Orbital angular momentum at small x revisited

Abstract

A bstract We revisit the problem of the small Bjorken- x asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised small- x helicity evolution derived recently in [1]. We relate the quark and gluon OAM distributions at small x to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the OAM distributions, we derive novel small- x evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of α s ln 2 (1/ x ) with α s the strong coupling constant). We solve these evolution equations numerically and extract the leading large- N c , small- x asymptotics of the quark and gluon OAM distributions, which we determine to be $$ {L}_{q+\overline{q}}\left(x,{Q}^2\right)\sim {L}_G\left(x,{Q}^2\right)\sim \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {\left(\frac{1}{2}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\uppi}}}, $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:msub> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> <mml:mo>~</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mfenced> <mml:mi>x</mml:mi> <mml:msup> <mml:mi>Q</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> <mml:mo>~</mml:mo> <mml:mo>∆</mml:mo> <mml:mi>Σ</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:mo>~</mml:mo> <mml:mo>∆</mml:mo> <mml:mi>G</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:mo>~</mml:mo> <mml:msup> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> <mml:mrow> <mml:mn>3.66</mml:mn> <mml:msqrt> <mml:mfrac> <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:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:math> in agreement with [2] within the precision of our numerical evaluation. (Here N c is the number of quark colors.) We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small- x region.

Topics & Concepts