Infra-red Finite Charge Propagation

The Coulomb gauge has a long history and many uses. It is especially useful in bound state applications. An important feature of this gauge is that the matter fields have an infra-red finite propagator in an on-shell renormalisation scheme. This is, however, only the case if the renormalisation point is chosen to be the static point on the mass-shell, p = (m, 0, 0, 0). In this letter we show how to extend this key property of the Coulomb gauge to an arbitrary relativistic renormalisation point. This is achieved through the introduction of a new class of gauges of which the Coulomb gauge is a limiting case. A physical explanation for this result is given. To appear in Modern Physics Letters A * Current address: Dept. of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849 In this letter we will study the propagation of a charged particle, such as an electron, in a mass-shell renormalisation scheme. To motivate this study, let us first recall the details of this calculation in a Lorentz gauge with gauge parameter ξ. Two renormalisation constants must be introduced: a mass shift, m → m − δm, and a fermion wave function renormalisation, ψ → √ Z2ψ. The mass shift, which we find by requiring the presence of a pole at the physical mass, is found to be gauge parameter independent as one would expect since the electron mass is physical. Problems arise when one now tries to demand that the residue of the pole be unity: the Z2 renormalisation constant depends on the unphysical gauge parameter ξ and has in general an infra-red divergence, which obscures the physical content of the theory. Another way to see that there is a problem is to consider the general form of the propagator around the mass shell. Renormalisation group arguments indicate that the one-loop renormalised propagator near the mass shell has the form (p −m2)β pμγ −m , with β = e2(−ξ − 3) 8π2 , (1) and we see that a pole structure emerges only in the Yennie gauge. However, even for this gauge the resulting propagator cannot be identified with that describing charge propagation. A gauge which appears not to have these problems is the Coulomb gauge (for various treatments of QED in this gauge see Ref.’s 6-9). Here one obtains the same mass shift as in covariant gauges and Z2 is infra-red finite. There is, however, an often unappreciated subtlety here: infra-red finiteness only holds if we are at the static point on the mass shell, i.e., if we demand p = (m, 0, 0, 0). Details of this calculation can be found in Sect. 6 of Ref. 10. The form of the propagator in Coulomb gauge near the mass shell is quoted in Ref. 5. Although this, like (1), is in general not a simple pole, in the static limit their formula indeed reduces to a pole. Such kinematical configurations naturally arise in many bound state problems. Indeed, even in QCD, the utility of this gauge in the calculation of the static inter-quark potential is well known (see, for example Ref. 11). A generalisation of the Coulomb gauge that allowed us to perform an on-shell renormalisation of the electron propagator at an arbitrary, relativistic point on the mass shell, i.e., for p = mγ(1, v) where γ = 1/ √ 1− v2, would improve our understanding of these fundamental topics, help extend the feasibility of such