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Three-Loop Anomalous Dimension of the Heavy Quark Pair Production Current in Non-Relativistic QCD

A. H. Hoang

TL;DR

The paper computes the NNLL non-mixing contributions to the anomalous dimension of the leading spin-triplet heavy-quark production current in vNRQCD, addressing the RG structure of near-threshold quark-antiquark production. It demonstrates that both soft and ultrasoft degrees of freedom must be present for all scales below the heavy mass and that the soft-ultrasoft renormalization scales must be correlated to maintain consistency. Analytic expressions for the NNLL non-mixing running are derived, with ultrasoft contributions shown to dominate and yield sizable shifts in the current, and an updated NNLL cross section for $e^+e^-$ annihilation at threshold is provided, including discussion of scheme-dependent subtleties. The results bolster the precision of RG-improved predictions for top-quark threshold phenomenology and serve as a nontrivial check of the vNRQCD framework.

Abstract

The three-loop non-mixing contributions to the anomalous dimension of the leading order quark pair production current in non-relativistic QCD are computed. It is demonstrated that the renormalization procedure can only be carried out consistently if the dynamics of both soft and the ultrasoft degrees of freedom is present for all scales below the heavy quark mass, and if the soft and ultrasoft renormalization scales are always correlated.

Three-Loop Anomalous Dimension of the Heavy Quark Pair Production Current in Non-Relativistic QCD

TL;DR

The paper computes the NNLL non-mixing contributions to the anomalous dimension of the leading spin-triplet heavy-quark production current in vNRQCD, addressing the RG structure of near-threshold quark-antiquark production. It demonstrates that both soft and ultrasoft degrees of freedom must be present for all scales below the heavy mass and that the soft-ultrasoft renormalization scales must be correlated to maintain consistency. Analytic expressions for the NNLL non-mixing running are derived, with ultrasoft contributions shown to dominate and yield sizable shifts in the current, and an updated NNLL cross section for annihilation at threshold is provided, including discussion of scheme-dependent subtleties. The results bolster the precision of RG-improved predictions for top-quark threshold phenomenology and serve as a nontrivial check of the vNRQCD framework.

Abstract

The three-loop non-mixing contributions to the anomalous dimension of the leading order quark pair production current in non-relativistic QCD are computed. It is demonstrated that the renormalization procedure can only be carried out consistently if the dynamics of both soft and the ultrasoft degrees of freedom is present for all scales below the heavy quark mass, and if the soft and ultrasoft renormalization scales are always correlated.

Paper Structure

This paper contains 11 sections, 37 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Method of the Computation
  3. Non-mixing contributions at NNLL order
  4. Ultrasoft non-mixing contributions at NNLL order
  5. Soft non-mixing contributions at NNLL order
  6. Solution of the anomalous dimension at NNLL order
  7. Production Cross Section at Threshold and Comparison
  8. Conclusion
  9. Convention for the potential operators ${\cal O}_k^{(1,T)}$
  10. 4-quark matrix elements of 6-field operators
  11. Collection of Wilson coefficients

Figures (5)

  • Figure 1: Three-loop current correlator diagrams and the counterterm diagram for the computation of the NLL anomalous dimension of $c_1$.
  • Figure 2: Four-loop graphs for the calculation of the ultrasoft non-mixing contributions of the NNLL anomalous dimension of $c_1$.
  • Figure 3: Counterterm graphs for the removal of subdivergences in graphs of Figs. \ref{['figusoftcorrelators']} and \ref{['figsoftcorrelators']}. Graphs with wave function renormalization constants are to be understood.
  • Figure 4: Four-loop graphs for the calculation of the soft non-mixing contributions of the NNLL anomalous dimension of $c_1$.
  • Figure 5: (a) Six-field operators that contribute to the soft running of ${\cal O}_{k,k1,k2}^{1,T}$ through the graphs in (b).