Low-energy theorems for nucleon-nucleon scattering at unphysical pion masses

The longest-range part of the nuclear force from the one-pion exchange governs the energy dependence of the scattering amplitude in the near-threshold region and imposes correlations between the coefficients in the effective range expansion. These correlations may be regarded as low-energy theorems and are known to hold to a high accuracy in the neutron-proton ${}^3{S}_1$ partial wave. We generalize the low-energy theorems to the case of unphysical pion masses and provide results for the correlations between the coefficients in the effective range expansion in this partial wave for pion masses up to $M_{\pi }\sim 400\text{MeV}$. We discuss the implications of our findings for the available and upcoming lattice-quantum-chromodynamics simulations of two-nucleon observables.

Figure 1

Cut structure of the partial-wave nucleon-nucleon scattering amplitude in the complex energy plane in the absence of electromagnetic interactions. The cross refers to a possible bound state.

Figure 2

Correlations between the inverse scattering length $a^{-1}$, effective range $r$, and the first three shape parameters $v_2,v_3$, and $v_4$ induced by the one-pion exchange interaction in the ${}^1{S}_0$ (left panel) and ${}^3{S}_1$ (right panel) channels. Solid rectangles, diamonds, open triangles, and circles correspond to the values of $r,v_2,v_3$, and $v_4$, respectively, extracted from the Nijmegen partial-wave analysis [53, 60]. The results in the ${}^3{S}_1$ partial wave correspond to the Blatt-Biedenharn parametrization of the $S$ matrix [61].

Figure 3

Quadratic polynomial regression fits to lattice-QCD data for the pion decay constant $F_{\pi }$, nucleon mass $m_N$, and the nucleon axial-vector coupling constant $g_A$. Lattice data for $F_{\pi }$ and $m_N$ $\left(g_A\right)$ correspond to the $2+1$ flavor simulations of the Budapest-Marseille-Wuppertal collaboration using Wilson fermions [66] and 2 and $2+1+1$ flavor simulations reported in Ref. [67] (Ref. [68]) using twisted mass fermion ensembles. For details of the simulations and analyses see the above-mentioned references. Solid circles without error bars show the experimentally measured values. The dashed lines depict our fit results to the lattice data and the experimental values while the shaded bands correspond to the $67\%$ confidence levels of the interpolations.

Figure 4

Correlations between the inverse scattering length $a^{-1}$, effective range $r$, and the first three shape parameters $v_2,v_3$, and $v_4$ in the ${}^3{S}_1$ partial wave induced by the one-pion exchange interaction. Various thick lines show the predictions of the LO LETs while light-shaded bands (hardly visible for small $M_{\pi })$ between thin lines depict the results of NLO LETs and reflect the estimated uncertainty from unknown $M_{\pi }$ dependence of the subleading short-range interaction as explained in the text.

Figure 5

(Left panel) Linear with $M_{\pi }$ interpolation of the quantity $M_{\pi }r$ in the ${}^3{S}_1$ partial wave according to Eq. (4.1) as suggested in Ref. [40]. (Right panel) Linear with $M_{\pi }^2$ interpolation of the quantity $M_{\pi }r$ in the ${}^3{S}_1$ partial wave according to Eq. (4.2). In both cases, solid squares refer to the value of $M_{\pi }r$ at the physical point.

Figure 6

NLO LET predictions for the pion mass dependence of the deuteron binding energy, the ratio ${\gamma }_d/M_{\pi }$, the ratio $a/r$, and the first three shape parameters in the ${}^3{S}_1$ partial wave assuming the linear $M_{\pi }$ dependence of the effective range specified in Eq. (4.1) and visualized in the left panel of Fig. 5. Dark-shaded bands show our estimation of the uncertainty of the NLO LETs because of the unknown $M_{\pi }$ dependence of the subleading short-range interaction specified in Eq. (3.1); light-shaded bands depict the uncertainty in the linear extrapolation of the effective range used as input, as shown in the left panel of Fig. 5.

Figure 7

NLO LET predictions for the pion mass dependence of the deuteron binding energy, the ratio ${\gamma }_d/M_{\pi }$, the ratio $a/r$, and the first three shape parameters in the ${}^3{S}_1$ partial wave assuming the linear $M_{\pi }^2$ dependence of the effective range specified in Eq. (4.2) and visualized in the right panel of Fig. 5. For notations see Fig. 6.