Dimensional regularization

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In theoretical physics, dimensional regularization is a particular way to get rid of infinities that occur when one evaluates Feynman diagrams in quantum field theory. One 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 parameterized as quantities proportional to $1 / ε$ whose coefficients must be cancelled by renormalization to obtain physical quantities.

The hypersurface area of a d-1 sphere of radius r 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 nonintegral dimensions like d=4-ε. When ε is positive, this integral converges and we take the limit as ε approaches zero.

This gives

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