Studies of nonresonant Higgs pair production at electron-proton colliders

The measurement of the Higgs quartic coupling modifier between a Higgs boson pair and a vector boson pair, $\kappa _{2V}$, is expected to be achieved from vector-boson fusion (VBF) production of a Higgs boson pair. However, this process involves another unmeasured parameter, the trilinear Higgs self-coupling modifier $\kappa _{\lambda }$. A sensitivity analysis should target both parameters. Since the LHC cannot avoid the gluon fusion pollution, which becomes severe for non-SM $\kappa _{\lambda }$, an electron-proton collider is more appropriate for the comprehensive measurement. In this regard, we study the VBF production of a Higgs boson pair in the $b\bar{b}b\bar{b}$ final state at the LHeC and FCC-he. Performing detailed analysis using the simulated dataset, we devise the search strategy specialized at the LHeC and FCC-he and give a prediction for the sensitivity to both $\kappa _{2V}$ and $\kappa _{\lambda }$. We find that the two electron-proton colliders have high potential: the LHeC has similar exclusion prospects as the HL-LHC; the FCC-he is extremely efficient, excluding the parameter space outside $\kappa _{2V} \in [0.8, 1.2]$ and $\kappa _{\lambda }\in [1, 2.5]$ at 95% C.L. for the total luminosity of $10$ ab$^{-1}$ and 10% uncertainty on the background yields.


Introduction
Albeit the absence of any signatures of the physics beyond the Standard Model (BSM), the journey to the final theory of the Universe will never stop. One important task to achieve the goal is to measure every coupling among the SM particles precisely, especially to the Higgs boson H. The Higgs coupling modifiers associated with a single Higgs boson have been observed to be SM-like at the LHC [1,2]. Their future projections at the high luminosity LHC (HL-LHC) expect the precisions at or below the percent level [3]. However, coupling modifiers involving a pair of Higgs bosons remain unmeasured, such as κ λ for the trilinear Higgs self-coupling and κ 2V for the quartic coupling between a Higgs boson pair and a vector boson pair. The κ λ shall be probed mainly from nonresonant Higgs boson pair (HH) production via gluon fusion: the triangle diagram mediated by the Higgs boson in the s-channel gives access to κ λ . It is found that if κ λ 1, non-trivial changes occur on both the shape and rate of the main kinematic distributions [4]. The current observed interval at the 95% confidence level (C.L.) is −5.0 < κ λ < 12.0 in the ATLAS analysis [5] and −11.8 < κ λ < 18.8 in the CMS analysis [6]. Several studies of the prospects for measuring κ λ at the HL-LHC and future colliders have been performed [7,8,9,10,11,12,13,14], which expect more stringent bounds.
The quartic coupling modifier κ 2V is much more challenging to measure at the LHC since the most efficient process, nonresonant HH production via vector boson fusion (VBF), has Email addresses: adil.hep@gmail.com (Adil Jueid), jinheung.kim1216@gmail.com (Jinheung Kim), soojinlee957@gmail.com (Soojin Lee), jhsong@konkuk.ac.kr (Jeonghyeon Song) very small cross-section of σ SM VBF (pp → HH j j) N 3 LO = 1.73 fb at √ s = 13 TeV [15] in addition to the huge SM backgrounds. The ATLAS collaboration performed the first search and excluded κ 2V < −0.76 and κ 2V > 2.90 at the 95% C.L. for κ V = 1 and κ λ = 1 [16]. The assumption of κ V = 1 is well motivated by the Higgs precision measurements at the LHC, but κ λ = 1 is questionable. The VBF production of HH, which also depends on κ λ via the H-mediated s-channel diagram, is susceptible to anomalous Higgs self-coupling (κ λ 1). For example, the cross-section for κ λ = 5 at the 14 TeV LHC is about twenty times that for κ λ = 1. More serious is the pollution from the gluon fusion production of HH associated with two jets, gg → HH j j [17,18]. This pollution also has the contribution from κ λ and greatly increases for κ λ 1. 1 Considering huge QCD uncertainties in the gluon fusion pollution [19], we expect an inevitable limitation to the precision measurement of κ 2V at the LHC.
Targeting the measurements of κ λ and κ 2V without the assumption about κ λ and thus the ambiguity of the gluon fusion pollution, we turn to two electron-proton colliders, the Large Hadron electron Collider (LHeC) [20,21,22] and the Future Circular Collider (FCC-he) [23]. The development of the energy recovery linac for the electron beam makes it possible to simultaneously operate the pp and e − p collisions. In particular, the LHeC has a bright outlook as its working group recently announced the default configuration and staging based on the cost estimation [22]. We find the following advantages of electronproton colliders in probing rare BSM events: • The pileup, which degrades the quality of the data for physics analyses, is very small even at the high luminosity option (∼ 10 34 /cm 2 /s): one expects about 0.1 (1) pileup collisions per event at the LHeC (FCC-he) while 150 at the LHC.
• The QCD backgrounds and the higher-order corrections are suppressed, providing a clean environment.
• The charged-current (CC) and neutral-current (NC) processes can be disentangled by tagging the outgoing neutrino (as large missing transverse energy) or electron. Independent measurements of κ 2W and κ 2Z are possible.
• The asymmetric initial state allows us to distinguish the forward and backward directions, which can increase the signal significance.
• High polarization of the electron beam, P e , is feasible, as large as ±80% [22]. The CC production cross-section increases by the factor of (1−P e ), while the NC cross-section does not change much.
For the configurations of FCC-he: E e = 60 GeV, E p = 50 TeV, we shall analyze the sensitivity of the LHeC and FCC-he to κ 2V and κ λ via the VBF production of HH through the CC channel. 2 Taking full advantage of the characteristics of the electronproton collider, we shall propose a search strategy which we believe is optimal for measuring κ 2V and κ λ . Finally, we will present the 95% C.L. exclusion in the (κ 2V , κ λ ) space, based on the detector-level analysis of the signals and the relevant backgrounds. The remainder of this letter is organized as follows.
In section 2, we discuss the formalism of Higgs boson pair production in e − p collisions within the κ-framework along with a discussion of the modeling of the signal and background processes. In section 3, we discuss the analysis strategy and present our results. We conclude in section 4.

Formalism and modeling for the signal and backgrounds
Based on the observed Higgs precision data via single Higgs production at the LHC, we assume that all the couplings to a single Higgs boson are the same as in the SM: where i and j are the SM particles. For renormalizable couplings to a Higgs boson pair, we consider where v 246 GeV. Note that κ 2V and κ λ parameterize the BSM interactions within the context of the non-linear effective field theory given by the electroweak chiral Lagrangian [26,27,28,29,30] 3 .
Aiming at the precision measurement of κ 2V and κ λ together, we focus on the pair production of Higgs bosons through the CC VBF interaction in the bbbb final state, where j f is a forward jet. There are three kinds of Feynman diagrams for this process, the contact one involving HHW + W − coupling, the s-channel involving HHH coupling, and the t, uchannels with the square of HW + W − coupling. The scattering amplitudes of W + W − → HH help us to understand the characteristics of the signal. As explicitly shown in Ref. [36], the longitudinally polarized W + L and W − L make an overwhelmingly dominant contribution. The corresponding amplitude, M LL , in the limit of √ s m H satisfies where we keep the notation of κ V to show its effects and θ * is the scattering angle in the center-of-mass frame of W + W − . Note that the effect of κ λ dominates in the small m HH (= √ s) region while that of κ 2V does in the high m HH region. First, at the parton level, we calculate the total crosssections of the signal by varying both κ 2V and κ λ . The calculations have been performed at leading order (LO) using MadGraph aMC@NLO with a modified UFO [37] model file for the Lagrangian in Eq. (3). Based on the current experimental bounds, we consider −1 ≤ κ 2V ≤ 3 [16] and −6 ≤ κ λ ≤ 12 [5,6]. The SM cross-section of the process pe − → HH j f ν e is very small: with the unpolarized electron beam, it is σ SM = 5.97 ab at the LHeC and σ SM = 233.77 ab at the FCC-he. Despite tiny SM signals, it is promising that the total cross-section rapidly increases when either κ 2V or κ λ deviates from their SM values: the two electron-proton colliders can exclude a large portion of the parameter space (κ 2V , κ λ ). To show this behavior, we present σ/σ SM in Fig. 1. It is clear to see that the deviation from κ 2V = 1 greatly increases the cross-section because it invalidates the cancellation of the longitudinal polarization enhancement, the first term of Eq. (5). The hypothesis of κ λ 1 also increases the signal cross-section, though less than that of κ 2V 1. Quantitatively, we have a tenfold increase of σ/σ SM if |κ 2V − 1| = |κ λ − 1| = 1. We also note that the same-sign κ 2V and κ λ yield constructive interference, explaining the negative slopes of the σ/σ SM contours: see the first two terms of Eq. (5). In detail, the LHeC and FCC-he show different shapes of the contours. As shall be demonstrated, the LHeC is more sensitive to κ λ than to κ 2V . The FCC-he has enough sensitivity to probe both. For the bbbb decay mode, the final state of the signal comprises at least four b-tagged jets, one light untagged jet, and large missing transverse energy (E miss T ). The main backgrounds 4 are the QCD multi-jets, diboson, tt, and single Higgs processes, all of which are associated with a forward jet j f and an electron neutrino ν e . In Table 1, we show the calculation of the LO cross-sections for the backgrounds at parton level using MadGraph aMC@NLO [38] with NNPDF31 lo parton distribution function (PDF) set [39]. Basic generator-level cuts were imposed on the parton-level objects like p j T > 5 GeV, ∆R j > 0.4, and |η j | < 10. The renormalization and factorization scales are set to be  The total cross-section of all the CC backgrounds is about 100 fb (751 fb) at the LHeC (FCC-he). The most dominant is the QCD production of bb j j, 5 where j refers to a light quark (including a charm quark) or a gluon. The second dominant backgrounds are from the production of a Z boson associated with another Z boson, the QCD bb, or a Higgs boson. The QCD production of four b quarks follows, and the production of a Higgs boson in association with bb is less critical. Finally, the contribution of a top quark pair production is smaller than the QCD 4b at the LHeC, but similar at the FCC-he. Important theoretical uncertainties arose from the scale variations, as shown in Table 1. PDF uncertainties are of order 1-2% for all the backgrounds. We close this section by summarizing the Monte Carlo event generation procedure. Initially, events for the signal and backgrounds are generated at LO using MadGraph aMC@NLO version 2.6.7. Parton luminosities were modeled with the NNPDF31 lo PDF set with α s (m 2 Z ) = 0.118. Setting the direction of the proton beam as forward, we convolute the partonic cross-sections with the PDFs in the LHAPDF6 library [41]. The decays of H, Z, and the top quark are modeled with MadSpin [42]. We confirmed that various kinematic distributions from on-shell samples using MadSpin well agree with those from the offshell samples. For the parton-showering and hadronization, we rely on Pythia6 [43] since Pythia8 does not support the LHE input in electron-proton collisions yet. To correctly model hadronization of the events, we modified the default Pythia6 setup. First, we switch off the lepton PDF by setting MSTP(11)=0. Second, we also switch off the QED initial state radiation for the electron beam by setting MSTP(61)=0. Finally, we switch off the negligible multiple-parton interactions, which saves a considerable amount of computing time. We use the default PDF at the Pythia6 level, CTEQ6l [44]. Fast detector simulation was performed using Delphes version 3.4.2 [45]. To match the particle efficiencies, momentum smearing, and isolation parameters with the default values in the Concept Design Report of the LHeC [22], we have performed minor modifications on the Delphes cards in the GitHub repository https:// github.com/delphes/delphes/tree/master/cards. Jets are clustered using the anti-k T algorithm [46] with a jet radius R = 0.4 in FastJet version 3.3.2 [47]. The b-tagging efficiency is set to be 70%. For the mistagging rates of the light and charm jets as a b jet, we adopted the default values in the above Delphes cards: at the LHeC, P j→b = 0.001 and P c→b = 0.05; at the FCC-he, P j→b = 0.001 and P c→b = 0.04 for |η| < 2.5, P j→b = 0.00075 and P c→b = 0.03 for 2.5 < |η| < 4.

Event selection
In this section, we update the ATLAS analysis strategy for the VBF production of HH [16], to optimize the signal significance at the LHeC and FCC-he. As summarized in Table 2, the event selections take the following steps: • Initial: The initial number of events, n 0 , is obtained from the full detector-level simulation. We consider the decays of H → bb, Z → bb, and both the semi-leptonic and hadronic decays of a top quark pair.
• 4b-tag: We require the presence of at least four b-tagged jets with p b T > 20 GeV and |η b | < 5. The acceptance times efficiency for the signal processes is around 10-16%, depending on the values of κ λ and κ 2V .
• Forward jet: We demand that at least one jet, untagged as a b jet, has p j f T > 20 GeV and 1.5 < η j f < 7. Note that the definition of being forward at asymmetric e − p colliders is different from that at the LHC. This selection reduces the signal events by about 20%, irrespective of the hypothesis of κ 2V and κ λ .

• Lepton veto:
We veto the events which contains an isolated lepton ( = e ± , µ ± ) with p T > 10 GeV and |η | < 5. The criteria of lepton isolation is required so that charged leptons from heavy hadron decays are not subject to this selection but their momenta are added to the hadronic jet if ∆R( , jet) < 0.2. Here ∆R ≡ ∆η 2 + ∆φ 2 . This selection is very effective in suppressing the NC backgrounds. The event yields for the signal and CC backgrounds remain almost the same.
• E miss T -cut: This selection consists of two requirements, E miss T > 40 GeV and |φ j f − φ E miss T | > 0.4. The latter removes the backgrounds with incorrectly measured E miss T . At this stage, the signal event yield is reduced by about 10%.

• Minimum D HH :
The mission here is to find two Higgs boson candidates from four b-tagged jets. There are three possible combinations for pairing two b-jets out of four, called the dijet. In each combination, we order two dijets according to their transverse momentum, and call them the 'leading' dijet and the 'sub-leading' dijet. Computing the angular separation of two b-jets inside each dijet system, ∆R lead and ∆R slead , we require where M 4b is the invariant mass of the four b-tagged jets. Among the pairings that satisfy Eq. (7), we choose the pairing with the smallest value of D HH as the final HH candidate. Here D HH is [16] where M lead dijet M slead dijet is the invariant mass of the leading (sub-leading) dijet system. The values of 116.5 GeV and 123.7 GeV are adopted to properly treat the energy loss in the semi-leptonic decays of the b-hadrons.
• X HH -cut: Finally, the signal region is defined by the following variable [16]: The ATLAS collaboration required X HH < 1.6 to maximize the LHC signal significance. To optimize the search at the LHeC and FCC-he, we present the differential cross-sections as a function of X HH for the LHeC and FCC-he in Fig. 2. The histograms in gray represent the total background distributions. We also show the signal results in six different hypotheses of (κ λ , κ 2V ) = {(−6, −1), (12,3), (0, 3), (1, 1), (−3, 0), (5, 0)} in green, blue, olive, red, purple, and cyan respectively. It is clear to see that the backgrounds are distributed in the high X HH region. We have calculated the signal significance, to be defined below, for different values of the upper-cut on X HH . We found that X HH < 3 (X HH < 2) at the LHeC (FCC-he) maximizes the signal significance, by which we   define the signal region. Note that especially at the LHeC, X HH < 3 allows significantly more data in the signal region than the LHC cut of X HH < 1.6, which partially offsets the weakness of the LHeC's having tiny signal events.

Results
In this section, we discuss the results of our analysis. After the full selection, the signal efficiency is about 2.0% for the LHeC and about 1.4% at the FCC-he, while the background efficiency is about O(10 −4 -10 −3 )% (see Table 2). To obtain the discovery potential, we compute the signal significance including the background uncertainty [48], defined by where N s is the number of signal events, N b is the number of background events, and δ b = ∆ bg N b is the uncertainty in the background yields. The numbers of the signal and background events are where L tot is the total integrated luminosity, X is the acceptance times efficiency for the process X in the signal region, and B X is the branching ratio of the decay X. Brief comments on the error estimation for the backgrounds are in order here. In principle, the background errors show different variation according to jet energy scale, the momentum smearing, b-tagging efficiency, jet energy resolution, and theoretical uncertainties.
Since the detailed study is beyond the scope of this work, we take two simple cases, ∆ bg = 10% and ∆ bg = 50%. 6 In Fig. 3, we display the expected exclusions on the plane of κ 2V and κ λ at the LHeC (upper panel) and the FCC-he (lower panel), corresponding to S > 2. We consider the electron beam polarization of P e = −80% and two cases of the background uncertainty, ∆ bg = 10% (solid) and ∆ bg = 50% (dashed). For the total integrated luminosity L tot , we take 1 ab −1 (olive) and 10 ab −1 (orchid) at the LHeC, and 0.1 ab −1 (olive), 1 ab −1 (orchid), and 10 ab −1 (blue) at the FCC-he. The common result of the LHeC and FCC-he is that the same-sign κ 2V and κ λ region is more strongly constrained because of the constructive interference discussed before.
In detail, the LHeC and FCC-he have different exclusion potential. In general, the LHeC has limitations in constraining κ 2V and κ λ because of its lower center-of-mass energy. Nevertheless, it can produce some meaningful results. If κ 2V = 1, the LHeC data with L tot = 1 ab −1 and ∆ bg = 10% can constrain κ λ as −3 κ λ 6, which is weaker than the HL-LHC prospect. If κ λ = 1, the LHeC data with L tot = 1 ab −1 and ∆ bg = 10% can exclude κ 2V −1 and κ 2V 3.2, which is compatible with the current bound on κ 2V ∈ [−0.66, 2.89] at 95% C.L. [16].  Considering the feasibility of the concurrent operation of the HL-LHC and LHeC, two colliders shall play a complementary role in probing κ 2V . In terms of the ratio of the cross-section of the CC VBF production of HH to the SM value, the LHeC with L tot = 1 ab −1 and ∆ bg = 10 (50)% can limit σ/σ SM 30 (35).
On the other hand, the FCC-he has high potential in probing both κ 2V and κ λ . For κ λ ∈ [0.1, 2.3] suggested by the HL-LHC prospect study [3], |κ 2V | 0.2 is to be excluded by the FCC-he data with L tot = 10 ab −1 and ∆ bg = 10%. Two important reasons for this high precision are higher signal cross-section and similar rejection rates of the SM backgrounds (see Table 2). At the FCC-he with ∆ bg = 10% (50%), we estimated conservative bounds on the ratio of the Higgs pair production cross section to the SM value as follows: (14) for L tot = 0.1 ab −1 , 3.5 (8) for L tot = 1 ab −1 , 1 (7) for L tot = 10 ab −1 .
Final comments on the role of higher electron beam energy in probing the HH process are in order here. Although it is practi- cal for the LHeC working group to choose E e = 50 GeV due to the cost issues, the physics gain from higher E e is more important than anything else. We found that setting E e = 120 GeV increases the background cross sections by a factor of 2.32 (1.82) at the LHeC (FCC-he). For the signal cross sections, the enhancement factor is 2.1 -2.7 at the LHeC and 4.4 -5.9 at the FCC-he, depending on the values of κ λ and κ 2V . Assuming similar efficiencies for both the signal and backgrounds to those in Table 2, we expect that the significance increases by a factor of 1.4 -1.8 (3.3 -4.4) at the LHeC (FCC-he). At the FCC-he, increasing E e into 120 GeV has almost the same effect as increasing the total luminosity tenfold. We strongly suggest that the FCC-he working group seriously consider the higher E e option.

Conclusions
Upon the current status where both the trilinear Higgs selfcoupling modifier (κ λ ) and the quartic coupling modifier between a Higgs boson pair and a vector boson pair (κ 2V ) are unmeasured, we consider two electron-proton colliders, the LHeC and FCC-he, in probing κ λ and κ 2V simultaneously. As a protonproton collider, the LHC cannot avoid the gluon fusion pollution in the VBF production of a Higgs pair, which becomes much worse for κ λ 1. At electron-proton colliders, the gluon fusion pollution is absent, and thus the charged-current VBF production of a Higgs boson pair can be solely measured if there is enough signal significance. With this motivation, we study the detailed phenomenology of pe − → HH jν e in the bbbb final state and suggest a search strategy at the LHeC and FCC-he based on the full simulation. Taking the default CDR values, we took E e = 50 (60) GeV and E p = 7 (50) TeV at the LHeC (FCC-he).
First, we calculated the parton-level cross-sections of the signal in the parameter space of (κ 2V , κ λ ) as well as all relevant backgrounds. Theoretical uncertainties from the variations of the scales and PDF are also calculated. Although the backgrounds are relatively manageable, the SM cross-section (κ 2V = κ λ = 1) is extremely small: without including the Higgs boson decays, σ SM = 5.97 ab at the LHeC and σ SM = 233.77 ab at the FCC-he for the unpolarized electron beam. It is very challenging to measure this process for the SM values of κ λ and κ 2V . What is hopeful is that a small deviation from κ λ = κ 2V = 1 greatly enhances the signal rate. The electron-proton collider can exclude a large portion of the (κ 2V , κ λ ) space.
We have completed the analysis with full simulations to devise an optimal strategy. We found that most of the current ATLAS search strategies for HH via VBF production apply to those at the LHeC and FCC-he. The key difference of ours is the cut on X HH , defined in Eq. (9). We found that the signal significance at the LHeC (FCC-he) is maximized by X HH < 3 (X HH < 2). A larger upper bound on X HH than the one for the LHC, X LHC HH < 1.6, increases the signal significance. As the final result, we calculated the expected exclusions on (κ 2V , κ λ ). The LHeC can play a meaningful role in probing κ 2V : the data with the total integrated luminosity of L tot = 1 ab −1 and the background uncertainty of ∆ bg = 10% can constrain −1 κ 2V 3.2 for κ λ = 1. The FCC-he has immense power in constraining both κ 2V and κ λ . If κ λ ∈ [0.1, 2.3] as the HL-LHC prospect, |κ 2V | 0.2 is to be excluded with L tot = 10 ab −1 and ∆ bg = 10%. We hope that this study would provide input to strongly support the future programs of electron-proton colliders which are capable of measuring two fundamental couplings, κ 2V and κ λ .