On the value of the non-perturbative field renormalization constant Z in gauge theories

TL;DR

This paper investigates the non-perturbative value of the field renormalization constant Z in gauge theories, showing it is generally non-zero unless massless or unstable particles are present, where Z=0 is universal.

Contribution

It challenges the common belief that Z is zero non-perturbatively, demonstrating conditions under which Z remains non-zero in interacting quantum field theories.

Findings

Z is not necessarily zero in non-perturbative regimes.

Presence of massless photons or unstable particles implies Z=0.

Results are valid within a framework requiring Hilbert space positivity.

Abstract

In the perturbative approach to quantum field theory it is common to replace the propagator $i (p^{2}-m_{0}^{2}+i\varepsilon )^{-1}$ for a scalar field by a similar expression, namely $iZ (p^{2}-m^{2}+i\varepsilon )^{-1}$, where the shift of the mass from $m_{0}$ to $m$ reflects the mass renormalization and the constant~$Z$ is the renormalized field strength (or wave-function). We argue that, contrary to general belief, the non-perturbative value of~$Z$ is not necessarily equal to zero in case the two-point function of an interacting quantum field theory is, as expected, more singular on the light-cone than the corresponding free field two-point function. If, however, (massless) photons or composite (unstable) particles are present, the condition $Z=0$ follows from two qualitatively different arguments, one being a theorem due to Buchholz, the other a criterion due to Weinberg. Hence, the condition $Z=0$ is, after all, a universal feature of realistic models of elementary particle physics, which include massless or unstable particles. The results hold within a natural framework which, in the case of gauge theories, requires Hilbert space positivity, and therefore the use of non-manifestly covariant gauges.