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Axion searches at colliders

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Pith's one-line read A photophilic axion, from Lagrangian to Belle II exclusion plot.

desk verdict Useful lecture notes for students, but the tutorial contains a genuine factor-of-two error in the a->γγ width that should be corrected before publication. read the letter →

arxiv 2505.00124 v1 pith:REPKUF3P submitted 2025-04-30 hep-ph

Michele Tammaro , Jure Zupan This is my paper · ORCID
classification hep-ph
keywords axion-likeparticlescollidersearchesphotophilicALPBelleIImissingenergysignaturesdisplacedverticesdarksectorportalslecturenotes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

These lecture notes argue that a collider search for axion-like particles can be carried end to end, from a two-line Lagrangian to an exclusion plot, and they demonstrate it for a photophilic ALP at an electron-positron collider. The paper walks through the production mechanisms, the decay width and boosted lifetime, and the four signal categories that emerge depending on where and how the ALP decays. It then reproduces the main ingredients of the Belle II search with 445 inverse picobarns of data, deriving reach curves for the invisible and resolved diphoton signatures with a simplified detector model. If the tutorial's recipe is right, a student or phenomenologist can estimate the sensitivity of a given e+e- experiment to a photophilic ALP without a full experimental simulation, and can see which signatures cover which regions of mass-coupling space.

What carries the argument

The load-bearing object is the simplified Belle II detector model combined with the analytic ALP kinematics. The vertex resolution $L_{\min}=0.14$ m and detector outer radius $L_{\max}=1.55$ m are compared with the boosted decay length $\ell_a = (|\vec p_a|/m_a)\,128\pi/(g_{a\gamma\gamma}^2 m_a^3)$; the electromagnetic calorimeter's $0.8^\circ$ angular resolution is compared with the opening angle $\Delta\theta \sim 4m_a/\sqrt{s}$. This comparison alone assigns each point in the mass-coupling plane to one of four signal classes. The statistical treatment is a $\chi^2$ test where resolved-signal limits use a 90% CL with $\Delta\chi^2 = 2.71$, and invisible-signal limits use the $N_{\rm signal}=3$ rule.

What would settle it

Compare the tutorial's Fig. 19 curves with the full Belle II exclusion contours from Ref. [82] at the same luminosity: if the resolved-channel contour disagrees by more than the expected reconstruction-efficiency factor in any mass bin, or if the invisible-channel contour excludes points the experiment does not, the simplified background treatment is the reason.

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Extended reading notes

Core claim

The central claim is that the expected signals of a photophilic ALP at an e+e- collider are completely determined, in their parametric behavior, by two formulas and one geometric comparison. The cross section for $e^+e^- \to \gamma a$ scales as $\frac{\alpha g_{a\gamma\gamma}^2}{32}(1+\cos^2\theta)(1 - m_a^2/s)^3$, and the decay width is $\Gamma(a\to\gamma\gamma)=g_{a\gamma\gamma}^2 m_a^3/(128\pi)$, so the boosted decay length is $\ell_a \sim (|\vec p_a|/m_a)\,128\pi/(g_{a\gamma\gamma}^2 m_a^3)$. Comparing $\ell_a$ to the detector's inner and outer radii, and comparing the typical photon opening angle, $\sim 4m_a/\sqrt{s}$, to the angular resolution, partitions the $(m_a, g_{a\gamma\gamma})$ plane into invisible, merged, displaced-resolved, and prompt-resolved regions. Using the Belle II geometry ($L_{\min}=0.14$ m, $L_{\max}=1.55$ m, angular resolution $0.8^\circ$, $0.25$ GeV energy threshold), an idealized background treatment, and a three-event discovery threshold, the paper produces approximate 90% CL exclusion curves for 445 inverse picobarns and for the full 50 inverse attobarns, mirroring the published Belle II analysis.

Load-bearing premise

The reach curves stand or fall on the assumption that a simplified detector model—inner radius 0.14 m, outer radius 1.55 m, 0.8 degree angular resolution, 0.25 GeV photon threshold—together with an idealized background treatment (zero background for the invisible channel, perfect background subtraction for the resolved channel, and a three-event discovery rule) reproduces the actual Belle II sensitivity.

Editorial extensions

If this is right

  • The resolved diphoton signature is the relevant probe for ALP masses from a few hundred MeV up to near $\sqrt{s}$, where the production cross section vanishes as $(1 - m_a^2/s)^3$.
  • The invisible single-photon signature covers low ALP masses and small couplings, where the decay length exceeds the detector size; its reach improves only with integrated luminosity, roughly as an order of magnitude when going from 445 inverse picobarns to 50 inverse attobarns.
  • Photon-fusion production has a larger cross section than associated production at Belle II, but it is discarded because the outgoing electrons remain in the uninstrumented forward region and the resulting signature suffers large beam-induced backgrounds.
  • Whether a given ALP is visible, displaced, merged, or invisible is fixed by the ratios $\ell_a/L_{\rm det}$ and $\Delta\theta/\Delta\theta_{\rm res}$, so the same classification applies to any e+e- detector with different radii and angular resolution.
  • For two benchmark flavor structures, rare kaon decays probe axion scales that differ by orders of magnitude: flavor-anarchic couplings reach $f_a \sim 10^{12}$ GeV, while loop-generated minimal-flavor-violation couplings reach only $f_a \sim 10^6$-$10^7$ GeV, or about 10 TeV for a gluon-only UV coupling.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same closed-form chain can be rerun for any other electron-positron machine by changing $\sqrt{s}$, $L_{\min}$, $L_{\max}$, and angular resolution, so the tutorial effectively gives a parametric calculator for photophilic-ALP reach rather than a single experiment-specific result.
  • The merged-photon region is classified but not quantified in the tutorial; a dedicated search using electromagnetic-shower shapes in that corner of parameter space would be a natural complement to the resolved and invisible analyses.
  • For a hadrophilic or leptophilic ALP, the same steps apply with the lifetime formulas of Section 4.3 replacing the two-photon width; the reach curves would be shaped by whichever decay channel dominates.
  • The idealized background treatment makes a concrete prediction: the full Belle II limits, which use a complete simulation, should lie close enough to the tutorial's curves to confirm that the simplified model captures the main sensitivity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. These lecture notes provide an introduction to axion and ALP searches at colliders. After reviewing basic collider physics and the landscape of dark-sector portals, the authors focus on the QCD axion and ALP phenomenology, including production and decay, and give a broad overview of search strategies. The final part is a worked tutorial that starts from the Lagrangian in Eq. (44), derives the associated-production cross section, the ALP decay width, and the expected detector signatures, and ends with Fig. 19, which shows approximate Belle II exclusion bounds for both the invisible and resolved γγ signatures. The tutorial is explicitly based on the published Belle II analysis of Ref. [82].

Significance. The paper fills a useful pedagogical niche: it takes the reader from first principles to a concrete, reproducible estimate of the Belle II sensitivity to photophilic ALPs, benchmarking each step against a real analysis. The main strengths are the clear step-by-step derivations (including the 2→2 phase-space counting, the χ2 statistics detour, and the explicit detector-geometry description), the explicit statement of the simplifications, and the transparent comparison with published data points in Fig. 21. If the numerical error identified below is corrected, the tutorial will be a valuable resource for students entering the field. The paper does not claim to derive new bounds; its value is didactic, and the declared approximations (zero background for the invisible channel, perfect background subtraction for the resolved channel, 3-event discovery rule) are acceptable for that purpose provided the underlying widths are correct.

major comments (1)
  1. [§5.3, Eq. (54)] Eq. (54) gives Γ(a→γγ) = g²_{aγγ} m³_a/(128π), but the standard result with the convention in Eq. (44) is Γ(a→γγ) = g²_{aγγ} m³_a/(64π). The inconsistency is internal: under the mapping g_{aγγ} = α C^{eff}_{γγ}/(2π f_a) in §4.3, Eq. (25) reproduces exactly the 64π result, while Eq. (54) is a factor of 2 smaller. The missing factor is the identical-particle phase-space factor 1/2!, which is not fully captured by the heuristic prefactor (4π/32π²) in Eq. (54). This error propagates directly into Eq. (56) for the proper lifetime, Eq. (57) for the boosted decay length, the numerical example in Eq. (58), and—through the survival probability P(L) in Eq. (72)—into the reach curves in Fig. 19. Since the tutorial’s stated purpose is to reproduce the quantitative sensitivity of Belle II, this factor must be corrected throughout §5.3 and §5.7–§5.8, and the affected numerical results and Fig. 19 should be updated.
minor comments (4)
  1. [§5.2, after Eq. (52)] The word “desribed” in the phase-space paragraph is a typo and should be “described”.
  2. [§5.3, Eq. (54)] The explanation of the prefactor in Eq. (54) would be clearer if the identical-particle factor 1/2! were mentioned explicitly; this is directly related to the factor-of-2 error and should be part of the corrected derivation.
  3. [§5.4, Eq. (60)] The sentence “the direction of the other photon in the rest frame is −(π−θ′)” is confusing; a photon back-to-back with a photon at polar angle θ′ has polar angle π−θ′ and an azimuthal angle shifted by π, so the notation with a minus sign should be revised.
  4. [§5.6] “The ECL barell” is a typo; it should be “the ECL barrel”.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the tutorial is benchmarked against external Belle II data, and the self-citations are background references rather than load-bearing inputs.

full rationale

This paper is a review with a pedagogical tutorial. The tutorial's derivation chain—Lagrangian Eq. (44), production cross-section Eq. (52), decay width Eq. (54), lifetime Eq. (56), survival probability Eq. (72), event rates Eqs. (73)-(74), and reach curves in Fig. 19—is self-contained in the sense that each quantity is computed from previous equations rather than fitted to the target reach. The reach is benchmarked against the published Belle II analysis Ref. [82] and against Belle II data points shown in Fig. 21, which is external evidence, not an input fitted into the prediction. The detector simplifications (L_min, L_max, angular resolution, energy threshold, zero or perfectly subtracted backgrounds, and the 3-event discovery rule) are explicitly declared approximations, and none of them is a parameter adjusted to force agreement with the Belle II limits; they are chosen before computing the reach. The self-citations (e.g., Refs. [1], [51], [56], [80]) appear as background references to prior work by one of the authors and do not carry the tutorial's derivation or the review's main claims. The internal factor-of-2 discrepancy between Eq. (25) and Eq. (54) is a correctness issue that propagates into lifetimes and reach, but it is not circularity: Eq. (54) is not defined in terms of the reach curves, nor is any fitted parameter renamed as a prediction. Overall, no load-bearing circular step was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a review and tutorial, so it relies on standard physics assumptions (ALP EFT, QCD axion mass, ChPT) and on the Belle II detector description. No new free parameters or invented entities are introduced.

assumptions (4)
  • domain assumption The ALP is a pNGB, so the most general dimension-5 Lagrangian contains only derivative couplings to fermions and anomalous couplings to gauge bosons, Eqs. (12)-(15).
    Assumed throughout the review; this is the standard ALP EFT.
  • domain assumption The QCD axion mass is generated by the QCD anomaly and given by m_a = m_pi f_pi / f_a, Eq. (2).
    Taken from Grilli di Cortona et al. (Ref [6]); not derived in the paper.
  • domain assumption The effective ALP-photon coupling C_γγ^eff in Eq. (26) includes NLO ChPT contributions from the gluon coupling and pi0-a mixing.
    Input from Ref [6] and Ref [72].
  • domain assumption For the resolved-signal background, the dominant SM process is e+ e- -> gamma gamma gamma, as stated in Sec. 5.8; Belle II detector parameters are as listed in Sec. 5.6.
    Used to compute the resolved-signal reach; the cross section is not derived.

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Cite this review

Pith. "Pith review of Axion searches at colliders." pith.science (2026). https://pith.science/paper/REPKUF3P

@misc{pith2026250500124,
  author       = {Pith},
  title        = {Pith review of: Axion searches at colliders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REPKUF3P}},
  note         = {Machine review of arXiv:2505.00124}
}
abstract

We provide an introduction to searches for axions and Axion-Like Particles (ALPs) at colliders. After covering the basics of collider physics, with a focus on production and detection of new particles, we give a rather broad introduction into searches for dark sectors at colliders. While our focus is on searches for light dark sectors, with the special emphasis on ALPs, we also contrast these with searches for heavy dark matter. In the final part of the notes we provide a ``tutorial'', a worked out example of a search for a photo-philic ALP in $e^+e^-$ collisions, reflecting the actual analysis that was performed at Belle-II with early data.

Figures

Figures reproduced from arXiv: 2505.00124 by the authors.

Figure 1
Figure 1. A sketch of the onion like structure of a multi-purpose particle physics detector used in 𝑝 𝑝 and 𝑒 + 𝑒 − collisions, with tracking components closest to the interaction point, followed by the electromagnetic and hadron calorimeters and finally the muon chambers. charged particles of energy 𝐸 moving on circular orbits of radius 𝑅. Neglecting proton mass, we have 𝐸 = 𝑄𝑅𝐵 = 15TeV 𝑄 𝑒 𝑅 5km 𝐵 10 Tesla . (5) Magnetic fi… view at source ↗
Figure 2
Figure 2. A schematics of a proton beam dump experiment, creating standard model (SM) particles and the ALP. SM particles get stopped in the shield, while a feebly interacting ALP can make it all the way to the detector, where it can be detected via its decays to the SM particles. where particles interact or decay. Further out are calorimeters that measure energy that is deposited either via electromagnetic or hadronic shower… view at source ↗
Figure 3
Figure 3. DM pair production in 𝑝 𝑝 collisions, where the examples of SM particles in the left bin can be a single hard photon (middle panel) or hard gluon (right panel). The single hard gluon hadronizes into a spray of SM hadrons (pions, kaons, etc) — a jet. neutrinos. The production of SM neutrinos in the collisions is thus an irreducible background to DM searches. Furthermore, missing energy can only be observed if there a… view at source ↗
Figures from the paper (18 more)
Figure 4
Figure 4. Figure 4: Representative diagrams for the DM production via 𝑠-channel (axial-)vector mediator (left panel), 𝑠-channel (pseudo-)scalar mediator (middle), and 𝑡-channel mediator (right). Reproduced from [1]. on-shell. The scattering is therefore described by a point-like interacti…
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Partonic 𝑠 → 𝑑𝑎 transition (left) induces a 𝐾 − → 𝜋 −𝑎 decay (right). a t C A tt s d V ∗ W ts ∼ λ 3 Vtd ∼ λ 3 a c C A cc s d V ∗ W cs ∼ 1 Vcd ∼ λ a u C A uu s d V ∗ W us ∼ λ Vud ∼ 1 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: At one loop the 𝑠 → 𝑑𝑎 transition is generated from diagonal couplings of 𝑎 to up quarks and the 𝑊 exchange, giving the MFV structure due to CKM matrix elements. The two questions that are phenomenologically very important are tied to the above discussion, and are • Ar…
Figure 8
Figure 8. Figure 8: The one loop generated MFV 𝑠 → 𝑑𝑎 amplitude due to ALP coupling to 𝑊 bosons. where 𝜇𝑡 = 𝑚𝑍, with 𝑥𝐹 = 𝑚 2 𝐹 /𝑚 2 𝑊, where 𝐹 = 𝑡, 𝑐, and the loop functions 𝑓𝐹 (𝑥) = 3 (1 − 𝑥 + log 𝑥) 2(1 − 𝑥) 2 + 1 4 , 𝑓𝑊 (𝑥) = 3𝑥 [1 − 𝑥 + 𝑥 log 𝑥] 2(1 − 𝑥) 2 . (19) All the Wilson coeff…
Figure 9
Figure 9. Figure 9: Different contributions to 𝐶 eff 𝛾𝛾 in eq. (26), from the UV couplings; the SM quarks and leptons running in the loop; the nonperturbative contribution from coupling to gluons; and from 𝑎 − 𝜋 0 mixing. while the diagonal couplings are of similar sizes 𝐶 𝑉 𝑡 𝑡 (ΛUV) ∼ 𝐶…
Figure 10
Figure 10. Figure 10: Left: Constraints on B (𝐾 + → 𝜋 +𝑋) with the 𝑋 particle decaying invisibly or escaping the detector as a function of 𝑋 mass up to the kinematic endpoint (vertical dotted line). Colored regions are excluded, from 𝐾 + → 𝜋 +𝑋 search [76] (red) and [77] (grey), and by 𝜋 0…
Figure 11
Figure 11. Figure 11: Bounds on the ALP coupling only to gluons in the UV (𝑁3 (ΛUV) ≠ 0 only). Reproduced from [74]. see that very high scales get probed. What is the reason for this? The first is that the kaon decay width, dominated by 𝐵𝑟(𝐾 + → 𝜇 + 𝜈𝜇) ≃ 64% and 𝐵𝑟(𝐾 + → 𝜋 +𝜋 0 ) ≃ 21%, i…
Figure 12
Figure 12. Figure 12: Typical shape of exclusions from beam dump experiments on ALP couplings to gluons 𝑁3/ 𝑓𝑎 as a function of ALP mass 𝑚𝑎. Dashed vertical line gives the kinematical endpoint. of a light ALP through its couplings to photons. Since the momentum exchange is small, the ALP “…
Figure 13
Figure 13. Figure 13: Production of ALPs in electron beam dumps from Primakoff process (left) and from electron bremsstrahlung (right). Electron beam dumps. In electron beam dumps the ALPs are produced in a Primakof process, where 𝑒 − in the electro-magnetic field of nucleus emits an ALP d…
Figure 14
Figure 14. Figure 14: Feynman diagrams for the ALP decay at tree level, with the associated Feynman rule for the vertex. In the remainder of this section we will illustrate the above procedure on a concrete example: the search for ALP–photon interactions at Belle II. The sequence of steps …
Figure 15
Figure 15. Figure 15: Feynman diagrams for the ALP production at 𝑒 + 𝑒 − colliders, assuming only coupling to two photons. The total cross section for 𝑒 + 𝑒 − → 𝛾𝑎 is obtained after a straightforward integration, which we leave as an exercise for the reader.2 The result is d𝜎 d cos 𝜃CM = 𝛼…
Figure 16
Figure 16. Figure 16: Left: The total cross section for ALP production in 𝑒 + 𝑒 − collisions at Belle II, via ALP-strahlung (blue) and photon fusion (orange). Right: the distribution of ALP momentum component along the beam axis, in the center of mass. a third particle in the final state i…
Figure 17
Figure 17. Figure 17: Feynman diagrams for the ALP decay at one-loop. In the lab frame, the conservation of linear momentum implies that the three particles, 𝑎 and 2𝛾, lie in a plane. In the ALP rest frame, the conservation of linear momentum thus implies that the two photons are back-to-b…
Figure 18
Figure 18. Figure 18: Different types of signals expected at Belle II in the parameter space spanned by 𝑚𝑎 and 𝑔𝑎𝛾𝛾. The hatched region indicates where the two photons in the displaced event cannot be resolved and become merged, see main text for details. Adapted from Ref. [85]. the direct…
Figure 19
Figure 19. Figure 19: Constraints in the parameters space of 𝑚𝑎 and 𝑔𝑎𝛾𝛾, from the invisible (blue) and resolved (orange) signatures, respectively. The dashed lines indicate the results rescaled to the full luminosity expected at Belle II. γ e − e − e − γ γ e + [PITH_FULL_IMAGE:figures/fu…
Figure 20
Figure 20. Figure 20: Feynman diagram for the main background process to the resolved photophilic ALP signature. 5.8 Experimental reach – the resolved signature We expect the resolved signature, 𝑒 + 𝑒 − → 𝛾(𝑎 → 𝛾𝛾), to be relevant for the high-mass part of the available ALP parameter space…
Figure 21
Figure 21. Figure 21: Data points measured at Belle II (black dots), compared with two simulations of the ALP signal. process 𝑒 + 𝑒 − → 𝛾𝛾𝛾, see fig. 20.7 Similarly to the photon-fusion production, it is faster to compute this cross section numerically, e.g., with MadGraph, than it is to c…

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