We investigate a possible CP violating effect in $e^+e^-$ annihilation into
$t\bar t$ top quark pairs. As an illustrative example, we assume the source of
the CP nonconservation is in the Yukawa couplings of a neutral Higgs boson
which contain both scalar and pseudoscalar pieces. One of the interesting
observable effects is the difference in production rates between the two CP
conjugate polarized $t\bar t$ states.Comment: 9 pages, 3 postscript figures not included but availabel upon
  request, CERN-TH.6658/9

Chang, Darwin

Keung, Wai-Yee

Phillips, Ivan

English

arXiv

We investigate a possible CP violating effect in $e^+e^-$ annihilation into $t\bar t$ top quark pairs. As an illustrative example, we assume the source of the CP nonconservation is in the Yukawa couplings of a neutral Higgs boson which contain both scalar and pseudoscalar pieces. One of the interesting observable effects is the difference in production rates between the two CP conjugate polarized $t\bar t$ states.We investigate a possible CP violating effect in $e~+e~-$ annihilation into $t\bar t$ top quark pairs. As an illustrative example, we assume the source of the CP nonconservation is in the Yukawa couplings of a neutral Higgs boson which contain both scalar and pseudoscalar pieces. One of the interesting observable effects is the difference in production rates between the two CP conjugate polarized $t\bar t$ states

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arXiv:hep-ph/9209267v1  23 Sep 1992CERN-TH.6658/92NUHEP–TH–92–17CP violation in top pair productionat an e+e− colliderDarwin Chang(1,2), Wai–Yee Keung(3,4)Ivan Phillips(1)(1)Department of Physics and Astronomy,Northwestern University, Evanston, IL 60208, USA.(2)Institute of Physics, Academia Sinica, Taipei, Taiwan, R.O.C.(3)Physics Department, University of Illinois at Chicago, IL 60680, USA(4)Theory Division, CERN, CH-1211, Geneva 23, SwitzerlandAbstractWe investigate a possible CP violating effect in e+e− annihilation into tt¯ top quarkpairs. As an illustrative example, we assume the source of the CP nonconservationis in the Yukawa couplings of a neutral Higgs boson which contain both scalar andpseudoscalar pieces. One of the interesting observable effects is the difference inproduction rates between the two CP conjugate polarized tt¯ states.CERN–TH.6658/92September 1992Since the top quark is widely believed to be within the reach of the present collidermachines, it is not unreasonable for theorists to imagine what we can learn from the topquark. The best place to study the top quark in detail is in an e+e− collider. One of thefacts one would like to learn from the discovery of the top quark is the origin of the stillmysterious CP violation. In this paper we investigate a way CP violation can manifest itselfin the top pair production of an e+e− collider.Among the various mechanisms of CP violation the one that may manifest itself mosteasily is the neutral Higgs mediated CP violation. Since the neutral Higgs couplings aretypically proportional to the quark mass, the large mass of the top quark naturally giveslarge couplings to the neutral Higgs bosons. In addition, the top quark, due to its shortlifetime, is widely believed to decay before it hadronizes. Therefore the information aboutits polarization may be preserved in its decay products. If that is the case, then one caninvestigate the source of CP nonconservation by measuring the CP violating observableinvolving a polarized top pair in the final state. This idea of detecting the rate asymmetrybetween different polarized states was recently pursued in Refs. [1,2]. For tt¯ productionthrough the virtual photonic or Z intermediate states, to the lowest order in the final statequark mass, the polarizations of the quarks are either tLt¯R or tRt¯L. (Note that we haveadopted the notation that t¯L is the antiparticle of tR and should be left handed.) These twomodes are CP self-conjugate. However since the top quark is heavy, there will also be largepercentage of tLt¯L and tRt¯R modes which are CP conjugates of each other. Therefore onecan consider the CP asymmetry in the event rate difference, N(tLt¯L)−N(tRt¯R).For detection of the asymmetry N(tLt¯L)− N(tRt¯R) [1,2], one assumes that the t quarkdecays semileptonically through the usual V − A weak interaction. Assuming that thehadronization time is much longer than the decay time [3], one can analyze polarizationdependence of its decay at the quark level. The top quark first decays into a b quarkand a W+ boson, which subsequently becomes l+ν. For heavy top quark, the W+ bosonproduced in top decay is predominantly longitudinal. Due to the V − A interaction, the bquark is preferentially produced with left-handed helicity. So the longitudinal W+ boson1is preferentially produced along the direction of the top quark polarization. Therefore theanti–lepton l+ produced in the W+ decay is also preferentially in that direction. In the restframe of the t, the angular distribution [4] of the produced l+ has the form 1+cosψ, with ψas the angle between l+ and the helicity axis of the t. Above the tt¯ threshold, the top quarkis produced with nonzero momentum. As a result of the Lorentz boost, the anti–leptonl+ produced in the decay of the right handed top quark tR has a higher energy than thatproduced in the decay of the left handed top quark tL. Similarly, the lepton l produced inthe decay of t¯L has a higher energy than that produced in the decay of t¯R. Consequently,in the decay of the pair tLt¯L the lepton from t¯L has a higher energy than the anti–leptonfrom tL; while in the decay of tRt¯R the anti–lepton has a higher energy. Therefore one canobserve N(tLt¯L)−N(tR t¯R) by measuring the energy asymmetry in the resulting leptons [5].In order to generate the asymmetries N(tLt¯L)−N(tRt¯R) it is necessary to include effectsof the final state interactions in order to escape from the hermiticity constraint at the tree–level due to the CPT theorem. In our case both CP violation and the final state effect areproduced by the same one–loop graphs.CP non–conservation occurs in the complex Yukawa coupling,LCPX = −(mt/v)t¯(APL + A∗PR)tH +BH(m2Z/v)ZνZν . (1)Here v = (√2GF )−12 ≃ 246 GeV. The complex coefficient A is a combination of model-dependent mixing angles. Simultaneous presence of both the real part AR = ReA andthe imaginary part AI = ImA guarantees CP asymmetry. For example, in the low energyregime, it can give rise to the electric dipole moment of elementary particles [6,7]. Here wewill show that CP nonconservation manifests itself in the event rate difference in colliderexperiments.The vertex amplitude ieΓj for the virtual γ∗ or Z∗ turning into t(p) and t¯(p′) isparametrized in the following expression:Γjµ = cjvγµ + cjaγµγ5 + cjdiγ5pµ − p′µ2mt+ · · · , j = γ, Z. (2)2We use the tree–level values for cv and ca. They arecγv =23, cγa = 0,cZv = (14− 23xW )/√xW (1− xW ) , (3)cZa = − 14/√xW (1− xW ) .The cd term is the the electric dipole moment factor. The spinor structure can be recast intoanother familiar form σµν(p+p′)νγ5/2mt. It is induced at the one–loop level as shown in theFig. 1. We are interested in the absorptive parts which can be easily calculated accordingto the Cutkosky rules. It can also be easily shown that, in the limit that the electron massis ignored, Imcjd are the only relevant one loop form factors for the CP violating quantitieswe are interested in.The leading contribution to Imcγd comes from the rescattering of the top quark pairthrough the Higgs–boson exchange.Imcγd = cγv(mtv)2ARAIt22piβ(1− h2β2log(1 +β2h2)) . (4)The dimensionless variables are defined by, t = mt/√s, z = mZ/√s, and h = mH/√s. ForImcZd , there is a similar contribution. In addition, there could be a contribution due to theZH intermediate state, Fig 1b, provided the kinematics is allowed.ImcZd =cZvcγvImcγd −αAIBcZv t22(1− xW )xWβ3[ββZ + (2t2 + 2t2h2 − 2t2z2 − h2)L] . (5)Here β2Z = 1 + h4 + z4 − 2z2 − 2h2 − 2h2z2, β2 = 1 − 4t2, and the logarithmic factorL = log(L−/L+) with L± = 1 − z2 − h2 ± ββZ . Other irrelevant terms, like the magneticmoments, are not listed in Eq.(2). Note that there is no CP violating contribution due to cZacoupling in the limit that the electron mass is ignored. Our expression in Eq.(5) agrees withthat in Ref. [8,9]. The amplitudes for the process e+e− → tt¯ of different helicities have beengiven in the literature [10]. Now, we can obtain the explicit CP asymmetry in the differenceof the production rates,δ ≡ [N(tLt¯L)−N(tRt¯R)]/N(tt¯; all)3=∑i=L,R 2β[(1− z2)cγv + ricZv ][(1− z2)Imcγd + riImcZd ]∑i=L,R(3− β2)[(1− z2)cγv + ricZv ]2 + 2β2r2i cZa 2. (6)Here ri = (12δi,L − xW )/√xW (1− xW ), which is the Z–coupling of the electron of differentchiralities. Typical values of δ are shown in Fig. 2.We can make use of this asymmetry parameter δ to illustrate the the difference in theenergy distributions of l+ or l− from the t or t¯ decays. The energy E0(l+) distribution ofa static t quark decay t → l+νb is very simple [4] in the narrow width ΓW approximationwhen mb is negligible.f(x0) =x0(1− x0)/D if m2W/m2t < x < 1,0 otherwise.(7)Here we denote the scaling variable x0 = 2E0(l+)/mt and the normalization factor D =16− 12(mW/mt)4 + 13(mW/mt)6. When the t quark is not static, but moves at a speed β withhelicity L or R, the distribution expression becomes a convolution,fR,L(x, β) =∫ x/(1−β)x/(1+β)f(x0)βx0 ± (x− x0)2x20β2dx0 . (8)Here x = 2E(l+)/Et. Similar distributions for the t¯ decay is related by CP conjugationat the tree–level. Using the polarization asymmetry formula in Eq.(6), we can derive anexpression for the difference in the energy distributions of l− and l+:1N[ dNdx(l+)− dNdx(l−)]= δ[fL(x, β)− fR(x, β)] . (9)Here distributions are compared at the same energy for the lepton and the anti–lepton,x(l−) = x(l+) = x = 4E(l±)/√s. The count N includes events with prompt leptons oranti–leptons from the top pair production. It is useful to compare Eq.(9) with that of theoverall energy distribution,1N[ dNdx(l−)+dNdx(l+)]=∑i=L,R 4βricZa [(1− z2)cγv + ricZv ][fR(x, β)− fL(x, β)]∑i=L,R(3− β2)[(1− z2)cγv + ricZv ]2 + 2β2r2i cZa 2+ fL(x, β) + fR(x, β) . (10)4Here we only keep the dominant tree–level contribution. Fig. 3 shows the overall promptlepton energy distribution of Eq.(10), and the ratio of the expressions in Eq.(9) and Eq.(10).In conclusion, we have shown that the CP violation in top pair production in e+e− anni-hilation can be at the level of 10−3. Experiments can measure the asymmetry in the energydistributions of prompt leptons and anti–leptons. An e+e− machine of high luminosity isneeded to accumulate several million prompt lepton events from the tt¯ production in orderto search for the CP nonconservation.D.C. wishes to thank the Theory Group at the Institute of Physics at Academia Sinicain Taipei, Taiwan and Department of Physics and Astronomy at University of Hawaii atManoa for hospitality while this work was in progress. This work is supported by grantsfrom Department of Energy and from National Science Council of Republic of China.5REFERENCES[1] C.R. Schmidt and M. E. Peskin, Phys. Rev. Lett. 69, 410 (1992).[2] D. Chang and W.–Y. Keung, Northwestern preprint, NUHEP–TH–92–14 (1992).[3] I. Bigi and H. Krasemann, Z. Phys. C7, 127 (1981); J. Ku¨hn, Acta Phys. Austr. Suppl.XXIV, 203 (1982); I. Bigi, Y. Dokshitzer, V. Khoze, J. Ku¨hn, and P. Zerwas, Phys.Lett. 181B, 157 (1986).[4] M. Jez´abek and J.H. Ku¨hn, Nucl. Phys. B320, 20, (1989).[5] Additional CP–odd observables in collider physics have been studied by G. Kane, G.A.Ladinsky, and C.–P. Yuan, Phys. Rev. D45, 124 (1991); J.F. Donoghue and G. Valencia,Phys. Rev. Lett. 58, 451 (1987); W. Bernreuther and O. Nachtmann, Phys. Rev. Lett.63, 2787 (1989); G. Valencia and A. Soni, Phys. Lett. B263, 517 (1991); A. Bilal,E. Masso´, and A. De Ru´jula, Nucl. Phys. 355, 549 (1991); D. Atwood and A. Soni,Brookhaven Report BNL–46892 (1991).[6] S. Weinberg, Phys. Rev. Lett. 63, 2333 (1989); Phys. Rev. D42, 860 (1990).[7] S. Barr and A. Zee, Phys. Rev. Lett. 65, 21 (1990); See also J. Gunion and R. Vega,Phys. Lett. 251B, 21 (1990); D. Chang, W.–Y. Keung, and T.C. Yuan, Phys. Lett.251B, 608 (1990); D. Chang, W.–Y. Keung, and T.C. Yuan, Phys. Rev. D43, R14(1991); R. Leigh, S. Paban, and R. Xu, Nucl. Phys. B352, 45 (1991).[8] W. Bernreuther, T. Schro¨der, and T.N. Pham, Phys. Lett. B279, 389 (1992).[9] W. Bernreuther, J.P. Ma, and T. Schro¨der, Heidelberg preprint,HD–THEP–92–30 (1992).[10] G. Kane, G.A. Ladinsky, and C.–P. Yuan, Phys. Rev. D45, 124 (1991). G6FIGURESFIG. 1. Feynman diagrams for the process e+e− → tt¯. The tree amplitude interferes with thoseone–loop amplitudes with (a) the final state interactions due to the exchange of a Higgs boson, or(b) the intermediate state production of the ZH bosons.FIG. 2. [N(tL t¯L) − N(tRt¯R)]/N(all tt¯) versus√s in the solid (dotted, dashed) curve forthe case that mH = 50 (150,200) GeV and mt = 100 GeV. The parameters are chosen to beAI = AR = B = 1.FIG. 3. The energy distribtuions of prompt leptons, for the case that mt = 120 GeV, mH = 50GeV,√s = 300 GeV, and AI = AR = B = 1. Case (a) for N−1[dN/dx(l+)+ dN/dx(l−)], and case(b) for [dN/dx(l+)− dN/dx(l−)]/[dN/dx(l+) + dN/dx(l−)].7

CP violation in top pair production at an e^+e^- collider

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