Small- x helicity evolution: An operator treatment

Abstract

We rederive the small-x evolution equations governing quark helicity distribution in a proton using solely an operator-based approach. In our previous works on the subject, the evolution equations were derived using a mix of diagrammatic and operator-based methods. In this work, we re-derive the double-logarithmic small-x evolution equations for quark helicity in terms of the "polarized Wilson lines", the operators consisting of light-cone Wilson lines with one or two non-eikonal local operator insertions which bring in helicity dependence. For the first time we give explicit and complete expressions for the quark and gluon polarized Wilson line operators, including insertions of both the gluon and quark sub-eikonal operators. We show that the double-logarithmic small-x evolution of the "polarized dipole amplitude" operators, made out of regular light-cone Wilson lines along with the polarized ones constructed here, reproduces the equations derived in our earlier works. The method we present here can be used as a template for determining the small-x asymptotics of any transverse momentum-dependent (TMD) quark (or gluon) parton distribution functions (PDFs), and is not limited to helicity.

Ultrarelativistic quark-nucleus scattering in a light-front Hamiltonian approach

“…However, in reality the quark carries a finite longitudinal momentum and therefore admits sub-eikonal effects. A variety of works using arXiv:2002.09757v1 [nucl-th] 22 Feb 2020 pQCD has studied sub-eikonal effects from different aspects, including helicity change of the quark, longitudinal momentum exchange, and finite length of the background field [26][27][28][29][30][31][32]. In this work, we treat the quark with finite energy and keep its interaction time with the nucleus finite, and we reveal a sub-eikonal effect through the evolution of the quark's transverse coordinate distribution from the non-perturbative aspect.…”

Helicity-dependent generalization of the JIMWLK evolution

“…The kernel for the flavor singlet helicity evolution of a test operator consisting of two Wilson lines in any irreducible representation (irrep) of the gauge group SU(N c ) (with one of them being polarized) is constructed in Sec. III A based on the calculation of helicitydependent evolution of polarized dipoles carried out in [5]. This kernel is successfully cross checked with the results of Bartels, Ermolaev and Ryskin [37] at the level of the ladder approximation in Sec.…”

“…where the Latin indices are i, j = 1, 2, ǫ ij ⊥ is a twodimensional Levi-Civita symbol, and F 12 is the gluon field strength tensor. As shown in [4,5], the transfer of helicity from the target shock wave to the probe in the gluon t-channel sector is proportional to this subeikonal operator β(x). Helicity can be transferred between the target and projectile via an (anti)quark exchange, as shown in the right two panels of Fig.…”

“…Functional derivatives and notations a. Polarized Wilson lines. Those are derived in [4,5] for the (anti)quarks and gluons scattering on a polarized target. They consist of light-cone Wilson lines with the sub-eikonal operator insertions shown in Fig.…”

“…A new feature of helicity-dependent distributions, compared to the unpolarized ones, is the use of Wilson lines beyond the eikonal approximation [4,5]. In order to be sensitive to the target helicity, the probe must interact with the target in a sub-eikonal way at least once.…”