Expansion of EYM amplitudes in gauge invariant vector space *

In the first-quantised worldline approach to quantum field theory, a longstanding problem has been to extend this formalism to amplitudes involving open fermion lines while maintaining the efficiency of the well-tested closed-loop case. In the present series of papers, we develop a suitable formalism for the case of quantum electrodynamics in vacuum (part one and two) and in a constant external electromagnetic field (part three), based on second-order fermions and the symbol map. We derive this formalism from standard field theory, but also give an alternative derivation intrinsic to the worldline theory. In this first part, we use it to obtain a Bern-Kosower type master formula for the fermion propagator, dressed with N photons, in terms of the "N -photon kernel," where offshell this kernel appears also in "subleading" terms involving only N − 1 of the N photons. Although the parameter integrals generated by the master formula are equivalent to the usual Feynman diagrams, they are quite different since the use of the inverse symbol map avoids the appearance of long products of Dirac matrices. As a test we use the N = 2 case for a recalculation of the one-loop fermion self energy, in D dimensions and arbitrary covariant gauge, reproducing the known result. We find that significant simplification can be achieved in this calculation by choosing an unusual momentum-dependent gauge parameter.

Section: The On-shell Casementioning

confidence: 99%

Worldline master formulas for the dressed electron propagator. Part I. Off-shell amplitudes

Ahmadiniaz

Guzman

Bastianelli

et al. 2020

“…(1.17) for a photon of momentum k with polarisation vector ε. Although this fact is even stated in some textbooks on QFT (see, e.g., [17]), in the standard Feynman diagram approach it is by no means trivial to actually achieve this rewriting explicitly for an arbitrary number of photons (see, e.g., [18]). Here we will show, both for the dressed scalar propagator D p p N as…”

Section: The On-shell Casementioning

confidence: 99%

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

Ahmadiniaz

Guzman

Bastianelli

et al. 2022

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.

“…In this paper, we will reconsider the computation of tadpole coefficients by using differential operators. Differential operators have played an important role in the area of scattering amplitude, for example, deriving the IBP relations and differential equations of Feynman integrals [11], relating tree-level amplitudes of different theories [12] and the expansion of Einstein-Yang-Mills amplitude [13,14].…”

Section: Jhep09(2021)081mentioning

confidence: 99%

“…The auxiliary vector R µ closely resembles the polarization vectors in the expansion of Einstein-Yang-Mills amplitude[13,14].…”

mentioning

confidence: 92%

Analytic tadpole coefficients of one-loop integrals

Feng

2021

Self Cite

One remaining problem of unitarity cut method for one-loop integral reduction is that tadpole coefficients can not be straightforward obtained through this way. In this paper, we reconsider the problem by applying differential operators over an auxiliary vector R. Using differential operators, we establish the corresponding differential equations for tadpole coefficients at the first step. Then using the tensor structure of tadpole coefficients, we transform the differential equations to the recurrence relations for undetermined tensor coefficients. These recurrence relations can be solved easily by iteration and we can obtain analytic expressions of tadpole coefficients for arbitrary one-loop integrals.