#### A tale of two portals: testing light, hidden new physics at future e + e − colliders

Received: April
physics at future e+e
Jia Liu 0 1 2 3
Xiao-Ping Wang 0 1 2 3
Felix Yu 0 1 2 3
0 Johannes Gutenberg University , 55099 Mainz , Germany
1 PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics
2 Open Access , c The Authors
3 [11] V. Prasad, H. Li and X. Lou, Search for low-mass Higgs and dark photons at BESIII , talk
We investigate the prospects for producing new, light, hidden states at a future e+e collider in a Higgsed dark U(1)D model, which we call the Double Dark Portal model. The simultaneous presence of both vector and scalar portal couplings immediately modi es the Standard Model Higgsstrahlung channel, e+e ! Zh, at leading order in each coupling. In addition, each portal leads to complementary signals which can be probed at direct and indirect detection dark matter experiments. After accounting for current constraints from LEP and LHC, we demonstrate that a future e+e have unique and leading sensitivity to the two portal couplings by studying a host of new production, decay, and radiative return processes. Besides the possibility of exotic Higgs decays, we highlight the importance of direct dark vector and dark scalar production at machines, whose invisible decays can be tagged from the recoil mass method.
Beyond Standard Model; Cosmology of Theories beyond the SM; Higgs
1 Introduction
Overview of the double dark portal model: simultaneous kinetic mixing
and scalar portal couplings
Neutral vector boson mixing
Scalar boson mixing
Dark matter interactions
Direct detection and indirect detection phenomenology and constraints
Direct detection and relic abundance
3.2 Indirect constraints from CMB, Gamma-ray and e measurements Collider phenomenology of the Double Dark Portal model and current constraints from LEP and LHC
Recoil mass method for probing new, light, hidden states
Modi cations to electroweak precision
LEP-I and LEP-II constraints
Modi cations to Higgs physics and LHC constraints
Modi cations to Drell-Yan processes
Radiative return processes and dark matter production at the LHC
Prospects for future colliders
Electroweak precision tests, Higgsstrahlung, and invisible Higgs decays at
future e+e
Production of new, light states at future e+e colliders
with new particle production
Z~K~ production
A~K~ production
Z~S production
Z~H0, H0 ! K~ Z~ exotic decay
A Two limiting cases for K~ , Z~, and A~ mixing
B Cancellation e ect in multiple kinetic mixing terms
C Annihilation cross sections
Searches for new, light, hidden states are strongly motivated from the overriding question
of determining the particle nature of dark matter. The possible couplings to such light
states, however, remain highly model-dependent. Because higher dimension operators are
expected to be suppressed in scattering processes at low energies, the most promising
couplings give marginal Lagrangian operators at dimension four. Along these lines, two
well-studied couplings are a new kinetic mixing term
between a new, light, hidden photon
and the hypercharge gauge boson and a new quartic Higgs portal coupling
HP between a
hidden charged scalar eld and the Standard Model Higgs eld.
In this work, we argue and demonstrate that both marginal couplings can be
simultaneously probed in future measurements of a high energy e+e collider. Such a collider is, of
course, very strongly motivated by a rich and diverse set of possible Higgs measurements,
with leading sensitivity to the total Higgs width, Higgs couplings to Standard Model (SM)
particles, exotic Higgs decays, and additional precision measurements of the top quark
mass and exotic Z boson decays if additional running conditions are a orded [1{4]. We
highlight that such a machine also has leading sensitivity to new, weakly coupled, hidden
sectors, which can be probed via both radiative return processes and exotic invisible and
semi-visible Higgs decays. We will show that these measurements are enabled because
of the expected high precision photon resolution in the electromagnetic calorimeter, the
exquisite reconstruction of charged leptons, and clean discrimination of exotic signals from
SM background processes.
Both of these marginal operators have been studied autonomously at electron colliders
in the hidden photon context [5{17] and the hidden scalar context [18{23]. Some works
study both operators in tandem [24{26] or adopt an e ective operator approach [27]. The
current status of light, sub-GeV hidden photon searches and future prospects is
summarized in ref. [28]. In contrast with previous studies, we focus on higher mass hidden photons
beyond the reach of B-physics experiments and beam-dump experiments. For illustrative
purposes, we show our projections to dark photons as light as 1 GeV to demonstrate the
complementarity with recent results from B-physics experiments such as BaBar [29]. In
addition, we will emphasize the unique capability of e+e
machines to reconstruct invisible
decays, which is a marked improvement over the reconstruction prospects at hadron colliders.
The lack of evidence for weakly interacting massive particles (WIMPs) in direct
detection (DD) experiments [30{33], increasingly strong constraints on thermal WIMPs from
indirect detection (ID) experiments [34{38], and non-observation of beyond the Standard
Model (BSM) missing transverse energy signatures at the LHC [39, 40], combine to an
increasing unease with the standard WIMP miracle paradigm. On the other hand, dark
matter coupled to kinetically mixed hidden photons su ers from strong direct detection
constraints (see, e.g., [41]). A consistent dark matter model must hence simultaneously
address the relic density mechanism and non-observation in the current experimental probes,
and thus minimal models either require nonthermal dark matter production in the early
universe, coannihilation channels [42{44], or resonant dark matter annihilation in order to
divorce the early universe dynamics from collider processes (see, e.g., [45]). Moreover, while
the nuclear recoil energy spectrum at direct detection experiments requires the dark matter
mass as input, colliders instead probe mediator masses if they are on-shell, which shows the
complementarity between both approaches. In our work, we will further demonstrate these
complementary aspects between dark matter experiments and hadron and lepton colliders
in the context of our dark matter model.
In section 2, we review the theoretical framework for the Double Dark Portal model,
which uni es the kinetic mixing portal and the scalar Higgs portal into a minimal setup
with dark matter. In section 3, we detail the phenomenology of the dark matter for
direct detection and indirect detection experiments.
We discuss the extensive collider
phenomenology of the model and review the current constraints from experiments at the
Large Electron-Positron (LEP) collider and the Large Hadron Collider (LHC) in section 4.
We then present the prospects for exploring new, light hidden states at a future e+e
machine in section 5 and conclude in section 6. In appendix A, we o er some detailed
discussion of limiting cases in our Double Dark Portal model for pedagogical clarity, and
we discuss a cancellation e ect in scattering processes via kinetic mixing in appendix B.
We also present the dark matter annihilation cross sections for charged SM
nal states
in appendix C.
Overview of the double dark portal model: simultaneous kinetic
mixing and scalar portal couplings
We begin with the Lagrangian of the Double Dark Portal Model,
+ (iD=
is the eld strength tensor for the U(1)D gauge boson,
is a dark Higgs scalar
eld with charge +1 under U(1)D, and
with charge +1 under U(1)D. We take
is the dark matter and a SM gauge singlet fermion
2D > 0, which trigger spontaneous
symmetry breaking of the SM electroweak symmetry and the U(1)D dark gauge symmetry,
The nonzero Higgs portal coupling, HP , induces mass mixing between the h and
which results in mass eigenstates H0 and S. Simultaneously, the kinetic mixing
result in an e ective mass mixing between the SM Z gauge boson and the K dark gauge
boson, which results in the mass eigenstates Z~ and K~ . The two marginal couplings,
HP , are commonly referred to as vector and scalar portals, respectively [46]. Because
the phenomenology of such portal couplings changes signi cantly when a light dark matter
particle is added, we call the Lagrangian in eq. (2.1) the Double Dark Portal (DDP) model.
We solve the Lagrangian in the broken phase after the Higgs and the dark Higgs obtain
their vacuum expectation values (vevs),
= p (vD + ) ;
H = p (vH + h) ;
by diagonalizing and canonically normalizing the kinetic terms for the electrically neutral
gauge bosons and diagonalizing their mass matrix. We can rewrite the Lagrangian using
matrix notation, with mass terms acting on the gauge basis vector ( W 3 B
In this breaking of SU(2)L
U(1)D ! U(1)em, the resulting eld strength tensors
of the individual neutral vectors corresponding to the gauge eigenstates W 3, B, and K all
have Abelian
eld strengths, while non-Abelian vector interactions are inherited from the
SU(2)L gauge boson
eld strength tensor. We will not explicitly write the non-Abelian
vector interactions in the following, but instead understand that they are correspondingly
modi ed when we perform the rescaling needed to canonically normalize the Abelian eld
strengths of the neutral vectors.1
Neutral vector boson mixing
To simplify the Lagrangian in the broken phase, we rst rotate by the tree-level SM weak
mixing angle, which reduces the mass matrix to rank 2 and correspondingly modi es the
kinetic mixing between the Abelian eld strengths. Explicitly, we sandwich R W RTW twice
in eq. (2.4), with
cW = cos W and sW = sin W , which gives
R W = B@
C B A ; SM CA
1We remark that the Stueckelberg mechanism [47, 48] provides an alternative mass generation for K~ ,
which we do not employ here. The collider phenomenology of a dark neutral gauge boson with mass arising
from the Stueckelberg mechanism is presented in ref. [48].
kinetic terms for the neutral gauge bosons, we use the successive transformations
U1 = B@
U2 = BB
2 A
(U1T ) 1(U2T ) 1I3U2 1U1 1 B@ A ; SM CA
2 U1 1 B@ A ; SM CA ;
where the kinetic terms are now canonically normalized and only one further unitary
rotation is needed to diagonalize the mass matrix. We remark that j j < cW is required to
ensure the kinetic mixing matrix in eq. (2.6) has a positive de nite determinant, which
allows U2 to remain non-singular. The nal Jacobi rotation required is
RM = B@
and the upper (lower) sign in tan M corresponds to mZ; SM > mK (mZ; SM < mK ). The
resulting non-zero mass eigenvalues are
2) + m2K )2 + 4m2Z; SMm2K 2t2
and the corresponding neutral vector basis is RMT U 1
mark that these are exact expressions valid for arbitrary .
1, we provide compact expressions for the masses and the corresponding gauge
elds in the mass basis. To O( 3),
m2K~ = m2K +
m2Z~ = m2Z; SM +
B A~ CA = RMT U2 1U1 1 B@ A ; SM CA
= BB
We note that this expansion for
1 is insu cient for mK ! 0 or mK ! mZ; SM. These
two limits are discussed in appendix A. Given that is small, the masses of K~ and Z~ are
altered only at the 2 level.
With the O( 3) expressions for the mass eigenstate vectors with canonically normalized
kinetic terms, we can now write down the corresponding currents associated with the mass
eigenstate vectors:
gZ ; SMJZ + eA ; SMJem + gDK J
= Z~
2m2K m2Z; SM + m4K )cW2 2
Again, the situation for mK ! 0 or mK ! mZ; SM is discussed in appendix A. From these
U(1)D sector correspondingly receives an O( ) dark charge mediated by Z~ .
expressions, we see explicitly that SM fermions, encoded via Jem and JZ , obtain an O( )
electric charge and an O( ) neutral weak charge mediated by K~ . Matter charged in the
The analysis of the scalar sector is simpler and follows previous discussions of scalar Higgs
portals in the literature (see, e.g. [49]). From eq. (2.2) and eq. (2.3), we have
2D =
2H =
2 HP vD2 :
The scalar mass eigenstates are then
is the scalar mixing angle. The scalar masses are
m2S; H0 =
We can thus reparametrize the scalar Lagrangian couplings
mS, mH0 , vD, vH , and
. The reparametrizations for 2 , 2H and
D
HP are given above,
while the reparametrization for D and
H =
D =
m2H0 + m2S + (m2H0
We also calculate the scalar interactions in the mass eigenstate basis H0 and S. The
cubic scalar interactions are
vD sin ) sin 2
+ vH sin ) sin 2 :
free parameters. We will restrict HP > 0 in our analysis, recognizing that HP < 0 and
H D can cause tree-level destabilization of the electroweak vacuum.
Lastly, the scalar-vector-vector interactions of K~ , Z~, S and H0 in the mass basis to
m2K m2Z;SM
K~ K~ H0
K~ K~ S :
bosons is changed not only by cos
coupling proportional to sin
and also 2 cos .
We reiterate that both
are theoretical parameters that must be constrained by
data, and hence a particular hierarchy between
would re ect model-dependent
assumptions. As a result, eq. (2.24) forms a consistent basis for determining the sensitivity
We can characterize the changes in the phenomenology of the Higgs-like H0 state as
a combination of modi ed SM-like production and decay modes and the opening of new
exotic production and decay channels. One main e ect of
is to suppress all of the SM
fermion couplings of the H0 state by cos , while the S state acquires Higgs-like couplings to
SM fermions proportional to sin . This feature also applies to the loop-induced couplings
to gluons and photons for H0 and S. On the other hand, the coupling between H0 to Z~
but also by 2 sin , while the S state acquires a Z~
In addition, if kinematically open, the H0 state can decay to pairs of S or pairs of K~ ,
as possible subsequent decays. These Higgs invisible
decays are also mimicked by the exotic H0 ! Z~K~ decay, when Z~ !
total invisible width of H0 is sensitive to a combination of di erent couplings and masses
in eq. (2.23) and eq. (2.24), further demonstrating the viability of the Double Dark Portal
model as a self-consistent theoretical framework for constraining Higgs observables. We
remark that we have not added a direct Yukawa coupling between
and , e.g. if
charge +2 under U(1)D, we would introduce a direct decay from S to dark matter and also
. As a result, the
split the Dirac dark matter into Majorana fermions [50].
Dark matter interactions
Finally, we will consider the DM interactions with the mass eigenstates of the gauge bosons
and scalars. The main observations can be obtained by recognizing that DM inherits its
couplings to SM particles via the JD current shown in eq. (2.15). Explicitly, the dark
matter particle Lagrangian reads
i @=
Direct detection and indirect detection phenomenology and constraints
The Double Dark Portal model presented in eq. (2.1) o ers many phenomenological
opportunities, including dark matter signals at direct detection, indirect detection, and collider
experiments and modi cations of electroweak precision and Higgs physics at colliders. We
remark that aside from the vacuum stability requirement on
HP and upper bound on j j,
the theory parameter space of the Double Dark Portal model is wide open and subject
only to experimental constraints. This vast parameter space has been extremely useful in
motivating searches for light, hidden mediators at high intensity, beam-dump experiments,
as reviewed in refs. [28, 46].
Our focus, however, is the O(10
its accompanying Higgs partner S, which will both dominantly decay to the dark matter
particle . This is readily motivated by considering m
K~ has an on-shell two-body decay to
< mK~ =2 and gD
, so that
gD, the SM gauge singlet scalar
All of these choices, however, can be reversed to give markedly di erent phenomenology.
pairs of SM charged fermions, as long as it is heavier than 2me. For very small , however,
the K~ lifetime can be long, leading to either displaced vertex signatures or missing energy
signatures. The lifetime and decay length of K~ can be estimated to be
100 GeV) scale for the K~ vector mediator and
= 0:9
L = c
1 GeV
open charge-weighted two-body SM
exotic decay of the SM-like Higgs will be strongly suppressed by multi-body phase space
gives an exotic Higgs decay, H0 ! SS, which is sensitive directly to
Given our mass hierarchy, the dominant collider signature from production of either K~
or S is missing energy from escaping
particles, while the relic density of
in our local dark
matter halo can be probed via nuclear recoils in terrestrial direct detection experiments
or through their annihilation products in satellite indirect detection experiments. We will
discuss the direct and indirect constraints from dark matter searches in the remainder of
this section and focus on the collider signatures for vector and scalar mediator production
in section 4 and section 5.
Dark matter direct detection experiments search for anomalous nuclear recoil events
consistent with the scattering of the dark matter halo surrounding Earth. Direct detection
scattering occurs via t-channel exchange of Z~ and K~ , as evident from the J
D and Jem
interactions shown in eq. (2.15). Because of the relative sign between the K~ and Z~ terms, dark
matter scattering proportional to g g
2 2 2 is naturally suppressed by extra 2 or Q2=m2K
D
factors, where Q is the momentum transfer scale, and the leading contribution is hence
proportional to e2g2 2
. This cancellation between K~ and Z~ mediators is generic, and we
outline the details in appendix B. As a result, the dominant DM-nucleon interaction for
direct detection is mainly from DM-proton scattering. With the SM and DM currents
from eq. (2.15), the DM-proton scattering cross-section is
2 10 GeV
p is the reduced mass of the dark matter
and the proton and e = p4 =137.
The cross-section p is calculated at leading order in
and vin, the incoming DM velocity,
and agrees with previous results when DM only interacts via t-channel K~ exchange with
strength proportional to the SM electromagnetic current [41].
Given that the momentum transfer in the propagator is smaller than gauge boson
all the vector boson contributions, where mV is the smaller of either gauge boson mass.
For DM direct detection, the momentum transfer is about Q2
m2K~ ;Z~, hence
the contribution induced by the JZ current cancels and we arrive at the same result in
insensitive to .
We can also motivate particular contours in the
vs. mK~ plane by considering rst
the requirement that the DM obtains the correct relic density and second the constraints
on the possible rates for DM annihilation to SM particles coming from indirect detection
We rst calculate the annihilation cross sections for
where f denotes a SM fermion. We focus on the region m
< mK~ , since the annihilation
! K~ K~ opens up otherwise and the dominant self-annihilation cross section is
In this setup, the annihilation cross section will be proportional to g2 2
D . We calculate
the annihilation in center of mass frame, and give the annihilation cross sections before
thermal averaging in appendix C. We perform the thermal averaging of the annihilation
cross section numerically according to ref. [51]. The annihilation cross section generally has
are from the s-channel resonant exchange of K~ and Z~, while the last one is due to maximal
mixing between K~ and Z~ when mK is close to mZ, SM, as discussed in appendix A.
gure 1, we show the direct detection constraints in the
vs. mK plane from
experiments LUX [33] with data from 2013 to 2016, PANDAX-II [32], and CRESST-II [31]
0:01, and the dark matter mass xed to 0:2mK , 0:495mK , 0:6 GeV, or 30 GeV.
LUX,CDMS-Lite,CRESST-II
gD = e
gD = 0.1
gD = 0.01
gD = e
gD = 0.1
gD = 0.01
LUX,CDMS-Lite,CRESST-II
gD = e
gD = 0.1
gD = 0.01
gD = e
gD = 0.1
gD = 0.01
LUX,CDMS-Lite,CRESST-II
LUX,CDMS-Lite,CRESST-II
II [31], as well as CDMSlite [30], shown in the
vs. mK plane for various choices of gD and m .
= 0:2mK (top left), m
= 0:495mK (top right), m
= 0:6 GeV (bottom left), and
requirement from the Planck collaboration [34]. Note that mK is approximately the mK~ mass
eigenvalue according to eq. (2.11).
The thermal relic abundance limit on
is given in
gure 1 using the
2 = 0:12
requirement from the Planck collaboration [34]. The dip around mK
mZ, SM re ects
increasing mixing between K~ and Z~. While for mK
is enhanced by the s-channel K~ resonance, thus the required
2m , the annihilation cross section
is very small. When 2m
mK~ < Tf , where Tf
thermal average will not bene t the resonance e ect anymore.
mK~ > 0, the
In recasting the direct detection limits, we recognize that the experiments assume
that the local DM density is
are identically meaningful only when the DDP model parameters give this assumed local
relic density. For other parameter space points, in particular for xed mK and varying
6= relic, with relic corresponding to h v
detection scattering events will be independent of . This is because the predicted local
In our recasting, however, we keep the local DM relic density
xed to 0:3 GeV=cm3
regardless of , in order to determine the sensitivity to the direct detection cross section.
For large , when the local DM relic density predicted in the DDP model is generally
underabundant, extra dark matter particles beyond the DDP model are needed, while for
small , the DM relic density is generally overabundant and extra annihilation channels
are typically needed.
Hence, the direct detection exclusion contours in each panel simply illustrate the
fracrelic density, relative to
constraint. When the DD contours are weaker than the relic density contours, the model
only minimally requires extra inert dark matter to make up the absent relic abundance.
When the DD contours are stronger than the relic density contours, an extra contribution
to the thermal relic annihilation cross section for
is required to satisfy the 0:3 GeV=cm3
assumption, and
is excluded by DD experiments as shown in the red shaded region.
In particular, for xed mK , the strengthening to the annihilation cross section can be
parametrized by the squared ratio of from the blue contour to
at the red contour.
We see that light DM masses are much less constrained, because of the
2 p=m2K~
suppression in eq. (3.3). For m
constraint on p
. We see that for heavy K~ and light , the direct detection sensitivity can
/ mK~ , the sensitivity on
generally follows the experimental
be weak, leaving signi cant parameter space to be probed by colliders. Interesting
parameter space also exists for m
Indirect constraints from CMB, Gamma-ray and e
After the relic abundance constraint, we next consider the constraints from cosmic
microwave background (CMB) observations. Measurements of the CMB generally give
constraints on DM annihilation or decay processes, which inject extra energy into the CMB and
thus delay recombination [52{57]. The constraint is calculated using the energy deposition
yield, fei , where i denotes a particular annihilation or decay channel and fe describes the
e ciency of energy absorption by the CMB from the energy released by DM in particular
channel. The constraint is expressed as
pann =
where the Planck experiment has constrained pann < 4:1
we sum all the SM fermion pair f f and W +W
parameters are plotted in gure 2 as shaded purple regions.
10 28 cm3 s 1 GeV 1 [34], and
channels in annihilation. The excluded
gD = e
gD = 0.1
gD = 0.01
gD = e
gD = 0.1
gD = 0.01
gD = e
gD = 0.1
gD = 0.01
gD = e
gD = 0.1
gD = 0.01
ments from dwarf galaxies [35, 36] and the inner galactic region [37] and e+
ux measurement
from AMS-02 [38]. The constraints are shown in the
vs. mK plane for m
= 0:2 mK (top left),
0:1 (solid), and 0:01 (dashed). Note that mK is approximately the mK~ mass eigenvalue according
to eq. (2.11).
The next constraints we consider are the gamma ray observations from Fermi-LAT and
MAGIC in dwarf galaxies [35, 36]. In ref. [35], Fermi-LAT gives constraints on e+e , +
, uu, bb and W +W
nal states, while in ref. [36], MAGIC has made a combined
analysis with Fermi-LAT and presented constraints on
, bb and W +W . The
computation of constraints on our model is straightforward, since we can calculate each
individual limit on
for each channel and take the most stringent constraint for each m
mass. The excluded parameter region is shaded by cyan in gure 2. Similarly, we consider
the gamma ray constraints from the inner Milky way [37]. This analysis sets conservative
constraints on various SM
nal states by using the inclusive photon spectrum observed by
the Fermi-LAT satellite. We apply their results by calculating the most stringent
annihilation pro le, assuming the Navarro-Frenk-White pro le for the DM density distribution
in the galactic center [58]. We can see in
gure 2 that the constraint from galactic center
region, shaded in orange, is much weaker than that from dwarf galaxies.
The last indirect detection constraint is based on e+ and e
data from the AMS-02
satellite [38]. We use the constraints from ref. [59] to set bounds on various SM
which mainly derive from the observed positron
ux. We again adopt the limits from the
strongest channel to constrain
for each mass parameter choice. We can see the constraint
from AMS-02 is the strongest at the largest m
masses in gure 2.
To summarize, in
gure 2, we see that the CMB constraint is strongest at small m ,
while AMS-02 is strongest at higher m . The constraint from gamma ray observations in
dwarf galaxies is very close to the CMB constraint. Meanwhile, the dips in gure 2 nicely
show the two s-channel resonances of K~ and Z~ as well as the maximal mixing peak between
K~ and Z~.
Collider phenomenology of the Double Dark Portal model and current
constraints from LEP and LHC
In this section, we give an overview of the possible probes of the Double Dark Portal Model
at both lepton and hadron colliders. While many separate searches have been performed
at LEP and LHC experiments in the context of either kinetic mixing or Higgs mixing
scenarios, we highlight the fact that a future e+e
machine must synthesize both e ects in
any given search. Hence, the Double Dark Portal model is a natural framework to study
light, hidden physics at a future e+e
The Double Dark Portal model motivates observable deviations in measurements of
both the SM-like H0 and the Z~ bosons, which test the scalar mixing angle
as well as the
kinetic mixing parameter . Notably, the primary SM Higgsstrahlung workhorse process
for nonzero
Higgs factory, e+e
! Zh, can deviate signi cantly from the SM expectation
or . For instance, nonzero
causes a well-known cos
suppression of the
vertex, but nonzero
gives an additional diagram with intermediate K~ , which
captured by a simple cos
rescaling of the H0Z~ Z~ vertex.
becomes on-shell when mK~ > mZ~ + mH0 . We remark that these e ects are not generically
In gure 3, we show the new possibilities for SM-like and dark scalar Higgsstrahlung
from the intermediate massive vector bosons K~ and Z~. We also show the radiative
return process for e+e
! A~K~ or A~Z~, and the diboson process e+e
! K~ Z~. All of these
processes give di erent signals at a future e+e
machine. If we also consider the
possibility of Z-pole measurements and Drell-Yan processes probing eq. (2.15), then we can
categorize the collider phenomenology of the Double Dark Portal model into four groups:
electroweak precision and Z-pole observables, Higgs measurements, Drell-Yan
measurements, and radiative return processes. We point out, however, that e+e
machines o er
unique opportunities for probing new, light, hidden particles by virtue of the recoil mass
method, which we discuss rst.
e−
e−
vector production and (bottom row) example new decay processes in the Double Dark Portal model
sensitive to the kinetic mixing
and scalar Higgs HP portal couplings. Note Z~, A~, and H0 are the
mass eigenstates corresponding to the SM-like Z, photon, and Higgs bosons, respectively.
Recoil mass method for probing new, light, hidden states
S and K~ masses. This p
new physics signal in the recoil mass distribution.
As long as they are kinematically accessible, both S and K~ can be produced in e+e
collisions in association with SM particles. Hence, even if they decay invisibly, the recoil
mass method can be used to probe the couplings sin
and , according to the
interactions from eq. (2.15) and eq. (2.24).
This is familiar from the leading e+e
sistent with a 125 GeV recoil mass gives a rate dependent only on the H0Z~ Z~
Higgsstrahlung production process, where the reconstruction of the Z~ ! `+` decay
con
We emphasize (see also ref. [60]) that this generalizes to any scattering process at an e+e
machine if visible SM states are produced in association with a new, light, hidden particle.
Moreover, sensitivity to the hidden states S and K~ can be improved by scanning over ps,
where the various production modes of Z~S,
K~ , and Z~K~ can be optimized for the di erent
s adjustment would be immediately motivated, for example, by a
The recoil mass method uses the knowledge that the center-of-mass frame for the e+e
collision is xed to be (ps; 0; 0; 0) in the lab frame, where p
s is the energy of the collider.
Hence, for an invisibly decaying
nal state particle X produced in association with a SM
state Y , four-momentum conservation requires
or equivalently,
EY =
mX =
If there are multiple visible states Yi, this generalizes to
X EYi =
mX =
where (P pi)2 is the total invariant mass of the Yi system. We see that studying the
di erential distribution of EY will show a characteristic excess at a given EY when X is
produced. Identifying this monochromatic peak is formally equivalent to nding a peak in
the recoil mass distribution, but we emphasize that these two distributions reconstructed
di erently at e+e
colliders. Speci cally, the recoil mass distribution uses both the
energy and total four-momentum of each detected SM particle, which the di erential energy
Higgsstrahlung process, the recoil mass method
distribution only requires calorimeter information.
In particular, for the Z~H0, Z~ ! `+`
requires measurements of each individual lepton four-momentum and the event-by-event
invariant mass m``. The resulting di erential distribution also includes o -shell
contributions and interference, giving a smeared peak in the recoil mass distribution whose width
is dominated by experimental resolution and not the intrinsic Higgs width. On the other
hand, in radiative return processes, both the recoil mass distribution and the photon
energy spectrum are only limited by the possible width of the recoiling new physics particle
and the photon energy resolution (see also [61, 62]).
For our studies, we assume both S and K~ have dominant decay widths to the dark
matter , which does not leave tracks or calorimeter energy deposits as it escapes. The
recoil mass technique, however, also readily probes both the K~ and S masses in numerous
production modes, when we produce K~ or S in association with a visible SM
For example, while the SM-like Z~ boson is a canonical choice to study Z~H0 events, we can
use the recoil mass technique in the radiative return process for A~K~ production to identify
the invisible decay of K~ . An even more striking possibility is to use the SM-like Higgs
boson, H0, as the recoil mass particle to probe K~ H0 production.
Modi cations to electroweak precision
We now consider the four categories of collider processes in turn. The rst set of observables
we consider are those from electroweak precision tests. In the Double Dark Portal model,
Z-pole observables will show deviations according to the new decay channel Z~ !
Z~ ! SK~ , sensitive to , shifts in the Z~ mass from the mixing with K~ , and deviations in
the weak mixing angle from the mixing between K~ , A~, and Z~. In particular, identifying
the Z~ mass eigenstate of the DDP as the 91:2 GeV Z boson studied by LEP, measurements
of the Z mass, total width, and the invisible decay to SM neutrinos give strong constraints
and the possibility of exotic decays. For mK~ < 10 GeV, both the visible and invisible
channels can be constrained by various experiments, as reviewed in ref. [28]. We thus focus
on the status and prospects for mK~ > 10 GeV.
LEP-I and LEP-II constraints
At LEP-II, contact operators (4 = 2)e
f were used to test for new physics, analogous
to angular distributions in dijet studies at the LHC. In the e+e
channel, the
corresponding bound on
because of maximal mixing, and again around p
Because the K~ decay width is dominated by K~
suppressed the branching ratio of K~ ! `+` .
constraint on
by matching the coe cient of the contact
m2~ )2 + m2~ 2K~ ). The
K K
is & 20 TeV [63]. Since the majority of the dataset was taken at a xed
is around O(0:1). There is sharper sensitivity for mK
200 GeV from resonant production.
, the sensitivity at resonance is
and gure 7 as \LEP-EWPT."
invisibly, e+e
As mentioned above, the mixing between ZSM and K leads to shifts in the Z~ mass
and couplings to SM fermions, leading to a constraint of
< 0:03 for mK~ < mZ using
a combination of electroweak precision observables [64]. The constraint is weakened for
Recently, the BaBar collaboration has published constraints on dark photons decaying
[29]. Their constraints directly map to our
parameter space and place strong limits on
. 0:001 for masses between 1 GeV to 10 GeV.
These are reproduced in gure 6 and gure 7, and labeled as \BaBar."
Although the canonical SM Higgs production channel e+e
LEP-II, the scalar mixing angle sin
can still be probed by the e+e
tion mode when S is kinematically accessible by LEP-II. For mS < 114 GeV, the
nonobservation of Higgs-like scalar decays constrains sin2
the S ! K~ K~ decay is turned o .
< O(0:01
0:1) [22], as long as
The LEP experiments have also searched for a low mass Higgs in the exotic Z ! HZ?
and H decaying invisibly, which excludes mH < 66:7 GeV if the
! Zh was ine ectual at
invisible branching fraction is 100% [18]. The ZH Higgsstrahlung process is also used
to push the mass exclusion to 114:4 GeV [19{21], although the intermediate mass range
between these two limits are not comprehensively covered. In our model, S ! K~ K~ is
the dominant decay when gD
the production cross section
(Z~S) are hence sin2
sin , and the decay branching fraction Z~ ! SZ~? and
Z~H0 Higgsstrahlung process in
Therefore, these limits apply to S as bounds on sin
in section 5. Note the constraint from the exotic Z~ ! SZ~? decay is much stronger than
gure 8 due to the high statistics of Z decays, and in the
suppressed compared to the SM rate.
and mS, which we will show in gure 8
and BR(K~ !
calculation we accounted for the subsequent decay branching fractions of BR(S ! K~ K~ )
Modi cations to Higgs physics and LHC constraints
H0 ! K~ K~ decays, have also given constraints on
With the era of precision Higgs characterization underway after the discovery of a Higgs-like
boson [65, 66], the ATLAS and CMS collaborations have provided the strongest constraints
on the possible mixing of the SM Higgs boson with a new gauge singlet . In addition,
searches for an invisible decay of the 125 GeV Higgs boson, sensitive to H0 ! SS or
HP , sin , and . The growing Higgs
dataset at the LHC continues to show no signi cant deviations from the SM expectation,
but the current sensitivity of the LHC experiments to our proposed signals is limited.
The most important constraint comes from the search for an invisible decay of the
H0 ! K~ Z~ decay widths at leading order in
125 GeV Higgs, where the Run 1 combination of ATLAS and CMS data constrains BR(h !
0:23 [67, 69]. We highlight, however, that this limit requires that the Higgs is
produced in the Z~H0 and vector boson fusion processes at SM rates, which is violated
in the DDP model. Moreover, in the DDP model, there are two possible direct invisible
decay H0 ! Z~K~
decays, H0 ! SS ! 4K~ ! 8
and H0 ! 2K~ ! 4 , in addition to the possible exotic
, which is often semi-visible. The H0 ! SS, H0 ! K~ K~ and
(H0 ! SS) = gD2 sin2
(H0 ! K~ K~ ) = gD2 sin2
(H0 ! K~ Z~) =
4m2S (m2H0 + 2m2S)2
The rst two decay widths are proportional to mK2 while the last one is proportional to
m2K , therefore the last one is usually much smaller comparing with the rst two when mK
is light. In the Higgs invisible studies, the experiments will constrain the rate for Higgs
invisible decays in the DDP model,
BR(H0 ! inv)
(H0 ! SS)BR2(S ! K~ K~ )BR4(K~ !
(H0 ! K~ K~ )BR2(K~ !
) + (H0 ! K~ Z~)BR(K~ !
BRienv =
= cos2
H0; tot = cos2
widths K~ ! f f , W +W
h; SM + (H0 ! SS) + (H0 ! K~ K~ ) + (H0 ! K~ Z~). The decay
can be found in the appendix of ref. [70], while the decay width
) =
The prediction for the invisible decay branching fraction of the 125 GeV Higgs is shown
in the left and middle panels of gure 4 in the sin
plane for mS = 50 GeV, mK =
mS = 50 GeV
mK = 20 GeV
gD = 0.01
mS = 3 mK
mS = 50 GeV
mK = 20 GeV
gD = e
in the sin
(Right panel) Exclusion regions in the sin
vs. mK plane from the search for an invisible decay of
the 125 GeV Higgs by ATLAS and CMS giving BRinv < 0:23 [67, 69], and projected reach from a
future e+e
machine giving BRinv < 0:005 [1{4].
refs. [67, 69], while the prospective sensitivity of BRinv < 0:005 is adopted from the estimate
limit can be lowered in combined ts, with more luminosity, or with other assumptions
about detector performance to the O(0:001) level [1, 2, 4].
In gure 4, we see that when gD is large, the sensitivity to sin
is much stronger than
, because the decay widths for H0 ! SS and H0 ! K~ K~ are much larger than H0 ! K~ Z~
due to light mK , as discussed previously. More importantly, for large gD, BR(S ! K~ K~ )
) are close to 100%. When gD < sin
or gD < , BR(S ! K~ K~ ) and
) will both be subdominant and result in the decrease of BRienv as in the
middle panel of gure 4. As shown in the right panel of gure 4, the constraint on sin
from invisible Higgs decays can be relaxed by making gD smaller.
These exotic decays can also give fully visible and semi-visible signatures [8, 71{75]
when the K~ or S particle decays to SM
nal states, which provide additional handles
for Higgs collider phenomenology. Those references concentrate on the scenario where
the decay to SM
nal states are dominant, e.g. mK~ < 2m . Therefore, such constraints
should be modi ed by the relevant branching ratios in the DDP model because the DDP
model includes a DM decay. If visible decays of K~ dominate, though, the search for
h ! 2a ! 4 [76] constrains the process H0 ! K~ K~ , and bounds 0HP . 0:01 for mK~ .
10 GeV [8], while the bound strengthens to 0HP . 0:001 [8] for mK~ & 10 GeV from
recasting the di erential distributions in 8 TeV h ! ZZ
! 4` data [77]. The coupling
0HP =
HP m2H0 =jm2H0
m2Sj, is roughly the same as
HP if mS is not close to or much
larger than mH0 . The high luminosity LHC (HL-LHC) with 3 ab 1 of 14 TeV luminosity
is expected to be sensitive to 0HP . (few)
the mK~ mass.
decay H0 ! Z~K~ , which gives sensitivity to
data is weak, with the strongest sensitivity for mK~
improvement at the HL-LHC is expected to reach
10 5 in this same channel, depending on
The four lepton
nal state has also been used to constrain the exotic
from (2.24). The current bound using 8 TeV
30 GeV giving
. 0:05, while the
. 0:01 [8]. These gains are mainly
limited by the statistics a orded by Higgs production rates. We remark that for very small
SM states, which provides a new set of challenges to trigger and detect at colliders. Current
exclusions and future prospects for displaced decays can be found, e.g., in refs. [8, 28, 46].
Modi cations to Drell-Yan processes
The K~ decay to SM
forbidden or gD
nal states can be dominant, if the decay to DM pairs is kinematically
. In this case, at the LHC, the Drell-Yan process pp ! Z~; K~ ! `+`
Drell-Yan data, while high mass K~ can be probed at the
can be used to constrain the kinetic mixing parameter , since this process has been studied
with exquisite precision by the ATLAS and CMS experiments. Both ATLAS and CMS
have searched for dilepton resonances at high mass, mK~ & 200 GeV, using 20 fb 1 of
8 TeV data [78, 79], which restricts
. 0:01 at mK~ = 200 GeV and weakens to
7 TeV data by CMS [82] and the corresponding sensitivity using 8 TeV data gives
stronger than the current electroweak precision constraints [8, 81].
The HL-LHC is expected to constrain
. 0:001 for mK~ between 10 and 80 GeV using
0:002 level for mK~ = 200 GeV
high mass dilepton resonances by ATLAS [83] also constrains the kinetic mixing parameter
. 0:04 for mK~ > 100 GeV [84], but this result is hampered by the small statistics.
We see that as long as K~ has an appreciable branching fraction to SM
nal states, in
particular leptons, the Drell-Yan process at the LHC and HL-LHC will provide stronger
sensitivity to
1, the decay
is dominant and the situation reverses, and then Drell-Yan constraints will not
compete with the electroweak precision observables. After rescaling by the appropriate
visible branching ratio, we plot these Drell-Yan constraints as the \LHC-DY" contours
in gure 6 and
Radiative return processes and dark matter production at the LHC
The radiative return process, e+e
s colliders by using an extra radiated photon to conserve four-momentum. At
hadron colliders, since the colliding objects are composite, dark matter production via
radiative return is more commonly known as monojet or monophoton processes, recognizing
the fact that the partonic center of mass energy is not constant on an event-by-event basis.
As a result, the visible decays K~ ! `+`
discussed in the Higgs to four leptons and the
Drell-Yan contexts are complemented by the LHC searches for dark matter production in
monojet and monophoton processes. We remark that in our DDP model, we will assume
in the above visible decay rates. Both ATLAS and CMS have searches for dark matter
production using 8 TeV data [85, 86], sensitive to mediator masses as low as 10 GeV [85].
The corresponding 13 TeV searches [39, 40] have yet to achieve the same sensivity at low
masses. In the DDP model, the K~ mediator is produced on-shell and decays dominantly
, and calculating the results for on-shell mediator production at the LHC, we obtain
X, enables on-shell production of new particles
. 0:07, similar to previous studies [87{89]. It is also possible to search for the dark
bremsstrahlung of K~ from the DM pair [90, 91], as a probe of , although these rates are
negligible in our model.
Prospects for future colliders
We have established that signi cant room remains to be explored in both the
portal couplings. We will now demonstrate that a future e+e collider, currently envisioned
HP
as a Higgs factory, will have leading sensitivity to probing both couplings simultaneously
through the production of new, light, hidden states K~ and S. The primary motivation
expected (e+e
250 GeV center-of-mass energy of such a collider is to optimize the
! Zh) SM Higgsstrahlung cross section, taking into account the possible
polarization of the incoming electron-positron beams. Such high energies, however, also
enable production of the new states K~ and S from radiative return processes, exotic Higgs
decays, and exotic Higgsstrahlung diagrams.
A few di erent variations exist for next-generation e+e
machines, namely the
International Linear Collider (ILC) [2], an e+e
Future Circular Collider (FCC-ee), which shares
the physics we discuss will only depend very mildly on the particular p
strong overlap with TLEP [3], or a Circular Electron-Positron Collider (CEPC) [4]. Since
s of the future
machine and possible polarization of the incoming electron and positron beams, we will adopt
beams as our reference machine
we also show future expectations for a possible p
s = 500 GeV machine with L = 5 ab 1
total integrated luminosity. Our work will complement and extend previous
sensitivity estimates made for various speci c collider environments, which we review rst.
Electroweak precision tests, Higgsstrahlung, and invisible Higgs decays at
future e+e
Because the SM Z and the dark vector K mix, the Z~ mass eigenstate develops a new
invisible decay channel, Z~ !
be accurately measured at a future e+e
, if kinematically allowed. This invisible decay width can
machine. At FCC-ee, for example, running on
the W +W
threshold using the radiative return process e+e
complemented by additional runs at p
s = 240 GeV (Lint = 10:44 ab 1) and p
(Lint = 15:2 ab 1),
s = 350 GeV
uncertainties, down to
0:001 [92, 93]. This leads to the possible constraint
this constraint because of theory uncertainties on the small-angle Bhabha-scattering cross
section remain too large [92]. This constraint also applies to the Z~ ! K~ S ! 6
decay, if mK~ + mS < mZ~. For a lepton collider running on the Z-pole, though, other
electroweak precision observables will have greatly enhanced precision. The combination
of improved electroweak precision observables can constrain
although the constraint is much weaker for mK~ > mZ~ [8].
. 0:004 for mK~ < mZ~,
Higgs factory is expected to have a precision measurement of the
Higgsstrahlung process e+e
! Zh, with accuracies ranging from O(0:3%
0:7%) expected,
10 ab 1 of luminosity [3, 4, 94]. These rates imply that the scalar mixing angle is
probed to sin
0:084, simply from the observation of the Higgsstrahlung process.
Aside from precision Higgs measurements, a future e+e
machine will have leading
sensitivity to an invisible decay of the 125 GeV Higgs. As reviewed in subsection 4.3, the
current constraint on the Higgs invisible decay branching ratio is BRinv < 0:23 [67, 69],
while the limit at FCC-ee is expected to be BRinv < 0:005 [3]. We have discussed the two
and H0 ! K~ K~
! 4 , as well as the irreducible
in subsection 4.3. We also show the corresponding sensivity
signal from H0 ! Z~K~ !
in the sin
vs. and sin
main invisible decays, H0 ! SS ! 8
vs. mK~ planes in gure 4. We again emphasize that these limits
can be stronger or weaker because of their dependence on gD, as seen in gure 4.
Production of new, light states at future e+e
As outlined in section 4 and shown in gure 3, many new possibilities open up for production
of new, light, hidden particles in the Double Dark Portal model. We can classify the new
physics processes into vector + scalar, radiative return, and massive diboson production
topologies. For small and scalar mixing angle , the leading processes are:
tional contribution from intermediate K~ that can interfere with Z~ exchange.
! Z~H0 : the usual Higgsstrahlung diagram is suppressed by cos2 , with an
addi! Z~S : this new process can be probed by the usual recoil mass method for
wellreconstructed Z~ decays, studying the entire recoil mass di erential distribution.
! K~ S : this exotic production process involves two non-standard objects, and is
dominantly produced via K~ , with a rate proportional to 2 cos2 . Since K~ and S
dominantly decay to dark matter, though, we would require an additional photon or
a visible decay of K~ or S in order to tag the event.
! A~K~ : the radiative return process produces K~ in association with a hard photon
, giving direct sensitivity to . We remark that the K~
been studied for mK~ between 10 GeV to 240 GeV at an p
decay has
s = 250 GeV machine with
10 ab 1 [10], giving
10 4, if the decay K~ ! `+` is assumed dominant.
! Z~K~ : the massive diboson pair production process also provides direct sensitivity
to , but measuring the rate precisely will pay leptonic branching fractions of the Z~.
! H0K~ : this very interesting scenario can be probed by using the 125 GeV SM-like
Higgs as a recoil candidate for the K~ heavy vector. The total rate gives sensitivity to
and highlights the power of considering the SM-like Higgs as a signal
probe for new physics.
Having identi ed the main production modes for the K~ and S states, we can match
them to decay topologies illustrated in the bottom row of gure 3. We also include the
underexplored decay H0 ! K~ Z~, which gives an exotic decay of the SM-like Higgs into the
SM-like Z~ boson and the hidden photon K~ sensitive to . As mentioned in section 3, we
focus on S ! K~ K~ ! 4 and K~ !
, and thus the dark portal couplings must be tested
by recoil mass techniques or mono-energetic photon spectra searches. We also demonstrate
the importance of these missing energy searches by explicitly considering leptonic decays
of K~ in the Z~K~ and
K~ processes as well as the fully inclusive recoil mass distribution
K~ production.
Since the workhorse SM Higgsstrahlung process has been studied extensively [2{4], we
use these previous results to recast the sensitivity for
. We also ignore the K~ S
production mode, since the dominant signature has nothing visible to tag the event. The
H0K~ process is interesting to consider for future work, but it requires optimizing the H0
decay channel to gain maximum sensitivity to the recoil mass of the rest of the event.
This leaves the Z~H0, Z~S,
K~ and Z~K~ processes as new opportunities to revisit or
study. We simulate each process using MadGraph5 v2.4.3 [95], Pythia v6.4 [96] for
showering and hadronization, and Delphes v3.2 [97] for detector simulation. Detector performance
parameters were taken from the preliminary validated CEPC Delphes card [98].
Backgrounds for each process are generated including up to one additional photon to account
for initial state and
nal state radiation e ects. Events are required to pass
preselection cuts of j j < 2:3 for all visible particles, while photons and charged leptons must
have E > 5 GeV, jets must have E > 10 GeV, and missing transverse energy must satisfy
missing energy signatures.
with new particle production
Z~K~ production
by g2 = 2. We also study Z~ ! `+`
D
The cross section for Z~K~ production is shown in gure 5 for various choices of mK and
. We see that Z~K~ production grows with 2, as expected. We consider both K~ ! `+`
with the SM branching fraction of 6:8% [99]. We
show the background cross sections for the corresponding 2`2
nal states after
the preselection cuts described in subsection 5.2 in table 1. The 2`2 background includes
a combination of Z
attributed to ZZ=Z
, ZZ and W +W
processes, while the 4` background is mainly
mZ j < 10 GeV and then
look for a peak in the recoil mass distribution, see eq. (4.2). For the 4` nal state, we identify
the Z-candidate from the opposite-sign, same- avor dilepton pair whose invariant mass is
closest to the Z mass and then study the invariant mass distribution of the remaining
dilepton pair. The K~ signal is tested for each signal mass point in the corresponding mass
distributions, and we draw 95% C.L. exclusion regions for each channel in the
between the 2`2
nal states is xed by choosing gD = e
0:3. We see that the
fully visible 4` nal state performs worse than the 2`2 signal selection, simply re ecting
the dominant signal statistics in the missing energy channel.
s =250 GeV
s =500 GeV
ϵ = 0.01
gD = 0.1
mK [GeV]
s =250 GeV
s =500 GeV
mK = 70 GeV
gD = 0.1
panel) . Solid lines correspond to e+e
beams, while dashed lines correspond to p
machines operating at p
s = 250 GeV with unpolarized
using eq. (2.11).
! A~K~ , for K~ !
, `+` , and inclusive
A~K~ production
We study the radiative return process, e+e
decays. While each search will use the same observable, namely a monochromatic peak in
the photon energy as in eq. (4.1), the di erent contributions of SM backgrounds in each
event selection will result in the best sensitivity for the K~ !
and signal regions are shown in table 1. For the inclusive decay of K~ , the background f f
decay. Background rates
is generated where f is a SM fermion, including neutrinos. As mentioned in subsection 5.2,
the visible energy distribution is technically equivalent to the recoil mass distribution, and
this equivalence is sharpest when the visible SM state is a single photon.
From the results in gure 6 and gure 7, we see that the most sensitive decay channel
, again re ecting the dominant statistics in this
nal state and the a ordable
The single photon in the background
generally comes from initial state radiation
and hence tends to be soft except when produced in the on-shell Z~ !
process. As
thus we have a
at sensitivity to
sensitivity. The rst is for mK
due to maximal K~
Z~ mixing, and the second is for mK
ps, when the production is
when mK < mZ~. There are two spikes in
mZ~, when the production cross section is greatly enhanced
enhanced by soft, infrared divergent photon emission. Note the exclusion can only reach
5 GeV because of the preselection cut on the photon energy.
Z~H0, H0 ! K~ Z~ exotic decay
The next process we consider is the exotic Higgs decay, H0 ! K~ Z~, with K~ !
! `+` . This Higgs exotic decay partial width, from eq. (4.6), is proportional to
K~ inclusive decay
0.929 (250 GeV)
0.545 (500 GeV)
0.055 (250 GeV)
0.023 (500 GeV)
23.14 (250 GeV)
8.88 (250 GeV)
12.67 (250 GeV)
4.38 (500 GeV)
3.45 (250 GeV)
2.92 (500 GeV)
0.87 (250 GeV)
0.87 (250 GeV)
mK~ j < 2:5 GeV
mZj < 10 GeV,
mK~ j < 2:5 GeV
mK~ j < 5 GeV
and E= > 50 GeV
mK~ j < 2:5 GeV
mSj < 2:5 GeV
given in the main text.
along with the most salient cuts to identify the individual signals. All background processes include
up to one additional photon to account for initial and
nal state radiation. Background rates are
2 cos2 , as long as mK~ . 34 GeV and sin
candidates balancing an invisible K~ particle, which we identify from the peak in the recoil
mass distribution. Our event selection cuts, summarized in table 1, require two pairs of
opposite sign and same avor lepton with invariant masses in a window around mZ~, and
the recoil mass from the four visible charged leptons should be in a window around the test
variable mK~ . The resulting sensitivity, as seen in
with the other K~ production processes, given the limited Higgs production statistics and
the suppression of the small leptomic decay branching ratio of Z~. We remark that this
decay can also be probed via H0 ! invisible searches using the SM rate for Z !
gure 6 and gure 7, is not competitive
is neglected. The signal process thus has 2 Z
was discussed in subsection 4.3.
Z~S production
Lastly, we can also probe the scalar mixing angle sin
in Z~S production. This search is
exactly analogous to the previous search at LEP-II for a purely invisible decaying Higgs [18],
where the visible Z~ ! `+`
Z~S cross section is proportional to sin2
decay is used to construct the recoil mass distribution. The
if we neglect , and
Z~S is shown in gure 8 for
K˜˜Z→2 2
LEP-EWPT
gD = e
K A inclusive
LHC-DY (HL-LHC)
mK[GeV]
s = 250 GeV
operator search, LEP electroweak precision tests (LEP-EWPT), BaBar K~ invisible decay search
machine data. Dashed lines indicate existing limits from the LEP e e+ ! ` `+ contact
(BaBar) and LHC Drell-Yan constraints (LHC-DY). The 3 ab 1 HL-LHC projection for
DrellYan constraints is also shown as a solid line. Note mK is approximately the mK~ mass eigenvalue
according to eq. (2.11).
= 0:1 and 0:01 at p
s = 250 GeV and p
and S decaying invisibly. The signal region is summarized in table 1 and
light mK using L = 5 ab 1 luminosity for p
focuses on selecting a dilepton Z candidate and reconstructing the recoil mass distribution
to identify the S peak. From this analysis, we nd that sin
= 0:03 can be probed for
would signi cantly improve on the current global t to Higgs data by ATLAS and CMS,
which constrains sin
< 0:33 [68]. This sensitivity also exceeds the projected LHC reach
< 0:28 (0:20) using 300 fb 1 (3 ab 1) data and critical reductions in theoretical
uncertainties [1]. We remark that improved sensivity can be obtained by varying the p
of the collider to maximize the (e+e
! Z~S) rate for the test S mass (see also ref. [60]).
K Z→ 2
LEP-EWPT
gD = e
mK[GeV]
LHC-DY (HL-LHC)
We summarize the sensitivity to
in di erent channels at a future e+e collider running
s = 250 (500) GeV with L = 5 ab 1 in
gure 9, and we compare the collider searches
with constraints from direct detection and indirect detection experiments. In
gure 9, the
dark green shaded region is the exclusion limit from the strongest of the e+e
searches presented in
gure 6 and gure 7. We also show the strongest limit from direct
detection and indirect detection experiments from
gure 1 and
gure 2, as well as the
contour satisfying the correct dark matter relic abundance measured by Planck [34]. While
the constraints from dark matter detection experiments depend sensitively on the dark
matter mass, the collider prospects are insensitive to the dark matter mass, as long as
the decay to
is kinematically allowed and gD
. We note that for mK around mZ~,
the best limit comes from the inclusive A~K~ search, which is insensitive to gD, while for
mK larger or smaller than mZ~, the best sensitivity comes from the monochromatic photon
search with E= .
On the other hand, the indirect detection sensitivity and the relic abundance contour
both change signi cantly with dark matter mass. When m
= 0:495mK , the dark matter
resonantly annihilates, improving the reach for indirect searches and dramatically lowering
s =250 GeV
s =500 GeV
gD=0.01
ϵ=0.001
50 100 150 200 250 300 350 400
mS [GeV]
H0 current global fit (LHC)
14 TeV, 300 fb-1
14 TeV, 3 ab-1
SZ→ ∄2 , 5 ab-1
s =250 GeV
s =500 GeV
mK < 0.5 mS
gD = e
ϵ= 0.001
using 5 ab 1 of e+e
Z~S, Z~ ! `+` search in the recoil mass distribution for invisible S decays in the sin
500 GeV as a function of mS, with sin
vs. mS plane
< 0:33 [68], future LHC projections of 0.28 (0.20) using 300 fb 1 (3 ab 1) luminosity [1],
and precision
(Zh) measurements constraining 0.084 (0.055) using 5 ab 1 (10 ab 1) [3, 4, 94].
We plot the excluded region from LEP searches for invisible low mass Higgs in ZS channel in
s = 250 GeV and
e-e+ Collider, 5 ab-1
s =250 GeV
s =500 GeV
gD = 0.01
LEP-EWPT
ID
s =250 GeV
s =500 GeV
gD = 0.01
searches (blue) in the
vs. mK plane. We choose gD = 0:01, m
= 0:2mK (left panel) and m
0:495mK (right panel). We also show the contours when
satis es the relic density measurement by
the Planck collaboration [34] as black dashed lines. The collider constraint is adapted from
and gure 7, taking into account the changes in the K~ branching fractions. We also include existing
constraints from LEP electroweak precision searches (LEP-EWPT) and the BaBar search for the
K~ invisible decay (BaBar).
the required
to satisfy the relic density measurement. During thermal freeze-out, the
nite temperature of the
velocity distribution gives a strong boost to the annihilation
cross section, and thus only very small
is needed. For m
= 0:2mK , however, the limits
from indirect detection exclude the relic abundance contour, and the parameter space is
instead characterized by an overabundance of the dark matter relic density. For this region
to satisfy the Planck bound, additional mediators or new dark matter dynamics controlling
the freeze-out behavior are needed. Direct detection experiments also lose sensitivity to
dark matter signals for light m , since the nuclear recoil spectrum is too soft to pass the
ducial energy threshold. In addition, the decreasing sensitivity for heavy m comes from
the fall o in the scattering cross section scaling as 2 p=m2~ , see eq. (3.3).
K
We also emphasize that the collider constraint is not sensitive to varying gD as long as
, which ensures the invisible decay of K~ dominates. Hence, the collider constraints
gure 6 and gure 7 and gure 9 are essentially unchanged, since changing gD from e
to 0:01 does not signi cantly change the invisible branching fraction, except for the trade
o between inclusive K~ decays and invisible K~ decays around mK
mZ~. On the other
hand, the direct detection and indirect detection rates scale with gD2, and thus collider
searches will have better sensitivity for small gD.
gure 9, we see that the prospective collider limits, corresponding to the radiative
return process e+e
! A~K~ , are expected to overtake the current bounds from direct
detection and indirect detection experiments. In the case where dark matter mass is light,
than the current limits, especially in the high mass region, and hence out of the reach of
next generation 1-ton scale direct detection experiments. For dark matter close to half
the mediator mass, m
o ers an attractive target parameter space for experimental probes. The projected e+e
sensitivity exceeds the current experimental sensitivity around mK
10 GeV and mK >
100 GeV, and while improvements in the dark matter experiments will also challenge the
open parameter space for mK
10 GeV, the striking sensitivity of e+e
radiative return
processes for mK > 100 GeV is expected to be unmatched. Thus, results from a future
collider will both complement and supersede the reach from dark matter searches,
stemming from its ability to produce directly the mediators of dark matter interactions.
Conclusion
We have presented a comprehensive discussion of the phenomenology of the Double Dark
Portal model, which addresses the simultaneous possibility of a kinetic mixing
with a scalar Higgs portal . We emphasize that these Lagrangian parameters are generic
in any U(1) extension of the SM when the additional gauge symmetry is Higgsed. An
additional motivation for considering such a U(1) extension is the fact that such a symmetry
readily stabilizes the lightest dark sector fermion , making this model a natural framework
to study possible dark matter interactions in tandem with updated precision Z and Higgs
constraints anticipated at future colliders. This study also demonstrates the ability of a
future e+e
matter and LHC experiments.
machine to produce new particles, which are not probed with the current dark
decay, K~
largely unexplored.
We work out the interactions in the mass eigenstate basis of neutral vector bosons
and Higgses. The direct detection limits for this model have been studied, along with
indirect detection constraints from CMB measurements, gamma ray measurements, and e
measurements, where we have explored both the non-resonant and resonant dark matter
parameter regions. For collider constraints, we discussed the existing bounds from by
electroweak precision and Z-pole observables, Higgs measurements, Drell-Yan measurements,
and radiative return processes. Previous constraints have mostly focused on the visible
! `+` , and leaving the prospects and sensitivity estimates for the invisible
We studied both the Higgs bremsstrahlung and radiative return processes for a future
e+e collider, emphasizing that a future lepton collider not only has vital Higgs precision
capabilities but also new possibilities for producing light new particles, K~ and S. Since
both K~ and S decays are dominantly invisible, the recoil mass method a orded by an e+e
machine is crucial. We also highlight that the recoil mass method can be simpli ed to a
monochromatic photon study in the case that the new particle is produced in the radiative
return process, which simpli es the search procedure and enhances the importance for
upcoming calorimeters to have a precise, high-resolution energy determination for photons.
The various Higgsstrahlung and radiative return processes we study are listed in table 1,
and we obtain the best sensitivity on
Higgsstrahlung process Z~S, respectively. from the radiative return process A~K~ and
In comparing
prospects, we analyzed the future collider reach with direct
detection, indirect detection and relic abundance sensitivities. The collider prospects are less
a ected by DM mass m , and surpass the other experimental probes for small gD. Since
K~ decays invisibly, the most relevant current constraints are from electroweak precision
measurements and LHC mono-jet searches, but they are not as strong as the radiative
return process A~K~ reach. Therefore, a future e+e
complementary sensitivity test of the DDP model.
collider provides an important and
projection is weaker than the direct Z~S search.
For sin , the best constraints come from studying the singlet bremsstrahlung process
Z~S, the Higgs invisible decay rate, and precision measurements of SM Higgs production
rates. We studied the Z~S process with S decaying invisibly for a future e+e
and estimated the sensitivity to be sin
0:03. This compares favorably with earlier
LEP studies for light mS, and readily provides leading sensitivity for heavy S. We also
recasted bounds using the Higgs invisible decay channel, where the current LHC constraint
is BRinv < 0:23 [67, 69] and the future e+e
collider reach is BRinv < 0:005 [3]. In the
DDP model, these bounds simultaneously constrain the three exotic processes, H0 ! SS,
H0 ! K~ K~ , and H0 ! Z~K~ when Z~ decays to neutrinos. While the constraints on sin
can be strong, these limits also depend sensitively on gD and are insigni cant for small
gD. The future
! Zh) precision measurement readily constrains cos2 , but this
In summary, the Double Dark Portal model predicts new dark sector particles, K~ ,
, whose vector and scalar portal interactions with the Standard Model can be
uniquely tested at a future e+e collider. We explicitly propose and study radiative return
and Higgsstrahlung processes to nd the invisible decays of the K~ and S mediators. An
additional bene t of the e+e search strategies discussed in this work is that, in the event
of a discovery, the K~ or S mass is immediately measured in the recoil mass distribution.
Hence, a future e+e
collider not only has exciting prospects for determining the precise
properties of the 125 GeV Higgs boson, but also has a unique and promising new physics
program founded on the production of new, light, hidden particles.
Acknowledgments
This research is supported by the Cluster of Excellence Precision Physics, Fundamental
Interactions and Structure of Matter (PRISMA-EXC 1098). FY would like to thank the
hospitality of the CERN theory group while this work was being completed. The work of
JL and XPW is also supported by the German Research Foundation (DFG) under Grants
No. KO 4820/1-1, and No. FOR 2239, and from the European Research Council (ERC)
under the European Union's Horizon 2020 research and innovation program (Grant No.
637506, \ Directions").
Two limiting cases for K~ , Z~, and A~ mixing
From subsection 2.1, we decompose the gauge eigenstate vectors into their mass eigenstate
components according to
B A ; SM CA = U1U2RM B@ A~~ C ;
@ A
where the expressions for U1, U2 and RM have been given in eq. (2.7), and eq. (2.9),
respectively. We will consider the two limiting cases, mK ! 0 and mK ! mZ;SM, and
study the corresponding changes for the kinetic and mass mixing matrices.
For mK ! 0, the gauge boson masses are
mA~ = mK~ = 0 ;
m2Z~ = m2Z; SM
and the eld rede nition is
U1U2 = BB
The Jacobi rotation RM , from eq. (2.9), is now ill-de ned in the lower right two-by-two
block, since A~ and K~ can be rotated into each other keeping both the kinetic terms and
masses unchanged. This simply re ects the residual unbroken U(1)em
metry. For RM = I3, the currents are
the SM fermions will generally have nonzero charges mediated by both A~0 and K~ 0, leading
to photon and dark photon-mediated electric and dark millicharges.
For mK ! mZ; SM, the masses of the three vector bosons are
mA~ = 0;
m2K~ ; Z~ = m2Z; SM
and the eld rede nition required, to O( 2), is
U1U2RM = p
1 B
where the top and bottom signs correspond to mK ! mZ; SM. We see that the mixing
between Z and K is nearly maximal, 45 , while the discontinuous behavior for mK below
and above mZ; SM re ects the level crossing in the mass eigenvalues. We remark that as long
mass matrix in eq. (2.8). If
becomes ill-de ned and the maximal mixing feature is lost.
Cancellation e ect in multiple kinetic mixing terms
We observe that the Z~ and K~ mediated couplings in eq. (2.15) show a cancellation e ect
when mediating DM interactions with SM fermions. This feature can be generalized to
the situation with multiple U(1) gauge groups with multiple kinetic mixing terms between
each other. Explicitly, we analyze the Lagrangian
where Kab = ab + O( )(1
ab) is the kinetic mixing matrix and M 2 is the diagonal mass
matrix, with a, b as indices. Then, we de ne the
eld rede nition matrix U such that
1 V T M 2V ;
i = giUikV~k; Ji ;
in the mass basis. As a result, scattering rates between two currents Ja and Jb (which
represent the corresponding fermion bilinears) are schematically
(gbJb ) UakUbk
q q =m2~ #
Uak(U T )kb, because these transformations are controlled by the diagonalization requirement
Annihilation cross sections
In this section, we present the annihilation cross sections for the processes
W +W , where f is a SM fermion. We focus on the case with m
the direct annihilation of dark matter to dark vectors K~ K~ opens up and does not depend on
. In this setup, the annihilation cross section is proportional to gD2 2. The diagrams include
s-channel K~ and Z~ exchange. The annihilation cross sections before thermal averaging are
< mK~ , since otherwise
48 s3=2c4W
m2~ )2 +m2~m2K~ (mK~ Z~
K Z
12c2W (s+2m`2)m2~ s(m2~
Z Z
m2~ )2 +mZ~mK~ ( mK~ Z~ +mZ~ K~ )( mZ~ Z~ +mK~ K~ )
K
! uu) =
144 s3=2c4W
17s+7m2u
m2~ )2 +m2~m2K~ (mK~ Z~
K Z
40c2W (s+2m2u)m2~ s(m2~
Z Z
m2~ )2 +mZ~mK~ ( mK~ Z~ +mZ~ K~ )( mZ~ Z~ +mK~ K~ )
K
! dd =
144 s3=2c4W
m2~ )2 +m2~m2K~ (mK~ Z~
K Z
m4K~ + m2Z~(m2Z~ +
2mZ~mK~ Z~ K~ + m2~ ( 2m2Z~ +
K
! ` `) =
m2~ )2 + m2~m2K~ (mK~ Z~
K Z
96 s3=2c4W
2mZ~mK~ Z~ K~ + m2~ (m2K~ +
K
where s is the Mandelstam parameter for the center-of-mass energy squared.
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