Searching for the charged-current non-standard neutrino interactions at the $e^{-}p$ colliders

Considering the theoretical constraints on the charged-current (CC) non-standard neutrino interaction (NSI) parameters in a simplified $W'$ model, we study the sensitivities of the Large Hadron electron Collider (LHeC) and Future Circular Collider-hadron electron (FCC-he) to the CC NSI parameters through the subprocess $e^{-}q \to W' \to v_{e}q'$. Our results show that the LHeC with $\mathcal{L} = 100\;{\rm fb}^{-1}$ is a little more sensitive to the CC NSI parameters than that of the high-luminosity (HL) LHC, and the absolute values of the lower limits on the CC NSI parameters at the FCC-he are smaller than those of the HL-LHC about two orders of magnitude.

I.

INTRODUCTION
The Standard Model (SM) has been a powerful method for studying particle physics.
However, the problem of neutrino oscillations [1] arising from mismatching flavor and mass eigenstates is one of the open questions that seems to be unexplainable within the traditional framework of the SM, highlighting the need for new physical theories that go beyond the SM (BSM) [2,3]. With the concentration of efforts in this area, the study of non-standard neutrino interactions (NSIs) is gradually emerging, as detailed in Refs. [4][5][6][7]. Many studies have constrained NSIs (both charged-current and neutral-current) to be of the order of O(10 −4 ) to O(10 −1 ), depending on the data used for the analysis and the model chosen [8][9][10][11][12][13][14]. NSIs are universal methods for studying the effects of new physics in neutrino oscillations. The neutrino production and detection processes involving non-standard interactions are usually associated with charged leptons and are referred to as charged current NSIs (CC NSIs). The experiments usually impose stricter constraints on the CC NSIs than those for the neutral-current NSIs (NC NSIs) [15,16].
The presence of NSIs not only has a great impact on the precision measurements of neutrino oscillation experiments, but also is an important platform for studying the impact of new physics in high-and low-energy collider experiments and leads to a very rich phenomenology [7][8][9][10][11][12][13][14][17][18][19][20][21][22]. In particular, the effect of the CC NSI in neutrino oscillation experiments depends on the strength of the new vector interaction, which strongly depends on the flavor structure of the CC NSI, leading to a degeneracy that is difficult to break by neutrino facilities [23]. However, the collider data can break the degeneracy because of its insensitivity to neutrino flavor and the signal will lead to the same observation for different flavors. So studying NSIs at the collider experiments are used as a complement to the neutrino oscillation experiments.
The new particles and the new interaction processes predicted by the new physics models are able to produce a wealth of physical phenomena in the high-energy scales, and it is entirely possible that the already operating and future high-energy colliders will detect these new physical signals in the future. In special, the new charged gauge boson W , a hypothetical heavy partner of the SM W gauge boson, is an important detection goal of future colliders, which can contribute to the CC NSI parameter as ∼ g 2 W /M 2 W in a simplified model framework. In this paper we will study the sensitivities of the e − p colliders to the CC NSI parameters via the subprocess e − q → W → v e q .
Currently, in order to combine the excellent performance of proton-proton collider and electron-electron collider, the e − p collider has been proposed, which can complement the proton ring with an electron beam and allow deep inelastic leptonic scattering (DIS) of electrons and protons at TeV energies. Its kinematics extends to higher scales relative to HERA [24][25][26][27][28][29][30][31]. Compared to the e + e − collider, the e − p collider allows DIS studies of the internal structure of nuclei. At the same time it is also more advantageous compared to the pp collider. The e − p collider can provide a cleaner background that suppresses the background of strong QCD interactions, providing high precision measurements. In addition, since the initial state is asymmetric, the backward and forward scattering can be disentangled, which can greatly increase the significance of the signal and get opportunities that cannot be observed in pp collider. Thus, the e − p collider not only becomes a good choice to complement the pp and e + e − colliders, but also provides distinctive ways to precision Higgs physics, top quark, electroweak physics as well as new physics beyond the SM [32].

II. THEORETICAL MODEL FRAMEWORK
NSIs are usually described by six-dimensional four-fermion operators [12,15,16,33], where for CC NSIs with quarks are given by effective Lagrangian Here G F is the Fermi constant and P Y is a chiral projection operator including P L or P R . The parameters qq Y αβ (Y = L or R) are dimensionless coefficients that quantify the strengthes of the new vector interactions. We have assumed that there are only left-handed neutrinos in above equation.
The constraints on the CC NSI parameters have been studied at the LHC [17,22], but relevant study at the e − p collider is currently not available. In this paper, we mainly study the possibility of detecting the CC NSI parameters induced by exchange of the new gauge boson W at the LHeC and FCC-he. Since the momentum transfers can be high to resolve further dynamics of the new physics in the collider, the impact of the CC NSI may not be simply described by the effective operators. In this paper, we focus on a simplified model with the CC NSI induced by exchange of a W boson. The effective Lagrangian can be written as [34,35], where W µ denotes the new force carrier with mass M W , g is the electroweak coupling constant, and V qq is the Cabbibo-Kobayashi-Maskawa (CKM) matrix element. q and q are up-type and down-type quarks. As long as the square of momentum transfer is much smaller than M 2 W , the effective parameters qq Y αβ can be approximatively written as For the subprocess e − q → v e q considered in this paper, there is α = β = e. If we assume that this process is flavor conservation, there are only two combinations, i.e. the ud and cs quarks. Ref. [36] has shown that the luminosity of the ud quarks is about two orders of magnitude higher than that of the cs quarks. If csY ee was at the same magnitude as udY ee , the contribution of ce − → ν e s is negligible, therefore we neglect the contribution from a none zero csY ee . In the following, udY ee is simplified as Y .
Ref. [17] TeV, 6 TeV and 7 TeV, rspectively. In the following MC analysis, we will also consider the theoretical constraints on the CC NSI parameters.
The dependencies of the factors α int and α Y nsi on M W can be fitted with the results of the MC simulation for different M W at the LHeC and FCC-he, which are listed in Table II. We find that the interference term plays an important role and its weight of the total cross-section can be compared to the measurement accuracy of the LHeC or FCC-he. The sensitivities of the LHeC and FCC-he to the CC NSI parameters have reached the region where interference effect should be considered.  In order to distinguish signals from all relevant backgrounds, the MadAnalysis5 [45] was used to analyze kinematic cuts of reconstruction-level events of the signals and backgrounds.
The process e − p → v e j is mediated by the exchange of W and W in the t-channel, according to M ∝ 1/(t − M 2 W,W ), M is inversely proportional to (t − M 2 W,W ), so when |t| becomes small, the amplitude is increased. The events are distributed in the region of large amplitude, so most of the background and signal events will be distributed in the region of small |t|. However, in comparison, M SM will increase much more than the M W , so a larger proportion of the background events will be distributed in the region of small |t|. In order to highlight the signal, the region of small |t| should be cut off. The smaller the transverse momenta of jets p j T , the smaller |t| will be, so regions with smaller p j T should be cut off. For the same reason, the most important variables that distinguish signals from background are also the missing transverse energy / E T and the visible transverse hadronic energy H T .
Therefore, the kinematic cuts chosen in this paper are p j T , / E T and H T . For simplicity of illustration, we show in Fig. 1   The improved cuts are shown in Table III. respectively. As can be seen from these tables, the background is strongly suppressed, while the signal still has good efficiency after all cuts are applied. To accurately obtain the sensitivities of the LHeC and FCC-he to the CC NSI parameters that can be estimated with the help of statistical significance (S stat ), defined as Where σ( Y ) is the total cross-section including the W contributions after cuts applied.
In Fig. 2 we plot the sensitivities of the 1.30 TeV LHeC with L = 10 fb −1 to the CC NSI parameters detected at the 2σ, 3σ, and 5σ levels for different M W , where the electron beam polarization is taken as P (e − ) = 0 and −90% in Fig. 2 (a)  at the 2σ, 3σ, and 5σ levels, respectively, as shown in Fig. 2 (b). It is obvious that when the integrated luminosity is larger, the L is tighter and more sensitive to the detection of new physical signals. We use the similar method to give the lower limits of the sensitivities of the √ s = 1.30 TeV LHeC with L = 100 fb −1 and P (e − ) = −90% to the CC NSI parameters, which are | L | ≥ 7.52 × 10 −5 , | L | ≥ 9.89 × 10 −5 and | L | ≥ 1.52 × 10 −4 at the 2σ, 3σ, and 5σ levels, respectively.
Details are shown in Fig. 3. It is obvious that the absolute values of the lower limits on the CC NSI parameters at the FCC-he are smaller than the results given at the LHeC about two orders of magnitude. All of the numerical results about the parameters Y derived in this paper satisfy the unitarity bounds and the W -decay constraints mentioned in Sec. II. | L | ≥ 6.78 × 10 −6 at the 2σ, 3σ, and 5σ levels, respectively, which are smaller than those of the HL-LHC about two orders of magnitude. In summary, studying NSI at the e − p colliders is a promising area for further research, and the e − p colliders have clean environment that will also benefit the physics program at the LHC to explore new physics beyond the SM.