Mass and width of the $\mathrm{\Delta }\left(1232\right)$ resonance using complex-mass renormalization

We discuss the pole mass and the width of the $\mathrm{\Delta }\left(1232\right)$ resonance to third order in chiral effective field theory. In our calculation we choose the complex-mass renormalization scheme (CMS) and show that the CMS provides a consistent power-counting scheme. In terms of the pion-mass dependence, we compare the convergence behavior of the CMS with the small-scale expansion.

Figure 1

Nonvanishing one-loop self-energy diagrams of the Delta resonance at $\mathcal{O}\left(q^3\right)$. The dashed, solid, and double lines correspond to the pion, nucleon, and Delta, respectively. We do not display the tadpole diagram with a vertex from ${\mathcal{L}}_{\pi \mathrm{\Delta }}^{\left(1\right)}$, because its contribution vanishes.

Figure 2

Contributions of the renormalized loop diagrams to the real part of the $\mathrm{\Delta }$-pole position (left panel) and the width (right panel) as a function of the pion mass. The solid (blue) lines denote the CMS and the dashed (red) lines the SSE.

Figure 3

Pion-mass dependence of the loop contribution to the real part of the $\mathrm{\Delta }$ pole. Left panel: CMS. Right panel: SSE. Solid (blue) curves denote the full result, dotted (black) curves the LO expansion, and dashed (red) curves the NLO expansion, respectively.

Figure 4

Pion-mass dependence of the loop contribution to the width. Left panel: CMS. Right panel: SSE. Solid (blue) curves denote the full result, dotted (black) curves the LO expansion, and dashed (red) curves the NLO expansion, respectively.

Figure 5

Pion-mass dependence of the width. The solid (blue) curve denotes the CMS result of Eq. (33), the dashed (red) curve denotes the SSE result with $\mathtt{g}=1.52$, and the dotted (black) curve denotes the LO-SSE result with $\mathtt{g}=1.13$. In each case the parameters are chosen such that the width at the physical mass $M_{{\pi }^{+}}=139.6$ MeV reproduces the experimental value of ${\mathrm{\Gamma }}_{\text{exp}}=100$ MeV.

Figure 6

Contributions of the renormalized loop diagrams to the real part of the $\mathrm{\Delta }$-pole position (left panel) and the width (right panel) as a function of the pion mass. The solid (blue) lines denote the CMS result with the Lagrangian of Ref. [56] [see Eqs. (4) and (5)], and the red (dashed) lines denote the CMS result with the Lagrangian of Ref. [49] [see Eq. (35)].