Consistency Conditions for 4-D Regularizations

Abstract

From the study of well-known divergent amplitudes we deduce three consistency conditions which are necessary and sufficient to eliminate ambiguities and symmetry violations. The conditions relate divergent integrals of the same degree of divergence and are automatically satisfied within the context of dimensional regularization. We show how the deduced consistency conditions can be satisfied in four-dimensional regularizations. An important conclusion of the present study is the possibility of working with 4-D regularization schemes which preserve the virtues of dimensional regularization avoiding, however, its restrictions.

To $${d}$$ d , or not to $${d}$$ d : recent developments and comparisons of regularization schemes

“…4.1 Introduction to IREG and electron self-energy at NLO Implicit regularization (ireg) is a regularization framework proposed by the end of the 1990s [73][74][75] as an alternative to well-known dimensional schemes. A main characteristic of the method is that it stays in the physical dimension of the underlying quantum field theory, avoiding, in principle, some of the drawbacks of ds such as the mismatch between fermionic and bosonic degrees of freedom which leads to the breaking of supersymmetry.…”

Systematic Implementation of Implicit Regularization for Multiloop Feynman Diagrams

“…It was shown in [15]- [20] that setting all surfaces terms to zero defines a constrained version of IReg (CIReg) and corresponds to invoking momentum routing invariance in the loops of a Feynman graph. This in turn is related to gauge invariance and it was shown that adopting CIReg is a sufficient condition to ensure gauge symmetry [31].…”

“…Therefore, the construction of an invariant regularization is justified and, in order to be as reliable as DReg (wherever DReg can be applied), it must be shown to comply with locality, Lorentz invariance, unitarity and causality. Recently, an invariant regularization framework (IReg) has been developed and shown to be consistent and symmetry preserving in several instances [15]- [31]. The essence of the method is to write the divergences in terms of loop integrals in one internal momentum which do not need to be explicitly evaluated.…”

Theγγdecay of the Higgs boson in FDR

“…In the usual framework, Dimensional Regularization (DR) [12], infinities arise at the intermediate steps of the calculation, forcing a huge analytic work in order to check all needed cancellations, before even starting to calculate the physically-relevant contribution. This has pushed the quest of alternative approaches in 4 dimensions [13][14][15][16][17][18]. In this context, Four Dimensional Regularization was proposed [19] as a method which is free of infinities, 4-dimensional and gauge-invariant at the same time.…”