Algebraic approach to solve $tbar{t}$ dilepton equations.pdf

a r X i v : h e p - p h / 0 5 1 0 1 0 0 v 1 7 O c t 2 0 0 5 Algebraic approach to solve tt? dilepton equations Lars Sonnenschein LPNHE, Universite?s Paris VI, VII (Dated: February 2, 2008. Submitted to Physical Review D) The set of non-linear equations describing the Standard Model kinematics of the top quark an- tiqark production system in the dilepton decay channel has at most a four-fold ambiguity due to two not fully reconstructed neutrinos. Its most precise solution is of major importance for measurements of top quark properties like the top quark mass and tt? spin correlations. Simple algebraic operations allow to transform the non-linear equations into a system of two polynomial equations with two unknowns. These two polynomials of multidegree eight can in turn be analytically reduced to one polynomial with one unknown by means of resultants. The obtained univariate polynomial is of degree sixteen. The number of its real solutions is determined analytically by means of Sturm’s theorem, which is as well used to isolate each real solution into a unique pairwise disjoint interval. The solutions are polished by seeking the sign change of the polynomial in a given interval through binary bracketing. PACS numbers: PACS29.85.+C I. INTRODUCTION In 1992 R. H. Dalitz and G. R. Goldstein have published a numerical method based on geometrical considerations to solve the system of equations describing the kinemat- ics of the tt? decay in the dilepton channel [1]. The prob- lem of two not fully reconstructed neutrinos - only the transverse components of the vector sum of their miss- ing energy can be measured - leads to a system of equa- tions which consists of as many equations as there are unknowns. Thus it is straight forward to solve the system of equations directly in contrast to a kinematic fit which would be appropriate in the case of an over-constrained problem or integration over the phase space of degrees of freedom in case of an under-constrained problem. E