An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 17.3
Quantum Chromodynamics — M. Peskin, D. Schroeder (Perseus, 1995)
Physics — QFT · qcd-parton-scattering
Problem
17.3 Quark-gluon and gluon-gluon scattering.
(a) Compute the differential cross section
dt^dσ(qqˉ→gg)
for quark-antiquark annihilation in QCD to the leading order in αs. This is most easily done by computing the amplitudes between states of definite quark and gluon helicity. Use explicit polarization vectors and spinors, for example,
ϵμ=21(0,1,i,0)
for a right-handed gluon moving in the +3^ direction. You need only consider transversely polarized gluons. By helicity conservation, only the initial states qLqˉR and qRqˉL can contribute; by parity, these two states give identical cross sections. Thus it is necessary only to compute the amplitudes for the three processes
qLqˉR→gRgR,qLqˉR→gRgL,qLqˉR→gLgL.
In fact, by CP invariance, the first and third processes have equal cross sections. After computing the amplitudes, square them and combine them properly with color factors to construct the various helicity cross sections. Finally, combine these to form the total cross section averaged over initial spins and colors.
(b) Compute the differential cross section
dt^dσ(gg→gg)
for gluon-gluon scattering. There are 16 possible combinations of helicities, but many of them are related to each other by parity and crossing symmetry. All 16 can be built up from the three amplitudes for
gRgR→gRgR,gRgR→gRgL,gRgR→gLgL.
Show that the last two of these amplitudes vanish. The first can be dramatically simplified using the Jacobi identity. When the smoke clears, only three of the 16 polarized gluon scattering cross sections are nonzero. Combine these to compute the spin- and color-averaged differential cross section.