Low-energy scattering and effective interactions of two baryons at $m_{\pi }\sim 450\mathrm{MeV}$ from lattice quantum chromodynamics

The interactions between two octet baryons are studied at low energies using lattice quantum chromodynamics (LQCD) with larger-than-physical quark masses corresponding to a pion mass of $m_{\pi }\sim 450\mathrm{MeV}$ and a kaon mass of $m_K\sim 596\mathrm{MeV}$. The two-baryon systems that are analyzed range from strangeness $S=0$ to $S=-4$ and include the spin-singlet and triplet $NN$, $\mathrm{\Sigma }N$ ($I=3/2$), and $\mathrm{\Xi }\mathrm{\Xi }$ states, the spin-singlet $\mathrm{\Sigma }\mathrm{\Sigma }$ ($I=2$) and $\mathrm{\Xi }\mathrm{\Sigma }$ ($I=3/2$) states, and the spin-triplet $\mathrm{\Xi }N$ ($I=0$) state. The corresponding $s$-wave scattering phase shifts, low-energy scattering parameters, and binding energies when applicable are extracted using Lüscher’s formalism. While the results are consistent with most of the systems being bound at this pion mass, the interactions in the spin-triplet $\mathrm{\Sigma }N$ and $\mathrm{\Xi }\mathrm{\Xi }$ channels are found to be repulsive and do not support bound states. Using results from previous studies of these systems at a larger pion mass, an extrapolation of the binding energies to the physical point is performed and is compared with available experimental values and phenomenological predictions. The low-energy coefficients in pionless effective field theory (EFT) relevant for two-baryon interactions, including those responsible for $SU(3)$ flavor-symmetry breaking, are constrained. The $SU(3)$ flavor symmetry is observed to hold approximately at the chosen values of the quark masses, as well as the $SU(6)$ spin-flavor symmetry, predicted at large $N_c$. A remnant of an accidental $SU(16)$ symmetry found previously at a larger pion mass is further observed. The $SU(6)$-symmetric EFT constrained by these LQCD calculations is used to make predictions for two-baryon systems for which the low-energy scattering parameters could not be determined with LQCD directly in this study, and to constrain the coefficients of all leading $SU(3)$ flavor-symmetric interactions, demonstrating the predictive power of two-baryon EFTs matched to LQCD.

Figure 1

Summary of the energy shifts extracted from LQCD correlation functions for all two-baryon systems studied in this work, together with the noninteracting energy shifts defined as $\mathrm{\Delta }E=\sqrt{m_1^2+|{\mathit{p}}_1{|}^2}+\sqrt{m_2^2+|{\mathit{p}}_2{|}^2}-m_1-m_2$, where $|{\mathit{p}}_1{|}^2=|{\mathit{p}}_2{|}^2=0$ corresponds to systems that are at rest (continuous line), $|{\mathit{p}}_1{|}^2=|{\mathit{p}}_2{|}^2=(\frac{2\pi }{L}{)}^2$ corresponds to systems which are either boosted or are unboosted but have back-to-back momenta (dashed line), and $|{\mathit{p}}_1{|}^2=0$ and $|{\mathit{p}}_2{|}^2=(\frac{4\pi }{L}{)}^2$ correspond to boosted systems where only one baryon has nonzero momentum (dash-dotted line). The points with no boost have been shifted slightly to the left, and the ones with boosts have been shifted to the right for clarity. Quantities are expressed in lattice units.

Figure 2

$k^{*}\cot \delta$ values as a function of the squared c.m. momentum $k^{*2}$ for all two-baryon systems studied in this work. The darker uncertainty bands are statistical, while the lighter bands show the statistical and systematic uncertainties combined in quadrature. The kinematic functions $c_{lm}^{\mathit{d}}(k^{*2};L)$, given by Eq. (12), are also shown as continuous and dashed lines. Quantities are expressed in lattice units.

Figure 3

$k^{*}\cot \delta$ values as a function of the c.m. momenta $k^{*2}$, along with the band representing the two- (yellow) and three-parameter (red) ERE for the two-baryon channels shown. The bands denote the 68% confidence regions of the fits. Quantities are expressed in lattice units.

Figure 4

The two-dimensional 68% and 95% confidence regions (C.R.) corresponding to the combined statistical and systematic uncertainty on the scattering parameters $a^{-1}$ and $r$ for all two-baryon systems that exhibit bound states, obtained from two-parameter ERE fits. The prohibited regions where the two-parameter ERE does not cross the $\mathcal{Z}$-function for given volumes (as well as the infinite-volume case) are denoted by hashed areas. Quantities are expressed in lattice units.

Figure 5

Summary of the inverse scattering length $a^{-1}$ (left panel), effective range $r$ (middle panel), and ratio $r/a$ (right panel) determined from the two-parameter ERE fit for the two-baryon systems analyzed. The background color groups the channels by the $SU(3)$ irreps they would belong to if $SU(3)$ flavor symmetry were exact (orange for 27, green for $\overline{\mathbf{10}}$, and blue for ${\mathbf{8}}_a$). Quantities are expressed in lattice units.

Figure 6

$k^{*}\cot \delta$ values as a function of the c.m. momenta $k^{*2}$, along with the bands representing the two- and three-parameter polynomial fits for two-baryon systems under the assumption that there is a smooth and monotonic behavior in $k^{*}\cot \delta$ as a function of $k^{*2}$ beyond the $t$-channel cut. Quantities are expressed in lattice units.

Figure 7

Extrapolation of the binding energies of different two-baryon systems, using the results obtained in this work and those at $m_{\pi }\sim 806\mathrm{MeV}$ from Ref. [31]. For comparison, the results with values obtained using one-boson-exchange models or $\chi \mathrm{EFTs}$ are also shown (and where needed, are shifted slightly in the horizontal direction for clarity).

Figure 8

LECs obtained by solving Eqs. (32) (upper panels) and (33) (lower panels) under the assumption of natural (left panels) and unnatural (right panels) interactions. The LECs of momentum-independent operators are in units of $[\frac{2\pi }{M_B}]$ and those of the momentum-dependent operators are in units of $[\frac{4{\pi }^2}{M_B^2}]$, where $M_B$ is the centroid of the octet-baryon masses. The gray-circle markers denote quantities that are extracted using the ERE parameters (method I), while black-square markers are those obtained from scattering lengths that are computed from binding momenta (method II).

Figure 9

The LO $SU(3)$ LEC $c^{(27)}$ (upper panels) and NLO $\cancel{SU(3)}$ LECs ${\mathit{c}}_{B_1{B}_2}^{\chi }$ (lower panels) under the assumption of natural (left panels) and unnatural (right panels) interactions, in units of $[\frac{2\pi }{M_B}]$, where $M_B$ is the centroid of the octet-baryon masses. The gray-circle markers denote quantities that are extracted using method I, while black-square markers show results obtained from method II. See the text for further details.

Figure 10

The leading $SU(6)$ LECs, $a$ (upper panels) and $b/3$ (lower panels), under the assumption of natural (left panels) and unnatural (right panels) interactions, in units of $[\frac{2\pi }{M_B}]$, where $M_B$ is the centroid of the octet-baryon masses. The gray-circle markers denote quantities extracted using the ERE parameters (method I), with the light pink band showing the averaged value, while black-square markers show results obtained from scattering lengths that are constrained by binding momenta (method II), with the dark pink band showing the averaged value.

Figure 11

The predicted (filled markers) LO $SU(3)$ coefficients $c^{(\text{irrep})}$ (upper panels) as well as Savage-Wise coefficients $c_i$ (lower panels) reconstructed from the $SU(6)$ relations are compared with the directly extracted LECs (empty markers) under the assumption of natural (left panels) and unnatural (right panels) interactions, in units of $[\frac{2\pi }{M_B}]$, where $M_B$ is the centroid of the octet-baryon masses. The gray-circle symbols denote quantities that have been extracted using the scattering parameters obtained from the ERE fit (method I), while black-square symbols denote those that are obtained from scattering lengths constrained by binding momenta (method II). The hashed background in the upper panels denotes coefficients whose values were used to constrain $a$ and $b$, and hence are not predictions.

Figure 12

The values of $k^{*2}$ for all systems analyzed in this work. Quantities are expressed in lattice units.

Figure 13

$k^{*2}\cot \delta$ values as a function of the c.m. momenta $k^{*2}$, together with bands representing the two-parameter ERE using all the energy levels (ground state $n=1$ and excited states $n=2$) in lighter yellow, or using just the ground state in darker yellow. Quantities are expressed in lattice units.

Figure 14

Comparison between the two-parameter ERE and the slope of $-\sqrt{-k^{*2}}$ at $k^{*2}=-{\kappa }^{(\infty )2}$, where ${\kappa }^{(\infty )}$ is taken from the $\mathit{d}=\{(0,0,0),(0,0,2)\}$ column of Table 6. Quantities are expressed in lattice units.

Figure 15

Comparison of the ground-state and first excited-state energies obtained in this work (blue circles) and from Ref. [42] (orange diamonds), labeled as NPLQCD 15. The figure shows results with statistical and systematic uncertainties combined in quadrature. Quantities are expressed in lattice units.

Figure 16

Ground-state energies for the $NN({}^1{S}_0)$ system computed on ensembles with $L=24$ (left) and $L=32$ (right), sorted by their weight. The weight of each individual fit is indicated by the level of transparency of each point (darker points have larger weight). The band shows the final result, obtained by combining the individual points with the corresponding weight according to Eq. (6), with statistical and systematic uncertainties combined in quadrature. To facilitate the comparison, the orange point in the right panel of each figure shows the result of Ref. [42], labeled as NPLQCD 15.

Figure 17

Normalized inverse covariance matrices computed for the $NN({}^1{S}_0)$ ground state with $L=24$ (top) and $L=32$ (bottom) for $\tau \in [4,11]$ l.u. using the mean and HL estimators applied to the effective energy function and correlation function.

Figure 18

Comparison of the effective energy-shift plots of the SP and SS correlation functions for the $NN({}^1{S}_0)$ $L=24$ (left panel) and $L=32$ (right panel) first excited states computed using the mean (dark green/red circles) and the HL estimator (light green/red squares, shifted horizontally for clarity). The bands show the results of this work and of Ref. [42], labeled as NPLQCD 15.

Figure 19

The bootstrap estimates of the variance ${\kappa }_2(C(\tau ))/C^2(\tau )$ and skewness ${\kappa }_3(C(\tau ))/C^2(\tau )$ for the SS correlation functions corresponding to the $NN({}^1{S}_0)$ first excited state with $L=24$ (green circles) and $L=32$ (red diamonds). The $L=32$ points have been shifted slightly along the $\tau$ axis for clarity.

Figure 20

Comparison of the 68% confidence region of the scattering parameters obtained using the energy levels extracted in this work (yellow area) and from Ref. [42] (gray area) with four different analyses. The regions include both statistical and systematic uncertainties combined in quadrature. The prohibited regions where the two-parameter ERE does not cross the $\mathcal{Z}$-function are the crossed areas. Quantities are expressed in lattice units.

Figure 21

Comparison of the 68% confidence region of the scattering parameters obtained in this work (yellow area), from Ref. [42] (gray area, labeled as NPLQCD 15), and predictions of low-energy theorems from Ref. [90] (LO and NLO results). The regions include both statistical and systematic uncertainties combined in quadrature. The prohibited regions where the two-parameter ERE does not cross the $\mathcal{Z}$-functions at given volumes or in the infinite-volume limit are denoted as hashed areas. Quantities are expressed in lattice units.

Figure 22

Single-baryon EMPs for the SP (blue squares) and SS (orange diamonds) source-sink combinations. The SS points have been slightly shifted along the horizontal axis for clarity.

Figure 23

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $NN({}^1{S}_0)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 24

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Sigma }N({}^1{S}_0)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 25

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Sigma }\mathrm{\Sigma }({}^1{S}_0)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 26

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Xi }\mathrm{\Sigma }({}^1{S}_0)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 27

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Xi }\mathrm{\Xi }({}^1{S}_0)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 28

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $NN({}^3{S}_1)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 29

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Sigma }N({}^3{S}_1)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 30

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Xi }\mathrm{\Xi }({}^3{S}_1)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.

Figure 31

The effective energy plots (upper panel of each segment) and the effective energy-shift plots (lower panel of each segment) for the $\mathrm{\Xi }N({}^3{S}_1)$ system at rest (left panels) and with boost $\mathit{d}=(0,0,2)$ (right panels) for the SP (blue circles) and SS (orange diamonds) source-sink combinations.