Kubo relations and radiative corrections for lepton number washout

The rates for lepton number washout in extensions of the Standard Model containing right-handed neutrinos are key ingredients in scenarios for baryogenesis through leptogenesis. We relate these rates to real-time correlation functions at finite temperature, without making use of any particle approximations. The relations are valid to quadratic order in neutrino Yukawa couplings and to all orders in Standard Model couplings. They take into account all spectator processes, and apply both in the symmetric and in the Higgs phase of the electroweak theory. We use the relations to compute washout rates at next-to-leading order in g, where g denotes a Standard Model gauge or Yukawa coupling, both in the non-relativistic and in the relativistic regime. Even in the non-relativistic regime the parametrically dominant radiative corrections are only suppressed by a single power of g. In the non-relativistic regime radiative corrections increase the washout rate by a few percent at high temperatures, but they are of order unity around the weak scale and in the relativistic regime.


Introduction and motivation
With the Standard Model (SM) of electroweak interactions being validated up to an energy scale of several hundred GeV by the LHC experiments, and the knowledge that the SM combined with the conventional Big Bang scenario can explain neither the baryon asymmetry of the Universe, nor the existence of dark matter, it is appealing to search for an answer to these cosmological questions from other than electroweak interactions. The physics associated with right-handed (RH) neutrinos, which can also account for the observed left-handed neutrino mass differences and mixing angles, is rich enough to conceivably solve both problems [1,2]. In this paper we concentrate on the baryon asymmetry aspect, and refer to the scenario as leptogenesis.
The original description of leptogenesis uses Boltzmann equations for the phase space density of the participating particles as a starting point. These are then integrated over momenta assuming kinetic equilibrium. The processes considered are the decays (1 → 2) and inverse decays (2 → 1) of the heavy RH neutrinos. In addition, so-called spectator processes, which proceed much faster than the expansion of the Universe, determine which quantities are in thermal equilibrium. Frequently also additional scattering processes (2 ↔ 2) and inverse decays have been included.
The 2 ↔ 2 processes are a part of the radiative corrections. To assess the theoretical uncertainty of the analysis it would be desirable to compute the complete next-to-leading order (NLO) radiative corrections. It is not clear how to do this consistently within a description based on Boltzmann equations. This has motivated several authors to search for a -1 -

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first principles description of leptogenesis, without already putting in the set of assumptions and approximations which are implicit to the Boltzmann equations. Among the strategies followed are Kadanoff-Baym and similar equations for Green's functions (cf. refs. [3][4][5] for recent work and references). Although the starting point is exact, it may be difficult to perform systematic NLO calculations in these settings.
In this paper we propose a different route towards a first principles understanding of leptogenesis. We first formulate a rather general non-equilibrium problem. We argue that it can be described by a simple set of ordinary differential equations. The coefficients in these effective equations are shown to be related to real-time correlation functions at finite temperature. We then focus on one of the coefficients, the dissipation of lepton minus baryon number n L−B , 1 which in the absence of expansion is described by We calculate the dissipation coefficient 2 γ L−B up to NLO in the SM couplings. It is shown, in particular, that the dominant NLO corrections are only suppressed by a single power of the gauge or top Yukawa couplings, and have a substantial relative influence even at temperatures much below the mass of the lightest right-handed neutrino.
In section 2 we describe the physical picture behind baryogenesis through leptogenesis. Section 3 contains a general analysis of the dissipation or washout rates of almost conserved charges, and relations of these rates to real-time correlation functions at finite temperature. We show that these rates factorize into a real-time spectral function and the inverse of a susceptibility matrix. In section 4 we specialize to the washout of lepton minus baryon number in extensions of the SM with right-handed neutrinos. We obtain a master formula, and leading-order (LO) and NLO results for its two ingredients, the spectral function and the susceptibility matrix. In section 5 we study the effect of the radiative corrections to the washout rate on the lepton asymmetry for one particular set of parameters. We summarize and conclude in section 6.
Notation. 4-vectors are denoted by lower-case italics and 3-vectors by boldface, and the metric is such that p 2 = (p 0 ) 2 − p 2 . For spatial integrals we use the notation x ≡ d 3 x, space-time integrals are denoted by x ≡ d 4 x. Spatial momentum integrals are written as

Physical picture
We start by outlining the physical picture for how our computation fits in a generic leptogenesis framework (a recent example can be found in ref. [6]). A key observation is that in leptogenesis the system is almost in thermal equilibrium. 3 Most physical quantities rapidly fluctuate thermally around their equilibrium values. The corresponding reactions are often referred to as "spectator processes" [7,8]. Other quantities relax to equilibrium on time scales much larger than the Hubble time, so that they can be considered conserved. A few have relaxation times of the order of the Hubble time. Only these have to be taken into account as dynamical degrees of freedom, and we will refer to them as "slow". What they are 1 Or closely related quantities, depending on the temperature under consideration. 2 Or, again depending on the temperature, the dissipation matrix. 3 With the exception of leptogenesis during reheating, which is a process far from equilibrium.  Examples of processes, up to O(g 2 ), through which a net lepton number can be "washed out". Arrowed, dashed, and wiggly lines correspond to leptons, scalars, and gauge fields, whereas RH neutrinos are denoted by a double line. Additional reactions (not shown) involve the generation of antileptons or processes mediated by SM Yukawa or Higgs couplings. depends on the Hubble rate and thus on the temperature. A non-equilibrium state is then characterized by deviations of the slowly relaxing quantities from their equilibrium values which are much larger than a typical thermal fluctuation.
One of the slowly relaxing quantities is L − B. It is violated by the Yukawa interactions between Standard Model leptons and right-handed (gauge singlet or "sterile") neutrinos N I , with Majorana masses M I . Therefore, if at some time in the evolution of the Universe L − B is non-zero, these interactions tend to reduce it (examples of processes are shown in figure 1). There can also be a source term for the lepton number if the number density of right-handed neutrinos deviates from thermal equilibrium and if their interactions violate CP. This would lead to baryogenesis through leptogenesis [1].
If the slow variables have only small deviations from equilibrium, we can linearize their equations of motion. For example, assuming that the only slow degrees of freedom are the lepton minus baryon number density n L−B and the right-handed neutrino phase space distribution f N (momentum, spin and flavour indices are suppressed and it is assumed that the entanglement of the right-handed neutrinos does not play a role), the evolution equations are of the form where D t is the appropriate time derivative in an expanding background, f eq is the equilibrium distribution, and terms of higher orders in deviations from equilibrium and in time derivatives have been neglected. 4 Within this effective description, the coefficients γ are independent of the values of the slow variables n L−B and f N − f eq and of the Hubble rate appearing in D t . In particular, they can be determined arbitrarily close to equilibrium and with vanishing Hubble rate, to any order in Standard Model couplings. This is the philosophy that underlies Kubo relations [9]. The coefficients γ (2) in eq. (2.1) are of O(h 2 ), where h denotes a generic neutrino Yukawa coupling. The terms containing γ (4) violate CP, and must contain additional Yukawa couplings with makes them O(h 4 ). If the terms omitted from eq. (2.1) are suppressed by increasingly high powers of h, and h is very small, we can make use of a perturbative expansion in h. At leading order in h 2 , γ N f eq agrees with the righthanded neutrino production rate of which a lot is known: it has been computed up to NLO in SM couplings in the non-relativistic [10][11][12] and relativistic [13,14] regimes, and to LO in the ultrarelativistic regime [15,16], which necessitates a resummation of the loop expansion.

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With a similar philosophy, the CP-violating source term γ (4) L−B,N f eq has been expressed in terms of a Green's function which could in principle be evaluated at NLO [17].
In the present paper, we are concerned with the coefficient γ L−B , which may be called the lepton minus baryon number washout or dissipation rate. Within the range of validity of eq. (2.1), it can be computed in a system with f N = f eq , D t = ∂ t , and assuming n L−B to be close to its equilibrium value. If f N = f eq , the rate D t n L−B changes, but the change originates from the other terms in eq. (2.1) rather than from a change of γ 3 General analysis

Conserved and approximately conserved charges
As a first step one has to identify the spectator processes. These processes conserve some charges, which by Noether's theorem are associated with the symmetries that the corresponding interactions respect. Some of these symmetries will be broken by the slow interactions, and the corresponding charges will no longer be conserved. If some symmetries remain unbroken, there are linearly independent charges Xā which are still conserved. 5 In addition, there are charges X a which together with the Xā form a linearly independent set, such that the original charges can be written as linear combinations of the X a and Xā. By definition the X a are no longer conserved. Their values can now depend on time and one finds equations of motion for them of the type in eq. (2.1). The choice of the non-conserved charges X a is not unique, because after adding a conserved charge they are still non-conserved. However, we consider a state in which the strictly conserved charges vanish. Then this ambiguity is irrelevant.
For computing the washout rates the expansion of the Universe can be ignored, as well as the interactions which are much slower than the expansion. We assume that the thermal expectation values of the X a vanish, X a eq = 0. Now we consider a non-equilibrium system in which the X a start with some non-zero values, which we assume to be much larger than their typical thermal fluctuation. The non-equilibrium state is completely specified by the values of X a . Thus, their time derivatives only depend on their values and on the temperature. For sufficiently small values one can expand dX a /dt in powers of X a , and keep only the linear term, It turns out that the coefficients γ ab can be factorized into two parts (cf. eq. (3.10)): to a real-time "spectral function" ρ ac (ω), and to the inverse of a static "susceptibility matrix", denoted by Ξ −1 cb . The latter contains similar information as the flavour matrix A introduced in ref. [18], or the coefficients c , c ϕ that appear widely in leptogenesis literature (cf. ref. [19]). We define these two parts in turn.

Kubo relations for washout rates
To determine the coefficients γ ab we proceed similarly to the general method 6 described in ref. [21]. We work to leading order in the neutrino Yukawa interaction (see below), but, if not stated otherwise, to all orders in other interactions. 5 We choose this set to be complete in the sense that there are no additional charges which are linearly independent of the Xā and which are conserved. In the actual analysis, it may be possible to choose the Xā such that not all of them need to be included explicitly, cf. the discussion below eq. (3.18). 6 For a recent application in relativistic field theory see ref. [20].

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Even in thermal equilibrium the values of the X a are not constant in time, but instead they fluctuate around their equilibrium values which we assume to be zero. For small frequencies these fluctuations can be described by an effective classical theory with an equation of motion similar to eq. (3.1). The only difference compared with eq. (3.1) is that on the right-hand side there is an additional gaussian noise term. This equation of motion can be solved to write the fluctuation of X a at time t in terms of its value at time zero plus the contribution from the noise. This can be used to compute the real-time correlation function where · · · denotes the thermal average over an ensemble in which the strictly conserved charges are zero and do not fluctuate. The effective description is classical, so the ordering inside the average does not play a role. Since the noise is gaussian it drops out when taking the noise average, and one obtains Taking the one-sided Fourier (or Laplace) transform we obtain where ω has a positive imaginary part. The effective description of the correlation function in eq. (3.2) is valid when ω ω UV , where ω UV is a characteristic frequency of the fluctuations of other, "faster" relaxation processes, the spectator processes. Now let ω be real and ω γ, where γ denotes the absolute value of the largest eigenvalue of γ ab . Then one can expand eq. (3.4) and finds On the other hand, the correlation function in eq. (3.2) can also be computed in the microscopic quantum field theory. By matching the results in the regime γ ω ω UV one can then obtain the coefficients γ ab , together with a consistency check on the functional form of the ω-dependence. In quantum field theory we define the correlation function as where the angular brackets indicate an average over a thermal ensemble in which the values of all strictly conserved charges are zero. Making use of text-book relations between frequencyspace correlators involving anticommutators and commutators, 7 the one-sided Fourier transform can now be expressed as

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where f B is the Bose distribution. Here the spectral function for bosonic operators X a and X b is defined as If time reversal T is a symmetry, this spectral function is real (cf. appendix B of ref. [22]). If one neglects the CP violation in the Standard Model and in the Yukawa interactions of the RH neutrinos (the latter is of O(h 4 )), CP is a symmetry, and so is T. In this approximation ρ ab is real. Then for real ω, Let us now compare eqs. (3.5) and (3.9). We are interested in ω ω UV < ∼ T , in which case we can approximate the expression in square brackets by T /ω. Matching the two expressions gives Here V is the spatial volume, and Ξ is a matrix of susceptibilities, As will be seen later on, Ξ ab is finite for V → ∞, and the same is true for γ ab . A consistency check is provided by whether ρ ac indeed has a 1/ω-tail at ω ω UV . The restriction on ω in eq. (3.10) limits the accuracy at which the γ ab can be defined. The γ ab are of order h 2 Λ, where h is a RH neutrino Yukawa coupling to be defined below, and Λ = max{πT, M I }. 8 A rough estimate gives a relative accuracy γ/ω UV ∼ h 2 modulo coupling constants which characterize spectator processes. It is therefore probably not meaningful to calculate the γ ab beyond leading order in h 2 . Of course, radiative corrections due to SM interactions can be computed and can be important.

The susceptibility matrix
In order to make use of eq. (3.10) we need to determine the susceptibility matrix Ξ defined in eq. (3.11), which turns out to require a little care (cf. ref. [23]). It is perhaps simplest to think of the problem in an ensemble in which the values of the strictly conserved charges Xā are zero and do not fluctuate. 9 We write the Hamilton operator as where H 0 describes all free particles as well as their X a -conserving interactions and H int is the interaction which violates X a -conservation. The susceptibilities are equal-time correlation functions, for which H int is a small perturbation. Since we only consider the leading order in h, the susceptibilities can be computed using H 0 , which commutes with the X a . In this JCAP05(2014)041 approximation the X a are conserved, and the ordering of the operators in eq. (3.11) does not matter.
To compute the susceptibilities start with a grand canonical partition function with chemical potentials µ A for all charges X A ∈ {X a , Xā}. Some of the Xā may be gauge charges. In this case the role of µā is played by the zero-momentum mode of the time component of the gauge field [23]. The thermodynamic potential Ω corresponding to fixed Xā = −∂Ω/∂µā is given by the Legendre transform Ω = Ω + µāXā. We are interested in Xā = 0, so we have Ω = Ω with the µā determined by 14) The required susceptibilities are In order to evaluate eq. (3.15) it is sufficient to expand Ω to second order, with the grand canonical susceptibilities Then eq. (3.14) reads Here we see which strictly conserved charges need to be included: the ones which are correlated with strictly conserved charges which are correlated with one of the X a . The remaining ones need not be taken into consideration. This is the case, e.g., for non-abelian gauge charges in the symmetric phase. Solving eq. (3.18) gives µā = − ξ −1 āb χb c µ c where the matrix ξ is defined by ξāb ≡ χāb . (3.19) Inserting this into eq. (3.15) we find 20) 4 Lepton number washout rate

Master formula
So far the discussion was general and did not make any use of the specifics of the interaction which breaks the X a -symmetry. In the basic leptogenesis scenario the SM is extended by -7 -

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adding right-handed neutrino fields N I with Majorana masses M I (we employ a basis in which the Majorana mass matrix is diagonal). In the simplest realization they interact with the SM particles via a Yukawa coupling to ordinary, left-handed lepton doublets i ≡ Li and the Higgs doublet ϕ as follows: Here ϕ ≡ iσ 2 ϕ * with the Pauli matrix σ 2 is the isospin conjugate of ϕ, and the Yukawa couplings are written as a matrix in flavour space, h = (h Ij ).
In the simplest case it is only the interaction in eq. (4.1) which is responsible for inducing slowly evolving processes. In more complicated cases some SM Yukawa interactions proceed at a similar rate, and these interactions have to be taken into account as well [24]. Here we restrict ourselves to the first situation. What the relevant conserved and quasi-conserved charges Xā and X a are, depends on the expansion rate of the Universe and thus on the temperature. One has to take into account all interactions which are much faster than the expansion. The quasi-conserved charges may contain many different fields, for instance both lepton and quark fields. For this reason it is convenient to compute the spectral function of the time derivativesẊ a of the charges rather than of the charges directly, and then use The operatorsẊ a only contain the fields which interact via L int . Furthermore,Ẋ a contains h explicitly. Since we compute only to leading order in h, the thermal average on the righthand side of eq. (4.2) can be taken in an ensemble with h = 0. Therefore we can re-express eq. (3.10) as where the subscript 0 indicates that h = 0 in the average, so that h only appears in the operators. Equation (4.3) has some similarity with Kubo formulas for transport coefficients [9,22], in particular for flavour diffusion (cf. ref. [25]). As already mentioned, the choice of the broken charges is not unique: adding some linear combination of the conserved ones we again obtain a charge which is not conserved. It is possible to choose the symmetries so that they only act on SM particles. Let the left-handed leptons transform as with Hermitian matrices T a . The interaction Lagrangian in eq. (4.1) is not invariant under this transformation. Following the usual steps to derive Noether's theorem one finds

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Given that the thermal average in eq. (4.3) is performed with h = 0, we can integrate out the RH neutrinos treating them as free fields. In a basis where the Majorana mass matrix is diagonal, a straightforward calculation 10 yields for eq. (4.3) where the trace refers to the spinor indices, and k 0 = E I ≡ (k 2 + M 2 I ) 1/2 . The prime is a derivative with respect to energy, and f F is the Fermi-Dirac distribution. We have introduced the spectral function for the composite operator of the SM fields that appears in eq. (4.6), where α, β are Dirac spinor indices. Note that for fermionic operators the spectral function is defined with anticommutators. Equations (4.7), (4.8), together with the expression for the Ξ matrix in eq. (3.15) for charges like those in eq. (4.5) constitute the main formal results of this paper. We stress again that these expressions are valid to any order in SM couplings.

Spectral function
We proceed to discussing how the real-time part of the washout rate, i.e. the weighted integral over ρ in eqs. (4.7) and (4.10), can be evaluated in practice.

Leading order in the non-relativistic regime
Consider first the LO contributions to ρ in the symmetric phase in a regime M I πT . In this case one can neglect the thermal masses of the SM particles 11 which can then be treated as massless and free. For timelike k (k 2 > 0) but with k 0 of either sign one obtains 10 For instance, making use of the imaginary-time formalism, (4.3) contains a 2-point correlator of the righthanded neutrino fields and of the composite operators to which they couple according to (4.6). The former is of the familiar form (/ k + MI )/(k 2 − M 2 I ), with k 0 → ikn; the mass in the numerator is projected out by the Dirac trace. The latter can be expressed in a spectral representation as Σ(pn, p) = dp 0 /(2π) ρ(p 0 , p)/(p 0 − ipn), where ρ is from (4.8). Matsubara sums can be carried out, and generate Fermi-Dirac distributions. A subsequent analytic continuation yields a retarded real-time correlator, and its cut yields the spectral function needed in (4.3). This contains structures like f F (EI + ω) − f F (EI ), which after taking the limit limω→0(. . .)/ω leave over f F (EI ). 11 Thermal masses have to be taken into account for MI < ∼ √ gT [14].

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where p 0 ≡ |p|, q 0 ≡ |q|, and the left chiral projector is defined as P L ≡ (1 − γ 5 )/2. For eq. (4.10) we get When M I πT , the Bose and Fermi distributions in the square brackets are exponentially suppressed with exp(−M I /T ) and can be neglected. Omitting terms of order exp(−M I /T ) also in f F (E I ), it is straightforward to perform the integrals, yielding Here K 1 is a modified Bessel function of the second kind. Once a LO susceptibility from section 4.3.1 is inserted, this reduces to a standard result, cf. eq. (4.26).

Next-to-leading order
In general the spectral function in eq. (4.8) contains two independent Dirac structures at finite temperature [26]. However the Dirac trace in eq. (4.7) is exactly the same as appears in the right-handed neutrino production rate, projecting out a particular linear combination of the Dirac structures. Assuming as before that ρ ijαβ (k) = δ ij ρ αβ (k), the production rate of flavour I reads (4.14) Comparing with eq. (4.10), it is seen that W differs from P only through a weight, −f F (E I ) versus f F (E I ). 12 Therefore, W can be extracted from known results for P. The extraction of W is particularly simple in the non-relativistic regime πT M I , where f F (E I ) = −f F (E I )/T . Displaying contributions involving the U Y (1), SU L (2) and SU(3) gauge couplings g 1 , g 2 and g 3 , the Higgs self-coupling λ, with the tree-level value λ = g 2 2 m 2 H /(8m 2 W ), as well as the top Yukawa coupling h t , the Dirac trace reads [11] ( k) βα ρ αβ (k) + ρ αβ (−k) = where Hereμ is the MS renormalization scale related to the neutrino Yukawa couplings appearing in eq. (4.9), and m ϕ is the thermal Higgs mass parameter, where m 2 0 < 0 is the vacuum value, and we assume m 2 ϕ > 0. Integrating over the momenta in eq. (4.10) yields corrections to eq. (4.13): If the temperature is increased to T > ∼ M I /4, which may be relevant e.g. for setting the initial conditions for leptogenesis, we leave the non-relativistic regime [14]. The NLO production rate in the relativistic regime (πT ∼ M I ) and the LO production rate in the ultrarelativistic regime (gT > ∼ M I ) are also known but only in numerical form [14,16]. 13 The results of refs. [14,16] for P(M I )/T 4 are plotted in figure 5 of ref. [14]. The results of refs. [14,16] for W(M I )/T 3 , obtained by changing the weight from f F (E I ) to −f F (E I ), are shown in figure 2. It is seen how the perturbative expansion breaks down and resummations are necessary for T > ∼ M I , and how the subsequent washout rate is strongly enhanced compared with a naive tree-level analysis ("LO" in the plot). It is also clear that the non-relativistic JCAP05(2014)041 expansion of eq. (4.20) breaks down for T > ∼ M I /4. (To be more precise, the non-relativistic expansion shows convergence only for T < ∼ M I /15, but the smallness of any loop corrections allows it to be used in practice up to somewhat higher temperatures [14].)

Susceptibility matrix
The susceptibilities as given by eq. (3.11) or eq. (3.15) measure the correlations in fluctuations of the slowly varying charges X a when the other charges Xā are constrained to vanish. The susceptibility matrix influences the lepton number washout rate as indicated in eq. (3.10), (4.7) and (4.9). Here we show how the susceptibilities can be computed in practice, first at leading order and then including corrections of O(g) and O(g 2 ). We consider the Standard Model with left-handed quarks (q ≡ q L ) and leptons ( ≡ L ); as well as right-handed up-type quarks (u ≡ u R ), down-type quarks (d ≡ d R ), and charged leptons (e ≡ e R ).

Leading order
The susceptibilities defined by eq. (3.11) are, at LO, determined by free field theory. It turns out that at NLO it is helpful to first evaluate the grand canonical potential Ω, and then extract the answer directly from the second relation in eq. (3.15). However, we start by showing how at LO the results can also be obtained from the 2-point correlators in eq. (3.17) through eq. (3.20).
For eq. (3.17) one has to compute fluctuations of charges, which for free fields are in one-to-one correspondence with particle number fluctuations. Start with the fluctuation Q 2 where Q is the difference of particle and antiparticle number of a single fermion species. By a single fermion species we mean either one chiral fermion or a single spin state of a Dirac fermion. For only one fermionic degree of freedom, we have the particle number fluctuation (∆N ) 2 = V T 3 /12. The fluctuations of particles and antiparticles are uncorrelated which implies Q 2 = V T 3 /6.
The fluctuations of the charges of a set of left-chiral fermion fields ψ i , ψ i = P L ψ i , are given by The free propagators are flavour diagonal which directly implies Q a Q b = V T χ ab with the susceptibilities We also need the fluctuation of the weak hypercharge Y ϕ of the Higgs field. For a single scalar particle species (including the antiparticle) of unit charge the charge fluctuation is Q 2 = V T 3 /3. Therefore, counting both isospin states, with the Higgs hypercharge y ϕ = 1/2.

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Consider now the simplest realistic case, with only one right-handed neutrino N 1 at a temperature T 10 13 GeV. Then only top Yukawa and gauge interactions are in equilibrium. All other SM Yukawa interactions, as well as strong and weak sphalerons can be neglected. It is then sufficient to consider only the left-handed leptons , the Higgs, the 3rd family quark doublet q 3 and the right-handed top t. Without H int we have a U(3) symmetry as well as two U(1) symmetries, generated by the baryon number B q 3 t carried by q 3 and t, and by the hypercharge Y q 3 tϕ carried by q 3 , t, and ϕ (we denote X AB ··· ≡ X A + X B + · · · ). 14 Using a U(3) transformation we choose the fields such that N 1 only couples to the of one family, which we denote by N 1 . Without H int the corresponding lepton number L N 1 is conserved. It is broken by H int , together with Y q 3 tϕ , leaving Y q 3 tϕ N 1 unbroken. Thus the only X a can be chosen as L N 1 , and the set of Xā consists of Y q 3 tϕ N 1 and B q 3 t . 15 If we arrange the charges in this order we obtain at LO Then from eq. (3.20) Ξ is just a number, Ξ = T 2 /4. In this scenario h Ii {T a , T c } ij h * Ij = 2|h 11 | 2 . Undoing the U(3) rotation, eq. (4.13) subsequently gives the LO result [6] (it corresponds to c = 1, c ϕ = 2/3 in the notation of ref. [19])

Next-to-leading order
We now include Standard Model interactions. The up-type, down-type, and charged lepton Yukawa couplings are denoted by h uij , h dij , h eij , respectively, where i, j ∈ {1, 2, 3} label families.
In order to determine the susceptibilities, it is convenient to first compute the pressure [27], P (T, µ), as a function of the temperature and the chemical potentials associated with all charges X A . The pressure determines the grand canonical potential through Ω = −P (T, µ)V . (4.27) One should include only interactions which are in thermal equilibrium (see the general discussion in sections 2 and 3). We assume this is the case for the gauge interactions, and, depending on the Hubble rate and thus on the temperature, some Standard Model Yukawa interactions.
If one collects all fermion fields in one big spinor ψ, the fermionic contribution to the conserved charges can be written as with hermitian matrices T A . Including the chemical potentials corresponds to adding a term ψγ 0 µψ with µ ≡ µ A T A to the Lagrangian. For simplicity we carry out the computation in 14 In principle one could have chosen the total hypercharge Y as one of the Xā. However, this would be rather inconvenient because there are a lot of conserved charges which are correlated with Y and which would then all have to be included in the set of XA. Note also that when employing (4.23) one has to keep in mind that the Ta may contain unit matrices in colour or weak isospin space which would contribute factors of N colour or N weak isospin to the trace (cf. section 4.3.2). 15 Without Hint the charges corresponding to the off-diagonal generators which mix N 1 with the other families are conserved as well, and they are also broken by Hint. Here we consider only the dissipation of LN 1 ; the evolution of diagonal and off-diagonal charges decouples at leading order. the symmetric phase of the electroweak theory. In this situation the T A commute with weak isospin rotations in addition to colour rotations. They are thus block diagonal and can be written as T A = T q A ⊗ 1 colour ⊗ 1 weak isospin + T u A ⊗ 1 colour + · · · . Correspondingly, the chemical potential matrix takes the form µ = µ q ⊗ 1 colour ⊗ 1 weak isospin + µ u ⊗ 1 colour + · · · , where µ q , µ u , . . . are matrices in family space. Then the fermion propagators are matrices in family space as well.

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The dependence of P (T, µ) on the chemical potentials is needed only up to quadratic order (cf. eq. (3.17)). The results of section 4.3.1 correspond to (unresummed) 1-loop contributions to the pressure. The Standard Model interactions enter at two loops. All 1-and 2-loop Feynman diagrams are displayed in figure 3. Since gauge interactions are flavour blind, the chemical potentials can be diagonalized. In the Lagrangian we include only Yukawa interactions which are in equilibrium. The terms included have to be invariant under the symmetry transformations generated by the X A . This implies relations between the T A and the in-equilibrium Yukawa couplings, where T ϕ A is simply a number. Multiplying by µ A one finds relations between the chemical potentials, (4.31) Yukawa couplings mediating reactions not in equilibrium have to be omitted. By making use of eqs. (4.31) and their hermitian conjugates, as well as substitutions of sum-integration variables, the 2-loop computation can be reduced to products of the following 1-loop sum-integrals: respect to µ Y , which then leads to Numerically, the correction appearing in the second structure of eq. (4.41) is 23%, and it is dominated by the term proportional to g 2 3 . Close to the electroweak crossover, where m ϕ ∼ g 2 T /π, the perturbative expansion associated with the Matsubara zero modes breaks down, and non-perturbative methods are needed for determining Ξ.
For T < ∼ 130 GeV, the sphaleron processes violating B + L are so slow that B is effectively conserved [30]. Then Ξ is a different 3×3 matrix from the above. In a narrow temperature range around T ∼ 130 GeV, both B and L i need to be treated as separate slow variables, and Ξ is a 4 × 4 matrix. For practical purposes it may be sufficient to solve separate 3-variable non-equilibrium problems in the regimes T > ∼ 130 GeV and T < ∼ 130 GeV and just match the solutions at T ∼ 130 GeV by requiring continuity. is the tree-level decay rate of N 1 , we have used K = 7. We have started the evolution with zero initial asymmetry and thermal N 1 -number density at T = M 1 . We have used the nonrelativistic approximation [6], and solved the evolution equations until T = M 1 /10, below which the asymmetry hardly changes any more. We find that the effect of the O(g) corrections to Ξ on the asymmetry is about 3%. The order O(g 2 ) corrections to Ξ and to ρ are 1.3% and 1%, and the total effect of the corrections on the final asymmetry is ∼ 5%.

Lepton asymmetry
If, in contrast, a scenario like in ref. [2] is considered, in which temperatures around the weak scale play a role and the dynamics takes place in the ultrarelativistic regime (M I T ), then it is clear from figure 2 and from the discussion below eq. (4.41) that effects of order 100% are to be expected. We have not carried out numerics for this scenario, however. In this paper we have obtained a relation, eq. (4.7), between the lepton number washout rate relevant for leptogenesis, and finite temperature equilibrium correlation functions. The washout rate factorizes into a real-time spectral function, and an inverse susceptibility matrix which is determined by equilibrium thermodynamics (cf. eq. (3.15)). This relation does not make use of any particle approximation, and is valid to all orders in Standard Model couplings and at any temperature. The main approximation made is that we have worked to order h 2 in neutrino Yukawa couplings, which should be a good approximation in many popular leptogenesis scenarios.
We have computed explicitly the spectral function and the susceptibility matrix to nextto-leading order (NLO) in Standard Model couplings for temperatures above the electroweak crossover temperature but below the mass M 1 of the lightest right-handed neutrino, i.e., when the right-handed neutrinos are non-relativistic (150 GeV < ∼ πT M 1 ). This is particularly relevant for leptogenesis in the strong washout regime.
We find that even in the non-relativistic regime there are corrections only suppressed by O(g). They originate from Higgs effects on the susceptibility matrix, cf. eqs. (4.37), (4.39), (4.41). In contrast, NLO corrections to the spectral function are of O(g 2 ) in this regime, cf. eq. (4.17). Numerically, the O(g) corrections are a few percent, except for temperatures close to the electroweak crossover, where they can be substantially larger.
In the relativistic regime M 1 < ∼ πT , the susceptibilities remain unmodified since they are insensitive to M 1 . In contrast, the spectral functions become increasingly sensitive to infrared corrections, and extensive resummations are needed for obtaining even the complete leading-order results for M 1 < ∼ gT . We have shown that fortunately, the results can be inferred, after minor modifications, from existing computations of the right-handed neutrino production rate [14,16]. Numerical results are shown in figure 2. The lepton number washout rate of the relativistic regime plays a role at the initial stage of the classic leptogenesis process, erasing some of the lepton asymmetry that is being generated when right-handed neutrinos are produced from the Standard Model plasma, and would also be relevant for scenarios in which the right-handed neutrino masses are at or below the weak scale ( [2] and references therein).