Beyond mean field at the pairing threshold: the gap-to-Tc ratio of Dirac fermions near four dimensions, where fluctuations add ζ(2) to η(2) (Paper P17)

An earlier paper of this series gave a closed mean-field ratio Δ₀/T_c = r(a) = R(a)1/a, R(a) = 2ᵃ⁺¹ η(a) Γ(1 + a/2)/Γ(1 − a/2), at the pairing threshold of a superconductor whose density of states grows like |ξ|^a, 0 < a < 2. We ask what survives beyond mean field and answer in three parts. (A) The two gap equations of that paper are, line by line, the large-N saddle-point equations of the Gross–Neveu model in D = a + 2 space-time dimensions; the threshold coupling a is the critical coupling, and the threshold ratio is the ratio M/T_c of the renormalised theory, for every coupling of the ordered phase. We prove that r(a) coincides with the closed formula of Inagaki, Kouno and Muta (1995) for D-dimensional four-fermion theories; the cases D = 2 and D = 3 give π e−γ and 2 ln 2. (B) Near D = 4 the order parameter fluctuates. In the Gross–Neveu–Yukawa model at its infrared fixed point in D = 4 − ε we obtain, at leading order in ε, the closed form (M/T_c)² = ε ρ_n(N) for the chiral Ising, XY and Heisenberg classes (n = 1, 2, 3) with N four-component Dirac fermions. The bosonic thermal tadpole brings ζ(2), the fermionic one η(2) = ζ(2)/2. For N → ∞, ρ_n(N) → 4η(2) = π²/3, which is exactly the mean-field edge r(a)² ∼ 4η(2)(2 − a): the two limits commute. The first correction is +(n + 2)π²/N, and ρ_n(N) > π²/3 for all N: fluctuations raise the gap-to-T_c ratio. (C) A descriptive comparison with lattice Monte Carlo and functional-renormalisation data for the chiral Ising class in D = 3 shows the same sign of the 1/N_f correction. All statements are classical; every numbered claim is checked by companion code. Each result carries a status label (proved, leading order, descriptive). The paper has boxes in plain words and for readers from physics and from mathematics, three figures and a glossary. The companion archive contains the check program (items A1–A7, B1–B8, E1–E4; 78 checks at 30 and 50 digits, all passed), the two reference logs and the figure script.