<p>We investigate the existence of bound states and resonances in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi><mo>,</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi><mo>,</mo><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems using HAL QCD interactions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mo>,</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi></math>. We employ the Gaussian expansion method to solve the complex-scaled Schrödinger equation and find no resonances or bound states in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems. We estimate the interaction between charmonium and nuclei, concluding that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> is likely to bind with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi mathvariant="normal">H</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><mi>He</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>4</mn></mrow></mmultiscripts></mrow></math>, and heavier nuclei. For the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> system, the lattice QCD <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>ϕ</mi><mi>N</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mrow></math> interaction is absent. We combine the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>p</mi></math> correlation function analysis and HAL QCD results in model A. We assume the spin-spin interactions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi></math> systems are inversely proportional to their masses in model B. Model A predicts a stronger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mo stretchy="false">(</mo><msub><mrow><mmultiscripts><mrow><mi>S</mi></mrow><mprescripts></mprescripts><none></none><mrow><mn>2</mn></mrow></mmultiscripts></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> interaction and permits a two-body bound state, whereas model B suggests the interaction is attractive but too weak to form a bound state. Both models predict bound states for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>0</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> systems. In model A, these states are deeply bound with binding energies exceeding 15 MeV and remain existent when considering parameter uncertainties. In contrast, these states are very loosely bound in model B, with binding energies below 1 MeV and an existent probability of about 60% when parameter uncertainties are considered. In both models, there exist very loosely bound <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> three-body states which resemble a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math>-d atom with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi></math> meson surrounding the deuteron, but their existence is sensitive to parameter uncertainties. No bound states or resonances are found in the isovector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math> system.</p>

China

Liang-Zhen Wen

Yao Ma

Germany

Lu Meng

Shi-Lin Zhu

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><mi>N</mi><mi>N</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>N</mi><mi>N</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mi>N</mi><mi>N</mi></math> systems based on HAL QCD interactions

We study a doubly bottomed tetraquark state <math><mo>(</mo><mi>b</mi><mi>b</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover><mo>)</mo></math> with quantum number <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><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></math>, denoted by <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math>, in lattice QCD with the nonrelativistic QCD (NRQCD) quark action for <math><mi>b</mi></math> quarks. Employing (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor gauge configurations at <math><mi>a</mi><mo>≈</mo><mn>0.09</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> on <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> lattices, we have extracted the coupled-channel potential between <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> and <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> in the HAL QCD method, which predicts an existence of a bound <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> below the <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> threshold. By extrapolating results at <math><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>410</mn></math>, 570, 700 MeV to the physical pion mass <math><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>140</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, we obtain a binding energy with its statistical error as <math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mtext>binding</mtext></mrow><mrow><mo>(</mo><mi>single</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mn>155</mn><mo>(</mo><mn>17</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and <math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mtext>binding</mtext></mrow><mrow><mo>(</mo><mi>coupled</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mn>83</mn><mo>(</mo><mn>10</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, where “coupled” means that effects due to virtual <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> states are included through the coupled channel potential, while only a potential for a single <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> channel is used in the analysis for “single.” A comparison shows that the effect from virtual <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> states is quite sizable to the binding energy of <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math>. We estimate systematic errors to be <math><mo>±</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> at most, which are mainly caused by the NRQCD approximation for <math><mi>b</mi></math> quarks.

Japan

Takafumi Aoki

Sinya Aoki

Takashi Inoue

Lattice study on a tetraquark state <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> in the HAL QCD method

Abstract We study decuplet baryons from meson–baryon interactions in lattice quantum chromodynamics (QCD), in particular, Δ and Ω baryons from P-wave I = 3/2 Nπ and I = 0  interactions, respectively. Interaction potentials are calculated in the HAL QCD method using 3-quark-type source operators at m$_{π}$ ≈ 410 MeV and m$_{K}$ ≈ 635 MeV, where Δ as well as Ω baryons are stable. We use the conventional stochastic estimate of all-to-all propagators combined with the all-mode averaging to reduce statistical fluctuations. We have found that the  system has a weaker attraction than the Nπ system while the binding energy from the threshold is larger for Ω than Δ. This suggests that an inequality  comes mainly from a smaller spatial size of a  bound state due to a larger reduced mass, rather than its interaction. Root-mean-square distances of bound states in both systems are small, indicating that Δ and Ω are tightly bound states and thus can be regarded qualitatively as composite states of three quarks. Results of binding energies agree with those obtained from temporal two-point functions within large systematic errors, which arise dominantly from the lattice artifact at short distances.

Kotaro Murakami

Yutaro Akahoshi

Takumi Doi

Kenji Sasaki(HAL QCDCollaboration)

Lattice quantum chromodynamics (QCD) studies on decuplet baryons as meson–baryon bound states in the HAL QCD method

Abstract We investigate how the derivative expansion in the HAL QCD method works to extract physical observables, using a separable potential in quantum mechanics, which is solvable but highly non-local in the coordinate system. We consider three cases for inputs to determine the HAL QCD potential in the derivative expansion: (1) energy eigenfunctions, (2) time-dependent wave functions as solutions to the time-dependent Schrödinger equation with some boundary conditions, and (3) a time-dependent wave function made by a linear combination of a finite number of eigenfunctions at low energy to mimic the finite volume effect. We have found that, for all three cases, the potentials provide reasonable scattering phase shifts even at the leading order of the derivative expansion, and they give more accurate results as the order of the expansion increases. By comparing the above results with those from the formal derivative expansion for the separable potential, we conclude that the derivative expansion is not a way to obtain the potential but a method to extract physical observables such as phase shifts and binding energies, and that the scattering phase shifts from the derivative expansion in the HAL QCD method converge to the exact ones much faster than those from the formal derivative expansion of the separable potential.

Koichi Yazaki

Derivative expansion in the HAL QCD method for a separable potential

We investigate the <math><mi>I</mi><mo>=</mo><mn>1</mn></math> <math><mi>π</mi><mi>π</mi></math> interaction using the HAL QCD method in lattice QCD. We employ the (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor gauge configurations on <math><mrow><msup><mrow><mn>32</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><mn>64</mn></mrow></math> lattice at the lattice spacing <math><mrow><mi>a</mi><mo>≈</mo><mn>0.0907</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>≈</mo><mn>411</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, in which the <math><mi>ρ</mi></math> meson appears as a resonance state. We find that all-to-all propagators necessary in this calculation can be obtained with reasonable precision by a combination of three techniques, the one-end trick, the sequential propagator, and the covariant approximation averaging (CAA). The nonlocal <math><mi>I</mi><mo>=</mo><mn>1</mn></math> <math><mi>π</mi><mi>π</mi></math> potential is determined at the next-to-next-to-leading order (<math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>LO</mi></mrow></math>) of the derivative expansion for the first time, and the resonance parameters of the <math><mi>ρ</mi></math> meson are extracted. The obtained <math><mi>ρ</mi></math> meson mass is found to be consistent with the value in the literature, while the value of the coupling <math><msub><mi>g</mi><mrow><mi>ρ</mi><mi>π</mi><mi>π</mi></mrow></msub></math> turns out to be somewhat larger. The latter observation is most likely attributed to the lack of low-energy information in our lattice setup with the center-of-mass frame. Such a limitation may appear in other P-wave resonant systems and we discuss possible improvement in future. With this caution in mind, we positively conclude that we can reasonably extract the <math><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>LO</mi></mrow></math> potential and resonance parameters even in the system requiring the all-to-all propagators in the HAL QCD method, which opens up new possibilities for the study of resonances in lattice QCD.

Emergence of the <math><mi>ρ</mi></math> resonance from the HAL QCD potential in lattice QCD

Abstract Employing an all-to-all quark propagator technique, we investigate kaon–nucleon interactions in lattice QCD. We calculate the S-wave kaon–nucleon potentials at the leading order in the derivative expansion in the time-dependent HAL QCD method, using (2+1)-flavor gauge configurations on  lattices with lattice spacing  fm and pion mass  MeV. We take the one-end trick for all-to-all propagators, which allows us to put the zero-momentum hadron operators at both source and sink and to smear quark operators at the source. We find a stronger repulsive interaction in the  channel than in the . The phase shifts obtained by solving the Schrödinger equations with the potentials qualitatively reproduce the energy dependence of the experimental phase shifts, and have similar behavior to previous results from lattice QCD without all-to-all propagators. Our study demonstrates that the all-to-all quark propagator technique with the one-end trick is useful for studying interactions in meson–baryon systems in the HAL QCD method, so we will apply it to meson–baryon systems which contain quark–antiquark creation/annihilation processes in our future studies.

S-wave kaon–nucleon potentials with all-to-all propagators in the HAL QCD method

Abstract We investigate the HAL QCD potential in   scattering using the hybrid method for all-to-all propagators, in which a propagator is approximated by low eigenmodes, and the remaining high-eigenmode part is stochastically estimated. To verify the applicability of the hybrid method to systems containing quark creationannihilation contributions such as the  meson, we calculate the   potential with the -flavor gauge configurations on a  lattice with lattice spacing  fm and  MeV, in which the  meson appears as a deeply bound state. While we find that the naive stochastic evaluations for quark creationannihilation contributions lead to extremely large statistical fluctuations, additional noise reduction methods enable us to obtain a sufficiently precise potential, which shows a strong attractive force. We also confirm that the binding energy and  obtained from our potential are roughly consistent with an existing  meson bound state, within the large systematic error associated with our calculation, whose possible origin is also discussed.

Tatsumi Aoyama

Takaya Miyamoto

Kenji Sasaki

The HAL QCD potential in the = 1 System With the Meson Bound State

The approximated partial-wave decomposition method for discrete data on a cubic lattice, developed by Misner, is applied to the calculation of <math><mi>S</mi></math>-wave hadron-hadron scatterings by the HAL QCD method in lattice QCD. We consider the Nambu-Bethe-Salpeter (NBS) wave function for the spin-singlet <math><msub><mi>Λ</mi><mi>c</mi></msub><mi>N</mi></math> system calculated in (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor QCD on a <math><mrow><mo>(</mo><mn>32</mn><mi>a</mi><mtext> </mtext><mtext> </mtext><mi>fm</mi><msup><mrow><mo>)</mo></mrow><mrow><mn>3</mn></mrow></msup></mrow></math> lattice with lattice spacing <math><mi>a</mi><mo>≃</mo><mn>0.0907</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and <math><msub><mi>m</mi><mi>π</mi></msub><mo>≃</mo><mn>700</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. We find that by Misner’s method the <math><mi>l</mi><mo>=</mo><mn>0</mn></math> component can be successfully extracted from the NBS wave function projected to the <math><msubsup><mi>A</mi><mn>1</mn><mo>+</mo></msubsup></math> representation of the cubic group, which contains small <math><mi>l</mi><mo>≥</mo><mn>4</mn></math> components. Furthermore, while the higher partial-wave components are enhanced so as to produce significant comb-like structures in the conventional HAL QCD potential if the Laplacian approximated by the usual second-order difference is applied to the NBS wave function, such structures are found to be absent in the potential extracted by Misner’s method, where the Laplacian can be evaluated analytically for each partial-wave component. Despite the difference in the potentials, the two methods give almost identical results on the central values and magnitude of statistical errors for the fits of the potentials, and consequently on the scattering phase shifts. This indicates not only that Misner’s method works well in lattice QCD with the HAL QCD method, but also that the contaminations from higher partial waves in the study of <math><mi>S</mi></math>-wave scatterings are well under control even in the conventional HAL QCD method. It will be of interest to study interactions in higher partial-wave channels in the HAL QCD method with Misner’s decomposition, where the utility of this new technique may become clearer.

Shinya Gongyo

Partial wave decomposition on the lattice and its applications to the HAL QCD method

Abstract A formalism is given to hermitize the HAL QCD potential, which needs to be non-Hermitian except for the leading-order (LO) local term in the derivative expansion as the Nambu– Bethe– Salpeter (NBS) wave functions for different energies are not orthogonal to each other. It is shown that the non-Hermitian potential can be hermitized order by order to all orders in the derivative expansion. In particular, the next-to-leading order (NLO) potential can be exactly hermitized without approximation. The formalism is then applied to a simple case of  scattering, for which the HAL QCD calculation is available to the NLO. The NLO term gives relatively small corrections to the scattering phase shift and the LO analysis seems justified in this case. We also observe that the local part of the hermitized NLO potential works better than that of the non-Hermitian NLO potential. The Hermitian version of the HAL QCD potential is desirable for comparing it with phenomenological interactions and also for using it as a two-body interaction in many-body systems.

Takumi Iritani

Hermitizing the HAL QCD potential in the derivative expansion

Abstract In this paper, we perform the first application of the hybrid method (exact low modes plus stochastically estimated high modes) for all-to-all propagators to the HAL QCD method. We calculate the HAL QCD potentials in the   scattering in order to see how statistical fluctuations of the potential behave under the hybrid method. All of the calculations are performed with the 2+1 flavor gauge configurations on a  lattice at the lattice spacing  fm and  MeV. It is revealed that statistical errors for the potential are enhanced by stochastic noises introduced by the hybrid method, which, however, are shown to be reduced by increasing the level of dilutions, in particular, that of space dilutions. From systematic studies, we obtain a guiding principle for a choice of dilution types/levels and a number of eigenvectors to reduce noise contamination to the potential while keeping numerical costs reasonable. We also confirm that we can obtain the scattering phase shifts for the   system by the hybrid method within a reasonable numerical cost; these phase shifts are consistent with the result obtained with the conventional method. The knowledge that we obtain in this study will become useful for the investigation of hadron resonances that require quark annihilation diagrams such as the  meson by the HAL QCD potential with the hybrid method.