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

This work employs the spectral reconstruction approach of ref. [1] to determine an inclusive rate in the 1 + 1 dimensional O(3) non-linear σ-model, analogous to the QCD part of e + e  −  → hadrons. The Euclidean two-point correlation function of the conserved current j is computed using Monte Carlo lattice field theory simulations for a variety of spacetime volumes and lattice spacings. The spectral density of this correlator is related to the inclusive rate for j → X in which all final states produced by the external current are summed. The ill-posed inverse problem of determining the spectral density from the correlation function is made tractable through the determination of smeared spectral densities in which the desired density is convolved with a set of known smearing kernels of finite width ϵ. The smooth energy dependence of the underlying spectral density enables a controlled ϵ → 0 extrapolation in the inelastic region, yielding the real-time inclusive rate without reference to individual finite-volume energies or matrix elements. Systematic uncertainties due to cutoff effects and residual finite-volume effects are estimated and taken into account in the final error budget. After taking the continuum limit, the results are consistent with the known analytic rate to within the combined statistical and systematic errors. Above energies where 20-particle states contribute, the overall precision is sufficient to discern the four-particle contribution to the spectral density.

United Kingdom

Maxwell Hansen

Austria

Michael Hansen

Agostino Patella

Italy

Nazario Tantalo

Inclusive rates from smeared spectral densities in the two-dimensional O(3) non-linear σ-model

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

Andrew Hanlon

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.

Graham Moir

Mike Peardon

String breaking by light and strange quarks in QCD

We study the John term of Fab Four cosmology in the presence of a scalar potential. We show here how this theory can describe a wide range of cosmological solutions. This theory has two general functions of the scalar field: the potential <math><mi>V</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></math> and the John coefficient function <math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>(</mo><mi>ϕ</mi><mo>)</mo></math>. We show that for very simple choices of those functions, we can describe an accelerated expansion, a radiation-dominated era, and a matter-dominated era. By means of simple modifications, it is also possible to describe nonsingular bouncing versions of those solutions and cyclic universes. We also address some quantum issues of that theory, showing that, for the most significant singular cases, the theory admits a classically well behaved quantization, even though the Hamiltonian has fractional powers in the momenta.

Brazil

Isaac Torres

Russia

Júlio C. Fabris

Oliver F. Piattella

Classical and quantum cosmology of Fab Four John theories

A novel proposal is outlined to determine scattering amplitudes from finite-volume spectral functions. The method requires extracting smeared spectral functions from finite-volume Euclidean correlation functions, with a particular complex smearing kernel of width <math><mi>ε</mi></math> which implements the standard <math><mi>i</mi><mi>ε</mi></math> prescription. In the <math><mi>L</mi><mo>→</mo><mi>∞</mi></math> limit these smeared spectral functions are therefore equivalent to Minkowskian correlators with a specific time ordering to which a modified Lehmann-Symanzik-Zimmermann reduction formalism can be applied. The approach is presented for general <math><mi>m</mi><mo>→</mo><mi>n</mi></math> scattering amplitudes (above arbitrary inelastic thresholds) for a single-species real scalar field, although generalization to arbitrary spins and multiple coupled channels is likely straightforward. Processes mediated by the single insertion of an external current are also considered. Numerical determination of the finite-volume smeared spectral function is discussed briefly and the interplay between the finite volume, Euclidean signature, and time-ordered <math><mi>i</mi><mi>ε</mi></math> prescription is illustrated perturbatively in a toy example.

CERN

Maxwell T. Hansen

Scattering amplitudes from finite-volume spectral functions

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