content_image/417355.jpg content_image/417354.jpg content_image/417353.jpg content_image/417356.jpg content_image/417357.jpg content_image/417358.jpg content_image/417359.jpg content_image/417360.jpg content_image/417352.jpg content_image/417350.jpg
gent integrals we firstly note the following: 

<img src='content_image/417355.jpg'>

where z and u are configuration variables, and use was made of the Fourier transforms from momen- tum to configuration space (appendix C). The ex- plicit iϵ dependence in configuration space will also be omitted in the following. 

Note the striking similarity between the above integral in z and equation (2). We can thus write the result immediately (the opposite sign as com- pared to (5) is due to the iϵ prescriptions in con- figuration and momentum space) 

<img src='content_image/417354.jpg'>

But now ˜ I$_{log}$ ( ˜ λ $^{2}$) is an infrared BDI and (4) has been used in order to eliminate µ 2 in favor of the infrared scale ˜ λ . 

Using this result and (7) in (6) we have 

<img src='content_image/417353.jpg'>

Here and henceforth the barred and the corre- sponding unbarred variables are related as defined in appendix C. 

Before closing this paragraph we emphasize that the extraction and classification of all infrared BDI follows the same strategy, of Fourier transforming the integrals, solving them in configuration space, and transforming the solution back to momentum space. As already remarked in [10], introducing the infrared scale in the Fourier transformed space has the convenient property of rendering this scale apriori independent from the ultraviolet scale. 

## III. INFRARED AND ULTRAVIOLET DIVERGENT INTEGRALS: SCALE SEPARATION 

Consider as a second example the following inte- gral that is both ultraviolet and infrared divergent 

<img src='content_image/417356.jpg'>

Multiplying and dividing by ( p − k ) 2 allows us to separate infrared and ultraviolet divergences, 

<img src='content_image/417357.jpg'>

where 

<img src='content_image/417358.jpg'>

<img src='content_image/417359.jpg'>

and 

<img src='content_image/417360.jpg'>

With Implicit Regularization rules it was shown in [26] (see also [27]) that 

<img src='content_image/417352.jpg'>

with λ being an ultraviolet scale and I ( n +1) log ( λ $^{2}$) ≡ ∫ Λ k 1 ( k $^{2}$− λ $^{2}$)2 ln n ( − k 2 λ 2 ) . Using this we easily evalu- ate 

<img src='content_image/417350.jpg'>

