  We compute the spin-dependent and spin-independent structure functions of the
forward virtual-photon Compton tensor of the proton at one loop using heavy
baryon effective theory including the Delta particle. We compare with previous
results when existing. Using these results we obtain the leading hadronic
contributions, associated to the pion and Delta particles, to the Wilson
coefficients of the lepton-proton four fermion operators in NRQED. The
spin-independent coefficient yields a pure prediction for the two-photon
exchange contribution to the muonic hydrogen Lamb shift, $\Delta E_{\rm
TPE}(\pi\&\Delta)=34(13)$ $\mu$eV. We also compute the charge, $\langle r^n
\rangle$, and Zemach, $\langle r^n \rangle_{(2)}$, moments for $n \geq 3$.
Finally, we discuss the spin-dependent case, for which we compute the
difference between the four-fermion Wilson coefficients relevant for hydrogen
and muonic hydrogen.
