Low-energy theorems for nucleon-nucleon scattering at $M_{\pi }=450$ MeV

We apply the low-energy theorems to analyze the recent lattice QCD results for the two-nucleon system at a pion mass of $M_{\pi }\simeq 450$ MeV obtained by the NPLQCD Collaboration. We find that the binding energies of the deuteron and dineutron are inconsistent with the low-energy behavior of the corresponding phase shifts within the quoted uncertainties and vice versa. Using the binding energies of the deuteron and dineutron as input, we employ the low-energy theorems to predict the phase shifts and extract the scattering length and the effective range in the ${}^{3}S_1$ and ${}^{1}S_0$ channels. Our results for these quantities are consistent with those obtained by the NPLQCD Collaboration from effective field theory analyses but are in conflict with their determination based on the effective-range approximation.

Figure 1

Linear with $M_{\pi }$ behavior of the effective range in units of pion mass $M_{\pi }r$ in the ${}^3\mathrm{S}_1$ partial wave conjectured in Ref. [19]. Red solid triangle corresponds to the NPLQCD result at $M_{\pi }\simeq 800$ MeV [18] while the shaded area shows the uncertainty of the suggested linear interpolation. The black square shows the empirical value of the effective range at the physical pion mass [31].

Figure 2

Graphical illustration of the functions $\mathrm{\Phi }$ and $L$ in Eq. (2.6), which correspond to convolutions of the pionic amplitude $T_{\pi }$ with the pointlike vertices as explained in the text.

Figure 3

Neutron-proton phase shifts (left panel) and the effective-range function (right panel) in the ${}^{3}S_1$ channel calculated on the lattice at $M_{\pi }\sim 450$ MeV [9] (filled black regions) in comparison with the predictions based on the LETs at LO (orange light-shaded bands) and NLO (blue dark-shaded and hatched blue light-shaded bands) using the NPLQCD result for the deuteron binding energy $B_d$ as input. The uncertainty at LO shown by the orange bands is entirely given by the uncertainty of $B_d$ in Eq. (3.1). The NLO dark-shaded (hatched light-shaded) bands correspond to the uncertainty in $B_d$ and the theoretical uncertainty of the LETs estimated via the variation of $\beta$ with $\delta \beta =0.5\left(\delta \beta =1.0\right)$ combined in quadrature. The grey light- and dark-shaded bands in the right panel depict the fit results of the lattice points of Ref. [9] based on the effective range approximation. The energy of the bound (virtual) states corresponds to the intersection points of the effective-range function $kcot{\delta }^{{(}^3{S}_1)}$ and the unitarity term $ik/M_{\pi }=\pm \sqrt{-{(k/M_{\pi })}^2}$, shown by the dotted line in the right panel, in the lower (upper) half-plane. The phase shift corresponds to the Blatt-Biedenharn parametrization of the $S$ matrix [43].

Figure 4

Correlations between the inverse scattering length $a^{-1}$, effective range $r$ and the binding energy in the ${}^{3}S_1$ partial wave induced by the one-pion exchange potential. The red solid and dashed magenta lines show the predictions of the LO LETs for $M_{\pi }{r}^{{(}^3{S}_1)}$ and $B_d$. The light-shaded bands between the red solid and dashed magenta lines visualize the predictions of the NLO LETs for $M_{\pi }{r}^{{(}^3{S}_1)}$ and $B_d$, respectively, and reflect the theoretical uncertainty estimated via the variation of $\beta$ with $\delta \beta =0.5$ as described in the text. The horizontal dotted lines specify the range of values for $B_d$ consistent with the lattice-QCD results of Ref. [9] for this observable. The solid dark-red circle (blue rectangle) shows the LO (NLO) LET predictions for the effective range. The open black circle gives the result for the inverse scattering length and effective range reported by the NPLQCD Collaboration [9] while the grey area around it shows the estimated uncertainty from that paper. All results correspond to the Blatt-Biedenharn parametrization of the S-matrix [43].

Figure 5

Neutron-proton phase shifts in the ${}^{3}S_1$ channel (left panel) and the mixing angle ${\overline{\epsilon }}_1$ (right panel) at $M_{\pi }\sim 450$ MeV based on the LETs at LO and NLO in comparison with the results obtained in Ref. [9] using the EFT formulation of Ref. [10], labeled as BBSvK, at LO (light-shaded band between pink dotted lines) and NLO (light-shaded band between blue dashed lines). The results for the ${}^3\mathrm{S}_1$ phase shift correspond to the Blatt-Biedenharn parametrization of the S-matrix [43] while the mixing parameter is shown for the Stapp parametrization to allow for the comparison with the results of Ref. [9]. For remaining notation see Fig. 3.

Figure 6

Two-nucleon phase shifts (left panel) and the effective-range function (right panel) in the ${}^{1}S_0$ channel calculated on the lattice at $M_{\pi }\sim 450$ MeV [9] in comparison with the predictions based on the LETs at LO and NLO using the NPLQCD result for the dineutron binding energy $B_{nn}$ as input. For notation see Fig. 3.

Figure 7

Two-nucleon phase shifts (left panel) and the effective-range function (right panel) in the ${}^{1}S_0$ channel calculated on the lattice at $M_{\pi }\sim 450$ MeV [9] in comparison with the predictions based on the LETs at NLO (blue shaded bands) using the scattering length in Eq. (3.14) as input. For remaining notation see Fig. 3.

Figure 8

Nucleon-nucleon effective range ($M_{\pi }r$) in the ${}^1\mathrm{S}_0$ (left panel) and ${}^3\mathrm{S}_1$ (right panel) partial waves predicted based on the next-to-leading order LETs using the bound state energies calculated on the lattice as input. Blue open triangles, orange crosses, green open circles and purple solid circles show $M_{\pi }r$ for the binding energies at $M_{\pi }\simeq 300$ MeV [8], $M_{\pi }\simeq 390$ MeV [7], $M_{\pi }\simeq 450$ MeV [9], and $M_{\pi }\simeq 510$ MeV [21], respectively. The uncertainty of our results for $M_{\pi }r$ is twofold: the smaller error bars reflect the uncertainty of the lattice results for the binding energies used as input while larger ones correspond to the theoretical uncertainty of the LETs estimated by setting $\delta \beta =1$ and the uncertainty of the lattice results added in quadrature. Red solid triangles correspond to the NPLQCD results for $M_{\pi }r$ at $M_{\pi }\simeq 800$ MeV [18]. The black squares show the empirical values of the effective range at the physical pion mass [31, 46].