Longitudinal polarization vector (${ϵ}_L^{\mu }=(|\mathbf{k}|,E\hat{\mathbf{k}})/m$)

For an on-shell massive vector with $k^{\mu }=(E,|\mathbf{k}|\hat{\mathbf{k}})$ in Minkowski metric signature $+---$, the displayed vector satisfies $k\cdot {ϵ}_L=0$ and ${ϵ}_L^2=-1$. It specifies the longitudinal one of the three physical polarization states. It is not a replacement for a final polarization sum when other states are unobserved.