Bounds on fifth forces from precision measurements on molecules

Highly accurate results from frequency measurements on neutral hydrogen molecules ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$ as well as the ${\mathrm{HD}}^{+}$ ion can be interpreted in terms of constraints on possible fifth-force interactions. Where the hydrogen atom is a probe for yet unknown lepton-hadron interactions, and the helium atom is sensitive for lepton-lepton interactions, molecules open the domain to search for additional long-range hadron-hadron forces. First principles calculations in the framework of quantum electrodynamics have now advanced to the level that hydrogen molecules and hydrogen molecular ions have become calculable systems, making them a search ground for fifth forces. Following a phenomenological treatment of unknown hadron-hadron interactions written in terms of a Yukawa potential of the form $V_5(r)=\beta \exp (-r/\lambda )/r$, current precision measurements on hydrogenic molecules yield a constraint $\beta <1.4\times {10}^{-8}\mathrm{eV}·\AA$ for long-range hadron-hadron interactions at typical force ranges commensurate with separations of a chemical bond, i.e., $\lambda \approx 1\AA$ and beyond. This corresponds to a constraint of $\beta /\alpha <{10}^{-9}$, where $\alpha$ represents the strength of the electromagnetic interaction, i.e., the fine-structure constant.

Figure 1

Calculated difference of the expectation values $\Delta Y_{\lambda }$ for different values of $\lambda$ for several ($v^{\prime }$, $v^{\prime \prime }=0$) vibrational energy separations in ${\mathrm{H}}_2$, ${\mathrm{D}}_2$, HD, and ${\mathrm{HD}}^{+}$ molecules.

Figure 2

Calculated expectation value $Y_{\lambda ,D_0}$ of a fifth-force contribution to the ${\mathrm{H}}_2$ dissociation limit $D_0$ for a range of $\lambda$ values. For reference the contributions $\Delta Y_{\lambda }$ for the ($v^{\prime }=1$, $v^{\prime \prime }=0$) band of ${\mathrm{H}}_2$ and the ($v^{\prime }=8$, $v^{\prime \prime }=0$) band of ${\mathrm{HD}}^{+}$ electronic ground state are plotted as well.

Figure 3

Limits on the strength of the coupling constant $\beta$ relative to the electromagnetic coupling constant $\alpha$ as a function of the interaction range $\lambda$. The alternate axis (right) expresses $|\beta |\hslash c$ in units of $\mathrm{eV}·\AA$, while the top axis indicates the mass $m_Y$ of the force-carrying particle corresponding to the range $\lambda$.