Probing the trilinear Higgs boson self-coupling via single Higgs production at the LHeC

at the LHeC Ruibo Li a,∗ Xiao-Min Shen a,† Bo-Wen Wang a,‡ Kai Wang a,§ and Guohuai Zhu a¶ a Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou, Zhejiang 310027, CHINA Abstract The determination of the Higgs self coupling is one of the key ingredients for understanding the mechanism behind the electroweak symmetry breaking. An indirect method for constraining the Higgs trilinear self coupling via single Higgs production at next-to-leading order (NLO) has been proposed in order to avoid the drawbacks of studies with double Higgs production. In this paper we study the Higgs self interaction through the vector boson fusion (VBF) process e−p → νehj at the future LHeC. At NLO level, we compute analytically the scattering amplitudes for relevant processes, in particular those induced by the Higgs self interaction. A Monte Carlo simulation and a statistical analysis utilizing the analytic results are then carried out for Higgs production through VBF and decay to bb̄, which yield for the trilinear Higgs self-coupling rescaling parameter κλ the limit [-0.28, 4.25] with 2 ab−1 integrated luminosity. If we assume 10% of the signal survives the event selection cuts, and include all the background, the constraint will be broadened to [-1.95, 5.93].


I. INTRODUCTION
A standard model (SM)-like Higgs boson has been discovered by the ATLAS and CMS collaborations at the CERN Large Hadron Collider (LHC) individually [1,2], which makes a milestone in particle physics. While it strongly supports the SM mechanism of spontaneous electroweak symmetry breaking (EWSB), by which all fermions and some of the vector bosons acquire their masses, the driven force of EWSB still remains mysterious. To better understand this problem, it is crucial to study the properties of the Higgs boson, e.g., to measure its mass, spin, CP properties and couplings [3][4][5][6]. From the second run of the LHC at 13 TeV, the ATLAS collaboration has recently reported the results of their measurements µ H→τ τ = 1.09 +0. 36 −0.30 and µ H→bb = 1.01 +0.20 −0.19 , with the integrated luminosities 36.1 fb −1 and 79.8 fb −1 , respectively [7,8]. These are significant improvements in Higgs precision physics.
However, the study of the Higgs self-coupling (λ) from the scalar potential V (Φ) is in a completely different situation. After EWSB the scalar potential takes a form with the trilinear (λ SM 3 = λ) and quartic (λ SM 4 = λ/4) self interactions: where Φ is the Higgs doublet field and h is the Higgs boson. At the LHC, double Higgs production as the standard process for determining the Higgs trilinear self coupling suffers from a small production rate and huge QCD backgrounds, and thus leads to large uncertainties even after the Run-II upgrade. The measurements of the γγbb final states by the CMS and ATLAS experiments yield and −8.2λ SM 3 < λ 3 < 13.2λ SM 3 , respectively [9,10]. For the bbbb production, the observed upper limit by ATLAS using the non-resonant Higgs pair production data is 13 times the SM value at 95% C.L [11]. There are also extensive phenomenological studies on determinning the trilinear Higgs self-coupling directly at the LHC [12][13][14][15][16][17][18][19][20][21], the future electron-positron collider [22][23][24][25], and future high energy hadron colliders [26][27][28][29][30][31][32][33][34], in which strict constraints are obtained with higher integrated luminosities and energies. On the other hand, an indirect method is proposed for constraining the Higgs self-coupling via single Higgs production at next-to-leading order (NLO) [24,[35][36][37][38][39]. The method relies on the account of one-loop electroweak radiative corrections to Higgs-strahlung and vector boson fusion (VBF) processes [40][41][42][43], and it has the potential of reaching a superior precision in the determination of the Higgs self-coupling, as compared to the determination via double Higgs production.
In view of the large QCD backgrounds interfering with the one-loop electroweak radiative corrections at the hadron-hadron collider, the Large Hadron electron Collider (LHeC) has been proposed as a deep inelastic scattering facility for the precision measurement of parton distributions and Higgs properties. LHeC as a relatively economic proposal is an upgrade based on the current 7 TeV proton beam of the LHC by adding one electron beam with 60-140 GeV energy [44], which could be tuned into a "Higgs factory" in which Higgs bosons are produced via VBF process. Thanks to the forward detector and reduction of QCD backgrounds in the e-p collider, the bottom Yukawa and trilinear Higgs self couplings could be measured precisely [45][46][47]. Therefore, we expect the LHeC to be a good facility for studying λ 3 via single Higgs production at NLO level.
The paper is organised as follows. In the next section, we discuss the one-loop contribution to single Higgs production, in particular that from processes via the trilinear Higgs self interaction and Higgs top quark Yukawa interaction, and calculate their scattering amplitudes analytically. In section III, we perform a Monto Carlo simulation for single Higgs produciton at the LHeC, produce the differential and total cross section, and carry out a statistical analysis to obtain constraints for λ 3 . Finally, we conclude in section IV.

II. THE ONE LOOP CORRECTION TO SINGLE HIGGS PRODUCTION AT THE LHEC
Given the tiny cross section of di-Higgs production [46], one could instead constrain the trilinear Higgs self-coupling λ 3 at the LHeC via the λ 3 induced loop corrections to the tree level single Higgs production process e − p → ν e hj shown in Fig.1. We parameterize the deviation of possible new physics from SM by a single parameter κ λ : where the physical Higgs field h has a zero vacuum expectation value (VEV), and λ SM 3 ≈ 0.13 is the Higgs trilinear self-coupling in the SM.
In the following, we shall identify various contributions up to NLO that are relevant for constraining the Higgs trilinear coupling.

C. Analytical result
In this section, we give the analytical result in both on-shell and MS schemes. We shall use MS scheme in our numerical simulation in the next section. Following [48], we take e, M H , M W , M Z , etc. as input parameters, and h to have a zero VEV. We do not show diagrams with Goldstone bosons in Figs.2 and 4, but for the convenience of the calculation we restore those diagrams and work in Feynman gauge.
We denote the momenta of the electron, incoming parton, Higgs boson, electron neutrino and outgoing parton by p 1−5 respectively. The Mandelstam variables are defined as S ij ≡ (p i + p j ) 2 , In this work the masses of u, c, d, s quarks are neglected. The CKM matrix elements V ub , V cb , V td , V ts are also taken to be zero.
We expand the amplitude M q of our process in powers of g W ≡ 4παe sin 2 θ W as where q stands for the incoming parton, α e is the fine structure constant and θ W ≡ arccos M W M Z is the Weinberg angle. The squared amplitude is then given by t+b + 14 ↔ 25 (4) The spin-summed fermion chains F where q = u, c,d,s,b is the incoming parton. The κ λ dependent term G λ reads Note that the contribution from κ λ dependent counter terms vanishes. G λ takes the same form in both OS and MS schemes, but the quantities on which it depends are generally renormalization scheme dependent. The contribution from top and bottom quarks, in the M b → 0 limit, is given by with contributions from counter terms [48] CT where the subscript t + b represents the contribution from the top and bottom quarks, D is the dimension of space-time and δZ e , δZ H , δZ W , δs, δM 2 W are renormalization constants in the onshell scheme. Detailed expressions of these renormalization constants can be found in [48] (Note that, at one loop level, all the renormalization constants in the above equations are independent of κ λ except for δZ H , whose contribution cancels out in the final result. ) B 0 , B 0 , C x (e.g. C 00 , C 1 , etc.) are scalar integrals and Here

III. MONTO CARLO SIMULATION
The squared amplitude in Eq.4 can be turned into the NLO cross section σ λ for the process e − p → ν e hj after integration over the phase space of the final states. We use Vegas algorithm implemented in Cuba library [53] to perform the numerical integration in our simulation at the parton-level. The following basic cuts are adopted: With beam energies being 7 TeV and 60 GeV for the proton and electron, the cross section of the process e − p → ν e hj is 80.16 fb at leading order. The contribution from the top and bottom quarks turns out to be -1.01 fb. The Vegas phase space integration is cross checked with Mad-Graph5 v2.6.5 [54], which gives the consistent result. In Fig.5, we show the cross section σ λ as a function of κ λ . The quadratic form can be traced back to the κ λ and (κ λ ) 2 terms in Eq.6. One way to show the significance of κ λ is via the differential distributions of characteristic kinematic variables, such as the azimuthal angle φ / E T j , the Higgs transverse momentum p h T , etc. Unfortunately, the discrimination between distributions for various processes relies heavily on the effect of threshold Sommerferld enhancement, which is absent in the case of loop corrections with the Higgs trilinear self-coupling [39]. This is very well illustrated in Fig.6 even when κ λ is varied in a very wide range. The distributions are normalized to reflect only the difference in shape. As there is little difference in shapes of distributions, we seek to identify the anomalous Higgs self interaction from SM processes using their normalizations. This can be done with the χ 2 method, in which the deviation between BSM and SM cross sections are described by where σ κ λ =1 is the production cross section of the e − p → ν e hj with κ λ = 1, while σ κ λ =1 contains the anomalous κ λ contribution. L is the integrated luminosity. We apply the χ 2 method to the H → bb decay channel for its large branch ratio (BR(h → bb) ≈ 58%) [55]. The results are shown in Fig.7, in which the solid blue, red and magnet curves correspond to integrated luminosities of 1, 2 and 3 ab −1 respectively. The limits on κ λ at 95% C.L. are shown in the figure and listed in Table.I. We find that κ λ is better constrained with the increase of the integrated luminosity. The most stringent limits on κ λ is [-0.10, 4.07], with L = 3 ab −1 .   [56,57], which is good enough for probing the Higgs self-coupling at NLO. If we assume 10% of the decays survives the event selection cuts [45], and use the background estimate from the same reference (where the 1-loop QCD correction to VBF processes at percent level [58] is not included, which has negligible effect on our result), the bounds in Ta ab −1 integrated luminosities respectively. These results can be improved if the background in the measurement could be further reduced.

IV. CONCLUSION
In this paper, we study the significance of the Higgs trilinear coupling through the single Higgs production process e − p → ν e hj at the LHeC. The analytical calculation is carried out up to one loop level for both λ 3 dependent and independent intermediate states. The analytical results are then used in the Monte Carlo simulation to produce numerically the cross section at various κ λ , which allows us to quantify the deviation of the cross section from the SM case (κ λ = 1) in a χ 2 statistic analysis. From this analysis we find that the 95% C.L. bound for κ λ is [-1.95, 5.93] with a 2 ab −1 integrated luminosity, after assuming a signal surviving ratio of 10%. This is a significant improvement compared with the current experimental result. We expect the result to be improved with more accurate measurement of Higgs decays.