quark parton modeli-galileo.phys.virginia.edu/~rjh2j/misc/rpv42i8p1285.pdf · 2005-11-07 · the...

Rep. Prog. Phys., Vol. 42, 1979. Printed in Great Britain

The quark parton modeli-

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Rutherford LaboratoryT, Chilton, Didcot, Oxon OX1 1 OQX, UK and Department of Physics, Argonne National Laboratory, Argonne, Illinois 60439, USA and Department of Physics, University of Virginia, Charlottesville, Virginia 22901) USA

Abstract

The evidence that high-energy collisions of subatomic particles reveal the presence of quarks inside the proton and neutron is reviewed. Low-energy electron scattering from nuclei is described with reference to the manner in which it reveals the presence of protons in the nucleus ; high-energy electron-proton interactions are then discussed and the presence of partons in the proton revealed by analogy. Then the basic ideas of the parton model are described and the evidence that partons are, in fact, quarks and electrically neutral gluons is discussed. The model is then applied to a variety of processes and is shown to correlate an enormous amount of data. Recent clues that the quark parton model is an approximation to a more fundamental theory are described, with particular reference to the possibility that quarks and gluons are the quanta of a field theory of strong interactions (quantum chromodynamics) analogous to the role that electrons and photons play in QED.

This review was received in September 1978.

t Work supported in part by United States Department of Energy. $ Permanent address.

0034-4885/79/081285 + 51 $05.00 0 1979 The Institute of Physics

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Contents Page

1. Introduction . . 1287 2. Electron scattering . . 1289

2.1. Electron-carbon scattering . . 1291 2.2. Electron-proton scattering . . 1292

3. The parton model in lepton-induced reactions . . 1293 3.1. Basic hypotheses . . 1293 3.2. Electron scattering in the parton model . 1294 3.3. Do electron scattering data support the parton model? . . 1297 3.4. Neutrino interactions . . 1301 3.5. Are the spin-4 partons quarks? . . 1306 3.6. Electron-positron annihilation . . 1309 3.7. Quark distribution functions and counting rules . . 1311 3.8. Inclusive production of hadrons . . 1315

4. Hadronic interactions at large momentum transfer . . 1321 4.1. Inclusive processes . . 1321 4.2. The Drell-Yan process . . 1324 4.3. t+h production . . 1326 4.4. Large-angle exclusive hadronic scattering . . 1327

5 . Particle production at small momentum transfer . . 1325 6. Modifications of the naive parton model . . 1329

6.2. Asymptotically free gauge theories and partons . . 1330 7. Conclusions . . 1332

6.1. Mass scale effects . . 1330

References . . 1333

The quark parton model 1287

1. Introduction

Four centuries before Christ, the Greeks had suggested that all matter was built from indivisible entities. They named these ‘atoms’.

In the seventeenth century the growth of chemistry showed that the majority of the known substances were compounds of a few basic elements. Dalton proposed that each element was made up from a different kind of atom, any quantity of a given element being a collection of the atoms of that element.

Many different elements were discovered. Using them as building blocks the phenomena of chemistry could be understood. Instrumental in this was Mendeleev’s phenomenological observation that the elements could be ordered in certain regular patterns-the periodic table. The subsequent discovery that atoms were not truly elementary but were built from electrons and nuclei gave a theoretical basis for Mendeleev’s table.

The electron can today still claim to be a truly elementary particle. The nucleus is not-it is composed from neutrons and protons. Nor are these the basic building blocks. Through the 1950s many and varied particles were found. Some of these were excited states of the proton and neutron (rather like there exist different energy levels of the hydrogen atom). Others appeared to have claim to aristocaracy just as much as the neutron (e.g. the E, A, E).

Gell-Mann (1961, see Gell-Mann and Ne’eman 1964, p l l ) noticed that the particles seemed to form patterns which are known as the ‘Eightfold Way’. These patterns can be thought of as the particle physics analogue of Mendeleev’s periodic table. Just as the periodic table emerges if atoms are made from electrons and nuclei, so the Eightfold Way emerges if the proton, neutron, X, etc, are all built from more fundamental entities called quarks (Gell-Mann 1964, Zweig 1964). Furthermore, just as the electron plus nucleus model of atoms generates the spectroscopy of hydrogen, so the quark model generates the spectroscopy of particles. A detailed description is given in a previous review in this journal by Hendry and Lichtenberg (1978), and Close (1978).

Returning to atomic physics, there is a second place where history has repeated itself. I n order to determine where the electrical charge within an atom was located, a- and P-particles were fired at the atoms. The 13-particles (electrons) are light and passed clean through with only small deflections. This caused Lenard to observe that atoms contain vast empty spaces. However, massive a-particles were seen to suffer violent collisions which scattered them through large angles. This caused Rutherford to suggest that the positive charge in an atom was localised in a massive compact nucleus. The energy and angular distribution of the a-particles suggested that they were scattered by a positively charged nucleus which is localised within a sphere of dimension 5 10-12 cm. This experiment can be legitimately regarded as the first time that the nucleus was ‘seen’. Subsequently, the proton was isolated in the laboratory.

History was repeated when in the 1960s very high-energy beams of electrons were fired at proton targets and were found to undergo violent collisions more frequently than would have been expected if the proton’s charge was distributed diffusely (Panofsky 1968). Instead, these violent collisions suggested that the proton’s charge

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was localised on discrete scattering centres (Bjorken 1967, Feynman 1969, Bjorken and Paschos 1969). The energy and angular distribution of the scattered electrons exhibit a correlation called ‘scaling’ (Bjorken 1969) which suggests that the scattering centres have no internal structure themselves. One says that they are ‘point-like’. This may mean either that we have indeed found the ultimate indivisible building blocks of matter, or that present accelerators are not powerful enough to reveal any substructure which these entities may have. Extrapolating from past experience, one may suspect that the latter will turn out to be the case.

These ‘parts’ of the proton that are being revealed in these experiments have been given the name ‘partons’. The data suggest that the partons may be of two kinds. The first kind are spin-3 fields which appear to be the quarks that are familiar from particle spectroscopy. The second kind are electrically neutral fields which have been called gluons. I t is suspected that these may be responsible for binding (gluing) the quarks together to form the proton.

Historically it has always been possible to isolate in the laboratory the constituents of any system under study: electrons can be removed from atoms, protons and neutrons from the nucleus, and so on. However, no quarks or gluons have ever been removed from the proton and manifested as free entities. If they had, then much of the work that has led to this review would probably not have taken place. It is because we have not isolated in the laboratory the ‘parts’ of the proton that so much effort and ingenuity has been required to discover, by indirect means, the quantum numbers of these ‘partons’ and the rules governing their interactions.

As the parton models of the proton were first taken seriously after the phenomenon of ‘scale invariance’ was discovered in electron scattering off protons (Panofsky 1968) we shall begin by reviewing electron scattering. We shall study electron scattering from a nucleus (the protons and neutrons are the partons of the nucleus) in order to illustrate the phenomena of interest in a relatively familiar context. This will lead naturally into the high-energy electron scattering from protons which is the point of departure for the modern parton models.

In $3 we shall review the accumulated evidence from electron and neutrino scattering data and electron-positron annihilation that supports the picture of a proton composed of quarks which possess a three-fold ‘colour’ degree of freedom and which interact by exchanging electrically neutral ‘gluons’. The momentum distribution of the quarks within the proton can be deduced from the data. I t appears that there is also a ‘sea’ of virtual quark-antiquark pairs in the proton. It is possible that these are particle-hole states produced from the vacuum, analogous to the vacuum polarisation of electron-positron pairs familiar in quantum electrodynamics. These quark pairs arise naturally in the recently developed field theory of quantum chromodynamics whose basic fermion fields are coloured quarks interacting via coloured vector gluons, analogous to the charged fermions interacting via photon exchange in QED.

This theory, which has many attractive and profound features, is currently under intense theoretical and experimental study. Its supporters claim that it may indeed be the correct theory of strong interactions and similarities between it and the unified theories of weak and electromagnetic interactions (Glashow 1961, Weinberg 1967, Salam 1967, ’t Hooft 1971a, b) suggest that a grand synthesis of these forces may be possible.

It is conjectured, but not yet fully proven, that quarks cannot exist in isolation in such a theory. As one pulls quarks apart the QCD force between them grows (like


an elastic band). I t is conjectured that the force will continue to grow indefinitely. I n such a case, the quarks will be ‘confined’. At very short distances the opposite effect is found. It has been proved that QCD predicts that the quarks in the proton appear to be (almost) like free particles when viewed by very energetic high momentum probes (i.e. short time and distance scales are resolved). In QCD this freedom is only found at asymptotically huge energies (‘asymptotic freedom’) (’t Hooft 1972 (see Politzer 1974, footnote 3) , Politzer 1973, 1974, Gross and Wilczek 1973, 1974).

The deviations from this asymptotically free behaviour are predicted to be so gradual as one comes to large, but finite, energies that, to a first approximation, one may regard the quarks as free even at present energies. This is essentially the idea of the parton model. I t is a straightforward way of correlating an enormous amount of data. Corrections to it that are necessary because the quarks are only free asymptotically can be investigated within the framework of quantum chromodynamics.

2. Electron scattering

An electron scatters from an atomic, nuclear or proton target by exchanging a photon with the target. The photon transfers energy and momentum from the electron to the target. The response of the target to this will depend upon the nagni- tude of the photon’s energy and momentum; for example, the target may recoil without being excited (elastic scattering), it might be excited into a resonant state or it might break up.

One can select events where the electrons have been scattered through some angle 6. If the incident electrons had energy E then one simply counts the number of events where the final electrons have energy E‘ and then plot the event rate, or cross section, d2a/d6 dE’. As E’ decreases, so the energy transferred to the target increases. In turn the target is excited to ever higher energy levels.

The qualitative behaviour of d2o/d6 dE‘ as E’ decreases is quite similar for atomic, nuclear and proton targets. Therefore we will look first at what happens when electrons scatter from a carbon nucleus. The observed behaviour is readily interpreted and gives clues as to how we might in turn interpret the very high-energy electron-proton scattering data and learn about the internal structure of the proton.

Before examining the data some kinematics should be discussed. The energy transfer is:

V Z E - E ’ (2.1) where E and E’ are the initial and final electron energies in the laboratory (defined as being the frame where the target is at rest). It is conventional to also use the invariant four-momentum squared of the photon :

q 2 = (k- k’)2- - 2EE’(1 -cos 6) = - 4EE’ sin2 ($0) (2.2) where these momenta are illustrated in figure 1. This quantity q2 is the invariant mass of the virtual photon exchanged. T o employ positive quantities it is conventional to define instead:

Q2zz -q2=4EE‘ sin2 (30). (2.3) These formulae have assumed that the electron energies E, E’ are both much larger than the electron mass. This is, in practice, an excellent approximation.

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Figure 1. Electron scattering by the exchange of one photon.

The target will, in general, be excited into a state with mass W. From the photon- target vertex in figure 1 we see that:

w2= ( p + q ) 2 = p2-k 2pq + 42 . (2 -4)

W'=M2+2Mv-Q'. (2 .5 )

Since p 2 3 M2 (the target mass squared) and, in the laboratory, p q = M v then:

Armed with these preliminaries, we can study the nuclear data (figure 2).

C U D

U

. c1 1 ,I ~:;i-~ias,tic I ~~,

E ' ( MeV)

, Elastic '1 I

( , , I /

' 141 , , , , , ,* : I* 1 1 , I ' t ,

50 100 150

Figure 2. Electron scattering from a carbon nucleus. The electron energies are 187 MeV and 194 Mev in (a) and (b). The scattering angle B is 80" in (a) and 135" in (b). Elastic scattering and resonance production enhancements are visible in (a) (low Q2), whereas the higher Qz data (b) are dominated by the quasi-elastic scattering from the nuclear constituents.


2.1. Electron-carbon scattering

Electrons with energy of about 190 MeV are scattered through 80" by the electromagnetic field of a 12C nucleus. If they are scattered elastically by the nucleus then :

W2= M2-+2Mv = Q2. (2.6)

From equation (2.3) for Q2 we see that elastic scattering occurs if:

E'=ME[E(l-cos O)+M]-l=186 MeV (2.7)

if E = 190 MeV, O = 80". The elastic scattering is clearly visible when E'= 186 MeV. As more energy is lost by the electron, so v increases and Wz increases. Carbon

resonances are excited and are indeed visible in the data when E'=180 or 177 MeV.

The elastic scattering and nuclear resonance excitations are both coherent responses of the whole nucleus. The nucleus is a composite of neutrons and protons (nucleons). Hence, if the energy and momentum transfer are correlated in a certain way then incoherent elastic scattering from the nuclear constituents can take place.

Elastic scattering from a proton occurs at a value of E' given by equation (2.7) but where M = Mproton= 940 MeV in place of the nuclear mass: this gives a value E' = 160 MeV. This is not shown in figure 2(a). For 135" scattering this phenomenon occurs at E' = 140 MeV. In the data a broad enhancement in the event rate is seen spread around this value of E' (figure Z(6)). These are indeed events due to elastic scattering from the nuclear constituents. If the nucleons had been perfectly at rest in the nucleus then there would have been a sharp peak at 140 MeV. I n practice, the nucleons have a Fermi momentum in the nucleus, and hence have a probability of moving towards or away from the incident photon and so give rise to the spreading around 140 MeV. This broad hump is known as the 'quasi-elastic peak'.

As more energy is given up by the electron so the protons in the nucleus can themselves be excited to resonance states and the inelastic production of pions occurs. These contribute to the data at the lowest values of E' shown in figure 2(b).

For future reference we should note that when E = 190 MeV and 8= 80" the typical order of magnitude of Q2 is around 0.06 GeVz. If we keep the same incident energy but now select events where the electron has been scattered through 135" then the typical order of magnitude of Q 2 is larger, about 0.1 GeV2. The data at this value of Q 2 (figure 2(b)) appear to be dominated by the quasi-elastic scattering from the nuclear constituents (E'? 140 MeV). There is very little sign of elastic nuclear scattering or coherent resonance excitation of the nucleus. Stated quantitatively, the area beneath the elastic peak is some 300 times smaller than that under the quasi-elastic peak, whereas in the 80" scattering, smaller Q2 data, the quasi- elastic peak and the genuine nuclear elastic peak were comparable in magnitude.

Physically what has happened is the following. As Q2 increases, the momentum striking the nucleus increases for any fixed W final-state mass. The probability that the nucleus stays together to coherently produce a final nuclear resonance state decreases. This decrease of elastic scattering as Q2 increases is typical of any composite system. The Q2 dependence of the system's response is known as the form factor of the system.

A nuclear form factor tends to die exponentially in Q2-it is very unlikely that

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if you hit a nucleus hard it will stay together. Equivalently one can picture this via the uncertainty principle-with increasing 2/Q2 the wavelength of the probe decreases. At small Q2 there is not sufficient resolution to see the nucleus as other than a point. Hence, coherent nuclear scattering occurs. However, once the Q2 is large enough that there is sufficient resolution to see the granular structures of the nucleus, the interaction will be over a short distance-a nucleon size, say-and so incoherent scattering off the constituents is more probable.

This indeed fits in quantitatively with the nuclear data. At Q2=0.1 GeV2 the photon is able to resolve the internal structure of the nucleus. However, a proton whose size is typically a length scale of about 10-13 cm still appears to be a point-like particle when viewed by such a photon. For this reason varying Q 2 from 0.05 GeV2 to 0.1 GeV2 has killed the elastic and coherent nuclear cross sections whereas the quasi-elastic scattering off the protons is, almost, Q2-independent. Over this range of Q 2 the nuclear constituents appear to be effectively point-like elementary particles. If one increases Q2 to the order of 1 GeV2 then the quasi-elastic peak begins to rapidly disappear. The resolution is now sufficient to probe the internal structure of the proton. The proton is seen not to be an elementary particle. It too has a form factor.

Now let us study what happens when the proton is itself used as a target.

2.2. Electron-proton scattering

A nucleus has a size typically N 10-12 cm. Hence energy and momentum transfers of the order of only some hundreds of MeV were needed to reveal the internal nuclear structure. T o reveal the structure in a proton better resolution is required. Hence, higher momentum transfers are called for.

T o probe inside the proton the MeV electron energies are replaced by beams with tens of GeV. Qualitatively similar features are seen in these data as were found in the nuclear case. The only essential difference is the increase in energy scale or, equivalently, resolving power.

Recall the nuclear case. The elastic nuclear peak died out rapidly as Q 2 was increased from 0.05 to 0.1 GeV2. The elastic scattering from the protons was almost independent of Q 2 over this range. Now irradiate proton targets with 20 GeV electrons. As Q 2 increases from 1 to 10 GeV2 the electron-proton eiastic scattering dies rapidly (see, for example, Perkins 1977)-analogous to the elastic nuclear scattering in the 0.1 GeV2 region. As the energy given up by the electron beam increases so proton resonances are produced. These coherent excitations of the proton also appear to die out as 8 2 is increased through the 1-10 GeV2 range. Finally, at the largest energy transfers is seen an analogue of the nuclear quasi-elastic peak and this appears to be Q2-independent through the range of Q2 of 1-10 GeV2.

From the experience with the nuclear example one may guess that this is quasi- elastic scattering from constituents of the proton. The range of 8 2 covered is not able to (yet) resolve any structure that these constituents may have. Hence, at the present level of resolution they appear to be point-like.

These point-like constituents, or parts of the proton, have been named ‘partons’. Given that there is evidence from the spectroscopy of particles to suggest that the particles, in particular the proton, are clusters of quarks then it is natural to suppose that these partons may be quarks. We will survey the evidence that supports this.


3. The parton model in lepton-induced reactions

3.1. Basic hypotheses

The essential assumption of the parton model is that when large energy and momentum ( v and 82) are transferred by currents to a nucleon, then the currents interact incoherently with quasi-free point-like constituents (partons) in the proton or neutron (nucleons).

Our nuclear experience suggests that the above may be true so long as v and Q2 are large enough to resolve the target structure but not so large that the constituent structure is revealed. For nucleon targets, this appears to be Q2 of about 1-10 GeV2. There are hints that when Q22.5 GeV2 the first signs of corrections to this naive parton model are beginning to appear. We shall return to these later and first con- centrate on that region of Q 2 where the constituents appear to be point-like and quasi-free.

Physically, at these large momentum transfers the time scale T which controls the interaction will be of order (Q2)-1/2. One hopes intuitively that for large enough Q2 (Q~-+.o modulo the above caveat) this time will be much less than the typical time scales of the interactions among the partons. Hence the large Q2 current sees a nucleon which is effectively a ‘frozen’ collection of quasi-free constituents.

This assumption works reasonably well when comparing the parton model with data over the limited range of Q2 described above. However, there is a strong feeling that quarks are point-like spin-& particles which interact by exchanging vector gluons analogous to the way that spin-+ electrons interact by vector photon exchange. If this is the case the above proposition will not be realised because, however large Q2 may be, and in turn however short T may be, there will always be residual quark- gluon interactions taking place at yet shorter time scales. The parton model dogma ignores this. T o a first approximation this is a posteriori justified by data; recent theoretical research has concentrated on the Q2-dependent corrections to the naive model that are expected to arise as a result of the field theoretic interactions of the quarks. For the bulk of this review we will be concerned with the naive model where the partons (quarks) are regarded as quasi-free in large v, Q2 (deep inelastic scattering) reactions. Corrections to this will be discussed in $6.2.

I n the atomic and nuclear case the target is shattered into its constituents in the deep inelastic region. Hence final-state interactions are negligible. The important difference in the case of proton targets is that the quarks appear to be permanently confined and so final-state interactions cause the quarks to reform into hadronic systems. If these interactions act on a time scale T’ T then they can still be neglected in calculating the total cross section. There is some theoretical justification for this; in particular, in a world of one space and one time dimension one can solve exactly an interacting field theory of quarks and gluons (two-dimensional quantum chromodynamics) in which the quarks are indeed confined while the naive parton model also works. Also there is an oft quoted classical analogy, namely that of a particle at one end of a slack elastic band whose other end is rigidly attached to some point. If the particle is struck then it recoils as if it were a free particle. Only later does the elastic band become taut and prevent the particle escaping. The total cross section is essentially the probability that something happens. I n this example this is the probability that the particle is struck and recoils ; the subsequent constraining by the tightening of the elastic band is irrelevant in the probability calculation (Llewellyn Smith 1977).

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Note that these remarks are only valid for the total cross section where final- state products are summed over. For any particular final state, e.g. elastic scattering, the final-state interactions are important (they create the particular state being studied). The greater the number of partons in the system then the greater the number of final-state interactions needed to reform the system after the current has interacted. This implies that the rate of fall-off of the elastic form factor with increasing Q 2 is faster the larger the number of constituents (Brodsky and Farrar 1973, 1975, Matveev et a1 1973). There is some support for this empirically (Sivers et a1 1976).

3.2. Electron scattering in the parton model

The scattered electron emits a current whose energy in the laboratory is v and whose mass squared, - Qz, is Lorentz-invariant (frame-independent). This current sees the target as an instantaneous cloud of free partons.

We can boost the whole system to a frame where the nucleon is moving with large (infinite) momentum. I n such a frame the target momentum will be much larger than its mass and so its four-momentum may be written:

Pp= (P; OT, P ) (3 ‘ 1)

where we use the convention p = [ t ; xy , x]. Hence for four-momenta:

(3.2)

If the target moves in the x direction with momentum p z then we may write the three-momentum of a parton as follows:

p = X P z + P T . (3.3)

Hence x is the ratio of the parton’s x component of momentum to that of the target and p~ is the component of the parton’s momentum that is transverse to the target’s momentum. It has conventionally been assumed that pr is bounded (to a few hundred MeV) as pz+ W . If one therefore neglects p~ and also neglects the parton and target masses in this infinite momentum limit, then x is also the ratio of the parton and target four-momenta?.

The partons will, in general, have a distribution of momenta. Let f i ( x ) be the probability of finding a parton of some type i carrying a fraction x of the target’s momentum in this frame. An electron scatters elastically from such a parton by exchanging a current with mass squared 8 2 and laboratory energy v. Let the cross section for this be (d2uIdQz dv)t.

The total cross section for electron scattering from the cloud of partons (i.e. the target) will be the sum of these individual contributions:

The left-hand side of this equation is experimentally measured and the data

t We shall neglect target and parton masses andpT throughout. Hence x can be equivalently regarded as the ratio of the momentum z components or as the ratio of four-momenta. A discussion of mass effects is in $6.1.


are usually presented for the two structure functions WI(V, Q2) and WZ(V, 82) which are defined by (Bjorken and Walecka 1966):

[cos2 3eW2(v, Q2)+2 sin2 +eWI(v, Q2)]. (3.5) d2a -4.rra2 E'

dQ2dv Q4 E ______

The kinematic quantities here are as illustrated in figure 1. Physically these two structure functions control the two degrees of freedom:

(i) The total magnitude of the cross section. (ii) Its angular, 0, dependence.

We can now see what the parton model predicts for these structure functions. The master equations are (3.13) and (3.14). The reader who is not interested in their derivation may prefer to proceed there directly.

The cross section for elastic electron-parton scattering may be written (S , Qz$ parton mass squared) :

where Sa is the total electron-parton centre-of-mass energy squared and et is the electric charge of that parton. Note that if p,, are the electron and parton four- momenta, then :

S i = ( p , + P p i ) 2 ~ 2 x p , P ~ XS (3.7)

where S is the electron-target CM squared energy. The structure of equation (3.6) follows on general grounds. The exchanged

photon generates the Q-4 and this takes care of the overall dimensions. Hence, only dimensionless quantities can accompany it. It has become conventional to denote the dimensionless ratio Q2/& by y . This is directly measurable experimentally because in the laboratory frame y is the fraction of the incident lepton energy that is transferred to the target, viz :

so long as E (that 92- 2Mvx is proved after equation (3.10)). The electromagnetic scale is set by a2 and the explicit parton squared charge determines the strength of the particular contribution for that parton.

Hence, we already see from equation (3.6) that the magnitude of the total cross section will be controlled by the magnitude of the squared charges of the partons. The relative importance of WI and Wz will be determined by the function gi(QZ/&) which controls the angular dependence, in particular the W2 cos2 &e and WI sin2 +e, contributions to the total cross section. The function ga(Q2/Sz) depends upon the spins of the partons. For spin 8 it is ~[1+(1-Q2/Si)2]=~[1+(1-y)2], a form familiar from the QED calculation of electron-muon elastic scattering. For spin 0 it is (1 - Q2/Si)2 = (1 -y)2 which is familiar from the cross section for scattering of electrons in a Coulomb field. T o explicitly make this correspondence note that y E Q2/& = +( 1 - cos e) where 0 is the electron-parton centre-of-mass scattering angle.

We now have all the tools necessary to predict the behaviours of W1, ~ ( v , Q2) in the parton model. If the parton is elastically scattered (figure 3) then its (squared)

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f

Figure 3. Inelastic lepton scattering in the parton model. The target is a cloud of partons with momentum probability distribution f (x). One of these partons is elastically scattered.

mass is the same before and after the current is absorbed. Hence:

and so (3 9)

(3.10)

Since Fq is a Lorentz invariant its magnitude can be found in any frame and it will be true in all frames. In the laboratory frame Pq= Mv and hence, in general, for elastic electron-parton scattering, Q2 N 2xMv ( M being the target mass).

Now use this to rewrite equation (3 .6) :

We can now compare this with the total cross section, equation (3.4)) whereby we find that in the parton model :

(3.12)

By rewriting the angular factors in equation (3.5) in terms of Q'/Si one can compare the cross section, equation ( 3 , 5 ) , with the parton model prediction at equation (3.12). This has the consequence that:

vW2( v, Q2)+F2(x= Q2/2Mv) = C ei2xft(x) (3.13) i

for either spin-0 or spin-; partons. For spin 0, W I Z 0, while for spin 8 : MWi( v, Q2")+Fl(xr QZ/ZMv) =F~(x ) /~x . (3.14)

The ratio of 2M'v and Qz is conventionally known as w(=2Mv/Q2). Hence the parton model predicts that for $xed w the structure functions will be independent of Qz. (This is the explicit version of our earlier remarks that loosely stated that the 'interaction' would be Q2-independent.)

Note that w-1, QZ/ZPq is a dimensionless ratio of Lorentz-invariant kinematic quantities. If elastic electron-parton scattering is the dynamical mechanism responsible for deep inelastic electron-nucleon interactions, then Qz/2Pq is also equal to x (the fraction of the target momentum that is carried by the struck parton, equation


(3.3)). For this reason it has become conventional to denote the kinematic ratio Q212Pq by x.

The x( = Q212Pq) dependence of the structure function F ~ x ) can be measured. This is then directly interpretable as a measure of the x (momentum) distributions of the partons in the target (equation (3.13)).

3.3. Do electron scattering data support the parton model?

What is the evidence that supports the idea that there are point-like spin-3 constituents in the proton? I n the early 1970s data were accumulated at w = 4 and the Q2 independence of vW2 is clearly seen in figure 4. This suggested that point-like constituents are indeed present, The support for their spin-& nature comes from comparing vW2 and W1. This is traditionally exhibited in a quantity:

R a ~ / q (3.15)

2 6 Q2( GeV12

0

Figure 4. Q2 independence o f vW2 when W =2Mv/Q2 is fixed at 4 and Q2 varies from 1 to 8 GeV2.

which is the ratio of the photoabsorbtion total cross sections for photons which have helicity zero (a~) or l ( a ~ ) . These two dynamical degrees of freedom cor- respond to the two degrees of freedom expressed by the two structure functions W1, vW2. The explicit relation among them is:

(3.16)

Hence, the vanishing of W1 for spin-0 partons corresponds to:

oL/aT = 00 (3.17)

while the relation Fz E 2xF1 for spin-3 partons corresponds to: Q2-m

OL/OT---~ 0. (3.18)

The data (figure 5) showed that OL/UT is indeed small and hence very little, if any, of the proton's charge is carried by spinless constituents. Spin 3 is clearly favoured.

This does not prove that spin 3 is the magic solution. One could undoubtedly choose higher spins and adjust couplings such that a ~ / a ~ came out small. However, until such time as one is forced into such an unmotivated game, Occam's razor

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Figure 5. Data on R= UL/UT (from Atwood e t U Z 1976).

certainly favours spin 8. Theoretically this is pleasing because we had a priori reasons to suspect that spin-4 quarlcs are present in the proton (Hendry and Lichtenberg 1978).

In the last year or two accumulating data (figure 9 in Hand (1977)) have begun to give information that enable refinements of the above first-order picture to be made. I n particular, there are several assumptions that were made in the above which are not exactly realised in the experiments. For example, we have assumed that Q 2 , v - tco . I n practice, they are finite and so it is not clear that it is a good approximation to ignore parton masses and transverse momenta. An illustration of the importance of worrying about their p~ comes if we consider the physical significance of UJ,/UT.

Consider spinless partons as an example. If the photon and parton momenta are collinear then a spin-0 parton cannot absorb a photon with helicity f 1. The magnitude of angular momentum projected along this axis will be 1 before the collision and zero after. This violates angular momentum conservation. Hence UT = 0 and R-tco. However, if the parton has a component of momentum transverse to the photon direction then the projection of its orbital angular momentum along the x axis (photon momentum) can allow the absorption to occur without violating angular momentum conservation. Hence UT#O. However, R will still be very large and so, presumably, we can still safely conclude that spin-0 partons are not significant empirically,

For spin-4 particles the inclusion of transverse momenta and parton mass effects yields (Feynmann 1972, Nachtman 1977, Close 1978):

(3.19)

Notice that if ( k T 2 ) is of the order of only 0.2 GeVz then the 4 in the numerator causes UL/UT to be quite different from zero if 82 is 1-5 GeV2. It is not entirely clear what m2 means in this formula. From spectroscopy studies it appears that m2~300 MeV for U, d flavoured quarks (such as occur in the proton). Hence m2 could also be of the order of 0.1 GeV2. Feynman (1972) suggests adding f A to the m2 in order to include binding energy correction uncertainties. Hence, neglecting


kT2 relative to Q2 in the denominator:

' J ~ - 4 ( k ~ ~ + m ~ + -- 41 'JT Q' at fixed U

(3.20)

Feynman suggests k ~ 2 ~ 0 . 2 5 GeV2 independent of x 'as in hadronic collisions' and m2 of the same order (or else one could not explain why kT2 is so small). This gives :

5 ~ ~ 2 & 44 -~

UT- Q' which is not inconsistent with the data.

Data in the last year have created some controversy. These are illustrated in figure 6. The value of R seems rather large, in particular at large x(=w- l ) . Many possibilities are being discussed. For example, is ( K T ) a function of x (Landshoff 1976, Close et a1 1977, Davis and Squires 1977, Gunion and Soper 1978, Hughes

Figure 6. Large Qz data on R (from Hand 1977). (a) as a function of x. (b) as a function of &2 (GeV2).

1978, Carlitz et a1 1977)? Does (h2) grow with Q 2 (Lam and Yan 1977, Kogut and Shigemitsu 1977, Politzer 1977, Fritsch and Minkowski 1978)? (There is some indication that this may be so in the high-energy production of lepton pairs pp+p+p-. . . , $4.2.) Are we seeing here the effects of interactions between the quarks (in a field theory of quarks and vector gluons, like QED, one can never com- pletely ignore these interactions except asymptotically as Q 2 + CO)? One must also bear in mind that the extraction of 'JL/5T from data is much more difficult than that of measuring, say, F1, 2 separately. One must take the difference of them to obtain OL/5T whereas, by taking data at very small or very large 8, one is, to a good approximation, measuring F2 and F1 directly (barring pathological behaviour in R). So the data on UL/UT should be treated with caution and an understanding of how they were obtained should be sought before attaching too much weight to one or other refinements of the basic model.

I n summary, 'JLl'JT data continue to support the idea that spin-fr chargedpartons are dominant.

More significant are the discoveries that while 'scaling' (the QZ independence at fixed x or U ) is a good first approximation to the data, it is violated by a few per

1300 F E Close

cent. In particular, it appears to be the case that for large x ( 2 $) Fz(x, 82) decreases as QZ increases while at small x it increases (figure 7).

A rise in Fz(x, 82) at small x is expected quite generally. For a fixed Q2, small x is large v and so one can produce new flavours of massive particles (charmed particles). This will cause an increase in the cross section as the threshold is traversed. It has been widely argued that the production of these massive states is suppressed at small Qz but not at high Q2 (i.e. Q2 must be much larger than typical squared masses of the charmed particles before their masses can be neglected and scaling occur). Therefore there is a slow approach to the Q2-independent regime above charm threshold. Hence the rise in Fz(x, Qz) there (Gounaris et a1 1975, Close et a1 1976).

I X

Figure 7. Scaling violation in F2(x, Q2) (from Perkins et a2 1977). Exact scaling would yield a flat line. 0 , ep, pp; 0, "p.

Alternatively this may be an indication that the naive free parton model is only an approximation to the truth over a limited range of 82. If we recall the nuclear example, we found there an apparent Q2 independence (scaling) in the quasi-elastic scattering from the constituents (nucleons) so long as Qz lay in the range 0 4 - 0 . 1 GeV2. When Q2 was increased beyond this, the current started to resolve the structure of these constituents. This led to a violation of scaling. Similarly one expects that the approximate scaling found in the range Q22: 1-5 GeVZ in the case of proton targets will be violated as the structure of the partons is resolved.

One possibility is that the partons (quarks) are built from pre-quarks which have a much smaller size than the quarks, analogous to the nucleus built from nucleons. If so, we are now seeing the beginning of a transition from the scaling behaviour at quark level to a new scaling regime when the pre-quarks are finally manifested. An alternative possibility is that the quarks interact with vector gluons (analogous to the interaction with photons in QED). Quark 'structure' in the form of virtual gluons and dressing with Q& pairs will be revealed continuously as Q2 increases and the quarks are probed at shorter distances. The rise in Fz(x, Qz) at small x and fall at large x as Q2 increases is not inconsistent with such a picture (Kogut and Susskind 1974, Llewellyn Smith 1975, Altarelli and Parisi 1977). This is discussed in $6.2.


As a result, current opinion is that at fixed x and Qz- 1-10 GeVz there is a slow Qz dependence and hence scaling is an approximate or transient phenomenon. If the proton had contained genuinely free point-like partons, then the scaling would presumably have been exact. T o the extent that we have found an island in the data where the scaling is almost exact then the parton model is a good idea to pursue in attempting to understand the message of the data. As data accumulate at higher energies and Qz so we will move further away from this island. For a while the parton model will still be a good first approximation, but we will increasingly reach into shorter distance scales where the parton model will probably be no more relevant than is, say, molecular physics to deep inelastic scattering off nuclei.

For the bulk of this review we shall stay within sight of the scaling island. What may lie over the horizon will be discussed later.

3.4. Neutrino interactions

3.4.1. Charge changing weak currents. The inelastic scattering of neutrinos yields data that complement and supplement the electron data. In the electron scattering the basic subprocess was the quasi-elastic electromagnetic scattering of electrons from the partons. Hence the structure functions were proportional to the sum of the squared charges of the partons. Neutrinos couple to the isospin of the partons. The neutrino and electron data together enable one to compare the isospin and charge couplings of the partons. I t was with the first neutrino data supplementing the earlier electron data that the first strong support for the identification of partons with quarks was obtained (for the most recent data, see Perkins (1977), Holder et al (1977), Berge et a1 (1977), Bosetti et a1 (1977) and the Proceedings of the 19th International Conference on High Energy Physics, Tokyo 1978).

T o illustrate this suppose that the partons are indeed quarks with flavours denoted U, d, s, c, . . . . Support for this will emerge in subsequent sections. The neutrino- quark interaction will be assumed to be described by the Fermi proposal of 1934 (Llewellyn Smith 1974):

G d2 &= ~ J,+(x) J h ( ~ ) (3.21)

where G= lO-5/mPz is the Fermi constant describing the strength of the interaction. The current Jh is:

JA(x) = c p ( x ) ~ ~ ( l - Y ~ ) P ( x ) + c e ( 4 Y A l - ~5>e(x>

+ zz(x)yA( 1 - ~ 5 ) [ d ( ~ ) COS 0, + S(X) sin e,] + E ( x ) Y ~ ( 1 - ~ 5 ) [ d ( ~ ) sin 0, - S(X) COS e,] + ... (3.22)

where 8,cz0.26 rad is the Cabibbo angle. Notice that just as the electromagnetic quark current was assumed to have the same form as that of electrons (i.e. point-like) so the above interaction has explicitly assumed that the quarks’ weak interactions have the same space-time structure as the leptons. Note that these assumptions are non-trivial. The fact that data support them suggests that there may be some profound relationship between leptons and quarks.

If we approximate 8,czO and are below threshold for charm production then 86

1302 F E Close

we can neglect the charmed quark and the basic processes in the neutrino interactions will be :

vd-t p-u ;d+ p+zi

va-t y-d ;U -+ p+d. (3 2 3 ) Analogous to equation (3.6) in the electromagnetic case, the elementary cross sections are :

(3.24)

(3.25)

where the g and g’ are dimensionless because the G2 has dimensions i l l - 4 in agreement with the dimensionality of the left-hand side.

4‘-* ri J==O -YL---2

Figure 8. v q and v 4 interactions. The solid arrows denote the spin projections.

Explicit calculation with the interaction of equations (3 -21) and (3.22) shows that g(y) 1 and g’(y)=(l - Y ) ~ . This dependence on y is easy to understand physically : it is a consequence of angular momentum conservation, together with the fact that helicity is conserved in the interaction. Consider the vQ or v Q interaction in their centre-of-mass frame where 2y= 1 - cos d (figure 8). The neutrino-quark interaction will have J , = 0 since both have helicity - 8 and are moving in opposite directions, For neutrino interacting with antiquark, whose helicity is + 8, the total J , = - 1. The emerging and p are right handed and left handed, respectively, and hence J,. = - 1 along the Z’ axis (oriented at 6’ relative to the original x axis). This is therefore forbidden when dcm = ~ ( y = 1) since J , = - 1 initially but + 1 finally. This angular momentum argument leads to a I dll’( d ) I 2 distribution?. Since dIl’(0) = &( 1 + cos d) = 1 - y, then the (1 - y)2 distribution occurs. In the case of the neutrino-quark interaction the net J z = 0 allows the process to occur at any angle; hence an isotropic distribution.

Notice that this derivation of (doldy)( FQ) N (1 - y)2 has depended crucially on helicity conservation. This is only valid if m, = 0, or in general if its energy is much larger than its mass. If these conditions are not met (e.g. massive quark production or x+O) then there is a probability that Jzj=O in the final state and hence a less

f The dmfm J ( 0) is the reduced rotation matrix element for total angular momentum J . &oJ is related to spherical harmonics by d m , o J = [4~/ (2J+ 1 ) ] 1 / 2 Y ~ m exp (-im$).


dramatic suppression ensues as y+l. At large x and for light quarks (U, d, s) this caveat can be safely neglected but, like the OL/OT discussion earlier, it is an example of an effect which could be manifested in precision experiments under the relevant kinematic conditions. In practice it would probably be lost among other competing corrections, like quark-gluon interactions, new flavour production thresholds, and possible right-handed currents for the production of further flavours of quark.

For our first look at the data we can ignore these caveats. Denoting the momentum (x) distribution of up, down quarks in a proton by u(x), d(x), then the charge symmetry of the strong interactions gives for the distributions in the neutron:

uN(x) = dP(x) d(x)

d y x ) = UP(X) = U(X)’ (3.26)

For a nuclear target containing an equal number of neutrons and protons the neutrinos probe the distributions:

dN(x) + dP(X) = U ( % ) + d(x) = Q(x) (3 -27)

and antineutrinos analogously probe &(x). Hence, on inserting the elementary cross sections into equation (3.12) we have:

d2a dx dy dQ2 dv

i

and similarly

(3.28)

(3.29)

From the success of the quark model in spectroscopy (Hendry and Lichtenberg 1978) one naturally first looks to see if the data support the picture of a three-quark nucleon. This would lead one to expect that (doldy)” w isotropic, (doldy): N (1 - y)2 and hence that aTOTALc/oTOTALv = 3.

T h e data are consistent with these predictions, at least for ~ 2 0 . 4 (Perkins 1975, 1977, Holder et aZl977, Berge et all977, Bosetti et aZl977). At small x there appears to be some need for antiquarks which is probably the reason that o P / a v is slightly larger than 9. Quantitatively a fit to the data yields:

J” @(x) N +“a J” xQ(.) (3.30)

which shows that antiquarks carry only a small percentage of the proton’s and neutron’s momentum.

An interesting byproduct is that the quarks and antiquarks do not carry all of the target momentum. If they did, then momentum conservation would give (Llewellyn Smith 1974) :

J” dxxQ(x) = 1 (3.31)

and hence from equation (3 .29) we would have a prediction for the absolute magnitude of the total cross section. This reads:

(3.32)

1304 F E Close

which quantitatively is: u'/E N 1.5 x 10-38 cm2 GeV-1. (3.33)

Empirically uY/E is only some SO% of this size. This can be blamed on equation (3.31) and instead one infers that only about 50% of the momentum is carried by quarks. The missing momentum is hypothesised to be carried by gluons which are neutral to both electromagnetic and weak interactions.

I n an interacting field theory of quarks and gluons (e.g. QCD) these have a natural interpretation as the field quanta that bind the nucleon together. An emerging picture of the nucleon is that three (valence) quarks carry 50% of the momentum and are the dominant partons at large x. These bind by exchanging gluons. Furthermore they can bremmstrahlung gluons which in turn produce pairs. A typical bremmstrahlung spectrum has a dx/x distribution, and hence the Q are found only at small values of x (Zacharov 1976).

Within the framework of QCD there has been much interest recently in attempting to calculate these x distributions explicitly (Buras and Gaemars 1978, Close and Sivers 1977, Gluck and Reya 1976). Qualitative and semi-quantitative successes appear to be emerging. However, I believe it is still a puzzle as to how the gluons manage to carry so much of the total momentum and yet do not seem to play a role in spectroscopy. Experimentally it would be interesting to know precisely how, and if, the gluon momentum fraction depends upon Qz.

3.4.2. Weak neutral currefits. In addition to weak interactions where a neutrino or antineutrino interacts with a target and turns into a charged lepton, there are the recently discovered 'neutral current' interactions (Hasert et a1 1973, Benvenuti et a1 1974) where the neutrino produces a neutral lepton. It is believed, but not yet proven, that this neutral lepton is t h e same neutrino that initiated the interaction. The space-time properties of the interaction are still not well understood, though they are consistent with the neutrino current being a mixture of vector and axial vector.

At the parton (quark) level the fundamental hadronic weak neutral current is hypothesised to be a mixture of vector and axial vector pieces (compare the charged current at equation (3.22)). This becomes:

JkNC(x> = q t ( X ) [ C ~ ' r ~ ( l + YS) f CL'YA(~ - y ~ ) ] q i ( x ) (3.34) i = u , d , 8 . c . . .

where CL, R are a priori arbitrary and control the relative importance of the left- and right-handed components as well as the overall strength of the interaction. If we consider a system of U and d quarks (neglecting s, c and antiquarks) then for an isoscalar target:

i=n, d

to be compared with equation (3.28) in the charged current case. If the v is replaced by 5 then CR++CL in the above, all other factors being unaltered.

Notice that the x and y dependencies have factorised both in the neutral and charged current cross sections. This therefore leads to the following ratios for the


total cross sections:

(3.36)

(3.37)

where a sum over i = U , d is implicit in CL, R (see equation (3.35)). I n the SU(2) x U( 1) model which unifies weak and electromagnetic interactions

(Glashow 1961, Salam 1967, Weinberg 1967) there is a parameter 8 which is defined by tan 8=g'/g, g' and g being the coupling constants of the SU(2) and U(1), respectively. This model predicts:

Rv=$-sin2 8+- 20 s11-14 . 8 (3.38) 27

R"$-sin2 8+- 20 sin4 8. (3.39) 9

Consequently Rv and RG are given as a function of this one parameter and so must lie on a curve in Rv, RC space (figure 9). The full curve is that applicable to our illustrative example where antiquarks have been ignored. If l0-15% of antiquarks are included then the dotted curve applies.

Data from various experiments are consistent with 0.2 6 sin2 8 < 0.4 (Blietschau et aZl977, Barish et aZl977, Benvenuti et aZl977, Steinberger 1977).

Figure 9. R v and RC in the Weinberg-Salam model as a function of 8.

1306 F E Close

3.5. Are the spin-4 partons quarks?

section (equation (3.5)) may be written: In terms of the kinematic variables x and y the electromagnetic interaction cross

(3.40)

uherc: scaling of MWl(v, Qz) and vWz(v, 8 2 ) has been assumed. For v(i)p+pL+X there is the similar form:

(3.41) d2o Gzs d x d y 27~ _ _ _ _ _ - [Fz(x)( 1 - y) + F1(x)xy2 T y( 1 - :y)xF3(x)l

where scaling has again been assumed and F are used to distinguish the weak structure functions from the electromagnetic, F. The - (+ ) sign in front of the 9 3 contribution is for v( i;) beams, respectively. This structure function is present as a result of the parity violation in the weak interactions v+p- and ;+p+.

This structure of the weak cross section implies that the most general angular distribution will have the form A+ By+ Cyz. However, in analysing the data, Berge et aZ(l977) and Bosetti et aZ(l977) have assumed that ZxPl((x) = 9 ~ ( x ) (which corresponds to equation (3 .18) for spin-& partons). As a result, the cross section simplifies to :

d2a 1 + ( 1 - y)2 ~ 1 - (1 -y)2 XF@) 2 f---- 2 F 2 ( x )

and hence is D + E( 1 - y)2 in form. In light of the UL/CTT data and discussion thereof in $3.3, it is clearly important to improve the v and i; data to the point that the general A +By + Cy2 structure can be studied and F 1 , 2 , 3 separated individually without assumed constraints being imposed. In particular, the parton model predicts that oL/aT is probably smaller for weak processes than electron scattering, justifying aposteriori the assumptions of the experimentalists (Close 1978). This is an interesting quantity to investigate.

Comparing equation (3.41) with the quark model form in equation (3.42) we can identifv :

(3.43)

and hence the x distributions of quarks and antiquarks can be compared. (More correctly, it is the distribution of left- and right-handed V 5 A elementary currents that are revealed.) The quark and antiquark distributions extracted by the CERN Gargamelle collaboration are shown in figure 10 as a function of xf(=x+M2/Q2). This clearly illustrates our earlier remarks that the 8 are dominantly at small x (Deden et aZ 1975).

The 9 2 structure functions in the quark model will be (if 0, = 0 and charm is ignored) : -

F2YP(x) = 2x[d(x) + C(x)]

FzV""(x) = 2x[u(x) + d(x)] (3.44)

(the factor of 2 is due to both vector and axial currents occurring with equal magnitude


0 0.2 0.4 0.6 1.0 X '

Figure 10. Quark and antiquark distributions (from Perkins 1977).

at the elementary level). The analogous structure functions in the electromagnetic case will be (again ignoring charm) :

F z ~ P ( x ) = $x[u(x) + f i (x)] + Bx[d(x) + a(%) + S ( X ) + S(x)]

F ~ e n ( 3 ) = +x[u(x) + ii(x)] + $x[d(x) + d(x)] + Bx[s(x) + S(X)]. (3.45)

These immediately imply that (Llewellyn Smith 1974):

F p + e n ( X ) 2 $ F:ZYP+Yn(x) (3.46)

the equality arising if ~ ( x ) , S(x) are negligible. The Gargamelle data suggest that this is well satisfied by equality €or x 2 0.4 and that as x+O the inequality is satisfied. I n the quark model this is consistent with the idea that strange quarks in the proton are found only in the QQ sea (the proton and neutron in the three-quark model are duu and udd states) and that the sea is dominantly at small x values.

The electromagnetic structure functions are bounded as a result of the fact that Q(x), Q(x)>O and so from equation (3.45) we have (Nachtmann 1971, 1972):

(3 $47)

The data are consistent with these bounds (Taylor 1975). As x- t l it is possible that the lower bound is saturated, in which case one infers that if a current interacts with a parton which is carrying all of the target's momentum, then that parton will be a U flavour in a proton and a d flavour in a neutron.

One can compare this phenomenon with neutrino data to verify if the same

1308 F E Close

conclusion on the large x flavour dependence is found there. For xZO.2 we can neglect antiquarks. Then from equations (3 .42) and (3 .44) at y = 0 we have :

(3 $48)

There is some evidence suggesting that this ratio is falling with increasing x, in line with the dominance of u(x+l) inferred from the electromagnetic data (Farrar et a1 1977).

If the partons are indeed quarks then sum rule constraints can be derived for the structure functions. The distribution functions h(x) of the partons must be consistent with the target’s quantum numbers. As examples, the proton has baryon number of unity and hence there will be three more quarks than antiquarks. This gives :

N(q) -N(q) = 3 = J” dx[u+ d +s+c- (E+d+F+E)]. (3.49)

The strangeness and charm of the proton are both zero, and hence:

O = J dx(s-F)=J” dx(c-2). (3.50)

Substituting into equation (3.49) we get a sum rule for F 3 :

J dxg3’P+Yn(x) E J dx[u + d - ( E + d)] = 3. (3.51)

This sum rule of Gross and Llewellyn Smith (1969) appears to be well satisfied empirically, a value of 3.2 2 0-6 being quoted (Cundy 1974, Deden et a1 1975). A word of caution is in order here. In all of the Gargamelle neutrino data the energy is rather low and, in particular at small x, the Q 2 is small. T o satisfactorily test the sum rule with large Q 2 data throughout, the higher energies available at CERN SPS and Fermilab will be required.

The electrical charge of proton and neutron constrain :

J” dx(u-E)=2

J” dx(d-d)=l (3.52)

(hence the net excesses of U over E and d over d i n the proton are 2 and 1, respectively). These yield the sum rule of Adler (1966):

(3.53)

which is very difficult to test experimentally as neutrino data are not yet accurate enough to study differences of structure functions.

Electromagnetic data are of sufficient quality that the difference of FzeP and Fzen can be studied. From equations (3.45) and (3.59) we find:

It is conventionally assumed that the sea is symmetric in the E and d flavours. If this is the case the integral on the right-hand side can be set to zero and so the integral over the structure functions will be +. The data give 0.28 & ? for the integral; the ‘?’ being the unknown contribution from high w (small x). If Regge behaviour is


relevant in this region then it is expected that Fep-Fenwx1/2 when xG0.1. If this is true empirically then the data are consistent with a value of Q for this integral, in line with the quark model prediction (Bloom 1973).

Field and Feynman (1977) have recently suggested that the ii and d do not have the same probability. The three-quark (valence) configuration for the proton is uud. If we add uzi and dd in the sea, then the presence of one more valence U and d implies that the Pauli exclusion principle will constrain the uii before the dd. This suggests that dd in the sea may be more probable than uii. Hence d > ii in equation (3.54) and so:

(3.55)

There has not yet been any quantitative estimate made for the Pauli exclusion effect on the sea.

The neutrino and electron scattering data are all consistent with the spin-4 partons being quarks and antiquarks. Further support will be found in electron- positron and hadronic interaction data. However, we should note that these data do not prove the quarks have fractional charges, etc. A particular example is the Han-Nambu (1965) quark model. Here, each flavour of quark occurs in three colours and has integer charges. The average charges of red, yellow and green quarks are 8 for U, -+ for d, -4 for s, etc. Hence, if one is below threshold for producing coloured states this model yields the same phenomenology as the 'standard' model. Moreover, in a gauge model built with these Han-Nambu quarks, Pati and Salam find that these integer charge quark partons respond to lepton beams as if they had fractional charges even above colour threshold. A crucial feature of this particular gauge model is that some of the gluons have charge and so can contribute directly to observables like UL/UT, Ij;(x), etc. This model has been described in detail in lectures by Pati (1976).

3.6. Electron-positron annihilation

test of parton model ideas. The prediction is that the ratio of cross sections : The production of hadrons by annihilating electrons and positrons is an important

(3.56)

is a constant, independent of the incident electron and positron energies, so long as the energies are high enough that prominent resonances and particle production thresholds are unimportant. This prediction is non-trivial since any individual hadronic production process tends to have a cross section that dies out rapidly as Q2 increases, e.g. seems to die like Q - 6 .

The data for this ratio (figure 11) do appear to show the predicted constancy at energies Ecm(=Ee++Ee-)k5 GeV. At energies between 3 and 5 GeV the Y, Y' resonances and charmed particle production thresholds are responsible for the non-constancy. At lower energies the data are unclear. It is possible that from about 2.5 to 3.5 GeV (apart from the Y resonance) the ratio is also constant, albeit a smaller constant than at the higher energies.

1310

R 6

0

W

1 ° ~ ~

2 -

F E Close

I'

I I I I 1 1 I 2 L 6 6

Ec m Figure 11. Data on R= o(e+e--+ hadrons)/(e+e--+ p+p-).

The parton model mechanism for this process is that e+e--+qq (quark and antiquark) and these quarks then fragment into the hadrons by some unknown mechanism. Then at high energies (Cabibbo et al 1970):

u(e+e--+hadrons) = u(e+e--+q&) (3.57)

i = u , d , e . . . = et%(e+e-+p+p-)

i

where the last equation follows because the only difference in the electromagnetic production of p+p- and 4g is in the charges of the quarks involved.

Hence not only does the model predict that R is constant, but its magnitude is also predicted, namely (Feynman 1972, Cabibbo et al 1970):

= 2 ei2 u( e+e--+hadrons) a(e+e-+ p*p-)

R S - i

(3.58)

for spin-i quarks as the fundamental partons. Below charm production threshold the U , d, s degrees of freedom are operative

whose squared charges are respectively 419, 119, 119. There are several reasons to believe that each of these flavours of quarks comes in three 'colours' (a further hidden degree of freedom) (Kendry and Lichtenberg 1978, Close 1978). Conse- quently :

(3 I59)

which is of the order of magnitude of the lower energy data. Improvement in these data are urgently needed in order to see how well this prediction works.

After having crossed the charm production threshold the charmed quark freedom is revealed. Hence:

(3.60)


One unit of the R data is due to production of heavy leptons, e+e-+r+r-. When this is removed from the data R is indeed consistent with the predicted magnitude.

If the coloured quarks interacted with coloured vector gluons (the theory of quantum chromodynamics analogous to quantum electrodynamics) then scaling violation is predicted. The value of R should decrease slowly with energy, ap- proaching the parton model predictions asymptotically from above.

All of this is remarkable support for the idea that quarks with charge 8 or - & of the proton charge, and having three colours, are at the root of things. It does confront us with the question as to why the quarks are not themselves produced but instead form the familiar, uncoloured, hadrons. The manner by which the hadrons are produced is also not understood. However, within the quark hypothesis there are predictions that can be made concerning the relative abundances of the various hadrons, e.g. T+ to T-. These are discussed in $3.8 and appear to be in good agreement with the data.

3.7. Quark distribution functions and counting rubs

We have seen various indications that for xk0.2 the U and d (valence) quark flavours are dominant and that antiquarks and strange or charmed quarks are negligible. In particular, a comparison of electron and neutrino data showed that strange quarks were unimportant at large x (equation (3.46)). The y distributions of V induced reactions showed that antiquarks are unimportant at large x (equation (3.29) et seq).

If we therefore suppose that only U(.) and d(x) are important for ~ 2 0 . 2 , then inverting equations (3 .45) yields :

(3 -61)

from which the x 2 0.2 distributions immediately are obtained. We can extend the analysis by adopting a picture where the nucleon contains

valence quarks, distributions uv(x) and dv(x), and a sea of quark-antiquark pairs. Suppose that the sea is an SU(3) singlet:

s(x)=S(x)=Zz(x)=d(x)= ((x)

(3.62) (Such a picture is reasonable if the three valence quarks are emitting and absorbing virtual gluons which pair produce Q and p in the sea.) Then we would have :

(3 .63)

(3 .64) F f P ( x ) - F p ( 4 = $x[u,(x) - d,(x)] Fzep(x) + Fzen(x) = & [ S ( U ~ ( ~ ) + dv(x)) + 24<(x)].

Hence, on comparing with the neutrino data:

(3.65)

1312 F E Close

From this we see that this ratio is not a very sensitive probe of the sea. Better is:

(3.66)

The quality of electromagnetic data enables the i(x) to be extracted best from:

(3-67)

Distribution functions extracted by Barger and Phillips (1974) are shown figure 12.

in

Figure 12. Valence and sea quark distribution functions (from Barger and Phillips 1974).

The discovery of charm has provided a new way of studying the sea. The basic weak production of charmed quarks is believed to be due to the fundamental quark process :

Ijct+p+(s COS 8 - d sin 8) (3.68)

where 8 is the Cabibbo angle and tan2 8 - 1/25. The lowest mass charmed particles decay weakly, a typical decay being semileptonic, e.g. D-tKpv. If this semileptonic branching ratio is B then 'dileptons' will be seen due to:

(3 -69)

where eS is the momentum carried by the strange quarks in the sea relative to the valence d quarks:

(3.70)

Hence the dilepton signal is split roughly equally between valence d and sea s quarks when a neutrino beam is used (figure 13(b)).


l o ) ibl Figure 13. Di-muon production via charm in the quark model. (a) fi induced. (b) v induced.

With an incident antineutrino beam, on the other hand, the dominant mechanism is (figure 13(a)):

(3.71)

The sea is necessarily involved and the resulting x distribution of the dileptons is immediately the x distribution of the S(x). The di-muon distributions are shown in figure 14 and are consistent with the’sea being confined to small x (Steinberger 1977).

The parton distribution functions extracted from these lepton production data have been used to compute rates for the production of lepton pairs with high invariant mass in proton-proton collisions (‘Drell-Yan annihilation’-§4.2). Barger and Phillips (1978) have recently inverted this procedure, combining the massive pair production and deep inelastic lepton data and extracting the non-strange parton sea components in the nucleon a(x, Q2)+d(x, Qz) at particular values of x and Q 2 , The quark distributions are written in terms of valence and sea distributions. Their analysis assumes SU(3) symmetry in the sea (G=d=S=S) and ignores charm (c=E=O). The values of in the various lepton production data sets and the

Figure 14. Observed x distribution of di-muons from fi and v induced reactions.

1314 F E Close

Drell-Yan data vary widely. If scaling is assumed (Q2 dependence ignored) then a best fit to the form x< = a( 1 - x)b yields:

xc = 0*2( 1 - 2)s. (3.72)

This fit is over the range 0.25 5 x 5 0.4 which is the ‘high-energy tail’ of the sea. However, if a recent datum from very high-energy inelastic muon scattering is employed, then :

(3 .73)

(at this small value of Qz charm can be neglected). Including this datum forces a steeper (1 - x) dependence :

F p ( x = 0.005 ; Q2 = 2 GeV2) = 0.36 i. O.O5t+x~(x+O, 8 2 = 2) = 0.27 2 0.04

~ (=0*26(1 -~)9 . (3 -74)

The primary criticism here is that data are being used for 2 6 Q2 d 130 GeV2 and over this range the Q2 dependence cannot be ignored. There is evidence that at small x, F z e P rises as Q2 increases. If the parametrisation of Perkins et a1 (1977) is used to take account of this Q2 dependence, viz:

xt;(x, Q 2 ) =f(x)(Q2)*-z (3.75)

then the effect on the Barger-Phillips analysis is to give :

x ~ ( x , Q2= 2) = 0*08(1- x)5 (3.76)

for the large x tail of the sea.

3.7.1. Counting rules. The interest in extracting the x dependence of the parton distribution functions at large x is related to the so-called ‘counting rules’ (Matveev ct a1 1973, Brodsky and Farrar 1973, 1975).

Blankenbecler and Brodsky (1974) have argued that for inclusive reactions one can formulate counting rules as follows. The elementary fields that take part will be called ‘active’, the remainder ‘passive’. Then as s, t, u+co :

(3.77)

where E = m22/s is fixed ( E being the ratio of the invariant squared mass of the produced system X to the total incident invariant momenta squared). Here:

(3.78)

The physical reason for this form is that for large N there will be suppression, while the ‘forbiddenness’ F increases as npassive increases due to these spectators using up the available phase space.

Consider ep+eX. For valence quarks the basic process if eq-+eq where nactive = 4 (initial and final electron and quark) while %passive (hadronic) = 2 (the two spectator quarks in the initial proton). This yields:

vw2(x+ qpN (1 - 4 3 (3 * 79)

for the valence quarks. One can use this relation to consider e+eq to determine


the x dependence of cj(x) as x+ l . Here npassive=4 (since qqqqcj is the minimal configuration) and hence :

vw~(x+l),w(l-X)7. (3.80)

These results have been used by Gunion (1974) and Farrar (1974) to suggest that as x + l :

qval(x) N ( 1 - x) qsea(x) (1 - 4 7 . (3.81)

Hence the relevance of the Barger and Phillips (1978) parametrisation. However, one can construct a sea where qsea(x) N (1 - x)5 (Close and Sivers 1977).

This is given by a diagram (figure 15) where the qcj annihilate into a gluon. Hence the basic picture is:

e + cjqiqpqq+ e + 444 (3.82)

and so the subprocess at work is ecj-teq. Here nactive=4, as always, whereas npassive = 3, yielding the (1 - x)5 in accord with explicit calculation. Hence one expects that:

(3.83) is the leading component.

qsea(x) N (1 - x ) ~

- Figure 15. (1 - x ) ~ contribution to the 4(x) sea.

The relative importance of the fifth and seventh power contributions will be controlled by the relative probability that c j can exist ‘in the wavefunction’ (seventh power) or be produced by gluons (fifth power).

As x + l there is some uncertainty as to whether the data follow (1 - x)3 or (1 - x)4 behaviour (Taylor 1975). I n part, this is due to pendence) in the data. Landshoff and Scott (1977) have pendence can be:

scaling violation (Qi de- pointed out that the de-

(3.84)

where ( kT2(x ) ) is the average squared momentum of the target momentum. I n a simple model they show that a for- (kTZ(x)) can cause Fz(x) to approximately behave as 0.85. Only for x>O*85 does the (1 -x)3 appear. Data do not exist at these extreme values of x. Their model does, at least, illustrate that it is only at the extreme limit x - t l that the naive counting rules might be manifested in the data.

partons transverse to the reasonable x dependence (1 - x)4 for x as large as

3.8. Inclusive production of hadrons Interesting tests of the quark parton model can be made by studying the inclusive

production of hadrons, h, in current-induced reactions like eN+ eh + anything, vN-tp-h + anything, e+e--+h + anything, etc. Here a particular hadron h is detected where h may be n-*, K*, p, for example. Other particles produced in the interaction are not detected and are included in the ‘anything’.

1316 F E Close

T o illustrate how the parton model is applied, consider the interaction of the current and a parton in the current-parton Breit frame (brick wall frame). I n this frame (illustrated in figure 16) the magnitudes of the parton's three-momentum and invariant mass squared (pppp z E 2 - p2) are preserved but the direction of the three-momentum is reversed. The nucleon has four-momentum Pa= (P, OT, P) (equation (3.1)) and is treated as a collection of independent point-like constituents (partons). A parton with momentum p i p = (xP, OT, &')(equation (3.3)) will have its three-momentum reversed and energy preserved when struck by a current with :

q= (0, 0, 0, - 2xP). (3.85)

After the interaction, the struck parton will be moving away from its fellows. The quark then produces hadrons by some unknown mechanism. One of these hadrons, h, is detected with momentum fraction x (i.e. with component XP in the direction of P, the nucleon target's momentum). I t is now assumed that the fragmentation of the quark q into the hadron h is independent of the earlier current interaction and of the quark's colour and depends only on the flavour q and the momentum fraction z. This quark fragmentation function is denoted by Dqh(x),

T c r g e :

B e f o r e

la)

C u r r e n t i r a g m e n i s

A f t e r

ibl Figure 15. (a) Current + parton Breit frame interaction. (b) The parton recoils and produces

observed hadron(s).

the probability that the quark produces a hadron h in interval dz about z being Dah(.) dx (Feynman 1972).

The total momentum of all the hadrons must be the same as the quark and so:

(3.86)

If the detected hadron h is a pion then isospin and charge conjugation invariance limit the number of independent D,h(x). Specifically, for a given value of z :

D,n'=Ddn-=D~n+=D-n- U

(3.87)

D,n' = D n- = D-n'= D-n-

and in each case DqRo = &(Dqn' + Dqn-). The implementation of these in various processes is illustrated below. The

derivation of equations (3.88)-(3.90) is almost obvious (see, for example, Close 1978).


I ' I I I I

(i) e+e-+h . . . .

I

The Df and Di are exhibited separately since the virtual photon has produced a parton-antiparton pair, either of which could have been the parent of the observed hadron.

(ii) ep+eh . . . . ___ &et2f(x)Qh(z> 1 d o TOT dz &es%(.>

(ep-teh . . . )=

where fi(x) are distribution functions for quark or antiquark flavours.

(iii) vp-tp-h . . . . d(x)Duh(z) + gc(X)Duyz) (vp+p-h.. . )= 1 d o ___

oTOT dz d(x) + @(x)

(3.89)

(3.90)

where we have approximated the Cabibbo angle Bc = 0 and ignored charm. Notice that the d quark has turned into a U quark before fragmenting; the 4 arising from an integral over dy and the (1 - y)2 distribution for v q interactions.

The neutrino data are, in principle, a direct measure of the D," since in equation (3.90) we have that:

and so Dun* E D,f

(3.91) 1 d o - (vp+p-rr*. . . )=Dun*=- - (vp+p++, . * ). 1 d o ___ UTOT dZ U dz

Hence the rr+/rr- ratio with a neutrino beam and the rr-/rr+ ratio for an antineutrino beam should be equal and independent of x. Data from CERN are in agreement with this (Cundy 1974) (see figure 17).

1

1318 F E Close

These data also show that:

y(x) = DU"+(x)/Du"-(x) (3.92)

is of the order of 3 for 0.3 < 26 0.7 rising for 2: > 0.7. It is intuitively reasonable that this ratio is greater than 1 because rf is ud in the simplest configuration. Hence a U quark can create rrf 'directly' (if a dd pair is produced) whereas a d quark requires both a iiu and dd to be produced so that the r+(ud) emerges.

Much more data are available for electron-induced reactions than neutrino. However, the analysis is more complicated because now all of the charged quarks can be contributed (equation (3.89)). Cleymans and Rodenberg (1974) made initially an analysis of o = 4 data, ignoring all but the valence quarks (U, d ) . Hence defining :

U

N"' ~ (x, x)=- do (eN-+en+X) (d:i2 ___ (eN-+en-X))-l (3.93) N" - dx du" one has

(3.94)

When W N 4, u(x>2: 2d (x ) and from the observed NK+/Nn-zz 2, Cleymans and Roden- berg deduced q ( x ) - 2 . 5 . This is in good agreement with the value found from the independent neutrino data. Dakin and Feldman (1973) extended this analysis to 3 5 w 5 40, 0-5 5 Qz 5 2.5 G e V and included the possible contribution from sea quarks, The effect was to slightly increase q(z) to around 3.0+0.6, in line with neutrino data.

From the observed tendency for Fzen(x)/FzeP(x)-+& as x + l we know that the U flavour dominates as x+ l . Hence we in turn expect that more rr+ are produced than rr- in this region and there is indeed some indication that the rr+/rr- ratio does increase as x+1, W+CO (figure 18).

The production from neutron targets is obtained by interchanging U and d in equation (3.94). Again, with q(z) = 3, good agreement with the data is found. Also, in general, as W+CO (x+O) the sea dominates and the nflr- ratio is predicted to become unity; the data agree with this (figure 18). Qualitatively this result at W+CO

is expected in many models, e.g. diffraction may be important here and this does not distinguish between r+ and rr-. At smaller ~ ( x - t l ) the dominance of rr+ over T-- from protons is well predicted by the model, but more generally one might qualitatively not be surprised at this excess of positive charge since the proton itself has positive charge. The real test is the predicted excess of positive charge being projected out of the neutron for ~ 2 4 . The data qualitatively support this but need to be improved to be fully convincing.

The observation that q(x) > 1 was qualitatively understood above by the fact that r+(ud) is more easily made from U than d because it already contains a U . It has been widely argued that as x+l where the hadron takes up nearly all of the quark's energy, then ~ ( x ) + c o . Data are not inconsistent with this (figure 17) but it is premature to claim genuine support for it. I n particular in this kinematic regime one has to be careful to ensure that quasi-exclusive channels are not contaminating the signal.

In e+e- annihilation equation (3.88) predicts that sdo/dx should scale in an energy range where SUTOT scales. Distributions for sdoldx at 3.0, 4.8 and 7.4 GeV do scale when z> 0.5. For x< 0.5 sdaldx rises as s rises. I n figure 19 we show sdo/dz


.- c 2

4 k

I .I I I 1 1 1 2 5 10 20 50 100

W

Figure 18. w dependence of &IT- production in ep and en interactions.

0 . 9 0

0.08 < z c 0.12

0

Figure 19. s da/dz for inclusive hadron production in e+e- annihilation as a function of energy and for various z bins.

1320 F E Close

plotted against E,, for various x bins. Scaling would imply that sdaldx should be independent of E,, for any fixed x. However, we knew that the total cross section does not scale since charm production threshold is crossed in this energy range causing SUTOT to rise. The charm pair production, near threshold, will yield decay products limited to x< 0.5 (this is a consequence of kinematics), and so the scaling violation should be limited to x < 0.5. This indeed appears to be the case. Hence the data appear to be a superposition of old (pre-charm) and new (charm).

If this is true, then the data at 3 GeV are due entirely to old physics and exhibit the scaling behaviour of the uds quark flavours. Hence one can analyse these data using equation (3.88). An initial orientation was made by Gilman (1975) who simpli- fied the analysis by considering only the contribution from the U flavour. This is a reasonable first approximation because the U flavour is the most probable flavour in a proton, and hence dominates the ep-teh , . . , and moreover has the biggest squared charge by a factor of four relative to the d and s flavours. As a consequence for ep-teh. , . one has:

while for e+e- annihilation at dQz5 3.5 GeV (where only the uds flavours contribute because one is still below charm production threshold), then ud dominance yields :

and so one predicts:

(e+e-+h* . . . ). (3 * 97) 1 do 1 do

* * *'== dz __ - (ep+eh* UTOT dx

This comparison is made in figure 20 and is excellent, especially at large x. At small x, where particles have low momenta, mass dependence cannot be ignored

0 1.0 z Figure 20. Comparison of hadron production in e+e- annihilation at 3 GeV (@) and in ep

scattering (0) (equation (3.97)).


(e.g. threshold effects can contribute in this region like efe-+nR with R some massive state). Hence, at small z one should not expect the naive parton model ideas to be applicable. At large x where mass effects may be ignored one expects that the parton model should apply and so the agreement of data with equation (3.97) is rather encouraging. In particular this is non-trivial. There is no a priori reason why a given hadron being produced from a proton should bear any resemblance in production characteristics to the case where it is produced from a virtual photon (as in e+e- annihilation).

Recently, Field and Feynman (1977) have extended this analysis and included d and s quarks. The model agrees with e+e-, ep and vp data.

4. Hadronic interactions at large momentum transfer

4.1. Inclusive processes

In deep inelastic lepton scattering and e+e- annihilation the fundamental point- like current-parton interaction was well defined. I n hadron-hadron collisions the interaction mechanism is more controversial. At high energies and small momentum transfers the basic mechanism appears to be the exchange of Regge trajectories between the interacting hadrons. The nature of the trajectory exchanged depends critically upon the quantum numbers of the participating hadrons. Indeed, most of the energy that is materialised in a hadronic collision goes into particles which have only a small momentum transverse to the initial collision axis. Typically ( p ~ ) - 0.3-0.4 GeV. A few hadrons are found to be produced at large p ~ . Although we say ‘few’, they are still orders of magnitude more plentiful than expected from an exponential tail of the small p~ production. It is generally agreed that these large p~ hadrons are the products of some fundamental short-distance parton interaction mechanism. The controversy has centred on precisely what this mechanism is. Quark-quark scattering (Berman et a1 1971, Field and Feynman 1977), quark- virtual meson scattering (e.g. the proton contains a virtual meson cloud) (Blanken- becler et al 1973, 1974), qq fusion (Landshoff and Polkinghorne 1973a, b, 1974) and quark interchange between the hadrons (Blankenbecler et a1 1972a, b, 1973) are but some of the mechanisms proposed. Nor need one restrict attention to quarks. If 50% of the proton’s momentum is carried by gluons, then gluon-quark and gluon-gluon interactions can also contribute to the large p~ hadron production.

Clearly this subject itself requires a major review in its own right. T o keep the present review finite we shall study a few general features of the models and data. A detailed review of the subject is that of Sivers et a1 (1976) where an extensive list of the experimental details can be found and data are exhibited.

In a typical large p~ inclusive process A + B+C + anything then p ~ = IpI sin 6’ (in figure 21) where p is the three-momentum of the detected particle C. Conven-

Figure 21. Kinematics in AB + C + anything.

1322 F E Close

tionally one meets the variable XT = 2 ~ ~ 1 4 ~ where s= ( P A + p ~ ) ~ , and where PA, B are the four-momenta of particles A and B. In the centre of mass of A and B:

and hence tu I s p ~ 2 . Also E = M 21s where M is the missing mass in A + B +C + anything. The parton model is hypothesised to be applicable when s, t, U, M2, p ~ 2 are all much larger than mA, B, c2, The fundamental dynamics of the scattering are then illustrated in figure 22. Implicitly assumed are that the fragmentations A-ta, B-tb, c+C, d-thadrons all scale, depend only upon the ratio of the longitudinal momenta of a / A , etc, and that the transverse momenta are limited. As a result any large p~ of the observed hadron C is due to the hard scattering in the elementary process ab-tcd.

1 I I

I I I

I I

1 r----------- r----------- I I I

1 J’dxf:(x) 1 I I JdzD~Ilizi

Figure 22. Constituent decomposition of AB -+ C + anything. The individual template contributions to the cross section’s functional form in the parton model are included.

If abcd are quarks then the fragmentation functions fi*t B(x) can be determined from deep inelastic lepton scattering data (if A, B are protons or neutrons). If A, B are mesons then some additional hypotheses are required if f i * l B(x) are to be parametrised.

The fragmentation of a quark c into the detected hadron C is given by DcC(z) which have already been extracted from data on inclusive hadron production by leptons ($3.8). One can test the hypothesis that abcd are quarks and that qq- fqq is the basic process by using the f & B(x) and D@(z) to compute the cross-section ratios for various inclusive processes, e.g. pp-+.rr+X, TOX, K+X, etc. A detailed investigation of this has been recently reported by Field and Feynman (1977).

We write the cross section:

E -=J” d a dxfa*(x)f dyfP(y)(.Z* (ab-xd) J” dzDcC(l/z) (4.2) d3P d3P

The quark parton model

and so

1323

t x - 6 xy+yz-+xx E) (4.3)

7T l ( s S

where a sum over all the flavour possibilities is implied. With the$+(Xi) extracted from lepton data and a hypothesis made for doldt (ab-tcd) the hadronic production rate is determined.

Independent of the quantum numbers of abcd some general remarks can be made already. Since ab -+ cd is necessarily coplanar then A + B-t C + anything will be coplanar if the p~ is limited in each of A+a, B+b, c+C. If x is the AB collision axis and x is the axis defining p~ then the p, distribution will yield a measure of non-coplanarity. For those events with large p, then the p, distribution appears to be well described by:

(4 * 4) d N - - exp ( - PYKPY)) dY

where (py> - 0.3-0.5 GeV. A jet of particles on the same side as the trigger particle C is expected and also

a jet on the opposite side should arise as a result of d'hadrons. Data suggest that other high p~ particles do follow the trigger and that a jet on the opposite side also exists. These points are discussed in detail by Sivers et a2 (1976). Hence it does appear that these basic phenomena may be present in the data and hence support the notion that a basic hard scattering ab+ cd is occurring.

As a first guess one expects that qq-tqq will occur. The stu dependenceof this is proposed to be similar to ee-+ee. Hence:

which follows on dimensional grounds, there being no other scale of dimension for the point-like quarks. This implies that:

Data on single-meson production at ISR and Fermilab do indeed seem to show a scaling behaviour of the form:

but N2: 8, not 4 (see the review by Sivers et aZ(1976)). This has led to suggestions that the naive 44+4q interaction with AT N 4 is not yet being revealed in the data. A first possibility is that quark-quark forces are masked at present energies by qAl- tqM where M is a meson component in the proton wavefunction (e.g. M is a coherent qp system in the sea). For this process the meson form factor 1 F(t) 2 - t -2 and so the required extra four powers of p~ are obtained, What is not clear is why attention should be restricted to elastic qM+qM and not qM-+qM* also. If this latter is allowed then the subprocess ab+cd will be the inelastic 4M-+4+ anything

1324 F E Close

in which event s - ~ , and hence pT-4, will again emerge. One possibility is that, although the hadronic collision A + B is at high energy, the subprocess a + b is still at relatively low energy on the quark scale and so coherent qM-fqM still dominates. At much higher energies the basic incoherent qq-tqq will emerge with its associated p T P 4 behaviour in AB+CX. (There is some indication that at the very largest p~ values ( p ~ > 6 GeV) PT-N may be more like N N 6 than the value of 8 for p~ 2 2 GeV. This may be an indication that a basic qq-fqq mechanism is being revealed with energy dependence governed by dimensional considerations (point-like quarks).)

Field and Feynman (1977) have investigated the particle production from qq-fqq in detail. Given that data appear to require N = 8 in PT-N, then four possibilities are suggested for the phenomenological behaviour of (doldt) (qq-+qq). In decreasing dependence on t these are:

s2+u2 1 (9 7 3' This is motivated by vector gluon exchange, as in ee-tee, with a form factor

F ( t ) - t-1.

(ii) ( s t3) -1 .

(iii) (s2t2)-1.

(iv) ~ - 4 ~

The authors point out that there is no reason to believe that daldt is as simple as any of these. All that these are proposed for is to try to obtain an adequate repre- sentation of the data, and hence the true doldt. Data on pp-fn-+X suggest that ( s t 9 - 1 is marginally the best description and Field and Feynman use this in the predictions for other hadronic processes at large p ~ .

Analogous discussion of large p~ data has been made in models using qM-tqM, constituent interchange, qp fusion, etc. It is still difficult to judge from a fit to data whether the particular model is being verified or whether the many unknown functions in the model have been adjusted to fit. Improved data on correlations among the charges, isospin and strangeness of the large p~ particles may ultimately show more clearly what is the fundamental mechanism. The specific models and their various consequences are described in some detail by Sivers et a1 (1976).

It has recently been shown that the basic naive qq-fqq hard scattering subprocess may be compatible with data if scaling violation is incorporated systematically. Field (1978) has pointed out that if the quarks have a transverse momentum distribution in the initial hadrons with (kT)hjq- 800 MeV, if the quark to hadron fragmentation has (kT)q j h N 440 MeV, and if the scaling violations of QCD are included then good agreement with the data ensues. Similar ideas have been investigated by Contogouris et a1 (1978).

4.2. The Drell- Yan process

Of particular current interest is the possible importance of qq fusion in hadronic interactions. This is proposed as the source of lepton pairs in pp+p+p-+anything; the fundamental subprocess here is qq-fpp and the s-2 behaviour of this (analogous to the efe-+p+p- process) shows up as an m-4 dependence in the mass distribution


of the lepton pairs. This is known as the Drell-Yan (1971) process and is of interest because it is a hadron-induced reaction for which the basic subprocess can be well identified and so the rate can be explicitly calculated in principle.

The cross section will be:

Hence there are two levels of prediction here. First is the predicted scaling behaviour m4 do/dm2-m3 da/dmN f(m2/s). Second, if A and B are both protons then the quark distribution functions faP(x) are already determinable from lepton scattering data. Hence, both the energy dependence and absolute size of the process can be predicted. A comparison of small s, small m2, data with large s, large m2, data is not inconsistent with the proposed scaling behaviour. An absolute computation of the cross section, however, requires accurate determination of the q(x) distribution in the proton. These are not yet too well determined from lepton data. Indeed it has recently been proposed (Barger and Phillips 1978) that the Drell-Yan annihilation data may be the best source of information on the p ( x ) distributions when combined with lepton data.

The uncertainty of the sea distributions in the proton can be avoided if np+p+p- . . . is studied. Here the pion contains an antiquark in its valence configuration, UB for n+, dti for n-, A direct test of qq annihilation being the p+p- source can be made by using an isoscalar target such as carbon. Since an isoscalar target contains equal abundance of U and d flavours then:

n+C+p+p- . . . mqs-m 1 4' n-C+p+p- a . .

- ( 4 . 9 )

This is because v+(ud) causes dd annihilation to be the p+p- source whereas n-(dti) yields Cu annihilation. These two rates are therefore proportional to the squared charges of the antiquarks involved in the annihilation.

In addition one can calculate the dependence of this ratio on X=M2/s. Note that X,Xg=fVrz[s and so at small values of M2/s, quarks and antiquarks from the sea of 7~ or nucleon will give the dominant contribution and hence the n+ and n- beams will have equal probabilities to produce the photon. Thus:

(4.10)

As X increases so the valence quarks increase their importance and the ratio of & will obtain if q4 annihilation is operative. The data do give some support to this (figure 23). At masses where prominent vector mesons are produced (p , 4, +) the ratio is unity, showing that here it is the meson rather than a photon which is the p+p- source. At the largest values of m21s the t ratio seems to be approached.

n+C+p+p-, . . m z / s 4

n-c+p+p-. . . -+ 1 . _ _ _

For an arbitrary nucleus with 01 protons and /3 neutrons then:

n++I*+I*-. . . - a+) + @(x) n-+p+p-. . . 4(au(x)+/3d(x))

- (4.11)

1326 F E Close

and so knowledge of d[u(x) is required to make predictions for this ratio. Only if a ~ , 6 does the & ratio emerge. As an example consider hydrogen (a= 1, ,8= 0). Then as x- t l where d/u (x)-+O the ratio will tend to zero (the TT+ cannot annihilate on a U quark). As x-++ where 2d N U the ratio will be 4. Good data on the n+p/rr-p ratio will provide an interesting independent extraction of u(x)/d(x) which may be compared with the quark distributions extracted from lepton-induced reactions.

4.3. $ production

With recent excitement generated by the discovery of the heavy $ particle (Aubert et al 1974, Augustin et a1 1974), there has been much interest in studying its production in hadronic interactions like pp-f $ + , . . . The $ is a bound state of charmed

Figure 23. The ratio of cross sections for di-muon production in v+ and v- carbon interactions as a function of the muon pair invariant mass. These data are at fixed S; hence small A4 corresponds to small M 2/S and conversely for large MPp.

quarks: $(cE). Mechanisms that come to mind for producing the $ in such an interaction are :

(i) Fusion of a charmed quark, c, in one proton with E in the other. (ii) Drell-Yan fusion of U , d or s in one with the corresponding antiquark from

(iii) Gluon-gluon fusion produces a x(cE) which produces $(cE) + y. If mechanisms (i) or (iii) were the whole story then the rates for PO+$. . . and

pp+$ . . . would be the same. If mechanism (ii) dominates then pp and ~p prodLic- tion of $ would swamp that from pp. The data (Corden et al 1977) lie in between these extremes so (ii) does not dominate. The absence of pp-+$DD.. . , with D(cii) a charmed meson, suggests that mechanism (i) is not important. There is some evidence from CERN that photons have been seen accompanying the ib, sup- porting gluon-gluon fusion. It appears that both mechanisms (ii) and (iii) are important (e.g. Gluck et a1 1978).

The discovery of Y (9.4 GeV) (Herb et a1 1977) will enable its production in hadronic collisions to be studied in detail. Comparison with the analogom production of $ will enable the dynamical mechanisms to be better understood in the not too distant future.

the other. Mixing the us (dd, sS) with cE produces the $,


4.4. Large-angle exclusive hadronic scattering

Matveev et a1 (1973), and independently Brodsky and Farrar (1973, 1975), showed that the energy dependence of the cross section for ab-tcd at fixed angle may be controlled by the number of constituents in the systems a, b, c and d. Explicitly:

(4.12)

where fab(t/s) is some arbitrary function of the scattering angle ( t i s N COS 0 - 1)) and N A is the number of elementary constituents in A, etc.

An illustrative example is ep-tep. Here each particle is elementary and hence N = 2. Consequently :

(4.13)

Explicit calculation of this cross section in QED yields this form. Analogously quark- quark elastic scattering will have a t-2 behaviour (see equation (4.16) and subsequent discussion),

In elastic electron-pion scattering one can define the pion form factor F,(t) by :

(4.14) d a 2 T d - (e.rr-te.rr) = -- [ F,(t) [ 2f(t/s). dt t2

The above counting rules will therefore predict that for a pion made of two constituents (qg) :

IF,(t)l wt-1. (4.15)

Data are not inconsistent with this (Brodsky and Chertok 1976). Analogously for the proton a t - 2 asymptotic behaviour is predicted for a three-quark system. This also appears to be supported by data (Taylor 1975). Data for the deuteron and helium nuclei ( I Sick 1978, private communication) will further test this picture.

Turning now to the hadronic arena one predicts that for meson (M) and baryon (B) scattering at large angles (e.g. 90"):

MM-tMM: N = 6

MB+MB: N = 8 BB-tBB: N=lO

(4.16)

as the slowest energy dependencies for each process since N@n= 2, Npin= 3 . Data on np+np and pp-fpp do seem to support these (Sivers et a1 1976).

Heuristically the reason for these dependencies upon the number of constituents is as follows. For elastic scattering, the target must not break up. If one constituent suffers a hard collision then it will recoil and the system will break up unless all the other constituents recoil along with the struck one. The larger the number of constituents, the smaller will be the chance that all of them manage to do this. Hence the rate at which elastic scattering dies with increasing momentum is faster the larger the number of constituents in the system. Clearly the power dependencies of these processes are interesting ways of 'measuring' the number of constituents in a system.

1328 F E Close

5. Particle production at small momentum transfer

Although the parton model was originally formulated with specific application to large momentum transfer processes, there has been an interesting extension of it to the small p~ area in the last two years. Specifically, there have been some nice applications to the small p~ production of hadrons.

The starting point was an empirical observation by Ochs (1977). He observed that the x (longitudinal momentum) distribution of fast mesons produced in proton- proton collisions mimics that of the valence quark x distributions as deduced from eN and vN data.

Now, at large p~ the hadrons are successfully hypothesised to result from quark fragmentation. If this was the case at small p~ then the convolution of the fragmentation function for producing the meson out of a quark with that for finding the quark in the proton, would cause the meson spectrum to fall more rapidly with x than observed. Das and Hwa (1977) then pointed out that at smallPT valence quarks and sea antiquarks are not much separated spatially (in contrast to high p~ where a quark is recoiling far away from its fellows). Hence, instead of producing hadrons by the costly (in an energy sense) fragmentation mechanism, a fast meson could be more efficiently produced by a fusion of the valence q and sea q. Hence the x of the meson would be:

x = x v + x s

and so, not only does one not lose momentum in a fragmentation of quark to meson, but one even gains a small amount from the positive antiquark momentum xs. , This directly leads to the observed similarity between the meson and valence quark x distributions.

Theoretically, these measurements now take on a new importance because they will be yielding information of the probability distribution for finding two partons (a quark and an antiquark) in the hadron wavefunction.

These ideas have been extended by DeGrand and Miettinen (1978). These authors note that if one studies fast meson production accompanied by a Drell-Yan trigger (figure 24) then one is measuring the probability of finding three partons simultaneously in the wavefunction of hadron A. Furthermore, since the Drell-Yan process tends to ‘eat up’ U quarks (the squared charge and the U probability dominance both work in favour of this) then there will be a deficiency of U flavour available for fusing into the detected meson M. Hence the n-+/n-- ratio, in the presence of a

# Figure 24. Quark-antiquark fusion producing a fast meson in the presence of a Drell-Yan

trigger.


Drell-Yan trigger, should be significantly lowered relative to that in the absence of this trigger.

Given the fusion mechanism then clear-cut predictions can be made for the x dependencies of production ratios like pp+n+/n-, pp+K+/K-, etc. The former will be related to u/d (x) distribution functions, and hence should rise slowly with x. The data are indeed in agreement with this and have been well fitted by Duke and Taylor (1978). The sea contributes to their fits and seems to be between (1 -x)7 and (1 -x)9. The K+/K- ratio will rise dramatically with x because this will be proportional to the ratio of valence to sea distributions :

pp+K+(uS) U& V pp+K-(sii) s,C, 5'

+--+- (x).

This dramatic rise is indeed seen and suggests that:

Y(x)-(l-x)-3. S

Significant new data have been reported on the ratio:

D D + n + n -

pp+n-rr-. . . where both pions are fast. The n+n- production involves U(%) and d(~)(n+(ud), n-(dii)) which are both available in the proton (uud) as valence quarks. The n-n-, on the other hand, involves d(x) twice. Only one d can be valence, and hence the other must come from the sea. Consequently:

The data (P Schlein 1978, private communication) exist only over the range 0.5 5 x 5 0.8 but are in excellent agreement with the K+/K- ratio as predicted. The similarity with K+/K- provides support for the parton model fusion mechanism for fast meson production at small PT.

Brodsky and Gunion (1978) have recently discussed this small p~ fragmentation in some detail. They show that the x dependencies of various particle production rates are rather sensitive probes of the underlying parton dynamics.

The most recent data are reviewed by Diebold (1978).

6. Modifications of the naive parton model

The original suggestion that deep inelastic data might exhibit the scaling phenomenon was made by Bjorken (1969) on the basis of a study of current algebra current commutators. The parton model was a physical realisation of this.

Recently it has become clear that Bjorken scaling is only approximately valid. This suggests that the naive parton model is a reasonable approximation to the true dynamics but that corrections to the picture are required. The parton model so far reviewed works well in the realm 15 825 10 GeV2 of the early data, The possible sources of scaling violation in the data include:

(i) Mass scales. At finite Q 2 the effect of parton or target masses may be non-

1330 F E Close

negligible. Furthermore the production of very heavy particles (like charm) may introduce explicit mass scales which cannot be ignored at large, but finite, Q2.

(ii) Asymptotically free gauge theories, such as quantum chromodynamics, are hypothesised to be the underlying quark dynamics. The quark-gluon interactions yield a slow (logarithmic in 82) violation of scaling.

6.1. Mass scale eflects

In $3.2 we neglected target and parton masses. The scaling in x then arose essentially from equation (3.3) (when combined with the assumed point-like current- quark interaction). If masses are included then equation (3.3) will not in general be realised. The relevant quantity now (neglecting parton transverse momenta) is (Frampton 1976) :

where A o , ~ , PO,^ are the energy and longitudinal momentum of the parton and target, respectively. For a target mass f j and parton mass m I (which is turned into mF by the absorbed current) then:

<E (A0 + K3)(pO f p3)-’

(6.1) ( Q 2 + m ~ ’ - m12)+ [(Q2+mF2- ,,2)2+41n12Q2]1/2

2M[v+(v2+Q2)1’2] E =

This variable has been derived by a variety of techniques by many people, including Georgi and Politzer (1976) and Barbieri et al (1976). If mF=mI it reduces to that of Nachtmann (1973). As Q2+w it reduces to x = Q22/2Mv. A detziled study of its effect on scaling violations and the data has been made by de Rujula et al(l977) and Barbieri et aZ(1976). These latter authors conclude that scaling violations found in the data cannot be attributed solely to the choice of E as against x and conclude that other scaling violations, such as those which arise in asymptotically free gauge field theories, must be present.

6.2. Asymptotically free gauge theories and partons

The observation of scaling violations, which are not due to mass effects, suggests that interaction effects are important. The discussions of $$3 and 4 clearly show evidence that the charge-carrying ‘parts’ of the proton are quarks. The momentum sum rule (equation (3.33) et seq) showed that about 50% of the proton’s momentum is carried by neutral gluons. If the quarks interact with the gluons then the scaling of the free quark parton model will be broken by the interaction terms.

,4n obvious consequence is that the gluons can dress the quarks with a virtual cloud of gluons and quark-antiquark pairs. As Q 2 increases, the external probing current will have increasing resolution. Thus whereas a low Q 2 current fails to resolve a quark parton, at higher Q2 it resolves it into a quark plus a gluon from the quark’s virtual cloud (figure 25(a)) (Kogut and Susskind 1974, Llewellyn Smith 1975, Ellis 1977). One can describe this (Ellis 1977) as a ‘low Q2 parton’ having become a ‘high Q 2 parton’ plus a gluon. The high Q 2 parton will necessarily have less momentum than the low 8 2 parton. Hence there will be a depletion of high momentum partons and an increase in the low momentum parton distribution as 8 2 increases. The structure function F2(x, 82) essentially measures the parton momentum distribution p(x, 82). Hence, as 82 increases the Fz(x, Q 2 ) will exhibit a behaviour like


Figure 25.

\ t %I., I \. I I __t &j-

&-ev' Gluon Gluon

The effects of improving resolution as Q2 increases. (a) A 'low Q2 quark' becomes 'a high Q2 quark' plus a gluon. (b) A gluon is resolved into a qq pair and hence contributes to the structure function at high Qz.

that in figure 26, i.e. scaling will be broken. The observed trend in the data is indeed of this form, viz depletion at large x and enhancement at small x (figure 7).

It seems that ~ ~ 0 . 2 5 is the critical point about which the depletion against enhancement switch occurs. Hence at this value of x the scaling is best seen. Coinci- dentally it was here that the accumulated early data were exhibited which provided the evidence for scaling.

In addition to the above there will be an enhancement at small x as Qz increases due to gluons being resolved (figure 25(b)). A low Q 2 photon does not interact with the electrically neutral gluon. At larger Q 2 it resolves the gluon into a quark and antiquark pair and interacts with one of them. This will lead to an increase in the structure functions at small x and will be interpreted as an increase in the sea at large Q 2 .

Figure 26. Q2 dependence of Fz(x , Q2) predicted in quark-gluon field theory.

1332 F E Close

Taken to its extreme, as Q 2 goes to infinity one will not be able to tell if it is proton or neutron that is the system under study. The valence quark distribution will be a delta function at x= 0 and the qa sea, with isospin zero, will dominate.

The scaling violations of quantum chromodynamics have been formulated in parton language by Altarelli and Parisi (1977).

7. Conclusions

It is now only ten years since Panofsky (1968) exhibited the data on high-energy inelastic electron-proton scattering that stimulated the interest in parton models.

Before that event it had been suspected that the proton was composed of quarks. This suspicion was based on detailed studies of hadron spectroscopy (Hendry and Lichtenberg 1978, Close 1978). The Panofsky data of 1968 provided the first direct clue that the proton was indeed a composite object containing point-like constituents. When combined with subsequent electron data and neutrino and antineutrino data, it became clear that the neutron and proton are indeed built from spin-; constituents with the quantum numbers of quarks.

During the past decade improved technology has enabled experimentalists to obtain data at energies an order of magnitude higher than those of 1968. The above parton model has given a good first-order description of all the subsequent data.

The advent of high-energy machines for annihilating electrons and positrons provided a new arena in the early 1970s where the model could be tested. The model again gave good agreement. These new data showed that the quarks have a degree of freedom called ‘colour’, in addition to their various flavours like strangeness and charm. This discovery tied in neatly with the development of the theory of quantum chromodynamics, a non-Abelian gauge theory of the strong interactions. This theory has many attractive features, not the least of which is the similarity with the Weinberg-Salam unified theory of weak and electromagnetic interactions, and which suggests that a grand unification of nature’s forces may be possible. A central ingredient of the theory is that quarks occur in three colours and interact by exchanging vector gluons (analogous to the photon in QED). The role of the colour seems con- firmed by the parton model predictions for e+e- annihilation data. Electron and neutrino scattering data suggest that gluons are present as well as quarks in the proton. This is encouraging, but it may still be premature to claim this as evidence for the gluons of QCD. If QCD is correct then its property of asymptotic freedom gives a raison d’etre for the earlier naive parton model where the quarks were hypothesised to be quasi-free.

The high-energy collisions of protons at Fermilab and the CERN ISR since about 1970 have provided a glimpse of the direct hard collisions between the partons. A wealth of data here are all well described to first order by the naive parton model.

When combined with the unified theory of weak and electromagnetic interactions the quark parton model has made testable predictions for neutral current weak interaction phenomena and for the production of charm and new heavy flavours. So far, the model has successfully passed these tests.

We have seen how a field theory of interacting quarks and gluons predicts violations of scaling in contrast to the perfect scaling of the free quarks (naive parton model). The pattern of scaling violation predicted can be motivated by a simple intuitive


picture (figures 25 and 26) and indeed appears to be realised in the highest energy lepton scattering data accumulated in the last two or three years. This suggests that QCD may indeed be the theoretical basis for the empirical successes of the naive parton model. Whether the theory will apply all the way to the shortest distance (Q~+co) or whether some new more fundamental layer of matter (pre-quarks) will be revealed, only time and much experimental effort will reveal. History may (or may not) be a good guide here.

Given the wealth of data that the model has helped correlate over a decade and the theoretical developments that it has directly and indirectly stimulated, then it is hard to decide what is the most significant datum that bears on the model and future development. Personally I find one discovery very profound. It seems that we have verified that (at least for the present generation) nature is built from elementary fermions : leptons and quarks. The electromagnetic Dirac current and V - A weak charged currents of the leptons have been known for many years. The discovery, that has been absorbed in the parton model, is that (apart from their fractional charges) the quarks have identical electromagnetic and weak currents to those of the leptons. There is no known reason why this had to be so but it is surely more than coincidence. The ultimate epitaph of the parton model may be that it provided the first pointer towards the existence of a profound relationship between quarks and leptons.

The deep inelastic phenomena, the development of the quark parton model and the discovery of charm have convinced all that the fundamental hadronic fermion constituents are quarks, and that quarks and leptons appear to be intimately related in some way. It is to the investigation of this relation that much theoretical effort will now turn. The development of even higher energy machines will provoke searches for new flavours and provide new tests of the parton model. The model has survived an explosion of data over its first ten years. I t will be interesting to see how well it survives the next ten and if, via QCD, we see its incorporation into a theory of strong interactions.

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