[Paper Review] Scheme Dependence and Multiple Couplings

I. Jack, H. Osborn|arXiv (Cornell University)|Jun 8, 2016

Particle physics theoretical and experimental studies9 references3 citations

TL;DR

This paper establishes a framework for scheme-dependent redefinitions in quantum field theories with multiple couplings, showing that one-particle reducible (1PR) contributions in beta-functions can be absorbed into antisymmetric parts of the anomalous dimension, which are physically irrelevant. The key result is a systematic method to identify scheme-invariant quantities in ${\cal N}=1$ supersymmetric theories up to four loops, enabling extraction of universal, scheme-independent contributions to the anomalous dimension and beta-function.

ABSTRACT

For theories with multiple couplings the perturbative $β$-functions for scalar, Yukawa couplings are expressible in terms of contributions corresponding to one particle irreducible graphs and also contributions which are one particle reducible depending on the anomalous dimension. Here we discuss redefinitions, or changes of scheme, which preserve this structure. The redefinitions allow for IPR contributions of a specific form, as is necessary to encompass the relation between MS and momentum subtraction renormalisation schemes. Many multiply 1PR terms in the transformed $β$-function are generated but these can all be absorbed into antisymmetric contributions to the anomalous dimensions which are essentially arbitrary and can be discarded. As an illustration the results are applied to the scheme dependence of the anomalous dimension, which determines the $β$-function, for ${\cal N}=1$ supersymmetric scalar fermion theories in four dimensions up to four loops.

Motivation & Objective

• To clarify the structure of beta-functions in theories with multiple couplings, particularly the role of one-particle reducible (1PR) contributions arising from field rescalings.
• To identify the class of coupling redefinitions that preserve the 1PI/1PR structure of perturbative beta-functions, ensuring consistency across renormalization schemes.
• To apply the formalism to ${\cal N}=1$ supersymmetric scalar-fermion theories, where the beta-function is fully determined by the anomalous dimension.
• To extract scheme-independent contributions to the anomalous dimension and beta-function at three and four loops using symmetry and consistency conditions.
• To demonstrate that certain transcendental constants (e.g., $\zeta(3)$, $\zeta(5)$) in four-loop results are scheme-invariant and reflect new topologies not reducible to lower-order structures.

Proposed method

• Derives the transformation law for beta-functions under infinitesimal coupling redefinitions, distinguishing 1PI and 1PR contributions via the anomalous dimension matrix $\gamma_{ij}$.
• Identifies that only 1PR contributions of the form $ (g\,c)^I $, with $c$ antisymmetric, are allowed in redefinitions to preserve the 1PI/1PR structure of $\beta^I$.
• Shows that antisymmetric parts of the anomalous dimension (arising from such 1PR terms) are unphysical and can be discarded, leaving only 1PI-based contributions.
• Applies the formalism to ${\cal N}=1$ supersymmetric theories, where the NSVZ beta-function form depends on a specific scheme, and identifies conditions under which this form is preserved.
• Uses consistency conditions and known results from [4] and [5] to derive scheme invariants via linear combinations of four-loop diagram coefficients.
• Constructs a minimal scheme where only specific diagram coefficients remain non-zero, isolating the scheme-independent parts.

Experimental results

Research questions

• RQ1 Which redefinitions of couplings preserve the 1PI/1PR structure of the beta-function in multi-coupling quantum field theories?
• RQ2 How can 1PR contributions in the beta-function be systematically absorbed or removed via redefinitions involving the anomalous dimension?
• RQ3 What are the scheme-invariant combinations of four-loop diagram coefficients in ${\cal N}=1$ supersymmetric theories?
• RQ4 Can the scheme-independent part of the anomalous dimension be extracted from lower-loop data using the $a$-theorem or other consistency conditions?
• RQ5 Do new topologies (e.g., $c_{3D}, c_{4K}$) lead to new transcendental constants that are inherently scheme-invariant?

Key findings

• The 1PR contributions to the beta-function are constrained to a specific form involving antisymmetric parts of the coupling redefinition, ensuring consistency with 1PI counterterm structure.
• Antisymmetric contributions to the anomalous dimension, which arise from such 1PR terms, are physically unobservable and can be discarded without loss of generality.
• At four loops, the scheme-invariant combinations include $c_{4A}-c_{4B}-c_{4C}+c_{4D}=0$, $2c_{4A}-2c_{4C}+c_{4E}=-\frac{1}{2}$, $c_{4A}+c_{4B}-2c_{4C}+2c_{4F}=\frac{1}{2}\zeta(3)-\frac{1}{2}$, and $c_{4G}=-\frac{5}{2}$.
• For non-planar diagrams, the invariants $2c_{4H}-c_{4I}=0$ and $c_{4H}+c_{4J}=-3\zeta(3)$ are scheme-independent, with $c_{4K}=-10\zeta(5)$ also scheme-invariant.
• The coefficients $c_1, c_2, c_{3C}, c_{4G}$ match the expansion of $((1+4x)^{1/2}-1)/4$, suggesting a possible all-order pattern, though no full derivation is yet available.
• The presence of $\zeta(3)$ in $c_{4F}$ and $\zeta(5)$ in $c_{4K}$ indicates that these are independent of lower-order structures and arise from genuinely new topologies.