New results in the quantum statistical approach to parton distributions

We will describe the quantum statistical approach to parton distributions allowing to obtain simultaneously the unpolarized distributions and the helicity distributions. We will present some recent results, in particular related to the nucleon spin structure in QCD. Future measurements are challenging to check the validity of this novel physical framework.


Basic procedure
Use a simple description of the PDF, at input scale Q 2 0 , proportional to [exp[(x − X 0p )/x] ± 1] −1 , plus sign for quarks and antiquarks, corresponds to a Fermi-Dirac distribution and minus sign for gluons, corresponds to a Bose-Einstein distribution. X 0p is a constant which plays the role of the thermodynamical potential of the parton p andx is the universal temperature, which is the same for all partons.
NOTE: x is indeed the natural variable, since all the sum rules we will use are expressed in terms of x

Basic procedure
Use a simple description of the PDF, at input scale Q 2 0 , proportional to [exp[(x − X 0p )/x] ± 1] −1 , plus sign for quarks and antiquarks, corresponds to a Fermi-Dirac distribution and minus sign for gluons, corresponds to a Bose-Einstein distribution. X 0p is a constant which plays the role of the thermodynamical potential of the parton p andx is the universal temperature, which is the same for all partons.
NOTE: x is indeed the natural variable, since all the sum rules we will use are expressed in terms of x From the chiral structure of QCD, we have two important properties, allowing to RELATE quark and antiquark distributions and to RESTRICT the gluon distribution: -Potential of a quark q h of helicity h is opposite to the potential of the corresponding antiquarkq −h of helicity -h, X h 0q = −X −h 0q . -Potential of the gluon G is zero, X 0G = 0.
New results from the quantum statistical approach to parton distributions -p. 5/37 The polarized PDF q ± (x, Q 2 0 ) at initial scale Q 2 0 For light quarks q = u, d of helicity h = ±, we take Note: q = q + + q − and ∆q = q + − q − (idem forq). Extra term is absent in ∆q and q v also in u − d orū −d.
The additional factors X h 0q and (X h 0q ) −1 are coming from TMD (see below) The polarized PDF q ± (x, Q 2 0 ) at initial scale Q 2 0 For light quarks q = u, d of helicity h = ±, we take Note: q = q + + q − and ∆q = q + − q − (idem forq). Extra term is absent in ∆q and q v also in u − d orū −d.
For gluons we use a Bose-Einstein expression given by xG(x, Q 2 0 ) = A G x b G exp(x/x)−1 , with a vanishing potential and the same temperaturex. For the polarized gluon distribution x∆G(x, Q 2 0 ) we take a similar expression at initial scale (positive for all x) New results from the quantum statistical approach to parton distributions -p. 6/37

Essential features from the DIS data
From well established features of u and d extracted from DIS data, we anticipate some simple relations between the potentials:

Essential features from the DIS data
From well established features of u and d extracted from DIS data, we anticipate some simple relations between the potentials: So we expect X + 0u to be the largest potential and X + 0d the smallest one. In fact, from our fit we have obtained the following ordering , in correspondance with ten free parameters for the light quark sector with some physical significance: * the four potentials X + 0u , X − 0u , X − 0d , X + 0d , * the universal temperaturex, * and b,b,b, b G ,Ã.

Very few free parameters
By performing a NLO QCD evolution of these PDF, we were able to obtain a good description of a large set of very precise data on F p 2 (x, Q 2 ), F n 2 (x, Q 2 ), xF νN 3 (x, Q 2 ) and g p,d,n 1 (x, Q 2 ), in correspondance with ten free parameters for the light quark sector with some physical significance: * the four potentials X + 0u , X − 0u , X − 0d , X + 0d , * the universal temperaturex, * and b,b,b, b G ,Ã.
We also have three additional parameters, A,Ā, A G , which are fixed by 3 normalization conditions . u −ū = 2, d −d = 1 and the momentum sum rule.
There are several additional parameters to describe the strange quark-antiquark sector and for the gluon polarization. We use the constraint s −s = 0. We note that potentials become smaller for heaviest quarks and since X − 0s > X + 0s , we will have ∆s < 0 like for d-quarks.

BNL-RHIC
Consider the processes − → p p → W ± + X → e ± + X, where the arrow denotes a longitudinally polarized proton and the outgoing e ± have been produced by the leptonic decay of the W ± boson. The helicity asymmetry is defined as Here σ h denotes the cross section where the initial proton has helicity h.
For W − production, the numerator of the asymmetry is found to be proportional to where θ is the polar angle of the electron in the c.m.s., with θ = 0 in the forward direction of the polarized parton. The denominator of the asymmetry has a similar form, with a plus sign between the two terms of the above expression. For W + production, the asymmetry is obtained by interchanging the quark flavors (u ↔ d).
We first show below the results of the calculations of the helicity asymmetries, versus the charged-lepton pseudo-rapidity and for a clear interpretation some explanations are required. At high negative η e , one has x 2 >> x 1 and θ >> π/2, so the first term above dominates and the asymmetry generated by the W − production is driven by ∆ū(x 1 )/ū(x 1 ), for medium values of x 1 . Similarly for high positive η e , the second term dominates and now the asymmetry is driven by −∆d(x 1 )/d(x 1 ), for large values of x 1 .

Transverse momentum dependence (TMD) of the PDF
How to introduce the TMD of the PDF ?
There are several possibilities Assume factorization and simple Gaussian behavior for the PDF and also for the fragmentation function A naive assumption which has no theoretical justification

(TMD) in the statistical approach
The parton distributions p i (x, k 2 T ) of momentum k T , must obey the momentum sum rule and also the transverse energy sum rule From the general method of statistical thermodynamics we are led to put p i (x, k 2 T ) in correspondance with the following expression where µ 2 is a parameter interpreted as the transverse temperature. So we have now the main ingredients for the extension to the TMD of the statistical PDF. We obtain in a natural way the Gaussian shape with NO x, k T factorization New results from the quantum statistical approach to parton distributions -p. 32/37

(TMD) in the statistical approach
The quantum statistics distributions for quarks and antiquarks read in this case because Y h 0q are the thermodynamical potentials chosen such that ln(1 + exp Y h 0q ) = kX h 0q , in order to recover the factors X h 0q , introduced earlier. Similarly forq we haveF (x) =Āx 2b−1 /kµ 2 . This determination of the 4 potentials Y h 0q can be achieved with the choice k = 3.05. Finally µ 2 will be determined by the transverse energy sum rule and one finds µ 2 = 0.198GeV 2 .