Fortschritte der Physik / Progress of Physics: Volume 34, Number 2 [Reprint 2022 ed.] 9783112613726, 9783112613719

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FORTSCHRITTE DER PHYSIK PROGRESS OF PHYSICS

Volume 34-1986 Number 2

Board of Editors

F. Kaschluhn A. Lösche R. Rompe

Editor-in-Chief F. Kaschluhn

Advisory Board A. M. Baldin, Dubna J . Fischer, Prague G. Höhler, Karlsruhe K. Lanius, Berlin J. topuszanski, Wroclaw A. Salam, Trieste D. V. Shirkov, Dubna A. N. Tavkhelidze, Moscow I. Todorov, Sofia J. Zinn-Justin, Saclay

CONTENTS:

W . - D . NOWAK

Review of Deep Inelastic Charged Lepton Scattering

AKADEMIE-VERLAG • BERLIN ISSN 0015 - 8208

Fortsohr. Phys., Berlin 34 (1986) 2, 67-117

57—117

Instructions to Authors 1. Only papers not published and not submitted for publication elsewhere will be accepted. 2. Manuscripts should be submitted in English, with an abstract in English. Two copies are desired. 3. Manuscripts should be no less than 30 and preferably no more than about 100 pages in length. 4. All manuscripts should be typewritten on one side only, double-spaced and with a margin 4 cm wide. Manuscript sheets should be numerated consecutively from "1" onwards. Footnotes should be avoided. 5. The titel of the paper should be followed by the author's name (with first name abbreviated), by the institution and its address from which the manuscript originates. 6. Figures and tables should be restricted to the minimum needed to clarify the text. They should be numbered consecutively and must be referred too in the text and on the margin. Figures and tables should be added to the manuscript on separate, consecutively numerated sheets. The tables should have a headline. Legends of figures should be submitted on a separate sheet. All figures should bear the author's name and number of figure overleaf. Photographs for half-tone reproduction should be in the form of highly glazed prints. Line drawings should be in a form suitable for reproduction. The lettering should be sufficiently large and bold to permit reduction. If requested, original drawings and photographs will be returned to the author upon publication of the paper. 7. Formulae should not be written so small and not with pencil. Separate lines for formulae are desirable. Si-units should be used. Letters in formulae are normally printed in italics, numbers in ordinary upright typeface. Underlining to denote special typefaces should be done in accordance with the following code: Italics: wavy underlined with pencil (only necessary for type written symbols in the text) Boldface italics (vectors): straight forward and wavy underlined Upright letters (all abbreviations like all units (cm, g, ...), all elements and particles (H, He, ..., n, p, ...), elementary mathematical functions like Re, Im, sin, cos, exp,...): black underlined Greek letters: red underlined Boldface Greek letters: red underlined twice Upright Greek letters (symbols of elementary particles): red and black underlined Large letters: underlined with pencil twice Small letters: overlined with pencil twice (This will be necessary for handwritten letters that do not differ in shape, as c G, k K, o 0, p P, s S, u U, v V, w W, x X, y 7, z Z). It will help the printer if position of subscripts and superscripts is marked with pencil in the following way: at, 6», Mv^, Mtj, Wn\ Please differentiate between following symbols: a, a.; oc, a , oo; a, d; c, C, c ; e, I; ¡S, e, £; k, K, x; x, x, X,x, X ; I, 1; o, 0, a, 0; p, q; u, U, \J ;v,v,V;

(2.1.1)

is shown schematically in fig. 1 (laboratory system). This interaction will be called scattering from now on. It can proceed via different mechanisms. For presently attainOutgoing muon T a r g e f nucleon (mass M)

( s c a t t e r i n g angle 0)

Incoming muon n (mass m) Hadronic final s t a t e (effective m a s s

X

W)

Fig. 1. Schematic view of the one-photon-exchange process in the laboratory system

k=ÍE',p")

kM^p) p=(M.O ) Pig. 2. Feynman diagram for the one-photon-exchange process

able lepton energies of some hundred GeV the dominating contribution is due to the electromagnetic interaction described by virtual photon exchange. The Feynman diagram of this One-Photon-Exchange (OPE) process is shown in fig. 2. The OPE amplitude can be written as a current-current ansatz [1] T

Here is

=

z

r i ,*>.

I™ = u(k') y„u(k) 1*

(2.i.2)

j.

(2.1.3)

60

W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

the electromagnetic lepton current with the Dirac spinors u(k) and u(k') of incoming and outgoing leptons, respectively, and the Dirac matrices The propagator e2jq2 describes the massless exchange particle and ^ " ( O ) is the electromagnetic hadron current where p and s denote momentum and spin of the target nucleon. The differential cross section of the inclusive OPE mechanism can be represented by (2.1.4) after summation over all possible hadronic final states X. Also, the spin states of the incoming lepton were averaged and those of the outgoing one were summed up. The leptonic tensor L^'ik, k') = IJ,,* describes the lepton photon vertex and can be completely calculated within Quantum Electrodynamics. The electromagnetic coupling constant is given within the usual convention h = c = l a s « = e2/4jr. Q2 = —q1 is the square of the four-momentum transfered by the photon. The hadronic tensor WMV describes the photon hadron vertex and will be known once there exists a theory of strong interactions. At present, it can be expressed by only two unknown functions Wx and W2 if general physical principles as Lorentz invariance as well as charge and parity conservation are applied. In this way from eq. (2.1.4) the following expression is obtained for the double differential OPE cross section: cos-}}.

=

(2.1.5)

Here E0 is the incoming lepton energy, E' and d are energy and scattering angle of the outgoing lepton and v is the energy fraction carried by the virtual photon. All these variables are defined in the laboratory system and illustrated in fig. 3. The unknown functions Wx and W2 are called structure functions. They carry information about an v(GeV)

- 0.5 ^.

Fig. 3. Illustration of kinematic variables in lepton nucleon scattering for E0 = 280 GeV

61

Fortschr. Phys. 3 4 (1986) 2

unpolarised nucléon excited by a virtual photon. For completeness it has to be mentioned that by scattering polarized leptons off polarized nucléons two more structure functions contribute to the cross section carrying information about the spin structure of the nucléon. 2.2.

Bjorken Scaling

In the years after 1967 the development of theoretical models on the nucleón structure was considerably stimulated by unexpected results of electron nucleón scattering experiments. They had measured for the first time the inelastic cross section in the socalled deep inelastic region, i.e. for relatively large values of Q2 and v. The experiments of H O F S T A D T E R and co-workers had already shown in the late fifties that the proton cannot be pointlike [2]. They measured a strong Q2 dependence of the elastic electron proton cross section relative to the cross section for a pointlike target. This led to the conclusion that the proton is an extended object scattering the incoming particles as a whole. The 1968 measurements on inelastic electron proton scattering showed quite unexpected results [3]. The observed Q2 dependence of the corresponding cross section ratio was very weak as if the incoming electron were scattered by a pointlike object. This experimental result opened a new level in studying the nucleón structure. Its interpretation in the framework of the parton picture, i.e. the hypothesis of pointlike proton constituents, will be discussed in the next section. In parallel there was another theoretical interpretation of the inelastic electron proton data. It was observed that the measured structure functions 2MW-i(Q2, v) and vW.¿{Q2, v) turned out to be independent of Q2 and v when drawn as a function of the dimensionless variable x = Q2j2Mv (M means nucleón mass). Basing on this experimental information B J O R K E N assumed in 1968 [4] that the new kinematical domain (Q2 up to 10 GeV 2 ) was already the beginning of an asymptotic region. He conjectured that the nucleón is not changing its structure anymore when it is deeper and deeper probed by the lepton. This would correspond to an independence of Q2 and v, of the nucleón structure functions. Bjorkens hypothesis on nucleón structure function scaling can be phrased as follows: In the limes of large momentum and energy transfer ( Q 2 ¡ M and v much larger than typical hadronic masses) the structure functions do not depend anymore on variables as length or mass. They approach finite and non-trivial functions which depend only on the dimensionless variable x = Q2j2Mv: lim

MW^Q2,

F^x)

(2.2.1)

F2(X).

(2.2.2)

v) =

x finite

lim VW2(Q2, F.Q'IMPIM xfinite

V) =

Then, the cross section from eq. (2.1.5) can be expressed as d2a dx dQ2

W 4TRX2 17 ~

-QT

W1

~

Mxy\

F2{X)

-2E¡)

—

y2\ +

2 F Á X )

~2~J

(2-2'3)

where y = v/E0 is the relative energy transfer to the target nucleón. For more than 5 years Bjorken's hypothesis was supported by the results of electron nucleón scattering experiments. First indications that there exist some deviations from the exact scaling behaviour, came in 1974 from high energy muon nucleón scattering experiments. These results were confirmed in 1975 by improved measurements with electrons. All attempts to re-establish the exact scaling behaviour modifying the scaling

62

W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

variable x gave no satisfying solutions [5]. Later, when the principle of asymptotic freedom was invented in developing Quantum Chromodynamics and applied to the description of deep inelastic processes a new era began in the understanding of nucléon structure functions. See chapter 3 for more details. 2.3.

Partons

Another interpretation of the unexpected inelastic electron proton results was invented by F E Y N M A N in 1968 [6]. This model explained the unexpected size of the cross section as well as the scaling behaviour of the structure functions in an intuitive fashion. The nucleón is considered to consist of: — pointlike — charged — quasi-free constituents (called partons) off which the incoming leptons are elastically scattered (cf. fig. 4).

Fig. 4. Parton picture of deep inelastic lepton nucléon scattering

The probability to find a parton of type i carrying a fraction x of the nucléon momentum is denoted by qi(x). The number of type ¿ partons is then given by: l Ni=jdxqi{x).

(2.3.3)

o

The deep inelastic lepton nucléon cross section as a non-coherent superposition of lepton parton cross sections can be written as follows : ¿ffLepton-Nucleon

=

l JP J gxq^Lepton-Parton t 0

_

(2.3.4)

Energy momentum conservation means for the parton momentum distribution xq^x) : \

Z f d x x q f a ) = 1.

i o

(2.3.5)

Inserting the explicit expression for the lepton parton cross section into eq. (2.3.4) shows that certain combinations of the parton momentum distribution functions xq^x) correspond directly to the structure functions of lepton nucléon scattering. Here the squares of the parton charges in units of the electron charge act as weights because the electron and muon parton interaction, respective^, is of electromagnetic nature. Independently from the parton spin one finds : F,.{x)=ZZl*xqi{x).

(2.3.6)

Fortschr. Phys. 84 (1986) 2

63

In contrast to F2 {x) the structure function Ft {x) depends on the parton spin. It is for SpinO:

F

Spin 1/2:

{ x ) =

t

2 x F

1

{ x )

0 =

(2.3.7) £

(2.3.8)

Z ^ x q ^ x ) .

i

This spin dependence of the structure function F^x) allows an experimental determination of the parton spin. The quantity R as introduced in the appendix represents the ratio of longitudinal to transversal photoabsorption cross section:

It is a direct measure of the parton spin because in the Bjorken limes x finite) it follows from (2.3.6), (2.3.7) and (2.3.8) : Spin 0 : R

( Q

2

/ M , v

M

;

-> oo

(2.3.10)

0.

(2.3.11)

Spin 1/2: i?

In case of parton spin 1/2 the comparison of (2.3.6) and (2.3.8) yields 2 x F

1

{ x )

=

F2(X)

-

(2.3.12)

which is known as CALLAN-Gitoss-Relation [7]. The physical meaning of the structure function F2 {x) in the parton picture becomes clear after integration over the parton momenta. With (2.3.3) it follows from (2.3.6) : ]l

/

— %

F

2

{ x ) dx

=

2J i

N i Z f .

(2.3.13)

This relation is known as Gottfried sum rule. It allows to measure the sums of squares of parton charges. In a similar way the Callan-Gross sum rule is obtained from eq. (2.3.6) using eq. (2.3.5): I ¡ F

o

2

( x ) d x =

{ Z

2

) .

(2.3.14)

This sum rule can be used to measure the average of the charge squared of partons in the nucleon. 2.4.

Quarks

Already in 1964 GELL-MANN [8] and ZWEIG [9] postulated independently the existence of three basic constituents of the nucleon, so-called quarks. They were assumed to be charged, structureless and non-selfinteracting particles of spin 1/2 distinguished by different quantum numbers. The three-quark model was the kej' for understanding the multiplet structure of hadrons and thus the beginning of hadron spectroscopy [10]. The identification of quarks and partons led in 1971 to the invention of the quark parton model (QPM) by KUTI and WEISSKOPF [11]. In this model a baryon is composed of 3 valence quarks defining the particle's quantum numbers and additionally, by a sea of virtual quark-antiquark pairs carrying the quantum numbers of the vacuum. All quarks have baryon number 1/3 whereas their electrical charge can be 1/3 or —2/3. The quark type is characterized by flavour quantum numbers.

64

W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

The existence of a fourth quark flavour (c = charm) was already proposed in 1964 basing on symmetry arguments. It was confirmed 10 years later by the discovery of the J If particle, a bound cc state with a mass of 3.1 GeV/c2 [12], The T-particle was found in 1977 at about 9.5 GeVT/c2 [13] being the indirect proof for the existence of the fifth quark flavour (b = bottom or b = beauty). Lepton quark symmetry requires the existence of a sixth quark flavour (t = top or t = truth) that possibly has been see recently in pp-annihilation with a mass of about 45 GeV [14]. Due to its rather large mass the 6-quark does not contribute significantly to the lepton nucleon cross section for incoming energies up to 100 GeV. Therefore, it is sufficient to consider a four-quark model when the quark distribution functions are discussed in the next sections. The quantum numbers of the first four quarks are shown in tab. 1. Table 1 Quantum numbers in the four-quark model

2.5.

quark type

baryonic number

isospin

isospin projection

u c d s.

1/3 1/3 1/3 1/3

1/2 0 1/2 0

1/2 0 -1/2 0

strangeness

0 0 0

-1

Charm

electr. charge

0

2/3 2/3 -1/3 -1/3

1

0 0

Structure Functions in the Quark Parton Model

In a four-quark model there are eight distribution functions u(x), u(x), d(x), d(x), s(x), s(x), c{x), c(x)

(2.5.1)

using the convention u(x) = qjx) and so on. For transparency the x dependence will be omitted from now on. The neutron quark content is (udd). Proton and neutron are related by isospin transformation d, i.e. un = dp = d;

dn = up = u

(2.5.2)

where the indices n and p mean neutron and proton, respectively. The sea contributions are by definition independent of the valence quark composition: sn

c" — cv = c = c.

_ SP = g = g.

(2.5.3)

The functions u and d can be split up into valence and sea parts the former being responsible for the quantum numbers: u = uv -f

d = dv + ds.

(2.5.4)

For the sea parts one has: us = w;

ds=d.

(2.5.5)

To reproduce the proton quantum numbers the quark distribution functions have to satisfy certain normalization conditions: i

I

• j {u — u) dx = J uvdx = 2, o o

(2.5.6)

Fortschr. Phys. 34 (1986) 2 l f (d o l I (s 0

65

d) dx = / dv dx = 1, o l = / (c 0

s)dx

(2.5.7)

c) dx = 0 .

(2.5.8)

The structure functions F2 for lepton scattering off free protons or neutrons can be expressed by quark distributions according to eq. (2.3.6) : F^

= x

(u + u) + 1

(d + d). + Ì

(s + g) + i

(c + c ) J ,

(2.5.9)

FJ*

=

(« + «) + j

(d + d) + j

(s + s) + i - (c + c ) J .

(2.5.10)

Scattering off isoscalar nuclear targets, i.e. if the nucleus contains the same number of protons and neutrons, it holds F2'» = j

(F2'P + F2">)

(2.5.11)

neglecting a possible mutual influence of the nucleons inside the nucleus. Collective effects inside nuclei will be discussed in chapter 5. Inserting eq. (2.5.9) and (2.5.10) gives : F2>» '= = ^

«

+ û + d + d + J

(s + g) +

(c + c ) J .

(2.5.12)

Introducing the sums of quark and antiquark contributions q=u

+ d + c + s

(2.5.13)

q =û

+ d + c + 5

(2.5.14)

yields a simpler representation for the structure functions of the isoscalar target. I t is useful for a later comparison with neutrino structure functions, as well :

ì*=A

F

* f +5 ~ I"{s+5 _c ~

( 2 - 5 - i 5 )

Quark distribution functions can be obtained by certain linear combinations of structure functions and cross sections, respectively. For presently attainable energies in charged lepton scattering one has to combine data from proton and neutron targets. The latter can be extracted from deuterium data applying certain corrections to be discussed in chapter 5.

2.6.

Neutrino Nucleon Scattering

This section deals with some aspects of weak interactions to facilitate a later comparison between electron/muon and neutrino data. Fig. 5 shows deep inelastic weak processes in a schematic way. In the lowest order of perturbation theory a virtual but massive vector boson (spin 1) is exchanged in analogy to the description of electromagnetic processes. According to the charge of the boson the exchanged current is called neutral (fig. 5 a) or charged (fig. 5b) weak current.

66

W.-D. Nowak: Deep Inelastic Charged Lepton Scattering _ _

a)

N

—

/

I.

rA

,,

—i

b)

Fig. 5. Schematic view of weak processes: (a) neutral current, (b) charged current

The kinematics of the One-Photon-Exchange process can be applied as well to the vector boson exchange. As a consequence of parity non-conservation in weak interactions the hadronic tensor contains an additional unknown function W3{Q2, v) that will be described b y a corresponding function F3(x) in the Bjorken limes. The scale invariant cross section is given b y : d*o GF2MEQ dx dy ~ „(1 + 0 W r f t

1

" 1 % ) '

f

x

F

A

X

)

-

±

i

^

-

I)1 (2.6.1)

Here the positive sign in front of the third term is for incoming neutrinos and the negative one for antineutrinos. GF is the Fermi constant. I n the most general case the functions Fu F2 and F3 are different for each of the processes vp, vn, vp, vn, i.e. there are 12 independent structure functions for each charged and neutral weak current, respectively. For flavour conserving charged weak currents follows from proton neutron isospin symmetry: Fivn=Fi;f>;

F?n=Fivi>

(2.6.2)

Another simplification is obtained by the quark parton model because the Callan-Gross relation (2.3.12) relates 2 xFx and F2. Thus there remain four independent functions only: Fjp = 2 x{u + d + s + c),

(2.6.3)

F2;P = 2 x{U + d + s + c),

(2.6.4)

xF/f

(2.6.5)

= 2 x(-u

+ d + s — c),

xFJP = 2x(u - d - s + c).

(2.6.6)

The flavour non-conserving charged currents and the neutral ones will not be discussed here. For an isoscalar target eq. (2.6.2) gives: FIRN

= J

(F{*

+

F R ) = J

{FI'N

+

FCP)

= Ff».

(2.6.7)

If eqs. (2.6.3) to (2.6.6) are inserted into eq. (2.6.7) and the sums of quark and antiquark contributions (2.5.13) and (2.5.14), respectively, are introduced together with the sea properties (2.5.3) it follows : F2*n = FjN

xF3"N = xFj«

= x(q + q),

= x{q - q) = x(uv + dv).

(2.6.8)

(2.6.9)

67

Fortschr. Phys. 34 (1986) 2

The comparison of eqs. (2.6.8) and (2.5.15) yields as a quark parton model relation: F2'N = A FJ>N-i^s+i-c-c). 18 o

(2.6.10)

The differences between different sea quark contributions are small and can be neglected for larger x values [x > 0.4). 3.

Quantum Chromodynamics

3.1.

Introductory Remarks

The development of Quantum Electrodynamics (QED) as the theory of electromagnetic interactions has shown that the experimental observations can be described in an extremely precise manner by a quantum field theory. In the beginning of the seventies serious attempts were made to develop a quantum field theory for strong interactions as well, basing on the principles of relativistic invariance, renormalisation, and local gauge invariance [15, 16]. An important step was the unification of two basic concepts, the concept of colour [17] that will be discussed below, and the quark parton concept [18] that has already been discussed. Another important step was the theoretical proof [19] that non-Abelian gauge theories (and only those [20]) show the property of asymptotic freedom. This was of great importance because it implied the prediction of logarithmic, i.e. small scaling violations confirmed experimentally already one year later. Today Quantum Chromodynamics (QCD) is considered to be the most prospective candidate for a field theory of strong interactions. QCD is built up quite similar to Quantum Electrodynamics, the only but essential difference is that they have different underlying symmetry groups. In QED the interaction is described by coupling the electromagnetic gauge field to the electric charges of the involved lepton fields. I n other words, the photon is considered to mediate the interaction. In QCD a colour charge is assigned to each quark field as a new degree of freedom. The strong interaction is mediated by a gluon field that couples to the colour charges of the involved quark fields. In both theories the gauge field (photon or gluon) is massless because of the unbroken symmetry of the fundamental fields (leptons or quarks, respectively). In QED the gauge field does not carry an (electric) charge — the underlying symmetry group is the abelian group Ul. In QCD however, the gauge field does carry a (colour) charge — the underlying symmetry group is the non-abelian group SU3C that describes the symmetry of the quark fields with respect to the colour quantum number. The requirement of local gauge invariance for QCD leads to an octet structure for the gluons. There are several reasons to believe in a colour triplet structure for quarks. It allows to bring the symmetry behaviour of the three-quark wave function in accordance with Pauli's principle [21]. Also, the experimental rate for the decay 7r° -» 2y can be explained assuming the quarks to appear in three different colours [22], The colour charge does not affect the flavour quantum number, i.e. strong interactions do not change quark flavours. There is again an analogy in QED — the electromagnetic interaction can not distinguish between electron and muon. A very important question not yet answered is whether the existence of colour charge can be proved experimentally. Up to now neither quarks nor gluons did show up directly, all observed hadrons are colour neutral. No dynamical reason is known yet for this experimental observation that has been generalized as confinement hypothesis: All observable fields and particles, respectively, are singlets with respect to the colour quantum number.

68

W.-D.

3.2.

Effective Coupling Constant

NOWAK

: Deep Inelastic Charged Lepton Scattering

In QED the presence of vacuum polarization effects leads to a screening of the electric charge. Thus the effective charge depends on the distance r in case of a static potential and on the momentum transfer Q2 in case of a scattering process. In other words, the charge eef( measured at large distances and small Q2, respectively, is smaller than the one measured at small distances and large Q2. The quark gluon interactions in QCD a) gluon-bremsstrahlung by a quark, b) quark-pair production by a gluon, have analogous processes in QED. In contrast to QED, the gauge bosons are charged in QCD. This gives the possibility of gluon self-interactions, e.g. c) 3-gluon interactions, which have no analogy in QED. They give rise to additional contributions to the vacuum polarization. The essential difference is that those correspond to an anti-screening of the colour charge [23], i.e. with increasing momentum transfer Q2 the effective colour charge gets smaller. In leading order, i.e. taking into account only the leading logarithms when summing up all terms of the perturbation expansion the Q2 dependence of the strong interaction coupling constant as{Q2) == ¡7fft/4jr is given by: (3.2.1)

Here ocs{q02) is the coupling strength measured at an arbitrary renormalization point q02 and (3.2.2) where Nf is the number of quark flavours. As can be seen from equation (3.2.2) each quark flavour contributes to the colour charge screening. However, only the existence of more than 16 quark flavours would compensate the increase of the colour charge caused by

t

INFRARED

a ¡5

a 8

oo a(Q2)/

/

/ QCD

caj E c FL) are shown separately

Fig. 13. Typical next-to-leading order prediction for i?QCD vs. x calculated at A = 300 MeV

3.6.

Higher Twist Processes

The deviations from Bjorken scaling as predicted by QCD are of purely logarithmic nature as far as virtual boson interactions with a single parton are concerned. I n the context of the operator product expansion [26] such processes are described by operators with leading twist (r = 2). Here twist means operator's dimension less spin. On the other hand, collective interactions which include more than one parton are described by higher twist operators (T = 4, 6, ...). Among these processes which are expected to have a power law dependence on Q2 are: — — — —

virtual boson interactions with a diparton, i.e. a diquark (qq) or a gluon-quark (gq); simultaneous interactions of the virtual boson with several partons; interactions of the scattered parton with other partons; resonant multi-parton states.

78

W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

At present there is no satisfactory theoretical solution yet to calculate the power corrections. Two attempts are based on the quark parton model. The first one takes the nucleón as a superposition of real collinear partons [42] whereas the second one considers the partons as virtual particles carrying a transverse momentum [43], Both approaches gave no firm predictions yet because they include only a part of the above quoted processes. In parallel there is a phenomenological approach to describe the power corrections [44]: F2(X, Q2) = Ff=2\x,

Q») j l +

H

-P +

+ -J.

(3.6.1)

Here the function F2 ra I* otí ,tí œ. -P çatí O ao © CÖ tí 'm otí O ft t» .tí ft © ao .2 -ptí t-i 3 o ® oS CÔ ft h 02 oFftH Tí ^ChG"ctí3 _o tí O cS 3 F©h ,— o s "tì

(G) 3*

92

W.-D.

NOWAK:

Deep Inelastic Charged Lepton Scattering

(iii) the internal properties of some or all bound nucleons are assumed to be different (increase of nucleón radius [73, 74], introduction of pions of other mesons forming resonant states together with nucleons [75]). Some of these predictions are illustrated in figs. 24(a), (b) and (c) according to the above classification. The presently best agreement with experimental data is obtained by a model basing on increased nucleón radius [73] but some other models are not much worse. To discriminate between different models needs more data allowing to study sumultaneously the dependence of the effect on x, Q2 and A. For one of the most recent models of the third group [119] the model parameters were fitted to both SLAC and EMC data [120]. As can be seen from the curves drawn in fig. 20 both data sets are well described. 5.3.

QCD Analysis oí Heavy Target Data

The knowledge of parton distribution functions of free nucleons is very important to predict cross sections for many processes at higher energies. In the prededing two sections it was shown that the parton distributions of a nucleón bound in a nucleus differ from those of a free nucleón. Therefore heavy target data cannot be used a priori to extract parton distribution functions of free nucleons. This might be true as well for the gluon distribution function that has been obtained so far only from heavy target data [77]. The presence of nuclear effects does not affect the validity of QCD analyses of lepton nucleón data taken with a heavy target [76]. A nuclear structure function FA can in general be represented as a superposition of nucleón structure functions FN: FA(x, Q2) = } dzfN(z) FN(xjAz, x/A

Q2).

(5.3.1)

Then, the nuclear structure function FA obeys exactly the same Altarelli-Parisi equations for the Q2 development as the nucleón structure function Fs. This is due to the independence on x and Q2, of the shape of the probability distribution fN{z) for nucleons inside the nucleus. There is also a simple way to understand the independence of the A determination on the nuclear binding effects. As discussed already in section 3.4 the QCD scale parameter is determined only by the change of the structure function with Q2, and thus practically independent of their shape at some starting value Q02 (cf. section 7.2). Therefore, the same A value will be deduced also from a slightly modified structure function as far as the modification, i.e. the influence of nuclear binding effects is independent of Q2. There is a small technical problem when a QCD analysis is performed for a nuclear structure function. The extension of the integration region up to x = A requires the starting parametrization for the structure function at Q(t2 to be given likewise. Fortunately, the presently missing knowledge about structure functions in the region 1 sS x A has little impact because of the strong damping by the splitting functions already for Closing the discussion on nuclear binding effects it has to be mentioned that the EMC effect was observed up to now only with incoming electrons and muons, respectively. The accuracy of presently available neutrino data is not sufficient yet to prove or disprove the observed phenomenon [77].

93

Fortschr. Phys. 34 (1986) 2

6.

Test of the Quark Parton Model

6.1.

Spin and Transverse Momentum of Quarks

The measurement of the photoabsorption cross section ratio R = aLjaT yields information on the quark spin. If a finite R is measured the quark spin can only be 1/2 (cf. eq. (2.3.11)). In this case R carries information on the transverse momentum of quarks. Its intrinsic component (eq. (A. 10)) is assumed to fall with 1/Q2 whereas the QCD order a 5 component is expected to have a strong x dependence (cf. fig. 13). A precise determination of R is difficult because absolutely normalised cross sections have to be compared at the same values of Q2 and x but at different E0. Results on R measured at SLAC in electron nucléon scattering are given in fig. 25(a). The QPM contribution (eq. (A.9)) shown as dotted line is as large as R ~ 0.1 at x = 0.2 because of a correspondingly low Q2 (1 < Q2 < 4 GeV2). The first non-zero QCD contribution is given by next-to-leading order, the predicted x dependence for Q2 = 6 GeV2 is shown as full line. Only for x < 0.3 the data points are consistent with the sum of QPM and QCD contribution. To get a better description of the data at large x one can introduce a third contribution due to a strong diquark contents of the nucléon. The corresponding x dependence calculated in QCD for twist = 4 [78] is shown as dashed line in fig. 25(a). The data are in good agreement with the sum of all three contributions. The muon proton data on R as measured by EMC [79] are shown in fig. 25(b). Due to the large average Q2{{Q2) = 55 GeV2 at x = 0.3) the QPM contribution is vanishing even at small x. The order rxs contribution as calculated at v = 100 GeV for A = 160 MeV is drawn as a solid line. As can be seen, the large error bars still prevent a discrimination

0

>

0PM QCD diquork

06 -

0

0.2

OA

0.6

° SLAC

ep

o CHI0

pp

0.8

-OA 0

0.1

0.2 X

Fig. 25

03

1.0

94

W.-D. NOWAK : Deep Inelastic Charged Lepton Scattering

CDHS o y-dependence • upper limifs

05

(K D 0.3 —I O 0.2 II

cc

0.1 0.0 0

.6

1.0

Fig. 25. Results for R ~ FL/2xFl from different lepton scattering experiments : (a) SLAC data on ep scattering, (b) EMC data on ¡j.p and [xFe scattering, (c) CDHS data on v scattering off iron, (d) CHARM data on v scattering off marble

between R = 0 and next-to-leading order. In the same figure are shown muon iron data on R measured with the EMC apparatus as well [80]. Within the error bars both data sets are consistent. Neutrino results are compatible with those from muon experiments. Fig. 25 (c) shows the CDHS results obtained from neutrino iron measurements using two different methods of data analysis. The first one covering the range 0.02 sS x g 0.6 cannot discriminate due to large error bars between R = 0 and next-to-leading order (dotted line) [81]. The second method yields upper limits on R with considerably smaller errors but it can be applied only for x > 0.4 where it agrees well with the prediction R 0. The integrated upper limit corresponding to {Q2) = 38 GeV2 is given by R 5S 0.039 ± 0.014 (stat.) ± 0.025 (syst.) [82], The CHARM data obtained in neutrino marble scattering [83] agree well with the above discussed results as can be seen from fig. 25(d). Summarising all R measurements the quark spin is uniquely determined to be 1/2. The average values on R are tabulated in tab. 3. The QCD next-to-leading order prediction R ~ 0 for x > 0.5 is confirmed whereas for smaller x the data cannot discriminate Table 3 Results on the average of B ~ FL/2xF1 Coll.

Year

Reaction

SLAC-MIT

1978

CHIO

1979

ep ed ep ed HP

CHIO

1979

WP

EMC

1981

WP

CDHS CDHS

1981 1982

vFe vPe

SLAC

1979

from different experiments 0.01

1 -T- 16 1 -r- 16 3 -j- 18 3 18 0.9 -i- 30

0.01 < x < 0.1

0.4 -^ 30

—

m

Special x-range — — — —

—

0.4 < x < 0.7

= 22.5

m = 20 w*{x) dx ~ 0.14 ± 0.02. o

(6.5.4)

The comparison of this number with the QPM value 5/18 shows that charged partons carry only about one half of the nucleon momentum. This experimental information was known prior to the development of QCD. Already in 1972 it was interpreted by Feynman [1] as first evidence for the existence of neutral-constituents within the nucleon. 6.6.

Number of Yalence Quarks

The neutrino structure function xF3vN directly measures the valence quark contents of the nucleon (eq. (2.6.9)). A sum rule is obtained by integrating this structure function and applying the normalisation conditions (2.5.6) and (2.5.7): i JjxF3(x)dx

= 3.

(6.6.1)

o This sum rule first derived by Gross and Llewellyn-Smith allows a measurement of the number of valence quarks per nucleon. As for the Gottfried sum rule the extrapolation to x 0 requires data in the low x region to keep the total error reasonably small.

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W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

The structure function xFs as measured in neutrino scattering off different nuclear targets is displayed in fig. 29 [77], There is good agreement concerning the Q2 dependence. Unresolved systematic differences up to 10% appear in the x dependence comparing CDHS [94] on one hand versus CHARM [95] and CCFRR [96] on the other hand. Cal-

0.3

0.1

"i M 11

7

0.5

m

1—i—r ri i n|' xcvN F f ( xtsj, a Z )

X=0.015

'1.0

¿ f

0.5

Ï

x=0.045

0.3 0.2

1.0 x=0.08

0.6

0.3

1.0 x-0.15

0.6

1.0

0.3

0.6

x=0.25

'1.0

0.3

0.1*

x=0.35 0.8

i

0.5

0.2 • •CCFRR(Fe) . o C D H S (fe) 0.2

0.2

i M A '^" »^ t.-ÍJ .. . >' . - .

0.3

x=0A5

Ô.5

r . V ^ t t o V i

x=0

-55

0.2 0.1

ka CHARMiCaCOj)

0.1

T

0.005

'T

k

.f

! W»T

10

IxL 100

x=0.65

I

1000

Q 2 (GeV 2 ) Fig. 29. 1.5 GeV2, W2 > 4 GeV2 [102, 103]. One fit was made per x bin parametrising F2(T=2) as a non-singlet function (7.2.1) and fitting the parameters a, ¡3 and y simultaneously with A and h4. No twist Sj 6 contributions were taken into account. The order e+7t°. In spite of the large uncertainties in the prediction (8.4) the minimal S U 5 model is almost ruled out by this experimental result.

Fortschr. Phys. 34 (1986) 2

111

Nevertheless, a considerably higher accuracy in the A measurement is very desirable to allow more stringent theoretical predictions to be calculated. The current high statistics BCDMS measurements of muon proton scattering are expected to allow a A determination with a total error of about 50 MeV. A result with similar accuracy could emerge from a possible future measurement of the non-singlet structure function difference F2p - F2n by EMC [124]. In the following years the experimental study of the nucleón structure will be pursued at higher energies. The next generation of lepton nucleón scattering experiments is in preparation at the FERMILAB Tevatron. A muon beam of 650 GeV maximum energy is expected to come into operation in 1986. Concerning quality and intensity it will be comparable with the CERN muon beam. The increase in energy by a factor of three will allow nearly the same increase in the maximum four-momentum transfer. One experiment, a large forward spectrometer equipped with a streamer chamber vertex detector, is being installed. The future accelerator/storage ring complex (UNK) at Serpukhov will have 3 TeV maximum proton energy. As possible future experiment on muon nucleón scattering [113] would push the maximum accessible Q2 by another factor of three to about 2000 GeV2. The technique of colliding beams will be used for the first time in electron nucleón scattering experiments at HERA. Interactions of 30 GeV electrons with 820 GeV protons will allow for a maximum Q2 of about 105 GeV2. However, the measurement of structure functions will be restricted to Q2 < 25000 GeV2 because the event rate is decreasing rapidly with increasing Q2 [114]. The future HERA experiments will resolve the nucleón structure down to 10~16 cm thus eventually probing the structure of quatks [115]. They may find first evidence for still smaller constituents of matter being already predicted by several theoretical models today [116].

Conclusions The first generation of deep'inelastic lepton scattering experiments starting in 1968 at the SLAC electron accelerator opened a new era in the understanding of the nucleón structure. The observed scaling behaviour of nucleón structure functions led to the development of the quark parton model (QPM), the basic assumptions of which were all confirmed by later experiments. In the second generation of experiments carried out some five years later at the FERMILAB muon beam small deviations from the exact scaling behaviour were found. This observation was in agreement with the small logarithmic scaling violations predicted at the same time by Quantum Chromodynamics (QCD), a candidate for the theory of strong interactions. The third generation of deep inelastic lepton scattering experiments came into operation in the late seventies using both muon and neutrino beams at CERN. These experiments allowed for the first time a quantitative test of the predictions of perturbative QCD. They all are consistent with asymptotic freedom as a basic property of QCD. However, testing in a unique way perturbative QCD predictions for deep inelastic lepton scattering is complicated by both experimental and theoretical problems. On one hand, the present measurement errors of the QCD scale parameter being mainly of systematic origin, are rather large especially because a logarithmically changing parameter has to be measured. On the other hand, there are several theoretical difficulties preventing a decisive confirmation of the predicted logarithmic Q2 evolution of the nucleón structure functions. The calculation of other than leading order predictions is rather difficult. For the time being the parameters of the perturbation expansion are known up to next-to-leading order, only. The longstanding and still unresolved problem

112

W.-D. NOWAK: Deep Inelastic Charged Lepton Scattering

of the best renormalisation scheme has lost its practical i m p o r t a n c e for deep inelastic scattering since the MS scheme was shown to allow for a quite fast convergence of the p e r t u r b a t i o n series. There is almost no theoretical understanding y e t of the higher twist effects being responsible for collective p a r t o n phenomena. T h e y are e x p e c t e d to drop with powers of Q2 and could, in principle, also fake a logarithmic Q2 dependence. T o d a y , the only experimental information on higher twist effects in deep inelastic scattering is a phenomenological fit to combined low Q2 electron and high Q2 m u o n p r o t o n d a t a . This fit shows the power corrections being small for Q 2 > 10 GeV 2 . Believing in this experim e n t a l result the high Q 2 lepton scattering e x p e r i m e n t s should be free of substantial higher twist contributions. Taking into a c c o u n t their statistical and s y s t e m a t i c errors all these e x p e r i m e n t s are compatible with a value of A = (150_1JQO) MeV for the QCD scale p a r a m e t e r . Summarizing all QCD tests performed so far in deep inelastic lepton s c a t t e r ing e x p e r i m e n t s it can be concluded t h a t the present e x p e r i m e n t a l a n d theoretical uncertainties are still too large t o consider Q u a n t u m C h r o m o d y n a m i c s as the confirmed t h e o r y of strong interactions. T h e r e c e n t l y discovered difference in the behaviour of nucleón s t r u c t u r e functions measured for bound and free nucleons, respectively, is now unanimously confirmed b y several experiments. However, to get unique results for small m o m e n t a of the nucleón constituents, one has to wait for more precise information t h a t will come from a presently running experiment. Also, m o r e d a t a on the a t o m i c n u m b e r dependence of this effect will be available. I t could allow a first discrimination a m o n g the huge n u m b e r of phenomenological models proposed so far. F u t u r e e x p e r i m e n t s on deep inelastic electron p r o t o n scattering planned for the late eighties with colliding b e a m s a t H E R A will probe the nucleón s t r u c t u r e down t o 10~ 1 6 cm. T h e y m a y observe for the first time a new substructure opening in this w a y possibly a new era in the understanding of the nucleón.

Acknowledgements For many years of good collaboration I am deeply indebted to my colleagues in the BCDMS group at J I N R and CERN. For repeated support and encouragement I want to thank Professors K. Lanius, I. A. Savin and A. Staude. I could benefit from the gratifying and stimulating cooperation with Dr. M. Klein. For patient support and numerous suggestions during the preparation of this paper I am grateful to Dr. H. Nowak. Special thanks are to Dr. P. Reimer for the proof-reading of the manuscript. Finally, I greatly acknowledge the friendly technical help in preparing this paper of Mrs. J. Ruiz, Y. Goidadin, M. d'Adhemar and C. Rigoni.

Appendix Ratio of Photoabsorption Cross Sections R =

aLlaT

T h e hadronic v e r t e x of the lepton nucléon scattering process can be described b y photoabsorption cross sections instead of structure functions. Denoting the cross section of a virtual p h o t o n having longitudinal or transversal polarization b y a L or a T , respectively, the cross section (2.1.5) can be written as follows [ 1 1 8 ] :

dQ2 dv

K * 2n Q2E2

1

1 -

e

{