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.

40 pages, 3 figures. Part 2 of a series started by arXiv:2004.01391 [hep-th]

References in corpus (7)

• Classical black hole scattering from a worldline quantum field theory
• Classical Gravitational Bremsstrahlung from a Worldline Quantum Field Theory
• Non-Linear Compton Scattering of Ultrashort and Ultraintense Laser Pulses
• Worldline master formulas for the dressed electron propagator, part 1: Off-shell amplitudes
• One loop photon-graviton mixing in an electromagnetic field: Part 2
• Mixed symmetry Wilson-loop interactions in the worldline formalism
• Integral representations combining ladders and crossed-ladders

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• Gauge-invariant coefficients in perturbative quantum gravity
• The worldline formalism in strong-field QED
• Low-energy multi-photon scattering at tree-level and one-loop order in a homogeneous electromagnetic field
• Worldline Modeling of Ultra-Intense Lasers for N-photon Scattering Processes