A non-iterative method for constraining masses in particle decays Ahmet Bing ¨ ul Department of Engineering Physics, University of Gaziantep, Gaziantep 27310, Turkey article info Article history: Received 26 June 2012 Received in revised form 19 July 2012 Accepted 19 July 2012 Available online 27 July 2012 Keywords: NIMCO Mass constraint Fast constraint Kinematic fit abstract To overcome the mass constraint problem of particle decays, a non-iterative method is developed as an alternative to relatively complicated and slow iterative methods. The new method can be applied to any two-body decay or a many-body decay which can be reduced to a two-body decay having well known daughter masses. By using a toy detector simulation and ALEPH full simulation data, the performance of the new method is compared with the traditional iterative chi-square method for several decay types. No significant difference is obtained between the two methods in terms of improvement in momentum resolution. However, the non-iterative method is found to be much faster than the chi-square method. & 2012 Elsevier B.V. All rights reserved. 1. Introduction In particle physics, mother particles are reconstructed from their decay products. If the natural width of the mother is very small then one can consider that the mother mass is a single scalar value called the nominal mass. However, the reconstructed momentum resolution of the mother is limited by the momentum and angular resolutions of the particle detector used. As a result, the invariant mass distribution of the daughters originating from such mother particles is smeared around the nominal mass instead of giving a single peak. In order to improve the momentum resolution of the mother particle candidate, one can apply a kinematic refit to the momen- tum vectors of the reconstructed daughters. To do that, the momenta of the daughters are adjusted such that their invariant mass ideally gives the nominal mass of the mother. Consider a mother particle of mass M denoted by 0 decaying to n daughters: 0-1 þ 2 þþ n: ð1Þ The invariant mass of the daughters, W, can be evaluated from W 2 ¼ X n i ¼ 1 ½9p i 9 2 þ m 2 i  1=2  ! 2  X n i ¼ 1 p i  ! 2 ð2Þ where p i and m i are the reconstructed momentum vector and the mass of the ith daughter, respectively. Traditionally refitting is performed by iterative methods. A common approach is to minimize a chi-square function of the form w 2 ¼½pf  T V 1 ½pf  ð3Þ where f is the vector of refitted momenta and V is the covariant matrix whose elements depend on detector resolutions. In the most general case with n decay daughters, one has 3n parameters in the refitting problem. These parameters are not entirely free since the model has to fulfill some special constraint(s). For instance, for one mass constraint, W is replaced by M and vector p is replaced by f in Eq. (2) from which one of the parameters is eliminated. Finally, minimizing Eq. (3) results in refitted values which are in general closer to the true values than the reconstructed values. Therefore, the momentum resolution of the mother is improved thanks to the chi-square method [1, 2]. It is clear that the minimiza- tion procedure becomes a difficult task for n 42 [3]. Another widely used method to solve such a minimization problem is known as the Lagrange Multipliers. In this method, a set of mass constraint functions of momenta are multiplied by Lagrange Multipliers, l j . These functions are added to Eq. (3) to construct an auxiliary function. Partial derivatives with respect to l j where j ¼ 1,2,y,q are also computed. For one mass constraint, q ¼ 1, the form of the auxiliary function to be minimized is Lðf , lÞ¼ w 2 þ lðM 2 W 2 Þ: ð4Þ The Lagrange Multipliers method is described in detail elsewhere [4]. This paper describes a fast non-iterative mass constraint method called NIMCO. The method was first developed in Ref. [5] for a mother particle decaying to massless particles such as p 0 -gg. 1 In this paper, the non-iterative method is generalized for particles Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/nima Nuclear Instruments and Methods in Physics Research A 0168-9002/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nima.2012.07.031 E-mail addresses: Ahmet.Bingul@cern.ch, aabingul@gmail.com 1 In this reference, there are actually a few methods which non-iteratively modify the photon energies and directions to obtain the true pion mass. The method used here was called ‘Empirical 3D’ for pions in Ref. [5]. However, we generalize the method and change the name to NIMCO in this paper. Nuclear Instruments and Methods in Physics Research A 693 (2012) 11–15