Paul Avery CBX 92–45 August 20, 1992

Applied Fitting Theory IV Formulas for Track Fitting

Section 1: Helix Equations of Motion Section 2: Useful Formulas for Track Fitting in a Solenoidal Field Section 3: Projections of helix including errors Section 4: Detector Misalignments and Track Fitting

This note introduces some mathematical algorithms that I have developed over the years for fitting and projecting tracks in solenoidal magnetic fields. I have spent several years developing these formulas but so far the documentation has existed only in handwritten notes and Fortran programs. Hopefully, this note will provide a useful reference for anyone interested in these details. Many of the equations have been implemented in the FTMONTE package, and I have made every effort to make sure that the formulas are correct, although there might be an error or two lurking somewhere. I have built in various cross checks in my programs to verify that the Fortran code is correct, especially the derivatives.

1. Helix equations of motion

1.1. Track representations

I use two track representations:

1. The C or canonical representation is used in track fitting; the 5 parameters describe the entire shape of the helix. α_{C} = (c, ϕ_{0}, D, λ, z_{0}), where c = 1/2R, R being the radius of curvature, ϕ_{0} is the ϕ of the track momentum at the distance of closest approach to the origin, D is the signed impact parameter in the x−y plane, λ = cot θ, where θ is the polar angle measured from the +z axis and z_{0} is the z of the track at the point of closest approach to the origin in the x − y plane.

2. The W representation is used in kinematic fitting and for finding the 4–momentum at a particular point on the helix. α_{W} = (p_{x}, p_{y}, p_{z}, E, x, y, z), i.e., a 4–momentum plus a point at which the 4–momentum is evaluated.

1.2. Equations of motion

I begin by defining the following useful symbols:

a = −0.2998BQ (B = mag. field, Q = charge)
p_{⊥} = \sqrt{p_{x}^{2} + p_{y}^{2}}
u = p_{x}/p_{⊥}
v = p_{y}/p_{⊥}
u_{0} = p_{0x}/p_{⊥} = cos ϕ_{0}
v_{0} = p_{0y}/p_{⊥} = sin ϕ_{0}
ρ = \frac{1}{R} R = radius of curvature

Assume that the particle has charge Q and is moving in a magnetic field of strength B. The particle moves along a helix with a signed radius of curvature R. R (and ρ) is positive for counterclockwise motion. The trajectory of a helix is governed by the following equations, valid when B is along z⃗.

p_{x} = p_{0x} cos ρs_{⊥} − p_{0y} sin ρs_{⊥}
p_{y} = p_{0y} cos ρs_{⊥} + p_{0x} sin ρs_{⊥}
p_{z} = p_{0z}
E = E_{0}
x = x_{0} + \frac{p_{0x}}{a} sin(ρs_{⊥}) − \frac{p_{0y}}{a} (1 − cos(ρs_{⊥}))
y = y_{0} + \frac{p_{0y}}{a} sin(ρs_{⊥}) + \frac{p_{0x}}{a} (1 − cos(ρs_{⊥}))
z = z_{0} + λs_{⊥}

where (x_{0}, y_{0}, z_{0}) is a known point on the helix, (p_{0x}, p_{0y}, p_{0z}, E_{0}) is its 4–momentum vector there and ρ = 2c = −0.2998BQ/p_{⊥} ≡ a/p_{⊥}, They are functions of s_{⊥}, the arc length in the x − y plane from (x_{0}, y_{0}, z_{0}) to the point x, y, z we are trying to find. If (x_{0}, y_{0}, z_{0}) is the point of closest approach to the origin, then p_{0x}, p_{0y}, p_{0z}, E_{0}, x_{0}, y_{0} and z_{0} can be written in terms of the 5 canonical parameters:

p_{0x} = p_{⊥}u_{0} = \frac{a}{2c} cos ϕ_{0}
p_{0y} = p_{⊥}v_{0} = \frac{a}{2c} sin ϕ_{0}
p_{0z} = p_{⊥}λ = \frac{a}{2c}λ
E_{0} = \sqrt{p_{⊥}^{2}(1 + λ^{2}) + m^{2}}
x_{0} = −Dv_{0} = −D sin ϕ_{0}
y_{0} = +Du_{0} = +D cos ϕ_{0}
z_{0} = z_{0}

It is easy to see that D is positive when the ϕ of the position is greater than that of the direction of motion at the point of closest approach. Note that r_{min} = |D| and that the point of closest approach, reached when s_{⊥} = 0 is just (x_{0}, y_{0}). The track reaches its maximum radius r_{max} = |D + 1/c| when ρs_{⊥} = π. This occurs at the point (x_{max}, y_{max}), where

\begin{aligned}x_{max} & = − (D + \frac{1}{c}) v_{0}, \\ y_{max} & = + (D + \frac{1}{c}) u_{0}. \end{aligned}

The center of the circle is found to be at

\begin{aligned}x_{c} & = − (\frac{1}{2c} + D) v_{0}, \\ y_{c} & = + (\frac{1}{2c} + D) u_{0}. \end{aligned}

The quantity 1 + ρD = 1 + 2cD is always positive; this fact turns out to be very useful when using the track parameters as we will see later.

A pair of conserved quantities can be derived from the (x, y, z) equations by substituting the equations for p_{x} and p_{y}:

p_{x} + ay = p_{0x} + ay_{0} = const
p_{y} − ax = p_{0y} − ax_{0} = const

1.3. Track coordinates as a function of radius

Sometimes it is useful to express ϕ, z and s_{⊥} as a function of r = \sqrt{x^{2} + y^{2}}:

sin(ϕ − ϕ_{0}) = \frac{rc + (D/r)(1 + cD)}{1 + 2cD} = \frac{D}{r} + \frac{(c/r)(r^{2} − D^{2})}{1 + 2cD},
cos(ϕ − ϕ_{0}) = ±\frac{\sqrt{(1 − D^{2}/r^{2})[(1 + cD)^{2} − r^{2}c^{2}]}}{1 + 2cD},
z − z_{0} = λs_{⊥},
sin cs_{⊥} = c\sqrt{\frac{r^{2} − D^{2}}{1 + 2cD}},

where the sign of cos(ϕ−ϕ_{)}) is chosen depending on whether the particle is on the outgoing (+1) or incoming (−1) branch of the circle. This ambiguity is fundamental; the particle always passes through the same radius going out and coming in.

The momentum and position coordinates at the radius r are given by

p_{x} = p_{0x}(1 − 2B^{2}) − p_{0y}2ϵB\sqrt{1 − B^{2}}
p_{y} = p_{0y}(1 − 2B^{2}) + p_{0x}2ϵB\sqrt{1 − B^{2}}
p_{z} = p_{0z}
x = x_{0} + \frac{u_{0}}{ρ} 2ϵB\sqrt{1 − B^{2}} − \frac{v_{0}}{ρ} 2B^{2}
y = x_{0} + \frac{v_{0}}{ρ} 2ϵB\sqrt{1 − B^{2}} + \frac{u_{0}}{ρ} 2B^{2}
z = z_{0} + λs_{⊥}
s_{⊥} = \{ \begin{matrix}(1/c) sin^{−1} B & for ϵ = +1 \\ (1/c)(π − sin^{−1} B) & for ϵ = −1 \end{matrix}

where B = c\sqrt{(r^{2} − D^{2})/(1 + 2cD)} = sin \frac{1}{2}ρs_{⊥} = sin cs_{⊥} and ϵ = +1(−1) corresponds to the outgoing (incoming) branch.

Similarly, we can calculate the angle α between the track direction and the radius vector

sin α = \frac{xp_{y} − yp_{x}}{rp_{⊥}} = rc − \frac{D}{r}(1 + cD),
cos α = \frac{xp_{x} + yp_{y}}{rp_{⊥}} = (1 + 2cD) cos(ϕ − ϕ_{0}) = ϵ\sqrt{(1 − D^{2}/r^{2})[(1 + cD)^{2} − r^{2}c^{2}]}.

Note that sin α ≃ rc and s_{⊥} ≃ (1/c) sin^{−1} rc for D ≪ r and cD ≪ 1. Also, the sign of ϵ can be determined if one knows the position and momentum components at any given point on the helix from the formula for cos α.

1.4. Computing track parameters from instantaneous position and momentum

This is similar to the classic problem of finding planetary orbital elements given the instantaneous position and velocity vectors. Assume we are given α_{W} = (p_{x}, p_{y}, p_{z}, E, x, y, z). First we find λ and c:

\begin{aligned}λ & = cot θ = p_{z}/\sqrt{p_{x}^{2} + p_{y}^{2}} ≡ p_{z}/p_{⊥} \\ c & = ρ/2 = −0.1499BQ/p_{⊥} ≡ a/2p_{⊥}. \end{aligned}

Take the equations of motion above and solve for sin ρs_{⊥}, cos ρs_{⊥}:

\begin{aligned}(p_{⊥} + aD) sin ρs_{⊥} & = ρ(xp_{x} + yp_{y}), \\ (p_{⊥} + aD) cos ρs_{⊥} & = p_{⊥} − ρ(xp_{y} − yp_{x}), \end{aligned}

which yields s_{⊥} and

\begin{aligned}cos ϕ_{0} & = \frac{p_{x} + ay}{T} , \\ sin ϕ_{0} & = \frac{p_{y} − ax}{T}, \\ D & = \frac{1}{a}[T − p_{⊥}] = \frac{−2(xp_{y} − yp_{x}) + a(x^{2} + y^{2})}{T + p_{⊥}}, \\ z_{0} & = z − λs_{⊥}, \end{aligned}

where T = \sqrt{p_{⊥}^{2} − 2a(xp_{y} − yp_{x}) + a^{2}(x^{2} + y^{2})}. The second form for D is useful for particles with small curvature. For the case when a = 0 (zero curvature), the equations simplify to

\begin{aligned}c & = ρ = 0 \\ cos ϕ_{0} & = p_{x}/p_{⊥} \\ sin ϕ_{0} & = p_{y}/p_{⊥} \\ D & = (yp_{x} − xp_{y})/p_{⊥} \\ λ & = p_{z}/p_{⊥} \\ z_{0} & = z − λs_{⊥} (s_{⊥} = \frac{xp_{x} + yp_{y}}{p_{⊥}}) \end{aligned}

2. Useful formulas for track fitting in a Solenoidal Field

I present here some useful formulas for calculating the quantities directly measured in the most common tracking devices and their partial derivatives with respect to the 5 helix parameters. These values are needed by the track fitting routines described in this document. I assume that the magnetic field is along the z axis. See the first section for the definition of some of the symbols used here.

The measurments needed are (1-2) the distance of closest approach to axial and stereo drift chamber wires, (3) z as measured by cathode strips in drift chambers, (4-5) the distance along r − ϕ and r − z silicon planes and (6-7) the distance along x and y in silicon disks located at fixed z positions. The Fortran code for these cases is stored in libFTLIB. The above quantities are computed in FTDRF1-7 and FTDF1-7, the analytic derivatives are calculated in FTDRV1-7 and the numerical derivatives (absolutely essential for checking the analytic expressions) are computed in FTDRN1-7.

2.1. Axial drift chamber wires

Consider an axial wire at position (x_{w}, y_{w}) and define (x, y, z) as the point on the helix which is closest to (x_{w}, y_{w}) (we need only consider the x−y motion) and call d_{w} the signed distance of closest approach of the track to the wire. Now use the fact that the normal to the circle at (x, y) lies along the radius vector and passes through (x_{w}, y_{w}). The x − y direction cosines of the track at that point are (u, v) = (p_{x}/p_{⊥}, p_{y}/p_{⊥}) so the normal to the circle has direction cosines (−v, u). Writing the equation of the normal as

\begin{aligned}x + d_{w}v & = x_{w}, \\ y − d_{w}u & = y_{w}, \end{aligned}

we get, after a little manipulation,

(1 + ρd_{w}) cos ρs_{⊥} = 1 − ρv_{0}Δ_{x} + ρu_{0}Δ_{y},
(1 + ρd_{w}) sin ρs_{⊥} = −ρu_{0}Δ_{x} − ρv_{0}Δ_{y},

where Δ_{x} = x_{0} − x_{w} and Δ_{y} = y_{0} − y_{w}. This allows us to solve for s_{⊥} and

d_{w} = \frac{1}{ρ} [\sqrt{1 + 2ρΔ_{w}} − 1] = \frac{2Δ_{w}}{1 + \sqrt{1 + 2ρΔ_{w}}} ≃ Δ_{w}(1 − \frac{1}{2}ρΔ_{w})

where Δ_{w} = u_{0}Δ_{y} − v_{0}Δ_{x} + \frac{1}{2}ρ(Δ_{x}^{2} + Δ_{y}^{2}) is a very good approximation to d_{w}. The sign of d_{w} is positive when the track has a larger ϕ than that of the wire.

The derivatives of d_{w} with respect to the 5 helix parameters are

\frac{∂d_{w}}{∂c} = (Δ_{x}^{2} + Δ_{y}^{2})(1 − ρΔ_{w}) − Δ_{w}^{2},
\frac{∂d_{w}}{∂ϕ_{0}} = −(Δ_{x}u_{0} + Δ_{y}v_{0})(1 + ρD)(1 − ρΔ_{w}),
\frac{∂d_{w}}{∂D} = (1 − ρΔ_{x}v_{0} + ρΔ_{y}u_{0})(1 − ρΔ_{w}),
\frac{∂d_{w}}{∂λ} = 0,
\frac{∂d_{w}}{∂z_{0}} = 0.

These reduce, when ρ = 0, to

\begin{aligned}\frac{∂d_{w}}{∂c} & = (Δ_{x}u_{0} + Δ_{y}v_{0})^{2}, \\ \frac{∂d_{w}}{∂ϕ_{0}} & = −(Δ_{x}u_{0} + Δ_{y}v_{0}), \\ \frac{∂d_{w}}{∂D} & = 1, \\ \frac{∂d_{w}}{∂λ} & = 0, \\ \frac{∂d_{w}}{∂z_{0}} & = 0. \end{aligned}

2.2. Stereo drift chamber wires

Stereo wires couple three equations containing terms cos ρs_{⊥}, sin ρs_{⊥} and s_{⊥}, which makes them impossible to solve analytically. Let z_{p} be the z coordinate of a point on the wire, tan τ be the tangent of the stereo angle τ and (x_{w}, y_{w}) be the coordinates of the wire at z_{p} = 0 ((r_{w}, ϕ_{w}) in polar coordinates). The coordinates of any point (x_{p}, y_{p}) on the wire are

x_{p} = x_{w} + z_{p} tan τ sin ϕ_{s},
y_{p} = y_{w} − z_{p} tan τ cos ϕ_{s},

where the CLEO drift chamber is wire in such a way that ϕ_{s} = ϕ_{w}. The (unnormalized) direction cosines of the wire and of the track at the point of closest approach to the wire are

η⃗_{w} = (tan τ sin ϕ_{s}, − tan τ cos ϕ_{s}, 1),
η⃗_{t} = (u, v, λ) = (u_{0} cos ρs_{⊥} − v_{0} sin ρs_{⊥}, v_{0} cos ρs_{⊥} − u_{0} sin ρs_{⊥}, λ).

The line which is normal to both the wire and the helix at its point of closest approach has direction cosines

u⃗_{d} = \frac{η⃗_{w} × η⃗_{t}}{|η⃗_{w} × η⃗_{t}|} = (v + λ′ cos ϕ_{s}, −u + λ′ sin ϕ_{s}, G_{1} tan τ)/u_{d}
u_{d} = \sqrt{1 + 2λ′H_{1} + λ^{′2} + G_{1}^{2} tan^{2} τ}

where u_{d} is a normalization factor and

λ′ = λ tan τ,
C_{1} = u_{0} cos ϕ_{s} + v_{0} sin ϕ_{s} = cos(ϕ_{0} − ϕ_{s}),
S_{1} = v_{0} cos ϕ_{s} − u_{0} sin ϕ_{s} = sin(ϕ_{0} − ϕ_{s}),
G_{1} = C_{1} cos ρs_{⊥} − S_{1} sin ρs_{⊥} = cos(ϕ_{0} − ϕ_{s} + ρs_{⊥}),
H_{1} = C_{1} sin ρs_{⊥} + S_{1} cos ρs_{⊥} = sin(ϕ_{0} − ϕ_{s} + ρs_{⊥}).

Just as in the axial wire case we write the equation of the normal:

x + d_{w}(v + λ′ cos ϕ_{s})/u_{d} = x_{p},
y − d_{w}(u − λ′ sin ϕ_{s})/u_{d} = y_{p},
z + d_{w}G_{1} tan τ/u_{d} = z_{p},

which can be seen to reduce to the axial case by setting the stereo angle τ to zero. Note that the line normal to the track and the wire lies outside the x − y plane. Inserting the equations of motion for x, y and z we get, after some manipulation, the equations we want to solve

(1 + ρd_{w}/u_{d}) cos ρs_{⊥} = +A_{1} − \frac{ρd_{w}}{u_{d}} λ′S_{1}
(1 + ρd_{w}/u_{d}) sin ρs_{⊥} = −A_{2} − \frac{ρd_{w}}{u_{d}} λ′C_{1}
z_{p} = z_{0} + λs_{⊥} − \frac{d_{w}}{u_{d}} G_{1} tan τ

where A_{1} = 1 − ρΔ_{x}v_{0} + ρΔ_{y}u_{0}, A_{2} = ρΔ_{x}u_{0} + ρΔ_{y}v_{0}, Δ_{x} = x_{0} − x_{w} − z_{p} tan τ sin ϕ_{s} and Δ_{y} = y_{0} − y_{w} + z_{p} tan τ cos ϕ_{s}.

Squaring, adding and collecting terms leads to the quadratic equation

\frac{d_{w}^{2}}{u_{d}^{2}} (1 − λ^{′2}) + \frac{d_{w}}{u_{d}} \frac{J}{ρ} − \frac{2Δ_{w}}{ρ} ,

where as before Δ_{w} = u_{0}Δ_{y} −v_{0}Δ_{x} + \frac{1}{2}ρ(Δ_{x}^{2} +Δ_{y}^{2}) and J = 1+λ′(A_{1}S_{1} −A_{2}C_{1}). Note that the definition of Δ_{w} here differs slightly from the axial case because it has z dependent terms in Δ_{x} and Δ_{y}. The solution is

d_{w} = \frac{u_{d}/ρ}{1 − λ^{′2}} J (−1 + \sqrt{1 + 2(1 − λ^{′2})ρΔ_{w}/J^{2}}) ,

= \frac{u_{d}}{J} \frac{2Δ_{w}}{1 + \sqrt{1 + 2(1 − λ^{′2})ρΔ_{w}/J^{2}}} ,

≃ \frac{u_{d}}{J} Δ_{w}(1 − \frac{1}{2}(1 − λ^{′2})ρΔ_{w}/J^{2}).

The first two expressions for d_{w} are exact but still depend on cos ρs_{⊥}, sin ρs_{⊥} and z_{p} ≃ z_{0} + λs_{⊥}. However, the latter expressions are all multiplied by terms proportional to tan τ so they only modify the final result slightly. For highest precision, I recommend determining s_{⊥} using the following procedure. First, treat the wire as axial and solve sin cs_{⊥} = c\sqrt{(r_{w}^{2} − D^{2})/(1 + 2cD)} for s_{⊥}. This yields z ≃ z_{0} + λs_{⊥} which can be used to get more precise values for Δ_{x} and Δ_{y}. These can in turn be used to obtain better values for cos ρs_{⊥} and sin ρs_{⊥} (and hence s_{⊥}) using the (1 + ρd_{w}/u_{d}) cos ρs_{⊥} and (1 + ρd_{w}/u_{d}) sin ρs_{⊥} equations above with Δ_{w} substituting for d_{w} in this first iteration. The values obtained for cos ρs_{⊥}, sin ρs_{⊥} and s_{⊥} are accurate enough to be plugged into the above equation for d_{w}. One can iterate this procedure once more if necessary (using the value of d_{w} just obtained to get precise values of cos ρs_{⊥}, sin ρs_{⊥} and z_{p}) but this is probably not necessary.

An alternative approach is to take the three coupled equations for cos ρs_{⊥}, sin ρs_{⊥} and z_{p} and expand the three unknowns s_{⊥}, z_{p} and d_{w} to first order. For this work it is useful to multiply the first two equations by cos ρs_{⊥} and sin ρs_{⊥} and add and subtract:

\begin{aligned}d_{w} & = \frac{u_{d}}{ρ} (A_{c} − 1) − d_{w}λ′H_{1}, \\ 0 & = −\frac{u_{d}}{ρ} A_{s} − d_{w}λ′G_{1}, \\ z_{p} & = z_{0} + λs_{⊥} − \frac{d_{w}}{u_{d}} G_{1} tan τ, \end{aligned}

where the terms A_{c} and A_{s} are defined by A_{c} = A_{1} cos ρs_{⊥} − A_{2} sin ρs_{⊥} and A_{s} = A_{1} sin ρs_{⊥} + A_{2} cos ρs_{⊥}. The solution is started by assuming axial wires, i.e., tan τ = 0. Collecting d_{w} terms and expanding to first order we get

\begin{aligned}δd_{w} & = \frac{(X_{u}A_{c} − X_{u} + u_{d}G_{1}λ′)δs_{⊥}}{1 + λ′H_{1}} , \\ δs_{⊥} & = \frac{Δ_{s} − G_{1}λ′δd_{w}}{u_{d} + ρd_{w} + X_{u}A_{s} + u_{d}H_{1}λ′} , \\ δz_{s} & ≃ λδs_{⊥}, \end{aligned}

where X_{u} = (G_{1}/u_{d}) [λ′ − H_{1} tan^{2} τ], Δ_{c} = (u_{d}/ρ)(A_{c} − 1) − d_{w}(1 + H_{1}λ′) and Δ_{s} = −(u_{d}/ρ)A_{s} − d_{w}G_{1}λ′.

To calculate the derivatives of d_{w} with respect to the five track parameters we use the approximate form of the d_{w} solution:

d_{w} ≃ \frac{u_{d}}{J} Δ_{w}(1 − \frac{1}{2}(1 − λ^{′2})ρΔ_{w}/J^{2}),

and ignore derivatives of u_{d} and J because they are multiplied by terms of order Δ_{w} tan τ which are very small for typical drift chambers. The derivatives with respect to c, ϕ_{0} and D are similar to the axial case except that for highest accuracy one must include the z_{p} term in the definitions of Δ_{x} and Δ_{y}. The derivatives of d_{w} can all be expressed in terms of those of Δ_{w}:

\frac{∂d_{w}}{∂α_{i}} = \frac{u_{d}}{J} (\frac{∂Δ_{w}}{∂α_{i}} (1 − ρΔ_{w}/J^{2}) − δ_{i1}Δ_{w}^{2}/J^{2}) .

The values of ∂Δ_{w}/∂α_{i} are calculated below, ignoring terms of order tan^{2} τ and ρd_{w} tan τ.

\frac{∂Δ_{w}}{∂c} = Δ_{x}^{2} + Δ_{y}^{2} + λ′ \frac{∂s_{⊥}}{∂c} (C_{1} + ρΔ_{n})
\frac{∂Δ_{w}}{∂ϕ_{0}} = −u_{0}Δ_{x} − v_{0}Δ_{y} + λ′ \frac{∂s_{⊥}}{∂ϕ_{0}} (C_{1} + ρΔ_{n})
\frac{∂Δ_{w}}{∂D} = 1 + λ′ \frac{∂s_{⊥}}{∂D} (C_{1} + ρΔ_{n})
\frac{∂Δ_{w}}{∂λ} = tan τ (s_{⊥} + λ\frac{∂s_{⊥}}{∂λ} ) (C_{1} + ρΔ_{n})
\frac{∂Δ_{w}}{∂z_{0}} = tan τ (1 + λ\frac{∂s_{⊥}}{∂z_{0}} ) (C_{1} + ρΔ_{n})

where Δ_{n} = −Δ_{x} sin ϕ_{s} + Δ_{y} cos ϕ_{s} and the derivatives with respect to s_{⊥} are given by

\frac{∂s_{⊥}}{∂c} = −\frac{2}{ρ}(D sin ρs_{⊥} − (x_{w}u + y_{w}v) + s_{⊥}),
\frac{∂s_{⊥}}{∂ϕ_{0}} = −(x_{w}v − y_{w}u),
\frac{∂s_{⊥}}{∂D} = − sin ρs_{⊥},
\frac{∂s_{⊥}}{∂λ} = −s_{⊥}H_{1} tan τ,
\frac{∂s_{⊥}}{∂z_{0}} = −H_{1} tan τ,

where u = u_{0} cos ρs_{⊥} − v_{0} sin ρs_{⊥} and v = v_{0} cos ρs_{⊥} + u_{0} sin ρs_{⊥}. ∂s_{⊥}/∂c = 0 at ρ = c = 0.

2.3. Cathode strips

Cathodes actually measure the z position of a hit on the nearest anode wire layer. Recall from the previous discussion of axial drift chamber layers

(1 + ρd_{w}) cos ρs_{⊥} = 1 − ρv_{0}Δ_{x} + ρu_{0}Δ_{y},
(1 + ρd_{w}) sin ρs_{⊥} = −ρu_{0}Δ_{x} − ρv_{0}Δ_{y},

where Δ_{x} = x_{0} − x_{w} and Δ_{y} = y_{0} − y_{w} and (x_{w}, y_{w}) is the coordinate of the corresponding anode wire. These equations can be solved trivially for s_{⊥} and z = z_{0} + λs_{⊥} (recall that 1 + ρd_{w} is always positive). For highest accuracy in calculating derivatives, it is necessary to solve for d_{w} using the results in the section on axial wires:

d_{w} = \frac{1}{ρ} [\sqrt{1 + 2ρΔ_{w}} − 1] = \frac{2Δ_{w}}{1 + \sqrt{1 + 2ρΔ_{w}}} ≃ Δ_{w}(1 − \frac{1}{2}ρΔ_{w})

where Δ_{w} = u_{0}Δ_{y} −v_{0}Δ_{x}+ \frac{1}{2}ρ(Δ_{x}^{2}+Δ_{y}^{2}) To find derivatives, we eliminate d_{w} by multiplying the first two equations by sin ρs_{⊥} and cos ρs_{⊥} and subtract to get

sin ρs_{⊥}(1 − ρv_{0}Δ_{x} + ρu_{0}Δ_{y}) + cos ρs_{⊥}(ρu_{0}Δ_{x} + ρv_{0}Δ_{y}) = 0

Setting as before A_{1} = 1 − ρΔ_{x}v_{0} + ρΔ_{y}u_{0} and A_{2} = ρΔ_{x}u_{0} + ρΔ_{y}v_{0} we get

\frac{∂z}{∂c} = −2\frac{λ}{ρ^{2}} \frac{ρs_{⊥}(1 + ρd_{w}) − sin ρs_{⊥}}{1 + ρd_{w}}
\frac{∂z}{∂ϕ_{0}} = −\frac{λ}{ρ} \frac{1 + ρd_{w} − (1 + ρD) cos ρs_{⊥})}{1 + ρd_{w}}
\frac{∂z}{∂D} = −\frac{λ sin ρs_{⊥}}{1 + ρd_{w}}
\frac{∂z}{∂λ} = s_{⊥}
\frac{∂z}{∂z_{0}} = 1

For the special case ρ = 0, s_{⊥} = −(u_{0}Δ_{x} + v_{0}Δ_{y}) and the derivatives reduce to

\begin{aligned}\frac{∂z}{∂c} & = 0 \\ \frac{∂z}{∂ϕ_{0}} & = λ(D − d_{w}) \\ \frac{∂z}{∂D} & = 0 \\ \frac{∂z}{∂λ} & = s_{⊥} \\ \frac{∂z}{∂z_{0}} & = 1 \end{aligned}

2.4. Silicon barrel (r − ϕ)

The silicon barrel detector is composed of silicon planes arranged in a polygonal fashion in ϕ around the origin. The planes are assumed to be parallel to z and have small overlaps at the edges where they meet. Assume that each plane is defined by its outward normal η⃗ = (cos β, sin β, 0) and any point on the plane x⃗_{c}. For this discussion it is convenient to take x⃗_{c} as the point at which the normal, proceeding from the origin, strikes the plane. The quantity measured by the electronics (up to a constant offset) is the distance along the plane in the r − ϕ direction from the point x⃗ where the track hits the plane and x⃗_{c}.

The equation of the plane is η⃗ ·(x⃗−x⃗_{c}) = 0 or x cos β +y sin β −Δ = 0, where Δ = η⃗ ·x⃗_{c} is the perpendicular distance from the plane to the origin. It is easier to solve the equation in terms of “local” coordinates (x_{β}, y_{β}, z_{β}) where the equation of the plane is x_{β} = 0. The transformation equations are

x_{β} = x cos β + y sin β − Δ,
y_{β} = y cos β − x sin β,
z_{β} = z,

The equations of the helix can then be written ρx_{0β} + u_{0β} sin ρs_{⊥} − v_{0β}(1 − cos ρs_{⊥}) = 0, where the rotated quantities are

u_{0β} = u_{0} cos β + v_{0} sin β = cos(ϕ_{0} − β),
v_{0β} = v_{0} cos β − u_{0} sin β = sin(ϕ_{0} − β),
x_{0β} = x_{0} cos β + y_{0} sin β − Δ = −Dv_{0β} − Δ,
y_{0β} = y_{0} cos β − x_{0} sin β = −Du_{0β}.

The quantity d_{t} lies in the plane and increases along some direction d̂ defined to be perpendicular to the orientation of the strips. Here we assume that the strips measure the distance along r − ϕ, i.e. d̂ = (− sin β, cos β, 0). For z measurements d̂ = (0, 0, 1) which will be exploited later.

d_{t} = d̂· (x⃗ − x⃗_{c}),

= −x sin β + y cos β

= y_{0β} + \frac{v_{0β}}{ρ} sin ρs_{⊥} + \frac{u_{0β}}{ρ} (1 − cos ρs_{⊥}),

= y_{0β} + \frac{u_{0β}}{ρ} − \frac{1}{ρ} cos(ϕ_{0} − β + ρs_{⊥}).

The equation of the plane is x_{β} = 0 or, writing it in terms of sin(ϕ_{0} − β + ρs_{⊥}),

0 = ρx_{0β} − v_{0β} + u_{0β} sin ρs_{⊥} + v_{0β} cos ρs_{⊥},
= ρx_{0β} − v_{0β} + sin(ϕ_{0} − β + ρs_{⊥}),

allowing d_{t} to be expressed as

\begin{aligned}d_{t} & = y_{0β} + \frac{1}{ρ}u_{0β} − \frac{1}{ρ}\sqrt{1 − (v_{0β} − ρx_{0β})^{2}}, \\ & = y_{0β} − \frac{2v_{0β}x_{0β} − ρx_{0β}^{2}}{u_{0β} + \sqrt{1 − (v_{0β} − ρx_{0β})^{2}}} , \end{aligned}

where the second form is useful when ρ is small. We can also solve for s_{⊥} using

\begin{aligned}sin ρs_{⊥} & = ρx_{0β}u_{0β} + ρ(y_{0β} − d_{t})v_{0β}, \\ cos ρs_{⊥} & = 1 − ρx_{0β}v_{0β} + ρ(y_{0β} − d_{t})u_{0β}. \end{aligned}

In the special case ρ = 0, d_{t} = y_{0β} − x_{0β}(v_{0β}/u_{0β}) = y_{0β} + v_{0β}s_{⊥}, where s_{⊥} = −x_{0β}/u_{0β}.

The derivatives of d_{t} with respect to the five helix parameters, ∂d_{t}/∂α_{i}, are best expressed in terms of ∂s_{⊥}/∂α_{i}. From the equation of the plane ρx_{0β} + u_{0β} sin ρs_{⊥} − v_{0β}(1 − cos ρs_{⊥}) = 0 we derive

\begin{aligned}\frac{∂s_{⊥}}{∂ρ} & = −\frac{1}{ρ} \frac{x_{0β} + s_{⊥}u_{β}}{u_{β}} , \\ \frac{∂s_{⊥}}{∂ϕ_{0}} & = \frac{1}{ρ} \frac{y_{0β} + (1/ρ)(u_{0β} − u_{β})}{u_{β}} = \frac{d_{t}}{u_{β}} , \\ \frac{∂s_{⊥}}{∂D} & = \frac{v_{0β}}{u_{β}} , \end{aligned}

where u_{β} = u_{0β} cos ρs_{⊥}−v_{0β} sin ρs_{⊥} = cos(ϕ_{0}+ρs_{⊥}+β) and v_{β} = v_{0β} cos ρs_{⊥}+u_{0β} sin ρs_{⊥} = sin(ϕ_{0} + ρs_{⊥} + β). For ρ = 0 these simplify to

\begin{aligned}\frac{∂s_{⊥}}{∂ρ} & = \frac{1}{2}s_{⊥}^{2} \frac{v_{0β}}{u_{0β}} , \\ \frac{∂s_{⊥}}{∂ϕ_{0}} & = \frac{y_{0β} + v_{0β}s_{⊥}}{u_{0β}} = \frac{d_{t}}{u_{0β}} , \\ \frac{∂s_{⊥}}{∂D} & = \frac{v_{0β}}{u_{0β}} , \end{aligned}

Now we can write the derivatives of d_{t} with respect to the five helix parameters:

\begin{aligned}\frac{∂d_{t}}{∂c} & = 2\frac{u_{β} − u_{0β}}{ρ^{2}} + 2 (\frac{s_{⊥}}{ρ} + \frac{∂s_{⊥}}{∂ρ} ) v_{β}, \\ \frac{∂d_{t}}{∂ϕ_{0}} & = Δ + v_{β} \frac{∂s_{⊥}}{∂ϕ_{0}} , \\ \frac{∂d_{t}}{∂D} & = u_{0β} + v_{β} \frac{∂s_{⊥}}{∂D} , \\ \frac{∂d_{t}}{∂λ} & = 0, \\ \frac{∂d_{t}}{∂z_{0}} & = 0. \end{aligned}

When ρ = 0, these simplify to

\begin{aligned}\frac{∂d_{t}}{∂c} & = u_{0β}s_{⊥}^{2} + 2v_{0β} \frac{∂s_{⊥}}{∂ρ} , \\ \frac{∂d_{t}}{∂ϕ_{0}} & = Δ + v_{0β} \frac{∂s_{⊥}}{∂ϕ_{0}} , \\ \frac{∂d_{t}}{∂D} & = u_{0β} + v_{0β} \frac{∂s_{⊥}}{∂D} , \\ \frac{∂d_{t}}{∂λ} & = 0, \\ \frac{∂d_{t}}{∂z_{0}} & = 0. \end{aligned}

2.5. Silicon barrel (r − z)

We use the same notation as for the silicon barrel in r − ϕ except that the quantity measured by the electronics (up to an offset) is z = d̂· (x⃗ − x⃗_{c}), where d̂ = (0, 0, 1). We must simultaneously solve the equation of the plane and the helix equations. This yields, as in the r − ϕ case (see the last section for notation)

\begin{aligned}0 & = ρx_{0β} + u_{0β} sin ρs_{⊥} − v_{0β}(1 − cos ρs_{⊥}), \\ z & = z_{0} + λs_{⊥}. \end{aligned}

Although it is possible to solve the first equation for s_{⊥}, which can be used to find z, it is simpler to use the formulas derived in the previous section for r−ϕ silicon measurements:

d_{t} = y_{0β} + \frac{1}{ρ}u_{0β} − \frac{1}{ρ}\sqrt{1 − (v_{0β} − ρx_{0β})^{2}},

= y_{0β} − \frac{2v_{0β}x_{0β} − ρx_{0β}^{2}}{u_{0β} + \sqrt{1 − (v_{0β} − ρx_{0β})^{2}}} ,

sin ρs_{⊥} = ρx_{0β}u_{0β} + ρ(y_{0β} − d_{t})v_{0β},
cos ρs_{⊥} = 1 − ρx_{0β}v_{0β} + ρ(y_{0β} − d_{t})u_{0β}.

where d_{t} is the distance along r − ϕ from the point where the normal, proceding from the origin, passes through the plane.

It is easy to express the partial derivatives in terms of ∂s_{⊥}/∂α_{i} defined in the last section:

\begin{aligned}\frac{∂z}{∂c} & = 2λ\frac{∂s_{⊥}}{∂ρ} , \\ \frac{∂z}{∂ϕ_{0}} & = λ\frac{∂s_{⊥}}{∂ϕ_{0}} , \\ \frac{∂z}{∂D} & = λ\frac{∂s_{⊥}}{∂D} , \\ \frac{∂z}{∂λ} & = s_{⊥}, \\ \frac{∂z}{∂z_{0}} & = 1. \end{aligned}

2.6. Silicon disk at fixed z position

This is a device placed at z = z_{D} measuring (up to an offset) a coordinate in the xy plane of the disk. As in the case of the silicon barrel, the equation of the plane is η⃗ · (x⃗ − x⃗_{c}) = 0, where x⃗_{c} = (0, 0, z_{D}). The quantity measured is x_{α} = d̂· (x⃗ − x⃗_{c}) where d̂= (cos α, sin α, 0) lies in the direction of increasing x_{α}. α = 0 (π/2) corresponds to x (y) measurements. We first define the rotated quantities

x_{α} = x cos α + y sin α,
y_{α} = y cos α − x sin α,
u_{α} = u cos α + v sin α,
v_{α} = v cos α − u sin α,

and similarly for x_{0α}, y_{0α}, u_{0α} and v_{0α}. From the helix equations we get s_{⊥} = (z_{D} − z_{0})/λ from which we compute

x_{α} = x_{0α} + \frac{u_{0α}}{ρ} sin ρs_{⊥} − \frac{v_{0α}}{ρ} (1 − cos ρs_{⊥}),

which reduces, when ρ = 0, to x_{α} = x_{0α} + u_{0α}s_{⊥}.

The derivatives for x_{α} are easily calculated:

\begin{aligned}\frac{∂x_{α}}{∂c} & = 2\frac{v_{0α} − v_{α} + ρs_{⊥}u_{α}}{ρ^{2}} , \\ \frac{∂x_{α}}{∂ϕ_{0}} & = −y_{α}, \\ \frac{∂x_{α}}{∂D} & = −v_{0α}, \\ \frac{∂x_{α}}{∂λ} & = −\frac{u_{α}}{λ} s_{⊥}, \\ \frac{∂x_{α}}{∂z_{0}} & = −\frac{u_{α}}{λ} . \end{aligned}

When ρ = 0, these derivatives become

\begin{aligned}\frac{∂x_{α}}{∂c} & = −s_{⊥}^{2}v_{0α}, \\ \frac{∂x_{α}}{∂ϕ_{0}} & = −y_{α}, \\ \frac{∂x_{α}}{∂D} & = −v_{0α}, \\ \frac{∂x_{α}}{∂λ} & = −\frac{u_{0α}}{λ} s_{⊥}, \\ \frac{∂x_{α}}{∂z_{0}} & = −\frac{u_{0α}}{λ} . \end{aligned}

3. Projections of helix including errors

It is frequently necessary to project a track to a particular wire or detector, for example, to decide if a measurement should be added to the fit or just to provide a reference impact point for a calorimeter. An important consideration is the errors of the projection. In this section I will derive the errors for the following kinds of track projections: (1) axial and stereo drift chamber wire, (2) cathode strip, (3) silicon barrel, (4) silicon disk, and (5) surface of sphere of radius r.

As before, α represents the five track parameters, V_{α} is the 5×5 covariance matrix and A represents the derivatives of the projected quantity with respect to the helix parameters. V_{d} is the covariance matrix of the projected quantities. A is either a 1 × 5 or 2 × 5 matrix, depending on the number of projected quantities. Calling generically the two projected quantities d_{1} and d_{2} (e.g., x and y), we get

A^{T} = ( \begin{matrix}∂d_{1}/∂c & ∂d_{2}/∂c \\ ∂d_{1}/∂ϕ_{0} & ∂d_{2}/∂ϕ_{0} \\ ∂d_{1}/∂D & ∂d_{2}/∂D \\ ∂d_{1}/∂λ & ∂d_{2}/∂λ \\ ∂d_{1}/∂z_{0} & ∂d_{2}/∂z_{0} \end{matrix} .)

The covariance matrix of d_{1} and d_{2} is then given by V_{d} = AV_{α}A^{T}.

The discussion is considerably simplified since most of the derivatives in A have already been calculated in Section 2.1–2.6. The only case we have not considered is projecting to a cylindrical surface of radius r.

3.1. Previous computations

1. Axial and stereo drift chamber wire (d_{w}, z). d_{w} and its derivatives were calculated for axial and stereo layers in Section 2.1 and 2.2 while the corresponding values for z were computed in Section 2.3.

2. Cathode strip (z). z and its derivatives were calculated in Section 2.3.

3. Silicon barrel (d_{t}, z). These quantities and their derivatives were calculated in Section 2.4 and 2.5.

4. Silicon disk (x, y). These quantities and their derivatives were calculated in Section 2.6.

3.2. Projecting to a cylindrical surface of radius r

The quantities we wish to calculate are ϕ and z, but the errors we want are δd_{t} = r sin θδϕ and δz, i.e., the errors in the r − ϕ and z distances. We exploit the formulas of Section 1.3 which express the helix coordinates as a function of r:

sin(ϕ − ϕ_{0}) = \frac{rc + (D/r)(1 + cD)}{1 + 2cD} = \frac{D}{r} + \frac{(c/r)(r^{2} − D^{2})}{1 + 2cD},
cos(ϕ − ϕ_{0}) = ϵ\frac{\sqrt{(1 − D^{2}/r^{2})[(1 + cD)^{2} − r^{2}c^{2}]}}{1 + 2cD},
z − z_{0} = λs_{⊥},
sin cs_{⊥} = c\sqrt{\frac{r^{2} − D^{2}}{1 + 2cD}},
s_{⊥} = \{ \begin{matrix}\frac{1}{c} sin^{−1} B & for ϵ = +1 \\ \frac{1}{c}(π − sin^{−1} B) & for ϵ = −1 \end{matrix}

where ϵ = +1(−1) corresponds to the outgoing (incoming) branch of the helix.

The derivative matrix (transposed) is

A^{T} = ( \begin{matrix}\frac{sin θ}{cos(ϕ−ϕ_{0})} \frac{r^{2}−D^{2}}{(1+2cD)^{2}} & \frac{1+cD}{1+2cD} \frac{sin cs_{⊥}}{c^{2}} − \frac{s_{⊥}}{c} \\ r sin θ & 0 \\ \frac{sin θ}{cos(ϕ−ϕ_{0})} \frac{1+2cD−2c^{2}(r^{2}−D^{2})}{(1+2cD)^{2}} & −\frac{sin cs_{⊥}}{c} (\frac{D}{r^{2}−D^{2}} + \frac{c}{1+2cD}) \\ 0 & s_{⊥} \\ 0 & 1 \end{matrix} )

which reduces, when c = ρ = 0, to

A^{T} = ( \begin{matrix}\frac{sin θ}{cos(ϕ−ϕ_{0})}(r^{2} − D^{2}) & −D\sqrt{r^{2} − D^{2}} \\ r sin θ & 0 \\ \frac{sin θ}{cos(ϕ−ϕ_{0})} & −\frac{D}{\sqrt{r^{2}−D^{2}}} \\ 0 & s_{⊥} \\ 0 & 1 \end{matrix} )

The covariance matrix for (δd_{t}, δz) is V_{d} = AV_{α}A^{T}.

4. Detector Misalignments and Track Fitting

Fitting with real detectors is complicated by the fact that they often have built in misalignments, causing the track to be projected to an incorrect position which in turn pulls the fitted parameters from their true values. The problem is often arises when several detectors are used together to fit a single track. If the misalignments are small enough to be approximated by linear parameters they can be incorporated very easily into the fit. Moreover, by fitting many hundreds or thousands of tracks, the parameters describing the misalignments can be extracted analytically from the pattern of residuals using a least squares technique. This method for determining misalignment constants was described in CBX 91–73.^{1}

We parametrize the misalignment parameters as three offsets along the coordinate axes, d_{x}, d_{y}, d_{z}, and three angles about these axes θ_{x}, θ_{y}, θ_{z}. The parameters are assumed to be small enough so that terms of second order, e.g., d_{x}θ_{y}, can be neglected. In this approximation the order in which the displacements and rotations are applied does not matter. A point (x, y, z) is shifted by the amounts

\begin{aligned}δx & = d_{x} + zθ_{y} − yθ_{z}, \\ δy & = d_{y} + xθ_{z} − zθ_{x}, \\ δz & = d_{z} + yθ_{x} − xθ_{y}. \end{aligned}

Each of these shifts has a linear effect on the measurement that can be expressed in terms of a matrix B (following the notation introduced in CBX 91–73). We express δy_{m}, the shift in the measured quantity (detector dependent), as

δy_{m} = B ( \begin{matrix}d_{x} \\ d_{y} \\ d_{z} \\ θ_{x} \\ θ_{y} \\ θ_{z} \end{matrix} )

where B must be determined separately for each detector as discussed below.

4.1. Axial drift chamber wire

The drift distance d_{w} to the wire is given by

d_{w} = \frac{1}{ρ} [\sqrt{1 + 2ρΔ_{w}} − 1] = \frac{2Δ_{w}}{1 + \sqrt{1 + 2ρΔ_{w}}} ≃ Δ_{w}(1 − \frac{1}{2}ρΔ_{w})

where Δ_{w} = u_{0}Δ_{y} − v_{0}Δ_{x} + \frac{1}{2}ρ(Δ_{x}^{2} + Δ_{y}^{2}), Δ_{x} = x_{0} − x_{w} and Δ_{y} = y_{0} − y_{w}. One can add the corrections directly to the wire position or add them after the fact with the B matrix. B is computed as

B^{T} = ( \begin{matrix}v_{w} \\ −u_{w} \\ 0 \\ zu_{w} \\ zv_{w} \\ −(x_{w}u_{w} + y_{w}v_{w}) \end{matrix} ) (1 − ρΔ_{w}).

where u_{w} = u_{0} + ρΔ_{y} and v_{w} = v_{0} − ρΔ_{x}.

4.2. Stereo drift chamber wire

As in the axial drift chamber case, one can apply the misalignment parameters directly to the wire position rather than add them after the fact as a linear correction. The first approach is complicated slightly by the fact that the direction cosines of the wire must be rotated.

To calculate the effect of the misalignments, we write down the equation of the drift distance d_{w} from Section 2.2

d_{w} ≃ \frac{u_{d}}{J} Δ_{w}(1 − \frac{1}{2}(1 − λ^{′2})ρΔ_{w}/J^{2}),

where Δ_{w} = u_{0}Δ_{y} − v_{0}Δ_{x} + \frac{1}{2}ρ(Δ_{x}^{2} + Δ_{y}^{2}), Δ_{x} = x_{0} − x_{w} − z_{p} tan τ sin ϕ_{s} and Δ_{y} = y_{0} − y_{w} + z_{p} tan τ cos ϕ_{s}. Applying the displacements and rotations of the misalignments allows B to be computed as follows

B^{T} = ( \begin{matrix}v_{w} \\ −u_{w} \\ − tan τ(u_{w} cos ϕ_{s} + v_{w} sin ϕ_{s}) \\ z_{p}u_{w} \\ z_{p}v_{w} \\ −(x_{w} + z_{p} tan τ cos ϕ_{s})u_{w} − (y_{w} − z_{p} tan τ sin ϕ_{s})v_{w} \end{matrix} ) \frac{u_{d}}{J} (1 − u_{d}ρΔ_{w}/J^{2}),

where u_{w} = u_{0} + ρΔ_{y} and v_{w} = v_{0} − ρΔ_{x} (see Section 2.2 for notation).

4.3. Cathode strips

Here the misalignment parameters are applied to the anode wire nearest to the cathode strips. The relevant equation of motion is z = z_{0} +λs_{⊥}, where s_{⊥} satisfies (see Section 2.3)

(1 + ρd_{w}) cos ρs_{⊥} = 1 − ρv_{0}Δ_{x} + ρu_{0}Δ_{y},
(1 + ρd_{w}) sin ρs_{⊥} = −ρu_{0}Δ_{x} − ρv_{0}Δ_{y},

where Δ_{x} = x_{0} − x_{w} and Δ_{y} = y_{0} − y_{w}, with (x_{w}, y_{w}) being the coordinate of the wire. As in the axial wire case it is probably simpler to add the corrections directly to the anode wire coordinates and then compute s_{⊥} and z. Nevertheless, we can add the corrections after the fact using B. First we eliminate d_{w} from the equations above and get an equation in s_{⊥} only

sin ρs_{⊥}(1 − ρv_{0}Δ_{x} + ρu_{0}Δ_{y}) + cos ρs_{⊥}(ρu_{0}Δ_{x} + ρv_{0}Δ_{y}) = 0.

Applying small changes to the axial wire (see the discussion above) allows the derivatives of s_{⊥} to be computed relative to each of the misalignment parameters, yielding

B^{T} = ( \begin{matrix}−λu/(1 + ρd_{w}) \\ −λv/(1 + ρd_{w}) \\ −1 \\ λzv/(1 + ρd_{w}) − y_{w} \\ λzu/(1 + ρd_{w}) + x_{w} \\ −λ(vx_{w} − uy_{w})/(1 + ρd_{w}) \end{matrix} ) (1 − ρΔ_{w}),

where u = u_{0} cos ρs_{⊥} − v_{0} sin ρs_{⊥}, v = v_{0} cos ρs_{⊥} + u_{0} sin ρs_{⊥} and d_{w} was computed in Section 2.1.

4.4. Silicon barrel

This case is a little more complicated than the previous ones because it is not as straightforward to apply the misalignments directly to the measurements. Misalignments are also more important for silicon because the induced changes in the measurements are of the same order of magnitude as the errors. From Section 2.4 the equation of the plane is η⃗ · (x⃗ − x⃗_{c}) = 0 or x cos β + y sin β − Δ = 0, where η⃗ = (cos β, sin β, 0) represents the direction cosines of the outward normal to the plane and x⃗_{c} is an arbitrary point on the plane which we choose to be the point where the normal, passing through the origin, strikes the plane. Δ = η⃗ · x⃗_{c} is the perpendicular distance from the plane to the origin.

We first consider the case where the measured quantity is the “transverse distance” in the r − ϕ plane defined by d_{t} = d̂ · (x⃗ − x⃗_{c}), where x⃗ is the point where the tracks hits the plane and d̂ is a unit vector lying along the direction of increasing d_{t}. For r − ϕ measurements d̂= (cos β, sin β, 0).

The effect of misalignments can be computed straightforwardly from the equations for d_{t} and the plane by applying the displacements and rotations to η⃗, d̂ and x⃗_{c} and including the change in s_{⊥}. Applying the misalignments to the quantities defining the plane yields

η′ = ( \begin{matrix}cos β \\ sin β \\ 0 \end{matrix} ) + ( \begin{matrix}−θ_{z} sin β \\ θ_{z} cos β \\ θ_{x} sin β − θ_{y} cos β \end{matrix} ) ,

d̂′ = ( \begin{matrix}− sin β \\ cos β \\ 0 \end{matrix} ) + ( \begin{matrix}−θ_{z} cos β \\ −θ_{z} sin β \\ θ_{x} cos β + θ_{y} sin β \end{matrix} ) ,

x′_{c} = ( \begin{matrix}Δ cos β \\ Δ sin β \\ 0 \end{matrix} ) + ( \begin{matrix}d_{x} − Δθ_{z} sin β \\ d_{y} + Δθ_{z} cos β \\ d_{z} + Δ(θ_{x} sin β − θ_{y} cos β) \end{matrix} ) .

The equation of the plane is modified to

x cos β + y sin β − Δ = d_{x} cos β + d_{y} sin β − (y cos β − x sin β)θ_{z} + z(θ_{y} cos β − θ_{x} sin β)

where (x, y, z) is the point where the track hits the plane. Since x, y and z are calculable in terms of s_{⊥} and the quantities on the right are small, we can expand the left side about the solution for the perfectly aligned case and obtain δs_{⊥}, the change in s_{⊥}. The solution is (see Section 2.4 for notation)

δs_{⊥} = \frac{d_{x} cos β + d_{y} sin β − (y cos β − x sin β)θ_{z} + z(θ_{y} cos β − θ_{x} sin β)}{u_{β}}.

To find the new measurement d′_{t} = d̂′ · (x⃗′ − x⃗′_{c}) we apply the equations above and get

d′_{t} = d_{t} + v_{β}δs_{⊥} − d_{y} cos β + d_{x} sin β + (x cos β + y sin β)θ_{z} + z(θ_{x} cos β + θ_{y} sin β),

from which we obtain B_{r−ϕ}

B_{r−ϕ}^{T} = ( \begin{matrix}(v_{β} cos β + u_{β} sin β)/u_{β} \\ −(u_{β} cos β − v_{β} sin β)/u_{β} \\ 0 \\ z(u_{β} cos β − v_{β} sin β)/u_{β} \\ z(v_{β} cos β + u_{β} sin β)/u_{β} \\ x(u_{β} cos β + v_{β} sin β)/u_{β} − y(v_{β} cos β − u_{β} sin β)/u_{β} \end{matrix} ) = ( \begin{matrix}v/u_{β} \\ −u/u_{β} \\ 0 \\ zu/u_{β} \\ zv/u_{β} \\ (xu − yv)/u_{β} \end{matrix} )

The misalignments can also be expressed in terms of shifts and rotations local to the silicon plane. The advantage is that the displacements decouple from the rotations (since the axis of rotation is now the center of the silicon) and the formulas for the measurement shifts are more compact and understandable. To begin, imagine the silicon plane rotated about the z axis by −β and then shifted so that its center is at (0, 0, 0) and its normal is (1, 0, 0). The local x_{β}, y_{β} and z_{β} axes are, respectively, along the silicon thickness, width and length. They are related to the global coordinates by

x_{β} = x cos β + y sin β − Δ
y_{β} = y cos β − x sin β
z_{β} = z
u_{β} = u cos β + v sin β
v_{β} = v cos β − u sin β

The three displacements along the local axes are

d_{xβ} = d_{x} cos β + d_{y} sin β
d_{yβ} = d_{y} cos β − d_{x} sin β + Δθ_{z}
d_{zβ} = d_{z}

while the three rotations can be written

θ_{xβ} = θ_{x} cos β + θ_{y} sin β
θ_{yβ} = θ_{y} cos β − θ_{x} sin β
θ_{zβ} = θ_{z}

With these definitions the equation for the plane becomes x_{β} = 0 and we get the simpified equations

\begin{aligned}δs_{⊥} & = (d_{xβ} − y_{β}θ_{zβ} + z_{β}θ_{yβ})/u_{β}, \\ d′_{t} & = d_{t} + v_{β}δs_{⊥} − d_{yβ} + z_{β}θ_{xβ}. \end{aligned}

For measurements along z the r − ϕ discussion applies except that the measured quantity is z = d̂· (x⃗ − x⃗_{c}) with d̂ = (0, 0, 1) When misalignments are applied, s_{⊥} changes as described above while d̂ moves to

d̂′ = ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) + ( \begin{matrix}−θ_{y} \\ θ_{x} \\ 0 \end{matrix} ) ,

so that the z measurement changes to

\begin{aligned}z′ & = z + λδs_{⊥} − d_{z} − xθ_{y} + yθ_{x} \\ & = z_{β} + λδs_{⊥} − d_{zβ} + Δθ_{yβ} − y_{β}θ_{xβ} \end{aligned}

which defines B_{z}

B_{z}^{T} = ( \begin{matrix}(λ/u_{β}) cos β \\ (λ/u_{β}) sin β \\ −1 \\ −z(λ/u_{β}) sin β + y \\ z(λ/u_{β}) cos β − x \\ (λ/u_{β})(x sin β − y cos β) \end{matrix} )

4.5. Silicon disk at fixed z position

This case is fairly easy since the disk sits at z = z_{D} in the xy plane (see Section 2.6). The equation of the plane is η⃗ · (x⃗ − x⃗_{c}) = 0, where η⃗ = (0, 0, 1) is the outward normal and x⃗_{c} = (0, 0, z_{D}). The quantity measured is x_{α} = d̂· (x⃗ − x⃗_{c}) where d̂ = (cos α, sin α, 0) lies in the direction of increasing x_{α}. Under the effect of misalignments, η⃗, d̂ and x⃗_{c} become

η′ = ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ) + ( \begin{matrix}−θ_{y} \\ θ_{x} \\ 0 \end{matrix} ) ,

d̂′ = ( \begin{matrix}cos α \\ sin α \\ 0 \end{matrix} ) + ( \begin{matrix}−θ_{z} sin α \\ −θ_{z} cos α \\ θ_{x} sin α − θ_{y} cos α \end{matrix} ) ,

x′_{c} = ( \begin{matrix}0 \\ 0 \\ z_{D} \end{matrix} ) + ( \begin{matrix}d_{x} + z_{D}θ_{y} \\ d_{y} − z_{D}θ_{z} \\ d_{z} \end{matrix} ) .

Since s_{⊥} = (z_{D} − z)/λ, the change in s_{⊥} from misalignments is δs_{⊥} = (d_{z} + yθ_{x} − xθ_{y})/λ, which yields the new value of x_{α}:

x′_{α} = x_{α} + u_{α}δs_{⊥} − d_{x} cos α − d_{y} sin α − z_{D}(θ_{y} cos α − θ_{x} sin α) + θ_{z}(y cos α − x sin α),

where u_{α} = u cos α + v sin α as defined in Section 2.6. These equations define B:

B^{T} = ( \begin{matrix}−d_{x} cos α \\ −d_{x} cos α \\ u_{α}/λ \\ z sin α + y(u_{α}/λ) \\ −z cos α − x(u_{α}/λ) \\ y cos α − x sin α \end{matrix} )

References

1. P. Avery, Applied Fitting Theory II: Determining Systematic Effects by Fitting, CBX 91–73.