Euler Factors at 2 in Neutrino Decoupling: (4/11)^{1/3}, a d-Dimensional Family, and Why α/π Appears in N_eff

Summary: After the neutrinos of the early universe decouple, electron–positron annihilation heats the photons, and the two temperatures end up in the ratio T_ν/T_γ = (4/11)^(1/3). For a number theorist the interesting part of this textbook result is where the rational numbers come from. The ratio 11/4 = 1 + (7/8)·2 contains 7/8 = 1 − 2^(−3), the Euler factor at 2 that separates η(4) from ζ(4), and the density ratio n_ν/n_γ = 3/11 contains in addition 3/4 = 1 − 2^(−2) at weight 3. We first give the same computation in d space dimensions. With d − 1 photon polarisations and 2^⌊(d+1)/2⌋ Dirac components one finds (T_γ/T_ν)^d = 1 + (1 − 2^(−d)) g_e/g_γ, and we prove that the tempting short form 3 − 2^(1−d) holds exactly for d ∈ {2, 3, 5}. We then turn to the effective number of neutrino species N_eff and ask why the fine-structure constant appears in it. Interactions in the electron–positron–photon plasma lower its entropy by the fraction (25/22) α/π, and this raises N_eff by (50/11) α/π − (160/(11√3)) (α/π)^(3/2) + … ≈ 0.0096 in the limit of instantaneous decoupling at T ≫ m_e. These formulas are classical (Lopez–Turner, Bennett et al.). What we add is an explanation of their shape. The two-loop pressure is a product of two thermal moments of weight 2, namely η(2)/2π² for an electron line and ζ(2)/2π² for a photon line. Since even zeta values are rational multiples of powers of π, the coefficient of α/π is rational. The √3 in the next term is the square root of the Debye mass m_D² = e²T²/3, again built from η(2), and only bosons contribute to this term, because fermionic Matsubara frequencies are odd multiples of πT and never vanish. Beyond this order the arithmetic changes character rather than stopping. At order e⁴ the coefficient, known in closed form since Arnold and Zhai, contains ζ′(−1)/ζ(−1), ζ′(−3)/ζ(−3) and log 2; by the functional equation these are first derivatives at the weights 2 and 4. At order e⁵ the constant is ψ(1/2) = −γ − 2 log 2, the finite part of the fermionic sum at the pole of ζ at s = 1. About three quarters of N_eff − 3 ≈ 0.044 come from the dynamics of incomplete decoupling, for which no closed form exists. As an outlook we show that the Euler factor at 2 is the case t = 1/2 of a twisted boundary condition: the twisted moments are Bernoulli polynomials B_2m(t), and at t = a/q the primes dividing q enter. No new physics and no new prediction for N_eff is claimed. Section 2 explains the physics from the beginning. The companion code (77 checks, Python with mpmath and sympy) reproduces every displayed number and identity. Euler factor - Riemann zeta function - Dirichlet eta function - even zeta values - pole at s = 1 - Euler's constant - functional equation - Matsubara frequencies - Fermi–Dirac statistics - neutrino decoupling - neutrino temperature - effective number of neutrino species - N_eff - finite-temperature QED - Debye mass - plasma frequency - Stefan–Boltzmann law in d dimensions - fine-structure constant - Bernoulli polynomials - Hurwitz zeta function - digamma function - Dirichlet L-functions - imaginary chemical potential