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k-Factorization and Impact Factors at Next-to-leading Level

M. Ciafaloni, D. Colferai

TL;DR

This work refines next-to-leading x k-factorization by explicitly defining a factorization scheme that separates the NL$x$ BFKL kernel from impact factors and extends it to colourless probes. It computes finite one-loop corrections to quark and gluon impact factors, showing they are universal and governed by a single coefficient ${\mathcal K}$, with connections to soft timelike splitting functions and DGLAP evolution. The authors apply the framework to partonic processes, deriving fragmentation vertices, real and virtual contributions, and demonstrate how scheme choices and energy-scale choices influence the finite parts. They outline how to extend the analysis to the two-loop NL$x$ kernel and emphasize the need to validate the all-orders factorization conjecture, potentially using group-theoretical methods to incorporate exact $s$-channel dynamics.

Abstract

We further analyse,at next-to-leading log(s) level,the form of k-factorization and the definition of impact factors previously proposed by one of us,and we generalize them to the case of hard colourless probes. We then calculate the finite one-loop corrections to quark and gluon impact factors and we find them universal,and given by the same K factor which occurs in the soft timelike splitting functions.

k-Factorization and Impact Factors at Next-to-leading Level

TL;DR

This work refines next-to-leading x k-factorization by explicitly defining a factorization scheme that separates the NL BFKL kernel from impact factors and extends it to colourless probes. It computes finite one-loop corrections to quark and gluon impact factors, showing they are universal and governed by a single coefficient , with connections to soft timelike splitting functions and DGLAP evolution. The authors apply the framework to partonic processes, deriving fragmentation vertices, real and virtual contributions, and demonstrate how scheme choices and energy-scale choices influence the finite parts. They outline how to extend the analysis to the two-loop NL kernel and emphasize the need to validate the all-orders factorization conjecture, potentially using group-theoretical methods to incorporate exact -channel dynamics.

Abstract

We further analyse,at next-to-leading log(s) level,the form of k-factorization and the definition of impact factors previously proposed by one of us,and we generalize them to the case of hard colourless probes. We then calculate the finite one-loop corrections to quark and gluon impact factors and we find them universal,and given by the same K factor which occurs in the soft timelike splitting functions.

Paper Structure

This paper contains 16 sections, 113 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. k-Factorization in dijet production
  3. Impact factors at NL$x$ accuracy
  4. Quark-parton collision
  5. Gluon-parton collision
  6. gb $\rightarrow$ ggb
  7. gb $\rightarrow$ q$\overline{\sf q}$b
  8. Collinear factorization and finite parts
  9. Discussion
  10. Squared matrix elements in the fragmentation region
  11. Polarization vectors
  12. $\sf gb\to ggb$ squared matrix element
  13. $\sf gb\to q\overline{q}b$ squared matrix element
  14. Calculation of ${\boldsymbol k}_1$ integrals
  15. Quark impact factor
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Double ${\boldsymbol k}$-factorization of dijet differential cross section.
  • Figure 2: Diagrammatic representation of the impact factor. In the sum over the possible final states, the outgoing parton index c denotes the gluon g for a = q while it denotes the quark q or the softer gluon for a = g.
  • Figure 3: Born diagram of high energy scattering.
  • Figure 4: Real gluon emission in central region.
  • Figure 5: Real gluon emission amplitude in quark a fragmentation region.
  • ...and 5 more figures