Width of $N\mathrm{\Delta }$ and $\mathrm{\Delta }\mathrm{\Delta }$ states

It is seen by a coupled-channel $NN$ scattering calculation that in the intermediate two-baryon $N\mathrm{\Delta }$ or $\mathrm{\Delta }\mathrm{\Delta }$ system the width of the state is greatly diminished due to the relative kinetic energy of the two baryons, since the internal energy of the particles, available for pionic decay, is smaller. Specifically, a state-dependent effect arises from the centrifugal barrier in $L\ne 0 N\mathrm{\Delta }$ or $\mathrm{\Delta }\mathrm{\Delta }$ systems. The double $\mathrm{\Delta }$ width can become even smaller than the free width of a single $\mathrm{\Delta }$. This has some bearing to the interpretation of the $d^{\prime }\left(2380\right)$ resonance recently discovered at COSY.

Figure 1

The ${}^5{S}_2\left(N\mathrm{\Delta }\right)$ wave function at energies 400, 578, and 765 MeV without the width (dashed, solid, and dotted curves, respectively). The dash-dotted curve has the width included at 578 MeV—the real part of the $N\mathrm{\Delta }$ wave function shown. In this case, the normalization is associated with the $NN$ wave function asymptotic form $u_{NN}\left(r\right)\sim sin(kr-\pi +{\delta }_2)$ and the wave function is dimensionless.

Figure 2

Basic kinematics of the $N\mathrm{\Delta }$ (left) and $\mathrm{\Delta }\mathrm{\Delta }$ (right) decays. The thick lines present the $\mathrm{\Delta }$'s. The relative momenta $q_{\left(i\right)}$ between the pion and the nucleon are associated with the $\mathrm{\Delta }$ internal energy (invariant mass) by $s_i=\sqrt{q_{\left(i\right)}^2+{\mu }^2}+\sqrt{q_{\left(i\right)}^2+M^2}$. Further dependencies on the relative baryon momentum $p$ are given in the text.

Figure 3

The widths of a representative selection of $N\mathrm{\Delta }$ states in $NN$ scattering: Curves as described in the text. The free width is the thick line above the others.

Figure 4

Probability distributions of momenta at $E_{\mathrm{lab}}=$ 578 MeV as described in the text; the solid curves as functions of the $N\mathrm{\Delta }$ momentum $p$ (left $y$ axis) and the dashed ones of the pion-nucleon momentum $q$ (right $y$ axis). The upper curves are for all main $N\mathrm{\Delta }$ contributions added together and the lower ones for the dominant ${}^5{S}_2\left(N\mathrm{\Delta }\right)$ alone ($p_{\max }=1.87{\mathrm{fm}}^{-1}$ and $q_{\max }=1.04{\mathrm{fm}}^{-1}$).

Figure 5

Argand diagrams of two $NN$ partial waves. Squares ${}^1{D}_2$ and full circles ${}^3{F}_3$ states. Also the unitarity circle is shown.

Figure 6

The accumulation of the phase shift of the ${}^3{F}_3$ state. The Reid potential (6) result (dashed) and the modification (10) (dotted). Coupling to the ${}^5{P}_3\left(N\mathrm{\Delta }\right)$ channel without (dash-dotted) and with the width (solid) as explained in the text. The second solid curve is the imaginary part of the phase. The data are the energy-dependent fit to $pp$ data from Ref. [33].

Figure 7

The widths of the ${}^7{S}_3\left(\mathrm{\Delta }\mathrm{\Delta }\right)$ state in $I=0NN$ scattering: The solid curve initiates from ${}^3{D}_3\left(NN\right)$ and the dashed one from ${}^3{G}_3\left(NN\right)$. The bullet shows the energy and width of the resonance reported, e.g., in Ref. [6].

Figure 8

Argand diagrams of two $NN$ partial waves with a coupling to two $\mathrm{\Delta }$'s. Squares ${}^3{D}_3$ and full circles ${}^3{G}_3$ states. The triangles have three coupled $\mathrm{\Delta }\mathrm{\Delta }$ states and additional attraction in each of these channels.