The reaction $\pi N\to \pi \pi N$ in chiral effective field theory with explicit $\Delta \left(1232\right)$ degrees of freedom

The reaction $\pi N\to \pi \pi N$ is studied at tree level up to next-to-leading order in the framework of manifestly covariant baryon chiral perturbation theory with explicit $\Delta \left(1232\right)$ degrees of freedom. Using total cross-section data to determine the relevant low-energy constants, predictions are made for various differential as well as total cross sections at higher energies. A detailed comparison of results based on the heavy-baryon and relativistic formulations of chiral perturbation theory with and without explicit $\Delta$ degrees of freedom is given.

Figure 1

LO graphs for the reaction $\pi N\to \pi \pi N$. Nucleons are denoted by solid lines; pions, by dashed lines. $\Delta$ is denoted by a double solid line. Crossed diagrams are not shown.

Figure 2

NLO graphs for the reaction $\pi N\to \pi \pi N$. A filled circle (filled square) denotes an insertion of the $c_i$ ($b_i$) vertices. Crossed diagrams are not shown.

Figure 3

Additional NLO graphs contributing to the reaction $\pi N\to \pi \pi N$ in the heavy-baryon framework. A hatched circle/square denotes a pure $1/m_N$ insertion. Crossed diagrams are not shown.

Figure 4

Kinematics and the definitions of angles ${\theta }_1$, ${\theta }_2$, and ${\varphi }_2$ (a), $\varphi$ (b), $\alpha$ and $\beta$ (c), and ${\varphi }^{\prime }$ (d).

Figure 5

Fit in HB$\chi$PT: anticorrelation between $b_4$ and $b_5$ ($g_1=2.27$).

Figure 6

Fit in $\chi$PT: anticorrelation between $b_4$ and $b_5$ (a) ($g_1=2.27$, $b_3=0.99$ GeV${}^{-1}$, $b_6=-0.61$ GeV${}^{-1}$) and between $b_3$ and $b_6$ (b) ($g_1=2.27$, $b_4=-6.37$ GeV${}^{-1}$, $b_5=11.34$ GeV${}^{-1}$).

Figure 7

Predictions for the total cross section up to $T_{\pi }\simeq 400\mathrm{M}\mathrm{e}\mathrm{V}$. Columns from left to right correspond to the $\Delta$-full covariant $\chi$PT, $\Delta$-less covariant $\chi$PT, $\Delta$-full HB$\chi$PT, and $\Delta$-less HB$\chi$PT predictions, respectively. Dashed and solid lines refer to LO (i.e., order $Q^1$ or ${\epsilon }^1$) and NLO (i.e., up to order $Q^2$ or ${\varepsilon }^2$) results. Energies used in the fit at NLO are below the vertical dotted line. Bands at NLO correspond to use of the KH and GW sets of LECs $c_i$.

Figure 8

NLO $\chi$PT predictions for the double-differential cross sections for ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ with respect to the kinetic energy and the solid angle of the outgoing ${\pi }^{+}$; see Eq. (38). Upper and lower panels correspond to the relativistic and heavy-baryon approaches, respectively. Light-gray (blue) and dark-gray (red) bands (the latter nearly shrink to lines) refer to $\Delta$-less and $\Delta$-full calculations, respectively. Bands correspond to use of the KH and GW sets of LECs $c_i$.

Figure 9

NLO relativistic $\chi$PT predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\theta }_2$ for $\sqrt{s}=1301\mathrm{M}\mathrm{e}\mathrm{V}$; see Eq. (43). Light-gray (blue) and dark-gray (red) bands refer to $\Delta$-less and $\Delta$-full calculations, respectively. Bands correspond to use of the KH and GW sets of LEC $c_i$.

Figure 10

Comparison of NLO relativistic $\Delta$-full and $\Delta$-less $\chi$PT predictions for the angular correlation functions in the ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ channel at fixed ${\theta }_1$ and ${\varphi }_2$ for $\sqrt{s}=1301\mathrm{M}\mathrm{e}\mathrm{V}$; see Eq. (43). For remaining notation see the caption to Fig. 9.

Figure 11

Comparison of NLO relativistic $\Delta$-full and $\Delta$-less $\chi$PT predictions for the single-differential cross sections with respect to $M_{\pi \pi }^2$ and $t$, respectively, between the two channels ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ (left) and ${\pi }^{+}p\to {\pi }^{+}{\pi }^{+}n$ (right); see Eq. (45). For remaining notation see the caption to Fig. 9.

Figure 12

Comparison of NLO relativistic $\Delta$-full and $\Delta$-less $\chi$PT predictions for the double-differential cross sections with respect to $M_{\pi \pi }^2$ and $t$ between the two channels ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ (left) and ${\pi }^{+}p\to {\pi }^{+}{\pi }^{+}n$ (right) for two different incoming pion energies; see Eq. (45). For remaining notation see the caption to Fig. 9.

Figure 13

Comparison of NLO relativistic $\Delta$-full and $\Delta$-less $\chi$PT predictions for the single-differential cross sections with respect to $cos\theta$ between the two channels ${\pi }^{-}p\to {\pi }^{+}{\pi }^{-}n$ (left) and ${\pi }^{+}p\to {\pi }^{+}{\pi }^{+}n$ (right) for two different incoming pion energies; see Eq. (45). For remaining notation see the caption to Fig. 9.