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Vertex reconstruction Generate 10,000 di-gamma events and change vertex position Regard the average value of all midpoints as reconstructed vertex position Vertex position(MC)(mm) Midpoint of di-gamma track(mm) （1，-1，2） （0.4145+0.0338， -0.3622+0.0315， 1.6639+0.1599） （1.5，-1.5，5） （0.6194+0.0348， -0.5618+0.0327， 4.2581+0.1634） （2，-2，8） （0.8808+0.0388， -0.8009+0.0340， 7.1029+0.1567）

Vertex position （1.5，-1.5，5）(mm) MeanX=0.6194+0.0348(mm) SigmaX=2.2382+0.0409(mm)

Vertex position （1.5，-1.5，5）(mm) MeanY=-0.5618+0.0327(mm) SigmaY=2.1172+0.0366(mm)

Vertex position （1.5，-1.5，5）(mm) MeanZ=4.2581+0.1634(mm) SigmaZ=11.6193+0.1425(mm)

Vertex reconstruction This method can quickly reconstruct vertex position Values of Z direction are acceptable Values of X and Y directions are bad

Primary vertex fitting on Kalman filter method Updating the vertex position and its covariance matrix step by step through adding a new track k x=(x,y,z) — the vertex position pk =(px,py,pz) — the 3-momentum of the k-th track, originating from the vertex x α0k — the k-th track measurement ~α= ˜α(x,p) — parameters of the k-th track

Primary vertex fitting on Kalman filter method Every step use least squares The parameters we know: xk−1 α0k (and their covariance matrix) The parameters to be estimated: xk pk The true vertex have no changes xk =xk−1 =x The k-th track is the function of xk and pk ,but it is nonlinear. A first order Taylor expansion. ~αk (xk,pk) ≈ ˜αe(xe,pe)+A(x−xe)+B(p−pe)=ce+Ax+Bp The χ2 can be written as a sum of two terms. Minimizing the χ2 and we can get the xk and pk χ2KF = (xk−xk−1)TC−1k−1(xk−xk−1)+(α0k−˜αk)TGk(α0k−˜αk)