Saturation effects in low-x DIS structure functions and related hadronic total cross sections

High-energy nucleon total cross sections are related to low-x DIS structure functions by using the additive quark model.


Abstract.
High-energy nucleon total cross sections are related to low-x DIS structure functions by using the additive quark model.
In the additive quark model, the hadron-hadron total cross section can be written as a product of the cross sections of the constituents, σ qq [1,2], e.g.
where n V is the number of valence quarks and n S (s) is that of sea quarks, their number increasing with energy. It was suggested in Refs. [1,2] that the increasing number of sea quarks is related to the Bjorken scaling-violating contribution to the deep inelastic lepton-hadron structure function (DIS SF), namely to the momentum fraction of the relevant quarks given by the integral over the DIS structure function F 2 (x, Q 2 ). In Ref. [1] a simple model for the DIS structure function, known at those times, was used, resulting in the following expression for the total cross section, compatible with the data where σ qq is a free parameter, Q 2 0 was fitted to the DIS data, and n V = 3 . In Ref. [2] the DIS SF was related to hadronic cross sections by means of finite-energy sum rules in Q 2 .
The number of quarks in a reaction can be calculated from the SF by means of sum rules. see e.g. [3,4].
In Ref. [5] following ansatz for the small-x singlet part (labelled by the upper index S , 0) of the proton structure function, interpolating between the soft (VMD, Pomeron) and hard (GLAP evolution) regimes was proposed: with the "effective power" and At small and moderate values of Q 2 , the exponent Δ(Q 2 ) (3.2) may be interpreted as a Q 2dependent "effective Pomeron intercept", as shown in Fig. 1.
The function f (Q 2 ) has been introduced in order to provide for the transition from the Regge behaviour, where f (Q 2 ) = 1, to the asymptotic solution of the GLAP evolution equation, where f (Q 2 ) = 1/2. In Ref. [5] the above singlet SF was appended by a non-singlet part, important at large values of x. The parameters were fitted to the DIS data in a wide range of x and Q 2 .The values of the fitted parameters are: A = 0.1623, a = 0.2916 GeV 2 , γ 2 = 0.01936, Q 2 0 = 0.1887 GeV 2 , Q 2 1 = 916.1 GeV 2 ; x 0 = 1, = 0.08, γ 1 + 2.4 were fixed (by QCD-related arguments). The resulting fits and more details can be found in Ref. [5].
The proton-proton total cross section is cast by integrating Eqs. performed numerically. The result is in reasonable agreement with the data on pp total cross sections, including those from the LHC.
The operator-product expansion beyond leading twist has diagrams in which two, three or four gluons may be exchanged in the t-channel such that these gluons may be viewed as recombining. This recombination could lead to gluon saturation. The colour-dipole framework also inspired a phenomenological model of saturation by Golec-Biernat-Wüsthoff (GBW), in which the onset of saturation is characterised as the transition from a "soft" to a "hard" scattering regime. This occurs along a "critical line" in the x B j , Q 2 plane.
Recently new results on low-x DIS parton distrubutions (PDF and SF) have appeared [6,7]; they show an intriguing change of regime towards smallest values of x (Fig. 2) -possible saturation effect?
We intend to investigate its impact on the asymptotic behaviour of the total cross sections.