Enhanced diHiggs signal from hidden scalar QCD at leading-order scale-symmetry limit

We develop an effective-model description arising from a recently proposed scale-invariant hidden scalar-QCD sector,which has been used to explain the dynamical origin of the electroweak scale. In addition to the previous works, our new effective model includes the dynamical scale-anomaly effect from the hidden QCD gluons, to explicitly break the classical-scale invariance at the level of an effective field theory, which is known as the leading-order scale-symmetry (LOSS). In the phenomenological analysis, the proposed model predicts a light composite dilaton composed of hidden scalar quarks and gluons with the mass around electroweak scale (around 280 GeV), and has only one input parameter, which is the mixing angle between the Higgs boson and the composite dilaton. Our result for the dilaton mass is in accord with the lattice simulation for scalar QCD, where the scalar-quark bound states acquire a large effective mass from the hidden gluon contribution. Furthermore, we predict several significant deviations from the SM, like the resonant and non-resonant diHiggs production cross sections (maximally about 6.8 times larger than the SM prediction), that could be directly tested at the high luminosity LHC. It is also the first study for the diHiggs production signal predicted from a scale(conformal)-invariant hidden sector, even from dark/hidden QCD. Our proposed effective model is thus significantly different than the conventional realization of scale-invariant hidden-scalar QCD without the scale anomaly effect, and can potentially provide a competitive explanation for many exotic phenomena beyond the standard model, such as new dark matter candidates and a strongly first-order electroweak phase transition.


Introduction
It is no doubt that the standard model (SM) meets with nearly all of the experiment results in high energy physics perfectly.However, from a theoretical point of view, the construction of the SM seems to be artificial: a negative Higgs mass term is introduced to generate a nonzero vacuum expectation value that finally breaks the electroweak (EW) symmetry.It is thus natural to ask why the EW symmetry has to be spontaneously broken at the EW scale (v ≃ 246 GeV)?What is the reason behinds this artifact?On the other hand, if the SM is a fundamental theory that is still useful even in the Planck scale, then the extremely large quadratic divergence from the quantum correction will eventually produce a highly-accurate fine tuning of the Higgs mass, which seems to be very unnatural in physics.Therefore, it would be natural to suspect that a more fundamental theory should exist beyond the SM, which should explain the genesis of EW scale in a more natural way and avoid the fine tuning problem.
To go beyond the SM, conventionally we should start from what we have confirmed in the experiment, like the existence of the dark matter (DM), neutrino oscillation, CP violation, etc.At the same time, it is also possible to get some indirect hints for the new physics by some strongly favored proposals even if no direct evidence for any of these scenarios was found.However, there're some ways to incorporate many of these proposals and find solutions related to well-known puzzles while maintaining a strong predictivity.A recently proposed hidden scalar QCD [1] is clearly one of them, with the classicalscale invariance incorporated as a solution for the fine-tuning problem, which possesses a minimal number of new parameters, but provides rich phenomenological results for the future experiments.
Since no direct evidence of new particles or strong deviation from the SM has been confirmed in the LHC experiment, it seems that the most natural way to extend the SM is to introduce a hidden sector, which only couples to the SM minimally.Among these hidden sector scenarios, the Higgs portal coupling to extra scalar fields is the simplest way.The hidden scalar QCD is such a proposal that the Higgs mass term is replaced by a Higgs portal coupling to some charged scalar fields S, so the EW symmetry breaking scale can be dynamically generated from a nonperturbative condensation effect of these scalar fields [1,2].After the replacement, this model maintains a classical-scale invariance in the Lagrangian level.Then no dimensionful parameters in the Lagrangian appear, and thus raise no quadratic divergence accounting for the hierarchy problems #1 .
In the theory of a hidden scalar QCD, the equilibrium of the potential will be reached when both the scalar field and the Higgs field get nonzero vacuum expectation value from the condensation of hidden scalar-field.Therefore, it is possible to explain the scalegenesis of the EW symmetry as the effect from the hidden scalar QCD.What's more, the hidden scalar QCD also naturally includes new DM candidate because of the possible existence of stable scalar hadron states [1,5].Furthermore, it is also possible to realize a stronglyfirst order EW phase transition and a detectable primordial gravitational waves signal, as long as the Higgs portal coupling can be large enough [6,7].Thus the hidden scalar QCD possesses a great potential to explain many exotic phenomena beyond the SM.However, initial investigations of this new model may still lack important ingredients.
To capture such a possibly missing point, we may make use of the powerful tools about the effective theory of fermionic QCD.In the fermionic QCD case, it is believed that the gluon condensation effect that arising from the nonperturbative-scale anomaly effect, can be very important for the low-energy effective theory of QCD, especially for the origin of hadron mass.Recently, based on this consideration, a chiral-scale effective theory has been built [8], where the scale anomaly was introduced as an essential effect to generate hadron mass.Combined with the classical-scale invariance, this minimal inclusion of the scale anomaly is known as the leading-order scale-symmetry (LOSS).
In this paper, we propose an effective model for hidden scalar QCD at the LOSS limit with the gluonic effect incorporated, which turns out to be consistent with the lattice simulations [9,10], as well as some nonperturbative analysis [11] on scalar QCD hadron spectra.It is demonstrated that our model gives a definite prediction to diHiggs signals at the LHC, in correlation with a significant deviation from the SM, and the Higgs coupling measurement.This result turns out to be a smoking-gun of the LOSS limit, which will open a new avenue for the phenomenological probe of the hidden scalar QCD scenario, other than those so far explored, such as DM and EW phase transition, and gravitational waves detectability.This paper is structured as follows: In Sec.II, we briefly review a classical-scale invarianthidden scalar QCD sector and its possible low-energy description governed by low-lying scalar QCD hadrons.In Secs.III and IV, we explicitly break the classical-scale symmetry in a minimal way by introducing the LOSS, and show the essential points of the scaleanomaly effect and the role that the gluon plays in detail.Then in Sec.V we make several #1 More rigorous argument on the fine-tuning problem based on the classical scale invariance by the renormalization group analysis has been made.See, e.g.[3,4].
phenomenological predictions from this model, such as the Higgs trilinear coupling enhancement, the non-resonant and the resonant diHiggs production cross section.Finally, Sec.VI devotes to summary of this paper and prospect on the LOSS for some future researches to the DM, as well as the EW phase transition.

A Scale-Invariant Hidden-Scalar QCD Sector
Let us begin with the review of a simple classical-scale invariant-SM extension.In some previous works [1,2], it's assumed that there exist some SM-singlet scalars (S a i ) strongly couple to each other via a hidden scalar QCD gauge interaction, with the hidden flavors i = 1, 2, ..., N f as well as the hidden QCD charges a = 1, 2, ..., N c under the hidden gauge group SU (N c ). Briefly speaking, this scenario is just an easy extension of the SM by replacing the Higgs mass term, with the interaction of a hidden scalar QCD sector plus a Higgs portal coupling, which can overall be described by the following Lagrangian: where D µ = ∂ µ − ig s G a µ (t a /2), g s is the strong coupling constant of the hidden scalar QCD, H is the SM Higgs doublet, G a µ and t a /2 correspond to the gauge fields and the generators of the hidden SU (N c ) symmetry respectively.
At the classical level, the whole system possesses an extra SU (N c ) gauge symmetry, an extra U (N f ) global symmetry, as well as the classical scale invariance.Due to the asymptotic freedom of the strong dynamics, at a low-energy scale, the gauge coupling g s grows up to be nonperturbatively large, driving the dynamical formation of an U (N f )invariant scalar-bilinear condensate #2 [2,12]: which breaks the classical-scale invariance spontaneously.It also induces a negative Higgs mass term proportional to λ HS S † S , and thus leads to the spontaneous breaking of EW symmetry.In this sense, the origin of EW and the classical-scale symmetry-breaking can be explained simultaneously by this hidden scalar QCD, as has extensively been worked on in many papers [1,2].At the same time as the scalar condensation takes place, the color confinement will also be triggered dynamically.It means that in the low-energy region, the degrees of freedom in the system will no longer be the colored scalars, but composite states like scalar mesons and baryons.Therefore, if the hidden scalar QCD scale is at least at the same order of the EW scale (v ≃ 246 GeV), we must construct a low-energy effective theory for those hidden hadrons when we are going to study physics at around the EW scale.
So far, there have been many attempts to establish a well low-energy effective theory for this kind of scale-invariant hidden scalar-QCD sector, such as a quark-meson model [1].The analysis was done based on assuming that the spontaneous breaking of the scale symmetry dominates at the low-energy description constructed from the quark-mesons, while the nonperturbative scale anomaly effect can be ignored.However, we find that it is this nonperturbative scale anomaly effect that should be emphasized to be a crucial ingredient in employing a low-energy description for hidden scalar QCD, which will be clearly demonstrated in the later sections.
Just like the fermionic QCD, for simplicity let us consider a linear sigma model as the low-energy description of the scale-invariant hidden scalar QCD.To this end, we can just simply apply the general symmetry argument to write down every possible term in the hidden sector at the low-energy scale.Now, scalar quarks are confined to form mesons and baryons, that can be denoted as S † i S j ∼ M ij and S i S † j S k ∼ B ijk , respectively.With the U (N f ) global symmetry and the classical scale invariance, the most general Lagrangian for these fields can be written as follows: where it is assumed that all λ's are positive, and the trace is taken over all flavor indices with the normalization for the U (N f ) generators, tr(T a T b ) = 1 2 δ ab .After expanding the field M in terms of χ and A 0 as: , we can redefine the coupling constants to be: As to the phenomenological consequence for the scalar baryon B and U (N f )-adjoint mesons A 0 , we will make give some comments in the later section, and hereafter will focus on the singlet scalar χ coupled to the Higgs field H. Now, comparing Eq.(2.4) with Eq.(2.1), and reminding ourselves with the previous analysis of the underlying scalar QCD dynamics, it's obvious that to have the appropriate spontaneous breaking of the scale invariance, a nonzero vacuum expectation value for the field χ (denoted as η) must exist in the low-energy scale, dynamically originated from the underlying scalar-bilinear condensation S † S .Furthermore, as we know, another nonzero vacuum expectation value should also exist for the Higgs field H at v ≃ 246 GeV.
Combining these two conditions together, we can require the following potential to reach a minimum simultaneously in the vacuum: (2.5) Then the stationary condition will be: which has the solutions: Not surprisingly, we find a flat direction along the η axis, where the vacuum expectation value of the χ field generates the nonzero vacuum expectation value of the Higgs field H, simultaneously triggering the spontaneous breaking of the EW symmetry as well as the scale symmetry.Around the vacuum, we may find out the physical spectra for scalars.To examine that, let us take the unitary gauge for the EW gauges, so that we can easily expand the potential by χ → η + χ and H → 1 √ 2 (0, v + h) T .After the spontaneous symmetry breaking, the linear terms vanish as it must be protected by the stationary condition.The quadratic terms are: with The last term reflects the mixing between H and χ.Actually, the determinant of this mixing matrix turns out to be zero, implying that a physical massless scalar boson, a massless dilaton χ appears, as the Nambu-Goldstone boson for the spontaneous-breaking of scale-invariance [14], which is also known as scalon in some literature [15].But we must note that actually the scale symmetry is merely an approximation and is assumed to be explicitly broken by the quantum corrections, which eventually produce a massive pseudo Nambu-Goldstone boson in the physical world, just like pions.As will be clarified later, a possible inclusion of the nonperturbative-explicit scale-symmetry breaking effect into this toy model will allow us to have a heavy enough massive composite"dilaton".
Actually, the nonperturbative-explicit breaking effect, i.e., the nonperturbative scale anomaly signaled by the gluon condensate, can be closely tied also with the scalar QCD hadron spectra: a decade ago, H. Iida, T. Takahashi, and H. Suganuma performed the first lattice simulation for the SU (3) c scalar QCD [9], showing that all of the scalar-quark hadrons obtain large quantum corrections dominantly from the gluons, i.e., the gluon condensate.Besides, a non-perturbative analysis with the Schwinger-Dyson formalism was also carried out a few years ago [10,11], which came to the conclusion that the mass of scalar-diquark bound state is dynamically generated from the nonperturbative-gluonic dressing-effect including the gluon condensate part.These results supports the gluonic effect to be one of the important missing keys to predicting the realistic scalar QCD hadron spectrum in the effective theory.
To properly incorporate such an important gluonic effect into the scalar QCD as the main mass-scale generator, in the present study we shall adopt some recent idea, named the chiral-scale effective model, developed as a low-energy description of the fermionic QCD [8].As will be clarified below, this idea can properly realize the important gluonic contribution playing the role of the mass-scale generator through the nonperturbatively explicit breaking of the scale invariance #3 .
3 Leading-Order Scale-Symmetry (LOSS) As we discussed above, it is essential to remove the strong constraint of the classical scale invariance to give a mass to the dilaton, and also to include a non-perturbative gluonic effect from the underlying dynamics.One possible access to achieve them is to introduce a small explicit breaking term characterized by a breaking parameter a arising from the scale anomaly, which is conventionally known as the leading order scale symmetry (LOSS) in the fermion QCD: [8].
where χ is a dilaton field operator, the breaking parameter a is actually an anomalous dimension that can be either positive or negative, η is the same as the conventional dilaton decay constant f χ , λ a is a dimensionless coupling constant and has been normalized by the vacuum expectation value of dilaton (η), L SI is the part that preserves the original classicalscale invariance.This Lagrangian then no longer possesses the classical-scale invariance, but can still have an approximate scale symmetry if the breaking parameter a is small enough.In that case, the scale symmetry is explicitly broken minimally and only at the dilaton potential, which is the reason why it is called the leading order scale symmetry (LOSS).
To see how this explicit breaking term is originated from the quantum gluonic corrections, we may neglect the effect of other fields at this moment, only concentrate on the self interaction of dilaton in Eq.(2.4), and derive the partially-conserved dilaton-current (PCDC) relation associated with the dilatation current D µ = θ µν x ν .Suppose now we have a potential: #3 In addition to the chiral-scale symmetry structure, the color confinement effect of these strongly coupled scalar fields can also be important, especially for the vacuum structure and phase transition.However, to discuss color confinement, we need more assumptions as well as lattice results to show that the scalar QCD behaves exactly the same as the fermionic QCD, as was discussed in [1].One recent paper [16] just tried to include the Polyakov loop effect into the effective model for this hidden scalar QCD and then found a significant enhancement to the energy density of the gravitational waves background produced from a first-order scale phase transition.So just like the fermionic QCD, the effective model for a scalar QCD is also extremely complicated, where the inclusion of color confinement can also be important within some context.However, this interesting issue is beyond scope of our present study.
then we can easily write down (the symmetric part of) the trace of the energy-momentum tensor that equals to the derivative of dilatation current: where δ D is an infinitesimal variation under the scale transformation.Since this potential has a minimum at χ = η, the vacuum expectation value of T µ µ is nonzero.We can thus replace one of η 2 with the dilaton mass in the true vacuum (m 2 χ = (a • λ a )(4 + a)η 2 ), and find: This is the PCDC relation for this potential.If we take into account the trace anomaly, the scaling dimension of T µ µ should equal that of the nontrivial gluon configuration (condensate): + a, which will be modified by an anomalous dimension a evaluated at some infrared scale, say, Λ HQCD ∼ O(1) TeV.Thus the parameter a surely corresponds to the anomalous dimension of gluon condensate.

Effective Model with LOSS
Now we are ready to incorporate the nonperturbative-scale anomaly effect to the original linear sigma model for the hidden scalar QCD and then observe what happens.
The dilaton potential in Eq.(3.2) together with the Higgs potential leads to new solutions to the stationary condition: where it is assumed that the explicit-breaking effect should be small enough to preserve an approximate scale invariance, so that terms coming with the anomalous dimension a can be expanded as χ a = 1 + a ln χ + O(a 2 ).Similar arguments can be repeated for the scalar-mass matrix having the mixing structure.After solving the mixing, the physical Higgs and dilaton fields (h ′ , χ ′ ) can be related to the original fields (h, χ) as Eventually, it turns out that only the formulas for the mass of the physical dilaton field χ ′ would be modified from the scale-invariant limit case to: where the new contribution to dilaton mass (∆m 2 χ ′ ) has been defined in the last line to emphasize the gluonic effect.
To numerically estimate the mass of this dilaton, we need to fix two more input parameters in the model setup.To demonstrate the crucial gluonic effect, we shall take a simple-minded ansatz: assume the dilaton mass is around the vacuum expectation value η.A similar situation can be seen in the current algebra argument applied to the ordinary QCD pion, arising as the pseudo Nambu-Goldstone boson for the spontaneous-chiral symmetry breaking, where the pion mass scale is nearly identical to the decay constant scale f π .Thus in the ideal limit where η = m χ ′ , by taking the small value for a, a = 0.1 as a reference point realizing the approximate scale invariance, we can numerically estimate the dilaton mass and values of other couplings to find where the physical Higgs mass (m h ′ ) and the EW scale (v) have been fixed to be about 125.09GeV and 246 GeV, respectively, and the mixing angle θ has been determined by cos 2 θ = 0.9 from the current upper bound on the Higgs coupling measurement at the LHC experiment [17].In particular, we note that these parameters predict ∆m 2 χ ′ ≃ 1.051v 2 , and thus This clearly shows that the mass of the dilaton composed of scalar bilinear is dominantly supplied by the gluon condensate, in agreement with the lattice results for scalar QCD [9][10][11].
Before closing this section, we want to emphasize that indeed, how to choose the value of anomalous dimension a is not important.As long as the value of a is small, we can always expand the anomalous term as χ a = 1 + a ln χ + O(a 2 ) and neglect all the higher order corrections.Then by making use of the stationary condition in Eq.(4.1), terms proportional to λ a can be replaced with λ χ , so we are only left with one possible combined variable (a • λ a ) entering into all the effective couplings.Actually, this striking fact also implies that even in evaluating cross sections involving the dilaton χ ′ , only the combination of (a • λ a ) can be observed and be determined from the experiment.Once we fix the free parameter m χ ′ in our model, this factor will also be fixed to a constant automatically.So in practice, the phenomenology of this model will not be affected by the value we choose for the small anomalous dimension a, which makes this model more predictive.

Phenomenological Predictions
In this section, we will discuss the specific phenomenological predictions made by this model.Because of the mixing between the dilaton and the Higgs boson, it's obvious that the dilaton can be generated at the LHC through the same process as we produce the Higgs boson, which would be dominated by gluon-gluon fusion production.Similarly, the dilaton decays into the SM particles immediately in the same way as the SM Higgs, signaled by diboson events including the diHiggs channel (if the dilaton has at least twice of the Higgs boson mass), and the diphoton event, etc.These cross sections can be easily calculated from this LOSS effective model, eventually resulting in the significant enhancement with respect to the SM's predictions.We will show that these predicted signals will be strong enough to be seen in the future collider experiments, which will be listed thoroughly in the following.
Now, let us vary the only input parameter sin 2 θ, from 0.05 to 0.1, and solve for the corresponding physical dilaton mass by resolving it to be identical to the vacuum expectation value of the original field χ (η).The result is shown in Figure 1.There we see that the physical dilaton mass is significantly greater than twice of the Higgs mass, kinematically permitting its decay into diHiggs.Next, we can calculate the trilinear couplings among the field χ ′ and h ′ .The definitions for the couplings we use are: Note that for these trilinear couplings, the sign of trigonometric function becomes important, as you can easily check in one of the expansions: Therefore, by varying the input mixing angle, the total decay width of dilaton (Γ χ ) will have an explicit dependence not only on the sin 2 θ, but also on the sign of cos θ. Figure 2 shows the numerical result for Γ χ .The list in Table 1 shows the numerical results for some particular choices for the angle θ including the sign ambiguity.Figure 3 shows the branching ratios for the dominant W W , hh, ZZ decay modes as a function of the mixing angle.cosθ>0(Orange) and cosθ<0(blue) Similarly, two possible solutions exist for the Higgs trilinear coupling λ hhh .Our results reveal a significant difference from the SM's prediction, as well as the prediction from the exactly scale-invariant model with a = 0.For simplicity, taking the input mixing angle to be sin 2 θ = 0.1, we can just compare the results for these three models in unit of the EW scale v.
In the SM, the straightforward calculation yields λ hhh v | SM ≃ 0.75.In the classicalinvariant SM-extension (i.e. a = 0), λ hhh v | a=0 ≃ 0.652 for cos θ > 0 and λ hhh v | a=0 ≃ −0.522 for cos θ < 0. So, the Higgs trilinear coupling is totally reduced in the classicalscale invariant-model by about 13% or 30% suppression with respect to the SM's value.Including the LOSS correction with a = 0.1, we find λ hhh v | LOSS ≃ 1.01 for cos θ > 0 and λ hhh v | LOSS ≃ −0.702 for cos θ < 0. Therefore, we can either maximally have about 34% enhancement to the SM's value, or have a nearly intact Higgs trilinear coupling in magnitude.From the above result it is easy to conclude that this result again reveals the importance of including the scale-anomaly effect coming from the radiative corrections of hidden gluons.If any deviation of the Higgs trilinear coupling can be found in the future collider experiment, then it could be a significant signal for probing the effective theory with LOSS.We now show some numerical results for the diHiggs production cross section at the LHC corrected by including the dilaton exchange contribution.The diHiggs signals are dominantly generated by the gluon-gluon fusion through a top quark box diagram, or a top quark triangle diagram combined with exchanging the Higgs boson or the dilaton.For simplicity, we only calculate these cross sections in the heavy top-quark limit (m t → ∞), which will turn out to give a good approximation for computation of the diHiggs cross section having the resonant term with the dilaton mass around 300 GeV, as predicted in our model.Combing all these diagrams, the total gluon-gluon fusion to diHiggs cross section (σ(gg → hh)) is: where b s is the well-known vertex function for the SM Higgs boson coupling to digluon which gives 2  3 in the heavy top-quark limit, α s ≃ 0.118 at the EW scale, ŝ is the total center of mass energy at the parton level, and Γ h ≃ 0.006 GeV is the total decay width of the Higgs boson predicted in the SM.Thus, the diHiggs cross section at the LHC is evaluated as: where f g/p denotes the parton distribution function (PDF) for gluon parton in the proton and x 1,2 are the Bjorken variables relating the ŝ to the center of mass energy for the protonproton collision s like ŝ = x 1 x 2 s.The kinematical cutoff parameter τ 0 (normalized by s) will be set to the threshold energy for the diHiggs production: τ 0 = (2m h ) 2 /s, and the factorization scale µ F will be taken as µ F = √ ŝ.
The LHC cross sections are computed by implementing the CTEQ6L1 parton distribution function [18] in Mathematica with the help of a PDF parser package, ManeParse 2.0 [19], The CUBA package [20] is used for numerical integrations.In numerical analysis, we have also taken into account the next-to leading order QCD corrections, parametrized as the K-factors: for the gg-hh type, K ≃ 1.9 and for the gg-h type, K ≃ 1.76, at around 300 GeV [21].
The diHiggs cross sections at the LHC with √ s = 13 TeV are thus computed to be plotted as a function of sin 2 θ in Fig. 4, where the sin θ ranged from 0.1 down to 0.05 monitors the currently accumulated luminosity relevant for the diHiggs event with L = 36.1 fb −1 , up to the prospected one realized at the high-lumnosity LHC (HL-LHC) with L = 3000 fb −1 , including some intermediate stages.First of all, we can observe the same destructive interference between the SM-box diagram and the triangle diagrams as in the SM.Remarkably, however, the predicted diHiggs cross section gets significantly larger than the SM prediction.For instance, with the currently maximal mixing allowed, i.e., sin 2 θ = 0.1, we have: which is approximately 6.8 times larger than the SM prediction.
The latest and most stringent upper bound (at 95% CL) for the non-resonant diHiggs production cross section comes from the diHiggs to b bb b channel in the ATLAS experiment with 36.1 fb −1 [22], which is about 434 fb (13 times the SM expectation).This bound has been incorporated as the red-solid horizontal line in Fig. 4. Just by naively scaling the current upper bound with respect to the luminosity, we can see from the figure that the LOSS prediction can be probed at the luminosity of about 130 fb −1 .In short, if any of the enhancement for the non-resonant diHiggs cross section will be found in the future, then it would be a strong supporting signal for this model, which can be explored at the HL-LHC with high sensitivity.
Finally, we have simulated the resonant dilaton production cross section.Because the dilaton total width is small enough as seen from Table 1, the dilaton-resonant cross section for generating AB particles in the final state can be evaluated safely by using the narrow ggF (pp → hh) as a function of sin 2 θ.The red horizontal lines are experimental upper bounds coming from non-resonant production, which have been scaled from the currently most stringent limit placed by the ATLAS group [22] to a various of luminosities to be accumulated in the near future, as described in the plot label and in the text.
width approximation (NWA): where the K factor (introduced as above) reads in a way appropriate to each channel, and η is the rapidity of the A-B final state system, cutoff by the pseudo rapidity Y B = − 1 2 ln(m 2 χ ′ /s).Table 2 shows the result for several resonant cross sections, including The table tells us that all the predicted cross section signals in this model are still at least one magnitude smaller than the current upper bounds in the LHC experiment [22][23][24].
Especially, for the resonant diHiggs production, recently the ATLAS group searching for diHiggs to 4b signals reported an excess for the resonant diHiggs production at around 280 GeV with a global significance of 2.3σ [22].In any case, it is expected that with the increasing of luminosity and improvement more on the detectability for the diHiggs signal as well as other channels as listed in Table 2, the present LOSS model would be probed through some nontrivial resonant signals at around 280 GeV dilaton mass.6 Summary and Future Prospect In conclusion, we constructed an effective model of a hidden scalar QCD sector at the leading-order scale-symmetry (LOSS) limit, which properly includes an crucial gluoncondensate effect as the nonperturbative-scale anomaly term, and is in accordance with the indications from the lattice simulation and other nonperturbative analysis of the scalar QCD.Remarkably, it turns out that the LOSS model predicts a massive enough dilaton (with the mass around 300 GeV) as well as a number of detectable cross sections.This effective model is significantly different than the conventional realization of scale-invariant hidden-scalar QCD without the scale anomaly effect.The difference is manifested especially in the Higgs-trilinear coupling-term, the enhancement of LHC diHiggs production cross section (at most about 6.8 time larger than the SM prediction), as well as the resonant cross sections.In this sense, our model can potentially provide a competitive explanation for many exotic signals beyond the SM, which can be directly tested, or presumably be excluded by the future collider experiments, such as the high luminosity-LHC.Finally, we want to emphasize that the existence of the rest of those composite scalars, B and A 0 as seen in Eq.(2.4), might potentially provide us with some new multi-component DM candidates.Moreover, the large enough portal coupling λ Hχ ≃0.138 (see Eq.(4.4)) also provides a possibility for a strongly first-order EW phase transition in the early universe.We leave a thorough discussion of such multicomponent DM scenario as well as the the consequences for the EW phase transition for future work.

Figure 1 .
Figure 1.The physical dilaton mass as a function of sin 2 θ

Figure 2 .
Figure 2. The total decay width Γ χ (in GeV) of the dilaton as a function of the mixing angle.

Figure 3 .
Figure 3. Branching ratios of the dominant WW, hh, ZZ decay modes.

Figure 4 .
Figure 4.The predicted diHiggs production cross section σ 13TeVggF (pp → hh) as a function of sin 2 θ.The red horizontal lines are experimental upper bounds coming from non-resonant production, which have been scaled from the currently most stringent limit placed by the ATLAS group[22] to a various of luminosities to be accumulated in the near future, as described in the plot label and in the text.

Table 1 .
Numerical results for total decay width and branching ratios of the dilaton.