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On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function

Einan Gardi

TL;DR

This work investigates perturbative quark distributions for on-shell heavy quarks and their all-order connections to heavy-quark fragmentation. It establishes, in the large-β0 limit, that the quark distribution equals the perturbative fragmentation function and shows that, in the Sudakov (x→1) region, the two are governed by the same Wilson-line–based exponent, enabling all-order exponentiation. The authors determine the two-loop anomalous dimension (NNLL accuracy) via both a Wilson-line calculation and extraction from fragmentation results, finding agreement and thus solidifying the shared Sudakov structure. While the distribution and fragmentation functions diverge in their DGLAP evolution away from large-x, the study reveals a deep, process-independent link through Wilson-line formalism and renormalon analysis, with implications for precision predictions in heavy-quark decays and a clean separation of perturbative and non-perturbative effects using DGE methods.

Abstract

I present new results on the quark distribution in an on-shell heavy quark in perturbative QCD and explore its all-order relations with heavy-quark fragmentation. I first compute the momentum distribution function to all orders in the large-beta_0 limit and show that it is identical to the perturbative heavy-quark fragmentation function in the same approximation. I then analyze the Sudakov limit of the distribution and the fragmentation functions using Wilson lines and prove that the corresponding Sudakov exponents in the non-Abelian theory are the same to any logarithmic accuracy. The anomalous dimension is then determined to two-loop order, corresponding to next-to-next-to-leading logarithmic accuracy in the exponent, in two ways: the first by extracting the singular terms from a recent calculation of the fragmentation function and the second by performing the two-loop Wilson-line calculation in configuration space. I find perfect agreement between the two.

On the quark distribution in an on-shell heavy quark and its all-order relations with the perturbative fragmentation function

TL;DR

This work investigates perturbative quark distributions for on-shell heavy quarks and their all-order connections to heavy-quark fragmentation. It establishes, in the large-β0 limit, that the quark distribution equals the perturbative fragmentation function and shows that, in the Sudakov (x→1) region, the two are governed by the same Wilson-line–based exponent, enabling all-order exponentiation. The authors determine the two-loop anomalous dimension (NNLL accuracy) via both a Wilson-line calculation and extraction from fragmentation results, finding agreement and thus solidifying the shared Sudakov structure. While the distribution and fragmentation functions diverge in their DGLAP evolution away from large-x, the study reveals a deep, process-independent link through Wilson-line formalism and renormalon analysis, with implications for precision predictions in heavy-quark decays and a clean separation of perturbative and non-perturbative effects using DGE methods.

Abstract

I present new results on the quark distribution in an on-shell heavy quark in perturbative QCD and explore its all-order relations with heavy-quark fragmentation. I first compute the momentum distribution function to all orders in the large-beta_0 limit and show that it is identical to the perturbative heavy-quark fragmentation function in the same approximation. I then analyze the Sudakov limit of the distribution and the fragmentation functions using Wilson lines and prove that the corresponding Sudakov exponents in the non-Abelian theory are the same to any logarithmic accuracy. The anomalous dimension is then determined to two-loop order, corresponding to next-to-next-to-leading logarithmic accuracy in the exponent, in two ways: the first by extracting the singular terms from a recent calculation of the fragmentation function and the second by performing the two-loop Wilson-line calculation in configuration space. I find perfect agreement between the two.

Paper Structure

This paper contains 15 sections, 83 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quark distribution in an on-shell heavy quark computed with a single dressed gluon
  3. Definition and calculation of the gluon emission cut
  4. Moments and evolution
  5. Sudakov logs in the quark distribution function
  6. Definitions and results
  7. Strictly factorized form and Wilson lines
  8. Evolution equation
  9. Relation between distribution and fragmentation in the Sudakov region
  10. Comments on the calculation of the Sudakov exponent to NNLO
  11. Conclusions
  12. Properties of Wilson lines
  13. One-loop integrals with a Borel--modified propagator
  14. The splitting function and the cusp anomalous dimension
  15. Calculation of diagram 11 in Ref. KM

Figures (3)

  • Figure 1: Minkowski space-time picture (vertical axis as time and horizontal axis as $x_3$) of the Wilson--line configuration $W[C_S](ip\cdot y\mu/m)$ of Eq. (\ref{['def_Wilson']}) representing the quark distribution function in an on-shell quark in the infinite mass limit (in the rest frame of this quark). The two figures describe the situation when $y^-$ is positive (l.h.s) or negative (r.h.s), where path--ordering on the lightlike segment $l_2$ from $0$ to $y$ corresponds to time--ordering and anti-time--ordering, respectively.
  • Figure 2: Minkowski space-time picture of the Wilson--line configuration in the second line of Eq. (\ref{['D_def_singular']}), i.e. $W^*[C_S](ip\cdot y\mu/m)=W[C_S](-ip\cdot y\mu/m)$, representing the perturbative fragmentation function in the infinite--mass limit (in the rest frame of the produced quark). The two figures describe the situation when $y^-$ is positive (l.h.s) or negative (r.h.s), where path ordering on the lightlike segment $l_2$ from $0$ to $y$ corresponds to time--ordering and anti-time--ordering, respectively; cf. Fig. \ref{['fig:dist']}.
  • Figure 3: One of the non-Abelian two-loop diagrams contributing to $\ln W[C_S]$ of Eq. (\ref{['def_Wilson']}). It corresponds to diagram 11 in Fig. 6 of Ref. KM (published version).