content_image/410222.jpg content_image/410218.jpg content_image/410217.jpg content_image/410216.jpg content_image/410213.jpg content_image/410211.jpg content_image/410210.jpg content_image/410221.jpg content_image/410220.jpg content_image/410219.jpg content_image/410215.jpg content_image/410214.jpg content_image/410209.jpg content_image/410212.jpg
In order to evaluate G$_{ir}$ we write it as 

<img src='content_image/410222.jpg'>

composed of three terms with integrals in k which for definiteness we name G$_{1}$ , G$_{2}$ and G$_{3}$ . The in- tegral 

<img src='content_image/410218.jpg'>

can be rewritten with the aid of the Fourier trans- forms in appendix C as: 

<img src='content_image/410217.jpg'>

The integral in z is divergent at large distances (in- frared) and finite at short distances. This integral is well known from two loop Feynman diagram cal- culations and was evaluated in momentum space [25]. Here we just display the result translated to configuration space 

<img src='content_image/410216.jpg'>

where ˜ I (2) $_{log}$( ˜ λ − $^{2}$) ≡ ∫ Λ d $^{4}$z (2 π ) 4 1 ( z $^{2}$− ˜ λ − $^{2}$)2 ln ( − z 2 ˜ λ 2 ) is another infrared logarithmic BDI. Therefore inte- grating over x leads to 

<img src='content_image/410213.jpg'>

Similarly the Fourier tramsform of 

<img src='content_image/410211.jpg'>

reads 

<img src='content_image/410210.jpg'>

so 

<img src='content_image/410221.jpg'>

Finally the Fourier transform of 

<img src='content_image/410220.jpg'>

is 

<img src='content_image/410219.jpg'>

which back to momentum space yields 

<img src='content_image/410215.jpg'>

Hence 

<img src='content_image/410214.jpg'>

To evaluate G$_{µ,finite}$ we need the following in- tegrals which are straightforward to compute [25] 

where the expression P ( j ) = P ( n ) means j with the same parity of n . Then 

<img src='content_image/410209.jpg'>

<img src='content_image/410212.jpg'>

