Implicit regularization beyond one-loop order: gauge field theories

Abstract

We extend a constrained version of Implicit Regularization (CIR) beyond one loop order for gauge field theories. In this framework, the ultraviolet content of the model is displayed in terms of momentum loop integrals order by order in perturbation theory for any Feynman diagram, while the Ward-Slavnov-Taylor identities are controlled by finite surface terms. To illustrate, we apply CIR to massless abelian Gauge Field Theories (scalar and spinorial QED) to two loop order and calculate the two-loop beta-function of the spinorial QED.

Symmetry preserving regularization with a cutoff

“…The connection with DREG is more transparent if one uses alternatively the µ ν → 1 µν 2 or (A6) substitution and takes different values determined by the degree of divergence in each term (A3), (A4), (A5). We stress that this new regularization holds without DREG as the substitutions (20), (21) and scalar integration with a cutoff are independent of DREG. The success of both regularizations based on the property that they fulfill the consistency conditions of gauge invariance and momentum shifting.…”

“…Constrained differential renormalization is useful in supersymmetric [34] and non-Abelian gauge theories, it fulfills Slavnov-Taylor identities at one and two loops [35]. Implicit regularization [20,21] requires the same conditions as we used and it was successfully applied to the Nambu-Jona-Lasinio model [20] and to higher loop calculations in gauge theory. It was shown that the conditions guarantee gauge invariance generally and the Ward identities are fulfilled explicitly in QED at two-loop order [21].…”

“…Implicit regularization [20,21] requires the same conditions as we used and it was successfully applied to the Nambu-Jona-Lasinio model [20] and to higher loop calculations in gauge theory. It was shown that the conditions guarantee gauge invariance generally and the Ward identities are fulfilled explicitly in QED at two-loop order [21]. In an effective composite Higgs model, the Fermion Condensate Model [36] oblique radiative corrections (S and T parameters) were calculated in DREG and with the improved cutoff, too, the finite results completely agree.…”

“…Under any regularized momentum integrals the identifications (20) or generalizations such as (21) are understood as a part of the regularization procedure for = 1 2 . For finite integrals (non divergent, for high enough ) the standard calculation automatically fulfills (20), (21). The connection with the standard substitution of free indices is discussed in Appendix A.…”

“…Another approach, later proved to be equivalent with the previous one [19], is the method of implicit regularization, in which a recursive identity (similar to Taylor expansion) is applied and all the dependence on the external momentum ( ) is transferred to finite integrals. The divergent integrals contain only the loop momentum, thus universal local counter terms can cancel the potentially dangerous symmetry violating contributions [20,21]. Gauge invariant regularization is implemented in the exact renormalization group method providing a cutoff without gauge fixing in [23].…”