 Spin-averaged electromagnetic parton tensor

For an on-shell massless quark with incoming momentum $k$ and outgoing momentum $k+q$, average the initial spin and sum the final spin. The gamma matrix trace identities give $S^{\mu \nu }=\frac{1}{2}\mathrm{Tr}[/k{\gamma }^{\mu }(/k+/q){\gamma }^{\nu }]=2[k^{\mu }(k+q{)}^{\nu }+k^{\nu }(k+q{)}^{\mu }-g^{\mu \nu }k\cdot q]$. On shell $2k\cdot q=Q^2=-q^2$, and defining $T^{\mu \nu }=-g^{\mu \nu }+q^{\mu }{q}^{\nu }/q^2$ and $\tilde{k}=k-(k\cdot q/q^2)q$ yields the displayed transverse decomposition. With the hadronic-tensor prefactor $1/(4\pi )$, one-particle Lorentz-invariant phase space gives $w_f^{\mu \nu }=(Q_f^2/2){\delta }_{+}((k+q{)}^2)S^{\mu \nu }$. The antiquark has the same unpolarized tensor and squared electric charge.