Implicit regularization beyond one-loop order: scalar field theories

Abstract

Implicit regularization (IR) has been shown as an useful momentum space tool for perturbative calculations in dimension specific theories, such as chiral gauge, topological and supersymmetric quantum field theoretical models at one loop level. In this paper, we aim at generalizing systematically IR to be applicable beyond one loop order. We use a scalar field theory as an example and pave the way for the extension to quantum field theories which are richer from the symmetry content viewpoint. Particularly, we show that a natural (minimal) renormalization scheme emerges, in which the infinities displayed in terms of integrals in one internal momentum are subtracted, whereas infrared and ultraviolet modes do not mix and therefore leave no room for ambiguities. A systematic cancelation of the infrared divergences at any loop order takes place between the ultraviolet finite and divergent parts of the amplitude for non-exceptional momenta leaving, as a byproduct, a renormalization group scale.

Systematic Implementation of Implicit Regularization for Multiloop Feynman Diagrams

“…It is important to note that only this type of divergence appears because linear and quadratic divergent integrals vanish for massless theories [29], [38].…”

“…In this theory, only graphs up to three external legs are divergent [10]. The graphs with one external leg have only quadratic divergences and these always vanish for massless theories [29]. Therefore, the graphs we deal with have only two or three external legs and correspond to the renormalization of the propagator and the vertex functions respectively.…”

On the equivalence between implicit regularization and constrained differential renormalization

“…This becomes essential if we are treating massless theories (see ref. [26]). It can be done by using the regularization independent relation…”

“…The mass parameter λ 2 is suitable to be used as the renormalization group scale, as it can be seen in refs. [23], [26]. After solving the finite part, we are left with…”

“…A non-trivial test in a supersymmetric model was performed, in which the anomalous magnetic moment of the lepton in supergravity was successfully calculated [27]. The extension of CIR to higher loop order has been implemented and has been applied in scalar [26] and gauge theories [31]. In what concerns higher order calculations, Differential Renormalization, in its original form, has been used with success in scalar and gauge theories.…”