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Top-pions and single top production at the HERA and THERA colliders
Nuclear Physics B 667 (2003) 349–358 www.elsevier.com/locate/npe
Top-pions and single top production at the HERA and THERA colliders Chongxing Yue a , Hongjie Zong b , Wei Wang a a Department of Physics, Liaoning Normal University, Dalian 116029, China b College of Physics and Information Engineering, Henan Normal University, Henan 453002, China
Received 9 May 2003; accepted 25 June 2003
Abstract The presence of physical top-pions in low-energy spectrum is an inevitable feature of topcolor scenario. We consider the contributions of the physical top-pions predicted by topcolor-assisted technicolor (TC2) models to the single top production via the t-channel process eq → et at the HERA and THERA colliders. We find that the neutral top-pion can generate large contributions to the process ec → et. In most of the parameter space, the production cross section is in the range of 1–6 pb. The signals and observability of the neutral top-pion can be studied in the HERA and THERA colliders. 2003 Elsevier B.V. All rights reserved. PACS: 14.80.-j; 14.65.Ha; 12.60.Nz
1. Introduction The top quark, with a mass of the order of the electroweak symmetry breaking (EWSB) scale, is singled out to play a key role in probing the new physics beyond the standard model (SM). The properties of the top quark could reveal information regarding flavor physics, EWSB mechanism, as well as new physics beyond the SM [1]. In particular, the anomalous top couplings, which affect top production and decay at high energy colliders as well as precisely measured quantities with virtual top contributions, offer a unique place for testing the SM flavor structure. In the SM, the anomalous top quark couplings tqV (q = u- or c-quarks and V = Z, γ or g gauge bosons), which are arised from the flavor changing (FC) interactions, vanish at E-mail address: [email protected] (C. Yue). 0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0550-3213(03)00553-4
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tree level but can be generated at the one-loop level. However, they are very suppressed by the GIM mechanism, which cannot be detected in the present and near future high energy experiments [2]. Hence, any signal indicating these types of couplings is evidence of new physics beyond the SM and will shed more light on flavor physics in the top quark sector. It has been suggested that the couplings of these types of couplings may be large in some extensions to the SM, such as supersymmetry or other models with multiple Higgs doublets [2,3] and dynamics EWSB models with new strong interactions of the top quark [4]. Single top production is very sensitive to the anomalous top quark couplings tqV . These couplings effectively appear in supersymmetry or in the scenario where new dynamics take place in the fermion mass generation. Thus, studying the contributions of the anomalous top quark couplings tqV to the single production of the top quark is of special interest. It will be helpful to test the SM flavor structure and new physics beyond the SM. This fact has lead to many studies involving single top production given by the tqV couplings in lepton colliders [5,6] and hadron colliders [7]. √ The HERA and THERA colliders [8] with the center-of-mass energy s = 320 GeV and 1000 GeV, respectively are the experimental facility where high energy electron– proton and positron–proton interactions can be studied. It is well known that in the SM, single top quark cannot be produced at an observable in these high energy colliders [9]. However, it has been shown that the HERA collider and THERA collider can provide a very good sensitivity on the anomalous top quark couplings tqV via single top production [10]. Single top production mediated by the FC interactions via the anomalous top quark couplings tqV (q = u- or c-quarks and V = γ or Z gauge bosons) is a t-channel process involving a massive final state top quark at the HERA and THERA collider experiments. Several model-independent studies of this type of single top production have appeared in the literature [11], which have shown that single top production is detectable, and the HERA and THERA colliders are powerful tools for searching for the anomalous top quarks couplings tqγ and tqZ. To completely avoid the problem arising from the elementary Higgs field in the SM, various kinds of dynamical EWSB model have been proposed, and among which the topcolor scenario is attractive because it explains the large top quark mass and provides possible EWSB mechanism. Topcolor-assisted technicolor (TC2) models [12], flavoruniversal TC2 models [13], top see-saw models [14], and the top flavor see-saw models [15] are four of such examples. All of these models propose that the scale of the gauge groups should be flavor non-universal. For example, SU(3) gauge group is flavor nonuniversal in TC2 models and U(1) gauge group is flavor non-universal in the flavoruniversal TC2 models. When one writes the non-universal interactions in the mass eigenbasis, it can induce the tree-level FC couplings. Thus the new particles, such as top-pions (πt± , πt0 ) predicted by TC2 models, have the tree-level FC couplings to the ordinary particles and may generate large contributions to the single top production in ep collisions at the HERA and THERA colliders. In this paper, we study the contributions of the top-pions to the single production of the top quarks in ep collisions at the HERA and THERA collider experiments in the context of TC2 models. Our aim is to investigate whether the t-channel process eq → et mediated by the FC couplings via the anomalous top quars vertices t–q–γ and t–q–Z can be used to
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detect the top-pions (πt0 , πt± ) and further probe new physics beyond the SM. We find that the contributions of the charged top-pions πt± to the single top production is very small. The neutral top-pion πt0 can generate significant contributions to the single top production via the process ec → et. In most of the parameter space, the production cross section is in the range of 1–6 pb. The virtual effects of the neutral top-pion πt0 on single top production can be detected in the HERA and THERA collider experiments. The paper is organized as follows: in Section 2 we give and discuss the anomalous top quark couplings tqV (V = γ or Z) arised from the top-pions πt± , πt0 . Their contributions to the single top production process ec → et are calculated in Section 3. The conclusions and discussions are given in Section 4.
2. Top-pions and the anomalous top quark couplings tqV In TC2 models [12], the TC interactions play a main role in breaking the electroweak gauge symmetry. The ETC interactions give rise to the masses of the ordinary fermions including a very small portion of the top quark mass, namely εmt with a model-dependent parameter ε 1. The main part of the top quark mass is dynamically generated by topcolor interactions at a scale of 1 TeV, which also make small contributions to EWSB. This means that the associates top-pions πt±,0 are not the longitudinal bosons W and Z, but are separately physically observable objects. The presence of physics top-pions in the lowenergy spectrum is an inevitable feature of topcolor scenario that purports to avoid finetuning[16]. The flavor diagonal couplings of top-poins to fermions can be written as [4,12, 16] 2 2 √ mt (1 − ε) νW − Ft 5 0 √ i t¯γ tπt + 2 t¯R bL πt+ + 2 b¯L tR πt− , (1) √ νW 2 Ft √ where νW = ν/ 2 = 174 GeV and Ft ≈ 50 GeV is the top-pion decay constant, which can be estimated from the Pagels–Stokar formula. For TC2 models, the underlying interactions, topcolor interactions, are non-universal and therefore do not possess the GIM mechanism. This is an essential feature of these kinds of models due to the need to single out top quark for condensate. The non-universal gauge interactions result in the FC coupling vertices when one writes the interactions in the mass eigen-basis. Thus, the top-pions have large Yukawa couplings to the third family quarks and can induce the tree-level FC scalar couplings [17,18] 2 − F2 νW t tc tt∗ mt t c∗ t t 0 ikU R kU L t¯L cR πt0 + ikU √ L kU R c¯L tR πt ν 2Ft W √ t c∗ bb √ t c bb∗ + − ¯ + 2 kU (2) R kDL c¯R bL πt + 2kU R kDL bL cR πt , where kU L(R) and kDL(R) are rotation matrices that diagonalize the up-quark and down+ + dia dia quark mass matrices MU and MD , i.e., kU L MU kU R = MU and kDL MD kDR = MD , + for which the CKM matrix is defined as V = kU L kDL . To yield a realistic form of the CKM V , it has been shown [17] that the top-pions mainly couple to the right-handed top
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(tR ) or charm (cR ) quark and thus the values of the coupling parameters can be taken as: tt bb tc tc 2 kU (3) kU kU R ≈ kDL ≈ 1, L ≈ 0, R 2ε − ε . √ tc In the following calculation, we will assume kU R = 2ε − ε2 and take ε as a free parameter, which is assumed to be in the range of 0.03–0.1 [4,12]. The relevant Feynman diagrams for the contributions of the neutral top-pion πt0 to the anomalous top quark couplings tcγ and tcZ via the tree-level FC couplings are shown in Fig. 1. Using Eqs. (1)–(3) and other relevant Feynman rules, we can give the effective forms of the anomalous top quark coupling vertices Ztc and γ tc: µ µ ΛZ t¯c = ie γ µ F1Z + F2Z γ 5 + Pt F3Z + F4Z γ 5 + Pcµ F5Z + F6Z γ 5 , (4) µ µ , Fiγ = FiZ 2 . Λ =Λ (5) γ t¯c
Z t¯c FiZ →Fiγ
Vt = 3 ,at =0
The form factors FiV are expressed in terms of two- and three-point standard Feynman integrals [19]. The expressions of FiV are given in Appendix A. The neutral top-pion πt0 can also generate the anomalous top quark couplings tuγ and tuZ via the tree-level FC coupling πt0 tu. However, it has been argued that the maximum flavor mixing occur between the third generation fermions and the second generation fermions, and the FC coupling πt0 t¯u is very small which can be neglected [18]. Hence we will ignore the anomalous top quark couplings tuγ and tuZ, and only calculate the contributions of the neutral top-pions πt0 to the production cross section of the t-channel process ec → et via the anomalous top quark couplings tcγ and tcZ in the following. The charged top-pions πt± can contribute to the anomalous top quark couplings tcγ and tcZ via the Feynman diagrams as depicted in Fig. 2. These diagrams are mediated
Fig. 1. Feynman diagrams for the contributions of the neutral top-pion πt0 to the anomalous top quark couplings tcγ and tcZ.
Fig. 2. Feynman diagrams for the contributions of the charged top-pions πt± to the anomalous top quark couplings tcγ and tcZ.
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by the FC couplings πt± bc. The internal fermion line must be bottom quark. Similar to calculation about Fig. 1, we can give the effective vertices Λ γ t¯c and Λ Z t¯c arising from Fig. 2. However, compared to Λγ t¯c and ΛZ t¯c given by Fig. 1, these effective vertices are very small and can be safely ignored [6]. So, in the following calculation, we will not consider the contributions of the charged top-pions πt± to the single top production via the process ec → et at the HERA and THERA colliders. 3. The neutral top-pion πt0 and single top production From above discussion, we can see that the neutral top-pion πt0 can induce large anomalous top quark couplings tcγ and tcZ. Hence, it is possible that πt0 generate significant contributions to the single top production at the HERA and THERA colliders via the t-channel process ec → et with the relevant Feynman diagrams shown in Fig. 3. Using the effective vertices ΛZ t¯c and Λγ t¯c given by Eqs. (4) and (5), we can obtain the cross section σˆ (ˆs ) of the subprocess ec → et which can be written as
1 1 d σˆ 2 2 † = | + |M | + 2 Re M M (6) |M Z γ Z γ dt 64π sˆ (m2c − sˆ) with MZ =
1 m2Z
−t
u¯ t ΛσZ t¯c uc u¯ e γσ (ve − ae γ5 ) ue ,
1 Mγ = u¯ t Λσγ t¯c uc u¯ e γσ ue . t Here ve =
e 2 −1 + 4SW , 4SW CW
(7) (8)
ae = −
e , 4SW CW
(9)
√ where sˆ is the center-of-mass energy of the subprocess ec → et in ep collisions. The total cross section σ (s) of the single top production at the HERA and THERA colliders can be obtained by folding the cross section σˆ with the charm-quark distribution function fc (x) in the proton: 1 σ= xmin
t+ fc (x) dx
d σˆ dt dt
(10)
t−
Fig. 3. Feynman diagrams for the single top production at the HERA and THERA colliders, due to the anomalous top quark couplings tcV (V = γ or Z).
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Fig. 4. The production cross section σ (s) contributed by πt0 as a function of mπt for ε = 0.02 (solid line), ε = 0.05 (dashed line), and ε = 0.08 (dotted line) at the HERA collider. m2 +m2
(m2 −ˆs )(ˆs −m2 )
c with sˆ = xs, xmin = t s e , t− = t sˆ and t− = tcut = −0.001 (GeV)2 . The parton distribution function fc (x) of the charm quark runs with the energy scale. In our calculation, we take CTEQ5 parton distribution [20] for fc (x). The cross sections σ (s) of the single top production at the HERA and THERA colliders are plotted in Figs. 4 and 5 respectively, as a function of the mass mπt of the neutral top pion πt0 for the parameter ε = 0.02 (solid line), 0.05 (dashed line) and 0.08 (dotted line). 1 2 = 0.2315, m = 175 GeV, m = 1.2 GeV, In our calculation, we have taken αe = 128.8 , SW t c mZ = 91.18 GeV and ΓZ = 2.49 GeV [21]. We can see from these two figures that the cross sections decrease with mπt increasing and ε decreasing. In all of the parameter space of TC2 models, the cross section of single top production at the THERA collider is large than that of at the HERA collider. For 0.02 ε 0.08 and 200 mπt 350 GeV, the cross sections are in the ranges of 0.5–3.7 pb and 0.86–6.4 pb for the HERA collider and THERA collider, respectively. In order to estimate√the number of the events, we consider two ep collider scenarios: the −1 HERA collider with s = 320 GeV √ and a yearly integrated luminosity of L = 160 pb and the THERA collider with s = 1000 GeV and a yearly integrated luminosity of L = 470 pb−1 [8]. The yearly production events of the single top production at the HERA and THERA colliders can be easily calculated. In most of the parameter space, there may be several hundreds of the single top events at the HERA collider and hundreds and up to thousands single top events at the THERA collider to be generated. For example, for mπt = 300 GeV and ε = 0.05, the HERA collider can generate 240 single top events and the THERA collider can generate 1250 single top events. Thus, the contributions of the
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Fig. 5. The production cross section σ (s) contributed by πt0 as a function of mπt for ε = 0.02 (solid line), ε = 0.05 (dashed line), and ε = 0.08 (dotted line) at the THERA collider.
neutral top pion πt0 to the single top production may be detected at the HERA collider or the THERA collider. The signals of πt0 can be studied at these colliders via the process ec → et.
4. Conclusions and discussions The anomalous top quark couplings tqV (q = u- or c-quarks and V = Z, or γ gauge bosons), which are arised from the FC interactions, offer an ideal place to search for new physics beyond the SM as they are very small in the SM. Single top production is very sensitive to the tqV couplings in the HERA and THERA colliders. Studying the contributions of the tqV couplings to the single production of the top quark will be helpful to test the SM flavor structure and new physics beyond the SM. In this paper, we study the contributions of the top-pions predicted by TC2 models to the single top production in ep collisions at the HERA and THERA colliders. Our results show. (1) Although the contributions of the up-quark distribution in the proton to the single top production is large than those of the charm-quark distribution, the contributions of the anomalous top quark couplings tuV given by the TC2 models to the single top production in ep collisions are much smaller than those of the tcV couplings and can be safely neglected. (2) The charged top-pions πt± can generate the anomalous top quark couplings tcV via the FC couplings πt± bc. However, compared with the tcV couplings generated by the
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neutral top-pion πt0 via the FC coupling πt0 tc, their values are negligibly small. The πt± cannot give significantly contributions to the production cross section of the single top quark at the HERA and THERA colliders. (3) The contributions of the neutral top-pion πt0 to the production cross section σ √ of the t-channel process ec → et increases with s increasing and mπt decreasing. For 200 mπt 400 GeV and ε = 0.05, the value of σ varies from 0.99 to 2.35 pb and from 1.7 to 4.1 pb at the HERA and THERA colliders, respectively. Thus, there may be several hundreds and up to thousands of the single top production events to be generated at these two colliders. We can study the signature and observability of πt0 via the process ec → et at the HERA collider or the THERA collider. TC2 models also predict the existence of the neutral cp-even state, called top-Higgs boson h0t , which is a t t¯ bound and analogous to the σ particle in low energy QCD. Its mass can be estimated in the Nambu–Jona-Lasinio model in the large Nc approximation and is found to be of the order mht ≈ mπt [18]. The main difference between πt0 and h0t is that h0t can couple to gauge boson pairs W W and ZZ at tree level, which is similar to that of the SM Higgs boson H 0 . Thus, the contributions of the top-Higgs h0t to the single top production at the HERA and THERA colliders are similar to those of πt0 . We find that, in most of the parameter space of TC2 models, the cross section of single top production given by the top-Higgs boson h0t varies in the range of 0.8–6.2 pb. Thus the top-Higgs boson h0t can also be detected at the HERA collider or the THERA collider via the process ec → et. The key feature of TC2 models is that a large part of the top quark mass is dynamically generated by topcolor interactions at a scale of order 1 TeV, which is flavor non-universal. To ensure that the top quark condenses and receives a large mass while the bottom quark does not, the topcolor gauge group is usually taken to be a strongly coupled SU(3) × U (1). The U (1) provides the difference that cause only top quark to condensate. Thus, TC2 models predict the existence of topcolor gauge bosons BµA and an extra U (1) gauge boson Z . Tree-level FCNC for the topcolor gauge bosons BµA and Z are generated when quarks fields are rotated to the mass eigenstate basis. The couplings of Z to ordinary fermions are non-universal and stronger for the third generation fermions, yielding potentially large top-charm couplings. Thus, the topcolor gauge bosons can generate large anomalous top couplings tcV (V = γ or Z), which may produce significant contributions to the single top production at the HERA and THERA collider via the process ec → et. However, the limits on the masses of the topcolor gauge bosons BµA and Zµ can be obtained via studying their effects on various experimental observables [4]. For example, Ref. [22] considered the bound placed by the electroweak measurement data on the gauge boson Z . They find that Z predicted by the TC2 models and the flavor universal TC2 models must be heavier than 1 TeV. If we assume MZ = MB 1 TeV, then we find that the production cross section is smaller than 1 × 10−2 pb in most of the parameter space of the TC2 models. Thus, it is very difficult to detect the topcolor gauge bosons via the single top production at the HERA and THERA colliders.
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Acknowledgements We thank C.-P. Yuan for pointing out that we should use the evolved parton distribution function of the charm quark to calculate the production cross section. This work was supported in part by the National Natural Science Foundation of China (90203005). Appendix A. The form factors in the effective vertices Z t¯c and γ t¯c for the neutral top-pion πt0 F1Z = g B0 + m2πt C0 − 2C24 + m2t (C11 − C12 ) − B0∗ − B1 vt , F2Z = g B0 + m2πt C0 − 2C24 − m2t (C11 − C12 ) + B0∗ + B1 − 4C24 at , F3Z = 2mt g(C21 + C22 − 2C23 )vt , F4Z = 2mt g(−C21 − C22 + 2C23 − 2C22 + C12 + C0 )at , F5Z = 2mt g(C22 − C23 + C12 )vt , F6Z = 2mt g(−C22 + C23 + C12 − 2C22 + 3C12 + 2C23 − 2C11 − C0 )at . Here
mt 1 g= √ 2 16π 2 Ft 1 vt = 1− 4SW CW
νw2 − Ft2 2
νw 8 2 SW , 3
KUt cR KUt t L∗ , at =
1 . 4SW CW
The expressions of two- and three-point scalar integrals Bn and Cij in this paper are [19]: √ Bn = Bn (− t, mt , mt ), Bn∗ = Bn (−pc , mπt , mt ), Bn = Bn (−pt , mπt , mt ), √ Cij = Cij (pt , − t, mπt , mt , mt ), √ C0 = C0 (pt , − t, mπt , mt , mt ), √ Cij = Cij (−pc , t, mt , mπt , mπt ), √ C0 = C0 (−pc , t, mt , mπt , mπt ).
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Top-pions and single top production at the HERA and THERA colliders Chongxing Yue a , Hongjie Zong b , Wei Wang a a Department of Physics, Liaoning Normal University, Dalian 116029, China b College of Physics and Information Engineering, Henan Normal University, Henan 453002, China
Received 9 May 2003; accepted 25 June 2003
Abstract The presence of physical top-pions in low-energy spectrum is an inevitable feature of topcolor scenario. We consider the contributions of the physical top-pions predicted by topcolor-assisted technicolor (TC2) models to the single top production via the t-channel process eq → et at the HERA and THERA colliders. We find that the neutral top-pion can generate large contributions to the process ec → et. In most of the parameter space, the production cross section is in the range of 1–6 pb. The signals and observability of the neutral top-pion can be studied in the HERA and THERA colliders. 2003 Elsevier B.V. All rights reserved. PACS: 14.80.-j; 14.65.Ha; 12.60.Nz
1. Introduction The top quark, with a mass of the order of the electroweak symmetry breaking (EWSB) scale, is singled out to play a key role in probing the new physics beyond the standard model (SM). The properties of the top quark could reveal information regarding flavor physics, EWSB mechanism, as well as new physics beyond the SM [1]. In particular, the anomalous top couplings, which affect top production and decay at high energy colliders as well as precisely measured quantities with virtual top contributions, offer a unique place for testing the SM flavor structure. In the SM, the anomalous top quark couplings tqV (q = u- or c-quarks and V = Z, γ or g gauge bosons), which are arised from the flavor changing (FC) interactions, vanish at E-mail address: [email protected] (C. Yue). 0550-3213/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0550-3213(03)00553-4
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tree level but can be generated at the one-loop level. However, they are very suppressed by the GIM mechanism, which cannot be detected in the present and near future high energy experiments [2]. Hence, any signal indicating these types of couplings is evidence of new physics beyond the SM and will shed more light on flavor physics in the top quark sector. It has been suggested that the couplings of these types of couplings may be large in some extensions to the SM, such as supersymmetry or other models with multiple Higgs doublets [2,3] and dynamics EWSB models with new strong interactions of the top quark [4]. Single top production is very sensitive to the anomalous top quark couplings tqV . These couplings effectively appear in supersymmetry or in the scenario where new dynamics take place in the fermion mass generation. Thus, studying the contributions of the anomalous top quark couplings tqV to the single production of the top quark is of special interest. It will be helpful to test the SM flavor structure and new physics beyond the SM. This fact has lead to many studies involving single top production given by the tqV couplings in lepton colliders [5,6] and hadron colliders [7]. √ The HERA and THERA colliders [8] with the center-of-mass energy s = 320 GeV and 1000 GeV, respectively are the experimental facility where high energy electron– proton and positron–proton interactions can be studied. It is well known that in the SM, single top quark cannot be produced at an observable in these high energy colliders [9]. However, it has been shown that the HERA collider and THERA collider can provide a very good sensitivity on the anomalous top quark couplings tqV via single top production [10]. Single top production mediated by the FC interactions via the anomalous top quark couplings tqV (q = u- or c-quarks and V = γ or Z gauge bosons) is a t-channel process involving a massive final state top quark at the HERA and THERA collider experiments. Several model-independent studies of this type of single top production have appeared in the literature [11], which have shown that single top production is detectable, and the HERA and THERA colliders are powerful tools for searching for the anomalous top quarks couplings tqγ and tqZ. To completely avoid the problem arising from the elementary Higgs field in the SM, various kinds of dynamical EWSB model have been proposed, and among which the topcolor scenario is attractive because it explains the large top quark mass and provides possible EWSB mechanism. Topcolor-assisted technicolor (TC2) models [12], flavoruniversal TC2 models [13], top see-saw models [14], and the top flavor see-saw models [15] are four of such examples. All of these models propose that the scale of the gauge groups should be flavor non-universal. For example, SU(3) gauge group is flavor nonuniversal in TC2 models and U(1) gauge group is flavor non-universal in the flavoruniversal TC2 models. When one writes the non-universal interactions in the mass eigenbasis, it can induce the tree-level FC couplings. Thus the new particles, such as top-pions (πt± , πt0 ) predicted by TC2 models, have the tree-level FC couplings to the ordinary particles and may generate large contributions to the single top production in ep collisions at the HERA and THERA colliders. In this paper, we study the contributions of the top-pions to the single production of the top quarks in ep collisions at the HERA and THERA collider experiments in the context of TC2 models. Our aim is to investigate whether the t-channel process eq → et mediated by the FC couplings via the anomalous top quars vertices t–q–γ and t–q–Z can be used to
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detect the top-pions (πt0 , πt± ) and further probe new physics beyond the SM. We find that the contributions of the charged top-pions πt± to the single top production is very small. The neutral top-pion πt0 can generate significant contributions to the single top production via the process ec → et. In most of the parameter space, the production cross section is in the range of 1–6 pb. The virtual effects of the neutral top-pion πt0 on single top production can be detected in the HERA and THERA collider experiments. The paper is organized as follows: in Section 2 we give and discuss the anomalous top quark couplings tqV (V = γ or Z) arised from the top-pions πt± , πt0 . Their contributions to the single top production process ec → et are calculated in Section 3. The conclusions and discussions are given in Section 4.
2. Top-pions and the anomalous top quark couplings tqV In TC2 models [12], the TC interactions play a main role in breaking the electroweak gauge symmetry. The ETC interactions give rise to the masses of the ordinary fermions including a very small portion of the top quark mass, namely εmt with a model-dependent parameter ε 1. The main part of the top quark mass is dynamically generated by topcolor interactions at a scale of 1 TeV, which also make small contributions to EWSB. This means that the associates top-pions πt±,0 are not the longitudinal bosons W and Z, but are separately physically observable objects. The presence of physics top-pions in the lowenergy spectrum is an inevitable feature of topcolor scenario that purports to avoid finetuning[16]. The flavor diagonal couplings of top-poins to fermions can be written as [4,12, 16] 2 2 √ mt (1 − ε) νW − Ft 5 0 √ i t¯γ tπt + 2 t¯R bL πt+ + 2 b¯L tR πt− , (1) √ νW 2 Ft √ where νW = ν/ 2 = 174 GeV and Ft ≈ 50 GeV is the top-pion decay constant, which can be estimated from the Pagels–Stokar formula. For TC2 models, the underlying interactions, topcolor interactions, are non-universal and therefore do not possess the GIM mechanism. This is an essential feature of these kinds of models due to the need to single out top quark for condensate. The non-universal gauge interactions result in the FC coupling vertices when one writes the interactions in the mass eigen-basis. Thus, the top-pions have large Yukawa couplings to the third family quarks and can induce the tree-level FC scalar couplings [17,18] 2 − F2 νW t tc tt∗ mt t c∗ t t 0 ikU R kU L t¯L cR πt0 + ikU √ L kU R c¯L tR πt ν 2Ft W √ t c∗ bb √ t c bb∗ + − ¯ + 2 kU (2) R kDL c¯R bL πt + 2kU R kDL bL cR πt , where kU L(R) and kDL(R) are rotation matrices that diagonalize the up-quark and down+ + dia dia quark mass matrices MU and MD , i.e., kU L MU kU R = MU and kDL MD kDR = MD , + for which the CKM matrix is defined as V = kU L kDL . To yield a realistic form of the CKM V , it has been shown [17] that the top-pions mainly couple to the right-handed top
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(tR ) or charm (cR ) quark and thus the values of the coupling parameters can be taken as: tt bb tc tc 2 kU (3) kU kU R ≈ kDL ≈ 1, L ≈ 0, R 2ε − ε . √ tc In the following calculation, we will assume kU R = 2ε − ε2 and take ε as a free parameter, which is assumed to be in the range of 0.03–0.1 [4,12]. The relevant Feynman diagrams for the contributions of the neutral top-pion πt0 to the anomalous top quark couplings tcγ and tcZ via the tree-level FC couplings are shown in Fig. 1. Using Eqs. (1)–(3) and other relevant Feynman rules, we can give the effective forms of the anomalous top quark coupling vertices Ztc and γ tc: µ µ ΛZ t¯c = ie γ µ F1Z + F2Z γ 5 + Pt F3Z + F4Z γ 5 + Pcµ F5Z + F6Z γ 5 , (4) µ µ , Fiγ = FiZ 2 . Λ =Λ (5) γ t¯c
Z t¯c FiZ →Fiγ
Vt = 3 ,at =0
The form factors FiV are expressed in terms of two- and three-point standard Feynman integrals [19]. The expressions of FiV are given in Appendix A. The neutral top-pion πt0 can also generate the anomalous top quark couplings tuγ and tuZ via the tree-level FC coupling πt0 tu. However, it has been argued that the maximum flavor mixing occur between the third generation fermions and the second generation fermions, and the FC coupling πt0 t¯u is very small which can be neglected [18]. Hence we will ignore the anomalous top quark couplings tuγ and tuZ, and only calculate the contributions of the neutral top-pions πt0 to the production cross section of the t-channel process ec → et via the anomalous top quark couplings tcγ and tcZ in the following. The charged top-pions πt± can contribute to the anomalous top quark couplings tcγ and tcZ via the Feynman diagrams as depicted in Fig. 2. These diagrams are mediated
Fig. 1. Feynman diagrams for the contributions of the neutral top-pion πt0 to the anomalous top quark couplings tcγ and tcZ.
Fig. 2. Feynman diagrams for the contributions of the charged top-pions πt± to the anomalous top quark couplings tcγ and tcZ.
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by the FC couplings πt± bc. The internal fermion line must be bottom quark. Similar to calculation about Fig. 1, we can give the effective vertices Λ γ t¯c and Λ Z t¯c arising from Fig. 2. However, compared to Λγ t¯c and ΛZ t¯c given by Fig. 1, these effective vertices are very small and can be safely ignored [6]. So, in the following calculation, we will not consider the contributions of the charged top-pions πt± to the single top production via the process ec → et at the HERA and THERA colliders. 3. The neutral top-pion πt0 and single top production From above discussion, we can see that the neutral top-pion πt0 can induce large anomalous top quark couplings tcγ and tcZ. Hence, it is possible that πt0 generate significant contributions to the single top production at the HERA and THERA colliders via the t-channel process ec → et with the relevant Feynman diagrams shown in Fig. 3. Using the effective vertices ΛZ t¯c and Λγ t¯c given by Eqs. (4) and (5), we can obtain the cross section σˆ (ˆs ) of the subprocess ec → et which can be written as
1 1 d σˆ 2 2 † = | + |M | + 2 Re M M (6) |M Z γ Z γ dt 64π sˆ (m2c − sˆ) with MZ =
1 m2Z
−t
u¯ t ΛσZ t¯c uc u¯ e γσ (ve − ae γ5 ) ue ,
1 Mγ = u¯ t Λσγ t¯c uc u¯ e γσ ue . t Here ve =
e 2 −1 + 4SW , 4SW CW
(7) (8)
ae = −
e , 4SW CW
(9)
√ where sˆ is the center-of-mass energy of the subprocess ec → et in ep collisions. The total cross section σ (s) of the single top production at the HERA and THERA colliders can be obtained by folding the cross section σˆ with the charm-quark distribution function fc (x) in the proton: 1 σ= xmin
t+ fc (x) dx
d σˆ dt dt
(10)
t−
Fig. 3. Feynman diagrams for the single top production at the HERA and THERA colliders, due to the anomalous top quark couplings tcV (V = γ or Z).
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Fig. 4. The production cross section σ (s) contributed by πt0 as a function of mπt for ε = 0.02 (solid line), ε = 0.05 (dashed line), and ε = 0.08 (dotted line) at the HERA collider. m2 +m2
(m2 −ˆs )(ˆs −m2 )
c with sˆ = xs, xmin = t s e , t− = t sˆ and t− = tcut = −0.001 (GeV)2 . The parton distribution function fc (x) of the charm quark runs with the energy scale. In our calculation, we take CTEQ5 parton distribution [20] for fc (x). The cross sections σ (s) of the single top production at the HERA and THERA colliders are plotted in Figs. 4 and 5 respectively, as a function of the mass mπt of the neutral top pion πt0 for the parameter ε = 0.02 (solid line), 0.05 (dashed line) and 0.08 (dotted line). 1 2 = 0.2315, m = 175 GeV, m = 1.2 GeV, In our calculation, we have taken αe = 128.8 , SW t c mZ = 91.18 GeV and ΓZ = 2.49 GeV [21]. We can see from these two figures that the cross sections decrease with mπt increasing and ε decreasing. In all of the parameter space of TC2 models, the cross section of single top production at the THERA collider is large than that of at the HERA collider. For 0.02 ε 0.08 and 200 mπt 350 GeV, the cross sections are in the ranges of 0.5–3.7 pb and 0.86–6.4 pb for the HERA collider and THERA collider, respectively. In order to estimate√the number of the events, we consider two ep collider scenarios: the −1 HERA collider with s = 320 GeV √ and a yearly integrated luminosity of L = 160 pb and the THERA collider with s = 1000 GeV and a yearly integrated luminosity of L = 470 pb−1 [8]. The yearly production events of the single top production at the HERA and THERA colliders can be easily calculated. In most of the parameter space, there may be several hundreds of the single top events at the HERA collider and hundreds and up to thousands single top events at the THERA collider to be generated. For example, for mπt = 300 GeV and ε = 0.05, the HERA collider can generate 240 single top events and the THERA collider can generate 1250 single top events. Thus, the contributions of the
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Fig. 5. The production cross section σ (s) contributed by πt0 as a function of mπt for ε = 0.02 (solid line), ε = 0.05 (dashed line), and ε = 0.08 (dotted line) at the THERA collider.
neutral top pion πt0 to the single top production may be detected at the HERA collider or the THERA collider. The signals of πt0 can be studied at these colliders via the process ec → et.
4. Conclusions and discussions The anomalous top quark couplings tqV (q = u- or c-quarks and V = Z, or γ gauge bosons), which are arised from the FC interactions, offer an ideal place to search for new physics beyond the SM as they are very small in the SM. Single top production is very sensitive to the tqV couplings in the HERA and THERA colliders. Studying the contributions of the tqV couplings to the single production of the top quark will be helpful to test the SM flavor structure and new physics beyond the SM. In this paper, we study the contributions of the top-pions predicted by TC2 models to the single top production in ep collisions at the HERA and THERA colliders. Our results show. (1) Although the contributions of the up-quark distribution in the proton to the single top production is large than those of the charm-quark distribution, the contributions of the anomalous top quark couplings tuV given by the TC2 models to the single top production in ep collisions are much smaller than those of the tcV couplings and can be safely neglected. (2) The charged top-pions πt± can generate the anomalous top quark couplings tcV via the FC couplings πt± bc. However, compared with the tcV couplings generated by the
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neutral top-pion πt0 via the FC coupling πt0 tc, their values are negligibly small. The πt± cannot give significantly contributions to the production cross section of the single top quark at the HERA and THERA colliders. (3) The contributions of the neutral top-pion πt0 to the production cross section σ √ of the t-channel process ec → et increases with s increasing and mπt decreasing. For 200 mπt 400 GeV and ε = 0.05, the value of σ varies from 0.99 to 2.35 pb and from 1.7 to 4.1 pb at the HERA and THERA colliders, respectively. Thus, there may be several hundreds and up to thousands of the single top production events to be generated at these two colliders. We can study the signature and observability of πt0 via the process ec → et at the HERA collider or the THERA collider. TC2 models also predict the existence of the neutral cp-even state, called top-Higgs boson h0t , which is a t t¯ bound and analogous to the σ particle in low energy QCD. Its mass can be estimated in the Nambu–Jona-Lasinio model in the large Nc approximation and is found to be of the order mht ≈ mπt [18]. The main difference between πt0 and h0t is that h0t can couple to gauge boson pairs W W and ZZ at tree level, which is similar to that of the SM Higgs boson H 0 . Thus, the contributions of the top-Higgs h0t to the single top production at the HERA and THERA colliders are similar to those of πt0 . We find that, in most of the parameter space of TC2 models, the cross section of single top production given by the top-Higgs boson h0t varies in the range of 0.8–6.2 pb. Thus the top-Higgs boson h0t can also be detected at the HERA collider or the THERA collider via the process ec → et. The key feature of TC2 models is that a large part of the top quark mass is dynamically generated by topcolor interactions at a scale of order 1 TeV, which is flavor non-universal. To ensure that the top quark condenses and receives a large mass while the bottom quark does not, the topcolor gauge group is usually taken to be a strongly coupled SU(3) × U (1). The U (1) provides the difference that cause only top quark to condensate. Thus, TC2 models predict the existence of topcolor gauge bosons BµA and an extra U (1) gauge boson Z . Tree-level FCNC for the topcolor gauge bosons BµA and Z are generated when quarks fields are rotated to the mass eigenstate basis. The couplings of Z to ordinary fermions are non-universal and stronger for the third generation fermions, yielding potentially large top-charm couplings. Thus, the topcolor gauge bosons can generate large anomalous top couplings tcV (V = γ or Z), which may produce significant contributions to the single top production at the HERA and THERA collider via the process ec → et. However, the limits on the masses of the topcolor gauge bosons BµA and Zµ can be obtained via studying their effects on various experimental observables [4]. For example, Ref. [22] considered the bound placed by the electroweak measurement data on the gauge boson Z . They find that Z predicted by the TC2 models and the flavor universal TC2 models must be heavier than 1 TeV. If we assume MZ = MB 1 TeV, then we find that the production cross section is smaller than 1 × 10−2 pb in most of the parameter space of the TC2 models. Thus, it is very difficult to detect the topcolor gauge bosons via the single top production at the HERA and THERA colliders.
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Acknowledgements We thank C.-P. Yuan for pointing out that we should use the evolved parton distribution function of the charm quark to calculate the production cross section. This work was supported in part by the National Natural Science Foundation of China (90203005). Appendix A. The form factors in the effective vertices Z t¯c and γ t¯c for the neutral top-pion πt0 F1Z = g B0 + m2πt C0 − 2C24 + m2t (C11 − C12 ) − B0∗ − B1 vt , F2Z = g B0 + m2πt C0 − 2C24 − m2t (C11 − C12 ) + B0∗ + B1 − 4C24 at , F3Z = 2mt g(C21 + C22 − 2C23 )vt , F4Z = 2mt g(−C21 − C22 + 2C23 − 2C22 + C12 + C0 )at , F5Z = 2mt g(C22 − C23 + C12 )vt , F6Z = 2mt g(−C22 + C23 + C12 − 2C22 + 3C12 + 2C23 − 2C11 − C0 )at . Here
mt 1 g= √ 2 16π 2 Ft 1 vt = 1− 4SW CW
νw2 − Ft2 2
νw 8 2 SW , 3
KUt cR KUt t L∗ , at =
1 . 4SW CW
The expressions of two- and three-point scalar integrals Bn and Cij in this paper are [19]: √ Bn = Bn (− t, mt , mt ), Bn∗ = Bn (−pc , mπt , mt ), Bn = Bn (−pt , mπt , mt ), √ Cij = Cij (pt , − t, mπt , mt , mt ), √ C0 = C0 (pt , − t, mπt , mt , mt ), √ Cij = Cij (−pc , t, mt , mπt , mπt ), √ C0 = C0 (−pc , t, mt , mπt , mπt ).
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