$\Lambda_b\to\Lambda^{(*)}\nu{\bar\nu}$ and $b\to s$ $B$ decays

This paper analyzes the baryonic $b\to s$ transition $\Lambda_b\to\Lambda^{(*)}\nu{\bar\nu}$ by combining constraints from mesonic $B$ decays to predict enhanced branching ratios and a new physics scale between 2.04 and 11.76 TeV, while also establishing a sum rule linking these baryonic and mesonic decay modes.

The Gist

Imagine the universe as a giant, complex machine where tiny particles called quarks are the gears and springs. Sometimes, these gears are supposed to stay in their own lane, but occasionally, they try to sneak into a different lane. This is called a "flavor-changing neutral current" (FCNC). In the Standard Model (the rulebook of physics), this is like a gear trying to change lanes only by taking a very long, winding detour through a tunnel (a "loop"). It's rare and slow.

However, scientists have noticed that some of these gears are changing lanes too fast or in unexpected ways. This suggests there might be a secret shortcut or a hidden mechanic (New Physics) helping them.

This paper is like a detective story where the author, Jong-Phil Lee, tries to solve this mystery by looking at two different crime scenes: Mesons (particles made of a quark and an anti-quark, like a married couple) and Baryons (particles made of three quarks, like a family of three).

Here is the breakdown of the investigation in simple terms:

1. The Mystery: The "B" Particle Anomalies

Scientists have been watching a specific particle, the B-meson (specifically the $B^+$ and $B^0$), decay into other particles.

• The Clue: When these B-mesons decay into a K-meson and a pair of muons (heavy electrons), the numbers don't quite match the rulebook. It's like a car odometer that says you drove 100 miles, but the map says you should have driven 120.
• The New Clue: Recently, scientists also looked at B-mesons decaying into neutrinos (ghostly particles that barely interact with anything). The data here is a bit messy, but it hints that something is boosting the rate of these decays.

2. The Detective's Strategy: The "Mirror" Effect

The author realizes that if there is a hidden mechanic helping the B-mesons (the "married couple"), that same mechanic should also be helping the $\Lambda_b$ baryon (the "family of three").

• The Analogy: Imagine you are trying to figure out how a secret elevator works in a building. You can't see the elevator shaft directly, but you see people taking the stairs (Mesons) faster than usual. You suspect the elevator is also helping the people in the penthouse (Baryons).
• The Goal: The paper uses the data from the "stairs" (Mesons) to predict what should happen in the "penthouse" (Baryons). If the prediction matches reality, it proves the elevator exists. If it doesn't, the theory is wrong.

3. The Investigation: Cracking the Code

The author uses a mathematical "fingerprint" called Wilson Coefficients. Think of these as the settings on a video game controller that determine how strong the "New Physics" force is.

• The author combines all the messy data from the B-mesons (the muon decays, the neutrino decays, and the angular spin of the particles).
• They run a massive simulation (a $\chi^2$ fit) to find the "best settings" that explain all the weird data at once.

4. The Big Discoveries

A. The "Ghost" Scale ($M_{NP}$)
The team calculated the size of the "hidden mechanic." They found that if this new physics exists, it operates at a scale of about 2 to 12 TeV (Tera-electronvolts).

• Analogy: It's like finding out the secret elevator is located in a basement that is 2 to 12 stories deep. This is a very specific range. It's deep enough that our current "shovels" (the Large Hadron Collider) might struggle to dig it out, but future, bigger shovels (like the High-Luminosity LHC or FCC-ee) might just reach it.

B. The "Magic" Prediction
The most exciting part is the prediction for the $\Lambda_b$ baryon.

• The Standard Model says the $\Lambda_b$ should decay into a $\Lambda$ particle and neutrinos at a certain rate (let's call it 100%).
• The Prediction: The author predicts that because of the hidden elevator, this rate will actually be 207% (more than double!) for the $\Lambda$ particle, and about 107% for the excited $\Lambda^*$ particle.
• Why it matters: This is a huge signal. If future experiments see the $\Lambda_b$ decaying twice as often as expected, it's a smoking gun for New Physics.

C. The "Sum Rule" (The Secret Handshake)
The paper found a beautiful mathematical relationship, a "Sum Rule."

• The Analogy: Imagine you have a recipe for a cake (Meson decay) and a recipe for a pie (Baryon decay). The author found that if you mix 30% of the cake recipe and 70% of the pie recipe, you get a perfect "New Physics" smoothie.
• This rule connects the behavior of the "married couple" (Mesons) and the "family" (Baryons). It's a very rare and powerful tool because it allows scientists to check their work in two different ways. If the Meson data says "A," the Baryon data must say "B" to satisfy the rule.

5. The Conclusion: What's Next?

The paper concludes that:

1. New Physics is likely: The data strongly suggests something beyond the Standard Model is happening.
2. The "Unparticle" Hint: The math suggests the hidden mechanic might not be a standard heavy particle, but something more exotic, like "unparticles" (a theoretical concept where particles don't have a fixed mass).
3. The Challenge: We need to build better particle colliders (like the FCC-ee) to produce enough $\Lambda_b$ baryons to see if they really decay at this super-fast rate.

In a nutshell:
The author looked at the weird behavior of one type of particle (Mesons) to predict the behavior of its cousin (Baryons). They found that if the weirdness is real, the cousin should be acting twice as crazy as we thought. This gives physicists a clear target to aim for in future experiments to finally find the "hidden elevator" of the universe.

Technical Summary

1. Problem Statement

The paper addresses the tension between Standard Model (SM) predictions and experimental data in flavor-changing neutral current (FCNC) processes involving the $b \to s$ transition. While recent LHCb data has resolved the anomaly in the lepton flavor universality ratios $R(K^{(*)})$, other observables such as the branching ratios of $B^+ \to K^+ \mu^+ \mu^-$ and the angular observable $P'_5$ in $B^+ \to K^{*+} \mu^+ \mu^-$ still show deviations from SM expectations.

Furthermore, recent measurements by Belle II indicate a significant excess in the branching ratio of $B^+ \to K^+ \nu \bar{\nu}$ compared to SM predictions, while $B^0 \to K^{*0} \nu \bar{\nu}$ remains consistent with the SM. The central question is whether these tensions extend to the baryonic sector, specifically the decays $\Lambda_b \to \Lambda^{(*)} \nu \bar{\nu}$. Unlike mesonic decays, baryonic decays offer different spin structures ($1/2 \to 1/2$ or $3/2$), distinct Lorentz structures, and the possibility of $\Lambda_b$ polarization, providing complementary probes for New Physics (NP).

2. Methodology

The author employs a model-independent phenomenological approach to analyze the interplay between mesonic and baryonic $b \to s$ transitions.

• Effective Hamiltonian & Wilson Coefficients:
  The analysis utilizes the effective Hamiltonian for $b \to s \ell^+ \ell^-$ and $b \to s \nu \bar{\nu}$ transitions. The relevant Wilson coefficients ($C_9, C_{10}$ for charged leptons and $C_{L,R}^{\nu}$ for neutrinos) are parametrized to include NP contributions:
  $$C_j = C_j^{SM} + C_j^{NP}, \quad \text{where} \quad C_j^{NP} \equiv N A_j \left(\frac{v}{M_{NP}}\right)^\alpha$$
  Here, $v$ is the SM vacuum expectation value, $M_{NP}$ is the NP scale, $A_j$ are coupling coefficients, and $\alpha$ is a free parameter.

  • $\alpha = 2$ corresponds to ordinary heavy mediators (e.g., $Z'$, Leptoquarks).
  • $\alpha \neq 2$ allows for "unparticle-like" degrees of freedom or non-standard scaling.

• Global $\chi^2$ Fit:
  A global fit is performed using experimental data and SM predictions for the following observables:

  • $R(K)$ and $R(K^*)$ (lepton universality ratios).
  • $Br(B_s \to \mu^+ \mu^-)$.
  • $Br(B^+ \to K^+ \mu^+ \mu^-)$ across various $q^2$ bins.
  • $P'_5(B^+ \to K^{*+} \mu^+ \mu^-)$.
  • $Br(B^+ \to K^+ \nu \bar{\nu})$ (Belle II data) and the upper limit on $Br(B^0 \to K^{*0} \nu \bar{\nu})$.

• Form Factors:
  The analysis incorporates form factors for $B \to K^{(*)}$ (FLAG and light-cone sum rules) and $\Lambda_b \to \Lambda^{(*)}$ (lattice QCD results), acknowledging that baryonic form factors are generally more reliable due to the stability of the $\Lambda$ baryon under strong interactions.

3. Key Contributions

• Complementarity of Baryonic Modes: The paper demonstrates that $\Lambda_b \to \Lambda^{(*)} \nu \bar{\nu}$ decays are highly sensitive to NP, particularly because they probe different combinations of Wilson coefficients compared to mesonic modes.
• Unified Parametrization: By using a unified parametrization for Wilson coefficients involving both charged leptons and neutrinos, the study links the constraints from the well-measured mesonic sector to the currently unobserved baryonic sector.
• Discovery of a Sum Rule: The author identifies a linear relationship (sum rule) connecting the ratios of branching ratios in the baryonic and mesonic sectors for $b \to s \nu \bar{\nu}$ transitions, analogous to the sum rules found in $b \to c$ semi-leptonic decays.

4. Key Results

• Best-Fit Parameters:

  • The global fit yields a best-fit value of $\alpha \approx 1.21$, suggesting that the data favors contributions that are not purely from ordinary heavy mediators ($\alpha=2$) but may involve unparticle-like degrees of freedom or other non-standard scaling.
  • The NP scale for ordinary heavy mediators ($\alpha=2$) is constrained to $2.04 \text{ TeV} \leq M_{NP} \leq 11.76 \text{ TeV}$ at the $1\sigma$ level. This window is notably narrow compared to some Leptoquark mass bounds.
  • The fit prefers negative values for $C_9^{NP}$ and specific non-zero values for $C_{L,R}^{\nu, NP}$ to explain the $B^+ \to K^+ \nu \bar{\nu}$ excess.

• Predictions for Baryonic Decays:

  • $\Lambda_b \to \Lambda \nu \bar{\nu}$: The predicted branching ratio is significantly enhanced compared to the SM.
    $$R(\Lambda)_{\nu\nu} \equiv \frac{Br(\Lambda_b \to \Lambda \nu \bar{\nu})}{Br(\Lambda_b \to \Lambda \nu \bar{\nu})_{SM}} \approx \mathbf{2.07}$$
    This implies the branching ratio is roughly 2.07 times the SM estimation.
  • $\Lambda_b \to \Lambda^* \nu \bar{\nu}$: The enhancement is much smaller.
    $$R(\Lambda^*)_{\nu\nu} \approx \mathbf{1.07}$$
    This decay mode remains close to the SM prediction.

• The Sum Rule:
  The analysis reveals a robust linear relation between the mesonic and baryonic ratios:
  $$R(\Lambda)_{\nu\nu} \approx 0.30 \, R(K)_{\nu\nu} + 0.69 \, R(K^*)_{\nu\nu}$$
  where $R(K)_{\nu\nu}$ and $R(K^*)_{\nu\nu}$ are the ratios of experimental to SM branching ratios for $B \to K^{(*)} \nu \bar{\nu}$. This relation holds despite the different underlying dynamics (FCNC vs. tree-level charged current in $b \to c$) and suggests a deep structural connection between the two sectors.

5. Significance

• Verification of New Physics: The prediction that $Br(\Lambda_b \to \Lambda \nu \bar{\nu})$ is roughly double the SM value provides a clear, testable signature for future colliders. If observed, it would strongly corroborate the NP interpretation of the $B^+ \to K^+ \nu \bar{\nu}$ anomaly.
• Experimental Prospects: While the High-Luminosity LHC (HL-LHC) may struggle to probe the upper end of the predicted $M_{NP}$ window ($\sim 11$ TeV) for vector Leptoquarks or $Z'$, future facilities like FCC-ee (capable of producing $130 \times 10^9$ $\Lambda_b$ particles) are well-positioned to verify these predictions.
• Theoretical Insight: The discovery of the sum rule (Eq. 56) offers a new tool for cross-checking experimental data between mesonic and baryonic sectors. Even if specific NP models change, the robustness of this relation could serve as a model-independent consistency check for future measurements.
• Robustness: The results remain stable even when excluding specific $q^2$ bins (like $[0.1, 0.98]$ GeV$^2$) that showed large local fluctuations, indicating the findings are not driven by a single data point but by the global trend of the dataset.

In conclusion, this paper establishes a strong theoretical link between the observed anomalies in mesonic $b \to s$ decays and specific, testable predictions for baryonic decays, highlighting $\Lambda_b \to \Lambda \nu \bar{\nu}$ as a prime candidate for discovering physics beyond the Standard Model.