An Introduction to Quantum Field Theory2.pdf - Part III Non-Abelian Gauge Theories Chapter 14 Invitation The Parton Model of Hadron Structure In Part II

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Unformatted text preview: Part III Non-Abelian Gauge Theories Chapter 14 Invitation: The Parton Model of Hadron Structure In Part II of this book, we explored the structure of quantum field theories in a formal way. We developed sophisticated calculational algorithms ( Chapter 10 ) , derived a formalism for the extraction of scaling laws and asymptotic behavior ( Chapter 12 ) , and worked out some of the consequences of spontaneously broken symmetry ( Chapter 1 1 ) . Much of this formalism turned out to have unexpected applications in statistical mechanics. However, we have not yet investigated its implications for elementary particle physics. To do so, we must first ask which particular quantum field theories describe the interactions of elementary particles. Since the mid- 1970s, most high-energy physicists have agreed that the elementary particles that make up matter are a set of fermions, interacting primarily through the exchange of vector bosons. The elementary fermions include the leptons ( the electron, its heavy counterparts µ and r, and a neu­ tral, almost massless neutrino corresponding to each of these species ) , and the quarks, whose bound states form the particles with nuclear interactions, mesons and baryons ( collectively called hadrons ) . These fermions interact through three forces: the strong, weak, and electromagnetic interactions. Of these, the strong interaction is responsible for nuclear binding and the inter­ actions of the constituents of nuclei, while the weak interaction is responsible for radioactive beta decay processes. The electromagnetic interaction is the familiar Quantum Electrodynamics, coupled minimally to all charged quarks and leptons. It is not clear that these three forces suffice to explain the most subtle properties of the elementary fermions-we will discuss this question in Chapter 20-but these three forces are certainly the most prominent. All three are now understood to be mediated by the exchange of vector bosons. The equations describing the electromagnetic interaction were discovered by Maxwell, and their quantum mechanical implications have been treated in de­ tail in Part I. The correct theories of the weak and strong interactions were discovered much later. By the late 1950s, studies of the helicity dependence of weak interaction cross sections and decay rates had shown that the weak interaction involves 473 474 Chapter 14 Invitation: The Parton Model of Hadron Structure a coupling of vector currents built of quark and lepton fields.* It was thus natural to assume that the weak interaction is due to the exchange of very heavy vector bosons, and indeed, such bosons, the W and Z particles, were discovered in experiments at CERN in 1982. But a complete theory of the weak interaction must include not only the correct couplings of the bosons to fermions, but also the equations of motion of the boson fields themselves, the analogue, for the W and Z, of Maxwell's equations. Finding the correct form of these equations was not straightforward, because Maxwell's equations prohibit the generation of a mass for the vector particle. The proper reconcili­ ation of the generalized Maxwell equations with the nonzero W and Z masses turned out to require incorporating into the theory a spontaneously broken symmetry. Chapters 20 and 21 treat this subject in some detail, describing the interplay of vector field theories with spontaneously broken symmetry. This interplay leads to new twists and new phenomena, beyond those dis­ cussed in our treatment of spontaneous symmetry breaking in Chapter 1 1 . A complete theory of the weak interaction also requires the simultaneous in­ corporation of the electromagnetic interaction, forming a unified structure as first hypothesized by Glashow, Weinberg, and Salam. On the other hand, it was for a long time completely obscure that a theory of exchanged vector bosons could correctly describe the strong interaction. Part of the mystery was that quarks do not exist as isolated species. Their existence, and eventually their quantum numbers, had to be deduced from the spectrum of observable strongly interacting particles. But, in addition, there were complications due to the fact that the strong interactions are strong. The Feynman diagram expansion assumes that the coupling constant is small; when the coupling becomes strong, a large number of diagrams are important. (if the series converges at all ) and it becomes impossible to pick out the contributions of the elementary interaction vertices. The crucial clue that the strong interactions have a vector character arose from what at first seemed to be just another mystery, the observation that the strong interactions turn themselves off when the momentum transfer is large, in a sense that we will now describe. Almost Free Partons In Section 5 . 1 we computed the cross section for the QED process e + e - ---> µ + µ - . We then remarked that the corresponding cross section for e + e - an­ nihilation into hadrons could be computed in the same way, using a simplis­ tic model in which the quarks are treated as noninteracting fermions. This method gives a surprisingly accurate formula for the cross section, capturing its most important qualitative features. But we deferred the explanation of this puzzle: How can a model of noninteracting quarks represent the behavior of a force that, under other circumstances, is extremely strong? * For an overview of weak interaction phenomenology, see Perkins ter 7, or any other modern particle physics text. (1987) , Chap­ Chapter 14 Invitation: The Parton Model of Hadron Structure 475 In fact, there are many circumstances in the study of the strong interaction at high energy in which this force has an unexpectedly weak effect. Historically, the first of these appeared in proton-proton collisions. At high energy, above 10 GeV or so in the center of mass, collisions of protons ( or any other hadrons ) produce large numbers of pions. One might have imagined that these pions would fill all of the allowed phase space, but, in fact, they are mainly produced with momenta almost collinear with the collision axis. The probability of producing a pion with a large component of momentum transverse to the collision axis falls off exponentially in the value of this transverse momentum, suppressing the production substantially for transverse momenta greater than a few hundred Me V. This phenomenon of limited transverse momentum led to a picture of a hadron as a loosely bound assemblage of many components. In this picture, a proton struck by another proton would be torn into a cloud of pieces. These pieces would have momenta roughly collinear with the original momentum of the proton and would eventually reform into hadrons moving along the collision axis. By hypothesis, these pieces could not absorb a large momentum transfer. We can characterize this hypothesis mathematically as follows: In a high-energy collision, the momenta of the two initial hadrons are almost lightlike. The shattered pieces of the hadrons, arrayed along the collision axis, also have lightlike momenta parallel to the original momentum vectors . This final state can be produced by exchanging momenta q among the pieces in such a way that, though the components of q might be large, the invariant 2 q is always small. The ejection of a hadron at large transverse momentum would require large ( spacelike ) q2, but such a process was very rare. Thus it was hypothesized that hadrons were loose clouds of constituents, like jelly, which could not absorb a large q2. This picture of hadronic structure was put to a crucial test in the late 1960s, in the SLAC-MIT deep inelastic scattering experiments. t In these ex­ periments, a 20 GeV electron beam was scattered from a hydrogen target, and the scattering rate was measured for large deflection angles, corresponding to large invariant momentum transfers from the electron to a proton in the tar­ get. The large momentum transfer was delivered through the electromagnetic rather than the strong interaction, so that the amount of momentum delivered could be computed from the momentum of the scattered electron. In models in which hadrons were complex and softly bound, very low scattering rates were expected. Instead, the SLAC-MIT experiments saw a substantial rate for hard scat­ tering of electrons from protons. The total reaction rate was comparable to what would have been expected if the proton were an elementary particle scat­ tering according to the simplest expectations from QED. However, only in rare cases did a single proton emerge from the scattering process. The largest part t For a description of these experiments and their ramifications, see J. I. Friedman, H. W. Kendall, and R. E. Taylor, Rev. Mod. Phys. 63, 573 ( 1991). 476 Chapter 14 Invitation: The Parton Model of Hadron Structure deep inelastic of the rate came from the region of phase space, in which the electromagnetic impulse shattered the proton and produced a system with a large number of hadrons. How could one reconcile the presence of electromagnetic hard scattering processes with the virtual absence of hard scattering in strong interaction pro­ cesses? To answer this question, Bjorken and Feynman advanced the following simple model, called the Assume that the proton is a loosely bound assemblage of a small number of constituents, called These include quarks ( and antiquarks , which are fermions carrying electric charge, and possibly other neutral species responsible for their binding. By assump­ tion, these constituents are incapable of exchanging large momenta q 2 through the strong interactions. However, the quarks have the electromagnetic inter­ actions of elementary fermions, so that an electron scattering from a quark can knock it out of the proton. The struck quark then exchanges momentum softly with the remainder of the proton, so that the pieces of the proton ma­ terialize as a jet of hadrons. The produced hadrons should be collinear with the direction of the original struck parton. The parton model, incomplete though it is, imposes a strong constraint on the cross section for deep inelastic electron scattering. To derive this con­ straint , consider first the cross section for electron scattering from a single constituent quark. We discussed the related process of electron-muon scat­ tering in Section and we can borrow that result. Since we imagine the reaction to occur at very high energy, we will ignore all masses. The square of the invariant matrix element in the massless limit is written in a simple form in Eq. parton model: ) partons. 5. 4 , 8 4 Q 2 ( s +u ) , � (14.1) 4 £2 where s, i, are the Mandelstam variables fo r the electron-quark collision and Qfori isa the electric charge of the quark in units of l ei- Recall from Eq. (5. 73) that , collision involving massless particles, s + i + = 0. Then the differential cross section in the center of mass system is 2_ l 8e4 Q� ( §2 + ii,2 ) da _ _ _ _ = __ 4 d cos OcM 28 16n £2 (14. 2 ) = no:: Q� ( 8 2 + � + £)2 ) . Or, since i = - s( l - cos OcM )/ 2, d� = 2 no:2 Q� ( s 2 + �s + £)2 ) (14.3) 82 t2 . dt (5. 7 1): 1 4 L:-- - � IMl2 = u A 2 A 2 spins u To make use of this result, we must relate the invariants s and i to ex­ perimental observables of electron-proton inelastic scattering. The kinematic variables are shown in Fig. The momentum transfer q from the electron 14.1. Chapter 14 Invitation: The Parton Model of Hadron Structure electron k ----< _ p+q _,___ _ 477 quark Figure 14.1. Kinematics of deep inelastic electron scattering in the parton model. can be measured by measuring the final momentum and energy of the elec­ tron, without using any information from the hadronic products. Since q/J. is a spacelike vector, one conventionally expresses its invariant square in terms of a positive quantity with Q, Q 2 - q2 . Then the invariant i is simply -Q 2 • (14. 4) = Expressing s in terms of measurable quantities is more difficult. If the collision is viewed from the electron-proton center of mass frame, and we visualize the proton as a loosely bound collection of partons ( and continue to ignore masses) , we can characterize a given parton by the fraction of the proton's total momentum that it carries. We denote this longitudinal fraction by the parameter e, with 0 < < For each species i of parton, for example, up-type quarks with electric charge there will be a function that expresses the probability that the proton contains a parton of type i and longitudinal fraction e. The expression for the total cross section for electron­ proton inelastic scattering will contain an integral over the value of � for the struck parton. The momentum vector of the parton is then p eP, where P is the total momentum of the proton. Thus, if k is the initial electron momentum, e 1. fi(e) Qi = + 2/3, = (14. 5 ) where s is the square of the electron-proton center of mass energy. Remarkably, e can also be determined from measurements of only the electron momentum, if one makes the assumption that the electron-parton scattering is elastic. Since the scattered parton has a mass small compared to sand Q2 , Thus e = x, where x = Q2 . 2p q (14. 6 ) (14. 7) . From each scattered electron, one can determine the values of and x for the scattering process. The parton model then predicts the event distribution Q2 Chapter 478 100 . 14 Invitation: The Parton Model of Hadron Structure 50, 13. 5 GeV 50, 16.l GeV 50 19. 5 GeV • • �· �.!! �� ·� ... .. " 10 -. "' • ' 0 0 /;. v \lo. "'h �� ��. . 0 . "" v 1000, 7. 0 GeV 10 , 9. 0 GeV 1000, 11. 0 GeV 100, 13. 5 GeV 100 , 15.2 GeV 10 , 17. 7 GeV 100, 19.4 GeV <>" v <> 1 0. 2 0. 4 x 0.6 0.8 1 Figure 14.2. Test of Bjorken scaling using the e-p deep inelastic scattering cross sections measured by the SLAC-MIT experiment, J. S. Poucher, et. al. , Phys. Rev. Lett. 3 2 , 1 18 (1974) . We plot d2 u/dxdQ2 divided by the factor ( 14.9) against x, for the various initial electron energies and scattering angles indicated. The data span the range 1 GeV 2 < < 8 GeV 2 . Q2 2 x-Q � x, fi (�), in the plane. Using the parton distribution functions evaluated at = and the cross-section formula we find the distribution (14. 3), (14. 8 ) fi (x) depend on the details of the structure of The distribution functions the proton and it is not known how to compute them from first principles. But formula ( 14.8) still makes a striking prediction, that the deep inelastic scattering cross section, when divided by the factor 1 + (1 - Q2 /xs) 2 (14. 9 ) Q4 to remove the kinematic dependence of the QED cross section, gives a quantity that depends only on x and is independent of Q 2 . This behavior is known as Bjorken scaling. Indeed, the data from the SLAC-MIT experiment exhibited Bjorken scaling to about 10% accuracy for values of Q above 1 GeV, as shown in Fig. 14. 2 . Bjorken scaling is, essentially, the statement that the structure of the proton looks the same to an electromagnetic probe no matter how hard the proton is struck. In the frame of the proton, the energy of the exchanged Chapter virtual photon is m 14 Invitation: The Parton Model of Hadron Structure p . q = Q2 q0 = --;;:2xm' 479 ( 14.10) where is the proton mass. The reciprocal of this energy transfer is, roughly, the duration of the scattering process as seen by the components of the pro­ ton. This time should be compared to the reciprocal of the proton mass, which is the characteristic time over which the partons interact. The deep inelastic regime occurs when » that is, when the scattering is very rapid com­ pared to the normal time scales of the proton. Bjorken scaling implies that, during such a rapid scattering process, interactions among the constituents of the proton can be ignored. We might imagine that the partons are approxi­ mately free particles over the very short times scales corresponding to energy transfers of a GeV or more, though they have strong interactions on longer time scales. q0 m, Asymptotically Free Partons The picture of the proton structure implied by Bjorken scaling was beautifully simple, but it raised new, fundamental questions. In quantum field theory, fermions interact by exchanging virtual particles. These virtual particles can have arbitrarily high momenta, hence the fluctuations associated with them can occur on arbitrarily short time scales. Quantum field theory processes do not turn themselves off at short times to reveal free-particle equations. Thus the discovery of Bjorken scaling suggested a conflict between the observation of almost free partons and the basic principles of quantum field theory. The resolution of this paradox came from the renormalization group. In · Chapter 12 we saw that coupling constants vary with distance scale. In QED and <jJ4 theory, we found that the couplings become strong at large momenta and weak at small momenta. However, we noted the possibility that, in some theories, the coupling constant could have the opposite behavior, becoming strong at small momenta or large times but weak at large momenta or short times. We referred to such behavior as asymptotic freedom. Section 1 3 . 3 dis­ cussed an example of an asymptotically free quantum field theory, the nonliri­ ear sigma model in two dimensions. The problem posed in the previous para­ graph would be resolved if there existed a suitable asymptotically free quan­ tum field theory in four dimensions that could describe the interaction and binding of quarks. Then, at least to some level of approximation, the strong in­ teraction described by this theory would turn off in large-momentum-transfer or short-time processes. At the time of the discovery of Bjorken scaling, no asymptotically free field theories in four dimensions were known. Then, in the early 1970s, 't Hooft, Politzer, Gross, and Wilczek discovered a class of such theories. These are the non-Abelian gauge theories: theories of interacting vector bosons that can be constructed as generalizations of quantum electrodynamics. It was subsequently shown that these are the only asymptotically free field theories 480 Chapter 14 Invitation: The Parton Model of Hadron Structure in four dimensions. This discovery gave the crucial clue for the construction of the fundamental theory of the strong interactions. Apparently, the quarks are bound together by interacting vector bosons (called gluons) of precisely this type. However, these gauge theories cannot precisely reproduce the expecta­ tions of strict Bjorken scaling. The differences between the free parton model and the quantum field theory model with asymptotic freedom appear when one moves to a higher level of accuracy in measurements of deep inelastic scattering and other strong interaction processes involving large momentum transfer. In an asymptotically free quantum field theory, the coupling con­ stant is still nonzero at any finite momentum transfer. In fact, the final evo­ lution of the coupling to zero is very slow, logarithmic in momentum. Thus, at some level, one must find small corrections to Bjorken scaling, associated with the exchange or emission of high-momentum gluons. Similarly, the other qualitative simplifications of hadron physics at high momentum transfer-for example, the phenomenon of limited transverse momentum in hadron-hadron collisions-should be only approximate, receiving corrections due to gluon ex­ change and emission. Thus the predictions of an asymptotically free theory of the strong interaction are twofold. On one hand, such a theory predicts quali­ tative simplifications of behavior at high momentum. But, on the other hand, such a theory predicts a specific pattern of corrections to this behavior. In fact, particle physics experiments of the 1970s revealed precisely this picture. Bjorken scaling was found to be only an approximate relation, show­ ing violations that correspond to a slow evolution of the parton distribu­ tions fi(x) over a logarithmic scale in Q 2 • The rate of particle ...
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