Iterated dimensional regularization

Question

Given a 2-loop divergent integral $\int F(q,p)\,\mathrm{d}p\mathrm{d}q$, can it be solved iteratively? I mean

1. I integrate over $p$ keeping $q$ constant
2. Then I integrate over $q$

In both iterated integrals I use dimensional regularization.

Can it be solved iteratively? I presented a paper to a teacher of mine about regularization of integrals using the Zeta regularization. He told me that for one dimensional integrals (or one loop integrals) it was fine, but that my method could not handle multi loop integrals. I argue back that you could apply the regularization method by introducing a regulator of the form

$$(a+qi)^{-s}$$

I could make the integral on each variable by iterated regularization, that is applying the algorithm iteratively.

EDIT: i think they have cheated me :) making up excuses not to put me atention

by the way , how can i insert math codes on my posts ?? is just setting $ at the beginning and the end of the equation ??

---

Edit: 19th of July

thanks :) your answer was quite useful :)

however can I not insert a term $(qi+a)^{-s} $ on each variable and then apply the regularization iteratively ??

I say so because I made a paper http://vixra.org/abs/1009.0047 to regularize integrals using the regulator $(q+1)^{-1}$ and tried to extend it to several variables, however that was my doubt, if i could apply my regularization scheme to every variable :) thanks again

I mean for a one dimensional integral $\int dx (x+a)^{m-s}$, I know how to regularize it by using the Euler-Maclaurin summation formula plus the Riemann Zeta function $ \zeta (s-m)$

Then for multiloop integrals, I had thought that i could introduce the $s$-regulators (x+a)^{-s}(y+a)^{-s}.. and so on on each variable, and then apply iterated integration... :)

so this is a resume of my method..

a) i know how to use Zeta regularization to get finite values for the integral $ \int dx(x+a)^{m-s} $ in terms of the Riemann Zeta function $ \zeta (s-k) $ with k=-1,0,1,....,m

b) for a more general 1-dimensional integral $ \int dx f(x)(x+a)^{-s}$ i add substract a Polynomial $K(x+a)(x+a)^{-s} $ to get a finite part and then regularize the divergent integrals $dx(x+a)^{m-s} $

c) for a more complicate 2-loop integral $ \iint dpdq F(q,p)(p+a)^{-s}(q+a)^{-s} $ to obtain a regularization of it i do the sema method , first on 'p' considering 'q' a constant and i treat it as a one dimensional integral over 'p' and then over 'q' by substracting a Polynoamials $ K(q,p+a)(p+a)^{-s} $ and so on

Answer 1

Assuming that you can actually do the dimensionally regularized integrals, then yes, you can integrate iteratively. Normally, two-loop and higher-loop integrals are quite difficult and you need good tricks like turning them into differential equations or using Mellin-Barnes parametrization of the propagators (or even just Feynman or Schwinger parametrizations). The new standard book on evaluating dimensionally regularized multiloop integrals is Feynman Integral Calculus by Vladimir Smirnov (2006). Also see the notes for the Father & Son lectures at Durham a couple of years ago.

Not only are multiloop integrals hard, but the regularization scheme becomes important once you leave one-loop. This is because you have to make a consistent renormalization / subtraction of counter-terms. Regularization schemes that modify the diagrams/integrals and not the original Lagrangian can lead to messy renormalization analysis. The simplest renormalization analyses are associated with regularizations that change the Lagrangian (or dimension) combined with a mass-independent minimal subtraction scheme.

Inserting terms like $(q^2)^s$ is a type of analytic regularization, which (like most regularizations) is easy at one-loop, but tricky at higher loops. Analytic regularization was extensively examined by Eugene Speer in the 60s and 70s.

Zeta regularization does only work at one-loop, but it has a generalization/extension called Operator Regularization that was studied by McKeon and friends. From memory, the calculations get a bit messy, although there is a new(ish) paper on using operator regularization and Feynman diagrams that is on my reading list (previous work used a more Schwinger-like, functional approach to loop calculations).

---

I hope that something I said in the above is useful and not too much is wrong. ;)