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Complex-momentum form factors from lattice QCD

This paper proposes a method to directly compute hadronic form factors at complex momenta from lattice QCD position-space correlators using a bilateral Laplace transform, thereby extending the accessible kinematic domain to improve precision in observables like the charge radius without requiring analytic continuation or solving inverse problems.

Original authors: Maxwell T. Hansen, Gurtej Kanwar

Published 2026-10-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Maxwell T. Hansen, Gurtej Kanwar

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

In the subatomic world, matter is not made of solid, indivisible spheres but of a dynamic, seething soup of particles and forces. At the heart of this complexity lies the strong nuclear force, the glue that binds quarks together to form protons, neutrons, and the pions that mediate interactions between them. To understand how these particles behave, physicists rely on a mathematical map called a form factor. Think of this map as a detailed profile of a particle's internal structure, revealing how its electric charge is distributed and how it responds when struck by another particle. For decades, scientists have tried to draw this map with increasing precision, but they have faced a stubborn limitation: they could only measure the particle's response to specific, discrete "kicks" or momentum transfers. It was as if they were trying to understand the shape of a complex object by only touching it at a few fixed points, leaving vast gaps in their knowledge.

This gap is particularly problematic when trying to understand the very center of the particle, where the charge radius is defined. Current methods often require guessing the shape of the curve between the measured points, introducing uncertainty. Furthermore, the standard tools of the trade, which use a mathematical framework called Euclidean space, struggle to see into the "timelike" region—a domain where particles can briefly transform into other forms of matter. This region is crucial for understanding how particles decay and interact, yet it has remained largely hidden from direct observation in computer simulations. The challenge has been to find a way to fill in the missing pieces of the map without having to solve an impossible puzzle or guess the answer.

A new approach, developed by researchers at the University of Edinburgh, offers a way to see the particle's structure in a continuous, fluid manner rather than at isolated points. Instead of forcing the particle to respond to standard, real-valued momentum transfers, the team proposed a method that allows the momentum to take on complex values. In the language of mathematics, this means introducing an imaginary component to the momentum, a concept that sounds abstract but has a very concrete effect on the computer simulations. By applying a specific mathematical transformation to the data collected from these simulations, the researchers found they could access a vast, continuous domain of the particle's behavior. This domain includes the elusive timelike region, allowing them to see how the particle behaves in energy ranges that were previously inaccessible to their standard tools.

The core of this discovery relies on the way particles fade away in computer simulations. In these digital experiments, the influence of a particle drops off exponentially as you move away from its source. The researchers realized that this rapid fading is a powerful feature, not a bug. It guarantees that a specific type of mathematical operation, known as a bilateral Laplace transform, will converge and produce a stable result, even when the momentum values are complex. By applying this operation directly to the position-space data—the raw coordinates of where particles are located in the simulation—they could extract the form factor at any point within a large, continuous region of the complex plane. This region extends from the familiar spacelike measurements into the timelike territory, covering a half-plane of possibilities that includes the critical point where the momentum transfer is zero.

To test whether this theoretical idea could work in practice, the team turned to a simplified model of particle physics known as the O(3) nonlinear sigma model. This model behaves similarly to the real world in many ways but is much easier to simulate on a computer. Using millions of computer-generated snapshots of this model, they applied their new method to calculate the form factor of a particle analogous to the pion. The results were striking. When they used the standard method of simply cutting off the simulation data at the edges of their computer box, the results were messy and unreliable, especially for the complex momentum values. However, when they used a more sophisticated technique to fit the long-distance tails of the data and extend them beyond the box, the method worked beautifully. The calculated form factors matched the known theoretical predictions with high precision, even in the difficult timelike regions.

The study demonstrates that it is possible to access the full, continuous structure of a particle's form factor without solving an inverse problem or analytically continuing the time coordinates of the simulation. This is a significant shift in how these calculations are performed. Previously, accessing the timelike region required reconstructing the entire spectrum of possible particle states, a process that is computationally expensive and prone to errors. The new method bypasses this by working directly with the position data, using the exponential decay of the particle's influence to its advantage. The researchers found that by carefully handling the finite size of their computer simulations, they could extract accurate results that span a continuous range of momentum transfers, including the sub-threshold timelike region where particles can momentarily exist as virtual states.

While the initial tests were performed on a simplified model, the implications for real-world physics are profound. The method provides a new bridge between the spacelike measurements, which are currently the gold standard in lattice quantum chromodynamics, and the timelike region, which is essential for understanding particle decays and resonances. It offers a way to improve the determination of fundamental properties, such as the charge radius of the pion, by providing a much denser set of data points around the zero-momentum limit. This could help resolve long-standing puzzles in particle physics, such as discrepancies in measurements of the proton's radius. The researchers also noted that the technique could be applied to other hadronic form factors and current insertions, potentially opening up new avenues for exploring the analytic structure of hadronic observables.

The path forward involves applying this technique to full-scale lattice quantum chromodynamics simulations, which describe the real world of protons and neutrons. This will require overcoming practical challenges, such as managing the statistical noise inherent in these complex calculations and ensuring that the finite size of the simulation boxes does not distort the results. The team suggests that using twisted boundary conditions, which effectively introduce a complex chemical potential, could be a viable way to implement this method in realistic simulations. If successful, this approach could transform how physicists map the internal structure of matter, turning a sparse collection of data points into a rich, continuous portrait of the subatomic world. The work stands as a proof of concept that the complex-momentum domain is not just a mathematical curiosity but a practical tool for uncovering the hidden details of the strong nuclear force.

Technical Summary: Complex-momentum form factors from lattice QCD

Problem Statement
Standard lattice QCD determinations of hadronic form factors (such as the pion electromagnetic form factor) typically rely on constructing three-point functions with discrete spatial momenta. In a finite periodic volume, these momenta are quantized as Fourier modes ( $p = 2\pi n/L$ ), restricting form factor measurements to a discrete set of spacelike momentum transfers ( $Q^2 \ge 0$ ). Extracting physical quantities like charge radii requires fitting data near $Q^2=0$ , which introduces model dependence and systematic uncertainties. While methods exist to access timelike kinematics ( $Q^2 < 0$ ) via spectral reconstruction or Lellouch-Lüscher relations, these often involve solving inverse problems or are limited to specific energy levels. There is a need for a method to access form factors continuously across the complex $Q^2$ plane, particularly in the region near $Q^2=0$ and in the sub-threshold timelike region, without solving inverse problems or analytically continuing Euclidean time coordinates.

Methodology
The authors propose a method to access form factors at complex-valued momenta directly from position-space Euclidean correlators. The core idea is to apply a bilateral Laplace transform (a Fourier transform evaluated at complex momenta) to the position-space data.

Infinite-Volume Formalism:
- The method relies on the exponential decay of hadronic correlators in position space. For a correlator $G(x)$ decaying as $e^{-m|x|}$ , the Fourier transform converges for complex momenta $p = k + i\kappa$ where $\|\text{Im } p\| < m$ .
- By analytically continuing the spatial momenta to complex values, the authors demonstrate that the ground-state saturation relations for two- and three-point functions remain valid within a specific convergence domain.
- The form factor $F_\pi(Q^2)$ is isolated by forming ratios of three-point to two-point functions (or ratios of three-point functions with different momentum configurations) at large Euclidean time separations. The complex momentum transfer $Q^2$ is determined by the complex momenta $p$ and $p'$ via the standard relativistic relation $Q^2 = (p'-p)^2 - (E(p')-E(p))^2$ .
- Accessible Domain:
  - For back-to-back kinematics ( $p = -p'$ ), the accessible region in the complex $Q^2$ plane is a parabola defined by $(\text{Im } Q^2)^2 < 16m_\pi^2(\text{Re } Q^2 + 4m_\pi^2)$ .
  - For general kinematics, the accessible domain expands to the entire half-plane $\text{Re } Q^2 > -4m_\pi^2$ . This includes the sub-threshold timelike region ( $-4m_\pi^2 < Q^2 < 0$ ) using purely imaginary momenta.
Finite-Volume Estimators:
- Direct application of complex Fourier transforms to periodic lattice data is problematic because the transform does not project onto a single finite-volume eigenstate; at large Euclidean time, the signal is dominated by the zero-momentum ground state due to the periodicity of the lattice.
- The authors propose two strategies to mitigate this:
  - Position-Space Fits and Extrapolation: Fit the long-distance tails of the position-space correlators to an infinite-volume functional form (e.g., based on spectral decomposition) and extend the data beyond the lattice volume before applying the complex Fourier transform. This effectively removes the "missing tail" contributions that cause finite-volume contamination.
  - Spatial Chemical Potential (Twisted Boundary Conditions): Implement complex twisted boundary conditions on the quark fields. This modifies the transfer matrix to directly support the desired complex momentum states. For the pion form factor, an isospin chemical potential can be used, which avoids the sign problem in QCD.

Key Results
The authors test the methodology numerically using the two-dimensional $O(3)$ nonlinear sigma model (NLSM), which possesses an exact analytic solution for the form factor and shares the mass-gap and isospin structure of the pion sector in QCD.

Numerical Validation: Using Monte Carlo data from lattices with spatial extents $L=128$ and $L=256$ , the authors computed the form factor for both real and complex momentum transfers.
Finite-Volume Effects: They demonstrated that a direct truncated Fourier transform of the periodic data yields poor results for complex momenta, converging to the ground-state energy rather than the desired complex-energy state.
Success of Tail Fitting: By fitting the position-space tails and replacing the data outside the lattice volume with the fitted infinite-volume form, the method successfully recovered the form factor. The results were stable across different lattice volumes ( $L=128$ vs $L=256$ ) and consistent with the analytic continuum solution within statistical uncertainties.
Comparison of Estimators: The study compared two ratio estimators: one using the two-point function in the denominator and one using a zero-momentum-transfer three-point function. Both yielded consistent results, though the latter showed specific advantages at $Q^2=0$ .

Significance and Claims
The paper claims to demonstrate a viable pathway to access hadronic form factors in a continuous complex domain without solving inverse problems or analytically continuing time coordinates.

Bridging Kinematic Regions: The method provides a natural intermediary between standard spacelike lattice calculations ( $Q^2 \ge 0$ ) and timelike spectral reconstruction methods ( $Q^2 < -4m_\pi^2$ ). It allows for the direct sampling of the form factor in the complex neighborhood of $Q^2=0$ and the sub-threshold timelike region.
Phenomenological Utility: Access to continuous complex data could improve determinations of charge radii and other derivatives at the origin. It also provides richer data for fits based on analyticity, such as conformal $z$ -expansions, and may aid in constructing dispersive representations.
Modest Outlook: The authors emphasize that while the infinite-volume formalism is established, the practical application to lattice QCD requires careful handling of finite-volume effects. They note that their numerical test was in a simplified 2D model and that a full lattice QCD implementation would need to address statistical noise, lattice artifacts, and the specific challenges of applying complex twisted boundary conditions to sea quarks (which currently lacks established techniques). The work is presented as a proof of concept and a proposal for future investigation rather than a completed solution for current lattice QCD calculations.

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