-dependent polarized parton distributions

PHYSICAL REVIEW D, VOLUME 58, 094017

x-dependent polarized parton distributions

Lionel E. GordonThomas Jefferson National Lab, Newport News, Virginia 23606;

Hampton University, Hampton, Virginia 23668;and Argonne National Lab, Argonne, Illinois 60439

Mehrdad GoshtasbpourShahid Beheshti University, Tehran, Iran

and Center For Theoretical Physics and Mathematics, AEOI, Tehran, Iran

Gordon P. RamseyLoyola University, Chicago, Illinois 60626

and Argonne National Lab, Argonne, Illinois 60439~Received 20 March 1998; published 28 September 1998!

Using QCD motivated and phenomenological considerations, we constructx dependent polarized partondistributions, which evolve under GLAP evolution, satisfy DIS data and are within positivity constraints. Eachflavor is done separately and the overall set can be used to predict polarization asymmetries for variousprocesses. We perform our NLO analysis strictly inx space. Small-x results and other physical considerationsare discussed.@S0556-2821~98!05819-6#

PACS number~s!: 13.88.1e, 13.60.Hb, 14.20.Dh

tatinastrillthrtotu

ans

-mhtta

the

on

d-n

vo

qu

onion

isteu-the

elsletein

-nd

ther

lu-fec-

in-

ge.andost

calted

I. INTRODUCTION

In light of recent polarized deep-inelastic-scattering dathere has been considerable interest in generax-dependent parton distributions for the spin-dependent cPhysics results have been extracted from the integrated sture functions@1–3#. The results indicate that there is stconsiderable uncertainty in the fraction of spin carried bygluons and sea quarks. Each of the analyses rely on ceassumptions to model the polarized distributions. One imptant way to test these assumptions is to generatex-dependent distributions and predict spin observables, sas the structure functions,g1 , for the proton, neutron anddeuteron, hard scattering cross sections for polarized hronic collisions and hadronic production of pions, kaons aheavy quark flavors. The structure function measurementg1 have been made, but the distributions at small-x are stillquite uncertain.

There are various sets ofx-dependent polarized distributions which extract the unknown parameters from assutions about the data@3–9#. Most of these are consistent witthe x-dependent data, but do not adequately addressphysical questions of compatibility with the integrated da~and hence the spin fractions of the partons! and the positiv-ity constraints for each flavor. The usual approach is to fitpolarized data directly with a given parametrization, and thcheck the integrals for agreement with the extracted fractiof spin carried by the quark flavors~or set normalizations tofit the spin fractions!. Often, either the valence is not consiered separately or the flavor dependence of the sea isconsidered.

Our approach is to establish a reasonable set of fladependent distributions at an initialQ0

2, motivated by physi-cal constraints and data, then evolve to arbitraryQ2 for usein predicting polarized observables. This approach is uni

0556-2821/98/58~9!/094017~11!/$15.00 58 0940

,ge.

uc-

eainr-hech

d-dof

p-

he

ens

ot

r-

e

in many aspects. The distributions begin with informatifrom the integrated distributions and impose normalizatand positivity constraints~all flavors, valence and sea! toensure that all of the spin information extracted from dataexplicitly contained in thex-dependent results. We generapolarized parton distributions from the unpolarized distribtions and well defined suitable assumptions, derived frommost recent polarized deep-inelastic-scattering~PDIS! datasets available@10–15#.

For the polarized sea, we assume a broken SU~3! model,to account for mass effects in polarizing the sea. Our modseparate out all flavors in the valence and sea for a companalysis of the flavor dependence of the spin fractionshadrons. We include charm via the evolution equations (Nf),at the appropriateQ2 of charm production, to avoid any nonempirical assumptions about its size. The entire LO aNLO analysis is done inx-space. Physically, the small-xbehavior is of the Regge type, consistent with data and otheoretical approaches@2#. The large-x behavior is compat-ible with the appropriate counting rules@16#.

We consider three distinct models for the polarized gons, which have a moderately wide range. Our choice eftively includes two separate factorization schemes: gaugevariant ~GI! and chiral invariant ~CI!. These are allphysically motivated models, whose overall size not larThe final parametrizations are easy to use, both in formformat. They are also in excellent agreement with the mrecent data.

II. THEORETICAL BACKGROUND

A. Polarized quark distributions

Any distributions that are to be used to predict physiobservables must be consistent with both existing, rela

© 1998 The American Physical Society17-1

tsnhairedethio

x-br-

-

ndero

a-

hic-t-

laso

th

nsatrg

ionns

y a-nedyan-

thee

rl-

t thedefi-ns-red.en-

of

talen

e-vel.

GORDON, GOSHTASBPOUR, AND RAMSEY PHYSICAL REVIEW D58 094017

data and certain fundamental theoretical assumptions. Incase of the polarized distributions, the spin information aapplies to hadronic structure must be implicitly included, athe appropriate kinematic behavior must be explicit, so tthey satisfy the fundamental constraints. This major requment covers both the theoretical and experimental consiations which are important. Thus, we wish to constructx-dependent polarized valence and sea quark distributsubject to the following physical constraints:

The integration overx should reproduce the values etracted from PDIS data, so that the fraction of spin carriedeach constituent is contained implicitly in the flavodependent distributions.

The distributions should reproduce thex-dependent polar-ized structure functions,g1

i , i 5p, n and d at the averageQ2

values of the data.The small-x behavior of g1(x) should fall between a

Regge quark-like power ofx and a gluon-dominated logarithmic behavior.

TheQ2 behavior of the quark distributions should be cosistent with the non-singlet and singlet next leading or~NLO! evolution equations, for the number of flavors apppriate to theQ2 range to be covered.

The positivity constraints are satisfied for all of the flvors.

The first two constraints build in compatibility with botthe integrated and thex-dependent polarized deep-inelastscattering~PDIS! data. The third and fourth conditions saisfy sound theoretical assumptions about both thex andQ2

kinematical dependence of the distributions. Finally, theconstraint is fundamental to our physical understandingpolarization.

Valence and sea quark assumptions

We construct the polarized valence distributions fromunpolarized distributions by imposing a modified SU~6!model @17,18#:

Duv~x![cosuD~x!Fuv~x!22

3dv~x!G ,

Ddv~x![cosuD~x!F21

3dv~x!G , ~2.1!

where the spin dilution factor is given by: cosuD[@11R0(12x)2/Ax#21. The R0 term is chosen to satisfy the Bjorkesum rule~BSR!, including the appropriate QCD correction@19#. In theQ2 region of the present PDIS data, we find thR0'2as/3. We may choose the unpolarized valence disbutions uv and dv as either the Martin-Roberts-Stirlin~MRS! set @20#,

[email protected]#,

[email protected]#,~2.2!

or the CTEQ@21# set,

09401

heitdt-r-

ens

y

-r-

tf

e

ti-

xuv~x!51.344x0.501~12x!3.689

[email protected]#,

xdv~x!50.640x0.501~12x!4.247

[email protected]#, ~2.3!

or equivalent.If we assume a model of the sea obtaining its polarizat

from gluon bremsstrahlung, then the polarized distributionaively would have a formDqf5xqf @22,23#. However, thepolarized deep-inelastic scattering data appear to implnegatively polarized sea@1#. If the integrated structure functions are to agree with data, then the normalization defiby hav[^Dq&/^xq& must be a part of the proportionalitbetween the polarized and unpolarized distributions. Mewhile, the positivity constraint, which requiresuDqu<q forall x, implies a functional form for the functionh(x). Thebasic idea of the bremsstrahlung model, coupled withimplications of the data motivate the following form for thflavor dependent polarized sea distributions:

Dqf~x![h f~x!xqf~x!, ~2.4!

whereq(x) is the x-dependent unpolarized distribution foflavor f . The functionh(x) is chosen to satisfy the normaization constraint

^Dqf&5E0

1

h f~x!xqf~x!dx[^hxqf&5hav^xqf&,

~2.5!

wherehav is extracted from data for each flavor@1#. Physi-cally, h(x) may be interpreted as a modification ofDq dueto unknown effects of soft physics at lowx. This motivates aform for h(x), which will be discussed later.

Positivity constraint

For the purposes of this analysis, we are assuming thasea quarks are effectively massless, so each quark has anite helicity state. In essence, this ignores higher twist traverse spin effects in the entire kinematic range consideThus, there is a probabilistic interpretation of the parton dsities, q, and the net parton distribution is given by:q5q↑1q↓. The total polarization is given as the differenceprobabilities of finding polarized↑ and ↓ partons in thenucleon:Dq[q↑2q↓. In a polarized↑ proton, this prob-ability for sea quarks should be less than that of the tounpolarized sea distribution, since every quark is in a givhelicity state. This is the positivity constraint: i.e,

uDq~x!u<q~x! ~2.6!

for all x.This is valid for the leading order~LO! x-dependent dis-

tributions which have a clear probabilistic interpretation, bcause there is no intrinsic scheme dependence at this leThe results in the next-to-leading-order~NLO! treatment de-pend only upon the Gribov-Lipatov-Altarelli-Parisi~GLAP!

7-2

eel-ityinenesthsr

tiotleanonin

t

du-wi

tr

sn

-ntly

thi-

twd

t ona

eolut

at

at

nar-nswo

I to

ed

hee-l

o-les,n-t

ls.

s

sis

alyge-

atg

n-a-

x-DEPENDENT POLARIZED PARTON DISTRIBUTIONS PHYSICAL REVIEW D 58 094017

evolution, so there is no problem with constraining the squarks using positivity atQ0

2 in either case. The valencquarks satisfy the positivity constraint by construction. Athough the probabilistic meaning gives rise to the positivconstraint, it is not clear what role the chiral and gaugevariant schemes have on this interpretation. In our treatmthe polarized gluon distribution is assumed small at thlow Q2 values, so the schemes are close enough sopositivity is unaffected. This provides a motivation to chooa zeroDG model—to investigate the gauge invariant factoization.

Evolution

The polarized partons are grouped in a linear combinawhich is a singlet of flavorSUf(3) group and a nonsinglelinear combination of that group. The singlet/non-singdesignations are useful for delineating certain evolutionfactorization properties of the quarks. These combinatican be related to the usual flavor decomposition, includthe valence, sea and gluons. The nonsinglet termDqNS is alinear combination of the tripleta3 axial charge and an octea8 axial charge of theSUf(3) symmetry. The singlet,DS isrelated to the axial charge,a0 and is normally associatewith the axial anomaly, which includes a gluonic contribtion. This depends upon the factorization scheme, whichbe discussed shortly.

The non-singlet term is dominated by the valence disbution, under polarized (Dus , Dds) symmetry. Its first mo-ment is scale (Q2) independent in LO. The singlet term haa definite physical interpretation in the gauge invariascheme, even though it is scale dependent. Since we areing to assume a smallDG in all of our models, this interpretation will be essentially valid, even in the chiral invaria~Adler-Bardeen! scheme, which separates out the anomaIn both cases, our distributions are constructed withinpositivity constraint. In the NLO analysis, the initial distrbutions are constructed atQ0

251.0 GeV2, where they satisfythe constraints that we have set. We assume that higher-effects are negligible compared to the quantities used totermine our distributions at this value ofQ0

2 @24#. The distri-butions are then evolved in NLO, completely independenany further considerations. In the evolution, the separatiosinglet and non-singlet is faithfully maintained. Thus, we csatisfy all of the important physical constraints.

Our parton distributions are evolved directly inx-spaceusing an iteration technique first suggested in@25# and weuse the splitting functions recently calculated in@26#. Thismethod requires less computer time than the ‘‘brute forctechnique recently used in@27# and it eliminates the need tinvert from n-moment space using the conventional evotion techniques. One advantage of our technique is thaavoids the need to know themeasuredstructure functionover the wholex-region, i.e., fromx50 to x51, which isnecessary if one is to extract Mellin moments from the d@28#. Details of the technique can be found in@29#.

B. The role of polarized gluons

Factorization

In PDIS, there are two factorization schemes which cbe used to represent the polarized sea distributions:

09401

a

-t,eat

e-

n

tdsg

ll

i-

tgo-

.e

iste-

fofn

’’

-it

a

nhe

gauge-invariant@30# @or modified minimal subtraction~MS!#and the chiral invariant@31# ~or Adler-Bardeen! schemes@32#. In the chiral-invariant~AB! scheme, the axial gluonanomaly term@33#, which depends upon the polarized gluodistribution, is separated out from the chiral invariant polized quark distributions. Since the measured distributiomust be gauge invariant, the relation between the tscheme dependent distributions is

Dq~x!GI5Dq~x!CI2Nfas

2pDG, ~2.7!

where the GI refers to the gauge-invariant scheme and Cthe chiral-invariant scheme. Thus, the size ofDG is relevantin the CI scheme. This, in turn, affects the averageh valuesextracted for each flavor. Thus,DG has an indirect bearingon the polarized sea distributions@1#.

There exists no empirical evidence that the polarizgluon distribution is very large at the relatively smallQ2

values of the data. In fact, even in the NLO evolution, tpolarized gluon distribution does not evolve significantly btweenQ251 andQ2510 GeV2, regardless of which modeis chosen. Data from Fermilab@34# indicate that it is likelysmall at theQ2 values of existing data. In addition, a theretical model of the polarized glue, based on counting ruimplies thatDG' 1

2 @16#. Other theoretical models substatiate this claim, as well@35,36#. However, since we do noknow the explicit size of theDG in this kinematic region, weperform our analysis using three distinct physical modeThe evolution ofDG is then done both in LO and NLO toinvestigate any possible NLO effects.

The first set ofh(x) functions, quoted in Table I, assumea moderately polarized glue:

[email protected]#,

~2.8!

using an unpolarized MRS glue, normalized to 0.50 or

DG~x!5xG~x!

[email protected]#, ~2.9!

for the CTEQ gluon distribution, consistent with the analyin @1#.

For the second polarized gluon model, we setDG50 todeterminehav and parametrize theh(x) accordingly. This isequivalent to the gauge-invariant scheme, since the anomterm vanishes. Any analysis which requires a gauindependent set of flavor dependent distributions~such as onthe lattice! should use this set of polarized distributionsQ0

2, evolved to the appropriateQ2 values. The correspondinh(x) functions are listed in Table II.

The third gluon model is motivated by an instantoinduced polarized gluon distribution, which gives a negtively polarized glue at small-x @37#. This modified distribu-tion ~normalized toDG520.23) is given by the best fit tothe curve in@37#:

[email protected] ln~x!#. ~2.10!

7-3

ef

onc-rnsete

r-

ncnel

sto

hd

-o--ere

ars

itargth

t

n-

he

oi

th

er

o

ichels

es,the

ibu-

lar-

n

tly

in-

vor

of

fla-all-.

c-, tocar-

thera-

e.ectingn

in-the-pa-

lesy

he


This would allow for instanton based non-perturbativefects at smallQ2.

Relation between the sea and glue in g1

The valence polarizations are rather well established frthe BSR and the polarized DIS experiments. The valequarks are dominant at largex and they give their polarization to the gluons through bremsstrahlung, which in tucreates sea polarization via pair-production. But thequarks share the momentum from the gluon which creathe pair, and thus, each constituent is at lowerx than theoriginal valence ‘‘parent.’’ This is consistent with the polaized sea being dominant at lowerx. The PDIS data implythat the sea is polarized opposite to that of the valequarks. The relative size of the negative sea polarizatioan indication as to whether gluon polarization is moderatpositive ~such asDG5xG, implying that polarization of Gcarries most of the spin of the proton!, nil, or negative. Weexpect a larger negatively polarized sea for the last two caas it must offset the positive anomaly term proportionalgluons in the chiral-invariant scheme and theJz5

12 sum rule

in either scheme. If the polarized sea is smaller~less nega-tive! or the polarized gluon is large, theg1

p curve will exhibita sharper rise at smallx. When future data are available witsmaller error bars, these scenarios can be better defineyield the correct sign and size ofDG. There may be a possibility to argue a positive proportionality of spin and mmentum of the sea ifDG is very large. A much larger polarized gluon distribution can imply either a smallnegatively polarized sea or even a slightly positive polarizsea. However, this would require a prohibitively large polized glue at these smallerQ2 values. We argue that this inot likely for the following reasons:

~1! When the light-cone wave functions at small-x areanalyzed, they implicate a negatively polarized sea@38#.

~2! In order for the sea to be entirely positive,DG wouldhave to be prohibitively large to satisfy the data. The orbangular momentum would have to be correspondingly lato satisfy the J51/2 sum rule. This would also disagree withe implications of the E704 data@34#.

~3! Our highly successful fits to thex-dependent data noonly indicate thatDG is likely moderate atQ2510 GeV2,but that the data at lowerQ2 is fit somewhat better with evesmallerDG. This is consistent with the evolution of the polarized gluon distribution. We also show in Sec. IV that tgrowth of DG presented by ABFR@4# is not likely, even inNLO. Thus, our present analysis further strengthens the pof a smallerDG at these lowerQ2 values.

~4! Most all other independent analyses agree withnegatively polarized sea.

We conclude that a positively polarized sea and a vlargeDG seem unlikely, given present data.

C. Extrapolation of data to small-x

Extrapolation ofg1i ( i 5p,n,d) to small-x is important

experimentally for determination of the integrated valuesthese structure functions. Theoretically, theg1 behavior at

09401

-

me

,ad

eisy

es

to

d-

le

nt

e

y

f

small-x is important to understanding the mechanisms whunderlie the physics in this region. There are various modthat attempt to explain the contributions to bothF2 andg1 atlow-x. The data appear to exhibit growth of these quantitibut since the error bars are somewhat large, most ofpredicted types of behavior cannot be ruled out@39#.

At large-x, the valence distributions dominateF2 andg1

and these are more well determined than the sea distrtions, which are more prevalent at small-x. Thus, one mustmake suitable assumptions about the behavior of the poized sea at low-x. Experimental analyses@14,12# have tendedto assume a relatively constant behavior and extrapolateg1

from its value at aboutx;1022 down tox50. A model byDonnachie and Landshoff@40# assumes that the Pomerocouples via vectorgm so thatg1 exhibits a logarithmic be-havior: g1; ln(1/x). Bass and Landshoff@41# analyze amodel of a two-gluon Pomeron which leads to a slighmore divergent behavior at small-x: g1;@112 ln(x)#. Ifnegative parity Pomeron cuts contribute to the spdependent cross section, a divergent behavior ofg1 results@42#, corresponding to the singular form:g1;1/x ln2(x).

We make no presumptions about the forms of the fladependent distributions at small-x, other than their relationto the unpolarized distributions. The parametrizationh f(x) in Eq. ~2.4! will be determined primarily from normal-ization and positivity constraints. Once the polarized seavors are generated, we can determine resulting the smxbehavior ofg1 and compare it to these theoretical models

III. PHENOMENOLOGY

A. Polarized quark flavors

In the work of@1#, the integrated polarized structure funtions were compared to the spin averaged distributionsestablish a comparison between the spin and momentumried by each flavor of quark. The results indicate thatrelation is flavor dependent, but the magnitudes of thesetios @hav in Eq. ~2.5!# are of the same order of magnitudAlthough this does not necessarily imply that there is a dirrelation between the two, it does provide a suitable startpoint for generating the polarized distributions from knowunpolarized distributions, which satisfy the data on spaveraged physical processes. This is the motivation forform in Eq.~2.4! for the polarized flavor-dependent distributions. The integrated data are satisfied by choosing therameters inh(x) to satisfy Eq.~2.5! and the positivity con-straints. We also wish to stay consistent with counting ruat large-x @16#. The small-x behavior can be controlled bthe functional form that we choose forh(x). All of theseconstraints are to be satisfied at some low value ofQ2, andthe evolution equations will ensure that positivity and tkinematical behavior stay consistent at allQ2.

A possible form forh(x), which gives flexibility in sat-isfying the constraints~2.5! and ~2.6! is h(x)5a1bxn. Weexpect the function to be decreasing withx, since the prob-lems with positivity ~for uhavu.1) occur at largex. We

7-4

pd

is

-si

ca

hels

t

er-henelar-ion

-

e

ost

ctedy

e

to

the

ny


chose not to modify the (12x) dependence in order to keethe counting rule powers in tact~insofar as the unpolarizedistributions do this! for the large-x behavior. We were ableto satisfy the positivity constraint at allx using this form forh(x).

In the following analysis, we assume the unpolarized dtributions in the CTEQ form: q(x)5A0xA1(12x)A2(11A3xA4). Then, using Eq.~2.4!, we generate the corresponding polarized distributions for each flavor. An analywith the MRS distributions yields similar results.

For the CTEQ distributions, we have

xq~x!51

[email protected]~12x!8.041~116.112x!

70.071x0.501~12x!8.041#, ~3.1!

where the~2! holds foru and the~1! for the d flavors. Thestrange sea has the parametrization

[email protected]~12x!8.041

3~116.112x!#. ~3.2!

Both of these sets account for theu, d asymmetry in theunpolarized sea.

The corresponding integrated polarized distributionsbe written in terms of beta functions,B(m,n), as

^Dq&5aA0B~A112,A211!

1aA0A3B~A11A412,A211!

2bA0B~A11n12,A211!

2bA0A3B~A11A41n12,A211! ~3.3!

for the CTEQ distributions. The integral^xq& can be simi-larly written and thus the restriction ona andb, correspond-ing to the normalization constraint~2.5! is a5hav2bl,where

l[B~A11n12,A211!1A3B~A11A41n12,A211!

B~A112,A211!1A3B~A11A412,A211!,

~3.4!

for the CTEQ unpolarized distributions.The positivity constraint in terms ofa andb is

TABLE I. h Values from data andh(x): DG5xG.

Quantity hu,d hs hu(x) hs(x)

SMC(I p) 22.0 21.6 22.8412.8Ax 22.2312.1AxE143(I p) 21.8 21.2 22.6412.8Ax 21.8312.1AxE154(I n) 21.5 20.6 22.3412.8Ax 21.2312.1AxHERMES(I n) 21.3 20.3 22.1412.8Ax 20.9312.1AxE143(I d) 21.6 20.8 22.4412.8Ax 21.4312.1AxSMC(I d) 22.4 22.3 23.2412.8Ax 22.9312.1Ax

09401

-

s

n

uDq~x!uq~x!

5uax1bxn11u<1, ~3.5!

for all xP@0,1#. In the following discussion, we will showhow h(x) is generated for the zero polarized gluon case. Tprocedure is virtually identical to the other two gluon modewhere the anomaly term is present.

When we choosea to satisfy the normalization constrain~2.5! and b to satisfy the positivity constraint~2.6!, thex-dependent polarized distribution for each flavor is detmined from ~2.4!. These conditions are independent of tset of unpolarized distributions that is used. However, omust be consistent by using the appropriate set of unpoized distributions which were used to determine the functh(x). Therefore, we seek the form

h~x!5~hav2bl!1bxn, ~3.6!

where bothb andn are chosen to satisfy the positivity constraint~3.5!. This form ofh(x) is motivated by~1! associat-ing this function with modifications ofDq by small-x phys-ics and ~2! keeping the (12x) dependence in tact to bconsistent with counting rule behavior at largex to the extentthat the unpolarized distributions have this desired form@16#.We can then both satisfy positivity~even at largex) andcontrol the smallx behavior, where the sea and glue are mprominent.

Choice ofh„x… at small-x

The polarized sea and the gluon distributions are expeto dominate in the small-x region. If we assume a stronglpolarized negative sea, withhav,21 then this defines arange ofb values which satisfy the positivity constraint. Thbehavior of our ansatzh(x)5a1bxn for 0,n,1 is suitedfor small-x dependence. When a certain small-x behavior isdesired, a series ofn values can be tried for the best fitsdata. In fact, when 0.2,n,1.0, it is easier to satisfy bothconstraints with appropriate choices ofa andb. These val-ues ofn allow a wider range ofa and b values. However,when 0,n,0.2, the range of possiblea andb values whichsatisfy the constraints gets very small and will not besame for all experimental yields ofhav . Thus, it makes itimpossible to find a uniform fit forh(x) for each flavor.Since h(x) should be only flavor dependent to have aphysical connection, we must choosen so thata andb willbe comparable for all experimental results.

TABLE II. h values from data andh(x): DG50.



7-5

e

l-

r-ith

it

u

re

foseto

f

ore

ib

ichd.

toting

ted

ach

try

turen.thein

ndsthemo-


Positivity of n is, in principle, not essential. If we choosa small negativen, we enhance the divergence ofg1 as x→0. This favors strong anti-polarization of sea at smalxand could give the sea some positive polarization for largex.However, it is virtually impossible to satisfy both the nomalization and positivity constraints simultaneously wnegativen and such a simple parametrization ofh. Thus, itdoes not appear to be advantageous to choosen to be nega-tive, especially since we have fit the data successfully wn5 1

2 .The ranges for possibleh(x) functions are given in

Tables I through III. The corresponding polarized distribtions satisfy the DIS data and the positivity constraint.

B. Input from DIS data

In Tables I through III, we summarize the integratedsults for each considered experiment, with the values forhavin each gluon model. The corresponding functional formeachh(x) is shown, which satisfies the constraints discusin the text. We chose two sets of data each, for the pro@10,14#, neutron@11,13# and deuteron@10,14#. These repre-sent the latest published data and are representative ogroups at SLAC, CERN and DESY.

Since our ultimate goal is to find a suitable set of flavdependenth(x) functions, which do not depend upon a spcific experimental result, we take a suitable average ofh(x)for each flavor to generate the polarized sea quark distrtions. There is enough flexibility in the choice ofa and b,

TABLE III. h values from data andh(x): Neg.DG.



09401

h

-

-

rdn

the

--

u-

given the experimental errors and the range of values whsatisfy positivity, so that all constraints are still satisfieNote from the tables that the range ofa values is not con-siderable, even when the values ofb are fixed. Our choice ofthe averaging procedure is further justified by our abilityreproduce the data from all of the experiments. The resulfunctionsh(x) for each gluon model are

Quantity hu,d(x) hs(x)

DG5xG 22.4912.8Ax 21.6712.1AxDG50 23.0313.0Ax 22.7112.9AxDG,0 23.2513.1Ax 23.3113.3Ax

IV. RESULTS AND DISCUSSION

A. Results for the polarized distributions

The polarized valence quark distributions are construcwith the assumptions made in Eqs.~2.1! and ~2.3!, with R0determined by the BSR. The overall parametrization for eof the polarized sea flavors, including theh(x) functions, theanomaly terms and the up-down unpolarized asymmeterm can be written~with the CTEQ basis! in the form:

Dqi~x!52Ax20.143~12x!8.041~12BAx!

[email protected]~x!#. ~4.1!

The values for the variables in Eq.~4.1! are given for eachflavor and each gluon model atQ0

251.0 GeV2 in Table IV.We have used these to calculate the polarized struc

functions, xg1(x), for the proton, neutron and deuteroThese are all compared with the corresponding data ataverageQ2 value for that data set. These plots are shownFigs. 1 through 4. In these figures, the solid line correspoto the small polarized gluon model, the dashed line tozero polarized glue and the dotted lines to the instantontivated gluon model.

TABLE IV. Parametrizations for polarized sea flavors atQ0251.0 GeV2.

Flavor DG A B P(x)

^Du&sea xG 0.317 1.124 20.278x0.64421.682x0.937(12x)23.368(114.269x1.508)

^Dd&sea xG 0.317 1.124 10.278x0.64421.682x0.937(12x)23.368(114.269x1.508)

^Ds& xG 0.107 1.257 23.351x0.937(12x)23.368(114.269x1.508)^Du&sea 0 0.386 0.990 20.278x0.644

^Dd&sea 0 0.386 0.990 10.278x0.644

^Ds& 0 0.173 1.070 0^Du&sea Neg 0.414 0.954 20.278x0.644210.49x1.143(12x)21.041(1

10.474 lnx)^Dd&sea Neg 0.414 0.954 10.278x0.644210.49x1.143(12x)21.041(1

10.474 lnx)^Ds& Neg 0.212 0.997 220.89x1.143(12x)21.041(110.474 lnx)

7-6

-sas

edeaa

eralcestri-u-ata.

ta

l

the

t- ree


In order to verify that our models forDG were reason-able, considering that the evolution governs theQ2 behaviorof the distributions, we evolvedDG(x,Q0

251) to Q2

51000 GeV2 for each model using both LO and NLO singlet evolution. Thex-behavior of the gluon distributions ishown in Figs. 5 through 7 at the appropriate orders of mnitude ofQ2. The corresponding integrated values for theevolved distributions are shown in Tables V and VI.

For comparison with other models of the polarizquarks, we show thex-dependent distributions of the valencand sea for each flavor in Figs. 8–11. The sea flavorsshown for each gluon model. Note that our results comp

FIG. 1. The polarized proton structureg1p as a function ofx at

fixed Q2 for three models ofDG compared to data, and highlighing smallx behavior.

FIG. 2. Same as Fig. 1 but forxg1p highlighting mediumx

behavior.

09401

g-e

rere

favorably with other models. There seems to be a genagreement about the shape of these distributions. Differenarise in the actual numerical values of the integrated disbutions. Both ourx-dependent and our integrated distribtions have been constructed to satisfy all of the present d

Physics implications

~1! All comparisons of our distributions with existing daare excellent, including Fig. 1, which showsg1 , as opposedto xg1 , accentuating the small-x behavior. The best overalfits occur with the moderate glue~model 1!. The zero gluemodel results are somewhat better for the neutron, where

FIG. 3. The polarized neutron distributionxg1n as a function ofx

at Q2510 GeV2 compared to data. The three curves are for thdifferent gluon models~see text!.

FIG. 4. Same as Fig. 3 forg1d .

7-7

thhe

p

o

asn

elaon-

b

le-

hlo

lu

u-

nt

ntic

ions

is-l-

d

so-ege,

thes

els,

asis-d-

ere,

vior

arkis

e

heut

ress

:


data are at lower averageQ2 values. This is consistent withthe Q2 evolution of the polarized gluon distribution.

~2! At small-x, the instanton gluon model predicts thatg1p

decreases slightly. However, considering the latitude indistribution, it is consistent with a constant behavior. Tdata appear to be rising in thisx region, contrary to thisimplication. Since the data are at averageQ2 of 10 GeV2,this seems to indicate that the gluons are not negativelylarized at such a relatively largeQ2. This is consistent withthe assumption that instantons are dominant at smallerx andQ2 values and are likely not a major contributor to the plarized glue at higherQ2 @37#.

~3! The polarized gluon distribution does not evolvelarge as BFR predict@4#, even with the moderate gluomodel. Assumption of such large polarization at these lowQ2 values is unfounded. In fact, data from E704 at Fermiindicate that it is likely more on the order of the moderatezero distribution. Even the NLO integrated polarized gluodo not evolve significantly different from the LO distributions.

~4! Our up and down valence distributions are comparato others. Ours is motivated from the physical SU~6! modelwith the BSR fixing the lone free parameter. It is compatibwith the u-valence domination at largex and has the appropriatex-dependent behavior at all otherx values.

~5! The u and d polarized sea distributions are not higdependent on the gluon model used to generate them. Hever, the polarized strange sea is quite sensitive to the gmodel and hence the anomaly term. This is discussedmore detail in@1#.

~6! The shape of thex-dependent polarized sea distribtions agrees with the analysis of Antonuccioet al. @38#. Theyexhibit Regge-like behavior at small-x and become slightlypositive at moderatex. Although it is not completely obviousfrom the figures, our sea distributions remain negative u

FIG. 5. Polarized gluon distribution as a function ofx at differ-ent Q2 values for theDG5xG input.

09401

is

o-

-

rbrs

le

yw-onin

il

aboutx;0.3 and then turn slightly positive. This is hiddeby the dominance of the valence quarks in this kinemarange, but indicates a consistency with physical expectatof the polarized sea.

B. Small-x behavior

For the SMC proton data with the CTEQ unpolarized dtributions and the positive gluon model, we find at smalxthat: g1

p;x20.19. Phenomenologically, this is due to the interplay between the sea distributions, with aDqi;x20.143

behavior in Eq.~3.1! and the gluons in the model, dominateby xG;x20.206 at small-x in Eq. ~2.9!. Physically, this isconsistent with Regge behavior, characteristic of the itriplet contributions tog1 . It does not have the steep rischaracteristic of the singlet behavior due to gluon exchanbut is slightly steeper than the quoted Regge intercept@43#.This could either be due to the uncertainty in the value ofRegge intercept@44# or to an interplay between the quarkand the logarithmic gluon exchange@41#. This gluon-sea in-terplay is also seen in the other polarized gluon modwhere the smaller~and negative! DG moderate the rise isg1

p .Extrapolating our results in Fig. 1 tox50.002, we can

compare to some of the models of small-x behavior dis-cussed in Sec. II. Our moderate gluon model would giveg1

p

a value of about 0.75 here, which is steeper than theA1intercept of20.14 for the isotriplet piece, but not as steepthe two-gluon model of the Pomeron. It is, however, constent with the vector coupling model of Donnachie and Lanshoff. The zero polarized gluon model givesg1

p a slightlyless steep slope, but is also consistent with this model. Hthe polarized sea dominatesg1

p at small-x. The instanton-motivated gluon model creates a relatively constant behafor g1

p .In our treatment, the polarized sea dominates the qu

contribution at small-x. Since our basic assumptionDq/q;x it follows that A1(x);x. Therefore, the relationg1(x)'F2(x)A1(x)/2x(11R) implies thatF2 and g1 havethe same behavior at smallx. The instanton motivated gluonmodel gives a constantg1

p behavior, which seems to disagrewith the apparent rise inF2 and, correspondingly,g1 . Thus,the small-x behavior of the data are not consistent with tnegativeDG model. It may therefore be possible to rule onegativeDG at theseQ2 values if the error bars ong1 can bereduced in future PDIS experiments. This does not addthe possibility for negativeDG at smallerQ2, where non-

TABLE V. Leading order polarized gluon evolution*xmin

1 DGdx.

Q2(GeV2) DG5xG DG50 Instanton

1 0.387 0.071 20.07610 0.651 0.107 10.045

100 0.736 0.167 10.1181000 0.794 0.211 10.182

7-8

r

ri

th

setoa

izean

a

fork

en

der-arer-

ier

not

lsange

ut.

:


perturbative effects are present.The neutron and deuteron structure functions appea

have more moderate behavior at small-x ~see Figs. 3 and 4!.In fact, g1

d asymptotically approaches zero, to within expemental errors. Since it is not clear whetherg1

n is negativelyincreasing or tending to zero, we cannot conclude whethere are cancellations ofg1

p and g1n at small-x to give this

moderate behavior tog1d or whether other nuclear effect

could be present. Similarly, we cannot distinguish betwethe gluon models with these data, as readily as the procase. All of the moderate gluon models seem to fit the dfairly well. A much larger DG would not provide goodagreement at low-x. More precise data at small-x could yieldmore exact conclusions.

V. CONCLUDING REMARKS

We have constructed a set of flavor-dependent polarparton distributions using QCD motivated assumptionsrecent PDIS data. The main advantages of our approachthat: both the gauge-invariant~GI! and chiral-invariant~CI!factorization are included, the positivity constraint holdsall flavors and the parametrizations include valence quaand all light flavors of the sea, in a form is easy to implemfor predicting polarized processes.

FIG. 6. Same as Fig. 5 for theDG50 input.

TABLE VI. Next-to-leading order polarized gluon evolution*xmin

1 DGdx.

Q2(GeV2) DG5xG DG50 Instanton

1 0.424 0.080 20.08210 0.653 0.119 10.047

100 0.751 0.183 10.1301000 0.811 0.229 10.190

09401

to

-

er

nn

ta

ddre

rst

There are a number of basic physical assumptions unlying these distributions. First, the valence distributionsSU~6! motivated and theDqv parametrizations are detemined using the Bjorken sum rule. The SU~3! sea symmetryis broken due to mass effects in polarizing the heavstrange quarks. Then, we generateDq from q under welldefined phenomenological assumptions. Our choice ofh(x)yields a small-x behavior which is Regge-like and a largexbehavior satisfying the counting rules. We have assumedunphysical largeDG and Lz , but have allowed an expliciinterplay betweenDS and DG via the anomaly in the CIfactorization. The three different polarized gluon modehave different physical bases and provide a reasonable r

FIG. 7. Same as Fig. 5 and Fig. 6 for the instanton gluon inp

FIG. 8. Polarized valence up quark (uv) distribution at lowQ2

for the different gluon models.

7-9

xren

e

-

o

talthete-

x-C

e

nnc-ifntse

ure.

the

.n-rt-ct


of possibilities, which can be narrowed down by future eperiments. These gluon models are consistent with theocal calculations involving quark models and assumptioabout the orbital angular momentum@16,35,36#.

The distributions exhibit success in fittingg1p,n,d both inx

dependence and the integral values:*01g1

p,n,ddx, since theseare built into the parametrizations. Evolution has been pformed in LO and NLO, with little significant difference inthe range 1 GeV2<Q2<10 GeV2. Differences start becoming apparent at theQ2 values of other experiments~around40– 50 GeV2). This will be discussed in more detail in@29#.

In Sec. III, we discussed the allowable range ofn andb inh(x), subject to the positivity constraint, witha fixed bynormalization to data. These two parameters are tightly cstrained together. Thus any variation inn, will restrict the

FIG. 9. Polarized valence down quark (dv) distribution at lowQ2 for the different gluon models.

FIG. 10. Polarized up and down sea (u1d) distribution at lowQ2 for the different gluon models.

09401

-ti-s

r-

n-

allowable values ofb, with the most flexibility for about12<n<1. The corresponding possible variation inb is compa-rable to the range ofa seen in Tables I through III for fixedvalues ofb. The variation ina is primarily due to the dif-ferenthav values, characteristic of the different experimenresults. This range is not significantly large, and sincepolarized sea is only a small part ofg1 , except perhaps asmall-x, the differences are not significant to the overall rsults we present here.

The results ofg1p at small-x imply that it may be possible

to narrow down the gluon size with more precise PDIS eperiments at small-x. Such experiments are planned at SLA~E155! and DESY~HERMES!. These would also refine thparametrizations by indicating the behavior ofg1

i at small-x.Comparisons of thex-dependent deuteron structure functiowith the corresponding proton and neutron structure futions could provide insight into possible nuclear effects,they are significant. There are various possible experimewhich would provide a better indication of the size of thpolarized gluon distribution. These include~1! one and twojet production ine2p andp2p collisions@45,46,47,48#, ~2!prompt photon production@18,49,50,51,52#, ~3! charm pro-duction @53# and ~4! pion production@47#. Groups at theBNL Relativistic Heavy Ion Collider ~RHIC! ~STAR!,SLAC ~E156!, CERN ~COMPASS! and DESY~HERA-NW )are planning to perform these experiments in the near futFor detailed explanations of these experiments, see@54#. Weare presently calculating the appropriate processes usingdistributions and gluon models presented here@29#.

ACKNOWLEDGMENTS

One of us~G.P.R.! would like to thank P. Ratcliffe and DSivers for useful discussions regarding the positivity costraint. This work was supported in part by the U.S. Depament of Energy, Division of High Energy Physics, ContraW-31-109-ENG-38.

FIG. 11. Polarized strange sea (s) distribution at lowQ2 for thedifferent gluon models.

7-10

u-

ys

,

y

ys

B

-

, see

7


@1# M. Goshtasbpour and G. P. Ramsey, Phys. Rev. D55, 1244~1997!.

@2# J. Ellis and M. Karliner, Phys. Lett. B341, 397 ~1995!.@3# T. P. Cheng and L.-F. Li, Phys. Rev. Lett.74, 2872 ~1995!;

Phys. Rev. D57, 344 ~1998!.@4# R. D. Ball, S. Forte, and G. Rudolfi, Nucl. Phys.B444, 287

~1995!; B449, 680E~1995!; B496, 337 ~1997!; R. Ball et al.,hep-ph/9707276; G. Altarelli, R. D. Ball, S. Forte, and G. Rdolfi, Acta Phys. Pol. B29, 1145~1998!.

@5# T. Gehrmann and W. J. Stirling, Z. Phys. C65, 461 ~1995!.@6# M. Gluck, E. Reya, M. Stratmann, and W. Vogelsang, Ph

Rev. D53, 4775~1996!.@7# J. Bartelski and S. Tatur, Z. Phys. C71, 595 ~1996!.@8# D. DeFlorian, O. A. Sampayo and R. Sassot, Phys. Rev. D57,

5803 ~1998!.@9# D. Indumathi, Z. Phys. C64, 439 ~1994!.

@10# K. Abe et al., Phys. Rev. Lett.78, 815 ~1997!; Phys. Rev. D54, 6620~1996!; Phys. Rev. Lett.74, 346~1995!; Phys. Lett. B364, 61 ~1995!; P. L. Anthonyet al., Phys. Rev. Lett.71, 959~1993!.

@11# K. Abe et al., Phys. Rev. Lett.79, 26 ~1997!.@12# P. L. Anthonyet al., Phys. Rev. D54, 6620~1996!.@13# K. Ackerstaffet al., Phys. Lett. B404, 383 ~1997!.@14# B. Adeva, et al., Phys. Lett. B320, 400 ~1994!; D. Adams

et al., ibid. 329, 399~1994!; 357, 248~1995!; Phys. Rev. D56,5330 ~1997!.

@15# B. Adevaet al., Phys. Lett. B369, 93 ~1996!.@16# S. J. Brodsky, M. Burkardt, and I. Schmidt, Nucl. Phys.B441,

197 ~1995!.@17# R. Carlitz and J. Kaur, Phys. Rev. Lett.38, 673 ~1977!; J.

Kaur, Nucl. Phys.B128, 219 ~1977!; F. E. Close, H. Osbornand A. M. Thomson,ibid. B77, 281 ~1974!.

@18# J.-W. Qiu, G. P. Ramsey, D. G. Richards, and D. Sivers, PhRev. D41, 65 ~1990!.

@19# S. A. Larin, F. V. Tkachev, and J. A. M. Vermaseren, PhRev. Lett.66, 862 ~1991!; S. A. Larin and J. A. M. Verma-seren, Phys. Lett. B259, 345 ~1991!; A. L. Kataev and V.Starshenko, Mod. Phys. Lett. A10, 235 ~1995!.

@20# A. D. Martin, R. G. Roberts, and W. J. Stirling, Phys. Lett.387, 419 ~1996!.

@21# H. L. Lai et al., Phys. Rev. D55, 1280 ~1997!; 51, 4763~1996!; W.-K. Tung, hep-ph/9608293.

@22# F. Close and D. Sivers, Phys. Rev. Lett.39, 1116~1977!.@23# P. Chiapetta and J. Soffer, Phys. Rev. D31, 1019~1985!.@24# E. Steinet al., Phys. Lett. B353, 107 ~1995!.@25# G. Rossi, Phys. Rev. D29, 852 ~1984!.@26# W. Vogelsang, Nucl. Phys.B475, 47 ~1996!; Phys. Rev. D54,

2023 ~1996!.

09401

.

s.

.

@27# M. Hirai, S. Kumano, and M. Mayama, Comput. Phys. Commun.108, 38 ~1998!.

@28# A. J. Buras, Rev. Mod. Phys.52, 199 ~1980!; E. Reya, Phys.Rep.69, 195 ~1981!.

@29# L. E. Gordon and G. P. Ramsey, ANL-HEP-PR-98-92.@30# G. Bodwin and J.-W. Qiu, Phys. Rev. D41, 2755~1990!.@31# Adler and W. Bardeen, Phys. Rev.182, 1517~1969!.@32# For a detailed discussion and comparison of these schemes

H.-Y. Cheng, Int. J. Mod. Phys. A11, 5109~1996!; Phys. Lett.B 427, 371 ~1998!.

@33# A. V. Efremov and O. V. Teryaev, JINR Report E2-88-28~1988!; G. Altarelli and G. G. Ross, Phys. Lett. B212, 391~1988!; R. D. Carlitz, J. C. Collins, and A. H. Mueller,ibid.214, 229 ~1988!.

@34# D. L. Adamset al., Phys. Lett. B336, 269~1994!; D. P. Gros-nick et al., Phys. Rev. D55, 1159~1997!.

@35# V. Barone, T. Calarco, and A. Drago, hep-ph/9801281.@36# I. Balitsky and X. Ji, Phys. Rev. Lett.79, 1225~1997!.@37# J. Blumlein and N. Kochelev,Deep Inelastic Scattering and

QCD, 5th International Workshop, AIP Conf. Proc. No.407~AIP, New York, 1997!, p. 875; DI IS97, Chicago, IL, edited byJ. Repond and D. Krakauer.

@38# F. Antonuccio, S. J. Brodsky, and S. Dalley, Phys. Lett. B412,104 ~1997!.

@39# F. Close and R. G. Roberts, Phys. Lett. B336, 257 ~1994!.@40# A. Donnachie and P. V. Landshoff, Z. Phys. C61, 139~1994!.@41# S. D. Bass and P. V. Landshoff, Phys. Lett. B336, 537~1994!.@42# A. H. Mueller and T. L. Trueman, Phys. Rev.160, 1306

~1967!; L. Galfi, J. Kuti, and A. Pathos, Phys. Lett. B31, 465~1970!; F. E. Close and R. G. Roberts, Phys. Rev. Lett.60,1471 ~1988!.

@43# R. L. Heimann, Nucl. Phys.B64, 429 ~1973!.@44# J. Ellis and M. Karliner, Phys. Lett. B213, 73 ~1988!.@45# M. Stratmann and W. Vogelsang, Z. Phys. C74, 641 ~1997!.@46# G. P. Ramsey, D. Richards, and D. Sivers, Phys. Rev. D37,

3140 ~1988!.@47# G. P. Ramsey and D. Sivers, Phys. Rev. D43, 2861~1991!.@48# L. E. Gordon, Phys. Rev. D57, 235 ~1998!.@49# L. E. Gordon and W. Vogelsang, Phys. Lett. B387, 629

~1996!.@50# L. E. Gordon and W. Vogelsang, Phys. Rev. D49, 170~1994!.@51# A. P. Contogouris and Z. Merebashvili, Phys. Rev. D55, 2718

~1997!.@52# C. Coriano and L. E. Gordon, Phys. Rev. D49, 170~1994!; 54,

781 ~1996!.@53# B. Baily, E. L. Berger, and L. E. Gordon, Phys. Rev. D54,

1896 ~1996!.@54# G. P. Ramsey, Prog. Part. Nucl. Phys.39, 599 ~1997!; Part.

World 4, 11 ~1995!.

7-11

-dependent polarized parton distributions

Documents