Table Of ContentTheJournal’snamewillbesetbythepublisher
DOI:willbesetbythepublisher
(cid:13)c Ownedbytheauthors,publishedbyEDPSciences,2016
6
1
0
2
n Progress on nuclear modifications of structure functions
a
J
5 S.Kumano1,2
2
1KEKTheoryCenter,InstituteofParticleandNuclearStudies,KEK,
] 1-1,Ooho,Tsukuba,Ibaraki,305-0801,Japan
h
2J-PARCBranch,KEKTheoryCenter,InstituteofParticleandNuclearStudies,KEK,
p
andTheoryGroup,ParticleandNuclearPhysicsDivision,J-PARCCenter,
-
p 203-1,Shirakata,Tokai,Ibaraki,319-1106,Japan
e
h
[ Abstract. Wereportprogressonnuclearstructurefunctions,especiallyontheirnuclear
modifications and anew tensor structurefunction for thedeuteron. Tounderstand nu-
1
clearstructurefunctionsisanimportantsteptowarddescribingnucleiandQCDmatters
v
fromlowtohighdensitiesandfromlowtohighenergiesintermsoffundamentalquark
9
and gluon degrees of freedom beyond conventional hadron and nuclear physics. It is
9
also practicallyimportant for understanding new phenomena in high-energy heavy-ion
4
6 collisionsatRHICandLHC.Furthermore, sincesystematicerrorsofcurrent neutrino-
0 oscillationexperimentsaredominatedbyuncertaintiesofneutrino-nucleusinteractions,
. suchstudiesarevaluableforfindingnewphysicsbeyondcurrentframework.Next,anew
1
tensor-polarizedstructurefunctionb isdiscussedforthedeuteron.Therewasameasure-
0 1
mentbyHERMES;however,itsdataareinconsistentwiththeconventionalconvolution
6
1 estimatebased on thestandard deuteron model withD-stateadmixture. Thisfact sug-
: geststhatanew hadronic phenomenon should existinthetensor-polarized deuteron at
v
highenergies,anditwillbeexperimentallyinvestigatedatJLabfromtheendof2010’s.
i
X
r
a
1 Introduction
Nuclearstructurefunctionsaredifferentfromcorrespondingonesofthenucleon. Suchnuclearmod-
ificationshavebeenmeasuredfromrelativelysmallx(∼0.005)tolargex(∼0.7)mainlyincharged-
leptondeepinelasticscattering(DIS).Physicsmechanismsforthenucleareffectsaredifferentineach
xregion[1]. Atsmall x,avirtualphotonbecomesqq¯ (orvectormesons),whichinteractswithanu-
cleusbystronginteractions. Itleadstonuclearshadowingphenomenaduetomultipleinteractionsin
anucleus. Atmedium x,therearenegativecontributionsfromnuclearbindingandpossibleinternal
nucleonmodifications. Themodificationsarepositiveatlarge x duetonucleon’sFermimotionina
nucleus. Althoughthe major nuclear-modificationmechanismsare known, we cannotcalculate the
structurefunctionsprecisely,typicallylessthan10%accuracy,bythetheoreticalmodels. Foractual
applicationto heavy-ionphysicsand neutrinoreactions, it is practically not appropriateto reply on
suchmodels. Forexample,neutrino-nucleuscrosssectionsneedtobecalculatedwithin5%accuracy
in future leptonic CP violation measurements [2]. Therefore, it is desirable to use global analysis
results as a modelfor the nuclear parton distribution functions(NPDFs) [3, 4], in the same way as
the globalanalysis PDFs are used for calculating cross sections at LHC to find physics beyondthe
standardmodel.
TheJournal’sname
TheNPDFsaredeterminedbyanalyzingworlddataonhigh-energynuclearreactions,including
charged-leptonDIS,neutrinoDIS,Drell-Yan,andhadronproductions.Currently,therearetwomajor
issuesintheNPDFstudies. Oneisthatgluonshadowingisnotdeterminedfromthecurrentmeasure-
ments[3,4],andtheotheristhatmodificationssuggestedbytheneutrinoDISdatacouldbedifferent
from the ones which are inferred from the charged-leptonDIS [5]. On the first point, the issue is
duepartlyto the fact thata lepton-nucleuscollider similar to HERA doesnotexistto observescal-
ingviolationatsmall x. However,thesituationcouldimprovebecauseLHCstartedproducingdata,
whicharesensitivetosmall-xphysics.Onthesecondpoint,somewhatconflictingresultsareobtained
amongdifferentanalysisgroups. Itisdesirabletoclarifythesituationbyanotherseriousanalysisto
discussthedetailsofanalysisconditionsanddatahandling.Inthisreport,wediscussthesituationof
thesestudiesinSec.2.
Asthesecondtopic,weexplaintensorstructurefunctionsofthedeuteronincharged-leptonDIS
withapolarizeddeuteron[6]. Thedeuteronstructureiswellknownatlowenergiesanditisdescribed
byaproton-neutronS-waveboundstatewithsmallD-waveadmixture.Sincethedeuteronisaspin-1
hadron, it has new polarized structure functions, b , b , b , and b in addition to the four structure
1 2 3 4
functions, F , F , g , and g , which exist for the spin-1/2 nucleon in the charged-leptonDIS. The
1 2 1 2
structurefunctionsb andb areleading-twistones,whicharerelatedbytheCallan-Grosslikerelation
1 2
2xb =b inthescalinglimit. Theb andb arehigher-twistones.
1 2 3 4
Thereareonlyafewb datameasuredbytheHERMEScollaboration[7].Althoughmuchaccurate
1
measurements are needed, there is already an interesting hint in the data toward a new discovery.
The most conventionalway to estimate b theoretically is to use a convolution model, where b is
1 1
calculatedbyanunpolarizedstructurefunctionofthenucleonwithnucleon-momentumdistributions
includingtheDwave. TheHERMESdataareanorder-of-magnitudelargerthanthisstandardmodel
predication.Itindicatesthatb cannotbeunderstoodbytheconventionalmodel,andanewhadronic
1
physicsshouldbe introducedfor interpretationof the HERMES data. Fortunately,an experimental
proposaltoJLabwasapprovedforb ,anditsmeasurementwillstartaroundtheyearof2019[8]. Itis
1
agoodopportunitytofindanewexotichadronicphenomenoninthesimplestnucleus,deuteron.The
deuterontensorstructureusedtoplayanimportantroleinlow-energynuclearphysics.Itisnowtime
tounderstandthetensorstructurein termsof quarkandgluondegreesof freedomwith newhadron
physics.WeexplainthiscurrentsituationasthesecondtopicinSec.3.
2 Nuclear partondistribution functions
Nuclearpartondistributionsaremodifiedfromtheonesforthenucleon. Thespaceislimitedforthis
article, so that we do not addressourselvesto physicsmechanisms. An interested reader may read
summary articles in Ref.[1]. We focus our discussions on the status of NPDFs and an application
to neutrinophysics. The NPDFs are determined by a globalanalysis of world data on high-energy
lepton-nucleus, proton-nucleus, and nucleus-nucleus reactions. The distributions are parametrized,
andoptimumparametersaredeterminedbyaχ2 analysis. Thereareafewgroupswhichinvestigate
theNPDFs. Therearetwotypesfortheparametrization. Oneistoobtainmodificationsfromtypical
nucleonic PDFs, and the other is to obtain the NPDFs directly. The latter one is a general way of
analysisonthesamefootingwiththenucleonicPDFanalysis,whereasitiseasiertoimposephysical
constrainsin the formerone becausethe determinedNPDFs could becomeunphysicaldistributions
due to the lack of data especially at extreme kinematical conditions. If we were to have sufficient
experimentaldata,bothresultsshouldagreewitheachother.
There are two major issues in the determination of NPDFs. First, nuclear modifications of the
gluondistributioncannotbefixedduetothelackofdatawhicharesensitivetothem. Second,some
6thInternationalConferenceonPhysicsOpportunitiesatanElecTron-IonCollider
inconsistencieswerepointedoutforthemodificationsbetweencharged-leptonandneutrinoscattering
processes[5],althoughsomeotheranalysisdonotshowsuchinconsistencyclearly.Inthefollowing,
wefirstexplainaconnectiontoneutrinophysicsanditispartiallyrelatedtothesecondissue.
PreciseNPDFsareneededforfindinganynewphysics
in high-energy nuclear reactions such as properties of Q2 (GeV2)
W2=4 GeV2
quark-gluonmattersin heavy-ioncollisionsatRHICand
LHC. There is another important application to neutrino
QE
physics.Thestatisticalerrorsinneutrino-oscillationmea-
surementsbecomesmaller andsmaller, buttheir system-
RES DIS
atic errors are still dominated by neutrino-nucleusinter-
actions, which is an obstacle fora new discovery. There 1
are three major kinematical regions as shown in Fig. 1:
quasi-elastic (QE), resonance (RES), and deep inelastic 0 2 ν (GeV)
scattering(DIS).Thereare notdefiniteboundariesin the
Figure 1. Kinematical regions of
sense that W2 and Q2 cut values depend on researchers.
neutrino-nucleus scattering. Here, QE,
However,a usualchoiceis the cutW2 ≥ 4 GeV2 for the
RES, and DIS indicate quasi-elastic, res-
DIS, and Q2 should be large enough, typically Q2 ≥ 1 onance,anddeepinelasticscattering.
GeV2. Thesethreeregionshavebeeninvestigatedbydif-
ferenttypesofphysicists. Inthecurrentneutrinoexperiments,exceptforthe ultra-highenergyIce-
Cube experiment, typical energies are from several hundred MeV to a few dozen GeV. Therefore,
all of these kinematicalcross sections should be precisely known for reducingthe systematical un-
certaintiesinneutrinooscillationmeasurements. ForfutureleptonicCPviolationmeasurements,the
crosssectionsneedtobeunderstoodwithinthe5%level.Aprojectisinprogresstoprovideacodeto
calculatethecrosssectioninanykinematicalrangebycombiningtheoriesofdifferentregionsatthe
J-PARCbranchoftheKEKtheorycenter[2]. Inthefollowing,wediscussonlytheDISstudiesand
suchaunificationeffortwillbeexplainedelsewhere.
Iftherewerenonuclearmodification,theNPDFsaresimpleadditionofprotonandneutroncon-
tributions. However,fromexperimentalmeasurements,nuclearmediumeffectsaretypically10-20%
dependingonanuclearsize. Suchmediumeffectsareparametrizedbythefunctionw inRef.[3]as
i
1
fA(x,Q2)=w(x,A,Z) Z fp(x,Q2)+Nfn(x,Q2) , (1)
i 0 i A i 0 i 0
h i
where p and n indicate the protonand neutron, A, Z, and N are mass number, atomic number, and
neutronnumber, i indicates a type of distribution (i = u , d , q¯, g), and Q2 is the initial Q2 scale.
v v 0
The functions w are expressed by a number of parameters, which are then determined by a global
i
analysis. The kinematicalrangeof x is0 < x < A fora nucleus. Therefore,thefunctionalformof
Eq.(1)cannotdescribetheregionx>1. However,thereisnoDISdatainsuchalarge-xregion,soit
isnotaseriousproblematthisstage. IntheHKNanalysis[3],thenuclearmodificationfunctionsare
expressedintermsoftheparameters,α,β,a,b,c,andd as
i i i i
1 a +bx+c x2+dx3
w(x,A,Z)=1+ 1− i i i i , (2)
i Aα! (1−x)β
at Q2 = 1 GeV2. Theparametersaredeterminedbyaglobalχ2 analysiswiththestandardDGLAP
0
evolutionequationincomparisonwiththeworlddata.
TheJournal’sname
Asanexample,nuclearmodificationsof40CaPDFs
1.2
andtheiruncertaintiesareshownforthenext-to-leading
Q2=1 GeV2
(NLO) order HKN parametrization [3] in Fig. 2 at 1.1
40Ca
Q2 =1GeV2. TherearesomedifferencesintheNPDFs wi(x) 1
fromothergroups[4]atlarge xespeciallyfortheanti-
0.9
quarkandgluondistributionsbecausetherearefewdata q
0.8 g uv
whichconstrainthem. Inaddition,largedifferencesex-
ist in the gluondistributionsat small x. The nucleonic 0.7
HKN-NLO
gluondistributionisconstrainedbythescalingviolation 0.6
0.001 0.01 x 0.1 1
ofF fromtheHERAep-colliderexperiments,whereas
2
there is no such a measurement for nuclei at small x. Figure 2. HKN nuclear modifications for
40CaatQ2=1GeV2.
However, the situation may improvein the near future
duetoLHCmeasurementswithheavyions. UsingtheobtainedNPDFstogetherwithDGLAPevo-
lutionequations,onecancalculatenuclearmodificationsofstructurefunctionsinneutrinoscattering.
Someinconsistencieswerereportedbetweenthecharged-leptonDISandneutrinoDISstructurefunc-
tions in Ref.[5]; however, the differencesare not clear in other groups [4]. Our analysis is now in
progressbyincludingneutrinodataforclarifyingthisissueasanindependenttrial.
3 New spin structurefunction b for deuteron
1
Therearefournewstructurefunctions,b ,b ,b ,andb ,forthespin-1hadronincharged-leptondeep
1 2 3 4
inelasticscattering.Theyaredefinedinthehadrontensoras[6]:
F ig ig
Wλfλi =−F gˆ + 2 pˆ pˆ + 1ǫ qλsσ+ 2 ǫ qλ(p·qsσ−s·qpσ)
µν 1 µν Mν µ ν ν µνλσ Mν2 µνλσ
1 1 1
−b r + b (s +t +u )+ b (s −u )+ b (s −t ), (3)
1 µν 2 µν µν µν 3 µν µν 4 µν µν
6 2 2
wherepandqarespin-1hadronandvirtual-photonmomenta,νistheenergytransfer,andr ,s ,t ,
µν µν µν
andu aredefinedbythepolarizationvectorofthespin-onehadronEµ inRef.[6]. Thespinvector
µν
sisexpressedbythepolarizationvectoras(s )µ =−iǫµναβE∗(λ )E (λ)p /Mwheretheinitialand
λfλi ν f α i β
finalpolarizationvectorsaredenotedasEµ(λ)andEµ(λ )withthespinstatesλ andλ ,andMisthe
i f i f
massofthespin-1hadron.Thecoefficientsofb ,b ,b andb aredefinedassymmetricunderµ↔ν
1 2 3 4
inEq.(3),andtheyvanishunderthespinaverage. Theyarealsodefinedsothatb andb aretwist-
1 2
two functions to satisfy the Callan-Gross type relation, 2xb = b . The b and b are higher-twist
1 2 3 4
structurefunctions. Thetwist-twostructurefunctionb isexpressedintermsofthetensor-polarized
1
partondistributionsδ qandδ q¯ as
T T
1 q+1+q−1
b (x,Q2)= e2 δ q(x,Q2)+δ q¯ (x,Q2) , δ q ≡q0− i i . (4)
1 2 i T i T i T i i 2
Xi h i
Here, i indicates the flavor of a quark, e is a quark charge, and qλ indicates an unpolarized-quark
i i
distribution in the hadronspin state λ. The tensor-polarizeddistribution is much differentfrom the
longitudinally-polarizedone∆q = q −q , where↑and↓indicatethepolarizationofaquarkalong
↑ ↓
thehadron-spindirection,inthesensethatitis“unpolarized"-quarkdistributioninatensor-polarized
spin-1hadron.
Thestructurefunctionb canbe calculatedby thestandardconvolutionpictureforthe deuteron
1
in terms of parton-momentumdistributions convolutedwith nucleon momentumdistributions. The
b is associated with tensorstructure ofthe deuteron,so thatD-wave admixtureshouldbe properly
1
considered for the convolution estimation. However, it is surprising to find an order-of-magnitude
6thInternationalConferenceonPhysicsOpportunitiesatanElecTron-IonCollider
differencebetweensuchastandard-modelestimateandexperimentaldataobtainedbytheHERMES
collaboration, although the data have large errors. There are other theoretical-model calculations;
however,wedonotstepintosuchtheoreticalmodelsinthisarticle,andwediscussprobabletensor-
polarizeddistributionsfromtheHERMESdata.
SuchaparametrizationwasstudiedinRef.[9]. Weconsiderthefollowingconstraintintheform
ofb sumrule[10],whichwasobtainedinapartonmodelanditissimilartotheGottfriedsum:
1
1
dxb (x)=0 + dx 4δ u¯(x)+δ d¯(x)+δ s¯(x) ,
Z 1 9Z T T T
h i
dx 1 2
[Fp(x)−Fn(x)]= + dx[u¯(x)−d¯(x)]. (5)
Z x 2 2 3 3Z
As the Gottfried-sum-ruleviolation created a field of flavor-symmetric light-antiquarkdistributions
(u¯ , d¯) [11], a finite b sum could indicate a finite tensor-polarized antiquark distribution, which
1
cannot be expected from ordinary theoretical ideas. For the time being as the first-step study, we
neglectsuchexoticcontributionsandconsiderthecondition dxb (x)=0forconstrainingthetensor
1
distributions. R
Asthefirsttrial,weconsiderthecasethatacertainfraction(δ w)ofunpolarizedPDFsistensor
T
polarized in the deuteron [9]: δ qD(x) = δ w(x)qD(x), δ q¯D(x) = α δ w(x)q¯D(x) for the tensor-
T iv T iv T i q¯ T i
polarized parton distributions. Here, we assume a common function δ w(x) for both quarks and
T
antiquarksexceptfor a differentoverallconstantα . Now, the problembecomeshow to determine
q¯
δ w(x). We assume that the valence-quarkdistributions should satisfy the sum dx(b ) = 0.
T 1 valence
Then,thefunctionδ w(x)shouldhaveanodeatleast,sowemaytaketheparamRetrizationδ w(x) =
T T
axb(1− x)c(x − x). Here, a, b, and c are the parametersdeterminedby the analysis, and the node
0
positionx isexpressedbytheotherparametersasx = dxxb+1(1−x)c(u +d )/ dxxb(1−x)c(u +
0 0 v v v
d ) duetothe sumrule. Sucha nodealso existsin b cRalculatedbythe convolutRionmodelwith the
v 1
D-stateadmixture.
Wetriedtwoscenariosdependingonwhetherornottheantiquarktensorpolarizationexists:
• Set1: Tensor-polarizedantiquarkdistributionsareterminated(α =0).
q¯
• Set2: Finitetensor-polarizedantiquarkdistributionsareallowed(α isaparameter).
q¯
The parameters are determined by a χ2 analysis in each case. Determined b and tensor-polarized
1
PDFsareshowninFigs. 3and4. Therearetwo curvesinFig. 3forthesets1and2. Itisobvious
fromthisfigure,withintherestrictionoftheusedparametrization,theHERMESdatacannotbeex-
plainedwithoutthe antiquarktensor polarization. Namely, χ2 is muchlarger for the set-1 analysis.
TheobtainedtensordistributionsareshowninFig. 4. Thedashedcurveindicatesthevalence-quark
distributionintheset1. Thesolidanddottedcurvesindicatethevalence-quarkandantiquarkdistri-
butionsintheset2. Thevalence-quarkdistributionisnegativeatlargex(>0.3)andthenitturnsinto
Figure3.Obtainedb andHERMESdata[9]. Figure4.Determinedtensor-polarizeddistributions[9].
1
TheJournal’sname
positiveatx≃0.2. Theantiquarkdistributionsareespeciallylargeatsmallx(<0.1).Sincethereare
onlyafewdatapointsatthisstage,therearelargeuncertaintiesforthedistributions.
Infuture,weneedtheoreticalandexperimentaleffortsonb physics.Onthetheoryside,weneed
1
tounderstandtheobtainedtensor-polarizeddistributionsinFig. 4byappropriatetheoreticalmodels.
The standard deuteronmodelwith the D-state admixturecannotexplain the distributionsbecause a
typicalconvolution-modelestimateisone-order-of-magnitudesmallerthanthevaluesinFig. 4. We
need to investigate more exotic hadronic mechanisms. At the same time, experimental efforts are
neededto measureb muchaccuratelybecausethere are onlya few data pointsas shownin Fig. 3
1
with the large errors. However, it is fortunate that the JLab b experiment was approved, and the
1
measurementwillstartinafewyears[8]. Ontheotherhand,therearepossibilitiesforinvestigating
the antiquarktensorpolarizationbypolarizedDrell-Yanprocesses, such as p+d~ → µ+µ− +X and
π+d~ → µ+µ− + X, at hadronfacilities, such as Fermilab-MI,RHIC, CERN-COMPASS, J-PARC,
GSI-FAIR,andU70[12]. Forexample,atensorspinsymmetryin pd~Drell-Yanisgivenby
e2q¯(x )δ q(x )
A (largex )≃ i i i 1 T i 2 , (6)
UQ0 F P e2q¯ (x )q(x )
i i i 1 i 2
so that δ q can be measured directly. It is similPar to the case that u¯/d¯ , 1 was clarified by the
T i
Drell-YanexperimentalthoughitwassuggestedbytheGottfried-sum-ruleviolation[11].
Acknowledgements
ThisworkwassupportedbyMinistryofEducation,Culture,Sports,ScienceandTechnology(MEXT)
KAKENHIGrantNo. 25105010.
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