1 Introduction

A vertex is called a secondary vertex or displaced vertex if it is outside the beam profile in a collider experiment or outside the target region in a fixed-target experiment. Secondary vertices arise in the following contexts:

Decays of unstable particles

The decaying particle is call the “mother” particle, the decay products are called the “daughter” particles. Depending on the lifetime and the momentum of the mother particle, the secondary decay vertex is displaced by a certain distance from its production point, which can be either a primary or itself a secondary vertex. The properties of the mother particles have to be inferred from the decay products. Finding decays is important for many types of physics analyses as well as for momentum scale calibration, by comparing the reconstructed mass of the mother particle to the known one.

Interactions in the detector material

Finding interaction vertices can be useful for mapping the amount and position of material in the detector. Two types of interactions are important in practice: photon conversions, i.e., pair production of electrons and positrons or muons in the electric field of a nucleus; and hadronic interactions of primary particles. A decay of a charged particles into a single charged daughter particle plus some neutral daughters looks just like a kink in the track. Kink finding is the subject of Sect. 6.4.3. The following subsections present some methods for finding decay and interaction vertices.

2 Decays of Short-Lived Particles

In the present context, short-lived particles are defined as particles that decay before they enter the first layer of the innermost tracking device. This includes B (beauty) and D (charmed) hadrons as well as τ leptons, which travel no more than a few millimeters before they decay.

The key to successful secondary vertex reconstruction is a precise knowledge of the primary vertex and the correct selection of the tracks of the decay products. In order to make sure that the primary vertex is not contaminated with secondary tracks, fake tracks or—at the LHC—pile-up tracks , the fit has to be made robust, either by removing outliers (Sect. 8.2.3), or by employing a robust fitting algorithm in the first place (Sects. 8.2.1 and 8.2.2). If there is prior information on the beam spot, it should be included into the primary vertex fit. The tracks removed from the primary vertex or down-weighted as outliers are by definition candidates for secondary tracks; see Sect. 7.3.3.

The selection of the secondary tracks depends on the physics goals of the analysis. If a specific decay channel is considered, the tracks can be selected according to kinematic criteria, for instance their type, charge, and momentum [1]. In addition, the distance of the track from the primary vertex in the transverse plane, called the transverse impact parameter d 0, can be used to verify that the selected tracks are not produced at the primary vertex. As the uncertainty of d 0 depends on the angle and the momentum of the track, the test is based on the standardized impact parameter s 0 = d 0∕σ[d 0], also called the significance of the impact parameter.

Secondary tracks can also be selected by their impact parameter alone. To this end, the impact parameter of a track is given a sign that is positive if the track intersects the trajectory of the decaying hadron or τ lepton downstream of the primary vertex [1,2,3]. As the trajectory is not known before the secondary vertex fit, it is approximated by the jet axis of the jet the decaying particle is embedded in. Only tracks with positive sign are candidates for secondary tracks.

The information contained in the impact parameter of a secondary track can be augmented by considering its functional relation with the azimuth angle of the track in the transverse projection, see Fig. 9.1. Let ϕ 1, …, ϕ m be the azimuth angles and d1 0, …, dm 0 the transverse impact parameters of m tracks emerging from the same secondary vertex. If the primary vertex is very close to the origin of the coordinate system, as it usually is, the following functional relation holds approximately:

$$\displaystyle \begin{gathered} {d^{\,0}}_i\approx\ell\hspace{0.5pt}\sin{}(\phi_i-\phi_{\mathrm{h}})\approx\ell\hspace{0.5pt}(\phi_i-\phi_{\mathrm{h}}),\ \; i=1,\ldots,m, \end{gathered} $$
(9.1)

where ℓ is the decay length of the decaying particle and ϕ h its azimuth angle. Thus in the (ϕ, d 0)-plane the points corresponding to secondary tracks lie on a straight line. This line can be found in various ways, for instance, by a Hough transform (Sect. 5.1.2), by histogramming the direction angles of segments connecting two points or by agglomerative clustering of the segments [3].

Fig. 9.1
figure 1

The functional relation between ϕ and d 0 of secondary tracks

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3 Decays of Long-Lived Particles

Long-lived particles such as neutral K-short mesons \(K^0_{\mathrm {S}}\) and Λ baryons decay in the tracker volume. Reconstruction of their decays serves as a powerful cross-check of the magnetic field map and the alignment, as their mass is very well known, and systematic errors in the field map as well as misalignment can lead to a bias in the momentum and invariant mass estimation. Candidates for the decay products can be identified by a large impact parameter and the fact that tracker hits are consistently missing up to a certain layer. In the search for 2-prong neutral decays, so-called V0s, pairs of tracks with opposite charge and small distance in space are combined to vertex candidates. Without particle identification, the Armenteros–Podolansky plot is a useful tool to decide between hypotheses about the mother particle from the kinematics of their decay products [4]. It is a scatter plot of the longitudinal momentum asymmetry versus the transverse momentum of the positive decay product, see Fig. 9.2. The figure shows that in some regions of the plot a unique decision may not be possible.

Fig. 9.2
figure 2

Armenteros–Podolansky plot for \(K^0_{\mathrm {S}}\) and \(\varLambda /\bar {\varLambda }\). (From [4], with permission by the author)

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The resolution of eventual ambiguities requires a kinematic fit with mass constraints (see Sect. 8.3) based on particle identification of the decay products. For example, in the case of \(K^0_{\mathrm {S}}\) versus \(\varLambda /\bar {\varLambda }\), charged pions have to be separated from protons.

If the decay vertices are allowed to have more than two charged outgoing particles, two-track vertices can serve as seed vertices [5] . A seed vertex is rejected if at least one of its tracks has hits between the seed vertex and the primary vertex. The seed vertices are then clustered to larger vertices by an agglomerative procedure.

4 Photon Conversions

The reconstruction of photon conversions is an integral part of photon reconstruction in general. While the energy of unconverted photons is measured directly in the electromagnetic calorimeter (ECAL), a converted photon is reconstructed from the charged lepton pair, mostly an e + e − pair, that is produced in the electric field of a nucleus. A characteristic feature of the conversion vertex is the fact that the two leptons are parallel to each other. As a consequence, the vertex position along the photon direction is not very well defined by the vertex fit alone. Two topical examples of the reconstruction of converted photons are presented in the following paragraphs.

In the ATLAS experiment (Sect. 1.6.1.2), the first step is the standard electron/positron reconstruction, briefly described in Sect. 10.2. Clusters in the ECAL are built and used to create regions of interest (ROIs). Inside an ROI track finding is modified such that up to 30% energy loss is allowed at each material layer. After loosely matching the tracks in the ROIs with the ECAL clusters, tracks with silicon hits are refitted with the Gaussian-sum filter (GSF, Sect. 6.2.3). Conversion finding is run on the loosely matched tracks that have a high probability to be electrons or positrons according to the transition radiation detector. A conversion may have one or two outgoing tracks. Double-track conversions are created when two tracks form a vertex that is consistent with the hypothesis that they have been produced from a massless particle. Single-track conversions look like tracks without hits before a certain layer. If there are several conversions matched to a cluster, double-track conversions are preferred over single-track conversions.

In CMS (Sect. 1.6.1.3) the reconstruction of photon conversions uses the full tracking [6]; see also Sect. 10.3. Electrons and positrons in particular are reconstructed by associating a track in the silicon tracker with a cluster in the ECAL [7]. Besides the standard seeding of tracks in the pixel detector, additional seeds are computed by extrapolating the electron/positron from the cluster toward the interaction vertex, using both charge hypotheses . Starting from these seeds, track candidates are built by the combinatorial Kalman filter (Sect. 5.1.7) and sent to track fit with the GSF (Sect. 6.2.3). Electron/positron candidates are then created from the association of a GSF track with a cluster in the ECAL. Track pairs of opposite charge sign are selected and filtered by various constraints: the two tracks must have small angular separation; the trajectories (helices) projected to the transverse plane must not intersect; and the presumptive vertex must be inside the tracker volume. Track pairs that pass these filters are sent to the vertex fit with the constraint that the tracks are parallel at the vertex; see Sect. 8.3. The track pair is kept if the p-value of the chi-square statistic indicates a good fit.

5 Hadronic Interactions

Knowledge of the material in a tracking device is important for understanding and tuning the track reconstruction algorithms. Reconstruction of hadronic interactions in the tracker material gives a precise map of the amount and location of active and inactive parts of the device. The hadronic interactions result in secondary vertices that have to be found. Whereas, in photon conversions, the vertex position along the photon direction is ill defined, the vertex position of hadronic interactions can be precisely estimated in all directions.

Tracks from secondary interaction vertices have a large impact parameter and may not be found by the standard track finder(s). In order to have the largest possible sample of such tracks, a special track finding pass can be implemented. Tracks from the primary vertex and short-lived decays are suppressed by a lower bound on the transverse impact parameter. Further quality cuts can be applied to select a reliable sample of secondary tracks. The following paragraphs describe examples from the experiments at the LHC.

In [8], a vertex finder is described that simultaneously finds all secondary vertices in an event in the ATLAS detector (Sect. 1.6.1.2) . It starts by finding all possible intersections of pairs of selected tracks, performs a vertex fit and applies a quality cut. Wrong combinations can be further suppressed by requiring that neither track has hits in the layers with a smaller radius than the vertex. Finally, nearby vertices are merged, and incompatibility between vertices sharing tracks are resolved via an incompatibility graph; see Sect. 5.3. For results on reconstruction efficiency and vertex resolution, see [8]. The search for hadronic interaction vertices can be complemented by the reconstruction of photon conversions; see Sect. 9.4.

A precision measurement of the structure of the CMS pixel detector (Sect. 1.6.1.3), including sensors and support material, and the beam pipe using hadronic interactions is reported in [9]. The track selection requires a minimal transverse momentum of 0.2 GeV∕c. For all pairs of selected tracks, their distance of closest approach is computed; if the distance is below a preset threshold, the midpoint of the points of closest approach is kept as a vertex candidate. These two-track vertices are then merged to larger vertices by agglomerative clustering , (Sect. 3.3.1). These vertices are sent to the adaptive vertex fit (AVF, Sect. 8.2.2).

The tracks associated to a fitted vertex are classified into incoming, outgoing, and merged tracks based on their transverse impact parameter and the number of hits before and after the vertex. At least three tracks have to be present, but not more than one incoming or merged tracks. Additional filters using the properties of the outgoing tracks serve to further reduce the number of fake interaction vertices. For results on resolution, efficiency and purity of the vertex reconstruction, see [9].

The Vertex Locator (VELO) of the LHCb experiment, see Sect. 1.6.1.4, was mapped by the reconstruction of secondary interaction vertices of hadrons produced in beam-gas collisions [10]. The data were collected during special runs in which helium or neon was injected into the collision region. The tracks used for the reconstruction of secondary vertices come from a modified track finding procedure that makes no prior assumption about the vertex position. Only well-reconstructed tracks with at least three 2D hits are selected. The secondary vertices are reconstructed from at least three tracks and have to be of good quality. They also must not be compatible with the primary vertex region. Only events with a single secondary vertex are used in the analysis. Results and the verification of the procedure with photon conversions to two muons can be found in [10].