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Baryon–baryon interactions at short distances: constituent quark model meets lattice QCD

  • Regular Article - Theoretical Physics
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Abstract

The interaction energies between two baryons at short distance in different flavor channels are calculated from the constituent quark model (CQM) and are compared with the recent lattice QCD (LQCD) results for baryon–baryon potentials at short distance. We consider the six-quark system with two strange quarks and focus on the quantum numbers, (Flavor, Spin) = (1, 0), (8, 1), (10, 1), (\(\overline{10}\), 1) and (27, 0). The interaction energy is defined by subtracting out isolated baryon masses and relative kinetic energy of two baryons from the total energy of a compact six-quark state. We introduce interaction energy ratio between different flavors as a useful measure to test the prediction of CQM. We find that the ratios in CQM show good agreement with those in LQCD, which indicates that the short range part of the baryon-baryon interaction can be understood qualitatively in terms of the Pauli principle and spin-dependent color interaction among constituent quarks.

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Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]

Notes

  1. The quark mass dependence in \(r_{0ij}\) is introduced to fit the hadrons masses not only with light flavors but also with charm and bottom in the s-wave [13].

  2. More explicitly, \(C_{\mathrm{f}}=\frac{1}{3} (p^2+q^2+3(p+q)+pq)\) with \((p,q)=(0,0)\) for flavor SU(3) singlet, (1,0) for triplet, (1,1) for octet, (2,2) for 27-plet, (3,0) for decuplet, and (0,3) for anti-decuplet. The same holds for \(C_{\mathrm{c}}\) in color SU(3). For the simplest case with \(N=1\), we have \((p,q)=(1,0)\) in both color and flavor, so that \(C_{\mathrm{f}}=C_{\mathrm{c}}=4/3\).

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Acknowledgements

The work by SHL was supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1901-04. This work by AP was supported by the Korea National Research Foundation under the grant number 2018R1D1A1B07043234. The work by T.H. was supported by JSPS Grant-in-Aid for Scientific Research (S), No. 18H05236. We thank H. Nemura for carefully reading the manuscript.

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Authors and Affiliations

  1. Department of Physics and Institute of Physics and Applied Physics, Yonsei University, Seoul, 03722, Korea

    Aaron Park & Su Houng Lee

  2. College of Bioresource Sciences, Nihon University, Kanagawa, 252-0880, Japan

    Takashi Inoue

  3. Quantum Hadron Physics Laboratory, RIKEN Nishina Center, Wako, 351-0198, Japan

    Takashi Inoue & Tetsuo Hatsuda

  4. Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS), RIKEN, Wako, 351-0198, Japan

    Tetsuo Hatsuda

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  1. Aaron Park
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  2. Su Houng Lee
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  4. Tetsuo Hatsuda
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Correspondence to Su Houng Lee.

Additional information

Communicated by Carsten Urbach

Appendix A: Beyond s-wave orbital for six-quark state in CQM

Appendix A: Beyond s-wave orbital for six-quark state in CQM

In the main text, we focus only on the s-wave orbitals for the 6-quark systems to extract the interaction \(V_{\mathrm{CQM}}\). When the two baryons overlap with each other, however, there arise four possible orbital states in general. They are characterized by the Young tableau as,

$$\begin{aligned} {[}3]_{O} \times [3]_{O}=[6]_{O} +[51]_{O} +[42]_{O} +[33]_{O} . \end{aligned}$$

Flavor-spin structure associated with each orbital state for color-singlet 6-quark is shown in the first row of Table 8. Then the color-spin matrix element \(\chi (F_{\ell })\) for \(N=6\) (defined in the first equality in Eq. (13)) can be evaluated as summarized in Table 8. The column for \([6]_O \otimes [33]_{FS}\) corresponds to the matrix elements employed in the text.

Table 8 The expectation value of color–spin interaction with respect to the 6-quark systems with mixed orbital symmetry in flavor SU(3) symmetric case. The elements with “P.f.” in the Table correspond to the Pauli forbidden states
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Park, A., Lee, S.H., Inoue, T. et al. Baryon–baryon interactions at short distances: constituent quark model meets lattice QCD. Eur. Phys. J. A 56, 93 (2020). https://doi.org/10.1140/epja/s10050-020-00078-z

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