Physics Letters B 764 (2017) 247–253 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Probing anomalous couplings using di-Higgs production in electron–proton collisions Mukesh Kumar a,∗, Xifeng Ruan b, Rashidul Islam c, Alan S. Cornell a, Max Klein d, Uta Klein d, Bruce Mellado b a National Institute for Theoretical Physics, School of Physics and Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, Johannesburg, Wits 2050, South Africa b University of the Witwatersrand, School of Physics, Private Bag 3, Wits 2050, South Africa c Department of Physics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700009, India d Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom a r t i c l e i n f o a b s t r a c t Article history: Received 25 September 2016 Received in revised form 11 November 2016 Accepted 19 November 2016 Available online 24 November 2016 Editor: G.F. Giudice Keywords: Higgs boson Extensions of Higgs sector Particle production A proposed high energy Future Circular Hadron-Electron Collider would provide sufficient energy in a clean environment to probe di-Higgs production. Using this channel we show that the azimuthal angle correlation between the missing transverse energy and the forward jet is a very good probe for the non-standard hhh and hhW W couplings. We give the exclusion limits on these couplings as a function of integrated luminosity at a 95% C.L. using the fiducial cross sections. With appropriate error fitting methodology we find that the Higgs boson self coupling could be measured to be g(1) hhh = 1.00+0.24(0.14) −0.17(0.12) of its expected Standard Model value at √ s = 3.5(5.0) TeV for an ultimate 10 ab−1 of integrated luminosity. © 2016 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The 125 GeV particle discovered by the ATLAS and CMS exper- iments [1–5] has been established as a spin-0 Higgs boson rather than a spin-2 particle [6,7]. The measurements of its couplings to fermions and gauge bosons are being updated constantly and the results confirm consistency with the expected Standard Model (SM) values [4,8–10]. However, to establish that a scalar doublet � does indeed break the electroweak (EW) symmetry spontaneously when it acquires a nonzero vacuum expectation value, v , requires a direct measurement of the Higgs boson self coupling, λ. The min- imal SM, merely on the basis of the economy of fields and inter- actions, assumes the existence of only one physical scalar, h, with J P C = 0++ . Although Ref. [8] has ruled out a pure pseudoscalar hypothesis at a 99.98% confidence limit (C.L.), the new particle can still have a small CP-odd admixture to its couplings. Theoretically, the Higgs boson self coupling appears when, as a result of electroweak symmetry breaking in the SM, the scalar * Corresponding author. E-mail addresses: mukesh.kumar@cern.ch (M. Kumar), xifeng.ruan@cern.ch (X. Ruan), rashidul.islam@cern.ch (R. Islam), alan.cornell@wits.ac.za (A.S. Cornell), Max.Klein@liverpool.ac.uk (M. Klein), Uta.Klein@liverpool.ac.uk (U. Klein), bruce.mellado.garcia@cern.ch (B. Mellado). http://dx.doi.org/10.1016/j.physletb.2016.11.039 0370-2693/© 2016 The Author(s). Published by Elsevier B.V. This is an open access artic SCOAP3. potential V (�) gives rise to the Higgs boson self interactions as follows: V (�) = μ2�†� + λ(�†�)2 → 1 2 m2 hh2 + λvh3 + λ 4 h4, (1) where λ = λSM = m2 h/(2v2) ≈ 0.13 and � is an SU (2)L scalar dou- blet. For a direct and independent measurement of λ we need to access double Higgs boson production experimentally. However, this path is extremely challenging and requires a very high inte- grated luminosity to collect a substantial di-Higgs event rate, and an excellent detector with powerful background rejection capabil- ities. On the theoretical side we need to also take into account all vertices involved in the process that are sensitive to the presence of new physics beyond the SM (BSM). There are various proposals to build new, powerful high en- ergy e+e− , e− p and pp colliders in the future. We have based our study on a Future Circular Hadron-Electron Collider (FCC-he) which employs the 50 TeV proton beam of a proposed 100 km circular pp collider (FCC-pp), and electrons from an Energy Recovery Linac (ERL) being developed for the Large Hadron Electron Collider (LHeC) [11,12]. The design of the ERL is such that the e− p and pp colliders operate simultaneously, thus optimising the cost and the physics synergies between e− p and pp processes. Such facilities would be potent Higgs research centres, see e.g. Ref. [13]. The LHeC and the FCC-he configuration are advantageous with respect to the Large le under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by http://dx.doi.org/10.1016/j.physletb.2016.11.039 http://www.ScienceDirect.com/ http://www.elsevier.com/locate/physletb http://creativecommons.org/licenses/by/4.0/ mailto:mukesh.kumar@cern.ch mailto:xifeng.ruan@cern.ch mailto:rashidul.islam@cern.ch mailto:alan.cornell@wits.ac.za mailto:Max.Klein@liverpool.ac.uk mailto:Uta.Klein@liverpool.ac.uk mailto:bruce.mellado.garcia@cern.ch http://dx.doi.org/10.1016/j.physletb.2016.11.039 http://creativecommons.org/licenses/by/4.0/ http://crossmark.crossref.org/dialog/?doi=10.1016/j.physletb.2016.11.039&domain=pdf 248 M. Kumar et al. / Physics Letters B 764 (2017) 247–253 Fig. 1. Leading order diagrams contributing to the process p e− → hhjνe with q ≡ u, c, ̄d, ̄s and q′ ≡ d, s, ̄u, ̄c respectively. Hadron collider (LHC) (or FCC-pp in general) in terms of (1) initial states are asymmetric and hence backward and forward scatter- ing can be disentangled, (2) it provides a clean environment with suppressed backgrounds from strong interaction processes and free from issues like pile-ups, multiple interactions etc., (3) such ma- chines are known for high precision measurements of the dynami- cal properties of the proton allowing simultaneous test of EW and QCD effects. A detailed report on the physics and detector design concepts can be found in the Ref. [11]. The choice of an ERL energy of Ee = 60 to 120 GeV, with an available proton energy E p = 50 (7) TeV, would provide a centre of mass (c.m.s.) energy of √ s ≈ 3.5(1.3) to 5.0(1.8) TeV at the FCC-he (LHeC) using the FCC-pp (LHC) protons. The FCC-he would have sufficient c.m.s. energy to probe the Higgs boson self coupling via double Higgs boson production. The inclusive Higgs produc- tion cross section at the FCC-he is expected to be about five times larger than at the proposed 100 km circular e+e− collider (FCC-ee). This article is organised as follows: We discuss the process to produce the di-Higgs events in an e− p collider and the most general Lagrangian with all relevant new physics couplings in Sec- tion 2. In Section 3 all the simulation tools and the kinematic cuts that are required to study the sensitivity of the involved couplings are given. Here we also discuss the details of the analyses that has gone into the study. In Section 4 there is a discussion on the valid- ity of the effective theory considered here. And finally we conclude and draw inferences from the analysis in Section 5. 2. Formalism In an e− p collider environment a double Higgs event can be produced through: (1) the charged current process, p e− → hhjνe , and (2) the neutral current process, p e− → hhj e− , if there are no new physics processes involved. The SM background will cloud each of the processes greatly, and it will be a formidable task to separate signal from backgrounds. Here we study the charged cur- rent process because the signal strength of this is superior to the neutral current process. Hence we show in Fig. 1 the Higgs boson pair production, at leading order, due to the resonant and non- resonant contributions in charged current deep inelastic scattering (CC DIS) at an e− p collider. As seen in Fig. 1, the di-Higgs pro- duction involves hhh, hW W and hhW W couplings. Note that the hW W coupling will be extensively probed at the LHC, where its value conforms to the value predicted by the SM [4,8,10]. Through vector boson fusion Higgs production mode at the LHC, a BSM analyses to determine the CP and spin quantum numbers of the Higgs boson has been studied in Refs. [14–17]. The authors of Ref. [18] have shown the sensitivity of new physics contributions in hW W couplings at e− p colliders through a study of the az- imuthal angle correlation for single Higgs boson production in p e− → hjνe with an excellent signal-to-background ratio based on the h → bb̄ decay channel. Since we do not have any direct measurement of the Higgs boson self coupling (hhh) and quar- tic (hhW W ) coupling, there can be several possible sources of new physics in the scalar sector. This article studies for the pro- posed FCC-he sensitivity of the Higgs boson self coupling around its SM value including BSM contributions by considering all pos- sible Lorentz structures. In order to make it a complete study we also retain the possibilities for hW W couplings that appear in the di-Higgs production modes. Following Refs. [18,19] the most general Lagrangian which can account for all bosonic couplings relevant for the phenomenology of the Higgs boson sector at the FCC-he are the three-point and four-point interactions involving at least one Higgs boson field. It can be written as: L(3) hhh = m2 h 2v (1 − g(1) hhh)h 3 + 1 2v g(2) hhhh∂μh∂μh, (2) L(3) hW W = −g [ g(1) hW W 2mW W μν W † μνh + g(2) hW W mW (W ν∂μW † μνh + h.c) + g̃hW W 2mW W μν W̃ † μνh ] , (3) L(4) hhW W = −g2 [ g(1) hhW W 4m2 W W μν W † μνh2 + g(2) hhW W 2m2 W (W ν∂μW † μνh2 + h.c) + g̃hhW W 4m2 W W μν W̃ † μνh2 ] . (4) Here g(i) (··· ), i = 1, 2, and g̃(··· ) are real coefficients corresponding to the CP-even and CP-odd couplings respectively (of the hhh, hW W and hhW W anomalous vertices), Wμν = ∂μWν − ∂ν Wμ and W̃μν = 1 2 εμνρσ W ρσ . In Eq. (2) g(1) hhh is parametrised with a multiplicative constant with respect to λSM as in Eq. (1). Thus the Higgs self coupling λ appears as g(1) hhhλSM in the expression for V (�). Clearly, in the SM g(1) hhh = 1 and all other anoma- lous couplings vanish in Eqs. (2)–(4). The Lorentz structures of Eqs. (2)–(4) can be derived from the SU (2)L ⊗ U (1)Y gauge in- variant dimension-6 operators given in Ref. [19]. The complete Lagrangian we work with is as follows: L =LSM +L(3) hhh +L(3) hW W +L(4) hhW W . (5) The most general effective vertices take the form: hhh = −6λv [ g(1) hhh + g(2) hhh 3m2 h (p1 · p2 + p2 · p3 + p3 · p1) ] , (6) hW −W + = gmW [{ 1 + g(1) hW W m2 W p2 · p3 + g(2) hW W m2 W (p2 2 + p2 3) } ημ2μ3 − g(1) hW W m2 W pμ3 2 pμ2 3 − g(2) hW W m2 W (pμ2 2 pμ3 2 + pμ2 3 pμ3 3 ) − i g̃hW W m2 W εμ2μ3μν pμ 2 pν 3 ] , (7) hhW −W + = g2 [{ 1 2 + g(1) hhW W m2 W p3 · p4 + g(2) hhW W m2 W (p2 3 + p2 4) } ημ3μ4 − g(1) hhW W m2 W pμ4 3 pμ3 4 − g(2) hhW W m2 W (pμ3 3 pμ4 3 + pμ3 4 pμ4 4 ) − i g̃hhW W m2 W εμ3μ4μν pμ 3 pν 4 ] . (8) M. Kumar et al. / Physics Letters B 764 (2017) 247–253 249 Table 1 Cross sections of signal and backgrounds in charged current (cc), neutral current (nc) and photo-production (photo) modes for Ee = 60 GeV and E p = 50 TeV, where j is light quarks and gluons. For this estimation we use basic cuts |η| ≤ 10 for light-jets, leptons and b-tagged jets, pT ≥ 10 GeV, �Rmin = 0.4 for all particles. And electron polarisation is taken to be −0.8. Process cc (fb) nc (fb) photo (fb) Signal: 2.40 × 10−1 3.95 × 10−2 3.30 × 10−6 bb̄bb̄ j: 8.20 × 10−1 3.60 × 10+3 2.85 × 10+3 bb̄ j j j: 6.50 × 10+3 2.50 × 10+4 1.94 × 10+6 Z Z j (Z → bb̄): 7.40 × 10−1 1.65 × 10−2 1.73 × 10−2 tt̄ j (hadronic): 3.30 × 10−1 1.40 × 10+2 3.27 × 10+2 tt̄ j (semi-leptonic): 1.22 × 10−1 4.90 × 10+1 1.05 × 10+2 hbb̄ j (h → bb̄): 5.20 × 10−1 1.40 × 100 2.20 × 10−2 h Z j (Z ,h → bb̄): 6.80 × 10−1 9.83 × 10−3 6.70 × 10−3 The momenta and indices considered above are of the same order as they appear in the index of the respective vertex . For exam- ple, in the vertex hW −W + the momenta of h, W − and W + are p1, p2 and p3 respectively. Similarly, μ2 and μ3 are the indices of W − and W + . Using the above effective field theory (EFT) approach a study has been performed as an example for di-Higgs production in vector boson fusion at the LHC in Ref. [19]. 3. Simulation tools and analysis We begin our probe of the sensitivity of these couplings by building a model file for the Lagrangian in Eq. (5) using Feyn- Rules [19], and then simulate the charged current double Higgs boson production channel p e− → hhjνe (see Fig. 1), with h further decaying into a bb̄ pair,1 in the FCC-he set up with √ s ≈ 3.5 TeV. Our analysis starts with optimising the SM di-Higgs signal events with respect to all possible backgrounds from multi-jet events, Z Z + jets, hbb̄ + jets, h Z + jets and tt̄ + jets in charged and neutral current deep-inelastic scattering (DIS) and in photo-production,2 taking into account appropriate b-tagged jets and a high perfor- mance multipurpose 4π detector. In Table 1 we have given an esti- mation of cross sections for signal and backgrounds considering all possible modes with basic cuts. We then investigate the limits on each coupling taking BSM events as the signal. For the generation of events we use the Monte Carlo event generator MadGraph5 [24] and the CTEQ6L1 [25] parton distribution functions. Fur- ther fragmentation and hadronisation are done with a customised Pythia-PGS3 [26]. The detector level simulation is performed with reasonably chosen parameters using Delphes4 [27] and jets were clustered using FastJet [28] with the anti-kT algorithm [29] using the distance parameter, R = 0.4. The factorisation and renormalisation scales for the signal simulation are fixed to the 1 In pp collider like the LHC, the main challenge of this search is to distinguish the signal of four bottom quarks in the final state (that hadronise into jets (b-jets)) from the QCD multijet backgrounds. Such challenges and difficulties are discussed and performed in ATLAS and CMS studies [20–22]. 2 We cross checked the modelling of photo-production cross sections from Mad- Graph5 by switching on the “Improved Weizsäcker–Williams approximation for- mula” described in Ref. [23] to give the probability of photons from the incoming electron, versus the expectation of the Pythia Monte Carlo generator. 3 In Pythia-PGS we modified several parameters in a way to use it for e− p collision and to get all required numbers of events demanded in each simulation. The coordinate system is set as for the HERA experiments, i.e. the proton direction defines the forward direction. The modifications have been successfully validated using neutral current DIS events and switched off QCD ISR. For e− p collisions mul- tiple interactions and pile-up are expected to be very small and are switched off in our studies. 4 For Delphes we used the ATLAS set-up with the modifications in the |η| ranges for forward and b-tagged jets up to 7 and 5 respectively with 70% tagging efficiency of b-jets as mentioned in the text. The resolution parameters for energy deposits in the calorimeters are based on the ATLAS Run-1 performance. Higgs boson mass, mh = 125 GeV. The background simulations are done with the default MadGraph5 dynamic scales. The e− polari- sation is assumed to be −80%. 3.1. Cut-based optimisation We base our simulation on the following kinematic selections in order to optimise the significance of the SM signal over all the backgrounds: (1) At least four b-tagged jets and one addi- tional light jet are selected in an event with transverse momenta, pT , greater than 20 GeV. (2) For non-b-tagged jets, the absolute value of the rapidity, |η|, is taken to be less than 7, whereas for b-tagged jets it is less than 5. (3) The four b-tagged jets must be well separated and the distance between any two jets, defined as �R = √ (�φ)2 + (�η)2, φ being the azimuthal angle, is taken to be greater than 0.7. (4) Charged leptons with pT > 10 GeV are re- jected. (5) For the largest pT forward jet J (the non-b-tagged jet after selecting at least four b-jets) η J > 4.0 is required. (6) The missing transverse energy, /E T , is taken to be greater than 40 GeV. (7) The azimuthal angle between /E T and the b-tagged jets are: ��/E T , leading jet > 0.4 and ��/E T , sub−leading jet > 0.4. (8) The four b-tagged jets are grouped into two pairs such that the distances of each pair to the true Higgs mass are minimised. The leading mass contains the leading pT -ordered b-jet. The first pair is required to be within 95–125 GeV and the second pair within 90–125 GeV.5 (9) The invariant mass of all four b-tagged jets has to be greater than 290 GeV. In the selections (described above) the b-tagging efficiency is assumed to be 70%, with fake rates from c-initiated jets and light jets to the b-jets of 10% and 1% respectively. Corresponding weights6 at a particular luminosity of 10 ab−1 for a signal, and all backgrounds with significance has been tabulated in Table 2. Sig- nificance at all stages of the cuts are calculated using the Poisson formula S = √ 2[(S + B) log(1 + S/B) − S], where S and B are the expected signal and background yields at a particular luminosity respectively. From Table 2 we recognise that selection on the forward jet in the FCC-he type machine plays a very significant role in distin- guishing the signal with respect to background. By selecting events with η J > 4.0, there is loss of 25% of signal events, while the total background loss is around 80%. The next significant cut on miss- ing energy (/E T > 40 GeV) is also very significant as due to this cut there is loss of 88% of events in total background, however, for the signal there is a loss of only 34% of events after forward jet selection. Furthermore the mass window cut for the invariant mass of two b-tagged jets, after the �φ/E T j > 0.4 selection, sig- nificantly reduces the total background events to 5%, only while the signal events remains at 40%. Efficient requirements on the invariant mass window of four b-jets are efficient, such that to re- duce backgrounds by 44% leads to a signal of 90% with respect to the previous two b-tagged jet mass window selection. And hence there is a 20% enhancement in the significance obtained in com- parison to the two different mass window selection criteria, and overall with respect to initial events this cut-based optimisation is 5 Among the four b-tagged jets, choices of pairing have been performed via ap- propriate selection of mass window, keeping in mind to reconstruct the Higgs boson mass, mh , in the signal as well as the Z -boson mass, mZ , in the backgrounds. We choose the pair in which the quadratic sum (m1 − mc ) and (m2 − mc ) is smallest, and in each mass mi , mass m1 has the largest pT b-jet, mc = (mh − m0) GeV, and normally m0 ≈ 20–40 GeV (which is not important, since the false pairing will have a much higher quadratic sum). 6 Here weights mean the number of expected events at a particular luminosity. The number of events of the photo-production of 4b + jets are derived using the efficiencies of the Monte Carlo samples due to the low statistics. The other back- grounds are obtained directly from the event selection. 250 M. Kumar et al. / Physics Letters B 764 (2017) 247–253 Table 2 A summary table of event selections to optimise the signal with respect to the backgrounds in terms of the weights at 10 ab−1. In the first column the selection criteria are given as described in the text. The second column contains the weights of the signal process p e− → hhjνe , where both the Higgs bosons decay to bb̄ pair. In the next columns the sum of weights of all individual prominent backgrounds in charged current, neutral current and photo-production are given with each selection, whereas in the penultimate column all backgrounds’ weights are added. The significance is calculated at each stage of the optimised selection criteria using the formula S =√ 2[(S + B) log(1 + S/B) − S], where S and B are the expected signal and background yields at a luminosity of 10 ab−1 respectively. This optimisation has been performed for Ee = 60 GeV and E p = 50 TeV. Cuts/Samples Signal 4b + jets 2b + jets Top Z Z bb̄H Z H Total Bkg Significance Initial 2.00 × 103 3.21 × 107 2.32 × 109 7.42 × 106 7.70 × 103 1.94 × 104 6.97 × 103 2.36 × 109 0.04 At least 4b + 1 j 3.11 × 102 7.08 × 104 2.56 × 104 9.87 × 103 7.00 × 102 6.32 × 102 7.23 × 102 1.08 × 105 0.94 Lepton rejection p� T > 10 GeV 3.11 × 102 5.95 × 104 9.94 × 103 6.44 × 103 6.92 × 102 2.26 × 102 7.16 × 102 7.75 × 104 1.12 Forward jet η J > 4.0 233 13007.30 2151.15 307.67 381.04 46.82 503.22 16397.19 1.82 /E T > 40 GeV 155 963.20 129.38 85.81 342.18 19.11 388.25 1927.93 3.48 �φ/E T j > 0.4 133 439.79 61.80 63.99 287.10 14.53 337.14 1204.35 3.76 m1 bb ∈ [95,125], m2 bb ∈ [90,125] 54.5 28.69 5.89 6.68 5.14 1.42 17.41 65.23 6.04 m4b > 290 GeV 49.2 10.98 1.74 2.90 1.39 1.21 11.01 29.23 7.51 Fig. 2. Azimuthal angle distributions, at Delphes detector-level, between missing transverse energy, /E T , and the forward jet, J, in the SM (including backgrounds) and with the anomalous hhh, hW W and hhW W couplings. The error bars are statistical. enhanced from a 0.04 to 7.51 significance. Here it is also impor- tant to mention that photo-production of the 4b final state is one of the main background with similar topological final states from Zh, where Z , h → bb̄, and is equally important. Hence choice of efficient selection criteria is too important to reduce these back- grounds. 3.2. Kinematic distributions and observable For our analysis we take ad hoc values of positive and negative couplings in such a manner that the production cross section does not deviate much from the SM value, and in particular modifica- tions in the shapes of the normalised azimuthal angle distribution between the missing transverse energy and the leading (forward) jet are studied, in addition to other kinematic distributions. Taking into account all the above criteria we study BSM mod- ifications in various differential distributions at the Delphes detector-level. This leads to the following observations: (1) pT has the usual tail behaviour, i.e. the number of events are more popu- lated in the higher pT region with respect to the SM for the chosen values of the anomalous couplings. (2) In cases of the η distri- butions: (a) For the forward jet, particularly for the couplings of hW W and hhW W vertices, the mean η is more central in the de- tector. The behaviour is similar if we increase the c.m.s. energy of the collider by increasing Ee to higher (> 60 GeV) values. For hhh couplings the η distribution remains the same as for the SM. (b) In case of b-tagged jets, for all values of anomalous couplings, the dis- tribution is populated around the value of η of the SM distribution. (3) For the specific observable of the azimuthal angle difference between missing transverse energy and the forward jet (�φ/E T J ) the shapes are clearly distinguishable from the SM. This behaviour is shown in Fig. 2, where the values of the couplings are ad hoc. However, these values are taken only for the purpose of illustra- tion, and in the limit of the couplings going to their SM values the shapes will coincide with the SM distributions. The specific char- acteristics of the curves also depend on the details of the selection requirements, but the qualitative differences could be seen at ev- ery selection step. The shape of the curves is due to the fact that all new physics couplings have a momentum dependent structure (apart from g(1) hhh) and positive or negative interference with SM events. We note that �φ/E T J is a novel observable and commands more focused and deeper analyses. In this regard one should follow the analysis (as performed in Ref. [30]) based on an asymmetry with two equal bins in �φ/E T J ≶ π/2, defined as A�φ/E T J = |A�φ>π/2| − |A�φ<π/2| |A�φ>π/2| + |A�φ<π/2| , (9) in which A is the yields obtained for the given ab−1 data after all of the selections, including both the signal and backgrounds. Table 3 shows the estimation of the asymmetry for a set of rep- resentative values of the couplings, shown in Fig. 2, along with the associated statistical uncertainty. Though the new physics cou- plings are representative, we can infer the sensitivities of these couplings with respect to the SM+Bkg estimation of asymmetry from Table 3, where g(1) hW W seems to have large fluctuations for both positive and negative choices of its values. Similarly the sen- sitivities of g(2) hhW W and g̃hhW W can also be noted, however g(1) hhW W is more sensible for the negative choice of its value. The study of the sensitivity of the non-standard couplings through this asym- M. Kumar et al. / Physics Letters B 764 (2017) 247–253 251 Table 3 Estimation of the asymmetry, defined in Eq. (9), and statistical error associated with the kinematic distributions in Fig. 2 at an integrated luminosity of 10 ab−1. The cross section (σ ) for the corresponding coupling choice is given in the last column with same parameters as in Table 1. Samples A�φ/E T J σ(fb) SM+Bkg 0.277 ± 0.088 g(1) hhh = 1.5 0.279 ± 0.052 0.18 = 2.0 0.350 ± 0.053 0.21 g(2) hhh = −0.5 0.381 ± 0.050 0.19 = 0.5 0.274 ± 0.024 0.74 g(1) hW W = −0.5 0.506 ± 0.022 0.88 = 0.5 0.493 ± 0.020 0.94 g(2) hW W = −0.02 0.257 ± 0.025 0.67 = 0.02 0.399 ± 0.040 0.33 g̃hW W = −1.0 0.219 ± 0.016 1.53 = 1.0 0.228 ± 0.016 1.53 g(1) hhW W = −0.05 0.450 ± 0.033 0.52 = 0.05 0.254 ± 0.029 0.68 g(2) hhW W = −0.03 0.462 ± 0.022 1.22 = 0.03 0.333 ± 0.018 1.46 g̃hhW W = −0.1 0.351 ± 0.020 1.60 = 0.1 0.345 ± 0.020 1.61 metry observable considering a kinematic distribution is basically corresponding to two bins by dividing the whole distribution in two halves with large bin-width. Moreover, this kind of study can be further appended with finer bin widths and a more efficient χ2 analysis (for example in Ref. [30]). However, these detailed analy- ses are beyond the scope of this article. 3.3. Exclusion limits through fiducial cross section as a function of luminosity Furthermore, we probe the exclusion limits on these couplings as a function of the integrated luminosity, with the log-likelihood method described in Ref. [31], using directly the fiducial inclu- sive cross section as an observable. In Fig. 3 we present exclusion plots at 95% C.L. for anomalous hhh, hW W and hhW W couplings, where the shaded areas are the allowed region. The exclusion limits are based on the SM ‘di-Higgs signal + backgrounds’ hy- potheses considering BSM contributions as the signal at the given luminosity. Each limit is given by scanning one coupling and fixing the other couplings to their SM value, where a 5% systematic un- certainty is taken into account on the signal and background yields respectively. From Fig. 3 our observations are as follows: (1) If the integrated luminosity exceeds 0.5 ab−1 g(1) hhh is restricted to be pos- itive. g(1) hhh is allowed to be within 0.7–2.5 when the integrated lu- minosity reaches 15 ab−1 as for values of 1 < g(1) hhh ≤ 2.1 the cross section is smaller than the SM di-Higgs production. (2) The g(2) hhh value is restricted to around 10−1. We only exclude the positive part of this coupling because its negative part has cancellations with the SM di-Higgs cross section. (3) The sensitivity for hW W couplings, namely g(1) hW W and g̃hW W , can be better probed at much lower energies and luminosity at the LHeC using the single Higgs boson production as shown in Ref. [18]. However, we have shown the sensitivity of g(2) hW W , which is not considered in Ref. [18], and it is of the order 10−2 in the allowed region. (4) One important as- pect of di-Higgs production in this type of collider is that one can measure the sensitivity of the hhW W couplings also. In our anal- ysis, since the CP-even (odd) coupling g(1) hhW W ( g̃hhW W ) has similar Lorentz structures, with the sensitivity of the exclusion plot hav- ing almost the same order of magnitude. However, the structure Fig. 3. The exclusion limits on the anomalous hhh (top panel), hW W (middle panel) and hhW W (lower panel) couplings at 95% C.L. as a function of integrated luminos- ity (shaded areas). Note that the allowed values of g(2) hhh and g(2) hW W are multiplied by 5 and 10 respectively to highlight their exclusion region, since the values are of the order 10−1. of g(2) hhW W allows a comparatively narrower region of values. The couplings belonging to both the hW W and hhW W vertices are strongly constrained because of their high production cross section at very low values of the couplings. By increasing the luminosity from 0.1–1 ab−1 the constraint on the couplings increases and its limits are reduced by a factor two. A further increase of the lumi- nosity will not change the results. All limits are derived by varying only one coupling at a time, as mentioned earlier. The exclusion limits on the couplings in this analysis are based on the constraints from an excess above the SM expectation while potential deficits from interference contributions are not sensitive yet to be used for limit settings. 3.4. Prospects at higher Ee and sensitivity of the Higgs self coupling Finally we discuss what happens once the electron energy Ee is increased to higher values, where we focus our analysis on a deter- mination of the SM Higgs self coupling, assuming no further BSM contributions. Without going into detail we can note that with in- creasing Ee (from 60 GeV to 120 GeV) the SM signal and dominant background production cross sections are enhanced by a factor of 2.2 and 2.1 respectively. As a result, the cut efficiency for the se- lection of four b-tagged jets and one forward jet is improved, but for the other cuts described previously (invariant mass, /E T , η J and �φ/E T j ) it remains very similar. This leads to an enhancement of the selected signal and dominant background events by a factor 2.5 and 2.6 respectively. Hence we would obtain the same statis- tical precision with only 40% of the luminosity of an Ee = 60 GeV beam when increasing the electron energy to 120 GeV. At an ul- timate integrated luminosity of 10 ab−1, increasing Ee from 60 to 120 GeV would increase the significance of the observed SM di- Higgs events from 7.2 to 10.6, obtained from a log-likelihood fit. This includes a 5% signal and background systematics mentioned earlier. For the SM Higgs boson self coupling, where the scaling factor is expected to be g(1) = 1, we perform an intelligent signal hhh 252 M. Kumar et al. / Physics Letters B 764 (2017) 247–253 injection test, which gives locally measured uncertainties for g(1) hhh . From this test the 1σ error band around the expected SM strength of this coupling is g(1) hhh = 1.00+0.24(0.14) −0.17(0.12) for Ee = 60(120) GeV. 4. Validity of EFT In the EFT-based approach for our analyses, the usual SM La- grangian is supplemented by higher-dimensional operators that parametrise the possible effects of new states assumed to appear at energies larger than the effective scales identified with mW (or equivalently v) by restricting the operators of dimension less than or equal to six. We have estimated the sensitivity of the in- volved coupling coefficients appearing in the effective Lagrangians in Eqs. (2)–(4) with the EW scale for the derivative terms. A de- tailed discussion with general couplings and mass scales with a higher dimension EFT Lagrangian can be found in Ref. [32]. For the processes at high energy, it is well known that an EFT approach provides an accurate description of the underlying new physics as long as the energies are below the new physics scale, �, and thus the limits on the couplings obtained in the above analyses shall degrade at scales higher than the EW scales (since for the fixed values of the couplings, the interference and pure BSM terms al- ways give low contributions in the cross section measurements for high values of scale choice). Also for O(1) values of anomalous couplings (apart from g(1) hhh) g(i) (··· ) , g̃(··· ) and TeV-scale momenta, one reaches the regime where the operators in Eqs. (2)–(4) may not be dominant, and operators with four and more derivatives may be equally important. In other words, the EFT behind these Lagrangian expansions breaks down. It would be important then to know how much the projected sensitivity depends on events that violate this EFT bound. With an EW precision test in Ref. [33], it is shown how an EFT’s reach deteriorates when only data below the cutoff scales are employed on the mass variable in the case of Drell–Yan processes at the LHC. A similar exercise can be performed in our case to estimate the deterioration of limits on anomalous tensorial couplings g(i) (··· ) and g̃(··· ) (the coupling coefficients which corresponds to momentum- dependent Lorentz structure) as a function of the cut-off scale. In this approach we put an upper cut on the di-Higgs invariant mass (m′ 4b)7 such that EFT-violating events (> m′ 4b) are cut away, and then we estimate by how much the projected sensitivity of g(2) hhh , g(1,2) hW W , g(1,2) hhW W and g̃hW W , g̃hhW W degrades with respect to their previous limits. In Fig. 4 we present the percentage of deteriora- tion of the exclusion limits of these anomalous effective couplings by selecting events below m′ 4b ∈ [0.35, 1] TeV for fixed luminos- ity of 1 ab−1 and 10 ab−1 at 95% C.L. It is apparent from Fig. 4 that the deterioration in the limits of these anomalous couplings is large for low values of the m′ 4b cut, because the effective cross sec- tion decreases (which is equivalent to the increase of the scale � of the tensorial couplings) with the decrease of the values of m′ 4b . Comparing the exclusion limits obtained in Fig. 3 we observed that at m′ 4b = 350 GeV the percentage of deterioration in g(2) hhh is more than 100%, while other hW W and hhW W couplings deteriorate by 60–80% on both upper and lower sides at 1 ab−1 and 10 ab−1. After 350 GeV a sudden decrease in degradation percentage can be noticed for m′ 4b = 400–450 GeV for all couplings. Furthermore 7 Note that in previous subsections we used the notation m4b for the lowest cut on di-Higgs invariant mass. Here we use m′ 4b to avoid confusion since for all anal- yses apart from the EFT validity we selected the events m4b > 290 GeV to suppress backgrounds and increase the overall significance. Here, to investigate the sensi- tivity of BSM tensorial couplings, we chose the events below m′ 4b cuts, keeping m4b > 290 GeV so that an one to one comparison can be performed. Fig. 4. Percentage of deterioration of exclusion limits of anomalous tensorial cou- plings (shown in Fig. 3) with respect to the upper di-Higgs invariant mass cut m′ 4b ≡ mcut 4b [in GeV] for fixed luminosity of 1 ab−1 (blue) and 10 ab−1 (red). The numbers in the vertical axis above (below) 0 is the degradation in the upper (lower) limits. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) around 500 GeV for g(2) hhh , it remains 18% while others are around 10%. Beyond a 650 GeV cut, all the couplings converge to the orig- inal value of limits obtained in our previous analyses, as shown in Fig. 3. 5. Summary and conclusions We conclude that the FCC-he, with an ERL energy of Ee ≥ 60 GeV and a proton energy E p = 50 TeV, would provide signif- icant di-Higgs event rates, and through this channel one can probe accurately the Higgs boson self coupling provided that integrated luminosities of more than 1 ab−1 may be achieved. Along with the Higgs self coupling one can search for any BSM signal through the measurement of the anomalous hhW W contributions. One inter- esting feature of this type of machine is recognised by identifying forward jets in the signal events where an appropriate selection, as shown for our study, reduces backgrounds efficiently around 80% with a loss of only 25% of signal events. Our work also shows that �φ/E T j is a very good observable for any new physics contribu- tions in the given channel. Estimation of an asymmetry observable in �φ/E T j for this kinematic distribution gives a preliminary idea of sensitivities of any new non-standard couplings. The limits on each coupling are set by measuring the observed event rate. But the asymmetry in �φ/E T j can provide more distinguishability of the new physics, especially cancelling many potential systematics, which is helpful to distinguish the signatures of each model. An exclusion limit with respect to luminosity for these couplings is studied, and a signal injection test shows the uncertainty of the Higgs self coupling around its expected SM strength. With all these analyses we infer that the order of sensitivities of all non-standard couplings considered for our study within most of the luminosity ranges are consistent with the adopted methodol- ogy of asymmetry observable, and exclusion limits through fiducial cross sections at 95% C.L. However, at luminosities ∼10–15 ab−1 M. Kumar et al. / Physics Letters B 764 (2017) 247–253 253 or higher, the method based on fiducial cross sections constrains the non-standard couplings more tightly. In addition to the fidu- cial cross sections, the drastic change in �φ/E T j shape for g(1) hhh around 2.0 with respect to the SM in Fig. 2 (left), suggest that us- ing further observables like �φ/E T j may significantly improve the sensitivity of BSM couplings in general. It is to be noted that the non-standard momentum dependent structures of the EFT breaks down at the TeV energy regime for couplings of O(1) and then additional derivative terms become relevant. Hence we also show the deteriorations in the limits of anomalous tensorial couplings for different regions of di-Higgs invariant mass (upper) cuts by an exclusion at fixed luminosity of 1 ab−1 and 10 ab−1 corresponding to 95% C.L. limits, with respect to the limits obtained using fiducial inclusive cross section as an observable. This method is used as an alternative approach to estimate the sensitivity of the scale depen- dent couplings in EFT and gives a probe to understand the regions where validity of EFT breaks down. Our studies show a unique capability and potential of FCC-he collider to probe the precision measurement of not only the Higgs boson self coupling but also other involved couplings with tenso- rial structure through di-Higgs boson production. Acknowledgements We acknowledge fruitful discussions within the LHeC Higgs group, especially with Masahiro Kuze and Masahiro Tanaka. RI ac- knowledges the DST-SERB grant SRlS2/HEP-13/2012 for partial fi- nancial support. References [1] G. Aad, et al., Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1–29, http://dx.doi.org/10.1016/j.physletb.2012.08.020, arXiv:1207.7214. [2] S. Chatrchyan, et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30–61, http://dx.doi. org/10.1016/j.physletb.2012.08.021, arXiv:1207.7235. [3] G. Aad, et al., Measurement of the Higgs boson mass from the H → γ γ and H → Z Z∗ → 4� channels with the ATLAS detector using 25 fb−1 of pp collision data, Phys. Rev. D 90 (5) (2014) 052004, http://dx.doi.org/10.1103/PhysRevD.90. 052004, arXiv:1406.3827. [4] V. Khachatryan, et al., Precise determination of the mass of the Higgs boson and tests of compatibility of its couplings with the standard model predic- tions using proton collisions at 7 and 8 TeV, Eur. Phys. J. C 75 (5) (2015) 212, http://dx.doi.org/10.1140/epjc/s10052-015-3351-7, arXiv:1412.8662. [5] G. Aad, et al., Combined measurement of the Higgs boson mass in pp col- lisions at √ s = 7 and 8 TeV with the ATLAS and CMS experiments, Phys. Rev. Lett. 114 (2015) 191803, http://dx.doi.org/10.1103/PhysRevLett.114.191803, arXiv:1503.07589. [6] S. Chatrchyan, et al., Study of the mass and spin-parity of the Higgs boson can- didate via its decays to Z boson pairs, Phys. Rev. Lett. 110 (8) (2013) 081803, http://dx.doi.org/10.1103/PhysRevLett.110.081803, arXiv:1212.6639. [7] G. Aad, et al., Evidence for the spin-0 nature of the Higgs boson using AT- LAS data, Phys. Lett. B 726 (2013) 120–144, http://dx.doi.org/10.1016/j.physletb. 2013.08.026, arXiv:1307.1432. [8] V. Khachatryan, et al., Constraints on the spin-parity and anomalous HVV cou- plings of the Higgs boson in proton collisions at 7 and 8 TeV, Phys. Rev. D 92 (1) (2015) 012004, http://dx.doi.org/10.1103/PhysRevD.92.012004, arXiv: 1411.3441. [9] G. Aad, et al., Study of the spin and parity of the Higgs boson in di- boson decays with the ATLAS detector, Eur. Phys. J. C 75 (10) (2015) 476, http://dx.doi.org/10.1140/epjc/s10052-015-3685-1, Erratum: Eur. Phys. J. C 76 (3) (2016) 152, http://dx.doi.org/10.1140/epjc/s10052-016-3934-y, arXiv: 1506.05669. [10] G. Aad, et al., Measurements of the Higgs boson production and decay rates and coupling strengths using pp collision data at √s = 7 and 8 TeV in the ATLAS experiment, Eur. Phys. J. C 76 (1) (2016) 6, http://dx.doi.org/10.1140/ epjc/s10052-015-3769-y, arXiv:1507.04548. [11] J.L. Abelleira Fernandez, et al., A large hadron electron collider at CERN: report on the physics and design concepts for machine and detector, J. Phys. G 39 (2012) 075001, http://dx.doi.org/10.1088/0954-3899/39/7/075001, arXiv:1206. 2913. [12] O. Bruening, M. Klein, The large hadron electron collider, Mod. Phys. Lett. A 28 (16) (2013) 1330011, http://dx.doi.org/10.1142/S0217732313300115, arXiv: 1305.2090. [13] J.L. Abelleira Fernandez, et al., On the relation of the LHeC and the LHC, arXiv: 1211.5102. [14] T. Plehn, D.L. Rainwater, D. Zeppenfeld, Determining the structure of Higgs cou- plings at the LHC, Phys. Rev. Lett. 88 (2002) 051801, http://dx.doi.org/10.1103/ PhysRevLett.88.051801, arXiv:hep-ph/0105325. [15] K. Hagiwara, Q. Li, K. Mawatari, Jet angular correlation in vector-boson fusion processes at hadron colliders, J. High Energy Phys. 07 (2009) 101, http://dx. doi.org/10.1088/1126-6708/2009/07/101, arXiv:0905.4314. [16] A. Djouadi, R.M. Godbole, B. Mellado, K. Mohan, Probing the spin-parity of the Higgs boson via jet kinematics in vector boson fusion, Phys. Lett. B 723 (2013) 307–313, http://dx.doi.org/10.1016/j.physletb.2013.04.060, arXiv:1301.4965. [17] C. Englert, D. Goncalves-Netto, K. Mawatari, T. Plehn, Higgs quantum numbers in weak boson fusion, J. High Energy Phys. 01 (2013) 148, http://dx.doi.org/10. 1007/JHEP01(2013)148, arXiv:1212.0843. [18] S.S. Biswal, R.M. Godbole, B. Mellado, S. Raychaudhuri, Azimuthal angle probe of anomalous H W W couplings at a high energy ep collider, Phys. Rev. Lett. 109 (2012) 261801, http://dx.doi.org/10.1103/PhysRevLett.109.261801, arXiv:1203. 6285. [19] A. Alloul, B. Fuks, V. Sanz, Phenomenology of the Higgs effective Lagrangian via FEYNRULES, J. High Energy Phys. 04 (2014) 110, http://dx.doi.org/10.1007/ JHEP04(2014)110, arXiv:1310.5150. [20] G. Aad, et al., Search for Higgs boson pair production in the bb̄bb̄ final state from pp collisions at √s = 8 TeV with the ATLAS detector, Eur. Phys. J. C 75 (9) (2015) 412, http://dx.doi.org/10.1140/epjc/s10052-015-3628-x, arXiv:1506. 00285. [21] G. Aad, et al., Searches for Higgs boson pair production in the hh → bbττ , γ γ W W ∗, γ γ bb, bbbb channels with the ATLAS detector, Phys. Rev. D 92 (2015) 092004, http://dx.doi.org/10.1103/PhysRevD.92.092004, arXiv:1509. 04670. [22] V. Khachatryan, et al., Search for resonant pair production of Higgs bosons decaying to two bottom quark antiquark pairs in proton proton collisions at 8 TeV, Phys. Lett. B 749 (2015) 560–582, http://dx.doi.org/10.1016/j.physletb. 2015.08.047, arXiv:1503.04114. [23] V.M. Budnev, I.F. Ginzburg, G.V. Meledin, V.G. Serbo, The two photon parti- cle production mechanism. Physical problems. Applications. Equivalent photon approximation, Phys. Rep. 15 (1975) 181–281, http://dx.doi.org/10.1016/0370- 1573(75)90009-5. [24] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, MadGraph 5: go- ing beyond, J. High Energy Phys. 06 (2011) 128, http://dx.doi.org/10.1007/ JHEP06(2011)128, arXiv:1106.0522. [25] J. Pumplin, D.R. Stump, J. Huston, H.L. Lai, P.M. Nadolsky, W.K. Tung, New generation of parton distributions with uncertainties from global QCD analy- sis, J. High Energy Phys. 07 (2002) 012, http://dx.doi.org/10.1088/1126-6708/ 2002/07/012, arXiv:hep-ph/0201195. [26] T. Sjostrand, S. Mrenna, P.Z. Skands, PYTHIA 6.4 physics and manual, J. High Energy Phys. 05 (2006) 026, http://dx.doi.org/10.1088/1126-6708/2006/05/026, arXiv:hep-ph/0603175. [27] J. de Favereau, C. Delaere, P. Demin, A. Giammanco, V. Lemaître, A. Mertens, M. Selvaggi, DELPHES 3, a modular framework for fast simulation of a generic collider experiment, J. High Energy Phys. 02 (2014) 057, http://dx.doi.org/ 10.1007/JHEP02(2014)057, arXiv:1307.6346. [28] M. Cacciari, G.P. Salam, G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896, http://dx.doi.org/10.1140/epjc/s10052-012-1896-2, arXiv:1111.6097. [29] M. Cacciari, G.P. Salam, G. Soyez, The Anti-k(t) jet clustering algorithm, J. High Energy Phys. 04 (2008) 063, http://dx.doi.org/10.1088/1126-6708/2008/04/063, arXiv:0802.1189. [30] S. Dutta, A. Goyal, M. Kumar, B. Mellado, Measuring anomalous W tb couplings at e− p collider, Eur. Phys. J. C 75 (12) (2015) 577, http://dx.doi.org/10.1140/ epjc/s10052-015-3776-z, arXiv:1307.1688. [31] G. Cowan, K. Cranmer, E. Gross, O. Vitells, Asymptotic formulae for likelihood- based tests of new physics, Eur. Phys. J. C 71 (2011) 1554, http://dx.doi.org/ 10.1140/epjc/s10052-011-1554-0, Erratum: Eur. Phys. J. C 73 (2013) 2501, http://dx.doi.org/10.1140/epjc/s10052-013-2501-z, arXiv:1007.1727. [32] G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, The strongly-interacting light Higgs, J. High Energy Phys. 06 (2007) 045, http://dx.doi.org/10.1088/1126- 6708/2007/06/045, arXiv:hep-ph/0703164. [33] M. Farina, G. Panico, D. Pappadopulo, J.T. Ruderman, R. Torre, A. Wulzer, Energy helps accuracy: electroweak precision tests at hadron colliders, arXiv:1609. 08157. http://dx.doi.org/10.1016/j.physletb.2012.08.020 http://dx.doi.org/10.1016/j.physletb.2012.08.021 http://dx.doi.org/10.1103/PhysRevD.90.052004 http://dx.doi.org/10.1140/epjc/s10052-015-3351-7 http://dx.doi.org/10.1103/PhysRevLett.114.191803 http://dx.doi.org/10.1103/PhysRevLett.110.081803 http://dx.doi.org/10.1016/j.physletb.2013.08.026 http://dx.doi.org/10.1103/PhysRevD.92.012004 http://dx.doi.org/10.1140/epjc/s10052-015-3685-1 http://dx.doi.org/10.1140/epjc/s10052-016-3934-y http://dx.doi.org/10.1140/epjc/s10052-015-3769-y http://dx.doi.org/10.1088/0954-3899/39/7/075001 http://dx.doi.org/10.1142/S0217732313300115 http://refhub.elsevier.com/S0370-2693(16)30702-X/bib4162656C6C656972614665726E616E64657A3A323031327479s1 http://refhub.elsevier.com/S0370-2693(16)30702-X/bib4162656C6C656972614665726E616E64657A3A323031327479s1 http://dx.doi.org/10.1103/PhysRevLett.88.051801 http://dx.doi.org/10.1088/1126-6708/2009/07/101 http://dx.doi.org/10.1016/j.physletb.2013.04.060 http://dx.doi.org/10.1007/JHEP01(2013)148 http://dx.doi.org/10.1103/PhysRevLett.109.261801 http://dx.doi.org/10.1007/JHEP04(2014)110 http://dx.doi.org/10.1140/epjc/s10052-015-3628-x http://dx.doi.org/10.1103/PhysRevD.92.092004 http://dx.doi.org/10.1016/j.physletb.2015.08.047 http://dx.doi.org/10.1016/0370-1573(75)90009-5 http://dx.doi.org/10.1007/JHEP06(2011)128 http://dx.doi.org/10.1088/1126-6708/2002/07/012 http://dx.doi.org/10.1088/1126-6708/2006/05/026 http://dx.doi.org/10.1007/JHEP02(2014)057 http://dx.doi.org/10.1140/epjc/s10052-012-1896-2 http://dx.doi.org/10.1088/1126-6708/2008/04/063 http://dx.doi.org/10.1140/epjc/s10052-015-3776-z http://dx.doi.org/10.1140/epjc/s10052-011-1554-0 http://dx.doi.org/10.1140/epjc/s10052-013-2501-z http://dx.doi.org/10.1088/1126-6708/2007/06/045 http://refhub.elsevier.com/S0370-2693(16)30702-X/bib466172696E613A32303136727773s1 http://refhub.elsevier.com/S0370-2693(16)30702-X/bib466172696E613A32303136727773s1 http://refhub.elsevier.com/S0370-2693(16)30702-X/bib466172696E613A32303136727773s1 http://dx.doi.org/10.1016/j.physletb.2012.08.021 http://dx.doi.org/10.1103/PhysRevD.90.052004 http://dx.doi.org/10.1016/j.physletb.2013.08.026 http://dx.doi.org/10.1140/epjc/s10052-015-3769-y http://dx.doi.org/10.1103/PhysRevLett.88.051801 http://dx.doi.org/10.1088/1126-6708/2009/07/101 http://dx.doi.org/10.1007/JHEP01(2013)148 http://dx.doi.org/10.1007/JHEP04(2014)110 http://dx.doi.org/10.1016/j.physletb.2015.08.047 http://dx.doi.org/10.1016/0370-1573(75)90009-5 http://dx.doi.org/10.1007/JHEP06(2011)128 http://dx.doi.org/10.1088/1126-6708/2002/07/012 http://dx.doi.org/10.1007/JHEP02(2014)057 http://dx.doi.org/10.1140/epjc/s10052-015-3776-z http://dx.doi.org/10.1140/epjc/s10052-011-1554-0 http://dx.doi.org/10.1088/1126-6708/2007/06/045 Probing anomalous couplings using di-Higgs production in electron-proton collisions 1 Introduction 2 Formalism 3 Simulation tools and analysis 3.1 Cut-based optimisation 3.2 Kinematic distributions and observable 3.3 Exclusion limits through fiducial cross section as a function of luminosity 3.4 Prospects at higher Ee and sensitivity of the Higgs self coupling 4 Validity of EFT 5 Summary and conclusions Acknowledgements References