[Paper Review] Analysis of discrepancies in Dalitz plot parameters in eta to 3 pion decay

Marián Kolesár|arXiv (Cornell University)|Sep 5, 2011

Particle physics theoretical and experimental studies2 references4 citations

TL;DR

This paper investigates discrepancies in Dalitz plot parameters for η→3π decay using resummed chiral perturbation theory (χPT), which systematically controls uncertainties from higher-order contributions. It finds no conclusive evidence for a theory-experiment discrepancy in the charged decay parameter $b$, but identifies a significant tension with the neutral decay parameter $α$, suggesting possible non-convergence of the chiral series due to large ππ rescattering bubble corrections at low energies.

ABSTRACT

We analyze the Dalitz plot parameters of eta to 3 pion decay in the framework of resummed chiral perturbation theory. This approach allows us to keep the uncertainties in the NNLO and higher orders under better control and estimate their influence. We cannot confirm the suspected discrepancy in the case of the charged decay parameter b, where even small uncertainties in higher orders could accommodate the difference. On the other hand, we find the experimental value of the neutral decay parameter alpha incompatible with an assumption of good convergence properties in the center of the Dalitz plot. We calculate pion-pion rescattering bubble corrections up to three loops and show that these might explain the discrepancy, especially for a low value of the pseudoscalar decay constant in the chiral limit. However, that could indicate a failure of convergence of the chiral series in this channel already at low energies around 500MeV.

Motivation & Objective

• To assess whether discrepancies between experimental Dalitz plot parameters and chiral perturbation theory (χPT) results stem from genuine theoretical failure or uncontrolled higher-order uncertainties.
• To investigate whether the observed tension in the neutral decay parameter $\alpha$ indicates a breakdown of chiral convergence in the η→3π decay channel.
• To evaluate the role of ππ rescattering bubble diagrams in generating the negative $\alpha$ value, particularly at low energies.
• To test the robustness of the NNLO χPT prediction by comparing it with uncertainty bands derived from resummed χPT and higher-order corrections.
• To determine whether the discrepancy in $\alpha$ could be explained by large $O(p^8)$ contributions from ππ rescattering, especially for small values of the pseudoscalar decay constant in the chiral limit.

Proposed method

• The study employs resummed chiral perturbation theory (resummed χPT), which treats higher-order corrections as systematic uncertainties rather than perturbative terms.
• It computes 1-, 2-, and 3-loop ππ rescattering bubble diagrams up to $O(p^8)$, including both leading and subleading $s$-channel terms.
• The method retains remainders from higher orders as non-perturbative corrections, allowing for uncertainty estimation without assuming perfect convergence of the chiral series.
• The amplitude is expressed in terms of 4-point Green functions $G_{ijkl}$, with isospin breaking effects included at leading order.
• The neutral decay amplitude is constructed from the symmetric sum of the charged amplitude over all permutations of Mandelstam variables.
• The scale dependence of the results is analyzed to assess the stability of the $O(p^8)$ contributions, with particular attention to the behavior near $\sqrt{s} \approx 500$–700 MeV.

Experimental results

Research questions

• RQ1 Does the discrepancy in the charged decay parameter $b$ between experiment and NNLO χPT stem from uncontrolled higher-order corrections or from a genuine failure of χPT?
• RQ2 Is the experimental value of the neutral decay parameter $\alpha$ incompatible with the assumption of good chiral convergence in the center of the Dalitz plot?
• RQ3 Can ππ rescattering bubble diagrams at $O(p^8)$ explain the negative sign of $\alpha$ observed experimentally, especially for small values of the pseudoscalar decay constant in the chiral limit?
• RQ4 Do the $O(p^8)$ contributions to $\alpha$ dominate over $O(p^6)$ terms, and do they indicate a breakdown of the chiral expansion at low energies (~500 MeV)?
• RQ5 Is the observed tension in $\alpha$ robust across different values of the free parameters $X$ and $Z$, or is it sensitive to specific low-energy constants?

Key findings

• The discrepancy in the charged decay parameter $b$ cannot be confirmed, as the experimental value lies within the uncertainty bands of the resummed χPT calculation, even for small corrections.
• The experimental value of the neutral decay parameter $\alpha$ is incompatible with the assumption of good chiral convergence in the center of the Dalitz plot, as defined by the resummed χPT framework.
• ππ rescattering bubble contributions at $O(p^8)$ can generate a negative $\alpha$, particularly for small values of the pseudoscalar decay constant in the chiral limit, suggesting a possible explanation for the discrepancy.
• The $O(p^8)$ contributions to $\alpha$ are larger in magnitude than the $O(p^6)$ terms, and their $s$-channel leading terms grow rapidly, indicating potential non-convergence of the chiral series around $\sqrt{s} \approx 500$–700 MeV.
• The scale dependence of the $O(p^8)$ contributions is benign, supporting the idea that ππ rescattering can be treated as a separate, physically motivated contribution rather than a unitarization artifact.
• The leading $s$-channel terms in $O(p^8)$ diagrams are responsible for the concavity in the amplitude, which is directly measured by $\alpha$, and these terms are not suppressed by powers of $F_0$.