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claude.bourrely@cpt.univ-mrs.fr

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In the quantum statistical parton distribution approach proposed more than one decade ago to describe the parton structure, new properties are now understood; in particular the relation between quarks and antiquarks leads to very specific properties. The simultaneous treatment of unpolarized and polarized parton distribution functions (PDFs) allows a determination of thermodynamical potentials (the master parameters of the model) which drive their behavior and consequently the behavior of the structure functions. The existence of a possible relation between the gluon and a

The main objective is to build a quark structure where constitutive elements can be understood through their parameters, which are easily associated with the quark properties. The first point to clarify is the choice of a statistical model. Taking the example of a proton at rest, which contains three quarks, a statistical treatment seems not justified due to the low number of elements. However, when a proton is accelerated in a collider the energy increase has not only an effect on the mass but also to creates a large number of

The statistical approach is characterized by thermodynamical potentials whose values are the master parameters; they drive not only the shape of the parton distribution functions (PDFs) but are found to control some specific properties of the structure functions. In order to introduce the maximum constraints, we decided to work from the beginning with helicity components, which are the building blocks of both the polarized and unpolarized PDFs; a unique situation in the domain. It is clear that the polarized data set is much smaller than the unpolarized one, and also that it is limited to a medium energy region; however, the experiments at the BNL Relativistic Heavy Ion Collider (RHIC) enlarge somehow this domain, but a large gap remains to reach energies available at the CERN Large Hadron Collider (LHC).

The objective of the paper is to discuss the consequence of the statistical approach on the quark structure, because from a collection of results obtained through the years one gets now a better global view.

In Sec.

The PDFs are the essential elements to evaluate scattering processes in QCD. In the absence of a theory they are usually parametrized with polynomials

Let us now describe the main features of the statistical approach. The fermion distributions are given by the sum of two terms, a quasi Fermi-Dirac function of helicity

With the above definitions, the diffractive term is the same for flavors

In the numerator of the non-diffractive parts of Eq. ^{1}

The PDF QCD evolution was done at next-to-leading order in the

To adopt a coherent scheme, it is natural to suppose that antiquarks must also contain a Fermi part analogous to the quarks and, in addition, a diffractive part that is the same as in the quarks. All these constraints lead to a general expression,

The other parameters are given by

From the results obtained in Eq.

It is easy to show that quark helicity PDFs increase with the potential values, while for antiquarks helicity PDFs increase when the potentials decrease.

As a consequence of the above hierarchy of potentials

We move on to mention more significant consequences concerning the helicity distributions that follow from Eqs.

Another important earlier prediction concerns the deep inelastic scattering (DIS) asymmetries, more precisely

BS15

There are two more important consequences which relate unpolarized and helicity distributions. Namely, for quarks

Now coming back to all the components

Similarly

Similarly if one considers the ratio

The ratios

To conclude, we predict a monotonic increase of the ratio

In the BS15 version of the model

Inside a proton at high energy, besides the presence of

For this purpose two new formulas are defined for the unpolarized and polarized gluons, and they play the role of a toy model at the input scale. In these formulas

We remark that although the two formulas

Also, in the expressions

A fit at NLO of unpolarized and polarized DIS experimental data gives in the case of BS15 parametrization

In this new fit the potentials read

The new results for the potential values are close to the previous ones

For the normalization constants one obtains

In Fig.

Left: Comparison of

One can ask the question if the previous formulation can be applied to another model. In the domain of polarized PDFs, the de Florian-Sassot-Stratmann-Vogelsang (DSSV) model

A more serious constraint on the polarized gluon can be obtained from the double-spin asymmetry in jet production

Parameters describing NLO (

For the polarized gluon one obtains a normalization coefficient

Comparison of

Now our purpose is to show that the polarized gluon discussed above offers a good exploratory domain for the parton structure. Beginning with the statistical model, it was natural to associate to the gluon a Bose-Einstein expression such that

It turns out that the extra multiplicative function

The toy model defined above proceeds along the same line: a polarized gluon is built in terms of a composite made of known physical functions, namely the PDFs associated with their probability. Figure

Quark

To summarize the discussion on the different expressions so far defined in

It is known that

Finally, I would like to present a new test of the toy gluon distribution in a pure hadronic reaction and compute the double-helicity asymmetry

Left: Inclusive differential cross section for

The purpose of this work was to show that a statistical model offers a unique framework to build a quark structure whose properties are clearly defined by parameters related to physical quantities in the PDF expressions. The thermodynamical potentials which are the master parameters generate definite properties of the quark PDFs, as confirmed by experimental structure functions.

This prediction results from the following characteristic features of the statistical approach:

The PDF helicity components defined by Fermi-Dirac expressions are the building blocks of the unpolarized and polarized PDFs.

The thermodynamical potentials satisfy a hierarchy relation given by Eq.

The expressions obtained between quark and antiquarks allow one to relate the behaviors of the ratios

A toy model has been defined for the gluon in terms of unpolarized and polarized quarks distributions that produces equivalent results to the original gluon parametrizations but with only one free normalization parameter. In addition this toy model gives for the gluon made with basic fermion helicity components a relation between unpolarized and polarized gluon distributions, which was not the case in the original version of the model.

It is clear that the model is able to explain a large set of unpolarized and polarized experimental deep inelastic scattering data. Of course the predictions which can be made in view of future experiments depend on the present values of the parameters, so it is a challenge for the model to be confirmed by new experiments.

To conclude, the statistical approach not only provides numerical PDF values compatible with experimental data but also gives a coherent model of the quark structure at the fundamental level of helicity distributions.