Probing Signatures of Right-Handed Neutrinos via $b \to s \nu \bar\nu$ Decays

Motivated by the recent Belle II measurement of $B^+\to K^+\nu\bar{\nu}$, this paper employs a model-independent effective field theory analysis to constrain right-handed neutrino interactions and predicts correlated enhancements in various $b\to s\nu\bar{\nu}$ branching fractions and polarization observables as testable signatures for future experiments.

The Gist

The Big Picture: A Mystery in the Particle World

Imagine the Standard Model as a giant, incredibly detailed instruction manual for how the universe's smallest building blocks (particles) are supposed to behave. For decades, this manual has been perfect. But recently, a new experiment called Belle II found a "glitch."

They watched a specific type of particle decay (a heavy "B-meson" turning into a lighter "K-meson" and some invisible particles). The manual predicted a certain number of these events should happen. Instead, they saw significantly more than expected. It's like a baker predicting 10 cookies will be eaten, but 27 disappear from the plate.

This paper asks: *What if the "invisible particles" aren't just the standard neutrinos we know, but something new called Right-Handed Neutrinos (RHNs)?*

The Detective Work: The "Invisible" Clue

In this experiment, the "missing" particles are neutrinos. Neutrinos are like ghosts; they pass through walls and detectors without leaving a trace. We only know they were there because energy seems to vanish from the scene.

The authors propose that if these ghosts are actually Right-Handed Neutrinos (a hypothetical type of particle that doesn't interact with the weak force in the same way as normal ones), they could explain why so many more events are happening than the "Standard Model" manual predicts.

The Investigation: Testing the Theory

The researchers didn't just guess; they built a mathematical "filter" (called an Effective Field Theory) to see if this Right-Handed Neutrino idea fits the data. They treated the new physics like a set of knobs (called Wilson Coefficients) that could be turned up or down to adjust the strength of the interaction.

They used two main pieces of evidence to tune these knobs:

1. The "Smoking Gun": The Belle II measurement of the $B \to K$ decay (the one with the excess).
2. The "Speed Limit": An older, stricter limit on a similar decay called $B \to K^*$ (where the K-meson is in a slightly excited state).

The Analogy: Imagine you are trying to find the right combination for a safe.

• The first clue ($B \to K$) tells you the combination is somewhere in a long, diagonal hallway.
• The second clue ($B \to K^*$) tells you the combination is in a different, oval-shaped room.
• The only place where the hallway and the room overlap are two small, isolated islands.

The paper found that the "Standard Model" (where the knobs are set to zero) is not on these islands. This means if the new data is correct, the universe must have these Right-Handed Neutrinos.

The Predictions: What Else Should We See?

If this theory is true, it doesn't just affect one type of decay. It's like turning on a faucet; water flows into every connected pipe. The authors predicted that if Right-Handed Neutrinos exist, we should see similar "floods" (enhancements) in other rare decays:

• $B_s \to \phi \nu \bar{\nu}$: A different heavy particle decaying.
• $\Lambda_b \to \Lambda \nu \bar{\nu}$: A heavy particle made of three quarks (a baryon) decaying.

They also looked at a specific measurement called the Longitudinal Polarization Fraction ($F_L$).

• The Analogy: Imagine a spinning top. The branching fraction (how often it happens) tells you how many tops are spinning. The polarization fraction tells you which way they are spinning.
• The paper found that while the "how many" might look similar for different theories, the "which way" (polarization) would change drastically if Right-Handed Neutrinos are involved. This acts as a unique fingerprint to confirm the theory.

The Conclusion: What's Next?

The paper concludes that:

1. The "Islands" Exist: There are specific settings for Right-Handed Neutrinos that explain the Belle II anomaly while respecting the limits set by the $B \to K^*$ data.
2. The Standard Model is Out: The "zero" setting (no new physics) is ruled out by the combined data.
3. Future Proof: To confirm this, scientists need to measure the other decays ($B_s \to \phi$ and $\Lambda_b \to \Lambda$) and the "spin direction" ($F_L$) at future experiments like Belle II, LHCb, and the proposed FCC-ee.

In short, the paper says: "The universe is acting weird in a way the old manual can't explain. If we assume these new 'Right-Handed' ghosts exist, the math works perfectly, and here is exactly what we need to look for next to prove it."

Technical Summary

Technical Summary: Probing Signatures of Right-Handed Neutrinos via $b \to s\nu\bar{\nu}$ Decays

Problem Statement
The Standard Model (SM) prediction for the rare flavor-changing neutral current (FCNC) decay $B^+ \to K^+ \nu\bar{\nu}$ has recently been challenged by the Belle II Collaboration, which reported a branching fraction approximately $2.7\sigma$ above the SM expectation. This discrepancy motivates a search for New Physics (NP) in $b \to s\nu\bar{\nu}$ transitions. While charged-lepton modes ($b \to s\ell^+\ell^-$) have been extensively studied, the neutrino modes offer a theoretically cleaner framework due to the absence of long-distance electromagnetic effects and reduced hadronic uncertainties. However, the presence of neutrinos in the final state leads to missing-energy signatures, making these decays sensitive probes for light, weakly interacting particles, such as Right-Handed Neutrinos (RHNs). The paper addresses the need to constrain NP interactions involving RHNs using the latest experimental data while predicting signatures for related decay modes.

Methodology
The authors perform a model-independent analysis within the Low-Energy Effective Theory (LEFT) framework.

• Effective Hamiltonian: The study extends the standard SM operator basis to include dimension-six vector operators involving RHNs. The effective Hamiltonian includes the SM left-handed operator ($O_L$) and two new NP operators: $O_{LR}^V = (\bar{s}\gamma^\mu P_L b)(\bar{\nu}\gamma^\mu P_R \nu)$ and $O_{RR}^V = (\bar{s}\gamma^\mu P_R b)(\bar{\nu}\gamma^\mu P_R \nu)$.
• Assumptions: The analysis assumes Lepton Flavor Universality (LFU), where the Wilson Coefficients (WCs) $C_{LR}^V$ and $C_{RR}^V$ are identical for all neutrino generations ($e, \mu, \tau$).
• Constraints: The allowed parameter space for the WCs is determined by minimizing a $\chi^2$ function constructed from:
  1. The Belle II measurement of $\mathcal{B}(B^+ \to K^+ \nu\bar{\nu})$ (combining hadronic-tagging and inclusive reconstruction).
  2. The Belle upper limit on $\mathcal{B}(B \to K^* \nu\bar{\nu})$.
• Observables: The study calculates and correlates the branching fractions of $B \to K\nu\bar{\nu}$, $B \to K^*\nu\bar{\nu}$, $B_s \to \phi\nu\bar{\nu}$, and $\Lambda_b \to \Lambda\nu\bar{\nu}$. Additionally, the longitudinal polarization fraction $F_L^{K^*}$ in $B \to K^*\nu\bar{\nu}$ is analyzed to probe the helicity structure of the interaction.

Key Results

• Parameter Space Constraints: The combined constraints from $B \to K\nu\bar{\nu}$ and $B \to K^*\nu\bar{\nu}$ exclude the SM point ($C_{LR}^V = C_{RR}^V = 0$) at the $1\sigma$ level. The analysis identifies two distinct "allowed islands" in the $(C_{LR}^V, C_{RR}^V)$ plane where RHN contributions are consistent with current data.
• Branching Fraction Correlations: The RHN scenario predicts sizeable enhancements in the branching fractions of all studied modes ($B \to K\nu\bar{\nu}$, $B \to K^*\nu\bar{\nu}$, $B_s \to \phi\nu\bar{\nu}$, and $\Lambda_b \to \Lambda\nu\bar{\nu}$). A strong positive correlation is observed between $\mathcal{B}(B^+ \to K^+ \nu\bar{\nu})$ and the other modes; any enhancement in the $K$ mode driven by RHNs necessitates corresponding enhancements in the $K^*$, $\phi$, and $\Lambda$ modes.
• Role of $F_L^{K^*}$: The longitudinal polarization fraction $F_L^{K^*}$ provides complementary sensitivity. In single-coefficient scenarios, $F_L^{K^*}$ remains close to the SM value. However, in the two-coefficient scenario, $F_L^{K^*}$ exhibits significant deviations (ranging from $\sim 0.3$ to $0.6$ within the experimental $1\sigma$ interval of the $K$ mode). This observable is sensitive to the relative sign and magnitude of $C_{LR}^V$ and $C_{RR}^V$, offering a way to distinguish operator structures that branching fractions alone cannot.
• Impact of $K^*$ Limits: The current upper limit on $\mathcal{B}(B \to K^* \nu\bar{\nu})$ is found to be highly restrictive, eliminating the bulk of the parameter space that would otherwise be allowed by the $K$ mode measurement alone.

Significance and Claims
The paper claims that the observed excess in $B^+ \to K^+ \nu\bar{\nu}$ can be accommodated by RHN interactions without violating existing constraints, provided the Wilson coefficients lie within the identified allowed islands. The primary significance of this work lies in:

1. Testable Signatures: It provides specific, correlated predictions for unmeasured or loosely constrained modes ($B_s \to \phi\nu\bar{\nu}$ and $\Lambda_b \to \Lambda\nu\bar{\nu}$) and the polarization fraction $F_L^{K^*}$.
2. Discriminating Power: It highlights that $F_L^{K^*}$ is crucial for disentangling the underlying RHN interaction structure, as it responds differently to the interference between left- and right-handed currents compared to total decay rates.
3. Experimental Outlook: The results are presented as testable signatures for current and future experiments, specifically Belle II, LHCb, and the proposed FCC-ee, which can further narrow the allowed parameter space or confirm the RHN hypothesis through simultaneous observation of these correlated enhancements.