Constraints on $L_{\mu }-L_{\tau }$ gauge interactions from rare kaon decay

A model with $L_{\mu }-L_{\tau }$ gauge symmetry is the least constrained model as a resolution to the disagreement of the muon anomalous magnetic moment between the theoretical predictions and the experimental results. In this paper, we discuss how well the $L_{\mu }-L_{\tau }$ model can be constrained by looking for decays of the charged kaon associated with a $L_{\mu }-L_{\tau }$ gauge boson. More concretely, we consider searches for single muon tracks from the decays of stopped charged kaon as in the E949 experiment. In our conservative estimation, we find that the favored parameter region for the muon anomalous magnetic moment can be tested by using a 10 times larger number of the stopped charged kaons and about a 100 times better photon rejection rate than the E949 experiment.

Figure 1

The parameter region which explains the muon anomalous magnetic moment within the $1\sigma$ (red) and the $2\sigma$ (pink) ranges. The dark blue, gray and cyan shaded regions are excluded by the neutrino trident production experiments [38], neutrino-electron scattering experiments [12, 13] and ${\mu }^{+}{\mu }^{-}$ pair production searches associated with $Z^{\prime }$ decaying into ${\mu }^{+}{\mu }^{-}$ [39] (Similar figure is given in [40]), respectively.

Figure 2

Left: The branching ratio of $K^{+}$ into $K^{+}\to {\mu }^{+}+{\nu }_{\mu }+Z^{\prime }$. Right: The differential branching ratio normalized by $g_{Z^{\prime }}^{-2}$ for given $Z^{\prime }$ masses.

Figure 3

Left: The single-muon acceptance in the E949 experiment after all the cuts are applied [42]. The band corresponds to the $1\sigma$ uncertainties of the single-muon acceptance. Right: The signal spectrum expected at the E949 experiment where the muon acceptance is assumed to be the lower one in the left panel. We also show the observed event numbers at the E949 experiment after tight photon veto cuts [42].

Figure 4

The 95% CL limit from the E949 experiment (the orange shaded region) overlaid on Fig. 1. We also show possible the improved sensitivities by assuming several setups as indicated in the figure. The green band around the expected exclusion limit for $N_K=10\times N_K^{\mathrm{E}949}$ and ${\epsilon }_{\gamma }={10}^{-2}\times {\epsilon }_{\gamma }^{\mathrm{E}949}$ corresponds to the $2\sigma$ statistical fluctuation of the expected exclusion limit.

Figure 5

Left: The fitted spectrum used to generate mock sample data sets (red line). One example of the mock data is also shown by red points. Right: The statistical fluctuations of the expected exclusion limits at 95% CL for various assumptions. The yellow shaded bands and the green bands correspond to the $1\sigma$ and the $2\sigma$ statistical fluctuations, respectively. The dashed lines show the mean expected exclusion limits. The black line is the observed exclusion limit obtained in the main text by using the E949 data.