Applying Bayesian inference to Galileon solutions of the muon problem

We derive corrections to atomic energy levels from disformal couplings in Galileon theories. Through Bayesian inference, we constrain the cutoff radii and Galileon scale via these corrections. To connect different atomic systems, we assume the various cutoff radii related by a one-parameter family of solutions. This introduces a new parameter $\alpha$ which is also constrained. In this model, we predict shifts to muonic helium of $\delta E_{{\mathrm{He}}^3}=1.97_{-1.87}^{+9.28}\mathrm{meV}$ and $\delta E_{{\mathrm{He}}^4}=1.69_{-1.61}^{+9.25}\mathrm{meV}$ as well as for true muonium, $\delta E_{\mathrm{TM}}=0.06_{-0.05}^{+0.46}\mathrm{meV}$.

Figure 1

Selected limits for $M$ as a function of $r_G$ with $\alpha =0$. The solid lines correspond to $1\sigma$ lower bounds, while the dashed lines are the mean values of the discrepancies in muonic hydrogen and muonic deuterium.

Figure 2

1D and 2D marginal PDFs for $\log (M)$, $\log (r_G)$, and $\alpha$ produced using the Galileon contributions from Table 1 to the transitions found in Tables 2, 3, 4, 5. $M$ is in units of MeV and $r_G$ is in units of m. Additionally plotted are the predictions for the $2s-2p$ Lamb shifts in ${\mu }^{-}{\mathrm{He}}^3$, ${\mu }^{-}{\mathrm{He}}^4$, and ${\mu }^{-}{\mu }^{+}$. In the 1D plots, the dashed lines correspond to the mean, $1\sigma$ credible regions. In the 2D plots, the contour regions are the $1\sigma$ and $2\sigma$ credible regions.