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\newcommand{\trmm}{m_{\perp}^2} \newcommand{\trpp}{p_{\perp}^2} \newcommand{\alp}{\alpha_s} \newcommand{\alps}{\alpha_s} \newcommand{\sqrts}{$\sqrt{s}$} \newcommand{\LO}{$O(\alpha_s^0)$} \newcommand{\Oa}{$O(\alpha_s)$} \newcommand{\Oaa}{$O(\alpha_s^2)$} \newcommand{\PT}{p_{\perp}} \newcommand{\JPSI}{J/\psi} \newcommand{\sh}{\hat{s}} %\newcommand{\th}{\hat{t}} \newcommand{\uh}{\hat{u}} \newcommand{\MP}{m_{J/\psi}} %\newcommand{\PO}{\mbox{l}\!\mbox{P}} \newcommand{\PO}{I\!\!P} \newcommand{\xbj}{x} \newcommand{\xpom}{x_{\PO}} \newcommand{\ttbs}{\char'134} \newcommand{\xpomlo}{3\times10^{-4}} \newcommand{\xpomup}{0.05} \newcommand{\dgr}{^\circ} \newcommand{\pbarnt}{\,\mbox{{\rm pb$^{-1}$}}} \newcommand{\gev}{\,\mbox{GeV}} \newcommand{\WBoson}{\mbox{$W$}} \newcommand{\fbarn}{\,\mbox{{\rm fb}}} \newcommand{\fbarnt}{\,\mbox{{\rm fb$^{-1}$}}} % % Some useful tex commands % \newcommand{\qsq}{\ensuremath{Q^2} } \newcommand{\gevsq}{\ensuremath{\mathrm{GeV}^2} } \newcommand{\et}{\ensuremath{E_t^*} } \newcommand{\rap}{\ensuremath{\eta^*} } \newcommand{\gp}{\ensuremath{\gamma^*}p } \newcommand{\dsiget}{\ensuremath{{\rm d}\sigma_{ep}/{\rm d}E_t^*} } \newcommand{\dsigrap}{\ensuremath{{\rm d}\sigma_{ep}/{\rm d}\eta^*} } % Journal macro \def\Journal#1#2#3#4{{#1} {\bf #2} (#3) #4} \def\NCA{\em Nuovo Cimento} \def\NIM{\em Nucl. Instrum. Methods} \def\NIMA{{\em Nucl. Instrum. Methods} {\bf A}} \def\NPB{{\em Nucl. Phys.} {\bf B}} \def\PLB{{\em Phys. Lett.} {\bf B}} \def\PRL{\em Phys. Rev. Lett.} \def\PRD{{\em Phys. Rev.} {\bf D}} \def\ZPC{{\em Z. Phys.} {\bf C}} \def\EJC{{\em Eur. Phys. J.} {\bf C}} \def\CPC{\em Comp. Phys. Commun.} \begin{titlepage} \noindent \hfill \bf{H1prelim-13-141} \rm and \bf{ZEUS-prel-13-003}\rm \\ \today \\ %Version: 0.6 \\ %Editors: \myref{mailto:voica@mail.desy.de}{V.~Radescu},\myref{mailto:a.cooper-sarkar1@physics.ox.ac.uk}{A.~Cooper-Sarkar}, \myref{mailto:hannes.jung@desy.de}{H.~Jung}, \myref{mailto:katarzyna.wichmann@desy.de}{K.~Wichmann} \\ %Referees: \\ %Comments by Deadline \vspace{2cm} \begin{center} \begin{Large} {\bf HERAPDF1.5LO PDF Set with Experimental Uncertainties} \vspace{2cm} H1 and ZEUS Collaborations \end{Large} \end{center} \vspace{2cm} \begin{abstract} This note presents the HERAPDF1.5 PDF set evolved to leading order (LO) $\alpha_s$ using DGLAP evolution equations. This LO PDF is particularly useful for Monte Carlo event generators, based on LO matrix elements plus parton showers. % and the simulation of higher order corrections via parton showers. Of special interest is this PDF set for the simulation of the parton shower and underlying event properties in calculations which match a NLO hard process calculation with parton showers and hadronsiation as done in POWHEG or MC@NLO. \end{abstract} \vspace{1.5cm} %\begin{center} %To be submitted to the LHAPD online repository %\end{center} \end{titlepage} % THE PAPER DRAFTS HAVE NO AUTHORLIST % % FOR PAPER ISSUED FOR THE FINAL READING % COPY THE AUTHOR AND INSTITUTE LISTS % INTO YOUR AREA % % from /h1/iww/ipublications/h1auts.tex % AND UNCOMMENT THE NEXT THREE LINES % %\begin{flushleft} % \input{h1auts} %\end{flushleft} \newpage \section{Introduction} \noindent %HERAPDF PDF presents a valuable alternative to other PDF sets %available for standard Monte Carlo generators such as PYTHIA or HERWIG, %which will require LO PDFs rather than the NLO ones. Parton densities evolved to leading order (LO) in $\alpha_s$ are essential for the proper simulation of parton showers (PS) and underlying event properties in LO+PS Monte Carlo (MC) event generators. %In the PS approach the parton densities %are used to simulate higher order parton emissions (usually in the leading $\log$ approximation) from %the hard scattering process to a low scale, typically of ${\cal O}(1\mbox{ GeV})$. A physical behavior of the parton densities at such low scales is essential, in particular the parton densities must be positive to represent a probability distribution to be used in a Monte Carlo event simulation. % A set of LO pdfs together with a corresponding PDF set evolved to NLO are of special interest to NLO + PS calculations like MC@NLO~\cite{Frixione:2002ik} and POWHEG \cite{Nason:2004rx,Frixione:2007vw}. In the light of the imminent restart of the LHC with upgraded proton energy beams, new tunes for the underlying events and minimum bias are needed using various MC generators. Since the higher energies at the LHC will cover measurements to lower values of the Bjorken-$x$ variable, the HERAPDF PDF sets with its special emphasis on the small-$x$ structure functions and its validity at small scales is important for equipping the MC generators with accurate PDFs that enable precise predictions of the underlying event properties and also for minimum bias events and the simulation of pile-up events. In this note we present the HERAPDF1.5 LO set based on the same settings as used for the HERAPDF1.5 NLO PDF set\cite{HERAPDF1.5NLO}, with the exception of the use of the LO DGLAP splitting kernel and, correspondingly, of a different value for the strong coupling constant. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{equation} %x_\pom = \frac{\qsq+M_X^2-t}{\qsq+W^2+M_p^2} \; . %\label{eq:xpom} %\end{equation} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Equation (\ref{eq:xpom}) gives ... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{figure}[hhh] % \begin{center} % \includegraphics[width=\textwidth]{d98-029f6} % \caption{ The average squared transverse momentum %$\langle p_t^{*2} \ %rangle$ of hadrons is shown as %a function of Feynman $x_F$,} % \label{fig:diff} % \end{center} %\end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Technical Description of the HERAPDF1.5 LO PDF set} The framework used in this QCD analysis is based on the HERAFitter project \cite{HERAFitter}, with evolution code as implemented in the QCDNUM package \cite{QCDNUM}. The results were cross checked by an independent framework referred to as the ZEUS Fitter \cite{ZEUSfitter}. The QCD fit settings are adopted from the previous HERAPDF fits to preliminary combined H1 and ZEUS HERA I+II data of inclusive deep-inelastic scattering used to extract HERAPDF1.5 NLO \cite{HERAPDF1.5NLO} and NNLO PDF \cite{HERAPDF1.5NNLO} sets. The experimental uncertainties of data are treated in the same way as in the HERAPDF1.5 NLO and NNLO fits. %with $130$ sources of systematic uncertainties %treated uncorrelated bin to bin, as well as correlated among four data sets of neutral current (NC) and charged current (CC) of $e^+p$ and $e^-p$ %scattering and with three sources of systematic uncertainties arising from the data combination procedure treated as correlated bin to bin and as well as %among data sets (referred to as $proced1$, $proced2$, $proced3$ in the results table). The PDFs parametrised at the starting scale of the evolution\footnote{chosen to be below the charm mass threshold as required by the QCDNUM package} of $Q^2_0=1.9$~\gevsq are the valence distributions $xu_v$ and $xd_v$, the gluon distribution $xg$, and the $x\bar{U}$ and $x\bar{D}$ distributions, where $x\bar{U} = x\bar{u}$ , $x\bar{D} = x\bar{d}+x\bar{s}$. The following functional form is used to parametrise them and is identical to the HERAPDF1.5 NLO set: \begin{eqnarray} x u_v(x) &=& A_{u_v} x^{B_{u_v}} (1-x)^{C_{u_v}} ( 1 + E_{u_v} x^2) \\ x d_v(x) &=& A_{d_v} x^{B_{d_v}} (1-x)^{C_{d_v}} \\ x \bar{U} (x) &=& A_{\bar{U}} x^{B_{\bar{U}}} (1-x)^{C_{\bar{U}}} \\ x \bar{D} (x) &=& A_{\bar{D}} x^{B_{\bar{D}}} (1-x)^{C_{\bar{D}}} \\ x g(x) &=& A_g x^{B_g} (1-x)^{C_g} \,. \label{eq:param} \end{eqnarray} where the normalization parameters ($A_{u_v}; A_{d_v}; A_g$) are constrained by quark counting and momentum sum rules. The $B$ exponents for the quark sea and valence distributions, respectively, are set equal, $B_{\bar{U}} = B_{\bar{D}}$ and $B_{u_v} = B_{d_v}$. %, such that there is a single $B$ parameter for % the quark sea distributions, as well as a single parameter $B$ for the valence quark distributions. The strange quark distribution at the starting scale is assumed to be a constant fraction of $\bar{D}$, $x\bar{s} = f_s x\bar{D}$, chosen to be $f_s = 0.31$ such that $\bar{s} \approx \bar{d}/2$. In addition, to ensure that $x\bar{u} = x \bar{d}$ as $x \to 0$, $A_{\bar{U}}=A_{\bar{D}}(1-f_s)$. This yields the same 13 free parameters as in the NLO fit. The PDFs are evolved using the DGLAP evolution equations at LO with the squared renormalisation and factorisation scales set to the squared momentum transfer of the NC or CC interaction, $Q^2$. The value for $\alpha_s(M_Z)$ has been chosen to be $0.13$, which yields the best level of agreement between data and the fit\footnote{ This value agrees with the value used by the CTEQ LO set \cite{CTEQ} of $\alpha_s(M_Z)=0.12978$, which is different from the value chosen by the MSTW LO set \cite{MSTW} of $\alpha_s(M_Z)=0.136$.}. As for previous HERAPDF PDF sets, the analysis is performed accounting for the charm and beauty quark masses in the Thorne-Roberts (TR) Variable Flavour Number Scheme \cite{RT}. %a check against the ACOT scheme is also performed as usual. The experimental uncertainties on the PDFs are determined using the $\Delta\chi^2 = 1$ criterion leading to uncertainties with a confidence level of $68\%$. The $\chi^2$ is defined as in \cite{ref:sasha}: \begin{equation} \chi^2=\sum_i\frac{\left[ \mu_i - m_i\left(1-\sum_j\gamma_j^i b_j\right) \right]^2}{\delta_{i,{\rm unc}}^2 m_i^2+\delta_{i,{\rm stat}}^2\mu_i m_i\left(1-\sum_ j\gamma_j^i b_j\right)} +\sum_j b_j^2 %+ \sum_i \ln \frac{\delta_{i,{\rm unc}}^2 m_i^2+\delta_{i,{\rm stat}}^2\mu_i m_i}{\delta_{i,{\rm unc}}^2 \mu_i^2+\delta_ %{i,{\rm stat}}^2\mu_i^2}, \label{eq:chi2} \end{equation} where $m_i$ is the theoretical prediction and $\mu_i$ is the measured cross section at point $i$, $(Q^2,x,s)$ with the relative statistical and uncorrelated systematic uncertainty $\delta_{i,{\rm stat}}$, $\delta_{i,{\rm unc}}$, respectively. % The values $\gamma_j^i$ denote the relative correlated systematic uncertainties and $b_j$ their shifts with a penalty term $\sum_j\!b^2_j$ added. % In %order to protect from a bias introduced by statistical fluctuations, %the expected rather than the observed number of events are used, with %the corresponding errors scaled accordingly. %This scaling of errors %introduces a $\ln$ term, coming from the likelihood transition to %$\chi^2$. Neglecting the $\ln$ term gives very similar results and %does not alter any of the conclusions. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{QCD Fit Results and Comparisons} The LO QCD fit to the preliminary combined H1 and ZEUS HERA I+II data\cite{HERAPDF1.5NLO} yields a reasonable total $\chi^2$ of $762$ for $664$ degrees of freedom, slightly worse than the NLO $\chi^2$ of $736$. % indicating a fair fit quality. %are summarized in table \ref{tab:results}. The choice of $\alpha_s(M_Z)=0.13$ was motivated through a scan procedure, where the data were refit with other choices of the strong coupling and the best value in terms of quality of fit was chosen. % %The dependence on the choice of $\alpha_s(M_Z)$ value is given in table \ref{tab:alphas}. The PDF parameters from the fit are listed in table \ref{tab:param}. The resulting PDF distribution plots for the starting scale as well as for momentum transfers of 10, 100 and 10000 GeV$^2$ can be found in Figs. \ref{fig:summary1.9}-\ref{fig:summary10000}. They are presented together with uncertainty bands reflecting the experimental uncertainties transferred from the data to the PDFs. In line with the restricted purpose of the LO sets, no further model or theoretical uncertainties were evaluated. % %%%%%%%%%%%%%%%%%%%%%%%%% % \begin{table}[!htb] % \begin{center} % \caption{\label{tab:results} $\chi^2$ values for the HERAPDF1.5 LO fit results to the preliminary HERA-I+II data.} % \begin{tabular}{l|c|c} % \hline % \hline % Data Set & partial $\chi^2$ & Nr. points \\ % \hline % NC cross section HERA-I+II H1-ZEUS combined e-p. & 237.57 & 182 \\ % NC cross section HERA-I+II H1-ZEUS combined e+p. & 430.74 & 412 \\ % CC cross section HERA-I+II H1-ZEUS combined e-p. & 42.44 & 41 \\ % CC cross section HERA-I+II H1-ZEUS combined e+p. & 50.38 & 39 \\ % \hline % Correlated & 1.18 & 3 (sources) \\ % Total & 762.30 & 664 (dof) \\ % \hline % \hline % Sys. Source & Sys. Shift (in $\sigma$) & Uncert. of Shift \\ % \hline % proced1 & -0.0125 & 0.2252\\ % proced2 & -0.9479 & 0.2633\\ % proced3 & -0.5260 & 0.2852\\ % \hline % \hline % \end{tabular} % \end{center} % \end{table} % %%%%%%%%%%%%%%%%%%%%%%%%% % \begin{table}[!htb] % \begin{center} % \caption{\label{tab:alphas} $\chi^2$ values for the LO fit to the HERA-I+II data using various values for strong couplings.} % \begin{tabular}{l|c} % \hline % \hline % $\alpha_s(M_Z)$ & Total $\chi^2$ \\ % \hline % 0.1176 & 791.86 \\ % 0.1200 & 780.93\\ % 0.1250 & 766.77 \\ % %0.12978 & 762.30 \\ % 0.1300 & 762.29 \\ % 0.1350 & 766.22 \\ % 0.1390 & 775.62 \\ % \hline % \hline % \end{tabular} % \end{center} % \end{table} % %%%%%%%%%%%%%%%%%%%%%%%%% % \begin{table}[!htb] % \begin{center} % \caption{\label{tab:param} PDF parameters from LO fit at the starting scale.} % \begin{tabular}{l|c|c} % \hline % \hline % PDF Param & Central Value & Uncertainty \\ % \hline % $B_g$ &-0.17 & 0.01 \\ % $C_g$ &5.85 & 0.34 \\ % $B_{u_v}$ &0.60 & 0.02 \\ % $C_{u_v}$ &4.22 &0.09 \\ % $E_{u_v}$ &10.38 &1.20 \\ % $C_{d_v}$ &3.69 &0.19 \\ % $C_{\bar{U}}$ & 2.80 & 0.40 \\ % $A_{\bar{D}}$ & 0.147 & 0.005 \\ % $B_{\bar{D}}$ & -0.198 & 0.005 \\ % $C_{\bar{D}}$ & 4.38 &0.84\\ % \hline % \hline % \end{tabular} % \end{center} % \end{table} % %%%%%%%%%%%%%%%%%%%%%%%%% Figures \ref{fig:ncepa}-\ref{fig:ccep} show LO cross section predictions obtained from the fitted PDFs, overlayed on the data used in the fits, for the neutral current and charged current $e^+p$ and $e^-p$ differential measurements. Good agreement is achieved. %\ref{fig:ncepb},\ref{fig:ncem}, \ref{fig:ccep}, %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/herapdf15lo_q2_1pt9.eps} % \includegraphics[width=\textwidth]{figures/herapdf15lo_q2_2.pdf} \caption{Summary of LO PDFs at $Q^2=1.9$ \gevsq. } \label{fig:summary1.9} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/herapdf15lo_q2_10.eps} \caption{Summary of LO PDFs at $Q^2=10.0$ \gevsq. } \label{fig:summary10} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/herapdf15lo_q2_100.eps} \caption{Summary of LO PDFs at $Q^2=100$ \gevsq. } \label{fig:summary100} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/herapdf15lo_q2_10000.eps} \caption{Summary of LO PDFs at $Q^2=10000$ \gevsq.} \label{fig:summary10000} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/ncepb_corrected_x1_new.eps} \caption{Neutral current $e^+p$ differential cross section measurements (data points, part I, for lower $Q^2$ bins) compared to predictions based on the HERAPDF1.5 LO PDF set with experimental uncertainties included in the predictions (contious bands). The dashed line indicates predictions for regions not included in the fit.} \label{fig:ncepa} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/ncepc_new.eps} \caption{Neutral current $e^+p$ differential cross section measurements (data points, part II, for higher $Q^2$ bins) compared to predictions based on the HERAPDF1.5 LO PDF set with experimental uncertainties included in the predictions (continous bands).} \label{fig:ncepb} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/ncem_new.eps} \caption{Neutral current $e^-p$ differential cross section measurements (data points) compared to predictions based on the HERAPDF1.5 LO PDF set with experimental uncertainties included in the predictions (continous bands).} \label{fig:ncem} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/ccem_new.eps} \caption{Charged current $e^-p$ differential cross section measurements (data points) compared to predictions based on the HERAPDF1.5 LO PDF set with experimental uncertainties included in the predictions (continous bands).} \label{fig:ccem} \end{center} \end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{figure}[] \begin{center} \includegraphics[width=\textwidth]{figures/ccep_new.eps} \caption{Charged current $e^+p$ differential cross section measurements (data points) compared to predictions based on the HERAPDF1.5 LO PDF set with experimental uncertainties included in the predictions (continous bands).} \label{fig:ccep} \end{center} \end{figure} The LO PDF set has been formatted to match the LHAPDF style, similarly to what was done for the HERAPDF1.5 NLO set, compatible with the LHAPDFv5 grid style. The LHAPDF grid contains $21$ members with member $0$ representing the central fit, while members $1-20$ correspond to the experimental uncertainties on the PDFs. The $20$ error PDFs should be treated two by two as the up and down excursions for each of the $10$ eigenvectors defined by the number of free PDF parameters in the fit, such that the symmetric error is calculated as the quadratic sum of the difference between the up and down eigenvectors divided by two. If asymmetric errors are desired equation $43$ of \cite{CHS} may be used. \section{Summary} We have extracted a HERAPDF1.5 LO PDF set based on the preliminary HERA I+II H1 and ZEUS combined NC and CC measurements, providing experimental uncertainties. The fit describes the data reasonably well. The set has been formatted to match the LHAPDF style, similarly to what was done for the HERAPDF1.5 NLO set, compatible with the LHAPDFv5 grid style. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Acknowledgements} We are grateful to the HERA machine group whose outstanding efforts have made this experiment possible. We thank the engineers and technicians for their work in constructing and maintaining the H1 detector, our funding agencies for financial support, the DESY technical staff for continual assistance and the DESY directorate for support and for the hospitality which they extend to the non DESY members of the collaboration. We would like to give credit to all partners contributing to the EGI computing infrastructure for their support for the H1 and ZEUS Collaborations. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} %\bibitem{Frixione:2002ik} %S.~Frixione and B.~R. 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This is of special interest for small $x$ processes and the simulation of parton shower as well as the underlying event. % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/glu19.pdf}\includegraphics[width=0.5\textwidth]{figures/glu6464.pdf} % \caption{Gluon PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:glu} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/uval19.pdf}\includegraphics[width=0.5\textwidth]{figures/uval6464.pdf} % \caption{Valence of $xu$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:uval} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/dval19.pdf}\includegraphics[width=0.5\textwidth]{figures/dval6464.pdf} % \caption{Valence of $xd$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:dval} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/Ubar19.pdf}\includegraphics[width=0.5\textwidth]{figures/Ubar6464.pdf} % \caption{Sea of $U$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:Ubar} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/Dbar19.pdf}\includegraphics[width=0.5\textwidth]{figures/Dbar6464.pdf} % \caption{Sea of $D$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:Dbar} % \end{center} % \end{figure} % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/ratglu19.pdf}\includegraphics[width=0.5\textwidth]{figures/ratglu6464.pdf} % \caption{Ratio plot for Gluon at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:ratglu} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/ratuval19.pdf}\includegraphics[width=0.5\textwidth]{figures/ratuval6464.pdf} % \caption{Ratio plot for Valence of $xu$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:ratuval} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/ratdval19.pdf}\includegraphics[width=0.5\textwidth]{figures/ratdval6464.pdf} % \caption{Ratio plot for Valence of $xd$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:ratdval} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/ratUbar19.pdf}\includegraphics[width=0.5\textwidth]{figures/ratUbar6464.pdf} % \caption{Ratio plot for Sea of $U$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:ratUbar} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/ratDbar19.pdf}\includegraphics[width=0.5\textwidth]{figures/ratDbar6464.pdf} % \caption{Ratio plot for Sea of $D$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:ratDbar} % \end{center} % \end{figure} % Comparison between HERAPDF1.5 LO, NLO, NNLO sets are shown in Figs. % ~\ref{fig:gluH}- \ref{fig:DbarH}. % %~\ref{fig:gluH},\ref{fig:uvalH}, \ref{fig:dvalH}, \ref{fig:UbarH}, \ref{fig:DbarH}. % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/Hglu19.pdf}\includegraphics[width=0.5\textwidth]{figures/Hglu6464.pdf} % \caption{Gluon at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:gluH} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/Huval19.pdf}\includegraphics[width=0.5\textwidth]{figures/Huval6464.pdf} % \caption{Valence of $xu$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:uvalH} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/Hdval19.pdf}\includegraphics[width=0.5\textwidth]{figures/Hdval6464.pdf} % \caption{Valence of $xd$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:dvalH} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/HUbar19.pdf}\includegraphics[width=0.5\textwidth]{figures/HUbar6464.pdf} % \caption{Sea of $U$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:UbarH} % \end{center} % \end{figure} % \begin{figure}[] % \begin{center} % \includegraphics[width=0.5\textwidth]{figures/HDbar19.pdf}\includegraphics[width=0.5\textwidth]{figures/HDbar6464.pdf} % \caption{Sea of $D$ PDF at $Q^2=1.9$ \gevsq (left) and at $Q^2=M_W^2$ \gevsq (right) } % \label{fig:DbarH} % \end{center} % \end{figure} \end{document}