Determining how strongly interacting two-particle systems respond to external currents is important for nuclear and particle physics. Electromagnetic probes reveal the internal structure of stable hadrons, but for unstable resonances, which live on the order of $10^{-23}$ seconds, standard approaches do not work. This work stands apart: there is almost no one nearby, the nearest neighbor is at a distance of 0.505, and the thematic group of the map includes quarks, kilonovae, magnetars, and neutron star mergers. The finite-volume formalism, developed earlier, promises to overcome this obstacle, but until now it had not been tested in a full QCD calculation.
A single lattice with 2+1 anisotropic Wilson fermions was used, with the temporal step 3.4 times finer than the spatial one, a pion mass of about 400 MeV, and a volume of $24^3$ × 128. Correlation functions were built using distillation with operators optimized from solving a generalized eigenvalue problem. Three-point correlators were measured for 20 π+π+ final states with four different total momenta. Current matrix elements were extracted by fitting over time slices, accounting for excited states, and then converted to the infinite-volume amplitude via the finite-volume formalism.
Twenty finite-volume matrix elements for the spatial component of the electromagnetic current were obtained. They agree with the prediction of the Ward–Takahashi identity with a total χ² = 42.7 for 20 data points (χ²/Ndata = 2.1). From these data, the infinite-volume amplitude $W_s$ and the energy-dependent form factor $A_s$ were reconstructed. Fitting $A_s$ with a constant gives m²_π $A_s$ = 6.7(3.3)×$10^{-4}$ with χ²/dof = 33.9/19 = 1.8, which is better than for the matrix elements themselves. The real and imaginary parts of $W_s$, determined from the lattice, match the Ward–Takahashi prediction at all energies and in all reference frames.
This work differs from its neighbors in that it is the first to implement the full cycle 'spectrum → matrix elements → infinite-volume amplitude' in lattice QCD. Neighboring studies either address spectroscopy or test the formalism in effective theories, but they do not provide an end-to-end result for two hadrons with a current. This is the first validation of an approach that had previously been checked only analytically and in simplified models. The method is now ready for application to systems with resonances and bound states, where the answer is not known in advance.
The next natural step is to move to nonzero momentum transfer, where the form factor carries information about the spatial distribution of charge and current. This will allow studying the structure of states such as a0(980), f0(980), and Tcc, whose nature remains controversial. It will also be necessary to include particles with spin and account for left-hand cuts to cover the deuteron and the dineutron. Extending the formalism to three or more particles will broaden the class of accessible processes.
The method will impact low-energy nuclear physics, where electroweak responses of two-nucleon systems are needed for interpreting neutrino experiments and searching for neutrinoless double beta decay. It is also important for hadron spectroscopy, as it allows distinguishing molecular states from compact quark states.
The immediate task is to compute form factors at nonzero momentum transfer for the same pion states. Then the method will be applied to channels with resonances, such as the ρ meson, and to two-nucleon systems.
This work is directly related to the unsolved problem of determining the internal structure of exotic hadrons and resonances from first principles. It also lays the foundation for computing nuclear matrix elements needed for quantitative interpretation of experiments searching for new physics.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
For the first time, the energy-dependent form factor for two-particle scattering π+π+ + γ → π+π+ has been determined from first principles in lattice QCD, at a pion mass of about 400 MeV. The finite-volume formalism, which relates the discrete spectrum and matrix elements of the electromagnetic current to the infinite-volume amplitude, was employed. Twenty finite-volume matrix elements were obtained, consistent with the Ward–Takahashi identity, and the infinite-volume amplitude W_s and form factor A_s were reconstructed.
Where its strength is
This work realizes for the first time the full cycle 'spectrum → matrix elements → infinite-volume amplitude' in lattice QCD with an electromagnetic current, validating the finite-volume formalism for unstable states. It opens the way to studying the structure of resonances and multi-hadron states from first principles.
Where it could go next
Worth trying
Why this may matter
This result could serve as a foundation for a systematic study of the electromagnetic structure of unstable hadrons, including resonances, from first principles of QCD, which is important for interpreting electron scattering experiments on hadrons and for understanding the internal dynamics of strong interactions.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.48, against a usual 0.64. It looks like few have drilled nearby.
Resonances live for about $10^{-23}$ seconds, which is a billion times shorter than the time it takes light to cross an atomic nucleus.