Helicity at small x: oscillations generated by bringing back the quarks
Yuri V. Kovchegov, Yossathorn Tawabutr
Abstract
A bstract We construct a numerical solution of the recently-derived large- N c & N f small- x helicity evolution equations [1] with the aim to establish the small- x asymptotics of the quark helicity distribution beyond the large- N c limit explored previously in the same framework. (Here N c and N f are the numbers of quark colors and flavors.) While the large- N c helicity evolution involves gluons only, the large- N c & N f 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- N c & N f limit lead to the x -dependence of the flavor-singlet quark helicity distribution which is well-approximated by $$ {\left.\Delta \Sigma \left(x,{Q}^2\right)\right|}_{\mathrm{large}\hbox{-} {N}_c\&{N}_f}\sim {\left(\frac{1}{x}\right)}^{\alpha_h^q}\cos \left[{\omega}_q\ln \left(\frac{1}{x}\right)+{\varphi}_q\right]. $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mfenced> <mml:mrow> <mml:mi>ΔΣ</mml:mi> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> </mml:mrow> </mml:mfenced> <mml:mrow> <mml:mtext>large</mml:mtext> <mml:mo>‐</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>&</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mfenced> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>h</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:msup> <mml:mo>cos</mml:mo> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>ω</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>ln</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>x</mml:mi> </mml:mfrac> </mml:mfenced> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>φ</mml:mi> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> </mml:mfenced> <mml:mo>.</mml:mo> </mml:math> The power $$ {\alpha}_h^q $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>h</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> </mml:math> exhibits a weak N f -dependence, and, for all N f values considered, remains very close to $$ {\alpha}_h^q\left({N}_f=0\right)=\left(4/\sqrt{3}\right)\sqrt{\alpha_s{N}_c/\left(2\pi \right)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mi>h</mml:mi> <mml:mi>q</mml:mi> </mml:msubsup> <mml:mfenced> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>/</mml:mo> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfenced> <mml:msqrt> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:mfenced> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msqrt> </mml:math> obtained earlier in the large- N c limit [2, 3]. The novel oscillation frequency ω q and phase shift φ q depend more strongly on the number of flavors N f (with ω q = 0 in the pure-glue large- N c 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 [456–7].