Quark-mass dependence of the $H$ dibaryon in $\mathrm{\Lambda }\mathrm{\Lambda }$ scattering

We study the quark mass dependence of the $H$ dibaryon in the strangeness $S=-2$ baryon-baryon scattering. A low-energy effective field theory is used to describe the coupled-channel scattering, in which the quark mass dependence is incorporated so as to reproduce the lattice QCD data by the HAL QCD collaboration in the SU(3) limit. We point out the existence of the Castillejo-Dalitz-Dyson pole in the $\mathrm{\Lambda }\mathrm{\Lambda }$ scattering amplitude below the threshold in the SU(3) limit, which may cause the Ramsauer-Townsend effect near the $N\mathrm{\Xi }$ threshold at the physical point. The $H$ dibaryon is unbound at the physical point, and a resonance appears just below the $N\mathrm{\Xi }$ threshold. As a consequence of the coupled-channel dynamics, the pole associated with the resonance is not continuously connected to the bound state in the SU(3) limit. Through the extrapolation in quark masses, we show that the unitary limit of the $\mathrm{\Lambda }\mathrm{\Lambda }$ scattering is achieved between the physical point and the SU(3) limit. We discuss the possible realization of the “$H$ matter” in the unphysical quark mass region.

Figure 1

Baryon masses with Eq. (32). Horizontal bars show the central values of the experimental data and the lattice results from Ref. [5].

Figure 2

Scattering lengths of the baryon-baryon scattering in the SU(3) limit by the lattice QCD simulation [5, 26] and by the EFT. Circles (solid lines), squares (dotted line), and crosses (dashed line) denote the lattice (EFT) results in the flavor 1 channel, 8 channel, and 27 channel, respectively. In the singlet channel, thick (thin) line represents the result in the contact (bare $H$) model.

Figure 3

$\mathrm{\Lambda }\mathrm{\Lambda }$ scattering amplitude in the contact model in the SU(3) limit.

Figure 4

$N\mathrm{\Xi }$ scattering amplitude in the contact model in the SU(3) limit.

Figure 5

$\mathrm{\Lambda }\mathrm{\Lambda }$ scattering amplitude at the physical point. Thick (thin) lines represent the results in the contact (bare $H$) model.

Figure 6

$N\mathrm{\Xi }$ scattering amplitude at the physical point. Same convention as Fig. 5.

Figure 7

$\mathrm{\Lambda }\mathrm{\Lambda }$ scattering length as a function of the interpolation parameter $x$ in Eq. (40). $x=0\left(x=1\right)$ corresponds to the HAL-1 data in the SU(3) limit (physical point). Thick (thin) lines represent the results in the contact (bare $H$) model.

Figure 8

Behaviors of the eigenenergies as functions of the interpolation parameter $x$ in Eq. (40). The bound (virtual) state is shown by the solid (dashed) line. Thick (thin) line represents the result in the contact (bare $H$) model.

Figure 9

Behaviors of the eigenenergies as functions of the interpolation parameter $x$ in Eq. (40). The real (imaginary) part is shown by the solid (dashed) line. Thick (thin) line represents the result in the contact (bare $H$) model. The dotted line stands for the energy of the $N\mathrm{\Xi }$ threshold.

Figure 10

Trajectories of the poles in the $\mathrm{\Lambda }\mathrm{\Lambda }$ scattering in the contact model with a variation of the interpolation parameter $x$ in Eq. (40). The arrows indicate the direction of the pole movement with the increase of the parameter $x$. The bound state (virtual and resonance state) pole is on the I-I-I (II-I-I) sheet. Bound and virtual poles are slightly shifted from the real axis for the purpose of illustration.

Figure 11

Behaviors of the energies of the CDD pole as functions of the interpolation parameter $x$ in Eq. (40). Thick (thin) line represents the result in the contact (bare $H$) model.

Figure 12

Unitary limit in the $m_{\pi }-m_K$ plane in units of GeV. Thick (thin) line represents the result in the contact (bare $H$) model. Diagonal dotted line represents the SU(3) limit.

Figure 13

Unitary limit in the $m_l-m_s$ plane in units of GeV. Thick (thin) line represents the result in the contact (bare $H$) model. Diagonal dotted line represents the SU(3) limit.