This page describes the formulae used project tracking errors to the ACD

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 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

  1. 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
  2. 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:

  1. Full Propagation:  track is kalman-propagated all the way back to the POCA and errors are calculated from the cov. matrix at the POCA
  2. 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.

 

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