Fit coefficients for $X_{\mathrm{max}}$ mean and variance parameterisations (SIBYLL/EPOS/QGSJet)

$X_{\mathrm{max}}$ moment and distribution parameterizations (CONEX-based) 

This Zenodo repository contains plain-text outputs used to build and validate parametrizations of the atmospheric depth of shower maximum, $X_{\mathrm{max}}$, derived from CONEX simulations.More details can be found on arXiv:

All files are provided separately for the three hadronic interaction models:- SIBYLL 2.3e- EPOS LHC-R- QGSJet III-01

Contents

1) CONEX moments tables: conex_*.xmaxTables of $X_{\mathrm{max}}$ moments directly extracted from the CONEX simulations.

For each hadronic interaction model and primary nucleus, the file reports (as a function of energy):- the mean $\langle X_{\mathrm{max}} \rangle$ and its statistical uncertainty,- the variance $\sigma^2(X_{\mathrm{max}})$ and its uncertainty.

Units- $\langle X_{\mathrm{max}} \rangle$: g cm$^{-2}$- $\sigma^2(X_{\mathrm{max}})$: g$^2$ cm$^{-4}$

2) CONEX binned distributions: conex_*.histBinned $X_{\mathrm{max}}$ distributions from the CONEX simulations.

- Range: $X_{\mathrm{max}} \in [0, 2000]$ g cm$^{-2}$- Number of bins: $N = 100$- Bin width: $\Delta X_{\mathrm{max}} = 20$ g cm$^{-2}$

The entries are raw counts per bin.

3) Mean $X_{\mathrm{max}}$ parametrization coefficients: xmax_mean_*_EVPS2026.txtBest-fit coefficients of the parametrization for the mean $X_{\rm max}$.

File format1. Row 1: best-fit parameters $(a_0, a_1, b_0, b_1)$2. Row 2: corresponding parameter uncertainties3. Row 3: $\chi^2$ and the number of degrees of freedom (ndf)

4) Variance parametrization coefficients: xmax_sigma2_*_EVPS2026.txtBest-fit coefficients of the parametrization for the variance $\sigma^2(X_{\rm max})$.

A keyword in the filename specifies the chosen functional form for $\sigma^2$, e.g.- `pol2` (polynomial)- `exp` (exponential)

File format1. Row 1: best-fit parameters     - Polynomial model: $(c_0, c_1, c_2, d_0, d_1, d_2)$     - Exponential model: $(f_0, f_1, g_0, g_1)$2. Row 2: corresponding parameter uncertainties3. Row 3: $\chi^2$ and ndf

5) Generalized Gumbel distribution coefficients: gumbel_params_*_EVPS2026.txtBest-fit coefficients of the generalized Gumbel parametrization of the full $X_{\rm max}$ distributions.

For each hadronic interaction model, the file reports the 21 coefficients entering the parametrizations of the three Gumbel parameters $(\mu, \beta, \lambda)$.

Coefficients are denoted as:- $p_k^{(i)}$ with $i \in \{\mu, \beta, \lambda\}$ and $k = 0, 1, 2$

Notes on usage

- All files are provided in plain-text format for ease of parsing in analysis pipelines.- Histogram files contain unnormalized counts; users should normalize as needed for likelihood fits or PDF construction.- The parametrization coefficient files include both best-fit values and uncertainties, plus goodness-of-fit information ($\chi^2$, ndf).

Suggested citation

If you use these data products or parametrizations in a publication, please cite:

Contact

For questions, bug reports, or requests (e.g., additional primaries/models or format clarifications), please contact the authors at [e-mail].