Revisiting Scalar Quark Hidden Sector in Light of 750-GeV Diphoton Resonance

In this short note, we revisit the model of a CP-even singlet scalar resonance proposed in arXiv:1507.02483, where the resonance appears as the lightest composite state made of scalar quarks participating in hidden strong dynamics. We show that the model can consistently explain the excess of diphoton events with an invariant mass around 750GeV reported by both the ATLAS and CMS experiments. We also discuss the nature of the charged composite states in the TeV range which accompany to the neutral scalar. Due to inseparability of the dynamical scale and the mass of the resonance, the model also predicts signatures associated with the hidden dynamics such as leptons, jets along with multiple photons at future collider experiments. We also associate the TeV-scale dynamics behind the resonance with an explanation of dark matter.


INTRODUCTION
Recently, both the ATLAS and CMS Collaborations reported intriguing excess events in the search for a high-mass resonance decaying into diphotons in 13-TeV pp collisions [1,2].
Among them, models of (pseudo) scalar resonances originating from hidden strong dynamics have gathered particular attention, with its production at the LHC and decay into photons being explained via the gauge interactions of the constituents of the singlet composite state [16][17][18][19][20][21][22][23][24]. In this paper, we want to point out that an existing model proposed in Ref. [25] can consistently account for the diphoton signal while evading constraints from other highmass resonance searches made at the 8-TeV LHC. This model was originally proposed to explain the excess at around 2 TeV in the searches for a diboson resonance in the ATLAS experiment [26]. As we will see, we can readily explain the 750-GeV resonance by lowering the dynamical scale and mass parameters in the model.
In this model, the scalar resonance appears as the lightest composite state under hidden strong dynamics at around the TeV scale. A peculiar feature of the model is that the hidden sector consists of scalar quarks, and the lightest composite state is not pseudo-Goldstone bosons. With this feature, the mass of the resonance should be in close proximity to the dynamical scale, unlike in the models where the resonance is identified with a pseudo Goldstone modes. As a result, the model predicts intriguing signatures associated with the hidden dynamics at the LHC such as leptons, jets and leptons with multiple photons as well as the existence of charged composite resonances in companion with the 750-GeV resonance.
We also associate the TeV-scale dynamics behind the resonance with an explanation of dark matter.

A SCALAR RESONANCE FROM HIDDEN DYNAMICS
In the model of Ref. [25], the scalar resonance, S with mass M S , couples to the gauge bosons in the Standard Model (SM) due to the SM gauge charges of the constituent hidden scalar quarks. Such interactions are described by the effective Lagrangian (see also Ref. [27] for earlier works): where, Λ 1,2,3 are suppression scales which are related to the dynamical scale of the hidden sector, and G, W and B are the field strengths of the SU (3) C , SU (2) L , and U (1) Y gauge bosons, respectively. These gauge fields are normalized so that their kinetic terms are given by with g s , g and g being the corresponding gauge coupling constants, and the superscripts a and i denoting the indices for the corresponding adjoint representations.
Through the effective interaction with the gluons and in the narrow width approximation, the scalar resonance is produced at the LHC via the gluon fusion process where τ = M 2 S /s and √ s denotes the center-of-mass energy of the proton-proton collision.
Using the of MSTW2008 parton distribution functions (PDF's) [28], we obtain where we fixed the factorization scale and the renormalization scale at µ = M S /2 for M S 750 TeV. 1 The partial decay widths of the scalar resonance are given by where s W ≡ sin θ W and c W = (1 − s 2 W ) 1/2 with θ W being the weak mixing angle. The masses of the W and Z bosons are neglected to a good approximation. Now let us discuss the model content and hidden dynamics that lead to the scalar resonance. Following Ref. [25], we consider a set of scalar fields Q's that carry both the hidden SU (N h ) and the SM gauge charges. The SU (N h ) interaction is assumed to become strong at a dynamical scale Λ dyn . Explicitly, we take N h = 5. The charge assignments of Q's are given in Table I. It should be noted that we assign the SM gauge charges to Q's in such a way that they form an anti-fundamental representation of the minimal SU (5) grand unified theory (GUT).
The bi-fundamental scalars are assumed to have masses, 2 m D,L : When the masses of the scalar quarks do not exceed Λ dyn , the lightest composite state is expected to be a CP -even neutral composite scalar that is a mixture of Q † L Q L , Q † D Q D , and a CP -even glueball. It should be emphasized here that the lightest neutral scalar is expected to be lighter than the other SM-charged composite states due to mixing, since the charged scalar composite fields are not accompanied by mixing partners. This situation should be compared with models with fermionic bi-fundamental representations where the lightest singlet appears as a Goldstone boson mode. In this case, one of the neutral Goldstone bosons becomes heavier than the SM-charged Goldstone bosons due to the chiral anomaly of the hidden gauge interaction. Thus, if we further take the mass parameter of the colored hidden quark, M D , larger than Λ dyn , no neutral Goldstone boson remains lighter than the SM-charged ones, such as the SU (2) L triplet Goldstone bosons. In our scalar quark model, on the other hand, we expect that the neutral scalar boson remains lighter than the SMcharged composite bosons even if we take m D larger than Λ dyn due to the mixing with the glueball. This feature may be important when we discuss the phenomenology of the charged composite states (see discussions at the end of this section).
In our analysis, we are interested in how the singlet S couples to the SM gauge bosons.
For this purpose, we parametrize the relative contributions of [Q † L Q L ] and [Q † D Q D ] by a mixing parameter θ Q : For example, the Q † D Q D content is expected to be suppressed for m D m L . A quantitative estimation of θ Q is, however, difficult due to the non-perturbative nature of the strong interaction. Hence we take θ Q as a free parameter in the following analysis. 3 For m D Λ dyn , the second contribution can be effectively regarded as the glueball contribution that couples to the gauge bosons through the Q D -loop diagrams (see also Ref. [30]).
To match the scalar resonance in the effective field theory onto the composite states, we rely on the Naive Dimensional Analysis (NDA) [31,32], leading to with a canonical kinetic term. The parameter κ represents O(1) uncertainties of the NDA.
Altogether, we obtain the effective interactions of S to the SM gauge bosons as As a result, the coefficients in the effective interactions given in Eq. (3) are given by Therefore, the production rates and the branching ratios are determined by two parameters, sin θ Q and Λ dyn , in this model. modes are about nine and three times larger than that of the γγ modes for most of the 3 We assume that the hidden strong dynamics does not cause spontaneous breaking of the SM gauge symmetries, although Q † Q is expected to be non-vanishing. In particular, the mass terms of Q's lead to linear terms of the SM singlet composite scalars, resulting in Q † Q = 0. parameter region. On the other hand, the branching ratio into gluons is suppressed compared to even that of γγ for small sin θ Q , as is evident from Eq. (16).
In the figure, we also take into account the decay of S into a pair of the 125-GeV Higgs bosons due to interactions between Q's and Higgs boson H, with λ L,D being coupling constants. These interactions induce an effective interaction between S and Higgs doublets, where we again use the NDA and reparameterize λ L,D and θ Q by λ. Through this operator, the resonance decays into a pair of Higgs bosons with a partial decay width: 4 In Fig. 1  Now, let us discuss the favored parameter region on the (sin θ Q , Λ dyn ) plane. Fig. 2 shows in blue curves the contours of the cross section of the diphoton signal at the 13-TeV LHC. 5 The searches for a resonance decaying into a pair of Higgs bosons have imposed an upper bound of σ(pp → S → hh) 39 fb [33], which can be satisfied in most of the parameter region in Fig. 2. 6 As discussed in Ref. [25], a similar quartic coupling between the SU (2) L composite triplet scalar and a pair of Higgs doublets leads to a non-vanishing vacuum expectation value of the composite triplet scalar. Due to electroweak precision constraints, the typical size of the quartic couplings should be at most O(0.1) for the model to be successful.
Since the composite scalar mass is expected to be at around Λ dyn , we find that an appropriate range of the mixing angle is sin θ Q respectively, which will be tested by the LHC Run-II experiments. 7 Before closing this section, let us comment on the SM-charged composite states predicted in this model. Since the hidden sector consists of Q D and Q L , the model predicts not only the singlet composite scalar, but also the charged composites: an SU (3) C octet, an SU (2) L triplet, and a bi-fundamental representation of SU (3) C × SU (2) L with a hypercharge of 5/6.
Due to the color charge of the octet scalar, it is directly produced through the SU (3) C gauge interaction at the LHC and decays into a pair of gluons. By the searches at the 8-TeV LHC, the production cross section of the octet scalar with a mass around 1 TeV is constrained to be around O(1) pb [40,42], which is much larger than the pair production cross section of the octet scalar [43] as well as the single production rate via Eq.
(3). It should be noted that the octet scalar mass is expected to be larger than that of S because The scalar in the bi-fundamental representation of SU (3) C ×SU (2) L requires special care, as it cannot decay into a pair of SM gauge bosons due to its charges. To make it decay promptly, we introduce one flavor of fermions under the hidden SU (5) gauge symmetry (ψ Q ,ψ Q ), which allow Q's to couple to the SM quarks and leptons,d R and L , via Here, y denotes a coupling constant and M Q the mass of the fermion ψ Q . 8 Through these interactions, the [Q † D Q L ] bound states decay intod † R + L + S 9 which is estimated to be roughly 7 If the coupling to the Higgs doublet in Eq. (18) is sizable, the above pretictions can be slightly modified. 8 We take M Q to be much larger than a TeV, so that they are not produced at the LHC. 9 The two-body decay width intod † R and L is suppressed by the mass of the masses of the quarks.
where M [Q † D Q L ] denotes the mass of the bound state. For M Q 10 4 GeV, the bound state decays promptly into down-type quarks and leptons and S which subsequently decays into jets, W W , ZZ, Zγ or γγ as discussed before.
For a larger M Q , e.g., M Q 10 7 GeV, the bound state can be stable within the detectors and give an striking signature. The lower mass limit put by the results of searches for heavy stable charged particles at CMS ranges up to 0.9-1 TeV, depending on the QED charges [44]. 10 It should also be noted that for M Q 10 8 − 10 9 GeV, the lifetime of the bound state becomes longer than O(1) second and spoils the success of the Big Bang Nucleosynthesis [47]. 11

DISCUSSIONS AND CONCLUSIONS
In this paper, we have revisited a model of scalar composite resonance that couples to the SM gauge bosons via the higher-dimensional operators proposed in Ref. [25]  The neutral scalar boson is accompanied by many charged bound states whose masses are also in the TeV regime. Therefore, we expect that the LHC Run-II experiments will discover a zoo of such particles around that scale. In particular, the bound state of [Q † L Q D ] has a striking signature of decaying into a lepton, a down-type quark and S, or it can even leave charged tracks inside the detector when the bound state is sufficiently stable. 10 When the mass of the scalar [Q † D Q L ] bound state is 1 TeV, the production cross section is 0.2 fb at 8 TeV [45] and 6 fb at 13 TeV [46]. 11 See also Ref. [48] for related discussions.
As a peculiar feature of this model, the mass of the lightest composite state is not separable from the dynamical scale of the hidden sector, as it is not protected for any symmetry reasons. Thus, the dynamical scale should be in close proximity to the composite mass, unlike again the models in which the 750-GeV resonance is identified as a pseudo-Goldstone boson. Therefore, we expect that the quark-like picture of the hidden sector emerges at a rather low energy in future collider experiments. For example, production of multiple partons in the hidden sector becomes possible and ends up with events of multiple jets, multiple leptons and multiple photons.
Before closing this paper, let us address an important question: "who ordered the 750-GeV resonance?" One ambitious answer is the dark matter candidate. In fact, as discussed in Ref. [25], this model has a good dark matter candidate: the lightest baryonic scalar This state is neutral under the SM gauge groups due to the choice of N h = 5. 12 It should be emphasized that the neutralness of the lightest baryonic state under the SM gauge group is one of the prominent features of this model. If, instead, the hidden sector consists of bifundamental fermions, the neutral baryonic state is expected to be heavier than the lightest but SM-charged baryonic state since the neutral baryonic state has a larger orbital angular momentum inside.
In the early universe, the baryonic scalars annihilate into a pair of lighter scalar nnonbaryonic composite states. The thermal relic abundance would be much lower than the observed dark matter density if the annihilation cross section (into S, glueballs, etc.) saturates the unitarity limit [50]. The abundance of B is roughly given by, where M B is the mass of B and F (M B ) denotes the form factor of the interactions of B with the lighter states. 13 By remembering M B Λ dyn (in particular when m D > Λ dyn ), it is 12 To make B stable, a (discrete) symmetry is required. We presume that such a symmetry is not broken spontaneously by the strong dynamics as long as m 2 D,L = 0. 13 We define the form factor in such a way that the interaction of B saturates the unitarity limit when F = 1. expected that the form factor is slightly smaller than 1. Therefore, the thermal relic abundance of B can be consistent with the observed dark matter density, although a quantitative estimation is difficult due to our inability to estimate the form factor precisely.
Finally, let us comment on the direct detection of the dark matter candidate. The coupling between Q's and the Higgs doublet in Eq. (18) also leads to a direct coupling between the scalar dark matter and the Higgs doublet, where λ B is of O(λ L,D ). 14 Thus, the dark matter interacts elastically with nuclei via the Higgs boson exchange resulting in a spin-independent cross section [51] σ SI = λ 2 where we have used the lattice result f N 0.326 [52]. Although this coupling is much smaller than the current limit σ SI 5 × 10 −44 cm 2 (M B /5 TeV) by the LUX experiment [53], it is within the reach of the proposed LUX-Zeplin (LZ) experiment [54], with details depending on the coupling constants and the dark matter mass.