Flavour effects in Resonant Leptogenesis from semiclassical and KadanoffBaym approaches
Abstract
Flavour effects play an important role in the statistical evolution of particle number densities in several particle physics phenomena. We present a fully flavourcovariant formalism for transport phenomena, in order to consistently capture all flavour effects in the system. We explicitly study the scenario of Resonant Leptogenesis (RL), and show that flavour covariance requires one to consider generically offdiagonal number densities, rank4 rate tensors in flavour space, and nontrivial generalization of the discrete symmetries , and . The flavourcovariant transport equations, obtained in our semiclassical framework, describe the effects of three relevant physical phenomena: coherent heavyneutrino oscillations, quantum decoherence in the chargedlepton sector, and resonant violation due to heavyneutrino mixing. We show quantitatively that the final asymmetry predicted in RL models may vary by as much as an order of magnitude between partially flavour offdiagonal treatments. A full fieldtheoretic treatment in the weaklyresonant regime, based on the KadanoffBaym (KB) equations, confirms that heavyneutrino oscillations and mixing are two distinct phenomena, and reproduces the results obtained in our semiclassical framework. Finally, we show that the quasiparticle ansaetze, often employed in KB approaches to RL, discard the phenomenon of mixing, capturing only oscillations and leading to an underestimate of the final asymmetry by a factor of order 2.
a]P. S. Bhupal Dev, Peter Millington, Apostolos Pilaftsis, Daniele Teresi
1 Introduction
Leptogenesis [1] is an elegant unifying framework for dynamically generating both the measured matterantimatter asymmetry in our Universe and the observed smallness of the light neutrino masses [2]. In scenarios of Resonant Leptogenesis (RL) [3, 4], this mechanism may be testable in foreseeable laboratory experiments. RL relies on the fact that the type asymmetry becomes dominant [5] and gets resonantly enhanced, when at least two of the heavy neutrinos have a small mass difference comparable to their decay widths [3]. This resonant enhancement allows a successful lowscale leptogenesis [4, 6], whilst retaining perfect agreement with the lightneutrino oscillation data. The level of testability is further extended in the scenario of Resonant Genesis (RL), where the final lepton asymmetry is dominantly generated and stored in a single lepton flavour [7, 8]. In such models, the heavy neutrinos could be as light as the electroweak scale [6], whilst still having sizable couplings to other chargedlepton flavours . Thus, RL scenarios may be directly testable at the energy frontier in the runII phase of the LHC [9], as well as in various lowenergy experiments searching for lepton flavour/number violation [10] at the intensity frontier.
Flavour effects in both heavyneutrino and chargedlepton sectors, as well as the interplay between them, can play an important role in determining the final lepton asymmetry in lowscale leptogenesis models (for a review, see e.g. [11]). These intrinsicallyquantum effects can, in principle, be accounted for by extending the classical flavourdiagonal Boltzmann equations for the number densities of individual flavour species to a semiclassical evolution equation for a matrix of number densities, analogous to the formalism presented in [12] for light neutrinos. This socalled ‘density matrix’ formalism has been adopted to describe flavour effects in various leptogenesis scenarios [13, 14, 15]. It was recently shown [16], in a semiclassical approach, that a consistent treatment of all pertinent flavour effects, including flavour mixing, oscillations and offdiagonal (de)coherences, necessitates a fully flavourcovariant formalism, in order to provide a complete and unified description of RL; for a summary, see [17]. In this flavourcovariant formalism, the resonant mixing of different heavyneutrino flavours and coherent oscillations between them are found to be two distinct physical phenomena, in analogy with the experimentallydistinguishable phenomena of mixing and oscillations in the neutral , ,  and meson systems [2].
One can go beyond the semiclassical ‘densitymatrix’ approach to leptogenesis by means of a quantum fieldtheoretic analogue of the Boltzmann equations, known as the KadanoffBaym (KB) equations [18] (for a review, see e.g. [19]). Such ‘firstprinciples’ approaches to leptogenesis [20] are, in principle, capable of accounting consistently for all flavour effects, in addition to offshell and finitewidth effects, including thermal corrections. However, it is often necessary to use truncated gradient expansions and quasiparticle ansaetze to relate the propagators appearing in the KB equations to particle number densities. Recently, using the novel perturbative formulation of thermal field theory developed in [21], it was shown [22] that quantum transport equations for leptogenesis can be obtained from the KB formalism without the need for gradient expansion or quasiparticle ansaetze, thereby capturing fully the pertinent flavour effects. Specifically, the source term for the lepton asymmetry obtained, at leading order, in this KB approach [22] was found to be exactly the same as that obtained in the semiclassical flavourcovariant approach of [16], confirming that flavour mixing and oscillations are indeed two physicallydistinct phenomena. The proper treatment of these flavour effects may have a significant effect upon the final lepton asymmetry, as compared to partially flavourdependent or flavourdiagonal limits, thereby altering the viable parameter space for models of RL and impacting upon the prospects of testing the leptogenesis mechanism.
The plan of these proceedings is as follows. In Section 2, we review the main features of our fully flavourcovariant formalism in the context of leptogenesis. In Section 3, we present the Markovian flavourcovariant transport equations for lepton and heavyneutrino number densities with arbitrary flavour content. We also discuss a numerical example to illustrate the full impact of the flavour offdiagonal effects within the context of an RL model. In Section 4, we derive the quantum transport equations relevant to the source term for the lepton asymmetry, following a welldefined perturbative loopwise truncation scheme, making comparison with the semiclassical approach discussed in Section 3. Our conclusions are given in Section 5.
2 Flavourcovariant formalism
We consider the leptondoublet field operators (with ) and righthanded Majorana neutrino field operators (with ), with arbitrary flavour content, transforming as follows in the fundamental representation of :
(1a)  
(1b) 
where and . The relevant neutrino Lagrangian is given by
(2) 
where is the isospin conjugate of the Higgs doublet and the superscript denotes charge conjugation. The Lagrangian (2) transforms covariantly under , provided the Yukawa couplings and Majorana mass matrix transform as
(3) 
In this flavourcovariant formalism, the planewave decompositions of the field operators are written in a manifestly flavourcovariant way, e.g.
(4) 
where , is the helicity index and , with being the chargedlepton mass matrix. Thus, flavour covariance requires the Dirac fourspinors and to transform as rank tensors in flavour space. The creation and annihilation operators , , and satisfy the equaltime anticommutation relations
(5) 
For the heavy Majorana neutrino creation and annihilation operators and , it is necessary to introduce the flavourcovariant Majorana constraint
(6) 
where are the elements of a unitary matrix , which transforms as a contravariant rank tensor under . Notice that in the mass eigenbasis.
Similar flavour rotations are forced by flavourcovariance under the discrete symmetry transformations and . This necessarily leads to generalized and transformations:
(7a)  
(7b)  
(7c) 
where is the lepton analogue of the heavyneutrino tensor .
We may now define the matrix number densities of the leptons and heavy neutrinos, which describe completely the flavour content of the system:
(8a)  
(8b)  
(8c) 
Here, is the infinite coordinate threevolume and is the macroscopic time, equal to the time interval between specification of initial conditions () and subsequent observation of the system (). The total number densities are obtained by tracing over helicity and isospin indices and integrating over the threemomenta. For the Majorana neutrinos, and are not independent quantities and are related by the generalized Majorana constraint (6).
3 Flavourcovariant semiclassical rate equations
We first derive a master equation governing the time evolution of the matrix number densities , as given in (8a)–(8c). By using the Liouvillevon Neumann and Heisenberg equations of motion and subsequently performing a WignerWeisskopf approximation in the Markovian limit [16], we find
(10) 
where and are respectively the free and interaction parts of the Hamiltonian and
(11) 
is the ensemble expectation value of the quantummechanical numberdensity operator , in which is the interactionpicture density operator. The first term on the RHS of (10), involving the free Hamiltonian, is responsible for flavour oscillations, whereas the second term contains the collision terms of the generalized Boltzmann equations. Explicitly, for the system of charged leptons and heavy neutrinos, we find [16]
(12a)  
(12b) 
The collision terms and involve the product of two new rank4 tensors in flavour space, namely, the statistical number density tensor and the absorptive rate tensor, whose appearance is necessary for the flavour covariance of the formalism [16]. The emergence of these rank4 tensors may be understood in terms of the unitarity cuts of the partial selfenergies, as was shown by an explicit calculation of the relevant transition amplitudes using a generalized optical theorem in [16]. The offdiagonal components of the rate tensor are responsible for the evolution of flavourcoherences in the system.
In the limit when two (or more) of the heavy Majorana neutrinos become degenerate, the type violation can be resonantly enhanced, even up to order one [3], due to the interference between the treelevel and selfenergycorrected decays. In this regime, finiteorder perturbation theory breaks down and one must resum the selfenergy corrections in order to account for the heavyneutrino mixing effects. In the semiclassical approach, we perform such resummation in an effective way by replacing the treelevel neutrino Yukawa couplings by their resummed counterparts in the transport equations. In the next section, this approach will be justified for the weaklyresonant regime of RL, where , by using a ‘firstprinciples’ fieldtheoretic approach. The explicit algebraic form of the resummed neutrino Yukawa couplings in the heavyneutrino mass eigenbasis can be found in [4]; the corresponding form in a general flavour basis may be obtained by the appropriate flavour transformation, i.e. , where in the mass eigenbasis [16].
In addition, we make the following reasonable approximations to simplify the flavourcovariant rate equations for RL: we (i) assume kinetic equilibrium, which is ensured by the presence of fast elasticscattering processes; (ii) work in the classicalstatistical regime; (iii) neglect thermal and chemicalpotential effects [6]; and (iv) neglect the mass splitting between different heavyneutrino flavours inside thermal integrals, using an average mass and energy , as is appropriate since the average momentum scale . In order to guarantee the correct equilibrium behaviour, we must also include the effect of the and scattering processes, with proper real intermediate state (RIS) subtraction [23, 4, 16]. As detailed in [16], it is necessary to account for thermal corrections in the RIS contributions, when considering offdiagonal heavyneutrino flavour correlations. Finally, it is important to include the effect of the chargedlepton Yukawa couplings, which are responsible for the decoherence of the charged leptons towards their wouldbe mass eigenbasis.
Taking into account the expansion of the Universe, we then obtain the following manifestly flavourcovariant rate equations for the normalized “even” number density matrix and “odd” number density matrices and (where , with being the photon number density) [16]:
Here, is the Hubble parameter at and we have defined the thermallyaveraged heavyneutrino energy matrix
(14) 
where counts the helicity degrees of freedom. In addition, and are respectively the “even” and “odd” thermallyaveraged rate tensors, describing heavyneutrino decays and inverse decays and written in terms of the resummed Yukawa couplings as
(15) 
where denotes the generalized conjugate and we have used the shorthand notation
(16) 
with the phasespace measure for the species given by
(17) 
In (13c), and respectively describe the washout due to and resonant scattering, and and govern the chargedlepton decoherence [16]. Lastly, for a Hermitian matrix , we have defined the flavourcovariant generalized real and imaginary parts
(18) 
which reduce to the usual real and imaginary parts in the heavyneutrino mass eigenbasis.
The flavourcovariant rate equations (13a)–(13c) provide a complete and unified description of the generation of the lepton asymmetry in RL, consistently describing the following physicallydistinct effects in a single framework:

Decoherence effects due to chargedlepton Yukawa couplings. Our description of these effects generalizes the analysis in [13] to an arbitrary flavour basis.
In order to illustrate the importance of the flavour effects captured only by the flavourcovariant rate equations (13a)–(13c), we consider an RL model, comprising an approximately symmetric heavyneutrino sector at the grand unification scale GeV, with masses [6]. The soft breaking mass term is taken to be of the form . At the electroweak scale, an additional mass splitting arises from the RG running, such that . In order to accommodate the smallness of the light neutrino masses in a technically natural manner, we also require the heavyneutrino Yukawa sector to possess an approximate leptonic symmetry. This results in the following structure for the heavyneutrino Yukawa couplings:
(19) 
where are arbitrary complex parameters. If the theory were to have an exact symmetry, i.e. if the breaking parameters were set to zero, the light neutrinos would remain massless to all orders in perturbation theory. For electroweakscale heavy neutrinos, we require , in order to be consistent with current lightneutrino mass bounds, and and , in order to protect the lepton asymmetry from washout effects.
Using the Yukawa coupling given by (19), we solve the rate equations (13a)–(13c) numerically to obtain the total lepton asymmetry in our flavourcovariant formalism. This is shown in Figure (a)a for a typical set of benchmark values for the Yukawa coupling parameters, as given in Figure (b)b, which is consistent with all current experimental constraints [16]. In Figure (a)a, the horizontal dotted line shows the value of required to explain the observed baryon asymmetry, whereas the vertical line shows the critical temperature , beyond which the electroweak sphaleron processes become ineffective in converting lepton asymmetry to baryon asymmetry. The thick solid lines show the evolution of for three different initial conditions, to which the final lepton asymmetry is insensitive as a general consequence of the RL mechanism in the strongwashout regime [6]. For comparison, Figure (a)a also shows various flavourdiagonal limits, i.e. when either the heavyneutrino (dashed line) or the lepton (dashdotted line) number density or both (dotted line) are diagonal in flavour space. Also shown (thin solid line) is the approximate analytic solution obtained in [16] for the case of a diagonal heavyneutrino number density. The enhancement of the lepton asymmetry in the fully flavourcovariant formalism (solid line), as compared to assuming a flavourdiagonal heavyneutrino number density (dashed line), is mainly due to coherent oscillations between the heavyneutrino flavours, leading to a factor of 2 increase. Finally, we observe that the predicted lepton asymmetry differs by approximately an order of magnitude between the two partially flavour offdiagonal treatments (dashed and dashdotted lines). This provides a striking illustration of the importance of capturing all pertinent flavour effects and their interplay by means of a fully flavourcovariant formulation for transport phenomena.

4 Flavourcovariant quantum transport equations
The semiclassical approach detailed in the preceding section has the advantage that it is constructed with physical observables, i.e. particle number densities, in mind. However, it has the disadvantage that quantum effects, such as finite particle widths, must be incorporated in an effective manner. For instance, in the latter example, we must subtract the RIS contributions from the collision terms [23], which would otherwise lead to double counting of decay and inversedecay processes. It is therefore desirable to seek a more firstprinciples description of transport phenomena, in which quantum effects are incorporated consistently from the outset.
Such a description is provided by the KadanoffBaym (KB) approach [18] (see also [24, 19]), constructed within the SchwingerKeldysh closedtime path (CTP) formalism of thermal field theory [25]. Therein, one arrives at systems of KB equations by partially inverting the SchwingerDyson equation of the 2PI CJT effective action [26]. Unfortunately, the KB equations describe the spacetime evolution of propagators and, as a result, it is necessary to use approximation schemes in order to obtain the quantum transport equations of particle number densities.
In order to avoid the technical complications of spinor fields, we will consider a scalar model of RL, see [22], comprising: two real scalar fields (), modelling heavyneutrinos of two flavours; one complex scalar field , modelling chargedleptons of a single flavour; and a real scalar field , modelling the Standard Model Higgs. The lepton number can be associated with the global symmetry of the complex scalar field .
The CTP formalism may be formulated in two ways, working either in the Heisenberg or interaction picture. In the former (see e.g. [27]), the density operator does not evolve in time, remaining fixed at the boundary time . As a result, the free propagators encode the initial conditions of the statistical ensemble, e.g.
(20) 
In this case, it is wellknown that there does not exist a welldefined perturbative expansion. This may be understood by considering the Taylor expansion of the exponential decay to equilibrium: . Any truncation of this expansion at a finite order in the decay rate leads to secular behaviour when [19]. This problem manifests in the FeynmanDyson series of the Heisenberg interpretation of the CTP formalism as pinch singularities [28], which result from illdefined products of delta functions with identical arguments. As a consequence, it is necessary to work with dressed propagators, and therefore, ansaetze are required in order to extract particle number densities. The most common is the quasiparticle approximation known as the KB ansatz:
(21) 
where approximates the matrix number density of spectrallydressed particles.
On the other hand and in strong contrast to the earlier literature, it was shown recently [21] that a perturbative framework of nonequilibrium thermal field theory is in fact viable, if we work instead in the interaction picture. Since the interactionpicture density operator evolves in time, being evaluated at a macroscopic time after the specification of the initial conditions, the free positivefrequency Wightman propagator becomes
(22) 
In the heavyneutrino mass eigenbasis and assuming spatial homogeneity, the free Wightman propagators then have the following explicit form:
(23) 
depending on the timedependent matrix number density of spectrallyfree particles. These number densities appear as unknown functions in the FeynmanDyson series, with their functional form being fixed only after the governing transport equations have been solved. Thus, the exponential decay to equilibrium is present implicitly in the free propagators of the theory, thereby avoiding the problem of secularity or pinch singularities, see [21].
Making a Markovian approximation and additionally setting , valid in the weaklyresonant regime, the free Wightman propagators in (4) reduce to
(24) 
written here in a singlemomentum representation and in a general flavour basis. The Markovian and homogeneous form of the free propagator in (24) should be compared (for ) with the KB ansatz of the dressed propagator in (21), wherein we note that their spectral structure is identical in spite of the fact that the latter should be fully dressed spectrally.
In coordinate space, the KB equations take the following generic form (see e.g. [24]):
(25) 
where is the d’Alembertian operator and indicates matrix multiplication in flavour space. The denotes the convolution
(26) 
which is performed over the hypervolume , bounded temporally from below and above by the boundary and observation times, respectively [21].
The KB equation (25) may be recast in a double momentumspace representation as follows:
(27) 
Here, denotes the weighted convolution integral
(28) 
where
(29) 
and
(30) 
By considering the Noether charge, see [21], the number density may be related unambiguously to the negativefrequency Wightman propagator via
(31) 
with and . Following [21], we may then translate (27) into the final rate equation for the number density
(32) 
where we use a compact notation
(33) 
The first two terms on the LHS of (4) are the drift terms and the latter two account for meanfield effects, including oscillations; the terms on the RHS describe collisions. It should be stressed that (4), obtained without employing a gradient expansion or quasiparticle ansatz, is valid to any order in perturbation theory and accounts fully for spatial inhomogeneity, nonMarkovian dynamics (memory effects) and flavour effects.
As identified in [21], the general rate equation in (4) may be truncated in a perturbative loopwise sense in two ways: (i) spectrally: by truncating the external leg, we determine what is being counted, e.g. inserting free propagators, we count spectrallyfree particles; (ii) statistically: by truncating the selfenergies, we determine the set of processes that drive the statistical evolution, e.g. inserting oneloop selfenergies, we include decay and inversedecay processes.
4.1 Heavyneutrino rate equations
Assuming spatial homogeneity and absorbing the principal part selfenergy into the thermal mass , we find the rate equation of the dressed heavyneutrino number density
(34) 
Herein, we have also neglected the commutator involving on the RHS of (4), since, in the weaklyresonant regime, it contains higherorder effects that are not relevant to this analysis.
Neglecting terms proportional to the lepton asymmetry, it is sufficient to approximate the chargedlepton and Higgs propagators, appearing in the heavyneutrino selfenergies, by their quasiparticle (narrowwidth) equilibrium forms with vanishing chemical potential, i.e.
(35)  
(36) 
Here, is the BoseEinstein distribution and is the thermal mass of species . We are then left with the nonMarkovian heavyneutrino selfenergies
(37) 
We now perform a WignerWeisskopf approximation by replacing by in all spacetime integrals. In the doublemomentum representation, this amounts to using the limit
(38) 
At the same time, we replace , absorbing the freephase evolution, which cancels that in the measure of (31) in the energyconserving limit. We then arrive at the Markovian rate equation for the dressed heavyneutrino number density