New scheme transformations and application to study scheme dependence of an infrared zero of the beta function in gauge theories

Abstract

We present two new one-parameter families of scheme transformations and apply these to study the scheme dependence of the infrared zero in the beta function of an asymptotically free non-Abelian gauge theory up to four-loop order. Our results provide a further quantitative measure of this scheme dependence, showing that for moderate values of the gauge coupling and the parameter specifying the scheme transformation, this dependence is relatively mild. We also remark on a generalized multi-parameter family of rational scheme transformations.

Integral formalism for the construction of scheme transformations in quantum field theory

“…Before doing this, we first illustrate how the method works with some scheme transformations that have already been studied in [3]- [7], which we showed to be acceptable for the analysis of a zero in a beta function located away from the origin in coupling constant space, in particular, an IR zero of the beta function of an asymptotically free non-Abelian gauge theory. Let us consider, for example, the scheme transformation (3.19) studied in [4].…”

“…We have listed these for comparative purposes in Table I. The first four lines of this table describe scheme transformations from [3]- [7], while the next seven lines describe new scheme transformations presented and analyzed in the present paper.…”

“…The previous works [3]- [7] addressed this task and studied applications at an IR zero of the beta function in an asymptotically free non-Abelian gauge theory. Early interest in such a zero had made use of the scheme-independent one-loop and two-loop coefficients and had noted the associated behavior of scaling with anomalous dimensions [10,11].…”

“…A procedure to assess the scheme dependence of the location of this zero in the beta function is to carry out the calculation first in a given scheme, obtain a result for the value of the IR zero at n-loop (nℓ) order, α IR,nℓ , then apply a scheme transformation, calculate the zero to this order in the transformed scheme, denoted α ′ IR,nℓ , and determine the fractional shift in the value. In a series of papers this program has been implemented [3]- [7]. Refs.…”

“…[14]- [16] carried out this analysis for a general gauge group and for fermions in both the fundamental representation and in the adjoint and rank-2 tensor representations. A particularly powerful approach uses scheme transformations that are dependent on an auxiliary parameter, r, with the property that as r → 0, they approach the identity; by varying r continuously away from r = 0, one can thus study the scheme dependence as a function of this continuous variable [3]- [7]. A valuable result from this program of higher-order perturbative computations of the values of IR fixed points of asymptotically free non-Abelian gauge theories is improvement in the accuracy of calculations of anomalous dimensions, such as the anomalous dimension of the fermion bilinear operator, evaluated at the IR fixed point, γ IR,nℓ ≡ γ nℓ (α IR,nℓ ).…”