eq_ℓ (2.16)) 𝑌⁰_Δ𝐿(∞)≃27𝜖16¹𝐾²₁𝑌^eq_𝑁1(0). (2.19)

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 𝑌_Δ𝐿(𝑧)1≈−𝑊(𝑧)𝜖₁𝑑𝑌_𝑁11𝑑𝑧≃−𝑊(𝑧)𝜖₁𝑑𝑌^eq_𝑁1=2𝑑𝑧𝑧𝐾₁𝜖₁𝑌^eq_ℓ, (2.20) where in the second approximation, we assume 𝑌_𝑁1≃𝑌^eq_𝑁1. The approximation no longer holds when 𝑌_Δ𝐿 freezes, and this happens when the washout decouples at 𝑧_𝑓, that is, 𝑊(𝑧_𝑓)<1. Hence, the final lepton asymmetry is given by (compare this to a more precise analytical approximation in [55]) 𝑌_Δ𝐿(∞)=2𝑧_𝑓𝐾₁𝜖₁𝑌^eq_ℓ=𝜋²6𝑧_𝑓𝐾₁𝜖₁𝑌^eq_𝑁1(0). (2.21) The freeze out temperature 𝑧_𝑓 depends mildly on 𝐾₁. For 𝐾₁=10–100 we have, for example, 𝑧_𝑓=7– 10. We also see that independently of initial conditions, in the strong regime 𝑌_Δ𝐿(∞) goes as 𝐾₁⁻¹.

2.4. Baryon Asymmetry from EW Sphaleron

The final lepton asymmetry 𝑌_Δ𝐿(∞) can be conveniently parametrized as follows: 𝑌_Δ𝐿(∞)=𝜖₁𝜂₁𝑌^eq_𝑁1(0), (2.22) where 𝜂₁ is known as the efficiency factor. In the weak washout regime (𝐾₁≪1) from (2.18) we have 𝜂₁=1(=(27/16)𝐾²₁<1) for thermal (zero) initial 𝑁₁ abundance. In the strong washout regime (𝐾₁≫1), from (2.21), we have 𝜂₁=(𝜋²/6𝑧_𝑓𝐾₁)<1.

If leptogenesis ends before EW sphaleron processes become active (𝑇≳10¹² GeV), the 𝐵−𝐿 asymmetry 𝑌_{Δ𝐵−𝐿} is simply given by 𝑌_{Δ𝐵−𝐿}=−𝑌_Δ𝐿. (2.23) At the later stage, the 𝐵−𝐿 asymmetry is partially transferred to a 𝐵 asymmetry by the EW sphaleron processes through the relation [57] 𝑌_Δ𝐵(∞)=28𝑌79^{Δ𝐵−𝐿}(∞), (2.24) 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 𝜖₁3=−116𝜋𝑌^†𝑌_11𝑗≠1Im𝑌^†𝑌²_𝑗1𝑀₁𝑀_𝑗. (2.25) Assuming three generations of RH neutrinos (𝑛=3) and using the Casas-Ibarra parametrization [60] for the Yukawa couplings 𝑌_𝛼𝑖=1𝑣𝐷_𝑚𝑁𝑅𝐷_𝑚𝜈𝑈^†_𝜈𝛼𝑖, (2.26) where 𝐷_𝑚𝑁=diag(𝑀₁,𝑀₂,𝑀₃), 𝐷_𝑚𝜈=diag(𝑚_𝜈1,𝑚_𝜈2,𝑚_𝜈3) and 𝑅 any complex orthogonal matrix satisfying 𝑅^𝑇𝑅=𝑅𝑅^𝑇=1, (2.25) becomes 𝜖₁3=−𝑀16𝜋𝑣¹²∑_𝑖𝑚_𝜈𝑖Im𝑅²_1𝑖∑_𝑖𝑚_𝜈𝑖𝑅_1𝑖². (2.27) Using the orthogonality condition ∑_𝑖𝑅²_1𝑖=1, we then obtain the Davidson-Ibarra (DI) bound [61] 𝜖₁≤𝜖^DI=3𝑀16𝜋𝑣¹²𝑚_𝜈3−𝑚_𝜈1=3𝑀16𝜋𝑣¹²Δ𝑚²_atm𝑚_𝜈1+𝑚_𝜈3, (2.28) where 𝑚_𝜈3 (𝑚_𝜈1) is the heaviest (lightest) light neutrino mass. Applying the DI bound on (2.22)–(2.24), and requiring that 𝑌_Δ𝐵(∞)≥𝑌^CMB_𝐵≃10⁻¹⁰, we obtain 𝑀₁0.1eV𝑚_𝜈1+𝑚_𝜈3𝜂₁^max𝑀₁≳10⁹GeV, (2.29) where the 𝜂₁^max(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,𝑚_𝜈1) plane and hence obtain bounds on 𝑀₁ and 𝑚_𝜈1. 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 𝑚_𝜈1≲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 𝑇_RH≳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 hierarchical 𝑀₁≪𝑀₂,𝑀₃. (3) Leptogenesis occurs in the unflavoured regime 𝑇≳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 neutrino 𝑁₁ maintain their coherence as flavour superpositions throughout the leptogenesis era, that is ℓ₁=∑_𝛼𝑐_𝛼1ℓ_𝛼 and ℓ^′₁=∑_𝛼𝑐^′∗_𝛼1ℓ_𝛼. Note that at the tree level the coefficients 𝑐 and 𝑐^′∗ are simply the Yukawa couplings: 𝑐_𝛼1=𝑌_𝛼1 and 𝑐^′∗_𝛼1=𝑌^∗_𝛼1. However it should be kept in mind that since CP is violated by loops, beyond the tree level approximation the antilepton state ℓ^′₁ is not the CP conjugate of the ℓ₁, that is, 𝑐^′_𝛼1≠𝑐_𝛼1.

The single flavour regime is realized only at very high temperatures (𝑇≳10¹²GeV) when both ℓ₁ and ℓ^′₁ 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 ℓ^′₁ decohere in their flavour components, and the dynamics of leptogenesis must then be described in terms of the flavour eigenstates ℓ_𝛼. 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 processes Γ_𝛼 (𝛼=𝑒,𝜇,𝜏) that are induced by the charged lepton Yukawa couplings ℎ_𝛼 become faster than the Universe expansion rate 𝐻(𝑇). An approximate relation gives [90, 91] Γ_𝛼(𝑇)≃10⁻²ℎ²_𝛼𝑇, (3.1) which implies that (in supersymmetric case, since ℎ_𝛼=𝑚_𝛼/(𝑣_𝑢cos𝛽), we have 𝑇≲𝑇_𝛼(1+tan²𝛽)) Γ_𝛼(𝑇)>𝐻(𝑇)when𝑇≲𝑇_𝛼, (3.2) where 𝑇_𝑒≃4×10⁴GeV, 𝑇_𝜇≃2×10⁹GeV, and 𝑇_𝜏≃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 Γ_𝜏(𝑀₁)>𝐻(𝑀₁) we also have Γ_𝜏(𝑀₁)>Γ_𝑁1(𝑀₁). However, there exists parameter space where Γ_𝜏(𝑀₁)>𝐻(𝑀₁) but Γ_𝜏(𝑀₁)<Γ_𝑁1(𝑀₁). This scenario was studied in [83].

3.2. The Effects on CP Asymmetry and Washout

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

(i) The leptons and antileptons are produced at different rates: 𝛾_𝑖≠𝛾_𝑖, (3.3) where 𝛾_𝑖≡𝛾(𝑁_𝑖→ℓ_𝑖𝐻) and 𝛾_𝑖≡𝛾(𝑁_𝑖→ℓ^′_𝑖𝐻^∗).

(ii) The leptons and antileptons produced are not CP conjugate states: CPℓ^′_𝑖=ℓ^′_𝑖≠ℓ_𝑖, (3.4) that is, due to loops effects they are slightly misaligned in flavour space.

We can rewrite the CP asymmetry for 𝑁_𝑖 decays from (2.6) as follows: 𝜖_𝑖𝛼=𝑃_𝑖𝛼𝛾_𝑖−𝑃_𝑖𝛼𝛾_𝑖𝛾_𝑖+𝛾_𝑖=𝑃_𝑖𝛼+𝑃_𝑖𝛼2𝜖_𝑖+𝑃_𝑖𝛼−𝑃_𝑖𝛼2≃𝑃⁰_𝑖𝛼𝜖_𝑖+Δ𝑃_𝑖𝛼2, (3.5) where terms of order 𝒪(𝜖_𝑖Δ𝑃_𝑖𝛼) and higher have been neglected. 𝑃_𝑖𝛼 is the projector from state ℓ_𝑖 into flavour state ℓ_𝛼 and Δ𝑃_𝑖𝛼=𝑃_𝑖𝛼−𝑃_𝑖𝛼. At tree level, clearly, 𝑃_𝑖𝛼=𝑃_𝑖𝛼≡𝑃⁰_𝑖𝛼 where the tree level flavour projector is given by 𝑃⁰_𝑖𝛼=𝑌_𝛼𝑖𝑌^∗_𝛼𝑖𝑌^†𝑌_𝑖𝑖. (3.6) 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 ∑_𝛼𝑃_𝑖𝛼=∑_𝛼𝑃_𝑖𝛼=1, when summing over flavour indices 𝛼, the second term vanishes ∑_𝛼Δ𝑃_𝑖𝛼=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 𝜖_𝑖𝛼 can be much larger and even of opposite sign from the total CP asymmetry 𝜖_𝑖. 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 𝑊_𝑖𝛼∝𝑃_𝑖𝛼𝛾_𝑖+𝑃_𝑖𝛼𝛾_𝑖≃𝑃⁰_𝑖𝛼𝑊_𝑖, (3.7) which results in a reduction of washout by a factor of 𝑃⁰_𝑖𝛼≤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 𝜖_𝑖=0 also becomes possible.

3.3. Classical Flavoured Boltzmann Equations

Here again we only consider leptogenesis from the decays and inverse decays of 𝑁₁. In this approximation, the BE for 𝑌_𝑁1 is still given by (2.13) while the BE for 𝑌_Δ𝐿𝛼 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]). 𝑑𝑌_Δ𝐿𝛼𝑑𝑧=𝜖_1𝛼𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1−𝑃⁰_1𝛼𝑊₁𝑌_Δ𝐿𝛼. (3.8) Notice that as long as 𝐿 violation from sphalerons is neglected (see Section 4) the BEs for 𝑌_Δ𝐿𝛼 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→𝜖_1𝛼 and 𝐾₁→𝐾_1𝛼≡𝑃⁰_1𝛼𝐾₁.

As an example let us assume that leptogenesis occurs around 𝑇∼10¹⁰GeV, that is, in the two-flavour regime. Due to the fast 𝜏 Yukawa interactions ℓ₁(ℓ^′₁) gets projected onto ℓ_𝜏(ℓ^′_𝜏) and a coherent mixture of 𝑒+𝜇 eigenstate ℓ_𝑒+𝜇(ℓ^′_𝑒+𝜇). For illustrative purpose, here we consider a scenario in which lepton flavour effects are most prominent. We take both 𝐾_1𝜏,𝐾_1𝑒+𝜇≫1, so that both 𝑌_Δ𝐿𝜏 and 𝑌_{Δ𝐿𝑒+𝜇} are in the strong regime. From (2.21) we can write down the solution: 𝑌_Δ𝐿(∞)=𝑌_Δ𝐿𝜏(∞)+𝑌_{Δ𝐿𝑒+𝜇}(∞)=𝜋²6𝑧_𝑓𝐾₁𝑌^eq_𝑁1(0)𝜖_1𝜏𝑃⁰_1𝜏+𝜖_1𝑒+𝜇𝑃⁰_1𝑒+𝜇≃𝜋²3𝑧_𝑓𝐾₁𝜖₁𝑌^eq_𝑁1(0)+𝜋²12𝑧_𝑓𝐾₁𝑌^eq_𝑁1(0)Δ𝑃_1𝜏𝑃⁰_1𝜏+Δ𝑃_1𝑒+𝜇𝑃⁰_1𝑒+𝜇, (3.9) where in the last line we have used (3.5). If 𝑃⁰_1𝜏≃𝑃⁰_1𝑒+𝜇, then since Δ𝑃_1𝜏+Δ𝑃_1𝑒+𝜇=0, the second term approximately cancels, and (3.9) reduces to 𝑌_Δ𝐿(∞)≃𝜋²3𝑧_𝑓𝐾₁𝜖₁𝑌^eq_𝑁1(0). (3.10) 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 𝑃⁰_1𝜏>𝑃⁰_1𝑒+𝜇 while Δ𝑃_1𝜏≪Δ𝑃_1𝑒+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, ℓ_𝛼𝐻→ℓ_𝛽𝐻, and can proceed, for example, via off-shell exchange of 𝑁_2,3. 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: 𝑑𝑌_Δ𝐿𝑑𝑧=𝜖₁𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1−𝑃⁰_1𝛼𝑊₁𝑌_Δ𝐿, (3.11) where 𝑃⁰_1𝛼=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 𝑁_2,3 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 in 𝑁₁ 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 of 𝐵−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 𝑌_Δ𝐻 [106]. This was omitted in Section 2 but in fact has to be added to the density asymmetry of the leptons 𝑌_Δℓ in weighting, for example, washouts from inverse decays. Equation (2.14) would then become 𝑑𝑌_Δ𝐿𝑑𝑧=𝜖₁𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1−2𝑌_Δℓ+𝑌_Δ𝐻𝑊₁, (4.1) where the factor of two in front of the washout term counts the leptons and Higgs gauge multiplicity. Clearly, in some regimes in which 𝑌_Δℓ and 𝑌_Δ𝐻 are not sufficiently diluted by interacting with other particles, this can have the effect of enhancing the washout rates and suppressing the efficiency.

In the study of spectator processes it is fundamental to specify the range of temperature in which leptogenesis occurs. This is because at each specific temperature 𝑇, 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≪𝑡_𝑈(𝑇); (2) much larger than the age of the Universe: 𝜏≫𝑡_𝑈(𝑇); (3) comparable with the Universe age: 𝜏∼𝑡_𝑈(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, ∑_𝐼𝜇_𝐼=∑_𝐹𝜇_𝐹, where 𝜇_𝐼 denotes the chemical potential of an initial state particle and 𝜇_𝐹 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: 2𝑌_Δ𝑄3+𝑌_Δ𝑡=0, (4.2) where 𝑌_Δ𝑄3 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 𝑌_Δ𝑡 is the density asymmetry of the SU(2)_𝐿 singlet top. Hypercharge is always conserved, yielding 𝑌_Δ𝑄3+2𝑌_Δ𝑡−𝑌_Δℓ+𝑌_Δ𝐻=0. (4.3) Finally, in terms of density asymmetries chemical equilibrium for the top-Yukawa-related reactions 𝜇_𝑄3+𝜇_𝐻=𝜇_𝑡 translates into 𝑌_Δ𝑄3+12𝑌_Δ𝐻=𝑌_Δ𝑡. (4.4) 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 𝑌_Δ𝐻=(2/3)𝑌_Δℓ. Moreover, given that only the LH lepton degrees of freedom are populated, we have 𝑌_Δ𝐿=2𝑌_Δℓ so that the coefficient weighting 𝑊₁ in (4.1) becomes 2(𝑌_Δℓ+𝑌_Δ𝐻)=(5/3)𝑌_Δ𝐿 and the washout is accordingly stronger.

With decreasing temperatures, more reactions attain chemical equilibrium, and accounting for spectator processes becomes accordingly more complicated. When the temperature drops below 𝑇∼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 Δ_𝜏=𝐵/3−𝐿_𝜏 and Δ_𝑒𝜇=𝐵/3−𝐿_𝑒𝜇. 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 Δ^′_𝑒𝜇 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 −𝑑𝑌_Δ𝛼𝑑𝑧=𝜖_𝛼𝐷₁𝑌_𝑁1−𝑌^eq_𝑁1−2𝑌_Δℓ𝛼+𝑌_Δ𝐻𝑊₁𝛼=𝜏,𝑒𝜇. (4.5) To rewrite these equations in a solvable closed form, 𝑌_Δℓ𝜏, 𝑌_{Δℓ𝑒𝜇}, and 𝑌_Δ𝐻 must be expressed in terms of the two charge densities 𝑌_Δ𝜏 and 𝑌_Δ𝑒𝜇. 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 𝑌_Δℓ𝛼 and 𝑌_Δ𝐻 to the charge densities 𝑌_Δ𝛼 can be expressed in terms of the 𝐴 matrix introduced in [67] 𝑌_Δℓ𝛼=𝐴^ℓ_𝛼𝛽𝑌_Δ𝛽(𝛼,𝛽=𝜏,𝑒𝜇) and 𝐶-vector 𝑌_Δ𝐻=𝐶^𝐻_𝛼𝑌_Δ𝛼 introduced in [69]. For the present case, with the ordering (𝑒𝜇,𝜏) they are [69] 𝐴^ℓ=1460196−24−9156,𝐶^𝐻=1()23041,56. (4.6) 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 𝑌_Δ𝛼. 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 the 𝑁₁ population, because decay and inverse decay rates are suppressed by a time dilation factor ∝𝑀₁/𝑇. The 𝑁₁=𝑁₁ particles can be produced by scatterings with the top quark in 𝑠-channel 𝐻-exchange 𝑞𝑡^𝑐→𝑁ℓ_𝛼 and 𝑞𝑡^𝑐→𝑁ℓ_𝛼, by 𝑡-channel 𝐻-exchange in 𝑞ℓ_𝛼→𝑁𝑡^𝑐, 𝑞ℓ_𝛼→𝑁𝑡^𝑐 and by 𝑢-channel 𝐻-exchange in ℓ_𝛼𝑡^𝑐→𝑁𝑞, 𝑡^𝑐ℓ_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.

(a)

(b)

(c)

Figure 2 

Diagrams for various 2↔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).

When the effects of scatterings in populating the 𝑁₁ 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 (ℎ²_𝑡𝜆⁴, 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 𝑁_𝑗. 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 as 𝑁ℓ_𝛼→𝐴𝐻𝑁𝐻→𝐴ℓ_𝛼 or 𝑁𝐴→ℓ_𝛼𝐻, 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 𝑚_𝐻(𝑇)≳𝑚_𝑞3,𝑡(𝑇)≫𝑚_ℓ(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 (𝑚_𝐻(𝑇)≈0.4𝑇), decays and inverse decays become kinematically forbidden in the temperature range in which 𝑚_𝐻(𝑇)−𝑚_ℓ(𝑇)<𝑀_𝑁1<𝑚_𝐻(𝑇)+𝑚_ℓ(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 𝛾^top_𝐻𝑠≡𝛾(𝑞₃𝑡↔ℓ𝑁₁) with the Higgs exchanged in the 𝑠-channel and of the sum of the 𝑡- and 𝑢-channel scatterings 𝛾^top_{𝐻𝑡+𝑢}≡𝛾(𝑞₃𝑁₁↔ℓ𝑡)+𝛾(𝑡𝑁₁↔ℓ𝑞₃) shows that the only corrections appearing at low temperatures, and thus more relevant, are for 𝛾^top_{𝐻𝑡+𝑢} (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(𝑀_𝑁1/𝑚_𝐻) from the quasimassless Higgs exchanged in the 𝑡- and 𝑢-channels, which disappears when the Higgs thermal mass 𝑚_𝐻(𝑇)∼𝑇∼𝑀_𝑁1 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 which 𝑁₁ 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 decaying 𝑁’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 𝑁_2,3 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, 𝑀_2,3≫𝑀₁, the related effects are Boltzmann suppressed by exp(−𝑀_2,3/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 −𝑖𝑆⁰_ℓ(𝑝,𝑚_ℓ)=(_╱𝑝−𝑚_ℓ+𝑖0⁺)⁻¹ and −𝑖𝐷⁰_𝐻(𝑝,𝑚_𝐻)=(𝑝²−𝑚²_𝐻+𝑖0⁺)⁻¹ acquire an additive term that is proportional to the particle density distribution 𝑛_ℓ,𝐻=[exp(𝐸_ℓ,𝐻/𝑇)±1]⁻¹: 𝛿𝑆_ℓ(𝑇)=−2𝜋𝑛_ℓ╱𝑝−𝑚_ℓ𝛿𝑝²−𝑚²_ℓ,𝛿𝐷_𝐻(𝑇)=+2𝜋𝑛_𝐻𝛿𝑝²−𝑚²_𝐻. (4.7) 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 −𝑛_ℓ+𝑛_𝐻−2𝑛_ℓ𝑛_𝐻 that vanishes when the final states thermal masses are neglected, because the Bose-Einstein and Fermi-Dirac statistical distributions depend on the same argument, 𝐸_ℓ=𝐸_𝐻=𝑀₁/2. As a consequence, the thermal corrections to the fermion and boson propagators (𝑛_ℓ and 𝑛_𝐻) and the product of the two thermal corrections (𝑛_ℓ𝑛_𝐻) 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 −𝑛_ℓ+𝑛_𝐻, which does not vanish even in the massless approximation. This would result in a further enhancement in the CP asymmetry from the thermal effects [127].

4.3.5. Particle Motion

Given that the decaying particle 𝑁₁ 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 𝐸_ℓ,𝐻=𝑀₁/2∓(𝑚²_𝐻−𝑚²_ℓ)/2𝑀₁. Since now 𝐸_ℓ≠𝐸_𝐻, the prefactor −𝑛_ℓ+𝑛_𝐻−2𝑛_ℓ𝑛_𝐻 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 𝑚_𝐻(𝑇)>𝑚_ℓ(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 𝜖_𝐻 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, 𝜖_𝐻 is nonvanishing only at temperatures 𝑇≳𝑇_𝐻∼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≳(𝑇_𝐻/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 at 𝑇≳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 of 𝑁_2,3 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 𝑚₁≪𝑚_∗ implies that 𝑁₁ washout effects are negligible. It is then clear that any asymmetry generated in 𝑁_2,3 decays can survive and becomes crucial for the success of leptogenesis. Another case in which it is intuitively clear that 𝑁_2,3 effects can be important is when the RH neutrino spectrum is compact, which means that 𝑀_2,3 have values within a factor of a few from 𝑀₁. Then 𝑁₁ and 𝑁_2,3 contributions to leptogenesis can be equally important and must be summed up. A model with compact RH neutrino spectrum in which 𝑁_2,3 dynamics is of crucial importance was recently discussed in [128].

It is less obvious that 𝑁_2,3 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 𝑁_2,3 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: 𝑚≫₂^/𝑚_∗,𝑚₁≫𝑚_∗. (4.8) Assuming also a strong hierarchy and that leptogenesis occurs thermally guarantees that [131] 𝑛_𝑁1𝑇∼𝑀₂≈0,𝑛_𝑁2𝑇∼𝑀₁≈0. (4.9) 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: ℓ₂=𝑌^†𝑌^−1/2_22𝛼𝑌_𝛼2ℓ_𝛼. (4.10) 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,ℓ^′₀) where, without loss of generality, ℓ^′₀ satisfies ⟨ℓ^′₀∣ℓ₂⟩=0. Then the asymmetry Δ𝑌_ℓ2 produced in 𝑁₂ decays decomposes into two components: Δ𝑌_ℓ0=𝑐²Δ𝑌_ℓ2,Δ𝑌_ℓ1=𝑠²Δ𝑌_ℓ2, (4.11) 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, Δ𝑌_ℓ0 is protected against 𝑁₁ washout. Consequently, a finite part of the asymmetry Δ𝑌_ℓ2 from 𝑁₂ decays survives through 𝑁₁ leptogenesis. A more detailed analysis [131] finds that Δ𝑌_ℓ1 is not entirely washed out, resulting in the final lepton asymmetry 𝑌_Δ𝐿=(3/2)Δ𝑌_ℓ0=(3/2)𝑐²Δ𝑌_ℓ2.

For 10⁹GeV ≲𝑀₁≲10¹²GeV, flavour issues modify the quantitative details, but the qualitative picture, and in particular the survival of a finite part of Δ𝑌_ℓ2, still holds. In contrast, for 𝑀₁≲10⁹GeV, the full flavour basis (ℓ_𝑒,ℓ_𝜇,ℓ_𝜏) is resolved, and thus there are no directions in flavour space where an asymmetry is protected, so that 𝑌_ℓ2 can be erased entirely. A dedicated study in which the various flavour regimes for 𝑁_1,2,3 decays are considered can be found in [132].

In conclusion 𝑁_2,3 leptogenesis cannot be ignored, unless one of the following conditions holds.(1) The reheat temperature is below 𝑀₂. (2) The asymmetries and/or the washout factors vanish, 𝜖_𝑁2𝜂₂≈0 and 𝜖_𝑁3𝜂₃≈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)neutrinos 𝑁₁ 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 Γ_𝑁2. 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/Γ_𝑁1 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 from 𝑡₀ 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 𝑍𝐽,𝐽^†=Tr𝒯exp𝑖_𝐶𝐽(𝑥)𝜙(𝑥)+𝐽^†(𝑥)𝜙^†(𝑥)𝑑⁴𝑥𝜌=Tr𝒯₊exp𝑖𝐽₊(𝑥)𝜙₊(𝑥)+𝐽^†₊(𝑥)𝜙^†₊(𝑥)𝑑⁴𝑥×𝒯₋exp−𝑖𝐽₋(𝑥)𝜙₋(𝑥)+𝐽^†₋(𝑥)𝜙^†₋(𝑥)𝑑⁴𝑥𝜌, (4.12) 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 field 𝜙 they are defined as 𝐺^>_𝜙𝑥,𝑦=−𝐺_𝜙⁻⁺𝑥,𝑦=−𝑖𝜙(𝑥)𝜙^†𝑦,𝐺^<_𝜙𝑥,𝑦=−𝐺_𝜙⁺⁻𝑥,𝑦=−𝑖𝜙^†𝑦𝜙(𝑥,𝐺_𝑡𝜙)𝑥,𝑦=𝐺_𝜙⁺⁺𝑥,𝑦=𝜃𝑥,𝑦𝐺^>_𝜙𝑥,𝑦+𝜃𝑦,𝑥𝐺^<_𝜙,𝐺𝑥,𝑦_𝑡𝜙𝑥,𝑦=𝐺_𝜙⁻⁻𝑥,𝑦=𝜃𝑦,𝑥𝐺^>_𝜙𝑥,𝑦+𝜃𝑥,𝑦𝐺^<_𝜙,𝐺𝑥,𝑦_𝑟𝜙𝑥,𝑦=𝐺^𝑡_𝜙−𝐺^<_𝜙=𝐺^>_𝜙−𝐺^𝑡_𝜙,𝐺^𝑎_𝜙𝑥,𝑦=𝐺^𝑡_𝜙−𝐺^>_𝜙=𝐺^<_𝜙−𝐺^𝑡_𝜙, (4.13) where the last two Green's functions are the retarded and advanced Green's functions, respectively, and 𝜃(𝑥,𝑦)≡𝜃(𝑡_𝑥−𝑡_𝑦) is the step function.

For a generic fermion field 𝜓 the six different propagators are analogously defined as 𝐺^>_𝜓𝑥,𝑦=−𝐺_𝜓⁻⁺𝑥,𝑦=−𝑖𝜓(𝑥)𝜓𝑦,𝐺^<_𝜓𝑥,𝑦=−𝐺_𝜓⁺⁻𝑥,𝑦=+𝑖𝜓𝑦𝜓(𝑥,𝐺_𝑡𝜓)𝑥,𝑦=𝐺_𝜓⁺⁺𝑥,𝑦=𝜃𝑥,𝑦𝐺^>_𝜓𝑥,𝑦+𝜃𝑦,𝑥𝐺^<_𝜓,𝐺𝑥,𝑦_𝑡𝜓𝑥,𝑦=𝐺_𝜓⁻⁻𝑥,𝑦=𝜃𝑦,𝑥𝐺^>_𝜓𝑥,𝑦+𝜃𝑥,𝑦𝐺^<_𝜓,𝐺𝑥,𝑦_𝑟𝜓𝑥,𝑦=𝐺^𝑡_𝜓−𝐺^<_𝜓=𝐺^>_𝜓−𝐺^𝑡_𝜓,𝐺^𝑎_𝜓𝑥,𝑦=𝐺^𝑡_𝜓−𝐺^>_𝜓=𝐺^<_𝜓−𝐺^𝑡_𝜓. (4.14) From the definitions of the Green's functions, one can see that the hermiticity properties 𝑖𝛾⁰𝐺_𝜓(𝑥,𝑦)^†=𝑖𝛾⁰𝐺_𝜓𝑦,𝑥,𝑖𝐺_𝜙(𝑥,𝑦)^†=𝑖𝐺_𝜙𝑦,𝑥 (4.15) are satisfied. For interacting systems, whether in equilibrium or not, one must define and calculate self-energy functions. Again, there are six of them: Σ^𝑡, Σ^𝑡, Σ^<, Σ^>, Σ^𝑟, and Σ^𝑎. The same relationships exist among them as for the Green's functions in (4.13) and (4.14), such as Σ^𝑟=Σ^𝑡−Σ^<=Σ^>−Σ^𝑡,Σ^𝑎=Σ^𝑡−Σ^>=Σ^<−Σ^𝑡. (4.16) 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: 𝐺𝐺=^𝑡±𝐺^<𝐺^>−𝐺^𝑡,ΣΣ=^𝑡±Σ^<Σ^>−Σ^𝑡, (4.17) 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: 𝐺𝑥,𝑦=𝐺⁰𝑥,𝑦+𝑑⁴𝑧₁𝑑⁴𝑧₂𝐺⁰𝑥,𝑧₁Σ𝑧₁,𝑧₂𝐺𝑧₂,𝑦, (4.18) 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 𝐺⁰ on the right in the interaction terms: 𝐺𝑥,𝑦=𝐺⁰𝑥,𝑦+𝑑⁴𝑧₃𝑑⁴𝑧₄𝐺𝑥,𝑧₃Σ𝑧₃,𝑧₄𝐺⁰𝑧₄,𝑦. (4.19) Equations (4.18) and (4.19) are the starting points to derive the quantum BE describing the time evolution of the CP-violating particle density asymmetries.

To proceed, one has to choose a form for the propagators. For a generic fermion 𝜓 (and similarly for scalars) one may adopt the real-time propagator in the form 𝐺^𝑡_𝜓(𝐤,𝑡_𝑥−𝑡_𝑦) in terms of the spectral function 𝜌_𝜓(𝐤,𝑘₀)𝐺^𝑡_𝜓𝐤,𝑡_𝑥−𝑡_𝑦=^+∞_−∞𝑑𝑘⁰𝑒2𝜋^{−𝑖𝑘0(𝑡𝑥−𝑡𝑦)}𝜌_𝜓𝐤,𝑘⁰×1−𝑓_𝜓𝑘⁰𝜃𝑡_𝑥−𝑡_𝑦−𝑓_𝜓𝑘⁰𝜃𝑡_𝑦−𝑡_𝑥, (4.20) where 𝑓_𝜓(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 Γ_𝜓(𝑘). For a fermion with chiral mass 𝑚_𝜓, one may safely choose ^𝜌_𝜓^{𝐤,𝑘0=𝑖}/^𝑘+𝑚_𝜓¹_{𝑘0+𝑖𝜖+𝑖Γ𝜓2−𝜔2𝜓(𝑘)}⁻¹_{𝑘0−𝑖𝜖−𝑖Γ𝜓2−𝜔2𝜓(𝑘)}^, (4.21) where 𝜔²_𝜓(𝑘)=𝐤²+𝑀²_𝜓(𝑇) and 𝑀_𝜓(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 ^𝐺>_𝜓^𝐤,𝑡_𝑥^−𝑡_𝑦^𝑖=−_2𝜔𝜓^𝑘+/^𝑚_𝜓^1−𝑓_𝜓^𝜔_𝜓^−𝑖Γ_𝜓^{𝑒−𝑖(𝜔𝜓−𝑖Γ𝜓)(𝑡𝑥−𝑡𝑦)}_+𝛾0^𝑘−𝑚_𝜓^𝛾0𝑓_𝜓^𝜔_𝜓^+𝑖Γ_𝜓^𝑒/^{𝑖(𝜔𝜓+𝑖Γ𝜓)(𝑡𝑥−𝑡𝑦),}_{𝐺<𝜓𝐤,𝑡𝑦−𝑡𝑥=𝑖2𝜔𝜓}^𝑘+/^𝑚_𝜓^𝑓_𝜓^𝜔_𝜓^+𝑖Γ_𝜓^{𝑒−𝑖(𝜔𝜓−𝑖Γ𝜓)(𝑡𝑥−𝑡𝑦)}_+𝛾0/^𝑘−𝑚_𝜓^{𝛾01−𝑓}_𝜓^𝜔_𝜓^−𝑖Γ_𝜓^{𝑒𝑖(𝜔𝜓+𝑖Γ𝜓)(𝑡𝑥−𝑡𝑦),} (4.22) where 𝑘⁰=𝜔_𝜓 and 𝑓_𝜓,𝑓_𝜓 denote the distribution function of the fermion particles and antiparticles, respectively. The expressions (4.22) are valid for 𝑡_𝑥−𝑡_𝑦>0.

The above definitions hold for the lepton doublets (after inserting the chiral LH projector 𝑃_𝐿), 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 neutrino 𝑁₁. To find its quantum BE we start from (4.18) for the Green's function 𝐺^<_𝑁1 of the RH neutrino 𝑁₁∶^𝑖→/^𝜕_𝑥^−𝑀₁^𝐺<_𝑁1^{𝑑𝑥,𝑦=−4𝑧−Σ𝑡}_𝑁1^{()𝐺𝑥,𝑧<}_𝑁1^{𝑧,𝑦+Σ<}_𝑁1^{()𝐺𝑥,𝑧𝑡}_𝑁1=𝑑^𝑧,𝑦_{3𝑧𝑡0𝑑𝑡𝑧Σ>𝑁1()𝐺𝑥,𝑧<𝑁1𝑧,𝑦−Σ<𝑁1()𝐺𝑥,𝑧>𝑁1.𝑧,𝑦} (4.23) Adopting the corresponding form for the RH neutrino propagator and the center-of-mass coordinates 𝑋≡𝑡,𝑋^→≡12𝑥+𝑦,𝜏,^→𝑟≡𝑥−𝑦, (4.24) one ends up with the following equation: 𝜕𝑓_𝑁1(𝐤,𝑡)𝜕𝑡=−2^𝑡₀𝑑𝑡_𝑧𝑑³𝐩(2𝜋)³12𝜔_ℓ(𝐩)12𝜔_𝐻()1𝐤−𝐩𝜔_𝑁1(𝐤)ℳ𝑁₁→ℓ𝐻²×𝑓_𝑁1(𝐤,𝑡)1−𝑓_ℓ(𝐩,𝑡)1+𝑓_𝐻(𝐤−𝐩,𝑡)−𝑓_ℓ(𝐩,𝑡)𝑓_𝐻(𝐤−𝐩,𝑡)1−𝑓_𝑁1(𝜔𝐤,𝑡)×cos_𝑁1(𝐤)−𝜔_ℓ(𝐩)−𝜔_𝐻(𝐤−𝐩)𝑡−𝑡_𝑧≃−2^𝑡₀𝑑𝑡_𝑧𝑑³𝐩(2𝜋)³12𝜔_ℓ(𝐩)12𝜔_𝐻()1𝐤−𝐩𝜔_𝑁1(𝐤)ℳ𝑁₁→ℓ𝐻²×𝑓_𝑁1(𝐤,𝑡)−𝑓^eq_ℓ(𝐩)𝑓^eq_𝐻()𝜔𝐤−𝐩×cos_𝑁1(𝐤)−𝜔_ℓ(𝐩)−𝜔_𝐻(𝐤−𝐩)𝑡−𝑡_𝑧. (4.25) 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, 𝑓_ℓ𝑓_𝐻≃𝑓^eq_ℓ𝑓^eq_𝐻. Elastic scatterings are typically fast enough to keep kinetic equilibrium. For any distribution function we may write 𝑓=(𝑛/𝑛^eq)𝑓^eq, where 𝑛 denotes the total number density. Therefore, (4.25) can be rewritten as 𝜕𝑛_𝑁1Γ𝜕𝑡=−_𝑁1(𝑡)𝑛_𝑁1+Γ_𝑁1(𝑡)𝑛^eq_𝑁1,⟨Γ_𝑁1(𝑡)⟩=^𝑡₀𝑑𝑡_𝑧𝑑³𝐤(2𝜋)³𝑓^eq_𝑁1𝑛^eq_𝑁1Γ_𝑁1(𝑡),Γ_𝑁1(𝑡)𝑑=2³𝐩(2𝜋)³ℳ(𝑁₁→ℓ𝐻)²2𝜔_ℓ2𝜔_𝐻𝜔_𝑁1cos𝜔_𝑁1−𝜔_ℓ−𝜔_𝐻𝑡−𝑡_𝑧,⟨Γ_𝑁1(𝑡)⟩=^𝑡₀𝑑𝑡_𝑧𝑑³𝐤(2𝜋)³𝑓^eq_𝑁1𝑛^eq_𝑁1Γ_𝑁1(𝑡),Γ_𝑁1(𝑡)𝑑=2³𝐩(2𝜋)³𝑓^eq_ℓ𝑓^eq_𝐻𝑓^eq_𝑁1ℳ(𝑁₁→ℓ𝐻)²2𝜔_ℓ2𝜔_𝐻𝜔_𝑁1cos𝜔_𝑁1−𝜔_ℓ−𝜔_𝐻𝑡−𝑡_𝑧, (4.26) where ⟨Γ_𝑁1(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: ^∞₀𝑑𝑡_𝑧cos𝜔_𝑁1−𝜔_ℓ−𝜔_𝐻𝑡−𝑡_𝑧=𝜋𝛿𝜔_𝑁1−𝜔_ℓ−𝜔_𝐻, (4.27) we find that the two averaged rates ⟨Γ_𝑁1⟩ and ⟨Γ_𝑁1⟩ coincide and we recover the usual classical BE for the RH distribution function 𝜕𝑛_𝑁1𝜕𝑡=−⟨Γ_𝑁1⟩𝑛_𝑁1−𝑛^eq_𝑁1,⟨Γ_𝑁1𝑑⟩=³𝐤(2𝜋)³𝑓^eq_𝑁1𝑛^eq_𝑁1𝑑³𝐩(2𝜋)³ℳ(𝑁₁→ℓ𝐻)²2𝜔_ℓ2𝜔_𝐻𝜔_𝑁1(2𝜋)𝛿𝜔_𝑁1−𝜔_ℓ−𝜔_𝐻. (4.28) 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 𝜕𝑛_Δ𝐿𝛼(𝑋)𝑑𝜕𝑡=−³𝑧^𝑡₀𝑑𝑡_𝑧TrΣ^>_ℓ𝛼(𝑋,𝑧)𝐺^<_ℓ𝛼(𝑧,𝑋)−𝐺^>_ℓ𝛼(𝑋,𝑧)Σ^<_ℓ𝛼(𝑧,𝑋)+𝐺^<_ℓ𝛼(𝑋,𝑧)Σ^>_ℓ𝛼(𝑧,𝑋)−Σ^<_ℓ𝛼(𝑋,𝑧)𝐺^>_ℓ𝛼(𝑧,𝑋). (4.29) Proceeding as for the RH neutrino equation one finds (including the moment only for the 1-loop wave contribution to the CP asymmetry 𝜖^𝛼_𝑤) ^{𝜕𝑛Δ𝐿𝛼𝜕𝑡=𝜖𝛼𝑤(𝑡)⟨Γ𝑁1⟩𝑛𝑁1−𝑛eq𝑁1,𝜖𝛼𝑤(𝑡)4=−⟨Γ𝑁1⟩3𝛽=1𝑌Im1𝛼𝑌1𝛽𝑌†𝛽2𝑌†𝛼2×𝑡0𝑑𝑡𝑧𝑡𝑧0𝑑𝑡2𝑡20𝑑𝑡1𝑒−Γ𝑁2(𝑡𝑧−𝑡2)𝑒−(Γℓ𝛽+Γ𝐻)}/^{(𝑡2−𝑡1)𝑑3𝐤()2𝜋3𝑓eq𝑁1𝑛eq𝑁1×𝑑3𝐩()2𝜋31−𝑓eqℓ𝛽(𝐩)+𝑓eq𝐻()𝐤−𝐩2𝜔ℓ𝛽(𝐩)2𝜔𝐻()𝜔𝐤−𝐩𝑁1(𝐤)𝑑3𝐪()2𝜋31−𝑓eqℓ𝛼(𝐪)+𝑓eq𝐻()𝐤−𝐪2𝜔ℓ𝛼(𝐪)2𝜔𝐻()𝜔𝐤−𝐪𝑁2(𝐤)𝜔×sin𝑁1𝑡−𝑡1+𝜔ℓ𝛽+𝜔𝐻𝑡1−𝑡2+𝜔𝑁2𝑡2−𝑡𝑧+𝜔ℓ𝛼+𝜔𝐻𝑡𝑧𝑀−𝑡×Tr}₁^𝑃_𝐿^╱𝑝𝑀₂^𝑞, (4.30) 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 ^{𝜖𝛼𝑤(𝑡)4=−⟨Γ𝑁1⟩3𝛽=1𝑌Im1𝛼𝑌1𝛽𝑌†𝛽2𝑌†𝛼2×𝑡0𝑑𝑡𝑧𝜔cos𝑁1−𝜔ℓ𝛼−𝜔𝐻𝑡−𝑡𝑧Γ2𝑁2+𝜔𝑁2−𝜔𝑁12Γℓ𝛽+Γ𝐻2+𝜔𝑁1−𝜔ℓ𝛽−𝜔𝐻2×𝑑3𝐤()2𝜋3𝑓eq𝑁1𝑛eq𝑁1Γℓ𝛽+Γ𝐻×2𝜔𝑁2−𝜔𝑁1sin2𝜔𝑁2−𝜔𝑁1𝑡𝑧2−Γ𝑁2𝜔sin𝑁2−𝜔𝑁1𝑡𝑧×𝑑3𝐩()2𝜋31−𝑓eqℓ𝛽(𝐩)+𝑓eq𝐻()𝐤−𝐩2𝜔ℓ𝛽(𝐩)2𝜔𝐻()𝜔𝐤−𝐩𝑁1(𝐤)×𝑑3𝐪}_()2𝜋3^{1−𝑓eqℓ𝛼(𝐪)+𝑓eq𝐻()𝐤−𝐪}_{2𝜔ℓ𝛼(𝐪)2𝜔𝐻()𝜔𝐤−𝐪𝑁}/_2(𝐤)^𝑀Tr₁^𝑃_𝐿^╱𝑝𝑀₂^𝑞. (4.31) 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) 𝜖_𝑤(𝑡)≃−Im𝑌𝑌^†2₁₂𝑌𝑌^†₁₁𝑌𝑌^†₂₂𝑀₁𝑀₂Γ_𝑁21(Δ𝑀)²+Γ²_𝑁2×2Δ𝑀sin²Δ𝑀𝑡2−Γ_𝑁2sin[Δ𝑀𝑡], (4.32) where Δ𝑀=(𝑀₂−𝑀₁). The CP asymmetry (4.32) is resonantly enhanced when Δ𝑀≃Γ_𝑁2, and at the resonance point it is given by 𝜖_𝑤(𝑡)1≃−2Im𝑌𝑌^†2₁₂𝑌𝑌^†₁₁𝑌𝑌^†₂₂(1−sin[Δ𝑀𝑡]−cos[Δ𝑀𝑡]). (4.33) 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 ⟨𝜖_𝑤1⟩=𝜏_p^𝑡_𝑡−𝜏p𝑑𝑡^′𝜖^W_𝑁1𝑡^′, (4.34) where 𝜏_p represents the typical timescale of the other processes relevant for leptogenesis, for example, the Δ𝐿=1 scatterings. If 𝜏_p≫1/Δ𝑀∼Γ_𝑁⁻¹₂, 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 𝜏_𝑝≲1/Δ𝑀∼Γ_𝑁⁻¹₂.

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 Γ_𝑁1≳Γ_𝑁2, 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: 𝜖_𝑣(𝑡)≃−Im𝑌𝑌^†2₁₂16𝜋𝑌𝑌^†₁₁𝑀₁𝑀₂2sin²𝑀₂𝑡2−Γ_𝑁2𝑀₂sin𝑀₂𝑡. (4.35) 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 of 1/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: 1𝑊=2𝑀_𝑝𝑞𝑁^𝑐_𝑝𝑁^𝑐_𝑞+𝜆_𝛼𝑝𝑁^𝑐_𝑝ℓ_𝛼𝐻_𝑢, (5.1) 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, 𝐻_𝑢, and 𝑁^𝑐 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 𝑁^𝑐 supermultiplet has scalar component 𝑁^∗ and fermion component 𝑁^𝑐_𝐿). Finally the SU(2)_𝐿 index contraction is defined as ℓ_𝛼𝐻_𝑢=𝜖_𝜌𝜎ℓ^𝜌_𝛼𝐻^𝜎_𝑢 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] 𝑌_Δ𝐵(∞)^MSSM𝑌_Δ𝐵(∞)^SM≈√2,2√strongwashout2(weakwashout). (5.2) 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, above 𝑇∼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 both SU(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 below 𝒪(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 ℎ_𝛼=𝑚_𝛼/(𝑣_𝑢cos𝛽).

The purpose of the following sections is twofold. We describe the chemical equilibrium conditions and conservation laws for MSSM in conjunction to SM. In Section 5.2 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 𝑀_𝑊=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 𝑚_𝑏,𝑓≪𝑇, and at first order in the chemical potential 𝜇_𝑏,𝑓/𝑇≪1: Δ𝑛_𝑏=𝑔_𝑏3𝑇²𝜇_𝑏,Δ𝑛_𝑓=𝑔_𝑓6𝑇²𝜇_𝑓. (5.3) For simplicity of notations, in the following we denote the chemical potentials with the same notation that labels the corresponding field: 𝜙≡𝜇_𝜙.(1)At scales much higher than 𝑀_𝑊, 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 by 𝑊_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, ℓ̃+𝑊_𝑅→ℓ, ℓ̃+𝐵_𝑅→ℓ, imply that all gauginos have the same chemical potential: −𝑔=𝑄−𝑄=−𝑊=ℓ−ℓ̃=−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 ̃𝐻𝑄,ℓ=𝑄,ℓ+𝑔,_𝑢,𝑑=𝐻_𝑢,𝑑+̃𝑔,𝑢,𝑑,𝑒=𝑢,𝑑,𝑒−𝑔. (5.4) The last relation, in which 𝑢,𝑑,𝑒≡𝑢_𝑅,𝑑_𝑅,𝑒_𝑅 denote the RH SU(2)_𝐿 singlets, follows, for example, from 𝑢^𝑐_𝐿=𝑢^𝑐_𝐿+𝑔 for the corresponding LH fields, together with 𝑢^𝑐_𝐿=−𝑢_𝑅, 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: _𝑖𝑌_Δ𝑄𝑖+2𝑌_Δ𝑢𝑖−𝑌_Δ𝑑𝑖−_𝛼𝑌_Δℓ𝛼+𝑌_Δ𝑒𝛼+𝑌_Δ𝐻_𝑢−𝑌_Δ𝐻_𝑑=0. (5.5)(4)When the reactions mediated by the lepton Yukawa couplings are faster than the Universe expansion rate (see Section 3.1 for the temperature regime when lepton Yukawa interactions are in thermal equilibrium), the following chemical equilibrium conditions are enforced: ℓ_𝛼−𝑒_𝛼+𝐻_𝑑+𝑔=0,𝛼=𝑒,𝜇,𝜏. (5.6) 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 ℓ_𝛼 to be independent quantities.(5)Reactions mediated by the quark Yukawa couplings enforce the following six chemical equilibrium conditions: 𝑄_𝑖−𝑢_𝑖+𝐻_𝑢+𝑔=0,𝑢_𝑖,𝑄=𝑢,𝑐,𝑡_𝑖−𝑑_𝑖+𝐻_𝑑+𝑔=0,𝑑_𝑖=𝑑,𝑠,𝑏. (5.7)(5.8)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: 𝑏_𝐿→𝑐_𝐿 and 𝑡_𝐿→𝑠_𝐿 imply 𝑄₂=𝑄₃; 𝑠_𝐿→𝑢_𝐿 and 𝑐_𝐿→𝑑_𝐿 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: 𝑄≡𝑄₃=𝑄₂=𝑄₁. (5.9) 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. 6_SE∶ Equilibration of the particle-sparticle chemical potentials 𝜇_𝜙=𝜇_𝜙 [149] is ensured when reactions like ℓℓ̃̃→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 𝑚_𝑔 and can be neglected in the limit 𝑚_𝑔→0.Furthermore, since the 𝜇 parameter of the 𝐻_𝑢𝐻_𝑑 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 Γ_𝑔∼𝑚²_𝑔/𝑇 and Γ_𝜇∼𝜇²/𝑇, are faster than the Universe expansion rate up to temperatures 𝑇≲5⋅10⁷𝑚_𝑔,𝜇500GeV^2/3GeV. (5.10)

The corresponding chemical equilibrium relations enforce the conditions: 𝐻𝑔=0,_𝑢+𝐻_𝑑=0. (5.11)(5.12)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 𝑂_EW=Π_𝛼(𝑄𝑄𝑄ℓ_𝛼) [151] and 𝑂_QCD=Π_𝑖(𝑄𝑄𝑢^𝑐_Li𝑑^𝑐_Li) [151–153]. Above the EW phase transition reactions induced by these operators are in thermal equilibrium, and the corresponding conditions read 9𝑄+_𝛼ℓ_𝛼=0,6𝑄−_𝑖𝑢_𝑖+𝑑_𝑖=0, (5.13)(5.14)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 Δ_𝛼≡𝐵/3−𝐿_𝛼 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 Δ_𝛼 charges normalized to the entropy density can be written as 𝑌_Δ𝛼1=33_𝑖2𝑌_Δ𝑄𝑖+𝑌_Δ𝑢𝑖+𝑌_Δ𝑑𝑖−2𝑌_Δℓ𝛼+𝑌_Δ𝑒𝛼−23𝑌_Δ𝑔. (5.15) 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 the 𝐴 matrix and 𝐶 vectors introduced, respectively, in [67, 69] that are defined as: 𝑌_Δℓ𝛼=𝐴^ℓ_𝛼𝛽𝑌_Δ𝛽,𝑌_Δ𝐻_𝑢,𝑑𝐻=𝐶^𝑢,𝑑_𝛼𝑌_Δ𝛼. (5.16) 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 𝑌_Δ𝑏=2𝑌_Δ𝑓 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 temperatures 𝑇≳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 𝐴^ℓ=1𝐻3×2148−90612012075−688287528−688,𝐶^𝑢=−𝐶𝐻_𝑑=−1()214837,52,52. (5.17) 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 𝑌_Δℓ𝛼+𝑌_Δℓ_𝛼̃ 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 𝑚_𝑔 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: 𝑚_𝑔→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 combinations 𝑅₂ and 𝑅₃ of 𝑅 and PQ, having, respectively, only SU(2)_𝐿 and SU(3)_𝐶 mixed anomalies, have been identified in [150]: 𝑅₂=𝑅−2𝑃𝑄,𝑅₃=𝑅−3𝑃𝑄. (5.18) The authors of [150] have also constructed the effective multifermions operators generated by the mixed anomalies: 𝑂_EW=Π_𝛼𝑄𝑄𝑄ℓ_𝛼𝐻_𝑢𝐻_𝑑𝑊⁴,𝑂_QCD=Π_𝑖𝑄𝑄𝑢^𝑐𝑑^𝑐_𝑖𝑔⁶. (5.19)(5.20) 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 𝑁^𝑐ℓ𝐻_𝑢 and the second anomaly free combination which is also an exact symmetry of the MSSM+seesaw in the NSE regime [89]: 5ℛ=3𝐵−𝐿+𝑅₂, (5.21) 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]: 𝜒_𝑞𝐿+𝜅_𝑞𝐿𝑅₃, (5.22) where, for example, 𝜅_𝑢^𝑐_𝐿=𝜅_𝑑^𝑐_𝐿=1/3 and 𝜅_𝑄𝐿=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 items 6_SE and 7_SE of the previous section have to be dropped, but new conditions arise.6_NSE: The conservation law for the ℛ charge yields the following global neutrality condition: ℛ_tot=_𝑓Δ𝑛_𝑓ℛ_𝑓+_𝑏Δ𝑛_𝑏ℛ_𝑏𝑁+Δ𝑛₁ℛ𝑁₁=𝑇²6_𝑖2𝑄_𝑖−5𝑢_𝑖+4𝑑_𝑖+2_𝛼ℓ_𝛼+𝑒_𝛼+5𝐻_𝑑−𝐻_𝑢𝑔𝑁+31−Δ𝑛₁=0. (5.23)The last terms in both lines of (5.23) correspond to the contribution to ℛ-neutrality from the lightest sneutrino asymmetry 𝑁Δ𝑛₁𝑁=𝑛₁𝑁−𝑛^∗₁ with charge ℛ𝑁₁=−ℛ_𝑁^𝑐=−1. Note that since in general 𝑁₁ 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 𝑌_Δ𝑁𝑁≡𝑌₁𝑁−𝑌^∗₁ (see Section 5.5).7_NSE: 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]: 3_𝑖𝑄_𝑖+_𝛼ℓ_𝛼+𝐻_𝑢+𝐻_𝑑+4𝑔=0,2_𝑖𝑄_𝑖−_𝑖𝑢_𝑖+𝑑_𝑖+6𝑔=0, (5.24)that replace (5.13) and (5.14).8_NSE: 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 ℎ_𝑢,𝑑→0 implies 𝑢 and 𝑑 decoupling, but 𝑄₁ decoupling is ensured only if also ℎ_𝑐,𝑠→0.) The conservation laws corresponding to these symmetries read 𝑇²63𝑞_𝑅+6𝑞_𝑅−𝑔+13𝑅_3tot𝑇=0,₂623𝑄_𝐿+6𝑄_𝐿+𝑔−23𝑅_3tot=0 (5.25)(5.26)and hold for 𝑞_𝑅=𝑢_𝑖,𝑑_𝑖 and 𝑄_𝐿=𝑄_𝑖 in the regimes when the appropriate Yukawa reactions are negligible. Note the factor of 2 for the 𝑄_𝐿 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 𝑁Δ𝑛₁, the total 𝑅₃ charge in (5.25) and (5.26) reads 𝑅_3tot=𝑇²682𝑔−3_𝑖2𝑄_𝑖+11𝑢_𝑖−4𝑑_𝑖+_𝛼16ℓ_𝛼+13𝑒_𝛼+16𝐻_𝑑−14𝐻_𝑢𝑁+Δ𝑛₁𝑅₃𝑁₁, (5.27)where 𝑅₃𝑁₁=−1. As regards the leptons, since they do not couple to the QCD anomaly, by setting ℎ_𝑒→0, a symmetry under chiral supermultiplet rotations is directly gained for the RH leptons implying Δ𝑛_𝑒+Δ𝑛_𝑒=0 and giving the condition: 2𝑒−3𝑔=0. (5.28)No analogous condition arises for the lepton doublets relevant for leptogenesis, since by assumption they remain coupled via Yukawa couplings to the heavy 𝑁’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 new 𝐶 vector to describe the gaugino number density asymmetry per degree of freedom in terms of the relevant charges: 𝑌_Δ𝑔𝑔=𝐶_𝑎𝑌_Δ𝑎,withΔ_𝑎=Δ_𝛼,Δ_𝑁. (5.29)

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

As an example, in the temperature range between 10⁸ 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 ℎ_𝑢,𝑑,ℎ_𝑒→0 we obtain 𝐴^ℓ=1,𝐶𝑔=9×162332−1981173398733987−825326634−14757114761−80552663414761−147571−8055−11(),𝐶𝐻162332163,165,165,−255_𝑢=−1(),𝐶𝐻1623323918,4713,4713,95_𝑑=1(),3×1623325413,9712,9712,−252 (5.30) where the rows correspond to (𝑌_Δ𝑒,𝑌_Δ𝜇,𝑌_Δ𝜏,𝑌_Δ𝑁). 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 𝑁Δ𝑛₁ to the global charges ℛ_tot in (5.23) and 𝑅_3tot 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 𝑌_Δ𝛼 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].) 𝑠𝐻𝑧𝑑𝑌_𝑁1𝑌𝑑𝑧=−^𝑁1𝑌^eq_𝑁1−1𝛾_𝑁1,𝑁𝑠𝐻𝑧𝑑𝑌₊𝑌𝑁𝑑𝑧=−⁺𝑌^eq𝑁₁𝛾𝑁−2₁,𝑠𝐻𝑧𝑑𝑌_Δ𝑁𝑌𝑑𝑧=−^Δ𝑁𝑌^eq𝑁₁𝛾𝑁₁−32𝛾𝑁_1𝑎𝐶𝑔_𝑎𝑌_Δ𝑎𝑌^eq_ℓ+⋯,𝑠𝐻𝑧𝑑𝑌_Δ𝛼𝑑𝑧=−𝜖_𝛼𝑌_𝑁1𝑌^eq_𝑁1−1𝛾_𝑁1+𝑌𝑁₊𝑌^eq𝑁₁𝛾𝑁−2₁+𝑃⁰_1𝛼𝛾_𝑁1+12𝛾𝑁_1𝑎𝐴^ℓ_𝛼𝑎+𝐶𝐻_𝑢𝑎𝑔+𝐶_𝑎𝑌_Δ𝑎𝑌^eq_ℓ, (5.31)(5.32)(5.33)(5.34) where 𝛾𝑁₁ is the corresponding thermally averaged decay rate for RH sneutrino 𝑁₁. In (5.32) we have introduced the overall sneutrino abundance 𝑌𝑁₊𝑁=𝑌₁𝑁+𝑌^∗₁, while 𝑌_Δ𝑁𝑁≡𝑌₁𝑁−𝑌^∗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 𝑌_Δ𝑎=(𝑌_Δ𝛼,𝑌_Δ𝑁) to the equilibrium density of a fermion with one degree of freedom 𝑌^eq_ℓ. 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 𝜖_𝛼. 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 𝑌_Δ𝑁 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 𝑌𝑁₁ and 𝑌𝑁^∗₁ in general differ. However, in this case recasting their equations in terms of two equations for 𝑌𝑁₊ and 𝑌_Δ𝑁 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 𝑌_Δ𝑁 is required to formulate properly the corresponding conservation law. As we have seen, this eventually results in 𝑌_Δ𝑁 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 (Section 6.2) 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 in 𝑁₁ 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 𝜖₁𝑀self−energy=−¹𝑀₂Γ_𝑁2𝑀₂𝑀²₂𝑀²₂−𝑀²₁𝑀²₂−𝑀²₁²+𝑀²₁Γ²_𝑁2Im𝜆^†𝜆²₁₂𝜆^†𝜆₁₁𝜆^†𝜆₂₂. (6.1) The resonance condition reads 𝑀₂−𝑀₁=Γ_𝑁22. (6.2) In this case 𝜖₁(≃1resonance)2Im𝜆^†𝜆²₁₂𝜆^†𝜆₁₁𝜆^†𝜆₂₂. (6.3)

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 of 𝑁₁ 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 Section 5. 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 the 𝐵 term implies that 𝑁 and 𝑁^† states mix to form two mass eigenstates with mass splitting proportional to 𝐵 itself. Furthermore when 𝐵∼Γ_𝑁, 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 𝜆₂^±,0, the U(1)_𝑌 gauginos 𝜆₁, and the three sleptons ℓ_𝛼̃ in the basis in which charged lepton Yukawa couplings are diagonal are given by −ℒ_soft=𝑀²𝑁^∗𝑁+𝐴𝑌_𝛼𝜖_𝑎𝑏𝑁ℓ^𝑎_𝛼̃𝐻^𝑏_𝑢+12𝑁+1𝐵𝑀𝑁+h.c.2𝑚₂𝜆₂^±,0𝑃_𝐿𝜆₂^±,0+𝑚₁𝜆₁𝑃_𝐿𝜆₁+h.c., (6.4) where for simplicity, proportionality of the bilinear and trilinear soft breaking terms to the corresponding SUSY invariant couplings has been assumed: 𝐵_𝑀=𝐵𝑀 and 𝐴_𝛼=𝐴𝑌_𝛼. The Lagrangian derived from (5.1) and (6.4) is characterized by only three independent physical phases: 𝜙_𝐴≡arg(𝐴𝐵^∗), 𝜙_𝑔2≡(1/2)arg(𝐵𝑚^∗₂) and 𝜙_𝑔𝑌≡(1/2)arg(𝐵𝑚^∗₁), which can be assigned to 𝐴 and to the gaugino coupling operators 𝑔₂,𝑔_𝑌, respectively. As mentioned earlier, a crucial role in soft leptogenesis is played by the 𝑁−𝑁^† mixing to form the mass eigenstates 𝑁₊=1√2𝑒^𝑖Φ/2𝑁+𝑒^−𝑖Φ/2𝑁^∗,𝑁₋𝑖=−√2𝑒^𝑖Φ/2𝑁−𝑒^−𝑖Φ/2𝑁^∗, (6.5) where Φ≡arg(𝐵𝑀) and the corresponding mass eigenvalues are 𝑀²_±=𝑀²+𝑀²±|𝐵𝑀|. Without loss of generality, we can set Φ=0, which is equivalent to assigning the phases only to 𝐴 and 𝑌_𝛼.

It has been pointed out that the CP asymmetries for the decays of 𝑁_± 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(𝑚³_soft/𝑀³) where 𝑚_soft 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 𝜖^𝑠_𝛼(𝑇)=𝜖_𝛼Δ^𝑠(𝑇),𝜖^𝑓_𝛼(𝑇)=−𝜖_𝛼Δ^𝑓(𝑇), (6.6) where 𝜖_𝛼 is the temperature independent term of ∼𝒪(𝑚_soft/𝑀) which contains contributions from the self-energy correction, vertex correction, and the interference between the two. In the limit 𝑇→0, we have Δ^𝑠(𝑇),Δ^𝑓(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|∼𝑚_soft/𝑀≳10⁻⁶ which gives 𝑀≲10⁹ GeV for 𝑚_soft∼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≪𝑚_soft to be resonantly enhanced. However it was shown in [195] that flavour effects can greatly enhance the efficiency, and eventually 𝐵∼𝑚_soft 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 is 𝜆≲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 Γ_𝜆∼𝜆²𝑇, becomes significant when it equals the expansion rate of the Universe, 𝐻∼𝑇²/𝑚_pl. Thus, the temperature of equilibration between LH and RH neutrinos is 𝑇∼𝜆²𝑚_pl∼(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⟩∼𝑣_𝑢. This breaking is communicated to the MSSM+N via extra, vector-like lepton doublet fields, 𝜙+𝜙, that have masses 𝑀_𝜙 much larger than 𝑣_𝑢. Consequently, the neutrino Yukawa couplings are suppressed by the small ratio ⟨𝜒⟩/𝑀_𝜙. The CP violation arises in the decays of the vector-like leptons, whereby Γ(𝜙→𝑁𝐻^𝑐_𝑢)≠Γ(𝜙→𝑁^𝑐𝐻_𝑢) and Γ(𝜙→𝐿𝜒)≠Γ(𝜙→𝐿^𝑐𝜒^𝑐). 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 of SU(2)_𝐿-triplet scalars 𝑇 [44, 208–211]. The relevant new terms in the Lagrangian are ℒ_𝑇=−𝑀²_𝑇|𝑇|²+12𝜆_𝐿𝛼𝛽ℓ_𝛼ℓ_𝛽𝑇+𝑀_𝑇𝜆_𝜙𝜙𝜙𝑇^∗+h.c.. (6.7) Here, 𝑀_𝑇 is a real mass parameter, 𝜆_𝐿 is a symmetric 3×3 matrix of dimensionless, complex Yukawa couplings, and 𝜆_𝜙 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: 𝜖_𝑇Γ≡2𝑇→ℓℓ−Γ𝑇→ℓℓΓ_𝑇+Γ_𝑇, (6.8) where the overall factor of 2 comes because the triplet scalar decay produces two (anti)leptons.

To calculate 𝜖_𝑇, 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 𝑚_II (𝑚_I) as the part of the light neutrino mass matrix that comes (does not come) from the contributions of the triplet scalar responsible for 𝜖_𝑇: 𝑚=𝑚_II+𝑚_I. (6.9) Then, assuming that the particles exchanged to produce 𝑚_I are all heavier than 𝑇, we obtain the CP asymmetry 𝜖_𝑇=1𝑀4𝜋𝑣^𝑇2_𝑢𝐵_𝐿𝐵_𝐻ImTr𝑚^†_II𝑚_ITr𝑚^†_II𝑚_II, (6.10) where 𝐵_𝐿(𝐵_𝐻) is the tree-level branching ratio to leptons (Higgs doublets). If these are the only decay modes, that is, 𝐵_𝐿+𝐵_𝐻=1, then 𝐵_𝐿/𝐵_𝐻=Tr(𝜆_𝐿𝜆^†_𝐿)/(𝜆_𝐻𝜆^†_𝐻), and there is an upper bound on the asymmetry 𝜖_𝑇≤1𝑀4𝜋𝑣^𝑇2_𝑢𝐵_𝐿𝐵_𝐻𝑖𝑚²_𝜈𝑖. (6.11) Note that, unlike the singlet fermion case, |𝜖_𝑇| increases with larger 𝑚_𝜈𝑖.

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→ℓℓ and 𝑇→𝜙𝜙 in order that this will happen [220](1) One of the two decay rates is faster than the 𝑇𝑇 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 of 𝑇→ℓℓ and 𝑇→𝜙𝜙.

Combining a calculation of 𝜂 with the upper bound on the CP asymmetry (6.11), successful leptogenesis implies a lower bound on the triplet mass 𝑀_𝑇 varying between 10⁹ GeV and 10¹² GeV, depending on the relative weight of 𝑚_II and 𝑚_I in the light neutrino mass.

Interestingly, in the supersymmetric framework, “soft leptogenesis” (see Section 6.2) 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 of SU(2)_𝐿-triplet fermions 𝑇^𝑎_𝑖 [234–236] (𝑖 denotes a heavy mass eigenstate while 𝑎 is an SU(2)_𝐿 index) with the Lagrangian ℒ_𝑇^𝑎=𝜆_𝑇𝛼𝑘𝜏^𝑎_𝜌𝜎ℓ^𝜌_𝛼𝜙^𝜎𝑇^𝑎_𝑘−12𝑀_𝑖𝑇^𝑎_𝑖𝑇^𝑎_𝑖+h.c. (6.12) Here 𝜏^𝑎 are the Pauli matrices, 𝑀_𝑖 are real mass parameters, and 𝜆_𝑇 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 and 𝜖𝜂 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 factor 𝜂 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 𝜂≈𝑀₁/10¹³GeVfor𝑚≪10⁻³eV. (6.13) 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 𝑀₁≳1.5×10¹⁰GeV. (6.14) 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 of 0𝜈𝛽𝛽 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 |𝑚_𝛽𝛽| 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 |𝑚_𝑒𝑒|≲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: supersymmetric 𝑅-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.