Helicity amplitudes for QCD with massive quarks

Alexander Ochirov

doi:10.1007/jhep04(2018)089

Abstract

The novel massive spinor-helicity formalism of Arkani-Hamed, Huang and Huang provides an elegant way to calculate scattering amplitudes in quantum chromodynamics for arbitrary quark spin projections. In this note we compute two families of tree-level QCD amplitudes with one massive quark pair and n − 2 gluons. The two cases include all gluons with identical helicity and one opposite-helicity gluon being color-adjacent to one of the quarks. Our results naturally incorporate the previously known amplitudes for both quark spins quantized along one of the gluonic momenta. In the all-multiplicity formulae presented here the spin quantization axes can be tuned at will, which includes the case of the definite-helicity quark states.

Highlights

We pay special attention to our conventions so that our results be consistent with the vast QCD literature
The spins of the quark and the antiquark remain unfixed throughout the calculations, which lets us specialize to the specific quarkspin projections considered previously [14] in the massless-spinor-based formalism [8,9,10]
We give a simple dictionary (5.4) between the two descriptions and compare our results with the literature. It shows that the new formalism incorporates the old one, the elegance of which suffered from the loss of the explicit littlegroup SU(2) symmetry

Summary

Spinor-helicity review

It is well-known that particles are defined as irreducible unitary representations of the Poincare group [15, 16]. The remaining labels of a one-particle state turn out to belong to a representation of its little group This subgroup of SO(1, 3) is crucial for understanding spin. It is defined through the Lorentz transformations that preserve the momentum pμ of the particle It corresponds to SO(2) for massless states or to SO(3) for massive ones. An important property of the SL(2, C) transformations (and the SU(2) ones) is that they preserve the antisymmetric form αβ = − αβ, i.e. the spinor product: Sαγ Sβδ γδ. This form allows to raise and lower both the spinor and massive-little-group indices at will. The massless and massive Weyl spinors comprise the spinor-helicity formalism [1,2,3,4,5,6, 11], while the Dirac spinors are helpful to connect it to the more traditional approaches

Massless Weyl spinors

Massive Weyl spinors

Dirac spinors and spin

Four-point amplitudes

All-multiplicity amplitudes

One-minus amplitudes with two quarks

Checks

Summary and discussion

A Massless spinor parametrizations

B Massive spinor parametrizations