We present a novel procedure for analyzing the lattice QCD spectrum via the finite-volume formalism to obtain constraints on multihadron scattering amplitudes at both real and complex energies. This approach combines a Bayesian reconstruction of the scattering amplitude on the real axis with Nevanlinna interpolation for analytic continuation to complex-valued energies. The method is nonparametric, inherently accounting for parametrization dependence within the uncertainty. We demonstrate the applicability of this approach using both toy data and real lattice QCD data in resonant systems from the Hadron Spectrum and the Baryon Scattering collaborations.

Switzerland

Miguel Salg

Fernando Romero-López

United States

William I. Jay

Bayesian analysis and analytic continuation of scattering amplitudes from lattice QCD

We perform a detailed comparison between three formalisms used in recent studies of $\textit{DD}$ scattering at heavier-than-physical pion masses, which aim to understand the properties of the doubly-charmed tetraquark, $$ {T}_{cc}^{+}(3875) $$ . These methods are the three-particle relativistic field theory (RFT) formalism, the two-body Lippmann-Schwinger (LS) equation with chiral effective field theory potentials, and the two-particle relativistic framework proposed by Baião Raposo and Hansen (BRH approach). In a simplified single-channel setting, we derive the conditions under which the infinite-volume integral equations from the RFT and BRH approaches reduce to the LS form. We present numerical examples showing that differences between these methods can be largely removed by adjusting short-range couplings. We also address a number of technical issues in the RFT approach.

Sebastian Dawid

Stephen Sharpe

Comparison of integral equations used to study $$ {T}_{cc}^{+} $$ for a stable $\textit{D}$

We study the scaling of meson-meson scattering amplitudes with the number of colors, $\textit{N}$ . We use lattice calculations in a theory with $\textit{N}$ = 4 degenerate flavors, with $\textit{N}$ = 3 – 6 and pion mass $\textit{M}$ ≈ 560 MeV. We focus on three different scattering channels, two of which have the same quantum numbers as some tetraquark candidates recently found at LHCb: the $$ {T}_{cs0}^0(2900) $$ , $$ {T}_{c\overline{s}0}^{++}(2900) $$ , $$ {T}_{c\overline{s}0}^0(2900) $$ and $$ {T}_{cs1}^0(2900) $$ states. Finite-volume energies are extracted using a large set of operators, containing two-particle operators with the form of two pions or two vector mesons, and local tetraquark operators. The resulting energy spectra is used to constrain the infinite-volume scattering amplitude by means of Lüscher’s quantization condition. We consider polynomial parametrizations of the phase shift, as well as one-loop chiral perturbation theory (ChPT) predictions. We find that our lattice results follow the expected $\textit{N}$ scaling and are sensitive to subleading $\textit{N}$ corrections. In addition, we constrain the scaling of different combinations of low-energy constants from matching to large $\textit{N}$ ChPT. The results for the channel corresponding to a $$ \left({\pi}^{+}{D}_s^{+}-{K}^{+}{D}^{+}\right) $$ state show evidence of a virtual bound state with energy $\textit{E}$ = 1.63(10)$\textit{M}$ for $\textit{N}$ = 3, while this pole disappears at $\textit{N}$ > 3. This may be connected to the exotic states found in experiment.

Germany

Spain

Jorge Baeza-Ballesteros

Pilar Hernández

The $\textit{ππ}$ scattering amplitude at large $\textit{N}$

We use lattice QCD calculations of the finite-volume spectra of systems of two and three mesons to determine, for the first time, three-particle scattering amplitudes with physical quark masses. Our results are for combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>+</mo></msup></math>, at a lattice spacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>a</mi><mo>=</mo><mn>0.063</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math>, and in the isospin-symmetric limit. We also obtain accurate results for maximal-isospin two-meson amplitudes, with those for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>2</mn><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> being the first determinations at the physical point. Dense lattice spectra are obtained using the stochastic Laplacian-Heaviside method, and the analysis leading to scattering amplitudes is done using the relativistic finite-volume formalism. Results are compared to chiral perturbation theory and to phenomenological fits to experimental data, finding good agreement.

Sebastian M. Dawid

Zachary T. Draper

Andrew D. Hanlon

Ben Hörz

Colin Morningstar

Stephen R. Sharpe

Sarah Skinner

QCD Predictions for Physical Multimeson Scattering Amplitudes

We study systems of two and three mesons composed of pions and kaons at maximal isospin using four CLS ensembles with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>≈</mo><mn>0.063</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, including one with approximately physical quark masses. Using the stochastic Laplacian-Heaviside method, we determine the energy spectrum of these systems including many levels in different momentum frames and irreducible representations. Using the relativistic two- and three-body finite-volume formalism, we constrain the two- and three-meson K matrices, including not only the leading <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>s</mi></math> wave, but also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi></math> waves. By solving the three-body integral equations, we determine, for the first time, the physical-point scattering amplitudes for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><msup><mi>π</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><msup><mi>K</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math> systems. These are determined for total angular momentum <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>1</mn><mo>+</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mn>2</mn><mo>−</mo></msup></math>. We also obtain accurate results for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><msup><mi>π</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><msup><mi>K</mi><mo>+</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>2</mn><msup><mi>K</mi><mo>+</mo></msup></math> phase shifts. We compare our results to chiral perturbation theory and to phenomenological fits.

Two- and three-meson scattering amplitudes with physical quark masses from lattice QCD

We develop a comprehensive framework for extracting the pole position and properties of the doubly-charmed tetraquark $$ {T}_{\textrm{cc}}^{+}(3875) $$ from lattice QCD data using the relativistic three-particle formalism. This approach incorporates the effect of the one-pion exchange diagram in and scattering, making it applicable at energies coinciding with the left-hand cut in the partial-wave projected amplitude. We present an example application of this framework to existing lattice QCD data at = 280 MeV. We solve the integral equations describing the reaction, use LSZ reduction to determine the corresponding amplitude, and find the values of the infinite-volume two- and three-body matrices that lead to agreement with lattice phase shifts within their uncertainties. Using these matrices in the three-particle quantization condition, we describe the finite- volume spectrum and find good agreement with the lattice QCD energies. Our results suggest that, at this pion mass, the tetraquark appears as a pair of subthreshold complex poles whose precise location strongly depends on the value of the three-particle matrix.

Finite- and infinite-volume study of

Understanding the behavior of dense hadronic matter is a central goal in nuclear physics as it governs the nature and dynamics of astrophysical objects such as supernovae and neutron stars. Because of the nonperturbative nature of quantum chromodynamics (QCD), little is known rigorously about hadronic matter in these extreme conditions. Here, lattice QCD calculations are used to compute thermodynamic quantities and the equation of state of QCD over a wide range of isospin chemical potentials with controlled systematic uncertainties. Agreement is seen with chiral perturbation theory when the chemical potential is small. Comparison to perturbative QCD at large chemical potential allows for an estimate of the gap in the superconducting phase, and this quantity is seen to agree with perturbative determinations. Since the partition function for an isospin chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>I</mi></msub></math> bounds the partition function for a baryon chemical potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mn>3</mn><msub><mi>μ</mi><mi>I</mi></msub><mo>/</mo><mn>2</mn></math>, these calculations also provide rigorous nonperturbative QCD bounds on the symmetric nuclear matter equation of state over a wide range of baryon densities for the first time.

Ryan Abbott

William Detmold

Marc Illa

Assumpta Parreño

Robert J. Perry

Phiala E. Shanahan

Michael L. Wagman

QCD Constraints on Isospin-Dense Matter and the Nuclear Equation of State

We generalize the relativistic field-theoretic three-particle finite-volume scattering formalism to describe generic DDπ systems in the charm C = 2 sector. This includes the isospin-0 channel, in which the recently discovered doubly-charmed tetraquark T  cc (3875)+ is expected to manifest as a pole in the DDπ → DDπ scattering amplitude. The formalism presented here can also be applied to lattice QCD settings in which the D * is bound and, in particular, remains valid below the left-hand cut in DD * scattering, thus resolving an issue in previous analyses of lattice-determined finite-volume energies.

United Kingdom

Maxwell Hansen

Incorporating DDπ effects and left-hand cuts in lattice QCD studies of the Tcc(3875)+

Recently, formalism has been derived for studying electroweak transition amplitudes for three-body systems both in infinite and finite volumes. The formalism provides exact relations that the infinite-volume amplitudes must satisfy, as well as a relationship between physical amplitudes and finite-volume matrix elements, which can be constrained from lattice QCD calculations. This formalism poses additional challenges when compared with the analogous well-studied two-body equivalent one, including the necessary step of solving integral equations of singular functions. In this work, we provide some non-trivial analytical and numerical tests on the aforementioned formalism. In particular, we consider a case where the three-particle system can have three-body bound states as well as bound states in the two-body subsystem. For kinematics below the three-body threshold, we demonstrate that the scattering amplitudes satisfy unitarity. We also check that for these kinematics the finite-volume matrix elements are accurately described by the formalism for two-body systems up to exponentially suppressed corrections. Finally, we verify that in the case of the three-body bound state, the finite-volume matrix element is equal to the infinite-volume coupling of the bound state, up to exponentially suppressed errors.

Raul Briceño

Andrew Jackura

Dimitra Pefkou

Electroweak three-body decays in the presence of two- and three-body bound states

Machine-learned normalizing flows can be used in the context of lattice quantum field theory to generate statistically correlated ensembles of lattice gauge fields at different action parameters. This work demonstrates how these correlations can be exploited for variance reduction in the computation of observables. Three different proof-of-concept applications are demonstrated using a novel residual flow architecture: continuum limits of gauge theories, the mass dependence of QCD observables, and hadronic matrix elements based on the Feynman–Hellmann approach. In all three cases, it is shown that statistical uncertainties are significantly reduced when machine-learned flows are incorporated as compared with the same calculations performed with uncorrelated ensembles or direct reweighting.

Aleksandar Botev

Denis Boyda

Daniel C. Hackett

Gurtej Kanwar

Sébastien Racanière

Danilo J. Rezende

Julian M. Urban

Applications of flow models to the generation of correlated lattice QCD ensembles