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

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

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28 April 2016

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

W. Schreiber* and

A.

W.Thomas

Department ofPhysics and Mathematical Physics, University ofAdelaide, Adelaide 500I,South Australia

J. T.

Londergan

Department

of

^Physics and Nuclear Theory Center, Indiana University, Bloomington, Indiana 47408 (Received 9April 1990)

Models ofthe long-distance behavior ofQCD can be used to make predictions for the twist-two piece ofthe current correlation function appearing ⁱⁿ deep-inelastic scattering. The connection be- tween the model predictions, valid at a four-momentum transfer squared Q' _ofthe order of ¹ (GeV/c)', and the data, collected at much higher

Q,

^{isprovided by} perturbative QCD. We present here the details ofa method, previously used for spin-averaged lepton scattering, applied to the spin-dependent case. We then use the method to make predictions for the spin structure functions

gtroron and ^gnenrron

I.

INTRODUCTION

For

some years now QCD has been the leading candi- date for a theory

of

the strong interaction. Perturbative QCD (PQCD) has provided us with aunifying framework for studying all high-energy scattering phenomena in- volving hadrons and, because

of

its property

of

asymptot- icfreedom, has given us an a posteriori justification

of

the popular 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 predictions for the nth moment

of

the structure functions, or equivalently (at least at leading order) the quark and gluon 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 the quark distribution under consideration and the

C„'s

are coefficient functions which give rise tothe _Q dependence

of

the moments tI(n,Q ). g ^is the running coupling con- stant while

g,

^is the coupling constant at the reference scale

p. y„and

^{/3 are the usual anomalous dimensions}

and the Callan-Symanzik functions, respectively. ^'

In perturbation theory one typically expands

C„, y„,

and

P

^{up to some power}

of

the coupling constant and evaluates them

to

this order. This was done forthe unpo- larized case by Gross and Wilczek, Politzer, and others up 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

and Sasaki. At next-to-leading order the coefticients

C„as

well as

y,

^{have been calculated by Kodaira et}

al. For

general 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. This is somewhat unfortunate as the experimentally measured quantities are the actual distributions themselves which typically vary ⁱⁿ accuracy as a function

of x

and are only measured down to some finite small

x,

with a subsequent extrapolation to

x =0.

^'

'"

_{Hence the extraction}

of

_the

moments issubject to some uncertainty and there _may be considerable correlation between the errors on different moments. Moreover, in the case

of

polarized scattering a great deal

of

attention has recently been given to the QCD evolution

of

the first moment

of

the _spin structure function

of

the proton, ^' while relatively little attention has 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 small value

of j

^g~&(x,Q

=

^10GeV )dx measured _by the Euro- pean Muon Collaboration (EMC) can be reconciled with the picture that at low energies the proton consists ex- clusively

of

three nonstrange valence quarks. ^'

In the case

of

unpolarized scattering, the evolution

of

the relevant distributions has been calculated in a variety

of

ways. Originally, at first order, Gross and others' reconstructed the nonsinglet quark distribution using asymptotic expansions

of

the anomalous dimensions and the inverse Mellin transform technique. Other methods have relied on parametrizations

of

the structure functions which have then been evolved by numerically solving' the integrodifferential Altarelli-Parisi' equations, by ^us- ing the method

of

Laguerre polynomials' or _{by using} particular parametrizations

of

the quark and glue distri- butions. In this paper ^we shall use the method

of

Gonzales-Arroyo et

al.

, the details

of

which can be found in a series

of

papers, starting with the second-order evolution

of

the unpolarized nonsinglet and leading to the 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

II.

THERECONSTRUCTION VIATHE BERNSTEIN POLYNOMIALS

We describe here the method

of

Refs. 6and 7,applied to the spin-dependent quark and glue distributions.

To

illustrate the method we take the case

of

the spin- dependent nonsinglet (which, because

of

parity invari- ance, isidentical to that

of

the spin-independent nonsing- let) and then give the straightforward generalization to the 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 want to proceed directly tothese 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 case in Refs. 6 and 7, ^we expect the generalization to second order to be straightforward. Higher-order e6'ects _may be important, because

of

both the considerable distance be- tween the model scale and the experimental momentum transfers, and because the model scale is rather close to the confinement scale _AQCD

For

these reasons our re- sults should be viewed as indicators

of

general trends which may be subject to some higher-order modifications.

Because

of

its speed the method described in

Sec. II

is particularly useful for comparing model predictions

of

structure functions, which apply typically at some low scale Q

=p,

^{to data} at higher Q ⁽ⁱⁿ our case _Q

=10

GeV

).

^This is because the model-dependent scale p, isa priori unknown and has to be obtained by fitting some characteristic prediction

of

the model to data. In

Ref. 21,

for example, the scale

of

a model where the quarks carry all the momentum

of

the proton is extracted from the known momentum fraction that they carry at _Q

=10

GeV . In

Sec. III

we shall present the results forapartic- ular model where we use the shape

of

the spin- independent valence distribution todetermine

p,

^enabling

us to make predictions for

g, (x,

Q ⁾

of

the proton and neutron.

whose nth moment

n—1

b, qNs( n, Q

)— :0x" 'b, qNs(x, g )dx (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) ^is the (lowest-order) anomalous dimension arising from the polarized quark-quark splitting function and _{Po is} the lowest-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 reconstruct the 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 property

f (x

⁾

= f

01^b,(y

x) f

^(y)dy

—

^.

A sufficient, but not necessary, choice for b,(y

— x)

_isthe

Dirac 5 function. We however use functions related to the Bernstein polynomials

b,(y

—x)=

lim

(N+1)

^' y

(1 —

_y)

)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 the}

behavior

of

a 5function in the range 0 _y

~

1 (it need not be symmetric in

x

_{and y} for our purposes, nor does it have 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 )=

lim

N,

k~

^oc

k/N~x

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

(N

+

^{1)) )v—k ( 1)l )}

(g2)

~qNs(y V )dy

i=0 (10)

As has been pointed out in Refs. 6and 7, itis dangerous tosum

Eq.

(10)numerically because

of

the oscillatory nature due to the appearance

of

the (

— 1)'.

_Because

of

_{the form}

of

_A

(1+k+1)

^it isnot 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.

'

X

lim

N,

k~

^oo

k/N~x

itis 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 Cis Euler's constant. p ^is dependent on the particular expansion and isa 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

and lim

N, kaz oo

k/N~x

(N+1)(

^N—k

k^~

)-0 X

k+I

t!(N —

_k

— t)! (p+k+t)

^t+

)[lna(p+k+l)]~

p—1

8(y

— x) x

I'(i +p)y

y

ln—

X

'i_+p^—1

sj-'a-' lny

f

o ^ds

I'(i+p+s)

X

S

(13) Hence we obtain

hqNs(x Q ⁾ 1dy

b„(x/y,

r)dqNs(y p2⁾ (14)

with

'(p—₁₎

b

(x/y, r)=

X

ln—

X

sj 'a ' ln—

cqq(i,^O) cqq(i,

j)

X

+

. ds

I

(i+p)

^.

I (j)

^{o 1}

(i+p+s)

S

(15)

By direct reevaluation

of

the moments

of Eq.

(14)it is easy to check that Eq.(5)isindeed satisfied.

For

the par- ticular case

of

the nonsinglet shown here, the expression simplifies considerably because the coefficients

c

(i,

j)

^are

0

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 method}

fast as one only needs toevaluate the b's once for a suit- able set

of x/y

and

r

values [this is

of

particular ^use for the singlet where the coefficients c (i,

j)

^{are not}

⁰

^for

j%0].

In the case

of

the singlet distributions the differential equation governing the evolution

of

its moments is given by

4 3 ¹

A

(n)= — — +

3 2

n(n+1)

n—1

— —

_n

—

₂ .

g

^—.

1

j

A

s(n)=3

ⁿ

—

₁

n(n+1)

(17)

4

n+2

3 n

(n+1)

Here

bg(n,

_Q )are the moments

of

the _glue distributions analogous to Eqs.(3) and (4). The ^A's are proportional to the 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

+ ——

_n

n+1 —

₂

j=1

.

+ j

(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

+w

a( ') a(p')

't+(n)

+

w2(n)

a(

₂

+w

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⁾

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):

bX(x, g )=

~dy _[b~

(x/y, r)bX(y,

p,

)+b«(x/y,

r)&g(y,

p )]

x

(20)

(21) and

hg

(x, g

⁾

= f

x y ^{[b ~}

(x

^{/y, r)AX(y,}

p)+

^,bgg(

x

^{/y, r)b g (y,}

p

⁾

],

(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 by

c

«&(i,

j

^{), csqk(i,}

j

^{), and} egg&(i,

j),

respectively. Here k

=

distinguishes the two sets

of

expansions necessary for each series

of

coefficients corresponding to the two distinct powers

t+

^{and t} appearing in Eq.

(18).

Choosing p

=

_{2, and using}

n—1

—.

=C+qt(n) =C+lnn-

1

~=)

J

²ⁿ

qq(")/( Po) ~ ~16u/27

1 8u

r

=

_jazzy

9z

—

_{76u 32u}²

81

+

81

—

¹

+O(z

^—3)

z2 t+(n)

w,

(n)r

⁺

=(az) "/

T

1 9

+ O((z,

lnaz)—₃)

(lnaz) 25z

( )

—

( )16M/27

t (n) 1 9

+O((z,

lnaz) )

(lnaz) 25z '+⁽ⁿ⁾ _4u/3

w2(n)r ⁺

=(az) "/

1

— +

z

— 37u+2u2

₁

+

^{1 4u 1 9}

+O((

¹ ) 3) z2 lnaz 15z2 (lnaz)2 25z2

w2(n)r ^{t (n)}

=(az)' "

1

—

^8u

+

9z

—

_{76u 32u 1 1 4u 1 9}

-3

+0

z, lnaz

81 81 ^z2 lnaz 15z2 (lnaz) 25z (23)

t+(n)

w3(n)r ⁺

=(az) "/

¹

lnaz

9 9u

10z S

'

_(lnaz¹ _{) 20z}²⁷²

+ O((z,

^{lnaz) )}

w3(n)r

=(az)' "

¹

lnaz 9 10z

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

of

order z . Terms

of

higher order are listed in the Ap- pendix. The rate

of

convergence

of

Eq. (23)is a function

of

lnr; more terms need to be kept

if

the evolution isover a greater range

of

_Q

. For

the value

of r

considered in the next section terms _up to order (z, lnaz) are more than sufficient. Because the expansions

(11)

^{are in} powers

of

1/(n

+p)

^{and 1/ln(n}

+p)

^{the accuracy}

of

^{the series}

(23) increases rapidly as n increases. Hence the method is most 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 that the expansions do not converge for

n=1.

There are several ways to overcome this. Because all higher mo- ments converge arbitrarily well one might choose to just add a6function at

x =0,

this being the only function that inAuences the first moment exclusively. Alternatively, and this is the course taken by us, one may add tothe ex- pansions (23) a term such asz

',

with cbeing some large number. Again only the first moment is aftected significantly. Because

of

the arbitrariness in the choice

of

cthis should be seen purely as a check that the distribu- tions for nonzero

x

are not sensitive to the problem with the singlet's first moment,

i.e.

, that the higher moments are not affected by this additional term. No significance should beattached to the predictions in the region that is affected by the term z

'.

_We will examine the effect

of

this term in the next section, after _briefly outlining a model that we shall use to illustrate the method that has been developed in this section.

III.

RESULTS

We now wish to apply the procedure developed in the previous section to a realistic problem. We shall start with 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 the model, by evolving the model prediction for the twist-two piece

of

the nonsinglet spin-independent valence distribu- tion

xq„(x, p )=xu, (x,p )+xd, (x,

}u ), and comparing the resultant

xq„(x,

Q ⁾with parametrizations

of

the data at _Q

=10

GeV

.

This enables us to make predictions for g~~(x,Q

=10

GeV )and

g}(x,

Q

=10

GeV ).

The model we use should describe the low-energy properties

of

the nucleons, such as

g~,

the magnetic mo- ment, the

X-6

mass splitting,

etc. For

this reason we shall use

MIT

bag wave functions for the nucleons, with the center-of-mass momentum projected out through the use

of

the Peierls-Yoccoz procedure. We shall assume that

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 wave function

of

the nucleon at rest while (pz~ is the wave function

of

the two-quark intermediate state with momentum _p2. The destruction operator for a quark

of

(lavor

f

^{and total angular momentum m is b}

f

^and

4

refers tothe corresponding spinor. The plus indicates the projection onto the

"good"

components and 1 1 refer to the two helicity projections:

i.e.

,

x r+ 1+x'@

+ 2 2 ^m

(Note that while

4

is an eigenstate

of

the total angular momentum operator,

41(+

are not.) Further details

of

the model can be found in

Ref.

25.

The

N-6

mass splitting isincorporated into the model by adjusting the diquark mass appearing in

Eq.

(24) [through p²⁺

=

_(pz

+

^m₂^)'

+ p, ],

depending on whether the 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 the Peierls-Yoccoz procedure does not produce an energy eigenstate, nor does it eliminate the center-of-mass momentum variable from the internal wave function

of

the nucleon. Furthermore, it has been pointed out that connected contributions to the twist-two piece

of

the scattering process also arise from four-quark intermedi- ate states, which are not considered in

Eq.

^{(24). As a} consequence

of

these deficiencies the first moment

of

the spin-independent quark distributions (summed over flavors) is not guaranteed to be precisely three, nor is the Bjorken sum rule satisfied. We shall overcome these problems phenomenologically _{by simply} adding in a term

of

the form

x

^'

(1 — x)

_tothe quark distributions in or- der tomimic the four-quark term. We note that this does not affect g^{&and g",above}

x=0. 3.

Note however that the vanishing

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

0

u'(x,

^}u')dx

=1. 53,

ul x p2dx=047

0

f

0^d1

(x,

^{p, )dx}

= 0. 37,

(25)

f

0

d}(x,

^{}M )dx}

=0.

⁶³

giving the value

of g„corresponding

to the Peierls-

Yoccoz

projected

MIT

bag model

1

g

„=

^bq3(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'=

₁₀GeV' (solid lines). The dashed and dotted lines correspond tothe Duke-Owens and Martin-Roberts-Stirling pa- rametrizations ofxu,^,(x,

g'=10 GeV'}+xd„(x, g'=10

GeV'), respectively (Figs.

I.

lb and

I.

2bofRef. 23).

0.6

0.15

P4

C3 p4

X

Lr4

c3X

0.2

0.10 (3

X

CL Chx 005

00 0.2 0.4 0.6

I

p.s 1.0

0

0.01 0.05 0.1 0.5 1.0

0.20

FIG.

3. The proton spin structure function g

f(x, g

^{)at both}

the model scale

Q'=p'

and

go=10

GeV'. The data were taken from Ref. 10.

0.15—

x p1p

C7l

c3X 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].

^The

solid and dashed lines correspond to the addition ofterms pro- portional to z ^' and z

',

respectively, to ensure the conver- gence ofthe 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 wave functions that are completely SU(6) symmetric. Clearly this is only an approximation to reality.

For

example, we have shown previously that a pion cloud reduces the in- tegral

of g~(x,

Q ⁾ by about

15%.

It should be noted however that the distributions themselves are different and, for the first moments

of

the nucleon structure func- tions

f

0

g, '"'(x, p )dx= f

0

[+

^—,',

bq3(x, p )+

^—,',

bs(x, p

⁾

+

^—,'b,

X(x, p') ]«

from the SU(6)-symmetric ones, in particular the neutron structure 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 difference while 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. ¹we show the evolved function xq,

(x,

_Q )and compare it with parametrized data at 10 GeV

.

As r decreases,

i.

e._,as Q /p increases, the momentum carried by the initial valence quarks is decreased and generates the rise

of

the quark and gluon sea contributions at low

x.

A value

of

r

=0.

3 (corresponding to

p=0.

⁵ GeV and

a, /2vr=0.

^{13 if}

A&cD=0.

² GeV) gives a reasonable ap- proximation to the data. We emphasize that using a different low-energy model, in particular one including a pion cloud, will change this value

of

r (in fact, because the pions tend to carry some

of

the proton's momentum, the distributions will be peaked at smaller

x

and so the fitted value

of r

should increase).

Before we turn to the predictions this enables us to make for the spin-dependent structure functions we need to make a comment about the effect

of

the anomaly.

0.03—

0.02—

TABLE

I.

Moments ofquark nonsinglet, singlet, and glue distributions. 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 ^' (seetext)^~ The numbers in parentheses do not have this additional term.

(3 0.01—

C CJl

Moment ~qNs(n Q

0.93 0.99 1.00 1.00

bX(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)

—0.01 0.01

I

0.05

I

0.1

I

0.5 1.0

FIG.

4. The predicted neutron _spin structure function g",(x,

g

⁾at both the model scale Q

=p

and

Q'=10

GeV'.

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 a contribution

of

order

0.

01to the sum rule. Furthermore, it has been shown' to depend on the regularization. We shall therefore neglect

it.

In

Fig.

2 we show the evolved quark singlet and glue distributions 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 is essentially 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 )corresponding to the quark distribution in

Fig.

2, as ^well as the

EMC

data for xg~i

(x,

Q

=

_{10GeV ).}

It

_is immediately evident that although the shape

of

the prediction for xg~i(x,_Q )

matches the data, the first moment is not reproduced for the 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 proposed effect

of

the anomaly).

Finally we turn to the prediction for xg",

(x,

Q ⁾ (Fig.

4). The nonzero value

of

this structure function has two origins: the glue splitting breaks the SU(6) symmetry

of

the wave function and the singlet and nonsinglet parts

of

the structure function evolve differently, giving rise to a nonzero neutron structure function at all

p WQ',

^ir- respective

of

whether the original wave function was SU(6) symmetric.

IV. CONCLUSIONS

We have presented a useful scheme for implementing the QCD evolution

of

spin-dependent quark and glue dis- tributions. The method is computationally fast because the details

of

the QCD evolution are independent

of

the distribution.

It

is therefore particularly useful in fitting model predictions to data. We have applied the pro- cedure to a model

of

the nucleon that includes the one gluon exchange responsible for the

N-6

mass splitting in a phenomenological way. This, as well as the QCD evo- lution itself, induces a nonzero neutron structure function

gi(x Q').

ACKNOWLEDGMENTS

The research

of J. T.L.

was supported in part _by the

U.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 528u

405 6561

2432Q 512Q 6561 19 683 CNS[5,

0] =

^{136u 41072u 57 664u}

27 10935 59049

19456u 4096u 177147 885735

—

_{16636u 264808u 19 329568u}

1701 32805 7971 615

184576u 38912u 16 384u 531 441 1594 323 23

914

845 cqq+[2

2]

⁹ cqq+[3

2] =27

cNs[6,

0] =

18u 25

27 9 91u 18u^~

c«+[3'3 25, c«+[, 2]=

_{5 25 25}

c

+[4, 3]=

^{1581 258u 5103}

250 125 ^'

'« "

^{2500 '}

cqq+qq+

[» 21= =27

5571

qq+[ ^&

]

₂₅₀

57u

5'

128u25

12Q 25 2064Q 246Q

125 125

cqq+

[5,4] =

¹¹⁷⁷²

625

4563Q

3159

12so ^'

'«+[

^'

]=

12so ^'

22u'

6u4

5 25

27 6709u 10009u²

5 250 450

c

_qq+

+[6,

_'

3]=

^{29 199}

500 cqq+

[6,4] =

^{201 111}

2000 444 447

«'

^' ₁₂SOO

41 553

'«=[

^'

]=

soooo

^'

59117Q 7493Q 750

+

375 105591u 4039u

2500 1250

12177Q 3125

156Q 125

cqq

[0, 0] =

_{1, cqqqq—}

[1,0] =

76u

+

^32u

81 81

been generated using the algebraic manipulations rou- tines in

Ref.

28. Terms up toorder (zlnaz) may be ob- tained from the authors [the factor

(n+p)

^~ in

Eq. (11)

isincluded in the coefficients listed here]:

CNS[0,

0]=

1, _CNS[1,

o]=—

Su 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 256u

27 729 2187

32 ²

c

[3 1]= +

Cqq—

c [3, 2]=—

²⁷

1078u 4p5 )

—

_{4u 1168u}

3 1215

128u 1215

9

47u

~

^56u

45 225

20 27

25 ^C

«- ''

₂₅

11 528u 24320 512u

6561 6561 19683

9 37EL 9EL

0

[

] 27 24M qg+

370 60

5+5

111u^~

[4

2) ¹⁷¹

~

^627u 50

5p

9621

10170

243

qR+ '

1OOO SOO '

«+

^' ₂₀₀₀

771 123u 1377

'«+"'

_{200 50} ^'

'«"'

1000 ^'

136u 41072u 57664u

27 10 935 59049

40960 885735

c [5, 0]=—

194560

+

177 147

1581

690

5103

qq-[ ]=-

250

+

125

qq-["]=-

₂₅₀₀

c

qg+

+[5, 1]=

⁶³

cqg~[5,

2]=—

156 553 20687u

4OOO

+

IOOO

673u 500

21u 749u 37u

100 36 5

2097

~

^{4969u 983u}

~

^33u

400 120 60 25

[ ] 4M+944M

4p5

c [5, 2]=—

²⁷

25

c [5, 4]=—

c [6 0]=—

11772 22230 625 1250 ^'

3159

1250 ^' 16 636u 264808u 19329568u

1701 32805 7971

615

1845760 38912@ 163840

531 441 1594 323 23

914

845 23432M 163616M

'

6O7S

9841S

189440 20480 98415 295245 27 36 949u 51988u

5 2250 6075

7424u 2816u 6075 55675

29 199 19517M 1744M 448EL

'

500 4500 1125 3375

201111 39747u

2ooa 2soo

444 447 41

0130

12500 12500 41553

'«[

^'

)= soooo

^'

cqg+[1,

1]=—

⁹

&0 ^'

9u 27

c«+ [2, 1] =,

₅ ^cqg~

[2, 2] =—

3580 625 5888u 1024u

10935 32805 383u 1312u 896u

75 675 6075

5571 64M 16M

«

^'

2sa

_{2s 37s}

1701 103731L

«'

^' _{1OO IOO} ¹⁵₂₀₀

0690

9620 290

+

75 50

230 871 234 531u 4841u 23u

2000 2000 250 50

3131

613

14589u 11673u

40 000 2500 5000

998163 433

4310

«+

^' _{SOOOO SOOOO}

391 473

qR+ '

4oooo c

~[6, 3]=—

cqg~

[6,4] =—

c

[1, 1]=

⁹

cqg—

[2 1] —

_4u

c [2 2] =27

5 '

«

^'

20'

[31)938u+

^16u

45 45

c

[3, 2]= =

⁷⁷¹

200

360 1377

1OOO '

[4

1) ²⁷

~

^{4u 304u}

5 15 405

171 406M 56EL

«

^' _{2S 7S 7S}

128u 1215

9621 243u 243

1000 125 ^{q~ '} 2000

63 307M 3172M 1216M 256M

5 75 3645 3645 10935^'

178929 5589u

«'

^'

2oaoo

_sooo ^'

349 191

IOOOOO '

c , [6, 1]=27-27u 6737u'

₁₅₀

+

^533u₁₈

' 74u'

₁₅

+

^6u₂₅

'

2097 400 156 553

«

^' _4ooo

178929

'«[' ]= 2oooo

c [6, 1] = — 27+

⁷²⁰

9728u 98415

15440 43840

6075 6561

20480 492 075

16763u 2372u 512u

1350 675 2025

703u 152u^~

50 125

918u 349 191

625

« 100000

c +[6, 2]=

²⁴³³

20 5036u

+

675

cgq+[6,

4] =

cg

+[6, 5]=

c«[1, 1]=

472 103 10000 338 211

25000 2 5

[ ]

¹⁵⁸⁴⁹⁷

1000

5779u

90

27149Q

150 58u 225

26351u 11701u 46u

300 1125 225

1379u 1297u^~

1250 1250

48159u 43497

12500 ^{' gg+ '}

10000

164u 4034u

9

+

405 1701

100

26176u 128u 18 225 2025

230 871 88909u 2152u

2000 1500 225

3131613 25779u 876u

40 000

1250 625

c

[6, 2]=—

998 163 17

7390 50000 12500

391 473

«['] 4oooo

^'

c, +[1, 1]=— —,

²

c,

^gq+

+[2,

⁷

1]= — —

⁶5

+,

^4u5 ⁷

c,

gq+

+[2,

⁷

2]= — —,

³ 64u

135

6 16u

=3

cgg

[3, 1] =

¹⁴

—

^584Q

+

^64Q

c

[3, 2]=

⁵²⁷150

—,

¹⁶⁰25 ^{' c}

[3, 3]=

¹⁵³ 568u

+

^{2944u 512u}

135 3645

10935

^'

679 584u 224u

50 135 675

764 1080 27

12s

i2s

^'

'« ["' ]

_soo '

62 7292u 92176u

67S 328OS

9472u 1024u 32805 98 415

14 182u 4u

«'

^' _s

4s

_s

214u 256u 8u

4s

+

is

^'

4 2 679 664u 74u

«'

^' _{SO 7S 7S}

764

1130

]= —, +, „+[ ]=—

₅₀₀²⁷

10 657u 44u 4u

405

+

9 15

62 3203u

4307 60 701u 6913u 44u

'

100 1350 675 75

321577 26 789u

9OOO 22SO

6730 1125

20 691 621u 38 799

5000 1250

'«+"

₂₅₀₀₀ ^'

527 820 153

gq+[ ^&

]

_{150 75} ^& gig+[ ^&

]

₂₅₀

20480 18 225

c, [5, 2]=

^{4307 112 366u}₆₀₇₅ ¹⁵₆₀₇₅^536u

9946u 6080~

1125 1125 321577

9000 20691 5000 c

[5, 3]=

c

[5, 4]=

cg~

[6, 1] =

4080 c

[5, 5]=

126 5852u 151568u

5 225 18225

13

3120 81920

177147 4 428 675 2433 42922u 220504u

20 675 18225

6400u 512u 6561 18225

158497 178423u 59456u 256u

1000 3375 10 125 1215

472103 20858u 1168u

10000 1875 1875

126

29030

S 2S

22253u^~ 225

338211 1971u 2SOOO

3i2S

22540 4040 80

81 135 75

43497

ioooo

^'

c

+[0, 0] =1,

c

+[1,0] = — 2u,

c_gg+

+[2, 0]= — +2u

9 25 28u 74u^~ 4u ³

3+9

³

css+[2,

gg+

1]=,

^c

+[2, 2]=-

c

+[3,0] =—

1581 321u 5103

250 125 ^{' gg+ '} 2500 ^'

c

+[5, 0]=—

^{124u 10577u~}

3

+

135

148u 4u

27 15

2881Q 81 c

+[3, 1]=

^{4u 8u}

5 15

c

+[3, 2]= —

²⁷25

+,

^28u25 ^cgg

+[3, 3]=—

²⁷25

1799u 4393u 74u 2u

90

162 9 3

4u 364u 8u

'

3 135 15

9 1346u 334u

5 225 225

c

[4, 3] =

¹⁵⁸¹

'" [

^'

]

₂₅₀₀5103 ^' 132Q

125

29 199 1399 291u

500 13500

102634u 20 714u

3375 10125

201

111

30 513u

2000 625

444447 50247u

12500 12500 6

= —

⁴¹⁵⁵³

50000

^' c

[2, 2]= 9

27 8u

25 25 ^' c_gg—

[3,

7

3]= =27

25 ⁷

[4 2]

^{9 292u}

+

^32u

5 225 225

4941Q 1250

76u^~ 512u³

9 135

16u 45 2528u ^272Q

225 225

c

+[5, 1]=

[5 2]

²⁷

+

^1474u

25 ^cgg

[5,3] =

27 236u 1472u 256u

25 75 2025 6075

2704u 64u

375 125

10769 egg+[5,

3] =—

375

c

+[5, 4]=—

c

+[6, 0]=—

c

+[6, 1]= — 4u—

27 54523u

5 1125

23896u 2025

41663u

810

158u

225 5571 7936Q

250

'

₃₇₅

11772 2574u

5

3159

625 625 ^' + '

1250 ^' 15 877u 168803u 2805899u

189

810

21870

2377u⁴ 74u

'

_4u

'

81 27 45

13418u 20 018u

675 1215

88u 8u

27 45

cgg

[5,4] =

¹¹⁷⁷²

3159

1404Q 625

csg

[6,2] =—

^{27 5858u 40904u~}

5 1125 18225

4736u 512u 18225 54675

29 199 98434u 13 504u

500 3375 3375

201111 14052u 752u

2000 625 625

10638u 3125 41553

«

^'

50000

1664Q 10125

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