The role of temperature dependent stringinspired CPT violating backgrounds in leptogenesis and the chiral magnetic effect
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Abstract
In a temperature dependent CPTViolating (CPTV) axial timelike background (induced by the KalbRamond tensor field of string theory) we discuss leptogenesis by solving the Boltzmann equation.The current work nontrivially modifies the framework of a previous phenomenological approach by the authors where the CPTV axial background was considered to be a constant (with no microscopic justification). The constant background approximation though is shown to capture the main phenomenological features of leptogenesis. On comparing our analysis to the related chiral magnetic effect for axial current condensates, we conclude that the KalbRamond field does not play the role of the chiral chemical potential needed for that effect.
1 Introduction and motivation
1.1 Microscopic (stringinspired) framework

First, we formulate the path integral, which involves a functional integration over the KR field strength H.
 We insist on the preservation of the Bianchi identity (8) at a quantum level, via the addition of appropriate counterterms (in a renormalisation group sense) order by order in perturbation theory. This guarantees the conservation of the “Htorsion charge ”\(Q = \int d^3 x \, \varepsilon _{ijk} H^{ijk}\), which is implemented in the pathintegral by adding a \(\delta \)function constraint in the form \(\delta \Big (\kappa ^{2}\, \varepsilon ^{\mu \nu \rho \sigma } \, \partial _{\mu }\, H_{\nu \rho \sigma }\Big ), \) and expressing the latter in terms of a (dimensionless) Lagrange multiplier field b(x), which eventually will correspond to the dual KR axion field:where the second equality has been obtained by partial integration, upon assuming that the KR field strength dies out at spatial infinity.$$\begin{aligned} \delta ({\kappa ^2}\,{\varepsilon ^{\mu \nu \rho \sigma }}\,{\partial _\mu }\,{H_{\nu \rho \sigma }})&= \int {{\mathcal {D}}b} \exp \left[ i\,{\kappa ^{  2}}\right. \nonumber \\&\left. \quad \times \int {{d^4}} x\sqrt{  g} \,b(x){\varepsilon _{\mu \nu \rho \sigma }}{\partial ^\mu }{H^{\nu \rho \sigma }}\right] \nonumber \\&= \int {{\mathcal {D}}b} \exp \left[  i\,{\kappa ^{  2}}\right. \nonumber \\&\left. \quad \times \int {{d^4}} x\sqrt{  g} \,{\partial ^\mu }b(x){\varepsilon _{\mu \nu \rho \sigma }}\,{H^{\nu \rho \sigma }}\right] , \end{aligned}$$(15)
 Integrating out the Hfield in the path integral with the action (14), we obtain a path integral over the Lagrange multiplier field b(x),$$\begin{aligned} Z =&\; \int \, {\mathcal {D}}g \, {\mathcal {D}}\psi \, {\mathcal {D}}{\bar{\psi }}\, {\mathcal {D}}b \, \exp [\imath {\tilde{S}}_{eff}], \nonumber \\ {{\tilde{S}}}_{eff} =&\; \dfrac{1}{2\kappa ^{2}}\int d^{4}x\sqrt{g}\,\Big (R + \dfrac{8}{3}\partial _{\sigma } b\, \partial ^{\sigma }b  \Omega \Big ) \nonumber \\&+ S_{Dirac}^{Free}  \int d^{4}x\sqrt{g}\partial _{\mu }b\, J^{5\mu }  \dfrac{3\kappa ^{2}}{16}\nonumber \\&\quad \times \int d^{4}x\sqrt{g}\,J^{5}_{\mu }J^{5\mu }\Big ]. \end{aligned}$$(16)
1.2 Choice of background

Consider (in the RobertsonWalker frame) KRaxion backgrounds \(\bar{b}(x)\) linear in cosmic time t, so that \({\dot{{\bar{b}}}} \) is constant. Such backgrounds rigorously exist in bosonic noncritical strings [10] and permit the decoupling of \(\bar{b}\) from \({\tilde{b}}\) (since the first term in the second line of (19) vanishes as a total derivative (on assuming that quantum fluctuations \({\tilde{b}}\) vanish rapidly at spacetime infinity). Upon restricting ourselves to the Hbackground terms of the total Lagrangian comprising of the sum of (1) and (19), the \(\partial {\bar{b}}\)\(J^5\) interaction term in (19) yields the CPTViolating axial background \(B_0\)term of the model discussed in [2, 3], which leads to leptogenesis. In this way one obtains a microscopic origin of \(B_0\) in the context of stringinspired models. However in our context the existence of such a background remains a postulate. The baxion background linear in time may not constitute exact solutions for superstrings in the presence of fermions. Moreover, even if it was an exact solution, it is not known whether one could fine tune the associated parameters so as to guarantee a \(B_0\) background (13) in the MeV or lower range. We note that in the scenario of [2] that a natural mass scale for such backgrounds is provided by the string scale \(M_s\) itself and \(M_s \gg \) MeV [10].
 In [2, 3], another possibility for obtaining a CPTV KR axion background, corresponding to a constant \(B_{0}\) (=\({\dot{{\bar{b}}}}\)), was proposed. This proposal involves fermionic axial condensates, that have been conjectured to occur at the freezeout epoch for the leptogenesis scenario of [2, 3]. Indeed, in the presence of fermions, the equations of motion for the KR background field \({\bar{b}}\) deduced from (16), is:In this proposal we assume a (constant) temporal chiral condensate (which respects the spatial isotropy of the universe),$$\begin{aligned} \partial _{\alpha }\Big [\sqrt{g}\Big (\dfrac{8}{3\kappa ^{2}}\partial ^{\alpha }\bar{b}  J^{5 \; \alpha }\Big )\Big ] = 0. \end{aligned}$$(20)Such a condensate may characterise fermions in the model except Majorana neutrinos [2], e.g. quarks in the SM sector; on expanding the current in (20) about the condensate (21), \(J^{5}_{0}\) = \(\langle J^{5}_{0}\rangle + \) quantum fluctuations, and on ignoring the fluctuations, we obtain from (20)$$\begin{aligned} 0 \ne \mathrm{const}. = \langle J^{05} \rangle = \langle \psi ^\dagger _i \, \gamma ^5 \psi _i \rangle .~ \end{aligned}$$(21)which allows a solution (cf. (13))$$\begin{aligned} \partial _{t}\Big [\sqrt{g}\Big (\dfrac{8}{3\kappa ^{2}}B^{0}  \langle J^{0\, 5} \rangle \Big )\Big ] = 0, \end{aligned}$$(22)implying a constant LV and CPTV axial background \(B^0\) (in the RobertsonWallker frame), as required for leptogenesis in the scenario of [2, 3]. In the currentera a plethora of precision measurements [18], imply that \(B_0 < 0.01\) eV (as well as much stronger constraints for the spatial components \(B_i  < 10^{31}\) GeV). In the scenario of [2], this can be guaranteed, if it is assumed that the chiral current condensate \(\langle J^{05}\rangle \), is destroyed at a temperature near the leptonasymmetry freezeout \(T \simeq T_D \simeq 10^5\) GeV (due to some unspecified physics beyond the SM). In that case, upon taking into account a RobertsonWalker spacetime with scale factor \(a(t) \sim T^{1}\) at high temperatures, we obtain from (22) a cooling ‘law’ \(B_0 \sim T^3\), for \(T \lesssim T_D\), which comfortably satisfies the above constraints in the current epoch [2]: the average current temperature of the universe is that of the Cosmic Microwave Background (CMB) radiation, \(T_0 \sim T_{\mathrm{CMB}} \simeq 0.23\) meV. Indeed, with such a cooling law, taking into account that at decoupling \(B_0(T_D \simeq 10^5~\mathrm{GeV}) ={{\mathcal {O}}}(0.1~\mathrm{MeV})\), one finds [2]: \(B_0 (T_0) = {{\mathcal {O}}}(10^{57})\) GeV.$$\begin{aligned} B^0 = {\dot{\bar{b}}} = \frac{3\kappa ^{2}}{8}\, \langle J_{0\, 5} \rangle = \mathrm{const.} \ne 0, \end{aligned}$$(23)
It is the purpose of this work to offer a new resolution to the above issues, by actually considering nonconstant backgrounds \(B_0\), obtained from the antisymmetric tensor field of string theory as described above. This does away with the twin requirements of formation and disappearance of axial current condensates in our earlier works [2, 3]. The cubic dependence of temperature for the background \(B_{0}\sim T^{3}\) all the way from temperatures around decoupling until the present day, is dictated by the equation of motion of the KRaxion field in the absence of an axialfermioncurrent condensate; this temperature dependence is sufficiently mild in the high temperature regime of interest, so that the conditions for leptogenesis considered in [3] are only slightly modified. We shall demonstrate that leptogenesis still occurs at decoupling temperatures of order \(T_D \simeq 100\) TeV, with the background field, though, smaller than that considered in [2]: \(B_0 (T_D) = {{\mathcal {O}}}(\mathrm{keV})\); we obtain for the currentepoch value \(B_0 (T_0) ={{\mathcal {O}}}(10^{59})\) GeV, which lies comfortably within the stringent current bounds of CPTV and LV [18].
Before proceeding, we would like to mention another important issue. Since our effective field theory appears to be one with a chiral chemical potential provided by the KR torsion field \(B_0\), on minimally extending the model by adding an external magnetic field, one would be tempted to conjecture that the conditions for the chiral magnetic effect (CME) [22] would be satisfied; however, this is not so. The \(B_0\) field is actually a fullyfledged axial background rather than a mere chiral chemical potential, and it is known that such backgrounds make no contributions to the CME [23, 24].^{6} The CME is connected with quantum (chiral [25]) anomalies, and, as we mentioned previously, in our stringinspired effective theory, the field \(B_0\) is associated with the KR Htorsion. It is well known [27, 28], that the contributions of the latter can be removed from the anomaly equation by an appropriate choice of a renormalisationgroup scheme. Hence, there should be no physical effects of the KR torsion field \(B_0\) on the anomaly equation, and thus on the CME.^{7} For completeness we shall give a fuller discussion of CME in the appendix.
2 Temperaturedependent background field \(\mathbf {B_0(T)}\)
3 Leptogenesis
A table showing the different values of the constant \(\Phi \) and the background field \(B_{0}\) with respect to different expansion points \(x_{P}\) at the decoupling value of \(x_{D} = 1\)
\(x_{P}\)  \(\dfrac{\Phi }{m_{N}}\)  \(\Phi \)(keV)  \(B_{0}(x_{D})\)(keV) 

0.50  \(3.6\times 10^{12}\)  0.36  0.36 
0.75  \(5.9\times 10^{12}\)  0.59  0.59 
0.90  \(7.4\times 10^{12}\)  0.74  0.74 
Compared to the case of constant \(B_0\) studied in [3], we observe that the LV and CPTV value of the background field \(B_0\) at decoupling \(T_D={\mathcal {O}}(100)\) TeV, which yields phenomenologically acceptable lepton asymmetry in the universe is smaller, is in the keV range.
4 Currentera magnitude of CPTV Background and Vacuum (Dark) Energy Contributions
A table showing the different values of the background field \(B_{0}\) in the temperature regimes where the quarks fall out of equilibrium (first line) and the value today (second line)
T(eV)  x  \(B_{0}\)(eV) 

\(173\times 10^{9}\)  578  \((1.9  3.8)\times 10^{6}\) 
\(2.4\times 10^{4}\)  \(4.2\times 10^{17}\)  \((4.9  10.0)\times 10^{51}\) 
These values indicate that the current value of the “torsion” LV and CPTV field \(B_0\) lies comfortably within the current bounds [18], \(B_0< 0.01\) eV and (for the spatial components) \(B_i < 10^{31}\) GeV; so even a boost by small velocities, due to a difference of the laboratory frame with respect to the cosmological frame, will still yield spatial components within the above limits.
5 Conclusions
In this work we have generalised our previous study of leptogenesis based on a constant LV and CPTV timelike axial background to the important case of a torsion background varying with the temperature of the early universe. The torsion is provided here by the antisymmetric tensor (KalbRamond) spinone field of the massless bosonic multiplet of closed string theory. The phenomenology of our leptogenesis though, remains largely unchanged from the constant background case, and is consistent with the stringent constraints from the current epoch on LV and CPTV, as well as with cosmological constraints on the vacuum energy density.
We should note that within the SME framework, there have been previous suggestions [30] for direct baryogenesis due to LV and CPTV terms in the the effective action, which induce “effective chemical potentials”, say for quarks. In the presence of a chemical potential, the equilibrium populations of quarks and antiquarks are already different within thermal equilibrium, since the phasespace distribution functions between particles and antiparticles are different. In principle, such scenarios in the SME context, can lead to alternative explanations for the observed matterantimatter asymmetry, provided that detailed mechanisms for freezeout of particle interactions are provided. However, in [30] no microscopic models leading to such SME Lagrangians have been provided, and moreover, for phenomenologically relevant baryogenesis, one needs nonminimal higher derivative LV and CPTV fermionic interactions with tensorial backgrounds in the pertinent SME Lagrangian.
By contrast, our model involves baryogenesis via leptogenesis, through a minimal coupling of fermions to the KR axion field b(x), which exists in the massless gravitational multiplet of the underlying string theory model; hence our model has a microscopic justification. Moreover, the induced CPTV arises naturally as a background solution which spontaneously breaks Lorentz symmetry, with the massless KR axion being the corresponding Goldstone boson [10]. Lorentz symmetry is considered as spontaneously broken, since the underlying string theory and the corresponding lowenergy effective actions (16) are both Lorentz (and CPT) invariant. According to the general discussion in [31] it is the breaking of Lorentz symmetry which in turn induces CPT violation . The leptogenesis in our model occurs through CP decays of Majorana neutrinos in the LV and CPTV KR axion background, as we have discussed. The related baryogenesis is assumed to take place through subsequent sphaleron processes in the SM sector of the effective theory, which violate both Baryon (B) and Lepton (L) number, but preserve the difference BL.
We would also like to compare briefly our results with those of the work of [33]. In that reference, an effective coupling between the lepton number current \(J_L^\mu \) and a timedependent massive axion field, has been introduced as a means of inducing leptogenesis through chiral anomalies. For a timedependent axion field, this coupling breaks time translation invariance and, thus, generates an effective chemical potential for leptons and antileptons. The presence of this effective chemical potential allows the generation of a lepton asymmetry by means of righthandedneutrino mediated \(\Delta \)L = 2 scattering processes. The model of [33] might be seen as constituting a particular realization of the “spontaneous baryogenesis” scenario of [34], in which a hypothesized neutral scalar field, called “thermion”, \(\varphi \), couples to the baryon current \(J_B^\mu \) (or any other current not orthogonal to it, such as the BL current) via a derivative coupling \(\partial _\mu \varphi J_B^\mu \). The thermion is assumed to develop a slowly varying time derivative, as the Universe cools, and this appears as an effective CPT breaking, which contributes to baryon asymmetry. In [34] the thermion has been connected with the Goldstone boson of an approximate global U(1) symmetry at a certain temperature; hence the thermion also has a small mass, like the axion field of [33].
Our KR axion b(x) exhibits a similar behaviour to the above axion and thermion fields, in that it has a nontrivial coupling to a fermion axial current, and is slowly varying in time, which is an essential feature of our study. However, in contrast to the scenarios of [33, 34], our model involves a massless axion that arises from the bosonic gravitational string multiplet, and the associated leptogenesis is due to tree level CP violating decays of RHN in the presence of Lorentz and CPT Violating backgrounds of the KR axion field. The derivative of the KR axion couples to the axial fermionic current \(J_\mu ^5\) of all fermion species, including RHN, as a consequence of the geometrical interpretation of the KR axion field as effective torsion [17]. However, we do not make explicit use of the chiral anomaly to calculate the lepton asymmetry; the coupling of the KR axion to the RHN current is nontrivial due to the gravitational anomalies that the RHN current is assumed to have  other gauge anomalies in extensions of the SM derived from string theory can also play a rôle, but the specific details of the anomalies are not relevant for our phenomenological scenarios.
Another important feature of our model for baryogenesis through leptogenesis is that a single RHN species suffices to produce phenomenologically relevant lepton asymmetry. However, if one insists on seesawlike scenarios for generating masses in the active neutrinos of the standard model, then at least two generations of RHN are needed. In such a case, it is interesting to examine whether there may be some resonant phenomena, within our CPTV scenario, in the case of nearly mass degenerate RHNs, which could enhance the induced lepton asymmetry (just as in the corresponding resonant leptogenesis models of [36].^{9}) If such resonant phenomena were in operation in extensions of our model with more flavours of (nearly degenerate) RHN, then, one would need much weaker CPTV KRaxion backgrounds to generate the observed matterantimatter asymmetry in the Universe, than the ones discussed in the current work and in [2, 3]. We plan to investigate such important issues in a future work.
Before closing, we would like to remark that during the leptogenesis era, there might be present primordial external magnetic fields, which can also lead to leptogenesis, however via mechanisms which are distinct from the one in our work [42]. In the presence of a chiral chemical potential \(\mu _5\), that is a difference of the chemical potentials \(\mu _L  \mu _R\) between left(L) and right(R)handed spinors, it is known that one has an induced electric current proportional to the magnetic field intensity, the phenomenon of CME [22]. In addition to primordial eras, of relevance to leptogenesis, systems such as neutron stars or a hot QCD quarkgluon (QG) plasma with external magnetic fields show the CME. Given that our temperature dependent axial background (torsion) field \(B_0(T)\) survives until today, and its coupling to a chiral fermion current in the effective action has the apparent form of a chiral chemical potential term, albeit temperature dependent, it is natural to examine whether the axial background \(B_0\) has any effect on the CME.
 1.
the fact that the phenomenon has its origin [22] in the chiral anomalies of quantum field theory [25]
 2.
the rôle of the \(B_0\) KR field as a torsion in the lowenergy string effective action
 3.
the wellknown result [27, 28] that torsion contributions to the anomaly equation are removable by the addition of appropriate local counterterms (in a renormalisation group sense) to the corresponding effective action. Physical effects, such as the CME, should thus be free from such ambiguities.
Footnotes
 1.
At high temperatures, above the spontaneous electroweak symmetry breaking, the charged Higgs fields \(h^\pm \) do not decouple from the physical spectrum, and play an important rôle in leptogenesis.
 2.
In string theory, in the presence of gauge and gravitational fields, the righthandside of (7) is modified by appropriate Chern–Simons threeforms, which lead to a nonzero righthand side of the Bianchi identity (8), expressing gauge and gravitational anomalies [5]. We shall not deal explicitly with such (higher derivative) terms here, as they are not directly relevant to our leptogenesis scenario.
 3.
In this and previous works, our conventions are as follows: spacetime metric signature \((+,,,)\), and we shall use the following representation of Dirac \(\gamma \)matrices: \(\gamma ^{\mu } = \begin{pmatrix} 0&{}\sigma ^{\mu }\\ {\bar{\sigma }}^{\mu }&{}0 \end{pmatrix}, \;\; \sigma ^{\mu } = \begin{pmatrix} \mathbb {1}\\ \sigma ^{\jmath } \end{pmatrix}, \;\; {\bar{\sigma }}^{\mu } = \begin{pmatrix} \mathbb {1}\\ \sigma ^{\jmath } \end{pmatrix} \), \(\gamma _{5} = \imath \, \gamma ^0\, \gamma ^1 \, \gamma ^2\, \gamma ^3 = \begin{pmatrix} \mathbb {1}&{}0\\ 0&{}\mathbb {1} \end{pmatrix}.\)
 4.The spin connection is given bywith \(\Gamma _{\mu \nu }^\lambda = \Gamma _{\nu \mu }^\lambda \) the standard Christoffel symbol.$$\begin{aligned} {\omega _\mu }^{ab} \equiv e_\nu ^{\;a}\left[ {{\partial _\mu }{e^{\nu b}} + \Gamma _{\;\mu \sigma }^\nu {e^{\sigma b}}} \right] , \end{aligned}$$
 5.
Note that the four fermion axial interaction terms in (19) are repulsive, and as such do not lead to condensate formation. Hence one could invoke other mechanisms, e.g. through the appropriate exchange of heavy states that may exist in string theory models. However, such models have not been elaborated further in [3].
 6.
Explicitly, constant backgrounds, unlike the chemical potentials, change the fermion dispersion relation. This is a crucial difference.
 7.
If the CME had been present then this would have opened up the possibility of a new mechanism for leptogenesis due to primordial magnetic fields.
 8.
In models based on string theory/Grand Unification (GUT) we note that \(H = {{\mathcal {O}}}(10^{4}10^{5})\, M_P\) during the GUT era, \(T\sim 10^{14}10^{16}\) GeV, after the inflation exit.
 9.
We also mention, for completion, that resonant phenomena, enhancing the induced CPT violation, are encountered in (the exotic) scenarios of quantumgravityinduced CPTV in models of entangled states of two (nearly degenerate in mass) neutral mesons propagating in a quantumfluctuating, stochastic and decoherening, spacetime gravitational (“foamy”) environment [41].
 10.
We remark at this point that the claimed CMElike effect in ref. [48], due to the axial component of the (generic) torsion, \(T_\mu = \epsilon _{\mu \nu \rho \sigma }K^{\nu \rho \sigma }\), with \(K^{\nu \rho \sigma }\) the contorsion tensor, arises because of an extension of the torsionful model considered in that work, which entails the replacement of the partial derivative \(\partial _\mu \) in the divergence of the axial current entering the anomaly equation by the quantity \(D_\mu = \partial _\mu  \imath \, T_\mu \). However, such a prescription is not available within the framework of our current work, where the anomaly equation has the form (A.7) (see appendix). The gravitational covariant derivative entering the divergence of the axial fourcurrent in the anomaly equation in a curved spacetime is necessarily torsion free, for symmetry reasons, as we will discuss in the appendix.
 11.
It was also argued in [22] that CME is also independent of the fermion mass, and hence the effect can also characterise massive fermions; however this latter statement may not be correct. There are subtleties if a finite fermion mass is present [24, 46]. At any rate, for our interests here, the CME will be studied for high temperatures, above the electroweak phase transition where the fermions are massless.
 12.
An alternative way to see this, is to observe that the torsion contributions to the gravitational part of the anomaly, (A.8), assume a total exterior derivative (closed) form. This implies that one can appropriately redefine the axial current by such torsion dependent terms [51], to arrive at a new gauge and Lorentzinvariant current, whose anomaly equation is torsion free.
 13.
Some important comments are in order at this point, for clarity and completeness. First of all, we stress that in this work we are concerned with a specific kind of (totally antisymmetric) torsion induced by the KR field, \({\mathbf {H}} = \mathbf {d B}\), for which one has the Bianchi identity \( {\mathbf {d}} \star {\mathbf {H}} = 0\), imposed exactly at a quantum level via the constraint (15) (conservation of the torsion charge) [17]. It is for this KR torsion that the anomaly has the form (A.7). For a generic torsion, however, defined (in the language of differential forms) as [13]: \({\mathbf {T}}^a= {\mathbf {d}} {\mathbf {e}}^a + {\overline{\omega }}^a_b \wedge {\mathbf {e}}^b = {\mathbf {K}}^a_b \wedge {\mathbf {e}}^b\), with \({\mathbf {K}}\) the contorsion tensor, there are [53] additional topological contributions to the index of the Dirac operator, and hence the chiral anomaly, which should be added on the righthandside of (A.7).These extra terms are proportional to the NiehYan topological density [56]: \( {{\mathcal {N}}} = {\mathbf {T}}^a \wedge {\mathbf {T}}_a  \mathbf {\overline{R}} ({\overline{\omega }})_{ab}\, \wedge {\mathbf {e}}^a \wedge {\mathbf {e}}^b = {\mathbf {d}} ({\mathbf {e}}^a \wedge {\mathbf {T}}_a). \) This is a divergent term, requiring proper regularisation, and in [53] it was claimed that any regulator dependence can be properly absorbed in rescalings of the vielbein, so that there are nontrivial contributions of torsion to the anomaly, coming from \({{\mathcal {N}}}\). However, there is currently a debate [57] as to whether such contributions can survive the removal of the regulator. (It is worth remarking though that, as a result of the total exterior derivative form of \({{\mathcal {N}}}\), its contribution to the anomaly shares a similar fate with that of the Htorsion (A.8.) Namely, it can be absorbed in a redefined current, depending on torsion, whose anomaly equation is torsion free [51]). Fortunately, for our stringinspired KR torsion model such ambiguities do not arise. The NiehYan invariant vanishes identically in this case, due to the Bianchi identity constraint (15); indeed, in terms of the torsion \({\mathbf {T}}^a\), this constraint can be expressed as \( 0 = {\mathbf {d}} \star {\mathbf {H}} = {\mathbf {d}}({\mathbf {e}}_a \wedge {\mathbf {T}}^a)= {{\mathcal {N}}}\), where we took into account that \(\star {\mathbf {H}} \propto {\mathbf {e}}_a \wedge {\mathbf {T}}^a \). Upon neglecting the NiehYan invariant, then, one can unambiguously find appropriate counterterms [27], taking into account (A.8), to express [28] the index of the torsionful Dirac operator, and thus the anomaly (A.7), in terms of torsionfree quantities (A.9), as discussed above.
 14.
In fact, it is only the gravitationalwave type fluctuations that contribute to the (torsionfree) Riemanncurvaturedependent part of the anomaly (A.9), which can then lead to interesting scenarios for leptogenesis, different from our approach here [58]. Moreover, graviton fluctuations in the \(R(\omega ) \, {\widetilde{R}}(\omega ) \) gravitational parts of the anomaly (A.9) might play an important rôle in radiative Majorana mass generation for the righthanded neutrinos, as explained in [59]. This occurs upon coupling the KR axion to ordinary axion fields, through kinetic mixing, with the ordinary axions coupling in turn to righthanded Majorana neutrinos via axionshiftsymmetry breaking Yukawa couplings. In this sense, the anomaly may be important in generating dynamically the mass scale for the right handed neutrinos \(m_N\) itself, that plays a pivotal rôle in our leptogenesis scenario [2, 3].
Notes
Acknowledgements
We thank L.C. Garcia de Andrade for discussions. The work of TB is supported by an STFC (UK) research (doctoral) studentship and that of NEM and SS is supported in part by STFC (UK) under the research grant ST/P000258/1. N.E.M. also acknowledges a scientific associateship (“Doctor Vinculado”) at IFICCSICValencia University (Spain).
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