Strong isospin symmetry breaking in light scalar meson production

Isospin symmetry breaking is discussed as a tool for studying the nature and production mechanisms of light scalar mesons. We are concerned with isospin breaking effects with an amplitude $\sim\sqrt{m_{\rm d}-m_{\rm u}}$ (instead of the usual $\sim m_{\rm d}-m_{\rm u})$, where $m_{\rm u}$ and $m{\rm _d}$ are the u and d quark masses, whose magnitude and phase vary with energy in a resonance-like way characteristic of the ${\rm K}\bar {\rm K}$ threshold region. We consider a variety of reactions that can experimentally reveal (or have revealed) the mixing of ${\rm a}^0_0(980)$- and ${\rm f}_0(980)$ resonances that breaks the isotopic invariance due to the mass difference between ${\rm K}^+$ and ${\rm K}^0$ mesons. Experimental results on the search for ${\rm a}^0_0(980)-{\rm f}_0(980)$ mixing in ${\rm f}_1(1285)\to {\rm f}_0(980)\pi^0\to\pi^+\pi^-\pi^0$ and $\eta(1405)\to {\rm f}_0(980)\pi^0\to\pi^+\pi^-\pi^0$ decays suggest a broader perspective on the isotopic symmetry breaking effects due to the ${\rm K}^+- {\rm K}^0$ mass difference. It has become clear that not only the ${\rm }a^0_0(980)-{\rm f}_0(980)$ mixing but also any mechanism producing ${\rm K}\bar {\rm K}$ pairs with a definite isospin in an S wave gives rise to such effects, thus suggesting a new tool for studying the nature and production mechanisms of light scalars. Of particular interest is the case of a large isotopic symmetry breaking in the $\eta(1405)$ $ \to$ ${\rm f}_0(980)\pi^0$ $\to$ $\pi^+\pi^-\pi^0$ decay due to the occurrence of anomalous Landau thresholds (logarithmic triangle singularities), i.e., due to the $\eta(1405)\to({\rm K}^*\bar {\rm K}+\bar {\rm K}^*{\rm K})\to({\rm K}^+{\rm K}^-+{\rm K}^0\bar {\rm K}^0)\pi^0\to {\rm f}_0(980)\pi^0\to\pi^+\pi^-\pi^0$ transition (where it is of fundamental importance that the ${\rm K}^*$ meson has a finite width).