EKO benchmarks are listed in the table below and are implemented in a separated tool ekomark. For each external program the evolution can be performed at LO, NLO, NNLO.

Available Benchmarks



Scale Variations



VFNS, FFNS (nf=4)




iterate-exact, iterate-expanded, truncated



iterate-exact, iterate-expanded, ordered-truncated, truncated

Les Houches Benchmarks

The benchmarking LHA reference is given by [G+02] (LO and NLO) and [D+05] (NNLO).

List of bugs in [G+02]

  • \(L_+ = 2(\bar u + \bar d)\)

  • head of table 1: \(\alpha_s(\mu_R^2 = 10^4~\mathrm{GeV}^2)=0.117574\) (FFN) - as pointed out by [D+05]

  • in table 3, part 3: \(xL_-(x=10^{-4}, \mu_F^2 = 10^4~\mathrm{GeV}^2)=1.0121\cdot 10^{-4}\) (wrong exponent) and \(xL_-(x=.1, \mu_F^2 = 10^4~\mathrm{GeV}^2)=9.8435\cdot 10^{-3}\) (wrong exponent)

List of bugs in [D+05]

  • in table 15, part 1: \(xd_v(x=10^{-4}, \mu_F^2 = 10^4~\mathrm{GeV}^2) = 1.0699\cdot 10^{-4}\) (wrong exponent) and \(xg(x=10^{-4}, \mu_F^2 = 10^4~\mathrm{GeV}^2) = 9.9694\cdot 10^{2}\) (wrong exponent)

  • in table 16: the distribution \(L_{m}\) is not vanishing despite the benchmark table being produced with the toy PDF, defined in 4.57, that has \(\bar{u} = \bar{d}\). This means that \(T_3\) is not correct. The whole column of \(L_{m}\) should be set to 0, sine at LO it is not possible to generate a difference between \(\bar{u}\) and \(\bar{d}\)

  • in table 17: same as before \(L_{m}\) is bugged.


lhapdf is the standard tool to store PDFs in Particle Physics. It provides a PDF dependent evolution method which can be compared with EKO applied to the same initial PDF.


APFEL [BCR14] is a tool aimed to the evolution of PDFs and DIS observables’ calculation (and FTDY as well). It has been used by the NNPDF collaboration up to NNPDF4.0

APFEL solves DGLAP numerically in x-space up to NNLO. QED evolution is also available. The programs provides 3 different strategies, and in various theory setups (FNS, SV, IC ) as shown in the table. As EKO, APFEL can be interfaced with lhapdf.


Pegasus [Vog05] is a tool aimed exclusively to the evolution of PDFs, it is written in Fortran. This program has been used to produce the LHA tables.

Pegasus solves DGLAP numerically in N-space up to NNLO. The programs provides 3 different strategies, with various FNS and Scale Variations as shown in the table. Pegasus takes as input a pdf with a fixed functional form and it’s not interfaced with lhapdf. Also the starting scale must be equal to the scale at which the reference value of \(\alpha_s\) is provided.