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Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations
High Energy Physics - Phenomenology · Submitted 16 Mar 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim By comparing with our numerical results, we are able to demonstrate the size, origin, and significance of subleading-$N_c$ corrections.
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The abstract, sentence by sentence
We systematically derive expressions for ten small-$x$ Transverse Momentum Dependent (TMD) distributions in the Gaussian approximation, three in the quark-gluon sector and seven in the gluon-gluon sector; for the general $SU(N_c)$ gauge group. The derived formulae depend on the logarithm of the dipole amplitude. Using the McLerran-Venugopalan model for the initial condition, we simulate the dipole amplitude as well as estimate all ten TMD distributions for $N_c \in \{2,3,4,5\}$. We compare these explicit numerical results with the derived expressions and find very good agreement for all studied values of $N_c$. Consequently, we study the scaling with $N_c$ of the TMD distributions. Thanks to that, we are able to derive the large-$N_c$ limit of the Gaussian approximation results and show that they agree with expressions obtained in the mean-field approximation. By comparing with our numerical results, we are able to demonstrate the size, origin, and significance of subleading-$N_c$ corrections. This work sets the stage for a systematic study of subleading corrections induced by the JIMWLK evolution in the rapidity evolution of the TMD distributions. Interestingly, we find an exact sum rule that relates all seven gluon-gluon TMD operators at $N_c = 3$ and arbitrary $x$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We systematically derive expressions for ten small-$x$ Transverse Momentum Dependent (TMD) distributions in the Gaussian approximation, three in the quark-gluon sector and seven in the gluon-gluon sector; for the general $SU(N_c)$ gauge group. | contribution 0.99 | 0.97 | 0.09 | asserted 1.00 |
| s1 | The derived formulae depend on the logarithm of the dipole amplitude. | result 0.74 | 0.87 | 0.04 | asserted 1.00 |
| s2 | Using the McLerran-Venugopalan model for the initial condition, we simulate the dipole amplitude as well as estimate all ten TMD distributions for $N_c \in \{2,3,4,5\}$. | method 0.97 | 0.96 | 0.05 | asserted 0.99 |
| s3 | We compare these explicit numerical results with the derived expressions and find very good agreement for all studied values of $N_c$. | result 1.00 | 0.97 | 0.17 | asserted 1.00 |
| s4 | Consequently, we study the scaling with $N_c$ of the TMD distributions. | contribution 0.88 | 0.93 | 0.06 | asserted 0.74 |
| s5 | Thanks to that, we are able to derive the large-$N_c$ limit of the Gaussian approximation results and show that they agree with expressions obtained in the mean-field approximation. | result 0.92 | 0.96 | 0.22 | asserted 1.00 |
| s6 | By comparing with our numerical results, we are able to demonstrate the size, origin, and significance of subleading-$N_c$ corrections. | result 0.79 | 0.96 | 0.16 | asserted 1.00 |
| s7 | This work sets the stage for a systematic study of subleading corrections induced by the JIMWLK evolution in the rapidity evolution of the TMD distributions. | future work 0.99 | 0.54 | 0.17 | asserted 0.82 |
| s8 | Interestingly, we find an exact sum rule that relates all seven gluon-gluon TMD operators at $N_c = 3$ and arbitrary $x$. | result 1.00 | 0.97 | 0.13 | asserted 1.00 |
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The paper
Original abstract
We systematically derive expressions for ten small-$x$ Transverse Momentum Dependent (TMD) distributions in the Gaussian approximation, three in the quark-gluon sector and seven in the gluon-gluon sector; for the general $SU(N_c)$ gauge group. The derived formulae depend on the logarithm of the dipole amplitude. Using the McLerran-Venugopalan model for the initial condition, we simulate the dipole amplitude as well as estimate all ten TMD distributions for $N_c \in \{2,3,4,5\}$. We compare these explicit numerical results with the derived expressions and find very good agreement for all studied values of $N_c$. Consequently, we study the scaling with $N_c$ of the TMD distributions. Thanks to that, we are able to derive the large-$N_c$ limit of the Gaussian approximation results and show that they agree with expressions obtained in the mean-field approximation. By comparing with our numerical results, we are able to demonstrate the size, origin, and significance of subleading-$N_c$ corrections. This work sets the stage for a systematic study of subleading corrections induced by the JIMWLK evolution in the rapidity evolution of the TMD distributions. Interestingly, we find an exact sum rule that relates all seven gluon-gluon TMD operators at $N_c = 3$ and arbitrary $x$.
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References
How the literature uses it
1 arXiv paper cite this in 11 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
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In this study, we built upon previous research on small-$x$ TMD distributions by incorporating evolution into our framework in [ref] .
Small-x TMD distributions: JIMWLK evolution and the Gaussian truncation · Discussion and prospects · 0.99
Uses its method or tool 1 paper
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The initial condition is generated according to the MV-model, and we refer to [ref] for further details.
Small-x TMD distributions: JIMWLK evolution and the Gaussian truncation · Implementation of the JIMWLK evolution · 0.87
What it cites · 37 sentences without a confident label
Uses the method or tool of 8 papers
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The methodology used is similar to that used in previous quadrupole calculations, employing a pictorial representation, as seen in, e.g. [ref] , or a more recent generalization at next-to-eikonal order [ref] .
2512.17848 · Introduction · 0.93
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The methodology used is similar to that used in previous quadrupole calculations, employing a pictorial representation, as seen in, e.g. [ref] , or a more recent generalization at next-to-eikonal order [ref] .
Universality of Unintegrated Gluon Distributions at small x · Introduction · 0.93
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The methodology used is similar to that used in previous quadrupole calculations, employing a pictorial representation, as seen in, e.g. [ref] , or a more recent generalization at next-to-eikonal order [ref] .
On multiple scatterings of mesons in hot and cold QCD matter · Introduction · 0.93
Compares with 5 papers
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First, we derive expressions for the Gaussian approximation of the TMD distributions, which turn out to agree with the explicit numerical evaluation for the McLerran–Venugopalan (MV) initial condition [ref] in a wide range of separations.
Green's Functions in the Color Field of a Large Nucleus · Introduction · 0.80
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First, we derive expressions for the Gaussian approximation of the TMD distributions, which turn out to agree with the explicit numerical evaluation for the McLerran–Venugopalan (MV) initial condition [ref] in a wide range of separations.
Computing Quark and Gluon Distribution Functions for Very Large Nuclei · Introduction · 0.80
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First, we derive expressions for the Gaussian approximation of the TMD distributions, which turn out to agree with the explicit numerical evaluation for the McLerran–Venugopalan (MV) initial condition [ref] in a wide range of separations.
Gluon Distribution Functions for Very Large Nuclei at Small Transverse Momentum · Introduction · 0.80
Cites as background 28 papers
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[For example, a lot of effort is currently being put into joint resummation of Collins–Soper–Sterman soft gluon emission corrections [ref] and small $x$ gluon emissions [ref] .] In the limit of low Bjorken $x$, TMD distributions are expressed in terms of operators evaluated against a target state described by a color glass condensate (CGC).
Dihadron correlations in small-$x$ DIS at NLO: transverse momentum dependent fragmentation · Introduction · 0.83
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[For example, a lot of effort is currently being put into joint resummation of Collins–Soper–Sterman soft gluon emission corrections [ref] and small $x$ gluon emissions [ref] .] In the limit of low Bjorken $x$, TMD distributions are expressed in terms of operators evaluated against a target state described by a color glass condensate (CGC).
Evolution of the transverse-momentum dependent gluon distribution at small $x$ · Introduction · 0.83
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[For example, a lot of effort is currently being put into joint resummation of Collins–Soper–Sterman soft gluon emission corrections [ref] and small $x$ gluon emissions [ref] .] In the limit of low Bjorken $x$, TMD distributions are expressed in terms of operators evaluated against a target state described by a color glass condensate (CGC).
The CSS Hamiltonian: high energy evolution of rapidity dependent observables · Introduction · 0.83
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