Pole properties of a resonance: When to subtract partial-decay widths to obtain the pole widths

When a resonance lies near the threshold of a heavier channel, an interesting feature can occur. The paradigmatic example employed here is the scalar isoscalar $f_0(980)$ resonance that couples to the lighter $\pi \pi$ and heavier $K\bar{K}$ channels. It is shown that the decay width is given by the sum or subtraction of the partial decay widths depending on whether the pole lies in the Riemann sheet that is contiguous with the physical one above or below the $K\bar{K}$ threshold, respectively. Next, we show that the usually disregarded renormalization of bare parameters in Flatté or energy-dependent Breit-Wigner parameterizations is essential to extract physical information. The compositeness of the $f_0(980)$ by using a Flatté parameterization matched to reproduce the pole properties obtained from Roy equations and other analytic constraints is evaluated.