Non-Coulombic corrections

At the level of accuracy required (less than about 1 meV), the effects beyond the Coulombic, nonrelativistic point-like description of the muonic molecules are important. Because of the heavy mass of the muon, muonic hydrogen atoms and molecules are sensitive to effects such as QED vacuum polarization and the nuclear charge form factor as well as hyperfine effects. However, it is the difference between the corrections to the atomic and molecular energy levels which affects the binding energy $\epsilon _{Jv}$,

$$\Delta \epsilon _{Jv} = \Delta E_{Jv}^{mol} - \Delta E_{1s}^{atom}.$$ (37)

Because the loosely bound $(d\mu t)_{11}$ states resemble a $\mu t$ atom and a weakly bound deuteron orbiting around it, the corrections $\Delta E_{Jv}^{mol}$ and $\Delta E_{1s}^{atom}$ are similar and partly cancel each other, hence the corresponding corrections $\Delta \epsilon _{Jv}$ are suppressed by a factor of about $\sim 10^{2}$ [132]. These corrections (without hyperfine effects) amount to about a -30 meV shift (towards deeper binding) in the $d\mu t$binding energy $\epsilon _{11}$ [4].

Hyperfine structure of the $\mu t$ and $d\mu t$ further introduces corrections for F=0 of +35.2 meV (S=0), and for F=1 of -8.3 meV (S=0), -14.9 meV (S=1), and -8.5 meV (S=2), where F is the total spin of $\mu t$ and S, that of $d\mu t$.