QCD I: Introduction, Tree Graph Predictions, and Jets
14.3 Hard scattering processes, QCD tree graphs, and jets
14.3.1 Introduction
The fundamental distinctive feature of non-Abelian gauge theories is that they are ‘asymptotically free’, meaning that the effective coupling strength becomes progressively smaller at short distances, or high energies (Gross and Wilczek 1973, Politzer 1973). This property is the most compelling theoretical motiva-tion for choosing a non-Abelian gauge theory for the strong interacmotiva-tions, and it enables a quantitative perturbative approach to be followed (in appropriate circumstances) even in strong interaction physics. This programme has in-deed been phenomenally successful, firmly establishing QCD as the theory of strong interactions, and now – in the era of the LHC – serving as a precision tool to guide searches for new physics.
A proper understanding of how this works necessitates a considerable de-tour, however, into the physics of renormalization. In particular, we need to understand the important cluster of ideas going under the general heading of the ‘renormalization group’, and this will be the topic of chapter 15. For the moment we proceed with a discussion of some simple tree-level applications of QCD, which provided early confrontation of QCD with experiment.
Let us begin by recapitulating, from a QCD-informed viewpoint, how the parton model successfully interpreted deep inelastic and large-Q2 data in terms of almost free point-like partons – now to be identified with the QCD quanta: quarks, antiquarks, and gluons.
14.3. Hard scattering processes, QCD tree graphs, and jets 87
In section 9.5 we briefly introduced the idea of jets in e+e− physics: two well collimated sprays of hadrons, apparently created as a quark–antiquark pair separate from each other at high speed. The angular distribution of the two jets followed closely the distribution expected from the parton-level
+ −
process e e → ¯qq. The dynamics at the parton level was governed by QED, but QCD is responsible for the way the emerging q and ¯q turn them-selves into hadrons, a process called parton fragmentation (it occurs for glu-ons too). We may think of it as proceeding in two stages. First, as the rapidly moving q and ¯q begin to separate, they develop perturbative show-ers of narrowly collimated gluons and quark–antiquark pairs. Then, as the partons separate further, the strength of the forces between them increases, becoming strongly non-perturbative at a separation of about 1 fm, and en-suring that the coloured quanta are all confined into hadrons. As yet we do not have a completely quantitative dynamical understanding of the sec-ond, hadronization, stage: it is implemented by means of a model. Nev-ertheless, we can argue that for the forces to be strong enough to produce the observed hadrons, the dominant processes in hadronization must involve small momentum transfers – that is, the exchange of ‘soft’ quanta. Thus the emerging hadrons are also well collimated into two jets, whose energy and angular distributions reflect the short-distance physics at the parton level.
This simple 2-jet picture will be extended in section 14.4, where we consider e e+ − → 3 jets.
A somewhat different aspect of parton physics arose in sections 9.2–9.3, where we considered deep inelastic electron scattering from nucleons. There the initial state contained one hadron. Correspondingly, one parton appeared in the initial state of the parton-level interaction, and the analysis required new functions measuring the probabilities of finding a particular parton in the parent hadron – the parton distribution functions. These too are beyond the reach of perturbation theory.
We may also consider, finally, hadron-hadron collisions. In this case, we need all three of the features we have been discussing: the parton distribu-tion funcdistribu-tions, to provide the intial parton-parton state from the two-hadron state; the perturbative short-distance parton-parton interaction; and the par-ton fragmentation process in the final state. These three parts to the process are pictured in figure 14.5. The identification and analysis of short distance parton-parton interactions provide direct tests of the tree-graph structure of QCD, and perturbative corrections to it.
This three-part schematization of certain features of hadronic interactions is useful, because although we cannot yet calculate from first principles ei-ther the parton distribution functions or the fragmentation process, both are universal. The quark and gluon composition of hadrons is the same for all processes, and so measurements in one experiment can be used to predict the results of others. We saw an example of this in the Drell–Yan process of section 9.4. As regards the fragmentation stage, this too will be universal, pro-vided one is interested in sufficiently inclusive aspects of the final state. The
14.3. Hard scattering processes, QCD tree graphs, and jets 89 the probability of observing jets, since the probability that a single hadron in a jet will actually carry most of the jet’s total transverse momentum is quite small (Jacob and Landshoff 1978; Collins and Martin 1984, Chapter 5). It is much better to surround the collision volume with an array of calorimeters which measure the total energy deposited. Wide-angle jetscan then be iden-tified by the occurrence of a large amount of total transverse energy deposited in a number of adjacent calorimeter cells: this is then a ‘jet trigger’. The importance of calorimetric triggers was first emphasized by Bjorken (1973), following earlier work by Berman, Bjorken and Kogut (1971). The applica-tion of this method to the detecapplica-tion and analysis of wide-angle jets was first reported by the UA2 collaboration at the CERN ¯pp collider (Banner et al.
1982). An impressive body of quite remarkably clean jet data was subse-quently accumulated by both the UA1 and UA2 collaborations (at√s= 546 GeV and 630 GeV), and by the CDF and D0 collaborations at the FNAL Tevatron collider (√s= 1.8 TeV).
For each event the total transverse energyE
ETis measured where EET=E
i
Eisinθi. (14.46)
Ei is the energy deposited in the ith calorimeter cell and θi is the polar angle of the cell centre; the sum extends over all cells. Figure 14.6 shows the EETdistribution observed by UA2: it follows the ‘soft’ exponential form for EET≤60 GeV, but thereafter departs from it, showing clear evidence of the wide-angle collisions characteristic of hard processes.
As we shall see shortly, the majority of ‘hard’ events are of two-jet type, with the jets sharing the EET approximately equally. Thus a ‘local’ trigger set to select events with localized transverse energy≥30 GeV and/or a ‘global’
trigger set at≥60 GeV can be used. At √s≥500–600 GeV there is plenty of energy available to produce such events.
The total√svalue is important for another reason. Consider the kinemat-ics of the two-parton collision (figure 14.5) in the ¯pp CMS. As in the Drell–Yan process of section 9.4, the right-moving parton has 4-momentum
x1p1=x1(P,0,0, P) (14.47) and the left-moving one
x2p2=x2(P,0,0,−P) (14.48) where P = √s/2 and we are neglecting parton transverse momenta, which are approximately limited by the observed <pT> value (∼0.4 GeV, and thus negligible on this energy scale). Consider the simple case of 900 scattering, which requires (for massless partons) x1 = x2, equal to x say. The total outgoing transverse energy is then 2xP =x√s. If this is to be greater than 50 GeV, then partons withx≥0.1 will contribute to the process. The parton distribution functions are large at these relatively smallxvalues, due to sea 88 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.5
Hadron-hadron collision involving parton-parton interaction followed by par-ton fragmentation.
three-part scheme is called factorization, and it has been rigorously proved for some cases. We shall return to factorization in section 15.7.
Let us turn now to some of the early data on parton-parton interactions in hadron-hadron collisions.
14.3.2 Two-jet events in pp ¯ collisions
How are short-distance parton-parton interactions to be identified experimen-tally? The answer is: in just the same way as Rutherford distinguished the presence of a small heavy scattering centre (the nucleus) in the atom: by look-ing at secondary particles emerglook-ing at large angles with respect to the beam direction. For each secondary particle we can define a transverse momentum pT = p sin θ where p is the particle momentum and θ is the emission angle with respect to the beam axis. If hadronic matter were smooth and uniform (cf the Thomson atom), the distribution of events in pT would be expected to fall off very rapidly at large pT values – perhaps exponentially. This is just what is observed in the vast majority of events: the average value of pT
measured for charged particles is very low (<pT> ∼0.4 GeV), but in a small fraction of collisions the emission of high-pT secondaries is observed. They were first seen (B¨usser et al. 1972, 1973, Alper et al. 1973, Banner et al.
1982) at the CERN ISR (CMS energies 30-62 GeV), and were interpreted in parton terms as previously indicated. Referring to figure 14.5, a parton from one hadron undergoes a short-distance ‘hard scattering’ interaction with a parton from the other, leading in lowest-order perturbation theory to two wide-angle partons, which then fragment into two jets.
We now face the experimental problem of picking out, from the enormous multiplicity of total events, just these hard scattering ones, in order to analyse them further. Early experiments used a trigger based on the detection of a single high-pT particle. But it turns out that such triggering really reduces
88 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.5
Hadron-hadron collision involving parton-parton interaction followed by par-ton fragmentation.
three-part scheme is calledfactorization, and it has been rigorously proved for some cases. We shall return to factorization in section 15.7.
Let us turn now to some of the early data on parton-parton interactions in hadron-hadron collisions.
14.3.2 Two-jet events in pp collisions¯
How are short-distance parton-parton interactions to be identified experimen-tally? The answer is: in just the same way as Rutherford distinguished the presence of a small heavy scattering centre (the nucleus) in the atom: by look-ing at secondary particles emerglook-ing at large angles with respect to the beam direction. For each secondary particle we can define a transverse momentum pT =psinθ where p is the particle momentum and θ is the emission angle with respect to the beam axis. If hadronic matter were smooth and uniform (cf the Thomson atom), the distribution of events in pT would be expected to fall off very rapidly at large pT values – perhaps exponentially. This is just what is observed in the vast majority of events: the average value ofpT
measured for charged particles is very low (<pT> ∼0.4 GeV), but in a small fraction of collisions the emission of high-pT secondaries is observed. They were first seen (B¨usser et al. 1972, 1973, Alper et al. 1973, Banner et al.
1982) at the CERN ISR (CMS energies 30-62 GeV), and were interpreted in parton terms as previously indicated. Referring to figure 14.5, a parton from one hadron undergoes a short-distance ‘hard scattering’ interaction with a parton from the other, leading in lowest-order perturbation theory to two wide-angle partons, which then fragment into two jets.
We now face the experimental problem of picking out, from the enormous multiplicity of total events, just these hard scattering ones, in order to analyse them further. Early experiments used a trigger based on the detection of a single high-pT particle. But it turns out that such triggering really reduces
14.3. Hard scattering processes, QCD tree graphs, and jets 89
the probability of observing jets, since the probability that a single hadron in a jet will actually carry most of the jet’s total transverse momentum is quite small (Jacob and Landshoff 1978; Collins and Martin 1984, Chapter 5). It is much better to surround the collision volume with an array of calorimeters which measure the total energy deposited. Wide-angle jets can then be iden-tified by the occurrence of a large amount of total transverse energy deposited in a number of adjacent calorimeter cells: this is then a ‘jet trigger’. The importance of calorimetric triggers was first emphasized by Bjorken (1973), following earlier work by Berman, Bjorken and Kogut (1971). The applica-tion of this method to the detecapplica-tion and analysis of wide-angle jets was first reported by the UA2 collaboration at the CERN ¯pp collider (Banner et al.
1982). An impressive body of quite remarkably clean jet data was subse-quently accumulated by both the UA1 and UA2 collaborations (at √s = 546 GeV and 630 GeV), and by the CDF and D0 collaborations at the FNAL Tevatron collider ( √s = 1.8 TeV).
For each event the total transverse energy E
ET is measured where EET = E
Ei sin θi. (14.46)
i
Ei is the energy deposited in the ith calorimeter cell and θi is the polar angle of the cell centre; the sum extends over all cells. Figure 14.6 shows the EET distribution observed by UA2: it follows the ‘soft’ exponential form for EET ≤ 60 GeV, but thereafter departs from it, showing clear evidence of the wide-angle collisions characteristic of hard processes.
As we shall see shortly, the majority of ‘hard’ events are of two-jet type, with the jets sharing the EET approximately equally. Thus a ‘local’ trigger set to select events with localized transverse energy √ ≥ 30 GeV and/or a ‘global’
trigger set at ≥ 60 GeV can be used. At s ≥ 500–600 GeV there is plenty of energy available to produce such events.
The total √s value is important for another reason. Consider the kinemat-ics of the two-parton collision (figure 14.5) in the ¯pp CMS. As in the Drell–Yan process of section 9.4, the right-moving parton has 4-momentum
x1p1 = x1(P, 0, 0, P) (14.47) and the left-moving one
x2p2 = x2(P, 0, 0, −P ) (14.48) where P = √ s/2 and we are neglecting parton transverse momenta, which are approximately limited by the observed <pT> value (∼ 0.4 GeV, and thus negligible on this energy scale). Consider the simple case of 900 scattering, which requires (for massless partons) x1 = x2, equal to x say. The total outgoing transverse energy is then 2xP = x s. If this is to be greater than √ 50 GeV, then partons with x ≥ 0.1 will contribute to the process. The parton distribution functions are large at these relatively small x values, due to sea
14.3. Hard scattering processes, QCD tree graphs, and jets 91
FIGURE 14.7
Two-jet event. Two tightly collimated groups of reconstructed charged tracks can be seen in the cylindrical central detector of UA1, associated with two large clusters of calorimeter energy depositions. Figure reprinted with per-mission from S Geer inHigh Energy Physics 1985, Proc. Yale Advanced Study Instituteeds M J Bowick and F Gursey; copyright 1986 World Scientific Pub-lishing Company.
FIGURE 14.8
Four transverse energy distributions for events with EET > 100 GeV, in the θ, φ plane (UA2, DiLella 1985). Each bin represents a cell of the UA2 calorimeter. Note that the sum of theφ’s equals 1800(mod 3600).
90 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.6
Distribution of the total transverse energy EET observed in the UA2 central calorimeter (DiLella 1985).
quarks (section 9.3) and gluons (figure 9.9), and thus we expect to obtain a reasonable cross section.
What are the characteristics of jet events? When EET is large enough (≥ 150 GeV), it is found that essentially all of the transverse energy is indeed split roughly equally between two approximately back-to-back jets. A typical such event is shown in figure 14.7. Returning to the kinematics of (14.47) and (14.48), x1 will not in general be equal to x2, so that – as is apparent in figure 14.7 – the jets will not be collinear. However, to the extent that the transverse parton momenta can be neglected, the jets will be coplanar with the beam direction, i.e. their relative azimuthal angle will be 1800 . Figure 14.8 shows a number of examples in which the distribution of the transverse energy over the calorimeter cells is analyzed as a function of the jet opening angle θ and the azimuthal angle φ. It is strikingly evident that we are seeing precisely a kind of ‘Rutherford’ process, or – to vary the analogy – we might say that hadronic jets are acting as the modern counterpart of Faraday’s iron filings, in rendering visible the underlying field dynamics!
We may now consider more detailed features of these two-jet events – in particular, the expectations based on QCD tree graphs. The initial hadrons provide wide-band beams of quarks, antiquarks and gluons2; thus we shall have many parton subprocesses, such as qq → qq, q¯q→ q¯q, q¯q→ gg, gg → gg, etc. The most important, numerically, for a p¯p collider are q¯q→ q¯q, gq → gq
2In the sense that the partons in hadrons have momentum or energy distributions, which are characteristic of their localization to hadronic dimensions.
90 14. QCD I: Introduction, Tree Graph Predictions, and Jets
FIGURE 14.6
Distribution of the total transverse energyEETobserved in the UA2 central calorimeter (DiLella 1985).
quarks (section 9.3) and gluons (figure 9.9), and thus we expect to obtain a reasonable cross section.
What are the characteristics of jet events? When EET is large enough (≥150 GeV), it is found that essentially all of the transverse energy is indeed split roughly equally between two approximately back-to-back jets. A typical such event is shown in figure 14.7. Returning to the kinematics of (14.47) and (14.48),x1 will not in general be equal tox2, so that – as is apparent in figure 14.7 – the jets will not be collinear. However, to the extent that the transverse parton momenta can be neglected, the jets will be coplanar with the beam direction, i.e. their relative azimuthal angle will be 1800. Figure 14.8 shows a number of examples in which the distribution of the transverse energy over the calorimeter cells is analyzed as a function of the jet opening angleθand the azimuthal angleφ. It is strikingly evident that we are seeing precisely a kind of ‘Rutherford’ process, or – to vary the analogy – we might say that hadronic jets are acting as the modern counterpart of Faraday’s iron filings, in rendering visible the underlying field dynamics!
We may now consider more detailed features of these two-jet events – in particular, the expectations based on QCD tree graphs. The initial hadrons provide wide-band beams of quarks, antiquarks and gluons2; thus we shall have many parton subprocesses, such as qq→qq, q¯q→q¯q, q¯q→gg, gg→gg, etc. The most important, numerically, for a p¯p collider are q¯q→q¯q, gq→gq
2In the sense that the partons in hadrons have momentum or energy distributions, which are characteristic of their localization to hadronic dimensions.
91 14.3. Hard scattering processes, QCD tree graphs, and jets
FIGURE 14.7
Two-jet event. Two tightly collimated groups of reconstructed charged tracks can be seen in the cylindrical central detector of UA1, associated with two large clusters of calorimeter energy depositions. Figure reprinted with per-mission from S Geer in High Energy Physics 1985, Proc. Yale Advanced Study Institute eds M J Bowick and F Gursey; copyright 1986 World Scientific Pub-lishing Company.
FIGURE 14.8
Four transverse energy distributions for events with EET > 100 GeV, in the θ, φ plane (UA2, DiLella 1985). Each bin represents a cell of the UA2 calorimeter. Note that the sum of the φ’s equals 1800 (mod 3600).
14.3. Hard scattering processes, QCD tree graphs, and jets 93
FIGURE 14.9
Two-jet angular distribution plotted against cosθ(Arnisonet al. 1985).
curve is the exact angular distribution predicted by all the QCD tree graphs – it actually follows the sin−4θ/2 shape quite closely.
It is interesting to compare this angular distribution with the one predicted on the assumption that the exchanged gluon is a spinless particle, so that the vertices have the form ‘¯uu’ rather than ‘¯uγμu’. Problem 14.4 shows that in this case the 1/ˆt2factor in the cross section is completely cancelled, thus ruling out such a model.
This analysis provides compelling evidence for elementary hard scatter-ing events proceedscatter-ing via the exchange of a massless vector quantum. It is
This analysis provides compelling evidence for elementary hard scatter-ing events proceedscatter-ing via the exchange of a massless vector quantum. It is