LHC constraints on color octet scalars

TL;DR

This paper analyzes LHC data to set constraints on color octet scalars in the MW model, finding that masses below 1 TeV are not excluded and that coupling constraints are generally weak.

Contribution

It provides the first comprehensive comparison of LHC experimental limits with the MW model's parameter space, clarifying the mass and coupling constraints.

Findings

Masses below 1 TeV are not excluded for color octet scalars.

Constraints on coupling constants are generally weaker than theoretical bounds.

The parameter η_D is an exception with stronger constraints.

Abstract

We extract constraints on the parameter space of the MW model by comparing the cross-sections for dijet, top-pair, dijet-pair, $t\bar t t \bar t$ and $b\bar b b \bar b$ productions at the LHC with the strongest available experimental limits from ATLAS or CMS at 8 or 13 TeV. Overall we find that masses below 1 TeV have not been excluded for color octet scalars as it is often claimed in the literature. The constraints that can be placed on coupling constants are typically weaker than those from existing theoretical considerations, with the exception of the parameter $\eta_D$.

Figures

Equations

$${\mathcal{L}}_Y=-{\eta }_U{e}^{i{\alpha }_U}{g}_{ij}^U{\bar{u}}_{Ri}{T}^A{Q}_j{S}^A-{\eta }_D{e}^{i{\alpha }_D}{g}_{ij}^D{\bar{d}}_{Ri}{T}^A{Q}_j{S}^{†A}+h.c,$$

$$V=\lambda {\left(H^{†i}{H}_i-\frac{v^2}{2}\right)}^2+2m_s^2\mathrm{Tr}{S}^{†i}{S}_i+{\lambda }_1{H}^{†i}{H}_i\mathrm{Tr}{S}^{†j}{S}_j+{\lambda }_2{H}^{†i}{H}_j\mathrm{Tr}{S}^{†j}{S}_i$$

$$+\left({\lambda }_3{H}^{†i}{H}^{†j}\mathrm{Tr}{S}_i{S}_j+{\lambda }_4{e}^{i{\varphi }_4}{H}^{†i}\mathrm{Tr}{S}^{†j}{S}_j{S}_i+{\lambda }_5{e}^{i{\varphi }_5}{H}^{†i}\mathrm{Tr}{S}^{†j}{S}_i{S}_j+\text{leavevmode}h.c.\right)$$

$$m_{S^{\pm }}^2=m_S^2+{\lambda }_1\frac{v^2}{4},$$

$$M_{S_R},{\lambda }_2,{\lambda }_4,{\eta }_U,{\eta }_D$$

$${\mathcal{L}}_{Sgg}$$

$$F_R$$

$$F_I$$

$$I_q(z)$$

$$f(z)$$

Full text

I Introduction

The search for new physics at the LHC has covered enormous ground so far, constraining the parameters of many possible new particles that appear in a variety of extensions of the standard model (SM). There are still exceptions, models that have not been carefully confronted with the LHC data, amongst them an extension of the scalar sector of the SM with a color octet electroweak doublet scalar. This model was introduced some time ago by Manohar and Wise (MW) Manohar:2006ga , and it is motivated by minimal flavor violation Manohar:2006ga .

There are several phenomenological papers concerning LHC studies of the MW model available, but all of them predate the LHC. The original MW paper already gave analytic expressions for the pair production of the new scalars through the dominant gluon fusion channel, finding production cross-sections that vary many orders of magnitude, between $\leavevmode\nobreak\ 50$ fb and $10^{-3}$ fb for $M_{S}$ between $1$ and $3$ TeV respectively at LHC14 Manohar:2006ga . Shortly after, Gresham and Wise considered the production of a single neutral scalar, a one loop process similar to the production of the Higgs boson, which becomes dominant over the two scalar production at around 1 TeV Gresham:2007ri . Gerbush et al Gerbush:2007fe considered pair production of scalars leading to final states with four heavy quarks (top or bottom) at the LHC concluding that discovery would be possible for masses up to about a TeV. Finally Arnold and Fornal considered constraints that can be imposed by studying high $p_{T}$ four-jet events at LHC Arnold:2011ra , and concluded that 10 fb*-1* would be enough to discover a scalar as heavy as 1.5 TeV at LHC14.

In this paper we partially address the existing gap in the extraction of LHC constraints for the MW model by comparing the cross section for production of a single neutral scalar and a pair of neutral scalars to limits obtained by ATLAS and CMS in the dijet, top-pair, dijet-pair, $t\bar{t}t\bar{t}$ and $b\bar{b}b\bar{b}$ channels. We defer the study of charged scalar production and their different decay modes to a future publication. We restrict this paper to a parton-level study because its main weakness is that there is no consistent set of LHC data that can be applied to this model.

A similar study for pseudo-scalar color octets, common to composite Higgs models, exists in the literature Belyaev:2016ftv .

II The Model

The MW model contains many new parameters, several of which have been studied phenomenologically before. In particular the new color octet scalars can have a large effect on loop level Higgs production and decay Manohar:2006ga and this has resulted in multiple studies of one-loop effective Higgs couplings He:2011ti ; Dobrescu:2011aa ; Bai:2011aa ; Cacciapaglia:2012wb ; Cao:2013wqa ; He:2013tia . They are also constrained by precision electroweak measurements Manohar:2006ga ; Gresham:2007ri ; Burgess:2009wm , flavor physics Grinstein:2011dz ; Cheng:2015lsa ; Martinez:2016fyd , unitarity and vacuum stability Reece:2012gi ; He:2013tla ; Cheng:2016tlc ; Cheng:2017tbn and other LHC processes Gerbush:2007fe ; Arnold:2011ra ; He:2011ws ; Kribs:2012kz .

In the MW model, the new field $S$ transforms as $(8,2,1/2)$ under the SM gauge group $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ and this gives rise to the gauge interactions responsible for its pair production at the LHC. Numerous new couplings appear in the self-interactions of the scalars as well as in the Yukawa couplings to fermions. The latter are restricted by minimal flavor violation to two complex numbers Manohar:2006ga ,

[TABLE]

where $Q_{i}$ are left-handed quark doublets, $S=S^{a}T^{a}\;(a=1,...,8)$ and the $SU(3)$ generators are normalized as ${\rm Tr}(T^{a}T^{b})=\delta^{ab}/2$. The matrices $g^{U,D}_{ij}$ are the same as the Higgs couplings to quarks, and $\eta_{U,D}$ along with their phases $\alpha_{U,D}$, are new overall factors. Non-zero phases signal $CP$ violation beyond the SM and contribute for example to the EDM and CEDM of quarks Manohar:2006ga ; Burgess:2009wm ; He:2011ws ; Martinez:2016fyd .

The most general renormalizable scalar potential is given in Ref. Manohar:2006ga and contains several terms. Of these, our study will only depend on the following (with $v\sim 246$ GeV the usual Higgs vacuum expectation value)

[TABLE]

The number of parameters in Eq.(2) can be reduced for our study as follows: first $\lambda_{3}$ can be chosen to be real by a suitable definition of $S$; then custodial $SU(2)$ symmetry can be invoked to introduce the relations $2\lambda_{3}=\lambda_{2}$ (and hence $M_{S^{+}}=M_{S_{I}}$) Manohar:2006ga and $\lambda_{4}=\lambda_{5}^{\star}$ Burgess:2009wm ; and finally, requiring CP conservation removes all the phases, $\alpha_{U}$, $\alpha_{D}$, and $\phi_{4}$.

After symmetry breaking, the non-zero vev of the Higgs field in Eq.(2) gives the physical $h$ its usual mass $m^{2}_{H}=2\lambda v^{2}$, and in addition it splits the octet scalar masses as,

[TABLE]

These relations, combined with the use of custodial and CP symmetries result in the following independent input parameters:

[TABLE]

The parameter $\lambda_{2}$ controls the split between the two neutral resonances $S_{I,R}$ and the parameter $\lambda_{4}$ controls the strength of scalar loop contributions to single neutral scalar production through gluon fusion. The parameters $\eta_{U,D}$ control respectively the strength of the $Stt$ and $Sbb$ interactions.

Under the above assumptions, the effective one-loop coupling $Sgg$ can be written in terms of two factors $F_{R,I}$ as,

[TABLE]

where $G^{A}_{\mu\nu}$ is the gluon field strength tensor and $\tilde{G}^{A\mu\nu}=(1/2)\epsilon^{\mu\nu\alpha\beta}G^{A}_{\alpha\beta}$.

At one-loop level these factors receive their main contributions from top-quark and scalar loops and are given by Gresham:2007ri

[TABLE]

We have already simplified Eq.(6) with the relations $\lambda_{5}=\lambda_{4}$, $m^{\pm}=m_{I}$, and we have allowed for the mass $m_{R}$, to be different from $m_{I}$ through $\lambda_{2}$ as discussed above. Throughout the calculation, the scalars $S_{R}$ and $S_{I}$ (when not in a loop), are taken to be on-shell, in keeping with the narrow width approximation. The loop functions $I_{q,s}$ and $f$ are well known and given by:

[TABLE]

From this it follows that the top-quark contribution starts as large as $I_{q}(1/4)=1/2$ for a resonance with twice the top-quark mass, and goes to zero as $m_{S}>>m_{t}$. Additionally, the factor $I_{s}(1)=\pi^{2}/9-1$, generates some suppression in contributions from scalar loops relative to top-quark loops. For this reason only relatively large values of $\lambda_{4}$ make a significant contribution to single scalar production. We have also included the bottom-quark loop, which is important only for regions of parameter space where $|\eta_{D}|>>|\eta_{U}|$. This situation is completely analogous to Higgs production in type II two Higgs doublet models where the $b$ quark contribution is important for $\tan\beta>>1$. We will consider this case for the dijet final state study.

III Confronting LHC searches

An early paper constraining new particles at the LHC introduced several benchmarks Han:2010rf , including a color octet scalar $S$, that have since been pursued by both ATLAS Aad:2014aqa and CMS Sirunyan:2016iap . Both these studies appear to rule out the new color octet scalars with masses up to 3 TeV. Unfortunately, the original benchmark does not cover interesting possibilities for additional color octet scalars like the MW model. This can be easily seen from their coupling to two gluons, $F_{R}=g_{s}\frac{\kappa_{S}}{\Lambda_{S}}$, which has been used with $\kappa_{S}=1/\sqrt{2}$ in the CMS study. Comparing this to $F_{R}$ in Eq.(6), one sees that realistic couplings in the MW model result in cross-sections that are three orders of magnitude below the benchmark.

To proceed with our numerical study we implement the Lagrangian of Eq.(1) and Eq.(6) in FEYNRULES Christensen:2008py ; Degrande:2011ua to generate a Universal Feynrules Output (UFO) file, and then feed this UFO file into MG5_aMC@NLO Alwall:2014hca . To extract constraints on the parameters of the new resonances we will compare our predicted cross-sections times branching ratios with existing LHC studies. We divide our study according to the different channels that dominate different regions of parameter space. We will use in particular the following channels: two jets; top pairs; four jets; four top-quarks; and four bottom quarks.

We illustrate all cases with production and decay of $S_{R}$ or $S_{R}S_{R}$ but the limits are identical for $S_{I}S_{I}$ and similar for $S_{I}$ provided we prevent the decays $S_{R,I}\to S^{\pm}W^{\mp}j$ (which is done by a choice of $\lambda_{2}$). If those decays are allowed, different limits apply from channels involving final state leptons which we do not consider in this paper. In the limit of $CP$ conservation that we consider here, the $S_{I}$ is a pseudo-scalar and therefore its gluon fusion production mechanism is a bit different from $S_{R}$, being independent of $\lambda_{4}$.

III.1 Search for resonances in dijets at LHC

This channel has been studied before in the context of the MW model for the Tevatron. A comparison with the dijet cross-section at the Tevatron revealed that signatures from the MW model are orders of magnitude below the QCD background and no constraint emerged for masses below 350 GeV with $\lambda_{4}=\lambda_{5}$ as large as 10 Burgess:2009wm . Existing LHC searches that can be used to compare the MW model with the two jets channel are: ATLAS-8 TeV Aad:2014aqa , ATLAS-13 TeV ATLAS:2016lvi , CMS-8 TeV Khachatryan:2016ecr , CMS-13 TeV Khachatryan:2015dcf . For example, ATLAS used its 8 TeV data to conclude that color octet scalars are ruled out below 2.8 TeV Aad:2014aqa , but as explained above, their benchmark color octet model, that of Ref. Han:2010rf , results in cross-sections many orders of magnitude larger than the MW model.

The dijet decay mode is dominant when one of the two neutral color octet resonances $S_{R}$ or $S_{I}$ is kinematically suppressed from decaying to the charged one through processes of the form $S_{R,I}\to S^{\pm}W^{\mp}j$, and its decay into top quark pairs is suppressed due to a small parameter $\eta_{U}$ (we will assume the mass is above threshold for top-pair production). Under these conditions the dominant decay rate is into two jets. We illustrate this scenario with the choices: $\eta_{U}=0$, which makes $B(S_{R}\to t\bar{t})\to 0$; and $\lambda_{2}=-5$, which places the mass, $M_{S_{I}}=M_{S^{\pm}}$ above $M_{S_{R}}$. The decay of $S_{R}$ in this case is almost 100% into two jets, with the dominant channels being $b\bar{b}$ and $gg$ as illustrated in Figure 1.

The left panel on Figure 1 corresponds to $\eta_{D}=10$ where the resonance couples most strongly to $b\bar{b}$ pairs. This coupling is weak enough that the resonance width does not exceed \Gamma/M\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}10^{-3}, a very narrow resonance which we compare to the experimental curve that assumes a width set at the detector resolution. This figure illustrates the dependence on $\lambda_{4}$, showing it is almost symmetric but with a slight enhancement (from constructive interference between quark and scalar loops) for negative values of $\lambda_{4}$.

The panel on the right in Figure 1 explores the dependence on $\eta_{D}$ for a fixed value $\lambda_{4}=-20$. We see how the dominant decay mode turns from $gg$ to $b\bar{b}$ for values of $\eta_{D}$ in the $2-4$ range. In this case $\Gamma/M$ can reach the percent level for \eta_{D}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}30.

The most restrictive data available for this channel comes from 13 TeV and we use it to extract bounds in Figure 2. The figure on the upper left panel, covers a resonance mass up to 1.6 TeV, and is best explored by CMS. For our comparison, we have used the same acceptance cuts as CMS, namely $p_{Tj}>30$ GeV, $|\eta_{j}|<2.5$, $|\Delta\eta_{jj}|<1.3$ and we show their results for resonances produced by (light) $q\bar{q}$ and $gg$ with 12.9 fb*-1* at 13 TeV CMS:2016wpz . We first fix $\eta_{D}=10$ and present model results for different values of $\lambda_{4}$. Figure 1 suggests that this type of resonance would be dominantly produced by a $b\bar{b}$ initial state so that it doesn’t match either of the two cases explicitly considered by CMS. For example, with $\lambda_{4}=10$ and $M_{S_{R}}=600$GeV, production from $b\bar{b}$ accounts for 98.7% of the cross-section, with only about 1.1% originating in gluon fusion. For $\lambda_{4}=30$, the contribution from gluon fusion has increased to about 10%.

From this comparison we conclude that for values of \lambda_{4}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}13, its perturbative unitarity bound He:2013tla , the cross-sections are too small for the LHC to place any meaningful limit. The figure also shows that the resonance mass is constrained to be larger than about $600\sim 800$ GeV for larger values of $\lambda_{4}$, when we ignore the perturbative unitarity constraint.

In the upper right panel we repeat the exercise using the results available for a higher range of resonance mass. We use in this case the ATLAS result with 15.7 fb*-1* ATLAS:2016lvi for a hypothetical resonance with a Gaussian distribution and a width set at the detector resolution (2-3% of the mass) and implement the ATLAS acceptance cuts $p_{Tj}>440$ GeV, sub-leading jet $p_{Tj}>60$ GeV, $|y^{*}|<0.6$ with $y^{*}=(y_{3}-y_{4})/2$. We see that the high energy region does not yet constrain this set of parameters.

On the bottom panels we repeat the comparisons, but this time we fix $\lambda_{4}=10$ and vary $\eta_{D}$ finding no constraints for values of $10<|\eta_{D}|<50$.

III.2 Search for resonances in top-pair production at LHC

A comparison with Tevatron cross-sections in the mass range $350-1000$ GeV found that this model could have a large enough cross-section to be relevant for $\lambda_{4}=\lambda_{5}\sim 75$ which is well beyond the perturbative unitarity bound Burgess:2009wm . In this section we use LHC data from ATLAS with 20.3 fb*-1* at 8 TeV Aad:2015fna and from CMS with 2.6 fb*-1* at 13 TeV CMS:2016zte ; CMS:2016ehh to obtain limits from top-pair production. Once the resonance mass is above the top-pair threshold, its branching ratio into $t\bar{t}$ will be completely dominant if we suppress decays into $Wj$ or $Zj$ by choosing $\lambda_{2}=-5$ as in the two-jet case and if $\eta_{U}$ is of order one. We perform two comparisons with data, in the first one we set $\eta_{U}=1$ and explore the sensitivity to $\lambda_{4}$ and in the second one we set $\lambda_{4}=10$ and explore the sensitivity to $\eta_{U}$. The $B(S\to t\bar{t})$ is at least 95% for all of the parameters studied (with the second most dominant decay mode being two gluons), and the width of the resonance is close to 1% of its mass for $\eta_{U}=1$ but grows to about 70% by the time $\eta_{U}=10$ so we do not consider any larger values. The value of $\eta_{D}$ is set to 0 for illustration but it is irrelevant for this channel.

In the left panel we compare to the ATLAS result Aad:2015fna for their benchmark spin 0 resonance produced by gluon fusion, which is the closest match to our discussion. Their resonance width is lower than 15% which is adequate for $|\eta_{U}|\sim 5$, when the width of $S_{R}$ is 17% of its mass. From the upper panel we see that m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}500 GeV is obtained for values of $\lambda_{4}$ within its perturbative unitarity limit. The bottom panel indicates that for $\eta_{U}=5$ the LHC constrains m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}800 GeV. For the larger value of $\eta_{U}=10$ the width of $S_{R}$ is much larger than that assumed by ATLAS, but if we still use this result, we find m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}1 TeV.

On the right hand panel we have used the CMS results CMS:2016zte presenting three of their benchmark studies: a generic $Z^{\prime}$ produced by $q\bar{q}$ with decays to $t\bar{t}$ and a width equal to 1% or 10% of its mass; and a KK gluon with about 94% branching ratio to top-pairs. Since none of these scenarios closely resembles $S_{R}$, we can only obtain a very rough limit of m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}700 GeV for large values of $\eta_{U}$ and no constraints for $\eta_{U}=1$.

III.3 Search for resonances in dijet pairs at LHC

The relevant mechanism for this channel corresponds to tree-level pair production of the neutral scalars through gluon fusion Dobrescu:2007yp , followed by decays into light quarks or gluons. The production cross-section depends only on the mass of the scalars and the different decay channels can be selected as before. NLO calculations for the pair production of scalar color octets exist GoncalvesNetto:2012nt ; Chivukula:2013hga ; Degrande:2014sta but we restrict ourselves to a LO calculation here, including a K-factor from GoncalvesNetto:2012nt . Scalar pair production becomes dominant over single scalar production when the one-loop diagrams responsible for the latter are suppressed. This can happen for low enough values of $\eta_{U}$, $\lambda_{4}$ and $\eta_{D}$ as we illustrate in Figure 4.

We begin with both scalars decaying to dijets, for which we suppress the decay into top-pairs with $\eta_{U}=0$ and the decay into $Wj$ with $\lambda_{2}=-5$. In the upper panels of Figure 5 we compare the model to published results from ATLAS at 7 TeV which cover the low mass region, $150-350$ GeV ATLAS:2012ds using their acceptance cuts $p_{Tj}>80$ GeV and $|\eta_{j}|<1.4$. For the mass region up to 1 TeV we compare in the middle panels with the CMS preliminary results at 7 TeV Chatrchyan:2013izb , in this case with the acceptance cuts given by $p_{Tj}>110$ GeV, $|\eta_{j}|<2.5$ and $\Delta R_{jj}>0.7$. Finally, in the bottom panels we compare the region below 1 TeV with the results from ATLAS with 15.4 pb*-1* at 13 TeV ATLAS:2016sfd corresponding to their benchmark scenario that most closely resembles our case, pair production of colorons. On the left panel of Figure 5 we fix $\eta_{D}=1$, for which Figure 1 tells us the jets will be dominantly gluon jets. The figure then explores the sensitivity to $\lambda_{4}$, finding that resonances as light as 500 GeV are still allowed for \lambda_{4}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}10, but that masses \mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}750 GeV are excluded if we allow for larger values of $\lambda_{4}$. In the right side panels we take $\lambda_{4}=10$ and vary $\eta_{D}$. Only the 13 TeV data constrains the model, finding in this case that resonances with m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}650 GeV are excluded with $\eta_{D}=1$. For larger $\eta_{D}$ the dominant decay is into four bottom quarks, which we treat as a separate case below.

III.4 Search for resonances in four top events

The $t\bar{t}t\bar{t}$, $b\bar{b}b\bar{b}$ or $b\bar{b}t\bar{t}$ final states have been discussed in the literature before. They have lower cross-sections than the respective heavy-quark pair production, but suffer from a much smaller QCD background and this makes them ideal to search for new resonances. They have studied both for pair produced resonances Beck:2015cga , as we do here, and for a new resonance produced in association with a heavy quark pair Han:2004zh .

As in the previous sub-section we consider pair production of scalars via their QCD couplings and adjust the model parameters so that they decay predominantly into top-pairs as we did in III.2. We thus fix $\lambda_{2}=-5$, \eta_{U}\mathrel{\vbox{\kern 0.0pt\hbox{>} \kern 0.0pt\hbox{\sim} }}0.1 and $\eta_{D}=0$. For LHC data we use the ATLAS results with 20.3 fb*-1* at 8 TeV Aad:2015kqa and their benchmark model of scalar gluon pair production with subsequent decays to two top quark anti-quark pairs. In the left side panel of Figure 6 we fix $\eta_{U}=0.1$ and explore the dependence on $\lambda_{4}$. We find that a new resonance with m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}800 GeV is excluded for values of $\lambda_{4}$ within their perturbative unitarity limits by the 8 TeV data. In the bottom panels we also compare with 13 TeV data from associated production of $S_{R}$ with a top quark pair, Figure 22 of ATLAS:2016btu . In this case a mass as large as $m_{S}\sim 850$ GeV is excluded. As $\lambda_{4}$ increases the constraints disappear because the decay of $S_{R}$ into two gluons becomes dominant over $t\bar{t}$ pairs. On the right side panel we fix $\lambda_{4}=10$ and allow $\eta_{U}$ to vary. As $\eta_{U}$ increases further the constraint at 8 TeV gets weaker because the resonance width becomes very large, going from 16% to 64% of the mass for $m_{S}=700$ GeV as $\eta_{U}$ is increased from 5 to 10, for example. For $\eta_{U}\sim 5$ the exclusion extends to m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}1 TeV with the 13 TeV data.

III.5 Search for resonances in four bottom events

Finally we consider pair production of scalars via their QCD couplings and adjust the model parameters so that they decay predominantly into bottom-pairs Bai:2010dj . We study this case by comparing it to ATLAS results from 13.3 fb*-1* at 13 TeV and their search for Higgs pair production in the $b\bar{b}b\bar{b}$ mode ATLAS:2016ixk . In the left panel of Figure 7 we set $\eta_{D}=1$ and vary $\lambda_{4}$. For values of |\lambda_{4}|\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}10 this LHC study excludes m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}1 TeV. At larger values of $\lambda_{4}$ the constraint disappears as the decay of $S_{R}$ into two gluons is dominant over $b\bar{b}$ pairs. On the right side panel we fix $\lambda_{4}=10$ and see that for 1\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}\eta_{D}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}5 the region m_{S}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}1 TeV is excluded. For lower values of $\eta_{D}$ the resonance decays to two gluons as can be seen in Figure 1 and the constraint disappears.

III.6 Search for resonances in $t\bar{t}b\bar{b}$ events

This channel is becoming accessible through searches for associated production of a new resonance with a pair of b-quarks from the preliminary results ATLAS:2016btu . No constraints result from this case yet as illustrated in Figure 8.

IV Summary

We have extracted constraints on the parameter space of the MW model by comparing the cross-sections for dijet, top-pair, dijet-pair, $t\bar{t}t\bar{t}$ and $b\bar{b}b\bar{b}$ production at the LHC with the strongest available experimental limits from ATLAS or CMS at 8 or 13 TeV. Our comparisons offer rough estimates as the experimental collaborations have not considered this particular model and their results are somewhat model dependent, for example through the parton process assumed to produce the hypothetical resonance.

Here we summarize of our main findings within the two main assumptions: $\lambda_{2}\sim-5$ to suppress decays involving leptons; and $\lambda_{4}=\lambda_{5}=10$ to remain perturbative.

• •

Without couplings to up-type quarks, $\eta_{U}=0$, the best bound ranges from $m_{S}>530$ GeV to $m_{S}>1060$ GeV for all values of $|\eta_{D}|$ as shown in Figure 9. Figure 4 indicates that in the perturbative range for $\lambda_{4}$, $S_{R}$ pair production dominates over single $S_{R}$ production explaining why the most stringent bound arises from production of two $b\bar{b}$ pairs. The very low sensitivity to the value of $\eta_{D}$ when it gets above $\sim 6$ is due to the saturation of $B(S_{R}\to b\bar{b})\sim 1$ that occurs in this region of parameter space. For very low values of \eta_{D}\mathrel{\vbox{\kern 0.0pt\hbox{<} \kern 0.0pt\hbox{\sim} }}1.3, the dominant decay mode of $S_{R}$ is into two gluons and the constraint $m_{S}>530$ GeV is placed by four jet production.

• •

Without couplings to down-type quarks, $\eta_{D}=0$, the best bound on $m_{S}$ is shown on the right panel of shown in Figure 9. It arises from the associated production of $S_{R}$ with a top-quark pair at 13 TeV, followed by $S_{R}$ decaying to a second top-quark pair. Constraints from $pp\to t\bar{t}$ at 8 TeV are almost competitive as shown in Figure 9, suggesting they may become more important when this channel is studied at 13 TeV.