Hyperfine structure in the H$_2^+$ and HD$^+$ molecular ions at $mα^6$ order

Abstract

A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M ) in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest. I. INTRODUCTIONHigh-resolution spectroscopy of the hydrogen molecular ions H + 2 and HD + may contribute significantly to the determination of fundamental constants such as the proton-electron mass ratio m p /m e [1]. A pure rotational transition in HD + has recently been measured with a relative uncertainty of 1.3 × 10 −11 [2]. The experimental accuracy of ro-vibrational transition frequencies is expected to reach a few parts per trillion in the near future using spectroscopy in the Lamb-Dicke regime [2][3][4] or in a Doppler-free geometry [5,6]. While information on fundamental constants is obtained from comparison of spin-averaged transition frequencies with theoretical predictions, the hyperfine splitting of ro-vibrational lines also allows for precise tests of theory.So far, the hyperfine structure of H + 2 and HD + has been calculated within the Breit-Pauli approximation [7, 8], taking into account the anomalous magnetic moment of the electron. All terms at orders mα 4 and mα 5 are included, so that the theoretical accuracy of the hyperfine coefficients is of order α 2 ∼ 5 × 10 −5 . Higherorder corrections to the largest coefficients, i.e. the spin-spin Fermi contact interaction, were later calculated in [9,10], which allowed to get excellent agreement with available RF spectroscopy data in H + 2 [11] at the level of ∼ 1 ppm. The following step to improve the hyperfine structure theory is to evaluate higher-order corrections to the next largest coefficients, i.e. the electron spin-orbit and spin-spin tensor interaction, starting with relativistic corrections at the mα 6 order.With this aim, we derive in the present work the complete effective Hamiltonian for the hydrogen molecular ions at the mα 6 and mα 6 (m/M ) orders, following the NRQED approach [12][13][14]. Then, we use it to calculate numerically the corrections to the electron spin-orbit interaction for a few transitions studied in ongoing experiments. The paper is organized as follows: in Secs. II and III, we recall the expression of the NRQED Lagrangian and associated interaction vertices. We then systematically derive the effective potentials, which are organized in three categories: tree-level interactions involving the exchange of a Coulomb or transverse photon (Sec. IV), terms due to retardation in the transverse photon exchange (Sec. V), and finally those coming from a seagull diagram with simul...