e\gamma$ and $\tau \rightarrow \mu(e)\gamma$. The stringent bounds for these decays are [@PDG] $$\begin{aligned}
\label{LFVbound}
B(\mu\rightarrow e\gamma) < 1.2\times10^{-11}, \quad B(\tau\rightarrow e\gamma) < 1.1\times10^{-7}, \quad B(\tau\rightarrow\mu\gamma) < 4.5\times10^{-8}\end{aligned}$$ at $90\%$ C.L. If the photon is on shell, we can write the invariant amplitude for the decays $l_{j} \rightarrow l_{i}\gamma$ in the form $$\begin{aligned}
\emph{A}_{l_{j}l_{i}\gamma} = \frac{ie}{8\pi}\epsilon^*_{\mu}(q)\bar{u}_{i}(p-q)\left[\sigma^{\mu\nu}q_{\nu}(c_{1}P_{L} + c_{2}P_{R})\right]u_{j}(p) \;\;, \end{aligned}$$ where the matrices $P_{L}$ and $P_{R}$ are defined by $P_{L} = (1 - \gamma_{5})/2$, $P_{R} = (1 + \gamma_{5})/2$ and $\sigma^{\mu\nu} = (i/2)[\gamma^{\mu},\gamma^{\nu}]$. The coefficients $c_{1}$ and $c_{2}$ were calculated in [@He:2002pva] and are $$\begin{aligned}
c_{1} = \frac{Y_{4}Y_{S2}m_{\mu}}{\pi m^2_{H}}H(z) + \frac{Y^2_{S2}m_{L+R,3}}{\pi m^2_{H}}G(z) \quad {\rm and} \quad c_{2} = \frac{Y_{4}Y_{S2}m_{\mu}}{\pi m^2_{H}}H(z) + \frac{Y^2_{4}m_{L+R,3}}{\pi m^2_{H}}G(z)\end{aligned}$$ with $$\begin{aligned}
H(z) &=& -\frac{1}{6(1 - z)}\left[1 + \frac{3}{1 - z} - \frac{6}{(1 - z)^2} - \frac{6z}{(1 - z)^3\ln{z}}\right] \;\;, \\
G(z) &=& -\frac{1}{1 - z}\left[1 + \frac{2}{1 - z} + \frac{2}{(1 - z)^2\ln{z}}\right] \;\;.\end{aligned}$$ The mixings between the charged leptons are ignored as before. If we take $m_{H} = 1$ TeV as the new physics scale and the limit of $m_{L+R,3} \approx m_{H}$, the anomalous muon magnetic moment and the branching ratio of $\mu\rightarrow e\gamma$ becomes $$\begin{aligned}
\Delta a_{\mu} \approx 5.1\times10^{-13}\end{aligned}$$ and $$\begin{aligned}
B(\mu\rightarrow e\gamma) \approx 