content_image/422719.jpg content_image/422720.jpg content_image/422721.jpg content_image/422722.jpg content_image/422723.jpg content_image/422724.jpg content_image/422726.jpg content_image/422725.jpg
with I$_{1}$ ( p ) we get 

<img src='content_image/422719.jpg'>

The other three integrals J$_{2}$ ( p ), J$_{3}$ ( p ) and J$_{4}$ ( p ) are evaluated in a similar fashion [36]. We only quote the results: 

<img src='content_image/422720.jpg'>

<img src='content_image/422721.jpg'>

<img src='content_image/422722.jpg'>

Finally J$_{1}$ ( p ) + J$_{2}$ ( p ) − J$_{3}$ ( p ) − J$_{4}$ ( p ) = J ( p ) leading to the result 

in agreement with the literature [28], [29]. 

It is important to say that the operations R and ˜ R that subtract ultraviolet and infrared di- vergences respectively [30, 31] could be defined as 

[36] The J 4 ( p ) calculation is easier if we first take a Fourier transform, evaluate J 4 ( x ) and then take the backward Fourier transform. The same thing can be done with J 1 ( p ), J 2 ( p ) and J 3 ( p ). 

<img src='content_image/422723.jpg'>

is done in other regularization schemes and could be used in the two examples above. However it is straightforward to see what these operations do within Implicit Regularization. They just remove the ultraviolet and infrared BDI’s. 

In the present contribution we have shown how the important steps which led to the construction of Implicit Regularization, initially used only in ultraviolet divergent integrals, can be adapted to handle infrared divergent integrals with the same degree of consistency. We have shown that an in- frared scale appears which is completely indepen- dent of the ultraviolet one. Moreover we show the consistency of the procedure for integrals involv- ing both divergences, in several testing non trivial examples. The results presented here, albeit being formal, are essential to answer several open ques- tions in the literature. A crucial step in this direc- tion is that Implicit Regularization allows to trace the infrared or ultraviolet origin of a basic loga- rithmic divergence and to associate to each kind of divergence independent scale relations. The inde- pendence of these scales is an important technical issue, the lack of which has prevented finding so- lutions to several important problems such as the NSVZ beta function. This work provides for an ad- equate framework within which this problem may be handled. 

## VI. APPENDIX A - VERIFICATION OF (32) 

Differentiating (32) we have: 

<img src='content_image/422724.jpg'>

with 

Shifting the variables we get 

<img src='content_image/422726.jpg'>

## V. CONCLUDING REMARKS 

<img src='content_image/422725.jpg'>

