$D^* \, \Xi$ state. For this we follow Ref. [@xiao] and consider the diagrams of Fig. \[pionX\].

![Box diagrams accounting for the $D \, \Xi \to D^* \, \Xi \to D \, \Xi$.[]{data-label="pionX"}](pionX.eps "fig:"){width="\textwidth"}\

As discussed in Ref. [@xiao], in addition to the $\pi$ exchange there is a contact term called Kroll-Ruderman in the $\gamma \, N \to \pi \, N$ (or $\rho \, N \to \pi \, N$) transition, then the four diagrams of Fig. \[pionX\] must be evaluated. They provide a $\delta V$ potential for $D \, \Xi \to D \, \Xi$ which can be evaluated by means of Eq. (40) of Ref. [@xiao], simply changing the masses of $B$, $B^*$ to $D$, $D^*$ and $N$ to $\Xi$. We have performed the calculation and, compared to the potential $V_{ij}$ from Eq.  and Table \[tabDij\] we find $\delta V / V \approx 0.012$ for the $\Xi \, D$ channel calculated at the energy of the pole around 3090 MeV (which is dominated by this channel, as shown in Table \[res1\] of the Results section), a correction of order $1\%$, which we safely neglect.

Results {#sec:res}
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We use the potential of Eq.  and the on-shell factorized Bethe-Salpeter equation [@nsd; 