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if one combines today s standard model of particle physics ( sm ) and that of cosmology , one finds inevitably that particles and their antiparticles annihilate at a very early moment in the evolution of the universe , leaving just radiation behind . the absence of a sizable matter - antimatter asymmetry at this epoch would imply that the universe as we know it could never form . the question about the origin of the observed asymmetry therefore represents a major challenge for modern physics .    in the sm baryon and lepton number are ( accidental ) global symmetries . if baryon number was also conserved in the early universe a dynamical emergence of the asymmetry would have been impossible . in grand - unified extensions ( guts ) of the sm baryon number ( and also lepton number ) is explicitly broken . according to past reasoning , this could provide a solution to the apparent discrepancy . in the class of ` gut - baryogenesis ' scenarios the matter - antimatter imbalance is generated by asymmetric decays of new super - heavy bosons . anomalous electroweak processes @xcite ( sphalerons ) which violate baryon and lepton number but conserve their difference essentially eliminated the prospects for gut - baryogenesis @xcite . at the same time , it inspired the now widely appreciated scenarios of ` electroweak baryogenesis ' @xcite and ` baryogenesis via leptogenesis ' @xcite . according to the latter scenario , the asymmetry is initially generated in the leptonic sector by the decay of heavy majorana neutrinos at an energy scale far above the electroweak scale . subsequently it is converted into the observed baryon asymmetry by sphalerons . the mass scale of the heavy majorana neutrinos required for leptogenesis  @xcite fits together very well with the mass - differences inferred from observations of solar- , atmospheric- and reactor - neutrino oscillations . we focus here on the conventional , but most popular , high - energy ( type - i ) seesaw extension : @xmath0 where @xmath1 are the heavy majorana fields , @xmath2 are the lepton doublets , @xmath3 is the conjugate of the higgs doublet , and @xmath4 are the corresponding yukawa couplings . the majorana mass term violates lepton number and the yukawa couplings can violate _ therefore the model fulfills essential requirements for baryogenesis @xcite . they can also be realized for more complicated sm extensions and a wide range of values for couplings and neutrino masses @xcite . in general the right - handed neutrinos do not necessarily get into thermal equilibrium and _ cp_-violating oscillations between them can contribute to the asymmetry . this effect of leptogenesis through neutrino oscillations @xcite is crucial for neutrino - minimal extensions of the sm ( @xmath5msm ) @xcite and poses interesting questions for non - equilibrium quantum field theory @xcite . in the considered scenario of thermal leptogenesis the heavy majorana neutrinos experience only a moderate deviation from thermal equilibrium at the time when the bulk of the asymmetry is produced . also , for a hierarchical mass spectrum , effects related to oscillations are negligible . the amount of the generated asymmetry is determined by the out of equilibrium evolution of the heavy majorana neutrinos . therefore , statistical equations for the abundance of the neutrinos and the generated asymmetry are needed . the conventional approach here follows the lines developed for gut - baryogenesis @xcite . cp_-violating amplitudes for the decay and scattering processes involving the heavy majorana neutrinos are computed in terms of feynman graphs at lowest loop order . they are used to build generalized boltzmann collision terms for these processes . each of them contributes to the evolution of the distributions of majorana neutrinos and leptons or , upon momentum integration , their entire abundances . however this approach is plagued by the so - called double - counting problem which manifests itself in the generation of a non - vanishing asymmetry even in thermal equilibrium . this technical issue is expression of the fact that the ` naive ' generalization of the collision terms is quantitatively inexact , and inconsistent in the presence of _ cp_-violation . after a real intermediate state ( or ris ) subtraction procedure and a number of approximations , it can be made consistent with fundamental requirements . nevertheless this pragmatic solution remains unsatisfactory . the requirement of unitarity guarantees a consistent approximation for the amplitudes , realized by the ris subtraction , if the statistical system is in thermal equilibrium . however , the deviation from equilibrium is a fundamental requirement for leptogenesis and it is not obvious how the equations have to be generalized for a system out of equilibrium . furthermore , the _ cp_-violation arises from one - loop contributions due to the exchange of virtual quanta . as such they seem to be beyond a boltzmann approximation . but the relevant imaginary part is due to intermediate states in which at least some of the particles are on - shell . these can also be absorbed or emitted by the medium and it is not obvious how such contributions enter the amplitudes . it is , however , clear that the influence of medium effects on the one - loop contributions enters directly the _ cp_-violating parameter and therefore the source for the lepton asymmetry . their size can be of the same order as that of the vacuum contributions . those questions can be addressed within a first - principle approach based on non - equilibrium quantum field theory ( neqft ) . several aspects of leptogenesis have already been investigated within this approach @xcite . the influence of medium effects on the generation of the asymmetry has been studied e.g.  in @xcite , and an analysis with special emphasis on off - shell effects was performed in  @xcite . the role of flavor effects as well as the range of applicability of the conventional approach to the analysis of flavored leptogenesis has been investigated in @xcite . the resonant enhancement of the lepton asymmetry has been addressed within a first - principle approach in @xcite . in addition , steps towards a consistent inclusion of gauge interactions have been taken  @xcite .    in this work we use the 2pi - formalism of neqft to derive boltzmann - like quantum kinetic equations for the lepton asymmetry . in particular , we show how two - body scattering processes that violate lepton number by two units and contribute to the washout of the asymmetry emerge within the 2pi - formalism . this approach treats quantum field theory and the out of equilibrium evolution on an equal footing and allows to overcome the conceptional difficulties inherent in the conventional approach . it allows us to obtain quantum - generalized boltzmann equations which include medium effects and which are free of the double - counting problem . in other words , the structure of the obtained quantum kinetic equations automatically ensures that the asymmetry vanishes in thermal equilibrium and no need for ris subtraction arises . the resulting equation for the lepton asymmetry @xmath6 is given by @xmath7 together with the ` effective amplitudes ' @xmath8 this is the main result of this paper . in eq . we introduced @xmath9 { { \ , , } } \end{aligned}\ ] ] with @xmath10 for fermions ( bosons ) . note that @xmath11 vanishes in equilibrium due to detailed balance . this ensures that the asymmetry vanishes in thermal equilibrium as mentioned before . the effective amplitudes contain medium effects ignored in the corresponding canonical expressions . we find that , in the amplitudes of the scattering processes medium effects are sub - dominant and can be neglected . the total decay amplitude of the majorana neutrino is barely affected as well . however , at high temperatures the available phase space shrinks when taking gauge interactions in the form of effective thermal masses of higgs and leptons into account . this leads to a suppression of the decay and scattering rates . since the _ cp_-violation appears as loop effect it is more sensitive to influences of the surrounding medium . even though there is a partial cancellation of the fermionic and bosonic contributions , the _ cp_-violating parameter is enhanced by medium effects . however , the thermal masses reduce the enhancement and turn it into suppression at high temperatures . we review the conventional approach to leptogenesis based on ris subtraction in section[conventionalapproach ] . in section [ risquantstat ] we demonstrate explicitly that _ in thermal equilibrium _ the success of this procedure is guaranteed by the requirement of unitarity . in section[rateequations ] we review the derivation of rate equations for total abundances and discuss in how far quantum statistical and medium corrections can be incorporated in the reaction densities . in section[neqftapproach ] we review the application of the 2pi approach of neqft to leptogenesis . equation and explicit expressions for the effective in - medium decay and scattering amplitudes are derived within this framework in section[majorana ] . we compare the results obtained within the 2pi - formalism to those of a conventional analysis with manual ris subtraction . in section[higgsdecay ] we derive rate equations and the _ cp_-violating amplitudes for higgs decay within the framework of neqft . finally , we summarize the results and present our conclusions in sec.[summary ] . the amount of produced asymmetry depends on the details of the non - equilibrium evolution of the majorana neutrinos as well as on the strength of _ cp_-violation . the latter is usually quantified by _ cp_-violating parameters@xcite : @xmath12 where @xmath13 and @xmath14 are the vacuum decay rates to a particle or anti - particle pair respectively.for a hierarchical mass spectrum @xmath15 can be computed perturbatively as the interference    tree - level , one - loop self - energy and one - loop vertex contributions to the decay of the heavy majorana neutrino . ]    of the tree - level , one - loop vertex @xcite and one - loop self - energy @xcite amplitudes in fig.[treevertexself ] . the contribution of the loop diagrams can be accounted for by effective yukawa couplings @xcite :    [ effective couplings ] @xmath16    where the loop function @xmath17 is defined as @xmath18{{\ , .}}\nonumber\end{aligned}\ ] ] the first term in eq . is related to the self - energy and the second term to the vertex contribution . the decay widths are proportional to the absolute values of the effective couplings , @xmath19 and @xmath20 respectively , where we have summed over flavors of the leptons and @xmath21 indices ( hence the factor @xmath22 ) in the final state . since the phase space for the decay into particles and antiparticles is the same , one gets for the _ cp_-violating parameter : @xmath23 let us note in passing that the divergence of the loop function for @xmath24 is not physical and can be removed by a resummation of the self - energy contribution @xcite . here we work in a regime where the mass splittings @xmath25 are large enough to render effects related to the enhancement of the self - energy contribution irrelevant ( non - resonant leptogenesis ) . we do not require a strictly hierarchical mass - spectrum , however .    to describe the statistical evolution of the lepton asymmetry one usually employs generalized boltzmann equations for the one - particle distribution functions of the different species @xcite . taking into account decay and inverse decay processes one writes for the distribution function of the leptons ( for a single flavor ) : @xmath26{{\ , , } } \label{eqn : collision terms decay}\end{aligned}\ ] ] where @xmath27 $ ] is the invariant phase space element , @xmath28 denotes spin degrees of freedom of @xmath29 , and @xmath30 is the covariant derivative . the corresponding equation for antileptons may be obtained by interchanging @xmath31 and @xmath32 . _ cpt_-invariance implies that @xmath33 and @xmath34 . furthermore , in _ thermal equilibrium _ detailed balance requires that @xmath35 . subtracting the two relations we find for the contribution of the ( inverse ) decay terms : @xmath36{{\ , .}}\end{aligned}\ ] ] if the decay amplitudes in square brackets differ , the right - hand side of eq . represents the ( non - zero ) _ cp_-violating source term for the asymmetry generation . the total asymmetry is given by the sum over all flavors and @xmath21 components : @xmath37 . neglecting the quantum - statistical terms , @xmath38 , and integrating eq . over the lepton phase space we obtain for its time derivative : @xmath39 where @xmath40 is the modified bessel function of the second kind , @xmath41 is the total tree - level decay width of @xmath29 , and the factor @xmath42 emerges from the sum over the majorana spin degrees of freedom in , see appendix[kinematics ] for more details . this implies that the source - term for the lepton asymmetry differs from zero even in equilibrium . on the other hand , combined with time translational invariance of an equilibrium state , _ cpt_-invariance requires the asymmetry to vanish in thermal equilibrium . thus , we arrive at an apparent contradiction . the generation of an asymmetry in equilibrium within the @xmath43-matrix formalism is a manifestation of the so - called double - counting problem . in vacuum an inverse decay immediately followed by a decay is equivalent to a scattering process where the intermediate particle is on the mass shell ( real intermediate state or ris ) .    two - body scattering process @xmath44 . both graphs contribute with all @xmath29 as intermediate states . note that we read ( b ) as t - channel contribution . ] thus , the same contribution is taken into account twice : once by the amplitude for ( inverse ) decay processes , and once by that for the @xmath45 scattering processes , see fig.[lhbarlbarh].(a ) . let us convince ourselves that this is indeed the case . including scattering processes we have for the distribution function of the leptons : @xmath46{{\ , , } } \end{aligned}\ ] ] where the dots denote the contribution of the ( inverse ) decay processes , the sum is over flavors and @xmath21 components of the antileptons and we have introduced @xmath47 to shorten the notation . in the unflavored regime , to which we restrict our analysis , the distribution functions of leptons of all flavors are equal . if the majorana neutrinos are close to equilibrium the difference between the distribution functions of the two spin degrees of freedom can be neglected as well . therefore , in the expression for the total asymmetry @xmath48 the summation over spin , flavor and @xmath21 components reduces to summation of the corresponding decay and scattering amplitudes . we will denote these sums over internal degrees of freedom by @xmath49 and call them effective amplitudes in the following . for the effective amplitude of @xmath50 scattering one obtains @xcite @xmath51\nonumber{{\ , , } } \end{aligned}\ ] ] where @xmath52 are the momenta of initial and final leptons respectively , and @xmath53 and @xmath54 are the usual mandelstam variables . the amplitude @xmath55 is obtained by interchanging @xmath56 . note that the loop corrections to the yukawas vanish for negative momentum transfer , i.e.  in the @xmath54-channel . for this reason the above scattering amplitude contains combinations of the yukawa couplings and their one - loop corrected counterparts . the propagators @xmath57 are given by @xmath58 where @xmath59 is the heaviside step function . the ris contribution appears for the flavor diagonal ( @xmath60 ) terms in the product of the @xmath53-channel amplitudes since only in this case the @xmath61 terms vanish simultaneously in both @xmath57 and @xmath62 . in other words : @xmath63 and a similar result for @xmath64 . using the definitions of the effective couplings and the expression for the _ cp_-violating parameter we find that @xmath65 . furthermore , for a small decay width , we can approximate the breit - wigner propagator by a delta - function using , @xmath66 ^ 2}=      2\pi\delta(\omega){{\ , , } } \end{aligned}\ ] ] where @xmath67 and @xmath68 in the considered case . the ris contribution to the scattering amplitude then takes the form @xmath69 where @xmath70 is the decay amplitude squared summed over all internal degrees of freedom ( and a similar expression for anti - particles ) . just as one would expect , it is proportional to the product of the corresponding inverse decay and decay amplitudes . the additional momentum dependence ( momenta of the leptons ) arises because the initial and final states contain fermions . close to thermal equilibrium @xmath71 and @xmath72 . neglecting the quantum - statistical terms we can write the ris contribution to the source - term as : @xmath73{{\ , .}}\end{aligned}\ ] ] taking into account that with maxwell - boltzmann distributions @xmath74 in the presence of the dirac - delta and performing the phase space integration using eq . we obtain a result _ identical _ to eq ..    to correct the double - counting _ in equilibrium _ we may therefore subtract the ris contribution from the scattering amplitude : @xmath75 and similarly for the conjugate process @xmath76 . at first sight it might seem that the ris subtracted scattering amplitudes @xmath77 do not contribute to the generation of the lepton asymmetry in equilibrium , @xmath78{{\ , , } } \end{aligned}\ ] ] but also can not compensate the asymmetry generated in equilibrium by the decay processes , see eq .. however , upon phase space integration the difference of the unsubtracted scattering amplitudes vanishes at leading order in @xmath4 . the remaining difference of the ris - amplitudes precisely compensates the contribution of the ( inverse ) decay processes . the ris subtracted scattering amplitude can be conveniently rewritten in terms of a ` ris subtracted propagator ' @xmath79 . motivated by eq . we define its diagonal components such , that they vanish upon integration over @xmath53 in the vicinity of the mass pole : @xmath80 ^ 2}{{\ , .}}\end{aligned}\ ] ] since the second of the expressions approaches the delta - function faster than the first it is common to write eq . in the form @xmath81 for @xmath82 there is no need to perform the ris subtraction and therefore @xmath83 . in the following we will also need the sum of the ris subtracted tree - level scattering amplitudes . it does not contribute to the generation of the asymmetry but plays a role for its washout . it is defined as @xmath84\nonumber\\      & = 4({{r}}_1 { { r}}_2 )      \textstyle\sum_{ij } m_i m_j\,\re ( h^\dagger h)^2_{ij } \nonumber\\      & \times \left[\,2{\cal p}_{ij}(s)+2p^{*}_i(t)p_j(t)\right.\nonumber\\      & + \left.p^{*}_i(s)p_j(t)+p^*_i(t)p_j(s)\,\right]{{\ , .}}\end{aligned}\ ] ] since it contains only the real part of @xmath85 this process is _ cp_-conserving .    two - body scattering process @xmath86 . ] a further important washout process is @xmath87 scattering which receives the @xmath54- and @xmath88-channel contributions , see fig.[llbarhbarh ] . by analogy with eq . it is convenient to introduce @xmath89 \nonumber\\      & = 2({{r}}_1 { { r}}_2)\textstyle\sum_{ij } m_i m_j\,\re(h^\dagger h)^2_{ij } \nonumber\\      & \times \left[2 p^{*}_i(t)p^{*}_j(t)+2 p^{*}_i(u)p^{*}_j(u)\right.\nonumber\\      & + \left . p^{*}_i(u ) p_j(t)+p^{*}_i(t ) p_j(u)\,\right]{{\ , .}}\end{aligned}\ ] ] since the intermediate majorana neutrino can not go on - shell in the @xmath54- and @xmath88-channel , there is no need to use the ris subtracted propagator in eq ..    above we have briefly reviewed the canonical approach to the computation of the lepton asymmetry , which is based on generalized boltzmann equations . boltzmann equations , according to conventional reasoning , describe scattering processes of particles which propagate freely over timescales large compared to the duration of individual interactions . this picture seems to be consistent with the use of @xmath43-matrix elements which are intended to describe transitions between asymptotically free initial and final states . however , in leptogenesis the crucial processes ( _ cp_-violating decays ) involve unstable particles which spoils this picture . in vacuum the amplitudes for such processes can be computed in terms of their feynman graphs . however the ` naive ' way of generalizing the boltzmann equation by multiplying the obtained amplitudes by the one - particle distributions of the initial states and integrating over phase space leads to inconsistent equations . the origin of this problem is that the obtained collision terms for particle decay and inverse decay in eq . miscount the rate of particle generation . in a short time - interval a finite number of unstable majorana neutrinos - formed by inverse decay of particles and antiparticles - decays immediately back to either particles or antiparticles . these contributions to particle generation are not included in eq . where the amplitudes are defined in terms of feynman graphs . for leptogenesis , in the presence of _ cp_-violation , it leads to inconsistent equations and must be corrected . since the missing contribution can be constructed as the rate of a two - body scattering process with on - shell intermediate state this issue can be addressed by the ris - subtracting procedure presented above . it modifies the amplitudes for two - body scattering in order to cure the problem which appears due to the collision terms for particle decay . however the corresponding correction appears at order @xmath90 , which is usually neglected . ] it is well known that unitarity has important consequences for baryogenesis and leptogenesis @xcite as it implies restrictions for the _ cp_-violating amplitudes . the issue of ris subtraction is as well tightly related to unitarity as has been mentioned in e.g.  @xcite . as noted in section[conventionalapproach ] , the use of ` naive ' boltzmann equations of the kind for unstable particles leads to problems such as the spurious asymmetry generation in the presence of _ cp_-violation in the decay of the heavy neutrinos . in this section we show explicitly that the success of the ris subtraction _ in thermal equilibrium _ is guaranteed by the unitarity of the @xmath43-matrix and how it can be generalized to include quantum - statistical terms . the approach to ris subtraction differs slightly from the one discussed in the previous section .    to illustrate it we work in thermal equilibrium , @xmath91 and @xmath92 , where @xmath93 subtracting from the boltzmann equation the corresponding equation for antiparticles , summing over internal degrees of freedom of the leptons and integrating with @xmath94 $ ] we obtain in thermal equilibrium : contribution which describes the dilution due to the expansion of the universe . ] @xmath95      { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{n}_i}^{eq}}\nonumber\\      & + 2 \int       { { { \,d}\pi_{{{\ell}}{\phi}{{\ell}}{\phi}}^{{{r}}_1 k_1 p_2 k_2 } } } ( 2\pi)^4\ ,      \delta({{r}}_1 + k_1 - p_2 - k_2 ) \nonumber\\      & \times \bigl [ { \xi^{'}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{{\ell}}{\phi } } }       - { \xi^{'}_{{{\ell}}{\phi}\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\bigr ]      { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{{\ell}}}^{eq}}{{f}_{{\phi}}^{eq}}\nonumber{{\ , .}}\end{aligned}\ ] ] we can exploit the unitarity of the @xmath43-matrix and _ cpt_-symmetry to obtain a requirement for a consistent approximation of the decay and scattering amplitudes . to this end we multiply eq . , which follows from the generalized optical theorem at order @xmath96 , by @xmath97 and integrate over @xmath98 . assuming maxwell - boltzmann equilibrium distributions we may use @xmath99 in the presence of the energy conserving dirac - delta on the right - hand - side : @xmath100{{f}_{{n}_i}^{eq}}\nonumber\\      & = -\int { { { \,d}\pi_{{{\ell}}{\phi}{{\ell}}{\phi}}^{{{r}}_1 k_1 p_2 k_2 } } }      ( 2\pi)^4{\delta({{r}}_1 + k_1 - p_2 - k_2 ) } \nonumber\\      & \hspace{15mm}\times\big [ { \xi^{'}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{{\ell}}{\phi } } }       - { \xi^{'}_{{{\ell}}{\phi}\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\big]{{f}_{{{\ell}}}^{eq}}{{f}_{{\phi}}^{eq}}{{\ , .}}\end{aligned}\ ] ] we see that imposing this as a condition for the scattering amplitudes will correctly yield @xmath101 if we neglect the quantum - statistical terms in . equation represents the zeroth - order term in an expansion about equilibrium . using eq . we can therefore obtain consistent equations at this order without the need to specify the detailed form of @xmath102 and @xmath103 . at higher order ( for washout contributions ) we also need to know the sum @xmath104 , see section[rateequations ] . we know from section[conventionalapproach ] that relation can be satisfied by subtracting ris contributions from the tree - level two - body scattering amplitudes and taking the zero width limit :    [ eqn : matrix element ris subtracted ] @xmath105    note that , strictly speaking , the ris terms in eq . include @xmath106 factors . however , upon the phase space integration in eq . the two expressions give identical results and are therefore equal in an average sense . it is obvious from comparison of eqs . and that the above definition of the ris subtracted scattering amplitudes is not sufficient to guarantee zero asymmetry in equilibrium if quantum - statistical terms are included . however this can be achieved if we replace the vacuum decay width in eq . by the thermal one @xcite : @xmath107      ( 1-{{f}_{{{\ell}}}^{eq}}+{{f}_{{\phi}}^{eq}}){{\ , .}}\end{aligned}\ ] ] using the identity @xmath108 and the fact that the ( inverse ) decay amplitudes are related by _ cpt_-symmetry we can rewrite the ris - contribution to the second term of eq . in the form : @xmath109\nonumber\\      & \times \int { { \,d}\pi^{{{\ell}}}_{{{r}}_2 } } { { \,d}\pi^{{\phi}}_{{{k}}_2}}(2\pi)^4\delta(q-{{r}}_2-{{k}}_2)\nonumber\\      & \hspace{5 mm } \times ( 1-{{f}_{{{\ell}}}^{eq}})(1+{{f}_{{\phi}}^{eq}})\bigl [ { \xi^{}_{{n}_i\rightarrow{{\ell}}{\phi } } }       - { \xi^{}_{{n}_i\rightarrow\bar{{{\ell}}}\bar{{\phi}}}}\bigr]{{\ , .}}\end{aligned}\ ] ] the integration over @xmath53 is trivial . the @xmath110 term ensures that after integration over @xmath111 the intermediate majorana neutrino is on - shell , @xmath112 . using @xmath113 together with the definition we can rewrite the second term of eq . as @xmath114 which cancels the factors coming from ris subtraction . the resulting expression reads    @xmath115      { ( 1-{{f}_{{{\ell}}}^{eq}})}{(1+{{f}_{{\phi}}^{eq}})}{{f}_{{n}_i}^{eq}}{{\ , , } } \end{aligned}\ ] ]    and cancels the first term on the right - hand side of eq .. since @xmath116 at @xmath117 the new ris subtracted source - term for the asymmetry vanishes in equilibrium .    the thermal width @xmath118 defined in eq . would also be obtained if one computes it using thermal cutting rules instead of the optical theorem ( which applies in vacuum ) , see appendix[generalizedoptth ] . we have seen that the unitarity of the @xmath43-matrix can be employed to generalize the concept of ris subtraction to rate equations which include quantum - statistical factors . as we shall see in section[higgsdecay ] , the majorana neutrino decay is at high temperature replaced by higgs decay if the higgs acquires a large effective thermal mass . in this case thermal cutting rules enforce relations between the amplitudes which can be used to obtain consistent equations , analogous to the optical theorem , see appendix[generalizedoptth ] . note again that in eq . we had to assume that the majorana neutrinos are in exact thermal equilibrium . for leptogenesis this is an inconsistent assumption since the deviation of their distribution from equilibrium realizes the third sakharov condition and drives the generation of the asymmetry . not surprisingly , the neqft approach leads to a ( slightly ) different result for the kinetic equations . however the differences between the two approaches enter only at an order beyond the usual approximation as we will discuss in the next section . in this section we review the derivation of rate equations , discuss in how far quantum - statistical and medium corrections can be incorporated , and compare the structure obtained when starting from the neqft result with the conventional form . solving a system of boltzmann - like equations in general requires the use of numerical codes capable of treating large systems of stiff differential equations for the different momentum modes  a cumbersome task if one wants to study a wide range of model parameters . in the context of baryogenesis , a commonly employed simplification is to approximate the boltzmann equations by the corresponding network of ` rate equations ' for number densities @xmath119 or abundances @xmath120 , where @xmath53 is the comoving entropy density . the resulting equations correspond to the hydrodynamical limit of the boltzmann kinetic equations , in the comoving frame of _ homogeneous _ frw space - time . to obtain evolution equations for @xmath121 in the conventional approach , i.e.  from eq . , we therefore integrate the corresponding boltzmann equations over @xmath122 $ ] to obtain , on the left - hand sides : @xmath123 where we have introduced the dimensionless inverse temperature @xmath124 and the hubble rate @xmath125 . in the homogeneous and isotropic universe the derivative of the quantity @xmath126 can be related to the divergence of the lepton - current @xmath127  a quantity which is particularly easy to access in the first - principles computation  by @xmath128 on the right - hand sides we get sums of integrated collision terms representing the effect of the different interactions . we separate contributions attributed to decays and scattering : @xmath129 the decay contributions @xmath130 to @xmath131 are very similar to the decay contributions @xmath132 to @xmath133 and we can treat them in the same way . reordering the contributions to @xmath134 we find @xmath135{{\ , , } } \nonumber\end{aligned}\ ] ] where the upper ( lower ) signs and arrows correspond to the rate equations for @xmath136 ( @xmath29 ) abundance and we defined @xmath137\nonumber{{\ , , } } \end{aligned}\ ] ] which corresponds to eq . , as well as @xmath138{{\ , .}}\end{aligned}\ ] ] we used _ cpt_-symmetry of the amplitudes in the derivation of eqs . and . later we will see that the second term in eq . appears also in the first - principle approach , compare eq . , while the terms in eqs . and are absent . this motivates the separation into ` regular ' and ` extra ' terms performed in eq .. for the contributions attributed to scattering we get : @xmath139{{\ , , } } \nonumber\end{aligned}\ ] ] with @xmath140{{\ , , } } \nonumber\end{aligned}\ ] ] corresponding to eq .. again , eq . does not appear in the first - principle approach . since in equilibrium the regular terms in each of eqs . and vanish by detailed balance we retain eq . in the sum of decay and scattering contributions . the latter vanishes as well in equilibrium if we adopt e.g.  eq . with thermal width for the ris subtracted amplitudes @xmath141 . out of equilibrium the last terms constitute a structural difference compared to the results obtained from first - principles . this difference carries over to the rate equations . we will therefore analyze these contributions separately . the computational advantage of rate equations over full boltzmann equations is maximized by a number of common approximations . in particular , assuming that all species are close to equilibrium and that the majorana neutrino distribution function @xmath142 is proportional to its equilibrium distribution for all values of the momentum @xmath143 . the temperature for all kinetic equilibrium distributions is set to a common value @xmath144 while finite deviations of the chemical potential with small @xmath145 are permitted . these approximations result for @xmath146 in a closed network of rate equations for the abundances of the form ( compare with @xcite ) :    [ rateequations1 ] @xmath147    where we have introduced    [ amplsqandepsdef ] @xmath148    the factor @xmath149 ( we neglect the thermal lepton masses here ) relates the chemical potential of the leptons to their number density , @xmath150 and the coefficient @xmath151 takes into account that in the sm the chemical potentials of leptons and higgs are related by @xmath152 with @xmath153 through equilibrium gauge , yukawa and sphaleron interactions @xcite . hence , the evolution of the abundances close to equilibrium is roughly governed by a few average quantities called _ reaction densities _ which describe decay and scattering processes . we will refer to @xmath154 , @xmath155 , @xmath156 as _ cp_-violating decay reaction density , decay reaction density and washout reaction density respectively . for comparison with standard results we want to maintain the form of eqs . and repeat their derivation from eqs . and to obtain expressions for the reaction densities which take the quantum statistical factors of the boltzmann equation into account . this is important in the present context because the thermal corrections to the _ cp_-violating parameter , to be derived later , are of a similar kind . to this end we use that the sm gauge and yukawa interactions keep the higgs and leptons very close to kinetic equilibrium : @xmath157 with a common temperature @xmath158 and chemical potentials @xmath159 , @xmath160 . we shall also use @xmath161 for the equilibrium distribution functions with zero chemical potential defined in eq ..    since a chemical potential with positive sign will appear for either the higgs or its antiparticle , we need to include at least the thermal mass of the higgs to be consistent . in the dense plasma gauge- , yukawa- and higgs self - interactions induce a large thermal higgs mass of about @xmath162 . with @xmath163 , the higgs can not acquire a condensate component . it is then safe to use a bose - einstein equilibrium distribution function to describe the distribution of the higgs particles . using that @xmath164 and hence , for a general decay collision term @xmath165 in the presence of the energy conserving dirac - delta , , @xmath166 and @xmath167 can be any species for which the above conditions apply . here we identify @xmath168 , @xmath169 , @xmath170 . ] @xmath171 , we may write : @xmath172\nonumber\\      & \times{(1-\xi^{n}{f^{}_{n}})}\frac{{{f}_{n}^{eq}}}{{(1-\xi^{n}{{f}_{n}^{eq}})}}{{\ , .}}\end{aligned}\ ] ] we can now expand the exponential in square brackets in the small quantity @xmath173 . if this quantity is tiny at all times the integral will not change much if we neglect quadratic and higher order terms . as well . ] for the zeroth - order ( first ) term in square brackets we use the linear expansion @xmath174 $ ] of the prefactor . the linear order ( second ) term in square brackets will appear preceded by just the zeroth - order factor @xmath175 with @xmath176 to write the results in a compact form we introduce decay reaction densities with quantum - statistical factors included : @xmath177 and @xmath178 where @xmath179 is the total majorana decay amplitude . similarly we define the scattering reaction densities as @xmath180 since @xmath181 refers here to a cp - symmetric ( tree - level ) amplitude squared we have @xmath182 if @xmath183 is symmetric as well .    with help of eq . we may separate the contributions to @xmath184 into terms proportional to @xmath185 , terms proportional to @xmath186 , or just proportional to @xmath187 ( see appendix[rateequationsapp ] for details ) :    [ eqn : quantum corrected boltzmann equation li result ] @xmath188    in addition we get with eq . for the extra term in eq . : @xmath189 equations describe the generation of a net asymmetry due to out of equilibrium decays of heavy majorana neutrinos . once @xmath187 has a non - zero value , there will be a slight difference in the decay rates to particles and antiparticles respectively which is not due to _ cp_-violation in the decay amplitude , but due to the presence of slightly different occupation numbers of leptons and higgs in the final states of the decays . at linear order this combined effect of blocking and stimulated emission is accounted for by eq .. depending on the ` typical ' sign of @xmath190 it can add to or diminish an existing asymmetry . finally , eq . describes washout due to inverse decays . in section[neqftapproach ] we will see that the functional dependence on @xmath191 in the integrated collision terms is the same as that encountered in the _ cp_-violating parameter @xmath15 itself . considering the last two terms in eq . we find for the scattering contributions : @xmath192 { { \ , , } } \end{aligned}\ ] ] where we defined the ` ris subtracted reaction density ' @xmath193 . if we adopt the amplitudes defined in eq . in the framework of ris subtraction , it is given by @xmath194      \gamma^{{{\ell}}{\phi}}_{\bar{{{\ell}}}\bar{{\phi}}}\bigg > { { \ , .}}\nonumber\end{aligned}\ ] ] note that the contribution proportional to @xmath90 is of higher order in @xmath195 . furthermore , we get for the extra term : @xmath196 here we used @xmath197 we have written eq . schematically in order to show how it compares to other washout terms . note that the extra terms indicate that there will be a slight difference between the equations obtained in the 2pi approach and those obtained with ris subtraction at finite temperature . comparing eqs . and we see that the first term in the former equation will cancel the latter contribution in thermal equilibrium ( @xmath198 ) if the decay contributions are summed up . the second term in eq . is due to quantum statistics . since it is proportional to @xmath199 it can be large only if @xmath15 is large ( as in the case of resonant leptogenesis ) . anticipating our knowledge about the structure obtained within neqft , we will ignore the extra term in what follows .    at the time being , everything is still exact with respect to deviations of @xmath200 from equilibrium . this distribution is necessarily distorted due to the fact that it is subject to conflicting equilibrium conditions corresponding to the decay into particles and antiparticles , by the effects of the expansion and , possibly , due to non - equilibrium initial conditions . in order to obtain the full momentum - dependent distribution function we would have to solve the corresponding full kinetic equations however @xcite .    to proceed we shall as usual assume that the deviation of the majorana neutrinos from equilibrium is small . this allows us to neglect the @xmath201 contribution and to replace @xmath202 in eq .. the extra terms cancel at this level of approximation up to the quantum - statistical term . in order to bring the remaining source - term eq . into the conventional form , we need to assume that the non - equilibrium distribution of the majorana neutrino is proportional to its equilibrium value ( with momentum independent prefactor ) @xmath203 with this approximation we can write @xmath204 the total contribution to the evolution equations for the lepton asymmetry is then given by @xmath205 i.e.  we obtain eq .. we see that , at this level of approximation , there are no contributions due to extra terms apart from those which cancel due to the ris - subtraction . quantitative differences can arise if the deviation of the majorana neutrinos from equilibrium is large or @xmath15 is of order @xmath206 . for the evolution of the majorana neutrino we obtain with appendix[rateequationsapp ] , similar to eq . , the decay contributions    [ eqn : quantum corrected boltzmann equation ni result ] @xmath207    and for the extra term in eq . : @xmath208 neglecting again @xmath201 and @xmath209 contributions we obtain @xmath210 i.e.  eq .. if higher order contributions are taken into account , we get a difference between the conventional equations and those derived in the 2pi - formalism . ignoring the contribution and the second term in eq . , which are due to quantum statistics , we obtain a contribution @xmath211 to @xmath133 . here the upper sign applies if the extra terms are included and the lower sign if not . this can therefore result in the inclusion of this term with wrong sign even if quantum statistics are neglected , compare e.g.  @xcite .    the reaction densities for decay , @xmath154 , @xmath212 , @xmath156 , and scattering , @xmath193 , @xmath213 , represent the hydrodynamical coefficients which govern the evolution of the number densities ( abundances ) . we will compute them numerically once the additional medium dependence of the amplitudes ( in particular the _ cp_-violating parameters ) has been derived . in addition , it is useful to define a thermally averaged _ cp_-violating parameter as @xmath214 which equals @xmath15 if it is momentum independent , such as in the zero temperature case , but will differ once thermal effects are included . this quantity is meaningful for the comparison with conventional results because it takes into account that the deviation of the majorana neutrino abundance from equilibrium , which appears in the source - term for the lepton abundance , is influenced by the ( _ cp_-conserving ) decay reaction density in the denominator .    inserting conventional vacuum amplitudes in eqs . with eqs . and and dropping quantum - statistical factors one obtains the conventional results for the reaction densities . for the readers convenience we quote them here . for the decay reaction density we obtain @xmath215 and @xmath216 , see appendix[kinematics ] . for the two - body scattering the reaction density is given by @xmath217 where @xmath218 is so - called reduced cross section : @xmath219 for the @xmath87 process it reads @xmath220 where we have replaced @xmath53 by @xmath221 and introduced dimensionless quantities @xmath222 and @xmath223 . the case @xmath60 is included in this expression in the limiting sense @xmath224 . note that eq . only contains the real part of @xmath225 . the contribution of the imaginary part vanishes because @xmath226 is antisymmetric with respect to @xmath227 whereas the sum in the curly brackets is symmetric under this transformation . the integration of eq . yields for the reduced ` ris subtracted cross section ' of the @xmath228 process : @xmath229 \left[(x - a_j)^2+(a_jc_j)^2\right ] } \right.\nonumber\\ & + 2\frac{x+a_i}{a_j - a_i}\ln\left(\frac{x+a_i}{a_i}\right)+ 2\frac{x+a_j}{a_i - a_j}\ln\left(\frac{x+a_j}{a_j}\right)\nonumber\\ & + \frac{x - a_i}{(x - a_i)^2+(a_ic_i)^2}\left[x-(x+a_j)\ln\left(\frac{x+a_j}{a_j }   \right)\right]\nonumber\\ & + \left.\frac{x - a_j}{(x - a_j)^2+(a_jc_j)^2}\left[x-(x+a_i)\ln\left(\frac{x+a_i}{a_i }   \right)\right]\right\}\nonumber{{\ , .}}\end{aligned}\ ] ] the reduced ` cross section ' is _ _ negative _ _ is not a physical cross section but denotes the contribution to the reaction density arising from the difference of the full and the ris term . we stress that all physical rates are manifestly positive , e.g. the washout term , to which @xmath230 yields a sub - leading correction that is relatively suppressed by yukawa couplings . see also @xcite . ] in the vicinity of the mass shells , @xmath231 . this is due to the @xmath232 term in the numerator of the ris subtracted propagator . note that because we have not approximated this term by the dirac - delta the structure of eq . is slightly different from the one usually used in the literature @xcite . in this section we briefly review the description of leptogenesis within non - equilibrium quantum field theory @xcite . this framework has been shown recently to be suitable for the derivation of quantum dynamic equations for the lepton asymmetry within a first - principle approach , and to incorporate medium , off - shell , coherence and possibly further quantum effects in a self - consistent way  @xcite . we continue these efforts by deriving consistent quantum corrected boltzmann equations that describe the generation and washout of the lepton asymmetry and include the ( inverse ) decay as well as scattering processes mediated by majorana neutrinos . the lepton asymmetry is given by the @xmath233-component of the expectation value of the lepton - current operator : @xmath234 it can be expressed in terms of the leptonic two - point function . we define the two - point functions for the higgs , lepton and majorana fields with time arguments attached to the closed time path ( ctp ) shown in fig.[fig : ctp ] by    [ twopointfunctions ] @xmath235    where the sub- and superscripts refer to @xmath21 and flavor indices and @xmath236 denotes time - ordering with respect to the ctp .    closed time path . ] we will frequently use matrix notation for the flavor indices , where e.g. @xmath237 denotes the flavor - matrix @xmath238 , etc . using the definition we find for the lepton - current : @xmath239{{\ , .}}\ ] ] two - point functions @xmath240 ( where @xmath241 stands for @xmath242 , @xmath243 or @xmath244 ) defined on the ctp can be decomposed into a _ spectral function _ @xmath245 and _ statistical propagator _ @xmath246 : @xmath247 the signum function is either @xmath248 or @xmath249 depending on whether @xmath250 or @xmath251 occur ` later ' on the contour @xmath252 . @xmath246 and @xmath245 encode information on the state and the spectrum of the system , respectively . for example , for the leptons they are given by @xmath253_- \rangle { { \ , , } } \\ { s^{\alpha\beta}_{ab\,\rho}}(x , y ) & = i \langle \left [ { { \ell}}_{\alpha}^a(x ) , \bar { { \ell}}_{\beta}^b(y ) \right]_+ \rangle { { \ , , } } \end{aligned}\ ] ] where @xmath254_\pm$ ] denote ( anti-)commutators . statistical and spectral functions of majorana neutrino and higgs can be expressed similarly , with @xmath255 and @xmath256 exchanged for bosons . although there are only two independent two - point functions for each species , it is convenient to introduce additional combinations of them , namely the _ wightman _ functions @xmath257 as well as _ retarded _ and _ advanced _ functions ,    [ retadvdef ] @xmath258    from the above definitions one can see that the difference of the retarded and advanced propagators gives the spectral one , whereas the sum yields the _ hermitian _ propagator @xmath259 :    [ hermdef ] @xmath260    finally , we will also need the _ cp _ conjugated propagators on the ctp :    [ cpconjugates ] @xmath261    here @xmath262 , @xmath263 and @xmath264 are the charge conjugation and parity matrices , respectively , and the transposition refers to spinor indices . _ cp _ conjugated statistical and spectral functions immediately follow from the above definition by inserting the decomposition . the time - evolution of the two - point functions is described self - consistently by the kadanoff - baym ( kb ) equations . these equations can be obtained from a variational principle using the so - called 2pi effective action  @xcite . the resulting equations of motion have the form of schwinger - dyson equations for the non - equilibrium propagators formulated on the ctp : @xmath265 here @xmath266 is the inverse of the _ full _ lepton propagator in coordinate space , and @xmath267 is the inverse of the _ free _ lepton propagator , @xmath268 the information about the interaction processes is encoded in the self - energies @xmath269 . they can be obtained by cutting one line of the 2pi contributions to the effective actions . the two- and three - loop contributions are presented in fig.[fig:2pi contributions ] . two- and three - loop contributions to the 2pi effective action and the corresponding contributions to the lepton self - energy . note that the propagator lines used here denote full resummed propagators in contrast to those employed in the previous feynman graphs . the contributions ( a ) and ( c ) to the 2pi effective action are known as ` setting - sun ' and ` mercedes ' diagrams respectively.,title="fig : " ] + two- and three - loop contributions to the 2pi effective action and the corresponding contributions to the lepton self - energy . note that the propagator lines used here denote full resummed propagators in contrast to those employed in the previous feynman graphs . the contributions ( a ) and ( c ) to the 2pi effective action are known as ` setting - sun ' and ` mercedes ' diagrams respectively.,title="fig : " ]    the kb equations can be obtained by convoluting the schwinger - dyson equation with the full propagator , which yields : @xmath270 here @xmath271 . after decomposing the resulting equation into statistical and spectral components , one obtains :    [ kblepton ] @xmath272    the equations for majorana and higgs propagators have a similar structure , with the klein - gordon instead of the dirac operator for the latter . the schwinger - dyson equations and the corresponding kadanoff - baym equations are formally very similar to the schwinger - dyson equation in vacuum . however , out of equilibrium the propagators depend not only on the relative coordinate @xmath273 , but also on the central coordinate @xmath274 , which makes their solution much more involved . in contrast to the schwinger - dyson equation in vacuum , the kb equations determine the spectral properties of the system including medium corrections , as well as the non - equilibrium dynamics of the statistical propagator self - consistently . since the latter represents the quantum field theoretical generalization of the classical particle distribution functions , kb equations can be seen as the quantum field theoretical generalizations of boltzmann equations .    as pointed out above , an equation of motion for the lepton asymmetry can be derived by considering the divergence of the lepton - current @xmath275 . using the kb equations one obtains and using the covariant derivative @xmath30 . as has been demonstrated in @xcite , this is the case for scalar fields . a manifestly covariant generalization of center and relative coordinates @xmath183 and @xmath53 to curved space - time can be found in @xcite . ] : @xmath276 \nonumber\\      & = g_w\ , i\int_0^{x^0 } { \mathscr{d}}^4z \ , { { \rm tr}}\bigl [ { \sigma^{\alpha\beta}_{\rho}}(x , z ) { s^{\beta\alpha}_{f}}(z , x ) \nonumber\\      & - { \sigma^{\alpha\beta}_{f}}(x , z ) { s^{\beta\alpha}_{\rho}}(z , x ) - { s^{\alpha\beta}_{\rho}}(x , z ) { \sigma^{\beta\alpha}_{f}}(z , x ) \nonumber\\      & + { s^{\alpha\beta}_{f}}(x , z ) { \sigma^{\beta\alpha}_{\rho}}(z , x ) \bigr ] { { \ , .}}\end{aligned}\ ] ] here summation over repeated indices is implicitly assumed . the two equations above represent the quantum generalization of the boltzmann equation for the lepton asymmetry . thus , they may be considered as the master equations for a quantum field theoretical treatment of leptogenesis @xcite . the dependence of the two - point functions on the relative coordinate @xmath53 is characterized by the hard scales like the majorana neutrino mass @xmath277 or the temperature @xmath144 of the surrounding plasma . in contrast to that , the variation with the central coordinate @xmath183 is given by the macroscopic time - evolution of the system , e.g. the hubble rate @xmath278 or the majorana decay rate @xmath279 . therefore , it is possible to perform an expansion in slow relative to fast time - scales , i.e. in powers of e.g. @xmath280 or @xmath281 . technically , this can be realized by a so - called gradient or derivative expansion with respect to @xmath183 , and a fourier transformation with respect to @xmath53 , known as wigner transformation , see appendix[selfenergy ] for more details . then , to leading order in the gradients , the evolution equation for the lepton asymmetry becomes markovian , and after some straightforward algebra , can be written as @xmath282\nonumber\\      & - \bigl[{\bar{\sigma}^{\beta\alpha}_{<}}(t,{{r } } ) { \bar{s}^{\alpha\beta}_{>}}(t,{{r } } ) -      { \bar{\sigma}^{\beta\alpha}_{>}}(t,{{r } } ) { \bar{s}^{\alpha\beta}_{<}}(t,{{r } } ) \bigr]\bigr\}{{\ , .}}\nonumber\end{aligned}\ ] ] note that it is possible to investigate higher orders in the derivative expansion systematically @xcite . in eq . all two - point functions are evaluated in wigner space , where @xmath283 is the _ physical _ momentum @xcite that corresponds to @xmath53 . for a spatially homogeneous system ( like frw ) the two - point functions depend only on the time coordinate @xmath284 , and on the momentum @xmath283 , because of spatial translational invariance . strictly speaking , this is true only in the rest frame of the medium ( comoving frame ) . in a general frame the two - point functions depend on @xmath285 , where @xmath286 is the four - velocity of the medium . the latter satisfies the normalization condition @xmath287 , and is given by @xmath288 in the medium rest frame .    in order to allow for a physical interpretation of eq we have written it such that the integration is over positive frequencies only , and expressed the lepton propagator and self - energy in terms of the wigner transformed wightman functions eq .. in thermal equilibrium , the wightman functions depend only on the momentum @xmath283 and satisfy the kubo - martin - schwinger ( kms ) relation @xmath289 for fermions / bosons , respectively . when inserting the kms relations for propagators and self - energies into eq . , one immediately finds that the divergence of the lepton - current vanishes in thermal equilibrium as it should ( see also @xcite ) . in other words , the quantum equation for the lepton asymmetry is in accordance with the third sakharov condition . we emphasize that it is _ not _ necessary to apply ris subtraction to obtain this result within the ctp approach @xcite . the four terms on the right - hand side of eq . may be interpreted as gain and loss terms of leptons and anti - leptons respectively @xcite . in particular , one may define generalized lepton distribution functions @xmath290 via the so - called kadanoff - baym ansatz @xmath291 thus the contribution on the right - hand side of eq . that contains @xmath292 corresponds to the lepton loss term , while the contribution proportional to @xmath293 represents the lepton gain term . analogous definitions relate the _ cp _ conjugate propagators with the anti - lepton distribution . note that the kms relations ensure that in equilibrium @xmath294 approaches the fermi - dirac distribution @xmath295 . the flavor off - diagonal components encode coherent flavor correlations @xcite . in the unflavored regime considered here @xmath296 and @xmath297 . in the quasiparticle ( qp ) approximation , the spectral function is given by @xmath298 where we assume that leptons obey conventional dispersion relation and @xmath299 is the effective thermal mass . these assumptions might be modified in the presence of a medium @xcite .    due to the presence of the dirac - delta - function in eq . the integration over @xmath300 in eq . is trivial and leaves only the integration over spatial momenta of on - shell leptons . therefore the right - hand side of eq . can be interpreted as a difference of two ( integrated ) boltzmann - like equations one for the particles and one for the antiparticles @xcite . according to the physical interpretation of eq . in terms of gain and loss terms , the wightman components of the lepton self - energy and of its _ cp _ conjugate are the analogs of the collision integrals . since we limit our analysis to the unflavored regime , it is convenient to perform the summation over the flavor indices : @xmath301 . then the one - loop contribution , see fig.[fig:2pi contributions ] ( b ) , takes the form : @xmath302 where @xmath303 . the explicit expression for the two - loop contribution is rather lengthy and it is convenient to split it into three distinct terms : @xmath304 the first term on the right - hand side reads @xmath305\nonumber { { \ , , } } \end{aligned}\ ] ] where we have introduced two functions containing loop corrections : @xmath306{{\ , , } } \end{aligned}\ ] ] and @xmath307 to shorten the notation . comparing eqs . and we see that they have a very similar structure . first , the integration is over momenta of the higgs and majorana neutrino and the delta - function contains the same combination of the momenta . second , both self - energies include one wightman propagator of the higgs field and one wightman propagator of the majorana field . upon the use of the kadanoff - baym ansatz the wightman propagators can be interpreted as cut - propagators which describe on - shell particles created from or absorbed by the plasma @xcite . on the other hand , the retarded and advanced propagators can be associated with the off - shell intermediate states . we therefore conclude that eqs . and describe ( inverse ) decays of the heavy neutrino into a lepton - higgs pair . the second term on the right - hand side of eq . contains two wightman propagators of the higgs field and one wightman propagator of the lepton field . the majorana propagator appears only in the intermediate state : @xmath308{{\ , .}}\end{aligned}\ ] ] we therefore conclude that this term describes lepton number violating scattering processes mediated by the heavy neutrino . finally the last term in eq . contains two wightman propagators of the majorana field and one of the lepton field , whereas the higgs field is in the intermediate state : @xmath309{{\ , .}}\end{aligned}\ ] ] therefore it can be identified with the higgs mediated scattering processes . these conserve lepton number and do not contribute to generation of the lepton asymmetry . the _ cp _ conjugate of the wigner transforms can be obtained using eq .. in practice this amounts to replacing the propagators by their _ conjugate and the couplings by their complex conjugate in the above expressions . for instance for the _ cp _ conjugate of the one - loop self - energy we find : @xmath310 expression for the _ cp _ conjugate of the two - loop lepton self - energy can be obtained in a similar way .    for the higgs propagators in the above self - energies we can also use the kadanoff - baym ansatz , @xmath311 and the simple quasiparticle approximation for the spectral function , @xmath312 where @xmath313 is the effective thermal mass . effects of the finite thermal higgs mass will be studied in section[higgsdecay ] . in this section we will analyze the lepton number and _ cp_-violating ( inverse ) decay of the majorana neutrino as well as the two - body scattering processes mediated by the heavy neutrino . in particular , we will derive expressions for the in - medium _ cp_-violating parameters , decay widths and scattering amplitudes . we will also explicitly demonstrate that the obtained equation for the lepton asymmetry is free of the double - counting problem .      in the previous section we have used the kadanoff - baym ansatz and quasiparticle approximation for the higgs and lepton fields . let us now assume that similar approximations also hold for majorana neutrinos . that is , we assume that in eqs . and the spectral function @xmath314 is diagonal in flavor space and can be approximated by @xmath315 and that it is related to the wightmann components via the kadanoff - baym ansatz : @xmath316 substituting eqs . and in eq . and making the above approximations we find after some algebra that the lepton - current can be represented in the form : @xmath317{{\ , , } } \end{aligned}\ ] ] where @xmath318 have been introduced in eq . and we have defined :    [ effectivedecayamplitudestree ] @xmath319{{\ , , } } \\ \label{effamplsebarlbarh }          { \xi^{t}_{\bar { { \ell}}\bar { \phi}\leftrightarrow { n}_i}}&\equiv           g_w ( h^\dagger h)_{ii } { { \rm tr}}[(\slashed{{{q}}}+m_i)p_l\slashed{{{r}}}\,]{{\ , .}}\end{aligned}\ ] ]    the superscript ` t ' stands for ` tree - level ' . the expression strongly resembles the boltzmann equation . therefore the functions @xmath320 and @xmath321 can be interpreted as effective in - medium amplitudes squared , summed over internal degrees of freedom , for the decays into leptons and antileptons respectively .    tree - level contribution . ] the two effective amplitudes can be replaced by the total decay amplitude and the _ cp_-violating parameter . we find that within the used approximations the resulting decay amplitude coincides with the outcome of the vacuum calculation , @xmath322 , and that @xmath323 .    in the presence of a nonzero lepton asymmetry @xmath324 and @xmath325 . therefore @xmath326 and this leads to a washout of the asymmetry . despite the fact that eq . correctly describes the ( leading - order ) washout processes , it fails to describe processes which generate lepton asymmetry : in the considered approximation @xmath323 because the _ cp_-violating effects , which are required to produce the asymmetry , are of fourth order in the yukawa couplings of the majorana neutrino . in eq . we have taken into account only terms quadratic in the coupling . in other words , this approximation corresponds to the tree - level approximation in the canonical approach . terms of higher order in the couplings emerge from three- and higher - loop contributions to the lepton self - energy , see eqs.- , as well as from expansion of the full majorana propagators entering the self - energies .      in order to define an effective cp - violating parameter and decay width that incorporate medium corrections we have to identify the quasiparticle excitations in the system . to perform this analysis we follow the discussion of the self - energy contribution within a toy - model as presented in @xcite . as has been demonstrated there , it is important to take the matrix structure of the majorana propagator in flavor space into account . our starting point is the schwinger - dyson equation for the majorana two - point function : @xmath327 let us split the self - energy into diagonal and off - diagonal components in flavor space and introduce a diagonal propagator @xmath328 defined by the equation : @xmath329 where @xmath330 is the free propagator and @xmath331 denotes the diagonal components of the self - energy . the poles of the diagonal propagator define the quasiparticle excitations . it can be shown that the dynamics of these is described by a boltzmann - like quantum kinetic equation .    inserting this decomposition into the schwinger - dyson equation we find , using matrix notation : @xmath332 where @xmath333 denotes the off - diagonal components of the self - energy and @xmath237 the full neutrino propagator including flavor - diagonal and flavor off - diagonal contributions . multiplying eq . by @xmath237 from the left , by @xmath334 from the right and integrating over the contour @xmath335 we obtain a formal solution for the full non - equilibrium propagator : @xmath336 after decomposing the propagators and self - energies into the spectral and statistical components , we can rewrite eq . in the form : @xmath337{{\ , .}}\end{aligned}\ ] ] here we are using the retarded and advanced propagators defined by eq . , so that the integration can be extended to the whole @xmath338-plane . using their definitions and eq . , we can also derive formal solutions for the retarded and advanced propagators : @xmath339 next we wigner transform eqs . and and perform the leading order gradient expansion as has been outlined in section[neqftapproach ] . combining both results , we find for the full statistical and spectral propagators and the corresponding causal two - point functions of the system in to equilibrium :    [ fullpropagator ] @xmath340    where all propagators and self - energies are evaluated at the same point @xmath341 in configuration space . we can express the full statistical and spectral propagators in terms of the diagonal ones and the off - diagonal self - energies , @xmath342 { \hat{\theta}^{}_{a}}{{\ , , } } \end{aligned}\ ] ] where @xmath343 and @xmath344 are defined by @xmath345 and @xmath346 respectively , with @xmath347 being the @xmath348 unit matrix in the dirac and flavor space of the @xmath349 generations . solution eq . reduces the dynamics of the full statistical and spectral propagators to the dynamics of two quasiparticle excitations . their masses , decay widths and _ cp_-violating parameters are determined by the medium and the abundances are described by the corresponding one - particle distribution functions . strictly speaking , the solution is valid only in thermal equilibrium . however , we assume that it also holds for small deviations from equilibrium .    to consistently analyze processes of the fourth order in the coupling one has to use so - called extended quasiparticle approximation ( eqp ) for the statistical propagator and spectral function @xcite . the eqp approximation represents the diagonal propagator as a sum of two terms : @xmath350 the first describes decay processes , whereas the second can be associated with scattering processes . inserting eq . into eq we get a solution for the resummed majorana propagator consistent up to the fourth order in the couplings : @xmath351{\hat{\theta}^{}_{a}}{{\ , .}}\end{aligned}\ ] ] the first term in the above formula describes majorana decay , see section[cpviolationindecay ] , whereas the remaining three terms describe the two - body scattering processes mediated by the majorana neutrino . these are discussed in section[majoranamediatedscattering ] .    using definition of the retarded and advanced two - point functions , eq . , and the schwinger - dyson equation for the diagonal propagators , eq . , we find that the causal propagators in eq . are given by @xmath352 splitting the retarded and advanced self - energies into the vector and scalar components we can write the solution of eq . in the form : @xmath353 where we have omitted flavor indices to shorten the notation and introduced @xmath354 from eq . we can extract the spectral and hermitian propagators . to leading order in the yukawas they read    [ shrhoapprox ] @xmath355    the on - shell condition is defined by @xmath356 . expanding @xmath357 to linear order in the yukawas we find : @xmath358 where @xmath359 is the medium - induced component of the hermitian self - energy in the on - shell renormalization scheme . in vacuum the on - shell condition is fulfilled for @xmath360 , i.e. @xmath361 is the physical vacuum mass . at non - zero temperatures the mass receives medium - induced corrections . to linear order in the yukawas the effective mass is given by @xmath362 . for a hierarchical mass spectrum , which we consider here , the contributions of the hermitian self - energy are always negligible and we will use @xmath363 and @xmath364 in the following . from eq . we can also deduce the effective width . to leading order in the yukawas it is given by @xmath365 . the minus sign in this definition ensures that the effective decay width is positive . one - loop contribution to the majorana self - energy is derived in appendix[selfenergy ] . in a _ cp_-symmetric medium it is given by @xmath366 l_{\rho}{{\ , .}}\end{aligned}\ ] ] therefore we can write the effective decay width in the form @xmath367 , where @xmath368 is the total vacuum decay width . for positive @xmath369 and @xmath370 the loop integral @xmath371 takes the form : @xmath372{{\ , .}}\end{aligned}\ ] ] for massless final states @xmath373 . therefore the definition of the effective decay width inferred in section[risquantstat ] from the requirement of successful ris subtraction is consistent with that implied by eq ..    for the eqp wightman propagators we can use the kadanoff - baym ansatz . as can be inferred from eq . , the corresponding spectral function reads @xmath374 substituting eq . into eq . we obtain @xmath375 where we have again omitted the flavor indices . the second and the third terms in eq . vanish on the mass shell and can be neglected . commuting @xmath376 and @xmath377 in the first term and again neglecting contributions which are tiny on the mass shell we finally obtain for the eqp spectral function : @xmath378 ^ 2 } { { \ , .}}\end{aligned}\ ] ] note that structures of eqs . and are very similar . furthermore , as follows from eq . , in the limit of vanishing decay width both of them approach the delta - function . however , for a small but _ finite _ decay width the eqp spectral function is a better approximation to the delta - function than eq .. therefore , we can approximate it by the usual expression , @xmath379 and at the same time keep finite - width terms in the diagonal propagators . to go beyond the tree - level approximation and take into account _ cp_-violating effects we need to consider contributions to the lepton self - energy that are of the fourth order in the yukawa couplings . one of them comes from expansion of the majorana propagator in the one - loop self - energy . substituting the decay term of eq . into eq . we can write it in the form : @xmath380 substituting eq . and its _ cp _ conjugate into eq . we find that the resulting contribution to the divergence of the lepton - current has precisely the form . however , the corresponding effective amplitudes are no longer equal :    [ effectivedecayamplitudesse ] @xmath381{{\ , , } } \\ \label{effamplsecpbarlbarh }          { \xi^{t}_{\bar{{\ell}}\bar{\phi}\leftrightarrow { n}_i } } &          + { \xi^{s}_{\bar{{\ell}}\bar{\phi}\leftrightarrow { n}_i}}\equiv           g_w { \textstyle\sum}_{mn}(h^\dagger h)^*_{mn}\nonumber\\          \times&{{\rm tr}}[{\bar{\theta}^{ni}_{r}}(t,{{q}})(\slashed{{{q}}}+m_i )          { \bar{\theta}^{im}_{a}}(t,{{q}})p_l\slashed{{{r}}}p_r\,]{{\ , .}}\end{aligned}\ ] ]    the matrices @xmath343 and @xmath344 are evaluated on the mass shell of the @xmath382th majorana neutrino . the bar denotes _ cp_-conjugation and the trace is over dirac indices . interference of tree - level and one - loop self - energy corrections . ] as compared to tree - level result it additionally contains interference of the tree - level and one - loop self - energy contributions to the majorana decay amplitude , see fig.[tpluss ] . for a hierarchical mass spectrum we can use the approximation @xmath383 and a similar approximation for @xmath344 . using furthermore eqs . and we find for the _ cp_-violating parameter : @xmath384 where @xmath283 and @xmath385 are on - shell momenta of the outgoing lepton and decaying majorana neutrino respectively . in vacuum @xmath386 and the _ cp_-violating parameter takes the form : @xmath387 the ` regulator ' in the denominator of eq . differs from the result @xmath388 found in @xcite by the ratio of the masses . for a hierarchical neutrino mass spectrum the ` regulator ' term is sub - dominant and this difference is numerically small . note also that although eq . does not diverge in the limit of vanishing mass difference the approximations made in the course of its derivation are not applicable for a quasidegenerate mass spectrum @xcite . for a consistent treatment of resonant enhancement within neqft we refer to  @xcite . the two - loop lepton self - energy is of the fourth order in the couplings to begin with . therefore , for a hierarchical mass spectrum one can safely neglect the off - diagonal components of the majorana propagators and replace @xmath244 by the eqp one : @xmath389 { { \ , .}}\end{aligned}\ ] ] substituting eq . and its _ cp _ conjugate into eq . we again find that the resulting contribution to the divergence of the lepton - current has the form . the corresponding effective amplitudes read    [ majoranavertexamplitudes ] @xmath390\\      & -g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ , { { \rm tr}}\bigr [ c v_{jj}(q , k)p_l\slashed p p_r\bigr]{{\ , , } } \nonumber\\      \label{effamplvertbarlbarh }      { \xi^{v}_{n_i \leftrightarrow \bar { { \ell}}\bar { \phi } } }       \equiv & -g_w(h^{\dagger}h)_{ij}^2\ m_i\ ,      { { \rm tr}}\bigr [ c v_{jj}(q , k)p_l\slashed p p_r\bigr]\\      & -g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ , { { \rm tr}}\bigl[\varlambda_{jj}(q , k )      c p_l\slashed pp_r\bigr]\nonumber{{\ , .}}\hphantom{aa}\end{aligned}\ ] ]    they describe interference of the tree - level and one - loop vertex contributions to the majorana decay amplitude , see fig.[interferencev ] .    interference of tree - level and one - loop vertex corrections . ]    to account for the contribution of the vertex correction to the decay width and the _ cp_-violating parameter we have to substitute the sum of @xmath391 and @xmath392 and a similar sum for the antiparticles into eq .. the vertex contribution to the decay amplitude is of fourth order in the coupling and is negligible compared to the tree - level term . since we assume the medium to be almost _ cp_-symmetric we can use , at leading order , _ cp_-symmetric two - point functions in the loop integrals @xmath393 and @xmath394 . then , at leading order in the yukawa couplings , we find for the vertex contribution to the _ cp_-violating parameter : @xmath395\nonumber{{\ , .}}\end{aligned}\ ] ] the quasiparticle approximation and the kb - ansatz enforce two of the intermediate lines of the vertex loop to be on - shell whereas the remaining line described by the hermitian part of the retarded and advanced propagators remains off - shell . the three lines in square brackets in eq . therefore correspond to different cuts through two of the three internal lines of the loop diagram fig.[treevertexself].(c ) . note also that only for one of the three internal lines the corresponding distribution function enters the result . the first possible cut described by the first line in square brackets corresponds to cutting the propagators of higgs and lepton . one can interpret this cut as decay of the majorana neutrino into a lepton - higgs pair which is followed by a subsequent @xmath54-channel scattering mediated by a virtual majorana neutrino . introducing @xmath396      m_i^2 { \boldsymbol{\mathcal{s}}^{ii}_{h}}({{k}}-{{r}}_2 ) { { \ , , } } \end{aligned}\ ] ] we can rewrite the first term in eq . in a form which strongly resembles the form of the self - energy _ cp_-violating parameter : @xmath397 in vacuum @xmath398 can be computed explicitly and we recover the well - known result @xcite : @xmath399{{\ , .}}\end{aligned}\ ] ] adding up eqs . and we obtain the canonical expression for the vacuum _ cp_-violating parameter , eq ..    if the intermediate majorana neutrino is much heavier than the decaying one then @xmath400 and therefore @xmath401 . in this case we can also neglect the ` regulator ' term in the denominator of eq .. in this approximation the two _ cp_-violating parameters have the same structure and their sum can be written in the form : @xmath402 note that the combination of the distribution functions that enters the self - energy and vertex _ cp_-violating parameters , see eqs . and , is the same as that of @xmath403 encountered in the derivation of the rate equations , see eq .. this result is in agreement with the findings of  @xcite using neqft and of @xcite based on imaginary - time thermal qft . note that older results featured a different dependence on the distribution functions , with an additional term quadratic in the one - particle distribution functions which is absent in eq . as well as in eq . : @xmath404 in @xcite it was demonstrated that the result obtained using thermal field theory can be reconciled with the result of neqft calculation once causal green s functions are used in the former . the two other cuts in eq . are proportional to @xmath405 and to @xmath406 respectively . they vanish in the zero temperature limit and are usually boltzmann - suppressed at finite temperatures , but can be relevant in specific cases @xcite . the quantities that enter the rate equations are the decay , washout and _ cp_-violating decay reaction densities . in the canonical approximation , i.e. when the quantum - statistical effects and effective masses of the higgs and leptons are neglected , they are given by eq .. if the thermal masses are neglected but the quantum - statistical effects are taken into account , there is an enhancement of the decay and washout reaction densities at high temperature , see fig.[gammadratio ] . decay and _ cp_-violating reaction densities with thermal lepton and higgs masses , @xmath407 , and with zero masses , @xmath408 , for the two majorana neutrinos @xmath409 and @xmath410 . the values are normalized to the corresponding reaction density in the conventional approximation @xmath411 . the thermal enhancement due to quantum - statistical factors is overcompensated by the phase space suppression due to thermal masses at high temperatures . note that we show only the self - energy contribution to the _ cp_-violating reaction densities . ]    however , the inclusion of the thermal masses turns this enhancement into a suppression at high temperatures . it is explained by the decrease of the decay phase space . at intermediate temperatures the thermal masses become small relative to the majorana mass and we observe a minor enhancement . for the _ cp_-violating reaction density we observe a very similar behavior . given that for a hierarchical mass spectrum most of the asymmetry is typically generated by the lightest majorana neutrino at @xmath412 , where @xmath40 is the washout parameter ( see appendix[numericalparameters ] ) , we expect the medium effects to induce a moderate enhancement of the total generated asymmetry . two - body scattering processes mediated by majorana neutrinos violate lepton number by two units and play an important role in the washout of the generated asymmetry . in this section we derive the effective scattering amplitudes using neqft . this is an important part of our results . the last three terms in eq . contain the wigner - transformed one - loop majorana self - energy : @xmath413 { { \ , , } } \hphantom{a}\end{aligned}\ ] ] see appendix[majoranaselfen ] for more details . combining them with eq . we find that their contribution to the divergence of the lepton - current contains two wightman propagators of leptons and two of the higgs field . as we have argued above , these correspond to initial and final states in the kinetic equations . therefore , we conclude that these terms describe scattering processes depicted in fig.[lhbarlbarh ] and fig.[llbarhbarh ] . as higgs and leptons are maintained close to equilibrium we can safely use the kadanoff - baym ansatz for their propagators in the majorana self - energy . inserting eq . into the scattering terms of eq . we can then split the majorana propagator into a lepton number conserving and lepton number violating part :    [ propmajapsp ] @xmath414 { { \ , , } } \\          \label{propmajapsp < }          { \mathscr{s}^{ij}_{<}}({{q}})&=- g_w\int { { \,d}\pi^4_{{{k } } } }           { { \,d}\pi^4_{{{r}}}}(2\pi)^4 \delta(q - p - k){\delta^{}_{\rho}}({{k } } )           { \mathbf{s}^{}_{\rho}}({{r } } ) \nonumber \\           & \times \bigl[f_{{\ell}}^{{r}}f_{\phi}^{{k}}{\mathscr{s}^{ij}_{lc}}({{q}},{{r } } ) + f_{\bar { { \ell}}}^{{r}}f_{\bar { \phi}}^{{k}}{\mathscr{s}^{ij}_{lv}}({{q}},{{r } } ) \bigr ] { { \ , , } } \end{aligned}\ ] ]    where we have defined :    [ lvlc ] @xmath415{{\ , , } } \nonumber \\          \label{lv }          & { \mathscr{s}^{ij}_{lv}}=(h^\dagger h)_{ji }           \bigl [ ( 1-\delta^{ij}){\mathcal{s}^{ii}_{r}}(q)p_r\slashed{{{r } } }           p_l { \mathcal{s}^{jj}_{a}}({{q } } ) \\           & + \frac{\delta^{ij}}{2 }          \big({\mathcal{s}^{ii}_{r } } ( q)p_r\slashed{{{r } } } p_l           { \mathcal{s}^{jj}_{r}}({{q } } ) + { \mathcal{s}^{ii}_{a}}(q)p_r          \slashed{{{r } } } p_l { \mathcal{s}^{jj}_{a}}({{q}})\big ) \bigr]{{\ , .}}\nonumber      \end{aligned}\ ] ]    here we neglected higher order terms coming from the matrices @xmath416 and @xmath417 . the first terms in eqs . and corresponds to the second term in eq . , whereas the remaining terms correspond to the last two terms in eq .. substituting the majorana propagators into the lepton - current together with the one - loop lepton self - energy we finally obtain @xmath418{{\ , .}}\end{aligned}\ ] ] note that the zeroth component of the momenta in the above equation can have both signs . the effective amplitudes of the lepton number conserving and lepton number violating processes read    [ traceap ] @xmath419{{\ , , } } \hphantom{aaaa}\\      \label{traceaplc }      a^{lc}({{q}},{{r}}_1,{{r}}_2)\equiv & ( h^\dagger h)_{ji}\ ,      { { \rm tr}}\bigl[{\mathscr{s}^{ij}_{lc}}(q,{{r}}_2 ) p_l \slashed{{{r}}}_1p_r\bigr]{{\ , .}}\hphantom{aaaa}\end{aligned}\ ] ]    the functions @xmath420 and @xmath421 are symmetric under the exchange of the momenta @xmath422 and @xmath423 . this implies that the contribution of @xmath424 in eq . the terms of @xmath425 diagonal in flavor space correspond to the ris - propagator . substituting eq . into eq . and taking the trace we find that it contains only the scalar components of the retarded and advanced propagators and is proportional to : @xmath426 ^ 2}\nonumber\\      & \approx 2m^2\frac{(q^2-m^2)^2-(m{{\boldsymbol \gamma}^{}_{}})^2}{\left[(q^2-m^2)^2      + ( m{{\boldsymbol \gamma}^{}_{}})^2\right]^2}{{\ , .}}\end{aligned}\ ] ] equation   differs from the canonical result only in that the vacuum masses and decay widths are replaced by thermal ones . for a hierarchical mass spectrum this difference can be safely neglected . introducing an analogue of the ris subtracted propagator , @xmath427 we can rewrite the lepton number violating effective amplitude in a compact form : @xmath428 next we perform the trivial integrations over the frequencies using the dirac - deltas in the quasiparticle spectral functions and . each dirac - delta can be decomposed into two terms , one with positive and one with negative frequency . therefore , the integration over the four frequencies gives rise to @xmath429 terms , but only six of them satisfy energy conservation ensured by the remaining delta - function . in a homogeneous and isotropic medium the one - particle distribution functions satisfy @xmath430 and the diagonal majorana propagators have the properties @xmath431 upon substitution of the resulting self - energy into eq . and the use of eqs . and the remaining six contributions in the lepton - current can be conveniently written as @xmath432{{\ , , } } \end{aligned}\ ] ] where we have defined the effective scattering amplitudes :    [ m1 ] @xmath433    the momenta of the majorana neutrinos are related to the momenta of the initial and final states by @xmath434 and @xmath435 . from eq . we see that the obtained amplitudes contain only @xmath436 and @xmath437 interference terms . indeed , in the products of the majorana propagators in eq . both of them depend on the same momentum . the missing cross terms emerge from the two - loop ( vertex ) contribution to the lepton self - energy . as we have mentioned in section[neqftapproach ] , within the discussed assumptions and approximations the second and third terms on the right - hand side of eq . describe scattering processes . since they are of the fourth order in the yukawas , we can replace the full majorana propagators by the diagonal propagators . the third term , eq . , corresponds to lepton number conserving processes and does not need to be discussed further . the second one , eq . , is given by @xmath438{{\ , .}}\end{aligned}\ ] ] we substitute eq . into the equation for the lepton - current and perform the steps preceding eq .. using furthermore relations we find @xmath439{{\ , , } } \end{aligned}\ ] ] where we have introduced    [ m2 ] @xmath440    and @xmath441 . the combinations of the momenta appearing in the products of the majorana propagators clearly indicate that the above amplitudes correspond to the interference terms of the @xmath53- , @xmath54- , and @xmath88-channel contributions .    combining eqs . and we obtain for the effective amplitude of @xmath442 scattering : @xmath443{{\ , , } } \end{aligned}\ ] ] whereas the effective amplitude of @xmath444 scattering reads @xmath445{{\ , .}}\end{aligned}\ ] ] to compare the obtained expressions with the vacuum results of the canonical computation eqs . and , we evaluate the retarded and advanced propagators at zero temperature . in vacuum @xmath446 . therefore for positive @xmath447 it follows from eq . that @xmath448^{-1}{{\ , .}}\end{aligned}\ ] ] in the vicinity of the mass shell of the respective majorana neutrino @xmath112 and we find @xmath449 for @xmath54- and @xmath88-channels the imaginary parts of eqs . and vanish , so that in the vacuum limit @xmath450 and @xmath451 is symmetric with respect to @xmath227 . the squares of the @xmath54- and @xmath88-channel propagators in eq . give identical contributions to the reduced cross section of @xmath87 process . therefore , upon substitution of eq . into eq . and the use of the @xmath227 symmetry , we recover for the reduced cross section the canonical result . relations imply that the off - diagonal components of @xmath452 and @xmath451 coincide . for the @xmath53-channel the diagonal components of @xmath453 are given in the vacuum limit by @xmath454 ^ 2}{{\ , .}}\end{aligned}\ ] ] thus , in the vicinity of the mass shell , @xmath455 , the canonical expression for the ris - subtracted propagator , eq . , coincides with the expression obtained from first principles . this justifies results of the earlier calculations . for the reduced cross section of @xmath228 scattering we recover eq ..    at finite temperatures @xmath456 is not zero even for @xmath54- and @xmath88-channels . in other words , the medium effects induce additional contributions to the effective decay amplitudes . however , these contributions are proportional to the coefficients @xmath457 . numerical analysis shows that for the two chosen sets of parameters , see appendix[numericalparameters ] , the additional correction typically do not have any sizable impact on the reaction densities .    a quantity relevant for the numerical analysis is the ratio @xmath458 . the dependence of this ratio on the dimensionless inverse temperature is presented in fig.[blzmnvsfdbe1 ] and fig.[blzmnvsfdbe2 ] . if the approximate expression is used , then the reaction density of @xmath228 scattering becomes negative for @xmath459 for the first set of the parameters whereas for the second set of the parameters it turns negative for @xmath460 . a qualitatively similar behavior has also been observed in @xcite . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point  1 in the approximation of massless leptons and higgs . shown are the reaction densities computed using the boltzmann approximation ( [ reactdenscanon ] ) ( thin lines ) , and taking into account the quantum - statistical effects , ( [ reactdensexact ] ) ( thick lines ) . the ( ris subtracted ) reaction densities @xmath193 may be negative as they are not the physical rates for the @xmath461 scattering process . ] this rather counter - intuitive result can be traced back to the behavior of the ris part of the effective amplitude which is negative in the vicinity of the mass shell . its sign is not fixed by physical requirements , since it constitutes a sub - leading contribution to the washout rate . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point  2 . compare fig.[blzmnvsfdbe1 ] . ]    as can be inferred from fig.[blzmnvsfdbe1 ] , for the first set of model parameters the quantum - statistical corrections render the ris subtracted reaction density of @xmath228 scattering positive in the whole range of temperatures . however , this is merely a numerical coincidence . for the second set of parameters , see fig.[blzmnvsfdbe2 ] , the reaction density of @xmath228 scattering remains negative for @xmath462 . it is nevertheless important to note that even in the region where the reaction density is negative the quantum - statistical corrections shift it upwards as compared to the result of the canonical computation .    as far as @xmath463 scattering is concerned , the quantum - statistical corrections enhance the reduced reaction density at high temperatures by about 50% . as the temperature decreases , the reaction density computed using eq . approaches the one computed using eq . as one would expect . a similar behavior is also observed for the reaction density of @xmath228 scattering .    in figs.[blzmnvsfdbe1 ] and [ blzmnvsfdbe2 ] we neglected the thermal masses of initial and final states . ratio of the reaction density computed using thermal masses with ( thick lines ) and without ( thin lines ) the quantum - statistical terms to the canonical one for the two sets of parameters . ]    to estimate the size of the mass corrections , in fig.[scatteringwithmasses ] we plot ratio of the reaction density of the @xmath87 scattering computed using thermal masses of the higgs and leptons with and without the quantum - statistical terms to the canonical one for the two sets of parameters . if the quantum - statistical effects are neglected , the thermal masses lead to a @xmath464 suppression of the reaction densities . on the other hand , an enhancement induced by the quantum - statistical terms to a large extent compensates the mass - induced suppression . as a result , the deviation from the canonical reaction density does not exceed @xmath465 in the whole range of temperatures .    to compare the relative importance of the ( inverse ) decay and scattering processes in fig.[washoutrdbm1 ] and fig.[washoutrdbm2 ] we also present the uniformly normalized reaction densities . washout reaction densities due to @xmath87 and @xmath228 scattering processes for benchmark point  1 in the approximation of massless leptons and higgs . for comparison the washout reaction densities for @xmath409 and @xmath410 ( inverse ) decays are shown as well . note that the normalization differs from the one used in fig.[blzmnvsfdbe1 ] and fig.[blzmnvsfdbe2 ] for the present choice of parameters contributions by @xmath461 scatterings are strongly suppressed by the smallness of the couplings . ]    in both cases we observe a qualitatively similar picture : for the chosen sets of parameters reaction densities of the scattering processes are strongly suppressed by the smallness of the yukawa couplings as compared to those of the decay processes . washout reaction densities due to @xmath87 and @xmath228 scattering processes as well as ( inverse ) decays for benchmark point  2 . compare fig.[washoutrdbm1 ] . ]    in the rate equation for the lepton asymmetry , see eq . , the total washout rate is given by a sum of reaction densities for decays and scattering . whereas @xmath156 and @xmath466 are positive , the ris subtracted reaction density @xmath467 can be negative at some temperatures . if the total washout rate would turn negative , it would lead to a spurious self - enhanced generation of the asymmetry . in figs.[washoutrdbm1 ] and [ washoutrdbm2 ] we show the sum of the reactions densities . for both parameter sets it is positive . this sum should always be positive . if the quantum - statistical terms are neglected this can be demonstrated explicitly . in the absence of the quantum - statistical corrections , the results of this section revert to the ones discussed in section[conventionalapproach ] . using the expression for the ris - propagator , eq . , we can rewrite the ris subtracted scattering amplitude as a difference of the unsubtracted one and the ris term . since the latter is proportional to @xmath468 the integration in eqs . and is trivial and we obtain after some algebra : @xmath469 the total washout rate is then given by @xmath470 and is positive as a sum of two positive functions . in other words , even though @xmath467 can be negative at some temperatures , the total washout rate is always positive . the form of the unsubtracted scattering reaction density that can be inferred from eq . is another manifestation of the double - counting . if we had not subtracted the ris contribution , we would have counted contributions of the inverse decay processes twice and ended up with an incorrect prediction for the generated asymmetry . in the preceding section we have approximately taken the gauge interactions into account in the form of effective masses of the higgs and leptons . the thermal masses are of order of @xmath471 and large enough to influence the values of the reaction densities quantitatively . in particular , the majorana neutrino decay can become kinematically forbidden when the sum of the masses of lepton and higgs exceeds the heavy majorana neutrino mass . for even higher temperatures ( with @xmath472 ) the higgs decay channel into a lepton - majorana pair becomes kinematically allowed instead and can contribute to the asymmetry since it violates _ cp_. for simplicity we do not take modified dispersion relations into account here , see  @xcite , but use the simple picture of temperature dependent thermal masses as an estimate : @xmath473 where we use the temperature dependent values of the @xmath21 , @xmath474 , top yukawa and higgs self - couplings @xmath475 , @xmath476 , @xmath477 and @xmath478 assuming a higgs mass of @xmath479 @xcite . we also ignore that the thermal mass of leptons might be better approximated by the ` asymptotic thermal mass ' @xmath480 in a kinematic regime in which their momentum is such that @xmath481 @xcite . we do also not take into account , in our quantitative analysis , the thermal corrections to the majorana neutrino masses as they are negligible compared to their vacuum masses . as they can influence the resonance in this case . ] tree level contribution and one - loop corrections to the ( anti-)higgs decay amplitude . the additional arrows illustrate the direction of momentum flow . ] the _ cp_-violating decay of the higgs arises from the interference of the tree - level , self - energy and vertex graphs depicted in fig.[higgsdecayplot ] . although in this case the decaying particle  the higgs doublet  is very close to thermal equilibrium due to the yukawa and gauge interactions of the standard model , the majorana neutrino in the final state may deviate from equilibrium , so that the third sakharov condition is fulfilled .    using the expression for the divergence of the lepton - current , eq . , we can extract the corresponding _ cp_-violating parameter . to calculate the self - energy contribution it is convenient to rewrite the one - loop lepton self - energy in the form : @xmath482 where the transposition is only in dirac space and we have used one of the properties of majorana propagator : @xmath483 in its _ cp _ conjugate the yukawa couplings are replaced by their complex conjugates and the propagators by the _ cp _ conjugate ones . similar to the case of the majorana decay , substituting eq . into eq . we can define effective higgs decay amplitudes :    [ higgsdecayamplitides ] @xmath484{{\ , , } } \hphantom{aa}\\          \label{higgsdecayamplitidesbarln }          { \xi^{}_{{\phi}\leftrightarrow \bar { { \ell}}{n}_i}}&\equiv           g_w\sum_{mn } ( h^\dagger h)^*_{mn}\nonumber\\          & \times { { \rm tr}}\left[{\bar{\theta}^{mi}_{r}}({{q } } )          ( \slashed{{{q}}}+m_i){\bar{\theta}^{in}_{a}}({{q } } )          p_r\slashed{{{r}}}p_l\right]{{\ , .}}\end{aligned}\ ] ]    the overall factor @xmath485 in eqs . comes from summation over the doublet components of the ( decaying ) higgs particle . to leading order in the couplings :    [ higgsamplitudes ] @xmath486{{\ , , } } \\ \label{antihiggsamplitude }          { \xi^{}_{{\phi}\leftrightarrow \bar{{\ell}}{n}_i } } & \approx          2g_w\bigl[(h^\dagger h)_{ii}({{r}}\cdot { { q}})\nonumber\\          & - { \textstyle\frac{g_w}{16\pi } }          \im ( h^\dagger h)_{ij}^2 m_i m_j { \boldsymbol{\mathcal{s}}^{jj}_{h } } ( { { r}}\ , l_\rho)\bigr]{{\ , , } } \end{aligned}\ ] ]    which , up to the relative sign in square brackets , coincides with the amplitudes . the corresponding _ cp_-violating parameter reads @xmath487 where @xmath283 and @xmath385 are the momenta of the on - shell final lepton and majorana neutrino with positive zeroth components respectively . the direction of momentum flow is as defined in fig.[higgsdecayplot ] . although @xmath371 in eqs . and is one and the same function , because of the different kinematic regimes the explicit result in terms of the distribution functions differs for the higgs decay : @xmath488{{\ , , } } \end{aligned}\ ] ] see appendix[majoranaselfen ] . note that our result is different from the one presented in @xcite . instead of the @xmath489 dependence , it is proportional to a sum , @xmath490 , of the two distribution functions . this dependence can also be obtained in the framework of real time thermal field theory using causal @xmath349-point functions , compare @xcite . the derivation within the kadanoff - baym formalism gives certainty concerning the sign of the contribution by higgs decay . cp_-violating parameter has an opposite sign relative to that for majorana neutrino decay . however , it is canceled by the relative sign in eq ..    to calculate the vertex contribution we use eq . and represent the two - loop self - energy in the form @xmath491 \nonumber{{\ , .}}\end{aligned}\ ] ] its _ cp _ conjugate again differs by the conjugation of the yukawas and propagators . substituting eq . and its _ cp _ conjugate into eq . we obtain for the corresponding effective amplitudes :    [ higgsvertexamplitudes ] @xmath492\nonumber\\      & \hspace{3mm}+g_w(h^{\dagger}h)_{ji}^{2}\,m_i\ ,       { { \rm tr}}\bigr[c v_{jj}(-q ,- k)p_l\slashed{{{r } } } p_r\bigr]{{\ , , } } \\ \label{effamplbarhbarln }      & { \xi^{v}_{{\phi}\leftrightarrow \bar { { \ell}}n_i } }       \equiv g_w(h^{\dagger}h)_{ij}^2\ m_i      { { \rm tr}}\bigr[c v_{jj}(-q ,- k)p_l\slashed{{{r } } } p_r\bigr]\nonumber\\      & \hspace{3mm}+g_w(h^{\dagger}h)_{ji}^{2}\ , m_i { { \rm tr}}\bigl[\varlambda_{jj}(-q ,- k )      c p_l\slashed{{{r } } } p_r\bigr]{{\ , .}}\end{aligned}\ ] ]    just like for the self - energy contribution we observe that the overall sign of the vertex contribution to the higgs decay amplitude is opposite to that in the majorana decay , compare eqs . and . the corresponding _ cp_-violating parameter reads @xmath493\nonumber{{\ , .}}\end{aligned}\ ] ] the last three lines of eq . correspond to the three possible cuts of the vertex graph . similarly to the majorana decay only two of the intermediate states can be on - shell and for only one of them the corresponding distribution function enters the result . the value of the vertex _ cp_-violating parameter depends on the temperature as well as on masses of the majorana neutrinos . for definiteness , let us assume a strongly hierarchical mass spectrum , @xmath494 . in this case contribution of the last two lines in eq . is strongly suppressed . integrating out the delta - functions we find for the contribution of the first cut : @xmath495 where the loop function @xmath496 is now defined as : @xmath497      m_i^2 { \boldsymbol{\mathcal{s}}^{ii}_{h}}({{k}}-{{r}}_2 ) { { \ , .}}\end{aligned}\ ] ] note that the vertex _ cp_-violating parameter has an opposite sign relative to that for majorana neutrino decay . similarly to the self - energy contribution we observe that the @xmath498 combination is replaced in by @xmath499 . for a milder mass hierarchy the two other cuts can become important . their contributions are proportional to @xmath500 and @xmath501 respectively . the first - principle computation gives for the higgs decay contribution to the evolution of the lepton current an expression similar to eq . : @xmath502{{\ , .}}\end{aligned}\ ] ] we do not discuss ` extra ' terms here which would arise from ` naive ' boltzmann equations . to write this as a rate equation we need to repeat the steps in section[rateequations ] which lead to eq .. for a general process @xmath503 ( where we allow for deviations from equilibrium in @xmath504 ) we have eq .. therefore we obtain , for higgs decay , the following contributions to the rate equations for lepton number and majorana neutrino abundance ( see appendix[rateequationsapp ] ) :    @xmath505    we introduced the decay and washout reaction densities , @xmath506 and @xmath507 , for higgs decay : @xmath508 where now @xmath509 . in complete analogy to eq . the total amplitude and the _ cp_-violating parameter for ( anti-)higgs decay are defined as    [ amplsqandepsdefhiggs ] @xmath510    similarly to the majorana neutrino decays we also define an averaged _ cp_-violating parameter as @xmath511 by comparing eqs . and we observe that , ignoring @xmath512 contributions , the difference to the majorana neutrino decay contributions amounts to the replacements @xmath513 and @xmath514 . we therefore obtain eq . with an opposite sign for the _ cp_-violating source term . this sign cancels the relative sign of the _ cp_-violating parameter such that majorana neutrino decay and higgs decay contribute effectively with same sign . similarly , for the contribution to the majorana neutrino rate equation : @xmath515 to compute the thermally averaged _ cp_-violating parameter we take into account the temperature dependent evolution of the lepton and higgs masses @xcite . the averaged _ cp_-violating parameter in the higgs decay and in the majorana decay as functions of the inverse temperature are presented in fig.[epsilongammahiggsbm1 ] and fig.[epsilongammahiggsbm2 ] .     averaged self - energy _ cp_-violating parameters for majorana neutrino and higgs decay for benchmark point  1 as a function of the inverse temperature . thin lines represent the value in the zero temperature limit . with conventional dispersion relations , the decay @xmath516 is active at @xmath517 but replaced at high temperatures ( @xmath518 ) by @xmath519 . the decay @xmath520 is active at @xmath521 but replaced at high temperatures ( @xmath522 ) by @xmath523 . ]    at very high temperatures the magnitude of @xmath524 can be much larger for higgs decay . averaged self - energy _ cp_-violating parameters for majorana neutrino and higgs decay for benchmark point  2 . see caption of fig.[epsilongammahiggsbm1 ] . ]    as the temperature decreases the _ cp_-violating parameter for the higgs decay approaches zero . this is explained by the shrinking of the available phase space in the loop integrals and . in this paper we have studied leptogenesis in the type - i seesaw extension of the standard model using the 2pi - formalism of non - equilibrium quantum field theory . the asymmetry generation can , in the case of thermal leptogenesis , be approximately described by rate equations . usually these statistical equations are treated as a ` black - box ' in the sense that their form is assumed given and model specific amplitudes are inserted by hand . indeed , this approach is supported by the observation that it describes the free decay in the zero temperature limit correctly and inherent inconsistencies ( namely the ` double - counting problem ' ) can be resolved in exact equilibrium . however , out of equilibrium it is not obvious whether the subtraction of real intermediate states works to all orders and how amplitudes computed in thermal field theory enter kinetic equations . these issues can be completely avoided in a systematic treatment within non - equilibrium quantum field theory .    only in recent years fundamental questions related to the non - equilibrium statistical description received more attention . the progress here is mainly based on the 2pi - formalism which is known to yield consistent quantum kinetic equations without double - counting . these equations can be reduced to a system of boltzmann - like kinetic equations for quasiparticles which can easily be compared to the conventional results . in the course of the derivation necessary approximations and the related physical assumptions have to be specified explicitly . therefore , this approach enables a deeper insight into the dynamics of the asymmetry generation .    in the conventional analysis the minimal set of interactions is obtained at order @xmath117 of the perturbative expansion . we have complemented existing analyses based on the 2pi - formalism by the computation of further processes which appear at this order . starting from a system of kadanoff - baym and ( equivalent ) schwinger - dyson equations for leptons and heavy majorana neutrinos we have derived boltzmann - like quantum - kinetic equations for the lepton asymmetry . they include ( inverse ) decays of the heavy majorana neutrinos as well as two - body scattering processes mediated by the heavy neutrinos : @xmath525 because all terms in this equation are proportional to @xmath318 , a combination of the distribution functions which vanishes in equilibrium , the obtained equations are free of the double - counting problem and no need for the real intermediate state subtraction arises . together with the systematic derivation of the effective decay and scattering amplitudes @xmath49 this is the main result of the present work . the individual amplitudes arise as combinations of different 2pi contributions . the vertex contribution to the _ cp_-violating decay amplitude is obtained as cut of the 2pi ` mercedes ' diagram . the same graph yields also the @xmath526 contributions to @xmath228 and the @xmath527 contribution to @xmath87 process . to extract the self - energy contribution , the off - diagonal elements of the majorana neutrino propagator have to be taken into account . in addition , an extended quasiparticle approximation needs to be employed in order to obtain the @xmath436 and @xmath437 contributions to @xmath228 as well as @xmath528 and @xmath437 contributions to @xmath87 scattering from the ` setting - sun ' diagram .    in the zero temperature limit the effective amplitudes reduce to the canonical ones . in particular , the form of the resulting amplitudes for @xmath228 scattering coincides with the ris subtracted amplitudes encountered in existing calculations . at finite temperatures the effective amplitudes receive thermal corrections . medium corrections to the majorana decay amplitudes into leptons and antileptons are @xmath117 . they are small compared to the tree - level vacuum contribution and are therefore negligible for the total decay width . on the other hand , they play an important role for the _ cp_-violating source - terms , which are proportional to the difference of the two amplitudes . we find that medium corrections to the _ cp_-violating parameter are linear in the particle number densities . although there is a partial cancellation of the bosonic and fermionic contribution , the _ cp_-violating parameter is enhanced . in the effective scattering amplitudes the medium corrections affect only the ` regulator ' term in the denominator of the breit - wigner propagators . due to the smallness of the majorana decay width , which is constrained by the light neutrino masses , numerically these corrections are very small in the case of non - degenerate majorana neutrinos .    taking sm interactions into account in the form of thermal lepton and higgs masses results in a suppression of the phase space for the majorana neutrino decay and the enhancement of the _ cp_-violating parameters is overcompensated . at even higher temperatures , when the effective higgs mass exceeds the majorana masses , the _ cp_-violating decay of the higgs into a lepton - majorana pair can become kinematically allowed instead . at these temperatures the averaged _ cp_-violating parameters for higgs decay exceeds that obtained for majorana decay in vacuum by orders of magnitude . the signs of the corresponding _ cp_-violating parameters are opposite but their contribution to the lepton asymmetry has the same sign ( at least in the limit of hierarchical majorana masses ) . these results are in qualitative agreement with earlier studies based on thermal field theory and may ultimately be compared to a full treatment of _ cp_-violating decays out of equilibrium , including gauge interactions . we have also derived the corresponding rate equations for abundances of the participating species . they are obtained as expansion in small deviations from equilibrium ( @xmath145 and @xmath529 ) and represent the hydrodynamical approximation of the boltzmann kinetic equations . as compared to the standard ( zero temperature ) result they are improved in that the obtained coefficients include medium corrections to the quasiparticle properties and take into account quantum - statistical effects . we compare with the result obtained if the amplitudes are computed in thermal quantum field theory and the ris subtraction is performed manually . we find that there are differences at higher order in the expansion parameters . the coefficients  reaction densities  reflect the interplay between the medium enhancement of the effective amplitudes and the phase space suppression induced by the thermal masses of higgs and leptons . at very low temperatures the reaction densities approach their canonical limit .    since for a hierarchical mass spectrum most of the asymmetry is typically generated by the lightest majorana neutrino at temperatures of the order and smaller than its mass , we expect a moderate enhancement of the total generated asymmetry is possible for a typical average to strong washout scenario . for a detailed phenomenological analysis it is necessary to include further phenomena such as flavour effects and @xmath530 scattering processes which contribute to the washout at @xmath531 . additional quantum effects beyond the present analysis are relevant for non - standard scenarios in which the majorana neutrinos have degenerate masses or if they are not as close to thermal equilibrium . this work has been supported by the german science foundation ( dfg ) under grant ka-3274/1 - 1 `` systematic analysis of baryogenesis in non - equilibrium quantum field theory '' and the collaborative research center 676 `` particles , strings and the early universe '' . t.f . acknowledges support by the imprs - ptfs . we thank s.  blanchet and m.  shaposhnikov for useful discussions . the reaction densities contain distribution functions of the initial and final states , which depend on the corresponding energies . therefore to compute the reaction densities we need to analyze the kinematics of the decay and scattering processes .      to compute the _ cp_-violating reaction density and the decay reaction density we need to evaluate the integral : @xmath532{{\ , .}}\end{aligned}\ ] ] for the washout reaction densities or reaction densities for higgs decay we have similar expressions . the integration over @xmath533 can be performed trivially and yields @xmath534 . using latexmath:[$|\vec k|=(\vec q\,^2 + \vec p\,^2 -     @xmath59 we remove the remaining dirac - delta and obtain @xmath536 the integration limits are given by @xmath537{{\ , , } } \end{aligned}\ ] ] where @xmath538 , @xmath539 and @xmath540 is the usual kinematical function . for an on - shell heavy neutrino @xmath360 . if @xmath541 then @xmath542 and the above expression simplifies to @xmath543 . on the other hand , if @xmath544 then @xmath545 and therefore @xmath546 . since the integration limits coincide in this case , the integral vanishes .    combining it with the integration over @xmath547 and using the isotropy of the medium we find @xmath548      { { \ , .}}\end{aligned}\ ] ] if quantum - statistical effects are neglected then both the _ cp_-violating parameter and the total tree - level decay amplitude are momentum independent and the integration can be performed analytically . we get : @xmath549 where @xmath42 is the number of the majorana spin degrees of freedom . similar results are obtained for higgs decay and washout reaction densities .      for @xmath461 scattering processes the reaction density is defined by @xmath550 to reduce it to a form suitable for the numerical analysis we insert an identity : @xmath551 into eq .. the resulting expression can be interpreted as a product of the inverse decay and decay amplitudes integrated over the ` mass ' and energy of the intermediate state : @xmath552 for the third line we will use eq .. for the last line it is more convenient to us a different representation . integrating out the delta - function we obtain for the last line in eq . : @xmath553 note that not all angles are kinematically allowed , see eq . below . as a product of lorentz - invariant quantities the integral is also lorentz - invariant . we can therefore boost to the center - of - mass frame where @xmath554 and @xmath555 . by energy - momentum conservation @xmath556 , where @xmath557 now . the angle integration can be partially reduced to integration over the mandelstam variable @xmath558 using the relation @xmath559 . note that the azimuthal angle @xmath560 is lorentz - invariant by itself . therefore , boosting back to the rest frame of the medium we can write the left - hand side of eq . in the form @xmath561 integrating furthermore over @xmath562 and using the fact the the integrand is independent of the orientation of @xmath563 , we finally obtain @xmath564 where @xmath565 . just like for particle decay , the integration limits @xmath566 are given by eq . but with @xmath567 and @xmath568 replaced by @xmath569 and @xmath570 respectively and the three - momentum given by @xmath571 . the range of integration over @xmath54 is given by @xmath572{{\ , .}}\end{aligned}\ ] ] in particular , for massless initial and finial states it reduces to @xmath573 and @xmath574 .    if the quantum - statistical effects are neglected then @xmath575 . the integration over @xmath576 can be easily performed in this case and , combined with the @xmath577 prefactor , gives @xmath578 . in the same approximation the last line of does not depend on the distribution functions and gives so - called ` reduced cross section ' : @xmath579 using eq . and integrating over @xmath580 we recover the usual expression for the scattering reaction density : @xmath581 to take the quantum - statistical effects into account we need to express energies of initial and final states in terms of the integration variables . by energy conservation @xmath582 . therefore @xmath576 and @xmath583 as well as the related momenta @xmath584 and @xmath585 are completely fixed by the second and third integration variables . next we consider the final states . by energy conservation @xmath586 . it remains to express @xmath587 in terms of the integration variables . let us choose the coordinates such that @xmath588 points along the @xmath589-axis and @xmath590 lies in the @xmath591-plane . then the momentum transfer @xmath592 also lies in the same plane . its components are given by @xmath593 where : @xmath594 the components of @xmath595 can be written in the form @xmath596 . then the angle between the vectors @xmath595 and @xmath592 is given by @xmath597 using energy - momentum conservation we can express @xmath587 in terms of this angle and the integration variables : @xmath598{{\ , .}}\end{aligned}\ ] ] note that for @xmath599 the difference under the square root in eq . can become negative for some angles . this means that such scattering angles are forbidden kinematically and should not be integrated over .    since eq . implicitly depends on @xmath600 it is convenient to use this angle as an integration variable instead of @xmath54 . the integration measure in eq . is then modified according to @xmath601 .    to calculate the scattering amplitudes we need the three mandelstam variables . @xmath53 is an integration variable . @xmath54 is given by @xmath602 the remaining one , @xmath88 , can be inferred from the mandelstam relation @xmath603 . the generalized optical theorem is a consequence of the unitarity of the @xmath43-matrix and can be seen as a consistency condition for the amplitudes to ensure conservation of probability . it can also be seen as a consequence of the cutkosky cutting rules @xcite for the computation of the discontinuities of feynman diagrams . as such it can be applied to unstable particles at any given order of perturbation theory . we may write it as @xmath604=\nonumber\\      & = \sum_{i}\left ( \prod_{i_l } \int { { { \,d}\pi^{}_{i_l}}}\right)(2\pi)^4{\delta(\sum_j k_j -\sum_{i_l } q_{i_l})}\nonumber\\      & \times{{\mathcal{m}}_{{a}\rightarrow{i}}}(\{k_i\},\{q_{i_l}\}){{\mathcal{m}}_{{b}\rightarrow{i}}}^*(\{p_i\},\{q_{i_l}\}){{\ , .}}\label{eqn : generalized optical theorem}\end{aligned}\ ] ] the amplitudes @xmath605 include all contributing diagrams ( at a given order of perturbation theory ) and the sum on the right - hand side is over all possible real intermediate states @xmath382 which contribute to @xmath605 . the generalized optical theorem can be exploited to see explicitly why the ris subtraction works . scattering at @xmath606 . the cuts through the internal lepton and higgs lines yield the @xmath436 , @xmath526 and @xmath437-contributions to @xmath76 . the cuts through single internal majorana lines yield the interference terms which contribute to the _ cp_-violating parameter . [ fig : gneral diagram bb - bb ] ]    to this end , we apply it to the forward scattering processes @xmath607 and @xmath608 , see fig.[fig : gneral diagram bb - bb ] , above the energy thresholds @xmath609 ( the contribution by @xmath410 real intermediate states etc .  can be addressed analogously ) and @xmath610 . we include all possible graphs up to @xmath606 . furthermore , we sum eq . over all internal degrees of freedom of initial and final states and absorb these in the ` effective amplitudes ' defined in section[conventionalapproach ] . we get for the process involving particles @xmath611 the amplitudes squared on the right - hand side contain the relevant graphs including the vertex and self - energy contributions , see fig.[treevertexself ] , whose interference terms lead to _ cp_-violation in the particle decay . in the same way one finds for @xmath612 : @xmath613 as a consequence of _ cpt _ we have @xmath614 . therefore the difference of the left - hand sides of eqs . and as well as that of the third terms on the right - hand - sides vanish . subtracting the right - hand sides we obtain @xmath615=\nonumber\\      & -\int { { { \,d}\pi^{{n}_1}_{{q}}}}(2\pi)^4{\delta({k}+{p}-{q } ) }      \big [ { \xi^{}_{{{\ell}}{\phi}\rightarrow{n}_1 } } -       { \xi^{}_{\bar{{{\ell}}}\bar{{\phi}}\rightarrow{n}_1}}\big ] ,      \label{eqn : generalized optical theorem applied at order four difference}\end{aligned}\ ] ] as a requirement for a consistent approximation of the amplitudes compatible with unitarity and _ cpt_. it is obvious that this can not be satisfied if the scattering amplitudes are in tree - level approximation meanwhile the decay amplitudes violate _ cp_. we show in section[risquantstat ] how eq . can be satisfied by replacing the two - body scattering amplitudes @xmath616 . the solution for @xmath617 amounts to subtracting the real intermediate state contributions from @xmath8 . in order to obtain an equivalent result for the higgs decay at high temperature ( i.e.  for @xmath618 ) we need to consider different processes since the amplitude @xmath619 can not be obtained as cut of the graphs in fig.[fig : gneral diagram bb - bb ] . one could try to draw and to cut graphs for @xmath620 scattering , but the obtained cuts are lepton number conserving and would drop out in the difference @xmath621 . instead the relevant contributions appear as ` thermal cuts ' of the @xmath54-channel contributions to @xmath622 depicted in fig.[fig : diagram bb - bb contribution ] . scattering at @xmath606 . due to medium effects the particles in the loop can be on - shell.[fig : diagram bb - bb contribution ] ]    since they exist only at finite density we need to use finite temperature ` circling rules ' , which can be seen as a generalization of eq . , to compute the imaginary part of causal @xmath349-point functions in the real time formalism @xcite : @xmath623 where ` not all ' means that not all @xmath624 should be circled at the same time . it was shown in @xcite that causal @xmath349-point functions are the ones relevant for the computation of _ cp_-violating parameters . the vertex @xmath625 with largest or smallest time is always circled . we can use this equation together with the circling rules given in @xcite to compute the imaginary parts of the graphs in fig.[fig : diagram bb - bb contribution ] and the _ cpt_-conjugated process @xmath626 . taking the difference of both we obtain , similar to eq . : @xmath627=\nonumber\\      & -\int { { { \,d}\pi^{{\phi}}_{{q}}}}(2\pi)^4{\delta({k}+{p}-{q})}\big [ { \xi^{}_{{{\ell}}{n}_1\rightarrow\bar{{\phi } } } }       - { \xi^{}_{\bar{{{\ell}}{n}_1}\rightarrow{\phi } } }      \big]{{\ , .}}\end{aligned}\ ] ] we can also use eq . to compute the thermal widths which cutoff the @xmath53- and @xmath54-channel resonances by cutting the self - energy graphs as shown in fig.[rate through thermal cutting ] .     the thermal width can be obtained using ( causal ) finite temperature cutting rules . ]    furthermore , the thermal _ cp_-violating parameters can be obtained using thermal cutting rules , see e.g.  fig.[fig : selfenergy loop corrections imaginary part causal products ]     and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ] +   and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ]   and @xmath628 can not be on - shell simultaneously for @xmath629 . contributions by graphs ( c - e ) are suppressed if the cut is through an internal majorana neutrino line.[fig : selfenergy loop corrections imaginary part causal products],title="fig : " ]    and @xcite . altogether , the concept of ris subtraction can be generalized to include quantum - statistical effects using thermal quantum field theory in the real time formalism and a complete set of reaction densities can be computed . note however that inconsistencies are inherent out of equilibrium and arise e.g.  at higher order in the expansion performed in section[rateequations ] . similar computations where performed in @xcite in the imaginary time formalism of thermal quantum field theory . in this appendix we present some detailed intermediate steps in the derivation of rate equations with quantum - statistical terms . to obtain these we need to assume that the system is close to thermal equilibrium . under certain conditions it is then possible to reduce boltzmann - like equations to a set of rate equations for the systems evolution . assuming that the detailed conditions given in section[rateequations ] are fulfilled and using eq . , we get for the integrated boltzmann equations with quantum - statistical terms :    @xmath630 \\ =           \int { { { \,d}\pi_{{n}_i{{\ell}}{\phi}}^{{{q}}{{r}}{{k } } } } }           ( 2\pi)^4\,{\delta({{q}}-{{r}}-{{k } } ) } \big\ { \pm & { \xi^{}_{{n}_i\rightleftarrows{{\ell}}{\phi } } }          ( 1-{f^{}_{{{\ell}}}})(1+{f^{}_{{\phi}}})\big [ \frac{{f^{}_{{n}_i } }          -{{f}_{{n}_i}^{eq}}}{{(1-{f^{}_{{n}_i}})}{{f}_{{n}_i}^{eq } } }          - ( e^{+\frac{\mu_{{{\ell}}}+\mu_{{\phi}}}{t}}-1)\big]\nonumber\\          - & { \xi^{}_{{n}_i\rightleftarrows\bar{{\ell}}\bar{\phi } } }          ( 1-{f^{}_{\bar{{{\ell}}}}})(1+{f^{}_{\bar{{\phi } } } } )          \big [ \frac{{f^{}_{{n}_i } }          -{{f}_{{n}_i}^{eq}}}{{(1-{f^{}_{{n}_i}})}{{f}_{{n}_i}^{eq } } }           - ( e^{-\frac{\mu_{{{\ell}}}+\mu_{{\phi}}}{t}}-1)\big]\big\ }           { ( 1-{f^{}_{{n}_i } } ) }          \frac{{{f}_{{n}_i}^{eq}}}{{(1-{{f}_{{n}_i}^{eq}})}}{{\ , .}}\nonumber      \end{aligned}\ ] ]    this expression is still exact with respect to deviations from equilibrium and , taking into account results before eq . , it has an obvious expansion in @xmath145 . we can see that there will be contributions proportional to @xmath631 , to @xmath145 and proportional to @xmath632 at linear order . the coefficients of these terms were introduced in section[rateequations ] ( eqs . and ) and dubbed reaction densities . we find for the @xmath633 expansion of each of the two terms in eq . , i.e.  for the general collision term of @xmath634 in eq . : @xmath635 where we defined the decay reaction densities as @xmath636 with @xmath637 . for a general process @xmath503 ( where we allow again for deviations from equilibrium in @xmath504 ) we find in the same way : @xmath638 where @xmath639 with , now , @xmath640 . note that these reaction densities will tend to zero in the zero temperature limit because of their dependence on the distribution functions . using the general result we obtain the contributions to eq . , and therefore to eq . , proportional to @xmath529 , @xmath641 contributions proportional to @xmath186 , @xmath642 or just proportional to @xmath187 : @xmath643 this leads immediately to the result in eq .. similarly , we get using eq . for the contributions by ( anti-)higgs decay to the rate equations : @xmath644 in addition we get for the extra terms in eqs . and :    @xmath645    this means that for @xmath646 only the first term and for @xmath133 the last two terms contribute . the terms proportional to the difference of @xmath647 and @xmath648 lead to contributions proportional to the _ cp_-violating parameter .    to reduce the two - body scattering terms in eq . we can use the same methods to find for the general expression , @xmath649{{\ , , } } \nonumber\end{aligned}\ ] ] the @xmath633 expansion : @xmath650\nonumber\\      = & \frac{\mu_c + \mu_d \pm ( \mu_a + \mu_b)}{t}\langle\gamma^{ab}_{ij}\rangle + ( 1\pm 1)\langle\gamma^{ab}_{ij}\rangle\nonumber\\      - & ( 1\pm 1)\langle ( \xi_a\frac{\mu_a}{t}{{f}_{a}^{eq}}+\xi_b\frac{\mu_b}{t}{{f}_{b}^{eq}}-\xi_i\frac{\mu_i}{t}{{f}_{i}^{eq}}-\xi_j\frac{\mu_j}{t}{{f}_{j}^{eq}})\gamma^{ab}_{ij}\rangle{{\ , , } } \nonumber\end{aligned}\ ] ] where the two - body scattering reaction density was introduced in eq .. the last expression results after first order expansion in @xmath145 . in this appendix we derive one - loop contribution to the self - energy of the majorana field as well as one and two - loop contributions to the self - energy of leptons . the 2pi effective action is defined as a functional of the one- and two - point functions consisting of an infinite sum of all 2pi vacuum diagrams @xcite . in practice , its expansion can be characterized in terms of the number of loops appearing in each diagram : @xmath651=\sum_{n } i\gamma_{2{\ensuremath{\,\mathrm{pi}}}}^{(n)}[{s^{}_{}},{\mathscr{s}^{}_{}},{\delta^{}_{}}]{{\ , .}}\end{aligned}\ ] ] the two lowest order contributions , @xmath652 and @xmath653 , relevant for leptogenesis are shown in fig.[fig:2pi contributions ] . their contributions to the 2pi action read    [ 2picontrib ] @xmath654{{\ , , } } \hphantom{aa}\\      \label{g3 }      i\gamma_{2{\ensuremath{\,\mathrm{pi}}}}^{(3)}= & \frac{1}{2}\int_{\cal c } d^4u\ d^4w\ d^4\eta\ d^4\xi \ { { \rm tr}}\big[h p_r      { \mathscr{s}^{}_{}}(u , w)\nonumber\\      & \times cp_r h^t { s^{t}_{}}(\eta , w)\epsilon { \delta^{*}_{}}(w,\xi)\epsilon h^ * p_lc \nonumber\\      & \times { \mathscr{s}^{}_{}}(\eta,\xi)p_l h^{\dagger } { s^{}_{}}(\xi , u)\ \epsilon { \delta^{*}_{}}(u,\eta ) \epsilon \big]{{\ , .}}\end{aligned}\ ] ]    in eq . the trace is taken over flavor , dirac and @xmath21 indices whereas the transposition only acts in flavor and dirac space .      by functional differentiation of the 2pi effective action with respect to the two - point function we obtain the corresponding self - energy which enters the schwinger - dyson equation : @xmath655}{\delta { s^{\ , t}_{\beta \alpha}}(y , x)}{{\ , .}}\end{aligned}\ ] ] here , flavor indices are shown explicitly whereas the @xmath21 and dirac structure is embodied implicitly in matrix notation . the resulting self - energy is given by a combination of the majorana , lepton and higgs propagators . since we do not consider the flavor effects and the early universe was in an @xmath21-symmetric state :    @xmath656    since @xmath657 the lepton self - energy also becomes diagonal : @xmath658 . furthermore , in the unflavored approximation it is convenient to sum over lepton flavors : @xmath659 . then the one- and two - loop order contributions to the lepton self - energy , see fig.[fig:2pi contributions ] , take the form :    @xmath660    eventually , it is the wightman components that we are interested in since they enter the gain- and loss terms on the right - hand side of eq .. therefore , we insert the usual decomposition of the propagators @xmath661 into the spectral and statistical parts , eq . , into the self - energies and . a formal decomposition of the self - energy in analogy to allows us to identify its spectral and statistical part and define the wightman components in coordinate space as @xmath662 . for the one - loop self - energy they read @xmath663 in the case of the two - loop contribution , eq . , the computation becomes slightly elaborate . the complication is due to the appearance of 32 different terms after inserting the decomposition for each of the five propagators into eq . as well as due to the two remaining integrations over the internal space - time arguments @xmath664 and @xmath665 . the decomposition makes the path - ordering explicit and allows us to convert the integration along the ctp into an integration along the positive branch . the 32 terms contain different combinations of the @xmath666-functions . these can be rewritten by using relations given in appendix c of @xcite . after some simple but lengthy algebra we obtain for the wightman components :    @xmath667{{\ , .}}\end{aligned}\ ] ]    specific approximations will allow us to interpret both expressions , eq . and , as describing decay , inverse decay and scattering processes of quasiparticles in the medium .    the expressions for the one- and two - loop self - energies given by eqs . and depend explicitly on two coordinates in four dimensional space - time . however , the self - energies which govern the gain- and loss term on the right - hand side of eq . are expressed in terms of phase space coordinates . let us therefore exchange the pair of space - time arguments @xmath668 for an equivalent set of center and relative coordinates , @xmath669 and @xmath670 . in contrast to thermal equilibrium , the out of equilibrium propagators depend not only on the relative coordinate @xmath53 but also on the center coordinate @xmath183 . performing a so - called wigner transformation @xcite , i.e. a fourier transformation with respect to the relative coordinate @xmath53 , we can trade the latter for a momentum space variable :    [ wt ] @xmath671    where we have used @xmath672 and @xmath673 , @xmath674 according to the various combinations appearing in eqs . and . note that the factor @xmath675 in the definition is conventional and makes the wigner transform of the spectral propagator a hermitian matrix . definitions of the wigner transforms of the advanced and retarded propagators coincide with that for the statistical propagator . the wigner transform of eq . is obtained straightforwardly : @xmath676 note that motivated by the homogeneity and isotropy of the early universe we only indicate time - dependence of the propagators and self - energy , @xmath677 . to obtain the wigner transform of eq . we will use an additional approximation : each of the wigner transforms of the propagators we replace by @xmath678 . this means that we neglect the variations of @xmath679 from the center coordinate @xmath680 at which the self - energy is evaluated . technically , it corresponds to a gradient expansion to lowest order and therefore disregards all memory effects . this can be compared to the ` stozahlansatz ' within the usual approach to the boltzmann equation . it is convenient to represent the resulting expression as a sum of three terms : @xmath681 the first term on the right - hand side of eq . corresponds to the wigner transform of lines one to three and four to six in eq . : @xmath682\nonumber { { \ , .}}\end{aligned}\ ] ] where we have introduced two functions containing loop corrections : @xmath683{{\ , , } } \end{aligned}\ ] ] and @xmath684 . as we will see , describes _ cp_-violating decay of the heavy majorana neutrino . the second term on the right - hand side of eq . is given by the wigner transform of the seventh line of eq . and describes lepton number violating scattering processes : @xmath685{{\ , .}}\end{aligned}\ ] ] finally the last term in eq . corresponds to the last line of eq . , @xmath686{{\ , , } } \end{aligned}\ ] ] and can be identified with lepton number conserving processes which do not contribute to generation of the lepton asymmetry . differentiating eq . with respect to the two - point function of the majorana neutrino and using definitions of the _ cp_-conjugate two - point functions we obtain for the majorana self - energy @xmath687{{\ , , } } \nonumber\end{aligned}\ ] ] where we have assumed the @xmath21 symmetry of the medium and neglected flavor effects . the factor @xmath22 in eq . comes from the summation over the @xmath21 indices . conjugate self - energy differs from eq . only in the propagators replaced by their _ conjugate counterparts and the couplings replaced by their complex conjugates .    from eq . we can read off the wightman components of the self - energy : @xmath688{{\ , .}}\end{aligned}\ ] ] its _ cp_-conjugate can be obtained by complex conjugating the couplings and replacing the two - point functions by their _ cp_-conjugates . to calculate amplitudes of the scattering processes we will need its wigner transform : @xmath689 { { \ , .}}\hphantom{aaaa}\end{aligned}\ ] ] from eq . we can deduce the wigner transform of the corresponding spectral self - energy : @xmath690{{\ , , } } \end{aligned}\ ] ] where we have introduced @xmath691{{\ , , } } \\      { \pi^{}_{\rho}}(t,{{q } } ) & \equiv       16\pi{\textstyle\int}{{\,d}\pi^4_{{{k } } } } { { \,d}\pi^4_{{{r } } } } ( 2\pi)^4{\delta({{q}}-{{k}}-{{r}})}\nonumber\\      & \times p\bigl[{\bar{\delta}^{}_{f}}(t,{{k}}){\bar{s}^{}_{\rho}}(t,{{r } } )      + { \bar{\delta}^{}_{\rho}}(t,{{k } } ) { \bar{s}^{}_{f}}(t,{{r}})\bigr]p{{\ , , } } \end{aligned}\ ] ] and @xmath692 . in the quasiparticle approximation the wigner transforms of the two - point functions of leptons and the higgs are given by eqs. and eqs. respectively . in a _ cp_-symmetric medium , which the early universe was to a very good approximation , @xmath693 and @xmath694 . the homogeneity and isotropy of the early universe furthermore imply , that there is no dependence on the momentum direction and the spatial central coordinate so that @xmath695 and @xmath696 . just like for scalars , in a _ cp_-symmetric medium the fermion two - point functions are related by @xmath697 and @xmath698 . as for the @xmath699 transformation , the terms in the lepton propagators which carry spinor structure are not invariant under it : @xmath700 since @xmath701 , eq . implies that in a homogeneous , isotropic and _ cp_-symmetric medium @xmath702 and @xmath703 are left and right projections of the same ` vector ' integral @xmath704 : @xmath705 l_{\rho}{{\ , .}}\end{aligned}\ ] ] explicit form of @xmath371 depends on the kinematic regime and is presented below .    to evaluate the decay amplitudes as well as amplitudes of the @xmath53-channel scattering processes we need to evaluate it for positive @xmath360 and @xmath706 . if @xmath707 is also positive then @xmath371 takes the form : @xmath708{{\ , .}}\end{aligned}\ ] ] to obtain eq . we have used the quasiparticle approximation for the two - point functions and integrated over the zeroth components of the momenta . integrating out the energy - momentum conserving delta - function we obtain for its lorentz components :    [ sigmavtherm ] @xmath709{{\ , , } } \end{aligned}\ ] ] where @xmath710 and @xmath711 . the integral functions @xmath712 are defined by @xmath713 where , in complete analogy with eq . , the integration limits are given by @xmath714{{\ , .}}\end{aligned}\ ] ] for positive @xmath369 and negative @xmath715 the components of @xmath371 are related to the ones above by @xmath716 and @xmath717 respectively . to evaluate amplitudes of the @xmath54- and @xmath88-channel processes we also need to calculate @xmath371 for negative square of the momentum transfer . in this case momentum - energy conservation ensures that @xmath718 and @xmath719 can not be positive or negative simultaneously . if they have different signs then , assuming homogeneity and isotropy of the medium and using relations , we find @xmath720\nonumber\\      + & { \delta({{q}}-{{k}}+{{r}})}[f_{\phi}(t,{{k } } )      + f_{\bar{{\ell}}}(t,{{r}})]\bigr\}{{\ , .}}\end{aligned}\ ] ] eq . implies that for negative square of the momentum transfer @xmath371 vanishes in vacuum . although it is in principle possible to retain the thermal masses of leptons and the higgs in the calculation , the resulting expressions are quite lengthy in this case . neglecting the thermal masses we obtain for the lorentz components of @xmath371 in this regime :    [ lrhonegmom ] @xmath721{{\ , , } } \end{aligned}\ ] ]    where the integral functions are given by @xmath722 note that in this regime @xmath723 and therefore the lower integration limit is positive .    to compute the scattering amplitude we need to calculate the product @xmath724 for the @xmath53-channel we find : @xmath725 whereas the corresponding expression for the @xmath54- and @xmath88-channels reads @xmath726 at low temperatures eq . is exponentially small and vanishes in the vacuum limit .    to analyze the higgs decay we need to evaluate the spectral loop integral in a region of the phase space where the effective higgs mass exceeds the sum of the majorana and lepton masses . using properties of the distribution functions under the @xmath727 transformation we find after some algebra from eq . : @xmath728.\end{aligned}\ ] ] just like in eq . , the integration is over the ( on - shell ) momenta of the higgs and lepton and the majorana momentum serves as a constraint . note that in this case we are interested only in the on - shell majorana momenta and therefore @xmath360 . after integrating out the delta - function we obtain a result similar to eq . : @xmath729{{\ , , } } \end{aligned}\ ] ] where the integral function is defined as : @xmath730 the integration limits are given by an expression similar to eq . but with @xmath731 replaced by @xmath732 . when the effective higgs mass approaches the kinematic limit , @xmath733 the upper integration limit approaches the lower one , and the integral vanishes to perform the quantitative analysis we need to specify the yukawa couplings . for simplicity we focus on the case of a very heavy third majorana neutrino @xmath734 ( msm ) . in this limit the yukawa couplings can be expressed in terms of the observed active neutrino masses and mixing angles and only one complex additional free parameter @xmath735 . where @xmath740 gev is the higgs vev and we have assumed normal hierarchy . in this case the physical neutrino masses are given by @xmath741 for illustration we choose the benchmark points @xmath742 and @xmath743 denoted by bm1 and bm2 in the plots . as masses of the right - handed neutrinos we choose @xmath744 gev and @xmath745 respectively . this choice of parameters is such that effects related to the resonant enhancement will be unimportant but contributions from both heavy majorana neutrinos can be relevant . note however that there are lower bounds on the washout parameters @xmath746 ( with ` equilibrium neutrino mass ' @xmath747 ) , see figs.[m1tildevsreomegaimomega ] and [ m2tildevsreomegaimomega ] . the freeze - out of the asymmetry will therefore typically occur late ( i.e.  @xmath752 ) which renders medium effects in general small . however for the qualitative issues discussed in this paper our preference is to specify a consistent set of parameters for which we can discuss the generation of the lepton asymmetry in terms of two heavy majorana neutrinos . this is of course not a general restriction for the employed techniques .
in this work we study thermal leptogenesis using non - equilibrium quantum field theory . starting from fundamental equations for correlators of the quantum fields we describe the steps necessary to obtain quantum kinetic equations for quasiparticles . 
 these can easily be compared to conventional results and overcome conceptional problems inherent in the canonical approach . beyond _ 
 cp_-violating decays we include also those scattering processes which are tightly related to the decays in a consistent approximation of fourth order in the yukawa couplings . 
 it is demonstrated explicitly how the s - matrix elements for the scattering processes in the conventional approach are related to two- and three - loop contributions to the effective action . 
 we derive effective decay and scattering amplitudes taking medium corrections and thermal masses into account . in this context 
 we also investigate _ 
 cp_-violating higgs decay within the same formalism . from the kinetic equations we derive rate equations for the lepton asymmetry improved in that they include quantum - statistical effects and medium corrections to the quasiparticle properties .