Connections between deep-inelastic and annihilation processes at next-to-next-to-leading order and beyond

Abstract

We have discovered 7 intimate connections between the published results for the radiative corrections, C K, to the Gross-Llewellyn Smith (GLS) sum rule, in deep-inelastic lepton scattering, and the radiative corrections, C R, to the Adler function of the flavour-singlet vector current, in e +e − annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED β-function and the zero-fermion-loop abelian terms in the 3-loop GLS sum rule. The combined effect of all 7 relations is to give the factorization of the 2-loop β-function in Δ R  C K C R −1 = β( a s a s {S 1C F a s + [S 2T F N F + S A C A + S F C F ]C F a 4 s , where a s = α s (μ 2 = Q 2)/4π is the MS coupling of an arbitrary colour gauge theory, and S = − 21 2 + 12ζ 32, S 2 = 326 3 − 304 3 ζ 3, S A = 629 2 + 884 3 ζ 3, S F = 397 6 + 136ζ 3−240ζ 5 specify the sole content of C K that is not already encoded in C R and β( a s ) = Q 2 d a s / dQ 2 at O( a 3 s) . The same result is obtained by combining the radiative corrections to Bjorken's polarized sum rule with those for the Adler function of the non-singlet axial current. We suggest possible origins of β in the “Crewther discrepancy”, Δ S, and determine Δ S /(β( a S )/ a s ) , to all orders in N F a s , in the large- N F limit, obtaining the entire series of coefficients of which S 1 and S 2 are merely the first two members.