Exotic colored scalars at the LHC

We study the phenomenology of exotic color-triplet scalar particles $X$ with charge $|Q|=2/3, 4/3,5/3,7/3,8/3$ and $10/3$. If $X$ is a non-singlet of $SU(2)_W$ representation, mass splitting within the multiplet allows for cascade decays of the members into the lightest state. We study examples where the lightest state, in turn, decays into a three-body $W^\pm jj$ final state, and show that in such case the entire multiplet is compatible with existing direct collider searches and indirect precision tests down to $m_X\sim250$~GeV. However, bound states $S$, made of $XX^\dag$ pairs at $m_S\approx2m_X$, form under rather generic conditions and their decay to diphoton can be the first discovery channel of the model. Furthermore, for $SU(2)_W$-non-singlets, the mode $S\to W^+W^-$ may be observable and the width of $S\to\gamma\gamma$ and $S\to jj$ may appear large as a consequence of mass splittings within the $X$-multiplet. As an example we study in detail the case of an $SU(2)_W$ quartet, finding that $m_X\simeq450$~GeV is allowed by all current searches.

The large hadron collider (LHC) search for new physics at or below the TeV scale is far from complete, even for strongly interacting particles. As concerns the commonly studied Standard Model (SM) extensions [1][2][3], the dedicated searches by CMS and ATLAS for new strongly interacting light degrees of freedom are covering a large part of the parameter space. However, new colored particles beyond these standard scenarios could still have unexpected phenomenology and, in this case, traditional LHC searches often lose much of their power. In this work we consider colored scalar states with exotic EM charges, with a focus on SU (2) W -non-singlets. Such particles, while being copiously produced at the LHC, could still be hiding undiscovered amidst the large QCD background. Three different paths can be pursued in the experimental search for these particles: 1. Direct collider searches for QCD continuum pair production of X Q , a colored particle with EM charge Q. Such searches are potentially effective, but depend on the decay modes of X Q and hence are model dependent.
2. Precision measurements of electroweak (EW) processes, constituting an indirect search for X Q .
3. Direct collider searches for S Q , the bound state formed out of X Q X † Q through Coulomb gluon exchange, with mass m S Q 2m X Q . S Q decays into diboson final states, with branching ratios that are determined to a large extent by the quantum numbers of X Q . For exotic states the consequent constraints are often less model dependent than continuum pair production searches (see e.g. [4,5]).
We pursue all three avenues in this work.
The paper is organized as follows. In Sec. II we present our theoretical framework and the relevant representations for our study. Sec. III details the experimental bounds from direct searches for continuum QCD pair production of X Q . In Sec. IV we discuss mass splittings within SU (2) W multiplets and the implications for cascade decays. In Sec. V we present a benchmark model. Sec. VI deals with the unique phenomenology of SU (2) W multiplets, and the footprint it might leave in indirect probes such as electroweak precision measurements (EWPM), Higgs couplings and the renormalisation group evolution of various couplings. In Sec. VII we study the QCD bound states formed out of X Q X † Q pairs, and the possible signatures at the LHC. We conclude in Sec. VIII. Various technical details are presented in the Appendices.

II. THEORETICAL FRAMEWORK
Consider a scalar X in the (R, n) Y representation of the SU (3) C × SU (2) W × U (1) Y gauge group. The Lagrangian is given by where D µ is the covariant derivative, determined by the quantum numbers (QN) of X, and H is the SM Higgs doublet, H ∼ (1, 2) +1/2 . The scalar potential V (H, X) has the form V (H, X) = V SM (H) + m 2 X X † X + λ X 2 (X † X) 2 + λ XH X † XH † H + λ XH (X † T a n X)(H † T a 2 H) , where T a n are the SU (2) W generators in the n representation. As we explore below, λ XH = 0 generates mass splitting between the various states X Q . Both λ XH = 0 and λ XH = 0 modify Higgs couplings to SM fermions and gauge bosons.
We comment that Eq. (2) is not the most general form possible for V (H, X). Additional X 4 couplings may arise e.g. for color triplets in a non-singlet SU (2) W representation. As long as these couplings are small compared to g 2 s ∼ 1, they are not essential in most of our analysis and we omit them here. As concerns the SU (3) C representation of X, we focus on color-triplets. This is a common starting point in many analyses, often considering quantum numbers similar to those of the SM quarks as occurs in supersymmetric models. The common lore is that first and second generation squarks are ruled out below 1.4 TeV while stops should be heavier than 900 GeV [6]. We study how this discussion is affected once exotic SU (2) W × U (1) Y representations are considered.
Other SU (3) C assignments have been studied in various contexts. For instance, supersymmetric models with Dirac gauginos introduce a color-octet scalar as the superpartner of the fermion which marries the gluino to form a Dirac fermion [7]. Color-sextets have been introduced in some models of grand unification [8,9]. Despite this interest, we keep our focus on R = 3 for concreteness, though we include generic representations R in some parts of the analysis where it does not introduce excess clatter.
The terms in L Y X break X number and thus control the decay of X to SM final states. With some abuse of notation, we refer to the terms in L Y X as Yukawa interactions. We maintain this terminology also to nonrenormalizable operators which, when the Higgs fields are replaced by their vacuum expectation values, lead to effective Yukawa couplings of X with SM fermions. A doublet or a triplet of SU (2) W can couple to a fermion pair in a renormalizable operator, while other representations of SU (2) W require higher dimensional operators for the decay of their members. The inclusion of effective operators truncates the validity of our model at some cut-off scale Λ. To avoid the need for low cut-off scale, we restrict our discussion to effective operators with mass dimension ≤ 6. This, in turn, leads us to consider n ≤ 5, and limits the possible hypercharge assignments for X.
In Table I we list all possible representations of X, for which we can find X-decay operators compatible with the restriction d ≤ 6 for L Y X . We also list the corresponding diquark and/or leptoquark X-number violating operators. We denote the SM left-handed doublets as Q and L, and the right-handed singlets as U, D and E. Throughout the analysis we will assume that, when several operators are available in Table I, only one of them exists while the others are absent or negligible. For brevity, we omit d ≤ 6 operators which include derivative interactions, as they introduce no new representations for X.  Colored particles are pair-produced at the LHC via initial state gluons. In this section we study the direct searches for continuum pair production of color triplet X Q . The EM charge Q dictates the possible decay modes and, subsequently, the experimental signatures. The SU (2) W quantum numbers are provisionally left out of the discussion. Table II summarizes the possible decay final states of X Q for a given charge. We distinguish between two different decay topologies: 1) fully hadronic, in which X Q decays to two jets and possibly also W bosons (we omit potential jjh and jjZ decay modes, as these are subdominant to an allowed jj decay), and, 2) lepto-quark signature, in which X Q decays to a lepton (possibly a neutrino) and a jet. Let us first analyze prompt signatures, highlighting the mass range 250 GeV ≤ m X Q ≤ 1000 GeV. For some X Q decay topologies, dedicated searches were carried out by ATLAS, CMS, or the Tevatron collaborations. These decay modes, along with the relevant searches, are summarized in Table III. However, some of the signatures we study have no dedicated experimental analysis. We identify relevant searches which are sensitive to these topologies and estimate the corresponding efficiencies for our signal. For this purpose we implement our model in FeynRules [10] and simulate the signal in MadGraph5 [11] using Pythia 8 [12,13] for showering and hadroniztion. Detector effects are simulated in Delphes [14] using the standard configuration. We stress that, for the recasted channels, our results should be taken as an estimation only. A detailed description of our recast procedure can be found in Appendices A, B and C.
Our findings are presented in Fig. 1(a) for the dijet decays, Fig. 1(b) for the jet and charged lepton signals, and Fig. 1(c) for the neutrino-jet topology. We also consider the case where a jet is replaced by heavy flavor quark. In each figure we show the current limit on the pair-production cross section times BR 2 , normalized to the NLO+NLL cross section of a scalar colored triplet taken from [15][16][17]. Presented this way, when a single mode dominates the decay (namely BR = 1), the y axis corresponds to the number of copies of the X representation that are experimentally allowed.   An important ingredient for collider phenomenology is the lifetime of X Q . Non-prompt decays are studied by the experimental collaborations in dedicated searches, leading to bounds in the ballpark of m X Q 700 − 900 GeV for color-triplet scalars. Refs. [18,19] analyzed displaced signatures in the context of RPV SUSY models. They find that X Q in the mass range of 100 − 1000 GeV, decaying to dijet, or to a jet and a charged lepton, or to a jet and a neutrino, would not be captured by the displaced-track searches if its mean-free path is less than 0.3 − 10 mm. While the exact number depends on the particle mass and decay mode, we conservatively use in the following 0.3 mm as an upper bound on a two-body decay length. We are not aware of any dedicated analysis for displaced signature of a three-(or four-) body final state. We estimate that the larger multiplicity of the final objects would increase the efficiency of these searches at high m X Q , while the low m X Q regime will suffer from the typically lower energy carried by each final object. Over all, we expect that the sensitivity is comparable to the other topologies, and so we consider cτ 1 mm for three-body decay. We then apply the following 'promptness' requirements on X Q decay rates: which translate into lower bounds on the Yukawa coupling of X Q to SM states. Concluding this section, we learn the following: • The lepto-quark topology is strongly constrained by direct searches. As can be seen in Fig. 1(b), none of the decay modes in this category allows for more than two states below m X Q 750 GeV.
• The neutrino-quark topology is subject to the standard SUSY searches for jet and missing energy. As can be seen in Fig. 1(c), the corresponding bounds on m X Q are even stronger than in the jl category.
• The hadronic decay modes are significantly less constrained by direct searches. This is expected given the large QCD backgrounds at the LHC.
• A W jj signature is poorly constrained by the LHC. As we show below, such topology could be the signature of multiple states which undergo cascade SU (2) W decays. This is an important gap in the LHC coverage for colored new particles which motivates dedicated searches for this decay topology.

IV. MASS SPLITTING AND CASCADE DECAYS
In general, two members of an SU (2) W multiplet with EM charges Q and Q are split in mass. Tree level mass splittings are induced by the λ XH term: Mass splittings also arise through electroweak gauge boson loops from the kinetic term (D µ X) † (D µ X) [54]: where . Assuming no fine-tuned cancelations between the tree and loop contributions, a mass splitting of at least O(100 MeV) between adjacent members of the multiplet (Q = Q + 1) is unavoidable. Much larger splittings are possible, depending on the value of λ XH . If the tree contribution dominates, the splitting can be of either sign, and the lightest colored scalar is the one with either the highest or the lowest Q.
The mass splitting between the members of an SU (2) W multiplet leads to W -mediated decays within the multiplet, X m → X m±1 W ∓( * ) . (Note that we change notations in this section from X Q to X m , with Q = m + Y .) For the three-body decay, X m → X m+1 f f , with massless fermions, we obtain If ∆M > m π , we have the two body decay X m → X m+1 π − , in which case For m = −1 we recover the results of Ref. [54]. We do not consider ∆M > m W .
To determine the phenomenological significance of these decays (for all but the lightest member of the multiplet), we need to compare their rate to those of the Yukawa mediated decays. We will do so in the next section. In the following we discuss the model example X ∼ (3, 4) +1/6 , containing a state with Q = +5/3 as the highest charge state. We assign X zero lepton number which, given our assumptions in Sec. II, forces X +5/3 to decay into the hadronic three body stated idj W + via the operator with Y QQ ij antisymmetric in the flavor indices i, j, and of dimension mass −1 . We consider two specific scenarios: • Case A: degenerate X Q states.
• Case B: non-degenerate X Q states.
We now show that these two cases exhibit distinct phenomenology.
For m X 8 TeV, the two-body decays of Eq. (10), where available, dominate over the three-body decays of Eq. (11). If the Y QQ term dominates the decay rates of all members of the quartet, then For i, j = 1, 2, we have three states decaying into a jj final state, and one state decaying into a W jj topology. This is allowed for m X 630 GeV. For i = 3, we have effectively 1.5 members decaying into jb and jt each. Looking at N ×BR 2 = 1.25 in Figure 1(a) we conclude that m X Q = 520 GeV is a viable possibility. We use this mass as our benchmark point in the following. To guarantee prompt decay of X +5/3 we impose B. Non-degenerate SU (2)W -quartet Mass splitting between the members of the quartet allow for fast cascade decays of the three heavier ones. In order to establish their phenomenological relevance one needs to compare the rate of these weak decays with the rate of the Yukawa mediated decay modes, which depend on the dimensional coupling Y QQ ij , Eqs. (10) and (11). The dominant terms need to induce prompt decays for all the members of the X multiplet. We distinguish between two cases: 1. X −4/3 is the lightest: In this case, either all states decay dominantly via their Yukawa coupling, or the Q = +5/3 state (and possibly also the Q = +2/3 and Q = −1/3 states) decay via W -mediated cascade decays. In either case, we have at least three color-triplet states decaying into two jets. The mass of the lightest state should then be similar to the mass considered in the degenerate quartet scenario.
2. X +5/3 is the lightest: In this case, X +5/3 decays to ad idj W + final state. As concerns the three heavier states, they can either decay into two jets, or cascade into the X +5/3 state. The latter would lead to effectively four states decaying to W qq in the final state, assuming the other cascade products are too soft to be detected (this is the case for a few GeV splitting). As far as direct searches for continuum pair production are concerned, we estimate the sensitivity of top-partner searches at the Tevatron [29] and find that, in this case, X +5/3 can be as light as 250 GeV. As we will see next, the direct searches for an X Q X † Q bound state place a stronger limit, of m X 450 GeV, with a corresponding lower bound on the Yukawa coupling, to ensure its prompt decay. Using Y QQ min as a convenient reference, and recalling that the two-body decay rate is faster than the three-body one for m X 8 TeV, a mass splitting of between two 'adjacent' members of the multiplet would effectively cause the four members of X(3, 4) +1/6 decay to W qq final states. The precise coefficient varies a little between the different SU (2) W members.
We therefore consider, for our second scenario, the following spectrum: which is the result of λ XH = 0.17.

VI. SU (2)W PHENOMENOLOGY
In this section we explore the distinct phenomenology of colored SU (2) W non-singlet scalars.

A. Electroweak precision measurements (EWPM)
Large mass splitting within an SU (2) W multiplet is constrained by EWPM. Specifically, it modifies the oblique T and S parameters [55], where the leading effect comes from generating the dimension six operators O T and O W B (see App. E for the definition of these operators). For an (R, n) Y representation, we have where in the second equation of each line we normalize to the quantum numbers of X(3, 4) +1/6 and to the value of λ XH which we use for case B in Sec. V. The EWPM constraints read (for U = 0) [56] T = 0.10 ± 0.07, S = 0.06 ± 0.09 , with correlation of ρ = 0.91. Using one dimensional χ 2 (λ ) function we find that |m X Q − m X Q±1 | 13 − 16 GeV is allowed around 450 GeV, where a positive (negative) λ XH implies that X +5/3 (X −4/3 ) is the lightest member of the multiplet. Clearly, EWPM allow the mass splitting we consider in case B.
In the limit of custodial symmetry, modifications to the EW vacuum polarization amplitude alter the oblique Y and W parameters [57]. These are primarily encoded in the dimension six operators O 2B , O 2W : These contributions to Y and W are below the current sensitivity of LEP (see, e.g., table 4 of [58]) and the LHC [59]. The values we take for the various coupling constants are listed in App. D.

B. Gauge coupling running
The presence of X ∼ (R, n) Y modifies the running of the gauge coupling constants. We describe this effect, at one-loop level, in App. D. In particular, high SU (2) W representations change significantly the running of α 2 . For instance, color-triplet in the quartet (or higher) representation of SU (2) W flips the sign of the SU (2) W beta function. In particular, for X(3, 5) Y , α 2 becomes non-perturbative at µ 10 15 GeV. Since the decay of X already requires some cut-off at a lower scale, this is insignificant to our study.
Additional probe for the running of EW gauge coupling is the differential distribution of Drell-Yan processes at various energies, as was proposed in Ref. [58]. Ref. [60] finds that for m ψ = 520 GeV, N ψ Q 2 ≥ 46 is excluded at the 2σ level, where N ψ is the number of copies of vector-like fermions transforming as ψ ∼ (3, 1) Q . This scenario would generate a 23% (50%) relative increase in the Drell-Yan rate at m = 1 (1.5) TeV, which excludes b X 2 ≤ −46. In our model example of Sec. V, b X 2 = −Rn(n 2 − 1)/36 = −5, clearly within bounds. A more recent analysis done in Ref. [59] yields the same conclusion.
C. Additional constraints SM Higgs couplings: Integrating out X generates dimension six effective operators involving the Higgs field. These, in turn, modify the Higgs couplings to fermions and gauge-bosons with respect to their SM values. LHC Higgs data constrain these modifications, resulting in bounds on the quartic couplings λ XH and λ XH . At present, EWPM induce stronger constraints on λ XH . The Higgs data do constrain λ XH , but this coupling is not directly relevant to our analysis. We present our numerical results of the Higgs data for X(3, 4) +1/6 in App. G, and the resulting minor effects on the various S → V V decays in App. I. Scalar quartic coupling running: In addition to modifying the SM Higgs couplings to fermions and gauge bosons, the presence of X changes the running of the SM Higgs couplings. We calculate these effects in App. H. We find that no dangerous runaway behavior is generated. The same conclusion holds for the X quartic coupling, and the mixed couplings λ XH and λ XH .

VII. QCD BOUND STATE
In the previous sections we obtained constraints from both direct and indirect probes on the existence of exotic colored scalars. The interesting result is that these constraints can be quite mild, allowing rather light colored scalars. For example, as demonstrated by the non-degenerate quartet scenario (case B in Sec. V), the data still allow four colored states with m X 250 GeV. In this section we study another way to discover light colored scalars, which might go first through the observation of their QCD bound state [4,5]. Moreover, constraints derived from bound state searches are less model dependent, in the sense that they do not depend on the decay mode of X.
A pair of X Q X † Q near threshold can form a QCD bound state, which we denote by S Q . If the decay rates of its constituents are smaller than Γ S Q , and its width is smaller than the respective binding energy, S Q can be seen as a resonance as it annihilates into SM particles. For a review we refer the reader to Ref. [61] and references therein. Heavy constituents exhibit Coulomb-like potential with a binding energy where n E is the excitation index (n = 1 is the ground state), α s is the strong coupling evaluated at the bound-state typical scale (for which we use, following Ref. [61], the Bohr radius) and C 2 (R) is the quadratic SU (3) Casimir of representation R, with C 2 (3) = 4/3. We assume that the resulting bound state is an SU (3) C singlet. The mass of S is m S = 2m X + E b . The condition that pair annihilation dominates the decay of S Q reads The RHS is well above the lower bounds in Eq. (3). In fact, (21) is fulfilled quite generally by the exotic states on which we focus the analysis. The argument goes as follows. Suppose that X decays into a two fermion final state, with effective coupling y. The condition (21) translates into y < 10 −2 . If the effective coupling comes from a dimension d operator, we have y =ŷ(v/Λ) d−4 , whereŷ is dimensionless and Λ is the scale of new physics. We assume perturbativity (ŷ 1), and a NP scale that is not very low (Λ 10 TeV). Then, for d = 6 operators, the condition is always fulfilled. For d = 5 operators it is not fulfilled only in a small region of parameter space where Λ 25 TeV andŷ > 0.4. Fully hadronic decays via renormalizable operators (d = 4) are possible only in a single case of SU (2) W non-singlet, that is X(3, 3) −1/3 , and even then the condition is fulfilled forŷ < 0.01. The condition (21) applies in all cases of dominant three body final state. We conclude that the search for bound states is truly a generic tool to look for exotic colored scalars [4]. The quantum numbers of X determine the gluon fusion (ggF) production cross section of S as well as its decay rates into pairs of vector bosons: gg, γγ, ZZ, Zγ and W W . Assuming that the X + X † production is dominated by ggF, and that there are no additional decay modes that give a significant contribution to the total width of S, then σ(pp → S) × BR(S → V 1 V 2 ) is predicted. The ggF partonic production cross section is given bŷ We convoluteσ with the partonic luminosity function s is the CoM energy. For the various two-body decay rates, we use (see [4] and references therein) where λ[x, y, z] is defined below Eq. (10), and ψ(0) is the joint wave function of X Q X † Q at the origin, which controls the probability to form a bound state, and is given by The full expressions for |M V1V2 | 2 can be found in App. I. We provide here the ratios between the different decay rates of S Q (with Q = m + Y ), denoting R Q X/Y = Γ(S Q → X)/Γ(S Q → Y ), and neglecting contributions proportional toλ m XH = λ XH − (m/2)λ XH and phase space suppressions: In the limit of small mass splitting, the various V 1 V 2 signals depend on the sum of the branching ratio of each member, rather than on the sum of R Q . They are the same if the total width of all the S Q members is equal, which is the case if the digluon mode dominates the total width. In Tab. IV we calculate the ratios between the different V 1 V 2 signals, summing over all S Q 's. Note that the running of the gauge coupling slightly modifies the numerical values of these ratios for various bound state masses. For concreteness, we quote these values at m S = 800 GeV, and denote We further specify, in Tab. IV, σ 13 γγ , the expected diphoton signal at the 13 TeV LHC for the various representations we consider, taking m S = 800 GeV. Bound state composed of SU (2) W non-singlet exhibit several interesting features, which we discuss next.

A. Diphoton signature
Interestingly, if X transforms in a large SU (2) W representation, its total width can be much larger than its partial width into gg. This can deplete the various S signals, in particular the S → γγ one. We demonstrate this effect in Fig. 2, where we show, for a given charge, the differences between the diphoton signal of an SU (2) W singlet to the one obtained from the highest SU (2) W representation listed in Table I. For the same charge Q we notice a dependence on the SU (2) W representation. The experimental upper bounds on σ γγ at 13 TeV translate into a lower bound on m S and, consequently, on m X . These bounds are effective: in fact, for SU (2) W singlets the bound is stronger than the bound from LHC direct continuum pair production searches in a large region of the parameter space. For instance, as discussed in the previous section, there are only very week bounds for an X +5/3 state from direct continuum pair production searches, while the search for diphoton resonance gives m X 5/3 600 GeV.   For higher SU (2) W representations, the bound state limits can be weaker than the ones from direct continuum searches, but have the advantage of being less model dependent. Consider, for example, the quartet X(3, 4) +1/6 . As discussed in the previous section, the lower bound on m X is very model dependent. It is around 800 GeV for decays into a leptoquark involving e or µ, but can be very weak for fully hadronic decays and reasonable mass splitting. Diphoton resonance searches set a solid bound of 450 GeV which is independent of these details of the model. Similar statements can be made for other high SU (2) W representations.

B. Distinct features of a bound state composed of SU (2)W -non-singlet constituents
If an X-onium S involves X that is an SU (2) W -non-singlet, then it might exhibit two features that would clearly distinguish it from the SU (2) W -singlet case: a large branching ratio into W + W − and an apparent large width. In this subsection we explain these two features.
Large BR(S → W + W − ): Observation of any diboson decay mode of S -γγ, W + W − , ZZ, Zγ -will help to close in on the representation of X. Our main focus is on cases where the S → W + W − decay rate is large. For the sake of concreteness, we examine whether R W W/γγ ≥ 10 is possible. Tab. IV shows five candidates. We list them by the order of the lower bound on their mass from diphoton searches: • (3, 2) +1/6 , with m S 500 GeV. We assume that all members of the X-multiplet are close enough in mass that they are observed as a single X-onium resonance. Another option would be separated signatures, in which, for example, a diphoton signal would come mainly from the |Q| high state, while the W + W − signature arises mainly from the |m| low state/s, possibly at different mass.
We note again that X ∼ (3, 4) +1/6 can be as light as 450 GeV only if X +5/3 is the lightest state and the mass splitting is large enough to let all the other states decay to it via three body decay. We further discuss this possibility in the next section, in the context of the second scenario we study.
Large apparent Γ S : The mass splitting between members of an SU (2) W multiplet may cause an apparent large width in the X-onium diphoton signal. To this end, it is important that the contribution to the diphoton events is not completely dominated by a single member of the multiplet. However, since the contribution of a particle of charge Q to the diphoton signal is proportional to Q 4 , a single member dominance is the case more often than not. For example, for the (3, 2) −5/6 multiplet, the contribution of the Q = −4/3 particle is 256 times larger than that of the Q = −1/3 particle. From the representations in Tab. I, only two could result in an apparent large diphoton width: • (3, 4) +1/6 , with σ The mass splitting between two extreme bound states of an SU (2) W n-tuplet is ∆m S −λ XH (n − 1)v 2 /(2m S ). Therefore, a quartic coupling of size would saturate an estimated 1% mass resolution of the diphoton signal (see, e.g. [62]). Such a small quartic coupling is allowed by EWPM and has no observed impact on Higgs couplings. Note that in order to understand whether the whole multiplet contributes to the resonance, or just the lightest member, one needs to make sure that the Wmediated decays within the multiplet, X m → X m±1 W ∓( * ) (Eqs. (6) and (7)), are not faster than the decay rate of S. This condition is generally satisfied below the m W threshold.

C. Back to our model examples
Let us now describe the phenomenology of the QCD bound state for our two benchmark scenarios of Sec. V.

Degenerate SU (2)W -quartet
In this scenario with m X = 520 GeV, the bound state has a mass m S = 1036 GeV, with possible small splitting between the various S Q states. It exhibits the following features: • γγ: Possible large apparent width in diphoton signals, with σ 13 γγ 0.25 fb.
In particular, a discovery of S with m S slightly above TeV is, in this case, within the reach of upcoming diphoton searches.

Non-degenerate SU (2)W -quartet
This is an example in which the bound state search is more powerful than the direct searches of X Q due to the lack of sensitivity for the three body final state W jj which would allow quartet as light as 250 GeV. Diphoton searches for S Q exclude m S ≤ 900 GeV, which corresponds to m X 450 GeV. At the 13 TeV with increased luminosity we expect a resonance which exhibits the following features: • γγ: Possibly two resolved diphoton resonances, with a total diphoton signal σ 13 γγ 0.58 fb.

VIII. SUMMARY AND CONCLUSIONS
The LHC search for new physics at or below the TeV scale is far from complete, even for strongly interacting particles. New particles might have surprising features, different from those predicted by the commonly studied extensions of the standard model. We studied the phenomenology of color-triplet scalar particles transforming in non-trivial representation of SU (2) W and potentially carrying exotic EM charges. Our main results are as follows.
• Color-triplet scalars (X), transforming in exotic representations of SU (2) W with masses at a few hundred GeV, are far from being experimentally excluded.
• Depending on the electromagnetic charges of such colored scalars, their dominant decay modes could be into three or four body final states. Some of these decay topologies, in particular the W ± jj one, are essentially unexplored by current analyses.
• In large parts of the parameter space, XX † for exotic X would form a QCD-bound state (S). It is easy to find examples where the observation of di-electroweak boson (e.g. diphoton) resonance at m S will precede the direct discovery of X.
• If X is an SU (2) W -non-singlet, the phenomenology of S might involve intriguing features, such as W W resonance at the same invariant mass as the diphoton resonance or somewhat removed from it, and a large apparent width for S.
• Two leptons: Searches for a final state containing two leptons, missing energy and jets have a potentially similar reach, but pay a higher price in the leptonic branching ratio of the W bosons. Therefore, they do not provide the best limits on our signal.
The search for first or second generation leptoquarks: The LQ searches typically suffers from a 25% reduction in the efficiency for our signal. This, together with the small leptonic W branching ratios, yield bounds that are insignificant. We note that a mixed (e ± j)(µ ∓ j) search, which is currently not done by the collaborations, may have better sensitivity due to lower expected background.
Searches for various states containing b jets: • The CMS 7 and 8 TeV analyses [34,67] search for heavy top-like quark (t ) decaying to W b final state. These searches might be sensitive to a W bj topology. Yet, as previously discussed, the t mass reconstruction weakens the reach of this search to the W bj topology. We again estimate this reduction to be between 30% and 50% and show the resulting bounds in Fig. 1. The same is done for the heavy bottom-like quark searches [27,39,40].
• The CMS RPV-SUSY search [27] forb → tj, whereb is the bottom squark, could have some sensitivity to W bj topology. However, it requires the reconstruction of t quarks which reduces significantly the sensitivity to our signal.
• SUSY stop searches, e.g. [68], look for a single lepton, missing energy and b-jets final state. We find these searches to be less sensitive than the heavy quark searches, as in the SUSY multi-jet searches with 1 lepton.
We conclude that the W jj decay mode is presently poorly constrained, irrespective of the flavor of the jets in the final state.
Precision cross-section measurements: Precision measurement of the tt and W + W − cross sections might probe best the low mass region of a (W + jj) (W − jj) signal. However, for m X ≥ 250 GeV we find that these are not sensitive even at multiplicity as high as n = 5; the argument goes as follows. We consider the NNLO-NNLL tt production cross section (see [69] and references therein), with m t = 172.5 GeV, and combine scale uncertainty and the uncertainty associated with variations of the PDF and α s (see [70][71][72][73]). At m X = 250 GeV, the production cross section for a quintuplet is below the theoretical uncertainty, assuming the efficiency of thett search to be 50% smaller than the efficiency for thett sample itself. This is a plausible estimate in the case of the W bj topology, and a conservative one for the W jj topology, even if we allow a large mistagging rate. Therefore, a quintuplet at 250 GeV is not constrained by the tt measurements. As for the W + W − cross section measurements, the relevant analyses veto on N j ≥ 1. Since our signal contains many jets in the final state, it would not contribute significantly to these measurements.
Appendix B: W + W + jj W − W − jj final state There are no dedicated searches for the four body W W jj decay mode, but other searches are potentially sensitive to it. For the fully hadronic final states and for the ones containing only one or two leptons, conclusions similar to those made for the W jj decay mode hold. However, for this topology, the most promising search strategy is to look for multilepton final states. The low SM background compensates for the branching ratio suppression of four W 's decaying leptonically.
We analyze the RPV multilepton CMS search [27,40] which does not rely on any missing energy cut. This analysis contains many exclusive signal regions, depending on the number of leptons, the presence of hadronically decaying τ , the presence of b jets, and the number of opposite-sign-same-flavor (OSSF) lepton pairs. We consider the low background regions, with four leptons, zero hadronic τ 's and 1 pair of OSSF leptons, summing over all S T bins. To be conservative, we allow the number of background events to fluctuate up by 95% C.L. and the number of signal events to fluctuate down by 95% C.L., assuming Poisson statistics. We take N sig = Lσ BR 4W →4 ,1OSSF with very high efficiency = 80% − 90%.
A somewhat stronger bound comes from the ATLAS analyses of Ref. [39]. For this, we consider the two overlapping signal regions, SR3L1 and SR0b1, with the corresponding bounds of σ SR3L1 ≤ 0.59 fb and σ SR0b1 ≤ 0.37 fb, set at 95% C.L.. (For the exact description of these signal regions we refer the reader to Ref. [39].) Since this search was specifically designed to be applicable to any SUSY RPV scenario, we assume the efficiency for our signal to be similar to the one quoted. We therefore use = 2% − 5%. The resulting limits are presented in Fig. 1. The hgg and hγγ couplings are computed using the Higgs effective low energy theory [77]: where Other couplings are computed by their definition in terms of the Wilson coefficients, for which we use the results of Refs. [74,78]. For our numerical results we use table 14 of [79] with B BSM = 0. We take as a concrete example the case of X ∼ (3, 4) +1/6 . The exact results, including EWPM constraints, are shown in Fig. 3. The constraints on λ XH and λ XH are rather mild and do not affect our conclusions.