text:
\section{\label{sec-0}Introduction}
So far no signal of new physics has been observed but its search is important in order to understand how to enlarge the Standard Model (SM) to solve
both the hierarchy and the flavor problems. One of the most significant observations requiring us to go beyond the Standard Model
is the assessment that neutrinos are massive and that they mix~\cite{Gonzalez-Garcia:2014bfa}.
Possible SM extensions aiming at incorporating massive neutrinos give rise to interesting collider signatures and open the door
to new phenomena such as lepton flavor violating (LFV) decays.

Currently, the search for manifestations of LFV constitutes a goal of
several experimental facilities dedicated to rare lepton decays, such as $\ell_\alpha \to \ell_\beta \gamma$ and $\ell\to \ell_\alpha\ell_\beta \ell_\gamma$, and to the neutrinoless
$\mu-e$ conversion in muonic atoms. One of the most stringent bounds from these searches is the one derived by the MEG Collaboration,
$\text{B}(\mu \to e \gamma) < 5.7 \times 10^{-13}$~\cite{Adam:2013mnn}, which is expected to be improved to a planned sensitivity of $6 \times 10^{-14}$~\cite{Baldini:2013ke}.
Moreover, the bound  $\text{B}(\mu \to eee)
< 1.0 \times 10^{-12}$, set by the SINDRUM experiment~\cite{Bellgardt:1987du}, is expected to be improved by the Mu3e experiment where a sensitivity $\sim 10^{-16}$ is planned~\cite{Blondel:2013ia}.
Limits on the $\tau$ radiative decays~\cite{Aubert:2009ag} and the three-body
decays of $\tau$~\cite{Hayasaka:2010np,Aushev:2010bq} appear to be less
stringent right now, but are likely to be improved at Belle II~\cite{Aushev:2010bq}, where the search for LFV decays of the $B$-meson will be made too~\cite{Bevan:2014iga}.
The most promising  developments regarding LFV  are those related to the $\mu-e$ conversion in nuclei. The present bound for the  $\mu^- \mathrm{Ti} \rightarrow e^- \mathrm{Ti}$ conversion rate is  $4.3\times
10^{-12}$~\cite{Dohmen:1993mp}, and the planned sensitivity is $\sim10^{-18}$~\cite{Alekou:2013eta}. Similar is the case for gold and aluminum~\cite{Bertl:2006up,Kuno:2013mha}.

Searches for  LFV are also conducted in high-energy experiments and a first bound on the Higgs boson LFV decay $h \to \mu \tau$ has been reported by the CMS Collaboration
 \cite{CMS:2014hha}. The LHCb Collaboration, instead, reported the bound $\text{B}(\tau \to 3 \, \mu)< 8.0 \times 10^{-8}$~\cite{Aaij:2013fia}, which is likely to be improved in the near future~\cite{PDG}.
 Notice also that they already improved the bounds on $\text{B}(B_{(s)} \to e \mu)$ by an order of magnitude~\cite{Aaij:2013cby}. \\

In this work we will focus on the indirect probes of new physics through the LFV processes of  neutral vector bosons, namely
$V\to \ell_\alpha\ell_\beta$, with $\ell_{\alpha,\beta}\in \{e,\mu, \tau\}$, and $V\in \{ \phi, \psi^{(n)}, \Upsilon^{(n)}, Z\}$, where $\psi^{(n)}$ stands for $J/\psi$ and its radial excitations, and similarly for $\Upsilon^{(n)}$.
Most of the research in this direction reported so far is related to the $Z\to \ell_\alpha\ell_\beta$ decay modes.
More specifically, the experimental bounds,  obtained at LEP are found to be   $\mathrm{B}(Z\to e^\mp \mu^\pm) < 1.7 \times 10^{-6}$~\cite{Abreu:1996mj}, $\text{B}(Z\to \mu^{\mp}\tau^{\pm})< 1.2\times 10^{-5}$~\cite{Akers:1995gz,Abreu:1996mj},
and $\text{B}(Z\to e^{\mp}\tau^{\pm})< 9.8 \times 10^{-6}$~\cite{Akers:1995gz,Adriani:1993sy}.
One of these bounds has been improved at LHC, namely $\text{B}(Z\to e^{\mp}\mu^{\pm})< 7.5\times 10^{-7}$~\cite{Aad:2014bca}. On the theory side, the $Z$ decays have been analyzed in the extensions of the SM involving
additional massive and sterile neutrinos that could mix with the standard (active) ones and thus give rise to the LFV decay rates~\cite{Mann:1983dv,Ilakovac:1994kj,Illana:1999ww}. A similar approach has been also adopted in Ref.~\cite{Abada:2014cca}, in the perspective of
a Tera-$Z$ factory FCC-ee~\cite{FCC-WG} for which a targeted sensitivity is expected to be $\text{B}(Z\to e^{\mp}\mu^{\pm}) \,\sim \, 10^{-13}$~\cite{Blondel:2014bra}.

Lepton flavor conserving decays of quarkonia have been measured to a high accuracy which can actually be used to fix the hadronic parameters (decay constants). Otherwise, one can use the results of
numerical simulations of QCD on the lattice, which are nowadays accurate as well~\cite{c-latt1,c-latt2,Lepage,RL}. The experimentally established bounds for the simplest LFV decays of quarkonia are~\cite{PDG}:
\begin{eqnarray}
&&\text{B}(\phi \to e \mu) < 2.0\times 10^{-6},   \qquad  \text{\cite{Achasov:2009en}}\nonumber \\
&&\hfill \nonumber\\
&&\text{B}(J/\psi \to e \mu) < 1.6\times 10^{-7}, \qquad \text{B}(J/\psi \to e \tau) < 8.3\times 10^{-6},  \nonumber \\
 &&\text{B}(J/\psi \to  \mu \tau) < 2.0\times 10^{-6} ,  \qquad  \text{\cite{Ablikim:2013qtm,Ablikim:2004nn}} \nonumber \\
&&\hfill \nonumber\\
&&\text{B}(\Upsilon \to  \mu \tau ) < 6.0\times 10^{-7} ,   \qquad \text{\cite{Love:2008ys}}   \nonumber \\
&&\hfill \nonumber\\
&& \text{B}(\Upsilon(2S) \to e \tau) < 8.3\times 10^{-6},\qquad  \text{B}(\Upsilon(2S) \to  \mu \tau) < 2.0\times 10^{-6}, \qquad \text{ \cite{Lees:2010jk}} \nonumber \\
&& \text{B}(\Upsilon(3S) \to e \tau) < 4.2\times 10^{-6},\qquad  \text{B}(\Upsilon(3S) \to  \mu \tau) < 3.1\times 10^{-6} , \qquad \text{ \cite{Lees:2010jk}}  \nonumber
\end{eqnarray}
where each mode is to be understood as $\text{B}(V \to \ell_\alpha\ell_\beta) = \text{B}(V \to \ell_\alpha^+ \ell_\beta^-)+ \text{B}(V \to \ell_\alpha^- \ell_\beta^+)$.

Despite the appreciable experimental work on the latter observables, only a few theoretical studies have been carried out so far.
The authors of Ref.~\cite{Nussinov:2000nm} applied a vector meson dominance approximation to $\mu \to 3 e$ and expressed the width of the latter process,
$\Gamma(\mu \to ee e)=\Gamma(\mu\to V e )\Gamma(V\to ee)$. Since the values of $\Gamma(V\to ee)$ are very well known experimentally~\cite{PDG}, the experimental
bound on $\Gamma(\mu \to 3 e)$ is then used to obtain an upper bound on the phenomenological coupling $g_{V\mu e}$, which is then converted to an upper bound on $\Gamma( V \to \mu e )$.
A similar approach has been used in Ref.~\cite{Gutsche:2009vp} where instead of $\mu \to eee$, the authors considered the $\mu-e$ conversion in nuclei ($N$), which they described in terms
of a product of couplings $g_{V\mu e}$ and $g_{V N N}$.
The latter could be extracted from the experimentally measured $\Gamma(V \to p\bar p)$, and with that knowledge the experimental upper bound on  $\text{R}(\mu \text{Ti}\to e\text{Ti})$ results in an upper bound
on $\Gamma( V \to \mu e )$.
A more dynamical approach in modeling the $V \to \ell_\alpha\ell_\beta$ processes has been made in a supersymmetric extension of the SM with type I seesaw~\cite{Sun:2012yq}.

Sterile fermions were proposed in various neutrino mass generation mechanisms, but the interest in their properties was further motivated by the reactor/accelerator
anomalies~\cite{reactor:I,Aguilar:2001ty,miniboone:I,gallium:I},  a possibility to offer a warm dark matter
candidate~\cite{Dodelson:1993je,Abazajian:2001nj,nu_WDM}, and by indications from the large scale structure
formation~\cite{general_structure,Kusenko:2009up,Abazajian:2012ys}.

Incorporating neutrino oscillations (masses and mixing~\cite{Gonzalez-Garcia:2014bfa}) into the  SM implies that the  charged current
is modified to
\begin{equation}\label{eq:cc-lag1}
- \mathcal{L}_\text{cc} = \frac{g}{\sqrt{2}} U^{\alpha i}
\bar{\ell}_\alpha \gamma^\mu P_L \nu_i  W_\mu^- + \, \text{c.c.}\,,
\end{equation}
$U$ being the leptonic mixing matrix,  $\alpha$ the flavor of a charged lepton, and
$i = 1, \dots, n_\nu$ denotes a physical neutrino state. If one assumes that  only three massive neutrinos are present,
the matrix $U$ corresponds to the unitary Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. In that situation the GIM mechanism makes the decay
rates B($V \to \ell_\alpha^\mp \ell_\beta^\pm$) completely negligible, $\lesssim10^{-50}$. That feature, however, can be drastically
changed in the presence of a non-negligible mixing with heavy sterile fermions.
In what follows we will consider such situations, derive analytical expressions for B($V \to \ell_\alpha\ell_\beta$), and discuss
a specific realization of the inverse seesaw mechanism, known as (2,3)-ISS~\cite{Abada:2014vea}. We will also discuss a
simplified model in which the effect of the heavy sterile neutrinos is described by one effective sterile neutrino state with non-negligible mixing with active neutrinos.~\footnote{In this work,
due to the tension between the most recent Planck results on extra light neutrinos (relics) and the reactor/accelerator  anomalies, we will consider the effect of (heavier) sterile neutrinos not
contributing as light relativistic degrees of freedom~\cite{Ade:2013zuv}. We will require our models to be compatible with current experimental data and constraints and to fulfill the so-called perturbative
unitary condition which puts a strong constraint on the models for the very heavy sterile fermion(s)~\cite{Chanowitz:1978mv}.}
Despite several differences, our approach is similar to the one discussed in Ref.~\cite{Ilakovac:1999md}, where the SM has been extended by new, heavy, Dirac neutrinos, singlets under $SU(2)\times U(1)$, and
applied to a number of low energy decay processes. Our sterile neutrinos are Majorana and we apply the approach to the leptonic decays of quarkonia for the first time.

The remainder of this paper is organized as follows: In Sec.~\ref{sec:obs} we formulate the problem in terms of a low energy effective theory of a larger theory which contains heavy sterile neutrinos,
we derive expression for B($V \to \ell_\alpha  \ell_\beta $) and compute the Wilson coefficients.  In Sec.~\ref{sec:ext} we briefly describe the specific models with sterile neutrinos which are used in this paper to produce our results presented
in Sec.~\ref{sec:results}. We finally conclude in Sec.~\ref{sec:concl}.

\section{LFV decay of Quarkonia - Effective Theory\label{sec:obs}}

In this section we formulate a low energy effective theory of the LFV decays of quarkonia of type $V\to \ell_\alpha^\pm\ell_\beta^\mp$, and express the decay amplitude in terms of the quarkonium decay constants and the corresponding Wilson coefficients.
The latter are then computed in the extensions of the SM which include the heavy sterile neutrinos.  We also derive the expression relevant to  $\Gamma(Z\to \ell_\alpha^\pm\ell_\beta^\mp)$.

\subsection{Effective Hamiltonian}

Keeping in mind the fact that we are extending the SM by adding sterile fermions, without touching the gauge sector of the theory, the decays of vector quarkonia,
$V (q) \to \ell_\alpha^\pm (p)\ \ell_\beta^\mp (q-p)$, can only occur through the photon and
the $Z$-boson exchange at tree level. In the lepton flavor conserving processes the $Z$-exchange terms are very small with respect to those arising from the electromagnetic interaction and are usually neglected. The generic effective Hamiltonian can be written as
\begin{equation}\label{Heff:V}
	\mathcal{H}_{\text{eff}} = {\cal Q}_Q \frac{e^2 g^2}{2 m_V^2} \, \bar{Q} \gamma_\mu Q\, \cdot\, \bar{\ell}_\alpha \left[C_{VL} \gamma^\mu P_L +C_{VR} \gamma^\mu P_R +  \frac{p^\mu}{m_W} (C_R P_R + C_L P_L)\right] \ell_\beta,
\end{equation}
where $ {\cal Q}_Q$ is the electric charge of the quark $Q$, $m_V$ is the mass of quarkonium $V$ which is dominated by the valence quark configuration $\bar Q Q$,~\footnote{We remind the reader that the ground vector meson $\bar s s$, $\bar c c$, $\bar b b$ states are $\phi$, $J/\psi$, $\Upsilon$, respectively, and the corresponding charges are  $ {\cal Q}_{s,b}=-1/3$ and  $ {\cal Q}_c=2/3$.}  $C_{VL,VR,L,R}$ are the Wilson coefficients,  $p$ is the momentum of one of the outgoing leptons, and $P_{L/R}= \frac{1}{2}(1 \mp \gamma_5)$. Contributions to the scalar (left and right) terms are suppressed by $m_{\alpha,\beta}/m_W$, where $m_{\alpha,\beta}$ are the charged lepton masses. In this section we will keep such terms so that our expressions can be useful to approaches in which the scalar bosons are taken  in consideration. For our phenomenological discussion, however, it is worth emphasizing that $C_{L,R,VR}\ll C_{VL}$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=.38]{fig1a.png}~\includegraphics[scale=.38]{fig1b.png}
\caption{{\footnotesize
Diagrams contributing the LFV decay of quarkonia $V\to \ell_\alpha \ell_\beta$. The blob in the first diagram is related to the penguin loop that generates the LFV, and the box diagram is particularly important to be included in the case of
$\Upsilon^{(n)}\to   \ell_\alpha \ell_\beta$ because of $V_{tb}\simeq 1$ and of the top quark mass, making the box diagram contribution to the Wilson coefficient significant.
 }}
\label{fig:1}
\end{center}
\end{figure}

Without entering the details of calculation it is easy to verify that the only relevant diagrams are those shown in Fig.~\ref{fig:1},
and therefore the structure of  the Wilson coefficients $C_i$ reads,
\begin{equation}
C_{i}=C^\gamma_{i} +  C^Z_{i} \frac{1}{\sin^2\theta_W\cos^2\theta_W}\frac{m_V^2}{m_V^2-m_Z^2} \frac{g_{V}^{\cal Q}}{{\cal Q}_Q}+C^{\mathrm{Box}}_{i} |V_{Qq}|^2\frac{1}{\sin^2\theta_W} \frac{m_V^2}{m_W^2} \frac{1}{{\cal Q}_Q},
\end{equation}
where $C^{\gamma,Z}_i$ are  the contributions arising from the photon and the $Z$-boson exchange, while $C_i^{\mathrm{Box}}$ comes from the box diagram that  involves the Cabibbo--Kobayashi-Maskawa
coupling $V_{Qq}$.~\footnote{The box diagram contribution to $V \to \ell_\alpha\ell_\beta$ in the case of $V=\Upsilon$ is dominated by the top quark ($|V_{tb}|\simeq 1$); for $V= \psi$ it is negligible because
the contribution of the $b$ quark is Cabibbo suppressed ($|V_{cb}|\simeq 0.004$) while the Cabibbo allowed one ($|V_{cs}|\simeq 0.99$) is suppressed by the strange quark mass; for $V=\phi$, the
contributions of the charm and top quarks are comparable but overall smaller than in the $\Upsilon \to \ell_\alpha\ell_\beta$ case.
} In the above expressions $g_V^Q=\frac{1}{2}I_3^Q-{\cal Q}_Q \sin^2\theta_W$.
The blob in the diagram shown in Fig.~\ref{fig:1} stands for the lepton loop diagrams that may contain one or two neutrino states and which, in the extensions of the SM involving a heavy neutrino sector, will give rise to the LFV decay  due to the effect of mixing which is parametrized by the matrix $U$ [see Eq.~(\ref{eq:cc-lag1})].
Separate contributions coming from different diagrams can be further reduced by factoring out the neutrino mixing matrix elements, namely
\begin{equation}
C_{i}^{\gamma,\mathrm{Box}}=\sum_{k=1}^{n_\nu} U_{\beta k} U_{\alpha k}^* C^{\gamma,\mathrm{Box}; k}_{i},  \quad \text{and} \quad C_{i}^Z=\sum_{k=1}^{n_\nu} U_{\beta k} U_{\alpha k}^* C^{Z, k}_{i}+\sum_{k=1}^{n_\nu}\sum_{j=1}^{n_\nu} U_{\beta k} U_{\alpha j}^* C^{Z, kj}_{i},
\end{equation}
where we see that the term involving two neutrino eigenstates appears only in the $Z$ coefficient because it is related to the vertex $Z \nu_k\nu_j$.
It is worth emphasizing that the tensor structure in Eq.~(\ref{Heff:V})
can be easily obtained from the coefficients $C_{L,R}$  by applying the Gordon identity.
Such contributions are $1/m_W$ suppressed, and thus completely negligible,  which is why we do not give explicit expressions for these coefficients.

Using the effective Hamiltonian (\ref{Heff:V}) and parameterizing the hadronic matrix as
\begin{eqnarray}
\langle 0 \vert  \bar Q \gamma_\mu Q \vert V(q,\sigma) \rangle = f_V m_V \varepsilon_\mu^\sigma\,,
\end{eqnarray}
where $f_V$ is the decay constant of a quarkonium $V$ with momentum $q$ and in a polarization state $\sigma$, we can write the decay rate as,
\begin{align}
\Gamma(V\to{\ell}_\alpha^- \ell_\beta^+ ) = \frac{8 \pi {\cal Q}_Q^2 \alpha^2}{3 m_V^3}  G_F^2 m_W^4 \left(\frac{f_V}{m_V}\right)^2  {\lambda^{1/2}(m_V^2,m_\alpha^2,m_\beta^2)} \phi_C,
\end{align}
with
\begin{equation}
\lambda(a^2,b^2,c^2)=[a^2-(b-c)^2][a^2-(b+c)^2],\label{lambdadef}\end{equation} and
\begin{align}
\phi_C=\left(-g^{\mu \nu}+\frac{q^\mu q^\nu}{m_V^2} \right) \mathrm{tr} \Big[(\slashed{q}-\slashed{p}+m_\beta) \cdot (C_{VL} \gamma^\mu P_L + C_{VR} \gamma^\mu P_R+C_L \frac{p^\mu}{m_W}P_L+C_R \frac{p^\mu}{m_W}P_R) \nonumber \\
\cdot (\slashed{p}-m_\alpha)\cdot (C_{VL}^* \gamma^\nu P_L + C_{VR}^* \gamma^\nu P_R+C_L^* \frac{p^\nu}{m_W}P_R+C_R^* \frac{p^\nu}{m_W}P_L)\Big],
\end{align}
which gives
\begin{align}
\phi_C=\frac{1}{4 m_V^2 m_W^2} &\Big{\lbrace}\lambda(m_V^2,m_\alpha^2,m_\beta^2)\Big[(m_V^2-m_\alpha^2-m_\beta^2)(|C_L|^2+|C_R|^2)-4\mathrm{Re} (C_L^* C_R) m_\alpha m_\beta \nonumber \\
&+4 m_W\mathrm{Re}(C_L^* (C_{VL} m_\beta+C_{VR} m_\alpha)+C_R^* (C_{VL} m_\alpha+ C_{VR} m_\beta))\Big] \nonumber\\
&+ 4 m_W^2 (|C_{VL}|^2+|C_{VR}|^2) \Big[2 m_V^4 - m_V^2(m_\alpha^2+m_\beta^2)- (m_\alpha^2-m_\beta^2)^2 \Big]\\
&+ 48 m_W^2 m_V^2 m_\alpha m_\beta   \mathrm{Re} (C_{VL}^* C_{VR}) \nonumber
 \Big{\rbrace}.
\end{align}
As we mentioned above, we consider in our framework $C_{VL}\gg C_{VR,R,L}$, and therefore we can write
\begin{align}
\Gamma (V\to \ell_\alpha^\pm\ell_\beta^\mp)=\frac{32\pi {\cal Q}_Q^2 \alpha^2}{3 m_V^3}  f_V^2 G_F^2 m_W^4   |C_{VL}|^2
 \lambda^{1/2}(m_V^2,m_\alpha^2,m_\beta^2)\Big{[}1-  \frac{(m_\alpha^2+m_\beta^2)}{2 m_V^2}-\frac{(m_\alpha^2-m_\beta^2)^2}{2 m_V^4}\Big{]},
\end{align}
where $\lambda(a^2,b^2,c^2)$ is given in Eq.~(\ref{lambdadef}). In this last expression we also used $\Gamma (V\to \ell_\alpha^\pm\ell_\beta^\mp) = \Gamma (V\to \ell_\alpha^+\ell_\beta^-)+\Gamma (V\to \ell_\alpha^-\ell_\beta^+)$.

Besides quarkonia we will also revisit the issue of adding extra species of sterile neutrinos to the decay of $Z\to \ell_\alpha^\pm\ell_\beta^\mp$. In that case the effective Hamiltonian can be written as
\begin{equation}
\mathcal{H}^Z_{\mathrm{eff}}= \frac{g^3}{2 \cos\theta_W} \bar{\ell}_\alpha \Big{[} D_{VL}\gamma^\mu P_L + D_{VR}\gamma^\mu P_R +D_{L}P_L+D_{R}P_R\Big{]}\ell_\beta Z^\mu,
\end{equation}
where the Wilson coefficients are now denoted by $D_i$ and take the form
\begin{equation}
	D_{i}=\sum_{k=1}^{n_\nu} U_{\beta k} U_{\alpha k}^* C^{Z, k}_{i}+\sum_{k =1}^{n_\nu}\sum_{j=1}^{n_\nu} U_{\beta k} U_{\alpha j}^* C^{Z, kj}_{i}\,.
\end{equation}
The decay rate in the similar limit, $D_{VL}\gg D_{VR,R,L}$, reads
\begin{align}
\label{gammaz}
\Gamma(Z\to{\ell}_\alpha^- \ell_\beta^+ ) = \frac{8 \sqrt{2}}{3 \pi m_Z} \frac{G_F^3 m_W^6}{\cos^2\theta_W} |D_{VL}|^2 \lambda^{1/2}(m_Z^2,m_\alpha^2,m_\beta^2) \left[1-\frac{(m_\alpha^2+m_\beta^2)}{2 m_Z^2}-\frac{(m_\alpha^2-m_\beta^2)^2}{2 m_Z^4}\right].
\end{align}

\subsection{Wilson coefficients}
Concerning the computation of the Wilson coefficients we stress again that our results are obtained in a theory in which the Standard Model is extended to include extra species of sterile fermions, without changing the gauge sector.
The origin  of the leptonic mixing matrix $U$ is model dependent and in order to be able to do a phenomenological analysis,  we will have to adopt a specific model which will be discussed in the next section.

The blob in the diagram shown in Fig.~\ref{fig:1} stands for a series of diagrams such as those displayed in Fig.~\ref{fig:2}. All of them, including the box diagram in Fig.~\ref{fig:1}, have been computed in the Feynman gauge and the results are collected in Appendix~A.
\begin{figure}[t!]
\hspace*{-11mm}\begin{tabular}{cccccc}
\includegraphics[scale=.28]{fig2a.png}&$\raise 7.4ex \hbox{=}$&\includegraphics[scale=.28]{fig2b.png}&$\raise 7.4ex \hbox{+}$&\includegraphics[scale=.28]{fig2c.png}&$\raise 7.4ex \hbox{+~\dots}$
  \end{tabular}
  \caption{{\footnotesize
Vertex diagrams contributing the LFV decays.
 }}
\label{fig:2}
\end{figure}
Here we focus on the most important contributions in the case of large masses of sterile (Majorana) neutrinos. Contributions to the Wilson coefficients coming from vertex diagrams can be divided into two pieces: those involving only one neutrino in the loop, $C_{VL}^{Z,\gamma}(x_i)$, where $x_i = m_i^2/m_W^2$, and those with two neutrinos in the loop, $C_{VL}^{Z }(x_i ,x_j)$. In the limit of large values of $x_{i,j} \gg 1$, we find the following behavior
\begin{align}\label{eq:Wils}
C_{VL}^{Z}(x_i)  \stackrel{x_{i}\gg 1}{\xrightarrow{\hspace*{9mm}}} &\,  \frac{5}{32\pi^2} \log x_i + \text{\small finite term} + {\cal O}(1/x_i) \sim \log x_i\, , \cr
 C_{VL}^{Z }(x_i ,x_i) \stackrel{x_{i}\gg 1}{\xrightarrow{\hspace*{9mm}}} &\,    \frac{C_{ii}}{64 \pi^2}\left\lbrace  \left( 2 x_i +3 -4 \log x_i\right)+x_i\left(\log x_i-\frac{7}{2}\right)\right\rbrace +\dots \cr  & \ \sim C_{ii}\ x_i \log x_i +  \dots
\end{align}

To illustrate the relative contribution of the different diagrams we fix the values of the coefficients $C_{ij}\equiv \displaystyle{\sum_{\alpha=e,\mu,\tau}} U_{\alpha i}^\ast U_{\alpha j} = 10^{-5}$, and plot  $|C_{VL}(x_i)-C_{VL}(0)|$ and $|C_{VL}^{Z}(x_i,2\ x_i)-C_{VL}^{Z}(0,0)|$ for the case of $\Upsilon \to \mu\tau$, cf. Fig.~\ref{fig:3}.~\footnote{Due to the unitarity of the mixing matrix $U$, the terms in the Wilson coefficients that do not depend on neutrino masses give a vanishing contribution after summing over all neutrino states. We thus subtract the constant terms in the plots in order to better appreciate the dependence on the neutrino masses. Notice also that $C_{ij}= 10^{-5}$ is in agreement with all constraints discussed in the text when the neutrino masses are below $\mathcal{O}(100)$ TeV.}  We see that only for very large masses the diagrams with two neutrinos in the loop become more important than those with one neutrino state.  We should stress that each contribution to $C_{VL}(x_i)$, i.e.  $C_{VL}^{\rm Box}(x_i)$ and $C_{VL}^{Z}(x_i)$, scales as $\log x_i$  for large values of $x_i$, except for $C_{VL}^{\gamma}(x_i)$ which goes to a constant in the same limit. That can also be seen in Fig.~\ref{fig:3} where in the left panel we show the dependence of the total $C_{VL}(x_i)$ on $x_i$ and in the right panel we show  $C_{VL}^{\gamma}(x_i)$ and its dependence on the mass of the initial decaying meson, $\phi$, $J/\psi$, and $\Upsilon$.  The contribution of sterile neutrinos to the LFV decay of $\Upsilon$ is larger than the one to lighter mesons, since the Wilson coefficients are also proportional to the mass of the initial particle.

\begin{figure}[h!]
\begin{center}
\hspace*{-4mm}\includegraphics[scale=.42]{fig3a.png}~\includegraphics[scale=.42]{fig3b.png}
\caption{{\footnotesize
In the left panel are shown $C_{VL}(x_i)$ and $C_{VL}^Z(x_i ,x_j)$, for $x_j=2 x_i$,  as functions of $m_i =  m_W \sqrt{x_i}$,  the mass of the heavy sterile neutrino propagating in the loops. For illustration purpose, the couplings $C_{ij}$ were fixed to a common value, $10^{-5}$, and the example corresponds to the $\Upsilon \to \mu\tau$ decay. Right panel:  $C_{VL}^{\gamma}(x_i)$  is plotted as a function of $m_i$ for the case of $V\to e\mu$ in three specific cases $V\in \{\phi, J/\psi,\Upsilon\}$. In both cases the value of functions at $x_{i,j}=0$ have been subtracted away. }}
\label{fig:3}
\end{center}
\end{figure}
 Before closing this section we should reiterate that our  Wilson coefficients have  been computed in the Feynman gauge. Since  all divergencies cancel out,  our results are finite and gauge invariant, as was already  observed in Refs.~\cite{Mann:1983dv,Ilakovac:1994kj,Illana:1999ww,Abada:2014cca}.

\section{SM in the presence of sterile fermions \label{sec:ext} }
With the expressions derived above, we now have to specify a model for lepton mixing (couplings) $U_{\alpha i}$ in  the presence of heavy sterile neutrinos propagating in the loops.
We opt for a minimal realization of the inverse seesaw mechanism for the generation of neutrino masses, which is nowadays rather well constrained by the available experimental data.
Furthermore, we will use a parametric model containing one effective sterile neutrino, which essentially mimics the behavior at low energy scales of mechanisms involving heavy sterile fermions.

\subsection{The (2,3)-inverse seesaw realization}
Among many possible realizations of accounting for massive neutrinos, the inverse seesaw mechanism (ISS)~\cite{ISS}
offers the possibility of accommodating
the smallness of the active neutrino masses for a comparatively  low seesaw scale, but still with natural $\mathcal{O}(1)$
Yukawa couplings, which renders  this scenario phenomenologically appealing. Indeed, depending on their masses and mixing with active neutrinos, the new  states can be
produced in collider and/or low energy experiments, and their
contribution to physical processes can be sizable.
 ISS, embedded in the SM, results in a mass term for neutrinos of the form
\begin{equation}
-\mathcal{L}_\text{mass}=\frac{1}{2}n_L^T C M n_L+\mbox{h.c.}\ ,
\end{equation}
where $C \equiv i \gamma^2 \gamma^0$ is the charge conjugation matrix and $n_L \equiv {\left(\nu_{L,\alpha},\nu_{R,i}^c,s_j\right)}^T$\!.  Here $\nu_{L,\alpha}$, $\alpha=e,\mu,\tau$ denotes the active (left-handed)  neutrino states of the SM, while $\nu_{R,i}^c$ ($i=1,  \#\nu_R$) and $s_j$ ($j=1, \# s$) are right-handed neutrino fields and  additional fermionic gauge singlets, respectively.  The  neutrino mass matrix $M$ then has the form
\begin{equation}\label{general-iss}
M \equiv
\left(
\begin{array}{ccc}
0 & d & 0 \\
d^T & 0 & n \\
0 & n^T & \mu
\end{array}
\right)\ ,
\end{equation}
where $d, n,\mu$ are complex matrices.~\footnote{It is in general possible to consider also a nonzero value for the central entry of the matrix (\ref{general-iss}), with elements at a mass scale similar to the one of $\mu$. These parameters, however, only affect neutrino masses and mixing at loop level~\cite{Dev:2012sg}, which is why we do not consider them here.}\\
The Dirac mass matrix $d$ arises from the Yukawa couplings to the SM Higgs boson, $\tilde{H} =i  \sigma^2 H$,
\begin{equation}
Y_{\alpha i} \overline{\ell_L^\alpha}\tilde{H }\nu_R^i+\text{H.c., }\,\,\,\,\,\,\ell_L^\alpha=\left(
\begin{array}{c}
\nu_{L}^\alpha \\
e^\alpha_L
\end{array}
\right)\ ,
\end{equation}
while the matrix $\mu$, instead, contains the Majorana mass terms for the sterile fermions $s_j$.
By assigning a leptonic charge  $L=+1$ to both $\nu_R$ and $s$, one makes sure that the off diagonal terms are lepton number conserving, while $s^T Cs$ violates the lepton number by two units.
Furthermore, the interesting feature of this seesaw realization is that the entries of  $\mu$ can be made small in order to accommodate for the ${\mathcal{O}}(\text{eV}$) masses of active neutrinos, with large Yukawa couplings.
This is not in conflict with naturalness since the lepton number is restored in the limit of $\mu \rightarrow 0$.~\footnote{In this work we consider configurations in which the entries in the above matrices fulfill a {\sl naturalness criterion}, $|\mu|\ll |d|<|n|$~\cite{Abada:2014vea}.}

Concerning the additional sterile states $\nu_R$ and $s$, since up to now there is no direct evidence for their existence and because they do not contribute to anomalies, their number is unknown.
In Ref.~\cite{Abada:2014vea}  it was shown that it is possible to construct several minimal distinct realizations of ISS, each reproducing  the correct neutrino mass spectrum and  satisfying all phenomenological constraints.
More specifically, it was shown that, depending on the number of additional fields, the neutrino mass spectrum obtained for each ISS realization is characterized by either two or three mass scales, one corresponding to $m_\nu \approx \mu\ d^2/n^2$ (light neutrino masses), one corresponding to the heavy mass eigenstates [the mass scale of the matrix $n$ of Eq.~(\ref{general-iss})], and finally an intermediate scale $\sim\mu$,  only present if $\# s > \#\nu_R$.
This allows us to identify  two truly minimal ISS realizations that comply with all experimental bounds, namely
the (2,2)-ISS model, which corresponds to the SM extended by two right-handed (RH) neutrinos and two additional sterile states, leading  to a three-flavor mixing scheme, and
the (2,3)-ISS realization, where the SM is extended by two RH neutrinos and three sterile states leading to a 3+1-mixing scheme. Interestingly, the lightest sterile neutrino with a mass around eV in the
(2,3)-ISS  can be used to explain the short baseline (reactor/accelerator) anomaly~\cite{reactor:I,Aguilar:2001ty,miniboone:I,gallium:I} if its mass lies around eV, or to provide a dark matter candidate if the lightest sterile state were in the keV range~\cite{Abada:2014zra}.

\subsection{A model with one effective sterile fermion}

Since the generic idea of obtaining a significant contribution to our observables applies to any model
in which the active neutrinos have sizable mixing with some additional singlet states (sterile fermions),
we can use an \textit{effective} model with three light active neutrinos plus one extra sterile neutrino.

The introduction of this extra state implies three new active-sterile mixing angles ($ \theta_{14}, \theta_{24}, \theta_{34}$), two extra Dirac $CP$ violating phases ($\delta_{14},\delta_{34}$) and one additional Majorana phase ($\phi_{41}$).
The lepton mixing matrix is then a product of six rotations times the Majorana phases, namely
\begin{eqnarray} \label{eq:3+1rot}
U &=& R_{34}(\theta_{34},\delta_{34}) \cdot R_{24}(\theta_{24}) \cdot R_{14}(\theta_{14},\delta_{14})
\cdot R_{23} \cdot R_{13} \cdot R_{12}  \cdot \rm diag(\phi_{21},\phi_{31},\phi_{41}) \nonumber \\
&=& R_{34}(\theta_{34},\delta_{34}) \cdot R_{24}(\theta_{24}) \cdot R_{14}(\theta_{14},\delta_{14})
\cdot U_{\rm PMNS} \cdot \rm diag(\phi_{21},\phi_{31},\phi_{41})\,,
\end{eqnarray}
where the rotation matrices $R_{34},R_{24},R_{14}$ can be defined as:
\begin{eqnarray} \label{eq:R}
R_{34}\ &=&\ \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
 0 & 0 & \rm cos \theta_{34} &\rm sin \theta_{34} \cdot e^{-i \delta_{34}}\\
 0 & 0 & -\rm sin \theta_{34} \cdot e^{i \delta_{34}}& \rm cos \theta_{34}
\end{array}%
\right)\,, \nonumber \\
R_{24}\ &=&\  \left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \rm cos \theta_{24}  & 0 & \rm sin \theta_{24}\\
0 & 0 & 1 & 0 \\
0 & - \rm sin \theta_{24}& 0 &  \rm cos \theta_{24} %
\end{array}%
\right)\,, \nonumber \\
R_{14}\ &=& \left(
\begin{array}{cccc}
\rm cos \theta_{14} & 0 & 0 & \rm sin \theta_{14} \cdot e^{-i \delta_{14}} \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
- \rm sin \theta_{14} \cdot e^{i \delta_{14}} & 0 & 0 &\rm cos \theta_{14} \\
\end{array}%
\right) \,.
\end{eqnarray}

In the framework of the SM extended by sterile fermion states, which
have a nonvanishing mixing with active neutrinos,
the Lagrangian describing the leptonic charged currents becomes
\begin{equation}\label{eq:cc-lag}
- \mathcal{L}_\text{cc} = \frac{g}{\sqrt{2}} U^{\alpha i}
\bar{\ell}_\alpha \gamma^\mu P_L \nu_i  W_\mu^- + \, \text{c.c.}\,,
\end{equation}
where $i = 1, \dots, n_\nu$ denotes the physical neutrino states,
and $\alpha = e, \mu, \tau$ are the flavors of the charged leptons.
In the case of the SM with three neutrino generations,  $U$ is the  PMNS matrix, while in the case of $n_\nu \geq 4$, the
$3\times 3$ submatrix ($\widetilde U_\text{PMNS}$) is not unitary anymore and one can parameterize it as
\begin{equation}\label{eq:U:eta:PMNS2}
U_\text{PMNS} \, \to \, \widetilde U_\text{PMNS} \, = \,(\mathbb{1} -\widetilde \eta)\,
U_\text{PMNS}\,,
\end{equation}
where $\widetilde \eta$ is a matrix that accounts for the deviation of $\widetilde
U_\text{PMNS}$ from unitarity~\cite{Schechter:1980gr,Gronau}, due to the presence of extra fermion states.
Many observables are sensitive to the
active-sterile mixing and their current experimental values can be used to constrain the $\widetilde \eta$ matrix~\cite{Antusch:2014woa}.

In order to express the deviation from unitarity in terms of a single parameter, we define
\begin{eqnarray}
\eta = 1- |\det \widetilde U_\text{PMNS}|  \,,
\end{eqnarray}
which, in the case of the extension of the SM by only one sterile fermion and in terms of the mixing angles defined above, reads
\begin{eqnarray}
\eta = 1 - \vert \cos\theta_{14} \cos\theta_{24}\cos\theta_{34}\vert \,.
\end{eqnarray}
\section{Results and discussion}\label{sec:results}

In this section we present and discuss our results.

Since the Wilson coefficients of the processes discussed here are proportional to the mass of the decaying particle, it is quite obvious that the most significant enhancement of B($V \to \ell_\alpha  \ell_\beta$) will occur for $V=\Upsilon$ and its radial excitations.
For this reason we will present plots of our results for this decay channel. Plots for other channels are completely similar which is why we do not display them. Before we discuss the impact of the active-sterile neutrino mixing on the LFV decay rates further, we first specify the constraints on parameters of
both of our models.

In Fig.~\ref{fig:4} (left panel), we plot the dependence of $\eta$ with respect to the  mass of the effective sterile neutrino $m_4$.
\begin{figure}
\begin{center}
\hspace*{-14mm}\begin{tabular}{cc}
\includegraphics[scale=.995]{etam4}&\includegraphics[scale=.97]{etam5} \cr
  \end{tabular}
  \caption{{\footnotesize
$\eta$ parameter, which  parametrizes the size of mixing between the active and heavy sterile states, is plotted vs the mass of the heavy sterile state. The gray points (left panel) correspond to solutions complying with all experimental data and constraints discussed in the text except for perturbative unitary condition~(\ref{pert-uni}), which we then applied to obtain  the region of dark-blue points.
In the case of the (2,3)-ISS model (right-panel),
we  further imposed constraints of Ref.~\cite{Antusch:2014woa} on the matrix $\widetilde \eta$, as well as the bound $B(\mu\rightarrow eee)<10^{-12}$, resulting in the bright-blue region of points.
 }}
\end{center}
\label{fig:4}
\end{figure}
Gray points in that plot are obtained by varying the mass of the lightest neutrino, $m_{\nu_e}\in (10^{-21}, 1)$~eV, and by
imposing the following constraints: (i) Neutrino data (masses and mixing angles) respect the normal hierarchy, with $\Delta m_{21}^2 = 7.5(2) \times 10^{-5}$~eV, and $\Delta m_{31}^2 = 2.46(5) \times 10^{-3}$~eV~\cite{Gonzalez-Garcia:2014bfa}. We checked to see that our
final results do not change in any significant manner if the inverse hierarchy is adopted. Furthermore, we vary the three mixing angles with the fourth neutrino by assuming $\theta_{i4}\in (0, 2\pi ]$, while keeping the other three mixing angles to
their best-fit values, namely $\sin^2 \theta_{12} =0.30(1)$,  $\sin^2 \theta_{23} =0.47(4)$,  $\sin^2 \theta_{13} =0.022(1)$~\cite{Gonzalez-Garcia:2014bfa}. (ii) The selected points satisfy the upper bound ${\rm B}(\mu \to e\gamma)< 5.7 \times 10^{-13}$~\cite{Adam:2013mnn}. (iii) The results for
$R_\pi = \Gamma(\pi \to e \bar \nu_e)/ \Gamma(\pi \to \mu \bar \nu_\mu )$, $R_K$, $\Gamma(W\to \ell\nu_\ell)$, and $\Gamma(Z\to \text{invisible})$, remain consistent with experimental findings.
We see that for all (heavy) sterile neutrino masses the unitarity breaking parameter is $\eta \lesssim 0.005$.
That parameter space is not compatible with the perturbative unitarity requirement, which for $m_{4}\gg m_W$ translates into~\cite{Ilakovac:1994kj},~\footnote{To write it in the form given in Eq.~(\ref{pert-uni}), we replaced $\alpha_W=g^2/(4\pi) = \sqrt{2} G_F m_W^2/\pi$. }
\begin{eqnarray}\label{pert-uni}
{G_F m_4^2 \over \sqrt{2}\pi}\sum_\alpha \vert U_{\alpha 4}\vert^2  < 1\,.
\end{eqnarray}
The resulting region, i.e. the one that satisfies constraints (i), (ii), (iii) and Eq.~(\ref{pert-uni}), is depicted by blue points (the dark region) in Fig.~\ref{fig:4}, where we see that  the parameter $\eta$ is indeed diminishing with the increase of the heavy sterile mass $m_4$. In other words, the decoupling of a very heavy sterile neutrino entails the unitarity of the $3\times 3$ submatrix $\widetilde U_\text{PMNS}$. Decoupling from active neutrinos for very large masses was also explicitly emphasized in Ref.~\cite{Alonso:2012ji}.
We should mention that, besides the above constraints, we also implemented the constraint coming from ${\rm B}(\mu \to eee) < 10^{-12}$~\cite{Bellgardt:1987du}, but it turns out that the present experimental bound does not bring any additional improvement.

By imposing the constraints (i) and Eq.~(\ref{pert-uni}) on the (2,3)-ISS model, we get a similar region of allowed (blue) points in the right panel of Fig.~\ref{fig:4}. A notable difference with respect to the situation with one effective sterile neutrino is that the region of very small mixing angles is excluded due to relations between the active neutrino masses and the active-sterile neutrino mixing, cf. Ref.~\cite{Abada:2014vea}. For very heavy $m_5$, on the other hand, the range of allowed $\eta$'s shrinks and eventually vanishes with $m_5\to \infty$.~\footnote{We recall that, in the (2,3)-ISS model, $m_4$ stands for the mass of the light sterile state whose impact on the decays discussed here is negligible [as seen from Eq.~(\ref{eq:Wils})], while $m_5 > m_4$ can be large and is important for B($V\to \ell_\alpha\ell_\beta$).}
Furthermore, we use the results of Ref.~\cite{Antusch:2014woa} which are derived in the minimal unitarity violation scheme in which the heavy sterile neutrino fields are integrated out, and therefore the observables computed in that scheme are functions of the deviation of PMNS matrix from unitarity only~\cite{MUV}. We adapt and apply them to our (2,3)-ISS model and get a region of the bright-blue points, as shown in the right panel of Fig.~\ref{fig:4}.
To further constrain the parameter space we find it useful to account for the experimental bound on ${\rm B}(\mu \to eee)<1\times 10^{-12}$, as is discussed in Refs.~\cite{Ilakovac:1994kj,mueee,Alonso:2012ji}.
This latter constraint appears to be superfluous in most of the parameter space, once the constraints of Eq.~(\ref{pert-uni}) and Ref.~\cite{Antusch:2014woa} are taken into account, except in the range $10\ \mathrm{TeV}\lesssim m_5 \lesssim 100\ \mathrm{TeV}$, where the bound ${\rm B}(\mu \to eee)<1\times 10^{-12}$ restricts the parameter space relevant to B($V \to e \mu$).

We also mention that we attempted implementing the constraints coming from various laboratory experiments, summarized in Ref.~\cite{Atre:2009rg}, but since those results only impact the region of relatively small sterile neutrino masses ($m_5 \lesssim 100$ GeV), they are of no relevance to the present study.
\begin{center}
\begin{figure}
\hspace*{-11mm}\begin{tabular}{cc}
\includegraphics[scale=.45]{Upsm4}&\includegraphics[scale=.45]{Upsm5} \cr
\includegraphics[scale=.45]{Zm4}&\includegraphics[scale=.45]{Zm5} \cr
  \end{tabular}
  \caption{{\footnotesize
${\rm B}(\Upsilon \to \mu \tau)$ and ${\rm B}(Z \to \mu \tau)$ are shown as functions of the heavy sterile neutrino(s) mass, and in both models considered in this paper.
The points are selected in such a way that the models are consistent with the constraints discussed in the text and shown in Fig.~\ref{fig:4}.
Dashed horizontal lines correspond to the current experimental upper bounds for these decay rates. Notice again that the mass of the heavy sterile neutrino is denoted as $m_4$ in the effective model, and $m_5$ in the (2,3)-ISS model because the latter contains a lighter sterile neutrino state, the impact of which  is negligible on the decay modes discussed here.
 }}
\label{fig:5}
\end{figure}
\end{center}

\begin{table}[ht!]
\renewcommand{\arraystretch}{1.5}
\centering{}%
\resizebox{\textwidth}{!}{
\begin{tabular}{|cc|ccc|ccc|}
\hline
$V$ & $\ell_\alpha \ell_\beta$ & $m_4 = 1$~TeV & $10$~TeV & $100$~TeV& $m_5 = 1$~TeV & $10$~TeV & $100$~TeV \\ \hline\hline
$\phi$ & $e\mu$  & $1 \times 10^{-24}$ & $5 \times 10^{-24}$ & $3 \times 10^{-24}$ & $1\times 10^{-23}$ & $6 \times 10^{-23}$ & $5 \times 10^{-23}$ \\ \hline
$J/\psi$ & $e\mu$  & $2 \times 10^{-21}$ & $3 \times 10^{-20}$ & $6 \times 10^{-21}$ & $2\times 10^{-20}$ & $9 \times 10^{-20}$ & $7 \times 10^{-20}$ \\
             & $e\tau$  & $5 \times 10^{-18}$ & $8\times 10^{-17}$ & $2\times 10^{-19}$ & $1 \times 10^{-19}$ & $3\times 10^{-18}$ & $1 \times 10^{-19}$ \\
             & $\mu\tau$  & $8 \times 10^{-18}$ & $6 \times 10^{-16}$ & $3 \times 10^{-20}$ & $4 \times 10^{-19}$ & $4 \times 10^{-18}$ & $8 \times 10^{-19}$ \\ \hline
$\psi (2S)$ & $e\mu$  & $9 \times 10^{-22}$ & $ 1.5 \times 10^{-20}$ & $3 \times 10^{-21}$ & $4 \times 10^{-21}$& $3\times 10^{-20}$ & $2 \times 10^{-20}$ \\
             & $e\tau$  & $5 \times 10^{-18}$ & $2 \times 10^{-17}$ & $9 \times 10^{-21}$ & $4\times 10^{-20}$ & $1\times 10^{-18}$ & $4 \times 10^{-20}$ \\
             & $\mu\tau$  & $8 \times 10^{-18}$ & $3 \times 10^{-17}$ & $1.2 \times 10^{-20}$ & $1 \times 10^{-19}$ & $1\times 10^{-18}$ & $2\times 10^{-19}$ \\ \hline
$\Upsilon$ & $e\mu$  & $7 \times 10^{-18}$ & $2 \times 10^{-17}$ & $6 \times 10^{-18}$ & $2 \times 10^{-19}$ & $2\times 10^{-17}$ & $2\times 10^{-17}$ \\
             & $e\tau$  & $5 \times 10^{-14}$ & $2 \times 10^{-13}$ & $9 \times 10^{-17}$ & $6\times 10^{-18}$ & $4\times 10^{-16}$ & $5\times 10^{-17}$ \\
             & $\mu\tau$  & $5\times 10^{-16}$  & $2.5\times 10^{-13}$ & $1.2\times 10^{-16}$ & $1\times 10^{-17}$& $8\times 10^{-16}$ &$3\times 10^{-16}$ \\ \hline
$\Upsilon(2S)$ & $e\mu$  & $5\times 10^{-18}$  & $5\times 10^{-18}$ & $1.5 \times 10^{-18}$ & $2 \times 10^{-19}$ & $2\times 10^{-17}$ & $2\times 10^{-17}$ \\
             & $e\tau$  & $1.8 \times 10^{-14}$ & $3 \times 10^{-14}$ & $3\times 10^{-18}$ & $8\times 10^{-18}$ & $5 \times 10^{-16}$ & $5\times 10^{-17}$ \\
             & $\mu\tau$  & $2 \times 10^{-16}$ & $2 \times 10^{-13}$ & $2 \times 10^{-17}$ & $2\times 10^{-17}$ & $8 \times 10^{-16}$ & $3 \times 10^{-16}$ \\ \hline
$\Upsilon(3S)$ & $e\mu$  & $1.5 \times 10^{-17}$ & $3 \times 10^{-17}$ & $1.5 \times 10^{-17}$ & $5 \times 10^{-19}$ & $5 \times 10^{-17}$ & $4\times 10^{-17}$ \\
             & $e\tau$  & $5.5 \times 10^{-14}$ & $3 \times 10^{-14}$ & $4 \times 10^{-17}$ & $2\times 10^{-17}$ & $1\times 10^{-15}$ & $1 \times 10^{-16}$ \\
             & $\mu\tau$  & $2 \times 10^{-15}$ & $2 \times 10^{-12}$ & $4 \times 10^{-17}$ &  $3 \times 10^{-17}$ & $2 \times 10^{-15}$ & $6\times 10^{-16}$ \\ \hline
$Z$ & $e\mu$  & $1.2\times 10^{-14}$ & $7\times 10^{-13}$ & $4 \times 10^{-13}$& $9\times 10^{-14}$& $8 \times 10^{-13}$ & $6 \times 10^{-13}$ \\
             & $e\tau$  & $2\times 10^{-10}$ & $9 \times 10^{-9}$ & $4 \times 10^{-13}$ & $7\times 10^{-13}$ & $4\times 10^{-11}$ & $2 \times 10^{-12}$ \\
             & $\mu\tau$  & $5.5\times 10^{-10}$  & $3.5\times 10^{-8}$ & $1.6\times 10^{-12}$ & $3\times 10^{-12}$& $6\times 10^{-11}$ &$1\times 10^{-11}$ \\ \hline
\end{tabular}
}
\caption{{\footnotesize{}\label{tab:results} Upper bound on ${\rm B}(V\to \ell_\alpha \ell_\beta)$ for three values of the mass $m_{4,5}$. The numbers in the three columns referring to $m_4$ are obtained by using the effective model discussed in the text, while the other three, referring to $m_5$, are results of the (2,3)-ISS model (also discussed in the text). }}
\end{table}
After having completed the discussion on several constraints, we present our results for branching fractions B($V \to \mu  \tau$) depending on the mass of heavy sterile neutrino(s). In Fig.~\ref{fig:5}
we plot our results for $V=\Upsilon$ and $V=Z$, for which the enhancement is more pronounced. Other cases of $V$ result in similar shapes but the upper bound becomes lower.
In Table~\ref{tab:results} we collect our results for three values of the heavy sterile neutrino(s) mass.

To better appreciate the enhancement of the LFV decay rates shown in Fig.~\ref{fig:5}, we emphasize that both of them are ${\rm B}(V \to \mu  \tau) < 10^{-50}$ in the absence of heavy sterile neutrinos.
Current experimental bounds in both cases are shown by dashed lines. Since those bounds are expected to improve in the near future, a possibility of seeing the LFV modes discussed in this paper might become realistic.
Conversely, an observation of the LFV modes $V \to \ell_\alpha  \ell_\beta$, with branching fractions significantly larger than the bounds presented in Table~\ref{tab:results} would be a way to disfavor many of
the models containing heavy sterile neutrinos as being the unique source of lepton flavor violation. In obtaining the bounds presented in Table~\ref{tab:results} we used masses and decay constants listed in Appendix~B.
In presenting our results (the upper bounds) for lepton flavor violating modes, we used the parameters from Ref.~\cite{Antusch:2014woa} which were determined at  90\% C.L.
For that reason, we treated all  other input data to  2 $\sigma$ as well. Therefore, our final results  in Table~\ref{tab:results} are also obtained at 2 $\sigma$ level.

Finally, we compare in Table~\ref{tab:compare} our upper bounds for the modes for which we could find predictions in the literature.
\begin{table}[h!!]
\renewcommand{\arraystretch}{1.5}
\centering{}%
\hspace*{-4mm}\begin{tabular}{|cccccc|}
\hline
Mode & Ref.~\cite{Nussinov:2000nm} & Ref.~\cite{Gutsche:2009vp} &  Ref.~\cite{Sun:2012yq}  & Eff. model & (2,3)-ISS \\ \hline
${\rm B}(\phi \to e\mu)$ & $<4\times 10^{-17}$  & $<1.3 \times 10^{-21}$ & $<5\times 10^{-20}$ & $< 5\times 10^{-24}$ & $<6\times 10^{-23}$ \\
${\rm B}(J/\psi \to e\mu)$ & $<4\times 10^{-13}$  & $<3.5 \times 10^{-13}$ & $<1.9\times 10^{-18}$ & $< 3\times  10^{-20}$ & $< 9\times 10^{-20}$ \\
${\rm B}(J/\psi \to \mu \tau )$ & $-$ & $-$ & $< 1.6 \times 10^{-7}$ & $< 6\times  10^{-16}$  & $< 4\times 10^{-18}$ \\
${\rm B}(\Upsilon \to e\mu)$ & $< 2\times 10^{-9}$  & $<3.8 \times 10^{-6}$ & $< 3.6\times 10^{-18}$ & $< 2\times  10^{-17}$ & $< 2\times 10^{-17}$ \\
${\rm B}(\Upsilon \to \mu \tau )$ & $-$ & $-$ & $< 5.3 \times 10^{-7}$ & $< 2.5\times  10^{-13}$  & $< 8\times 10^{-16}$ \\
${\rm B}(Z \to e\mu)$ & $< 5\times 10^{-13}$  & $< 8 \times 10^{-15}$ & $-$ & $< 7\times  10^{-13}$ & $< 8\times 10^{-13}$  \\ \hline
\end{tabular}\caption{{\footnotesize{}\label{tab:compare} Upper bounds ${\rm B}(V\to \ell_\alpha \ell_\beta)$: Comparison of the results reported in the literature with the bounds
obtained in this work by using two different models (the last two columns). The bounds for other similar decay modes that have not been discussed in the literature can be found in
Table~\ref{tab:results}. }}
\end{table}

\section{Conclusions}\label{sec:concl}

In this paper we discussed the enhancement of the LFV decays of flavorless vector bosons,  $V \to \ell_\alpha  \ell_\beta$, with $V\in \left\{\phi, \psi^{(n)},\Upsilon^{(n)},Z\right\}$, induced by a mixing between the active and sterile neutrinos.
The enhancement grows with the mass of the heavy sterile neutrino(s), as can be seen from the mass dependence of the Wilson coefficients that we explicitly calculated. We find that the most significant diagram
that gives rise to the LFV decay amplitudes is the one coming from the $Z\nu \nu$ vertex, which suggests a steady growth of the decay rate with the mass of the sterile neutrino(s). In the physical amplitude, however, the region of very large
mass of the sterile neutrino(s) is suppressed as the decoupling takes place, i.e. mixing between the active and sterile neutrinos rapidly falls.

We illustrated the enhancement of ${\rm B}(V \to \ell_\alpha  \ell_\beta)$ in two scenarios: a model with one effective sterile neutrino that mimics the effect of a generic extensions of the SM including heavy sterile fermions,
and in a minimal realization of the inverse seesaw scenario compatible with current observations.
Our results for upper bounds on  ${\rm B}(V \to \ell_\alpha  \ell_\beta)$  [$V\in \left\{\phi,J/\psi, \psi(2S),\Upsilon(1S),\Upsilon(2S),\Upsilon(3S),Z\right\}$] are still considerably smaller than the current experimental bounds (when available),
but that situation might change in the future as more experimental research will be conducted at Belle II, BESIII, LHC, and hopefully at FCC-ee (TLEP).
If one of the decays studied here is observed and turns out to have a branching fraction larger than the upper bounds reported here, then sources of LFV other than those coming from mixing with heavy sterile neutrinos must be accounted for.

\section*{Acknowledgements}
We gratefully acknowledge a partial support from the European Union, FP7 ITN INVISIBLES (Marie Curie Actions, PITN-\-GA-\-2011-\-289442). M.L. thanks M.B. Gavela for the interesting comments and discussions.

\newpage
\section*{Appendix A: Wilson Coefficients}

In this Appendix we present detailed expressions for the Wilson coefficients.
All computations have been made in the Feynman gauge.
Contributions coming from the penguin and self-energy diagrams are shown in Fig.~\ref{fig:AA}, whereas the box diagrams are shown in Fig.~\ref{fig:B}.

\begin{center}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=.44]{figA1.png}&\includegraphics[scale=.44]{figA2.png} \cr
\includegraphics[scale=.44]{figA3.png}&\includegraphics[scale=.44]{figA4.png} \cr
\includegraphics[scale=.44]{figA5.png}&\includegraphics[scale=.44]{figA6.png} \cr
\includegraphics[scale=.44]{figA7.png}&\includegraphics[scale=.44]{figA8.png} \cr
\includegraphics[scale=.44]{figA9.png}&\includegraphics[scale=.44]{figA10.png} \cr
  \end{tabular}
  \caption{{\footnotesize
Penguin and self-energy diagrams contributing the LFV decay in Feynman gauge.
 }}
\label{fig:AA}
\end{figure}
\end{center}

\begin{figure}
\begin{tabular}{cc}
\includegraphics[scale=.4]{figB1.png}&\includegraphics[scale=.4]{figB2.png} \cr
\includegraphics[scale=.4]{figB3.png}&\includegraphics[scale=.4]{figB4.png} \cr
  \end{tabular}
  \caption{{\footnotesize
Box diagrams contributing the LFV decay $\Upsilon^{(n)}\to \ell_\alpha\ell_\beta$ in Feynman gauge.
 }}
\label{fig:B}
\end{figure}

We use the standard notation, $x_i=m_i^2/m_W^2$, $x_t=m_t^2/m_W^2$, $x_q=q^2/m_W^2= m_V^2/m_W^2$, and write
\begin{equation}
C_{VL}^r=\displaystyle\sum_{i,j=1}^{n_\nu} U_{\beta i} U_{\alpha j}^* C^{r, ij}_{VL} (x_i,x_j),
\end{equation}
where $r\in\lbrace\gamma,Z,\text{box}\rbrace$. The coefficients $C^{r, ij}_{VL}$ related to $\gamma$ and the box contributions are diagonal, $C^{r, ij}_{VL}= \delta_{ij}C^{r, i}_{VL}$, while
those related to the $Z$ penguins can also involve a coupling to two different neutrinos, since the $3\times 3$ mixing matrix is no longer unitary.
We therefore separate the diagonal and nondiagonal parts of the corresponding coefficient $C^{Z, ij}_{VL}=\delta_{ij} C^{Z,i}+\widehat{C}^{Z,ij}$, where the second term depends on the parameter $C_{ij}$ defined by
\begin{align}
C_{ij} = \sum_{\alpha={e,\mu,\tau}} U_{\alpha i}^* U_{\alpha j},
\end{align}
which, in the presence of sterile neutrinos, is generally different from $\delta_{ij}$.
Furthermore, from the plots presented in the body of the present paper we see that the region of $m_{4,5}\gg m_V$ is particularly interesting because there occurs the enhancement
of the LFV decay rate. For the sake of clarity we thus expand our expressions in $x_q$ and present here only the dominant terms. We also neglected, in the denominators of the loop integrals, the external momenta since they are
negligible with respect to heavy neutrino masses.
Therefore, up to terms ${\cal O}(x_q^2)$, our results read:
\begin{align}
\label{cvgamma}
	C_{VL}^{\gamma,i} (x_i) = -\frac{1}{16\pi^2} + x_q\frac{-43 x_i^3 + 108 x_i^2 +6 (5 x_i-6)x_i^2 \log x_i-81 x_i+16}{288 \pi^2 (x_i-1)^4} ,
\end{align}

\begin{align}
	C_{VL}^{Z,i}(x_i) =  &  \frac{-1+12 x_i-11 x_i^2+10 x_i^2 \log x_i }{64 \pi^2 (x_i-1)^2} +   \cos^2 \theta_W   C_{VL}^{\gamma,i} (x_i)  ,
\end{align}

\begin{align}
	\widehat{C}_{VL}^{Z,ii}&(x_i,x_i) =C_{ii} \frac{(x_i-2) \left(3 \left(x_i^2-1\right)+2
   (x_i-4) x_i \log x_i\right)}{128 \pi^2 (x_i-1)^2}\\
	&-x_q C_{ii}\frac{(x_i-1) (x_i (x_i (2 x_i-47)+25)+14)+6 (x_i (12 x_i-13)+2) \log x_i}{1152 \pi^2 (x_i-1)^4},\nonumber
\end{align}

\begin{align}
	\widehat{C}_{VL}^{Z,ij}(x_i,x_j)&= \frac{\sqrt{x_i x_j} C_{ij}^*}{64 \pi^2} \Big{[}\frac{x_i (x_i-4)}{(x_i-1)(x_i-x_j)}\log x_i+\frac{x_j (x_j-4)}{(x_j-1)(x_j-x_i)}\log x_j-\frac{3}{2}\Big{]} \\
	&+\frac{C_{ij}}{64 \pi^2}\Big{[} \frac{2 x_i^2 (x_j-1)}{(x_i-1)(x_i-x_j)}\log x_i+\frac{2 x_j^2 (x_i-1)}{(x_j-1)(x_j-x_i)}\log x_j+3\Big{]}\nonumber \\
	&+\frac{x_q}{192 \pi^2} \Big{\lbrace} \sqrt{x_i x_j} C_{ij}^*\Big{[} \frac{x_i^2(x_i-3 x_j+2 x_i x_j)}{(x_i-1)^2(x_i-x_j)^3}\log x_i+\frac{x_j^2(x_j-3 x_i+2 x_i x_j)}{(x_j-1)^2(x_j-x_i)^3}\log x_j\nonumber\\
	&-\frac{x_i^3(x_j-1)-x_j^3(x_i-1)+x_i x_j (x_i-x_j)}{(x_i-1)(x_j-1)(x_i-x_j)^3}\Big{]}+ 2 C_{ij} \Big{[} \frac{x_i^2(3 x_i^2 +3x_j^2+3x_j-x_i-8x_i x_j)}{(x_i-1)^2(x_i-x_j)^3}\log x_i \nonumber \\
	&+\frac{x_j^2(3 x_j^2 +3x_i^2+3x_i-x_j-8x_i x_j)}{(x_j-1)^2(x_j-x_i)^3}\log x_j - \frac{8 x_i^2 x_j-8 x_i x_j^2-x_i^3 x_j+x_i x_j^3-2x_i^3+2x_j^3}{(x_i-1)(x_j-1)(x_i-x_j)^3}\Big{]}\Big{\rbrace},\nonumber
\end{align}

\begin{align}
\label{cvbox}
	C_{VL}^{\mathrm{Box},i} = \frac{1}{256 \pi^2}
	 \left\lbrace \frac{[ x_i(x_t-8)+4]x_t^2 \log x_t}{(x_t-1)^2(x_i-x_t)}+\frac{[ x_t(x_i-8)+4]x_i^2 \log x_i}{(x_i-1)^2 (x_t-x_i)}+\frac{7 x_i x_t-4}{(x_i-1)(x_t-1)}\right\rbrace.
\end{align}

\section*{Appendix B: Formulas and hadronic quantities}

In this Appendix we collect the expressions used to constrain the parameters of the models discussed in the present paper, as well as the values of the masses and decay constants used in our numerical analysis.
In the expressions below we used the value of $G_F=G_\mu = 1.166 \times 10^{-5} \ {\rm GeV}^{-2}$, as extracted from $\mu \to e\nu_\mu\bar\nu_e$.
In our scenarios, in which we extended the neutrino sector by adding heavy sterile neutrinos, the Fermi constant becomes $G_F=G_\mu/\sqrt{ \sum_{i,j} |U_{ei}|^2  |U_{\mu j}|^2}$. For the models used in this paper, we checked to see that
$G_F = G_\mu$ remains an excellent approximation.

\begin{itemize}
\item \underline{$\mu \to e\gamma$}: We use the experimentally established upper bound ${\rm B}(\mu\to e\gamma) < 5.7\times 10^{-13}$, and the expression~\cite{Ilakovac:1994kj}
\begin{align}
{\rm B}(\mu \to e\gamma) &= {\sqrt{2} G_F^3 s_W^2 m_W^2 \over 128 \pi^5 \Gamma_\mu} m_\mu^5 \vert U_{\mu 4}^\ast U_{e4} G_\gamma (x_4)\vert^2\,, \cr
G_\gamma (x) & = - { 2 x^3+5x^2 -x\over 4(1-x)^3 }
- {  3 x^3  \over 2(1-x)^4 } \log x\ ,
\end{align}
to get one of the most significant constraints in this study. Notice that we use $s_W^2=1-m_W^2/m_Z^2$, and we kept the dominant contribution with $x_4$.
\item \underline{$W \to \ell_\alpha \nu$}: Combining the measured ${\rm B}(W\to e\nu )=0.1071(16)$ and ${\rm B}(W\to \mu\nu)=0.1063(15)$, with the expression
\begin{align}
{\rm B}(W \to \ell_\alpha \nu) = {\sqrt{2} G_F m_W \over 24 \pi  \Gamma_W} \sum_{j=1}^4 \lambda(m_\alpha^2,m_j^2,m_W^2) \left(
2 - { m_\alpha^2+m_j^2\over m_W^2}- { (m_\alpha^2-m_j^2)^2\over m_W^4}
\right) \vert U_{\alpha j}^2\vert  ,
\end{align}
we further restrain the possible values of $m_4$ while varying the mixing angles in the largest possible range.
\item  \underline{$\Delta r_{K,\pi}= R_{K,\pi}^{\rm exp.}/R_{K,\pi}^{\rm SM} - 1$}: The ratio of the leptonic decay widths of a given meson $P$, $R_P=\Gamma(P\to e \nu_e )/\Gamma(P\to \mu \nu_\mu )$ was  recently shown to be quite restrictive on
the possible values of $m_{4,5}$ and $\eta$~\cite{Abada:2013aba}.  The most significant constraints actually come from $\Delta r_{\pi}= 0.004(4)$ and  $\Delta r_{K}= -0.004(3)$, and the corresponding formula reads,
\begin{align}
\Delta r_P= - 1 + {m_\mu^2 (m_P^2-m_\mu^2)^2\over m_e^2 (m_P^2-m_e^2)^2} {\displaystyle{\sum_i} \vert U_{ei}\vert^2 \left[ m_P^2 (m_{\nu_i}^2 + m_e^2) -  (m_{\nu_i}^2 - m_e^2)^2 \right] \lambda^{1/2}(m_P^2,m_{\nu_i}^2 ,m_e^2)
\over \displaystyle{\sum_i} \vert U_{\mu i}\vert^2 \left[ m_P^2 (m_{\nu_i}^2 + m_\mu^2) -  (m_{\nu_i}^2 - m_\mu^2)^2 \right] \lambda^{1/2}(m_P^2,m_{\nu_i}^2 ,m_\mu^2)
}.
\end{align}
\item \underline{$Z \to \nu \nu$}: To saturate the experimental ${\Gamma}(Z\to \text{invisible} )=0.499(15)$~GeV, we sum over the kinematically available channels involving active and sterile neutrinos,
\begin{align}
\Gamma(Z \to \nu \nu) =& \sum_{i,j} \left(1-\frac{\delta_{ij}}{2}\right)  {G_F \over 12\sqrt{2} \pi m_Z}\lambda^{1/2}(m_Z^2,m_i^2,m_j^2) |C_{ij}|^2 \nonumber\\
                                              &\times  \left[  2 m_Z^2-m_i^2-m_j^2 - 6m_im_j - {(m_i^2-m_j^2)^2\over m_Z^2}\right]\,.
\end{align}
\item \underline{$\mu \to e e e$}: We use the experimental upper bound ${\rm B}(\mu \to e e e) <1 \times 10^{-12}$~\cite{Bellgardt:1987du}, and the expression~\cite{Ilakovac:1994kj}
\bee
{\rm B}(\mu \to eee) &=& \frac{G_F^4 m_W^4 }{6144 \pi^7}\frac{m^5_\mu}{\Gamma_\mu}\non
&&  \left\{ 2 \left|\frac{1}{2}F^{\mu eee}_{\rm Box}+F^{\mu e}_Z-2\sin^2\theta_W (F^{\mu e}_Z-F^{\mu e}_\gamma)\right|^2+4 \sin^4\theta_W \left|F^{\mu e}_Z-F^{\mu e}_\gamma\right|^2 \right. \non
&& \left.		+ 16 \sin^2\theta_W \mathrm{Re} \left[	(F^{\mu e}_Z +\frac{1}{2}F^{\mu eee}_{\rm Box})	{G^{\mu e}_\gamma}^* 			\right]		- 48 \sin^4\theta_W \mathrm{Re}\left[	(F^{\mu e}_Z-F^{\mu e}_\gamma)	{G^{\mu e}_\gamma}^* 			\right] \right. \non
&& \left.
		+32 \sin^4\theta_W |G^{\mu e}_\gamma|^2\left[		\ln \frac{m^2_\mu}{m^2_{e}} -\frac{11}{4}		\right]		\right\},
\eee
with the loop functions $F^{\mu eee}_{\rm Box},F^{\mu e}_Z,F^{\mu e}_\gamma,G^{\mu e}_\gamma$ defined in~\cite{Alonso:2012ji}.

\end{itemize}

Finally, the values of hadronic quantities not discussed in the body of the paper but used in our numerical analysis are listed in Table~\ref{tab:costs}.~\footnote{
Notice that the ratio of decay constants $f_{\psi(2S)}/f_{J/\psi}$ has been obtained from the corresponding (measured) electronic widths and the expression $\Gamma (\psi_n\to e^+e^-)= 16 \pi \alpha_{\rm em}^2 f_{\psi_n}^2/(27 m_{\psi_n}^2) $.
 }

\begin{table}[h!!]
\renewcommand{\arraystretch}{1.5}
\centering{}%
\hspace*{-4mm}\begin{tabular}{|ccc|ccc|}
\hline
Quantity & Value & Ref. & Quantity & Value & Ref.  \\ \hline
$m_\phi$ & $1.0195$~GeV  & \cite{PDG} & $f_\phi$ & $241(18)$~MeV &  \cite{hpqcd-2} \\
$m_{J/\psi}$ & $3.0969$~GeV  & \cite{PDG} & $f_{J/\psi}$ & $418(9)$~MeV &  \cite{c-latt1} \\
$m_{\psi(2S)}$ & $3.6861$~GeV  & \cite{PDG} & $f_{\psi(2S)}/f_{J/\psi}$ & $0.713(16)$ &  \cite{PDG} \\ \hline
$m_{\Upsilon}$ & $9.460$~GeV  & \cite{PDG} & $f_{\Upsilon}$ & $649(31)$~MeV &  \cite{Lepage} \\
$m_{\Upsilon(2S)}$ & $10.023$~GeV  & \cite{PDG} & $f_{\Upsilon(2S)}$ & $481(39)$~MeV &  \cite{Lepage} \\
$m_{\Upsilon(3S)}$ & $10.355$~GeV  & \cite{PDG} & $f_{\Upsilon(3S)}$ & $539(84)$~MeV &  \cite{RL} \\ \hline
\end{tabular}\caption{{\footnotesize{}\label{tab:costs} Masses and decay constants used in numerical analysis. }}
\end{table}

\newpage

redpajama_set_name: RedPajamaArXiv


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text:
Ex-BioShock dev to create interactive autobiography via a Doom 2 mod
By Joe Donnelly 2016-08-23T14:36:17.235Z
The first instalment of JP LeBreton's Autobiographical Architecture due this year.
JP LeBreton is an ex-Irrational, 2K Marin and Double Fine developer who is also a self-confessed lover of Doom. Since 2007, his main wish has been to go it alone as an independent developer "making things only [he] can make," with "[his] own limits the only compromise," so he says on his website. His latest project, Autobiographical Architecture, is one such venture—an interactive memoir of sorts created using the Doom engine.
Its first chapter is heading to Itch.io later this year, and this is its announcement trailer:
Previously, in an impressive effort that combined his previous work with said love of Doom, LeBreton recreated BioShock's Arcadia and Farmer's Market levels in Doom 2 after being partly inspired by TIGSource's Bootleg Demakes Competition.
Using a Doom level editor named SLADE, he chose these particular zones as he was their primary designer. "It's a monster of a level, crammed full of weird little BioShock-to-Doom transmutations and symbolism," he says. "If you're a fan of either game, I hope you enjoy it." You can check that out over here.
LeBreton's Autobiographical Architecture Doom 2 mod is set to launch its first chapter at some point this year on Itch.io.
Thanks, Eurogamer.
JP LeBreton

redpajama_set_name: RedPajamaCommonCrawl


text:
Ipoh Community Forums > Bangsa Malaysia > Bangsa Malaysia
A Matter Of Protection, Not Privilege
Bangsa Malaysia Berikan pendapat anda bagi mewujudkan BANGSA MALAYSIA!
\r\n	\r\n	 \r\n	\r\n	09-12-2006, 08:00 PM\r\n	\r\n	\r\n	\r\n
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\r\n	 \r\n	A Matter Of Protection, Not Privilege\r\n
By Malik Imtiaz Sarwar
\nSaturday, December 2, 2006
\nI believe the time has come for us to ask difficult questions about this country and who we are when we say we are Malaysians. In the same vein, I believe that it is time for Malaysians to stop offering avenues of escape to those responsible for our well being and our collective future. We must confront the fact that something is very wrong at the heart of Malaysia.
\nLike many others, I was angered by some of the speeches delivered at the recent UMNO general assembly. As I considered the matter further, I began to appreciate that we may have benefited from having heard the sentiments expressed. Despite the feeble attempts to subsequently soften and explain away the obviously painful impact of the speeches, the truth had revealed itself; those who claim a virtual monopoly to lead this nation are racial supremacists and proud of it. All others, it would seem, are incidental to the vision of supremacist ideology as it was played out at the assembly.
\nThe attempts at damage control were pitiful for the fact that they appear to have been aimed not so much at the healing of the community as a whole but more at attempting to exculpate the party. In this way, such efforts added insult to injury. The suggestion that the sentiments expressed were intended only for the party was implausible, disingenuous and clearly an attempt to avoid responsibility. From the outset it was obvious that this assembly, like others, would be heavily publicized. Like in previous years, the print media would be giving the event maximum coverage. Additionally, the assembly was to be broadcast live. These were matters known to all concerned and, I would say, factored heavily in how they positioned themselves. The extreme racism, for that was what it was, was put on display for the world to see as a miscalculated show of strength.
\nThe justifications, offered as they have been instead of the apologies and humility we all deserve, have not gone far in convincing Malaysians that the racial extremism displayed is not an accurate reflection of how the UMNO, and as such by its own declaration the leadership, sees things.
\nAnd the way it sees things is obvious; while all of us are equal, some of us are more equal than others. I do not say this to incite disaffection. I have Malay friends who were equally disgusted by the goings on at the assembly. They show me that the attitudes celebrated at the assembly are not universally those of the Malay community at large.
\nInstead, I say this to lay foundation for the question of whether the Federal Constitution does provide for a privileged existence in the manner suggested at the general assembly.
\nThe constitutional provision pivotal to any discussion of this subject is article 153. It refers to the 'special position' of the Malays and the natives of Sabah and Sarawak and declares that it is the responsibility of the Yang Dipertuan Agong to safeguard these communities and the legitimate interests of other communities. It does not describe this 'special position' as a privileged status.
\nThe founders of the nation did not at any point in time contemplate the creation of a two-tiered society such as some of some might have us believe. They contemplated the possibility that due to historical factors there might be a need to introduce certain measures so as not to allow for the dislocation or marginalizing of certain communities. This was not intended to translate into a promotion of particular communities to the detriment of others.
\nFor this reason, article 153 recognises the legitimate interests of other communities. Further, it provides the means to ensure an adequate balance in the way protection measures are deployed. It is in this light that provision is made for reservation of positions in the public service and of scholarships, exhibitions and other similar educational or training privileges or special facilities given or accorded by the Federal Government and permits and licences where such are required under federal law. This balance is guaranteed further by the express declaration that the provision is not aimed at empowering the restriction of business or trade solely for the purpose of the protective measures. In short, there is no basis for wholesale reservations or quotas.
\nPut another way, despite suggestions to the contrary the Federal Constitution does not lend itself to any notions of a privileged existence for any community. It does however envisage a protected status for the Malays and the indigenous which allows for selective measures to be taken fairly and reasonably to a particular end. Any policy of the Federal Government aimed at achieving this end, whether the National Economic Policy, the New Development Policy or otherwise, cannot be seen as vesting greater rights than those contemplated under Article 153. Any other reading would allow for the kinds of abuses that are apparent.
\nThe analysis offered above is not a difficult one. Article 153 is clear. It lends itself to affirmative action where such action is needed. Political expediency has resulted in the provision being mischaracterized and used to particular ends. This in itself is not surprising as politicians will operate as politicians do. What is of interest to me is how and why the characterization has been permitted by Malaysian society to perpetuate to the extent that as we come into our 50th year as an independent nation, the 'two-tier' notion has become so entrenched in our social landscape. And why is it that even as the notion continues to divide us, we insist on describing what happens in the country as being a part of a democratic process?
\nOne of the principal reasons for this sad state of affairs is the nurturing of a climate of fear. Laws that allow for preventive detention without trial like the Internal Security Act, that stifle free expression such as the Sedition Act and the Printing Presses and Publications Act and which impede necessary access to vital information such as the Official Secrets Act have kept many Malaysians in the dark and in fear. If they have not been directly threatened or attacked, these laws have allowed the unscrupulous to demonize concerned Malaysians as being anti-Malay or anti-constitutional. They have allowed for the perversion of the social contract theory.
\nAdditionally, a conservative Judiciary which has shied away from developing a strong civil rights tradition in our legal system has perhaps unwittingly allowed for the concomitant undermining of the rule of law. This has weakened our sense of right and wrong.
\nIn this way, policy stereotypes have been permitted to develop and in effect become the law. Our claims to social integration are a hollow boast. Harmony cannot be found at the bottom of the gun barrel that supremacist Malay thinking is pointing at all others including non-extremist Malays.
\nIf UMNO is serious about making amends for what transpired at the general assembly, let us hear then from its leaders of a plan by which the Federal Government intends to take us back to the spirit of protection and the dismantling of the 'privilege' mindset. Let us hear an admission of how the climate of fear has led to the development of racist tendencies that no amount of platitudes will help us deal with.
\nUntil then, Malaysians should be forgiven for thinking that they have arrived at the beginning of the end.
\nSource: Disquiet
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Saturday, March 05, 2005Liberal voice refuses to be hushed up
\nLawyer braves Muslims\' rage by pushing for interfaith dialogue
\nBy Carolyn Hong
\nStraits Times
\nKUALA LUMPUR - HIS boyish appearance often surprises those who expect lawyer Malik Imtiaz Sarwar to look much older, or more like a rabid activist.
\nHe is 35 and looks younger, but can seem older as he tends to speak in the full sentences of a lawyer used to thinking on his feet, and with a certain wariness.
\nHe seems rabid only because he sticks to his beliefs. His language is temperate and considered.
\nMr Malik Imtiaz is one of the new liberal voices of Malaysia shaking up the conservative Muslim mainstream.
\nNot surprisingly, he has often been the target of Muslim anger, most recently for leading an initiative by non-governmental organisations to set up an interfaith dialogue that will go beyond merely reaffirming universal values.
\nHis idea was to have an Interfaith Commission that would tackle the tough issues and religious conflicts, an initiative seen by many Muslims as challenging the supremacy of Islam.
\nThe criticism grew into a torrent last week.
\n\'To me, the criticism has brought into focus the underlying societal discourse on religion,\' Mr Malik Imtiaz told The Straits Times.
\nIt highlighted how strongly the Muslim community felt about the position of Islam, and the overall reluctance to rock the delicately balanced boat of inter-religious relations.
\nAnd it showed the difficulties in moving dialogue beyond the safe ground to tackle potentially divisive issues without causing unease.
\nThis is not the first time that Mr Malik Imtiaz has faced the wrath of Muslims.
\nNeither is he the first person to be in the line of fire.
\nSome years ago, writer and political scientist Farish Noor was the main target of anger, while the women\'s group Sisters in Islam never managed to find acceptance within mainstream Islamic thought.
\nIn 2002, Mr Malik Imtiaz, Dr Farish and Ms Zainah Anwar of Sisters in Islam were unceremoniously lumped together, along with a few other social commentators, in a complaint that they were allegedly humiliating Islam.
\nThe complaint was lodged with the Malay Rulers Council by a group of religious scholars.
\nBut the government did not pursue the matter.
\nLike the other liberal voices, Mr Malik Imtiaz tends to rely on intellectual arguments to convince people of his cause, and eschews emotional appeal.
\nHis forte is the law, and his views are very much human rights-driven, or as his critics say, \'legalistic\'.
\n\'I am only doing what I do because I am a citizen of Malaysia. From a personal standpoint, as a Muslim, I can believe that Islam is supreme, but from the standpoint of society, I must recognise the rights of others to their religion,\' he said.
\nMr Malik Imtiaz\'s ideas on human rights were honed during his studies at the International Islamic University of Malaysia. This year, he obtained a master\'s in international human rights from Oxford University in Britain.
\nHis ideals were planted at a young age.
\nHis father founded the humanities school of Universiti Sains Malaysia in Penang, and is now a professor of theatre studies at the University of Malaya.
\n\'His own battle is saving the traditional Malay performing arts,\' said Mr Malik Imtiaz.
\nThe Interfaith Commission is not his first attempt to use the law to clarify positions on Islamic issues. He had earlier represented Muslims who wanted to convert from Islam, a greatly sensitive issue in Malaysia.
\nHe was also part of the legal team challenging the legality of hudud, the Islamic penal code incorporating Quranic punishments, when it was enacted by the states controlled by Parti Islam SeMalaysia.
\nHe has now turned his energies to reviving the National Human Rights organisation.
\nHe likes his nights out in glitzy Bangsar as well.
\nAs he says, everyone has the right to live as he pleases - within the law, of course.
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ipohan
I believe the time has come for us to ask difficult questions about this country and who we are when we say we are Malaysians. In the same vein, I believe that it is time for Malaysians to stop offering avenues of escape to those responsible for our well being and our collective future. We must confront the fact that something is very wrong at the heart of Malaysia.
Like many others, I was angered by some of the speeches delivered at the recent UMNO general assembly. As I considered the matter further, I began to appreciate that we may have benefited from having heard the sentiments expressed. Despite the feeble attempts to subsequently soften and explain away the obviously painful impact of the speeches, the truth had revealed itself; those who claim a virtual monopoly to lead this nation are racial supremacists and proud of it. All others, it would seem, are incidental to the vision of supremacist ideology as it was played out at the assembly.
The attempts at damage control were pitiful for the fact that they appear to have been aimed not so much at the healing of the community as a whole but more at attempting to exculpate the party. In this way, such efforts added insult to injury. The suggestion that the sentiments expressed were intended only for the party was implausible, disingenuous and clearly an attempt to avoid responsibility. From the outset it was obvious that this assembly, like others, would be heavily publicized. Like in previous years, the print media would be giving the event maximum coverage. Additionally, the assembly was to be broadcast live. These were matters known to all concerned and, I would say, factored heavily in how they positioned themselves. The extreme racism, for that was what it was, was put on display for the world to see as a miscalculated show of strength.
The justifications, offered as they have been instead of the apologies and humility we all deserve, have not gone far in convincing Malaysians that the racial extremism displayed is not an accurate reflection of how the UMNO, and as such by its own declaration the leadership, sees things.
And the way it sees things is obvious; while all of us are equal, some of us are more equal than others. I do not say this to incite disaffection. I have Malay friends who were equally disgusted by the goings on at the assembly. They show me that the attitudes celebrated at the assembly are not universally those of the Malay community at large.
Instead, I say this to lay foundation for the question of whether the Federal Constitution does provide for a privileged existence in the manner suggested at the general assembly.
The constitutional provision pivotal to any discussion of this subject is article 153. It refers to the 'special position' of the Malays and the natives of Sabah and Sarawak and declares that it is the responsibility of the Yang Dipertuan Agong to safeguard these communities and the legitimate interests of other communities. It does not describe this 'special position' as a privileged status.
The founders of the nation did not at any point in time contemplate the creation of a two-tiered society such as some of some might have us believe. They contemplated the possibility that due to historical factors there might be a need to introduce certain measures so as not to allow for the dislocation or marginalizing of certain communities. This was not intended to translate into a promotion of particular communities to the detriment of others.
For this reason, article 153 recognises the legitimate interests of other communities. Further, it provides the means to ensure an adequate balance in the way protection measures are deployed. It is in this light that provision is made for reservation of positions in the public service and of scholarships, exhibitions and other similar educational or training privileges or special facilities given or accorded by the Federal Government and permits and licences where such are required under federal law. This balance is guaranteed further by the express declaration that the provision is not aimed at empowering the restriction of business or trade solely for the purpose of the protective measures. In short, there is no basis for wholesale reservations or quotas.
Put another way, despite suggestions to the contrary the Federal Constitution does not lend itself to any notions of a privileged existence for any community. It does however envisage a protected status for the Malays and the indigenous which allows for selective measures to be taken fairly and reasonably to a particular end. Any policy of the Federal Government aimed at achieving this end, whether the National Economic Policy, the New Development Policy or otherwise, cannot be seen as vesting greater rights than those contemplated under Article 153. Any other reading would allow for the kinds of abuses that are apparent.
The analysis offered above is not a difficult one. Article 153 is clear. It lends itself to affirmative action where such action is needed. Political expediency has resulted in the provision being mischaracterized and used to particular ends. This in itself is not surprising as politicians will operate as politicians do. What is of interest to me is how and why the characterization has been permitted by Malaysian society to perpetuate to the extent that as we come into our 50th year as an independent nation, the 'two-tier' notion has become so entrenched in our social landscape. And why is it that even as the notion continues to divide us, we insist on describing what happens in the country as being a part of a democratic process?
One of the principal reasons for this sad state of affairs is the nurturing of a climate of fear. Laws that allow for preventive detention without trial like the Internal Security Act, that stifle free expression such as the Sedition Act and the Printing Presses and Publications Act and which impede necessary access to vital information such as the Official Secrets Act have kept many Malaysians in the dark and in fear. If they have not been directly threatened or attacked, these laws have allowed the unscrupulous to demonize concerned Malaysians as being anti-Malay or anti-constitutional. They have allowed for the perversion of the social contract theory.
Additionally, a conservative Judiciary which has shied away from developing a strong civil rights tradition in our legal system has perhaps unwittingly allowed for the concomitant undermining of the rule of law. This has weakened our sense of right and wrong.
In this way, policy stereotypes have been permitted to develop and in effect become the law. Our claims to social integration are a hollow boast. Harmony cannot be found at the bottom of the gun barrel that supremacist Malay thinking is pointing at all others including non-extremist Malays.
If UMNO is serious about making amends for what transpired at the general assembly, let us hear then from its leaders of a plan by which the Federal Government intends to take us back to the spirit of protection and the dismantling of the 'privilege' mindset. Let us hear an admission of how the climate of fear has led to the development of racist tendencies that no amount of platitudes will help us deal with.
Until then, Malaysians should be forgiven for thinking that they have arrived at the beginning of the end.
Source: Disquiet
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Spring 2020-Table of Contents/
Integrating Art Into Pediatric Healthcare
Ozzy and artist Sadie Wilcox with two of his large-scale spin art creations prepared for public exhibition at the Berkeley Art Museum.
Integrating Art Into Pediatric Health Care: A Collaborative Approach
ACLP Bulletin | Spring 2020 | VOL. 38 NO. 2
Sadie Wilcox, MFA, with Jen Johnson-Geisse, MA, CCLS
UCSF Benioff Children's Hospital Oakland, Oakland, CA
The Artist-in-Residence office at UCSF Benioff Children's Hospital Oakland is located in the historic wing of the main hospital. A small window on the west side of my office looks out on the hospital courtyard. I catch a glimpse of a young girl in the courtyard below. She is slowly ambling down the ramp, a physical therapist at her side. Her mother is trailing behind them, pausing for a moment to watch her daughter's slow and deliberate descent. A teenage boy sits on a wooden bench on the west side of the courtyard. He basks in the sun. His grandmother sits at his side, eyes closed. An enormous magnolia tree rises behind them, branches reaching wide. The white flowers above them bloom skyward.
The Artist-In-Residence program is part of the child life and creative arts therapy department. Unlike traditional Artist-in-Residence programs within museums, universities, or civic institutions that are short term in nature, in our hospital the Artist-in-Residence title refers to a permanent staff position. The program was founded in 2004 with donor funds, supplemented by an initial grant from the Society for Arts in Healthcare. The Artist-in-Residence role offers an arts-informed perspective to complement the work of child life services and the interdisciplinary medical team.
My office is flanked by built-in shelves on both sides. I have labeled bins of acrylic paint, colored pencils, liquid watercolors, sketch books, patterned paper, plastic stencils, and adaptive art tools of all shapes and sizes. My digital art supplies include a stack of iPads, a small video camera, a laptop, and robotic drawing kits. Books on child life, art therapy, expressive arts, trauma-informed practice, and pediatric palliative care line the shelves above the desk. My clinical training includes trauma-informed practice and family-centered care, and my approach is multidisciplinary. I am a member of the unionized staff, and have held this position for more than a decade. Polaroid photos of former patients I've worked with fill the walls around my workspace.
Child life specialist Jen Johnson-Geisse, MA, CCLS, knocks on the door and enters the office. She holds a printed census in her hand. She pulls up a chair, and we review the list of current patients on the pediatric rehabilitation unit. We discuss and reflect on the expressed needs of each patient, one by one. Jen and I have worked in tandem on many patient cases over the years.
In one such instance, Jen emerged from a single-occupancy hospital room and approached me from across the corridor. She described her work with a newly admitted teen, Ozzy*, who had sustained an abdominal injury causing him to be placed in a medically induced coma. When he awoke, doctors determined that he had experienced significant damage to his central and peripheral nervous system. At the time of his admission to our unit, Ozzy had lost most of his physical functioning and was unable to speak. It eventually became clear that his cognitive ability was unimpaired and he understood everything happening around him. Jen also learned that Ozzy had a long-standing interest in art. Given his injury and hospitalization, he was no longer able to access his previous methods of artmaking.
We began to discuss the ways in which Jen and I could align our individual skill sets to optimize Ozzy's care. We identified our primary clinical goal as facilitating outlets for emotional and creative expression while providing comprehensive family support. In order to achieve this goal, we explored a range of therapeutic activities that would offer Ozzy a sense of agency and empowerment during his long-term hospital admission. We agreed that a creative arts intervention, developed in close consultation with Ozzy and his family, could be an ideal point of entry.
Jen led me to Ozzy's room across the hall and we knocked on the door before entering. Ozzy had his back to us, facing the window. Squeezing behind the power wheelchair and circling around the IV pole, we located ourselves across from Ozzy. Jen greeted him and introduced me. I offered a few examples of ways that previous patients had engaged with the arts. Ozzy responded with interest and we began to explore creative possibilities.
Ozzy experimented with multiple approaches to art-making during his hospital stay, but was most responsive to an adaptive spin art session that Jen facilitated in his hospital room. Spin art is an expressive modality that offers a processoriented approach. The medium provides ample opportunities for creative choice, control, and innovative adaptation. To create spin art, a square piece of paper is secured to a portable spin art machine. With the flip of a manual switch, the platform rotates at a high speed. The artist pours paint onto the paper and the centrifugal force draws the paint outward, creating original designs. Small-scale spin art machines are a low-cost, easy point of entry for children and teens, and they can be adapted to accommodate a range of accessible switches.
Following his initial engagement in spin art, Ozzy and his mother proposed the idea of working on a larger scale. We immediately took the opportunity to follow their lead. I began to work with Ozzy's mother, who put me in contact with the robotics club at his school, a club he had participated in prior to his injury. In response, the robotics club offered to build him a large-scale spin art machine for use at the hospital. I communicated directly with the club about infection control and storage size requirements in order to build a device specifically designed for our medical setting.
When the customized machine was built, Jen was able to collaborate with Ozzy's physical and occupational therapists, who set up a mobile arm and grip-support system. This allowed Ozzy to utilize his own body to manipulate and apply the art materials with significantly less assistance. Over time, he also began to utilize a form of "yes" or "no" response through head movement. This evolved to the use of an automated voice to indicate an auditory "yes" or "no" response.
During our first collaborative session with the large-scale spin art machine, Jen and I took turns offering acrylic paint bottles one by one. Ozzy indicated a "yes" or "no" reply in response to each color choice. We repeated the process and he indicated his next color. He was deliberate, thoughtful, and self-assured in the curation of his painting palette.
After selecting his color combination, Ozzy decided on the location of color on the page. He then determined the amount of paint based on the length of time it was poured. He was able to activate the spinning of the device using adaptive switches initiated by subtle movements, including turning his head, rotating his wrist, and bending his fingers.
Ozzy soon expressed interest in sharing this creative resource with other patients. In response, Jen and I began addressing the emergent clinical goals of supporting Ozzy's socialization with peers, engaging in community-building activities, and reinforcing his sense of identity within the hospital. Jen offered large-scale spin art sessions as a group activity. I worked with two teen patients who filmed and edited their own video about the spin art process, with Ozzy positioning the camera on his lap to serve as the cameraman. Patients of all ages began to share their own spin art creations with hospitalized peers across multiple units. The large-scale spin art was featured on our weekly patient television show, the CHO Show, aired live to every hospital room. I was also able to arrange an opportunity for Ozzy to display his spin art at the University of California Berkeley Art Museum. The opening of the art exhibit attracted a full house at the museum, with Ozzy's family members and interdisciplinary medical team in attendance.
Group art session with Child Life Specialist Jen Geisse that inspired the decision to make t-shirts for Ozzy's interdisciplinary medical team. One of Ozzy's signature designs is pictured on the bottom right.
I arrived at the Berkeley Art Museum with an iPad in hand. Jen and I knew that Ozzy would be unable to attend the exhibit in person, so we coordinated to allow him to attend via video conference. While I walked through the museum, filming the exhibition as I went, the video was screened live in the teen lounge at the hospital. Ozzy's family members were at the exhibition and congratulated him over the live video feed. Museum attendees could watch Ozzy's spin art activity at the hospital, which was facilitated by our child life staff. Jen was available in the teen lounge to assist Ozzy with the video conference and help him interact nonverbally with those attending his exhibition throughout the evening. His virtual presence had a powerful impact on the museum attendees viewing his work on display.
As Ozzy approached discharge and prepared to return home, he and his mother suggested the idea of a farewell spin art session where he would be able to make wearable artwork on t-shirts for members of his medical team. This final session served as an important transitional intervention for Ozzy to say goodbye to his primary team of clinicians and leave his mark on the hospital community.
While working in tandem, Jen and I regularly come together to talk about next steps in a creative intervention. Our aim is to work in partnership with the patient and their family members. We believe that artwork can serve as a powerful medium to share the first-person perspective of the patient, while also connecting with a community of hospitalized peers. In Ozzy's case, spin art was not only an accessible tool that enabled creative expression and empowerment in the inpatient setting, but it also provided a way for Ozzy to foster deep connections between his interdisciplinary hospital team and home community. Ozzy has since returned home from the hospital, but his large-scale spin art machine continues to be a valuable resource for new patients and families.
Looking out my office window, Jen and I pause as a helicopter lands on the nearby helipad. The wind picks up. The stretcher crosses the parking lot. We can see a small child tucked under the blanket. The medical team surrounds the child as they enter the back entrance of the hospital. Jen and I take a brief pause to acknowledge the new family entering the emergency department below, before turning our attention back to the next child on our list.
Special thanks and appreciation to Bayshore Christian Ministries for designing and constructing the large-scale spin art machine.
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