A team of physicists has achieved the first determination of an energy-dependent form factor for a two-hadron scattering state, using Quantum Chromodynamics (QCD). This breakthrough represents a crucial step towards understanding the electromagnetic structure of multi-hadron systems and resonances from first principles. Specifically, the study calculated the QCD contribution to the forward electromagnetic amplitude of the reaction $\pi^+\pi^+ + \gamma \to \pi^+\pi^+$ using lattice QCD with a pion mass of approximately 400 MeV.

The difficulty of this calculation lies in the fact that lattice QCD simulations are performed in a finite Euclidean spacetime, where asymptotic scattering states are not directly accessible. To overcome this limitation, the researchers applied a finite-volume formalism that allows for the extraction of infinite-volume scattering amplitudes from finite-volume data. This method requires two main components: the discrete finite-volume energy spectrum and finite-volume matrix elements of the electromagnetic current. The team calculated three-point correlation functions coupling $\pi^+\pi^+$ finite-volume states to extract the corresponding electromagnetic matrix elements.

By combining these new results with the previously determined energy spectrum, the researchers were able to constrain the infinite-volume $\pi^+\pi^+ + \gamma \to \pi^+\pi^+$ amplitude in the forward limit. Using constraints from Lorentz symmetry, unitarity, and analyticity, this amplitude could be described in terms of a single real-valued, energy-dependent two-hadron form factor. The resulting amplitude and form factor agree with the Ward-Takahashi identity across all energies and moving frames considered, providing the first QCD validation of this finite-volume approach.

This work not only validates a key methodology for studying hadronic interactions but also opens a promising path towards first-principles studies of the electromagnetic structure and electroweak responses of resonances and multi-hadron bound states. The ability to calculate these properties directly from QCD is fundamental for a deeper understanding of the strong force and the structure of matter. Next steps will include applying this methodology to more complex systems and to pion masses closer to physical values.