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Flavor Singlet Contribution to the Structure Function $g_1$ at Small-x
High Energy Physics - Phenomenology · Submitted 1 Mar 1996
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Double logarithmic contributions of the type $(\alpha_s \ln ^2 (1/x))^k$ which are not included in the GLAP evolution equations are shown to give a power-like rise at small-x which is much stronger than the extrapolation of the GLAP expressions.
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The abstract, sentence by sentence
The singlet contribution to the $g_1(x,Q^2)$ structure function is calculated in the double-logarithmic approximation of perturbative QCD in the region $x \ll 1$. Double logarithmic contributions of the type $(\alpha_s \ln ^2 (1/x))^k$ which are not included in the GLAP evolution equations are shown to give a power-like rise at small-x which is much stronger than the extrapolation of the GLAP expressions. The dominant contribution is due to the gluons which, in contrast to the unpolarized case, mix with the fermions also in the region $x \ll 1$. The two main reasons why the small-x behavior of the double logarithmic approximation is so much stronger than the usual GLAP evolution are: the larger kinematical region of integration (in particular, no ordering in transverse momentum) and the contributions from non-ladder diagrams.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The singlet contribution to the $g_1(x,Q^2)$ structure function is calculated in the double-logarithmic approximation of perturbative QCD in the region $x \ll 1$. | contribution 0.96 | 0.90 | 0.09 | asserted 1.00 |
| s1 | Double logarithmic contributions of the type $(\alpha_s \ln ^2 (1/x))^k$ which are not included in the GLAP evolution equations are shown to give a power-like rise at small-x which is much stronger than the extrapolation of the GLAP expressions. | result 1.00 | 0.82 | 0.71 | asserted 1.00 |
| s2 | The dominant contribution is due to the gluons which, in contrast to the unpolarized case, mix with the fermions also in the region $x \ll 1$. | result 0.72 | 0.72 | 0.12 | asserted 1.00 |
| s3 | The two main reasons why the small-x behavior of the double logarithmic approximation is so much stronger than the usual GLAP evolution are: the larger kinematical region of integration (in particular, no ordering in transverse momentum) and the contributions from non-ladder diagrams. | interpretation 0.96 | 0.74 | 0.14 | asserted 1.00 |
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The paper
Original abstract
The singlet contribution to the $g_1(x,Q^2)$ structure function is calculated in the double-logarithmic approximation of perturbative QCD in the region $x \ll 1$. Double logarithmic contributions of the type $(\alpha_s \ln ^2 (1/x))^k$ which are not included in the GLAP evolution equations are shown to give a power-like rise at small-x which is much stronger than the extrapolation of the GLAP expressions. The dominant contribution is due to the gluons which, in contrast to the unpolarized case, mix with the fermions also in the region $x \ll 1$. The two main reasons why the small-x behavior of the double logarithmic approximation is so much stronger than the usual GLAP evolution are: the larger kinematical region of integration (in particular, no ordering in transverse momentum) and the contributions from non-ladder diagrams.
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References
How the literature uses it
40 arXiv papers cite this in 139 sentences
Papers citing it · 63 sentences without a confident label
Contests it 3 papers
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In [ref] we identify DLA diagram contributions not included by the authors of [ref] in their treatment of the problem.
Helicity Evolution at Small x: Summary of Recent Developments · Solution of the Large-$N_c$ Helicity Evolution Equations · contest 0.79
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Let us note again that the results shown in Eqs. [ref] and [ref] disagree with the earlier conclusions regarding the small-$x$ asymptotics of helicity distributions reached in [ref] .
From Parton Saturation to Proton Spin: the Impact of BFKL Equation and Reggeon Evolution · Small-$x$ asymptotics for quark and gluon helicity distributions · contest 0.73
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In either kinematic region, the non-ladder gluon $k_1 - k_2$ is hard, thus contributing a hard (cut) vertex correction in an apparent violation of the absence of such gluons in the IREE argued in [ref] .
Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept · Comparison of some diagrams in BER IREE and in the shock wave approach · contest 0.80
Uses its data or values 2 papers
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The first determination of the parameter $b$ was achieved in [ref] b≈3.664 √(α_sN_c/2π) The square-root is the hallmark of the double logarithmic resummation $e^√(α_sln^21/x)=1/x^√(α_s)$.
Spin Structure of the Nucleon: Overview · Helicity and orbital angular momentum at small-$x$ · 0.99
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This form was first derived by Bartels, Ermolaev, and Ryskin (BER) [ref] via their use of infrared evolution equations (IREE), and they numerically computed an intercept of $α_h = 3.66√(α_sN_c2π)$ in the pure-gluon case and $α_h = 3.45√(α_sN_c2π)$ for $N_f = 4$ and $N_c = 3$ [ref] .
First study of polarized proton-proton scattering with small-$x$ helicity evolution · Introduction · 0.82
Compares with it 18 papers
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The gluon-gluon polarized anomalous dimensions of [ref] and [ref] , when expanded as power series in $$, agree to the three known loops, and disagree beyond that.
Helicity Evolution at Small $x$: Quark to Gluon and Gluon to Quark Transition Operators · Introduction · 0.99
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While the numerical solution from [ref] appeared to be in good quantitative agreement with the pure-glue limit of the earlier work by BER [ref] , the analytical solution found in [ref] revealed differences between KPS-CTT evolution and BER.
Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\& N_f$ · Introduction · 0.97
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We solved these equations numerically showing that the resulting asymptotics of the gluon and flavor-singlet quark helicity distributions, along with the $g_1$ structure function, are given by small_x_asymp and agree with that found by BER [ref] in the pure-glue case.
Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated · Conclusions and Outlook · 1.00
Cites it as background 16 papers
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In recent years, an effort to develop small-$x$ evolution equations for helicity PDFs has been underway [ref] , building in part on Refs. [ref] .
First analysis of world polarized DIS data with small-$x$ helicity evolution · Introduction · 0.81
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The latter possibility received a lot of attention in the literature, starting with the groundbreaking work by Bartels, Ermolaev and Ryskin (BER) [ref] , which studied the small-$x$ asymptotics of the $g_1$ structure function along with $ΔΣ$ and $ΔG$ employing the infrared evolution equations (IREE) approach pioneered in [ref] .
Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated · Introduction · 0.88
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In recent years, evolution equations describing the quark and gluon helicity distributions at small Bjorken $x$ have been derived in [ref] (see also [ref] for earlier calculations based on a different method).
Small-$x$ Helicity Evolution: an Operator Treatment · Introduction · 0.93
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