Direct $CP$ violation in $D^\pm \to π^\pm π^+ π^-$ with $a_0^0(980)-f_0(980)$ mixing

1. Problem Statement

The paper addresses the search for Direct CP violation (CPV) in the three-body decay of charged D mesons ($D^\pm \to \pi^\pm\pi^+\pi^-$). While CP violation is well-established in the neutral Kaon system and observed in B meson decays (notably by LHCb), CPV in the charm sector is predicted to be very small within the Standard Model (SM).

• The Gap: Previous experimental results (LHCb, BESIII) have not found significant localized or overall CP asymmetries in $D^\pm \to \pi^\pm\pi^+\pi^-$.
• The Hypothesis: The authors propose that isospin-breaking effects, specifically the mixing between the scalar resonances $a_0^0(980)$ and $f_0(980)$, could act as a mechanism to enhance CP asymmetry in the Dalitz plot, similar to how $\rho-\omega$ mixing enhances CPV in other channels.
• The Goal: To quantify the impact of $a_0^0(980) - f_0(980)$ mixing on the direct CP asymmetry in $D^\pm \to \pi^\pm\pi^+\pi^-$ and determine if this mechanism can produce observable effects.

2. Methodology

The study employs a theoretical framework combining Naive Factorization with a rigorous treatment of resonance mixing.

• Mixing Mechanism Formalism:

  • The authors model the $a_0^0(980) - f_0(980)$ mixing using a full propagator matrix derived from summing chain transitions (e.g., $a_0 \to f_0 \to \dots \to a_0$).
  • The mixing arises primarily from $K\bar{K}$ loop diagrams, which break isospin symmetry due to the mass difference between charged ($K^\pm$) and neutral ($K^0$) kaons.
  • They define mixing intensities ($\xi_{fa}$ and $\xi_{af}$) representing the transition probabilities between $f_0(980) \to a_0^0(980)$ and vice versa. These are calculated using Flatté parametrization for the propagators and integrated over the relevant invariant mass regions.

• Decay Amplitude Calculation:

  • The total decay amplitude for $D^\pm \to \pi^\pm\pi^+\pi^-$ is constructed by summing contributions from intermediate resonances $f_0(980)$ and $a_0^0(980)$, incorporating the mixing terms.
  • The weak decay amplitudes ($D \to f_0 \pi$ and $D \to a_0 \pi$) are calculated using the Naive Factorization Approach.
  • The effective Hamiltonian includes tree operators ($O_{1,2}$) and QCD penguin operators ($O_{3-6}$).
  • Scalar Meson Structure: The paper treats $f_0(980)$ and $\sigma(600)$ as mixed states of $s\bar{s}$ and $n\bar{n}$ (where $n\bar{n} = (u\bar{u} + d\bar{d})/\sqrt{2}$). The mixing is governed by an angle $\theta$, defined by:
    $$|f_0(980)\rangle = |s\bar{s}\rangle \cos\theta + |n\bar{n}\rangle \sin\theta$$
  • The authors assume a dominant $q\bar{q}$ structure rather than tetraquark or molecular configurations for the calculation of decay constants and form factors.

• CP Asymmetry Definition:

  • The direct CP asymmetry ($A_{CP}$) is calculated as the difference between the squared amplitudes of the particle and antiparticle decays, normalized by their sum.
  • The analysis focuses on the differential CP asymmetry as a function of the invariant mass of the $\pi^+\pi^-$ pair ($\sqrt{s}$).

3. Key Contributions

• Quantification of Mixing Intensities: The authors calculated the integrated mixing intensities $\bar{\xi}_{fa}$ and $\bar{\xi}_{af}$ using parameters from various theoretical models (e.g., $q\bar{q}$, tetraquark, $K\bar{K}$ models) and experimental data (SND, KLOE, BES). They found that $\bar{\xi}_{af}$ (transition $a_0 \to f_0$) is generally larger and more favorable for experimental observation, reaching values in the percent range (up to ~27% in some models).
• Dependence on Mixing Angle ($\theta$): A critical contribution is the demonstration that the magnitude of CP violation is highly sensitive to the internal quark structure of the $f_0(980)$, specifically the mixing angle $\theta$ between $s\bar{s}$ and $n\bar{n}$.
• Resonance Interference: The paper explicitly shows how the interference between the $a_0^0(980)$ and $f_0(980)$ amplitudes, mediated by isospin breaking, creates a distinct signature in the Dalitz plot near the $f_0(980)$ mass region.

4. Results

• Mixing Intensities:
  • The differential mixing intensity satisfies $\xi_{fa} > \xi_{af}$, but the integrated intensity $\bar{\xi}_{af}$ is found to be larger than $\bar{\xi}_{fa}$ across most models.
  • Values for $\bar{\xi}_{af}$ range from roughly 0.5% to 27% depending on the input model parameters, confirming the effect is phenomenologically significant.
• CP Asymmetry ($A_{CP}$):
  • The differential $A_{CP}$ varies rapidly and changes sign near the $f_0(980)$ resonance mass.
  • Pure $s\bar{s}$ state ($\theta = 0^\circ$): $A_{CP}$ is small, ranging from $-4.40 \times 10^{-4}$ to $1.52 \times 10^{-4}$.
  • Pure $n\bar{n}$ state ($\theta = 90^\circ$): $A_{CP}$ is even smaller ($\sim 10^{-5}$).
  • Mixed States: The asymmetry is significantly enhanced when $f_0(980)$ contains a substantial $n\bar{n}$ component.
    • For $\theta \approx 25.1^\circ$ (consistent with QCD sum rules and $f_0$ data), $A_{CP}$ reaches $-20.21 \times 10^{-4}$ to $7.55 \times 10^{-4}$.
    • For $\theta \approx 27^\circ$, the range is $-10.29 \times 10^{-4}$ to $3.76 \times 10^{-4}$.
  • The enhancement factor is roughly 10 to 20 times larger than the pure $s\bar{s}$ case, pushing the asymmetry into the $10^{-3}$ regime.

5. Significance

• Theoretical Insight: The paper establishes that isospin-breaking mixing between scalar mesons is a potent mechanism for enhancing CP violation in charm decays, comparable to the $\rho-\omega$ mixing effect in B decays.
• Experimental Guidance: The results suggest that future high-statistics experiments (like LHCb and BESIII) should focus on the invariant mass region near the $f_0(980)$ resonance in the $D^\pm \to \pi^\pm\pi^+\pi^-$ Dalitz plot.
• Probing Scalar Structure: The strong dependence of $A_{CP}$ on the mixing angle $\theta$ implies that precise measurements of CP asymmetry in this channel could serve as a new probe to determine the internal quark structure (mixing angle) of the $f_0(980)$ meson, distinguishing between $q\bar{q}$, tetraquark, or molecular interpretations.
• New Physics Potential: By providing a robust SM prediction for enhanced CPV in this specific channel, the study sets a baseline. Any deviation from these predicted ranges could signal physics beyond the Standard Model.

In conclusion, the authors demonstrate that the $a_0^0(980) - f_0(980)$ mixing mechanism is a crucial factor that must be included in amplitude analyses of D and B meson decays to correctly interpret CP violation data.