Convergence properties of $$\eta \rightarrow 3\pi $$ η → 3 π decays in chiral perturbation theory

Constraints on low energy QCD parameters from $$\eta \rightarrow 3\pi $$ η → 3 π and $$\pi \pi $$ π π scattering

“…As we have shown in [27], we argue that the four point Green functions relevant for the η → 3π amplitude (see (4.1) below) do not necessarily have large contributions beyond next-to-leading order and a reasonably small higher order remainder is not in contradiction with huge corrections to the decay widths. The widths do not seem to be sensitive to the details of the Dalitz plot distribution, but rather to the value of leading order parameters-the chiral decay constant, the chiral condensate and the difference of u and d quark masses, i.e.…”

“…However, as we have discussed in detail in [27], we have not found the convergence of the theory in the case of the slopes reliable enough to include all the experimentally measured Dalitz plot parameters into the analysis. To stay on the conservative side, we used the lowest order parameter a in the charged channel only.…”

“…However, as discussed in [27], the results depend on the value of the constants L 1 -L 3 only very weakly.…”

“…The details of the calculations with explicit formulas can be found in [27]. We only summarize the basic steps here, which closely follow the procedure outlined in [35].…”

“…Such a strictly derived amplitude has an incorrect analytical structure due to the leading order masses in loops, cuts and poles being in unphysical positions. We have developed several ways to account for this in [35], explicit formulas for the η → 3π case are listed in [27]. The simplest approach is to exchange the LO masses in unitarity corrections and chiral logarithms for physical ones.…”