Projections for Neutral Di-Boson and Di-Higgs Interactions at FCC-he Collider

As a high energy e-p collider, FCC-he, has been recently proposed with sufficient energy options to investigate Higgs couplings. To analyse the sensitivity on the Higgs boson couplings, we focus spesifically on the CP-even and CP-odd Wilson coefficients with $hhZZ\:$and $hh\gamma\gamma\:$ four-point interactions of Higgs boson with Effective Lagrangian Model through the process $e^{-}p\to hhje^{-}$ . We simulate the related processes in FCC-he, with 60 GeV and 120 GeV $e^{-}$ beams and 50 TeV proton beam collisions. We present the exclusion limits on these couplings both for 68% and 95% C.L. in terms of integrated luminosities.


I. INTRODUCTION
The discovery of the Higgs boson [1,2] and the consistency of Higgs measurements by ATLAS and CMS [3,4] brought up all available Higgs production and decay channels to an utmost importance level. Of these channels, arguably the most important ones are the Higgs-self coupling (λ) and the anomalous couplings, since it will show a direct evidence of the Electroweak Symmetry Breaking (EWSB) mechanism [5] which is expected to work as predicted by Standard Model (SM).
Over the years, extensive studies have shown that it is quite challenging to observe the Yukawa couplings of Higgs boson to other fermions even with the correction algorithms at the LHC through gluon-fusion process due to the enormous SM background [6]. Although Vector Boson Fusion (VBF) processes are accesible at the LHC [7], there are studies suggest that it is more feasible to accomplish this task using linear colliders [8] or through epcollisions [9]. Consequently, searching for Higgs decays at future colliders became relatively important just because they bring unique opportunities to fully cover SM scalar sector [10].
To study anomalous couplings, di-Higgs boson production through charged current (CC) mechanisms are well studied in Large Hadron Electron Collider (LHeC) and Future Circular Collider (FCC-he) [11] expressing that neutral current (NC) mechanisms have a potential to enhance the overall Higgs boson signal efficiency. In addition, for the completeness of the studies based on the Higgs Effective Lagrangian Model [14], it is quite promising to study on di-higgs productions via four-point interaction vertices since it contains aspects for both new physics and SM Higgs studies. However a complete understanding of Higgs sector can open doors to new physics and particles. Likewise, it has recently been reported that if additional scalar bosons exist, they can be interpreted in the effective theory approach leaving signatures in the final states with a pair of invisible χ particles that are proposed to be the dark matter candidate [15]. According to this approach, newly proposed heavy Higgs boson eventually decays to a Higgs boson and a pair of χ, causing a distortion in the p T distribution that are compatible with the observations at LHC [16].
Here, it is considered the electron -proton collision variant of the FCC-he that is proposed to build on the same site with LHC, as the future extension of the LHeC. In FCC-he, construction of an Energy Recovery Linac is proposed to deliver electrons with energies ranging from E e = 60 GeV to E e = 120 GeV , while a proton beam is provided by a 100 km circular beam pipe and has an maximum proton energy of E p = 50 T eV .
The main idea in this letter is that by analysing neutral four-point interactions in FCChe, one can get rid of a part of the SM background and get a better detection efficiency by electron tracks involved in the final state which can be reconstructed efficiently. We studied Higgs boson couplings at neutral four-point interaction vertices through Wilson coefficients within the Higgs Effective Lagrangian Model. The outline of our paper has been prepared as in the following: In section 2, we basically reveal the related Lagragian terms and their phenomenological interpretations as well as the assumptions of our case. In section 3 and 4, we discuss event productions for signal and background processes respectively. In section 5, we explain applied event selection criterias and statistical analysis of data that we obtained from simulation tools. Finally in section 5, we present our results and exclusion limits for obtaining related coefficients in FCC-he collisions.

II. HIGGS EFFECTIVE LAGRANGIAN (HEL) MODEL
Since the details of the Higgs sector is not trivial, an effective field theory (EFT) that covers all related interactions at a given scale, but not the others that play a role at significantly different scales, might be a good approach. In EFT models, particularly interactions at much higher energies than the energy scale of interest are ignored. So that the underlying physics event at energies below the new physics scale can be described precisely. In this letter, we studied on the exclusive Higgs Effective Lagrangian (HEL) Model, that is valid above a Λ scale around TeV order, makes possible to include dimension-six operators with free parameters, namely, Higgs self-couplings, Yukawa couplings and Wilson coefficients. In this approach, the complete Lagrangian is handled by SM Lagrangian and supplemented higher dimensional operators which are assumed to appear at energies larger than the effective scale. L, the most general gauge-invariant total Lagrangian, can be expressed as in the followings with Wilson coefficientsc i and independent operators O i of dimension less than or equal to six.
After EWSB, the Higgs sector can be expressed as; where numbers in superscript denotes the set of interactions of a Higgs boson with a vector boson pair. Related Lagrangians can specifically be rewritten as follows for the mass basis.
Here, tilde operator denotes the CP-violating terms, while all other non-tilde terms are CP-conserving. One can also consider other neutral four-point interactions such as di-higgs and di-gluon or quartic-self-interaction of Higgs. But these processes are shown to give no events at FCC-he collider. Therefore we can describe the general Lagrangian that we are working on as L = L SM +L hγγ +L (4 ) hhzz +L (4 ) hhγγ . Several different representations of couplings in Eq. (3-7) are available via FCNC notation [17]. In principle, we concentrate on gauge basis representations of couplings with Wilson coefficients as in Table 1 -2 and take the same notation as explicitly described in [19]. From Table 2, one can see that g hhγγ (g hhγγ ) strictly corresponds to terms with onlyc γ (c γ ) coefficient, while g hhzz (g hhzz ) indirectly corresponds to terms with coefficientsc HB ,c HW ,c γ ,c W (c HB ,c HW ,c γ ,c W ) for the first two orders, respectively. And for the third order of g hhzz , it is seen an explicit dependence tō c T ,c H ,c γ . To scan over these parameters, we explain our strategy in the next section with the case-spesific assumptions. To understand physical analysis of EFT explicitly, one must build SILH (Strongly-Interacting Light Higgs) Lagrangian in terms of indepent operators as shown in Eq. (1) and described in Ref [18]. One can then discuss the relative effect of Table I: Corresponding couplings of a Higgs boson and a pair of neutral bosons in the mass and gauge basis for Eq. (3) as in Ref [14].  [14].

Mass Basis Gauge Basis
Mass Basis g hhγγghhγγ ,g hhzz g the various operators on physical observables through Wilson coefficients. However, SILH Lagrangian includes only CP-conserving operators multiplied with Higgs related fields. For completeness, one should also add a CP-violating Lagrangian as in Eq. (1) which has the same interactions with SILH Lagrangian but rewritten with CP-violating coefficients (c HB ,c HW ,c γ ,c W ) and operators. One of the naive ways of estimating these coefficient values has been made by power counting after expanding the effective Lagrangian in the number of fields and derivatives at tree level. According to power counting for the related terms that we are interested in, one can estimates; where Here, we are searching for both Z bosons and γ as mediators that together forms the NC processes. One can name each subprocesses as Vector Boson Fusion (VBF) with Z boson and photo fusion (PF) with γ mediators, respectively. Due to gauge invariant structure of HEL model, one can not actually separate event productions for Z boson and γ mediators.
However, it is possible to minimize the contribution of one of the subprocesses setting related Wilson coefficients as below. The corresponding couplings of PF process are well known with already studied in letters Ref [12,13]  (1)c B andc W are suppressed, since they should be order of m 2 W/M 2 where M is the typical mass scale of the new physics sector. [14] (2)c 6 is also supressed for our case, since the corresponding production cross section gives no events at FCC-he. (3) Constraints from the electroweak precision parameters suggest thatc T ,c W ,c B should be order of 10 −3 according to [22]. results [23].
(5)c HW andc HB that are expected to be order of 10 −3 tend to cancel each other at the Z-pole.
We investigated e − p → hhje − process both for Z boson and γ as mediators for the  [27] with a size parameter of ∆R = 0.5 by using FastJet package [28].
From Fig.2 and 3, one can see that a cross section scan overc H ,c γ parameters for the processes where hhZZ and hhγγ vertices involved. It is trivial that PF signal has higher cross sections if the Wilson coefficient is aroundc γ 1. On the other hand, hhZZ vertex has an asymmetric large sensitivity toc γ coefficient as shown in Fig.2b. In Table 3, we present the event counts for signal and background processes where both VBF and PF signals are independently produced.

IV. BACKGROUND PRODUCTION
Although it is highly supressed in the phase space, one can produce events for e − p →  Table 3. Third contribution is obtained by electroweak neutral productions such as ZZ / ZH + 1jet + 1 lepton as shown in Table 3.
Due to the basic transverse momentum cut applied at 20 GeV for low-pt jets, gluon jets and a small portion of quark jets have been removed. Both signal and background events are produced by setting the factorization and renormalization scales at 125 GeV with standard NN23LO1 parton distribution function set. In productions, b-tagging efficiency is assumed to be %60 and considered 1% of light-jets faking the leptons while for c-quark jets, the same fake rate is %10.

V. EVENT SELECTION AND ANALYSIS
Event selection criteria: (1) Four b-tagged jets and a light jet is selected with p T > 20GeV .
(2) |η| < 5 for all jets and |η| < 2.5 for leptons applied. In Fig.4 and 5, we present the kinematic distributions in comparison with background and SM processes through a Z boson mediator while c H = 0.1 . Fig. 4 shows that the forward and sub jets in signal have a separable transverse momentum than the background jets as expected, while the pseudo-rapidity distributions (Fig.5) behave similar for both signal and background jets. For outgoing electron, one can see that the signal distribution slightly deviates to the negative region, while background signal locates at zero pseudo-rapidity.
For the higher values ofc H (orc γ in PF process), the higher negative deviation for the mean of pseudo-rapidity distributions are observed. In Fig.6, it is seen the reconstructed contribution comes from the b-jets created by di-Higgs decays. As a significant evidence for coupling seperations, azimuthal angle distribution (∆φ) between lepton and forward jet for VBF and PF signals are shown in Fig.7 and Fig.8, respectively for √ s ≈ 5 T eV . Similarly for √ s ≈ 3.5 T eV , same distributions are obversed with a factor ∼ 0.286 in the event count.
About the shape of the distribution of ∆φ, we observed that this interference is strictly dependent on the coefficients and detector parameters. For evaluating the limits in Fig.9, we calculated statistical significances and followed the methods described in Ref. [29] It's also worthwile to comment on the event selection criteria: First three conditions consist of almost default cuts for event production. Fourth condition has higher p T cuts that can be seen directly from

VI. CONCLUSION
In this letter, we have investigated the sensitivity on the Higgs boson couplings and Wilson coefficients in productions through NC mechanism (in Fig. 1) for FCC-he. Since the process is possible through both Z boson and γ mediators, one should take into account the interference of VBF and PF processes with the right parameter set. We observed that ∆φ variable which is strictly affected by detector parameters, is a key to separate interferences of both VBF and PF signals. It is observed that di-Higgs production through NC mechanism has a major sensitivity toc γ andc H coefficients within the considerations of electroweak precision measurements. FCC-he collider can coverc H (c γ ) coefficients as in Fig.9 through NC processes with integrated limunosities up to 3 (50) ab −1 respectively. On the other hand, one can reveal the corresponding Higgs couplings by obtaining limits of Wilson parametrization that is involved in Higgs productions at a specific limunosity. Thus, we present g hhγγ /g hhγγ , g hhzz andg hhzz in Fig. 10 that shows the required limunosities to discover these couplings. For EFT approach, it is known that above the new physics scale, Λ, Lagrangian expansions will be unconvinced and limits on the couplings will deteriorate rapidly. One can see from the Wilson parametrization that the coefficients can naively be expressed in terms of M,  Total systematic uncertainties are roughly extrapolated from LHC data in percentage.
overall mass scale. However, couplings such asg hhγγ ,g hhzz that have degrading sensitivities because of the higher order dependences, have deterioration of limits such that, at higher energies, deviations from the original limit in percentage getting lower. Thus, according to recent mass limits on heavy particles, one can see that the deterioration of limits cannot deviate above %10. Although similar detailed searches at the LHC are avaliable to set limits on corresponding Higgs couplings, it is possible to obtain high precision on the couplings using FCC-he advantages in center of mass energy and background. (b) Figure 10: Required integrated luminosities to obtain the limits on the corresdonding couplings (a) g hhγγ /g hhγγ (b) g hhzz andg hhzz for 60 GeV and 120 GeV electron beam energy options at FCC-he where the shaded areas are not allowed assumingc T ,c W ,c B = 10 −3 and all other Wilson coefficients are zero. Total systematic uncertainties are roughly extrapolated from LHC data in percentage.