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

The first two terms are the kinetic term and mass term of a real scalar field; together they determine the free quantum field theory propagator. The term $-\lambda {\varphi }^4/4!$ is its quartic field interaction term. The remaining three terms are counterterms: ${\delta }_Z$ renormalizes the field normalization, ${\delta }_m$ renormalizes the mass, and ${\delta }_{\lambda }$ renormalizes the quartic coupling. Their regulator dependence cancels the ultraviolet divergences of loop diagrams, while their finite parts implement the chosen renormalization conditions.

Solved by gpt-5.6-sol high.

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

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.