This repository contains the C++ codes used to benchmark the inclusive deep-inelastic-scattering (DIS) structure functions at $\mathcal{O}(\alpha_S^3)$, i.e. N³LO, as implemented in apfel++ and Hoppet. The benchmark is presented in:

• V. Bertone and A. Karlberg, Benchmark of deep-inelastic-scattering structure functions at $\mathcal{O}(\alpha_S^3)$, arXiv:2404.15711,

along with the suite of Matplotlib scripts used to produce the plots shown in that paper and many others.

In order to compile the code, both apfel++ and Hoppet must be installed. See:

• apfel++,
• Hoppet,

for the respective installation instructions.

To compile the benchmark codes, simply go into the code folder and type:

make [-j]

This will produce three executables. They all use the same PDF initial condition of section 1.3 in http:https://arxiv.org/abs/hep-ph/0204316.

In the code folder, running the code:

./StructureFunctionsJoint

will print on the screen comparison tables between apfel++ and Hoppet for different values of $x_{\rm B}$ and $Q$ and for all perturbative orders from LO to N³LO. On top of the structure functions, the tables also show a few PDF combinations. Results on screen are presented as |apfel++ / Hoppet - 1|. This code will also dump into the plots folder the files: StructureFunctions_N0LO.dat, StructureFunctions_N1LO.dat, StructureFunctions_N2LO.dat, and StructureFunctions_N3LO.dat that can be used to produce a number of plots (see below). Moreover, also the files: table_N0LO_APFEL.tex, table_N1LO_APFEL.tex, table_N2LO_APFEL.tex, table_N3LO_APFEL.tex, table_N0LO_HOPPET.tex, table_N1LO_HOPPET.tex, table_N2LO_HOPPET.tex, and table_N3LO_HOPPET.tex will be created in the tables folder.

Similarly to the code above, running the code:

./TabulateStructureFunctions

will print on screen comparison tables for both apfel++ and Hoppet, this time showing the values of structure functions and, again, a few PDF combinations.

Finally, running the code:

./ScaleVariations

will produce files in the plots folder for the structure function $F_2^{\rm NC}$ evaluated at multiple values of the renormalisation and factorisation scales $\mu_{\rm R}$ and $\mu_{\rm F}$. The files produced are: F2NC_Scale_Variations_N0LO.dat, F2NC_Scale_Variations_N1LO.dat, F2NC_Scale_Variations_N2LO.dat, and F2NC_Scale_Variations_N3LO.dat and can be used to produce plots.

The plots folder, where some of the codes described above dump part of their output, also contains some python scripts that utilise matplotlib to produce plots in .pdf format. The scripts are:

• PerturbativeConvergence.py
• ScaleVariations.py
• StructureFunctions_N0LO.py
• StructureFunctions_N1LO.py
• StructureFunctions_N2LO.py
• StructureFunctions_N3LO.py

and are responsible for the production of all of the plots contained in the plots folder.

The tables folder contains the benchmark tables presented in the paper in LaTeX format. The file tables.tex collects the table files produced by the StructureFunctionsJoint code and can be compiled with LaTeX producing this file.

• Valerio Bertone: [email protected]
• Alexander Karlberg: [email protected]