Pion-nucleon scattering to $\mathcal{O}(p^3)$ in heavy baryon SU(3) chiral perturbation theory

We calculate the complete $T$ matrices of pion-nucleon ($\pi N$) scattering to the third order in heavy baryon SU(3) chiral perturbation theory. The baryon mass in the chiral limit $M_0$ and the low-energy constants are determined by fitting to phase shifts of $\pi N$ and the experimental octet-baryon masses simultaneously. By using these constants, we obtain the pion-nucleon sigma terms, ${\sigma }_{\pi N}=(34.57\pm 11.85)\mathrm{MeV}$ . We also find a very small strangeness content of the proton, $y\simeq 0$. The scattering lengths and the scattering volumes are predicted, which turn out to be in good agreement with those of other approaches and the available experimental data. The contributions from the third-order amplitudes are discussed in detail. We find that the contributions from the internal kaon lines of one-loop diagrams and the counterterms of the third order are sizeable. In addition, the issue of convergence is also discussed.

Figure 1

Tree diagrams in the heavy baryon approach. Dashed lines represent Goldstone bosons and solid lines represent octet baryons. The heavy dots refer to vertices from ${\mathcal{L}}_{\varphi B}^{(2)}$ and the filled squares refer to insertions from ${\mathcal{L}}_{\varphi B}^{(3)}$. Diagrams with crossed meson lines are not shown.

Figure 2

Nonvanishing one-loop diagrams contributing at third chiral order. Diagrams with self-energy correction on external pion or nucleon lines are not shown.

Figure 3

Fits and predictions for the WI08 phase shifts versus the pion laboratory momentum $|{\mathit{p}}_{\mathrm{lab}}|$ in pion-nucleon ($\pi N$) scattering. The solid lines (fit 1), dashed lines (fit 2), and dotted lines (fit 3) are our results, and the black dots denote the WI08 solutions. Note that the dashed and dotted lines almost coincide with each other. Fitting for all $\pi N$ waves are the data in the range of 50–100 MeV for fit 1 and 50–150 MeV for fits 2 and 3. For higher and lower energies, the phase shifts are predicted.

Figure 4

Contributions involving different mesons’ ($\pi$, $K$, $\eta$) internal lines from one-loop diagrams at third chiral order are shown as the real part of the phase shifts. The dashed, dotted, dash-dotted, thick-solid, and thin-solid lines denote the contributions from mesons’ ($\pi$, $K$, $\eta$) internal lines and the total contributions from one-loop diagrams of the SU(3) $\mathrm{HB}\chi \mathrm{PT}$ and SU(2) $\mathrm{HB}\chi \mathrm{PT}$ [20], respectively.

Figure 5

Contributions from the third-order amplitudes are shown as the real part of phase shifts. The dashed, dotted, dash-dotted, and solid lines denote the contributions from counterterms, one-loop diagrams, relativistic corrections, and total amplitudes, respectively.

Figure 6

Convergence properties for the $1/M_0$ expansion of $\pi N$ phase shifts. The dashed, dotted, and solid lines denote the $1/M_0$, $1/M_0^2$, and the total of their corrections, respectively.

Figure 7

Convergence properties for the $\pi N$ phase shifts. The dashed, dotted, and dashed-dotted lines denote the first, second, and third order, respectively. The solid lines give the sum of the first-, second-, and third-order contributions.