$L_{\mu }-L_{\tau }$ gauge-boson production from lepton flavor violating $\tau$ decays at Belle II

$L_{\mu }-L_{\tau }$ gauge boson ($Z^{\prime }$) with a mass in the MeV to GeV region can resolve not only the muon $g-2$ excess, but also the gap in the high-energy cosmic neutrino spectrum at IceCube. It was recently proposed that such a light gauge boson can be detected during the Belle II experiment with a luminosity of $50{\mathrm{ab}}^{-1}$ by the $e^{+}{e}^{-}\to \gamma +\cancel{E}$ process through the kinetic mixing with the photon, where the missing energy $\cancel{E}$ is from the $Z^{\prime }\to \overline{\nu }\nu$ decays. We study the phenomenological implications when a pair of singlet vectorlike leptons carrying different $L_{\mu }-L_{\tau }$ charges are included, and a complex singlet scalar (${\varphi }_S$) is introduced to accomplish the spontaneous $U(1{)}_{L_{\mu }-L_{\tau }}$ symmetry breaking. It is found that the extension leads to several phenomena of interest, including (i) branching ratio (BR) for $h\to \mu \tau$ can be of the order of ${10}^{-3}$; (ii) ${\varphi }_S$-mediated muon $g-2$ can be of the order of $10\times {10}^{-10}$; (iii) BR for $\tau \to \mu {\varphi }_S^{*}\to \mu Z^{\prime }{Z}^{\prime }$ can be ${10}^{-8}$, and (iv) kinetic mixing between the $Z^{\prime }$ boson and photon is sensitive to the relative heavy lepton masses. The predicted BRs for $\tau \to (3\mu +\cancel{E},5\mu )$ through the leptonic $Z^{\prime }$ decays can reach a level of ${10}^{-9}$, in which the results fall within the sensitivity of the Belle II in the search for the rare tau decays.

Figure 1

Sketched Feynman diagram for the Higgs- and ${\varphi }_S$-mediated muon $g-2$.

Figure 2

Contours for $\mathrm{BR}(h\to \mu \tau )$ in units of ${10}^{-3}$ (dashed) and $\mathrm{\Delta }{a}_{\mu }^{h+{\varphi }_S}$ in units of ${10}^{-10}$ (solid) as a function of $a_R$ and $a_L$, where $(v_S,m_S)=(80,8)\mathrm{GeV}$ in plot (a) and $(v_S,m_S)=(140,2)\mathrm{GeV}$ in plot (b) are taken.

Figure 3

Contours for $\mathrm{\Delta }{a}_{\mu }^{{\varphi }_S}$ (in units of ${10}^{-10}$) as a function of $\mathrm{BR}(h\to \mu \tau )$ and $v_S$, where $m_S=8\mathrm{GeV}$ and $a_L=a_R$ are used in plot (a), $m_S=2\mathrm{GeV}$ and $a_L=-a_R$ are used in plot (b), and the dashed line is the upper limit from CMS [37].

Figure 4

Contours for $\mathrm{\Delta }{a}_{\mu }$ with (a) $m_S=8\mathrm{GeV}$ and $a_L=a_R$ and (b) $m_S=2\mathrm{GeV}$ and $a_L=-a_R$ as a function of $m_{Z^{\prime }}$ and $g_{Z^{\prime }}$, where the results bounded by the dot-dashed, dashed, and solid lines denote the $Z^{\prime }$, ${\varphi }_S$, and $Z^{\prime }+{\varphi }_S$ contributions, respectively.

Figure 5

Contours for $\mathrm{BR}(\tau \to \mu Z^{\prime }{Z}^{\prime })$ in units of ${10}^{-8}$ (dot-dashed) and $\mathrm{\Delta }{a}_{\mu }^{{\varphi }_S}$ in units of ${10}^{-10}$ (solid) with $m_{Z^{\prime }}=0.25\mathrm{GeV}$ as a function of $\mathrm{BR}(h\to \mu \tau )$ and $v_S$, where plot (a) denotes $m_S=8\mathrm{GeV}$ and $a_L=a_R$, and the plot (b) is $m_S=2\mathrm{GeV}$ and $a_L=-a_R$.

Figure 6

Sketched Feynman diagram for the kinetic mixing of $\gamma$ and $Z^{\prime }$, where the leptons inside the loop include ${\ell }^{\prime }=\mu ,\tau ,{\tau }^{\prime }$, and ${\tau }^{\prime \prime }$.

Figure 7

Left panel: $\epsilon$ for $m_{L2}>m_{L1}$ in Eq. (36) as a function of $E_{\gamma }$, where $E_{\gamma }=(s-q^2)/(2\sqrt{s})$, and $\sqrt{s}=10.58\mathrm{GeV}$ and $m_{L1}=0.7\mathrm{TeV}$ are used; the solid, dashed, dotted, and dot-dashed lines denote the results of $m_{L2}=(0.7,0.9,1.1,1.5)\mathrm{TeV}$, respectively. The horizontal lines are the same situations but for the case of ${\epsilon }_{\nu e}=\mathrm{\Pi }(0)$. The contributions of $m_{L1}=m_{L2}=0.7\mathrm{TeV}$ are the same as for those without vectorlike leptons. Right panel: the legend is the same as that in the left panel, but for $m_{L1}>m_{L2}$.