Lattice QCD study of the $H$ dibaryon using hexaquark and two-baryon interpolators

We present a lattice QCD spectroscopy study in the isospin singlet, strangeness $-2$ sectors relevant for the conjectured $H$ dibaryon. We employ both local and bilocal interpolating operators to isolate the ground state in the rest frame and in moving frames. Calculations are performed using two flavors of $O(a)$-improved Wilson fermions and a quenched strange quark. Our initial point-source method for constructing correlators does not allow for bilocal operators at the source; nevertheless, results from using these operators at the sink indicate that they provide an improved overlap onto the ground state in comparison with the local operators. We also present results, in the rest frame, using a second method based on distillation to compute a Hermitian matrix of correlators with bilocal operators at both the source and the sink. This method yields a much more precise and reliable determination of the ground-state energy. In the flavor-SU(3) symmetric case, we apply Lüscher’s finite-volume quantization condition to the rest-frame and moving-frame energy levels to determine the $S$-wave scattering phase shift, near and below the two-particle threshold. For a pion mass of 960 MeV, we find that there exists a bound $H$ dibaryon with binding energy $\mathrm{\Delta }E=(19\pm 10)\mathrm{MeV}$. In the 27-plet (dineutron) sector, the finite-volume analysis suggests that the existence of a bound state is unlikely.

Figure 1

Quark lines for correlators using point sources. The vertical lines indicate that either the single or the two different end points of the quark lines are summed over the sink time slice.

Figure 2

Quark lines showing an example contraction for correlators using distillation. The vertical lines indicate that the two start and two end points for the quark lines are summed over the source and sink time slices, respectively.

Figure 3

Ground-state effective energy for the singlet (top) and 27-plet (bottom) on ensemble E1. The legend indicates the operators used at the sink. Solid green diamonds denote the energies determined from the Hermitian $2\times 2$ GEVP, while the solid blue and red circles represent the results extracted from the non-Hermitian $2\times 2$ matrix. Open red circles correspond to the effective energy obtained from a single correlation function with a narrow-smeared hexaquark operator at the source and a two-baryon operator with ${\mathit{p}}_1={\mathit{p}}_2=0$ at the sink. The plots on the right show the plateau region with the fitted energy levels. The horizontal line represents the value of $2m_{\mathrm{\Lambda }}$, with the uncertainty denoted by the grey band.

Figure 4

Ground-state effective energy for the singlet (top) and 27-plet (bottom) on ensemble E5 for which SU(3) symmetry is broken. The assignment of a particular multiplet to the energy levels was done on the basis of the dominant overlaps with the interpolating operator. The explanation of symbols is similar to that of Fig. 3.

Figure 5

Top: Effective energies for different multiplets for the SU(3)-symmetric case (ensemble E1). The right panel shows the effective energy difference relative to the $\mathrm{\Lambda }\mathrm{\Lambda }$ threshold. Bottom: Energy levels and relative difference for the SU(3)-broken case (ensemble E5). As in Fig. 4, the assignment of flavor multiplets to the energy levels was done by means of the dominant overlaps with the interpolating operator. All results have been obtained using distillation. Colored bands indicate the fitted values across the relevant time interval.

Figure 6

Comparison of the effective energy difference between the singlet ground state and two noninteracting $\mathrm{\Lambda }\mathrm{s}$, computed with point-to-all propagators and distillation, respectively, for ensemble E1. Open blue circles derive from the same non-Hermitian $2\times 2$ correlator matrix that was used to determine the ground state energy in the singlet channel, represented by the blue points in the top panel of Figure 3.

Figure 7

SU(3) singlet scattering phase shift on ensemble E1. The legend indicates the moving frame $\mathit{D}=\frac{L}{2\pi }\mathit{P}$; the data point labeled $[000{]}^{*}$ was obtained using distillation; and the others were obtained using point-source data. The grey line and its error band indicate the effective-range-expansion fit, and the orange dashed curve corresponds to $p\cot \delta =-\sqrt{-p^2}$. The horizontal error bar shows the intersection between the grey line and the orange dashed curve, which is translated to a binding energy in the label above it.

Figure 8

SU(3) singlet scattering phase shift on ensemble A1. The legend indicates the moving frame $\mathit{D}=\frac{L}{2\pi }\mathit{P}$, and the orange dashed curve corresponds to $p\cot \delta =-\sqrt{-p^2}$.

Figure 9

SU(3) 27-plet scattering phase shift on ensemble E1. The legend indicates the moving frame $\mathit{D}=\frac{L}{2\pi }\mathit{P}$; the data point labeled $[000{]}^{*}$ was obtained using distillation; and the others were obtained using point-source data. The orange dashed curve corresponds to $p\cot \delta =-\sqrt{-p^2}$.

Figure 10

Summary of our results for the binding energy of the $H$ dibaryon. Green and blue colors refer to the SU(3)-symmetric and broken cases, respectively. Full circles and open diamonds represent results obtained in the rest frame using point sources and distillation, respectively. Crosses denote estimates for the binding energy extracted from the finite-volume analysis of results in different frames as described in Sec. 4.

Figure 11

Comparison of our results in Eqs. (23)–(25) to the estimates quoted by NPLQCD [14, 18, 20] and HAL QCD [22, 23]. Green and blue symbols refer to the SU(3)-symmetric and broken cases, respectively. The data point marked by a star denotes the result in infinite volume.

Figure 12

Left: Effective masses for the $\mathrm{\Lambda }$ hyperon in the SU(3)-symmetric case (ensemble E1). Right: Effective masses for octet baryons, i.e., the nucleon, $\mathrm{\Lambda }$, $\mathrm{\Sigma }$, and $\mathrm{\Xi }$ in the SU(3)-broken case (ensemble E5). Diamonds and open circles denote results obtained using distillation and point sources, respectively. For the latter, the narrow smearing width was used (see Sec. 2b).