Helicity evolution at smallx: Flavor singlet and nonsinglet observables

Abstract

We extend our earlier results for the quark helicity evolution at small x [1] to derive the small-x asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the g1 structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject [1], performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-x evolution equations by employing the large-Nc limit. All evolution equations resum double-logarithmic powers of αs ln 2 (1/x) in the polarization-dependent evolution along with the single-logarithmic powers of αs ln(1/x) in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin [2] using the infra-red evolution equations. Our numerical solution of the linearized large-Nc evolution equations for the flavor singlet case is presented in the accompanying Letter [3] and is further discussed here.

Small x Asymptotics of the Quark and Gluon Helicity Distributions

“…in the flavor-singlet case [3]. In the above and throughout this paper, we use light-front coordinates x ± ≡ 1 √ 2 (x 0 ±x 3 ), denote transverse vectors (x 1 ⊥ , x 2 ⊥ ) by x and their magnitudes by x T ≡ |x|, and indicate differences in transverse coordinates by the abbreviated notation x 10 ≡ x 1 − x 0 .…”

“…The small-x asymptotics of the gluon helicity can be determined in a similar way [5]. Start with the operator definition of the gluon dipole helicity TMD [7] g…”

“…To obtain a closed system of equations one has to invoke either the large-N c or the large-N c &N f limits (the latter would also require the quark exchange contribution added to the polarized Wilson lines). In the large-N c limit, with DLA accuracy, the polarized evolution equations are [1,3] …”

“…The polarized dipole G depends upon an auxiliary function Γ, termed the "neighbor dipole amplitude", in which further evolution is constrained by the lifetime of an adjacent dipole. Equations (11) were solved numerically in [2] and analytically in [4] for the high-energy asymptotics yielding…”

“…Below we will review our determination of the small-x asymptotics of the quark helicity, and then present the determination of the small-x asymptotics for the gluon helicity distributions. This talk and these proceedings are based on [1][2][3][4][5].…”