Worldline master formulas for the dressed electron propagator. Part 2. On-shell amplitudes

Abstract

In the first part of this series, we employed the second-order formalism and the “symbol” map to construct a particle path-integral representation of the electron propagator in a background electromagnetic field, suitable for open fermion-line calculations. Its main advantages are the avoidance of long products of Dirac matrices, and its ability to unify whole sets of Feynman diagrams related by permutation of photon legs along the fermion lines. We obtained a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the “N-photon kernel,” where this kernel appears also in “subleading” terms involving only N − 1 of the N photons.In this sequel, we focus on the application of the formalism to the calculation of on-shell amplitudes and cross sections. Universal formulas are obtained for the fully polarised matrix elements of the fermion propagator dressed with an arbitrary number of photons, as well as for the corresponding spin-averaged cross sections. A major simplification of the on-shell case is that the subleading terms drop out, but we also pinpoint other, less obvious simplifications.We use integration by parts to achieve manifest transversality of these amplitudes at the integrand level and exploit this property using the spinor helicity technique. We give a simple proof of the vanishing of the matrix element for “all +” photon helicities in the massless case, and find a novel relation between the scalar and spinor spin-averaged cross sections in the massive case. Testing the formalism on the standard linear Compton scattering process, we find that it reproduces the known results with remarkable efficiency. Further applications and generalisations are pointed out.

Obtaining fully polarised amplitudes in gauge invariant form

“…To illustrate this approach we recall some simple examples from [7,8]. To begin, note that the symbol map implies the kernel is naturally written in terms of elements of the even sub-algebra of the Dirac representation of the Clifford algebra, so we can decompose (…”

“…where f is the dual field strength tensor. Useful choices for the reference vectors include r i = p and r i = p + m 2 2p •ki k i -see [8] for more details and identities allowing for A and C to be constructed from B µν on-shell.…”

“…In [8] these are used to construct the fully polarised amplitude, the spin-summed amplitude with fixed photon helicities and the un-polarised amplitude. Indeed, there a particularly simple "diagonal" formula was found for the spin / polarisation summed amplitudes:…”

“…Yet in the same papers that developed this approach [1,2], Feynman hinted at an alternative, first quantised formulation of field theory that later evolved into the worldline formalism [3] -for reviews see [4,5]. Historical applications within this alternative framework focussed on calculating the effective action [3] or multi-loop amplitudes [6] and the worldline approach was only recently extended to type of tree-level amplitudes we highlight here [7,8]. In this contribution we report the salient formulae describing the fermion propagator dressed by N -photons and describe how to extract manifestly gauge invariant amplitudes for multi-photon emission or absorption from the line.…”

Light bending from eikonal in worldline quantum field theory

“…Dressed propagators have been developed in a worldline representation for a variety of models, see e.g. [2][3][4][5][6][7][8][9].…”