TheB Factory experiments have deployed several imple- mentations of vertex fits. A complete description of these algorithms is outside the scope of this book. In the fol- lowing we sketch the formalism of a generic minimumχ2
vertex algorithm. A pedagogical introduction to vertex fit- ting can be found in the lectures by P. Avery (Avery, 1991, 1998).
To start, we consider a collection ofN charged tracks and use a χ2 minimization algorithm to determine the
best vertex out of which they emerge. Once that is done, the vertex can be improved by adding neutral particles, enforcing mass constraints to the in-going or some of the outgoing composite particles, and requiring consistency of the vertex location with the collider luminous region. The goodness of a fit is measured by testing the compatibility of the minimum χ2 with the expected probability distri-
bution of a χ2 with the relevant number of degrees of
freedom.
Following the notation in (Fruhwirth, 1987) we denote the reconstructed helix parameters of track i by pi and
the corresponding covariance matrix byVi. Given a set of
N outgoing tracks each labeled with an indexi, theχ2of
the vertex can be generically written as
χ2 = N i=1 [pi−hi(x,qi)] T V−1 i [pi−hi(x,qi)] (6.3.1)
wherex is a 3D vector representing the fitted vertex po- sition,qi is the fitted momentum vector of the outgoing
track and hi, the measurement model, is a function ofx
and qi that expresses the parameters of the helical tra-
jectory of the charged particle emerging from the vertex with momentumqi.
The solution to the vertex fit is the set of parameters
ξ≡(x,q1. . .qN) that minimizes theχ2. In case the func-
tionhi is linear in the parametersξ, the solution can be
expressed generically as ξ = ξ0− d2χ2 dξ2 (ξ0) −1 dχ2 dξ (ξ0) (6.3.2)
whereξ0 is an arbitrary starting point forξ. The inverse
of the second derivative matrix on the right hand side is also half the covariance matrix forξ. If the derivative ofhi
is denoted byHi, this leads to the well known expression
for the linear least squares estimator,
ξ = ξ0−C i HTiV− 1 i [pi−hi(x,qi)] (6.3.3)
with the covariance matrix C = i HTiVi−1Hi −1 . (6.3.4) For vertex fits to helix trajectories the functionhi is not
linear and hence its derivative Hi not constant. In that
case the minimum is obtained by starting from a suitable expansion point ξ0 and iteratively applying Eq. (6.3.2)
until a certain convergence criterion is met, usually a min- imum change in theχ2.
There are two flavors of measurement models for tracks in vertex fits: If the parameters pi are helix parame-
ters, the measurement model is given by the inverse of Eq. (6.2.1) and Eq. (6.2.2) above. Alternatively, the track parameters can also be translated into position and mo- mentum space using Eq. (6.2.1) and Eq. (6.2.2). In this case the measurement model is trivial, but has one dimen- sion more than the original five parameter helix. Further- more, since the transformation only applies to a particular point on the helix, it needs to be repeated if the vertex position estimate changes between iterations.
The number of degrees of freedom of the computedχ2
isNDOF ≡2N−3, i.e.the difference between the num-
ber of measurements, 5N (5 helix parameters per track) and the number of fitted parameters 3(N+ 1) (3 vertex coordinates and 3 momentum components per track). As- suming that the uncertainties on the track parameters are correctly estimatedi.e.that they are representative of the RMS of the error distribution, the minimum χ2 follows
the probability distribution of a χ2 variate with N
DOF
degrees of freedom whose expectation value equalsNDOF.
A goodness of fit requirement is usually derived from χ2
andNDOF to retain the acceptableN-prong verticese.g.
in the selection of event data samples.
The vertex fitting formalism can be extended with ad- ditional constraints, such as prior knowledge of the vertex position (for example from knowledge of the interaction point, IP) or the known mass of the decaying particle. Such constraints always take the form of a constraint equa- tion
f(ξ) = 0. (6.3.5) A distinction can be made between exact constraints and constraints that have an associated uncertainty. The latter are sometimes called ‘χ2constraints’. Mass constraints are
usually (but not always) implemented as exact constraints while IP constraints are an example of aχ2constraint. Ex-
act constraints can be implemented by using a Lagrange multiplier. They add a term to the χ2
Δχ2 = λf(
ξ) (6.3.6)
where the Lagrange multiplier λ is treated as an addi- tional parameter in the vertex fit. An alternative (more efficient) method to deal with exact constraints is dis- cussed in (Hulsbergen, 2005). For one-dimensional con- straints with an uncertaintyσtheχ2contribution is
Δχ2 = f(ξ)2
σ2 . (6.3.7)
This expression can be generalized to more than one di- mension by writing it in a matrix notation. Note that each independent constraint adds one degree of freedom to the χ2.
The vertex fit can also be extended to include re- constructed neutral particles. Photons reconstructed as calorimeter clusters do not add position information to the vertex, but they contribute to the momentum, and affect theχ2minimization once mass constraints are ap-
plied.
Several vertex fits are implemented in sequence to re- construct decay trees that involve more than one decay vertex, e.g. B → DX transitions. Such decay trees are usually reconstructed by starting from the most down- stream vertex and working towards the mother of the de- cay trees: first fit the Dvertex, then use the result to fit the B (this approach is sometimes called leaf-by-leaf fit- ting). Other more global associations of constraints are implemented for decay trees with leaves or branches with many neutral particles (Hulsbergen,2005).
The vertex fits applied in theB Factory experiments are essentially extensions of the scheme above – see in par- ticular (Tanaka,2001) for Belle and (Hulsbergen,2005) for
BABAR. Implementations of the vertex fitting algorithm differ both in the parameterization of the problem and in the way the χ2 is minimized. As outlined above, tracks
can be parameterized in terms of helix coordinates or (lo- cally) in terms ofCartesian coordinates. The latter leads to simpler expressions for derivatives, but may lead to slower convergence because derivatives vary more rapidly along the track.
For the minimization both the globalχ2fit technique
described above and the Kalman filter are used. Even for algorithms that seemingly use the same minimization scheme, the implementations may differ. To our knowl- edge, the most efficient method to fit tracks to a common vertex is the algorithm developed by Billoir, Fruhwirth, and Regler (1985), presented in slightly different from in (Fruhwirth, 1987). This algorithm was extended with a mass constraint in (Amoraal et al.,2013).
Not all algorithms are applicable to all vertexing prob- lems. The general leaf-by-leaf approach for decay tree fit- ting cannot easily be applied to the reconstruction ofe.g. K0
S → π
0π0 or B0 → K0
Sπ
0. For these types of decay
trees a ‘global’ decay tree fit can be used (Hulsbergen,
f1 =0.76±0.02 μ1 = 1.0±0.5 σ1 =41.4±0.9 μ2 = 2.9±2.4 σ2 = 118±6 f1 =0.84±0.03 μ1 =0.01±0.01 σ1 =0.97±0.02 μ2 =0.21±0.06 σ2 =1.96±0.10 Δz residual (cm) even ts Δz pull even ts
Figure 6.3.1. Residual (left) and pull (right) of the decay vertexzposition of reconstructedB0→J/ψ K0Scandidates in
aBABARsimulated data sample. Fits to a double Gaussian are superimposed.
2005). The latter also has the advantage that one has ac- cess to the vertex-constrained parameters of all particles in the decay tree. However, this algorithm computes a single covariance matrix for all of the parameters in the decay tree, making it noticeably slower than a leaf-by-leaf approach. TheCPU consumption of vertex algorithms is often a concern because of the combinatoric background in the reconstruction and selection of composite particles. A strict control on the accuracy of the vertex recon- struction is mandatory for the B Factory experiments where the primary goal is to determine time-dependent CP asymmetries from the distance between two vertices. This is illustrated in Figure 6.3.1 which shows the resid- uals and pull20 for the decay vertex z position of recon-
structed B0 → J/ψ K0
S (J/ψ → μ
+μ−) candidates from
a sample of simulated data taken fromBABAR. The ver- tex resolution depends on the topology of the decay and the direction and momenta of the final state particles and especially on whether the K0
S particles decays inside or
outside the vertex detector volume. These effects are ac- counted for in the per-event reconstruction uncertainty, the estimate of which is computed by the vertex fit al- gorithm. Due to spread in the estimated uncertainty, the vertex resolution is not a Gaussian distribution. However, the pull distribution is reasonably Gaussian with an RMS value close to unity, indicating that the uncertainties are correctly estimated.
For this decay thezresidual distribution has an RMS of about 70μm. A double Gaussian fit returns a core com- ponent, which corresponds to about three quarters of the distribution, with a standard deviation equal to 40μm. The resolution in the transverse coordinates is compara- ble to that inz: about 50μm.
Figure 6.3.2 shows the reconstructed mass of B± → J/ψ K±decays in data, fromBABAR, fitted both with and without a mass constraint on the J/ψ → μ+μ− decay.
The mass constraint improves the accuracy of the derived J/ψ momentum and this leads to a large improvement in theB±invariant mass resolution. The improvement in mass resolution is comparable to what one would obtain
20
A ‘pull’ is a residual divided by its estimated uncertainty. See also Section 11.5.2.
) 2 mass (GeV/c ± B 5.2 5.25 5.3 5.35 5.4 ) 2 Events/(3 MeV/c 0 200 400 600 800 1000 1200 1400 no constraint mass con. ψ J/
Figure 6.3.2.Distribution of the reconstructed invariant mass ofB±→J/ψ K±decays in BABARdata with and without en- forcement of a mass constraint on the J/ψ → μ+
μ− decay vertex leaf.
by considering theB±-J/ψ mass difference instead of the B± mass. However, the advantage of applying the mass- constrained vertex fit is that the resolution on both the vertex position and on theB momentum are improved.