Dimensional regularization

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 is thought of as a complex number. It often turns out that the integrals extrapolated to a general dimension converge. The divergences are then parameterized as quantities proportional to $1/\epsilon$ 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

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

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

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

See also

• Regularization (physics)
• Pauli-Villars regularization