In order to evaluate G$_{ir}$ we write it as composed of three terms with integrals in k which for definiteness we name G$_{1}$ , G$_{2}$ and G$_{3}$ . The in- tegral Finally the Fourier transform of can be rewritten with the aid of the Fourier trans- forms in appendix C as: 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 which back to momentum space yields Hence 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 Similarly the Fourier tramsform of 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 reads G$_{µ,finite}$ = ln ( λ 2 ¯ ˜ λ 2 ) ∫ Λ k k$_{µ}$ k $^{4}$( p − k ) 2 ln 2 ( − k 2 λ 2 ) + ∫ Λ k k$_{µ}$ k $^{4}$( p − k ) 2 ln 3 ( − k 2 λ 2 ) (28) ˜ G$_{2}$ ( x ) = i (4 π $^{2}$) x 2 { ˜ I (3) $_{log}$( ˜ λ − $^{2}$) + b [ 4 − 2 ln ( − x 2 ˜ λ 2 ) + ln 2 ( − x 2 ˜ λ 2 ) − 1 3 ln 3 ( − x 2 ˜ λ 2 ) ]} , G$_{2}$ ( p ) = ∫ Λ k 1 k $^{4}$( p − k ) 2 ln 2 ( − k 2 ¯ ˜ λ 2 ) (21) ∫ k k$_{µ}$ k $^{4}$( p − k ) 2 ln n ( − k 2 λ 2 ) = b p$_{µ}$ p 2 n ∑ j =0 , P ( j )= P ( n ) n ! j ! ln j ( − p 2 λ 2 ) , (27) G$_{1}$ ( p ) = − 1 p 2 ˜ I (2) $_{log}$( ˜ λ − $^{2}$) + b p 2 { 1 2 ln 2 ( − p 2 ¯ ˜ λ 2 ) + ln ( − p 2 ¯ ˜ λ 2 )} , (20) G$_{ir}$ = ln 2 ( λ 2 ¯ ˜ λ 2 ) G$_{1}$ − 2 ln ( λ 2 ¯ ˜ λ 2 ) G$_{2}$ + G$_{3}$. (26) G$_{3}$ ( p ) = 1 p 2 [ $_{−}$˜ I (4) $_{log}$( ˜ λ − $^{2}$) − ˜ I$_{log}$ ( ˜ λ − $^{2}$) ζ (3) ] + b p 2 { − 4 ζ (3) + 6 ln ( − p 2 ¯ ˜ λ 2 ) +3 ln 2 ( − p 2 ¯ ˜ λ 2 ) + ln 3 ( − p 2 ¯ ˜ λ 2 ) + 1 4 ln 4 ( − p 2 ¯ ˜ λ 2 ) } . (25) − (2 π ) 4 [ ˜ I (2) $_{log}$( ˜ λ − $^{2}$) + b ln ( − x 2 ˜ λ 2 ) − b 2 ln 2 ( − x 2 ˜ λ 2 ) ] G$_{1}$ ( p ) = i (4 π $^{2}$)3 ∫ Λ d $^{4}$x e ipx x 2 ∫ Λ d $^{4}$z ln( − z 2 ˜ λ $^{2}$) z $^{2}$( z + x ) $_{2}$. (19) G$_{1}$ ( p ) = ∫ Λ k 1 k $^{4}$( p − k ) 2 ln ( − k 2 ¯ ˜ λ 2 ) (18) ˜ G$_{3}$ ( x ) = − i (4 π $^{2}$) x 2 { ˜ I (4) $_{log}$( ˜ λ − $^{2}$) + b [ 6 ln ( − x 2 ˜ λ 2 ) − 3 ln 2 ( − x 2 ˜ λ 2 ) + ln 3 ( − x 2 ˜ λ 2 ) − 1 4 ln 4 ( − x 2 ˜ λ 2 ) ]} − i 4 ζ (3) (4 π $^{2}$) x 2 { ˜ I$_{log}$ ( ˜ λ − $^{2}$) − b ln ( − x 2 ˜ λ 2 ) + 2 b } (24) G$_{3}$ ( p ) = ∫ Λ k 1 k $^{4}$( p − k ) 2 ln 3 ( − k 2 ¯ ˜ λ 2 ) (23) G$_{2}$ ( p ) = 1 p 2 ˜ I (3) $_{log}$( ˜ λ − $^{2}$) + b p 2 { 4 + 2 ln ( − p 2 ¯ ˜ λ 2 ) + ln 2 ( − p 2 ¯ ˜ λ 2 ) + 1 3 [ ln 3 ( − p 2 ¯ ˜ λ 2 ) + 4 ζ (3) ] } . (22) G$_{ir}$ = ln 2 ( λ 2 ¯ ˜ λ 2 ) ∫ Λ k 1 k $^{4}$( p − k ) 2 ln ( − k 2 ¯ ˜ λ 2 ) − 2 ln ( λ 2 ¯ ˜ λ 2 ) ∫ Λ k 1 k $^{4}$( p − k ) 2 ln 2 ( − k 2 ¯ ˜ λ 2 ) + ∫ Λ k 1 k $^{4}$( p − k ) 2 ln 3 ( − k 2 ¯ ˜ λ 2 ) , (17) so is 4