# Flavour Covariant Transport Equations:

an Application to Resonant Leptogenesis

###### Abstract

We present a fully flavour-covariant formalism for transport phenomena, by deriving Markovian master equations that describe the time-evolution of particle number densities in a statistical ensemble with arbitrary flavour content. As an application of this general formalism, we study flavour effects in a scenario of resonant leptogenesis (RL) and obtain the flavour-covariant evolution equations for heavy-neutrino and lepton number densities. This provides a complete and unified description of RL, capturing three distinct physical phenomena: (i) the resonant mixing between the heavy-neutrino states, (ii) coherent oscillations between different heavy-neutrino flavours, and (iii) quantum decoherence effects in the charged-lepton sector. To illustrate the importance of this formalism, we numerically solve the flavour-covariant rate equations for a minimal RL model and show that the total lepton asymmetry can be enhanced by up to one order of magnitude, as compared to that obtained from flavour-diagonal or partially flavour off-diagonal rate equations. Thus, the viable RL model parameter space is enlarged, thereby enhancing further the prospects of probing a common origin of neutrino masses and the baryon asymmetry in the Universe at the LHC, as well as in low-energy experiments searching for lepton flavour and number violation. The key new ingredients in our flavour-covariant formalism are rank-4 rate tensors, which are required for the consistency of our flavour-mixing treatment, as shown by an explicit calculation of the relevant transition amplitudes by generalizing the optical theorem. We also provide a geometric and physical interpretation of the heavy-neutrino degeneracy limits in the minimal RL scenario. Finally, we comment on the consistency of various suggested forms for the heavy-neutrino self-energy regulator in the lepton-number conserving limit.

###### keywords:

Flavour Covariance, Discrete Symmetries, Transport Equations, Resonant Leptogenesis^{†}

^{†}journal: Nuclear Physics B

MAN/HEP/2014/01, IPPP/14/20, DCPT/14/40

April 2014

###### Contents

- 1 Introduction
- 2 Flavour Diagonal Boltzmann Equations
- 3 Flavour Covariant Transport Equations
- 4 Application to Resonant Leptogenesis
- 5 Minimal Resonant -Genesis Model
- 6 Numerical Examples
- 7 Conclusions
- A Asymmetry in the -conserving Limit
- B Flavour Covariant Helicity Amplitude Formalism
- C Generalized Optical Theorem
- D Form Factors for LFV Decay Rates

## 1 Introduction

The observed matter-antimatter asymmetry in the Universe and the observation of non-zero neutrino masses and mixing (for a review, see pdg ()) provide two of the strongest pieces of experimental evidence for physics beyond the Standard Model (SM). Leptogenesis Fukugita:1986hr () is an elegant framework that satisfies the basic Sakharov conditions Sakharov:1967 (), dynamically generating the observed matter-antimatter asymmetry. According to the standard paradigm of leptogenesis (for reviews, see e.g. Pilaftsis:1998pd (); Buchmuller:2005eh (); Davidson:2008bu (); Blanchet:2012bk ()), there exist heavy Majorana neutrinos in minimal extensions of the SM, whose out-of-equilibrium decays in an expanding Universe create a net excess of lepton number (), which is reprocessed into the observed baryon number () through the equilibrated -violating electroweak sphaleron interactions Kuzmin:1985mm (). In addition, these heavy SM-singlet Majorana neutrinos (with ) could explain the observed smallness of the light neutrino masses by the seesaw mechanism seesaw1 (); seesaw2 (); seesaw4 (); seesaw5 (); seesaw6 (). Hence, leptogenesis can be regarded as a cosmological consequence of the seesaw mechanism, thus providing an attractive link between two seemingly disparate pieces of evidence for new physics at or above the electroweak scale.

In the original scenario of thermal leptogenesis Fukugita:1986hr (), the heavy Majorana neutrino masses are typically close to the Grand Unified Theory (GUT) scale, GeV, as suggested by natural GUT embedding of the seesaw mechanism seesaw2 (); seesaw4 (); seesaw5 (). In a ‘vanilla’ leptogenesis scenario Buchmuller:2004nz (), where the heavy neutrino masses are hierarchical (), the solar and atmospheric neutrino oscillation data impose a lower limit on GeV Davidson:2002qv (); Buchmuller:2002rq (); Hambye:2003rt (); Branco:2006ce (). As a consequence, such leptogenesis models are difficult to test in foreseeable laboratory experiments. Moreover, these high-scale thermal leptogenesis scenarios, when embedded within supergravity models of inflation, could potentially lead to a conflict with the upper bound on the reheating temperature of the Universe, – GeV, required to avoid overproduction of gravitinos whose late decays may otherwise spoil the success of Big Bang Nucleosynthesis Khlopov:1984pf (); Ellis:1984eq (); Ellis:1984er (); Kawasaki:1994af (); Cyburt:2002uv (); Kawasaki:2004qu (); Kawasaki:2008qe (). In general, it is difficult to build a testable low-scale model of leptogenesis, with a hierarchical heavy neutrino mass spectrum Pilaftsis:1998pd (); Hambye:2001eu ().

A potentially interesting solution to the aforementioned problems may be obtained within the framework of resonant leptogenesis (RL) Pilaftsis:1997dr (); Pilaftsis:1997jf (); Pilaftsis:2003gt (). The key aspect of RL is that the heavy Majorana neutrino self-energy effects Liu:1993tg () on the leptonic -asymmetry become dominant Flanz:1994yx (); Covi:1996wh () and get resonantly enhanced, even up to order one Pilaftsis:1997dr (); Pilaftsis:1997jf (), when at least two of the heavy neutrinos have a small mass difference comparable to their decay widths. As a consequence of thermal RL, the heavy Majorana neutrino mass scale can be as low as the electroweak scale Pilaftsis:2005rv (), while maintaining complete agreement with the neutrino oscillation data pdg ().

A crucial model-building aspect of RL is the quasi-degeneracy of the heavy neutrino mass spectrum, which could be obtained as a natural consequence of the approximate breaking of some symmetry in the leptonic sector. In minimal extensions of the SM, there is no theoretically or phenomenologically compelling reason that prevents the singlet neutrino sector from possessing such a symmetry and, in fact, in realistic ultraviolet-complete extensions of the SM, such a symmetry can often be realized naturally. For instance, the RL model discussed in Pilaftsis:1997dr (); Pilaftsis:1997jf () was based on a lepton symmetry in the heavy neutrino sector, motivated by superstring-inspired GUTs Mohapatra:1986aw (); Nandi:1985uh (); Mohapatra:1986bd (). The small mass splitting between the heavy neutrinos was generated by approximate breaking of this lepton symmetry via GUT- and/or Planck-scale-suppressed higher-dimensional operators. The RL model discussed in Pilaftsis:2003gt () was based on the Froggatt-Nielsen (FN) mechanism Froggatt:1978nt () in which two of the heavy Majorana neutrinos, having opposite charges under , naturally had a mass difference comparable to their decay widths. There is a vast literature on other viable constructions of RL models, e.g. within minimal extensions of the SM Xing:2006ms (); Hambye:2006zn (); Blanchet:2009bu (); Iso:2010mv (); Okada:2012fs (); Haba:2013pca (), with approximate flavour symmetries Ellis:2002eh (); Araki:2005ec (); Cirigliano:2006nu (); Chun:2007vh (); Babu:2007zm (); Branco:2009by (), with variations of the minimal type-I seesaw Ma:1998dx (); Albright:2003xb (); Hambye:2003rt (); Hambye:2000ui (); Asaka:2008bj (); Blanchet:2009kk (), within GUTs Akhmedov:2003dg (); Albright:2004ws (); Majee:2007uv (); Blanchet:2010kw (), within the context of supersymmetric theories Dar:2003cr (); Allahverdi:2004ix (); Hambye:2004jf (); West:2004me (); West:2006fs (), and in extra-dimensional theories Pilaftsis:1999jk (); Gherghetta:2007au (); Eisele:2007ws (); Gu:2010ye (); Bechinger:2009qk (). There also exist other variants of the RL scenario, such as radiative RL Felipe:2003fi (); Turzynski:2004xy (); Branco:2005ye (); Branco:2006hz () and soft RL Grossman:2003jv (); D'Ambrosio:2003wy ().

In another important variant of RL, a single lepton-flavour asymmetry is resonantly produced by out-of-equilibrium decays of heavy Majorana neutrinos of a particular family type Pilaftsis:2004xx (); Deppisch:2010fr (). This mechanism uses the fact that the sphaleron processes preserve, in addition to , the individual quantum numbers Khlebnikov:1988sr (); Harvey:1990qw (); Dreiner:1992vm (); Cline:1993vv (); Laine:1999wv (), where is the SM family index and is the lepton asymmetry in the th family. Therefore, it is important to estimate the net baryon number created by sphalerons just before they freeze out. In particular, a generated baryon asymmetry can be protected from potentially large washout effects due to sphalerons if an individual lepton flavour is out of equilibrium. We refer to such scenarios of RL as resonant -genesis (RL). In this case, the heavy Majorana neutrinos could be as light as the electroweak scale Pilaftsis:2005rv () and still have sizable couplings to other charged-lepton flavours . This enables the modelling of minimal RL scenarios Deppisch:2010fr () with electroweak-scale heavy Majorana neutrinos that could be tested at the LHC Dev:2013wba (), while being consistent with the indirect constraints from various low-energy experiments at the intensity frontier deGouvea:2013zba ().

Flavour effects play an important role in determining the final
lepton asymmetry in RL models. There are two kinds of flavour effects,
which are usually ignored in vanilla leptogenesis scenarios,
namely: (i) heavy neutrino flavour effects, assuming that the final
asymmetry is produced dominantly by the out-of-equilibrium decay of
only one (usually the lightest) heavy neutrino, with negligible
contributions from heavier species; and (ii) charged-lepton
flavour effects, assuming that the flavour composition of the lepton
quantum states produced by (or producing) the heavy neutrinos can be
neglected and all leptons can be treated as having the same flavour.
Neglecting (i) can be justified in
‘vanilla’ scenarios, because the asymmetries due to the heavier
Majorana neutrinos are usually suppressed in the hierarchical limit . Moreover, even if a sizable asymmetry is produced by
these effects, it is washed out by the processes involving the
lightest heavy neutrino Buchmuller:2004nz ().^{1}^{1}1There is an
exception to this case depending on the flavour structure of the
neutrino Yukawa couplings, when the contribution from the
next-to-lightest heavy neutrino decay could be
dominant DiBari:2005st (); Vives:2005ra (). However, for
quasi-degenerate heavy neutrinos, as in the RL case, the flavour effects
due to the neutrino Yukawa couplings do play an important
role Pilaftsis:2004xx (); Endoh:2003mz (). In fact, a sizable lepton
asymmetry can be generated through -violating
oscillations of sterile neutrinos Akhmedov:1998qx (); Asaka:2005 (); Shaposhnikov:2008pf (); Canetti:2012kh (); Drewes:2012ma (), which is then communicated to the SM lepton sector
through their Yukawa couplings.

On the other hand, the lepton flavour effects, as identified in (ii) above,
are related to the
interactions mediated by charged-lepton Yukawa
couplings Barbieri:1999ma (). Depending on whether these
interactions are in or out of thermal equilibrium at the leptogenesis
scale, the predicted value for the baryon asymmetry could get
significantly modified, as already shown by various partially
flavour-dependent treatments Abada:2006ea (); Abada:2006fw (); Nardi:2006fx (); Blanchet:2006be (); De Simone:2006dd (); Pascoli:2006ie ().^{2}^{2}2Similar
partial flavour effects have also been considered for other variants
of leptogenesis models, e.g. with type-II seesaw Abada:2008gs (); Felipe:2013kk (); Sierra:2014tqa () and soft leptogenesis Fong:2008mu (). The
lepton flavour effects can be neglected only when the heavy neutrino mass
scale GeV, in which case all the
charged-lepton Yukawa interactions are out-of-equilibrium and the
quantum states of all charged-lepton flavours evolve coherently,
i.e. effectively as a single lepton flavour, between their production
from and subsequent inverse decay . Here,
is the
lepton doublet (with flavour index ) and is the SM Higgs doublet. For GeV, the -lepton Yukawa interactions
are in thermal equilibrium, and hence, the lepton quantum states are
an incoherent mixture of -lepton and a coherent superposition of
electron and muon. Finally, for GeV, since the
muon and electron Yukawa interactions are also in equilibrium,
their impact on the final lepton asymmetry must be taken into
account in low-scale RL models. Note that flavour effects also play an
important role in the collision terms describing
scatterings that involve Yukawa and gauge interactions, as well as
and scatterings mediated by heavy
neutrinos Pilaftsis:2005rv ().

Therefore, a flavour-covariant formalism is required, in order to consistently capture all the flavour effects, including flavour mixing, oscillations and (de)coherence. These intrinsically quantum effects can be accounted for by extending the classical Boltzmann equations for number densities of individual flavour species to a semi-classical evolution equation containing a matrix of number densities, analogous to the formalism presented in Sigl:1993 () for light neutrinos. Following this approach, a matrix Boltzmann equation in the lepton flavour space was obtained in Abada:2006fw (); De Simone:2006dd (). Similar considerations were made in Blanchet:2011xq () to include heavy neutrino flavour effects in a hierarchical scenario. However, in RL scenarios, the interplay between heavy-neutrino and lepton flavour effects are important. With these observations, a fully flavour-covariant treatment of the quantum statistical evolution of all relevant number densities, including their off-diagonal coherences, is entirely necessary. This is the main objective of this long article.

To this end, we derive a set of general flavour-covariant transport equations for the number densities of any population of lepton and heavy-neutrino flavours in a quantum-statistical ensemble. This set of transport equations are obtained from a set of master equations for number density matrices derived in the Markovian approximation, in which quantum ‘memory’ effects are ignored (see e.g. Bellac ()). We demonstrate the necessary appearance of rank-4 tensor rates in flavour space that properly account for the statistical evolution of off-diagonal flavour coherences. This novel formalism enables us to capture three important flavour effects pertinent to RL: (i) the resonant mixing of heavy neutrinos, (ii) the coherent oscillations between heavy neutrino flavours, and (iii) quantum (de)coherence effects in the charged-lepton sector. In addition, we describe the structure of generalized flavour-covariant discrete symmetry transformations , and , ensuring definite transformation properties of the transport equations and the generated lepton asymmetries in arbitrary flavour bases. Subsequently, we obtain a simplified version of the general transport equations in the heavy-neutrino mass eigenbasis, but retaining all the flavour effects. We further check that these rate equations reduce to the well-known Boltzmann equations in the flavour-diagonal limit.

To illustrate the importance of the effects captured only in this flavour-covariant treatment, we consider a minimal low-scale RL scenario in which the baryon asymmetry is generated from and protected in a single lepton flavour Pilaftsis:2004xx (). As a concrete example, we consider a minimal model of resonant -genesis (RL) Pilaftsis:2004xx (), involving three quasi-degenerate heavy neutrinos, at or above the electroweak scale, with sizable couplings to the electron and muon, while satisfying all the current experimental constraints. We show that the final lepton asymmetry obtained in our flavour-covariant formalism can be significantly enhanced (by roughly one order of magnitude), as compared to the partially flavour-dependent limits.

We should emphasize that our flavour-covariant formalism is rather general, and its applicability is not limited only to the RL phenomenon. The flavour-covariant transport equations presented here provide a complete description of the leptogenesis mechanism in all relevant temperature regimes. In addition, this formalism can be used to study other physical phenomena, in which flavour effects may be important, such as the evolution of multiple jet flavours in a dense QCD medium in the quark-gluon plasma (see e.g. Blaizot:2013vha ()), the evolution of neutrino flavours in a supernova core collapse (see e.g. Zhang:2013lka ()), or the scenario of -violation induced by the propagation of neutrinos in gravitational backgrounds Mavromatos:2012ii (). We have also developed a flavour-covariant generalization of the helicity amplitude technique, and a generalized optical theorem in the presence of a non-homogeneous background ensemble, which may find applications in non-equilibrium Quantum Field Theory (QFT).

It is worth mentioning here that there have been a number of studies (see e.g. Buchmuller:2000nd (); De Simone:2007rw (); De Simone:2007pa (); Cirigliano:2007hb (); Garny:2009qn (); Cirigliano:2009yt (); Beneke:2010dz (); Anisimov:2010dk (); Garbrecht:2011aw (); Garny:2011hg (); Frossard:2012pc (); Iso:2013lba ()), aspiring to go beyond the semi-classical approach to Boltzmann equations in order to understand the transport phenomena from ‘first principles’ within the framework of non-equilibrium QFT. Such approaches are commonly based on the Schwinger-Keldysh Closed Time Path (CTP) formalism Schwinger:1961 (); Keldysh:1964 (). This real-time framework allows one to derive quantum field-theoretic analogues of the Boltzmann equations, known as Kadanoff-Baym equations Kadanoff:1962 (), obtained from the CTP Schwinger-Dyson equation and describing the non-equilibrium time-evolution of the two-point correlation functions. The Kadanoff-Baym equations are manifestly non-Markovian, accounting for the so-called ‘memory’ effects that depend on the history of the system. These equations can, in principle, account consistently for all flavour and thermal effects. However, one should note that in order to define particle number densities and solve the Kadanoff-Baym equations for their out-of-equilibrium evolution (as e.g. in the context of leptogenesis), particular approximations are often made. These specifically include quasi-particle approximation and gradient expansion in time derivatives Bornath:1996zz (). Moreover, the loopwise perturbative expansion of non-equilibrium propagators are normally spoiled by the so-called pinch singularities Altherr:1994fx (), which are mathematical pathologies arising from ill-defined products of delta functions with identical arguments. Recently, a new formalism was developed for a perturbative non-equilibrium thermal field theory Millington:2012pf (), which makes use of physically meaningful particle number densities that are directly derivable from the Noether charge. This approach allows the loopwise truncation of the resulting transport equations without the appearance of pinch singularities, while maintaining all orders in gradients, thereby capturing more accurately the early-time non-Markovian regime of the non-equilibrium dynamics. An application of this approach to study the impact of thermal effects on the flavour-covariant RL formalism presented here lies beyond the scope of this article.

The rest of the paper is organized as follows: in Section 2, we review the main features of the flavour-diagonal Boltzmann equations. In Section 3, we derive a set of general flavour-covariant transport equations in the Markovian regime. In Section 4, we apply the formalism developed in Section 3 to a generic RL scenario and derive the relevant flavour-covariant evolution equations for the heavy-neutrino and lepton-doublet number densities. In Section 5, we present a geometric understanding of the degeneracy limit in minimal RL scenarios and also discuss an explicit model of RL. In Section 6, we present numerical results for three benchmark points, which illustrate the impact of flavour off-diagonal effects on the final lepton asymmetry. We summarize our conclusions in Section 7. In A, we comment on different forms of the self-energy regulator used in the literature to calculate the leptonic -asymmetry in RL models and check their consistency in the -conserving limit. In B, we develop a flavour-covariant generalization of the helicity amplitude formalism and describe the flavour-covariant quantization of spinorial fields in the presence of time-dependent and spatially-inhomogeneous backgrounds. In C, we justify the tensorial flavour structure of the transport equations introduced in Section 3, by means of a generalization of the optical theorem. Finally, in D, we exhibit the form factors relevant for the lepton flavour violating decay rates discussed in Section 6.

## 2 Flavour Diagonal Boltzmann Equations

The time-evolution of the number density of any particle species can be modelled by a set of coupled Boltzmann equations (see e.g. kolb ()). Adopting the formalism described in Kolb:1980 (); Luty:1992un (), this may be written down in the generic form

(2.1) |

where the drift terms on the left-hand side (LHS) arise from the covariant hydrodynamic derivative and include the dilution of the number density due to the expansion of the Universe, parametrized by the Hubble expansion rate . The right-hand side (RHS) of (2.1) comprises the collision terms accounting for the interactions that change the number density . Here, we have summed over all possible reactions of the form or , in which the species can be annihilated or created, respectively. If the species is unstable, it can occur as a real intermediate state (RIS) in resonant processes of the form , which must be properly taken into account in order to avoid double-counting of this contribution from the already considered decays and inverse decays in the Boltzmann equations Kolb:1980 (). At this point, it is important to note that the formalism leading to (2.1) neglects both the coherent time-oscillatory terms, describing particle oscillations between different flavours, and off-diagonal correlations in the matrix of number densities , corresponding to the annihilation of a particle species and the correlated creation of a particle species . For this reason, we refer to (2.1) as a set of flavour-diagonal Boltzmann equations.

It is useful to summarize the notation and definitions used in (2.1). Firstly, the Hubble expansion rate in the early Universe is given as a function of the temperature by kolb ()

(2.2) |

where GeV is the Planck mass and is the number of relativistic degrees of freedom at temperature . Throughout our discussions, all species are assumed to be in kinetic (but not necessarily chemical) equilibrium. In this case, the number density of a particle species is given by

(2.3) |

where is a short-hand notation for the three-momentum integral, the sign in the denominator corresponds to particles obeying Bose-Einstein (Fermi-Dirac) quantum statistics, is the relativistic energy of the species , being its rest mass, is the total degeneracy factor of the internal degrees of freedom, and being the degenerate helicity and degenerate isospin degrees of freedom respectively, and is the temperature-dependent chemical potential, encoding the deviation from local thermodynamic equilibrium. It will prove convenient in our later discussions to define an in-equilibrium number density as the limit in (2.3). We note however that the true equilibrium number density will depend on the equilibrium chemical potential , which may not be zero in general.

There are two limits of (2.3) of interest here: (i) the Maxwell-Boltzmann classical statistical limit in which we can drop the term in the denominator of (2.3), giving

(2.4) |

where is the th-order modified Bessel function of the second kind; (ii) the relativistic limit () in which case

(2.5) |

where for bosons (fermions), and is the Riemann zeta function, with .

Following Pilaftsis:2003gt (), we define the -conserving collision rate for a generic process and its -conjugate as

(2.6) |

where we have used the shorthand superscript to denote conjugation, and

(2.7) |

Here, the squared matrix element is summed, but not averaged, over the internal degrees of freedom of the initial and final multiparticle states and . We have introduced an abbreviated notation in (2.7) for the phase-space integrals over and . The phase-space measure for the multiparticle state , containing particles, is defined as

(2.8) |

where and are the usual Dirac delta and Heaviside step functions, respectively, and is a symmetry factor in the case that the multiparticle state contains identical particles. In a -conserving theory, the -conserving collision rates must obey . Analogous to (2.6), a -violating collision rate can be defined as Pilaftsis:2003gt ()

(2.9) |

which obeys , following invariance.

The relevant Boltzmann equations for describing leptogenesis are those involving the number densities (with of the heavy Majorana neutrinos, (with ) of the lepton-doublets and of their conjugates. When solving the coupled system of first-order differential equations (2.1) for , and , it is convenient to introduce a new variable . In the radiation-dominated epoch, relevant to the production of lepton asymmetry, is related to the cosmic time via the relation , where

(2.10) |

is the Hubble parameter (2.2) at , assuming only SM relativistic degrees of freedom. We also normalize the number density of species to the number density of photons, defining , with given by (2.5) for and , i.e.

(2.11) |

With these definitions, we write down the flavour-diagonal Boltzmann equations (2.1) in terms of the normalized number densities of heavy neutrinos and the normalized lepton asymmetries as follows Pilaftsis:2005rv ():

(2.12) | ||||

(2.13) |

where is the normalized equilibrium number density of the heavy neutrinos, obtained using (2.11) and (2.4) with . The various collision rates appearing in (2.12) and (2.13) can be readily understood from the general definitions in (2.6) and (2.9); their explicit expressions in terms of the Yukawa couplings will be given in Section 2.2. Here we have included only the dominant contributions arising from the decays and inverse decays of the heavy neutrinos, proportional to the rate , and the resonant part of the and scatterings, proportional to and respectively. We ignore the sub-dominant chemical potential contributions from the right-handed (RH) charged-lepton, quark and the Higgs fields, as well as the Yukawa and gauge scattering terms Pilaftsis:2005rv ().

Note that for the collision rate pertinent to the heavy neutrino decay in (2.12), we have summed over the lepton flavours, and similarly, for the charged-lepton rate equation (2.13), we have summed over the heavy neutrino flavours; therefore, these are still designated as flavour-diagonal Boltzmann equations. The -odd collision rate in (2.13) can be expressed in terms of the flavour-dependent leptonic -asymmetries and the -even collision rate , as follows: , where is defined in terms of the partial decay widths and their -conjugates :

(2.14) |

where is the total decay width of the heavy Majorana neutrino . Since we are interested in the heavy neutrino decay for temperatures above the electroweak phase transition, where the SM Higgs vacuum expectation value (VEV) vanishes, only the would-be Goldstone and Higgs modes of the -doublet contribute predominantly to the partial decay widths and the total decay width in (2.14).

### 2.1 Resummed Effective Yukawa Couplings

The physical -violating observable defined in (2.14) receives contributions from two different mechanisms (see Figure 1): (i) -type violation due to the interference between the tree-level and absorptive part of the self-energy graphs in the heavy-neutrino decay, and (ii) -type violation due to the interference between the tree-level graph and the absorptive part of the one-loop vertex. This terminology is in analogy with the two kinds of violation in the -system (for reviews, see pdg (); kabir ()), where represents the indirect violation through – mixing, while represents the direct violation entirely due to the decay amplitude.

The contribution of the self-energy diagrams to the -asymmetry
can in principle be calculated using an effective Hamiltonian
approach, similar to that applied for the
-system kabir (). However, the heavy
neutrinos, being unstable particles, cannot be described by the
asymptotic (free) in- and out-states of an -matrix
theory Veltman:1963 (). Instead, their properties can be inferred
from the transition matrix elements of
scatterings of stable particles, and by identifying the resonant part
of the amplitude that contains the RIS
contributions only. This allows one to perform an effective
resummation of the heavy-neutrino self-energy diagrams contributing to
the -type -asymmetry Pilaftsis:1997jf (); Buchmuller:1997yu (); Pilaftsis:2003gt ().^{3}^{3}3For other effective
approaches within the framework of perturbative field theory,
see Flanz:1996fb (); Flanz:1994yx (); Liu:1993ds (); Covi:1996fm (); Sarkar:1998 (); Rangarajan:1999kt (); Anisimov:2005hr (). However, for a critical appraisal of the existing approaches,
see A.

Neglecting the charged-lepton and light neutrino masses, the absorptive part of the heavy Majorana neutrino self-energy transitions can be written in a simple spinorial structure, as follows:

(2.15) |

where are the left- and right-chiral projection operators respectively, and is the absorptive transition amplitude, summed over all charged-lepton flavours running in the loop:

(2.16) |

Here is the Yukawa coupling of the heavy neutrino with the lepton-doublet , and the caret ( ) denotes the fact that (2.16) was derived in a basis in which the heavy Majorana neutrino mass matrix is diagonal. The tree-level decay width of the heavy Majorana neutrino is related to the diagonal transition amplitude by

(2.17) |

To account for unstable-particle mixing effects between the heavy Majorana neutrinos, we define the one-loop resummed effective Yukawa couplings, denoted by (bold-faced Latin) , and their -conjugates , related to the matrix elements and respectively. This formalism captures all dominant effects of heavy neutrino mixing and -violation, and has been shown Pilaftsis:2003gt () to be equivalent to an earlier proposed resummation method Pilaftsis:1997jf () based on the Lehmann-Symanzik-Zimmermann reduction formalism LSZ (). Working in the heavy neutrino mass eigenbasis, the resummed effective Yukawa couplings are given by Pilaftsis:2003gt (); Pilaftsis:2008qt ()

(2.18) |

where is the usual Levi-Civita anti-symmetric tensor, is a shorthand notation used here for brevity, and

(2.19) |

All the transition amplitudes in (2.18) are evaluated on-shell
with . The
respective -conjugate resummed effective Yukawa couplings
can be obtained from (2.18) by
replacing the tree-level Yukawa couplings with their
complex conjugates .^{4}^{4}4Note that in general, whereas for the tree-level Yukawa couplings,
by -invariance of the Lagrangian. We will neglect
the one-loop corrections to the proper vertices ,
whose absorptive parts are numerically insignificant in RL. The
partial decay widths and
appearing in (2.14) can now be expressed in terms of the
effective Yukawa couplings and
, and the flavour-dependent absorptive
transition amplitudes , as
follows:

(2.20) |

Note the explicit dependence of the absorptive transition amplitudes on the effective Yukawa couplings in (2.20). The total decay width of the heavy neutrino is thus obtained by summing over all lepton flavours:

(2.21) |

Replacing by the tree-level Yukawa coupling in (2.21), we can reproduce the tree-level decay width given by (2.17). Substituting (2.20) in (2.14), the flavour-dependent leptonic -asymmetry in RL can be written as

(2.22) |

Note that (2.22) encodes both - and -type asymmetries, although we simply denote it by for brevity. The analytic results for both types of -asymmetry and their -conserving limits for a simplified case will be discussed in A.

### 2.2 Analytic Solutions

It is instructive to derive approximate analytic solutions of the Boltzmann equations (2.12) and (2.29). To do this, we express (2.12) in terms of the non-equilibrium deviation parameter , thus obtaining

(2.23) |

where the K-factors, defined by , determine the depletion of the lepton asymmetry due to inverse decays. In deriving (2.23), we have used the analytic expression for the total collision rate pertinent to the heavy neutrino decay

(2.24) |

In the kinematic regime , (2.23) has an approximate attractor solution

(2.25) |

independent of the initial conditions.

The collision rates for the and scatterings are given by Deppisch:2010fr ()

(2.26) | ||||

(2.27) |

where we have used the narrow-width approximation (NWA) for the resummed heavy neutrino propagators in the pole-dominance region, i.e.

(2.28) |

since we are only interested in the resonant part of these scatterings in the RL case. Separating the diagonal RIS contributions from the off-diagonal terms in the sum, (2.13) can be rewritten as Pilaftsis:2005rv ()

(2.29) |

where is the heavy neutrino decay branching ratio, and denote the RIS-subtracted collision rates, which can be obtained from (2.26) and (2.27) taking . Including only the important RIS-subtracted collision rates proportional to , and neglecting the terms proportional to (for ) which are numerically small for the minimal RL scenarios Deppisch:2010fr (), we can simplify (2.29) to

(2.30) |

where we have introduced a flavour-dependent parameter

(2.31) |

Using the attractor solution (2.25) in the kinematic regime , (2.30) can be written as

(2.32) |

where is the total leptonic -asymmetry stored in a given lepton flavour , and is the effective washout parameter due to scatterings mediated by heavy neutrinos. Note that if we only consider the diagonal terms representing the RIS contributions in the sum in (2.31), reaches its maximum value, i.e. . On the other hand, in the -conserving limit, vanishes at a rate at least equal to that of (see A). In the regime , the total lepton asymmetry, dominated by -type mixing effects, can be approximated by the analytic solution to (2.32):

(2.33) |

up to a point , beyond which the lepton asymmetry freezes out and approaches a constant value Deppisch:2010fr ().

### 2.3 Observed Lepton Asymmetry

Having obtained the net lepton asymmetry , the next step is to convert it to the asymmetry in the total baryon-to-photon ratio via -violating sphaleron interactions. In a sphaleron transition, an and -singlet neutral object from each generation of the SM is created out of the vacuum Kuzmin:1985mm (); 'tHooft:1976up (); Klinkhamer:1984di (); Dimopoulos:1978kv (). The operator responsible for sphaleron transitions can be written as

(2.34) |

where is the family index; are the colour indices; are the isospin indices; and is the quark doublet. The operator is invariant under both gauge transformations and flavour rotations. For the case of our interest, the latter freedom can be used to make the charged lepton Yukawa matrix positive and diagonal. Above the electroweak phase transition, all the SM processes, including the sphaleron interactions in (2.34), are assumed to be in full thermal equilibrium, which leads to the following relations among their chemical potentials Harvey:1990qw ():

(2.35) |

where stands for all vector bosons, and for up and down-type quark singlets, and for lepton singlets in the SM. The total chemical potentials of the baryonic and leptonic doublet fields are then given by

(2.36) |

Using the relations (2.36) into (2.4), in the linear
approximation of , we obtain the conversion of the total
lepton asymmetry stored in the SM lepton-doublet to the baryon
asymmetry^{5}^{5}5Note that since we are converting the asymmetry
stored in the lepton-doublet, the conversion coefficient derived
here is different from 28/51 used elsewhere (see
e.g. Deppisch:2010fr ()), which corresponds to the total
lepton asymmetry, including the RH leptons.

(2.37) |

assuming a rapid sphaleron transition rate . This is valid at temperatures , where is the critical temperature for the electroweak phase transition, given at one loop by Cline:1993bd ()