Baryon-Meson Sum Rule for $b \to s \nu\bar\nu$

This paper establishes a model-independent sum rule linking the branching fractions of $\Lambda_b \to \Lambda \nu\bar\nu$ and $B \to K^{(\ast)} \nu\bar\nu$ decays, which remains exact despite numerous Wilson coefficients and shares numerical coefficients with the $b\to c$ semileptonic counterpart, thereby providing a powerful tool to discriminate new-physics scenarios involving left-handed neutrinos.

The Gist

Imagine the universe is a giant, high-stakes detective story. Physicists are the detectives, and subatomic particles are the suspects. Recently, they've noticed something strange happening in the "bottom quark" neighborhood (a heavy particle that decays into lighter ones). Specifically, when a bottom quark turns into a strange quark, it sometimes sneaks away with invisible neutrinos.

This paper, written by a team of physicists, introduces a brilliant new "rule of thumb" that connects two different types of crime scenes: Meson cases and Baryon cases.

Here is the breakdown in simple terms:

1. The Two Crime Scenes

In the world of particle physics, particles come in families.

• The Mesons (The "B" family): Think of these as lightweight, two-person teams (like a quark and an anti-quark). A famous example is the B-meson, which can decay into a K-meson (a lighter partner) plus invisible neutrinos.
• The Baryons (The "Lambda" family): Think of these as heavier, three-person teams (like three quarks stuck together). A famous example is the Lambda-b baryon, which can decay into a Lambda baryon plus invisible neutrinos.

For a long time, scientists studied these two families separately. They measured how often the Mesons did their thing and how often the Baryons did theirs, but they didn't have a direct way to link the two.

2. The "Magic Mirror" Rule

The authors of this paper discovered a Sum Rule. In everyday language, this is like finding a magic mirror that reflects one crime scene onto the other with perfect precision.

They found that if you know exactly how often the B-meson decays into a K-meson (specifically the vector version, $K^*$), you can instantly calculate exactly how often the Lambda-b baryon will decay into a Lambda baryon, without needing to know the messy details of the new physics causing it.

The Analogy:
Imagine you have two different types of vending machines:

• Machine A (Mesons) sells "K-candies."
• Machine B (Baryons) sells "Lambda-cookies."

Usually, you'd think the machines work independently. But this paper says: "Hey, if Machine A sells 100 K-candies, Machine B must sell exactly 74 Lambda-cookies, and if Machine A sells 25 K-candies, Machine B must sell 26 Lambda-cookies."

The ratio is fixed by the laws of physics (specifically, the math of how these particles spin and interact), not by the specific "new physics" causing the decay.

3. Why is this a Big Deal?

The "Invisible" Problem:
Neutrinos are ghosts. They pass through everything and leave no trace. When a particle decays into a neutrino, the detector sees "missing energy." It's like a magician pulling a rabbit out of a hat, but the rabbit is invisible. You know the rabbit is gone, but you can't see it.

The "Model-Independent" Superpower:
Usually, to predict what happens, scientists have to guess which new theory is correct (e.g., "Is it a new heavy particle? Is it a dark matter particle?"). This paper says: You don't need to guess.

Because of this Sum Rule, if the Belle II experiment (a giant particle detector in Japan) measures the Meson decay ($B \to K^*$), they can immediately predict what the Baryon decay ($\Lambda_b \to \Lambda$) should be, assuming only "left-handed" neutrinos are involved (which is the standard assumption).

4. The "Smoking Gun" for New Physics

This is where it gets exciting.

• Scenario A: The experiments measure the Meson decay, use the Sum Rule to predict the Baryon decay, and then measure the Baryon decay. They match.

  • Conclusion: Everything is working as expected. The "ghosts" are behaving normally.

• Scenario B: The experiments measure the Meson decay, predict the Baryon decay, but when they measure the Baryon, it doesn't match.

  • Conclusion: BINGO! The Sum Rule has been broken. This means our assumption was wrong. It implies the existence of "Right-handed" neutrinos (a type of ghost we haven't seen before), or some other exotic new physics that breaks the rules of the game.

5. A Fun Coincidence

The authors also noticed a funny mathematical coincidence. The numbers in this rule (0.26 and 0.74) are almost identical to a similar rule used for a different type of decay involving heavy quarks ($b \to c$). It's like finding that the recipe for a chocolate cake is mathematically identical to the recipe for a vanilla cake, even though they taste different. It suggests a deep, hidden symmetry in the universe's code.

Summary

This paper provides a universal translator between two different types of particle decays.

1. Measure the easier-to-see Meson decay.
2. Calculate the Baryon decay using the new "Sum Rule."
3. Compare with the actual Baryon measurement.

If they match, we are good. If they don't, we have found a crack in the Standard Model, likely pointing to new types of neutrinos or dark matter. It turns a difficult experimental challenge into a powerful, model-independent test for the future of physics.

Technical Summary

1. Problem Statement

The paper addresses the need for robust, model-independent probes of New Physics (NP) in flavor-changing neutral current (FCNC) transitions, specifically the $b \to s\nu\bar{\nu}$ process.

• Context: Rare decays like $B \to K^{(*)}\nu\bar{\nu}$ are theoretically clean (free from long-distance hadronic uncertainties) and sensitive to short-distance NP. Recent Belle II data shows a $2.7\sigma$ excess in $B^+ \to K^+\nu\bar{\nu}$, while $B \to K^*\nu\bar{\nu}$ remains unmeasured with only upper limits.
• Gap: While mesonic decays ($B \to K^{(*)}\nu\bar{\nu}$) are being actively studied, the corresponding baryonic decay $\Lambda_b \to \Lambda\nu\bar{\nu}$ offers complementary information due to its distinct hadronic structure but is experimentally more challenging to measure (requiring high-luminosity $e^+e^-$ colliders or LHCb with specific reconstruction).
• Challenge: The effective Hamiltonian for $b \to s\nu\bar{\nu}$ contains up to 18 independent Wilson Coefficients (WCs) (accounting for 3 neutrino generations and left/right-handed currents). Establishing a direct relationship between baryonic and mesonic observables in the presence of such a large parameter space is non-trivial.

2. Methodology

The authors employ a low-energy Effective Field Theory (EFT) framework to derive a sum rule connecting baryonic and mesonic branching fractions.

• Effective Hamiltonian: They define the Hamiltonian assuming decoupled right-handed neutrinos (or focusing on the left-handed sector), parameterized by 18 WCs ($C_{ij}^{L,R}$).
  $$H_{eff} \propto \sum_{i,j} \left[ (\delta_{ij} + C_{ij}^L) O_{ij}^L + C_{ij}^R O_{ij}^R \right]$$
• Signal Strengths: They define signal strengths ($\mu$) as the ratio of the measured branching fraction to the Standard Model (SM) prediction for:
  • $B \to K\nu\bar{\nu}$ (Pseudoscalar meson)
  • $B \to K^*\nu\bar{\nu}$ (Vector meson)
  • $\Lambda_b \to \Lambda\nu\bar{\nu}$ (Baryon)
  • Longitudinal polarization fraction $F_L(K^*)$.
• Form Factors: The calculations utilize refined form factors for $B \to K$, $B \to K^*$, and $\Lambda_b \to \Lambda$ transitions, parameterized using the Bourrely-Caprini-Lellouch (BCL) $z$-series expansion ($N=3$). Uncertainties are propagated using correlation matrices derived from lattice QCD and sum rules.
• Derivation Logic:
  1. Due to parity conservation in QCD, the axial-vector current contribution to $B \to K$ vanishes, simplifying the expression for $\mu(B \to K\nu\bar{\nu})$ to depend solely on the sum of squared coefficients $|C_{ij}^+|^2$ (where $C^\pm = \delta + C_L \pm C_R$).
  2. The expressions for $\mu(B \to K^*\nu\bar{\nu})$ and $\mu(\Lambda_b \to \Lambda\nu\bar{\nu})$ are linear combinations of $|C_{ij}^+|^2$ and $|C_{ij}^-|^2$.
  3. Since these observables depend on only two independent vector space components ($\sum |C^+|^2$ and $\sum |C^-|^2$), any three observables must satisfy a linear relation (a sum rule), regardless of the specific values of the 18 WCs.

3. Key Contributions

• Derivation of a Robust Sum Rule: The paper derives an exact linear relation between the signal strengths of the baryonic and mesonic modes:
  $$\mu(\Lambda_b \to \Lambda\nu\bar{\nu}) = (0.26 \pm 0.12) \mu(B \to K\nu\bar{\nu}) + (0.74 \mp 0.12) \mu(B \to K^*\nu\bar{\nu})$$
  • Key Insight: This relation holds exactly even with 18 independent Wilson coefficients, provided right-handed neutrinos are decoupled (or the analysis is restricted to the left-handed sector). The coefficients are determined purely by kinematics and form factors.
• Numerical Identity with $b \to c$ Transitions: The authors highlight a remarkable finding: the coefficients $(0.26, 0.74)$ are numerically identical to those found in the $b \to c\tau\bar{\nu}$ semileptonic sum rule (which are $1/4$ and $3/4$ in the heavy quark limit). This suggests a deeper mathematical structure where the observables span a 2D vector space, making the sum rule independent of neutrino flavor violation.
• Additional Constraints: They derive a second sum rule linking the $K^*$ polarization fraction $F_L(K^*)$ to the branching fractions, providing a consistency check for experimental data.
• Model-Independent Prediction: The work demonstrates that once $B \to K^*\nu\bar{\nu}$ is measured, the branching fraction of $\Lambda_b \to \Lambda\nu\bar{\nu}$ can be predicted without assuming a specific New Physics model.

4. Results

• Theoretical Predictions: The updated SM predictions for the branching fractions are:
  • $\mathcal{B}(B^0 \to K^0\nu\bar{\nu})_{SM} \approx 4.37 \times 10^{-6}$
  • $\mathcal{B}(B^0 \to K^{*0}\nu\bar{\nu})_{SM} \approx 7.91 \times 10^{-6}$
  • $\mathcal{B}(\Lambda_b \to \Lambda\nu\bar{\nu})_{SM} \approx 8.01 \times 10^{-6}$
• Current Data Analysis: Using the latest Belle II measurement of $B^+ \to K^+\nu\bar{\nu}$ (which shows a $2.7\sigma$ excess), the authors plot the allowed regions for $\Lambda_b \to \Lambda\nu\bar{\nu}$ and $F_L(K^*)$.
  • The current data implies an upper bound $\mu(\Lambda_b \to \Lambda\nu\bar{\nu}) \leq 2.6$.
  • The sum rule allows for a "cross-check": if future measurements of $B \to K^*\nu\bar{\nu}$ and $\Lambda_b \to \Lambda\nu\bar{\nu}$ violate this linear relation, it would definitively signal physics beyond the standard left-handed neutrino framework (e.g., light right-handed neutrinos, lepton number violation, or light dark particles).
• Inclusive vs. Exclusive: The paper notes that the baryonic branching fraction is approximately equal to the inclusive $B \to X_s\nu\bar{\nu}$ rate, consistent with previous findings in $b \to c$ transitions.

5. Significance

• Discriminating Power: The sum rule serves as a powerful tool to discriminate between New Physics scenarios. A violation of the derived relation would immediately rule out models involving only left-handed neutrino interactions.
• Experimental Guidance: It provides a clear target for future experiments (Belle II, LHCb, FCC-ee, CEPC). Specifically, it suggests that measuring $B \to K^*\nu\bar{\nu}$ is sufficient to predict the $\Lambda_b \to \Lambda\nu\bar{\nu}$ rate, guiding the search strategy for baryonic modes which are harder to detect.
• Theoretical Robustness: The result is mathematically rigorous and independent of the specific values of the Wilson coefficients, relying only on the dimensionality of the operator space and QCD parity conservation. This makes it a "smoking gun" test for the nature of neutrino couplings in FCNCs.
• Broader Implications: The numerical coincidence with the $b \to c$ sum rule suggests that heavy quark symmetry arguments might extend to heavy-to-light transitions in ways previously unappreciated, potentially unifying the understanding of semileptonic decays across different flavor sectors.

In summary, this paper establishes a fundamental, model-independent link between baryonic and mesonic rare decays, turning the measurement of one into a precise prediction for the other, thereby offering a stringent test for the existence of right-handed neutrinos and other exotic new physics.