Right-Handed Neutrino Production at Finite Temperature: Radiative Corrections, Soft and Collinear Divergences
aa r X i v : . [ h e p - ph ] F e b TUM-HEP-875-13TTK-13-02
Right-Handed Neutrino Production at FiniteTemperature:Radiative Corrections, Soft and CollinearDivergences
Bj¨orn Garbrecht ♠ , ♦ , Frank Glowna ♠ , ♦ and Matti Herranen ♣ , ♦♠ Physik Department T70, James-Franck-Straße,Technische Universit¨at M¨unchen, 85748 Garching, Germany ♣ Niels Bohr International Academy, Niels Bohr Institute and Discovery Center,Blegdamsvej 17, DK-2100 Copenhagen, Denmark ♦ Institut f¨ur Theoretische Teilchenphysik und Kosmologie,RWTH Aachen University, 52056 Aachen, Germany
Abstract
The production and decay rate of massive sterile neutrinos at finite temperaturereceives next-to-leading order corrections from the gauge interactions of lepton andHiggs doublets. Using the Closed-Time-Path approach, we demonstrate that theperturbatively obtained inclusive rate is finite. For this purpose, we show that soft,collinear and Bose divergences cancel when adding the tree-level rates from 1 ↔ ↔ ↔ ontents Introduction
Scattering processes between particles at finite temperature play an important role inEarly Universe Cosmology in view of applications such as Baryogenesis and the produc-tion of Dark Matter. Besides, certain properties of hot and dense strongly interactingmatter, such as the photon production rate and transport coefficients rely on the knowl-edge of these scattering rates.It turns out useful to distinguish between the scatterings of four massless particles,which are at leading order restricted to 2 ↔ ↔ ↔ t -channel exchange of a massless particle, leading to a Coulombdivergence at tree level. This divergence is mitigated by the screening in the plasma,and technically, it can be removed by a resummation of thermal self-energies withinthe propagator of the mediating particle [1–3]. When one of the scattering partnersis massive, this divergence no longer occurs, but instead, there are soft and collineardivergences.The soft and collinear divergences are of the same type as e.g. those familiar fromQCD corrections to the quark pair production from an off-shell photon. While theproduction rate for quark pairs and a soft or collinear gluon becomes non-perturbativebelow a certain transversal momentum scale between the gluon and the emitting quark,the inclusive cross section for pair production with and without gluons is perturbativelywell defined, due to the cancellation of soft and collinear divergences between tree-leveland loop diagrams. In the following, we refer to the soft and collinear divergencescollectively as infrared (IR) divergences.It is natural to consider the same problem at finite temperature. Recently, thismatter has received attention in the context of Leptogenesis [3–6]. The quantity ofinterest is the relaxation rate of sterile right-handed neutrinos N toward equilibriumfrom decays and inverse decays of Higgs bosons and leptons. When the mass M of N issmall compared to the temperature T , radiative corrections are of leading importance,because the 1 ↔ M ≫ T ,radiative corrections are subdominant to the leading 1 ↔ M ≫ T corresponds to strong washout, which is in type-Isee-saw scenarios perhaps favoured in the light of the observed mass-scale of the activeneutrinos. Whether the 1 ↔ M ≪ T or not, we refer to the radiative corrections from Standard Model gauge interactions asNLO in this paper. The problem of the NLO corrections to N production for M ≫ T has recently been resolved [5, 6], making use of the fact that Maxwell statistics is a goodapproximation in the non-relativistic regime and that one may assume that the scale ofthe three momentum of N is much below the temperature T . Related to this matterare also works on the Drell-Yan production of dileptons from a strongly interactingplasma [7–10], and some useful calculational techniques are developed in Ref. [11, 12],3here the spectral function of hot Yang-Mills theory is studied.However, it still remains interesting to demonstrate the cancellation of IR divergencesbetween tree-level and loop corrections at NLO, keeping full account of relativistic effectsand of quantum statistics. This is the main result of this work. The proof of thecancellation of the IR divergences that is presented here relies partly on analyticallyisolating them from the integrals (for the wave-function corrections) and partly on arearrangement of the integrand such that only integrable singularities remain (for thevertex corrections). Therefore, this proof is already suggestive of a method of numericallycalculating the production rate, a task that we will pursue in a separate work.In order to compute the relaxation rate for N at finite temperature, we choose touse the Closed-Time-Path (CTP) method [13, 14]. In this formulation, Schwinger-Dysonequations can be derived [15–17] (a particular subset of which are known as the Kadanoff-Baym equations) that describe the time evolution of the Quantum-Field-Theoreticalnon-equilibrium system. For this reason, the CTP approach has recently been appliedto Leptogenesis in different parametric regimes, and various new effects and correctionshave been derived [18–34] . In the CTP formulation, the relaxation rate is directlyrelated to the spectral (or anti-hermitian) self energy. As we assume that the fieldsthat participate in gauge interactions are in thermal equilibrium (which applies to allfields within the loop diagrams that contribute to the self energy of N in the presentapproximations), we notice that the relaxation rate can be obtained from equilibriumfield theory as well. Here, we choose the CTP approach because of its connection tonon-equilibrium field theory formulations for Leptogenesis. Moreover, the relation of thespectral self energy to the relaxation rate can be easily inferred from the kinetic equationsthat are derived from the Schwinger-Dyson equations, ( cf. Ref. [24]), and finally, theCTP approach provides very efficient Feynman rules that are perhaps easier to applythan certain finite-temperature cutting rules used in Ref. [5].The methods that we employ in order to demonstrate the cancellation of IR di-vergences are very explicit, which has the disadvantage that they rely on a number oftechnical details on the evaluation of phase-space and loop integrals. On the other hand,the explicit demonstration of the cancellation readily suggests a method of practicalevaluation of the neutrino relaxation rate to be pursued in a future work. In order tocommunicate the main points of this work to a reader who is not interested in the tech-nical details and to give a first overview to a reader who is at least partly interested inthese details and their reproduction, we provide in Section 2 a summary of our methodand of the cancellation of the IR divergences, that gives reference to the main resultsthat are worked out in the technical Sections. We begin Section 2 with a part containinggeneral prerequisites and remarks about the CTP approach and the neutrino relaxationrate (Section 2.1). As suggested by the different diagrammatic contributions to the relax-ation rate, which also lead to different calculational methods, we further present separateparts about wave-function type (Section 2.2) and vertex-type (Section 2.3) corrections.In Section 3, we present the main results for the IR- and ultraviolet-finite neutrino pro-duction rate, and the the full calculational details are then presented in Section 4 for thewave-function corrections and in Section 5 for the vertex corrections. We conclude with4ection 6.
We are interested in the production rate of right-handed neutrinos N , or more precisely,in their relaxation rate toward thermal equilibrium. The self energy for N is given by / Σ N . At one-loop order, it originates from Yukawa interactions Y (that we choose hereto be real) with lepton doublets ℓ and Higgs-boson doublets φ . Besides, / Σ N receiveshigher order corrections due to the SU(2) and U(1) gauge interactions in the symmetricElectroweak phase at high temperatures. In particular, the model we consider is givenby the Lagrangian L = L SM + 12 ¯ ψ N (i /∂ − M ) ψ N − Y ¯ ψ ℓ φ † P R ψ N − Y ¯ ψ N φP L ψ ℓ , (1)where L SM is the Lagrangian of the Standard Model, ψ N,ℓ are the spinors associated with N and ℓ and P L , R = (1 ∓ γ ) /
2. (Group indices of SU(2) and antisymmetric tensors thatmay be necessary to form invariant products are suppressed throughout this paper.) Onthe spinor associated with N , we impose the Majorana condition ψ cN = C ¯ ψ TN = ψ N ,where C is the charge-conjugation matrix. The gauge couplings are given by by g for SU(2) and g for U(1), and moreover, we denote by g w = 2 the dimension of thefundamental doublet representation of SU(2). As the Higgs doublet and the leptondoublet have the same Electroweak charges, it is useful to define the recurring factor G = 34 g + 14 g . (2)Besides, there are sizeable corrections from top-quark loops and self-interactions of theHiggs boson. While we do not include the latter explicitly, their treatment should bestraightforward, given the discussion of the wave-function corrections in Sections 2.2and 4.Since we assume the leptons, Higgs bosons and gauge bosons to be in thermal equi-librium, the relaxation term in the kinetic equation for the right handed neutrinos isproportional to their deviation from equilibrium [24]: ddt f N ( k ) = 14 Z dk π sign( k )tr (cid:2) iΣ / >N ( k )i S 0, becauseof the t -channel exchange of massless fermions [1–3]. We will deal with this matter in aseparate publication [35].The Yukawa coupling between the right-handed neutrino N , the lepton doublet ℓ andthe Higgs doublet φ gives rise to the leading-order contribution to the self energy of theright-handed neutrino, that is represented diagrammatically in Figure 1, h i / Σ abN ( p ) i LO = g w Y Z d p (2 π ) (cid:0) i S abℓ ( p + k )i∆ abφ ( − k ) + C [i S baℓ ( − p − k )] t C † i∆ baφ ( k ) (cid:1) , (4)where a, b = ± are CTP indices (we follow the conventions used in Ref. [16]) and t standsfor a transposition of Dirac matrices. (We indicate the various contributions to / Σ N bysuperscripts on square brackets, for the sake of readability. Notice that we deviate fromthe conventions of Ref. [16] in the detail that we define the fermionic self-energies as / Σ, while in Ref. [16] the same quantities are defined without a slash. ) The tree-level Here it is understood that / Σ may involve chiral projection matrices P L , R , for instance for left-chiral h / Σ A N ( p ) i LO can be found in Ref. [24].The main purpose of this work is to provide a method for calculating the NLOcontributions. These may be categorised in two-particle-reducible wave-function typecorrections (superscript WV, Figure 3) and two-particle-irreducible vertex-type correc-tions (superscript VERT, Figure 4). Due to the t -channel divergences, it turns out thatthe two-particle-reducible contributions diverge for M → 0, and instead, a two-particle-irreducible one-loop diagram with resummed propagators should be calculated [35].In all our calculations, we denote by p µ = ( M, , , 0) (5)the momentum of the right-handed neutrino in the Centre of Mass System (CMS). Therelative motion with respect to the plasma is accounted for by a plasma vector u µ = 1 M (˜ p , ˜ p ) , (6)where ˜ p and ˜ p are energy and three-momentum of the right-handed neutrino in theplasma frame (where the plasma is at rest). Moreover we note thattr[ /p / Σ abN ( p )] = tr[ /p / Σ baN ( − p )] , (7)which follows from the relation (A8), such that our choice for p with p > The wave-function type contributions can be written as h / Σ A N ( p ) i WV = g w Y Z d k (2 π ) h i S (1) >ℓ ( p + k )i∆ >φ ( − k ) + i S >ℓ ( p + k )i∆ (1) >φ ( − k ) i − < ↔ > , (8)where i S (1) ℓ and i∆ (1) φ are the one-loop corrections to the propagators, i.e. the self-energieswith external legs ( cf. Figure 2). The particular diagrams contributing to [ / Σ N ] WV are therefore two-particle reducible, as apparent from the diagrammatic expression for[ / Σ N ] WV in Figure 3. It would be a simple matter to use fully resummed propagatorsrather than the single-loop insertions, however it is interesting to verify that perturbationtheory is well defined without resorting to a resummation. fermions / Σ = P R γ µ Σ µ . Figure 2: The amputated diagrams in this Figure represent the gauge contributions to / Σ ℓ (first diagram) and Π φ (the sum of the second and third diagram). When thesediagrams are not amputated, they correspond to i S (1) ℓ (first) and i∆ (1) φ (sum of secondand third). We refer to the first and second diagram as sunset diagrams, to the thirdas seagull diagram. Due to the CTP Feynman-rules, the seagull diagram contributes toΠ Hφ , but not to Π A φ .The rate can be decomposed into a contribution F from gauge-boson radiation fromthe lepton and a contribution B from gauge-boson radiation from the Higgs boson, suchthat tr h /p / Σ A N ( p ) i WV = F + B , (9a) F = g w Y Z d k (2 π ) tr h /p (cid:16) i S (1) >ℓ ( p + k )i∆ >φ ( − k ) − i S (1) <ℓ ( p + k )i∆ <φ ( − k ) (cid:17)i , (9b) B = g w Y Z d k (2 π ) tr h /p (cid:16) i S >ℓ ( p + k )i∆ (1) >φ ( − k ) − i S <ℓ ( p + k )i∆ (1) <φ ( − k ) (cid:17)i . (9c)The diagrams that represent [ / Σ N ] WV and its decomposition in F and B are shown inFigure 3.In order to explain how the cancellation of IR divergences works, we focus on emis-sions of gauge radiation from the Higgs boson, that is captured by B . Radiation from thelepton can be treated along the same lines, but is technically slightly more involved dueto the spinor structure. In Section 5, we present all the necessary details on gauge radi-ation from the lepton. The one-loop correction to the scalar propagator can be writtenas i∆ (1) <,>φ ( k ) =2∆ A (1) φ ( k ) f <,>φ ( k · u ) , (10a) f <φ ( k · u ) = f φ ( k · u ) , f >φ ( k · u ) = 1 + f φ ( k · u ) , (10b)where ∆ A (1) φ ( k ) is the one-loop correction to the spectral function, cf. Figure 1 andEqs. (A1). As we assume that thermal equilibrium is forced by the gauge interactions,throughout this paper, we take for the distributions of Higgs bosons and gauge bosons f φ and f A the Bose-Einstein distribution and for the distribution of leptons f ℓ the Fermi-Dirac distribution. Nonetheless, we keep the subscripts A , φ and ℓ in order to indicatethe origin of the individual distributions. 8 + ++ + Figure 3: The self-energy term [ / Σ N ] WV . The first and second terms constitute thecontribution F , the third to sixth the contribution B to tr[ /p / Σ A N ] WV .Now in thermal equilibrium, the one-loop correction to the spectral function can beexpressed as [36, 37]∆ A (1) φ ( k ) = 12 (cid:0) [∆ Rφ ( k )] [iΠ Hφ + Π A φ ] − [∆ Aφ ( k )] [iΠ Hφ − Π A φ ] (cid:1) (11)= 12 (cid:0) [∆ Rφ ( k )] − [∆ Aφ ( k )] (cid:1) iΠ Hφ + 12 (cid:0) [∆ Rφ ( k )] + [∆ Aφ ( k )] (cid:1) Π A φ , where Π H, A φ are the hermitian and spectral self-energies of the Higgs boson with a gaugeboson in the loop. [ Cf. Eqs. (A1) for the definition of spectral and hermitian two-pointfunctions.] In principle, it is important to assume here thermal equilibrium, because oth-erwise, there would also occur the product of a retarded and an advanced propagator, i.e. R,Aφ = i k ± i sign k ε , (12)such that in the distributional sense,[∆ Rφ ( k )] − [∆ Aφ ( k )] = − π i X ± δ ( k ± | k | ) (cid:18) k ddk − k (cid:19) , (13a)[∆ Rφ ( k )] + [∆ Aφ ( k )] = 2 k . (13b)Using above identities together with the expressions for the tree-level propagators (A6,A7),the one-loop corrections due to real and virtual gauge bosons attaching to the Higgs bo-son are B = g w Y Z d k (2 π ) tr h /p i S >ℓ ( p + k )i∆ (1) >φ ( − k ) − /p i S <ℓ ( p + k )i∆ (1) <φ ( − k ) i (14)= g w Y Z d k (2 π ) X k = − p ±| k | p (1) A φ ( k ) [ f ℓ (( p + k ) · u ) + f φ ( k · u )]= g w Y Z d k (2 π ) X k = − p ±| k | p " π X ± ′ δ ( k ± ′ | k | ) (cid:18) k ddk − k (cid:19) Π Hφ ( k )+ 2 k Π A φ ( k ) × [ f ℓ (( p + k ) · u ) + f φ ( k · u )] . In order to identify the IR-divergent contributions, it is useful to split the self-energyinto parts that are present for T = 0 with a superscript vac and parts that vanish for T = 0 with a superscript T = 0:Π H, A φ = Π H, A , vac φ + Π H, A ,T =0 φ , (15)The self-energies Π Hφ and Π A φ give rise to two different contributions to B . To thosefrom Π Hφ , we refer to as wave-function contributions, and to those from Π A φ as scattering10ontributions. Notice that the scattering contributions include besides the tree-level2 ↔ ↔ B resulting from this splittingand explain how the IR divergences present in the particular contributions can be seento cancel eventually: • Finite temperature contributions: The contributions from Π H, A ,T =0 φ to Eq. (14) leadto IR-finite integrals. It is well known that the hermitian part of the thermal self-energy Π H,T =0 φ is finite, cf. the explicit expression (27) below. Therefore, it givesrise to a finite contribution to the integral (14). For the contribution from Π A ,T =0 φ ,we show that we can express Π A ,T =0 φ ( k ) = k h ( k ), where h ( k ) is a continuousfunction, cf. Eqs. (36) below. Therefore, the first order singularity at | k | = k (or,equivalently, | k | = p / 2) is integrable in the principal value sense. • Vacuum wave-function contributions: It is well-known that the vacuum self-energyΠ H, vac φ for a scalar field with a gauge boson and a scalar boson in the loop is IR-divergent. This divergence can be regulated by a fictitious gauge-boson mass λ .More precisely, the λ dependence is ∝ log λ and given by Eqs. (24b,25) below. • Real gauge boson emission (scatterings) in the vacuum: On the other hand, Π A , vac φ ( k ) ∝ k , such that the resulting singularity at | k | = k is again first order. Now however,this gives an IR-divergent contribution to the integral, because Π A , vac φ ( k ) /k is notcontinuous for k = 0. Therefore, we introduce for this term the gauge boson mass λ as well. The contribution from Π A , vac φ to B adds to the rate of 1 → N . Because Π A φ ( k ) /k itself corresponds to a decay rate of a scalar ofmass-square k into a massless scalar and a gauge boson of mass λ , there is akinematic threshold that can be expressed in terms of a Heaviside ϑ -function, cf. Eq. (28) below. As a result, this contribution to the integral has an IR divergence ∝ log λ as well. When we perform an integration by parts, we can isolate the IRdivergence and compare it with the one from wave-function corrections. • Cancellation of IR divergences from wave-function and scattering corrections: Oncethe IR divergences are isolated in terms ∝ log λ by performing the integral overthe δ function in Eq. (14) (for the virtual contributions from Π H, vac φ ) or throughintegration by parts (for the real contributions from Π A , vac φ ), one can see that thedependences on log λ cancel in the total result. For this purpose, compare Eqs. (33)and (38). This can be viewed as a consequence of the fact that ϑ ( k − λ ) π dd log λ Π H, vac φ ( k ) = sign( k )Π A , vac φ ( k ) , (16)which is no accident, because Π H, vac φ and Π A , vac φ are real and imaginary part of thesame analytic self-energy. 11he contributions from gauge-boson radiation from the lepton are discussed in detailin Section 4.2. For the vacuum contributions to the spectral and hermitian self-energies,the discussion goes along the same lines as for the radiation from the Higgs boson. Inaddition, there are IR divergences when the thermal part of the hermitian self-energy isevaluated on shell, which are matched by IR-divergent contributions from the thermalpart of the spectral self-energy. We note here that these thermal contributions whichlead to IR divergences can be identified with the self energies that one obtains in thehard thermal loop (HTL) approximation. Otherwise, the method of demonstrating thecancellation of these divergences is very similar to the one applied to the cancellation ofthe IR-divergent vacuum-contributions. The method relying on the relation (16) between the spectral and hermitian self-energiesthat we employ for the two-particle-reducible wave-function contributions obviously can-not be applied to the present case of the vertex diagram, Figure 4. One may again usea gauge-boson mass in order to regulate the soft and collinear divergences. Due to themultiple integration boundaries that appear for virtual corrections and real emissions,one yet faces the problem of matching the particular contributions that lead to a cancel-lation of the IR divergences, i.e. of the dependence on the fictitious gauge-boson mass.This task can be facilitated by transforming the integrals and by arranging and addingthe particular integrands in such a manner that there remains a single integrand thatmanifestly evaluates to a finite answer. This is the method pursued in the present work.One could then either go back to evaluate the particular divergent contributions with aregulating gauge-boson mass, since it is then clear that the dependence on the regulatoreventually cancels from the finite result. Alternatively, one may directly integrate thetotal integrand without introducing an IR regulator, what again leads to a finite result.In order to accomplish the task of arranging the particular contributions to the vertex-type self-energy into a manifestly IR-finite integral, we focus on the reduced Wightmanself-energy tr[ /p i / Σ Figure 5. • The second term yields a correction to the 1 ↔ /p i / Σ Eqs. (77) for the parametrisation in terms of the momenta chosen in Fig-ure 6.] In addition, as we are working in a homogeneous finite-temperature background,there eventually remain two non-trivial angular integrations. The kinematic constraintsthat are forced within the CTP approach by the on-shell delta functions within theWightman and statistic propagators then imply that the integrand for tr[ /p i / Σ Parametrise the integral tr[ /p i / Σ 0) to thequadrant where x, y > • Now add all contributions. The resulting integrand J total ( x, y ), Eq. (116b), onlycontains integrable singularities (Section 5.5). • The integral is not yet convergent for large values of x , y , what corresponds toan ultraviolet (UV) divergence. We obtain a convergent integral by subtractinga term that corresponds to the vertex correction to the vacuum decays N → ℓφ weighted by the thermal distributions for ℓ and φ , ¯ J vert , vac , Eq. (130a). As thiscontribution is IR divergent, we cancel it by adding a correspondingly weightedrate for 1 → J sca , vac , Eq. (132). The subtracted contributions must beadded again to the final results, but for these particular terms, we can performthe integration over dx dy analytically and thus isolate and renormalise the UVdivergence (Section 5.6).The point stating the presence of integrable singularities only requires a proof, thatwe present in Section 5 and that at least in parts elucidates how the cancellation of IRdivergences works in the finite-temperature background. The various contributions to theintegrand obviously factorise into kinematic and quantum statistical terms. We performan expansion of the kinematic factors and the arguments of the statistical functions thatapplies to the collinear fringes, where either | x | ≪ M or | y | ≪ M as well as close tothe point of the soft divergence, where both | x | , | y | ≪ M . Using these expansions, inSection 5.5, we demonstrate the following points: • On the collinear fringes, where x = 0 or y = 0, the total integrand takes finitevalues and is hence integrable, as it is expressed by Eqs. (118) and (119). • The point x = y = 0 requires special care, since the soft gauge-boson singularitycoincides with the Bose divergence of the gauge-boson distribution function. Weshow that the integrand behaves 1 / p x + y for ( x, y ) → 0, such that this isolatedsingularity is integrable as well. A technical point is here that contributions fromdifferent regions of the angular integrations must be averaged. • The divergence from the Higgs boson distribution functions is integrable in theprincipal value sense. 16n order to show the cancellation of the singularities on the collinear fringes, we need toassume that Higgs bosons, leptons and gauge bosons are in thermal equilibrium. As aconsequence, the collinear splitting processes are in equilibrium as well, what leads to thedetailed balance relations (117), that are essential in order to show that the integrand isfinite on the fringes. In this Section, we present the final results for the neutrino relaxation rate in terms ofmanifestly IR- and UV-finite integrals. The explicit expressions for the various termsare given in the subsequent Sections. The total relaxation rate in the plasma frame is[ cf. Eq. (3)]12˜ p tr h / ˜ p / Σ A N (˜ p ) i = 12˜ p (cid:18) tr h / ˜ p / Σ A N (˜ p ) i LO + tr h / ˜ p / Σ A N (˜ p ) i WV + tr h / ˜ p / Σ A N (˜ p ) i VERT (cid:19) . (19)Expressions for the leading order term h / Σ A N (˜ p ) i LO can be found in Ref. [24]. The wave-function-type contributions are given by tr h /p / Σ A N ( p ) i WV = ( B wv , vac + B sca , vac ) + B wv ,T =0 + B sca ,T =0 (20)+ (cid:0) F vac , col , wv + F vac , col , sca (cid:1) + (cid:0) F HTL , col , wv + F HTL , col , sca (cid:1) + F vac , fin + F HTL , fin + F T =0 , and where the B terms correspond to radiation from the Higgs boson and F to radiationfrom the fermion. The various wave function, scattering, vacuum, collinear and HTL(sub)contributions are isolated and collected in a way that the IR divergences cancelwithin each of the parentheses. The UV divergences cancel among the vacuum partsagainst the counter terms. The expressions for the various B and F contributions aregiven in Section 4. Due to Lorentz covariance: tr h / ˜ p / Σ A N (˜ p ) i = tr h /p / Σ A N ( p ) i . f eq N ( p · u )tr (cid:2) /p / Σ A N ( p ) (cid:3) VERT = − g w GY π ) π Z d ( ϕ − ψ ) Z − d cos ̺ ∞ Z dx ∞ Z dy J totalUV ( x, y ) − g w GY π ) Z − d cos ̺ (cid:0) ¯ J vert , vac + ¯ J sca , vac (cid:1) − M π Y δY Z − d cos ̺ f ℓ ( E + ) f φ ( E − ) , (21)with J totalUV ( x, y ) = 14 (cid:16) J sca ( x, y ) + J sca ( − x, − y ) + J sca ( − x, y ) + J sca ( x, − y ) (22)+ J vert1+ ( x, y ) + J vert1 − ( x, y ) + J vert2+ ( x, y ) + J vert2 − ( x, y )+ J vert3+ ( x, y ) + J vert3 − ( x, y ) −J sca , vac ( x, y ) − J vert , vac ( x, y ) + x ↔ y (cid:17) + ϕ → ϕ + π , where the J sca ( x, y ) terms correspond to 2 ↔ ↔ indicated in Figure 6, and J vert1 ± ( x, y ), J vert2 ± ( x, y ) and J vert3 ± ( x, y ) correspond to vertex corrections with an on-shell gauge boson,Higgs boson and lepton, respectively. The vacuum-type contributions J sca , vac ( x, y ) and J vert , vac ( x, y ) are subtracted within J totalUV ( x, y ) to make it UV finite (in addition to beingmanifestly IR finite), and they are cancelled by the vertex counter-term δY . The Fermi-Dirac distribution f eq N in the front of Eq. (21) results from using the KMS relation (A4).The explicit expressions for the integrands are presented in Section 5. In addition to thekinematic x, y -variables, the integrands depend non-trivially also on the angles ϕ − ψ and ρ depicting the orientation between loop momenta and the plasma vector u , and E ± are given by Eq. (82). We now present the technical details that are omitted in Section 2.2, where a morequalitative overview of the present approach to the regulation and cancellation of the IRdivergences is provided. We have recast the integration area to positive x, y > 0, hence the explicit minus signs in thearguments of contributions corresponding to regions II and III. λ , thevacuum and finite-temperature contributions to the hermitian self-energy areΠ H, vac φ ( p ) = − G Z d q (2 π ) Z d k (2 π ) (2 π ) δ ( p − k − q )( p + k ) (23a) × (cid:20) PV 1 k πδ ( q − λ ) + PV 1 q − λ πδ ( k ) (cid:21) +4 G Z d k (2 π ) i k − λ + i ε , Π H,T =0 φ ( p ) = − G Z d q (2 π ) Z d k (2 π ) (2 π ) δ ( p − k − q )( p + k ) (23b) × (cid:20) PV 1 k πδ ( q − λ ) f A ( | q · u | ) + PV 1 q − λ πδ ( k ) f φ ( | k · u | ) (cid:21) +4 G Z d k (2 π ) πδ ( k − λ ) f A ( k ) , where G as defined by Eq. (2) encompasses the gauge coupling constants and u is theplasma vector defined in Eq. (6). The first of the integrals for Π H, vac φ and Π H,T =0 φ cor-respond to the sunset diagrams, the second of the integrals to the seagull diagrams ( cf. Figure 2).A possible way of evaluating Π H, vac φ is to go to the frame where p = and to introducea momentum cutoff Λ. Then, ∂ Π H, vac φ ( p , ) ∂ ( p ) = G π (cid:18) log Λ λ + log 2 + 58 (cid:19) , (24a) ⇒ Π H, vac φ ( p ) = G π p (cid:18) log Λ λ + C (cid:19) + C + p δZ φ + δm φ , (24b)where δZ φ and δm φ are field-strength and mass counterterms. Alternatively, of course,an effectively equivalent result may be obtained using other regularisation procedures, e.g. dimensional regularisation in 4 − ǫ dimensions, such that the leading terms in p areΠ H, vac φ ( p ) = G π p (cid:18) log µλ + 12 ∆ ǫ + 12 (cid:19) + p δZ φ , (25)where ∆ ǫ = 2 ǫ − γ E + log(4 π ) . (26)Notice that, as it is well known, the seagull graph vanishes when evaluated using dimen-sional regularisation. For both regularisation procedures, the UV divergences should19e cancelled by the field-strength renormalisation δZ φ . In the following, we effectivelyaccount for this by replacing Λ → ¯Λ, which takes a finite, renormalisation-scheme de-pendent value.For the part that vanishes as we take T → 0, we set λ = 0 and obtainΠ H,T =0 φ ( p ) = G T − G p π | p | (27) × Z d | k | (cid:20) log (cid:12)(cid:12)(cid:12)(cid:12) p − | k | p + 2 | k || p | p − | k | p − | k || p | (cid:12)(cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12) p + 2 | k | p + 2 | k || p | p + 2 | k | p − | k || p | (cid:12)(cid:12)(cid:12)(cid:12) (cid:21) f B ( | k | ) . This is IR finite, such that it is indeed justified to take λ = 0 for this contribution.For the spectral self-energy, we perform the split into vacuum and finite-temperatureparts accordingly. Notice that due to the CTP Feynman-rules, the spectral self-energyonly receives sunset and no seagull contributions. The zero-temperature part isΠ A , vac φ ( k ) = − G k − λ π k − λ k ϑ ( k − λ )sign k ≈ − G π k ϑ ( k − λ )sign k , (28)where for the approximation, we neglect numerator terms of order λ . The Heaviside ϑ -function occurs due to the mass threshold of the would-be process of a scalar boson ofmass square k decaying into a massless scalar boson and a gauge boson of mass λ . (Theminus sign in front of the rate is not problematic, because an on-shell scalar particlecannot actually change its mass by radiating a gauge boson when the gauge symmetryis unbroken.) For the finite temperature contribution, we obtainΠ A ,T =0 φ ( k ) = G π k | k | | k | − β log 1 − e β k | k | − e β k −| k | ! for k ≥ , (29a)Π A ,T =0 φ ( k ) = G π k | k | k − β log 1 − e β | k | + k − e β | k |− k ! for k < . (29b)We aim to calculate the correction B [defined in Eq. (14)] by gauge-boson emissionfrom the Higgs boson to the relaxation rate of right-handed neutrinos. The term thatdepends on Π Hφ can be identified as the wave-function correction (superscript wv), theterm depending on Π A φ as the correction from 2 ↔ ↔ B = B wv + B sca . (30)For the following calculations, we choose M = p > 0, for definiteness, and note that B p . The wave-function correction is given by B wv = g w Y Z d k (2 π ) X k = − p ±| k | p π X ± ′ δ ( − p ± | k | ± ′ | k | ) 14 k (31) × (cid:18) ∂∂k − k (cid:19) Π Hφ ( k ) [ f ℓ (( p + k ) · u ) + f φ ( k · u )] , where we have made use of Eq. (13a). Clearly, we obtain only contributions for ± = ± ′ = + when p > 0. According to our decomposition above, this can be written as asum of a term depending on Π H, vac and Π H,T =0 , B wv = B wv , vac + B wv ,T =0 . (32)The contribution B wv ,T =0 can be evaluated numerically, while B wv , vac inherits the IRdivergence from Π H, vac φ , Eq. (24b). We can cast it to the form ( | p | = 0) B wv , vac = − g w Y GM π log ¯Λ λ Z d Ω [ f ℓ (( p + k ) · u ) + f φ ( k · u )] . (33)The scattering correction is B sca = g w Y Z d k (2 π ) X k = − p ±| k | p ( k ) Π A ( k ) [ f ℓ (( p + k ) · u ) + f φ ( k · u )] . (34)Following the splitting of Π A φ into vacuum and T = 0 contributions, we decompose aswell B sca = B sca , vac + B sca ,T =0 . (35)First, we show that B sca ,T =0 yields a finite result when λ → 0. For this purpose, notethat for small | k − | k || , one may expand for k > A ,T =0 φ ( k ) = G π k | k | " | k | − β log (cid:12)(cid:12)(cid:12)(cid:12) − e β k | k | (cid:12)(cid:12)(cid:12)(cid:12) + 2 β log (cid:18) β k − | k | (cid:19) + k − | k | · · · (36a)and for k < A ,T =0 φ ( k ) = G π k | k | " k − β log (cid:12)(cid:12)(cid:12)(cid:12) − e β k | k | (cid:12)(cid:12)(cid:12)(cid:12) + 2 β log (cid:18) β | k | − k (cid:19) + | k | − k · · · . (36b)Therefore, within B sca ,T =0 , we can integrate over the singularity at k = 0 in the principalvalue sense. 21ext, we demonstrate that B sca , vac depends on λ in a way that cancels the λ depen-dence within B wv , vac . For this purpose, we evaluate ( p = ) B sca , vac = − g w Y Z d k (2 π ) X k = − p ±| k | p ∓ | k | G π ϑ ( p ∓ p | k | − λ )sign k (37) × [ f ℓ (( p + k ) · u ) + f φ ( k · u )] =: B sca , vac+ + B sca , vac − . While for B sca , vac − , we can set λ = 0 and perform the integral numerically, we proceedwith B sca , vac+ as B sca , vac+ = g w Y G π Z d Ω ( M − λ /M ) Z k d | k | (2 π ) M − | k | [ f ℓ (( p + k ) · u ) + f φ ( k · u )] (38)= g w Y G π Z d Ω(2 π ) ( − (cid:2) | k | log( M − | k | ) ( f ℓ (( p + k ) · u ) + f φ ( k · u )) (cid:3) | k | = ( M − λ /M ) | k | =0 + M/ Z d | k | log( M − | k | ) ∂∂ | k | k ( f ℓ (( p + k ) · u ) + f φ ( k · u )) ) . Therefore, the terms ∝ log λ cancel between B wv and B sca . We emphasise that thiscancellation works out, because Π A , vac φ and Π H, vac φ are imaginary and real part of thesame analytic self-energy, which implies relation (16).While these expressions are already defined within the text above, we finally collectthe explicit expressions for the various IR-finite contributions: B sca , vac − = g w Y Z d k (2 π ) M + 2 | k | G π [ f ℓ (( p + k ) · u ) + f φ ( k · u )] , (39)where k = − M − | k | , B sca ,T =0 = g w Y Z d k (2 π ) X k = − M ±| k | M ( k ) Π A ,T =0 ( k ) [ f ℓ (( p + k ) · u ) + f φ ( k · u )] (40)with Π A ,T =0 given by Eqs. (29) and B wv ,T =0 = g w Y Z d k (2 π ) πδ ( M − | k | ) GT M (cid:18) M ddk (cid:19) [ f ℓ (( p + k ) · u ) + f φ ( k · u )] , (41)where k = − M + | k | = − M/ 2. In the last expression, we have substituted Π H,T =0 ( k ),Eq. (27) and have made use of the fact that the non-HTL part of Π H,T =0 ( k ) [ i.e. the22ntegral term in Eq. (27)] and its derivative with respect to k vanish for k = 0. Wealso note that B wv ,T =0 corresponds to the phase-space suppression due to the asymptoticthermal Higgs boson mass (note the minus sign from the statistical factors in the squarebrackets, as k < We now adapt the same strategy as for radiation from the scalar propagator to radiationfrom the fermionic one. As a complication, besides the cancellation between scatteringand wave-function corrections from vacuum loops, there is also a cancellation betweenscattering and wave-function hard thermal loop (HTL) contributions.We express the leptonic self-energies as / Σ ℓ = P R γ µ Σ ℓµ . In the approximation ofmassless particles in the loop, the spectral self-energy is given byΣ A ℓ ( k ) = GT π | k | I (cid:18) k T , | k | T (cid:19) , (42a)Σ A iℓ ( k ) = GT π | k | (cid:20) k | k | I (cid:18) k T , | k | T (cid:19) − ( k ) − k | k | T I (cid:18) k T , | k | T (cid:19)(cid:21) k i | k | , (42b)where I ( y , y ) = I vac0 ( y , y ) + I T =00 ( y , y ) , (43a) I T =00 ( y , y ) = − ϑ ( y − ( y ) ) y − y − ϑ (( y ) − y ) y sign( y ) (43b)+ log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y + y ) ( y − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − e ( y + y ) − e ( y − y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,I vac0 ( y , y ) = ϑ (( y ) − y ) y sign( y ) , (43c) I ( y , y ) = I vac1 ( y , y ) + ¯ I T =01 ( y , y ) + I HTL1 ( y , y ) , (43d)¯ I T =01 ( y , y ) = − ϑ ( y − ( y ) ) ( y ) − ϑ (( y ) − y ) | y | y (43e)+Re h x (log(1 + e x ) − log(1 − e x − y )) + Li ( − e x ) − Li (e x − y ) i x = ( y + y ) x = ( y − y ) ,I HTL1 ( y , y ) = ϑ ( y − ( y ) ) π , (43f) I vac1 ( y , y ) = 12 ϑ (( y ) − y ) | y | y . (43g)The bar on ¯ I T =01 ( y , y ) indicates that from this finite temperature term, the HTL con-tribution, which is purely thermal as well, is subtracted.23o this end, we also need the HTL-type contribution to the hermitian self-energyΣ H, HTL0 ℓ = GT | k | log (cid:12)(cid:12)(cid:12)(cid:12) k + | k | k − | k | (cid:12)(cid:12)(cid:12)(cid:12) , (44a)Σ H, HTL iℓ = GT k k i | k | log (cid:12)(cid:12)(cid:12)(cid:12) k + | k | k − | k | (cid:12)(cid:12)(cid:12)(cid:12) − GT k i | k | , (44b)and the vacuum contribution / Σ H, vac ℓ ( k ) = G π P R /k log λ Λ + /kδZ ℓ , (45)or, in dimensional regularisation, / Σ H, vac ℓ ( k ) = G π P R /k (cid:18) log λµ − 12 ∆ ǫ (cid:19) + /kδZ ℓ . (46)The term δZ ℓ is a field-strength renormalisation that cancels the divergences as Λ → ∞ or ǫ → 0, but is scheme dependent otherwise. In the following, we again implement itseffect by replacing Λ → ¯Λ, where ¯Λ is finite. The non-HTL contribution to the finite-temperature hermitian self-energy / Σ H,T =0 is IR finite and can be expressed in terms ofa one-dimensional integral, a classic result that can be found in Ref. [39].The leading correction to the fermionic spectral function is S (1) A ℓ = − (cid:16) i S (0)R ℓ i / Σ R ℓ i S (0)R ℓ − i S (0)A ℓ i / Σ A ℓ i S (0)A ℓ (cid:17) (47)= 12i (cid:16) i S (0)R ℓ / Σ Hℓ i S (0)R ℓ − i S (0)A ℓ / Σ Hℓ i S (0)A ℓ (cid:17) − (cid:16) i S (0)R ℓ / Σ A ℓ i S (0)R ℓ + i S (0)A ℓ / Σ A ℓ i S (0)A ℓ (cid:17) , and the corrections to the Wightman functions arei S (1) <ℓ = 2 S (1) A ℓ ( − f ℓ ) , (48a)i S (1) >ℓ = 2 S (1) A ℓ (1 − f ℓ ) . (48b)In the following, we attach the same superscripts as for the functions I , to variousquantities, in order to indicate which loop terms they originate from.24he correction to the neutrino relaxation rate follows from Eq. (9b) F = tr h /p / Σ A ℓ ( p ) i = g w Y Z d k (2 π ) tr h /p i S (1) >ℓ ( k )i∆ >φ ( p − k ) − /p i S (1) <ℓ ( k )i∆ <φ ( p − k ) i (49)= g w Y Z d k (2 π ) X k = p ±| k | ∓ | p − k | tr h /p S (1) A ℓ ( k ) i × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]=: F vac , fin + F vac , col + F HTL , fin + F HTL , col + F T =0 , where the particular contributions to F are defined below.The Dirac trace is evaluated astr[ /pS (1) A ℓ ( k )] = "(cid:18) i k + i sign k ε (cid:19) − (cid:18) i k − i sign k ε (cid:19) (50) × (cid:2) − ( p + k ) k · iΣ Hℓ ( k ) + k p · iΣ Hℓ ( k ) (cid:3) + "(cid:18) i k + i sign k ε (cid:19) + (cid:18) i k − i sign k ε (cid:19) × (cid:2) − ( p + k ) k · Σ A ℓ ( k ) + k p · Σ A ℓ ( k ) (cid:3) . As a consequence of Lorentz invariance, Σ A ,H, vac µ ( k ) ∝ k µ , which may be verifiedexplicitly by inspection of Eqs. (42,43,45). Therefore, we define / Σ A ,H, vac ℓ ( k ) = P R /k ˆΣ A ,H, vac ℓ ( k ) , (51)such that ˆΣ A , vac ℓ ( k ) = G π ϑ ( k − λ )sign( k ) , (52a)ˆΣ H, vac ℓ ( k ) = G π log λ ¯Λ . We have included here the threshold condition form the gauge-boson mass λ , that isneeded for the infrared regularisation. We obtain h − p + k ) k · Σ A ,H, vac ℓ ( k ) + 4 k p · Σ A ,H, vac ℓ ( k ) i = − k p + ( k ) ) ˆΣ A ,H, vac ℓ ( k ) . (53)The term ∝ ( k ) gives a finite contribution to F vac , fin and can be evaluated straightfor-wardly. The term ∝ p k gives rise to F vac , col = F vac , col , wv + F vac , col , sca . The wave-functioncontribution F vac , col , wv originates from Σ H, vac ℓ and the scattering correction F vac , col , sca A , vac ℓ . These terms individually contain collinear divergences that are regulatedby the gauge-boson mass λ and that cancel when added to F vac , col , as we demonstratenow. We find F vac , col , wv = g w Y Z d k (2 π ) | k | πδ ( p − | k | ) 14 k ∂∂k G π p k log λ ¯Λ (54) × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]= − g w Y GM π log ¯Λ λ Z d Ω [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] , where k = p − | k | . The scattering corrections yield F vac , col , sca = g w Y Z d k (2 π ) | k | X k = p ±| k | ∓ p k G π ϑ ( k − λ )sign k (55) × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]=: F vac , col , sca+ + F vac , col , sca − . Again, we analytically isolate the logarithmic dependence on λ , which is for the − contribution F vac , col , sca − = g w Y G π Z d Ω p − λ p Z | k | d | k | p p − | k | [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] (56)= − g w Y G π Z d Ω ((cid:2) log( M − | k | ) M | k |× [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] (cid:3) | k | = M − λ M | k | =0 + M Z d | k | log( M − | k | ) ∂∂ | k | M | k | [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] ) . The logarithmic dependence on λ is therefore cancelled cancelled with F vac , col , wv , Eq. (54).This is again a consequence of the fact that / Σ A ,H, vac are the anti hermitian and hermitianparts of the same analytic self energy, evaluated at the two-particle branch cut.The HTL-type contributions are decomposed into terms originating from / Σ Hℓ and / Σ A ℓ as well, F HTL = F HTL , wv + F HTL , sca . We observe that k · Σ A , HTL ( k ) = 0 and k · Σ H, HTL ( k )is finite for k = 0, whereas p · Σ H, HTL ( k ) has a logarithmic divergence for k → ∝ p · Σ H, HTL ( k ) and ∝ p · Σ A , HTL ( k ) in Eq. (50) therefore leadto collinearly divergent contributions to F HTL , and we denote these by F HTL , col , wv and26 HTL , col , sca , whereas the terms ∝ k · Σ H, HTL ( k ) and ∝ k · Σ A , HTL ( k ) are finite and referredto as F HTL , fin , wv and F HTL , fin , sca , such that F HTL , col = F HTL , col , wv + F HTL , col , sca , (57a) F HTL , fin = F HTL , fin , wv + F HTL , fin , sca . (57b)The finite terms can be integrated in a straightforward manner, while the collinearlydivergent contributions must be regularised, with a cancellation of the dependence onthe regulator in the sum of wave-function and scattering contributions. In order toregulate these terms, we set the gauge-boson mass λ = 0, but we introduce a leptonmass m ℓ . Then, we find that F HTL , col , wv = − g w Y Z d k (2 π ) | k | πδ ( p − | k | − q k + m ℓ ) (58) × k ∂∂k k p · Σ H, HTL ℓ ( k ) [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]= g w Y GT π log m ℓ M Z d Ω [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]and F HTL , col , sca = − g w Y Z d k (2 π ) | k | k − m ℓ ) k p · Σ A , HTL ℓ ( k ) (59) × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]= g w Y GT π Z d Ω ((cid:2) log (cid:0) m ℓ + (2 | k | − M ) M (cid:1) × (cid:2) − f ℓ ( k · u ) + f φ (( p − k ) · u ) (cid:3)(cid:3) | k | = ∞| k | = M − ∞ Z M d | k | log (cid:0) m ℓ + (2 | k | − M ) M (cid:1) ∂∂ | k | [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] ) . Notice that there is no + contribution from the sum over ± in Eq. (49), because Σ HTL A ( k )vanishes for positive k . The logarithmic dependence on m ℓ therefore cancels whenadding scattering and wave-function corrections, Eqs. (59) and (58). Once more, theprefactors for the cancellation match because Σ HTL A ,H ( k ) are anti hermitian and her-mitian part of an analytic self-energy evaluated at the branch cut. Notice, that theHTL-type contributions diverge logarithmically when p → 0. In that situation, oneshould use the resummed lepton propagator [1–3, 35].The result for F T =0 [that is defined as the contribution to F from / Σ H,T =0 and from / Σ A ,T =0 , which in turn results from replacing I → I T =00 and I → ¯ I T =01 in Eq. (42)] can27e obtained by integrating over the pole at k = 0 in the principal value sense, what canbe verified by checking the limiting behaviour of I T =00 ( y , y ) and ¯ I T =01 ( y , y ) for y → y , i.e. that these functions are continuous at that point.For completeness, we again list the explicit expressions for the various IR-finite terms: F vac , fin , wv = 0 , (60) F vac , fin , sca = g w Y Z d k (2 π ) | k | X k = M ±| k | ∓ G π ϑ ( k ) sign( k ) (61) × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] , F vac , col , sca+ = − g w Y Z d k (2 π ) | k | MM + 2 | k | G π [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] k = M + | k | , (62) F HTL , fin , wv = − g w Y Z d k (2 π ) | k | πδ ( M − | k | ) (cid:18) − k ddk + 14 k (cid:19) (63) × (cid:16) M + k − k (cid:17) GT − f ℓ ( k · u ) + f φ (( p − k ) · u )] , where k = M , F HTL , fin , sca =0 , (64) F T =0 = g w Y Z d k (2 π ) | k | ( (65) X k = M ±| k | ∓ k ) h p + k ) k · Σ A ,T =0 ℓ ( k ) − k p · Σ A ,T =0 ℓ ( k ) i × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )]+2 πδ ( M − | k | ) h (cid:18) k ddk − k (cid:19) (cid:16) ( p + k ) k · Σ H,T =0 ℓ ( k ) − k p · Σ H,T =0 ℓ ( k ) (cid:17) × [1 − f ℓ ( k · u ) + f φ (( p − k ) · u )] i k = M ) , where F vac , fin = F vac , fin , wv + F vac , fin , sca . These expressions can be substituted into Eq. (20),in order to obtain the production rate for right-handed neutrinos. Notice also that thesign of F HTL , fin , wv is consistent with the phase-space suppression due to the asymptoticthermal lepton mass. 28 Vertex Contribution The 2 ↔ ↔ N follow fromthe first term of Eq. (18),tr[ /p i / Σ 29e integrate first over dr , dq and d cos ϑ using the three on-shell delta-functions andthen change the integration variables | r | and | q | back to r and q , i.e. the integrationmeasure changes as Z d q (2 π ) d r (2 π ) πδ (cid:18)(cid:16) p − q − r (cid:17) (cid:19) πδ (cid:18)(cid:16) p q − r (cid:17) (cid:19) πδ (cid:0) r − λ (cid:1) (71) → X r = ±| r | q ±′| q | π ) π Z dϕ π Z dψ Z − d cos ̺ Z I q q d | q | Z I r r d | r | √ r + λ | q | | q || r |→ 18 1(2 π ) π Z dϕ π Z dψ Z − d cos ̺ Z I q dq Z I r dr , where ̺ denotes the angle between q and u . More specifically, we choose to parametrisethe angular dependences as r = | r | sin ϑ sin ϕ sin ϑ cos ϕ cos ϑ , q = | q | , u = | ˜ p | M sin ̺ sin ψ sin ̺ cos ψ cos ̺ . (72)The integration domains I are best determined from distinguishing between thevarious kinematic situations, which are sketched in Figure 6. Besides, we take withinthese regions a positive integration measure, such that we do not obtain explicit minussigns from the Jacobians. The regions follow from the three on-shell constraints. Noticethat these conditions may be combined to obtain | q | = 12 q M + 4 q − M r + λ , (73)and when using this as a relation for cos ϑ , (cid:16) p q − r (cid:17) = (cid:18) M q (cid:19) − (cid:18) M q (cid:19) r + | q || r | cos ϑ + λ − q = 0 . (74)The relation − ≤ cos ϑ ≤ λ = 0 this inequalityreduces to: (cid:18) r − M (cid:19) (cid:18) q − r (cid:19) ≥ . (75)Besides, we note that the on-shell constraints imply that the invariant momentumsquares of the off-shell propagators are (cid:16) p q + r (cid:17) = M (cid:0) q + r (cid:1) , (76a) (cid:16) p − q + r (cid:17) = M (cid:0) − q + r (cid:1) . (76b)30his suggests a change of variables to x = q + 12 r , (77a) y = − q + 12 r , (77b)which has the Jacobian − 1. The integration regions in terms of these variables areindicated in Figure 6 as well. The collinear divergences are located on the fringes of theintegration region where either x or y vanish, soft divergences are at the coincident point x = y = 0.We can therefore recast the scattering contributions to the neutrino production astr[ /p i / Σ 132 1(2 π ) π Z d ( ϕ − ψ ) Z − d cos ̺ Z I x dx Z I y dy (78) × sign( x + y )sign (cid:18) M − x (cid:19) sign (cid:18) M − y (cid:19) PV − M + 4 M x + 2 M y + 4 xyxy × f ℓ (( p/ − q − r/ · u ) f φ (( p/ q − r/ · u ) [1 + f A ( − r · u )] . The statistical functions can be evaluated using above relations. In particular, the scalarproducts that appear in the arguments are given by p · u = ˜ p , (79a) q · u = 12 ( x − y ) ˜ p M − | q | | ˜ p | M cos ̺ , (79b) r · u =( x + y ) ˜ p M − | x + y | | ˜ p | M [cos ̺ cos ϑ + sin ̺ sin ϑ cos( ϕ − ψ )] . (79c)with | q | as in Eq. (73) and cos ϑ given by Eq. (74). Close to the collinear edges, where31 = 0 ∨ y = 0, we can evaluate (cid:16) p − q − r (cid:17) · u ≈ (cid:18) − xM (cid:19) (˜ p + | ˜ p | cos ̺ sign( M − x )sign( M − y )) (80a)+ √ xyM | ˜ p | sin ̺ cos( ϕ − ψ ) , (cid:16) p q − r (cid:17) · u ≈ (cid:18) − yM (cid:19) (˜ p − | ˜ p | cos ̺ sign( M − x )sign( M − y )) (80b)+ √ xyM | ˜ p | sin ̺ cos( ϕ − ψ ) ,r · u ≈ x + yM ˜ p + x − yM | ˜ p | cos ̺ sign( M − x )sign( M − y ) (80c) − √ xyM | ˜ p | sin ̺ cos( ϕ − ψ ) . Note that Eqs. (80) are understood as leading order expansions and hence the ∼ √ xy corrections are applicable only when both x ∼ y ≈ 0. As we aim for rearranging thevarious contributions to the vertex-type correction into a manifestly finite integral, wehave dropped here the dependence on the regulating gauge-boson mass λ . We have alsolet cos ̺ → sign( M − x )sign( M − y ) cos ̺ for later convenience.In order to arrange for an IR-finite integral, where the collinear divergences on thefringes x = 0 and y = 0 cancel, it is useful to define the integrand J sca ( x, y ) = K sca ( x, y ) f ℓ (( p/ − q − r/ · u ) f φ (( p/ q − r/ · u ) (81) × [1 + f A ( − r · u )] S ( x, y ) ≈ | x |≪ M ˙ ∨| y |≪ M − K sca ( x, y ) f ℓ (cid:18)(cid:18) − xM (cid:19) E s (cid:19) f φ (cid:18)(cid:18) − yM (cid:19) E − s (cid:19) × f A (cid:18) xM E s + 2 yM E − s (cid:19) S ( x, y ) , where s ≡ sign( M − x )sign( M − y ), E ± ≡ 12 (˜ p ± | ˜ p | cos ̺ ) , (82)and S ( x, y ) is a step function defining the support of the integral in accordance withFigure 6: S ( x, y ) = ϑ ( x ) ϑ ( y ) ϑ ( M/ − x − y ) + ϑ ( − x ) ϑ ( − y ) (83)+ (cid:0) ϑ ( x ) ϑ ( − y ) + ϑ ( − x ) ϑ ( y ) (cid:1) ϑ ( x + y − M/ . K sca ( x, y ) =sign( x + y )sign (cid:18) M − x (cid:19) sign (cid:18) M − y (cid:19) PV − M + 4 M x + 2 M y + 4 xyxy . (84)In the approximate form for J sca , we have made the restriction to | x | ≪ M ˙ ∨ | y | ≪ M ,where ˙ ∨ denotes the exclusive or, which justifies dropping the ∼ √ xy corrections. We aregoing to use this expression in order to show the cancellation of collinear divergences. Forthe soft divergence at x = y = 0, there occurs a complication due to the Bose divergenceof the gauge boson distribution, and we will provide a separate discussion. We now bring the vertex contribution with an on-shell gauge boson to a form that can bematched with the contributions from scatterings. It is useful to reparametrise in Eq. (18) k → p − q , q → q − k , (85)such that the second term in Eq. (18) turns intotr[ /p i / Σ < ( p )] vert1 =(3 g + g ) Y Z d k (2 π ) d q (2 π ) [ − k − q ] ν tr h /p (86) × i S Hℓ ( p − q ) γ µ i S <ℓ ( p − k )i∆ <φ ( k )i∆ Hφ ( q )i∆ Fµν ( q − k ) i . We perform the integrations over dk and dq making use of the on-shell delta-functions.When we define k ± = ( ±| k | , k ) , q ± = ( k ± ω, q ) , ω = p ( q − k ) + λ , (87)we obtaintr[ /p i / Σ < ( p )] vert1 =4 g w GY π ) X ± π Z d ( ϕ − ψ ) Z − d cos ̺ ∞ Z λ dω q max Z q min d | q | | q | M (88) × PV (cid:20) M [( M/ ∓ ω ) − q ] [( M/ ± ω ) − q ] − M ( M/ ∓ ω ) − q − M ( M/ ± ω ) − q (cid:21) × [1 + 2 f A ( | ( q ± − k + ) · u | )] f ℓ (( p − k + ) · u ) f φ ( k + · u ) , where q max , min = (cid:12)(cid:12)(cid:12)(cid:12) M ± √ ω − λ (cid:12)(cid:12)(cid:12)(cid:12) . (89)33ere, we parametrise the angular dependences as q = | q | sin ϑ sin ϕ sin ϑ cos ϕ cos ϑ , k = | k | , u = | ˜ p | M sin ̺ sin ψ sin ̺ cos ψ cos ̺ . (90)In order to evaluate the distribution functions, we use k + · u = 12 (˜ p − | ˜ p | cos ̺ ) = E − , (91a)( p − k + ) · u = 12 (˜ p + | ˜ p | cos ̺ ) = E + , (91b) q ± · u = ˜ p (cid:18) ± ωM (cid:19) − | ˜ p | | q | M (cos ϑ cos ̺ + sin ϑ sin ̺ cos( ϕ − ψ )) , (91c)cos ϑ = q + M − ω + λ M | q | . (91d)Next, we change variables to x = − M "(cid:18) M − ω (cid:19) − q , y = 12 M "(cid:18) M ω (cid:19) − q . (92)Notice that x = 0 corresponds to the pole of the lepton (Higgs) propagator for ± → +( ± → − ) and that vice versa, y = 0 corresponds to the pole of the lepton (Higgs)propagator for ± → − ( ± → +). The inverse transformation gives | q | = 12 p M + 4 M ( x − y ) + 4( x + y ) , (93a) ω = x + y , (93b)and the Jacobian is − M/ | q | . We express the vertex contributions as (we take λ → /p i / Σ < ( p )] vert i =4 g w GY π ) π Z d ( ϕ − ψ ) Z − d cos ̺ Z ∞ dy Z ∞ dx (94) × (cid:0) J vert i + ( x, y ) + J vert i − ( x, y ) (cid:1) , where for the present terms J vert1 ± ( x, y ) = K vert1 ± ( x, y ) [1 + 2 f A ( | ( q ± − k + ) · u | )] f ℓ ( E + ) f φ ( E − ) , (95)34ith K vert1 ± ( x, y ) = − PV (cid:20) ϑ ( ∓ ) M + 2 M x − M yxy + ϑ ( ± ) M − M y + M xxy (cid:21) . (96)In this form, we can directly compare with the terms from real emissions. Wheneither or both, x , y are close to zero, we can approximate | q | ≈ | M + 2 x − y | , (97)( q ± − k + ) · u ≈ ± ˜ p x + yM − | ˜ p | cos ̺ x − yM − √ xyM | ˜ p | sin ̺ cos( ϕ − ψ ) . (98)Expanding the arguments of the distribution functions close to the surfaces x =0 ˙ ∨ y = 0, we obtain J vert1 ± ( x, y ) ≈ K vert1 ± ( x, y ) (cid:20) f A (cid:18) xM E ∓ + 2 yM E ± (cid:19)(cid:21) f ℓ ( E + ) f φ ( E − ) . (99) We now consider the third term of Eq. (18), which using the reparametrisation (85)becomes tr[ /p i / Σ 12 + x − yM (cid:19) ( ± ˜ p + | ˜ p | cos ̺ ) − √ xyM | ˜ p | sin ̺ cos( ϕ − ψ ) . (108)36sing these approximations, we find for x ≪ M ˙ ∨ y ≪ MK vert , appr2+ ( x, y ) =PV 4 M ( x + y )( M + 2 x − y ) , (109a) K vert , appr2 − ( x, y ) =PV (cid:20) M ( M + x − y ) xy − MM + 2 x − y (cid:21) sign( x − y ) , (109b) J vert2 ± ( x, y ) ≈ K vert , appr2 ± ( x, y )sign( M − y ) (109c) × (cid:20) f φ (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) x − yM (cid:12)(cid:12)(cid:12)(cid:12) E ± (cid:19)(cid:21) f ℓ ( E + ) f φ ( E − ) . The third term of Eq. (18) in the reparametrisation (85) istr[ /p i / Σ 0, we makeuse of the detailed balance relations applicable to collinear splittings f φ ( E ) + f φ ( E ) f φ ( αE ) + f φ ( E ) f A ((1 − α ) E ) = f φ ( αE ) f A ((1 − α ) E ) , (117a) f ℓ ( E ) − f ℓ ( E ) f ℓ ( αE ) + f ℓ ( E ) f A ((1 − α ) E ) = f ℓ ( αE ) f A ((1 − α ) E ) . (117b)It then follows that for x → xy (cid:0) J sca ( x, y ) + J sca ( − x, y ) + J vert1 − ( x, y ) + J vert2 − ( x, y ) (cid:1) → , (118a) xy (cid:0) J sca ( − y, − x ) + J vert1 − ( y, x ) + J vert3 − ( y, x ) (cid:1) → y → xy (cid:0) J sca ( x, y ) + J sca ( x, − y ) + J vert1+ ( y, x ) + J vert3 − ( y, x ) (cid:1) → , (119a) xy (cid:0) J sca ( − y, − x ) + J vert1+ ( x, y ) + J vert2 − ( x, y ) (cid:1) → . (119b)The sum J sca ( x, y ) + J sca ( − x, y ) in Eq. (118a) makes it sure that the cancellation withvertex contributions takes place on the whole collinear edge x = 0 (for both y < M/ y ≥ M/ y = 0 in accordance with the kinematicregions, cf. Figure 6. In order to obtain the manifestly IR-finite result for the vertex-typepart of the neutrino production rate (116a), we need to add these contributions to thoseterms, that are integrable from the outset and finally symmetrise x ↔ y , in order tomake sure that the sum of all divergent terms cancels on both edges ( x = 0 and y = 0).Furthermore, we have symmetrised over the polar angles.The criteria (118,119) are in general not sufficient in order to show the integrabilityat the point x = y = 0, because the detailed balance relations only hold on the collinearfringes where x = 0 ˙ ∨ y = 0. We must therefore show in addition that xy J total ( x, y ) van-ishes on each trajectory for ( x, y ) → 0. The problematic terms are the Bose divergencesin J sca and J vert1 ± . When using Eqs. (81,95) together with the approximations for thearguments of the distribution functions (80,98), we notice the cancellation of the diver-gences in xy ( J sca ( x, y ) + J vert1 ± ( x, y )). There is however a finite remainder that moreoverdepends on the direction of approach of ( x, y ) toward (0 , x , y and √ xy in the arguments of the distribu-tion functions (80,98). For this finite term to cancel as well, we must add I sca ( − x, − y ),symmetrise over x and y (as it is already necessary for the cancellation of the collineardivergences) and moreover, add the contributions from positive and negative values ofcos( ϕ − ψ ). In summary, it follows thatlim x,y → xy J total ( x, y ) = 0 , (120)such that J total ( x, y ) grows more slowly than 1 / ( x + y ), and therefore it is integrableat x = y = 0. In fact, we have verified by numerical evaluation that J total ( x, y ) ∼ / p x + y for ( x, y ) → x, y ) = (0 , M/ 2) and ( x, y ) =( M/ , x → 0. Inspecting the approxima-tion (109) for J vert2 ± , we notice that the y integration can be performed at y = M/ ∝ /x in J sca (sign( M/ − y ) x, y ), Eq. (81). Therefore, we can perform the d ∆ y integral over J total ( x, M/ − ∆ y ) + J total ( x, M/ y ) as well (∆ y > x times this integral vanishes for x → 0, which is why the remaining x integration yields a finite result as well. We have also checked numerically that for( x, y ) → (0 , M/ x ( y − M/ J total ( x, y ) → , (121)39nd moreover that J total ( x, y ) ∼ / p x + ( y − M/ irrespective of the limiting di-rection, which further confirms the integrability at (0 , M/ 2) and ( M/ , 0) by x ↔ y symmetry. Furthermore, we have verified that the Bose divergences in J vert2 ± on theline y = M/ x away from fringe x = 0 are actually cancelled in that ( y − M/ − x ) J total ( x, y ) → J total ( x, y ) → finite for y → M/ x , and similarly for x ↔ y . The vertex-type correction (116a) to the neutrino production contains UV divergencesthat need to be isolated and subtracted in order to obtain a remainder that is suitable fornumerical integration. The UV divergences originate from the vertex contributions. Inparticular, the integrands (95,104,111) are not exponentially suppressed for large valuesof x or y because of the constant term in the square brackets. The contributions arisingfrom this constant term are easily interpreted as the vacuum vertex-corrections to the2 → ℓ, φ → N or ¯ ℓφ † → N , weighted by the statistical distributions for ℓ and φ . These UV-divergent contributions are analytically calculable, and are given bythe the following sum of terms: J vert , vac ( x, y ) = X ± (cid:0) K vert1 ± ( x, y ) + K vert2 ± ( x, y ) + K vert3 ± ( x, y ) (cid:1) f ℓ ( E + ) f φ ( E − ) . (122a)Now, we need to subtract this expression from the integrand that we aim to evaluatenumerically, while adding terms that balance the collinear divergences that this subtrac-tion may induce. The vacuum-vertex corrections should only have IR divergences thatcancel with those for real emissions in 1 → ↔ J vertUV ( x, y ) should not yield collinearlydivergent contributions, which can be verified as xy ( J vert , vac ( x, y ) + J vert , vac ( y, x )) = 0 for ( x = 0 ∧ y > M/ ∨ ( y = 0 ∧ x > M/ . (123)This is no longer true for Region I. In order to cancel collinear divergences there, wedefine J sca , vac ( x, y ) = K sca ( x, y ) ϑ ( x + y − M/ f ℓ ( E + ) f φ ( E − ) , (124)which we subtract in addition. This is just 1 → J vert , vac ( x, y ), and it has the desired propertythat xy ( J sca , vac ( x, y ) + J vert , vac ( x, y ) + x ↔ y ) = 0 for x = 0 ∨ y = 0 . (125)Now the integrand J totalUV ( x, y ) = J total ( x, y ) − (cid:0) J sca , vac ( x, y ) + J vert , vac ( x, y ) + x ↔ y (cid:1) (126)40s IR and UV finite and suitable for numerical evaluation.Finally, the subtracted terms must be calculated analytically and then added again.For this purpose, we note that the virtual T = 0 corrections to the decay rate, evaluatedwith a cutoff ω < Λ [from the parametrisations in Eqs. (88,101)] aretr[ /p Σ A N ( p )] vert T =0 = 4 g w GY π ) M (cid:20) − λM + π − λM + log Λ M − 32 + log 2 (cid:21) (127)+ Y δY M π . Here, δY is a counter term that must remove the logarithmic divergence in Λ and thefurther details of which are determined by a particular renormalisation scheme.Alternatively, we can use dimensional regularisation in order to calculate the reducedvertex function V ( p ) =2 g w GY µ ǫ Z d − ǫ q (2 π ) − ǫ tr[ /p ( /p − /q ) γ µ ( /p − /k )]( k + q ) µ [( p − q ) + i ε ][( q − k ) − λ + i ε ][ q + i ǫ ] + Y δY M (128)=2 g w GY M π (cid:20) − λM + π − λµ + 2 log Mµ − ∆ ǫ − (cid:21) + Y δY M . In this form, δY must cancel the 1 /ǫ divergence and may otherwise again be renormalisation-scheme dependent. Integrating over the T = 0 phase space, we obtain the zero-temperaturespectral-function for the right-handed neutrinotr[ /p Σ A N ( p )] vert T =0 = Z d k (2 π ) d q (2 π ) (2 π ) δ ( p − k − q )2 πδ ( k )2 πδ ( q ) V ( p ) = V ( p )8 π (129)= 4 g w GY π M (cid:20) − λM + π − λµ + 2 log Mµ − ∆ ǫ − (cid:21) + Y δY M π . Note the factors from the two different diagrammatic contributions in the CTP approach(cut on the left or on the right). Comparing with the prefactors in Eq. (94) and usingthe definitions (A1), we can conclude that¯ J vert , vac = ∞ Z dx ∞ Z dy J vert , vac ( x, y ) , (130a)¯ J vert , vac ( x, y ) = − M (cid:20) − λM + π − λµ + 2 log Mµ − ∆ ǫ − (cid:21) (130b) × f ℓ ( E + ) f φ ( E − ) . λ , which cancel with corre-sponding terms in the vacuum vertex contribution (130a).Next, we need to obtain a matching expression for J sca , vac ( x, y ), which we haveintroduced in order to cancel the IR-divergent λ -dependence. For this purpose, weevaluate the integral M Z λ M dx M − x Z λ x dy (cid:20) − M xy + 4 My + 2 Mx + 4 (cid:21) = 2 M (cid:20) − λM + π Mλ − M (cid:21) , (131)where the boundaries of integration derive from the condition − ≤ cos ϑ ≤ J sca , vac = ∞ Z dx ∞ Z dy J sca , vac ( x, y ) (132)=2 M (cid:20) − λM + π Mλ − M (cid:21) f ℓ ( E + ) f φ ( E − ) . The final expression for the reduced self-energy istr[ /p i / Σ The results of this work show that the relaxation rate of right-handed neutrinos N toward thermal equilibrium is finite to NLO in perturbation theory, i.e. when includingthe leading corrections from Standard Model gauge radiation. As we perform a fully42elativistic calculation, we generalise earlier results that are based on non-relativisticapproximations [5, 6] that are valid when M ≫ T . In particular, the relative motionof the neutrino N with respect to the plasma is kept general here and the full quantumstatistical Bose-Einstein and Fermi-Dirac effects are accounted for.We have developed two somewhat different methods in order to handle the wave-function- and the vertex-type corrections. The treatment of the wave-function correctionsis greatly facilitated, because well known and relatively simple analytical expressions forthe thermal self-energies are available. We can isolate the IR divergences in terms oflogarithmic dependences on the regulating gauge-boson mass λ for both, the scatteringcontributions that rely on the spectral self-energy and the wave-function contributionsoriginating from the hermitian self-energy. Eventually, we find that the IR divergencescancel [Eq. (33) with Eq. (38), Eq (54) with Eq. (56) and Eq (58) with Eq. (59)] becausethe hermitian and the spectral self-energies are the real and imaginary parts of the sameanalytic function evaluated at the two-particle branch cut.For the two-particle irreducible vertex-type corrections, the situation appears morecomplicated. Rather than isolating the IR divergences in analytic expressions for vari-ous contributions, we therefore manipulate these in such a manner that we obtain theintegrand (116b) that is manifestly free from non-integrable IR and Bose divergences.Since the methods that we use in order to demonstrate the cancellation are veryexplicit and as we also discuss the subtraction of UV divergences, they readily suggesta method for analytic or numerical evaluation of the relaxation rates of right-handedneutrinos N . In a future work, it would be of interest to verify the results of Refs. [5, 6]by taking the non-relativistic limit of the present results and moreover, to perform anumerical evaluation of the production rate of N without non-relativistic approximations.While the present work provides a method for evaluating the relaxation rate of right-handed neutrinos N , that derives from tr[ /p / Σ A N ], cf. Eq. (3), the rates of non-equilibrium CP -violation from decays and inverse decays of N typically rely on different componentsof the self energy [24, 29, 31–33]. Another future task would therefore be the evaluationof CP -violating rates, in generalisation of the methods presented here.While we have focused in the present work on the production of right-handed neutri-nos, the basic topology of the self-energy diagrams that describe the decay and scatteringrates and their NLO corrections is obviously shared with diagrams for other high-energyreactions in finite-temperature backgrounds. For that reason, it may be useful to for-mulate our method in terms of master integrals in such a way that the results could bemore directly applied to other situations with different kinds of particles in the loops(different Lorentz tensor structures). One may investigate for example the processes thatunderlay the transport coefficients used in calculations for Electroweak Baryogenesis orcorrections to the production rate of Dark Matter particles. It will therefore be interest-ing to explore the possibilities of applying the present methods to additional topics inEarly Universe Cosmology. 43 cknowledgements The authors acknowledge support by the Gottfried Wilhelm Leibniz programme of theDFG, by the DFG cluster of excellence ‘Origin and Structure of the Universe’ and bythe Alexander von Humboldt Foundation. Appendix We follow the conventions of Ref. [16]. The relations between the CTP two-point func-tions, ( G stands for a propagator ∆ or S or for a self-energy Π or / Σ) are given by: G A = G T − G > = G < − G ¯ T (advanced) , (A1a) G R = G T − G < = G > − G ¯ T (retarded) , (A1b) G H = 12 ( G R + G A ) (Hermitian) , (A1c) G A = 12i ( G A − G R ) = i2 ( G > − G < ) (anti-Hermitian, spectral) , (A1d) G F = 12 ( G > + G < ) = 12 ( G T + G ¯ T ) (statistic) . (A1e)In terms of the functions bearing CTP indices, these can be expressed as G T = G ++ , G ¯ T = G −− (time ordered, anti time-ordered) , (A2a) G < = G + − , G > = G − + (Wightman) . (A2b)The Wigner transform is defined by G ( p, x ) = Z d r e i pr G ( x + r/ , x − r/ . (A3)Under spatially homogeneous conditions, there is no dependence on the average spatialcoordinate x and furthermore, we suppress the explicit average time coordinate t = x .Equilibrium Green functions or self energies observe the Kubo-Martin-Schwinger (KMS)relation G > ( p ) = ± e βp · u G < ( p ) , (A4)where the plus sign applies to bosonic, the minus sign to fermionic two-point functions,and u is the plasma four-velocity. 44he tree-level equilibrium propagators for the lepton doublet that we use here arei S <ℓ ( p ) = − πδ ( p )sign( p ) P L p/f ℓ ( p · u ) , (A5a)i S >ℓ ( p ) = 2 πδ ( p )sign( p ) P L p/ (1 − f ℓ ( p · u )) , (A5b)i S Tℓ ( p ) = P L i p/p + i ε − πδ ( p ) P L p/f ℓ ( | p · u | ) , (A5c)i S ¯ Tℓ ( p ) = − P L i p/p − i ε − πδ ( p ) P L p/f ℓ ( | p | ) , (A5d)where f ℓ is the Fermi-Dirac distribution. The generalisation to non-equilibrium distribu-tions can be found in Ref. [24]. We suppress explicit SU(2) indices, which are contractedwith that of the Higgs doublet φ in the usual manner, using the two-dimensional anti-symmetric tensor.Similarly, we use the scalar equilibrium propagatorsi∆ <φ ( p ) = 2 πδ ( p )sign( p ) f φ ( p · u ) , (A6a)i∆ >φ ( p ) = 2 πδ ( p )sign( p )(1 + f φ ( p · u )) , (A6b)i∆ Tφ ( p ) = i p + i ε + 2 πδ ( p ) f φ ( | p · u | ) , (A6c)i∆ ¯ Tφ ( p ) = − i p − i ε + 2 πδ ( p ) f φ ( | p · u | ) , (A6d)where f φ is a Bose-Einstein distribution. The gauge field propagators arei∆ <µν ( p ) = 2 πδ ( p )sign( p )( − g µν ) f A ( p · u ) , (A7a)i∆ >µν ( p ) = 2 πδ ( p )sign( p )( − g µν )(1 + f A ( p · u )) , (A7b)i∆ Tµν ( p ) = − i g µν p + i ε + 2 πδ ( p )( − g µν ) f A ( | p · u | ) , (A7c)i∆ ¯ Tµν ( p ) = − − i g µν p − i ε + 2 πδ ( p )( − g µν ) f A ( | p · u | ) , (A7d)and f A is a Bose-Einstein distribution.As a consequence of the Majorana condition, the self energy of the right-handedneutrinos inherits the property / Σ N ( x, y ) = C / Σ tN ( y, x ) C † , (A8)where the transposition t acts here on both the CTP and the Dirac indices, cf. Ref. [29]for a more detailed discussion that also includes the mixing of several right-handedneutrinos and the effect of non-zero chemical potentials for the leptons ℓ .45 eferences [1] P. B. Arnold, G. D. Moore and L. G. Yaffe, “Transport coefficients in hightemperature gauge theories. 1. Leading log results,” JHEP (2000) 001[hep-ph/0010177].[2] P. B. Arnold, G. D. Moore and L. G. Yaffe, “Photon emission from quark gluonplasma: Complete leading order results,” JHEP (2001) 009 [hep-ph/0111107].[3] D. Besak and D. Bodeker, “Thermal production of ultrarelativistic right-handed neutrinos: Complete leading-order results,” JCAP (2012) 029[arXiv:1202.1288 [hep-ph]].[4] A. Anisimov, D. Besak and D. Bodeker, “Thermal production of relativistic Ma-jorana neutrinos: Strong enhancement by multiple soft scattering,” JCAP (2011) 042 [arXiv:1012.3784 [hep-ph]].[5] A. Salvio, P. Lodone and A. Strumia, “Towards leptogenesis at NLO: the right-handed neutrino interaction rate,” JHEP (2011) 116 [arXiv:1106.2814 [hep-ph]].[6] M. Laine and Y. Schroder, “Thermal right-handed neutrino production rate in thenon-relativistic regime,” JHEP (2012) 068 [arXiv:1112.1205 [hep-ph]].[7] R. Baier, B. Pire and D. Schiff, “Dilepton production at finite temperature: Per-turbative treatment at order α s ,” Phys. Rev. D (1988) 2814.[8] T. Altherr and T. Becherrawy, “Cancellation Of Infrared And Mass Singularities InThe Thermal Dilepton Rate,” Nucl. Phys. B (1990) 174.[9] Y. Gabellini, T. Grandou and D. Poizat, “Electron - Positron Annihilation In Ther-mal Qcd,” Annals Phys. (1990) 436.[10] T. Altherr, P. Aurenche and T. Becherrawy, “On Infrared And Mass SingularitiesOf Perturbative Qcd In A Quark - Gluon Plasma,” Nucl. Phys. B (1989) 436.[11] M. Laine, A. Vuorinen and Y. Zhu, “Next-to-leading order thermal spectral func-tions in the perturbative domain,” JHEP (2011) 084 [arXiv:1108.1259 [hep-ph]].[12] Y. Zhu and A. Vuorinen, “The shear channel spectral function in hot Yang-Millstheory,” arXiv:1212.3818 [hep-ph].[13] J. S. Schwinger, “Brownian motion of a quantum oscillator,” J. Math. Phys. (1961) 407.[14] L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Zh. Eksp. Teor.Fiz. (1964) 1515 [Sov. Phys. JETP (1965) 1018].4615] E. Calzetta and B. L. Hu, “Nonequilibrium Quantum Fields: Closed Time PathEffective Action, Wigner Function and Boltzmann Equation,” Phys. Rev. D (1988) 2878.[16] T. Prokopec, M. G. Schmidt and S. Weinstock, “Transport equations for chiralfermions to order h bar and electroweak baryogenesis. Part 1,” Annals Phys. (2004) 208 [hep-ph/0312110].[17] T. Prokopec, M. G. Schmidt and S. Weinstock, “Transport equations for chiralfermions to order h-bar and electroweak baryogenesis. Part II,” Annals Phys. (2004) 267 [hep-ph/0406140].[18] W. Buchmuller and S. Fredenhagen, “Quantum mechanics of baryogenesis,” Phys.Lett. B , 217 (2000) [hep-ph/0004145].[19] A. De Simone and A. Riotto, “Quantum Boltzmann Equations and Leptogenesis,”JCAP (2007) 002 [hep-ph/0703175].[20] M. Garny, A. Hohenegger, A. Kartavtsev and M. Lindner, “Systematic approachto leptogenesis in nonequilibrium QFT: vertex contribution to the CP-violatingparameter,” Phys. Rev. D (2009) 125027 [arXiv:0909.1559 [hep-ph]].[21] M. Garny, A. Hohenegger, A. Kartavtsev and M. Lindner, “Systematic approach toleptogenesis in nonequilibrium QFT: self-energy contribution to the CP-violatingparameter,” Phys. Rev. D (2010) 085027 [arXiv:0911.4122 [hep-ph]].[22] A. Anisimov, W. Buchm¨uller, M. Drewes and S. Mendizabal, “Leptogenesis fromQuantum Interference in a Thermal Bath,” Phys. Rev. Lett. (2010) 121102[arXiv:1001.3856 [hep-ph]].[23] M. Garny, A. Hohenegger, A. Kartavtsev, “Medium corrections to the CP-violatingparameter in leptogenesis,” Phys. Rev. D81 (2010) 085028. [arXiv:1002.0331 [hep-ph]].[24] M. Beneke, B. Garbrecht, M. Herranen and P. Schwaller, “Finite Number DensityCorrections to Leptogenesis,” Nucl. Phys. B (2010) 1 [arXiv:1002.1326 [hep-ph]].[25] M. Beneke, B. Garbrecht, C. Fidler, M. Herranen and P. Schwaller, “Flavoured Lep-togenesis in the CTP Formalism,” Nucl. Phys. B (2011) 177 [arXiv:1007.4783[hep-ph]].[26] M. Garny, A. Hohenegger and A. Kartavtsev, “Quantum corrections to leptogenesisfrom the gradient expansion,” arXiv:1005.5385 [hep-ph].[27] B. Garbrecht, “Leptogenesis: The Other Cuts,” Nucl. Phys. B847 (2011) 350-366.[arXiv:1011.3122 [hep-ph]]. 4728] A. Anisimov, W. Buchmuller, M. Drewes and S. Mendizabal, “Quantum Leptogen-esis I,” Annals Phys. (2011) 1998 [arXiv:1012.5821 [hep-ph]].[29] B. Garbrecht and M. Herranen, “Effective Theory of Resonant Leptogenesis in theClosed-Time-Path Approach,” Nucl. Phys. B (2012), 17. [arXiv:1112.5954 [hep-ph]].[30] M. Garny, A. Kartavtsev and A. Hohenegger, “Leptogenesis from first principles inthe resonant regime,” Annals Phys. (2013) 26 [arXiv:1112.6428 [hep-ph]].[31] B. Garbrecht, “Leptogenesis from Additional Higgs Doublets,” Phys. Rev. D (2012) 123509 [arXiv:1201.5126 [hep-ph]].[32] M. Drewes and B. Garbrecht, “Leptogenesis from a GeV Seesaw without MassDegeneracy,” arXiv:1206.5537 [hep-ph].[33] B. Garbrecht, “Baryogenesis from Mixing of Lepton Doublets,” Nucl. Phys. B (2013) 557 [arXiv:1210.0553 [hep-ph]].[34] T. Frossard, M. Garny, A. Hohenegger, A. Kartavtsev and D. Mitrouskas, “System-atic approach to thermal leptogenesis,” arXiv:1211.2140 [hep-ph].[35] B. Garbrecht,P. Schwaller, F. Glowna, in preparation.[36] T. Altherr, “Resummation of perturbation series in nonequilibrium scalar field the-ory,” Phys. Lett. B (1995) 325 [hep-ph/9407249].[37] B. Garbrecht and M. Garny, “Finite Width in out-of-Equilibrium Propagators andKinetic Theory,” Annals Phys. (2012) 914 [arXiv:1108.3688 [hep-ph]].[38] P. Millington and A. Pilaftsis, “Perturbative Non-Equilibrium Thermal Field The-ory,” arXiv:1211.3152 [hep-ph].[39] H. A. Weldon, “Effective Fermion Masses of Order gT in High Temperature GaugeTheories with Exact Chiral Invariance,” Phys. Rev. D26