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\begin{document}

\title{The Art of Global Fitting for Parton Distributions%
\footnote{Presented at the Ringberg Workshop ``New Trends in HERA Physics,'' Sept 29,2003}}
\author{Jon Pumplin}
\affiliation{Department of Physics and Astronomy, Michigan State University,
East Lansing, MI 48824 USA}

\maketitle

\input Ring03Pumplin.figs

\section{Introduction}
High energy hadrons interact through their quark and gluon constituents.
The interactions become weak at short distances thanks to the asymptotic 
freedom property of Quantum Chromodynamics, which allows perturbation 
theory to be applied to a rich variety of experiments.
The nonperturbative nature of the proton for single hard interactions 
is thus characterized by Parton Distribution Functions (PDFs) $f_a(Q,x)$ 
of momentum scale $Q$ and light-cone momentum fraction $x$. The evolution 
in $Q$ is determined perturbatively by QCD renormalization group equations,
so the non-perturbative physics can be characterized by functions of $x$ 
alone at a fixed small $Q_0$.  The ongoing project to extract those 
functions from experiment is the subject of this talk.

The goal of the CTEQ global fitting group (like that of our friendly 
rivals MRST) is to extract the universal non-perturbative parton 
densities of the proton---or potentially of the neutron or 
nucleus---from a large variety of hard-scattering experiments, 
based on three pillars: 
\begin{itemize}
\item
{\bf Factorization} $\Rightarrow$ 
Short distance and long distance are separable (see Fig.\ \ref{fig:factorize});
\item
{\bf Asymptotic Freedom} $\Rightarrow$ 
Hard scattering processes are perturbatively calculable; 
\item
{\bf DGLAP Evolution} $\Rightarrow$ 
PDFs are characterized by functions of $x$ at a fixed 
small $Q_0$, with the PDFs at all higher $Q$ being 
determined from these by renormalization group evolution.
\end{itemize}
\figFactorize
The goals of this work are to test the consistency of QCD theory 
with the collective body of experiments; 
to test the consistency of individual experiments with QCD and the 
others; 
to explore the range of uncertainties of the estimated PDFs; 
and to make state-of-the-art extracted PDFs widely available, 
since they are needed to interpret all experiments 
that use hadron beams or targets---HERA, RHIC, Tevatron, LHC, 
and non-accelerator.

The kinematic range of available data, after 
cuts to suppress effects at low $Q$ and low $W$, are shown in 
Fig.\ \ref{fig:kinematics}.
The data cover a wide range of scales.  They are tied 
together by the DGLAP evolution equation and by the fact that the 
PDFs are universal.  Consistency or inconsistency between 
different processes, and between different data points for 
the same process can be observed only by applying QCD to 
tie them together in a global fit.
Future data from HERA, Tevatron run II (W, Z production), HERA II, 
and LHC will dramatically extend the range and 
accuracy of this global fit.
\figkinematics

\section{Some details of the CTEQ6 Global analysis}

{\bf Input from Experiment:}
$\sim 2000$ data points with $Q > 2  \, {\rm GeV}$, 
$W > 3.5  \, {\rm GeV}$ from $e$, $\mu$, $\nu$ DIS; 
lepton pair production (DY); 
lepton asymmetry in $W$ production; 
high $p_{T}$ inclusive jets;
$\alpha_s(M_Z)$ from LEP.

{\bf Input from Theory:}
NLO QCD evolution and hard scattering.

{\bf Parametrization at $Q_0$:} Use the form 
$A_{0} \, x^{A_{1}} \, (1-x)^{A_{2}} \, e^{A_{3}x}
(1+A_{4}x)^{A_{5}}$ for
$u_v = u - \bar{u}$, $d_v = d - \bar{d}$, $\bar{u} + \bar{d}$, and $g$.

{\bf Assumptions based on lack of information:}
$s = \bar{s} = 0.4 \, (\bar{u} + \bar{d})/2$ at $Q_0$; no intrinsic $b$ or $c$.

{\bf Procedure:}
Construct effective 
$\chi_{\mathrm{global}}^{2} =\sum_{\rm expts}\chi_{n}^{2}$, including published 
systematic error correlations.  Minimize $\chi_{\mathrm{global}}^{2}$ to 
obtain ``Best Fit'' PDFs.

{\bf Uncertainty estimates:}
Use the variation of $\chi_{\mathrm{global}}^{2}$ in neighborhood of the minimum 
to estimate uncertainty limits as the region of parameter space where 
$\chi^2 < \chi^2({\rm BestFit}) + T^2$ with $T \approx 10$. 
This ``Tolerance Factor'' $T \sim 10$ is quite different from 
the traditional value $1$ from Gaussian statistics, because of 
unknown systematic errors in theory and experiments.  It can be estimated 
from the apparent inconsistencies between experiments when they are combined 
in the global fit.

A note on the details of parametrization at $Q_0$:
For $d_{\rm val}$, $u_{\rm val}$, $\bar{u} + \bar{d}$ or $g$, we use the general form
\[
x f(x,Q_{0})=A_{0} \, x^{A_{1}} \, (1-x)^{A_{2}} \, e^{A_{3}x}
(1+e^{A_{4}}x)^{A_{5}}
\]
This corresponds to 
$$
\frac{d}{dx}\ln\left(xf\right)
=\frac{A_{1}}{x}-\frac{A_{2}}{1-x}
+\frac{c_{3}+c_{4}x}{1+c_{5}x}
$$
i.e., we add a 1:1 Pad\'{e} approximation term to the singular terms of 
the time-honored $A_{0} \, x^{A_{1}} \, (1-x)^{A_{2}}$ parametrization,
which is based on Regge theory and quark counting rules.
A sufficiently flexible parametrization is important;
but for convergence, there must not be too many ``flat directions.'' 
For that reason, some of the parameters are frozen at arbitrary values 
for some flavors.

To measure a set of continuous PDF functions at $Q_0$ on the basis of a 
finite set of data points would appear to be an ill-posed mathematical problem.
However, this difficulty is not so severe as might be expected since the actual 
predictions of interest that are based on the PDFs are discrete quantities.
In particular, fine-scale structure in $x$ in the PDFs at $Q_0$ tend to be 
smoothed out by evolution in $Q$.  They correspond to flat directions in $\chi^2$ 
space, so they are not accurately measured; but they have little effect on the 
applications of interest.

Some representative Best Fit parton distributions from the analysis are shown 
in Figs\ \ref{fig:pdfQlow}, \ref{fig:pdfQhigh}.
One sees that valence quarks dominate for $x \to 1$, and 
the gluon dominates for $x \to 0$, especially at large $Q$.
\figpdfQlow
\figpdfQhigh

\section{Including Experimental Systematic Errors}

The simplest definition of chi-squared,
\[
\chi_{0}^{2} = \sum_{i=1}^{N}
\frac{\left(D_{i}-T_{i}\right)^{2}}
{\sigma_{i}^{2}}
\qquad \qquad \left\{\begin{array}{l}
D_{i}=\ \mbox{data} \\ T_{i}=\ \mbox{theory} \\
\sigma_{i}=\ \mbox{``expt.\,error''}
\end{array}\right.
\]
works for random Gaussian errors,
\[
D_{i} = T_{i} + \sigma_{i}r_{i}
\qquad \mbox{with} \qquad
P(r) = \frac{e^{-r^{2}/2}}{\sqrt{2\pi}} \;.
\]

When correlated experimental systematic errors have been estimated, the 
errors become
\[
D_{i} = T_{i}(\textcolor{Blue}{a})
+\alpha_{i} \, r_{{\rm stat},i}
+ \sum_{k=1}^{K} \textcolor{Red}{r_{k}} \, \beta_{ki} \;,
\]
where $\alpha_{i}$ is the `standard deviation' of the
random uncorrelated error, and 
$\beta_{ki}$ is the `standard deviation' of the
$k$\,th (completely correlated!) systematic error on $D_{i}$.

In that case the parameters $\{r_{k}\}$ that control the corrections 
for systematic errors join the original PDF shape parameters $\{a_{\lambda}\}$
as parameters for which the global $\chi^2$ is to be minimized.
In effect, the other experiments in the global fit are used to determine 
the unknown systematic error coefficients.  (This is an extension of a 
familiar situation in which the overall normalization of an experiment may be  
determined by fitting to others if it is not well measured directly.)


To take into account the systematic errors, we define
\[
\chi^{\prime{2}}(a_{\lambda},r_{k})
=\sum_{i=1}^{N} \, \frac
{\left(D_{i}-\sum_{k} r_{k} \, \beta_{ki}-T_{i}\right)^{2}}
{\alpha_{i}^{2}}
+\sum_{k} \, r_{k}^{2},
\]
and minimize with respect to $\{r_{k}\}$.
The result is
\[
\widehat{r}_{k}=\sum_{k^{\prime}} \left(A^{-1}\right)_{kk^{\prime}}
B_{k^{\prime}},
\qquad
\mbox{(systematic shift)}
\]
where
\bea
A_{kk^{\prime}} &=& \delta_{kk^{\prime}} + \sum_{i=1}^{N}
\frac{\beta_{ki}\beta_{k^{\prime}i}}{\alpha_{i}^{2}}
\\
B_{k} &=& \sum_{i=1}^{N} \, \frac
{\beta_{ki}\left(D_{i}-T_{i}\right)}
{\alpha_{i}^{2}}.
\eea
Since the dependence of $\chi^2$ on the $\widehat{r}_{k}$'s is 
quadratic, we are able to minimize with respect to those 
parameters explicitly for any set of PDF shape parameters 
$\{a_{\lambda}\}$.
We then minimize the 
remaining $\chi^{2}(a)$ with respect to the PDF shape 
parameters $\{a_{\lambda}\}$.  Then the 
$\{a_{\lambda}\}$ determine $f_{i}(x,Q_{0}^{2})$.
Meanwhile, $\{\widehat{r}_{k}\}$ are
the optimal ``corrections'' for systematic errors;
i.e., systematic shifts to be applied to the data points
to bring the data from different experiments into compatibility,
within the framework of the theoretical model.

\figZEUSlowx
\figzsh
An example of how this works is shown in Fig.\ \ref{fig:ZEUSlowx}:
The data points include the estimated corrections for systematic errors, i.e., 
the central values plotted have been shifted 
by an amount that is consistent with the estimated systematic errors, 
where the systematic error parameters are determined using other experiments 
via the global fit.  The error bars are statistical errors only.
Fig.\ \ref{fig:zsh} shows that the systematic error parameters come out of 
order $1$ as they should.  Fig.\ \ref{fig:histZEUS} shows that the 
residuals defined as (data-theory)/error also come out with the expected 
distribution.  In contrast, Fig.\ \ref{fig:histZEUSnocor} shows that if 
the systematic shifts are not included, the fit becomes bad.
\fighistZEUS
\fighistZEUSnocor

\section{Uncertainties in PDFs}

There are several {\bf Sources of uncertainty:}
\begin{itemize}
\item Experimental errors included in $\chi^2$
\item Unknown experimental errors 
\item Parametrization dependence
\item Higher-order corrections \& Large Logarithms
\item Power Law corrections (``higher twist'')
\end{itemize}

There are some {\bf Fundamental difficulties:}
\begin{itemize}
\item
Good experiments run until systematic errors dominate, so
the magnitude of remaining systematic errors involves 
guesswork. 
\item
Systematic errors of the theory and their correlations 
are even harder to guess.
\item
Quasi--ill-posed problem: we must determine continuous functions 
from a discrete data set.  (Because of the smoothing effect of 
DGLAP evolution, this is not as impossible as it sounds.)
\item
Some combinations of variables are unconstrained, e.g., 
$s - \bar{s}$ before NuTeV data.
\end{itemize}

There are several {\bf Approaches to estimating the uncertainty.}
In all of the uncertainty methods, we continue to use $\chi^2$ 
as a measure of the quality of the fit; 
but vary weights assigned to the 
experiments to estimate the range of acceptable fits, rather 
than relying on the classical $\Delta \chi^2 = 1$.

\fighistdanfig
The essence of the Uncertainty Problem can be seen in 
the Fig.\ \ref{fig:danfig}, which shows some hypothetical 
measurements of a single parameter $\theta$.  Suppose 
the quantity $\theta$ has been measured by two different
experiments, or extracted using two different approximations
to the True Theory.  What would you quote as the Best Fit 
and the Uncertainty?  The disagreements are of course not so 
obvious in the many-parameter global fit.  However, the 
disagreements can be probed in one dimension by, for 
example, studying the variations of the Best Fit that 
result from assigning different weights to different 
experiments, or to different kinematic regions etc.  Much 
of this is discussed in our papers; and more is work in progress.

\section{Outlook}

Parton Distribution Functions are a necessary infrastructure 
for precision Standard Model studies and New Physics searches 
at hadron colliders and at experiments using hadron targets.  
Some issues that were discussed, but not necessarily included 
in this writeup due to space limitations:
\begin{itemize}

\item 
PDFs of the proton are increasingly well measured.

\item 
Useful tools are in place to estimate the uncertainty of PDFs and 
to propagate those uncertainties to physical predictions.
There is adequate agreement between various 
methods for estimating the uncertainty:
 \begin{itemize}
  \item ``Hessian Method'' based on the eigenvectors of the error matrix
  \item ``Lagrange Multiplier Method'' based on finding the uncertainty 
         on a predicted quantity by studying the variation of $\chi^2$ 
         as a function of that quantity
  \item systematic reweighting of experiments (work in progress with John Collins)
  \item random reweighting of experiments:  a variant of the ``well known'' 
        statistical bootstrap method
 \end{itemize}

\item 
The ``Les Houches Accord'' interface makes it easy to handle the large 
number of PDF solutions that are needed to characterize uncertainties. 
[hep-ph/0204316]

\item 
Improvements in the treatment of heavy quark effects are in progress, 
and together with neutrino experiments they will allow improved flavor 
differentiation.

\item 
Since PDFs summarize some fundamental nonperturbative physics of the proton,
they should be considered a challenge to be computed! 
(Low moments of meson PDFs have indeed been calculated in lattice gauge theory.)

\item 
Other non-perturbative methods, e.g. for $s(x) - \bar{s}(x)$, may be helpful.

\item 
HERA and Fermilab run II data will provide the next major experimental 
steps forward, followed by LHC.

\item 
Theoretical improvements such as resummation to use direct photon and 
W transverse momentum data will be useful.

\item 
In view of possible isospin breaking, and the importance of 
nuclear shadowing \& anti-shadowing effects, 
HERA measurements on deuterons would be highly welcome.

\end{itemize}


I thank the organizers of Ringberg '03 for an excellent workshop in 
an excellent setting.  I thank my collaborators on this work, 
D.\ Stump, W.K.\ Tung, J.\ Huston, 
H.\ Lai, P.\ Nadolsky, F.\ Olness, S.\ Kuhlmann, J.\ Owens; S.\ Kretzer,
and J.\ Collins for many valuable discussions.

\end{document}
