Abstract
The running of the QCD coupling is incorporated into the infrared evolution equations for the flavour structure function g1. The explicit expressions for g1 including the total resummation of the double-logarithmic contributions and accounting for the running coupling are obtained. We predict that asymptotically g1∼x−ΔS, with the intercept ΔS≈0.86, which is a factor of 2 larger than the non-singlet intercept ΔNS. The impact of the initial quark (δq) and gluon (δg) densities on the sign of g1 at x⪡1 is discussed and explicit expressions relating δq and δg are obtained.
Highlights
The flavour singlet structure function g1 has been the object of intensive theoretical investigations
First g1 was calculated in Refs. [1],[2], where the LO DGLAP evolution equations[2],[3] were used
As a matter of fact, these contributions become very important in the small x region and should be accounted for to all orders in αs. Such total resummation was done in Refs. [7] for the non-singlet component, g1NS, of g1 and in Ref. [8] for the singlet one. These calculations were done in the double-logarithmic approximation (DLA) and it was shown that g1 has the power-like (Regge) behaviour ∼ x−∆ when x → 0
Summary
The flavour singlet structure function g1 has been the object of intensive theoretical investigations. Despite this agreement, it is known that from a theoretical point of view the DGLAP equations are not supposed to work well at x ≪ 1 because the expressions for the anomalous dimensions and the coefficient functions in these equations account only for a finite part of the NLO (∼ 1/ωn+1), with n = 1, 2 (see [6]). As a matter of fact, these contributions become very important in the small x region and should be accounted for to all orders in αs Such total resummation was done in Refs. [9] are valid only when x is large (x ∼ 1), while on the contrary in the small-x region, in a general ladder rung i αs depends on the virtuality of the “horizontal” gluon, (ki − ki−1)2
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