Basic Idea of Measurements in Particle Collisions

Abstract

Some of the readers of this book may have seen a so-called “event display”, visualisation of a particle collision. An example from the ATLAS experiment is given in Fig. 2.1. There, what we see are many curves from a particular point, which indicates the location of two particles colliding with each other. The curves emerging from the point are charged tracks, and the traces of charged particles are identified from detector responses. Also seen are many colourful boxes, which look like histograms, with the direction of the height of the histograms pointing to a radial direction. These indicate the amount of energy from particles produced via the collision, measured in particle detectors.

2.1 Observables of Particle Scattering

Some of the readers of this book may have seen a so-called “event display”, visualisation of a particle collision. An example from the ATLAS experiment is given in Fig. 2.1. There, what we see are many curves from a particular point, which indicates the location of two particles colliding with each other. The curves emerging from the point are charged tracks, and the trace of charged particles are identified from detector responses. Also seen are many colourful boxes, which look like histograms, with the direction of the height of the histograms pointing to a radial direction. These indicate the amount of energy from particles produced via the collision, measured in particle detectors.

Fig. 2.1

Reprinted under the Creative Commons Attribution 4.0 International license from  [1] © 2017 CERN for the benefit of the ATLAS Collaboration. The black lines (or curves, more precisely) represent the charged tracks for the two electron candidates. The yellow curves correspond to other charged tracks. The red and purple boxes indicate energies measured in calorimeter units

Event display for an event, where the invariant mass of the two electrons found in the event is one of the highest among the events (up to 2016) taken by the ATLAS experiment.

We like to extract the properties of the physics processes of collision through such measurements. Ultimately, we like to know the property of the underlying interactions, i.e., how the particles are scattered and the final state particles are produced at the level of a few elementary particles involved. As you know, however, we have no way to know exactly how the elementary particles are interacting since it is through a process obeying quantum mechanics. All we can do experimentally is to measure the physical observables as precisely as possible in order to obtain the distribution of the observables. Such observables include the species of the particles produced by the collisions and the energy, momentum, and possibly the quantum numbers of the particles. Then we identify the underlying interaction of particles by comparing theoretical predictions of various interactions.

Here, we explain how such a correspondence between the observable and the underlying process is realised by taking again an example of the reaction $e^+ e^- \rightarrow e^+ e^-$ (Fig. 2.2), in $e^+ e^-$ colliders. In the experiment, we can measure the existence of the final state particles, $e^+$ and $e^-$, as well as their energy and momentum.

Fig. 2.2

Diagram for a reaction $e^+ e^- \rightarrow e^+ e^-$. a the detail of the interaction is left unspecified. b An example of the lowest order Feynman diagram

Let us suppose that you have no knowledge of quantum electrodynamics (QED). Then you do not know how the electrons exchange force when they scatter, i.e. how the force is mediated. We often draw interactions of particles schematically in diagrams. The diagram is drawn with a blob at the vertex of four fermions, in this case either electron or positrons. The interaction of the objects inside the blob is not “visible”: although we can guess that some particles are exchanged to mediate the scattering force, they cannot be observed directly. All we can observe are the “observables”, the particles in the final state.

Now we like to know what is happening in the blob through experimental measurements of the scattering. If we observe just one event of the interaction $e^+ e^- \rightarrow e^+ e^-$, there is not much more information than that such a kind of interaction exists. However, if we observe many events with an $e^+ e^-$ pair in the final state, we can also measure the distributions of the final state particles. This would give much more information on the interaction. The distribution can be angular, energy or, more generally, the (four-)momentum distributions. Not only the shape of the distribution but also the integrated number of events for a given intensity of collisions (the luminosity, see the next section) tell us the “strength” of the interaction, i.e. how often such interactions should occur. For simple interactions like $e^+ e^- \rightarrow e^+ e^-$, the angular and momentum distributions allow us to extract the shape of the potential between two scattering particles through appropriate transformation. For more complicated interactions like collisions with many particles produced, inclusive or exclusive distributions (see Sect. 2.3) of a particular type of particles can be compared to theoretical predictions. In this way, the blob of the diagram is uncovered and the Lagrangian of the interaction could be constructed, even though the interpretation is often limited by both experimental uncertainties from the measurement and uncertainties of the theoretical predictions.

2.2 Cross Section and Luminosity

The strength of the scattering has the dimension of $\mathrm{L^2}$ (area), where $\textrm{L}$ represents the dimension in length. This can be understood if we take an example of scattering in classical mechanics with one of the objects standing still (a target). Suppose that you hit the target of a certain size by throwing an object (a projectile) moving in a certain direction (Fig. 2.3a). If you assume that the object can hit only when they contact each other, and assume that the size of the projectile is negligible, the probability to hit the target is proportional to the size viewed from the direction where the projectile is running. More precisely, it is proportional to the cross section $\sigma$ of the object on the plane perpendicular to the projectile direction. For that reason, we also call the size of the interaction area the cross section. Whether the projectile goes inside the interaction area or not depends on the distance between the centre of the target and the line where the projectile flies. The distance between the target centre and the trajectory of the projectile at the infinite distance projected to the area of the target in the plane perpendicular to the projectile momentum is called the impact parameter b (see Fig. 2.3b).

Fig. 2.3

a Target and projectile. b Scattering angle $\theta$ and impact parameter b, where a projectile particle is injected towards a target particle

For point masses that interact remotely, which is the case for the elementary particles, the projectile and target are scattered infinitesimally even if their impact parameter is very large, if the force is long-distance, i.e. propagating in the infinite distance. This implicates that the cross sections for such an interaction are well defined as a function of a property of the scattering, such as the scattering angle $\theta$ in Fig. 2.3. Differential cross sections can be defined using the variable: $d\sigma /d\theta$ in this case.

The differential cross-section formula can be calculated using theory if the interaction is known. If we observe any deviation in such a distribution from the prediction, this tells us that the particle is not point-like or there exist some new type of interaction.

The number of observed scatterings for a given cross section can be calculated if we know the “strength” of the projectile beam. One needs to give as many particles in an area of as small cross section as possible, in order to maximise the number of events. This means that the particle flow density $(1/a)dN_p/dt$ is one of the parameters to define the number of interactions. Here, a is the area where the projectile is injected, i.e. the size of the beam of the projectile. $N_p$ gives the number of projectiles passing through the area. Then the number of interactions N per unit time is given as $dN/dt = \sigma \cdot (1/a) dN_p/dt$. You see that the cross section gives the correct dimension to deduce the number of interactions.

To obtain the number of collisions, we need to also express the number of particles in a target object via the density of the target. For the colliding beam, the corresponding number would be the density of the “target beam”. For the fixed target, one should count the number of target particles within the projectile beam size a. Suppose that the depth of the target is D and the target number density per volume is $n_t$. The total number of targets $N_t$, which could be scattered by the projectile beam, is $N_t = Da n_t$. It is useful to express the target density in terms of the mass density of material $\rho$ when we discuss the interaction of a particle with material such as that consisting of detectors. The number of the target is then expressed as

$$ N_t = Da \frac{\rho N_A}{A} \, , $$

where the targets are nucleus and $\rho$ given is in g/cm$^3$, A is the mass number given in g/mol, and $N_A$ is the Avogadro number. The number of the interactions for a still target (for the case of the fixed target experiment) integrated over time is reduced to

$$\begin{aligned} N = \frac{N_p N_t \sigma }{a} = \frac{N_p D \sigma \rho N_A}{A} \, , \end{aligned}$$

assuming the target is larger than the beam size.

The number of collisions for the collider can be obtained by replacing the number of targets with the number of the beam particle. For the beam as a target, it is easier to use the number of target beam particles per unit time $dN_t/dt$ since the target is also not a still object. The beam particles are bunched in RF buckets (see Sect. 3.2) for almost all colliders. The number of particles per unit time crossing a given plane perpendicular to the beam $dN_{\textrm{beam}}/dt$ is given as $f_{\textrm{coll}} \cdot n_{\textrm{bunch}}$ where $f_{\textrm{coll}}$ is the number of bunch collisions per unit time and $n_{\textrm{bunch}}$ is the number of particles in a bunch. In an ideal situation where both the lateral bunch size a (dimension$=\mathrm{L^2}$) and the longitudinal distribution of particles inside the bunch are the same for both the target beam and the projectile beam, the frequency of collisions is given as

$$ \frac{dN}{dt} = \sigma \cdot f_{\textrm{coll}}\frac{n_1 n_2}{a} \, , $$

where $n_1$ and $n_2$ are the numbers of particles in a bunch for beam 1 and beam 2—we no longer distinguish the target and projectile beams at this point. This implies that it is convenient to define the instantaneous luminosity L,

$$ L \equiv f_{\textrm{coll}}\frac{n_1 n_2}{a} \, . $$

Note that the dimension of L is $\mathrm{L^{-2}T^{-1}}$. Then the number of collisions $N_{\textrm{coll}}$ for a given period with the cross section $\sigma$ can be obtained as

$$N_{\textrm{coll}} = \sigma \times \int {L dt} \, , $$

where $\int {L dt}$ is called integrated luminosity.

2.3 Identifying Processes Through Measurements of Final State Particles

In this section, we discuss the way to identify if the observed events are indeed the ones of concern. You may regard the identification of events is just as simple as counting the number of events with a given final state, if we can safely assume that the detector works well enough to identify the particles. There are, however, a few points yet when we consider the “definition of the signal events.”

Let us start with the example of $e^+ e^- \rightarrow e^+ e^-$. To see if an event is classified to this category or not, first we need to identify the type of the final state particles through the measurements by detectors. We call this procedure particle identification or particle ID. We discuss the technical detail of the particle ID in Chap. 6. Here, we simply assume that the final state particles are identified at certain probabilities. You would then count the number of particles of interest. In this example, you would request that the events should have one electron and one positron in the final state.

A few questions come along: we may wonder if we should request certain criteria in energy or momentum for the electron and positron. Also, we need to decide if we allow any other particle(s) in the final state. For example, since what we like to measure is the $e^+ e^- \rightarrow e^+ e^-$, you may like to limit yourself to select the events which have only one pair of $e^+ e^-$ and no additional particles are observed. In order to have such a selection well-defined, we need to consider the following things.

Firstly, the electrons can emit soft photons induced by an internal electromagnetic field induced by the interacting electron and positron themselves. This probability is very high for the electrons, whose mass is very small. Since such soft photons are anyhow not efficiently observed by the detector, experimentally you can define that the event contains only an $e^+e^-$ pair observed in the final state by requesting the other detector responses not associated with either of the electron or positron to be below certain thresholds (typically slightly above the noise level of the detectors). A photon collinearly emitted in the direction of the electron would still be difficult to separate from the parent electron. In that case, such collinear photons are often treated as a part of the parent electron and then the measured energy and momentum are considered to be those of the primary electron.

The second reason is that none of the particle detectors using accelerators have $4\pi$ coverage in the solid angle: the detector is not completely hermetic. This means that we may miss a part of final state particles even if it is hard enough to be detected since we may have holes in the detector in that direction. For example, the probability for an $e^+e^-$ collision to emit a photon in the direction of the incoming electron or positron is very high, again because of the small mass of the electron. Such events are called initial state radiation (ISR) events. It is not possible to catch the photon if the emission is at a very small angle from the incoming beam direction since the accelerator should accommodate the beam with a beam pipe with the finite diameter of typically more than a few centimetres. The photon escapes from the detector through the beam pipe.

Therefore, all we can do to select an event is to impose criteria which look like an $e^+e^-$ final state. For an $e^+e^-$ collider where the laboratory frame coincides with the centre-of-mass system of the two beams, which have the same energy, an example of such criteria would be

• a pair of electron-like particles in opposite charge, both within the angular region $\theta _{\textrm{min}} + \varDelta< \theta < \pi - (\theta _{\textrm{min}} + \varDelta )$. Here, $\theta$ is the polar angle with the z direction defined as the incoming direction of one of the beams, $\theta \,\textrm{min}$, which is the polar angle of the detector boundary of the hole to accommodate the beam pipe, and $\varDelta$ is a margin to be taken so that the observed particles are enough away from the boundary;

• no other track nor cluster observed in other parts of the detector with their momentum or energy greater than $0.05 E_{\textrm{beam}}$, where $E_{\textrm{beam}}$ is the beam energy; and

• each electron fulfils $E_{\textrm{elec}} > 0.8 E_{\textrm{beam}}$, where $E_{\textrm{elec}}$ is the energy of the electron or the positron.

The first criterion makes sure that the events are well contained within the angular coverage of the detector $\theta _{\textrm{min}}< \theta < \pi - \theta _{\textrm{min}}$. The second criterion removes the events where extra particles with significant energies on top of the signal electron and positron exist within the acceptance of the detector. The threshold $0.05 E_{\textrm{beam}}$ may vary depending on the experimental condition. The last criterion reduces the probability that the $e^+ e^-$ pair is produced in association with some other particles that escape from detection. This requirement would remove effectively events from so-called two-photon processes (see Fig. 2.4) where an $e^+e^-$ pair is produced in the final state and both electron and positron lose significant energies.

Fig. 2.4

The two-photon process in $e^+ e^-$ collisions. The blob indicates an interaction of incoming $\gamma \gamma$, emitted from $e^+$ or $e^-$, and outgoing final-state particles (multi-particle state). The final state often consists of many hadrons since the photon-photon collision has coupling either to a quark or a neutral vector meson, which contains a $q\bar{q}$ pair

You may like to add further criteria to constrain the process to increase the fraction of the process in your mind among the event sample, so that the interpretation of thus defined cross section becomes more intuitive. This kind of event selection is called exclusive event selection. The above-given example would be called measurements of exclusive $e^+ e^-$ production.

Another way to investigate the underlying physics of $e^+e^-$ collisions through $e^+e^-$ final state is to define the selection criteria as simple enough. An extreme would be to just select an $e^+ e^-$ pair, both of which are energetic enough, like

• a pair of electron-like particles in opposite charge, both within the angular region $\theta _{\textrm{min}}< \theta < \pi - \theta _{\textrm{min}}$;

• each electron fulfils $E_{\textrm{elec}} > 0.3 E_{\textrm{beam}}$, where $E_{\textrm{elec}}$ is the energy of the electron.

This allows you to select events which include other particles in the final state; for example, the selected events contain, with a high probability, the process from Fig. 2.4. This kind of event selection is called inclusive event selection. The benefit of the inclusive event selection is that it is indeed “well defined” in terms of the theoretical prediction. For measurements with exclusive event selections, the modelling of the soft emission of many particles in the theoretical prediction (Monte Carlo simulation) is often not easy since it involves higher orders of the perturbative calculation. On the other hand, the total production rate, which is one of the measurements with inclusive event selections, is often calculated well since certain techniques exist to sum up all the contributions of final states.

The inclusive measurements are often performed when one would not know the number of particles produced in the reaction, either it is not measured or it is not easy to measure. Examples are jet production in hadron-hadron collisions, e.g. $pp \rightarrow n\times \textrm{jet} + X$, where $n \ge 1$ and X represents a part of the final state with any number of particles, or in deep-inelastic scattering $eN \rightarrow eX$ where N is a nucleon.

2.4 Event Acceptance and Efficiency

The event selection criteria will reduce the number of events you take from the process of your concern. The remaining number of events becomes smaller if the event selection becomes more exclusive, which is necessary if the amount of contribution from background processes is to be reduced. The solid angle coverage of your detector also gives a hard limit on the detection possibility. All these will cause the reduction in acceptance, defined as

$$ (\textrm{acceptance}) = \frac{N_{\textrm{sel}}}{N}, $$

where $N_{\textrm{sel}}$ denotes the number of events passing the selection criteria, imposed on true four-momenta of particles, while N is the number of events from the processes in concern. Here, the true four-momentum means that it is used in theoretical calculation without detector smearing. This may be available in event generators (see Chap. 7).

The acceptance is often a very small number (like $\ll O(10^{-1})$) if the event selection is exclusive and the denominator is defined as all the events from the process. Instead, the acceptance may become closer to unity if it is defined for differential cross sections at a given point in the phase space both for the denominator and numerator. As an example, for the exclusive event selection criteria given above for the $e^+e^-$ final state, the cross section may be defined as double-differential cross sections $d^2\sigma /dE_1 d\theta _1$, where $E_1$ and $\theta _1$ are the energy and angle of the highest energy electron. The acceptance for events with either $\theta _1 < \theta _{\textrm{min}} + \varDelta , \theta _1 > \pi - (\theta _{\textrm{min}} + \varDelta )$, or $E_1 < 0.8 E_{\textrm{beam}}$ is zero. That means that the differential cross sections corresponding to these kinematic regions are also zero. But for other regions, one would expect the acceptance is closer to unity than the average acceptance over all possible kinematic regions of the processes in concern. In this way, we can avoid a large extrapolation factor from $N_{\textrm{sel}}$ to N.

Since there is no detector with 100% detection efficiency, we necessarily lose events by the inefficiencies of the detectors. We often treat the loss of this effect to the acceptance separately from the geometrical acceptance and call it efficiency, defined as

$$ (\textrm{efficiency}) = \frac{N_{\textrm{det}\bigcap \textrm{sel}}}{N_{\textrm{sel}}} ,$$

where ${N_{\textrm{det}\bigcap \textrm{sel}}}$ denotes the number of events passing the selection criteria imposed on quantities obtained from measurements, while ${N_{\textrm{sel}}}$ is the number of events passing the selection criteria imposed on true four-momenta of particles.

Note that a detailed definition of acceptance and efficiency may be different from what is given above and may depend on physics analysis or literature.

With the acceptance A and efficiency $\epsilon$ defined, the number of signal events $N_{\textrm{sig}}$ for collisions with integrated luminosity $L_{\textrm{int}}$ can be derived from

$$ N_{\textrm{sig}} = L_{\textrm{int}} \sigma A \epsilon . $$

The cross section can be obtained from this equation.

In most measurements, you cannot ignore the presence of the background events. The $N_{\textrm{sig}}$ should be replaced with $N_{\textrm{sig}} = N_{\textrm{obs}} - N_{\textrm{bkgd}}$, where $N_{\textrm{obs}}$ is the observed number of events and $N_{\textrm{bkgd}}$ is the number of background events. The estimation of background events is time-consuming in data analysis, which is discussed later.

Note that it is often difficult to know $L_{\textrm{int}}$ precisely enough in collider experiments, especially for hadron colliders. Also the deviation in detector calibration from its truth value causes shift in the number of observed events, often rather uniformly across other kinematical variables (e.g. angles). The normalisation of a cross section, or differential cross sections, is important for extracting the strength of the interaction such as extraction of coupling constant and higher order effect on perturbation calculation. The normalisation is, however, not necessary when extracting the physics quantities from the shape of the distribution, such as the mass spectra and the spin of the exchanged particles. A normalised distribution is used to extract physics quantities for such purposes.

Fig. 2.5

a A schematic view of a soft pp collision with multi-hadronic final state. b An example of a hard pp collision with two high-$p_{\textrm{T}}$ partons in the final state

2.5 Nature of Hadron-Hadron Collisions and Kinematic Variables

Hadron-hadron collisions are realised by two high-energy stable hadron beams brought into collisions. Only protons or anti-protons have been used in modern high-energy accelerators in practice. The proton is not an elementary particle; instead, it consists of partons, i.e. quarks and gluons. The processes that undergo in hadron-hadron collisions are categorised into two: ${ soft}$ and ${ hard}$ interactions.

In soft collisions (Fig. 2.5a), the constituent of a proton is not resolved during the interaction of, for example, two protons. This occurs when the exchanged particle during the interaction does not carry high momentum. Such an interaction cannot be described perturbatively by quantum chromodynamics (QCD) since the strong coupling constant $\alpha _S$ is in fact too strong for low-energy interactions. Figure 2.6 shows the behaviour of $\alpha _S(\mu ^2)$, where $\mu$ represents the energy scale of the interaction, e.g. the four-momentum of the exchanged particle. Since $\alpha _S$ becomes much larger than $O(10^{-1})$ when the momentum transfer is similar to or smaller than $\varLambda _{QCD} \simeq 200$ MeV, a perturbative expansion based on the number of partons is no longer possible there. In such a situation, the partons are bound strongly and the nucleons move collectively. Individual partons are no longer visible. We often call such an object “quark matter”, like a fluid consisting of quarks and gluons, which binds the quarks together.

Fig. 2.6

Reprinted under the Creative Commons Attribution 4.0 International license from [2] © 2018 CERN, for the ATLAS Collaboration. The data are taken from measurements from hadron and ep colliders. The curve indicates the solution of the renormalisation-group equation using $\alpha _S$ obtained from the results indicated by red (solid circle) points

The strong coupling constant $\alpha _S$ as a function of the renormalisation scale $\mu = Q$.

The soft interaction would look, therefore, like two pancake-like composite objects moving and crossing across. The collision of such objects may be simplified as follows: the two objects interact with each other if the compound has an overlap with the other composite material, and do not interact if the impact parameter is larger than the diameter of the objects. It is expected that the cross section for such an interaction is constant as a function of the centre-of-mass (CM) energy of the collision, assuming that the size of the proton is approximately independent of the interaction. Measurements at various CM energies show that the cross section rises with energy, but very slowly.

On the other hand, the parton (here denoting the parton A) in a proton can resolve the parton (the parton B) of the other proton when the momentum transfer of the exchanged particle is much larger than $\varLambda _{\textrm{QCD}}$ (see Fig. 2.5b). If the impact parameter of the parton A with respect to the parton B is so small, the field produced by the parton B is strong so that the interaction may occur at a very high energy regime. In other words, the particle exchanged in such an interaction may carry very high momentum with a short wavelength. This allows the partons inside hadrons to be resolved. The $\alpha _S$ gets much smaller such that the interaction can be described by the perturbative calculation. In such hard interactions, the lowest order in perturbation, i.e. 2-to-2 interaction becomes dominant; two high-momentum (i.e. hard) partons are produced in the final state.

In order to give the kinematic feature of hadronic collisions, let us define the coordinate commonly used for hadron-hadron collisions. We choose the system where the z-axis is along the beam line, with the positive direction of the z-axis being the running direction of the beam A (like the proton beam for $p\bar{p}$ colliders or one of the proton beam rotating counter-clockwise for pp colliders). The x-axis is then chosen towards the centre of the accelerator ring. The y-axis is defined so as to form a right-hand coordinate system. The transverse plane is the xy plane perpendicular to the beam direction. The transverse momentum is defined as $p_{\textrm{T}}= \sqrt{p_x^2 + p_y^2} = p \sin {\theta }$ where $\theta$ is the polar angle of the coordinate system given above. We often use $(p_{\textrm{T}}, \phi )$ instead of $p_x$ and $p_y$, where $\phi$ is the azimuthal angle.

A soft-interaction event is characterised by an absence of particles with high $p_{\textrm{T}}$, while at least a few high-$p_{\textrm{T}}$ particles are produced in the hard collisions. The reason to use $p_{\textrm{T}}$ comes from the nature of the hadron-hadron collisions. The hard collision occurs between two partons, whose energies are not equal even if the hadron beam energies are the same. The momentum of the parton A (B) can be expressed as $p_{\mathrm{A(B)}} = x_{\mathrm{A(B)}} p_{\textrm{beam}}$ using a momentum fraction $x_{\mathrm{A(B)}}$, which is defined as the ratio of the parton momentum involved in the hard collision to the momentum of a hadron to which the parton belongs. In general $x_{\textrm{A}} \ne x_{\textrm{B}}$, meaning that the centre-of-mass frame of the two partons involved in the hard collision is boosted against the centre-of-mass frame of the two beams, which corresponds to the laboratory frame for symmetric colliders. Therefore, the only component of the momentum preserved in the hard collision is the one in the transverse plane. The longitudinal component of the centre-of-mass system of the partons A and B cannot be determined unless the values of $x_{\textrm{A}}$ and $x_{\textrm{B}}$ are obtained by other experimental quantities.

Now we may like to determine the third coordinate component $p_z$ of the momentum of the parton-parton collision system. However, it is not always possible to measure the third coordinate precisely in hadron-hadron collisions, since a large fraction of the longitudinal momentum is lost through particles entering in the beam pipe, and some of the lost particles may have emerged from the parton-parton collision. The variable $E-p_z$ of an event, instead, can well be measured even if we lose particles lost in the beam pipe in $+z$, since $E-p_z$ contribution from such particles with very small scattering angle, escaping the beam pipe, is almost zero. Similarly, $E+p_z$ is also well measured even if we lose particles collinear to the $-z$ direction. Constructing variables using $E-p_z$ and $E+p_z$ of a system (event or a part of the event), or a particle, would therefore be determined with certain accuracy. The most common and convenient choice is to use the rapidity y, defined as

$$ y = \ln \sqrt{\frac{E + p_z}{E - p_z}} = \ln \frac{E + p_z}{m_\textrm{T}}, \ \ m_\textrm{T} \equiv \sqrt{m^2 + p_{\textrm{T}}^2} . $$

A good property of the rapidity is that the difference in rapidity between two four-momenta is preserved under a Lorentz boost. This also means that the Lorentz boost can be calculated by adding the rapidity of the boost vector.

In a limit where a particle is massless, the rapidity is equal to pseudorapidity $\eta$, defined as

$$\begin{aligned} \eta = - \ln \tan (\theta /2). \end{aligned}$$

$\eta$ is a good approximation of y in modern high-energy collider experiments, where the particles from hard collisions are produced at $> O(10)$ GeV, which is much higher than typical mass of long-lived final state particles like electron, muon, pion, or kaon. At $\theta = \pi /2$, $d\eta /d\theta = 1$, i.e. $\varDelta \eta$ corresponds exactly to $\varDelta \theta$.

Since the Lorentz transformation of the rapidity is additive, it expresses well the Lorentz-invariant phase space of final state particles. The phase space for a particle is

$$\begin{aligned} d^4 p \delta (p^2 - m^2) = d^3\textrm{p}/E = \pi dy d{p}_{T}^2 \, , \end{aligned}$$

where m is the mass of the particle and $\textrm{p}$ is the three-momentum of the particle while p denotes the four-momentum. This means that the particle is uniformly produced in y if the particle is equally distributed in the phase space. A differential cross section is, therefore, expressed often as $d\sigma /dy$ instead of $d\sigma /d\theta$. The latter is more commonly used in non-relativistic collisions, fixed target experiments, or $e^+e^-$ collisions.

2.6 Structure of Hadrons and Parton Density Function

The hard process as introduced in the previous section regards a hadron-hadron collision as a scattering of one parton from the parent hadron A of its momentum fraction $x_{\textrm{A}} = p_{\textrm{A}}/p_{\mathrm{beam_A}}$, with another parton from the other parent hadron B, $x_{\textrm{B}} = p_{\textrm{B}}/p_{\mathrm{beam_B}}$. For high-energy collisions where the mass of the partons can be ignored, the centre-of-mass energy of the two partons A and B, $\sqrt{\hat{s}}$, is given as $\sqrt{x_{\textrm{A}}x_{\textrm{B}}s}$.

In order to estimate the cross section of hard collisions, one needs to know the “luminosity” of such partons, i.e. number of partons in the incoming beam particles, in order to convert the luminosity of hadron-hadron collisions to the luminosity of parton-parton collisions. The number density would depend on x of the parton since a hadron is a composite particle consisting of partons of various momenta. Now a slight complication is that the number of partons with a given x also depends on the wavelength of the probe. This is explained qualitatively as follows.

One needs an electron microscope to see the structure of viruses since the virus is smaller than the wavelength of the visible light. The electron beam energy of the electron microscope ($\gg$ keV) is much larger than that of the visible light so that one can resolve the fine structure of the virus. Similarly, if we like to see the structure of hadrons, it is necessary to use a probing beam, whose wavelength is much shorter than the size of the hadron itself $\simeq 1\,\textrm{fm} \sim 200\,\textrm{MeV}$. Practically, what probes the structure of hadrons is not the beam itself but rather a particle coupling to the partons, e.g. photons for the electron beam, or gluons, or quarks for hadron beams.

Now, what is known is that more partons (the structure of hadrons) are seen as the wavelength of probe particles gets shorter, as schematically drawn in Fig. 2.7. In this figure, an electron as a projectile collides with a quark inside a target proton. Such an interaction is called deep-inelastic scattering (DIS). The scatter exchanges a virtual photon $\gamma ^{*}$, which mediates the force and probes the quarks inside the proton with a short wavelength if $Q^2 \equiv -(p_e' - p_e)^2$ is large, since the wavelength is $\propto \sqrt{1 / Q^2}$. When the partons start to feel the external force from the electron beam, an energetic parton may radiate another parton before coupling to the virtual photon, since the coupling constant involved in the parton radiation, $\alpha _S$, is larger than the electromagnetic coupling. This behaviour of the parton radiation is very well described by a theoretical framework based on perturbative QCD, for example, by the DGLAP equation (for review, see, for example, Ref. [3] or more in detail in Ref. [4]). The equation, with experimental data from lepton-hadron scattering experiments, tells that partons should increase with $Q^2$, except for very high-x partons, which give momentum to low-x partons through the parton radiation.

Fig. 2.7

A schematic drawing of the $ep \rightarrow eX$ reaction where a $\gamma ^{*}$ is exchanged between the proton and the electron. The large blob in the left figure indicates the area that can be probed by the virtual photon with a long wavelength. As the wave becomes shorter like in the right figure, more parton radiation may be visible

As a consequence, the number of partons, or the parton density function (PDF) $f_i$, where i denotes the type of the parton (gluon or quark flavour), depends not only on x but also on the wavelength of the probe: $f_i = f_i(x, Q^2)$. For low-$x (< 0.1)$ regime, the number of partons increases logarithmically with $Q^2$. It also shows rapid increase as x of the parton gets lower $(x \ll 0.1)$, $f_i \propto x^{-\lambda }$ where $\lambda$ is typically 0.2–0.4. An example of the parton density functions may be found in the review section for the structure function in the Particle Data Group review [5] and references therein.

For hadron-hadron collisions, a projectile is a hadron and the exchanged force for the scattering is propagated also by a parton. The $Q^2$ of such collisions needs $p_z$ of the scattered parton, which is not well reconstructed. Instead, $p_{\textrm{T}}^2$ of the hard-scattered parton is used as the probing scale when estimating the parton density. The cross section of such hard collisions can be expressed as a product of the parton densities of the partons A and B and the scattering cross section $\mathrm{AB \rightarrow CD}$ where the parton indices C and D are the two outgoing partons, as shown in Fig. 2.8:

$$ \sigma \propto \sum _{q, g}{f_{\textrm{A}}(x_{\textrm{A}}, \mu _F^2) f_{\textrm{B}}(x_{\textrm{B}}, \mu _F^2)\sigma _{\mathrm{AB \rightarrow CD}}(\hat{s}, \mu ^2)}. $$

The probing scale is called a factorisation scale $\mu _F$, which is $p_{\textrm{T}}$ in this case.

Fig. 2.8

A schematic diagram showing how hard scattering cross section is factorised into

This formula assumes that a parton from a hadron collides with a parton from the other hadron. This picture may not be valid if the number of partons inside a hadron is very large (in particular if the partons are from very low-x) or the scattering cross section is very large (e.g. due to large $\alpha _S$ in low-$p_{\textrm{T}}$ regime). For such cases, more than one parton pair may cause scatterings. If the number of scatterings becomes so many, the interaction may not be described anymore by perturbative QCD. They may have to be described, at least partially, by a theoretical framework for soft interactions, which treats the entire hadron as one body for the interaction.