^a^ainstitutetext: Indian Institute of Technology Hyderabad, Kandi, Sangareddy-502285, Telengana, India

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

Priyotosh Bandyopadhyay a Saunak Dutta a Anirban Karan [email protected] [email protected] [email protected]

Abstract

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). The effects of non-monochromatic photons on the zeros of amplitude under laser backscattering and equivalent photon approximation have also been addressed.

^†^†preprint: IITH-PH-0003/20

1 Introduction

Leptoquarks are proposed particles that couple to quarks and leptons simultaneously, and hence carry non-zero baryon number as well as lepton number. They emerge naturally in various extensions of the Standard Model (SM), such as Pati-Salam model pati-salam , GUT based on $SU(5)$ or $SO(10)$ GUT1 ; GUT2 ; GUT3 , extended technicolor models technicolor1 ; technicolor2 , etc. These colour-triplet electromagnetically charged bosons (spin zero or one) could be singlet, doublet or triplet under $SU(2)_{L}$ group Rev1a ; Rev2a ; Rev3a ; Rev4a ; Rev5a ; Rev6a . Detection of leptoquark would be a signal for the unification of matter fields. Anomalies observed in the lepton flavour universality ratios $R_{K}$ , $R_{K*}$ , $R_{D}$ , $R_{D*}$ related to rare B decays exptRdRk1 ; exptRdRk2 ; exptRdRk3 ; exptRdRk4 ; exptRdRk5 and the deviations in the measurements of angular observables from their theoretical estimates can be addressed using several leptoquark models. Some of these models can explain observed discrepancy in muon $g-2$ g-21 ; g-22 and also accommodate the excess of $2.4\sigma$ in a Higgs decay branching fraction to $\mu\tau$ at 8 TeV with 19.7 fb^-1 luminosity htomutau . Because of their great importance in elucidating several issues of flavour physics Motiv1a ; Motiv1b ; Motiv1c ; Motiv1d ; Motiv1e ; Motiv1f ; Motiv1g ; Motiv2a ; Motiv3a ; Motiv4a ; Motiv4b ; Motiv4c ; Motiv4d ; Motiv4e ; Motiv4g ; Motiv5a ; Motiv6a ; Motiv7a ; Motiv8a ; Motiv9a ; Motiv10a ; Motive11 ; Motive12a ; Motive12b ; Motive13a ; Motive13b ; Motive13c ; Motive13d ; Motive13e ; Motive13f ; Motive14 ; Motive15 ; Motive16a , leptoquarks have been studied in literature in gory details during last few decades Rev1a ; Rev2a ; Rev3a ; Rev4a ; Rev5a ; Rev6a ; ModS1a ; ModS2a ; ModS2b ; ModS2c ; ModS3a ; ModS4a ; ModS5a ; ModS6a ; ModS6b ; ModS7 ; ModS8a ; ModS8b ; Exp1a ; Exp10a ; Exp13a ; Exp13b . In parallel, numerous searches for leptoquarks have been performed in different colliders Exp2a ; Exp2b ; Exp3a ; Exp5a ; Exp6a ; Exp7a ; Exp8a ; Exp9a ; Exp11a ; Exp12a ; Exp14a ; Exp15a ; LEP1 ; LEP2 .

On the other hand, the phenomenon of RAZ (radiation amplitude zero) was first discussed for $q_{i}\bar{q}_{j}\to W^{\pm}\gamma$ process at $pp$ or $p\bar{p}$ collider in order to probe the magnetic property of $W$ -boson RAZ1 . 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 RAZ2 ; RAZ2p ; RAZ3 ; RAZ4 ; RAZ5 ; RAZ7 ; RAZ8 ; RAZ9 ; RAZ10 ; RAZ11 ; RAZ12 ; RAZ13 ; RAZ14 ; RAZ15 ; RAZ16 ; RAZ17 ; RAZ18 ; RAZ19 ; RAZ19a ; RAZ20 ; RAZ21 ; RAZ22 ; RAZ23 ; RAZ24 ; RAZ25 ; RAZ26 ; RAZ27 ; RAZ27p ; RAZ28 ; RAZ29 ; RAZ30 ; RAZ31 ; RAZ32 . In non-Abelian theories the tree-level amplitudes¹¹1The word “amplitude” in this context is synonymous to $|\mathcal{M}|^{2}$ where $\mathcal{M}$ is the matrix element for a given process. 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 RAZ2 ; RAZ2p . 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 RAZ3 . The general criterion for tree-level single photon amplitude to vanish is that $\displaystyle\Big{(}\frac{p_{j}\cdot k}{Q_{j}}\Big{)}$ must be same for all the external particles (other than photon) involved in the process RAZ3 where, $p_{j}^{\mu}$ and $Q_{j}$ are the four momentum and charge of $j$ th external particle and $k^{\mu}$ is the four momenta of photon. For $2\to 2$ scattering processes with photon in final state, this condition reduces to:

\cos\theta^{*}=\frac{Q_{f_{2}}-Q_{f_{1}}}{Q_{f_{2}}+Q_{f_{1}}}

(1)

where, $Q_{f_{1}}$ and $Q_{f_{2}}$ are the charges for the incoming particles $f_{1}$ and $f_{2}$ and $\theta^{*}$ 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. $\sqrt{s}$ .

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 EPhC1a ; EPhC2a ; EPhC2b ; EPhC2c ; EPhC2d ; EPhC2e ; EPhC3a ; EPhC3b ; EPhC3c ; EPhC4a . 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 EphLQ1 ; EphLQ2 ; EphLQ3 . 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 RAZLQ1 ; RAZLQ2 . 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) 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 $e$ - $\gamma$ colliders depends on the mass of the leptoquark as well as $\sqrt{s}$ along with the electric charge RAZLQ3 . Nevertheless, the general condition for tree-level single photon amplitude being zero RAZ3 still remains valid.

In this paper we have analysed all kinds of leptoquarks that are going to be produced at $e$ - $\gamma$ 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 and electron 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. Furthermore, we have studied the effects of non-monochromatic photons on the zeros of angular distribution under laser backscattering and equivalent photon approximation schemes considering the current experimental limitations of electron-photon colliders.

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 $e$ - $\gamma$ 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. Sec. 5 deals with effects of non-monochromatic photons on the zeros of differential distribution. Finally, we conclude in sec. 6.

2 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 electron-photon collider to get the mathematical expression for the differential distribution of this process. We consider the process $e^{-}\gamma\to q\,\phi^{c}$ where $q$ is a quark and $\phi$ is a leptoquark (the sign ‘ $c$ ’ indicates charge conjugate), for which there are three possible tree-level Feynman diagram, as shown in fig. 1.

Refer to caption — Figure 1: Feynman diagrams for $e^{-}\,\gamma\to q\,\phi$

2.1 Scalar Leptoquark

If the leptoquark $\phi$ is a scalar one, the matrix elements for the respective diagrams are as follows:

		$\displaystyle\mathcal{M}_{1}^{S}=\bar{u}(p_{q})\,(-iY_{L}^{eq}P_{L}-iY_{R}^{eq}P_{R})\,\frac{i}{(\not{p}_{e}+\not{p}_{\gamma})}\,(ie\gamma^{\mu})\,u(p_{e})\,\epsilon_{\mu}^{\gamma}\leavevmode\nobreak\ ,$		(2)
		$\displaystyle\mathcal{M}_{2}^{S}=\bar{u}(p_{q})\,(-ieQ_{q}\gamma^{\mu})\,\frac{i}{(\not{p}_{q}-\not{p}_{\gamma})}\,(-iY_{L}^{eq}P_{L}-iY_{R}^{eq}P_{R})\,u(p_{e})\,\epsilon_{\mu}^{\gamma}\leavevmode\nobreak\ ,$		(3)
		$\displaystyle\mathcal{M}_{3}^{S}=\bar{u}(p_{q})\,(-iY_{L}^{eq}P_{L}-iY_{R}^{eq}P_{R})\,u(p_{e})\,\frac{i}{(p_{q}-p_{e})^{2}-M_{\phi}^{2}}\,[ie(1+Q_{q})(2p_{e}^{\mu}-2p_{q}^{\mu}+p_{\gamma}^{\mu})]\,\epsilon_{\mu}^{\gamma}\leavevmode\nobreak\ .$		(4)

where $p_{e}^{\mu}$ , $p_{\gamma}^{\mu}$ and $p_{q}^{\mu}$ are the four momenta of the particles electron, photon and the produced quark respectively, $Y_{L,R}$ are $3\times 3$ matrices describing the couplings of leptoquark with left-handed and right-handed leptons and quarks respectively, $e$ denotes the charge of positron, $Q_{q}$ signifies the charge of $q$ quark in the unit $e$ , $M_{\phi}$ indicates the mass of leptoquark, $\epsilon_{\mu}^{\gamma}$ is the polarization of the photon and $P_{L,R}\equiv(1\mp\gamma^{5})/2$ . 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:

\begin{split}\sum_{spin}|\mathcal{M}^{S}|^{2}=&\frac{e^{2}\,[(Y_{L}^{eq})^{2}+(Y_{R}^{eq})^{2}]\Big{[}(s-M_{\phi}^{2})(1-\cos\theta)+2sQ_{q}\Big{]}^{2}}{s\,(s-M_{\phi}^{2})\,(1-\cos\theta)\,\Big{[}s(1+\cos\theta)+M_{\phi}^{2}(1-\cos\theta)\Big{]}^{2}}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times\Big{[}(s-M_{\phi}^{2})^{2}(1+\cos\theta)^{2}+4M_{\phi}^{4}\Big{]}\end{split}

(5)

where, $s=(p_{e}+p_{\gamma})^{2}$ and $\theta$ is the angle between electron and the leptoquark or equivalently photon and the quark $q$ .

2.2 Vector Leptoquark

Now, if the leptoquark $\phi$ be a vector particle, the matrix elements will get modified in the following way:

		$\displaystyle\mathcal{M}_{1}^{V}=\epsilon^{\gamma}_{\nu}\,\epsilon^{\phi}_{\mu}\,\bar{u}(p_{q})\,(-iY_{L}^{eq}\gamma^{\mu}P_{L}-iY_{R}^{eq}\gamma^{\mu}P_{R})\,\frac{i}{(\not{p}_{e}+\not{p}_{\gamma})}\,(ie\gamma^{\nu})\,u(p_{e})\leavevmode\nobreak\ ,$		(6)
		$\displaystyle\mathcal{M}_{2}^{V}=\epsilon^{\gamma}_{\nu}\,\epsilon^{\phi}_{\mu}\,\bar{u}(p_{q})\,(-ieQ_{q}\gamma^{\nu})\,\frac{i}{(\not{p}_{q}-\not{p}_{\gamma})}\,(-iY_{L}^{eq}\gamma^{\mu}P_{L}-iY_{R}^{eq}\gamma^{\mu}P_{R})\,u(p_{e})\leavevmode\nobreak\ ,$		(7)
		$\displaystyle\mathcal{M}_{3}^{V}=\epsilon^{\gamma}_{\nu}\,\epsilon^{\phi}_{\mu}\,\bar{u}(p_{q})\,(-iY_{L}^{eq}\gamma^{\rho}P_{L}-iY_{R}^{eq}\gamma^{\rho}P_{R})\,u(p_{e})\,\frac{i}{(p_{q}-p_{e})^{2}-M_{\phi}^{2}}\,$
		$\displaystyle\qquad\;\;[ie(1+Q_{q})\{(2p_{e}^{\nu}-2p_{q}^{\nu}+p_{\gamma}^{\nu})g_{\mu\rho}+(p_{q}^{\rho}-p_{e}^{\rho}-2p_{\gamma}^{\rho})g_{\mu\nu}+(p_{\gamma}^{\mu}-p_{e}^{\mu}+p_{q}^{\mu})g_{\nu\rho}\}]\leavevmode\nobreak\ .$		(8)

Here, $\epsilon^{\phi}_{\mu}$ is polarization vectors for the vector leptoquark. After taking the spin and polarization sum of initial and final state particles²²2It should be noted that: $\displaystyle\sum_{polarization}\epsilon_{\mu}^{\phi*}\epsilon_{\nu}^{\phi}=\Big{(}-g_{\mu\nu}+\frac{p_{\phi\mu}p_{\phi\nu}}{M_{\phi}^{2}}\Big{)}$ where $p_{\phi}^{\mu}$ is the four-momentum of the leptoquark., the modulus squared matrix element becomes:

\begin{split}\sum_{spin}|\mathcal{M}^{V}|^{2}=&\frac{2e^{2}\,[(Y_{L}^{eq})^{2}+(Y_{R}^{eq})^{2}]\Big{[}(s-M_{\phi}^{2})(1-\cos\theta)+2sQ_{q}\Big{]}^{2}\,}{s\,(s-M_{\phi}^{2})\,(1-\cos\theta)\,\Big{[}s(1+\cos\theta)+M_{\phi}^{2}(1-\cos\theta)\Big{]}^{2}}\\ &\qquad\qquad\qquad\qquad\times\Big{[}\{s(1-\cos\theta)+M_{\phi}^{2}(1+\cos\theta)\}^{2}+4(s-M_{\phi}^{2})^{2}\Big{]}\end{split}

(9)

The differential cross-section for this process turns out to be:

\frac{d\sigma}{d\cos\theta}=\frac{s-M_{\phi}^{2}}{32\pi s^{2}}\bigg{(}\frac{3}{4}\sum_{spin}|\mathcal{M}^{(S,V)}|^{2}\bigg{)}

(10)

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. (5) and (9) that the differential cross-section vanishes iff:

(s-M_{\phi}^{2})(1-\cos\theta^{*})+2sQ_{q}=0\quad\implies\quad\cos\theta^{*}=1+\frac{2\,Q_{q}}{[1-(M_{\phi}^{2}/s)]}=f(Q_{q},M_{\phi}^{2}/s)\leavevmode\nobreak\ ,

(11)

since all the other terms are positive quantities. This also follows from the general condition for tree-level single photon amplitude to vanish RAZ3 :

\frac{p_{e}.p_{\gamma}}{-1}=\frac{p_{q}.p_{\gamma}}{Q_{q}}=\frac{p_{\phi}.p_{\gamma}}{Q_{\phi}}

(12)

where $Q_{\phi}$ is the charge of leptoquark in unit of $e$ and can be expressed as: $Q_{\phi}=-(1+Q_{q})$ . However, the striking difference between single photon emission with two body final state and this process is that $\cos\theta^{*}$ in the former case does not depend on the mass of fourth particle as well as the centre of momentum energy (as shown in Eq. (1)) after neglecting the masses of fermions, whereas $\cos\theta^{*}$ in the later scenario does depend on the mass of leptoquark and $\sqrt{s}$ (as can be seen from Eq. (11)). The variation of $\cos\theta^{*}$ with increasing centre of momentum energy ( $\sqrt{s}$ ) for different masses of leptoquark has been shown in fig. 2; the left panel depicts the variation for production of a leptoquark associated with a down-type quark, while the right panel describes the same with a up-type anti-quark. It can also be observed from Eq. (11) that $\cos\theta^{*}$ approaches $(1+2Q_{q})=(Q_{q}-Q_{\phi})$ amyptotically when $\sqrt{s}>>M_{\phi}$ . For the vanishing amplitude to be inside the physical region, the condition that must satisfy is:

Q_{q}<0\quad\text{ and }\quad\frac{M_{\phi}}{\sqrt{s}}\leq\sqrt{-Q_{\phi}}

(13)

which in turn would imply that

-1<Q_{\phi}<0\leavevmode\nobreak\ .

(14)

It should be noted that instead of quark $q$ , if the leptoquark is produced with an anti-quark $\bar{q}$ , all the expressions for that process can be achieved by replacing $\bar{u}(p_{q})$ with $\bar{v}(p_{\bar{q}})$ and $Q_{q}$ with $Q_{\bar{q}}$ in the equations from Eq. (2) to Eq. (13) where $Q_{\bar{q}}$ is the charge of $\bar{q}$ in unit of $e$ .

All of the leptoquarks Rev1a ; Rev2a ; Rev3a ; Rev4a ; Rev5a ; Rev6a , that can be produced at $e$ - $\gamma$ collider, have been listed in table 2.2. Here, $\Psi_{q}$ , $\Psi_{l}$ are quark and lepton doublets whereas $q_{u}$ , $q_{d}$ and $l_{e}$ are fields for $u$ -quark, $d$ -quark and electron respectively. The transpose $T$ acts on $SU(2)$ indices only. $S_{3}^{ad}$ and $U_{3}^{ad}$ denote scalar and vector triplet respectively in the adjoint representation of SU(2); they are defined as:

S_{3}^{ad}=\begin{pmatrix}\frac{S_{3}^{+1/3}}{\sqrt{2}}&S_{3}^{+4/3}\\ S_{3}^{-2/3}&-\frac{S_{3}^{+1/3}}{\sqrt{2}}\end{pmatrix}\quad\text{and}\quad U_{3}^{ad}=\begin{pmatrix}\frac{U_{3}^{+2/3}}{\sqrt{2}}&U_{3}^{+5/3}\\ U_{3}^{-1/3}&-\frac{U_{3}^{+2/3}}{\sqrt{2}}\end{pmatrix}.

[Uncaptioned image] — Table 1: The values of $\cos\theta^{*}$ for production of different leptoquarks at $e^{-}\gamma$ collider.

\hlineB 5 LQ	Y	$Q_{em}$	Interaction	Process	$\cos\theta^{*}$
\hlineB 5 Scalar Leptoquarks
$S_{1}$	$\nicefrac{{2}}{{3}}$	$\nicefrac{{1}}{{3}}$	$\overline{\Psi}_{q}^{c}\,P_{L}\,i\sigma_{2}\Psi_{l}\,S_{1}$ ,	$\bar{u}\,\Big{(}{S_{1}^{\nicefrac{{+1}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-2}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
			$\bar{q}_{u}^{c}\,P_{R}\,l_{e}\,S_{1}$	$\bar{u}\,\Big{(}{S_{1}^{\nicefrac{{+1}}{{3}}}}\Big{)}^{c}$
$\widetilde{S}_{1}$	$\nicefrac{{8}}{{3}}$	$\nicefrac{{4}}{{3}}$	$\bar{q}_{d}^{c}\,P_{R}\,l_{e}\,\widetilde{S}_{1}$	$\bar{d}\,\Big{(}{\widetilde{S}_{1}^{\nicefrac{{+4}}{{3}}}}\Big{)}^{c}$	——
$\vec{S}_{3}$	$\nicefrac{{2}}{{3}}$	$\nicefrac{{4}}{{3}}$		$\bar{d}\,\Big{(}{S_{3}^{\nicefrac{{+4}}{{3}}}}\Big{)}^{c}$	——
		$\nicefrac{{1}}{{3}}$	$\overline{\Psi}_{q}^{c}\,P_{L}\,(i\sigma_{2}\,S_{3}^{ad})\,\Psi_{l}$	$\bar{u}\,\Big{(}{S_{3}^{\nicefrac{{+1}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-2}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
		$\nicefrac{{-2}}{{3}}$		——	——
$R_{2}$	$\nicefrac{{7}}{{3}}$	$\nicefrac{{5}}{{3}}$	$\overline{\Psi}_{q}\,P_{R}\,R_{2}\,l_{e}$ ,	$u\,\Big{(}{R_{2}^{\nicefrac{{+5}}{{3}}}}\Big{)}^{c}$	——
		$\nicefrac{{2}}{{3}}$	$\bar{q}_{u}\,P_{L}\,(R_{2}^{T}\,i\sigma_{2})\,\Psi_{l}$	$d\,\Big{(}{R_{2}^{\nicefrac{{+2}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-1}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
$\widetilde{R}_{2}$	$\nicefrac{{1}}{{3}}$	$\nicefrac{{2}}{{3}}$	$\bar{q}_{d}\,P_{L}\,(\widetilde{R}_{2}^{T}\,i\sigma_{2})\,\Psi_{l}$	$d\,\Big{(}{\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-1}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
		$\nicefrac{{-1}}{{3}}$		—	—
\hlineB 5 Vector Leptoquarks
$V_{2\mu}$	$\nicefrac{{5}}{{3}}$	$\nicefrac{{4}}{{3}}$	$\overline{\Psi}_{q}^{c}\,\gamma^{\mu}P_{R}\,(i\sigma_{2}\,V_{2\mu})l_{e}$	$\bar{d}\,\Big{(}{V_{2\mu}^{\nicefrac{{+4}}{{3}}}}\Big{)}^{c}$	——
		$\nicefrac{{1}}{{3}}$	$\bar{q}_{d}^{c}\,\gamma^{\mu}P_{L}\,(V_{2\mu}^{T}\,i\sigma_{2})\,\Psi_{l}$	$\bar{u}\,\Big{(}{V_{2\mu}^{\nicefrac{{+1}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-2}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
$\widetilde{V}_{2\mu}$	$\nicefrac{{-1}}{{3}}$	$\nicefrac{{1}}{{3}}$	$\bar{q}_{u}^{c}\,\gamma^{\mu}P_{L}\,(\widetilde{V}_{2\mu}^{T}\,i\sigma_{2})\,\Psi_{l}$	$\bar{u}\,\Big{(}{\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-2}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
		$\nicefrac{{-2}}{{3}}$		—	——
$U_{1\mu}$	$\nicefrac{{4}}{{3}}$	$\nicefrac{{2}}{{3}}$	$\overline{\Psi}_{q}\,\gamma^{\mu}P_{L}\,\Psi_{l}\,U_{1\mu}$	$d\,\Big{(}{U_{1\mu}^{\nicefrac{{+2}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-1}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
			$\bar{q}_{d}\,\gamma^{\mu}P_{R}\,l_{e}\,U_{1\mu}$	$d\,\Big{(}{U_{1\mu}^{\nicefrac{{+2}}{{3}}}}\Big{)}^{c}$
$\widetilde{U}_{1\mu}$	$\nicefrac{{10}}{{3}}$	$\nicefrac{{5}}{{3}}$	$\bar{q}_{u}\,\gamma^{\mu}P_{R}\,l_{e}\,\widetilde{U}_{1\mu}$	$u\,\Big{(}{\widetilde{U}_{1\mu}^{\nicefrac{{+5}}{{3}}}}\Big{)}^{c}$	——
$\vec{U}_{3\mu}$	$\nicefrac{{4}}{{3}}$	$\nicefrac{{5}}{{3}}$	$\overline{\Psi}_{q}\,\gamma^{\mu}P_{L}\,U_{3\mu}^{ad}\,\Psi_{l}$	$u\,\Big{(}{U_{3\mu}^{\nicefrac{{+5}}{{3}}}}\Big{)}^{c}$	——
		$\nicefrac{{2}}{{3}}$		$d\,\Big{(}{U_{3\mu}^{\nicefrac{{+2}}{{3}}}}\Big{)}^{c}$	$f(\nicefrac{{-1}}{{3}},\nicefrac{{M_{\phi}^{2}}}{{s}})$
		$\nicefrac{{-1}}{{3}}$		—	—
\hlineB 5

\hlineB 5 Lepto-	Bench-	$M_{\phi}$	$Y_{L}^{11}$	$Y_{L}^{22}$	$Y_{L}^{33}$	$Y_{R}^{11}$	$Y_{R}^{22}$	$Y_{R}^{33}$
quarks	mark	in
	points	GeV
$(S_{1}^{\nicefrac{{+1}}{{3}}})^{c}$ ,	BP1	70	0.035	0.04	0.035	0.03	0.03	0.03
$(R_{2}^{\nicefrac{{+5}}{{3}}})^{c}$ ,	BP2	650	0.1	0.1	0.1	0.1	0.1	0.1
$(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}$	BP3	1500	0.1	0.1	0.1	0.1	0.1	0.1
$(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}$ ,	BP1	70	0.07	0.07	0.1	—	—	—
$(S_{3}^{\nicefrac{{+4}}{{3}}})^{c}$ ,	BP2	650	0.07	0.07	0.1	—	—	—
$(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}$ ,	BP3	1500	0.07	0.07	0.1	—	—	—
$(U_{3\mu}^{\nicefrac{{+5}}{{3}}})^{c}$		1500
$(V_{2\mu}^{\nicefrac{{+4}}{{3}}})^{c}$	BP1	70	0.05	0.05	0.1	0.1	0.1	0.1
	BP2	650	0.05	0.05	0.1	0.1	0.1	0.1
	BP3	1500	0.05	0.05	0.1	0.1	0.1	0.1
\hlineB 5

\hlineB 5 Bench-mark points	$\sqrt{s}$ in TeV	Cut	Signal	Back-ground	Signi-ficance
BP1	0.2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	5870.1	43725.0	26.4
		cut1+ $(-0.2)\leq\cos\theta_{\ell j}\leq 1$	5549.6	32989.8	28.3
	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	80.3	319.4	4.0
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	50.9	114.2	4.0
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	33.6	219.8	2.1
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	19.4	44.2	2.4
BP2	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	27.0	2003.6	0.6
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	20.8	129.1	1.7
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	11.99	1660.7	0.3
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	10.8	167.5	0.8
BP3	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	19.4	1061.6	0.6
		cut1+ $(-0.9)\leq\cos\theta_{\ell j}\leq 1$	13.8	391.5	0.7
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	7.6	815.0	0.3
		cut1+ $(-0.8)\leq\cos\theta_{\ell j}\leq 1$	7.2	254.7	0.4
\hlineB 5

\hlineB 5 Bench-mark points	$\sqrt{s}$ in TeV	Cut	Signal	Back-ground	Signi-ficance
BP1	0.2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	24719.2	43725.0	94.5
		cut1+ $(-0.2)\leq\cos\theta_{\ell j}\leq 1$	23448.8	32989.8	98.7
	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	365.7	319.4	13.9
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	251.7	114.2	13.2
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	148.9	219.8	7.8
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	96.2	44.2	8.1
BP2	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	757.4	2003.6	14.4
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	585.5	129.1	21.9
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	362.6	1660.7	8.1
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	329.5	167.5	14.8
BP3	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	440.9	1061.6	11.4
		cut1+ $(-0.9)\leq\cos\theta_{\ell j}\leq 1$	311.2	391.5	11.7
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	187.8	815.0	5.9
		cut1+ $(-0.8)\leq\cos\theta_{\ell j}\leq 1$	180.0	254.7	8.6
\hlineB 5

\hlineB 5 Bench-mark points	$\sqrt{s}$ in TeV	Cut	Signal	Back-ground	Signi-ficance
BP1	0.2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	8237.3	43725.0	36.1
		cut1+ $(-0.2)\leq\cos\theta_{\ell j}\leq 1$	7812.9	32989.8	38.7
	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	132.7	319.4	6.2
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	99.3	114.2	6.8
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	53.6	219.8	3.2
		cut1+ $(0.9)\leq\cos\theta_{\ell j}\leq 1$	38.0	44.2	4.2
BP2	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	40.4	2003.6	0.9
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	31.4	129.1	2.5
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	21.2	1660.7	0.5
		cut1+ $0\leq\cos\theta_{\ell j}\leq 1$	19.6	167.5	1.4
BP3	2	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	20.4	1061.6	0.6
		cut1+ $(-0.9)\leq\cos\theta_{\ell j}\leq 1$	14.4	391.5	0.7
	3	$\|M_{lj}-M_{\phi}\|\leq 10$ GeV	9.7	815.0	0.3
		cut1+ $(-0.8)\leq\cos\theta_{\ell j}\leq 1$	9.3	254.7	0.6
\hlineB 5

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

Abstract

1 Introduction

2 Theoretical formalism

2.1 Scalar Leptoquark

2.2 Vector Leptoquark

3 Mass and coupling

4 Leptoquark models and simulation

4.2.2 Leptoquark $(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}$

4.2.3 Leptoquark $({R}_{2}^{\nicefrac{{+5}}{{3}}})^{c}$

4.2.4 Leptoquarks $(S_{3}^{\nicefrac{{+4}}{{3}}})^{c}$

4.3 Vector leptoquarks

4.3.1 Leptoquark $(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}$

4.3.2 Leptoquark $(V_{2\mu}^{\nicefrac{{+4}}{{3}}})^{c}$

4.3.3 Leptoquark $(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}$

4.3.4 Leptoquark $(U_{3\mu}^{\nicefrac{{+5}}{{3}}})^{c}$

5 Effects of non-monochromatic photons

6 Conclusion

7 Acknowledgement

References

\hlineB 5 Photon	Cross-section in fb
	$(S_{1}^{\nicefrac{{+1}}{{3}}})^{c}$ , BP1 $\sqrt{s}=0.2$ TeV	$(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}$ , BP2 $\sqrt{s}=2$ TeV	$(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}$ , BP3 $\sqrt{s}=2$ TeV	$(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}$ , BP1 $\sqrt{s}=0.2$ TeV
Laser back- scattering	688.20	4.87	11.11	3337.54
EPA	101.42	0.81	0.40	486.94
Monochromatic	430.24	2.89	14.84	2127.02
\hlineB 5

\hlineB 5 Model	Bench-mark points	$\sqrt{s}$ in TeV	Photon	Cut	Signal	Back-ground	Signi- ficance
\hlineB 5 $(S_{1}^{\nicefrac{{+1}}{{3}}})^{c}$	BP1	0.2	Laser Back-	cut1	18518.7	36417.9	79.0
			scattering	cut1+cut2	17499.5	18173.8	92.65
			EPA	cut1	2863.6	1964.4	41.2
				cut1+cut2	2051.6	1038.2	36.9
			mono-	cut1	11133.6	43725.0	47.5
			chromatic	cut1+cut2	10537.8	32989.8	50.5
$(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}$	BP2	2	Laser Back-	cut1	62.7	2078.4	1.4
			scattering	cut1+cut2	48.1	326.1	2.5
			EPA	cut1	11.7	390.2	0.6
				cut1+cut2	7.0	230.8	0.5
			mono-	cut1	27.0	2003.6	0.6
			chromatic	cut1+cut2	20.8	129.1	1.7
$(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}$	BP3	2	Laser Back-	cut1	131.8	446.3	5.5
			scattering	cut1+cut2	93.4	246.0	5.1
			EPA	cut1	4.5	86.1	0.5
				cut1+cut2	3.5	79.7	0.4
			mono-	cut1	144.4	1061.6	4.2
			chromatic	cut1+cut2	102.2	391.5	4.6
$(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}$	BP1	0.2	Laser Back-	cut1	168135.3	36417.9	371.8
			scattering	cut1+cut2	157708.8	18173.8	376.0
			EPA	cut1	24697.2	1964.4	151.3
				cut1+cut2	17881.0	1038.2	130.0
			mono-	cut1	102107.7	43725.0	267.4
			chromatic	cut1+cut2	96573.2	32989.8	268.3
\hlineB 5

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

Abstract

1 Introduction

2 Theoretical formalism

2.1 Scalar Leptoquark

2.2 Vector Leptoquark

3 Mass and coupling

4 Leptoquark models and simulation

4.2.2 Leptoquark (R~2+2/3)c(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}

4.2.3 Leptoquark (R2+5/3)c({R}_{2}^{\nicefrac{{+5}}{{3}}})^{c}

4.2.4 Leptoquarks (S3+4/3)c(S_{3}^{\nicefrac{{+4}}{{3}}})^{c}

4.3 Vector leptoquarks

4.3.1 Leptoquark (U1​μ+2/3)c(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}

4.3.2 Leptoquark (V2​μ+4/3)c(V_{2\mu}^{\nicefrac{{+4}}{{3}}})^{c}

4.3.3 Leptoquark (V~2​μ+1/3)c(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}

4.3.4 Leptoquark (U3​μ+5/3)c(U_{3\mu}^{\nicefrac{{+5}}{{3}}})^{c}

5 Effects of non-monochromatic photons

6 Conclusion

7 Acknowledgement

References

4.2.2 Leptoquark $(\widetilde{R}_{2}^{\nicefrac{{+2}}{{3}}})^{c}$

4.2.3 Leptoquark $({R}_{2}^{\nicefrac{{+5}}{{3}}})^{c}$

4.2.4 Leptoquarks $(S_{3}^{\nicefrac{{+4}}{{3}}})^{c}$

4.3.1 Leptoquark $(U_{1\mu}^{\nicefrac{{+2}}{{3}}})^{c}$

4.3.2 Leptoquark $(V_{2\mu}^{\nicefrac{{+4}}{{3}}})^{c}$

4.3.3 Leptoquark $(\widetilde{V}_{2\mu}^{\nicefrac{{+1}}{{3}}})^{c}$

4.3.4 Leptoquark $(U_{3\mu}^{\nicefrac{{+5}}{{3}}})^{c}$