New explicit expressions for Dirac bilinears

Cédric Lorcé

doi:10.1103/physrevd.97.016005

Abstract

We derive new explicit expressions for the Dirac bilinears based on a generic representation of the massive Dirac spinors with canonical polarization. These bilinears depend on a direction $n$ in Minkowski space which specifies the form of dynamics. We argue that such a dependence is unavoidable in a relativistic theory with spin, since it originates from Wigner rotation effects. Contrary to most of the expressions found in the literature, our ones are valid for all momenta and canonical polarizations of the spinors. As a by-product, we also obtain a generic explicit expression for the covariant spin vector.

Highlights

One often faces the problem of computing amplitudes which are expressed as MΓ 1⁄4 wε0 ðp0; S0ÞΓwεðp; SÞ; ð1Þ
Contrary to most of the expressions proposed in the literature, our results are valid for all momenta and canonical polarizations
The explicit dependence on an external four-vector arises as a consequence of the use of a canonical polarization basis and can be understood in terms of Wigner rotation effects

Summary

Introduction

One often faces the problem of computing amplitudes which are expressed as MΓ 1⁄4 wε0 ðp0; S0ÞΓwεðp; SÞ; ð1Þ where Γ is a 4 × 4 matrix in Dirac space, wþðp; SÞ 1⁄4 uðp; SÞ and w−ðp; SÞ 1⁄4 vðp; SÞ are the free positive and negative-energy spinors, and p (p0) is the initial (final) fourmomentum of the particle with mass m (m0). An old question is whether the amplitude MΓ can be expressed as an analytic and covariant function depending on the vectors p, p0, S, and S0. The main motivation for this is that such a covariant expression greatly helps the calculations in practical cases and allows one to develop efficient numerical methods, see e.g. Since these distributions are expressed as Fourier transforms of MΓ with respect to the four-momentum transfer Δ 1⁄4 p0 − p, explicit expressions for the Dirac bilinears

Objectives

Discussion

Conclusion