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arxiv: 2604.19997 · v1 · submitted 2026-04-21 · ✦ hep-ph

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Introduction to transverse momentum imaging

Andrea Signori
Authors on Pith no claims yet

Pith reviewed 2026-05-10 01:34 UTC · model grok-4.3

classification ✦ hep-ph
keywords transverse momentum dependent distributionshadron structureQCDparton modelelectron-ion colliderTMD PDFsimaging
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The pith

Transverse momentum dependent functions provide a framework for three-dimensional imaging of hadrons in QCD.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The notes introduce transverse momentum dependent parton distribution functions as a tool to access the three-dimensional structure of hadrons. They explain how these functions encode both longitudinal momentum fractions and transverse momenta of partons inside protons and other hadrons. The material is organized to support lectures at graduate schools focused on hadron physics and the upcoming electron-ion collider program. A sympathetic reader would see this as establishing the basic language and relations needed to interpret data that go beyond the usual one-dimensional parton distributions.

Core claim

These lecture notes establish the definitions, evolution equations, and phenomenological applications of transverse momentum dependent parton distributions, showing how they extend the standard parton model to include transverse momentum and thereby enable momentum-space imaging of hadrons.

What carries the argument

Transverse momentum dependent parton distribution functions (TMD PDFs), which describe the probability density for finding a parton with given longitudinal momentum fraction and transverse momentum inside a hadron.

If this is right

  • Data from the electron-ion collider can be interpreted in terms of three-dimensional parton distributions rather than one-dimensional ones.
  • Spin and orbital angular momentum contributions inside hadrons become accessible through TMD observables.
  • Factorization theorems allow separation of hard scattering from soft transverse-momentum effects in high-energy processes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same TMD framework could be applied to other high-energy processes beyond the electron-ion collider, such as Drell-Yan at the LHC.
  • Connections to generalized parton distributions might allow unified three-dimensional imaging in both momentum and position space.
  • Phenomenological extractions of TMDs from existing data can serve as benchmarks before new collider runs begin.

Load-bearing premise

The reader already knows the basics of quantum chromodynamics and the collinear parton model.

What would settle it

A measurement at an electron-ion collider that shows transverse-momentum distributions inconsistent with the predicted evolution and factorization properties of TMD functions.

Figures

Figures reproduced from arXiv: 2604.19997 by Andrea Signori.

Figure 1
Figure 1. Figure 1: FIG. 1: Contraction of the leptonic and hadronic tensors in DIS. The lines crossing the cut represent on-shell states [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Diagrammatic interpretations of the operators contributing to the DIS hadronic tensor. Diagram (a) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: A physical observable can be expanded in powers of the coupling constant [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The hadronic tensor [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Leading twist approximation for the SIDIS hadronic tensor as a trace of the TMD parton correlation [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Diagrams contributing to the gauge link for DIS-like processes up to order [PITH_FULL_IMAGE:figures/full_fig_p029_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Representations of staple-like Wilson lines running from 0 to [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
read the original abstract

This set of notes complements the lectures and recitation sessions discussed in the following graduate schools: HUGS at Jefferson Lab (years 2018, 2019, 2021), the International School and Workshop on Probing Hadron Structure at the Electron-Ion Collider at ICTS (2024), Frontiers in Nuclear and Hadronic Physics at GGI (2025), and the International Workshop and School on Hadron Structure and Strong Interactions at Nanjing University (2025).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. This manuscript is a set of lecture notes providing an introduction to transverse momentum dependent distributions and imaging in hadrons. It is explicitly designed to complement lectures and recitations at the HUGS graduate schools (2018, 2019, 2021), the ICTS school on probing hadron structure at the EIC (2024), the GGI school on nuclear and hadronic physics (2025), and the Nanjing workshop on hadron structure (2025). No new scientific results, derivations, or predictions are claimed.

Significance. These notes address a pedagogical need in the QCD and hadron-structure community, particularly as preparation for research at the Electron-Ion Collider. Well-structured lecture notes on TMDs can help graduate students master the parton-model extensions, factorization theorems, and experimental observables that are central to current and future programs; their value lies in clarity and accessibility rather than novelty.

minor comments (1)
  1. The abstract lists the supported schools but does not outline the specific topics or organization of the notes themselves; adding a brief table of contents or topic list would improve usability for readers who did not attend the lectures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of these lecture notes. We are pleased that the pedagogical focus and structure were recognized as addressing a clear need in the community, particularly in preparation for EIC-related research. We have no major comments to address, as none were raised.

Circularity Check

0 steps flagged

Lecture notes with no derivations or predictions

full rationale

This document is explicitly presented as lecture notes to accompany graduate-school sessions on transverse-momentum-dependent distributions. It advances no novel scientific claims, derivations, or results whose internal consistency can be examined for circularity. No equations, predictions, or load-bearing steps are offered that could reduce to fitted parameters, self-citations, or self-definitional constructs. The content assumes standard QCD and parton-model background and remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No new free parameters, axioms, or invented entities are introduced because the document is introductory lecture notes rather than original research.

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Reference graph

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