Optimization of the SHiP Spectrometer Tracker geometry 1 using the Bayesian Optimization with Gaussian Processes

. One of the most important aspects of data processing at SHiP [1] ex-7 periments is tracks pattern recognition. The purpose of the SHiP Spectrometer 8 Tracker (SST) is e ﬃ cient reconstruction of charged particle tracks originating 9 from decays of neutral New Physics objects. The reconstruction performance 10 strongly depends on the tracker design and should be considered as an objective 11 to deﬁne the best SST geometry parameters. In this study the SHiP Spectrom-12 eter Tracker geometry optimization using Bayesian optimization with Gaussian 13 processes in considered. The study have been done on MC data. The ﬁrst results 14 of the optimization are also considered.

tector is demonstrated in Fig. 1. SHiP is a detector with a fixed target. A proton beam from 38 the CERN SPS accelerator with energy of 400 GeV bumps into it. The target [8] is designed 39 to maximize production of charm and beauty hadrons and photons. It is followed by the 5m 40 hadron absorber that absorbs all particles except weakly interacting ones: muons, neutrinos, 41 and particles of new physics. The Muon Shield [9,10] is used to reduce muon flux in the 42 detector acceptance. The shield is a system of 6 magnets with a total length of 35m. The av-43 erage magnetic field is 1.7T. The Scattering and Neutrino Detector (SND) [11,12] is located 44 after the muon shield. The goal of the detector is to study neutrino physics and to detect light 45 dark matter candidates.  The SHiP Spectrometer Tracker has 2 straw tube stations before the magnet and 2 stations 53 after it. Each station has 4 views: 2 Y-views with straw tubes oriented parallel to X-axis and 54 U, V-views rotated by a small angle, +α or −α, around the Z axis. A view has 2 planes with 55 2 straw tubes layers in each plane as it is shown in Fig 2. The view geometry defines the 56 particle track recognition [13] quality. The geometry is defined by the following parameters: 57 straw pitch, Z shift between layers, Z shift between planes, Z shift between views, Y offset 58 between layers, Y offset between planes and angle α between Y and U, V views. The goal of 59 this study is to find optimal values of these parameters. 60 * e-mail: mikhail.hushchyn@cern.ch

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GP regression [3,5] is used for the spectrometer geometry optimization as described in the 62 next section. Consider an objective function y = f (x) and a set of In 63 the GP concept, the sequence of observations can be represented as a sample from a Gaussian 64 distribution: where N(0, K) represents multivariate Gaussian (or normal) distribution with zero mean 66 and covariance matrix K. For a new point (x n+1 , y n+1 ) a similar relation with the new covari-67 ance matrix K can be written: where the covariance coefficients k i j characterize the correlation between different points 69 of the process: for nearby points x i and x j the observables y i and y j are strongly correlated 70 than for distant points. Here squared exponential covariance function is used to define coef-71 ficients k i j : where θ defines the smoothness of the objective function.

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The mean and standard deviation of the objective function approximation, µ(x n+1 ) and 74 σ(x n+1 ), are defined by the conditional distribution: The optimal θ value in the covariance matrix K is found based on the likelihood function 76 optimization: log p(y|X, θ) = − 1 2 Equations for µ(x n+1 ) and σ(x n+1 ) are used during Bayesian optimization to approximate 78 the objective function based on known observations. steps which are repeated until the optimum is found: , fit a GP regression model to get µ(x n+1 ) and σ(x n+1 ) 85 of the objective function approximation.

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• Optimize the Expected Improvement (EI) acquisition function [14] based on the regression model for sampling the next point: • Sample the next observation (x n+1 , y n+1 = f (x n+1 )).

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The goal of this study is to find the optimal geometry of the   For each combination of the parameters a set of 500 events of HNL → µπ decay is 92 generated using FairShip [15]. Then, a track pattern recognition algorithm [11] is applied  Table 2: Values of the spectrometer geometry parameters. differences in the range of 1-2 cm. These three parameters define size of holes between 104 the straw tubes where a particle can pass through without leaving any hit. So, the smaller 105 holes the better for the tracks pattern recognition. Z shifts between planes and views, and