Diboson Resonance as a Portal to Hidden Strong Dynamics

We propose a new explanation for excess events observed in the search for a high-mass resonance decaying into dibosons by the ATLAS experiment. The resonance is identified as a composite spin-$0$ particle that couples to the Standard Model gauge bosons via dimension-5 operators. The excess events can be explained if the dimension-5 operators are suppressed by a mass scale of ${\cal O}(1$-$10$) TeV. We also construct a model of hidden strong gauge dynamics which realizes the spin-$0$ particle as its lightest composite state, with appropriate couplings to Standard Model gauge bosons.


I. INTRODUCTION
Recently, the ATLAS Collaboration reported excess events in the search for a high-mass resonance decaying into dibosons which subsequently decay hadronically [1]. The excess peaks at the diboson invariant mass around 2 TeV. For an integrated luminosity of 20 fb −1 in the 8-TeV LHC run, the local significances of the excess events are 3.4 σ, 2.6 σ, and 2.9 σ when they are interpreted as the decay of the resonance into W Z, W W and ZZ, respectively.
In this paper, we propose another promising possibility where the resonance is identified as a composite spin-0 neutral particle that couples to the Standard Model (SM) gauge bosons via dimension-5 operators. As we will show, the excess events can be explained if dimension-5 operators are suppressed by a mass scale of O(1-10) TeV.
We also construct a model of hidden strong dynamics which realizes the above-mentioned spin-0 particle as its lightest composite state, with appropriate couplings to the SM gauge bosons. In this case, the composite spin-0 particle consists of bi-fundamental scalars under the hidden and the SM gauge symmetries. The mass of the resonance as well as the suppres- The organization of the paper is as follows. In section II, we discuss whether a spin-0 resonance can explain the excess events when it couples to the SM gauge bosons via dimension-5 operators. In section III, we propose a model of hidden strong dynamics which yields a composite spin-0 particle with appropriate couplings to the SM gauge bosons. Discussions and conclusions are given in the last section.

II. EFFECTIVE FIELD THEORY OF DIBOSON RESONANCE
In our proposal, the diboson resonance will eventually be identified as the lightest composite scalar boson in a hidden sector with strong dynamics. The hidden sector couples to the SM sector through fields charged under both the hidden and the SM gauge symmetries. We assume that the hidden strong dynamics exhibits confinement at a dynamical scale around a few TeV, Λ dyn ∼ O(1) TeV, leaving a spin-0 particle as the lightest state, which couples to the SM gauge bosons via higher dimensional operators. Before elucidating explicit models of the hidden strong dynamics, let us discuss how the observed excess events can be explained by a composite spin-0 particle using the effective field theory approach.
Let us consider an effective field theory which consists of its lightest neutral scalar boson S and SM particles, after integrating out heavier degrees of freedom. Effective interactions of S with the SM gauge bosons are given by where Λ is a suppression scale. Here, G, W and B denote the field strengths of the SM gauge bosons of the SU (3) c , SU (2) L , and U (1) Y groups, respectively, with the superscripts a and i being the indices for the corresponding adjoint representations. The field strengths are normalized so that their kinetic terms are given by, where g s , g and g are the corresponding gauge coupling constants. The coefficients κ 3,2,1 are of O(1) and encapsulate details of the strong dynamics.

A. Production cross section of the scalar resonance
Through the above effective interactions, the parton-level production cross section of S via the gluon fusion process is given bŷ in the narrow width approximation, as suggested by the result of ATLAS experiment. In Eq. (3), M S denotes the mass of the scalar boson S andŝ the square of the partonic centerof-mass energy. The partial decay width of S into a pair of gluons is After convolution with the parton distribution function (PDF) of the gluon inside the proton, f g , the total production cross section in the proton-proton collision becomes where τ = M 2 S /s and √ s = 8 TeV. For M S 2 TeV, the luminosity function where we have used the PDF's of MSTW2008 [18] and For comparison, we also give the corresponding values for √ s = 13 TeV: ∂L gg ∂τ 6.6 (5.7) , 1 s ∂L gg ∂τ 15 pb (13 pb) .
The production cross section of S becomes about ten times larger at LHC Run-II.

B. The diboson excess at the LHC
The partial decay widths of the scalar boson S into the other gauge bosons, W , Z and A (photon), are given by where Γ S denotes the total decay width of S.
In the analysis of Ref. [1], excess is observed in each of the W Z, W W and ZZ channels.
At this point, however, the observed signals can be explained by purely an excess in W W  [19,20] from the semi-leptonic channel searches for M S 2 TeV. One caveat here is, however, that the W and Z bosons from the decays of S are in the transverse modes. This feature can make some slight differences in selection efficiencies from the ones estimated in Ref. [1], where the W and Z bosons are assumed to be in the longitudinal modes. Besides, higher-order QCD corrections to the production cross section (the so-called K-factor), can be sizeable. 4 With these reasons, we are satisfied with concentrating on the parameter space where the leading order cross section σ W W + σ ZZ = O(1-10) fb. For a more accurate estimate of the viable parameter range in the effective field theory, we will need higher-order corrections as well as detailed calculations that are beyond the scope of this paper.
In Fig. 1, we show a contour plot of the total production cross section of S that decays into W W and ZZ pairs on the plane of 1 − B gg and Γ S for M S = 2 TeV. In the figure, the gray region is disfavored by the requirement of a narrow resonance: Γ S 100 GeV for the light gray are and 200 GeV for the darker gray area. The pink region is excluded by the constraint from the dijet channel, i.e., σ(p + p → S → g + g) 100 fb [22,23]. The figure shows that the required cross section of O(1-10) fb can be realized in the parameter space where B W W + B ZZ = O(10)% without any conflict with the narrow width approximation or the dijet constraint.
It should be emphasized that the suppressed couplings to the SM fermions are one of the striking features of the composite scalar resonance where the composite scalar couples to the SM fermions via the mixing to the Higgs bosons. These features should be compared with W /Z bosons or generic composite scalar resonance (see e.g. [25]). Therefore, our model is free from the constraints of dilepton mode searches, σ(p + p → S → + ) 1 fb [26,27] 4 See e.g., Ref. [21] for a discussion on the K-factor for the Higgs production, although their analysis cannot be directly applied to the S production here since the effective field theory is valid up to O(Λ), whereas the Higgs effective field theory is valid only for a Higgs mass below twice of the top-quark mass.
Decay branching ratios of S into different modes as a function of κ 2 /κ 3 . Solid curves are drawn by assuming κ 1 = κ 2 . The branching ratios of the γγ and Zγ modes for κ 1 = 0 are also drawn for comparison, while the other modes are barely changed when κ 1 = 0. and the decay into a Higgs boson and a Z boson, σ(p + p → S → Z + h) 7 fb [28].
It is also noted that a spin-0 resonance can decay into a pair of photons. This feature should be contrasted with models where the diboson resonance is interpreted as a massive spin-1 particle whose decay into a pair of photons is forbidden by the Landau-Yang theorem [29,30]. Therefore, an observation of excess in the diphoton mode will be a smoking gun signal of models with a spin-0 resonance. In Fig. 2, we show the branching ratios of all the allowed diboson modes as a function of κ 2 /κ 3 , the ratio of effective coupling strengths in the weak and strong interactions. The solid curves are drawn under the assumption that We also show the branching ratios of the γγ and Zγ mode for κ 1 = 0 as a comparison. Therefore, the branching ratio of S decaying into two photons is expected to be Moreover, the angular distribution of the W and/or Z bosons in the rest frame of the resonance with respect to the colliding direction can be used to diagnose the spin nature of the resonance. The spin-0 resonance in our model would result in a uniform distribution, while models with spin-1 resonance should predict a parabolic distribution.
So far, we have not discussed the sizes of the coefficients κ i in Eq. (1). Since the decay widths are controlled by M S and the coefficients, we can estimate the required sizes of them as a function of Γ S and B gg . In Fig. 3, we draw the contours of on the plane of B gg and Γ S , thereby giving us some rough ideas about the corresponding dynamical scales (see Eqs. (4), (9), (10) and (13)). In Eq. (16), we have used the approximation that g 0. The figure shows that an appropriate cross section, σ W W + σ ZZ = O(1-10) fb, is obtained for those parameters in the range of O(1-10) TeV. It is also noted that the parameter space with B W W + B ZZ ∼ 90% is disfavored since a rather low suppression scale is required to explain the excesses (see also Eq. (21)). As these coefficients are intimately related to the dynamical scale of the hidden strong dynamics behind the scalar composite field S, this result suggests that the strong dynamics is also at around the TeV scale.

III. HIDDEN STRONG DYNAMICS
In the above discussion, we have shown that the diboson excess events reported by the the SM gauge charges. The charge assignments of Q's are given in Table. I. As an explicit example, we take N c = 5 (see discussions at the end of this section), though most of the following discussions can be applied to different choices of the hidden gauge group. We assign the SM gauge charges to Q's in such a way that they form an anti-fundamental representation of the SU (5) GU T gauge group, the minimal SU (5) grand unified theory (GUT). In the following, Q L,D denote the bi-fundamental scalars.
Let us assume that the bi-fundamental scalars have masses, m D,L , so that When these masses are smaller than the dynamical scale, m L,D Λ dyn , the lightest composite state is expected to be generally a mixture of composite mesons consisting of a pair of Q and Q † and a glueball. In our analysis, we assume that the lightest scalar state is dominated by the neutral meson states 4 where θ Q parameterizes the relative contents of Q † D Q D and Q † L Q L . For example, the Q † D Q D content is expected to be suppressed for m D m L , although it is difficult to estimate θ Q quantatively due to the non-perturbative nature of the interaction. 5 In the following, we assume that the hidden strong dynamics does not cause spontaneous breaking of the SM gauge symmetries.
Using the Naive Dimensional Analysis (NDA) [31,32], the scalar boson S is matched to the composite fields by where κ is an O(1) coefficient within the uncertainty of the NDA. As a result, we obtain the 4 The possibility of the glueball-dominated scenario is discussed in Appendix A. 5 Here we naively assume that the lightest singlet scalar corresponds to the singlet under SU By comparing with Eq. (1), we can then identify the coefficients used in the previous section: As discussed in the previous section, the scales Λ/κ 1,2,3 are required to be of O(1-10) TeV to account for the diboson excess (see Fig. 3). On the other hand, the mass of S, M S 2 TeV, is expected to be of O(Λ dyn ). These conditions are simultaneously satisfied for κ ∼ O(1), consistent with the NDA.
In Fig. 4, we show various contours on the plane of Λ/κ 2 and Λ/κ 3 , the two of which are related to κ/(4πΛ dyn ) and θ Q via Eq. (21). 6 The figure reconfirms that the cross section keeping the total width of S sufficiently narrow, as indicated by the green region. 7 As the figure shows, it is preferred to have a smaller value for κ 3 /κ 2 tan θ Q . This can be readily achieved when the mass of Q D is larger than that of Q L .
In the figure, we also show contours of the cross section of the diphoton channel, σ γγ , which is about 5-10% of σ W W + σ ZZ . The light green region is excluded by the constraints on the diphoton channel, σ γγ 0.3 fb [24], which is one of the most constraining channels at LHC Run-I. By remembering that the production cross section of S is enhanced by a factor of ten at LHC Run-II (see Eq. (8)), it is possible to test this model by searching for the diphoton signals.
So far, we have not included couplings between the bi-fundamental scalars and the Higgs 6 The effective field theory is controlled by two parameters, κ/(4πΛ dyn ) and θ Q , in this dynamical model.
In particular, the branching ratio of each S decay mode is solely determined by θ Q . 7 The total cross section σ W W +σ ZZ is slightly smaller than the one shown in Fig. 1  bosons, such as, where λ represents a coupling constant and Γ A = 1 or the Pauli matrix, Γ A = σ i , for Q L and Γ A = 1 for Q D .If we allow such interactions, the resonance also decay into a pair of Higgs bosons, which alter the total decay width of Γ S as well as the branching ratios, in particular the ratio of the diphoton mode. So far, the resonance decay into pair of the higgs is not severely constrained [33]. In this paper, we simply assume that the direct couplings between Q's and the Higgs bosons are somewhat suppressed.
We also comment on the constraints from electroweak precision measurements. The most dangerous effect is from the interaction term in Eq. (22) with Γ A = σ i for Q L . Below the dynamical scale of the hidden strong dynamics, we obtain the effective interaction where T i is the composite triplet scalar. After the Higgs field obtains a vacuum expectation value v EW , the triplet is also induced to have a vacuum expectation value, where M T is the mass of the triplet. As long as λ < ∼ O(1), the constraint from the T parameter can be evaded. Contributions to the S, T , U parameters by quantum corrections [34] are also suppressed by the dynamical scale and hence small. The octet scalar is pair produced via QCD processes and singly produced via dimension-5 operators coupling to the gluons. So far, the production cross section of the octet scalar is constrained to be smaller than about 100 fb for a mass around 2 TeV [22,23]. In both production processes, this constraint is evaded.
On the other hand, the triplet scalar is produced via the Drell-Yan process and immedi- where y denotes some coupling constant and M denotes the mass of the fermion ψ Q . 9 Through these interactions, the Q † D Q L bound states immediately decay into a pair ofd R and L . With a sufficiently short lifetime, there is no stringent constraint on the bi-fundamental representation of SU (3) c × SU (2) L with a mass of O(1) TeV.
Before closing this section, let us comment on the baryonic states of the hidden SU (5) gauge interaction. The lightest baryonic scalar is given by, which is neutral under the SM gauge groups. This neutrality of B is the reason why we have chosen N c = 5 for the hidden strong gauge interaction. It should be noted that the lightest baryonic state is stable due to an approximate U (1) symmetry. 10 Therefore, the baryonic scalar serves as a good candidate for dark matter.
At the early universe, the baryonic scalars annihilate into a pair of light scalar composite fields. The thermal relic abundance is expected to be much lower than the observed dark matter density if the annihilation cross section saturates the unitarity limit [35]. However, the mass of the lightest baryonic scalar is higher than the dynamical scale. Thus, the effective coupling between the light scalar composites and baryon dark matter can be suppressed by 9 By taking M much larger than a TeV, these additional fermions cannot be produced at the LHC. 10 Here we assume that the U (1) symmetry is not spontaneously broken by the strong gauge dynamics.
form factors, which leads to a somewhat suppressed annihilation cross section. If this is the case, the observed dark matter density may be explained by the thermal relic density of the baryonic dark matter in the model.
The strong dynamics also predicts heavier composite modes. The heavier composite states are expected to decay into the light scalar composite or the lightest baryon state by emitting S,d R and L , so that there is no stringent constraint on them.

IV. CONCLUSIONS AND DISCUSSIONS
In this paper, we have proposed a new explanation for the excess events observed in the search for a high-mass resonance decaying into dibosons by the ATLAS experiment. The resonance is identified as a composite spin-0 particle coupling to the Standard Model gauge bosons via dimension-5 operators. We find that the reported excess can be explained if the dimension-5 operators are suppressed by a mass scale of O(1-10) TeV. As a notable feature of our model, the resonance decays into a pair of photons, which is absent in proposals of interpreting the resonance as a spin-1 particle.
We have also constructed a model of hidden strong dynamics which realizes the spin-0 particle as its lightest composite state, with appropriate couplings to the Standard Model gauge bosons. In this scenario, the composite spin-0 particle consists of bi-fundamental scalars of the hidden and the Standard Model gauge symmetries. The mass of the resonance as well as the suppression scale of O(1-10) TeV are achieved when the hidden strong dynamics exhibits confinement at a dynamical scale of O(1) TeV. Along with the neutral scalar boson, the dynamical model predicts many charged particles whose masses are also in the TeV regime. Therefore, we expect in this model that the LHC Run-II experiment will discover a zoo of particles around that scale.
A natural question about the diboson resonance at the TeV scale is "who ordered that?" One possible answer is the dark matter. In our model, for example, there is a dark matter candidate in the hidden sector with a mass in the TeV regime. In conjunction with the anthropic arguments, the dynamical scale at the TeV regime may be justifiable. If the scale of the resonance is related to the origin of the electroweak scale, the TeV scale of the resonance may again be justifiable by anthropic arguments.
If, on the other hand, the dibsoson resonance is not directly related to either the dark matter or the electroweak scale, the resonance at the TeV scale provides a strong counterexample to the anthropic arguments. In such a case, the TeV scale of the resonance needs to be explained for its own sake by, for example, supersymmetry. Interestingly, the dynamical model in section III has almost an identical structure to the supersymmetric model in Ref. [36], where the dynamical sector was introduced to achieve the observed Higgs boson mass in the MSSM with soft supersymmetry breaking masses in the TeV regime. We will discuss the diboson resonance in the supersymmetric model in a separate work.
where H A µν (A = 1 − 24) denote the field strengths of SU (5) gauge bosons, which are again normalized to have Below the mass scale of Q's, the hidden sector ends up with a strongly interacting pure Yang-Mills theory. At around the dynamical scale Λ dyn , the gauge coupling constant in the hidden sector becomes strong, i.e., g H ∼ 4π, and confinement is expected to occur. In this case, the scalar glueball S becomes the lightest state. To match the scalar glueball S to HH in Eq. (A1), we again use the NDA: H Aµν H A µν = 4πκ Λ 3 dyn S . seems contradicting with the NDA expectation. Therefore, it is unlikely that the excess can be explained by the glueball state.