AlpsML

ML analysis (XGBoost + SHAP + MCMC) of axion-like particle (ALP) likelihoods in non-universal UV models with alpaca, targeting the ~2 GeV ALP explanation of the Belle II B⁺ → K⁺νν̄ excess. A χ² surrogate is trained over the full alpaca observable sector (FCNC meson decays K⁺ → a π⁺, B⁺ → K⁺ a, …, plus visible channels, meson mixing and leptonic/radiative decays), and the posterior is explored with emcee.

→ Open notebook: notebooks/AlpsML.ipynb

Index

• Physics
• Notebook ↗
• Repository structure
• Pipeline overview
• Performance
• Setup
• Notes

Physics

Axion-like particles (ALPs) emerge from the spontaneous breaking of a global U(1)_PQ symmetry at scale f_a. Their couplings to SM particles are not free parameters: in any UV-complete model they are entirely determined by the Peccei–Quinn (PQ) charges assigned to the SM fermion representations.

From PQ charges to ALP couplings

Two rules connect the UV input to the observable IR couplings:

1. Couplings to fermions equal the PQ charge of that fermion (up to a sign from the Yukawa structure).
2. Couplings to gauge bosons (gluons, photons, W/Z) are linear combinations of the PQ charges, fixed by the chiral anomaly coefficients of the PQ current with each gauge group:

$$ g_{aGG} \propto \sum_f T(R_f),Q_f^{\rm PQ}, \qquad g_{a\gamma\gamma} \propto \sum_f Q_f^{\rm em,2},Q_f^{\rm PQ}. $$

This is the physical advantage over scanning over independent IR couplings: the five PQ charges plus f_a fully determine the ALP phenomenology, respecting the UV structure of the theory.

Non-universal model

The charges are generation-dependent for the SU(2)_L doublets — only the third generation of q_L and l_L carries a non-zero PQ charge — while the right-handed singlets are universal across generations. In alpaca:

from alpaca.uvmodels import PQChargedModel
import numpy as np

model = PQChargedModel('non-universal model', {
    'qL': [0, 0, pq_qL],   # left-handed quark doublet: charge only in 3rd gen
    'lL': [0, 0, pq_lL],   # left-handed lepton doublet: charge only in 3rd gen
    'uR': pq_uR,            # right-handed up quarks: universal
    'dR': pq_dR,            # right-handed down quarks: universal
    'eR': pq_eR,            # right-handed charged leptons: universal
})
couplings = model.get_couplings({}, 4 * np.pi * fa)

get_couplings propagates the five PQ charges through the anomaly equations and returns the full set of IR couplings used by alpaca to compute the χ² over its full observable sector — the rare meson decays (K⁺ → a π⁺, K⁰_L → a π⁰, B⁺ → K⁺ a, B⁰ → K⁰ a, B⁺ → a π⁺) together with visible channels (e.g. B → K μμ), meson mixing and radiative/leptonic decays.

The scan is therefore over seven physically meaningful parameters — the six UV inputs plus the ALP mass:

Parameter	Description	Range
log_fa	log₁₀ of the PQ scale fa (GeV)	[6, 7.5]
pq_qL	PQ charge of the 3rd-gen left-handed quark doublet	[−1, 1]
pq_lL	PQ charge of the 3rd-gen left-handed lepton doublet	[−1, 1]
pq_uR	PQ charge of right-handed up quarks	[−1, 1]
pq_dR	PQ charge of right-handed down quarks	[−1, 1]
pq_eR	PQ charge of right-handed charged leptons	[−1, 1]
ma	ALP mass (GeV)	[1.7, 2.2]

The ma window brackets the ~2 GeV particle preferred by the Belle II B⁺ → K⁺νν̄ excess. In the MCMC the flat box priors are supplemented by an informative Gaussian prior on log_fa, N(6.8, 0.4), matching the reference ALP analysis; the generation range of log_fa is capped at 7.5 so it coincides with the MCMC prior box.

Notebook

notebooks/AlpsML.ipynb — full pipeline: dataset → XGBoost → SHAP → MCMC

Repository structure

AlpsML/
├── notebooks/
│   └── AlpsML.ipynb       # full pipeline: dataset → XGBoost → SHAP → MCMC
├── outputs/paper/         # generated (gitignored)
│   ├── datasets/          # dataset_alps_uv_v2.csv, posterior_samples_uv_v2.csv, …
│   ├── figures/           # corner_plot_uv_v2*.png, SHAP*.png, learning_curve_uv_v2.png
│   └── models/            # modelo_alps_{clf,reg}_v2.json (XGBoost), best_params_{clf,reg}_v2.json
└── requirements.txt

Pipeline overview

• (1) Dataset generation: 8 000 χ² evaluations sampled via a 7-D Latin Hypercube (log_fa, the five PQ charges pq_qL, pq_lL, pq_uR, pq_dR, pq_eR, and the ALP mass ma), computed with alpaca over its full observable sector and parallelized across cores. The training target is a sigmoid of the Δχ² relative to the dataset minimum, with the allowed/excluded boundary at Δχ² ≈ 10.
• (2) XGBoost surrogates (two-stage strategy): two models tuned with Optuna, with distinct roles. The classifier (CLF) learns the sigmoid target over the whole space and defines the allowed/excluded boundary; the regressor (REG) learns the raw χ² only inside the allowed region, where the physically relevant structure lives.
• (3) SHAP interpretability: feature importance and dependence plots on the classifier, ranking which PQ charges (and ma) control the phenomenology.
• (4) MCMC (two-surrogate posterior): sampling with emcee of log p = −½·χ²(REG) + log-prior − softplus wall(CLF). The likelihood comes from the raw-χ² regressor; the classifier only keeps the walkers out of the region where the regressor would extrapolate. A corner plot of derived physical observables (|c_V^sb|, |c_A^μμ|, |c_G|, cτ, BR(B⁺ → K⁺a)) is computed with alpaca on a posterior subsample.

All steps run as independent cells in notebooks/AlpsML.ipynb. outputs/ is fully regenerable from the notebook and is gitignored.

Performance

Reference figures from this repo's benchmark (single CPU core, cold caches):

	XGBoost surrogate	Exact alpaca	Speed-up
Per evaluation	≈ 3.5 µs	≈ 1.85 s	≈ 5×10⁵
8 000-point training set	≈ 0.03 s	≈ 4 core-h	≈ 5×10⁵

The notebook includes a benchmark cell that reproduces this measurement; the exact branch runs in a cold subprocess so alpaca's internal caches cannot distort the timing. The posterior favours m_a ≈ 1.8 GeV and f_a in the 10⁶–10⁷ GeV range, in agreement with previous analyses of the Belle II excess.

Setup

python -m venv .venv
source .venv/bin/activate
pip install -r requirements.txt
jupyter lab notebooks/AlpsML.ipynb

Notes

• SHAP figures and the learning curve are not saved automatically by the notebook (only plt.show()). The versions committed under outputs/figures/ were exported manually from Jupyter.
• alpaca-alps is the public package for UV models; see github.com/alpaca-physics/alpaca.