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\begin{document}
\preprint{TTP13-004}

\title{Higgs decay to goldstini and its observability at the LHC}

\author{Tao Liu}
\affiliation{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe
  Institute of Technology (KIT), D-76128 Karlsruhe, Germany}

\author{Lin Wang}
\affiliation{Institut f\"ur Theoretische Teilchenphysik, Karlsruhe
  Institute of Technology (KIT), D-76128 Karlsruhe, Germany}

\author{Jin Min Yang}
\affiliation{State Key Laboratory of Theoretical Physics,
      Institute of Theoretical Physics, Academia Sinica, Beijing 100190,
      China}

\begin{abstract}
If supersymmetry is broken independently in multiple sectors with
different scales, a number of goldstinos will be generated. One
linear combination of these goldstinos is massless and eaten by the
gravitino, while the orthogonal combinations acquire a tree level
mass and become the physical states named goldstini ($G'$). Compared
to the gravitino, such goldstini could couple more strongly to the
visible fields and lead to some exotic phenomenology. In this note
we first check the goldstini couplings in some GMSB models and find
that the goldstini-photon-neutralino interaction may be very small
while the goldstini-Z-neutralino and  goldstini-Higgs-neutralino
interactions may be sizable. This can induce a new decay mode for
the Higgs boson: $h\to G'+\chi \to Z+2G'$. Then we study the
observability of this decay channel at the LHC and find that it is
not accessible at the finished 8 TeV run (25 fb$^{-1}$) or 14 TeV
run with 100 fb$^{-1}$, but could be observed at the high luminosity
LHC (14 TeV, 1000-3000 fb$^{-1}$).
\end{abstract}
\pacs{14.80.Da,14.80.Ly,12.60.Jv}

\maketitle

\section{Introduction}
Supersymmetry (SUSY) is a consistent extension of Poincare symmetry in
quantum field theory and can solve the hierarchy
problem in the standard model due to the vanishing of quadratic
divergence. However, in a realistic particle theory, SUSY must be broken.
Usually it is assumed that SUSY breaking happens in a hidden sector and then
is transmitted to low energy fields via certain mechanisms. This will
generate a massless goldstino which acts as the
longitude component of gravitino in supergravity theory.
%%(which acquires a mass $m_{3/2}=F/(\sqrt{3}M_{planck})$.
If SUSY breaking happens in only one sector, the interaction
of the gravitino is given by \cite{Fayet:1977vd,Dreiner:2008tw}
\begin{eqnarray}\label{one}
\mathcal{L}_{int}&=&\frac{1}{F}(\partial_\mu G^\alpha
J^\mu_\alpha+h.c.),\nonumber \\
J^\mu&=&\sigma^\nu\bar\sigma^\mu\psi D_\nu\phi^*-i\sigma^\mu\bar\psi
F_\phi+i\frac{1}{2\sqrt{2}}\sigma^\nu\bar\sigma^\rho\sigma^\mu\bar\lambda
F_{\nu\rho}+\frac{1}{\sqrt{2}}\sigma^\mu\bar\lambda D ,\label{eqn1}
\end{eqnarray}
where ($\phi$, $\psi$, $F_\phi$) are the boson, fermi and $\theta^2$
components of a chiral superfield; ($\lambda,A_\mu, D$) are
the fermi, gauge and $\theta^2\bar\theta^2$ components of a vector superfield;
$F_{\nu\rho}$ denotes the strength tensor of the gauge field $A_\mu$
and $D_\mu$ is the corresponding covariant derivative (
throughout this paper we use the two component weyl notation
for the fields).
The non-derivative couplings can be obtained after integrating by parts
and using the equation of motion. From Eq.(\ref{eqn1}) we see that
the gravitino does not play an important role in low energy phenomenology
unless the SUSY breaking scale $F$ is sufficiently low.

However, if SUSY is broken in multiple sectors, some exotic
phenomenology will be generated
\cite{Cheung:2010mc,Cheung:2010qf,Craig:2010yf,Cheng:2010mw,Mawatari:2011jy,Argurio:2011hs,Thaler:2011me,Cheung:2011jq,Argurio:2011gu}.
In such a scenario, each sector breaks
SUSY independently at a scale $F_i$ and gives a
goldstino $\eta_i$. One linear combination of $\eta_i$ is
massless and eaten by the gravitino, while the orthogonal
combinations named goldstinis acquire a tree level mass
$m_{G^\prime}=2m_{3/2}$ in SUGRA \cite{Cheung:2010mc} (possible
loop corrections to the goldstini mass are also considered
\cite{Cheung:2010mc,Argurio:2011hs}). The true goldstino (longitudinal
component of gravitino) has an interaction in Eq.(\ref{eqn1}) with
$F=\sqrt{\sum_i F_i^2}$.
Unconstrained by the supercurrent, the interaction of goldstini can
be quite different and large enough to have intriguing phenomenology
at colliders. For example, in the framework of gauge mediated SUSY
breaking (GMSB), the goldstini can lead to some final states which
are softer and more structured at colliders \cite{Argurio:2011gu}.
It can also serve as a dark matter candidate \cite{Cheng:2010mw}.
Additionally, as noted in \cite{Thaler:2011me}, the goldstini may
couple 'strongly' (much stronger than gravitino) with the lightest
observable-sector supersymmetric particle (LOSP), which then may
lead to some exotic decay channels for the Higgs boson, e.g.,
the Higgs may decay to the LOSP plus a goldstini.

Given the importance and urgency of Higgs physics, any possible
exotic decays of the Higgs boson should be scrutinized. As is well
known, a Higgs-like particle around 125 GeV has been observed by
ATLAS and CMS collaborations \cite{125ATLAS,125CMS}, which motivated
numerous theoretical studies, especially the enhanced diphoton decay
has been intensively studied in various new physics models, such as
the low energy SUSY \cite{susy}, the little Higgs model \cite{lht},
the two-Higgs-doublet model \cite{2hdm}, the Higgs triplet model
\cite{triplet}, the models with extra dimensions \cite{extrad} and
other Higgs extensions \cite{hrrmodel}. In this note we focus on the
effects of goldstini and examine the exotic Higgs decay mode $h\to
\chi+G'$ followed by $ \chi \to Z+G'$ under the condition
$m_Z<m_\chi<m_h$ ( here $\chi$ is the ordinary lightest neutralino
which is assumed to be the LOSP, $G^\prime$ denotes the goldstini).

We will make
a brief review on goldstini and explain why we focus on this channel
in Section II. Then we take a model-independent way to study the
Higgs decay $h\to \chi+G'$ followed by $ \chi \to Z+G'$
at the LHC in Section III. Finally,
we give our conclusion in Section IV.

\section{Theoretical motivations}
\subsection{A brief review on goldstini}
 For simplicity we assume that there are only two
sequestered sectors which break SUSY spontaneously. Following the
arguments in \cite{Thaler:2011me}, each of them can be parameterized
in a non-linear way:
\begin{eqnarray}
X_i=\frac{\eta_i^2}{2F_i}+\sqrt{2}\theta\eta_i+\theta^2F_i,
\end{eqnarray}
where $\eta$ is the so-called goldstino. Due to the
non-renormalization theorem of superpotential, visible sector only
obtains SUSY breaking information through non-trivial K$\ddot{a}$hler
potential $K$ and gauge kinetic functions $f$. After integrating out
hidden sector fields, $X_i$ couple to single species visible fields
$\Phi$ as
\begin{eqnarray}
K&=&\Phi^+\Phi\sum_i\frac{m_{\phi,i}^2}{F_i}X_i^+X_i,\\
f_{ab}&=&\frac{1}{g_a^2}\delta_{ab}\left(1+\sum_i\frac{2m_{a,i}}{F_i}X_i\right),
\end{eqnarray}
where $g_a$ denote the gauge coupling constants, $m_{\phi,a}$ are
respectively the soft masses for the scalar of the chiral superfield
and gauginos.
The trilinear and bilinear soft terms between multiple fields could also
be constructed, but since they are subleading and model-dependent in the
contribution to interactions between goldstino and visible sector,
we do not consider them here. Substituting the expression of $X_i$
into the above formula gives the Lagrangian up to order $1/F_i$:
\begin{eqnarray}
\mathcal
{L}=&-&\sum_im_{\phi,i}^2\phi^*\phi
+\sum_i\frac{m_{\phi,i}^2}{F_i}\eta_i\psi\phi^* \nonumber\\
~~~~&-&\frac{1}{2}\sum_i
m_{a,i}\lambda^a\lambda^a
-\sum_i\frac{im_{a,i}}{\sqrt{2}F_i}\eta_i\sigma^{\mu\nu}\lambda^aF^a_{\mu\nu}
+\sum_i\frac{m_{a,i}}{\sqrt{2}F_i}\eta_i\lambda^aD^a\label{gold}.
\end{eqnarray}
The mass eigenstates can be obtained by a rotation of $\eta_i$:
\begin{align}
G=\cos\theta\eta_1+\sin\theta\eta_2, \quad
G^\prime=-\sin\theta\eta_1+\cos\theta\eta_2,
\end{align}
where $\theta$ is defined by $\tan\theta=F_2/F_1$.
It is easy to see that the Weyl spinor $G$ is related to SUSY breaking
scale $F=\sqrt{F_1^2+F_2^2}$ while $G^\prime$ vanish in the
non-linear form. Then the interaction Lagrangian becomes
\begin{eqnarray}
\mathcal{L}_G&=&\frac{m_\phi^2}{F}G\psi\phi^*
-\frac{im_a}{\sqrt{2}F}G\sigma^{\mu\nu}\lambda^aF^a_{\mu\nu}+
\frac{m_a}{F}G\lambda^aD^a  \label{LG},\\
\mathcal{L}_{G^\prime}&=&\frac{\tilde{m}_\phi^2}{F}G^\prime\psi\phi^*
-\frac{i\tilde{m}_a}{\sqrt{2}F}G^\prime\sigma^{\mu\nu}\lambda^aF^a_{\mu\nu}+
\frac{\tilde{m}_a}{F}G^\prime\lambda^aD^a \label{Leta},
\end{eqnarray}
with the parameter $m$ and $\tilde{m}$ defined as
\begin{align}
m_{\phi/a}=m_{\phi/a,1}+m_{\phi/a,2}, \quad
\tilde{m}_{\phi/a}=-m_{\phi/a,1}\tan\theta+m_{\phi/a,2}\cot\theta.
\end{align}
$\mathcal{L}_G$ is just the non-derivative version of
Eq.(\ref{eqn1}) and the couplings are proportional to the
soft masses as expected. For goldstinis, their interactions with
visible fields could be enhanced as long as $\tilde{m}$ is larger
than $m$. Note
that if $D^a$ get a vacuum expectation value, there will be small
mixing between $\lambda^a$ and $\eta_i$. In most cases this mixing
is very small, so we can omit it safely.

The mass of goldstini is given at tree level by
$m_{G^\prime}=2m_{3/2}$ due to the intrinsic property of SUGRA.
Additionally, there could be corrections to such a tree-level mass,
which, however, are model-dependent \cite{Cheung:2010mc,Argurio:2011hs}.
In our analysis we assume
that both $m_{G}$ and $m_{G^\prime}$ are very
smaller than the Higgs mass.

\subsection{Couplings between goldstini and Higgs boson}
Without the extra goldstinis, the interaction of goldstino ($G$)
with the lightest neutralino ($\chi$) is given by
\begin{align}
\mathcal{L}=\frac{y_1}{\sqrt{2}F}\chi
Gh+\frac{y_2}{2\sqrt{2}F}\chi\sigma^{\mu\nu}
GF_{\mu\nu}^{\gamma}+\frac{y_3}{2\sqrt{2}F}\chi\sigma^{\mu\nu}
GZ_{\mu\nu}+\frac{y_4}{\sqrt{2}F}\bar\chi\bar\sigma^{\mu}GZ_\mu+h.c.
\end{align}
where $F^\gamma_{\mu\nu}$ is the photon field strength, $Z_\mu$ and
$Z_{\mu\nu}$ are respectively the $Z$-boson field and its field strength.
In the minimal supersymmetric standard model (MSSM), the above
parameters $y_{1,2,3,4}$ are given by
\begin{eqnarray}
y_1&=&-N_{11}^{-1}m_1m_Z\sin\theta_W\sin(\alpha+\beta)
+N_{21}^{-1}m_2m_Z\cos\theta_W\sin(\alpha+\beta)\nonumber\\
&&+N_{31}^{-1}(B_\mu\cos\alpha-m_{H_d}^2\sin\alpha)
+N_{41}^{-1}(-B_\mu\sin\alpha+m_{H_u}^2\cos\alpha),\\
y_2&=&-2im_\chi(N_{11}^*\cos\theta_W+N_{12}^*\sin\theta_W),\\
y_3&=&-2im_\chi(-N_{11}^*\sin\theta_W+N_{12}^*\cos\theta_W)
-2im_Z(N_{13}^*\cos\beta-N_{14}^*\sin\beta),\label{ZG1} \\
y_4&=&im_Z^2(-N_{11}^*\sin\theta_W+N_{12}^*\cos\theta_W)
+im_Zm_\chi(N_{13}^*\cos\beta-N_{14}^*\sin\beta).\label{ZG2}
\end{eqnarray}
where $N_{1i}$ denote the mixing between the lightest neutralino
and the gauginos or higgsinos.
We see that unless the SUSY breaking scale $F$ is small
enough, such interactions of goldstino
could hardly affect the 125 GeV Higgs.

Naively speaking, the interactions of the goldstini can be obtained
from the above expression with the goldstino $G$ replaced by the
goldstini $G'$ and each soft mass replaced by its corresponding
$\tilde{m}$. Then it is clear that for $\tilde{m}\gg m$ the
interactions of the goldstini can be significantly stronger than the
goldstino. However, there are some subtle differences which deserve
attention.

Next, let us scrutinize the concrete low energy interactions of the
goldstini. The first one is its interaction with Higgs and
neutralino: $\frac{\tilde{y}_1}{\sqrt{2}F}\chi G^\prime h$. From
Eq.(\ref{Leta}) we can obtain the coefficient of this interaction:
\begin{align}
\tilde{y}_1=-N_{11}^{-1}\tilde{m}_1m_Z\sin\theta_W\sin(\alpha+\beta)
+N_{21}^{-1}\tilde{m}_2m_Z\cos\theta_W\sin(\alpha+\beta)\nonumber\\
+N_{31}^{-1}(\tilde{B}_\mu\cos\alpha-\tilde{m}_{H_d}^2\sin\alpha)
+N_{41}^{-1}(-\tilde{B}_\mu\sin\alpha+\tilde{m}_{H_u}^2\cos\alpha).
\end{align}
Similarly, the interaction with photon and neutralino takes a
form of $\frac{\tilde{y}_2}{2\sqrt{2}F}\chi\sigma^{\mu\nu}G^\prime
F_{\mu\nu}^\gamma$ with
\begin{align}\label{new}
\tilde{y}_2= -2im_\chi\left[\frac{\tilde{m}_{1}}{m_{1}}
N_{11}^*\cos\theta_W+\frac{\tilde{m}_2}{m_2}N_{12}^*\sin\theta_W\right].
\end{align}
The interaction with $Z$-boson and neutralino is somewhat complicated.
But based on the experience obtained above, some hints could be obtained.
From the expression of $y_{3/4}$, it can be easily found that
the $Z$-boson can be divided into two parts: one is its transverse
component proportional to
$(-N_{11}^*\sin\theta_W+N_{12}^*\cos\theta_W)$, and the other is the
longitude component proportional to
$(N_{13}^*\cos\beta-N_{14}^*\sin\beta)$.
The interaction of goldstini with the transverse part of the $Z$-boson
is almost the same as photon because they come from the same origin
$\frac{im_{a,i}}{\sqrt{2}F_i}\eta_i\sigma^{\mu\nu}\lambda^aF^a_{\mu\nu}$.
After multiplying a factor $\tilde{m}_1/m_{1}$ to
$N_{11}^*\cos\theta_W$ and a factor $\tilde{m}_2/m_2$ to
$N_{12}^*\sin\theta_W$ in Eqs.(\ref{ZG1},\ref{ZG2}), we can get
the coefficient of this transverse coupling. For the longitude
component of the $Z$-boson, since it
has some relations with the tilded mass parameters of the Higgs,
such as $\tilde{m}_{h_{u,d}}$ and $\tilde{\mu}$, we can
not get the same simple result. In \cite{Argurio:2011gu}
a free factor $K_{Z_L}$ is introduced to connect the longitudinal
interaction between $G$ and $G^\prime$. Although we do not
know the exact formula for $K_{Z_L}$, it must be a function of
$\tilde{m}_\phi/m_\phi$ from naive arguments and dimensional
analysis (here ${m_\phi}$ denotes the soft Higgs parameters).
The details of the coupling between $Z$-boson and goldstini can
be found in the appendix of \cite{Thaler:2011me}.

From the above analysis we see that the $G^\prime-\gamma-\chi$
coupling is proportional to $\tilde{m}_a$ (tilded gaugino masses)
while the $G^\prime-Z-\chi$ and
 $G^\prime-h-\chi$ couplings are proportional to $\tilde{m}_{a,\phi}$.
So, if  $\tilde{m}_a$ are very small while $\tilde{m}_\phi/m_{\phi}$
is much larger than one, we can get a new scenario in which the
Higgs decay $h\to \chi G'\to ZG'G'$ is more sizable than
  $h\to \chi G'\to \gamma G'G'$.
Note that from the viewpoint of model-building, this scenario could be
easily realized in a two-sector messenger with $F_1\gg F_2$.
If the second sector preserves $R$-symmetry, it will give no
contribution to the gaugino masses.
Besides, in some concrete GMSB models, especially the
direct gauge mediation \cite{Komargodski:2009jf},
 the gaugino masses are usually suppressed by a factor $F_i^2/M_i^4$
due to the vacuum structure of superfields. So it is common to have
approximately vanishing $\tilde{m}_a$.
Additionally, in the case of stimulate SUSY
breaking \cite{McCullough:2010wf}, it is quite natural to
split $F_1$ and $F_2$.

\section{Higgs decay to goldstini at the LHC}
From the analysis in the preceding section,
we see that in some GMSB models
the goldstini couplings  $G^\prime-h-\chi$  and  $G^\prime-Z-\chi$
can be sizable while the coupling $G^\prime-\gamma-\chi$ can be
much suppressed. This scenario can lead to the
Higgs decay $h\to \chi G'\to ZG'G'$ which will be studied in
this section.

\subsection{The rate of Higgs decay to goldstini}
We take an effective way to study the Higgs decay $h\to \chi G'\to ZG'G'$.
The effective Lagrangian is given by
\begin{align} \mathcal
{L}_{eff}=\frac{m^2}{F}[g_{h\chi}h\chi G^\prime + \frac{g_{\chi
Z_{1}}}{m}G^\prime\sigma^{\mu\nu}Z_{\mu\nu}\chi + g_{\chi Z_{2}}
\bar{G}^\prime\bar{\sigma}^{\mu}Z_{\mu}\chi + h.c.] \label{eff}
\end{align}
Here a mass parameter $m$ is introduced to make the couplings
$g_{h\chi}$, $g_{\chi Z_1}$ and  $g_{\chi Z_2}$ dimensionless and
the Lagrangian takes the similar form as in \cite{monophoton} which
studied the signal of monophoton plus missing energy from the Higgs
decay. Note that for $m_{\chi}>m_h$, the lightest neutralino can
decay to the Higgs boson which may lead to many boosted Higgs bosons
at the LHC \cite{Howe:2012xe}.

For the Higgs decay $h\to ZG'G'$, if it proceeds through
an off-shell neutralino, i.e.,  $h\to \chi^* G'\to ZG'G'$,
its rate is small.
But if the neutralino can be on-shell, the decay
 $h\to \chi G'\to ZG'G'$ may be sizable.
In Fig.~\ref{fig1} we show the partial widths for
 $h\to ZG'G'$ and $ \chi\to ZG'$. We see that only for
an on-shell neutralino (lighter than the Higgs boson and
heavier than the $Z$-boson) can the decay $h\to ZG'G'$ be
sizable.

%%%%%%%%%%%%%%% Fig.1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htbp]
\includegraphics[width=3.2in,height=2.4in]{fig1a.eps}%
\hspace{-0.25in}%
%\vspace{0.01cm}
\includegraphics[width=3.2in,height=2.4in]{fig1b.eps}%
\hspace{0in}%
\vspace{-0.5cm} \caption{The partial widths
of  $h\to ZG'G'$ and  $ \chi\to ZG'$ versus the neutralino mass.
Here we choose $m = 0.1\sqrt{F}$ with $\sqrt{F} = 1.5$ TeV,
and $g_{h\chi}=g_{\chi Z_1}=g_{\chi Z_2}=1$.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Note that there are already some constraints on the whole SUSY
breaking scale $\sqrt{F}$ through the interaction of gravitino
(usually photon is also involved). The current lower bound is around 300 GeV
\cite{Acosta:2002eq,Achard:2003tx} and it is expected that the
LHC could push it to about 1.6 TeV \cite{Brignole:1998me}.
The most sensitive
process to $\sqrt{F}$ is the decay channel $h\rightarrow
\chi G^\prime$ whose branching ratio is proportional to $1/F$.
In our calculation we fix $\sqrt{F}=1.5$ TeV and $m=0.1\sqrt{F}$,
and assume all the dimensionless couplings to be unity.

About the decay of the neutralino $\chi$, it may have
some other modes, e.g., decay to SM particles if R-parity
is violated or decay to a light $U(1)_X$ gaugino \cite{Baryakhtar:2012rz}.
In our calculation we assume that $\chi$ only decays to $ZG^\prime$.
If there are other decay modes, the corresponding signal
should be multiplied by the branching ratio $Br(\chi\rightarrow ZG^\prime)$.
Additionally, the decay length of the nuetralino with energy E is
approximately
$\Gamma^{-1} \sqrt{(E^2-m_\chi^2)/m_\chi^2}\sim
10^{-10}cm$, so the neutralino will decay inside the detector.

\subsection {Signal and background}
With the goldstini couplings in Eq.(\ref{eff}), the final state
$ZG'G'$ can come from three processes at the LHC, as shown in
Fig.~\ref{fig2}. Generally, the processes (a) and (b) will generate
$Z$-boson with low transverse momentum, while (c) with $t$-channel
squark exchange may generate a $Z$-boson with large transverse
momentum \cite{Brignole:1998me}. Additionally, heavier squarks which
are consistent with the result of LHC will suppress the $t$-channel
contributions, so we will not consider the process (c).
%%%fig.2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\epsfig{file=fig2.eps,width=16cm,height=3.5cm} \vspace{0.25cm}
\caption{Feynman diagrams for the production of $ZG'G'$ at the LHC
induced by the goldstini couplings in Eq.~(\ref{eff}).}
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We require the $Z$-boson produced in association with goldstini to
decay to electrons or muons. Therefore, the signal is a pair of
electrons or muons and low missing transverse momentum, i.e., $pp
\to Z + {\large \not} \! E_{t} \to l^{+}l^{-} + {\large \not} \!
E_{t}$ ($l = e,\mu$). Some studies about such mono-Z process have
been performed in \cite{invisible
z'decay,mono-z-dm,Carpenter:2012rg}, where the mono-Z has a large
transverse momentum. In our signal, the $Z$-boson has small
transverse momentum because it comes from the decay of the
neutralino.

For the $s$-channel process shown in Fig.2(a),
the Higgs boson can be produced on-shell and then decay to the
$Z$-boson plus missing energy. For such a production we can
get the high order corrections by multiplying a $k$-factor.
At the LHC, the leading order Higgs
boson production in the SM is from the gluon-gluon fusion. We
calculate the Higgs boson production cross section at $\sqrt{s}=14$
TeV with CTEQ6M parton distribution functions. We set the
renormalization scale $\mu_{R}$ and factorization scale $\mu_{F}$ at
the Higgs mass of 125 GeV. The other relevant parameters
are set as
\begin{eqnarray}
m_{t}=173.3 ~{\rm GeV},  ~~m_{b}=4.67 ~{\rm GeV},
~~m_{Z}=91.188 ~{\rm GeV},~~m_{W}=80.4 ~{\rm GeV}
\end{eqnarray}
We consider the higher order corrections and follow \cite{monophoton} to
get $\sigma_h^{NNLO}=49.99$~pb at the next-to-next-to leading order
with next-to-next-to leading logarithm resummation.

For the $Z$-boson $s$-channel process shown in Fig.2(b),
the neutralino can be produced on-shell and then decay
to $Z$-boson and $G^\prime$. So far no high
order corrections have been calculated for this process
and we calculate this production at the leading order.

The main SM backgrounds are
\begin{eqnarray}
 pp &\rightarrow& Z Z \rightarrow l^{+}l^{-}\nu\bar{\nu} , \\
 pp &\rightarrow& W^{+} W^{-} \rightarrow l^{+}\nu  l^{-}\bar{\nu} , \\
  pp &\rightarrow& Z j \rightarrow l^{+}l^{-} j ,\\
  pp &\rightarrow& t \bar{t} \rightarrow
 b\bar{b}l^{+}l^{-}\nu\bar{\nu} ,\\
 pp &\rightarrow& Z W^{\pm} \rightarrow l^{+}l^{-}l^{\pm}\nu .
\end{eqnarray}
Obviously, the first two processes are our backgrounds. Because
they proceed through the electroweak interaction,
their cross sections are expected to be relatively small at the LHC.
The third process can mimic our signal with the missing jet,
which happens for $|\eta_j| > 4$ \cite{monophoton}.
For the forth process to mimic our signal, the two $b$-jets
must be missing, which is less likely. So we will not consider it.
For the last process, one lepton must be missing, similar to the third
process. Since its cross section is much smaller than the third process,
we neglect it in our calculation.
Therefore, in our analysis we only consider the first three processes.
In our calculation we use MadGraph5 \cite{MG5} for both the signal
and background.

\subsection {Numerical results}
Before giving the numerical result on the signal and backgrounds,
we summarize our selection criteria on the final state:
\begin{equation}
P_{T}^{l} > 20~GeV,~~~ {\large \not} \! E_{T} > 20~GeV,~~~|\eta_{l}|
< 2.5,~~~|\eta_{j}| > 4.0,~~~\Delta R \equiv \sqrt{\Delta
\eta^{2}+\Delta \phi^{2}} > 0.4 ,\label{criteria}
\end{equation}
where the separation $\Delta R$ is for the leptons or jet in the
final state with $\Delta \eta$ being the pseudo-rapidity difference
and $\Delta \phi$ being the azimuthal angle difference.
In  Table I we present the number of events for
the LHC with $\sqrt{s} = 14$ TeV and 10 $fb^{-1}$ of integrated luminosity.
In our calculation of the signal,
the mass parameters in Eq.~(\ref{eff}) are fixed as
$m_{\chi}=110$ GeV, $m_{G^{\prime}}=0$
(the goldstini is much lighter than the Higgs, so we set it to zero for simplicity),
 $m=0.1\sqrt{F}$,
$\sqrt{F}=1.5$ TeV, while all the dimensionless couplings
are fixed to unity.

From Table \ref{table 1} we see that the signal is overwhelmed by the
backgrounds. The largest background is from $Z+jet$ production, which
is about three orders above the signal.
As expected, the cut on the invariant dilepton mass
 $|m_{ll}-m_{Z}|<5$ GeV can suppress the $W^{+}W^{-}$ background efficiently.
Since in our signal the missing transverse momentum is relatively low,
the cut on the missing $E_T$ can further suppress the backgrounds.

%%%%%%%%%%%%%%%%%%%%%%%%%%%  table 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{The number of signal and background events
 for the LHC with $\sqrt{s} = 14$ TeV and 10 $fb^{-1}$
of integrated luminosity.}
\vspace{0.2cm}
\begin{tabular}{||c||c|c|c|c||c|c|c||}
 \hline \hline
 \multirow{2}{*}{\minitab{c}{$\sqrt{s} = 14$ TeV \\($10fb^{-1}$)}}  &  \multicolumn{4}{c||}{Background (B)}  &  \multicolumn{3}{c||}{Signal (S)}  \\
\cline{2-8}
 &  $Z Z$ $\rightarrow l^{+}l^{-} \nu\bar{\nu}$ &  $W^{+} W^{-} \rightarrow l^{+}\nu  l^{-}\bar{\nu}$  &  $Z j$ $\rightarrow l^{+}l^{-} j$  &
 $B_{tot}$  &  $S_{h}$  &  $S_{Z}$  &  $S_{tot}$ \\
 \hline
 no cut  &  964  &   12598  &  37950  &  51512  &  72  &  21  &  93 \\
 \hline
 $|m_{ll}-m_{Z}|<5$ GeV   &  834  &  900   &  33000  &  34734   &  58   &  18  &  76  \\
\hline
 ${\large \not} \! E_{T} < 30$ GeV  &  133  &  214   &  20700  &  21047   &  58   &  4  &  62  \\
 \hline
 \hline
\end{tabular}
\label{table 1}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Finally, we show in Table II the signal significance
for different integrated luminosity.
Since the contribution from the $s$-channel $Z$-boson process
is much smaller than the Higgs process (as shown in Table \ref{table 1}),
in Table II we only consider the Higgs process so that
the result could be interpreted as the product of the Higgs production rate,
the Higgs decay branching ratio and the neutralino decay branching ratio.
We see that only for the high luminosity LHC with 1000-3000 fb$^{-1}$
can the signal be observable.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%   table 2   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\caption{Same as Table I, but showing the event number of the signal
$pp \to h \to \chi G' \to Z G' G' \to l^{+}l^{-} + E_{t}$
and its significance for the LHC with $\sqrt{S} = 14$ TeV
and different luminosity. }
\vspace{0.2cm}
\begin{tabular}{||c|c|c|c|c|c|c||}
 \hline \hline
 $\sqrt{S} = 14$ TeV  &  100 $fb^{-1}$  &  500 $fb^{-1}$  &  800 $fb^{-1}$  & 1000 $fb^{-1}$ &  2000 $fb^{-1}$  & 3000 $fb^{-1}$  \\
 \hline
 $S_{[no~cut]}$ &  720   &   3600   &  5760  &  7200  &  14400  &  21600     \\
 \hline
 $S_{[passing~all~cut]}$  &  580  &  2900   &  4640  & 5800  &  11600  &  17400 \\
\hline
 $S$/$\sqrt{ S+B}$  &  1.3  &  2.8  &  3.6  &  4.0  &  5.6  &  7.0   \\
 \hline
 \hline
\end{tabular}
\label{table 2}
\end{table}

\section{Conclusion}
Compared with the gravitino, the goldstini can couple more strongly
to the visible fields and thus induce some interesting
phenomenology. In this note we considered the effects of the
goldstini on the Higgs phenomenology. We found that in some GMSB
models the goldstini has approximately vanishing interaction with
photon and the lightest neutralino, but the corresponding coupling
with the $Z$-boson is sizable. This could induce the mono-$Z$ decay
of the Higgs boson ($h\to G^\prime+\chi \to Z+2G^\prime$) which
signals mono-Z and missing $E_{t}$ at the LHC. Then in an effective
model we studied the observability of this decay at the LHC. From a
Monte Carlo simulation of the signal and background, we found that
it is not accessible at the finished 8 TeV run (25 fb$^{-1}$) or 14
TeV run with 100 fb$^{-1}$, but could be observed at the high
luminosity LHC (14 TeV, 1000-3000 fb$^{-1}$). Although at the LHC it
is so challenging to detect this exotic decay channel of the Higgs
boson, it is worth hunting because such a scenario may naturally
exist in GMSB with multi SUSY breaking sectors.

\section*{Acknowledgments}
We would like to thank Olivier Mattelaer for his great help on
MadGraph and Lei Wu for useful discussions. Lin Wang acknowledges
Prof.~Johann H. K\"{u}hn and Prof.~Matthias Steinhauser for their
warm hospitality. This work is supported by DFG through SFB/TR 9
``Computational Particle Physics'' and by the National Natural
Science Foundation of China under grant Nos. 11275245, 10821504 and
11135003.

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\end{document}
