ar X iv : h ep - p h / 98 07 57 3 v 1 3 1 Ju l 1 99 8 FERMILAB -

Unlike other examples of fermion pair production in e+e− collisions, we show that top-quark pairs are produced in an essentially unique spin configuration in polarized e+e− colliders at all energies. Since the directions of the electroweak decay products of polarized top-quarks are strongly correlated to the top-quark spin axis, this unique spin configuration leads to a distinctive topology for top-quark pair events which can be used to constrain anomalous couplings to the top-quark. A significant interference effect between the longitudinal and transverse W-bosons in the decay of polarized topquarks is also discussed. Dedicated to Sid Drell on the occasion of the “Sid Drell Symposium” held at the Stanford Linear Accelerator Center, Stanford University, California on July 31, 1998. At an ee collider with center of mass energy from 400 to 1000 GeV, top quark [1] pair production will be very different than fermion pair production at any previous ee machine for the following reasons: the photon (γ), Z-boson and their interference will contribute approximately equally to the production and the top quark pairs will be produced at non-ultra-relativistic speeds. At present and previous ee colliders fermion pair production is either in the ultra-relativistic regime (LEP, SLC and TRISTAN) and/or completely dominated by the photon contribution (PETRA, PEP, CESR, SPEAR and CEA). This marked difference in the production mechanism leads to significantly different correlations, both angular and in spin, of the top and anti-top quarks compared with any previous fermion pairs produced in ee collisions. Moreover, even the spin correlations are measurable for top quark pair production because the top quark decays [2] before QCD effects de-correlate its spin and the decay products of a top quark are highly correlated with the direction of its spin i.e. the top quark behaves more like a charged lepton than one of the light quarks in this regard. The matrix element for e−Le + R → tst̄s̄ is M ∼ e 2 S v̄(ē)γγLu(e) ū(t, s)γμ{ fLLγL + fLRγR }v(t̄, s̄) (1) where γR,L = (1±γ5) 2 , √ S is the center of mass energy and s and s̄ are the spins of the top and anti-top quarks. The fIJ ’s are the sum of the photon and Z-boson products of couplings to the fermions corrected for the difference in the propogators. For top production the fIJ ’s have a very weak S dependence and are given by fLL = 1.22− 1.19 fLR = 0.418− 0.434 fLR fLL = 0.343− 0.365 (2) from threshold to ultra-high energies. Without the Z-boson couplings, fLL = fLR = 2/3 so that the contribution from the Z-boson is constructive for fLL and destructive for fLR. This difference between fLL and fLR and the fact top quark pairs are non-ultra-relativistic is what makes top pair production different than other examples of fermion pair production. 2 We will first consider the limit where fLR = 0. In the ultra-high energy limit, where the mass of the top quark, mt, can be neglected √ S >>> 2mt, the matrix element is M ∼ e fLL(1 + cos θ), (3) where θ is the scattering angle. In this limit the top quark is purely left-handed and the anti-top quark is purely right-handed, e−Le + R → tLt̄R. At threshold, √ S ∼ 2mt, the matrix element is simply M ∼ e fLL (4) and the direction of the top and anti-top quark spins are opposite the electron momentum or equivalently in the direction of the positron momentum. At intermediate energies the matrix element interpolates between these two extremes M ∼ e fLL(1 + β cos θ), (5) where β is the ZMF speed of the top quarks. What are the directions of the top and anti-top quark spins? In the rest frame of the top quark there are three natural possibilities for the direction of the top spin; the electron, the positron or the anti-top quark momentum directions. The threshold result excludes the anti-top quark momentum direction, which is undefined at threshold, leaving only the electron or positron momentum direction. Similarly the natural possibilities for the anti-top quark spin, in its rest frame, are the electron or positron momentum directions. The correct choice is that the top quark spin vector is in the direction of the positron momentum in the top quark rest frame and the anti-top quark spin vector is in the direction opposite that of the electron momentum in the anti-top quark rest frame. This spin basis has been called the beamline basis [3] and smoothly interpolates the required basis from threshold to ultra-high energies. An obvious question is, why is the spin of the top quark associated with the positron and the anti-top associated with the electron