Hyperfine Splitting in True Muonium to $\mathcal{O}(m_\mu\alpha^6)$: Two Photon Annihilation Contribution from Other Flavors

TL;DR

This paper analytically calculates the two-photon-annihilation contribution to true muonium hyperfine splitting from electron and tau loops at order $m_bcb1^6$, providing precise results for energy shifts and decay rates.

Contribution

It presents the first analytical computation of the $e$ and $ au$ loop contributions to true muonium hyperfine splitting at order $m_bcb1^6$, including decay rate calculations.

Findings

Hyperfine splitting contribution: -793.93 MHz for n=1.

Decay rate contribution: 3840.74 MHz for n=1.

Results extended to other leptonic bound states.

Abstract

The two-photon-annihilation contribution to the true muonium hyperfine splitting arising from $e$ and $\tau$ loops is obtained analytically at order $m_\mu\alpha^6$. The contribution to the hyperfine splitting is $-2.031092873 m_\mu\alpha^6/n^3\pi^2=-793.926988/n^3$ MHz. The contribution to the triplet true muonium decay rate has also been obtained and was found to be $9.825708266 m_\mu\alpha^6/n^3\pi^2=3840.737698/n^3$ MHz. Additional results have been computed for other purely leptonic bound states.