Constraints on new physics from decays of polarized $Λ_b^0$ baryons at the FCC-ee

This paper demonstrates that utilizing the unique polarization of $\Lambda_b^0$ baryons produced in $Z^0$ decays at the FCC-ee, despite comparable statistical sensitivity to LHCb Upgrade II for individual observables, significantly improves constraints on the Wilson coefficients $C_{9^{(\prime)}}$ and $C_{10^{(\prime)}}$ through a broader set of accessible angular observables.

The Gist

Imagine you are a detective trying to solve a mystery about the fundamental rules of the universe. For years, you've been looking at clues left behind by tiny particles called "b-quarks" as they decay (fall apart). You've noticed that the clues don't quite match the rulebook (the Standard Model), suggesting there might be a secret agent—New Physics—hiding in the shadows.

This paper is a proposal for how to catch that secret agent using a future, super-powerful microscope called the FCC-ee.

Here is the breakdown of the detective work, explained with some everyday analogies:

1. The Mystery: The "Spinning Top" vs. The "Rolling Ball"

In the past, at the Large Hadron Collider (LHC), scientists studied these particles like rolling balls. They were produced randomly, spinning in every direction, so when they fell apart, the debris scattered in a messy, hard-to-read pattern. It was like trying to read a book where the pages were shuffled randomly.

However, the proposed FCC-ee (Future Circular Collider) is different. It acts like a spinning top factory. When it creates these specific particles (called $\Lambda_b^0$ baryons), they don't just roll; they spin in a very specific, predictable direction (they are "polarized").

The Analogy: Imagine trying to figure out the shape of a hidden object by throwing sand at it.

• At the LHC: The object is tumbling in the wind. The sand hits it from all sides, and you can't tell its shape clearly.
• At the FCC-ee: The object is held still and spinning on a specific axis. When the sand hits it, the pattern of the spray tells you exactly what the object looks like.

Because the FCC-ee particles are "held still" (polarized), scientists can measure 34 different angles of the debris, compared to only 10 angles available at the LHC. It's like going from a blurry black-and-white photo to a high-definition 3D hologram.

2. The Crime Scene: The "Four-Body" Breakup

The specific crime being investigated is a decay chain:

1. A heavy particle ($\Lambda_b^0$) breaks into a lighter particle ($\Lambda$) and two muons (heavy electrons).
2. The lighter particle ($\Lambda$) immediately breaks into a proton and a pion.

This is a "four-body" breakup. The paper uses a complex mathematical map (Equation 1) to track exactly where every piece flies. Because the original particle was spinning, the way the pieces fly depends on the "secret rules" (called Wilson Coefficients) governing the decay.

3. The Detective Tools: The "IDEA" Detector

To catch these clues, the paper proposes using a detector called IDEA. Think of this as a high-tech, multi-layered security camera system.

• It has a "silicon pixel" eye to see exactly where particles start.
• It has a "crystal calorimeter" to measure their energy.
• It has "muon chambers" to catch the heavy electrons that sneak through everything else.

The researchers used a computer simulation (like a video game engine) to pretend they were running this experiment with 6 trillion collisions. They had to filter out "noise"—other particles that look similar but aren't the real clue. They found that while background noise exists, their "topological" filters (checking if the particles flew in a straight line from a specific point) were very good at cleaning up the scene.

4. The Results: Catching the "Secret Agent"

The main goal is to pin down the values of the "Secret Agents" (the Wilson Coefficients, specifically $C_9$ and $C_{10}$). These numbers tell us how strongly the particles interact with the forces of nature.

• The Old Way (LHC): Using only the 10 angles from unpolarized particles, the detectives get a blurry picture. They can guess the agent's location, but there's a lot of uncertainty.
• The New Way (FCC-ee): By adding the 24 new angles made possible by the polarization, the picture snaps into focus.

The Big Reveal:
The paper shows that even though the FCC-ee might not produce more total particles than the LHC, the fact that they are polarized gives them much more information per particle.

• It's like having a witness who can only say "I saw a car" (LHC) versus a witness who can say "I saw a red car, driving north, with a dent on the left side, speeding at 60mph" (FCC-ee).
• The combination of all 34 angles allows scientists to constrain the "Secret Agents" much tighter, potentially revealing if the Standard Model is wrong and New Physics is real.

5. The Bottom Line

This paper is a "toy model" (a proof-of-concept). It says: "If we build this machine and look at these spinning particles, we will get a much clearer view of the laws of physics than we ever could before."

It doesn't promise to find New Physics tomorrow, but it promises that if New Physics is hiding in the shadows of these decays, the FCC-ee is the flashlight that will finally reveal it. The polarization of the particles is the key that unlocks the door to a deeper understanding of the universe.

Technical Summary

Here is a detailed technical summary of the paper "Constraints on new physics from decays of polarized $\Lambda_b^0$ baryons at the FCC-ee."

1. Problem and Motivation

• Context: Heavy hadron electroweak penguin decays (specifically $b \to s \ell^+ \ell^-$) are a primary probe for physics beyond the Standard Model (SM). Tensions between SM predictions and experimental measurements (e.g., in $B^0 \to K^{*0} \mu^+ \mu^-$) suggest potential new physics.
• The Limitation of Hadron Colliders: At the LHC, $\Lambda_b^0$ baryons are produced unpolarized due to the strong production of $b\bar{b}$ pairs. This limits the accessible angular observables to a subset that does not fully disentangle the helicity structures of the underlying effective operators (Wilson coefficients).
• The Opportunity at FCC-ee: The Future Circular Collider (FCC-ee), operating as a high-luminosity $Z^0$ factory, produces $\Lambda_b^0$ baryons with significant longitudinal polarization ($P_\parallel \approx -0.45$). This polarization, combined with the clean $e^+e^-$ environment, allows access to a much richer set of angular observables (34 coefficients vs. ~10 at the LHC), offering a more nuanced disentanglement of Wilson coefficients $C_9^{(\prime)}$ and $C_{10}^{(\prime)}$.

2. Methodology

The study employs a "toy" angular analysis using simulation samples to estimate the sensitivity of the FCC-ee to the decay $\Lambda_b^0 \to \Lambda(\to p\pi^-)\mu^+\mu^-$.

• Simulation Framework:

  • Event Generation: $Z^0 \to q\bar{q}$ events generated with Pythia. Signal and background decays modeled with EvtGen.
  • Detector Model: The IDEA detector concept (silicon pixel vertex detector, drift chambers, crystal calorimeters, etc.) is simulated using the DELPHES fast simulation framework.
  • Dataset: Assumes a total integrated luminosity corresponding to $6 \times 10^{12}$$Z^0$ bosons.
  • Polarization: Longitudinal polarization set to $P_\parallel = -0.45$ (based on LEP measurements).

• Angular Distribution:

  • The analysis utilizes the full 5-dimensional angular phase space ($\vec{\Omega} = \cos\theta_\parallel, \cos\theta_\mu, \cos\theta_h, \phi_\mu, \phi_h$).
  • The differential decay rate is expanded into 34 angular coefficients ($K_1$ to $K_{34}$). The first 10 are independent of polarization; the remaining 24 depend on the polarization angle $\theta_\parallel$.

• Selection and Backgrounds:

  • Signal: $\Lambda_b^0 \to \Lambda(\to p\pi^-)\mu^+\mu^-$.
  • Main Background: $B^0 \to K_S^0(\to \pi^+\pi^-)\mu^+\mu^-$. Due to similar topology and higher $B^0$ production fraction, this is the dominant background (approx. 11% of signal yield without PID).
  • Selection Cuts:
    • Flight distance requirements ($FD(\Lambda_b^0) > 100 \times IP$; $FD(\Lambda) > 10$ mm).
    • Invariant mass windows: $m(p\pi\mu\mu)$ within 100 MeV/$c^2$ of $\Lambda_b^0$; $m(p\pi)$ within 20 MeV/$c^2$ of $\Lambda$.
    • Analysis Bins: Two bins in dimuon invariant mass ($q^2$) are used to avoid resonance regions:
      1. Between $\phi$ and $J/\psi$.
      2. Above $\psi(2S)$ up to the kinematic limit.
    • Low $q^2$ region is excluded due to photon pole instabilities and complex resonance backgrounds.

• Efficiency Modeling:

  • A 6-dimensional efficiency model $\varepsilon(\vec{\Phi})$ is constructed using a sum of Legendre polynomials and Fourier terms.
  • Coefficients are determined via the method of moments on phase-space simulation.
  • Validation is performed using a Boosted Decision Tree (BDT) to ensure the model is indistinguishable from the true simulation (AUC $\approx 0.5035$).

• Fitting Strategy:

  • A maximum likelihood fit is performed on 1000 toy datasets to extract the 34 angular coefficients.
  • Systematic uncertainties are assessed by varying particle identification ($p-\pi$ separation) and checking for mismodeling.
  • Wilson Coefficient Extraction: The extracted angular coefficients are fed into the flavio package to constrain the real and imaginary parts of $C_9, C_{10}, C_9', C_{10}'$.

3. Key Contributions

1. First FCC-ee Sensitivity Study for Polarized $\Lambda_b^0$: Provides the first quantitative estimate of angular observables for $\Lambda_b^0 \to \Lambda \mu^+ \mu^-$ at the FCC-ee, specifically leveraging the unique polarization available at the $Z$-pole.
2. Full Angular Analysis: Demonstrates the feasibility of measuring all 34 angular coefficients, whereas current LHCb analyses are limited to a subset due to lack of polarization.
3. Background Characterization: Detailed analysis of backgrounds, particularly the $B^0 \to K_S^0 \mu^+ \mu^-$ contamination, and the development of selection strategies to suppress combinatorial backgrounds from light quarks and heavy flavor resonances.
4. Efficiency Parametrization: Development and rigorous validation of a high-dimensional efficiency model necessary for unbiased angular fits in a polarized environment.

4. Results

• Yield: The projected signal yield is ~38,895 candidates. This exceeds the expected yield of LHCb Upgrade I but is slightly lower than the projected LHCb Upgrade II yield.
• Angular Observables:
  • Statistical uncertainties for individual observables are comparable to (or slightly larger than) LHCb Upgrade II projections due to the lower total yield, but the systematic uncertainties are well-controlled.
  • Most observables are statistically limited. Some coefficients ($K_4, K_5, K_6, K_{14}, K_{23}, K_{32}$) show larger systematic uncertainties even with perfect particle identification.
  • Particle misidentification ($p-\pi$ separation) becomes negligible for separations $> 2\sigma$.
• Wilson Coefficients ($C_9, C_{10}, C_9', C_{10}'$):
  • Unpolarized vs. Polarized: Fitting only the first 10 (unpolarized) observables yields sensitivity similar to current LHCb projections.
  • Significant Improvement: Including the 24 polarization-dependent observables leads to a significant improvement in the constraints on the Wilson coefficients.
  • Specific Sensitivity:
    • The real parts of $C_9$ and $C_{10}$ are constrained by a combination of all observables.
    • The primed coefficients ($C_9', C_{10}'$) are strongly constrained by specific observables ($K_3, K_4, K_8$), but the addition of polarization-dependent data further tightens these constraints.
  • The study confirms that the polarization information breaks degeneracies that exist in unpolarized analyses, allowing for a more precise determination of the underlying New Physics parameters.

5. Significance

• Complementarity to LHC: While the LHCb Upgrade II will have higher statistics, the FCC-ee offers a unique qualitative advantage through polarization. This allows for the measurement of observables that are inaccessible at hadron colliders, providing a crucial cross-check and a more complete picture of the $b \to s \ell^+ \ell^-$ transition.
• New Physics Reach: The enhanced sensitivity to Wilson coefficients, particularly the ability to disentangle $C_9$ from $C_{10}$ and their primed counterparts, significantly improves the potential to identify or rule out specific New Physics models (e.g., $Z'$ bosons, leptoquarks) that might explain current flavor anomalies.
• Feasibility Demonstration: The paper proves that the IDEA detector concept, combined with advanced analysis techniques (full angular fits, efficiency modeling), is capable of exploiting the unique physics potential of the FCC-ee $Z$-pole run for heavy flavor physics.

In conclusion, the paper establishes that the FCC-ee will provide a unique and powerful laboratory for flavor physics, where the polarization of $\Lambda_b^0$ baryons transforms the angular analysis from a limited probe into a comprehensive tool for constraining the Standard Model and searching for new physics.