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Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems

K. G. Chetyrkin, B. A. Kniehl, M. Steinhauser

TL;DR

The paper addresses the challenge of heavy-quark decoupling in MSbar QCD by introducing decoupling constants $\zeta_g$ and $\zeta_m$ that match the full and effective theories at a heavy-flavour threshold. It develops a formalism to compute these constants via vacuum tadpole integrals, obtaining three-loop expressions, and shows how low-energy theorems relate the Higgs couplings to gluons and light quarks to derivatives of these constants, enabling four-loop predictions when combined with known four-loop RG functions. The authors derive explicit MSbar and OS expressions for the coefficient functions $C_1$ and $C_2$ (and $C_{1\gamma}$ for photons) up to ${\cal O}(\alpha_s^4)$, linking heavy-quark effects to observable Higgs processes such as $gg\to H$ and $\gamma\gamma H$. The framework provides a systematic, RG-consistent approach to threshold matching, with implications for precise Higgs phenomenology at the LHC and beyond, and extends to QED decoupling as a cross-check.

Abstract

If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively. Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals. This allows us to evaluate zeta_g and zeta_m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively. Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.

Decoupling Relations to O(alpha_s^3) and their Connection to Low-Energy Theorems

TL;DR

The paper addresses the challenge of heavy-quark decoupling in MSbar QCD by introducing decoupling constants and that match the full and effective theories at a heavy-flavour threshold. It develops a formalism to compute these constants via vacuum tadpole integrals, obtaining three-loop expressions, and shows how low-energy theorems relate the Higgs couplings to gluons and light quarks to derivatives of these constants, enabling four-loop predictions when combined with known four-loop RG functions. The authors derive explicit MSbar and OS expressions for the coefficient functions and (and for photons) up to , linking heavy-quark effects to observable Higgs processes such as and . The framework provides a systematic, RG-consistent approach to threshold matching, with implications for precise Higgs phenomenology at the LHC and beyond, and extends to QED decoupling as a cross-check.

Abstract

If quantum chromodynamics (QCD) is renormalized by minimal subtraction (MS), at higher orders, the strong coupling constant alpha_s and the quark masses m_q exhibit discontinuities at the flavour thresholds, which are controlled by so-called decoupling constants, zeta_g and zeta_m, respectively. Adopting the modified MS (MS-bar) scheme, we derive simple formulae which reduce the calculation of zeta_g and zeta_m to the solution of vacuum integrals. This allows us to evaluate zeta_g and zeta_m through three loops. We also establish low-energy theorems, valid to all orders, which relate the effective couplings of the Higgs boson to gluons and light quarks, due to the virtual presence of a heavy quark h, to the logarithmic derivatives w.r.t. m_h of zeta_g and zeta_m, respectively. Fully exploiting present knowledge of the anomalous dimensions of alpha_s and m_q, we thus calculate these effective couplings through four loops. Finally, we perform a similar analysis for the coupling of the Higgs boson to photons.

Paper Structure

This paper contains 8 sections, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Formalism
  3. Decoupling relations
  4. Low-energy theorems for the $ggH$ and $q\bar{q}H$ interactions
  5. Low-energy theorem for the $\gamma\gamma H$ interaction
  6. Discussion and conclusions
  7. Results for the gauge group SU($N_c$)
  8. Decoupling relations for the quark and gluon fields

Figures (4)

  • Figure 1: Typical three-loop diagrams pertinent to $\Sigma_V^{0h}(0)$ and $\Sigma_S^{0h}(0)$. Solid, bold-faced, and loopy lines represent massless quarks $q$, heavy quarks $h$, and gluons $G$, respectively.
  • Figure 2: Typical three-loop diagrams pertinent to $\Pi_G^{0h}(0)$, $\Pi_c^{0h}(0)$, and $\Gamma_{G\bar{c}c}^{0h}(0,0)$. Bold-faced, loopy, and dashed lines represent heavy quarks $h$, gluons $G$, and Faddeev-Popov ghosts $c$, respectively.
  • Figure 3: $\mu^{(5)}$ dependence of $\alpha_s^{(5)}(M_Z)$ calculated from $\alpha_s^{(4)}(M_\tau)=0.36$ and $M_b=4.7$ GeV using Eq. (\ref{['eqrgea']}) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eqs. (\ref{['eqdecg']}) and (\ref{['eqzetagos']}) at the respective orders.
  • Figure 4: $\mu^{(5)}$ dependence of $m_c^{(5)}(M_Z)$ calculated from $\mu_c=m_c^{(4)}(\mu_c)=1.2$ GeV, $M_b=4.7$ GeV, and $\alpha_s^{(5)}(M_Z)=0.118$ using Eq. (\ref{['eqmas']}) at one (dotted), two (dashed), three (dot-dashed), and four (solid) loops in connection with Eqs. (\ref{['eqdecm']}) and (\ref{['eqzetamos']}) at the respective orders.