On the rigidity of back-to-back top quark pairs in e^+e^- annihilation

We consider the effect of gluon radiation on the energy of top/antitop quarks and on the anticollinearity of top-antitop quark pairs produced in $e^+e^-$ annihilation. Our results are presented in terms of the $E_q$-dependence of the $t\bar tg$ cross section and the dependence on the cosine of the opening angle $\theta_{12}$ between top and antitop for a center of mass energy of $\sqrt{q^2}=500 GeV$. We then go on to determine mean values for the top quark's energy as well as its longitudinal and transverse projections, and for the deviation of $\sin\theta_{12}$ and $\cos\theta_{12}$ from the anticollinearity limits $\sin\theta_{12}=0$ and $\cos\theta_{12}=-1$. For a center of mass energy of $500 GeV$ we obtain $<E_q>=248.22 GeV$, $<E_L>=247.24 GeV$ and $<E_T>=4.70 GeV$. Thus, at this energy gluon radiation causes a total average energy loss of 0.71% of the top quark's energy. The average energy loss in the longitudinal direction is 1.06% and the average energy gain in the transverse direction is 1.88%. These percentage figures go up to 3.77%, 5.19% and 6.06%, respectively, at $1000\GeV$. For the mean of the acollinearity angle $\bar\theta_{12}=180^0-\theta_{12}$ we obtain $<\bar\theta_{12}>=1.25^0$ at $500 GeV$, the value of which goes up to $4.62^0$ at $1000 GeV$. From an analysis of the transverse momentum of the top we find that the mean transverse momentum of the top stays close to the mean total momentum of the gluon in the energy range from threshold to $1000 GeV$ showing that the gluon momentum has a large mean transverse component in this energy range.