LHC constraints on gauge boson couplings to dark matter

Collider searches for energetic particles recoiling against missing transverse energy allow to place strong bounds on the interactions between dark matter (DM) and standard model particles. In this article we update and extend LHC constraints on effective dimension-7 operators involving DM and electroweak gauge bosons. A concise comparison of the sensitivity of the mono-photon, mono-W, mono-Z, mono-W/Z, invisible Higgs-boson decays in the vector boson fusion mode and the mono-jet channel is presented. Depending on the parameter choices, either the mono-photon or the mono-jet data provide the most stringent bounds at the moment. We furthermore explore the potential of improving the current 8 TeV limits at 14 TeV. Future strategies capable of disentangling the effects of the different effective operators involving electroweak gauge bosons are discussed as well.


I. INTRODUCTION
Because of their potential connection to DM, searches for / E T represent one of the main lines of LHC research. These searches can be categorised based on the type of SM particles that recoil against DM. By now, ATLAS and CMS have considered a plethora of different final states in DM searches containing jets of hadrons, gauge bosons, heavy quarks and even the Higgs boson (see e.g. [1] for a recent review of the experimental status).
In most cases these studies are performed in the context of an effective field theory (EFT) which correctly captures the physics of heavy particles mediating the interactions between DM and SM fields, if the mediators are heavy enough to be integrated out. Below we will consider the effective Lagrangian which contains the following four SU (2) L ×U (1) Y gaugeinvariant dimension-7 operators Here Λ represents the scale of new physics at which the higher-dimensional operators (1) are generated, i.e. the scale where the messenger particles are removed as active degrees of freedom. The DM particle χ can be both a Dirac or a Majorana fermion and is the U (1) Y SU (2) L field strength tensor, whileB µν = 1/2 µνλρ B λρ (W i µν = 1/2 µνλρ W i,λρ ) denotes its dual and g 2 is the weak coupling constant. * Electronic address: andreas.crivellin@cern.ch † Electronic address: u.haisch1@physics.ox.ac.uk ‡ Electronic address: Anthony.Hibbs@physics.ox.ac.uk The operators introduced in (2) appear in models of Rayleigh DM (see for instance [2][3][4]). They are special in the sense that, up to dimension 7, they are the only effective interactions which lead to velocity-suppressed annihilation rates of DM to photon pairs [5][6][7]. While the sensitivity of future direct detection experiments may allow to set novel bounds on the Wilson coefficients C B (Λ) and C W (Λ) for heavy DM particles with m χ > ∼ 1 TeV once loop effects are taken into account [8], in the case of light DM the leading and for CB(Λ) and CW (Λ) the only restrictions arise and will continue to arise from collider searches involving large amounts of / E T . In fact, the DM-SM interactions (2) have been constrained using 7 TeV and 8 TeV LHC data on invisible decays of the arXiv:1501.00907v1 [hep-ph] 5 Jan 2015 Higgs boson in the VBF mode [9] as well as the / E T + Z [10,11], the mono-photon [12] and the / E T +W [13] channel.
The main goal of this article is twofold. First, to update the existing constraints by taking into account the latest mono-photon [14,15], / E T + W/Z (→ hadrons) [16] and VBF h → invisible [17] searches. Second, to extend the studies [9][10][11][12][13] by considering in addition the / E T +W (→ leptons) channel [18,19] as well as the newest mono-jet data [20]. An assortment of Feynman diagrams that lead to the / E T signatures investigated in the following are displayed in Fig. 1.
Our analysis shows that depending on the choice of parameters, either the mono-photon or the mono-jet data give rise to the strongest restrictions at present. By combining the information on all available channels we are thus able to derive bounds on the coefficients C k (Λ)/Λ 3 in (1) that improve on the existing limits. Building upon [21], we furthermore demonstrate that measurements of the jet-jet azimuthal angle difference in / E T + 2j events may be used to disentangle whether the DM bilinearχχ couples more strongly to the combination B µν B µν (W i µν W i,µν ) or the product B µνB µν (W i µνW i,µν ) of field strength tensors. Similar ideas have also been brought forward in [9].
The outline of this article is as follows. In Sec. II we review the existing LHC searches for / E T signatures that we will use to constrain the effective interactions (2). In Sec. III we derive the restrictions on the parameter space by combining all individual search modes, commenting also on how future measurements may improve these limits. This section contains in addition a discussion of the azimuthal angle correlations between the two jets in the / E T + 2j channel. Our conclusions are presented in Sec. IV.

II. SEARCH CHANNELS
In this section we list the various cuts and the values of the fiducial cross section (σ fid ) of each individual / E T channel. This information will be used in the next section to set limits on the coefficients C k (Λ)/Λ 3 appearing in the effective Lagrangian (1).

A. Mono-photon signal
We begin with the mono-photon signal, which has recently been searched for by both CMS [14] and AT-LAS [15]. Since the former search leads to the stronger restrictions, we employ the CMS results, which are based on 19.6 fb −1 of 8 TeV data. The relevant cuts are where η γ denotes the pseudorapidity of the photon. The CMS collaboration performs the measurement in six dif-ferent signal regions with a varying cut on the transverse momentum of the photon (p T,γ ). Note that due to the higher-dimensional nature of the operators (2), the / E T + γ signal has a rather hard p T,γ spectrum. As a result, we find that the most severe cut of p T,γ > 700 GeV gives the strongest bounds on the parameter space in our case. The corresponding 95% confidence level (CL) limit on the fiducial cross section reads

B. Mono-Z signal
In the case of the / E T + Z (→ + − ) search channel, we use the ATLAS results [11], that utilise 20.3 fb −1 of 8 TeV data. The selection criteria relevant to our analysis are p T, > 20 GeV , |η | < 2.5 , m ∈ [76, 106] GeV , Here m , η and p T, denote the invariant mass, the pseudorapidity and the transverse momentum of the dilepton system, respectively. The ATLAS analysis defines four signal regions with different lower / E T thresholds. As it turns out, in the considered case the requirement / E T > 350 GeV gives rise to the best bounds. Including Z-boson decays to both electrons and muons ( = e, µ), the AT-LAS experiment obtains for this / E T cut the following 95% CL bound C. Mono-W signal Both ATLAS [18] and CMS [19] have searched for a mono-W signal in the leptonic decay mode. We find that the ATLAS search for the µν µ final state, which uses 20.3 fb −1 of 8 TeV data, gives the strongest constraints, and thus we consider only this channel. The most important experimental cuts are where m T is the transverse mass which depends on the angle ϕ µ / E T between the p T,µ and the / E T vectors. AT-LAS sets bounds on σ fid for three different m T cuts, and like in the case of the mono-photon signal, we observe that the strongest restriction of m T > 843 GeV provides the best limits on the interactions (2). At 95% CL the bound on the corresponding fiducial signal cross section is given by The ATLAS search [16] looks for a / E T + W/Z signal, where the W or Z boson decays hadronically. This analysis is based on 20.3 fb −1 of 8 TeV data, jet candidates are reconstructed using the Cambridge/Aachen (C/A) algorithm [22] with a radius parameter R = 1.2 and subjected to a mass-drop filtering procedure [23]. Events are required to have at least one C/A jet with Here m j refers to the mass of the large-radius jet, while √ y = min (p T,j1 , p T,j2 ) (∆φ j1j2 ) 2 + (∆η j1j2 ) 2 /m j is a measure of the momentum balance of the two leading subjets j 1 and j 2 contained in the C/A jet. The 95% CL limits on the fiducial cross section depend also on the imposed / E T threshold, and it turns out that the stronger of the two cuts, i.e. / E T > 500 GeV, provides the most stringent constraints. In this case, the relevant limit on the fiducial cross section is

E. Mono-jet signal
One can also use mono-jet events to constrain the operators in (2), since the corresponding searches allow for the presence of a secondary jet. Here we will employ the newest CMS results [20], which make use of 19.7 fb −1 of 8 TeV data. Like CMS, we reconstruct jets using an anti-k t algorithm [24] with radius parameter R = 0.5. The relevant selection cuts are where ∆φ j1j2 is the azimuthal separation of the two leading jets. Another important selection criterion is the imposed jet-veto [25], which rejects events if they contain a tertiary jet with p T,j3 > 30 GeV and |η j3 | < 4.5. The CMS measurement is performed for seven different / E T regions, and we find that for the considered interactions the highest sensitivity is obtained for / E T > 500 GeV. The corresponding 95% CL limit on the fiducial cross section reads

F. VBF invisible Higgs-boson decays
Last but not least, we consider the results of the CMS search for invisible decays of the Higgs boson in the VBF channel [17], which uses a 8 TeV data sample, corresponding to an integrated luminosity of 19.5 fb −1 . Jets are reconstructed employing an anti-k t clustering algorithm with R = 0.5, and subject to the following requirements The missing-energy cut is / E T > 130 GeV and a central jet-veto is imposed to any event that has a third jet with p T,j3 > 30 GeV and a pseudorapidity between those of the two tagging jets. For these cuts, CMS obtains the following 95% CL bound on the fiducial cross section

III. NUMERICAL RESULTS
In order to determine the cross section for the / E T signals associated to the effective operators (2), we have implemented each of them in FeynRules [26], generating a UFO output [27]. The actual event generation has been performed at leading order with MadGraph 5 [28] utilising CTEQ6L1 parton distributions [29]. Partonshower effects and hadronisation corrections have been included by means of PYTHIA 8 [30] and jets constructed using FastJet 3 [31]. Our Monte Carlo (MC) implementation has been validated by reproducing the numerical results of [11,12] within theoretical uncertainties. These errors have been assessed by studying the scale ambiguities of our results. We have used the default dynamical scale choice of MadGraph 5, varying the scale factor in the range [1/2, 2]. We find that the predictions for the mono-photon, / E T + Z (→ + − ) and / E T + W (→ µν µ ) cross sections calculated in this way vary by around ±15%, while in the case of the / E T + W/Z (→ hadrons), the mono-jet and the VBF h → invisible signal, relative differences of about ±20% are obtained. Note that these errors are smaller than those found in [25,32,33], since all the tree-level / E T cross sections considered in our work do not explicitly depend on α s . The quoted uncertainties thus reflect only the ambiguities related to the change of factorisation scale, but not renormalisation scale.
, mono-jet (blue) and VBF h → invisible (grey) searches. The width of the bands reflect the associated scale uncertainties.
impact of scale variations. For C B (Λ) = 1, C W (Λ) = 0, one observes that the mono-photon search [14] provides the strongest constraints in most of the parameter space. Numerically, we find that the scale Λ has to satisfy Λ > ∼ 540 GeV for m χ < ∼ 100 GeV in order to meet the 95% CL requirement (4). In the case C B (Λ) = 0, C W (Λ) = 1, on the other hand, the latest mono-jet data [20] impose the leading restrictions. At 95% CL, the inequality (12) translates into a lower limit of Λ > ∼ 600 GeV for DM masses below 100 GeV. The shown limits also hold in the case that CB(Λ) = 1, CW (Λ) = 0 or CB(Λ) = 0, CW (Λ) = 1 and C B (Λ) = C W (Λ) = 0, while for Majorana DM the constraints on Λ would be stronger by around 12%. Note finally that / E T + W (→ µν µ ) searches do not provide any constraint on scenarios with C W (Λ) = CW (Λ) = 0.
To better understand the restrictions imposed by the various search channels, we consider the Feynman rules associated to the effective operators O B and O W entering (1). In momentum space, the resulting interactions between pairs of DM particles and SM gauge bosons take the form where p i (µ i ) denotes the momentum (Lorentz index) of the vector field V i and for simplicity the spinors associated with the DM fields have been dropped. In terms of the sine (s w ) and cosine (c w ) of the weak mixing angle and the Wilson coefficients C B (Λ) and C W (Λ), the couplings g ViVj read These results do not coincide with the expressions reported in [10,12,13]. From (16) we see that in the coupling g AA of DM to two photons, the Wilson coefficients C B (Λ) enters compared to C W (Λ) with a relative factor of c 2 w /s 2 w 3.3. On the other hand, in the case of the coupling between DM and Z-boson pairs g ZZ , the dependence on s w and c w is reversed compared to g AA . These properties explain why the limit on the new-physics scale Λ from mono-photon / E T + Z (→ + − ), / E T + W/Z (→ hadrons), mono-jet and VBF h → invisible searches is stronger (weaker) in the upper panel than in the lower panel of Fig. 2.
A second important feature worth noting is that channels with leptons in the final state typically lead to weaker restrictions on the parameter space than modes involving hadrons. This is a simple consequence of the fact that the electroweak SM gauge bosons dominantly decay hadronically. Numerically, one has Br (Z → + − ) 7% and Br (W → µν µ ) 11%, while Br (Z → hadrons) 70% and Br (W → hadrons) 68% [34]. The strong suppression of the leptonic decay widths overcompensates the higher detection efficiencies of final states involving leptons, and as a result the LHC searches for / E T + hadrons are superior to those looking for / E T + leptons signals. Our third observation is that the latest mono-jet data are evidently more constraining than the recent VBF h → invisible search. While these analyses explore the same final state, i.e. / E T +2j, they probe quite different parts of the phase space. In fact, the selection criterion that has the biggest impact in our study is the rather loose missing transverse energy cut of / E T > 130 GeV imposed in the VBF h → invisible search. This selection is tailored for a Higgs boson of 125 GeV, but fares less well if one tries to probe higher-dimensional operators of the form (2). Since the operators O B (OB) and O W (OW ) produce a rather hard / E T spectrum, more severe / E T requirements allow for a cleaner separation between signal and SM background.

B. Dependence on two Wilson coefficients
Until now we have have studied the constraints on the new-physics scale Λ as a function of the DM mass m χ , keeping the values of the high-scale Wilson coefficients fixed. In the panels of Fig. 3 we instead show contours of constant Λ in the C B (Λ)-C W (Λ) plane. In all plots we employ m χ = 100 GeV and set CB(Λ) = CW (Λ) = 0. The first noticeable feature of the shown predictions is that only the mono-photon signal depends more strongly on C B (Λ) than C W (Λ), while for all the other / E T channels the situation is reversed. Second, with the exception of the mono-photon case, one observes that the major axes of the elliptic contours in all panels are almost aligned with the C W (Λ) axes. This means that interference effects between contributions arising from O B and O W are small in all of these cases. The third important property following from the colour shading of the depicted results is that currently either the newest monophoton or the mono-jet data provide the leading bounds in the entire C B (Λ)-C W (Λ) plane. This feature is further illustrated by the upper panel in Fig. 4. In this plot the overlaid numbers indicate the search strategy that contributes the best sensitivity on Λ at each point, with 1 and 5 corresponding to the mono-photon and monojet channel, respectively. One sees that if the ratio of Wilson coefficients satisfies |C B (Λ)/C W (Λ)| > ∼ 1.3 then the limit (4) gives rise to the strongest constraint, while in the remaining C B (Λ)-C W (Λ) plane the bound (14) is most restrictive. The Λ contours obtained by combining all available / E T channels are depicted in the lower panel of Fig. 4.
Finally, we note that the values of Λ that can be excluded with the current data are low compared to typical LHC energies. In order to go beyond the EFT description, one has to specify a ultraviolet (UV) completion, where the operators in (4) arise from a renormalisable theory after integrating out the heavy degrees of freedom mediating the interactions. UV-complete models that generate the operators O B and O W through loops of states charged under U (1) Y and/or SU (2) L have been proposed in [3] and their LHC signatures have been studied in [4]. If these new charged particles are light, the high-p T gauge bosons that participate in the / E T processes considered here are able to resolve the substructure of the loops. This generically suppresses the cross sections compared to the EFT predictions [32], and thus will weaken the bounds on the interaction strengths of DM and the electroweak gauge bosons to some extent. Furthermore, the light charged mediators may be produced on-shell in pp collisions, rendering direct LHC searches potentially more restrictive than / E T searches. Making the above statements precise would require a study of a concrete UV completion.

C. Future sensitivity
It is also interesting to explore how the reach on the new-physics scale Λ might improve at the 14 TeV LHC.
In what follows, we will only consider the mono-jet signal, applying the event selection criteria that have been used in the sensitivity study by ATLAS [35]. These read and jets are reconstructed using an anti-k t algorithm with R = 0.4. Events with a third jet of p T,j3 > 50 GeV and |η j3 | < 3.6 are vetoed and the missing transverse energy cut that we employ is / E T > 800 GeV. Note that compared to (11) the p T,j1 , p T,j2 and / E T thresholds are increased both to avoid pile-up and to enhance the signalover-background ratio. In order to determine the limits on the scale Λ, we take σ fid pp → Z (→νν) + j = 5.5 fb [35], assuming a total systematic uncertainty on the SM background of 5%. For the choice C B (Λ) = 0, C W (Λ) = 1 and CB(Λ) = CW (Λ) = 0, we find that with 25 fb −1 of data, corresponding to the first year of running after the LHC upgrade to 14 TeV, one may be able to set a 95% CL bound of Λ > ∼ 1.3 TeV for m χ < ∼ 100 GeV. Compared to the present limit, this corresponds to an improvement of the bound on Λ by more than a factor of 2. With 300 fb −1 and 3000 fb −1 of accumulated data, we obtain instead Λ > ∼ 1.4 TeV. These numbers make clear that at 14 TeV the sensitivity of / E T + j searches will rather soon be limited by systematic uncertainties associated to the irreducible SM background. To what extent this limitation can be evaded by an optimisation of the mono-jet searches and/or an improved understanding of the pp → Z (→νν) + j channel, would require a dedicated study. Such an analysis is beyond the scope of this work.

D. Analysis of jet-jet angular correlations
So far we have analysed only observables that are insensitive to whether the / E T signal is generated by an insertion of the effective operator O B (O W ) or OB (OW ). This ambiguity can however be resolved by measuring the azimuthal angle difference ∆φ j1j2 of forward jets produced in / E T + 2j events [9,21]. Besides the cuts (17), we impose the following VBF-like selection requirements in our analysis η j1 · η j2 < 0 , ∆η j1j2 > 2 , m j1j2 > 1100 GeV . (18) Here the cut on the pseudorapidity separation helps to sculpt the angular correlations between the tagging jets, while the di-jet invariant mass threshold improves the signal-over-background ratio.
In order to understand why the operators O B (O W ) and OB (OW ) lead to different jet-jet angular correlations, one has to consider their Feynman rules. In the case of the operators containing regular field strength tensors this has already been done in (15), while for their dual counterparts we obtain with g ViVj given in (16). The selection cuts (18) emphasise the parts of the phase space where the external partons experience only a small energy loss and the momentum components of the tagging jets in the beam direction are much greater than those in the transverse plane. In this limit the structure of the pp → / E T + 2j matrix elements is straightforward to work out [36]. In the case of the effective operator O W , one gets for instance M W ∼ J µ1 1 J µ2 2 (g µ1µ2 p 1 · p 2 − p 1 µ1 p 2 µ2 ) ∼ p T,j1 · p T,j2 , while for OW one arrives instead at MW ∼ µ1µ2ν λ J µ1 1 J µ2 2 p ν 1 p λ 2 ∼ p T,j1 × p T,j2 . Here J i and p i denote the currents and momenta of the electroweak gauge bosons that partake in the scattering. These simple arguments imply that the ∆φ j1j2 spectrum corresponding to O W should be enhanced for collinear tagging jets, ∆φ j1j2 = 0, while for ∆φ j1j2 = π/2 it should show an approximate zero. In the case of OW , on the other hand, the ∆φ j1j2 distribution should have a dip if the two jets are collinear, ∆φ j1j2 = 0, or back-to-back, ∆φ j1j2 = π. Note that the above arguments do not depend on the chirality of the DM current. This means that O B , O W , OB, OW and the operators obtained from (2) by replacinḡ χχ withχγ 5 χ lead to very similar jet-jet angular correlations, as we have explicitly verified.
In Fig. 5 we plot the ∆φ j1j2 spectra for the choices The given S/ √ B values imply that running the LHC for a couple of years at 14 TeV should provide a sufficient number of events to analyse the jet-jet angular correlations. To quantify this statement, we use a toy MC and generate event samples for both signals and background corresponding to 300 fb −1 and 3000 fb −1 of luminosity. The resulting differential cross sections are then fitted to [37] 1 σ dσ d∆φ j1j2 = 2 n=0 a n cos (n∆φ j1j2 ) .
The coefficient a 0 is fixed by the normalisation of the ∆φ j1j2 spectrum, and the ratio r 1 = a 1 /a 0 turns out to be rather insensitive to which type of higher-dimensional interactions is considered. In contrast, the combination r 2 = a 2 /a 0 is a measure of the CP nature of the interactions that lead to the 2j final state (see e.g. [36,37] become fairly independent of EFT assumptions [21]. In Fig. 6 we present the results of our toy MC. The upper panel (lower panel) corresponds to 300 fb −1 (3000 fb −1 ) of LHC data collected at 14 TeV. The expected azimuthal angle distributions for the signal plus background predictions are coloured blue (red) for O W (OW ). For comparison, the SM-only result (grey) divided by a factor of 3 is also shown. The solid curves illustrate the best fits to (20), restricting the rapidity separation ∆φ j1j2 to the range [0, 2.5]. For 300 fb −1 of data, we obtain for r 2 the central values and uncertainties (r 2 ) W +SM = 0.15 ± 0.10 , (r 2 )W +SM = −0.45 ± 0.14 , In the case of 3000 fb −1 of luminosity, we find instead (r 2 ) W +SM = 0.18 ± 0.03 , We observe that for O W (OW ) the combination r 2 is indeed positive (negative). Defining a significance as s k = (r 2 ) k+SM − (r 2 ) SM /(∆r 2 ) k+SM , we get from (21) the values s W = 2.7 and sW = −2.4, while (22) leads to s W = 10.3 and sW = −6.8. Our toy MC study corresponding to 300 fb −1 (3000 fb −1 ) of data hence suggest that a distinction between the azimuthal angle distributions of O W and OW at the 5σ (17σ) level should be possible at the 14 TeV LHC. We emphasise that our toy study assumes a perfect detector and that we have not optimised the cuts (18) to achieve the best significance. Once the data is on tape, it will become an experimental issue of how stringent the VBF-like selections can be made to extract the most information on the jet-jet angular correlations for a given limited sample size.

IV. CONCLUSIONS
In this article we have studied LHC constraints on effective dimension-7 operators that couple DM to the SM electroweak gauge bosons and emphasised the complementarity of different / E T searches for constraining the associated Wilson coefficients. Focusing on the interactions that induce only velocity-suppressed annihilation rates, we have combined the information on all individual search modes that are available after LHC run-1. In this way we are able to derive bounds on the new-physics scale Λ that exceed all previous limits. Our studies show that at present, depending on the choice of parameters, either mono-photon or mono-jet searches provide the most severe constraints on the considered dimension-7 interactions. For DM masses m χ < ∼ 100 GeV and Wilson coefficients |C k (Λ)| 1, the existing 8 TeV LHC searches allow to exclude values of Λ below about 600 GeV at 95% CL. The improved reach of / E T analyses in 2015 and beyond is also studied, finding that with 25 fb −1 of 14 TeV data, LHC mono-jet searches should be able to improve the latter bound to approximately 1.3 TeV. Beyond this point further progress will be hindered by the imperfect understanding of irreducible SM backgrounds such as pp → Z (→νν) + j. Finding ways to overcome these limitations will be crucial to exploit the full physics potential of / E T searches to be carried out at later stages of the LHC.
We have furthermore emphasised that given the large statistics expected at the phase-1 and phase-2 upgrades of the 14 TeV LHC, / E T searches should be able to not only determine integrated, but also differential cross sections. From the theoretical point of view, such normalised distributions have the clear advantage, that compared to the total cross sections theoretical uncertainties are reduced and that the obtained predictions depend only weakly on the assumptions underlying the EFT description. As an example we have explored the prospects to measure jet-jet angular correlations in / E T + 2j events. Taking into account the pseudorapidity correlations of the two tagging jets, the resulting distributions in the azimuthal angle separation ∆φ j1j2 exhibit the relative strength of CP-even and CP-odd interactions of DM with gauge boson pairs. Our toy MC studies indicate that already with 300 fb −1 of data a distinction between the new-physics and the SM-only hypotheses can be achieved at a statistically significant level, and that the sensitivity of the discussed searches is greatly improved by going to 3000 fb −1 of luminosity. A more precise determination of the analysing power, including systematic uncertainties would require a full detector simulation, which is beyond the scope of the present article. We however believe that it is imperative that the ATLAS and CMS collaborations direct some activity towards the study of differential distributions of final states like / E T + 2j.