The Singlet Contribution to the Structure Function $g_1(x,Q^2)$ at Small $x$

A. Vogt; J. Bl\"umlein

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The Singlet Contribution to the Structure Function $g_1(x,Q^2)$ at Small $x$

J. Blümlein, A. Vogt

High Energy Physics - Phenomenology · Submitted 6 Jun 1996

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.63 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.14 · compares with prior results 0.23 · mentions code or data availability 0.02.

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The resummation of $O(\alpha_s^{l+1} \ln^{2l} x)$ terms in the evolution equation of the singlet part of $g_1(x,Q^2)$ is carried out. The corresponding singlet evolution kernels are calculated explicitely. The leading small-$x$ contribution to the three-loop splitting function matrix is determined in the $\overline{\rm MS}$ scheme. Relations are derived for the case of ${\cal N} = 1$ supersymmetric Yang--Mills field theory. Numerical results are presented for the polarized singlet and gluon densities, and the structure functions $g_1^{\, p}(x,Q^2)$ and $g_1^{\, n}(x,Q^2)$. They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The resummation of $O(\alpha_s^{l+1} \ln^{2l} x)$ terms in the evolution equation of the singlet part of $g_1(x,Q^2)$ is carried out.	contribution 0.70	0.93	0.17	asserted 1.00
s1	The corresponding singlet evolution kernels are calculated explicitely.	contribution 0.58	0.94	0.07	asserted 1.00
s2	The leading small-$x$ contribution to the three-loop splitting function matrix is determined in the $\overline{\rm MS}$ scheme.	result 0.79	0.93	0.20	asserted 1.00
s3	Relations are derived for the case of ${\cal N} = 1$ supersymmetric Yang--Mills field theory.	contribution 0.54	0.90	0.05	asserted 1.00
s4	Numerical results are presented for the polarized singlet and gluon densities, and the structure functions $g_1^{\, p}(x,Q^2)$ and $g_1^{\, n}(x,Q^2)$.	result 0.89	0.94	0.14	asserted 1.00
s5	They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.	contribution 0.42	0.93	0.14	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.88.

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Original abstract

The resummation of $O(\alpha_s^{l+1} \ln^{2l} x)$ terms in the evolution equation of the singlet part of $g_1(x,Q^2)$ is carried out. The corresponding singlet evolution kernels are calculated explicitely. The leading small-$x$ contribution to the three-loop splitting function matrix is determined in the $\overline{\rm MS}$ scheme. Relations are derived for the case of ${\cal N} = 1$ supersymmetric Yang--Mills field theory. Numerical results are presented for the polarized singlet and gluon densities, and the structure functions $g_1^{\, p}(x,Q^2)$ and $g_1^{\, n}(x,Q^2)$. They are compared for different assumptions on the non--perturbative input distributions, and the stability of the results against presently unknown subleading contributions is investigated.

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References

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Cited by 8 Pith papers

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The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case
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The paper presents the first complete set of next-to-next-to-leading-order splitting functions for longitudinally polarized parton densities in perturbative QCD.
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The high-energy OPE for deep inelastic scattering is extended to sub-eikonal order, yielding new impact factors and evolution equations for polarized structure functions at small x.
Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables
hep-ph 2016-10 conditional novelty 6.0 of 10

The large-Nc flavor non-singlet helicity evolution equation yields an intercept sqrt(alpha_s N_c/pi) that matches Bartels-Ermolaev-Ryskin, while the flavor-singlet intercept comes out about 35 percent below theirs.

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The Polarized Three-Loop Anomalous Dimensions from On-Shell Massive Operator Matrix Elements
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The authors recompute the T_F-proportional pieces of the polarized three-loop QCD anomalous dimensions using massive operator matrix elements and find full agreement with the published 2014 results.
The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
hep-ph 2021-11 accept novelty 4.0 of 10

An independent off-shell operator-matrix-element calculation reproduces the known three-loop polarized singlet anomalous dimensions and finds a small-x mismatch with infrared-evolution predictions.

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16 arXiv papers cite this in 33 sentences

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Compares with it 4 papers

The splitting functions extracted in [ref] also completely agreed with the first three loops predicted by BER [ref] , but all showed minor disagreements beginning at the four-loop level [ref] — the same situation as encountered in the solution of the large-$N_c$ helicity evolution equations [ref] .

Analytic Solution for the Helicity Evolution Equations at Small $x$ and Large $N_c\&N_f$ · Introduction · 0.93
These splitting functions completely agree with the results of BER IREE expansion (obtained by applying the large-$N_c & N_f$ limit to Eq. (14) in [ref] ).

Helicity Evolution at Small $x$: Quark to Gluon and Gluon to Quark Transition Operators · Summary of our results up to three loops · 0.96
Therefore, the agreement between our evolution and the small-$x$ limit of spin-dependent DGLAP equation in the gluon sector has been verified to three loops, the same order as the IREE of [ref] .

Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated · Cross-check against the spin-dependent DGLAP evolution · 0.90

Cites it as support 1 paper

Subleading terms dominate over the leading order effects, cf. [ref] .

The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements · Comparison to the literature · 0.99

Cites it as background 4 papers

We finally consider the small $x$ behaviour of the corrections calculated in the present paper.[For the unpolarized case see [ref] .] The leading order small $x$ resummation for the polarized flavor non–singlet and singlet contributions were studied in [ref] .

$O(\alpha_s^2$) Polarized Heavy Flavor Corrections}to Deep-Inelastic Scattering at $Q^2 \gg m^2$ · Discussion · 0.85
It led to helicity phenomenology developed in [ref] .

Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\& N_f$ · Introduction · 0.88
Such analyses based on the BER formalism had been performed in Refs. [ref] , before the LCOT approach was developed.

Gluon Double-Spin Asymmetry in the Longitudinally Polarized $p+p$ Collisions · Introduction · 0.83

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