Hyperfine structure of the excited state $1s_{1/2}^{\left(e\right)}2s_{1/2}^{\left(\mu \right)}$ of the muonic helium atom

The recoil, vacuum polarization, and electron vertex corrections of first and second orders in the fine structure constant $\alpha$ and the ratio of electron to muon and electron to $\alpha$-particle masses are calculated for the hyperfine splitting of the $1s_{1/2}^{\left(e\right)}2s_{1/2}^{\left(\mu \right)}$ state of muonic helium atom $\left(\mu e_2^4\text{He}\right)$ on the basis of perturbation theory. We obtain a total result for the muonically excited state hyperfine splitting $\Delta {\nu }^{\text{hfs}}=4295.66$ MHz, which improves previous calculations, taking additional corrections into account, and a more accurate treatment of the electron vertex contribution.

Figure 1

Two photon-exchange amplitudes in the electron-muon hyperfine interaction.

Figure 2

The electron vertex corrections. The dashed line represents the Coulomb photon. The wavy line represents the hyperfine part of the Breit potential. $\tilde{G}$ is the reduced Coulomb Green's function.

Figure 3

Vacuum polarization effects. The dashed line represents the Coulomb photon. The wavy line represents the hyperfine part of the Breit potential. $\tilde{G}$ is the reduced Coulomb Green's function.

Figure 4

Vacuum polarization effects in second-order PT. The dashed line represents the first part of the potential $\Delta H$ (3). The wavy line represents the hyperfine part of the Breit potential.

Figure 5

Nuclear structure effects in second-order PT. The bold point represents the nuclear vertex operator. The wavy line represents the hyperfine part of the Breit potential.