Vacuum polarization

Photon vacuum polarization is the one-particle-irreducible photon two-point function generated by charged-particle loops.

For one unit-charge Dirac field in $d=4-2ϵ$, define the insertion as $i{\Pi }_{\mu \nu }=i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu })\Pi$. The one-loop ultraviolet pole is ${\Pi }_{\mathrm{UV}}=-e^2/(12{\pi }^2ϵ)$. The Ward identity cancels any apparent divergent vector mass term. With photon counterterm insertion $-i(q^2{g}_{\mu \nu }-q_{\mu }{q}_{\nu }){\delta }_3$, this gives ${\delta }_3=-e^2/(12{\pi }^2ϵ)$ in the minimal subtraction scheme.

For $J_h^{\mu }={\sum }_f{Q}_f{\bar{q}}_f{\gamma }^{\mu }{q}_f$ without a factor of $e$, use $i\int e^{iqx}⟨T{J}_h^{\mu }(x)J_h^{\nu }(0)⟩d^{4}x=(q^{\mu }{q}^{\nu }-q^2{g}^{\mu \nu }){\Pi }_J(q^2)$. The inclusive tensor is $W_J^{\mu \nu }=2\pi {\rho }_J(s)(q^{\mu }{q}^{\nu }-s{g}^{\mu \nu })$. A massless free-quark calculation gives ${\rho }_J=N_c{\sum }_f{Q}_f^2/(12{\pi }^2)$; strong interactions alter this function.