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R. Frühwirth et al.
13.2.4 Kinematic Fitting
Kinematic fitting imposes physical constraints on the particles participating in an
interaction and thereby improves the measured track momenta and positions. At the
same time hypotheses about the interaction and the participating particles can be
tested.
13.2.4.1 Lagrange Multiplier Method
The most commonly used method of imposing constraints on the measured tracks is
by way of Lagrange multipliers [141]. Let ˜
q = ( ˜
q 1 , . . . , ˜
q n ) T be the unconstrained
estimated parameters of a set of n tracks, along with their joint information matrix
G = diag(G 1 , . . . , G n ) = V −1 . The r functions describing the constraints can be
written as g(q) = 0. Taylor expansion around a suitable point ˘
q yields the linearized
equation
˘
g + D(q − ˘
q) = 0,
where D is the Jacobian of g with respect to q, evaluated at ˘
q, and ˘
g = g( ˘
q). The
obvious expansion point is ˘
q = ˜
q. The constrained track parameters ¯
q i are obtained
by minimizing the objective function
M(q, λ) = (q − ˜
q)
T G(q − ˜
q) + 2λ
T
˘
g + D(q − ˘
q)
with respect to q and λ. λ is a vector of r unknowns, the Lagrange multipliers. The
solution is
¯
q = ˜
q − V D
T ¯
λ, with ¯
λ = G D
˘
g + D( ˜
q − ˘
q)
and G D = (DV D
T )
−1 .
The covariance matrix ¯
V and the χ 2 statistic are given by
¯
V = V − V D
T G D DV , χ
2
= ¯
λ
T G
−1
D
¯
λ = ¯
λ
T
˘
g + D( ˜
q − ˘
q)
.
If required, the constraint function g can be re-expanded at the new point ˘
q = ¯
q,
and the constrained track parameters can be recomputed.
The Jacobian D depends both on the parametrization of the tracks and on the
type of constraint to be imposed. For kinematic constraints it is often convenient
to choose a parametrization that uses physically meaningful quantities. In [142]
it is proposed to use the four-momentum and a point in space, i.e., q =
(p x , p y , p z , E, x, y, z). With this parametrization the following constraints can be
formulated in a straightforward manner (for further examples see [142]).
R. Frühwirth et al.
13.2.4 Kinematic Fitting
Kinematic fitting imposes physical constraints on the particles participating in an
interaction and thereby improves the measured track momenta and positions. At the
same time hypotheses about the interaction and the participating particles can be
tested.
13.2.4.1 Lagrange Multiplier Method
The most commonly used method of imposing constraints on the measured tracks is
by way of Lagrange multipliers [141]. Let ˜
q = ( ˜
q 1 , . . . , ˜
q n ) T be the unconstrained
estimated parameters of a set of n tracks, along with their joint information matrix
G = diag(G 1 , . . . , G n ) = V −1 . The r functions describing the constraints can be
written as g(q) = 0. Taylor expansion around a suitable point ˘
q yields the linearized
equation
˘
g + D(q − ˘
q) = 0,
where D is the Jacobian of g with respect to q, evaluated at ˘
q, and ˘
g = g( ˘
q). The
obvious expansion point is ˘
q = ˜
q. The constrained track parameters ¯
q i are obtained
by minimizing the objective function
M(q, λ) = (q − ˜
q)
T G(q − ˜
q) + 2λ
T
˘
g + D(q − ˘
q)
with respect to q and λ. λ is a vector of r unknowns, the Lagrange multipliers. The
solution is
¯
q = ˜
q − V D
T ¯
λ, with ¯
λ = G D
˘
g + D( ˜
q − ˘
q)
and G D = (DV D
T )
−1 .
The covariance matrix ¯
V and the χ 2 statistic are given by
¯
V = V − V D
T G D DV , χ
2
= ¯
λ
T G
−1
D
¯
λ = ¯
λ
T
˘
g + D( ˜
q − ˘
q)
.
If required, the constraint function g can be re-expanded at the new point ˘
q = ¯
q,
and the constrained track parameters can be recomputed.
The Jacobian D depends both on the parametrization of the tracks and on the
type of constraint to be imposed. For kinematic constraints it is often convenient
to choose a parametrization that uses physically meaningful quantities. In [142]
it is proposed to use the four-momentum and a point in space, i.e., q =
(p x , p y , p z , E, x, y, z). With this parametrization the following constraints can be
formulated in a straightforward manner (for further examples see [142]).