PUBLISHED VERSION

A. W. Schreiber, A. W. Thomas, and J. T. Londergan QCD evolution of the spin structure functions of the neutron and proton Physical Review D, 1990; 42(7):2226-2236

© 1990 The American Physical Society

Originally published by American Physical Society at:-

http://dx.doi.org/10.1103/PhysRevD.42.2226

http://hdl.handle.net/2440/98468

PERMISSIONS

http://publish.aps.org/authors/transfer-of-copyright-agreement

permission 4.11.2015

“The author(s), and in the case of a Work Made For Hire, as defined in the U.S. Copyright Act, 17 U.S.C. §101, the employer named [below], shall have the following rights (the “Author Rights”):

3. The right to use all or part of the Article, including the APS-prepared version without revision or modification, on the author(s)’ web home page or employer’s website and to make copies of all or part of the Article, including the APS-prepared version without revision or modification, for the author(s)’ and/or the employer’s use for educational or research purposes.”

28 April 2016

PHYSICAL REVIEW 0 VOLUME 42, NUMBER 7 1 OCTOBER 1990

QCD evolution of the spin structure functions of the neutron and proton

A. W. Schreiber* and A. W. ThomasDepartment of Physics and Mathematical Physics, University ofAdelaide, Adelaide 500I, South Australia

J. T. LonderganDepartment ofPhysics and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47408

(Received 9 April 1990)

Models of the long-distance behavior of QCD can be used to make predictions for the twist-two

piece of the current correlation function appearing in deep-inelastic scattering. The connection be-tween the model predictions, valid at a four-momentum transfer squared Q' of the order of 1

(GeV/c)', and the data, collected at much higher Q, is provided by perturbative QCD. We presenthere the details of a method, previously used for spin-averaged lepton scattering, applied to thespin-dependent case. We then use the method to make predictions for the spin structure functionsgtroron and gnenrron

I. INTRODUCTION

For some years now QCD has been the leading candi-date for a theory of the strong interaction. PerturbativeQCD (PQCD) has provided us with a unifying frameworkfor studying all high-energy scattering phenomena in-volving hadrons and, because of its property of asymptot-ic freedom, has given us an a posteriori justification of thepopular parton model. '

At finite Q moreover PQCD predicts logarithmic de-viations from exact scaling. These are usually derived us-

ing the renormalization group which makes predictionsfor the nth moment of the structure functions, orequivalently (at least at leading order) the quark andgluon distributions, of the form

tI ( n, Q') = A „C„(Q /p', g, ) +higher twists,

where

C„(Q'/p', g, ) =C„(l, g )exp —I dg y„(g)/)33(g) . (2)

Here 3„ is a matrix element which is dependent on thequark distribution under consideration and the C„'s arecoefficient functions which give rise to the Q dependenceof the moments tI (n, Q ). g is the running coupling con-stant while g, is the coupling constant at the referencescale p. y„and /3 are the usual anomalous dimensionsand the Callan-Symanzik functions, respectively. '

In perturbation theory one typically expands C„, y„,and P up to some power of the coupling constant andevaluates them to this order. This was done for the unpo-larized case by Gross and Wilczek, Politzer, and othersup to first order ' and later up to second order by Flora-tos, Ross, and Sachrajda, and others.

For polarized scattering the calculations to leading or-der have been done by Ahmed and Ross and by Ito andSasaki. At next-to-leading order the coefticients C„aswell as y, have been calculated by Kodaira et al. Forgeneral n, the anomalous dimensions have not been cal-

culated beyond leading order.The works mentioned above give direct predictions

only for the moments of the distribution functions. Thisis somewhat unfortunate as the experimentally measuredquantities are the actual distributions themselves whichtypically vary in accuracy as a function of x and are onlymeasured down to some finite small x, with a subsequentextrapolation to x =0. ' '" Hence the extraction of themoments is subject to some uncertainty and there may beconsiderable correlation between the errors on differentmoments. Moreover, in the case of polarized scattering agreat deal of attention has recently been given to theQCD evolution of the first moment of the spin structurefunction of the proton, ' while relatively little attentionhas been paid to the consistent evolution of the other mo-ments. " As we have previously pointed out, ' considera-tion of the latter in fact makes it unlikely that the smallvalue of jg~&(x, Q = 10 GeV )dx measured by the Euro-pean Muon Collaboration (EMC) can be reconciled withthe picture that at low energies the proton consists ex-clusively of three nonstrange valence quarks. '

In the case of unpolarized scattering, the evolution ofthe relevant distributions has been calculated in a varietyof ways. Originally, at first order, Gross and others'reconstructed the nonsinglet quark distribution usingasymptotic expansions of the anomalous dimensions andthe inverse Mellin transform technique. Other methodshave relied on parametrizations of the structure functionswhich have then been evolved by numerically solving'the integrodifferential Altarelli-Parisi' equations, by us-ing the method of Laguerre polynomials' or by usingparticular parametrizations of the quark and glue distri-butions. In this paper we shall use the method ofGonzales-Arroyo et al. , the details of which can befound in a series of papers, starting with the second-orderevolution of the unpolarized nonsinglet and leading tothe more complicated case of the singlet.

In Sec. II we present their method applied to the polar-ized case. Because the anomalous dimensions for the po-larized singlet have not been calculated at second order

42 2226 1990 The American Physical Society

42 QCD EVOLUTION OF THE SPIN STRUCTURE FUNCTIONS OF. . . 2227

II. THE RECONSTRUCTION VIA THE BERNSTEINPOLYNOMIALS

We describe here the method of Refs. 6 and 7, appliedto the spin-dependent quark and glue distributions. Toillustrate the method we take the case of the spin-dependent nonsinglet (which, because of parity invari-ance, is identical to that of the spin-independent nonsing-let) and then give the straightforward generalization tothe singlet. The main result of this paper is the calcula-tion of sets of coefficients, given in Eq. (23) and in the Ap-pendix. The reader familiar with the method may wantto proceed directly to these results.

We suppose that we have a nonsinglet quark distribu-tion

~qNs(x Q )=qNS(x Q ) qNs(x Q (3)

we restrict ourselves to leading order only. As is the casein Refs. 6 and 7, we expect the generalization to secondorder to be straightforward. Higher-order e6'ects may beimportant, because of both the considerable distance be-tween the model scale and the experimental momentumtransfers, and because the model scale is rather close tothe confinement scale AQCD For these reasons our re-sults should be viewed as indicators of general trendswhich may be subject to some higher-order modifications.

Because of its speed the method described in Sec. II isparticularly useful for comparing model predictions ofstructure functions, which apply typically at some lowscale Q =p, to data at higher Q (in our case Q =10GeV ). This is because the model-dependent scale p, is apriori unknown and has to be obtained by fitting somecharacteristic prediction of the model to data. In Ref. 21,for example, the scale of a model where the quarks carryall the momentum of the proton is extracted from theknown momentum fraction that they carry at Q =10GeV . In Sec. III we shall present the results for a partic-ular model where we use the shape of the spin-independent valence distribution to determine p, enablingus to make predictions for g, (x, Q ) of the proton andneutron.

whose nth moment

n —1b,q Ns( n, Q )—: x" 'b, qNs(x, g )dx0

(4)

obeys the evolution equation (1). To first order this yields(n)/(2P )

qq 0

bqNs(n, 12 ) . (5)bqNs(n, g )= a(O )

a()M')

Here qN!s('(x, Q ) is the distribution of quarks with heli-city equal (opposite) to that of the parent hadron, a(Q )

is the strong coupling constant at the scale Q, Azq(n) isthe (lowest-order) anomalous dimension arising from thepolarized quark-quark splitting function and Po is thelowest-order coeScient in the expansion of the Callan-Symanzik function and is given by, for SU(3) color,

33 2f—(6)

where f is the number of flavors. In order to reconstructthe distribution hqNs(x, Q ) from b, qNs(n, Q ) we write

1

bqNs( Q ) b'(y )l))qNs(y Q )dy (7)0

where b, (y —x) must have the property1f (x )

= f b, (y x)f (y)dy—.0

A sufficient, but not necessary, choice for b, (y —x) is theDirac 5 function. We however use functions related tothe Bernstein polynomials

b, (y —x)= lim'

y (1—y)(N+1)

)v, k-~ (N —k)!k!k /N -~x

(N+ 1)) N —k( 1)lyk+l

lim Y . (9).v, k-~ k! l 0 1!(N—k —1)!k/N ~x

By integrating over y with a test function y', or other-wise, it can readily be shown that 2)),(y —x) exhibits thebehavior of a 5 function in the range 0 y ~ 1 (it need notbe symmetric in x and y for our purposes, nor does ithave to behave like a 5 function for y outside this inter-val). Using Eqs. (9) and (5) in (7), we obtain

b.qNs(x, g )= limN, k~ oc

k /N~x

), l &- k + 1 ) /( 2PO )

(N + 1)) )v —k( 1)l ) (g2)

~qNs(y V )dyi=0

(10)

As has been pointed out in Refs. 6 and 7, it is dangerous to sum Eq. (10) numerically because of the oscillatory naturedue to the appearance of the (

—1)'. Because of the form of A (1+k+1) it is not possible to do the sum in Eq. (10)analytically. However, if one uses the expansion

—A (n)/(2') c(i,j)r(p + )'+n[ln)a (n +p)]~

(N+1)! '-"k.

'

XlimN, k~ oo

k/N~x

it is in fact possible to perform the sum analytically for each individual term in Eq. (11). In Eq. (11), r =[a(Q )la(p )],and a, p, and p are arbitrary constants; later on we shall use a =e and p=2, where C is Euler's constant. p isdependent on the particular expansion and is a function of r. We then find that

(—1)' y

+'6)(y —x) x y

1!(N—k —1)! (p +k+1)'+)' I (i+p)y y x

2228 A. W. SCHREIBER, A. W. THOMAS, AND J. T. LONDERGAN 42

and

limN, kaz ook/N~x

(N+1)( N —k

k ~ )-0Xk+I

t!(N —k —t)! (p+k+t) t+ )[lna(p+k+l)]~

p —1

8(y —x) xI'(i +p)y y

ln—X

'i +p —1sj-'a-' lny

X

f dso I'(i+p+s)

S

(13)

Hence we obtain

hqNs(x Q ) b„(x/y, r)dqNs(y p )1dy 2 (14)

with

'(p —1)

Xb (x/y, r)= ln—

X

sj 'a ' ln—Xcqq(i, O) cqq(i, j)+ . ds

I (i+p) . I (j) o 1 (i+p+s)

S

(15)

By direct reevaluation of the moments of Eq. (14) it iseasy to check that Eq. (5) is indeed satisfied. For the par-ticular case of the nonsinglet shown here, the expressionsimplifies considerably because the coefficients c (i,j) are0 for jAO. The details of the QCD evolution are con-tained in the b coefficients and are independent of the in-put distributions hqNs(y, p ). This makes the methodfast as one only needs to evaluate the b's once for a suit-able set of x/y and r values [this is of particular use forthe singlet where the coefficients c (i,j) are not 0 forj%0].

In the case of the singlet distributions the differentialequation governing the evolution of its moments is givenby

4 3 1A (n)= ——+

3 2 n(n+1)n —

1———2 g —.n .

1 j

A s(n)=3 n —1

n(n+1)(17)

4 n+23 n (n+1)

Here bg(n, Q ) are the moments of the glue distributionsanalogous to Eqs. (3) and (4). The A's are proportional tothe anomalous dimensions and for three Aavors are given

KX(n, g )

d lng hg(n, g )

(g2) A ( qq) nAqs(n) (I},P(n Q )

A (n) A (n) bg(n, g )

3 2 4 " '1

A (n)=3 —+ —— —2 +2 n n+1 . jj=1

(16)Equation (16) has the solution

' t+(n)

bX„(g )= w, (n)a( )

a(p')

t(n)'+w2(n)

a( ')a(p')

b,X„(p )

t+(n)

+ —w3(n)a(Q')a(p')

't ~(n)

Qg„(g~}= —w4(n) 2+wa( ')

a(p')' t+(n)

+ w2(n) 2+wa(

a(p')

't (n)

+w3(n)a( ')a(p. ')

t (n)a(Q')

4(n)a(p')

t (n)a(g')a(p')

bg„(p ),

t4),X„(p )

bg„(p'),

where

A (n)(~)=—1+, w (n)=AA (n)/R(n},

w4(n)=b, A (n)/R (n), r+(n)= —,'[ —A+(n)+R (n)](19)

42 QCD EVOLUTION OF THE SPIN STRUCTURE FUNCTIONS OF. . . 2229

and

A+(n)=AAq~(n)+bA (n), R(n)=+A (n)+4bA (n)bA (n) .

Finally, for the singlets we obtain the following expressions analogous to (14):

~dybX(x, g )= [b~ (x/y, r)bX(y, p, )+b«(x/y, r)&g(y, p )]x

(20)

(21)

and

hg (x, g ) =f [b~(x /y, r )AX(y, p)+, bgg ( x /y, r) b g (y, p )],

x y(22)

1 + ~ ~ ~

12n

one arrives at the expansions

where b (x/y, r), b (x/y, r), and b (x/y, r) are equivalent to b (x/y, r) given in Eq. (15) with c k(i,j ) replaced byc «&(i,j ), csqk(i,j ), and egg&(i,j), respectively. Here k = distinguishes the two sets of expansions necessary for eachseries of coefficients corresponding to the two distinct powers t+ and t appearing in Eq. (18).

Choosing p =2, and usingn —1

1—.=C+qt(n) =C+lnn-

~=) J 2n

qq(")/( Po) ~ ~ 16u/271

8ur = jazzy

9z

—76u 32u 2

+81 81

—+O(z )1 —3

z2

t+(n)w, (n)r + =(az) "/

T

1 9 + O((z, lnaz) )—3

(lnaz) 25z

( )—

()16M/27t (n) 1 9 +O((z, lnaz) )

(lnaz) 25z

'+ (n) 4u/3w2(n)r + =(az) "/1 — +

z

—37u+2u2 1 + 1 4u 1 9 +O(( 1 ) 3)z2 lnaz 15z2 (lnaz)2 25z2

w2(n)r =(az)' "1 — +t (n) 8u

9z

—76u 32u 1 1 4u 1 9 -3+0 z, lnaz81 81 z2 lnaz 15z2 (lnaz) 25z

(23)

t+(n)w3(n)r + =(az) "/ 1

lnaz9 9u

10z S '1 27

2+ O((z, lnaz) )

( lnaz ) 20z

w3(n)r =(az)' " 1

lnaz

910z

4u 1

5z (lnaz)

27 —3+ O((z, lnaz) )20z

t+(n)w4(n)r + =(az) " 1

lnaz

—2 +sz

—6 4Q 1

2 (lnaz)+O((z, lnaz) )sz'

)16M/27 1

lnaz

2 6+sz s

16u 1

4S 2 2+O((z, lnaz) ) . .

sz'

Here a =e /, z =n +2, u =lnr =in[a(Q2)/ct(p2)],and O((z, lnaz) ) imply that terms of order (lnaz)have been omitted. Any such terms are also at least oforder z . Terms of higher order are listed in the Ap-pendix. The rate of convergence of Eq. (23) is a functionof lnr; more terms need to be kept if the evolution is overa greater range of Q . For the value of r considered inthe next section terms up to order (z, lnaz) are morethan sufficient. Because the expansions (11)are in powersof 1/(n +p) and 1/ln(n +p) the accuracy of the series(23) increases rapidly as n increases. Hence the method ismost accurate at large x, and indeed the number of terms

that are to be kept is primarily determined by the pre-cision that is required as x approaches 0.

Finally, for the polarized singlet only, it turns out thatthe expansions do not converge for n=1. There areseveral ways to overcome this. Because all higher mo-ments converge arbitrarily well one might choose to justadd a 6 function at x =0, this being the only function thatinAuences the first moment exclusively. Alternatively,and this is the course taken by us, one may add to the ex-pansions (23) a term such as z ', with c being some largenumber. Again only the first moment is aftectedsignificantly. Because of the arbitrariness in the choice of

2230 A. W. SCHREIBER, A. W. THOMAS, AND J. T. LONDERGAN 42

c this should be seen purely as a check that the distribu-tions for nonzero x are not sensitive to the problem withthe singlet's first moment, i.e., that the higher momentsare not affected by this additional term. No significanceshould be attached to the predictions in the region that isaffected by the term z '. We will examine the effect ofthis term in the next section, after briefly outlining amodel that we shall use to illustrate the method that hasbeen developed in this section.

III. RESULTS

We now wish to apply the procedure developed in theprevious section to a realistic problem. We shall startwith a model prediction for the proton and neutron spin-dependent quark distributions at some, in principle un-

known, scale p . This scale is determined, within themodel, by evolving the model prediction for the twist-twopiece of the nonsinglet spin-independent valence distribu-tion xq„(x,p )=xu, (x,p )+xd, (x,}u ), and comparingthe resultant xq„(x,Q ) with parametrizations of the dataat Q =10 GeV . This enables us to make predictionsfor g~~(x, Q =10 GeV ) and g}(x,Q =10 GeV ).

The model we use should describe the low-energyproperties of the nucleons, such as g~, the magnetic mo-ment, the X-6 mass splitting, etc. For this reason weshall use MIT bag wave functions for the nucleons, withthe center-of-mass momentum projected out through theuse of the Peierls-Yoccoz procedure. We shall assumethat hs(x, )=bg(x, (u )=0.

b,u(x, }u ) and hd(x, p ) are given by

d p2~qf(x, p')=2~& f &(~ —«I —p2+)(l&pzl& f+'+(0)lp=0&l' —l&p2lb f'p'+(0)lp=0&l')

m

(24)

Here ~p=0) is the Peierls-Yoccoz projected wavefunction of the nucleon at rest while (pz~ is the wavefunction of the two-quark intermediate state withmomentum p2. The destruction operator for a quark of(lavor f and total angular momentum m is b f and 4refers to the corresponding spinor. The plus indicates theprojection onto the "good" components and 1 1 refer tothe two helicity projections: i.e.,

x r+ 1+x'@+ 2 2 m

(Note that while 4 is an eigenstate of the total angularmomentum operator, 41(+ are not. ) Further details ofthe model can be found in Ref. 25.

The N-6 mass splitting is incorporated into the modelby adjusting the diquark mass appearing in Eq. (24)[through p 2+ = (p z +m 2

)' +p, ], depending on whetherthe state is in the mixed symmetric or mixed antisym-metric part of the spin-Aavor wave function. We em-phasize that the model is not entirely satisfactory as thePeierls-Yoccoz procedure does not produce an energyeigenstate, nor does it eliminate the center-of-massmomentum variable from the internal wave function ofthe nucleon. Furthermore, it has been pointed out thatconnected contributions to the twist-two piece of thescattering process also arise from four-quark intermedi-ate states, which are not considered in Eq. (24). As aconsequence of these deficiencies the first moment of thespin-independent quark distributions (summed overflavors) is not guaranteed to be precisely three, nor is theBjorken sum rule satisfied. We shall overcome theseproblems phenomenologically by simply adding in a termof the form x ' (1—x) to the quark distributions in or-der to mimic the four-quark term. We note that this doesnot affect g &

and g", above x=0.3. Note however that thevanishing of the support above x=1 is guaranteed be-cause of the t) function in Eq. (2~) which ensures energy-momentum conservation explicitly. For a bag of radius

R=0.8 fm the model predicts the following normaliza-tions at the scale p:

f u'(x, }u')dx =1.53,0

ul x p2dx=0470

f d 1(x,p, )dx =0.37,0

(25)

f d}(x,}M )dx =0.630

giving the value of g„corresponding to the Peierls-Yoccoz projected MIT bag model

1

g „= b q3(x, p')dx = 1.32 (26)

1.2

0 0.2 0.4 0.6 0.8 1.0

FIG. 1. xu, (x, Q')+xd„(x, g'} at the model scale Q =p.'and at Q'= 10 GeV' (solid lines). The dashed and dotted lines

correspond to the Duke-Owens and Martin-Roberts-Stirling pa-rametrizations of xu, , (x,g'=10 GeV'}+xd„(x,g'=10 GeV'),respectively (Figs. I.lb and I.2b of Ref. 23).

42 QCD EVOLUTION OF THE SPIN STRUCTURE FUNCTIONS OF. . . 2231

0.6

0.15

P4C3 p4X

Lr4

c3X

0.2

0.10(3X

CLChx 005

00 0.2 0.4 0.6

I

p. s 1.0

00.01 0.05 0.1 0.5 1.0

0.20 FIG. 3. The proton spin structure function g f(x,g ) at boththe model scale Q'=p' and go=10 GeV'. The data were takenfrom Ref. 10.

0.15—

x p1pC7lc3X

0.05 ':I

00 0.2 0.4 0.6

I

0.8 1.0

FIG. 2. The quark singlet (a) and polarized glue (b) distribu-tions at Q'=p' and Q'=10 GeV' [note that Ag(x, p') =0]. Thesolid and dashed lines correspond to the addition of terms pro-portional to z ' and z ', respectively, to ensure the conver-gence of the first moment as described in the text. The dotted-dashed line does not have this term added in.

=0.22(0.00) . (27)

These are the same values as those expected with wavefunctions that are completely SU(6) symmetric. Clearlythis is only an approximation to reality. For example, wehave shown previously that a pion cloud reduces the in-tegral of g~(x, Q ) by about 15%. It should be notedhowever that the distributions themselves are different

and, for the first moments of the nucleon structure func-tions

f g, '"'(x,p )dx= f [+—,', bq3(x, p )+ —,', bs(x, p )0 0

+ —,' b, X(x,p') ]«

from the SU(6)-symmetric ones, in particular the neutronstructure function is not zero everywhere. For the pur-pose of this paper we shall assume that

R =0.8 fm, m""'"""""=750 (950) MeV

and M=938 MeV. The difference between the two di-quark masses is fixed because of the N-5 mass differencewhile their absolute value has been treated as an adjust-able parameter.

We now turn to the QCD evolution of the above mod-el. In Fig. 1 we show the evolved function xq, (x, Q ) andcompare it with parametrized data at 10 GeV . As rdecreases, i.e., as Q /p increases, the momentum carriedby the initial valence quarks is decreased and generatesthe rise of the quark and gluon sea contributions at low x.A value of r =0.3 (corresponding to p=0.5 GeV anda, /2vr=0. 13 if A&cD=0.2 GeV) gives a reasonable ap-proximation to the data. We emphasize that using adifferent low-energy model, in particular one including apion cloud, will change this value of r (in fact, becausethe pions tend to carry some of the proton's momentum,the distributions will be peaked at smaller x and so thefitted value of r should increase).

Before we turn to the predictions this enables us tomake for the spin-dependent structure functions we needto make a comment about the effect of the anomaly.

0.03—

0.02—

TABLE I. Moments of quark nonsinglet, singlet, and gluedistributions. All moments are in percent of expected ones.The singlet moments correspond to those obtained from Eq. (23)with the first moment readjusted by a term z ' (see text) ~ Thenumbers in parentheses do not have this additional term.

(3

0.01—C

CJl

Moment ~qNs(n Q

0.930.991.001.00

b X(n, g')

1.00 (0.99)0.99 (1.00)1.00 (1.00)1.00 (1.00)

hg (n, g')

1.00 (0.40)1.00 (0.84)0.99 (0.95)0.99 (0.98)

—0.010.01

I

0.05I

0.1I

0.5 1.0

FIG. 4. The predicted neutron spin structure functiong", (x, g ) at both the model scale Q =p and Q'=10 GeV'.

2232 A. W. SCHREIBER, A. W. THOMAS, AND J. T. LONDERGAN 42

Given the scale p obtained above, one finds that the first

moment of the glue distribution at Q = 10 GeV is about

0.8. It has been proposed that the anomaly produces an

additional contribution to the first moment of the singlet

of the form' 'f ~ Q g (

—1 Q2)

9 2a

With the above value of hg (n =1,Q ) it only gives acontribution of order 0.01 to the sum rule. Furthermore,it has been shown' to depend on the regularization. Weshall therefore neglect it.

In Fig. 2 we show the evolved quark singlet and gluedistributions at Q =10 GeV, as well as the variation ex-pected at low x due to the arbitrariness of the term insert-ed to reproduce the first moment. The region x ~0.1 isessentially unaffected by this term and hence reliable.This is also evident from the moments of the distribu-tions, shown in Table I.

In Fig. 3 we show xg~i(x, Q =10 GeV ) correspondingto the quark distribution in Fig. 2, as well as the EMCdata for xg~i (x, Q = 10 GeV ). It is immediately evidentthat although the shape of the prediction for xg~i(x, Q )

matches the data, the first moment is not reproduced forthe ratio of coupling constants considered here. A reduc-tion of this ratio, translating into a smaller scale p, de-creases the first moment (if one includes the proposedeffect of the anomaly).

Finally we turn to the prediction for xg", (x, Q ) (Fig.4). The nonzero value of this structure function has twoorigins: the glue splitting breaks the SU(6) symmetry ofthe wave function and the singlet and nonsinglet parts ofthe structure function evolve differently, giving rise to anonzero neutron structure function at all p WQ', ir-respective of whether the original wave function wasSU(6) symmetric.

IV. CONCLUSIONS

We have presented a useful scheme for implementingthe QCD evolution of spin-dependent quark and glue dis-tributions. The method is computationally fast becausethe details of the QCD evolution are independent of thedistribution. It is therefore particularly useful in fittingmodel predictions to data. We have applied the pro-cedure to a model of the nucleon that includes the onegluon exchange responsible for the N-6 mass splitting ina phenomenological way. This, as well as the QCD evo-lution itself, induces a nonzero neutron structure functiongi(x Q').

ACKNOWLEDGMENTS

The research of J.T.L. was supported in part by theU.S. National Science Foundation under Contract No.NSF-PH Y88-805640.

CNs[2, 0]=—

cNS[3, 0]=—

cNs[4 0]=

76u 32u ~

81 81

40u 608u —256u+27 729 2187

1078u 11 528u405 6561

2432Q 512Q

6561 19 683

CNS[5, 0]= 136u 41 072u 57 664u27 10 935 59 049

19 456u 4096u177 147 885 735

—16 636u 264 808u 19 329 568u1701 32 805 7 971 615

184 576u 38 912u 16 384u531 441 1 594 323 23 914 845

cqq+[2 2] cqq+[3 2]9 =27

cNs[6, 0]=

18u25

27 9 91u 18u ~

c«+[3'3 25, c«+[,2]=5 25 25

c +[4,3]= 1581 258u 5103250 125

' '« " 2500 '

cqq+ [»21==27qq+

5571qq+[ & ] 250

57u 128u5' 2512Q

25

2064Q 246Q

125 125

cqq+ [5,4]= 11 772625

4563Q 315912so ' '«+[ ' ]=

12so '

22u' 6u4

5 2527 6709u 10 009u 2

5 250 450

c +[6,3]= 29 199qq+ ' 500

cqq+ [6,4]= 201 1112000

444 447«' '12 SOO

41 553'«=[ ' ]=soooo '

59 117Q 7493Q750

+375

105 591u 4039u2500 1250

12 177Q

3125

156Q

125

cqq [0,0]= 1, cqq [1,0]=qq—

76u + 32u81 81

been generated using the algebraic manipulations rou-tines in Ref. 28. Terms up to order (z lnaz) may be ob-tained from the authors [the factor (n+p) ~ in Eq. (11)is included in the coefficients listed here]:

SuCNS[0, 0]=1, CNS[ 1,o]=—

9

APPENDIX

We list below the higher-order coefficients occurring in

Eq. (23). The asymptotic expansions given here have

c [2,1]=, c [2,2]=—25

'

40u 608u 256u27 729 2187

42 QCD EVOLUTION OF THE SPIN STRUCTURE FUNCTIONS OF. . . 2233

32 2

c [3 1]= +Cqq—

c [3,2]=—27

1078u4p5

)—4u 1168u

3 1215128u1215

9 47u ~ 56u45 225

20 2725 «- ''

25C

11 528u 24320 512u6561 6561 19 683

9 37EL 9EL

0

[ ]27 24M

qg+370 605+5

111u ~

50[4 2)

171 ~ 627u5p

9621 10170 243qR+ '

1OOO SOO' «+ ' 2000

771 123u 1377'«+"' 200 50 ' '«"' 1000 '

136u 41 072u 57 664u27 10935 59 049

40960885 735

c [5,0]=—

194560+177 147

1581 690 5103qq-[ ]=-

250+

125qq-["]=-

2500c +[5,1]= 63

qg+

cqg~[5, 2]=—

156 553 20 687u4OOO

+IOOO

673u500

21u 749u 37u100 36 5

2097 ~ 4969u 983u ~ 33u400 120 60 25

[ ]4M+944M

4p5

c [5,2]=—2725

c [5,4]=—

c [6 0]=—

11 772 22230625 1250 '

31591250 '

16 636u 264 808u 19 329 568u1701 32 805 7 971 615

184 5760 38 912@ 16 3840531 441 1 594 323 23 914 845

23 432M 163 616M '6O7S 9841S

18 9440 2048098 415 295 245

27 36 949u 51 988u5 2250 6075

7424u 2816u6075 55 675

29 199 19 517M 1744M 448EL '500 4500 1125 3375

201 111 39 747u2ooa 2soo

444 447 41 013012 500 12 500

41 553'«[ ' )=soooo '

cqg+[1, 1]=—9&0

'

9u 27c«+ [2, 1]=, cqg ~ [2,2]=—

5

3580625

5888u 1024u10 935 32 805

383u 1312u 896u75 675 6075

5571 64M 16M« ' 2sa 2s 37s

1701 10 3731L«' '1OO IOO

15 0690200

9620 290+75 50

230 871 234 531u 4841u 23u2000 2000 250 50

3 131 613 14 589u 11 673u40 000 2500 5000

998 163 433 4310«+ '

SO OOO SO OOO

391 473qR+ '

4o ooo

c ~[6,3]=—

cqg ~ [6,4]=—

c [1,1]= 9

cqg —[2 1] c [2 2]—4u =27

5' « ' 20'

[31)938u+ 16u45 45

c [3,2]== 771200

360 13771OOO

'

[4 1)27 ~ 4u 304u5 15 405

171 406M 56EL« '

2S 7S 7S

128u1215

9621 243u 2431000 125 q~ ' 2000

63 307M 3172M 1216M 256M

5 75 3645 3645 10 935 '

178 929 5589u«' ' 2oaoo sooo '

349 191IOOOOO

'

c ,[6,1]=27-27u 6737u' + 533u ' 74u' + 6u '150 18 15 25

2234 A. W. SCHREIBER, A. W. THOMAS, AND J. T. LONDERGAN

2097400

156 553« '

4ooo

178 929'«[' ]=2oooo

c [6, 1]= —27+ 720

9728u98 415

15440 438406075 6561

20480492 075

16 763u 2372u 512u1350 675 2025

703u 152u ~

50 125

918u 349 191625 « 100000

c +[6,2]= 243320

5036u+675

cgq+[6, 4]=

cg +[6,5]=

c«[1,1]=

472 10310000

338 21125 0002

5

[ ]158 497

1000

5779u90

27 149Q

150

58u225

26 351u 11 701u 46u300 1125 225

1379u 1297u ~

1250 1250

48 159u 43 49712 500

' gg+ ' 10000

164u 4034u9

+405

1701100

26 176u 128u18 225 2025

230 871 88 909u 2152u2000 1500 225

3 131 613 25 779u 876u40 000 1250 625

c [6,2]=—

998 163 17 739050000 12500391 473«['] 4oooo '

c,+[1,1]=——,2

c,+[2, 1]=——+, c,+[2,2]=——,6 4u 3gq+ 7

5 57 gq+ 7

64u135

6 16u =3

cgg [3,1]= — +14 584Q 64Q

c [3,2]= —,c [3,3]=527 160 153150 25

'

568u + 2944u 512u135 3645 10935 '

679 584u 224u50 135 675

764 1080 2712s i2s ' '« ["' ] soo '

62 7292u 92 176u67S 328OS

9472u 1024u32 805 98 415

14 182u 4u«' 's 4s s

214u 256u 8u

4s+

is '

4 2679 664u 74u«' 'SO 7S 7S

764 1130]=—, +, „+[ ]=—27500

10 657u 44u 4u

405+

9 15

62 3203u

4307 60 701u 6913u 44u '100 1350 675 75

321 577 26 789u9OOO 22SO

67301125

20 691 621u 38 7995000 1250 '«+" 25 000 '

527 820 153gq+[ & ] 150 75

& gig+[ & ] 250

2048018 225

c, [5,2]= 4307 112 366u 15 536u6075 6075

9946u 6080~1125 1125

321 5779000

20 6915000

c [5,3]=

c [5,4]=

cg~ [6, 1]=

4080c [5,5]=

126 5852u 151 568u5 225 18 225

13 3120 81920177 147 4 428 675

2433 42 922u 220 504u20 675 18 225

6400u 512u6561 18 225

158 497 178 423u 59 456u 256u1000 3375 10 125 1215

472 103 20 858u 1168u10000 1875 1875

126 29030S 2S

22 253u ~

225338 211 1971u2S OOO 3i2S

22540 4040 8081 135 75

43 497ioooo '

42 QCD EVOLUTION OF THE SPIN STRUCTURE FUNCTIONS OF. . . 2235

c +[0,0]=1, c +[1,0]= —2u,

c +[2,0]=— +2ugg+

925

28u 74u ~ 4u 33+9 3

css+[2, 1]=, c +[2,2]=-gg +

c +[3,0]=—

1581 321u 5103250 125

' gg+ ' 2500 '

c +[5,0]=—124u 10 577u ~

3+

135

148u 4u

27 15

2881Q81

c +[3,1]= 4u 8u

5 15

c +[3,2]=— +, c +[3,3]=—27 28u 2725 25 gg 25

1799u 4393u 74u 2u

90 162 9 3

4u 364u 8u '3 135 15

9 1346u 334u5 225 225

c [4,3]= 1581

5103'" [ ' ] 2500 '

132Q

125

29 199 1 399 291u500 13 500

102 634u 20 714u3375 10 125

201 111 30 513u2000 625

444 447 50 247 u

12 500 12 500

6 = —41 55350000 '

c [2,2]= 9

27 8u

25 25'

c [3,3]==27gg

— 7

257

[4 2]9 292u + 32u5 225 225

4941Q1250

76u ~ 512u 3

9 13516u

45

2528u 272Q

225 225

c +[5,1]=

[5 2]27 + 1474u25

cgg [5,3]=

27 236u 1472u 256u25 75 2025 6075

2704u 64u375 125

10769375egg+[5, 3]=—

c +[5,4]=—

c +[6,0]=—

c +[6,1]=—4u—

27 54 523u5 1125

23 896u2025

41 663u810

158u225

5571 7936Q250

'375

11 772 2574u5

3159625 625

' + ' 1250'

15 877u 168 803u 2 805 899u189 810 21 870

2377u 4 74u ' 4u '81 27 45

13 418u 20 018u675 1215

88u 8u

27 45

cgg [5,4]= 11 772

3159

1404Q

625

csg [6,2]=—27 5858u 40 904u ~

5 1125 18 225

4736u 512u18 225 54 675

29 199 98434u 13 504u500 3375 3375

201 111 14 052u 752u2000 625 625

10 638u3125

41 553« ' 50000

1664Q

10 125

Present address: NIKHEF-K, P.O. Box 41882, 1009 DB Arn-sterdam, The Netherlands.

'G. Altarelli, Phys. Rep. 81, 1 (1982); A. Buras, Rev. Mod.Phys. 52, 199 (1980);T. Muta, Foundations of Quantum Chromodynamics (Lecture Notes in Physics, Vol. 5) (WorldScientific, Singapore, 1987).

~E. C. G. Stueckelberg and A. Peterman, Helv. Phys. Acta 26,499 (1953);M. Gell-Mann and F. E. Low, Phys. Rev. 95, 1300(1954).

D. Gross and F. Wilczek, Phys. Rev. D 8, 3633 (1973); 9, 980(1974).

4H. D. I. Politzer, Phys. Rep. 14, 129 (1974); H. Georgi and H.

2236 A. W. SCHREIBER, A. W. THOMAS, AND J. T. LONDERGAN 42

D. I. Politzer, Phys. Rev. D 9, 416 (1974).~E. Floratos, D. Ross, and C. Sachrajda, Nucl. Phys. B129, 66

(1977); B139, 545(E) (1978);B152, 493 (1979);Phys. Lett. 80B,269 (1979); 87B, 403(E) (1980); D. R. T. Jones, Nucl. Phys.B75, 531 (1974); M. Calvo, Phys. Rev. D 15, 730 (1977); A. deRujula, H. Georgi, and H. Politzer, Ann. Phys. (N.Y.) 103,315 (1977); W. Bardeen, A. Buras, D. Duke, and T. Muta,Phys. Rev. D 18, 3998 (1978);A. Zee, F. Wilczek, and S. Trei-man, ibid. 10, 2881 (1974); I. Hinchliffe and C. LlewellynSmith, Nucl. Phys. B128, 93 (1977); G. Altarelli, R. Ellis, andG. Martinelli, ibid. B143, 521 (1978).

A. Gonzales-Arroyo et a/. , Nucl. Phys. B153, 161 (1979);B159, 51 (1979).

7A. Gonzales-Arroyo et al. , Nucl. Phys. 8166, 429 (1980).M. Ahmed and G. G. Ross, Phys. Lett. 56B, 385 (1975); Nucl.

Phys. B111,441 (1976); H. Ito, Frog. Theor. Phys. 54, 555

(1975); K. Sasaki, ibid. 54, 1816 (1975).J. Kodaira et al. , Phys. Rev. D 20, 627 (1979); Nucl. Phys.

B159,99 (1979);J. Kodaira, ibid. B165, 129 (1980).J. Ashman et al. , Phys. Lett. B 206, 364 (1988); Nucl. Phys.B328, 1 (1989)~

'M. J. Alguard et al. , Phys. Rev. Lett. 37, 1261 (1976); 41, 70(1978);G. Baum et al. , ibid. 45, 2000 (1980);51, 1135 (1983).

' G. Altarelli and G. G. Ross, Phys. Lett. B 212, 391 (1988); R.D. Carlitz et al. , ibid. 214, 229 (1988); A. V. Efremov and O.V. Terjaev, Dubna Report No. E2-88-287, 1988 (unpub-

lished); R. L. Jaffe, Phys. Lett. B 193, 101 (1987); R. L. Jaffe

and A. V. Manohar, Nucl. Phys. B337, 509 (1990);G. Bodwinand J. Qiu, Phys. Rev. D 41, 2755 (1990);S. Forte, Phys. Lett.B 224, 189 (1989);Nucl. Phys. B331, 1 (1990).

' G. Altarelli and W. J. Stirling, Particle World 1, 40 (1989);M.

Gliick, E. Reya, and W. Vogelsang, Nucl. Phys. B329, 347(1990).

' A. W. Schreiber, A. W. Thomas, and J. T. Londergan, Phys.Lett. B 237, 120 (1990).

~5M. Gluck and E. Reya, Z. Phys. C 43, 679 (1989).' D. Gross, Phys. Rev. Lett. 32, 1071 (1974); G. Parisi, Phys.

Lett. 43B, 207 (1973); A. de Rujula, H. Georgi, and H. Pol-itzer, Phys. Rev. D 10, 2141 (1974).N. Cabibbo and R. Petronzio, Nucl. Phys. B137, 395 (1978).

' G. Altarelli and G. Parisi, Nucl. Phys. B126, 298 (1977).' W. Furmanski and R. Petronzio, Nucl. Phys. B195, 237

(1982); G. P. Ramsey, J. Comp. Phys. 60, 97 (1985).A. Buras, Nucl. Phys. B125, 125 (1977); A. Buras and K. Gae-mers, ibid. B132, 249 {1978);X. Song and J. Du, Phys. Rev. D40, 2177 (1989).

'M. Gliick et al. , Z. Phys. C 41, 667 (1989).C. Lopez and F. Yndurain, in Pade Approximants and TheirApplications, edited by P. Graves-Morris (Academic, New

York, 1973).K. Charchula et al. , ZEUS Report No. 89-121 (unpublished).

24R. E. Peierls and J. Yoccoz, Proc. Phys. Soc. A70, 381 (1957).A. I. Signal and A. W. Thomas, Phys. Lett. B 211, 481 (1988);Phys. Rev. D 40, 2832 (1989); A. W. Thomas, Cargese Insti-tute on Hadrons and Hadronic Matter (Plenum, New York, in

press); A. W. Schreiber, A. I. Signal, and A. W. Thomas (in

preparation).F. E. Close and A. W. Thomas, Phys. Lett. B 212, 227 (1988).A. W. Schreiber and A. W. Thomas, Phys. Lett. B 215, 141(1988).S. Wolfram, Mathematica (Addison-Wesley, New York, 1988).