Quantum Boltzmann Equations and Leptogenesis

The thing to know: De Simone and Riotto derive the quantum Boltzmann equations for leptogenesis from the CTP formalism and get an explicit time-dependent CP asymmetry, Eqs. (34)-(36), that oscillates on the 1/ΔM timescale. That is the new content, and it is a real contribution. The CTP machinery itself is standard (their own ref. [21]), but nobody had written the asymmetry as a function of time with the memory integral intact. The Markovian limit correctly recovers the classical Boltzmann equation (Eq. 30) and the standard constant asymmetry (Eq. 39), which is a meaningful consistency check against the literature, not a circular step. The paper is also honest about its approximations: constant lepton and Higgs damping rates, no thermal channel opening, no numerical estimate of the effect, all stated and the numerical study explicitly deferred. That is fine for a framework paper and I would not hold it against them at this stage. The soft spot the stress-test flags is real and lands. Going from Eq. (34) to Eq. (35), the paper drops Γ_ℓ + Γ_H from the denominator; that is the narrow-width replacement of the cosine by a delta function. In resonant leptogenesis ΔM ~ Γ_N2 and Γ_ℓ ~ Γ_N2, so the width is comparable to the oscillation scale and the factorization is marginal at best. The paper never states the condition. This does not kill the qualitative claim, since the asymmetry is still time-dependent and still oscillates on 1/ΔM, but the amplitude and precise form of Eqs. (35)-(36) are not controlled in the regime the paper is aimed at. That is a genuine gap, distinct from the acknowledged factorization assumption in §4. Two smaller things. The introduction motivates the whole exercise with flavour-matrix asymmetries and off-diagonal quantum correlations, but the derivation delivers a diagonal n_Li equation; the matrix structure is never written down. And Eq. (38) for the vertex contribution is obtained by dividing the wave result by 2, which is asserted rather than derived. Both are minor relative to the main result. Who it is for: people working on resonant and soft leptogenesis who need a reference for the time-dependent asymmetry. It deserves a serious referee, mainly to force the narrow-width condition to be stated and the resonant-regime validity checked.

Eq. (34) is reduced to Eq. (35) by replacing the cosine factor cos[(ω_N1 − ω̄_ℓi − ω̄_H)(t − t_z)] with an energy-conserving delta function while retaining the sin²[ΔMt/2] and sin[ΔMt] factors that carry the claimed time dependence. This step requires the N1 on-shell width, set by the lepton and Higgs damping rates Γ_ℓ + Γ_H appearing in the denominator of Eq. (34), to be small compared to the scale ΔM on which the retained factors vary. In resonant leptogenesis ΔM ~ Γ_N2, and the lepton damping rate is typically Γ_ℓ ~ (λλ†)M/(8π) ~ Γ_N2 for comparable Yukawa couplings, so Γ_ℓ + Γ_H ≳ ΔM. In this regime the delta function is broad compared to the retained oscillation, the factorization underlying Eq. (35) does not follow, and the amplitude and timescale of the claimed time dependence are not reliably determined. The paper does not state or check this condition. This is distinct from the acknowledged assumption in §4 that the distribution functions can be pulled out of the time integrals: even granting that factorization, the delta-function replacement in Eq. (34) is a separate uncontrolled step in the resonant regime the paper targets.

The paper's qualitative claim — that the CP asymmetry is time-dependent and oscillates on the 1/ΔM timescale — is plausible and follows from the CTP formalism. The time average of Eq. (35) correctly reduces to the standard resonant-leptogenesis result, and the vertex-diagram limit Eq. (38) averages to the literature value Eq. (39). The reader's CONDITIONAL verdict is appropriate. My concern is a specific technical reason for that verdict: the step from Eq. (34) to Eq. (35) requires a separation of timescales (Γ_ℓ + Γ_H ≪ ΔM) that is not guaranteed in the resonant regime the paper targets, and the paper does not state or check it. The reader identified the pulling-out-of-distributions assumption in §4, which is related but not identical; both are separation-of-timescales issues, but the delta-function replacement in Eq. (34) is a separate uncontrolled step that directly affects the quantitative form of the claimed time dependence. The paper's acknowledged limitations (no numerical estimate of the effect, neglected thermal channel opening) are real but do not by themselves break the qualitative claim; the concern I raise goes to the quantitative reliability of Eq. (35), which is the paper's concrete realization of the central claim.

The paper provides a first-principles CTP derivation of non-Markovian kinetic equations for leptogenesis, which is a valuable formal result and a natural starting point for future quantitative studies of memory and off-shell effects. The observation that the CP asymmetry oscillates on a timescale set by the heavy-state mass splitting is physically well motivated and potentially important for resonant and soft leptogenesis, where the oscillation and damping timescales are comparable. The recovery of the standard classical Boltzmann equation and the standard CP asymmetry in the appropriate Markovian limit is a meaningful consistency check against external benchmarks. The paper is formal and does not present numerical estimates of the impact on the final baryon asymmetry, but it sets the stage for such work.

The paper asks how the universe's lepton asymmetry is actually produced, and answers with a quantum kinetic theory rather than the classical Boltzmann bookkeeping usually assumed. Working in the closed time-path (Schwinger–Keldysh) formalism, the authors derive equations for the abundance of the lightest right-handed neutrino and for the lepton-doublet asymmetry in which the familiar energy-conserving delta functions are replaced by retarded time integrals and cosines. The resulting equations are non-Markovian: what happens at a given moment depends on the prior history of the plasma, and off-shell processes contribute alongside on-shell ones. The payoff is that the CP asymmetry itself becomes a function of time, oscillating with a period set by the mass splitting of two nearly degenerate heavy neutrinos and collapsing to the standard constant only when every other process in the plasma is slower than that oscillation. This matters most for resonant and soft leptogenesis, where the splitting is comparable to the decay width and the two timescales cannot be separated.

What carries the argument. The engine is the closed time-path (Schwinger–Keldysh) formulation of non-equilibrium quantum field theory, in which the time contour runs forward and back and each field is doubled, giving six propagators related by $G^r = G^t - G^< = G^> - G^{¯t}$ and the analogous self-energy relations. Dyson equations for the two-by-two matrix propagator are differentiated to produce a transport equation for the lepton current, $∂n_{L_i}/∂t = -∫ d^3z ∫_0^t dt_z Tr[Σ^>_{ℓ_i}G^<_{ℓ_i} - G^>_{ℓ_i}Σ^<_{ℓ_i} + G^<_{ℓ_i}Σ^>_{ℓ_i} - Σ^<_{ℓ_i}G^>_{ℓ_i}]$, whose time integral over the past is the mathematical locus of the memory effects. Quasiparticle propagators with finite width supply the damping factors $e^{-Γ|t_x-t_y|}$ that cut off the time integrations, and the CP-violating source is extracted from the two-loop

Core claim. The central claim is that the CP asymmetry used in leptogenesis is not a fixed number but a function of time whose value at any instant depends on the previous history of the system. For the wave-diagram contribution, the authors obtain $ǫ^W_{N_1}(t) = -[Im(λλ^†)^2_{12}/((λλ^†)_{11}(λλ^†)_{22})](M_1/M_2)[Γ_{N_2}/((ΔM)^2+Γ_{N_2}^2)](2ΔM sin^2[ΔM t/2] - Γ_{N_2} sin[ΔM t])$, with $ΔM = M_2 - M_1$. The asymmetry starts at zero, grows, and oscillates on a timescale $1/ΔM$; at the resonance point $ΔM ≃ Γ_{N_2}$ it takes the closed form proportional to $1 - sin[ΔM t] - cos[ΔM t]$. Only when the other leptogenesis processes have a characteristic time $τ_p ≫ 1/ΔM$ do the oscillating terms average away and reproduce the constant asymmetry quoted in the literature. The authors argue that when $Γ_{N_1} ≳ Γ_{N_2}$ this averaging is illegitimate and the time dependence must be kept, which is precisely the regime of resonant and soft leptogenesis. A parallel result is given for the vertex diagram, where the oscillation timescale is $∼M_2$ and averaging is safe.

The premise it rests on. The derivation pulls the particle distribution functions outside the retarded time integrals on the assumption that their relaxation timescales are much longer than the kernel timescales, and it sets the particle damping rates to zero except where they regulate the resonance, so the separation of a clean time-dependent asymmetry from the evolving abundances depends on that timescale hierarchy holding.

If the decay width of the lightest right-handed neutrino is comparable to or larger than that of its near-degenerate partner, the standard constant CP asymmetry is not the right object to insert into the Boltzmann equations; the oscillating form must be used instead.

The literature value is recovered as a limiting case, so the paper delimits when earlier treatments are valid rather than discarding them: averaging is justified exactly when the other relevant rates are slow compared with $1/ΔM$.

Because the quantum kernels are cut off by exponential damping and oscillate, the effective time integration range shrinks, which the authors expect to reduce washout rates and keep the system out of equilibrium longer than the classical treatment predicts.

The formalism provides a systematic route to a matrix of lepton asymmetries in flavour space, with diagonal flavour asymmetries and off-diagonal quantum correlations, which is what a full treatment of the flavour transition at $T ∼ 10^{12}$ GeV requires.

A numerical integration of the non-Markovian equations, which the authors explicitly leave for future work, would show whether the oscillating asymmetry shifts the final baryon asymmetry by an amount comparable to the experimental precision on $η_B$ or only by a modest factor.

Because the size of the effect is controlled by $ΔM/Γ_{N_2}$, the time-dependent asymmetry could serve as a sharp discriminator: a near-degenerate pair produces a measurably different yield than a hierarchical pair with the same zero-temperature asymmetry, giving an indirect handle on the heavy-neutrino mass splitting.

The assumed vanishing initial asymmetries at $t=0$ sit uneasily with a formalism built to remember history; if any asymmetry predates the onset of leptogenesis, the same memory kernels would propagate it into the late-time result, which suggests the initial-condition dependence deserves its own study.

The same retarded-kernel structure appears in electroweak baryogenesis and in quantum transport in nuclear collisions, so the reduction of washout rates found here may have a counterpart in those settings that could be checked against existing transport calculations.

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review