How Does Dimensional Regularization Simplify Integrals in Quantum Field Theory?

[mhill]

how does dimensional regularization work ?

i see , how can you calculate integrals in d-dimensions of the form

$$\int d^{d} k F( \vec k )$$ ??

and for other cases , let us suppose we have the integral

$$\lim_{\varepsilon\rightarrow 0^+}\int \frac{dp}{(2\pi)^{4-\varepsilon}} \frac{2\pi^{(4-\varepsilon)/2}}{\Gamma\left(\frac{4-\varepsilon}{2}\right)} \frac{p^{3-\varepsilon}}{\left(p^2+m^2\right)^1}$$

there is no way this integral can be calculated

 

[Avodyne]

It works by analytic continuation. Consider the integral

$$\int_0^\infty dx\,{x^n\over x^2+1}.$$

For $-1<{\rm Re}\,n<1$, the integral converges, and the result is

$${\pi/2\over\cos(n\pi/2)}.$$

We now define this to be the value of the integral for all $n$.