In the string-inspired running-vacuum cosmology of Basilakos, Mavromatos and Solà Peracaula ('Do we Come from a Quantum Anomaly?', Int. J. Mod. Phys. D 28 (2019) 1944002), the observed matter-antimatter asymmetry is produced by an unconventional leptogenesis driven by a Lorentz- and CPT-violating background of the Kalb-Ramond axion, itself a relic of a gravitational-anomaly (Chern-Simons) condensate formed during inflation. The mechanism requires heavy sterile right-handed neutrinos with Majorana masses of order 10510^{5}-10710^{7} GeV, but the essay postulates these masses as inputs and explicitly declines to derive them, pointing only qualitatively to an older torsion/axion-mixing mechanism. The open problem is to derive, rather than postulate, these sterile-neutrino masses from controlled microphysics while remaining consistent with seesaw active-neutrino masses, the Higgs-mass naturalness bound mN≲108m_N \lesssim 10^{8} GeV, the required leptogenesis yield, and the Kalb-Ramond-axion dark-matter abundance. No controlled microscopic derivation of these masses is known: existing estimates are parametric, UV-sensitive, and require coupling products of order 10−1010^{-10}-10−710^{-7} left unexplained.

Background

Leptogenesis is the standard mechanism by which a lepton-number asymmetry, generated in the early Universe and partially reprocessed by electroweak sphaleron transitions (thermal transitions that convert lepton number into baryon number), produces the observed baryon asymmetry. In the most studied ('type-I seesaw') realization, heavy sterile right-handed neutrinos (RHNs) — gauge-singlet Majorana fermions whose large masses mNm_N also generate the light active-neutrino masses through the seesaw relation mν∼∣y∣2v2/mNm_\nu \sim |y|^2 v^2/m_N with Higgs vacuum expectation value vv — decay out of equilibrium to produce the asymmetry. The origin of the RHN Majorana mass scale is a standalone question in beyond-Standard-Model physics, and in string theory calculable mechanisms for it (for example D-brane instanton effects in specific compactifications, as in Ibáñez and Uranga, JHEP 0703 (2007) 052) exist but are highly model-dependent.

The framework studied here is the string-inspired running-vacuum model (StRVM). The running vacuum model (RVM) posits that the vacuum energy density 'runs' with powers of the Hubble rate, unifying inflation and dark energy; the string-inspired variant realizes this through the massless bosonic string multiplet. Its key extra field is the Kalb-Ramond (KR) axion b(x)b(x), the pseudoscalar dual of the totally antisymmetric torsion field strength of the effective four-dimensional theory. The KR axion couples anomalously to the gravitational Chern-Simons (Pontryagin) density RR~R\tilde{R}, the topological scalar built from the Riemann tensor and its dual, whose nonzero expectation value signals a gravitational anomaly; chiral primordial gravitational waves during inflation can form a gravitational-anomaly condensate that sources a constant-dilution-rate axion background b˙=const\dot{b}=\text{const}. This background survives (is not diluted by) inflation, spontaneously violates Lorentz and CPT symmetry, and biases RHN decays in the early radiation era so that a lepton asymmetry freezes out at a temperature of order mNm_N, without the departure from thermal equilibrium required in standard leptogenesis. Requiring the observed baryon asymmetry fixes the combination of mNm_N, the Higgs-portal Yukawa coupling yy, and the background amplitude: Basilakos, Mavromatos and Solà find mN∼105m_N \sim 10^{5}-10710^{7} GeV with y∼10−5y \sim 10^{-5}, a window also compatible with the naturalness bound mN≲108m_N \lesssim 10^{8} GeV that keeps one-loop RHN corrections to the Higgs mass under control.

What the framework does not supply is where mNm_N comes from. Because the KR axion couples only derivatively (shift-symmetric), it cannot by itself generate a chirality-changing mass. Mavromatos and Pilaftsis (Phys. Rev. D 86 (2012) 124038) exhibited a loophole: if the KR axion kinetically mixes with other axions from the string spectrum, and those axions couple to sterile neutrinos through chirality-changing Yukawa interactions (plausibly instanton-generated), then two- and three-loop gravitational diagrams generate an effective RHN Majorana mass. However, its magnitude is highly model dependent — the original authors quote possible ranges from keV to multi-TeV — and scales as a high power of the ultraviolet cutoff, MR∝y~a γ Λ6M_R \propto \tilde{y}_a\,\gamma\,\Lambda^{6}, with axion Yukawa coupling y~a\tilde{y}_a and kinetic-mixing parameter γ\gamma. The companion analyses of the StRVM program (Basilakos-Mavromatos-Solà, PoS CORFU2018 (2019) 044 and Phys. Rev. D 101 (2020) 045001) show parametric compatibility with the leptogenesis window for cutoff Λ\Lambda near the Planck scale, but only at the price of coupling products quoted as ∣y~aγ∣∼10−10|\tilde{y}_a\gamma| \sim 10^{-10}-10−710^{-7} that are inserted by hand, and without embedding in any concrete compactification. The question is therefore whether a controlled microscopic derivation of the required mass window exists or can be constructed within this framework.

Problem Statement

Within the string-inspired running-vacuum framework of anomaly-induced leptogenesis, derive — rather than postulate — the Majorana masses of the sterile right-handed neutrinos from a controlled microscopic mechanism, in which the relevant couplings and mass parameters are computed in a setting with a physical ultraviolet scale and an explicit matching onto the four-dimensional effective theory with quantified uncertainties, yielding a mass for the lightest sterile neutrino — the state whose mass sets the freeze-out temperature — in the window mN∼105m_N \sim 10^{5}-10710^{7} GeV indicated by the observed baryon asymmetry and the freeze-out analysis, with the relevant couplings derived rather than tuned by hand (computed from the microscopic theory or explained by a symmetry or selection rule, not chosen to reproduce the target; in particular not relying on an unexplained product ∣y~aγ∣∼10−10|\tilde{y}_a\gamma| \sim 10^{-10}-10−710^{-7}). The derivation must be simultaneously consistent with: (i) type-I-seesaw active-neutrino masses given the associated Higgs-portal Yukawa couplings; (ii) the Higgs-mass naturalness bound mN≲108m_N \lesssim 10^{8} GeV; (iii) the undiluted KR-axion background dynamics required for Lorentz-/CPT-violating leptogenesis; and (iv) consistency with the identification of the Kalb-Ramond or mixed axion as the dark matter, i.e. with an axion potential UbU_b whose scale reproduces the observed dark-matter abundance (the detailed microphysical origin of UbU_b is a separate open issue of the framework and need not itself be solved here, but the derived mass mechanism must not be incompatible with it). The expected mass window and its uncertainty follow from the microscopic parameters rather than being treated as a free input, and the derived mass is consistent with the observed baryon asymmetry.

References

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