CONNECTIONS BETWEEN DEEP-INELASTIC AND ANNIHILATION PROCESSES AT NEXT-TO-NEXT-TO-LEADING ORDER AND BEYOND

We have discovered 7 intimate connections between the published result s for the radiative corrections, C(K), to the Gross-Llewellyn Smith (G LS) sum rule, in deep-inelastic lepton scattering, and the radiative c orrections, C(R), to the Adler function of the flavour-singlet vector current, in e+e- annihilation. These include a surprising relation bet ween the scheme-independent single-electron-loop contributions to the 4-loop QED beta-function and the zero-fermion-loop abelian terms in th e 3-loop GLS sum rule. The combined effect of all 7 relations is to gi ve the factorization of the 2-loop beta-function in DELTA(S) = C(K)C(R ) - 1 = beta(a(s)BAR)/a(s)BAR {S1C(F)a(s)BAR + [S2T(F)N(F) + S(A)C(A) + S(F)C(F)]C(F)a(s)2BAR} + O(a(s)4BAR), where a(s)BAR = alpha(s)(mu2 = Q2)/4pi is the MSBAR coupling of an arbitrary colour gauge theory, an d S1 = -21/2 + 12zeta3, S2 = 326/3 - 304/3zeta3, S(A) = -629/2 + 884/3 zeta3, S(F) = 397/6 + 136zeta3- 240zeta5 specify the sole content of C (K) that is not already encoded in C(R) and beta(a(s)BAR) = Q2da(s)BAR /dQ2 at O(a(s)3BAR). The same result is obtained by combining the radi ative corrections to Bjorken's polarized sum rule with those for the A dler function of the non-singlet axial current. We suggest possible or igins of beta in the ''Crewther discrepancy'', DELTA(S), and determine DELTA(S)/(beta(a(s)BAR/a(s)BAR), to all orders in N(F)a(s)BAR, in the large-N(F) limit, obtaining the entire series of coefficients of whic h S1 and S2 are merely the first two members.