Final state interactions and Khuri-Treiman equations in,ν → 3π decays

Abstract

Using extended Khuri-Treiman equations, we evaluate the final state interactions due to two-pion rescatterings to the decays η → π 0 π + π − and η → π 0 π 0 π 0 . As subtraction to the dispersion relation we take the one-loop chiral perturbation theory result of Gasser and Leutwyler. The calculated corrections are moderate and amount to about 14% in the amplitude at the center of the decay region. A careful analysis of the errors inherent to our approach is given. As a consequence, the experimental rate of the decay can only be reproduced if the double quark mass ratio Q −2 ≡ m d −mu ms−m · m d +mu ms+m is increased from the usual value of 1/(24.1) 2 to 1/(22.4 ± 0.9) 2 . We have also calculated the ratio of the rates of the two decays and various Dalitz Plot parameters. In particular, the linear slope a in the charged decay is different from the one-loop value and agrees better with experiment.

Decayπ0→γγto next to leading order in chiral perturbation theory

“…There is additional evidence that R is overestimated by applying Dashen's theorem, and that comes from the decay η → π + π − π 0 . At lowest order the decay rate, which is proportional to (m d − m u ) 2 , is found to be a mere 66 eV At NLO in the chiral expansion the rate is increased to 167 ± 50 eV [16], while dispersive analyses [25,26] give 209 ± 56 eV, which is still substantially below the experimental value.…”

Measurement of the η→3π0 Dalitz plot distribution with the WASA detector at COSY

“…The origin of the non-uniform density distribution in the 3π 0 Dalitz plot are in part pion-pion interactions in the final state. In contrast to the perturbative calculations, as soon as the rescattering effects are considered to infinite orders via unitarization of the decay amplitude using dispersion relations [10,11] or iteration of the Bethe-Salpeter equation [12] the sign of α turns out to be in accordance with experiment. The calculations indeed predict negative values for α in agreement with the experimental results which are summarized in Table 1 together with the different theoretical predictions.…”

Measurement of the η→3π0 slope parameter α with the KLOE detector

“…The predicted values show differences -due to large cancellations in the amplitude computation -even quoting in some case a positive sign for α contrary α Crystal Ball (2001) [8] -0.031 ± 0.004 CELSIUS WASA [9] -0.026 ± 0.014 WASA at COSY [10] -0.027 ± 0.009 Crystal Ball at MAMI-B [11] -0.032 ± 0.003 Crystal Ball at MAMI-C [12] -0.032 ± 0.003 ChPT / LO 0.000 ChPT / NLO [3] 0.015 ChPT / NLO + unit. corrections [4] -0.014 ÷ -0.007 U(3) ChPT + Bethe Salpeter [6] -0.031 ± 0.003 ChPT / NNLO [7] 0.013 ± 0.032 to the experimental evidence. A precise measurement of α therefore poses a significant constraint to theoretical models.…”

“…One loop (NLO) calculations which include the π − π rescattering effects [3] predict higher rates but still below the observed values. Some improvements are obtained by computing unitary corrections [4] to NLO using a dispersion relation for the decay amplitude derived by Khuri and Treiman [5]. Recently, more advanced calculations have become available.…”