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A. T.Meneguzzo September 2006 Drift Tubes Chambers : A track parameters Fit ( position, angle , s , vdrift ) for the Phi view A track parameters Fit ( position, angle, s~T0) for Theta and Phi view The code developed in Orca and updated in CMSSW for 2D DriftTube local segments finding is a 2 parameter fit ( position and angle) of the 1D segments and assumes a well known timing of the track with respect to the trigger and a well known relation between time and position with respect to the wire ( drift velocity). These conditions should be verified at pp and in general with bunched generated tracks but is not the case for cosmic data. For analyzing Cosmic Ray Data from autotriggered L1A of the commissioning a 3 parameters fit ( position, angle, s=vdrift *T0) has been developed and used in the analysis of commissioning data and is available in CMSSW. For Cosmic Ray Data of the MTCC with magnetic field a 4 parameters fit have been tested for the best track parameters finding with the aim to find position, angle , s=vdrift *T0, vdrift that is for checking the apparent variation of drift velocity with magnetic field . Here the formula of the minimization and the application to the DT is reported. Input : The 1D segments (points) associated to a track are already assumed known with the left right ambiguity resolved from a preliminary track finding algorithm. The standard 2D (4D) segments in DT chambers with LinearDriftAlgo reconstructed are used assuming a constant drift velocity ( let call v0drift ) and a hit resolution sigma 0 . Usual values are v0drift = 0.00543 cm/ns and sigma0= 0.0500 cm for MTCC and commissioning data and sigma0= 0.0200 cm for pp (MC) or TB . Nomenclatura : i point i of the segment , y wire,i position of the i wire in the drift direction y,i reconstructed position wire I in the drift direction x i position of the layer of i wire in the direction normal to the layer t i,digi the reconstructed drift time of wire i li indicate the drift direction that is the right or left cell side of the track with respect the i wire ( l i= +1 or li= -1). Goal and Method We want to find the best line y= a+ b * x interpolating the points (x i , y i ) y= a+ b * x (b is the tangent of the angle with respect to the normal of the wire plane of the track in the selected view). For each track with enough hits, it is possible to perform a minimization of the residuals assuming as parameters a and b and vdrift and t0 , that is of the quantities S = iyi –y i,pred)2 = i( [y wire,i + (t i,digi-t0) * vdrift * li ] - a+ b * x i )2 (1) Application of the method to the found tracks in the code . ti,digi = t(tdc)i - t0i drift time= ti = t i,digi - t0 generally t(tdc)I = tdc time ti,digi = tdc time corrected for the time difference between the wires in a MiniCrate (the t0i fix values depending only on the MiniCrate cables lenght and measured with Test Pulses (*). Expression (1) can be developed in a different way xi= coordinate of the point i in the direction normal to the wire planes yi are recorded directly by the 2D segments ( 4Dsegments). yi = y wire,I + ti * vdrift*li (reconstructed position in the drift direction of the hit of wire i from t i ) yi are recorded directly by the 2D segments ( 4Dsegments). Defining the quantity yic = y wire,I + ti * v0drift*li tic = t(tdc)i-t0i-tTrig = t i,digi-tTrig0 S = iyi –y i,pred)2 = i{ (y wire,i + (t i,digi-( tTrig0+t0) *(v0drift+ vdrift )* li ) - a+ b * x i }2 S = iyi –y i,pred)2 = i{ (y wire,i + (t i,digi-( tTrig0+t0) *(v0drift+ vdrift )* li ) - a+ b * x i }2 S = iyi –y i,pred)2 = i{ (y wire,i + (t i,digi- tTrig0) *(v0drift+ vdrift ) * li +t0 *(v0drift+ vdrift )* li ) - a+ b * x 2 i } S = iyi –y i,pred)2 = i{ ( [y wire,i + (t i,digi- tTrig0) *v0drift]]+ (t i,digi- tTrig0) * li * vdrift + t0 *(v0drift+ vdrift )* li ) - a+ b * x i }2 S = iyi –y i,pred)2 =i{ (y ci - tic * li vdrift - li *vdrift *t0 ) – a + b * x i }2 S = iyi –y i,pred)2 =i{ (y ci - tic * li vdrift - li *vdrift * li *t0 * ) – a + b * x i }2 S = iyi –y i,pred)2 = i{ (y ic - tic* li vdrift - s * li ) – a + b * x i }2 with s =c= t0 * vdrift It is possible to minimizing with respect the 4 parameters vdrift , s, a , b dS/da= -1/2 i{ (y ic - tic* li vdrift - s * li ) – a + b * x i } dS/db= -1/2 i{ [(y ic - tic* li vdrift - s * li ) – a + b * x i ] * x i } dS/d(s) = -1/2 i{ [(y ic - tic* li vdrift - s * li ) – a + b * x i ] * li } dS/d(vdrift) = -1/2 i{ [(y ic - tic* li vdrift - s * li ) – a + b * x i ] * tic* li } and have a system of 4 linear equations in the 4 parameters ( with s=c ), n= yi –tili * vdrift –li * c - n a + b xi = 0 yixi –xitili * vdrift –xili * c - xia + b xi *xi)= 0 yili –litili * vdrift –lili * c - xia + b xi *li)= 0 yili tic –litili tic * vdrift –lili tic * c - xi tic a + b xi *li tic = 0 * At first approximation t0 = tTrig0 + t0 that is the common L1A latency tTrig0 and some extra term +t0 . The tTrig is the mean latency of the L1A trigger signal with respect to a track passing on a wire at the middle of its length; the tTrig accuracy is ~1-2 ns at TB, pp, and a ~flat distribution 25ns wide for MTCC and commissioning data. The t0 term depends on the event: position of the track with respect to the FE read out ( signal propagation delay) and of the time of the muon with respect to the clock ( the L1A has a 25 ns step and cosmic are almost uniformly distributed. Available: A) Application of the method inside CMSSW : - available for the drift linear algorithm - on each reconstructed segment , before being updated , on request , a 3 parameter fit is performed and from the computed s the equivalent t0 is associated to the segment with seg- >setT0(t0 ). - Then the drift times of all computed points in the normal update of the segment are recomputed (compute algo) and the new track with all the error recomputed. Up to now, no upgrade of the error is applied after the minimization (that should be upgraded in order to have reasonable errors onthe segment parameters computation). - When t0 computation is not required, the t0 is at zero so no effect . B) Application of the method with the 4 parameters vdrift , s, a , b computation, with a stand alone code on 2D/4D segments output , available on request.