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larization which works in four dimensions. Given the spectacular success of Dimensional Regulariza- tion a natural extension has been tried: Dimen- sional Reduction. This generalization is known to work well at the one loop level. At two loops it has also been tried [14] but its extension to higher loops do not preserve supersymmetry automatically. A successful theory in four dimensions has been con- structed around the same time as Implicit Regular- ization: Differential Regularization [15–22]. The main difference is that the latter works in configu- ration space while the former works directly in mo- mentum space. Recently the equivalence of both at one loop level was shown [23, 24]. Implicit Regular- ization has been consistently extended to n-loops in gauge theories [25] while Differential Regular- ization has not yet. 

The purpose of this paper is to show that in- frared divergences as well as ultraviolet ones can be treated in a symmetrical way, which gives us further confidence in the method, in addition to an elegant formulation for massless theories. 

## II. FORMAL ANALOGIES IN HANDLING ULTRAVIOLET AND INFRARED DIVERGENCES 

The main ideia of Implicit Regularization is based on the recursively use within Feynman in- tegrals [34] of the algebraic identity 

in order to isolate the divergent pieces as basic di- vergent integrals (BDI) which are independent of the external momenta p$_{i}$ and do not need to be evaluated. 

Let us look at the logarithmically ultraviolet di- vergent integral: 

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where p stands for the external momentum and m for the mass of the theory. In the following we drop 

[34] We assume that the integrals are regularized - by an im- plicit regularization - in such a way that formal manipu- lations of the integrand are possible. 

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the iϵ in the propagators, but it is tacitly assumed that they depend on them. Applying (1) once, we obtain 

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Hereafter the superscript Λ designates that the in- tegral is regularized and k ≡ d $^{4}$k (2 π ) $_{4}$. 

∫ ∫ The first term on the right hand side is inde- pendent of the physical momentum and belongs to the class of BDI. This one is called I$_{log}$ ( m $^{2}$) and we leave it (and also the other BDI’s) in this in- tegral form in order to avoid explicit contact with specific regularizations. The second term is finite and can be evaluated in usual ways. 

Note that the mass m is still part of I$_{log}$ . If the theory is massive, it may be left there and used as a renormalization group scale. However if the theory is massless, m has a completely different meaning. It represents an infrared regulator which has to be set to zero in the end of the calculation. In this case it is mandatory to substitute the mass in I$_{log}$ by an arbitrary scale λ which will play the role of the renormalization scale. For this purpose we make use of another mathematical identity: 

where b ≡ i (4 π ) $_{2}$. 

Inserting this relation in (3) we obtain 

in terms of the arbitrary parameter λ . An im- portant observation should be made at this point. The structure of the finite parts of (3) involves log- arithms, precisely what one needs to cancel the in- frared regulator in favor of λ . 

Now let us proceed with an integral which illus- trates the appearance of an infrared BDI in Im- plicit Regularization: 

By power counting U is infrared divergent and ultraviolet finite. In order to be able to use all the mathematics developed for ultraviolet diver- 

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