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We propose new experiments with high sensitivity to a possible spatial ortemporal variation of the electron-to-proton mass ratio $\mu \equiv m_e/m_p$.We consider a nearly-degenerate pair of molecular vibrational levels, whereeach state is associated with a different electronic potential. The change inthe splitting between such levels, with respect to a change in $\mu$, can belarge both on an absolute scale and relative to the splitting. We demonstratethe existence of such pairs of levels in Cs$_2$. The narrow spectral linesachievable with ultracold Cs$_2$ in these long-lived levels make this systempromising for future searches for small variations in $\mu$.

Enhanced sensitivity to variation of $m_{e}/m_{p}$ in molecular spectra