Rescattering-induced $D\to SS$ weak decays

This paper investigates two-body non-leptonic $D\to SS$ weak decays, demonstrating that long-distance triangle rescattering processes mediated by pion exchange dominate over negligible short-distance contributions, and provides theoretical predictions for branching fractions of specific channels to guide future experimental studies at BESIII, Belle(-II), and LHCb.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor where tiny particles called mesons are constantly colliding, breaking apart, and reforming. This paper is a detailed study of a very specific, rare dance move involving a heavy dancer (a D meson) trying to split into two light, scalar partners (two S mesons).

Here is the story of what the researchers found, explained simply:

The Problem: The "Silent" Dance

Usually, when a heavy particle decays, it does so through a direct, "short-distance" interaction. Think of this like a dancer suddenly snapping their fingers to change partners. In most cases, this is the main way the dance happens.

However, the researchers discovered that for this specific type of dance (turning into two scalar mesons), the "finger-snapping" method is broken. The physics rules say the probability of this happening directly is so close to zero that it's effectively silent. If you only looked at the direct snap, you would predict that this dance move never happens.

The Solution: The "Roundabout" Detour

If the direct path is blocked, how does the dance happen? The paper argues that the particles take a long, winding detour called Final State Interactions (FSI).

Imagine you want to get from Point A to Point B, but the direct bridge is out. Instead, you take a bus to a nearby town, get off, walk through a park, hop on a different bus, and finally arrive at your destination. In the subatomic world, this is called rescattering.

1. The First Leg: The heavy D meson first decays into two different, intermediate particles (like a pion and an eta meson).
2. The Collision: These two intermediate particles bump into each other.
3. The Swap: During this bump, they exchange a tiny messenger particle (a pion) and transform into the two scalar mesons we wanted to see in the first place.

The paper calls this a "triangle rescattering" process because if you draw the path of the particles on a piece of paper, it looks like a triangle.

The Key Players

The researchers focused on specific "dancers":

• The Start: Heavy D mesons ($D_s^+$, $D^+$, and $D^0$).
• The Finish: Pairs of light scalar mesons, specifically combinations like $\sigma^0 a_0$ (a mix of two specific types of scalar particles).
• The Mechanism: The "triangle" where the particles bounce off each other via pion exchange (like two people tossing a ball back and forth to change their positions).

The Results: How Often Does It Happen?

The team did the math to predict how often this "detour" dance occurs. They found that while the direct path is dead, the detour path is actually quite lively:

• $D_s^+ \to \sigma^0 a_0^+$: This happens about 1 time in every 100 decays. This is a surprisingly high number for such a complex process.
• $D^+ \to \sigma^0 a_0^+$: This happens about 1 time in every 1,000 decays.
• $D^0 \to \sigma^0 a_0^0$: This is rarer, happening about 1 time in every 100,000 decays.

They also looked at a different pair ($D_s^+ \to f_0 a_0^+$). This one is much harder to do because the "dance floor" is too small (the particles are too heavy to fit comfortably in the space available). It's like trying to fit a large sofa through a tiny doorway. Even with the detour, it only happens about 3 or 4 times in every 10,000 attempts.

Why This Matters

The paper concludes that if scientists at major labs (like BESIII, Belle-II, or LHCb) look for these specific particle pairs, they will find them.

The discovery is important because it proves that the "long-distance" detour (rescattering) is the dominant force here, not the direct "short-distance" snap. It's like realizing that in a crowded city, the fastest way to get somewhere isn't always the straight line; sometimes, you have to take the scenic route through the neighborhood to get there.

In short: The paper predicts that heavy particles can turn into two specific light scalar particles, but only if they take a complex, multi-step detour involving a collision and a swap, rather than doing it directly. The math says this happens often enough to be seen in experiments.

Technical Summary

Technical Summary: Rescattering-induced $D \to SS$ Weak Decays

Problem Statement
The paper addresses the theoretical challenge of explaining two-body non-leptonic $D \to SS$ weak decays, where $S$ represents light scalar mesons ($a_0/a_0(980)$, $f_0/f_0(980)$, or $\sigma_0/f_0(500)$). Standard short-distance topologies, specifically $W$-boson emission, are strongly suppressed in these channels due to the vanishing or near-zero scalar decay constants ($f_S$). Consequently, naive theoretical expectations predict negligible branching fractions, making these decays difficult to observe. However, analogous decays involving scalar mesons and pseudoscalars or vectors (e.g., $D_s^+ \to a_0 \pi$) have been measured with branching fractions significantly larger than short-distance predictions. This discrepancy suggests that long-distance final-state interactions (FSI) may play a dominant role, a mechanism that remains largely unexplored for $D \to SS$ transitions.

Methodology
The authors investigate the hypothesis that long-distance triangle rescattering processes provide the dominant contribution to $D \to SS$ decays. The methodology involves:

1. Formalism Construction: The decay amplitudes are modeled as a two-step process. First, the $D$ meson undergoes a tree-level weak decay into intermediate two-body states (e.g., $D \to \pi\eta^{(\prime)}$, $D \to a_1\eta$, or $D \to K\bar{K}$). Second, these intermediate states rescatter via meson exchange (pion or kaon exchange) to form the final scalar-scalar ($SS$) pair.
2. Diagrammatic Analysis: The study identifies specific triangle loop diagrams. For $D_s^+ \to \sigma_0 a_0^+$, the leading mechanisms are $D_s^+ \to \pi\eta^{(\prime)} \to \sigma_0 a_0^+$ and $D_s^+ \to a_1\eta \to \sigma_0 a_0^+$. Similar pathways are analyzed for $D^+$ and $D^0$ decays, as well as the Cabibbo-allowed $D_s^+ \to f_0 a_0^+$ channel.
3. Amplitude Calculation: The authors calculate the weak decay amplitudes for the initial states using $SU(3)_f$ flavor symmetry and topological amplitudes ($T, C, A, E$). The strong decay vertices (e.g., $a_1 \to \sigma_0 \pi$, $\sigma_0 \to \pi\pi$) are parameterized by coupling constants extracted from experimental data.
4. Loop Integration: The triangle loop integrals are evaluated using a phenomenological form factor to regularize the ultraviolet divergence. The cutoff parameters ($\Lambda_\pi, \Lambda_K$) are constrained by fitting available experimental branching fractions of related decays (e.g., $D \to \pi a_0$).
5. Numerical Analysis: Using Wolfenstein parameterization for CKM matrix elements and specific values for form factors and coupling constants, the authors compute the total branching fractions, including interference effects between different rescattering paths.

Key Contributions

• First Systematic Prediction: This work provides the first theoretical predictions for $D \to SS$ decay channels, specifically $D \to \sigma_0 a_0$ and $D_s^+ \to f_0 a_0^+$.
• Identification of Dominant Mechanisms: The study establishes that short-distance contributions are effectively zero and that long-distance triangle rescattering is the necessary mechanism to generate observable rates.
• Role of $a_1$ Rescattering: A novel contribution identified is the rescattering via the intermediate axial-vector meson $a_1(1260)$ (i.e., $D \to a_1\eta \to \sigma_0 a_0$). The authors find this channel provides a significant enhancement, often exceeding the contributions from $\pi\eta^{(\prime)}$ rescattering.
• Kinematic Constraints: The paper explicitly addresses the phase space suppression for the $D_s^+ \to f_0 a_0^+$ mode due to the near-threshold condition ($m_{D_s} \simeq m_{f_0} + m_{a_0}$).

Results
The calculated branching fractions for the rescattering-induced decays are:

• $D_s^+ \to \sigma_0 a_0^+$: $\mathcal{B} = (1.0 \pm 0.2^{+0.1}_{-0.2}) \times 10^{-2}$. This value is comparable to the measured branching fraction of $D_s^+ \to \rho^0 a_0^+$.
• $D^+ \to \sigma_0 a_0^+$: $\mathcal{B} = (1.1 \pm 0.2^{+0.1}_{-0.2}) \times 10^{-3}$.
• $D^0 \to \sigma_0 a_0^0$: $\mathcal{B} = (0.9 \pm 0.2^{+0.2}_{-0.3}) \times 10^{-5}$.
• $D_s^+ \to f_0 a_0^+$: $\mathcal{B} = (3.4 \pm 0.3^{+0.4}_{-0.9}) \times 10^{-4}$. Despite kinematic suppression, this rate is predicted to be within reach of current experiments.

The analysis confirms that the $a_1\eta$ intermediate state contributes significantly, with $D_s^+ \to a_1\eta \to \sigma_0 a_0^+$ yielding a branching fraction of $\approx 5.6 \times 10^{-3}$, which is several times larger than the contribution from $\pi\eta^{(\prime)}$ rescattering alone.

Significance
The paper concludes that rescattering-induced $D \to SS$ decays are promising channels for experimental investigation. The predicted branching fractions, particularly for $D_s^+ \to \sigma_0 a_0^+$ and $D_s^+ \to f_0 a_0^+$, are large enough to be observable at facilities such as BESIII, Belle(-II), and LHCb. The observation of these decays would serve as a direct signal of significant long-distance final-state interaction effects, validating the triangle rescattering mechanism as a crucial component in understanding heavy meson decays into light scalar mesons. Furthermore, these results may offer insights into the internal structure of light scalar mesons, although the primary focus of this work is the dynamical mechanism of the decay rather than resolving the scalar meson composition debate.