We derive relations between finite-volume matrix elements and infinite-volume decay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-Luscher re
lation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes using lattice QCD. Unlike for two particles, even in the simplest approximation, one must solve integral equations to obtain the physical decay amplitude, a consequence of the nontrivial finite-state interactions. We first derive the result in a simplified theory with three identical particles, and then present the generalizations needed to study phenomenologically relevant three-pion decays. The specific processes we discuss are the CP-violating $K to 3pi$ weak decay, the isospin-breaking $eta to 3pi$ QCD transition, and the electromagnetic $gamma^*to 3pi$ amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic $g-2$.
We discuss a method to construct hadronic scattering and decay amplitudes from Euclidean correlators, by combining the approach of a regulated inverse Laplace transform with the work of Maiani and Testa. Revisiting the original result, we observe tha
t the key observation, i.e. that only threshold scattering information can be extracted at large separations, can be understood by interpreting the correlator as a spectral function, $rho(omega)$, convoluted with the Euclidean kernel, $e^{- omega t}$, which is sharply peaked at threshold. We therefore consider a modification in which a smooth step function, equal to one above a target energy, is inserted in the spectral decomposition. This can be achieved either through Backus-Gilbert-like methods or more directly using the variational approach. The result is a shifted resolution function, such that the large $t$ limit projects onto scattering or decay amplitudes above threshold. The utility of this method is highlighted through large $t$ expansions of both three- and four-point functions that include leading terms proportional to the real and imaginary parts (separately) of the target observable. This work also presents new results relevant for the un-modified correlator at threshold, including expressions for extracting the $N pi$ scattering length from four-point functions and a new strategy to organize the large $t$ expansion that exhibits better convergence than the expansion in powers of $1/t$.
Focusing on three-pion states with maximal isospin ($pi^+pi^+pi^+$), we present the first non-perturbative determination of an energy-dependent three-hadron scattering amplitude from first-principles QCD. The calculation combines finite-volume three-
hadron energies, extracted using numerical lattice QCD, with a relativistic finite-volume formalism, required to interpret the results. To fully implement the latter, we also solve integral equations that relate an intermediate three-body K matrix to the physical three-hadron scattering amplitude. The resulting amplitude shows rich analytic structure and a complicated dependence on the two-pion invariant masses, represented here via Dalitz-like plots of the scattering rate.
Quantum computing may offer the opportunity to simulate strongly-interacting field theories, such as quantum chromodynamics, with physical time evolution. This would give access to Minkowski-signature correlators, in contrast to the Euclidean calcula
tions routinely performed at present. However, as with present-day calculations, quantum computation strategies still require the restriction to a finite system size, including a finite, usually periodic, spatial volume. In this work, we investigate the consequences of this in the extraction of hadronic and Compton-like scattering amplitudes. Using the framework presented in Phys. Rev. D101 014509 (2020), we quantify the volume effects for various $1+1$D Minkowski-signature quantities and show that these can be a significant source of systematic uncertainty, even for volumes that are very large by the standards of present-day Euclidean calculations. We then present an improvement strategy, based in the fact that the finite volume has a reduced symmetry. This implies that kinematic points, which yield the same Lorentz invariants, may still be physically distinct in the finite-volume system. As we demonstrate, both numerically and analytically, averaging over such sets can significantly suppress the unwanted volume distortions and improve the extraction of the physical scattering amplitudes.
The leading finite-volume and thermal effects, arising in numerical lattice QCD calculations of $a^{text{HVP,LO}}_mu equiv (g-2)^{text{HVP,LO}}_mu/2$, are determined to all orders with respect to the interactions of a generic, relativistic effective
field theory of pions. In contrast to earlier work based in the finite-volume Hamiltonian, the results presented here are derived by formally summing all Feynman diagrams contributing to the Euclidean electromagnetic-current two-point function, with any number of internal pion loops and interaction vertices. As was already found in our previous publication, the leading finite-volume corrections to $a^{text{HVP,LO}}_mu$ scale as $exp[- m L]$ where $m$ is the pion mass and $L$ is the length of the three periodic spatial directions. In this work we additionally control the two sub-leading exponentials, scaling as $exp[- sqrt{2} m L]$ and $exp[- sqrt{3} m L]$. As with the leading term, the coefficient of these is given by the forward Compton amplitude of the pion, meaning that all details of the effective theory drop out of the final result. Thermal effects are additionally considered, and found to be sub-percent-level for typical lattice calculations. All finite-volume corrections are presented both for $a^{text{HVP,LO}}_mu$ and for each time slice of the two-point function, with the latter expected to be particularly useful in correcting small to intermediate current separations, for which the series of exponentials exhibits good convergence.
We present a generalization of the relativistic, finite-volume, three-particle quantization condition for non-identical pions in isosymmetric QCD. The resulting formalism allows one to use discrete finite-volume energies, determined using lattice QCD
, to constrain scattering amplitudes for all possible values of two- and three-pion isospin. As for the case of identical pions considered previously, the result splits into two steps: The first defines a non-perturbative function with roots equal to the allowed energies, $E_n(L)$, in a given cubic volume with side-length $L$. This function depends on an intermediate three-body quantity, denoted $mathcal{K}_{mathrm{df},3}$, which can thus be constrained from lattice QCD input. The second step is a set of integral equations relating $mathcal{K}_{mathrm{df},3}$ to the physical scattering amplitude, $mathcal M_3$. Both of the key relations, $E_n(L) leftrightarrow mathcal{K}_{mathrm{df},3}$ and $mathcal{K}_{mathrm{df},3}leftrightarrow mathcal M_3$, are shown to be block-diagonal in the basis of definite three-pion isospin, $I_{pi pi pi}$, so that one in fact recovers four independent relations, corresponding to $I_{pi pi pi}=0,1,2,3$. We also provide the generalized threshold expansion of $mathcal{K}_{mathrm{df},3}$ for all channels, as well as parameterizations for all three-pion resonances present for $I_{pipipi}=0$ and $I_{pipipi}=1$. As an example of the utility of the generalized formalism, we present a toy implementation of the quantization condition for $I_{pipipi}=0$, focusing on the quantum numbers of the $omega$ and $h_1$ resonances.
Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $mathbf{2}+mathcal{J}tomathbf{2}$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactio
ns. Working in a finite cubic volume with periodicity $L$, we derive a $1/L$ expansion of the matrix element through $mathcal O(1/L^5)$ and find that it is governed by two universal current-dependent parameters, the scalar charge and the threshold two-particle form factor. We confirm the result through a numerical study of the general formalism and additionally through an independent perturbative calculation. We further demonstrate a consistency with the Feynman-Hellmann theorem, which can be used to relate the $1/L$ expansions of the ground-state energy and matrix element. The latter gives a simple insight into why the leading volume corrections to the matrix element have the same scaling as those in the energy, $1/L^3$, in contradiction to earlier work, which found a $1/L^2$ contribution to the matrix element. We show here that such a term arises at intermediate stages in the perturbative calculation, but cancels in the final result.
A relation is presented between single-hadron long-range matrix elements defined in a finite Euclidean spacetime, and the corresponding infinite-volume Minkowski amplitudes. This relation is valid in the kinematic region where any number of two-hadro
n states can simultaneously go on shell, so that the effects of strongly-coupled intermediate channels are included. These channels can consist of non-identical particles with arbitrary intrinsic spins. The result accommodates general Lorentz structures as well as non-zero momentum transfer for the two external currents inserted between the single-hadron states. The formalism, therefore, generalizes the work by Christ et al.~[Phys.Rev. D91 114510 (2015)], and extends the reach of lattice quantum chromodynamics (QCD) to a wide class of new observables beyond meson mixing and rare decays. Applications include Compton scattering of the pion ($pi gamma^star to [pi pi, K overline K] to pi gamma^star$), kaon ($K gamma^star to [pi K, eta K] to K gamma^star$) and nucleon ($N gamma^star to N pi to N gamma^star$), as well as double-$beta$ decays, and radiative corrections to the single-$beta$ decay, of QCD-stable hadrons. The framework presented will further facilitate generalization of the result to studies of nuclear amplitudes involving two currents from lattice QCD.
Recently, a framework has been developed to study form factors of two-hadron states probed by an external current. The method is based on relating finite-volume matrix elements, computed using numerical lattice QCD, to the corresponding infinite-volu
me observables. As the formalism is complicated, it is important to provide non-trivial checks on the final results and also to explore limiting cases in which more straightforward predications may be extracted. In this work we provide examples on both fronts. First, we show that, in the case of a conserved vector current, the formalism ensures that the finite-volume matrix element of the conserved charge is volume-independent and equal to the total charge of the two-particle state. Second, we study the implications for a two-particle bound state. We demonstrate that the infinite-volume limit reproduces the expected matrix element and derive the leading finite-volume corrections to this result for a scalar current. Finally, we provide numerical estimates for the expected size of volume effects in future lattice QCD calculations of the deuterons scalar charge. We find that these effects completely dominate the infinite-volume result for realistic lattice volumes and that applying the present formalism, to analytically remove an infinite-series of leading volume corrections, is crucial to reliably extract the infinite-volume charge of the state.
In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $textbf 2 + mathcal J to textbf 2$ transition amplitudes. We also give various details relevant for the implementati
on of this formalism in a realistic numerical lattice QCD calculation. Particular focus is given to the role of single-particle form factors in disentangling finite-volume effects from the triangle diagram that arise when $mathcal J$ couples to one of the two hadrons. This also leads to a new finite-volume function, denoted $G$, the numerical evaluation of which is described in detail. As an example we discuss the determination of the $pi pi + mathcal J to pi pi$ amplitude in the $rho$ channel, for which the single-pion form factor, $F_pi(Q^2)$, as well as the scattering phase, $delta_{pipi}$, are required to remove all power-law finite-volume effects. The formalism presented here holds for local currents with arbitrary Lorentz structure, and we give specific examples of insertions with up to two Lorentz indices.
