Probing the minimal $U(1)_X$ model at future electron-positron colliders via the fermion pair-production channel

The minimal $U(1)_X$ extension of the Standard Model (SM) is a well-motivated new physics scenario, where anomaly cancellation dictates new neutral gauge boson ($Z^\prime$) couplings with the SM fermions in terms of the $U(1)_X$ charges of the new scalar fields. We investigate the SM charged fermion pair-production process for different values of these $U(1)_X$ charges at future $e^-e^+$ colliders: $e^+e^-\to f\bar f$. Apart from the standard $\gamma$ and $Z$-mediated processes, this model features additional $s$-channel (or both $s$ and $t$-channel when $f=e^-$) $Z^\prime$ exchange which interferes with the SM processes. We first estimate the bounds on the $U(1)_X$ coupling $(g^\prime)$ and the $Z^\prime$ mass $(M_{Z^\prime})$ considering the latest dilepton and dijet constraints from the heavy resonance searches at the LHC. Then using the allowed values of $g^\prime$, we study the angular distributions, forward-backward $(\mathcal{A}_{\rm{FB}})$, left-right $(\mathcal{A}_{\rm{LR}})$ and left-right forward-backward $(\mathcal{A}_{\rm{LR, FB}})$ asymmetries of the $f\bar{f}$ final states. We find that these observables can show substantial deviations from the SM results in the $U(1)_X$ model, thus providing a powerful probe of the multi-TeV $Z^\prime$ bosons at future $e^+e^-$ colliders.


I. INTRODUCTION
Although the Standard Model (SM) is on a solid theoretical foundation and has been tested experimentally to great accuracy, it cannot explain the observations of nonzero neutrino masses, dark matter relic density and matter-antimatter asymmetry in the Universe [1].
These empirical evidences and other theoretical considerations indicate the necessity for an extension of the SM.
A simple beyond the SM (BSM) scenario that can in principle address some of the abovementioned issues is to extend the SM gauge group by an additional U (1) gauge symmetry.
The associated neutral gauge boson (known as Z ) has been extensively studied in the literature due to its wide range of phenomenological aspects; see Refs. [2,3] for reviews.
There are many ultraviolet-complete scenarios, where the Z boson naturally arises, such as in the Left-Right symmetric models [4][5][6], and in theories of grand unification based on SO(10) [7,8] and E 6 [9,10]. The Z bosons also inevitably appear in the gauge-Higgs unification scenario where the Higgs boson is identified with a part of the fifth-dimensional component of the gauge fields, and the Kaluza-Klein (KK) excited modes of the photon and Z boson become Z bosons [11][12][13][14][15][16]. Dedicated searches for the Z boson have been previously carried out at LEP [17] and Tevatron [18,19], but the most stringent bounds on the Z mass and coupling currently come from the LHC dilepton searches [20,21], which also supersede the low-energy electroweak constraints [22].
In this paper we investigate the future e + e − collider prospects of a general but minimal U (1) X extension of the SM where, in addition to the SM particles, three generations of right-handed neutrinos (RHNs) and a SM-singlet U (1) X Higgs field are included. The U (1) X charge assignment for the fermions in this scenario is generation-independent which makes the model free from all gauge and mixed gauge-gravitational anomalies. Reproducing the Yukawa structure of the SM, one finds that the U (1) X symmetry can be identified as the linear combination of the U (1) Y in SM and the U (1) B−L gauge groups [23][24][25][26]. Hence the U (1) X scenario is the generalization of the U (1) B−L extension of the SM [27,28].
Due to the presence of the Z boson with modest to large couplings to SM fermions under the gauged U (1) X extension, the model shows a variety of interesting features at the e + e − colliders. In particular, the general charge assignment of the particles after the anomaly cancellations leads to potentially large parity violation in the fermion couplings and distinct interference effects in the process e − e + → ff (where f stands for the SM fermions). We investigate this process for both leptonic and hadronic final states, by analyzing the crosssections as well as different kinematic observables, including the forward-backward asymmetry (A FB ), left-right asymmetry (A LR ) and left-right forward-backward asymmetry (A LR,FB ).
We show that even if the Z boson is sufficiently heavy and off-shell (even inaccessible at the LHC), large deviations from the SM expectations in the angular distributions, forwardbackward asymmetries, left-right asymmetries and left-right forward-backward asymmetries can be seen at the proposed e − e + colliders. 1 We consider various center-of-mass energy values √ s = 250 GeV, 500 GeV, 1 TeV and 3 TeV to illustrate this effect. Furthermore, we take special care for the e − e + → e − e + Bhabha process, which can proceed via either s or t-channel Z boson, in addition to the SM γ and Z exchanges. Here we study the deviations in differential and total cross sections, and in left-right asymmetry from the SM results, which are then compared with the theoretically estimated statistical errors.
It is worth noting here that to obtain the bounds on the Z boson at the LHC, the CMS and ATLAS collaborations use the so-called sequential SM where the couplings of the Z boson with the fermions are exactly same as those of the SM Z boson [30]. In our U (1) X scenario, we reinterpret these bounds, properly taking into account the appropriate Z branching ratios to dileptons, and obtain the updated limits on the U (1) X gauge coupling (g ) as a function of the Z mass, which are then used in our numerical analysis for e + e − → ff .
The paper is organized as follows. We discuss the U (1) X model, model parameters and the constraints on the g in Sec. II. We study different observables related to e − e + → f f scattering process for f = e in Sec. III. We discuss the Bhabha scattering in Sec. IV. Some discussion on usefulness of the kinematic variables is given in Sec. V. We finally conclude the paper in Sec. VI.

II. THE U (1) X MODEL
The model we consider here is based on the gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y ⊗ U (1) X . The particle content is shown in Table I. In addition to the SM particles, three generations of the RHNs are introduced to cancel the gauge and mixed gauge-gravity anomalies. 1 Similar consequences have been predicted in the SO(5)×U (1)×SU (3) gauge-Higgs unification formulated in the Randall-Sundrum warped space in which the KK modes of the photon, Z boson, and Z R boson play the role of Z bosons [29].
Gauge group  There also exists a SM-singlet scalar Φ which generates the Majorana mass term for the RHNs after the U (1) X symmetry breaking. The RHNs couples to the SM lepton ( L ) and Higgs (H) doublets to generate the Dirac Yukawa couplings that go into the seesaw mechanism for neutrino masses [31][32][33][34][35]. To introduce the fermion mass terms and the flavor mixings, the Yukawa interaction can be written as (1) whereH ≡ iτ 2 H * (τ 2 being the second Pauli matrix). The U (1) X charges of all the particles are shown in Table I after solving the gauge and mixed gauge-gravity anomalies [25] and using the Yukawa interaction from Eq. 1. We see that x H = 0 and x Φ = 1 will reproduce the B − L scenario. From the structure of the individual charges we can confer that the U (1) X gauge group can be considered as a linear combination of the U (1) Y and U (1) B−L gauge groups. The U (1) X gauge coupling g is a free parameter of our model which appears as either g x H or g x Φ in the interaction Lagrangian. Without the loss of generality we fix x Φ = 1 in this paper. As a result x H acts as an angle between the U (1) Y and U (1 The renormalizable Higgs potential of the model is given by In the limit of small λ , the mixing between the scalar fields H and Φ is negligible, so they can be analyzed separately [25,36]. After the electroweak and U (1) X symmetry breaking the scalar fields H and Φ develop their vacuum expectation values (VEVs) At the potential minimum where the electroweak scale is marked with v 246 GeV, v Φ is considered to be a free parameter with v 2 Φ v 2 . After the symmetry breaking, the mass of the U (1) X gauge boson (Z ) can be expressed as The U (1) X VEV governs the Majorana mass term for the RHNs from the fifth term of the Eq. 1 and the electroweak VEV generates the Dirac neutrino mass term from the fourth term of Eq. 1. They can be written as 2 v respectively. Hence the full neutrino mass mixing can be written as Diagonalizing Eq. 5 the light neutrino mass can be generated as m ν −m D m −1 N m T D in the seesaw limit [31][32][33][34][35].

A. Z interactions with fermions
Due to the presence of the general U (1) X charges (q f L,R x ) shown in Table I, the Z interactions with the SM quarks (q) and leptons ( ) can be written as where P L and P R are the left and right projection operators (1 ∓ γ 5 )/2 respectively. Using Eq. 6 we can calculate the partial decay widths of Z into the SM fermions. For charged fermions, we get where N c = 3 (1) is a color factor for the quarks (leptons) and g f L(R) g , x H , x Φ is the coupling of the Z with left (right) handed charged fermions, which depends on the U (1) X charges. The partial decay width of the Z into a pair of single-generation light neutrinos can be written as The partial decay width of the Z into a pair of RHNs can be written as However, in this analysis we assume for simplicity that the decay of the Z into a pair of RHNs is kinematically disallowed because m N > M Z . 2 Using the partial decay widths of Z from Eqs. 7 and 8 we show the variation in total decay width of the Z (Γ), normalized by g 2 , as a function of x H in the left panel of Fig. 1 for TeV and x Φ = 1. The branching ratios of Z into single-generation SM fermions are shown in the right panel of Fig. 1 Table I.

B. Collider bounds
The U (1) X charges of the particles for different values of x H with x Φ = 1 are given in where g 2 /4π is taken to be 1 by convention, δ ef = 1 (0) for f = e (f = e), η AB = ±1 or 0, and Λ f ± AB is the scale of the contact interaction, having either constructive (+) or destructive (−) interference with the SM processes e + e − → ff [44]. Following Ref. [45] we calculate the Z exchange matrix element for our process as Machine √ s 95% CL lower limit on M Z /g (in TeV) where x and x e are the U (1) X charges of e L and e R respectively, and similarly, x f L and x f R are the U (1) X charges of f L and f R respectively, all of which can be found in Table II.
Matching Eqs. 10 and 11 we evaluate the following bound on M Z as considering M 2 Z s where √ s = 209 GeV for LEP-II. Using Eq. 12, we can translate the LEP bounds on Λ f ± AB reported in Ref. [17] to the bounds on M Z /g as a function of x H , as shown in Fig. 2 by the grey-shaded region. We use the 95% confidence level (CL) limits on Λ ± from Ref. [17] for both hadronic and leptonic channels, where for the latter, we assume universality in the contact interactions. Moreover, for any given x H value, we consider all possible chirality structures, i.e. AB = LL, RR, LR, RL, V V and AA. The exclusion contour shown in Fig. 2 is obtained by taking the boundary of the most stringent bounds. Using the same procedure, we also estimate the prospective reaches at the ILC with √ s = 250 GeV, 500 GeV and 1 TeV from the Λ f ± AB values reported in Ref. [46], as represented by red dotted, purple dashed and green dot-dashed lines respectively in Fig. 2.
Our results are summarized in Table III for some benchmark values of x H to be used in our subsequent analysis.
We can easily translate the limits on shown by the unshaded magenta dot-dashed, dashed and dotted lines for √ s = 250 GeV, 500 GeV and 1 TeV, respectively.
For comparison, we also calculate the hadron collider bounds in the M Z − g plane for different x H values by recasting the current ATLAS and CMS search results for Z in both dilepton [20,21] and dijet [47,48] channels, as shown in Fig. 3 by various shaded regions.In each case, we calculate the Z -mediated production cross section in our model (σ Model ) for a given M Z with fixed x H , properly taking into account the modified branching ratios, and compare it to the observed 95% CL limit on the cross section (σ Obs. ) to derive an upper bound on the coupling strength where g Model is the coupling considered to calculate σ Model . The FeynRules file of the model can be found in [49] . For the dilepton channel, we consider the electrons and muons combined to derive the limits shown in Fig. 3. We also consider the future high-luminosity regions are ruled out by the current experimental data from LEP-II [17], and LHC dilepton [20,21] and dijet [47,48] searches. The future HL-LHC [50], as well as the ILC prospects (this work), are also shown as unshaded curves for comparison. The middle panel with x H = 0 is the B − L case. To illustrate our point, for the rest of this paper we will consider a specific benchmark value of M Z = 7.5 TeV, which is just beyond the LHC reach. From In Table III, we have used ILC with different √ s options just as a representative for future e + e − machines. Our analysis in this work is equally valid for other e + e − collider proposals.
For completeness, we summarize in Fig. 4 the expected run time, total integrated luminosity and the center-of-mass energy options for four future e + e − collider proposals currently being discussed, namely, FCC-ee [51], CEPC [52], ILC [53] and CLiC [54]. In the following, we will generically consider the possibilities of √ s = 250 GeV, 500 GeV, 1 TeV and 3 TeV, all with L int = 1 ab −1 . which affect their interactions with Z . Since the U (1) X charge and gauge coupling are assumed to be family-universal, we will consider the representative case of f = µ for leptonic final states, and f = b and t respectively for the down-type and up-type quark final states in the process e − e + → f f . Note that in a realistic detector environment, the top quarks can only be identified by their decay products, i.e. bottom quarks and W bosons (which are further characterized depending on whether they decay leptonically or hadronically).
However, for simplicity, we restrict our study of the Z effect in the e − e + → f f process to parton level only, which already illustrate the main points we want to emphasize, and moreover, all the numerical results presented here can be understood analytically. A full detector-level simulation, including systematic effects, detector efficiency for the leptons and misidentification of jets or leptons, is beyond the scope of the current work, and will be pursued elsewhere. Such a detailed study will be more relevant when the actual e + e − collider is built.
We capture the Z effects in the process e − e + → f f by considering several kinematic observables, as described below.

A. Differential cross section
Let us first consider the differential scattering cross sections for the processes e − L e + R → f f and e − R e + L → f f , which can be respectively written as where θ is the scattering angle, m f is the final state fermion mass and β = 1 − 4m 2 f s . In the high energy collider limit when m f √ s, we obtain β → 1. In Eqs. 14 and 15 we use x H Interaction Z contribution observable for From Table II  top row, second column, whereas both s|q LL | and s|q LR | contribute in the bb and tt cases.
In case of x H = 1 the coupling between d R and Z vanishes; therefore, the Z contributions from s|q LL | and s|q RL | only are observed in the e − e + → bb process. At x H = 2 all the s|q XY | quantities contribute to the Z resonance in e − e + → f f because in this case all the charged fermions have non-vanishing couplings with Z . 3 The effects of Z on the q XY observables depending on the x H values are summarized in Table IV.

B. Total cross section
An important advantage of lepton colliders is that the incoming beams can be polarized.
Let us consider the polarized electron and positron beams with the polarization fractions P e − and P e + respectively. The differential scattering cross section of the process e − e + → f f can be written as where P eff = P e − −P e + 1−P e − P e + is the effective polarization, and the differential cross sections dσ LR d cos θ and dσ RL d cos θ have been defined in Eqs. 14 and 15 respectively. From Eq. 17 we calculate the total cross section by integrating over the scattering angle as where θ max depends upon the experiment. Theoretically using cos θ max = 1 we get where Furthermore considering m f √ s we get The statistical error of the cross section ∆σ stat (P e − , P e + ) is given by where N = L int σ (P e − , P e + ) is the total number of signal events. The deviation of the total fermion pair-production cross section can be written as To study the effect of beam polarization on the cross section, we consider three polar- We first consider the e − e + → µ − µ + process which is shown in the top panels of Fig. 7 for different x H values. As shown in Table IV, for x H = −1 there is no interaction between e R and Z , so the cross sections and deviations will have the BSM effect only from q LL .
Similarly, for x H = −2 there is no interaction between L and Z ; thus the only BSM effect comes from q RR . These features are manifest in the e − e + → µ − µ + cross sections, which only slightly deviate from the SM case for these x H values, except exactly at the resonance. On the other hand, for x H = 1 and 2 the BSM contributions will come from all q XY amplitudes to create larger deviations in the total cross sections by widening the resonance.
We perform similar analyses for e − e + → bb and e − e + → tt processes which are shown in the middle and bottom panels of Fig. 7 respectively. The nature of the total cross section is the same in these two cases; however, differences appear for different x H values, as can be seen from Table IV. The bb process will be uniquely affected at x H = 1 as there is no interaction between Z and d R . As for the tt final state, we include the top quark mass of 172 GeV, which is why the cross section goes down when √ s approaches this value from above and we only consider √ s ≥ 350 GeV for this process.
As shown in Fig. 7 Table IV.
To see the effect of other polarization choices on the total cross section given by Eq. 23, we now set P e + = 0 and study the variation of the deviation ∆ σ as a function of P e − in its entire theoretically-allowed range of −1 ≤ P e − ≤ 1, as shown in Fig. 8  Thus the statistical error decreases with increasing cross sections (or increasing √ s) for a fixed luminosity.
In the µ + µ − process ∆ σ can reach up to 2.7% for P e − = 0.8 and 1.5% for P e − = −0.8 at √ s = 250 GeV with x H = 2. The deviations for other x H are comparatively small. At √ s = 500 GeV these values become 11% and 6% at P e − = 0.8 and −0.8 respectively for x H = 2. At the same √ s these values become 2.8% and 2.5% respectively for x H = 1.
These deviations gradually increase with √ s, while the statistical error decreases, as can be seen by comparing the different columns in Fig. 8. We find that for some of the x H and P e − values, the deviations can be larger than the statistical error, and hence, observable at future colliders.
At √ s = 250 GeV, ∆ σ for bb is roughly below 1% for all x H when P e − = 0.8 and where the cross sections in the forward (σ F ) and backward (σ B ) directions can be defined by taking the limits of the θ integration in Eq. 18 as [0, + cos θ max ] and [− cos θ max , 0] respectively. For m f √ s and cos θ max = 1, Eq. 25 is reduced to where the coupling dependent quantities B 1 and B 2 can be defined as The integrated FB asymmetry from Eq. 25 is shown in Fig. 9  (P e − , P e + ) 3 4 For other x H values, the BSM effects come from all q XY combinations. In Eqs. 29  For the e − e + → bb process, we notice that for x H = −1 the BSM contributions come from q LL and q LR in the FB asymmetry, whereas for x H = 1 the BSM contributions come from q LL and q RL . The expressions for these charges can be written as In this case x H = −2 will affect the interaction between electron and Z which will be reflected in the nature of A FB . The nature of the asymmetries for bb process is shown in the middle panels of Fig. 9  in the bottom panels of Fig. 9 for different values of x H and different sets of polarizations.
The deviation of the FB asymmetry from the SM result can be defined as ∆ A FB is shown in Fig. 10  estimated statistical error represented by gray-shaded band has been calculated as where N F (B) = L int σ F (B) (P e − , P e + ) is the number of events in the forward (backward) direction. The different columns correspond to √ s = 250 GeV (except for tt), 500 GeV, 1 TeV and 3 TeV. The nature of the deviations shown here can be understood from Eqs. 29-32.
The BSM contributions in differential FB asymmetry for different fermions for different x H are guided by Table IV.
for P e − < 0 and |P e − | > |P e + |. Hence we find that Eq. 37 is related to Eq. 36 by The nature of differential LR asymmetry from Eq. 37 is shown in Fig. 11 for µ + µ − (top  Table IV. According to that, the differential LR asymmetry for x H = −2 and −1 for the e − e + → µ + µ − process from Eq. 36 can be written as The results are shown in the top panels of Fig 11. For the bb process we find the BSM contributions for different x H in the differential LR asymmetry in terms of q XY following Table IV. Using Eq. 37, the differential LR asymmetry of this process for x H = −1 and 1 can be written as The results are shown in the middle panels of Fig 11. Similarly, we study the LR asymmetry for e − e + → tt process according to Table IV and Eq. 37 for different x H . We find that for all three final states, the size of the differential LR asymmetry increases with the increase in √ s from the SM prediction (solid black line). In the U (1) X case, it is governed by different couplings of the left and right-handed fermions with Z , as summarized in Table IV.
The amount of deviation from the SM in the differential asymmetries can be defined as We show these deviations using M Z = 7.5 TeV in Fig. 12 where N LR = L int σ LR and N RL = L int σ RL .
From Fig. 12, we find that the deviation in differential LR asymmetry for e − e + → µ + µ − process for x H = −2 is slightly above the theoretically estimated statistical error for cos θ > 0.37; however, for x H = −1 it is within the range of the statistical error. For x H = 1 and 2, the deviations vary between 5% − 3% and 10% − 8% respectively for −1 ≤ cos θ ≤ 1.
The deviations increase with √ s. The differential LR asymmetry is negative for x H = 2 at √ s = 3 TeV which is reflected in the deviation of the differential LR asymmetry as well.
The deviation in differential LR asymmetry as a function of cos θ for the process e − e + → bb has a singularity. This is because the differential LR asymmetry for bb process in the for √ s = 3 TeV for the SM. Around these angles the differential LR asymmetry for the bb process is very high and rapidly grows towards 100%. As for the tt process, at √ s = 500 GeV the differential LR asymmetry is greater than 6% at x H = 2 for cos θ > 0. The deviation is around 3% for x H = 1 for cos θ > 0. For the rest of the choices of x H it stays within the theoretically estimated statistical error. With the increase in √ s, the deviation increases with cos θ; however, x H = −1 stays within the theoretically estimated statistical error throughout.

E. Integrated left-right asymmetry (A LR )
We also calculate the integrated A LR by integrating Eq. 35 over the scattering angle: In terms of the gauge couplings of the fermions, we can write Eq. 45 as In the limit m f √ s, Eq. 46 is reduced to The observable integrated A LR as a function of the electron and positron beam polarizations is given by for P e − < 0 and |P e − | > |P e + |. This is related to Eq. 45 by The integrated LR asymmetries from Eq. 46 for e − e + → µ + µ − , bb and tt as a function of √ s are shown in Fig. 13 with M Z = 7.5 TeV and for different x H values. The BSM A LR e + e -→ μ + μ - contributions depending on the x H charges are governed according to process and x H = −2 and −1 can be written as In case of the other x H charges the BSM contributions come from all q XY .
From Eq. 46 and Table IV we can write the integrated LR asymmetry for e − e + → bb process at x H = −1 and 1 as The BSM contributions are from the |q XY | quantities. For x H = 2 the contribution will come from all |q XY |. Similar behavior will be observed for the tt process where the BSM contributions will be associated with the choices of x H following Table IV. From Fig. 13, we see that for all the fermion pair-production processes, the x H = −1 case is close to the SM, whereas other x H charges can lead to significant differences with respect to the SM prediction.
The amount of deviation from the SM in the integrated LR asymmetry can be defined as Δ ALR e + e -→ μ + μ - Δ ALR e + e -→ bb In terms of the gauge interactions of the fermions, we write A LR,FB as For m f √ s the differential LR-FB asymmetry can be written as The observable LR-FB asymmetry can be written as A LR,FB (P e − , P e + , cos θ) = σ(P e − , P e + , cos θ) + σ(−P e − , −P e + , − cos θ) − σ(−P e − , −P e + , cos θ) + σ(P e − , P e + , − cos θ) σ(P e − , P e + , cos θ) + σ(−P e − , −P e + , − cos θ) + σ(−P e − , −P e + , cos θ) + σ(P e − , P e + , − cos θ) , for P e − < 0 and |P e − | > |P e + |. The relation between A LR,FB (cos θ) in Eq. 55 and A LR,FB (P e − , P e + , cos θ) in Eq. 58 is given by The differential LR-FB asymmetry defined in Eq. 57 as a function of cos θ for M Z = 7.5 TeV is shown in Fig. 15  For the e − e + → µ − µ + process, it starts to become noticeable from √ s = 250 GeV depending on x H and cos θ. The differential LR-FB asymmetry for bb process also follows the same behavior from √ s = 1 TeV. For tt process the asymmetry parameter starts to become different from the SM results from √ s = 500 GeV depending on x H and cos θ. The LR-FB asymmetry involves the couplings of the Z with the SM charged fermions which contain BSM effects governed by Table IV. From Eqs. 56, 57 and using Table IV the differential LR-FB asymmetry for the e − e + → µ − µ + process for x H = −2 and −1 can be written as A LR,FB (cos θ) x H =−1 2 cos θ 1 + cos 2 θ In case of e − e + → bb process, the differential LR-FB asymmetries for x H = −1 and 1 can be written as For x H = 2 all q XY contribute in the differential LR-FB asymmetry in the bb process.
Similar behavior is observed in case of e − e + → tt process depending on the choices of x H and following Table IV. The nature of the differential LR-FB asymmetry is governed by the term cos θ 1+cos 2 θ . In case of µ + µ − process the BSM effect is nominal for x H = 1 and 2 at √ s = 250 GeV for larger values of | cos θ|; however, with the increase in √ s the differential LR-FB asymmetry becomes prominently different from the SM results. Similar behavior can be observed for bb and tt processes depending on √ s and θ.
The deviation in the differential LR-FB asymmetry from the SM can be defined as This is shown as a function of cos θ in Fig. 16  (n 3 + n 2 ) √ n 1 + √ n 4 + (n 1 + n 4 ) √ n 3 + √ n 2 where (n 1 , n 2 , n 3 , n 4 ) = (N LRF , N RLF , N LRB , N RLB ), N iF = L int σ i ([0, cos θ]) and N iB = L int σ i ([− cos θ, 0]) with (i = LR, RL). ∆ A LR,FB is a ratio between the two differential LR-FB quantities. As a result the model independent quantity cos θ 1+cos 2 θ gets canceled from the numerator and denominator. Therefore the deviations in the differential LR-FB asymmetry are independent of cos θ. The variation with respect to x H involves the BSM effects from different q XY following Table IV. We also comment from Fig. 15 that A LR,FB (cos θ) is an anti-symmetric function of cos θ, hence the integrated A LR,FB will be zero. Therefore for the LR-FB asymmetry, only the differential variables are useful to study.
From Fig. 16 we obtain that µ − µ + process can have a sizable deviation around 3.2% and 8.4% for x H = 1 and 2 at √ s = 250 GeV respectively from the SM. At √ s = 500 GeV the deviation increases up to 6.2%, 14% and 38% fro x H = −2, 1 and 2 respectively.
The corresponding deviations become orders of magnitude higher at √ s = 1 TeV and 3 TeV respetively. We notice similar behavior for bb and tt processes, however, the deviations depend on √ s and x H . We have noticed that the deviation is negative in some cases where the observable quantity is sub-dominant over the SM case.

IV. OBSERVABLES FOR THE BHABHA SCATTERING PROCESS
For f = e in the process e − e + → f f , we get the Bhabha scattering which has both s-channel and t-channel contributions from neutral vector bosons. In the SM, the Bhabha scattering is induced by γ and Z-mediated channels, whereas in the U (1) X model an additional contribution from the Z boson is present. These three channels also interfere due to presence same initial and final sates. The coupling between Z and the electron contains the U (1) X charge. As a result the effect of x H will be manifest in the Bhabha scattering.
For the longitudinally polarized initial states the differential scattering cross section can be written as The corresponding differential scattering cross sections can be written as where s, t and u are the Mandelstam variables given by s = (E e − + E e + ) 2 , t = −s sin 2 θ 2 and u = −s cos 2 θ 2 , and E e + , E e − are the incoming electron and positron energies respectively. The quantities q s(t) are the corresponding s (t)-channel propagators. The propagators for the s-channel process can be written as and those for the t channel process are Here e = √ 4πα, α = 1 137 , g L , g R are the left and right-handed couplings of the electron with the Z boson, and g L , g R are the left and right-handed couplings of the electron with the Z boson, respectively. Using the above expressions, we define which are plotted in Fig. 17 for the SM (top left) and also for the U (1) X model with different there is no coupling between e R and Z .

A. Differential and integrated cross sections
The differential scattering cross section from Eq. 66 can be written as where the three terms correspond to the s-channel, t-channel and interference between them, respectively. Explicitly, The deviation from the SM for the differential and integrated scattering cross sections can respectively be written as The estimated statistical error can theoretically be calculated as ∆σ(P e − , P e + , − cos θ min , + cos θ max ) = σ √ L int σ .

B. Differential and integrated LR asymmetries
The e − e + → e − e + process contains t-channel scattering; hence the forward scattering dominates. Therefore the FB asymmetry A FB is not a well-measured quantity for Bhabha scattering. On the other hand, the LR asymmetry can be measured when the initial electron and /or positron is longitudinally polarized.
The LR asymmetry of the differential cross section for 1 ≥ P − ≥ 0 and 1 ≥ P + ≥ −1 can be written as (1 + P − P + ) 2s 2 |q t (s, cos θ) LR | 2 + (1 − P − P + ) The LR asymmetry will vanish if both the initial states are unpolarized. The integrated LR asymmetry of the polarized cross sections can be given as The deviation from the SM in the differential and integrated LR asymmetries can be written as respectively. The theoretically estimated statistical error can be estimated as where N 1 = L int σ(P e − = −P − , P e + = −P + ) and N 2 = L int σ(P e − = +P − , P e + = +P + ).

(89)
The asymmetries are beyond the range of the theoretically estimated statistical error for cos θ > 0 and x H = 2 for both sets of polarizations at √ s = 250 GeV. The results for x H = −2 and 1 are also outside the range of the statistical error; however, the difference is not large. The deviations in the asymmetry from the SM result become more prominent for larger √ s for both cos θ < 0 and cos θ > 0.
The integrated LR asymmetry is shown in Fig. 21 for two polarizations (P − , P + ) = Tab. III and Fig. 2 estimated from Ref. [46] are also shown for comparison.

V. DISCUSSION
The various kinematic observables discussed in this work can be used as a post-discovery tool to distinguish between different x H charges in our chiral U (1) scenario where the Z differently interacts with the left and right-handed fermions. To illustrate this point, let us take the e + e − → µ + µ − process as an example and consider the deviation of the integrated left-right asymmetry ∆A LR from the SM prediction. This is plotted in Fig. 22 Fig. 2 and Tab. III) derived from the limits on the effective scales from Ref. [46]. We note that while a simple recasting of the contact interaction analysis in Ref. [46] gives a slightly larger reach for M Z /g as compared to ∆A LR by itself, the latter can be used as a precision tool to probe x H once a deviation in the total cross-section is seen for a given M Z /g . The other fermion-pair final states considered in previous sections (bb, tt, e + e − ) give similar results as in Fig. 22. In principle, all the other kinematic variables discussed here can be combined into a multi-variate analysis which could potentially enhance the sensitivity reach in M Z /g as well, but this is beyond the scope of the current work.

VI. CONCLUSION
We have shown that the general U (1) X scenario can be effectively probed via the fermion pair production process at future e − e + colliders, even when the associated Z boson is well beyond the kinematic reach of the colliders. This will be possible by precisely measuring the deviations of the differential and integrated scattering cross sections, as well as the FB, LR and LR-FB asymmetries, from their SM-predicted values. In particular, since the asymmetries are the ratios of (differential or integral) cross sections, their deviations from the SM values highly depend on the U (1) X charges. In fact, we observe significant deviations from the SM for several choices of the charge x H considering the limits on the U (1) X gauge coupling depending on M Z . Hence we expect that FB, LR and LR-FB asymmetries can be successfully probed in e − e + colliders to test and characterize multi-TeV Z bosons coupling differently to left-and right-handed fermions.