Hadronic electromagnetic-current spectral density
= Hadronic electromagnetic-current spectral density
{title2=$\rho_J(s)=\operatorname{Im}\Pi_J(s)/\pi$}
For $J_h^\mu=\sum_fQ_f\bar q_f\gamma^\mu q_f$ without a factor of $e$, use $i\int e^{iqx}\langle T J_h^\mu(x)J_h^\nu(0)\rangle d^4x=(q^\mu q^\nu-q^2g^{\mu\nu})\Pi_J(q^2)$. The inclusive tensor is $W_J^{\mu\nu}=2\pi\rho_J(s)(q^\mu q^\nu-sg^{\mu\nu})$. A massless free-quark calculation gives $\rho_J=N_c\sum_fQ_f^2/(12\pi^2)$; strong interactions alter this function.