Helicity evolution at small <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>x</mml:mi></mml:math>: Revised asymptotic results at 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> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>N</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:math>
Daniel Adamiak, Yuri V. Kovchegov, Yossathorn Tawabutr
Abstract
We present a numerical solution of the revised version [J. High Energy Phys. 07 (2022) 095] of the small-$x$ helicity evolution equations from [J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)] at large ${N}_{c}$ and ${N}_{f}$. (Here ${N}_{c}$ and ${N}_{f}$ are the numbers of quark colors and flavors, respectively.) The evolution equations are double-logarithmic in the Bjorken $x$ variable, resumming powers of ${\ensuremath{\alpha}}_{s}{\mathrm{ln}}^{2}(1/x)$ with ${\ensuremath{\alpha}}_{s}$ the strong coupling constant. The large-${N}_{c}&{N}_{f}$ evolution we consider includes contributions of small-$x$ quark emissions and is thus more realistic than the large-${N}_{c}$ one, which only involves gluon emissons. The evolution equations [J. High Energy Phys. 07 (2022) 095, J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)] are written for the so-called ``polarized dipole amplitudes,'' which are related to the helicity distribution functions and the ${g}_{1}$ structure function. Unlike the previously reported solution [J. High Energy Phys. 08 (2020) 014] of the earlier version of helicity evolution equations at large ${N}_{c}&{N}_{f}$ [J. High Energy Phys. 01 (2016) 072, Phys. Rev. D 99, 054032 (2019)], our solution does not exhibit periodic oscillations in $\mathrm{ln}(1/x)$ for ${N}_{f}<2{N}_{c}$, while only showing occasional sign reversals. For ${N}_{f}=2{N}_{c}$, we report oscillations with $\mathrm{ln}(1/x)$, similar to those found earlier in [J. High Energy Phys. 08 (2020) 014]. We determine the intercept of our evolution for ${N}_{f}<2{N}_{c}$ as well as the parameters of the oscillatory behavior for ${N}_{f}=2{N}_{c}$. We compare our results to the existing resummation [Z. Phys. C 72, 627 (1996)] and finite-order calculations for helicity-dependent quantities in the literature.