We present a lattice QCD investigation of isoscalar tetraquark systems involving bottom quarks with explicit flavor content <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>b</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>s</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math>. In the doubly bottom sector, the study focuses on axial-vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> quantum numbers, whereas in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>s</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math> channel, both axial-vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> and scalar <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> quantum numbers are investigated in search of signatures for possible tetraquark bound states. The calculations are performed on four ensembles with dynamical quark fields up to the charm quark generated by the MILC Collaboration, with lattice spacings ranging from approximately 0.058 to 0.12 fm, and at different values of the valence light quark mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mrow><mi>u</mi><mo>/</mo><mi>d</mi></mrow></msub></math>, corresponding to pseudoscalar meson masses, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mrow><mi>p</mi><mi>s</mi></mrow></msub><mo>=</mo><mn>0.5</mn></math>, 0.6, and 0.7 GeV. The energy eigenvalues in the finite volume are determined by applying a variational procedure to correlation matrices constructed from two-meson interpolating operators and diquark-antidiquark operators. Continuum extrapolated elastic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave scattering amplitudes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><msup><mi>B</mi><mo>*</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><msup><mi>B</mi><mo>*</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><mi>B</mi></math> are extracted from the ground-state eigenenergies following a finite-volume analysis à la Lüscher. The chiral and continuum extrapolated binding energy estimates for the isoscalar axial-vector doubly bottom tetraquark <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> from the extracted elastic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><msup><mrow><mi>B</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi></mrow></math>-wave scattering amplitudes is found to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><msub><mi>E</mi><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></msub><mo stretchy="false">(</mo><msup><mn>1</mn><mo>+</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>116</mn><msubsup><mo stretchy="false">(</mo><mrow><mo>−</mo><mn>36</mn></mrow><mrow><mo>+</mo><mn>30</mn></mrow></msubsup><mo stretchy="false">)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>s</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math> system, no statistically significant deviations were observed in the ground-state energies from the respective elastic threshold energies, leading to no conclusive evidence for any bound states.

India

Bhabani Sankar Tripathy

Nilmani Mathur

M. Padmanath

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>b</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mi>s</mi><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover></math> tetraquarks from lattice QCD using two-meson and diquark-antidiquark variational basis

We present the ground state energy spectra of two-flavored heavy dibaryons in the spin-singlet channel. In particular, the ground state masses of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ω</mi><mrow><mi>l</mi><mi>l</mi><mi>Q</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>l</mi><mi>l</mi><mi>Q</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="normal">Ω</mi><mrow><mi>Q</mi><mi>l</mi><mi>Q</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>l</mi><mi>l</mi><mi>Q</mi></mrow></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ω</mi><mrow><mi>Q</mi><mi>l</mi><mi>Q</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>Q</mi><mi>l</mi><mi>Q</mi></mrow></msub></math> states are computed and compared with their respective lowest noninteracting energy levels, where the flavor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> denotes charm and bottom quarks, and the other flavor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>l</mi><mo>∈</mo><mrow><mi>s</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>b</mi></mrow></math> represents strange, charm and bottom, respectively. Considering their valence quark structures, these hadrons could be thought of as the heavy flavor analogs of spin-singlet nucleon-nucleon states. The gauge configurations employed in this study are highly improved staggered quark ensembles with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> flavors, generated by the MILC collaboration, at four lattice spacings, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>a</mi><mo>=</mo><mn>0.1207</mn></math>, 0.0888, 0.0582, and 0.0448 fm. The aforementioned states are also computed at different quark masses, between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>s</mi></msub><mo>≤</mo><msub><mi>m</mi><mi>l</mi></msub><mo>≤</mo><msub><mi>m</mi><mi>b</mi></msub></math>, including at unphysical heavy quark masses, to explore the quark mass dependence of any possible binding. For the dibaryon states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ω</mi><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msub><mo>,</mo><msub><mi mathvariant="normal">Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>b</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>b</mi></mrow></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>b</mi></mrow></msub><msub><mi mathvariant="normal">Ω</mi><mrow><mi>b</mi><mi>b</mi><mi>c</mi></mrow></msub></math>, we find a clear evidence of an energy level below their respective noninteracting energy levels. In addition, for these dibaryons at quark masses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>c</mi></msub><mo>&lt;</mo><msub><mi>m</mi><mi>l</mi></msub><mo>≤</mo><msub><mi>m</mi><mi>b</mi></msub></math>, a trend is found where the gap, between the lowest energy levels and the respective lowest noninteracting levels, increases as the quark mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>l</mi></msub></math> increases. We also study the heavy quark spin-symmetry and its breaking for these heavy dibaryons.

Japan

Parikshit M. Junnarkar

Spectrum of two-flavored spin-zero heavy dibaryons in lattice QCD

Exploring nuclear physics through the fundamental constituents of the strong force—quarks and gluons—is a formidable challenge. While numerical calculations using lattice quantum chromodynamics offer the most promising approach for this pursuit, practical implementation is arduous, especially due to the uncontrollable growth of quark-combinatorics, the so-called Wick-contraction problem of nuclei. We present here two novel methods providing a state-of-the art solution to this problem. In the first, we exploit randomized algorithms inspired from computational number theory to detect and eliminate redundancies that arise in Wick contraction computations. Our second method explores facilities for automation of tensor computations—in terms of efficient utilization of specialized hardware, algorithmic optimizations, as well as ease of programming and the potential for automatic code generation—that are offered by new programming models inspired by applications in machine learning (e.g., tensorflow). We demonstrate the efficacy of our methods by computing two-point correlation functions for Deuteron, Helium-3, Helium-4, and Lithium-7, achieving at least an order of magnitude improvement over existing algorithms with efficient implementation on GPU-accelerators. Additionally, we discover an intriguing characteristic shared by all the nuclei we study: specific spin-color combinations dominate the correlation functions, hinting at a potential connection to an as-yet-unidentified symmetry in nuclei. Moreover finding them beforehand can reduce the computing time further and substantially. Our results, with the efficiency that we achieved, suggest the possibility of extending the applicability of our methods for calculating properties of light nuclei, potentially up to <math><mrow><mi>A</mi><mo>∼</mo><mn>12</mn></mrow></math> and beyond.

Debsubhra Chakraborty

Piyush Srivastava

Arpith Kumar

Nuclear correlation functions using first-principle calculations of lattice quantum chromodynamics

We present a lattice QCD study of the elastic <math><mi>S</mi></math>-wave <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> scattering in search of tetraquark candidates with explicitly exotic flavor content <math><mi>b</mi><mi>c</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> in the isospin <math><mi>I</mi><mo>=</mo><mn>0</mn></math> and <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup></math> channel. We use four lattice QCD ensembles with dynamical <math><mi>u</mi><mo>/</mo><mi>d</mi></math>, <math><mi>s</mi></math>, and <math><mi>c</mi></math> quark fields generated by the MILC collaboration. A nonrelativistic QCD Hamiltonian, including improvement coefficients up to <math><mi>O</mi><mo>(</mo><msub><mi>α</mi><mi>s</mi></msub><msup><mi>v</mi><mn>4</mn></msup><mo>)</mo></math>, is utilized for the bottom quarks. For the rest of the valence quarks we employ a relativistic overlap action. Five different valence quark masses are utilized to study the light quark mass dependence of the <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> scattering amplitude. The finite volume energy spectra are extracted following a variational approach. The elastic <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> scattering amplitudes are extracted employing Lüscher’s prescription. The light quark mass dependence of the continuum extrapolated amplitudes suggests an attractive interaction between the <math><mover><mi>B</mi><mo>¯</mo></mover></math> and <math><mi>D</mi></math> mesons. At the physical pseudoscalar meson mass (<math><msub><mi>M</mi><mrow><mi>p</mi><mi>s</mi></mrow></msub><mo>=</mo><msub><mi>M</mi><mi>π</mi></msub></math>) the <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> scattering amplitude has a subthreshold pole corresponding to a binding energy of <math><mo>−</mo><mn>39</mn><msubsup><mo>(</mo><mrow><mo>−</mo><mn>6</mn></mrow><mrow><mo>+</mo><mn>4</mn></mrow></msubsup><mo>)</mo><msubsup><mo>(</mo><mrow><mo>−</mo><mn>18</mn></mrow><mrow><mo>+</mo><mn>8</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> with respect to the <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> threshold. The critical <math><msub><mi>M</mi><mrow><mi>p</mi><mi>s</mi></mrow></msub></math> at which the <math><mi>D</mi><mover><mi>B</mi><mo>¯</mo></mover></math> scattering length diverges and the system becomes unbound corresponds to <math><msubsup><mi>M</mi><mrow><mi>p</mi><mi>s</mi></mrow><mo>*</mo></msubsup><mo>=</mo><mn>2.94</mn><mo>(</mo><mn>15</mn><mo>)</mo><mo>(</mo><mn>5</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. This result can hold significant experimental relevance in the search for a bound scalar <math><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> tetraquark, which could well be the next “doubly heavy” bound tetraquark to be discovered with only weak decay modes.

Archana Radhakrishnan

Study of the isoscalar scalar <math><mi>b</mi><mi>c</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> tetraquark <math><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> with lattice QCD

We report a lattice QCD study of the heavy-light meson-meson interactions with an explicitly exotic flavor content <math><mi>b</mi><mi>c</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math>, isospin <math><mi>I</mi><mo>=</mo><mn>0</mn></math>, and axial-vector <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> quantum numbers in search of possible tetraquark bound states. The calculation is performed at four values of lattice spacing, ranging from <math><mo>∼</mo><mn>0.058</mn></math> to <math><mo>∼</mo><mn>0.12</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, and at five different values of valence light quark mass <math><msub><mi>m</mi><mrow><mi>u</mi><mo>/</mo><mi>d</mi></mrow></msub></math>, corresponding to pseudoscalar meson mass <math><mrow><msub><mrow><mi>M</mi></mrow><mrow><mi>ps</mi></mrow></msub></mrow></math> of about 0.5, 0.6, 0.7, 1.0, and 3.0 GeV. The energy eigenvalues in the finite volume are determined through a variational procedure applied to correlation matrices built out of two-meson interpolating operators as well as diquark-antidiquark operators. The continuum limit estimates for <math><mi>D</mi><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> elastic <math><mi>S</mi></math>-wave scattering amplitude are extracted from the lowest finite-volume eigenenergies, corresponding to the ground states, using amplitude parametrizations supplemented by a lattice spacing dependence. Light quark mass <math><msub><mi>m</mi><mrow><mi>u</mi><mo>/</mo><mi>d</mi></mrow></msub></math> dependence of the <math><mi>D</mi><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> scattering length (<math><msub><mi>a</mi><mn>0</mn></msub></math>) suggests that at the physical pion mass <math><msubsup><mi>a</mi><mn>0</mn><mrow><mi>phys</mi></mrow></msubsup><mo>=</mo><mo>+</mo><mn>0.57</mn><msubsup><mo>(</mo><mrow><mo>−</mo><mn>5</mn></mrow><mrow><mo>+</mo><mn>4</mn></mrow></msubsup><mo>)</mo><mo>(</mo><mn>17</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>fm</mi></math>, which clearly points to an attractive interaction between the <math><mi>D</mi></math> and <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> mesons that is strong enough to host a real bound state <math><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub></math>, with a binding energy of <math><mrow><mo>−</mo><mn>43</mn><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>7</mn></mrow><mrow><mo>+</mo><mn>6</mn></mrow></msubsup><mo>)</mo><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>24</mn></mrow><mrow><mo>+</mo><mn>14</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> with respect to the <math><mi>D</mi><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> threshold. We also find that the strength of the binding decreases with increasing <math><msub><mi>m</mi><mrow><mi>u</mi><mo>/</mo><mi>d</mi></mrow></msub></math> and the system becomes unbound at a critical light quark mass <math><msubsup><mi>m</mi><mrow><mi>u</mi><mo>/</mo><mi>d</mi></mrow><mo>*</mo></msubsup></math> corresponding to <math><msubsup><mi>M</mi><mrow><mi>ps</mi></mrow><mo>*</mo></msubsup><mo>=</mo><mn>2.73</mn><mo>(</mo><mn>21</mn><mo>)</mo><mo>(</mo><mn>19</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>.

Bound Isoscalar Axial-Vector <math><mi>b</mi><mi>c</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> Tetraquark <math><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> from Lattice QCD Using Two-Meson and Diquark-Antidiquark Variational Basis

We investigate the different decompositions of the angular momentum in QCD for a relativistic spin <math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math> composite state, namely a quark dressed with a gluon. We use light-front Hamiltonian perturbation theory, and in the light-front gauge, use the two-component framework by eliminating the constrained degrees of freedom. We also investigate the different decompositions of the angular momentum at the level of two-dimensional densities in the front form, including the effect of the so-called potential term. In this work, we consider the contribution coming from the quark part of the energy-momentum tensor. We contrast the different decompositions and also compare with other calculations in the literature. We also present the gravitational form factor related to the antisymmetric part of the energy-momentum tensor.

Ravi Singh

Sudeep Saha

Asmita Mukherjee

Angular momentum distribution of a quark dressed with a gluon

We report the first lattice QCD study of the heavy dibaryons in which all six quarks have the bottom (beauty) flavor. Performing a state-of-the-art lattice QCD calculation we find clear evidence for a deeply bound <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msub><mtext>−</mtext><msub><mrow><mi>Ω</mi></mrow><mrow><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msub></mrow></math> dibaryon in the <math><mrow><mmultiscripts><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow></math> channel, as a pole singularity in the <math><mi>S</mi></math>-wave <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msub><mtext>−</mtext><msub><mrow><mi>Ω</mi></mrow><mrow><mi>b</mi><mi>b</mi><mi>b</mi></mrow></msub></mrow></math> scattering amplitude with a binding energy <math><mrow><mo>−</mo><mn>81</mn><mo>(</mo><msubsup><mrow><mtext> </mtext></mrow><mrow><mo>−</mo><mn>16</mn></mrow><mrow><mo>+</mo><mn>14</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. With such a deep binding, Coulomb repulsion serves only as a perturbation on the ground state wave function of the parametrized strong potential and may shift the strong binding only by a few percent. Considering the scalar channel to be the most bound for single flavored dibaryons, we conclude this state is the heaviest possible most deeply bound dibaryon in the visible universe.

Germany

Strongly Bound Dibaryon with Maximal Beauty Flavor from Lattice QCD

We present results of the first lattice QCD calculation of three-flavored heavy dibaryons both in the flavor-symmetric and in the antisymmetric channels. These dibaryons have spin zero, and the three-flavored states are constructed using various possible combinations of quark flavors with at least one of them as the charm (<math><mi>c</mi></math>) or the bottom (<math><mi>b</mi></math>) quark i.e., namely, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>u</mi><mi>d</mi><mi>c</mi><mi>u</mi><mi>d</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>u</mi><mi>d</mi><mi>b</mi><mi>u</mi><mi>d</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>c</mi><mi>s</mi><mi>b</mi><mi>c</mi><mi>s</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi><mi>s</mi><mi>l</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>s</mi><mi>l</mi><mi>c</mi><mi>s</mi><mi>l</mi><mo>)</mo></mrow></math>n,m, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi><mi>s</mi><mi>l</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>s</mi><mi>l</mi><mi>b</mi><mi>s</mi><mi>l</mi><mo>)</mo></mrow></math> and <math><msub><mi>H</mi><mrow><mi>b</mi><mi>c</mi><mi>l</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>c</mi><mi>l</mi><mi>b</mi><mi>c</mi><mi>l</mi><mo>)</mo></math>; <math><mrow><mi>l</mi><mo>∈</mo><mi>u</mi></mrow></math>, <math><mrow><mi>d</mi></mrow></math>. We compute the ground state masses of these dibaryons and the calculations are performed on three <math><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>+</mo><mn>1</mn></math> HISQ gauge ensembles of the MILC Collaboration, with lattice spacings <math><mi>a</mi><mo>=</mo><mn>0.1207</mn></math>, 0.0888 and 0.0582 fm. A relativistic overlap action is employed for the valence light and charm quarks and a nonrelativistic QCD Hamiltonian with improved coefficients is used for the bottom quarks. Unlike the doubly heavy tetraquarks, deuteron-like heavy dibaryons and dibaryons with only bottom quarks, for which lattice QCD calculations have predicted deeply bound strong-interactions-stable states, for these <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>u</mi><mi>d</mi><mi>c</mi><mi>u</mi><mi>d</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>u</mi><mi>d</mi><mi>b</mi><mi>u</mi><mi>d</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>c</mi><mi>s</mi><mi>l</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>s</mi><mi>l</mi><mi>c</mi><mi>s</mi><mi>l</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mi>b</mi><mi>s</mi><mi>l</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>s</mi><mi>l</mi><mi>b</mi><mi>s</mi><mi>l</mi><mo>)</mo></mrow></math> dibaryons we do not find any such deeply bound state at the physical quark masses. However, for <math><msub><mi>H</mi><mrow><mi>b</mi><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mi>b</mi><mi>c</mi><mi>s</mi><mi>b</mi><mi>c</mi><mi>s</mi><mo>)</mo></math>, our results indicate the presence of an energy level <math><mn>29</mn><mo>±</mo><mn>24</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> below the lowest two-baryon elastic threshold, which could be relevant for its future experimental searches. Moreover, we find that the energy difference between the lowest state and the lowest elastic threshold, which could well be interpreted as the binding energy for such heavy dibaryons (<math><msub><mi>H</mi><mrow><mi>b</mi><mi>c</mi><mi>l</mi></mrow></msub></math>), increases with the increase of quark masses (<math><msub><mi>m</mi><mi>l</mi></msub><mo>&gt;</mo><msub><mi>m</mi><mi>s</mi></msub></math>). Taken together, our findings indicate the possibility of the existence of the physical <math><msub><mi>H</mi><mrow><mi>b</mi><mi>c</mi><mi>s</mi></mrow></msub></math> dibaryon while all other physical three-flavored dibaryons are much closer to their thresholds suggesting either they are weakly bound or unbound, resolving which requires further detail study. Our results also point that the binding of a dibaryon configuration becomes stronger with the increase of its valence quark masses which suggests an interesting aspect of strong interaction dynamics at multiple scales.

Study of three-flavored heavy dibaryons using lattice QCD

The Roper state is extracted with valence overlap fermions on a <math><mn>2</mn><mo>+</mo><mn>1</mn></math>-flavor domain-wall fermion lattice (spacing <math><mrow><mi>a</mi><mo>=</mo><mn>0.114</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>330</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>) using both the sequential empirical Bayes (SEB) method and the variational method. The results are consistent, provided that a large smearing-size interpolation operator is included in the variational calculation to have better overlap with the lowest radial excitation. The SEB and variational calculation with large smearing size are also carried out for an anisotropic clover lattice with similar parameters (spatial lattice spacing <math><msub><mi>a</mi><mi>s</mi></msub><mo>=</mo><mn>0.12</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and pion mass <math><msub><mi>m</mi><mi>π</mi></msub><mo>=</mo><mn>396</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>) and obtain consistent results. However, these calculations with clover fermions give a Roper mass of <math><msub><mi>m</mi><mi>R</mi></msub><mo>=</mo><mn>1.92</mn><mo>(</mo><mn>6</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, while the same approach with overlap fermions finds the Roper <math><mrow><mo>≈</mo><mn>280</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> lower, at <math><msub><mi>m</mi><mi>R</mi></msub><mo>=</mo><mn>1.64</mn><mo>(</mo><mn>9</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, for identical valence pion mass. The fact that the prediction of the Roper state by overlap fermions is consistently lower than those of clover fermions, chirally improved fermions, and twisted-mass fermions over a wide range of pion masses has been dubbed a “Roper puzzle.” To understand the origin of this difference, we study the hairpin <math><mi>Z</mi></math>-diagram in the isovector scalar meson (<math><msub><mi>a</mi><mn>0</mn></msub></math>) correlator in the quenched approximation. The lack of quark loops in the quenched approximation turns the <math><msub><mi>a</mi><mn>0</mn></msub></math> correlator negative; giving rise to a ghost “would-be” <math><mi>η</mi><mi>π</mi></math> state. Comparing the <math><msub><mi>a</mi><mn>0</mn></msub></math> correlators for valence clover and overlap fermions, at a valence pion mass of 290 MeV, on three quenched Wilson-gauge lattices, we find that the spectral weight of the ghost state with clover fermions is smaller than that of the overlap at <math><mi>a</mi><mo>=</mo><mn>0.12</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and 0.09 fm—the ratios of the Wilson ghost-state magnitudes (correlator minima) are about half of those of overlap—whereas, the whole <math><msub><mi>a</mi><mn>0</mn></msub></math> correlators of clover and overlap at <math><mi>a</mi><mo>=</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> coincide within errors. This suggests that chiral symmetry is restored for clover at <math><mi>a</mi><mo>≤</mo><mn>0.06</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> and that the Roper mass should agree between clover and overlap fermions toward the continuum limit. We conclude that the present work supports a resolution of the “Roper puzzle” due to <math><mi>Z</mi></math>-graph type chiral dynamics. This entails coupling to higher components in the Fock space (e.g., <math><mi>N</mi><mi>π</mi></math>, <math><mi>N</mi><mi>π</mi><mi>π</mi></math> states) to induce the effective flavor-spin interaction between quarks as prescribed in the chiral quark model, resulting in the parity-reversal pattern as observed in the experimental excited states of <math><mrow><mi>N</mi></mrow></math>, <math><mrow><mi>Δ</mi></mrow></math> and <math><mi>Λ</mi></math>.

United States

Mingyang Sun

China

Ying Chen

Gen Wang

Andrei Alexandru

Shao-Jing Dong

Terrence Draper

Jacob Fallica

Ming Gong

Frank X. Lee

Anyi Li

Jian Liang

Keh-Fei Liu

Yi-Bo Yang

Roper state from overlap fermions

We report the first lattice quantum chromodynamics (QCD) study of deuteronlike (<math><mrow><mi>n</mi><mi>p</mi></mrow></math>-like) dibaryons with heavy quark flavors. These include particles with the following dibaryon structures and valence quark contents: <math><mrow><msub><mrow><mi>Σ</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>u</mi><mi>u</mi><mi>c</mi><mi>u</mi><mi>c</mi><mi>c</mi><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi></mrow></msub><msub><mrow><mi>Ω</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>s</mi><mi>s</mi><mi>c</mi><mi>s</mi><mi>c</mi><mi>c</mi><mo>)</mo></mrow></math>, <math><msub><mi>Σ</mi><mi>b</mi></msub><msub><mi>Ξ</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>u</mi><mi>u</mi><mi>b</mi><mi>u</mi><mi>b</mi><mi>b</mi><mo>)</mo></math>, <math><msub><mi>Ω</mi><mi>b</mi></msub><msub><mi>Ω</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>s</mi><mi>s</mi><mi>b</mi><mi>s</mi><mi>b</mi><mi>b</mi><mo>)</mo></math>, and <math><msub><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>b</mi></mrow></msub><msub><mi>Ω</mi><mrow><mi>c</mi><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>c</mi><mi>b</mi><mi>c</mi><mi>b</mi><mi>b</mi><mo>)</mo></math>, and with spin (<math><mi>J</mi></math>) parity (<math><mi>P</mi></math>), <math><msup><mi>J</mi><mi>P</mi></msup><mo>≡</mo><msup><mn>1</mn><mo>+</mo></msup></math>. Using a state-of-the art lattice QCD calculation, after controlling relevant systematic errors, we unambiguously find that the ground state masses of dibaryons <math><msub><mi>Ω</mi><mi>c</mi></msub><msub><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>s</mi><mi>s</mi><mi>c</mi><mi>s</mi><mi>c</mi><mi>c</mi><mo>)</mo></math>, <math><msub><mi>Ω</mi><mi>b</mi></msub><msub><mi>Ω</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>s</mi><mi>s</mi><mi>b</mi><mi>s</mi><mi>b</mi><mi>b</mi><mo>)</mo></math>, and <math><msub><mi>Ω</mi><mrow><mi>c</mi><mi>c</mi><mi>b</mi></mrow></msub><msub><mi>Ω</mi><mrow><mi>c</mi><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>c</mi><mi>c</mi><mi>b</mi><mi>c</mi><mi>b</mi><mi>b</mi><mo>)</mo></math> are below their respective two-baryon thresholds, suggesting the presence of bound states that are stable under strong and electromagnetic interactions. We also predict their masses precisely. For dibaryons <math><msub><mi>Σ</mi><mi>c</mi></msub><msub><mi>Ξ</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mi>u</mi><mi>u</mi><mi>c</mi><mi>u</mi><mi>c</mi><mi>c</mi><mo>)</mo></math>, and <math><msub><mi>Σ</mi><mi>b</mi></msub><msub><mi>Ξ</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub><mo>(</mo><mi>u</mi><mi>u</mi><mi>b</mi><mi>u</mi><mi>b</mi><mi>b</mi><mo>)</mo></math>, we could not reach to a definitive conclusion about the presence of any bound state due to large systematics associated with these states. We also find that the binding of these dibaryons becomes stronger as they become heavier in mass. This study also opens up the possibility of the existence of many other exotic nuclei, which can be formed through the fusion of heavy baryons, similar to the formation of nuclei of elements in the periodic table.

Parikshit Junnarkar

Deuteronlike Heavy Dibaryons from Lattice Quantum Chromodynamics