Investigating the Production of Leptoquarks by Means of Zeros of Amplitude at Photon Electron Collider

Leptoquarks are one of the possible candidates for explaining various anomalies in flavour physics. Nonetheless, their existence is yet to be confirmed from experimental side. In this paper we have shown how zeros of single photon tree-level amplitude can be used to extract information about leptoquarks in case of $e$-$\gamma$ colliders. Small number of standard model backgrounds keep the signal clean in this kind of colliders. Unlike other colliders, the zeros of single photon amplitude here depend on $\sqrt{s}$ as well as the mass of leptoquark along with its electric charge. We perform a PYTHIA based simulation for reconstructing the leptoquark from its decay products of first generation and estimating the background with luminosity of 100 fb$^{-1}$. Our analysis is done for all the leptoquarks that can be seen at $e$-$\gamma$ collider with three different masses (70 GeV, 650 GeV and 1 TeV) and three different centre of momentum energy (200 GeV, 2 TeV and 3 TeV).

On the other hand, the phenomenon of RAZ (radiation amplitude zero) was first discussed for q iqj → W ± γ process at pp or pp collider in order to probe the magnetic property of W -boson [67]. This phenomenon has been studied extensively in literature for various BSM models like supersymmetry, leptoquarks, other gauge theories, etc and physics behind its occurrence has also been scrutinized . In non-Abelian theories the tree-level amplitudes 1 for single photon emission processes, which is the sum generated by attaching photon to the internal and external particles in all possible ways, can be factorized into two parts: a) the first part contains the combination of generators of the gauge group, various kinematic invariants, charges and other internal symmetry indices, whereas b) the second part corresponds to the actual amplitudes of the Abelian fields containing the dependence on the spin or polarization indices [68,69]. The first factor goes to zero in certain kinematical zones depending on the charge and four momenta of external particles and forces the single-photon tree amplitudes to vanish [70]. The general criterion for tree-level single photon amplitude to vanish is that p j · k Q j must be same for all the external particles (other than photon) involved in the process [70] where, p µ j and Q j are the four momentum and charge of jth external particle and k µ is the four momenta of photon. For 2 → 2 scattering processes with photon in final state, this condition reduces to: where, Q f 1 and Q f 1 are the charges for the incoming particles f 1 and f 2 and θ * is the angle between photon and f 1 in the centre of momentum (CM) frame at which RAZ occurs provided that the masses of colliding particles are negligible with respect to total energy of the system, i.e. √ s. Linear colliders in the range of a few hundred GeV to 1.5 TeV are going to be build in near future. These colliders can provide the possibility for studying electron-photon interactions at very high energy [99][100][101][102][103][104][105][106][107][108]. Using modern laser technology, high energetic photons with large luminosity can be prepared through laser backscattering for this kind of studies. Since very few SM processes contribute to the background for these electron-photon colliders, they can reveal clean signals of leptoquarks through zeros of tree-level single photon amplitude [109][110][111]. In this paper we have studied this possibility in detail. The phenomena of RAZ in various leptoquark models has already been described in literature in context of e-p colliders where leptoquark is expected to be produced associated with a photon or the it undergoes radiative decays [112,113]. Though our scenario looks quite similar to it, there arises great difference between these two colliders while considering the position of zero amplitude in the phase space. It is evident from Eq. (1.1) that RAZ for e-p colliders occurs at some particular angle between the photon and the quark which depends only on the electric charge of electron and the quark; however, we show that the same angle for zero amplitude at e-γ colliders depends on the mass of the leptoquark as well as √ s along with the electric charge [114]. Nevertheless, the general condition for tree-level single photon amplitude being zero [70] still remains valid.
In this paper we have analysed all kinds of leptoquarks that are going to be produced at e-γ colliders for three different masses (70 GeV, 650 GeV and 1 TeV) with three different centre of momentum energy (200 GeV, 2 TeV and 3 TeV). Though leptoquark with light mass seems to be ruled out, most of these analysis assumes coupling of leptoquark to single generation of quark and lepton, whereas, the results from UA2 and CDF collaboration show that there is still room for low mass leptoquark with sufficiently small couplings and appropriate branching fractions to different generations of quarks and leptons. On the other hand, the bounds on couplings and branching fractions of higher mass leptoquarks are more relaxed. The leptoquark will eventually decay to a lepton and a quark, and hence we it will produce a mono-lepton plus di-jet signal at detector. In a PYTHIA based analysis, we reconstruct the leptoquark from the invariant mass of the lepton and one jet. Then we look for the angle between the reconstructed leptoquark (or the other jet) and photon and construct the angular distribution which should match with the theoretical estimates. Observation of the zeros of this distribution at the theoretically predicted portion of phase space would indicate the presence of some leptoquarks.
The paper is disposed in the following way. In the next section (sec. 2) we describe the theoretical approach to the production of scalar as well as vector leptoquarks at e-γ collider and find the conditions for the zeros of angular distribution. The experimental constrains on the mass, coupling and branching fractions of the leptoquarks have been summarised in sec. 3. In sec. 4, we describe the simulation set up, choice of benchmark points and centre of momentum energies, production cross sections and branching fractions of the leptoquarks and PYTHIA based simulation for different types of leptoquarks produced at the electron-photon collider. Finally, we conclude in sec. 5.

Theoretical formalism
In this section, we develop the theoretical formalism for the production of a leptoquark (more precisely anti-leptoquark) associated with a quark or an anti-quark at the electronphoton collider to get the mathematical expression for the differential distribution of this process. We consider the process e − γ → q φ c where q is a quark and φ is a leptoquark (the sign 'c' indicates charge conjugate), for which there are three possible tree-level Feynman diagram, as shown in fig. 1.

Scalar Leptoquark
If the leptoquark φ is a scalar one, the matrix elements for the respective diagrams are as follows: where p µ e , p µ γ and p µ q are the four momenta of the particles electron, photon and the produced quark respectively, Y L,R are 3 × 3 matrices describing the couplings of leptoquark with lefthanded and right-handed leptons and quarks respectively, e denotes the charge of positron, Q q signifies the charge of q quark in the unit e, M φ indicates the mass of leptoquark, γ µ is the polarization of the photon and P L,R ≡ (1 ∓ γ 5 )/2. Here, we have deliberately neglected the masses of electron and the quark since they would have insignificant effects in determining the zero of amplitude involving production of very heavy leptoquark for all practical purposes unless the produced quark is top. Therefore, after taking the spin and polarization sum of initial and final state particles, the modulus squared matrix element for this mode becomes: where, s = (p e + p γ ) 2 and θ is the angle between photon and the quark q.

Vector Leptoquark
Now, if the leptoquark φ be a vector particle, the matrix elements will get modified in the following way: Here, φ µ is polarization vectors for the vector leptoquark. After taking the spin and polarization sum of initial and final state particles 2 , the modulus squared matrix element becomes: The differential cross-section for this process turns out to be: Here, the one fourth factor comes because of the average over initial state spins and polarizations; on the other hand, the factor three indicates the number of colour combinations available in the final state. Now, it is evident from the Eqs. (2.4) and (2.8) that the differential cross-section vanishes iff: since all the other terms are positive quantities. This also follows from the general condition for tree-level single photon amplitude to vanish [70] : where Q φ is the charge of leptoquark in unit of e and can be expressed as: However, the striking difference between single photon emission with two body final state LQ Y Q em Interaction Process cos θ * Scalar Leptoquarks Table 1. The values of cos θ * for production of different leptoquarks at e − γ collider.   The measurement of R−ratio from PEP and PETRA constrains the scalar leptoquarks to have M φ 15 − 20 GeV [50] in a model-independent way depending on the charges of them only where they are assumed to be pair-produced in the decay of a virtual photon. Measurement from AMY [51] provides M φ ≥ 22.6 GeV for scalar leptoquarks and similar bound for vector ones too. The LEP constrains M φ ≥ 44 GeV [52,53] with the coupling to Z 0 to be 1 /3 sin 2 θ w assuming the pair-production of leptoquarks from Z 0 and further decay of them into jets and two leptons. For decay into first two generations of quarks and leptons, this lower bound is almost independent of branching fraction; however for third generation it depends slightly. UA2 provides the relation between lowest allowed mass and the branching ratio of the leptoquark [54]. Assuming 50% branching to first generation, di-electron+ di-jet channel gives M φ ≥ 58 GeV, electron+ / p T +di-jet channel shows M φ ≥ 60 GeV and combination of them provides M φ ≥ 67 GeV. However, 100% branching to first generation will exclude the mass lower than 74 GeV. DELPHI concludes M φ ≥ 77 GeV [55], but their analysis assumes large coupling for leptoquark-lepton-quark (λ ≥ e). CDF and D0 suggest the mass of leptoquarks to be greater than 113 GeV and 126 GeV [56] respectively, on first and second generation of leptoquarks. Several bounds from meson decays, meson-antimeson mixing, lepton flavour violating decays, lepton-quark universality, g − 2 of muon and electron, neutrino oscillation and other rare processes have been presented in Ref. [12,36,[115][116][117]. If the leptoquark couples to left handed quarks and leptons of first generation only, then according to pdg [117]

Mass and coupling
1T eV ); however, the constraints change for the second generation as . This analysis is done for leptoquark induced four-fermion interaction. Results from ATLAS and CMS [57][58][59] rule out leptoquarks with mass upto 1500 GeV for first and second generation leptoquarks with 100% branching and 1280 GeV for 50% branching. In the fig. 3, we show the plots for branching fraction against the mass of leptoquark from Tevatron and LHC. In the top left panel, data from D0 has been presented, where the brown (obliquely meshed) region represents the disallowed mass range for leptoquark from LEP experiment and the greenish (horizontally meshed) and bluish (vertically meshed) areas indicate the excluded portions for the mass of first and second generation leptoquarks from two-electron plus two-jet and two-muon plus two-jet channels at D0. The rest three plots are from LHC for three generations of leptoquarks. The continuous black line signify the observed limit whereas the green and yellow areas indicate 1σ and 2σ regions. The black, blue and red portions with dashed line inside show theoretical predictions with branching (β) to be 100%, 50% and 10% respectively. Nevertheless, all these analyses have been done assuming that one leptoquark couples to quark and lepton from one generation only. The scenario changes drastically if branching for a leptoquark to quarks and leptons of all the generations are kept open.

Leptoquark models and simulation
For our purpose, we choose four leptoquarks of different charges from scalar sector and same from the vector sector separately. For every leptoquark scenario, we have studied three different benchmark points (with mass 70 GeV, 2 TeV and 3 TeV respectively and different couplings), each of which has been scrutinised at three distinct energy scale (200 GeV, 2 TeV, 3 TeV). The couplings have been picked out in such a way that they lie inside the allowed region, as shown in fig. 3. For low mass leptoquark we use the data from D0, which allows around 25% branching to first and second generations of quarks and leptons at M φ = 70 GeV. For the heavy leptoquark scenarios, one should look at the graphs from ATLAS and CMS. There is no data for ATLAS beyond the mass range 500 GeV > M φ > 1.5 TeV; similarly CMS probe the mass range for leptoquark to be 300    Table 3. Production cross-sections for the chosen leptoquarks at e-γ collider for the benchmark points listed in table 2 at centre of momentum energies to be 200 GeV, 2 TeV and 3 TeV.
The benchmark points used in our analysis for different leptoquarks are described in table 2. It should be kept in mind that R 2 , S 3 , V 2µ and U 1µ do not have any coupling to right-handed leptons. The production cross-sections and branching fractions for all the leptoquarks under consideration have been put together at table 3 and 4 respectively. The tree-level cross-sections and branching fractions have been calculated using CalcHEP 3.7.5 [118]. It should be noticed that the mass of the leptoquark being higher than the centre of momentum energy, the scenarios BP2 and BP3 can not be explored at √ s =200 GeV. On the other hand, top being heavy than the leptoquarks of BP1 case, it will not get produced by decay of the later one. The production cross-sections for the vector modes are in general higher than that of the scalar modes which happens mainly because of two reasons. Firstly, vector leptoquarks couple to the vector currents giving rise to very different distribution from the scalar case. Secondly, any vector leptoquark has three states of polarizations which enhance the production cross-section.

Branching fraction Modes
Branching fraction   Table 4. Branching fractions of the leptoquarks for the given benchmark points.
The zeros of amplitude shows up for the leptoquarks having charges −1 /3 and −2 /3 only since the other ones fail to satisfy Eq. (2.13). The zeros for all these scenarios have been merged in table 5. It should be noted that unlike BP2 and BP3 at √ s = 200 GeV, leptoquark of 1.5 TeV mass (BP3) and charge −1 /3 gets produced at √ s = 2 TeV; but it does not show the zero in distribution since the ratio of its mass squared to s is larger than its charge violating the condition in Eq. (2.12). It should also be noticed that due to low mass of letoquark in BP1, cos θ * reaches the asymptotic value of ±1 /3 at √ s = 2 TeV and 3 TeV in both the cases of Q φ being −1 /3 and −2 /3. In the next few sections, we discuss the kinematical distributions leading to appropriate cuts and final states. Later, we present the signal and background number for those final states for different centre of momentum energies at the integrated luminosity of 100 fb −1 .

Benchmark
Values of cos θ * for zeros of (dσ/d cos θ) at different √ s Table 5. Values of cos θ * corresponding to zeros of differential cross-section for production of leptoquark at different centre of momentum energy for various benchmark points.

Simulation set up
For the simulation in electron-photon collider we implement the scenarios in SARAH 4.13.0 [119]. Later models files are generated for CalcHEP 3.7.5 which is used for signal and background event generation. The generated events have then been simulated with PYTHIA 6.4 [120]. The simulation at hadronic level has been performed using the Fastjet-3.2.3 [121] with with the CAMBRIDGE AACHEN algorithm. For this, the jet size have been selected to be R = 0.5, with the following criteria: • Calorimeter coverage: |η| < 4.5.
• Minimum transeverse momentum of each jet: p jet T,min = 20.0 GeV; jets are ordered in p T .
• No jet should be accompanied by a hard lepton in the event.
• Jet-lepton isolation ∆R lj > 0.4 and lepton-lepton isolation∆R ll > 0.2 • Selected leptons are hadronically clean, i.e, hadronic activity within a cone of ∆R < 0.3 around each lepton should be less than 15% of the leptonic transeverse momentum, i.e. p had T < 0.15p lep T within the cone.
Prepared with this set up, we analyse different leptoquark scenarios and plot the required invariant mass for jet and lepton and their angular correlations. The would guide us to choose the kinematical cuts appropriately.
The leptoquark will eventually decay into a quark (or antiquark) and a lepton providing mono-lepton plus di-jets signal at the electron photon collider. The SM background for this process, shown in fig. 4, is governed by eight Feynman diagrams for each generation of quark-antiquark pair mediated by photon and Z-boson (neglecting the one with Higgs boson propagator since its coupling with electron is very small).While plotting against the invariant mass of lepton-jet pair (M j ), the background gives a continuum, whereas the signal shows a peak at M φ . So, to reconstruct the leptoquark, we first put a cut constraining (M j ) to deviate from M φ by 10 GeV at most, which is denoted as "cut1" in all the signal background analysis table.. Next, to distinguish the daughter jet produced by the decay of leptoquark, we apply an angular cut on the angle between the lepton and each of the jets depending on the boost of the leptoquark. If the three momentum of the leptoquark becomes small, the path of the daughter jet will make an obtuse angle with the final state lepton providing negative values of cos θ j , whereas for a highly boosted leptoquark, it makes an acute angle with the lepton giving positive valued cos θ j . To enhance the significance, we choose the angular cut in such a way that the background reduces conspicuously without much change in the signal event. TeV) are allowed for the production of 70 GeV leptoquark associated with a light jet. As discussed in last paragraph, the leptoquark produced at √ s = 200 GeV will not be boosted highly and hence, we apply the angular cut as −0.2 ≤ cos θ j ≤ 1, which increases the significance from 47.5σ to 50.5σ. But for √ s equal to 2 TeV and 3 TeV the leptoruark will be very highly boosted; so we put an angular cut of 0.9 ≤ cos θ j ≤ 1 that changes the significance from 6.8σ to 6.4σ and 3.7σ to 3.9σ, respectively. In case of BP2, centre of momentum energy of 200 GeV is forbidden for the production of 650 GeV leptoquark. For the rest of two values of √ s, the leptoquark will be moderately boosted. So, an angular cut of 0 ≤ cos θ j ≤ 1 has been employed for both the cases. It elevates the significance from 8.1σ to 14.5σ and 4.1σ to 8.8σ for √ s to be 2 TeV and 3 TeV, respectively. On the other hand, for BP3 also, real leptoquark gets produced at 2 TeV and 3 TeV centre of momentum energy. At √ s = 2 TeV, the produced leptoquark of mass 1.5 TeV moves very slowly and hence an angular cut of −0.9 ≤ cos θ j ≤ 1 has been implemented which enhances the significance to 7.7σ from 8.2σ. Similarly, at √ s = 3 TeV, also a slow leptoquark gets produced for BP3. So, we put an angular cut of −0.8 ≤ cos θ j ≤ 1 which enhances the significance to 5.4σ from 3.5σ.  Table 6. Signal background analysis for leptoquark (S +1 /3 ) c with luminosity 100 fb −1 at e-γ collider.

Bench
In fig. 5, we present the detailed pictorial description of our PYTHIA simulation with 10 5 number of events and luminosity of 100 fb −1 . The graphs are arranged in the same order like in table 6. In the left panel, the number of events has been plotted against the invariant mass of electron and jet for both signal and background at different centre of mass energies for the three benchmark points. The greenish (aqua) regions indicate the SM background whereas, the purple regions signify the signal events. As expected, the signal events peak around the masses of leptoquarks. On the other hand, the number of events against the cosine of angle between the final state electron and the two jets has been plotted in the right panel for same benchmark points with same √ s. While the blue and green lines represent the background events, the yellow and red lines depict the signal events. These plots justify our choice of cuts for invariant mass and the angle between final state lepton and the two jets. If any of the two jets passes those two cuts, we identify that as signal event.   In the fig. 6, the differential cross-section has been delineated against the cosine of the angle between initial state electron and the leptoquark (or equivalently, the angle between photon and the quark that is produced associated with the leptoquark) at different centre of momentum energies for various benchmark points. The green (ragged) lines portray the simulated data with hundred bins within the range −1 < cos θ < 1 whereas, the brown (smooth) lines represent the theoretical predictions given by Eq. (2.9). The plots are arranged in the order of table 6. The left and right plots at the top in BP1 row are for 200 GeV and 2 TeV centre of momentum energies respectively while the third one is for 3 TeV. In BP2 row, the first and second plots are done for 2 TeV and 3 TeV centre of momentum energies respectively. Likewise, for BP3 also the plots for √ s valued 2 TeV and 3 TeV are presented in the left and right panel of the third row. As can be seen, the angular distribution in each graph vanishes at some point except the first one in third row which fails to satisfy the condition described by Eq. 2.12. The positions of zeros can be verified from the left column ( titled "Qq = −2 /3 or Q φ = −1 /3") of table 5.

Leptoquark
The signal-background analysis for ( R +2 /3 2 ) c with luminosity of 100 fb −1 has been rendered in table 7. For BP1, the cut on invariant mass of lepton-jet pair shows significances of 26.4σ, 4.0σ and 2.1σ respectively, at three different values of centre of momentum energy; after applying the angular cuts, as described in case of (S +1 / 1 ) c , the significances become 28.3σ, 4.0σ and 2.4σ respectively. In case of BP2, only significances of 0.6σ and 0.3σ are achieved by cut1 at 2 TeV and 3 TeV centre of momentum energies respectively, which increase to 1.7σ and 0.8σ respectively after implementation of the angular cut 0 ≤ cos θ j ≤ 1. For BP3 with 2 TeV energy, the significances reached by the two cuts are 0.6σ and 0.7σ and the same for 3 TeV energy are 0.3σ and 0.4σ respectively. It should be noticed that the significances are quite low in case of ( R +2 /3 2 ) c compared to (S +1 / 1 ) c especially with BP2 and BP3, and hence escalation in luminosity is essential for amelioration of the statistics. ) c with luminosity 100 fb −1 at e-γ collider. fig. 7 where the brown (even) and green (uneven) lines signify the theoretical estimates and simulated data respectively. The plots are arranged in the same order as of table 7. It can be observed that the distribution in every graph comes to zero at some point of phase space. The positions of zeros can be verified from the right column ( titled "Q q = −1 /3 or Q φ = −2 /3") of table 5.

Leptoquark
The PYTHIA analysis for leptoquark (R +5 /3 2 ) c has been presented in table 8. The cut on M j provides significances of 94.5σ, 13.9σ and 7.8σ for the signal events at three centre of momentum energies in case of BP1 which change to 98.7σ, 13.2σ and 8.1σ respectively after using suitable angular cuts on cos θ j . For BP2, signal events are produced with significances 14.4σ and 8.1σ at 2 TeV and 3 TeV centre of momentum energies respectively, any they get increased to 21.9σ and 14.8σ after applying the angular cut as 0 ≤ cos θ j ≤ 1. For BP3 at √ s = 2 TeV, the significances become 11.4σ and 11.7σ after implementation of the two cuts and the same become 5.9σ and 8.6σ respectively for √ s = 3 TeV.  The fig. 8 describes the differential distribution with respect to the cosine of angle between leptoquark and initial state electron in this scenario. The plots are arranged in the order of table 8. As the earlier cases the green (jagged) and brown (smooth) lines indicate the simulated data with 100 bins and the theoretical expectation for various benchmark points at different centre of momentum energy respectively. Unlike the other two cases, the angular distributions never vanish inside the physical region since this leptoquark does not satisfy Eq. (2.13).

Leptoquarks
The signal-background analysis for ( S +4 /3 3 ) c with luminosity of 100 f b −1 has been rendered in table 9. For BP1, the cut on invariant mass of lepton-jet pair shows significances of 36.1σ, 6.2σ and 3.2σ at centre of momentum energies to be 200 GeV, 2 TeV and 3 TeV respectively. The angular cuts modify these significances to become 38.7σ, 6.8σ and 4.2σ respectively. In case of BP2, the significances achieved by cut1 at 2 TeV and 3 TeV centre of momentum energies are 0.9σ and 0.5σ only, which increase to 2.5σ and 1.4σ respectively after implementation of the angular cut 0 ≤ cos θ j ≤ 1. For BP3 with 2 TeV energy, the significances reached by the two cuts are 0.6σ and 0.7σ respectively, which change to 0.3σ and 0.7σ at √ s to be 3 TeV. In this case also the significances are quite low compared to (S +1 /3 1 ) c especially with BP2 and BP3. Increase in luminosity is needed for improvement of the statistics.  Table 9. Signal background analysis for leptoquark (S +4 /3 3 ) c with luminosity 100 fb −1 at e-γ collider. Fig. 9 shows the comparison between theoretical expectation and PYTHIA simulated data for for the production of (S +4 /3 3 ) c . The plots are arranged in the order of table 9. As the earlier cases the green (uneven) and brown (even) lines indicate the simulated data with 100 bins and the theoretical expectation for various benchmark points at different centre of momentum energy respectively. In this case also no zero of differential distribution in any of the diagrams is found since its charge is smaller than −1 unit.   In table 10, we summarise the signal background analysis for the vector singlet leptoquark GeV, the invariant mass cut of 10 GeV gives 44.7σ significance and further application of the angular cut of (−0.2) ≤ cos θ j ≤ 1 changes it to 46.9σ. For BP1 at centre of momentum energies to be 2 TeV and 3 TeV, the significances after the first cut are 119.8σ and 120.8σ respectively. In these cases, the signal events after the first cut are so large in number relative to the background events that the angular cut becomes obsolete. In case of BP2, the cut on M j produce signal events with significances 9.0σ and 11.3σ for the two values of √ s, which get enhanced to 15.6σ and 19.0σ respectively after constraining the angle θ j within the limit 0 ≤ cos θ j ≤ 1. For BP3 at √ s = 2 TeV, the angular cut increases the significance to 4.6σ from 4.2σ. Likewise, for BP3 at √ s = 2 TeV, the angular cut increases the significance to 3.4σ from 2.2σ. Angular distribution for this case has been limned in fig. 10 where the brown (smooth) and green (ragged) lines signify the theoretical estimates and simulated data respectively. The plots are arranged in terms of benchmark points and centre of momentum energy according to the table 10. All the curves show zero certainly at some points of phase space which matches with the right column of table 5. The signal background analysis for leptoquark (V +4 /3 2µ ) c has been shown in table 11. For BP1, the significances of leptoquark production is very high (506.2σ, 631.5σ and 635.8σ) at all the three values of √ s and angular cuts become almost obsolete. For BP2, the significances after first cut are 26.2σ and 32.1σ which get enhanced to 33.1σ and 40.0σ respectively after the second cut at the two different values of √ s. For BP3 at 2 TeV centre of momentum energy the significances after the two cuts are 3.5σ and 3.9σ respectively. At 3 TeV centre of momentum energy for the same benchmark point, the significances after the two cuts become 4.5σ and 6.7σ respectively.  Figure 11. Angular distribution for the production of (V +4 /3 2µ ) c at various centre of momentum energies for different benchmark points, arranged in the order of table 11. The brown (smooth) and green (jagged) lines indicate the theoretical expectations the PYTHIA simulated data.

Leptoquark
In fig. 11, we show the angular distribution for the production of leptoquark (V +4 /3 2µ ) c associated with an antiquarkd for all the three benchmark points at different centre of momentum energies as described in table 11. As before, the brown (even) and green (uneven) lines signify the theoretical expectations and the PYTHIA simulated data respectively. In this case also, no zero in any of the plots is found.  2µ ) c at 100 fb −1 luminosity. In this case also the significance for production of the leptoquark is quite high after the first cut for BP1 (267.4σ, 130.6σ and 128.8σ respectively) and hence the second cuts become unimportant. For BP2, the significances after the invariant mass cut are 5.0σ and 2.8σ which get improved to 10.2σ and 6.4σ respectively after the angular cut for 2 TeV and 3 TeV centre of momentum energies respectively. For BP3 at √ s = 2 TeV, the significance goes to 4.9σ from 4.4σ after implementing the angular cut of (−0.8) ≤ cos θ j ≤ 1. For √ s = 3 TeV the corresponding change in significance is from 2.1σ to 4.2σ.
In fig. 12, we show the differential distribution for the production of this leptoquark. We ordered the graphs in the same way as in table 12. The brown (smooth) and green (coarse) lines signify the theoretical estimates and the simulated data respectively. As expected the distributions at different centre of momentum energies for various benchmark points go to zero at different points of phase space except the plot at the left panel in third row. The positions of zeros can be verified from the left column ( titled "Qq = −2 /3 or Q φ = −1 /3") of table 5.  We present the PYTHIA analysis for leptoquark (U +5 /3 3µ ) c in table 13 with a luminosity of 100 fb −1 . By putting a cut on the invariant mass of lepton jet pair, we get the signals with very high significances (602.2σ, 648.7 and 648.8σ) in case of BP1 for all the three centre of momentum energies. The angular cut in this case becomes inessential. For BP2, the leptoquark can be reconstructed with the significances 18.8σ and 21.1σ at √ s to be 2 TeV and 3 TeV respectively. Using the angular cut, the significances can be upgraded to 26.5σ and 29.5σ respectively. For BP3 at 2 TeV, the cut of 10 GeV on M j around the mass of leptoquark provides 4.6σ significance for the signal events whereas the angular cut of (−0.9) ≤ cos θ j ≤ 1 enhances it to 5.1σ. For same benchmark point at 3 TeV, significance for the signal events goes to 5.9σ from 3.9σ after applying the angular cut of (−0.8) ≤ cos θ j ≤ 1.  cosθ → (dσ/dcosθ) in fb → Figure 13. Angular distribution for the production of (U +5 /3 3µ ) c at various centre of momentum energies for different benchmark points, arranged in the order of table 13. The brown (smooth) curves indicate the theoretical expectations whereas the green (jagged) lines signify the PYTHIA simulated data.

Benchmark points
In fig. 13, we show the angular distribution for the production of leptoquark (U +5 /3 3µ ) c associated with a u-quark for all the three bench mark points at different centre of momentum energies. The brown (even) and green (uneven) lines signify the theoretical expectations and the PYTHIA simulated data respectively. In this case also, no zero in any of the distributions is found.

Conclusion
In conclusion, we have studied zeros of single photon tree-level amplitude at the e-γ collider producing a leptoquark associated with a quark (or antiquark). Unlike other colliders, we find that the position of zeros of single photon tree-level amplitude in this case does depend on the centre of momentum energy as well as the mass and charge of leptoquark. The cosine of angle between leptoquark and initial state electron, at which zero of the angular distribution happens, approaches ±1 /3 asymptotically depending on the charge of the produced leptoquark for very high value of √ s with respect to the mass of leptoquark. No zero in the differential distribution can be found for leptoquarks having charges smaller than -1 unit. In a PYTHIA based analysis we look for both light and heavy leptoquarks at both low and high energy scales. Light leptoquarks having small couplings to quarks and leptons of all generation is not completely ruled out by Tevatron. In our simulation, we reconstruct the leptoquark from the lepton-jet pair and then study the differential distribution against the cosine of the angle between it and the initial state electron which matches with the theoretical expectation.