Published for SISSA by Springer
Received: June 7, 2016
Revised: July 19, 2016 Accepted: August 30, 2016 Published: September 5, 2016
JHEP09(2016)018
Constraints on Z[prime] models from LHC dijet searches and implications for dark matter
Malcolm Fairbairn,a John Heal,a Felix Kahlhoeferb and Patrick Tunneya
aPhysics, Kings College London,
Strand, London, WC2R 2LS, U.K.
bDESY,
Notkestrae 85, D-22607 Hamburg, Germany
E-mail: mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected] , mailto:[email protected]
Web End [email protected]
Abstract: We analyse a combination of ATLAS and CMS searches for dijet resonances at run I and run II, presenting the results in a way that can be easily applied to a generic Z[prime] model. As an illustrative example, we consider a simple model of a Z[prime] coupling to quarks and dark matter. We rst study a benchmark case with xed couplings and then focus on the assumption that the Z[prime] is responsible for setting the dark matter relic abundance.
Dijet constraints place signi cant bounds on this scenario, allowing us to narrow down the allowed range of dark matter masses for given Z[prime] mass and width.
Keywords: Beyond Standard Model, Cosmology of Theories beyond the SM
ArXiv ePrint: 1605.07940
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP09(2016)018
Web End =10.1007/JHEP09(2016)018
Contents
1 Introduction 1
2 Limits on generic Z[prime] models from dijet resonance searches at the LHC 22.1 Dijet event generation 32.2 Compatibility of a dijet signal with LHC data 42.3 Results 7
3 Constraints on a leptophobic Z[prime] coupling to DM 93.1 Bounds for xed couplings 103.2 Combining di-jet bounds and relic density 123.2.1 Relic density constraints for a xed width 143.2.2 Dijet bounds on the DM coupling 163.2.3 Dijet bounds on the DM mass 16
4 Conclusions 18
A Tabulated bounds on gq 20
B Speci c modi cations of micrOMEGAs 20
1 Introduction
Resonant structures in the invariant mass distribution of dijet events are amongst the most generic signatures for \exotic" new physics at the LHC, since any new heavy particle produced in the s-channel at hadron colliders can decay back into a pair of jets. Searches for dijet resonances are therefore a high priority at both ATLAS and CMS and have been among the rst searches carried out at a centre-of-mass energy of 13 TeV [1, 2]. These searches are complemented by earlier searches at 8 TeV [3, 4], as well as a dedicated search for dijet resonances with an invariant mass below 1 TeV at CMS based on a novel data scouting technique [5]. Among the many models probed by such searches are Randall-Sundrum (RS) gravitons [6], excited quarks [7, 8] and models with a leptophobic Z[prime] [9{11]
(see [12] for the implications of these searches on more generic Z[prime] models).
The latter model of a massive spin-1 boson that couples predominantly to quarks has received a signi cant amount of interest in the context of dark matter (DM) production at hadron colliders. The reason is that a leptophobic Z[prime] can have large couplings to the
DM particle and thereby mediate the interactions that keep DM in thermal equilibrium in the early Universe. We can then hope to experimentally probe these interactions with a range of di erent DM search experiments. Two sensitive probes of such scenarios are
{ 1 {
JHEP09(2016)018
direct detection experiments searching for evidence of DM-nucleus scattering and searches for missing energy at the LHC [13{28]. In fact, DM models with a leptophobic Z[prime] have served to inspire a class of so-called simpli ed DM models, which are now commonly used to optimise LHC searches for DM [29, 30].
However, as emphasised in a number of previous works [13, 25, 31, 32], DM models with a massive spin-1 mediator cannot only be probed by conventional DM searches, but also by direct searches for the mediator, in particular searches for dijet resonances. Due to the presence of invisible decays of the mediator, the width of resonance may be broadened, making it harder to distinguish the signal from the smoothly-falling QCD backgrounds. The purpose of the present work is to derive combined limits on such models by re-analysing all available LHC searches for dijet resonances.
Previous combinations of dijet constraints have either focussed on narrow resonances [13], resonances that decay exclusively into quarks [10] or on individual DM models with speci c choices of couplings [25, 31]. The present work aims to take a largely model-independent approach, so that our results can be applied to a range of di erent Z[prime] models.
For this purpose, we take the width of the Z[prime] as a free parameter, which we allow to be as large as /mZ[prime] = 0.3. The resulting bound on the Z[prime]-quark coupling can then be applied to models where the Z[prime] decays exclusively to quarks (for example if couplings to DM are negligible or absent, or if decays of the Z[prime] into DM available are kinematically forbidden), to models where the width of the Z[prime] is given exclusively in terms of its couplings to quarks and DM (as is the case in simpli ed DM models) and to models where additional unobserved decay channels may be present that further broaden the Z[prime] width. To illustrate our approach, we show how the resulting dijet constraints can be applied to simpli ed DM models. Moreover, we develop a new method to combine dijet constraints with information on the relic abundance of DM to determine those regions of parameter space where thermal freeze-out via a Z[prime] is incompatible with constraints from the LHC.
The outline of our work is as follows: we rst perform a combined analysis of searches for dijet resonances from the CMS and ATLAS experiments at both 8 and 13 TeV in section 2. We present the results of this analysis in terms of an upper limit on the Z[prime]-
quark coupling as a function of the Z[prime] mass and width, which applies to a wide range of Z[prime] models irrespective of what other couplings and particles are present in the model. In section 3 we then show how these results may be applied to a speci c model, namely a model with a Z[prime] mediator coupling to quarks and DM. We rst consider a simpli ed model similar to the one presently employed by the LHC collaborations and then combine such a model with relic density constraints to place limits on the coupling and the mass of the DM particle. Additional technical details are provided in the appendices.
2 Limits on generic Z[prime] models from dijet resonance searches at the LHC
In this section we describe our technique for the re-analysis of searches for dijet resonances from the LHC run I and II and present the resulting combined limits. Our basic approach can be divided into three separate steps. We rst implement a fully general Z[prime] model in a Monte Carlo generator to produce dijet events at the relevant centre-of-mass energies
{ 2 {
JHEP09(2016)018
and apply the selection cuts corresponding to the various analyses. We then compare the predicted distributions of the dijet invariant mass to the experimental data, employing the same strategy as the experimental collaborations to model the background contribution. Finally, we combine the experimental tension from all data sets in a statistically consistent way in order to determine the largest signal strength that can still be compatible with all experimental data. The details of each of these steps are discussed in the following subsections.
2.1 Dijet event generation
The rst step in our analysis is to generate dijet events resulting from a new Z[prime] mediator with mass 500 GeV mZ[prime] 4 TeV, which interacts with the standard model quarks q
via the vectorial coupling gq. To generate this signal, we add the following terms to the SM Lagrangian:
Lkin =
14F [prime] F [prime] +
JHEP09(2016)018
1
2m2Z[prime] Z[prime]Z[prime] , (2.1)
Lint = gqZ[prime]
Xq q q , (2.2)
where F [prime] = @Z[prime] @ Z[prime]. Although we do not specify any other couplings of the Z[prime],
such couplings may in principle be present.1 We therefore do not calculate the total decay width of the Z[prime] in terms of its quark coupling but instead take to be a free parameter of the model.
An important advantage of taking and gq as independent parameters is that the shape of the dijet invariant mass distribution depends only on the two parameters mZ[prime] and
, whereas the total magnitude of the signal is proportional to g4q irrespective of whether the Z[prime] is produced on-shell or o -shell.2 We can therefore generate events for di erent values of mZ[prime] and and a xed value of gq and apply the simple rescaling / g4q to obtain
signal predictions for the full three-dimensional parameter space.
Our simulation of dijet events is carried out using a pipeline of the publicly available software packages FeynRules v1.6.11 [33], MadGraph v3.2.2 [34], Pythia v8.186 [35, 36] and FastJet v3.0.5 [37]. First, we implement the model Lagrangian in FeynRules to calculate the Feynman rules and generate a UFO model le [38]. In MadGraph we then generate matrix elements for all processes involving a virtual Z[prime] and a pair of u, d, s, c or b quarks in the nal state.
The output from MadGraph is interfaced with Pythia, which we use both as a Monte Carlo event generator and to simulate showering and hadronisation. For our simulations, we use the CTEQ5L parton distribution function [39]. We neglect next-to-leading order corrections, which are expected to lead to somewhat larger cross sections [40], so we give a
1In particular, we assume at this point that the Z[prime] has vectorial couplings to quarks. We will discuss below how our results can be applied to models with axial couplings to quarks and to a combination of vectorial and axial couplings.
2Note in particular that we allow the unphysical situation that the branching ratio into quarks, given by q q/ , can become larger than unity. This does not pose a problem as long as the resulting constraints are only evaluated for physical combinations of gq and .
{ 3 {
R [notdef] [notdef] pTmin Ref.
ATLAS 13 TeV 0.4 < 2.4 50 GeV [1] CMS 13 TeV 1.1 < 2.5 30 GeV [2]
ATLAS 8 TeV 0.6 < 2.8 50 GeV [3] CMS 8 TeV 1.1 < 2.5 30 GeV [4]
CMS 8 TeV (low mjj) 1.1 < 2.5 30 GeV [5]
Table 1. Jet parameters chosen for the anti-kT algorithm for the ve experimental searches. The radius parameter is de ned as R =
p( )2 + ( )2 with the azimuthal angle. For the CMS analyses, jets are rst reconstructed with a radius parameter of 0.5 (0.4) at 8 TeV (13 TeV) and are then combined into two fat jets with radius parameter 1.1.
conservative bound. The resulting nal states are clustered with FastJet using the anti-kT algorithm [41]. We choose the jet parameters (cone-size R, maximum pseudorapidity and minimum transverse momentum pTmin of the jet) to match those adopted by each experiment under consideration (see table 1).
Once the jets have been reconstructed, we apply the experimental selection cuts outlined in table 2. For each event that passes these cuts we calculate the invariant mass of the dijet system. Rather than performing a full detector simulation, we approximate the uncertainties in reconstructing the energy and momentum of the jet events arising from detector performance by applying a Gaussian smearing to the invariant dijet mass. For this purpose, we take the detector resolution in both ATLAS and CMS to be
(mjj) = 1.8GeV
qmjj/GeV , (2.3)
which was determined by tting the smeared signals to shapes given by the CMS experiment [4] for a RS graviton benchmark model. The smeared invariant masses are then binned according to the bin sizes given by the di erent experiments and the resulting histograms are converted into di erential cross sections dZ[prime] /dmjj.
2.2 Compatibility of a dijet signal with LHC data
Once the dijet invariant mass distributions have been generated, the second step is to determine the compatibility of such a signal with observations at the LHC. For this purpose, we follow the approach of the experimental collaborations and assume that the SM background can be described by a smooth function of the form:
dSM
dmjj =
P0 (1 mjj/ps)P1 (mjj/ps)P2+P3
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log(mjj/ps) , (2.4)
where the parameters Pi are determined by tting the function to the data.3 The total dijet invariant mass distribution is then given by d/dmjj = dSM/dmjj + dZ[prime] /dmjj, where the rst term depends on the unknown parameters Pi, while the second term depends on the assumed values for mZ[prime] , and gq.
3While the CMS analyses at 8 and 13 TeV and the ATLAS analysis at 8 TeV allow all four parameters to vary, the ATLAS analysis at 13 TeV xes P3 = 0.
{ 4 {
mjj [notdef] jj[notdef] additional Ref.
ATLAS 13 TeV > 1.1 TeV < 1.2 pT,j1 > 440 GeV and pT,j2 > 50 GeV [1] CMS 13 TeV > 1.2 TeV < 1.3 pT,j1 > 500 GeV or HT > 800 GeV [2]
ATLAS 8 TeV > 250 GeV < 1.2 { [3] CMS 8 TeV > 890 GeV < 1.3 { [4]
CMS 8 TeV (low) > 390 GeV < 1.3 { [5]
Table 2. Experimental cuts adopted by the ve experimental searches. pT,j1 refers to the transverse momentum of the leading jet, while pT,j2 refers to the subleading jet. HT is the scalar sum of all jet pT for jets with pT > 40 GeV and [notdef] [notdef] < 3, and jj refers to the rapidity separation of the leading
and subleading jets.
To compare the model prediction to experimental data, we calculate the usual ~2 test statistic
~2 =
Xi
where the index i denotes the bin number in a given experiment, di is the observed di erential cross section with corresponding (statistical) error i and si is the predicted signal containing both the SM contribution and the new-physics signal. We now x the unknown parameters Pi by nding the minimum of the ~2 distribution with respect to these parameters (called ^
~2).
We now want to place an upper bound on the magnitude of the new-physics signal, beyond which the sum of signal and background are incompatible with the data. For this purpose, we employ a ~2 method. We rst calculate ^
~2 in the absence of a contribution
from the Z[prime] mediator (called ^
For ~2 < 0, the data actually prefers a non-zero contribution from the Z[prime] mediator. Positive values of ~2, on the other hand, are disfavoured by the data. In such a case, we can calculate the p-value, i.e. the probability to observe at least as large a value of ~2 from random uctuations in the data as
P = 1 CDF 1, ~2
[parenrightbig]
is the cumulative distribution function for the ~2 distribution with one degree of freedom.4
As discussed above, the new-physics signal is proportional to g4q. As we increase gq, keeping mZ[prime] and xed, we will reach the point where ~2 becomes so large that P becomes unacceptably small. For P < 5%, we can exclude the corresponding value of
4We note that the ~2 test statistic as we de ne it does not exactly follow a ~2-distribution. The reason is that we take ^
~20 to be the value of ^
~2 for gq = 0 rather than nding the value of gq that actually minimises
~2 (called gq,0) in order to avoid the problem that the data may prefer a negative signal contribution. Since ^
~2(0) ^
~2(gq,0), our de nition yields slightly smaller values for ~2 than the one obtained from minimising ^
~2 with respect to gq. Using a ~2-distribution to calculate the p-value therefore means that we slightly overestimate the p-value and consequently place more conservative bounds. We have veri ed that the error made by this approximation is small by determining the actual distribution of ~2 from a Monte Carlo simulation for speci c parameter points.
{ 5 {
JHEP09(2016)018
di si i
2, (2.5)
~20) and then de ne ~2(mZ[prime] , , gq) = ^
~2(mZ[prime] , , gq) ^
~20.
(2.6)
where CDF 1, ~2
[parenrightbig]
^
1
0.100
0.010
0.001
A[pb]
BR
Our limitCMS observed limit CMS expected limit
1000 1500 2000 2500 3000 3500 4000
mRS [GeV]
Figure 1. Comparison of our method for setting limits from dijet data (blue, dashed) with those from the CMS experiment [4] (green, solid) presented in terms of the cross section times acceptance, for the benchmark model of an RS graviton [6].
gq at the 95% con dence level. If P > 5%, the value of gq cannot be excluded by the experiment under consideration, but it may still be excluded by the combination of results from several experiments. Such a combination is necessarily model-dependent in the sense that it requires an assumption on the ratio of the production cross section of the resonance at 8 TeV and 13 TeV. Since we use a Z[prime]-model to generate dijet events, our combination is valid for any resonance produced dominantly from light quarks (with equal couplings to all avours).
For a given signal hypothesis, we can follow the procedure described above to obtain a value of ~2 for each experiment under consideration. Crucially, the parameters Pi are tted independently for each experiment. Since the two CMS searches at 8 TeV are not statistically independent, we use ref [4] for mZ[prime] 1 TeV and ref [5] for smaller Z[prime] masses.
We can then simply add up the individual contributions to ~2 to obtain ~2total. This test statistic is again expected to approximately follow a ~2-distribution with one degree of freedom, so we can calculate the combined p-value with eq. (2.6).
Validation based on the RS-graviton model. To validate our limit-setting procedure, we have applied our analysis chain to the RS graviton model [6], which is used as a ducial benchmark by CMS [4]. The model contains two free parameters, namely the mass of the RS graviton mRS and the curvature of the ve-dimensional bulk k/ ~
MPl where ~
MPl is
the reduced Planck mass. The latter is taken to be k = 0.1 ~
MPl, which fully determines the width and the couplings relevant for the generation of dijet events for a given value of mRS.
We adopt these values to calculate the total production cross section and to generate dijet invariant mass distributions. We then multiply the resulting distributions with a rescaling factor in order to determine the largest signal strength that is still compatible with data. Applying the resulting rescaling factor to the total cross section then yields an upper bound on the production cross section as a function of the RS graviton mass. Figure 1 shows the comparison of the bounds we obtain with the results from the CMS analysis. We conclude that our approach yields good agreement with the results from the CMS collaboration over a wide range of masses.
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2.3 Results
Having described the strategy for deriving bounds on the Z[prime]-quark coupling gq, we now present the results of our analysis in a way that can be applied to a wide range of Z[prime] models. We consider mZ[prime] masses between 500 GeV and 4 TeV in steps of 50 GeV. For each mediator mass, we consider six di erent widths: 1%, 2%, 5%, 10%, 20% and 30% (in units of mZ[prime] ). These values were chosen to best sample the variation in the constraints for di erent widths. In particular, we note that for a Z[prime] width smaller than 1% of mZ[prime] , the shape of the dijet invariant mass distribution is dominated by the detector resolution and therefore becomes independent of . For each combination of mZ[prime] and we then determine the largest value of gq that is compatible with the experimental data at 95% con dence level (called gq,95%).
While the resulting values of gq,95% typically depend on mZ[prime] in a non-monotonic way (due to di erent random uctuations from bin to bin), the dependence on is typically very smooth, with larger values of corresponding to larger values of gq,95% (i.e. weaker bounds). We can make use of this observation to interpolate between the values of
considered in our simulation. Speci cally, we nd that it is possible to t gq,95% for xed mZ[prime] using a function of the form
gq,95%(mZ[prime] , Z[prime] )4 = a(mZ[prime] ) [parenleftbigg]
Z[prime]
mZ[prime]
JHEP09(2016)018
b(mZ[prime] ) + c(mZ[prime] ) , (2.7)
where the values of a, b and c are listed in table 3 in appendix A as a function of mZ[prime] . We show gq,95% as a function of mZ[prime] and as obtained from the interpolation functions in the left panel of gure 2.
For mZ[prime] [lessorsimilar] 1.5 TeV dijet constraints are able to exclude values of gq between 0.1 (for a narrow width) and 0.3 (for a broad width). For larger masses, these bounds become somewhat weaker and reach up to gq,95% 0.6 for mZ[prime] 4 TeV and /mZ[prime] > 0.2. We
observe that rather weak bounds are obtained for mZ[prime] 1.6{1.7 TeV. The reason is that
in this mass range all four experiments see an upward uctuation in the data, so that the observed bound is weaker than the expected one (see also [42, 43]).5
Since we have consistently treated and gq as independent parameters, our results can be applied to any Z[prime] model (with universal vector-like couplings to quarks) by applying the following procedure:
1. For given Z[prime] mass and given couplings of the Z[prime] to all other particles in the theory, calculate the total decay width .
2. Look up gq,95% for this value of and the assumed Z[prime] mass.
3. If gq,95% is larger than the assumed Z[prime]-quark coupling, the parameter point is allowed. Otherwise, it is excluded at 95% con dence level.
5This pattern is driven by the ATLAS 8 TeV data set and is most pronounced for very broad widths. The largest preference for a non-zero contribution from a Z[prime] is found for mZ[prime] = 1.7 TeV, /mZ[prime] = 0.3 and gq = 0.55. The local signi cance of this excess is 3.3 for ATLAS alone and 3.8 for the combination of all data sets.
{ 7 {
While the procedure detailed above applies to Z[prime] models with universal vector couplings to all quarks, it is also possible for us to constrain more complicated models. For this purpose, we make use of the narrow-width approximation (NWA), which is valid as long as the width of the Z[prime] is small compared to its mass (typically /mZ[prime] < 0.3). The NWA states that the cross section for the production of dijet events via a resonance factorises into the production cross section of the resonance and the probability for this resonance to decay into a pair of jets:
(pp ! Z[prime] ! jj) = (pp ! Z[prime]) [notdef] BR(Z[prime] ! jj) , (2.8) where BR(Z[prime] ! jj) = (Z[prime] ! jj)/ = 5 g2q mZ[prime] /(4 ). In the model we consider the
Z[prime] production cross section is proportional to g2q, with a constant of proportionality that depends on the Z[prime] mass and the centre-of-mass energy. Consequently, the dijet signal in each experiment is proportional to g2q times the relevant branching ratio:
(pp ! jj) / g2q [notdef] BR(Z[prime] ! jj) . (2.9) Indeed, this relation is also correct for the di erential cross section, i.e. the shape of the dijet invariant mass distribution is independent of the coupling gq for xed mediator mass and width.
This observation motivates a di erent way of presenting our results, namely to place an upper bound on the combination [13]
j g2q [notdef] BR(Z[prime] ! jj) . (2.10) As discussed above, j is proportional to g4q for xed . We can then calculate the upper bound on j at 95% con dence level, called j95%, by evaluating j for gj,95%. We emphasise that it is perfectly acceptable for this calculation to yield a branching ratio larger than unity. In this case the conclusion would simply be that the experimental bounds cannot exclude any value of gq compatible with the chosen value of .
Our results for j95% are shown in the right panel of gure 2. The advantage of this approach is that j95% can also be used to constrain models beyond the one considered here.
In particular, our analysis can be applied to the following cases:
For a Z[prime] with both vector (gVq ) and axial (gAq) couplings to quarks, the production
cross section for a Z[prime] is proportional to (gVq )2 + (gAq)2. In such a model, one should therefore calculate
(gVq )2 + (gAq)2
[bracketrightbig]
BR(Z[prime] ! jj) and compare the result to j95% as
shown in the right panel of gure 2.6
The Z[prime] production is typically dominated by up and down quarks in the initial state.
Consequently, for a Z[prime] with di erent couplings to the three generations, one can obtain an approximate bound by calculating g21 [notdef] BR(Z[prime] ! jj), where g1 is the
coupling to the rst generation, and comparing the result to the bound on g2q [notdef] BR(Z[prime] ! jj) (i.e. j95%) shown in gure 2.
6For a Z[prime] with purely axial couplings to quarks, one can also directly compare gAq to gq,95% shown in left panel of gure 2.
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Upper bound on gq (95% CL)
Upper bound on j (95% CL)
0.30
0.30
0.6
0.1
0.25
0.3
0.25
0.07
0.5
0.20
0.20
0.2
0.25
0.005 0.01
0.02
/m Z '
0.15
0.45
/m Z '
0.15
0.10
0.2
0.4
0.10
0.04
0.35
0.1
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0.05
0.15
0.3
0.05
0.1
0.02
0.00 500 1000 1500 2000 2500 3000 3500 4000
0.00 500 1000 1500 2000 2500 3000 3500 4000
mZ' [GeV]
mZ' [GeV]
Figure 2. Bounds on gq (left) and j g2q [notdef] BR(Z[prime] ! jj) (right) from a combination of ATLAS
and CMS dijet searches at 8 TeV and 13 TeV at 95% con dence level as a function of the Z[prime] mass and width.
For the convenience of the reader, we provide a plain text version of j95% as a function of mZ[prime] and in the supplementary material accompanying this paper. The advantage of these bounds is that they are independent of other interactions that may be present in a given Z[prime] model.
This discussion concludes the general analysis of dijet constraints. The remainder of the paper is dedicated to a speci c application of the results shown above, which serves as an illustration for the procedure described above and uses this procedure in order to constrain a model of particular interest.
3 Constraints on a leptophobic Z[prime] coupling to DM
We now show how the results from the previous section can be applied to a speci c model. For this purpose, we consider a simple model of a leptophobic Z[prime] coupling to DM (see [44{ 47]), which is similar in spirit to the spin-1 s-channel simpli ed DM model discussed in refs. [18, 20, 23, 26, 27, 29, 30]. Assuming DM to be a Majorana fermion , the interactions between the SM and the dark sector are de ned by the following Lagrangian:
Lkin = i 2
@
1
2mDM
14F [prime] F [prime] +
1
2m2Z[prime] Z[prime]Z[prime] (3.1)
Lint =
1
2gADMZ[prime]
5 gqZ[prime]
Xq q q . (3.2)
The Majorana nature of the DM particle ensures that it can only have an axial coupling to the Z[prime], which signi cantly reduces constraints on the model from direct detection experiments. On the SM side, the couplings of the Z[prime] are assumed to be purely vectorial, which is consistent with the assumption that the Z[prime] couples neither to leptons nor to the
{ 9 {
SM Higgs [48]. We do not specify the additional dark Higgs necessary to generate the Z[prime] mass and the DM mass [48], assuming that this particle is su ciently heavy and su -ciently weakly mixed with the SM Higgs to be irrelevant for LHC phenomenology. While additional heavy fermions are needed to cancel anomalies, these can be colour-neutral and vector-like with respect to the SM gauge group, making them very di cult to observe at the LHC (see [12] for a discussion of anomaly-free models).
The resulting decay widths are then given by7
(Z[prime] ! q
q) = mZ
mZ[prime]
where we have assumed mZ[prime] > 2mt, 2mDM. The equations above enable us to calculate the total decay width , which is required in order to apply the dijet bounds derived above.
3.1 Bounds for xed couplings
The model introduced above has four free parameters (the two masses mZ[prime] and mDM and the two couplings gq and gDM). Since kinematic distributions at the LHC depend more sensitively on the masses than on the couplings, it is interesting as a rst step to study LHC constraints for xed couplings and varying masses. This approach is consistent with the most common way of presenting LHC constraints on DM simpli ed models.
Note however that our model is not identical to any of the simpli ed models presently used by the ATLAS and CMS collaborations, because we consider a Majorana DM particle and a di erent coupling structure (in our model the Z[prime] has vector couplings to quarks and axial couplings to DM). Using a Majorana fermion instead of a Dirac fermion leads to an invisible width smaller by a factor of two giving slightly stronger bounds from dijet searches. The di erent coupling structure is not expected to signi cantly alter dijet bounds which typically depend on (gVq )2 + (gAq)2 (see above). It does, however, change the relic density constraint compared to the one obtained for the simpli ed models used by the experimental collaborations.
Following the recommendations from [30, 49], we consider the case qq = 0.25, gDM = 1. For these couplings the width of the Z[prime] varies between 2.5% (for m[prime]Z < 2mDM, 2mt) and4.3% (for m[prime]Z 2mDM, 2mt) of its mass. For each combination of mZ[prime] and mDM, we
calculate the width and then read o the largest allowed value for gq from gure 2. Whenever gq,95% < 0.25, the parameter point is excluded by our combined dijet bounds.
The results of this analysis are shown in gure 3 with the dijet excluded regions shown in red. We nd that Z[prime] masses between 500 GeV and 1600 GeV are excluded irrespective of the value of mDM. For mZ[prime] between 1600 GeV and 3 TeV, the model is excluded for heavy DM particles, such that the invisible branching ratio of the Z[prime] is small and decays
7The pre-factor 1/(24) for the decay into DM results from the fact that there are two identical particles in the nal state.
{ 10 {
[prime] g2q 4
s1 4m2q m2Z[prime]
1 + 2 m2q m2Z[prime]
3/2, (3.4)
[parenrightBigg]
(3.3)
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(Z[prime] ! ) =
24 (gADM)2 [parenleftbigg]
1
4m2DM m2Z[prime]
2000
1000
perturbativity bound
Wh2=0.12
m DM[GeV]
gDM=1
gq=0.25
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0 0 1000 2000 3000 4000 mZ[GeV]
Figure 3. Excluded regions of parameter space in the mass-mass plane for xed couplings, following the recommendations of the DM LHC working group [49]. The region in red is excluded by our combined dijet analysis at the 95% con dence level while the green lines represent parameter points which reproduce the observed relic density of DM in the Universe. In the grey region perturbative unitarity is violated.
into dijets dominate. These bounds are somewhat stronger than the ones found by the individual experiments due to our combined analysis.
For comparison, we also show the parameter region (in grey) where the coupling of the DM particle to the longitudinal component of the Z[prime] violates perturbative unitarity [25, 48] and the masses for which the assumed couplings reproduce the observed DM relic abundance, h2 = 0.12 [50], calculated using micrOMEGAs v4.1.8 [51]. Details on the relic density calculation can be found in the following subsection.
We nd that there are only two regions where the relic density is compatible with dijet constraints: a low-mass region with mZ[prime] < 500 GeV and mDM < 150 GeV and a high-mass region with mZ[prime] > 3 TeV and mDM > 1200 GeV.8 This conclusion is, however, obviously dependent on our choice of couplings. For example, smaller values of gq would reduce the production cross section of the resonance, while larger values of gDM would increase the invisible branching ratio (provided mDM < mZ[prime] /2), so that dijet constraints could be signi cantly relaxed.
To study whether the intermediate mass range can be made compatible with dijet constraints for di erent choices of couplings, one could in principle repeat the analysis from above for many di erent combinations of gq and gDM (or simply scan the entire parameter space). Instead, we will take a more systematic approach and develop a new method that can be used to establish the compatibility of relic density constraints and dijet searches across the entire parameter space of our model (a similar comparison between LHC searches
8We emphasise that in the low-mass region there may be additional dijet constraints from previous hadron colliders, as well as constraints from dijet resonances produced in association with SM gauge bosons [11, 25]. Moreover, this region of parameter space is tightly constrained by mono-X searches, in particular searches for jets in association with missing transverse energy [52{55]. These searches are very sensitive to Z[prime] masses below about 1 TeV, but lose sensitivity very quickly towards larger masses, where dijet constraints can still be sensitive.
{ 11 {
for dilepton resonances and the DM relic abundance in the context of gauged B L has
been performed in [56]).
3.2 Combining di-jet bounds and relic density
Out of the four-dimensional parameter space of our model, we are particularly interested in those combinations of masses and couplings for which the thermal freeze-out of the DM particle can reproduce the relic abundance
h2 = 0.1188 [notdef] 0.0010 (3.5) which is the result from Planck CMB observations combined with Baryon Acoustic Oscillations, supernova data and H0 measurements [50]. We will approximate the relic density as h2 = 0.12 in the rest of this work.
We emphasise that the relic density requirement can be relaxed if the dark sector consists of multiple components or if the thermal history of the Universe is non-standard. Nevertheless, it is certainly of interest to consider those parameters for which the simplest assumptions are already su cient to match observations. If these parameters can be excluded experimentally, the model would require additional ingredients in the dark sector (such as additional annihilation channels, additional stable states or a mechanism to produce additional entropy after DM freeze-out), which by itself would be an important conclusion.
The remainder of this section focusses on how to reduce the parameter space of our model by imposing the relic density constraint. We rst discuss some general aspects of the relic density calculation and then introduce a convenient set of free parameters that can be used to combine the relic density requirement with dijet constraints. Finally, we apply the dijet constraints from above to place bounds on the simple thermal freeze-out scenario.
To rst approximation, we can obtain the relic density by calculating the cross section for DM annihilation into a pair of quarks, !q q, and expanding the result in terms of
the relative velocity v of the two DM particles:
!q q v a + b v2 + O(v4) . (3.6)
The relic abundance is then approximately given by
h2 [similarequal] 1.07 [notdef] 109 GeV1
xfo
MPlpg (a + 3b/xfo)
JHEP09(2016)018
, (3.7)
where xfo 20{30 is the ratio of the DM mass and the freeze-out temperature and g
80{90 is the number of relativistic degrees of freedom during freeze-out. For our model, we nd a = 0 (due to the Majorana nature of DM) and
b = 3 (gDM)2 g2q
12
(m2q + 2 m2DM)(1 m2q/m2DM)1/2
(m2Z[prime] 4m2DM)2 + ( mZ[prime] )2[bracketrightbig]. (3.8)
For mDM mZ[prime] /2, the denominator in eq. (3.8) becomes very small and DM an
nihilation receives a resonant enhancement. In this case, an expansion in terms of the
{ 12 {
1
1
mZ=1 TeV, mDM=200 GeV
mZ=1 TeV, mDM=300 GeV
mZ=1 TeV, mDM=400 GeV
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mZ=2 TeV, mDM=400 GeV
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gq
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gDM
Figure 4. Curves of constant relic density h2 = 0.12 in the plane of the two couplings for xed masses of the dark matter particle and a mediator mass of 1 TeV (left) and 2 TeV (right).
velocity of the two DM particles is insu cient for an accurate calculation of the relic density and numerical methods are needed. We therefore calculate the relic density using micrOMEGAs 4.1.8 [51], including two modi cations under the instruction of the authors (see appendix B).9
For given mZ[prime] , mDM and gDM we can then numerically determine the value of gq that is required to reproduce the observed relic density.10 As long as mDM is well below the resonance region, i.e. mDM mZ[prime] /2, eq. (3.8) implies that the annihilation cross section
is proportional to g2q g2DM m2DM/m4Z[prime]. Therefore it is always possible to x gq in such a
way that the observed value of h2 is matched and the solution is always unique. In the resonance region, the annihilation cross section is proportional to g2q g2DM/ , which is still a monotonic function of gq so that any solution is unique. However, since the expression g2q g2DM/ remains nite for gq ! 1, it is possible that no solution exists. In short, as long
as gDM is large enough, there will always be a unique value of gq that reproduces the relic abundance.
We therefore obtain a function gq(mZ[prime] , mDM, gDM), which is illustrated in gure 4 as a function of gDM for various xed values of mZ[prime] and mDM. The resulting curves have the following features:
1. Since the annihilation cross section grows monotonically with the DM mass (for xed couplings and mediator mass), the lines for di erent DM masses never cross, i.e. smaller values of mDM always require larger couplings.
2. For su ciently small DM masses, the curves are hyperbolas (gq / 1/gDM), whereas
for larger values of mDM, the curves are steeper at small gDM and atter at large gDM due to the resonance e ects discussed above.
We will make use of these properties below to choose a particularly convenient set of free parameters for the analysis of our model.
9We thank Alexander Pukhov for providing us with these modi cations and for his help in the implementation.
10The width is determined internally by micrOMEGAs in a self-consistent way.
{ 13 {
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3.2.1 Relic density constraints for a xed width
Having constructed the function gq(mZ[prime] , mDM, gDM) from the relic density requirement, one could now simply proceed to scan the remaining three-dimensional parameter space.
One subtlety arises, however, from the fact that the width | and therefore the bound from dijet constraints | depends on all three parameters in a non-trivial way. For example, for xed mZ[prime] and mDM one would naively expect stronger dijet constraints for smaller gDM corresponding to larger gq (implying both a larger production cross section of the resonance and a larger branching fraction into dijets). However, if at the same time increases, it is conceivable that dijet constraints are weakened su ciently to evade experimental bounds and that in fact larger values of gq are less constrained than smaller couplings.
To avoid this complication, we take both mZ[prime] and as free parameters. As shown in gure 2, for xed values of these two parameters we can always place an unambiguous upper bound on gq. A second important advantage of this approach is that mZ[prime] and are the two parameters that are most directly observable at the LHC. While the DM mass is very di cult to measure at the LHC and coupling constants can only be inferred in the context of a speci c model, an observation of a new resonance in the dijet channel would immediately enable us to determine the mass and the width of the mediator from the invariant mass distribution.11 To be able to directly interpret such an observation in the context of the present model, it therefore makes sense to construct all bounds in terms of these two most apparent observables.
In order to treat the width as a free parameter, we need to determine those combinations of mDM, gDM and gq that reproduce the observed relic density while at the same time matching the required width. For this purpose we rst of all observe from eqs. (3.3){(3.4) that for xed mZ[prime] and mDM < mZ[prime] /2 the total width is an ellipse in the couplings.12 We can now consider ellipses of constant width in the same gq-gDM-plane used in gure 4 to study the relic density constraints. Since the relic density curve is convex while the constant-width curve is concave, the two curves will have either exactly two intersects or zero intersects (neglecting those special cases where the two curves just touch at exactly one point). In other words, for xed values of mZ[prime] , and mDM, there is either no combination of gq and gDM that reproduces the relic density constraint or there are two separate solutions corresponding to the desired value of . Whenever there are two solutions, we de ne Solution I to be the one with larger gDM (and therefore smaller gq) and Solution II to be the one with smaller gDM (larger gq). Two examples are shown in gure 5.
As noted above, increasing the value of mDM will shift the relic density curve towards smaller couplings. Conversely, the constant-width curve will be shifted to larger couplings (due to the larger phase-space suppression of Z[prime] ! ). This means that for each value of
mZ[prime] and there is a minimum value of mDM, called mDM,min(mZ[prime] , ), such that there is no solution for mDM < mDM,min and two solutions for mDM > mDM,min. Increasing mDM beyond mDM,min will shift Solution I to larger values of gDM and smaller values of gq and
11This argument assumes that the width of the resonance is large compared to the detector resolution. Nevertheless, for a narrow resonance it is still possible to determine the mass and place an upper bound on the width.
12For mDM mZ[prime] /2, the total width is independent of gDM and hence a straight line gq = const.
{ 14 {
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Wh
2 = 0.12 (
mZ
= 1 TeV,
m
DM
= 300 GeV)
G
= 50 GeV (
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0.1
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0 0 1 2 3
gDM
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Figure 5. Examples of how we nd pairs of couplings that satisfy the relic density constraint (blue) for a given xed width (red).
0.00 500 1000 1500 2000 2500 3000 3500 4000
0.30
mDM,min /mZ'
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/m Z '
0.15
0.10
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0.00 500 1000 1500 2000 2500 3000 3500 4000
mZ' [GeV]
mZ' [GeV]
Figure 6. Left: the minimum value of the DM mass (in units of mZ[prime] ) required in order to simultaneously satisfy the relic density constraint and reproduce the assumed Z[prime] width. For smaller DM masses, the relic density curve and the constant width curve do not intersect in the gDM-gq
plane (see gure 5). Right: the smallest value of gDM that can reproduce the observed relic density and the assumed width. This value corresponds to Solution II for mDM slightly below mZ[prime] /2.
vice versa for Solution II. As mDM approaches mZ[prime] /2, Solution I will yield arbitrarily large values of gDM and thus ultimately violate the perturbativity bound gDM < p4. Solution II, on the other hand, will approach a small but non-zero minimum value of gDM, called
gDM,min.13 Figure 6 shows both mDM,min and gDM,min as a function of mZ[prime] and .
With these considerations in mind, we can now eliminate either mDM or gDM in
favour of and proceed with either (mZ[prime] , , gDM), where gDM,min < gDM < p4, or with (mZ[prime] , , mDM), where mDM,min < mDM < mZ[prime] /2. While in the rst case we nd a single value of gq for each set (mZ[prime] , gDM) compatible with the relic density constraint and consistent with the required width, the second case yields two separate solutions as discussed above. We discuss both possibilities below, as they each o er di erent physical insights.
13Note that in fact the resonant enhancement of the annihilation cross section is maximal (and hence the coupling gDM is minimal) for mDM slightly below mZ[prime] /2. We determine the DM mass corresponding to gDM,min numerically.
{ 15 {
0.30
0.25
0.20
0.15
0.10
0.05
0.00 500 1000 1500 2000 2500 3000 3500 4000
mZ' [GeV]
Figure 7. Lower bound on the DM coupling gDM from the combination of the relic density constraint and LHC dijet searches. In the orange shaded region, gDM,min becomes non-perturbative,
i.e. all perturbative values of gDM are excluded by LHC dijet searches.
3.2.2 Dijet bounds on the DM coupling
We have shown above that for xed mZ[prime] and smaller values of gDM correspond to larger values of gq. Using gure 2 to read o the upper bound on gq from dijet searches thus allows us to place a lower bound on gDM. This lower bound on gDM is shown in gure 7. Wherever no bounds from dijet searches can be placed, we simply show the smallest value of gDM for which the relic density curve and the constant-width curve intersect (called gDM,min above). If on the other hand the lower bound from dijet searches is so strong that it requires gDM to be larger than p4, we conclude that it is impossible to nd perturbative values of gq and gDM such that the width and the relic density can be reproduced without violating dijet constraints. The corresponding regions are shaded in orange in gure 7.
We observe that rather large values of gDM are required in order to avoid dijet constraints. While the consistency of the relic density requirement and the assumed width only required gDM,min 0.1{0.3 (see gure 6), dijet constraints require gDM > 1 in almost the entire parameter space that we consider. For large Z[prime] width, even larger values of gDM are required in order to reduce the branching ratio of the Z[prime] into dijets. For mZ[prime] [lessorsimilar] 1.5TeV and /mZ[prime] [greaterorsimilar] 0.2 as well as for 1.7 TeV [lessorsimilar] mZ[prime] [lessorsimilar] 3.3 TeV and /mZ[prime] [greaterorsimilar] 0.25, all perturbative values of gDM that reproduce the relic abundance are excluded by dijet searches. For larger Z[prime] masses, LHC dijet searches lose sensitivity, but signi cant improvements in this mass range can be expected from upcoming runs of the LHC at 13 TeV.
3.2.3 Dijet bounds on the DM mass
Let us nally present our results from a complementary perspective by taking mDM as a
free parameter and determining both gDM and gq from the relic density constraint and the requirement of a constant width. As discussed above, for each value of mDM we obtain two separate solutions, with Solution I (II) corresponding to larger (smaller) gDM. For each of
{ 16 {
gDM,min
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)1/2
g DM ,min>(4 )1/2
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mDM,min
m DM/m Z '
Dijetexcluded
(m DM ,max<m DM ,min)
0.00 0.05 0.10 0.15 0.20 0.25 0.30
/mZ' [GeV]
Figure 8. Maximum (blue, dashed) and minimum (orange) allowed value of the DM mass as a function of the mediator width for mZ[prime] = 1TeV using Solution I (larger values of gDM). The dotted orange line indicates the bound on mDM,min in the absence of LHC dijet constraints (see gure 6).
the two solutions, we can directly read o from gure 2 whether the parameter point is excluded by the combined dijet constraints that we have derived above. These exclusion limits in turn allow us to determine the allowed range of DM masses as a function of mZ[prime]
and . We now discuss the two di erent solutions in turn.
As noted above, for Solution I (i.e. larger values of gDM), the DM coupling increases with the DM mass. The requirement to have a perturbative coupling, gDM < p4, therefore
gives an upper bound on mDM, called mDM,max. For some values of mZ[prime] and we nd that Solution I yields a non-perturbative value of gDM for all values of the DM mass, so for these combinations of Z[prime] mass and width only Solution II is of interest.
Conversely, for Solution I smaller DM masses correspond to larger values of gq. Since large values of gq are excluded by LHC dijet searches, we can use the LHC bounds to place a lower limit on mDM, called mDM,min.14 The combination of the perturbativity requirement and LHC dijet searches therefore yield a range of permitted dark matter masses [mDM,min, mDM,max], which satisfy all of our constraints. In other words, for mDM
in this range, it is possible to nd values of gq and gDM that yield the observed relic abundance and are consistent with all other constraints that we consider.
Figure 8 shows one example, where we have xed the Z[prime] mass to 1 TeV and show mDM,min (orange) and mDM,max (blue, dashed) as a function of . To illustrate the impact of dijet searches, we also show the value of mDM,min that one obtains solely from the
consistency of relic density and constant width (orange, dotted). For large values of we nd that mDM,max < mDM,min, i.e. all perturbative solutions are excluded by LHC dijet
searches. In the speci c case under consideration, this occurs for /mZ[prime] [greaterorsimilar] 0.19.
Figure 9 shows the largest allowed DM mass (left) and the smallest allowed DM mass (right) as a function of mZ[prime] and . The plots can be read by picking a point on the plane(i.e. xing mZ[prime] and ) and then reading of mDM,max and mDM,min from the two panels to nd the range of permitted dark matter masses [mDM,min, mDM,max] that satisfy all
constraints. Those combinations of mZ[prime] and for which mDM,max < mDM,min are shaded
14Note that even if LHC dijet searches are not constraining, there is always a lower limit on mDM from the requirement that the relic density curve and the constant-width curve intersect in the gq-gDM plane.
{ 17 {
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mDM,max/mZ' (larger gDM solution)
Only non-perturbative solutions
Dijet excluded (mDM,max < mDM,min)
0.45
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Dijet excluded (mDM,max < mDM,min)
0.4
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0.05
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0.00 500 1000 1500 2000 2500 3000 3500 4000
0.00 500 1000 1500 2000 2500 3000 3500 4000
mZ' [GeV]
mZ' [GeV]
Figure 9. Maximum (left) and minimum (right) allowed value of the DM mass as a function of the mediator mass and width using Solution I (larger values of gDM).
in orange. The grey region indicates those combinations of Z[prime] mass and width for which no perturbative solutions can be found. As expected, the orange shaded region is identical to the one found in gure 7.
Turning now to Solution II, we note that for this solution perturbativity constraints will typically be less important (because we consider smaller values of gDM), while dijet constraints will be more important (because the corresponding values of gq are larger).15 Compared to the previous solution, the situation is now reversed: the requirement of perturbativity may raise mDM,min, while dijet constraints will lower mDM,max. We show the maximum and minimum allowed DM masses for Solution II in gure 10.
As expected, we nd dijet constraints (shown in orange) to be signi cantly stronger than for Solution I. For large width, the entire range 500GeV mZ[prime] 3500GeV is excluded
by dijet constraints. For mZ[prime] 1200 GeV the dijet bounds extend down to very narrow
resonances. As discussed above, dijet bounds are particularly weak around 1600 GeV, due to an intriguing upward uctuation in the data. Finally, we note that we can always nd a value of the DM mass such that Solution II corresponds to perturbative couplings (so the grey shaded region from gure 9 is absent).
4 Conclusions
We have presented a combination of all available searches for dijet resonances at the LHC in the context of a generic Z[prime] model. Taking the width of the resonance and its coupling to quarks as independent parameters allows us to obtain constraints that apply irrespective of whether the Z[prime] decays exclusively into quarks or dominantly into other states. The results of this analysis, summarised in gure 2 and table 3, are provided in such a way that they can be easily used to constrain a range of di erent models.
15For the widths that we are considering, /MZ[prime] 0.3, gq is always less than unity, so we never run into
problems with the perturbativity of gq.
{ 18 {
mDM,max/mZ' (smaller gDM solution)
mDM,min/mZ' (smaller gDM solution)
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0.00 500 1000 1500 2000 2500 3000 3500 4000
0.00 500 1000 1500 2000 2500 3000 3500 4000
mZ' [GeV]
mZ' [GeV]
Figure 10. Maximum (left) and minimum (right) allowed value of the DM mass as a function of the mediator mass and width using Solution II (smaller values of gDM).
As a speci c illustration of our approach, we have applied our constraints to a Z[prime] that couples to quarks and dark matter (DM), similar in spirit to a DM simpli ed model with a spin-1 s-channel mediator. It is straight-forward to map our constraints onto the parameter plane showing DM mass versus mediator mass for xed couplings, which is conventionally used to present LHC results from missing energy searches. We show that for the typical choice of couplings (gDM = 1, gq = 0.25), dijet searches can exclude the range 500GeV < mZ[prime] < 3TeV for almost all values of the DM mass (see gure 3). These ndings suggest that future searches for simpli ed DM models should focus on smaller values of gq and larger gDM, which would relax constraints from searches for dijet resonances while still allowing for sizeable interactions between DM and quarks.
Finally, we have focussed on the special case that the Z[prime] mediates the interactions of DM and quarks responsible for thermal freeze-out, so that one of the parameters of the model can be eliminated by the requirement to reproduce the observed relic abundance. We have constructed a novel way of studying this set-up by making explicit the parameters that can be directly probed by searches for dijet resonances, i.e. the mass and the width of the Z[prime]. The remaining free parameter can then be taken to be either the DM coupling ( gure 7)
or the DM mass ( gures 9 and 10). We nd that for very broad widths ( /mZ[prime] [greaterorsimilar] 0.25) and Z[prime] masses below about 3 TeV, LHC searches already exclude the possibility that the DM-quark interactions mediated by the Z[prime] are responsible for setting the DM relic abundance.
Furthermore, these gures provide a useful tool for interpreting future searches for dijet resonances at the LHC. Should an excess be seen in such a search, the mass and the width of the resonance can be determined from the data in a model-independent way. One can use these gures to look up whether the new state could conceivably act as the mediator into the dark sector. If a solution to the relic density requirement exists, the plots then provide the allowed ranges of the DM mass and coupling. Presently there is still ample room for such an interpretation, so there is much to be learned from the upcoming LHC data at 13 TeV.
{ 19 {
Acknowledgments
We thank Antonio Boveia, Caterina Doglioni and Sungwon Lee for answering our questions on the various LHC dijet searches and Alexander Pukhov for signi cant help with micrOMEGAs. FK is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe. MF and JH acknowledge support from the STFC. MF and PT are funded by the European Research Council under the European Unions Horizon 2020 program (ERC Grant Agreement no.648680 DARKHORIZONS).
A Tabulated bounds on gq
We provide the numerical values of the dijet constraints obtained in section 2 in table 3.
B Speci c modi cations of micrOMEGAs
In this appendix we detail two modi cations to CalcHEP [57], which is used by micrOMEGAs to calculate the cross sections for DM pair annihilation. Both modi cations are necessary in order to correctly treat the width of the Z[prime] close to the resonance (i.e. for mDM mZ[prime] /2).
The rst modi cation is necessary to avoid numerical instabilities leading to kinks in the curves of constant relic density as a function of gDM and gq for xed mDM and mZ[prime]
(see gure 4). These kinks arise due to the way the Breit-Wigner (BW) propagator is implemented in CalcHEP. The standard BW distribution for a particle with mass m and momentum q is given by
[notdef]M[notdef]2 /
1(q2 m2)2 + m2 2
, (B.1)
where the term involving removes the divergence as the particle becomes on-shell (q2 ! m2). However, since the width is a sum of diagrams of varying orders, its presence in eq. (B.1) can spoil gauge invariance. For this reason CalcHEP implements the BW formula as a piecewise function over three regions: formula B.1 is used with a non-zero width for
|q2 m2[notdef] < R m , where R is an arbitrary number that is by default xed to 2.7. For |q2 m2[notdef] > pR2 + 1 m , on the other hand, the width in eq. (B.1) is set to zero. In the
intermediate region the width is replaced by a function of q2 that interpolates between the two cases. For fairly large widths, as considered in the present work, this interpolation procedure can lead to kinks in the relic density calculation. To remove such kinks one can simply increase the value of R from its default value by changing the value of the variable BWrange [58]. We have found that R = 100 is su cient to remove the kinks in our plots.16
Furthermore, as pointed out in ref. [31], the width of the resonance depends in general on the momentum transfer q2, i.e. = (q2). For the case of narrow widths ( /m 1),
eq. (B.1) gives a good approximation, because is only relevant for q2 m2 and one can
therefore approximate (q2 = m2). Since we consider widths as large as 30% in this
16Another e ect that could lead to kinks in the relic density curves was due to a typographical mistake in micrOMEGAs 4.1.8, which has been xed in the most recent version.
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JHEP09(2016)018
MZ[prime] [GeV] 10 [notdef] a(MZ
500 2.4295 2.208 0.0588 550 1.3808 1.760 0.0712 600 0.7648 1.452 0.0000 650 0.5251 1.496 0.0584 700 0.4153 1.389 0.0000 750 0.4266 1.375 0.0000 800 0.4865 1.386 0.0000 850 1.2889 2.047 0.2376 900 0.3078 1.259 0.0000 950 0.7027 1.729 0.0540 1000 0.5892 1.341 0.0743 1100 0.4600 1.183 0.0000 1200 0.3674 1.334 0.0000 1300 1.4714 1.879 0.0809 1400 1.8096 1.723 0.1545 1500 4.4052 1.920 0.1901 1600 13.7015 1.989 0.5733 1700 5.4250 1.468 0.0000 1800 5.1603 1.729 0.2606 1900 4.8469 1.751 0.3736 2000 4.7523 1.629 0.3762 2100 3.3313 1.425 0.0000 2200 3.9147 1.458 0.1891 2300 4.9732 1.550 0.3758 2400 4.9159 1.588 0.4994 2500 3.3318 1.450 0.3996 2600 3.5345 1.509 0.3922 2700 3.9016 1.565 0.4383 2800 3.2388 1.440 0.4402 2900 2.6469 1.318 0.6291 3000 3.0428 1.315 0.7375 3100 3.5767 1.326 1.1560 3200 2.6266 1.129 0.6442 3300 4.0536 1.286 0.6851 3400 6.1825 1.421 1.3730 3500 3.7765 1.162 0.5939 3600 5.0627 1.262 1.3046 3700 6.5994 1.307 2.1257 3800 6.8087 1.191 3.0885 3900 5.6611 0.936 0.0000 4000 9.5274 1.061 0.0000
Table 3. Numerical values of the t to the constraint on the quark coupling as a function of width outlined in eq. (2.7).
{ 21 {
[prime] ) b(MZ[prime] ) 1000 [notdef] c(MZ
[prime] )
JHEP09(2016)018
work, it is however appropriate to modify the BW formula in order to take the momentum dependence of the width into account.
Following appendix A of [31], we can approximately capture the momentum dependence by setting (q2) = pq2
m (q2 = m2). This substitution yields
[notdef]M[notdef]2 /
1
(B.2)
for the shape of the BW resonance. This modi cation can be implemented by editing the function prepDen used in the CalcHEP code.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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{ 25 {
JHEP09(2016)018
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SISSA, Trieste, Italy 2016
Abstract
Abstract
We analyse a combination of ATLAS and CMS searches for dijet resonances at run I and run II, presenting the results in a way that can be easily applied to a generic Z' model. As an illustrative example, we consider a simple model of a Z' coupling to quarks and dark matter. We first study a benchmark case with fixed couplings and then focus on the assumption that the Z' is responsible for setting the dark matter relic abundance. Dijet constraints place significant bounds on this scenario, allowing us to narrow down the allowed range of dark matter masses for given Z' mass and width.
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