The $$D_s^+ \rightarrow a_0(980) e^+ \nu _e$$ reaction and the $$a_0(980)-f_0(980)$$ mixing

We perform a study of the $$D_s^+ \rightarrow a_0(980)(f_0(980)) e^+ \nu _e$$ D s + → a 0 ( 980 ) ( f 0 ( 980 ) ) e + ν e reactions investigating the different sources of isospin violation which make the production of the $$a_0(980)$$ a 0 ( 980 ) possible. We find that loops involving kaons in the production mechanism provide a source of isospin violation since they do not cancel due to the different mass of charged and neutral kaons, but we also find that the main source comes from the breaking of isospin in the meson-meson transition T matrices, which contain information on the nature of the low lying scalar mesons. The reaction is thus very sensitive to the nature of the $$a_0(980)$$ a 0 ( 980 ) and $$f_0(980)$$ f 0 ( 980 ) resonances. Our results are consistent with the present upper bound for $$a_0(980)$$ a 0 ( 980 ) production and only a factor three smaller, indicating that future runs with more statistics should find actual numbers for this reaction from where we can learn more about the origin of the scalar resonances and their nature.

A phenomenological analysis on isospin-violating decay of X(3872)

In a molecular scenario, we investigate the isospin-breaking hidden charm decay processes of X(3872), i.e., $$X(3872) \rightarrow \pi ^+ \pi ^- J/\psi $$ X ( 3872 ) → π + π - J / ψ , $$X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi $$ X ( 3872 ) → π + π - π 0 J / ψ , and $$X(3872)\rightarrow \pi ^0\chi _{cJ}$$ X ( 3872 ) → π 0 χ cJ . We assume that the source of the strong isospin violation comes from the different coupling strengths of X(3872) to its charged components $$D^{*+} D^-$$ D ∗ + D - and neutral components $$D^{*0 } {\bar{D}}^0$$ D ∗ 0 D ¯ 0 as well as the interference between the charged meson loops and neutral meson loops. The former effect could fix our parameters by using the measurement of the ratio $$\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- \pi ^0 J/\psi ]/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$ Γ [ X ( 3872 ) → π + π - π 0 J / ψ ] / Γ [ X ( 3872 ) → π + π - J / ψ ] . With the determined parameter range, we find that the estimated ratio $$\Gamma [X(3872) \rightarrow \pi ^0 \chi _{c1}/\Gamma [X(3872) \rightarrow \pi ^+ \pi ^- J/\psi ]$$ Γ [ X ( 3872 ) → π 0 χ c 1 / Γ [ X ( 3872 ) → π + π - J / ψ ] is well consistent with the experimental measurement from the BESIII collaboration. Moreover, the partial width ratio of $$\pi ^0 \chi _{cJ}$$ π 0 χ cJ for $$J=0,1,2$$ J = 0 , 1 , 2 is estimated to be $$1.77{-}1.65:1:1.09{-}1.43$$ 1.77 - 1.65 : 1 : 1.09 - 1.43 , which could be tested by further precise measurements of BESIII and Belle II.

The cross section of $$e^+e^-\rightarrow \Lambda \overline{\Sigma }{}^0+\text {c.c.}$$ as a litmus test of isospin violation in the decays of vector charmonia into $$\Lambda \overline{\Sigma }{}^0+\text {c.c.}$$

Under the aegis of isospin conservation, the amplitudes in Born approximation, i.e., considering the only one-photon-exchange mechanism, of the decay $$\psi \rightarrow \Lambda \overline{\Sigma }{}^0+\text {c.c.}$$ ψ → Λ Σ ¯ 0 + c.c. , where $$\psi $$ ψ is a vector charmonium, and of the reaction $$e^+e^-\rightarrow \Lambda \overline{\Sigma }{}^0+\text {c.c.}$$ e + e - → Λ Σ ¯ 0 + c.c. at the $$\psi $$ ψ mass, are parametrized by the same electromagnetic coupling. It follows that, the modulus of such a coupling can be extracted from the data on the two observables: the decay branching fraction and the annihilation cross section. By considering the first two vector charmonia, $$J/\psi $$ J / ψ and $$\psi (2S)$$ ψ ( 2 S ) , it is found that, especially in the case of $$\psi (2S)$$ ψ ( 2 S ) , there is a substantial discrepancy between the values of the modulus of the same electromagnetic coupling extracted from the branching ratio and the cross section. We propose, as a possible explanation for such a disagreement, the presence in the decay amplitude of isospin-violating contributions driven by two different mechanisms, that, however, appear to be more favored in the $$\psi (2S)$$ ψ ( 2 S ) than in the $$J/\psi $$ J / ψ decays.

a0(980)–f0(980) mixing in χc1 decay

We investigate the isospin breaking in the ${\chi _{c1}} \to {\pi ^0}{\pi ^ + }{\pi ^ - }$ and ${\chi _{c1}} \to {\pi ^0}{\pi ^0}\eta $ decays and its relation to the a0(980) – f0(980) mixing, which was measured by the BESIII Collaboration. The isospin violation is introduced through the use of different masses for the charged and neutral kaons, either in the propagators of pairs of mesons created in the χc1 decay, or in the propagators inside the T matrix. We find that the most important effect in the total amplitude is the isospin breaking inside the T matrix which is essential to get a good agreement with the experimental measurement of the mixing.