Baryon-baryon interactions and spin-flavor symmetry from lattice quantum chromodynamics

Lattice quantum chromodynamics is used to constrain the interactions of two octet baryons at the $SU(3)$ flavor-symmetric point, with quark masses that are heavier than those in nature (equal to that of the physical strange quark mass and corresponding to a pion mass of $\approx 806\mathrm{MeV}$). Specifically, the $S$-wave scattering phase shifts of two-baryon systems at low energies are obtained with the application of Lüscher’s formalism, mapping the energy eigenvalues of two interacting baryons in a finite volume to the two-particle scattering amplitudes below the relevant inelastic thresholds. The leading-order low-energy scattering parameters in the two-nucleon systems that were previously obtained at these quark masses are determined with a refined analysis, and the scattering parameters in two other channels containing the $\mathrm{\Sigma }$ and $\mathrm{\Xi }$ baryons are constrained for the first time. It is found that the values of these parameters are consistent with an approximate $SU(6)$ spin-flavor symmetry in the nuclear and hypernuclear forces that is predicted in the large-$N_c$ limit of QCD. The two distinct $SU(6)$-invariant interactions between two baryons are constrained for the first time at this value of the quark masses, and their values indicate an approximate accidental $SU(16)$ symmetry. The $SU(3)$ irreps containing the $NN({{}^{1}S}_0)$, $NN({{}^{3}S}_1)$ and $\frac{1}{\sqrt{2}}({\mathrm{\Xi }}^{0}n+{\mathrm{\Xi }}^{-}p)({{}^{3}S}_1)$ channels unambiguously exhibit a single bound state, while the irrep containing the ${\mathrm{\Sigma }}^{+}p({{}^{3}S}_1)$ channel exhibits a state that is consistent with either a bound state or a scattering state close to threshold. These results are in agreement with the previous conclusions of the NPLQCD collaboration regarding the existence of two-nucleon bound states at this value of the quark masses.

Figure 1

The single-baryon EMPs for the SP (blue) and SS (pink) source-sink combinations. The center and right panels present the same EMPs as in the left panel, rescaled to focus on the plateau region. The bands correspond to a correlated single-exponential fit to the SP and SS correlation functions, and obtain the mass of the baryon, $M_B$. The inner bands represent the statistical uncertainty of the fits, while the outer bands correspond to the statistical and systematic uncertainties combined in quadrature. The systematic uncertainty encompasses the variation of the fit window, as described in the text, with the longest time interval considered shown in the plots. The additional systematic resulting from multiple analyses is included in the bands. All quantities are expressed in lattice units (l.u.).

Figure 2

The EMPs of two baryons at rest (upper panel) and with $\mathbf{d}=(0,0,2)$ (lower panel) in the 27 irrep for the SP (blue) and SS (pink) source-sink combinations (the upper panel of each segment), as well as the EMP (the lower panel of each segment) corresponding to the ratio of the SS two-baryon correlation function and the square of the SS single-baryon correlation function. The bands correspond to one-exponential fits to the SS/SS correlation function ratios and obtain the energy shift $\overline{\mathrm{\Delta }E}=E_{BB}-2M_B$. The inner bands represent the statistical uncertainty of the fits, while the outer bands correspond to the statistical and systematic uncertainties combined in quadrature. The systematic uncertainty encompasses the variation of the fit window, as described in the text, with the longest time interval considered shown in the plots. The additional systematic resulting from multiple analyses is included in the bands. All quantities are expressed in lattice units (l.u.).

Figure 3

The EMPs of two baryons at rest (upper panel) and with $\mathbf{d}=(0,0,2)$ (lower panel) in the $\overline{10}$ irrep for the SP (blue) and SS (pink) source-sink combinations (the upper panel of each segment), as well as the EMP (the lower panel of each segment) corresponding to the ratio of the two-baryon correlation function and the square of the single-baryon correlation function, the former with the SP (or SS as indicated) and the latter with the SS source-sink combinations. The bands correspond to one-exponential fits to the SP/SS (or SS/SS as indicated) ratios of correlation functions and obtain the energy shifts $\overline{\mathrm{\Delta }E}=E_{BB}-2M_B$. See the caption of Fig. 2 for more details.

Figure 4

The EMPs of two baryons at rest (upper panel) and with $\mathbf{d}=(0,0,2)$ (lower panel) in the 10 irrep for the SP (blue) and SS (pink) source-sink combinations (the upper panel of each segment), as well as the EMP (the lower panel of each segment) corresponding to the ratio of the two-baryon correlation function and the square of the single-baryon correlation function, the former with the SP and the latter with the SS source-sink combinations. The bands correspond to one-exponential fits to the SP/SS ratio of correlation functions and obtain the energy shifts $\overline{\mathrm{\Delta }E}=E_{BB}-2M_B$. See the caption of Fig. 2 for more details.

Figure 5

The EMPs of two baryons at rest (upper panel) and with $\mathbf{d}=(0,0,2)$ (lower panel) in the $8_A$ irrep for the SP (blue) and SS (pink) source-sink combinations (the upper panel of each segment), as well as the EMP (the lower panel of each segment) corresponding to the ratio of the two-baryon correlation function and the square of the single-baryon correlation function, the former with the SP and the latter with the SS source-sink combinations. The bands correspond to one-exponential fits to the SP/SS ratios of correlation functions and obtain the energy shifts $\overline{\mathrm{\Delta }E}=E_{BB}-2M_B$. See the caption of Fig. 2 for more details.

Figure 6

The shifts in the energy of the two-baryon systems in the 27, $\overline{10}$, 10 and $8_A$ irreps from that of two noninteracting baryons at rest in the three lattice volumes, i.e., $\overline{\mathrm{\Delta }E}=E_{\mathrm{BB}}-2M_{\mathrm{B}}$. Energies are expressed in lattice units (l.u.). Different columns correspond to different volumes and boosts, as indicated.

Figure 7

A comparison of the SS EMPs of two-baryon channels at rest belonging to the four irreps, 27, $\overline{10}$, 10 and $8_A$, for the lowest-lying states ($n=1$) in the lattice volumes $L=24$ l.u. (dark magenta), $L=32$ l.u. (dark blue) and $L=48$ l.u. (green) in the left panels, and for the second lowest-lying states ($n=2$) in the lattice volumes $L=24$ l.u. (dark magenta) and $L=32$ l.u. (dark blue) in the right panels. The points from different volumes in the panels on the left have been slightly shifted in the time direction for display purposes. The light-blue band corresponds to twice the mass of the baryon and its uncertainty in the $32^3\times 48$ ensemble. Quantities are expressed in lattice units (l.u.).

Figure 8

$k^{*}\cot \delta$ values in the two-baryon channels belonging to the four irreps 27, $\overline{10}$, 10 and $8_A$, obtained by solving Eq. (2) at the corresponding values of the square of the CM momentum of the two baryons, ${k^{*}}^2$. The functions on the left-hand side of this equation for $l=m=0$, i.e., $\frac{2}{\sqrt{\pi }L}{\mathcal{Z}}_{00}^{\mathbf{d}}[1;(k^{*}L/2\pi {)}^2]$, are also shown at the corresponding volumes and CM boost momenta. The thick points cover the statistical uncertainty in the results, while the thin points cover the statistical and systematic uncertainties combined in quadrature. Quantities are expressed in lattice units (l.u.).

Figure 9

$k^{*}\cot \delta$ versus the square of the CM momentum of the two baryons, ${k^{*}}^2$, in the 27 irrep. The bands represent fits to the two and three-parameter EREs. Quantities are expressed in lattice units (l.u.).

Figure 10

$k^{*}\cot \delta$ versus the square of the CM momentum of the two baryons, ${k^{*}}^2$, in the $\overline{10}$ irrep. The bands represent fits to the two and three-parameter EREs. Quantities are expressed in lattice units (l.u.).

Figure 11

$k^{*}\cot \delta$ versus the square of the CM momentum of the two baryons, ${k^{*}}^2$, in the 10 irrep. The bands represent fits to the two and three-parameter EREs. Quantities are expressed in lattice units (l.u.).

Figure 12

$k^{*}\cot \delta$ versus the square of the CM momentum of the two baryons, ${k^{*}}^2$, in the $8_A$ irrep. The bands represent fits to the two and three-parameter EREs. Quantities are expressed in lattice units (l.u.).

Figure 13

The 68% (red) and 99% (green) confidence ellipses (C.E.) corresponding to the full uncertainty (statistical and systematic combined in quadrature) of the inverse scattering length and effective range in channels belonging to each of the four irreps as denoted in the label of the plots.

Figure 14

A comparison of the values of the inverse scattering length, $a^{-1}$, effective range, $r$, the first shape parameter, $P$, and the ratio $r/a$, obtained from fits to the two-parameter ($N_{\mathrm{ERE}}=2$) and three-parameter ($N_{\mathrm{ERE}}=3$) EREs, in channels belonging to the four different irreps.

Figure 15

The coefficients, $a$ and $b/3$, of the leading $SU(6)$ effective interactions obtained by fitting pairs of inverse scattering lengths in the channels belonging to each of the four $SU(3)$ irreps considered in this work. The left panel uses a renormalization scheme relevant for unnatural interactions given in Eq. (14) with $\mu =m_{\pi }$. The right panel corresponds to a tree-level expansion of the scattering amplitudes with natural interactions giving rise to Eq. (14) with $\mu =0$. Note the different plot ranges in the two panels. The pink bands represent a combined constant fit to all five different determinations of $a$ and $b/3$ in each case. The couplings are expressed in units of $[\frac{2\pi }{M_B}]$, with $M_B$ being the baryon mass in this calculation, expressed in lattice units.

Figure 16

A comparison of the coefficients of the LO $SU(3)$-symmetric interactions. The left panel corresponds to the unnatural case with $\mu =m_{\pi }$, while the right panel represents the natural case with $\mu =0$, corresponding to a tree-level expansion of the scattering amplitudes. The coefficients are expressed in units of $[\frac{2\pi }{M_B}]$, with $M_B$ being the baryon mass in this calculation, expressed in lattice units.

Figure 17

Diagrammatic representation of the 27, $8_S$ and 1 irreps resulting from the $SU(3)$ decomposition of the product of two octet baryons, along with the corresponding two-baryon states with $J=0$. Strangeness decreases from top to bottom in the diagrams, while the third component of isospin increases from left to right. Mixed states that are colored alike have the same total isospin and strangeness quantum numbers.

Figure 18

Diagrammatic representation of the 10, $\overline{10}$ and $8_A$ irreps resulting from the $SU(3)$ decomposition of the product of two octet baryons, along with the corresponding two-baryon states with $J=1$. Strangeness decreases from top to bottom in the diagrams, while the third component of isospin increases from left to right. Mixed states that are colored alike have the same total isospin and strangeness quantum numbers.

Figure 19

The values of ${k^{*}}^2$ obtained in this work for two-baryon systems in the 27, $\overline{10}$, 10 and $8_A$ irreps, expressed in lattice units (l.u.). The circle and square symbols correspond to $\mathbf{d}=(0,0,0)$ and (0, 0, 2), respectively, and the agreement within uncertainties in each pair of data points demonstrates the source independence of this calculation.

Figure 20

$k^{*}\cot \delta$ versus the square of the CM momentum of the two baryons, ${k^{*}}^2$, along with the bands representing fits to two-parameter EREs obtained from (i) only the ground states ($n=1$) and (ii) from both the ground states ($n=1$) and the first excited states ($n=2$). The plots demonstrate the consistency of the EREs between negative and positive ${k^{*}}^2$ regions in all channels. Quantities are expressed in lattice units (l.u.).

Figure 21

The two-parameter ERE is compared with the tangents to the ($-\sqrt{-{k^{*}}^2}$) curve at values of ${k^{*}}^2=-{{\kappa }^{(\infty )}}^2$, with ${\kappa }^{(\infty )}$ values given in Table 3. The plots verify that all the identified bound states in this work are consistent with the criterion in Eq. (d1) within uncertainties. Quantities are expressed in lattice units (l.u.).