This page
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describes the formulae used project tracking errors to the ACD
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General framework
Errors on Point of Closest Approach (POCA) between a track and a point
These are written up in greater detail (and with real formulae, instead of text-based approximations) here: formulae.pdf
General framework
The tracker has a 4x4 representation (x_tkr) which uses a pair of slope-intercepts (x_0,y_0,m_x, m_y) to parametrize the position as a function of z. This is great for a bunch of planes normal to the z axis, but really not so nice for planes normal to the x and y axes. The first order of business is to transform the track parameterization to a 5x5 representation (x_acd)which uses a Point at Z=0 and a Vector (x_0,y_0, v_x,v_y,v_z) to parameterize the positions by arclength along the track (s).
We can the transform the covarience matrix by using the 5x4 track representation derivative matrix A = d x_acd / d x_tkr:
C_acd (5x5) = A C_tkr (4x4) A_T
If we then use the track parameterization to calculate a [point of closest approach (POCA) or an intersection point (p), the covarience on that point can be found using 5x3 solution derivative matrix
B = d p / d A_acd
with the covarience given by:
C_p (3x3) = B C_acd (5x5) B_T
In the case where we care about the distance of closest approach to a point or a ray, we can project along the vector of closest approach J (the vector from the POCA to the point or ray in question).
Then we find the error on the distance of closest approach:
C_doca (1x1) = J C_p(3x3) J_T
In the case where we care about the intersection of a track with a plane, we can rotate into the plane using the rotation matrix R.
C_plane (3x3) = R C_p(3x3) R_T
and take only the upper-left 4 entries to get the 2x2 error ellipse on the surface of the plane.
Errors on POCA between a track and a point
Call the point q, and the track (x,v) the POCA occurs at an arclength
s = (x - q) . v
therefore the poca is
p = x + sv
and the vector of closest approach is:
J = q-p = q - x + sv
The full derivative matrix (B) is given in the attached write up.
Errors Errors on POCA between a track and a ray
Call the ray (r,q) and the track (x,v), the POCA occurs at an arclength along the track
s = [b(r.(x-q) - v.(x-q)] / [1-b*b]
and an arclength along the ray:
t = [r.(x-q) - b(v.(x-q))] / [1-b*b]
where b = v.r
Then the poca and vector of closest approach are:
p = x + sv
J = (x-q) + sv - tr
The full derivative matrix (B) is given in the attached write up.
Errors on point where a track crosses a plane
Given a plane defined by a point q and a rotation matrix R, the intersection occurs at:
s = [ (x-q).R_z ] / [ v. R_z ]
Then the intersection point in the global frame is:
p = x + sv
Of course we can get the intersection point in the local frame l:
l = R p + q
The full derivative matrix (B) is given in the attached write up.
Some issues with the Error Projection
- They depend on the Kalman fitter hypothesis. Describing a 8GeV proton as a 100MeV electron (or vice-versa) is not going to give accurate errors
- They rely on knowing the material along the track path. For low energy particle the errors can blow up very quickly if there is a lot of material between the head of the track and the ACD
We distinguish between to ways of calculating the error projections:
- Full Propagation: track is kalman-propagated all the way back to the POCA and errors are calculated from the cov. matrix at the POCA
- Projection: errors are taken at the head of the track projected out at the POCA.
The errors from method 1) will alway be larger than the error from 2). For low-energy tracks the difference can be quite large.