The three-loop QED contributions to the $g-2$ of charged leptons with two internal fermion loops and a class of Kamp\'e de F\'eriet series

Understand

The three-loop QED mass-dependent contributions to the $g-2$ of each of the charged leptons with two internal closed fermion loops, sometimes called $A^{(6)}_3\left(\frac{m_1}{m_2}, \frac{m_1}{m_3}\right)$ in the $g-2$ literature, is revisited using the Mellin-Barnes (MB) representation technique.

• Results for the muon and $\tau$ lepton anomalous magnetic moments $A^{(6)}_{3,\mu}$ and $A^{(6)}_{3,\tau}$, which were known as series expansions in the lepton mass ratios up to the first few terms only, are extended to their exact expressions.
• The contribution to the anomalous magnetic moment of the electron $A^{(6)}_{3,e}$ is also explicitly given in closed form.
• In addition to this, we show that the different series representations derived from the MB representation collectively converge for all possible values of the masses.