Deacay and scattering amplitudes are not the same

We study the lowest lying $π^+ π^-$ resonance R, bound by the long range Coulomb potential and destabilized by short range strong interactions mediating the dominant decay into two neutral pions. Parametrizing the corresponding partial decay width by $Γ(R \to 2 π^0) = 2/9 α^3 \sqrt{m_{π^+}^2 - m_{π^0}^2} (Δa_{D} m_{π^+})^2$ the quantity $Δa_{D}$ depends within QCD & QED on four basic parameters : $Λ_{QCD}, m_{u} + m_{d}, α, m_{d}- m_{u}$. We are interested here only in the limit where the last two parameters vanish $Δa_{D}^{0} = Δa_{D} (Λ_{QCD}, m_{u} + m_{d}, α= 0, m_{d} - m_{u} = 0)$. While isospin invariance implies $Δa_{D}^{0} = a_{D}^{I = 0} - a_{D}^{I = 2}$ it is shown that beyond the first order expansion in the strong interaction the identities $a_{D}^{I} = a^{I}$, where $a^{I}$ denote the strong interaction scattering lengths, do not hold.

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