Statistical description of the proton spin with a large gluon helicity distribution

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Statistical description of the proton spin with a large

gluon helicity distribution

Claude Bourrely, Jacques Soffer

To cite this version:

Claude Bourrely, Jacques Soffer. Statistical description of the proton spin with a large gluon helicity

distribution. Physics Letters B, Elsevier, 2015, 740, pp.168-171. �10.1016/j.physletb.2014.11.044�.

�hal-01230963�

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Statistical

description

of

the

proton

spin

with

a

large

gluon

helicity

distribution

Claude Bourrely

a

,

Jacques Soffer

b

a_{Aix-MarseilleUniversité,DépartementdePhysique,FacultédesSciencessitedeLuminy,13288Marseille cedex 09,France} b_{PhysicsDepartment,TempleUniversity,1835N,12thStreet,Philadelphia,PA 19122-6082,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received29August2014

Receivedinrevisedform17November2014 Accepted23November2014

Availableonline2December2014 Editor:J.-P.Blaizot

Keywords:

Gluonpolarization Protonspin

Statisticaldistributions

The quantumstatisticalpartondistributionsapproachproposedmorethanonedecadeagoisrevisited byconsideringalargersetofrecentand accurateDeepInelastic Scattering(DIS)experimental results. It enablesus toimprovethe descriptionofthe databy meansof anew determinationofthe parton distributions.Wewillseethatalargegluonpolarizationemerges,givingasignificantcontributiontothe protonspin.

©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/3.0/).FundedbySCOAP3_.

1. Introduction

Several years ago a new set of parton distribution functions (PDF) was constructedin theframework ofa statisticalapproach of the nucleon [1]. For quarks (antiquarks), the building blocks arethehelicitydependentdistributionsq±

(

x

)

(q

¯

±

(

x

)

). Thisallows to describe simultaneously the unpolarized distributions q

(

x

)

=

q+

(

x

)

+

q−

(

x

)

andthehelicitydistributions



q

(

x

)

=

q+

(

x

)

−

q−

(

x

)

(similarly forantiquarks).At theinitial energyscale,these distri-butions are given by the sumoftwo terms,a quasi Fermi–Dirac function and a helicity independent diffractive contribution. The flavor asymmetry for the light sea, i.e. d

¯

(

x

)

>

u

¯

(

x

)

, observed in the data is built in. This is simply understood in terms of the Pauli exclusion principle, basedon the fact that the proton con-tains two up-quarks and only one down-quark. The chiral prop-ertiesofQCD lead tostrongrelations betweenq

(

x

)

andq

¯

(

x

)

.For example,itisfoundthatthewellestablished result



u

(

x

)

>

0 im-plies



u

¯

(

x

)

>

0 and similarly



d

(

x

)

<

0 leadsto

¯

d

(

x

)

<

0. This earlier prediction was confirmed by recent data. In addition we foundtheapproximateequalityoftheflavorasymmetries,namely

¯

d

(

x

)

− ¯

u

(

x

)

∼ ¯

u

(

x

)

−¯

d

(

x

)

.Concerningthegluon,theunpolarized distribution G

(

x

,

Q2

0

)

is given in terms of a quasi Bose–Einstein

function,withonlyonefreeparameter,andforsimplicity,wewere assuming zero gluon polarization, i.e.



G

(

x

,

Q₀2

)

=

0, at the

ini-E-mailaddress:jacques.soffer@gmail.com(J. Soffer).

tial energy scale Q₀2. As we will see below, the newanalysis of a larger set ofrecent accurate DISdata, has enforced usto give up thisassumption. It leads to an unexpected large gluon helic-ity distributionandthisis themajor point, whichis emphasized inthisletter.Inourprevious analysisallunpolarizedandhelicity light quark distributions were depending upon eightfree parame-ters, which were determined in 2002(see Ref. [1]),from a next-to-leading(NLO)fitofasmallsetofaccurateDISdata.Concerning thestrange quarksandantiquarks distributions,thestatistical ap-proachwas applied usingslightlydifferentexpressions,withfour additionalparameters[2].Sincethefirstdeterminationofthefree parameters, new testsagainst experimental (unpolarized and po-larized) data turned out to be very satisfactory, in particular in hadronicreactions,asreportedinRefs.[3–5].

In this letter, after a brief review of the statistical approach, wewillonlygivesome elementsofthenewdeterminationofthe partondistributions,butwewillfocusonthegluonhelicity distri-bution,a fundamentalcontributiontotheprotonspin.

2. Basicreviewonthestatisticalapproach

Letusnowrecallthe mainfeatures ofthestatisticalapproach forbuildingupthePDF,asopposetothestandardpolynomialtype parameterizationsofthePDF,basedonReggetheoryatlow x and oncountingrulesatlarge x.Thefermiondistributionsaregivenby thesumoftwoterms,aquasiFermi–Diracfunctionandahelicity independentdiffractivecontribution:

http://dx.doi.org/10.1016/j.physletb.2014.11.044

0370-2693/©2014TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/3.0/).Fundedby SCOAP3_.

xqh



x

,

Q₀2



=

AqX h 0qxbq exp

[(

x

−

Xh_0q

)/

x

¯

] +

1

+

˜

Aqxb˜q exp

(

x

/

x

¯

)

+

1

,

(1) xq

¯

h



x

,

Q₀2



=

¯

Aq

(

X−0qh

)

−1xb¯q exp

[(

x

+

X−_0qh

)/

¯

x

] +

1

+

˜

Aqxb˜q exp

(

x

/

x

¯

)

+

1

,

(2)

attheinputenergyscaleQ2

0

=

1 GeV2.Wenotethatthediffractive

term is absent in the quark helicity distribution



q and in the quarkvalencecontributionq

− ¯

q.

In Eqs. (1), (2) the multiplicative factors Xh

0q and

(

X−

h

0q

)

−1 in

thenumeratorsofthenon-diffractivepartsofthe q’sandq’s

¯

dis-tributions, imply a modification of the quantum statistical form, wewereledtoproposeinordertoagreewithexperimentaldata. The presence of these multiplicative factors was justified in our earlierattempttogeneratethetransversemomentumdependence (TMD)[6],whichwasrevisitedrecently[7].Theparameterx plays

¯

theroleofauniversaltemperature and X±_0q arethetwo thermody-namicalpotentials of thequark q,withhelicity h

= ±

.Notice the changeofsignofthepotentialsandhelicityfortheantiquarks.1

Foragivenflavorq thecorrespondingquarkandantiquark dis-tributions involve eight free parameters: X_0q±, Aq, A

¯

q, A

˜

q, bq, b

¯

q and b

˜

q. It reduces to seven since one of them is fixed by the valence sum rule,



(

q

(

x

)

− ¯

q

(

x

))

dx

=

Nq, where Nq

=

2

,

1

,

0 for u

,

d

,

s, respectively. Forthe light quarks q

=

u

,

d, the total num-beroffreeparametersisreducedtoeight bytaking,asinRef.[1], Au

=

Ad, A

¯

u

= ¯

Ad, A

˜

u

= ˜

Ad,bu

=

bd,b

¯

u

= ¯

bd andb

˜

u

= ˜

bd.Forthe strangequarkandantiquarkdistributions,thesimplechoicemade inRef.[1]wasimprovedinRef.[2],butheretheyareexpressedin termsofseven freeparameters.

Forthegluonsweconsidertheblack-bodyinspiredexpression

xG



x

,

Q₀2



=

AGx

bG

exp

(

x

/

x

¯

)

−

1

,

(3)

aquasiBose–Einsteinfunction,withbGbeingtheonlyfree param-eter,since AG isdetermined by themomentum sumrule.Inour earlierworks [1,4], we were assuming that, at the input energy scale,thepolarizedgluon,distributionvanishes,so

x



G



x

,

Q₀2



=

0

.

(4)

However in ourrecent analysis ofa much larger setof very ac-curate unpolarizedandpolarized DISdata,we mustgive up this simplifyingassumption.Wearenowtaking



G



x

,

Q₀2



=

P

(

x

)

G



x

,

Q₀2



,

(5)

where P

(

x

)

isexpressedin termsoffour freeparameters, as fol-lows P

(

x

)

= ¯

AGxb¯G

/



cG

+

xdG



.

(6)

Wemusthave

|

P

(

x

)

|

≤

1,toinsurethatpositivityissatisfied. Itisclearthatwedon’thaveaseriousjustificationofthe func-tionalform of



G

(

x

)

. However the above expression shows that it is strongly related to G

(

x

)

, since it is constructed by means ofaBose–Einstein distributionwithzeropotential. A simpler ex-pressionwouldbe P

(

x

)

=

Axb_{,buttheadditionaltermx}dG _inthe

denominatorisneededinordertogetareasonablefitofthedata aswewilldiscussbelow.

TosummarizethenewdeterminationofthePDFinvolvesa to-taloftwentyone freeparameters:inadditiontothetemperaturex

¯

1 _{Atvariancewithstatisticalmechanicswherethedistributionsareexpressedin}

termsoftheenergy,hereoneusesx whichisclearlythenaturalvariableentering inallthesumrulesofthepartonmodel.

andtheexponentbG ofthegluondistribution,wehaveeight free parameters for the light quarks

(

u

,

d

)

, seven free parameters for the strange quarks and four free parameters forthe gluon helic-itydistribution.Theseparametersweredeterminedfromanext-to leading order (NLO) fit of a large set of accurate DIS data, (the unpolarizedstructure functionsF₂p,n,d

(

x

,

Q2

)

,thepolarized struc-turefunctionsg₁p,n,d

(

x

,

Q2

)

,thestructurefunctionxF ν₃N

(

x

,

Q2

)

in

νN DIS,

etc...)atotalof2140experimentalpointswithanaverage

χ

2

_/

_{pt of 1.33. Forthe polarizedstructure functions, itis slightly}

better since for a total of 271 experimental points, we have an average

χ

2

_/

_{pt of 1.21. Althoughthefull details ofthesenew}

re-sultswill be presentedin a forthcomingpaper [8], we justwant tomakeageneralremark.Bycomparingwiththeresultsof2002 [1], we have observed a remarkable stability of some important parameters,thelightquarkspotentialsX±_0uandX_0d±,whose numer-icalvaluesarealmostunchanged.Thenewtemperatureisslightly lower. As a result the main features of the newlight quark and antiquarkdistributionsareonlyhardlymodified,whichisnot sur-prising,since our2002PDFsethasproven tohavearathergood predictivepower.

We now turn to the gluon helicity distribution which is the mainpurposeofthisletter.

3. Thegluonhelicitydistributionandtheprotonspin

The gluon helicity distribution



G

(

x

,

Q2

)

of the proton is a fundamentalphysicalquantityforourunderstandingoftheproton spin.Itsintegraloverx,



G

(

Q2

)

,maybeinterpreted,inthe light-cone gauge A+

=

0,asthe gluon spincontribution tothe proton spin[9].If

Σ(

Q2

₎

_{denotesthetotalsumofquarkandantiquark}

helicitycontributionsandLq,q¯,G

(

Q2

)

arethequark,antiquarkand gluonorbitalangularmomentumcontributions,theprotonhelicity sumrulereads

1

/

2

=

1

/

2

Σ



Q2



+ 

G



Q2



+

Lq



Q2



+

Lq_¯



Q2



+

LG



Q2



.

(7)

Intheabovesumrule

Σ(

Q2

₎

_{iscertainlythecontributionwhich}

is bestknown, witha typical value

∼

0

.

3 according to Refs. [10, 11] and also fromour own result. This contribution is therefore toosmalltosatisfythesumruleanditiscrucialtofindouthow much the gluon contributes to the proton spin, a long standing problem.

ThenewdeterminationofthePDFleads,inthegluonsector,to thefollowingparameters:

AG

=

36

.

778

±

0

.

085

,

bG

=

1

.

020

±

0

.

0014

,

¯

AG

=

0

.

731

±

0

.

001

,

b

¯

G

=

5

.

214

±

0

.

250

,

cG

=

0

.

006

±

0

.

0005

,

dG

=

6

.

072

±

0

.

35

,

(8) andthenewtemperatureis

¯

x

=

0

.

090

±

0

.

002.WedisplayinFig. 1 the gluon helicity distribution versus x atthe initial scale Q₀2

=

1 GeV2 andQ2

=

10 GeV2.Attheinitialscaleitissharplypeaked around x

=

0

.

4,butthisfeaturelessens aftersome evolution.We find that P

(

x

)

=

0

.

731x5.210

_/(

_x6.072

₊

₀

_.

₀₀₆

₎

_{, which is such that}

0

<

P

(

x

)

<

1 for0

<

x

<

1,sopositivityissatisfiedandinaddition thegluonhelicitydistributionremainspositive.

AsalreadymentionedthetermxdG _{playsanimportantrole.We}

found that it hasa strong effect on the quality of the fit ofthe polarizedstructurefunctionssincethechi2increasessubstantially whendGdecreases.Thechi2whichis328fordG

=

6

.

072,becomes 337fordG

=

5,368fordG

=

4

.

5 and465fordG

=

4.Itsvaluealso affectstheshapeofthegluonhelicitydistribution,whichbecomes largertowardsthesmallerx-values,forsmallerdG.

Fig. 1. ThegluonhelicitydistributionxG(x,Q2_{)versusx,forQ}2_{=1 GeV}2

(dashed curve)andQ2_{=10 GeV}2

(solidcurve),withthecorrespondingerrorband.

Fig. 2. G(x,Q2_)/G(x,Q2_{) versus x,for Q}2_{=2 GeV}2 _{(solidcurve)and Q}2₌

10 GeV2_{(dashedcurve).ThedataarefromHERMES[12]andCOMPASS[13].}

We display



G

(

x

,

Q2

_)/

_G

₍

_x

_,

_Q2

₎

_{in Fig. 2 for two Q}2 _values

and some data points [12,13] suggestingthat the gluon helicity distributionispositiveindeed.Accordingtotheconstraintsofthe countingrules thisratio should goto 1 when x

=

1, butwe ob-servethatthisisnotthecasehere,sinceforexampleattheinitial scaleP

(

x

=

1

)

=

0

.

726.Insomeotherparameterizationsinthe cur-rentliterature,thisratiogoestozero,sincethelargex behaviorof x



G

(

x

)

is

(

1

−

x

)

β with

β



3[10,11].Clearlyoneneedsabetter knowledgeof



G

(

x

,

Q2

_)/

_G

₍

_x

_,

_Q2

₎

_forx

_>

₀

_.

_2.

Letus now examine the consequences of thisnew gluon he-licity distribution, with a rather strong Q2 dependence, on the proton helicity sum rule Eq. (7). By considering only the quark, antiquark andgluon helicity densities, we display in Fig. 3 their

Fig. 3. Totalcontributionofthequark,antiquarkΣ(x,Q2_{)andgluonG(x,Q}2₎

helicitydensities,totheprotonhelicitysumruleversusthelowerlimitofthe inte-gralxmin,fordifferentQ2values.

Fig. 4. Solidcurve:Ourpredicteddouble-helicityasymmetryAjet_LLforjetproduction at BNL–RHICinthenear-forwardrapidityregion,versus pT andthedatapoints

fromRef.[14].Dashedcurve:TheapproximateexpressionEq.(9)withk=1/7.

contributions versus thelower limit ofthe integral xmin, for

dif-ferent Q2 _{values. For increasing Q}2 _{they slowly saturate the}

sum rule, allowing a decreasing positive contribution to the or-bital angular momentum, which definitely must change sign for Q2

=

1000 GeV2.

Finally, we would like to test our new gluon helicity dis-tribution in a pure hadronic reaction. In a very recent pa-per, the STAR Collaboration at BNL–RHIC has reported the ob-servation, in one-jet inclusive production, of a non-vanishing positive double-helicity asymmetry Ajet_LL for 5

≤

pT

≤

30 GeV, in the near-forward rapidity region [14]. We show in Fig. 4 our prediction2 _{compared with these high-statistics data points}

2 _{WearegratefultoProf.W.Vogelsangforprovidinguswiththecodetomake}

and the agreement is very reasonable. There is a simple way to understand the trend of this double-helicity asymmetry. In this kinematic region, where the jet has a pseudo-rapidity

η

close to zero and a moderate pT, the dominant subprocess is uG

→

uG, so we can write the following approximate expres-sion Ajet_LL

=

k



G

(

xT

)

G

(

xT

)

·



u

(

xT

)

u

(

xT

)

,

(9)

where xT

=

2pT

/

√

s and k is a normalization factor such that 0

≤

k

≤

1. It exhibits the strong correlation of Ajet_LL on the sign and magnitude of



G and the validity of this approximation is clearlyshowninFig. 4.

This new STAR data has been used recently by the DSSV Collaboration [15] to perform a new global polarized fit which leads them to extract also a rather large positive gluon helic-ity distribution (see also the recent NNPDF Collaboration paper [16]).

Althoughwe cannot yet firmly claim the discovery of a large positive gluon helicity distribution, giving a significant contribu-tion to the proton spin, these new results are strongly suggest-ing that we may have reached a benchmark in our knowledge ofthenucleonstructure.Otherindependentprocessessensitiveto



G

(

x

,

Q2

)

mustbeinvestigatedandinparticularin pp collisions,

the di-jetproduction atforward rapidity isnow strongly consid-eredatBNL–RHIC.

References

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