Abstract
We construct a numerical solution of the recently-derived large-Nc&Nf small-x helicity evolution equations [1] with the aim to establish the small-x asymptotics of the quark helicity distribution beyond the large-Nc limit explored previously in the same framework. (Here Nc and Nf are the numbers of quark colors and flavors.) While the large-Nc helicity evolution involves gluons only, the large-Nc&Nf evolution includes contributions from quarks as well. We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of x. Our numerical results in the large-Nc&Nf limit lead to the x-dependence of the flavor-singlet quark helicity distribution which is well-approximated by1ΔΣxQ2large‐Nc&Nf∼1xαhqcosωqln1x+φq.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {\\left.\\Delta \\Sigma \\left(x,{Q}^2\\right)\\right|}_{\\mathrm{large}\\hbox{-} {N}_c\\&{N}_f}\\sim {\\left(\\frac{1}{x}\\right)}^{\\alpha_h^q}\\cos \\left[{\\omega}_q\\ln \\left(\\frac{1}{x}\\right)+{\\varphi}_q\\right]. $$\\end{document}The power {alpha}_h^q exhibits a weak Nf-dependence, and, for all Nf values considered, remains very close to {alpha}_h^qleft({N}_f=0right)=left(4/sqrt{3}right)sqrt{alpha_s{N}_c/left(2pi right)} obtained earlier in the large-Nc limit [2, 3]. The novel oscillation frequency ωq and phase shift φq depend more strongly on the number of flavors Nf (with ωq = 0 in the pure-glue large-Nc limit). The typical period of oscillations for ∆Σ is rather long, spanning many units of rapidity. We speculate whether the oscillations we find are related to the sign variation with x seen in the strange quark helicity distribution extracted from the data [4–7].
Highlights
A weak N√f -dependence, and, for all Nf values considered, 0) = (4/ 3) αsNc/(2π) obtained earlier in the large-Nc remains limit [2, 3]
We construct a numerical solution of the recently-derived large-Nc&Nf smallx helicity evolution equations [1] with the aim to establish the small-x asymptotics of the quark helicity distribution beyond the large-Nc limit explored previously in the same framework. (Here Nc and Nf are the numbers of quark colors and flavors.) While the largeNc helicity evolution involves gluons only, the large-Nc&Nf evolution includes contributions from quarks as well
We speculate whether the oscillations we find are related to the sign variation with x seen in the strange quark helicity distribution extracted from the data [4,5,6,7]
Summary
The equations (3.2) are self-contained over the following region in (s10, η)-plane: η ∈ [0, ηmax], η−ηmax ≤ s10 ≤ η, where ηmax is some arbitrary positive upper value of the η-range This is the region where we will solve them numerically. For i = j we have Qj(j−1) = 1 in the last term of eq (3.5): this is determined by the initial conditions Including this additional contribution from a single point in the i, j grid does not significantly affect the numerical solution. Writing each equation in (3.3) in this recursive form allows for a numerical evaluation with one fewer layer of loops, resulting in much shorter computation time for smaller step sizes, ∆η and ∆s, and for a larger η-range, defined by η ∈ [0, ηmax].
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