Pauli–Villars regularization


In theoretical physics, Pauli–Villars regularization is a procedure that isolates divergent terms from finite parts in loop calculations in field theory in order to renormalize the theory. Wolfgang Pauli and Felix Villars published the method in 1949, based on earlier work by Richard Feynman, Ernst Stueckelberg and Dominique Rivier.
In this treatment, a divergence arising from a loop integral is modulated by a spectrum of auxiliary particles added to the Lagrangian or propagator. When the masses of the fictitious particles are taken as an infinite limit one expects to recover the original theory.
This regulator is gauge invariant due to the auxiliary particles being minimally coupled to the photon field through the gauge covariant derivative. It is not gauge covariant, though, so Pauli–Villars regularization cannot be used in QCD calculations. P–V serves as an alternative to the more favorable dimensional regularization in specific circumstances, such as in chiral phenomena, where a change of dimension alters the properties of the Dirac gamma matrices.
Gerard 't Hooft and Martinus J. G. Veltman invented, in addition to dimensional regularization, the method of unitary regulators, which is a Lagrangian-based Pauli–Villars method with a discrete spectrum of auxiliary masses, using the path-integral formalism.

Examples

Pauli–Villars regularization consists of introducing a fictitious mass term. For example, we would replace a photon propagator , by, where can be thought of as the mass of a fictitious heavy photon, whose contribution is subtracted from that of an ordinary photon.