Study of Higgs Effective Couplings at Electron-Proton Colliders

We perform a search for beyond the standard model dimension-six operators relevant to the Higgs boson at the Large Hadron Electron Collider (LHeC) and the Future Circular Hadron Electron Collider (FCC-he). With a large amount of data (few ab$^{-1}$) and collisions at TeV scale, both LHeC and FCC-he provide excellent opportunities to search for the BSM effects. The study is done through the process $e^-p \to h j \nu_e$ where the Higgs boson decays into a pair of $b \bar{b}$ and we consider the main sources of background processes including a realistic simulation of detector effects. For the FCC-he case, in some signal scenarios to obtain an efficient event reconstruction and to have a good background rejection, jet substructure techniques are employed to reconstruct the boosted Higgs boson in the final state. In order to assess the sensitivity to the dimension-six operators, a shape analysis on the differential cross sections is performed. Stringent bounds are found on the Wilson coefficients of dimension-six operators with the integrated luminosities of 1 ab$^{-1}$ and 10 ab$^{-1}$ which in some cases show improvements with respect to the high-luminosity LHC results.

So far, the Standard Model (SM) of particle physics has been found to be a successful theory describing nature up to the scale of electroweak. However, there are reasons to believe that the SM is not the ultimate theory of particle physics at the TeV scale. The Higgs boson discovery by the LHC experiments [1,2] has been a milestone in understanding the mechanism of electroweak symmetry breaking (EWSB). After that, one of the main goals would be to precisely measure the Higgs boson properties that would provide the possibility of searching for new physics effects beyond the SM.
Given the absence of any signature of new physics in the present data, one can parametrize the effects of beyond the SM in an effective field theory (EFT) expansion. This approach is a powerful tool which parameterizes possible new physics effects via a systematic expansion in a series of higher-dimensional operators composed of SM fields [3,4]. The operators are composed of all possible combinations of SM fields respecting the SU (3) c × SU (2) L × U (1) Y gauge symmetries and Lorentz invariance. In the EFT approach, potential deviations from the SM could be described 3 using the following Lagrangian: where the O i is the ith dimension-six operator, Λ is the scale at which new physics is expected to appear and c i s are arbitrary Wilson coefficients. These dimension-six operators have been listed and studied in Refs. [3][4][5][6]. There have been a lot of studies to probe these operators and so far a lot of attention has been paid to constrain these operators that can be found in Refs. .
The aim of this study is to explore Wilson coefficients of dimension-six operators, as described in Refs. [32][33][34], contributing to the Higgs production in association with a jet and a neutrino at the LHeC and FCC-he [35][36][37][38]. The LHeC is a proposed deep inelastic electron-nucleon scattering (DIS) machine which has been designed to collide electrons with an energy from 60 GeV to possibly 140 GeV, with protons with an energy of 7 TeV. The future circular collider (FCC) has the option of colliding electrons to protons with the electron energy E e = 60 GeV and the proton energy of E p = 50 TeV. The inclusive Higgs boson production cross section at the high-energy FCC-he is expected to be about five times larger than at the future proposed high-energy and high-luminosity electronpositron collider TLEP/FCC-ee [39]. In comparison with the LHC or FCC-hh, the LHeC or FCC-he have the advantages of providing a clean environment with small background contributions from QCD strong interactions. Furthermore, no effects of pile-up and multiple interactions exist in these machines and they are able to provide precise measurements of the proton structure, electroweak and strong interactions.
The remaining part of this paper is structured as follows: In Section II, we present the theoretical framework of our analysis by recalling the relevant aspects of the effective SM in which dimensionsix operators are considered. In this section, we review the higher-dimensional operators and highlight the operators contributing to the Higgs boson production processes at the LHeC and FCC-he. Section III describes the details of our analysis including the simulation tools and analysis strategy for both LHeC and FCC-he. We will explain the event selection criteria and statistical method by which we obtain the constraints on the Wilson coefficients. The analysis strategy for the LHeC collider and its sensitivity to the dimension-six operators have been presented in Section IV.
Section V is dedicated to present the sensitivity of the FCC-he collider to the related Wilson coefficients in the hjν e process. Our results and the constraints on the Wilson coefficients are given in Section VI. Comparisons with the LHC bounds are made also in this section. Finally, Section VII presents the summary and conclusions.

II. THEORETICAL FRAMEWORK
The SM effective Lagrangian can be obtained by including higher-dimensional operators that takes into account the new physics effects beyond the SM which may appear at the energy scale much larger than the SM energy scale. Under the assumption of baryon and lepton number conservation and keeping only dimension-six operators, the most general invariant Lagrangian can be constructed from the SM fields. We concentrate on the dimensionsix interactions of the Higgs boson, fermions, and the electroweak gauge bosons in the strongly interacting light Higgs (SILH) basis conventions which can be written as [32,33,40]: wherec i coefficients are dimensionless Wilson coefficients, and O i are dimension-six operators made up of SM fields. The first term in the effective Lagrangian of Eq. (2) is the SM Lagrangian, L SM .
The second term ∆L F 1 in Eq. (2) addresses the interactions between two Higgs fields and a pair of quarks or leptons. This term has the following form: The third term ∆L F 2 of the effective Lagrangian in Eq. (2) contains the interactions of a pair of quark or lepton, a Higgs field, and a gauge boson. This term reads: Finally, the last term of this Lagrangian corresponds to the Higgs field which is the part of a strongly interacting light Higgs sector (SILH). The ∆L SILH term can be expressed as: where Φ is a weak doublet containing the Higgs boson field, and G µν , B µν , W µν are the strong and electroweak field strength tensors.
derivative. In Eq.5, λ is the Higgs boson quartic coupling and v is the vacuum expectation value In the electron-proton colliders, the Higgs bosons are produced through two main channels.
The Higgs boson can be produced either via charged current: e − q → Hq ν e or neutral current e − q → eHq. The leading order diagram for the production of a Higgs boson in the electron-proton collisions for the charged current process is depicted in Fig. 1. There are already several studies on different aspects of the Higgs boson production via charged and neutral currents in the electronproton collisions [41][42][43][44][45][46][47][48][49]. In Ref. [45], it has been shown that the production cross section of the charged current process is larger than the neutral current process by a factor of a around five for the energy of the incoming electron 140 GeV and the proton energy of 7 TeV. In this work, our focus is on the charged current production process due to its larger production cross section. Also, this process has a clean signature as it comprises of a significant missing transverse energy and an energetic jet which tends to be forward. The concentration of this analysis is on the Higgs boson decay into a pair of bottom quarks because of its large branching fraction.
The present work is dedicated to consider the effects of L eff presented in Eq. 2 on the Higgs boson production through charged current process e − q → Hq ν e . The contributions originating from other possible effective operators are neglected for simplicity. The representative Feynman diagrams for e − q → Hq ν e are displayed in Fig. 2. The vertices which receive contributions from the L eff are shown by filled circles. It is remarkable that the SM tree level contribution does not have any dependency on the momenta of the involved particles, while considering L eff enters momentumdependent interactions in the calculations. This leads to changes in the production cross sections as well as the shape of differential distributions. In this paper, differences in the shapes of distributions is used to constrain the involved Wilson coefficients in this process.
The e − p → hjν e process is sensitive to the following set of L eff parameters: The cross section of e − p → hjν e process is found to be almost insensitive to parametersc l ,c eW , c d ,c u ,c uW ,c dW . This is because of very small Yukawa couplings of light quarks and electron. As a result, our analysis is restricted to the remaining seven parameters: An interesting way to represent the effective Lagrangian from the experimental and phenomenological point of view is the effective Lagrangian in the mass basis. In particular, it has been found to be an applicable approach in the electroweak precision tests (EWPT) studies. The anomalous Higgs interactions in the mass basis has been presented in Ref. [33]. The relation between the mass basis couplings and the dimension-six coefficients which are involved in this analysis are given in Table I.
There are already many studies to constrain the Wilson coefficients discussed above in different colliders using various channels which can be found in references [7-25, 28, 50-54]. Although the  Mass basis Gauge basis vc HL obtained limits on some of the coefficients in the previous studies are tight, we are going to examine possible improvements for these limits in the future high-energy electron-proton colliders via a careful investigation of the Higgs production mechanism in the framework of effective field theory.
In the next section, the details of simulation for probing the effective Lagrangian using e − p → hjν e process in the future LHeC and FCC-he colliders will be discussed.
The chain we have used to perform the generation and simulation of the signal and background processes are described is this section. The full set of interactions generated by the dimensionsix operators mentioned in the Higgs Effective Lagrangian L SILH of Eq. (5), ∆L F 1 in Eq. (3) and ∆L F 2 in Eq. (4) have been implemented in FeynRules [55,56] and the model is imported to a Universal FeynRules Output (UFO) module [33,57]. Then, the UFO model files have been inserted in the MadGraph5-aMC@NLO [58,59] Monte-Carlo (MC) event generator to calculate the cross sections and generate the signal events. The CTEQ6L1 PDF set [60] is used to describe the proton structure functions. The renormalization and factorization scales are set to be dynamical in The next-to-leading order QCD correction to the signal process e − p → hjν e is found to be small [61]. Therefore, in this work the k-factor for the signal is assumed to be one. , respectively [36]. The b-tagging efficiency is assumed to be 60% while mis-tag probabilities of 10% and 1% for c-quark jets and light-quark jets are considered, respectively [36].
The tracker of the LHeC detector is expected to cover pseudorapidity range up to 3.0 [36].
Therefore, the b-tagging performance is valid up to |η b−jet | < 3. For the light-jets, the calorimeter coverage is considered to be |η light−jet | < 5. In the next sections, we will present the analysis strategies for LHeC and FCC-he separately in more details. As mentioned before, we consider the LHeC with the electron energies of 60 GeV and 50 TeV protons.

IV. LHEC SENSITIVITY
In this section, we present the analysis strategy and the results for the LHeC. The strategy for choosing the basic cuts are similar to the one proposed in [36]. In the reconstructed distribution of the Higgs boson mass, the mass peak is lower than the right Higgs boson mass because of the energy carried by the neutrino from the b-quark decays.  The cross sections (in fb) after each cut for the signal, SM production of Higgs boson via hjν e process, and the main backgrounds processes are presented in Table II where g * is the coupling constant of the heavy degrees of freedom with the SM particles. Additional suppression factors appear in the case that an operator is generated at loop level. An upper bound can be put on the new mass scale M * using the fact that the underlying theory is strongly coupled by setting g * = 4π. Assumingc = O(1), we find This upper bound is not violated in this analysis as we have M Higgs,j < 1 TeV.

A. Sensitivity estimate
This subsection is dedicated to estimate the sensitivity of e − p → hjν e process to the Wilson coefficients. The sensitivity are obtained using a χ 2 analysis over all bins of ∆ Ep Z distribution. The ∆ Ep Z variable is defined as: In Fig. 4, we show the the expected normalized distribution of ∆ Ep Z for the signal and the main sources of background processes after applying all cuts presented in Table II. As it can be seen, the shapes ∆ Ep Z signal withc H = 0.1 is quite different from the sum of all background processes. As a result, it is a useful distribution to obtain the exclusion limits on the Wilson coefficients defined in Eq.3 and Eq.5.   Table II.
To set upper limits at the 95% CL, we use a χ 2 criterion from the distribution of ∆ Ep Z defined as:  Table III and Table V with the electron energies of 60 GeV and 140 GeV. As an example, LHeC would be able to constrainc H by more than one order of magnitude with respect to the LHC in high luminosity regime.

V. FCC-HE SENSITIVITY
In this section, the sensitivity of the FCC-he to the related Wilson coefficients in the hjν e process is studied. As we mentioned before, FCC-he employs the 50 TeV proton beam of a proposed circular proton-proton collider.  where p b ( pb) is the momentum of the b(b)-quark and the bottom quark mass has been neglected.
One can express the opening angle θ bb versus the parent mass Higgs boson and the momenta of the b-andb-quark. Using kinematic relations, the angular separation of a bb pair produced in a Higgs boson decay can be also written as: where p T is the transverse momentum of the Higgs boson, x and 1−x are the momentum fractions of the b andb quarks. Figure 6 shows the distributions of the Higgs boson momentum and transverse momentum for the signal scenario ofc HW = 0.1 for the LHeC and FCC-he. As it can be seen, at the FCC-he Higgs bosons reside at large values of momentum and p T .
For the Higgs bosons with substantial momentum and p T , from Eq.(12) and Eq.(13) it is expected that the angular separation of the Higgs boson decay products decreases. Figure 7 shows the normalized distribution of ∆R between two b-quarks from the Higgs boson decay for the FCC-he.
We present the distributions for two for two signal scenariosc H = 0.1 andc HW = 0.1. The plot clearly confirms that for the signal scenario ofc HW , a considerable fraction of Higgs bosons are produced in the boosted regime while this is not valid for the signal scenario ofc H . As a result, the high-p T Higgs bosons produce a collimated jet with substructure.
Because of the small angular separation between two b-jets from the Higgs decay and large boost, the common jet reconstruction with a cone size of ∆R = 0.4 − 0.5 would not be usable for most of the signal events with non-zero value ofc HW . An alternative method of fat jet algorithm is applied [71] for these boosted events.
To reconstruct the signal events with two boosted b-jets in the final state, first we reconstruct the fat jets by using the Cambridge/Aachen (CA) jet algorithm [72,73] assuming a jet cone size of R = 1.2. Then, in order to identify the boosted Higgs boson, the methods described in the fat jet reconstruction algorithm [71] is done as explained in the following. In the beginning, a reconstructed fat jet J is split into two sub-jets J 1 , J 2 with the masses m J 1 > m J 2 . Then, the method requires a significant mass drop of m J 1 < µ M D m J with µ = 0.667. It should be mentioned that µ M D is an arbitrary parameter that shows the mass drop degree. Also, to avoid of including high p T light-jets, two sub-jets are required to be symmetrically split. This requires the two sub-jets to satisfy: where y cut is a parameter of the algorithm which determines the limit of asymmetry between two sub-jets and p 2 T, J 1 and p 2 T, J 2 are the square of the transverse momentum of each sub-jet. Finally, if the above criteria are not satisfied, the algorithm takes J = J 1 and returns to the first step for performing decomposition. The explained algorithm for boosted object reconstruction has been implemented in the FastJet package [66] by which our analysis is done.  Tables III, IV and V. In Tables III and IV,    The results for LHeC and FCC-he at very high integrated luminosities are presented in Table V.
The bounds are given for maximum achievable integrated luminosities of 1 ab −1 and 10 ab −1 for the LHeC and FCC-he, respectively. Based on this analysis for the FCC-he with E e = 60 GeV for an integrated luminosity of 10 ab −1 , the sensitivity to the Wilson coefficients is much better than the other options analyzed in this study, and in some cases is better than the ones expected to be achieved by the HL-LHC with an integrated luminosity of 3000 fb −1 .    Table IV showing a great sensitivity and in some cases improvements are expected with respect to the potential constraints for the LHC [20,23]. We also show that the FCC-he collider with E e = 60 GeV and with an integrated luminosity of L = 10 ab −1 or even with 3 ab −1 would be able to probe the Wilson coefficients of dimension-six operators of the Higgs boson (especiallyc H ,c HW andc W couplings) beyond the HL-LHC.