Angular Distribution of Gamma Rays Produced in Proton-Proton Collisions

Annika H. G. Peter; John F. Beacom; Jung-Tsung Li; Spencer Griffith

REVIEW 3 major objections 5 minor 87 references

Angular Distribution of Gamma Rays Produced in Proton-Proton Collisions

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper provides the joint energy-angle yield of gamma rays from proton-proton collisions, from the pion threshold to 100 GeV, and shows that the gamma rays are measurably non-collinear at the low energies that dominate astrophysical…

desk verdict A genuinely useful public resource for gamma-ray modelers, held back by the fact that the new angular dimension is never independently validated. read the letter →

arxiv 2412.08726 v2 pith:67NYFZWS submitted 2024-12-11 hep-ph astro-ph.HE

Spencer Griffith , John F. Beacom , Jung-Tsung Li , Annika H. G. Peter This is my paper · ORCID

classification hep-phastro-ph.HE

keywords gamma-rayproductionproton-protoncollisionsangulardistributioncosmicrayshadronicinteractionspiondecayastrophysicalgammaMonteCarlosimulations

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

The paper's target is the joint energy-angle gamma-ray yield of a single proton-proton collision, $d^2N_\gamma/dE\,d\Omega$, over proton kinetic energies from threshold to 100 GeV and photon energies above 100 MeV. Its central claim is that collinearity—the assumption that gamma rays follow the parent proton's direction—fails badly in exactly the regime where much of the astrophysical yield lives, so modellers need the correlated distribution rather than the angle-integrated spectrum alone. The authors provide that distribution as public tables with a software tool, validate their angle-integrated spectra against established results, and add an approximate analytic form. This matters because off-axis jets, the Sun, and the Moon all produce observable gamma rays whose interpretation depends on the off-axis contribution, which is largest at low proton energy and low photon energy where existing treatments are least reliable.

What carries the argument

The carrying mechanism is a particle-transport Monte Carlo code calibrated to accelerator data, used to simulate inelastic proton-proton collisions and the subsequent decays of pions and other secondaries, with gamma-ray energy binned logarithmically ($\Delta\log E_\gamma=0.125$) and angle in $5^\circ$ bins. The paper's central objects are the per-interaction joint yield $F_{p,\gamma}(E_p,E_\gamma,\theta)$ and its dimensionless form $x\,d^2N_\gamma/dx\,d\Omega$ with $x=E_\gamma/E_p$, chosen to exploit approximate Feynman scaling. The ancillary analytic fit, $f(x,\theta,E_p)=A(x,E_p)\exp[-B(x,E_p)\theta^{I(E_p)}]$, captures the exponential suppression with angle and resolves the peak position through the $e^{-A_1 x^2}-e^{-A_3 x}$ factor. The critical kinematic ingredient is that pion decay produces a flat-in-log photon spectrum centered near $E_\gamma=E_\pi/2$, so backward-emitted photons fall below the 100 MeV threshold and the forward beaming cone widens as energy drops.

What would settle it

Measure the gamma-ray yield per collision in a fixed energy-angle bin, for example near $E_p=5.6$ GeV, $E_\gamma \sim 200$ MeV, and $\theta \sim 60^\circ$, using an independent hadronic simulation or a dedicated accelerator measurement; if the measured or independently computed yield disagrees with the tables beyond the estimated hadronic uncertainties, the angular component of the central claim fails.

Watch

Extended reading notes

Core claim

The core claim is that gamma-ray production in $pp$ collisions is measurably non-collinear up to primary energies of tens of GeV, and that the full double-differential yield per inelastic collision, expressed as $E_\gamma F_{p,\gamma}(E_p,E_\gamma,\theta)$, can be tabulated with enough accuracy to replace the collinear approximation. The paper shows that near threshold the yield extends beyond $90^\circ$ with low-energy photons, that the spectral shape depends sharply on viewing angle, and that even at high $E_p$ a low-energy off-axis component persists. It further claims that by roughly 65 GeV a $10^\circ$ cone contains 90% of the $\geq100$ MeV gamma rays, establishing 100 GeV as the energy above which the old collinear formulas suffice. The compiled tables, the analytic fit, and the validation against angle-integrated spectra together constitute the deliverable.

Load-bearing premise

The angular shape of the simulated secondary production is trusted as accurate; the paper validates only the angle-averaged spectra against independent calculations, so a systematic error in the off-axis yield would invalidate the tables even if the total spectra match.

Editorial extensions

If this is right

Above roughly 65 GeV, a $10^\circ$ cone contains 90% of the $\geq100$ MeV yield, so for higher-energy systems the collinear approximation remains valid and the new tables are not needed.
For a power-law cosmic-ray spectrum like $\propto E_p^{-2.2}$, protons near threshold dominate, so a large fraction of the observed gamma rays are emitted at significant angles; jet-cloud models that assume collinearity will mispredict off-axis spectra.
The gamma-ray spectrum shape varies with viewing angle, so in principle a measured spectrum can be inverted to determine the direction of an unseen cosmic-ray beam.
Solar gamma rays from cosmic rays reflected by magnetic fields require the joint distribution to predict the disk-versus-annulus brightness pattern; the paper sketches how the brightness moves from a thin limb ring to a full disk as photon energy decreases.
The analytic fit keeps maximum error under about 40% over the first decade above threshold (away from sharp pion-multiplicity thresholds), making the distribution cheap enough for repeated use in astrophysical simulations.

Reading between the lines

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

If the tables are correct, lunar gamma rays are a clean test: the Moon lacks a magnetic field, so only gamma rays emitted at large angles can escape, making the observed limb profile a direct read-off of the off-axis yield.
A natural extension is to generate analogous tables for proton-helium and alpha-proton collisions rather than a simple nuclear enhancement factor, because intranuclear cascades could alter the angular spread at low energy.
The exponential form of the analytic fit suggests a compact universal scaling in $\theta^{I(E_p)}x$ that could be tested across independent Monte Carlo codes; if it holds, the distribution reduces to a one-dimensional family of curves.
The analogy drawn to off-axis neutrino beams suggests the same tables could be applied to studies of beam-dump and collider neutrino fluxes where parent meson direction matters.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a useful phenomenological resource, not a physics breakthrough. What is actually new is a public FLUKA-based database and Python tool for d2Nγ/dE dΩ in pp collisions from threshold to 100 GeV, plus an honest 29-parameter analytic fit. The angle-integrated comparison to Kafexhiu et al. is the right sanity check and it passes. The paper also deserves credit for shipping tables and code, for the cone-containment plots that quantify the onset of collinearity, and for explicitly stating where the fit fails.

The stress-test note lands, though not fatally. The central new quantity is the angular shape from FLUKA, and that shape is never checked against an independent model or experimental data. Agreement after integrating over angle constrains only dN/dE; a too-narrow or too-wide pion angular distribution with compensated total yield would pass Figure 4. FLUKA's calibration to hadronic data is cited, but no pion angular distribution comparison is shown. Since the double-differential yield is the advertised product, this is a real gap. The tables also carry no quoted statistical or systematic uncertainties, so users cannot tell how much risk they are taking in the non-collinear regime. A quantitative comparison to Karlsson and Kamae would partially address this; the paper only discusses that prior model qualitatively. The fit itself is openly a fit, so there is no circularity problem, and the ncut choice is disclosed and reasonable.

One place the stress test is too harsh: this is not a load-bearing flaw in the sense that the whole enterprise is wrong. FLUKA's track record plus the angle-integrated agreement mean the tables are probably usable with caution. But the paper slightly overstates its own support when it leans on FLUKA's general calibration without testing the specific double-differential quantity being delivered.

The paper is for modelers working on off-axis jets, solar gamma rays, and lunar emission with Fermi-LAT; they will want this tool. It deserves a serious referee. The main revision requests should be uncertainty estimates on the tables and either an explicit angular validation or a clearly scoped caveat that the angular shapes inherit FLUKA's unquantified systematics. I would accept it with major revisions rather than desk-reject.

Referee Report

3 major / 5 minor

Summary. This paper calculates the double-differential gamma-ray yield d²Nγ/dE dΩ from inelastic proton-proton collisions for incident proton energies from 1.33 to 100 GeV, using FLUKA simulations for pion production and secondary decays. The authors present the results as public tables and a Python tool, validate the angle-integrated yields against Kafexhiu et al. and check the inclusive π0 cross section, provide an analytic fit over the first decade above threshold with quantified errors, and discuss applications to off-axis jets and solar gamma rays.

Significance. If the tabulated results are accurate, the paper fills a genuine gap: the commonly used models provide dNγ/dE only, while the earlier angular treatments either cover different energy regimes or are less accessible. The public database and Python tool are a useful community resource for Fermi-LAT-era modeling, and the paper is explicit that the analytic formula is an ad-hoc fit with quantified errors rather than a prediction. The main risk is that the angular dimension itself, which is the central deliverable, is not independently validated against experiment or another angular model. The paper's angle-integrated agreement with Kafexhiu et al. is necessary but not sufficient to certify the off-axis and low-energy regime for which the tables are intended.

major comments (3)

[Sec. III, Fig. 4] The only quantitative validation shown is the solid-angle-integrated yield compared with Kafexhiu et al.; this constrains dN/dE only. The abstract's deliverable is d²N/dE dΩ, and the angular shape of pion production is never checked against experiment, against the earlier model of Karlsson & Kamae [45], or against an alternative Monte Carlo. Because off-axis and low-energy users rely precisely on the angular shape, the paper should add a direct comparison of angular distributions (e.g., π0 or γ-ray yields at representative Ep) or otherwise provide a quantitative estimate of FLUKA's angular uncertainty. Without this, the central product is not yet supported.
[Sec. III (public tables and GitHub database)] The primary tabular results carry no quoted statistical or systematic uncertainties per bin. The analytic fit errors are quantified in Sec. IV, but the tables themselves are presented without error bars, so the reader cannot assess the significance of discrepancies in the non-collinear regime. Please state the number of simulated events per energy and provide at least a global systematic uncertainty estimate, for example from FLUKA version dependence or from comparison with an alternative code.
[Sec. IV, Eqs. (15)-(16) and Fig. 7] The analytic fit is restricted to 0°–90° and to the first decade above threshold, with maximum errors of roughly 40% and larger errors above ~13.3 GeV. This is clearly stated in the text, but the abstract's advertised range of 'the pion production threshold to 100 GeV' could mislead users into applying the formula outside its fitted domain. The text should state prominently that the analytic formula is only for the first decade and that the tables are the intended product for the full range.

minor comments (5)

[Sec. III, Eq. (9)] In the sentence following Eq. (9), 'Fp,γ(Ep, Eγ, θ), defined similarly to Fp,γ(Ep, Eπ, θ)' should refer to Fp,π in the second occurrence; the pion index is lost.
[Sec. III, Eq. (12)-(13)] The cutoff ncut depends on xmin = 100 MeV/Ep, which varies with primary energy; please state this dependence explicitly when defining ncut.
[Sec. III, Fig. 2] The caption notes that x ranges and colorbar ranges vary among panels, which makes quantitative cross-panel comparison difficult; consider adding a fixed reference scale or contour labels.
[Sec. V.A, Eq. (21)-(22) and Fig. 9] The power-law integration in the jet example uses an arbitrary high-energy cutoff at 1000 GeV; please state in the text or caption that the displayed spectra may depend on this cutoff choice.
[Sec. III, simulation setup] The GitHub repository URL is given, but the text does not state the exact FLUKA version or release used to generate the tables; because FLUKA results can change between releases, this information should be included with the tables.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central FLUKA-based tables are independent simulated outputs, and the analytic formula is an openly labeled fit with quantified errors.

full rationale

The paper's central deliverable, d2Nγ/dE dΩ from 1.33 to 100 GeV, is generated by FLUKA simulations of proton-proton collisions and pion/secondary decays (Sec. III), not derived from the analytic formula. The comparison with Kafexhiu et al. (Fig. 4) checks the angle-integrated yield against an independent model built from other Monte Carlos and data, providing external support for the energy spectrum; the lack of an independent check of the angular shape is a validation gap, not a circular reduction. The analytic formula in Sec. IV is explicitly introduced as a fit ('we fit our numerical results ... with an ad-hoc functional form'), with the residuals quantified by Eq. (14), so the 29 constants are not disguised predictions. Citations to the authors' own solar-gamma-ray papers (e.g., Refs. [17, 26, 29-31]) are contextual applications and do not support the FLUKA calculation itself, and there is no imported uniqueness theorem or ansatz smuggled in by citation. No equation reduces to its own input by construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The ledger is dominated by trust in FLUKA and by the fit choices. The paper is transparent that the analytic formula is an ad hoc fit and that the low-yield cutoff is arbitrary, so these entries are recorded as disclosed assumptions rather than hidden ones.

free parameters (2)

Analytic fit coefficients Aij, Bkl, Im = See Table II; 29 constants
These constants are obtained by fitting the FLUKA double-differential tables with the ad hoc form in Eqs. (15)-(19), so they are a compact encoding of the simulated yields, not derived quantities.
Yield cutoff ncut = Set by Eq. (13); chosen so 100 interactions would over-predict one gamma ray if every bin were at ncut
The cutoff defines which bins are treated as zero when computing fit errors (Eqs. (12)-(14)); it is an arbitrary but disclosed choice that affects the reported accuracy of the formula.

assumptions (6)

domain assumption FLUKA's hadronic interaction models correctly reproduce the double-differential gamma-ray yield of proton-proton collisions over the studied energy range.
All numerical results in Sec. III are FLUKA output; the only external validation is the angle-integrated comparison to Kafexhiu et al., so the angular shape is assumed accurate.
domain assumption The inelastic pp cross section and its energy dependence used for normalization (Fig. 1) are correct.
The cross section from PDG and Dermer is used in Sec. III to convert yields per inelastic collision and to compare with Kafexhiu et al.
domain assumption Feynman scaling in x = Eγ/Ep is a sufficiently good organizing variable over 1.33-100 GeV.
The presentation and fit use this scaling; the paper notes it is approximate and is approached only at high Ep.
domain assumption Gamma rays below 100 MeV can be ignored for the Fermi-LAT applications considered.
The 100 MeV threshold is imposed in Sec. II and truncates all results; it is appropriate for Fermi-LAT but not for other instruments.
ad hoc to paper The functional form f = A exp(-B theta^I) is flexible enough to represent the simulated yields in the fitted range.
Sec. IV states the form is not theoretically motivated and was selected by inspecting data slices; the fit is not a physics derivation.
ad hoc to paper Bins below ncut contribute negligibly to likely applications and can be zeroed.
Eqs. (12)-(14) define this cutoff, and the paper cautions it may not suit every application, e.g., lunar gamma rays.

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Pith. "Pith review of Angular Distribution of Gamma Rays Produced in Proton-Proton Collisions." pith.science (2026). https://pith.science/paper/67NYFZWS

@misc{pith2026241208726,
  author       = {Pith},
  title        = {Pith review of: Angular Distribution of Gamma Rays Produced in Proton-Proton Collisions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/67NYFZWS}},
  note         = {Machine review of arXiv:2412.08726}
}

abstract

Accurate modeling of how high-energy proton-proton collisions produce gamma rays through the decays of pions and other secondaries is needed to correctly interpret astrophysical observations with the Fermi-LAT telescope. In the existing literature on cosmic-ray collisions with gas, the focus is on the gamma-ray yield spectrum, $d N_\gamma/dE$. However, in some situations, the joint energy and angular distribution can be observed, so one needs instead $d^2 N_\gamma/dE \, d\Omega$. We provide calculations of this distribution over the energy range from the pion production threshold to $100~{\rm GeV}$, basing our results on FLUKA simulations. We provide the results in tabular form and provide a Python tool on GitHub to aid in utilization. We also provide an approximate analytic formula that illuminates the underlying physics. We discuss simplified examples where this angular dependence can be observed to illustrate the necessity of taking the joint distribution into account.

Figures

Figures reproduced from arXiv: 2412.08726 by the authors.

**Figure 2.** FIG. 2. Double-differential gamma-ray yields for a representative selection of primary energies, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Gamma ray yields integrated over solid angle to give [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: FIG. 5. Angular cone (defined by the half-angle off the beam [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Gamma-ray yield above 100 MeV in different angular ranges relative to the beam direction, [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: FIG. 7. Analytic function modeling [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: FIG. 9. Gamma-ray spectrum for a target illuminated by [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 8.** Figure 8: FIG. 8. Parameters used in Eqs. ( [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

Discussion (0). Continue with ORCID to comment.

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