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Production of excited electrons at TESLA and CLIC based e gamma colliders

We analyze the potential of TESLA and CLIC based electron-photon colliders to search for excited spin-1/2 electrons. The production of excited electrons in the resonance channel through the electron-photon collision and their subsequent decays to leptons and electroweak gauge bosons are investigated. We study in detail the three signal channels of excited electrons and the corresponding backgrounds through the reactions e gamma -->e gamma, e gamma -->eZ and e gamma -->nu W. Excited electrons with masses up to about 90% of the available collider energy can be probed down to the coupling f=f'=0.05(0.1) at TESLA(CLIC) based e gamma colliders.


I. INTRODUCTION
In order to explain the fundamental aspects of the standard model (SM) such as the number of fermion generations and fermion mass spectrum compositeness models are expected to be a good candidate. The replication of three fermionic generations of known quarks and leptons implies composite structures made up of more fundamental constituents. The existence of such quark and lepton substructure leads one to expect a rich spectrum of new particles with unusual quantum numbers. A possible signal of excited states of quarks and leptons as predicted by composite models [1,2] would supply convincing evidence for a new substructure of matter. All composite models of fermions have an underlying substructure which is characterized by a scale Λ. In such models, the known light fermions would be the ground state spectrum of the excited fermions.
In order to have an agreement between the precise measurements of electron and muon g − 2 and theoretical predictions for chiral couplings, the compositeness scale Λ is expected to be less than 10 TeV [3]. The absence of electron and muon electric dipole moments implies the chiral properties of the excited leptons. A right-handed excited lepton should couple to only left-handed components of the corresponding lepton. Excited leptons may be classified by SU(2) × U(1) quantum numbers and they are assumed to be both left-and right-handed weak isodoublets.
Experimental lower limits for the excited electron mass are given as m ⋆ > 200 GeV in [4], and m ⋆ > 306 GeV in [5]. The higher limits are derived from indirect effects due to e ⋆ exchange in the t-channel and depend on transition magnetic coupling between e and e ⋆ .
In this work, resonant production of excited electron in the s-channel and its subsequent decay modes e ⋆ → eγ, e ⋆ → νW , e ⋆ → eZ are considered. In addition, we include the contributions coming from the excited electron in the t-channel. In order to probe excited electrons, we examine the potential of TESLA and CLIC based eγ colliders with the main parameters given in Table I. The production cross section and decays of excited electrons are calculated using an effective Lagrangian which depends on a compsiteness scale Λ and on free parameters f and f ′ .

II. EFFECTIVE LAGRANGIAN
The Lagrangian describing the transition between ordinary and excited leptons should respect to chiral symmetry in order to protect the light leptons from acquiring radiatively a large anomalous magnetic moment. The excited leptons (l ⋆ ) can couple to leptons (l) and electroweak gauge bosons through the SU(2) × U(1) invariant effective interaction La- where the W µν and B µν represent the field strength tensors of SU(2) and U(1) gauge fields.
The τ and Y are the corresponding gauge group generators; g and g ′ are gauge coupling constants. The parameters f and f ′ associated to the gauge groups SU(2) and U(1) depend on compositeness dynamics and they describe the effective changes from the SM coupling constants g and g ′ . In the physical basis the Lagrangian (1) can be rewritten in more explicit where the first term in the paranthesis is a purely diagonal U(1) term and vanishes for the coupling f = f ′ . It contains only triple vertices with The second term in (2) is a non-abelian part which involves triple as well as quartic terms From Eq. (2), the vertex factor for excited lepton (l ⋆ ) interacting with lepton (l) and gauge bosons (V = γ, Z, W ) can be obtained as follows where q is the gauge boson momentum. The couplings f V are defined by where Q f and I 3L are the charge of excited lepton and the weak isospin, respectively; and θ W is the weak mixing angle.

III. DECAY WIDTHS
Decay widths of excited electrons in the individual channels e ⋆ → eγ, e ⋆ → eZ, and e ⋆ → νW are given by For m ⋆ ≫ m Z,W , total decay width of excited electron is given by The branching ratios for the excited electron decay channels are described as follows where l is electron or neutrino. We choose the parameters either f = f ′ or f = −f ′ in our calculations in order to reduce the number of free parameters. For the case the coupling of the photon to excited neutrinos (electrons) vanishes. We display the decay widths and branching ratios for excited electrons in Fig. 1. The decay widths of excited electrons, for the accessible mass range, could be comparable with the detector resolution at TESLA and CLIC based eγ colliders. As can be seen from Fig For an immediate estimation, the cross section for the signal can be well approximated with the Breit Wigner formulaσ for the narrow decay widths where Γ i and Γ f are the initial and final state decay widths, respectively. Here, x =ŝ/s where √ŝ being the center of mass energy of the subprocess. In order to obtain the total cross sections for the signal and background, without the narrow width approximation, we use the following formula where σ( s) is obtained from the well-known matrix element calculation from the Feynman diagrams given in Figs. 2 and 3. Here, x min = m 2 ⋆ /s. The high energy photon spectrum f γ (x) obtained from the Compton backscattering is given by where x 0 = 4.82, x max = x 0 /(1 + x 0 ) and N=1.84 [11]. The production cross section of excited electron in three modes, taking f = f ′ = 1 and Λ=m ⋆ , are given in Fig. 4 and 5 for TESLA and CLIC based eγ colliders at the center of mass energies of √ s = 911 GeV and √ s = 2733 GeV, respectively. From Fig. 4 and Fig. 5, we get the following information; the νW channel gives higher cross section than the others however there is an ambiguity with the neutrino in this channel. Therefore, the photon channel gives a more promising result because of its simple kinematics. Excited electrons can be produced copiously at TESLA and CLIC based eγ colliders. The cross sections and the numbers of signal events are shown in Table II  In order to get one particle inclusive cross sections for the production of a particle of transverse momentum p T and rapidity y, we use the following standard procedure. The differential cross section for the process eγ → eZ with respect to the transverse momentum p T of outgoing electron is given by where E a and E b are the incoming particle energies.ŝ andt are the Lorentz invariant Mandelstam variables.
The p T distribution of W boson in eγ collision at rapidity y for the process eγ → νW is given by T is the definition for the transverse mass of W boson.
The production of a photon in eγ collision at rapidity y and transverse momentum p T for the process eγ → eγ can be found by replacing m W = 0 in (21).
In the Eqs. In order to estimate the number of events for the signal and background in a chosen p T window, we integrate the transverse momentum distribution around the half of each excited electron mass point (m ⋆ /2 − m 2 V /2m ⋆ ) in the interval of transverse momentum resolution ∆p T . Here ∆p T is approximated as ≈ 10 GeV for m ⋆ = 500 GeV and ≈ 20 GeV for m ⋆ = 1500 GeV for a generic detector. In order to calculate signal significance, we use the integrated luminosity for TESLA and CLIC based eγ colliders with L = 9.4 × 10 4 pb −1 and L = 9 × 10 4 pb −1 , respectively [12].
In order to quantify the potential of TESLA and CLIC based eγ colliders to search for excited electron, we define the statistical significance (SS) where σ total and σ back is the total cross sections for signal+background and background inside chosen p T window, respectively. We calculate the value of SS for different couplings assuming f = f ′ and requiring the condition SS > 5 for the signal observability. We find from the Tables IV-VII that  In this study, we have taken into account that the excited electrons interact with the Standard Model particles through the effective Lagrangian (1). This may be a conservative assumption because it is possible for excited fermions couple to ordinary quarks and leptons via contact interactions originating from the strong constituents dynamics. In this case, the decay widhts can be enhanced [14].
In conclusion, we have presented the results of excited electron production with subsequent decays into three decay channels. The excited electrons can be produced at TESLA and CLIC based eγ colliders up to kinematical limit at each channel due to the smooth photon energy spectrum. We find that excited electron with mass 500(750) GeV can be