Viability and distinguishability of the linked inflation–spontaneous-CP-violation framework of Hashimoto, Risdianto and Suematsu (2021): multi-field dynamics, quantum-gravity-induced U(1) breaking, and the joint viable parameter space

A 2021 standard-model extension links, through one singlet-scalar sector with nonminimal gravitational coupling, cosmic inflation to the origin of CP violation: spontaneous CP breaking supplies the CKM and PMNS phases via vector-like-fermion mixing, the Peccei–Quinn mechanism solves the strong CP problem, and the same sector drives slow-roll inflation, reheating, and low-scale leptogenesis thermalized by vector-like-fermion scattering. The original analysis relied on a single-field potential-valley approximation, neglected quantum-gravity-induced explicit U(1)-breaking contributions to the singlet mass, checked perturbativity only at benchmark points, and identified no experimental discriminator. The open problem is whether this framework is globally viable and experimentally distinguishable once these approximations are removed: full multi-field inflaton dynamics including Einstein-frame kinetic mixing through reheating, quantified sensitivity of the axion sector to explicit U(1)-breaking operators, an RG-improved global scan of the parameter space under all simultaneous constraints, and identifying signatures that distinguish the mechanism from other low-scale leptogenesis scenarios.

Background

In the Standard Model (SM), CP symmetry (the product of charge conjugation C and parity P) is violated by an irreducible phase in the Cabibbo–Kobayashi–Maskawa (CKM) quark-mixing matrix, and the same is expected in the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) neutrino-mixing matrix; the origin of these phases is unexplained. A separate CP issue is the strong CP problem: the CP-odd QCD vacuum angle $\bar\theta$ is experimentally bounded to $|\bar\theta|<10^{-10}$ by neutron electric-dipole-moment searches, an unexplained smallness. The Peccei–Quinn (PQ) mechanism solves this by promoting $\bar\theta$ to a dynamical axion field, at the cost of a global $U(1)_{\mathrm{PQ}}$ symmetry whose spontaneous breaking at a scale $f_a$ must not occur with too large axion fluctuations during inflation, or CMB isocurvature limits are violated. If instead CP is an exact symmetry of nature broken only spontaneously, the Nelson–Barr mechanism can generate CKM-like phases while keeping $\det(M_q)$ real so that $\bar\theta$ stays zero at tree level; Hashimoto, Risdianto and Suematsu (Phys. Rev. D 104 (2021) 075034) realized such a model with vector-like fermions (fermions whose left- and right-handed components transform identically under the SM gauge group, here $D$, $E$ and singlets $N_j$) mixing with ordinary quarks and leptons, plus an inert doublet scalar $\eta$ and singlet scalars $\sigma$ and $S$, under a global $U(1)\times Z_4$ symmetry. After the singlets acquire vacuum values $w$ and $u$, the lepton sector reduces to the scotogenic model, in which neutrino masses arise radiatively at one loop and the lightest $Z_2$-odd neutral scalar (here the inert doublet) is a weakly interacting dark-matter (DM) candidate.

For cosmology, the singlet scalars are given nonminimal couplings $\xi_\sigma|\sigma|^{2}R$ and $\xi_S|S|^{2}R$ to the Ricci scalar curvature $R$, as in Higgs inflation. In the Einstein frame (obtained by a conformal transformation that makes the gravitational term canonical), two-field potentials of this type generically possess a potential valley: a field-space direction along which the mixing angle $\varphi=\arctan(\tilde S/\tilde\sigma)$ is constant and the dynamics reduce to a single effective inflaton $\tilde\chi$. The paper studies the $\tilde S$-dominated valley, where the Einstein-frame kinetic-term mixing between the two canonically normalized fields is negligible during slow roll, and finds Higgs-inflation-like predictions for the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$. Reheating proceeds in two stages: preheating, resonant particle production at the inflaton's zero crossings when its quartic potential makes the mode equations approximately conformally invariant, followed by perturbative decays that set the reheating temperature $T_R$; requiring all couplings to stay perturbative up to the Planck scale bounds $T_R$ from above. Baryogenesis proceeds via leptogenesis: the lightest right-handed neutrino $N_1$ decays produce a CP-asymmetric lepton number converted to baryons by sphalerons (electroweak nonperturbative processes that violate $B+L$ and convert lepton number into baryon number), but unlike the ordinary seesaw scenario (where the Davidson–Ibarra bound requires $M_{N_1}\gtrsim 10^{9}\,\mathrm{GeV}$), the $N_1$ abundance is built up thermally by scattering of the vector-like fermions, allowing viable leptogenesis even at $T_R$ well below $10^{9}\,\mathrm{GeV}$. Because the PQ symmetry is broken during inflation, the axion isocurvature fluctuation (a CMB fluctuation uncorrelated with the inflaton's adiabatic perturbation) constrains the axion fraction $R_a$ of DM; the paper shows consistency provided the inert-doublet neutral, not the axion, dominates the DM abundance.

The authors explicitly flag four unfinished parts of this program, which together determine whether the framework stands as a consistent, testable whole: (i) the multi-field dynamics were never treated beyond the constant-$\varphi$ valley and neglected kinetic mixing, so the reheating and isocurvature conclusions rest on a single-field effective description whose post-inflation validity is unestablished; (ii) explicit $U(1)$-breaking higher-dimension operators, for example quantum-gravity effects contributing to the $\sigma$ mass, were not computed or bounded, leaving the axion mass and the stability of $\bar\theta=0$ unquantified; (iii) the parameter space consistent simultaneously with the DM relic abundance, direct-detection limits, leptogenesis, inflationary observables, and perturbativity/vacuum stability was sampled only at benchmarks, without an RG-improved global scan (RG = renormalization group, the running of couplings with energy scale) — in particular, how high $T_R$ can be raised before perturbativity or vacuum stability fails is unknown; (iv) no observable that could experimentally distinguish this vector-like-fermion-scattering leptogenesis from other low-scale scenarios was identified, and the authors note that a supersymmetric extension could give a possibility to escape the gravitino problem (the gravitino, the spin-$3/2$ superpartner of the graviton in supersymmetric models, is dangerously overproduced at high reheating temperature, which bounds $T_R$).

Problem Statement

For the standard-model extension of Hashimoto, Risdianto and Suematsu (Phys. Rev. D 104 (2021) 075034) — in which one CP-even singlet sector with nonminimal gravitational coupling simultaneously supplies slow-roll inflation, spontaneous CP violation inducing the CKM and PMNS phases through vector-like-fermion mixing, the Peccei–Quinn solution of the strong CP problem, scotogenic neutrino masses with inert-doublet dark matter, and low-scale leptogenesis thermalized by vector-like-fermion scattering — determine whether the framework is globally viable and experimentally distinguishable once the approximations of the original analysis are removed. This requires, as jointly load-bearing parts of one viability-and-testability determination: (i) treating the inflaton sector as a genuine multi-field system, including the Einstein-frame kinetic mixing, through the end of inflation and reheating, and establishing whether the single-field valley conclusions for the reheating temperature and the axion isocurvature bound survive in the post-inflation regime where the constant-mixing-angle valley decomposition breaks down; (ii) quantifying the sensitivity of the $\sigma$/axion mass spectrum and the stability of $\bar\theta=0$ to explicit $U(1)$-breaking higher-dimension operators, in particular contributions to the $\sigma$ mass induced by quantum-gravity effects; (iii) mapping, via an RG-improved analysis, the parameter region that simultaneously satisfies the dark-matter relic abundance, direct-detection limits, successful leptogenesis, the inflationary observables including the isocurvature constraint, and perturbativity and vacuum stability up to the Planck scale — including the maximal reheating temperature the model allows; and (iv) identifying observables that could distinguish this thermalization-by-vector-like-fermion-scattering leptogenesis from other low-scale leptogenesis scenarios, constructing the supersymmetric extension if that is required for distinctive phenomenology (for example relative to gravitino constraints on the reheating temperature). A single well-supported negative result on any part that falsifies global viability or distinguishability resolves the umbrella question in the negative, while a positive resolution requires all four parts.

References

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