The low-lying energy spectrum of the static-colour-source-anti-source system in a vacuum containing light and strange quarks is computed using lattice QCD for a range of different light quark masses. The resulting levels are described using a simple model Hamiltonian and the parameters in this model are extrapolated to the physical light-quark masses. In this framework, the QCD string tension is found to be <math><msqrt><mrow><mi>σ</mi></mrow></msqrt><mo>=</mo><mn>445</mn><msub><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow><mrow><mi>stat</mi></mrow></msub><msub><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow><mrow><mi>sys</mi></mrow></msub></math> MeV.

Germany

John Bulava

Francesco Knechtli

Vanessa Koch

United States

Colin Morningstar

Ireland

Michael Peardon

The quark-mass dependence of the potential energy between static colour sources in the QCD vacuum with light and strange quarks

A lattice QCD computation of the coupled channel <math><mi>π</mi><mi>Σ</mi><mi>–</mi><mover><mi>K</mi><mo>¯</mo></mover><mi>N</mi></math> scattering amplitudes in the <math><mi>Λ</mi><mo>(</mo><mn>1405</mn><mo>)</mo></math> region is detailed. Results are obtained using a single ensemble of gauge field configurations with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical quark flavors and <math><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>200</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><msub><mi>m</mi><mi>K</mi></msub><mo>≈</mo><mn>487</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. Hermitian correlation matrices using both single baryon and meson-baryon interpolating operators for a variety of different total momenta and irreducible representations are used. Several parametrizations of the two-channel scattering <math><mi>K</mi></math>-matrix are utilized to obtain the scattering amplitudes from the finite-volume spectrum. The amplitudes, continued to the complex energy plane, exhibit a virtual bound state below the <math><mi>π</mi><mi>Σ</mi></math> threshold and a resonance pole just below the <math><mover><mi>K</mi><mo>¯</mo></mover><mi>N</mi></math> threshold.

Bárbara Cid-Mora

Andrew D. Hanlon

Ben Hörz

Daniel Mohler

Joseph Moscoso

Amy Nicholson

Fernando Romero-López

Sarah Skinner

André Walker-Loud

Lattice QCD study of <math><mi>π</mi><mi>Σ</mi><mo>−</mo><mover><mi>K</mi><mo>¯</mo></mover><mi>N</mi></math> scattering and the <math><mi>Λ</mi><mo>(</mo><mn>1405</mn><mo>)</mo></math> resonance

This Letter presents the first lattice QCD computation of the coupled channel <math><mrow><mi>π</mi><mi>Σ</mi><mtext>−</mtext><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>N</mi></mrow></math> scattering amplitudes at energies near 1405 MeV. These amplitudes contain the resonance <math><mi>Λ</mi><mo>(</mo><mn>1405</mn><mo>)</mo></math> with strangeness <math><mi>S</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> and isospin, spin, and parity quantum numbers <math><mrow><mi>I</mi><mo>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo></mrow></math>. However, whether there is a single resonance or two nearby resonance poles in this region is controversial theoretically and experimentally. Using single-baryon and meson-baryon operators to extract the finite-volume stationary-state energies to obtain the scattering amplitudes at slightly unphysical quark masses corresponding to <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>≈</mo><mn>200</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>≈</mo><mn>487</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, this study finds the amplitudes exhibit a virtual bound state below the <math><mi>π</mi><mi>Σ</mi></math> threshold in addition to the established resonance pole just below the <math><mover><mi>K</mi><mo>¯</mo></mover><mi>N</mi></math> threshold. Several parametrizations of the two-channel <math><mi>K</mi></math> matrix are employed to fit the lattice QCD results, all of which support the two-pole picture suggested by SU(3) chiral symmetry and unitarity.

Two-Pole Nature of the <math><mi>Λ</mi><mo>(</mo><mn>1405</mn><mo>)</mo></math> Resonance from Lattice QCD

Elastic nucleon-pion scattering amplitudes are computed using lattice QCD on a single ensemble of gauge field configurations with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical quark flavors and <math><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>200</mn><mspace width="0.25em"></mspace><mtext>MeV</mtext></math>. The s-wave scattering lengths with both total isospins <math><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> and <math><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> are inferred from the finite-volume spectrum below the inelastic threshold together with the <math><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> p-wave containing the <math><mi>Δ</mi><mo>(</mo><mn>1232</mn><mo>)</mo></math> resonance. The amplitudes are well-described by the effective range expansion with parameters constrained by fits to the finite-volume energy levels, enabling a determination of the <math><mi>I</mi><mo>=</mo><mn>3</mn><mo>/</mo><mn>2</mn></math> scattering length with statistical errors below 5%, while the <math><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math> scattering length is somewhat less precisely evaluated. Systematic errors due to excited states and the influence of higher partial waves are controlled, providing a step toward future computations down to physical light quark masses with multiple lattice spacings and volumes.

Pavlos Vranas

Elastic nucleon-pion scattering at = 200 MeV from lattice QCD

We study the interactions of systems of two and three nondegenerate mesons composed of pions and kaons at maximal isospin using lattice QCD, specifically π + K +, π + π + K + and K + K + π +. Utilizing the stochastic LapH method, we determine the spectrum of these systems on two CLS N  f  = 2 + 1 ensembles with pion masses of 200 MeV and 340 MeV, and include many levels in different momentum frames. We constrain the K matrices describing two- and three-particle interactions by fitting the spectrum to the results predicted by the finite-volume formalism, including up to p waves. This requires also results for the π + π + and K + K + spectrum, which have been obtained previously on the same configurations. We explore different fitting strategies, comparing fits to energy shifts with fits to energies boosted to the rest frame, and also comparing simultaneous global fits to all relevant two- and three-particle channels to those where we first fit two-particle channels and then add in the three-particle information. We provide the first determination of the three-particle K matrix in π + π + K + and K + K + π + systems, finding statistically significant nonzero results in most cases. We include s and p waves in the K matrix for π + K + scattering, finding evidence for an attractive p-wave scattering length. We compare our results to Chiral Perturbation Theory, including an investigation of the impact of discretization errors, for which we provide the leading order predictions obtained using Wilson Chiral Perturbation Theory.

Zachary Draper

Andrew Hanlon

Stephen Sharpe

Interactions of πK, ππK and KKπ systems at maximal isospin from lattice QCD

We study two- and three-meson systems composed either of pions or kaons at maximal isospin using Monte Carlo simulations of lattice QCD. Utilizing the stochastic LapH method, we are able to determine hundreds of two- and three-particle energy levels, in nine different momentum frames, with high precision. We fit these levels using the relativistic finite-volume formalism based on a generic effective field theory in order to determine the parameters of the two- and three-particle K-matrices. We find that the statistical precision of our spectra is sufficient to probe not only the dominant s-wave interactions, but also those in d waves. In particular, we determine for the first time a term in the three-particle K-matrix that contains two-particle d waves. We use three N  f  = 2 + 1 CLS ensembles with pion masses of 200, 280, and 340 MeV. This allows us to study the chiral dependence of the scattering observables, and compare to the expectations of chiral perturbation theory.

Tyler Blanton

Spain

Interactions of two and three mesons including higher partial waves from lattice QCD

We report on the first application of the stochastic Laplacian Heaviside method for computing multiparticle interactions with lattice QCD to the two-nucleon system. Like the Laplacian Heaviside method, this method allows for the construction of interpolating operators which can be used to construct a set of positive-definite two-nucleon correlation functions, unlike nearly all other applications of lattice QCD to two nucleons in the literature. It also allows for a variational analysis in which optimal linear combinations of the interpolating operators are formed that couple predominantly to the eigenstates of the system. Utilizing such methods has become of paramount importance to help resolve the discrepancy in the literature on whether two nucleons in either isospin channel form a bound state at pion masses heavier than physical, with the discrepancy persisting even in the SU(3)-flavor-symmetric point with all quark masses near the physical strange quark mass. This is the first in a series of papers aimed at resolving this discrepancy. In the present work, we employ the stochastic Laplacian Heaviside method without a hexaquark operator in the basis at a lattice spacing of <math><mrow><mi>a</mi><mo>≈</mo><mn>0.086</mn></mrow></math> fm, lattice volume of <math><mrow><mi>L</mi><mo>=</mo><mn>48</mn><mi>a</mi><mo>≈</mo><mn>4.1</mn></mrow></math> fm and pion mass <math><mrow><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>714</mn></mrow></math> MeV. With this setup, the observed spectrum of two-nucleon energy levels strongly disfavors the presence of a bound state in either the deuteron or dineutron channel.

Dean Howarth

Japan

Enrico Rinaldi

Chia Cheng Chang

Christopher Körber

Evan Berkowitz

Denmark

M.A. Clark

Wayne Tai Lee

Two-nucleon <math><mi>S</mi></math>-wave interactions at the SU(3) flavor-symmetric point with <math><mrow><msub><mi>m</mi><mrow><mi>u</mi><mi>d</mi></mrow></msub><mo>≈</mo><msubsup><mi>m</mi><mi>s</mi><mtext>phys</mtext></msubsup></mrow></math>: A first lattice QCD calculation with the stochastic Laplacian Heaviside method

The energy spectrum of a system containing a static quark anti-quark pair is computed for a wide range of source separations using lattice QCD with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical flavours. By employing a variational method with a basis including operators resembling both the gluon string and systems of two separated static mesons, the first three energy levels are determined up to and beyond the distance where it is energetically favourable for the vacuum to screen the static sources through light- or strange-quark pair creation, enabling both these screening phenomena to be observed. The separation dependence of the energy spectrum is reliably parameterised over this saturation region with a simple model which can be used as input for subsequent investigations of quarkonia above threshold and heavy-light and heavy-strange coupled-channel meson scattering.

United Kingdom

Graham Moir

Mike Peardon

String breaking by light and strange quarks in QCD

The elastic <math><mi>I</mi><mo>=</mo><mn>1</mn></math> p-wave ππ scattering amplitude is calculated together with the isovector timelike pion form factor using lattice QCD with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical quark flavors. Wilson clover ensembles generated by the Coordinated Lattice Simulations (CLS) initiative are employed at four lattice spacings down to <math><mi>a</mi><mo>=</mo><mn>0.05</mn><mspace width="0.2em"></mspace><mtext>fm</mtext></math>, several pion masses down to <math><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>200</mn><mspace width="0.2em"></mspace><mtext>MeV</mtext></math>, and spatial volumes of extent <math><mi>L</mi><mo>=</mo><mn>3.1</mn><mtext>–</mtext><mn>5.5</mn><mspace width="0.2em"></mspace><mtext>fm</mtext></math>. The set of measurements on these ensembles, which is publicly available, enables an investigation of systematic errors due to the finite lattice spacing and spatial volume. The ππ scattering amplitude is fit on each ensemble by a Breit–Wigner resonance lineshape, while the form factor is described better by a thrice-subtracted dispersion relation than the Gounaris–Sakurai parametrization.

Christian Andersen

The = 1 Pion–pion Scattering Amplitude and Timelike Pion Form Factor From = 2 + 1 lattice QCD

The elastic <math><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>, s- and p-wave kaon-pion scattering amplitudes are calculated using a single ensemble of anisotropic lattice QCD gauge field configurations with <math><msub><mrow><mi>N</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> flavors of dynamical Wilson-clover fermions at <math><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>230</mn><mtext> MeV</mtext></math>. A large spatial extent of <math><mi>L</mi><mo>=</mo><mn>3.7</mn><mtext> fm</mtext></math> enables a good energy resolution while partial wave mixing due to the reduced symmetries of the finite volume is treated explicitly. The p-wave amplitude is well described by a Breit–Wigner shape with parameters <math><msub><mrow><mi>m</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow></msub><mo>/</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>3.808</mn><mo>(</mo><mn>18</mn><mo>)</mo></math> and <math><msubsup><mrow><mi>g</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>K</mi><mi>π</mi></mrow><mrow><mi>BW</mi></mrow></msubsup><mo>=</mo><mn>5.33</mn><mo>(</mo><mn>20</mn><mo>)</mo></math> which are insensitive to the inclusion of d-wave mixing and variation of the s-wave parametrization. An effective range description of the near-threshold s-wave amplitude yields <math><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mo>−</mo><mn>0.353</mn><mo>(</mo><mn>25</mn><mo>)</mo></math>.

Ruairí Brett

Jacob Fallica

Determination of - and -wave = 1/2 Scattering Amplitudes in = 2 + 1 lattice QCD