Global decay chain vertex fitting at Belle II

Abstract

In this paper we report the implementation of a global vertex fitting algorithm within the Belle II analysis software environment, which was originally developed for BaBar (Hulsbergen, 2005). We explore the impact of global vertex fitting algorithms for flavour physics analyses with the Belle II detector at the SuperKEKB $e^{+}{e}^{-}$ collider, such as in the reconstruction of final states with neutral particles, and in fits with geometrical constraints from SuperKEKB’s nano-beam interaction region. The algorithm is compared to the standard vertex fitting algorithm employed by the Belle experiment. We have developed the fitting framework to utilise the EIGEN library for linear algebra operations, reducing the computation time for vertex fitting operations by an order of magnitude over previous methods. This has a significant impact on physics analysis computing efficiency, where vertex fitting over large combinations of final state particles is one of the most CPU intensive operations at Belle II.

Introduction

Particle vertex fitting techniques are widely used in particle and nuclear physics. Beyond the suppression of background, applications range from the improvement of particle momentum resolution (under the assumption they originate from some vertex point), to the determination of the presence of intermediate particles, and the precision determination of decay vertex positions. One can, for example, combine the measurements of two charged pion tracks originating from the decay of a $K_{\mathrm{S}}^0$ to extract the decay vertex position, flight length, and four-vector and their uncertainties. By performing a kinematic fit, one obtains an improvement of the pion track momenta and can use the ${\chi }^2$ of the fit result to suppress background.

In order to construct more complex decay topologies, one usually combines cascades of these fits starting with long lived stable particles such as electrons, muons, pions and photons, forming intermediate resonances and finally the full decay of interest. For example, in the decay $B^0\to J∕\psi K_{\mathrm{S}}^0$, where $J∕\psi \to {\mu }^{+}{\mu }^{-}$ and $K_{\mathrm{S}}^0\to {\pi }^{+}{\pi }^{-}$, one would first fit the $J∕\psi$ and $K_{\mathrm{S}}^0$ candidates and then use these to construct the $B^0$ candidates, as depicted in Fig. 1a and 1b. However, this only works well if the final state particles are charged and leave traces in the tracking detectors. Neutral particles cannot be tracked in Belle II; they are only measured by their energy deposition in the calorimeter. Single layer crystal calorimeters do not offer directional information on where the particle associated to an energy deposition originated from. That means that the decay vertex cannot be extracted from a fit that exploits only the calorimeter information. In order to obtain the momentum vector, it could be assumed that the particle originates from the primary interaction point and travels directly into the cluster’s centre of gravity. This can introduce a large bias on the momentum direction. Consider, for example, the decay $B^0\to J∕\psi K_{\mathrm{S}}^0$, where the kaon instead decays to neutral final states, $K_{\mathrm{S}}^0\to {\pi }^0{\pi }^0$, and the pions decay, ${\pi }^0\to \gamma \gamma$, as displayed in Fig. 2. Weakly decaying intermediate particles such as $K_{\mathrm{S}}^0$ mesons have flight lengths of up to tens of centimetres at Belle II. Thus for a neutral particle in such a particle decay chain, the assumption that it originates from the primary $e^{+}{e}^{-}$ collision is not sufficient. Furthermore, decays of particles into an intermediate state and a charged particle, i.e. $D^{\ast +}\to D^0{\pi }^{+}$, form a decay vertex that is only constraint by one measurement and thus susceptible to measurement errors.

The method we present in this paper overcomes these issues by globally fitting the entire decay tree in a single fit, taking into account all intermediate particles, extracting all involved particle’s four-momenta, vertex positions, flight lengths and their covariance matrices, using a Kalman Filter as described in Ref. [1]. We use the software environment of Belle II and rely on the C++template library EIGEN [2] for matrix operations, which provides fast execution times for the fit algorithm. We furthermore present physics applications of the fitter with Belle II Monte Carlo samples and study performance characteristics.

The vertex fitting algorithm described in this paper was developed for the analysis software framework of Belle II. The Belle II experiment takes place at the asymmetric $e^{+}{e}^{-}$ collider, SuperKEKB. SuperKEKB provides a beam energy slightly above the mass of the $\Upsilon \left(4\mathrm{S}\right)$ resonance $(10.58\mathrm{GeV})$ at an instantaneous luminosity of $8\cdot 10^{35}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$. The $\Upsilon \left(4\mathrm{S}\right)$ resonance decays into pairs of $B$-mesons just above production threshold, hence this type of experiment is referred to as a $B$-factory. The asymmetric beam energy gives the $B$-meson a relativistic boost along a direction close to the detector’s axis of symmetry, increasing its flight length in the lab frame, which makes it possible to study the time evolution of $B$ decays — a key observable in the study of CP symmetry violation.

The Belle II detector has a cylindrical structure designed to study the decays of $B$- and $D$-mesons, $\tau$-leptons and other processes produced in $e^{+}{e}^{-}$ collisions. Six layers of silicon vertex detectors (2 layers of silicon pixels (PXD), and 4 layers of double sided silicon detectors (SVD)) are located in the central volume of the detector, designed to accurately track the flight paths of charged particles. The following layers are, a central drift chamber (CDC) used to measure track trajectories within a solenoid magnetic field, Cherenkov light based particle identification devices surrounding the CDC in the barrel (TOP) and forward regions (ARICH), followed by the CsI(Tl) electromagnetic calorimeter (ECL). The outermost layers are composed of a magnet solenoid and a $K_{\mathrm{L}}^0$ and muon detector system (KLM), which is also used as the flux return yoke of the magnet. The magnetic field is aligned along the detector’s axis of symmetry. For a more detailed description of the detector see Ref. [3].

Fig. 3, Fig. 4 show an event display depicting simulated particles traversing the Belle II detector. In the decay $\Upsilon \left(4\mathrm{S}\right)\to {\bar{B}}^0{B}^0$, one meson decays as ${\bar{B}}^0\to D\omega \pi \pi$, and the other as $B^0\to K_{\mathrm{S}}J∕\psi$, with $J∕\psi \to {\mu }^{+}{\mu }^{-}$ and $K_{\mathrm{S}}^0\to {\pi }^0{\pi }^0$ with ${\pi }^0\to \gamma \gamma$. Fig. 3 depicts the full detector geometry and Fig. 4 shows a close-up of the inner vertex detectors. In this example we show that the decay vertex of the $K_{\mathrm{S}}^0$ can be highly displaced.