Higgs couplings and Naturalness in the littlest Higgs model with T-parity at the LHC and TLEP

Motivated by the recent LHC Higgs data and null results in searches for any new physics, we investigate the Higgs couplings and naturalness in the littlest Higgs model with T-parity. By performing the global fit of the latest Higgs data, electroweak precise observables and Rb measurements, we find that the scale f can be excluded up to 600 GeV at 2σ confidence level. The expected Higgs coupling measurements at the future collider TLEP will improve this lower limit to above 3 TeV. Besides, the top parnter mass mT+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_{T_{+}} $$\end{document} can be excluded up to 880GeV at 2σ confidence level. The future HL-LHC can constrain this mass in the region mT+<2.2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_{T_{+}}<2.2 $$\end{document} TeV corresponding to the fine-tuning being lager than 1%.


Introduction
The discovery of a Higgs boson [1][2][3][4] by the ATLAS [5] and CMS [6] collaborations at the LHC marks a milestone of an effort that has been ongoing for almost half a century and opens up a new era of particle physics. The existing measurements [7][8][9][10] and the global fits to the ATLAS and CMS Higgs data within remarkable precision [11][12][13][14][15][16][17][18][19][20] agree with the standard model (SM) predictions. This conclusion is consistent with the ATLAS and CMS null results in searches for any new physics. However, the experiments of cold dark matter [21] and neutrino oscillations [22] cannot be explained in the framework of the SM so that they are supposed to provide obvious evidence for the new physics beyond the SM. In particular, the facts that the SM can be an effective theory valid all the way up to the Planck scale and there is no symmetry protecting the scalar masses lead to the naturalness problem, i.e., why the Higgs boson mass is of the order of the electroweak scale and not driven by the radiative corrections to the Planck scale, remains unanswered.
Since the discovery of the Higgs boson the fine-tuning problem has become even more intriguing. Among many new physics models, Little Higgs models based on a collective symmetry breaking can provide a natural explanation of the fine-tuning by constructing the Higgs as a pseudo-goldstone boson. The littlest Higgs (LH) model [23][24][25][26] is an economical approach to implement the idea of the little Higgs theory. However, due to the large corrections to the electroweak precision observables (EWPO) from the mixing of the SM gauge bosons and the heavy gauge bosons, the original LH model is severely constrained by precision electroweak data. This constraint can be relaxed by introducing the discrete symmetry T-parity, which is dubbed as littlest Higgs model with T-parity(LHT) [27][28][29][30].
With current data, all properties of the observed Higgs-like particle turn out to be in rough agreement with expectations of the SM [31][32][33][34][35], but there are still some rooms for the new physics [36,37], which may be ultimately examined at the LHC-Run2 and the future Higgs factories [38][39][40][41][42][43][44][45][46][47][48][49]. Since top partner is naturally related to the Higgs physics and plays an important role in the naturalness problem, one can obtain constraints from the Higgs data [50][51][52][53][54][55]. In this work, we will discuss the Higgs couplings and the naturalness problem in the LHT model at the LHC and Triple-Large Electron-Positron Collider (TLEP) [56,57] by performing a global fit of the latest Higgs data, R b and oblique parameters, and give the current and future constraints to the LHT parameters.

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Recently, some similar works have been carried out in refs. [58][59][60][61]. Different from these papers, we perform a state-of-the-art global fit to obtain the indirect constraints on the breaking scale and the top partner with a comprehensive way. This method was widely used in the fit of the SM to the electroweak precision data. So, it will be also meaningful to explore what might happen in the LHT model with a global fit at future colliders. By building an overall likelihood function for the constraints from the EWPO, R b measurements and Higgs data, we can obtain a well-defined statistical results of the exclusion limit on the breaking scale. More importantly, we obtain the exclusion limit on the top partner mass, which is obvious absent in other papers. This paper is organized as follows. In section 2, we give a brief description of the LHT model. In section 3, we present the calculation methodology and the numerical results at the LHC and the TLEP. Finally, we draw our conclusions in section 4.

A brief review of the LHT model
The LHT model is a non-linear σ model based on the coset space SU(5)/SO(5), where the spontaneous symmetry breaking is realised at the scale f via the vacuum expectation value (VEV) of an SU(5) symmetric tensor Σ, given by (2.1) The VEV of Σ 0 breaks the gauged subgroup [SU(2) × U(1)] 2 of SU(5) down to the SM electroweak SU(2) L × U(1) Y , which leads to new heavy gauge bosons W ± H , Z H , A H . After the EWSB, their masses up to O v 2 /f 2 are given by with g and g ′ being the SM SU(2) and U(1) gauge couplings, respectively. In order to match the SM prediction for the gauge boson masses, the VEV v needs to be redefined via the functional form where v SM = 246 GeV is the SM Higgs VEV. Under the unbroken SU(2) L × U(1) Y the Goldstone boson matrix Π is given by where H is the little Higgs doublet (h + , h) T and Φ is a complex triplet under SU(2) L which forms a symmetric tensor (2.5)

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φ 0 and φ P are both real scalars, whereas the φ ++ and φ + are complex scalars. The other Goldstone bosons are the longitudinal modes of the heavy gauge bosons and therefore will not appear in unitary gauge. The mass of Φ can be given by where all components of the triplet are degenerate at the order we are examining.
When T-parity is implemented in the quark sector of the model, we require the existence of mirror partners with T-odd quantum number for each SM quark. We denote the up and down-type mirror quarks by u i H and d i H , where i(i = 1, 2, 3) is the generation index. After the EWSB, their masses up to O v 2 /f 2 are given by where κ i are the diagonalized Yukawa couplings of the mirror quarks. One can notice that the down-type mirror quarks have no interactions with the Higgs.
In order to stabilize the Higgs mass, an additional T-even heavy quark T + is introduced to cancel the large one-loop quadratic divergences caused by the top quark. Meanwhile, the implementation of T-parity requires a T-odd mirror partner T − with T + . The T-even quark T + mix with the SM top-quark and leads to a modification of the top quark couplings relatively to the SM. The mixing can be parameterized by dimensionless ratio R = λ 1 /λ 2 , where λ 1 and λ 2 are two dimensionless top quark Yukawa couplings. This mixing parameter can also be used by x L with Considering only the largest corrections induced by EWSB, their masses up to O v 2 /f 2 are then given by The corrections to the Higgs couplings of the other two generations of T-even (SM-like) up-type quarks up to O v 4 SM /f 4 are given by For the T-even (SM-like) down-type quarks and charged leptons, the Yukawa interaction have two possible constructions [62]. The corresponding corrections to the Higgs couplings with respect to their SM values up

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One can notice that Case B predicts a stronger suppression for the down-type fermion couplings to the Higgs boson.
The naturalness of the model can be quantified by how much the contributions from the heavy states (δµ 2 ) exceed the observed value of the Higgs mass squared parameter (µ 2 obs ) [23]: (2.14) Here m h is the Higgs boson mass. In the LHT model, the dominant negative log-divergent contribution to the Higgs mass squared parameter comes from the top quark and its heavy partner T + loops [23] where Λ = 4πf is the cut-off of the nonlinear sigma model, λ t is the SM top Yukawa coupling and m T + is the mass of the heavy top partner.

Calculations and numerical results
In our numerical calculations, we take the SM input parameters as follows [63]: Our global fit is based on the frequentist theory. For a set of observables O i (i = 1 . . . N ), the experimental measurements are assumed to be Gaussian distributed with the mean value O exp i and error σ exp i . The χ 2 can be defined as where σ i is the total error both experimental and theoretical. The likelihood L ≡ exp − χ 2 i for a point in the parameter space is calculated by using the χ 2 statistics as a sum of individual contributions from the latest experimental constraints. The confidence regions are evaluated with the profile-likelihood method from tabulated values of δχ 2 ≡ −2 ln(L/L max ). In three dimensions, 68.3% confidence regions (corresponding to 1σ range) are given by δχ 2 = 3.53 and 95.0% confidence regions (corresponding to 2σ range) are given by δχ 2 = 8.02.
Under few assumptions involving mainly flavour independence in the mirror fermion sector, the LHT model can be parametrised by only three free parameters, i.e., the scale f , the ratio R and the Yukawa couplings of the mirror quarks κ j . Considering the recent constraint from the searches for the monojet, we require the lower bound on the Yukawa couplings of the mirror quarks are κ j ≥ 0.6 [61]. We scan over these parameters within the following ranges [58][59][60][61]64] where we assume the three generations κ j are degenerate. The couplings of the UV operators are set as c s = c t = 1. The likelihood function L is constructed from the following constraints: 1. EWPO: these oblique corrections can be described in terms of the Peskin-Takeuchi S, T and U parameters [65]. 3. Higgs data. The experimental results are given in terms of signal strengths µ(X; Y ), which is defined as the ratio of the observed rate for Higgs process X → h → Y relative to the prediction for the SM Higgs, µ(X; Y ) ≡ σ(X)BR(h→Y ) σ(X SM )BR(h SM →Y ) . We confront the modified Higgs interactions and the one-loop contribution of the new particles in the LHT model with the available Higgs data. We calculate the χ 2 values by using the public package HiggsSignals-1.2.0 [69,70], which includes 81 channels from the LHC and Tevatron and these experimental data are listed in ref. [71]. In our calculations, the Higgs mass m h is fixed as 126 GeV. Note that for the Higgs data, the HiggsSignals has provided the calculation of χ 2 , where both experimental (systematic and statistical) uncertainties as well as SM theory uncertainties are included.
In figure 1, we show the results of the global fit to the above three kinds of constraints in the plane of R versus f for Case A and Case B, respectively. We can see that the lower bound on the symmetry breaking scale at 95% C.L. is The constraints are stronger than the electroweak precision constraints in ref. [64], which is because the main constraint here comes from the Higgs data. For the top partner mass, we can see that the combined indirect constraints can exclude m T + at 95% C.L. up to It's worth noting that they are stronger than the lower bound set by the ATLAS direct searches for the SU(2) singlet top partner, m T > 640 GeV [72]. Our study may play a complementary role to the direct searches in probing top partner. Facility The expected precision for the Large Hadron Collider High-Luminosity Upgrade (HL-LHC) and the TLEP are assumed in table 1, which comes from the table 14 and table 16 of the Higgs working group report [73].
In the LHT model, the loop-induced couplings hgg and hγγ can receive contributions from both the modified couplings and the new particles. The decay h → gg can be corrected by the modified htt coupling and the loops of top partner T + and T-odd mirror quarks. In addition to these corrections involved in the decay h → gg, the decay h → γγ can be also corrected by the modified hW W coupling and the loops of W H , φ + , φ ++ . Besides, the couplings hcc, hss, hbb, hZZ are also modified, they can exert an effect on our fit.
In figure 2 and figure 3, we show the shifts of the Higgs couplings hV V , htt, hgg, hγγ for the above samples in the 2σ range. In order to investigate the observability, we compare  them with the corresponding expected measurement uncertainties of the Higgs couplings in table 1 at HL-LHC with a luminosity of 3000 fb −1 . The value of the fine-tuning for each point is also calculated by using the eq. (2.14). From figure 2 and figure 3, we can have some observations as follows: 1. The values of the fine-tuning for the samples are cornered to be smaller than about 6% by the above global fit.
2. For the Higgs couplings hV V and htt, they are suppressed by the high order factor O v 2 /f 2 . The deviation of the Higgs couplings g hV V from the SM predictions are at percent level and the deviation of the Higgs coupling g htt from the SM prediction can reach over 10%.
For the loop-induced couplings g hgg and g hγγ , on one hand they are corrected by the high order factor, on the other hand they are corrected by the loop contributions of the new particles. For the effects of these loop diagrams, there are cancelation  between t(W L ) and the corresponding partner T + (W H ) so that the effective g hgg and g hγγ couplings are reduced. The deviation of the Higgs coupling g hγγ from the SM prediction is at percent level, that is because the dominant contribution to the coupling g hγγ comes from the W L (W H ) over the t(T + ).  to negative modification of the relevant couplings with respect to the SM. Besides, the non-linear expansion of the model field suppresses these couplings at the order 3. In figure 2 and figure 3, we attempt to show the expected constraints from the future individual Higgs coupling meaurements on the top partner and naturalness at the HL-LHC. The couplings hV V and htt are modified at the order O v 2 /f 2 , which can determine the scale f and help us understand the nature of the Higgs boson in the LHT model. Apart from this, the coupling hgg can provide the information for the cancelation between t and the corresponding partner T + , while the coupling hγγ can provide the information for the cancelation between W L and the corresponding partner W H . So, we can see that the individual Higgs coupling meaurements can help us understand the different parts of the LHT model. 4. The future measurements of the g hgg coupling at the HL-LHC will be able to exclude the m T + < 2.2 TeV, which corresponds to the fine-tuning being lager than about 1%. However, other expected measurements, such as g hV V , g htt and g hγγ couplings, can only improve the limits for the top partner mass mildly.
In figure 4, we present the prospect of improving the constraints on the scale f at a possible future Higgs factory TLEP with √ s = 240 GeV. In our fit, the χ 2 can be defined as where µ i represents the signal strength prediction from the LHT model and σ i represents the 1σ uncertainty i.e. the expected measurement precision at the TLEP. We use the Snowmass Higgs working group results to simply estimate the exclusion JHEP10(2014)047 limits. Given that the super-high luminosity of 10000 fb −1 can be achieved at the TLEP, we assume that all the measured Higgs couplings will be the same as the SM predictions with the expected measurement uncertainties in table 1. From the figure 4, we can see that the lower bound on the scale f will be pushed up to 3.1 TeV for Case A and 3.25 TeV for Case B at 95% C.L..

Conclusions
In this paper, we investigated the Higgs couplings and naturalness in the LHT model under the available constraints from the current Higgs data and the EWPO. By performing the global fit, we find that the scale f can be excluded up to 670 GeV for Case A and 600 GeV for Case B at 2σ level. The precise measurements of the Higgs couplings at the future collider TLEP will constrain this limit to above 3 TeV. Besides, the top partner mass m T + can be excluded up to 980 GeV for Case A and 880 GeV for Case B at 2σ level. This mass can be constrained in the region m T + < 2.2 TeV at the HL-LHC corresponding to the fine-tuning being lager than 1%.