# Higgsphobic and fermiophobic Z’ as a single dark matter candidate

Nan Chen1<EMAIL_ADDRESS>Ying Zhang2<EMAIL_ADDRESS>Qing Wang1,3,4<EMAIL_ADDRESS>1Department of Physics, Tsinghua
University, Beijing 100084, P.R. China
2School of Science, Xi’an Jiaotong University, Xi’an 710049, P.R. China
3Center for High Energy Physics, Tsinghua University, Beijing 100084, P.R.
China
4Collaborative Innovation Center of Quantum Matter, Beijing 100084, P.R. China
Giacomo Cacciapaglia5<EMAIL_ADDRESS>Aldo Deandrea5,6
<EMAIL_ADDRESS>Luca Panizzi7<EMAIL_ADDRESS>5Universit$\acute{e}$ de Lyon, F-69622 Lyon, France; Universit$\acute{e}$ Lyon
1, Villeurbanne
CNRS/IN2P3, UMR5822, Institut de Physique Nucl$\acute{e}$aire de Lyon F-69622
Villeurbanne Cedex, France
6 Institut Universitaire de France, 103 boulevard Saint-Michel, 75005 Paris,
France
7School of Physics and Astronomy, University of Southampton, Highfield,
Southampton SO17 1BJ, UK

###### Abstract

A spin-1 Z’ particle as a single dark matter candidate is investigated by
assuming that it does not directly couple to the Higgs boson and standard
model fermions and does not mix with the photon and Z boson. The remaining
dominant vertices are quartic $Z^{\prime}Z^{\prime}ZZ$ and
$Z^{\prime}Z^{\prime}W^{+}W^{-}$, which can induce effective
$Z^{\prime}Z^{\prime}q\bar{q}$ couplings through standard-model gauge-boson
loops. We discuss constraints from the cosmological thermal relic density, and
direct and indirect-detection experiments, and find that a dark Z’ can only
exist above the W boson mass threshold, and the effective quartic coupling of
$Z^{\prime}Z^{\prime}VV$ is bounded in the region of $10^{-3}\sim 10^{-2}$.

###### pacs:

95.35.+d, 14.70.Pw, 12.60.Cn

##  I Introduction

After the discovery of the 125 GeV Higgs boson, the standard model (SM) of
particle physics has become a complete theory; within the SM, the remaining
task is the precision measurements of various Higgs properties, in particular
its couplings, and to further narrow down the possible parameter space of new
physics. Although the hierarchy and meta-stable vacuum problems remain for the
SM Higgs theoretically, for new physics beyond the SM the absence of new
physics signals at the LHC to date implies that extensions of the SM still
only need to rely on the traditional particle physics facts of non-zero
neutrino masses and baryon asymmetry. Given these circumstances, the presence
of dark matter (DM) in our Universe becomes an even more important leading
empirical evidence for the existence of new physics, because no SM particle
can account for DM. Cosmology and astrophysics tell us that almost 85$\%$ of
matter in our universe is dark, i.e., neutral, non-luminous and non-baryonic.
The fact that the abundance of DM is comparable to that of ordinary visible
matter seems to imply that DM may have the same or similar origins and
properties as ordinary matter. If we accept the conclusion of quantum field
theory (QFT) that all matter should be made of particles, then an unambiguous,
non-gravitational signal of DM must appear in particle physics experiments.
This has driven the particle physics community to try harder to unravel DM’s
still enigmatic properties.

Because details of the particle properties of DM are lacking, the best
investigative strategy for theorists is to try to cover as much ground as
possible. Considering that QFT classifies particles according to their spin
(even or odd half-integers), elementary particles discovered so far all have
low spins. Most DM candidates discussed so far in the literature have been
assumed to be spin-0 scalars ScalarDM1 ; *ScalarDM2; *ScalarDM3; *ScalarDM4;
*ScalarDM5; *ScalarDM6; *ScalarDM7 or spin-1/2 spinors SpinorDM3 ; *SpinorDM4.
Whereas a scalar DM has relatively simple structure and provides possible
intimate interplay with the 125 GeV Higgs, a spinor DM extends the traditional
observation that matter is composed of spin-1/2 particles. The heavy sterile
neutrinos SpinorDM1  and the lightest neutralino SpinorDM2  in supersymmetric
models are DM candidates belonging to this type. Apart from scalar and spinor
DM, the next level of higher-spin candidate particles comprise spin-1 vectors.
If we limit ourselves to the simplest vector particle scenario in particle
physics, a single extra neutral vector particle, usually denoted by Z’, is
sufficient. We shall discuss this possibility in the present paper. A higher
spin case, spin-3/2 DM, has been discussed in Ref.spin3over2 .

A vector particle Z’ can be viewed as a gauge boson that mediates an extra
$U(1)$ gauge force beyond the conventional SM strong $SU(3)_{c}$ force and
electroweak $SU(2)_{L}\otimes U(1)_{Y}$ forces. For as yet unknown reasons,
this additional $U(1)$ gauge symmetry is spontaneously broken, thus yielding a
massive Z’. SM plus Z’ is a minimal and well motivated generalisation of the
SM; many new-physics models have such a $Z^{\prime}$ boson (for details see
review Refs. LangackerRMP2008 ; *LangackerPRL2008; *Han; *Zwirner) as a
necessary constituent and remnant for new-physics interactions. Before July 4,
2012, the Higgs was the superstar of particle physics searches and a Z’ only
played a supporting role. With the discovery of the 125 GeV Higgs, a Z’ now
becomes one of the hot new-physics candidate particles and the LHC is actively
searching for it in various channels, with the model-dependent lower mass
bound already reaching the TeV-energy region depending on the final state it
is assumed to decay into. Now, if we further take the Z’ as a DM candidate
thus changing the Z’ from visible to invisible, the interactions between the
invisible Z’ and SM particles will be strongly reduced, and the corresponding
search strategies (such as direct detection, indirect detection, and collider
experiment) will change with respect to those for visible Z’. Various
constraints must therefore be re-examined.

In the literature, the invisible Z’ has been intensively discussed as a
messenger between the visible sector (which contains the SM particles) and a
hidden sector (to which DM belongs) DMZ1 ; *DMZ2; *DMZ3; *DMZ4; *DMZ5; *DMZ6;
*DMZ7; *DMZ8; *DMZ9; *DMZ10; *DMZ11; *DMZ12: in such a scenario, the SM
particles can be either charged under the additional gauge symmetry or not. In
the event that SM particles are neutral with respect to the extra $U(1)$
symmetry, the interaction occurs via effective operators connecting directly
Z’ to the SM sector. The simplest case is the kinetic mixing terms between the
SM hypercharge field strength and the new Abelian field strength mixing . The
underlying reason in adopting Z’ as a portal to the hidden sector stems from
the traditional mediating role of gauge bosons. In this type of DM models,
there are too many unknowns concerning the hidden sector, a situation that is
not helpful in DM searches. In this paper, we consider an alternative simple
approach by ignoring the conventional messenger role of Z’, and instead treat
it as pure matter. This approach is similar to the minimal darkon model
ScalarDM1 ; *ScalarDM2; *ScalarDM3; *ScalarDM4; *ScalarDM5; *ScalarDM6 where
SM is minimally expanded with the addition of a dark scalar (SM+D), except now
we replace the scalar darkon D with a single vector DM candidate Z’. The
change from the traditional Z’ portal model to our present single dark Z’
approach is similar to that from the Higgs portal model (where a scalar is
taken as a messenger between the visible and hidden sectors) HiggsPortal  to
the darkon model. After the reduction, because of the unique choice of DM
candidate, we can ignore the uncertainties arising from the arbitrary hidden
sector in the traditional Z’ or Higgs portal models. The difference in the
present approach with respect to scalar DM is that our dark Z’ is a vector
particle, which behaves not like a scalar or Higgs boson, but very much like a
Z boson of SM and will have relatively complex interaction structure owing to
its polarisation. A spin-1 dark matter candidate appears in models with one
extra dimensions KK5  and has been widely studied in this context *KKDM1;
*KKDM2. Note that this is not a generic prediction of extra dimensions, as in
higher then 5 dimensions the candidate is a scalar *KK6a; *KK6b; *KK6c, and a
scalar is again found in models of pseudo-Goldstone Higgs in warped space
frigerio  and technicolor sannino .

This paper is organised as follows. In Section II, in terms of the model-
independent extended electroweak chiral Lagrangian and the six assumptions
needed to keep Z’ dark, we determine the necessary operators that couple our
dark Z’ to SM particles. In Section III, we calculate the relic density
produced from our single dark Z’, derive a constraint on the effective
coupling of the dark Z’ pair to $W$ or $Z$ pairs. Section IV looks at the
direct-detection constraint, where we compute the SM gauge-boson-loop-induced
$Z^{\prime}Z^{\prime}\bar{q}q$ vertex and discuss direct detection. Section V
examines indirect-detection constraints and includes discussions of the
Pamela, AMS02, and FermiLAT experiments. In Section VI, we discuss the
combined results and some other possible DM related issues. Section VII
presents a summary. Some necessary results for Section II are to be found in
Appendix A.

##  II $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)$ theory and Z’ as a DM candidate

To make our investigation general, we start from a model-independent effective
extended electroweak chiral Lagrangian (EEWCL) proposed in Ref.CL ,

$\displaystyle\mathcal{L}_{\mathrm{EEWCL}}=\mathcal{L}_{2}+\mathcal{L}_{4}+\cdots,$
(1)

where $\mathcal{L}_{2n}$ for $n=1,2,3,\ldots$ is the $p^{2n}$-order of EEWCL
with Z’ and all SM gauge fields, plus four necessary Goldstone bosons
described by a two-by-two unitary matrix field $\hat{U}$. The symmetry of the
Lagrangian is $SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)$, which will be
spontaneously broken to $U(1)_{\mathrm{em}}$. The SM Higgs field and the
fermion fields are not included in the above Lagrangian: in fact, couplings to
the Higgs are not required by the symmetry of the model and, if present, would
reproduce a model of Higgs portal DM. This implies that we ignore the possible
direct (or tree-order) couplings between Z’ and Higgs (a dark Z’ with direct
coupling to the Higgs has been discussed in Ref.ZprimeHiggs ) or between Z’
and SM fermions (a dark Z’ coupling directly to SM fermions has been discussed
in Ref.Yu:2011by ). These are the first two assumptions we adopt for our dark
Z’. These higgsphobic and fermiophobic dark Z’ assumptions simplify our theory
significantly, and we take it as the first step in our investigation. Although
at tree level we can ignore the explicit Higgs and fermion couplings, loops
can still induce effective couplings. We shall carefully discuss these loops
in Section IV.

The $p^{2}$-order Lagrangian $\mathcal{L}_{2}$ is CL

$\displaystyle\mathcal{L}_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}f^{2}\mathrm{tr}[\hat{V}_{\mu}\hat{V}^{\mu}]+\frac{1}{4}\beta_{1}f^{2}\mathrm{tr}[T\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]+\frac{1}{4}\beta_{2}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]+\frac{1}{4}\beta_{3}f^{2}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}],$
(2)

where $T\equiv\hat{U}\tau_{3}\hat{U}^{\dagger}$,
$\hat{V}_{\mu}\equiv(D_{\mu}\hat{U})\hat{U}^{\dagger}$, and $\tau_{3}$ is the
Pauli matrix. The covariant derivative is

$\displaystyle
D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-i\hat{U}\frac{\tau_{3}}{2}g^{\prime}B_{\mu}-i\hat{U}(\tilde{g}^{\prime}B_{\mu}+g^{\prime\prime}X_{\mu})I,$
(3)

where $W_{\mu}\equiv\frac{\tau_{i}}{2}W^{i}_{\mu}$, $B_{\mu}$, and $X_{\mu}$
are the $SU(2)_{L}$, $U(1)_{Y}$, and $U(1)$ gauge fields, respectively, and
$g,g^{\prime},g^{\prime\prime}$ are the corresponding coupling constants;
$\tilde{g}^{\prime}$ is a special Stueckelberg coupling. $W_{\mu}$, $B_{\mu}$,
and $X_{\mu}$ are gauge eigenstates, and the Z’ discussed in this paper is the
physical state after diagonalization. In $\mathcal{L}_{2}$, $\beta_{1}$ and
$\beta_{2}$ are mass mixing parameters.

The $p^{4}$-order Lagrangian $\mathcal{L}_{4}$ is composed of three termsCL

$\displaystyle\mathcal{L}_{4}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{K}+\mathcal{L}_{B}+\mathcal{L}_{A},$ (4)

for which the kinetic term $\mathcal{L}_{K}$ is

$\displaystyle\mathcal{L}_{K}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{2}\mathrm{tr}[W_{\mu\nu}W^{\mu\nu}]-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}.$
(5)

The normal term $\mathcal{L}_{B}$ is

$\displaystyle\mathcal{L}_{B}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\alpha_{1}gg^{\prime}B_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+\frac{i}{2}\alpha_{2}g^{\prime}B_{\mu\nu}\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+i\alpha_{3}g\mathrm{tr}[W^{\mu\nu}[\hat{V}^{\mu},\hat{V}^{\nu}]]$
(6)
$\displaystyle+\alpha_{4}\mathrm{tr}[\hat{V}_{\mu}\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}\hat{V}^{\nu}]+\alpha_{5}\mathrm{tr}[\hat{V}_{\mu}\hat{V}^{\mu}]\mathrm{tr}[\hat{V}^{\nu}\hat{V}_{\nu}]+\alpha_{6}\mathrm{tr}[\hat{V}_{\mu}\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}^{\nu}]$
$\displaystyle+\alpha_{7}\mathrm{tr}[\hat{V}_{\mu}\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\nu}]+\frac{1}{4}\alpha_{8}g^{2}\mathrm{tr}[TW_{\mu\nu}]\mathrm{tr}[TW^{\mu\nu}]+\frac{i}{2}\alpha_{9}g\mathrm{tr}[TW^{\mu\nu}]\mathrm{tr}[T[\hat{V}_{\mu},\hat{V}_{\nu}]]$
$\displaystyle+\frac{1}{2}\alpha_{10}\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}^{\nu}]\mathrm{tr}[T\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]+\alpha_{11}g\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[T\hat{V}_{\mu}]\mathrm{tr}[\hat{V}_{\nu}W_{\rho\lambda}]$
$\displaystyle+\alpha_{12}g\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[\hat{V}^{\nu}W_{\mu\nu}]+\alpha_{13}gg^{\prime}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}\mathrm{tr}[TW_{\rho\lambda}]+\alpha_{14}g^{2}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[TW_{\mu\nu}]\mathrm{tr}[TW_{\rho\lambda}]$
$\displaystyle+\alpha_{15}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\nu}]+\alpha_{16}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[\hat{V}_{\nu}\hat{V}^{\nu}]+\alpha_{17}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}\hat{V}^{\nu}]$
$\displaystyle+\alpha_{18}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}^{\nu}]+\alpha_{19}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}\hat{V}^{\nu}]+\alpha_{20}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\nu}]$
$\displaystyle+\alpha_{21}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[\hat{V}_{\nu}\hat{V}^{\nu}]+\alpha_{22}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[\hat{V}_{\nu}]\mathrm{tr}[T\hat{V}^{\nu}]+\alpha_{23}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}_{\nu}]\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[\hat{V}^{\nu}]$
$\displaystyle+gg^{\prime\prime}\alpha_{24}X_{\mu\nu}\mathrm{tr}[TW^{\mu\nu}]+g^{\prime}g^{\prime\prime}\alpha_{25}B_{\mu\nu}X^{\mu\nu}+\alpha_{26}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T[\hat{V}_{\rho},\hat{V}_{\lambda}]]$
$\displaystyle+ig^{\prime}\alpha_{27}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]B_{\rho\lambda}+ig\alpha_{28}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[TW_{\rho\lambda}]$
$\displaystyle+g\alpha_{29}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[\hat{V}_{\nu}W_{\rho\lambda}]+ig^{\prime\prime}\alpha_{30}\epsilon^{\mu\nu\rho\lambda}X_{\mu\nu}\mathrm{tr}[T[\hat{V}_{\rho},\hat{V}_{\lambda}]]+ig^{\prime\prime}\alpha_{31}X_{\mu\nu}\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]$
$\displaystyle+g^{\prime\prime}\alpha_{32}\epsilon^{\mu\nu\rho\lambda}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]X_{\rho\lambda}+\alpha_{33}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[T[\hat{V}^{\mu},\hat{V}^{\nu}]]+g^{\prime}g^{\prime\prime}\alpha_{34}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}X_{\rho\lambda}$
$\displaystyle+gg^{\prime\prime}\alpha_{35}\epsilon^{\mu\nu\rho\lambda}X_{\mu\nu}\mathrm{tr}[TW_{\rho\lambda}]+ig^{\prime}\alpha_{36}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]B^{\mu\nu}+ig\alpha_{37}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]\mathrm{tr}[TW^{\mu\nu}]$
$\displaystyle+g\alpha_{38}\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[\hat{V}^{\nu}W_{\mu\nu}]+g^{\prime\prime}\alpha_{39}\mathrm{tr}[\hat{V}_{\mu}]\mathrm{tr}[T\hat{V}_{\nu}]X^{\mu\nu}+ig\alpha_{40}\mathrm{tr}[\hat{V}^{\mu}]\mathrm{tr}[T\hat{V}^{\nu}W_{\mu\nu}]$

among which $\alpha_{1},\alpha_{8},\alpha_{24},\alpha_{25}$ are kinetic mixing
parameters; $\alpha_{12}\sim\alpha_{14}$, $\alpha_{30}$,
$\alpha_{33}\sim\alpha_{40}$ are associated with CP-violation terms. The
anomalous term $\mathcal{L}_{A}$ is

$\displaystyle\mathcal{L}_{A}=\alpha_{42}g^{2}\epsilon^{\mu\nu\rho\lambda}tr[W_{\mu\nu}W_{\rho\lambda}]+\alpha_{43}g^{\prime
2}\epsilon^{\mu\nu\rho\lambda}B_{\mu\nu}B_{\rho\lambda}+{g^{\prime\prime}}^{2}\alpha_{44}\epsilon^{\mu\nu\rho\lambda}X_{\mu\nu}X_{\rho\lambda}~{}~{}~{}~{}~{}$
(7)

With the exception of the kinetic term $\mathcal{L}_{K}$ in $\mathcal{L}_{4}$,
the $\alpha_{i}$ ($i=1,\cdots,14$) correspond to terms appearing in the
conventional electroweak chiral Lagrangian (EWCL) CLold  without Z’,
$\alpha_{j}$ ($j=15,\cdots,44$) correspond to terms in the EEWCL involving Z’.
In the rest of the paper, we shall ignore the CP-violation terms and the
anomaly terms. This constitutes our third and fourth assumptions. Our fifth
assumption is to forbid possible mixing between the Z’ and the electroweak
bosons $\gamma,Z$ in order to keep the Z’ dark. This implies that the gauge
eigenstate $X_{\mu}$ can be identified with the physical state
$Z^{\prime}_{\mu}$. No mass mixing requires $\beta_{2}=0$, and no kinetic
mixing leads to $\alpha_{24}=\alpha_{25}=0$. Furthermore, we need to set to
zero the Stueckelberg coupling $\tilde{g}^{\prime}=0$. At this point it is
important to stress that there are mixing parameters which do not involve the
Z’: $\beta_{1}$ is a mass mixing term among SM gauge bosons, thus it will
induce a correction to the $\rho$ parameter ($T$ parameter), while
$\alpha_{1}$ and $\alpha_{8}$ induce kinetic mixings, thus generating a
contribution to the $S$ parameter. The remaining $\beta_{3}$ generates a
contribution to the mass of the Z’, which is given by

$M_{\rm Z^{\prime}}=g^{\prime\prime}f\sqrt{1-2\beta_{3}}.$ (8)

With the above five assumptions, the $p^{4}$-order EEWCL (4) in the unitary
gauge gives the following Lagrangian up to quartic couplings:

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}V_{\mu\nu}V^{\mu\nu}-\frac{1}{2}W^{+}_{\mu\nu}W^{-\mu\nu}+iC_{V-+}V_{\mu\nu}W^{+\mu}W^{-\nu}+iC_{+V-}(W^{+}_{\mu\nu}W^{-\mu}V^{\nu}-W^{-}_{\mu\nu}W^{+\mu}V^{\nu})$
(9)
$\displaystyle+iC_{V_{1}V_{2}V_{3}}V^{\mu\nu}_{1}V_{2\mu}V_{3\nu}+D_{++--}W^{+}_{\mu}W^{+\mu}W^{-}_{\nu}W^{-\nu}+D_{+-+-}W^{+}_{\mu}W^{-\mu}W^{-}_{\nu}W^{+\nu}$
$\displaystyle+D_{+-V_{1}V_{2}}W^{+}_{\mu}W^{-\mu}V_{1\nu}{V_{2}}^{\nu}+D_{+V_{1}-V_{2}}W^{+}_{\mu}V_{1}^{\mu}W^{-}_{\nu}V_{2}^{\nu}+D_{V_{1}V_{2}V_{3}V_{4}}V_{1\mu}V_{2}^{\mu}V_{3\nu}V_{4}^{\nu}.$

Here $V_{i}$ denotes the neutral gauge bosons $Z$, $\gamma$, and $Z^{\prime}$,
and the various $C$ and $D$ coefficients in terms of the $\alpha_{i}$ and
$\beta_{i}$ coefficients are given in Appendix A. Note that, with our fifth
assumption of no mass mixing, $C_{V_{1}V_{2}V_{3}}$ vanishes.

In order to keep the Z’ stable, we need to impose the vanishing of vertices
that are linear in the Z’ field; this constitutes our sixth assumption. For
the triple couplings $C_{Z^{\prime}-+}$ and $C_{+Z^{\prime}-}$ to vanish, we
need to set $\alpha_{31}=0$; there is then no triple coupling involving
$Z^{\prime}$ (note that without the triple coupling $C_{Z^{\prime}VV}$ with
$V=\gamma,Z,W^{\pm}$, the longitudinal $W$ and $Z$ scattering will not involve
$Z^{\prime}$ at tree level and that imposes no unitarity constraint on the Z’
couplings). Left with four Z’-independent triple couplings, one $C_{+\gamma-}$
has fixed coefficients, which only depend on SM couplings, whereas the other
three $C_{\gamma-+},C_{Z-+}$, and $C_{+Z-}$ are free, corresponding to
independent coefficients $\alpha_{2},\alpha_{3}$, and $\alpha_{9}$. For
quartic couplings, $D_{+-ZZ^{\prime}}=0$ leads to $\alpha_{16}=0$,
$D_{+Z-Z^{\prime}}=D_{+Z^{\prime}-Z}=0$ leads to $\alpha_{17}=0$, and
$D_{Z^{\prime}ZZZ}=-g_{Z}^{3}g^{\prime\prime}(2\alpha_{15}+\alpha_{16}+\alpha_{17})=0$
further leads to $\alpha_{15}=0$. (Note that if the Stueckelberg coupling
$\tilde{g}^{\prime}$ does not vanish, it will also generate nonzero
$D_{+-ZZ^{\prime}}$, $D_{+Z-Z^{\prime}}$, $D_{+-AZ^{\prime}}$,
$D_{+A-Z^{\prime}}$.) We are finally left with 16 nonzero quartic couplings.
Among them, $D_{+-\gamma\gamma}$ and $D_{+\gamma-\gamma}$ also have fixed
coefficients and are not free. The other four $D_{+-\gamma Z}$,
$D_{+\gamma-Z}$ and $D_{+Z-\gamma}=D_{+\gamma-Z}$ only rely on $\alpha_{3}$
and then are related to the triple vertex. The remaining 11 nonzero quartic
couplings
$D_{++--},D_{+-+-},D_{+-ZZ},D_{+-Z^{\prime}Z^{\prime}},D_{+Z-Z},D_{+Z^{\prime}-Z^{\prime}}$,
$D_{ZZZZ},D_{Z^{\prime}Z^{\prime}ZZ},D_{Z^{\prime}ZZ^{\prime}Z},D_{Z^{\prime}Z^{\prime}Z^{\prime}Z},D_{Z^{\prime}Z^{\prime}Z^{\prime}Z^{\prime}}$
are free, corresponding to the 11 independent coefficients $\alpha_{4}$ to
$\alpha_{7}$, $\alpha_{10}$, and $\alpha_{18}$ to $\alpha_{23}$. In Table I,
we list details of all the triple and quartic couplings.

Table 1: List of triple and quartic couplings. couplings | exist in SM (modified by) | independent | control by | vanishing condition
---|---|---|---|---
$C_{\gamma-+}$ | yes ($\alpha_{2,3,9})$ | - | - | -
$C_{Z-+}$ | yes ($\alpha_{2,3,9}$) | - | - | -
$C_{Z^{\prime}-+}$ | no | yes | $\alpha_{31}~{}\&~{}\tilde{g}^{\prime}$ | $\alpha_{31}=\tilde{g}^{\prime}=0$
$C_{+\gamma-}$ | yes (not modified) | - | - | -
$C_{+Z-}$ | yes ($\alpha_{3}$) | - | - | -
$C_{+Z^{\prime}-}$ | no | - | $\tilde{g}^{\prime}$ | $\tilde{g}^{\prime}=0$
$C_{V_{1}V_{2}V_{3}}$ | no | - | - | always
$D_{++--}$ | yes ($\alpha_{3,4,9}$) | - | - | -
$D_{+-+-}$ | yes ($\alpha_{3,4,5,9}$) | - | - | -
$D_{+-\gamma\gamma}$ | yes (not modified) | - | - | -
$D_{+-ZZ}$ | yes ($\alpha_{3,5,7}$) | - | - | -
$D_{+-Z^{\prime}Z^{\prime}}$ | no | yes | $\alpha_{21}$ | -
$D_{+-\gamma Z}$ | yes ($\alpha_{3}$) | - | - | -
$D_{+-\gamma Z^{\prime}}$ | no | - | $\tilde{g}^{\prime}$ | $\tilde{g}^{\prime}=0$
$D_{+-ZZ^{\prime}}$ | no | yes | $\alpha_{16}~{}\&~{}\tilde{g}^{\prime}$ | $\alpha_{16}=\tilde{g}^{\prime}=0$
$D_{+\gamma-\gamma}$ | yes (not modified) | - | - | -
$D_{+Z-Z}$ | yes ($\alpha_{3,4,6}$) | - | - | -
$D_{+Z^{\prime}-Z^{\prime}}$ | no | yes | $\alpha_{19}$ | -
$D_{+\gamma-Z}$ | yes ($\alpha_{3}$) | - | - | -
$D_{+\gamma-Z^{\prime}}=D_{+Z^{\prime}-\gamma}$ | no | - | $\tilde{g}^{\prime}$ | $\tilde{g}^{\prime}=0$
$D_{+Z-\gamma}=D_{+\gamma-Z}$ | yes | - | - | -
$D_{+Z-Z^{\prime}}=D_{+Z^{\prime}-Z}$ | no | yes | $\alpha_{17}~{}\&~{}\tilde{g}^{\prime}$ | $\alpha_{17}=\tilde{g}^{\prime}=0$
$D_{ZZZZ}$ | no | yes | $\alpha_{10}$ | -
$D_{Z^{\prime}ZZZ}$ | no | yes | $\alpha_{15}$ | $2\alpha_{15}+\alpha_{16}+\alpha_{17}=0$
$D_{Z^{\prime}Z^{\prime}ZZ}$ | no | yes | $\alpha_{20}$ | -
$D_{Z^{\prime}ZZ^{\prime}Z}$ | no | yes | $\alpha_{18}$ | -
$D_{Z^{\prime}Z^{\prime}Z^{\prime}Z}$ | no | yes | $\alpha_{22}$ | $\alpha_{22}=0$
$D_{Z^{\prime}Z^{\prime}Z^{\prime}Z^{\prime}}$ | no | yes | $\alpha_{23}$ | -

Given the six assumptions stated above:

  * (i)

no direct coupling to the Higgs;

  * (ii)

no direct coupling to fermions;

  * (iii)

no CP violating terms in the EEWCL;

  * (iv)

no anomalous terms in the EEWCL;

  * (v)

no kinetic nor mass mixing terms;

  * (vi)

no single Z’ couplings;

we are finally left with four $Z^{\prime}$-dependent quartic vertices each
involving two $Z^{\prime}$ fields,

$\displaystyle D_{+-Z^{\prime}Z^{\prime}}$ $\displaystyle\equiv$
$\displaystyle g_{1}=4g^{2}g^{{}^{\prime\prime}2}(\alpha_{5}+\alpha_{21})$
(10) $\displaystyle D_{+Z^{\prime}-Z^{\prime}}$ $\displaystyle\equiv$
$\displaystyle g_{2}=4g^{2}g^{{}^{\prime\prime}2}(\alpha_{4}+\alpha_{19})$
(11) $\displaystyle D_{Z^{\prime}Z^{\prime}ZZ}$ $\displaystyle\equiv$
$\displaystyle
g_{3}=g_{Z}^{2}g^{{}^{\prime\prime}2}(\alpha_{5}+2\alpha_{7}+4\alpha_{20}+2\alpha_{21})$
(12) $\displaystyle D_{Z^{\prime}ZZ^{\prime}Z}$ $\displaystyle\equiv$
$\displaystyle
g_{4}=4g_{Z}^{2}g^{{}^{\prime\prime}2}(\alpha_{4}+\alpha_{6}+2\alpha_{18}+\alpha_{19})$
(13)

where $g_{Z}^{2}=g^{2}+{g^{\prime}}^{2}$. Here we limit ourselves to vertices
with up to 4 particles: the EEWCL does contain Z’ vertices with more gauge
boson, however their physical effect is subleading and we will not consider
them any further. The above couplings do contribute to electroweak precision
tests at loop level: such contributions are log–divergent, and their
contribution can be absorbed in the tree–level contributions $\beta_{1}$ and
$\alpha_{1,8}$, whose values are therefore strongly constrained. In the
following we will therefore not consider bounds from precision tests, as they
do not give unique indication on the size of the quartic couplings listed
above. To keep the number of free parameters to a minimum, we shall take in
our analysis a unique coupling constant $g_{0}$ and consider five different
arrangements of coupling constants as follows:

  * •

universal case: $g_{1}\\!=\\!g_{2}\\!=\\!4g^{2}g_{0}$,
$g_{3}\\!=\\!\frac{3}{8}g_{4}\\!=\\!\frac{3}{2}g_{Z}^{2}g_{0}$, or
$\alpha_{4}\\!=\\!\alpha_{5}\\!=\\!\alpha_{19}\\!=\\!\alpha_{21}\\!\equiv\\!\frac{g_{0}}{2g^{{}^{\prime\prime}2}}$,
$\alpha_{6}\\!=\\!\alpha_{7}\\!=\\!\alpha_{18}\\!=\\!\alpha_{19}\\!=\\!\alpha_{20}\\!=0$

  * •

case 1: $g_{1}=4g^{2}g_{0}$, $g_{3}=\frac{3}{2}g_{Z}^{2}g_{0}$,
$g_{2}=g_{4}=0$, or
$\alpha_{5}=\alpha_{21}\equiv\frac{g_{0}}{2g^{{}^{\prime\prime}2}}$,
$\alpha_{4}=\alpha_{6}=\alpha_{7}=\alpha_{18}=\alpha_{19}=\alpha_{20}=0$

  * •

case 2: $g_{2}=4g^{2}g_{0}$, $g_{4}=4g_{Z}^{2}g_{0}$, $g_{1}=g_{3}=0$, or
$\alpha_{4}=\alpha_{19}\equiv\frac{g_{0}}{2g^{{}^{\prime\prime}2}}$,
$\alpha_{5}=\alpha_{6}=\alpha_{7}=\alpha_{18}=\alpha_{20}=\alpha_{21}=0$

  * •

case 3: $g_{3}=3g_{Z}^{2}g_{0}$, $g_{1}=g_{2}=g_{4}=0$, or
$\alpha_{7}=\alpha_{20}\equiv\frac{g_{0}}{2g^{{}^{\prime\prime}2}}$,
$\alpha_{4}=\alpha_{5}=\alpha_{6}=\alpha_{18}=\alpha_{19}=\alpha_{21}=0$

  * •

case 4: $g_{4}=6g_{Z}^{2}g_{0}$, $g_{1}=g_{2}=g_{3}=0$, or
$\alpha_{6}=\alpha_{18}\equiv\frac{g_{0}}{2g^{{}^{\prime\prime}2}}$,
$\alpha_{4}=\alpha_{5}=\alpha_{7}=\alpha_{19}=\alpha_{20}=\alpha_{21}=0$.

In the following sections, we shall mainly focus on the above five cases and
derive information on the unique coupling $g_{0}$ when discussing possible
constraints from various experiments. Some possible exceptions are also
discussed.

##  III Relic density constraint of dark Z’

In the standard Cosmology picture ($\Lambda$CDM), it is assumed that the DM
particles are in thermal equilibrium with the other SM particles via various
fundamental processes such as $Z^{\prime}Z^{\prime}\rightarrow\bar{P}P$ where
$P$ is any SM particle. In the high-temperature Early Universe, DM particles
were kept in thermal equilibrium as long as the reaction rate, scaled by the
temperature, was faster than the expansion rate $H$ (the Hubble parameter) of
the Universe. The Universe cooled down as it continued to expand. At
temperatures around which the reaction rate fell below the expansion rate $H$,
the DM particles began to decouple from the thermal bath. The DM particles
will continue to annihilate into SM particles up until the point when they no
longer encounter one another. The remaining DM particles will then become the
relics that we can observe today. The two possible annihilation processes for
our dark Z’ are shown in Fig. 1.

Figure 1: Two annihilation processes of $Z^{\prime}$ to SM weak gauge bosons.

Their annihilation rates to SM weak gauge bosons are

$\displaystyle\sigma_{W}\;v$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-\frac{1}{r_{W}^{2}}}}{9\times 64\pi
m_{Z^{\prime}}}\\{[(224r_{W}^{4}+112r_{W}^{2}+136)g_{1}^{2}-(160r_{W}^{4}+80r_{W}^{2}+176)g_{1}g_{2}+(152r_{W}^{4}+128r_{W}^{2}+96)g_{2}^{2}]$
(14)
$\displaystyle+[(88r_{W}^{4}-40r_{W}^{2}+56)g_{1}^{2}-(\frac{304}{3}r_{W}^{4}+32r_{W}^{2}+\frac{32}{3})g_{1}g_{2}+(24r_{W}^{4}+\frac{112}{3}r_{W}^{2}-\frac{4}{3})g_{2}^{2}]v^{2}\\}$
$\displaystyle\sigma_{Z}\;v$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-\frac{1}{r_{Z}^{2}}}}{18\times 64\pi
m_{Z^{\prime}}}\\{[(224r_{Z}^{4}+112r_{Z}^{2}+136)g_{3}^{2}-(160r_{Z}^{4}+80r_{Z}^{2}+176)g_{3}g_{4}+(152r_{Z}^{4}+128r_{Z}^{2}+96)g_{4}^{2}]$
(15)
$\displaystyle+[(88r_{Z}^{4}-40r_{Z}^{2}+56)g_{3}^{2}-(\frac{304}{3}r_{Z}^{4}+32r_{Z}^{2}+\frac{32}{3}g_{3}g_{4}+(24r_{Z}^{4}+\frac{112}{3}r_{Z}^{2}-\frac{4}{3})g_{4}^{2}]v^{2}\\},$

in which $r_{W}=m_{Z^{\prime}}/m_{W}$ and $r_{Z}=m_{Z^{\prime}}/m_{Z}$, and
$v$ is the relative velocity of the colliding DM particles. In above result,
due to the non-relativistic characteristics of our dark Z’, we have taken
expansion in terms of powers of $v$ up to the order of $v^{2}$. Note that for
these annihilations to arise, the mass of Z’ should be heavier than the mass
of the SM weak gauge bosons. If the Z’ mass is lighter than the SM weak boson
mass, we instead use the loop-induced effective $Z^{\prime}Z^{\prime}q\bar{q}$
vertices given in the next section. The corresponding annihilation rate is

$\displaystyle\sigma\;v$ $\displaystyle=$
$\displaystyle\frac{v^{2}}{6}\bigg{[}\frac{1}{6\pi}{\displaystyle\sum_{f}}(\frac{K_{V,f}}{\sqrt{2}})^{2}c_{f}\sqrt{1-\frac{m_{f}^{2}}{M^{2}_{Z^{\prime}}}}M^{2}_{Z^{\prime}}(2+\frac{m_{f}^{2}}{M^{2}_{Z^{\prime}}})+\frac{1}{3\pi}{\displaystyle\sum_{f}}(\frac{K_{VA,f}}{\sqrt{2}})^{2}c_{f}(1-\frac{m_{f}^{2}}{M^{2}_{Z^{\prime}}})^{3/2}M^{2}_{Z^{\prime}}\bigg{]},$
(16)

where $K_{V,f}$ and $K_{VA,f}$ are the effective couplings introduced in the
next section in Eq. (19), and $c_{f}$ is the color index, which is 1 for
leptons and 3 for quarks.

The relic density is calculated by solving the Boltzmann equation in the
standard approximation procedure Kolb:1981hk ,

$\Omega_{\mathrm{WIMP}}h^{2}=\frac{1.07\times
10^{9}}{m_{pl}}\frac{x_{F}GeV^{-1}}{\sqrt{g_{*S}}}\frac{1}{a+3b/x_{F}},$ (17)

where $h$ is the scaled Hubble constant, $x_{F}=m_{Z^{\prime}}/T_{F}$ with
$T_{F}$ the freezing temperature, $m_{pl}=1.22\times 10^{19}GeV$, $g_{*S}$ the
total number of effectively relativistic degrees of freedom at freeze-out
temperature, and $a$ and $b$ are parameters in the expansion $\sigma
v=a+bv^{2}+\mathcal{O}(v^{4})$. The freeze-out temperature parameter $x_{F}$
can be evaluated by numerically solving the following equation:

$x_{F}=\ln\left[c(c+2)\sqrt{\frac{45}{8}\frac{gm_{Z^{\prime}}m_{pl}(a+6b/x_{F})}{2\pi^{3}\sqrt{g_{*S}}x_{F}^{1/2}}}\right],$
(18)

where $g=3$ is the number of degrees of freedom for the $Z^{\prime}$ DM, and
$c$ is a numerical constant usually taken equal to $1/2$. With DM mass ranging
from GeV to TeV, $x_{F}\approx 25$ and remains essentially constant. In our
numerical analysis, we demand that the resulting relic density be less than
the measured value from PLANCK $\Omega_{\mathrm{WIMP}}h^{2}=0.1199\pm 0.0027$
at $68\%$CL PLANCK , which leads to constraints for the effective coupling
constants $g_{i},\ i=1,2,3,4$ and the dark Z’ mass $M_{Z^{\prime}}$. The
result of the five different coupling arrangements introduced at the end of
Sec. II for $M_{Z}^{\prime}>100$GeV is shown in Fig. 2, where we have used
result (14) and (15) to perform our calculation. Note that because the
ordinate is logarithmically scaled, the diagram would be unable to show
clearly the possible deviations of several percent from the experiment, and
hence are not plotted.

Figure 2: Predicted unique coupling constant $g_{0}$ as a function of DM mass
$m_{Z^{\prime}}\geq 100$GeV fixed by the observed relic density. Each color of
the curves represents different arrangements of coupling constant $g_{i}$. The
allowed region is located above the curve.

For dark $Z^{\prime}$ masses below the W mass threshold, we instead use Eq.
(16) to perform our estimation; the results are shown in Fig. 3.

Figure 3: Same as for Fig. 1 but with DM mass $m_{Z^{\prime}}\leq 80$ GeV.

Note that in the intermediate region $m_{W}/2<M_{Z^{\prime}}<m_{W}$, the
three–body process $Z^{\prime}Z^{\prime}\to WW^{\ast}(ZZ^{\ast})$, where one
of the two gauge bosons is off–shell, is also relevant: we expect it to
smoothly interpolate between the results in Fig. 2 and 3. Nevertheless, below
the $WW$ threshold, the coupling is required to be very large, and this region
is excluded by direct–detection experiments, as we will show in the next
section.

##  IV Effective quark vertex and direct detection of dark Z’

In this section, we discuss the direct-detection constraints on dark Z’.
Direct-detection experiments are designed to measure the recoil energy of the
atomic nuclei following DM elastic scattering. DM-quark interactions will
naturally induce DM-nucleon interactions, and the latter further induce DM-
nucleus interactions. Such interactions may be detected in underground direct-
detection experiments.

As our second assumption does not allow $Z^{\prime}$ to directly couple to SM
fermions (fermiophobic), it can only couple indirectly to quarks through the
SM gauge-boson loop depicted in Fig. 4.

Figure 4: Dark matter scattering from quarks through SM gauge-boson loops.

This loop diagram leads to finite effective vector-vector and vector-axial-
vector interaction vertices Yu:2011by :

$\mathcal{L}_{eff}=\Sigma_{q}\frac{K_{V,q}}{\sqrt{2}}(Z^{\prime*}_{\nu}i\overleftrightarrow{\partial}_{\mu}Z^{\prime\nu})\bar{q}\gamma^{\mu}q+\Sigma_{q}\frac{K_{VA,q}}{\sqrt{2}}(Z^{\prime*}_{\nu}i\overleftrightarrow{\partial}_{\mu}Z^{\prime\nu})\bar{q}\gamma^{\mu}\gamma_{5}q.$
(19)

The finiteness of this loop-induced $Z^{\prime}Z^{\prime}\bar{q}q$ vertex
implies that our fermiophobic assumption is consistent by itself. Furthermore,
the structure of the above result for the effective vertices only generates
spin-independent (SI) interaction. Hence there will be no constraint from the
cross-section of the spin-dependent interaction.

Note that the velocity of DM near the Earth is considered to be $v\approx
0.001c$; for low-energy SI interactions, only the vector interaction survives.
The cross-section is then given by

$\sigma_{V,Z^{\prime}N}=\frac{m_{N}^{2}m_{Z^{\prime}}^{2}}{\pi(m_{Z^{\prime}}+m_{N})^{2}}(\frac{K_{V,N}}{\sqrt{2}})^{2},$
(20)

where

$\displaystyle K_{V,p}=2K_{V,u}+K_{V,d},\quad K_{V,n}=K_{V,u}+2K_{V,d}$ (21)
$\displaystyle\frac{K_{V,q}}{\sqrt{2}}=\frac{(g_{1}+g_{2})(c_{q}^{2}+c_{q}^{\prime
2})}{32\pi^{2}m_{W}^{2}}+\frac{(g_{3}+g_{4})(c_{q}^{2}+c_{q}^{\prime
2})}{32\pi^{2}m_{Z}^{2}}$ (22)

and $c_{q}$ and $c_{q}^{\prime}$ for $q=u,d$ are coefficients associated with
SM couplings

$c_{u}=\frac{ig}{4\cos\theta_{W}}(1-\frac{8}{3}\sin^{2}\theta_{W}),\quad
c_{d}=\frac{ig}{4\cos\theta_{W}}(-1+\frac{4}{3}\sin^{2}\theta_{W}),\quad
c_{u}^{\prime}=\frac{ig}{4\cos\theta_{W}},\quad
c_{d}^{\prime}=-\frac{ig}{4\cos\theta_{W}}.$ (23)

The curves for the predicted cross-section $\sigma$ vs $m_{Z^{\prime}}$ with
different coefficient settings are shown in Fig. 5. The upper bounds set by
the XENON100 2012 data XENON100 , the latest LUX result LUX  and SuperCDMS
result SuperCDMS  are also plotted in Fig. 5.

Figure 5: Predicted spin-independent WIMP-proton cross-sections for dark Z’.
The solid line is for the universal case, dashed line for case 1, dotted line
for case 2, short dotted line for case 3, and dash-dotted line for case 4. For
each case, the black lines correspond to $g_{0}=0.5$, magenta lines to
$g_{0}=0.1$, and orange lines to $g_{0}=0.02$. Also, the upper bounds set by
XENON100, LUX and SuperCDMS are marked in blue, red and purple respectively.

We next discuss some of these results:

  * -

Because the unique coupling $g_{0}$ is constrained to order $10^{-2}$ in the
typical few tens to few hundreds GeV region (see Fig. 5), it is natural to ask
if we go beyond the limitations set for the five arrangements and consider
arbitrary $g_{i}$ couplings; is there a possibility to enhance the value of
the coupling? To examine this issue, note that Eq. (22) gives $K_{V,q}$, which
then determines the final cross-section $\sigma$; it depends on $g_{1}+g_{2}$
and $g_{3}+g_{4}$. If we take $g_{1}\approx-g_{2}=4g^{2}g_{0}$, $g_{3}\approx-
g_{4}=4g_{Z}^{2}g_{0}$, then we achieve an arbitrary small $K_{V,q}$ and
subsequently an arbitrary small scattering cross-section. For example,
$g_{2}=-0.9g_{1}$, $g_{4}=-0.9g_{3}$, and $g_{1}=g_{3}$ leads to
$K_{V,q}\propto 0.1g_{1}$, which is already smaller by an order of magnitude
than the five cases already derived in Fig. 5. Hence $\sigma\propto 0.01g_{1}$
is less than those used in Fig. 5 by two orders of magnitudes and is
equivalent to enhancing the constraint for $g_{0}$ by two order of magnitudes;
i.e., it relaxes the constraint for $g_{0}$ from $10^{-2}$ to $10^{0}$,
because experimentally we must fix the cross-section. Therefore, we do have
flexibility in the couplings in relaxing the constraints on $g_{0}$ (or
$\alpha_{i}$) from direct detection. In the next section, we shall see for
indirect detection that this scenario does not arise as there exists no such
coupling space.

  * -

As we have introduced in our theory the one-loop-induced
$Z^{\prime}Z^{\prime}\bar{q}q$ vertex (19), one may inquire of the role of the
one-loop-induced $Z^{\prime}Z^{\prime}h$ vertex depicted in Fig. 6. One can
easily check that this loop diagram is logarithmically divergent. This implies
that our first higgsphobic assumption, introduced in Sec.II, by which we
ignore the direct coupling between Z’ and Higgs is not consistence by itself.
To cancel such a divergence, we have to introduce into the theory the tree-
level vertex, $Z^{\prime}Z^{\prime}h$, which violates our higgsphobic
assumption.

Figure 6: Dark matter annihilating into Higgs through SM gauge boson loops.

Furthermore, once we have such a $Z^{\prime}Z^{\prime}h$ vertex, no matter
whether it is a tree-level one added to the theory by hand or a loop-induced
one, it can further decay via the Higgs-mediated s-channel (Fig. 7) into a
fermion pair, which then leads to further corrections to the effective
$Z^{\prime}Z^{\prime}\bar{q}q$ vertex given by Eq. (19).

Figure 7: Dark matter annihilating to form a quark pair through the effective
$Z^{\prime}Z^{\prime}h$ vertex.

The reason that we do not consider this correction is as follows: suppose we
abandon our first assumption by introducing the tree-level
$Z^{\prime}Z^{\prime}h$ vertex into our theory. The $Z^{\prime}Z^{\prime}h$
vertex then includes two contributions: one is a tree-level term, and the
other a loop term (Fig. 6). After renormalisation, i.e., cancelling the loop-
induced divergence (Fig. 6) by the tree-level term, the remaining finite part
should not be very much different with respect to the finite part of Fig. 6.
This finite part of the $Z^{\prime}Z^{\prime}h$ vertex via the Higgs-mediated
s-channel (Fig. 7) further leads to an effective
$Z^{\prime}Z^{\prime}\bar{q}q$ vertex, which is of scalar type
$Z^{{}^{\prime}2}\bar{q}q$ or pseudo-scalar type
$Z^{{}^{\prime}2}\bar{q}\gamma_{5}q$. These vertices have different vertex
structures in comparison with the vector and axial vector vertices given by
Eq. (19). Only the scalar vertex survives and it contributes to the nucleon
cross-section similar to that in (20), but has an extra factor proportional to
$m_{q}^{2}/m_{Z^{\prime}}^{2}\sim 10^{-8}$ with $m_{q}$ the mass of the u or d
quark. This factor results from the replacement of the derivative-type vector
coupling of (20) with the non-derivative-type scalar coupling
$Z^{\prime}Z^{\prime}\bar{q}q$ (see Ref.Yu:2011by ). It is this suppression
factor that allows us to ignore the loop-induced and tree-level
$Z^{\prime}Z^{\prime}h$ vertices. Although our higgsphobic assumption is not
consistent by itself, ignoring it only creates a very minor correction and
thus we can still approximately adhere to it.

##  V Indirect detection of dark Z’

In addition to the direct DM searches at underground laboratories, indirect
searches look for DM annihilations or decay products in the atmosphere. These
particles, which include neutrinos, gamma rays, positrons, and antiprotons,
can be detected in cosmic-ray experiments. In this section, we discuss three
of the latest experiments.

###  V.1 PAMELA experiment

We first follow the procedure in Yu:2011by  to obtain constraints for Z’ using
the anti-proton to proton flux ratio $\bar{p}/p$ measured by the satellite-
borne experiment PAMELA Adriani:2010rc . The tree-level annihilations
$Z^{\prime}Z^{\prime}\rightarrow ZZ,WW$ contribute to the generation of the
$\bar{p}/p$ signal via the hadronic decays of the W and Z bosons ppbar .
Together with the antiproton-to-proton flux ratio data of PAMELA, we can
derive constraints for each of our five cases. The result is shown in Fig. 8,
where the allowed region is below each curve.

Figure 8: Upper bounds on the coupling constant $g_{0}$ from the PAMELA
$\bar{p}/p$ flux ratio.

It should be noted that the loop-induced vertex Z’Z’qq given by Eq. (19) can
also contribute to the generation of the $\bar{p}/p$ signal. Nevertheless, our
computations show that the constraint is very weak and we shall not discuss it
here.

###  V.2 AMSII experiment

Recently, the AMSII collaboration has announced a new measurement of the
cosmic-ray positron fraction AMS02 . We discuss DM annihilation in view of
these measurements and derive constraints on our dark Z’ matter mass and the
universal case for coupling constant $g_{0}$. We consider the annihilation
channels $Z^{\prime}Z^{\prime}\longrightarrow W^{+}W^{-}$ and
$Z^{\prime}Z^{\prime}\longrightarrow ZZ$ as in the discussion for the relic
density in the previous section, and use the same method in AMS02explanation
to set conservative limits by requiring that the predicted positron flux
remains smaller than the measured flux over all energies. For simplicity, we
assume a standard Navarro-Frenk-White (NFW) density profile Navarro:1996gj ,
i.e., $\bar{J}\Delta\Omega\approx 1$. The result is graphed in Fig. 9, which
shown that it is not competitive with the PAMELA bounds.

Figure 9: Upper bounds from AMSII cosmic-ray positron fraction spectrum.

###  V.3 FermiLAT experiment

Photons from DM annihilations in the centre of the galaxy provide another
source of an indirect signal. As dark Z’ cannot annihilate into photon pairs
directly (we have checked that even by including one-loop corrections, the
$Z^{\prime}Z^{\prime}\gamma$ vertex still vanishes), we can only detect
continuum photon signals. The differential flux of the $\gamma$-rays observed
on Earth from DM annihilation is as follows,

$\frac{d\Phi}{dE_{\gamma}}=(5.5\times
10^{-10}s^{-1}cm^{-2})\frac{dN}{dE_{\gamma}}(\frac{\langle\sigma
v\rangle}{pb})(\frac{100GeV}{m_{\chi}})^{2}\bar{J}\Delta\Omega.$ (24)

Again, as in the previous subsection, we have assumed the DM distribution to
follow the NFW profile. The annihilation cross-section is

$\langle\sigma v\rangle\approx a+b\langle v^{2}\rangle=a+2b\bar{v}^{2},$ (25)

where $\bar{v}=270km/s$. The simple analytic fit is as follows
Bergstrom:1997fj

$\frac{dN}{dE_{\gamma}}=\frac{dN}{m_{\chi}dx}=\frac{1}{m_{\chi}}\frac{a_{0}}{x^{1.5}}e^{-b_{0}x},$
(26)

where $x=E_{\gamma}/m_{\chi}$, $a_{0}=0.73$, and $b_{0}=7.76$ for $W/Z$
bosons. We consider the universal case with $g_{0}$ fixed by the relic density
for different DM masses; the predicted $\gamma$-ray spectra is shown in Fig.
10.

Figure 10: Predicted $\gamma$-ray spectra for the annihilation of dark Z’ in
the universal case and with a NFW density profile. The FermiLAT observation
data are also presented.

From it we can see the photon energy flux is about 3 orders of magnitude lower
than the experiment data. To explain the discrepancy with the data, we will
need an enhancement of 300 to 8000. The required boost factors are obtained by
fitting the data and the result is shown in Fig. 11.

Figure 11: Comparison of predicted dark Z’ signals for different masses and
boost factors with FermiLAT observation data.

Recently, $\gamma$-ray observations of 25 Milky-Way dwarf spheroidal satellite
galaxies from four years of FermiLat data was reported in Ackermann:2013yva .
We can use the constraint for the channel $Z^{\prime}Z^{\prime}\rightarrow
W^{+}W^{-}$ in Ackermann:2013yva  to give upper limits to coefficients $g_{1}$
and $g_{2}$. The result is shown below in Fig. 12.

Figure 12: Upper limits to the dark-Z’ coupling constant $g_{0}$ for the
universal case, case 1, and case 2 from FermiLAT $\gamma$-ray data.

With the above discussion of the five $g_{0}$ cases, we may ask a similar
question to that of the last section: if we go beyond the five limit cases
discussed above and consider arbitrary $g_{i}$ couplings, is there a
possibility to enhance the value of the couplings? To answer this question,
note that the contributions to cross-section for indirect detection are given
by Eq. (14) and Eq. (15), in which the terms proportional to $v^{2}$ play very
little role as $v^{2}$ is small. We can demand that the remaining terms take
minimum values (which result in the smallest indirect-detection cross-section)
to fix the coupling. For $m_{Z^{\prime}}=100~{}GeV$ (the result changes little
in the range 100 GeV$<m_{Z^{\prime}}<$1 TeV), we find two minima:

$\displaystyle g_{2}\approx 0.521g_{1},~{}~{}~{}g_{3}=g_{4}=0\hskip
28.45274pt\mbox{or}\hskip 28.45274ptg_{1}=g_{2}=0,~{}~{}~{}g_{4}\approx
0.5207g_{3}.$ (27)

This result gives the same sign for $g_{1}$ and $g_{2}$, and for $g_{3}$ and
$g_{4}$. This differs from the result in the direct detection discussed in the
last section, where we obtain opposite signs for $g_{1}$ and $g_{2}$ and for
$g_{3}$ and $g_{4}$. Further, in plotting the constraint for these two extreme
cases (Fig. 13), we find that the PAMELA experiment still yields constraint
$g_{0}<3\times 10^{-2}$. This shows that, all in all, the constraint on the
coupling is rather robust.

Figure 13: Upper bounds from the PAMELA experiment on the coupling constant
$g_{0}$ for dark Z’ in two extreme cases compared with the universal case.

##  VI Combined result and other DM related issues

As the Z’ is the only DM particle in our theory, we can discuss its mass
$M_{Z^{\prime}}$ and unique coupling $g_{0}$ in a $g_{0}-M_{Z^{\prime}}$ plot.
For simplicity, we only discuss the universal case. Combining all effective
phenomenological constraints from the last three sections, we obtain Fig. 14,
where the shaded region is excluded by the different experiments.

Figure 14: Combined constraints on coupling constants $g_{0}$ of dark Z’ for
the universal case.

The allowed region has a the left boundary at roughly $M_{Z^{\prime}}\sim 100$
GeV: below this mass scale, the relic density constraint from Fig. 3 already
contradicts data from the XENON100 and LUX experiments. The lower bound above
$M_{Z^{\prime}}=100$ GeV is the red curve from the relic density (Fig. 2),
while the upper bound is from the PAMELA $\bar{p}/p$ flux ratio constraint
(Fig. 8). We find $g_{0}$ is restricted to the region of $10^{-3}\sim
10^{-2}$. As discussed in the previous sections, this bound is quite solid,
and it does depend mildly on the configuration of couplings.

Considering the low-energy region below the threshold of the W boson mass, the
relic density from Fig. 3 offers a very strong constraint, which combined with
the direct-detection results kills any possibility of the existence of a dark
Z’ in this low mass region. If we simply ignore this fact, then a small dark
Z’ mass is allowed in this lower energy region. Currently, though, tensions
exist among the different direct-detection experiments in the low-energy
region. For example, null results from CDMS Ge, XENON, LUX and SuperCDMS
challenge the CoGeNT/DAMA and latest CDMS SI results. Because different
experiments use different detection materials with different numbers of
proton/neutrons, isospin-violating DM may weaken these tensions. It was
demonstrated that isospin-violating DM with $f_{p}/f_{n}\approx$-$0.7$ may
alleviate the problem Feng:2011vu . For our dark Z’, $f_{q}$ is given as
follows:

$\displaystyle\frac{f_{q}}{\sqrt{2}}=\frac{(g_{1}+g_{2})(c_{q}^{2}+c_{q}^{\prime
2})}{32\pi^{2}m_{W}^{2}}+\frac{(g_{3}+g_{4})(c_{q}^{2}+c_{q}^{\prime
2})}{32\pi^{2}m_{Z}^{2}}$ (28) $\displaystyle f_{p}=2f_{u}+f_{d},\hskip
56.9055ptf_{n}=f_{u}+2f_{d}.$ (29)

One can check that

$\frac{f_{u}}{f_{d}}=\frac{c_{u}^{2}+c_{u^{\prime}}^{2}}{c_{d}^{2}+c_{d^{\prime}}^{2}},$
(30)

which further leads to

$f_{p}/f_{n}\approx 0.93.$ (31)

Clearly, the ratio is completely unrelated to the coupling constants
$g_{1},g_{2},g_{3}$, and $g_{4}$. Our dark Z’ itself cannot give
$f_{p}/f_{n}<0$ (adding in Higgs boson may bring new vertices and change the
rate). Indeed, it was recently observed that the LUX LUX  and SuperCDMS
SuperCDMS  data are in strong contrast tension1 ; *tension2; *tension3 with
the CDMS-Si signal result CDMS-Si , even for isospin- violating DM (IVDM)
where the ratio of the proton-to-neutron couplings can be maximally
“xenophobic”. Another possibility to reconcile this tension is to consider an
inelastically scattering DM model or an exothermic DM model ExothermicDM .
However, to realise inelastic scattering, we need at least two DM states. As
our simplest dark Z’ model only has one state, the present dark Z’ cannot
produce inelastic scattering. This statement can be weakened in models where
the Z’ belongs to a triplet, i.e. it is accompanied by a W’: this can be
achieved in a chiral lagrangian with an extra SU(2) symmetry Wprime .
Remaining in our framework, we obtain the result that even if we ignore the
strong constraint allowing our dark Z’ to survive into the low-energy region,
it still cannot reduce the tension among those observed possible DM signals
with other null result experiments.

Finally, we need to discuss the impact of our dark Z’ model on other
experiments in particle physics. First, following our second and sixth
assumptions, there are no mixing of the Z’ with the Z and photon and no linear
couplings of the Z’. This implies that there are no tree level bounds on the
Z’ couplings and mass from electroweak precision measurementsEWprecision .
However, loops of the $Z^{\prime}Z^{\prime}WW$ and $Z^{\prime}Z^{\prime}ZZ$
couplings do generate corrections to both $T$ and $S$ parameters. Such loop
corrections are logarithmical divergent, and they lead to the renormalisation
of the parameters $\beta_{1}$ and $\alpha_{1,8}$ in the EEWCL. Therefore, the
new physics effects must be encoded in the renormalised value of those
parameters, and no robust and model independent bound on the couplings of the
Z’ can be extracted. Furthermore, the allowed window for the coupling $g_{0}$
provides enough suppression to evade eventual bounds from the finite
contribution of the loops. Another issue regards the discovery potential at
the LHC: as the only tree level couplings at the order we are interested in
involve heavy gauge bosons, the only channels where the Z’ can be produced are
production in association with a gauge boson ($pp\to
Z^{\prime}Z^{\prime}(W,Z)$) and vector boson fusion production ($pp\to
Z^{\prime}Z^{\prime}jj$). One may therefore look for a single $W/Z$ signal or
monojet. However, these channels suffer from large backgrounds, and we checked
that the small allowed values of the couplings ($g_{0}\sim 10^{-3}$–$10^{-2}$)
would lead to cross sections that are too small to be detected at the LHC. We
therefore conclude that the minimal model we study in this paper is not
accessible at the LHC.

##  VII Summary

In this paper, we have investigated a rarely discussed possibility where a Z’
boson is the sole DM candidate.
We considered an extended chiral Lagrangian with an additional U(1) gauge
symmetry, with the following additional assumptions:

  1. (i)

dark Z’ is higgsphobic, i.e., it does not directly couple to the Higgs,

  2. (ii)

dark Z’ is fermiophobic, i.e., it does not directly couple to SM fermions,

  3. (iii)

there is no CP violating Z’ couplings,

  4. (iv)

there is no anomalous Z’ couplings,

  5. (v)

dark Z’ does not mix with $\gamma$ and the $Z$ boson, and

  6. (vi)

there is no Z’ interaction linear in the Z’ field.

The remaining quartic vertices, $Z^{\prime}Z^{\prime}ZZ$ and
$Z^{\prime}Z^{\prime}W^{+}W^{-}$, then dominate the Z’ physics, which has four
independent effective coupling constants $g_{1},g_{2},g_{3},g_{4}$. We found
that the mass of this dark Z’ is not allowed below the W boson mass threshold,
due to a combination of strong constraints from the relic density and those
from direct-detection experiments. For mass $M_{Z^{\prime}}>100$GeV, from the
relic density and direct and indirect-detection experiments where effective
$Z^{\prime}Z^{\prime}q\bar{q}$ couplings are induced from SM gauge-boson
loops, we produce five different coupling scenarios that are in the region
$10^{-3}$–$10^{-2}$ (for the universal case, the result is given in Fig. 14).
This range of coupling can be relaxed beyond the five cases analyzed for
direct-detection experiments, but cannot be changed for indirect-detection
experiments. To improve FermiLAT $\gamma$-ray spectra by our dark Z’, we
require a boost factor from 300 to 8000. We checked that even if our dark Z’
mass lies within the low-energy region, it cannot reduce tensions among the
observed possible DM signals with other null-result experiments. The bounds we
extracted are therefore rather robust and model-independent.

## Acknowledgments

This work was supported by the National Science Foundation of China (NSFC)
under Grant No. 11075085, National Basic Research Program of China (973
Program) under Grant No. 2010CB833000, the Specialized Research Fund for the
Doctoral Program of High Education of China No. 20110002110010, France China
Particle Physics Laboratory (FCPPL) financial support, and the Tsinghua
University Initiative Scientific Research Program. GC and AD also acknowledge
partial support from the Labex-LIO (Lyon Institute of Origins) under grant
ANR-10-LABX-66. AD is partially supported by Institut Universitaire de France.

##  Appendix A List of Couplings

$\displaystyle C_{\gamma-+}$ $\displaystyle=$
$\displaystyle-\frac{g^{3}g^{\prime}}{g_{Z}}(\frac{1}{g^{2}}+\alpha_{2}+\alpha_{3}+\alpha_{9})$
$\displaystyle C_{Z-+}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{g_{Z}}[1-g^{\prime
2}\alpha_{2}+g^{2}(\alpha_{3}+\alpha_{9})]$ $\displaystyle C_{Z^{\prime}-+}$
$\displaystyle=$ $\displaystyle-2{g}^{2}g^{\prime\prime}\alpha_{{31}}$
$\displaystyle C_{+\gamma-}$ $\displaystyle=$
$\displaystyle-\frac{gg^{\prime}}{g_{Z}}$ $\displaystyle C_{+Z-}$
$\displaystyle=$ $\displaystyle-\frac{g^{2}}{g_{Z}}-g^{2}g_{Z}\alpha_{3}$
$\displaystyle C_{+Z^{\prime}-}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle C_{V_{1}V_{2}V_{3}}$ $\displaystyle=$ $\displaystyle 0$ (32)

Here, $g_{Z}\equiv\sqrt{g_{0}^{2}+g_{1}^{2}}$.

$\displaystyle D_{++--}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{2}+g^{4}(\alpha_{3}+\frac{\alpha_{4}}{2}+\alpha_{9})$
$\displaystyle D_{+-+-}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}}{2}+g^{4}(-\alpha_{3}+\frac{\alpha_{4}}{2}+\alpha_{5}-\alpha_{9})$
$\displaystyle D_{+-\gamma\gamma}$ $\displaystyle=$
$\displaystyle-\frac{g^{2}g^{\prime 2}}{g_{Z}^{2}}$ $\displaystyle D_{+-ZZ}$
$\displaystyle=$
$\displaystyle-\frac{g^{4}}{g_{Z}^{2}}-2g^{4}\alpha_{3}+g^{2}g_{Z}^{2}(\alpha_{5}+\alpha_{7})$
$\displaystyle D_{+-Z^{\prime}Z^{\prime}}$ $\displaystyle=$ $\displaystyle
4{g}^{2}{g^{\prime\prime}}^{2}(\alpha_{{5}}+\alpha_{21})$ $\displaystyle
D_{+-\gamma Z}$ $\displaystyle=$
$\displaystyle-\frac{2g^{3}g^{\prime}}{g_{Z}^{2}}-2g^{3}g^{\prime}\alpha_{3}$
$\displaystyle D_{+-\gamma Z^{\prime}}$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle D_{+-ZZ^{\prime}}$ $\displaystyle=$
$\displaystyle-2{g}^{2}g_{Z}g^{\prime\prime}\alpha_{{16}}$ $\displaystyle
D_{+\gamma-\gamma}$ $\displaystyle=$ $\displaystyle\frac{g^{2}g^{\prime
2}}{g_{Z}^{2}}$ $\displaystyle D_{+Z-Z}$ $\displaystyle=$
$\displaystyle\frac{g^{4}}{g_{Z}^{2}}+2g^{4}\alpha_{3}+g^{2}g_{Z}^{2}(\alpha_{4}+\alpha_{6})$
$\displaystyle D_{+Z^{\prime}-Z^{\prime}}$ $\displaystyle=$ $\displaystyle
4{g}^{2}{g^{\prime\prime}}^{2}(\alpha_{{4}}+\alpha_{{19}})$ $\displaystyle
D_{+\gamma-Z}$ $\displaystyle=$
$\displaystyle\frac{g^{3}g^{\prime}}{g_{Z}^{2}}+g^{3}g^{\prime}\alpha_{3}$
$\displaystyle D_{+\gamma-Z^{\prime}}$ $\displaystyle=$ $\displaystyle
D_{+Z^{\prime}-\gamma}=0$ $\displaystyle D_{+Z-\gamma}$ $\displaystyle=$
$\displaystyle
D_{+\gamma-Z}=\frac{g^{3}g^{\prime}}{g_{Z}^{2}}+g^{3}g^{\prime}\alpha_{3}$
$\displaystyle D_{+Z-Z^{\prime}}$ $\displaystyle=$ $\displaystyle
D_{+Z^{\prime}-Z}=-{g}^{2}g_{Z}g^{\prime\prime}\alpha_{{17}}$ $\displaystyle
D_{ZZZZ}$ $\displaystyle=$
$\displaystyle\frac{1}{4}g_{Z}^{4}(\alpha_{4}+\alpha_{5}+2\alpha_{6}+2\alpha_{7}+2\alpha_{10})$
$\displaystyle D_{Z^{\prime}ZZZ}$ $\displaystyle=$ $\displaystyle-
g_{Z}^{3}g^{\prime\prime}(2\alpha_{15}+\alpha_{{16}}+\alpha_{17})$
$\displaystyle D_{Z^{\prime}Z^{\prime}ZZ}$ $\displaystyle=$
$\displaystyle{g^{\prime\prime}}^{2}g_{Z}^{2}(\alpha_{{5}}+2\alpha_{7}+4\alpha_{20}+2\alpha_{21})$
$\displaystyle D_{Z^{\prime}ZZ^{\prime}Z}$ $\displaystyle=$ $\displaystyle
4{g^{\prime\prime}}^{2}g_{Z}^{2}(\alpha_{{4}}+\alpha_{6}+2\alpha_{18}+\alpha_{19})$
$\displaystyle D_{Z^{\prime}Z^{\prime}Z^{\prime}Z}$ $\displaystyle=$
$\displaystyle-4{g^{\prime\prime}}^{3}g_{Z}(\alpha_{{16}}+\alpha_{17}+2\alpha_{22})$
$\displaystyle D_{Z^{\prime}Z^{\prime}Z^{\prime}Z^{\prime}}$ $\displaystyle=$
$\displaystyle
4{g^{\prime\prime}}^{4}(\alpha_{{4}}+\alpha_{5}+2\alpha_{19}+2\alpha_{21}+4\alpha_{23})$

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