Claude Bourrely

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Statistical Polarized Parton Distributions at Large-x
Claude Bourrely
To cite this version:
Claude Bourrely. Statistical Polarized Parton Distributions at Large-x. Structure of the Nucleon at
Large Bjorken x, American Institute of Physics Conference Proceedinds, 2005, Marseille, France. vol.
747, p. 57-62. ￿hal-00128478￿
Statistical Polarized Parton Distributions at
Large-x1
Claude BOURRELY
Centre de Physique Théorique (UMR 6207)2,
CNRS-Luminy case 907,
13288 Marseille cedex 9, France
Abstract. We present a new set of polarized parton distributions constructed in a statistical physical
picture of the nucleon. The chiral peoperties of QCD lead to precise relations between quarks and
antiquarks distributions and the importance of the Pauli exclusion principle is also emphasized. The
model involves a small number of free parameters which are determined by a global next-to-leading
order QCD analysis of unpolarized and polarized deep-inelastic scattering data. We discuss the
implications of experimental data on the structure of polarized distributions.
INTRODUCTION
An experimental study of deep-inelastic scattering (DIS) of leptons on hadrons during
the last twenty years has produced measurements of cross section and structure func-
tions over a large kinematic domain, and as a consequence a better knowledge of the
partons distributions functions (PDF) in the unpolarized an polarized cases. Although,
we have no theory to compute the PDF, a large variety of phenomlogical functions has
been proposed in the literature [1], and their success in describing experimental data has
provided a better explanation of the hadrons structure and atest of pQCD.
The aboveclassicPDF are generally parametrized as a polynomial in the Bjorken
variablex at an input scaleQ0, in order to introduce more physical aspects in the de-
scription of PDF, we propose a new statistical approach where the nucleon is viewed
as a gas of massless partons (quarks, antiquarks, gluons) inequilibrium at a given tem-
perature in a finite size volume. Moreover, in contrast with ot er parametrizations, we
consider from the beginning that the fundamental entities ar polarized parton distribu-
tions defined by their helicities. In the process of buildingour PDF, we take into account
experimental informations well established by now. For insta ce, data exhibit a clear
evidence for a flavor-asymmetric light sea,i.e., d̄ > ū, which can be understood in terms
of the Pauli exclusion principle, based on the fact that the proton contains twou quarks
and only oned quark. The signs of the polarized light quarks distributions are essentially
well established,∆u> 0 and∆d < 0, while we just begin to uncover a flavor asymmetry,
1 Talk presented at HIX2004, The 2nd International Workshop on the Structure of the Nucleon at Large
Bjorken x, Marseille, 26-28 July 2004.
2 UMR 6207 is Unité Mixte de Recherche du CNRS and of Universités Aix-Marseille I and Aix-Marseille
II and of Université du Sud Toulon-Var, laboratoire affilié àla FRUMAM.
for the polarized light sea, namely∆ū 6= ∆d̄.
CONSTRUCTION OF THE STATISTICAL PDF
Our motivation is to use the statistical approach to build up: qi ,∆qi, q̄i,∆q̄i ,G and∆G,
by means of a very small number of free parameters. We proposea imple de-
scription of the parton distributionsp(x), at an input energy scaleQ20, proportional
to 1/
[
exp[(x−X0p)/x̄]±1
]
, theplussign for quarks and antiquarks, corresponds to a
Fermi-Dirac distribution and theminussign for gluons, corresponds to a Bose-Einstein
distribution. Here,X0p is a constant which plays the role of thethermodynamical
potentialof the partonp and x̄ is theuniversal temperature, which is supposed to be
the same for all partons. Since quarks carry a spin-1/2, it isnatural to consider that the
basic distributions areq±i (x), corresponding to a quark of flavori and helicity parallel
or antiparallel to the nucleon helicity. Clearly one has :qi = q+i +q
−
i and∆qi = q
+
i −q−i
and similarly for antiquarks and gluons.
From the chiral structure of QCD, we have two important propeties which allow to
relate quark and antiquark distributions and to restrict the gluon distribution :
• The potential of a quarkqhi of helicity h is opposite to the potential of the corre-
sponding antiquark ¯q−hi of helicity -h,
Xh0q = −X−h0q̄ . (1)
• The potential of the gluonG is zero, X0G = 0.
From well established features of theu andd quark distributions extracted from DIS
data, we anticipate some simple relations between the potentials:
• u(x) dominates overd(x), therefore one can expect:X+0u+X
−
0u > X
+
0d +X
−
0d
• ∆u(x) > 0, thereforeX+0u > X
−
0u
• ∆d(x) < 0, thereforeX−0d > X
+
0d .
So we expectX+0u to be the largest thermodynamical potential andX
+
0d the smallest one.
In fact, we have the following orderingX+0u > X
−
0d ∼ X
−
0u > X
+
0d, these inequalities are
consistent with the previous determinations of the potentials [2]. By using Eq. (1), this
ordering leads immediately to some important consequencesfor antiquarks, namely,
d̄(x) > ū(x), the flavor symmetry breaking which also follows from the Pauli exclusion
principle, as recalled above. This was already confirmed by the violation of the Gottfried
sum rule [3, 4]. Notice that since ¯u+(x)∼ d̄+(x), we have∆ū(x)−∆d̄(x)∼ d̄(x)− ū(x),
so the flavor symmetry breaking is almost the same for unpolarized and polarized
distributions.
Let us now complete the description of our parametrization.The smallx region is
characterized by a rapid rise of the distributions asx→ 0, so they should be dominated by
a universal diffractive term, flavor and helicity independet, coming from the Pomeron
universality. Therefore we must add a term of the formÃxb̃/[exp(x/x̄)+1], whereb̃< 0
andÃ is a normalization constant. In summary, for the light quarks q = u, d of helicity
h = ±, at the input energy scaleQ20 = 4GeV2, we take
xqh(x,Q20) =
AXh0qx
b
exp[(x−Xh0q)/x̄]+1
+
Ãxb̃
exp(x/x̄)+1
, (2)
and similarly for the light antiquarks
xq̄h(x,Q20) =
Ā(X−h0q )
−1x2b
exp[(x+X−h0q )/x̄]+1
+
Ãxb̃
exp(x/x̄)+1
. (3)
For the strange quarks and antiquarks, we take the particular hoice
xs(x,Q20) = xs̄(x,Q
2
0) =
1
4
[xū(x,Q20)+xd̄(x,Q
2
0)] , (4)
x∆s(x,Q20) = x∆s̄(x,Q
2
0) =
1
3
[x∆d̄(x,Q20)−x∆ū(x,Q20)] . (5)
This choice gives rise to a large negative∆s(x,Q20). The charm quarksc , both unpolar-
ized and polarized, are set to zero atQ20 = 4GeV
2. Finally concerning the gluon distri-
bution, as indicated above, we use a Bose-Einstein expression given by
xG(x,Q20) =
AGxb̃+1
exp(x/x̄)−1 , (6)
with a vanishing potential and the universal temperature ¯x. For the sake of completeness,
we also need to specify the polarized gluon distribution where we take the particular
choice,x∆G(x,Q20) = 0.
To summarize our parametrization involves a total ofeightfree parameters
x̄ ,X+0u ,X
−
0u ,X
−
0d ,X
+
0d ,b , b̃ andÃ .
These parameters, are determined at NLO by a fitting procedure described in Ref. [5, 6].
We have defined the PDF in terms of Fermi functions at the inputscaleQ20, does
this property still persist at any givenQ2. We have proven in Ref. [7] that the statistical
PDF can be approximated by a linear combination of Fermi functio s which depend
explicitely onx andQ2. In the next section we will concentrate on the large x results
obtained at NLO with the statistical PDF.
EXPERIMENTAL TESTS FOR (UN)-POLARIZED DIS
We first considerµ p andep DIS for which several experiments have yielded a large
number of data points on the structure functionF p2 (x,Q
2). In Fig. 1 (left) we compare our
calculations with measurements made by NMC, BCDMS, ZEUS andH1 collaborations,
FIGURE 1. Left, FP2 for fixedQ
2 as a function ofx. Right,xFνN3 as a function ofQ
2 for fixedx bin .
for fixed Q2 and largex values. We have also a quite good description of the neutron
structure functionFn2 (x,Q
2) data [5]. The high statisticνN DIS data from CCFR/NuTeV
allows to extract thexFνN3 (x,Q
2) structure function, which is successfully compared to
our results on Fig. 1 (right), we have also a very good description of the ZEUS and H1
data in the range 1500≤ Q2 ≤ 3105 [8, 9].
Since our approach is based on the direct construction of quarks and antiquarks PDF
of a given helicity, we immediately obtained∆qi and∆q̄i for each flavor. We display in
Fig. 2 (left) the polarized structure functionsgp,n,d1 at largex, from different experiments
on proton, deuterium and helium targets, evolved at a fixed value Q2 = 5GeV2. Thex
dependence is in fair agreement with our results. We notice,in particular, forgn1, the
perfect agreement with the 3 very precise data points recently measured at Jefferson Lab
(Jlab-E99).
In Fig. 2 (right) the asymmetriesAp,n1 are displayed with the world data atQ
2 = 4GeV2
and restricted tox > 0.1. An interesting feature is the behavior of the asymmetries
whenx → 1, in our model we find thatAp,n1 < 1 due to the fact that∆u/u = 0.77 and
∆d/d = −0.46. For theAn1 asymmetry measured by JLab-E99 atx = 0.33,0.47,0.6,
our model agrees with their data at the two extreme points, but at x = 0.47 its value is
above by 1σ . We have also computed thegp,n2 asymmetries using the Wandzura-Wilczek
relation, a comparison with data from SLAC and Jefferson Lab. show a reasonable
agreement owing to the experimental errors.
Recently, HERMES collaboration has obtained a set of polarized PDF atQ2 =
2.5GeV2, (see L. De Nardo, these proceedings) a comparison with our results is dis-
played in Fig. 3 (left), the agreement is satisfactory within t e experimental errors.
Let us comme back to the important question of the flavor asymmetry of the light
antiquarks. Our determination of ¯u(x,Q2) and d̄(x,Q2) is perfectly consistent with the
violation of the Gottfired sum rule, for which we foundIG = 0.2493 forQ2 = 4GeV2.
FIGURE 2. gp,n,d1 andA
p,n
1 for fixedQ
2 as a function ofx.
FIGURE 3. Left, comparison between polarized PDF and Hermes data, Right, x(∆ū−∆d̄) compared
with Hermes data atQ2 = 2.5GeV2 solid line andQ2 = 54GeV2 dashed line. .
Nervertheless there remains an open problem with thex distribution of the ratiod̄/ū
for x > 0.2, in connection with the E866/NuSea collaboration. They have found at
Q2 = 54GeV2 that d̄/ū is decreasing abovex > 0.2 while in our model this ratio
is, according to the Pauli principle, above 1 for any value ofx. One possibility to
clarify the situation is to measure the ratio of the unpolarized cross sections for the
production ofW± in pp collisions, which will directly probe the behavior of̄d/ū
ratio. As an illustration, we have computed this ratio usingMRST2002 PDF, where
d̄/ū < 1 for x > 0.2, and compare with our result, a significant gap aty = 0 exists. An
other test is to consider the polarized differencex[∆ū(x)−∆d̄(x)], taking into account
the properties of the polarized PDF in our model, we can show that if d̄/ū > 1 for
x > 0.2, the previous difference is positive, in the opposite caseit b comes negative.
The HERMES collaboration has released some values of this difference shown in Fig. 3
(right), however, no definite conclusion can be drawn due thelarge errors, nevertheless,
abovex > 0.1, it seems that the trend of data favor positive values. In the figure, the
solid curve corresponds toQ2 = 2.5GeV2, the dashed curve toQ2 = 54GeV2, showing a
small variation. An extended measurement is scheduled by the E906 collaboration (see
A. Towell, these proceedings).
Let us mention an other way to obtain a good flavor separation by measuring the
parity-violating asymmetry inW± production at RHIC-BNL. Calculations in the statis-
tical model at
√
s= 500GeV near a rapidity= +1 show thatAPVL (W
+)∼∆u/u and that
APVL (W
−)∼ ∆d/d (evaluated atx= 0.435). Similarly, neary=−1,APVL (W+)∼−∆d̄/d̄
andAPVL (W
−) ∼−∆ū/ū (see Ref. [5] for more details).
We have presented a few samples of the predictions obtained with the statistical
model, a full set of results is available in Ref. [9].
REFERENCES
1. A detailed description of PDF models : CETQ, GRV, MRS and others, can be found inhttp:
//durpdg.dur.ac.uk/hepdata/pdf.html.
2. C. Bourrely, F. Buccella, G. Miele, G. Migliore, J. Sofferand V. Tibullo,Z. Phys.C 62, 431 (1994);
C. Bourrely and J. Soffer,Phys. Rev.D 51, 2108 (1995); C. Bourrely and J. Soffer,Nucl. Phys.B
445, 341 (1995).
3. K. Gottfried,Phys. Rev. Lett.18, 1154 (1967).
4. New Muon Collaboration, M. Arneodoet al., Phys. Rev.D 50, R1 (1994) and references therein.
5. C. Bourrely, F. Buccella and J. Soffer,Eur. Phys. J.C 23, 487 (2002).
6. C. Bourrely, F. Buccella and J. Soffer,Mod. Phys. Lett.18, 771 (2003)
7. C. Bourrely,Numerical approximation of statistical parton densities functions.http://www.
cpt.univ-mrs.fr/~bourrely/research/bbs-dir/pdflib/partondf.pdf (inter-
nal report).
8. C. Bourrely, F. Buccella and J. Soffer, in preparation
9. C. Bourrely, F. Buccella and J. Soffer,A statistical approach for unpolarized and polarized
parton distributions and fragmentations : a comparison with experiments.http://www.cpt.
univ-mrs.fr/~bourrely/research/bbs-dir/pdflib/gallery.pdf (internal re-
port).


Statistical Polarized Parton Distributions at Large-x

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Statistical Polarized Parton Distributions at Large-x

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