MSTW 2008 PDFs
==============
The enclosed Fortran, C++ and Mathematica code provides public access
to the parton distribution functions (PDFs) published in:
A. D. Martin, W. J. Stirling, R. S. Thorne and G. Watt,
"Parton distributions for the LHC",
Eur. Phys. J. C63 (2009) 189-285
[arXiv:0901.0002 [hep-ph]].
Please reference this paper if you use the PDFs for a publication.
Unpack the code tarball with "tar zxvf mstw2008code.tar.gz" to give the
following files:
README : This short documentation file.
mstwpdf.f : Fortran code to access PDF grids.
mstwpdf.h : C++ class header file.
mstwpdf.cc : C++ class implementation file.
example.f : Fortran example code.
example.cc : C++ example code.
Makefile : Compilation using Make utility.
alphaS.f : Fortran code for alphaS.
mstwpdf.m : Mathematica package to access PDF grids.
example.nb : Mathematica example notebook.
alphaS.tm : Mathlink template for alphaS code.
alphaS.c : Mathlink C wrapper around Fortran code for alphaS.
diff3all.sh : Compare output of Fortran, C++ and Mathematica example code.
Unpack the grids tarball with "tar zxvf mstw2008grids.tar.gz". The PDF
grids from the best fit at LO, NLO and NNLO are stored in files named
Grids/mstw2008lo.00.dat
Grids/mstw2008nlo.00.dat
Grids/mstw2008nnlo.00.dat
The grids are provided for
10^-6 <= x <= 1 and 1 GeV^2 <= Q^2 <= 10^9 GeV^2
with an extrapolation invoked for smaller x or smaller/larger Q^2. As usual,
it is always the momentum density that is returned, that is, x times the number
density. The values of the heavy quark masses used and the values of alphaS
obtained from the fits are stored in the grid files. For consistency, the same
values should be used in all calculations involving the PDFs. Code is provided
for the alphaS definition used in the fitting code: this differs from the
alphaS definition used previously by MRST. In particular, the input alphaS
value can be given at any reference scale, and there is no Lambda_{QCD}^{nf=4}
parameter as in the previous MRST definition. Note that the PDFs (and alphaS)
are discontinuous at NNLO at the heavy quark thresholds Q^2 = mc^2 and
Q^2 = mb^2. To account for this, Q^2 grid points are chosen at mc^2, mc^2+eps,
mb^2, mb^2+eps (with eps = 10^-6 GeV^2). If the PDFs are requested for
mc^2 < Q^2 < mc^2+eps the value at mc^2+eps is used, and similarly if
mb^2 < Q^2 < mb^2+eps the value at mb^2+eps is used.
The eigenvector PDF sets are stored in files named
Grids/mstw2008lo.90cl.ii.dat
Grids/mstw2008nlo.90cl.ii.dat
Grids/mstw2008nnlo.90cl.ii.dat
and
Grids/mstw2008lo.68cl.ii.dat
Grids/mstw2008nlo.68cl.ii.dat
Grids/mstw2008nnlo.68cl.ii.dat
where ii = 01 to ii = 40 label the eigenvector sets and ii = (2*n-1) and
ii = (2*n) correspond, respectively, to the '+' and '-' directions of the nth
eigenvector (n = 1 to 20). In producing the eigenvector PDF sets, the
tolerance parameter T, i.e. the square root of the change in global chi^2
when moving along an eigenvector direction starting from the global minimum,
is determined by a new dynamic procedure such that each individual data set
included in the global fit is described within its 90% or one-sigma (68%)
confidence level (C.L.) limits. The values of the tolerance T, and also the
corresponding distance t moved along each eigenvector direction, are stored in
the grid files. (In the ideal quadratic case, t = T.) The user is therefore
free to choose which of the two error sets to use to obtain PDF uncertainties,
or indeed to rescale the uncertainties to obtain a different tolerance T.
However, we recommend the use of the conservative 90% C.L. PDF uncertainties
as the default choice. Note that these 90% and 68% C.L. limits are defined
according to "hypothesis-testing" criteria and therefore the values of the
tolerance T differ from the values of T = 1.64 (90% C.L.) and T = 1 (68% C.L.)
expected according to the more usual "parameter-fitting" criteria. Previous
MRST eigenvector PDF sets have used a fixed value of T = sqrt(50) corresponding
to an estimated 90% C.L. limit. Also, previous MRST uncertainties were
determined for fixed data set normalisations, whereas we now allow data set
normalisations to go free when determining PDF uncertainties. Full details
of the new dynamic procedure for determining PDF uncertainties are given in
Section 6 of the paper mentioned above.
Corresponding Fortran code for calculating proton (and neutron) structure
functions in neutral-current deep-inelastic scattering (only photon exchange),
i.e. F_2 and F_L including their heavy quark (c,b) components, is available
on request. Note that the previous MRST structure function code, available
from HEPDATA, should not be used with the MSTW PDFs due to the different
implementation of the general-mass variable flavour number scheme (GM-VFNS).
Fortran and C++ example programs
--------------------------------
To compile the Fortran and C++ example programs, check that the correct
Fortran and/or C++ compilers are enabled in the Makefile, then just type
"make". Of course, the compilation can also be done by hand, e.g.
g77 example.f mstwpdf.f alphaS.f -o example
Then run the programs with:
./example
./example_cpp
The C++ example program contains a wrapper to the Fortran alphaS code, and
therefore requires a Fortran compiler. If this is not available, comment out
the corresponding parts of the C++ example program. The Fortran and C++
example programs should give identical output. Verify this with e.g. for
the gluon:
diff xg_vs_x.dat xg_vs_x_cpp.dat
diff xg_vs_q2.dat xg_vs_q2_cpp.dat
Note that the numbering scheme for the flavour "f" now follows the PDG
convention (apart from the gluon has f=0, not 21) used in e.g. LHAPDF
and differs from the numbering scheme used in previous versions of
the MRST code. See the example programs for usage.
Mathematica package
-------------------
For the first time a Mathematica package is provided to access the PDFs. The
grids are interpolated using exactly the same method adopted in the Fortran
and C++ code, rather than using Mathematica's own interpolation routines. This
ensures that identical results will be obtained for the PDFs using the Fortran,
C++ or Mathematica interfaces. The Mathematica example notebook will output
files in an identical format to the Fortran and C++ example programs. Check
these with e.g.
diff xg_vs_x.dat xg_vs_x_math.dat
diff xg_vs_q2.dat xg_vs_q2_math.dat
A script diff3all.sh is provided to allow a check of all output files from the
Fortran, C++ and Mathematica example programs.
The Fortran alphaS code can be called from Mathematica using the Mathlink
interface. The binary alphaS.exe should be compiled from alphaS.c,
alphaS.tm and alphaS.f. Under Linux, this can be done with e.g.
mcc -o alphaS.exe alphaS.c alphaS.tm alphaS.f -lg2c
(Use -lg2c with g77 and -lgfortran with gfortran.) Alternatively, compile in
steps by checking the variables in the Makefile then doing "make alphaS.exe".
Updates
-------
5th January 2009. First public version.
22nd January 2009. Minor changes to code and grids:
- "alphaSnfmax" added in header of grid files. Gives maximum number of
flavours used in alphaS evolution. Interpolation codes modified accordingly
to read this in. This might be useful in future for automatic initialisation
of alphaS when using PDFs generated in a Fixed Flavour Number Scheme.
- Minor changes made when extrapolating for small x or large Q^2 beyond the
grid points. Original code extrapolated ln(xf) for xf > 0 and xf itself
for xf <= 0. This causes numerical problems if xf is positive but very
small (e.g. close to the nodes in xsv, xcv, xbv), so now only extrapolate
in ln(xf) if xf > 10^-3, otherwise extrapolate in xf itself.
- Mathematica version: now allow any numeric values of x and q as input to
xf[ih,x,q,f], not only floating-point numbers as in the previous version.
14th July 2010. Minor extension to allow wider range of heavy-quark masses.
- Previously 1.5 < m_c^2 < 2.5 GeV^2 and 12 < m_b^2 < 26 GeV^2.
- Now allow 1 < m_c^2 < 4 GeV^2 and 10 < m_b^2 < 40 GeV^2.
- Impose minimum anomalous dimension for low Q^2 < 1 GeV^2 extrapolation.
- Avoid using Fortran 90 intrinsic function "len_trim"
since not supported by all Fortran 77 compilers.
- Steps to build "alphaS.exe" used for Mathlink interface added to Makefile.
(None of these changes affects any numerical results obtained using the
previous grids other than possibly with the Q^2 < 1 GeV^2 extrapolation.)
25th January 2011. Fix "NaN" bug for q <= m_c when m_c^2 < 1.25 GeV^2.
Comments to Graeme.Watt(at)cern.ch.