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Proton and neutron electromagnetic form factors from lattice QCD in the continuum limit

This study computes the proton and neutron electromagnetic form factors in the continuum limit using $N_\mathrm{f}=2+1+1$ twisted mass lattice QCD at physical pion masses, employing multi-state fits and advanced noise-reduction techniques to precisely determine electric and magnetic radii and magnetic moments with controlled statistical and systematic uncertainties.

Original authors: Constantia Alexandrou, Simone Bacchio, Giannis Koutsou, Bhavna Prasad, Gregoris Spanoudes

Published 2026-07-01

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CC BY 4.0

Original authors: Constantia Alexandrou, Simone Bacchio, Giannis Koutsou, Bhavna Prasad, Gregoris Spanoudes

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

Imagine the proton and neutron not as solid, smooth marbles, but as bustling, fuzzy clouds of tiny, energetic particles called quarks. These clouds are held together by a force so strong it's like a rubber band that never breaks. Scientists want to understand exactly how these clouds are shaped, how big they are, and how they react when you "poke" them with a magnetic or electric field.

This paper is like a high-resolution 3D map of those clouds, created by a team of scientists using a super-powerful digital microscope called Lattice QCD.

Here is a breakdown of what they did and found, using simple analogies:

1. The Digital Sandbox (Lattice QCD)

To study these particles, the scientists couldn't just look at them in a lab; they had to build a digital universe.

The Grid: Imagine a giant 3D chessboard (a grid) filling all of space. The particles (quarks) live on the intersections of this grid.
The Simulation: They ran a massive computer simulation on this grid to see how the particles move and interact.
The "Physical" Setting: In the past, these simulations were like playing a video game on "Easy Mode" with the wrong physics settings. This team tuned their simulation to "Real Life Mode," using the exact masses of the particles found in nature. This means they didn't have to guess how the results would change if they tweaked the settings later.

2. The Three Lenses (Continuum Limit)

One problem with digital grids is that they are made of squares, not smooth lines. If the squares are too big, the picture looks blocky and blurry.

The Solution: The team didn't just use one grid size. They used three different grid sizes (like taking photos with a low-resolution camera, a medium one, and a high-definition one).
The Magic Trick: By comparing the results from all three, they could mathematically "smooth out" the blockiness to see what the particles look like in a perfectly smooth, real-world universe. This is called taking the "continuum limit."

3. The "Ghost" Problem (Disconnected Contributions)

When calculating how the proton reacts to a magnetic field, there are two types of interactions:

The Direct Path (Connected): Like a ball bouncing off a wall. This is easy to calculate.
The Ghost Path (Disconnected): Imagine the ball briefly turning into a cloud of smoke, swirling around, and then turning back into a ball. In physics, this is a "quark loop" that appears and disappears. It's incredibly hard to calculate because it's like trying to measure the weight of a ghost.
The Fix: The team used advanced statistical tricks (like "hierarchical probing," which is like using a very organized flashlight to scan a dark room) to reduce the "noise" and see these ghostly loops clearly.

4. The "Excited State" Noise

When you poke a proton, it doesn't just sit there; it wiggles and vibrates.

The Problem: If you measure the proton too quickly after poking it, you might catch it mid-wiggle, which gives you a blurry picture of its true shape.
The Fix: The team waited longer and longer after the "poke" (increasing the time separation) to let the proton settle down into its calmest, most stable state (the "ground state"). They used complex math to filter out the wiggles and isolate the true shape.

5. What They Found (The Results)

By combining all these techniques, they measured the "shape" and "size" of the proton and neutron with high precision. Think of these as the radius (how big the cloud is) and the magnetic moment (how strong its magnet is).

Proton Size: They found the proton's electric radius is about 0.86 femtometers (a femtometer is one-quadrillionth of a meter).
Neutron Size: Surprisingly, the neutron (which has no electric charge) still has a shape. Its electric radius is slightly negative (meaning the charge is distributed in a specific way), and its magnetic radius is about 0.91 femtometers.
Magnetic Strength: They measured how strongly these particles act like magnets. The proton's magnetic moment is about 2.85, and the neutron's is about -1.82.

6. Why This Matters

The paper claims that their results match what we see in real-world experiments (like smashing particles together in accelerators).

The "Radius Puzzle": There has been a long-standing mystery in physics about the exact size of the proton (some experiments say it's bigger, some say smaller). This paper's result sits right in the middle of those conflicting measurements, helping to clarify the picture.
Validation: Because they used a method that requires no "guessing" (no extrapolation from fake masses), their results serve as a gold-standard benchmark. It proves that our digital models of the universe are accurate enough to predict the behavior of matter from first principles.

In short: The team built a perfect digital twin of the proton and neutron, smoothed out the digital graininess, filtered out the background noise, and measured their size and magnetic strength. Their measurements are precise, match real-world data, and help solve a decades-old mystery about the size of the proton.

Technical Summary: Proton and Neutron Electromagnetic Form Factors in the Continuum Limit

Problem Statement
The electromagnetic form factors of the proton and neutron are fundamental quantities describing the internal structure of hadrons. While proton form factors are well-measured experimentally, neutron form factors remain less precise due to the lack of free neutron targets, requiring indirect extraction from electron-deuteron or electron-helium scattering. Lattice Quantum Chromodynamics (QCD) offers a first-principles approach to compute these quantities. However, previous lattice calculations often faced limitations: they either lacked a continuum extrapolation, relied on chiral extrapolations from unphysical quark masses, or did not include disconnected contributions (isoscalar terms) with sufficient precision. This work addresses these gaps by computing the form factors directly at the physical pion mass and taking the continuum limit using ensembles with physical quark masses.

Methodology
The authors employ Lattice QCD with $N_f=2+1+1$ twisted mass clover-improved fermions. The simulation utilizes three gauge ensembles with lattice spacings of $a=0.080$ fm, $0.068$ fm, and $0.057$ fm, all tuned to physical quark masses (the "physical point").

Correlation Functions: Nucleon matrix elements are extracted from two- and three-point correlation functions. To suppress excited state contamination, the authors utilize multiple sink-source time separations ranging from 0.5 fm to 1.5 fm.
Excited State Analysis: A rigorous multi-state fit analysis is performed. Unlike standard plateau fits, this approach allows for different excited state energies in the two-point and three-point functions, acknowledging that the electromagnetic current may couple differently to multi-particle states (e.g., pion-nucleon states) than the interpolating field.
Disconnected Contributions: The isoscalar contributions, which involve quark loops (disconnected diagrams), are computed using high-statistics techniques to mitigate stochastic noise. These include low-mode deflation, color-spin dilution, and hierarchical probing. The disconnected contributions are computed using a local vector current, which requires renormalization, while connected contributions use the conserved vector current.
Renormalization: Renormalization factors for flavor non-singlet and singlet vector operators are determined using the RI'-MOM scheme. The analysis confirms that the singlet and non-singlet factors coincide, simplifying the renormalization of the isoscalar current.
Extraction of Observables: The form factors are extracted using both dipole and $z$ -expansion Ansätze. The authors employ a "one-step" approach, fitting the $Q^2$ -dependence and the continuum extrapolation (in $a^2$ ) simultaneously. Model averaging is used to combine results from different fit ranges and parameterizations, assigning weights based on the Akaike Information Criterion (AIC) to derive final statistical and systematic errors.

Key Contributions

Continuum Limit at Physical Point: This is one of the first calculations to achieve the continuum limit for nucleon electromagnetic form factors using ensembles with physical pion masses, thereby eliminating the need for chiral extrapolations.
Comprehensive Excited State Control: The use of multi-state fits with varying time separations and the allowance for distinct excited state energies in two- and three-point functions provides a more robust extraction of ground-state matrix elements compared to previous studies.
Inclusion of Disconnected Contributions: The study includes a high-statistics analysis of disconnected contributions to the isoscalar form factors, utilizing advanced noise-reduction techniques (hierarchical probing, deflation) to achieve precision comparable to connected contributions.
Systematic Error Estimation: The paper provides a detailed breakdown of systematic uncertainties arising from fit ranges, model dependence (dipole vs. $z$ -expansion), and continuum extrapolation, utilizing model averaging to quantify these effects.

Results
The authors present the electric and magnetic radii and magnetic moments for the proton and neutron in the continuum limit. The results are quoted with statistical and systematic errors (the latter including fit and continuum extrapolation uncertainties):

Proton Electric Radius: $\sqrt{\langle r^2_E \rangle_p} = 0.860(38)(23)$ fm.
Neutron Electric Radius: $\langle r^2_E \rangle_n = -0.147(48)$ fm $^2$ .
Proton Magnetic Radius: $\sqrt{\langle r^2_M \rangle_p} = 0.870(53)(15)$ fm.
Neutron Magnetic Radius: $\sqrt{\langle r^2_M \rangle_n} = 0.913(67)(19)$ fm.
Proton Magnetic Moment: $\mu_p = 2.849(92)(52)$ .
Neutron Magnetic Moment: $\mu_n = -1.819(76)(29)$ .

Additionally, the authors compute the Zemach and Friar radii, which are relevant for hyperfine splitting calculations:

Proton Zemach Radius: $r_Z^p = 1.091(34)$ fm.
Neutron Zemach Radius: $r_Z^n = -0.0405(81)$ fm.

The results for the radii and magnetic moments are consistent with Particle Data Group (PDG) values within the combined statistical and systematic errors. The calculated proton electric radius carries an uncertainty that overlaps with both the larger values from historical electron-proton scattering and the smaller values from muonic hydrogen spectroscopy.

Significance and Claims
The paper claims that these results provide a benchmark for lattice QCD formulations, validating non-perturbative calculations against experimental data. By computing the form factors directly at the physical point and in the continuum limit, the study demonstrates the capability of lattice QCD to evaluate hadron structure quantities that are less well known experimentally, particularly for the neutron.

The authors modestly note that while their results agree with experimental data, the precision of the proton electric radius is currently limited by statistical and systematic errors. They suggest that increasing the accuracy of individual ensembles and including ensembles with smaller lattice spacings and larger physical volumes (to access momenta closer to $Q^2=0$ ) would be necessary to further reduce these errors and potentially resolve discrepancies between different experimental determinations of the proton radius. The work also serves as a validation of the methodology for including disconnected contributions and performing rigorous excited state analyses in lattice QCD.