Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Renormalization and Running of Quark Mass and Field in the Regularization Invariant and MS-bar Schemes at Three and Four Loops

K. G. Chetyrkin, A. Retey

TL;DR

This work delivers a comprehensive analysis of quark mass and field renormalization across schemes, deriving explicit NNNLO conversion factors between MS-bar and the lattice-friendly RI/RI' schemes using a massless three-loop quark propagator expansion. The findings reveal that the NNNLO mass-conversion terms are sizable at typical lattice scales (e.g., 2 GeV), stressing the need to perform conversions at higher scales (around 3 GeV) for reliable MS-bar masses of light quarks. Additionally, the authors compute the four-loop anomalous dimensions in RI/RI' and observe an absence of zeta(4) in these schemes, a feature argued to be more physical than an artifact of MS-bar. They also present the RG-invariant mass hat m_q and its four-loop relation to RI masses, enabling scale- and scheme-independent comparisons. Overall, the results have important implications for precision quark-mass determinations from lattice QCD and for understanding scheme dependence in high-order perturbative QCD.

Abstract

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the MS-bar scheme with the corresponding quantities defined in any other scheme. By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the MS-bar quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (RI) scheme which is more suitable for lattice QCD calculations. The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV --- the typical normalization scale employed in lattice simulations. Thus, in order to get a precise prediction for the MS-bar masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. We also compute two more terms in the high-energy expansion of the MS-bar renormalized quark propagator. The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the MS-bar mass of the charmed quark. As a by-product of our calculations we determine the four-loop anomalous dimensions of the quark mass and field in the Regularization Invariant scheme. Finally, we discuss some physical reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions.

Renormalization and Running of Quark Mass and Field in the Regularization Invariant and MS-bar Schemes at Three and Four Loops

TL;DR

This work delivers a comprehensive analysis of quark mass and field renormalization across schemes, deriving explicit NNNLO conversion factors between MS-bar and the lattice-friendly RI/RI' schemes using a massless three-loop quark propagator expansion. The findings reveal that the NNNLO mass-conversion terms are sizable at typical lattice scales (e.g., 2 GeV), stressing the need to perform conversions at higher scales (around 3 GeV) for reliable MS-bar masses of light quarks. Additionally, the authors compute the four-loop anomalous dimensions in RI/RI' and observe an absence of zeta(4) in these schemes, a feature argued to be more physical than an artifact of MS-bar. They also present the RG-invariant mass hat m_q and its four-loop relation to RI masses, enabling scale- and scheme-independent comparisons. Overall, the results have important implications for precision quark-mass determinations from lattice QCD and for understanding scheme dependence in high-order perturbative QCD.

Abstract

We derive explicit transformation formulae relating the renormalized quark mass and field as defined in the MS-bar scheme with the corresponding quantities defined in any other scheme. By analytically computing the three-loop quark propagator in the high-energy limit (that is keeping only massless terms and terms of first order in the quark mass) we find the NNNLO conversion factors transforming the MS-bar quark mass and the renormalized quark field to those defined in a ``Regularization Invariant'' (RI) scheme which is more suitable for lattice QCD calculations. The NNNLO contribution in the mass conversion factor turns out to be large and comparable to the previous NNLO contribution at a scale of 2 GeV --- the typical normalization scale employed in lattice simulations. Thus, in order to get a precise prediction for the MS-bar masses of the light quarks from lattice calculations the latter should use a somewhat higher scale of around, say, 3 GeV where the (apparent) convergence of the perturbative series for the mass conversion factor is better. We also compute two more terms in the high-energy expansion of the MS-bar renormalized quark propagator. The result is then used to discuss the uncertainty caused by the use of the high energy limit in determining the MS-bar mass of the charmed quark. As a by-product of our calculations we determine the four-loop anomalous dimensions of the quark mass and field in the Regularization Invariant scheme. Finally, we discuss some physical reasons lying behind the striking absence of zeta(4) in these computed anomalous dimensions.

Paper Structure

This paper contains 15 sections, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Scheme dependence of quark mass and field
  3. Generalities
  4. Regularization Invariant scheme versus $\overline{\mathrm{MS}}$
  5. Three-loop quark propagator in $\overline{\hbox{MS}}$-scheme
  6. $\hbox{RI}$ scheme versus $\overline{\hbox{MS}}$ and $\hbox{MOM}$ schemes
  7. Three Loop Conversion functions
  8. Four Loop Quark Anomalous Dimensions
  9. NNNLO relation for the $\mathrm{RI}$ quark mass and the $\mathrm{RGI}$ mass $\hat{m}_q$
  10. Conclusions
  11. Acknowledgments
  12. Note
  13. Quark Propagators
  14. Conversion Functions
  15. Anomalous Dimensions

Figures (2)

  • Figure 1: The ratio of the $\mathrm{RI}$ and $\mathrm{MOM}$ scheme conversion functions $C_{2,l}^{ \mathrm{MOM} }/C_{2,l}^{ \mathrm{RI} }$ as functions of $-\mu^2/m^2$. Shown are the coefficients in the expansion in $a_s/\pi$ as expansions to order $z$ to $z^5$ ($z^2$ for 3 loops). Note that in Landau gauge $C_2^? = 0$ , for 2 loops some numeric values for the exact mass dependence are shown as well.
  • Figure 2: The ratio of the $\mathrm{RI}$ and $\mathrm{MOM}$ scheme conversion functions $C_{m,l}^{ \mathrm{MOM} }/C_{m,l}^{ \mathrm{RI} }$ as functions of $-\mu^2/m^2$. Shown are the coefficients in the expansion in $a_s/\pi$ as expansions to order $z$ to $z^5$ ($z^2$ for 3 loops). For 1 and 2 loops some numeric values for the exact mass dependence are shown as well.