A parametrized Kalman filter for fast track fitting at LHCb
Pierre Billoir, Michel De Cian, Paul André Günther, Simon Stemmle
AA parametrized Kalman filter forfast track fitting at LHCb
P. Billoir , M. De Cian , P. A. G¨unther , S. Stemmle , † LPNHE, Sorbonne Universit´e, Paris Diderot Sorbonne Paris Cit´e, CNRS/IN2P3, Paris, France Institute of Physics, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland Physikalisches Institut, Ruprecht-Karls-Universit¨at Heidelberg, Heidelberg, Germany † Author was at institute at time work was performed.
Abstract
We present an alternative implementation of the Kalman filter employed for trackfitting within the LHCb experiment. It uses simple parametrizations for the ex-trapolation of particle trajectories in the field of the LHCb dipole magnet andfor the effects of multiple scattering in the detector material. A speedup of morethan a factor of four is achieved while maintaining the quality of the estimatedtrack quantities. This Kalman filter implementation could be used in the purelysoftware-based trigger of the LHCb upgrade.
Submitted to Computer Physics Communications a r X i v : . [ phy s i c s . i n s - d e t ] J a n Introduction
The LHCb experiment is a dedicated heavy flavour physics experiment at the LHC focusingon the study of hadrons containing b and c quarks [1]. Due to the high luminosity at theLHC and the high proton-proton interaction cross section, a sophisticated trigger systemis needed to reduce the rate of collisions saved for offline analysis. During Runs 1 and2 of the LHC, this trigger system consisted of a hardware stage, reducing the rate from40 MHz to 1 MHz, followed by a two-stage software trigger. In the latter, the full trackingsystem was read out and a partial (first stage) and full (second stage) event reconstructionwere performed [2]. Both software stages included a fit of selected track candidates usinga Kalman filter to extract their parameters and to reject fake tracks. In addition, thesoftware trigger allowed an online calibration and alignment of the detector [3].During Run 3 of the LHC, LHCb will be provided with a factor five higher luminositycompared to Run 2. In this scope, most of the subdetectors are currently being replacedor upgraded [4–7] and a new trigger strategy has been developed [8]. The hardwaretrigger will be removed and a two-stage, fully software-based trigger will process thefull 30 MHz of bunch-crossing rate. In the first stage, tracks with a high transversemomentum ( p T ) and primary vertices will be reconstructed. These objects are used toselect events with displaced topologies typical for b -hadron and c -hadron decays, and toselect high- p T objects from decays of heavy vector bosons. In the second stage, a fullevent reconstruction will be performed, without any requirement on the p T and includingparticle identification. A large number of exclusive and several universal event selectionsbased on the decay topology will be applied.In LHCb, track reconstruction is split into a pattern recognition and a Kalman filtering [9, 10] stage. During pattern recognition, sets in each subdetector are constructed fromsignals that potentially result from the passage of a single charged particle. Simpleparametrizations are used throughout this procedure as it is only concerned with findingthe right sets of signals and not to provide the best estimate of the track parameters.During the filtering stage, an estimate for the track parameters is calculated, and faketracks are rejected. Given that the output of the filtering stage is used for physics selectionsthe best possible precision needs to be achieved, hence an (extended) Kalman filter is usedfor track fitting. Ideally, Kalman filtering of the track candidates is already performedduring the first trigger stage. However, the Kalman filter which was used during Run 1and 2 in LHCb, in the following called default Kalman , is significantly too slow. It relieson lookup tables for the magnetic field and the material distribution of the detector [11],so-called maps . In addition it uses Runge-Kutta methods to solve the differential equationsnecessary to propagate the particle through the regions with an inhomogeneous magneticfield. Accessing the values in the lookup table and solving the differential equations aretime consuming and prohibit the usage of the current Kalman filter in the first stage ofthe upgraded trigger system. This conclusion is independent of the choice of computingarchitecture (CPU or GPU) which is used for the first trigger stage.In this paper, a fully parametrized version of the Kalman filter in LHCb, called The nominal bunch-crossing frequency of the LHC is 40 MHz, however empty and non-collidingbunches reduce this to a collision frequency of 30 MHz at LHCb. arametrized Kalman , is presented. It obtains precise values of track parameters and trackquality variables, while relying on neither computationally costly extrapolation methodsnor material or magnetic field maps. The LHCb detector [1] is a single-arm forward spectrometer covering the pseudorapidityrange 2 < η <
5. Its Run 3 configuration includes a high-precision tracking systemconsisting of a silicon-pixel vertex detector surrounding the pp interaction region [5](VELO), a large-area silicon-strip detector (Upstream Tracker (UT)) [7] located upstreamof a dipole magnet with a bending power of about 4 Tm [12], and three stations ofscintillating-fibre detectors (SciFi) [7] placed downstream of the magnet. Different typesof charged hadrons are distinguished using information from two ring-imaging Cherenkovdetectors [6, 13]. Photons, electrons and hadrons are identified by a calorimeter systemconsisting of an electromagnetic and a hadronic calorimeter [6,14]. Muons are identified bya system composed of alternating layers of iron and multiwire proportional chambers [6,15].Given the lack of collision data at this point for Run 3, simulation is required tomodel the effects of the detector response, the detector acceptance and the imposedselection requirements. In the simulation, pp collisions are generated using Pythia [16]with a specific LHCb configuration [17]. Decays of unstable particles are describedby
EvtGen [18], in which final-state radiation is generated using
Photos [19]. Theinteraction of the generated particles with the detector, and its response, are implementedusing the
Geant4 toolkit [20] as described in Ref. [21].
In the following, the Kalman filter formalism and its application in the LHCb trackreconstruction is outlined. During Kalman filtering, the information from measurementsat detector planes is successively combined to obtain optimal estimates of the trackparameters. The track is represented as a set of states at fixed z -positions , which aretypically detector layers. Each of theses states is given by x = ( x, y, t x , t y , qp ) and thecorresponding covariance matrix P , where t x and t y are the slopes with respect to the z axis, q the charge of the particle in units of the electron charge and p its absolutemomentum.The Kalman filter procedure needs an estimate of a state as a starting point. Filteringis then a repeated application of two steps. Firstly, the current state is extrapolatedto the next detector layer, and secondly, the extrapolated state is updated using themeasurement in this layer. If the track has no associated measurement in this layer, theupdate step is omitted. These steps can be formalized as follows: given the state ( x k − | k − , P k − | k − ) at position z k − , the extrapolated state ( x k | k − , P k | k − ) at position z k is givenby x k | k − = f k ( x k − | k − ) , (1) P k | k − = F k P k − | k − F Tk + Q k , (2) The detector coordinate system is chosen such that the z -axis is parallel to the beam line and chargedparticles are deflected in the direction of the x -axis. f k ( x ) is given by five individual mappings f k =( f xk , f yk , f t x k , f t y k , f qp k ). This leads to the transport matrix F k as F ijk = ∂f ik ∂x j . (3)The noise matrix Q k accounts for uncertainties of the extrapolation, e.g. due to scatteringat the material of the detector layers or the material in between.The extrapolated state is then combined with the measurement m k in the respectivedetector layer to obtain the new state estimate at the position z k , x k | k and P k | k , usingthe following steps: r k = m k − H k x k | k − , (4) S k = H k P k | k − H Tk + R k , (5) K k = P k | k − H Tk S − k , (6) x k | k = x k | k − + K k r k , (7) P k | k = ( − K k H k ) P k | k − . (8)Here H k projects the estimated state vector to the measurement space in order to allowa calculation of the residual r k . The covariance matrix of this residual is given by S k andis combined with the covariance matrix of the state to obtain the Kalman gain K k . Thelatter defines then how the estimated state is modified by the residual. The variance ofthe residual is given by R k .Starting at the most upstream measurement, the measurements are successively addedand the track parameters updated until the last detector layer is reached. The sameprocedure is repeated starting at the most downstream measurement and successively in-cluding more upstream measurements. This yields two sets of states at every measurementposition, which can be combined to obtain the respective optimal state.The quality of a track can be estimated by its χ value. The value at eachmeasurement is given by: χ k = χ k − + r Tk P − k | k r k , (9)and χ is then simply χ k after all measurements have been added using the combined,optimal states.The optimal state estimates and the measurement information can also be used toremove measurements that show a large separation from the fitted trajectory by having alarge contribution to the χ value. They are therefore likely to be wrongly associatedto the respective track, and are so-called outliers . Once an outlier is removed, all Kalmanfilter steps are performed again. This procedure can be repeated until the maximumallowed number of outliers are removed, or no more outliers are present.The above formalism is also the basis of the Kalman filter that is currently used fortrack fitting in the LHCb experiment. The extrapolation functions f k are based on mapsof the magnetic field along the trajectory and numerical models for the extrapolations.Their complexities range up to a fifth-order Runge-Kutta method. The noise matrices Q k are obtained by a dedicated model for the multiple scattering and a map of the materialtraversed by the particle. 3n the parametrized Kalman filter presented in this paper, these two costly steps arereplaced by simple parametrizations. The extrapolation functions f k are given by analyticexpressions that allow a fast evaluation and calculation of the derivatives in Equation 3.The noise matrices Q k depend on the momentum of the particle and are parametrized bya few parameters per extrapolation step.An important difference with respect to the default Kalman filter is the treatment ofenergy loss due to the interaction with the detector material. While the multiple scatteringis taken directly into account, the energy loss is not part of the extrapolation functions f k , i.e. f qp k is the unity transformation. This shortcoming is compensated by choosing themomentum of the state vectors to represent the momentum at the moment of productionof the particle. Thereby, the extrapolation functions also take this initial momentum asinput and thus indirectly take into account all energy loss that happened on average upto the respective detector layer. The only caveat being that qp after the filtering is onlythe best representation of the true value at the production point of the particle. Depending on the strength of the magnetic field and the typical distance between detectorlayers, different empirical analytical functions for the extrapolation are used.Inside the VELO, where the magnetic field is very weak, these functions and the noisematrix are given by: f ( x ) = f x ( x ) f y ( x ) f t x ( x ) f t y ( x ) f qp ( x ) = x + 0 . t x + f t x ( x )]∆ zy + t y ∆ zt x + p V0 qp ( z + p V1 )∆ zt yqp (10)and Q = (cid:0) ˜ p V1 ∆ z (cid:1) Q t x t x p V2 √ Q xx Q t x t x (cid:0) ˜ p V1 ∆ z (cid:1) Q t y t y p V3 √ Q yy Q t y t y p V2 √ Q xx Q t x t x (cid:16) ˜ p V0 (cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:17) p V3 √ Q yy Q t y t y (cid:16) ˜ p V0 (cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:17)
00 0 0 0 0 , (11)where ∆ z is the extrapolation distance along the z -direction and z the initial or final z coordinate for a downstream or upstream extrapolation, respectively. The parameters p V0 , p V1 and ˜ p V0 to ˜ p V3 are the same for all upstream and downstream extrapolations inside theVELO. They are determined using simulated B s → φφ decays within the LHCb softwareframework, where φ → K + K − . This simulated sample allows to create a dataset D ,containing pairs of states representing two consecutive measurements of one track insidethe VELO. In addition to the true state parameters obtained from the simulation, alsoan extrapolation of each state to the z position of the respective other state is includedin the dataset. Such extrapolation is based on the default extrapolation algorithm in4HCb [11]. This dataset allows tuning the parameters employing a minimization of thefollowing likelihood-inspired function: (cid:89) D (cid:104) G (cid:16) f s ( x ) − x s , (cid:112) Q ss (cid:17) + c (cid:105) . (12)Here, G ( x, σ x ) is a normalized Gaussian distribution centered around 0 with width σ x .The two states of each dataset entry are represented by x and x , and the variable s isone of the state variables, s ∈ { x, t x , y, t y } . The positive empirical constant c is chosen tobe small with respect to the amplitude of the Gaussian function and softens the impactof outliers.In a first step, the extrapolation functions f x to f t x are tuned individually, taking intoaccount that f x depends on the previously determined parameters for f t x . These tuningminimizations employ the state vector x that is obtained by the extrapolation of thestate vector x . This choice improves the precision of the parametrized extrapolation, byremoving the effect of multiple scattering that would be present if instead the true statewas chosen for x .In a second step, the parameters of the extrapolation functions are fixed, and aminimization of the following function is performed: (cid:89) D (cid:104) G (cid:16) f d ( x ) − x d , f t d ( x ) − x t d , (cid:112) Q dd , (cid:112) Q t d t d , Q dt d / (cid:112) Q dd Q t d t d (cid:17) + c (cid:105) . (13)Here, G ( x, y, σ y , σ y , ρ ) is a normalized two dimensional Gaussian distribution centeredaround 0 with widths σ x and σ y and a correlation factor ρ . The variable d is either x or y .In this minimization, the true state vector x is used in order to get the correct estimateof the parameters for the respective elements of the noise matrix Q .Inside the UT and the SciFi detector stations, the magnetic field is significantlystronger than inside the VELO and higher order terms are needed for the extrapolationfunctions: f ( x ) = x + (cid:2) p T3 t x + (1 − p T3 ) f t x ( x ) (cid:3) ∆ zy + (cid:2) p T5 t y + (1 − p T5 ) f t y ( x ) (cid:3) ∆ zt x + (cid:104) p T0 qp + p T1 ( qp ) + p T2 y qp (cid:105) ∆ zt y + p T4 qp t x y | y | qp . (14)The noise matrix is given in full analogy to Equation 11 with the parameters ˜ p T0 to˜ p T3 , where T either stands for the UT or the SciFi detector. These parameters and theparameters p T0 to p T4 are individually determined on simulation for every step from onedetector layer to the next and for the upstream and downstream extrapolation separately.The same strategy as for the tuning of the parameters related to the extrapolation insidethe VELO is followed.For the long extrapolations between the different tracking subdetectors, more sophisti-cated parametrizations are necessary. In the case of the step between the VELO and theUT, where the magnetic field is still weak, the extrapolation is based on two equations.5he first describes the change in momentum along the x -direction of the particle:∆ p x = p t x, UT (cid:113) t x, UT + t y, UT − t x, V (cid:113) t x, V + t y, V = q (cid:90) (d l × B ) x , (15)where t x/y, UT and t x/y, V are the state variables at the first UT detector layer and the lastmeasurement inside the VELO, respectively. The right hand side of the equation consistsof an integral of the magnetic field along the trajectory of the particle. Note that theintegral expression is simply a parameter which was fitted for on the dataset. The secondingredient for the extrapolation is to model the effect of the magnetic field as a singlekink of the trajectory at a certain z -position z mag between the VELO and the UT: x UT = x V + ( z mag − z V ) t x, V + ( z UT − z mag ) t x, UT , (16)where z V and z UT are the positions of the states inside the VELO and the UT, respectively.Equation 15 can be solved for t x, UT and Equation 16 is then employed to get anexpression for x UT . The unknowns in these expressions are parametrized as a function ofthe state variables inside the VELO: t y, UT = t y, V + p S0 qp t x, V y V | y V | (17) (cid:90) (d l × B ) x = p S1 + p S2 z V + p S3 t y, V (18) z mag = p S4 + p S5 z V + p S6 z + p S7 t y, V . (19)In addition, the y -position of the extrapolated state is given by: y UT = y V + (cid:2) p S8 t y, V + (1 − p S8 ) t y, UT (cid:3) ∆ z, (20)where ∆ z is defined as the difference between z UT and z V . The noise matrix is defined inanalogy to Equation 11 with the parameters ˜ p S0 to ˜ p S3 . These parameters and the parameters p S0 to p S8 are individually determined for the upstream and downstream extrapolation. Thesame strategy as for the tuning of the parameters related to the extrapolation inside theVELO is followed.The extrapolation from the UT to the SciFi detector is more delicate because it isdone over a distance of more than 5 meters through a strong magnetic field. Moreover,this field is far from uniform - in particular, it varies rapidly in the upper and lowerregions, close to the magnet yoke. To ensure a good quality of the global track fit, theerror on the extrapolation should be well below the other sources of error, mainly multiplescattering. The chosen solution is an expansion of the magnetic deviation in powers of q/p .The parametrization aims at giving good precision for charged particles used in physicsanalyses, that is for trajectories which roughly come from the origin.To do so, the ideal direction ( t x , t y ) as the one of a particle of charge q , momentum p , starting from the origin and hitting the UT detector layer in a given point ( x, y ) isdefined. As a good approximation, we can take t x = x/z + B q/p , t y = y/z , where B isproportional to the integrated field between the origin and the UT. The deviations fromthe ideal direction, δt x = t x − t x , δt y = t y − t y , are small, so only a first order expansionin δt x , δt y is considered. Corrections of higher order would be negligible compared tomultiple scattering errors. 6inally, a polynomial expansion in q/p for the ideal direction is built, and a correctionin δt x , δt y with coefficients which are themselves polynomials of q/p is added: f x ( x ) = x + t x ∆ z + K (cid:88) k =1 A xk ( x, y ) (cid:18) qp (cid:19) k + K (cid:88) k =1 ( B xk ( x, y ) δt x + C xk ( x, y ) δt y ) (cid:18) qp (cid:19) k , (21)where the first two terms are the straight line extrapolation, and the next ones thecurvature correction. Similar expressions are used for the other state parameters f y ( x ), f t x ( x ), f t y ( x ). The degrees of expansion K and K are tuned for each parameter toobtain the required precision. In practice K = 9, K = 7 for f x and f t x and K = 7, K = 5 for f y and f t y are used.The dependence on x, y of the coefficients A uk , B uk , C uk , with u = x, y, t x , t y , is describedthrough a tabulation on a grid of 50 ×
50 points regularly spaced on the rectangle definedby | x/z | ≤ . | y/z | ≤ .
25, by steps ∆ X , ∆ Y . In order to avoid a systematic convexitybias of a bilinear interpolation, the values at x, y are computed by a quadratic interpolationbetween the tabulated values at the six closest points on the grid: if ( X, Y ) is the closestone, these values are: F = ( X, Y ), F +0 = F ( X + ∆ X, Y ), F − = F ( X − ∆ X, Y ), F = F ( X, Y + ∆ Y ), F − = F ( X, Y − ∆ Y ), and F ε x ε y = F ( X + ε x ∆ X, Y + ε y ∆ Y ),where ε x and ε y are the signs of ξ = ( x − X ) / ∆ X and ψ = ( y − Y ) / ∆ Y , respectively.With these notations the interpolation formula for a quantity F is given by : F ( x, y ) = F + F d ξψ + (cid:0) ( F +0 − F − ) ξ + ( F − F − ) ψ + ( F +0 + F − − F ) ξ + ( F + F − − F ) ψ (cid:1) / F d = ε x ε y ( F + F ε x ε y − F ε x − F ε y ) . (23)The tabulated values are obtained using the standard Runge-Kutta method of order4, with 20 values of q/p in the range ( − /p min , /p min ), with p min = 3000 MeV/ c and apolynomial fit in q/p . As a consequence, they do not give a reliable result for momentabelow p min . Another limitation are the larger errors on the edges of the acceptance,especially for | t y | (cid:39) .
25, where the field has strong spatial variations.
A sample of simulated proton-proton collisions that include a B s → φφ , φ → K + K − decayis used to compare the reconstruction quality of the parametrized and the default Kalmanfilter. The extrapolation of the most upstream state estimate to the beam line is the samein both filters and is based on a simplified material map of the detector [11]. Therefore, notthe state near the beam line, but the state at the most upstream measurement is employedfor the comparison of the two Kalman filters. Although only tracks with measurements ineach of the subdetectors are considered for this study, this is in principle not a requirementfor operating the parameterized Kalman filterFigure 1 compares the resolution of the momentum, the x -position and the slope t x as a function of the true momentum of a particle. Since the position and slope arenearly exclusively determined by the measurements in the VELO, where only a very weakmagnetic field is present, the parametrizations of the parametrized Kalman filter aresufficient to obtain results comparable to the default Kalman filter in these variables. In7 p [ MeV/ c ] δ p / p ParametrizedDefault
LHCb simulation 0 20000 40000 60000 80000 p [ MeV/ c ] δ x [ mm ] ParametrizedDefault
LHCb simulation p [ MeV/ c ] × − δ t x ParametrizedDefault
LHCb simulation
Figure 1: Comparison of the resolution in simulation in (top left) momentum, (top right) x -position and (bottom) slope t x between the default and parametrized Kalman filter. Theresolution is represented by the root mean square of the residual distribution when comparingto the true value. contrast, the momentum estimate strongly depends on the extrapolations in regions withstrong magnetic field. There, especially at momenta below 10 GeV/ c , an up to 20% worseresolution is observed for the parametrized Kalman filter.The Kalman filter does not only provide an estimate of the state parameters, but alsoa corresponding covariance matrix. In Figure 2 the pull distributions of the estimatedmomentum, x -position and slope t x for the parametrized Kalman filter are shown. In allthree cases, good uncertainty estimates are visible. However, in analogy to the observationsmade for the resolution, the pull distribution of the momentum features slightly morepronounced tails.Besides the estimate of the state near the beam line, which is used for the reconstructionof charged particles, an important output of the Kalman filter is the fit quality describedby the χ per degrees of freedom N dof . In Figure 3, this quantity is shown for theparametrized Kalman filter for real tracks coming from a particle and fake tracks consistingof random combinations of clusters. In addition, the real track efficiencies and fake trackrejection rates are shown for both Kalman filter versions when applying upper boundson this quantity. The parametrized Kalman filter shows a slightly worse but overallcomparable performance in separating the two track classes.The fitted tracks are combined to reconstruct B s → φφ candidates. Figure 4 shows theinvariant mass distribution of candidates based on the two Kalman filter versions. A single8 p pull E v e n t s / . DataFit µ = − . ± . σ = 1 . ± . . ± . x pull E v e n t s / . DataFit µ = − . ± . σ = 0 . ± . . ± . t x pull E v e n t s / . DataFit µ = 0 . ± . σ = 0 . ± . . ± . Figure 2: Pull distributions of the momentum, x -position and slope t x estimates of theparametrized Kalman filter at the most upstream measurement. The given values correspond tothe mean, width and root mean square of a Gaussian function that is fitted to the distribution. χ /N dof E n t r i e s / . Real tracksFake tracks
LHCb simulation
Fake track rejection rate R e a l t r a c k e ffi c i e n c y DefaultParametrized
LHCb simulation
Figure 3: Track quality estimate, χ /N dof , in simulation for the parametrized filter (left).Fake tracks are shown in red and real tracks in black. Real track efficiency and fake trackrejection for the parametrized and default Kalman filter (right).
300 5400 m ( K + K − K + K − ) [ MeV/ c ] C a nd i d a t e s / ( M e V / c ) DataFitLHCb simulationDefault 5300 5400 m ( K + K − K + K − ) [ MeV/ c ] C a nd i d a t e s / ( M e V / c ) DataFitLHCb simulationParametrized
Figure 4: Reconstructed B s mass in simulated B s → φφ decays for the parametrized and thedefault Kalman filter. Fit projections are overlaid. Gaussian distribution and a first order polynomial are employed to model the signal peakand the combinatorial background, respectively. This yields nearly identical estimatedmass resolutions of 12 . c and 12 . c for the default and the parametrizedKalman filter, respectively.In order to compare the timing performance of the parametrized Kalman filter andthe default Kalman filter, throughput studies on a machine with two Intel(R) Xeon(R)Silver 4214 processors were performed. Simulated proton-proton collisions were used inorder to mimic the situation or real data taking. Depending on the configuration of theoutlier removal strategy, an overall speedup factor between 4 and 5.5 with respect to thedefault Kalman filter was achieved. The largest speedup is achieved when no iterationsfor the outlier removal are performed. Singling out the calculation steps of the Kalmanfilter, i.e. neglecting the part of the algorithms where the measurement information isconstructed, the speedup factor is even larger and ranges from 5.7 to 10.In the case of the parametrized Kalman filter, and singling out again the calculationstep of the Kalman filter, 50% of the time is spent extrapolating the states between thedetector layers. Here, the extrapolation between the UT and the SciFi constitutes thebiggest component with a relative fraction of 40%. The remaining Kalman filter steps,consisting of updating the states with the cluster information and the combination ofupstream and downstream filtered states, are responsible for 16% and 14% of the timespent, respectively. The extrapolation to the beam line, which is based on the defaultLHCb extrapolation algorithm, is responsible for the remaining 20% of the time budget. We presented an alternative implementation of a Kalman filter for the LHCb experiment.Based on simple parametrizations of material effects and the extrapolation through themagnetic field of the detector, this algorithm achieves a significant speedup with respectto the current implementation, while retaining comparable quality of the track parameters.In the future, further improvements of the parametrizations might allow an even betterestimate of the track parameters and a subsequent speedup. Ideas currently underdiscussion include for example an analytic parametrization of the x and y dependence10f the parameters employed in the extrapolation from the UT to the SciFi detector anda better account for the limited acceptance of low momentum particles. The versionpresented in this document or a future implementation might therefore be well suited forthe usage in the LHCb software trigger system for Run 3 of the LHC. Acknowledgements
We would like to thank the LHCb RTA team for supporting this publication and reviewingthe work. M. De Cian acknowledges support from the Swiss National Science Foundationgrant “Probing right-handed currents in quark flavour physics”.
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