C is the center of the circle. P is the point of closest approach to the beamline O. The data fields of a PixelRecTrack are summarized in Table 1.
Notation | Realization, comment |
---|---|
q | charge() electric charge; positives are bent to the right, negatives to the left |
b_{r} | transverseImpactParameterSignedByTrajCenter() signed radial distance, OP positive if the beamline is outside of the circle, negative otherwise |
b_{z} | zImpactParameter() z coordinate of P |
cotθ | cotTheta() |
p_{T} | pT() |
φ | phi() direction of the particle trajectory at P in the transverse plane |
Table 1: Data fields of a PixelRecTrack.
Radius of the circle, obtained from p_{T}
R = p_{T} / 0.003 B
Direction of the vector CP
χ = φ + q π/2
Coordinates of C
X = - (R+b_{r}) cos χ
Y = - (R+b_{r}) sin χ
The direction of the vector C_{1} C_{2} pointing from the center of the first to the center of the second circle is ψ_{0}. Depending of the relative placement of two circles they will have a pair of closest points or two intersections.
The circles are disjoint (R_{12} > R_{1}+R_{2}). The direction of the closest points and the smallest distance is
ψ_{1} = ψ_{0}
ψ_{2} = ψ_{0} + π
Δ r = R_{12} - (R_{1}+R_{2})
One circle contains the other (R_{12} < |R_{1} - R_{2}|)
The direction of the closest points and the smallest distance is
ψ_{1} = ψ_{2} =
ψ_{0} if R_{1 }> R_{2}
ψ_{0} + π otherwise
Δ r = |R_{1} - R_{2}| - R_{12}
The cirles intersect (R_{12} < R_{1}+R_{2} and R_{12} > |R_{1} - R_{2}|)
The smallest distance Δ r = 0.
The direction of the intersection(s) is
γ =
arccos [(R_{1}^{2} - R_{2}^{2} + R_{12}^{2}) / (2 R_{1} R_{12})]
ψ_{1,i} = ψ_{0} ± γ
ψ_{2,i} = atan2(Y_{1}+R_{1} sin ψ_{1,i} - Y_{2},
X_{1}+R_{1} cos ψ_{1,i} - X_{2})
The azimuthal angle Δ ψ with respect to P is
Δ ψ = ψ - χ + k 2π
where k is chosen such that -π < Δ ψ < π. For a valid track q Δ ψ < 0 must hold. Using the equation of the helix
z = b_{z} - R q Δ ψ cotθ
The closest point of a circle I is thus given by
I(X + R cos ψ, Y+R sin ψ, b_{z} - R q Δ ψ cot θ)
The distance of closest points or intersections in z direction is given by
Δ z = |z_{2} - z_{1}|
The presumed production vertex r is the midpoint of line segment I_{1}I_{2}. The momentum components of a particle at the presumed production vertex can be obtained by
p_{x} = p_{T} q sinψ
p_{y} = -p_{T} q cosψ
p_{z} = p_{T} cotθ
A neutral mother particle can be formed if the two tracks have opposite electric charge. The momentum vector and the distance of linear trajectory of the neutral mother particle from the primary vertex is
p = p1 + p2
r = (r1 + r2)/2
b = | r - p ( p r )/p^{2} |
The resulting distances are summarized in Table 2. They can be later used for cuts.
Notation | Comment |
---|---|
Δ r | Smallest distance in the transverse plane |
Δ z | Distance of closest points or intersections in z direction |
r | Distance of the production vertex from the primary vertex |
b | Distance of trajectory of the neutral mother particle from the primary vertex in three dimensions |
Table 2: Resulting distances.
A ntuple with 1000 special events have been generated and simulated with OSCAR. Each event has the following primary particle composition:
Every particle has p_{T} = 1 GeV / c, the p_{L} is in the interval [-0.5,0.5] GeV / c emitted isotropically.
For v0 finding the cuts are Δ r < 0.1 cm, Δ r < 0.1 cm, r > 0.4 cm and b < 0.1 cm.
Armenteros plot with the predictions for K_{S}^{0} (red) and Λ (blue). The q_{T} cut for removing photon conversions is indicated with the green line. The ellipses are broadened by factor 1/β in α because the particles are relatively slow.
Particles with two mass hypotheses (photons removed by q_{T} cut):
K_{S}^{0} mass spectrum with O (10 MeV / c^{2}) resolution:
Λ mass spectrum with O (5 MeV / c^{2}) resolution:
Distribution of the distance of the production vertex from the primary vertex, for photon conversions and hadrons. It is clear that photons convert in the first pixel barrel layers while the hadrons show an exponential-like decay scheme.
Legend
K_{S}^{0} decay:
Λ decay:
Photon conversion:
The V0Finder class (experimental)
-- FerencSikler - 05 Apr 2006