HCAL ECAL
4.2 Event Vertices
transverse to the beam line (x-y), and z0 as the distance along the beam line (z- axis) from the closest pixel primary vertex, if present, or the beam spot, otherwise.
δd0 and δz0 come from the covariance matrix of the track fit, while σd0 and σz0 are parametrized as functions of the trackpT. Tracks that fail the selection are discarded, and their associated hits are reused in the next iteration.
In each iteration, the CTF repeats the above four steps: the seed generation, the track search, the track fit, and the track selection. It performs six iteration passes in total. Individual CTF iterations differ mainly in the requirements on the seed- generation hits, and the criteria on the track selection, as well as the collection of available hits that contains only those that have not been associated with any tracks yet.
In addition to the general CMS tracking, the same algorithm with different con- figurations is used to produce other specialized track collections. These include the tracks reconstructed at the High Level Trigger (HLT), electron tracks and tracks in the muon systems. The HLT configuration emphasizes speed over precision. For example, it stops the track finding sooner, when enough hits have been found. We discuss the specialized muon and electron tracking below in the respective Sections 4.7 and 4.6.
to the beam spot, the number of pixel and strip hits, and the value of the normalized χ2 of their track fit.
In 2010, the number of pileup interactions per bunch crossing was relatively low. A simple gap algorithm [99] was sufficient to assign tracks to vertices since the individual vertices were relatively well separated along the beam direction (z-axis) compared to the longitudinal impact parameter uncertainties of the tracks. The tracks were sorted according to thez-coordinates of the points of their closest approach to the beam spot.
When a distance in thez-coordinates between any two closest tracks reached a value greater than a given large gap, it defined a boundary separating the tracks into groups belonging to different vertices.
Thanks to the great performance of the LHC in delivering higher instantaneous luminosities, the number of pileup interactions increased significantly in 2011. This required the introduction of a different, more sophisticated, algorithm for the track clustering: deterministic annealing (DA) [100]. Similarly to the gap algorithm, it is based on a clustering according to the z-coordinate of the points of their closest approach to the nominal beam spot position, as defined in the next section. It is inspired by an analogy to statistical mechanics. The collection of tracks is treated as a thermodynamical system. The assignment of the tracks to vertices is not unique.
It is ‘soft’ with each trackiat zTi with an uncertaintyσi having a certain probability pik of originating from a given vertex k at zkV. The χ2 pull of the distance of the track i from the vertex k along the z-axis plays the role of the potential energy Eik = (zTi −zkV)2/σi2. The vertex locationszkVand weightswkat a given “temperature”
T are given by minimizing what is the equivalent of the system’s free energy in statistical thermodynamics (using natural units in which the Boltzmann constant is k = 1). It is defined as [78]:
F =−T X
i
pilogX
k
wke−Eik/T, (4.1)
where the pi are the track weights reflecting their compatibility with the beam spot and the wk are the vertex weights. The track-to-vertex assignment probabilities pik
are then given by:
pik = wke−Eik/T P
lwle−Eil/T. (4.2)
The track-to-vertex assignments are “soft” in the sense that a track may be compatible with multiple vertices. They become “hard” in the limit T →0 at which each track corresponds to exactly one vertex ziT=zkV and there is as many vertices as tracks.
The DA algorithm starts at a very highT with a single vertex prototypek = 1 and equal track assignments pik = 1 all compatible with the vertex. As the temperature decreases, the minimum of F (4.1) reaches a saddle point at a critical temperature given by:
Tck= 2 P
i pipik
σ2i Eik
P
i pipik
σ2i . (4.3)
At that temperature, the single vertex is split, a new vertex prototype is added and the vertex positions zkV and vertex weights ρk are re-optimized.
The starting temperature is chosen to be above the first critical temperature calculated for pik = ρk = 1. It is decreased at every step by a cooling factor of 0.6 until a minimal temperature of Tmin = 4 is reached. This value of minimal temperature corresponds to an optimal compromise between the power to resolve two nearby vertices and the danger of accidentally splitting vertices due to outliers in the measurements of the track ziT.
To assign the tracks uniquely to vertices, the annealing is then continued down to T = 1 without further vertex splitting. During this cool-down, an outlier rejection is used to reduce the bias from tracks that are displaced by more than 4-σ from the nearest vertex candidate. This completes the clustering of the selected tracks by the DA algorithm.
Finally, an adaptive vertex fitter [101] is used to estimate the vertex parameters for vertex prototypes with at least two tracks. These include:
• the vertex position in all three dimensions,
• the vertex fit covariance matrix,
• the weights ωi ∈[0, 1] of the associated tracks describing their statistical com- patibility with the vertex (0 meaning incompatible, 1 meaning perfectly com- patible), and
• the number of degrees of freedom defined as:
ndof =−3 + 2
nXtracks
i
ωi, (4.4)
where the sum runs over all tracks assigned to the vertex and ωi is the weight of the ith track.
The number of degrees of freedom is a good estimator of the total number of tracks compatible with the interaction region and is therefore a useful quantity to identify events with real proton-proton inelastic collisions.