Invisible Z′ and dark matter: LHC vs LUX constraints

We consider a simple, yet generic scenario in which a new heavy Z′ gauge boson couples both to SM fermions and to dark matter. In this framework we confront the best LHC limits on an extra gauge boson Z′ to the constraints on couplings to dark matter from direct detection experiments. In particular we show that the LHC searches for resonant production of dileptons and the recent exclusion limits obtained by the LUX collaboration give complementary constraints. Together, they impose strong bounds on the invisible branching ratio and exclude a large part of the parameter space for generic Z′ models. Our study encompasses many possible Z′ models, including SSM, E6-inspired or B-L scenario.

The paper is organized as follows. After a summary of the model under consideration and its effective low energy limit, we establish in section II the relation between the invisible branching ratio of a Z and the direct detection cross sections (spin-dependent and spinindependent). We combine the exclusion limits set by the direct detection experiments with the LHC bounds on Z mass and couplings in section III. Some prospects are discussed in the last section.
2 Invisible Z decay rates and direct detection cross sections

The model
While the existence of extra U(1) gauge symmetries may be motivated in various ways [1], we will try to be as generic as possible in the present work. To do so, we parametrize as follows the couplings of a Z gauge bosons to the SM quarks and leptons on one hand and to a dark matter particle χ on the other hand, which we take to be fermionic for concreteness, 1 Here V f D and V χ D (A f D and A χ D ) are the vectorial (axial) couplings for the SM fermions f and DM particle χ respectively (V χ D = 0 if χ is a Majorana particle) and we have introduced an overall gauge coupling g D . Since the gauge coupling g D , as well as V f,χ D and A f,χ are a priori free parameters, our formalism can then describe all the possible Z realisations in terms of this five quantities. It is anyway possible to fix the combinations g D V f D and g D A f D by specifying the Z model (e.g. SSM, E 6 , B-L, . . . ) as it is shown in table I. The partial widths of the Z can be straightforwardly determined as function of these four fundamental parameters as: where i = f or χ, c f is the number of color of the SM fermion f and c χ = 1. From the microscopic Lagrangian (2.1), we can deduce the effective low energy interaction of the dark matter to a nucleon. The XENON [30] or LUX [31] experiments are supposed to observe dark matter collisions on nuclei, which are translated as couplings to nucleons. These processes occur at velocities of the order of magnitude of that of the Sun in the galactic halo, i.e. 300 km/s so that the energy transferred in such collisions is JHEP03(2014)134 Figure 1. Feynman diagrams contributing to the width of the Z boson and the scattering cross section of the dark matter on a nucleon.
below the MeV scale for typical dark matter masses. We can thus integrate out the Z from the Lagrangian (2.1) to get the effective Lagrangian describing interactions between a χ and, say, a proton p, with a similar expression for the neutron. Here ∆ p i is the spin content of the quark i in the proton, which may be extracted from lepton-proton scattering data. The values we use are taken from the latest determination of the light quark contributions [41] As already discussed above, the combinations g D V f D and g D A f D are fixed in a given fundamental theory. For illustration, we report in table I their values in some popular realizations of Z models, while a more exhaustive list can be found e.g. in [2].
For the sake of concreteness, we will mainly in this paper refer to the so-called sequential model (noted SSM in the sequel) [1], for which the Z has the same couplings to SM fermions as the SM Z boson. A possible realization of the SSM appears in constructions with extra dimensions at the weak scale, but in the present work we merely take it as a benchmark model. Other realizations like (B-L) or E 6 Z do not change drastically our conclusion. Having set the Z model, the two parameters left free are the χ − Z couplings, namely V χ D and A χ D . We will show in the next section that V χ D is severely constrained by SI direct detection experiments, like XENON100 [30] or LUX [31]. On the contrary, A χ D can only be tested by SD direct detection experiments, like PICASSO [32], which are much 2 Another description of this formalism can be found in [42][43][44]. less constraining. It is thus convenient to define a parameter

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which determines the nature of the coupling of the Z with dark matter, α = 0 corresponding to a purely vectorial coupling, whereas large α correspond to a mostly axial coupling. Although α is of order unity for a generic model, a vector-like DM candidate would have α = 0 while α = ∞ for a Majorana DM candidate. For concreteness we will present our results for some definite values, covering the range 0 ≤ α ≤ 1000.

Direct detection
From (2.3) we can derive the elastic scattering cross section of χ with a nucleon N , which takes place through t-channel exchange of a Z , as illustrated in figure 1. In the case of SI scattering we may write the effective nucleon scattering cross section as where Z and A are the charge number and mass number of of the nucleus that contains the nucleon and µ χN is the nucleon-dark matter reduced mass. Notice that direct detection experiments give their limits on χ − N elastic scattering cross section assuming isopsin symmetry, σ χp = σ χn = σ χN , where p, n stands for the proton and neutron respectively. This explains the presence in eq. (2.6) of a factor that depends on the atomic number

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Z and mass number A of the given nucleus. 3 Similarly for Spin Dependent (SD) elastic scattering we obtain S A p and S A n being the proton and neutron contribution to the nucleus spin. Combining eqs. (2.2), (2.6) and (2.7) we can eliminate the parameters V χ D and A χ D , and express the invisible branching ratio of the Z as function of physical observables, like the spin dependent and spin independent scattering cross sections: 4 It is convenient to re-express Br χ in terms of α (cf. (2.5)), . The expression (2.10), which establishes a relation between the direct detection cross section and the invisible branching ratio of the Z , is our main point. It implies that the exclusion limits on σ SI χN set by XENON100 or LUX give an upper limit on the dark coupling to the Z and thus an upper limit to its invisible width. The model dependence is entirely encapsulated in V f D and A f D , which correspond to the SM fermions couplings to the extra U(1) gauge group.

LHC limits
In this section we reexamine the current limits established at the LHC. They rely on searches of dileptons [28,29] or dijet [45,46] resonances, and monojet or single photons with missing transverse energy [47,48]. In this work we will mainly rely on the limits actually set by the LHC collaborations. We are assuming a generic setup in which the Z couples both to leptons and quarks; as a consequence, among resonance searches, the ones of dileptons turn to be the most constraining. 5 Regarding the searches of missing transverse energy, we notice that most of the analysis of the LHC collaborations are based on an effective operator approach, namely assuming that new heavy degrees of freedom (the Z our case) are integrated out. This approximation may not be appropriate in general (see for instance [49]). In our analysis we have estimated possible modification of the limits when the effective approach is not valid while more quantitative results will require a dedicated study. We will notice anyway that in the example considered below, the monojet searches give subdominant constraints.
The existence of an invisible Z branching ratio weakens the current LHC limits. Indeed, at the partonic level the cross section for σ(pp → Z → ) has the following Breit-Wigner profile near the resonance, where Γ Z is defined in eq. (2.2). For a narrow resonance, Γ Z M Z , the Breit-Wigner peak can be approximated by a Dirac delta, As Γ Z ∝ g 2 D we see that the resonant cross-section scales as g 2 D . Now, the LHC exclusion limits do not take into account the possibility of decay of a Z into some invisible sector, i.e. it is assumed that Br χ = 0. To implement an invisible Z width, we first write the visible width as Γ SM which is the quantity used by the experiments in reporting their limits. Then with Γ Z = Γ χ Z +Γ SM Z . Thus, if all other things are kept constant, the resonant cross section into dileptons is diminished by a factor of (1 − Br χ ). We will furthermore assume that g D may vary freely for a generic Z model (while g D = g reproducing the ATLAS/CMS limits for the SSM). Taking into account the dependence of the cross section in g D and a non-zero Br χ in a generic Z model, this implies a simple rescaling of the resonant cross section 6 In figure 2 we have applied this scaling relation to reinterpret the ATLAS exclusion limit in presence of an invisible decay width of the Z . The new limits have been reported as function of the Z mass and for different values of g D . The couplings of the Z with SM fermions have been assumed to be the ones of the SSM since this is, at the moment, the most constrained Z model. Thus our limits can be regarded as being conservative. For instance, we can see that, for g D = 0.3, a Z with a mass of 800 GeV is still allowed by the LHC data, provided its invisible branching ratio is large enough, Br χ > ∼ 40%.

Results
In order to confront the limits on an invisible Z from resonant dilepton production at the LHC to direct searches for dark matter, we concentrate in this work on spin independent scattering and in particular on the results from the LUX collaboration [31]. The LUX detector, which is operated at the Sanford Underground Research Facility, is made of a 6 Notice that the narrow width approximation is not always appropriated for some Z realizations, like in particular the SSM, since it may occur that the decay width of the Z exceeds the detector resolution. In such a case the detector effect can be taken into account by convoluting the Breit-Wigner expression with a Gaussian factor [52], which depends on the width of the Z and the detector resolution. As a consequence the relation (2.14) should be modified accordingly. We have taken into account this factor in our numerical analysis. dual-phase xenon time-projection chamber. With an exposure of 85.3 live-days the LUX collaboration has set the strongest bounds on the SI elastic scattering cross section of dark matter, σ SI χN . For a given dark matter candidate mass, we can translate the exclusion limit set by LUX experiment into an upper bound on the invisible branching ratio of the Z using eq. (2.10). We present the result as limits in the plane (m χ , M Z ) in figures 3 and 4, respectively for for a purely vectorial coupling (α = 0) and for a mixture of vector and axial couplings (α = 10).
Fixing M Z and m χ , the figures give the allowed range of invisible branching ratios. For instance, for M Z = 5 TeV and m χ = 100 GeV, the LUX limits forbid an invisible branching ratio of the Z larger than 10 % for a purely vectorial coupling. Indeed, a larger invisible branching ratio would mean a larger coupling of the Z with the DM and thus a larger σ SI χN . Clearly, for a heavier Z , the invisible branching may be larger. Although the two plots share similar features, we notice that a non-zero axial couplings relaxes the limits on the constraints on the invisible branching ratio, since LUX is not very sensitive to such couplings which lead to spin dependent interactions. We further notice that very light dark matter candidate weaker in this case. On the contrary, for heavier dark matter candidates, and in particular for masses between roughly 50 and 100 GeV, the limits on the invisible branching ratio are very severe, even for a quite heavy Z . Indeed, this mass range corresponds to the region where LUX is the most sensitive. At the same time, if the invisible branching is small it should be easier to see the Z at the LHC. It is thus of interest to combine both analyses. We compare the exclusion limits from the ATLAS collaboration with the ones set by LUX, expressed in terms of the resonant dilepton production and of the Z mass and for g D = g and g D = 0.3 in figures 5 and 6 respectively. One clearly understands from these figures the complementarity between the LHC and direct detection searches: for a given Z mass, the LHC sets an upper limit on the production cross section while, in our framework, LUX sets a lower limit. This is because a smaller dilepton production corresponds to larger invisible branching ratio, and thus larger coupling to DM. We then understand easily that the LUX constraint gets weaker if the coupling g D is smaller. A similar conclusion holds if there is substantial axial coupling.
The dependence of the direct detection exclusion limits on the dark mass is also illustrated in figures 5 and 6. For light dark matter (8 GeV for example), the constraints from direct detection are weaker, thus allowing more invisible branching, and a weaker limit on the Z mass. On the contrary, for a 50 GeV dark matter, the LUX experiment substantially disfavors the presence of an invisible branching ratio, in the case of purely vectorial couplings, such that the LUX line practically overlaps with the prediction of the JHEP03(2014)134 Figure 5. Observational limit (black dashed lines) with 1 − 2σ uncertainties (green-yellow band) on the production cross section times branching ratio of decay into leptons for a Z prime particle as reported by ATLAS collaboration. The blue and red lines represent the LUX exclusion limits for g D = g and two values of the DM mass, namely m χ = 8 and 50 GeV, assuming pure vectorial (solid lines), namely α = 0, and substantially pure axial (dashed lines), α = 10 3 . The black dash-dotted line represent the prediction of σB for the SSM. As discussed in the text this coincides with the LUX exclusion for m χ = 50 GeV and pure vectorial couplings since for this value of the mass only a very small invisible branching ratio is allowed. dilepton production cross section for the SSM without dark matter. In all cases, the allowed parameter range, for a given Z realization and DM mass, corresponds to the region which is simultaneously below the LHC observational limit (black dashed line in the plots) and above the corresponding LUX exclusion line.
It is of interest to present the above constraints in the plane σ SI − m χ , see figure 7. There we show again the fact that, for a given Z mass, dilepton searches put a lower bound on the Z coupling to dark matter (which is quantified by V χ for fixed g D ). By the same token, a larger V χ leads to a larger SI cross section, which may be excluded by direct searches. This is illustrated for two different (M Z ,g D ) pairs. The features in this figure are pretty clear. Obviously there is no constraint from resonant production for M Z > 2m χ , since in this case the Z is fully visible. The spike for m χ M Z /2 is due to phase-space suppression, which requires a larger value of the coupling to dark matter to be compatible with dilepton searches (the values of V χ quoted in the figure are for m χ M Z /2). Of course there are no constraints from dileptons if the Z is too heavy to be produced at the LHC (e.g. for M Z > ∼ 2.8 TeV for g D = g), but a lighter Z is a priori possible provided it has a larger coupling to an invisible sector. However this coupling cannot be arbitrarily large. Imposing g 2 D V 2 D 4π gives a lower bound on the Z mass, which we will refer to as a unitarity bound in the sequel.
Combining all our preceding results, we give in the figure 8 the parameter space allowed when combining ATLAS and LUX limits in the plane (m χ , M Z ) and in the case of the JHEP03(2014)134  SSM. All the points below the lines are excluded by our analysis. We see that a large region does not survive the constraints, as expected from the discussion above. For instance one may conclude from this figure that it is excluded that a Z lighter than about 2-3 TeV couples to a dark matter candidate with mass 50 GeV. This rather strong conclusion holds for any natural U(1) model studied here, since we have considered variations of order unity for the vector and axial couplings the heaviest DM candidates is simply due to a threshold effect as Br χ → 0. For light dark matter (m χ 10 GeV) direct detection experiments allow for a larger invisible branching, and thus the LHC bound on the Z mass can be much weaker. Similarly, for non-zero α, the constraints on invisible branching from LUX are weaker, and so the mass of the Z may also be smaller.
In the region of the plots where the constraints from both direct detection and dileptons resonance are weak, one may expect that other limits become relevant. In the plot, we show the LEP limits on (non-resonant) dileptons production (dashed horizontal line), which for g D = g in the SSM model, corresponds to M Z 1.7 TeV [1,51] (for other couplings, one may use the fact that this limits scales likes g 4 D /M 4 Z ). We have also reported the unitarity bound, M Z 1.8 TeV (solid horizontal line), which for the SSM is more constraining than the LEP limit. 7 Notice that these two constraints are particularly strong because we have chosen g D = g and could be relaxed by considering a weaker couplings of the Z to leptons. We finally report our estimate of the limit from monojet searches, M Z 0.4 TeV (dotdashed line) for g 2 D V 2 D = 4π. This bound is weaker than the two other bounds. However, because of the possibility of a resonant Z , notice that our estimate is stronger than the ones reported in e.g. [47,48], which rely on an effective operator analysis. 8 Moreover, the limit from monojet searches depends on the interplay between the coupling of the Z with DM and leptons, so that the monojet constraint could be relevant in other realizations and should not be a priori neglected.

Conclusion and Prospects
We have considered a very generic scenario in which a Z gauge boson couples both to the Standard Model fermions and to a dark matter candidate, which belongs to some invisible sector. We have encompassed many realizations existing in the literature by using a simple parametrization of the couplings of the Standard Model and dark sector to the Z . We have studied the complementarity between the searches for Z at LHC and those for dark matter by direct detection experiments. For the sake of illustration, we have confronted on one hand the constraints on the resonant production of dileptons at the LHC and on the other hand the exclusion limits on spin independent elastic scattering set by the LUX collaboration. Our main point is illustrated in figures 5-7, where we show how a Z with substantial invisible branching ratio is constrained by the current direct detection experiments, and LUX in particular. The model dependent character of this statement is illustrated through the parametric dependence in an universal coupling parameter g D , and a parameter noted α that parametrizes the sensitivity to spin independent scattering collisions in direct detection experiments (see eq. (2.5)).
The final result of our analysis is shown in figure 8 in the case of a natural sequential model, in which all the Standard Model fermions have the same couplings as that to the Z boson. It shows that an electroweak scale dark matter candidate is essentially excluded in such a model, except for heavy Z (M Z 2 TeV). We also remark that we have been agnostic regarding the mechanism at the origin of the abundance of dark matter. Standard thermal production through annihilations mediated by Z leads in general to an overabundance of dark matter in the parameter space allowed by our limits. The correct value of the dark matter relic density might be achieved thanks to additional interactions in the hidden sector or due to a non-standard cosmological evolution. In a forthcoming work, we will extend the phenomenological analysis of partially invisible Z models, including the dark matter abundance and several completions of the collider searches.
Note added. During the completion of this work the paper [53], discussing the possible complementarity between detection strategies for dark Z models, has appeared. The authors consider the specific case of a leptophobic Z . Our analysis is more general and can be applied in principle to any U(1) extension of the standard model completed by a dark coupling.

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ERC advanced grant Higgs@LHC. The work of M.T. and B.Z. is supported by the IISN, an ULB-ARC grant and by the Belgian Federal Science Policy through the Interuniversity Attraction Pole P7/37.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.