Elastic and inelastic pion-nucleon scattering to fourth order in chiral perturbation theory

We extend our previous study of elastic pion-nucleon scattering in the framework of chiral perturbation theory by performing a combined analysis of the reactions $\pi N\to \pi N$ and $\pi N\to \pi \pi N$. The calculation is carried out to fourth order in the chiral expansion using the heavy-baryon approach and the covariant formulation supplemented with a modified version of the extended on-mass-shell renormalization scheme. We demonstrate that a combined fit to experimental data in both channels leads to a reduced amount of correlations between the low-energy constants. A satisfactory description of the experimental data in both channels is obtained, which is further improved upon including tree-level contributions of the $\mathrm{\Delta }$(1232) resonance. We also explore a possibility of using the empirical information about $\pi N$ subthreshold parameters obtained recently by means of the Roy-Steiner equations to stabilize the fits.

Figure 1

Tree-level graphs contributing to the reaction $\pi N\to \pi \pi N$. The black, gray, white blobs denote insertions of the $c_i/d_i/e_i$ vertices, whereas the black diamonds denote insertions of the $l_i$ vertices. Dashed and solid lines refer to pions and nucleons, respectively. Crossed diagrams are not shown.

Figure 2

One-loop graphs of the tadpole type contributing to the reaction $\pi N\to \pi \pi N$. For notation see Fig. 1.

Figure 3

One-loop graphs of the self-energy type contributing to the reaction $\pi N\to \pi \pi N$. For notation see Fig. 1.

Figure 4

Transition from leading to next-to-leading order loop graphs. For notation see Fig. 1.

Figure 5

Transition from $\pi N\to \pi \pi N$ graphs to $\pi N\to \pi N$ graphs. The shaded blob denotes any possible interaction. For notation see Fig. 1.

Figure 6

Leading-order $\mathrm{\Delta }$ pole diagrams, where the double solid line refers to the $\mathrm{\Delta }$. For notation see Fig. 1.

Figure 7

Reduced ${\chi }_{\pi N}^2/{\chi }_{\pi \pi N}^2$ (with theoretical error) and ${\overline{\chi }}_{\pi N}^2/{\overline{\chi }}_{\pi \pi N}^2$ (without theoretical error) for fits up to various maximum energies $T_{\pi ,\pi \pi N}$. The blue, red, green bars denote the results for the HB-$NN$, HB-$\pi N$, Cov counting, respectively.

Figure 8

Change of the LECs at $Q^3$ over the maximum fit energy $T_{\pi ,\pi \pi N}$.

Figure 9

Change of the LECs at $Q^4$ over the maximum fit energy $T_{\pi ,\pi \pi N}$.

Figure 10

Change of the LECs at $Q^4$ over the maximum fit energy $T_{\pi ,\pi \pi N}$.

Figure 11

Predictions for the $\pi N\to \pi NS$ and $P$ waves up to $T_{\pi }=100\mathrm{MeV}$ with the LECs in Table 1 taken as input. Columns from left to right correspond to the predictions in the HB-$NN$, HB-$\pi N$ and covariant counting, respectively. The orange, pink, and red (dotted, dashed, and solid) bands refer to the $Q^2, Q^3$, and $Q^4$ results including theoretical uncertainties, respectively.

Figure 12

Predictions for the $\pi N\to \pi NS$ and $P$ waves up to $T_{\pi }=100\mathrm{MeV}$ with the LECs in Tables 2 and 3 taken as input. Columns from left to right correspond to the predictions in the HB-$NN$, HB-$\pi N$ and covariant counting, respectively. The orange, pink, and red (dotted, dashed, and solid) bands refer to the $Q^2, Q^3$, and $Q^4$ results including theoretical uncertainties, respectively.

Figure 13

Predictions for the $\pi N\to \pi \pi N$ total cross sections up to $T_{\pi }=400\mathrm{M}\mathrm{e}\mathrm{V}$. The energies used in the fit are on the left of the vertical dotted lines. For remaining notation see the caption of Fig. 12.

Figure 14

Predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\theta }_2$ for $T_{\pi }=280\mathrm{M}\mathrm{e}\mathrm{V}$. The bottom, middle, top panel correspond to the HB-$NN$, HB-$\pi N$ and covariant counting. The orange, pink, and red (dotted, dashed, and solid) bands refer to the $Q^2, Q^3$, and $Q^4$ results including theoretical uncertainties, respectively.

Figure 15

Predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\theta }_2$ for $T_{\pi }=280\mathrm{M}\mathrm{e}\mathrm{V}$. The bottom, middle, top panel correspond to the HB-$NN$, HB-$\pi N$, and covariant counting. The orange, pink, and red (dotted, dashed, and solid) bands refer to the $Q^2, Q^3$, and $Q^4$ results including theoretical uncertainties, respectively.

Figure 16

Predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\varphi }_2$ for $T_{\pi }=280\mathrm{M}\mathrm{e}\mathrm{V}$. For remaining notation see Fig. 14.

Figure 17

Predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\varphi }_2$ for $T_{\pi }=280\mathrm{M}\mathrm{e}\mathrm{V}$. For remaining notation see Fig. 14.

Figure 18

Predictions for the single-differential cross sections with respect to $M_{\pi \pi }^2$ for the channel ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$. For remaining notation see Fig. 12.

Figure 19

Predictions for the single-differential cross sections with respect to $t$ for the channel ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$. For remaining notation see Fig. 12.

Figure 20

Predictions for the $\pi N\to \pi \pi N$ total cross sections up to $T_{\pi }=400\mathrm{M}\mathrm{e}\mathrm{V}$. Columns from left to right correspond to the predictions in the HB-$NN$, HB-$\pi N$, and covariant counting, respectively. The orange, pink, and red (dotted, dashed, and solid) bands refer to the $Q^2+{\delta }^1, Q^3+{\delta }^1$, and $Q^4+{\delta }^1$ results including theoretical uncertainties, respectively.

Figure 21

Reduced ${\chi }_{\pi N}^2/{\chi }_{\pi \pi N}^2$ (with theoretical error) and ${\overline{\chi }}_{\pi N}^2/{\overline{\chi }}_{\pi \pi N}^2$ (without theoretical error) for fits including leading $\mathrm{\Delta }$-pole contributions up to various maximum energy $T_{\pi ,\pi \pi N}$. The blue, red, green bars denote the results for the HB-$NN$, HB-$\pi N$, Cov counting, respectively.