Constraints on Mediator Coupled to Heavy Quarks from LHC Data

We apply LHC data to constrain a simplified extension of the Standard Model containing a new spin-1 mediator $R$, which does not couple to first generation quarks, and a spinor dark matter particle $\chi$. We recast ATLAS and CMS searches for final states containing one or more jet(s) + missing $E_T$, with or without $b$ tags, as well as searches for di-jet resonances with $b$ or $t$ tagging. We find that LHC constraints on the axial vector couplings of the mediator are always stronger than the unitarity bound, which scales like $m_R/m_t$. If $R$ has a sizable invisible branching ratio, the strongest LHC bound on both vector couplings and axial vector coupling comes from a di-jet + missing $E_T$ search with or without double $b$ tag. These bounds are quite strong for $m_R<1$ TeV, even though we have switched off all couplings to valence quarks. Searches for a di-jet resonance with double $b$ tag lead to comparable bounds with the previous results even if $R \rightarrow \chi \bar \chi$ decays are allowed; these are the only sensitive LHC searches if the invisible branching ratio of $R$ is very small or zero.


Introduction
Simpli ed models of particle dark matter o en need a mediator coupling the dark matter particle χ to some particles in the Standard Model (SM). Models where the mediator couples to both quarks and leptons are strongly constrained by LHC searches for + − resonances, where stands for a charged lepton [ -]. This motivates the investigation of "leptophobic" models, where the mediator does not couple to leptons. In case of a spin−1 mediator R, universal couplings to all quarks are o en assumed. If R has a sizable branching ratio into invisible nal states, which is generally true if m R > 2m χ , the allowed vector and axial vector couplings are then strongly constrained by mono-jet searches [ , ] unless m R is well above 1 TeV. For mediator mass between 1 and 2.5 TeV, searches for di-jet resonances [ , ] perform even better. Additionally, the constraints from spin-dependent and spin-independent interactions in direct detection experiments imposes strong constraints on couplings to rst generation quarks [ ]; these bounds scale like m R .
In our previous study [ ], which applies LEP data to probe the low m R region, we therefore switched off all couplings to rst generation quarks and axial vector couplings to second generation quarks in order to avoid an excess in direct DM detection experiments. Axial vector couplings lead to spin-dependent contributions to the scattering cross section, which also receive a sizable contribution from strange quarks, whereas vector couplings lead to spin-independent contributions which only probe u and d quarks in the nucleon [ ]. Moreover, the non-zero couplings to other quarks are still available to generate a sizable annihilation rate to explain the observed dark matter relic density through thermal freeze-out. By switching off couplings to rst generation quarks, and hence to all valence quarks, we greatly reduce the cross sections for pp scattering processes with an R boson in the intermediate or nal state. The published bounds from the LHC experiments, which assume equal couplings of R to all quarks, are therefore no longer valid.
The goal of this article is to estimate the LHC constraints on this model. We showed in ref.
[ ] that LEP data impose strong constraints only for m R < 10 GeV, and become entirely insensitive for m R > 70 GeV. Here we therefore focus on scenarios with m R ≥ 10 GeV. The relevant searches we exploit are similar to those that constrain scenarios with avor-universal couplings of R: mono-jet + / E T searches, di-jet + / E T searches and di-jet resonance searches. By switching off couplings to light quarks, we increase the branching ratio for R → bb or tt decays. Since in background events most jets originate from light quarks or gluons, b or t tagging can increase the signal to background ratio even for avoruniversal couplings of R, and should be even more helpful in our case.
The reminder of this article is organized as follows. In Sec. , we brie y describe the Lagrangian of the simpli ed model containing a leptophobic mediator, which does not couple to rst generation quarks. The application to the relevant LHC data is discussed in Sec. . The LEP result and the tightest unitarity condition from top quark are compared to the LHC exclusion limits we estimate. Finally, Sec. contains our summary and conclusions.

The Simpli ed Model . Lagrangian and Free Parameters
A spinor dark sector particle (DSP) and a new spin−1 mediator connecting DSP to SM particles are introduced to extend the SM. Therefore the total Lagrangian is given by: Since we use MadGraph [ ] to generate the Monte Carlo Events, the kinetic terms in the Lagrangian follow the default convention in MadGraph. The mediator part of the Lagrangian is thus: In order to allow both vector and axial-vector couplings the DSP should be a Dirac fermion, because Majorana fermions cannot have a vector interaction. Again using MadGraph convention, the corresponding piece of the Lagrangian is Finally, the interaction terms are In this model DSPs can scatter off nucleons via R exchange. The corresponding spinindependent and spin-dependent cross sections are to leading order in perturbation theory [ ]: The coef cients a n/p and f n/p depend on different combinations of couplings: ∆u (p) = ∆d (n) = 0.84 ± 0.02 ; ( ) ∆u (n) = ∆d (p) = −0.43 ± 0.02 ; ∆s (p) = ∆s (n) = −0.09 ± 0.02 .
Eqs.( ) show that setting g V u, d = 0 suf ces to make the leading order spin-independent cross sections on protons and neutrons vanish. On the other hand, eqs.( ) show that g A u, d, s = 0 is needed in order to "switch off" the leading order spin-dependent cross sections; weak SU (2) invariance then implies g A c = 0 as well. This leaves us with seven free parameters: m R and m χ . However, since R does not couple to leptons signals involving missing transverse energy / E T require a pair of DSPs in the nal state. Since SM Z boson couple to all quarks, nal states with an R boson replaced by an invisible decaying Z boson will always contribute to (and indeed o en dominate) the background to these signals. Clearly the signal can only compete with this background from on-shell Z bosons if on-shell R → χχ decays are possible. The relevant quantity is then the branching ratio for these decays, rather than the couplings g V χ and g A χ separately. Moreover, the DSP mass m χ also affects the signal only through this branching ratio. This observation implies that replacing the Dirac DSP χ by a complex scalar φ is trivial, since again only the branching ratio for R → φφ decays is relevant in that model. Turning to quark couplings, we assume all non-vanishing couplings to be equal. In case of axial vector couplings, this can again be motivated by SU (2) invariance. This would still allow different, non-vanishing second and third generation vector couplings, but we set them equal for simplicity. Note that the case g V s = g V c = 0 would give very similar results as the scenario with non-vanishing axial vector couplings. The reason is that contributions to the relevant matrix elements from g V q and g A q differ only by terms of the order m q /Q, where Q is the energy scale of the process. Since the parton distribution function for top quarks is still very small at the energies we are interested in, and top tagging turns out to be quite inef cient, the relevant quark is the b quark, and m b /Q 1 for all cases of interest to us. The main difference between vector and axial vector couplings is therefore that in the former case couplings to second generation quarks are included, while these couplings vanish in the latter case.
Finally we are therefore le with four relevant free parameters: g V q , g A q , Br(R → χχ) and m R . Since the parton distribution functions for second generation quarks in the proton are signi cantly larger than those for third generation (basically, b) quarks, for xed size of the nonvanishing couplings we expect much smaller total cross sections for the case g V q = 0, g A q ≡ g q than for the case g A q = 0, g V q ≡ g q . On the other hand, scenarios with g V = 0 should have higher ef ciency for b tagging, which is required in some searches. .

Perturbativity and Unitarity Conditions
We will use leading order tree-level diagrams in the simulation. Therefore, the relevant couplings should not be too big, so that the perturbation theory is reliable. We impose the simple perturbativity condition where Γ R is the total decay width of the mediator. The partial width for R → ff decay, where f is some fermion, is: Since we use Br(R → χχ) as free parameter instead of m χ and g V,A χ , the perturbativity condition can be written as In the next section, we will only discuss the mass range where the bounds for vector or axial vector couplings are smaller than 2; this satis es the perturbativity condition.
Another important theoretical constraint on the parameters in the Lagrangian originates from demanding that unitarity in scattering amplitudes is preserved [ ]. This constraints the axial vector couplings between the mediator and fermions: ( ) Due to the assumption of universal axial vector couplings to b and t quarks, the strongest constraint always comes from the much heavier top, and becomes quite strong for light mediator: For example, for m R = 10 GeV, g A should be smaller than 0.08. In contrast, for m R > 275 GeV the unitarity constraint becomes weaker than the perturbativity condition.

Application to LHC Data
In this section, we recast various LHC searches to constrain the model introduced in section , including a mono-jet In order to simulate the events and recast the analysis, we use FeynRules [ ] to encode the model and generate an UFO le [ ] for the simulator, MadGraph [ ] to generate the parton level events, PYTHIA [ ] for QCD showering and hadronization, DELPHES [ ] to simulate the ATLAS and CMS detectors, and CheckMATE [ , ] to reconstruct and b-tag jets, to calculate kinematic variables, and to apply cuts. We note that the toolkit CheckMATE uses a number of additional tools for phenomenology research [ -].
Let us rst discuss nal states involving missing E T . These are o en categorized as "mono-jet + / E T " and "multijet + / E T " nal states. However, the "mono-jet" searches also allow the presence of at least one additional jet. On the other hand, "multijet" searches do indeed require at least two jets in the nal state. These signals thus overlap, but are not identical to each other.
As remarked in the Introduction, missing E T in signal events always comes from invisibly decaying mediators, R → χχ. Since multijet searches require at least two jets in the nal state, we use MadGraph to generate parton-level events with a χχ pair plus one or two partons (quarks or gluons) in the nal state. The former process only gets contributions from the le diagram in g. plus its crossed versions, including the contribution from gq → Rq. Note that R has to couple to the initial quark line in this case. We use parton distribution functions (PDFs) with ve massless avors; the mass of the corresponding quarks should be set to 0 in order to avoid the inconsistency with massless evolution equations (DGLAP equations). The b−quark PDF is nonzero, but it is still considerably smaller than those of rst generation quarks. The contribution from this diagram, which is formally of leading order in α S , is therefore quite small, especially for scenarios with g V q = 0 where R only couples to third generation quarks.
If we allow the nal state to contain two partons in addition to the DSPs, there are contributions with only light quarks or gluons in the initial state; an example is shown in the middle of g. , but there are several others. These diagrams are higher order in α S , but they receive contributions from initial states with much larger PDFs than those contributing to the rst diagram. It is thus not clear a priori which of these contributions will be dominant for a given set of cuts.
There is one additional complication. At the parton level, events with one and two partons in the nal state are clearly distinct. However, once we include QCD showering, which is handled automatically by PYTHIA, the distinction becomes less clear. In particular, a single parton event with an additional gluon from showering can no longer be distinguished from a certain two parton event without additional gluon. Naively adding contributions with one and two partons in the nal state before showering can therefore lead to double counting. Similarly, if one of the nal-state quarks shown in the middle diagram of g. has small p T , the diagram can be approximated by g → qq splitting followed by gq → Rq production. This contribution is already contained in the crossed version of the le diagram of g. , via the scale-dependent PDF of q, so simply adding these diagrams again leads to double counting. MadGraph avoids both kinds of double counting by using the "MLM matching" algorithm [ ]. Of course, showering can add more than one additional parton; indeed, we nd signi cant rates for nal states with up to four jets (having transverse energy E T ≥ 35 GeV each). Examples of Feynman diagrams contributing to mono-jet + / E T (le ), di-jet + / E T (center) and di-jet resonance (right) nal states; in the former two cases it is assumed that the mediator R decays into two dark sector particles, which escape detection, whereas in the latter case R is assumed to decay into a quark antiquark pair. The diagram to the right is unique (with different initial states contributing), and the one on the le is unique up to crossing; however, many additional diagrams, with different combinations of partons in the initial and nal states and different propagators, contribute to R+di-jet production.
Searches for nal states leading to large missing E T are typical cut-and-count analyses, where the nal state is de ned by cuts on the type and number of nal state objects (in particular, leptons and jets with or without b−tag) and on kinematic quantities (in particular, the transverse momenta or energies of the jets and the missing E T ). The experiments themselves designed these cuts, and estimated the expected number of surviving SM background events. The comparison with the actually observed number of events a er cuts then allows to derive upper bounds on the number of possible signal events. We pass our simulated signal events through CheckMATE, which applies the same cuts (including detector resolution effects), and compares the results with the upper bounds obtained by the experiments.
The second kind of search we consider are searches for di-jet resonances. The leadingorder signal diagram is shown on the right in g. . In this case the nal state contains no partons besides the mediator R; for g V q = 0, only bb initial states contribute, whereas for nonvanishing vector couplings also ss and cc initial states contribute. Of course, the le and middle diagrams shown in g. also contribute to this signal if R decays into a qq pair. However, in this case one has to add two powers of α S in order to access initial states including only light quarks or gluons. Moreover, if all nal state transverse momenta are small, which maximizes the cross section, the contribution from the middle diagram is actually already included in the right diagram, via double g → qq splitting. The le and middle diagrams should therefore only be included in inclusive R production when a full NLO or even NNLO calculation is performed, which is beyond the scope of this work.
Note also that resonance searches are not cut-and-count analyses. The analyses still use a set of basic acceptance cuts, in this case on the (pseudo-)rapidities and transverse momenta of the two leading jets. The bound on resonance production is then obtained by tting a smooth function to the di-jet invariance mass distribution, which is assumed to be dominated by backgrounds, and computing the limit on a possible additional contribution peaked at a certain value (basically, the mass of the resonance). The current version of CheckMATE does not include comparison with this kind of searches. However, Check-MATE does allow to estimate the ef ciency with which our signal events pass the acceptance cuts. This allows to derive the constraints from resonance searches on our model, as follows.
The most sensitive di-jet resonance search we found is that of ref.
[ ], which requires a double b−tag in the nal state. This paper presents the resulting upper bounds for a couple of models. One of them is quite similar to ours, but assumes universal couplings to all quarks; this leads to a greatly enhanced resonance production cross section, and a somewhat reduced branching ratio into bb pairs, compared to our model. The paper also gives the cut ef ciency for the model with universal couplings. We, therefore, recast their cuts and compare the cut ef ciencies of their model and our models in order to estimate the bound for our model through the following rescaling: ( ) Here σ max, ours is the largest allowed cross section for our model, σ max, exp is the largest allowed cross section in the original experimental analysis, exp is the selection ef ciency of the model in the paper, and ours is the selection ef ciency of our model. Finally, we cannot easily reproduce the top tagging required in the di-top resonance search [ ]. However, even if we assume 100% ef ciency for the di-top tag, the resulting bound is much weaker than our recast of [ ] described in the previous paragraphs. We therefore do not show this bound in our summary plot.
The results of our analyses are summarized in g. . The thin solid lines in the top-le corner show the bounds we derived [ ] from analyses of older ALEPH searches for four jet nal states at the e + e − collider LEP; note that these bounds are valid for m R < 2m χ . The solid straight line is the unitarity bound ( ) applied to the top mass; recall that it applies only to axial vector couplings. (Since top quarks could not be produced at the LEP collider, in [ ] we only considered the unitarity constraints involving m b and m χ .) The other results shown in g. are new. The dashed curves show the bounds on the square of the coupling of R to quarks times the branching ratio for invisible R decays which we derived from the most sensitive jet(s) plus missing E T searches, for pure axial vector couplings (red, upper curve) and pure vector couplings (green, lower curve); the right frame shows the corresponding bounds on the signal cross section, de ned as the total cross section for the on-shell production of a mediator R times the invisible branching For g A = 0 only vector couplings g V q are allowed with q = s, c, b, t, while for g V = 0 only axial vector couplings g A q are allowed with q = b, t. LHC missing E T results are from the combination of mono-jet and multi-jet analyses. The right frame shows the upper bound on the total cross sections from the missing E T analyses. ratio of R. It is important to note that these constraints are only signi cant in our model if on-shell R → χχ decays are allowed, i.e. they constrain a region of parameter space that is complementary to that analyzed in ref.
The dot-dashed curves in the le frame show the bounds on the square of the coupling of R to quarks times the branching ratio for R → qq decays that result from searches for dijet resonances, again separately for pure axial vector couplings (purple, upper curve) and pure vector couplings (blue, lower curve). The relevant analysis by the ATLAS collaboration [ ] is sensitive only to m R ≥ 600 GeV.
The difference between the constraints on vector and axial vector couplings is almost entirely due to the additional coupling to s and c quarks that we allow only for the former, as discussed in Sec. . . In particular, we see that the constraint from the bb resonance search is much stronger for the model with vector couplings.
In the le frame of Fig. the curves depicting the bounds from searches for nal states containing / E T evidently lie below the ones showing bounds from di-jet resonance searches, except for the scenario with pure vector coupling at m R 2 TeV. However, this is somewhat misleading, since the dashed curves show bounds on g 2 q ·Br(R → χχ), while the dot-dashed curves shows bounds on g 2 q · [1 − Br(R → χχ)]. For m R ≥ 1 TeV the two sets of constraints on the coupling are actually comparable if Br(R → χχ) 0.3 (0.1), for pure vector (axial vector) coupling; for even smaller invisible branching ratio of R, the bb resonance search imposes the stronger constraint in this large m R region. We note that for m 2 R m 2 t and g χ = g q , i.e. equal coupling of the mediator to the DSP and to heavy quarks, the invisible branching ratio of R is below 1/7 (1/13) for pure axial vector (vector) coupling, the differ-ence being due to the different number of accessible qq nal states.
Within the missing E T searches the best bound on g V q for m R < 1.4 TeV is from ref.
[ ], a double b tagged multi-jet + / E T analysis, while ref.
[ ], a general multi-jet + / E T analysis, is the most sensitive one for m R ≥ 1.4 TeV; this change of the most sensitive analysis explains the structure in the dark green curves at that m R , which is most visible in the right frame. In contrast, the strongest bound on g A q is always from ref.
[ ] with double b tag, which also determines the bound on the vector coupling for m R < 1.4 TeV. This explains why the bound on the coupling is actually very similar in both cases: the required double b tag means that the contribution from partonic events containing only s or c quarks, which only exists in the case of vector coupling, has very small ef ciency, since the b tag requirement can only be satis ed though mistagging, or through additional b quarks produced in hard showering. As a result the bound on the total cross section, shown in the right frame, is much weaker for pure vector coupling, since the coupling to s and c quarks greatly increases the total cross section while contributing little to the most sensitive signal.
As noted above, we also derived constraints on our model from mono-jet searches. The most sensitive analysis has been published in [ ], and does not require any avor tagging. The resulting constraint on the vector coupling is only slightly weaker than that shown in Fig. , while the constraint on the axial vector coupling is not competitive. Since no avor tagging is required, the large contribution from s or c quarks in the initial and nal states has similar ef ciency as contributions with b quarks, and greatly strengthens the limit on the vector coupling.

Conclusions
In this study, we discuss a model containing a Dirac fermion χ as dark matter candidate as well as a spin−1 mediator R. We assume that R has vanishing couplings to rst generation quarks and vanishing axial vector coupling to second generation quarks, thereby easily satisfying constraints from direct dark matter searches. By assuming vanishing couplings to leptons the otherwise most sensitive LHC searches, based on analyses of + − nal states where stands for a charged lepton, are evaded as well. Due to the vanishing couplings to light quarks, and hence to all valence quarks in the proton, the R production rate at the LHC is considerably smaller than for the more commonly considered scenarios with (essentially) universal couplings to all quarks.
Nevertheless LHC data impose quite strong constraints on the model if the branching ratio for invisible R decays is sizable, which requires m R > 2m χ . The best LHC bound then always comes from searches for nal states containing jets plus missing E T . Our CheckMATE-based recast of these analyses leads to an upper bound on the product of the squared coupling and the invisible branching ratio of R of 10 −3 for m R ≤ 200 GeV. This weakens to 0.01 (1) for m R = 600 GeV (2 TeV), see Fig. . Searches for invisibly decaying mediators have traditionally been framed as "mono-jet" searches (which allow additional jets in the nal state, as mentioned above), and have been interpreted assuming equal (vector or axial vector) couplings to all quarks [ , ]. For pure axial vector couplings these bounds are actually weaker than ours if m R ≤ 600 GeV. Since the signal need only contain a single hard jet, and no b−tagging is used, one needs a very strong cut on the missing E T to suppress the background; for m R < ∼ 1 TeV this leads to a much worse cut ef ciency than the most sensitive analysis we use, which requires two tagged b−jets plus missing E T . For m R < ∼ 600 GeV this search may thus also impose tighter bounds on the model with universal couplings. Nevertheless the bound on g 2 q times the invisible branching ratio from mono-jet searches in the model with universal coupling becomes signi cantly stronger than ours for larger m R , by about one order of magnitude for m R = 1.5 TeV.
For m R ≥ 0.6 TeV roughly comparable bounds on the product of the squared coupling and the branching ratio of R into qq quarks can be derived from an ATLAS search for bb resonances. Searches for generic di-jet or tt resonances yield much weaker constraints on our model. Generic di-jet resonance searches at the TeV LHC become sensitive only at a resonance mass above . TeV or so. The resulting bounds on mediators with unsuppressed couplings to valence quarks are quite strong. For example, for m R = 1.5 TeV the ATLAS analysis [ ] gives a bound on the squared universal coupling to quarks in a leptophobic model that is about two orders of magnitude stronger than our bound from bb resonant searches in the model with vector couplings, which in turn is a factor of about 3 stronger than the analogous bound in the model with axial vector couplings.
We thus see that both in the missing E T and in the resonance searches switching off the couplings to rst generation quarks greatly weakens the limits on the couplings for m R > 1 TeV, less so for smaller mediator masses.
Since the energy scale of these reactions (e.g. the missing E T , or m R in the resonance searches) is much larger than the masses of the relevant quarks, the matrix elements for vector and axial vector couplings are almost the same. Unless m R m χ for equal coupling strengths the branching ratio for invisible R → χχ decays will be larger for pure vector coupling than for pure axial vector coupling; however, this effect is absorbed by interpreting the relevant constraints as upper bounds on the product of the squared coupling times the invisible branching ratio, as we did in the above discussion.
LHC searches lose sensitivity to our model if m R > 2 TeV, or if m R < 0.6 TeV and m R < 2m χ . Probing signi cantly higher values of m R would require higher center-ofmass energies; since all relevant searches are background-limited, increasing the luminosity will increase the reach only slowly. If on-shell R → χχ decays are not possible, missing E T searches at the LHC are essentially hopeless in our model. The reason is that in this case a signal which is of second order in the couplings of the mediator has to compete with SM signals that are rst order in electroweak couplings, in particular the production of Z and W bosons which decay into neutrinos. * For m R < 70 GeV the old LEP experiments have some sensitivity, but the resulting bound is not very strong [ ]. Straightforward dijet resonance searches at the LHC are not possible for m R much below 0.6 TeV, since the trigger rate would be too high. One might consider previous hadron colliders, in particular the Tevatron. However, these earlier colliders were pp colliders, where the bb background includes contributions where both initial-state quarks are valence quarks; recall that in our model the signal does not receive contributions from such initial states.
A probably more promising approach is to consider nal states containing an additional hard "tagging jet" besides the mediator R. Both ATLAS [ ] and CMS [ ] have presented bounds on rather light di-jet resonances using this trick, which is also employed in the "monojet" searches. Unfortunately these searches are currently not easy to recast, since they use "fat jet" substructure techniques. In any case, in order to gain sensitivity to our model this technique would probably have to be combined with b−tagging, which proved * In case of universal couplings to all quarks the "monojet" analyses [ , ] do exclude a small region of parameter space with m R /2 < m χ < ∼ 200 GeV for a vector mediator, but not for an axial vector mediator.
crucial for deriving useful constraints from di-jet resonance searches at m R > 600 GeV. An analysis of this kind should be able to probe deep into the parameter space with m R < 600 GeV and m R < 2m χ .  [ ] Matteo Cacciari and Gavin P. Salam. Dispelling the N 3 myth for the k t jet-nder. Phys. Lett., B : -, .