Higgs-dilaton(radion) system confronting the LHC Higgs data

We consider the Higgs-dilaton(radion) system using the trace of energy-momentum tensor ($T_{~\mu}^\mu$) with the full Standard Model (SM) gauge symmetry $G_{\rm SM} \equiv SU(3)_c \times SU(2)_L \times U(1)_Y$, and find out that the resulting phenomenology for the Higgs-dilaton(radion) system is distinctly different from the earlier studies based on the $T_{~\mu}^\mu$ with the unbroken subgroup $H_{\rm SM} \equiv SU(3)_c \times U(1)_{\rm em}$ of $G_{\rm SM}$. After electroweak symmetry breaking (EWSB), the SM Higgs boson and dilaton(radion) will mix with each other, and there appear two Higgs-like scalar bosons and the Higgs-dilaton mixing changes the scalar phenomenology in interesting ways. The signal strengths for the $gg$-initiated channels could be modified significantly compared with the SM predictions due to the QCD scale anomaly and the Higgs-dilaton(radion) mixing, whereas anomaly contributions are almost negligible for other channels. We also discuss the self-couplings and the signal strengths of the $126$ GeV scalar boson in various channels and possible constraints from the extra light/heavy scalar boson. The Higgs-dilaton(radion) system considered in this work has a number of distinctive features that could be tested by the upcoming LHC running and at the ILC.


I. INTRODUCTION
Scale symmetry has been an interesting subject both in formal quantum field theory and in particle physics phenomenology [1]. The most important example is the scale invariance (Weyl invariance or conformal invariance) in string theory, which is nothing but 2-dimensional quantum field theories for the string world sheet in target space-time of spacetime dimensionality d. The condition of vanishing quantum scale anomaly constrains possible perturbative string theories to be defined only in d = 26 spacetime for bosonic string theory and d = 10 spacetime for superstring theory. However, implementing scale symmetry to particle physics has not been so successful compared with string theory for various reasons.
First of all, scale symmetry is always broken by quantum radiative corrections through renormalization effects. Even if we start from a theory with classical scale symmetry (namely, no dimensional parameters in L classical ), the corresponding quantum theory always involves hidden scales, the cutoff scale (Λ) in cutoff regularization or Pauli-Villa regularization, and the renormalization scale µ in dimensional regularization. In either case, scale symmetry is explicitly broken by quantum effects, and scale symmetry is anomalous. If the couplings do not run because of vanishing β function, we would have truly scale invariant (or conformal symmetric) theory, and N = 4 super Yang-Mills theory is believed to be such an example.
Secondly scale symmetry may be spontaneously broken by some nonzero values of dimensionful order parameters due to some nonperturbative dynamics, very often involving some strong interaction. For example, we can consider massless QCD with classical scale invariance. In this case there could be nonzero gluon condensate G a µν G aµν ∼ Λ 4 G 2 and chiral condensate qq ∼ Λ 3 qq , where new scales Λ G 2 and Λq q are generated dynamically and they would be roughly order of the confinement scale Λ QCD . Since scale symmetry is spontaneously broken, there would appear massless Nambu-Goldstone (NG) boson, which is often called dilaton related with dilatation symmetry. If scale symmetry were not anomalously broken by quantum effects, dilaton could be exactly massless. However scale symmetry is usually broken explicitly by renormalization effects, dilaton would acquire nonzero mass which is related with the size of quantum anomaly, in a similar way to the pion as a pseudo Nambu-Goldstone boson in ordinary QCD. If the dilaton mass is too large compared with the spontaneous scale symmetry breaking scale, it is not meaningful to talk about dilaton as a pseudo NG boson. On the other hand, if dilaton is light enough, then we can use the nonlinear realization of scale symmetry with built-in quantum scale anomaly. Whether dilaton can be light enough or not is a very difficult question to address. The answer would depend on the underlying theories with classical scale symmetry, without which we cannot say for sure about pseudo NG boson nature of dilaton.
Let us note that there have been longtime questions about generating the masses of (fundamental) particle only from quantum dynamics. A good example is getting proton mass from massless QCD. Since the contributions of current quark masses to proton mass are negligible, we can say that proton mass is mostly coming from quantum dynamics between (almost massless) quarks and gluons. Another well known example is radiative symmetry breakingá la Coleman-Weinberg mechanism [2]. In fact, a number of recent papers address generating particle masses along this direction. There are two different ways to getting mass scales from scale invariant classical theories: one from new strong dynamics in a hidden sector [3][4][5] and the other by CW mechanism [6][7][8][9][10][11][12][13]. If there are no mass parameters in classical Lagrangian, the theory would have classical scale symmetry. And all the mass scales would have been generated by quantum effects, either nonperturbatively or perturbatively.
Before the Higgs boson was discovered, dilaton (denoted as φ in this paper) has been considered an alternative to the Higgs boson [14][15][16][17][18] from time to time, since dilation couplings to the SM fields is similar to the SM Higgs field at classical level, except that the overall coupling scale is given by the dilaton decay constant f φ instead of the Higgs vacuum expectation value (VEV) v. At quantum level, dilaton has couplings to the gauge kinetic functions due to the quantum scale anomaly [19], a distinct property of dilaton which is not shared by the SM Higgs boson. The radion [20] in Randall-Sundrum (RS) model [21,22] has the similar properties as the dilaton, in that it couples to the trace of energy-momentum tensor too just like the dilaton [20,[23][24][25] [67].
The other modes are consistent with the SM predictions, but within a large uncertainty.
The effective interaction Lagrangian for a dilaton φ to the SM field can be derived by using nonlinear realization: χ = e φ f φ [1]. With the trace of the energy momentum tensor, which is the divergence of dilatation current, the interaction terms which are linear in φ cast We argue that this form of dilaton interaction to the SM fields may not be proper, since only the unbroken subgroup of the SM gauge symmetry has been imposed on T µ µ . If we imposed the full SM gauge symmetry on T µ µ , the more proper form of the dilaton couplings to the SM should be described by Eq. (3) below.
The SM Lagrangian is written as where G, f and H denote the SM gauge fields, fermions and Higgs field in a schematic way. In this form, scale symmetry is explicitly broken by a single term, µ 2 H H † H in the SM. Also quantum mechanical effects break scale symmetry anomalously. In the end, the trace of energy-momentum tensor of the SM, which measures the amount of scale symmetry breaking, is given by This form of T µ µ respects the full SM gauge symmetry It is the purpose of this paper to analyze the Higgs-dilaton system using the dilaton couplings to the SM fields which respects the full SM gauge interactions, and compare the results with the most recent LHC data on the Higgs boson. In Sec. II, we derive the effective Lagrangian for dilaton coupled to the SM fields, and derive the interactions between them.
Then we perform phenomenological analysis in Sec. III, comparing theoretical predictions

A. Model Lagrangian
Let us assume that there is a scale invariant system where scale symmetry is spontaneously broken at some high energy scale f φ , with the resulting Nambu-Goldstone boson which is called dilaton φ. In terms of χ(x) ≡ e φ(x)/f φ , the Lagrangian for the SM plus a dilaton would be written as where S(x) is the conformal compensator, which is put to 1 at the end of calculation.
Keeping the linear term in φ, we recover the Eq. (1) with T µ µ being given by Eq. (3). Note that the dilaton coupling to the SM fields in this work is different from other works in the literature. In most works, the dilaton is assumed to couple to the SM fields in the broken phase with unbroken local SU (3) c × U (1) em symmetry. However if scale symmetry breaking occurs at high energy scale, it would be more reasonable to assume that the dilaton couple to the SM Lagrangian as given in the above form with the full SM gauge symmetry The ground state of the potential for the classical Lagrangian is given by either H = 0, χ = 1 for the unbroken EW phase, and H = (0, v/ √ 2) T , φ =φ for EWSB into U (1) em , ignoring the contributions from the vacuum expectation values of the scale anomaly, such as G a µν G aµν etc. The vanishing tadpole conditions for the correct vacua are given by We have used the µ 2 = −µ 2 H > 0, for convenience. From these two conditions, one can derive which solves forφ for given µ 2 , λ, f φ and m 2 φ . Note that the Higgs VEV v is fixed by the weak gauge boson masses m W and m Z to be 246 GeV.
We will consider the EWSB vacuum, and calculate the (mass) 2 matrix for the field fluctuation around the VEV: H = (0, (v + h(x))/ √ 2) T andφ + φ. Note that rescaling of the quantum fluctuation φ aroundφ is necessary, i.e. φ eφ f φ → φ. After rescaling the mass matrix should be where we definem One can diagonalize this matrix by introducing two mass eigenstates H 1 and H 2 and the mixing angle α between the two states, with the following transformation: Here we use the basis  Now the interaction Lagrangian between dilaton and the SM fields can be derived in terms of H 1 and H 2 .

B. Interaction Lagrangian for dilaton(radion) and the SM Fields
In this subsection, we derive the interaction Lagrangian between the dilaton(radion) and the SM fields both in the interaction and in the mass eigenstate basis.
Let us first discuss the interactions of the dilaton(radion) with the SM fermions and the SM Higgs boson with the full G SM : with s α ≡ sin α and c α = cos α. The first equality is in the interaction basis, whereas the second one is in the mass basis. Note that there is no direct coupling of the dilaton(radion) Note that there is no proper limit where the earlier result (14)  It should be the full SM gauge symmetry G SM = SU (3) C × SU (2) L × U (1) Y rather than its unbroken subgroup H SM = SU (3) C × U (1) em that has been widely used in earlier literature, when we consider new physics at EW scale and the new physics scale is not known [69].
The same argument applies to other interactions of the dilaton(radion) with the SM gauge bosons or the SM Higgs boson. We list them below for completeness: The β functions for the SM gauge groups are listed in the Appendix A for convenience. The SM Higgs field h will interact with gluons or photons in just as in the standard model case, and we have to add these to the above interaction Lagrangian.
The offshoot of our approach is that the dilaton φ mixes with the SM Higgs boson h, and couples to the SM fields through quantum scale anomaly in addition to the classical scale symmetry breaking term, i.e. µ 2 H H † H. Since the dilaton φ and the SM Higgs boson h mix with each other to make two scalar bosons H 1 and H 2 , their couplings to the SM fermions will be reduced by a universal amount due to the mixing effects [55], while their couplings to the SM gauge bosons, especially to gluons, could be further modified by quantum scale anomalies. This observation has a very tantalizing implication for Higgs signals at the LHC, which will be elaborated in the following. Since there are two scalar bosons, we takes one of them to be 126 GeV resonance that was observed recently at the LHC. Since the dilaton(radion) φ coupling to the trace anomaly of the SM fields (Eq. (3)) are distinctly different from the interactions between the SM fields and other singlet scalar bosons appearing in various extensions of the SM [55], phenomenological consequences of the Higgs-dilaton mixing are analyzed separately in this paper.

A. Analysis Strategy
Compared with the SM Higgs boson, the Higgs-dilaton system considered in this paper has only two more parameters (m φ and f φ ), which makes phenomenological analysis feasible. For a given (m φ , f φ ), we calculate the signal strength of each scalar boson into a specific final state:   , which was larger than the SM value in the previous analysis [56]. The enhancement in diphoton mode is still there in the ATLAS report, and also in the ZZ * mode with less significance.
Considering the current situation of conflicting data on H → γγ, we consider two separate cases reported by CMS and ATLAS collaborations. For each case, we perform the χ 2 analysis and select the parameter sets within the 3σ range around the each χ 2 minimum [57]. Let us start with the case that heavier H 2 is the observed 126 GeV boson. As mentioned in the previous section, we put the constraints of 3σ range around the minimum χ 2 . In addition to that, we also consider the experimental constraint for the light scalar particle that is determined by the LEP experiment [58].
Considering these three constraints, the allowed parameter region is shown in Fig.1 in the (m H 1 , sin α) plane. The colored columns denote the signal strengths. As noted in Sec. III A, the dilaton production from gluon fusion (gg → φ) can be enhanced due to the QCD scale anomaly and thus can compete with the SM Higgs production from gluon fusion (gg → h).
Therefore the Higgs signal strengths depend mainly on the production channels rather than the decay channels, and we present the γγ channel only in Fig. 1. One can see that the ggF initiated process can be modified significantly compared to the SM value. On the other hand, the VBF initiated one is suppressed by the mixing angle only, so that its signal strength is always smaller that one. Also note that the allowed region for the mixing angle α is highly constrained around α ∼ − π 2 . This means that the observed 126 GeV boson is largely SM-like and the extra light mode is dilaton-like, namely H 2 h and H 1 φ. Even though the mixing angle is close to −π/2 and H 2 h, rather large modification is possible from the mixing with the dilation through the tuning of the input parameters m φ and f φ .
There should be an extra light scalar mode H 1 whose mass is constrained to be in the range m H 1 ∼ [58, 104] GeV, which is a prediction of our model.
Since the model has only two more input parameters (m φ , f φ ), some observables are highly correlated, which make the generic signals of the model. In Fig.2 to prefer the larger value of m H 1 [70].
Triple and quartic couplings for the H 2 (m H 2 = 126 GeV) are completely determined within this model, making distinct discriminators for this model. In the allowed parameter region, the predictions for triple and quartic couplings of the SM-like Higgs boson H 2 are shown in Fig.3. One can see that triple and quartic couplings are suppressed compared with the SM values, depending on the H 1 mass. Especially for the triple coupling it gives relative minus sign compared to the SM value, which would result in the constructive interference between the box diagram and the triangle diagram with the s−channel H 2 propagator, and thus increase the H 2 pair production in gg → H 2 H 2 [59]. In addition, we observe a strong correlation between triple and quartic couplings, which is presented in Fig.4. Along with the H 1 mass, the triple and quartic couplings are highly inter-related. This will be the strong distinctive signal for testing the model, which could be probed at the upcoming LHC run and at the ILC.  In this case, the allowed range for the mixing angle α is severely restricted around the SM values, α 0 (see Fig.5).
For both cases the signal strengths are very close to 1, the SM values. This means that the experimental data strongly favor the SM case and the dilaton should be heavy enough to decouple from the theory. Compared to the SM, only the surviving region for the heavier scalar mass m H 2 is relatively relaxed compared with the constraints on the SM-like Higgs boson, which can be expected because of the mixing between the SM Higgs boson h and the dilation φ depending on the (m φ , f φ ) parameter values. As a result, other observables as triple and quartic couplings are also strongly restricted around the SM values.
As a result, unlike the Case I, it is not sufficient just to look into the observed H 1 (m H 1 = 126 GeV) sector to discriminate the model from the SM, since the model is pointing to the almost exact SM values for it. The heavier scalar boson mass is constrained to be larger than ∼ 367 GeV from the Higgs signal strengths of the observed 126 GeV boson and the heavier Higgs searches (see Fig. 5). This is a distinctive feature of our model compared with the SM. So the more detailed studies on the possible extra heavy scalar boson are necessary in the future 14 TeV LHC and tentative International Linear Collider (ILC) to test this model more completely.

IV. CONCLUSIONS
In this letter, we considered the SM coupled with some spontaneously broken scale symmetric sector with light dilaton (pseudo Nambu-Goldstone boson) (or the radion in the RS scenario) using the SU ( On the other hand, if we identify the observed scalar particle with mass 126 GeV as light mode H 1 , with the constraints upon the extra heavy SM-like scalar mode searched by CMS and ATLAS, the remaining parameter sets become severely confined around the SM expectations. This means that it is not enough to discriminate the model from the SM just by looking into the 126 GeV sector. In this case, the more detailed study on the extra heavy mode will be necessary to test the model completely.