arXiv subjects

Andrew W. Jackura

Publications and source records attributed to Andrew W. Jackura.

At least 19 recordsLinked to original sources

arXiv2026

Isotensor $πππ$ scattering with a $ρ$ resonant subsystem from QCD

This work presents a lattice quantum chromodynamics (QCD) determination of $πππ$ scattering amplitudes for the isospin-2 channel with angular momentum and parity $J^{P} = 1^+$. The calculation is performed using unphysically heavy light-quark masses, corresponding to a pion mass of $m_π \approx 400$~MeV, for which the $ρ$ meson manifests as a narrow resonance. The analysis employs a previously developed formalism that non-perturbatively relates the finite-volume spectra of two- and three-pion systems to their infinite-volume scattering amplitudes. We combine earlier lattice QCD results for $ππ$ systems with isospins 1 and 2 with new determinations of the isospin-2 three-pion finite-volume spectrum, obtained for three cubic, periodic lattice volumes with box lengths ranging from $\approx 4/m_π$ to $6/m_π$. These combined data constrain the low-lying partial waves of the infinite-volume three-pion K matrix, from which we solve a set of coupled integral equations to extract the corresponding scattering amplitudes.

hep-lat↗

arXiv2026

An improved partial-wave projection of the one-particle exchange in relativistic three-body scattering

A critical component of relativistic three-body scattering amplitudes is the one-particle exchange (OPE) process, wherein a single particle is exchanged between incoming and outgoing two-body subsystems. We present an improved, finite-sum expression for the partial-wave OPE that is valid for three massive, spinless particles in arbitrary angular momentum configurations. A collection of analytic and numerical checks demonstrate that our finite-sum expression reproduces the known properties of the partial-wave OPE, including its threshold scaling behavior and singularity structure. Compared with previous work, this result more efficiently handles the rapid proliferation of partial waves contributing to any total angular momentum $J$, enabling the construction of robust wavesets for three-body amplitude analyses.

hep-ph↗

arXiv2026

Resolving the structure of bound states using lattice quantum field theories

This work presents the first lattice calculation of a two-to-two particle matrix element of a local current. This exploratory calculation is performed using a leading-order pionless effective field theory of two nucleons in a finite 3D spatial volume, where the Hamiltonian can be diagonalized exactly for moderate volumes. By considering a range of couplings where the theory supports a deuteron-like bound state, we determine the finite-volume spectra and matrix elements of the conserved local vector current. Using the Lüscher formalism, we constrain the infinite-volume, purely hadronic amplitude for this theory. Using previously derived formalism, we then map the finite-volume matrix elements to scattering amplitudes describing a reaction coupling two-particle states via a current insertion, $\2+\Jc \to \2$. We then use a recently derived relation between this class of amplitudes and the bound-state elastic form factor to directly constrain the infinite-volume form factor. By varying over a range of values of the coupling of the theory, we explore the effects of this analysis for deep-bound states and shallow-bound states. We reproduce the expected result that for deep bound states, the finite-volume formalism is largely unnecessary, while for shallow bound states, it is absolutely critical to obtain a sensible result. We present a detailed outline of the analysis of this class of matrix elements, including the determination of the charge radius of the bound state. In the shallow bound state limit, we find good agreement with the prediction stemming from the anomalous threshold.

hep-lat↗

arXiv2025

Symmetrizing relativistic three-body partial wave amplitudes

S matrix principles and symmetries impose constraints on three-particle scattering amplitudes, which can be formulated as a class of integral equations for their partial wave projections. However, these amplitudes are typically expressed in an asymmetric basis, where one of the initial and final state particles is singled out, and all quantum numbers are defined relative to this spectator. In this work, we show how to construct symmetric partial wave amplitudes, which have been symmetrized over all possible spectator combinations, using their asymmetric counterparts and sets of recoupling coefficients. We derive these recoupling coefficients for arbitrary angular momentum and isospin for arbitrary systems of spinless particles with SU(2) flavor symmetry. We propose a simple intensity observable suitable for visualizing the structure of three-body dynamics in Dalitz distributions. Finally, we provide some numerical examples of Dalitz distributions relevant for future lattice QCD calculations of $3π$ systems and provide numerical evidence that the symmetrization procedure is consistent with expected symmetries of Dalitz plots.

hep-ph↗

arXiv2024

Finite-volume quantization condition from the $N/D$ representation

We propose a new model-independent method for determining hadronic resonances from lattice QCD. The formalism is derived from the general principles of unitarity and analyticity, as encoded in the $N/D$ representation of a partial-wave two-body amplitude. The associated quantization condition relates the finite-volume spectrum to the infinite-volume numerator, $\mathcal{N}$, used to reconstruct the scattering amplitude from dispersive relations. Unlike the original Lüscher condition, this new formalism is valid for energies coinciding with the left-hand cuts from arbitrary one- and multi-particle exchanges.

hep-lat↗

arXiv2024

Partial-wave projection of relativistic three-body amplitudes

We derive the integral equations for partial-wave projected three-body scattering amplitudes, starting from the integral equations for three-body amplitudes developed for lattice QCD analyses. The results, which hold for generic three-body systems of spinless particles, build upon the recently derived partial-wave projected one-particle exchange, a primary component of the relativistic framework proven to satisfy S matrix unitarity. We derive simplified expressions for factorizable short-distance interactions, $\mathcal{K}_{3}$, in two equivalent formalisms - one symmetric under particle interchange and one asymmetric. For the asymmetric case, we offer parameterizations useful for amplitude analysis. Finally, we examine toy models for $3π$ systems at unphysically heavy pion masses with total isospins 0,1, and 2.

hep-ph↗

arXiv2023

Partial-wave projection of the one-particle exchange in three-body scattering amplitudes

As the study of three-hadron physics from lattice QCD matures, it is necessary to develop proper analysis tools in order to reliably study a variety of phenomena, including resonance spectroscopy and nuclear structure. Reconstructing the three-particle scattering amplitude requires solving integral equations, which can be written in terms of data-constrained dynamical functions and physical on-shell quantities. The driving term in these equations is the so-called one-particle exchange, which leads to a kinematic divergence for particles on-mass-shell. A vital component in defining three-particle amplitudes with definite parity and total angular momentum, which are used in spectroscopic studies, is to project the one-particle exchange into definite partial waves. We present a general procedure to construct exact analytic partial wave projections of the one-particle exchange contribution for any system composed of three spinless hadrons. Our result allows one full control over the analytic structure of the projection, which we explore for some low-lying partial waves with applications to three pions.

hep-ph↗

arXiv2023

Evolution of Efimov States

The Efimov phenomenon manifests itself as an emergent discrete scaling symmetry in the quantum three-body problem. In the unitarity limit, it leads to an infinite tower of three-body bound states with energies forming a geometric sequence. In this work, we study the evolution of these so-called Efimov states using relativistic scattering theory. We identify them as poles of the three-particle $S$ matrix and trace their trajectories in the complex energy plane as they evolve from virtual states through bound states to resonances. We dial the scattering parameters toward the unitarity limit and observe the emergence of the universal scaling of energies and couplings -- a behavior known from the non-relativistic case. Interestingly, we find that Efimov resonances follow unusual, cyclic trajectories accumulating at the three-body threshold and then disappear at some values of the two-body scattering length. We propose a partial resolution to this "missing states" problem.

nucl-th↗

arXiv2022

Three-pion effects in $K^0-\bar{K}^0$ mixing

The rate of mixing between a neutral kaon and an anti-kaon ($K^0-\bar{K}^0$) is given, in part, by a long-range matrix element, defined with two insertions of the weak Hamiltonian separated by physical, Minkowski time evolution. For physical quark masses, the kaon mass lies above the two- and three-pion thresholds and, as a result, this long-range matrix element receives contributions from intermediate on-shell $2π$ and $3π$ states. These contributions cannot easily be captured in a finite Euclidean spacetime, meaning that such matrix elements are not directly accessible via lattice QCD. In this talk, we present a strategy for combining quantities that can be extracted in numerical lattice QCD calculations in order to reproduce the physical, infinite-volume long-range amplitude for $K^0-\bar{K}^0$. The key novelty relative to published work is that we fully include the effects of three-particle states that were previously neglected. The strategy is built on existing formalism for long-range matrix elements with two-particle intermediate states, together with the relativistic-field-theory finite-volume formalism for extracting three-hadron weak decays.

hep-lat↗

arXiv2022

Prospects for $γ^\star γ^\star \to ππ$ via lattice QCD

The $γ^\star γ^\star \to ππ$ scattering amplitude plays a key role in a wide range of phenomena, including understanding the inner structure of scalar resonances as well as constraining the hadronic contributions to the anomalous magnetic moment of the muon. In this work, we explain how the infinite-volume Minkowski amplitude can be constrained from finite-volume Euclidean correlation functions. The relationship between the finite-volume Euclidean correlation functions and the desired amplitude holds up to energies where $3π$ states can go on shell, and is exact up to exponentially small corrections that scale like $\mathcal{O}(e^{-m_πL})$, where $L$ is the spatial extent of the cubic volume and $m_π$ is the pion mass. In order to implement this formalism and remove all power-law finite volume errors, it is necessary to first obtain $ππ\to ππ$, $πγ^\star \to π$, $γ^\star \toππ$, and $ππγ^\star \toππ$ amplitudes; all of which can be determined via lattice quantum chromodynamic calculations.

hep-lat↗

arXiv2022

Three-body scattering and quantization conditions from $S$ matrix unitarity

Two methodologies have been presented in the literature which connect relativistic three-particle scattering amplitudes with lattice QCD spectra -- the ``relativistic effective field theory'' approach and the ``finite-volume unitarity'' method. While both methods have been shown to be equivalent in various works, it has not been shown how to arrive at the relativistic effective field theory results directly from $S$ matrix unitarity. In this work, we provide a simple proof of the relativistic effective field theory form of the scattering equations directly from unitarity. Motivated by the finite-volume unitarity approach, we then postulate a set of quantization conditions which relate the finite-volume energy spectra to the $K$ matrices which drive the short-distance physics in the scattering equations, obtaining all previously known results for three identical particles. This work also presents new relations which provide a new pathway to generalize the results to arbitrary systems.

hep-lat↗

arXiv2022

Novel approaches in Hadron Spectroscopy

The last two decades have witnessed the discovery of a myriad of new and unexpected hadrons. The future holds more surprises for us, thanks to new-generation experiments. Understanding the signals and determining the properties of the states requires a parallel theoretical effort. To make full use of available and forthcoming data, a careful amplitude modeling is required, together with a sound treatment of the statistical uncertainties, and a systematic survey of the model dependencies. We review the contributions made by the Joint Physics Analysis Center to the field of hadron spectroscopy.

hep-ph↗

arXiv2022

Snowmass white paper: Need for amplitude analysis in the discovery of new hadrons

We highlight the need for the development of comprehensive amplitude analysis methods to further our understanding of hadron spectroscopy. Reaction amplitudes constrained by first principles of $S$-matrix theory and by QCD phenomenology are needed to extract robust interpretations of the data from experiments and from lattice calculations.

hep-ph↗

arXiv2022

Two-current transition amplitudes with two-body final states

We derive the on-shell form of amplitudes containing two external currents with a single hadron in the initial state and two hadrons in the final state, denoted as $1+\mathcal{J}\to 2+\mathcal{J}$. This class of amplitude is relevant in precision tests of the Standard Model as well as for exploring the structure of excited states in the QCD spectrum. We present a model-independent description of the amplitudes where we sum to all orders in the strong interaction. From this analytic form we are able to extract transition and elastic resonance form factors consistent with previous work as well as a novel Compton-like amplitude coupling a single particle state to a resonance. The results also hold for reactions where the one-particle state is replaced with the vacuum, namely $\mathcal{J}\to 2+\mathcal{J}$ amplitudes. We also investigate constraints placed upon the formalism for the case of a conserved vector current in the form of the Ward-Takahashi identity. The formalism presented here is valid for currents of arbitrary Lorentz structure and quantum numbers with spinless hadrons where any number of two-particle intermediate channels may be open. When combined with the appropriate finite-volume framework, this work facilitates the extraction of physical observables from this class of amplitudes via lattice QCD calculations.

hep-lat↗

arXiv2021

Connecting Matrix Elements to Multi-Hadron Form-Factors

We discuss developments in calculating multi-hadron form-factors and transition processes via lattice QCD. Our primary tools are finite-volume scaling relations, which map spectra and matrix elements to the corresponding multi-hadron infinite-volume amplitudes. We focus on two hadron processes probed by an external current, and provide various checks on the finite-volume formalism in the limiting cases of perturbative interactions and systems forming a bound state. By studying model-independent properties of the infinite-volume amplitudes, we are able to rigorously define form-factors of resonances.

hep-lat↗

arXiv2020

On-shell representations of two-body transition amplitudes: single external current

This work explores scattering amplitudes that couple two-particle systems via a single external current insertion, $2+\mathcal{J}\to 2$. Such amplitudes can provide structural information about the excited QCD spectrum. We derive an exact analytic representation for these reactions. From these amplitudes, we show how to rigorously define resonance and bound-state form-factors. Furthermore, we explore the consequences of the narrow-width limit of the amplitudes as well as the role of the Ward-Takahashi identity for conserved vector currents. These results hold for any number of two-body channels with no intrinsic spin, and a current with arbitrary Lorentz structure and quantum numbers. This work and the existing finite-volume formalism provide a complete framework for determining this class of amplitudes from lattice QCD.

hep-lat↗

arXiv2020

Solving relativistic three-body integral equations in the presence of bound states

We present a systematically improvable method for numerically solving relativistic three-body integral equations for the partial-wave projected amplitudes. The method consists of a discretization procedure in momentum space, which approximates the continuum problem with a matrix equation. It is solved for different matrix sizes, and in the end, an extrapolation is employed to restore the continuum limit. Our technique is tested by solving a three-body problem of scalar particles with an $S$ wave two-body bound state. We discuss two methods of incorporating the pole contribution in the integral equations, both of them leading to agreement with previous results obtained using finite-volume spectra of the same theory. We provide an analytic and numerical estimate of the systematic errors. Although we focus on kinematics below the three-particle threshold, we provide numerical evidence that the methods presented allow for determination of amplitude above this threshold as well.

hep-lat↗

arXiv2020

Consistency checks for two-body finite-volume matrix elements: II. Perturbative systems

Using the general formalism presented in Refs. [1,2], we study the finite-volume effects for the $\mathbf{2}+\mathcal{J}\to\mathbf{2}$ matrix element of an external current coupled to a two-particle state of identical scalars with perturbative interactions. 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.

hep-lat↗