Quantum statistical effects on parton distribution scaling behaviour

Starting from the old idea that Fermi statistics for quarks play a fundamental role to explain some features of hadron structure, we study the modification of the scaling behaviour of parton distributions due to quantum statistical effects. In particular, by following an interesting formal analogy which holds between the Altarelli-Parisi evolution equations, in leading-log approximation, and a set of Boltzmann equations, we generalize the evolution equations to take into account Pauli exclusion principle and gluon induced emission. Submitted to Phys. Lett. B ∗On leave of absence from the Dipartimento di Scienze Fisiche, Università di Napoli, Italy. 1 Deep inelastic experiments seem to be an inexhaustible source of information on the hadronic structure and continue to considerably improve our understanding of strong interaction dynamics. A measurement of proton and neutron F2(x) structure function performed by the NMC Collaboration at CERN [1] suggests a rather large SU(2) flavour breaking in the sea quark [2]. In particular they have obtained a determination, at very small x, for the difference IG(x) = ∫ 1 x dy y [F μp 2 (y)− F μn 2 (y)] = 1 3 ∫ 1 x dy [u(y) + ū(y)− d(y)− d̄(y)] , (1) finding IG(0.004) = 0.227± 0.007 . Thus, by extrapolating down to x = 0, they have estimated IG(0) = 0.240± 0.016 . (2) This result represents a relevant violation of the Gottfried sum rule [3], which would predict IG(0) = 1/3. Moreover, from (2) we get d̄− ū = ∫ 1 0 dx [d̄(x)− ū(x)] ∼ 0.14 . (3) However, the inequality d̄ > ū was already argued many years ago by Field and Feynman [4] on pure statistical basis. They suggested that in the proton the production from gluon decays of uū-pairs with respect to dd̄-pairs would be suppressed by Pauli principle because of the presence of two valence u quarks but of only one valence d quark. Assuming this point of view, the experimental result (2) naturally leads to the conclusion that quantum statistical effects play a sensible role in parton dynamics and that, in particular, parton distribution functions are affected by them. In this picture one may also easily account for the known dominance at high x of u-quarks over dquarks, whose characteristic signature is the fast decreasing of the ratio F n 2 (x)/F p 2 (x) in this regime. In recent papers [5] this idea has been developed, succeeding in making reasonable assumptions for various polarized parton distributions in terms of unpolarized ones, explaining the observed violation of Ellis-Jaffe sum rule [6], and giving a possible solution to the spin crisis problem [7]. Finally, using Fermi-Dirac or Bose-Einstein inspired form for parton distribution functions, a rather good agreement has been obtained with the experimental data on structure functions [8]. The aim of this letter is to study the role of quantum statistical effects, namely Pauli exclusion principle and induced gluon emission, in the Q-evolution of parton distributions. We will show that there is a quite close analogy between the well-known 2 Altarelli-Parisi (A-P) evolution equations [9], in leading-log approximation, and a set of Boltzmann equations written for a dilute system of particles. This analogy will guide us in finding the generalized scaling when also quantum statistics have been taken into account. As well-known, the logarithmic dependence on Q of the parton distribution momenta, predicted in the framework of perturbative QCD, has a simple and beautiful interpretation in terms of evolution equations for parton distribution functions. At leading-log level, the A-P equations can be written in the following way d dt pA(x, t) = αs(t) 2π ∫ 1 x dy y ∑