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LHAPDF manual

LHAPDF version 5 User Guide ¹

Contents

1  Introduction
2  Installing LHAPDF
    2.1  Version 4.1 onwards
    2.1  Version 4.0 and earlier
3  Interfacing LHAPDF with a Code
    3.1  Using the LHAPDF routines directly
    3.2  Using the LHAGLUE inteface
    3.3  Whether to use .LHpdf or .LHgrid files?
4  Multiset Initialization with Version 5
    4.1  How many sets can be initialised
    4.2  Using multiset initialization with LHAglue
    4.3  Using multiset initialization with native LHAPDF routines
5  C++ wrapper
    5.1  Description

Appendices

Abstract

The Les Houches Accord PDF (LHAPDF) interface package is designed to work with PDF sets. A PDF set can consist of many individual member PDFs. While the interpretation of the member PDFs depends on the particular set, the LHAPDF interface is specifically designed to accommodate PDFs with uncertainties. For PDFs with uncertainties the PDF set represents one ``fit'' to the data. The individual member of a PDF set are needed to calculate the PDF uncertainty of the observable.

All PDF sets are defined through external files. This means that in many cases a new set can be added by simply downloading its file while the LHAPDF interface code does not change. The user has, since LHAPDF version 2, the option of using parameter files (.LHpdf) and doing the evolution "on the fly" or of using the interpolation grid files (.LHgrid) as supplied by the PDF set authors. The evolution code is not part of LHAPDF. Currently, QCDNUM [1] is the default evolution code fully interfaced with LHAPDF. Other evolution codes can easily be interfaced with the LHAPDF.

Since version 3 the LHAGLUE package ² has been included as an alternative, and PDFLIB-like, inteface to the LHAPDF code.

The major changes in version 4 are the additional of the photon and pion PDF sets and the access to the QCD parameters Lamdba4/5 quoted by the authors of the PDF sets as applicable to that set.

Version 4.1 introduces a more standard "configure; make; makeinstall" installation procedure using gnu autoconf tools with and has some small changes for compatibility with f90/95 compilers.

Version 5 introduces multiset initialisation whereby several PDFs can be initialised a the beginning and then used interchangeably without incurring the cpu penalty of re-initialization.

1  Introduction

The Les Houches Accord Parton Density Function interface originated at the Les Houches 2001 workshop [2] in the PDF working group to enable the usage of Parton Density Functions with uncertainties in a uniform manner. When PDFs with uncertainties are considered, a ``fit'' to the data no longer is described by a single PDF. Instead in its most flexible implementation, a fit is represented by a PDF set consisting of many individual members. Calculating the observable for all the PDF members enables one to reconstruct the uncertainty on the observable. The LHAPDF interface was made with this in mind and manipulates PDF sets.

The LHAPDF interface can be viewed as a successor to PDFLIB [3]. Many improvements were added, to list some of the features:

• The handling of PDF sets to enable PDF fits that include uncertainties.
• All PDF sets are defined through external files in either parameterized or grid form. The parameterized files are compact compared to the grid files, but the latter allow faster usage when multiple PDF sets are called frequently and also allow older PDF sets to be made available in LHAPDF. New PDF sets can be defined by constructing the PDF defining parameter or grid files. The actual LHAPDF code does not have to be changed unless, in the grid file case, the authors of the PDFs use new interpolation methods.
• The LHAPDF code is modular and default choices can be easily altered.

Note that the current ``best fit'' PDFs can be viewed as sets with one member PDF and can be easily defined through the PDF set external file definition. Alternatively one can group these ``fits'' is single sets (e.g. MRST98 set) as they often represent best fits given different model assumptions and as such reflect theoretical modeling uncertainties.

As mentioned above, using LHAPDF to determine the PDF uncertainty on an observable will involve evaluating N different PDFs in a single set. How to use the N predictions of the observable depends on the method used in the PDF set for propagation of errors. During the fitting of the PDFs many approximations can be made. An overview is given in the Les Houches proceedings [2]. The relevant section is reproduced here. http://projects.hepforge.org/lhapdf/manual

2  Installing LHAPDF

The method of installing changed between versions 4.0 and 4.1. These are desribed below:

2.1   LHAPDF Installation (4.1 onwards)

First download the required gzipped tar file (eg lhapdf-v.r.tar.gz) from the downloads section (either with or without the PDFsets as required). Then do the following:-

cd lhapdf-v.r to change directory (v.r = version.revision, eg 4.1).

If you have root priviledge and want the installed files to go by default into /usr/local then do:

./configure

If you do not have root priviledge and/or want the files installed into a different directory then do:

./configure --prefix=/path/to/directory

Then do:

./make

./make install

The following directories/files should now have been installed in your selected installation directory.

From version 5.2.3 onwards additionally the following files are installed:

From version 5.3.0 onwards additionally the following files are installed:

If you wish to install the various components into different directories than above, the following options to ./configure (as well as --prefix=) can be used:

--bindir=DIR --datadir=DIR --libdir=DIR

More details are given in the INSTALL file in the downloaded package.

2.2   LHAPDF Installation (up to version 4.0)

Installing LHAPDF is simple and can be accomplished in two ways:

1. Download all the LHAPDF files (including LHAGLUE) in one go from the download page on the web site. Un-tarring this file will create a directory LHAPDFv3 with all the files and makefiles in a subdirectory structure beneath this. There is a Makefile in this directory which will create (with the 'make' or 'make all' command) a library 'libLHAPDF.a', in this same directory, which can then be linked to external programmes.

2. Alternatively the programme code and input parameter/grid files can be downloaded separately.

The Makefile has other options for compiling only the code relevant to the EVLCTEQ (make evlcteq) evolution code programmes. This is included mainly for compatibility with LHAPDF version 1 and in general it is better to compile the complete code as this will allow different PDF sets from the different authors and also the different type of input files (parameter or grid) to be used seamlessly without recompiling any code.

The 'make clean' command will remove the .o files from the directories once the installation has been successfully accompished.

Thus in its simplest form

download the file 'LHAfullv4.0'
tar -xvf LHAfullv4.0.tar
cd LHAPDFv4
make
make clean

3  Interfacing LHAPDF with a Code

Since version 3 there are two ways of interfacing the LHAPDF code into a user program.

1) Using the LHAPDF routines directly.
2) Using the LHAGLUE interface.

3.1  Using the LHAPDF routines directly

The interface of LHAPDF with an external code is easy. We will describe the basic steps sufficient for most applications. In this section we describe single set intitalization. In a following section we describe how to use multiset initialization introduced in version 5. The function calls described here will not be altered in any way in future versions. Including the LHAPDF code into a program involves three steps:

1. First one has to setup the LHAPDF interface code:

   call InitPDFset(name)

     It is called only once at the beginning of the code. The string variable name is the fully qualified file name of the external PDF file (i.e. including specific directory path) that defines the PDF set. For the default evolution code QCDNUM it will either calculate or read from file the LO/NLO splitting function weights. The calculation of the weights might take some time depending on the chosen grid size. However, after the first run a grid file is created. Subsequent runs will use this file to read in the weights so that the lengthy calculation of these weights is avoided. The file depends on the grid parameters and flavor thresholds. This means different PDF sets can have different grid files. The name of the grid file is specified in the PDF setup file. Appendix B gives a table showing the names of all the available files.

   As emphasised above, when using call InitPDFset the parameter name is the full path qualified name of the set. From version 4.1 onwards and additional routine:

   call InitPDFsetByName(name)

   has been added which uses the "lhapdf-config" script generated during the LHAPDF installation to determine and use the correct path so that in this case name is only the specific PDFset name itself (eg MRST2001nlo.LHpdf).

Table 1: The flavor enumeration convention used in the LHAPDF interface.

As long as the member PDF is not changed (by the call InitPDF of step 2) the evolution calls will always use the same PDF member.

A different subroutine call

call evolvePDFp(x,Q,P2,IP2,f)

is available for the photon PDFs where Q is the QCD scale in GeV, P2 is the vitruality of the photon in GeV², which should by 0 for an on-shell photon, and IP2 is the parameter to evaluate the off-shell anomalous component. x and f are as above. Note that all inputs and outputs are defined in DOUBLE PRECISION

A few additional calls listed below can be useful:

3.2  Using the LHAGLUE interface

The LHAGLUE interface ² has been designed as a PDFLIB look alike interface to LHAPDF.

The interface contains four subroutines (similar to PDFLIB) and can be used seamlessly by Monte Carlo generators interfaced to PDFLIB or in stand-alone mode.

• For initialization (called once):

  call PDFSET(parm,value)

• For the proton and pion structure functions:

  call STRUCTM(X,Q,UPV,DNV,USEA,DSEA,STR,CHM,BOT,TOP,GLU)

• For the photon structure function:

  call STRUCTP(X,Q2,P2,IP2,UPV,DNV,USEA,DSEA,STR,CHM,BOT,TOP,GLU)

• For statistics on PDF under/overflows:

  call PDFSTA

Of course any of the LHAPDF routines, except the initialization routines InitPDFset and InitPDF, described in the previous subsection can also be used, for example to return the value of alphas (alphasPDF(Q)) or print the file description (GetDesc()).

The interface can be invoked 3 ways depending on the value of parm(1) provided by the user when calling PDFSET(parm,value)

• for PYTHIA: PARM(1).EQ.'NPTYPE'
  (this is set automatically by PYTHIA)
• for HERWIG: PARM(1).EQ.'HWLHAPDF'
  (set by the user)
  e.g. in the main program like this:
  AUTPDF(1) = 'HWLHAPDF'
  AUTPDF(2) = 'HWLHAPDF'
• for Stand-alone: PARM(1).EQ.'DEFAULT'
  (can be used for PDF studies or interfacing with a new generator.)

The LHAPDF set/number is selected depending on the value of:

• PYTHIA: ABS(MSTP(51)) - proton
• PYTHIA: ABS(MSTP(53)) - pion
• PYTHIA: ABS(MSTP(55)) - photon


• HERWIG: ABS(INT(value(1))


• Stand-alone: ABS(INT(value(1))

There are other CONTROL switches which determine how the interface operates. The location of the LHAPDF library has to be specified as described below, the rest are optional.If the user does nothing, sensible defaults are active. To change the behaviour the corresponding values of parm() should be set to the values given below.

1) Location of the LHAPDF library of PDFs (pathname):

From version LHAPDF v4.1 onwards, and the LHAglue routines distributed with it, the location of the PDFsets data files is set automatically using the "lhapdf-config" script, provided that the precribed installation instructions have been used.

Alternatively from version 5.0 onwards the PDFsets path can also be specified

(a) with the call

• CALL SetPDFPath('/path/to/PDFsets')

(b) with the environmental variable

• $LHAPATH

The maximum length of the path is set to 232. Should this need to be increased then it can be changed in the include file "src/pathsetup.inc", followed by a complete recompilation.

For previous version (4.0 and earlier) the following describes the situation:

• uses common block COMMON/LHAPDFC/LHAPATH
  • If the user does nothing => default path is subdir 'PDFsets' of the current directory (can be a real subdir OR a soft link to the real location)
  • The user sets LHAPATH in COMMON/LHAPDFC/LHAPDFC - before calling PDFSET

The following use the common block

CHARACTER*20 LHAPARM(20)
REAL*8 LHAVALUE(20)
COMMON/LHACONTROL/LHAPARM,LHAVALUE

which can be used directly in the user's program.

Alternatively the various flags can be set and reset using the CALL SetLHAPARM(string) function call, thus removing the need to include the common block in the user's program.

2) Collect statistics on under/over-low requests for PDFs outside their validity ranges in X and Q**2.

• LHAPARM(16).EQ.'NOSTAT' ==> No statistics (faster)
• LHAPARM(16).NE.'NOSTAT' ==> Default: collect statistics

• call SetLHAPARM('NOSTAT') ==> No statistics (faster)
• call SetLHAPARM('16') ==> Default: collect statistics

• call PDFSTA at end to print out statistics

3) Option to use the values for the strong coupling alpha_s as computed in LHAPDF

• LHAPARM(17).EQ.'LHAPDF' ==> Use alpha_s from LHAPDF
• LHAPARM(17).NE.'LHAPDF' ==> Default (same as LHAPDF v1/v3)

• call SetLHAPARM('LHAPDF') ==> Use alpha_s from LHAPDF
• call SetLHAPARM('17') ==> Default (same as LHAPDF v1/v3)

4) Extrapolation of PDFs outside the LHAPDF validity range given by [Xmin,Xmax] and [Q2min,Q2max]:

• Default ==> PDFs "frozen" at the boundaries.
• LHAPARM(18).EQ.'EXTRAPOLATE' ==> Extrapolate PDFs at own risk
  (Warning - crazy values can be returned!)

• call SetLHAPARM('EXTRAPOLATE') ==> Extrapolate PDFs at own risk
• call SetLHAPARM('18') ==> Default: PDFs "frozen" at the boundaries.

5) Printout of initialization information in PDFSET (by default):

• LHAPARM(19).EQ.'SILENT' ==> No printout (silent mode).
• LHAPARM(19).EQ.'LOWKEY' ==> Print 5 times (almost silent).

• call SetLHAPARM('SILENT') ==> No printout (silent mode).
• call SetLHAPARM('LOWKEY') ==> Print 5 times (almost silent).
• call SetLHAPARM('19') ==> Default: Print everything

6) Double Precision values of qcdl4 and qcdl5 relevant to the selected PDF are available in the COMMON block

• COMMON/W50512/QCDL4,QCDL5 ==> as in PDFLIB.

3.3  Whether to use the ".LHpdf" or ".LHgrid" files ?

• .LHpdf files - the PDFs are generated "on-the-fly" using an evolution package (eg QCDNUM or EVLCTEQ) from a parameterisation in X at an initial Q0 value. The form of the parameterisation is given by the PDF authors.
• .LHgrid files - the PDFs are generated by interpolation inside the pre-calculated tables produced the PDF authors and with the same interpolation codes used by the authors. This is closer to the methods used in PDFLIB.

The advantage of the .LHpdf files is that they are much smaller than the equivalent .LHgrid files.

Whichever the user chooses the method of using the LHAPDF and LHAGLUE routines is exactly the same.

4  Multiset initialization with Version 5 Version 5 introduces the concept holding more than one PDF set initialized in memory at the same time. This is implemented to speed up the running of programmes where it is necessary to switch frequently between PDF sets, for example to use nucleon,photon and possibly pion PDF sets together. Speed has been shown to be a particular issue when using the .LHpdf sets.

4.1  How many sets can be initialised? This is determined by the parameter nmxset in the file src/parmsetup.inc. It is set as a default to 3 but can be set to whatever the user wishes. Obviously the larger it is the larger will be the size of the executable programme. When changing this parameter it is necessary to recompile all the routines since the file is included in most of them. To do this do make clean followed by make then make install

4.2  Using multiset initialization with LHAglue The LHAglue in version 5 program will do multiset initialization automatically with no changes to the users routines. Of course the parameter nmxset in the file src/parmsetup.inc described in the previous subsection the has to be large enough for the maximum number of sets.

4.3  Using multiset initialization with the native LHAPDF routines Version 5 contains a set of subroutines which perform the same functions as those described in section 3.1 (using LHADF routines dirtectly) but with the additional parameter nset. This allows specific PDF sets/members to be assigned to nset and used interchangeably once they have been intialized. The table below lists those routines and their use.

call InitPDFsetM(nset,name) or call InitPDFsetByNameM(nset,name)	Called once per PDF set to allocate nset to a specific set
call InitPDFM(nset,mem)	Called once per PDF set/member to allocate the member number (mem) to nset.
The above two calls are each made only ONCE per run no matter how many times a PDF set is used thereafter
call evolvePDFM(nset,x,Q,f)	Returns the PDF momentum density (f ) for the nucleon or pion PDF set initialised as nset for the given x and Q.
call evolvePDFpM(nset,x,Q,P2,IP2,f)	Returns the PDF momentum density (f ) for the photon PDF set initialised as nset for the given x,Q,P2and IP2.
call GetLam4M(nset,mem,qcdl4) call GetLam5M(nset,mem,qcdl5)	Return the values of qcdl4 and qcdl5 for the member number mem of the set nset. Note that the values should be used with care due to the differing defintions of lambda.
call GetXminM(nset,mem,xmin) call GetXmaxM(nset,mem,xmax) call GetQ2minM(nset,mem,q2min) call GetQ2maxM(nset,mem,q2max)	Return the minimum and maximum values of x and Q**2 for the member number mem of the set nset (version 5.3.0 onwards)
call numberPDFM(nset,Nmem)	Returns as Nmem the number of PDF members in the set nset:(excluding the special ``best fit'' member, i.e. the member numbers run from 0 to Nmem)
call GetOrderPDFM(nset,order)	To get the evolution order of the PDFs in set nset. The integer variable order is 0 for Leading Order, 1 for Next-to-Leading Order, etc.
call GetOrderAsM(nset,order)	To get the evolution order of aS in set nset. The integer variable order is 0 for Leading Order, 1 for Next-to-Leading Order, etc.
call GetRenFacM(nset,muf)	Gives the ratio of the renormalization scale over the factorization scale used in the ``fit'' for set nset. The double precision variable muf contains the ratio. Usually muf is equal to unity.
call GetDescM(nset)	This call will print the PDF description of set nset to the default output stream.
call GetQmassM(nset,nf,mass)	The mass mass is returned for quark flavor nf for set nset. The quark masses are used in the aS evolution.
call GetThresholdM(nset,nf,Q)	The flavor threshold Q is returned for flavor nf for set nset. If Q=-1d0 the flavor is not in the evolution (e.g. the top quark is usually not included in the evolution). If Q=0d0 the flavor is parametrized at the parametrization scale. For positive non-zero values of Q the value is set to the flavor threshold at which the PDF starts to evolve.
call GetNfM(nset,nfmax)	For the PDF sets of type .LHpdf returns the number of flavor thresholds in the PDF evolution for set nset. Usually the returned value for nfmax is equal to five as the top quark is usually not considered in the PDFs For the PDF sets of the .LHgrid type it returns -1.

5  The C++ wrapper class LHAPDFWrap

5.1  Description

// constructors LHAPDFWrap();
  // std c'tor d'tor
LHAPDFWrap(char *name); // (uses initPDFSetByName)
  // typical constructor with pdfset 'name', where 'name' is the name
  // of the data file of the desired set.
LHAPDFWrap(char *name, int memset); // (uses initPDFSetByName)
  // typical constructor with pdfset 'name' and subset 'memset'
  // 'name' is the name of the pdf set and 'memset' the subset member number. // intialisers void initPDFSet(char *name);
  // initialises the pdfset by full name (including the path directory)
void initPDFSetByName(char *name);
  // initialises the pdfset by name only
void initPDF(int memset);
  // selects the pdf subset out of the initialised pdf distribution // methods for nucleon pdfs std::vector< double > xfx(const double &x;, const double &Q;);
  // returns a vector xf(x, Q) with index 0 < i < 12.
  // 0..5 = tbar, ..., ubar, dbar;
  // 6 = g;
  // 7..12 = d, u, ..., t
double xfx(const double &x;, const double &Q;, int fl);
  // returns xf(x, Q) for flavour fl - this time the flavour encoding
  // is as in the LHAPDF manual...
  // -6..-1 = tbar,...,ubar, dbar
  // 1..6 = duscbt
  // 0 = g // methods for pion pdfs ($P2 and ip are ) std::vector< double > xfxp(const double &x;, const double &Q;, const double &P2;, int ip);
  // returns a vector xf(x, Q) with index 0 < i < 12.
  // 0..5 = tbar, ..., ubar, dbar;
  // 6 = g;
  // 7..12 = d, u, ..., t
double xfxp(const double &x;, const double &Q;, const double &P2;, int ip, int fl);
  // returns xf(x, Q) for flavour fl - this time the flavour encoding
  // is as in the LHAPDF manual...
  // -6..-1 = tbar,...,ubar, dbar
  // 1..6 = duscbt
  // 0 = g // methods for nuclear pdfs (&a; is the atomic number) std::vector< double > xfxa(const double &x;, const double &Q;, const double &a;);
  // returns a vector xf(x, Q) with index 0 < i < 12.
  // 0..5 = tbar, ..., ubar, dbar;
  // 6 = g;
  // 7..12 = d, u, ..., t
double xfxa(const double &x;, const double &Q;, const double &a;, int fl);
  // returns xf(x, Q) for flavour fl - this time the flavour encoding
  // -6..-1 = tbar,...,ubar, dbar
  // is as in the LHAPDF manual...
  // -6..-1 = tbar,...,ubar, dbar
  // 1..6 = duscbt
  // 0 = g // methods for photon pdfs (MRST2004qed.LHgrid) std::vector< double > xfxphoton(const double &x;, const double &Q;);
  // returns a vector xf(x, Q) with index 0 < i < 14.
  // 0..5 = tbar, ..., ubar, dbar;
  // 6 = g;
  // 7..12 = d, u, ..., t
  // 13 = photon
double xfxphoton(const double &x;, const double &Q;, int fl);
  // returns xf(x, Q) for flavour fl - this time the flavour encoding
  // is as in the LHAPDF manual...
  // -6..-1 = tbar,...,ubar, dbar
  // 1..6 = duscbt
  // 0 = g
  // 7 = photon // other methods void getDescription();
  // prints a brief description of the current pdf set to stdout
int numberPDF();
  // return the number of subsets available in the current distribution.
double alphasPDF(double Q);
  // return the alphas used by the current pdf.
int getOrderPDF();
  // returns the order (LO, NLO, NNLO) of the current pdf
int getOrderAlphaS();
  // return is order (LO, NLO, NNLO) of alpha_S.
double getQMass(int f);
  // returns quark mass used for flavour f.
double getThreshold(int f);
  // returns the threshold for flavour f.
int getNf();
  // returns the number of flavours used in the current pdf set.
double getLam4(int m);
  // returns the value of qcd lambda4 for member m
double getLam5(int m);
  // returns the value of qcd lambda5 for member m
double getXmin(int m);
  // returns the value of Xmin for member m
double getXmax(int m);
  // returns the value of Xmax for member m
double getQ2min(int m);
  // returns the value of Q2min for member m
double getQ2max(int m);
  // returns the value of Q2max for member m
void extrapolate();
  // sets the flag to extrapolate beyond the x and Q2 limits of the pdf set. // Equivalent methods for multiset pdf use LHAPDFWrap(int nset, char *name);
LHAPDFWrap(int nset, char *name, int memset);

void initPDFSetM(int nset, char *name);
void initPDFSetByNameM(int nset, char *name);
void initPDFM(int nset, int memset);

std::vector< double > xfxM(int nset, const double &x;, const double &Q;);
double xfxM(int nset, const double &x;, const double &Q;, int fl);

std::vector< double > xfxpM(int nset, const double &x;, const double &Q;, const double &P2;, int ip);
double xfxpM(int nset, const double &x;, const double &Q;, const double &P2;, int ip, int fl);

std::vector< double > xfxaM(int nset, const double &x;, const double &Q;, const double &a;);
double xfxaM(int nset, const double &x;, const double &Q;, const double &a;, int fl);

std::vector< double > xfxphotonM(int nset, const double &x;, const double &Q;);
double xfxphotonM(int nset, const double &x;, const double &Q;, int fl);

void getDescriptionM(int nset);
int numberPDFM(int nset);
double alphasPDFM(int nset, double Q);
int getOrderPDFM(int nset);
int getOrderAlphaSM(int nset);
double getQMassM(int nset, int f);
double getThresholdM(int nset, int f);
int getNfM(int nset);
double getLam4M(int nset, int m);
double getLam5M(int nset, int m);
double getXminM(int nset, int m);
double getXmaxM(int nset, int m);
double getQ2minM(int nset, int m);
double getQ2maxM(int nset, int m);

Appendices

A  PDF set numbers and names

Notes:

• When using the LHAGLUE initialization method:
• The columns headed .LHpdf and .LHgrid give the set numbers to use with LHAGLUE

• When Using the direct LHAPDF initialization routines:
• The .LHpdf and .LHgrid columns show the availablity of the respective files
• The File Name and Member columns give the names to use in the direct LHAPDF initialization routines.
• .LHpdf or .LHgrid has to be appended to the File Name depending the availability of that file (as indicated in the table) and wishes of the user.

Proton PDFs
r>

PDF set	.LHpdf	.LHgrid	File Name	Member	Xmin	Xmax	Q2min GeV2	Q2max GeV2
CTEQ6m (central value)	10000	10050	cteq6m	0	10-6	1	1.69	108
CTEQ6 (40 error sets)	10001-10040	10051-10090	cteq6 | cteq6mE	1-40	10-6	1	1.69	108
CTEQ6l (LO fit/NLO alphas)	10041	-	cteq6l	0/1	10-6	1	1.69	108
CTEQ6ll (LO fit/LO alphas)	10042	-	cteq6ll	0/1	10-6	1	1.69	108
CTEQ61 (central value)	10100	10150	cteq61	0	10-6	1	1.69	108
CTEQ61 (40 error sets)	10101-10140	10151-10190	cteq61	1-40	10-6	1	1.69	108
CTEQ6AB (20 sets, variable alphas)	-	10250-10269	cteq6AB	0-19	10-6	1	1.69	108
CTEQ65 (40 sets)	-	10350-10390	cteq65	0-40	10-7	1	1.69	1010
CTEQ65c (6 sets)	-	10450-10456	cteq65c	0-6	10-7	1	1.69	1010
CTEQ5m (Standard MSbar)	-	19050	cteq5m	0/1	10-5	1	1.00	108
CTEQ5m1 (updated CTEQ5m)	-	19051	cteq5m1	0/1	10-5	1	1.00	108
CTEQ5f3 (3-flav-DIS)	-	19053	cteq5f3	0/1	10-5	1	1.00	108
CTEQ5f4 (4-flav-DIS)	-	19054	cteq5f4	0/1	10-5	1	1.00	108
CTEQ5d (Standard DIS)	-	19060	cteq5d	0/1	10-5	1	1.00	108
CTEQ5l (LO fit)	-	19070	cteq5l	0/1	10-5	1	1.00	108
CTEQ4m (Standard MSbar)	-	19150	cteq4m	0/1	10-5	1	2.56	108
CTEQ4d (Standard DIS)	-	19160	cteq4d	0/1	10-5	1	2.56	108
CTEQ4l (LO fit)	-	19170	cteq4l	0/1	10-5	1	2.56	108
MRST2001nlo (Standard MSbar)	20000	20050	MRST2001nlo	0/1	10-5	1	1.25	107
MRST2001nlo (lower $\alpha_S$)	20002	20052	MRST2001nlo	2	10-5	1	1.25	107
MRST2001nlo (higher $\alpha_S$)	20003	20053	MRST2001nlo	3	10-5	1	1.25	107
MRST2001nlo (Jet Fit)	20004	20054	MRST2001nlo	4	10-5	1	1.25	107
MRST2001lo (LO fit)	-	20060	MRST2001lo	0/1	10-5	1	1.25	107
MRST2001nnlo (NNLO fit)	-	20070	MRST2001nnlo	0/1	10-5	1	1.25	107
MRST2001E (central value)	20100	20150	MRST2001E	0	10-5	1	1.25	107
MRST2001E (30 error sets)	20101-20130	20151-20180	MRST2001E	1-30	10-5	1	1.25	107
MRST2002nlo (Standard MSbar)	20200	20250	MRST2002nlo	0/1	10-5	1	1.25	107
MRST2002nnlo (NNLO fit)	-	20270	MRST2002nnlo	0/1	10-5	1	1.25	107
MRST2003cnlo (NLO - restricted )	20300	20350	MRST2003cnlo	0/1	10-3	1	10.0	107
MRST2003cnnlo (NNLO - restricted)	-	20370	MRST2003cnnlo	0/1	10-3	1	7.0	107
MRST2004nlo (Standard MSbar)	20400	20450	MRST2004nlo	0/1	10-5	1	1.25	107
MRST2004FF3lo (fixed flavour)	-	20452	MRST2004FF3lo	0/1	10-5	1	1.25	107
MRST2004FF4lo (fixed flavour)	-	20454	MRST2004FF4lo	0/1	10-5	1	1.25	107
MRST2004FF3nlo (fixed flavour)	-	20456	MRST2004FF3nlo	0/1	10-5	1	1.25	107
MRST2004FF4nlo (fixed flavour)	-	20458	MRST2004FF4nlo	0/1	10-5	1	1.25	107
MRST2004qed (photon)	-	20460	MRST2004qed	0/1	10-5	1	1.25	107
MRST2004nnlo (NNLO fit)	-	20470	MRST2004nnlo	0/1	10-5	1	1.25	107
MRST2006nnlo (NNLO fit)	-	20550-20580	MRST2006nnlo	0-30	10-6	1	1.0	109
MRST98 (central gluon/alphas)	29000	-	MRST98	0/2	10-5	1	1.25	107
MRST98 (lower gluon)	29001	-	MRST98	1	10-5	1	1.25	107
MRST98 (higher gluon)	29003	-	MRST98	3	10-5	1	1.25	107
Fermi2002_100 (101 sets)	30100-30200	-	Fermi2002_100	0-100	10-6	1	1.00	1010
Fermi2002_1000 (1001 sets)	31000-32000	-	Fermi2002_1000	0-1000	10-6	1	1.00	1010
Alekhin_100 (101 sets)	40100-40200	-	Alekhin_100	0-100	10-6	1	1.00	1010
Alekhin_1000 (1000 sets)	41000-41999	-	Alekhin_1000	0-999	10-6	1	1.00	1010
Alekhin2002 (LO - cent val)	-	40350	a02m_lo	0	10-7	1	0.8	2 x 108
Alekhin2002 (NLO - cent val)	-	40450	a02m_nlo	0	10-7	1	0.8	2 x 108
Alekhin2002 (NNLO - cent val)	-	40550	a02m_nnlo	0	10-7	1	0.8	2 x 108
Alekhin2002 (LO VFN 17 sets)	-	40351-40367	a02m_lo	1-15	10-7	1	0.8	2 x 108
Alekhin2002 (NLO VFN 17 sets)	-	40451-40467	a02m_nlo	1-15	10-7	1	0.8	2 x 108
Alekhin2002 (NNLO VFN 17 sets)	-	40551-40567	a02m_nnlo	1-15	10-7	1	0.8	2 x 108
Botje_100 (101 sets)	50100-50200	-	Botje_100	0-100	10-6	1	1.00	1010
Botje_1000 (1000 sets)	51000-51999	-	Botje_1000	0-999	10-6	1	1.00	1010
ZEUS2002 (VFN/TR cent value)	60000	-	ZEUS2002_TR	0	10-6	1	0.3	2 x 105
ZEUS2002 (ZM cent value)	60100	-	ZEUS2002_ZM	0	10-6	1	0.3	2 x 105
ZEUS2002 (FF cent value)	60200	-	ZEUS2002_FF	0	10-6	1	0.3	2 x 105
ZEUS2005 (ZJ cent value)	60300	-	ZEUS2005_ZJ	0	10-6	1	0.3	2 x 105
ZEUS2002 (VFN/TR 22 sets)	60001-60022	-	ZEUS2002_TR	1-22	10-6	1	0.3	2 x 105
ZEUS2002 (ZM 22 sets)	60101-60122	-	ZEUS2002_ZM	1-22	10-6	1	0.3	2 x 105
ZEUS2002 (FF 22 sets)	60201-60222	-	ZEUS2002_FF	1-22	10-6	1	0.3	2 x 105
ZEUS2005 (ZJ 22 sets)	60301-60322	-	ZEUS2005_ZJ	1-22	10-6	1	0.3	2 x 105
H12000 (nlo msbar cent value)	-	70050	H12000ms	0	5.7 x 10-5	1	1.5	106
H12000E (nlo msbar 20 sets)	-	70051-70070	H12000msE	1-20	5.7 x 10-5	1	1.5	106
H12000 (nlo dis cent value)	-	70150	H12000dis	0	5.7 x 10-5	1	1.5	106
H12000E (nlo dis 20 sets)	-	70151-70170	H12000disE	1-20	5.7 x 10-5	1	1.5	106
H12000 (lo cent value)	-	70250	H12000lo	0	5.7 x 10-5	1	1.5	106
H12000E (lo 20 sets)	-	70251-70270	H12000loE	1-20	5.7 x 10-5	1	1.5	106
GRV98 (nlo msbar)	-	80050	GRV98nlo	0	10-9	1	0.8	2 x 106
GRV98 (nlo dis)	-	80051	GRV98nlo	1 d	10-9	1	0.8	2 x 106
GRV98 (lo)	-	80060	GRV98lo	0	10-9	1	0.8	2 x 106
Unused	90000-99999	90050-9995+	-	-	-	-	-	-

Pion PDFs

PDF set	.LHpdf	.LHgrid	File Name	Member	Xmin	Xmax	Q2min GeV2	Q2max GeV2
OW-P Set 1 LO	-	211	OWPI	0/1	5 x 10-3	0.9998	4.0	2 x 103
OW-P Set 2 LO	-	212	OWPI	2	5 x 10-3	0.9998	4.0	2 x 103
SMRS-P 1 NLO	-	231	SMRSPI	1	10-5	0.9998	5.0	1.31 x 106
SMRS-P 2 NLO	-	232	SMRSPI	0/2	10-5	0.9998	5.0	1.31 x 106
SMRS-P 2 NLO	-	233	SMRSPI	3	10-5	0.9998	5.0	1.31 x 106
GRV-P HO NLO	-	251	GRVPI1	0/1	10-5	0.9998	0.3	106
GRV-P HO LO	-	252	GRVPI0	0/1	10-5	0.9998	0.25	106
ABFKW-P Set 1 NLO	-	261	ABFKWPI	0/1	10-3	0.9998	2.0	108
ABFKW-P Set 2 NLO	-	262	ABFKWPI	2	10-3	0.9998	2.0	108
ABFKW-P Set 3 NLO	-	263	ABFKWPI	3	10-3	0.9998	2.0	108

Photon PDFs

PDF set	.LHpdf	.LHgrid	File Name	Member	Xmin	Xmax	Q2min GeV2	Q2max GeV2
DO-G Set 1 LO	-	311	DOG0	0/1	10-5	0.9	10.0	104
DO-G Set 2 NLO	-	312	DOG1	0/1	10-5	0.9	10.0	104
DG-G Set 1 LO	-	321	DGG	0/1	10-5	0.9998	1.0	104
DG-G Set 2 LO	-	322	DGG	2	10-5	0.9998	1.0	50.0
DG-G Set 3 LO	-	323	DGG	3	10-5	0.9998	20.0	500.0
DG-G Set 4 LO	-	324	DGG	4	10-5	0.9998	200.0	104
LAC-G Set 1 LO	-	331	LACG	0/1	10-4	0.9998	4.0	105
LAC-G Set 2 LO	-	332	LACG	2	10-4	0.9998	4.0	105
LAC-G Set 3 LO	-	333	LACG	3	10-4	0.9998	1.0	105
GAL-G LO	-	334	LACG	4	10-4	0.9998	4.0	105
GS-G NLO	-	341	GSG1	0/1	5 x 10-4	0.9998	5.3	108
GS-G LO Set 1	-	342	GSG0	0/1	5 x 10-4	0.9998	5.3	108
GS-G LO Set 2	-	343	GSG0	2	5 x 10-4	0.9998	5.3	108
GS-G-96 NLO	-	344	GSG961	0/1	5 x 10-4	0.9998	5.3	108
GS-G-96 LO	-	345	GSG960	0/1	5 x 10-4	0.9998	5.3	108
GRV-G 1HO DIS NLO	-	351	GRVG1	1	10-5	0.9998	0.3	106
GRV-G HO DIS NLO	-	352	GRVG1	0/2	10-5	0.9998	0.3	106
GRV-G LO	-	353	GRVG0	1	10-5	0.9998	0.25	106
GRS-G LO	-	354	GRVG0	0/2	10-5	0.9998	0.6	5 x 104
ACFGP-G HO NLO	-	361	ACFGPG	1	1.37 x 10-3	0.9986	2.0	5.5 x 105
ACFGP-G HO-mc NLO	-	362	ACFGPG	2	1.37 x 10-3	0.9986	2.0	5.5 x 105
ACFGP-G HO NLO	-	363	ACFGPG	0/3	1.37 x 10-3	0.9986	2.0	5.5 x 105
WHIT-G 1 LO	-	381	WHITG	0/1	10-3	0.9998	4.0	2.5 x 103
WHIT-G 2 LO	-	382	WHITG	2	10-3	0.9998	4.0	2.5 x 103
WHIT-G 3 LO	-	383	WHITG	3	10-3	0.9998	4.0	2.5 x 103
WHIT-G 4 LO	-	384	WHITG	4	10-3	0.9998	4.0	2.5 x 103
WHIT-G 5 LO	-	385	WHITG	5	10-3	0.9998	4.0	2.5 x 103
WHIT-G 6 LO	-	386	WHITG	6	10-3	0.9998	4.0	2.5 x 103
SaS-G 1D (ver.1) LO	-	391	SASG	1	10-5	0.9998	0.36	5 x 104
SaS-G 1M (ver.1) LO	-	392	SASG	2	10-5	0.9998	0.36	5 x 104
SaS-G 2D (ver.1) LO	-	393	SASG	3	10-5	0.9998	4.0	5 x 104
SaS-G 2M (ver.1) LO	-	394	SASG	4	10-5	0.9998	4.0	5 x 104
SaS-G 1D (ver.2) LO	-	395	SASG	5	10-5	0.9998	0.36	5 x 104
SaS-G 1M (ver.2) LO	-	396	SASG	0/6	10-5	0.9998	0.36	5 x 104
SaS-G 2D (ver.2) LO	-	397	SASG	7	10-5	0.9998	4.0	5 x 104
SaS-G 2M (ver.2) LO	-	398	SASG	8	10-5	0.9998	4.0	5 x 104

B  Examples

The example tar file can be downloaded from the website. Untarring the file will create the directory Examples which contains the source code of the 3 examples together with the makefile to generate the executables. Note that the pathnames in the makefile have to be edited for proper execution. When running an example for the first time with the evolution code QCDNUM a grid has to be calculated. This might take some time depending on the grid size. Running with the PDF set after that the initialization time is much shorter as everything can be read in from an external file.

B.1  Example 1: A PDF table

The executable example.x is made by the command make 1. The first example code simply runs through all 100 PDF members of the Alekhin set [6], printing out a_S(M_Z) and the gluon PDF momentum densities at a few (x,Q) points:

  
      program example1
      implicit real*8(a-h,o-z)
      character name*64
      real*8 f(-6:6)
*
      name='cteq61.LHpdf'
      call InitPDFsetByName(name)
*
      QMZ=91.18d0
      write(*,*)
      call numberPDF(N)
       do i=0,N
         write(*,*) '---------------------------------------------'
         call InitPDF(i)
         write(*,*) 'PDF set ',i
         write(*,*)
         a=alphasPDF(QMZ)
         write(*,*) 'alpha_S(M_Z) = ',a
         write(*,*)
         write(*,*) 'x*up'
         write(*,*) '   x     Q=10 GeV     Q=100 GeV    Q=1000 GeV'
         do x=0.1d0,0.95d0,0.1d0
            Q=10d0
            call evolvePDF(x,Q,f)
            u1=f(2)
            Q=100d0
            call evolvePDF(x,Q,f)
            u2=f(2)
            Q=1000d0
            call evolvePDF(x,Q,f)
            u3=f(2)
            write(*,*) x,u1,u2,u3
         enddo
      enddo
*
     end

B.2  Example 2: Calculating Uncertainties for Monte Carlo PDF sets

Warning: only use the following on the Botje99, Fermi02 and Alekhin00 sets!

The method os NOT appiclable to other error sets such as MRST and CTEQ

The executable example.x is made by the command make 2. The second example calculates the average value of the gluon PDF and its standard deviation for several x values at Q=100 GeV using the Botje set [7]. Note that the program calculates the quantities in two manners. First by placing the loop over the PDF members inside the loop over the parton momentum fractions. Second, by exchanging this order and storing the first and second moments in an array depending on the x value. The second method is considerably faster in computing time as each PDF member is only initialized once.

Also calculated is the correlation of the gluon momentum PDF between x=0.001 and x=0.01 and the correlation between the strange quark momentum PDF at x=0.001 and a_S(Q) at Q=10 GeV.

      program example2
      implicit real*8(a-h,o-z)
      character*32 name
      real*8 f(-6:6),mom1(9),mom2(9)
*
      Q=100d0
      name='../PDFsets/Botje_100.LHpdf'
      call InitPDFset(name)
*
      call numberPDF(Nmem)
      write(*,*)
      write(*,*) 'Calculating the gluon momentum PDF average <g>'
      write(*,*) 'and standard deviaton SD(g) for several x values'
      write(*,*) 'at Q=100 GeV'
      write(*,*)
      write(*,*) '1. The slow way:'
      write(*,*) 
      write(*,*) '   x       <g>         SD(g)'
      do x=0.01d0,0.095d0,0.01d0
         gmom1=0d0
         gmom2=0d0
         do i=1,Nmem
            call InitPDF(i)
            call evolvePDF(x,Q,f)
            gmom1=gmom1+f(0)
            gmom2=gmom2+f(0)**2
         enddo
         av=gmom1/Nmem
         sd=sqrt(gmom2/Nmem-av**2)
         write(*,*) x,av,sd
      enddo
      write(*,*) 
      write(*,*) '2. The fast way:'
      write(*,*) 
      write(*,*) '   x       <g>         SD(g)'
      do i=1,9
         mom1(i)=0d0
         mom2(i)=0d0
      enddo
      do i=1,Nmem
         call InitPDF(i)
         ic=0
         do x=0.01d0,0.095d0,0.01d0
            call evolvePDF(x,Q,f)
            ic=ic+1
            mom1(ic)=mom1(ic)+f(0)
            mom2(ic)=mom2(ic)+f(0)**2
         enddo
      enddo
      ic=0
      do x=0.01d0,0.095d0,0.01d0
         ic=ic+1
         av=mom1(ic)/Nmem
         sd=sqrt(mom2(ic)/Nmem-av**2)
         write(*,*) x,av,sd
      enddo
*
      Q=10d0
      x1=0.001d0
      x2=0.01d0
      write(*,*)
      write(*,*) 'Calculating the normalized correlation coefficient'
      write(*,*) '<g1g2> between g(x=0.001) and g(x=0.01) and'
      write(*,*) '<sAlpha> between the strange quark momentum PDF'
      write(*,*) 'at x=0.001 and alpha_S(Q) for Q=10 GeV'
      write(*,*)
      avg1=0d0
      avg2=0d0
      avs=0d0
      avAs=0d0
      sdg1=0d0
      sdg2=0d0
      sds=0d0
      sdAs=0d0
      Cg1g2=0d0
      CsAs=0d0
      j=3
      do i=1,Nmem
         call InitPDF(i)
         As=alphasPDF(Q)
         call evolvePDF(x1,Q,f)
         g1=f(0)
         s=f(j)
         call evolvePDF(x2,Q,f)
         g2=f(0)
         avAs=avAs+As
         avg1=avg1+g1
         avg2=avg2+g2
         avs=avs+s
         sdAs=sdAs+As**2
         sdg1=sdg1+g1**2
         sdg2=sdg2+g2**2
         sds=sds+s**2
         CsAs=CsAs+s*As
         Cg1g2=Cg1g2+g1*g2
      enddo
      avAs=avAs/Nmem
      avs=avs/Nmem
      avg1=avg1/Nmem
      avg2=avg2/Nmem
      sdAs=sdAs/Nmem-avAs**2
      sds=sds/Nmem-avs**2
      sdg1=sdg1/Nmem-avg1**2
      sdg2=sdg2/Nmem-avg2**2
      CsAs=CsAs/Nmem-avs*avAs
      Cg1g2=Cg1g2/Nmem-avg1*avg2
      write(*,*) '<g1g2>    = ',Cg1g2/sqrt(sdg1*sdg2)
      write(*,*) '<sAlpha> = ',CsAs/sqrt(sds*sdAs)
      end

B.3  Example 3: A Convenient Wrap

The executable example.x is made by the command make 3. The third example prints out the same as the first example. However it uses a wrapper around the LHAPDF code. The single call wrapper takes care of the initialization calls. Note that the number of PDF members is only known after the first call to the subroutine parden. Also the a_S value is only known after the first call to the subroutine parden for a specific member PDF.

      program example3
      implicit real*8(a-h,o-z)
      real*8 f(-6:6)
*
      QMZ=91.71d0
      Nmem=100
      write(*,*)
      do i=1,Nmem
         write(*,*) '---------------------------------------------'
         write(*,*) 'PDF set ',i
         write(*,*)
         write(*,*) 'x*Gluon'
         write(*,*) '   x     Q=10 GeV     Q=100 GeV    Q=1000 GeV'
         do x=0.01d0,0.095d0,0.01d0
            Q=10d0
            call parden(x,Q,f,i)
            g1=f(0)
            Q=100d0
            call parden(x,Q,f,i)
            g2=f(0)
            Q=1000d0
            call parden(x,Q,f,i)
            g3=f(0)
            write(*,*) x,g1,g2,g3
         enddo
         a=alphasPDF(QMZ)
         write(*,*)
         write(*,*) 'alpha_S(M_Z) = ',a
         write(*,*)
      enddo
*
      end
*
      subroutine parden(x,Q,f,imem)
      implicit none
      character*32 name/'../PDFsets/Alekhin_100.LHpdf'/
      real*8 x,Q,f(-6:6),alfas
      integer imem,init/0/,lmem/-1/
      save init,lmem
*
      if (init.eq.0) then
         init=1
         call InitPDFset(name)
      endif
      if (imem.ne.lmem) then
         lmem=imem
         call InitPDF(lmem)
      endif
      call evolvePDF(x,Q,f)
      return
*
      end

B.4  Example 4: Using LHAGLUE to do Example 1

This program illustrates using the LHAGLUE routines ro repeat Example 1.

      program example4
c
c using lhaglue to do example 1
c
      implicit real*8(a-h,o-z)
      character*20 parm(20)
      double precision value(20)
*
      parm(1)='DEFAULT'
      call SetLHAPARM('SILENT')
c      parm(19)='SILENT'
      val = 10100
      do i=0,40
      value(1)=val+i
        write(*,*) '---------------------------------------------'
        call pdfset(parm,value)
        if (i.eq.0) call getdesc()
        qmz = 91.71d0
        a = alphasPDF(qmz)
        print *,'Alpha_s(Mz)=',a
         write(*,*) 'x*up'
         write(*,*) '   x     Q2=10 GeV     Q=100 GeV    Q=1000 GeV'
         do x=0.1d0,0.95d0,0.1d0
            Q=10.0d0
            call structm(x,q,upv,dnv,usea,dsea,str,chm,bot,top,glu)
            g1=upv+usea
            Q=100d0
            call structm(x,q,upv,dnv,usea,dsea,str,chm,bot,top,glu)
            g2=upv+usea
            Q=1000d0
            call structm(x,q,upv,dnv,usea,dsea,str,chm,bot,top,glu)
            g3=upv+usea
            write(*,*) x,g1,g2,g3
         enddo
      enddo
*
      end

B.4  Example 5: Using Multiset Initialisation

This program illustrates multiset initialisation LHAglue/LHAPDF.

      program example5
c
c using lhaglue for 3 different PDF sets proton/photon/pion
c using lhaglue for 3 different PDF sets proton/photon/pion
c using setpdfpath to define the path
c using PDFsta for statistics
c
      implicit real*8(a-h,o-z)
      character*20 parm(20)
      character name*64
      double precision value(20),g(3)
      integer inset(3)
      data inset/20200,391,231/
      common/W50513/xmin,xmax,q2min,q2max
*
      parm(1)='DEFAULT'

      call setlhaparm('SILENT')
      call setpdfpath('../PDFsets')
      qmz = 91.180d0
      ip2 = 0
      p2 = 0.0d0
      do j=1,3
        print *,'PDF set ',j,' is PDF set number ',inset(j)
        value(1)=inset(j)
        call pdfset(parm,value)
        call getdescm(j)
        call getlam4m(j,0,xlam4)
        call getlam5m(j,0,xlam5)
        a=alphasPDF(QMZ)
        print *,'PDF number, alpha_s(mz), xmin, xmax, q2min, q2max, lambda4, lambda5'
        print *,value(1),a,xmin,xmax,q2*
      call PDFsta
      end
wmin,q2max,xlam4,xlam5
      write(*,*) '---------------------------------------------'
      enddo
      Q=100.0d0
      print *,'x*gluon at Q = 100 GeV'
      print *,'PDF sets: ',(inset(j),j=1,3)
      write(*,*) '---------------------------------------------'
      do x=0.1d0,0.95d0,0.1d0
       do j=1,3
        value(1)=inset(j)
        call pdfset(parm,value)
        if(value(1).ge.300.and.value(1).le.399) then
          call structp(x,q2,p2,ip2,upv,dnv,usea,dsea,str,chm,bot,top,glu)
        else
          call structm(x,q,upv,dnv,usea,dsea,str,chm,bot,top,glu)
        endif
            g(j)=glu
       enddo
         write(*,*) x,(g(j),j=1,3)
      enddo
*
      call PDFsta
      end

References

[1]

M. Botje, QCDNUM version 16.12, preprint ZEUS-97-066,
http://www.nikhef.nl/ ~ h24/qcdnum.

[2]

S. Alekhin, M. Botje, W. Giele, J. Pumplin, F. Olness, G. Salam and A. Vogt,
Physics at TeV Colliders II Workshop, Les Houches, France, May 2001.

[3]

H. Plothow-Besch, Comput. Phys. Commun. 75, 396 (1993).

[4]

W. Giele, S. Keller and D. Kosower, hep-ph/0104052;
W. Giele and S. Keller, hep-ph/9803393 [Phys. Rev. D58, 1997, 094023].

[5]

W. Giele and S. Keller, hep-ph/0104053.

[6]

S. I. Alekhin, hep-ph/0011002, [Phys. Rev. D63, 2001, 094022].

[7]

M. Botje, hep-ph/9912439, [Eur. Phys. J. C14, 2000, 285].

Footnotes:

¹W. Giele, Fermilab, giele@fnal.gov and M.R. Whalley, Durham U, m.r.whalley@durham.ac.uk

²Developed by D. Bourilkov and C. Group, University of Florida hep-ph/0305126.

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On 5 Apr 2002, 11:34.