III. - DIFFUSION INÉLASTIQUE DES LEPTONSAN ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTION AND THE PARTON MODEL

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III. - DIFFUSION INÉLASTIQUE DES LEPTONSAN ELEMENTARY INTRODUCTION TO DEEP

INELASTIC ELECTROPRODUCTION AND THE PARTON MODEL

A. de Rújula

To cite this version:

A. de Rújula. III. - DIFFUSION INÉLASTIQUE DES LEPTONSAN ELEMENTARY INTRODUC- TION TO DEEP INELASTIC ELECTROPRODUCTION AND THE PARTON MODEL. Journal de Physique Colloques, 1971, 32 (C3), pp.C3-77-C3-86. �10.1051/jphyscol:1971311�. �jpa-00214592�

Ill. - DIFFUSION INELASTIQUE DES LEPTONS

AN ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTIBN AND THE PARTON MODEL (*)

A. DE RUJULA (**I

CERN, Geneva

Rksumk. - Nous prksentons de maniere simple une Ctude de la diffusion profondement ink- lastique d'electrons sur des protons, a haute energie et grande transference de moment. Les sujets que Yon couvre sont la cinematique, les donnkes exp&imentales, l'invariance d'echelle et la pheno- mknologie du processus en termes du modde des ⁽⁽ partons ⁾⁾ du nucleon.

Abstract. - A simple review of the subject of inelastic electron-nucleon scattering at high energy and momentum transfer is given. The topics discussed are the kinematics, the experimental data, scale invariance and the phenomenology of the process in terms of the basic ideas of the parton model of the nucleon.

1. Introduction. - An enormous amount of interest and expenditure of paper has been trigerred by the beautiful and well advertised results of the recent MIT-SLAC collaboration on high energy electron scattering from hydrogen and deuterium targets [I], [2].

These were one-arm ( c inclusive ^D) experiments in which onIy the scattering angle and energy of the electrons are measured. Not observing the final individual hadrons results is an enormous simplifica- tion of the theoretician's work. Only very recently have coincidence measurements been reported [3].

Some related experiments-inelastic neutrino scatter- ing 141, e + e- annihilation a t high energies 151, ^p pair production by protons on heavy targets [6], photo- production [7] have become comparably fashionable.

The common feature of these processes is the study of the hadron structure in extreme kinematical conditions by means of weak or electromagnetic probes that are supposed to be well known chirurgical tools.

In this version of the talk we shall only cover the electroproduction process and its description in terms of Feynman's parton model. The connection with related processes and the relevance of other models will be discussed by different speakers in this Ren- contre.

The actual talk was somehow shorter than its written version, mainly due to the excellent weather and snow conditions at MCribel-les-Allues. No attempt is made to give references to the innumerable excellent reviews of the subject, nor to compete with them in any respect.

in the electric charge, the amplitude for the process eN -+ e r is described by the diagram of figure 1 where the notations for the momenta of the particles are specified. The index r stands for the unobserved final hadrons.

FIG. 1. - One photon exchange approximation and kinematics of electron-nucleon scattering.

According to the usual Feynman rules the amplitude can be written as

2. Description of electroproduction in the one- x < r I I P, 0 > (1) photon exchange approximation. - The lowest order

where V,, is the hadronic electromagnetic current. The

(*) Review talk at the Rencontres de Moriond, Meribel-les- transition probability is proportional t o Allues, France (1971).

(**) On leave from ^J. E. N., Madrid. I A 1' K e4 la" w a y / q 4 (2)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971311

RUJULA

where I,, is an explicit lepton tensor :

l,, = - 1 C ^{u(k', -} A') y, u(k, A) G(k, 1) 7, u(k', 1') = 2 a,at

= 2(k, k: + ^{k, kh} + ^(m: - k. k') g,,) (3) and Wpv corresponds to the hadron vertex :

x (2 7cI3 aS(p + q - P,) . (4) The tensor W,, is hermitian by construction (W,, = w,,). The definition of W,, incorporates the phase space and spin sum over the unobserved final hadron states

0

It is not possible to give an explicit form of W,,, since one lacks a complete theory of strong interac- tions. However, one can construct W,, in its most general form. In so doing, let us consider and respect the following requisites :

i) covariance : W,, is a Lorentz second rank tensor;

ii) current conversation (PV, = 0) ; iii) parity (P) conservation ;

iv) invariance of the theory under time reversal (7').

i) At the hadron vertex only the four vectors p and q are available, since individual final particles are unobserved. Hence, the general form of W,, is :

where the i factors have been placed in such a way that the hermiticity of W,, (W,, = )w: implies the reality of the ⁽⁽ structure functions ⁾⁾ Wj. The W' s are scalar functions of two of the independent invariants that can be constructed with p and q. It has become custo- mary to use q2 [8], the invariant squared momentum transfer, and v = pq. In the lab. frame v is proportional to the energy transfer from the lepton line

Another variable in use is the total invariant hadron mass M

M 2 = ( p f q ) 2 = m 2 + 2 v + q 2 . (6) ii) Current conservation constrains the hadron matrix elements to satisfy

and W,, consequently :

From (8) it follows that W6 = 0 and two equations that allow, for instance, to express W4,5 in terms of W,,,. Adopting the customary sign conventions we have that

The functions Wl ,,,, are dimensionless in our norma- lization (< p I ^p' > ⁼ (2 7 ~ ) ~ 2 ~6~ (p - p'), %U = 2 m).

iii) Parity conservation implies W3 = 0, since W, multiplies the only Lorentz pseudotensor in (9).

However, W3 automatically vanishes in the contrac- tion Iw W,,. This is no surprise since we have written a normal )) lepton current under P : W3 would (to lowest order in cl) give a pseudoscalar contribution to the cross-section, which is a scalar quantity.

iv) Under time reversal a T-even tensor ought to satisfy Wpv = W;,,(T~, Tq) where Tinverts the spacial components of four-vectors. The condition implies W, = 0, a result already obtained from current conservation. This means that one cannot (to lowest order in cl and summing over the target nucleon spin) detect the violation of T by a conserved current. The same considerations as in the case of the parity cons- traint apply to the contraction I"" W,,.

To summarize, the conditions of Lorentz invariance, current conservation and parity conservation, imply that the hadron vertex is described by two independent scalar functions W1,,(v, q2). The conservation of the lepton current further implies that the terms propor- tional to q in W,, drop from the cross-section formula so that, for practical purposes, we are left with

There are still extra model-independent constraints on the structure functions 191, usually referred to as

<< positivity conditions D, stemming from the definition

of W,,. A simple way to obtain them for Wl and W2 is to consider the most general vector v = ap + ^{pg at}

the hadron vertex. The explicit form of W,, implies that v, Wwv v: >, 0. The inequality constrains the structure functions to be positive semi-definite and to satisfy the relation

The total cross-section at an energy E(E1) of the incoming (outgoing) electrons is then

AN ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTION

do mEEf do

-- - - - - dE'dB d 1 q2 ( dv

- - - 4 E' m sin4 812 ^a2 -- C0S2 ( ^{W2 + 2 W, tg2}

where 8 is the laboratory scattering angle and the lepton mass has been neglected. The W, ,, functions currently used by experimentalists have dimensions of [Mass]-' and are l/m times the dimensionless ones we use.

A different set of hadronic structure functions is also of common use [lo]. They are introduced by viewing the lepton vertex as an explicitly known source of virtual photons, the experiment determining their cross-section on the nucleon target. There are three possible polarization directions 8, for a virtual photon of momentum q = (q,, 0, 0, q,) satisfying the gauge condition 4.8 = 0, namely

(13a)

8, = (CIS, 0 ~ 0 , qo)/ J q

FIG. 2. - The (v, q 2 ) plane in eIectroproduction.

E: = (0, 1, + ^{i, o)/} 42. (13b)

The corresponding longitudinal and right and left- iii) Fixed missing mass lines, i. e., the one corres- handed transverse cross-sections, defined as being ponding to the excitation of a narrow I-esonance (111)- proportional to e2 E, 8: WwY can be explicitly related The missing mass is measured along the M 2 axis of

to W , ,, : figure 2.

K iv) Fixed scattering angle : straight lines like VI.

W, ⁼ l s, (14a) Line IV corresponds to forward scattering or photo- 471. a production (q2 = 0) and V t o backward scattering

K ^(- q2 = max.). The separation of W, and W2 in the

(14b) cross-section formula is made by measuring at different E and 8 such that the same (v, q2) point corresponds to The constant K is arbitrarily fixed to be different values of tg2 012.

The shaded part of figure 2 is the physical region for K = ( M ~ - m 2 ) / 2 m , electroproduction at a given E. The dashed region, measurable at high energies, is the so-called c< deep corresponding to the flux of virtuaI photons being inelastic region ,,.

equal to the flux of real photons that would produce a

final hadron state of mass M. 4. Experimental data. - The results of the SLAC experiments have constituted a double surprise : 3. Topography >> of the structure functions. - It

is very useful to visualize the different parts of the i) Pointlike behaviour in the deep inelastic region.

kinematical plane (,,, q2), represented in figure 2. Consider the cross-section (12) integrated over some Keeping in mind the relations small range of final hadron mass AM, and amputated

of the (( trivial N factors - q2 = - (k - kt)' = 4 EE' sin2- e

2 4 = 1

e ^{~ E I} ~ S Z (16)

= 4 E E - - sin2-

3 2 (15a)

v = p . q = m(E - E') (15b) where do/dQ (Mott) is the elastic cross-section on a massive structureless target :

M 2 = ( p + q ) 2 = m 2 + q 2 + 2 v (15c)

we consider the following regions. - do (Mott) = a2 cos2 012

dS2 4 E' sin4 812 ' (17) i) Elastic scattering (M2 = m2). The straight line I

of figure 2. Figure 3 [ l 1 ] schematically represents the experi-

ii) Inelastic threshold ( ~ 2 2(m + m11)2). ~ h , line mental results for t as a function of q2 for fixed missing q2 + 2 v ^1:2 mm, denoted by I1 in figure 2. Inelastic M2-

scattering occurs to its right. - Line I, that drops approximately like q - 8 for big

C3-80 A. DE RUJULA

FIG. 3 . - Momentum transfer dependence of the hadron vertex at fixed missing mass.

q2 corresponds to elastic scattering [(( dipole ⁾⁾ form factors F ( ~ ' ) -- q - 4 ] .

- Line 11, analogous to I would correspond to the excitation of a nuclear isobar, integrated over its width.

- By increasing M 2 one obtains lines like I11 ( M

-

-

The above-mentioned facts somehow suggest that upon energetically breaking the nucleon in deep inelastic events, one is hitting point-like (structureless) components. A possible analogy would be the scatter- ing of low energy electrons by atoms, elastic when the atom is left in the ground state, resonant when it is raised to an excited level. Upon increasing the incident energy the process becomes the point-like electron- electron scattering, ⁽⁽ deeply inelastic ⁾⁾ from the ionised atom's point of view.

ii) Scale invariance. This was an enormous surprise, since it had been predicted on theoretical grounds.

According to the current algebra arguments of Bjorken [12], in the limit where v and - q2 tend to infinity, their ratio o r - 2 v/q2 being fixed, one would have that

F,,, being non-trivial functions of just one variable o. The Bjorken limiting direction is denoted VII in figure 2.

Consider a hypothetical theory of massless particles, where nothing sets a mass scale. In such a theory dimensioned magnitudes like v or q2 have no sense, only their dimensionless ratio o ⁼ 2 vl- q2 might be meaningful. The scale of the momenta in the defini- tion (10) of the dimensionless tensor W,, ought to be set by one of the variables v or q2 and not by an arbi- trary mass :

no mass

w,, ^{= -} w ^{1 g,,} + ^w2

*

-

The dimensionless functions F, and F2 in (17) can only depend on o. If nature approaches such a simple dimensional description as v, q2 S m2 one would have that, in the Bjorken limit

which are precisely the expressions (18) conjectured by Bjorken.

If the baryon spectrum turned out to be bounded in mass, may be one could expect masses t o become irre- levant at very high energy and momentum transfer, and scaling to occur [13]. The experimental surprise was that scaling seems to take place from surprisingly low values of q2 and M 2 . This means that plotting for instance vW, as a function of o in the mentioned kinematical region, the experimental points approxi- mately fall on a single line, although v W 2 is in principle a function of two variables. Figure 4 schematically

FIG. 4. - Schematic dependence of vWz/m2 on w, in the deep inelastic region.

shows the trend of v W 2 . Two close experimental points like the ones in the figure might correspond to very different values of v and q2.

More refined ways to display the eventual experi- mental scaling and to distinguish different scaling variables [14] have been devised by Nachtmann [I51

A N ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTION C3-81 and Chen [16]. They are discussed by the first author

in this Rencontre.

5. The parton model. - The language of partons was originally introduced by Feynman [17] in an attempt to reach a simple understanding of electro- production processes and has subsequently been applied to other electromagnetic and weak processes, as well as to purely hadronic ones. Like all models of hadrons proposed to date, the parton model has suffered a steady growth of complication and might eventually survive just as a mnemotechnic tool.

FIG. 5. - Parton model description of electroproduction.

The essence of the model is an application of a classical impulse approximation to the conjectured parts of the nucleon (partons).

The impulse approximation is customarily applied to composite targets like atoms or nuclei when the energy transfer from the scattered particle is big compared to the binding energies of the constituents of the target.

Then, it is argued, the building blocks of the scatterer can be treated as free and independent uncorrelated targets. The trouble with the application of an impulse approximation to the nucleon is two-fold : first, the eventual fundamental constituents are unknown ; second, if they exist, they are presumably strongly bound together. However, there are hints that whatever stuff builds the nucleon wave function, its high momen- tum components are somehow strongly suppressed : when a nucleon is cc broken ^)> in a high energy hadronic collision a forward jet of particles results, the particles shaken off the nucleon have exponentially damped small transverse momenta ( 1 k, 1 2 0.4 GeV). A

possible analogy could be the breaking of a deuteron by a high energy photon : if the high momentum compo- nents of the deuteron's wave function are unimportant, the spectator nucleon will preferentially recoil with a small momentum. The analogy might be misleading, because the deuteron components are known.

The trick of Feynman consists in using the above- mentioned fact to choose a particular class of frames of reference where the impulse approximation to the nucleon might be valid, thereby enormously simplifying the description of the process.

In a frame where its momentum is asymptotically big( ( p 1 -+ a), the nucleon, roughly speaking, is contracted to a very thin disc and 'frozen (because of the relativistic time dilation) into one of the possible configurations of its constituent partons. If the momen- tum and energy transfers are big, the photon is a projectile highly localized in space and time. The electron cc sees ⁾⁾ the instantaneous charge distribution of the nucleon. More explicitly, the starting hypotheses of the parton model are the following :

1) THE TRANSVERSE MOMENTA OF THE PARTONS IN THE INFINITE MOMENTUM FRAME ARE NEGLIGIBLE RELA- TIVE TO THEIR LONGITUDINAL MOMENTA (parallel to the _.- carrier nucleon's momentum).

2) T H E PHOTON-PARTON INl%RACTION IS POINTLIKE.

- The second assumption is suggested by the SLAC data. It gives physicists a breath till the day they are faced with a new (sub-sub-nuclear) spectroscopy.

From the first assumption one can argue the appli- cability of an impulse approximation in given kinema- tical conditions. Let us define x i as the fractional longitudinal (11 p) momentum of the ith parton : pi = x i p (the index i = 1, ..., N labels the partons in a given N parton configuration). The values of x i are restricted to the interval 0 < x i < 1, since the limiting values correspond to one or all but one of the partons having a vanishing longitudinal momentum, a situation which is excluded by hypothesis ( I ^pi I I pf I).

Let z (life) be the lifetime of a particular parton configuration with N partons of energies Ei, and z (int) the time during which the interaction takes place, both computed, for instance, in the infinite momentum electron-nucleon center of mass system (c. m. s.) [18]

z (int)

-

( E - ^{' m .} 2 v + q2

1 -

z (life)

-

i

C3-82 A. DE RUJULA with the usual definition of s [ s = (p + ^{k)' z 4} 1 p 1'

is asymptotically big]. We have used p2 $- m2 and, in agreement with assumption I), neglected partons with very small xi [<< wee partons >>with ^X: q (m: + pT,)/s]

[19]. A condition for the application of an impulse approximation is z (life) % z (int) (binding energy

< transferred energy), namely

2

q 2 + 2 ^J, = - 42(0 - 1) ^{Pil +} m: - m2 . (20)

i Xi

This requires masses and transverse momentum be negligible relative t o - q2 and v [20], o not being allowed to approach its limiting value of 1 (exclusion of the elastic and resonance regions). We shall see below that in the parton picture o -+ co corresponds to.

x i + 0 for the interacting parton : the model might not be applicable to the region of high v, fixed q2 (Regge region, direction VIII, Fig. 2). The eventual validity of the impulse approximation is therefore limited to the Bjorken asymptotic region (VII in Fig. 2).

It is no surprise that the partons can be treated as free in a reference frame where the lifetime of the nucleon states is dilated. This is because in a second quantized theory interactions correspond to the exchange of quanta (new partons). The transverse momenta being relatively small, the exchanges take place over times that are big compared to the interac- tion time. The free particle behaviour of the partons is also used to argue that the final state interactions between the scattered and the pass-by partons do not affect the explicit calculations (See Fig. 5). The sum over all final states T' (assumed to be a complete set at a given M2) results in an expression from which the effect of the final state interaction drops, if a free particle normalization is used for the parton states [21].

Although the final state interactions can be neglected for calculational purposes, it is necessary to consider them in theories where the partons are particles (like quarks) that have up to now been reluctant to show up.

The difference between Fig. 5a and 5b explains why.

To summarize, the parton model of deep inelastic electron-nucleon scattering uses the following recipe.

The rtucleon in the infinite momentum frame can be treated as a one-dimensional stationary gas of free pointlike constitutents that interact incoherently with the

electron.

The recipe makes explicit calculations possible, as we shall see in detail in Section 7.

6. The Drell-Levy-Yan (DLY) parton model. - The consistency of the parton picture has been checked in a series of most instructive papers by Drell, Levy and Yan [22]. They study order by order in strong interac- tions a field-theoretical model with bare protons and pions as partons, interacting via the usual G $ ~ , I)@

coupling. A cut-off in transverse momentum is made right from the beginning.

Consider the electromagnetic vertex < r I J, I p >.

In first order in G the state r ^{is a p - z} system and two

diagrams are possible (see Fig. 6a, b). The pions are ^{FIG. 6.} - Graphs of the Drell-Levy-Yan model to lowest order in strong interactions.

AN ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTION C3-83 taken as neutral for the sake of the discussion. An

explicit calculation within DLY's model shows that the contribution of diagram 6b is negligible relative to 6a in the Bjorken limit. In any given frame of reference the covariant Feynman diagram 6a can be split into two non-covariant (old fashion perturbation theory) time ordered diagrams (6c, d). In diagram 6d the pho- ton first splits into a parton-antiparton pair. DLY have shown that in a particular class of infinite momentum frames (in which the photon has go -- ^{q, cz 11} I p I ^; the electron-nucleon c. m. s. consi- dered so far belongs to this class), the contribution of 6d is negligible relative to 6a. The seemingly non- relativistic picture applies, in which the nucleon first dissociates into its constituents and then interacts with the photon. It gives the full covariant answer, since the rest of the contributions are negligible. Moreover, in the Bjorken limit and in the reference frames just mentioned, the over-all energy momentum conserva- tion boils down to the conservation at the elementary photon-parton vertex and the applicability of the impulse approximation follows. These results are valid order by order in perturbation theory, and might be extended to the case of charged pions.

A delicate point in the model is the fact that, due to the transverse momentum cut-off, the final state interac- tions between the unperturbed partons and the parton

<( hit D by the photon are vanishingly small in each

order of perturbation theory : the diagram of figure 5b is dominant. Eventual extensions of the model to partons with odd properties like quarks, would then be excluded. A possible remedy could be that the pertur- bation series sum of vanishingly small contributions added up to a significant non-zero result. This pheno- menon takes place for the wee partons of a Q3 theory, needing no cut-off, studied by Chang and Yan [23].

EXPLICIT CALCULATIONS WITHIN THE PARTON MODEL.

- Let us recall or introduce the following definitions : i) xi : the longitudinal share of total momentum by the i-th parton (pi = xi p). The four-momentum relation p i = x i p does also hold, if the masses are negligible relative to I ^p 1 and the wee xi values are excluded.

ii) ~ i , ~ ( q ~ , v, x) : Structure functions describing the photon-parton electromagnetic vertex.

iii) pN : probability of a given N parton configura- tion. N is not necessarily finite and includes the extra quantum numbers eventually necessary to label a state.

iv) fi(x) : probability for the i-th parton (in the configuration N) to have a share x of total four- momentum.

Since the W functions represent probabilities, and these are added incoherently in the model (see the recipe of the previous Section), the total W,,, of the nucleon is :

where we have summed the contributions of all partons in all configurations, with the appropriate weight functions. In the normalizations we use, there are 2 E particles per unit volume ; the E/Ei -- l/xi factor in (21) ensures proper counting of the partons, such that each one-parton contribution is counted once in the one- nucleon state.

The tensor describing the photon-parton vertex is, in analogy with (4) :

w;,= C < p , I V , I p f > < ~ f l V v I P i > x

s p i n p ;

where the momenta are labelled like in figure 5a. The assumed pointlike character of the partons permits to write the elementary vertices in an explicit manner :

= Qi u(pi) y, u(pj) (spin one-half partons) (23a)

= Q i ( ~ i + PI), (spin zero partons) (23b)

= Qi(pi + pi), + q [spin dependent terms]

(in general). (23~) Substituting the above expressions into (22) and iso- lating the W,,, contributions one gets [24]

independently of spin, and similar (though spin- dependent) results for W,. The 6 function in (24) stems from the elastic character of the photon-parton scattering. It allows to give an interpretation to the scaling variable o = - 2 v/q2 = l/x : the inverse of w is the fractional momentum of the scattered parton.

Substituting the parton Wi functions into (21) gives the results :

(any parton spin) (25a)

(spin 4 partons) (25 b)

= 0 ,, R - ~ ⁼ 0

(spin 0 partons) ( 2 5 ~ )

(higher spin partons) . (25d) The above expressions are scale invariant. This is no surprise since masses have been neglected and a pointlike coupling has been used for the partons (a form factor would necessarily have involved a mass for dimensional reasons).

C3-84 A. DE RUJULA The results for the ratio of the virtual photon cross-

sections had previously been obtained by Callan and Gross [25] on current algebra grounds (R = 0 for quark currents, R - I = 0 for the algebra of fields commutation relations). It is amusing to recover for spin partons the results of the quark model algebra.

Experiment, in the deep inelastic region where W , and W2 have been separated, seems to be consistent with a constant R -- 0.18. This is compatible with spin

5 partons playing a predominant role.

It is also extremely simple to derive sum rules [26]

from the parton model. Consider, for instance, the integral

Inserting the value (25a) of F,(w) gives

= ) ^{' dx p, Q? hN(x) =}

x

O N i N i

where it has been taken into account that f r(x) is a normalized probability distribution.

Experiment does not yet provide a clear-cut answer to whether F,(o) is constant or decreases for high values of w. However, the difference FI(w) - F:(o) is, within large experimental errors [2], compatible with a decreasing function of w for high w (see Fig. 7).

FIG. 7. - Difference of the vWz/m2 structure functions for proton and neutron, extracted from hydrogen and deuterium

data (radiative corrections only preliminary).

Complemented with some Regge lore (and thus making rather obscure the meaning of the errors) the data give

A = I , - I , = 0.19 + ^0.08

where a decrease of Fz - Fi o-', a = 4 has been used in the unmeasured region ( o 5 10). The 40 %

relative error in the total result is of the same order as the relative error in the measured region contributing to A .

Consider now a particular parton model where the nucleons are made of three << valence ^B quarks with the traditional non integer charges and a cc sea )) of any neutral SU(2) symmetric stuff (quark-antiquark pairs, gluons and what not) [27], [28].

Proton = (ppn) + sea Neutron = (pnn) + sea .

Taking (27) into account the model immediately gives I = 3. If one is ready to consider this result as suggestive of a contradiction with experiment [29] all models with singled-out valence quarks are excluded.

On the other hand, these models have to be made unduly complicated on grounds of a second sum rule for F2(w) [30] in the sense of requiring a considerable quantity of neutral gluons [28] or the addition of configurations where the nucleon does not obey the parton picture [31]. A more general parton model where no valence quarks are singled out [32] might still survive and is discussed by Nachtmann elsewhere in these Proceedings.

Notwithstanding the normalization difficulties encountered when trying to be too explicit, simple valence quark parton models are excellent mnemotech- nical tools and still have a say, as the following qualita- tive considerations [29], [31] show.

In trying to understand the shape of the function F,P(o) - F:(o) let us make the extra hypothesis that In the average the total proton momentum in the infinite momentum frame is equally distributed bet- ween the partons : < x > ⁼ 1/N in an N parton configuration. Since x = 1/o within the model, at a fixed o one preferentially tests nucleon configurations with N = o partons. The distinction between proton and neutron might be difficult for electrons to tell in the maremagnum of high N states : FI - Fi should tend to zero for big w . A maximum of F i - F: will occur around w = N = 3 where the proton and neutron look most different, the configurations with just the valence quarks being enhanced. If one adds to the above speculations the fact that F2(1) = 0, because of the vanishing phase space, one is ready to draw a guess [27] of the functional shape of F; - F:. The triangles in figure 7, taken from a mise au point by Weisskopf [33], show the results. The over-all nor- malization has to be fixed with an ad hoc prescrip- tion [31]. The separate agreement of F$ and F; with experiment is also qualitatively good. The model reproduces at the point o = N = 3 the simplest quark model result (nucleon = 3 quarks, i. e., F;/F; = 3) in beautiful agreement with the data.

Conclusions. - The parton model of deep inelastic electron-nucleon scattering gives a simple under- standing as well as good mnemotechnics of the process.

Facts like scaling and pointlike cross-sections are built in. The model provides grounds for suspecting an

AN ELEMENTARY INTRODUCTION TO DEEP INELASTIC ELECTROPRODUCTION C3-85 important role of half-spin constituents of the nucleon,

with a small average square charge (fractional charges ?).

Some results for quark constituents give strikingly good results, i. e., the relation of the proton and nucleon structure functions (v W,) at o = 2 v / - q2 ⁼ 3.

Explicit models become unpleasantly complicated when they are asked t o provide precise numerical agreement with the data. This is not particularly surprising, since a detailed understanding of the nucleon structure would be a major step forward in the solution of strong interaction dynamics, a problem of whose difficulty we are by now well aware. The parton model is a set of very interesting qualitative arguments

and working rules, with n o pretention t o be a complete theory. As emphasized by Callan in these Rencontres, should the parton model results and predictions (i. e., for neutrino scattering) turn out t o be good, physicists would be faced with the most difficult problem of unveiling a complete consistent theory underlying the results.

Acknowledgements. - I am indebted t o C . H.

Llewellyn Smith for many discussions that triggered my interest o n the subject. I d o also thank J. S. Bell for his intense criticism, that helped much in improving the manuscript, and J. Prentki for reading it.

References and footnotes [1 ] BLOOM (E. D.) et al., Phys. Rev. Letters, 1969,23,935.

[2] BLOOM (E. D.) et al., SLCA-PUB-796, 1970.

[3] BERKLEMAN (K.) et a]., Cornell preprint CLNS-21, 1970.

[4] BUDAGOV (I.) et al., Phys. Letters, 1969, 30B, 364.

[5] See for instance :

GOURDIN (M.), CERN preprint TH. 1 238, 1970, and references therein.

[6] Report of LEDERMAN (L.) to the 4th International Symposium on Electron and Photon Interactions at High Energies, 1969, Daresbury Nuclear Phy- sics Laboratory, Lancashire, England, 1969.

[7] Reports of WALKER (R. L.), LUKELSMEYER (K.) and SILVERMAN (A.), same Proceedings as Ref. [6].

[a] q2 is spacelike in the process and negative in our metric : q2 = q?) - 4.2.

[9] DONCEL (M.) and DE RAFAEL (E.), I. H. E. S. Preprint, 1971.

1101 HAND (L.), Proceedings of the 3rd International Symposium on Electron and Photon Interaction at High Energies, SLAC, 1967 (Clearing House of Federal Scientific and Technical Information, Washington, D. C., 1968).

[ I l l TAYLOR (R. E.), Proceedings of Ref. [6].

El21 BJORKEN (J. D.), Phys. Rev., 1969, 179, 1547.

[13] Other scaling mechanisms have been devised, where the boundedness of the hadronic spectrum is replaced by an increase of the density of states.

See for instance :

LANDSHOFF (P. V.) and POLKINGHORNE (J. C.), Nuel.

Phys., 1970, B 19, 432.

1141 BLOOM (E. D.) and GILMAN (F. J.) haye discussed the relevance of a d~fferent scallng varlable w' = w + ^m*/q2 that coincides with win Bjorken's limit.

BLOOM (E. D.) and GILMAN (F. J.), SLAC-PUB-779, 1970.

[15] NACHTMANN (O.), Orsay Preprint LPTHE 70/49,1970.

[16] CHEN (F. C.), CERN Preprints TH. 1285 and 1298, 1971.

[I71 FEYNMAN (R. P.), Phys. Rev. Letters, 1969, 23,1415 and in Proceedings of the 3rd High Energy Colli- sion Conference at Stony Brook., Gordon and Breach, 1970.

[IS] The frame dependence of the parton picture has been thouroughly studied by DRELL, LEVY and YAN in their particular model (see Section 4). In studying the hadron structure one might think that other reference frames like the gamma-nucleon c. m. s.

are more natural, since the electron is only an

ct external device)). As discussed in Section 6, there frames are bad in the sense that Z diagrams where the photon splits into parton-antiparton pairs, that later interacts with the nucleon, need be

considered. The time-ordered picture in which the nucleon first splits into its constituents, that are then probed by the photon scattering, does not give the full answer to the problem and the parton picture fails.

[19] FEYNMAN has argued that partons with a wee fraction of longitudinal momentum might play a central role in high energy hadron-hadlon collisions. No mass scale is relevant to the hard partons and their cross-sections might be expected to vanish like l/s, on dimensional grounds. This is not the case for the wee partons, whosemomentaarenot assympto- tically big. Their cross-sections might stay cons- tant at high energies.

[20] The production of quarks is energetically allowed in quark parton models when - q2, v % mi. How- ever, one can invoke strong binding between quarks to prevent them from getting rid of their compa- nlons.

[21] Denoting

l i > = i p ~ , p z ,..., p ~ >and If'>=IpI1,~6, ..., P I N >

the parton states of figure 4a, we have, for the W,, tensor describing that particular diagram :

where the normalization condition < f 1 f > ⁼ 1 for free states and the completeness condition

Z I T > < ^T I = 1 have been used.

[22] DRELL (S. D.), LEVY (D. J.) and YAN (T. M.), Phys.

Rev. Letters, 1969,22, 744 ; Phys. Rev., 1969, 187, 2159 ; Phys. Rev., 1970, D 1, 1035 ; Phys. Rev., 1970, D 1, 1617.

YAN (T. M.) and DRELL (S. D.), Phys. Rev., 1970, D 1, 2404 ;

DRELL (S. D.) and YAN (T. M.), SLAC-PUB-808, 1970, to be published in Ann. Phys.

[23] CHANG (S. J.) and YAN (T. M.), SLAC-PUB-793,1970.

[24] In our covariant normalization convetionns

[25] CALLAN (C. G.) and GROSS (D. J.), Phys. Rev. Letters, 1969,22, 156.

[26] A thorough.review of sum rules is given in : LLEWELLYN SMITH (C. H.), CERN Preprint TH. 1188,

1970, where earlier references can be sought.

C3-86 A. DE RUJULA [27] BJORKEN (J. D.) and PASCHOS (E. A.), Phys. Rev., 1969,

185, 1975.

[28] LLEWELLYN SMITH (C. H.), NucI. Phys., 1970, B 17,277.

[29] LLEWELLYN SMITH (C. H.), SLAC-PUB-843, 1970.

[30] The other sum rule refers to the faster converging Fz(w) dw . Making the extra assumption that in an N particle configuration the partons have on average an equal share of the proton infinite momentum

gives J = p~ (z) 1 ^{Q? =} the average qua-

N i

dratic parton charge. The values of J come out too

big in a three-quark parton model and even in a model with valence quarks plus an SU(3) symme- tricsea of ijq pairs 1271. Extra neutral stuff is needed.

The situation would worsen for integrally charged partons.

[3 11 WEISSKOPF (V. F.), DESY-MIT collaboration lectures, 1970, assumes that there are states of the nucleon that do not satisfy the parton model assumptions.

In the calculations they would only affect over-all normalizations, since their interaction with elec- trons would involve the usual rapidly decreasing form factors.

1321 NACHTMANN (O.), Orsay Preprint LPTHE 71/12,1970.

[33] WEISSKOPF essentially uses the model of BJORKEN and PASCHOS with 3 quarks + qq infinite sea. The probability of an N particle configuration used by the earlier authors ( p ~

-

follow from the simplest statistical population of states.