⚛️ nuclear theory

Femtoscale imaging of the proton with Ioffe-time distributions

This paper proposes a novel space-time description of proton structure using impact-parameter Ioffe-time distributions, which enables the first lattice QCD calculation of Ioffe-time-dependent mean squared proton radii and the subsequent extraction of the Compton form factor.

Original authors: Robert G. Edwards, Joe Karpie, Christopher Monahan, Kostas Orginos, Anatoly Radyushkin, David Richards, Eloy Romero, Savvas Zafeiropoulos

Published 2026-08-26

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CC BY 4.0

Original authors: Robert G. Edwards, Joe Karpie, Christopher Monahan, Kostas Orginos, Anatoly Radyushkin, David Richards, Eloy Romero, Savvas Zafeiropoulos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Inside every proton, the building block of atomic nuclei, lies a chaotic and tightly bound world of quarks and gluons. These particles do not sit still; they zip around at nearly the speed of light, interacting with a force so powerful that it cannot be easily calculated with standard math. For decades, physicists have tried to map this internal landscape, asking not just what particles are inside, but exactly where they are and how they move. To do this, they use high-energy collisions to probe the proton, much like shining a light into a dark room to see what is inside. However, the data from these experiments often comes in a form that is difficult to translate into a clear picture of the proton's shape. The challenge has been to connect the abstract numbers measured in a lab to a concrete image of the proton's three-dimensional structure in space and time.

A team of researchers has now taken a significant step toward solving this puzzle by creating a new way to visualize the proton's interior. Instead of trying to reconstruct the proton's shape directly from the messy data of particle collisions, they used a powerful computer simulation based on the fundamental laws of physics to calculate how the proton's constituents are distributed. They developed a method that treats the proton's structure as a series of snapshots in time, linking the position of a particle inside the proton to the specific duration of its interaction with a probe. This approach allowed them to measure the size of the proton's internal structure with a new kind of precision, revealing how the distribution of matter changes depending on how long the interaction lasts.

The researchers focused on a concept called the "Ioffe-time," which represents the duration of the interaction between a probe and a particle inside the proton. Imagine taking a photograph of a moving object; the clarity of the image depends on how fast the camera's shutter opens and closes. In the same way, the researchers realized that the proton's internal structure looks different depending on the "shutter speed" of the interaction. By calculating how the proton's size changes as this interaction time varies, they could map out the proton's transverse radius—the distance of quarks from the center of the proton in the direction perpendicular to its motion. They found that for shorter interaction times, the proton appears smaller and more compact, while for longer interaction times, the distribution of matter spreads out, revealing a wider structure.

To achieve this, the team performed complex calculations using a technique called lattice Quantum Chromodynamics, which breaks space and time into a grid to simulate the behavior of quarks and gluons. They analyzed data from these simulations to determine the average squared radius of the proton at different interaction times. Their results showed a clear trend: the proton's internal structure is not static but dynamic, changing its effective size based on the timescale of the observation. They compared their findings with a popular theoretical model known as the Goloskokov-Kroll model and found that while the model predicted a similar trend, the actual calculated values from their simulation were smaller. This difference suggests that the heavy quarks used in their simulation might lead to a smaller cloud of particles surrounding the core, a detail that future studies will need to explore further.

Perhaps most importantly, the team demonstrated a new way to extract a quantity called the Compton form factor, which is a key piece of information that experiments measure but is notoriously difficult to calculate from first principles. Traditionally, scientists had to solve a difficult mathematical puzzle to get from the raw data to this form factor, often relying on assumptions that could introduce errors. The new method bypasses this difficult step entirely. By using the interaction time as a direct variable, they were able to calculate the Compton form factor straight from their simulation data. The results they obtained matched the general shape of the predictions from the Goloskokov-Kroll model, providing a strong validation of their new approach.

This work offers a fresh perspective on the proton, moving away from abstract momentum fractions and toward a more intuitive picture of space and time. By defining the proton's structure in terms of where particles are and how long they interact, the researchers have created a framework that is easier to interpret and more directly connected to physical reality. Their findings suggest that the proton is a fluid, evolving system where the perceived size depends on the timescale of the observation. As future experiments at facilities like the Electron Ion Collider gather more data, this new method will provide a crucial tool for comparing those observations with the fundamental laws of nature, helping to paint a more precise and accurate picture of the matter that makes up our universe.

Technical Summary: Femtoscale imaging of the proton with Ioffe-time distributions

Problem Statement
Understanding the three-dimensional structure of hadrons, specifically how quarks and gluons are distributed within protons, is a central objective of modern nuclear physics. While Generalized Parton Distributions (GPDs), $H(x, \xi, t)$ , provide a comprehensive description of hadron structure by correlating longitudinal momentum fraction ( $x$ ), skewness ( $\xi$ ), and momentum transfer ( $t$ ), extracting them from experimental data is challenging. The relationship between experimentally measurable Compton Form Factors (CFFs) and GPDs involves an ill-posed inverse problem, particularly due to singularities in the integral relations at $x = \pm \xi$ . Furthermore, traditional lattice QCD calculations often rely on extracting GPDs in momentum fraction space, which can be computationally difficult and subject to model-dependent constraints.

Methodology
The authors propose a novel space-time description of hadron structure using Impact-Parameter Ioffe-time Distributions (IP-ITDs). This approach utilizes the Ioffe-time variable $\nu = P \cdot z$ (where $z$ is a space-like separation) as a proxy for interaction time, rather than relying solely on the momentum fraction $x$ .

Definitions: The paper defines the IP-ITD, $h(\nu, b_T)$ , as the Fourier transform of the impact-parameter distribution $h(x, b_T)$ with respect to $x$ . This links the transverse spatial distribution ( $b_T$ ) to the interaction time ( $\nu$ ).
Lattice QCD Inputs: The study utilizes lattice QCD calculations of pseudo-Generalized Ioffe-time Distributions (pseudo-GITDs) from a specific ensemble ( $N_f = 2+1$ clover-improved Wilson fermions, $a=0.094$ fm, $m_\pi = 358$ MeV). These pseudo-distributions depend on the space-like interval $z^2$ .
Data Analysis: To extract physical quantities, the authors employ Gaussian Process Regression (GPR) to model the dependence of the pseudo-Double Distributions (DDs) on $(\beta, \alpha, t)$ . This framework allows for robust uncertainty quantification and enforces the polynomiality condition of GPDs intrinsically.
Derivation of CFFs: A key theoretical derivation establishes a direct relationship between the Compton Form Factor (CFF) and the GITD. By utilizing the odd component of the GITD, $H^{(+)}(\nu, \xi\nu, t)$ , the authors derive integral relations (Eqs. 16 and 17) that express the real and imaginary parts of the CFF as Fourier-like integrals over $\nu$ . This approach amounts to a new forward problem, bypassing the need to explicitly extract the $x$ -dependent GPD $H(x, \xi, t)$ before calculating the CFF.
Radius Calculation: The Ioffe-time-dependent transverse radius, $r_H^2(\nu)$ , is calculated by taking the derivative of the GITD with respect to $t$ at $t=0$ .

Key Contributions

Introduction of IP-ITDs: The paper introduces the concept of Impact-Parameter Ioffe-time distributions, offering a frame-independent, intuitive space-time picture of the proton where the transverse spatial distribution is mapped against the interaction time.
First Lattice Extraction of CFFs: The authors perform the first extraction of the Compton Form Factor from lattice calculations. This is achieved by integrating the lattice-determined GITDs using the newly derived relations, avoiding the intermediate step of reconstructing the full $x$ -dependent GPD.
Ioffe-Time Dependent Radii: The paper provides the first numerical estimates of the Ioffe-time-dependent mean squared proton radii from lattice QCD.
Theoretical Framework: The work establishes a rigorous link between the experimentally accessible CFF and the lattice-calculable GITD, demonstrating that the CFF can be viewed as a Fourier transform of the CP-odd GITD component.

Results

Transverse Radii: The calculated transverse radii $r_H(\nu)$ show that shorter interaction times correspond to smaller transverse distributions, while longer interaction times probe wider transverse regions. This trend is consistent with the behavior of transverse distributions narrowing as the momentum fraction $x$ increases. The results are compared with the phenomenological Goloskokov-Kroll (GK) model; while the lattice results show smaller radii (potentially due to heavier quark masses in the simulation), the qualitative trends agree.
CFF Extraction: The extracted CFFs from the lattice data exhibit a qualitative shape consistent with the CFFs derived from the GK model. The analysis highlights that determining CFFs at low $\xi$ requires data at large $\nu$ , where the GITD reaches a minimum, indicating a need for future systematic control over large momentum extrapolations.
Validation: The lattice data for the imaginary part of the GITD at various $\xi$ values show consistent slopes at $\nu=0$ , validating the polynomiality constraints and the quality of the data.

Significance
The paper claims that this work provides a "natural and intuitive representation" of the three-dimensional structure of hadrons by shifting the focus from momentum fraction space to a space-time description. By deriving a direct expression for CFFs in terms of GITDs, the authors formulate a new forward problem that avoids the ill-posed inverse problem of extracting GPDs from CFFs. This approach allows for a more direct comparison between lattice QCD results and experimental observables (DVCS data). The authors assert that working directly in Ioffe-time space enables radii to be determined from reliable interpolations, offering higher fidelity than momentum fraction space. Ultimately, this methodology complements the experimental programs at Jefferson Lab and the future Electron Ion Collider (EIC) by providing a first-principles, frame-independent tool to map the femtoscale structure of the proton.

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