content_image/441655.jpg content_image/441657.jpg content_image/441663.jpg content_image/441662.jpg content_image/441665.jpg content_image/441656.jpg content_image/441661.jpg content_image/441654.jpg content_image/441658.jpg content_image/441659.jpg content_image/441660.jpg
is evaluated to be 

<img src='content_image/441655.jpg'>

Therefore collecting just the finite parts we have 

<img src='content_image/441657.jpg'>

## IV. CONSISTENCY IN THE EVALUATION OF FINITE INTEGRALS WITH INFRARED AND ULTRAVIOLET DIVERGENCES IN INTERMEDIATE STEPS 

As a third example, let us consider the two loops massless master diagram in four dimensions J ( p ): 

<img src='content_image/441663.jpg'>

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 

<img src='content_image/441662.jpg'>

which implies that 

It is not difficult to show that, for α an arbitrary positive constant, 

<img src='content_image/441665.jpg'>

<img src='content_image/441656.jpg'>

<img src='content_image/441661.jpg'>

in Implicit Regularization with an additional log- arithm [35], which in turn implies that 

<img src='content_image/441654.jpg'>

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 

<img src='content_image/441658.jpg'>

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: 

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 

[35] This identity is argued in [29] to be a four dimensional version of the integration by parts method described in [30, 31]. 

<img src='content_image/441659.jpg'>

<img src='content_image/441660.jpg'>

