Quantitative baryogenesis from the QCD axion's damped, time-varying strong-CP violation: computing the baryon yield coupled to low-scale baryon-number violation across the axion parameter space

The QCD axion, introduced to solve the strong-CP problem, generically produces two epochs of large CP violation in the early universe: an era in which the effective strong-CP parameter $\bar{\theta}$ stays of order unity down to temperatures of order a few GeV, and a subsequent non-equilibrium relaxation epoch in which damped axion oscillations deliver time-varying CP-violating pulses with an effective magnitude estimated by the Dirichlet series $\eta(3/4)\approx 0.651$ (Krauss, arXiv:2201.01447). Whether these axion-sourced CP-violating epochs can quantitatively generate the observed baryon asymmetry, when coupled to explicit baryon-number-violating processes in a concrete particle-physics model, has never been computed. The open problem is to perform that computation: derive the baryon yield produced by the damped, time-varying $\bar{\theta}(t)$ (and the preceding $\bar{\theta}=O(1)$ era) coupled to baryon-number-violating interactions active at $T\sim 0.1$–$2$ GeV, across the viable axion-decay-constant range with a controlled treatment of the temperature-dependent axion mass, and determine whether any observationally viable parameter region reproduces the observed asymmetry or whether the mechanism fails quantitatively.

Background

The strong-CP problem is the unexplained smallness of the effective CP-violating parameter of quantum chromodynamics (QCD), $\bar{\theta}$, which is bounded by neutron electric-dipole-moment searches to $|\bar{\theta}|\lesssim 10^{-10}$ today. The Peccei–Quinn (PQ) mechanism solves this dynamically by introducing a pseudo-scalar field, the axion $a$, whose vacuum value dynamically drives $\bar{\theta}\approx a/f_a$ to zero; $f_a$ is the PQ symmetry-breaking scale, bounded from below to roughly $f_a\gtrsim 10^{10}$ GeV by astrophysics. In the early universe the axion field is displaced from its minimum (the 'misalignment' mechanism), and it begins coherent damped oscillations when the Hubble expansion rate $H$ (the inverse age of the universe) drops to the axion mass $m_a$. Below QCD confinement (chiral symmetry breaking at $\Lambda_{\mathrm{QCD}}\sim O(1)$ GeV) the axion mass is fixed; above it, in the dilute instanton gas approximation (DIGA — the approximation that treats topological QCD fluctuations, which generate the axion mass, as a sparse gas of instantons), the mass falls steeply with temperature as $m_a(T)\sim m_a(0)T^{-4}[\ln T]^{3}$. Krauss (arXiv:2201.01447, 2022; a preprint with no journal publication) observed two consequences. First, the DIGA scaling shifts the oscillation-onset temperature to $T\propto f_a^{-1/6}$ instead of $f_a^{-1/2}$, so that for all viable $f_a$ the axion has not yet relaxed at $T\sim 2$ GeV: $\bar{\theta}$ — a large source of CP violation outside the neutron-dipole bound, which applies only after the axion relaxes — remains greater than $\sim 10^{-7}$ (and of order unity for $f_a<10^{12}$ GeV) until well below the electroweak epoch. Second, once oscillations begin, the axion amplitude redshifts as $t^{-3/4}$ during radiation domination, so the oscillating $\bar{\theta}(t)$ does not average to zero: on timescales $\sim H^{-1}$ it acts as short, damped, non-equilibrium pulses of CP violation, whose effective magnitude the paper estimates by the alternating Dirichlet series $\sum_n(-1)^{n-1}n^{-3/4}=\eta(3/4)\approx 0.651$ relative to the pre-oscillation value. Sakharov's conditions for baryogenesis — baryon-number violation, departure from thermal equilibrium, and CP violation — would then be met by the axion's relaxation itself, provided some explicit baryon-number-violating interactions are still active at $T\sim 0.1$–$2$ GeV. Such low-scale baryon-number violation is highly constrained (nucleon decay, neutron–antineutron oscillation, flavor bounds), but model frameworks for baryogenesis below the electroweak scale exist (Krauss and Trodden, Phys. Rev. Lett. 83, 1502 (1999); Servant, Phys. Rev. Lett. 113, 171803 (2014), where strong CP violation drives baryogenesis in a two-flavor color-superconducting phase). The nearest quantitatively developed relatives are: axiogenesis (Co and Harigaya, Phys. Rev. Lett. 124, 111602 (2020)), where baryon asymmetry arises from a rotating PQ field with electroweak sphalerons (thermal transitions that violate $B+L$, the sum of baryon and lepton number) still in equilibrium — a different CP source at a different epoch; spontaneous baryogenesis from axions with generic couplings (Domcke, Ema, Mukaida and Yamada, JHEP 08 (2020) 096) — baryon production driven by the effective chemical potential of a time-dependent field — whose transport framework is the natural tool here but whose worked cases assume baryon-number-violating interactions in equilibrium at the oscillation epoch, which for the QCD axion begins far below the sphaleron freeze-out temperature of $\sim 130$ GeV; and QCD-transition baryogenesis scenarios (Croon, Howard, Ipek and Tait, Phys. Rev. D 101, 055042 (2020)). On the input side, the temperature dependence of the axion mass near the QCD transition is controlled by lattice QCD only up to $T\lesssim 600$ MeV (Borsanyi et al., Nature 539, 69 (2016); Borsanyi et al., Phys. Lett. B 752, 175 (2016)); the DIGA regime relevant for oscillation onset at higher temperatures remains without controlled lattice input. What no existing work provides is the quantitative link: the baryon yield actually produced when the axion's damped, time-varying $\bar{\theta}(t)$ is coupled to explicit low-scale baryon-number-violating dynamics in a concrete model.

Problem Statement

The problem, as left open by the source, is a single coupled quantitative question: can the QCD axion's early-universe CP violation produce the observed baryon asymmetry $Y_B\approx 8.7\times 10^{-11}$? Concretely, in an internally consistent particle-physics model containing the QCD axion (decay constant $f_a\gtrsim 10^{10}$ GeV) and explicit baryon-number-violating interactions that remain active at temperatures $T\sim 0.1$–$2$ GeV, compute from the coupled axion–baryon-number transport dynamics the baryon yield generated (i) by the epoch in which $\bar{\theta}$ remains large (of order unity for $f_a<10^{12}$ GeV) down to $T\sim 2$ GeV, and (ii) by the subsequent non-equilibrium relaxation in which the damped, time-varying $\bar{\theta}(t)$ delivers CP-violating pulses of effective magnitude $\eta(3/4)\approx 0.651$ times the pre-oscillation value. The computation must be carried out as a function of $f_a$ across the viable parameter space, using a justified treatment of the temperature-dependent axion mass $m_a(T)$ — including the intermediate regime where $m_a(T)$ evolves between the dilute-instanton-gas behavior and the low-temperature fixed mass, which the source only treats in the simplified $f_a\sim 10^{16}$ GeV case — and must determine whether any region consistent with laboratory and astrophysical bounds on the baryon-number-violating operators, and with survival of the generated asymmetry against washout (erasure by the inverse baryon-number-violating reactions), reproduces the observed asymmetry. The resolution is either a viable parameter region with the corresponding predicted yield and its model-dependence, or a quantitative demonstration that a specific ingredient fails (the CP-odd bias induced in the chosen baryon-number-violating rates is too small, the interactions are out of equilibrium at the relevant epochs, or washout and laboratory bounds exclude every viable corner). The problem is posed within a single, explicitly stated benchmark model for the baryon-number-violating dynamics — or within a named finite set of such models — with the sweep over $f_a$ taken across the viable parameter space of the stated model or models. The source also asks, as a further open-ended extension, whether the same time-varying CP violation leaves observable residual effects on cosmological or astrophysical parameters beyond baryogenesis; the outcome is either a positive identification of such an imprint or a quantified search that finds no observable imprint within stated assumptions.

References

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