Evolution equations for truncated moments of the parton distributions

Abstract

We derive evolution equations for the truncated Mellin moments of the parton distributions. We find that the equations have the same form as those for the partons themselves. The modified splitting function for n-th moment P ′ (n, x) is x n P (x), where P (x) is the well-known splitting function from the DGLAP equation. The obtained equations are exact for each n-th moment and for every truncation point x0 ∈ (0; 1). They can be solved with use of standard methods of solving the DGLAP equations.This approach allows us to avoid the problem of dealing with the unphysical region x → 0. Furthermore, it refers directly to the physical values -moments (rather than to the parton distributions), what enables one to use a wide range of deep-inelastic scattering data in terms of smaller number of parameters.We give an example of an application.

Cut moments approach in the analysis of DIS data

“…The authors obtained the nondiagonal differential evolution equations, in which the nth truncated moment couples to all higher ones. Later on, diagonal integro-differential DGLAP-type evolution equations for the single and double truncated moments of the parton densities were derived in [11] and [12,13], respectively. The main finding of the truncated CMM approach is that the nth moment of the parton density also obeys the DGLAP equation, but with a rescaled evolution kernel P ′ (z) = z n P (z) [11].…”

“…Later on, diagonal integro-differential DGLAP-type evolution equations for the single and double truncated moments of the parton densities were derived in [11] and [12,13], respectively. The main finding of the truncated CMM approach is that the nth moment of the parton density also obeys the DGLAP equation, but with a rescaled evolution kernel P ′ (z) = z n P (z) [11]. The CMM approach has already been successfully applied, e.g., in spin physics to derive a generalization of the Wandzura-Wilczek relation in terms of the truncated moments and to obtain the evolution equation for the structure function g 2 [13,14].…”

Measurement of Inclusive Proton Double-Spin Asymmetries and Polarized Structure Functions

“…A rigorous calculation of these terms has been made by Dokshitzer, Gribov, Lipatov, Altarelli and Parisi [15] by solving for the crosssections in terms of dΩ or dp T = d(p sin θ), and integrating in Q 2 down to an infrared divergence limit Q 2 0 . 18 The results for the quark and gluon spin contributions are the DGLAP equations (also and α s (t) is the strong coupling constant, which is a "running" function of t. 19 The splitting functions ∆P AB (where A, B = g or q for gluons or quarks) are defined as ∆P AB ≡ P A+B+ − P A−B+ (1.108) where the "+" and "−" represent the helicities of the quarks/gluons in question, and P can be interpreted as the probability for a coupling between the quarks/gluons taking place. 20 These are…”

“…1.105 and 1.106 are cumbersome, so, using q to represent a parton (gluon or quark) distribution, these are often written in the condensed notation [18] dq(x, t) dt = α s (t) 2π (∆P ⊗ q)(x, t) (1.113) where the ⊗ symbol represents a convolution of the ∆P operators with the parton distributions.…”

“…However, this is enough information to apply the DGLAP equations, for example, to obtain information about the accuracy of pQCD regarding the Q 2 evolution of the structure functions [18] [26]. In practice, the entire range 0 ≤ x ≤ 1 cannot be completely known due to experimental limitations, especially as x → 0 (where infinite beam energies would be needed), so truncated moments are used in the reconstruction:…”