Iterated dimensional regularization

[zetafunction]

let be the 2-loop integral

$$\iint d^{k}pd^{k}qF(q,p)$$

where k is the dimension so we regularize it by dimensional regularization

the my idea is the following

i integrate oper ''q' considering p is constant to get $$F(p,e^{-1} )$$

here e is the parameter inside k=4-e

after dimensional regularization over q i have an integral over 'p' , so i apply dimensional regularization again to obtain a regularized value

can this be done ?? i mean can we apply dimensional regularization iteratively over each variable ??

 

[ardie]

not quiet. if the integral is doable or not will depend on what the function F(q,p) represents. of course to do an integral over n variables you have to perform the integral n times, but you also need n-1 expressions relating your n-variables all together. so if you have a relation that will translate F(q,p) in terms of F(k(p),p), then you can do the integral over dp first taking q to be a constant.

 

[zetafunction]

but can it be done iteratively ??

for example i make dimensional regularization over 'p' keeping 'q' constant and afterwards i make use of dimensional regularization over q to get a finite result.