is evaluated to be in Implicit Regularization with an additional log- arithm [35], which in turn implies that Therefore collecting just the finite parts we have Since J ( p ) is finite, the divergences must cancel each other. This is demonstrated in appendix B showing the consistency of the method. So in the results below we will consider just the finite pieces of the integrals. We start by evaluating As a third example, let us consider the two loops massless master diagram in four dimensions J ( p ): The integral in l , let us call it I$_{1}$ ( p ), is infrared divergent, The infrared scale ¯ ˜ λ 2 was introduced inside the log- arithm to simplify the calculation separating I$_{1}$ ( p ) into two pieces: J ( p ) is finite and can be evaluated for instance using the Gegenbauer technique [28]. In appendix A it is shown within Implicit Regulazation that which implies that It is not difficult to show that, for α an arbitrary positive constant, Both integrals have been already calculated, the first is proportional to U , eq. 9, the second is G$_{1}$ ( p ), 20. Finally putting the ultraviolet result (5) together J ( p ) = ∫ Λ k 1 k $^{2}$( k − p ) 2 ∫ Λ l 1 ( l − p ) 2 l 4 ln ( − l 2 α 2 ) + ∫ Λ k 1 k $^{2}$( k − p ) 2 ∫ Λ l 1 l $^{2}$( l − p ) 4 ln ( − l 2 α 2 ) − ∫ Λ k,l 1 k $^{2}$l$^{2}$( k − l ) $^{2}$( l − p ) 4 ln ( − l 2 α 2 ) − ∫ Λ k,l 1 ( k − p ) $^{2}$( k − l ) $^{2}$( l − p ) 2 l 4 ln ( − l 2 α 2 ) ≡ J$_{1}$ ( p ) + J$_{2}$ ( p ) − J$_{3}$ ( p ) − J$_{4}$ ( p ) . (35) G$_{µ,finite}$ = b p$_{µ}$ p 2 { ln ( λ 2 ¯ ˜ λ 2 ) [ 2 + ln 2 ( − p 2 λ 2 )] + 6 ln ( − p 2 λ 2 ) + ln 3 ( − p 2 λ 2 ) } . (29) = − b { 4 ζ (3) + 2 [ 1 + 4 3 ζ (3) ] ln ( λ 2 ¯ ˜ λ 2 ) + ln 2 ( λ 2 ¯ ˜ λ 2 ) − 1 12 ln 4 ( λ 2 ¯ ˜ λ 2 ) } . (30) G ( ¯ ˜ λ, λ ) = G$_{uv}$ + p $^{2}$G$_{ir}$ − 2 p $^{µ}$G$_{µ,finite}$ J$_{1}$ ( p ) = ∫ Λ k 1 k $^{2}$( k − p ) 2 ∫ Λ l 1 ( l − p ) $^{2}$l4 ln ( − l 2 α 2 ) . (36) I$_{1}$ ( p ) = ∫ Λ l 1 ( l − p ) $^{2}$l4 ln ( − l 2 α 2 ¯ ˜ λ 2 ¯ ˜ λ 2 ) . I$_{1}$ ( p ) = ln ( ¯ ˜ λ 2 α 2 ) ∫ Λ l 1 ( l − p ) $^{2}$l4 + ∫ Λ l 1 ( l − p ) $^{2}$l4 ln ( − l 2 ¯ ˜ λ 2 ) . (37) ∫ Λ k,l 1 l $^{4}$( k − p ) $^{2}$( k − l ) $^{2}$( l − p ) 2 = ∫ Λ k,l 1 k $^{2}$l$^{4}$( k − p ) $^{2}$( l − p ) $_{2}$. (33) ∫ Λ k,l ∂ ∂k$_{µ}$ { ( k − l )$_{µ}$ k $^{2}$l$^{2}$( k − p ) $^{2}$( k − l ) $^{2}$( l − p ) 2 } = 0 (32) J ( p ) = ∫ k ∫ l 1 k $^{2}$l$^{2}$( k − p ) $^{2}$( k − l ) $^{2}$( l − p ) $_{2}$. (31) IV. CONSISTENCY IN THE EVALUATION OF FINITE INTEGRALS WITH INFRARED AND ULTRAVIOLET DIVERGENCES IN INTERMEDIATE STEPS ∫ Λ k,l ∂ µ { ( k − l )$_{µ}$ ln( − l $^{2}$/α$^{2}$) k $^{2}$l$^{2}$( k − p ) $^{2}$( k − l ) $^{2}$( l − p ) 2 } = 0 , (34) [35] This identity is argued in [29] to be a four dimensional version of the integration by parts method described in [30, 31]. 5