V0Finder < CMS

---+++!!V0 Finder

%TOC%

---++++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()= %BR% electric charge; positives are bent to the right, negatives to the left |
 | _b<sub>r</sub>_        | =transverseImpactParameterSignedByTrajCenter()= %BR% signed radial distance, _OP_ %BR% positive if the beamline is outside of the circle, negative otherwise |
 | _b<sub>z</sub>_        | =zImpactParameter()= %BR% _z_ coordinate of _P_ |
 | cot&theta; | =cotTheta()= |
 | _p<sub>T</sub>_        | =pT()= |
 | &phi;       | =phi()= %BR% 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<sub>T</sub>_

 _R_ = _p<sub>T</sub>_ / 0.003 B

Direction of the vector *CP*

 &chi; = &phi; + q &pi;/2

Coordinates of _C_

 X = - (R+b_r) cos &chi; %BR%
 Y = - (R+b_r) sin &chi;

---+++++Two circles

The direction of the vector *C<sub>1</sub> C<sub>2</sub>* pointing from the center of the
first to the center of the second circle is &psi;<sub>0</sub>. Depending of the relative
placement of two circles they will have a pair of closest points or two
intersections.

*The circles are disjoint* (R<sub>12</sub> > R<sub>1</sub>+R<sub>2</sub>).
       The direction of the closest points and the smallest distance is
       
        &psi;<sub>1</sub> = &psi;<sub>0</sub> %BR%
        &psi;<sub>2</sub> = &psi;<sub>0</sub> + &pi; %BR%
        &Delta; r = R<sub>12</sub> - (R<sub>1</sub>+R<sub>2</sub>)

*One circle contains the other* (R<sub>12</sub> < |R<sub>1</sub> - R<sub>2</sub>|) %BR%
       The direction of the closest points and the smallest distance is
 
        &psi;<sub>1</sub> = &psi;<sub>2</sub> = %BR%
          &psi;<sub>0</sub>      if R<sub>1 </sub>> R<sub>2</sub> %BR%
          &psi;<sub>0</sub>  + &pi; otherwise %BR%
        &Delta; r = |R<sub>1</sub> - R<sub>2</sub>| - R<sub>12</sub>

*The cirles intersect* (R<sub>12</sub> < R<sub>1</sub>+R<sub>2</sub> and R<sub>12</sub> > |R<sub>1</sub> - R<sub>2</sub>|) %BR%
       The smallest distance &Delta; r = 0.
       The direction of the intersection(s) is
 
        &gamma; = 
          arccos [(R<sub>1</sub><sup>2</sup> - R<sub>2</sub><sup>2</sup> + R<sub>12</sub><sup>2</sup>) / (2 R<sub>1</sub> R<sub>12</sub>)] %BR%
        &psi;<sub>1,i</sub> = &psi;<sub>0</sub> &plusmn; &gamma; %BR%
        &psi;<sub>2,i</sub> = atan2(Y<sub>1</sub>+R<sub>1</sub> sin &psi;<sub>1,i</sub> - Y<sub>2</sub>,
                                                          X<sub>1</sub>+R<sub>1</sub> cos &psi;<sub>1,i</sub> - X<sub>2</sub>)

---+++++Distance in _z_

The azimuthal angle &Delta; &psi; with respect to _P_ is

 &Delta; &psi; = &psi; - &chi; + _k_ 2&pi;

where _k_ is chosen such that -&pi; < &Delta; &psi; < &pi;.
For a valid track _q_ &Delta; &psi; < 0 must hold.  Using the equation of the
helix

 z = b<sub>z</sub> - R  q &Delta; &psi; cot&theta;

The closest point of a circle _I_ is thus given by

 I(X + R cos &psi;, Y+R sin &psi;, b<sub>z</sub> - R q &Delta; &psi; cot &theta;)


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

 &Delta; z = |z<sub>2</sub> - z<sub>1</sub>|

---+++++Neutral mother particle

The presumed production vertex r is the midpoint of line segment
I<sub>1</sub>I<sub>2</sub>.  The momentum components of a particle at the presumed production
vertex can be obtained by

 p<sub>x</sub> =  p<sub>T</sub> q  sin&psi; %BR%
 p<sub>y</sub> = -p<sub>T</sub> q  cos&psi; %BR%
 p<sub>z</sub> = p<sub>T</sub> cot&theta; %BR%

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* %BR%
 *r* = *r1* + *r2* %BR%
 b = | *r* - *p* ( *p* *r* )/p<sup>2</sup> |

---+++++Cuts

The resulting distances are summarized in Table 2. They
can be later used for cuts.

| *Notation* | *Comment* |
| &Delta; _r_ | Smallest distance in the transverse plane |
| &Delta; _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 &pi;<sup>+</sup> and 1 &pi;<sup>-</sup>
   * 1 K<sub>S</sub><sup>0</sup>
   * 1 &Lambda;
   * 10 &gamma;

Every particle has p<sub>T</sub> = 1 !GeV / _c_, the p<sub>L</sub> is in the interval [-0.5,0.5] !GeV / _c_ emitted isotropically.

For v0 finding the cuts are &Delta; r < 0.1 cm, &Delta; r < 0.1 cm, r > 0.4 cm and b < 0.1 cm.

*Armenteros plot* with the predictions for K<sub>S</sub><sup>0</sup> (red) and &Lambda; (blue). The _q<sub>T</sub>_ cut for removing photon conversions is indicated with the green line. The ellipses are broadened by factor 1/&beta; in &alpha; because the particles are relatively slow.

%ATTACHURL%/v0_fig4.png

*Particles with two mass hypotheses* (photons removed by _q<sub>T</sub>_ cut):

%ATTACHURL%/v0_fig1.png

*K<sub>S</sub><sup>0</sup> mass spectrum* with _O_ (10 !MeV / _c<sup>2</sup>_) resolution:

%ATTACHURL%/v0_fig2.png

*&Lambda; mass spectrum* with _O_ (5 !MeV / _c<sup>2</sup>_) resolution:

%ATTACHURL%/v0_fig3.png

*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.

%ATTACHURL%/v0_fig5.png

---++++Event gallery

Legend
   * %GREEN% Primary vertex%ENDCOLOR%: green open circle.
   * %RED%RecHits%ENDCOLOR%: red open boxes.
   * %RED%PixelRecTracks%ENDCOLOR%: red lines connecting the !RecHits.
   * *Trajectory* of the neutral mother: thick black arrow.
   * %BLUE%Daughter trajectories%ENDCOLOR%: blue helices.

K<sub>S</sub><sup>0</sup> decay:

%ATTACHURL%/event_k0.png

&Lambda; decay:

%ATTACHURL%/event_la.png

Photon conversion:

%ATTACHURL%/event_ph.png

---++++Sources

The V0Finder class (experimental)

   * [[%ATTACHURL%/V0Finder.cc][V0Finder.cc]]
   * [[%ATTACHURL%/V0Finder.h][V0Finder.h]]

-- Main.FerencSikler - 05 Apr 2006
This topic: CMS > TrackerSoftware > V0Finder
Topic revision: r1 - 2006-04-05 - FerencSikler
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