$\Xi_b \to \Xi$ form factors from lattice QCD and Standard-Model predictions for $\Xi_b \to \Xi \mu^+\mu^-$ and $\Xi_b \to \Xi \gamma$ decays

This paper presents the first lattice QCD calculation of $\Xi_b \to \Xi$ form factors using 2+1 flavor domain-wall fermions and an anisotropic clover action for the bottom quark, which are then utilized to provide Standard-Model predictions for the branching fractions and angular observables of the rare decays $\Xi_b^- \to \Xi^- \gamma$ and $\Xi_b^- \to \Xi^- \mu^+\mu^-$.

The Gist

Imagine the universe is a giant, complex machine built from tiny, invisible Lego bricks called quarks. Most of the time, these bricks snap together to form stable structures like protons and neutrons. But sometimes, a heavy, unstable brick (called a bottom quark) decides to change into a lighter one (a strange quark). When this happens, it's like a heavy, clunky engine part suddenly transforming into a sleek, lightweight one.

This transformation doesn't happen in a vacuum; it's a messy, chaotic process that spits out other particles. Physicists are obsessed with watching these transformations because they are the perfect place to look for "ghosts in the machine"—signs of new physics that the current rulebook (the Standard Model) doesn't explain.

Here is what this paper does, broken down into simple concepts:

1. The Problem: The "Black Box" of Heavy Baryons

Scientists have been studying a specific type of heavy particle called the $\Lambda_b$ baryon (a heavy Lego structure made of three bricks). They know exactly how it behaves when it changes. But there's a cousin particle called the $\Xi_b$ baryon. It's very similar, but slightly different because of how its internal bricks are arranged.

Until now, the $\Xi_b$ was a bit of a mystery. We knew it existed, and we knew it could decay, but we didn't have a precise map of how it does it. It's like knowing a car can drive from New York to Boston, but not knowing the exact speed limits, road conditions, or fuel efficiency for that specific route. Without this map, we can't tell if the car is driving "normally" or if there's a secret shortcut (New Physics) being used.

2. The Tool: The "Quantum Super-Computer"

To get this map, the authors used Lattice QCD. Imagine space and time not as a smooth, continuous sheet, but as a giant 3D grid (like a giant chessboard made of invisible cubes).

• The Simulation: They put their heavy $\Xi_b$ particle on this grid and ran a massive computer simulation to watch how it changes into a lighter $\Xi$ particle.
• The Challenge: In the real world, these particles are fuzzy and jittery. In the simulation, the "grid" has a specific size. If the grid squares are too big, the picture is blurry. If the simulation uses "heavy" pions (another type of particle used to stabilize the math), the results are off.
• The Fix: The authors ran this simulation on four different grids (some fine, some coarse) and with different weights for the internal particles. They then used a clever mathematical trick (called a modified z-expansion) to stitch all these blurry, different-sized pictures together into one crystal-clear, high-definition image.

3. The Map: "Form Factors"

The result of their simulation is a set of numbers called Form Factors.

Think of a Form Factor as a detailed instruction manual for the transformation. It tells you:

• How much "push" (momentum) is transferred during the change.
• How the internal structure of the particle stretches or squishes.
• The probability of the particle taking a specific path.

Before this paper, we had to guess these numbers for the $\Xi_b$ particle, often assuming it behaved exactly like its cousin, the $\Lambda_b$. This paper is the first time anyone has actually calculated these numbers from first principles using the laws of quantum mechanics.

4. The Prediction: Checking the Rulebook

Once they had their precise "instruction manual" (the form factors), they used it to predict what should happen in two specific scenarios:

1. The Radiative Decay ($\Xi_b \to \Xi \gamma$): The particle sheds a photon (a particle of light) as it changes.
2. The Rare Decay ($\Xi_b \to \Xi \mu^+ \mu^-$): The particle changes and spits out a pair of muons (heavy electrons).

The authors calculated exactly how often these events should happen if the Standard Model is 100% correct. They also calculated specific "angles" at which the new particles fly out.

5. Why This Matters: The Detective Work

Why do we care? Because for a long time, there have been "anomalies" (glitches) in how other heavy particles behave. Some experiments suggest the universe is breaking the rules.

• The Analogy: Imagine you are a detective trying to solve a crime. You have a suspect (New Physics) who might be breaking the law. You have a witness (the $\Lambda_b$ particle) who says, "I saw something weird."
• The New Clue: Now, you have a second witness (the $\Xi_b$ particle). If the second witness tells a story that matches the first one, it's just a coincidence. But if the second witness tells a different story that contradicts the rulebook, it's a smoking gun for new physics.

By providing the first precise map for the $\Xi_b$, this paper gives experimentalists at the LHCb (a giant particle detector at CERN) a target to aim for. When they look at their data, they can now say: "The theory says it should happen this many times. Did we see that? If not, we might have found a crack in the Standard Model."

Summary

In short, this paper is like drawing the first accurate topographical map of a previously uncharted mountain.

• The Mountain: The $\Xi_b$ particle decay.
• The Map: The Form Factors calculated via super-computers.
• The Goal: To see if hikers (experimentalists) find the path they expect, or if they stumble upon a hidden cave (New Physics) that changes everything we know about the universe.

The authors found that, for now, the mountain looks exactly as the Standard Model predicted. But now, we have a map precise enough to spot even the tiniest deviation in the future.

Technical Summary

1. Problem Statement

Rare decays of $b$-hadrons, specifically the flavor-changing neutral current (FCNC) transitions $b \to s\ell^+\ell^-$ and $b \to s\gamma$, are critical probes for physics beyond the Standard Model (BSM). While significant attention has been paid to mesonic decays (e.g., $B \to K^{(*)}\mu^+\mu^-$) and the baryonic decay $\Lambda_b \to \Lambda \mu^+\mu^-$, the baryonic sector remains less explored.

• The Gap: There was a lack of first-principles theoretical predictions for the $\Xi_b \to \Xi$ transitions. Previous estimates relied on Light-Cone Sum Rules (LCSR) at low momentum transfer ($q^2$), $pQCD$ frameworks, or $SU(3)$ flavor symmetry extrapolations from $\Lambda_b$ data, all of which carry large uncertainties, particularly at high $q^2$.
• The Need: Precise Standard Model (SM) predictions for the differential branching fractions and angular observables of $\Xi_b^- \to \Xi^- \mu^+ \mu^-$ and the radiative decay $\Xi_b^- \to \Xi^- \gamma$ are required to interpret upcoming experimental data from LHCb and to constrain Wilson coefficients (specifically $C_7, C_9, C_{10}$) in the effective Hamiltonian.

2. Methodology

The authors performed the first lattice Quantum Chromodynamics (QCD) calculation of the vector, axial-vector, and tensor form factors governing these transitions.

Lattice Setup:

• Ensembles: Calculations were performed on four $2+1$ flavor domain-wall fermion ensembles generated by the RBC and UKQCD collaborations.
  • Lattice Spacings ($a$): Ranging from $0.073$ fm to $0.111$ fm.
  • Pion Masses ($m_\pi$): Ranging from $230$ MeV to $430$ MeV (covering the physical point via extrapolation).
• Quark Actions:
  • Light/Strange Quarks: Domain-wall fermions (preserving chiral symmetry).
  • Bottom Quark: Anisotropic clover action, tuned non-perturbatively to reproduce $B_s$ meson properties.
• Correlation Functions:
  • Computed 3-point functions with a wide range of source-sink separations ($t/a = 4$ to $20$) to isolate ground-state contributions.
  • Used "all-mode-averaging" (AMA) to improve statistical precision, combining exact and sloppy (reduced iteration) samples.
  • Extracted form factors from ratios of 3-point to 2-point correlation functions designed to cancel overlap factors and time dependencies.

Extrapolation and Analysis:

• Chiral-Continuum-Kinematic Extrapolation: The lattice data were fitted to the physical limit ($a \to 0, m_\pi \to m_{\pi,phys}$) using modified $z$-expansions (BGL-type).
  • Constraints: The fit incorporated dispersive bounds (unitarity constraints) and asymptotic behavior constraints (sum rules derived from perturbative QCD) to control uncertainties across the full kinematic range.
  • Order of Expansion: The order of the $z$-expansion ($N$) was increased until the central values and uncertainties of the form factors stabilized. The authors determined that $N=5$ was sufficient for stability.
• Systematic Uncertainties: Included renormalization factors (mostly non-perturbative), $O(a)$ improvement coefficients, finite-volume effects, isospin breaking, and perturbative matching uncertainties.

3. Key Contributions

1. First Lattice Determination: This work provides the first ab initio lattice QCD determination of the full set of $\Xi_b \to \Xi$ form factors (vector $f_{+,0,\perp}$, axial-vector $g_{+,0,\perp}$, and tensor $h_{+, \perp}, \tilde{h}_{+, \perp}$).
2. Methodological Advance: The authors successfully applied dispersive bounds to baryonic form factor fits. This allowed them to extend the $z$-expansion order higher than previous baryonic studies (like $\Lambda_b \to \Lambda$) without introducing uncontrolled priors, resulting in robust uncertainties over the entire semileptonic region.
3. Comprehensive Kinematic Coverage: Unlike previous LCSR calculations limited to low $q^2$, this lattice calculation provides precise results at high $q^2$ (near $q^2_{max}$), which are crucial for the $\Xi_b \to \Xi \mu^+ \mu^-$ decay rate.

4. Key Results

Form Factors:

• The form factors were determined with controlled uncertainties in the physical limit.
  • High $q^2$ ($q^2 \approx q^2_{max}$): Uncertainties are approximately 3–4% for vector and axial-vector form factors, and 6–7% for tensor form factors.
  • Low $q^2$ ($q^2 = 0$): Uncertainties are larger, around 25%, due to the extrapolation distance from the lattice data points.
• Comparison: The results show reasonable agreement with continuum calculations (e.g., Rui et al., 2025) and LCSR, though the lattice central values tend to be slightly lower than some previous predictions.
• SU(3) Symmetry: The results confirm that $\Xi_b \to \Xi$ form factors satisfy $SU(3)$ flavor symmetry relations with $\Lambda_b \to p$ and $\Lambda_b \to \Lambda$ form factors up to expected symmetry-breaking corrections.

Standard Model Predictions:
Using the calculated form factors, the authors derived SM predictions for:

1. Radiative Decay ($\Xi_b^- \to \Xi^- \gamma$):
   • Predicted Branching Fraction: $\mathcal{B}(\Xi_b^- \to \Xi^- \gamma) = (2.9 \pm 1.6) \times 10^{-5}$.
   • This is consistent with the current LHCb upper limit ($< 1.3 \times 10^{-4}$) but significantly lower than the prediction in Ref. [93] (which was $\approx 3 \times 10^{-4}$).
2. Semileptonic Decay ($\Xi_b^- \to \Xi^- \mu^+ \mu^-$):
   • Provided differential branching fractions ($d\mathcal{B}/dq^2$) and angular observables ($F_L$ and $A_{FB}^\ell$) across the kinematic range (excluding charmonium resonances).
   • The central values for the differential branching fraction are approximately 25% lower than previous predictions (Ref. [85]), though consistent within combined uncertainties.

5. Significance

• New Physics Sensitivity: The $\Xi_b \to \Xi$ transitions offer a unique testing ground for BSM physics. Being baryonic decays, they are sensitive to different Dirac structures and polarization effects compared to mesonic decays. The initial $\Xi_b$ polarization and the parity-violating weak decay of the final $\Xi$ allow for new angular observables.
• Experimental Guidance: With LHCb actively analyzing $\Xi_b^- \to \Xi^- \mu^+ \mu^-$ data, these precise SM predictions are essential for identifying potential anomalies. The discrepancy in the tensor form factor predictions for the radiative decay highlights the importance of lattice results in resolving theoretical tensions.
• Theoretical Robustness: By utilizing dispersive bounds and high-order $z$-expansions, this work sets a new standard for precision in lattice calculations of heavy-to-light baryon transitions, reducing reliance on model-dependent extrapolations.

In conclusion, this paper bridges a critical gap in the theoretical understanding of rare $b$-baryon decays, providing the necessary non-perturbative inputs to interpret current and future experimental data from the LHC and potentially future colliders.