Regularization in quantum field theory

Regularization modifies divergent loop integrals by introducing an auxiliary parameter. The regulator is removed after its dependence has been absorbed into counterterms.

Cutoff regularization restricts loop momenta to $|\ell |<\Lambda$. The ultraviolet cutoff $\Lambda$ makes individual integrals finite while displaying power and logarithmic ultraviolet divergences explicitly.

Dimensional regularization analytically continues loop integrals from an integer spacetime dimension to $d=d_0-ϵ$. Ultraviolet logarithms then appear as poles in $1/ϵ$.

The minimal subtraction scheme chooses counterterms that remove only the poles in the dimensional regulator $ϵ$, without additional finite terms.