### V0 Finder

#### Mathematical desciption

##### Intersection of helices

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
br `transverseImpactParameterSignedByTrajCenter()`
positive if the beamline is outside of the circle, negative otherwise
bz `zImpactParameter()`
z coordinate of P
cotθ `cotTheta()`
pT `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 pT

R = pT / 0.003 B

Direction of the vector CP

χ = φ + q π/2

Coordinates of C

X = - (R+br) cos χ
Y = - (R+br) sin χ

##### Two circles

The direction of the vector C1 C2 pointing from the center of the first to the center of the second circle is ψ0. Depending of the relative placement of two circles they will have a pair of closest points or two intersections.

The circles are disjoint (R12 > R1+R2). The direction of the closest points and the smallest distance is

ψ1 = ψ0
ψ2 = ψ0 + π
Δ r = R12 - (R1+R2)

One circle contains the other (R12 < |R1 - R2|)
The direction of the closest points and the smallest distance is

ψ1 = ψ2 =
ψ0 if R1 > R2
ψ0 + π otherwise
Δ r = |R1 - R2| - R12

The cirles intersect (R12 < R1+R2 and R12 > |R1 - R2|)
The smallest distance Δ r = 0. The direction of the intersection(s) is

γ = arccos [(R12 - R22 + R122) / (2 R1 R12)]
ψ1,i = ψ0 ± γ
ψ2,i = atan2(Y1+R1 sin ψ1,i - Y2, X1+R1 cos ψ1,i - X2)

##### Distance in _z

The azimuthal angle Δ ψ with respect to P is

Δ ψ = ψ - χ + k

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 = bz - 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, bz - R q Δ ψ cot θ)

The distance of closest points or intersections in z direction is given by

Δ z = |z2 - z1|

##### Neutral mother particle

The presumed production vertex r is the midpoint of line segment I1I2. The momentum components of a particle at the presumed production vertex can be obtained by

px = pT q sinψ
py = -pT q cosψ
pz = pT 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 )/p2 |

##### Cuts

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.

#### Results

A ntuple with 1000 special events have been generated and simulated with OSCAR. Each event has the following primary particle composition:

• 1 π+ and 1 π-
• 1 KS0
• 1 Λ
• 10 γ

Every particle has pT = 1 GeV / c, the pL 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 KS0 (red) and Λ (blue). The qT 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 qT cut):

KS0 mass spectrum with O (10 MeV / c2) resolution:

Λ mass spectrum with O (5 MeV / c2) 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.

#### Event gallery

Legend

• Primary vertex: green open circle.
• RecHits: red open boxes.
• PixelRecTracks: red lines connecting the RecHits.
• Trajectory of the neutral mother: thick black arrow.
• Daughter trajectories: blue helices.

KS0 decay:

Λ decay:

Photon conversion:

#### Sources

The V0Finder class (experimental)

-- FerencSikler - 05 Apr 2006

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Topic revision: r3 - 2007-11-11 - FerencSikler

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