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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.
We present a Monte Carlo based analysis of the combined world data on polarized lepton-nucleon deep-inelastic scattering at small Bjorken $x$ within the polarized quark dipole formalism. We show for the first time that double-spin asymmetries at $x<0
Analytical study of the rapidity distribution of the final state particles in deep inelastic scattering at small x is presented. We separate and analyse three sources of particle production: fragmentation of the quark-antiquark pair, accompanying coh
Transverse momentum dependent (TMD) distributions at small x exhibit a rich infinite twist structure that encompasses the leading twist (partonic) distributions as well as the physics of gluon saturation. Progress to further the connection between th
We elaborate on the dichotomy between the description of the semileptonic decays of heavy hadrons on the one hand and the semileptonic decays of light hadrons such as neutron $beta$ decays on the other hand. For example, almost without exception the
The nuclear modification factor $R_{pA}(p_T)$ provides information on the small-$x$ gluon distribution of a nucleus at hadron colliders. Several experiments have recently measured the nuclear modification factor not only in minimum bias but also for