Meson-baryon scattering to one-loop order in heavy baryon chiral perturbation theory

We calculate the T-matrices of pseudoscalar meson octet-baryon scattering to one-loop order in SU(3) heavy baryon chiral perturbation theory. The pertinent combinations of low-energy constants are determined by fitting to phase shifts of $\pi N$ and $KN$ scattering and the corresponding data. By using these low-energy constants, we obtain the strong phase shift difference of $\pi \mathrm{\Lambda }$ scattering at the $\mathrm{\Xi }$ mass, ${\delta }_P-{\delta }_S=(8.8\pm 0.2)$ degrees, in agreement with experimental results. We find that the phase shifts in the $S_{01}(\pi \mathrm{\Sigma })$, $S_{01}(K\mathrm{\Xi })$, $S_{01}(\overline{K}N)$, and $S_{11}(\overline{K}\mathrm{\Sigma })$ waves are so strong that resonances may be generated dynamically in all these channels. We also predict the scattering lengths and make comparisons with the results obtained from the threshold T-matrices in heavy baryon chiral perturbation theory and the method of covariant infrared regularization. The issue of convergence is also discussed in detail.

Figure 1

Tree diagrams contributing at first and second chiral order. Dashed lines represent Goldstone bosons and solid lines represent octet baryons. The heavy dots refer to vertices from ${\mathcal{L}}_{\varphi B}^{(2)}$. 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 meson or baryon lines are not shown.

Figure 3

Fits and predictions for the WI08 and SP92 phase shifts in $\pi N$ and $KN$ scatterings versus the pion and kaon laboratory momenta, respectively. Fitting in all $\pi N$-waves except for $P_{11}$-wave and $KN$-waves of isospin $I=1$ are the data between 50 and 90 MeV, while fitting $P_{11}$-wave data from 30 to 70 MeV and $KN$-waves of isospin $I=0$ data at $p_{\mathrm{lab}}=(110,120,130,140,150)\mathrm{MeV}$. For higher and lower energies, the phase shifts are predicted. The dotted lines denote the results from SAID [43]. The solid lines refer to our calculations in SU(3) $\mathrm{HB}\chi \mathrm{PT}$ and the dashed lines denote the results in SU(2) $\mathrm{HB}\chi \mathrm{PT}$ for $\pi N$ scattering.

Figure 4

Predictions for the pion-baryon phase shifts versus the pion laboratory momentum. The dashed, dotted, dash-dotted and solid lines denote the first-, second-, third-order and their sum contributions, respectively. The dotted lines denote the results from WI08 solution [43].

Figure 5

Predictions for the kaon-baryon phase shifts versus the kaon laboratory momentum. The notation for lines is the same as in Fig. 4. The dotted lines denote the results from SP92 solution [43].

Figure 6

Predictions for the antikaon-baryon phase shifts versus the antikaon laboratory momentum. The notation for lines is the same as in Fig. 4.

Figure 7

Predictions for the eta-baryon phase shifts versus the eta laboratory momentum. The notation for lines is the same as in Fig. 4.

Figure 8

Contributions involving different mesons ($\pi$, $K$, $\eta$) internal lines from one-loop diagrams at third chiral order are shown as the real part of amplitudes multiplied by the center-of-mass momentum. The dashed, dotted, dash-dotted and solid lines denote the contributions from mesons ($\pi$, $K$, $\eta$) internal lines and total one-loop diagrams, respectively.