The charged-current non-standard neutrino interactions at the LHC and HL-LHC

A series of new physics scenarios predict the existence of the extra charged gauge boson W (cid:48) , which can induce the charged-current (CC) non-standard neutrino interactions (NSIs). Considering the theoretical constraints on the CC NSI parameters from the partial wave unitarity and the W (cid:48) decays, we discuss the constraints and sensitivities on the CC NSI parameters (cid:15) qq (cid:48) Y αβ ( or µ ) via the process pp → W (cid:48) → (cid:96)ν at the LHC and high-luminosity (HL) LHC experiments by the Monte-Carlo (MC) analysis. We ﬁnd that the interference eﬀects play an important role and the LHC data can give strong constraints on the CC NSI parameters. Compared with the 13 TeV LHC with L = 139 fb − 1 , the sensitivities given by the 14 TeV LHC with L = 3 ab − 1 can be strengthened to about one order of magnitude.


INTRODUCTION
The Standard Model (SM) has achieved great success in describing elementary particles and interactions.Numerous experimental studies have verified its predictions with very high accuracy.However, the SM does not reveal the origin of the neutrino mass.Strong evidence show that neutrinos with different flavors cannot oscillate with each other without small mass differences.The origin of neutrino mass requires new physics beyond the SM (BSM) [1,2], many examples of which share the common feature of emerging effective non-standard neutrino interactions (NSIs) between neutrinos and matter fields [3][4][5][6][7].NSIs are generally divided into two types: the neutral-current (NC) NSI and charged-current (CC) NSI.The NC NSI mainly affects neutrino propagation in matter, whereas the CC NSI is associated with neutrino production and detection processes (for recent reviews, see Refs.[5][6][7][8]).These new interactions lead to rich phenomenology in neutrino oscillation experiments and high-and low-energy collider experiments.Very strong constraints on the NSIs have been obtained, see, for example, Refs.[4,[9][10][11][12][13][14][15][16][17][18].The experiments generally impose more strict restrictions on the CC NSI than on the NC NSI [19,20].
Recently, numerous phenomenological studies have been conducted on NSIs.For example, the new gauge boson Z can introduce NC NSIs [12,[21][22][23][24][25][26].Refs.[12,[24][25][26] studied the constraints on NC NSIs in the context of a simplified Z model using mono-jet signals at the Large Hadron Collider (LHC).Many BSMs predict the existence of W , such as the left-right symmetry model [27,28], little Higgs models [29], and models with extra dimensions [30,31], which can lead to CC NSIs.Previous searches for the W boson at the LHC have been carried out by the ATLAS and CMS collaborations with data collected at the center-of-mass (c.m.) energy √ s = 7 TeV [32,33], 8 TeV [34,35], and 13 TeV [36,37].The most sensitive channels are eν and µν production, with the constraints to date set in Refs.[38,39], which can be translated to constraints on CC NSIs.The LHC experiments have further promoted the theoretical research on W properties [40][41][42].Although there are other new particles that can also cause the CC NSI, such as charged scalar particles (for example, see [14]), in this paper, we focus on the simplified W model.
It has been shown that there is a well-known 'degeneracy' between the parameter spaces of NSIs in neutrino oscillation experiments because the effects of NSIs strongly depend on the flavor structure and oscillation channel being studied [43].Although it is difficult to break the degeneracy at neutrino facilities, the LHC plays a complementary role in the study of NSIs [4,11,14,44], and it is believed to be able to break the degeneracy because the effects of NSIs at the LHC do not distinguish between different neutrino flavors [25,26].
Furthermore, the LHC is sensitive to both vector-and axial-vector interactions, whereas neutrino oscillation experiments are not sensitive to the latter.
Thus far, various studies have been completed on the detection of W at the LHC, and searching for this type of new particle will continue to be important at the future LHC with higher luminosities (HL-LHC).Based on a Monte-Carlo (MC) simulation, the expected constraints on the CC NSI parameters via the process pp → + ν are studied with an emphasis on the effects of the interference between the CC NSI and the SM.Moreover, the sensitivities at future runs of the LHC are also discussed.
The remainder of this paper is organized as follows: In Sec.II, we consider the theoretical constraints on the simplified W model and further on the CC NSI parameters qq Y αβ (α = β = e or µ) in terms of partial wave unitarity and W decay.The sensitivities of the process pp → + ν to the parameters udY αβ at the current LHC and future runs of the LHC are studied in Secs.III and IV, respectively.Our conclusions and discussions are given in Sec.V.

II. THEORETICAL CONSTRAINTS ON THE SIMPLIFIED W MODEL AND NSI PARAMETERS
NSIs are new vector interactions between neutrinos and matter fields, induced by either a vector or scalar mediator.They can be parameterized in terms of the low-energy effective four-fermion Lagrangian [19,20,44,45].CC NSIs with quarks are given by the effective Lagrangian where G F is the Fermi constant, α and β are lepton flavor indices, α, β ∈ {e, µ, τ }, and P Y is a chiral projection operator (P L or P R ).We assume that there are only left-handed neutrinos in the above equation.The parameters qq Y αβ are dimensionless coefficients that quantify the strengths of the new vector interactions.According to Hermiticity, qq Y αβ = qq Y * βα .The CC NSI might change the production of neutrinos through its effects on processes such as muon decay and inverse beta decay [46,47].
Because the momentum transfer can be very high to resolve further dynamics of new physics at the LHC, the influence 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 the exchange of a W boson.The effective Lagrangian can be written as [48][49][50] where W µ denotes the new force carrier with a mass M W , and g is the electroweak coupling constant.q = {u, c, t} and q = {d, s, b} are up-type and down-type quarks, respectively.V qq is the Cabbibo-Kobayashi-Maskawa(CKM) matrix element, in which the non-diagonal terms contribute to flavor changing neutral current (FCNC) processes [51], and the contributions are very small and negligible.For simplicity, one can also absorb gV qq / √ 2 into the coupling parameters A qq Y and B αβ L .In a simplified model framework, the relationship between Eq. (1) and Eq. ( 2) can be thought of as , and qq Y αβ = qq Y αβ when s → 0, and However, it can also be considered that the effective operator in Eq. (1) corresponds to the leading order of expansion of qq Y αβ (s) in s/M 2 W , whereas Eq. ( 2) also collects the contributions of the higher dimensional operators corresponding to the higher orders of s/M 2 W .In principle, there are also flavor violating NSI couplings.Neutrino oscillation and scattering experiments are found to have tight constraints on diagonal NSIs, whereas off-diagonal NSIs are primarily constrained by various low-energy processes, such as atomic parity violation and charged-lepton flavor violation (cLFV) [14].Because only diagonal NSIs are relevant for the process pp → ν, we concentrate on the α = β case in this paper.

A.
Unitarity constraints from the In a weakly coupled renormalizable theory, the high-energy behavior of the scattering amplitude of bosons under the perturbation calculation is expected to respect unitarity.It has been shown that the constraint on M Z can be obtained using partial wave unitarity bounds derived from the f f → V 1 V 2 process, where V represents vector bosons, including SM W, Z bosons and also the new gauge bosons W , Z , and f and f are fermions and anti-fermions [52].The helicity amplitudes can be expanded as [53,54] M fσ 1 fσ 2 →V 1,λ 3 V 2,λ 4 = 16π where m 1 = σ 1 − σ 2 and m 2 = λ 3 − λ 4 are the helicity differences of fermions and vector bosons, respectively.d J m 1 ,m 2 are Wigner d-functions [53], φ and θ are the azimuth and zenith angles of V 1 in the rest frame of the fermions whose Z-axis points in the direction of f .The partial wave unitarity bound is |T J | ≤ 1 [55,56], where T J is the coefficient of partial wave expansion.As noted in Ref. [52], we only need to consider processes with the longitudinal , whose amplitudes are denoted as M (−+) In the following, we only consider tree-level processes, including t-channel, u-channel, and s-channel processes.The corresponding Feynman diagrams are shown in Fig. 1.There is also an schannel scalar exchange diagram, which only contributes to M (++) Partial wave unitarity should be satisfied for arbitrary s.When s → ∞, a finite |T J | implies A 1 = 0, which results in a set of 'sum rules' [52,57], where V 1,2,3 and f 1,2,3,4 are allowed vector bosons and fermions in the Feynman diagrams in Fig. 1, respectively.
For the scattering amplitudes of the process have non-zero coupling with fermions and other gauge bosons, and the existence of Z is necessary to maintain the perturbative unitarity of all amplitudes.Therefore, triple gauge couplings involving Z and Z -fermion couplings must be involved in the calculation of the Feynman diagrams shown in Fig. 1.There are enough 'sum rules' to solve all unknown couplings of Z using Eq. ( 6), that is, the Z -fermion couplings are automatically fixed when the W -fermion couplings are fixed [52].Certainly, the W − W and Z − Z mixings would modify the relevant SM couplings slightly, which can affect the amplitudes of the process It has been shown that the correction effect is less than 1% [52,58]; hence, for simplicity, we neglect it in our calculations.With the help of Eq. ( 6), the amplitudes corresponding to the Feynman diagrams in Fig. 1 are written as Eq.(A.1).
Using the relationship |T J | ≤ 1, one can obtain a constraint on A qq Y or B αα L from each of the above processes.All of these constraints should be satisfied, therefore, we concentrate on the tightest ones, which depend on the free parameters M W and M Z .In the case of 0.2 TeV < M Z < 2 TeV [25] and s → ∞, the tightest bounds on A qq Y or B αα L are given in Eq. (A.2).The relationship between M W and the coupling parameters according to Eq. (A.2) are shown in Fig. 17.It can be seen that in the wide range of M W , the tightest bound on A qq L originates from the process u d (dū) → W Z. Similarly, the tightest bound on B αα L mainly arises from the process e − e + → W Z. For simplicity, we only use the above bounds.For A qq R , the two tightest bounds (depending on M W ) are both considered.Consequently, the perturbative unitarity constraints on the parameters qq Y αα are qq L αα By using Eq. ( 1), the unitarity bounds on qq Y αα can be directly obtained from the process f f → f f .However, because the W model corresponds to the substitution qq Y αα → qq Y αα (s), where the latter is just the 'form factor unitarization' widely used in the study of SM EFT [59][60][61], that is, unitarity is guaranteed with this substitution; therefore, the unitarity bounds from the process f f → f f are not considered.

B. Constraints from W decays
From Eq. ( 2), the total decay width of the W boson at leading order can be written as Without loss of generality, using the case of A ud L = 0, B ee L = 0 as an example, when the mass of fermions are negligible compared with M W , The first inequality takes the equals sign at For simplicity, taking udY ee as an example, the maximally allowed udY ee as a function of M W is shown in Fig. 2 for different values of the ratio Γ W /M W .The unitarity bounds given by Eq.( 7) are also shown in Fig. 2. It can be seen that with additional assumptions on Γ W /M W , that is, for some values of Γ W /M W and M W , tighter constraints can be obtained.For larger M W , the perturbative unitarity constraints become stronger than those from W decays.If we take Γ W /M W = 0.1 [25,26], qq R αα TeV, and 7 TeV, respectively.The theoretical constraints given in Fig. 2 are used as a reference for the MC studies in the next section.It can be seen from Fig. 2 that the larger the mass of the W boson, the tighter the constraint on udL ee .However, with fixed Γ W /M W , the W contributions to the process pp → e + ν e typically decrease with increasing M W ; therefore, the larger the value of M W , the larger the value of udL ee needed to observe the signal at the LHC.However, udL ee must satisfy the unitarity constraint given by Eq. ( 7).In the following MC analysis, we take into account the theoretical constraints on the NSI parameters qq Y αα , set Γ W /M W = 0.1, and assume that the mass M W is from 1 TeV to 7 TeV.

III. EXPECTED CONSTRAINTS ON udY AT THE LHC
A simple and efficient way to search for the gauge boson W at the LHC is through its single s-channel resonance and subsequent leptonic decays [62].Because the main components of protons are u and d quarks, the contribution of the u d process is considerably larger than that of the cs process.It has been noted that the luminosity of u d quarks is two orders of magnitude larger than that of cs quarks [63].If csY are at the same magnitude as udY , the contribution of cs → + ν is negligible compared with u d → + ν ; therefore, we neglect the contribution from non zero qq Y with qq = ud.In this paper, we consider the subprocess where is e or µ, which corresponds to udY ee or udY µµ .Fig. 3 shows the Feynman diagram of the subprocess u d → + ν ( = e or µ) induced by W exchange at the LHC.In numerical estimation, the Lagrangian described by Eq. ( 2) is implemented using FEYNRULES [64][65][66], and a Universal Feynrules Output (UFO) file is generated and taken into the MadGraph5_aMC@NLO toolkit [67,68] with the standard cuts A fast detector simulation is applied using Delphes [69] with the CMS detector card.
The parton distribution functions are taken as NNPDF2.3[70,71].To highlight the signal from the background, the kinematical features of the signal and background are studied using MadAnalysis5 [72].The processes in the SM with the same final states are considered as the background.At tree level, there is only one type of background in which + ν are from the SM W boson.The main difference between the signal and the background is that for signal events, the leptons are from a W boson with a considerably lager mass M W than that of the SM W boson.As a consequence, the transverse momenta of charged leptons for signal events are generally larger than those for background events.p T has also been used in previous studies to highlight the signal of W .The normalized distributions of p T for pp → e + ν e are shown in Fig. 4. (a).It can be seen that p T is generally smaller than 360 GeV for background events, whereas the signal events have large p T tails.In this paper, we only keep events with p T ≥ 360 GeV.For the same reason, the missing transverse energy / E T for the signal events is also typically larger than those for the background events.The normalized distributions of / E T are shown in Fig. 4. (b).The background events are dominantly distributed in the region / E T < 460 GeV, which is not the case for the signal events.In this paper, we only keep events with / E T ≥ 460 GeV.The normalized distributions for the process pp → µ + ν µ are shown in Fig. 5.When searching for the signal of the W boson with an unknown mass, the cuts are applied uniformly.However, the event selection strategies can be further improved.If the signal is not found, the goal should be to constrain the parameter for different M W .For this purpose, the event selection strategy can be different for different  Because the masses of fermions are negligible, the interference between qq L and qq R is neglected in this study for simplicity.To focus on the sensitivities of the process on , we assume is a real number in numerical estimations.The cross-sections of the process pp → + ν including the W contributions can be parameterized as σ IN T ( ) = α int × originates from the interference between the SM and W contribution, and σ Y N SI represents the contribution from only W exchanges with σ Y N SI ( ) = α Y nsi × 2 .After the event selection strategy, 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 √ s = 13 TeV, which are listed in Table I.For M W = 2 ∼ 7 TeV, σ ee SM = 0.0034 pb and σ µµ SM = 0.0027 pb (σ ee SM = 0.021 pb and σ µµ SM = 0.027 pb for M W = 1 TeV owing to different event selection strategies).Taking pp → e + ν e as an example, the cross-sections of the process pp → e + ν e after cuts for different M W are shown in Fig. 6.The numerical results fit the bilinear functions in Eqs.(11) and ( 12) very well.Fig. 6 shows that the interference plays an important role in σ L ( ).Moreover, as shown Fig. 6 and presented in the next section, the sensitivity of the process pp → + ν on W at the LHC has already reached the region where the interference effect should be considered.The expected constraints on are estimated with the help of the statistical significance defined as where σ SM and σ qq Y are the SM cross-section and total cross-section including the W contributions after cuts are applied, respectively.When qq L and qq R are both non zero, For S stat = 2, 3, and 5, the upper limits on the udL ee and udR ee plane are shown in Fig. 7.For clarity, in Fig. 7, we use the signal significance of events that exceed the SM.The constraints are approximate eccentric ellipses, indicating the importance of the interference term.To study the sensitivities, it is assumed that only one of qq L and qq R is non zero.
For M W in the range of 1 TeV ∼ 7 TeV, the expected constraints on udY at the 2σ, 3σ, and 5σ confidence levels (CLs) are shown in Table II.In Figs. 8 and 9, we show the expected constraints on udL(R) ee and udL(R) µµ for different M W at the 13 TeV LHC with L = 139 fb −1 .The tightest theoretical constraints discussed in Sec.II are also presented in Figs. 8 and 9.The results of udY satisfy the unitarity bounds generated by Eq. ( 7) and satisfy the constraints given by Eq. ( 10).From Figs. 8 and 9, we find that for different M W , the expected constraints are similar for udL .This can be understood by looking at Table I; α int are at the same order of magnitude.Therefore, when the interference term is dominant, the expected constraints are insensitive to M W .For the same reason, the positive and negative expected constraints are different for udL .This is not the case for udR .Because there is no interference, the expected constraints on udR are generally looser than those on udL .Except for the case of M W = 1 TeV, the expected constraints are looser when M W is larger because the cross-section decreases rapidly with increasing M W .For M W = 1 TeV, the mass of the W boson is closer to that of the SM W boson; therefore, the event selection strategy is less efficient.As a consequence, even though the cross-section of the signal is larger, a stronger expected constraint cannot be reached compared with the case of M W = 2 TeV.It is interesting to discuss the expected constraints for special cases.In the case of the µν µ channel at the 2σ CL (we choose the largest lower bounds from Refs.[36][37][38][39]).By considering the interference effect, these results can be further improved.

IV. SENSITIVITIES OF THE HL-LHC TO udY
In this section, sensitivities to udY in future runs of the LHC with higher luminosity, known as the HL-LHC, are investigated.We assume that the HL-LHC runs at √ s = 14 TeV with integrated luminosities of 300 fb −1 , 1 ab −1 , and 3 ab −1 [74,75].Taking the process pp → e + ν e as an example, the normalized distributions of p T and E / T for the signal and background are shown in Fig. 11.According to Fig. 11, for different masses (1 TeV ∼ 7 TeV), we choose the same cuts such that p T ≥ 300 GeV and / E T ≥ 280 GeV for pp → e + ν e and p T ≥ 300 GeV and / E T ≥ 340 GeV for pp → µ + ν µ .The cross-sections after cuts are also fitted using Eqs.(11) and ( 12), σ ee SM = 0.022 pb, and σ µµ SM = 0.016 pb.The results are shown in Table III and Fig. 12.The upper limits on the udL ee and udR ee plane at the 14 TeV LHC with L = 300 fb −1 are shown in Fig. 13.As shown in Figs. 14 and 15, udY satisfies the unitarity bounds generated by Eq. ( 7) and is also within the constraints given by Eq. (10).Comparing these figures with Figs. 8 and 9, the HL-LHC is more sensitive to the W boson than the LHC.The results for higher luminosities, L = 1 ab −1 and 3 ab −1 , are also investigated and shown in Fig. 16.Compared with those at the 13 TeV LHC with L = 139 fb −1 , the expected constraints can be strengthened to approximately an order of magnitude for the 14 TeV LHC with L = 3 ab −1 .(right panel).

V. CONCLUSIONS AND DISCUSSIONS
Many extensions of the SM predict the presence of charged heavy gauge bosons that might be found at the LHC, which are commonly referred to as the W boson and can induce the CC NSI.In this paper, the contributions of W bosons to the process pp → ν are investigated.Before MC simulation, we first consider the theoretical constraints on the simplified W model and further on qq Y from two perspectives: the partial wave unitarity and W decays.Our numerical calculation shows that interference effects play a vital role, and as a consequence, the expected constraints on L are not less mass-dependent than in previous studies.The event selection strategy is then studied.With the help of S stat , the expected constraints on are estimated.For the integrated luminosities considered in this paper, the expected constraints are insensitive to the nature of neutrinos (Dirac or Majorana) [76].
To date, there have been many studies on the constraints on NC NSI.For example, Ref. [26] proposed that the LHC sensitivity to NC NSIs is ≤ 2 × 10 −3 for M Z = 2 TeV, and the result of Ref. [25] shows that in the simplified Z model, the upper limits on are 0.042 and 0.0028 corresponding to Z with M Z = 0.2 and 2 TeV, respectively.However, up to our knowledge, studies on the new gauge boson W mainly focus on the constraints on its mass and couplings, and there are few studies on the CC NSI in the context of a simplified W model, especially at the current and future LHC.We focus on the expected 2 TeV in the electron channel [36] and M W > 4.9 TeV in the muon channel [38], at the 2σ CL using the SSM, which is looser than the constraints 6.3 TeV and 6.5 TeV in our paper.Moreover, the expected constraints at the 14 TeV LHC with L = 3 ab −1 can further narrow down the expected constraints to one order of magnitude from those at the 13 TeV LHC.We propose that the interference effects are non-negligible and should be considered in future studies.

FIG. 3 .
FIG. 3. Leading-order (LO) Feynman diagram for the single production of the W boson and subsequent leptonic decays.

MFIG. 4 .
FIG. 4. For the process pp → e + ν e , the normalized distributions of p T and E / T for the signal and background events.The cases for A ud R = 0 are shown in (a,b), while the cases for A ud L = 0 are shown in (c,d).
and the cross-section of the process pp → e + ν e for different M W .The case of M W = 1 TeV is not shown.
which corresponds to the SSM[73], the expected constraints on can be translated to the lower bounds on M W .The results are shown in Fig.10.However, because the MC simulation is carried out for fixed M W , if the lower bound on M W is larger than the value of M W used in the simulation, such value should be ruled out.Consequently, an expected constraint can be set on M W .The expected constraints of the process pp → e + ν e on M W for S stat = 2, 3, and 5 are M W > 6.3 TeV, M W > 5.2 TeV, and M W > 3.8 TeV, respectively.The expected constraints of the process pp → µ + ν µ on M W for S stat = 2, 3, and 5 are M W > 6.5 TeV, M W > 5.4 TeV, and M W > 3.9 TeV, respectively.The results of the LHC experiments are M W > 6.0 TeV for the eν e channel and M W > 5.0 TeV for

1 FIG. 10 .
FIG. 10.For |A qq Y | = |B αα L | = 1, the expected constraints on M W obtained by events generated at M W = m.M W on the left hand side of the dashed-dotted line is ruled out.The intersection of the dashed-dotted lines and the dotted lines gives the lower bound for S stat = 2 as an example.

FIG. 14 .
FIG. 14. Same as Fig. 8 but for udL ee (left panel) and udR ee (right panel) at the 14 TeV HL-LHC

TABLE I .
After the event selection strategy, α int and α Y nsi are fitted for different M W at

TABLE II .
Expected constraints on udY detected at the 2σ, 3σ, and 5σ confidence levels (CLs).

TABLE III .
Same as Table I but for the HL-LHC with √ s = 14 TeV.