Probing topcolor-assisted technicolor from lepton flavor violating processes in photon-photon collision at ILC
aa r X i v : . [ h e p - ph ] A p r arXiv:1002.0659 Probing topcolor-assisted technicolor from lepton flavor violatingprocesses in photon-photon collision at ILC
Guo-Li Liu ∗ Physics Department, Zhengzhou University, Henan, 450001, China;Kavli Institute for Theoretical Physics China,Academia Sinica, Beijing 100190, China
Abstract
In topcolor-assisted technicolor models (TC2) the hitherto unconstrained lepton flavor mixinginduced by the new gauge boson Z ′ will lead to the lepton flavor violating productions of τ ¯ µ , τ ¯ e and µ ¯ e in photon-photon collision at the proposed International Linear Collider (ILC). Through acomparative analysis of these processes, we find that the better channels to probe the TC2 is theproduction of τ ¯ µ or τ ¯ e which occurs at a much higher rate than µ ¯ e production due to the largemixing angle and the large flavor changing coupling, and may reach the detectable level of the ILCfor a large part of the parameter space. Since the rates predicted by the Standard Model are farbelow the detectable level, these processes may serve as a sensitive probe for the TC2 model. PACS numbers: 14.65.Ha 12.60.Jv 11.30.Pb ∗ E-mail:[email protected] . INTRODUCTION Since in the Standard Model (SM) the lepton flavor violating (LFV) interactions areextremely suppressed, any observation of the LFV processes would serve as a robust evidencefor new physics beyond the SM. These LFV processes, which have been searched in variousexperiments [1–3], can be greatly enhanced in new physics models like supersymmetry [4, 5]and the topcolor-assisted models (TC2) [6–8]. Such enhancement can be several orders tomake them potentially accessible at future collider experiments.Due to its rather clean environment, the proposed International Linear Collider (ILC) willbe an ideal machine to probe new physics. In such a collider, in addition to e + e − collision,we can also realize γγ collision with the photon beams generated by the backward Comptonscattering of incident electron- and laser-beams. The LFV interactions in TC2 model willinduce various processes at the ILC, such as the productions of τ ¯ µ , τ ¯ e and µ ¯ e via e + e − . Itis noticeable that the productions of τ ¯ µ , τ ¯ e and µ ¯ e in γγ collision have not been studied inthe framework of the TC2. Such LFV productions in γγ collision may be more importantthan in e + e − collision [8] collision since these productions are a good probe for new physicsbecause it is essentially free of any SM irreducible background. It is also noticeable thatall these LFV processes at the ILC involve the same part of the parameter space of theTC2. Therefore, it is necessary to comparatively study all these processes to find out whichprocess is best to probe the TC2 model.We in this work will study the LFV processes γγ → ℓ i ¯ ℓ j ( i = j and ℓ i = e, µ τ ) inducedby the extra U (1) gauge boson Z ′ in TC2 models. We calculate the production rates tofigure out if they can reach the sensitivity of the photon-photon collision of the ILC.The work is organized as follows. We will briefly discuss the TC2 model in Section II,giving the new couplings which will be involved in our calculation. In Section III and IV wegive the calculation results and compare with the results in the SUSY. Finally, the conclusionis given in Section V. II. ABOUT TC2 MODEL
There are many kinds of new physics scenarios predicting new particles, which can leadto significant LFV signals. For example, in the minimal supersymmetric SM, a large ν µ − ν τ Z ′ , which are prediced by various specificmodels beyond the SM, can lead to the large tree-level flavor changing(FC) couplings. Thus,these new particles may have significant contributions to some LFV processes [10].The key feature of TC2 models [6] and flavor-universal TC2 models [11] is that thelarge top quark mass is mainly generated by topcolor interactions at a scale of order 1TeV . The topcolor interactions may be flavor non-universal (as in TC2 models) or flavor-universal (as in flavor-universal TC2 models). However, to tilt the chiral condensation inthe t ¯ t direction and not form a b ¯ b condensation, all of these models need a non-universalextended hypercharge group U (1). Thus, the existence of the extra U (1) gauge bosons Z ′ is predicted. These new particles treat the third generation fermions (quarks and leptons)differently from those in the first and second generations, namely, couple preferentially tothe third generation fermions. After the mass diagonalization from the flavor eigenbasis intothe mass eigenbasis, these new particles lead to tree-level FC couplings. The flavor-diagonalcouplings of the extra U (1) gauge bosons Z ′ to ordinary fermions, which are related to ourcalculation, can be written as [6, 12]: L = − g { tan θ ′ { (¯ e L γ µ e L + 2¯ e R γ µ e R + ¯ µ L γ µ µ L + 2¯ µ R γ µ µ R ) + cot θ ′ (¯ τ L γ µ τ L + 2¯ τ R γ µ τ R ) } · Z ′ µ (1)where g is the ordinary hypercharge gauge coupling constant. θ ′ is the mixing angle andtan θ ′ = g √ πK where K is the coupling constant.The flavor-changing couplings of the extra U (1) gauge bosons Z ′ to ordinary fermions,which are related to our calculation, are given in the followings: [6, 12]: L = − g { K µe (¯ e L γ µ µ L + 2¯ e R γ µ µ R ) + k τµ (¯ τ L γ µ µ L + 2¯ τ R γ µ µ R ) + k τe (¯ τ L γ µ e L + 2¯ τ R γ µ e R ) } · Z ′ µ , (2)where k µe , k τe and k τµ are the flavor mixing factors. Since the new gauge boson Z ′ couplespreferentially to the third generation, the factor K µe are negligibly small, so in the followingestimation, we will neglect the µ − e mixing, and consider only the flavor changing couplingprocesses γγ → τ ¯ µ and γγ → τ ¯ e .Note that the difference between the Z ′ τ ¯ µ and Z ′ τ ¯ e couplings lies only in the flavor mixingfactor K τµ and K τe and the masses of the final state leptons. Since the non-universal gauge3oson Z ′ treats the fermions in the third generation differently from those in the first andsecond generations and treats the fermions in the first same as those in the second generation,so in the following calculation, we will assume K τµ = K τe . Then what makes the discrepancyof the cross sections of the two channels γγ → τ ¯ µ and γγ → τ ¯ e is only the masses of thefinal state particles. Considering the large mass M Z ′ > M τ = M µ = M e = 0 in the following discussion, i.e., assuming the cross sections of the twochannels γγ → τ ¯ µ and γγ → τ ¯ e are equal to each other. III. CALCULATION
The Feynman diagram of the LFV processes γγ → ℓ i ℓ j ( i = j and ℓ i = e, µ, τ ) inducedby the extra U(1) Z ′ is shown in Fig. 1. There are only t- and u- channel contributions, thelatter not shown in Fig 1, but We can calculate them by exchanging the two photons. ν k l j l j l i µ k l j l j Z ′ ( a ) ν k l i l j k l i l j Z ′ l j l i ( b ) ν k l i µ k l j l j Z ′ ( c ) ν k µ k l j l j l j l j l i Z ′ ( d ) ν k l i µ k l i l j l j Z ′ l j ( e ) ν k l j µ k l j l j Z ′ l i l j ( f ) FIG. 1: Feynman diagrams contributing to the process γγ → ℓ i ℓ j in TC2 models. Note that there is no s-channel contribution to the LFV processes. As we know, in the SMproduction of on-shell Z boson at a photon-photon collider (or Z decays into γγ ) is strictlyforbidden by angular momentum conservation and Bose statistics, which is the predict of thefamous Laudau-Yang Theorem. This theorem is still effective to our case, since that the tworeal photons cannot be in a state with angular momentum J = 1 regardless of on-shell oroff-shell bosons, so the s-channel contribution with two real photons to an extra Z ′ vanishesautomatically.The electroweak gauge bosons γ and Z can not couple to τ ¯ e , µ ¯ e and τ ¯ µ , so we neednot consider the interference effects between the γ , Z and Z ′ on the cross section of the4rocess γγ → ℓ i ¯ ℓ j ( i = j and ℓ i = e, µ, τ ). In TC2 models the gauge invariant amplitude of γγ → τ ¯ µ (¯ e ) induced by the extra boson Z ′ is given by M = 12 ¯ u τ Γ µν P L v µ ǫ µ ( λ ) ǫ ν ( λ ) (3)where the Γ µν is defined same as that in [14]. These amplitudes contain the Passarino-Veltman one-loop functions, which are calculated by using LoopTools [15].Since the photon beams in γγ collision are generated by the backward Compton scatteringof the incident electron- and the laser-beam, the events number is obtained by convolutingthe cross section of γγ collision with the photon beam luminosity distribution: N γγ → ℓ i ¯ ℓ j = Z d √ s γγ d L γγ d √ s γγ ˆ σ γγ → ℓ i ¯ ℓ j ( s γγ ) ≡ L e + e − σ γγ → ℓ i ¯ ℓ j ( s ) (4)where d L γγ / d √ s γγ is the photon-beam luminosity distribution and σ γγ → ℓ i ¯ ℓ j ( s ) ( s is thesquared center-of-mass energy of e + e − collision) is defined as the effective cross section of γγ → ℓ i ¯ ℓ j . In optimum case, it can be written as [16] σ γγ → ℓ i ¯ ℓ j ( s ) = Z x max √ a zdz ˆ σ γγ → ℓ i ¯ ℓ j ( s γγ = z s ) Z x max z /xmax dxx F γ/e ( x ) F γ/e ( z x ) (5)where F γ/e denotes the energy spectrum of the back-scattered photon for the unpolarizedinitial electron and laser photon beams given by F γ/e ( x ) = 1 D ( ξ ) (cid:20) − x + 11 − x − xξ (1 − x ) + 4 x ξ (1 − x ) (cid:21) (6)with D ( ξ ) = (1 − ξ − ξ ) ln(1 + ξ ) + 12 + 8 ξ − ξ ) . (7)Here ξ = 4 E e E /m e ( E e is the incident electron energy and E is the initial laser photonenergy) and x = E/E E with E being the energy of the scattered photon moving along theinitial electron direction. The definitions of parameters ξ , D ( ξ ) and x max can be found inRef.[16]. In our numerical calculation, we choose ξ = 4 . D ( ξ ) = 1 .
83 and x max = 0 . IV. NUMERICAL RESULTS AND DISCUSSIONS
As for the involved SM parameter, we take [17] m µ = 0 .
106 GeV , m τ = 1 .
777 GeV , m b = 4 . , α = 1 / , sin θ W = 0 .
223 (8)5he TC2 parameters concerned in this process are K τe , K τµ , K eµ , K and the mass ofthe extra gauge boson M ′ Z . K eµ is very small, about 10 − , we will not consider the e − µ conversion processes. In our calculation, we have assumed K τµ = K τe ≃ λ ≃ .
22 [12, 19],which λ is the Wolfenstein parameter [18]. It has been shown that the vacuum tilting (thetopcolor interactions only condense the top quark but not the bottom quark), the couplingconstant K should satisfy certain constraint, i.e. K ≤ Z ′ mass M ′ Z can be obtained via studying its effects on various experimental observables [12]. Ref.[20],for example, has been shown that to fit the electroweak mearsurement data, the Z ′ mass M ′ Z must be larger than 1 TeV. As numerical estimation, we choose the center-of-mass energy √ s = 500 and 1000 GeV, to observe the different behavior in the two energy area, and takethe M ′ Z and K as free parameters. Finally, Note that the charge conjugate ¯ τ µ ( e ) productionchannel are also included in our numerical study. -2 -1 M Z , (GeV) s (f b ) √ s =500GeV ------ K = . K = . K = . (a) -1 M Z , (GeV) s (f b ) √ s =1000GeV ------ K = . K = . K = . (b) FIG. 2: The cross section σ of the LFV process γγ → τ ¯ µ (¯ e ) as a function of the gauge boson Z ′ mass M Z ′ for K = 0 .
2, 0 . . √ s = 500 GeV (b) √ s = 1000 GeV . In Fig. 2 we plot the production cross section σ of the LFV process γγ → ℓ i ¯ ℓ j as a functionof M Z for three values of the parameter K : K = 0 .
2, 0 .
6, and 1 .
0. We can see from Fig.2that the production cross section σ increases as K increasing and strongly suppressed bylarge M Z ′ . This situation is slightly different from the result of e + e − → ℓ i ¯ ℓ j in [8], in whichfrom Fig.1 we can see the cross section of e + e − → ¯ µτ increases with K decreasing. Thereason is that the Z ′ τ ¯ τ coupling involved in the process γγ → τ ℓ i ( ℓ i = e or µ ) is proportionalto 1 / tan θ ′ ∼ √ K , while the Z ′ e + e − contains tan θ ′ ∼ √ K and tan θ ′ <<
1. We can feelfrom this point the spirit of the technicolor models: to give the natural top quark mass, the6 -2 -1 K s (f b ) √ s =500GeV ------ M Z , = T e V M Z , = . T e V M Z , = . T e V (a) -1 K s (f b ) √ s =1000GeV ------ M Z , = T e V M Z , = . T e V M Z , = . T e V (b) FIG. 3: The cross section σ of the LFV process γγ → τ ¯ µ (¯ e ) as a function of the parameter K with the gauge boson Z ′ mass M Z ′ = 1 , 1 . . √ s = 500 GeV (b) √ s = 1000 GeV . third generation is singled out from the former two ones, so that it always shows distinctfeatures.The background for γγ → e ¯ τ comes from γγ → τ + τ − → τ − ν e ¯ ν τ e + , γγ → W + W − → τ − ν e ¯ ν τ e + and γγ → e + e − τ + τ − [5], and we make kinematical cuts [21]: | cos θ ℓ | < . p ℓT >
20 GeV ( ℓ = e, µ ), to enhance the ratio of signal to background. With these cuts, thebackground cross sections from γγ → τ + τ − → τ − ν e ¯ ν τ e + , γγ → W + W − → τ − ν e ¯ ν τ e + and γγ → e + e − τ + τ − at √ s = 500 GeV are suppressed respectively to 9 . × − fb, 1 . × − fb and 2 . × − fb (see Table I of [21]). To get the 3 σ observing sensitivity with 3 . × fb − integrated luminosity [22], the production rates of γγ → τ ¯ e, τ ¯ µ after the cuts mustbe larger than 2 . × − fb [21]. We see from Fig.2 that under the current bounds from ℓ i → ℓ j γ [3] and µ → e [10], the LFV couplings in TC2 models can still large enough toenhance the productions γγ → e ¯ τ , µ ¯ τ to the 3 σ sensitivity and may be detected in the futureILC colliders. Finally note that we in Fig.2 only show the results of the channels with the τ lepton in the final states, i.e., γγ → τ ¯ µ , τ ¯ e .Fig.3 shows that the cross section of the LFV processes as a function of K for M ′ Z = 1,1 .
5, 2 . K increasing.We also show the cross sections of γγ → ℓ i ¯ ℓ j as a function of center-of-mass energy √ s of the ILC in Fig.4. We see that with the increasing of the center-of-mass energy, thecross sections of these processes are not compressed, instead of becoming larger. This is7 -2 -1 √ s (GeV) ------ s (f b ) K = 0.2 M Z , = T e V M Z , = . T e V M Z , = . T e V (a) -1 √ s (GeV) ------ s (f b ) K = 0.2 M Z , = T e V M Z , = . T e V M Z , = . T e V (b) FIG. 4: The dependence of the cross section σ of the LFV process γγ → τ ¯ µ (¯ e ) on the center-of-massenergy √ s for M Z ′ = 1, 1 .
5, and 2 . k = 0 .
2, (b) k = 0 . different with the results in [8], since, as mentioned above, the contribution of the γγ → ℓ i ¯ ℓ j are the the results of t- and u-channels, while in the processes e + e − → ℓ i ¯ ℓ j , the s-channel contribution decreases with the increasing √ s when the center-of-mass energy ofthe processes arrives at the critical value[8]. Actually, we can also feel the larger crosssection with larger center of mass from Fig.2 and Fig.3. Table I: Theoretical predictions for the ℓ i ¯ ℓ j ( i = j ) productions at γγ collision at the ILC. SUSYand TC2 predictions are the optimum values. The collider energy is 500 GeV. SUSY TC2 σ ( γγ → τ ¯ µ ) O (10 − ) fb 1 fb σ ( γγ → τ ¯ e ) O ( < − ) fb 1 fb σ ( γγ → µ ¯ e ) O ( < − ) fb 10 − fbAs discussed in the former sections, motivated by the fact that any process that is for-bidden or strongly suppressed within the SM constitutes a natural laboratory to search forany new physics effects, LFV processes have been the subject of considerable interest in theliterature. It turns out that they may have large cross sections, much larger than the SMones, within some extended theories such as the R-parity violating MSSM [5] and the TC2models. However, in the R-parity violating MSSM, as discussed in [5], the LFV couplingby the exchange of the squark is λ ijk ∼ − , much smaller than that of the TC2 models( K τµ ( e ) ∼ . γγ → ℓ i ℓ j is about 2 − V. CONCLUSION
We have performed an analysis for the TC2-induced LFV productions of τ ¯ µ and τ ¯ e via γγ collision at the ILC. We found that in the optimum part of the parameter space, theproduction rate of γγ → τ ¯ µ (¯ e ) can reach 1 fb. This means that we may have 100 eventseach year for the designed luminosity of 100 fb − /year at the ILC. Since the SM value ofthe production rate is completely negligible, the observation of such τ ¯ µ (¯ e ) events would bea robust evidence of TC2. Therefore, these LFV processes may serve as a sensitive probe ofTC2. Acknowledgement
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