Production and decays of a light $\phi^0$ in the LRTH model under the LHC Higgs data

In this paper we study the production and decays of a light pseudoscalar boson $\phi^0$ with $m_{\phi^0} \leq m_h/2$ appeared in the left-right twin Higgs (LRTH) model, and explore its phenomenological consequences when the latest LHC Higgs data are taken into account. We found that (a) the decay rate $Br(h\to \phi^0\phi^0)$ can be as large as $80\%$ and can suppress significantly the visible $\gamma\gamma$ signal rate, but the latest LHC Higgs data put a strong constraint on it: $Br(h \to \phi^0\phi^0) \leq 30\%$ at $3\sigma$ level; (b) the $p$-value of the LRTH model is around $0.6$, smaller than that of the SM in most of the parameter space and approaches the SM value $0.8$ for a sufficiently large $f$ parameter; (c) the neutral pseudoscalar $\phi^0$ dominantly decay into $b\bar{b}$ and the decay rate $Br(\phi^0\to b\bar{b})$ can be larger than $80\%$ for $m_{\phi^0}\leq 60$ GeV, and the second main decay mode is $\phi^0\to \tau^{+}\tau^{-}$ with a branching ratio about $14\%$; and (d) at the future $e^-e^+$ collider with $\sqrt{s}=250$ GeV, the processes $e^{+}e^{-}\to Zh\to Z(\phi^0\phi^0)\to Z(4b,2b2\tau)$ are promising for discovering such a light pseudoscalar $\phi^0$.


I. INTRODUCTION
The discovery of a neutral Higgs boson with a mass around 125 GeV at CERN's Large Hadron Collider (LHC) has been confirmed by the ATLAS and CMS collaborations [1][2][3][4][5][6], which heralds the beginning of a new era of Higgs physics. So far the observed signal strengths, albeit with large experimental uncertainties, consistent with the Standard Model (SM) predictions [5,6]. However, the SM suffers from the so-called gauge hierarchy problem and cannot provide a dark matter candidate.
During past three decades, many new physics (NP) models beyond the SM have been constructed by extending the Higgs sector in the SM, such as the Supersymmetric (SUSY) models [7], large extra-dimensions [8], two-Higgs doublet models (2HDM) [9], and little Higgs models [10][11][12] etc. Very recently, the twin Higgs models have been proposed [13][14][15][16] as a solution to the little hierarchy problem. Here we focus on the left-right twin Higgs (LRTH) model which is implemented with the discrete symmetry being identified with left-right symmetry [17,18]. In the LRTH model, several physical Higgs bosons are still left after the spontaneous symmetry breaking. Another additional discrete symmetry is introduced under an odd SU(2) L doubletĥ while the other fields are even. The lightest particle in the neutral componentsĥ 0 2 is stable and can be a candidate for weakly interacting massive particle (WIMP) dark matter. Besidesĥ 0 2 , the LRTH model predicts the SM-like Higgs boson h and other three scalars: φ 0 and φ ± . The neutral φ 0 is a pseudoscalar and thus there are no φ 0 W + W − and φ 0 ZZ couplings at the tree level, which makes the φ 0 rather special. The particle spectrum and collider signatures of the LRTH model have been widely studied, for example, in Refs. [19][20][21][22][23][24][25][26][27].
In a recent paper [28], we studied the properties of the SM-like Higgs boson h, calculated the new physics contributions to the decays h → (γγ, Zγ, τ τ, W W * , ZZ * , τ τ ) in the LRTH model, performed a globe fit to the current LHC Higgs data, and found that all the signal rates are suppressed when NP contributions are taken into account, while the LRTH prediction for R γγ agrees well with the CMS measurement R γγ = 0.77 ± 0.27 at 1σ level. In this paper, we will study the production and decays of the light pseudo-scalar φ 0 and to draw the possible constraints from currently available LHC Higgs data. If this neutral φ 0 were lighter than half of the SM-like Higgs boson h, i.e. m φ 0 < m h /2, the new decay channel h → φ 0 φ 0 will be opened with a sizable branching ratio. Because the 125 GeV SM Higgs decay width is small ( the measured value is about 4 MeV), such an exotic decay mode can suppress greatly the visible signals of h and would have important phenomenological consequences [29][30][31][32]. We know that the current bound on the branching ratios to exotic states is still weak: a branching fraction as large as ∼ 60% is allowed at the 2σ C.L. [5,6]. If SM couplings are assumed, the universal Higgs fits constrain the invisible branching fraction to be less than 25% at 95% C.L. [33][34][35], which still leaves appreciable scope for such an exotic decay mode. Thus, we will investigate the constrains of the latest LHC Higgs data on the properties of such a light pseudoscalar φ 0 in the LRTH model. We will also study the possibility of detecting such a light boson φ 0 at high energy colliders. This paper is organized as follows. In the next section, we briefly review the LRTH model and study the possible decay modes for a light pseudoscalar boson φ 0 . In Sec. III, we investigate the decay branching ratios of h → φ 0 φ 0 and perform a fit using the latest LHC Higgs data. We study the possibility of detecting such a light pseudoscalar at the LHC experiments in section IV. Finally, we present our conclusion in Sec.V.

II. THE LEFT-RIGHT TWIN HIGGS MODEL
A.
Outline of the LRTH model This model is based on the global U(4)×U(4) symmetry with a locally gauged subgroup [13][14][15][16][17]. The twin symmetry is identified as the left-right symmetry which interchanges L and R, implying that the gauge couplings of SU(2) L and SU(2) R are identical (g 2L = g 2R = g 2 ). Two Higgs fields, H andĤ, are introduced and each transforms as (4,1) and (1,4), respectively. They are written as where H L,R andĤ L,R are two component objects which are charged under the SU (2) The global U(4) 1 (U(4) 2 ) symmetry is spontaneously broken down to its subgroup U(3) 1 (U(3) 2 ) with non-zero vacuum expectation values(VEV): Each spontaneously symmetry breaking yields seven Nambu-Goldstone bosons, which can be parameterized as where π are the corresponding Goldstone fields. N is a neutral real pseudoscalar, C and C * are a pair of charged complex scalar fields. (h 1 , h 2 ) is the SM SU(2) L Higgs doublet. Accordingly,Ĥ is parametrized in the same way by its own Goldstone boson matrixπ, which containsN,Ĉ andĥ = (ĥ + 1 ,ĥ 0 2 ). The original gauge symmetry SU(2) L × SU(2) R × U(1) B−L is broken down to the SM U(1) Y , six out of the 14 Goldstone bosons are respectively eaten by the SM gauge bosons W ± and Z, and additional gauge bosons W ± H , and Z H with masses of TeV order. Then we are left with the SM-like physical Higgs boson h, one neutral pseudoscalar φ 0 , a pair of charged scalar φ ± , and an odd SU(2) L doubletĥ = (ĥ + 1 ,ĥ 0 2 ) which only couples to the gauge boson sector. The lightest particle inĥ is stable and thus can be a candidate for WIMP dark matter, which have been studied for example in Refs. [21,22].
The covariant kinetic terms of Higgs fields can be written as [19] where the covariant derivative is D µ = ∂ µ − ig 2 W µ 2 − ig 1 n B−L W µ B−L , and where g 1 and g 2 are the gauge couplings for U(1) B−L and SU(2) L,R , n B−L = 1 is the charge of the field under U(1) B−L . In the LRTH model, a pair of vector-like quarks (U L , U R ) are introduced to cancel the one-loop quadratic divergence of Higgs mass induced by the top quark. The relevant Lagrangian can be written as [19] where The details of the LRTH model as well as the particle spectrum, Feynman rules, and some phenomenology analysis have been given for example in Ref. [19]. Here we will focus on the properties of the light pseudoscalar φ 0 .
B. The mass and decay of the light scalar φ 0 In the LRTH model, the soft left-right symmetry breaking terms, so called µ−term, can generate mass for the light φ 0 : The mass of φ 0 and new scalar self-interactions are given by [19] GeV is the electroweak scale. Once f is fixed, the scalê f can be determined from the electroweak symmetry breaking condition. In generalf is larger f about 5 times or more [19,20] and here we setf = 5f as a rough estimate.
From the expression of m 2 φ 0 in Eq. (9), one can see that the value of m 2 φ 0 depend on two parameters µ r and f . The value of µ r cannot be too large, since the fine-tuning of the SM-like Higgs boson mass m h will become severe for larger µ r [19]. Assuming 4 ≤ µ r ≤ GeV and 500 ≤ f ≤ 1500 GeV, one finds the upper constraint on m φ 0 : m φ 0 ≤ 62.5 GeV. In other words, the new decay channel h → φ 0 φ 0 can be opened in the parameter space of the LRTH model considered here. The lower limit of m φ 0 , say m φ 0 > 7 GeV, comes from the non-observation of the decay Υ → γ + X 0 [36,37]. As illustrated in Fig. 1, the upper limit m φ 0 ≤ 62.5 GeV are guaranteed when the values of (f, µ r ) are in the light-dark region in the f − µ r plane. The rare decays of Z → ff φ 0 and Z → φ 0 γ have been studied in Ref. [38]. In the LRTH model, the decays φ 0 → gg, γγ are mediated by the one loop Feynman diagrams involving the top quark and the new heavy quark T . The leading order decay widths can be written as [39] where The explicit expressions of the relevant couplings y t and y T are of the form where the mixing angles S L,R and C L,R are with . The mass of the top quark and new heavy T -quark are therefore can also be written as [19] The parameter y in Eqs. (14)(15)(16)(17) denotes the top quark Yukawa coupling, and can be determined by fitting the measured value of m t according to Eq. (17) for given values of the new physics parameters f and M.
For φ 0 → f ifi decays with f i the leptons and/or light quarks, the decay width can be written as: (1) for f i being a quark (lepton). It is easy to see that the decays of φ 0 to those light final state fermions, such as f i = (e, µ, u, d, s), are strongly suppressed due to the severe helicity suppression ( ∝ m 2 i ), and therefore can be neglected safely.
In the LRTH model, consequently, the five major decay modes of φ 0 are φ 0 → bb, cc, τ + τ − , gg and γγ. The m φ 0 -dependence of the branching ratios, assuming f = 500 GeV and M = 150 GeV, are illustrated in Fig.2a. The Fig. 2b shows the M-dependence of the branching ratio of the dominant φ 0 → bb decay for fixed f = 500 and 1500 GeV. For φ 0 → ff decays, furthermore, their decay rates in the LRTH model are strongly suppressed by a factor of v 2 /(2f 2 ) ≤ 0.12 when compared with those of the SM Higgs boson decays H → ff . From Fig. 2 one can see that: 1. The dominant decay mode of the light pseudoscalar φ 0 is φ 0 → bb. In the considered region of 500GeV ≤ f ≤ 1500GeV , the value of the branching ratio Br(φ 0 → bb) is about 81% for m φ 0 = 50 GeV, and has a rather weak dependence on the variations of the parameters f and M.
2. The partial width into cc is smaller than that into τ + τ − , this is because we use the running mass of the quarks evaluated at the scale m φ 0 to calculate the Yukawa coupling. In the allowed parameter spaces, 3. Br(φ 0 → gg) becomes large along with the increase of m φ 0 , which can reach 30% for m φ 0 = 200 GeV. This is due to the enhancement from the contribution of heavy T -quark, which is non-decoupled in the triangle loops. In our calculations, we take the SM-like Higgs mass as m h = 125.5 GeV. The SM input parameters relevant in our study are taken from [40]. The free parameters in the LRTH model relevant for this work are f , M and m φ 0 . Following Ref. [19], we here also assume that the values of the free parameters f and M are in the ranges of while 8 ≤ m φ 0 ≤ 62.5 GeV according to the analysis in previous section.
For m h ≥ 2m φ 0 , the new decay channel h → φ 0 φ 0 will open and the partial decay width is given by where g hφ 0 φ 0 is the coupling of hφ 0 φ 0 vertex. The open of this new decay mode, consequently, can suppress greatly the visible signals of the boson h at the LHC. Thus, the major decay modes of the SM-like Higgs boson h in the LRTH model become now: where W * /Z * denoting the off-shell charged or neutral electroweak gauge bosons. The branching ratio of h → φ 0 φ 0 can be written as where Γ LRTH (h) denotes the total decay width of SM-like Higgs boson h for m φ 0 > m h /2 in the LRTH model, which has been studied in Refs. [28,41]. In Fig. 3 we show the f -dependence of the decay width Γ(h → φ 0 φ 0 ) and the branching ratio Br(h → φ 0 φ 0 ) for M = 150 GeV and three typical values of m φ 0 : m φ 0 = 40 ± 20 GeV. One can see that both the decay width and decay rates for h → φ 0 φ 0 decay becomes smaller rapidly along with the increase of the parameter f . This is because the couplings of hφ 0 φ 0 is proportional to the suppression factor of (v/f ) 2 . For m φ = 40 GeV, we find 2% ≤ Br(h → φ 0 φ 0 ) ≤ 70% for 500 ≤ f ≤ 1500 GeV. For the case of f = 500 GeV, the decay width Γ(h → φ 0 φ 0 ) can be as large as 16 MeV and thus can suppress greatly the branching ratios for other decay modes of the SM-like Higgs boson h: such as the phenomenologically very interesting h → γγ decay.

B. h → γγ decay in the LRTH model
For the SM Higgs diphoton decay, the measured signal strength as reported by ATLAS [5] and CMS collaboration [6] are rather different, but these results are still consistent with the SM expectation within 2σ level due to rather large errors. If the excess (deficit) seen by ATLAS (CMS) were eventually confirmed by the near future LHC measurements, the extra NP contributions would be help to understand such excess or deficit [42][43][44][45][46]. At the LHC, the Higgs single production is dominated by the gluon-gluon fusion (ggF) process. The hadronic production cross section σ(gg → h) has a strong correlation with the decay width Γ(h → gg). Other main production processes of the Higgs boson include vector-boson fusion (VBF), associated production with a W/Z boson (VH) and associated production with a tt pair (ttH). For m h = 125.5 GeV, the production cross sections for each production channels at LHC have been given for example in Ref. [47]. In the LRTH model, the production rate of h → γγ normalized to the SM values is generally defined as In Fig. 4 we plot R γγ versus f for m φ 0 = 50 GeV and M = 0, 150 GeV, respectively. It can be seen from Fig. 4 that ratio R γγ in the LRTH model is always smaller than unit, and will approach one (the SM prediction) for a large f . On the other hand, one can see that the ratio R γγ is insensitive to the variation of the mixing parameter M. Since the ATLAS diphoton data is above the SM value by about 2σ, the predicted rate in the LRTH model is always outside the 2σ range of the ATLAS data. But the theoretical prediction for R γγ in the LRTH model is in good agreement with the current CMS data within 1σ error for f ≥ 600 GeV. The key point here is the large difference between the central values reported by ATLAS and CMS respectively. Further improvement of the R γγ measurements from both ATLAS and CMS collaboration are greatly welcome and will play the key role in constraining the new physics models beyond the SM.
In Fig. 5 we show the contours of R γγ in f -m φ 0 plane and f -Br plane for R γγ ≥ 0.5, 0.7, and 0.9, respectively. One can see that the assumption R γγ ≥0.7 will indicate f ≥ 700 GeV for m φ 0 = 60 GeV, but leads to a limit f ≥ 900 GeV for m φ 0 = 30 GeV. From Fig. 5b, it is easy to see that one can draw strong constraint on the exotic decay rate Br(h → φ 0 φ 0 ) from the measured Higgs diphoton rate. A limit of R γγ ≥ 0.7, for example, can result in a strong constraint Br(h → φ 0 φ 0 ) ≤ 26%.
When fitting the various observable, we considered the correlation coefficients given in Ref. [68] due to the independent data for different exclusive search channels by two collaborations. The global χ 2 function is defined as: where σ 2 ij = σ i ρ ij σ j ,μ i and σ are the measured Higgs signal strengths and their 1σ error, ρ ij is the correlation matrix, µ i is the corresponding theoretical predictions in terms of the LRTH parameters. The details about the statistical treatment are presented in Appendix A.
In Fig. 6a we plot χ 2 versus f for M = 150 GeV and m φ 0 = 20, 40 and 60 GeV, respectively. One can see that the value of χ 2 of the LRTH model is larger than that for SM for most of parameter space of f and approaches the SM value for a sufficiently large f . For a light pseudoscalar φ 0 , for example setting m φ 0 = 20 GeV, the Higgs data will

Channel
Signal strengthμ i LRTH predictions µ i f=800 f=1000 f=1200 ATLAS [5,[57][58][59][60][61][62]  lead to effective constraint on the value of the parameter f : f ≥ 1000 (900) GeV at the 2σ (3σ) level. In Fig. 6b we plot the p-values versus m φ 0 for M = 150 GeV and f = 800, 1000 and 1200 GeV, respectively. We note that the goodness of the fit in the SM, measured by the p−value, is about 0.80, which means that the SM has a chance of 80% to be the true interpretation of the data. One can see that the p-value become smaller for the LRTH model in large part of its parameter space, and approaches the SM value for a sufficiently large f . For m φ 0 = 40 GeV and f = 1000(1200) GeV, its p-value is about 0.63(0.75).
In Fig. 7 we plot the contours of χ 2 for Br(h → φ 0 φ 0 ) against the parameter f . One can see that the current LHC Higgs data can put strict constraint on the exotic decay h → φ 0 φ 0 : for example, Br(h → φ 0 φ 0 ) should be less than 30% at 3σ level.

IV. PHENOMENOLOGY OF A LIGHT φ 0
When the decay h → φ 0 φ 0 is open, the decays h → φ 0 φ 0 → 4b, 2b2τ or 4τ are the major promising channels to detect such a light pseudoscalar at the LHC experiments. As demonstrated in Ref. [69], the process pp → W/Zh → l + 4b + X (l denotes one lepton and X denotes anything) may provide a clean signature out of the backgrounds for a light Higgs boson. Following the suitable cuts, the signal rate depends on an overall scaling factor which determines the cross section of the process V h → V 4b at the LHC [69,70]. In the LRTH model, [19]. The DELPHI Collaboration [71] has made model-independent searches for the process e + e − → Zh → ZAA → Z + 4b with A a pseudoscalar particle. However, the experimental upper bound on C 2 4b is relaxed for this model (C 2 4b ≥ 1 for m h = 110 GeV and m A = 12 GeV), and it is the same case in the simplest little Higgs (SLH) model [72]. In Fig. 8 we plot the factor C 2 4b versus the parameter f in the LRTH model. One can see that, for f = 500 GeV and m φ 0 = 20 GeV, the value of C 2 4b can be as large as 0.5. However, it is smaller than 0.2 after considering the bound of global fit at 3σ level. Noticing that the value of C 2 4b is directly proportional to the factor y 2 V = (1 − v 2 /(6f 2 )) 2 in the LRTH model and thus becomes larger for a large f . For the process pp → W/Zh → l + 4b + X, the authors of Refs. [69,70] have shown that the cut on invariant mass of the four bottom quarks can suppress efficiently the relevant backgrounds. It is worth of mentioning that Cheung et al. studied the h → ηη decay [69], calculated the total signal and background cross sections at parton level in the SLH model with C 2 4b = 0.16 [69], and found a significance S/ √ B = 3.7 for a luminosity of 30 fb −1 . Of course, a much higher luminosity is needed to discover such a light scalar. For example, even for C 2 4b = 0.11 in the SLH model, the significance S/ √ B can be increased from 1.4 to 4.4 for a luminosity of 300 fb −1 . Considering the LHC Higgs data bound at 3σ level, we estimate the value of C 2 4b is approximately 0.19 in the LRTH model (C 2 4b ≃ 0.3 × 0.8 2 ). Therefore, we hope that by using the suitable cuts, the possible signatures of the light scalar in the LRTH model may be detected via the process pp → V h → V 4b at the LHC with a high luminosity of 300 fb −1 . Certainly, detailed confirmation of the observability of the signals would require Monte-Carlo simulations of the signals and backgrounds, which is beyond the scope of this paper.
The light scalar φ 0 can also be produced associated with the SM-like Higgs h at the International Linear Collider (ILC), which has been studied in Ref. [73]. The numerical results show that the resonance production cross section can be significantly enhanced at the high energy linear collider with √ s ≃ m Z H . On the other hand, the properties of SM-like Higgs h can be precisely measured through the Zh associated production at the linear collider [74][75][76]. Here we calculate the cross sections of the process e + e − → hφ 0 and e + e − → Zh → Z(φ 0 φ 0 ) → Z(4b, 2b2τ ) at an electron-positron collider with √ s = 250 GeV, as shown in the Fig. 10. As shown in Fig. 9, the associated production rate Br(e + e − → hφ 0 ) is smaller than the order of 10 −2 fb at √ s=250 GeV, which can hardly be utilized to search for the light scalar h. However, the production cross sections of processes e + e − → Zh → Z4b and e + e − → Zh → Z(φ 0 φ 0 ) → Z(2b2τ ) can reach 120 fb and 20 fb respectively, as illustrated in Fig. 10. Certainly, the cross sections would become smaller when we consider the global fit bound at 3σ level (reduced about two thirds). Since these signals are free of the SM background, such production process may contribute the light scalar discovery at an electron-positron collider.

V. CONCLUSIONS
The LRTH model predicts one neutral pseudoscalar particle φ 0 , which may be lighter than half of the Higgs boson mass. In this work we focus on the case of m φ 0 < m h /2 so that the new decay mode h → φ 0 φ 0 can be open. In this work, we firstly calculated the decay widths and the branching ratios of the h → φ 0 φ 0 decay , as well as the major decay modes of the φ 0 itself: such as φ 0 → (bb, cc, τ + τ − ) and φ 0 → (gg, γγ) decays. We then examined the f , M and m φ 0 -dependence of the decay widths and corresponding branching ratios, and checked the possible constraints on the LRTH model from the latest LHC Higgs data on such a possibility. We performed a global fit by using 20 sets of the measured Higgs signal strengths as reported by ATLAS and CMS collaboration for γγ, ZZ * , W W * , bb and τ + τ − channels. We also studied the detection of φ 0 at future electron-positron collider experiments.
From our numerical calculations and the phenomenological analysis we found the following points: 1. Without the LHC constrains, the branching ratio of the decay h → φ 0 φ 0 can be as large as 80% and it can suppress significantly the visible γγ signal rate. The current LHC Higgs data for the γγ channel can place strong limit on such a decay: for example, Br(h → φ 0 φ 0 ) ≤ 26% for R γγ ≥ 0.7.
2. The p-value of the SM Higgs boson is 0.80, which means that the SM is a reasonably good fit to the Higgs data. In the LRTH model, its p-value is smaller than that of the SM in most of the parameter space and approaches the SM value for a sufficiently large f parameter.
3. The latest LHC Higgs data constrain the branching ratio Br(h → φ 0 φ 0 ) to be less than 30% at 3σ level.