REVIEW
Transverse momentum dependence in gluon distribution and fragmentation functions
High Energy Physics - Phenomenology · Submitted 29 Sep 2000 · 21 references
Pith's verdict: The paper that fixes the gluon TMD basis, and it holds up — the only real complaint is that the Section V bounds cover the T-even functions only, which the abstract does not flag.
T0 · AI review
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Referee report
The paper
The paper claims
The paper classifies the leading-power gluon correlator for a spin-1/2 hadron into eight transverse-momentum-dependent distributions and derives positivity bounds from a four-by-four spin-density matrix.
The reading
What carries the argument
The load-bearing object is the gauge-invariant gluon correlator $Gamma^{ij}(x,mathbf{k}_T)=int dk^-,Gamma^{+i;+j}(k;P,S)$, built from two field strengths joined by a light-cone gauge link. The paper decomposes it using the hadron momentum $P$, spin vector $S$, and the two light-like directions $n_+$ and $n_-$; the leading-power part is the piece with two transverse indices and no minus index. A circular-polarization basis turns the correlator into a $4times4$ matrix whose entries are the eight distributions, and the positivity of a density matrix converts the statement that probabilities are non-negative into the bounds. The gauge-link discussion supplies the bridge between the formal matrix and the correlator that actually enters a hard cross section.
Extended reading notes
Core claim
The central result is a complete leading-power parameterization of the gluon-gluon correlator for a spin-1/2 hadron in terms of eight transverse-momentum-dependent functions: $G$, $Delta G_L$, $G_T$, $Delta G_T$, $H^perp$, $Delta H^perp_L$, $Delta H^perp_T$, and $Delta H_T$. After integrating over the minus component of the gluon momentum, the correlator $Gamma^{ij}(x,mathbf{k}_T)$ is organized by hadron polarization and by the transverse indices $i,j$; the eight functions are the coefficients of the allowed transverse tensors and vectors. The same functions form a $4times4$ matrix in the product of gluon helicity and nucleon spin space, and positivity of its diagonal elements yields $|Delta G_L(x,mathbf{k}_T^2)|le G(x,mathbf{k}_T^2)$, $|Delta G_T^{(1)}|le (|mathbf{k}_T|/2M)G$, and $|H^{perp(1)}|le G$. For fragmentation the analogous matrix includes T-odd functions, so the bounds become coupled inequalities such as $|Delta hat G_T^{(1)}|^2+|hat G_T^{(1)}|^2le (|mathbf{k}_T|^2/4M_h^2)hat G^2$.
Load-bearing premise
The bounds rest on treating the gauge-invariant gluon correlator as a positive density matrix in the $A^+=0$ gauge, which requires the gauge links to close; the paper explicitly assumes that hadronic matrix elements with transverse gluon fields at infinite light-cone separation vanish.
Editorial extensions
If this is right
- Any leading-power gluon contribution to an azimuthal asymmetry in a process with two hadrons can be written as a product of these eight distributions, or their fragmentation analogues, and a hard coefficient.
- Model estimates of gluon spin effects can be checked against the inequalities without computing the full correlator.
- The T-odd fragmentation functions $hat G_T$, $Delta hat H^perp_L$, $Delta hat H^perp_T$, and $Delta hat H_T$ can be nonzero because final-state hadrons break time-reversal symmetry, so they are available to generate single-spin asymmetries.
- The twist-three functions classified in the appendix provide the subleading terms needed when the hard scale is not asymptotically large.
Reading between the lines
- If the positivity argument survives the gauge-link caveat, it gives a cheap consistency test for future global fits: an extracted gluon transverse-momentum distribution outside the bound would point to a breakdown of the density interpretation or to neglected link-dependent effects.
- The same matrix method should extend to spin-1 hadrons and to off-forward gluon correlators, where the larger polarization space would produce additional inequalities.
- Because the gauge-link path differs between leptoproduction and Drell-Yan scattering, the T-odd gluon functions may change sign between those processes, similar to the quark Sivers effect; measuring that sign change would separate link-dependent from intrinsic spin effects.
Original abstract
We investigate the twist two gluon distribution functions for spin 1/2 hadrons, emphasizing intrinsic transverse momentum of the gluons. These functions are relevant in leading order in the inverse hard scale in scattering processes such as inclusive leptoproduction or Drell-Yan scattering, or more general in hard processes in which at least two hadrons are involved. They show up in azimuthal asymmetries. For future estimates of such observables, we discuss specific bounds on these functions.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We investigate the twist two gluon distribution functions for spin 1/2 hadrons, emphasizing intrinsic transverse momentum of the gluons. These functions are relevant in leading order in the inverse hard scale in scattering processes such as inclusive leptoproduction or Drell-Yan scattering, or more general in hard processes in which at least two hadrons are involved. They show up in azimuthal asymmetries. For future estimates of such observables, we discuss specific bounds on these functions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We investigate the twist two gluon distribution functions for spin 1/2 hadrons, emphasizing intrinsic transverse momentum of the gluons. | contribution 1.00 | 0.89 | 0.10 | asserted 0.80 |
| s1 | These functions are relevant in leading order in the inverse hard scale in scattering processes such as inclusive leptoproduction or Drell-Yan scattering, or more general in hard processes in which at least two hadrons are involved. | background 1.00 | 0.07 | 0.05 | asserted 0.99 |
| s2 | They show up in azimuthal asymmetries. | background 0.78 | 0.12 | 0.08 | asserted 1.00 |
| s3 | For future estimates of such observables, we discuss specific bounds on these functions. | contribution 0.59 | 0.72 | 0.21 | asserted 0.68 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/0009343 by the authors.
References
How later papers used it
136 later papers cite it
2 later papers by other authors are listed below: 1 built on it and 1 used its method or tool. 1 more paper by the same authors is listed: 1 built on it and 1 used its method or tool. It appears under more than one label.
Builds on it 2 of 5 papers with this label · 1 by the same authors
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Extended this paper's positivity bounds for gluon TMDs for spin-1/2 hadrons to spin-1 hadrons by applying the same strategy.
Mapping the internal structure of hadrons through color and spin effects (2018) · 0.99 · paraphrased: the licence does not allow quoting
By the same authors
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Extended this paper's positivity bounds on gluon TMDs for spin-1/2 hadrons to spin-1 hadrons by applying the same strategy.
Positivity bounds on gluon TMDs for hadrons of spin $\le$ 1 (2017) (same authors) · 1.00 · paraphrased: the licence does not allow quoting
Uses its method or tool 2 of 5 papers with this label · 1 by the same authors
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Defined a quantity following this paper, with k+ = x P+.
Gluon Double-Spin Asymmetry in the Longitudinally Polarized $p+p$ Collisions (2024) · 0.82 · paraphrased: the licence does not allow quoting
By the same authors
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Chose the same basis for the matrix G as this paper, using circular gluon polarizations.
Positivity bounds on gluon TMDs for hadrons of spin $\le$ 1 (2017) (same authors) · 0.95 · paraphrased: the licence does not allow quoting
How the literature uses it
124 arXiv papers cite this in 212 sentences
Papers citing it · 129 sentences without a confident label
Builds on it 4 papers
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Positivity bounds for gluon TMDs were derived in [ref] for spin-$12$ hadrons and, by applying the same strategy, we extend their analysis to spin-$1$ hadrons.
Mapping the internal structure of hadrons through color and spin effects · Positivity bounds · 0.99
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Positivity bounds on gluon TMDs were studied in [ref] for spin-$1/2$ hadrons and, by applying the same strategy, we extend here this analysis to spin-$1$ hadrons.
Positivity bounds on gluon TMDs for hadrons of spin $\le$ 1 · Positivity bounds on gluon distributions · 1.00
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[ref] by parametrizing both the gluon-gluon and Wilson loop correlators for unpolarized, vector polarized, as well as tensor polarized hadrons.
Gluon and Wilson loop TMDs for hadrons of spin $\leq$ 1 · Parametrizations of gluonic TMD correlators · 1.00
Uses its method or tool 5 papers
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We choose the same basis for the matrix $G$ as in [ref] , namely we use circular gluon polarizations, given by $|±⟩ = ∓ 1/√(2) (|x⟩ ±i |y⟩)$.
Positivity bounds on gluon TMDs for hadrons of spin $\le$ 1 · Bounds on transverse momentum dependent functions · 0.95
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We can decompose the gluon fragmentation correlator and define eight leading-twist TMD FFs of the gluon through the following projection [ref] [equation] and the four T-odd gluon TMD FFs can be projected into [equation] Here, we have used the symmetric transverse tensor $δ_T^i j =-g_T^i j$ with $g_T^i j = g^i j-n_+^i n_-^j-n_+^j n_-^i$, the anti-symmetric transverse tensor $ϵ_T^i j$ with $ϵ_T^1 2=1$, and a symmetrization operator $Ŝ$ for a generic tensor $O^ij$ which is defined as: [equation]
T-odd transverse momentum dependent gluon fragmentation functions in a spectator model · Analytic calculation of the T-odd FFs of a spin-1/2 hadron · 0.85
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The correlators $Φ^[U]_g$, describing the proton $→$ gluon transitions, are defined in terms of matrix elements of the gluon field strengths $F^μν(0)$ and $F^μν(ξ)$ on the light front (LF) $ξ·n≡0$ [ref] , [equation] with $n≡n_-$ for gluon $a$ with momentum $p_a$ and $n≡n_+$ for gluon $b$ with momentum $p_b$.
Linear polarization of gluons and Higgs plus jet production at the LHC · Theoretical framework · 0.81
Compares with it 2 papers
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We emphasize again that our approach naturally involves two separate tri-gluon correlation functions $T_G^(f)(x, x)$ and $T_G^(d)(x, x)$, whereas only one Sivers function occurs in the study of [ref] (see also [ref] ).
Accessing tri-gluon correlations in the nucleon via the single spin asymmetry in open charm production · Introduction · 0.87
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Note that our convention for the definitions of both TMDs is slightly different from those in [ref] , with our definition of the linearly polarized TMD following that in [ref] .
Weizs\"acker-Williams Gluon Helicity Distribution and Inclusive Dijet Production in Longitudinally Polarized Electron-Proton Collisions · A Tale of Two Gluon Helicity Distributions · 0.93
Cites it as background 61 papers
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The TMD correlator $Γ$ can be parametrised in terms of gluon TMDs (cf. Refs. [ref] ).
Associated production of a dilepton and a $\Upsilon(J/\psi)$ at the LHC as a probe of gluon transverse momentum dependent distributions · General Structure of the Cross Section · 0.85
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On the other hand, the dipole-type gluon Sivers function $x f_1T^⊥g(x,)$ is defined through the following correlator [ref] 2/P^+∫v^- ^2 /(2π)^3 ^x P^+ v^- ^-· ⟨P | tr[F^i+(-v/2) ^[+]_[-v/2,v/2] F^i+(v/2)^[-]†_[-v/2,v/2]]|P ⟩ = x f_1^g(x,) - (×)/M_p x f_1T^⊥g(x,) ,
Gluon Sivers function from forward exclusive $\chi_{c1}$ photoproduction on unpolarized protons · Exclusive $γp →χ_c1 p$ cross section · 0.87
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In this work, we consider unpolarized initial hadrons, there are two TMD gluon distributions $f_1^g$ and $h_1^⊥g$ [ref] , which represent unpolarized and linearly polarized gluon distributions in the unpolarized hadron, respectively.
Probing transverse momentum dependent gluon distribution functions from hadronic quarkonium pair production · Introduction · 0.82
Discussion
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