Analytic solution for the revised helicity evolution at small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math> and large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>N</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math>: 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 by Cougoulic et al. [J. High Energy Phys. 07 (2022) 095], based on the earlier work of Kovchegov et al. [J. High Energy Phys. 01 (2016) 072; Phys. Rev. D 99, 054032 (2019)]. 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 ${\ensuremath{\alpha}}_{s}{\mathrm{ln}}^{2}(1/x)$ with ${\ensuremath{\alpha}}_{s}$ 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 and for the ${g}_{1}$ structure function, with their leading small-$x$ asymptotics given by $\mathrm{\ensuremath{\Delta}}\mathrm{\ensuremath{\Sigma}}(x,{Q}^{2})\ensuremath{\sim}\mathrm{\ensuremath{\Delta}}G(x,{Q}^{2})\ensuremath{\sim}{g}_{1}(x,{Q}^{2})\ensuremath{\sim}{(\frac{1}{x})}^{{\ensuremath{\alpha}}_{h}}$, where the exact analytic expression we obtain for the intercept ${\ensuremath{\alpha}}_{h}$ can be approximated by ${\ensuremath{\alpha}}_{h}=3.66074\sqrt{\frac{{\ensuremath{\alpha}}_{s}{N}_{c}}{2\ensuremath{\pi}}}$. Our solution also yields an all-order (in ${\ensuremath{\alpha}}_{s}$) resummed small-$x$ anomalous dimension $\mathrm{\ensuremath{\Delta}}{\ensuremath{\gamma}}_{GG}(\ensuremath{\omega})$ 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 et al. (BER) [Z. Phys. C 72, 627 (1996)] 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 ${\ensuremath{\alpha}}_{h}^{\mathrm{BER}}=3.66394\sqrt{\frac{{\ensuremath{\alpha}}_{s}{N}_{c}}{2\ensuremath{\pi}}}$. We speculate on the origin of this disagreement.

Topics & Concepts