Non-leading contributions to the asymptotic behavior of the quark form factor

Abstract

We have explicitly calculated, to fourth-order in perturbation theory, the first non-leading contributions to the quark ( p, m)-quark ( p, m)-color-singlet current ( q = p′ − p) form factor in the region − q 2 ⪢ p 2 ≈ p′ 2 ⪢ m 2. These contributions, together with the leading ones, combine into the simple form F 1 = exp{ B 1 [ p; p′; g( k 2/ M 2; g M )]} where B 1 [ p; p′; g( k 2/ M 2; g M )] is the lowest-order contribution calculated replacing the renormalized coupling g M by the invariant running charge g( k 2/ M 2; g M ) = g M + β 0 g M 3 log k 2/ M 2 + O( g M 5), β 0 being the one-loop β-function coefficient and k the internal gluon momentum. This result, if true to all order and for all non-leading terms, implies that the asymptotic behavior of the form factor cannot be determined unless both the short-distance and long-distance behavior of g( k 2) are known.