Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept

Abstract

We construct an exact analytic solution of the revised small-$x$ helicity evolution equations derived recently. The equations we solve are obtained in the large-$N_c$ limit (with $N_c$ the number of quark colors) and are double-logarithmic (summing powers of ${\alpha }_s{\ln }^2(1/x)$ with ${\alpha }_s$ the strong coupling constant and $x$ the Bjorken $x$ variable). Our solution provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure function, with their leading small-$x$ asymptotics given by \begin{align} ΔΣ(x, Q^2) \sim ΔG (x, Q^2) \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{α_h} , \notag \end{align} where the exact analytic expression we obtain for the intercept ${\alpha }_h$ can be approximated by ${\alpha }_h=3.66074\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}$. Our solution also yields an all-order (in ${\alpha }_s$) resummed small-$x$ anomalous dimension $\Delta {\gamma }_{GG}(\omega )$ which agrees with all the existing fixed-order calculations (to three loops). Notably, our anomalous dimension is different from that obtained in the infrared evolution equation framework developed earlier by Bartels, Ermolaev, and Ryskin (BER), with the disagreement starting at four loops. Despite the previously reported agreement at two decimal points based on the numerical solution of the same equations, the intercept of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ given by a different analytic expression from ours with the numerical value of ${\alpha }_h^{BER}=3.66394\sqrt{\frac{{\alpha }_s{N}_c}{2\pi }}$. We speculate on the origin of this disagreement.

arXiv comment22 pages, 3 figures

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Cited by in corpus (6)

• Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\&N_f$
• Global analysis of polarized DIS & SIDIS data with improved small-$x$ helicity evolution
• Quark TMDs from back-to-back dijet production at forward rapidities in pA collisions beyond eikonal accuracy in the CGC
• Color Glass Condensate meets High Twist Expansion
• Parton model contributions as next-to-eikonal corrections to the dipole factorization of DIS and SIDIS at low $x_{Bj}$
• Proton spin from small-$x$ with constraints from the valence quark model