MC-TH-2003-09

[-0.15cm] hep-ph/0309342

[-0.15cm] October 2003

Resonant Leptogenesis

[1.5cm] Apostolos Pilaftsis and Thomas E. J. Underwood

[0.3cm] Department of Physics and Astronomy, University of Manchester,

Manchester M13 9PL, United Kingdom

ABSTRACT

We study the scenario of thermal leptogenesis in which the leptonic asymmetries are resonantly enhanced through the mixing of nearly degenerate heavy Majorana neutrinos that have mass differences comparable to their decay widths. Field-theoretic issues arising from the proper subtraction of real intermediate states from the lepton-number-violating scattering processes are addressed in connection with an earlier developed resummation approach to unstable particle mixing in decay amplitudes. The pertinent Boltzmann equations are numerically solved after the enhanced heavy-neutrino self-energy effects on scatterings and the dominant gauge-mediated collision terms are included. We show that resonant leptogenesis can be realized with heavy Majorana neutrinos even as light as TeV, in complete accordance with the current solar and atmospheric neutrino data.

PACS numbers: 11.30.Er, 14.60.St, 98.80.Cq

## 1 Introduction

The recent results from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite have dramatically improved the accuracy of many cosmological parameters [1], thus signalling a new era of precision cosmology. For the first time, the baryon–to–photon ratio of number densities has been measured to the unprecedented precision of less than 10%. The reported value for is [1]

(1.1) |

where and are the number densities of the net baryon number and photons at the present epoch, respectively.

Many theoretical models have been suggested in the literature [2, 3] in order to explain the presently small but non-zero value of that quantifies the so-called cosmological Baryon Asymmetry in the Universe (BAU). One of the most attractive as well as field-theoretically consistent scenarios of baryogenesis is the one proposed by Fukugita and Yanagida [4]. In this model, out-of-equilibrium -violating decays of singlet neutrinos with Majorana masses considerably larger than the critical temperature –200 GeV produce initially an excess in the lepton number . This excess in is then converted into the observed asymmetry through -violating sphaleron interactions [5, 6], which are in thermal equilibrium for temperatures ranging from up to GeV [7, 8, 9]. Many studies have been devoted to analyze in detail this scenario of baryogenesis through leptogenesis [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

In the last few years, the on-going neutrino experiments, mainly at Super-K [34] and SNO [35], have been able to address another important question in particle and astro-particle physics [36]. Their analyses have offered overwhelming support to the theoretical idea that the ordinary neutrinos have tiny but non-zero masses and mixings [37], thereby enabling them to oscillate from one type of lepton to another [38]. In the Standard Model (SM) neutrinos are strictly massless. An economical as well as natural solution to this problem can be achieved by augmenting the SM field content with right-handed (singlet) neutrinos. By the same token, bare Majorana masses that violate the lepton number by two units are allowed to be added to the Lagrangian. The scale of these singlet masses is rather model-dependent and may range from about 1 TeV in Left-Right Symmetric [39, 40] or certain E [41] models up to GeV in typical Grand Unified Theories (GUTs) such as SO(10) [42, 43] models. From the low-energy point of view, the large Majorana masses present in the complete neutrino mass matrix give rise to a kind of a seesaw mechanism [44], through which the phenomenologically favoured values for neutrino masses of order 0.1 eV and smaller can be explained without unnaturally suppressing the Yukawa couplings of the theory.

One of the central questions that several articles have been addressing recently is to which extent the afore-mentioned heavy Majorana neutrinos can be responsible for both the observed BAU and the neutrino oscillation data, including possible data from other non-accelerator experiments. In this context, it has been found [19, 20] that if the heavy singlet neutrinos have an hierarchical mass spectrum, a lower bound of about – GeV on the leptogenesis scale can be derived. In the derivation of this lower bound, the size of the leptonic asymmetry between the heavy Majorana neutrino decay into a lepton doublet and a Higgs doublet , , and its respective charge and parity (CP) conjugate mode, , plays a key role. In other words, the larger the leptonic CP asymmetry, the smaller the lower bound on the leptogenesis scale becomes.

As is shown in Fig. 1 in a Feynman–diagrammatic way, there are two one-loop graphs that contribute to the CP-violating leptonic asymmetry. In particular, the interference of the tree-level decay amplitude with the absorptive parts of the one-loop self-energy and vertex graphs violates CP and hence gives rise to a non-vanishing leptonic asymmetry. These self-energy and vertex contributions are often termed in the literature [45, 32] - and -types of CP violation, respectively. Unlike -type [4, 10, 11], -type CP violation can be considerably enhanced [12, 13, 14] through the mixing of two nearly degenerate heavy Majorana neutrinos.

The fact that -type CP violation can become several orders of magnitude larger than -type CP violation might raise concerns on the validity of perturbation theory. Indeed, finite-order perturbation theory breaks down if two heavy Majorana neutrinos become degenerate. However, based on a field-theoretic approach that consistently resums all the higher-order self-energy-enhanced diagrams, it has been shown in [14] that the leptonic CP asymmetry is not only analytically well-behaved, but it can also be of order unity if two of the heavy Majorana neutrinos have mass differences comparable to their decay widths. Because of this resonant enhancement of the leptonic asymmetries, we call this scenario of leptogenesis resonant leptogenesis.

An immediate consequence of resonant leptogenesis is that the singlet mass scale can be drastically lowered to TeV energies [14, 32]. However, these previous studies have not considered possible limits that may arise from the presently better constrained light-neutrino sector. In this paper we will analyze the scenario of resonant leptogenesis in light of the current solar and atmospheric neutrino data [46, 47]. In particular, we will show that resonant leptogenesis can occur with heavy Majorana neutrinos even as light as TeV, within the framework of light-neutrino scenarios with normal or inverted mass hierarchy and large - and - mixings, namely within schemes currently suggested by neutrino oscillation data.

Our predictions for the BAU are obtained after numerically solving a network of Boltzmann Equations (BEs) related to leptogenesis. In our analysis, we include the dominant collision terms that account for scatterings involving the SU(2) and U(1) gauge bosons. Furthermore, the resonantly enhanced CP-violating as well as CP-conserving effects on the scattering processes thanks to heavy-neutrino mixing are taken into account. To the best of our knowledge, these two important contributions to the BEs have not been considered in the existing literature before.

The proper description of the dynamics of unstable particles and their mixing phenomena is a subtle issue within the context of a field theory. To deal with this problem, one is compelled to rely on resummation approaches to unstable particles that consistently maintain all desirable field-theoretic properties, such as gauge-invariance, analyticity and unitarity [48, 49]. In this context, a resummation approach to unstable particle mixing in decay amplitudes was developed in [14] which preserves CPT invariance and unitarity [50, 32].

In this paper we address another important issue related to the proper subtraction of the so-called real intermediate states (RIS) from the -violating scattering processes that result from the exchange of unstable particles in the -channel. Such a subtraction is necessary in order to avoid double-counting in the BE’s [51] from the already considered decays and inverse decays of the unstable particles, namely those associated with heavy Majorana neutrino decays. By examining the analytic properties of the pole and residue structures [49, 50] of a resonant -violating scattering amplitude, we can identify the part of the amplitude that contains RIS contributions only. We find that the so-derived resonant amplitude exhibits the very same analytic form with the one obtained with an earlier proposed resummation method [14]. Since the present derivation does not rely on resorting to a kind of Lehmann–Symanzik–Zimmermann (LSZ) reduction formalism [52] for the decaying unstable particle, it offers therefore a firm and independent support to the earlier treatment presented in [14].

The paper is organized as follows: in Section 2 we discuss the generic structure of a heavy Majorana-neutrino model that possesses a low singlet scale and predicts nearly degenerate heavy Majorana neutrinos. Employing the Froggatt–Nielsen (FN) mechanism [53], we also put forward a generic texture for the light neutrino mass matrices that enable an adequate description of the present solar and atmospheric neutrino data. In Section 3 we address field-theoretic issues that arise from the proper subtraction of RIS from the -violating scattering processes. In particular, we explicitly demonstrate how the resonant part of the scattering amplitude is intimately related to the resummed decay amplitude derived earlier by means of an LSZ-type resummation approach [14]. Analytic formulae related to the general case of three heavy-Majorana-neutrino mixing are given in Appendix A. In Section 4 we derive the relevant network of BE’s for resonant leptogenesis, where the gauge-mediated collision terms and the resonantly enhanced CP-violating as well as CP-conserving contributions to scatterings due to heavy neutrino mixing are taken into account. Analytic expressions of reduced cross-sections for all relevant scattering reactions are presented in Appendix B. Our conclusions are summarized in Section 5.

##
2 Low-Scale Heavy Majorana-Neutrino Model

and
Neutrino Data

In this section, we first set up our conventions by briefly reviewing the low-energy structure of a minimally extended SM that includes heavy Majorana neutrinos. We then put forward a generic scenario that predicts nearly degenerate heavy Majorana neutrinos at the TeV scale and can naturally be realized by means of the FN mechanism [53]. In this generic scenario, the light-neutrino sector admits the Large Mixing Angle (LMA) Mikheyev–Smirnov–Wolfenstein (MSW) [38] solution and so may explain the solar neutrino data through a large - mixing. The light-neutrino sector also allows for a large --mixing to account for the atmospheric neutrino anomaly. Another property of our generic scenario is that it leads to a mass spectrum for the light neutrinos, denoted as (with the mass convention ), with normal or inverted hierarchy, depending on whether the lightest physical neutrino has predominantly a or a component. In particular, the generic scenario can accommodate the phenomenologically favoured neutrino-mass differences [46, 47]:

(2.1) |

at the 3 confidence level, with and .

A minimal, symmetric realization of a model with heavy Majorana neutrinos can be obtained by adding to the SM field content one right-handed (singlet) neutrino per family , with . The leptonic sector of this minimal model consists of the fields:

(2.2) |

where the obvious labelling, , will be employed. At temperatures larger than the critical temperature associated with the electroweak phase transition, the -dependent vacuum expectation value (VEV) of the SM Higgs doublet vanishes, i.e. . This is the epoch where a possible leptonic asymmetry created by out-of-equilibrium heavy Majorana-neutrino decays can be actively reprocessed into the BAU through the equilibrated -violating sphaleron interactions.

At this epoch relevant to leptogenesis, the dynamics of the early Universe is usually described by a Lagrangian in the unbroken gauge-symmetric phase of the theory. In this unbroken phase, the Lagrangian of the leptonic sector of the model under study may conveniently be expressed as

(2.3) |

with

(2.4) | |||||

(2.5) | |||||

(2.6) |

In the above, , and describe the kinetic terms, the Yukawa sector and the Majorana masses of the model, respectively. In addition, is the isospin conjugate of the Higgs doublet , where is the usual Pauli matrix, and the superscript denotes charge conjugation.

In the unbroken phase of the theory, only the singlet neutrinos are massive. Their physical masses can be found by diagonalizing the 3-by-3 singlet Majorana mass matrix in (2.6). The matrix is symmetric and in general complex, and can be diagonalized by means of a unitary transformation

(2.7) |

where is a -dimensional unitary matrix and denote the 3 physical masses of the heavy Majorana neutrinos , ordered as . Correspondingly, the flavour states and are related to the mass eigenstates through

(2.8) |

where and . Note that and do not transform independently of one another under a unitary rotation. In the physical basis, the Yukawa leptonic sector reads

(2.9) |

where a four-component chiral representation for all fermionic fields should be understood. In (2.9), is a diagonal positive matrix and is related to through a bi-unitary transformation: is a 3-by-3 unitary matrix that transforms the left-handed charged leptons to their corresponding mass eigenstates. Our computations of the leptonic asymmetries and collision terms relevant to leptogenesis will be based on the Lagrangian (2.9). , where

Having set the stage, it is now instructive to discuss the possible flavour structure of low singlet-scale models with nearly degenerate heavy Majorana neutrinos. Such a class of models may be constructed by assuming that lepton-number violation (and possibly baryon-number violation) occurs at very high energies at the GUT scale – GeV, or even higher close to the Planck scale GeV through gravitational interactions. On the other hand, operators that conserve lepton number are allowed to be at the TeV scale.

Since our interest is to resonant leptogenesis, the following sufficient and necessary conditions under which leptonic asymmetries of order unity can take place have to be satisfied by the model under discussion [14]:

(2.10) |

for a pair of heavy Majorana neutrinos . In (2.10), are the decay widths, which at the tree level are given by

(2.11) |

In the following, we present a rather generic scenario that minimally realizes the above requirements and still has sufficient freedom to describe the neutrino data. Our generic scenario is based on the FN mechanism [53]. Specifically, we introduce two FN fields, and , with opposite U(1) charges, i.e. . Under U(1), the following charges for the right-handed neutrinos are assigned:

(2.12) |

In addition, all other fields, including charged leptons, are singlets under U(1). Then, the singlet mass matrix assumes the generic form:

(2.13) |

where and
. In (2.13), sets up the scale of the leptonic symmetry
,^{1}^{1}1Similar textures of may result from
E theories [41], where the lepton numbers are
approximately broken [54]. while represents the scale of
violation. It is conceivable that these two scales may be
different from one another. For the case of our interest, it is TeV, while is considered to be many orders of magnitude
larger close to .

The FN mechanism also determines the strength of the Yukawa couplings. After spontaneous symmetry breaking (SSB), the resulting Dirac-neutrino mass matrix has the generic form

(2.14) |

where is a matrix containing the neutrino Yukawa couplings, expressed in the positive and diagonal basis of the respective charged-lepton Yukawa couplings.

If one assumes that and , a rather simple pattern for the mass matrices and emerges. In this case, the mass spectrum of the generic scenario under investigation contains one super-heavy Majorana neutrino, with a mass , and two nearly degenerate heavy Majorana neutrinos , with and a mass difference . Since it is , it can be readily seen that one of the crucial conditions for resonant leptogenesis in (2.10), i.e. , can naturally be satisfied within our generic framework.

In the above exercise, one should bear in mind that the FN mechanism can only give rise to an order-of-magnitude estimate of the different entries in the mass matrices and . Moreover, since our focus will be on the neutrino sector of this minimal model of resonant leptogenesis, we will not attempt to explain the complete quark- and charged-lepton-mass spectrum of the SM by analyzing all possible solutions through the FN mechanism. Such an extensive study is beyond the scope of the present article and may be given elsewhere.

We will now explicitly demonstrate that the mass textures stated in (2.13) and (2.14) can lead to viable light-neutrino scenarios, when the latter are confronted with the present solar and atmospheric neutrino data. To further simplify our discussion, we assume that the super-heavy neutrino decouples completely from the light-neutrino spectrum. As a result, to leading order in the FN parameters and , the 3-by-3 light-neutrino mass-matrix may be cast into the form:

(2.15) |

Here, are the neutrino Yukawa couplings in the weak basis described after (2.14). Note that effects due to the mass degeneracy of the heavy Majorana neutrinos contribute terms to . As long as , these sub-leading terms do not affect the light-neutrino mass spectrum and hence they can be safely neglected. The scenarios which we will address numerically in Section 4.3 are compatible with these limits on and .

Let us now present a concrete example by considering the following set of Yukawa couplings (given in units of ):

(2.16) |

For our illustrations, we also neglect the existence of possible CP-odd phases in the Yukawa couplings. Then, the light-neutrino mass matrix exhibits the structure

(2.17) |

It is not difficult to see that the above light-neutrino mass matrix can be diagonalized by large - and - mixing angles, i.e. and . Instead, the - mixing angle is estimated to be small, i.e. , as is suggested by the CHOOZ experiment [55, 47]. Furthermore, the physical light-neutrino masses derived from are approximately given by

(2.18) |

In deriving the last step of (2.18), we have used the fact that and . Observe that although our approach here has been different, the light-neutrino masses in (2.18) still obey the known seesaw mass relation [44], and scale independently of . In particular, one can easily check that (2.18) is compatible with the observed light-neutrino mass differences stated in (2.1). Even though the present example realizes a light-neutrino mass spectrum with normal hierarchy, an inverted hierarchy can easily be obtained by appropriately rearranging the Yukawa couplings in (2.16).

## 3 Subtraction of RIS and Leptonic Asymmetries

In this section we wish to address an important issue related to the proper subtraction of the so-called real intermediate states (RIS), e.g. heavy Majorana neutrinos , from the -violating scattering processes. As we will see in Section 4, such a subtraction is necessary in order to avoid double-counting in the BEs [51] from the already considered decays and inverse decays of the unstable heavy Majorana neutrinos. By studying the analytic properties of the pole and the residue structures of a resonant -violating scattering amplitude, we are able to identify the resonant part of a amplitude that contains RIS contributions only. The so-derived resonant amplitude can then be shown to exhibit the very same analytic form with the one obtained with an earlier proposed resummation method [14]. Another important result of our considerations is that we can define one-loop resummed effective Yukawa couplings that capture all dominant effects of heavy Majorana-neutrino mixing and CP violation.

### 3.1 Approach to the Subtraction of RIS

Let us first consider the simple scattering process , mediated by a single heavy-neutrino exchange . This exercise will help us to demonstrate our approach to subtracting the RIS part of an amplitude. The more realistic case of resonant leptogenesis with two heavy Majorana neutrinos will be discussed later on. To keep things at an intuitive level, we assume throughout this section that all particles involved in this process are scalar, e.g. scalar neutrinos or sneutrinos that are predicted in supersymmetric theories [56]. Nevertheless, we will discuss the complications that may arise in our considerations from the spinorial nature of the lepton and heavy neutrino fields. The -channel contribution to the scattering amplitude due to a single -exchange reads:

(3.1) |

where the Breit–Wigner-like propagator has been obtained by summing up an infinite series of heavy sneutrino self-energies . The dispersive part of the self-energy , which has been omitted here, can be suppressed by renormalization at the resonant region (see also our discussion below). Instead, its absorptive part is essential to obtain an analytically well-behaved amplitude at , where is the total decay width of the heavy sneutrino .

As we will see in Section 4.1, out-of-equilibrium constraints on the heavy-(s)neutrino width () imply (). In this kinematic regime, the so-called pole-dominance or narrow-width approximation constitutes a very accurate approach to subtract the RIS part from the squared matrix element . According to the pole-dominance approximation, we have

(3.2) | |||||

where and are the usual Dirac and step functions, respectively. Notice that in (3.2) only one residue related to the physical pole at is considered by means of the Cauchy theorem. Substituting (3.2) into , i.e. after squaring (3.1), we can uniquely isolate the RIS part for this process:

(3.3) |

This last result is fully consistent with the one presented in [51]. As we will discuss below, however, the above approach of identifying the proper RIS part of the squared amplitude becomes more involved in the presence of two strongly-mixed unstable particles.

The above derivation of the RIS component of the squared amplitude was based on the assumption that the heavy neutrino is a scalar particle. The spinorial nature of introduces further complications. It naively violates the factorized form of (3.3), and can no longer be written as a product of a production and a decay squared amplitude, i.e. it is not proportional to and to . Instead, we find

(3.4) | |||||

where is the on-shell 4-component spinor, with , and the trace is understood to act on the spinor space. The RIS squared amplitude can be written in the factorized form (3.3), only after we perform a Fierz rearrangement of the spinors and have integrated over the phase space of the initial and final states in the calculation of the corresponding reduced cross-section (see also our discussion in Appendix B). Then, after the phase-space integrations, the only non-vanishing Lorentz structure that survives is of the parity-even form: , where and are mass dependent constants. Based on this observation, it can be shown that the final result is fully equivalent to (3.3), and amounts to substituting into (3.4):

(3.5) |

It is important to note here that the above spin de-correlated subtraction of RIS carrying spin can always be carried out independently of the number of the exchanged particles in the -channel, such as heavy neutrinos, provided the resonant amplitude itself can be written as a sum of single-pole resonance terms that have the simple factorized form of (3.1). We will elucidate this point below, while deriving the RIS squared amplitude due to the exchange of two heavy Majorana neutrinos.

Let us therefore turn our attention to the case of two heavy Majorana neutrinos and . Analytic expressions for unstable particle-mixing effects with three heavy neutrinos are given in Appendix A. Again, we initially assume that the heavy neutrinos and are scalar particles, i.e. sneutrinos , but we will discuss in Section 3.2 the complications originating from their spinorial nature. As is shown in Fig. 2, the -dependent part of the amplitude may conveniently be expressed as

(3.6) |

In (3.6), represent the vertices () that include the wave-functions of the initial and final states. Analogously with the single heavy-sneutrino case described above, (with ) are the corresponding -propagators obtained by resumming an infinite series of heavy sneutrino self-energy graphs [14]:

(3.7) | |||||

where . We assume that the heavy sneutrino self-energies have already been renormalized in the on-shell (OS) scheme, i.e. they satisfy the properties:

(3.8) |

where indicates that only the dispersive part of the self-energies must be considered. Further details on OS renormalization in scalar theories may be found in [50].

To technically facilitate our discussion, it is convenient to introduce the following abbreviations:

(3.9) |

With the above definitions, the resummed -propagators in (3.7) can now be expressed in the simplified forms:

(3.10) |

In addition, we introduce the quantity

(3.11) |

In the OS scheme, with all contributions from unitarity cuts neglected, we obtain the known relation for the residues of the diagonal propagators: . However, is in general complex, but UV finite at order .

The two complex pole positions associated with the heavy sneutrinos are determined by the equation , where is given in (3.1). Since each resummed propagator given in (3.10) contains two complex poles at , it can be expanded about as follows:

(3.12) | |||||

(3.13) | |||||

(3.14) | |||||

(3.15) |

where the ellipses denote off-resonant terms which are non-singular at . Notice that we have retained the -dependent analytic form for the residues in the above complex pole expansion by virtue of the Cauchy theorem. Substituting (3.12)–(3.15) into the -channel amplitude in (3.6) and neglecting off-resonant terms yields

(3.16) |

with

(3.17) |

Here, it is important to remark that the expressions in (3.17) become identical at to the resummed decay amplitudes derived in [57], using an LSZ-type reduction formalism. Instead, in the present approach, the corresponding resummed decay amplitude can be obtained by studying the analytic structure of the residues of the complete resonant scattering amplitude, in which the unstable heavy sneutrinos are described as intermediate states in the -channel. The fact that these two approaches lead to identical results provides a firm support for the validity of the method developed in [14].

The RIS squared amplitude pertinent to the propagation of and can now be identified as

with . In (3.1), () and () are evaluated at (). Observe that up to higher-order wave-function renormalization effects, the RIS squared amplitude (3.1) for two unstable particles is very analogous to the corresponding one (3.3) derived for one single resonance. Although we will not address this issue in detail here, we simply note that our subtraction approach of isolating the RIS part of a squared amplitude can be extended to more than two unstable particles. The key observation to be made here is that such a generalization is possible, since the -channel dependent amplitude in the pole-dominance approximation can always be expressed as a sum of products of resummed vertices [cf. (3.17)] and Breit–Wigner propagators with single complex poles [cf. (3.1) and (3.16)].

One might worry that the subtraction approach described above may not
be applicable for the case of our interest with overlapping
resonances, i.e. for .
However, we should realize that the particles associated with the
complex poles, e.g. , of a transition amplitude have a
completely different thermal history, because of their many
decoherentional collisions with the other particles in the thermal
bath. On the other hand, the so-called quantum memory effects are
expected to play a relevant role only when the decay widths
or the mass difference are much
smaller than the Hubble parameter governing the expansion rate of
the early Universe at . In the former case, one
also finds that are weakly thermalized [20].
Otherwise, our subtraction approach not only takes into account the
part of the squared amplitude associated with the RIS, but also
provides a consistent description of the incoherent properties of the
heavy neutrinos ^{2}^{2}2For instance, within the context of thermal
leptogenesis, the use of a time-integrated CP-asymmetry formula, very
analogous to the one applied for a coherently oscillating
-system, leads to an erroneous incorporation of the
decoherentional properties of the thermal bath..

###
3.2 Resummed Effective Yukawa Couplings and

Leptonic Asymmetries

Until now in this section, the heavy Majorana neutrinos were mainly treated as scalar particles. However, our approach described above for subtracting the RIS from the squared amplitude of the process carries over very analogously to a strongly-mixed fermionic system, including the spinorial nature of the heavy Majorana neutrinos and .

To see the above point, we first introduce an abbreviated form for the one-loop corrected inverse -propagator matrix:

(3.19) |

where denote the self-energy transitions , renormalized in the OS scheme, and is the 4-momentum of . With the aid of these newly-introduced spinorial functions (3.19), the resummed -propagators are given by (suppressing the argument everywhere)