All-Orders Evolution of Parton Distributions: Principle, Practice, and Predictions

Abstract

Parton distribution functions (PDFs) are defining expressions of hadron structure. Exploiting the role of effective charges in quantum chromodynamics, an algebraic scheme is described which, given any hadron's valence parton PDFs at the hadron scale, delivers predictions for all its PDFs (unpolarized and polarized) at any higher scale. The scheme delivers results that are largely independent of both the value of the hadron scale and the pointwise form of the charge; and, inter alia , enables derivation of a model-independent identity that relates the strength of the proton's gluon helicity PDF, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msubsup> <mml:mi>G</mml:mi> <mml:mi>p</mml:mi> <mml:mi>ζ</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , to that of the analogous singlet polarized quark PDF and valence quark momentum fraction. Using available data fits and theory predictions, the identity yields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> <mml:mi mathvariant="normal">GeV</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1.48</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mn>10</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . It furthermore entails that the measurable quark helicity contribution to the proton spin is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msubsup> <mml:mover accent="true"> <mml:mi>a</mml:mi> <mml:mo>∼</mml:mo> </mml:mover> <mml:mrow> <mml:mn>0</mml:mn> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>ζ</mml:mi> <mml:mi mathvariant="normal">C</mml:mi> </mml:msub> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>0.32</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , thereby reconciling contemporary experiment and theory.