Ultraviolet and Infrared Divergences in Implicit Regularization: a Consistent Approach H. G. Fargnoli ( a $^{)}$, ∗ A. P. Baˆ eta Scarpelli, † L. C. T. Brito ( b $^{)}$, ‡ B. Hiller ( c $^{)}$, § Marcos Sampaio ( a $^{)}$, ¶ M. C. Nemes ( a $^{)}$, ∗∗ and A. A. Osipov ( c,d ) †† (a) Universidade Federal de Minas Gerais - Departamento de F´ısica - ICEx P.O. BOX 702, 30.161-970, Belo Horizonte MG - Brazil (b) Universidade Federal de Lavras - Departamento de Ciˆencias Exatas P. O. BOX 3037, 37.200-000, Lavras MG - Brazil (c) Departamento de Fsica, Centro de Fsica Computacional, Faculdade de Ciˆencias e Tecnologia, Universidade de Coimbra, P-3004 516 Coimbra - Portugal and (d) Dzhelepov Laboratory of Nuclear Problems, JINR 141980 Dubna, Russia (Dated: November 8, 2018) Implicit Regularization is a 4-dimensional regularization initially conceived to treat ultraviolet di- vergences. It has been successfully tested in several instances in the literature, more specifically in those where Dimensional Regularization does not apply. In the present contribution we extend the method to handle infrared divergences as well. We show that the essential steps which rendered Implicit Regularization adequate in the case of ultraviolet divergences have their counterpart for infrared ones. Moreover we show that a new scale appears, typically an infrared scale which is completely independent of the ultraviolet one. Examples are given. PACS numbers: 11.10.Gh, 11.15.Bt, 11.30.Qc sive ones we are able to introduce an arbitrary scale which has been shown to play the role of a renormalization scale. Now a word about in- frared divergences is in order. In cases when the theory is infrared safe such divergences will ap- pear in the pertubative calculation within Implicit Regularization. We have shown in this case that an infrared regulator can be used to manipulate the integral and then an algebraic identity which replaces this infrared regulator by an arbitrary scale (the renormalization scale) restore the in- frared safeness. In the case that genuine infrared divergences appear, this process cannot be applied and as we show in this contribution, an infrared scale emerges which is independent and can be completely distinguished from the ultraviolet one. This result is an important technical step in the direction of solving the NSVZ beta function prob- lem [7–10]; d) and last, but not least, we discuss symmetry preservation. As shown in [11] a diagrammatic proof to all orders of the preservation of gauge sym- metry in QED is possible if and only if shifts in the integrations of internal loops are allowed. In sev- eral instances involving non-abelian gauge theories and supersymmetric ones we have also shown that symmetries will be preserved if and only if surface terms originated from shifts in perturbative inte- grals are all set to zero [12, 13]. Of course Implicit Regularization is not the only attempt throughout the years to construct a regu- c) in what regards massless theories or even mas- b) the technique we employ does not, in any step of the calculation, modify the original integrand. Us- ing a mathematical algebraic relation we are able to separate divergent from finite parts in a clear way; a) all calculations are performed in the dimension specific to a given theory (in the present work we deal with four dimensions), which opens the way to solve problems beyond the scope of Dimensional Regularization; Among the essential properties of this regular- ization which allowed for its success [2–6] some are worth stressing: The Implicit Regularization technique is a scheme proposed in [1] with the main purpose to provide for a clear separation between the diver- gent content of physical amplitudes (regularization dependent) and its finite physical content (regular- ization independent). arXiv:1001.1543v2 [hep-th] 4 Dec 2010 I. INTRODUCTION $^{∗}$hfargnoli@fisica.ufmg.br $^{†}$scarp@fisica.ufmg.br $^{‡}$lctbrito@fisica.ufmg.br $^{§}$brigitte@teor.fis.uc.pt $^{¶}$msampaio@fisica.ufmg.br $^{∗∗}$mcnemes@fisica.ufmg.br $^{††}$osipov@nu.jinr.ru