Quark-Hadron Duality in Structure Functions

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by  A. Fantoni

EPJ manuscript No.
(will be inserted by the editor)
Quark-Hadron Duality in Structure Functions
Alessandra Fantoni1, Nicola Bianchi1 and Simonetta Liuti2
1
Laboratori Nazionali di Frascati dell’INFN,Via E. Fermi40, 00044 Frascati (RM), Italy
5
2
University of Virginia, Charlottesville, Virginia 22901, USA
0
0
2 Received: date/ Revised version: date

n
a Abstract. Thequark-hadrondualityisstudiedinasystematicwayforpolarizedandunpolarizedstructure
J functions, by taking into account all the available data in the resonance region. In both cases, a precise
4 perturbative QCD based analysis to the integrals of the structure functions in the resonance region has
1 beendone:nonperturbativecontributionshavebeendisentangledandthehighertwistcontributionshave
2
been evaluated. A different behavior for the unpolarized and polarized structure functions at low Q has

1 been found.
v
6 PACS. PACS-key quark-hadronduality – PACS-key structure functions
2
1
1
1 Introduction ranges,extendinge.g.oversingleresonances,onecanrefer
0
to local duality.
5
0 Understanding the structure and interaction of hadron in Although the duality between quark and hadron de-
/ terms of the quark and gluon degrees of freedom of QCD scriptionsis, in principle, formally exact,how this reveals
h
itself specifically indifferent physicalprocessesandunder
p is one of the unsolved problems of the Standard Model
different kinematical conditions is the key to understand
- of nuclear and particle physics. At present it’s not possi-
p theconsequencesofQCDforhadronicstructure.Thephe-
ble todescribe the physicsofhadronsdirectly fromQCD,
e nomenon of duality is quite general in nature and can be
however it is known that it should just be a matter of
h studied in a variety of processes, such as DIS, e+e− an-
: convenience the choice of describing a process in terms of
v quark-gluonorhadronicdegreesoffreedom.This concept nihilation into hadrons, and hadron-hadron collisions, or
i semi-leptonic decays of heavy quarks. With the advent of
X iscalledquark-hadronduality.Athighenergies,wherethe
both more detailed studies of soft scales and confinement
interactionsbetweenquarksandgluonsbecome weakand
r [2], and higher precision measurements covering a wide
a quark can be considered asymptotically free, an efficient
range of reactions, it is now becoming possible to investi-
description of phenomena is possible in terms of quarks.
gate the role of duality in QCD as a subject per se.
At low energies, where the effects of confinement become
large, it is more efficient to work in terms of collective
degrees of freedom, the physical mesons and baryons. In
these terms, it’s clear that the duality between the quark 2 Kinematical variables
and hadron descriptions reflects the relationship between
confinementandasymptoticfreedom,andisintimatelyre- Besides the scaling variable x, other variables have been
latedtothenatureofthetransitionfromnon-perturbative used in the literature to study duality and a number of
(low energy) to perturbative QCD (high energy). parametrizations based on these variables have been pro-
The concept of duality was introduced for the first posed that reproduce in an effective way some of the cor-
timebyBloomandGilman[1]indeepinelasticscattering rections to the perturbative QCD calculations. The most
′ ′ ′
(DIS). They noticed an equivalence between the smooth extensively used variables are: x = 1/ω , where ω =
x dependence of the inclusive structure function at large 1/x+M2/Q2,originallyintroducedbyBloomandGilman
Q2 and the average over W2 of the nucleon resonances. in order to obtain a better agreement between DIS and
Furthermore, this equivalence appeared to hold for each theresonanceregion;ξ =2x/(1+(1+4x2M2/Q2)1/2)[3],
resonance,overrestrictedregionsinW.Basedonthis ob- originallyintroducedtotakeintoaccountthe targetmass
servations, one can refer to global duality if the average, effects; xw =Q2+B/(Q2+W2−M2+A), A and B be-
defined e.g. as the integral of the structure functions, is ingfittedparameters,usedinRefs.[4,5].Theseadditional
takenoverthe whole resonanceregion1≤W2 ≤4GeV2. variables include a Q2 dependence that phenomenologi-
If, however, the averaging is performed over smaller W2 callyabsorbssomeofthescalingviolationsthatareimpor-
tant at low Q2. In Fig.1 their behavior vs. x is compared
Send offprint requests to: A. Fantoni fordifferentvaluesofQ2.Fromthefigureonecanseethat
2 Alessandra Fantoni,Nicola Bianchi, Simonetta Liuti: Quark-HadronDuality in StructureFunctions
thenumeratorisevaluatedusingtheexperimentaldatain
1.3
o the resonance region,while the one at the denominator is
ati x /x (Q2=20 GeV2) calculated from parametrizations that reproduce the DIS
R w
x /x (Q2=20 GeV2) behavior of the data at large Q2. The ratios have been
1.2
x’/x (Q2=20 GeV2) calculatedin unpolarizedandpolarizedcases.Ithas been
found [13] that quark-hadron duality has not been ful-
filled by using solely the parton distribution functions up
1.1
to NLO in both the unpolarized and polarized structure
functionsF andg .Howeveritwaspossibletoseeadiffer-
2 1
1 entbehaviorbetweenRunpol andRpol. Inthe unpolarized
case the ratio is increasing with Q2, but for the polarized
case the situation is different: while at low Q2 the ratio
0.9 is significantly below unity and shows a strong increase
with Q2, athigher Q2 the ratio derivedfromHERMESis
aboveunity and it appears to be weakly dependent of Q2
x /x (Q2= 2 GeV2)
0.8 w within error bars. The situation is different with the use
x /x (Q2= 2 GeV2) of the phenomenological fits to DIS data [14,15,5]. Since
x’/x (Q2= 2 GeV2) these phenomenologicalparametrizationsare obtained by
0.7 fitting deep-inelastic data evenin the lowQ2 region,they
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x can implicitly include non-perturbative effects and this
may explain the “observation of duality”. It becomes re-
′
Fig.1.Ratiobetweenthethreevariablesx,ξandxW defined ally important to understand the contribution of these
in thetext and theBjorken variable x as a function of x. non-perturbative effects.
′
bycalculatingF inξ andx,oneeffectively“rescales”the
2 4 Size of non-perturbative contributions
structurefunctiontolowervaluesofx,inaQ2 dependent
way, namely the rescaling is larger at lower Q2. In the In order to understand the nature of the remaining Q2
analysisreportedinthe following,thexvariablehasbeen
dependencethatcannotbedescribedbyNLOpQCDevo-
usedandallthecorrectionshavebeenappliedonebyone. lution, the effect of target mass corrections and large x
resummation have been studied. As mentioned early, the
analysis was performed by using x as an integration vari-
able, which avoids the ambiguities associated to the us-
3 Analysis of data age of other ad hoc kinematical variables. Standard in-
put parametrizationswith initial scaleQ2 =1 GeV2 have
o
A quantitative analysis of the Q2 dependence of quark- been used. Once both effects have been subtracted from
hadrondualityinbothpolarizedandunpolarizedepscat- thedata,andassumingthevalidityofthetwistexpansion,
tering is presented. All current data in the resonance re- one can interpret any remaining discrepancy of the ratio
gion, 1 ≤ W2 ≤ 4 GeV2, have been taken into account. from unity in terms of higher twist.
For the unpolarized case it has been used the data ob-
tained at Jefferson Lab in the range 0.3 ≤ Q2 ≤ 5 GeV2
4.1 Target Mass Corrections (TMC)
[6], and the data from SLAC ([7] and references therein)
for Q2 ≥ 4 GeV2. For the polarized case there are only
TMC are necessary to take into account the finite mass
few experimental data in the resonance region.One set is
of the initial nucleon. They are corrections to the leading
part of the E143 data [8], and it corresponds to Q2 =0.5
twist (LT) part of the unpolarized structure function F .
and 1.2 GeV2. Another set is the one from HERMES [9, 2
ForQ2largerthan≈1GeV2,TMCaretakenintoaccount
10] in the range 1.2≤Q2 ≤12 GeV2.
through the following expansion [16]:
In the polarized case the Q2 dependence originates
fwrohmichththeestSrLuActCurgelofubnacltaionnalFys1isa[n1d1]frpoamramtheetrriaztaitoioRn,hfoars F2TMC(x,Q2)= ξx2γ23F2∞(ξ,Q2)+6xQ32Mγ42 Z 1 dξξ′2′ (1)
been used. For the asymmetry A , it was used a power ξ
1
law fit to the world DIS data at x >0.3, A = x0.7, as whereF∞isthestructurefunctionintheabsenceofTMC.
1 2
already shown in Ref. [9]. This parametrization of A is Followingthe originalsuggestionof[17], onlyterms up to
1
constrainedto 1 atx=1andit does notdepend onQ2,as orderM2/Q2 arekeptintheexpansion,soastominimize
indicated by experimental data in this range [12]. ambiguities in the behavior of F at x≈1. Although this
2
The full procedure of the analysis is described in [13]. proceduredisregardspartonoff-shelleffectsthatmightbe
The quark-hadron duality in DIS is studied by consider- important in the resonance region [18,19], it’s important
ing the ratio of the integrals of the structure functions to emphasize its power expansion character, and setting
integrated in a defined x-range, corresponding to the W as a limiting condition for its validity, that the inequality
range of the resonance region. The structure function in x2M2/Q2 <1 be verified [20], Q2 ≃ 1.5 GeV2.
Alessandra Fantoni,Nicola Bianchi, Simonetta Liuti: Quark-HadronDuality in StructureFunctions 3
4.2 Large x Resummation (LxR) LT
I
/
LxR effects arise formally from terms containing powers res
I
ofln(1−z),z beingthelongitudinalvariableintheevolu- =
tion equations, that are present in the Wilson coefficient ol
p
functions C(z). The latter relate the parton distributions un 1
T
to e.g. the structure function F2, according to: LR
α 1
FNS(x,Q2)= s dzC (z)q (x/z,Q2), (2)
2 2π Z NS NS
Xq x
where it has been considered only the non-singlet (NS)
contributionto F since only valence quarksdistributions
2
are relevant in the present kinematics. The logarithmic Data/NLO
terms in C (z) become very large at large x, and they -1
NS 10 Data/NLO+TMC
need to be resummed to all orders in α . This can be ac-
S
Data/NLO+TMC+LxR
complished by noticing that the correct kinematical vari-
able that determines the phase space for the radiation of
gluonsatlargex,is W2 =Q2(1−z)/z,insteadofQ2 [21,
22]. As a result, the argumentof the strongcoupling con- 1 10
stant becomes z-depfendent: α (Q2) → α (Q2(1−z)/z) Q2 [GeV2]
S S
([23] and references therein). In this procedure, however,
an ambiguity is introduced, related to the need of con- LT
1
tinuing the value of α for low values of its argument,
S
i.e. for z very close to 1 [24]. The size of this ambiguity G / es
could be of the same order of the HT corrections. Nev- r1 1
ertheless, the present evaluation is largely free from this G=
problem because of the particular kinematical conditions ol
p
intheresonanceregion.Inthisanalysis,infact,thestruc- T
L
ture functions have been studied at fixed W2, in between R
T1d4.uh3≤riesDWsrioeslfe2itaneb≤tnalsen4tgfholGeereatoVhmfe2bn.eiogxCnuto-rinptaysceetriiqtnouunαrebnSoatf,ltyHaivnTQed2ctreoeirnnnmdtcrrseie.rbasusttehisoisnwspitrhocxe-. 10 -1 DDDaaatttaaa///NNNLLLOOO++TTMMCC+LxR
All the effects described in the present section are sum-
marized in Fig.2, where the ratio between the resonance
region and the ’DIS’ one is reported for the unpolarized
1 10
andforthepolarizedcase:thenumeratorisobtainedfrom Q2 [GeV2]
theexperimentaldata,whilethedenominatorincludesthe
different components of the present analysis, one by one. Fig. 2. Ratiobetweentheintegralsofthemeasured structure
functions and thecalculated ones plotted as a function of Q2.
ForunpolarizedscatteringithasbeenfoundthatTMC
Thecalculation includesonebyonetheeffectsofNLOpQCD
andLxRdiminish considerablythe spaceleft forHT con-
(squares), TMC (open circles) and LxR (triangles), The top
tributions.ThecontributionofTMCislargeatthelargest
panelreferstotheunpolarizedcase,whilethebottompanelto
values of Q2 because these correspondalso to large x val-
thepolarized one.
ues.Moreover,the effectofTMCis largerthanthe oneof
LxR.ThelowestdatapointatQ2 ≈0.4GeV2hasbeenex-
cludedfromtheanalysisbecauseofthehighuncertaintyin
both the pQCD calculation and the subtraction of TMC.
Similarly,inpolarizedscatteringtheinclusionofTMC
and LxR decreases the ratio RLT. However, in this case ing .Furthermore,afulltreatmentoftheQ2 dependence
pol 2
these effects are included almost completely within the wouldrequirebothamoreaccurateknowledgeoftheratio
errorbars.Clearly,dualityisstronglyviolatedatQ2 <1.7 R in the resonance region, and a simultaneous evaluation
GeV2. ofg .Thepresentmismatchbetweentheunpolarizedand
2
Thedifferencebetweenunpolarizedandpolarizedscat- polarized low Q2 behavior might indicate that factoriza-
tering atlowQ2 canbe attributed e.g. to unmeasured,so tion is broken differently for the two processes, and that
far,Q2 dependenteffects,bothintheasymmetry,A ,and the universality of quark descriptions no longer holds.
1
4 Alessandra Fantoni,Nicola Bianchi, Simonetta Liuti: Quark-HadronDuality in StructureFunctions
5 Size of Higher Twist (HT) corrections oftheHTintheresonanceregionisattributedtoTMC,in
[29]the contributionofTMCissmallandthesuppression
Thediscrepancyfromunityoftheratiosalreadypresented is dominated by LxR. In other words, the Q2 behavior in
is interpreted in terms of HTs. In Figs. 3,4 the question the DIS and resonance regions seems to be dominated by
of the size of the HT corrections is addressed explicitely. different effects.
For F , they are defined as:
2
H(x,Q2)=Q2 F2res(x,Q2)−F2LT (3) 6 Conclusions
(cid:0) (cid:1)
C (x)= H(x,Q2) ≡Q2F2res(x,Q2)−F2LT (4) Apreciseanddetailedanalysisofallpublishdatainreso-
HT FpQCD(x/Q2) FLT nanceregionhasbeenpresented,withtheaimofstudying
2 2
the quark-hadronduality in unpolarized and polarizedep
A similar expression is assumed for g . C is the so-
1 HT scattering. A pQCD NLO analysis including target mass
called factorized form obtained by assuming that the Q2
correctionsandlargex resummationeffects wasextended
dependences of the LT and of the HT parts are similar
to the integrals of both unpolarized and polarized struc-
and therefore they cancel out in the ratio. Although the
ture functions in the resonance region. Both effects have
anomalous dimensions of the HT part could in principle
been quantifiedand disentangledfor the firsttime. In the
be different, such a discrepancy has not been found so
present analysis [13], duality is satisfied if the pQCD cal-
far in accurate analyses of DIS data. The HT coefficient,
culations agree with the data, modulo higher twist con-
C has been evaluated for the three cases listed also in
HT tributionsconsistentwiththetwistexpansion.Adifferent
Fig.2,namelywithrespecttotheNLOpQCDcalculation,
behaviorforunpolarizedandpolarizedstructurefunctions
to NLO+TMC and to NLO+TMC+LxR. The values of
has been found, andduality seemstronglyviolatedin the
1+CHT/Q2 are plotted in Fig.3 (upper panel) as a func- lattercaseforQ2 <1.7GeV2 Thediscrepancyoftheratio
tion of the averagevalue of x for each spectrum. One can
from unity has been interpreted in terms of HTs. While
see that the NLO+TMC+LxR analysis yields very small
the size of the HT contributions is comparable in both
values for CHT in the whole range of x. Furthermore, the polarized and unpolarized scattering at larger x and Q2
extracted values are consistent with the ones obtained in values, at low x and Q2 large negative non-perturbative
Ref.[20]using adifferentmethod,howeverthe presentex-
contributions have been found only in the polarized case.
traction method gives more accurate results. Because of The present detailed extraction of both the Q2 depen-
the increasedprecisionof our analysis,we are able to dis-
dence and the HTs in the resonance region establishes a
entangle the different effects from both TMC and LxR.
backgroundforunderstandingthetransitionbetweenpar-
In the polarized case (lower panel) the HTs are small
tonic and hadronic degrees of freedom. In particular, it
within the given precision, for Q2 > 1.7 GeV2, but they
seems to be detecting a region where the twist expansion
appear to drop dramatically below zero for lower Q2 val-
breaks down, and at the same time, the data seem to be
ues. The inclusion of TMC and LxR renders these terms still far from the Q2 → 0 limit, where theoretical predic-
consistentwithzeroatthelargerQ2values,butitdoesnot
tions can be made [30]. More studies addressing this re-
modify substantiallytheir behavioratlowerQ2.Itshould
gionwill be pursued in the future, some of which are also
alsobe noticedthat,byparametrizingthe structurefunc-
mentionedin[20,31].Abreakdownofthetwistexpansion
tions as in Eq.(3), it is assuming that all of the non-
can be interpreted in terms of the dominance of multi-
perturbative (np) contributions are included in O(1/Q2)
parton configurations over single parton contributions in
twist-4 terms. These are in fact the largest type of devia-
the scattering process. In order to confirm this picture it
tions froma pQCDbehavior,to be expectedatQ2 values
will be necessary to both extend the studies of the twist
of the order of few GeV. Only from accurate analyses us- expansion, including the possible Q2 dependence of the
ing a larger number of more precise data, would one be HT coefficients [32,33] and terms of order O(1/Q4), and
able to distinguish among different np behaviors. From a
to perform duality studies in semi-inclusive experiments.
comparisonwithresultsofratioincludingphenomenologi-
calparametrizations[13]thatincludessomeoftheseextra
npbehaviorsit’spossibletosee,however,thattheireffect
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Alessandra Fantoni,Nicola Bianchi, Simonetta Liuti: Quark-HadronDuality in StructureFunctions 5
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L 2.5 2.5
res / I = Ipol2.225 NNNALLLlOOOek++hTTinMM (CCT+MLCx)R resLTI / I2.225 FBABiColxeteDkjdehM (iWnnS o 2(( T TnaMnoM aTCClyM))siCs)
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u
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R
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1.25 1.25
1 1
0.75 0.75
0.5 0.5
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T Fig. 4. Comparison of the HT coefficient displayed in Fig.3,
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1 NLO withotherextractions.Thetrianglesandsquaresarethesame
~
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