# Novel Scenarios for Majorana Neutrino Mass Generation and Leptogenesis from Kalb-Ramond Torsion.

###### Abstract

The Kalb-Ramond (KR) antisymmetric tensor field arises naturally in the gravitational multiplet of string theory. Nevertheless, the respective low-energy field theory action, in which, for reasons of gauge invariance, the only dependence on the KR field is through its field strength, constitutes an interesting model per se. In this context, the KR field strength also acts as a totally antisymmetric torsion field, while in four space-time dimensions is dual to an (KR) axion-like pseudoscalar field. In this context, we review here first the rôle of quantum fluctuations of the KR axion on the generation of Majorana mass for neutrinos, via a mixing with ordinary axions that may exist in the theory as providers of dark matter candidates. Then we proceed to discuss the rôle of constant in time (thus Lorentz violating) KR torsion backgrounds, that may exist in the early Universe but are completely negligible today, on inducing Leptogenesis by means of tree-level CP violating decays of Right Handed Massive Majorana neutrinos in the presence of such H-torsion backgrounds. Some speculations regarding microscopic D-brane world models, where such scenarios may be realised, are also given.

Novel Scenarios for Majorana Neutrino Mass Generation and Leptogenesis from Kalb-Ramond Torsion.

Nikolaos E. Mavromatos^{†}^{†}thanks: Speaker. ^{†}^{†}thanks: Currently also at: Theory Division, Department of Physics, CERN, CH 1211 Geneva 23, Switzerland.
This work is supported in part by the London Centre for Terauniverse Studies (LCTS), using funding from the European Research Council via the Advanced Investigator Grant 267352, and by STFC (UK) under the research grant ST/L000326/1.

Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s College London, Strand, London WC2R 2LS, UK

E-mail: Nikolaos.Mavromatos@kcl.ac.uk

\abstract@cs

## 1 Introduction and Summary

The discovery [1] of the Higgs boson at the CERN Large Hadron Collider (LHC) in 2012 constitutes an important milestone for the Ultra-Violet (UV) completion of the Standard Model (SM), verifying that a Higgs-like mechanism can explain the generation of most of the particle masses in the SM. Necvertheless, the origin of the small neutrino masses still remains an open issue. One such mechanism providing a rather natural explanation of the observed smallness of the light neutrino masses is the so-called see-saw mechanism [2], which necessitates the Majorana nature of the light (active) neutrinos and postulates the presence of heavy right-handed Majorana partners of mass . The right-handed Majorana mass is usually considered to be much larger than the lepton or quark masses. The origin of has been the topic of several extensions of the SM in the literature, within the framework of quantum field theory [2, 3] and string theory [4]. However, up to now, there is no experimental evidence for right-handed neutrinos or, in fact, for any extension of the SM, although some optimism of discovering supersymmetry in the current round of LHC (operating ultimately at 14 TeV energies) exists among particle physicists [5].

Until therefore such extensions of the SM are discovered, it is legitimate to search for alternative mechanisms for neutrino mass generation, that keep the spectrum of SM intact, except perhaps for the existence of right handed neutrinos that are allowed. Such minimal, non supersymmetric extensions of the Standard Model with three in fact right-handed Majorana neutrinos complementing the three active left-handed neutrinos (termed MSM), have been proposed [6], in a way consistent with current cosmology. Such models are characterised by relatively light right-handed neutrinos, two of which are almost degenerate, with masses of order GeV, and a much lighter one, almost decoupled, with masses in the keV range, which may play the role of warm dark matter. The right-handed neutrinos serve the purpose of generating, , through seesaw type mechanisms, the active neutrino mass spectrum, consistent with observed flavour oscillations. However, there are no suggestions for microscopic mechanisms for the generation of the right-handed neutrino mass spectrum in such scenarios.

Motivated by these facts we first review in this talk, in section 2, an alternative proposal for dynamical generation of Majorana mass for neutrinos propagating in space-time geometries with quantum-fluctuating torsion [7]. Microscopically this torsion may be provided by the spin-one antisymmetric tensor (Kalb-Ramond (KR)) field [8] of the gravitational multiplet of a string [9, 10], but the low energy theory of the KR torsion is an interesting field theory model that can be studied in its own right. In this scenario, the totally antisymmetric part of the torsion couples, via the gravitational covariant derivative, to all fermions in a way that the resulting interaction resembles that of the Lorentz and CPT-Violating pseudovector background with the axial fermion current in the Standard Model Extension (SME) of Kostelecky and collaborators [11]. The generation of (right-handed, sterile) neutrino masses in that case proceeds, as we shall review below, via chiral anomalous three-loop graphs of neutrinos interacting with the totally antisymmetric torsion quantum field. In four space-time dimensions, the latter is represented as an axion field [9, 10], whose mixing with ordinary axion fields, that in turn interact with the Majorana right-handed neutrinos via chirality changing Yukawa couplings, is held responsible for the right-handed Majorana neutrino mass generation through the aforementioned anomalous graphs. In the second part of the talk, in section 3, we discuss non-trivial backgrounds of the KR torsion field, constant in cosmic time, and thus Lorentz-violating, whose coupling to the fermion axial current induces extra sources of CP violation, which in the case of right-handed Majorana neutrinos assumed abundant in the early Universe, may lead to Leptogenesis [12, 13]. Finally, in the last part of the talk, section 4, instead of conclusions, I present some speculations as to how the above scenarios of Leptogenesis, which require relatively large backgrounds of the KR torsion in early epochs of the Universe, but negligible today, due to phenomenological reasons, may be realised in some microscopic string/brane models of the Universe, in which our four space-time dimensional brane world propagates in a bulk punctured by populations of point-like D-brane defects, interacting with right-handed neutrinos [14]. Some technical aspects of our approach are outlined in an Appendix.

## 2 Neutrino Mass Generation due to Quantum (Kalb-Ramond) Torsion

Let us for concreteness
consider Dirac fermions in a torsionful space-time [15]. The extension to the Majorana case is straightforward.
The relevant
action reads ^{1}^{1}1The second term in the right-hand-side of (2) is usually not written in flat space, as its contribution is equal to the first term plus a surface integral. However, the situation is different when spacetime is not flat. In fact the second term is needed in order preserve unitarity, allowing for the cancellation of an anti-hermitean term involving the trace of the spin-connection coefficients .:

(2.0) |

where , is the covariant derivative (including gravitational and gauge-field connection parts (the latter represented by ), in case the fermions are charged). The overline above the covariant derivative, i.e. , denotes the presence of torsion,

(2.0) |

In the formula above is the generator of the Lorentz group representation on four-spinors, while is the torsionful spin connection

(2.0) |

with the generalised Christoffel symbols in the presence of torsion.

The torsionful spin connection can be decomposed as: , where is the contorsion tensor. The latter is related to the torsion two-form via [15, 10]: . The presence of torsion in the covariant derivative in the action (2) leads, apart from the standard terms in manifolds without torsion, to an additional term involving the axial current

(2.0) |

The relevant part of the action reads:

(2.0) |

where is the dual of T: .

We next remark that the torsion tensor can be decomposed into its irreducible parts [15], of which is the pseudoscalar axial vector: , with . This implies that the contorsion tensor undergoes the following decomposition:

(2.0) |

where includes the trace vector and the tensor parts of the torsion tensor.

The gravitational part of the action can then be written as: where , with the hatted notation defined in (2).

In a quantum gravity setting, where one integrates over all fields, the torsion terms appear as non propagating fields and thus they can be integrated out exactly. The authors of [10] have observed though that the classical equations of motion identify the axial-pseudovector torsion field with the axial current, since the torsion equation yields

(2.0) |

From this it follows , leading to a conserved “torsion charge” . To maintain this conservation in quantum theory, they postulated at the quantum level, which can be achieved by the addition of judicious counter terms. This constraint, in a path-integral formulation of quantum gravity, is then implemented via a delta function constraint, , and the latter via the well-known trick of introducing a Lagrange multiplier field . Hence, the relevant torsion part of the quantum-gravity path integral would include a factor

where and the non-propagating S field has been integrated out. The reader should notice that, as a result of this integration, the corresponding effective field theory contains a non-renormalizable repulsive four-fermion axial current-current interaction, characteristic of any torsionful theory [15].

The torsion term, being geometrical, due to gravity, couples universally to all fermion species, not only neutrinos. Thus, in the context of the SM of particle physics, the axial current (2) is expressed as a sum over fermion species

(2.0) |

In theories with chiral anomalies, like the quantum electrodynamics part of SM, the axial current is not conserved at the quantum level, due to anomalies, but its divergence is obtained by the one-loop result [16]:

(2.0) | |||||

We may then partially integrate the second term in the exponent on the right-hand-side of (2) and take into account (2.0). The reader should observe that in (2.0) the torsion-free spin connection has been used. This can be achieved by the addition of proper counter terms in the action [10], which can convert the anomaly from the initial to . Using (2.0) in (2) one can then obtain for the effective torsion action in theories with chiral anomalies, such as the QED part of the SM:

(2.0) |

A concrete example of torsion is provided by string-inspired theories, where the totally antisymmetric component of the torsion is identified with the field strength of the spin-one antisymmetric tensor (Kalb-Ramond (KR) [8]) field , where the symbol denotes antisymmetrization of the appropriate indices. The string theory effective action depends only on as a consequence of the “gauge symmetry” that characterises all string theories. Indeed, in the Einstein frame, to first order in the string Regge , the bosonic part of the low-energy effective action (in four large target-space-time dimensions) is given by [9]

(2.0) |

where , with the four-dimensional (gravitational) constant, is the string mass scale, the (dimensionless) compactification volume in units of the Regge slope of the string and is a dilaton potential.

It can be shown [9] that the terms of the effective action up to and including quadratic order in the Regge slope parameter , of relevance to the low-energy (field-theory) limit of string theory, which involve the H-field strength, can be assembled in such a way that only torsionful Christoffel symbols, appear:

(2.0) |

where is the ordinary, torsion-free, symmetric connection, and is the gravitational constant. In four space-time dimensions, the dual of the H-field is indeed the derivative of an axion-like field,

(2.0) |

where is a pseudoscalar field, entirely analogous to the field above, hence the use of the same letter to describe it. It is termed the Kalb-Ramond (KR) (or “gravitational”) axion field, to distinguish from the ordinary axion fields that can play the rôle of dark matter in particle physics models.

We mention at this point that background geometries with (approximately) constant background torsion, where Latin indices denote spatial components of the four-dimensional space-time, may characterise the early universe. In such cases, as we shall discuss in section 3, the H-torsion background constitutes extra source of CP violation, necessary for lepotogenesis, and through Baryon-minus-Lepton-number (B-L) conserving processes, Baryogenesis, and thus the observed matter-antimatter asymmetry in the Universe [12, 13]. Today of course any torsion background should be strongly suppressed, due to the lack of any experimental evidence for it. Scenarios as to how such cosmologies can evolve so as to guarantee the absence of any appreciable traces of torsion today can be found in [13] and will be briefly reported upon in sections 3 and 4.

In the remainder of this section, we shall consider the effects of the quantum fluctuations of torsion, which survive the absence of any torsion background. An important aspect of the coupling of the torsion (or KR axion) quantum field to the fermionic matter discussed above is its shift symmetry, characteristic of an axion field. Indeed, by shifting the field by a constant: , the action (2.0) only changes by total derivative terms, such as and . These terms are irrelevant for the equations of motion and the induced quantum dynamics, provided the fields fall off sufficiently fast to zero at space-time infinity. The scenario for the anomalous Majorana mass generation through torsion proposed in [7], and reviewed here, consists of augmenting the effective action (2.0) by terms that break such a shift symmetry. To illustrate this last point, we first couple the KR axion to another pseudoscalar axion field . In string-inspired models, such pseudoscalar axion may be provided by the string moduli [17] and play the rôle of a dark matter candidate. The proposed coupling occurs through a mixing in the kinetic terms of the two fields. To be specific, we consider the action (henceforth we restrict ourselves to right-handed Majorana neutrino fermion fields)

(2.0) | |||||

where is the charge-conjugate right-handed fermion , is the axial current of the four-component Majorana fermion , and is a real parameter to be constrained later on. Here, we have ignored gauge fields, which are not of interest to us, and the possibility of a non-perturbative mass for the pseudoscalar field . Moreover, we remind the reader that the repulsive self-interaction fermion terms are due to the existence of torsion in the Einstein-Cartan theory. The Yukawa coupling of the axion moduli field to right-handed sterile neutrino matter may be due to non perturbative effects. These terms break the shift symmetry: .

It is convenient to diagonalize the axion kinetic terms by redefining the KR axion field as follows: . This implies that the effective action (2.0) becomes:

(2.0) |

Thus we observe that the field has decoupled and can be integrated out in the path integral, leaving behind an axion field coupled both to matter fermions and to the operator , thereby playing now the rôle of the torsion field. We observe though that the approach is only valid for otherwise the axion field would appear as a ghost, i.e. with the wrong sign of its kinetic terms, which would indicate an instability of the model. This is the only restriction of the parameter . In this case we may redefine the axion field so as to appear with a canonical normalised kinetic term, implying the effective action:

(2.0) | |||||

Evidently, the action in (2.0) corresponds to a canonically normalised axion field , coupled both to the curvature of space-time, à la torsion, with a modified coupling , and to fermionic matter with chirality-changing Yukawa-like couplings of the form .

The mechanism for the anomalous Majorana mass generation is shown in Fig. 1. We may now estimate the two-loop Majorana neutrino mass in quantum gravity with an effective UV energy cut-off . Adopting the effective field-theory framework of [18], the gravitationally induced Majorana mass for right-handed neutrinos, , is estimated to be:

(2.0) |

In a UV complete theory such as strings, the cutoff and the Planck mass scale are related.

It is interesting to provide a numerical estimate of the anomalously generated Majorana mass . Assuming that , the size of may be estimated from (2.0) to be

(2.0) |

Obviously, the generation of is highly model dependent. Taking, for example, the quantum gravity scale to be GeV, we find that is at the TeV scale, for and . However, if we take the quantum gravity scale to be close to the GUT scale, i.e. GeV, we obtain a right-handed neutrino mass keV, for the choice . This is in the preferred ballpark region for the sterile neutrino to qualify as a warm dark matter [6].

In a string-theoretic framework, many axions might exist that could
mix with each other [17]. Such a mixing can give rise to reduced UV
sensitivity of the two-loop graph shown in Fig. 1. In such cases, the anomalously generated Majorana mass
may be estimated to be:

for , and
for .

It is then not difficult to see that three axions with next-to-neighbour mixing as discussed above would be sufficient to obtain a UV finite result for at the two-loop level. Of course, beyond the two loops, will depend on higher powers of the energy cut-off , i.e. , but if , these higher-order effects are expected to be subdominant.

In the above -axion-mixing scenarios, we note that the anomalously generated Majorana mass term will only depend on the mass-mixing parameters of the axion fields and not on their masses themselves, as long as . As a final comment we mention that the values of the Yukawa couplings may be determined by some underlying discrete symmetry [19], which for instance allows two of the right-handed neutrinos to be almost degenerate in mass, as required for enhanced CP violation of relevance to leptogenesis [13], or in general characterises the MSM [6]. These are interesting issues that deserve further exploration.

We stress at this point that, although above we kept our discussion general, and included right-handed neutrinos in the spectrum of our models, assuming that the Yukawa-oridinary-axion interactions with them are the dominant ones, nevertheless our mechanism applies also to models with no right-handed neutrinos. Indeed, all one has to do in such cases is to assume that the ordinary-axion-Yukawa interactions occur in the active neutrino sector directly. At any rate, as in the next section our interest lies on producing Leptogenesis using (heavy) right handed neutrinos, from now on we do assume the existence of such heavy particles. However, as we shall argue below, for producing sufficient Leptogenesis using a KR torsion background in the early Universe, we need only one generation of heavy right handed Majorana (RHM) neutrinos. It goes without saying that the extension of our scenario to three RHM neutrino species, e.g. as required in seesaw models [2, 3], is straightforward.

## 3 Kalb-Ramond-Torsion Backgrounds in the Early Universe and Leptogenesis

In this section we consider first the interacrtion of massive fermion species , , of mass , with non-trivial gravitational and Kalb-Ramond H-torsion backgrounds, which we assume characterise the early Universe [12, 13]. From the considerations in the previous section 2, it becomes clear that the physical content of the effects of an H-torsion background on Leptogenesis is contained in the following fermionic action (we start with Dirac fermions, the generalisation to Majorana ones, of relevance to us here, is straightforward and will be considered later on):

(3.0) |

where summation over the species index is implied, , and the axial vector is defined by

(3.0) |

In this last step we have used (2), (2.0) and the symmetry of the torsion-free Christoffel symbol. In the special cases of interest here, either flat (Minkowski) or Robertson-Walker space-times (which do not contain off-diagonal metric elements mixing temporal and spatial components), the axial vector is non-trivial and constitutes just the dual of the torsion tensor

(3.0) |

In four space-time dimensions where is the Kalb-Ramond (KR) axion field (2.0).

For a bosonic string theory (with four uncompactified dimensions) in non-trivial cosmological backgrounds, a world-sheet description has been provided by a sigma model that can be identified with a Wess-Zumino-Witten type conformal field theory [20].This construction has led to exact solutions (valid to all orders in the Regge slope, ) for cosmological bosonic backgrounds with non-trivial metric, antisymmetric tensor and dilaton fields. Such solutions, in the Einstein frame, consist of (i) a Robertson-Walker metric with a scale factor where is the cosmic time, (ii) a dilaton field that scales as and (ii) a KR axion field scaling linearly with the cosmic time, with denoting the background value of (cf. (2.0). The resulting background axial vector has only a non-trivial temporal component

(3.0) |

and

(3.0) |

in the Robertson-Walker frame.

In the absence of fermions, such constant H-torsion background in a Robertson-Walker space time do not constitute solutions equations of motion derived from the perturbative (in ) action (2.0) (cf. Appendix). However, in the presence of fermions coupled to the torsion -field as in (3), the four-dimensional low-energy effective action gives the following equations of motion for the graviton and antisymmetric tensor:

(3.0) |

where denotes higher order terms in in the gravitational part of the action, is the stress-energy tensor of fermionic matter and . There is of course an equation of motion for the dilaton which provides additional constraints for the background. In order to simplify the analysis we will assume a constant dilaton below. A full analysis is given in the Appendix.

It is conceivable, as argued in [13], that in the presence of high temperature and densities of fermions (relevant for the early universe), a condensate of the temporal component of the fermion axial current may be formed, in which case one may have (Lorentz-violating) perturbative fixed points corresponding to a constant -torsion. Indeed, from the equation for the antisymmetric tensor field (assuming a constant dilaton), (3), we observe that it can be solved upon using the pseudoscalar dual field defined in (3.0):

(3.0) |

where is a constant of proportionality. From the above equations (in truncated form) it is clear that the fermion condensate can be a source of torsion . Hence the non-perturbative solution (3.0), derived in [20] is still qualitatively valid since, from (3.0), we have

(3.0) |

where runs over appropriate fermion species.

In [21] a calculation in strong coupling gauge theory supported the formation of axial vector fermion condensates. At weak gauge coupling the condensates cease to form. The gauge coupling and string coupling are related in string model building of the fundamental interactions [9]. If the dilaton, rather than being a constant becomes more negative with time (as in the explicit solutions from bosonic string theory considered in [20]), the string coupling and gauge coupling decrease with time; so there will be a time (and a critical value of , ) when the gauge coupling will be too weak to support a condensate. This is of course qualitative: currently we can only speculate that the value of is achieved in the era of leptogenesis. For a fundamental mechanism we would need to be quantitative but this has not been achieved as yet. Nevertheless, we note that in late era of the Universe, when the axial current condensate becomes vanishingly small, from (3.0) we find (upon ignoring (as subleading) the higher order terms), that the rate of change of the field diminishes with the cosmic time as the cube of the scale factor

(3.0) |

At this stage we notice that there is another challenging problem at the microscopic level related to the cosmological constant and the need for fine-tuning, a generally unsolved problem at present. The time dependent pseudoscalar, with constant rate (3.0) induces a vacuum energy term of the type of a positive cosmological constant once fluctuations around the background are allowed (cf. Appendix, Eq. (4.0)). There are ways by means of which such positive contributions can be cancelled by negative vacuum energy contributions; for instance, in brane world models propagating in higher-dimensional bulk space times, one may obtain negative contributions on the brane vacuum energy from the bulk dynamics [22, 14], as we shall discuss in section 4. Such negative contributions do not constitute an instability in the presence of the H-torsion constant background as they are cancelled by it. The real issue, however, is how, once the fermion condensates have disappeared, the brane vacuum energy acquires the present tiny value consistent with observational cosmology. One possible way to achieve this is associated with the dynamics of time dependent dilatons, on which we make some speculative remarks in the last part of this section. Hence, at present we do not have a microscopic derivation but rather a microscopic motivation for postulating our H-torsion background. Of course such issues do not apply to the neutrino mass generation mechanism of section 2, which assumes zero H-torsion backgrounds.

A final remark before moving onto a discussion of leptogenesis concerns the path integration of quantum fluctuations of the antisymmetric torsion background. Such fluctuations need to be included for an accurate estimate of the energy budget of the universe. Indeed, as we have seen in section 2, such a process leads to repulsive axial current four fermion itneractions in the action that may contribute to the energy budget, especially in eras where condensation of the fermion axial current occurs. To take formal account of such flucgtuations, a split of the field can be made explicit into background and quantum fluctuation parts, , where the background satisfies (3.0). The result for the relevant factor of the path integral after integration over the quantum fluctuations reads

(3.0) |

where is the action in the presence of the background torsion, given by the sum of (2.0), and (3). For a more detailed discussion we refer the reader to ref. [13]. The presence of a fermion axial condensate of the type (3.0) implies in general a generalised background for the fermions in a Hartree-Fock approximation

(3.0) |

where, on account of our previous discussion (cf. (3.0), (3.0)) only temporal components of will be non zero, with . The reader should notice that the sum over fermion species , does not include Majorana neutrinos that yield zero contributions to the condensate [13]. Hence, it is the other fermion fields of the SM, such as quarks, that could contribute to such condensates.

We commence our discussion on H-torsion-background induced leptogenesis by considering a minimal extension of the Standard Model, with one right-handed massive (of mass ) Majorana neutrino field in the presence of constant axial backgrounds, (3.0). The right-handed neutrino sector of such a model is described by

(3.0) |

where is the Majorana field with mass (that can be itself generated dynamically by the fluctuations of the torsion field in the way described in section 2), is a lepton field of the SM, with a generation index, and the adjoint of the Higgs field is defined by the relation , with are SU(2) group indices. We note that, since our primary motivation here is to identify the axial background field with the totally antisymmetric part of a torsion background, one should also consider the coupling of the axial field to all other fermions of the SM sector, (=leptons, quarks) via a universal minimal prescription. Hence, the coupling with all fermionic species is the same : . Specifically, as we have seen above, the identification of the torsion background with a homogeneous and isotropic cosmological Kalb-Ramond field in a string-theory-inspired model will lead to axial backgrounds with non-trivial temporal components only

(3.0) |

Since in SM the leptons have definite chirality , the Yukawa interactions can be rewritten as

(3.0) |

Using the properties of the charge conjugation matrix and the Majorana condition, it is again seen to be equivalent to

(3.0) |

It should be noted that the two hermitian conjugate terms in the Yukawa Lagrangian are also CPT conjugate. This is to be expected on the basis of the CPT theorem. In fact CPT violation is introduced only by interactions with the background field. In the absence of the background, the squared matrix elements obtained from tree level diagrams for the two decays would be the same [23]. From the form of the interaction Lagrangian in Eq. (3.0), it is straightforward to obtain the Feynman rules for the diagrams giving the decay of the Majorana particle in the two distinct channels. It also allows us to use positive frequency spinors both for the incoming Majorana particle and for the outgoing leptons.

Let us now turn to the study of the tree-level decay processes of a Majorana right-handed neutrino into leptons and Higgs fields, depicted in (the left panel of) fig. 2. For the qualitative purposes of the talk we assume that the massive right-handed neutrino is initially at rest. The more general case is studied in ref. [13]. For the channel the decay rate is

(3.0) |

while, for the other channel, , the respective decay rate reads

(3.0) |

The reader should notice that the decay rate of one process is obtained from the other upon flipping the sign of . The total decay rate is

(3.0) |

It is worthwhile observing that this mechanism can produce a lepton asymmetry even with only one right-handed neutrino, whereas the standard leptogenesis scenario [24] requires at least three generations. Moreover, the occurrence of leptogenesis here is just due to decay processes at tree level, since the required violation is introduced by the background field that enters in the external lines of Feynman diagrams.

The decay process goes out of equilibrium when the total decay rate drops below the expansion rate of the Universe, which is given by the Hubble constant [25]

(3.0) |

Here is the effective number of degrees of freedom of all elementary particles and is the Planck mass. From the last equation one can estimate the decoupling temperature , in terms of the unknown parameters , and , is

(3.0) |

In order for the inverse decay to be suppressed by the Boltzmann factor, we have to impose the further requirement that when (delayed decay mechanism [25, 24]). From this condition one can determine a lower bound for the mass . In fact we are lead to the following inequality

(3.0) |

where . If we require that the bound is satisfied for all values of we get

(3.0) |

For us the Yukawa coupling is a free parameter. If we assume , , we get an order of magnitude estimate for the lower bound of the right-handed Majorana neutrino mass

(3.0) |

Clearly in this mechanism for leptogenesis the low-mass right-handed neutrinos of [6] do not fit.

The lepton number density produced can then be estimated in the following way: by assumption all the neutrinos are at rest before the decay; hence the branching ratios of the decays are given by and . The decay of a single neutrino produces the lepton number

(3.0) |

Multiplying this quantity by the initial abundance of right-handed Majorana neutrinos at the temperature one gets an approximate estimate of the lepton number density. The density of the Majorana neutrinos is given by