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QCD Decoupling at Four Loops

K. G. Chetyrkin, J. H. Kühn, C. Sturm

TL;DR

This work computes the four-loop decoupling function for the QCD gauge coupling across a heavy-quark threshold, reducing the theoretical uncertainty in the evolution of $\alpha_s$ through quark thresholds and refining the determination of $\alpha_s(M_Z)$ from low-energy data. It also leverages a low-energy theorem to derive the Higgs–gluon coupling $C_1$ at four loops and provides a partial five-loop prediction, connecting decoupling, RG evolution, and Higgs phenomenology. The results are validated by independent methods and supported by extensive multiloop computational techniques, including vacuum tadpole integrals and master integrals. Overall, the paper tightens the link between high- and low-energy QCD physics and improves precision for Standard Model predictions involving $\alpha_s$ and Higgs processes.

Abstract

We present the matching condition for the strong coupling contant alpha_s at a heavy quark threshold to four loops in the modified minimal subtraction scheme. Our results lead to further decrease of the theoretical uncertainty of the evolution of the strong coupling constant through heavy quark thresholds. Using a low energy theorem we furthermore derive the effective coupling of the Higgs boson to gluons (induced by a virtual heavy quark) in four- and (partially) through five-loop approximation.

QCD Decoupling at Four Loops

TL;DR

This work computes the four-loop decoupling function for the QCD gauge coupling across a heavy-quark threshold, reducing the theoretical uncertainty in the evolution of through quark thresholds and refining the determination of from low-energy data. It also leverages a low-energy theorem to derive the Higgs–gluon coupling at four loops and provides a partial five-loop prediction, connecting decoupling, RG evolution, and Higgs phenomenology. The results are validated by independent methods and supported by extensive multiloop computational techniques, including vacuum tadpole integrals and master integrals. Overall, the paper tightens the link between high- and low-energy QCD physics and improves precision for Standard Model predictions involving and Higgs processes.

Abstract

We present the matching condition for the strong coupling contant alpha_s at a heavy quark threshold to four loops in the modified minimal subtraction scheme. Our results lead to further decrease of the theoretical uncertainty of the evolution of the strong coupling constant through heavy quark thresholds. Using a low energy theorem we furthermore derive the effective coupling of the Higgs boson to gluons (induced by a virtual heavy quark) in four- and (partially) through five-loop approximation.

Paper Structure

This paper contains 9 sections, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. Decoupling for the Gauge Coupling Constant
  4. Vacuum Integrals
  5. Decoupling at four loops: results
  6. Coupling of the Higgs boson to gluons
  7. Application: $\alpha_s(M_Z)$ from $\alpha_s(M_{\tau})$
  8. Summary and Conclusions
  9. $\zeta_g$ for the heavy quark mass in the on-shell scheme

Figures (2)

  • Figure 1: Analytically known master integrals
  • Figure 2: Master integrals where only a few terms of their $\epsilon$-expansion are known analytically. The solid (dashed) lines denote massive (massless) propagators. The three numbers in brackets $(n_1,n_2,n_3)$ are decoded as follows: $n_1$ is the maximal power of the spurious pole in $\epsilon$ which might appear in front of the integral, $n_2$ is the maximal power of the spurious pole in $\epsilon$ which happens to enter into the decomposition of the bare decoupling constant $\zeta_g^0$ in terms of the master integrals, ${n_3}$ is the maximal analytically known power of the $\epsilon$-expansion of the same integral as determined in Schroder:2005va and confirmed in Chetyrkin:2005.