Dimensional regularization

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In theoretical physics, dimensional regularization is a method to tentatively render divergent integrals in the evaluation of Feynman diagrams finite. Once the divergent quantities are properly controlled, mathematical manipulations are made legitimately.

Dimensional regularization assumes that the spacetime dimension is not four but rather d which need not be an integer. It often turns out that the integrals extrapolated to a general dimension converge. The divergences are then parametrized as quantities proportional to $1 / ε$ whose coefficients must be cancelled by renormalization to obtain physical quantities.

The volume of a unit d-1 sphere is $\frac{2\pi^{d/2}}{\Gamma\left(\frac{d}{2}\right)}$ where Γ is the gamma function when d is a positive integer. We can assume by fiat that this equation also holds when d isn't an integer.

If we wish to evaluate a loop integral which is logarithmically divergent in 4 dimensions, like

$\int\frac{d^dp}{(2\pi)^d}\frac{1}{\left(p^2+m^2\right)^2}$

we first generalize this equation to an arbitrary number of dimensions, including non-integral dimensions like d=4-ε. When ε is positive, this integral converges and we take the limit as ε approaches zero.

This gives

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