[Paper Review] Dispersive analysis of $\mathbf\eta ightarrow 3 \pi$

Gilberto Colangelo, Stefan Lanz+2 more|arXiv (Cornell University)|Jul 31, 2018

Particle physics theoretical and experimental studies6 references4 citations

TL;DR

This paper presents a dispersive analysis of the $ar{\eta} \to 3\pi$ decay using unitarity, causality, and chiral symmetry constraints to determine the quark mass ratio $Q^2 = (m_s^2 - m_{ud}^2)/(m_d^2 - m_u^2)$. By fitting the charged channel Dalitz plot data and incorporating precise $\pi\pi$ phase shifts, the analysis yields $Q = 22.1(7)$, with uncertainties covering experimental, theoretical, and input noise sources, indicating small higher-order corrections in chiral perturbation theory.

ABSTRACT

The dispersive analysis of the decay $\\eta\ o3\\pi$ 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 $\\Gamma_{\\eta\ o3\\pi^0}/ \\Gamma_{\\eta\ o\\pi^+\\pi^-\\pi^0}$ 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.

Motivation & Objective

• To determine the quark mass ratio $Q^2 = (m_s^2 - m_{ud}^2)/(m_d^2 - m_u^2)$ with high precision using dispersive methods.
• To assess the role of higher-order chiral corrections by comparing the result with lattice QCD data.
• To test the validity of current algebra formulae for meson masses by evaluating their higher-order corrections.
• To provide a theoretically consistent prediction for the neutral $\eta \to 3\pi^0$ Dalitz plot distribution and branching ratio.
• To quantify uncertainties from experimental data, $\pi\pi$ phase shift noise, and isospin-breaking effects.

Proposed method

• Constructing a dispersive representation of the $\eta \to 3\pi$ amplitude using unitarity and causality, with subtraction constants determined from experimental Dalitz plot data.
• Applying dispersion relations with a minimal number of subtractions, where final-state $\pi\pi$ scattering dominates the dynamics via known phase shifts.
• Using chiral perturbation theory (ChPT) up to two loops to match the dispersive representation and constrain subtraction constants.
• Incorporating isospin-breaking effects via a kinematic map and one-loop corrections, including electromagnetic and quark mass differences.
• Fitting the KLOE data for $\eta \to \pi^+\pi^-\pi^0$ using theoretical constraints from Nambu-Goldstone boson symmetry and chiral symmetry.
• Performing error propagation by combining Gaussian uncertainties from data, noise in $\pi\pi$ phase shifts, and theoretical uncertainties in isospin breaking and chiral constraints.

Experimental results

Research questions

• RQ1 What is the precise value of the quark mass ratio $Q = (m_s^2 - m_{ud}^2)^{1/2}/(m_d^2 - m_u^2)^{1/2}$, and how do uncertainties from experiment, phase shifts, and theory affect it?
• RQ2 To what extent do higher-order chiral corrections modify the current algebra formulae for meson masses, and is the result consistent with lattice QCD?
• RQ3 How well do the theoretical constraints from chiral symmetry and unitarity predict the Dalitz plot distribution of the neutral $\eta \to 3\pi^0$ decay?
• RQ4 What is the strength of the cusp effect in the $\eta \to 3\pi^0$ decay, and how is it related to the quark mass ratio?
• RQ5 How do isospin-breaking corrections, particularly from electromagnetic and quark mass differences, affect the dispersive amplitude and the determination of subtraction constants?

Key findings

• The quark mass ratio is determined as $Q = 22.1(7)$, with the uncertainty encompassing experimental, phase shift noise, and theoretical uncertainties.
• Theoretical predictions for the Dalitz plot distribution of $\eta \to 3\pi^0$ and the branching ratio $\Gamma_{\eta \to 3\pi^0}/\Gamma_{\eta \to \pi^+\pi^-\pi^0}$ are in excellent agreement with experimental data.
• The error budget for the subtraction constants is dominated by Gaussian errors from data and phase shift noise, with isospin-breaking corrections contributing negligibly.
• The result implies that higher-order chiral corrections to current algebra formulae for meson masses are small, contrary to some recent lattice QCD results.
• The imaginary parts of the subtraction constants are estimated with a precision of $\delta\beta_0 = 0.24$, $\delta\gamma_0 = 6.6$, $\delta\delta_0 = 2.6$, $\delta\beta_1 = 0.23$, $\delta\gamma_1 = 2.1$.
• The dispersive representation successfully reproduces the KLOE Dalitz plot data for $\eta \to \pi^+\pi^-\pi^0$ when theoretical constraints are applied, confirming the robustness of the method.