CP-violating Higgs-gauge boson couplings in $H\nu \bar{\nu}$ production at three energy stages of CLIC

A phenomenological study of CP-violating dimension-six operators via the $e^+e^-\to\nu \bar{\nu} H$ process is performed in a model-independent Standard Model effective field theory framework at all energy stages of CLIC using the updated baseline integrated luminosities. All signal and relevant background events are generated in MadGraph and passed through PYTHIA for parton showering and hadronization at parton level. Detector effects are considered via tuned CLIC detector cards in Delphes. Since we reconstruct the Higgs boson from a pair of b-jets, limits on CP-violating dimension-six couplings are obtained at three $b$-tagging working points: tight, medium and loose defined in the CLIC Delphes card for all three energy stages of CLIC. Our best 95 \% C.L. limits at the loose working point (90 \% b-tagging efficiency) on $\tilde c_{HW}$ and $\tilde c_{HB}$ are $[-7.0\times10^{-3};7.0\times10^{-3}]$ and $[-3.0\times10^{-2};3.0\times10^{-2}]$, respectively at the 3 TeV energy stage of CLIC with an integrated luminosity of 5.0 ab$^{-1}$. Considering a 0.3 \% systematic uncertainty from possible experimental sources worsens the limits on these couplings by a factor of two.

for each stage of CLIC. The sensitivity estimations of CP-violating dimension-6 Higgs-Gauge boson couplings are given in section 4. Finally, we conclude and compare our obtained limits with current experimental results in section 5.

Effective Operators
It is well known that all operators which define quark and lepton fields along with a single Higgs doublet field interacting via an SU(3) C × SU(2) L × U(1) Y SM gauge symmetry are restricted to have mass dimension of four or less in the SM Lagrangian ( L SM ). In the Effective Field Theory (EFT) approach, the SM Lagrangian is extended with higher-dimensional operators having coefficients of inverse powers of mass (Λ), and hence are suppressed if this mass is large compared to the reachable experimentally energies; where Λ is the new physics scale, O i are the dimension-six operators, and the coefficients c i are dimensionless parameters of the new physics coupling to the SM particles. It is also noted that we ignore the dimension-5 operators responsible for generating Majorana neutrinos and are only concerned with the extended Lagrangian with dimension-6 operators. The most general gauge-invariant dimension-6 Lagrangian L can be expressed in a convenient basis of independent operators O i using normalized Wilson coefficients asc i = c i /Λ 2 that are free parameters [50][51][52]. In this work, we consider the dimension-6 CP-violating interactions of the Higgs boson and electroweak gauge boson in the SILH basis as [52]: where the dual field strength tensors are defined by and Φ is the Higgs field containing a single SU(2) L doublet of fields; B µν = ∂ µ B ν − ∂ ν B µ and W µν = ∂ µ W k ν − ∂ ν W k µ + gε i jk W i µ W j ν are the electroweak field strength tensor and G µν is the strong field strength tensors; g , g and g s denote 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 and σ k are the Pauli matrices. The SM EFT Lagrangian (Eq.(2)) containing the Wilson coefficients in the SILH basis of dimension-6 CP-violating 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. The effective couplings in gauge basis defined as dimension-6 operators are given in Table 1.
The parametrization of Ref. [52] which is based on the formulation given in Ref. [51] is considered in our analysis. Since it chooses to remove two fermionic invariants while retaining all the bosonic m Wc HW operators, the parametrization is not complete as explained in Ref. [53,54]. The main purpose of this paper is to estimate the direct sensitivity toc HW ,c HB andc γ couplings without considering higherorder electroweak effects. For this purpose, the effects of the dimension-6 CP-violating operators on Hνν production mechanism in e + e − collisions are investigated using the Monte Carlo simulations with leading order in MadGraph5_aMC_v2.6.3.2@NLO [55]. The described CP-violating operators in the effective Lagrangian of Eq.(2) are implemented into the MadGraph5_aMC@NLO based on FeynRules [56] and the UFO [57] framework. The cross sections of e + e − → ννH process at generator level as a function ofc HW ,c HB andc γ couplings for three center of mass energy stages of CLIC; 380 GeV, 1.5 and 3 TeV are given in Fig.1. The quoted cross sections do not include the effects of initial state radiation (ISR) and beamstrahlung. In this figure, we vary dimension-6 CP-violating operators individually and calculate the contributions to the corrections from new physics. Since one coefficient at a time is varied in the calculation of cross section, only quartic contributions are taken into account. It can be easily seen that the contribution of thec HW coupling to the SM increases with center of mass energy even in a small value region for the e + e − → ννH process.

Signal and Background Analysis
In this section, we investigate the sensitivity of CLIC at three center of mass energy stages for CPviolatingc HW ,c HB ,c γ effective couplings through the e + e − → ννH process and relevant backgrounds assuming Higgs boson decays to a pair of b-quarks. The effective CP-violating dimension-6 couplings and SM contribution (S + B H ) in the e + e − → ννH process are taken into account. The values of CPviolating dimension-6 couplings considered in this study are very small so the total decay width under influence of these operators are close to the SM expectations. Therefore, the influence of CP-violating dimension-6 couplings on the branching ratio of H → bb is neglected in the analysis. The following relevant backgrounds are included in the analysis. i) The same final state as the considered signal process including only SM contribution is the e + e − → ννH process, which is labelled B H . ii) The production of two Z bosons is labeled as B ZZ , considering one Z decaying to bb while the other decays to νν. iii) The W boson pair production is labeled as B WW , considering one W decaying to bb while the other decays to lν. iv) B tt is the pair production of the top quark process i.e. , e + e − → tt in which one of the top quark (anti- Figure 2: The pseudo-rapidity versus transverse momentum distribution of leading (b1) and sub-leading (b2) b-tagged jets for SM e + e − → ννH process with defined WP (90 % b-tagging efficiency) in CLIC Delphes card at √ s=380 GeV The hadronic decay channel of the Z boson in the e + e − → ννZ process is taken into account and labelled B Zνν . As shown in Ref [45], one can expect to see significant contribution to the background due to eγ and γγ collisions. In our analysis framework, we generate events via Madgraph which does not include photons from Beamstrahlung. Therefore, we neglect these backgrounds in our analysis. The cross section of the considered backgrounds in our analysis are given in Table 2. All signal and background events (500k for each) are generated in MadGraph5_aMC@NLO and passed through PYTHIA 8.2 for parton showering, hadronization and decay of unstable particles [58]. We use the Delphes 3.4.1 [59] for a fast simulation of detector response with tuned CLIC detector cards [60,62]. There are three cards, designed for each center-of-mass energy stage of CLIC: √ s =380 GeV, 1.5 TeV and 3 TeV. Some properties of the cards are as follows. Jets are clustered with the Valencia Linear Collider (VLC) algorithm [63,64] in exclusive mode with a fixed number of jet (N = 2, 3, 4, 5, 6 where N corresponds to the number of partons expected in the tree level final state) and five different cone size parameters (R = 0.5, 0.7, 1.0, 1.2, 1.5) with γ=β =1 using FastJet [65]. The b-tagging efficiency and misidentification rates implemented in these cards are discussed in Ref. [66,67] where the three working points (WP) are defined; the tight WP (50 % btagging efficiency), medium WP (70 % b-tagging efficiency), and loose WP (90 % b-tagging efficiency). Misidentification rates for three working points are given as a function of energy and pseudorapidity. For example; In a bin where E 500 GeV and 1.53 < |η| 2.09, misidentification rates are 3 × 10 −3 , 9 × 10 −3 and 5 × 10 −2 for the tight, the medium and the loose WP, respectively. In our analysis, we picked N jets =2 and R=1.0 for three energy stages with the three b-tagging working points, tight, medium and loose. Then, all events are analyzed by using the ExRootAnalysis utility [68] with ROOT 6.16. [69].

√ s = 380 GeV
In addition to initial jet clustering (i.e, N jets =2 and R=1.0), events having no charged leptons are selected for further analysis (Cut-0). In order to separate signal and background events we use the following kinematical cuts: i) In exclusive mode, we have two jets which are obtained from subsequent decay of Higgs boson, tagged as b-tagged jets. The b-tagged jet with the highest transverse momentum (p T ) is labeled as b1 and the one with lower p T as b2 (Cut-1). The phase space of b-tag jets for the SM background process with the same final state as signal at b-tagging efficiency working points (90 %) defined in the CLIC Delphes card are shown in Fig.2. The transverse momentum and pseudo-rapidity of b1 and b2 for signal (forc HW =0.1 benchmark point) and all relevant background processes taking the loose b-tagging working point are shown in Fig.3. ii) We select a region in phase space where the transverse momentum of b1 is p b1 T > 50 GeV and b2 has p b2 T > 30 GeV, and the pseudo-rapidity of the btagged jets is |η b1,b2 | ≤ 2.0. This cut suppresses B ZZ and B Zνν backgrounds. For signal and background processes, distributions of the missing energy transverse ( E T ), scalar sum of the transverse energy (H T ), the invariant mass and the transverse momentum of the reconstructed Higgs-boson from two b-jets are depicted in Fig.4. Subsequent cuts can be determine from these figures: The missing energy transverse  Table 3. The numbers of events after each cut are shown in Table 4 Table 4 are normalized to the cross section of each process times the integrated luminosity, L int = 1.0 ab −1 .

TeV and 3 TeV
The analysis of √ s = 380 GeV is repeated for a 1.5 and 3 TeV center of mass energy of CLIC with L int = 2.5 ab −1 and L int = 5.0 ab −1 , respectively. For the signal (c HW = 0.1) and all relevant background processes taking the loose b-tagging working point at √ s = 1.5 TeV, the distributions of transverse momentum and pseudo-rapidity of b1 and b2 are shown in Fig.6 while the missing energy transverse ( E T ) and scalar sum of the transverse energy (H T ) are given in Fig.7. Both of these figures are normalized  to the cross section of each process times the integrated luminosity, L int = 2.5 ab −1 . We only modified Cut-3 to H T > 100 GeV as shown in Table 3 at √ s = 1.5 TeV. Since similar distributions to Fig.6 and 7 have been observed at √ s = 3 TeV, we implemented the same cuts used in the √ s = 1.5 TeV analysis.  Fig.8-Fig.9 corresponding to 1.5 and 3 TeV center of mass energies, respectively. After applying the final cut, which requires the transverse momentum of bb system to be greater than 75 GeV, we obtained the normalized number of events for signals and relevant SM backgrounds. The total normalized number of events in the existence of effective couplings (c HW =0.1,c HB =0.3 andc γ =0.3) and all relevant backgrounds are given in Table 5.

Sensitivity of Higgs-Gauge Boson Couplings
The sensitivities to CP-violating dimension-6 Higgs couplings are obtained by a χ 2 criterion method with systematic error, defined by   where N NP is the total number of events in the presence of effective couplings (S) and total SM backgrounds (B tot ), N B tot is the total number of events only coming from SM backgrounds, defined as B tot = B H + B ZZ + B WW + B tt + B Zνν , and ∆ i = δ 2 sys + 1 N B i is the combined systematic (δ sys ) and statistical error in each bin. In this study, we concentrate on obtaining 95 % confidence level (CL) limits of thec HW ,c HB ,c γ couplings via the e + e − → ννH signal process at CLIC with the center of mass energies at three stages √ s= 380 GeV, 1.5 TeV, 3 TeV, and the integrated luminosities L int = 1.0 ab −1 , 2.5 ab −1 , 5.0 ab −1 respectively. Since we study the H → bb decay channel, b-tagging plays an important role in our analysis. To see the effect, we present the comparison of b-tagging efficiencies with three working points of 50 %, 70 %, 90 % for the first stage center of mass energy of CLIC (CLIC-380) in the left panel of the Fig.10. This figure emphasizes that the sensitivity of CLIC increases with the increase of btagging efficiencies, resulting in a better limit with the loose working point (90 % b-tagging efficiency). We measure the Hνν cross section in the channel H → bb after b-tagging with statistical uncertainty of 1.67 % in the first stage of CLIC for an integrated luminosity of 1 ab −1 at √ s = 380 GeV, assuming unpolarised beam and loose WP (90%) of b-tagging efficiency. In the higher energy CLIC stages for integrated luminosity 2.5 ab −1 at √ s = 1.5 TeV and 5 ab −1 at √ s = 3 TeV, the statistical uncertainties are 0.26 % and 0.15 %, respectively. In the right panel of Fig.10, we plot obtained 95 % C.L. limits at 90 % working point of b-tagging efficiency for all three stages of CLIC and the recent High-Luminosity (HL-LHC) projections on these limits [23]. The HL-LHC projection limit onc γ = [−0.6 × 10 −3 ; 0.6 × 10 −3 ] is reported via pp → h → γγ process which is sensitive to this coupling. However, we obtain better limits onc HW ,c HB than HL-LHC projection limits. At 3 TeV energy stage of CLIC, the sensitivities ofc HW and c HB couplings are [−7.0 × 10 −3 ; 7.0 × 10 −3 ] and [−3.0 × 10 −2 ; 3.0 × 10 −2 ] with integrated luminosity of 5.0 ab −1 , respectively. Our limits onc HW ,c HB at √ s = 3 TeV with L int = 5 ab −1 are one order of magnitude better than HL-LHC projected limits and also better than observed current experimental limit oñ c HW (assumingc HW =c HB ) measured in the two-photon final state using 36.1 fb −1 of proton-proton collisions at √ s = 13 TeV by the ATLAS experiment at the Large Hadron Collider [8]. We also recomputed the bounds including systematic uncertainties at 95 % C.L.. In the case of 0.3 % systematic uncertainty from possible experimental sources as in Ref. [45], the constraint onc HW andc HB at the highest energy stage of CLIC with L int = 5.0 ab −1 are [−9.97 × 10 −3 ; 9.97 × 10 −3 ] and [−4.18 × 10 −2 ; 4.18 × 10 −2 ]. These bounds are lower than the experimental current limits and HL-LHC projected limits even at the first stage of CLIC. We should note that we neglect the theoretical uncertainties and only show the potential sensitivity of experimental reach. However, including theoretical systematic uncertainty is very likely to worsen our results.
The validity of the EFT can be tested with the relation between the new physics scale and the Wilson coefficients of the dimension-six operators as follows where g * is the coupling constant of the heavy degrees of freedom with the SM particles. An upper bound on the new physics scale using g * = 4π and obtained limits onc HW andc HB are 36.94 TeV and 17.84 TeV, respectively. This upper bounds are within the range of EFT.

Conclusions
For a better understanding of the new physics beyond the SM in the Higgs sector, among the proposed future colliders, CLIC is an attractive option that has a clean environment. In this paper, we have emphasized the effects of CP-violating dimension-6 operators defined by an SM EFT Lagrangian approach via the e + e − → ννH process for three energy stages of CLIC ( √ s = 380 GeV, 1.5 TeV, 3 TeV) and integrated luminosities (L int = 1.0 ab −1 , 2.5 ab −1 , 5.0 ab −1 ). We have presented the kinematical distributions of signal and relevant backgrounds; transverse momentum and rapidity of b-tagged quarks, missing energy transverse, scalar sum of the transverse energy and the invariant mass and transverse momentum of the Higgs boson, reconstructed from a pair of b-quarks (with 90 % b-tagging efficiency). In order to obtain limits on the CP-violating dimension-6 couplings at each energy stage of CLIC, we focused on the transverse momentum of the reconstructed Higgs boson at three working points of b-tagging efficiency considering realistic detector effects with tuned CLIC detector cards designed for each center of mass energy stage in a cut-based analysis. The e + e − → ννH process is more sensitive toc HW andc HB couplings than the other CP-violating dimension-six couplings at three energy stages of CLIC. The obtained sensitivity of couplings at 95 % C.L. of thec HW andc HB in all energy stages of CLIC are better than both HL-LHC projected and observed current experimental limits. As a conclusion, CLIC with three energy stages will offer advantages to probing the couplings of Higgs with SM particles that appear in the new physics beyond the SM scenarios.