The $B^+ \to K^+ \nu \bar \nu$ decay as a QCD axion search: comparing reinterpretation approaches

This paper demonstrates that the four-fold discrepancy in recent limits on the $B^+ \to K^+ a$ decay arises from the choice of kinematic variables, showing that fine-grained binning in the reconstructed di-neutrino invariant mass ($q^2_{\rm rec}$) is essential for resolving narrow axion signals while BDT-based approaches dilute sensitivity, thereby advocating for the publication of likelihoods in physical variable spaces to ensure robust reinterpretability.

The Gist

Imagine you are a detective trying to find a very specific, tiny clue hidden inside a massive, chaotic crime scene. In this story, the "crime scene" is data collected by the Belle II experiment, which studies how certain particles (called B-mesons) decay. The "tiny clue" is a hypothetical particle called the QCD axion—a ghostly, invisible particle that scientists hope exists but have never seen.

Two different detective teams recently looked at the same pile of evidence to find this axion. However, they came up with very different conclusions: one team said, "We can rule out the axion existing up to a very high level of certainty," while the other team said, "Our limits are about four times weaker."

This paper explains why they got different answers. It turns out, the difference wasn't because one team was better at math or had better equipment. It was because they chose to look at the evidence through different "lenses."

The Two Lenses: A High-Res Photo vs. A Blurry Sketch

The "High-Res" Approach (The Stronger Limit)
One team decided to look at the data using a very fine-grained map. Imagine trying to find a specific needle in a haystack. If you look at the haystack in huge, blurry chunks, the needle gets lost in the hay. But if you look at the haystack inch-by-inch, you can spot the needle immediately because it sits in a specific, tiny spot.

In physics terms, this team looked at a variable called $q^2_{rec}$ (a measure of the energy of the invisible particles) with 21 tiny bins (slices).

• The Axion Signal: If an axion exists, it would show up as a sharp, concentrated spike in this energy map, right near zero.
• The Result: Because they used 21 slices, they could see this sharp spike clearly, separated from the "noise" (background particles). This gave them a very strong, sensitive limit.

The "Blurry Sketch" Approach (The Weaker Limit)
The other team used a map that was pre-packaged by the experimental collaboration. This map was designed to find a different type of signal (a smooth, broad cloud of particles called neutrinos).

• The Problem: This map only had 3 huge bins for the energy variable.
• The Result: When they tried to find the sharp axion spike, it got squashed into one of those three giant bins. Inside that bin, the axion signal was drowned out by the massive amount of background noise. It was like trying to hear a whisper in a stadium full of people shouting; the whisper (axion) got lost in the roar (background).

The "Filter" That Wasn't Built for This Job

The second team also used a special filter called a BDT (Boosted Decision Tree). Think of this as a security guard trained to spot a specific type of criminal (the neutrino signal).

• The guard is excellent at spotting the neutrino criminal.
• However, the axion looks completely different from the neutrino.
• Because the guard was only trained on the neutrino, they don't know how to spot the axion. In fact, the guard might even accidentally ignore the axion because it doesn't look like the criminal they were trained to catch.

The paper shows that this filter adds almost no help in finding the axion. It's like using a metal detector to find a wooden chair; the tool is great for metal, but useless for wood.

Why the "Blurry" Team Wasn't Just Being Cautious

You might wonder: "Maybe the second team was just being more careful with their mistakes?"
The authors checked this. They found that even if you add more uncertainty to the second team's method (making them even more cautious), it doesn't explain the huge gap in results.

• In fact, adding more "safety nets" (systematic errors) usually makes the limits tighter (better), not worse.
• The main reason for the weak limit is simply the blurry map and the wrong filter.

The "Dual-Probe" Superpower

The paper highlights a cool feature of the "High-Res" approach: it acts like a dual-probe.

• Because the axion signal looks so different from the background, the team can measure the axion without needing to know exactly how much background noise there is.
• The "Blurry" approach, however, gets confused. If the background noise changes slightly, their axion limit changes drastically. They lose the ability to be independent.

The Big Lesson for Science

The authors conclude with a recommendation for all future experiments.
When scientists publish their data, they often give us a "summary" optimized for the specific thing they were looking for (like the neutrino). But if other scientists want to use that data to look for something totally different (like the axion), that summary is often too blurry or uses the wrong tools.

The Recommendation:
Experimental teams should publish their data in two ways:

1. The "optimized" version for their specific search (using the BDT filters).
2. The "raw" version in physical variables (the fine-grained map), so that anyone can use it to hunt for different kinds of new physics without losing sensitivity.

In short: If you want to find a needle in a haystack, don't use a map that only shows three giant piles of hay. You need a map that shows the individual strands of straw.

Technical Summary

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

Problem Statement
Two recent independent analyses of Belle II data regarding the decay $B^+ \to K^+ \nu\bar{\nu}$, when reinterpreted as a search for the two-body decay $B^+ \to K^+ a$ (where $a$ is a light invisible particle such as a QCD axion), yield limits on the branching fraction $\mathcal{B}(B^+ \to K^+ a)$ that differ by a factor of approximately four. The first approach [1] utilizes public data to reconstruct kinematic variables analytically, deriving the strongest existing bounds. The second approach [4] relies on a published model-agnostic likelihood [5] and yields significantly weaker limits. The authors investigate the origins of this discrepancy, hypothesizing that it stems from the choice of kinematic variable space and the granularity of the data projections used in the reinterpretation.

Methodology
The paper employs a comparative analysis of two distinct statistical frameworks applied to the same underlying Belle II data and simulation:

1. Fine-Grained $q^2_{\text{rec}}$ Approach (Ref. [1]): This method utilizes the reconstructed di-neutrino invariant mass ($q^2_{\text{rec}}$) distribution with approximately 21 bins over the kinematic range $[-1, 25] \text{ GeV}^2$. It treats the background normalization as the dominant systematic uncertainty (estimated at $\sim 1\%$) and analytically maps the true kinematics to reconstructed variables.
2. Coarse $q^2_{\text{rec}} \times \eta(\text{BDT}^2)$ Approach (Ref. [4]): This method uses the published likelihood from Ref. [5], which projects data into a 2D space of four bins in a transformed BDT output ($\eta(\text{BDT}^2)$) and three coarse bins in $q^2_{\text{rec}}$ (boundaries at $\{-1, 4, 8, 25\} \text{ GeV}^2$). The BDT variable was trained specifically to separate $B^+ \to K^+ \nu\bar{\nu}$ signal from background.

The authors conduct a series of five numerical tests to isolate the effects of variable granularity, BDT axis relevance, systematic uncertainties, and the treatment of Standard Model (SM) parameters:

• Tests 1 & 2: Vary the background-normalization error (from 1% to 50%) in the fine-grained approach to test the robustness of the axion limit and the profile likelihood for $B^+ \to K^+ \nu\bar{\nu}$.
• Test 3: Directly rerun the published likelihood of Ref. [4] with varying constraints on the $B^+ \to K^+ \nu\bar{\nu}$ scaling factor ($\mu_{\nu\bar{\nu}}$) and shape systematics to establish a baseline.
• Test 4: Construct a dedicated "hybrid" analysis applying the Ref. [1] statistical framework to the coarse $q^2_{\text{rec}} \times \eta(\text{BDT}^2)$ binning of Ref. [4]. This isolates the impact of the kinematic axes choice alone.
• Test 5: Examine the profile likelihood for $B^+ \to K^+ \nu\bar{\nu}$ within the hybrid analysis to compare shape separation capabilities.

Key Contributions and Results

• Resolution as the Key Figure of Merit: The primary driver of the sensitivity difference is the resolution in $q^2_{\text{rec}}$. For a QCD axion ($m_a \simeq 0$), the signal is monochromatic at $q^2=0$, mapping to a structured distribution near $q^2_{\text{rec}} \simeq 0$. The coarse 3-bin structure of the Ref. [4] likelihood subsumes this signal into a single bin dominated by the $B^+ \to K^+ \nu\bar{\nu}$ background, diluting the signal. The fine-grained 21-bin structure of Ref. [1] resolves the signal structure, significantly improving sensitivity.
• Inadequacy of the BDT Axis: The variable $\eta(\text{BDT}^2)$ is optimized for $B^+ \to K^+ \nu\bar{\nu}$, not for the narrow $B^+ \to K^+ a$ signal. Since the axion produces a monochromatic kaon while $\nu\bar{\nu}$ produces a continuous spectrum, the BDT axis carries little discriminating power for the axion and may even suppress signal events that do not resemble the $\nu\bar{\nu}$ topology.
• Systematic Uncertainties are Conservative: The authors demonstrate that the omission of subleading shape systematics in the Ref. [1] approach does not artificially tighten the limit. Including these systematics (as done in Ref. [4]) actually lowers the limit by $\sim 20\%$ by allowing the fit more freedom to accommodate the $B^+ \to K^+ \nu\bar{\nu}$ shape, leaving less room for the axion signal. Thus, the Ref. [1] bound is conservative.
• Loss of "Dual-Probe" Feature: In the fine-grained $q^2_{\text{rec}}$ approach, the axion limit is statistically decoupled from the $B^+ \to K^+ \nu\bar{\nu}$ measurement (the "dual-probe" feature). In the coarse $q^2_{\text{rec}} \times \eta(\text{BDT}^2)$ space, the axion limit varies by a factor of roughly two depending on whether the $B^+ \to K^+ \nu\bar{\nu}$ rate is fixed or allowed to float, indicating a loss of this robustness.
• Quantification of Sensitivity Loss: The hybrid analysis (Test 4) confirms that the change in kinematic axes alone accounts for a degradation in sensitivity by a factor of roughly five at low axion masses. The remaining difference between the hybrid analysis and Ref. [4] is attributed to the inclusion of full shape systematics in Ref. [4], which further lowers the limit.

Significance and Claims
The paper concludes that the choice of kinematic variable space is decisive for the reinterpretation of experimental data for narrow invisible signals. Likelihoods dominated by BDT variables trained on a specific signal topology (like $B^+ \to K^+ \nu\bar{\nu}$) are of limited utility for recasting measurements into signals with appreciably different shapes (like $B^+ \to K^+ a$).

The authors advocate that experimental collaborations publish likelihood projections in physical variable spaces (such as fine-grained $q^2_{\text{rec}}$) alongside BDT-based likelihoods. This practice would maximize the reinterpretability of measurements for a broad range of new-physics scenarios, particularly those involving narrow resonances or signals that differ kinematically from the training data. The paper does not propose new experimental proposals but calls for a modification in data release strategies to facilitate more robust theoretical reinterpretations.