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Mass Corrections to the Vector Current Correlator

K. G. Chetyrkin, R. Harlander, J. H. Kuehn, M. Steinhauser

TL;DR

This work develops and implements an automated, large-momentum expansion method to compute mass corrections to the three-loop vector current correlator $\Pi(q^2)$, obtaining terms up to $(m^2/q^2)^6$ and translating them into the $e^+e^-\to$ hadrons cross-section ratio $R(s)$. The authors provide explicit, colour-separated results for the polarization function and the corresponding $R(s)$, analyze scheme dependence between MSbar and pole masses, and validate the expansion against known analytic results and Padé approximations across energy ranges. The methodology combines subgraph generation, Taylor expansion in small parameters, and evaluation with MINCER and MATAD, enabling reliable predictions from the high-energy region toward threshold. The findings demonstrate good agreement with prior results and extend the applicability of mass-correction expansions to near-threshold energies.

Abstract

Three-loop QCD corrections to the vector current correlator are considered. The large momentum procedure is applied in order to evaluate mass corrections up to order $(m^2/q^2)^6$. The inclusion of the first seven terms to the ratio $R=σ(e^+e^- \to hadrons)/σ(e^+e^- \to μ^+μ^-)$ leads to reliable predictions from the high energy region down to relatively close to threshold.

Mass Corrections to the Vector Current Correlator

TL;DR

This work develops and implements an automated, large-momentum expansion method to compute mass corrections to the three-loop vector current correlator , obtaining terms up to and translating them into the hadrons cross-section ratio . The authors provide explicit, colour-separated results for the polarization function and the corresponding , analyze scheme dependence between MSbar and pole masses, and validate the expansion against known analytic results and Padé approximations across energy ranges. The methodology combines subgraph generation, Taylor expansion in small parameters, and evaluation with MINCER and MATAD, enabling reliable predictions from the high-energy region toward threshold. The findings demonstrate good agreement with prior results and extend the applicability of mass-correction expansions to near-threshold energies.

Abstract

Three-loop QCD corrections to the vector current correlator are considered. The large momentum procedure is applied in order to evaluate mass corrections up to order . The inclusion of the first seven terms to the ratio leads to reliable predictions from the high energy region down to relatively close to threshold.

Paper Structure

This paper contains 5 sections, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Large momentum procedure
  3. Mass corrections to $\Pi(q^2)$
  4. $\sigma(e^+e^-\to {\rm hadrons})$
  5. Summary

Figures (5)

  • Figure 1: Possible topologies for the subgraphs of an NO-type diagram.
  • Figure 2:
  • Figure 3: $R_i^{(2)}, i=A,{\it NA},l,F$ for $\mu^2 = s$ including successively higher orders in $m^2/s$. The same notation as in Fig. \ref{['rvx.ps']} is adopted. The semi-analytical result is not shown.
  • Figure 4: $R_{\it NA}^{(2)} - R_{g}^{(2)}(\xi = 4)$ over $x = 2m/\sqrt{s}$. Dotted: semi-analytical result; dashed: mass terms up to $(m^2/s)^5$; solid: mass terms up to $(m^2/s)^6$.
  • Figure 5: Comparison between the analytical result without 4-particle contribution ($R^{(2)}_{F,virt}$, dotted line) and approximate result with terms up to order $(m^2/s)^5$ (dashed) and $(m^2/s)^6$ (solid line).