Fock Space Distributions, Structure Functions, Higher Twists and Small x

TL;DR

Addressing DIS at small x, the paper develops an EFT in which the gluon field is treated classically and sea quarks arise as O(αs) corrections. By solving the Dirac equation in this background and performing color averaging over higher rapidities, it derives explicit expressions that relate quark structure functions to the underlying gluon distribution via the function tildeγ. It provides all-twist expressions for the current-current correlator W^{μν} and shows that, at large Q^2 for light quarks, the results reduce to leading-twist DGLAP-like evolution, recovering F2 and the Callan-Gross limit; heavy-quark structure functions at small x are also explicitly obtained. The framework thus connects small-x structure functions to gluon dynamics in a high-density regime and sets the stage for exploring nonlinear saturation effects beyond leading twist.

Abstract

We compute quark structure functions and the intrinsic Fock space distribution of sea quarks in a hadron wavefunction at small x. The computation is performed in an effective theory at small x where the gluon field is treated classically. At $Q^2$ large compared to an intrinsic scale associated with the density of gluons $μ^2$, large compared to the QCD scale $Λ^2_{QCD}$, and large compared to the quark mass squared $M^2$, the Fock space distribution of quarks is identical to the distribution function measured in deep inelastic scattering. For $Q^2 \le M^2$ but $Q^2 >> μ^2$, the quark distribution is computed in terms of the gluon distribution function and explicit expressions are obtained. For $Q^2 \le μ^2$ but $Q^2 >> Λ_{QCD}^2$ we obtain formal expressions for the quark distribution functions in terms of the glue. An evaluation of these requires a renormalization group analysis of the gluon distribution function in the regime of high parton density. For light quarks at high $Q^2$, the DGLAP flavor singlet evolution equations for the parton distributions are recovered. Explicit expressions are given for heavy quark structure functions at small x.

Figures (4)

• Figure 1: Diagrammatic representation of the propagator in Eq. 48.
• Figure 2: Polarization tensor with arbitrary number of insertions from the classical background field. The wavy lines are photon lines, the solid circle denotes the fermion look and the dashed lines are the insertions from the background field (see Fig. 1). The imaginary part of this diagram gives $W^{\mu\nu}$.
• Figure 3: Cut diagrams corresponding to the imaginary part of $W^{\mu\nu}$.
• Figure 4: In the singular gauge representation for the propagator (see Eq. \ref{['singprop']} and Fig. 1), multiple, higher twist contributions from the classical gluon background field to the current--current correlator (imaginary part of LHS) is equivalent to the imaginary part of RHS.