A Statistical Treatment to the Evolution Equation for Quark Distributions

The evolution of polarized quark distribution functions is taken into account the gluon emission and absorption, quark pair production and annihilation processes and treated by a statistical method which provides quark distribution functions at small Q2. The evolution function of quark distributions is derived and solved by numerical methods. Then the Q2 dependent and αs dependent structure functions F2(x,Q 2) are obtained and the example of it, the proton structure function, is calculated, whose increase and decrease tendency with αs at different x is consistent with experiments. This is also an evidence that the statistical methods for the nucleon is reasonable. Following our statistical model proposed in [1, 2], we would like to extend our discussion to the Q dependence of the quark distribution functions, but only one part of the evolution equations in QCD. In the early discussion, we didn’t take into account the detailed processes involving gluons, for example, gluon emission and absorption by quarks. By a combination of effective field theory [1, 3, 4, 5] and statistical methods [2, 6, 7, 8, 9, 10, 11, 12], the quark distribution at small Q is obtained as follows [2], q(x) = 3MRx 2π ∫ M 2 Mx 2 1 exp( T ) + 1 dE E2 , (1) where, T,R,M are the nucleon’s temperature, radius and mass, μ is the quark’s potential. When Q is big, in the e-p deep inelastic scattering, the electron can probe many other detailed processes other than γq → q [13, 14, 15, 16, 17]. We assume that the dominant detailed process involves a photon and the total spin in the process is conserved. Therefore, for a quark with spin parallel to the total spin of the nucleon q ↑, we consider the following detailed processes: 1. gluon emission: γq ↑→ q ↑ g, 2. gluon absorption: q ↑ g → q ↑ γ, 3. pair production: γg → q ↑ q ↓, 4. pair annihilation: q ↑ q ↓→ γg, while for q ↓, they are: 1.gluon emission: γq ↓→ q ↓ g,