Transverse Tau Polarization in Decays of the Top and Bottom Quarks in the Weinberg Model of CP Non-conservation*

We show that the transverse polarization asymmetry of the r-lepton in the decay t + bra is extremely sensitive to CP violating phases arising from the charged Higgs exchange in the Weinberg model of CP non-conservation. Qualitatively, the polarization asymmetries are enhanced over rate or energy asymmetries by a factor of x % x O(100). Thus for optimal values of the parameters the method m, requires e lo4 top pairs to be observable rather than lo7 needed for rate or energy asymmetries. We also examine r polarization in b decays via b -+ cur and find that it can also be very effective in constraining the CP violation parameters of the extended Higgs sector. Submitted to Physical Review Letters. * Work supported by the Department of Energy, contract DE-AC03-76SF00515. .* he pph /9 30 32 68 16 M ar 1 99 3 Stanford Linear Accelerator Center March 2001 SLAC-PUB-6083 Transverse Tau Polarization in Decays of the Top and Bottom Quarks in the Weinberg Model of CP Non-conservation D. Atwood*, G. Eilam† and A. Soni‡ * Dept. of Physics, SLAC, Box 4349 Stanford CA 94309 USA. † Dept. of Physics, Technion, Haifa, Israel ‡ Dept. of Physics, Brookhaven National Laboratory, Upton NY 11973 USA. ABSTRACT We show that the transverse polarization asymmetry of the τ -lepton in the decay t→ bτν is extremely sensitive to CP violating phases arising from the charged Higgs exchange in the Weinberg model of CP non-conservation. Qualitatively, the polarization asymmetries are enhanced over rate or energy asymmetries by a factor of ≈ mt mτ ≈ O(100). Thus for optimal values of the parameters the method requires ≈ 10 top pairs to be observable rather than 10 needed for rate or energy asymmetries. We also examine τ polarization in b decays via b→ cντ and find that it can also be very effective in constraining the CP violation parameters of the extended Higgs sector.We show that the transverse polarization asymmetry of the τ -lepton in the decay t→ bτν is extremely sensitive to CP violating phases arising from the charged Higgs exchange in the Weinberg model of CP non-conservation. Qualitatively, the polarization asymmetries are enhanced over rate or energy asymmetries by a factor of ≈ mt mτ ≈ O(100). Thus for optimal values of the parameters the method requires ≈ 10 top pairs to be observable rather than 10 needed for rate or energy asymmetries. We also examine τ polarization in b decays via b→ cντ and find that it can also be very effective in constraining the CP violation parameters of the extended Higgs sector. Due to its mass scale the top quark represents a unique probe for addressing to the long-standing issue of CP violation. While the FNAL Tevatron is expected to produce enough top quarks to establish its existence, the hadronic and e − e− colliders under construction and being proposed should facilitate studies related to CP. In this context, the search for optimal experimental strategies is clearly important and requires extended phenomenological studies. In the Standard Model (SM) CP violation effects in the top quark are too small to be observable 1 . However, it is difficult to see why the Kobayashi-Maskawa 2 (KM) phase should be the only source for CP non-conservation. Extensions of the SM almost invariably lead to additional CP violating phases. Indeed the observed baryon asymmetry in the universe is often used to argue that additional sources of CP beyond the KM phase are a necessity 3 . We are therefore motivated to continue our investigation of CP non-conservation caused in the production 4 and decays 5,6 of the top quark in one popular extension of the SM 7 , namely the Weinberg Model 8 (WM). This model leads to large, possibly observable, dipole moments 9 and rather sizable partially integrated rate asymmetry 5 (PIRA) in the semi-leptonic modes t→ bτντ . In this work we show that the transverse polarization of the τ in these semileptonic transitions is extremely sensitive to CP violation effects due to the extended Higgs sector. Thereby, with optimal values of the parameters in the WM, requiring only a few thousand top quark pairs rather than ≥ 10 needed for the rate, the triple correlation or the energy asymmetries 5,6,10 . Thus the polarization effects become accessible not only to SSC/LHC, where the estimates are for 10 − 10 top pairs/year but also to an electron-positron linear collider with an anticipated rate of about 10/yr. Clearly this approach can only be used if the top detectors are capable of measuring the τ polarization. We have also examined the effectiveness of τ polarization as a possible signal of CP violation in semi-leptonic b decays due to the extended Higgs sector. Our finding is that one of the transverse polarization asymmetries is useful in b decays as well as it can lead to stringent constraints on the parameters of the Higgs model with about 10 B mesons. The semi-leptonic decay of the top quark proceeds through W exchange and Higgs exchange graphs (See Fig. 1a and 1b). The CP violating phase is in the Higgs exchange. Since mt > mW , the dominant contribution of the W-graph is from on shell W-bosons. 1 Thus Fig 1a possesses both dispersive and absorptive pieces, and indeed both can have significant resonance enhancement, as we shall see below 1,5,6 . The absorptive piece of Fig 1a interferes with the CP violating Higgs phase to contribute to observables (e.g. rate asymmetry) that are CP-odd, TN even 11 . Here TN denotes naive time reversal where the spins and momenta of all particles are reversed but initial and final states are not interchanged. On the other hand observables such as momentum triple correlations that are CP-odd, TN -odd receive contribution from the dispersive part of Fig. 1a. Let us now qualitatively try to understand why the transverse polarization is much more sensitive compared to energy or rate asymmetries. Recall that PIRA (or energy asymmetry) goes as 12 mτ/m 2 H . It is important to understand the sources for the two powers of mτ . One of these originates from the Yukawa couplings at the Hτντ vertex. The second power of mτ originates from the trace over the lepton loop resulting from the interference between the W and the Higgs. This trace goes as: Tr[γμ(p/τ +mτ)(1 − γ5)p/ν ] = 4mτp μ ν (1) Clearly the mτ in this trace arises because one is summing over the spins of the final τ . So the mτ in the above trace can be avoided provided we do not sum over the spins of the τ . Thus we arrive at CP violating transverse polarization asymmetries of the τ that are larger by ≈ mt/mτ ≈ 100 compared to PIRA requiring therefore ≈ 100 fewer top pairs. Thus the transverse polarization asymmetries may well be observable with ≈ 10 tt̄ pairs rather than ≈ 10 needed for PIRA. We now briefly recapitulate some aspects of the Weinberg Model that are relevant to this work 13 . The model consists of three Higgs doublets enabling it to possess a CP violating Higgs sector without the flavor changing neutral currents. It therefore has two charged Higgs states H± 1 and H ± 2 . For simplicity we will assume that H ± 1 has a much higher mass than H± 2 (mH) so that we need to consider effects due to H ± 2 only. The Lagrangian coupling H2 to quarks and leptons is given 13 by: L = g √ 2 H 2 (ūi mdj mW UdPRdjV KM ij + ūi mui mW UuPLdjV KM ij + ν̄i mli mW UlPRljδij) +H.C. (2) where Ul =− c1s2s3 + c2c3e s1s2 2 Uu = c1c2s3 − s2c3e s1c2 Ud = s1s3 c1 (3) and V KM is the Kobayashi-Maskawa matrix. We denote si = sin(θi) and ci = cos(θi) where θi and δ are parameters of the Higgs potential. The CP violation which we consider is proportional to combinations of these couplings such as Vul ≡ Im(U∗ u Ul). In the rest frame of the τ lepton, let us define a reference frame where the momentum of the top quark is in the −x direction, the y direction is defined to be in the decay plane such that the y component of the b momentum is positive and the z axis is defined by the right hand rule. In the limit that the τ mass is small, (WM) can give rise to two kinds of CP violating polarization asymmetries. These are AY = τ(↑) − τ(↓) + τ−(↑) − τ−(↓) τ+(↑) + τ+(↓) + τ−(↑) + τ−(↓) AZ = τ(↑) − τ(↓) − τ−(↑) + τ−(↓) τ+(↑) + τ+(↓) + τ−(↑) + τ−(↓) (4) where for AY (AZ) the arrows indicate the spin up or down in the direction y (z). While both AY and AZ are CP odd, being TN even, AY needs an interaction phase; AZ is odd under TN and does not need an interaction phase 14 . Thus AY (AZ) is proportional to absorptive (dispersive) part of Fig. 1a. For our analysis of top decay, we will ignore the masses of the b quark and the τ lepton (wherever that is a good approximation). Thus let us define the invariants: s = 2pν · pτ t = 2pτ · pb u = 2pb · pν (5) as well as the quantities yW = ΓW/m 2 t , λ = s/m 2 t and xi = m 2 i /m 2 t for i = b, τ , H and W . The angle θ is defined to be the angle between the W momentum and the τ momentum in the W rest frame. Consider now the case of the CP violating polarization in the x− y plane. This gives rise to a polarization asymmetry in the direction V = spb − tpν √ stu (6)