# Iterated dimensional regularization

• zetafunction
In summary, the conversation discusses the concept of dimensional regularization in the context of the 2-loop integral. The idea is to integrate over the variable 'q' while keeping 'p' constant, and then apply dimensional regularization over 'p' to obtain a regularized value. However, it is noted that the feasibility of this approach depends on the function F(q,p) and whether it can be expressed in terms of F(k(p),p). The possibility of applying dimensional regularization iteratively over each variable is also mentioned, but it is emphasized that this would require n-1 expressions relating the n-variables in order to perform the integral.
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 ??

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.

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.

## What is iterated dimensional regularization?

Iterated dimensional regularization is a mathematical technique used in quantum field theory to regulate and renormalize divergent integrals. It involves performing a series of integrations over different dimensional spaces to obtain a finite result.

## Why is iterated dimensional regularization used?

Iterated dimensional regularization is used because it allows for a more systematic and efficient way of dealing with divergences in quantum field theory calculations. It also preserves the symmetries of the theory and can be applied to a wide range of physical problems.

## How does iterated dimensional regularization work?

Iterated dimensional regularization works by first expressing a divergent integral in terms of a sum of integrals over different dimensional spaces. Then, the integrals are evaluated using analytical techniques, such as the Mellin-Barnes representation, to obtain a finite result. The final result is obtained by summing over all the dimensions.

## What are the advantages of using iterated dimensional regularization?

One of the main advantages of iterated dimensional regularization is that it can handle a wider range of divergences compared to other regularization techniques. It also preserves the symmetries of the theory, which is important for making predictions that are consistent with experimental results.

## Are there any limitations to iterated dimensional regularization?

Although iterated dimensional regularization is a powerful technique, it is not a complete solution to all the problems in quantum field theory. It can still encounter difficulties when dealing with certain types of divergences, and in some cases, additional techniques may be needed to obtain finite results. Additionally, it may not always be straightforward to apply iterated dimensional regularization to complex physical problems.

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