Leptogenesis from heavy righthanded neutrinos in CPT violating backgrounds
Abstract
We discuss leptogenesis in a model with heavy righthanded Majorana neutrinos propagating in a constant but otherwise generic CPTviolating axial timelike background (motivated by string theory). At temperatures much higher than the temperature of the electroweak phase transition, we solve approximately, but analytically (using Padé approximants), the corresponding Boltzmann equations, which describe the generation of lepton asymmetry from the treelevel decays of heavy neutrinos into Standard Model leptons. At such temperatures these leptons are effectively massless. The current work completes in a rigorous way a preliminary treatment of the same system, by some of the present authors. In this earlier work, lepton asymmetry was crudely estimated considering the decay of a righthanded neutrino at rest. Our present analysis includes thermal momentum modes for the heavy neutrino and this leads to a total lepton asymmetry which is bigger by a factor of two as compared to the previous estimate. Nevertheless, our current and preliminary results for the freezeout are found to be in agreement (within a \(\sim 12.5\%\) uncertainty). Our analysis depends on a novel use of Padé approximants to solve the Boltzmann equations and may be more widely useful in cosmology.
1 Introduction and motivation
At first sight, the asymmetry (1) (and the result (2)) appears to be in conflict with fundamental properties of relativistic quantum field theories, which form the basis of our phenomenology of elementary particles. Specifically, in flat spacetime, any unitary and local Lorentz invariant quantum field theory, which respects unitarity and locality, should be described by a Lagrangian that is invariant under CPT transformations where C denotes charge conjugation, T denotes reversal in time and P denotes parity (spatial reflection) transformations. This is the celebrated CPT theorem [3]. For the physics of the early universe based on any Lorentz invariant quantum field theory, such a theorem implies that matter and antimatter should be created in equal amounts after the Big Bang. If such is the case, the universe today would be filled with radiation, as a result of matterantimatter annihilation processes, in conflict with (2).

Baryon (B) number violation.

Charge (C) and chargeparity (CP) symmetries need to be broken.

Chemical equilibrium does not hold during an epoch in the early universe, since chemical equilibrium washes out asymmetries.
Unfortunately, within the framework of the Standard Model (SM), although Sakharov’s axioms can be qualitatively reproduced, especially because one has both B and CP violation in the quark sector, the resulting baryon asymmetry is several orders of magnitude smaller than the observed one (1) [17, 18, 19]. There are several ideas that go beyond the SM (e.g. grand unified theories, supersymmetry, extra dimensional models etc.) and provide extra sources of CP violation, necessary for yielding the observed magnitude for the asymmetry. Some of these attempts, involve the elegant mechanism of baryogenesis via leptogenesis, in which a lepton asymmetry is generated first, by means of decays of right handed sterile neutrinos to SM particles; the lepton asymmetry is subsequently communicated to the baryon sector by means of sphaleron processes which violate both Baryon (B) and Lepton (L) numbers, but preserve the difference BL [20, 21, 22, 23, 24, 25, 26]. Heavy sterile neutrinos, through the seesaw mechanism [27, 28, 29, 30, 31], play another essential rôle in particle physics, since they provide a natural explanation for the existence of three light neutrinos with masses small compared to other mass scales in the SM), as suggested by observed neutrino oscillations [32, 33]. Fine tuning and some ad hoc assumptions are involved though in such scenarios, especially in connection with the magnitude of the CP violating phases and the associated decay widths. Consequently the quest for a proper understanding of the observed BAU still requires further investigation.
In the scenario of Sakharov it is assumed that CPT symmetry holds in the very early universe and this leads to the equal production of matter and antimatter. CPT invariance is regarded as fundamental since it is a direct consequence of the celebrated CPT theorem [3]. However, it is possible that some of the assumptions in the proof of the CPT theorem do not hold in the early universe, leading to violations of CPT symmetry. Sakharov has stated that nonequilibrium processes are necessary for BAU in CPT invariant theories. If the requirement of CPT is relaxed, the necessity of nonequilibrium processes can be dropped . In a lowenergy version of quantum gravity Lorentz invariance and unitarity are likely to emerge since not all degrees of freedom are accessible to a lowenergy observer. Lorentz invariance violation has been singled out in Ref. [34] as a fundamental reason for inducing CPT violation (CPTV) and vice versa. (However, such claims have been disputed in [35, 36], through counterexamples of Lorentz invariant systems, which violate CPT through relaxation, for example, of locality.) In our work we will consider Lorentz invariance violating (LV) backgrounds in the early universe as a form of spontaneous violation of Lorentz and CPT symmetry.
If LV is the primary source of CPTV, then the latter can be studied within a local effective field theory framework, which is known as the Standard Model Extension (SME) [37]. The latter provides the most general parametrization for studying the phenomenology of Lorentz violation in a plethora of physical systems, ranging from cosmological probes, to particle and precision atomic physics systems. For the current era of the universe [38, 39, 40, 41] very stringent upper bounds on the potential amount of Lorentz and CPT violation have been placed by such systems. However, under the extreme conditions present in the very early universe, such violations could be significantly stronger than in the present era (where they could be extremely suppressed (or absent), in agreement with current stringent constraints).^{1} In a previous work [43] we presented a phenomenological model for generating a lepton asymmetry via CPTV in the early universe. The model was based on a specific extension of the SM, involving massive Majorana righthanded neutrinos (RHN), propagating on a Lorentz and CPTV, constant in time, axial vector background coupling to fermions. The latter could be traced back to a specific configuration of a cosmological Kalb–Ramond (KR) antisymmetric tensor field [44] that appears in the gravitational multiplet of string theory [45, 46, 47, 48, 49], and plays the rôle of torsion in a generalised connection, although we do not restrict ourselves to such an identification.^{2} The involvement of sterile RHN in the model is physically motivated primarily by the need to provide a natural explanation for the light neutrino masses of the SM sector. The lightest RHN may also have a potential role as (warm) dark matter candidates [25, 26, 51, 52, 53]. However, in our CPTV models sterile neutrinos responsible for leptogenesis have masses in the \(10^5\) GeV range or higher [43] and so cannot be considered as dark matter.
In [43] we only gave a qualitative and rather crude estimate of the induced CPTV lepton asymmetry, based on the decaying right handed Majorana neutrino being at rest. In this way it was possible to estimate the lepton asymmetry, without following the standard procedure of solving the appropriate Boltzmann equation that determines correctly the asymmetry value at decoupling of RHN. In the early universe the heavy righthanded neutrinos are not at rest but have a MaxwellBoltzmann momentum distribution. The purpose of this article is to properly take into account this momentum distribution in the calculation of the lepton asymmetry.
The structure of the article is as follows: in the next Sect. 2 we review the model of [43] and an earlier estimate of the CPTVbackground induced lepton asymmetry, which shall be compared with the much more accurate result of the present article, obtained by solving the appropriate Boltzmann equations analytically. We commence our analysis by considering the lepton asymmetry associated with the decays of the RHN into charged leptons. In Sect. 3, we construct the appropriate system of Boltzmann equations in the presence of a weak CPTV axial background involved in the problem, and compare it with the standard CP violating case [20, 21, 22, 23, 24, 25, 26]. In Sect. 4, we solve the Boltzmann equations using Padé approximants [54], which is an approximation popular in several fields of physics, ranging from statistical mechanics to particle physics and quantum field theory [55, 56, 57, 58, 59, 60]. In this way, we manage to compute the induced lepton asymmetry at RHN decoupling analytically, avoiding numerical treatment. It should be remarked, that setting up and solving such a system of differential equations is a highly nontrivial and algebraically complicated task. Our analytical results agree (within \(\sim 12.5\%\) accuracy) with our earlier preliminary estimates of the freezeout point, as outlined, in [43]. In view of this, we consider our system of Boltzmann equations as providing another efficient use of Padè approximants, this time with relevance to cosmology. The lepton asymmetry that we find in our analytic treatment is slightly larger (by a factor of about 2) than the estimate of [43]; this is to be expected, since nonzero momentum modes of the RHN have been included. In Sect. 5 we complete our analysis by including the contributions to the Boltzmann equations and the lepton asymmetry coming from the decays of the RHN into the neutral Higgs and active neutrinos. Our calculations show that the resultant lepton asymmetry increases by a factor \(\sim 2\) compared to the one based on the RHN decays to charged leptons only. Conclusions and outlook are given in Sect. 6. A review of the formalism and derivations of the corresponding decay amplitudes and thermally averaged rates used in the Boltzmann equations, are presented in several Appendices.
2 Review of the CPT violating model for leptogenesis
In what follows, we shall first calculate the lepton asymmetry based only on the decay channels (9), involving charged leptons in the final stage. In Sect. 5 we shall include the neutral decay channels (10), into active neutrino and neutral Higgs. As we will demonstrate, the complete lepton asymmetry is increased by a factor of 1.98 as compared to the contribution from the charged channels (9) alone (the case considered in the estimate of [43]). It will turn out that the estimate of [43]) for the lepton asymmetry is of the same order of magnitude as the one derived in our current accurate treatment, thus providing an a posteriori justification of the simplified analysis of [43].
In order to get a physically correct and more accurate estimate of the induced lepton asymmetry, the relevant Boltzmann equation needs to be studied in detail, since the heavy righthanded neutrinos are not at rest, but characterised by the Maxwell–Boltzmann momentum distribution in the early universe. This requires a good approximation for the thermally averaged decay rates (9) of all the relevant processes and will be the subject of the current article. As the Boltzmann equations associated with the leptogenesis scenario advocated here and in [43] involve appropriately averaged thermal rates of the decays (9), we develop in Appendix 7.D the relevant formalism (for \(B_0/m_N \ll 1\)); the formalism will be used in the next Sect. 3 to set up the pertinent system of Boltzmann equations. We shall often borrow methods and techniques from the standard case of CPT conserving RHNinduced leptogenesis, where the CPTV background \(B_0\) is absent, but there is CP violation in the lepton sector [20, 21, 24]. In the current article we shall closely follow the formalism outlined in [24].
3 Setting up the Boltzmann equations for leptogenesis in the presence of CPTV backgrounds
 (i)the heavy neutrino abundance in units of entropy density (cf. (26)), and averaged over helicities \(\lambda =\pm \,1\):and$$\begin{aligned} \bar{Y}_{N} \equiv \; \dfrac{Y_{N}^{()} + Y_{N}^{(+)}}{2} \end{aligned}$$(32)
 (ii)the leptonasymmetry for the processes (9), defined in terms of the lepton abundances:where we took into account that the asymmetry is generated between the leptons of helicity \(\lambda = 1\) and the antileptons of helicity \(\lambda = +1\), since these are the only decays for the heavy neutrino (9), for each of which helicity is conserved. There will be no asymmetry between leptons of helicity \(\lambda = +1\) and antileptons of helicity \(\lambda = 1\) and so \(Y_{l^{}}^{(+)}  Y_{l^{+}}^{()} = 0\). Moreover, all of the negative helicity lepton abundance \(Y_{l^{}}\) comes from the decay of the negative helicity heavy neutrino. The same argument for the antilepton positive helicity abundance generated by the positive helicity heavy neutrinos. These imply the second of the relations (33).$$\begin{aligned} \mathcal {L} \equiv&\; Y_{l^{}}^{()}  Y_{l^{+}}^{(+)} = 2\Big [\bar{Y}_{l^{}}  \bar{Y}_{l^{+}}\Big ], \nonumber \\ \bar{Y}_{l} \equiv&\; \dfrac{Y_{l}^{()} + Y_{l}^{(+)}}{2} = \frac{Y_{N}^{()} + Y_{N}^{(+)}}{2} = \bar{Y}_{N}, \end{aligned}$$(33)
3.1 Heavyrighthandedneutrino abundance Boltzmann equation
3.2 Lepton asymmetry Boltzmann equation
4 Solutions to the system of Boltzmann equations
In this section we derive approximate analytic solutions of the system of Boltzmann equations (56), (62), which will allow us to compute the lepton asymmetry induced by the CPTV background in our model. So far we have derived equations for the RHN and lepton asymmetry (cf. (56) and (62) respectively) for high temperatures, \(z < 1\). However, we are eventually interested in solutions of the corresponding Boltzmann equations at the RHN decoupling temperatures (13), (15), where \(z \sim 1\) [43]. We shall attempt to extrapolate our results above to this case, by performing a Taylor expansion of the series solutions to these differential equations. The expansion takes place around an arbitrarily chosen point in the interval \(0< z < 1\), where the solution is valid, taking proper account of the (thermodynamic equilibrium) boundary conditions for the abundances as \(z \rightarrow 0\) (see Appendix 7.C), which fixes the integration constants characterising the solutions. In our analysis below, we take, as a Taylor expansion point, the midpoint of the interval \((0, 1),z=0.5\) .
To extrapolate the solutions to the regime \(z \simeq 1\), we shall use a Padé approximation [54]. As well known, a Padé expansion can accelerate the convergence of an asymptotic expansion or, for a series, turn a divergence into a convergence. It is widely used for producing in solving approximately complicated problems in several fields of physics, ranging from statistical mechanics to particle physics and quantum field theory [55, 56, 57, 58, 59, 60]. Here we present another useful application of the method in cosmology. We outline the general concepts of the Padé approximants method and the specific algorithm used in our computation in this work in Appendix 8.
4.1 Solution to the heavyneutrino Boltzmann equation
4.2 Solution to the lepton asymmetry Boltzmann equation
In this subsection, we proceed with the substitution of the previous result onto the Boltzman equation (62) and proceed with its solution, which will allow for a determination of the lepton asymmetry.
4.3 Series solutions of the Boltzmann equations
From either (82) or (96), we obtain that phenomenologically relevant leptogenesis in our system, in the sense of (21), is achieved for \(B_0/m_N ={\mathcal O}(10^{9}10^{8})\), which is in the same approximate range as the estimate of [43], but here the result includes all the nonzero momentum modes of the heavy neutrino. This implies that for \(m_N ={{\mathcal {O}}}(100)\) TeV, we must have a \(B_0\) in the range \(B_0 \sim 0.11~\mathrm{MeV}\) for leptogenesis to lead to the observed baryogenesis via the BL conserving sphaleron processes.
Comparing the freezeout points between the two approximate methods (82) and (95), we observe agreement with only 12.5 % uncertainty, indicating stability of the freezeout point in the region around one. This completes our analysis. Perhaps as we mentioned earlier, a full numerical solution will yield a freezeout point closer to the qualitative value of [43], although we should emphasize that the above approximate analyses have yielded results in this respect that are of the same order of magnitude. This adds confidence to the efficient application of Padé approximant method to our cosmological problem.
5 Inclusion of the neutral Higgs portal
In the system of Boltzmann equations we will now include the contributions from the decays of the RHN into a neutral Higgs field \(h^0\) and an (active) neutrino \(\nu \) of the SM sector (see Eq. (10)). As we shall demonstrate below, the computed (complete) lepton asymmetry has an approximate factor of two compared with the one based only on the charged lepton decay channels (9) .
5.1 The complete heavy neutrino Boltzmann equation
5.2 The complete lepton asymmetry Boltzmann equation
5.3 Integrating factor solutions of the complete Boltzmann equations
5.4 Series solutions of the complete Boltzmann equations
6 Conclusions and outlook
In this work we have completed the analysis presented in an earlier work [43] by computing the lepton asymmetry generated due to the decays of heavy righthanded neutrinos in the presence of a CPTV axial vector background with only temporal components \(B_0 \ne 0\) in the early universe through an analytic (but approximate) solution of the corresponding algebraic system of Boltzmann equations. In [43] we only presented a heuristic estimate of the generated asymmetry. In order to facilitate the comparison of our detailed analysis with the heuristic order of magnitude estimates of [43] we have first assumed the active neutrino to be purely Majorana as in [43] and concentrated only on the decays of RHN into charged leptons. The inclusion of the decays to neutral Higgs and light neutrinos has also been done here, with the (expected) result that the total asymmetry is increased by a factor of about 2 as compared to the chargedleptoncase.
in agreement with the estimate (22) of [43]. In our analysis we assumed Yukawa couplings of order \(y \sim 10^{5}\) in the Higgs portal term (6), that couple the righthanded neutrino to the SM sector of the model. This prompted us to ignore higher order terms of order \( \vert y \vert ^4 \sim 10^{20} \ll B_0/m_N\), which a posteriori was proved to be a selfconsistent result, due to the smallness of the \(B_0/m_N\) (114), required for the observed baryon asymmetry today (through leptogenesis).
implying a ratio \(B_0/m_N\) which is (less than) an order of magnitude smaller than (22). The slight increase of the freezeout point does not affect the order of magnitude of the asymmetry nor that of the Yukawa coupling of the Higgs portal; hence in order of magnitude there is qualitative agreement with the estimates of [43].
Although our analysis has been generic in not specifying the microscopic origin of the CPTV background, nonetheless some microscopic scenarios originating from string theory have been presented in [43]. According to these scenarios the background is identified with the dual of the Kalb–Ramond antisymmetric tensor field strength, \(\epsilon _{\mu \nu \rho \sigma } \, H^{\nu \rho \sigma }\), which in a fourdimensional spacetime is equivalent to the derivative of a pseudoscalar field b(x) (Kalb–Ramond axion), \(\partial _\mu b\). Nevertheless such an identification is not binding. However, if it is made, then within the context of realistic brane/string models the pressing question concerns the microscopic mechanism, for the transition from a relatively large value (in the Robertson–Walker frame) of the constant \(B_0 \ne 0\) CPTV background in early eras of the (string) universe, required for leptogenesis, to a very weak background today, compatible with the very stringent limits of CPTV in the current era [38, 39, 40, 41]. Some speculations have been presented in [43] but detailed microscopic mechanisms, compatible with the rest of astroparticle phenomenology, including the open issue of the smallness of the (observed) cosmological constant (or dark energy) today, are still lacking and will be the subject of future investigations.
Nevertheless, we believe that the scenario for baryogenesis through leptogenesis presented initially in [43] and completed here, is an attractive, relatively simple one, which deserves further investigations, within the context of appropriate microscopic models (not necessarily within the framework of string/brane theory). We hope to come back to such studies in the near future. Another important aspect of our current work is the demonstration of the efficiency of the Padé approximant method [54] in solving Boltzmann equations, thus adding yet another successful example of this method, this time of relevance to cosmology.
Footnotes
 1.If one considers, for instance, quark fields in some Lorentz and CPTV backgrounds (such as those allowed by the SME formalism), it is possible to induce baryogenesis, as a consequence of the fact that the LV and CPTV effects induce “chemical potentials” for the quarks [42]. This leads directly to baryogenesis, given that in the presence of a chemical potential \(\mu \), the populations of quarks and antiquarks are already different within thermal equilibrium, since the particle and antiparticle phasespace distribution functions \(f(E,\mu ), f({\overline{E}}, {\overline{\mu }})\), with E the energy (and an overline over a quantity denoting that of an antiparticle) are different (in the presence of a chemical potential, \(\mu \), for a particle, the antiparticle has a chemical potential of opposite sign \({\overline{\mu }} = \mu \). In SME models, of course, even the magnitudes of \({\overline{\mu }}\) and \({\overline{E}}\) may be different from those of particles, as a consequence, for example, of different dispersion relations between particles and antiparticles). All these cause a difference in the corresponding equilibrium populations[where the \(+ ()\) will denote a fermionic (bosonic) (anti)particle]. In principle, such scenarios can lead to alternative explanations for the observed matterantimatter asymmetry, provided that detailed mechanisms for freezeout of particle interactions in this SME context are provided.Unfortunately, so far, microscopic models leading to such SME lagrangians and related phenomena have not been provided.$$\begin{aligned} ~ f(E,\mu )= & {} [\mathrm{exp}(E\mu )/T)\pm 1]^{1}, \quad f(\overline{E},\bar{\mu })\nonumber \\= & {} [\mathrm{exp}(\bar{E}\overline{\mu })/T)\pm 1]^{1}, \end{aligned}$$(3)
 2.
In four spacetime dimensions, the field strength of the KR field is dual to a massless pseudoscalar (axionlike) field. In the recent literature [50] axionbased approaches to leptogenesis, involving an effective CPTviolating coupling between the (temporal component of the) lepton number current and the time derivative of a (timedependent) axion field (which is quite different from our KR axion which couples to axial fermion currents), have been proposed. This interaction breaks time translation invariance and, thus, generates an effective chemical potential which differentiates between leptons and antileptons. The presence of this effective chemical potential allows for the generation of a lepton asymmetry by means of RHNmediated \(\Delta L = 2\) scattering processes in that model.
 3.
We do not specify here or in [43] the mechanism by which the heavy righthanded neutrinos acquire their mass. Exotic scenarios may be at play here [61], in which the quantum fluctuations of the Kalb–Ramond \(H_{\mu \nu \rho }\) field (equivalently the axion field b(x) in four spacetime dimensions) are allowed to mix with ordinary axions, via kinetic mixing, and thus may be responsible for radiative generation of the righthanded Majorana neutrino mass, as a result of Yukawa coupling interactions of the ordinary axion with such righthanded neutrinos. In such a case, one may arrange that such masses are non trivial in the high temperature regime of the decoupling of the righthanded neutrinos, even if the rest of the SM fields are massless at such temperatures.
 4.
Scattering processes \(l \, l \rightarrow {\bar{h}} {\bar{h}}\) or \(l \, h \rightarrow {\bar{l}} \, {\bar{h}} \), are of higher order in the Yukawa coupling y and hence are suppressed in our case, although such processes are equally important in standard CPT invariant, CP violating leptogenesis, with more than one species of righthanded neutrinos, as they are of the same order as the CP violating oneloop graphs [24].
 5.
Such an assumption is non trivial and depends on the microscopic model considered. For instance, in terms of braneworld scenarios for the background \(B_0\) [43], where the latter is derived from a cosmological Kalb–Ramond axion field b(t), such an assumption is justified by requiring a cancellation of the constant in time kinetic energy density of the field b by the (negative) dark energy of the higherdimensional bulk. After the decoupling, where the string/brane Universe undergoes a phase transition, the dark energy falls off with the temperature sufficiently rapidly, so as today it reaches the value measured by cosmological observations. We shall not discuss such details in the current article.
Notes
Acknowledgements
NEM wishes to thank the University of Valencia and IFIC for a Distinguished Visiting Professorship, during which the current work has been completed. The work of TB is supported by an STFC (UK) research studentship and that of NEM and SS is supported in part by STFC (UK) under the research Grant ST/P000258/1.
Supplementary material
References
 1.P.A.R. Ade et al. [Planck Collaboration], Astron. Astrophys. 571, A16 (2014). arXiv:1303.5076 [astroph.CO]
 2.D.N. Spergel et al. [WMAP Collaboration], Astrophys. J. Suppl. 148, 175 (2003). arXiv:astroph/0302209
 3.R.F. Streater, A.S. Wightman, PCT, spin and statistics, and all that’ (Princeton Univ. Pr., Princeton, 2000), p. 207Google Scholar
 4.A.D. Sakharov, Zh Pisma, Eksp. Teor. Fiz. 5, 32 (1967)Google Scholar
 5.A.D. Sakharov, JETP Lett. 5, 24 (1967)ADSGoogle Scholar
 6.A.D. Sakharov, Sov. Phys. Usp. 34, 392 (1991)ADSCrossRefGoogle Scholar
 7.A.D. Sakharov, Usp. Fiz. Nauk 161, 61 (1991)CrossRefGoogle Scholar
 8.W. Buchmuller, P. Di Bari, M. Plumacher, Ann. Phys. 315, 305 (2005). https://doi.org/10.1016/j.aop.2004.02.003. arXiv:hepph/0401240 ADSCrossRefGoogle Scholar
 9.S. Davidson, E. Nardi, Y. Nir, Phys. Rep. 466, 105 (2008). https://doi.org/10.1016/j.physrep.2008.06.002. arXiv:0802.2962 [hepph]ADSCrossRefGoogle Scholar
 10.A. Pilaftsis, J. Phys. Conf. Ser. 171, 012017 (2009). arXiv:0904.1182 [hepph]
 11.A. Pilaftsis, J. Phys. Conf. Ser. 447, 012007 (2013). https://doi.org/10.1088/17426596/447/1/012007 CrossRefGoogle Scholar
 12.S. Biondini et al., arXiv:1711.02864 [hepph]
 13.A.G. Cohen, D.B. Kaplan, A.E. Nelson, Ann. Rev. Nucl. Part. Sci. 43, 27 (1993). https://doi.org/10.1146/annurev.ns.43.120193.000331. arXiv:hepph/9302210 ADSCrossRefGoogle Scholar
 14.M. Trodden, Rev. Mod. Phys. 71, 1463 (1999). https://doi.org/10.1103/RevModPhys.71.1463. arXiv:hepph/9803479 ADSCrossRefGoogle Scholar
 15.A. Riotto, M. Trodden, Ann. Rev. Nucl. Part. Sci. 49, 35 (1999). https://doi.org/10.1146/annurev.nucl.49.1.35. arXiv:hepph/9901362 ADSCrossRefGoogle Scholar
 16.W. Buchmuller, arXiv:0710.5857 [hepph]
 17.V.A. Kuzmin, V.A. Rubakov, M.E. Shaposhnikov, Phys. Lett. 155B, 36 (1985). https://doi.org/10.1016/03702693(85)910287 ADSCrossRefGoogle Scholar
 18.M.B. Gavela, P. Hernandez, J. Orloff, O. Pene, Mod. Phys. Lett. A 9, 795 (1994). https://doi.org/10.1142/S0217732394000629. arXiv:hepph/9312215 ADSCrossRefGoogle Scholar
 19.M.B. Gavela, P. Hernandez, J. Orloff, O. Pene, C. Quimbay, Nucl. Phys. B 430, 382 (1994). https://doi.org/10.1016/05503213(94)004102. arXiv:hepph/9406289 ADSCrossRefGoogle Scholar
 20.M. Fukugita, T. Yanagida, Phys. Lett. B 174, 45 (1986)ADSCrossRefGoogle Scholar
 21.M.A. Luty, Phys. Rev. D 45, 455 (1992)ADSCrossRefGoogle Scholar
 22.A. Pilaftsis, Phys. Rev. D 56, 5431 (1997). https://doi.org/10.1103/PhysRevD.56.5431. arXiv:hepph/9707235 ADSCrossRefGoogle Scholar
 23.W. Buchmuller, R.D. Peccei, T. Yanagida, Ann. Rev. Nucl. Part. Sci. 55, 311 (2005). https://doi.org/10.1146/annurev.nucl.55.090704.151558. arXiv:hepph/0502169 ADSCrossRefGoogle Scholar
 24.A. Strumia, in Particle physics beyond the standard model (Proceedings, Summer School on Theoretical Physics, 84th Session, Les Houches, France, August 1–26, 2005) ed. by D. Kazakov, S. Lavignac, J. Dalibard (Elsevier, Amsterdam, 2006). arXiv:hepph/0608347
 25.M. Shaposhnikov, I. Tkachev, Phys. Lett. B 639, 414 (2006). https://doi.org/10.1016/j.physletb.2006.06.063. arXiv:hepph/0604236 ADSCrossRefGoogle Scholar
 26.M. Shaposhnikov, Subnucl. Ser. 47, 167 (2011). https://doi.org/10.1142/9789814374125.0008 Google Scholar
 27.P. Minkowski, Phys. Lett. B 67, 421 (1977)ADSCrossRefGoogle Scholar
 28.M. GellMann, P. Ramond, R. Slansky, in Supergravity, ed. by D.Z. Freedman, P. van Nieuwenhuizen (NorthHolland, Amsterdam, 1979)Google Scholar
 29.T. Yanagida, in Proc. of the Workshop on the Unified Theory and the Baryon Number in the Universe, ed. by O. Sawada, A. Sugamoto (Tsukuba, Japan, 1979)Google Scholar
 30.R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44, 912 (1980)ADSCrossRefGoogle Scholar
 31.J. Schechter, J.W.F. Valle, Phys. Rev. D 22, 2227 (1980)ADSCrossRefGoogle Scholar
 32.M.C. GonzalezGarcia, M. Maltoni, Phys. Rep. 460, 1 (2008). arXiv:0704.1800 [hepph] and references thereinADSCrossRefGoogle Scholar
 33.D.V. Forero, M. Tortola, J.W.F. Valle, Phys. Rev. D 86, 073012 (2012). https://doi.org/10.1103/PhysRevD.86.073012. arXiv:1205.4018 [hepph]ADSCrossRefGoogle Scholar
 34.O.W. Greenberg, Phys. Rev. Lett. 89, 231602 (2002). arXiv:hepph/0201258 ADSCrossRefGoogle Scholar
 35.M. Chaichian, A.D. Dolgov, V.A. Novikov, A. Tureanu, Phys. Lett. B 699, 177 (2011). https://doi.org/10.1016/j.physletb.2011.03.026. arXiv:1103.0168 [hepth]ADSCrossRefGoogle Scholar
 36.M. Chaichian, K. Fujikawa, A. Tureanu, Eur. Phys. J. C 73(3), 2349 (2013). https://doi.org/10.1140/epjc/s1005201323492. arXiv:1205.0152 [hepth]ADSCrossRefGoogle Scholar
 37.D. Colladay, V.A. Kostelecky, Phys. Rev. D 58, 116002 (1998). arXiv:hepph/9809521 ADSCrossRefGoogle Scholar
 38.V.A. Kostelecky, N. Russell, Rev. Mod. Phys. 83, 11 (2011). arXiv:0801.0287 [hepph]ADSCrossRefGoogle Scholar
 39.N.E. Mavromatos, Hyperfine Interact. 228(1–3), 7 (2014). arXiv:1312.4304 [hepph]
 40.V.A. Kostelecky, M. Mewes, Phys. Rev. Lett. 99, 011601 (2007). arXiv:astroph/0702379 ADSCrossRefGoogle Scholar
 41.M. Das, S. Mohanty, A.R. Prasanna, Int. J. Mod. Phys. D 22, 1350011 (2013). arXiv:0908.0629 [astroph.CO]ADSCrossRefGoogle Scholar
 42.O. Bertolami, D. Colladay, V.A. Kostelecky, R. Potting, Phys. Lett. B 395, 178 (1997). arXiv:hepph/9612437 ADSCrossRefGoogle Scholar
 43.M. de Cesare, N.E. Mavromatos, S. Sarkar, Eur. Phys. J. C 75(10), 514 (2015). https://doi.org/10.1140/epjc/s100520153731z. arXiv:1412.7077 [hepph]ADSCrossRefGoogle Scholar
 44.M. Kalb, P. Ramond, Phys. Rev. D 9, 2273 (1974)ADSCrossRefGoogle Scholar
 45.D.J. Gross, J.H. Sloan, Nucl. Phys. B 291, 41 (1987)ADSCrossRefGoogle Scholar
 46.R.R. Metsaev, A.A. Tseytlin, Nucl. Phys. B 293, 385 (1987)ADSCrossRefGoogle Scholar
 47.M.C. Bento, N.E. Mavromatos, Phys. Lett. B 190, 105 (1987)ADSMathSciNetCrossRefGoogle Scholar
 48.M.J. Duncan, N. Kaloper, K.A. Olive, Nucl. Phys. B 387, 215 (1992)ADSCrossRefGoogle Scholar
 49.I. Antoniadis, C. Bachas, J.R. Ellis, D.V. Nanopoulos, Nucl. Phys. B 328, 117 (1989)Google Scholar
 50.A. Kusenko, K. Schmitz, T.T. Yanagida, Phys. Rev. Lett. 115(1), 011302 (2015). https://doi.org/10.1103/PhysRevLett.115.011302. arXiv:1412.2043 [hepph]ADSCrossRefGoogle Scholar
 51.M. Lindner, A. Merle, V. Niro, JCAP 1101, 034 (2011). Erratum: [JCAP 1407, E01 (2014)]. https://doi.org/10.1088/14757516/2011/01/034. https://doi.org/10.1088/14757516/2014/07/E01. arXiv:1011.4950 [hepph]
 52.A. Merle, V. Niro, JCAP 1107, 023 (2011). https://doi.org/10.1088/14757516/2011/07/023. arXiv:1105.5136 [hepph]ADSCrossRefGoogle Scholar
 53.C.R. Argelles, N.E. Mavromatos, J.A. Rueda, R. Ruffini, JCAP 1604(04), 038 (2016). https://doi.org/10.1088/14757516/2016/04/038. arXiv:1502.00136 [astroph.GA]ADSCrossRefGoogle Scholar
 54.G.A.Baker, P. GravesMorris, Pade Approximants, ed. by C. U. Press (Encyclopedia of Mathematics and its Applications, 1996)Google Scholar
 55.For a representative sample of references see (and references therein, or citations thereof): M.A. Samuel, G. Li, E. Steinfelds, Phys. Lett. B 323, 188 (1994). https://doi.org/10.1016/03702693(94)902909
 56.M.A. Samuel, G. Li, E. Steinfelds, Phys. Rev. D 48, 869 (1993). https://doi.org/10.1103/PhysRevD.48.869 ADSCrossRefGoogle Scholar
 57.J.R. Ellis, M. Karliner, M.A. Samuel, E. Steinfelds, arXiv:hepph/9409376
 58.J.R. Ellis, E. Gardi, M. Karliner, M.A. Samuel, Phys. Lett. B 366, 268 (1996). https://doi.org/10.1016/03702693(95)013261. arXiv:hepph/9509312 ADSMathSciNetCrossRefGoogle Scholar
 59.J.R. Ellis, I. Jack, D.R.T. Jones, M. Karliner, M.A. Samuel, Phys. Rev. D 57, 2665 (1998). https://doi.org/10.1103/PhysRevD.57.2665. arXiv:hepph/9710302 ADSCrossRefGoogle Scholar
 60.S.J. Brodsky, J.R. Ellis, E. Gardi, M. Karliner, M.A. Samuel, Phys. Rev. D 56, 6980 (1997). https://doi.org/10.1103/PhysRevD.56.6980. arXiv:hepph/9706467 ADSCrossRefGoogle Scholar
 61.N.E. Mavromatos, A. Pilaftsis, Phys. Rev. D 86, 124038 (2012). arXiv:1209.6387 [hepph]ADSCrossRefGoogle Scholar
 62.E.W. Kolb, S. Wolfram, Nucl. Phys. B 172, 224 (1980)ADSCrossRefGoogle Scholar
 63.S. Weinberg, Cosmology (Oxford University Press, Oxford, 2008)zbMATHGoogle Scholar
 64.E.W. Kolb, M.S. Turner, The Early Universe. Front. Phys. 69, 1 (1990)ADSMathSciNetzbMATHGoogle Scholar
 65.S. Weinberg, Phys. Rev. Lett. 42, 850 (1979)ADSCrossRefGoogle Scholar
 66.G. Nagy, Ordinary differential equations (Michigan State University, East Lansing, 2017)Google Scholar
 67.M.L. Boas, Mathematical methods in the physical sciences, 3rd edn. ISBN 9780471365808Google Scholar
 68.M. Abramowitz, I.A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series, vol. 55 (United States Department of Commerce, National Bureau of Standards, Dover Publications,Washington D.C., 1983). ISBN 9780486612720. LCCN 6460036Google Scholar
 69.N.E. Mavromatos, S. Sarkar, Eur. Phys. J. C 73(3), 2359 (2013). https://doi.org/10.1140/epjc/s1005201323590. arXiv:1211.0968 [hepph]ADSCrossRefGoogle Scholar
 70.C.M. Bender, S. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGrawHill, New York, 1977)zbMATHGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}