Odd Zeta Values in Critical Exponents: Graph Periods, Group Factors and the Gamma Function

Summary: Near a continuous phase transition – water at its critical point, a magnet at its Curie temperature, liquid helium at the λ-point – physical quantities obey power laws whose exponents η, ν, ω are universal: they depend only on the dimension d and on the symmetry group O(n) of the order parameter, not on the material. In the expansion around d = 4, with d = 4 − 2ε, the coefficients of these exponents are rational functions of n times odd zeta values ζ(3), ζ(5), ζ(7), ζ(9), powers of π, and from six loops on multiple zeta values. Why these numbers appear is not usually asked. We make the arithmetic bookkeeping explicit, using the six-loop results of Kompaniets and Panzer for general n and Schnetz's seven-loop results for n = 1. First, the loop order chooses the weight: the L-loop coefficient of the β-function has weight 2L − 3, and the coefficient of εᵏ has weight 2k − 3 in 1/ν and ω and 2k − 5 in η, up to ε⁷. Second, the top weight is a sum of graph periods times group factors: for L = 3, …, 6 the top-weight part of the β-function equals (−1)^(L+1)·48(L+2)·Σ_G P(G)/|Aut G| times an explicit polynomial in n, and we prove how it passes to the exponents; the only n in the denominators is (n+8)ʲ, the location of the fixed point. Third, π is not a new constant: in the hatted zeta values of Baikov and Chetyrkin all three exponents are free of π to the computed order. For large n we prove this to all orders. The leading coefficient η₁ is a rational function times Γ(1−2ε)/(Γ(1+ε)Γ(1−ε)³), and we show that a product of Gamma values ∏ Γ(1−λᵢε)^(wᵢ) with Σ wᵢλᵢ = 0 is π-free in hatted zeta values if and only if its arguments come in reflection pairs λ ↔ 1−λ of opposite weight; the generating function of the hatting is 1 − cosh((λ−½)t)/cosh(t/2). In η₁ the odd zeta values appear as 2ᵏη(k), with the Dirichlet eta function, i.e. through the Euler factor of the prime 2. The paper is written for number theorists; Section 2 explains the physics. No new physics is claimed. A final section proposes, as the author's hypothesis and in continuation of P5, an arithmetic reading of universality: the exponents are counts of graph cycles and of closed index loops, weighted by periods, and the critical point is where these counts balance. The companion code (76 checks) reproduces every coefficient and numerical statement; it downloads the published data from arXiv and verifies their checksums.