The nonleptonic decays of double-charmed baryon $Ω_{cc}^{+}$ within the nonrelativistic quark model

This paper investigates the two-body nonleptonic decays of the double-charmed baryon $\Omega_{cc}^{+}$ using a nonrelativistic quark model with pole model assumptions for nonfactorizable amplitudes, predicting that several Cabibbo-favored and singly Cabibbo-suppressed decay modes have branching fractions up to a few percent, making them promising discovery channels for future experiments like LHCb and Belle II.

The Gist

Imagine the subatomic world as a bustling, chaotic city made of tiny building blocks called quarks. Most of the time, these blocks stick together in threes to form baryons (like protons and neutrons). Usually, these baryons have one "heavy" quark (like a charm quark) and two "light" ones.

But in this paper, the author, Yu-Shuai Li, is investigating a very rare and special citizen of this city: the $\Omega_{cc}^+$ baryon. Think of this particle as a "double-decker" house where two of the heavy charm quarks are living together with one strange quark. It's a unique family structure that physicists have been hunting for a long time.

Here is the story of the paper, broken down into simple concepts:

1. The Mystery of the Missing Family Member

Scientists have already found the "twin" of this particle (called $\Xi_{cc}^{++}$), but the $\Omega_{cc}^+$ is still hiding. It's like looking for a specific cousin at a huge family reunion; you know they exist because the family tree says so, but you haven't spotted them in the crowd yet.

The problem is that this particle is unstable. It doesn't last long. It decays (breaks apart) almost instantly. To find it, we need to know exactly how it breaks apart so we can look for the specific debris it leaves behind.

2. The "Recipe" for Breaking Apart

The paper focuses on nonleptonic decays. In plain English, this means the particle is breaking apart into other particles (baryons and mesons) without creating any electrons or neutrinos.

There are two main ways this happens, and the author had to calculate the "probability" (or branching fraction) for each:

• The "Direct" Route (Factorizable): Imagine a charm quark deciding to change its identity and shooting out a new particle. This is the easy, predictable path.
• The "Handshake" Route (W-Exchange): This is the tricky part. Imagine two quarks inside the particle swapping places or exchanging a "message" (a W boson) with each other before breaking apart. This is messy, hard to calculate, and often ignored in simpler models.

The Author's Innovation:
Most previous studies tried to guess the messy "Handshake" route using rough estimates. This author decided to do it properly. They used a Non-Relativistic Quark Model, which is like using a very detailed, high-resolution map of the particle's interior. Instead of assuming the particles are simple points, they solved complex equations (Schrödinger's equation) to understand exactly how the quarks are moving and vibrating inside the baryon.

3. The "Pole Model" Analogy

To handle the messy "Handshake" (W-exchange) part, the author used something called the Pole Model.

Think of the decay process like a ball rolling down a hill.

• The Hill: The energy of the particle.
• The Destination: The final broken pieces.
• The Poles: Imagine there are temporary "stepping stones" (intermediate particles) the ball can land on before reaching the bottom.

The author calculated how likely the particle is to land on these specific stepping stones (like $\Xi_c$ or $\Lambda_c$) before finishing its journey. This allowed them to accurately predict the "messy" contributions that other models often miss.

4. The Big Discovery: Where to Look

After doing all this heavy math, the author produced a "Wanted Poster" for the $\Omega_{cc}^+$. They listed the specific ways this particle is most likely to break apart.

• The "Golden" Channels: They found that the particle is most likely to decay into a specific mix of a charmed baryon and a pion or kaon (like $\Omega_{cc}^+ \to \Omega_c^0 \pi^+$).
• The Surprise: One specific decay path, $\Omega_{cc}^+ \to \Omega_c^0 K^+$, turned out to be surprisingly common. Even though it's usually considered a "suppressed" (unlikely) path, the author found that the "Handshake" (W-exchange) and "Direct" routes actually work together to boost its probability. It's like two people pushing a car from opposite sides; instead of canceling out, they push it forward together!

5. Why This Matters

The paper concludes with a clear message for experimentalists (the people running giant machines like LHCb and Belle II):

"Don't just look everywhere. Look specifically at these channels we identified. If you see a spike in the data for $\Omega_{cc}^+ \to \Omega_c^0 K^+$ or the pion modes, you have found the double-charmed baryon!"

Summary

In short, this paper is a theoretical treasure map.

• The Problem: We know the $\Omega_{cc}^+$ exists, but we can't find it.
• The Solution: The author built a super-accurate model of how this particle works, including the messy parts others ignored.
• The Result: They identified the best "clues" (decay channels) for experimentalists to follow.
• The Goal: To finally catch this elusive particle, complete the family tree of heavy baryons, and understand the fundamental forces of nature a little better.

It's like the author saying, "I've calculated the exact footprints this ghost leaves behind. If you look right here, you'll catch it."

Technical Summary

Here is a detailed technical summary of the paper "The nonleptonic decays of double-charmed baryon $\Omega_{cc}^+$ within the nonrelativistic quark model" by Yu-Shuai Li.

1. Problem Statement

The discovery and characterization of double-charmed baryons are crucial for completing the heavy-flavor baryon family. While the $\Xi_{cc}^{++}$ has been observed, the existence and properties of the $\Xi_{cc}^+$ remain unconfirmed due to conflicting experimental data (SELEX vs. LHCb/Belle/BaBar) and a significant mass discrepancy. The $\Omega_{cc}^+$ (quark content $ccs$) remains unobserved.
Since ground-state double-charmed baryons ($J^P = 1/2^+$) cannot decay via strong or electromagnetic interactions, they decay solely via the weak interaction. A major challenge in theoretical predictions for these decays is the accurate calculation of nonfactorizable amplitudes, particularly those arising from W-exchange diagrams. Previous models often relied on oversimplified wave functions (e.g., Gaussian types) or flavor symmetry assumptions that introduce significant model dependence. This paper aims to provide a robust theoretical prediction for the two-body nonleptonic decays of $\Omega_{cc}^+$ to guide future experimental searches at facilities like LHCb and Belle II.

2. Methodology

The authors employ a Nonrelativistic Quark Model (NRQM) framework combined with the Pole Model to evaluate decay amplitudes.

• Wave Functions: To reduce sensitivity to arbitrary wave function choices, the authors solve the Schrödinger equation using a nonrelativistic quark potential model.
  • The potential includes one-gluon exchange, linear confinement, hyperfine interactions, and spin-orbit terms (color-magnetic and Thomas-precession).
  • The spatial wave functions are expanded using the Gaussian Expansion Method (GEM) with a geometric progression of basis parameters, rather than simple Gaussian ansatzes.
  • The system treats the double-charmed baryon as a bound state of a $(cc)$ cluster and a light quark ($s$), while single-charmed baryons are treated as a light-quark cluster and a charm quark.
• Decay Amplitudes:
  • Factorizable Amplitudes ($T, C$): Calculated under the naive factorization assumption, involving baryon transition matrix elements and meson decay constants.
  • Nonfactorizable Amplitudes ($C', E_{1,2}, E'$): Evaluated using the Pole Model. This approach accounts for intermediate baryon states ($B_m$) in the $s$-channel and $u$-channel.
    • Parity-Conserving (PC) amplitudes are dominated by $J^P = 1/2^+$ ground-state poles.
    • Parity-Violating (PV) amplitudes arise mainly from low-lying $J^P = 1/2^-$ poles.
  • The effective Hamiltonian includes Wilson coefficients ($c_1, c_2$) and CKM matrix elements.
• Classification: Decays are categorized into:
  • Cabibbo-Favored (CF): $c \to su\bar{d}$ or $cd \to su$.
  • Single Cabibbo-Suppressed (SCS): $c \to du\bar{d}$, $cd \to du$, etc.
  • Doubly Cabibbo-Suppressed (DCS): $c \to du\bar{s}$, etc.

3. Key Contributions

• Improved Wave Functions: The primary technical contribution is the use of exact baryon wave functions derived from solving the Schrödinger equation with a sophisticated potential model, significantly reducing model dependence compared to previous studies using simplified Gaussian wave functions.
• Comprehensive Pole Model Analysis: The study systematically evaluates nonfactorizable contributions from W-exchange diagrams by considering intermediate poles from both single-charmed baryons ($\Lambda_c, \Sigma_c, \Xi_c$) and double-charmed baryons ($\Xi_{cc}, \Omega_{cc}$).
• Interference Effects: The paper explicitly analyzes the interference between factorizable and nonfactorizable amplitudes, revealing cases where they constructively or destructively interfere, which drastically alters branching fractions.

4. Key Results

The authors calculated decay amplitudes, branching fractions ($\mathcal{B}$), and up-down asymmetry parameters ($\alpha$) for various two-body modes, assuming a lifetime of $\tau_{\Omega_{cc}^+} = 128$ fs.

• Cabibbo-Favored (CF) Decays:
  • $\Omega_{cc}^+ \to \Omega_c^0 \pi^+$: Purely factorizable ($T$-diagram). Predicted $\mathcal{B} \approx 4.41\%$.
  • $\Omega_{cc}^+ \to \Xi_c^+ \bar{K}^0$: Significant nonfactorizable contributions. Factorizable and nonfactorizable parts interfere destructively. Predicted $\mathcal{B} \approx 6.11\%$.
  • $\Omega_{cc}^+ \to \Xi_c^{\prime+} \bar{K}^0$: Small nonfactorizable contribution. Predicted $\mathcal{B} \approx 3.21\%$.
• Single Cabibbo-Suppressed (SCS) Decays:
  • Notable Finding: The decay $\Omega_{cc}^+ \to \Omega_c^0 K^+$ is predicted to have a branching fraction of $\approx 5.09\%$. This is comparable to CF modes because the large factorizable amplitude is reinforced by constructive pole contributions.
  • Other SCS modes with significant rates include $\Omega_{cc}^+ \to \Sigma_c^{++} K^-$ ($\approx 1.04\%$), $\Omega_{cc}^+ \to \Xi_c^{\prime+} \eta$ ($\approx 1.62\%$), and $\Omega_{cc}^+ \to \Lambda_c^+ \bar{K}^0$ ($\approx 0.88\%$).
• Doubly Cabibbo-Suppressed (DCS) Decays:
  • These modes are generally suppressed by CKM elements, with branching fractions typically in the range of $10^{-4}$to$10^{-3}$.
• Comparison with Other Models:
  • The results for $\Omega_{cc}^+ \to \Omega_c^0 \pi^+$ are consistent with Refs. [32–34] but show a PC amplitude roughly twice as large as Ref. [34].
  • For $\Omega_{cc}^+ \to \Xi_c^+ \bar{K}^0$, the destructive interference found here aligns with Refs. [33, 35].
  • The paper notes that Final State Interaction (FSI) effects, which are not fully included in the pole model, could significantly enhance channels like $\Omega_{cc}^+ \to \Lambda_c^+ \pi^0$ (predicted as 0 here but potentially $\sim 10^{-5}$ with FSI).

5. Significance

• Discovery Channels: The study identifies $\Omega_{cc}^+ \to \Omega_c^0 \pi^+$, $\Omega_{cc}^+ \to \Omega_c^0 K^+$, and $\Omega_{cc}^+ \to \Xi_c^{\prime+} \eta$ as the most promising discovery channels for the $\Omega_{cc}^+$ baryon. The unexpectedly high branching fraction for the SCS mode $\Omega_{cc}^+ \to \Omega_c^0 K^+$ makes it a prime target for LHCb and Belle II.
• Theoretical Insight: The work highlights the critical role of nonfactorizable W-exchange diagrams and pole contributions in charmed baryon decays, demonstrating that they cannot be neglected even in modes dominated by factorizable amplitudes.
• Model Dependence Reduction: By utilizing wave functions derived from a rigorous potential model rather than phenomenological Gaussian fits, the results offer a more reliable baseline for future experimental comparisons and theoretical refinements.

In conclusion, this paper provides a comprehensive and theoretically robust prediction for the weak decays of the $\Omega_{cc}^+$, offering specific, high-yield channels that experimentalists should prioritize to confirm the existence and properties of this elusive double-charmed baryon.