Positive-energy on-shell delta function identity ($\frac{\delta (\omega -E)}{E}=2\delta (E^2-{\omega }^2),\omega >0$)

For $E>0$ and the positive-energy branch, the root Jacobian of the Dirac delta function gives this identity. In a massless collinear parton approximation, it produces $\delta (x-\xi )/(P\cdot q)$.