Constraints on Higgs effective couplings in $H\nu \bar{\nu}$ production of CLIC at 380 GeV

The potential of the $e^+e^-\to\nu \bar{\nu} H$ process in the first stage of CLIC considering center of mass energy of 380 GeV and assuming the baseline integrated luminosity of 500 fb$^{-1}$ is examined to probe CP conserving dimension-six operators in a model-independent Standard Model effective field theory framework. In the analysis, a detailed fast simulation on $e^+e^-\to\nu \bar{\nu} H$ signal processes and dominant backgrounds are performed including parton showering with PYTHIA and detector simulation based on ILD type detector with DELPHES in MadGraph. The obtained best limits on the $\bar{c}_{HB} $, $\bar{c}_{W}=- \bar{c}_{B}$ and $\bar{c}_{HW} $ are $[-4.82;1.53]\times10^{-2}$, $[-5.11;4.13]\times10^{-3}$ and $[-6.58;5.55]\times10^{-3}$, respectively.


I. INTRODUCTION
The recent Large Hadron Collider (LHC) discovery of a scalar particle with 125 GeV which is compatible with Standard Model (SM) Higgs boson predicted by Brout-Englert-Higgs symmetry breaking mechanism opens up a gateway to search for physics beyond the SM [1,2]. But, an evidence for new physics beyond the SM using analysis of combined ATLAS and CMS data for probing the couplings of Higgs boson has not been observed yet. Possible deviation from the SM predictions of Higgs boson couplings would imply the presence of new physics involving massive particles that are decoupled at energy scales much larger than the Higgs sector energies being probed [3]. The SM Effective Field Theory (EFT) is a well-known model independent method for investigation of any deviation from SM [4,5]. The origin of this method is based on all new physics contributions to the SM described by a systematic expansion in a series of high dimensional operators beyond the SM fields. All high dimensional operators conform to SU (3) C × SU (2) L × U (1) Y SM gauge symmetry. The dimension-6 operators play an important role in the framework since they match to ultraviolet (UV) models which are simplified by the universal one-loop effective action. There have been many analyses for constraints on SM EFT operators with available data from LHC-Run 1 [6][7][8][9][10][11][12][13] and with electroweak precision measurements provided from previous accelerator, namely Large Electron Positron (LEP) [14][15][16][17]. Especially, the prediction on dimension-6 operators have been examined in many rewarding studies at High Luminosity LHC (HL-LHC) [18][19][20] and future e + e − colliders [21][22][23][24][25][26][27][28][29].
The precision measurements of Higgs boson couplings with the other SM particles at the LHC and planned future colliders will give us detailed information about its true nature. The future multi-TeV e + e − colliders with extremely high luminosity and clean environment due to the absence of hadronic initial state, would give access to precise measurement, especially for the Higgs couplings. The Compact Linear Collider (CLIC) is one the mature proposed linear colliders with centre of mass energies from a few hundred GeV up to 3 TeV [30]. The first energy stage of CLIC operation was chosen to be √ s=380 GeV, with the predicted integrated luminosity of 500 f b −1 .
The primary motivation of this stage is the precision measurements of SM Higgs properties and also the model independent Higgs couplings to both fermions and bosons [30,31].
In this study, we focus on the analysis of e + e − → ννH production process in order to assess the projection of the first energy stage of the CLIC on the CP-conserving dimension-6 operators involving the Higgs and gauge bosons (W ± , γ, Z) defined by an SM EFT Lagrangian in the next section.

II. EFFECTIVE OPERATORS
The well known SM Lagrangian ( L SM ) involving renormalizable interactions is suppressed by higher dimensional operators in SM EFT approach. All these operators parametrised by an energy scale of non-observed states assumed larger than vacuum expectation value of Higgs field (v). A few different operator bases are presented in the literature, we consider SM EFT operators as the strongly interacting light Higgs Lagrangian (L SILH ) in bar convention [18,32,33]. Assuming the baryon and lepton number conservation, the most general form of dimension-6 effective Lagrangian including Higgs boson couplings that keep SM gauge symmetry is given as follows; wherec i are normalized Wilson coefficients that are free parameters. In this work, we consider the dimension-6 CP-conserving interactions of the Higgs boson and electroweak gauge boson in SILH basis as [33]: where λ represents the Higgs quartic coupling; y u , y d and y l are the 3 × 3 Yukawa coupling matrices in flavor space; g , g and g s denotes coupling constant of U (1) Y , SU (2) L and SU (3) C gauge fields, respectively; the generators of SU (2) L in the fundamental representation are given by T 2k = σ k /2, σ k being the Pauli matrices; Φ is Higgs field contains a single SU (2) L doublet of fields; and G µν is the strong field strength tensors; and the Hermitian derivative operators defined as, The SM EFT Lagrangian (Eq.(2)) containing the Wilson coefficients in the SILH bases of dimension-6 CP-conserving operators can be defined in terms of the mass eigenstates after electroweak symmetry breaking (Higgs boson, W, Z, photon, etc.) as follows where W µν , Z µν and F µν are the field strength tensors of W -boson, Z-boson and photon, respectively; m H represent the mass of the Higgs boson; the effective couplings in gauge basis defined as dimension-6 operators are given in Table I in which a H (g H ) coupling is the SM contribution to the Higgs boson to two photons (gluons) vertex at loop level.
We use the parametrization in Ref. [33] based on the formulation given in Ref. [32] in our analysis. The parametrization is not complete as described in detail in section 3 of Ref. [34] and also Ref. [35]. It chooses to remove two fermionic invariants while retaining all the bosonic operators. This choice assumes completely unbroken U (3) flavor symmetry of the UV theory where the coefficient of these operators are unit matrices in flavor space. Therefore, we assume flavor diagonal dimension-six effects. It is sufficient for the purpose of this paper in which we do not consider higher order electroweak effects but only claim a sensitivity study forc W ,c B ,c HW ,c HB andc γ couplings.
We have used the Monte Carlo simulations with leading order in MadGraph5_aMC@NLO [36] involving effect of the dimension-6 operators on Hνν production mechanism in e + e − collisions. The effective Lagrangian of the SM EFT in Eq.(2) is implemented into the MadGraph5_aMC@NLO based on FeynRules [37] and UFO [38] framework. In this study, we focus on searching for the dimension-6 Higgs-gauge boson couplings via e + e − → ννH process as shown in Fig.1. This process is sensitive to Higgs-gauge boson couplings; g hzz , g hww , g hzγ , and the couplings of a quark or lepton pair and one single Higgs field;ỹ u ,ỹ d ,ỹ l in the mass basis. In the gauge basis, e + e − → ννH process is sensitive to the seven Wilson coefficients:c W ,c B ,c HW ,c HB ,c H ,c γ andc T related to Higgs-gauge boson couplings and also effective fermionic couplings. Due to the small Yukawa couplings of the first and second generation fermions, we neglect the effective fermionic couplings. We setc W +c B andc T to zero in all our calculations since the linear combination ofc W +c B andc T strongly constrained from the electroweak precision test of the oblique parameters S and T . The cross sections of e + e − → ννH process as a function ofc W ,c B ,c HW ,c HB andc γ couplings are shown in Requiring missing energy transverse ( E T ), no charged leptons and at least 2 jets with their transverse momenta (p j T ) greater than 20 GeV and pseudo-rapidity (η j ) between -2.5 and 2.5 are the pre-selection of the event to be further analysed. The energy resolution of jets for |η j | 3 is assumed to be The momentum resolution for jets as a function of p j T and η j is Jets are clustered with the anti-k t algorithm [44] using FastJet [45] where a cone radius is used as R = 0.5. In order to select the signal and background events, the following kinematic cuts and requirements are applied; i) requiring at least two jets tagged as the b-jet which significantly suppress the light-quark jet backgrounds. These two b-jets are used to reconstruct Higgs bosonmass. ii) One of the b-tagged jets with the highest p T is defined as b1 while the other is b2 with lower p T . Fig. 3 shows p T distributions of b1 and b2 of signal (forc HW =0.05) and all relevant background processes versus reconstructed Higgs boson-mass from b1 and b2 (M b,b ). As it can be seen in Fig. 3, the b 1 with p b1 T > 50 GeV, b 2 with p b2 T >30 GeV and pseudo-rapidity of the b-tagged jets to be |η b1,b2 | 2.0 are considered to reduce B ZZ and B Zνν . In ILD detector card, both btagging efficiency and misidentification rates are given as function of jet transverse momentum.
For the transverse momentum of leading jet (b1) ranging from 50 GeV to 180 GeV, b-tagging efficiency is between 64% and 72 %, c-jet misidentification rate is 17%-20%, and misidentification rate of light jet 1.2%-1.76%. The missing transverse energy ( E T ) and scalar transverse energy Finally, the reconstructed invariant mass of Higgs-boson from two b-jet is selected to be in the range 92 GeV < M rec inv (b1, b2) < 136 GeV. The kinematic distributions for each processes are normalized to the number of expected events which is defined to be the cross section of each processes times integrated luminosity with L int =500 fb −1 .  Effects of the cuts used in the analysis can be seen from the Table II which

IV. SENSITIVITY OF HIGGS-GAUGE BOSON COUPLINGS
We calculate the sensitivity of the dimension-6 Higgs-gauge boson couplings in e + e − → ννH process by applying χ 2 criterion with and without a systematic error. The χ 2 function is defined as follows where and 3000 fb −1 (LHC-3000) [18] in Fig. 6. We see that CLIC-380 results would be significantly more sensitive toc HW and somewhat sensitive toc W = −c B whereas sensitivity toc HB is comparable with expected LHC results. The prediction on the limits for the future lepton colliders; ILC [24,29] of an integrated luminosity L int =300 fb −1 at the center of mass energy √ s = 500 GeV, FCC-ee [24] for L int =10 ab −1 at √ s = 240 GeV, CEPC [46] for L int =5 ab −1 at √ s = 240 GeV are also shown in This bound is four times lower than the obtained limits without systematic uncertainties. fb −1 [24,29], FCC-ee for L int =10 ab −1 at √ s = 240 GeV [24], CEPC for L int =5 ab −1 at √ s = 240 GeV [46] at Higgs decay in e + e − → ννH signal process and the other dominant backgrounds. The e + e − → ννH process is more sensitive toc HW couplings than the other dimension-six couplings at first energy stage of CLIC. Our results show that a CLIC with √ s = 380 GeV, L int =500 fb −1 will be able to probe the dimension-six couplings of Higgs-gauge boson interactions in e + e − → ννH process especially forc HW couplings at scales beyond the HL-LHC bounds while they become competitive with thec HB ,c W = −c B couplings.