Single top and Higgs associated production in the minimal $B-L$ model at the LHC

In this paper, we study the single top production in association with a Higgs boson in the $U(1)_{B-L}$ extension of the Standard Model at the LHC. We calculate the production cross sections of the processes $pp\rightarrow thX(h=H_{1},H_{2};X=j,b,W)$ in this model. Then we further study the observability of the process $pp\rightarrow tH_{2}j$ through $pp\rightarrow t(\rightarrow q\bar{q'} b)H_{2}(\rightarrow 4\ell)j$. We find that the systematic significance can be improved obviously, but it is still challenging for the 14 TeV LHC with high-luminosity to detect this signal.


I. INTRODUCTION
In July 2012, a Higgs-like resonance with mass m h ∼ 125 GeV has been caught by the ATLAS and CMS experiments at the Large Hadron Collider (LHC) [1]. So far, all the measurements of the discovered new particle [2] are well compatible with the scalar boson predicted by the Standard Model(SM) [3].
It is well known that the SM cannot be the final theory of nature. Theoretically, successful explanation of some problems, such as the hierarchy problem, requires new physics beyond the SM near the TeV scale. Experimentally, the solid evidence for neutrino oscillation is one of the firm hints for new physics. The minimal extension of the SM is that the SM gauge groups are augmented by a U(1) B−L symmetry, where B and L represents the baryon number and lepton number respectively. The B − L gauge symmetry can explain the presence of three right-handed neutrinos and provide a natural framework for the seesaw mechanism [4].
In addiction, it's worth noting that B − L symmetry breaking takes place at the TeV scale, hence giving rise to new and interesting TeV scale phenomenology.
Concerning the probe of new physics through the Higgs boson, the Yukawa couplings play an important role in probing the new physics. The top quark is the heaviest particle discovered and owns the strongest Yukawa coupling. The top quark Yukawa coupling is speculated to be sensitive to the electroweak symmetry breaking (EWSB) mechanism and new physics. The tth production process is a golden channel for directly probing the top Yukawa coupling, however, this process cannot provide the information on the relative sign between the coupling of the Higgs to fermions and to vector bosons. As a beneficial supplement, the thj production process can bring a unique possibility [5] and many relevant works have been carried out [6].
The U(1) B−L model predicts heavy neutrinos, a TeV scale extra neutral gauge boson and an additional heavy neutral Higgs, which makes the model phenomenologically rich. The heavy Higgs state mixes with the SM Higgs boson so that some Higgs couplings are modified and this effect can also influence the process of single top and Higgs associated production.
Besides, the process of single top and heavy Higgs associated production deserves attention, which is equally important for understanding the EWSB and probing new physics. By performing the detailed analysis on this process may provide a good opportunity to probe the U(1) B−L model signal.
The paper is structured as follows. In Sec.II we review the U(1) B−L model related to our work. In Sec.III we first calculate the production cross sections of the single top and h(= H 1 , H 2 ) associated production at the LHC, then explore the observability of t-channel process pp → tH 2 j through pp → t(→ qq ′ b)H 2 (→ 4ℓ)j by performing a parton-level simulation.
Finally, we make a summary in Sec.IV. In this model, the most general gauge-invariant and renormalisable scalar Lagrangian can be expressed as with the scalar potential given by From the mass terms in the scalar potential, the mass matrix between the two Higgs bosons in the basis (H, χ) can be given by The mass eigenstates are related via the mixing matrix  where the mixing angle α (− π 2 < α < π 2 ) satisfies .
The masses of the physical Higgs bosons H 1 and H 2 are given by where H 1 and H 2 are light SM-like and heavy Higgs bosons, respectively.
To complete the discussion on the Lagrangian, we write down the Yukawa term, which in addition to the SM terms has interactions involving the right-handed neutrinos N R , where H = iσ 2 H * and i, j runs from 1∼3. The vacuum expectation value (VEV) of the χ field breaks the B − L symmetry and generates the Majorana masses for the right handed neutrinos and the Dirac masses for the light neutrinos.
In terms of the mixing angle α, the couplings of H 1 and H 2 with the fermions and gauge bosons can be expressed as follows where f denotes the SM fermions, s W = sin θ W with θ W is the usual Weinberg angle.

III. NUMERICAL RESULTS AND DISCUSSIONS
For the single top and Higgs associated production, the three processes of interest are characterized by the virtuality of the W boson in the process [8]: where there is emission of a real W boson. In the U(1) B−L model, the lowest-order Feynman diagrams of the t-channel process pp → tH 1 (H 2 )j(j = b) are shown in Fig.1, the s-channel process pp → tH 1 (H 2 )b are shown in Fig.2 and the W -associated production channel process Fig.3. We can see that the Feynman diagrams for these processes are the same as the corresponding SM processes. Moreover, the conjugate processes where t is replaced byt have been included in our calculations.
We compute the cross sections by using CalcHEP 3.6.25 [9] with the parton distribution function CTEQ6L [10], and set the renormalization scale µ R and factorization scale µ F to The SM input parameters are taken as follows [11]: In our calculations, the relevant U(1) B−L model parameters are the mixing parameter α FIG. 3: Lowest-order feynman diagrams for pp → tH 1 (H 2 )W − in the U (1) B−L model. and the heavy Higgs mass m H 2 . Considering the constraints in Refs. [12,13], we choose the parameter space as follows: 0.01 < sin α < 0.4, 250GeV≤ m H 2 ≤1000GeV.
A. Single top and H 1 associated production In Fig.4, we show the production cross sections of the processes pp → tH 1 j, pp → tH 1b   respectively. In order to see the influence of the heavy Higgs mass m H 2 on the production cross sections, we take m H 2 = 250, 500, 750, 1000GeV as example. We can see that the cross sections increase with increasing sin α, which is because the heavy Higgs H 2 couplings in Eq. (8) are proportional to sin α so that the cross sections are proportional to sin 2 α.

C. Observability of pp → tH 2 j
The t-channel process dominates amongst these three production modes at the LHC, so we will explore the observability through the t-channel pp → tH 2 j at 14 TeV LHC in the following section. The three most dominant decay modes of the heavy Higgs H 2 are W W, H 1 H 1 and ZZ [14]. Though the branching fraction of H 2 → ZZ is smaller than the branching fractions of H 2 → W W and H 2 → H 1 H 1 , the ZZ signal is much easier to separate from SM backgrounds. For the ZZ decay modes, the leptonic decay mode of ZZ offer the cleanest possible signatures though the di-jet and semi-leptonic decay modes of ZZ are larger. This leptonic decay mode has been studied in the heavy Higgs production at the LHC and it found that a heavy Higgs boson of mass smaller than 500 GeV can be discovered at the LHC with high-luminosity (HL-LHC) [13]. In our work, we concentrate on the channel Fig.7, where H 2 decays to two Z bosons and the two Z bosons subsequently decay to four leptons. The signal is characterised by 3 jet + b jet + 4ℓ (10) where j denotes the light jets and ℓ = e, µ. The largest background for this process comes from the tZZj production mode that will generate the same final state. including the decay chain with hadronic top quark, leptonic Z boson decay and Higgs decay We generate the signal and background events with with MadGraph5 [15] and perform the parton shower and the fast detector simulations with PYTHIA [16] and Delphes [17]. To simulate b-tagging, we take moderate single b-tagging efficiency ǫ b = 0.7 for b-jet in the final state. Follow the analysis on tth signature by ATLAS and CMS collaborations [18] at the LHC Run-I, the events are selected to satisfy the criteria as follows: p ℓ T > 10 GeV, |η ℓ | < 2.5 p j T > 25 GeV, |η j | < 5. Due to the small signal cross section, this process has a low signal-to-background ratio S/ √ B at the LHC. In this case, we will focus on enhancing the systematic significance S/B.
Considering the transverse momentum of the leptons have little effect on the signal-tobackground ratio and the systematic significance, we don't use them as selection cuts here.
After analysis, we will adopt the following two cuts, the relevant normalized distributions of the kinematic variables for m H 2 = 250 GeV, sinα = 0.3 with respect to the background are shown in Fig.8.
Firstly, we impose the cut H T < 380 GeV to separate signal from background, where H T (= hadronic particles p T ) is the total transverse hadronic energy. This cut can improve both the signal-to-background ratio S/ √ B and the systematic significance S/B.
After that, we apply the invariant mass of the four lepton system to further isolate the signal and let M 4l lie in the range m H 2 ± 20 GeV. We can see that the signal-to-background ratio S/ √ B is improved and the systematic significance S/B is enhanced obviously.
The cut-flow cross sections of the signal and background for 14 TeV LHC are summarized in Table.I. After all cuts above, we can see that the systematic significance S/B is substantially improved. For the HL-LHC with a final integrated luminosity of L = 3000fb −1 , the signal-to-background ratio S/ √ B can reach 1.6σ and systematic significance S/B can reach 2.86 for m H 2 = 250 GeV, sinα = 0.3. Unfortunatel, we can see that the number of signal events is very small because of the small leptonic branching ratio of the Z boson, which will be a trouble for detecting this signal at the LHC.

IV. SUMMARY
In the minimal B − L extension of the SM, we investigated the single top and Higgs associated production at the LHC. We computed the production cross sections of the processes pp → tH 1 (H 2 )X(X = j, b, W ) for 8, 14 TeV LHC and displayed the dependance of the cross sections on the relevant U(1) B−L model parameter. Moreover, we investigated the observability of process pp → tH 2 j followed by the decays t → qq ′ b and H 2 (→ ZZ → ℓ + 1 ℓ − 1 ℓ + 2 ℓ − 2 ) at 14 TeV LHC for m H 2 = 250 GeV, sinα = 0.3. We performed a simple parton-level simulation and found that it is challenging for the 14 TeV LHC and future HL-LHC with the integrated luminosity L = 3000fb −1 to observe the effect of the process pp → tH 2 j through this final state. So, we have to expect a collider with higher energy and higher luminosity to probe this effect. Maybe, a 100 TeV proton-proton collider with integrated luminosities of 3 ab −1 ∼ 30ab −1 can provide us a potential opportunity [19].