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Dipole factorization for DIS at NLO I: Loop correction to the photon to quark-antiquark light-front wave-functions

Guillaume Beuf

TL;DR

The paper computes the complete one-loop corrections to the γ* → q q̄ light-front wave-function in the dipole factorization framework at small x, addressing both momentum-space and mixed (dipole) space representations. It identifies and corrects a flaw in a previous unitarity-based approach, showing that explicit loop calculations are necessary to obtain consistent NLO contributions, including UV and low-x divergences. The results demonstrate that, in mixed space, the NLO corrections acquire a universal form independent of photon polarization and virtuality, which strengthens the dipole picture’s applicability at NLO. A follow-up work combines these γ* → q q̄ results with γ* → q q̄ g corrections to deliver full NLO DIS structure functions F2 and FL at low x, and discusses the scheme-dependence of rational terms and potential extensions to massive quarks.

Abstract

The one-loop QCD corrections to the light-front wave-function for the quark-antiquark Fock state inside a transverse or longitudinal off-shell photon are explicitly calculated, both in full momentum space and in mixed space (a.k.a. dipole space). These results provide one of the main contributions to virtual NLO corrections to many DIS observables (inclusive or not) in the dipole factorization formalism at low Bjorken x. In a follow-up article, these one-loop corrections are combined with earlier results on the wave-function for the quark-antiquark-gluon Fock state, in order to get the full set of NLO corrections to the DIS structure functions $F_2$ and $F_L$ in the dipole factorization formalism, valid at low Bjorken x.

Dipole factorization for DIS at NLO I: Loop correction to the photon to quark-antiquark light-front wave-functions

TL;DR

The paper computes the complete one-loop corrections to the γ* → q q̄ light-front wave-function in the dipole factorization framework at small x, addressing both momentum-space and mixed (dipole) space representations. It identifies and corrects a flaw in a previous unitarity-based approach, showing that explicit loop calculations are necessary to obtain consistent NLO contributions, including UV and low-x divergences. The results demonstrate that, in mixed space, the NLO corrections acquire a universal form independent of photon polarization and virtuality, which strengthens the dipole picture’s applicability at NLO. A follow-up work combines these γ* → q q̄ results with γ* → q q̄ g corrections to deliver full NLO DIS structure functions F2 and FL at low x, and discusses the scheme-dependence of rational terms and potential extensions to massive quarks.

Abstract

The one-loop QCD corrections to the light-front wave-function for the quark-antiquark Fock state inside a transverse or longitudinal off-shell photon are explicitly calculated, both in full momentum space and in mixed space (a.k.a. dipole space). These results provide one of the main contributions to virtual NLO corrections to many DIS observables (inclusive or not) in the dipole factorization formalism at low Bjorken x. In a follow-up article, these one-loop corrections are combined with earlier results on the wave-function for the quark-antiquark-gluon Fock state, in order to get the full set of NLO corrections to the DIS structure functions and in the dipole factorization formalism, valid at low Bjorken x.

Paper Structure

This paper contains 36 sections, 239 equations, 3 figures.

Table of Contents

  1. Introduction
  2. On the problems with the unitarity prescription of ref. Beuf:2011xd
  3. Quark-antiquark components at LO
  4. Quark self-energy loop within an off-shell Fock state
  5. Transverse photon: quark-antiquark Fock component at one loop
  6. Diagram A for transverse photon
  7. Diagrams with instantaneous lines for transverse photon
  8. Diagrams with instantaneous quark line
  9. Diagram with instantaneous gluon line
  10. Diagram 1 for transverse photon
  11. Summing the graphs
  12. Longitudinal photon: quark-antiquark Fock component at one loop
  13. Diagram A for longitudinal photon
  14. Diagram 1 for longitudinal photon
  15. Diagram 3 for longitudinal photon
  16. ...and 21 more sections

Figures (3)

  • Figure 1: Tree-level contribution to the $q\bar{q}$ Fock component inside an incoming transverse photon.
  • Figure 2: Left: LFWF (at a given order in perturbation theory) for a Fock state ${\cal F}$ inside a physical incoming state $i$. A quark is singled out from the Fock state ${\cal F}$, as ${\cal F}=q_0+\hat{\cal F}$. Right: One-loop quark self-energy diagram correcting the contribution shown on the left.
  • Figure 3: One-gluon-loop diagrams contributing to the light-front wave-function of the $q\bar{q}$ Fock component inside an incoming transverse photon.