Transverse decomposition of an unpolarized hadronic tensor ($W^{\mu \nu }=-T^{\mu \nu }{F}_1+{\tilde{P}}^{\mu }{\tilde{P}}^{\nu }{F}_2/(P\cdot q)$)

For electromagnetic deep inelastic scattering with an unpolarized target, Lorentz invariance, parity and current conservation leave two independent deep-inelastic structure functions. With $T^{\mu \nu }=g^{\mu \nu }-q^{\mu }{q}^{\nu }/q^2$ and $\tilde{P}=P-(P\cdot q/q^2)q$, both tensor structures are transverse to $q$. Terms with a free $q$ index disappear on contraction with a conserved leptonic tensor, although such terms can still be necessary to make the full hadronic tensor transverse.