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Deep inelastic scattering near the endpoint in soft-collinear effective theory

Junegone Chay, Chul Kim

TL;DR

This work uses soft-collinear effective theory to factorize deep inelastic scattering near the endpoint (x→1) into a hard coefficient, a jet function, and soft/collinear matrix elements forming the parton distribution function. Through a two-step SCET matching (SCET_I→SCET_II) and careful zero-bin subtraction, the authors derive explicit factorization formulas for F1 and F_L, relate endpoint PDFs to a convolution of a collinear piece with a soft Wilson line, and show that the parton distribution evolves via the Altarelli-Parisi kernel even at the endpoint. Moments of the structure functions decompose into a product of moments of the jet function, soft Wilson line, and collinear matrix elements, enabling perturbative radiative corrections to be computed and resummed (LL accuracy). The analysis clarifies the role of soft gluon emission in PDFs near the endpoint and sets the stage for extensions to flavor-singlet channels and other high-energy processes where endpoint factorization is relevant.

Abstract

We apply the soft-collinear effective theory (SCET) to deep inelastic scattering near the endpoint region. The forward scattering amplitude, and the structure functions are shown to factorize as a convolution of the Wilson coefficients, the jet functions, the parton distribution functions. The behavior of the parton distribution functions near the endpoint region is considered. It turns out that it evolves with the Altarelli-Parisi kernel even in the endpoint region, and the parton distribution function can be factorized further into a collinear part and the soft Wilson line. The factorized form for the structure functions is obtained by the two-step matching, and the radiative corrections or the evolution for each factorized part can be computed in perturbation theory. We present the radiative corrections of each factorized part to leading order in alpha_s, including the zero-bin subtraction for the collinear part.

Deep inelastic scattering near the endpoint in soft-collinear effective theory

TL;DR

This work uses soft-collinear effective theory to factorize deep inelastic scattering near the endpoint (x→1) into a hard coefficient, a jet function, and soft/collinear matrix elements forming the parton distribution function. Through a two-step SCET matching (SCET_I→SCET_II) and careful zero-bin subtraction, the authors derive explicit factorization formulas for F1 and F_L, relate endpoint PDFs to a convolution of a collinear piece with a soft Wilson line, and show that the parton distribution evolves via the Altarelli-Parisi kernel even at the endpoint. Moments of the structure functions decompose into a product of moments of the jet function, soft Wilson line, and collinear matrix elements, enabling perturbative radiative corrections to be computed and resummed (LL accuracy). The analysis clarifies the role of soft gluon emission in PDFs near the endpoint and sets the stage for extensions to flavor-singlet channels and other high-energy processes where endpoint factorization is relevant.

Abstract

We apply the soft-collinear effective theory (SCET) to deep inelastic scattering near the endpoint region. The forward scattering amplitude, and the structure functions are shown to factorize as a convolution of the Wilson coefficients, the jet functions, the parton distribution functions. The behavior of the parton distribution functions near the endpoint region is considered. It turns out that it evolves with the Altarelli-Parisi kernel even in the endpoint region, and the parton distribution function can be factorized further into a collinear part and the soft Wilson line. The factorized form for the structure functions is obtained by the two-step matching, and the radiative corrections or the evolution for each factorized part can be computed in perturbation theory. We present the radiative corrections of each factorized part to leading order in alpha_s, including the zero-bin subtraction for the collinear part.

Paper Structure

This paper contains 12 sections, 140 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Kinematics
  3. Operators in SCET near the endpoint region
  4. Factorization of $F_1 (x,Q)$
  5. parton distribution function
  6. moment analysis
  7. Radiative corrections
  8. Factorization of $F_L (x,Q)$
  9. Conclusion
  10. Zero-bin subtraction before the usoft factorization
  11. Discontinuity in $\mathrm{SCET}_{\mathrm{I}}$ and $\mathrm{SCET}_{\mathrm{II}}$
  12. Anomalous dimension of $J_{\mu}^{(1b)}$

Figures (12)

  • Figure 1: (a) Feynman diagram for the forward scattering amplitude in DIS in $\rm{SCET_I}$ and (b) the prescription of the (u)soft Wilson lines.
  • Figure 2: An energetic parton comes out of the proton with the momentum $y n\cdot P$. It emits soft gluons with momentum $\kappa$, and the momentum of the hard parton before the hard scattering becomes $yn\cdot P -\kappa$.
  • Figure 3: Feynman rules for the collinear operator $O_c^q$. (a) the tree-level operator (b) the operator with a collinear gluon with incoming momentum $q^{\mu}$.
  • Figure 4: Radiative corrections for the quark distribution operator at one loop.
  • Figure 5: The Feynman diagram describing the forward scattering amplitude $T_2 (x,Q)$.
  • ...and 7 more figures