Are two nucleons bound in lattice QCD for heavy quark masses? Consistency check with Lüscher’s finite volume formula

On the basis of Lüscher’s finite volume formula, a simple test (consistency check or sanity check) is introduced and applied to inspect the recent claims of the existence of the nucleon-nucleon ($NN$) bound state(s) for heavy quark masses in lattice QCD. We show that the consistency between the scattering phase shifts at $k^2>0$ and/or $k^2<0$ obtained from the lattice data and the behavior of phase shifts from the effective range expansion (ERE) around $k^2=0$ exposes the validity of the original lattice data; otherwise, such information is hidden in the energy shift $\mathrm{\Delta }E$ of the two nucleons on the lattice. We carry out this consistency check for all the lattice results in the literature claiming the existence of the $NN$ bound state(s) for heavy quark masses and find that (i) some of the $NN$ data show a clear inconsistency between the behavior of ERE at $k^2>0$ and that at $k^2<0$, (ii) some of the $NN$ data exhibit a singular behavior of the low-energy parameter (such as the divergent effective range) at $k^2<0$, (iii) some of the $NN$ data have the unphysical residue for the bound-state pole in the $S$ matrix, and (iv) the rest of the $NN$ data are inconsistent among themselves. Furthermore, we raise a caution of using the ERE in the case of the multiple bound states. Our finding, together with the fake plateau problem previously pointed out by the present authors, brings a serious doubt on the existence of the $NN$ bound states for pion masses heavier than 300 MeV in the previous studies.

Figure 1

Effective energy shift $\mathrm{\Delta }{E}_{NN}^{\mathrm{eff}}(t)=E_{NN}^{\mathrm{eff}}(t)-2m_N^{\mathrm{eff}}(t)$ in the $NN({{}^{1}S}_0)$ channel (left) and the $NN({{}^{3}S}_1)$ channel (right) at $m_{\pi }=0.51\mathrm{GeV}$, $L=4.3\mathrm{fm}$, and $a\simeq 0.09\mathrm{fm}$, from the smeared source (blue squares) and the wall source (red circles) with the nonrelativistic operator. Here $E_{NN}^{\mathrm{eff}}(t)$ and $m_N^{\mathrm{eff}}(t)$ are the effective energy of $NN$ and the effective mass of $N$, respectively. The black solid line represents the fit to the plateau of data in Ref. [25], in which $\mathrm{\Delta }{E}_{NN}^{\mathrm{eff}}(t)$ was calculated from the same smeared source on the same gauge configurations but with smaller statistics. These figures are adapted from Ref. [33].

Figure 2

The energy shifts $\mathrm{\Delta }E$ on the $L/a=24$ and 32 lattices, in the $NN({{}^{1}S}_0)$ channel (left) and the $NN({{}^{3}S}_1)$ channel (right), from CalLat2017 (red circles) and from NPL2013 (blue squares). Inner and outer error bars represent the statistical errors and statistical and systematic errors added in quadrature, respectively. Red solid (open) circles for CalLat2017 are obtained from a zero (nonzero) displaced two-nucleon source operator in the center of mass system. Blue solid (open) squares for NPL2013 are obtained from a zero displaced one in the center of mass system (the $n=2$ boosted system).

Figure 3

The relation between $k\cot {\delta }_0(k)/m_{\pi }$ and $(k/m_{\pi }{)}^2$ in the infinite volume for $NN({{}^{1}S}_0)$ (upper left) and $NN({{}^{3}S}_1)$ (upper right) with $m_{\pi }=0.14\mathrm{GeV}$. The red solid lines denote empirical ERE relations, and the black solid lines are the condition for the bound-state pole. In the upper right figure, the bound state is identified as the solid black point. The lower panels show the relation between $k\cot {\delta }_0(k)/m_{\pi }$ and $(k/m_{\pi }{)}^2$ on finite volumes. The colored dashed lines represent Lüscher’s formula for each finite volume $L$. Realized on each volume are the discrete points which satisfy both Lüscher’s formula and the ERE relation, as shown by open squares, up and down triangles, and diamonds for $L=12$, 14, 18, and 24 fm, respectively.

Figure 4

Schematic illustration for the system with the pole satisfying the physical condition (left) or the pole having the unphysical residue (right). The red solid lines denote $k\cot {\delta }_0(k)$ obtained by fitting the lattice QCD data at finite volumes. The bound state is identified as the crossing point of $k\cot {\delta }_0(k)$ and the bound-state condition (black solid line), as indicated by the black solid circle. In the left (right) figure, the behavior of $k\cot {\delta }_0(k)$ near the bound-state pole is consistent (inconsistent) with the condition Eq. (6), and thus the bound-state pole has a physical (unphysical) residue in the $S$ matrix.

Figure 5

(Left) The $k\cot {\delta }_0(k)$ (the red line) for a three-dimensional square-well potential. The radius of the potential is denoted by $b$, and the potential depth is chosen so that there exist two bound states. The black solid line is the condition for the bound-state poles, which are denoted by the black solid circles. Note that $k\cot {\delta }_0(k)$ diverges between two bound states. (Right) Illustration of a misuse of ERE beyond its convergence radius, where the left crossing point between the red line and black line violates the condition (6).

Figure 6

$k\cot {\delta }_0(k)/m_{\pi }$ as a function of $(k/m_{\pi }{)}^2$ for $NN({{}^1\mathrm{S}}_0)$ (left) and $NN({{}^3\mathrm{S}}_1)$ (right) of NPL2015. Black dashed lines correspond to Lüscher’s formula for each finite volume, while the black solid line represents the bound-state condition that $-\sqrt{-(k/m_{\pi }{)}^2}$. Upper panels show the data at $(k/m_{\pi }{)}^2<0$, while lower ones include the data at $(k/m_{\pi }{)}^2>0$. Light blue bands correspond to ERE with statistical and systematic errors added in quadrature given in NPL2015, obtained from data $k^2>0$ and the binding energy in the infinite volume; this is called ${\mathrm{ERE}}_{k^2>0,BE}$ in the text. Light red bands in lower panels correspond to the ERE obtained by using data at $k^2<0$ on finite volumes; this is called ${\mathrm{ERE}}_{k^2<0}$ in the text. The red (blue) lines in the middle of the red (blue) bands correspond to the best fits.

Figure 7

$k\cot {\delta }_0(k)/m_{\pi }$ as a function of $(k/m_{\pi }{)}^2$ for $NN({{}^1\mathrm{S}}_0)$ (left) and $NN({{}^3\mathrm{S}}_1)$ (right) of YKU2011. Black dashed lines correspond to Lüscher’s formula for each volume, while the black solid line represents $-\sqrt{-(k/m_{\pi }{)}^2}$. EREs corresponding to ${\mathrm{ERE}}_{k^2>0,BE}$ and ${\mathrm{ERE}}_{k^2<0}$ are shown by the light blue band and light red band, respectively, with statistical and systematic errors added in quadrature.

Figure 8

$k\cot {\delta }_0(k)/m_{\pi }$ as a function of $(k/m_{\pi }{)}^2$ for $NN({{}^1\mathrm{S}}_0)$ (left) and $NN({{}^3\mathrm{S}}_1)$ (right) for data on each volume from YIKU2012, together with YIKU2012’s infinite volume extrapolation. Black dashed lines correspond to Lüscher’s formula for each finite volume, while the black solid line represents $-\sqrt{-(k/m_{\pi }{)}^2}$. NLO ERE fits to finite volume data are shown by red lines, together with light red bands corresponding to statistical and systematic errors added in quadrature.

Figure 9

The same as Fig. 8, but from YIKU2015. Red lines correspond to NLO ERE fits.

Figure 10

The same as Fig. 8, but from NPL2012. Red lines correspond to the NLO ERE fits. The red dashed line represents Lüscher’s formula for $L/a=20$. Lattice data around $(k/m_{\pi }{)}^2=0$ at $L/a=20$ [49] are located way out of the plot region of the figures.

Figure 11

The inverse scattering length $a_0^{-1}$ (blue dashed line) and the effective range $r_0$ (red solid line) as a function of $(Kb{)}^2$. The first bound state appears at $(Kb{)}^2=(\pi /2{)}^2$ and the second one at $(Kb{)}^2=(3\pi /2{)}^2$.

Figure 12

The $kb\cot {\delta }_0(k)$ as a function of $(kb{)}^2$ are shown by colored lines. The black solid lines denote the condition for the bound states, and the solid circles correspond to the poles. (a) Weak repulsion. (b) Weak attraction. (b’) Weak attraction together with Lüscher’s formula at $L/b=2$ (3) by the red (blue) thin dashed line, where open squares are finite volume spectra. (c) Moderate attraction. (d) Strong attraction having the 2nd pole at $(Kb{)}^2=23$.

Figure 13

The same as Fig. 2 but for the excited state ($\mathrm{\Delta }E>0$) from CalLat2017 (red circles) and NPL2013 (blue triangles) in the center of mass system.

Figure 14

(Upper) The same as Fig. 8, but from NPL2013. (Lower) The same as upper figures but with data from excited states. Red bands correspond to ${\mathrm{ERE}}_{k^2>0,BE}$ given in NPL2013 with statistical and systematic errors added in quadrature.

Figure 15

(Upper) The same as Fig. 8, but from CalLat2017. Red bands correspond to NNLO ERE given in CalLat2017. (Lower) The same as upper figures but with data from excited states.

Figure 16

$k\cot {\delta }_0(k)/m_{\pi }$ as a function of $(k/m_{\pi }{)}^2$ in NPL2012 for $\mathrm{\Xi }\mathrm{\Xi }({{}^1\mathrm{S}}_0)$ (left) and $\mathrm{\Lambda }\mathrm{\Lambda }({{}^1\mathrm{S}}_0)$ (right). Red lines correspond to the NLO ERE fit using two volumes. The red dashed line represents Lüscher’s formula for $L/a=20$, while the corresponding lattice data around $(k/m_{\pi }{)}^2=0$ [49] are located way out of the plot region of the figures.