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.

Germany

John Bulava

Bárbara Cid-Mora

United States

Andrew D. Hanlon

Ben Hörz

Daniel Mohler

Colin Morningstar

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

We present the first lattice investigation of coupled-channel   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  and   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  scattering in the J  PC  = 0++ and 2++ channels. The scattering matrix for partial waves l = 0, 2 and isospin zero is determined using multiple volumes and inertial frames via Lüscher’s formalism. Lattice QCD ensembles from the CLS consortium with m  π  ≃ 280 MeV, a ≃ 0.09 fm and L/a = 24, 32 are utilized. The resulting scattering matrix suggests the existence of three charmonium-like states with J  PC  = 0++ in the energy region ranging from slightly below 2 mD  up to 4.13 GeV. We find a so far unobserved   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  bound state just below threshold and a   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$  resonance likely related to χ  c0(3860), which is believed to be χ  c0(2P). In addition, there is an indication for a narrow 0++ resonance just below the   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  threshold with a large coupling to   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$  and a very small coupling to   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ D\overline{D} $$ . This resonance is possibly related to the narrow X(3915)/χ  c0(3930) observed in experiment also just below   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {D}_s{\overline{D}}_s $$ . The partial wave l = 2 features a resonance likely related to χ  c2(3930). We work with several assumptions, such as the omission of J/ψω, η  c  η and three-particle channels. Only statistical uncertainties are quantified, while the extrapolations to the physical quark-masses and the continuum limit are challenges for the future.

Slovenia

Sasa Prelovsek

Sara Collins

M. Padmanath

Stefano Piemonte

Charmonium-like resonances with JPC = 0++, 2++ in coupled   <math> <mi>D</mi> <mover> <mi>D</mi> <mo>¯</mo> </mover> </math>  $$ \mathrm{D}\overline{\mathrm{D}} $$ ,   <math> <msub> <mi>D</mi> <mi>s</mi> </msub> <msub> <mover> <mi>D</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {\mathrm{D}}_{\mathrm{s}}{\overline{\mathrm{D}}}_{\mathrm{s}} $$  scattering on the lattice

In the simulation of QCD with <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> flavors of Wilson fermions, the positivity of the fermion determinant is generally assumed. We present evidence that this assumption is in general not justified and discuss the consequences of this finding.

Stefan Schaefer

Remarks on strange-quark simulations with Wilson fermions

We study isospin-1 <math><mrow><mi>P</mi></mrow></math>-wave <math><mi>π</mi><mi>π</mi></math> scattering in lattice QCD with two flavors of <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math> improved Wilson fermions. For pion masses ranging from <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>265</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> to <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>437</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, we determine the energy spectrum in the center-of-mass frame and in three moving frames. We obtain the scattering phase shifts using Lüscher’s finite-volume quantization condition. Fitting the dependence of the phase shifts on the scattering momentum to a Breit-Wigner form allows us to determine the corresponding <math><mi>ρ</mi></math> mass <math><msub><mi>m</mi><mi>ρ</mi></msub></math> and <math><msub><mi>g</mi><mrow><mi>ρ</mi><mi>π</mi><mi>π</mi></mrow></msub></math> coupling. By combining the scattering phase shifts with the decay matrix element of the vector current, we calculate the timelike pion form factor, <math><msub><mi>F</mi><mi>π</mi></msub></math>, and compare the results to the Gounaris-Sakurai representation of the form factor in terms of the resonance parameters. In addition, we fit our data for the form factor to the functional form suggested by the Omnès representation, which allows for the extraction of the charge radius of the pion. As a further application, we discuss the long-distance behavior of the vector correlator, which is dominated by the two-pion channel. We reconstruct the long-distance part in two ways: one based on the finite-volume energies and matrix elements, and the other based on <math><msub><mi>F</mi><mi>π</mi></msub></math>. It is shown that this part can be accurately constrained using the reconstructions, which has important consequences for lattice calculations of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment.

United Kingdom

Felix Erben

Jeremy R. Green

Hartmut Wittig

Rho resonance, timelike pion form factor, and implications for lattice studies of the hadronic vacuum polarization

We present a lattice QCD study of charmonium resonances and bound states with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> and <math><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> near the open-charm threshold, taking into account their strong transitions to <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math>. Vector charmonia are the most abundant in the experimentally established charmonium spectrum, while recently LHCb reported also the first discovery of a charmonium with likely spin three. The <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> scattering amplitudes for partial waves <math><mi>l</mi><mo>=</mo><mn>1</mn></math> and <math><mi>l</mi><mo>=</mo><mn>3</mn></math> are extracted on the lattice by means of the Lüscher formalism, using multiple volumes and inertial frames. Parametrizations of the scattering amplitudes provide masses and widths of the resonances, as well as the masses of bound states. CLS ensembles with <math><mn>2</mn><mo>+</mo><mn>1</mn></math> dynamical flavors of nonperturbatively <math><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></math> improved Wilson quarks are employed with <math><msub><mi>m</mi><mi>π</mi></msub><mo>≃</mo><mn>280</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, a single lattice spacing of <math><mi>a</mi><mo>=</mo><mn>0.09</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and two lattice spatial extents of <math><mi>L</mi><mo>=</mo><mn>24</mn></math> and 32. Two values of the charm quark mass are considered to examine the influence of the position of the <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> threshold on the hadron masses. For the lighter charm quark mass we find the vector resonance <math><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo></math> with mass <math><mi>m</mi><mo>=</mo><mn>3780</mn><mo>(</mo><mn>7</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and coupling <math><mi>g</mi><mo>=</mo><mn>16.0</mn><msubsup><mo>(</mo><mrow><mo>−</mo><mn>0.2</mn></mrow><mrow><mo>+</mo><mn>2.1</mn></mrow></msubsup><mo>)</mo></math> (related to the width by <math><mi>Γ</mi><mo>=</mo><msup><mi>g</mi><mn>2</mn></msup><msup><mi>p</mi><mn>3</mn></msup><mo>/</mo><mn>6</mn><mi>π</mi><msup><mi>m</mi><mn>2</mn></msup></math>). Both quantities are consistent with their experimental values, <math><msup><mi>m</mi><mi>exp</mi></msup><mo>=</mo><mn>3773.13</mn><mo>(</mo><mn>35</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><msup><mi>g</mi><mi>exp</mi></msup><mo>=</mo><mn>18.7</mn><mo>(</mo><mn>9</mn><mo>)</mo></math>. The vector <math><mi>ψ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></math> appears as a bound state with <math><mi>m</mi><mo>=</mo><mn>3666</mn><mo>(</mo><mn>10</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The charmonium resonance with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> is found at <math><mrow><mi>m</mi><mo>=</mo><mn>3831</mn><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>16</mn></mrow><mrow><mo>+</mo><mn>10</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, consistent with the <math><mi>X</mi><mo>(</mo><mn>3842</mn><mo>)</mo></math> recently discovered by LHCb. At our heavier charm-quark mass the <math><mi>ψ</mi><mo>(</mo><mn>2</mn><mi>S</mi><mo>)</mo></math> as well as the <math><mi>ψ</mi><mo>(</mo><mn>3770</mn><mo>)</mo></math> are bound states and the <math><mi>X</mi><mo>(</mo><mn>3842</mn><mo>)</mo></math> remains a resonance. We stress that all quoted uncertainties are only statistical, while lattice spacing effects and the approach to the physical point (for the light and strange quarks) still need to be explored. This study of conventional charmonia sets the stage for more challenging future studies of unconventional charmoniumlike states.

Charmonium resonances with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> and <math><msup><mn>3</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup></math> from <math><mover><mi>D</mi><mo>¯</mo></mover><mi>D</mi></math> scattering on the lattice

The comparison of the theoretical and experimental determinations of the anomalous magnetic moment of the muon <math><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mo>)</mo><mi>μ</mi></msub></math> constitutes one of the strongest tests of the Standard Model at low energies. We compute the leading hadronic contribution to <math><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mo>)</mo><mi>μ</mi></msub></math> using lattice QCD simulations employing Wilson quarks. Gauge field ensembles at four different lattice spacings and several values of the pion mass down to its physical value are used. We apply the <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math> improvement program with two discretizations of the vector current to better constrain the approach to the continuum limit. The electromagnetic current correlators are computed in the time-momentum representation. In addition, we perform auxiliary calculations of the pion form factor at timelike momenta in order to better constrain the tail of the isovector correlator and to correct its dominant finite-size effect. For the numerically dominant light-quark contribution, we rescale the lepton mass by the pion decay constant computed on each lattice ensemble. We perform a combined chiral and continuum extrapolation to the physical point, and our final result is <math><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>hvp</mi></mrow></msubsup><mo>=</mo><mo>(</mo><mn>720.0</mn><mo>±</mo><mn>12</mn><mo>.</mo><msub><mrow><mn>4</mn></mrow><mrow><mi>stat</mi></mrow></msub><mo>±</mo><mn>9.</mn><msub><mrow><mn>9</mn></mrow><mrow><mi>syst</mi></mrow></msub><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>10</mn></mrow></msup></mrow></math>. It contains the contributions of quark-disconnected diagrams, and the systematic error has been enlarged to account for the missing isospin-breaking effects.

Antoine Gérardin

Marco Cè

Georg von Hippel

Harvey B. Meyer

Konstantin Ottnad

Jonas Wilhelm

Leading hadronic contribution to <math><mrow><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mrow><mo>)</mo></mrow><mrow><mi>μ</mi></mrow></msub></mrow></math> from lattice QCD with <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> flavors of <math><mrow><mi>O</mi><mo>(</mo><mi>a</mi><mo>)</mo></mrow></math> improved Wilson quarks

We present high-precision results from lattice QCD for the mass splittings of the low-lying charmonium states. For the valence charm quark, the calculation uses Wilson-clover quarks in the Fermilab interpretation. The gauge-field ensembles are generated in the presence of up, down, and strange sea quarks, based on the improved staggered (asqtad) action, and gluon fields, based on the one-loop, tadpole-improved gauge action. We use five lattice spacings and two values of the light sea-quark mass to extrapolate the results to the physical point. An enlarged set of interpolating operators is used for a variational analysis to improve the determination of the energies of the ground states in each channel. We present and implement a continuum extrapolation within the Fermilab interpretation, based on power-counting arguments, and thoroughly discuss all sources of systematic uncertainty. We compare our results for various mass splittings with their experimental values, namely, the 1S hyperfine splitting, the 1P-1S splitting and the P-wave spin-orbit and tensor splittings. Given the uncertainty related to the width of the resonances, we find excellent agreement between our results and the mass splittings derived from the hadron masses measured in the laboratory.

Carleton DeTar

Andreas S. Kronfeld

Song-haeng Lee

James N. Simone

Splittings of low-lying charmonium masses at the physical point

The spectrum of charmonium resonances contains a number of unanticipated states along with several conventional quark-model excitations. The hadrons of different quantum numbers <math><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup></mrow></math> appear in a fairly narrow energy band, where <math><msup><mi>J</mi><mi>P</mi></msup></math> refers to the spin-parity of a hadron in its rest frame. This poses a challenge for lattice QCD studies of (coupled-channel) meson-meson scattering aimed at the determination of scattering amplitudes and resonance pole positions. A wealth of information for this purpose can be obtained from the lattice spectra in frames with nonzero total momentum. These are particularly dense since hadrons with different <math><msup><mi>J</mi><mi>P</mi></msup></math> contribute to any given lattice irreducible representation. This is because <math><msup><mi>J</mi><mi>P</mi></msup></math> is not a good quantum number in flight, and also because the continuum symmetry is reduced on the lattice. In this paper we address the assignment of the underlying continuum <math><msup><mi>J</mi><mi>P</mi></msup></math> quantum numbers to charmonia in flight using a <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></math> Coordinated Lattice Simulations ensemble. As a first step, we apply the single-hadron approach, where only interpolating fields of quark-antiquark type are used. The approach follows techniques previously applied to the light meson spectrum by the Hadron Spectrum Collaboration. The resulting spectra of charmonia with assigned <math><msup><mi>J</mi><mi>P</mi></msup></math> will provide valuable information for the parametrization of (resonant) amplitudes in future determinations of resonance properties with lattice QCD.

Andreas Schäfer

Simon Weishaeupl

Identifying spin and parity of charmonia in flight with lattice QCD

We perform an exploratory lattice QCD simulation of D D ¯ $$ D\overline{D} $$ scattering, aimed at determining the masses as well as the decay widths of charmonium resonances above open charm threshold. Neglecting coupling to other channels, the resulting phase shift for D D ¯ $$ D\overline{D} $$ scattering in p -wave yields the well-known vector resonance ψ (3770). For m π = 156 MeV, the extracted resonance mass and the decay width agree with experiment within large statistical uncertainty. The scalar charmonium resonances present a puzzle, since only the ground state χ c 0 (1 P ) is well understood, while there is no commonly accepted candidate for its first excitation. We simulate D D ¯ $$ D\overline{D} $$ scattering in s -wave in order to shed light on this puzzle. The resulting phase shift supports the existence of a yet-unobserved narrow resonance with a mass slightly below 4 GeV. A scenario with this narrow resonance and a pole at χ c 0 (1 P ) agrees with the energy-dependence of our phase shift. Further lattice QCD simulations and experimental efforts are needed to resolve the puzzle of the excited scalar charmonia.

Austria

C. Lang

Luka Leskovec

Vector and scalar charmonium resonances with lattice QCD