Relative magnitudes of electron versus proton spin‑independent SME couplings

Determine the relative magnitudes of the spin‑independent Standard Model Extension tensor couplings E^e_{ij} and E^p_{ij} for electrons and protons in the non‑relativistic SME Hamiltonian, specifically those entering the electron Schrödinger equation for the hydrogen molecular ion H2+ and its antimatter counterpart via the suppressed proton term proportional to (m_e^2/m_p^2) E^p_{ij}, in order to assess whether this proton contribution can be neglected or must be retained when analyzing rovibrational spectra and deriving bounds on Lorentz and CPT violation.

Background

In the Born–Oppenheimer framework extended to include Lorentz and CPT violation, the spin‑independent SME tensors E^e_{ij} (electron sector) and E^p_{ij} (proton sector) contribute both to the electron and nucleon Schrödinger equations. The electron equation acquires a term involving E^e_{ij} plus a proton‑sector contribution suppressed by the mass ratio m_e^2/m_p^2.

Although the proton term appears numerically suppressed, the authors caution that the relative sizes of SME couplings across particle species are not known a priori, which impacts whether the proton contribution in the electron equation can be dropped. Establishing these relative magnitudes is necessary for interpreting constraints from rovibrational spectroscopy and for deciding which SME contributions are phenomenologically relevant in H2+ and anti‑H2−.