= Solution
For an unpolarized electromagnetic target, <Lorentz covariance> and <parity symmetry in quantum field theory> permit a symmetric <hadronic tensor> built from $g^{\mu\nu}$, $P^\mu$ and $q^\mu$, where $P=P_H$. Its surviving general form is
$$
W^{\mu\nu}=a g^{\mu\nu}+bP^\mu P^\nu+cq^\mu q^\nu+d(P^\mu q^\nu+q^\mu P^\nu).
$$
The <totally antisymmetric tensor> structure $\epsilon^{\mu\nu\rho\sigma}P_\rho q_\sigma$ is excluded by electromagnetic <parity symmetry in quantum field theory> for this unpolarized target. The other possible antisymmetric structure, $P^\mu q^\nu-q^\mu P^\nu$, cannot be transverse for generic scattering kinematics and hence is excluded by <current conservation>. <Current conservation> gives the <Ward identities> $q_\mu W^{\mu\nu}=q_\nu W^{\mu\nu}=0$, imposing
$$
b\nu+dq^2=0,\qquad a+cq^2+d\nu=0,\qquad \nu=P\cdot q.
$$
Two scalar functions remain. Taking $\widetilde P^\mu=P^\mu-\nu q^\mu/q^2$ gives the transverse basis
$$
\boxed{W_H^{\mu\nu}=\left(-g^{\mu\nu}+\frac{q^\mu q^\nu}{q^2}\right)F_1(x,q^2)+\widetilde P^\mu\widetilde P^\nu\frac{F_2(x,q^2)}{\nu},\qquad x=-\frac{q^2}{2\nu}.}
$$
These are the <deep-inelastic structure functions>; $x$ is <Bjorken x>. At fixed target mass, the two independent <Lorentz scalar> invariants can equivalently be chosen as $x,q^2$. In this paper $\nu=P\cdot q$ has dimensions of mass squared, rather than the alternative convention $\nu=P\cdot q/M$ used for energy transfer.
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