Which Axion Couplings Admit Heavy-Fermion Completions?

Abstract

Which axion couplings can heavy particles generate, and what unavoidable contribution to gauge running does their realization require? This theoretical paper gives an exact answer for specified heavy-fermion classes coupled to the Standard Model gauge group with its Z6 quotient. It determines the full set of three gauge coefficients produced by arbitrarily many coherently wound complex Dirac representations: every response is a nonnegative integer combination of eight basic vectors. Allowing Majorana masses in real representations changes the answer to six generators. Neither result requires a cutoff on representation size or a measurement of a gravitational coupling. The classification separates globally consistent effective couplings from those realizable by the chosen matter class. It also establishes a limit: multiplying any nonnegative allowed lattice vector by four gives a coherent Dirac realization, and multiplying by two suffices when real Majorana masses are allowed. Absolute normalization therefore matters; coupling ratios alone cannot provide these exclusions. For every anomaly produced by primitive masses of either winding sign, the complete running frontier follows from at most 70 explicit candidates for Dirac matter or nine when real Majorana masses are allowed. It gives every minimum of a nonnegative weighted running cost and tests all running upper bounds simultaneously. For the response (3,0,0), three attaining sectors describe the full colour-hypercharge tradeoff, including a mixed-cost optimum missed by considering only its endpoints. The frontier gives an attained one-loop scale ceiling uniform over a bounded mass window. A further differential inequality controls the two-loop gauge and Yukawa contributions under stated coupling and matrix-norm bounds. An explicit benchmark remains excluded in both matter classes, although it passes global quantization and componentwise triangle tests. A finite running budget also restricts unknown integer normalizations, allowing their complete enumeration. The classification is then connected to published Standard Model gauge determinations through top-scale matching and an enlarged input interval. Requiring the couplings to remain below 0.12 up to 10^18 GeV, with the stated Yukawa bounds, gives a concrete heavy-threshold constraint. For the matched response (17,-15,-354), at least one heavy threshold must exceed 3.20 x 10^8 GeV; the corresponding triangle-only test gives approximately 3.10 x 10^7 GeV. The stronger condition persists under the reported input variations and assumed favorable endpoint shifts. It is a conditional consistency bound, not a measured axion mass limit. Complete coefficient sets quantify the size of the improvement. At a heavy-mass ceiling of 10^8 GeV, the 11,861 responses admitted by global quantization and the triangle test reduce to 11,749 Dirac or 11,807 Dirac-Majorana candidates. The additional exclusion is small but nonzero. Results for 24 choices of validity scale, mass ceiling and endpoint allowance are included. Six published KSVZ running benchmarks are reproduced under their original assumptions, providing a like-for-like comparison with existing calculations. The paper projects every permitted coefficient vector before comparing with photon information. The measurement-anchored example gives 398 values of E/N instead of the lattice test's 400. One of the two exclusions disappears when a larger endpoint allowance is admitted, even though the original full response ray remains excluded. This distinction prevents more information from being attributed to a photon measurement than it contains. Physical comparison still requires QCD matching, its uncertainties and the unmeasured amplitude sign. The paper also exhibits two low-dimensional heavy sectors with identical three-component anomaly response and identical one-loop gauge running. Their two-loop gauge coefficients differ by an explicit matrix. With a common 2 TeV threshold, their gauge-only trajectories lie on opposite sides of the selected ultraviolet colour-coupling ceiling. This separate calculation illustrates why a response generator cannot replace the particle spectrum in a higher-order model assessment. Further results bound the charged representation content using gauge data alone, reconstruct a primitive irreducible source when a gravitational coefficient is added, and describe finite coherent full-spectrum fibres. Directional defect transport and a compact-CHC construction are applications with separate assumptions. The central classification does not require accepting CHC. The classification is also applied to actual particle spectra. From a published 141-representation decay-operator list, the paper constructs all 19,612 one- and two-multiplet spectra in a stated primitive-winding class and retains their distinct two-loop matrices. At common masses of 10^10, 10^12 and 10^14 GeV, respectively 263, 442 and 1,009 pass the specified gauge-only perturbativity condition. These are finite-class counts, not claims that the models satisfy every phenomenological constraint. A selected dimension-six-decaying triplet is then followed through number-density evolution, energy transfer, axion oscillations and entropy production. Obtaining the dark-matter density by changing the initial axion angle does not preserve a pre-existing baryon asymmetry through a large entropy release. The example requires an initial baryon yield about 0.254 and fails a specified initial-yield ceiling of 0.01. A low maximum temperature avoids this obstruction for the same spectrum under explicit production and PQ assumptions. Reference isocurvature and reheating-energy conditions leave a nonempty interval for the inflationary Hubble scale. This is a conditional cosmological application, not a complete inflationary construction or a primordial-element likelihood. The scalar and nuclear conditions are also examined separately. For one Higgs doublet and one complex PQ scalar, an exact classical criterion distinguishes a bounded potential from one whose desired mixed vacuum is the global minimum. Four coupled one-loop examples expose the additional dependence on scalar and Yukawa couplings. A full PRIMAT nuclear-network calculation gives helium, deuterium and lithium abundances for a specified standard post-decay history, with correlated errors from 2,000 nuclear-rate and neutron-lifetime draws. A common-history result states when different completions can share that calculation. Neither classical stability nor a passed abundance comparison substitutes for matching decay products, quantum vacuum decay or finite-temperature evolution. The results are intended for axion model building, effective-theory matching and inverse problems involving heavy matter. They classify response existence and sharp conditional running constraints, not unique spectra or complete phenomenological viability. The two-loop obstruction uses continuous threshold matching and a bound on the sum of heavy Yukawa matrix products, with no heavy-light mixing. Finite matching shifts and higher orders require separate control. No experimental detection is claimed. An explicit radiative-decay calculation also connects leading photon and electron spectra to thermal burning, nonthermal photodisintegration and a redshifted neutrino distribution. Continuing the full nuclear network through the overlap removes a 0.5% deuterium offset caused by premature freeze-out. A separate radial PQ effective potential gives quantum and thermal tunnelling actions, checked by shooting and independent collocation. The added neutral particle and scalar operators are stated assumptions; these results do not determine full neutrino collision evolution, hadronic cascades for every coloured spectrum, or a complete multi-field vacuum-decay rate. A specified late-time radiative population also yields reproducible abundance constraints over continuous parameter intervals. Positive comparison equations enclose all trajectories in a declared photon-rate family. Joint nuclear-input calculations, finite-sample calibration and observational errors give sufficient compatible and excluded bands for a 100 MeV neutral particle initialized at 1 keV. Across lifetimes from 300,000 to 3,000,000 seconds, a number ratio at most 9.5e-11 has a permitted mean prediction, whereas ratios from 4.304e-9 through 1e-8 are excluded within the bounded-rate model. The rate envelope is a stated robustness assumption, not an experimentally calibrated uncertainty for every omitted process. Production, neutrino collisions, full backreaction and UV vacuum survival are separate questions. A further example makes the relic abundance a consequence of the particle model. A 205 MeV electron-portal scalar reheats the plasma and produces a 100 MeV neutral dipole fermion. Evolving neutrino number and energy exchange determines the entropy and photon multiplication; scalar decay and inverse dipole production fix the relic abundance. Nine conditional nuclear lifetime slices connect that source to abundance comparisons, late radiation and CMB spectral heating. The scalar potential and radial loop controls use the same masses and couplings. This supplies a reproducible production-to-observable example, not a unique CHC model or a proof of complete cosmological survival. Momentum-resolved neutrino/nuclear transport, the joint perturbation likelihood and the full scalar effective action remain distinct requirements. The archive contains the manuscript, editable LaTeX source, exact response and reconstruction tools, finite proof certificates, numerical cosmology, nuclear-transport and scalar programs, licensed inputs, pinned original-author PRIMAT/ACROPOLIS/CosmoTransitions/nudec_BSM sources, five executed companion notebooks, independent tests, machine-readable results and a guide to applying the criteria. Primary literature is cited for established mechanisms and numerical inputs. Numerical