$\tau$- and $\mu$-physics in a general two Higgs doublet model with $\mu -\tau$ flavor violation

Motivated by the recent CMS excess in a flavor violating Higgs decay $h\to \mu \tau$ as well as the anomaly of muon anomalous magnetic moment (muon $g-2$), we consider a scenario where both the excess in $h\to \mu \tau$ and the anomaly of muon $g-2$ are explained by the $\mu -\tau$ flavor violation in a general two Higgs doublet model. We study various processes involving $\mu$ and $\tau$, and then discuss the typical predictions and constraints in this scenario. Especially, we find that the prediction of $\tau \to \mu \gamma$ can be within the reach of the Belle II experiment. We also show that the lepton nonuniversality between $\tau \to \mu \nu \overline{\nu }$ and $\tau \to e\nu \overline{\nu }$ can be sizable, and hence the analysis of the current Belle data and the future experimental improvement would have an impact on this model. Besides, processes such as $\tau \to \mu l^{+}{l}^{-}$ $(l=e,\mu )$, $\tau \to \mu \eta$, $\mu \to e\gamma$, $\mu \to 3e$, and the muon electric dipole moment can be accessible, depending on the unknown Yukawa couplings. On the other hand, the processes like $\tau \to e\gamma$ and $\tau \to e{l}^{+}{l}^{-}$ $(l=e,\mu )$ could not be sizable to observe because of the current strong constraints on the $e-\mu$ and $e-\tau$ flavor violations. Then we also conclude that contrary to the $h\to \mu \tau$ decay mode, the lepton flavor violating Higgs boson decay modes $h\to e\mu$ and $h\to e\tau$ are strongly suppressed, and hence it will be difficult to observe these modes at the LHC experiment.

Figure 1

A Feynman diagram for neutral Higgs boson contributions to the muon $g-2$. A photon is attached somewhere in the charged lepton line.

Figure 2

Numerical result for $\delta a_{\mu }$ as a function of $c_{\beta \alpha }$ and $\mathrm{BR}(h\to \mu \tau )$ for $m_A=250\mathrm{GeV}$ (upper figure) and 350 GeV (lower figure). Regions where the muon $g-2$ anomaly in Eq. (13) is explained within $\pm 1\sigma$, $\pm 2\sigma$, and $\pm 3\sigma$ are shown. Here we determine the mass spectrum of heavy Higgs bosons assuming ${\lambda }_4={\lambda }_5=0.5$ in Eq. (10). We have assumed ${\rho }_e^{\mu \tau }{\rho }_e^{\tau \mu }<0$ with ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$ to obtain the positive contribution to $\delta a_{\mu }$.

Figure 3

Numerical result for $\delta a_{\mu }$ as a function of $m_A$ and $c_{\beta \alpha }$ assuming $\mathrm{BR}(h\to \mu \tau )=0.84\%$. Regions that explain the muon $g-2$ anomaly in Eq. (13) within $\pm 1\sigma$, $\pm 2\sigma$, and $\pm 3\sigma$ are shown. Here we determine the mass spectrum of heavy Higgs bosons as a function of $m_A$ assuming ${\lambda }_4={\lambda }_5=0.5$ in Eq. (10). We have assumed ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$ to obtain the positive contribution to $\delta a_{\mu }$, and the Yukawa couplings ${\rho }_e^{\mu \tau (\tau \mu )}$ are fixed to realize $\mathrm{BR}(h\to \mu \tau )=0.84\%$.

Figure 4

Numerical result for $\delta a_{\mu }$ as a function of $m_A$ and ${\lambda }_5$ assuming $\mathrm{BR}(h\to \mu \tau )=0.84\%$ with the fixed $c_{\beta \alpha }(=-0.008)$. Here we have assumed ${\lambda }_4={\lambda }_5$ and ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$.

Figure 5

Some of the Feynman diagrams that contribute to $\tau \to \mu \gamma$ processes at the one-loop level (left figure) which are induced by $y_{\varphi \tau \tau }^e(\varphi =h,H,A)$ Yukawa couplings and at the two-loop level (right figure) which are Barr-Zee–type contributions and induced by the third generation fermions via $y_{\varphi 33}^f(f=u,d,e)$ Yukawa couplings. Diagrams where the fermion chiralities are flipped also contribute. We also have a Barr-Zee–type two-loop contribution involving the $W$-loop, which is not shown here.

Figure 6

$\mathrm{BR}(h\to \mu \tau )$ vs $\mathrm{BR}(\tau \to \mu \gamma )$. The red and green lines correspond to the cases with $c_{\beta \alpha }=-0.012$ ($m_A=250\mathrm{GeV}$) and $c_{\beta \alpha }=-0.007$ ($m_A=350\mathrm{GeV}$), respectively.

Figure 7

Numerical result for $\mathrm{BR}(\tau \to \mu \gamma )$ as a function of ${\rho }_e^{\tau \tau }$ and ${\rho }_u^{tt}$. Lines for $\mathrm{BR}(\tau \to \mu \gamma )/{10}^{-8}=0.1$ and 4.4 (current experimental limit) are shown. Here we have assumed $m_A=350\mathrm{GeV}$ with ${\lambda }_4={\lambda }_5=0.5$, $c_{\beta \alpha }=-0.007$, and $\mathrm{BR}(h\to \mu \tau )=0.84\%$ with ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$. We note that for this parameter set, $\delta a_{\mu }=2.2\times {10}^{-9}$, which explains the muon $g-2$ anomaly within the $1\sigma$.

Figure 8

Numerical result for $\mathrm{BR}(\mu \to e\gamma )$ as a function of ${\rho }_e^{\tau e}$ and ${\rho }_e^{e\tau }$. Lines for $\mathrm{BR}(\mu \to e\gamma )/{10}^{-13}=5.7$ (current limit), 1.0, 0.1, and 0.01 are shown. Here we have assumed that $m_A=350\mathrm{GeV}$ with ${\lambda }_4={\lambda }_5=0.5$, $c_{\beta \alpha }=-0.007$, and $\mathrm{BR}(h\to \mu \tau )=0.84\%$ with ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$, and extra Yukawa couplings ${\rho }_f$ other than ${\rho }_e^{\mu \tau (\tau \mu )}$ and ${\rho }_e^{\tau e(e\tau )}$ are negligible. We note that for this parameter set, $\delta a_{\mu }=2.2\times {10}^{-9}$.

Figure 9

Numerical result for $\mathrm{BR}(\mu \to e\gamma )$ as a function of ${\rho }_e^{\mu e}$ and ${\rho }_e^{e\mu }$. Lines for $\mathrm{BR}(\mu \to e\gamma )/{10}^{-13}=5.7$ (current limit), 1.0, 0.1, and 0.01 are shown. Here we have assumed that $m_A=350\mathrm{GeV}$ with ${\lambda }_4={\lambda }_5=0.5$, $c_{\beta \alpha }=-0.007$, and extra Yukawa couplings ${\rho }_f$ other than ${\rho }_e^{\mu \tau (\tau \mu )}$ and ${\rho }_e^{\mu e(e\mu )}$ are negligible.

Figure 10

Numerical result for a correction to $\tau \to \mu \nu \overline{\nu }$, $\delta$ given in Eq. (39) as a function of $c_{\beta \alpha }$ and $\mathrm{BR}(h\to \mu \tau )$ in the same parameter set of Fig. 2. Black solid lines for $\delta ={10}^{-5},{10}^{-4},{10}^{-3},{10}^{-2}$ (from left to right) are shown for $m_A=250\mathrm{GeV}$ (upper figure) and $m_A=350\mathrm{GeV}$ (lower figure). Here we also show the region where the muon $g-2$ anomaly is explained within $1\sigma$ (light green area) and $2\sigma$ (light blue areas). Here we have assumed that the extra Yukawa couplings ${\rho }_f$ other than ${\rho }_e^{\mu \tau (\tau \mu )}$ are negligible.

Figure 11

$\mathrm{BR}(\tau \to 3\mu )$ (red curve) and $\mathrm{BR}(\tau \to \mu ee)$ (green curve) as a function of ${\rho }_e^{ll}$ ($l=\mu$ for $\tau \to 3\mu$ and $l=e$ for $\tau \to \mu ee$). Here we have assumed that $c_{\beta \alpha }=-0.007$, $m_A=350\mathrm{GeV}$ with ${\lambda }_4={\lambda }_5=0.5$ and $\mathrm{BR}(h\to \mu \tau )=0.84\%$ with ${\rho }_e^{\mu \tau }=-{\rho }_e^{\tau \mu }$.

Figure 12

$\mathrm{BR}(\mu \to 3e)$ as a function of ${\rho }_e^{ee}$ and ${\rho }_e^{\mu e}$. Here we have assumed that ${\rho }_e^{\mu e}={\rho }_e^{e\mu }$, $c_{\beta \alpha }=-0.007$, and $m_A=350\mathrm{GeV}$ with ${\lambda }_4={\lambda }_5=0.5$.