2.1 Particle physics
2.1.2 The top quark at hadron colliders like the LHC
Most of the content of this section is covered by Bettini [7] or Wagner [10].
If further resources are used, they are cited explicitly.
Hadron colliders
At a hadron collider, as the name suggests, hadrons (protons in the case of the LHC) are accelerated to almost speed of light and collide at an interaction point in a detector that identifies and measures the properties of the products of the collision (event). At the LHC, two proton beams are accelerated to 6.5 TeV each in opposite directions such that the center of mass energy of the collision, the length of the total four-momentum, is equal 13 TeV. For a collision with equal but opposite momenta, the rest frame, the inertial system in which the center of mass is at rest, is the laboratory.
As mentioned above, protons are no fundamental particles. Besides three valence quarks (2 up quarks and a down quark), a proton consists of many
Figure 2.2: Slice of the CMS-detector with trajectories of particles and their interac-tions with the detector. The muon is a positively charged antimuon, the charged pion a positively charged hadron and the electron’s charge is negative [11].
quark-antiquark pairs (the sea quarks) that are generated and destroyed spontaneously and gluons exchanging energy between them.
Therefore, for an accelerated proton, the total momentum is divided among its partons (all constituents of the proton) in a non-deterministic way. In general, one finds that the gluons carry about half of the total momentum of the proton and are very common among the partons carrying a small fraction of the total momentum. So intuitively, there are many gluons within the proton that carry a small fraction of the total momentum. A collision of two protons is actually a collision of two partons, one from each proton.
Such a collision results in a lot of particles and energy emerging from it and the detector is designed to identify them and measure their properties as accurately as possible. The following paragraphs summarize roughly what the most important components of the CMS detector are and what they do.
For more details about the CMS detector see CMS Collaboration [11] and Cerrito [12] for more details about the physics used in detectors.
Figure 2.2 shows a slice of the CMS detector. It is shaped like a rotational symmetric cylinder with respect to the beam axis, consisting of the cylinder barrel together with an end cap on each of the two ends. The shown slice illustrates a part of the barrel and the components used to detect and measure the properties of various decay products. The end caps contain
the same instruments arranged such that particles that move rapidly in the beam direction can be detected there in a similar way as the particles in the barrel.
The silicon tracker consists of millions of pixel detectors having side lengths smaller than about a tenth of a millimeter each. A highly-energetic charged particle leaves a trace of activated pixels in the silicon tracker that can be reconstructed to see its path through the tracker.
The purpose of calorimeters (electromagnetic and hadronic calorimeters) measure the total energy of electrons and photons (ECAL) or hadrons (HCAL). A particle entering a calorimeter will start to form a shower of many more particles within the calorimeter (the dark blue clusters in the green/yellow regions in Figure 2.2) that will eventually pass all their energy to the calorimeter. In some sense, every created particle is the origin of a smaller subshower, so this chain reaction terminates fast enough such that the total shower is mostly contained in the calorimeter. Measuring the total energy of all particles of a shower yields the energy of the initial particle.
The superconducting solenoid is responsible for a strong magnetic field that curves the trajectories of all charged particles. Due to the magnetic field, charge and momentum of particle are related via the radius of the curve. In addition, the sign of the charge can be read of by the direction of the magnetic force. For this orientation of the magnetic field relative to the viewer, positive particles move clockwise while negative particles move counter-clockwise within the magnet. Outside of the solenoid, the magnetic field changes the direction, therefore, the direction is the other way around, too. The radius of the magnet is chosen such that most of the instruments can be placed inside of it to minimize inaccuracies introduced by interactions with the solenoid.
Muons are detected in the most outer part of the detector, the muon cham-bers, because they can pass all the inner layers without showering. Almost all the other particles do not reach that far out. The muon chambers work similar to another layer of trackers that detect the path of the particle passing through them.
Since the interactions in a hadron collider happen on parton level, the rest frame of a collision is not necessarily the laboratory frame. Instead, the rest frame is (approximately) z-boosted due to the different fractions of the proton momentum carried by the interacting partons. Unfortunately, it is not possible to know how exactly this system moves with respect to the
detector. Therefore, it is desirable to use a coordinate system that is invariant under Lorentzz-boosts [2].
The Cartesian coordinates in a detector are usually defined such that the z-axis is tangent to the counter-clockwise beam and they-axis points upwards.
To satisfy the common right hand rule, the x-axis has to point towards the center of the accelerator ring.
This definition implies that the x-y-plane is perpendicular to the beam and therefore, movement in this plane is invariant under Lorentzz-boost.
Since no direction is favoured physically (because gravity is negligible) and detectors are usually designed to be rotational symmetric with respect to thez-axis, it is natural to use polar coordinates for thex-y-plane. Therefore, movement in this plane is described by the two variables pT = qp2x+p2y andϕ∈ [−π,π], the azimuthal angle between thex-axis and the momentum vector.
The polar angle θ is not invariant under z-boosts, so another coordinate to measure the movement in z-direction is needed. The rapidity has the desirable property to transform additively such that differences are invari-ant underz-boost. Unfortunately, it is a kinematic quantity influenced by momentum and energy and not only by the direction. The purely geomet-ric pseudo rapidity on the other side is defined as η = −ln
tanθ2 and therefore, only depends on the direction. For massless particles, rapidity and pseudo-rapidity agree, but this is not true for particles with mass [13].
Instead, the rapidity converges to the pseudo-rapidity for momenta much larger than the mass (highly relativistic particles) as it is the case for the majority of decay products [2].
We have that:
px = pT·cosϕ py = pT·sinϕ pz = pT·sinhη
Since the total transverse momentum vanishes (approximately) before the collision, this should still be the case after the collision, but weakly interact-ing particles like neutrinos escape the detector with almost no reaction and therefore, their momentum is not measured. The constraint of vanishing
g Figure 2.3: Feynman diagrams of lowest order (two vertices) for top pair production
at hadron colliders.
transverse momentum can be used to obtain another pair of variables that describe the total transverse momentum of the weakly interacting particles called~pmissT . It is the negative sum of all measured momenta. The absolute value of this vector is also called missing transverse energy (MET/EmissT ) [2].
The top quark
One especially interesting particle is the top quark. Having a mass of about 173 GeV (about the mass of a gold atom) makes it the heaviest fundamental particle, even heavier than the Higgs boson (about 125 GeV rest mass) and by far the heaviest fundamental fermion compared to the bottom quark at the second place having a rest mass of ”only” about 4 GeV.
Figure 2.3 shows the two-vertex-processes contributing to top pair pro-duction at hadron colliders. At a proton-proton collider specifically, the antiquarks must be sea quarks while the quark can be either a valence or a sea quark. In case of the LHC at 13 GeV center of mass energy about 90% of the top pairs are created via gluon fusion [2]. Thus, the valence quarks only contribute little to the top pair production. At the Tevatron, the proton-antiproton collider where the top quark was discoverd, about 80-90%
of the produced top pairs were quark anti-quark events.
Another noteworthy property of the top is connected to its decays. Firstly, its average lifetime is about two orders of magnitude smaller than the time scale on which the strong interaction operates. This implies that the top quark almost certainly decays before being able to form any kind of non-fundamental particle like mesons and baryons. Therefore, the measurements of top quark properties are not ”disturbed” by any kind of bound states involving the top. Secondly, the by far most dominant decay channel is
Figure 2.4: Diagram showing the cone structure of jets together with the secondary vertex for bjets [14].
t→W+b happening about 99.8% of the time. Similarly, the antitop decays mostly via ¯t→W−b¯ before forming hadrons.
Another handy property of this decay is that bottom quarks (and antibottom quarks) admit special properties, too. Since they are heavy particles, they decay quickly as well.
As stated above, free color charged particles are not observed. Whenever a quark or gluon emerges from a collision, it is the origin of a jet, i.e. multiple hadrons moving in a cone in roughly the same direction. Often, this is pictured as ”tension/high energy density” in some kind of potential of the strong force that is resolved via the creation of new particles that form color neutral bound states (hadrons). This results in a lot of particles hitting the detector and making measurements more difficult and imprecise since there are a lot of particles carrying the energy of the single particle the shower emerged from.
If there is a jet containing a bottom quark in the detector, the bottoms can travel a small distance before decaying. After that short amount of time, the bottom quark will decay into another quark and aW boson. That creates a secondary vertex and yields a jet within the jet. This is the key observation that led to b tagging of jets to determine whether a bottom quark was contained in a jet. This is pictured in Figure 2.4.
Therefore, analyzingtt¯events can be done by analyzing the simultaneous decay of a top and an antitop which can be even further simplified by considering only the by far most common decay respectively. But this is where things start to become a bit more complicated. TheW+ (andW− in a similar way, particles and antiparticles change roles) can decay leptonically into an anti-lepton and its neutrino or hadronically in an up-type quark and a down-type antiquark.
This results in 3 different decay channels for the decay of att¯pair. Those are:
• dilepton: BothWs decay leptonically. This yields 2 bjets, 2 neutrinos and 2 leptons.
• lepton+jets: One W decays leptonically and the other hadronically.
This yields 2 bjets, 2 jets and 1 neutrino.
• all hadronic/alljets: BothW decay hadronically. This yields 2 bjets and 4 jets.
Often, decays including the tau are excluded from the categorization above and considered separately because of the various decay channels of the tau, but we will consider them as part of the dileptonic decays.
For every process one would like to observe, there might be other processes that have the same final state and therefore, measured data might look similar to the one we are interested in, since we can only measure the final state of a collision event. Those other possibilities are called background for this process. When using data measured at a collider experiment, we have to somehow separate signal and background to analyze particular particles and interactions.
Due to the non-deterministic nature of particle physics events, statistical methods are widely used. In order to obtain results that are as precise as possible, it is favourable to have many events available such that statistical uncertainties decrease.
The number of events occurring per unit time interval is characterized as the product of the two quantities Landσwhereσ is the cross section and L is the luminosity.
The cross section contains information that is specific for a process, e.g. two protons colliding, forming a tt¯pair and decaying in the dilepton channel.
The luminosity summarizes all the properties of the experiment, e.g. the
all-hadronic
electron+jets
electron+jets muo n+jets
muon+jets
tau+jets
tau+jets
e
µ
eτ
eτ µτ
µτ ττ
e
+µ
+τ
+ud cs e
–cs ud τ
–µ
–Top Pair Decay Channels
W decay
e
µ
eeµµ
di lep to ns
Figure 2.5: This graphic shows the relative frequency of final states fort¯t-events.
The labels containing the abbreviation of multiple quarks represent the by far most common quark combinations in jets originating in top pair decays [15]
number of particles per beam, the width of the beam and how often the beams cross at the interaction point.
When considering one specific collider, a high rate of occurrence corresponds to a high cross section. Thus, it would be nice if a particularly interesting decay has a small background, i.e. comparably few or rarely occurring events with the same signature in the detector and a high cross section and therefore, occur often.
Figure 2.5 shows the relative frequency (branching ratio) of the most dom-inant possible decay channels. As one can see, the dilepton is the rarest, followed by lepton+jets (when as usual excluding taus) and a lot of events decay hadronically. In fact, the background of the full hadronic decay chan-nel is pretty big, because a lot of processes can produce jets, e.g. gluon
radiation which yields an additional jet. On the other hand, the background for the dilepton channel is comparably small.
To summarize, we find these practical properties of the decay channels:
• dilepton: small background yields clean event samples, but has a comparably small cross section.
• lepton+jets: moderate amount of background and cross section.
• all hadronic: high cross section, but also high background
Furthermore, every decay channel admits some difficulties to face in the kinematic reconstruction events, which tries to determine the kinematic properties of the involved particles. The all hadronic channel yields a large amount of jets, but it is not clear, which jet originated from whichW boson.
In addition, it is not possible to decide whether a bjet contained a bottom quark or an antibottom quark. These ambiguities have to be resolved in the analysis. Every event containing a lepton (and hence a neutrino) will have missing energy since neutrinos do not interact with matter strongly enough to be measured in the detector.
This is just a shallow and mostly phenomenological introduction to the stan-dard model but it should be sufficient to follow the physical consequences arising from it.