Helicity evolution at small x: quark to gluon and gluon to quark transition operators

A bstract We include the quark to gluon and gluon to quark shock-wave transition operators into the small Bjorken- x evolution equations for helicity in the flavor-singlet channel derived earlier in [1–3]. While such transitions do not affect the large- N c version of the evolution equations for helicity, the large- N c & N f equations are affected. ( N c and N f are the numbers of quark colors and flavors, respectively.) We derive the corresponding corrected large- N c & N f equations for the polarized dipole amplitudes contributing to the flavor-singlet quark and gluon helicity distributions in the double-logarithmic approximation (DLA), resumming powers of α s ln 2 (1 /x ) with α s the strong coupling constant. We solve these equations iteratively and extract the polarized splitting functions up to four loops. We show that our splitting functions agree with the fixed-order perturbative calculations up to and including the existing three-loops results [4–7]. Similar to the large- N c helicity evolution in the shock-wave approach [8], our large- N c & N f small- x splitting functions agree with those obtained in the infrared evolution equations framework from [9, 10] up to three loops, but appear to slightly disagree at four loops.