Semileptonic and nonleptonic weak decays of bottom baryons $\Omega^{(*)}_{b}$

This paper investigates the semileptonic and nonleptonic weak decays of bottom baryons $\Omega_b$ and $\Omega^*_b$ using three-point QCD sum rules to calculate form factors, decay widths, and branching ratios, thereby providing theoretical predictions to test Standard Model validity and explore new physics in heavy baryonic systems.

The Gist

Imagine the universe is built out of tiny, fundamental Lego bricks called quarks. Usually, these bricks snap together in groups of three to build larger structures called baryons (which include protons and neutrons). Most of the time, these bricks are light and fast. But sometimes, nature builds a "heavy" version of these structures by swapping in a giant, heavy brick called a bottom quark.

This paper is a theoretical investigation into two specific heavy Lego structures: the $\Omega_b$ and the $\Omega_b^*$. Think of them as heavy-duty trucks made of three specific bricks: two "strange" bricks and one "bottom" brick. The only difference between the two is how they are twisted or spun (their "spin").

The authors wanted to figure out exactly how these heavy trucks fall apart or transform into other things. In the world of particle physics, this falling apart is called decay. They looked at two main ways these trucks decay:

1. The "Leaky Pipe" Decay (Semileptonic): Imagine the heavy truck is leaking a fluid. In this scenario, the heavy bottom brick inside the truck transforms into a lighter "charm" brick. As it changes, it shoots out a stream of invisible particles (a lepton and a neutrino). The paper calculates exactly how fast this leak happens and how much "pressure" (energy) is involved.
2. The "Explosion" Decay (Nonleptonic): Imagine the truck doesn't just leak; it explodes into two pieces. The heavy truck transforms into a lighter truck, and in the process, it spits out a brand new, smaller object (a meson, which is like a two-brick structure). This is like a heavy truck crashing and turning into a smaller car plus a flying tire.

How They Did It: The "Shadow" Method

The authors couldn't build these heavy trucks in a lab to watch them fall apart because they are incredibly rare and short-lived. Instead, they used a mathematical tool called QCD Sum Rules.

Think of this method like trying to figure out the shape of a hidden object by looking at its shadow.

• The Shadow (Theoretical Side): They used complex math based on the fundamental laws of physics (Quantum Chromodynamics) to calculate what the "shadow" of the decay should look like. They considered the interactions of the quarks and the "glue" holding them together.
• The Object (Physical Side): They also calculated what the decay should look like if they treated the particles as solid, real objects with specific masses and spins.
• Matching: By making the "shadow" match the "object," they could deduce the hidden details of the process. Specifically, they calculated Form Factors.

What are Form Factors?
Imagine you are trying to describe how a sponge absorbs water. You can't just say "it absorbs water." You need a number that tells you how it absorbs it at different speeds. Form factors are those numbers. They describe the internal "sponginess" or structure of the heavy baryons as they change. The paper calculated these numbers for the first time for these specific particles.

What They Found

Using these calculated numbers, the authors predicted:

• How fast these heavy trucks decay (the decay width).
• How often they decay into specific types of particles (branching ratios).
• They looked at different "flavors" of the particles they might turn into, such as pions, kaons, or D-mesons (which are like different types of smaller Lego blocks).

They found that while some decay paths are very rare, others are more likely. For example, the $\Omega_b$ truck is more likely to turn into a lighter truck plus a specific type of meson (like a pion or a D-meson) than others. They also calculated the ratio of decays involving heavy tau particles versus lighter electrons or muons, which helps test if our current understanding of physics is correct.

Why This Matters

The paper concludes that these calculations are like a roadmap for future experiments.

Scientists at massive particle colliders (like the LHC at CERN) are currently smashing particles together to find these heavy trucks. The authors are saying, "We have done the math to predict exactly what these trucks should look like when they break apart. If you see these specific patterns in your detectors, you will know you have found the $\Omega_b$ or $\Omega_b^*$."

They hope that by comparing their mathematical predictions with real-world data, scientists can:

1. Confirm the internal structure of these heavy particles.
2. Check if the Standard Model of physics (our current rulebook) is perfect or if there are cracks in it that hint at "New Physics" (unknown forces or particles).

In short, this paper is a detailed theoretical manual that tells experimentalists exactly what to look for when hunting for these rare, heavy cosmic Lego trucks.

Technical Summary

Technical Summary: Semileptonic and Nonleptonic Weak Decays of Bottom Baryons $\Omega_b$ and $\Omega_b^*$

Problem and Motivation
The theoretical and experimental analysis of bottom baryons offers a critical environment for probing hadronic structures, strong interaction dynamics, and testing the Standard Model (SM) of particle physics. While the existence of heavy baryons containing at least one $b$-quark has been predicted by the quark model, and several states (such as $\Omega_b$ and its excited states) have been identified by collaborations like D0, CDF, and LHCb, a comprehensive understanding of their weak decay properties remains an active area of research. Specifically, the weak transitions of singly $b$-heavy baryons, $\Omega_b$ (spin-1/2) and $\Omega_b^*$ (spin-3/2), into charmed baryons ($\Omega_c$ and $\Omega_c^*$) require precise theoretical inputs. These inputs are essential for determining CKM matrix elements, investigating CP violation, and searching for New Physics (BSM) effects. The primary challenge lies in calculating the non-perturbative hadronic form factors that govern these transitions, which cannot be derived directly from perturbative QCD due to color confinement.

Methodology
This study employs the three-point QCD sum rules method, a well-established non-perturbative approach, to investigate the semileptonic and nonleptonic weak decays of $\Omega_b$ and $\Omega_b^*$. The methodology involves the following steps:

1. Correlation Functions: The authors construct three-point correlation functions involving the interpolating currents of the initial baryon ($\Omega_b$ or $\Omega_b^*$), the final baryon ($\Omega_c$ or $\Omega_c^*$), and the weak transition current ($b \to c$).
2. Dual Representation:
   • Phenomenological Side: The correlation function is expressed in terms of hadronic degrees of freedom, parameterized by decay constants, masses, and Lorentz-invariant form factors ($F_i$ and $G_i$). The contributions of excited states and the continuum are suppressed using the quark-hadron duality assumption and Borel transformation.
   • QCD Side: The correlation function is calculated in the deep Euclidean region using the Operator Product Expansion (OPE) up to dimension six. This side incorporates perturbative contributions and non-perturbative vacuum condensates (quark, gluon, and quark-gluon condensates).
3. Sum Rules Derivation: By matching the Lorentz structures of the phenomenological and QCD sides, sum rules are derived for the transition form factors.
4. Numerical Analysis: The stability of the results is ensured by defining working intervals for auxiliary parameters, including Borel mass parameters ($M^2, M'^2$), continuum thresholds ($s_0, s'_0$), and the mixing parameter $\beta$ (related to the interpolating current structure). The form factors are fitted to a specific function of $q^2$ across the physical region.
5. Decay Width Calculation:
   • Semileptonic: The fitted form factors are used to compute helicity amplitudes and subsequently the decay widths for $\Omega_b^* \to \Omega_c \ell \bar{\nu}_\ell$ and $\Omega_b \to \Omega_c^* \ell \bar{\nu}_\ell$ in $e, \mu,$ and $\tau$ channels.
   • Nonleptonic: Using the naive factorization approximation, the form factors (evaluated at $q^2 = m_{meson}^2$) are employed to calculate two-body nonleptonic decay widths where a pseudoscalar ($\pi, K, D, D_s$) or vector ($\rho, K^*, D^*, D_s^*$) meson is emitted.

Key Contributions and Results

• Form Factors: The study provides the first determination of the invariant form factors for the $\Omega_b^* \to \Omega_c$ and $\Omega_b \to \Omega_c^*$ transitions within the QCD sum rules framework. The results include eight form factors ($F_1$ to $F_4$ and $G_1$ to $G_4$) for each transition, fitted as functions of $q^2$.
• Semileptonic Decay Widths:
  • For $\Omega_b^* \to \Omega_c \ell \bar{\nu}_\ell$, the decay widths are calculated for all lepton channels. The ratio $R_{\Omega_b^*} = \Gamma(\tau)/\Gamma(e/\mu)$ is found to be $0.21 \pm 0.02$.
  • For $\Omega_b \to \Omega_c^* \ell \bar{\nu}_\ell$, the decay widths and branching ratios are presented. The branching ratio for the $\tau$ channel relative to $e/\mu$ is $R_{\Omega_b} = 0.14 \pm 0.02$. The total branching fractions for the $e$ and $\mu$ channels are approximately $10.7\%$ and $10.6\%$, respectively.
• Nonleptonic Decay Widths: The paper presents the first theoretical predictions for the two-body nonleptonic decays of $\Omega_b^*$ and $\Omega_b$ into various meson-baryon final states.
  • For $\Omega_b^* \to \Omega_c M$, the dominant modes include $\Omega_c \pi^-$, $\Omega_c D_s^-$, $\Omega_c \rho^-$, and $\Omega_c D_s^{*-}$.
  • For $\Omega_b \to \Omega_c^* M$, the dominant modes include $\Omega_c^* \pi^-$, $\Omega_c^* D_s^-$, $\Omega_c^* \rho^-$, and $\Omega_c^* D_s^{*-}$.
• Comparisons: The calculated form factors for $\Omega_b \to \Omega_c^*$ are compared with results from the light-front quark model, revealing discrepancies in most cases, suggesting a need for further theoretical investigation. The decay width for $\Omega_b \to \Omega_c^* \ell \bar{\nu}_\ell$ is compared with various other theoretical approaches, showing consistency within uncertainties.

Significance and Claims
The authors claim that this work provides valuable theoretical inputs for future experimental investigations at facilities such as LHCb and Belle. Specifically:

• The calculated branching ratios and decay widths for $\Omega_b$ semileptonic decays suggest that the $e$ and $\mu$ channels are the most probable candidates for experimental identification.
• In the nonleptonic sector, the study identifies specific decay modes (e.g., $\Omega_b \to \Omega_c^* \pi^-$, $\Omega_b \to \Omega_c^* D_s^-$, $\Omega_b \to \Omega_c^* \rho^-$) as accessible candidates for observation.
• For the $\Omega_b^*$ baryon, while strong decays are dominant, the study argues that semileptonic and specific nonleptonic weak transitions (e.g., $\Omega_b^* \to \Omega_c \pi^-$, $\Omega_b^* \to \Omega_c D_s^-$) are expected to have observable contributions.
• The results are intended to assist in examining SM predictions, constraining QCD parameters, and exploring potential New Physics effects in heavy baryonic decays. The paper emphasizes that the significant progress in experimental luminosity and energy (e.g., LHC at $\sqrt{s}=13.6$ TeV) makes the observation of these predicted channels increasingly likely.