[Paper Review] Multi-Meson Model applied to $D^+ o K^+ K^- K^+$

Rafael Aoude, P. Camargo Magalhaes+2 more|arXiv (Cornell University)|Apr 11, 2016

Particle physics theoretical and experimental studies1 references3 citations

TL;DR

This paper proposes a novel Multi-Meson Model (Triple-M) for the $D^+ \to K^+K^-K^+$ decay, replacing the conventional isobar model by deriving amplitude components—nonresonant, $f_0(980)K^+$, and $\phi K^+$—from chiral effective field theory with theoretically constrained parameters. The model naturally incorporates full multi-particle final-state topology without two-body decomposition, offering a theoretically grounded alternative for analyzing large LHC data sets with improved unitarity and dynamical consistency.

ABSTRACT

Matrix elements of weak currents involving light multi-meson states are important in many hadronic decays of both heavy leptons and heavy mesons. In this paper we focus on the latter case where the current size of the data set demands better models. The specific case of three-kaon weak matrix elements is considered and expressed as a relatively simple structure, which generalizes naturally the concept of form factor. We propose a model for the decay $D^+ o K^+ K^- K^+$ as an alternative to isobar model, with free parameter predicted by the theory to be fine-tuned by a fit to data. An important qualitative outcome is that we encompass naturally all final states topologies, which involve necessarily proper multi-particle structures and cannot be decomposed into simpler two-body processes. This aspect represents a significant improvement when compared to isobar model, often employed in analyses of heavy-meson decay data.

Motivation & Objective

• To develop a theoretically grounded alternative to the widely used isobar model for analyzing three-body heavy meson decays.
• To address the limitations of ad hoc nonresonant amplitude parametrizations in current data fits, especially in large LHC data samples.
• To incorporate full multi-particle final-state topology naturally, without decomposing into two-body intermediate states.
• To provide a framework where resonance contributions and phases are theoretically constrained, reducing reliance on free fit parameters.
• To lay the foundation for future inclusion of three-body final-state interactions via unitary extensions.

Proposed method

• Formulates the decay amplitude as a sum of three theoretically derived components: nonresonant $KKK$, $f_0(980)K^+$, and $\phi K^+$, based on chiral effective field theory.
• Uses a quark-level annihilation diagram as the dominant topology, with matrix elements derived from axial vector current matrix elements between vacuum and three-kaon states.
• Applies coupled-channel dynamics to dress the $f_0(980)$ and $\phi$ resonances with $K\bar{K}$ and $\pi\rho$ loops, ensuring unitarity and proper energy dependence.
• Derives the amplitude using Feynman diagrams and self-energy corrections, with key propagators like $D_{\phi}^{\pi\rho}$ and $D_{f_0}$ encoding coupled-channel effects.
• Parametrizes the amplitude in terms of only three physical parameters: $f_0$ mass and couplings to pseudoscalars, to be tuned by data fits.
• Constructs the full amplitude as a coherent sum of tree-level contributions from multiple diagrams, with phase-space and kinematic invariants properly handled.

Experimental results

Research questions

• RQ1 How can a theoretically consistent amplitude model be constructed for $D^+ \to K^+K^-K^+$ that avoids the ad hoc parametrization of nonresonant components?
• RQ2 Can a multi-meson model based on chiral effective field theory reproduce the dominant topologies of three-kaon decays without relying on the isobar approximation?
• RQ3 What is the role of coupled-channel effects in shaping the $f_0(980)$ and $\phi$ resonance contributions in the Dalitz plot distribution?
• RQ4 How does the Triple-M model compare to the isobar model in describing the full topology of three-body decays, especially in the presence of broad scalar resonances?
• RQ5 Can the model be extended to include three-body final-state interactions in a unitary and computationally feasible way?

Key findings

• The Triple-M model provides a theoretically motivated, unitary amplitude structure for $D^+ \to K^+K^-K^+$, with all components derived from chiral effective theory.
• The model naturally incorporates all final-state topologies without decomposing into two-body intermediate states, overcoming a key limitation of the isobar model.
• The nonresonant, $f_0(980)K^+$, and $\phi K^+$ components are dynamically linked through self-energy corrections and coupled-channel effects, ensuring proper energy dependence.
• The amplitude is expressed in terms of only three free parameters—$f_0$ mass and couplings—offering a minimal yet comprehensive fit framework.
• The model's structure is consistent with unitarity and can be extended to include three-body final-state interactions via Faddeev-like or Kunitomi-Triemann approaches.
• The formalism is generalizable to other decays such as $D_s^+ \to K^+K^-K^+$ and $B^+ \to K^+K^-K^+$, with the same underlying topology and dynamics.