Srednicki — Quantum Field Theory · Problem 81.5

The core idea here is that summing over helicities eliminates the explicit spin dependence, leaving you with a Lorentz-invariant result that depends only on the kinematic invariants. For this quark-antiquark to gluon pair process, you'll use the completeness relation for polarization vectors (or spinors) to collapse the helicity sum into traces of products of Mandelstam variables. The key insight is that the squared amplitude simplifies dramatically when you sum over all possible spin configurations—what looked complicated becomes a clean polynomial in $s$, $t$, and $u$.

The core idea here is that summing over helicities eliminates the explicit spin dependence, leaving you with a Lorentz-invariant result that depends only on the kinematic invariants. For this quark-antiquark to gluon pair process, you'll use the completeness relation for polarization vectors (or spinors) to collapse the helicity sum into traces of products of Mandelstam variables. The key insight is that the squared amplitude simplifies dramatically when you sum over all possible spin configurations—what looked complicated becomes a clean polynomial in $s$, $t$, and $u$.