AdS/QCD approach to the scale-invariant extension of the standard model with a strongly interacting hidden sector

In this paper, we revisit a scale-invariant extension of the standard model (SM) with a strongly interacting hidden sector within AdS/QCD approach. Using the AdS/QCD, we reduce the number of input parameters to three, {\it i.e.} hidden pion decay constant, hidden pion mass and $\tan\beta$ that is defined as the ratio of the vacuum expectation values (VEV) of the singlet scalar field and the SM Higgs boson. As a result, our model has sharp predictability. We perform the phenomenological analysis of the hidden pions which is one of the dark matter (DM) candidates in this model. With various theoretical and experimental constraints we search for the allowed parameter space and find that both resonance and non-resonance solutions are possible. Some typical correlations among various observables such as thermal relic density of hidden pions, Higgs boson signal strengths and DM-nucleon cross section are investigated. We provide some benchmark points for experimental tests.


I. INTRODUCTION
Although the SM-like Higgs boson has been discovered at the Large Hadron Collider (LHC) [1,2], there are still a number of questions that call for physics beyond the SM (BSM): (i) the origin of the mass of Higgs particle or the origin of weak scale, (ii) the nature of non-baryonic dark matter (DM), (iii) the origin of neutrino masses and mixing, (iv) matter-antimatter asymmetry of the universe, to name a few.
The first question is often phrased as hierarchy problem, that addresses why the electroweak (EW) scale v H = 246 GeV is much smaller than the Planck scale. One of the nice ways to understand this is through quantum dimensional transmutation, which explains why the proton mass is much suppressed compared with the Planck mass in Quantum Chromodynamics (QCD) [3,4]. Technicolor (TC) provides an answer in this way, but the naive version of it is strongly disfavored by the electroweak precision test (EWPT) [5].
Since the observation of W. Bardeen [6], softly broken scale invariance has been considered as a possible solution for the hierarchy problem. If the model is scale invariant at classical level, no dimensionful parameters are allowed and the scale symmetry is broken only logarithmically through scale-anomaly. Many authors have studies this type of models where the EW symmetry is dynamically broken via dimensional transmutation in the hidden sector with new confining strong interactions [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], or Coleman-Weinberg mechanism .
Some of the present authors have proposed a scale-invariant extensions of the SM with a strongly interacting hidden sector, namely hidden QCD models [7][8][9][10][11]. At the classical level, the scale invariance is imposed so that all dimensionful parameters are forbidden in the classical Lagrangian. The Higgs mass term arises at quantum level through the dimensional transmutation driven by asymptotically free gauge theories in the hidden sector. Hidden sector couples to the Higgs through singlet scalar field only and there are stable or long-lived particles (lightest hidden mesons and hidden baryons) that can make good DM candidates.
In those works, hidden QCD sector was studied in the chiral effective Lagrangian approach and non-perturbative parameters were estimated by naive dimensional analysis. Then the same model was analyzed in the Nambu-Jona-Lasinio (NJL) approach in Ref. [14,18].
In this paper, we consider the same model using another approximation method, the AdS/QCD [70,71], in order to analyze non-perturbative strong dynamics in the hidden QCD models. First we reformulate the hidden QCD sector in terms of the linear sigma model, in which the sigma and pi mesons are effective degrees of freedom. We consider a linear sigma model coupled with a scale-invariant Higgs-singlet sector and analyzed the masses and mixing between the SM Higgs boson, a singlet scalar messenger and the sigma meson. In the AdS/QCD we successfully reduce the number of free parameters by matching the mass spectra of the lightest scalar, vector and axial vector mesons.
Next, we apply this model to dark matter phenomenology. In our model, since the hidden quarks do not couple to any U(1) gauge fields, the hidden pions cannot decay through the U(1) anomaly and are found to be stable lightest particles coupling weakly with the SM fields. Hence the hidden pions become candidates of the weakly interacting massive particle (WIMP) DM. With the free parameters reduced by the AdS/QCD, we identify the parameter space that satisfies the recent observations. Then we study the distinctive features of the allowed parameter region and also some typical correlations among various observables. We address on the possible signatures of the model that can be further scrutinized in the future experiments such as LHC Run-II, ILC and so forth. This paper is organized as follows. In Sec. II, we revisit the original hidden QCD models [7,11] by reformulating the hidden QCD sector with the linear sigma model. Then in Sec. III, we apply the idea of the AdS/QCD to the linear sigma model described in Sec. II. In Sec. IV, numerical results on the Higgs and the dark matter phenomenologies are presented. Then Sec. V is devoted to summary and discussions.

II. THE MODEL
Scale-invariant extension of the SM with a strongly interacting hidden sector contains the SM fields plus a singlet scalar S and a scale-invariant hidden QCD sector [8][9][10][11]. The corresponding Lagrangian is given by Hereafter we consider the case in which N h,c = 3 and N h,f = 2, for which we can use the known results from the hadronic system with π, ρ and σ mesons. Then λ Q = diag(λ Qu , λ Qd ) and for simplicity we assume the hidden quarks have isospin symmetry λ Qu ∼ λ Qd . In such a case the low-energy effective theory of the hidden QCD is described by the pi meson triplets and the sigma meson. It would be written in the form of a linear sigma model where Σ and π represents sigma and pi meson fields.
We parameterize the VEVs and fluctuations of scalars as To minimize the potential energy with where tan β ≡ v S /v H and F π ≡ v σ . Since a off-diagonal part M 2 hs satisfies M 2 hs = −M 2 hh / tan β, the Higgs-singlet mixing can be large when tan β is small. λ σ and µ σ are traded with pion mass M π and sigma meson mass M σσ by The couplings λ H , λ S are given by where v H = 246 GeV. Since one of the physical scalar should be the Higgs boson with mass where we have parameterized M σσ = ξ σ F π .
The mixing matrix is defined as At this stage we have four free parameters: v S , v σ ≡ F π , M π and M σσ (or ξ σ ). To reduce the number of free parameters, in particular, to relate the M σσ with F π , we use a holographic treatment of the hidden QCD.

III. ADS/QCD ANALYSIS
In the AdS/QCD [70,71], the hidden QCD sector is described by SU (2) L ⊗SU (2) R gauge theory on AdS 5 space with metric where L 0 ≤ z ≤ L 1 and L is the curvature radius of AdS 5 . L 1 breaks the conformal symmetry in the infrared (IR) regime, while one can take L 0 to be arbitrary small, L 0 → 0.
The non-perturbative breaking of chiral symmetry is regarded as the spontaneous breaking of SU (2) L ⊗ SU (2) R symmetry by the VEV of the bulk scalar Φ which is a bi-doublet Then the 5D bulk Lagrangian is given by 5), and the bulk mass parameter M 2 Φ = −3/L 2 is chosen so as to relate the bulk scalar field Φ with the dimension-three operatorqq. The profile of the VEV is obtained by solving the zero-mode equation of motion. We have where c 1 and c 2 can be written in terms of the value of v at boundaries from boundary conditionsM Here nonzero c 2 corresponds to the spontaneous breaking of the chiral symmetry in the IR, while the boundary condition at z = L 0 corresponds to the explicit breaking of chiral symmetry. The boundary condition at z = L 1 is induced by the scalar potential localized on z = L 1 boundary. The boundary interaction is After the symmetry breaking SU (2) L × SU (2) R → SU (2) V , vector-and axial-vector gauge where V (n) , A (n) correspond to hadronic vector and axial-vector currents, respectively.
The bulk scalar Φ is decomposed into Φ = (v + Φ S )e iΦ P /v and gauge fixing conditions are and Φ P,S (x, z) have the following KK expansions: where S (n) and P (n) are scalar and pseudo-scalar hadronic states, respectively. In particular P (0) corresponds to the pion.
In the QCD the two point correlators for the scalars and pseudoscalars are defined as where the two-point correlator can be obtained from the generating function S according to whereŝ andp s are the scalar and pseudoscalar external sources coupled to QCD: According to the AdS/CFT correspondence, the generating function S is obtained by integrating bulk fields restricted to a given Ultraviolet(UV)-boundary value which play the role of the external sources coupled to QCD. For the 5D scalar field we have (α is a constant which will be determined in the matching of correlation in UV as α = √ 3 [71]) or explicitly, Since the quark masses are given by M q = λ Q v S , the singlet scalar fluctuation s can be related to the scalar source term asŝ This correspondence can be used to obtain the couplings of s to the meson states.
This AdS/QCD model has five relevant free parameters: M q , L 1 , M 5 , ξ and λ. M q is traded with the pion mass M π by the Gell-Mann-Oaks-Renner relation M 5 is related with the beta-function of the QCD and we fix where we consider the case N c,h = 3. L 1 is related with the mass of the first KK state of V µ which corresponds to the rho meson mass by M V (1) =ρ 2.4/L 1 . The value of ξ can be fixed by adjusting the mass of the first KK of A µ and the first KK vector meson mass with m a 1 (1260) = 1230 ± 40 MeV and m ρ(770) = 770 MeV, which yields [70] ξ 4.
The pion decay constant F π is written in terms of L 1 , ξ,Ñ c,h as when ξ 1. Here Γ(x) is the gamma function. With L 1 = 320 MeV and N h,c = 3 one has Now we fix the value of λ b . In the original paper the author estimated λ b = 10 −2 − 10 −3 and identified the lightest scalar meson as a 0 (980). In the present study, we regard sigma meson as lightest scalar resonance state, S (1) = σ. In the AdS/QCD, since the wave functions there are no direct interactions between meson states with source term. In the AdS/QCD, source-pion-pion interactions are given by where M Sn , F Sn and G nππ is the mass, decay constant and the S n (∂ µ π) 2 coupling. These terms arise due to the S n −ŝ mixing. In particular, the first term of r.h.s. of eq. (37) is induced by σ − π − π coupling through the σ-source mixing. We assume that F S 1 = F σ , M S 1 = M σσ and that the mixing is given by Together with the Gell-Mann-Oaks-Renner relation eq. (29) we obtain In the AdS/QCD, and Q Q −18F 3 π is obtained for N h,c = 3, ξ = 4. M Sn and F Sn is obtained by formulas summarized in the Appendix. We find numerically that eq. (39) is satisfied when (See fig. 1), and hence we obtain a relation (See fig. 2) When we take a scale normalized by L −1 1 = 320 MeV so that we have M V 1 = 2.4/L 1 = m ρ(770) = 770 MeV, we obtain F π = 87 MeV, M σ ∼ 450 MeV ( fig. 2). This result well agree with the experimental bound 400 MeV ≤ m f 0 (500) ≤ 550 MeV [72], if we identify f 0 (500) as the sigma meson.

IV. HIDDEN PION PHENOMENOLOGY
As a result of the previous sections, we now have three free parameters in this model: M π , F π and v S = v H tan β. For numerical analysis, we scan the three-dimensional parameter space (M π , F π , tan β). Note that the SM-like Higgs boson with mass 125 GeV is termed as H, and extra scalar particles as H 1 and H 2 with M H1 < M H2 . We have considered several theoretical and experimental constraints.
Let us first consider the theoretical constraints. From the stability of the potential, the dimensionless couplings should satisfy the relation, which are translated into the following relation with the minimization conditions Since the strongly interacting hidden sector are now treated as linear sigma model, the symmetry breaking condition in the σ-sector would constrain the value of µ 2 σ to be positive. With the help of eq. (10), the condition reads We also adopt the perturbativity bound on λ S with our definition of the Lagrangian: Experimental constraints considered in the analysis are listed in the following: • Signal strength for the SM Higgs boson [73,74], µ = 1.00 ± 0.13 [73].
• Neutrino signals through the DM capture by the Sun, mostly from Super-Kamiokande for upward muon flux [80].
We apply 2σ bounds with these experimental constraints except for the relic density, for which we use the measured value as an upper bound. This is because there could be additional contributions from hidden baryons to DM thermal relic density, which we do not include in this paper. We vary the F π up to 2 TeV and the ranges for M π and tan β are fixed with eq. (44) and eq. (45). We use micrOMEGAs [87] for evaluating DM-related observables.
The result of scanning is depicted in fig. 3. Here we see that there is definite lower bound for F π , around 100 GeV. Also tan β is bounded from below, tan β 0.7, mainly because of the perturbativity of λ S since it can be written in the form So if tan β is too small, λ S will have very large value, above the perturbativity bound. The As a distinctive observable, we show the deviation of the triple Higgs coupling from the SM prediction in fig. 5. The triple Higgs coupling in the model can reach ∼ 85% of the SM prediction for relatively small M π . If we take larger values for M π and F π , it approaches to the SM one and cannot be a distinctive observable. Especially when M π is larger than 1 TeV, the triple Higgs coupling is very close to the SM prediction and the deviation cannot be detected.
Let us consider another observables. To be more specific, we separate the cases as the SM-like Higgs resonance and the other cases. fig. 6 shows the correlations between the Higgs signal strengthμ and DM-nucleon cross section in Higgs resonance case. Deviation of µ from 1 is generated by the mixing angle |V h0 | 2 in eq. (13). In this case, the SM-like Higgs boson can decay to a pair of DM's and these decay modes contribute to the Higgs invisible Color contours represent the hidden pion decay constant F p i. cross section. We can understand this by the fact that the deviation of the signal strength is determined entirely by non-zero mixing angles among the SM-like Higgs and other extra scalar particles. The more they are mixed, the more wide the channel between the visible and hidden sector is open. We show in the right plot the same correlation with relic density contours. The variation of the relic density is caused by the small variation of M π and F π as long as M π is close to ∼ M H /2. In both plots we apply 2σ bound for the signal strengtĥ µ such asμ ≥ 0.74.
The same correlation is shown in fig. 7 for the other cases than the SM-like Higgs resonance. As mentioned before, the cases include light and heavy scalar resonances and non-resonance solutions. In this plot, the color contour represents the mass of the DM.
Unlike the Higgs resonance case, relatively large values of the signal strength are favored, with generically larger values of DM-nucleon cross section. Note that there is an upper limit forμ. This bound is originated from the h − σ mixing, that should be different from zero for avoiding the overclosure of the universe by the DM.  Table I. We classify the points as (A)

V. SUMMARY AND CONCLUSION
In this paper, we have analyzed the scale-invariant extension of the SM with vectorlike confining gauge theory in the hidden sector by using the AdS/QCD proposed in Refs. [70,71]. The model contains the singlet scalar field that connects the confining hidden sector and the scale-invariant SM sector. Hidden sector fermions develop nonzero chiral condensates and generate the linear term in the potential of the singlet scalar field S. As a result, the singlet scalar field S develops a nonzero VEV and it provides the tachyonic mass term for the SM Higgs field. Therefore the origin of the EWSB in the SM sector lies in the new strong dynamics in the hidden sector.
We have used the AdS/QCD approach to describe non-perturbative dynamics of the hidden QCD sector. By the AdS/QCD, strongly interacting SU (3)  Higgs signal strengths and non-observation of another scalar particles, etc. By scanning the three-dimensional parameter space (M π , F π , tan β), we found that the non-resonance solutions are also possible in addition to the resonance solutions. We also considered various correlations among the experimental observables. For example, there is the correlation between the Higgs signal strengthμ and DM-nucleon cross section, and also betweenμ and the triple SM-like Higgs coupling. Especially for the latter, we found that their values and correlations behave differently depending on whether hidden pions have the SM-like Higgs resonance or not. Though the Higgs signal strengthμ has been measured quite precisely and seems to be consistent with the SM prediction, there is still room for the physics beyond the SM as discussed in this paper. If the Higgs signal strengthμ is measured more precisely, according to the sharp correlations we found, we can give peculiar predictions on the DM properties and others such as triple Higgs coupling etc. This could be seen in the benchmark points we presented at the end of the analysis.  The masses of scalar resonances are determined by finding the poles, b(p E ) = 0 or and corresponding residues gives the scalar decay constants . (A6)