Factorizing the hard and soft spectator scattering contributions for the nucleon form factor F₁ at large Q²

Abstract

In a previous paper (Phys. Rev. D 83, 093005 (2011)) we suggested the factorization formula for the nucleon form factors which consists of the sum of two contributions describing the hard and soft spectator scattering, and we provided a description of the soft rescattering contribution for the FF F₁ in terms of convolution integrals of the hard and hard-collinear coefficient functions with the appropriate soft matrix elements. In the present paper we investigate the soft spectator scattering contribution for the FF F₁. We focus our attention on the factorization of the hard-collinear scale $\sim Q\Lambda$ corresponding to transition from SCET-I to SCET-II. We compute the leading-order jet functions and find that the convolution integrals over the soft fractions are logarithmically divergent. This divergency is the consequence of the boost invariance and does not depend on the model of the soft correlation function describing the soft spectator quarks. Using as example a two-loop diagram we demonstrated that such a divergency corresponds to the overlap of the soft and collinear regions. As a result one obtains a large rapidity logarithm which must be included in the correct factorization formalism. We conclude that a consistent description of the factorization for F₁ implies the end-point collinear divergencies in the hard and soft spectator contributions, i.e. convolution integrals with respect to collinear fractions are not well defined. Such scenario can only be realized when the twist-3 nucleon distribution amplitude has specific end-point behavior which differs from one expected from the evolution of the nucleon distribution amplitude. Such behavior leads to the violation of the collinear factorization for the hard spectator scattering contribution. We suggest that the soft spectator scattering and chiral symmetry breaking provide the mechanism responsible for the violation of collinear factorization in the case of form factor F₁. In spite of complexities of the SCET factorization it can be very useful for a phenomenological analysis of hard exclusive reactions. The basis for such approach is provided by the universality of the SCET-I form factors which can appear in different hard processes. We show that using the so-called physical subtraction scheme SCET factorization in some cases allows to perform the systematical analysis of the hadronic processes in the range of moderate values of $Q^{2}\sim 5--20$ GeV^2 where the hard-collinear scale $\sim Q\Lambda$ is still not large.