Semileptonic neutral current decays of $\Xi_b$ with dileptons or dineutrinos in the final state

This paper presents a perturbative QCD analysis of semileptonic $\Xi_b$ decays mediated by flavor-changing neutral currents, calculating form factors and predicting measurable branching fractions and angular observables that could help determine CKM matrix element ratios and constrain Wilson coefficients to address long-standing anomalies in $B$ meson decays.

The Gist

Imagine the universe as a giant, bustling city where tiny particles are the citizens. Most of the time, these citizens follow strict traffic laws written in the "Standard Model" rulebook. But every now and then, a citizen breaks a rule in a very specific, rare way: a heavy "bottom" quark (let's call him Mr. Bottom) suddenly transforms into a lighter "strange" or "down" quark without changing its electric charge. This is called a Flavor-Changing Neutral Current (FCNC).

In the Standard Model, this is like a VIP trying to sneak through a locked door that only opens once in a blue moon. Because it's so rare, it's the perfect place to look for "ghosts"—signs of new, invisible physics that we haven't discovered yet.

For decades, scientists have been watching mesons (particles made of two quarks, like a couple) do this sneaky transformation. They've found some strange glitches in the data, like a traffic light turning green when it should be red. But to be sure it's not just a glitch in the camera, they need to check if the same thing happens to baryons (particles made of three quarks, like a trio).

This paper is all about checking the "trio" citizens, specifically a heavy baryon called $\Xi_b$ (pronounced "Zee-b"), to see if it can pull off the same sneaky transformation.

The Mission: Two Types of Sneaky Transitions

The authors are looking at two specific scenarios where Mr. Bottom transforms:

1. The Dilepton Exit: Mr. Bottom turns into a lighter baryon (like $\Xi$, $\Sigma$, or $\Lambda$) and shoots out a pair of charged particles (like an electron and a positron, or a muon and an antimuon).
2. The Dineutrino Exit: Mr. Bottom transforms and shoots out a pair of invisible ghosts (neutrinos).

The Toolkit: The "PQCD" Calculator

To predict how often these events happen, the authors use a sophisticated mathematical tool called Perturbative Quantum Chromodynamics (PQCD).

• The Analogy: Imagine trying to predict the exact path of a leaf falling from a tree in a hurricane. It's chaotic. But if you zoom in on the leaf's tiny fibers and calculate the wind hitting each one, you can get a good estimate. PQCD is that microscope. It breaks the complex interaction of quarks down into smaller, manageable pieces to calculate the "shape" of the transformation.

The Key Ingredients: "Form Factors"

The biggest challenge in these calculations is the "Form Factors."

• The Analogy: Think of the baryon as a squishy, shape-shifting jellyfish. When it transforms, it doesn't just snap into a new shape; it stretches and contorts. The "Form Factors" are the measurements of exactly how much it stretches and twists at every stage of the transformation.
• The authors calculated 10 different ways this jellyfish can stretch (vector, axial-vector, tensor, etc.). They used a technique called z-expansion to map these shapes from the easy-to-calculate low-energy zone all the way to the high-energy zone, creating a complete map of the transformation.

The Findings: What Did They See?

1. The "Measurable" Future
They predict that the decay where Mr. Bottom turns into a $\Xi$ baryon and a pair of muons happens about 3 times out of every million attempts.

• Why it matters: This is right on the edge of what the LHCb experiment (a giant particle detector at CERN) can currently see. It's like predicting a specific rare bird will land on your porch tomorrow. If the LHCb sees it, great! If they see it more or less often than predicted, it's a smoking gun for new physics.

2. The "Ghost" Channels (Neutrinos)
They also looked at the invisible neutrino channels. Because neutrinos don't interact with matter, they are hard to catch. However, the math says these decays might actually happen more often than the charged particle ones (up to 1 in 100,000 times).

• The Twist: These channels are "cleaner." In the charged particle version, there's a lot of background noise (like static on a radio) from other particles interfering. The neutrino version is like a clear, silent channel. If we can detect these, they give us a very pure view of the underlying physics.

3. The "Spin" Dance (Angular Observables)
The authors didn't just count how many times the decay happens; they looked at how the particles fly out.

• The Analogy: Imagine a spinning top breaking apart. Do the pieces fly straight out, or do they spiral? Do they prefer to go left or right?
• They calculated specific "angles" of the debris. These angles act like a fingerprint. If the Standard Model is correct, the fingerprint looks a certain way. If "New Physics" (like a hidden force or a new particle) is interfering, the fingerprint gets distorted.
• They found that some of these angles are very sensitive to the "Wilson Coefficients" (the numbers that define the strength of the forces). If these numbers are off, it means the rulebook needs a rewrite.

The Big Picture: Solving the Mystery

For years, scientists have been puzzled by "anomalies" in the behavior of mesons (the couples). The data suggested the laws of physics might be slightly different for electrons than for muons (a violation of "Lepton Flavor Universality").

This paper says: "Let's check the trios ($\Xi_b$ baryons) to see if the same rule-breaking happens there."

• If the $\Xi_b$ baryons behave exactly like the mesons, it strengthens the case that there is a new, universal force at play.
• If they behave differently, it might mean the anomalies in the mesons were just a fluke or a specific quirk of that particle type.

Conclusion

In simple terms, this paper is a detailed instruction manual and a weather forecast for a very rare particle event. The authors have built a high-precision map of how a heavy baryon transforms into lighter particles. They are telling experimentalists at the LHC: "Look here, at these specific angles and frequencies. If you see what we predicted, the Standard Model holds. If you see something else, we might have just found the door to a new universe."

It's a bridge between the messy, complex reality of particle collisions and the clean, mathematical laws that govern them, offering a fresh perspective on one of the biggest mysteries in modern physics.

Technical Summary

1. Problem Statement

Flavor-Changing Neutral Current (FCNC) transitions involving $b$-quark decays ($b \to s$ and $b \to d$) are highly suppressed in the Standard Model (SM) and serve as sensitive probes for New Physics (NP). While extensive studies have been conducted on mesonic decays (e.g., $B \to K^{(*)}\ell^+\ell^-$), baryonic decays remain less explored despite offering unique insights into the helicity structure of effective Hamiltonians due to the half-integer spin of baryons.

• The Gap: There is a lack of comprehensive, systematic theoretical predictions for rare semileptonic decays of the $\Xi_b$ baryon, specifically for the channels $\Xi_b \to (\Xi, \Sigma, \Lambda)(\ell^+\ell^-, \nu\bar{\nu})$.
• The Challenge: Calculating the hadronic transition form factors for heavy-to-light baryon transitions is non-trivial. Furthermore, the dineutrino channels ($\nu\bar{\nu}$) are theoretically "clean" but experimentally challenging, while dilepton channels suffer from long-distance effects due to charmonium resonances ($J/\psi, \psi(2S)$).

2. Methodology

The authors employ the Perturbative Quantum Chromodynamics (PQCD) framework to perform a systematic analysis.

• Effective Hamiltonian: The analysis is based on the SM effective Hamiltonian for $b \to Q\ell^+\ell^-$ and $b \to Q\nu\bar{\nu}$ transitions ($Q=s, d$), incorporating Wilson coefficients $C_7, C_9, C_{10}$ (for dileptons) and $C_L$ (for dineutrinos).
• Form Factor Calculation:
  • The hadronic matrix elements are parametrized using 12 dimensionless form factors (vector, axial-vector, tensor, and pseudotensor currents).
  • Using the equations of motion, these are reduced to 10 independent form factors.
  • The PQCD approach is used to calculate these form factors in the large-recoil region (low $q^2$), utilizing light-cone distribution amplitudes (LCDAs) for the initial ($\Xi_b$) and final ($\Xi, \Sigma, \Lambda$) baryons. The Gegenbauer model is selected for the LCDAs due to its stability and inclusion of SU(3) flavor symmetry breaking.
  • To cover the entire kinematic range, the PQCD results at low $q^2$ are extrapolated to high $q^2$ using a $z$-expansion parametrization.
• Angular Analysis:
  • The decay chain $\Xi_b \to B_f(\to B_h P)\ell^+\ell^-$ is analyzed using the helicity-amplitude formalism for unpolarized $\Xi_b$ baryons.
  • Finite lepton mass effects ($m_\ell \neq 0$) are retained.
  • The four-fold differential decay distribution is decomposed into angular coefficients ($K_i$) expressed via 10 transversity amplitudes.
• Observables: The study computes:
  • Branching fractions ($\mathcal{B}$).
  • Lepton Flavor Universality (LFU) ratios ($R_\mu, R_\tau$).
  • Angular observables: Longitudinal polarization fraction ($F_L$), leptonic/hadronic/combined forward-backward asymmetries ($A_{FB}^\ell, A_{FB}^h, A_{FB}^{h\ell}$).

3. Key Contributions

• First Systematic Study: This is the first detailed investigation of semileptonic FCNC $\Xi_b$ decays involving dineutrino final states within the PQCD framework.
• Comprehensive Form Factors: The authors provide the first complete set of PQCD predictions for vector, axial-vector, tensor, and pseudotensor form factors for $\Xi_b \to \Xi, \Sigma, \Lambda$ transitions, including their $q^2$ dependence and endpoint behaviors.
• Combined Analysis: The paper uniquely compares dilepton and dineutrino channels, leveraging the $SU(2)_L$ symmetry relations between them to provide complementary constraints on Wilson coefficients.
• Model Validation: The study verifies that the calculated form factors satisfy heavy-quark symmetry limits (HQET) at zero recoil and soft-collinear effective theory (SCET) limits at large recoil, ensuring theoretical consistency.

4. Key Results

A. Form Factors

• The form factors increase steadily with $q^2$.
• At zero recoil ($q^2_{max}$), the results satisfy the endpoint relations predicted by HQET (e.g., $f_0 \approx f_\perp \approx g_t$).
• The predictions differ significantly from previous QCD Sum Rule (QCDSR) calculations, particularly for the $\Xi_b \to \Sigma$ transition, where the authors' results align better with heavy-quark symmetry expectations (e.g., suppression of subleading form factors).

B. Branching Fractions

• Dilepton Modes:
  • $\Xi_b \to \Xi \ell^+\ell^-$: $\mathcal{B} \sim 10^{-6}$ (measurable by LHCb).
  • $\Xi_b \to (\Sigma, \Lambda) \ell^+\ell^-$: $\mathcal{B} \sim 10^{-8}$ (suppressed by $|V_{td}/V_{ts}|^2$).
  • Long-distance effects (charmonium resonances) create sharp peaks but are experimentally vetoed; the integrated branching fractions outside these regions are only slightly enhanced.
• Dineutrino Modes:
  • Branching fractions are generally larger than dilepton modes due to larger Wilson coefficients.
  • $\mathcal{B}(\Xi_b \to \Xi \nu\bar{\nu}) \sim 10^{-5}$ (summed over 3 flavors), comparable to Belle II's $B^+ \to K^+ \nu\bar{\nu}$ measurement.
  • These channels are free from long-distance hadronic uncertainties.

C. Angular Observables & LFU

• LFU Ratios:
  • $R_\mu = \mathcal{B}(\mu\mu)/\mathcal{B}(ee) \approx 1.00$, confirming SM universality. Any deviation $> \text{few}\%$ would signal NP.
  • $R_\tau = \mathcal{B}(\tau\tau)/\mathcal{B}(ee) \approx 0.6$, driven by phase-space suppression and spin-flip effects.
• Zero Crossings:
  • The leptonic forward-backward asymmetry ($A_{FB}^\ell$) exhibits a zero-crossing point at $q^2_0 \approx 4.0 \text{ GeV}^2$. This position is highly sensitive to Wilson coefficients and serves as a robust NP probe.
  • For $\tau$ leptons, the zero crossing is kinematically inaccessible, leaving $A_{FB}^\tau$ negative throughout.
• Polarization:
  • The longitudinal polarization fraction $F_L$ behaves differently for neutrinos vs. electrons. In the large-recoil limit, $F_L \to 1$ for neutrinos (due to the absence of photon-penguin contributions), whereas it peaks and then drops for electrons.

5. Significance

• New Physics Probes: The angular observables (especially zero-crossings and polarization fractions) are largely independent of hadronic uncertainties, making them powerful tools to constrain Wilson coefficients and detect deviations from the SM.
• Complementarity: The study provides a crucial cross-check for the anomalies observed in $B$-meson decays. If NP exists, it should manifest consistently in both mesonic and baryonic sectors.
• Experimental Guidance: The paper identifies $\Xi_b \to \Xi \ell^+\ell^-$ and $\Xi_b \to \Xi \nu\bar{\nu}$ as the most promising channels for near-future observation at LHCb and future high-luminosity $e^+e^-$ colliders (FCC-ee, CEPC).
• Theoretical Benchmark: The detailed form factor predictions and angular distributions serve as a standard reference for future experimental analyses and theoretical refinements (e.g., lattice QCD inputs for high $q^2$).

In conclusion, this work establishes a robust theoretical foundation for exploring rare $\Xi_b$ decays, offering precise predictions that can help resolve long-standing anomalies in flavor physics and potentially uncover physics beyond the Standard Model.