Factorizing the hard and soft spectator scattering contributions for the nucleon form factor F_1 at large Q^2

TL;DR

This paper analyzes the factorization of nucleon form factor F_1 at large Q^2, revealing divergences in soft spectator scattering and proposing that these lead to a breakdown of collinear factorization due to end-point behaviors.

Contribution

It demonstrates the presence of end-point divergences in soft spectator contributions and links them to the violation of collinear factorization for F_1 at high momentum transfer.

Findings

Logarithmic divergences in convolution integrals over soft fractions.

Overlap of soft and collinear regions causes rapidity logarithms.

Collinear factorization breaks down due to specific end-point behaviors.

Abstract

We investigate the soft spectator scattering contribution for the FF $F_{1}$. We focus our attention on 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 large rapidity logarithm which must be included in the correct factorization formalism. We conclude that a consistent description of the factorization for $F_{1}$ 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 case of form factor $F_{1}$.