Hadronic tensor ($W^{\mu \nu }$)

The inclusive product of hadronic-current matrix elements, summed over final states with a Lorentz-invariant phase-space measure. Electromagnetic current conservation makes both indices transverse to the exchanged four-momentum.

For an absorbed spacelike photon, use $q^2=-Q^2<0$, $\nu =P\cdot q>0$, $x=Q^2/(2\nu )$ and the physical momentum delta function ${\delta }^{(4)}(P+q-p_X)$. The displayed invariant final mass is compatible with deep inelastic scattering for $0<x<1$. Using $P-q$ with the same positive-$x$ definition instead gives $M^2-Q^2(1+1/x)$, which becomes negative in the Bjorken scaling limit and has no physical hadronic spectral support. Reversing the photon convention requires reversing the associated scalar-product and scaling-variable conventions too.

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.

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.

The two scalar coefficients in the unpolarized electromagnetic hadronic tensor. Extra polarization or parity-violating currents can require further coefficients.

In the massless-target convention, the Callan-Gross relation sets this coefficient to zero at leading order in the parton model.