An Analysis of Momentum Fractions of Quarks and Gluons in a Model of Proton

Abstract

Momentum sum rule can be used as an inequality to estimate the lower and upper bounds of the momentum fractions of quarks and gluons in a model of proton valid in a limited x range. We compute such bounds in a self-similarity based model of proton structure function valid in the range 6.2 × 10 −7 ≤ x ≤ 10 −2 . The results conform to the asymptotic QCD expectations.

Momentum fractions carried by quarks and gluons in models of proton structure functions at small x

“…we evaluate I 1 and I 2 numerically and then calculate x q by using Eqn (25), for a few representative values of Q 2 (GeV 2 ). For the same values of Q 2 , x q is calculated numerically by using Eqn (22). In Eqn (22), e i is the fractional electric charges of quarks and anti quarks.…”

“…For the same values of Q 2 , x q is calculated numerically by using Eqn (22). In Eqn (22), e i is the fractional electric charges of quarks and anti quarks. If we assume their flavored dependence and take number of flavors n f = 4, we obtain…”

“…However, the phenomenological analysis has one inherent limitation: Due to the negative value of D 3 , Eqn(6) develops a singularity at x 0 ∽ 0.019 [21,22], as it satisfies the condition 1…”

“…In Ref [22], only the 1st term of the infinite series is taken into account without taking into account the convergence property and their Q 2 -dependence. Below, we address to this point.…”

“…The main reason behind the work of Ref [22] is that it is more reasonable to study the model in the phenomenologically allowed range of x than going beyond it, as the model [14] has a singularity at x ∼ 0.019, outside the range of validity.…”