Erratum: $S$-wave kaon-nucleon interactions from lattice QCD at the physical point [Phys. Rev. D 113, 054506 (2026)]

We use the following interpolating operators for kaons:

and for nucleons:

where $N=(p,n)$ and $q=(u,d)$, and $\alpha$ is the spinor indices. Note that all the quark operators in these equations are local.

The $KN$ four-point correlation function for isospin $I(=0,1)$ is given by

where $(KN)^{(I)}_{\alpha}({\bf r},t+t_{0})$ is the $KN$ sink operator defined as

and $\bar{K}^{0}(t_{0})$ and $\bar{p}_{\alpha}(t_{0})$ are the wall source operators, where local quark source operators in Eqs. (6) and (7) are replaced with the wall source operators. Note that the baryon operators at the sink and source in Eq. (8) share the same spin indices, since the spin flipping does not occur in the S-wave scatterings.

We use $(2+1)$-flavor gauge configurations,“HAL-conf-2023”, generated with the Iwasaki gauge action and the non-perturbatively $\order{a}$-improved Wilson quark action with stout smearing at $\beta=1.82$ and $c_{SW}=1.11$ on a $96^{4}$ lattice [12]. The corresponding lattice spacing and size are $a\approx 0.084~\textrm{fm}$ and $L\approx 8.1~\textrm{fm}$, respectively. The hopping parameters of the ensemble in our calculation are $\kappa_{ud}=0.126117$ and $\kappa_{s}=0.124902$, which corresponds to $m_{\pi}\approx 137~\textrm{MeV}$. The periodic boundary condition is imposed in all four directions. We use 360 gauge configurations which are picked up one per five trajectories across three independent runs. On each configuration, we perform measurements for 96 temporal source locations together with averages over forward-backward propagations, rotating the temporal direction 4 times by hypercubic symmetry. Therefore, the total number of the measurements is $360\times 96\times 2\times 4=276,480$. The statistical errors are estimated by the jackknife method with a bin size of 40 configurations. By comparing with a bin size of 20 configurations, we confirm that the bin-size dependence is small.

We also average the four-point correlation functions over spin indices to increase statistics, as the LO potential is spin-independent. In order to obtain the S-wave components, we employ the partial wave decomposition by the Misner method [30] with $n_{\textrm{max}}=4$, $l_{\textrm{max}}=4$, and $\Delta/a=1.0$, after projecting the four-point correlation functions onto the A${}^{+}_{1}$ representation of the cubic group. However, at the three shortest points on the lattice, $r=0$, $a$, and $\sqrt{2}a$, the LO potentials are computed using the correlation functions without the Misner method.

The kaon and nucleon masses are determined from the corresponding two-point correlation functions. We fit a cosh function in a temporal range $25\leq t/a\leq 39$ for kaon and a single exponential function in $14\leq t/a\leq 20$ for nucleon, leading to $m_{K}=502.0(2)~\textrm{MeV}$ and $m_{N}=942.3(3.4)~\textrm{MeV}$, respectively, which are used in subsequent analysis for the LO potentials and scattering observables.

Figure 1 presents the LO potentials extracted at $t/a=12\textrm{--}16$, which do not suffer from large statistical uncertainties. No significant time dependence is observed in the potentials, which indicates that the LO term in the derivative expansion is dominant and that contamination from the inelastic scattering states is suppressed.

The potential in both $I=1$ and $0$ channels has a repulsive core with a range of about $1~\textrm{fm}$. Between two channels, the range of the core is larger for $I=1$. In addition, the $I=0$ potential possesses an attractive pocket of a few MeV at intermediate distances.

We make a comparison of the previous HAL QCD results at heavier pion masses, both isospin channels at $m_{\pi}\approx 570~\textrm{MeV}$ [28] and only the $I=1$ channel at $m_{\pi}\approx 705~\textrm{MeV}$ [27] ¹¹ 1 There is another result in the HAL QCD method at $m_{\pi}\approx 870~\textrm{MeV}$ [26]. We exclude this work in the comparison, however, since the potentials suffer from a large finite-volume effect.. A visual comparison of the results in the present study with those calculated at $m_{\pi}\approx 570~\textrm{MeV}$ is provided in Appendix A. The potentials in previous studies share the same qualitative feature as the present ones, while the range of the repulsive core is smaller in the previous studies. This indicates that the repulsion is weaker at heavier quark masses, suggesting that the color-magnetic interaction between quarks, that scales inversely with respect to the quark mass, is responsible for the repulsion at short distances. On the other hand, no significant difference is found at long distances.

We then perform the uncorrelated fit to the obtained potentials. We employ the following two types of fit functions: a four-range Gaussian as

and a three-range Gaussian plus a two-pion-exchange (TPE) term

where $b^{I}_{1}<b^{I}_{2}<b^{I}_{3}<b^{I}_{4}$ and $d^{I}_{1}<d^{I}_{2}<d^{I}_{3}$, and $\beta$ is fixed to $2.0~\textrm{fm}^{-2}$, a value used in, e.g., Ref. [31] ²² 2 We cannot take $\beta$ as a fit parameter since the fit results strongly depend on the initial value of $\beta$. . The TPE term is associated with a phenomenological two-pion exchange process via $K^{*}N$ intermediate states [32] ³³ 3 The vector-meson exchange term is not considered in this analysis since it is short-ranged and cannot be distinguished with the interaction process at the level of quarks.. The fit using $V^{\textrm{4G}}_{I}(r)$ works well for both isospin channels. The fit parameters are listed in Table 1 and the corresponding fit curves are shown by teal bands in Fig. 2. For $V^{\textrm{3GTPE}}_{I}(r)$ in the $I=1$ channel, no reasonable fit results can be obtained for some jackknife samples, while the resulting coefficient $\alpha^{I=1}$ is negligibly small for the others, which suggests that the TPE term is not suitable to describe its long-range part. For $I=0$, on the other hand, the function $V^{\textrm{3GTPE}}_{I=0}(r)$ produces good fit results. The parameters are shown in Table 2, and the corresponding curve is depicted by orange bands in Fig. 2. However, the range of the third Gaussian term ($d^{I=0}_{3}$) is comparable with that of the TPE term, $(2m_{\pi})^{-1}\approx 0.72~\textrm{fm}$, which indicates that the TPE term is not enough to reproduce the long-range part of the $I=0$ potential. These results suggest that the coupling of the TPE is likely to be very small since it is suppressed by $(m_{K^{\ast}}m_{N})^{-1}$ with the $K^{*}$ meson mass $m_{K^{\ast}}$.

We solve the Schrödinger equations with the fitted potentials to extract phase shifts of the S-wave $KN$ scatterings. We first notice that two fits for $I=0$ data using $V^{\textrm{4G}}_{I=0}(r)$ and $V^{\textrm{3GTPE}}_{I=0}(r)$ lead to almost identical results on the phase shifts, of which the details will be discussed in the next subsection. We therefore present results obtained from $V^{\textrm{4G}}_{I}(r)$ in both $I=1$ and $0$ channels hereafter. In addition, for visual clarity, we show the scattering results only for $t/a=12$, $14$, and $16$ unless otherwise stated, although we have confirmed that the potentials for the other time slices yield scattering results consistent within statistical uncertainties.

Fig. 3 shows results of phase shifts as a function of the kaon momentum in the laboratory frame $P_{\textrm{lab}}$. In the figure, we show data at $P_{\rm lab}\leq 300$ MeV by bands with dark colors, while those at $P_{\rm lab}>300$ MeV with light colors and stripe pattern, in order to remind readers of a fact that results in the higher momentum region contain larger systematics due to non-locality of the potentials neglected in our LO analysis. The value $P_{\textrm{lab}}=300~\textrm{MeV}$ lies between the first and second excited states in a non-interacting $KN$ system in our finite spatial volume of $(8.1{\rm fm})^{3}$. For comparisons, experimental results (in Refs. [33, 34, 35] for $I=1$ and Refs. [36, 37] for $I=0$) and results from the partial wave analysis in the energy-dependent approach [38, 39, 40] are also shown in the figure.

Our results of the phase shifts monotonically decrease for $I=1$, due to the pure repulsive interaction. Such behaviors are also seen in all experimental results as well as the partial-wave analysis, though they are larger in magnitude than ours. Possible systematic uncertainties in our calculation that might cause this difference are discussed in the next subsection.

For $I=0$, phase shifts from the potential are nearly zero within errors below $P_{\textrm{lab}}\sim 180~\textrm{MeV}$ and start decreasing above it, which is probably implied by a combination of the repulsion with the small attractive pocket. Two experimental results [36, 37] disagree with each other in both sign and magnitude, and three partial-wave analyses show quantitatively different $P_{\rm lab}$ dependence among them. The lack of low energy experimental data for the $I=0$ phase shifts seems to be a reason for these disagreements. Our results lie between two experimental results. Moreover, the phase shifts at $t/a=14$ are consistent with the two of partial-wave analysis [38, 39] near the threshold.

In both isospin channels, phase shifts start from 0 degrees at the threshold (with the convention that phase shifts approach zero at infinitely large momentum), indicating an absence of bound states according to Levinson’s theorem. Above the threshold, they show neither sudden rise nor crossing $90$ degrees, which is a typical behavior of resonances. These observations lead to the conclusion that the $\Theta^{+}(1540)$ pentaquark does not exist in S-wave $KN$ systems.

To evaluate scattering parameters, we calculate $p^{\ast}\cot{\delta_{0}}$ with $p^{\ast}$ being the relative momentum in the center-of-mass frame, of which the inversions, $(p^{\ast}\cot{\delta_{0}})^{-1}$, are shown in Fig. 4. The experimental results [33, 34, 35, 36, 37] and the results from the partial-wave analysis [38, 39, 40] are also depicted in the figure. The scattering length $a_{0}$ and effective range $r_{\mathrm{eff}}$ defined as

are given by

where numbers in the first (second) parentheses are statistical (systematic) errors. Systematic errors are estimated from differences between the central value at $t/a=14$ and that within $t/a=12\textrm{--}16$. The effective range for $I=0$ is not shown here since statistical errors are too large relative to the small central value. We notice that the scattering length corresponds to the $y$-intercept in Fig. 4. In addition, since $(p^{\ast}\cot\delta_{0}(p^{\ast}))^{-1}=a_{0}-\frac{1}{2}a_{0}^{2}r_{\mathrm{eff}}(p^{\ast})^{2}+\order{(p^{\ast})^4}$, positive (negative) sign of the slope at the threshold in the figure indicates negative (positive) sign of the effective range. As a comparison of the scattering lengths, we show in Fig. 4 the summary results of various analyses based on the experimental data [16] ⁴⁴ 4 The summary results include those in Ref. [38], in which the results of $(p^{\ast}\cot\delta_{0}(p^{\ast}))^{-1}$ are shown as dashed lines in Fig. 4., that $a_{0,\textrm{exp}}^{I=1}\in[-0.33,-0.28]~\textrm{fm}$ for $I=1$ and $a_{0,\textrm{exp}}^{I=0}\in[-0.042,0.18]~\textrm{fm}$ for $I=0$.

The scattering length is negative for $I=1$ and nearly zero within errors for $I=0$. Both are qualitatively consistent with the summary results [16] and the results in the partial-wave analysis [38, 39, 40], though the magnitude of our result for $I=1$ is a little smaller. On the other hand, the effective range for $I=1$ is negative, indicating a positive slope in Fig. 4, which is opposite to the summary results $r_{\textrm{eff},\textrm{exp}}^{I=1}\in[0.32,0.5]~\textrm{fm}$ but consistent with the partial-wave analyses that exhibit positive slopes around the threshold in the figure [38, 39]. A discrepancy in $(p^{\ast}\cot\delta_{0}(p^{\ast}))^{-1}$ for $I=1$ between our results and experimental data, existing only above $P_{\textrm{lab}}=140~\textrm{MeV}$ ($(p^{\ast}/m_{\pi})^{2}=0.44$), as seen in Fig. 4, seems to cause a difference in the scattering length.

To make a more direct comparison of our lattice results and experimental data, we plot in Fig. 5 the S-wave component of the total $KN$ cross section obtained from lattice QCD, together with experimental results at $P_{\textrm{lab}}\leq 506~\textrm{MeV}$ [33, 41, 42, 34, 35]. While the total cross section obtained in experiments contains all partial waves, the one for $I=1$ is supposed to be dominated by the S-wave component at low $P_{\textrm{lab}}$, since the differential cross section for $K^{+}p\to K^{+}p$ scatterings has a small dependence on the angle except for forward scatterings, which diverges due to the Coulomb repulsion (see, e.g., Fig.1 in Ref. [35]).

Our result on the $I=1$ S-wave cross section, which is $6.3\pm 0.3~\textrm{mb}$ at the threshold and slowly decreases with respect to $P_{\textrm{lab}}$, deviates from the experimental total cross section, which is larger than $10~\textrm{mb}$ at $P_{\textrm{lab}}<300~\textrm{MeV}$, though some data in Refs. [33, 34] are still consistent with ours within $2$–$3$ $\sigma$. This deviation is also related to discrepancies in the $I=1$ phase shifts. In the next subsection, we will discuss possible sources for this difference.

The $I=0$ S-wave cross section, on the other hand, is nearly zero within errors below $P_{\textrm{lab}}\sim 180~\textrm{MeV}$, and becomes nonzero but less than $1~\textrm{mb}$ above $P_{\textrm{lab}}\sim 180~\textrm{MeV}$. This behavior is consistent with the recent (unitarized) chiral perturbation analysis [44, 14], which claims that experimental data above $P_{\textrm{lab}}=300~\textrm{MeV}$ [41, 42] are mainly explained by the P-wave rather than the S-wave component. The P-wave dominance has also been proposed by the analysis to experimental data [45, 36, 46, 37, 47, 48, 38]. While it is likely that the $I=0$ scattering amplitudes are dominated by the P-wave rather than the S-wave, it is necessary to calculate the P-wave $KN$ interactions in lattice QCD for the definite conclusion. As this calculation is challenging on the physical point due to considerable statistical and systematic errors, it is worth performing it even with heavier quark masses in future.

In this subsection, we discuss possible systematic uncertainties in our calculation that may lead to discrepancies observed between experimental results and ours, particularly in the $I=1$ channel.

We begin with the finite-volume effect on the interacting potential. The half of the spatial extension, $L/2\approx 4.03~\textrm{fm}$, is much larger than the interaction range, which is less than $2.0~\textrm{fm}$ as estimated from Fig. 1. Therefore, the finite-volume effect to our potentials is negligible.

We next examine the systematic uncertainty associated with the fit ansatz to potential. In Sec. IV.1, we introduced the four-range Gaussian Eq. (10), denoted by $V^{\textrm{4G}}_{I}(r)$, and three-range Gaussian plus two-pion exchange term Eq. (11), denoted by $V^{\textrm{3GTPE}}_{I}(r)$. To examine the fit function dependence, we additionally employ a three-range Gaussian plus a pion-exchange term with an arbitrary power

Note that the case $n=2$ is equivalent to $V^{\textrm{3GTPE}}_{I}(r)$. For $I=0$, in addition to $V^{\textrm{4G}}_{I}(r)$ and $V^{\textrm{3GTPE}}_{I}(r)$, we find that $V^{\textrm{3GGPE}}_{I,n}(r)$ with $n=3$ provides good fit parameters. For $I=1$, as discussed in Sec. IV.1, we were able to obtain reasonable fit results for $V^{\textrm{4G}}_{I}(r)$ while we were not for $V^{\textrm{3GTPE}}_{I}(r)$. In addition to $V^{\textrm{4G}}_{I}(r)$, we identify two further fit functions that works well: $V^{\textrm{3GGPE}}_{I,n}(r)$ with $n=4$ and a three-range Gaussian without the pion-exchange term. Hereafter, the latter is referred to as $V^{\textrm{3G}}_{I}(r)$.

The fit results for each ansatz are shown in Fig. 6, and the corresponding $(p^{\ast}\cot{\delta_{0}(p^{\ast})})^{-1}$ obtained from the fitted potential are displayed in Fig 7. For both isospin channels, while the fitted potentials exhibit slight differences at middle and long distances, $0.8\lesssim r\lesssim 2.5~\textrm{fm}$, the resulting phase shifts are consistent within statistical uncertainties. This indicates that the choice of fit ansatz does not significantly affect scattering observables in both channels. However, we note that there remains room for improvement in the examination of the model dependence; to quantify such an effect in a more rigorous manner, it is necessary to take an approach that does not rely on specific functional assumptions, such as one based on Bayesian inference. Establishing such a framework is left in future.

We then investigate systematics associated with contamination of inelastic states and the LO approximation to potentials. These effects typically appear as the time dependence of potentials. As seen in Fig. 1, however, such dependence is negligibly small. Furthermore, as a consistency check, we compute the phase shifts at the ground-state energies on a finite volume using the Lüscher’s formula [20], which is shown as brown curves in Fig. 4. The ground-state energy is evaluated from the $KN$ temporal correlation functions with optimized operators. For details, see Appendix B. We also evaluate the scattering lengths from the ground-state energies on a finite volume via the $1/L$ expansion [49] as

where $c_{1}=-2.837297$ and $c_{2}=6.375183$. We obtain $a^{I=1}_{0,\textrm{FV}}=-0.261(164)~\textrm{fm}$ and $a^{I=0}_{0,\textrm{FV}}=+0.122(243)~\textrm{fm}$, which should be compared with $a^{I=1}_{0}=-0.226(5)(^{+5}_{-0})~\textrm{fm}$ and $a^{I=0}_{0}=+0.028(61)(^{+3}_{-26})~\textrm{fm}$, respectively, obtained from the potentials. The FV analysis and the potential method give consistent results on the phase shifts, while errors in the FV analysis are too large to establish the absence of inelastic contributions or non-locality effects. In future studies, it is important to explicitly quantify the non-locality effect for the high momentum region shown by light color and stripes in Figs. 3, 4, and 5, by performing the next-to-leading-order analysis in the derivative expansion for the potentials.

We also investigate an effect due to a small difference in the quark masses between our setup and nature. Since an extra simulation with different quark masses is beyond the scope of this paper, we have performed the following analysis. When we derive the LO potential and solve the Schrödinger equation, we use the same R-correlator but with the isospin-averaged values of kaon and nucleon masses in nature, $m_{K}=495.64~\textrm{MeV}$ and $m_{N}=938.92~\textrm{MeV}$, instead of $m_{K}=502.0(2)$ MeV and $m_{N}=942.3(3.4)$ MeV in our setup. We find no considerable change in the phase shifts, which suggests that the phase shifts are insensitive to the small differences of quark masses in kinematics.

Lasts but not least, we consider the lattice discretization effect [50]. Again, an extra simulation with a different lattice spacing is beyond the scope of this paper. To estimate how the lattice artifact at short distances affects scattering observables for $I=1$, we artificially enhanced the coefficient of the Gaussian term with the shortest range, namely $a^{I=1}_{1}$ in Eq. (10), by five times larger than the original value. The modified potential provides almost the same phase shifts as the original ones, indicating that the discretization error at short distances does not affect our results significantly. While there still remains a possibility that the lattice artifact non-trivially affects potentials at intermediate distances, such an examination of the lattice discretization effect is left to future studies.