Parton helicities at arbitrary x and Q2 in double-logarithmic approximation

TL;DR

The paper tackles the problem of describing spin-dependent hadronic processes at high energies by deriving explicit expressions for the parton helicities $h_q(x,Q^2)$ and $h_g(x,Q^2)$ using the double-logarithmic approximation (DLA) and Infra-Red Evolution Equations (IREE). It develops interpolation formulas valid at arbitrary $x$ and $Q^2$ by combining DL resummation with DGLAP inputs and compares Collinear and KT Factorizations, arguing that KT Factorization is required when parton orbital angular momenta contribute to the nucleon spin. The work analyzes small-$x$ asymptotics, showing that $h_q$, $h_g$, and $g_1$ share the same leading intercept $\omega_0$ in DL, and discusses how DGLAP can fail to generate Regge-type small-$x$ behavior without careful treatment of initial densities. It provides explicit DL expressions for helicities, discusses color-octet contributions, and presents a self-consistent, IR-cutoff–aware framework for spin-dependent observables applicable in both factorization schemes, including OAM effects through KT Factorization.

Abstract

Description of spin-dependent hadronic processes at high energies in terms of parton helicities is a both effective and technically convenient means. In the present paper, we obtain explicit expressions for the parton helicities when either Collinear or KT forms of QCD Factorization are used. Starting our studies with calculation of the helicities in the double-logarithmic approximation (DLA) in the region of small x and large Q^2, we generalize the results in order to obtain formulae valid at arbitrary x and Q^2. We argue against using Collinear Factorization, when the parton orbital angular momenta are accounted for, and prove that KT Factorization should be used instead. We also consider in detail the small-x asymptotics of the parton helicities, compare them with the DGLAP-asymptotics in LO,NLO, etc and prove that the DGLAP asymptotics are less singular at small x than the Regge asymptotics

Figures (2)

• Figure 1: Constructing IREE for amplitude $A$ of $2 \to 2$- parton scattering, using factorization of the softest partons with momentum $k$. Letters $s,t$ in the l.h.s. denote the standard Mandelstam variables. The first term in the r.h.s. stands for the Born contribution. The blobs denote that radiative corrections in DLA are accounted for. The blobs on graphs (b,c) (marked with horizontal strips) are $t$-channel color octets while the other blobs are color singlets. The letters on the blobs stand for IR cut-offs. Graphs (a,b,c) correspond to factorization of non-ladder partons (Option (i)) whilst factorization of ladder ones (Option (ii)) is shown on graphs (d) and (e).
• Figure 2: Representation of the parton helicities through convolutions of perturbative (upper blobs) and non-perturbative (lower blobs) components in any of the forms of QCD Factorization. Waved lines denote gluons with momenta $q$, the thick straight lines attached to the lowest blobs correspond to the hadron with momentum $p$, the other straight lines attached to the upper blobs denote quarks with momenta $q$. The cuts with respect to $s = (p + q)^2$ on each graph are implied.