Two nucleon systems at $m_{\pi }\sim 450\mathrm{MeV}$ from lattice QCD

Nucleon-nucleon systems are studied with lattice quantum chromodynamics at a pion mass of $m_{\pi }\sim 450\mathrm{MeV}$ in three spatial volumes using $n_f=2+1$ flavors of light quarks. At the quark masses employed in this work, the deuteron binding energy is calculated to be $B_d=14.4_{-2.6}^{+3.2}\mathrm{MeV}$, while the dineutron is bound by $B_{nn}=12.5_{-5.0}^{+3.0}\mathrm{MeV}$. Over the range of energies that are studied, the S-wave scattering phase shifts calculated in the ${}^1{S}_0$ and ${{}^{3}S}_1-{{}^{3}D}_1$ channels are found to be similar to those in nature, and indicate repulsive short-range components of the interactions, consistent with phenomenological nucleon-nucleon interactions. In both channels, the phase shifts are determined at three energies that lie within the radius of convergence of the expansion, allowing for constraints to be placed on the inverse scattering lengths and effective ranges. The extracted phase shifts allow for matching to nuclear effective field theories, from which low-energy counterterms are extracted and issues of convergence are investigated. As part of the analysis, a detailed investigation of the single hadron sector is performed, enabling a precise determination of the violation of the Gell-Mann–Okubo mass relation.

Figure 1

Cosh EMPs for the ${\pi }^{\pm }$ in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 2

Cosh EMPs for the $K^{\pm }$ in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 3

EMPs for the nucleon in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 4

EMPs for the $\mathrm{\Lambda }$ in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 5

EMPs for the $\mathrm{\Sigma }$ in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 6

EMPs for the $\mathrm{\Xi }$ in the $L=24$ (left panel), $L=32$ (center panel), $L=48$ (right panel) lattice volumes, respectively. In ascending order, the momenta are $\mathbf{P}=2\pi \mathbf{n}/L$ with $|\mathbf{n}{|}^2=0$, 1, 2, 3, 4, 5.

Figure 7

The volume dependence of the ${\pi }^{+}$ (left panel) and $K^{\pm }$ (right panel) masses. Energies (l.u.) in the $L=24$, 32 and 48 lattice volumes are shown as the blue, yellow and red points, respectively, while the results of fits to these results of the form given in Eq. (4) are shown by the shaded regions with the inner (outer) band denoting the statistical (statistical and systematic combined in quadrature) uncertainties.

Figure 8

The volume dependence of the $N$, $\mathrm{\Lambda }$, $\mathrm{\Sigma }$ and $\mathrm{\Xi }$ masses. Energies (l.u.) in the $L=24$, 32 and 48 lattice volumes are shown as the blue, yellow and red points, respectively, while fits to these results are shown by the gray, shaded regions with the inner (outer) band denoting the statistical (statistical and systematic combined in quadrature) uncertainties.

Figure 9

Dispersion relations of the ${\pi }^{\pm }$, $K^{\pm }$. The results in the $L=24$, 32 and 48 lattice volumes are shown as the blue, yellow and red points, respectively, while fits to these results are shown by the gray curves.

Figure 10

Dispersion relations of the octet baryons. The results in the $L=24$, 32 and 48 lattice volumes are shown as the blue, yellow and red points, respectively, while fits to these results are shown by the gray curves.

Figure 11

EMPs for the energy difference between the deuteron and twice the nucleon in the $L=24$ (left panel), $L=32$ (center panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions. The extracted binding energies are given in Table 6.

Figure 12

The region in $c_1-B_d^{(\infty )}$ parameter space defined by ${\chi }^2\to {\chi }_{\min }^2+1$. The inner region is defined by the statistical uncertainty, while the outer region is defined by the statistical and systematic uncertainties combined in quadrature.

Figure 13

EMP associated with the energy difference between the $\mathbb{E}$ ($j_z=\pm 1$) and ${\mathbb{A}}_2$ ($j_z=0$) deuteron states with boost vector $\mathbf{d}=(0,0,1)$ in the $L=32$ ensemble, along with fits to the plateau region. The energy difference depends upon the mixing parameter ${\epsilon }_1$.

Figure 14

The pion-mass dependence of the deuteron binding energy calculated with LQCD. The NPLQCD anisotropic-clover result is from Ref. [11], the Yamazaki et al. results are from Refs. [15, 21] and the NPLQCD isotropic-clover results are from this work and Ref. [14]. The black disk corresponds to the experimental binding energy.

Figure 15

Ek2Ps for the lowest-lying continuum ${{}^{3}S}_1-{{}^{3}D}_1$ NN states near $k^{*}=2\pi /L$ in the $L=24$ (left panel), $L=32$ (center panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions.

Figure 16

Ek2Ps for the continuum ${{}^{3}S}_1-{{}^{3}D}_1$ NN states near $k^{*}=2\sqrt{2}\pi /L$ in the $L=32$ (left panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions.

Figure 17

Ek2Ps for the spin-averaged continuum ${{}^{3}S}_1-{{}^{3}D}_1$ NN states with $\mathbf{d}=(0,0,1)$ near $k^{*}=0$ in the $L=32$ ensemble, along with the fit to the plateau region.

Figure 18

Scattering in the ${{}^{3}S}_1-{{}^{3}D}_1$ coupled channels. The left panel shows $k^{*}\text{cot}{\delta }_{1\alpha }/m_{\pi }$ as a function of $k^{*2}/m_{\pi }^2$, while the right panel shows the phase shift as a function of momentum in MeV, assuming that ${\delta }_{1\beta }$ and the D-wave and higher partial-wave phase shifts vanish. The thick (thin) region of each result corresponds to the statistical uncertainty (statistical and systematic uncertainties combined in quadrature). The black circle in the right panel corresponds to the known result from Levinson’s theorem, while the dashed-gray curve corresponds to the phase shift extracted from the Nijmegen partial-wave analysis of experimental data [61].

Figure 19

Scattering in the ${{}^{3}S}_1-{{}^{3}D}_1$ coupled channels below the start of the t-channel cut, $k^{*2}<m_{\pi }^2/4$, assuming that ${\delta }_{1\beta }$ and the D-wave and higher partial-wave phase shifts vanish. The left panel shows a solid region corresponding to linear fits associated with the statistical uncertainty and the statistical and systematic uncertainties combined in quadrature. The right panel shows the scattering parameters, $1/a^{({{}^{3}S}_1)}$ and $r^{({{}^{3}S}_1)}$ determined from fits to scattering results below the t-channel cut. The solid circle corresponds to the experimental values.

Figure 20

EMPs for the dineutron in the $L=24$ (left panel), $L=32$ (center panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions. The extracted binding energies are given in Table 8.

Figure 21

The region in $c_1-B_{nn}^{(\infty )}$ parameter space defined by ${\chi }^2\to {\chi }_{\min }^2+1$. The inner region is defined by the statistical uncertainty, while the outer is defined by the statistical and systematic uncertainties combined in quadrature.

Figure 22

EMPs for the energy difference between the dineutron and the deuteron in the $L=24$ (left panel), $L=32$ (center panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions.

Figure 23

The pion-mass dependence of the dineutron binding energy calculated with LQCD. The NPLQCD anisotropic-clover result is from Ref. [11], the Yamazaki et al. results are from Refs. [15, 21] and the NPLQCD isotropic-clover results are from this work and Ref. [14]. The black disk corresponds to the location of the near-bound state at the physical quark masses.

Figure 24

Ek2Ps for the lowest-lying continuum ${{}^{1}S}_0$ NN state near $k^{*}=2\pi /L$ in the $L=24$ (left panel), $L=32$ (center panel) and $L=48$ (right panel) ensembles, along with fits to the plateau regions.

Figure 25

Ek2P for the lowest-lying continuum ${{}^{1}S}_0$ NN state near $k^{*}=2\sqrt{2}\pi /L$ in the $L=32$ ensemble, along with the fit to the plateau region.

Figure 26

Ek2P for the continuum ${{}^{1}S}_0$ NN states with $\mathbf{d}=(0,0,1)$ near $k^{*}=0$ in the $L=32$ ensemble, along with the fit to the plateau region.

Figure 27

Scattering in the ${{}^{1}S}_0$ channel. The left panel shows $k^{*}\text{cot}{\delta }^{({{}^{1}S}_0)}/m_{\pi }$ as a function of $k^{*2}/m_{\pi }^2$, while the right panel shows the phase shift as a function of momentum in MeV. The thick (thin) region of each result corresponds to the statistical uncertainty (statistical and systematic uncertainties combined in quadrature). The black circle in the right panel corresponds to the known bound-state result from Levinson’s theorem, while the dashed-gray curve corresponds to the phase shift extracted from the Nijmegen partial-wave analysis of experimental data [61].

Figure 28

Scattering in the ${{}^{1}S}_0$ channel below the start of the t-channel cut, $k^{*2}<m_{\pi }^2/4$. The left panel shows the linear fit with the darker and lighter shaded regions associated with the statistical uncertainty and the statistical and systematic uncertainties combined in quadrature. The right panel shows the scattering parameters, $1/a^{({{}^{1}S}_0)}$ and $r^{({{}^{1}S}_0)}$, determined from fits to scattering results below the t-channel cut. The solid circle corresponds to the experimental values.

Figure 29

The left panel shows the LQCD ${{}^{1}S}_0$ scattering phase shift along with the KSW NNEFT fits at LO, NLO and NNLO. At LO there is one parameter that is fit to recover the dineutron pole, giving the red-shaded region, at NLO there is one additional fit parameter, giving the blue-shaded region, and at NNLO there is a further fit parameter, giving the green-shaded region. The darker (lighter) shaded regions correspond to the statistical (statistical and systematic uncertainties combined in quadrature). The right panel is a scatter plot of the central values of the extracted NNLO fit parameters, ${\xi }_{1,4}$ over the $1\sigma$ range of the dineutron pole. The red- (orange-) shaded region corresponds to the statistical (statistical and systematic uncertainties combined in quadrature).

Figure 30

The ${{}^{3}S}_1-{{}^{3}D}_1$ coupled-channel scattering phase shift, ${\delta }_{1\alpha }$, along with BBSvK fits at LO and ${\mathrm{NLO}}^{*}$. At LO there is one parameter that is fit to recover the deuteron pole, giving the red-shaded region, while at ${\mathrm{NLO}}^{*}$ there is one additional fit parameter, giving the blue-shaded region. The darker (lighter) shaded regions correspond to the statistical (statistical and systematic uncertainties combined in quadrature).

Figure 31

The left panel shows the ${{}^{3}D}_1$ scattering phase shift, ${\overline{\delta }}^{({{}^{3}D}_1)}$, in the Stapp parametrization [78] along with BBSvK fits at LO and ${\mathrm{NLO}}^{*}$, while the right panel shows the mixing parameter, ${\overline{\epsilon }}_1$. The darker (lighter) shaded regions correspond to the statistical (statistical and systematic uncertainties combined in quadrature).