More corrections to the sixth-order anomalous magnetic moment of the muon

The contribution to the sixth-order muon anomaly from second-order electron vacuum polarization is determined analytically to orderm _e/m _μ. The result, including the contributions from graphs containing proper and improper fourth-order electron vacuum polarization subgraphs, is $$begin{gathered} left( {frac{alpha }{pi }} right)^3 left{ {frac{2}{9}log ^2 } right.frac{{m_mu }}{{m_e }} + left[ {frac{{31}}{{27}}} right. + frac{{pi ^2 }}{9} - frac{2}{3}pi ^2 log 2 hfill left. { + zeta left( 3 right)} right]log frac{{m_mu }}{{m_e }} + left[ {frac{{1075}}{{216}}} right. - frac{{25}}{{18}}pi ^2 + frac{{5pi ^2 }}{3}log 2 hfill left. { - 3zeta left( 3 right) + frac{{11}}{{216}}pi ^4 - frac{2}{9}pi ^2 log ^2 2 - frac{1}{9}log^4 2 - frac{8}{3}a_4 } right] hfill + left[ {frac{{3199}}{{1080}}pi ^2 - frac{{16}}{9}pi ^2 log 2 - frac{{13}}{8}pi ^3 } right]left. {frac{{m_e }}{{m_mu }}} right} hfill end{gathered} $$ . To obtain the total sixth-order contribution toa _μ?a _e, one must add the light-by-light contribution to the above expression.