N3LO cross sections calculator
A tool suite to calculate up to N3LO in QCD various cross sections at hadron colliders:
- Neutral Drell-Yan p p(pbar) --> gamma*/Z + X (--> l+ l- + X)
- Charged Drell-Yan p p(pbar) --> W+/W- + X (--> l+ nu_l / l- ~nu_l + X)
- Higgsstrahlung p p(pbar) --> W+/W- H + X
- Higgsstrahlung p p(pbar) --> Z H + X
- Higgs production via gluon fusion g g --> H in the Born-improved approximation
- Bottom-quark fusion Higgs production b bbar --> H + X in the five-flavor scheme
Prerequisites:
- A C++11 compatible C++ compiler.
- Python 3.5 or higher.
- The LHAPDF library installed on your computer.
The code requires GNU Scientific Library (gsl) version 2.6 or higher. The library is shipped with the code.
Compilation of the code:
This is done in several steps. First untar the archive with
$ tar xzf n3loxs.tar.gzand then enter the main directory with
$ cd ./n3loxsNow compile the GSL library using the shell script makegsl.sh:
$ ./makegsl.shOnce it is done, please make sure that the LHAPDF config program is
correctly assigned in the Makefile. If you have a custom
installation of LHAPDF on your computer, update the Makefile:
LHAPDFCONFIG = [absolute path to your LHAPDF installation]/bin/lhapdf-configAfter this check, simply enter the following command to create all the subprograms:
$ ./make allThe main executable is the Python script n3loxs. It accesses the
subprograms located in the directory subprograms. If you change the
location of the main executable, you have to update the 51st line of
the script so that it accesses correctly the subprograms:
maindir=[put here the absolute path of the directory n3loxs on your computer in single quotes]The program calculates hadronic cross sections for various processes up to N3LO in QCD:
- Drell-Yan (DY) processes are calculated for an off-shell gauge boson (photon/Z and W bosons) at a given virtuality Q, that is chosen as the central scale by default. The user can choose between calculating the differential cross section Q^2*dxs/dQ^2 at a given Q, or calculating the integrated cross section in a bin range Qmin <= Q <= Qmax.
- The Higgs-strahlung processes (WH and ZH productions) are inclusive calculations and the central scale by default is the sum of the Higgs and gauge boson masses. Only DY-type contributions are taken into acccount for the Higgs-strahlung processes, which are by far the dominant contributions. There is also an alternative version where the central scale is dynamical and by default the invariant mass M_HW or M_HZ.
- The gluon fusion process (ggH) is an inclusive calculation, in the so-called Born-improved high top-mass limit (Born-improved HTL), where the calculation is performed in an effective theory in which the top-quark is decoupled, matched to the full Standard Model. The predictions are rescaled to the exact leading order (one-loop) result including the top-quark mass. The code allows the user to get results either in the pure HTL (no rescaling) or in the Born-improved HTL, and also to choose between the on-shell (OS) scheme or the MSbar scheme for the top-quark mass. The central scale is by default muF = muR = MH/2.
- The bottom-quark fusion pocess (bbH) is an inclusive calculation in the five-flavor scheme (5FS) for which the central scale by default is muF = (MH+2*mbpole)/4, muR = MH. The bottom-quark mass used in the Yukawa coupling is taken in the MSbar scheme.
The program accepts up to 3 arguments on the command line, as well as an optional flag:
-lattice lattice: The lattice size that is used for the integration (integer), by default taken to be lattice=1000.-seed seed: The seed that is used to initialize the pseudo-random-number generator (integer). By default taken to be seed=1.--filename filename: The name of the input file (see below). By default this isn3loxs_parameters.in.--scaleor--7point: An optional flag to calculate 15 different predictions for the renormalization scale varied between 0.5 and 2 times the central scales of the process (--scaleflag); or an optional flag to calculate the seven-point scale variation around the central scales of the process (--7pointflag). They are mutually exclusive.
The user can type ./n3loxs --help or ./n3loxs -h to display
informations about these command-line arguments.
The program uses an input file for the physical parameters, by default
this is n3loxs_parameters.in. Please note that it is possible to use
a custom input file (as stated above), but this file needs to have the
same structure as the default input file. The following parameters can
be modified:
process
An integer to select the process to be studied. The program allows for the following options:
- 1 is for the differential cross section Q^2*dxs/dQ^2 in neutral Drell-Yan production
- 2 is for the differential cross section Q^2*dxs/dQ^2 in charged W+ Drell-Yan production
- 3 is for the differential cross section Q^2*dxs/dQ^2 in charged W- Drell-Yan production
- 4 is for the inclusive cross section in W+ H Higgs-strahlung production with a fixed scale
- 5 is for the inclusive cross section in W- H Higgs-strahlung production with a fixed scale
- 6 is for the inclusive cross section in Z H Higgs-strahlung production with a fixed scale
- 7 is for the inclusive cross section in 5FS bbH process at a fixed scale
- 8 is for the inclusive cross section in ggH process at a fixed scale.
- 9 is for the neutral Drell-Yan production cross section in an invariant mass windows between Qmin and Qmax for the invariant lepton pair Q
- 10 is for the charged W+ Drell-Yan production cross section in an invariant mass windows between Qmin and Qmax for the invariant lepton pair Q
- 11 is for the charged W- Drell-Yan production cross section in an invariant mass windows between Qmin and Qmax for the invariant lepton pair Q
- 12 is for the inclusive cross section in W+ H Higgs-strahlung production with a dynamical scale
- 13 is for the inclusive cross section in W- H Higgs-strahlung production with a dynamical scale
- 14 is for the inclusive cross section in Z H Higgs-strahlung production with a dynamical scale
PDFset
Name of the PDF set to be used in the calculation. Default:
PDF4LHC15_nnlo_mc.
PDFnum
Member of the PDF set to be used in the calculation. Default: 0.
order
Up to which order in QCD the calculation shall be performed. 0
stands for LO, 1 stands for NLO, 2 stands for NNLO, 3 stands for
N3LO. Default: 3.
collider
The calculation can be performed for a p-p (0, LHC-type) or a p-pbar
(1, Tevatron-type) collider. Default: 0.
energy
The hadronic center-of-mass energy of the collider, in TeV. Default:
13.0.
Q
The value, in GeV, for the virtuality of the gauge bosons in the
Drell-Yan processes (only relevan for these DY processes). Default:
100.0.
Qmin
The value, in GeV, for the minimum virtuality of the gauge bosons in
the Drell-Yan processes when calculated in a bin range. In this case
the value for Q is ignored. Default: 80.0.
Qmax
The value, in GeV, for the maximum virtuality of the gauge bosons in
the Drell-Yan processes when calculated in a bin range. In this case
the value for Q is ignored. Default: 90.0.
muf0
The value, in GeV, of the user-defined central factorization scale
muF0. If muf0=-1, then muF0 is internally fixed to the default
central scale of the chosen process. Default: -1.
xmuf
Floating-point coefficient rescaling the central factorization scale
muF0, so that muF = xmuf*muF0 where muF0 stands for the central
factorization scale of the chosen process. Default: 1.0.
mur0
The value, in GeV, of the user-defined central renormalization scale
muR0. If mur0=-1, then muR0 is internally fixed to the default
central scale of the chosen process. Default: -1.
xmur
Floating-point coefficient rescaling the central factorization scale
muR0, so that muR = xmur*muR0 where muR0 stands for the central
renormalization scale of the chosen process. This parameter is
ignored if the flags --scale or --7point are used. Default:
1.0.
MW
The value, in GeV, of the W-boson mass. Default: 80.398.
GammaW
The value, in GeV, of the W-boson total decay width. Default: 2.085.
MZ
The value, in GeV, of the Z-boson mass. Default: 91.1876.
GammaZ
The value, in GeV, of the Z-boson total decay width. Default: 2.4952.
MH
The value, in GeV, of the Higgs boson mass. Default: 125.09.
Mt
The value, in GeV, of the OS top-quark mass. Default: 172.5.
Mb
The value, in GeV, of the OS bottom-quark mass. Default: 4.58.
mt(mt)
The value, in GeV, of the MSbar top-quark mass at the scale of the
MSbar top-quark mass. Default: 162.7.
mb(mb)
The value, in GeV, of the MSbar bottom-quark mass at the scale of the
MSbar bottom-quark mass. Default: 4.18.
vev
The value, in GeV, of vacuum expectation value. Default: 246.221.
1/alpha
The value of the inverse of the fine-structure constant. Default: 137.035999084.
mt_scheme
The calculation of the ggH process can be performed in the OS
scheme (0) or in the MSbar scheme (1), as far as the top quark is
concerned. Default: 1.
htl_flag
The results for the ggH process can be given in the pure HTL (0) or
for Born-improved predictions (1). Default: 1.
ncdy_flag
The neutral Drell-Yan process can be calculated either with only the
off-shell photon contribution (0) or with both off-shell Z and
photon contributions (full process, 1). Default: 1.
The CKM parameters are hard-coded in the
include file constants.h located in the include directory at the
moment. The user can modify these parameters, however it requires a
new compilation of the code for the modification to be taken into
account.
The program produces an output file containing the cross sections up to the desired order in QCD, including the numerical error of the integration. The factorization and renormalization scales used for the calculation are also reported.
Please note that the PDF used in the
calculation is the same throughout the whole evaluation of the
program: When asking e.g. for an N3LO calculation, the LO, NLO, and
NNLO results are not using the corresponding PDFs, but simply the one
provided by the user in the input file n3loxs_parameters.in. The
evolution of the strong coupling constant and of the MSbar masses are
consistently done at the given QCD order.
The user should cite the publication describing this computer program,
[1] J. Baglio, C. Duhr, B. Mistlberger, and R. Szafron, "Inclusive Production Cross Sections at N3LO". arXiv:2209.XXXX.
The program uses a quasi-Monte-Carlo (QMC) integration as implemented by
[2] S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, and J. Schenk, "A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec". Comp. Phys. Commun. 240 (2019) 120. DOI: 10.1016/j.cpc.2019.02.015, arXiv:1811.11720.
Their implementation can be found at this link.
The calculations underlying the Drell-Yan processes have been described in the following references,
[3] C. Duhr, F. Dulat, and B. Mistlberger, "The Drell-Yan cross section to third order in the strong coupling constant". Phys. Rev. Lett. 125 (2020) 172001. DOI: 10.1103/PhysRevLett.125.172001, arXiv:2001.07717.
[4] C. Duhr, F. Dulat, and B. Mistlberger, "Charged Current Drell-Yan Production at N3LO". JHEP 11 (2020) 143. DOI: 10.1007/JHEP11(2020)143, arXiv:2007.13313.
[5] C. Duhr and B. Mistlberger, "Lepton-pair production at hadron colliders at N3LO in QCD". JHEP 03 (2022) 116. DOI: 10.1007/JHEP03(2022)116, arXiv:2111.10379.
The user should refer to the following papers when using the program for the bottom-quark fusion process,
[6] C. Duhr, F. Dulat, and B. Mistlberger, "Higgs Boson Production in Bottom-Quark Fusion to Third Order in the Strong Coupling". Phys. Rev. Lett. 125 (2020) 051804. DOI: 10.1103/PhysRevLett.125.051804, arXiv:1904.09990.
[7] C. Duhr, F. Dulat, V. Hirschi, and B. Mistlberger, "Higgs production in bottom quark fusion: matching the 4- and 5-flavour schemes to third order in the strong coupling". JHEP 08 (2020) 017. DOI: 10.1007/JHEP08(2020)017, arXiv:2004.04752.
When using the program for the gluon fusion process, at least the following references should be cited,
[8] C. Anastasiou, C. Duhr, F. Dulat, E. Furland, T. Gehrmann, F. Herzog, and B. Mistlberger, "Higgs boson gluon-fusion production at threshold in N3LO QCD". Phys. Lett. B 737 (2014) 325-328. DOI: 10.1016/j.physletb.2014.08.067, arXiv:1403.4616.
[9] B. Mistlberger, "Higgs boson production at hadron colliders at N3LO in QCD". JHEP 05 (2018) 028. DOI: 10.1007/JHEP05(2018)028, arXiv:1802.00833.
but the user is reminded that many other papers relevant for the desired process should also be referenced (e.g. LO, NLO, NNLO calculations for example).
- Julien Baglio (@jubaglio)
- Claude Duhr
- Bernhard Mistlberger
- Robert Szafron