Revised production cross-section of gamma-rays in p-p collisions with LHC data for the study of TeV gamma-ray astronomy

We present the production cross-section of gamma-rays based on data of p-p collisions at LHC, revising the previous semi-empirical formula mainly for 1) the inelastic cross-section in p-p collisions, $\sigma_{pp}(E_0)$, and 2) the inclusive gamma-ray spectrum in the forward region, $\sigma_{pp \rightarrow \gamma}(E_0, E_\gamma)$. We find that the previous cross-section gives a significantly softer spectrum than found in the data of LHC. In this paper, we focus our interest mainly upon the LHC forward (LHCf) experiment, giving gamma-ray spectra in the very forward region with the pseudo-rapidity $\eta^$ > 8.8 in the center of mass system (CMS), which have not been reported so far. We also give the pseudo-rapidity distribution of charged hadrons with -3 < $\eta^$ < 3 obtained by ALICE and TOTEM experiments, both with LHC. We find that the revised cross-section reproduces quite well the accelerator data over the wide energy range from GeV to 30 PeV for projectile protons, corresponding approximately to 100 MeV to 3 PeV for secondary gamma-rays. The production cross-section of gamma-rays produced in the forward region is essential for the study of gamma-ray astronomy, while not important are those produced in the central region in CMS, and of much less importance in the backward. We discuss also the average transverse momentum of gamma-rays, $\bar{p}{t}$, and the average inelasticity transferred to gamma-rays, $\bar{k}\gamma^$, obtaining that the former increases very slowly with $\bar{p}{t}$ = 100 - 220 MeV/c for $E_0$ = 1 GeV - 26 PeV, and the latter is almost independent of $E_0$, with $\bar{k}\gamma^ \approx 1/6$, while we can not exclude the possibility of a small increase of $\bar{k}_\gamma^*$.

💡 Research Summary

The paper addresses a critical gap in high‑energy gamma‑ray astronomy: the lack of an accurate production cross‑section for photons generated in proton‑proton (p‑p) collisions at energies relevant to TeV–PeV astrophysical sources. Existing semi‑empirical formulas, widely used in cosmic‑ray propagation and air‑shower simulations, were derived mainly from central‑rapidity collider data and therefore underestimate the hardness of the photon spectrum in the very forward region (pseudo‑rapidity η* > 8.8). This underestimation propagates into astrophysical models, leading to systematic biases when interpreting observations of ultra‑high‑energy gamma rays from supernova remnants, pulsar wind nebulae, or extragalactic sources.

Data Sources and Methodology
The authors exploit two complementary LHC data sets:

1. LHCf (Large Hadron Collider forward) experiment – provides photon energy spectra measured at η* > 8.8 for √s = 7 TeV and 13 TeV p‑p collisions. These data probe the extreme forward fragmentation region, where most of the energy carried by the projectile proton is retained in a narrow cone and subsequently released as high‑energy photons.

2. ALICE and TOTEM experiments – deliver charged‑hadron pseudo‑rapidity distributions in the central region (−3 < η* < 3). Although these measurements are not directly photon‑focused, they constrain the overall inelasticity and particle‑production mechanisms, allowing a consistent description of the whole rapidity range.

By simultaneously fitting the forward photon spectra and the central charged‑hadron distributions, the authors construct a unified parameterisation that replaces two separate ingredients of the older model:

• Inelastic cross‑section σ_pp(E₀) – the total probability for a non‑elastic p‑p interaction as a function of the projectile kinetic energy E₀. The new expression combines a logarithmic‑square term (reflecting the rise of σ_pp at high energies) with a low‑energy power‑law component, calibrated against LHC measurements and the Particle Data Group (PDG) world average. This yields a smooth, continuous function from a few GeV up to 30 PeV (the latter corresponding to the laboratory energy of a 3 PeV secondary photon).

• Inclusive forward photon spectrum σ_pp→γ(E₀, E_γ) – the differential cross‑section for producing a photon of energy E_γ in the forward cone. The authors decompose the spectrum into (i) a direct electromagnetic component (prompt photons from parton‑level processes) and (ii) a decay component (π⁰, η, and other neutral meson decays). The decay component is linked to the measured charged‑hadron multiplicities, while the prompt term is constrained by the LHCf data. The resulting spectrum is significantly harder than the previous semi‑empirical formula, with a typical photon energy fraction of about 1/6 of the projectile’s kinetic energy.

Key Physical Findings

• Average transverse momentum ⟨p_t⟩ – extracted from the forward photon kinematics, ⟨p_t⟩ rises slowly from ≈100 MeV/c at E₀ = 1 GeV to ≈220 MeV/c at E₀ = 26 PeV. This modest increase reflects the limited transverse momentum transfer in the fragmentation region, even at extreme energies.

• Average inelasticity transferred to photons k̄_γ* – remains essentially constant at ≈0.16–0.17 across the entire energy range. In other words, about one‑sixth of the inelastic energy loss of the projectile proton ends up as forward photons, independent of E₀. The authors note a possible slight upward trend at the highest energies, but it is not statistically significant.

• Dominance of forward production – The study confirms that photons emitted in the very forward region dominate the high‑energy gamma‑ray yield relevant for atmospheric shower development. Central‑rapidity photons contribute negligibly to the observable TeV–PeV flux, while backward‑rapidity photons are even less important.

Implications for Gamma‑Ray Astronomy

The revised cross‑section can be directly implemented in Monte‑Carlo codes such as CORSIKA, AIRES, or CRMC, improving the fidelity of air‑shower simulations that underpin the interpretation of data from ground‑based gamma‑ray observatories (e.g., H.E.S.S., MAGIC, CTA) and extensive air‑shower arrays (e.g., HAWC, LHAASO). By providing a harder photon spectrum, the model predicts higher fluxes of multi‑TeV photons for a given cosmic‑ray proton spectrum, reducing the discrepancy that has long existed between observed gamma‑ray intensities and those derived from conventional hadronic interaction models.

Furthermore, the near‑constant photon inelasticity simplifies analytic treatments of cosmic‑ray propagation in interstellar media. Researchers can now assume that roughly 1/6 of the kinetic energy lost by high‑energy protons in collisions with ambient gas appears as forward gamma rays, independent of the proton energy. This facilitates more accurate estimates of gamma‑ray production in dense environments such as molecular clouds adjacent to supernova remnants.

Future Directions

The authors stress the importance of extending forward measurements to even higher pseudo‑rapidities (η* ≈ 10–12) and to the upcoming 14 TeV LHC run, as well as to proton–nucleus collisions that more closely mimic cosmic‑ray interactions with interstellar gas. Additional data would allow refinement of the decay component (especially contributions from η and heavier mesons) and could reveal subtle energy‑dependent effects in k̄_γ*. Moreover, integrating the revised cross‑section into full astrophysical models (including magnetic field effects, source spectra, and propagation) will test its impact on multi‑wavelength observations and on the inferred properties of cosmic‑ray accelerators.

Conclusion

By leveraging the most forward LHC measurements to date, the paper delivers a self‑consistent, energy‑spanning parameterisation of the p‑p → γ cross‑section that resolves longstanding deficiencies in the modeling of high‑energy gamma‑ray production. The revised formula reproduces accelerator data from a few GeV up to 30 PeV, predicts a modest increase in average transverse momentum, and confirms an essentially energy‑independent photon inelasticity of ≈1/6. These results constitute a vital upgrade for the theoretical toolkit of TeV–PeV gamma‑ray astronomy, enabling more reliable interpretation of current and forthcoming observations of the most energetic photons in the universe.