Abstract

The dispersive analysis of the decay eta rightarrow 3pi is reviewed and thoroughly updated with the aim of determining the quark mass ratio Q^2=(m_s^2-m_{ud}^2)/(m_d^2-m_u^2). With the number of subtractions we are using, the effects generated by the final state interaction are dominated by low energy pi pi scattering. Since the corresponding phase shifts are now accurately known, causality and unitarity determine the decay amplitude within small uncertainties – except for the values of the subtraction constants. Our determination of these constants relies on the Dalitz plot distribution of the charged channel, which is now measured with good accuracy. The theoretical constraints that follow from the fact that the particles involved in the transition represent Nambu–Goldstone bosons of a hidden approximate symmetry play an equally important role. The ensuing predictions for the Dalitz plot distribution of the neutral channel and for the branching ratio varGamma _{eta rightarrow 3pi ^0}/ varGamma _{eta rightarrow pi ^+pi ^-pi ^0} are in very good agreement with experiment. Relying on a known low-energy theorem that relates the meson masses to the masses of the three lightest quarks, our analysis leads to Q=22.1(7), where the error covers all of the uncertainties encountered in the course of the calculation: experimental uncertainties in decay rates and Dalitz plot distributions, noise in the input used for the phase shifts, as well as theoretical uncertainties in the constraints imposed by chiral symmetry and in the evaluation of isospin breaking effects. Our result indicates that the current algebra formulae for the meson masses only receive small corrections from higher orders of the chiral expansion, but not all of the recent lattice results are consistent with this conclusion.

Highlights

  • Our world is almost isospin symmetric: the up and the down quarks can be freely interchanged inside hadrons almost without any observable consequence

  • Kubis and Meißner, correctly observe that while terms of order2 are negligible, there are a number of effects which scale as e2(mu − md ) and should be taken into account, like real and virtual photon corrections to the purely strong amplitude, and and most importantly, effects related to the pion mass difference, which are in particular responsible for the presence of cusps in the Dalitz plot of η → 3π 0

  • The two solutions exclusively differ in the values of the subtraction constants: while those relevant for the matching solution are given in Eq (3.11), the fit to the KLOE data is characterized by Eq (5.4)

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Summary

Introduction

Our world is almost isospin symmetric: the up and the down quarks can be freely interchanged (or replaced by any linear combination of them) inside hadrons almost without any observable consequence. If one were able to accurately calculate the proportionality factor – the modulus squared of the transition amplitude between the η and a three-pion state mediated by the third component of the scalar isovector quark bilinear – a measurement of the decay rate would provide a determination of this quark mass difference This approach has been adopted before, but both, recent improved measurements of the differential decay rates as well as progress on the theory side call for an updated and improved analysis. Decays into three particles are not accessible to lattice calculations yet, but both the effective field theory approach and dispersion relations can be and have been used to analyze these processes As it turns out, the main difficulty concerns the evaluation of rescattering effects among the pions in the final state.

Isospin
Polynomial ambiguities
Elastic unitarity
Phase shifts
Integral equations
Taylor invariants
2.10 Nonrelativistic expansion
Chiral perturbation theory
Imaginary parts at two loops
Matching the dispersive and one-loop representations
Adler zero at one loop
Neutral decay mode
Isospin breaking corrections
Kinematics
Isospin breaking at one loop
Self-energy effects
Applying the kinematic map to the one-loop representation
Correcting the dispersive solutions for isospin breaking effects
Experiment
Dispersive fits to the KLOE data without theoretical constraints
Theoretical constraints
Error analysis
Imaginary parts of the subtraction constants
Dalitz plot coefficients of our central solution
Comparison with the nonrelativistic effective theory
Anatomy of the two-loop representation
Final state interaction at two loops
Contribution from the low-energy constants at NNLO
Branching ratio
Dispersive representation of the Dalitz plot distribution
Z -distribution
M-distribution
X nmax max n d Xn
Polynomial approximation
Strength of the cusps
Dispersive analysis of the MAMI data
Mass difference between charged and neutral kaons
Determination of the quark mass ratio Q
Chiral expansion of the meson masses
10.1 Dispersive approaches
10.2 Nonrelativistic effective field theory
11 Summary and conclusions
A.1: Analytic continuation in Mη
A.2: Contribution from the horseshoe
B.1: Elastic unitarity
B.2: Branch cuts generated by kaons and η-mesons
Findings
Mπ2 3 Δηπ
Full Text
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