Semileptonic decay of $\Lambda_b^0 \to \Lambda_c (2860)^+/\Lambda_c(2625)^+\ell^-\overline{\nu}_\ell$ within QCD light-cone sum rules

This paper utilizes QCD light-cone sum rules to calculate transition form factors and predict the branching fractions for the semileptonic decays $\Lambda_b^0 \to \Lambda_c(2860)^+\ell^-\overline{\nu}_\ell$ and $\Lambda_b^0 \to \Lambda_c(2625)^+\ell^-\overline{\nu}_\ell$, with the latter's result validating the method against experimental data to provide theoretical guidance for future measurements of the former.

The Gist

Imagine the subatomic world as a bustling, high-energy construction site where particles are constantly being built, broken apart, and rebuilt. This paper is a detailed blueprint for understanding one specific, complex demolition and reconstruction project: the decay of a heavy particle called the Lambda-b baryon ($\Lambda_b^0$).

Here is the story of what the authors did, explained without the heavy math.

The Main Event: A Heavy Particle's Transformation

Think of the $\Lambda_b^0$ as a very heavy, unstable truck carrying a "bottom" quark. In the world of particle physics, heavy things don't stay heavy for long; they want to shed weight. In this specific scenario, the truck sheds its heavy bottom quark and transforms into a slightly lighter "charm" quark, while also spitting out a pair of invisible particles (a lepton and a neutrino).

The tricky part is what the truck turns into. Usually, it turns into a standard, stable version of a charm baryon. But in this paper, the authors are looking at two specific, "excited" versions of the destination:

1. $\Lambda_c(2625)^+$: A slightly heavier, vibrating version of the charm baryon.
2. $\Lambda_c(2860)^+$: An even heavier, more energetic version.

These "excited" versions are like a car engine that is revving loudly and shaking before settling down. They are unstable and short-lived.

The Challenge: Measuring the "Shape" of the Change

To understand how likely this transformation is to happen (the "branching fraction"), physicists need to know the form factors.

The Analogy: Imagine you are trying to predict how much water flows through a pipe when you squeeze it. The "flow" is the decay, but the "pipe" isn't a simple cylinder; it's a complex, squishy shape that changes as you squeeze it. The form factors are the mathematical map that tells you exactly how that pipe squishes and stretches at every moment of the transition. Without this map, you can't calculate how much water (or in this case, how often the decay happens) will flow.

The Tool: QCD Light-Cone Sum Rules

The authors used a sophisticated mathematical tool called QCD Light-Cone Sum Rules (LCSR) to draw this map.

The Analogy: Think of the $\Lambda_b$ particle as a complex machine made of smaller gears (quarks) held together by springs (gluons). You can't see the gears directly while the machine is running. Instead, the authors used LCSR as a "shadow puppet" technique.

• They looked at the "shadows" cast by the machine's internal parts (the light-cone distribution amplitudes).
• They used these shadows to reconstruct the internal mechanics of the machine.
• By doing this, they could calculate the "squishiness" (the form factors) of the transition from the heavy truck to the two different excited destination cars.

The Results: Two Different Destinations

1. The Known Destination ($\Lambda_c(2625)^+$)
The authors calculated the decay path to the $\Lambda_c(2625)^+$.

• The Check: They compared their calculated "flow rate" (branching fraction) with real-world data collected by experiments (like the CDF collaboration) and other theoretical models.
• The Verdict: Their numbers matched the real-world data and other theories very well. This is like building a bridge and finding that your engineering calculations perfectly match the actual weight it can hold. This success proves their "shadow puppet" method is reliable.

2. The Unknown Destination ($\Lambda_c(2860)^+$)
This is the new territory. The $\Lambda_c(2860)^+$ is a particle that hasn't been studied as thoroughly in this specific type of decay.

• The Prediction: Since their method worked for the known destination, they used the exact same blueprint to predict the decay rate for the $\Lambda_c(2860)^+$.
• The Result: They provided the first theoretical prediction for how often this specific decay happens. They essentially handed experimentalists a "wanted poster" with a predicted frequency, telling them, "Look for this event happening at this rate."

Why This Matters (According to the Paper)

The paper doesn't claim this will lead to new medicines or energy sources. Instead, its value is purely in precision and prediction:

• Validation: They proved their mathematical tool works for complex, spinning particles (spin-3/2).
• Reference: They gave experimental physicists a specific target number to look for when they analyze data from particle colliders. If the experiments find the decay happening at the rate they predicted, it confirms our understanding of how the strong force (the glue holding particles together) works in these high-energy scenarios.

In short, the authors built a precise mathematical map for a rare particle transformation, verified it against a known landmark, and then used that map to chart the course for a new, unexplored destination.

Technical Summary

Technical Summary: Semileptonic Decay of $\Lambda_b^0 \to \Lambda_c(2860)^+/\Lambda_c(2625)^+ \ell^- \bar{\nu}_\ell$ within QCD Light-Cone Sum Rules

Problem Statement
This work addresses the theoretical description of weak semileptonic decays of the bottom baryon $\Lambda_b^0$ into excited charm baryons, specifically the transitions $\Lambda_b^0 \to \Lambda_c(2860)^+$ and $\Lambda_b^0 \to \Lambda_c(2625)^+$. While previous studies within the QCD light-cone sum rules (LCSR) framework have successfully modeled transitions to ground states ($1/2^+ \to 1/2^+$) and negative-parity excited states ($1/2^+ \to 1/2^-$), there is a need to investigate processes involving final-state baryons with higher spin-parity quantum numbers, specifically $1/2^+ \to 3/2^\pm$. The $\Lambda_c(2625)^+$ ($3/2^-$) and $\Lambda_c(2860)^+$ ($3/2^+$) represent such cases. Although the $\Lambda_b^0 \to \Lambda_c(2625)^+ \ell^- \bar{\nu}_\ell$ channel has been measured experimentally, the $\Lambda_b^0 \to \Lambda_c(2860)^+ \ell^- \bar{\nu}_\ell$ channel remains largely unexplored theoretically and experimentally.

Methodology
The authors employ the QCD light-cone sum rules (LCSR) method to calculate the transition form factors. The calculation proceeds through the following steps:

1. Correlation Function Construction: The analysis begins with the correlation function of the hadron weak transition, $\Pi_{\mu\nu}(p, q) = i \int d^4x e^{ip'\cdot x} \langle 0 | T \{ \eta_\mu(x), j_\nu(0) \} | \Lambda_b(p) \rangle$. Here, $\eta_\mu$ is the interpolating current for the spin-$3/2$ final baryon, and $j_\nu$ is the standard $V-A$ weak current for the $b \to c$ transition.
2. Hadronic Representation: On the hadronic level, the correlation function is expressed in terms of the physical states. A critical technical aspect of this work is the separation of Lorentz structures. Since the interpolating current for a spin-$3/2$ particle can couple to both spin-$1/2$ and spin-$3/2$ states, the authors explicitly isolate the terms associated with the spin-$3/2$ final state (characterized by Lorentz indices $\mu$ and $\nu$) while discarding contributions from the spin-$1/2$ background.
3. QCD Representation: On the QCD level, the correlation function is calculated using the operator product expansion (OPE) near the light-cone. This involves contracting the heavy quark fields and expressing the result in terms of the light-cone distribution amplitudes (LCDAs) of the initial $\Lambda_b^0$ baryon. The authors utilize twist-3 LCDAs for the $\Lambda_b^0$, adopting specific wave function models from previous literature.
4. Sum Rule Extraction: By applying quark-hadron duality, performing a Borel transformation to suppress higher excited states and continuum contributions, and introducing a threshold parameter $s_0$, the authors derive sum rules for the form factors.
5. Phenomenological Application: The calculated form factors, initially valid near $q^2 \approx 0$, are extrapolated to the full physical kinematic region using the "z-series" expansion method. These form factors are then integrated to compute the differential decay widths and total branching fractions for decays involving electrons, muons, and tau leptons.

Key Contributions

• Extension of LCSR to Spin-3/2 Transitions: The paper extends the LCSR framework to handle transitions from a spin-$1/2$ initial baryon to spin-$3/2$ final baryons with both positive ($3/2^+$) and negative ($3/2^-$) parity.
• Form Factor Calculation: The authors provide explicit expressions and numerical values for the eight independent form factors governing these transitions ($f^V_i, g^A_i$ for $3/2^+$ and $f^A_i, g^V_i$ for $3/2^-$).
• Prediction for $\Lambda_c(2860)^+$: The work presents the first theoretical prediction for the semileptonic branching fractions of $\Lambda_b^0 \to \Lambda_c(2860)^+ \ell^- \bar{\nu}_\ell$.
• Validation via $\Lambda_c(2625)^+$: The methodology is validated by calculating the branching fraction for $\Lambda_b^0 \to \Lambda_c(2625)^+ \ell^- \bar{\nu}_\ell$ and comparing it with existing experimental data and other theoretical models.

Results

• Branching Fractions:
  • For $\Lambda_b^0 \to \Lambda_c(2625)^+ \ell^- \bar{\nu}_\ell$, the calculated branching fractions are approximately $8.25 \times 10^{-3}$ (for $e$), $8.12 \times 10^{-3}$ (for $\mu$), and $0.64 \times 10^{-3}$ (for $\tau$). These results are consistent with the experimental measurement of $1.3^{+0.6}_{-0.5}\%$ (PDG) and align well with other theoretical approaches like Light-Front Quark Models (LFQM) and Covariant Confined Quark Models (CCQM), though they differ slightly from some LCSR and HQSS results.
  • For $\Lambda_b^0 \to \Lambda_c(2860)^+ \ell^- \bar{\nu}_\ell$, the predicted branching fractions are $4.32 \times 10^{-3}$ (for $e$), $4.18 \times 10^{-3}$ (for $\mu$), and $0.0975 \times 10^{-3}$ (for $\tau$).
• Form Factors: The numerical analysis provides the $q^2$ dependence of the form factors, fitted using the $z$-series expansion parameters listed in Tables 2 and 3 of the paper.

Significance and Claims
The authors state that the consistency of their results for the $\Lambda_c(2625)^+$ channel with experimental data and other theoretical predictions validates the reliability of their LCSR approach for spin-$3/2$ transitions. Consequently, the primary significance of this work lies in providing a theoretical reference for the $\Lambda_b^0 \to \Lambda_c(2860)^+ \ell^- \bar{\nu}_\ell$ decay channel. The authors position their results as a guide for future experimental measurements of this specific decay mode, which has not yet been observed. The paper does not propose new experimental setups or claim to resolve fundamental physics beyond the Standard Model, but rather focuses on refining the theoretical understanding of heavy baryon weak decays within the QCD framework.