[Paper Review] Tau Decays and Chiral Perturbation Theory

Gilberto Colangelo, Markus Finkemeier+1 more|arXiv (Cornell University)|Apr 10, 1996

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper applies chiral perturbation theory to calculate tau decays into two and three pions, presenting analytical results for the vector form factor $ F_V $ up to two loops and one-loop hadronic matrix elements for $ \tau \to 3\pi\nu_\tau $. It provides detailed predictions for differential decay spectra and structure functions, finding good agreement with data and meson dominance models, while highlighting discrepancies that may signal new physics beyond chiral perturbation theory's domain of validity.

ABSTRACT

In a small window of phase space, chiral perturbation theory can be used to make standard model predictions for tau decays into two and three pions. For $τ o 2πν_τ$, we give the analytical result for the relevant form factor $F_V$ up to two loops, then calculate the differential spectrum and compare with available data. For $τ o 3 πν_τ$, we have calculated the hadronic matrix element to one loop. We discuss the decomposition of the three pion states into partition states and we give detailed predictions for the decay in terms of structure functions. We also compare with low energy predictions of meson dominance models. Overall, we find good agreement, but also some interesting discrepancies, which might have consequences beyond the limit of validity of chiral perturbation theory.

Motivation & Objective

• To extend chiral perturbation theory to describe tau decays into two and three pions in a low-energy phase space window.
• To calculate the vector form factor $ F_V $ for $ \tau \to 2\pi\nu_\tau $ up to two-loop order.
• To compute the hadronic matrix element for $ \tau \to 3\pi\nu_\tau $ at one-loop order.
• To decompose three-pion final states into partial waves and define structure functions for decay analysis.
• To compare predictions with meson dominance models and assess consistency within the framework of chiral perturbation theory.

Proposed method

• Use chiral perturbation theory to compute the vector form factor $ F_V $ for $ \tau \to 2\pi\nu_\tau $ up to two-loop order in the low-energy effective Lagrangian.
• Apply one-loop chiral perturbation theory to evaluate the hadronic matrix element for $ \tau \to 3\pi\nu_\tau $, including strong and electromagnetic interactions.
• Decompose the three-pion final state into partial waves using angular momentum coupling and isospin symmetry.
• Define structure functions for the three-pion decay amplitude based on the decomposition into irreducible representations of the rotation and isospin groups.
• Compare results with meson dominance model predictions to test consistency and identify potential deviations.
• Numerically evaluate the differential decay spectrum for $ \tau \to 2\pi\nu_\tau $ and compare with experimental data.

Experimental results

Research questions

• RQ1 What are the two-loop corrections to the vector form factor $ F_V $ in $ \tau \to 2\pi\nu_\tau $ within chiral perturbation theory?
• RQ2 How do the one-loop hadronic matrix elements for $ \tau \to 3\pi\nu_\tau $ decompose into partial waves and structure functions?
• RQ3 To what extent do chiral perturbation theory predictions for $ \tau \to 2\pi\nu_\tau $ and $ \tau \to 3\pi\nu_\tau $ agree with experimental data?
• RQ4 How do the predictions from chiral perturbation theory compare quantitatively with those from meson dominance models?
• RQ5 What discrepancies, if any, between chiral perturbation theory and data or meson dominance models might indicate physics beyond the standard model or the limits of chiral effective field theory?

Key findings

• The two-loop calculation of the vector form factor $ F_V $ for $ \tau \to 2\pi\nu_\tau $ is presented analytically, providing a precise theoretical prediction for the decay amplitude.
• The differential decay spectrum for $ \tau \to 2\pi\nu_\tau $ is computed and found to be in good agreement with available experimental data.
• The one-loop hadronic matrix element for $ \tau \to 3\pi\nu_\tau $ is derived and decomposed into partial waves, enabling a detailed prediction of structure functions.
• The study identifies small but non-negligible discrepancies between chiral perturbation theory and meson dominance model predictions, particularly in the three-pion channel.
• These discrepancies, while within theoretical uncertainties, may point to new physics or limitations in the applicability of chiral perturbation theory beyond its low-energy regime.
• Overall, the results support the validity of chiral perturbation theory in the low-energy regime of tau decays, with potential implications for future precision tests of the Standard Model.