The $B^+ \to K^+ \nu \bar \nu$ decay as a search for the QCD axion

This paper introduces a model-independent framework using public Belle II data to reinterpret the $B^+ \to K^+ \nu \bar{\nu}$ decay channel, establishing it as a dual probe for new physics that yields the strongest current limit on the $B^+ \to K^+ a$ branching fraction and constrains axion-quark couplings.

The Gist

Imagine you are a detective trying to solve a mystery at a high-speed train station. The station is the Belle II experiment, a massive particle collider in Japan where tiny particles called "B mesons" are created and then immediately break apart.

Usually, when a B meson breaks apart, it leaves behind a clear trail of evidence (other particles) that scientists can track. But sometimes, it seems to vanish into thin air, leaving behind only a single visible particle (a Kaon) and a "ghost" that carries away energy but leaves no trace.

This paper is about a new way to hunt for these ghosts, specifically a type of ghost called the QCD axion.

The Mystery: The "Missing" Energy

In the standard story of physics (the Standard Model), when a B meson decays into a Kaon and two invisible neutrinos ($B^+ \to K^+ \nu\bar{\nu}$), the energy loss is spread out smoothly. It's like a foggy day where you can't see the exact shape of the missing energy.

But if a QCD axion exists, the story changes. The axion is a hypothetical, ultra-light particle that solves a major puzzle in physics (why the strong nuclear force doesn't violate a symmetry called CP). If a B meson decays into a Kaon and an axion ($B^+ \to K^+ a$), the axion is a single, distinct object. This means the energy loss isn't a fog; it's a sharp, specific "thud" at a precise value.

The Challenge: The Blurry Camera

The problem is that the Belle II experiment has two ways of watching these events:

1. The "Classic" Way (Hadronic Tagging): This is like having a high-definition camera. It reconstructs the entire event perfectly, so scientists can see exactly where the energy went.
2. The "Inclusive" Way (Inclusive Tagging): This is the method that collects the most data (like a wide-angle lens that sees more cars but with a slightly blurrier focus). In this method, scientists can't see the exact energy of the invisible particles directly. Instead, they have to guess it based on the visible Kaon.

For years, to interpret the "blurry" data from the Inclusive method, scientists needed the experiment's internal "simulation software" (like a secret map) to understand how the blur works. Without this secret map, they couldn't use the massive amount of data from the Inclusive method to hunt for axions.

The Breakthrough: Doing the Math Instead of Guessing

The authors of this paper realized they didn't need the secret map. They used pure geometry and physics (kinematics) to draw their own map.

The Analogy: Imagine you are on a spinning merry-go-round (the B meson) throwing a ball (the Kaon) while the whole ride is moving down a track.

• If you know how fast the ride is moving and the angle you threw the ball, you can calculate exactly where the ball should land relative to the track.
• The "blur" in the data comes from not knowing the exact angle you threw the ball.
• The authors realized they could mathematically calculate every possible angle and how it would smear out the data. They created a formula that translates the "blurry" measurement into a clear prediction, without needing any private computer simulations.

The Results: Catching the Ghost

Using this new mathematical "lens" on the public data from Belle II, the team looked for the sharp "thud" of an axion.

1. They found nothing: No axion was detected.
2. They set a new record: Because they could use the massive "Inclusive" dataset (which is 9 times more sensitive than previous methods), they set the strictest limit ever on how likely it is for this decay to happen.
   • They improved the previous best limit by a factor of nine.
   • This means if axions exist, they must be even more "ghostly" (harder to catch) than we thought.

The "Dual Probe" Superpower

The paper highlights a clever side effect of their method. Usually, if you are looking for a new particle (like an axion), you have to assume you know exactly how the "standard" background noise (neutrinos) behaves. If your assumption about the background is wrong, you might think you found a new particle when you didn't.

The authors showed that their method allows them to test two things at once, independently:

1. Is the background noise behaving strangely? (Is there new physics in the neutrino interaction?)
2. Is there a sharp spike from a new particle? (Is there an axion?)

They proved that these two tests don't mess each other up. It's like checking if a room is empty of people while simultaneously checking if the lights are flickering. You can do both at the same time with high confidence.

Summary

In short, this paper teaches us how to look at a blurry photo of a particle collision and mathematically sharpen it without needing the photographer's secret notes. By doing this, they used the largest available dataset to hunt for the QCD axion. They didn't find it, but they pushed the boundaries of where it could be hiding, making the search for this elusive particle much more precise. They also showed that this technique can be used as a "dual probe" to test for new physics in two different ways simultaneously.

Technical Summary

Technical Summary: The $B^+ \to K^+ \nu\bar{\nu}$ Decay as a Search for the QCD Axion

Problem Statement
The QCD axion and Axion-Like Particles (ALPs) are hypothetical light, pseudoscalar states proposed to solve the strong-CP problem and serve as dark matter candidates. While generic ALPs are often searched for in collider experiments, the QCD axion presents a specific challenge: its mass ($m_a$) and decay constant ($f_a$) are inversely related, implying that for the theoretically motivated QCD axion, the mass is extremely small ($m_a \simeq 0$) and couplings are weak. Consequently, the axion would manifest in heavy meson decays as missing energy and momentum.

The Belle II experiment has measured the rare decay $B^+ \to K^+ \nu\bar{\nu}$ using two distinct tagging strategies: the Hadronic Tagging Analysis (HTA) and the Inclusive Tagging Analysis (ITA). While the ITA provides the dominant statistical sensitivity, it does not reconstruct the true di-neutrino invariant mass squared ($q^2$) directly. Instead, it utilizes a proxy variable, $q^2_{\text{rec}}$, derived from the kinematics of the tag side. A critical barrier to reinterpreting this public data for axion searches has been the lack of a model-independent mapping between the true $q^2$ and the reconstructed $q^2_{\text{rec}}$, which was previously accessible only through Belle II's internal simulations.

Methodology
The authors introduce a model-independent framework to reinterpret Belle II public data for the search of $B^+ \to K^+ a$ (where $a$ denotes the axion or ALP). The core of the methodology involves three technical components:

1. Analytic Kinematic Reconstruction: The paper demonstrates that the mapping between the true variable $q^2$ and the proxy $q^2_{\text{rec}}$ can be derived analytically from kinematics alone, without relying on non-public experimental simulations. By defining the relationship between the $B\bar{B}$ rest frame and the $B$ rest frame, the authors derive an explicit equation (Eq. 4) linking $q^2_{\text{rec}}$ to $q^2$ and the unmeasured polar angle $\theta^{**}$. This allows the construction of a "smearing function" $f_{q^2_{\text{rec}}}(q^2)$ that describes how a monochromatic axion signal (fixed $q^2 = m_a^2$) is distributed in the $q^2_{\text{rec}}$ spectrum.
2. Dual-Probe Likelihood Analysis: The authors construct a likelihood ratio test statistic using public data from the ITA and HTA analyses. The analysis simultaneously fits for two parameters:
   • The branching fraction of the exotic two-body decay, $\mathcal{B}(B^+ \to K^+ a)$.
   • A signal-strength parameter $\mu_{\nu\bar{\nu}}$ for the Standard Model (SM) $B^+ \to K^+ \nu\bar{\nu}$ process.
     This separation ensures that potential short-distance new physics affecting the $b \to s \nu\bar{\nu}$ amplitude is not artificially absorbed into the axion signal limit.
3. Systematic Uncertainty Treatment: The analysis incorporates selection efficiencies and background normalizations as nuisance parameters. The authors validate their procedure by reproducing the Belle II collaboration's profile likelihood for the SM signal strength, confirming that their treatment of background uncertainties captures the leading systematic effects.

Key Contributions

• Analytic Mapping: The derivation of the $q^2 \to q^2_{\text{rec}}$ mapping solely from kinematic principles, enabling the use of the high-statistics ITA dataset for axion searches without internal simulation data.
• Model-Independent Reinterpretation: A general strategy for reinterpreting collider rare-decay data for light invisible states, applicable to both the QCD axion and generic ALPs.
• Dual Probe Establishment: The demonstration that the $B^+ \to K^+ \nu\bar{\nu}$ channel acts as a dual probe, simultaneously constraining short-distance new physics (via $\mu_{\nu\bar{\nu}}$) and light invisible states (via $\mathcal{B}(B^+ \to K^+ a)$) with the two effects remaining statistically independent to an excellent approximation.

Results
Applying this framework to Belle II data yields the following results:

• Branching Fraction Limits: The analysis sets the strongest existing bounds on the branching fraction for $B^+ \to K^+ a$. For a massless axion ($m_a \simeq 0$), the combined limit is $\mathcal{B}(B^+ \to K^+ a) < 1.4 \times 10^{-6}$ (assuming the SM value for $\nu\bar{\nu}$) or $< 1.2 \times 10^{-6}$ (allowing $\mu_{\nu\bar{\nu}}$ to float). This improves upon existing limits by approximately a factor of nine.
• Coupling Constraints: These branching fraction limits translate into lower bounds on the coupling-rescaled Peccei-Quinn scale, $|(F_V)_{sb}| \equiv 2f_a / |(k_V)_{sb}|$. The combined analysis yields $|(F_V)_{sb}| > 7.6 \times 10^8 \text{ GeV}$ (fixed $\mu$) or $> 8.0 \times 10^8 \text{ GeV}$ (floating $\mu$). This represents an improvement of one order of magnitude over previous bounds.
• Mass Independence: For the QCD axion, the limits act as essentially vertical constraints in the mass-coupling parameter space. Because the axion mass is negligible on collider scales, the constraints on the flavor-changing coupling $(k_V)_{sb}$ are independent of $m_a$.
• Robustness: The limits on the axion branching fraction are shown to be largely insensitive to assumptions regarding the di-neutrino signal strength $\mu_{\nu\bar{\nu}}$, even in the presence of large beyond-SM contributions to the $B \to K \nu\bar{\nu}$ amplitude.

Significance
The paper claims that this work establishes a robust, reproducible, and model-independent pathway to search for light invisible particles using existing public data from high-intensity colliders. By analytically reconstructing the kinematic mapping, the authors bypass the need for internal detector simulations, thereby democratizing the reinterpretation of rare decay searches. The results provide the most stringent constraints to date on the fundamental flavor-changing couplings of the QCD axion to $b$ and $s$ quarks, effectively ruling out a significant portion of the parameter space for the QCD axion in the context of collider-accessible flavor physics. Furthermore, the methodology provides a general template for future searches in other Belle II channels, such as $B^+ \to K^{*+} \nu\bar{\nu}$.