# Gauge extensions of supersymmetric models and hidden valleys

Mingxing Luo and Sibo Zheng
Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University, Hangzhou 310027, P. R. China.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

Supersymmetric models with extended group structure beyond the standard model
are revisited in the framework of general gauge mediation. Sum rules for
sfermion masses are shown to depend genuinely on the group structure, which
can serve as important probes for specific models. The left-right model and
models with extra $U(1)$ are worked out for illustrations. If the couplings of
extra gauge groups are small, supersymmetric hidden valleys of the scale
$10-100$ GeV can be naturally constructed in companion of a TeV-scale
supersymmetric visible sector.

Supersymmetric Phenomenology, Gauge Mediation

##  1 Introduction

Recently, a general method has been proposed to calculate soft terms in gauge
mediation models (GGM) [1]. It turns out that all soft terms in a specific
model can be determined by a few parameters, which encode the information of
the hidden sector. One obtains two sum rules [2] which make a distinctive
feature in $R$-symmetry breaking gauge mediation, in comparsion with gravity
mediation [3, 4], if the breaking of supersymmetry is communicated to the
visible sector only by standard model gauge interactions. In principle, the
hidden sector can be either weakly or strongly coupled. And the formalism is
valid for both direct and non-direct gauge mediation.

In this paper, we will reconsider the gauge extended supersymmetric model
(ESM) using the GGM formalism. The ESM can easily be constructed in deformed
ISS theories [14]. In these models, the unbroken global symmetry $G_{0}$ in
the hidden sector is larger than $G_{SM}$. If there are extra gauge structures
beyond the standard model (SM) and the corresponding symmetry in $G_{0}$ is
weakly gauged at the supersymmetric breaking scale $M_{SUSY}$, extra
interactions in addition to those of $G_{SM}$ yield modifications to the soft
terms, among other things. In general, they modify the sum rules in [2]. The
extra gauge structure is assumed to be spontaneously brokon at an intermediate
scale between $M_{SUSY}$ and $M_{EW}$, with the corresponding gauge boson of
masses in the order of a few TeV. At the electro-weak scale, only the SM is
left in the visible sector. Such arrangement is similar to the
$Z^{\prime}$-mediated supersymmetric breaking [9, 10] in spirit.

To be concrete, we will consider the abelian case where the gauged part of
$G_{0}$ is $G_{SM}\times U(1)^{\prime}$, and the non-abelian supersymmertic
left-right model where the gauged part of $G_{0}$ is $SU(3)_{c}\times
SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. We find that the sum rules in [2]
cannot be retained in either cases. The resulting modifications are easily
obtained and dependent on the couplings of extra gauge sector. If the extra
gauge couplings are comparable with the ones in the SM, the sum rules are
broken significantly. If these couplings are small enough, these sum rules can
survive as approximations. These analysis can be directly applied to theories
with more sophisticated gauge structures, with similar conclusions.

The small coupling case is then used to construct models with a particular
type supersymmetric hidden valley models [17]. In particular, we will
construct a model in which the extra $U(1)^{\prime}$ communicates between the
supersymmetry breaking sector and a hidden valley sector, which generates
supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden valley sector to the visible sector.
If the $U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at
$M_{SUSY}$, the soft terms in the hidden valley are two or four orders of
magnitude smaller than those in the visible sector, which implies that an
$\mathcal{O}(1-10)$ TeV-scale visible sector is accompanied by an
$\mathcal{O}(10-100)$ GeV-scale hidden valley.

The rest of the paper is organized as follows. In section II, we briefly
review and comment on the GGM formalism. In section III, we discuss
supersymmetric models with group structure beyond SM. Particular attention
will be payed to the sum rules for sfermion masses. In section IV, we propose
a class of supersymmetric hidden valleys with $U(1)^{\prime}$. Finally we
conclude in section V.

##  2 Review and comments on GGM

In this section, we briefly review the GGM formalism, with an emphasis on
calculations of soft terms and sum rules for sfermion masses in the MSSM. The
gauge current of the hidden sector is a real linear superfield,

$\displaystyle{}\bar{D}^{2}\mathcal{J}=D^{2}\mathcal{J}=0$ (2.1)

with $\partial_{\mu}j^{\mu}=0$, as required by the current conservation
condition. The two-point correlator of $\mathcal{J}$ can generally be written
as,

$\displaystyle<\mathcal{J}(p,\theta,\bar{\theta})\mathcal{J}(p^{\prime},\theta^{\prime},\bar{\theta}^{\prime})>=2\pi^{4}\delta^{4}(p+p^{\prime})I(p,p^{\prime},\theta,\theta^{\prime})$
(2.2)

(2.1) and (2.2) can be generally solved, by a set of real functions
$B_{1/2},C_{a}$ [5]. The Fourier transforms of the correlators in momentum
space are,

$\displaystyle{}\langle J(p)J(-p)\rangle$ $\displaystyle=$
$\displaystyle\tilde{C}_{0}(p^{2}/M^{2};\,M/\Lambda)$ $\displaystyle\langle
j_{\alpha}(p)\bar{j}_{\dot{\alpha}}(-p)\rangle$ $\displaystyle=$
$\displaystyle-\sigma_{\alpha\dot{\alpha}}^{\mu}p_{\mu}\tilde{C}_{1/2}(p^{2}/M^{2};\,M/\Lambda)$
$\displaystyle\langle j_{\mu}(p)j_{\nu}(-p)\rangle$ $\displaystyle=$
$\displaystyle-(p^{2}\eta_{\mu\nu}-p_{\mu}p_{\nu})\tilde{C}_{1}(p^{2}/M^{2};\,M/\Lambda)$
$\displaystyle\langle j_{\alpha}(p)j_{\beta}(-p)\rangle$ $\displaystyle=$
$\displaystyle\epsilon_{\alpha\beta}M\tilde{B}_{1/2}(p^{2}/M^{2})$ (2.3)

If SUSY is unbroken, $\tilde{C}_{0}=\tilde{C}_{1/2}=\tilde{C}_{1}$,
$\tilde{B}_{1/2}=0$. If supersymmetry is broken, these relations do not hold
in general, as now $(Q_{\alpha}+Q^{\prime}_{\dot{\alpha}})I\neq 0$ and
$(\bar{Q}_{\alpha}+\bar{Q}^{\prime}_{\dot{\alpha}})I\neq 0$.

The gauge current superfield acts as a source for visible vector superfield
via the coupling,

$\displaystyle{}\mathcal{L}_{int}=2g\int
d^{4}\theta\mathcal{J}V+\dots=g(JD-\lambda
j-\bar{\lambda}\bar{j}-j^{\mu}V_{\mu})+\cdots,$ (2.4)

Note that in writing (2.4) the Wes-Zumino gauge has been chosen for the vector
superfield. Integrating out the messenger sector, we obtain the effective
Lagrangian for the gauge supermultiplet,

$\displaystyle\delta\mathcal{L}_{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g^{2}\tilde{C}_{0}(0)D^{2}-g^{2}\tilde{C}_{1/2}(0)i\lambda\sigma^{\mu}\partial_{\mu}\bar{\lambda}-{1\over
4}g^{2}\tilde{C}_{1}(0)F_{\mu\nu}F^{\mu\nu}$ (2.5) $\displaystyle-$
$\displaystyle{1\over 2}g^{2}(M\tilde{B}_{1/2}(0)\lambda\lambda+c.c.)+\dots$

This gives contributions to the gaugino and sfermion masses, respectively,

${}M_{r}=g_{r}^{2}M\tilde{B}_{1/2}^{(r)}(0),~{}~{}~{}~{}~{}~{}\tilde{m}_{f}^{2}=\sum_{r=1}^{r=3}g_{r}^{4}c_{2}(f;r)A_{r}$
(2.6)

where $c_{2}(f;r)$ is the Casimir of the representation $f$ under the $r$
gauge group and

$\displaystyle{}A_{r}=-\int\frac{d^{4}p}{(2\pi)^{4}}\frac{1}{p^{2}}\left(3\tilde{C}_{1}^{(r)}(p^{2}/M^{2})-4\tilde{C}_{1/2}^{(r)}(p^{2}/M^{2})+\tilde{C}_{0}^{(r)}(p^{2}/M^{2})\right)$
(2.7)

Note that the $\mu$ and $B\mu$ terms are model dependent. They can not be
determined unless more assumptions on the Higgs sector and the hidden sector
are made.

Now a few comments are in order for (2.6). First, one can choose the
superfield formalism at the starting point (2.4). The correlator of vector
superfields takes a simple form
$<\mathcal{V}\mathcal{V}>=\delta^{4}(\theta-\theta^{\prime})/p^{2}$ with the
gauge fixing parameter ${\xi}=1$. The wave function renormalization
$\mathcal{Z}_{Q}$ in the Kahler potential $\int
d^{4}\theta\tilde{Q}e^{-2\mathcal{V}}Q$ yields exactly the soft sfermion
masses in (2.6). Furthermore, the tri-linear $A$ terms in superpotential can
be obtained by replacing $Q$ with canonically normalized $Q^{\prime}$,

$\displaystyle{}Q^{\prime}=\left(1-\frac{1}{2}\mathcal{Z}|_{\theta^{2}}\theta^{2}-\frac{1}{2}\mathcal{Z}|_{\theta^{2}}\bar{\theta}^{2}\right)Q,$
(2.8)

Second, vector superfileds in (2.4) are usually massive after the
corresponding gauge symmetries are broken. If $m_{V}>>M_{SUSY}$, they can be
integrated out and will play a minor role in the communication of
supersymmetry breaking. If $m_{V}\sim M_{SUSY}$, the effects of $m_{V}$ need
then to be taken into account. At the leading order, one would have

$\displaystyle{}A^{\prime}_{r}=-\int\frac{d^{4}p}{(2\pi)^{4}(p^{2}-m_{V}^{2})}\left(\frac{3p^{2}}{p^{2}-m_{V}^{2}}\tilde{C^{\prime}}_{1}^{(r)}(p^{2}/M^{2})-4\tilde{C^{\prime}}_{1/2}^{(r)}(p^{2}/M^{2})+\tilde{C^{\prime}}_{0}^{(r)}(p^{2}/M^{2})\right)$

In this paper, we will assume $m_{V}<<M_{SUSY}$ for simplicity and (2.7) will
be used.

The positivity of $A_{r}$ has been proved in F-term supersymmetry breaking
with $F\ll M^{2}$ [6, 5], where $A_{r}$ can be written as a derivative term.
From (2.7), one sees that there are three independent functions for all
sfermion masses in a generation. Thus, there are at least two independent
sfermion masses relations, or sum rules,

$\displaystyle{}Tr\left(Y\tilde{m}_{f}\right)=0,~{}~{}~{}~{}Tr\left((B-L)\tilde{m}_{f}\right)=0$
(2.10)

These sum rules are one of the most distinctive features in such a gauge
mediation setting, in comparsion with other mediation mechanisms.

##  3 Sum rules in ESM

In this section, we will discuss supersymmetric models with gauge groups
beyond the SM ones $G_{SM}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}$. We will
start with the simple extension with an extra abelian $U(1)^{\prime}$, then
move on to non-abelian gauges. In particular, we will concentrate on the left-
right symmetric model, though our analysis can be easily generalized to any
theories with more elaborated groups, with similar conclusions.

###  3.1 Abelian case

Since most of phenomenological results in this section are independent of the
details in the hidden sector, we will not address the issue of realizations of
these gauge structure in this section. In literature, there have been
extensive efforts to construct viable models. For example, gauge extended
models in ISS-like theories have been discussed before [11, 12, 13]. Theories
with similar gauge structures in the hidden sector can be found in [14], where
the ISS superpotential is deformed by $W_{def}$.111 Strongly coupled ISS-like
SQCD theories can be described by weakly coupled magnetic dual theories at low
energy scale. The magnetic theories have superpotentials of the same structure
as that of generalized O’Raifeartaigh models. According to the general proof
in generalized O’Raifeartaigh models [15], the $R$-symmetry must be
spontaneously broken when $W_{def}$ comes from a set of singlet fields with
$R$-charges of neither zero or two. On the other hand, the hidden sector
discussed in section 4 will be in another paradigm [8], instead of direct
gauge mediation. Partly, it is because that there are generally unacceptable
light gauginos or LHC unaccessible heavy sfermions in direct gauge mediation,
as discussed in [21].

Here, we assume that there is an extra abelian $U(1)^{\prime}$ in both the
hidden and the visible sectors. The soft terms can be obtained by calculations
similar to the ones in the previous section. The $U(1)^{\prime}$ introduces
extra $C^{\prime}_{a}$’s (thus $A^{\prime}$) and $\tilde{B}^{\prime}_{1/2}$,
which modify the sfermion and gaugino masses

$\displaystyle{}\delta\tilde{m}^{2}_{f_{i}}=\frac{3}{5}g^{\prime
4}q^{2}_{i}A^{\prime},~{}~{}~{}~{}~{}~{}~{}\delta\tilde{M}_{\lambda_{i}}=g^{\prime
2}M\tilde{B}^{\prime}_{1/2}.$ (3.1)

where $q_{i}$ are the $U(1)^{\prime}$ charges of fermions and $g^{\prime}$ is
the gauge coupling. Putting everything together, the soft masses are,

$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{Q}\\\ m^{2}_{U}\\\ m^{2}_{D}\\\
m^{2}_{L}\\\ m^{2}_{E}\\\
\end{array}\right)=\frac{1}{60}\left(\begin{array}[]{ccccc}80&45&1&36q^{2}_{Q}\\\
80&0&16&36q^{2}_{U}\\\ 80&0&4&36q^{2}_{D}\\\ 0&45&9&36q^{2}_{L}\\\
0&0&36&36q^{2}_{E}\\\
\end{array}\right)\left(\begin{array}[]{c}g_{3}^{4}A_{3}\\\ g_{2}^{4}A_{2}\\\
g_{Y}^{4}A_{Y}\\\ g^{{}^{\prime}4}A^{\prime}\\\ \end{array}\right)$ (3.16)

So the sum rules in the previous subsection is not valid in general. Instead,
one has

$\displaystyle{}Tr\left(Y\tilde{m}_{f}\right)$ $\displaystyle=$
$\displaystyle\frac{3}{5}g^{\prime
4}(q^{2}_{Q}-2q^{2}_{U}+q^{2}_{D}-q^{2}_{L}+q^{2}_{E})A^{\prime}$ (3.17)
$\displaystyle Tr\left((B-L)\tilde{m}_{f}\right)$ $\displaystyle=$
$\displaystyle\frac{3}{5}g^{\prime
4}(2q^{2}_{Q}-2q^{2}_{U}-q^{2}_{D}-2q^{2}_{L}+q^{2}_{E})A^{\prime}$ (3.18)

Without the $U(1)^{\prime}$ interaction, one gets back the original sum rules
(2.10). One the other hand, there are five soft masses and four independent
$A$’s in (3.16), from which one can deduce one sum rule for the sfermion
masses,

$\displaystyle{}0$ $\displaystyle=$
$\displaystyle\left(q^{2}_{U}-q^{2}_{D}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{Q}+\left(-q^{2}_{Q}+q^{2}_{D}+q^{2}_{L}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{U}$
(3.19) $\displaystyle+$
$\displaystyle\left(q^{2}_{Q}-q^{2}_{U}-q^{2}_{L}+\frac{2}{3}q^{2}_{E}\right)m^{2}_{D}-\left(q^{2}_{U}-q^{2}_{D}-\frac{1}{3}q^{2}_{E}\right)m^{2}_{L}$
$\displaystyle+$
$\displaystyle\frac{1}{3}\left(q^{2}_{Q}+q^{2}_{U}-2q^{2}_{D}-q^{2}_{L}\right)m^{2}_{D}$

Now we have a few comments on these results:

  * •

The sum rule $Tr(Ym^{2})=0$ holds provided that,

$\displaystyle{}q_{E}=q_{D}=0,~{}~{}~{}q^{2}_{Q}=2q^{3}_{U}+q_{L}^{2}$ (3.20)

while the other sum rule $Tr((B-L)m^{2})=0$ holds provided that,

$\displaystyle{}q_{Q}=q_{U}=0,~{}~{}~{}q^{2}_{D}=q^{2}_{E}-2q_{L}^{2}$ (3.21)

From (3.20) and (3.21), one can see the original sum rules (2.10) cannot be
retained at the same time, except for all $U(1)^{\prime}$ charges being set to
zero.

  * •

If the visible and the hidden sectors are assumed to be anomaly free
separately, neither sum rules in (2.10) can be retained.

  * •

If the coupling $g^{\prime}$ is substantially smaller than the SM couplings in
magnitude, (2.10) hold approximately.

The spontaneous breaking of $U(1)^{\prime}$ can be similar to the usual
$U(1)$’s without supersymmetry. One can introduce standard model singlets $S$
to trigger the breaking and extra exotic singlets to cancel the anomalies [9].
In particular, $S$ can obtain vacuum expectation value by radiative
corrections, provided that Yukawa couplings between $S$ and exotic singlets is
large enough.

###  3.2 Non-abelian case

We now move on to the discussion of extra non-abelian gauge groups. For
concreteness, we will consider the left-right model with
$G_{0}=SU(3)_{c}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ [16], which
breaks into $G_{SM}$ via $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}$ at
some scale $M_{R}$. For simplicity, we will assume that $M_{R}<<M_{SUSY}$,
though our general results do not depend on this assumption. The analysis can
be easily generalized to gauge groups of higher ranks.

The $U(1)_{B-L}$ charges can be easily read from their $U(1)_{Y}$ charges.
Explicitly, coupling $g_{Y}$ and charges $q_{Y}$ are determined by
$g_{R},g_{B-L}$ via

$\displaystyle{}g_{Y}=\frac{g_{R}g_{B-L}}{g_{R}^{2}+g_{B-L}^{2}},~{}~{}~{}~{}~{}~{}q_{Y,i}=T^{3}_{R,i}+\tilde{q}_{i}.$
(3.22)

It is straightforward to get the masses for soft sfermions,

$\displaystyle{}\left(\begin{array}[]{c}m^{2}_{Q}\\\ m^{2}_{P}\\\ m^{2}_{L}\\\
m^{2}_{E}\\\
\end{array}\right)=\frac{1}{60}\left(\begin{array}[]{ccccc}80&45&0&1\\\
80&0&45&1\\\ 0&45&0&9\\\ 0&0&0&36\\\
\end{array}\right)\left(\begin{array}[]{c}g_{3}^{4}A_{3}\\\ g_{L}^{4}A_{L}\\\
g_{R}^{4}A_{R}\\\ g_{B-L}^{4}A_{B-L}\\\ \end{array}\right)$ (3.35)

where $P=(U,D)$ carries quantum numbers of
$(\mathbf{\bar{3}},\mathbf{1},\mathbf{2},\frac{1}{6})$.

Since fermions in the visible sector fits into spinor representations of
$SO(10)\supset G_{0}$, it is anomaly free. So the hidden sector must be
anomaly free also. Generally, there can be chiral matters $S_{i}$ with quantum
numbers $(\mathbf{1},\mathbf{1},\mathbf{2},q_{S_{i}})$ $(i\geq 1)$ and $M_{j}$
with quantum numbers $(\mathbf{1},\mathbf{2},\mathbf{1},q_{M_{j}})$ $(j\geq
0)$ in the hidden sector. The quantum numbers $q$ are constrained by the
anomaly free conditions. Specifically,

$\displaystyle{}SU(2)_{R}-SU(2)_{R}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{(doublet,S)}\tilde{q}_{i}=0$ $\displaystyle
SU(2)_{L}-SU(2)_{L}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{(doublet,M)}\tilde{q}_{i}=0$ $\displaystyle
U(1)_{B-L}-U(1)_{B-L}-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{i=(Q,S,M)}\tilde{q}^{3}_{i}=0$ $\displaystyle Graviton-
Graviton-U(1)_{B-L}$ $\displaystyle:$
$\displaystyle\sum_{i=(Q,S,M)}\tilde{q}_{i}=0$ (3.36)

Other anomaly free conditions are automatically satisfied by the charge
assignments in (3.22). We note that

  * •

The sum rules (2.10) are both broken. Actually, they are modified to be

$\displaystyle{}Tr(Ym^{2}_{\tilde{f}})=\frac{3}{4}m^{2}_{E},~{}~{}~{}~{}~{}Tr((B-L)m^{2}_{\tilde{f}})=\frac{1}{2}m^{2}_{E}$
(3.37)

These two equations are independent of specific contents of the hidden sector.
Thus, they can serve as important probes of left-right supersymmetric models.

  * •

If $SU(2)_{R}\times U(1)_{B-L}\rightarrow U(1)_{Y}$, with masses of gauge
bosons $(A_{+},~{}A_{-},~{}A_{0})$ near $M_{SUSY}$, the $A_{r}$’s in (2.7)
need to be replaced by those in (2). There are then six free parameters
$(A_{3},A_{2},A_{Y},A_{+},A_{-},A_{0})$ and five sfermion masses. This implies
(3.37) is modified again in this case.

The constraints (3.2) can be satisfied by proper assignments of charges
$q_{S_{i}}$ and $q_{M_{j}}$. At least one $S$ is needed to break $G_{0}$ into
$G_{SM}$.

Other extensions of group structure beyond SM induce corresponding sum rules,
some of which can be independent of details of the hidden sector, which serve
as generic probes of such theories.

##  4 Supersymmetric hidden valleys

Usually the hidden sector is assumed to be very heavy. Actually, a light
hidden sector cannot be ruled out if its communication with the visible sector
is sufficiently suppressed. Scenarios of light hidden sectors with small
coupling with the visible sector has been recently advocated and dubbed as
hidden valleys [17].

In $U(1)$ theories, one always has $\beta_{g^{\prime}}>0$ and the
corresponding couplings decrease with the decrease of energy. It is thus
possible that the effects from $U(1)^{\prime}$s are tiny at the electro-weak
scale due to renormalization group flows. In addition, the couplings between
the visible sector and the $U(1)^{\prime}$s are suppressed further by the
massive gauge boson $m_{Z^{\prime}}$’s. So the existence of extra
$U(1)^{\prime}$s cannot be ruled out by present experiments. Naturally, extra
$U(1)^{\prime}$s has been proposed to communicate the hidden valley sector to
the visible sector [17].

Here we will construct a model in which the extra $U(1)^{\prime}$ communicates
between the supersymmetry breaking sector and a hidden valley sector, which
generates supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden sector to the visible sector. We will
see that if the $U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at
$M_{SUSY}$, the soft terms in the hidden valley are two or four orders of
magnitude smaller than those in the visible sector. That is to say, an
$\mathcal{O}(1-10)$ TeV-scale visible sector is accompanied by an
$\mathcal{O}(10-100)$ GeV-scale hidden valley.

To be concrete, we will consider a class of models with the following
symmetries and particle contents,

(4.6)

Specifically,

  * •

The theory is composed of three parts. The hidden sector is composed of a
spurion $X$ referred to be SUSY-breaking sector and an $SU(n_{v})$ gauge
theory with $v$-quarks in its bi-fundamental representations. The $v$-sector
is referred to as hidden valley. The messenger sector contains the
$\Phi_{i}$’s, which are neutral under $SU(n_{v})$ but charged under
$G_{SM}\times U(1)^{\prime}$. The visible sector contains gauged
$U(1)^{\prime}$ extension of group structure beyond $G_{SM}$ below
$\sqrt{F_{X}}$.

  * •

The gauge symmetry is $SU(n_{v})\times G_{SM}\times U(1)^{\prime}$. Shown in
Table 1 are also the quantum numbers and representations of chiral matters.

If the SUSY-breaking sector is realized in the scheme of direct gauge
mediation, there will be unacceptable light gauginos or LHC unaccessible heavy
sfermions in general [21]. Thus, we turn to the old paradigm [8] to realize
supersymmetry and R-symmetry breaking. In such a scheme, it is not necessary
to construct the hidden sector explicitly. One simply assumes that a singlet
spurion $X$ is responsible for supersymmetry breaking and $X$ almost
determines all the phenomenological features. For explicit SUSY-breaking
sectors that induce such a spurion X, see [8] and reference therein.

The Lagrangian for the model in the table reads,

$\displaystyle{}\mathcal{L}=\int d^{2}\theta W+\int d^{4}\theta K+\int
d^{2}\theta\left(\mathcal{W}_{MSSM}^{2}+\mathcal{W}^{{}^{\prime}2}+\mathcal{W}_{h}^{2}\right)$
(4.7)

where

$\displaystyle{}W$ $\displaystyle=$
$\displaystyle\lambda_{ij}X\bar{\Phi}_{i}\Phi_{j},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}X=M+F_{X}\theta^{2}$
(4.8) $\displaystyle K$ $\displaystyle=$
$\displaystyle\left(\Phi^{\dagger}_{i}e^{-2V_{MSSM}-2V^{\prime}}\Phi_{i}+Q_{m}^{\dagger}e^{-2V_{MSSM}-2V^{\prime}}Q_{m}+q_{j}^{\dagger}\
e^{-2V_{h}-2V^{\prime}}q_{j}+T_{\pm}^{\dagger}\
e^{-2V^{\prime}}T_{\pm}\right)$

where $Q_{m}$ denote the chiral matter superfields in SSM sector, $V_{h}$ and
$\mathcal{W}_{h}$ denote the vector and spinor superfield of hidden valley
gauge theory respectively. $T_{\pm}$ are responsible for triggering the
spontaneously breaking of $U(1)^{\prime}$.

Figure 1: Gauge mediation with (right) and without (left) extra
$U(1)^{\prime}$. The black line indicates gauge mediation due to $G_{SM}$
while the dashed ones due to $U(1)^{\prime}$.

One can see any two of the three sectors can communicate through the gauged
$U(1)^{\prime}$ group theory, as shown in Figure.1. Both the messenger sector
and the visible sector contain $G_{SM}$ gauge interactions, which dominate the
communication between them. But the hidden valley communicates with others
only via the $U(1)^{\prime}$.

It is straightforward to work out the soft masses in both the visible sector
and the hidden valley ($\lambda_{ij}=\delta_{ij}=1$). Generically,

$\displaystyle{}\tilde{m}^{2}_{Q_{i}}$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{3}\frac{g_{a}^{4}N_{mess}}{(16\pi^{2})^{2}}C_{2}(f_{i},a)\left(\frac{F_{X}}{M}\right)^{2}+\mathcal{O}(g^{\prime
4})$ $\displaystyle M_{\lambda_{a}}$ $\displaystyle=$
$\displaystyle\frac{g_{a}^{2}N_{mess}}{(16\pi^{2})}\left(\frac{F_{X}}{M}\right)+\mathcal{O}(g^{\prime
2})$ (4.10)

in the visible sector and

$\displaystyle{}\tilde{m}^{2}_{q_{i}}=\frac{3g^{\prime 4}N_{mess}q^{\prime
2}_{i}}{5(16\pi^{2})^{2}}\left(\frac{F_{X}}{M}\right)^{2},~{}~{}~{}~{}~{}\tilde{m}^{2}_{T_{\pm}}=\frac{3g^{\prime
4}N_{mess}q^{\prime
2}_{\pm}}{5(16\pi^{2})^{2}}\left(\frac{F_{X}}{M}\right)^{2}$ (4.11)

in the hidden valley and $T_{\pm}$ chiral superfields respectively, $N_{mess}$
is the number of messengers. Finally, the gaugino mass of $U(1)^{\prime}$
vector superfield reads,

$\displaystyle{}M_{\lambda_{\bar{V}}}=\frac{g^{\prime
2}N_{mess}}{(16\pi^{2})}\left(\frac{F_{X}}{M}\right)$ (4.12)

Below the scale where $U(1)^{\prime}$ is spontaneously broken at scale
$\Lambda$ with mass $M_{V^{\prime}}=4g^{\prime 2}\Lambda^{2}$, the
$V^{\prime}$ vector superfield can be integrated out, leaving following
couplings in the effective theory at leading order,

$\displaystyle{}-\frac{1}{4\Lambda^{2}}\int
d^{4}\theta\left((1+\tilde{m}^{2}_{T}\theta^{4})(\sum_{m}q^{\prime}_{m}Q^{\dagger}_{m}e^{V_{MSSM}}Q_{m}+\sum_{j}q^{\prime}_{j}q^{\dagger}_{j}e^{V_{h}}q_{j})^{2}\right)$
(4.13)

where $q^{\prime}_{m}$ are the $U(1)^{\prime}$ charges of chiral matters in
MSSM. (4.13) also induces mixing couplings between operators in MSSM and
hidden valley, which are suppressed by the $U(1)^{\prime}$ gauge bosons mass.

The tree level Higgs masses $m_{H_{u}}$ and $m_{H_{d}}$ in the visible sector
are similar to (4),

$\displaystyle\tilde{m}^{2}_{H_{u,d}}=\sum_{a=1}^{2}\frac{g_{a}^{4}N_{mess}}{(16\pi^{2})^{2}}C_{2}(H_{u,d},a)\left(\frac{F_{X}}{M}\right)^{2}\sim\tilde{m}^{2}_{Q}$
(4.14)

As is well-known, the tree-level lightest Higgs mass $m_{h}$ is always lighter
than $m_{Z}$, no matter explicit values of $m_{H_{u}}$ and $m_{H_{d}}$. This
contradicts experimental observations but $m_{h}$ can be lifted over $m_{Z}$
by taking loop corrections into account. On the other hand, $m_{h}$ may be
further lifted by including higher dimensional couplings in (4.13).
Explicitly, the correction to potential in visible sector reads,

$\displaystyle{}\delta
V=q^{\prime}_{H_{u}}\tilde{v}H^{\dagger}_{u}H_{u}+q^{\prime}_{H_{d}}\tilde{v}H^{\dagger}_{d}H_{d}+\epsilon_{1}(H_{u}^{{\dagger}}H_{u})^{2}+\epsilon_{2}(H_{d}^{{\dagger}}H_{d})^{2}+\epsilon_{3}(H^{{\dagger}}_{d}H_{u})^{2}$
(4.15)

where

$\displaystyle\tilde{v}$ $\displaystyle=$
$\displaystyle\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}\sum_{j}q^{\prime}_{\tilde{H}_{j}}|<\tilde{H}_{j}>|^{2},$
$\displaystyle\epsilon_{1}$ $\displaystyle=$ $\displaystyle q^{\prime
2}_{H_{u}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}},$ $\displaystyle\epsilon_{2}$
$\displaystyle=$ $\displaystyle q^{\prime
2}_{H_{d}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}$ $\displaystyle\epsilon_{3}$
$\displaystyle=$ $\displaystyle(q^{\prime
2}_{H_{u}}+q^{\prime}_{H_{u}}q^{\prime}_{H_{d}}+q^{\prime
2}_{H_{d}})\frac{\mu^{2}}{4\Lambda^{2}}+q^{\prime}_{H_{u}}q^{\prime}_{H_{d}}\frac{\tilde{m}^{2}_{T}}{4\Lambda^{2}}$
(4.16)

Here $<\tilde{H}_{j}>$ are VeVs of scalars in hidden valley. Linear
approximations $F_{H_{u}}\simeq-\mu H_{d}^{{\dagger}}$ and
$F_{H_{d}}\simeq-\mu H_{u}^{{\dagger}}$ have been used in above calculations.
For typical parameters $\Lambda\sim 10^{3}$GeV, $\tilde{m}_{T}\sim 10-100$GeV,
$<\tilde{H}_{j}>\sim 100$GeV and $\mu\sim 200$GeV, the corrections to lightest
higgs bosons are dominated by $\epsilon_{3}$,

$\displaystyle\delta_{\epsilon_{3}}m^{2}_{h}\simeq\epsilon_{3}v^{2}\sim\mathcal{O}(10~{}GeV)^{2}$
(4.17)

This correction is independent of $\tan\beta$. Thus it contributes
significantly at the large $\tan\beta$ limit, as other contributions are
usually proportional to $1/\tan\beta$ [22].

Figure 2: Spectra and decay chains of the supersymmetric hidden valley with
$F_{X}/M\sim 10^{5}$ GeV. The dashed lines refer to particles that decay into
jets/leptons. $\lambda_{\bar{V}}$ and VSSP represent the next-lightest
supersymmetric particles in the visible and hidden sectors respectively.

From (4) and (4.11), One sees that the soft masses in supersymmetric hidden
valley are two or four order of magnitude smaller than those in visible
sector. Typically, they are in the order of $10-10^{2}$ GeV, while soft masses
of the visible sector and $m_{Z^{\prime}}$ are usually around TeV. For such
mass parameters, decay chains can be expected between the visible and hidden
valley sectors. In most decaying processes, jets/lepton pairs will be
generated, as shown in Figure.2. Phenomenologically, the generations of
$v$-quarks, the decay widths and their signals at colliders follow the general
pattern discussed in [18].

Finally, we outline the phenomenological features in the visible sector:

  * •

As worked out in section 3.1, the sum rules in the visible sector are expected
to hold approximately. Notice that (most of) results in section 3 are
independent of the SUSY-breaking sector.

  * •

The gaugino of $U(1)^{\prime}$ vector superfield is the next-lightest
supersymmetric particles (NSSP) in the visible sector if $\mid q_{+}\mid$ is
larger than $1/\sqrt{0.6N_{mess}}$. Otherwise, $T$-scalars are NSSP. When
sfermion and SM gaugino masses taken to be LHC accessible $\mathcal{O}(1)$
TeV, NSSP is around $10-100$ GeV.

  * •

At the large $\tan\beta$ limit, higher dimensional couplings arising from
(4.13) in MSSM are the main sources to correct the Higgs spectra, which can
substantially uplift the lightest Higgs masses across the lower bound at LEPII
in the typical parameter space.

It would be interesting to construct a single hidden sector, which
spontaneously breaks supersymmetry and $R$-symmetry, but has desired unbroken
gauge symmetry and a hidden valley sector. One possible realization could be
an ISS-like theory with partially unbroken gauge symmetry [12].

##  5 Conclusions

In this paper, we have analyzed supersymmetric models with extended group
structure beyond the standard model in the framework of general gauge
mediation. We have concentrated on the sum rules for sfermion masses, and they
are shown to depend genuinely on the group structure, which can serve as
important probes of the specific model. In particular, they are rather
different from those in models with SM gauge group (2.10). For definiteness,
the left-right model and models with extra $U(1)$ has been worked out in
details. When the couplings of extra gauge groups are smaller than those in
the SM, the sum rules in (2.10) hold approximately.

We have constructed a model in which the extra $U(1)^{\prime}$ communicates
between the supersymmetry breaking sector and a hidden valley sector, which
generates supersymmetry breakings in the latter. Simultaneously, the same
$U(1)^{\prime}$ communicates the hidden sector to the visible sector. If the
$U(1)^{\prime}$ coupling is of the order $10^{-1}-10^{-2}$ at $M_{SUSY}$, soft
terms in the hidden valley are a few orders smaller than those in the visible
sector, which imply an $\mathcal{O}(1-10)$ TeV-scale visible sector is
accompanied by an $\mathcal{O}(1-100)$ GeV-scale hidden valley. Also, extra
higher dimensional couplings help to uplift the mass of the lightest Higgs
particle. The model conforms to the stringent constraints from LEP and other
precision experiments, as the communication between the visible and hidden
valley sectors is suppressed by the massive gauge bosons $m_{Z^{\prime}}$, in
addition to the smallness of the gauge coupling.

## Acknowledgement

This work is supported in part by the National Science Foundation of China
(10425525) and (10875103).

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