Setup of a Drift Tube Muon Tracker and Calibration of Muon ...

the z-coordinate z op,0 of the first hit in the detector is used. This coordinate isdetermined by looping over all hits thus finding the smallest z. For each projectionin the x − z and y − z plane the following 2D parametrization can be used:b x and b y can be determined as follows:x op = s x z op + b x ,y op = s y z op + b y .x op,0 = s x z op,0 + b x ⇒ b x = x op,0 − s x z op,0 ⇔ x op = s x z op + x op,0 − s x z op,0 ,y op,0 = s y z op,0 + b y ⇒ b y = y op,0 − s y z op,0 ⇔ y op = s y z op + y op,0 − s y z op,0 .Applying the coordinate transformation, one gets∆y − y = s x (∆x − x) + x op,0 − s x z op,0 ⇒ y = s x x − s x ∆x − x op,0 + s x z op,0 + ∆y,−∆z + z = s y (∆x − x) + y op,0 − s y z op,0 ⇒ z = −s y x + s y ∆x + y op,0 − s y z op,0 + ∆z.In Borexino, the track parametrizationy = α + βx,z = γ + δyis used. Comparing the parameters leads toα = −s x ∆x − x op,0 + s x z op,0 + ∆y,β = s x ,γ = s y ∆x + y op,0 − s y z op,0 + ∆z,δ = −s y .For an easy comparison of tracks, the polar and azimuth angles θ and ϕ are used.They can easily be obtained:ϕ = tan −1 (β),( ) 1θ = tan −1 .δ cos ϕOne can assume that muons originating from the CNGS beam have a direction closeto the beam axis. Using the definition above, the azimuth angle ϕ can obtain valuesbetween − π 2 and π 2which is the angle in the horizontal plane towards the beamdirection. For the polar angle, values between 0 and π are realistic. Therefore, θhas to be corrected by π for negative values:⎧ ( )( )11⎪⎨ tan −1 for tan −1 ≥ 0δ cos ϕδ cos ϕθ = ( )( )11⎪⎩ tan −1 + π for tan −1 < 0δ cos ϕδ cos ϕ108