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

Abstract

Metastable pionic helium ($\ensuremath{\pi}{\mathrm{He}}^{+}$) is a three-body atom composed of a helium nucleus, an electron occupying the 1$s$ ground state, and a negatively charged pion ${\ensuremath{\pi}}^{\ensuremath{-}}$ in a Rydberg state with principal and orbital angular momentum quantum numbers of $n\ensuremath{\sim}\ensuremath{\ell}+1\ensuremath{\sim}16$. We calculate the spin-independent energies of the $\ensuremath{\pi}\phantom{\rule{0.16em}{0ex}}{{}^{3}\mathrm{He}}^{+}$ and $\ensuremath{\pi}\phantom{\rule{0.16em}{0ex}}{{}^{4}\mathrm{He}}^{+}$ isotopes in the region $n=15$--19. These include relativistic and quantum electrodynamics corrections of orders ${R}_{\ensuremath{\infty}}{\ensuremath{\alpha}}^{2}$ and ${R}_{\ensuremath{\infty}}{\ensuremath{\alpha}}^{3}$ in atomic units, where ${R}_{\ensuremath{\infty}}$ and $\ensuremath{\alpha}$ 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 ${\ensuremath{\pi}}^{\ensuremath{-}}$ and the radiative and Auger decay rates of the states are also calculated. Some states $(n,\ensuremath{\ell})=(16,15)$ and $(17,16)$ retain nanosecond-scale lifetimes against ${\ensuremath{\pi}}^{\ensuremath{-}}$ absorption into the helium nucleus. We propose the use of laser pulses to induce ${\ensuremath{\pi}}^{\ensuremath{-}}$ transitions from these metastable states to states with large ($\ensuremath{\sim}{10}^{11}$ s${}^{\ensuremath{-}1}$) Auger rates. The $\ensuremath{\pi}{\mathrm{He}}^{2+}$ ion that remains after Auger emission of the 1$s$ 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 ${\ensuremath{\pi}}^{\ensuremath{-}}$. 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. A resonance curve is obtained from which the $\ensuremath{\pi}{\mathrm{He}}^{+}$ transition frequency can in principle be determined with a fractional precision of ${10}^{\ensuremath{-}8}--{10}^{\ensuremath{-}6}$ provided the systematic uncertainties can be controlled. By comparing the measured $\ensuremath{\pi}{\mathrm{He}}^{+}$ frequencies with the calculated values, the ${\ensuremath{\pi}}^{\ensuremath{-}}$ mass may be determined with a similar precision. The $\ensuremath{\pi}{\mathrm{He}}^{+}$ will be synthesized by allowing a high-intensity ($>{10}^{8}$ s${}^{\ensuremath{-}1}$) beam of ${\ensuremath{\pi}}^{\ensuremath{-}}$produced by a cyclotron to come to rest in a helium target. The precise time structure of the ${\ensuremath{\pi}}^{\ensuremath{-}}$ beam is used to ensure a sufficient rate of coincidence between the resonant laser pulses and the $\ensuremath{\pi}{\mathrm{He}}^{+}$ atoms.