Subtleties in triangle loops for $D_s^{+}\to {\rho }^{+}\eta \to {\pi }^{+}{\pi }^0\eta$ in $a_0(980)$ production

We address a general problem in the evaluation of triangle loops stemming from the consideration of the range of the interaction involved in some of the vertices, as well as the energy dependence of the width of some unstable particles in the loop. We find sizeable corrections from both effects. We apply that to a loop relevant to the $D_s^{+}\to {\pi }^{+}{\pi }^0\eta$ decay, and find reductions of about a factor of 4 in the mass distribution of invariant mass of the $\pi \eta$ in the region of the $a_0(980)$. The method used is based on the explicit analytical evaluation of the $q^0$ integration in the $d^{4}q$ loop integration, using Cauchy’s residues method, which at the same time offers an insight on the convergence of the integrals and the effect of form factors and cutoffs.

Figure 1

Triangle loop diagram for the $D_s^{+}$ decay to ${\pi }^{+}{\pi }^0\eta$. The momenta of the particles are shown in parenthesis.

Figure 2

Real and imaginary parts of the amplitude $t_1$ in Eq. (13) with and without the $\theta (·)$ function.

Figure 3

Real and imaginary parts of the amplitude $t_2$ in Eq. (13) with and without the $\theta (·)$ function.

Figure 4

Real and imaginary parts of the amplitude $t_1$ in Eq. (13) without the $\theta (·)$ function, taking a $\rho$ width energy dependent or constant, see Eq. (15).

Figure 5

Ratio $R=|t_1+t_2{|}^2[\mathrm{with}\theta (·)]/|t_1+t_2{|}^2[\mathrm{without}\theta (·)]$, using a constant $\rho$ width (continuous line) in both, numerator and denominator, an energy dependent width (dashed line), and, with a constant $\rho$ width in the denominator and an energy dependent width in the numerator (dot-dashed line).

Figure 6

$d\mathrm{\Gamma }/d{M}_{\mathrm{inv}}({\pi }_0\eta )$ as a function of $M_{\mathrm{inv}}({\pi }_0\eta )$, with constant and energy dependent ${\mathrm{\Gamma }}_{\rho }$. The $|t_{\eta \pi ,\eta \pi }{|}^2$ coming from the scattering amplitude should also multiply both expressions, but it is also omitted here to show the relative effects, as done in the other figures.