ALPaca: Computational Tools for ALP Searches

• ALPaca is a suite of open-source computational tools for axion-like particle (ALP) phenomenology, enabling detailed studies in fixed-target and meson decay experiments.
• The 2019 Alpaca Monte Carlo generator enhances fixed-target analysis by accurately modeling coherent proton–nucleus ALP production beyond the traditional Equivalent Photon Approximation.
• The 2025 ALPaca framework integrates effective field theory, running, matching, and chiral perturbation theory to deliver comprehensive meson decay and flavour observable predictions.

Searching arXiv for the relevant ALPaca papers and related terminology. ALPaca is the name used in axion-like-particle phenomenology for open-source computational tools that target different parts of the ALP search program. In the literature, the term encompasses both the 2019 Alpaca Monte Carlo generator for coherent proton–nucleus production in fixed-target experiments and the 2025 ALPaca framework, explicitly expanded as “ALP Automatic Computing Algorithm,” for meson decays and flavour observables. Taken together, these tools represent two complementary layers of ALP phenomenology: exact production and detector acceptance in beam-based searches, and EFT-to-hadron-level prediction for prompt, displaced, and invisible decays in meson processes (Harland-Lang et al., 2019, Alda et al., 25 Jul 2025).

1. Nomenclature and scope

The published literature uses closely related spellings—“Alpaca” and “ALPaca”—for software intended to study axion-like particles. The 2019 code was introduced as “the Alpaca Monte Carlo generator” in the context of coherent scattering of protons on nuclei and fixed-target discovery prospects (Harland-Lang et al., 2019). The 2025 framework was presented as ALPaca, with the explicit expansion “ALP Automatic Computing Algorithm,” in a study of meson decays that combines effective field theory and ultraviolet models, running and matching effects across energy scales, non-perturbative QCD corrections via chiral perturbation theory, and experimental signatures classified as prompt, displaced, and invisible (Alda et al., 25 Jul 2025).

This naming overlap has a straightforward technical consequence: ALPaca is not a single monolithic code covering all ALP searches. Rather, the name designates a small software lineage within ALP phenomenology. One branch specializes in coherent $p+N\to p+N+a$ production and detector-level acceptance for fixed-target experiments; the other organizes low-energy flavour physics, decay widths, and likelihood construction for meson decays. This suggests that the shared name reflects continuity of purpose—precision ALP phenomenology—rather than identity of implementation.

Framework	Domain	Core functionality
Alpaca	Fixed-target coherent production	Exact $p+N\to p+N+a$, decay geometry, $a\to\gamma\gamma$
ALPaca	Meson decays and flavour observables	EFT/UV input, RGE, matching, $\chi$PT, prompt/displaced/invisible signatures

2. The 2019 Alpaca generator for fixed-target production

The 2019 Alpaca generator was developed to improve the calculation of ALP production in the coherent process $p+N\to p+N+a$, going beyond the usual Equivalent Photon Approximation. In the $p$–nucleus centre-of-mass frame, the paper gives a fully differential cross section in terms of the virtual-photon density matrices $\rho_i^{\alpha\beta}(q_i)$, the $\gamma\gamma\to a$ amplitude $$M_{\mu\nu}=g_{a\gamma\gamma}\,\epsilon_{\mu\nu\sigma\rho}q_1^\sigma q_2^\rho$$, exact photon virtualities, and form factors for both the proton and the nucleus. The proton is modeled with standard dipole Sachs form factors, while the nucleus is described through an electric form factor $F(Q^2)$ obtained from the Woods–Saxon charge density (Harland-Lang et al., 2019).

A central result of that work is methodological rather than merely numerical. The calculation does not assume $p+N\to p+N+a$0 or $p+N\to p+N+a$1, and therefore corrects the regions in which naive EPA treatments fail. The paper states that, at beam energies $p+N\to p+N+a$2 and/or $p+N\to p+N+a$3, the EPA can misestimate the cross section by up to $p+N\to p+N+a$4–$p+N\to p+N+a$5 orders of magnitude, while in the high-energy regime $p+N\to p+N+a$6 Alpaca reproduces the EPA within $p+N\to p+N+a$7 (Harland-Lang et al., 2019). This directly addresses a common misconception that EPA-based production estimates are uniformly adequate across fixed-target parameter space.

The generator is a stand-alone Fortran Monte Carlo program. It samples the phase space $p+N\to p+N+a$8 by adaptive Monte Carlo, reconstructs the ALP four-momentum $p+N\to p+N+a$9, solves the on-shell conditions for the outgoing proton and nucleus, and rejects points outside physical phase space. It then computes the ALP proper decay length, boosts to the lab frame, generates the decay vertex along the beamline, simulates $a\to\gamma\gamma$0 in the ALP rest frame, boosts the photons to the lab, and applies user-defined cuts such as $a\to\gamma\gamma$1, $a\to\gamma\gamma$2, $a\to\gamma\gamma$3, $a\to\gamma\gamma$4, $a\to\gamma\gamma$5, and $a\to\gamma\gamma$6 (Harland-Lang et al., 2019).

For sensitivity estimates, the expected signal yield is written as

$a\to\gamma\gamma$7

where $a\to\gamma\gamma$8 is the geometric and kinematic acceptance and $a\to\gamma\gamma$9 is built from the number of protons on target and the target properties. Requiring, for example, $\chi$0 gives a $\chi$1 CL exclusion curve in the $\chi$2 plane (Harland-Lang et al., 2019).

3. The 2025 ALPaca framework for meson decays

The later ALPaca framework addresses a broader flavour-physics problem. Its starting point is the most general dimension-$\chi$3 ALP EFT at a UV scale $\chi$4, invariant under $\chi$5 up to anomaly and a soft-breaking mass $\chi$6. In the notation of the paper,

$\chi$7

Below the electroweak scale, after running and matching, the theory is written as

$\chi$8

The Wilson coefficients evolve according to

$\chi$9

and at $p+N\to p+N+a$0 the ALP–gluon coupling is removed by a chiral rotation, after which the framework matches onto the $p+N\to p+N+a$1 chiral Lagrangian and diagonalizes kinetic and mass mixing between $p+N\to p+N+a$2 and the neutral mesons (Alda et al., 25 Jul 2025).

This construction is intended to make meson-decay predictions consistent across scales. The analysis explicitly accounts for running and matching effects across energy scales, includes non-perturbative QCD corrections via chiral perturbation theory, and extends the $p+N\to p+N+a$3PT form factors above $p+N\to p+N+a$4 to $p+N\to p+N+a$5 through vector-meson dominance and data-driven factors $p+N\to p+N+a$6 (Alda et al., 25 Jul 2025). The practical significance is that ALP widths, branching ratios, and flavour-changing amplitudes are not treated as isolated low-energy parameters but as quantities derived from a UV-to-IR chain.

The decay-width module includes standard channels such as

$p+N\to p+N+a$7

and

$p+N\to p+N+a$8

while hadronic decays are treated with $p+N\to p+N+a$9PT for $p$0, perturbative QCD for $p$1, and interpolation in the intermediate region (Alda et al., 25 Jul 2025).

4. Computational organization and observable construction

ALPaca’s internal organization follows the physics flow. The code structure described in the paper comprises a Model-Definition module for EFT Wilson coefficients or UV-model choices, an RGE-Evolution module, a Matching module, a Decay-Width module, a Meson-Decay module, a Signature module, an Observable-Likelihood module, and a Benchmark-Scan and Plotting module (Alda et al., 25 Jul 2025). The data structures are correspondingly conventional: Python dictionaries for couplings, NumPy arrays for scales and observables, and a SciPy ODE solver for renormalization-group evolution.

The meson-decay component implements rates such as

$p$2

and

$p$3

together with $p$4 and weak-$p$5PT insertions. Experimental realization is handled by the Signature module, which computes

$p$6

These probabilities are then combined with branching ratios and experimental bounds in a $p$7 or profile-likelihood analysis (Alda et al., 25 Jul 2025).

The 2019 Alpaca generator solves a different computational problem but in a structurally analogous way. It also exposes user-defined inputs for beam energy, target nucleus, number of protons on target, ALP mass and coupling grids, geometry and cuts, and Monte Carlo parameters, and it can output weighted or unweighted events in HEPEVT, Les Houches, or HepMC formats (Harland-Lang et al., 2019). A plausible implication is that the two tools occupy adjacent layers of the same phenomenological workflow: one predicts production and detector acceptance in fixed-target beamlines, while the other predicts decay observables and flavour constraints from EFT or UV input.

5. Benchmark models, signatures, and the Belle II case study

The 2025 ALPaca analysis considers both flavour-universal and flavour-non-universal benchmarks. The flavour-universal set includes DFSZ-like and KSVZ-like constructions. The flavour-non-universal set includes top-philic models, flaxion models with $p$8, and non-universal DFSZ models in which different Higgs couplings are assigned to the first and second generations versus the third (Alda et al., 25 Jul 2025). This model menu is important because the same low-energy signature can originate from tree-level flavour violation, loop-induced effects, or anomaly couplings, and ALPaca is designed to keep those possibilities explicit rather than collapsing them into a single effective parameterization.

The framework organizes signatures into prompt, displaced, and invisible categories. In experimental terms, prompt decays correspond to $p$9, displaced decays occur when $\rho_i^{\alpha\beta}(q_i)$0, and invisible signatures arise when $\rho_i^{\alpha\beta}(q_i)$1 (Alda et al., 25 Jul 2025). This categorization is not merely descriptive; it determines how theoretical branching ratios are mapped to exclusion likelihoods.

A dedicated application is the Belle II anomaly in $\rho_i^{\alpha\beta}(q_i)$2. The paper states that Belle II sees a $\rho_i^{\alpha\beta}(q_i)$3 excess in this channel and that BaBar provides compatible bounds on $\rho_i^{\alpha\beta}(q_i)$4 and $\rho_i^{\alpha\beta}(q_i)$5. Reinterpreting the data in terms of an invisible two-body decay $\rho_i^{\alpha\beta}(q_i)$6, the combined analysis yields a lower bound $\rho_i^{\alpha\beta}(q_i)$7 at $\rho_i^{\alpha\beta}(q_i)$8 CL. In the EFT interpretation based on

$\rho_i^{\alpha\beta}(q_i)$9

the fit prefers $\gamma\gamma\to a$0 with $\gamma\gamma\to a$1. Among the UV benchmarks, only the non-universal DFSZ model with $\gamma\gamma\to a$2 and $\gamma\gamma\to a$3 simultaneously produces a tree-level flavour-violating coupling large enough and $\gamma\gamma\to a$4 meter; other models require a small $\gamma\gamma\to a$5 to remain invisible (Alda et al., 25 Jul 2025).

6. Limitations, interpretation, and outlook

The two ALPaca/Alpaca tools are powerful precisely because each is specialized, but that specialization also defines their limits. The 2019 generator is optimized for coherent $\gamma\gamma\to a$6–$\gamma\gamma\to a$7 production and fixed-target detector geometry; it is not a general-purpose flavour code (Harland-Lang et al., 2019). Conversely, the 2025 ALPaca framework provides a unified treatment of RGE, matching, $\gamma\gamma\to a$8PT, vector-meson dominance, meson decays, and experimental signature classification, but the paper explicitly states current limitations: no dedicated beam-dump or astrophysical routines, and no collider production beyond $\gamma\gamma\to a$9 (Alda et al., 25 Jul 2025).

This also clarifies two common misunderstandings. First, ALPaca is not synonymous with a single public package covering every ALP search channel. Second, “open-source ALP phenomenology” does not imply a purely EFT-level treatment. The later ALPaca framework was built specifically to combine EFT Wilson coefficients and UV models, while the earlier Alpaca generator was built to replace oversimplified production estimates with exact kinematics and realistic form factors (Harland-Lang et al., 2019, Alda et al., 25 Jul 2025).

The stated development path of the 2025 framework includes integration of beam-dump production, astrophysical bounds, and interfacing with general event generators such as MadGraph (Alda et al., 25 Jul 2025). If that program is realized, a plausible implication is that the historical split between the fixed-target Alpaca generator and the meson-decay ALPaca framework could narrow, producing a more unified software ecosystem for ALP searches across flavour, beam-dump, and collider regimes.