Past exam of the gauge field theory course of the University of Cambridge / 2014 / Paper 1 / 3 / b / Solution

Loop corrections in perturbative quantum field theory integrate over unbounded internal four-momentum. For example, after a Wick rotation, a four-dimensional integral with large-momentum behaviorhas a logarithmic ultraviolet divergence. This is sensitivity to short distances, distinct from a possible low-momentum infrared divergence.

Regularization in quantum field theory first makes the expression mathematically well defined, for example with a cutoff $\Lambda$ or dimensional regularization in $d=4-2ϵ$. A regulator is an intermediate prescription rather than a measured physical parameter; a hard momentum cutoff can obscure gauge identities, so their preservation or restoration must be checked. Introduce bare fields and parameters, such as ${\varphi }_0=Z_{\varphi }^{1/2}\varphi$, and split the regulated Lagrangian density into a renormalized part and local counterterms. For instance, wavefunction, mass and coupling counterterms cancel the corresponding divergent parts of Green functions order by order.

Renormalization conditions fix finite parameters through specified observables or a subtraction prescription. Once these physical inputs are held fixed, predictions must not depend on the arbitrary regulator. In a renormalizable quantum field theory, a finite set of operator coefficients suffices to absorb ultraviolet divergences at every perturbative order, with the allowed structures constrained by the theory's symmetries. Gauge theories impose relations among counterterms through identities such as the Ward-Takahashi identity; one cannot choose every diagram's subtraction independently.

The remaining dependence on a chosen renormalization scale describes running couplings and is organized by the renormalization group. An effective field theory can instead include higher-dimension operators suppressed by a high scale. Infinitely many are allowed in principle, but only finitely many contribute at a fixed accuracy in the low-energy expansion. Renormalization is therefore a controlled relation between parameters and predictions at different scales, rather than an instruction to discard every infinity without fixing physical inputs.