Probing anomalous $WW\gamma$ Triple Gauge Bosons Coupling at LHeC

The precision measurement of the $WW\gamma$ vertex at the future Large Hadron electron Collider (LHeC) at CERN is discussed in this paper. We propose to measure this vertex in the $e^{-} p\to e^{-}W^{\pm}j$ channel as a complement to the conventional charged current $\nu_{e}\gamma j$ channel. In addition to the cross section measurement, $\chi^{2}$ method studies of angular variables provide powerful tools to probe the anomalous structure of triple gauge boson couplings. We study the distribution of the well-known azimuthal angle between the final state forward electron and jet in this vector-boson fusion (VBF) process. On the other hand, full reconstruction of leptonic $W$ decay opens a new opportunity to measure $W$ polarization that is also sensitive to the anomalous triple gauge boson couplings. Taking into consideration the superior determination of parton distribution functions~(PDFs) based on future LHeC data, the constraints of $\lambda_{\gamma}$ and $\Delta\kappa_{\gamma}$ might reach up to $\mathcal{O}(10^{-3})$ level in the most ideal case with the 2--3~ab$^{-1}$ data set, which shows a potential advantage compared to those from LHC and LEP data.


I. INTRODUCTION
A Standard Model (SM) like Higgs of 125 GeV has been discovered by the ATLAS and CMS collaborations at the CERN Large Hadron Collider (LHC) [1,2], while other hints at physics beyond the SM (BSM) have not shown up at the current LHC run. On the other hand, there still exist many open questions that have been driving the studies of BSM physics in the last three decades.
For instance, neither the mass of the Higgs boson nor the driving force of electroweak symmetry breaking (EWSB) is explained within the SM. Therefore, precision measurements of known channels, which include precision measurement of Higgs and triple or quartic couplings of electroweak gauge boson (TGCs/QGCs) as well as rare processes of heavy flavor mesons, play an important role in the indirect probe of BSM physics.
Several electron-positron colliders such as FCC-ee, ILC and CEPC have been recently proposed as "Higgs factories" for the precision measurement of Higgs couplings and properties. Beside these lepton colliders, there's another relatively economic proposal for the Large Hadron electron Collider (LHeC), which is an upgrade based on the current 7 TeV proton beam of the LHC by adding one electron beam of 60-140 GeV [3]. LHeC as a deep inelastic scattering (DIS) facility can improve the measurement of parton distribution at larger x at TeV range significantly which is crucial for future high-energy hadron colliders. A recent proposal of turning the machine into a Higgs factory, in which the Higgs bosons are produced via vector-boson fusion (VBF), has come out. Because of significant reduction of the QCD background and VBF forward jet tagging, the bottom quark Yukawa can be measured via h → bb [4]. In addition, by measuring via production instead of from decay, the LHeC has apparent advantages in studying anomalous V V h coupling.
At the same time, there also exist several studies on anomalous TGCs (aTGCs) at these Higgs factories. At the LHeC, the TGCs can be directly probed via single γ/Z and single W production [5][6][7][8]. In this work, we focus on the e − p → e − W ± j process because in this channel leptonic W decay could provide its polarization information as an additional handle. That is to say, one can further use cos θ W , which is defined with the moving direction of decay product and the W boson itself, to distinguish contributions from anomalous couplings. This serves as a useful complementary channel to the aTGC study in single γ/Z production measurement, in which the total cross section and azimuthal angle distribution are usually used.
In principle, the e − p → e − W ± j process contains both diagrams with the W W Z vertex and diagrams with the W W γ vertex, which interfere with each other. However, due to large suppres-sion from Z boson mass, the results are actually insensitive to anomalous W W Z couplings [8].
Therefore, we set the anomalous W W Z couplings to zero and use the results in this study as a direct constraint on the anomalous W W γ vertex. This paper is organized as follows. In the next section, we discuss the physics argument of proposed differential distributions and the current status of aTGC measurement. In section III we discuss the phenomenology of this collider search, which includes event selection and reconstruction, W polarization analysis and azimuthal angle correlation analysis. In section IV, we give the numerical analysis results with the χ 2 method. In the last section, we give a brief conclusion.

II. ATGC AND W POLARIZATIONS
As stated in the introduction, we focus on e − p → e − W ± j which provides additional information on W -polarization as a handle besides the known azimuthal angle dependence ∆φ ej . To measure the W polarization, we choose the muonic decay subchannel, We neglect the electronic and hadronic decay channel to avoid combinatory backgrounds and additional irreducible backgrounds. Detailed discussion on this can be found in section III. The diagrams contributing to e − p → e − µ + ν µ j are shown in Fig. 1. If one computes the single TGC-only diagram as in Fig. 1 (a), the longitudinal polarized W dominates and the cross section is huge. On the other hand, it is well known that the gauge symmetry unitarizes the scattering amplitudes which ensures the consistency condition of theories for spin-1 vector boson. In a theory with exact gauge symmetry such as QED, the requirement that current associated with the gauge symmetry is covariantly conserved leads to the result that the longitudinal-polarized component of massless vector-boson cancels and does not contribute to physical processes as Ward-Takahashi Identity,   The triple gauge boson vertices with anomalous contributions could be generally parametrized by effective Lagrangian as [8,9] respectively. The charge (C) and parity (P) conjugate properties of the terms in Eq. (3) are as follows. g V 4 violates C and CP , g V 5 violates C and P but preserves CP , andκ V andλ V are P and CP violating. The rest of the couplings g 1,V , κ V , and λ V are both C and P conserving.
There're only five C and P conserving aTGCs because electromagnetic gauge symmetry requires g 1,γ = 1. We can reduce two of them for independency because of the relations λ γ = λ Z and [10][11][12]. So the only independent aTGCs are ∆g 1,Z , ∆κ γ and λ γ , which should vanish in the SM. where Λ is in TeV [13]. The cutoff scale Λ is larger than 3 TeV for aTGC sensitivity better than scattering also sets unitarity breaking scales from the present aTGC bound, but they're all in the several TeV range [14]. Therefore, LHeC collision energy is safe from violating scattering unitarity and its high sensitivity to aTGC would improve the unitarity bound for future energy frontier experiments.
In Table I

A. Event selection and signal production
In this section, we discuss the collider phenomenology of aTGC measurement through the e − p → e − W ± j → e − ± ν j process and use MadGraph5 v2.4.2 [18] for a parton-level analysis of the measurements. There are four different leptonic channels. For = e + , the e + e − pair from processes with neutral boson decay would be additional backgrounds that we want to avoid. For = e − , the mistagging rate between the electron from W boson decay and the scattered beam electron is 7%, if we assume the electron from W decay takes the smaller rapidity value. On the other hand, neutral current deep inelastic scattering events in the e − channel are potential sources of backgrounds as well. For = µ − , its signal production rate would be smaller than in the µ + channel because of the parton distribution of proton (uud) at the e − p collider. Thus among all the leptonic channels, we expect the µ + channel to be more sensitive to aTGCs than others. With respect to W hadronic decay channel, we need to consider e − + 3j with a 30.53 pb production cross section as the final state, which is approximately two orders over the leptonic decay channel because of huge QCD processes. When E e = 60 GeV, we checked the dijet from W decay would not appear as a single fat jet. One can set the dijet-invariant mass cut and forward jet tagging as a means to reduce QCD backgrounds and extract electroweak processes, but the cross section is still O(pb) level despairing to probe tiny aTGC contributions. Moreover, because of the jet substructure, we cant define the polar angle between decay product jets and W boson. Therefore, the W boson polarization information we focus on could no longer be used.
In Fig.3 we plot the total cross sections σ tot of the e − p → e − µ + ν µ j process. The basic cuts are where and j mean leptons and jets in the final state, respectively. Off-shell W + contribution is also taken into account for the respect of gauge invariance, though the result is actually dominated by the on-shell W + contribution. The production cross section in the SM is 0.120 pb while small aTGC contributes only O(f b). One can see the σ tot increases monotonically with ∆κ γ and the absolute value of λ γ within the parameter region allowed by current experiments, but this is not yet enough to probe tiny aTGC contributions. Therefore, the kinematic differential distributions are to be used as an indirect probe of the anomalous couplings. We would demonstrate this idea by studying the cos θ µW variable in W boson decay and use it for its polarization information. In addition, the azimuthal angle ∆φ ej , which was used to measure the CP nature of Higgs couplings [19], would be used as well. We turn to more detailed discussion on cos θ µW and ∆φ ej distributions in the e − p → e − µ + ν µ j process with nonvanishing λ γ and ∆κ γ . For concreteness, θ µW is defined as the angle between the decay product µ + in the W + rest frame and W + direction in the collision rest frame. ∆φ ej is the angle between scattered beam electron and parton on the azimuthal plane. In Fig.4 and Fig.5, we show cos θ µW and ∆φ ej distributions varying with λ γ and ∆κ γ when E e = 60 GeV, where the red, blue, green, purple and black lines correspond to the λ γ /∆κ γ = −1, −0.1, 0.1, +1 and 0 (SM) respectively.
According to the semiquantitive description of the e − p → e − W + j process with the helicity technique [8], the aTGC λ γ leads to a significant enhancement in the transverse polarization fraction of the W boson, while ∆κ γ leads to a similar enhancement in the longitudinal component fraction. This could be seen from the cos θ µW distribution in Fig.4. The black line shows that µ + tends to move in direction opposite of the W + boson when there's no aTGC contribution. In the left panel, qualitative change, that the peak moves from cos θ µW = −1 to cos θ µW = 1 as the aTGC terms dominate, could be seen in the red/purple lines. In the meantime, the peak in the right panel moves to cos θ µW 0 due to larger contribution from longitudinal-polarized W when ∆κ γ is contributing. In both panels, the distributions for |λ γ /∆κ γ | = 0.1 are quite similar to the SM distribution, indicating we have to use a more precise method, e.g. the χ 2 method, to measure tiny but nonzero aTGC values. respectively for E e = 60 GeV.
On the other hand, the ∆φ ej distribution would show a peak at ∆φ ej = π without contribution from aTGCs. That is to say, in the SM the scattered e − and jet are dominantly back-to-back on the azimuthal plane. Just like cos θ µW , the ∆φ ej would present a deviation from the SM in its distribution with λ γ and ∆κ γ , as is shown in Fig.5. We also notice that when |λ γ | is large (λ γ = ±1), the shape of the ∆φ ej distribution depends on the sign of λ γ : (i) λ γ = +1, the ∆φ ej distribution has two peaks at ∆φ ej = 0/π as part of the e − and jet now move in the same direction on the azimuthal plane; (ii) λ γ = −1, the maximum of distribution shifts to around ∆φ ej = π 2 . there're always two solutions for the invisible neutrino. One way to distinguish them is to assume W decay products would move in the same direction and have a small angular difference. Then the solution with momentum more parallel with the muon is used to reconstruct the W boson.
In addition, we could also get a single accurate solution for the invisible neutrino by combining energy and z-direction momentum conservation conditions to cancel unknown Bjorken x dependence. Splitting the final states into two parts, the invisible neutrino with p µ νµ and the others(e − , µ + and jet) with p µ e jµ , after a bit more algebra which is shown in the Appendix A, we have where p T νµ is the transverse momentum of the neutrino i.e the missing transverse energy / E T , E e is the energy of the initial electron. This avoids the ambiguity of two solutions.
Another kinematic method is the recoil mass, which was used in the Higgs-strahlung process at the e + e − collider [20]. The final states could be separated into two parts: a scattered electron-jet system with p µ e j and all remaining particles with p µ X called the recoil system. Then, we have where M X is the recoil mass,ŝ is the partonic collision energy square, and E q and E e are the energy of initial parton and electron. Since the process we study gets a large contribution from on-shell W channels, we could simply choose the W boson itself as the recoil system and get a relation of Bjorken x with the known input With this relation, one could solve for the invisible neutrino because z-direction momentum conservation condition is now available. The explicit procedure is shown in the Appendix A. This method works well for events with an on-shell W , but leads to certain deviation for other backgrounds. By the way, the above analysis are based on the definition that the z direction is the electron beam moving direction. The reconstructed partonic collision energy distributions are shown in Fig.6 through the above relation to confirm the validity of the recoil mass method. with Pythia 6.420 [21]. The detector simulation is with Delphes 3.3.0 [22].
The VBF final state consists of only one forward energetic quark. However, the additional jets due to gluon radiation are still inevitable although most of them are soft or colinear to the final-state quark. Therefore, one would need criteria for correct forward jet tagging, for instance, with jet energy or jet rapidity, etc. In this study, for simplicity, we only select the events with one forward jet and veto all the others with a second or more hard jets (P T j > 20 GeV) to minimize the mistag rate of jets. Under these criteria, a full simulation including Pythia 6.420 and Delphes 3.3.0 approximately results in a 30% survival probability.

IV. RESULTS
Without real data, it is always difficult to do a comprehensive analysis of uncertainties. For instance, one of the leading theoretical uncertainties of SM prediction is the PDF variation. We estimate this contribution in the cross section measurement is 0.6% with NNPDF23 nlo as 0119 sets.
On the other hand, one of the purposes of the LHeC is to provide precision measurements of valence quark distributions. The striking improvement of PDF determinations would lead to a dramatic reduction in the above uncertainty by a factor of three to four in the O(10 −2 ) x region of our processes [3]. Therefore, aTGC contributions would not be submerged by the PDF uncertainty and one could combine them for the constraints. We expect this has only an insignificant effect on aTGC constraints and therefore neglect the PDF uncertainty in the following study.
In the meantime, we set / E T > 20 GeV to avoid pileup errors because of the low transverse energy basic cut before constraining the aTGC bounds. This additional cut results in about 87% survival probability. Since the lepton/neutrino p T depends on the polarization of the W boson, the / E T cut certainly affects the cos θ µW distribution. Those events with a neutrino moving in the direction opposite of the W boost direction are likely to be cut away by this cut which corresponds to the cos θ µW toward 1. We expect it to give a minor improvement on the results.
To illustrate the feature of the two kinematic distributions proposed above, we adopt the χ 2 method for large event numbers by assuming that the best-fitting aTGC values of future data equal zero [23].
where N BSM i and N SM i are the numbers of events in the ith bin for the differential distributions with and without aTGCs. In this χ 2 method, we use ten bins to analysis the distributions and take 95% C.L. bounds as the aTGC values. Single-parameter fitting results at parton level are shown in Table II with two electron beam energy options and L = 1 ab −1 integrated luminosity. Two aTGC parameter bounds are pushed to a few O(10 −3 ) level in the most ideal case when there's an upgrade for E e = 140 GeV. In the best measurement channel, we find that ∆φ ej would impose stringent constraints on both λ γ and ∆κ γ . The other observable cos θ µW , however, could put a tight bound on ∆κ γ but fails to constrain λ γ . Moreover, the µ + channel is indeed more sensitive to aTGCs than the µ − channel as we have discussed in section III.
parameter variable µ + decay, E e = 60 GeV µ + decay, E e = 140 GeV   The above results are all obtained via pure partonic level study which is certainly unrealistic.
However, as we have discussed in the previous section, the criteria of vetoing a second or more hard jets minimizes the mistag rate and only gives about 30% survival probability. Therefore, to achieve the same results in a full simulation with Pythia and Delphes, one expects about threefold integrated luminosity.

V. CONCLUSION
We find in the e − p → e − µ + ν µ j subchannel, the sensitivity to λ γ and ∆κ γ could reach O(10 −3 ) when L = 1 ab −1 based on the χ 2 method at parton level with the expectation of more precise PDFs at the future LHeC, while in a full simulation the integrated luminosity needs to be increased to 2-3 ab −1 to be consistent with the result. Furthermore, the same result might be reached with approximately half integrated luminosity if we combined the µ + and µ − channels. From the results in Table II and Fig.7, we could see a significant improvement compared to the present LHC and LEP bounds. Therefore, the measurement of the e − p → e − W ± j process at the LHeC would provide a promising opportunity to probe aTGCs and improve our knowledge of the gauge sector.
For future aTGC measurement, we expect complementary studies with different electron beam polarizations and more realistic detector-level analysis to be helpful.
With regard to more technical analysis methods, we may further consider the joint distribu-tion of ∆φ ej and W boson polorization, which could be realized by dividing each ∆φ ej bin into three sub-bins corresponding to three W boson polarization states with fractions f L , f R , andf 0 respectively [24]. On the other side, these polarization fractions are also able to be calculated by decomposing the cos θ µW distribution in Legendre polynomails of cos θ µW .
Finally, it is noteworthy that the kinematic methods in event reconstruction, through which one could retrieve z-direction momentum conservation condition despite of the ignorance of initial state parton and final state neutrino momentums. We believe the kinematic methods are useful for future measurements of processes with / E T at this ep collider.  where E e = p z e and E P = −p z P ; In the final state, p µ e : (E e , p x e , p y e , p z e ) p µ j : (E j , p x j , p y j , p z j ) p µ µ + : (E µ , p x µ , p y µ , p z µ ) p µ νµ : (E νµ , p x νµ , p y νµ , p z νµ ).

Method 1: Energy-momentum conservation
We split the final states into two parts: the invisible neutrino with p µ νµ and the others(e − , µ + , and jet) with p µ e jµ . Then, we have p z e µj = p z e + p z µ + p z j .

Method 2: Recoil mass
First, the final states could be separated into two parts: a scattered electron-jet system with p µ e j and all remaining particles with p µ X called the recoil system. Since the process we study gets a large contribution from on-shell the W channels, we could simply choose the W boson itself as the recoil system. In the partonic level, we have p µ q ≡ xp µ P : (E q , 0, 0, p z q ) p µ e j ≡ p µ e + p µ j : (E e j , p x e j , p y e j , p z e j ) (A5) p µ X ≡ p µ q + p µ e − p µ e j : (E X , −p x e j , −p x e j , p z X ), where q is the parton from the initial proton and x is the unknown Bjorken parameter. Then we can calculate the partonic collision energy squareŝ of this process, as well as computeŝ through the final-state particles: s = (p µ X + p µ e j ) 2 = M 2 X + M 2 e j + 2 E x E e j + (p x e j ) 2 + (p y e j ) 2 − p z X p z e j = M 2 X + M 2 e j + 2 E e j (E q + E e − E e j ) − p z e j (p z q + p z e − p z e j ) + (p x e j ) 2 + (p y e j ) 2 = M 2 X − M 2 e j + 2E e j (xE P + E e ) − 2p z e j (xp z P + p z e ) where M X andM e j are invariant masses of the recoil system and scattered electron-jet system respectively i.e. M 2 X = p µ X · p Xµ , M e j = p µ e j · p e jµ . E X = E q + E e − E e j , and p z X = p z q + p z e − p z e j have been used in the above derivation process. Finally, combining Eqs. (A5) and (A6), and substituting the W boson mass M W for the recoil mass M X , we can get the unknown Bjorken x: (A6) = (A7) ⇒ M 2 W = 4xE e E P + M 2 e j − 2E e j (xE P + E e ) + 2p e j z (xp z P + p z e ) ⇒ x = M 2 W − M 2 e j + 2E e (E e j − p z e j ) 2E P (2E e − E e j − p z e j ) , (2E e − E e jµ − p z e jµ = 0) (A8) So, it is easy to get the z direction momentum of the invisible neutrino: