Proposed method for laser spectroscopy of pionic helium atoms to determine the charged-pion mass

Metastable pionic helium ($\pi {\mathrm{H}\mathrm{e}}^{+}$) is a three-body atom composed of a helium nucleus, an electron occupying the $1s$ ground state, and a negatively charged pion ${\pi }^{-}$ in a Rydberg state with principal- and orbital angular momentum quantum numbers of $n\sim \mathrm{\ell }+1\sim 16$. We calculate the spin-independent energies of the $\pi {{}^3\mathrm{H}\mathrm{e}}^{+}$ and $\pi {{}^4\mathrm{H}\mathrm{e}}^{+}$ isotopes in the region $n=15$–19. These include relativistic and quantum electrodynamics corrections of orders $R_{\mathrm{\infty }}{\alpha }^2$ and $R_{\mathrm{\infty }}{\alpha }^3$ in atomic units, where $R_{\mathrm{\infty }}$ and $\alpha$ denote the Rydberg and fine structure constants. The fine-structure splitting due to the coupling between the electron spin and the orbital angular momentum of the ${\pi }^{-}$, and the radiative and Auger decay rates of the states are also calculated. Some states $(n,\mathrm{\ell })=(16,15)$ and $(17,16)$ retain nanosecond-scale lifetimes against ${\pi }^{-}$ absorption into the helium nucleus. We propose to use laser pulses to induce ${\pi }^{-}$ transitions from these metastable states, to states with large ($\sim {10}^{11}$ s${}^{-1}$) Auger rates. The $\pi {\mathrm{H}\mathrm{e}}^{2+}$ ion that remains after Auger emission of the $1s$ electron undergoes Stark mixing with the $s$, $p$, and $d$ states during collisions with the helium atoms in the experimental target. This leads to immediate nuclear absorption of the ${\pi }^{-}$. The resonance condition between the laser beam and the atom is thus revealed as a sharp spike in the rates of neutrons, protons, deuterons, and tritons that emerge…(continued)