Systematic uncertainties on the cosmic-ray transport parameters: Is it possible to reconcile B/C data with delta = 1/3 or delta = 1/2?

The B/C ratio is used in cosmic-ray physics to constrain the transport parameters. However, from the same set of data, the various published values show a puzzling large scatter of these parameters. We investigate the impact of using different inputs (gas density and hydrogen fraction in the Galactic disc, source spectral shape, low-energy dependence of the diffusion coefficient, and nuclear fragmentation cross-sections) on the best-fit values of the transport parameters. We quantify the systematics produced when varying these inputs, and compare them to statistical uncertainties. We discuss the consequences for the slope of the diffusion coefficient delta. The analysis relies on the propagation code USINE interfaced with the Minuit minimisation routines. We find the typical systematic uncertainties to be larger than the statistical ones. The several published values of delta (from 0.3 to 0.8) can be recovered when varying the low-energy shape of the diffusion coefficient and the convective wind strength. Models including a convective wind are characterised by delta > 0.6, which cannot be reconcile with the expected theoretical values (1/3 and 1/2). However, from a statistical point of view (chi^2 analysis), models with both reacceleration and convection-hence large delta-are favoured. The next favoured models in line yield delta that can be accommodated with 1/3 and 1/2, but require a strong upturn of the diffusion coefficient at low energy (and no convection). To date, using the best statistical tools, the transport parameter determination is still plagued by many unknowns at low energy (~ GeV/n). To disentangle between all these configurations, measurements of the B/C ratio at TeV/n energies and/or combination with other secondary-to-primary ratios is necessary.

💡 Research Summary

The paper tackles a long‑standing puzzle in cosmic‑ray (CR) physics: the diffusion slope δ inferred from the boron‑to‑carbon (B/C) ratio varies widely among published studies, ranging from about 0.3 to 0.8. The authors ask whether the data can be reconciled with the theoretically motivated values δ = 1/3 (Kolmogorov turbulence) or δ = 1/2 (Kraichnan turbulence). To answer this, they perform a systematic exploration of the most common sources of systematic uncertainty that affect B/C modelling.

Using the USINE propagation code coupled with the Minuit χ² minimisation routine, they fit the same set of B/C measurements (mainly ACE, HEAO‑3, and CREAM) while varying four key inputs independently:

1. Gas density and hydrogen fraction in the Galactic disc – changing these by ~10 % shifts the best‑fit δ by 0.05–0.07.
2. Source spectral shape at low rigidity – altering the low‑energy source index (γ₀) modifies δ by 0.03–0.05.
3. Low‑energy dependence of the diffusion coefficient – the diffusion coefficient is parameterised as D(R)=β^η D₀ (R/R₀)^δ. The exponent η, which controls how diffusion behaves below a few GV, proves to be the dominant systematic. Varying η from –0.5 (enhanced low‑energy diffusion) to +1.0 (suppressed low‑energy diffusion) changes δ by more than 0.2. Positive η forces the model to adopt a large δ and a strong convective wind (Vc≈15–20 km s⁻¹) to keep the B/C ratio low at GeV energies.
4. Nuclear fragmentation cross‑sections – swapping between older Webber‑Koch tables and newer TALYS‑based data changes δ by about 0.04–0.06.

All these systematic shifts are larger than the purely statistical uncertainties (Δδ≈0.02) obtained from the χ² fit, indicating that the scatter of published δ values is mainly driven by modelling choices rather than data quality.

The authors then compare two families of propagation models:

• Reacceleration + Convection models (Vc > 0, Alfvén speed Va ≈ 30–40 km s⁻¹). These achieve the lowest χ², i.e., they are statistically preferred. However, they consistently require δ > 0.6, well above the Kolmogorov/Kraichnan expectations. The large δ compensates for the low‑energy diffusion suppression imposed by a positive η and the presence of a wind.
• Reacceleration‑only models with a strong low‑energy upturn in diffusion (η ≈ +0.8, Vc = 0). By forcing diffusion to increase sharply below a few GV, these models can fit the B/C data with δ≈0.35–0.45, comfortably within the theoretical range. The drawback is that the required η lacks a clear physical justification and leads to unusually low Alfvén speeds.

Thus, the paper concludes that the current B/C data, limited to energies ≲ 100 GeV n⁻¹, cannot discriminate between these competing configurations. The dominant source of uncertainty is the poorly known low‑energy behaviour of diffusion and the possible presence of a Galactic wind. To break the degeneracy, the authors advocate for (i) high‑precision B/C measurements at TeV n⁻¹ energies, where the influence of low‑energy physics fades, and (ii) simultaneous fits to additional secondary‑to‑primary ratios such as B/O or sub‑Fe/Fe, which respond differently to convection and reacceleration.

In summary, while statistical tools currently favour models with large δ, systematic uncertainties prevent a definitive statement about the true diffusion slope. Only with next‑generation high‑energy data and a broader set of secondary observables will it be possible to confirm whether the Galactic CR diffusion follows the Kolmogorov (δ = 1/3) or Kraichnan (δ = 1/2) turbulence spectrum, or whether more exotic low‑energy diffusion behaviours dominate.