Sixth-Order Contribution toZ3in Finite Quantum Electrodynamics

If internal photon propagators in the vacuum polarization tensor are replaced by $\frac{1}{q^2}$, the divergent part of ${Z_3}^{-1\mathrm{}}$ in quantum electrodynamics is ${\left({Z_3}^{-1\mathrm{}}\right)}_{\mathrm{div}}=\frac{{\alpha }_0}{2\pi }\left[\frac{2}{3}+\frac{{\alpha }_0}{2\pi }-\frac{1}{4}{\left(\frac{{\alpha }_0}{2\pi }\right)}^2\right]\ln \frac{M^2}{m^2}$ to sixth order. The simple nature and negative sign of the last term encourage the search for a closed form for ${\left({Z_3}^{-1\mathrm{}}\right)}_{\mathrm{div}}$ which vanishes for ${\alpha }_0>0\mathrm{}$.