Higher derivative terms in the $\pi \mathrm{\Delta }N$ interaction: Some phenomenological consequences

In this paper, we implement the use of a $\pi N\mathrm{\Delta }$(1232) vertex interaction containing both first- and second-order derivative terms, as required by renormalization and power-counting considerations. As was previously shown, both interactions present quantization shortcomings but can be used in a pertubative calculation. Our results indicate that the usual $\pi$ derivative plus the spin-3/2 gauge invariant (derivative also in the $\mathrm{\Delta }$ field) should be included in amplitude calculations, as also all higher derivative interactions respecting chiral invariance. We show that both interactions make essentially the same resonant contribution to the elastic ${\pi }^{+}p$ cross section, so changing the ratio between both coupling constants amounts to a correction of the background. The elastic ${\pi }^{+}p$ cross section up to 300 MeV changes only mildly when that ratio is changed, but the total ${\pi }^{-}p$ scattering, which has poor fit within both interactions separately, can be much improved in the same energy range by tuning the ratio between both coupling constants.

Figure 1

Amplitude of pion ($\pi$)–nucleon ($N$) scattering split in $s$ or pole (left) and $u$ or cross (right) contributions for $\mathrm{\Delta }$ resonance.

Figure 2

Background non-resonant contributions to the $\pi N$ amplitude.

Figure 3

Elastic ${\pi }^{+}p$ and ${\pi }^{-}p$ total and elastic cross sections. (a) Fittings obtained with the interaction $I_1$ ($\kappa =0)$ and $I_2$ ($\kappa =1)$ for ${\pi }^{+}p$ and results for ${\pi }^{-}p$ (total and elastic), keeping the $\kappa =0$ fitted parameters with other values of $\kappa$ to improve the description. (b) Refitting obtained with $I_1+I_2$ ($\kappa =0.45$) for ${\pi }^{+}p$ and reproduction of ${\pi }^{-}p$ with the obtained parameters.