A new search for the variation of fundamental constants using the rovibrational levels and isotope effects of the magnesium fluoride molecule

The recently demonstrated methods for cooling and trapping diatomic molecules offer new possibilities for precision searches in fundamental physical theories. Here, we propose to study the variations of the fine-structure constant (α = e²/(ℏc)) and the proton-to-electron mass ratio (μ = m_p/m_e) with time by taking advantage of the nearly degenerate rovibrational levels in the electronic states of the magnesium fluoride (MgF) molecule. Specifically, due to the cancellation between the fine-structure splitting and the rovibrational intervals in the different MgF natural isotopes, a degeneracy occurs for A² Π_3/2 (v′ = 0, J′ = 18.5, −) and A²Π_1/2 (v″ = 0, J″ = 20.5, −). We find that using the nearly degenerate energy level of such states can be 10⁴ times more sensitive than using a pure rotational transition to measure the variations of α and μ. To quantify the small gap between A²Π_3/2 (v′ = 0, J′ = 18.5, −) and A² Π_1/2 (v″ = 0, J″ = 20.5, −), special transitions of choice are feasible: ${\mathrm{X}}^2{\Sigma }_{1/2}^{+}$ (v = 0, J = 19.5, +) to A²Π_3/2 (v′ = 0, J′ = 18.5, −) and ${\mathrm{X}}^2{\Sigma }_{1/2}^{+}$ (v = 0, J = 19.5, +) to A²Π_1/2 (v″ = 0, J″ = 20.5, −). In addition, we estimate the frequency uncertainties caused by the narrow linewidth, Zeeman shift, Stark shift, Doppler broadening and blackbody radiation.