Higgs-strahlung production process e^+ e^- \to Z h at the future Higgs factory in the Minimal Dilaton Model

We investigate the Higgs-strahlung production process e^+ e^- \to Z h at the future Higgs factory such as TLEP by including radiative corrections in the Minimal Dilaton Model (MDM), which extends the SM by one singlet scalar called dilaton. We consider various theoretical and experimental constraints on the model, and perform fits to the Higgs data taken from ATLAS, CMS and CDF+D0. Then for the 1\sigma surviving samples, we calculate the MDM predictions on the inclusive production rate \sigma(e^+e^-\to Zh) at the 240-GeV Higgs factory, and also the signal rates of e^+e^-\to Zh with the Higgs boson decaying to b\bar b and \gamma\gamma. We have following observations: (1) In the heavy dilaton scenario, the deviation of \sigma(e^+e^-\to Zh) from its SM prediction can vary from -15\% to 85\%, which mainly arises from the modification of the tree-level hZZ coupling and also the radiative correction induced by possibly large Higgs self-couplings. (2) The processes e^+e^-\to Zh at the Higgs factory and pp\to hh at 14-TeV LHC are complementary in limiting the MDM parameter space. Requiring the deviation of \sigma(e^+e^-\to Zh) from its SM prediction to be less than 1\% and that of \sigma(p p \to h h) to be less than 50\%, \tan \theta_S in the MDM will be limited to be -0.1<\tan\theta_S<0.3, and the deviations of the signal rates are constrained to be |R_{b\bar b}|<2\% and |R_{\gamma\gamma}|<7\%. Especially, the Higgs self-coupling normalized to its SM prediction is now upper bounded by about 4. (3) In the light dilaton scenario, the deviation of \sigma(e^+e^-\to Zh) may reach -7\%, and requiring its size to be less than 1\% will result in 0<\tan\theta_S<0.1, and -10\%


I. INTRODUCTION
In July 2012, the discovery of a new boson with mass around 125 GeV at the CERN Large Hadron Collider (LHC) [1,2] marked a great triumph in the history of particle physics.
With the growingly accumulated data, the properties of this newly discovered boson are in excellent agreement with those of the Higgs boson predicted by the Standard Model (SM), including the further measurements of its spin and parity quantum numbers [3,4].
However, up to now, there is no evidence to establish whether the Higgs sector contains only one Higgs doublet. Instead, the Higgs-like resonance with mass about 125 GeV can also be well explained in many new physics models, such as low energy supersymmetric models [5,6] and the dilaton models [7].
So far various Higgs couplings to SM fermions and vector bosons based on the current LHC data still have large uncertainties. Taking the hZZ coupling as an example, the measured value normalized to its SM prediction is 1.43 ± 0.33(stat) ± 0.17(syst) for ATLAS result and 0.92 ± 0.28 for CMS result [8]. Nevertheless, at the future High Luminosity LHC (HL-LHC) with 300 fb −1 (3000 fb −1 ) integrated luminosity, the precision of the C hZZ measurement is expected to reach 4 − 6% (2 − 4%) [8]. Compared with the hadron collider, the future e + e − collider may have a stronger capability in the C hZZ measurement through the Higgs-strahlung production e + e − → Zh. For example, at the proposed International Linear Collider (ILC) with collision energy up to 1TeV and luminosity up to 1000f b −1 , the precision may be improved to be near 0.5% [8]. And an even more remarkable precision of 0.05% may be achieved at the recently proposed Triple-Large Electron-Positron Collider (TLEP) [8], which is a new circular e + e − collider operated at √ s = 240 GeV with 10 4 f b −1 integrated luminosity [9]. The story of the Higgs self-coupling, however, is quite different. By now such a coupling has basically not been constrained by the current Higgs data, while on the other hand, it can be quite large in some new physics models such as the Minimal Dilaton Model (MDM) [10][11][12]. Obviously, the next important task of experimentalists is to determine the coupling size as precise as possible, which is essential in reconstructing the Higgs potential and consequently determining the mechanism of the electro-weak symmetry breaking. At both the LHC and the ILC, the Higgs self-coupling can be measured directly through the Higgs pair production [13][14][15]. The recent studies suggest that a precision of 50% for the coupling can be obtained through pp → hh → bbγγ at the HL-LHC with an integrated luminosity of 3000 fb −1 [8,16], and it may be further improved to be around 13% at the ILC with collision energy up to 1TeV [8].
One interesting feature of the Higgs-strahlung production e + e − → Zh is that, while at tree level its rate is solely determined by the C hZZ coupling, at loop level the rate also depends on the Higgs self-coupling and may be significantly altered by such a coupling.
This brings us the possibility that apart from the direct Higgs pair production, the Higgs self-coupling may also be measured indirectly from the process e + e − → Zh with the e + e − collision energy below the di-Higgs threshold. As shown in [17], given that the inclusive cross section σ(e + e − → Zh) is measured with a precision of 0.4% at the TLEP [9], the Higgs self-coupling may be constrained to an accuracy of 28%.
In this work we take the MDM as an example to investigate the Higgs-strahlung production e + e − → Zh by including radiative corrections. We scan the MDM parameters by considering various experimental and theoretical constraints. Then for the surviving samples we calculate their predictions on σ(e + e − → Zh) at the 240-GeV TLEP, and investigate to what extent the parameters will be constrained given the future precision of the cross section measurement. Noting that more observables will be helpful to further limit the parameter space, we also perform a study on the signals of the Higgs-strahlung production with the Higgs boson decaying to γγ or bb. We note that similar study has been done in supersymmetric models [18].
This work is organized as follows. In Sec. II, we briefly review the MDM and experimental and theoretical constraints on it. Then we calculate σ(e + e − → Zh) by including radiative corrections and discuss the capability of the Higgs factory to determine the model parameters in Sec. III. Finally, we summarize our conclusions in Sec. IV.

II. THE MINIMAL DILATON MODEL
The MDM is an extension of the SM by introducing a linearized singlet dilaton field S and a vector-like top partner T with the same quantum number as the right-handed top quark. The low energy effective Lagrangian of the MDM is given by [10][11][12] where L SM is the part of the SM Lagrangian without the Higgs potential, M represents the scale of a certain strong dynamics in which the fields T and S are involved, q 3L is the SU ( Higgs field H will mix with each other, which can be parameterized by the Higgs-dilaton mixing angle θ S as with f and v = 246 GeV being the vacuum expectation values (vev) of S and H respectively, h and s denoting the mass eigenstates of the Higgs boson and the dialton, and φ 0 and φ + representing the Goldstone bosons. Similarly, q u 3L and T will mix to form mass eigenstates t and t ′ so that If θ S , f and physical masses m h , m s are taken as the input of the theory, one can re-express the dimensionless parameters λ S , κ and λ H as follows [12] In this case, the trilinear interactions among h, s, φ 0 and φ ± are given by with η ≡ v f , and the normalized couplings of h and s with Z or φ 0 are given by In the following we differentiate two scenarios according to the dilaton mass [12]: • Heavy dilaton scenario: m s > m h . An important feature of this scenario is that the trilinear Higgs self-coupling C hhh may be very large.
• Light dilaton scenario: m s < m h 2 ≃ 62 GeV. In this scenario, the Higgs exotic decay h → ss is open with a possible large branching ratio, while C hhh /SM is around at either 1 or 0.
For each parameter point of these scenarios, we impose the constraints similar to what we did in [12], which are given by (1) Vacuum stability of the scalar potential and absence of the Landau pole up to 1TeV.
(2) Bounds from the search for Higgs-like scalar at LEP, Tevatron and LHC.
(3) m t ′ ≥ 1TeV as suggested by the LHC search for top quark partner [19] and constraints from the precision electroweak data [10]. With this constraint, we have cos θ L > 0.97 and consequently C htt /SM ≃ cos θ S [12].
(4) Constraints from the measured Higgs properties. In implementing this constraint, we use the combined data (22 sets) from ATLAS, CMS and CDF+D0 and perform a fit with the same method as that in [20][21][22]. We obtained χ 2 min = 18.66 in the MDM, which is less than χ 2 for the SM (χ 2 SM = 18.79), and paid particular attention to 1σ samples in the fit.
As shown in [12], parameter points satisfying the above constraints will predict cos θ S > 0.92, and C ht ′ t ′ /C SM htt < 0.1. As will be seen below, this feature is beneficial for our analysis.

III. CALCULATIONS AND NUMERICAL RESULTS
In the SM, the radiative corrections to the Higgs-strahlung production process e + e − → Zh come from the Z boson self-energy, the vertex corrections to Ze + e − , he + e − , ZZh and Zγh interactions, and also box contributions [23,24]. The full calculation of these corrections is quite complex (e.g. more than sixty diagrams need to be caculated) and it was shown recently that the total weak correction is 5% for m h = 125GeV and √ s = 240GeV [25].
About the corresponding corrections in the MDM, we have following observations • Although the contribution induced by the Higgs self-coupling is only 2% in the SM [25], it is potentially large in the MDM since the self-couplings among the scalars may be greatly enhanced [12]. In this work, we will focus on such a contribution.
• The correction mediated by t ′ quark can be safely neglected since t ′ is heavy and meanwhile C ht ′ t ′ is relatively small.
• For loops that involves the sZZ interaction and meanwhile do not involve possible large self-couplings among the scalars, their contributions are negligible since the dilaton is highly singlet dominated.
• For the rest contributions, they can be obtained from the corresponding SM results in [23] by scaling with a factor of either cos 3 θ S or cos θ S . We find by detailed calculation that the size of the former contribution, i.e. that obtained by the scaling factor of cos 3 θ S , is −0.4% in the SM, and the latter contribution is 3.4%.
Based on these observations, we conclude that the deviation of the inclusive production rate σ(e + e − → Zh) from its SM prediction, which is generally called genuine new physics contribution, is given by where σ LOOP MDM and σ LOOP SM are the cross sections at one loop level in the MDM and the SM respectively, σ 0 SM is the SM prediction on the cross section at tree level, and δσ scalar MDM denotes the correction induced by the self-couplings among the scalars with the corresponding diagrams given in Fig.1. Note that the first term on the right hand of the second equation corresponds to the tree-level contribution, which differs from its SM prediction due to the modified hZZ coupling by a factor cos θ S . Also note that the constraints have required cos θ S > 0.92, so the deviation R mainly arises from the modification of the tree-level hZZ coupling and δσ scalar MDM . In this work, we take m Z = 91.19GeV, α = 1/128 [26] and m h = 125GeV, and fix the e + e − collision energy at 240 GeV. We obtained σ 0 SM (e + e − → Zh) = 236f b, which is in accordance with the result in [9]. In our calculations of δσ scalar MDM , we adopt the Feynman-'t Hooft gauge, and therefore the diagram involving the Goldstone fields must be considered.
Moreover, we note from Fig.1 that, except for the dilaton mass, the masses of the particles in the loops are fixed, and meanwhile, since the dilaton coupling with Z boson is very small due to its singlet-dominated nature, its induced contribution should be relatively small if C hss or C hhs is not much larger than C hhh . These features imply that δσ scalar MDM or R can be expressed in a semi-analytic way, which is given by In above equation, the second term on the right side reflects the interference between the tree-level contribution and the correction from the self-couplings, the third represents the pure self-coupling contribution which can not be neglected if C hhh /SM ≫ 1, and the fourth term can be safely neglected given cos θ S > 0.92. For the results presented below, we obtain the value of δσ scalar MDM by exact calculation, and we checked that for nearly all the surviving samples, Eq. (12)

A. Numerical results in the heavy dilaton scenario
For the heavy dilaton scenario, we consider the constraints listed in Sect. II and scan the relevant parameters in the following ranges like what we did in [12] 1 ≤ η −1 < 10, 130 GeV < m s < 1 TeV, | tan θ S | < 2, 1TeV < m t ′ < 3TeV.
Then we investigate the properties of the 1σ samples, which satisfy χ 2 − χ 2 min ≤ 2.3. In Fig.2 we project the 1σ samples on the plane of C hhh /SM versus cos θ S and also show some lines corresponding to specific values of R calculated from Eq. (12). One can learn the following features: • Due to the small coefficients of the second and third terms in Eq. (12), a given value of R in Eq. (12) corresponds to a very prolate ellipse on the whole plane of C hhh /SM versus cos θ S after neglecting the term proportional to sin 2 2θ S . For cos θ S > 0.92, the   ellipse curves turn out to be nearly straight lines in Fig.2.
• As indicated by Eq. (12), the tree-level modification of the hZZ coupling is to decrease the inclusive rate, while the effect of the correction induced by the self-couplings is to increase the rate. For the 1σ samples considered, the deviation R varies from −15% to 85%. Such possible large deviation is due to a large uncertainty in determining the hZZ coupling from current Higgs data as well as currently a very weak constraint on the self-couplings.
Obviously, if R is moderately large, two loop or higher order corrections should also be taken into account.
With the upgraded energy and luminosity of the LHC, C hhh may be measured directly through the Higgs pair production since it affects the production rate through the parton process gg → h * → hh. As shown in Fig.6 of [12], for C hhh /SM 2.5 in the heavy dilaton scenario of the MDM, the normalized cross section σ(pp → hh)/SM at the 14-TeV LHC increases monotonically as C hhh /SM becomes larger. In order to compare the effect of the Higgs self-coupling at the LHC with that at the future Higgs factory, we show the correlation of σ(pp → hh) at the 14-TeV LHC with σ(e + e − → Zh) at 240-GeV TLEP in be moderately large, changing from −15% to 5%. These features tell us that the processes pp → hh and e + e − → Zh are complementary in limiting the parameters of the MDM.
In order to investigate the capability of the future experiments to detect the parameter space of the MDM, we assume a measurement precision of 1.0% for σ(e + e − → Zh) at 240 GeV [9] and 50% for σ(pp → hh) at 14 TeV [8,16]. Then we show the allowed parameter region on η −1 −tan θ S plane in the right panel of Fig.4 by requiring the 1σ samples to further satisfy |R| < 1.0% and |σ(pp → hh)/SM − 1| < 50%. For comparison, we also show the 1σ samples in the left panel of Fig.4 without the requirement. Fig.4 indicates that tan θ S is allowed to be within −0.4 < tan θ S < 0.4 and −0.1 < tan θ S < 0.3 before and after the requirement respectively. Furthermore, we checked that, after imposing the requirement, the number of the 1σ samples in our random scan is reduced by more than 80%, and now C hhh satisfies 0.98 ≤ C hhh /SM ≤ 4.4. This reflects the great power of the future experiments in limiting the MDM.
Since the MDM parameters can still survive in a fairly wide region after considering the measurement of the inclusive production rate at future Higgs factory, we need to consider more observables to limit the model. So we also investigate the signal rates of e + e − → Zh → Zbb, Zγγ. Similar to R, we define the deviations of the signal rates from their SM predictions by where Br M DM (h → bb) and Br SM (h → bb) denote the branching ratio of h → bb in the MDM and the SM respectively, and similar notation is applied for h → γγ. In the heavy dilaton scenario, R bb and R γγ can be approximated by [12] R bb ≃ (R + 1.05) · cos 2 θ S Γ bb where Γ SM and Γ bb SM denote respectively the total width of the Higgs boson and the partial width of h → bb in the SM. Note that the above approximations are good only for a sufficiently large R, but anyhow, they are helpful to understand our results. In Fig.5, we project the samples in the right panel of Fig.4 on the plane of R γγ versus R bb for different values of η tan θ S . This figure indicates that R bb is basically constrained in the range of |R bb | < 2%, while |R γγ | can maximally reach 7%. Considering that the expected precisions of measured σ · BR(h → bb) and σ · BR(h → γγ) at 240-GeV TLEP can reach the level of 0.2% and 3.0% respectively [8], one can expect that by the measurement of the bb and γγ signal rates, one can get additional information about η tan θ S if the MDM is a correct theory.

B. Numerical results in the light dilaton scenario
In the light dilaton scenario we scan following parameter ranges by considering the constraints listed in Sec. II 1 ≤ η −1 < 10, 0 GeV < m s < 62 GeV, | tan θ S | < 2, 1TeV < m t ′ < 3TeV, (17) and investigate the properties of the 1σ samples, which are now defined by χ 2 − χ 2 min ≤ 1.0 [12]. Compared with the heavy dilaton scenario, the light dilaton scenario has two distinct features. One is the Higgs exotic decay h → ss is open with a possible large branching ratio. So this scenario is more tightly constrained by current Higgs data. And the other is the Higgs self-coupling strength C hhh /SM is relatively small, around at either 1 or 0. As a result, the deviation R mainly comes from the modified hZZ coupling, so R ≃ cos 2 θ S − 1. In Fig.6 we project the 1σ samples on the plane of deviation R versus cos θ S . As expected, the size of the deviation R monotonically decreases as cos θ S approach 1, and it can maximally reach 7%. This figure also shows that there are two separated regions of R. We checked that it is due to the discontinuousness of C hhh /SM, that is, the upper region corresponds to C hhh /SM ≃ 1, while the lower region corresponds to C hhh /SM ≃ 0.
Adopting the same analysis as Fig.4, we show the 1σ samples projected on the plane Fig.7, where the left panel shows all 1σ samples, while for comparison the right panel shows samples that further satisfy the requirement |R| < 1.0%.
Here we do not consider the deviation of σ(pp → hh) because it is very small in the light dilation scenario [12]. Fig.7 clearly shows that the MDM parameter space in the light dilaton scenario is also strongly constrained resulting in 0 < tan θ S < 0.1, in contrast with −0.24 < tan θ S < 0.2 without the requirement of |R| < 0.1. Moreover, we checked that after the requirement, the number of the 1σ samples in the left panel of Fig.7 is reduced by more  Similar to what we did in the heavy dilaton scenario, we also investigate the signal deviations R bb and R γγ , which can now be expressed as where Γ ss is the width of h → ss in the MDM. In Fig.8 we show the relationship between R γγ and R bb , and their dependence on η tan θ S . From this figure we can see that R γγ and R bb follow a nearly linear correlation since now η tan θ S is very small, i.e. |η tan θ S | < 0.035. One can also see that even with the requirement |R| < 1%, R bb and R γγ may reach −10%. This is because the branching ratio of h → ss may still be moderate large under the constraint of current Higgs data. Note that generally |R γγ | is slightly larger than |R bb |, which can be understood by the positiveness of tan θ S in Eq. (19).

IV. SUMMARY AND CONCLUSION
In this work, we intend to investigate the capability of the future Higgs factory such as TLEP in detecting the parameter space of the MDM, which extends the SM by one singlet scalar called dilaton. For this end, we calculate the Higgs-strahlung production process e + e − → Zh at the future Higgs factory by including radiative corrections in the model. We consider various theoretical and experimental constraints on the model, such as the vacuum stability, absence of Landau pole, the electro-weak precision data and the LHC search for Higgs boson, and perform fits to the Higgs data taken from ATLAS, CMS and CDF+D0. Then for the 1σ surviving samples, we investigate the MDM predictions on the inclusive production rate σ(e + e − → Zh) at the 240-GeV Higgs factory, and also the signal rates of e + e − → Zh with the Higgs boson decaying to bb and γγ. We have following observations: (1) In the heavy dilaton scenario, the deviation of σ(e + e − → Zh) from its SM prediction can vary from −15% to 85%, which mainly arises from the modification of the tree-level hZZ coupling and also the radiative correction induced by possibly large Higgs self-couplings. (2) The processes e + e − → Zh at the Higgs factory and pp → hh at 14-TeV LHC are complementary in limiting the MDM parameter space. Requiring the deviation of σ(e + e − → Zh) from its SM prediction to be less than 1% and that of σ(pp → hh) to be less than 50%, tan θ S in the MDM will be limited to be −0.1 < tan θ S < 0.3, the deviations of the signal rates are constrained to be |R bb | < 2% and |R γγ | < 7%, and the Higgs selfcoupling normalized to its SM prediction is upper bounded by about 4.