Gluon distribution function inside the nucleon and critical tests of asymptotically free gauge theories

P H Y S I C A L R E V I E W D V O L U M E 1 7 , N U M B E R 3 1 F E B R U A R Y 1 9 7 8

Gluon distribution function inside the nucleon and critical tests of asymptotically free gauge theories?

Wu-Ki Tung Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616

(Received 28 July 1977)

In the framework of asymptotically free gauge theories, we express the q dependence of each (Nachtmann) moment of the structure function v W, in terms of independently measurable quantities plus one unknown-the gluon matrix element. Comparison of the theoretical expressions with deep-inelastic scattering data (i) provides stringent tests of the usefulness of gauge theories (quantum chromodynamlcs) in contemporary hadron physics, and (ii) offers the natural way to measure the gluon distribution inside the nucleon (within the framework of such theories). The procedure relies on an explicit study of the precise connection between basic gauge-field-theoretic formulas and the parton distribution functions, including that of the gluon. We also suggest a practical method to deduce the latter from available data.

I, INTRODUCTION

Non-Abelian gauge-field theories of the asymptotically f ree genre [quantum chromodynamics (QCD)] a r i s e from attempts to provide a theoretical basis for the observed (approximate) scaling behavior of deep-inelastic scattering, a s well a s to unify theories of weak, electromagnetic, and strong interactions. In spite of such attractive theoretical motivations, the true relevance of these theories nzust be determzned by clearly for- nzz~lated pkenomenological tests. The a r e a where the predictions of asymptotically f ree gauge theo- r i e s (AFGT) a r e the most direct and specific con- cerns the pattern of deviation from scaling in deep-inelastic scattering.' Because this is the natural region of applicability of perturbation expansion for these theories, the predictions a r e very definite and the assumptions needed for their derivation a r e both clean and simple.

In this paper we focus attention on theoretical predictions concerning moments of the best measured structure function, vW,. We write out explicitly the full q 2 dependence of these moments in the framework of AFGT.' We exhibit (for the nonexperts) how these results reduce to the familiar quark-parton language and define the te rm "gluon distribution function." We then show that by a straightforward rearrangement of te rms, one may express the q2 dependence of each moment in t e rms of only one unknown constant-the moment of the gluon distribution (plus other independently measurable quantities). This resltlt p e m i t s rather strzngent tests on the zcsefulness of fke AFGT franzework and allows one lo detern~ine the glzcon distribution function inside the ~zzrcleon.

Because the accurate and extensive SLAC-MIT data, on which these analyses must mainly rely,

ry out a complete numerical study in this paper. We try to demonstrate, however, that existing eN data when properly assembled and when supple- mented by recent (and forthcoming) very-high-energy ~ A V data, should be sufficient to begin provid- ing answers to the crucial issues raised in the previous paragraph. The main content of this paper consists of elucidation of known basic results of AFGT.' Our emphasis is on clarifying the precise connection betzoeen AFGT and parton-nzodel distribution fiozctiotzs (for readers other than the few theoretical experts) and denzonstmting bozo the "l~nknozrns" of the theory can be sgste~?zatically elirrzinated in favor of independently nzeaslwable q~rantities. Thereby we exhibit the salient features of the very specific predicted q2 dependence of the moments and se t the stage for cri t ical tests of the basic theory and for "measuring" the fundamental parameters of the theory.

The central results of this paper a r e in Sec. IV, centered on Eq. (18). The main ideas which lead to the sharpening of the theoretical predictions and the possibility of measuring the gluon distribution function a r e explained in the paragraphs between Eq. (17) and the end of that section. Sections I1 and I11 provide a brief, pedagogical, and precise r e - view of the AFGT formalism a s applied to deep- inelastic scattering. Sections V to VIII elaborate on the ideas of Sec. IV and provide preliminary and illustrative sample analyses of data. Section M explains how these contents relate to existing l i terature on scaling-violation calculations based on AFGT and outlines apraclical procedure to deduce the gluon distribution.

11. SUMMARY OF THEORETICAL FORMALISM AND STATEMENT OF ASSUMPTIONS

a r e not available in uniformly assembled and read- The deep-inelastic s tructure functions represent ily retrievable form, we have not been able to car - independent components of the forward Compton

17 - G L U O N D I S T R I B U T I O N F U N C T I O N I N S I D E T H E N U C L E O N . . . 739

amplitude

In field theories, the product of two current operators which appear on the right-hand side can be analyzed in terms o f Wilson's operator-product ex- pans'ion.' The operators which enter this expansion are organized according to their "spin" n (i.e., Lorentz irreducible tensor rank) and "twist" 7 (i.e., naive dimension minus 2). The contribution f rom spin-n operators can be projected out exactly by the method of ~achtmann.' In particular, for the best-known structure function vW2, the following moment integral i s controlled entirely by spin-n operators2:

Here q2, V , x = q ' / 2 ~ ~ v are the familiar variables for deep-inelastic scattering and

The large-q2 behavior of M(n,q2) can be analysed in the AFGT framework using the renormalization- group method.' The most important contribution to M(n,q2) comes from operators o f the lowest naive dimension ("twist" 7 = 2). The relative contributions from higher-twist operators contain powers of p</q2, where p; i s some fixed hadronic mass scale. It has been argued3 (but not proved) that p, i s of the order of 300-500 MeV. Thus, (p;/q2) may be small for , say, q2 > 3 Gev2. Al- though the unknown n dependences of these higher- twist contributions3 complicates the picture somewhat, it i s generally considered reasonable to assume that, for q2> 3 G ~ V ' and for moderate values of n , one can keep only the "leading contributions" from twist-2 operators. W e shall adopt this as our f irst working hypothesis. We then have'

The indices a , p sum over the various twist-2 operators: a = 0 refers to the gluon operator, where- as a! = i (= 1,2, . . . , m) refer to quark operators of the flavor variety i. (m = number of flavors.) y ( n , g3 i s the ( m + 1) x (m + 1) "anomalous dimension" matrix. Both the exponential factor contain- ing y ( n , a and C t ( 9 arise f rom the coefficient function of the Wilson expansion and can be calcu- lated in power series of the "effective coupling

constant" g ( q 2 ) . For large q2, the relevant expansion parameter is4

where A i s a filndarneletal constant of the theory. The last factor in Eq. (4), ~ ' ( n , go2) are the ( re - duced) matrix elements of the twist-2 operators (labeled by P and n ) in between the physical target states. They are strong-interaction matrix elements and are not calculable except for such spec- ial operator as the energy-momentum tensor. Be- ing renormalized matrix elements, these quanti- t ies have to be defined with respect to an arbitrar- ily chosen renormalization point 9:. This fact i s explicitly displayed in Eq. (4) and has a direct bearing on the parton-model interpretations to be discussed later.

Previous semiquantitative analyses of deep-inelastic scattering data5-7 reveal that A 2 0.5 GeV. Hence, 3 / 4 a 2 < 0.1 for q2, q< > 3 G e p . Conse- quently, in Eq. (4) one can replace C",(g(q2)) and y(n ,g(q '2)) by their lowest-order perturbation values. This approximation is our second working hypothesis. Its validity can be checked independently by the analysis described in Sec. V .

These two na t~~ra l approximations of AFGT (or QCD) lead to

&I(n,q2) =c;(z= ~ ) ( e ' " ' ( " ) ) ~ ~ ~ ~ ( n , q ~ ) , (6)

where C",(g= 0) = (0,Q:,Q22,. . . ,Qm2; for a =0,1 ,2 , . . . , m, respectively) ( Q i i s the charge of the ith quark); s i s the q2-dependent variable,

s = ln[ln(q2/A2)/ln(q,2/A2)] ; (7)

and h(n) i s the numerical matrix:

with 9 9 -- h g g ( n ) (- (n + l ) ( n + 2) (n - 1)n

3 m "

+ 1 + 4 2 i ) , h f f ( n ) = Gm (- 1 = 2 1

6 A,(~)=G,(- (n + l ) ( n + 2)

1 2 ' g f ( n ) = ~ m (= + -) 7

710 W U - K I T C N G - 17

and

111. RELATION TO PARTON MODEL AND CLUON DISTRIBUTION

We now display the connection of these results to the familiar parton-model language. Consider Eq. (6) at the point q = qo2. Since s ( q 2 = 9:) = 0 , the exponential (wave-function renormalization) matrix reduces, by definition, to the unit matrix for all n. Therefore,

This i s to be compared with the familiar parton- model formula

where u ' @ ) (tii) i s the probability distribution of the ithquark (antiquark) carryingxfraction of the target momentum. These two equations suggest that one can interpret Ai(n,q:) as the moments of the parton distributionfunctionx(u '+iZi). Stnce in the AFGT fmmework, A' (n, q;) depend on the renornzaliza- tion point 4 2 , the parton distribution funcfions ac- quire meaning only at the given 9 2 . However, as the value of q: i s arbitrary, this only implies that the parton distribution functions u' depend on two variables, i.e., x and 9 2 , rather than just one, i.e., x , as originally suggested by strict Bjorken scaling.' This i s how, in a precisely defined way, the AFGT formalism reduces to the conventional quark-parton model at any given q2 = q;.

A notable feature of Eq. (10) [which i s the q2=qo2 case of Eq. (6)] i s that the gluon matrix element AO(n, q:) does. not appear on the right-hand side. The reason for this i s not because Ao(n,q:) i s zero (or small), but rather because the mixing matrix e-'" being diagonal at q =qo2, couples A0 to the bare " charge" of the gluon C o ( g = 0) ullzich i s zero. In this way, we are reminded of the well- known parton-model lore that the gluon distribution is not probed in these experiments. However, it i s very important to realize that in the AFGT formalism this i s strictly a matter of definition applicable only at the selected renorrrlalization point 4 =4:.

Going back to Eq. (6), the exponential matrix does mix AO(n,q;) with ~ ' ( n , q : ) and leads to non- zero contribution of the gluon matrix element to M(n, q2) for general q2. Therefore, Ao(n, q;) can be "measured" through its influence on the q2 dependence of the experimentally measzcrable moment functions M(n,q2). By analogy with the cor-

responding quark matrix elements, we can interpret AO(n,q;) as moments of the gluon distribution function at q =qo2. The normalization of this distribution function can be determined by the energy- momentum sum rule':

2 Ai(2,q;) - 2A0(2,q,2) = 1 . (12) I

I f we denote by ~ ( x , q , ' ) the probability for a gluon to carry x fraction of the longitudinal momentum of a nucleon in the infinite-momentum frame, then Eq. (12) suggests

Note that although for clarity we used the simple x-moment definition (13), one can always substi- tute the precise Nachtmann definition, Eq. (2) , to take into account nucleon mass e f fec ts . Eq~~ation (13) (or rather its intlerse) defznes the "glzton dis- tribz~lion firnctton" ~ ( x , q , ' ) .

N. PHENOMENOLOGICAL FORMULAS

For the purpose of determining AO(n, q;), we now work out the explicit q2 dependence of M(n,q2) in terms of the dynamical "unknowns" A' and A'. To that end, we need the matrix S which diagonalizes A(n):

where

and

The matrices S and S-' are given in the Appendix. Now Eq. (6) can be written, using C,, S , and A,

already given:

17 - G L U O N D I S T R I B U T I O N F U N C T I O N I N S I D E T H E N U C L E O N . . . 74 1

Here ( Q 2 ) = (l/bz)Ci Q i 2 i s the average charge squared of the quarks.

We call attention to the following features of Eq. (17):

( i ) All the q2 dependence enters through the variable s which i s defined by Eq. ( 7 ) . The three terms with distinct q2 dependences correspond to contributions from two mixed singlet operators (with respect to quark "flavor") and the combined total nonsinglet operators (which all have the same anomalous dimension A,,);

( i i ) The coefficients to the q2-dependent factors consist of theoretically calculable constants (the A's and Q's) together with dynatnical unknowns ~ O ( n , q , ~ ) , the gluon matrix element, C Ai(n,q;) , the singlet quark matrix element, and hi Qi"Ai(n, q;), the familiar parton combination, Eq. (10);

( i i i) At q2=q;, s = 0 , all the exponential factors reduce to 1, Ao from the f irst two terms cancel out, and C i ~ ' from all three terms also add to zero. W e are then le f t with iM(n, 4;) =xi Q ' ' A ~ ( ~ , go2) , which i s , of course, just Eq. (10);

( iv) Since A- < A N s <A+, the f irst term is expected to dominate at extremely large values of s (q2) . For this reason, it i s often referred to as the "leading However, for the q2 range of current ~n teres t , the expo?zerttial factors are all slozuly varying, and the three terms are actually of similar order of magnitude-a point of consid- erable practical consequence. In addition, for n 2 4, A- hNS for all practical purposes. As a result, except for n= 2, there are only two terms with distinct q2 dependence.

Because of points (iii) and (iv) discussed above, it i s much more advantageous to regroup Eq. (17) as

where the coefficient functions, both exactly calculable, are

Equation (18) will be the foc~ns of our subsequent discussions. Since M(n,q;) can be obtained from experiment, the list of unknowns reduces, at this stage, to A0 and xi A'. W e note however, the quark matrix elements A' are probed directly at fixed q2 in the combination which enters M(n,q;),

S

FIG. 1. The relative magnitudes of the three terms of Eq. (18 ) are illustrated in this figure. Since ~ ( n , q ~ ' ) , ~ ' ( n , ~ , , ~ ) , C i ~ i ( , , q o 2 ) can, a priori, be of similar order of magnitude and they are the dynamical unknozons, we show the corresponding coefficient functions which are exactly calculable in these theories. Plotted are e ' S X ~ s 61' ; Xn ( s ) , and Y , (s ) as functions of s (q2) for n = 2, 4 , and 6 . For each n notice the order-of-magnitude difference between these three coefficient functions within the plotted range. [Over truly asymptotic s (which will be well beyond any conceivable experiment), X , (s) and Yn (s) will approach e-S Ns for most n and over- shoot e ' S X ~ ~ for n=2.]

Eq. ( l o ) , and in other combinations from neutrino and antineutrino scattering. Thus, information on A' (n, q;) can be obtained independently by conventional parton-model analyses applied to data at a fixed q2 range, say qZ=q;. With that information, the q2 variation of ~ ( n , q ' ) as exhibited in Eq. (18) i s controlled by only one unknown number-AO(n, q:), the gluon matrix element. Therefore, we have very specific predictions for the q2 variation of M(n,q2) which provide clear-cut tests of the theoretical framework and o f f e r an unambiguous way to measure the gluon distribution function.

To gain some idea as to the practicality of carrying out this program, we plot in Fig. 1 the functions e-shNS, X,(s), and Yn(s) for n=2 ,4 ,6 and rn (number of quark flavor) = 4. The results displayed there show, for each n , a clear separation in the orders of magnitude o f the three terms on the right-hand side of Eq. (18). The first term, M(n,

74 2 W U - K I T U N G - 17

q t ) e - S A ~ S , i s the dominant te rm for most practical values of s (q2) . The gluon matrix element contribution i s of the order 1% in the range of s shown. The quark matrix element contribution i s much smal ler still, being of order 0.1-1% in the same range.

The root of this c lear separatimz in the nzagni- tudes of the variozcs te rms lies in asymptotic f ree- don2 itself. Because the q2 dependence in these theories i s very mild, all the e-SX factors a r e reasonably close to l over a moderate range in q2. A glance a t Eq. (19) clearly shows then that both X,(s) and Y,,(s) will be small. Since, in addition, in these theories A - = Xff = hNS (n 2 4) to a high de- gree of accuracy, one can also see why Y,(s) i s "second order" in smallness. The last argument does not really apply to n = 2. However, Fig. 1 shows the numbers happen to work out reasonably well even for that case.

This c lear separation of orders of magnitude means that one can practically ignore the third te rm in Eq. (18) in phenomenological analyses. The general trend of the q2 dependence of M(n,q2) i s determinedg by ~ ( n , q , ~ ) e - ' ' ~ s ; systematic deviations frorn the lat ter a r e controlled by A0(n,q:) rrndprovides the means to determine the gluon dis- tvibution function. We should note, however, that this interesting feature holds only for a certain limited range of q2. At truly asymptotic q2, Eq. (17) should provide the more natural description with the e-SX- t e rm being dominant over the others. Present evidence, however, i s that we a r e f a r f rom the true asymptotic region-since the

nonsinglet t e rms in Eq. (17) for nucleon targets (as well a s the vW3 t e rm in neutrino scattering) a r e known to be substantial. Nonetheless, the two basic assumptions of Sec. 11, which a r e the only ones needed to derive Eq. (18), may well hold. (The smallness of 5 can be checked self-consis- tently, c.f. Sec. V; the possible effects of higher- twist operators has been discussed in Ref. 3.) It i s therefore quite remarkable that we a r e able to a r - r ive a t such specific predictions on the q2 variation of hl(n,q2) a s well a s unambiguous means to measure A0(n,qo2) a s described above even the asy??lptotic region i s obviously not yet in sight.

V. DETERMINATION O F THE RUNNING COUPLING

CONSTANT

In order to compare Eq. (18) with experiment, we need to know ~ ) l ( n , q ~ ) a s a function of q2 not of s. The transformation from s to q2 depends on the basic parameter A . Within the scheme outlined in this paper, A can be determined by analyzing the n = 2 moment. This i s because for n = 2 the gluon matrix element A0(2,q,2) i s related to the singlet quark matrix element C,A' through the energy - momentum sum rule, Eq. (12),

Conventional parton-model a n a l y s e ~ ~ " ~ suggest C i A i ( 2 , q O 2 ) - 5Wo a t go2- 4 G~v ' . Hence, -2A0(2, q t = 4 Gev2) = 0.5. We can write down the explicit formula for the n = 2 case of Eq. (18):

where again, s = ln[ln(q2/A2)/ln(q~/A2)], rn i s the number of quark flavors, and (Q2) the average of squares of the charges of the quarks. For m= 4, (Q2) =%. For go2= 4 Gev2, we have

Eq. (22) provides a one-parameter (i.e., A) expression fo r the q2 evolution o f ~ ( 2 , q ' ) . [,Vf(2,q2) i s essentially the a r e a under the vW, v s x curve, although the precise Nachtmann definition, Eq. (2), involves some slight modification.] As explained previously, the right-hand side i s not sensitive to reasonable variations in the input C i ~ ' . In Fig. 2 we plot the theoretically predicted M(2,q2) a s a

function of q2 for three values of A. We chose q,2 = 3 . 5 G ~ V ' and used M(n,q;) from smooth f i ts to vEr2 measured by the ep and ~p experiment^.""^ If the theory i s viable, the choice of q ; value should be immaterial, for A i s the only substantial parameter .

By comparing these predicted q2 dependences of M(2,q2) [or ones that one can obtain with improved M(2,qO2), A0(2,q,2), and Ci.A'(2,qO2), in the future] with measured values of A1(2,q2) a t various q2 one can expect to answer two important questions:

(i) Can the q2 variation prescribed by AFGT provide a viable description of the measured 1M(2,q2)?

(ii) If so, what i s the value of R which determines the fundamental (running) coupling constant

G L U O N D I S T R I B U T I O N F U N C T I O N I N S I D E T H E N U C L E O N . . .

t VI. MEASURING THE GLUON DISTRIBUTION

0.14 4 8 12 16 20 24 28 32

q 2 ( G ~ v 3 FIG. 2 . The predicted q 2 dependence for the "area"

under the v W 2 vs x curve according to the Nachtmann definition Eq. (2 ) for n = 2 . Three assumed values of A, the parameter which determines the size of the running coupling constant, cf. Eq. (5), a r e shown.

Wlth the value of A obtalned from either of the analyses discussed above, Eq. (18) specifies the q2 variation of M(n, q2) for n i 2 in terms of AO(n, q:) alone. It can be used to test the theory and to measure the dynamical "unknown" AO(n, 4:). To illustrate this point we plot in Fig. 3 M(4,q2) vs q 2 for two choices of ~ ~ ( 4 , q , ~ ) with each of two possible A-the solid lines correspond to A0(4,q,2) = 0, the dashed lines correspond to A0(4, 9:) = -+CiA'(4,qo2) with the lat ter taken from conventional parton models.1° These choices of A0 a r e used merely for illustrative purposes. The A0(4, 9:) = 0 case corresponds to a gluon distribution G(x, q;) sharply peaked a t the small-x end (with respect to the combined quark distribution function) a s commonly assumed. On the other hand, -2A0(n,q:) =C1Ai(n,q,2) (for most n) would im- ply ~ ( x , q ; ) and Ciui(x,q;) a r e of similar shape. It is, of course, of tremendous interest to know the true gluon distribution. In practice, therefore, one should try to fit Eq. (18) with A0 as the free parameter.

In Fig. 4 we make similar plots for the sixth mo-

[cf. Eq. (5)] of these gauge theories? Estimates on the value of A already exist in the

Th ey a r e based on analyses of the q 2 variation of vW2 itself using additional a s - sumptions. The value of A one obtains depends on those additional input which among other things in- clude an assumed gluon distribution. Recently, an improved method using only experimentally measurable quantities a s input has been proposed.14 The value of A obtained by applying this method to the analysis of data should be relatively unambiguous. This work is currently underway.15 More details of these alternative approaches a r e given in Sec. M.

The two methods of determining A a r e related, but not completely equivalent. The n = 2 moment analysis makes explicit use of the energy sum rule. The structure-function analysis does not use the latter, but implicitly assumes that, a t least for certain x range, taking the inverse Mellin transform of Eq. (4) gives reasonable results. It would be interesting to see if the two methods yield the same result,

Existing data in the published literature a r e not quite adequate for carrying out reliably the n = 2 moment analysis a s outlined here. We shall dis- cuss some preliminary comparisons in Sec. VLII. [ ~ o t e added in proof. For a recent attempt, in- corporating some g2 correction effects, see H. D. Politzer, Nucl. Phys. B122, 237 (1977).] -

1 1

4 8 12 16 20 24

q2 ( G ~ v ?

FIG. 3. The predicted q Z dependence of ~ ( 4 , ~ ~ ) from the n = 4 case of Eq. (18) for two values of A and two illustrative A ' ( 4 , q : ) . The solid lines correspond to A 4 , q O 2 ) = 0 , the dashed lines to - 2 ~ ~ ( 4 , q ~ ~ ) =%(A ( ( n , q A . With A determined f rom independent analyses (see Sec. V), comparison of measured ~ ( 4 , ~ ~ ) with similar curves based on Eq. (18) will allow a stringent tes t of AFGT and a measurement of ~ O ( 4 , q ~ ' ) , the gluon matrix element.

W U - K I T U N G 17 -

FIG. 4. Same as Fig. 3 except for the case n = 6.

ment M(6,q2). The qualitative features o f the curves shown here are very similar to the corresponding ones for the n = 4 case.

I f one succeeds in measuring AO(n,q:) for several n (say n = 2,4,6,. . . ) by the procedure described here, it i s a simple matter to construct the general shape of the gluon distribution function G(x ,qOZ). For instance, one can assume certain smooth functional forms for ~ ( x , q : ) , with free parameters to be determined by the known moments. Of course, the more moments o f ~ ( x , q : ) one "measures," the finer the details of this function one can determine.

VII. THEORETICAL ISSUES

We have stated explicitly the predictions of AFGT on the q 2 variation of the moment integrals of the vW2 structure function. These are given in terms o f the fundamental running coupling constant of the theory plus one unknown gluon matrix element for each moment. The derivation of these results, i s direct. Only two basic approximations are invoked. ( c f . Sec. 11). Both are at the very heart of AFGT, In fact, these simplifications are precisely the features which motivated AFGT and made these theories theoretically appealing. In this sense, the predictions are f i rm and of fer critical tests of the theory.

I f the theoretical predictions can give a reasonable description of data, we will have a viable theoretical framework for many lepton-hadron

processes. As formulated in detail in the previous sections, we will have a precisely defined set of parton distributions-for the gluon us well as for quarks. These will be functions of x and q; (the latter quality i s the arbitrary starting point for the q2-variation analysis and where the "distribution functions" are defined in AFGT). These functions, together with the running coupling constant are determined by deep-inelastic eN, FAT, and vN scattering data. They can then be used as input in (renormalization-group-improved) parton-model calculations for other lepton-hadron processes as we11.16

On the other hand, since the theoretical predictions are very specific, it i s possible that the experimentally measured q 2 variation of the moments does not entirely agree with theoretical (one-parameter) predictions. In that case, we would have f i rm evidence that either AFGT i s not the underlying theory for strong-interaction physics or the present energy range is not high enough for its basic approximations to be applicable. In either case, one would have lost tlze "explanation" of the observed afiproxinlate scattering belzat~ior of the structure functions which nzolit~uted and supported the growth of AFGT. One may try to rem- edy the situation by attempting to calculate terms which are o f higher order in the running coupling constant. (For the Wilson-expansion coefficient functions, this i s possible in principle, but very complicated in practice.) On the other hand, the contributions due to higher-twist operators are not calculable with our present state o f k n ~ w l e d g e . ~ Hence, for all practical purposes, AFGT would not have much predictive power i f these e f fec ts have to be included. Without its foundation in deep-inelastic scattering, the other less direct ap- plications of AFGT to hadron physics would be on shaky ground, too. Therefore, i f one contends that the present energy range i s not high enough for the application of AFGT, one may have to say: for all its theoretical uttractiz!eness, its time lzas not yet conle.

VIII. PRELIMINARY ANALYSIS OF DATA AND AN APPEAL

Attempts2 have been made before to analyze the q 2 dependence of the moments of vW2. The results were inconclusive. Several developments dictate, however, that this should not be the case now. ( i ) First, the series of SLAC-MIT experiments" on ep and ed scattering have further extended the ki- nematic region of coverage and increased their statistics and accuracy. The combined data from all these experiments encompass a substantial


range of q2 for almost all x > 0.2. (ii) Second, the structure function for x < 0.2 which was virtually unknown before (except for very smal l q2) has since been measured in the pN experiments a t ~ e r m i l a b . ' ~ The new results enhance considerably one's confidence in the evaluation of moments since, unlike before, virtually no extrapolation (of any consequence) i s needed in the evaluation (with perhaps the exception of n = 2). The reason for this i s that the effective factor xn-2 in the inte- grand for the nth moment renders the integral completely insensitive to the values of vW2 a t very small x (say x <0.02) where it i s not yet measured for q2 > 3 G~v'. This same fact implies that the accuracy of the (n > 2) moment integrals will not be much affected by the relatively large e r r o r s a s - sociated with the small-x range (say, 0.02 < x < 0.2) that these latest very high-energy experiments do cover. Needless to say, the situation with respect to this point will improve considerably when the second generation of high-energy pN experiments come along; (iii) The las t incen- tive for attempting a new moment analysis i s the realization, a s emphasized in this paper, that the theoretical predictions from AFGT a r e extremely specific-much more so than what one would expect. One can gain a crucial f i r s t quantitative test and study of this general framework (AFGT o r QCD, a s one prefers) by comparing theory with experiment.

In spite of these strong motivations, we have un- fortunately not been able to ca r ry out a full analysis in this paper for two reasons: (i) First , the evaluation of the n = 2 moment i s s t i l l strongly affected by the uncertainties on the measured vW2 function a t small x . TO do a definitive comparison, one will probably have to wait for the second generation of pN experiments with improved muon beam a t CERN SPS. (Before that i s done, information of A can, of course, be obtained by the in- direct methods of Refs. 5, 6, 14, 15.) A preliminary study of the n = 2, moment with existing data i s presented below; (ii) Although the evaluation of the higher moments (n + 4) does not suffer from this drawback since the results depend mainly on the ra ther accurate and extensive SLAC-MIT data," we encounter a different (and seemingly un- necessary) difficulty. This s tems from the fact that these experiments, covering various ranges in x and q2, were car r ied out over an extended period of time by a t leas t two basic groups with substantial turnover in active participants." As a result, only selected data a r e tabulated and published by each group, and these a r e usually presented in different variables and formats. A systematic and reliable evaluatzon of the moments does not seem possible without a uniform and com-

plete compilation of all available data (which does not seem to exist at present).

Cl'e strongly urge than an effort be made toward the goal of giving a studied, conzplete map of vW2 as measured at SLAC so that detailed information on this important structure function will be auail- able not only for this analysis, but also for conr- parison with any other useful ideas or models that can shed light on the structure of strong-interaction physics.

Without the benefit of such information a t present, we present below a simplified analysis to illustrate the points we a r e trying to advocate and to show what can be expected in a more complete analysis.

we concentrate on the proton v W2 function. We evaluate the n = 2 , 4 , 6 moments of vW2 starting from the very recent parameterizations of this function obtained by the Chicago-Harvard-Illinois- Oxford (CHIO) group,'' which f i t their new data a t small x together with some of the latest SLAC- MIT data in the medium-x range." These para- metrizations a r e presented in four q 2 ranges1': 2-4, 4-8, 8-15, and 15-30 all in (GeV/c)'. We compare the curves obtained from these paramet- rizations with independent SLAC-MIT data a t la rger values of x." For the f i r s t two q 2 bins, we found some discrepancy between the two. We cor- rec t this by defining modified curves which go through the data points in the relevant q 2 and x ranges. We then evaluate the n = 2 , 4 , 6 moments of these modified curves using the Nachtmann definition, Eq. (2). This crude procedure does not allow an estimate of the relevant e r r o r s (which a r e , of course, quite crucial in a definitive analy- s is) . As pointed out previously, the e r r o r s for the n = 2 moment a r e expected to be sizable, but those f o r n = 4 , 6 should not be large. Fo r the lat ter two cases, the principal limitation in the present evaluation comes f rom the crude q 2 bin- ning which does not do justice to the detailed and accurate data f rom SLAC on which these integrals main1 y depend.

The results of these evaluations a r e presented in Fig. 5, where each horizontal line spanning the width of the relevant q 2 bin represents the value of a particular moment. No vertical e r r o r s a r e indicated for reasons explained above. All three moments a r e seen to be decreasing functions of q2-a feature demanded by th is type of theoretical framework. It i s obviously dangerous to try to draw quantitative conclusions without a comprehensive study of data including a proper e r r o r analysis. With this caution one may notice that the trends of the q dependence for the n = 4 , 6 moments seem to be more l inear than (inversely) logarithmic in q a s demanded by theory. It i s

W U - K I T U N G

Fig. 5. (a) Preliminary evaluation of M(2,q2) from readily available data. These results should be compared with the theoretical curves of Fig. 2. A definitive conclusion can not be reached until errors of the measurements at small x region a r e reduced, the data at larger x are more uniformly compiled, and a meaningful error analysis incorporated. (b) Preliminary evaluation of Ji'(4, q2) from readily available data. A detailed calculation of all available data with proper error analysis should already be feasible (cf. Sec. VEI) if a proper compilation of data existed. Such an evaluation should be compared with the theoretical predictions of Fig. 3 . (c) Same as (b) except for the case n = 6.

17 - G L U O N D I S T R I B U T I O N F U N C T I O N I N S I D E T H E N U C L E O N . . . 74 7

obviously of vital in te res t to have th i s point c lea red up by a m o r e careful study. We believe the optimal use of all SLAC data can provide the needed information to carry out this task. [ ~ o t e added in proof. A m o r e detailed evaluation of the moments using available ep, ed, p p , and pd da ta has been done very recently by H. Anderson (unpublished). (However, systematic e r r o s in the da ta a r e not yet taken into account.) We thank P r o - f e s s o r Anderson for communicating to u s his results.]

Measuring the gluon mat r ix e lements and the gluon distribution, on the other hand, will be a much h a r d e r task a s can be seen f r o m Figs. 3 and 4. It can be c a r r i e d out only if very accura te information on the moments and a good determination of A f i r s t exist. Th is naturally l eads to a discussion of alternative but more practical methods of measuring A and G(x , qO2).

IX. RELATION TO OTHER ANALYSES O F DEVIATIONS FROM SCALING

AND CONCLUSIONS

It is well known that the predictions of AFGT a r e mos t d i rec t on the moments of the deep-in- e last ic-scat ter ing s t ruc ture functions.' That fact we exploited to the ex t reme in the foregoing sections. F o r pract ical reasons , however, com- par i sons with experiments a r e e a s i e r to c a r r y out i f one can make predictions on the q 2 dependences of the s t ruc ture functions themselves. Parisi17 showed how t h i s can be done (given the s t ruc ture function a t one q 2 = q t ) if the Wilson expansion for each moment w a s dominated by only one (twist-%) operator . His method trivially g e n e r a l i z e ~ ~ . ~ . ~ s ~ ~ * ~ ~ to the c a s e of severa l opera - t o r s provided initial dis t r ibut ions of quarks and gluons (instead of s t ruc ture functions) a r e given a t q 2 =q02. Almost a l l ana lyses of q 2 dependences of s t ruc ture functions in the AFGT framework (and, in fact, other renormalizable theor ies , a s well) make use of this method. The p r i c e s to pay f o r this shortcut a r e two well-known caveats: (i) the inversion of, say, our Eq. (17), into a n equation f o r vW2 i s valid only if the truncation in the Wilson expansion [leading to Eqs. (4), (6), (17) ] i s uniform in n-an assumption which i s mos t probably false in its strict form, and; ( i i ) although one has s o m e good educated g u e s s e s on the initial quark distributions,1° the input initial gluon distribution h a s been a mat te r of pure conjecture [aside f r o m the value of i t s second moment a s determined f rom Eq. (21)].

There have been many discussions in the l i t e ra - tu re concerning point (i) abovea3 Although there does not exis t a definitive resolution of that difficulty, i t is generally believed that, away f rom

the diffractive (very smal l x) and the resonance- production (x - 1) regions, the theoretical p r e - dictions on the q dependences of the s t ruc ture functions may be b e l i e ~ a b l e . ~

In a m o r e positive development, it h a s been shown recently14 that caveat number (ii) alluded to above i s only a n ar t i f ice of the generally adopted calculational method. In fact , a natural generalization of P a r i s i ' s method leads to a n equation which needs, a s input, only direct ly measurable quantities.14 In o ther words, the need of an assumed gluon distribution function i s superflzlous. T h i s new approach i s being applied to a complete re-analysis of a l l the ep , ed, pp, and pd data.14-l5

The reason for mentioning these other developments h e r e is to point out the complementary nature of th i s a l ternat ive approach to the moment analysis descr ibed in detail in th i s paper. With the somewhat dubious assumption of inverse Mellin t rans form (but no need of a conjecture on the gluon distribution), we can use the convention- a l analyses to calculate the genera l t r e n d s of q 2 variation of the s t ruc ture functions, compare with data in the i r a l ready available fo rm, and deduce a favored value of the running coupling constant g (o r equivalently A). On the o ther hand, in o r d e r t o t e s t the basic viability o r a t l e a s t the usefulness of th i s generally accepted theoretical f ramework (AFGT o r QCD, whichever you may pre fe r to call i t) , one ought to c a r r y out the moment ana lyses in a careful way. Cont ra ry to the belief of many people, the d i scuss ions in th i s paper indicate that th i s l a s t task is within o u r reach-if we earnest ly a s s e m b l e a l l the available data in the optimal way (and perhaps t r y to fill in the few missing gaps in the needed information by the planned experiments in the near future). Considering the s takes a t hand, the effor ts will be well worth it .

Finally, we would like to comment on the pro- spect of measuring the gluon distribution function. One of our main themes in th i s paper i s that AO(n, qO2) and hence G(X,~ , ' ) ( for any q;), although not direct ly coupled to weak o r electromagnetic c u r r e n t s , can be naturally in fe r red f r o m the mea- s u r e d q 2 dependences of moment functions h l (n ,q2) . The numerical s tudies of Sec. VIII indicate , however, th i s pr is t ine approach would r e - q u i r e very accura te data, which perhaps a r e not ye t real izable in the very n e a r fu r tu re . We would l ike to point out, however, that the use of a hybrid method of the two approaches described in this section will yield valuable first information on G(x, q2). T h i s would involve: (i) fitting all available data on the s t r u c t u r e functions using the inverse Mellin t r a n s f o r m approach ulithout an assumed gluon distribution14.15 in o r d e r to get the

748 W U - K I T U N G

best parameters A etc.; (ii) calculating the q 2 dependences of the various momentsM(n,q2) by Eq. (2); and (iii) determining AO(n, q 2 ) by the method of Sec. VI; and (iv) constructing G ( x , q 2 ) from the inverse of Eq. (13). This program i s being carried out.20

ACKNOWLEDGMENT

The author i s grateful for very useful discussions with H. Anderson, P. Johnson, H. D. Politzer, and S. Sarkar.

APPENDIX

The matrix S which diagonalizes the anomalous- dimension matrix X was used to transform Eqs. (8), (6) to Eqs. (14), (17), respectively. Although it i s straightforward to work out i t s form from the basic papers,' we give the explicit expression here for the nonexperts. Let

and correspondingly for 8 = s-'. Then, -

Soo=-XNS+h_ , Soo=(X--A+)- ' , -

S ~ ~ = S ~ , = X ~ , , s ~ ~ = s ~ ~ = ( x - - hbg)/mhgf(h-- A + ) , -

s ' ,=~x, , , S O l = ( ~ - - ~ , , ) / m ~ f f ( ~ - - h + ) ,

s ' , = s ' , = X + - A,,, ~ ' ~ = ~ ~ ~ = l / m ( ~ + - h - ) , -

s k 0 = 0 , sO,=o.

For our purposes, the remaining unspecified mat- r ix elements a re subjected to the following con- straints but a r e otherwise arbitrary:

We must, of course, always have

*Work supported by the National Science Foundation. 'H. Georgi and H. D. Politzer, Phys. Rev. Dz, 416

(1974); D. Gross andF. Wilczek, ibid. 9, 980 (1974). 20. Nachtmann, Nucl. Phys. @, 237 (1973); E, 455

(1974); V. Baluni and E.'Eichten, Phys. Rev. Lett. 37, 1181 (1976).

3 ~ . Georgi and H. D. Politzer, Phys. Rev. D s , 1829 (1976); A. De Rhjula, H. Georgi, and H. D. Politzer, ibid. 15, 2495 (1977); R. K. Ellis et a l . , Phys. Lett. 64B, 97 (1976); G. Barbieri et a1 ., Nucl. Phys. =, - 50 (1976), D. Gross et a1 ., Phys. Rev. D 15, 2486 (1977).

%he same quantity (with an extra factor of s ) i s often written a s

where jt2 i s an arbitrary renormalization point. The relation between the two parametrization i s ci(p2) = [12n/(33 - 2m)lPn(p2/~2)]" k

5 ~ . De Rfijula at a1 ., Phys. Lett. %, 428 (1976); Ann. Phys. (N.Y.) 103, 315 (1977).

'M. Gliick and E. Reya, Phys. Rev. D 14, 3034 (1976); 16, 3242 (1977).

7 ~ > . Johnson and Wu-Ki Tung, Nucl. Phys. z, 270 (1977). The value fo r A obtained in this work may be too low, for reasons t o be explained in Ref. 13.

8 ~ . Feynman, Photon-Hadron Interactions (Benjamin, New York, 1972).

$ ~ f we kept only the first t e r m on the right-hand side of Eq. (18), we would have the equivalent of the "no- gluon" approximation of De Rhjula et a1 ., Ref. 5. The above discussion shows that this i s a reasonably good

approximation. It is interesting t o note, however, that the stated reason for the i r approximation in Ref. 5-that A0(n,q:) a r e assumed t o be very small for n 3 4 because ~ & , q $ ) i s believed t o be sharply peaked near x -0-is perhaps beside the point. The detailed studies shown here indicate that the first t e r m of Eq. (18) i s the dominant one for any limited, but large (by experimental standards), range of q 2 by virtue of asymptotic freedom. For reasons already mentioned in the text, X,, (s) and Y, ( s ) a r e small , independent of the shapes of ~ ( x , q ' ~ ) a n d ~ ~ ( x , q $ ) .

"s. Pakvasa, D. Parashar , and S. F. Tuan, Phys. Rev. D g , 2124 (1974); V. Barger and R. Phillips, Nucl. Phys. E, 269 (1974); R. R e l d and R. Feyn- man, Wys . Rev. D 15, 2590 (1977).

"G. Miller et a l . , Phys. Rev. D 5 , 528 (1972); A . Bodek et a1 ., Phys. Rev. Lett. 30, 1087 (1973); J. S. Poucher et a1 ., ibid. 32, 118 (1974); S. Stein et a1 ., Phys. Rev. D 12, 1884 (1975); E. M. Riordan et a1 ., SLAC Report No. SLAC-Pub-1634, 1975 (unpublished), and (unpublished).

l2y. Watanabe et a1 ., Phys. Rev. Lett. 2, 898 (1975); C. Chang et a l . , ibid. 35, 901 (1975); H. Anderson et a1 ., ibid. 2, 1422 (1976); H. Anderson et a1 ., ibid. 38, 1450 (1977).

131'he value of A obtained in Ref. 7 appears t o be too small . That analysis was based on the inverse-Mel- lin-transformed version of Eq. (17) for deuteron tar - gets, neglecting the nonsinglet (last) t e rm. The las t approximation should be a good one a s (for the four- flavored version of AFGT, at least) , the nonsinglet t e rm (being pure charm minus strangeness) i s expected to be small . That fact notwithstanding, the analysis of


the present paper shows, however, the two singlet sults .) t e r m s (which Ref. 7 kept) should conspire t o make the 15p. Johnson, S. Sarkar, and WU-Ki Tung (unpublished). order-of-magnitude estimates of Eq. (18) true-a fact 1 6 ~ o r instance, e* e' collisions and lepton production in which was not recognized in that previous analysis. high-energy hadronic collisions at high t ransverse mo- This resulted in an overestimate of the q 2 growth of the menta. structure function vW2 at small x values, which forced "G. Par is i , Phys. Lett. e, 207 (1973); E, 367 a choice of small A. (1974).

14p. Johnson and Wu-Ki Tung, Phys. Rev. D E , 2769 18G. Pa r i s i and R. Petronzio, Phys. Lett. s, 331 (1977). (The unpublished report version of this paper (1976). contained an inadvertent e r r o r in the computer pro- "A. Buras, Nucl. Phys. =, 125 (1977). gram which resulted in mistakes in the numerical re- "P. Johnson and Wu-Ki Tung (unpublished).

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