\documentclass[12pt]{article} %\usepackage{twoside} \usepackage{epsfig} \usepackage{amsmath} \usepackage{hhline} \usepackage{amssymb} \usepackage{times} \usepackage{cite} \usepackage{rotating} %\usepackage{amstex} \newcommand{\rb}[1]{\raisebox{1.5ex}[-1.5ex]{#1}} \newcommand{\F}{$ F_{2}(x,Q^2)\:$} \newcommand{\FL}{$ F_{L}(x,Q^2)\:$} \newcommand{\V}{$ V(x,Q^2)\,$} \newcommand{\Vc}{$ V\,$} \newcommand{\A}{$ A(x,Q^2)\,$} \newcommand{\Ac}{$ A\,$} \newcommand{\Fc}{$ F_{2}\,$} \newcommand{\Q}{$ Q^{2}\,$} \newcommand{\gv}{GeV$^2\,$} \newcommand{\as}{$\alpha_s\,$} \newcommand{\amz}{$\alpha_s(M_Z^2)\,$} \newcommand{\bs}{\overline{s}} \newcommand{\bu}{\overline{u}} \newcommand{\bd}{\overline{d}} \newcommand{\bq}{\overline{q}} \newcommand{\FLc}{$ F_{L}\,$} \newcommand{\xg}{$xg(x,Q^2)\,$} \newcommand{\xgc}{$xg\,$} \newcommand{\ipb}{pb$^{-1}\,$} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\PO}{I\!\!P } \newcommand{\XP}{x_{{I\!\!P}/{p}}} \newcommand{\TOSS}{x_{{i}/{\PO}}} \newcommand{\un}[1]{\mbox{\rm #1}} \newcommand{\pdsi}{$\partial \sigma_r / \partial \log y\,$} \newcommand{\pdff}{$\partial F_{2} / \partial \log Q^{2}\,$ } \newcommand{\llam}{$\lambda(x,Q^2)\:$} \newcommand{\lam}{$\lambda(Q^2)\:$} %\renewcommand{\topfraction}{1.0} %\renewcommand{\bottomfraction}{1.0} %\renewcommand{\textfraction}{0.0} \newlength{\dinwidth} \newlength{\dinmargin} \setlength{\dinwidth}{21.0cm} \textheight23cm \textwidth15.5cm \setlength{\dinmargin}{\dinwidth} \setlength{\unitlength}{1mm} \addtolength{\dinmargin}{-\textwidth} \setlength{\dinmargin}{0.5\dinmargin} \oddsidemargin -1.0in \addtolength{\oddsidemargin}{\dinmargin} \setlength{\evensidemargin}{\oddsidemargin} \setlength{\marginparwidth}{0.9\dinmargin} \marginparsep 8pt \marginparpush 5pt \topmargin -38pt \headheight 12pt %\headsep 30pt \footheight 10pt \footskip 22pt \parskip 3mm plus 2mm minus 2mm \setlength{\unitlength}{1mm} % e.p needed for figures %\special{!userdict begin /bop-hook{gsave 200 30 translate % 65 rotate /Times-Roman findfont 216 scalefont setfont % 0 0 moveto 0.93 setgray ( DRAFT) show grestore}def end} % \begin{document} \pagestyle{empty} \begin{titlepage} \noindent \begin{center} {\it {\large version of \today}} \\[.3em] \begin{small} \begin{tabular}{llrr} %Submitted to & \multicolumn{3}{r}{\footnotesize Electronic Access: {\it http://www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline %Submitted to & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \hline Submitted to & & & \epsfig{file=/h1/www/images/H1logo_bw_small.epsi,width=2.cm} \\[.2em] \hline \multicolumn{4}{l}{{\bf International Europhysics Conference on High Energy Physics}, July~12,~2001,~Budapest} \\ {\bf EPS 2001:} & Abstract: & {\bf 833} &\\ & Parallel Session & {\bf 1,2} &\\ & Plenary Session & {\bf 1,2} &\\[.7em] \multicolumn{4}{l}{{\bf XX International Symposium on Lepton and Photon Interactions}, July~23,~2001,~Rome} \\ {\bf LP 2001:} & Abstract: & {\bf 494} &\\ & Plenary Session & {\bf S08} &\\ \hline & \multicolumn{3}{r}{\footnotesize {\it www-h1.desy.de/h1/www/publications/conf/conf\_list.html}} \\[.2em] \end{tabular} \end{small} \end{center} \vspace*{2cm} \begin{center} \begin{Large} {\bf A Measurement of the Rise \\ of {\boldmath \F} Towards Low {\boldmath $x$} \\} \vspace*{1.cm} \end{Large} {\Large H1 Collaboration} \end{center} \date{today} %=======================abstract============================== \begin{abstract} \noindent A measurement of the derivative $(\partial \ln F_2 / \partial \ln x)_{Q^2} \equiv -\lambda(x,Q^2)$ of the proton structure function $F_2$ is presented in the low $x$ domain of deeply inelastic positron--proton scattering. For $ 0.01\geq x\geq 5\cdot 10^{-5}$ and $Q^2\geq 1.5\,{\rm GeV}^2$, $\lambda(x,Q^2)$ is found to be independent of $x$ and to rise linearly with $\ln Q^2$. % From a measurement of the derivative %$(\partial \ln F_2 / \partial \ln x)_{Q^2} \equiv -\lambda(x,Q^2)$ the rise %of the proton structure function is shown to be continuous towards low %$x > 5 \cdot 10^{-5}$ in the deeply inelastic scattering domain, the %function $\lambda$ showing no significant dependence on $x$ for $x < 0.01$ %but a linear rise with $\ln Q^2$. \end{abstract} %% \end{titlepage} % %\begin{flushleft} % \input{h1auts} %\end{flushleft} \newpage % %-------------------------------------------------------------------- % %\section{} The inclusive cross section for deeply inelastic lepton-proton scattering is governed by the proton structure function \F. Because of the large centre-of-mass energy squared, $s \simeq 10^5 $~GeV$^2$, the $ep$ collider HERA has accessed the region of low Bjorken $x$, $x > Q^2/s > 10^{-5}$ for four-momentum transfers squared $Q^2 > 1 $ GeV$^2$, and \Fc was observed to rise towards low $x$~\cite{f292}. Perturbative Quantum Chromodynamics (QCD) provides a rigorous and successful theoretical description of the $Q^2$ dependence of \F. Although the $x$ dependence of \Fc\ is not predicted in QCD, in the limit of small $x$ the evolution equations~\cite{dglap} can be approximately solved in the double asymptotic limit to leading order~\cite{das} and \Fc\ can be represented as \begin{equation} F_2(x,Q^2) \sim x^{-\lambda(x,Q^2)}. \end{equation} Below the deeply inelastic region, for $Q^2 < 1 $~GeV$^2$, Regge theory~\cite{regge} predicts that $\lambda$ approaches a value given by the Pomeron intercept, $\lambda = \alpha_{\PO}(0)-1 \simeq 0.08$, independently of $x$ and $Q^2$~\cite{dl}. \\ % %%%%%% % Very recently the H1 Collaboration has presented~\cite{paper} a new measurement of \F in the kinematic range $3 \cdot 10^{-5} \le x \le 0.2$ and $1.5 \le Q^2 \le 150 $ GeV$^2$ based on data taken in the years 1996/97 with a positron beam energy $E_e = 27.6$ GeV and a proton beam energy $E_p = 820$ GeV. The high accuracy of these data allows for the first time the function $\lambda$ to be measured as a two-dimensional function of $x$ and $Q^2$ from the derivative \begin{equation} -\left( \frac{\partial \ln F_2(x,Q^2)}{\partial \ln x}\right)_{Q^2} = \lambda(x,Q^2). \end{equation} Use of this quantity for investigating the behaviour of \Fc at low $x$ has been advocated e.g. in~\cite{Navel}. The derivative is extracted in the full kinematic range using \Fc data points for adjacent $x$ bins at fixed $Q^2$ taking into account the full error correlations and correcting for finite bin size. %The values are quoted in Table~\ref{tabder}. As can be seen in Figure~\ref{lamx}, in this low $x$ region \llam becomes independent of $x$ for $x \leq 0.01$ to within the experimental accuracy. Thus there is no experimental evidence that the steepness of the rise of \Fc with decreasing $x$ is tempered throughout the measured deeply inelastic kinematic range. In perturbative QCD the observed behaviour of $(\partial \ln F_2 / \partial \ln x)_{Q^2}$ is very well described, see Figure~\ref{lamx}. At low $x$, for $Q^2 > 3$ GeV$^2$, this behaviour is driven solely by the gluon field, since quark contributions to the scaling violations of \Fc\ are negligible~\cite{paper}. At larger $x$ the transition to the valence-quark region causes a strong dependence of $\lambda$ on $x$ as is indicated by the QCD curves in Figure~\ref{lamx}. \\ % %%%% % Figure~\ref{lamq2} displays the measured derivative as a function of $Q^2$ for different $x$ values. It is well described by the NLO QCD calculation. The derivative is observed to rise approximately logarithmically with $Q^2$. For $x \leq 0.01$ its behaviour can be well described by a universal function \lam. The function \lam is determined from a fit of $F_2(x,Q^2) = c(Q^2) \cdot x^{-\lambda(Q^2)}$, restricted to the region $x \leq 0.01$ where $\lambda$ is now explicitly assumed to be independent of $x$. This fit yields values $c(Q^2) \simeq 0.18$, approximately independent of $Q^2$. The result for \lam is %quoted in Table~\ref{tablam} and illustrated in Figure~\ref{lammax}. It is more accurate than data hitherto published by the H1 Collaboration \cite{h1svx} and by the ZEUS Collaboration \cite{zeuslam}. As can be seen, $\lambda(Q^2)$ rises approximately linearly with $\ln Q^2$. The dependence can be represented as $\lambda(Q^2) = a \cdot \ln Q^2/\Lambda^2$, see Figure~\ref{lammax}. The coefficients are $a=0.0481 \pm 0.0013(\rm stat)\pm 0.0037(\rm syst)$ and $\Lambda = 292 \pm 20(\rm stat) \pm 51(\rm syst)$~MeV, obtained for $Q^2 \geq 3.5$ GeV$^2$ with a $\chi^2$ per degree of freedom of 1.0 using statistical errors only. When extrapolating \lam into the lower $Q^2$ region it takes the value of $0.08$ at $Q^2 = 0.45$~GeV$^2$. To summarise, the derivative $-(\partial \ln F_2 / \partial \ln x)_{Q^2} = \lambda$ is observed to be independent of $x$ for $x \leq 0.01$. This implies that the rise of \Fc towards low $x$, $\partial F_2 / \partial x$, is continuous and proportional to $F_2/x\:$ throughout the covered deeply inelastic domain. At low $x$, \lam rises linearly with $\ln Q^2$. %\vspace{1cm} % {\bf Acknowledgements} %===================== \normalsize \noindent We are very grateful to the HERA machine group whose outstanding efforts made this experiment possible. We acknowledge the support of the DESY technical staff. We appreciate the substantial effort of the engineers and technicians who constructed and maintain the detector. We thank the funding agencies for financial support of this experiment. We wish to thank the DESY directorate for the support and hospitality extended to the non-DESY members of the collaboration. % %\newpage \begin{thebibliography}{99} % % ----------introduction ------------------------------------- % \bibitem{f292} %\cite{Abt:1993cb} I.~Abt {\it et al.} [H1 Collaboration], %``Measurement of the proton structure function F2 (x, Q**2) in the low x region at HERA,'' Nucl.\ Phys.\ B {\bf 407} (1993) 515. \\ %%CITATION = NUPHA,B407,515;%% %\cite{Derrick:1993ft} M.~Derrick {\it et al.} [ZEUS Collaboration], %``Measurement of the proton structure function F2 in e p scattering at HERA,'' Phys.\ Lett.\ B {\bf 316} (1993) 412. %%CITATION = PHLTA,B316,412;%% % \bibitem{dglap} %\cite{Dokshitzer:1977sg} Y.~L.~Dokshitzer, %``Calculation Of The Structure Functions For Deep Inelastic Scattering And E+ E- Annihilation By Perturbation Theory In Quantum Chromodynamics. (In Russian),'' Sov.\ Phys.\ JETP {\bf 46} (1977) 641 [Zh.\ Eksp.\ Teor.\ Fiz.\ {\bf 73} (1977) 1216]. %%CITATION = SPHJA,46,641;%% %\cite{Gribov:1972rt} V.~N.~Gribov and L.~N.~Lipatov, %``E+ E- Pair Annihilation And Deep Inelastic E P Scattering In Perturbation Theory,'' Yad.\ Fiz.\ {\bf 15} (1972) 1218 [Sov.\ J.\ Nucl.\ Phys.\ {\bf 15} (1972) 675]. %%CITATION = YAFIA,15,1218;%% %\cite{Gribov:1972ri} V.~N.~Gribov and L.~N.~Lipatov, %``Deep Inelastic E P Scattering In Perturbation Theory,'' Yad.\ Fiz.\ {\bf 15} (1972) 781 [Sov.\ J.\ Nucl.\ Phys.\ {\bf 15} (1972) 438]. %%CITATION = YAFIA,15,781;%% %\cite{Altarelli:1977zs} G.~Altarelli and G.~Parisi, %``Asymptotic Freedom In Parton Language,'' Nucl.\ Phys.\ B {\bf 126} (1977) 298. %%CITATION = NUPHA,B126,298;%% % \bibitem{das} %\cite{DeRujula:1974rf} A.~De Rujula, S.~L.~Glashow, H.~D.~Politzer, S.~B.~Treiman, F.~Wilczek and A.~Zee, \\ %``Possible Nonregge Behavior Of Electroproduction Structure Functions,'' Phys.\ Rev.\ D {\bf 10} (1974) 1649; \\ %%CITATION = PHRVA,D10,1649;%% %\cite{Ball:1994du} R.~D.~Ball and S.~Forte, %``Double asymptotic scaling at HERA,'' Phys.\ Lett.\ B {\bf 335} (1994) 77 [hep-ph/9405320]. %%CITATION = HEP-PH 9405320;%% %\cite{DeRoeck:1996nb} %A.~DeRoeck, M.~Klein and T.~Naumann, %``On the Asymptotic Behaviour of F_2(x,Q~2),'' %Phys.\ Lett.\ B {\bf 385} (1996) 411 %[hep-ph/9605276]. \\ %%CITATION = HEP-PH 9605276;%% % \bibitem{regge} T.~Regge, Nuov.\ Cim. {\bf 14} (1959) 951, Nuov.\ Cim.\ {\bf 18} (1960) 947; \\ G.~Chew, S.~Frautschi and S.~Mandelstam, Phys.\ Rev.\ {\bf 126} (1962) 1202. % \bibitem{dl} %\cite{Donnachie:1994it} A.~Donnachie and P.~V.~Landshoff, %``Proton structure function at small q**2,'' Z.\ Phys.\ C {\bf 61} (1994) 139 [hep-ph/9305319]. %%CITATION = HEP-PH 9305319;%% % \bibitem{paper} %\cite{Adloff:2000qk} C.~Adloff {\it et al.} [H1 Collaboration], %``Deep-inelastic inclusive e p scattering at low x and a determination of alpha(s),'' hep-ex/0012053. %%CITATION = HEP-EX 0012053;%% % \bibitem{Navel} %\cite{Navelet:1994qp} H.~Navelet, R.~Peschanski and S.~Wallon, %``On the singular behavior of structure functions at low x,'' Mod.\ Phys.\ Lett.\ A {\bf 9} (1994) 3393 [hep-ph/9402352]. %%CITATION = HEP-PH 9402352;%% % \bibitem{h1svx} %\cite{Aid:1996au} S.~Aid {\it et al.} [H1 Collaboration], %``A Measurement and QCD Analysis of the Proton Structure Function $F_2(x,Q~2)$ at HERA,'' Nucl.\ Phys.\ B {\bf 470} (1996) 3 [hep-ex/9603004]. \\ %%CITATION = HEP-EX 9603004;%% %\cite{Adloff:1997mf} C.~Adloff {\it et al.} [H1 Collaboration], %``A measurement of the proton structure function F2(x,Q**2) at low x and low Q**2 at HERA,'' Nucl.\ Phys.\ B {\bf 497} (1997) 3 [hep-ex/9703012]. %%CITATION = HEP-EX 9703012;%% % \bibitem{zeuslam} %\cite{Breitweg:1999dz} J.~Breitweg {\it et al.} [ZEUS Collaboration], %``ZEUS results on the measurement and phenomenology of F2 at low x and low Q**2,'' Eur.\ Phys.\ J.\ C {\bf 7} (1999) 609 [hep-ex/9809005]. %%CITATION = HEP-EX 9809005;%% % \end{thebibliography} % ---------------------- all figures -------------------- %\input{papfigs107} % % derivative vs. x in q2 bins % \begin{figure}[ht] \epsfig{file=H1prelim-01-141.fig1.eps,width=15.cm} \caption{ Measurement of the function \llam: the inner error bars represent the statistical experimental uncertainty, while the full error bars include the systematic uncertainty added in quadrature; the solid curves denote the NLO QCD fit to the H1 cross section data described in~\cite{paper}; the dashed curves represent the extrapolation of the QCD fit below $Q^2 = 3.5 $ GeV$^2$.} \protect\label{lamx} \end{figure} %% %% derivative vs q2 in x bins %% \begin{figure}[ht] \epsfig{file=H1prelim-01-141.fig2.eps,width=15.cm} \caption{ Measurement of the function \llam: the inner error bars represent the statistical experimental uncertainty, while the full error bars include the systematic uncertainty added in quadrature; the solid curves denote the NLO QCD fit to the H1 cross section data described in~\cite{paper}; the minimum $Q^2$ value of cross section data included in this fit is $Q^2 = 3.5 $ GeV$^2$. } \protect\label{lamq2} \end{figure} % % lambda vom maxe % \begin{figure}[ht] \epsfig{file=H1prelim-01-141.fig3.eps,width=15.cm} \caption{ Determination of the exponent \lam from a fit $F_2 = cx^{-\lambda}$ to the H1 data~\cite{paper} for $x \leq 0.01$; the inner error bars illustrate the statistical uncertainties, the full error bars represent the statistical and the systematic uncertainties added in quadrature. The solid line is a fit $\lambda = a \ln Q^2/\Lambda^2$, see text, using data for $Q^2 \geq 3.5$ GeV$^2$} \protect\label{lammax} \end{figure} % \end{document} % %----------------------------------------------------------------- % all the tables at the end %----------------------------------------------------------------- % %------------------- %-- d lnF2/d lnx part1 | %------------------- % \begin{table}[h] \begin{tabular}{cc} \begin{scriptsize} \begin{tabular}[t]{|r|l|c|r|r|r|} \hline & & $ -\partial \ln F_2 /$ & & & \\ \rb{$Q^2$} & \multicolumn{1}{|c|}{\rb{$x$}} & $ \partial \ln x$ & \rb{$\delta_{sta}$} & \rb{$\delta_{sys}$} & \rb{$\delta_{tot}$} \\ \hline 1.5 & 0.000041 & 0.225 & 0.052 & 0.131 & 0.141 \\ 1.5 & 0.000065 & 0.146 & 0.101 & 0.180 & 0.206 \\ 2.0 & 0.000065 & 0.274 & 0.029 & 0.089 & 0.093 \\ 2.0 & 0.000105 & 0.090 & 0.036 & 0.085 & 0.092 \\ 2.0 & 0.000165 & 0.211 & 0.043 & 0.116 & 0.124 \\ 2.0 & 0.000260 & 0.093 & 0.040 & 0.096 & 0.104 \\ 2.0 & 0.000410 & 0.022 & 0.047 & 0.124 & 0.132 \\ 2.0 & 0.000750 & 0.276 & 0.030 & 0.080 & 0.086 \\ 2.0 & 0.00210 & 0.161 & 0.020 & 0.057 & 0.060 \\ 2.5 & 0.000065 & 0.199 & 0.034 & 0.068 & 0.076 \\ 2.5 & 0.000105 & 0.158 & 0.026 & 0.053 & 0.059 \\ 2.5 & 0.000165 & 0.242 & 0.031 & 0.084 & 0.089 \\ 2.5 & 0.000260 & 0.170 & 0.030 & 0.082 & 0.088 \\ 2.5 & 0.000410 & 0.126 & 0.032 & 0.088 & 0.093 \\ 2.5 & 0.000650 & 0.177 & 0.032 & 0.084 & 0.090 \\ 2.5 & 0.00119 & 0.106 & 0.021 & 0.058 & 0.061 \\ 2.5 & 0.00329 & 0.211 & 0.013 & 0.057 & 0.058 \\ 3.5 & 0.000105 & 0.174 & 0.030 & 0.065 & 0.071 \\ 3.5 & 0.000165 & 0.163 & 0.031 & 0.050 & 0.059 \\ 3.5 & 0.000260 & 0.174 & 0.030 & 0.071 & 0.077 \\ 3.5 & 0.000410 & 0.136 & 0.032 & 0.054 & 0.063 \\ 3.5 & 0.000650 & 0.216 & 0.031 & 0.065 & 0.072 \\ 3.5 & 0.00105 & 0.152 & 0.031 & 0.063 & 0.070 \\ 3.5 & 0.00190 & 0.186 & 0.021 & 0.039 & 0.044 \\ 3.5 & 0.00525 & 0.210 & 0.012 & 0.050 & 0.051 \\ 5.0 & 0.000165 & 0.159 & 0.035 & 0.055 & 0.065 \\ 5.0 & 0.000260 & 0.192 & 0.033 & 0.065 & 0.073 \\ 5.0 & 0.000410 & 0.262 & 0.036 & 0.052 & 0.063 \\ 5.0 & 0.000650 & 0.251 & 0.034 & 0.049 & 0.060 \\ 5.0 & 0.00105 & 0.129 & 0.033 & 0.051 & 0.061 \\ 5.0 & 0.00165 & 0.145 & 0.038 & 0.072 & 0.081 \\ 5.0 & 0.00299 & 0.209 & 0.022 & 0.044 & 0.049 \\ 5.0 & 0.00849 & 0.227 & 0.013 & 0.042 & 0.044 \\ 6.5 & 0.000165 & 0.115 & 0.046 & 0.068 & 0.082 \\ 6.5 & 0.000260 & 0.268 & 0.036 & 0.047 & 0.059 \\ 6.5 & 0.000410 & 0.210 & 0.039 & 0.073 & 0.082 \\ 6.5 & 0.000650 & 0.193 & 0.038 & 0.049 & 0.062 \\ 6.5 & 0.00105 & 0.255 & 0.037 & 0.047 & 0.060 \\ 6.5 & 0.00165 & 0.121 & 0.042 & 0.058 & 0.072 \\ 6.5 & 0.00299 & 0.195 & 0.024 & 0.037 & 0.044 \\ 6.5 & 0.00849 & 0.253 & 0.014 & 0.040 & 0.043 \\ 8.5 & 0.000260 & 0.188 & 0.042 & 0.054 & 0.068 \\ 8.5 & 0.000410 & 0.206 & 0.042 & 0.048 & 0.064 \\ 8.5 & 0.000650 & 0.230 & 0.042 & 0.064 & 0.077 \\ 8.5 & 0.00105 & 0.237 & 0.042 & 0.049 & 0.064 \\ 8.5 & 0.00165 & 0.220 & 0.047 & 0.054 & 0.072 \\ 8.5 & 0.00260 & 0.292 & 0.045 & 0.058 & 0.073 \\ 8.5 & 0.00475 & 0.162 & 0.028 & 0.040 & 0.049 \\ 8.5 & 0.0132 & 0.253 & 0.017 & 0.045 & 0.048 \\ 12.0 & 0.000410 & 0.174 & 0.020 & 0.052 & 0.056 \\ 12.0 & 0.000650 & 0.330 & 0.036 & 0.067 & 0.076 \\ 12.0 & 0.00105 & 0.243 & 0.047 & 0.052 & 0.070 \\ 12.0 & 0.00165 & 0.147 & 0.053 & 0.059 & 0.079 \\ 12.0 & 0.00260 & 0.276 & 0.050 & 0.053 & 0.073 \\ 12.0 & 0.00475 & 0.247 & 0.032 & 0.038 & 0.049 \\ 12.0 & 0.0132 & 0.245 & 0.019 & 0.043 & 0.047 \\ 15.0 & 0.000410 & 0.164 & 0.020 & 0.060 & 0.064 \\ 15.0 & 0.000650 & 0.227 & 0.018 & 0.047 & 0.050 \\ \hline \end{tabular} \begin{tabular}[t]{|r|l|c|r|r|r|} \hline & & $ -\partial \ln F_2 /$ & & & \\ \rb{$Q^2$} & \multicolumn{1}{|c|}{\rb{$x$}}& $ \partial \ln x$ & \rb{$\delta_{sta}$} & \rb{$\delta_{sys}$} & \rb{$\delta_{tot}$} \\ \hline 15.0 & 0.00105 & 0.296 & 0.018 & 0.074 & 0.076 \\ 15.0 & 0.00165 & 0.266 & 0.021 & 0.055 & 0.059 \\ 15.0 & 0.00260 & 0.238 & 0.021 & 0.045 & 0.050 \\ 15.0 & 0.00410 & 0.199 & 0.024 & 0.050 & 0.056 \\ 15.0 & 0.00750 & 0.292 & 0.016 & 0.041 & 0.044 \\ 15.0 & 0.0210 & 0.206 & 0.013 & 0.057 & 0.059 \\ 20.0 & 0.000650 & 0.227 & 0.019 & 0.048 & 0.052 \\ 20.0 & 0.00105 & 0.234 & 0.019 & 0.041 & 0.045 \\ 20.0 & 0.00165 & 0.284 & 0.022 & 0.082 & 0.085 \\ 20.0 & 0.00260 & 0.293 & 0.022 & 0.051 & 0.055 \\ 20.0 & 0.00410 & 0.206 & 0.025 & 0.049 & 0.055 \\ 20.0 & 0.00750 & 0.289 & 0.016 & 0.038 & 0.041 \\ 20.0 & 0.0210 & 0.246 & 0.012 & 0.063 & 0.064 \\ 25.0 & 0.000650 & 0.244 & 0.027 & 0.055 & 0.061 \\ 25.0 & 0.00105 & 0.291 & 0.020 & 0.045 & 0.050 \\ 25.0 & 0.00165 & 0.246 & 0.024 & 0.081 & 0.084 \\ 25.0 & 0.00260 & 0.244 & 0.024 & 0.047 & 0.053 \\ 25.0 & 0.00410 & 0.348 & 0.027 & 0.053 & 0.059 \\ 25.0 & 0.00650 & 0.274 & 0.028 & 0.051 & 0.058 \\ 25.0 & 0.0119 & 0.273 & 0.020 & 0.056 & 0.060 \\ 25.0 & 0.0329 & 0.178 & 0.016 & 0.060 & 0.062 \\ 35.0 & 0.00105 & 0.335 & 0.025 & 0.049 & 0.055 \\ 35.0 & 0.00165 & 0.334 & 0.026 & 0.047 & 0.054 \\ 35.0 & 0.00260 & 0.213 & 0.026 & 0.077 & 0.081 \\ 35.0 & 0.00410 & 0.295 & 0.030 & 0.051 & 0.059 \\ 35.0 & 0.00650 & 0.283 & 0.030 & 0.048 & 0.057 \\ 35.0 & 0.0105 & 0.293 & 0.033 & 0.070 & 0.077 \\ 35.0 & 0.0190 & 0.261 & 0.026 & 0.042 & 0.049 \\ 35.0 & 0.0525 & 0.194 & 0.023 & 0.076 & 0.079 \\ 45.0 & 0.00165 & 0.374 & 0.031 & 0.053 & 0.061 \\ 45.0 & 0.00260 & 0.270 & 0.028 & 0.043 & 0.052 \\ 45.0 & 0.00410 & 0.262 & 0.033 & 0.068 & 0.075 \\ 45.0 & 0.00650 & 0.342 & 0.035 & 0.049 & 0.060 \\ 45.0 & 0.01050 & 0.277 & 0.037 & 0.058 & 0.069 \\ 45.0 & 0.0190 & 0.325 & 0.029 & 0.055 & 0.062 \\ 45.0 & 0.0525 & 0.263 & 0.025 & 0.056 & 0.061 \\ 60.0 & 0.00260 & 0.379 & 0.033 & 0.054 & 0.063 \\ 60.0 & 0.00410 & 0.356 & 0.038 & 0.081 & 0.089 \\ 60.0 & 0.00650 & 0.246 & 0.040 & 0.051 & 0.064 \\ 60.0 & 0.0105 & 0.340 & 0.042 & 0.061 & 0.074 \\ 60.0 & 0.0165 & 0.311 & 0.053 & 0.070 & 0.088 \\ 60.0 & 0.0299 & 0.225 & 0.039 & 0.052 & 0.065 \\ 60.0 & 0.0849 & 0.255 & 0.035 & 0.064 & 0.073 \\ 90.0 & 0.00410 & 0.245 & 0.043 & 0.063 & 0.076 \\ 90.0 & 0.00650 & 0.363 & 0.044 & 0.081 & 0.092 \\ 90.0 & 0.0105 & 0.310 & 0.047 & 0.061 & 0.077 \\ 90.0 & 0.0165 & 0.392 & 0.058 & 0.081 & 0.100 \\ 90.0 & 0.0299 & 0.291 & 0.042 & 0.043 & 0.060 \\ 90.0 & 0.0849 & 0.340 & 0.039 & 0.035 & 0.052 \\ 120.0 & 0.00650 & 0.410 & 0.068 & 0.066 & 0.095 \\ 120.0 & 0.0105 & 0.252 & 0.061 & 0.107 & 0.124 \\ 120.0 & 0.0165 & 0.498 & 0.074 & 0.102 & 0.126 \\ 120.0 & 0.0260 & 0.161 & 0.079 & 0.095 & 0.124 \\ 120.0 & 0.0475 & 0.267 & 0.065 & 0.063 & 0.091 \\ 120.0 & 0.132 & 0.312 & 0.056 & 0.091 & 0.106 \\ 150.0 & 0.0260 & 0.545 & 0.150 & 0.158 & 0.218 \\ 150.0 & 0.0475 & 0.408 & 0.122 & 0.102 & 0.159 \\ 150.0 & 0.132 & 0.308 & 0.101 & 0.096 & 0.139 \\ \hline \end{tabular} \end{scriptsize} \end{tabular} % \caption{ Measurement of the derivative $-(\partial \ln F_2 / \partial \ln x)_{Q^2}$ at fixed $Q^2$. The systematic uncertainties take the correlations between adjacent $x$ bins into account yielding a quoted total error as the squared sum of the statistical and systematic uncertainties. The uncertainties are given as absolute values.} \protect\label{tabder} \end{table} \newpage % %%%table lambda from fit % \begin{table}[h] \centering \begin{tabular}{|r|c|c|c|} \hline $Q^2$ & $ \lambda$ & $\delta_{sta}$ & $\delta_{tot}$ \\ \hline 1.5 & 0.204 & 0.037 & 0.095 \\ 2.0 & 0.159 & 0.003 & 0.015 \\ 2.5 & 0.169 & 0.002 & 0.009 \\ 3.5 & 0.179 & 0.002 & 0.007 \\ 5.0 & 0.196 & 0.003 & 0.008 \\ 6.5 & 0.202 & 0.004 & 0.009 \\ 8.5 & 0.223 & 0.004 & 0.010 \\ 12.0 & 0.240 & 0.004 & 0.011 \\ 15.0 & 0.250 & 0.002 & 0.010 \\ 20.0 & 0.260 & 0.003 & 0.011 \\ 25.0 & 0.274 & 0.004 & 0.014 \\ 35.0 & 0.286 & 0.005 & 0.016 \\ 45.0 & 0.302 & 0.007 & 0.017 \\ 60.0 & 0.332 & 0.011 & 0.024 \\ 90.0 & 0.304 & 0.022 & 0.040 \\ \hline \end{tabular} % \caption{ Exponent \lam of $F_2 = cx^{-\lambda}$ from a fit to H1 \Fc data~\cite{paper} restricted to $x \leq 0.01$. The total uncertainties are obtained by adding all sources of \Fc uncertainty in quadrature. The uncertainties are given in absolute values.} \label{tablam} \end{table} \newpage \end{document} %