1. workreport

2. 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）
（ ， ， ）
（1.5，-1.5，5）
（ ， ， ）
（2，-2，8）
（ ， ， ）

3. Vertex position （1.5，-1.5，5）(mm)
MeanX= (mm)
SigmaX= (mm)

4. Vertex position （1.5，-1.5，5）(mm)
MeanY= (mm)
SigmaY= (mm)

5. Vertex position （1.5，-1.5，5）(mm)
MeanZ= (mm)
SigmaZ= (mm)

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

7. 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

8. 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)