Two body nonleptonic decays of $\Omega_{b}\rightarrow \Omega_{c}$ beyond tree level

This paper investigates the two-body nonleptonic decays of $\Omega_{b}$ into $\Omega_{c}$ accompanied by pseudoscalar or vector mesons using the naive factorization approach, calculating decay amplitudes across various topologies to determine rates and branching ratios for comparison with other theoretical predictions and experimental data.

The Gist

Imagine the subatomic world as a bustling, chaotic construction site. In this universe, heavy particles called baryons are like massive, complex trucks built from smaller bricks called quarks.

This paper is about a specific, rare event: a heavy truck called the $\Omega_b$ (Omega-b) breaking apart and transforming into a slightly lighter truck called the $\Omega_c$ (Omega-c), while simultaneously spitting out a smaller, fast-moving car called a meson (like a pion, kaon, or D-meson).

Here is the breakdown of what the scientists did, explained through simple analogies:

1. The Big Picture: Why Study This?

Think of the Standard Model (our current rulebook for physics) as a massive instruction manual. Sometimes, the manual predicts things perfectly, but other times, we see "glitches" or unexpected behaviors in the universe (like why there is more matter than antimatter).

To find these glitches, physicists watch heavy particles decay (break apart). The $\Omega_b$ is a very heavy, rare truck. Watching it turn into an $\Omega_c$ is like watching a master magician perform a complex trick. If the trick doesn't go exactly according to the manual's instructions, it might mean there is "New Physics" hiding in the shadows—something we haven't discovered yet.

2. The Mechanism: How Does the Truck Break?

The paper focuses on non-leptonic decays.

• Leptonic decays are like a truck dropping off a passenger (a lepton) and a bag of luggage. It's relatively clean.
• Non-leptonic decays are messier. The truck breaks apart, and the pieces rearrange themselves into only other trucks and cars (hadrons). It's like a demolition derby where the debris instantly reassembles into new vehicles. This is much harder to predict because the "debris" (quarks) interacts strongly with each other via the Strong Force (like sticky glue).

3. The Three Ways the Trick Happens (Topologies)

The scientists looked at three different ways this transformation could happen, using a method called "Naive Factorization." Imagine this as trying to predict the outcome of a complex dance by looking at the individual steps of the dancers separately.

• The Tree-Level (The Direct Path):

  • Analogy: The driver of the $\Omega_b$ truck (the bottom quark) simply swaps places with a passenger, and a new car (the meson) is born directly from this swap. This is the most common, straightforward way the decay happens.
  • In the paper: This is the "Tree" diagram. It's the dominant force.

• The Color-Suppressed Path (The Internal Shuffle):

  • Analogy: Imagine the driver tries to swap places, but the new car has to be built inside the truck's engine room first before it can be ejected. It's a bit more cramped and difficult, so it happens less often.
  • In the paper: This is the "Color-Suppressed" diagram. It's like a traffic jam inside the particle.

• The Penguin Path (The Sneaky Detour):

  • Analogy: This is the most complex. The driver doesn't just swap; they take a secret, circular detour through a "loop" of virtual particles (like a ghostly shortcut) before the new car appears. In physics, these are called Penguin diagrams (named because the loop looks a bit like a penguin).
  • In the paper: These are rare, but they are crucial. Even though they happen rarely, they carry a special "phase" (a timing difference) that can create CP Violation.
  • Why CP Violation matters: It's the difference between a clock running forward and backward. If the "Penguin" path makes the truck break differently than its mirror-image twin, it helps explain why the universe is made of matter and not empty space.

4. The Calculation: Doing the Math

The authors didn't just guess; they did heavy lifting with math:

1. The Blueprint (Hamiltonian): They wrote down the rules of the interaction using "Wilson Coefficients" (numbers that tell us how strong each force is).
2. The Ingredients (Form Factors): To calculate the speed and direction of the new trucks, they needed to know how "squishy" or "stiff" the $\Omega_b$ and $\Omega_c$ are. They used a method called QCD Sum Rules (a way to estimate the properties of particles using the fundamental laws of the strong force) to get these numbers.
3. The Result: They calculated the Decay Rate (how fast it happens) and the Branching Ratio (the percentage of the time it happens compared to other ways the truck could break).

5. The Findings: What Did They Discover?

• The Numbers: They predicted exactly how often the $\Omega_b$ turns into an $\Omega_c$ plus a specific meson (like a pion, kaon, or D-meson).
• The Surprise: While the "Tree" path is the main event, the "Penguin" and "Color-Suppressed" paths, though small, are not zero. They add up to a significant amount.
• The Comparison: They compared their results with other scientists' predictions. Their numbers generally matched up well, but their method was more complete because they included all three "dance steps" (Tree, Color-Suppressed, and Penguin) rather than just the main one.

6. Why Should You Care?

This paper is a reference manual for future experiments.
Experiments like LHCb (at CERN) are currently smashing particles together to find these rare decays. When they see a real $\Omega_b$ decay in their detectors, they will look at this paper to say, "Okay, the theory predicts it should happen this often. If we see it happening more or less than this, we might have found a crack in the Standard Model!"

In summary:
These scientists built a detailed map of a very rare, messy traffic accident in the subatomic world. They accounted for the direct crash, the internal shuffling, and the sneaky detours. Their map helps experimentalists know exactly what to look for, potentially leading to the discovery of new laws of physics that explain the very existence of our universe.

Technical Summary

Here is a detailed technical summary of the paper "Two body nonleptonic decays of $\Omega_b \to \Omega_c$ beyond tree level" by Neishabouri, Azizi, and Moshfegh.

1. Problem Statement

The paper addresses the theoretical understanding of non-leptonic two-body decays of the singly heavy baryon $\Omega_b$ into the charmed baryon $\Omega_c$ accompanied by a pseudoscalar ($P$) or vector ($V$) meson.

• Context: While the decay channel $\Omega_b \to \Omega_c \pi$ has been experimentally observed (with $3.3\sigma$significance), other modes involving different mesons ($K, D, D_s$, and their vector counterparts) remain unobserved or require further theoretical prediction to guide future experiments (e.g., LHCb).
• Gap in Literature: Previous theoretical studies often focused exclusively on tree-level (color-allowed) contributions. However, non-leptonic decays involve complex QCD dynamics where color-suppressed and penguin (loop) topologies can significantly influence decay rates and are essential for calculating CP-violating asymmetries. There was a lack of comprehensive analysis incorporating all these topological contributions for the $\Omega_b \to \Omega_c$ system.
• Goal: To calculate the decay amplitudes, rates, and branching ratios for eight specific decay channels ($\Omega_b \to \Omega_c \pi, K, D, D_s$ and their vector partners $\rho, K^*, D^*, D_s^*$) by including tree-level, color-suppressed, and penguin diagrams.

2. Methodology

The authors employ the Naive Factorization Approach combined with QCD Sum Rules (QCDSR) for form factors.

A. Effective Hamiltonian and Topologies

The decay is described using an effective weak Hamiltonian ($H_{eff}$) derived by integrating out the heavy $W$ boson. The analysis considers three distinct topological diagrams:

1. Tree-level (T): External $W$-emission (color-allowed). Dominant contribution.
2. Color-Suppressed (C): Internal $W$-emission. Involves color rearrangement via Fierz transformations, suppressed by a factor of $1/N_c$.
3. Penguin (P): Loop diagrams (QCD and electroweak penguins). While their contribution to the total rate is small, they introduce necessary weak phases for CP violation studies.

The study covers transitions driven by $b \to c\bar{u}d$, $b \to c\bar{u}s$, and $b \to c\bar{c}s$ quark-level processes.

B. Factorization and Amplitudes

Using the naive factorization hypothesis, the hadronic matrix element $\langle \Omega_c M | H_{eff} | \Omega_b \rangle$ is separated into:

1. Meson Decay Constants: Matrix elements coupling the meson current to the vacuum (e.g., $\langle M | \bar{q}\gamma_\mu(1-\gamma_5)q' | 0 \rangle$).
2. Baryon Transition Form Factors: Matrix elements describing the $\Omega_b \to \Omega_c$ transition.

The amplitude is expressed in terms of effective Wilson coefficients ($a_1, a_2, a_4, \dots$) which combine the original Wilson coefficients ($C_i$) and color factors.

C. Form Factors

The non-perturbative baryonic transition form factors ($F_{1,2,3}$ and $G_{1,2,3}$) are taken from the authors' previous work [42], calculated using QCD Sum Rules up to dimension-6 condensates. These form factors are parameterized as a function of momentum transfer $q^2$ and evaluated at $q^2 = m_{meson}^2$.

D. Numerical Inputs

The calculations utilize:

• Wilson Coefficients: Calculated at Next-to-Leading Order (NLO) at the scale of the $b$-quark mass.
• CKM Matrix Elements: Standard values from the Particle Data Group.
• Decay Constants: For pseudoscalar and vector mesons ($f_\pi, f_K, f_D, f_{D_s}$, etc.).
• Masses: Precise masses for $\Omega_b, \Omega_c$, and the participating mesons.

3. Key Contributions

1. Comprehensive Topological Analysis: Unlike previous works that often restricted analysis to tree-level diagrams, this study explicitly calculates and separates the contributions of tree-level, color-suppressed, and penguin topologies for all eight channels.
2. Beyond Tree-Level Corrections: The paper quantifies the impact of sub-dominant diagrams. It demonstrates that while color-suppressed and penguin contributions are numerically small individually, their interference with the dominant tree amplitude is non-negligible for precise branching ratio predictions.
3. CP Violation Potential: By including penguin diagrams (which carry different weak phases compared to tree diagrams), the study lays the groundwork for future calculations of Direct CP asymmetries ($A_{CP}$) in these baryonic decays.
4. Systematic Comparison: The results are compared against a wide range of existing theoretical models (Light-Front, Covariant Constituent Quark Model, Non-Relativistic Quark Model), highlighting discrepancies arising from different form factor parameterizations and topological assumptions.

4. Results

The authors present numerical results for decay widths ($\Gamma$) and branching ratios ($Br$) for the following channels:

• Pseudoscalar: $\Omega_b \to \Omega_c \pi, K, D, D_s$
• Vector: $\Omega_b \to \Omega_c \rho, K^*, D^*, D_s^*$

Key Findings:

• Branching Ratios: The calculated branching ratios are generally in good agreement with other theoretical predictions, though variations exist due to different form factor inputs.
  • Example: $Br(\Omega_b \to \Omega_c D_s) \approx 15.4 \times 10^{-3}$ and $Br(\Omega_b \to \Omega_c D_s^*) \approx 18.6 \times 10^{-3}$.
  • Example: $Br(\Omega_b \to \Omega_c \pi) \approx 4.7 \times 10^{-3}$.
• Topology Impact:
  • For channels like $\Omega_b \to \Omega_c \pi$ and $\rho$, the decay is purely tree-level (color-allowed).
  • For channels involving $K, K^*, D_s, D_s^*$, color-suppressed contributions add a small but distinct percentage to the total rate.
  • Penguin contributions are found to be very small (e.g., $\sim 0.003 \times 10^{15}$ GeV for $D$ mesons) but are crucial for the phase structure required for CP violation.
• Uncertainties: The primary source of uncertainty in the results stems from the form factors and the decay constants. The authors provide error bars reflecting these uncertainties.

5. Significance

• Experimental Guidance: The predicted branching ratios provide essential benchmarks for ongoing and future experiments (particularly LHCb) to search for and measure these decay modes. The observation of these channels will help validate the theoretical framework of heavy baryon decays.
• Testing the Standard Model: Precise measurements of these rates and the subsequent CP asymmetries (enabled by the inclusion of penguin diagrams in this work) serve as rigorous tests of the CKM mechanism and the interplay between weak and strong interactions in the baryon sector.
• Probing Baryon Structure: The study contributes to the understanding of the internal structure and lifetime of the $\Omega_b$ baryon, a member of the single-heavy spin-1/2 baryon family that is less explored than mesons.
• Methodological Robustness: By combining naive factorization with high-precision QCDSR form factors and a full topological decomposition, the paper offers a robust theoretical reference for non-leptonic heavy baryon decays.

In conclusion, this work provides the most complete theoretical analysis of $\Omega_b \to \Omega_c$ non-leptonic decays to date, moving beyond simple tree-level approximations to include the full spectrum of QCD topological contributions, thereby enhancing the predictive power for future high-energy physics experiments.