These are the PDFs described in
the paper hep-ex/0208023 (DESY-02-105)
ZEUS-NLOQCD
fit paper
The figures from this paper are collected separately below
Fig.1
F2 as a function x : low Q2 fixed target and ZEUS in colour
Fig.2
F2 as a function of Q2: fixed target and ZEUS in colour
F
ig.3 HIgh Q2 NC data in color
Fig.4
High Q2 CC data in color
Fig.5b
ZEUS-S parton distributions compared to MRST2001 and CTEQ6 in colour
Fig.6
ZEUS-S uv plus fractional error band
Fig.7
ZEUS-S dv plus fractional error band
Fig.8
ZEUS-S sea plus fractional error band
Fig.9
ZEUS-S gluon plus fractional error band
Fig.10
ZEUS-S sea and glue compared
Fig.11a
ZEUS-S glue as a function of Q2
Fig.11b
ZEUS-S sea as a function of Q2
Fig.12
ZEUS-S FL prediction
Fig.13
ZEUS-O glue and sea compared
Fig.14
ZEUS-O valence distributions
special
version of figs 6/7 combined ZEUS-S valence distributions for comparison
to fig 14
Fig.15
ZEUS-S extrapolated to low Q2 compared to BPT and SVX F2 data in
colour
F
ig.16
ZEUS-S extraploated to low Q2 FL predictions
A .ps file describing how to access the ZEUS PDFs is given here
Brief
write-up of how to access ZEUS-2002 PDfs
BUT it is also summarised below:
The ZEUS PDFs are sets of PDFs with errors extracted from a global NLO-QCD fit in the DGLAP formalism to ZEUS data and fixed target data The DGLAP equations were solved at NLO in the \msbar\ scheme with the renormalisation and factorisation scales chosen to be $Q^2$. The standard ZEUS fit (ZEUS-S) was done using the Thorne-Roberts heavy quark scheme- (Thorne Roberts Variable Flavour Number Scheme -TRVFN) but PDFs are also available using a Zero Mass Variable Flavour Number Scheme (ZMVFN) and a Fixed Flavour Number Scheme (FFN) with 4-flavours. For each of these schemes the PDFs and their uncorrelated (statistical plus uncorrelated systematic) error bands, correlated systematic error bands, and total error bands are available. For most purposes, only the total errors would be needed.
These parton distributions are publically
available in various formats:
1)Eigenvector
PDF sets: a central PDF parameter set plus
eigenvector PDF parameter sets from which the errors may be calculated
easily.
2)Covariance
matrices: a central PDF parameter set plus
covariance matrices
3)PDF
grids: as $x,Q^2$ grids of PDF values for
for the parton momentum distributions: gluon, sea, $xu_v$, $xd_v$, $x\bar{u}$,
$x\bar{d}$, $x\bar{s}$, $x\bar{c}$, $x\bar{b}$. Grids of various structure
functions and reduced cross-sections are also available.
Eigenvector PDF sets
Eigenvector PDF sets represent the most efficient and compact way of storing the information on the PDF errors. The technique was first suggested by CTEQ (hep-ph/0101032) and CTEQ6 errors are given in this form (hep-ph/0201195). MRST have recently issued their PDF parameters in this form (hep-ph/0211080).
The errors on the parton distribution parameters are encapsulated in the error matrix (covariance matrix) of the fit. This may be diagonalised and its eigenvalues represent the squared errors on the combinations of parameters which are the eigenvectors. The results of the fit can be summarised as one central PDF set ($S$) and $2* N_{pdf}$ eigenvector PDF sets for the errors, where $N_{pdf}$ is the number of free PDF parameters. For the ZEUS-S fit $N_{pdf}=11$. These eigenvector PDF sets represent excursions up and down along each of the $N_{pdf}$ eigenvector directions by an amount equal to the error on the corresponding combination of the original PDF parameters, i.e the square-root of corresponding eigenvalue. These eigenvector PDF sets are labelled $S_i^+$ and $S_i^-$ for each of the $i= 1, N_{pdf}$ eigenvalues.
The error on a quantity $F(S)$, which is a function of the PDF parameters, (for example, such a quantity could be a PDF distribution, a structure function or a cross-section) is then calculated from
\Delta F^2 = \Sigma_i ((F(S_i^+) - F(S_i^-)/2)^2
i.e. the value of $F$ is calculated from the parameter sets $S_i^+$ and $S_i^-$, exactly as it is for the central set, and then the difference between its value for these two sets gives the error on $F$ due to eigenvector $i$. These are then simply added in quadrature.
These eigenvector PDF sets are supplied in data files as follows
For the FFN heavy quark scheme
parcen_ff.dat
is the 11 free PDF parameters at their central values
instat_ff.datis
a file with 22 sets of such 11 parameters in the order $S_i^+$, then $S_i^-$
for the $i=1, 11$ eigenvector directions; for uncorrelated (statistical
plus uncorrelated systematic) errors
insys_ff.datis
a file with 22 sets of such 11 parameters in the order $S_i^+$, then $S_i^-$
for the $i=1, 11$ eigenvector directions; for correlated systematic errors
intot_ff.datis
a file with 22 sets of such 11 parameters in the order $S_i^+$, then $S_i^-$
for the $i=1, 11$ eigenvector directions; for total errors
For the ZMVFN scheme the
files carry the same names with _ff repleaced
by _zm
parcen_zm.dat
instat_zm.dat
insys_zm.dat
intot_zm.dat
and for the TRVFN scheme
the files carry the same names with _ff
replaced
by _tr
parcen_tr.dat
instat_tr.dat
insys_tr.dat
intot_tr.dat
In order to use these PDF sets it
is necessary to know how the parton distributions were parameterised and
used for the ZEUS fits.
The programme qcd_results.f
illustrates how the PDF sets are used. This programme constructs the PDFs
from the PDF parameters and evolves them in $Q^2$. It also performs the
convolutions necessary to calculate many functions of the PDF parameters
which are of interest like structure functions, and reduced cross-sections.
Abrief description of what goes into this is given below.
The parton distribution functions
(PDFs) are parameterised at $Q^2_0$ by the form
xf(x) = p_1 x^(p_2) (1-x)^(p_3)(
1 + (p_5). x)
The NLO DGLAP equations are used
to evolve the parton distributions to all values of $Q^2$. The input scale
is $Q^2_0 = 7 GeV}^2$, but backward evolution can be performed to fit lower-$Q^2$
data. The PDFs give a good description of data down to $Q^2 \sim 1GeV^2,
but note that the low-$x$ gluon becomes negative for $Q^2 < 1.8 GeV^2$.
The parton momentum distributions
parameterised are: u-valence, $xu_v(x)$; d-valence, $xd_v(x)$; total sea,
$xS(x)$; gluon, $xg(x)$; and the difference between the $d$ and $u$ contributions
to the sea, $x\Delta=x(\bar d-\bar u)$. The total sea at $Q^2_0$ is made
from the flavours up, $xu_{ sea}(x)$, down, $xd_{sea}(x)$, strange, $xs_{sea}(x)$
and charm, $xc_{ sea}(x)$, as follows
\[ xs_{sea}(x) = 0.2xS(x) \]
\[ xu_{sea}(x) = 0.4xS(x)-0.5xc_{sea}(x)-x\Delta(x)
\]
\[ xd_{sea}(x) = 0.4xS(x)-0.5xc_{sea}(x)+x\Delta(x)
\]
where the symbols $u_{sea}$, $d_{sea}$,
$s_{sea}$, $c_{sea}$ include both quark and antiquark contributions to
the sea for each flavour. The suppression of the strange sea to $20\%$
of the total sea is consistent with neutrino-induced dimuon data from CCFR.
The charmed sea is treated according to the chosen heavy quark scheme.
The following parameters
were fixed:
$p_1$ for $xu_v$ and $xd_v$
were fixed through the number sum-rules and $p_1$
for $xg$ was fixed through the momentum sum-rule;
$p_2=0.5$ was fixed for both valence
distributions, since there is little information
on the low-$x$ valence shapes. Allowing this parameter to vary produces
values which are consistent with $0.5$;
The only free parameter for the
$x\Delta$ distribution is its normalisation, $p_1$,
because there is insufficient information on its shape when using only
DIS data Thus, $p_2(\Delta)=0.5$, $p_3(\Delta)=p_3({\rm Sea})+2$ were fixed
and $p_5(\Delta)=0$; the normalisation $p_1(\Delta)$ was found to be compatible
with the measured value of the Gottfried sum-rule;
For the gluon distribution, $p_5$
was set to zero, since this parameter was
found to be highly correlated to the $p_3$ parameter of the gluon. Allowing
this parameter to vary in the fit produced values which are consistent
with zero.
There are thus 11 free parameters
in the ZEUS-S fit. The value of $\alpha_s(M_Z)
=0.118$ is fixed and $\alpha_s(Q^2)$ is calculated to 2-loop accuracy.
The evolution was performed using
the program QCDNUM
and the evolution equations were written in terms of quark-flavour singlet
and non-singlet distributions (made from the sea and valence quark distributions)
and the gluon momentum distribution. The parton distributions must be convoluted
with coefficient functions in order to calculate structure functions and
cross-sections. The coefficient functions are specific to the heavy-quark
formalism used. The routines of QCDNUMcan
be used to perform this convolution for the FFN
and ZMVFN
schemes.
For the TRVFN
scheme additional routines are needed (rt.f
and jacksmith.f).
rt.f
jacksmith.f
The table gives the central values of the 11 free fit
parameters of the TRVFN fit, and their errors. The values of the fixed
parameters and the PDF parameters which are functions of the fitted parameters
are also given.
The first uncertainty given derives from statistical
and other uncorrelated sources and the second uncertainty is the additional
contribution from correlated systematic uncertainties. The numbers in parentheses
were derived from the fitted parameters through the number and momentum
sum-rules
| p_1 | p_2 | p_3 | p_5 | |
| xu_v | (1.69) | 0.5 | 4.00 \pm 0.01 \pm 0.08 | 5.04 \pm 0.09 \pm 0.64 |
| xd_v | (0.96) | 0.5 | 5.33 \pm 0.09 \pm 0.48 | 6.2 \pm 0.4 \pm 2.3 |
| xS | 0.603 \pm 0.007 \pm 0.048 | -0.235 \pm 0.002 \pm 0.012 | 8.9 \pm 0.2 \pm 1.2 | 6.8 \pm 0.4 \pm 2.0 |
| xg | (1.77) | -0.20 \pm 0.01 \pm 0.04 | 6.2 \pm 0.2 \pm 1.2 | 0 |
| x\Delta | 0.27 \pm 0.01 \pm 0.06 | 0.5 | (10.9) | 0 |
The programme qcd_results.f
constructs the PDFs from the PDF parameters read in from the data files.
It evolves PDFs in $Q^2$ and performs the convolutions necessary
to calculate many functions of the PDF parameters which are of interest
like structure functions, and reduced cross-sections. These are output
onto $x,Q^2$ grids of central value and up and down errors.
qcd_results.f consists of
a main routine which the user may alter and various subroutines which
should NOT be changed . The order in which they are called should also
be respected. The routine is commented to allow the user to find
his/her way through it easily.
The length of time which it takes
to execute depends on the size of the grid. The default grid in the programme
is 61*61- but code for the 161*161 fine grid used for the PDF grids (supplied
below) is also available in the routine in the comment lines. Execution
time also depends on whether or not full errors are required- the error
calculation takes 22 times as long as the calculation for the central values.
Finally the Thorne-Roberts coefficient functions take more time to compute
than those for FFN or ZMVFN- so for checking the default has been set to
FFN.
This
programme needs QCDNUM and the Thorne Roberts routines (rt.f
and
jacksmith.f),
as well as various cernlib/mathlib functions.
The following linux link command puts these together
f77 -g -o\ \ qcdres\ \ qcd\_results.f\
\ rt.f -L. -lqcdnum1612 -L/cern/pro/lib -lmathlib -lpacklib -lkernlib -lgrafX11
-L/usr/X11R6/lib -lX11
Covariance matrices
For some users it may be more convenient to use a single
central PDF set and the original covariance matrices of the fit. In this
case the error on a quantity $F(S)$ is calculated from
\Delta F^2 = \Sigma_{ij} (\delta F/ \delta p_i ) V_{ij}
(\delta F/ \delta p_j)
where $V_{ij}$ is the covariance matrix. Such covariance
matrices are available for the three different heavy quark schemes
For the FFN heavy quark scheme
parcen_ff.dat
is the 11 free PDF parameters at their central values
statcov_ff.datis
the 11*11 covariance matrix for uncorrelated (statistical plus uncorrelated
systematic) errors
syscov_ff.datis
the 11*11 covariance matrix for correlated systematic errors
totcov_ff.datis
the 11*11 covariance matrix for total errors
For the ZMVFN scheme the
files carry the same names with _ff repleaced
by _zm
parcen_zm.dat
statcov_zm.dat
syscov_zm.dat
totcov_zm.dat
and for the TRVFN scheme
the files carry the same names with _ff
replaced
by _tr
parcen_tr.dat
statcov_tr.dat
syscov_tr.dat
totcov_tr.dat
In order to use these PDF sets and covariance matrices it is necessary to know how the parton distributions were parameterised and used for the ZEUS fits. This is explained above in the section on Eigenvector PDF sets.
PDF grids
For some purposes it may be easier to use $x,Q^2$ grids
of PDF values or structure functions, reduced cross-sections etc. Such
grids are available for the FFN, ZMVFN and TRVFN heavy quark schemes. For
each of these schemes grids are available only for total errors. (Grids
for the
uncorrelated (statistical plus uncorrelated systematic)
and correlated systematic errors can be generated from the eigenvector
PDF sets using qcd_results.f)
For each function $F$ of the PDF parameters, the information for each point of the $x,Q^2$ grid is supplied as:$F$, $F + \Delta F$ and $F - \Delta F$.
The grids are 161*161 points in $Q2,x$ constructed with
the $Q2,x$ values given by
DO I=0,160
IF(I.LE.72)THEN
Q2=10**(5D0/120D0*I)-7D-1
ELSE
Q2=10**(2D0/88D0*(I-72)+3D0)
ENDIF
DO J=0,160
IF(J.LE.80)THEN
X=10**(6D0/120D0*J-6D0)
ELSE
X=10**(2D0/80D0*(J-80)-201D-2)
\ENDIF
such that the grid ranges from $0.3$ to $10^5$ in $Q^2$
and $10^{-6}$ to $0.98$ in $x$, with logarithmic spacing which is designed
to become finer at high $x$ and high $Q^2$.
The PDFs available are: gluon, sea,
xu_v, xd_v, x\bar{u}, x\bar{d},x\bar{s}, x\bar{c}, x\bar{b}
For the FFN heavy quark scheme;
upval_ff_tot.dat
is
the grid for the total error on the $xu_v$ distribution
dnval_ff_tot.datis
the grid for the total error on the $xd_v$ distribution
gluon_ff_tot.datis
the grid for the total error on gluon distribution
seaqk_ff_tot.datis
the grid for the total error on sea distribution
ubar_ff_tot.datis
the grid for the total error on the $x\bar{u}$ distribution
dbar_ff_tot.datis
the grid for the total error on the $x\bar{d}$ distribution
sbar_ff_tot.datis
the grid for the total error on the $x\bar{s}$ distribution
Note that there is no $x\bar{c}$ or $x\bar{b}$ distribution
for the FFN scheme.
For the ZMVFN scheme the
grids carry the same names with $\_ff$ replaced by $\_zm$,
upval_zm_tot.dat
dnval_zm_tot.dat
gluon_zm_tot.dat
seaqk_zm_tot.dat
ubar_zm_tot.dat
dbar_zm_tot.dat
sbar_zm_tot.dat
cbar_zm_tot.dat
is the grid for the total error on the $x\bar{c}$ distribution
bbar_zm_tot.dat
is the grid for the total error on the $x\bar{b}$ distribution
and for the TRVFN scheme the
grids carry the same names with $\_ff$ replaced by $\_rt$.
upval_tr_tot.dat
dnval_tr_tot.dat
gluon_tr_tot.dat
seaqk_tr_tot.dat
ubar_tr_tot.dat
dbar_tr_tot.dat
sbar_tr_tot.dat
cbar_tr_tot.datis
the grid for the total error on the $x\bar{c}$ distribution
bbar_tr_tot.datis
the grid for the total error on the $x\bar{b}$ distribution
The structure functions available currently are those for $\gamma$ exchange in the $e^{\pm}$ NC processes: $F_2^{\gamma}$, $F_L^{\gamma}$, $F_2^{charm}$ and those for $\gamma$ and $Z_0$ exchange in the $e^{\pm}$ NC processes: $F_2, $F_L$, $xF_3$
For the FFN heavy quark scheme;
f2_ff_tot.datis
the grid for the total error on $F_2 ^{\gamma}(e p NC)$
fl_ff_tot.datis
the grid for the total error on $F_L^{\gamma} (e p NC)$
f2c_ff_tot.datis
the grid for the total error on $F_2^{charm} (e p NC)$
f2nc_ff_tot.datis
the grid for the total error on $F_2 (e p NC)$
flnc_ff_tot.datis
the grid for the total error on $F_L (e p NC)$
xf3nc_ff_tot.datis
the grid for the total error on $xF_3 (e p NC)$
For the ZMVFN scheme the
grids carry the same names with $\_ff$ replaced by $\_zm$,
f2_zm_tot.dat
fl_zm_tot.dat
f2c_zm_tot.dat
f2nc_zm_tot.dat
flnc_zm_tot.dat
xf3nc_zm_tot.dat
and for the TRVFN scheme
the grids carry the same names with $\_ff$ replaced by $\_rt$.
f2_tr_tot.dat
fl_tr_tot.dat
f2c_tr_tot.dat
f2nc_tr_tot.dat
flnc_tr_tot.dat
xf3nc_tr_tot.dat
The reduced cross-sections available are: $\tilde{\sigma}(e^+p CC)$, $\tilde{\sigma}(e^-p CC)$, $\tilde{\sigma}(e^+p NC)$, $\tilde{\sigma}(e^-p NC)$, for HERA running with lepton beam energy $27.5$GeV, proton beam energy $920$GeV.
For the FFN scheme;
epnc_ff_tot.dat
is the grid for the reduced cross-section $\tilde{\sigma}(e^+p NC)$,
emnc_ff_tot.datis
the grid for the reduced cross-section $\tilde{\sigma}(e^-p NC)$
BEware: do not use the FFN scheme for the CC cross-sections-
the $W$ exchange is not correctly a handled. Use ZM or TRVFN.
For the ZMVFN scheme the
grids carry the same names with $\_ff$ replaced by $\_zm$,
epnc_zm_tot.dat
emnc_zm_tot.dat
epcc_zm_tot.dat
emcc_zm_tot.dat
and for
the TRVFN scheme they carry the same names with $\_ff$ replaced
by $\_rt$.
epnc_tr_tot.dat
emnc_tr_tot.dat
epcc_tr_tot.dat
emcc_tr_tot.dat
An example of the use of such grids is given in the PAW
kumacs
qerr_x.kumac
emccerr.kumac
which need the routine f2qcdfine.f
A more sophisticated interface routine to these grids
is available from k.long@ic.ac.uk