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

28 Mar 2023, 16:50
20m
Centennial Room ABC (MSU Kellogg Center)

Centennial Room ABC

MSU Kellogg Center

Parallel talk WG2: Small-x, Diffraction and Vector Mesons WG2

Speaker

Jeremy Borden

Description

We consider the novel small-$x$ helicity evolution equations previously derived using the light-cone operator treatment (LCOT) [1,2]. In the double logarithmic approximation (summing powers of $\alpha_s\ln^2(1/x)$) and in the large-$N_c$ limit, the evolution yields a closed system of equations for which we construct an analytic solution. This solution can then provide small-$x$, large-$N_c$ expressions for the flavor-singlet quark and gluon helicity PDFs and TMDs along with the $g_1$ structure function, with their leading small-$x$ asymptotics given by
\begin{align}
\Delta \Sigma (x, Q^2) \sim \Delta G (x, Q^2)
\sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{\alpha_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 the fixed-order calculations to the existing three-loop order. Notably, our anomalous dimension slightly disagrees at 4 loops with that obtained in the infrared evolution equation framework by Bartels, Ermolaev, and Ryskin (BER) [3] (the latter also agrees with the existing 3-loop calculations).
Despite the previously reported agreement at the two decimal points [2], the intercepts of our large-$N_c$ helicity evolution and that of BER disagree beyond that precision, with the BER intercept at large $N_c$ being equal to $\alpha_h^{BER} = 3.66394 \, \sqrt{\frac{\alpha_s \, N_c}{2 \pi}}$. We speculate on the origin of this disagreement.

[1] Y. V. Kovchegov, D. Pitonyak and M. D. Sievert, Helicity Evolution at Small-x, JHEP 01 (2016) 072, [1511.06737].
[2] F. Cougoulic, Y. V. Kovchegov, A. Tarasov and Y. Tawabutr, Quark and gluon helicity evolution at small x: revised and updated, JHEP 07 (2022) 095, [2204.11898]
[3] J. Bartels, B. I. Ermolaev and M. G. Ryskin, Flavor singlet contribution to the structure function G(1) at small x, Z. Phys. C 72 (1996) 627–635, [hep-ph/9603204].

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