Study of Scheme Transformations to Remove Higher-Loop Terms in the $\beta$ Function of a Gauge Theory

Abstract

Since three-loop and higher-loop terms in the $\beta$ function of a gauge theory are scheme-dependent, one can, at least for sufficiently small coupling, carry out a scheme transformation that removes these terms. A basic question concerns the extent to which this can be done at an infrared fixed point of an asymptotically free gauge theory. This is important for quantitative analyses of the scheme dependence of such a fixed point. Here we study a scheme transformation $S_{R,m}$ with $m\ge 2$ that is constructed so as to remove the terms in the beta function at loop order $\ell =3$ to $\ell =m+1$, inclusive. Starting from an arbitrary initial scheme, we present general expressions for the coefficients of terms of loop order $\ell$ in the beta function in the transformed scheme from $\ell =m+2$ up to $\ell =8$. Extending a previous study of $S_{R,2}$, we investigate the range of applicability of the $S_{R,3}$ scheme transformation in an asymptotically free SU($N_c$) gauge theory with an infrared zero in $\beta$ depending on the number, $N_f$, of fermions in the theory. We show that this $S_{R,3}$ scheme transformation can only be applied self-consistently in a restricted range of $N_f$ with a correspondingly small value of infrared fixed-point coupling. We also study the effect of higher-loop terms on the beta function of a U(1) gauge theory.

15 pages

References in corpus (10)

• Eliminating the Renormalization Scale Ambiguity for Top-Pair Production Using the Principle of Maximum Conformality
• The Renormalization Scale-Setting Problem in QCD
• Higher-Loop Corrections to the Infrared Evolution of a Gauge Theory with Fermions
• Large N flavor beta-functions: a recap
• An Analysis of Scheme Transformations in the Vicinity of an Infrared Fixed Point
• Scheme Transformations in the Vicinity of an Infrared Fixed Point
• Higher-Loop Structural Properties of the $\beta$ Function in Asymptotically Free Vectorial Gauge Theories
• Higher-Loop Calculations of the Ultraviolet to Infrared Evolution of a Vectorial Gauge Theory in the Limit $N_c\to \infty$, $N_f\to \infty$ with $N_f/N_c$ Fixed
• The relation between the QED charge renormalized in MSbar and on-shell schemes at four loops, the QED on-shell beta-function at five loops and asymptotic contributions to the muon anomaly at five and six loops
• Comparison of Some Exact and Perturbative Results for a Supersymmetric SU($N_c$) Gauge Theory

Cited by in corpus (35)

• The QCD Running Coupling
• Infrared Zero of $\beta$ and Value of ${\gamma }_m$ for an SU(3) Gauge Theory at the Five-Loop Level
• Study of Possible Ultraviolet Zero of the Beta Function in Gauge Theories with Many Fermions
• Higher-Order Scheme-Independent Series Expansions of ${\gamma }_{\bar{\psi }\psi ,IR}$ and ${\beta }_{IR}^{\prime }$ in Conformal Field Theories
• Scheme-Independent Series Expansions at an Infrared Zero of the Beta Function in Asymptotically Free Gauge Theories
• Scheme-Independent Calculation of ${\gamma }_{\bar{\psi }\psi ,IR}$ for an SU(3) Gauge Theory
• Conformal Behavior at Four Loops and Scheme (In)Dependence
• Generalized Scheme Transformations for the Elimination of Higher-Loop Terms in the Beta Function of a Gauge Theory
• Higher Loop Corrections to the Infrared Evolution of Fermionic Gauge Theories in the RI' Scheme
• Infrared Fixed Points in the minimal MOM Scheme
• New Scheme Transformations and Application to Study Scheme Dependence of an Infrared Zero of the Beta Function in Gauge Theories
• Infrared Fixed Point Physics in $\mathrm{SO}(N_c)$ and $\mathrm{Sp}(N_c)$ Gauge Theories
• $A_k\bar{F}$ Chiral Gauge Theories
• Physics of the Non-Abelian Coulomb Phase: Insights from Padé Approximants
• Renormalization-Group Flows and Fixed Points in Yukawa Theories
• Scheme-Independent Calculations of Anomalous Dimensions of Baryon Operators in Conformal Field Theories
• Higher-Order Scheme-Independent Calculations of Physical Quantities in the Conformal Phase of a Gauge Theory
• On the Question of an Ultraviolet Zero of the Beta Function of the $\lambda ({\vec{\varphi }}^2{)}_4^2$ Theory
• Analysis of a Zero of a Beta Function Using All-Orders Summation of Diagrams
• An Integral Formalism for the Construction of Scheme Transformations in Quantum Field Theory
• On the Question of a Possible Infrared Zero in the Beta Function of the Finite-$N$ Gross-Neveu Model
• Renormalization-Group Evolution of Chiral Gauge Theories
• Renormalization-Group Behavior of ${\varphi }^3$ Theories in $d=6$ Dimensions
• General Properties on Applying the Principle of Minimum Sensitivity to High-order Perturbative QCD Predictions
• Scheme-Independent Series for Anomalous Dimensions of Higher-Spin Operators at an Infrared Fixed Point in a Gauge Theory
• Study of the Renormalization-Group Evolution of $\mathcal{N}=1$ Supersymmetric Gauge Theories Using Padé Approximants
• Study of the Question of an Ultraviolet Zero in the Six-Loop Beta Function of the O($N$) $\lambda |\vec{\varphi }{|}^4$ Theory
• Renormalization of QCD in the interpolating momentum subtraction scheme at three loops
• Multiple Couplings and Renormalization Scheme Ambiguities
• Effect of Scheme Transformations on a Beta Function with Vanishing One-Loop Term
• Duality in a Supersymmetric Gauge Theory From a Perturbative Viewpoint
• Anomalous Dimensions at an Infrared Fixed Point in an SU($N_c$) Gauge Theory with Fermions in the Fundamental and Antisymmetric Tensor Representations
• Search for an Ultraviolet Zero in the Seven-Loop Beta Function of the $\lambda {\varphi }_4^4$ Theory
• Study of the Ultraviolet Behavior of an O($N$) $|\vec{\varphi }{|}^6$ Theory in $d=3$ Dimensions
• A Comparative Study of Criticality Conditions for Anomalous Dimensions using Exact Results in an $\mathcal{N}=1$ Supersymmetric Gauge Theory