text:
\section{Introduction}

Very recently, the Belle Collaboration observed an $\Omega$ excited state in the $\Xi^0 K^-$ and $\Xi^- K^0_S$ invariant mass distributions~\cite{Yelton:2018mag}. Its  mass and width are determined to be $M = 2012.4 \pm 0.7 \pm 0.6$ MeV and $\Gamma = 6.4^{+2.5}_{-2.0}  \pm 1.6$ MeV. The existence of such $\Omega$ excited states with a mass around 2000 MeV has already been predicted by various models, such as quenched quark models~\cite{Capstick:1986bm,Loring:2001ky,Pervin:2007wa,Faustov:2015eba}, the Skyrme model~\cite{Oh:2007cr}, and lattice gauge theory~\cite{Engel:2013ig}. On the other hand, the extended quark models~\cite{Yuan:2012zs,An:2013zoa,An:2014lga}, where the instanton-induced quark-antiquark pair creation or Nambu-Jona-Lasino interaction was employed, predicted  $\Omega$ states with negative parity but lower masses, the lowest $\Omega$ state lying around 1800 MeV, about 200 MeV lower than those predicted in Refs.~\cite{Capstick:1986bm,Loring:2001ky,Pervin:2007wa,Faustov:2015eba,Oh:2007cr,Engel:2013ig}. One of the reasons is that the $\Omega$ states in Refs.~\cite{Yuan:2012zs,An:2013zoa,An:2014lga} have large five-quark components,

In Refs.~\cite{Kolomeitsev:2003kt,Sarkar:2004jh,Si-Qi:2016gmh}, the interactions of the $\bar{K}\Xi(1530)$ and $\eta \Omega$ coupled channels were investigated in the chiral unitary approach. An $\Omega$ excited state with a mass around 2012 MeV and  $J^P =3/2^-$ can be dynamically generated  by use of a reasonable subtraction constant.

After the observation of the $\Omega(2012)$, its two-body strong decays were studied within the chiral quark model in Ref.~\cite{Xiao:2018pwe}, where it was shown that the newly observed $\Omega(2012)$ could be assigned to the $J^P = 3/2^-$ three quark state.  In Ref.~\cite{Aliev:2018syi} the mass and residue of the $\Omega(2012)$ were calculated by employing the QCD sum rule method with the conclusion that the $\Omega(2012)$ could be a $1P$ orbitally excited $\Omega$ state with $J^P =3/2^-$.  The analysis of Ref.~\cite{Aliev:2018syi} was extended in Ref.~\cite{Aliev:2018} to study the $\Omega(2012) \to K^- \Xi^0$ decay. In Ref.~\cite{Polyakov:2018mow},  the authors performed a  flavor $SU(3)$ analysis  and concluded that the preferred $J^P$ for the $\Omega(2012)$ is also $3/2^-$. In Refs.~\cite{Valderrama:2018bmv,Lin:2018nqd}, its strong decay modes were studied assuming that the $\Omega(2012)$ is a $\bar{K}\Xi(1530)$ hadronic molecule.

In this work, we take the chiral unitary approach and assume that the $\Omega(2012)$ is a dynamically generated state from the $\bar{K} \Xi(1530)$ and $\eta \Omega$ interactions. The coupling of the $\Omega(2012)$ to $\bar{K}\Xi(1530)$ is then obtained from the residule at the pole position. We then
calculate its decay into $K \Xi$ via a triangle diagram. We also calculate the three-body partial decay width of the $\Omega(2012)$ into $K\Xi\pi$.
 The total decay width compares favorably  with the experimental data~\cite{Yelton:2018mag} and agrees with other theoretical approaches.

 This work is organized as follows. In Section II, we briefly explain the chiral unitary approach and calculate the two and three body decays of the $\Omega(2012)$. Results
 and discussions are shown in Section III, followed by a short summary in Section IV.

 \section{Formalism}

\subsection{$\Omega(2012)$ as a $\bar{K}\Xi(1530)$ and $\eta \Omega$ molecula}

The mass of the $\Omega(2012)$ is slightly below the $\bar{K} \Xi(1530)$ threshold. It is natural to treat it as a $\bar{K} \Xi(1530)$ molecular state, dynamically generated from the interaction of the coupled channels $\bar{K}\Xi(1530)$ and $\eta\Omega$ in the isospin $I=0$ sector. However, the possibility to be an $I=1$ molecule  cannot be excluded~\cite{Polyakov:2018mow}. Within the chiral unitary approach, the interaction of the coupled channels $\bar{K} \Xi(1530)$ and $\eta \Omega$ in the strangeness $-3$ and isospin 0 was first studied in Ref.~\cite{Sarkar:2004jh}, where a pole at $(2141-i38)$ MeV was found with a natural subtraction constant $a_{\mu} = -2$ and a renormalization scale $\mu = 700$ MeV. Later,  it was explicitly shown  in Ref.~\cite{Si-Qi:2016gmh} that the pole position of the $3/2^-$ $\Omega$ state can shift by varying $a_{\mu}$. If we take $a_{\mu} = -2.5$ and $\mu=700$ MeV, we obtain a pole at $z_R = (2012.7, i0)$ MeV, which can be associated to the newly observed $\Omega$(2012) state~\cite{Yelton:2018mag}.  In the cutoff regularization scheme, the
corresponding momentum cutoff is $\Lambda= $726 MeV, which seems to quite natural as well.

The couplings of the bound state to the coupled  channels, $\bar{K}\Xi(1530)$ (channel 1) and $\eta \Omega$ (channel 2), can be obtained from the residue of the
scattering amplitude at the pole position  $z_R$,
\begin{eqnarray}
T_{ij} = \frac{g_{ii}g_{jj}}{\sqrt{s} - z_R},
\end{eqnarray}
where $g_{ii}$ is the coupling of the state to the $i$-th channel. One finds
\begin{eqnarray}
g_{11} = 1.65, ~~~~~~~~~~g_{22} = 2.80.
\end{eqnarray}

Then one can write down the effective interaction of $\Omega(2012)\bar{K}\Xi(1530)$ ($\equiv \Omega^* \bar{K} \Xi^*$) and $\Omega(2012)\eta \Omega$ ($\equiv \Omega^* \eta \Omega$),
\begin{eqnarray}
v_{\Omega^* \bar{K} \Xi^*} &=& \frac{g_{11}}{\sqrt{2}} \bar{\Xi}^{* \mu} \Omega^*_{\mu} \phi_{\bar{K}}, \\
v_{\Omega^* \eta \Omega} &=& g_{22} \bar{\Omega}^{\mu} \Omega^*_{\mu} \phi _\eta.
\end{eqnarray}

It is worth to mention that the $g_{\Omega^*(2012)\bar{K}\Xi(1530)}$  obtained in Ref.~\cite{Lin:2018nqd} with the assumption that the $\Omega^*(2012)$ is a pure $S$-wave $\bar{K} \Xi(1530)$ hadronic molecule with spin-parity $3/2^-$ is about 1.97, which is different from ours  by a mere $16\%$, and the difference is only $2.6\%$ in terms of $g^2_{\Omega^*(2012)\bar{K}\Xi(1530)}$. For the $\Omega^*(2012) \to \bar{K} \pi \Xi$ three-body decay, since only the $\bar{K} \Xi(1530)$ component contributes at  tree level and the partial decay width is proportional to $g^2_{\Omega^*(2012)\bar{K}\Xi(1530)}$, our three-body decay width is almost the same as that of Ref.~\cite{Lin:2018nqd}.

\subsection{The $\Omega(2012) \to \bar{K} \Xi$ and $\bar{K}\Xi\pi$ decays}

In the present work, we assume that the $\Omega(2012)$ ($\equiv \Omega^{*-}$) exists  and has a mass as that reported by the Belle Collaboration, and study its strong decays to the two-body final state $\bar{K}\Xi$ assuming  that it is a molecular state of $\bar{K}\Xi(1530)$ and $\eta\Omega$, as predicted by the chiral unitary approach~\cite{Sarkar:2004jh}. Then, the $\Omega(2012) \to \bar{K} \Xi$ decay can proceed through the triangular diagrams as shown in Fig.~\ref{mku}[(a)-(e)], where the $\Sigma$, $\Lambda$, $\rho$, $\omega$, $\phi$, and $K^{*}$ exchanges are considered. Note that by considering those diagrams of Fig.~\ref{mku} we can easily show that the partial decay widths of $\Omega^{*-} \to K^- \Xi^0$ and $\Omega^{*-} \to K^0 \Xi^-$ are the same and we will explicitly calculate the decay of $\Omega^{*-} \to K^- \Xi^0$ in the following.   Compared with the decays to the two final states, the contribution to the three-body decay $\Omega(2012) \to \bar{K} \Xi \pi$ only comes from the $\bar{K}\Xi(1530)$ component.  The decay width of the $\Xi(1530)$ listed in PDG is around 9 MeV and the partial decay width $\Xi(1530)\to\Xi\pi$ is the largest and almost saturates its total decay width.  Therefore, we only compute the three-body decay through the decay of the $\Xi(1530)$ and the simplest Feynman diagram is shown in Fig.~\ref{mku}[(f)].   Considering the quantum numbers and phase space, in addition to
the final state $K^{-}\Xi^{0}\pi^{0}$ shown in Fig.~\ref{mku}[(f)], the final states $\bar{K}^{0}\Xi^{-}\pi^{0}$, $\bar{K}^{0}\Xi^{0}\pi^{-}$, and  $K^{-}\Xi^{-}\pi^{+}$  should also be calculated.

\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.45]{Dlambdala.eps}
\caption{Feynman diagrams contributing to the decays of the $\Omega(2012)$  to $\bar{K}\Xi$ and $\bar{K}\Xi\pi$.}\label{mku}
\end{center}
\end{figure}

In order to evaluate the decay amplitudes of the diagrams shown in Fig.~\ref{mku}, we need  the following effective Lagrangians to calculate the relevant vertices~\cite{Xie:2010zza,Nakayama:2006ty,Huang:2018wgr},
\begin{align}
&{\cal{L}}_{\Xi\Lambda{}K}=\frac{ig_{\Xi{}\Lambda{}K}}{m_{\Lambda}+m_{\Xi}}\partial^{\mu}\bar{K}\bar{\Lambda}\gamma_{\mu}\gamma_5\Xi+H.c.,\\
&{\cal{L}}_{\Xi\Sigma{}K}=\frac{ig_{\Xi{}\Sigma{}K}}{m_{\Sigma}+m_{\Xi}}\partial^{\mu}\bar{K}\bar{\Sigma}\cdot\tau\gamma_{\mu}\gamma_5\Xi+H.c.,\\
&{\cal{L}}_{\Xi^{*}\Sigma{}K}=\frac{f_{\Xi^{*}\Sigma{}K}}{m_{K}}\partial^{\nu}\bar{K}\bar{\Xi}^{*}_{\nu}\gamma_5\tau\cdot\Sigma+H.c.,\\
&{\cal{L}}_{\Xi^{*}\Lambda{}K}=\frac{f_{\Xi^{*}\Lambda{}K}}{m_{K}}\partial^{\nu}\bar{K}\bar{\Xi}^{*}_{\nu}\gamma_5\Lambda+H.c.,\\
&{\cal{L}}_{KK\rho}=ig_{\rho{}KK}[\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K]\vec{\tau}\cdot{}\vec{\rho}^{\mu},\\
&{\cal{L}}_{KK\omega}=ig_{\omega{}KK}[\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K]\omega^{\mu},\\
&{\cal{L}}_{KK\phi}=ig_{\phi{}KK}[\bar{K}\partial_{\mu}K-\partial_{\mu}\bar{K}K]\phi^{\mu},\\
&{\cal{L}}_{\eta{}KK^{*}}=ig_{\eta{}KK^{*}}[\bar{K}\partial_{\mu}\eta-\partial_{\mu}\bar{K}\eta]K^{*\mu},\\
&{\cal{L}}_{\rho\Xi\Xi^{*}}=i\frac{g_{\rho\Xi\Xi^{*}}}{m_{\rho}}\bar{\Xi}^{*\mu}\gamma^{\nu}\gamma_5[\partial_{\mu}\vec{\tau}\cdot\vec{\rho}_{\nu}-\partial_{\nu}\vec{\tau}\cdot\vec{\rho}_{\mu}]\Xi+H.c.,\\
&{\cal{L}}_{\omega\Xi\Xi^{*}}=i\frac{g_{\omega\Xi\Xi^{*}}}{m_{\omega}}\bar{\Xi}^{*\mu}\gamma^{\nu}\gamma_5[\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}]\Xi+H.c.,\\
&{\cal{L}}_{\phi\Xi\Xi^{*}}=i\frac{g_{\phi\Xi\Xi^{*}}}{m_{\phi}}\bar{\Xi}^{*\mu}\gamma^{\nu}\gamma_5[\partial_{\mu}\phi_{\nu}-\partial_{\nu}\phi_{\mu}]\Xi+H.c.,\\
&{\cal{L}}_{K^{*}\Xi\Omega}=i\frac{g_{K^{*}\Xi\Omega}}{m_{K^{*}}}\bar{\Omega}^{\mu}\gamma^{\nu}\gamma_5[\partial_{\mu}K^{*}_{\nu}-\partial_{\nu}K^{*}_{\mu}]\Xi+H.c.,\\
&{\cal{L}}_{\pi\Xi\Xi^{*}}=\frac{g_{\pi\Xi\Xi^{*}}}{m_{\pi}}\bar{\Xi}\partial^{\mu}\vec{\tau}\cdot\vec{\pi}\Xi^{*}_{\mu}+H.c\label{eq33}.
\end{align}
Within the $SU(3)$ symmetry, we determine the coupling constants to be $g_{\eta\bar{K}\bar{K}^{*}}=\sqrt{3}g_{\pi\bar{K}\bar{K}^{*}}=5.23$~\cite{Ronchen:2012eg}, $g_{\phi\Xi\Xi^{*}}=g_{\rho\Xi\Xi^{*}}=g_{\omega\Xi\Xi^{*}}=\frac{1}{\sqrt{6}}g_{\rho\Delta{}N}=-2.48$~\cite{Gao:2010ve}, and $g_{K^{*}\Xi\Omega}=-7.01$~\cite{Gao:2010ve}.  The coupling constant $g_{\pi\Xi\Xi^{*}}=2.24$($g_{\pi\Xi\Xi^{*}}=2.04$) is evaluated using Eq.~(\ref{eq33}) and the partial decay width $\Gamma_{\Xi^{*}\to\Xi\pi}=\Gamma_{\Xi^{*}}=9.1$ MeV($\Gamma_{\Xi^{*}\to\Xi\pi}=\Gamma_{\Xi^{*}}=9.5$ MeV) in the $\Xi^{*}$ rest frame. The masses of the particles needed in the present work are listed in Table~\ref{table-mass}.
The other coupling
constants are taken from Refs.~\cite{Xie:2010zza,Nakayama:2006ty,Huang:2018wgr} and are listed in Table~\ref{table0}.

\begin{table}[h!]
\centering
\caption{Masses of the particles needed in the present work (in units of MeV).}\label{table-mass}
\begin{tabular}{ccccccc}
\hline\hline
$\pi^{\pm}$~& $\pi^{0}$   & ~$\eta$   & ~$\rho$      &~$\omega$          & ~$\phi$           & ~$K^{\pm}$                                    \\ \hline
139.57     ~&  134.98     & ~547.86   & ~775.26      &~782.65            & ~1019.46          & ~493.68                                               \\  \hline \hline
$K^{0}$    ~& $K^{*}$     & ~$\Xi^{0}$& ~$\Xi^{-}$   & ~$\Xi^{0}(1530)$  & ~$\Xi^{-}(1530)$               \\ \hline
497.61     ~& 891.76      & ~1314.86  & ~1321.71     & ~1531.80          & ~1535.00                       \\  \hline \hline
\end{tabular}
\end{table}

\begin{table}[h!]
\centering
\caption{Values of the effective meson-baryon  and meson-meson coupling constants.}\label{table0}
\begin{tabular}{ccccccc}
\hline\hline
$g_{KK\phi}$&~~~~~~$g_{KK\rho}$ & ~~~~~~$g_{KK\omega}$ & ~~~~~~$g_{\Xi^{*}\Sigma{}K}$ &~~~~~~$g_{\Xi^{*}\Lambda{}K}$& ~~~~~~$g_{\Xi\Sigma{}K}$       & ~~~~~~$g_{\Xi\Lambda{}K}$                                   \\ \hline
-3.02    &~~~~~~  -3.02        & ~~~~~-3.02           & ~~~~~~3.22                   &~~~~~~5.58                   & ~~~~~~-13.26                   & ~~~~~~3.37                                                 \\  \hline \hline
\end{tabular}
\end{table}

With above effective interaction Lagrangians and the coupling constants, we obtain the decay amplitudes for $\Omega^{*}(2012)\to{}K^{-}\Xi^{0}$ and $\Omega^{*}(2012)\to{}\bar{K}\Xi\pi$, corresponding to the diagrams shown in Fig.~\ref{mku},

\begin{align}
{\cal{M}}_{\Sigma^{-}}&=\frac{\sqrt{2}g_{11}g_{\Xi^{0}\Sigma^{-}K^{-}}g_{\Xi^{*0}\Sigma^{-}K^{-}}}{m_{K^{-}}(m_{\Sigma^{-}}+m_{\Xi^{0}})}\int{}\frac{d^4q}{(2\pi)^4}\bar{u}(p_1)k\!\!\!/_2\gamma_{5}\frac{1}{q\!\!\!/-m_{\Sigma^{-}}+i\epsilon}\nonumber\\
                      &\times\gamma_5{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)p_2^{\nu}\frac{1}{k_2^2-m^2_{K^{-}}+i\epsilon},\label{eq1}
\end{align}
\begin{align}
{\cal{M}}_{\Sigma^{0}}&=\frac{g_{11}g_{\Xi^{0}\Sigma^{0}\bar{K}^{0}}g_{\Xi^{*-}\Sigma^{0}K^{-}}}{\sqrt{2}m_{K^{-}}(m_{\Sigma^{0}}+m_{\Xi^{-}})}\int{}\frac{d^4q}{(2\pi)^4}\bar{u}(p_1)k\!\!\!/_2\gamma_{5}\frac{1}{q\!\!\!/-m_{\Sigma^{0}}+i\epsilon}\nonumber\\
                      &\times{}\gamma_5{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)p_2^{\nu}\frac{1}{k_2^2-m^2_{K^{-}}+i\epsilon},\label{eq2}
\end{align}
\begin{align}
{\cal{M}}_{\Lambda}&=\frac{g_{11}g_{\Xi^{0}\Lambda\bar{K}^{0}}g_{\Xi^{*-}\Lambda{}K^{-}}}{\sqrt{2}m_{K^{-}}(m_{\Lambda}+m_{\Xi^{-}})}\int{}\frac{d^4q}{(2\pi)^4}\bar{u}(p_1)k\!\!\!/_2\gamma_{5}\frac{1}{q\!\!\!/-m_{\Lambda}+i\epsilon}\nonumber\\
                      &\times{}\gamma_5{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)p_2^{\nu}\frac{1}{k_2^2-m^2_{K^{-}}+i\epsilon},\label{eq3}
\end{align}
\begin{align}
{\cal{M}}_{\rho^{0}/\omega/\phi}&=\frac{-ig_{11}g_{\Xi^{*}\Xi\rho/\omega/\phi}g_{\rho/\omega/{\phi}\bar{K}\bar{K}}}{\sqrt{2}m_{\rho/\omega}}\int{}\frac{d^4q^{'}}{(2\pi)^4}\bar{u}(p_1)\gamma^{\mu}\gamma_5\nonumber\\
                           &\times{}{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)(q^{'}_{\nu}g_{\mu\alpha}-q^{'}_{\mu}g_{\nu\alpha})(k_1^{\beta}-p_2^{\beta})\nonumber\\
                           &\times\frac{1}{k_2^2-m^2_{K^{-}}+i\epsilon}\frac{-g^{\alpha\beta}+q^{'\alpha}q^{'\beta}/m^2_{\rho/\omega}}{q^{'2}-m^2_{\rho/\omega/\phi}+im_{\rho/\omega/\phi}\Gamma_{\rho/\omega/\phi}},\label{eq4}
\end{align}

\begin{align}
{\cal{M}}_{\rho^{-}}&=\frac{-i\sqrt{2}g_{11}g_{\Xi^{*}\Xi\rho}g_{\rho{}\bar{K}\bar{K}}}{m_{\rho}}\int{}\frac{d^4q^{'}}{(2\pi)^4}\bar{u}(p_1)\gamma^{\mu}\gamma_5\nonumber\\
                    &\times{}{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)(q^{'}_{\nu}g_{\mu\alpha}-q^{'}_{\mu}g_{\nu\alpha})(k_1^{\beta}-p_2^{\beta})\nonumber\\
                    &\times\frac{1}{k_2^2-m^2_{K^{-}}+i\epsilon}\frac{-g^{\alpha\beta}+q^{'\alpha}q^{'\beta}/m^2_{\rho}}{q^{'2}-m^2_{\rho}+im_\rho\Gamma_{\rho}},\label{eq5}
\end{align}

\begin{align}
{\cal{M}}_{K^{*-}}&=\frac{-i g_{22}g_{\eta{K^{-}}K^{*-}}g_{\Omega^{-}K^{*-}\Xi^{0}}}{m_{K^{*-}}}\int{}\frac{d^4q^{'}}{(2\pi)^4}\bar{u}(p_1)\gamma^{\mu}\gamma_5\nonumber\\
                    &\times{}{\cal{P}}^{\nu\eta}u^{\eta}_{\Omega^{*}}(k_0)(q^{'}_{\nu}g_{\mu\alpha}-q^{'}_{\mu}g_{\nu\alpha})(k_1^{\beta}-p_2^{\beta})\nonumber\\
                    &\times\frac{1}{k_2^2-m^2_{\eta}+im_{\eta}\Gamma_{\eta}}\frac{-g^{\alpha\beta}+q^{'\alpha}q^{'\beta}/m^2_{K^{*}}}{q^{'2}-m^2_{K^{*}}+im_{K^*}\Gamma_{K^*}}\label{eq6},
\end{align}

\begin{align}
{\cal{M}}&_{\bar{K}(k_2)\Xi(p_3)\pi(p_4)}=f_I\frac{ig_{\pi\Xi\Xi^{*}}g_{\Omega^{*}\bar{K}\Xi^{*}}}{m_{\pi}}F(k^2_1)\bar{u}(p_3)p_3^{\nu}{\cal{P}}^{\nu\eta}u_{\eta}(k_0),
\end{align}
where
\begin{align} {\cal{P}}^{\nu\eta}=\frac{k\!\!\!/_1+m_{\Xi^{*}}}{k_1^2-m^2_{\Xi^{*}}+im_{\Xi^{*}}\Gamma_{\Xi^{*}}}[-g^{\nu\eta}+\frac{1}{3}\gamma_{\nu}\gamma_{\eta}+\frac{2}{3m^2_{\Xi^{*}}}k_{1\nu}k_{1\eta}+\frac{1}{3m_{\Xi^{*}}}(\gamma^{\nu}k_{1\eta}-\gamma^{\eta}k_{1\nu})],
\end{align}
with $q=k_2-p_1=p_2-k_1$, and $q^{'}=k_2-p_2=p_1-k_1$. We take $\Gamma_{\Xi^{*0}} = 9.1$ MeV, $\Gamma_{\Xi^{*-}} = 9.9$ MeV, $\Gamma_{\rho}= 149.1$ MeV, $\Gamma_{\omega} = 8.5$ MeV, $\Gamma_{\eta}=1.3$ MeV, $\Gamma_{\phi}=4.2$ MeV, and $\Gamma_{K^*} = 50.3$ MeV.  For the three-body final states $K^{-}\Xi^{0}\pi^{0}$ and  $\bar{K}^{0}\Xi^{-}\pi^{0}$, the isospin factors are $f_I=1$ and $-1$, respectively.  The isospin factor is $f_I=\sqrt{2}$ for all the other three-body final states.   To take into account the finite size of hadrons, for each vertex, we multiply a form factor $F(k_1^2)$ of the following form~\cite{Lin:2018nqd}
\begin{align}
F(k_1^2)=\frac{\Lambda^4}{(m^2-k_1^2)^2+\Lambda^4}\label{eqff},
\end{align}
where $m$ is the mass of the exchanged particle and $k_1$ is its momentum, with the cutoff $\Lambda$ varying from 0.8 GeV to 2.0 GeV.

In order to avoid  ultraviolet divergence in the triangle diagrams,  we take the three momentum truncation method to  compute the amplitudes.
In the rest frame of the $\Omega^*$, the relevant momenta are defined as follows:
\begin{align}
k_0&=(M,0,0,0),~~~~p_1=(E_{\Xi},0,0,p_{cm}),~~~~~p_2=(E_{K},0,0,-p_{cm}),\\
q&=(q_0,q_1\sin\theta,0,q_1\cos\theta)=q^{'},
\end{align}
and we can rewrite the $\int{}d^4q^{(')}=\int{}dq_0q_1^2dq_1d\cos{}\theta{}d\phi$ with $q_0 \epsilon{}(-\infty,\infty)$, $q_1\epsilon{}(0,\Lambda)$, $\cos\theta\epsilon(-1,1)$, and $\phi\epsilon{}(0,2\pi)$, where $\Lambda$ is a free parameter and is also taken to vary from 0.8 GeV to 2.0 GeV.

The partial decay width of the $\Omega^{*} \to \bar{K} \Xi$ and $\Omega^{*} \to \bar{K} \Xi\pi$ in its rest frame are given by
\begin{eqnarray}
d\Gamma_{\Omega^{*} \to \bar{K} \Xi}    & = & \frac{1}{32\pi^2} \overline{|{\cal M}|^2}  \frac{p}{M^2}d\Omega,\\
d\Gamma_{\Omega^{*}\to \bar{K} \Xi\pi} & = & \frac{1}{(2\pi)^5}\frac{1}{16M^2} \overline{|{\cal M}|^2} |\vec{p}_3^{*}||\vec{k}_2|dm_{\pi\Xi}d\Omega^{*}_{p_3}d\Omega_{k_2},
\end{eqnarray}
where $M$ is the mass of the $\Omega(2012)$, while $p$ is the module of the $\Xi$ (or $\bar{K}$) three-momentum in the rest frame of the $\Omega(2012)$. The $(|\vec{p}_3^{*}|,\Omega_{p_3}^{*})$ is the momentum and angle of the particle $\Xi$ in the rest frame of $\Xi$ and $\pi$, and $\Omega_{k_2}$ is the
angle of the $\bar{K}$ in the rest frame of the decaying particle.  The $m_{\pi\Xi}$ is the invariant mass for $\pi$ and $\Xi$ and $m_{\pi}+m_{\Xi}\leq{}m_{\pi\Xi}\leq{}M-m_{\bar{K}}$.
The averaged squared amplitude is then
\begin{eqnarray}
 \overline{|{\cal M}|^2}  &=& \frac{1}{4} \sum_{s_{\Omega^{*}}} \sum_{s_\Xi} |{\cal M}|^2,
\end{eqnarray}
where
\begin{eqnarray}
  {\cal M}_{\Omega^{*} \to \bar{K} \Xi} &=& {\cal M}_{\Sigma/\Lambda} + {\cal M}_{\rho/\omega/\phi} + {\cal M}_{K^*},\\
  |{\cal M}|^2_{\Omega^{*} \to \bar{K} \Xi\pi}&=&|{\cal M}|^2_{K^{-} \Xi^{0}\pi^{0}}+|{\cal M}|^2_{K^{-} \Xi^{-}\pi^{+}}+|{\cal M}|^2_{\bar{K}^{0} \Xi^{0}\pi^{-}}+|{\cal M}|^2_{\bar{K}^{0} \Xi^{-}\pi^{0}}.
\end{eqnarray}

\section{Numerical results and discussions}

As explained in the previous section, the triangle diagrams are regularized with a sharp momentum cutoff $\Lambda$, which is taken to be the same as those appearing in the form factor, $F(k_1^2)$.  Because the triangle diagrams are
ultraviolet divergent, our two-body decay width will depend on the value of the cutoff. Therefore,  it  is important to check whether one can obtain a decay width consistent with the experimental data using a reasonable value for the cutoff.

In Fig.~\ref{t-mass} we show the total decay width of $\Omega(2012) \to \bar{K} \Xi$ as a function of the cutoff parameter $\Lambda$. Note that the $\Omega(2012) \to \bar{K} \pi\Xi$ three-body decay does not depend on the cut-off parameter $\Lambda$, but it  depends weakly on the parameter $\Lambda_0$ as  in Ref.~\cite{Lin:2018nqd}.  We can see that the $\bar{K}\Xi(1530)$ component provides the dominant contribution to the partial decay width of the $\bar{K}\Xi$ two-body channel.   The $\eta\Omega$ contribution to the $\bar{K}\Xi$ two-body channel is very small.  However, the interference between them is still sizable and increases with the cutoff parameter $\Lambda$.

In Refs~\cite{Lin:2018nqd,Polyakov:2018mow},  the three-body decay was emphasized,  while  our result shows that two-body $\bar{K} \Xi$ decay width can become larger than the $\bar{K}\pi \Xi$ three-body decay width when $\Lambda$ is larger than $1.65$ GeV.  We note that
the  hyperon exchange and $\eta\Omega$ component contribution are not considered in   Ref.~\cite{Lin:2018nqd}.   More specifically, our three-body partial decay width, $\sim3.0$ MeV,  is close to the estimate of Ref.~\cite{Lin:2018nqd}, but smaller than that of Ref.~\cite{Polyakov:2018mow}, about10 MeV.  We note that our total decay width and that of
Ref.~\cite{Valderrama:2018bmv} agree with the experimental data.   The difference is  that Ref.~\cite{Valderrama:2018bmv} assumes that the $\Omega^{*}(2012)$ is a pure
$\Xi(1530)\bar{K}$ molecule and invokes some power counting arguments to calculate its two-body decay width.  Indeed, our study shows that the contribution  from the $\eta\Omega$ component is small.   Note that the molecule picture  is different from the $qqq$ picture of the  chiral quark model~\cite{Xiao:2018pwe} and light cone QCD sum rules~\cite{Aliev:2018syi,Aliev:2018}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{width-total.eps}
\caption{(Color online) Total decay width of the $\Omega(2012)$ as a function of the parameter $\Lambda$. The cyan error bands correspond to the experimental decay width~\cite{Yelton:2018mag}.}
\label{t-mass}
\end{center}
\end{figure}

The contribution of the $\bar{K}\Xi(1530)$ component includes
two parts: i) $\Sigma$ and $\Lambda$ exchanges (fig.~\ref{mku} (a) and (b)); ii) $\rho$, $\phi$, and $\omega$ meson exchanges
(fig.~\ref{mku} (c) and (d)). The relative importance of these two mechanisms
to the $\Omega^* \to \bar{K} \Xi$ decay is shown
in Fig.~\ref{fig3}. One can see that the contribution
from $\Sigma$ and $\Lambda$ exchanges is larger than those from
$\rho$, $\phi$, and $\omega$ meson exchanges for the cutoff range studied.

\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{gamrwidth-vh.eps}
\caption{(Color online) Decomposed contributions to the decay width of the $\Omega(2012)$ into $K\Xi$ as a function of the parameter $\Lambda$ }
\label{fig3}
\end{center}
\end{figure}

\section{summary}
In summary, we have studied the $\bar{K} \Xi$ decay of the newly observed $\Omega^{*}(2012)$ assuming that the $\Omega(2012)$ is a dynamically generated state with
spin-parity $3/2^-$ from the coupled channel interactions of $\bar{K}\Xi(1530)$ and $\eta \Omega$ in $S$-wave.
Take $\alpha_{\mu}=-2.5$ and $\mu=0.7$ GeV,  we obtained a pole at $M=2012.7$ MeV, and associated it
to the newly observed $\Omega^{*}$(2012).   With the coupling constants between the  $\Omega^{*}(2012)$ and its components calculated from the residue at the pole position, $g_{\Omega^{*}\bar{K}\Xi^{*}}$=1.65 and $g_{\Omega^{*}\eta\Omega}$=2.80, we obtained the partial decay widths of the $\bar{K}\Xi$ final
state through triangle diagrams in an effective Lagrangian approach.   In such a picture, the decay $\Omega^{*}(2012)\to\bar{K}\Xi$
occurs by exchanging $\Sigma,\Lambda$ hadrons and $\rho,\phi,\omega$,and $K^{*}$ vector mesons.  The contribution to the three-body decay $\Omega(2012) \to \bar{K} \Xi \pi$
only comes from the $\bar{K}\Xi(1530)$ component.

 It was shown that the calculated total decay width of the $\Omega^{*}(2012)$ is in fair agreement with the experimental data, thus supporting the assignment of its spin-parity as $3/2^-$, In addition, we showed that the $\eta\Omega$ channel plays a less relevant role.

The present work should be viewed as a natural extension of the works of Refs.~\cite{Si-Qi:2016gmh,Sarkar:2004jh}, where the chiral unitary approach
was employed to dynamically generate such an exited $\Omega^*$. The present work showed indeed
that the chiral unitary approach can provide a satisfactory explanation of not only the mass but also the decay width
of the $\Omega(2012)$, consistent with the experimental data.

\section*{Acknowledgements}

JJX and LSG thank Qiang Zhao and Bing-Song Zou for valuable discussions. This work is partly supported by the National Natural
Science Foundation of China under Grants No. 11522539,
No.1 1735003, and No. 11475227, the fundamental Research Funds for the Central
Universities and the Youth Innovation Promotion Association
CAS No. 2016367.


text:
\section{Mass of the $\tau$ lepton}
\label{sec:Tau_Mass}

\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=.42\textheight,width=.49\textwidth]{figures/TauMass.eps}
\end{center}
\caption{Measurements and average value of $m_\tau$.}
\label{fig:Tau_Mass}
\end{figure}

The mass of the $\tau$ lepton has recently been measured by the BaBar~\cite{Aubert:2009ra} and Belle~\cite{Abe:2006vf} experiments
using the end-point technique from the pseudo-mass distribution in $\tau^-\to\pi^-\pi^-\pi^+\nu_\tau$ decays,
as well as by the KEDR experiment~\cite{Shamov:2009zz} from a study of the $\tau^+\tau^-$ cross-section around the production threshold.
In Figure~\ref{fig:Tau_Mass}, we present the measurements and average values of $m_\tau$.

\section{$\tau$ Branching Fractions:}
\label{sec:Tau_BR}

Average values of the $\tau$ branching fractions~\footnote{charge conjugate $\tau$ decays are implied throughout}
are obtained from a global unitarity constrained fit.
We account for correlations due to common dependence of
normalization on the $\tau-$pair cross-section~\cite{Banerjee:2007is}
and assumed branching fractions for the background modes.
The detector-specific systematics uncertainties are considered to be fully correlated between
measurements of the same experiment.

We use 131 measurements from non-B-factory experiments,
which includes the set used in the global fit performed by the PDG~\cite{PDG_2010}.
The measurements from non-B-factories include
     37 measurements from ALEPH,
      2 measurements from ARGUS,
      1 measurement from CELLO,
     36 measurements from CLEO,
      6 measurements from CLEO3,
     14 measurements from DELPHI,
      2 measurements from HRS,
     11 measurements from L3,
     19 measurements from OPAL, and
      3 measurements from TPC.
Finally, we include the following measurements from the B-factories (BaBar and Belle):
\begin{itemize}
\item 12 measurements from BaBar:
\begin{eqnarray*}
{{\cal{B}}(\tau^- \to \mu^- \bar{\nu}_\mu \nu_\tau)}/{{\cal{B}}(\tau^- \to e^- \bar{\nu}_e \nu_\tau)}~\cite{Aubert:2009qj},\\
{{\cal{B}}(\tau^- \to \pi^- \nu_\tau)}/{{\cal{B}}(\tau^- \to e^- \bar{\nu}_e \nu_\tau)}~\cite{Aubert:2009qj},\\
{{\cal{B}}(\tau^- \to  K^-  \nu_\tau)}/{{\cal{B}}(\tau^- \to e^- \bar{\nu}_e \nu_\tau)}~\cite{Aubert:2009qj},\\
{\cal{B}}(\tau^- \to K^- \pi^0 \nu_\tau)~\cite{Aubert:2007jh}\\
{\cal{B}}(\tau^- \to \bar{K}^0 \pi^- \nu_\tau)~\cite{Aubert:2008an}\\
{\cal{B}}(\tau^- \to \bar{K}^0 \pi^- \pi^0 \nu_\tau)~\cite{Paramesvaran:2009ec}\\
{\cal{B}}(\tau^- \to \pi^- \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Aubert:2007mh}\\
{\cal{B}}(\tau^- \to  K^-  \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Aubert:2007mh}\\
{\cal{B}}(\tau^- \to  K^-  \pi^-  K^+ \nu_\tau)~\cite{Aubert:2007mh}\\
{\cal{B}}(\tau^- \to  K^-   K^-   K^+ \nu_\tau)~\cite{Aubert:2007mh}\\
{\cal{B}}(\tau^- \to  3h^- 2h^+ \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Aubert:2005waa}\\
{\cal{B}}(\tau^- \to  2\pi^- \pi^+ \eta \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Aubert:2008nj}
\end{eqnarray*}
\item and 10 measurements from Belle:
\begin{eqnarray*}
{\cal{B}}(\tau^- \to h^- \pi^0 \nu_\tau)~\cite{Fujikawa:2008ma}\\
{\cal{B}}(\tau^- \to \bar{K}^0 \pi^- \nu_\tau)~\cite{Epifanov:2007rf}\\
{\cal{B}}(\tau^- \to \pi^- \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Lee:2010tc}\\
{\cal{B}}(\tau^- \to  K^-  \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0))~\cite{Lee:2010tc}\\
{\cal{B}}(\tau^- \to  K^-  \pi^-  K^+ \nu_\tau)~\cite{Lee:2010tc}\\
{\cal{B}}(\tau^- \to  K^-   K^-   K^+ \nu_\tau)~\cite{Lee:2010tc}\\
{\cal{B}}(\tau^- \to  \pi^- \pi^0 \eta \nu_\tau)~\cite{Inami:2008ar}\\
{\cal{B}}(\tau^- \to  K^- \eta \nu_\tau)~\cite{Inami:2008ar}\\
{\cal{B}}(\tau^- \to  K^- \pi^0 \eta \nu_\tau)~\cite{Inami:2008ar}\\
{\cal{B}}(\tau^- \to  \bar{K}^0 \pi^- \eta \nu_\tau)~\cite{Inami:2008ar}
\end{eqnarray*}
\end{itemize}

All of these 153 measurements are expressed as a linear function
of the form $(\sum_i \alpha_i P_i )\over(\sum_j \beta_j P_j )$
of ``base'' branching fractions $(P_i)$,
which are chosen such that they sum up to unity.
The results of the fit are shown in Table~\ref{tab:TauGlobalFit},
which has $\chi^2/\mathrm{ndof} = 143.4/117$ (CL = 4.9\%).

\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=.4\textheight,width=.49\textwidth]{figures/compare_with_NoBB.eps}
\end{center}
\caption{Normalized difference of B-factory measurements w.r.t average of non-B-factory measurements.}
\label{fig:comp_NoBB}
\end{figure}

The differences between the global fit performed by PDG and us are:
\begin{itemize}
\item We expand the list of base modes from 31 to 37 to account for recent
  measurements by the B-factories with significant precision.
\item We use the original correlation matrix published by
 ALEPH~\cite{Schael:2005am} between hadronic modes instead of between
 pionic modes.
\item We use updated background estimates to adjust the B-factory measurements.
\item We avoid applying the PDG-style scale factors to all our measurements.
However, the BaBar and Belle measurements of ${\cal{B}}({\tau^- \to
  K^-K^+K^-\nu_\tau})$ are $5.4~\sigma$ apart. We scale the errors on
these 2 measurements only, following the PDG procedure for
single-quantity averaging.
\end{itemize}

\begin{table}[!hbtp]
\begin{tabular}{l| c} \hline
Base       modes                                                    & Branching fractions \\
(from $\tau^-$ decay)                                               & (in \%)\\ \hline
\multicolumn{2}{c}{leptonic modes} \\ \hline
$e^- \bar{\nu}_e \nu_\tau$                                            & 17.833 $\pm$  0.040 \\
$\mu^- \bar{\nu}_\mu \nu_\tau$                                        & 17.408 $\pm$  0.038 \\
\hline
\multicolumn{2}{c}{non-strange modes} \\ \hline
$\pi^- \nu_\tau$                                                      &10.831 $\pm$  0.051 \\
$\pi^- \pi^0 \nu_\tau$                                                &25.531 $\pm$  0.090 \\
$\pi^- 2\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0)$                           &9.278 $\pm$ 0.097 \\
$\pi^- 3\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0)$                           & 1.046 $\pm$ 0.074 \\
$h^- 4\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0,\eta)$                        & 0.107 $\pm$ 0.039 \\
$K^- K^0 \nu_\tau$                                                    & 0.160 $\pm$ 0.016 \\
$K^- \pi^0 K^0 \nu_\tau$                                              & 0.162 $\pm$ 0.019 \\
$\pi^- K_S^0 K_S^0 \nu_\tau$                                          & 0.024 $\pm$ 0.005 \\
$\pi^- K_S^0 K_L^0 \nu_\tau$                                          & 0.119 $\pm$ 0.024 \\
$\pi^- \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0,\omega)$               & 8.983 $\pm$ 0.050 \\
$\pi^- \pi^- \pi^+ \pi^0 \nu_\tau ~(\mathrm{ex.~}K^0,\omega)$         & 2.751 $\pm$ 0.069 \\
$h^- h^- h^+ 2\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0,\omega,\eta)$         & 0.097 $\pm$ 0.036 \\
$h^- h^- h^+ 3\pi^0 \nu_\tau$                                         & 0.032 $\pm$ 0.003 \\
$\pi^- K^- K^+ \nu_\tau$                                              & 0.144 $\pm$ 0.003 \\
$\pi^- K^- K^+ \pi^0 \nu_\tau$                                        & 0.006 $\pm$ 0.002 \\
$3h^- 2h^+ \nu_\tau ~(\mathrm{ex.~}K^0)$                              & 0.082 $\pm$ 0.003 \\
$3h^- 2h^+ \pi^0 \nu_\tau ~(\mathrm{ex.~}K^0)$                        & 0.020 $\pm$ 0.002 \\
$\pi^- \pi^0 \eta \nu_\tau$                                          & 0.139 $\pm$ 0.007 \\
$\pi^- \omega \nu_\tau$                                              & 1.959 $\pm$ 0.064 \\
$h^- \pi^0 \omega \nu_\tau$                                          & 0.409 $\pm$ 0.042 \\
\hline
\multicolumn{2}{c}{strange modes} \\ \hline
$K^- \nu_\tau$                                                       & 0.697 $\pm$ 0.010 \\
$K^- \pi^0 \nu_\tau$                                                 & 0.431 $\pm$ 0.015 \\
$K^- 2\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0)$                            & 0.060 $\pm$ 0.022 \\
$K^- 3\pi^0 \nu_\tau ~(\mathrm{ex.~}K^0,\eta)$                       & 0.039 $\pm$ 0.022 \\
$\bar{K}^0 \pi^- \nu_\tau$                                          & 0.831 $\pm$ 0.018 \\
$\bar{K}^0 \pi^- \pi^0 \nu_\tau$                                    & 0.350 $\pm$ 0.015 \\
$\bar{K}^0 \pi^- 2\pi^0 \nu_\tau$                                   & 0.035 $\pm$ 0.023 \\
$\bar{K}^0 h^- h^- h^+ \nu_\tau$                                     & 0.028 $\pm$ 0.020 \\
$K^- \pi^- \pi^+ \nu_\tau ~(\mathrm{ex.~}K^0,\omega)$                & 0.293 $\pm$ 0.007 \\
$K^- \pi^- \pi^+ \pi^0 \nu_\tau ~(\mathrm{ex.~}K^0,\omega,\eta)$     & 0.041 $\pm$ 0.014 \\
$K^- \phi \nu_\tau (\phi \to KK)$                                   & 0.004 $\pm$ 0.001 \\
$K^- \eta \nu_\tau$                                                 & 0.016 $\pm$ 0.001 \\
$K^- \pi^0 \eta \nu_\tau$                                           & 0.005 $\pm$ 0.001 \\
$\bar{K}^0 \pi^- \eta \nu_\tau$                                     & 0.009 $\pm$ 0.001 \\
$K^- \omega \nu_\tau$                                              & 0.041 $\pm$ 0.009 \\
\hline
Sum of strange modes                                              & 2.8796 $\pm$ 0.0501 \\ \hline
Sum of all modes                                                  & 100.00              \\ \hline
\end{tabular}\\[2pt]
\caption{Results of unitarity constrained fit.}
\label{tab:TauGlobalFit}
\end{table}

The three-kaon decays had not been observed before the B-factory era.
For the remaining 20 B-factory measurements,
Figure~\ref{fig:comp_NoBB} shows a histogram of the normalized difference
((B-factory value minus averaged non-B-factory value)/estimated uncertainty in the difference).
The average difference between the two sets of measurements is -0.98 $\sigma$
(-1.26 $\sigma$ for the 9 Belle measurements and -0.75 $\sigma$ for the 11 BaBar measurements).
Although this systematic trend is yet to be understood,
the magnitude of discrepancy is less than the value of reported by the PDG~\cite{PDG_2010}.

\section{Tests of Lepton Universality}
\label{sec:Tau_LU}

From the unitarity constrained fit, we obtain
$
{{\cal{B}}({\tau^- \to \mu^- \overline{\nu}_\mu \nu_\tau})}/
{{\cal{B}}({\tau^- \to e^- \overline{\nu}_e  \nu_\tau})}$
=
$0.9762\, \pm\,   0.0028$,
which includes a correlation of $18.33\%$
between the branching fractions.
This yields a value of $\left( \frac{g_\mu}{g_e} \right)$ =
$1.0019\, \pm\, 0.0014$,
which is consistent with the Standard Model (SM) value.

Using the world averaged mass, lifetime and meson decay rates~\cite{PDG_2010}, we determine
$\left( \frac{g_{\tau}}{g_{\mu}} \right)$ =
$0.9966\, \pm\, 0.0030$ $(0.9860\, \pm\, 0.0073)$
from the pionic and kaonic branching fractions,
with a correlation of $13.10\%$.
Combining these results, we obtain
$\left( \frac{g_{\tau}}{g_{\mu}} \right)$ = $0.9954\, \pm\, 0.0029$,
which is consistent with ($1.6~\sigma$ below) the SM expectation.

We also test lepton universality between $\tau$ and $\mu$ ($e$),
by comparing the averaged electronic (muonic) branching fractions of
the $\tau$ lepton with the predicted branching fractions from measurements
of the $\tau$ and $\mu$ lifetimes and their respective masses~\cite{PDG_2010}.
This gives
$\left( \frac{g_{\tau}}{g_{\mu}} \right)$ = $1.0011\, \pm\, 0.0021$
and
$\left( \frac{g_{\tau}}{g_{e}} \right)$ = $1.0030\, \pm\, 0.0021$.
The correlation co-efficient between the determination of
$\left( \frac{g_{\tau}}{g_{\mu}} \right)$ from electronic branching
fraction with the ones obtained from pionic and kaonic branching fractions
are $48.16\%$ and  $21.82\%$, respectively.
Averaging the three values, we obtain
$\left( \frac{g_{\tau}}{g_{\mu}} \right)$ = $1.0001\, \pm\, 0.0020$,
which is consistent with the SM value.
In Figure~\ref{fig:TauLU}, we compare these determinations
with the values obtained from pion~\cite{PiToMu},
kaon~\cite{Antonelli:2010yf} and W decays~\cite{Alcaraz:2006mx}.

\begin{figure}[!hbtp]
\begin{center}
\begin{minipage}{.42\textwidth}
\includegraphics[height=.3\textheight]{figures/gmuge.eps}
\includegraphics[height=.3\textheight]{figures/gtaugmu.eps}
\includegraphics[height=.3\textheight]{figures/gtauge.eps}
\end{minipage}
\end{center}
\caption{Measurements of lepton universality from W, kaon, pion and tau decays.}
\label{fig:TauLU}
\end{figure}

\section{Measurement of $|V_{us}|$}
\label{sec:Tau_Vus}

\newcommand{\Vudvalue} {\ensuremath{0.97425\, \pm\, 0.00022}\xspace}

Using the unitarity constraint on the first row of the CKM matrix
and $\ensuremath{|V_{ud}|}\xspace = \Vudvalue$~\cite{Hardy:2008gy}, one gets $\ensuremath{|V_{us}|}\xspace = 0.2255\, \pm\, 0.0010$.
Here we present 3 extractions for $|V_{us}|$ using
${\cal{B}}({\tau^-\to K^-\nu_\tau})$,
${{\cal{B}}({\tau^- \to K^-\nu_\tau})}/
{{\cal{B}}({\tau^- \to \pi^-\nu_\tau})}$,
and inclusive sum of $\tau$ branching fractions having net strangeness of unity in the final state.

We use the value of kaon decay constant $f_K = 157\, \pm\, 2 \ensuremath{\mathrm{Me\kern -0.1em V}}\xspace$
obtained from Lattice calculations~\cite{Follana:2007uv},
and our value of ${\cal{B}}({\tau^-\to K^-\nu_\tau}) =$
$\frac{G^2_F f^2_K \ensuremath{|V_{us}|}\xspace^2 m^3_{\tau} \tau_{\tau}}{16\pi\hbar} \left (1 - \frac{m_K^2}{m_\tau^2} \right )^2 S_{EW},$
where $S_{EW} = 1.0201\, \pm\, 0.0003$~\cite{Erler:2002mv},
to determine $\ensuremath{|V_{us}|}\xspace= 0.2204\, \pm\, 0.0032$,
which is consistent with ($1.5~\sigma$ below) the value from CKM unitarity.

We extract $\ensuremath{|V_{us}|}\xspace$ using $\ensuremath{|V_{ud}|}\xspace$, Lattice calculation of the ratio of decay constants
$f_K/f_\pi = 1.189\, \pm\, 0.007$~\cite{Follana:2007uv} and
the long-distance correction $\delta_{LD} = (0.03\, \pm\, 0.44)\%$,
estimated using corrections to $\tau\to h\nu_\tau$ and $h \to \mu\nu_\mu$~\cite{LongDistance},
for the ratio
$$\frac{{\cal{B}}({\tau^- \to K^-\nu_\tau})}{{\cal{B}}({\tau^- \to \pi^-\nu_\tau})}
=
\frac{f_K^2 |V_{us}|^2}{f_\pi^2 |V_{ud}|^2}
 \frac{\left( 1 -  \frac{m_K^2}{m_\tau^2} \right)^2}{\left( 1 -  \frac{m_\pi^2}{m_\tau^2} \right)^2} (1+\delta_{LD}),$$
where short-distance electro-weak corrections cancel in this ratio
measured to be
${{\cal{B}}({\tau^- \to K^-\nu_\tau})}/{{\cal{B}}({\tau^- \to \pi^-\nu_\tau})}$
$=$ $ 0.0644\, \pm\, 0.0009$.
This includes a correlation of $-0.49\%$
between the branching fractions, and yields $\ensuremath{|V_{us}|}\xspace = 0.2238\, \pm\, 0.0022$,
which is also consistent with ($0.7~\sigma$ below) $\ensuremath{|V_{us}|}\xspace$ from CKM unitarity.

The total hadronic width of the \ensuremath{\tau}\xspace normalized to the electronic branching fraction,
$R_{\rm{had}} = {\cal{B}}_{\rm{had}}/{\cal{B}}_{\rm{e}}$,
can be written as $R_{\rm{had}} = R_{\rm{non-strange}} + R_{\rm{strange}}$.
We can then measure
$$|V_{us}| = \sqrt{R_{\rm{strange}}/\left[\frac{R_{\rm{non-strange}}}{|V_{ud}|^2} -  \delta R_{\rm{theory}}\right]}.$$

Here, we use $\ensuremath{|V_{ud}|}\xspace = \Vudvalue$~\cite{Hardy:2008gy}, and $\delta R_{\rm{theory}} = 0.240\, \pm\, 0.032$~\cite{Gamiz:2006xx}
which contributes to an error of $0.0010$ on $|V_{us}|$.
We note that this error is equivalent to  half the difference between calculations of $|V_{us}|$
obtained using fixed order perturbation theory (FOPT) and
contour improved perturbation theory (CIPT) calculations of $\delta R_{\rm{theory}}$~\cite{Maltman:2010hb},
and twice as large as the theoretical error proposed in Ref.~\cite{Gamiz:2007qs}.

As in Ref.~\cite{Davier:2005xq}, we improve upon the estimate of
electronic branching fraction by averaging its direct measurement
with its estimates of $(17.899\, \pm\, 0.040)\%$ and $(17.794\, \pm\, 0.062)\%$
obtained from the averaged values of muonic branching fractions and
the averaged value of the lifetime of the
\ensuremath{\tau}\xspace lepton = $(290.6\, \pm\, 1.0) \times 10^{-15} ~\mathrm{s}$~\cite{PDG_2010},
assuming lepton universality and taking into account the correlation between the leptonic branching fractions.
This gives a more precise estimate for the electronic branching fraction: ${\cal{B}}^{\rm{uni}}_{\rm{e}}$ = $ (17.852\, \pm\, 0.027)\%$.

Assuming lepton universality, the total hadronic branching fraction
can be written as: ${\cal{B}}_{\rm{had}} = 1 - 1.972558 ~ {\cal{B}}^{\rm{uni}}_{\rm{e}}$,
which gives a value for the total \ensuremath{\tau}\xspace hadronic width normalized to
the electronic branching fraction as $R_{\rm{had}} = 3.6291\, \pm\, 0.0086$.

Non-strange width is $R_{\rm{non-strange}} = R_{\rm{had}} - R_{\rm{strange}}$,
where the value for the strange width $R_{\rm{strange}}$ = $0.1613\, \pm\, 0.0028$
is obtained from the sum of the strange branching fractions
with the unitarity constrained fit as listed in Table~\ref{tab:TauGlobalFit}.
This gives a value of  $\ensuremath{|V_{us}|}\xspace$ $=$ $0.2174\, \pm\, 0.0022$,
which is $3.3~\sigma$ lower than the CKM unitarity prediction.

\begin{figure}[!hbtp]
\begin{center}
\includegraphics[height=.38\textheight,width=.49\textwidth]{figures/Vus.eps}
\end{center}
\caption{Measurements of $\ensuremath{|V_{us}|}\xspace$ from kaon, hyperon and tau decays.}
\label{fig:Vus}
\end{figure}

Summary of these $\ensuremath{|V_{us}|}\xspace$ values are plotted in Figure~\ref{fig:Vus},
where we also include values from hyperon and kaon decays~\cite{Antonelli:2010yf}.

\section{Search for lepton flavor violation in $\tau$ decays}
\label{sec:Tau_LFV}

\begin{figure*}[!hbtp]
\begin{center}
\includegraphics[height=.4\textheight]{figures/TauLFV_UL_2010001.eps}
\end{center}
\caption{Status of searches for lepton flavor violation in $\tau$ decays.}
\label{fig:Tau_LFV}
\end{figure*}

The status of searches for lepton flavor violation in $\tau$ decays is
summarized in Figure~\ref{fig:Tau_LFV}. A table of these results and
the corresponding references are provided on the HFAG-Tau web site.

redpajama_set_name: RedPajamaArXiv