A Demonstration of One-Loop Scheme Independence in a Generalised Scalar Exact Renormalization Group (original) (raw)

Exact scheme independence at one loop

Journal of High Energy Physics, 2002

The requirement that the quantum partition function be invariant under a renormalization group transformation results in a wide class of exact renormalization group equations, differing in the form of the kernel. Physical quantities should not be sensitive to the particular choice of the kernel. We demonstrate this scheme independence in four dimensional scalar field theory by showing that, even with a general kernel, the one-loop beta function may be expressed only in terms of the effective action vertices, and thus, under very general conditions, the universal result is recovered.

Zeta Functions, Renormalization Group Equations, and the Effective Action

Physical Review Letters, 1998

We demonstrate how to extract all the one-loop renormalization group equations for arbitrary quantum field theories from knowledge of an appropriate Seeley-DeWitt coefficient. By formally solving the renormalization group equations to one loop, we renormalization group improve the classical action, and use this to derive the leading-logarithms in the one-loop effective action for arbitrary quantum field theories.

Using the renormalization group

Universality and Renormalization, 2007

In computing quantum effects, it is necessary to perform a sum over all intermediate states consistent with prescribed initial and final states. Divergences arising in the course of evaluating this sum forces one to "renormalize" parameters characterizing the system. An ambiguity inherent in this rescaling is parameterized by a dimensionful parameter µ 2 which serves to set a scale for the process. Requiring that the explicit and implicit dependence of a physical quantity on µ 2 conspire to cancel leads to the so-called "renormalization group" equation [1-10]. It has proved possible to extract a lot of useful information from this equation; we will enumerate a number of these in this report.

Essential renormalisation group

2021

We propose a novel scheme for the exact renormalisation group motivated by the desire of reducing the complexity of practical computations. The key idea is to specify renormalisation conditions for all inessential couplings, leaving us with the task of computing only the flow of the essential ones. To achieve this aim, we utilise a renormalisation group equation for the effective average action which incorporates general non-linear field reparameterisations. A prominent feature of the scheme is that, apart from the renormalisation of the mass, the propagator evaluated at any constant value of the field maintains its unrenormalised form. Conceptually, the scheme provides a clearer picture of renormalisation itself since the redundant, non-physical content is automatically disregarded in favour of a description based only on quantities that enter expressions for physical observables. To exemplify the scheme’s utility, we investigate the Wilson-Fisher fixed point in three dimensions at...

Systematic Differential Renormalization to All Orders

Annals of Physics, 1994

We present a systematic implementation of differential renormalization to all orders in perturbation theory. The method is applied to individual Feynman graphs written in coordinate space. After isolating every singularity which appears in a bare diagram, we define a subtraction procedure which consists in replacing the core of the singularity by its renormalized form given by a differential formula. The organization of subtractions in subgraphs relies on Bogoliubov's formula, fulfilling the requirements of locality, unitarity and Lorentz invariance. Our method bypasses the use of an intermediate regularization and automatically delivers renormalized amplitudes which obey renormalization group equations.

Two-loop RGE of a general renormalizable Yang-Mills theory in a renormalization scheme with an explicit UV cutoff

Journal of High Energy Physics, 2016

We perform a systematic one-loop renormalization of a general renormalizable Yang-Mills theory coupled to scalars and fermions using a regularization scheme with a smooth momentum cutoff Λ (implemented through an exponential damping factor). We construct the necessary finite counterterms restoring the BRST invariance of the effective action by analyzing the relevant Slavnov-Taylor identities. We find the relation between the renormalized parameters in our scheme and in the conventional MS scheme which allow us to obtain the explicit two-loop renormalization group equations in our scheme from the known two-loop ones in the MS scheme. We calculate in our scheme the divergences of two-loop vacuum graphs in the presence of a constant scalar background field which allow us to rederive the two-loop beta functions for parameters of the scalar potential. We also prove that consistent application of the proposed regularization leads to counterterms which, together with the original action, combine to a bare action expressed in terms of bare parameters. This, together with treating Λ as an intrinsic scale of a hypothetical underlying finite theory of all interactions, offers a possibility of an unconventional solution to the hierarchy problem if no intermediate scales between the electroweak scale and the Planck scale exist.

A manifestly gauge invariant exact renormalization group

Arxiv preprint hep-th/9810104, 1998

We construct a manifestly gauge invariant Exact Renormalization Group for SU (N) Yang-Mills theory, in a form suitable for calculations without gauge fixing at any order of perturbation theory. The effective cutoff is incorporated via a manifestly realised spontaneously broken SU (N |N) gauge invariance. Diagrammatic methods are developed which allow the calculations to proceed without specifying the precise form of the cutoff structure. We confirm consistency by computing for the first time both the one and two loop beta function coefficients without fixing the gauge or specifying the details of the cutoff. We sketch how to incorporate quarks and thus compute in QCD. Finally we analyse the renormalization group behaviour as the renormalized coupling becomes large, and show that confinement is a consequence if and only if the coupling diverges in the limit that all modes are integrated out. We also investigate an expansion in the inverse square renormalized coupling, and show that under general assumptions it yields a new non-perturbative approximation scheme corresponding to expanding in 1/ΛQCD.

A gauge invariant exact renormalisation group. (I)

Nuclear Physics B, 2000

A manifestly gauge invariant continuous renormalization group flow equation is constructed for pure SU (N ) gauge theory. The formulation makes sense without gauge fixing and manifestly gauge invariant calculations may thus be carried out. The flow equation is naturally expressed in terms of fluctuating Wilson loops, with the effective action appearing as an integral over a 'gas' of Wilson loops. At infinite N , the effective action collapses to a path integral over the trajectory of a single particle describing one Wilson loop. We show that further regularization of these flow equations is needed. (This is introduced in part II.)

The beta function of N= 1 SYM in differential renormalization

Journal of High Energy …, 2002

Using differential renormalization, we calculate the complete two-point function of the background gauge superfield in pure N=1 Super Yang-Mills theory to two loops. Ultraviolet and (off-shell) infrarred divergences are renormalized in position and momentum space respectively. This allows us to reobtain the beta function from the dependence on the ultraviolet renormalization scale in an infrared-safe way. The two-loop coefficient of the beta function is generated by the one-loop ultraviolet renormalization of the quantum gauge field via nonlocal terms which are infrared divergent on shell. We also discuss the connection of the beta function to the flow of the Wilsonian coupling.