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\begin{document} % change natbib spacing between numbers in citations \makeatletter \def\NAT@space{} \makeatother \begin{titlepage} \noindent September 2010 \noindent Editors: A.~Cooper-Sarkar, S.~Glazov, K.~Lipka, R.~Pla\v{c}akyt\.{e}, V.~Radescu \\ H1-prelim-10-143, ZEUS-prel-10-019 \vspace*{3.5cm} \begin{center} \begin{Large} {\bfseries QCD Analysis of Combined H1 and ZEUS $\boldsymbol{F_2^{c\bar{c}}}$ Data } \vspace*{2cm} H1 and ZEUS Collaborations \end{Large} \end{center} \vspace*{2cm} \begin{abstract} \noindent A next-to-leading order QCD analysis is performed to the preliminary combination of the H1 and ZEUS \ftcc\ measurements together with the published HERA inclusive neutral and charged current cross sections. Different models in variable flavour number scheme were used\ for the heavy flavour treatment. The fits are used to estimate the optimal value of the charm quark mass parameter \mct\ within a given heavy flavour scheme. Depending on the scheme, the optimal values of \mct\ range between $1.26$~GeV and $1.68$~GeV, and are determined with a precision of $0.04$~GeV including statistical, model and parameterisation uncertainties. The parton distribution functions determined using the above heavy quark schemes at their optimal values of \mct\ are further used to predict the $W^{\pm}$ and $Z$ production cross sections at the LHC. Good agreement %on the $W^{\pm}$ and $Z$ cross sections between these predictions for the $W^{\pm}$ and $Z$ cross sections is observed which allows to reduce the uncertainty due to the heavy flavour treatment, to below $1.0\%$. \end{abstract} \vspace*{1.5cm} \begin{center} %{\slshape To be Shown at Trento 2010} \end{center} \end{titlepage} %\begin{flushleft} % \input{h1auts} %\end{flushleft} %\newpage %\tableofcontents %\newpage \section{Introduction} %\input{intro} \label{sec:intro} The combined H1 and ZEUS inclusive $e^{\pm}p$ Deep Inelastic Scattering (DIS) cross sections from HERA I and the next-to-leading order (NLO) Quantum Chromodynamic (QCD) analysis have been recently published~\cite{h1zeus:2009wt}. In this set, the combination of the Neutral and Charged Current (NC and CC) data allows for the following PDF separation: gluon $xg(x)$; valence $xu_v(x)$, $xd_v(x)$ as well as $u-$ and $d$ type sea $x\bar{U}(x)= x\bar{u}(x)$, $x\bar{D}(x) = x \bar{d}(x) + x\bar{s}(x)$ densities at the evolution starting scale below the charm quark mass. The heavy quark densities are calculated following various implementations of the variable flavour number schemes (VFNS). % %Recently published combined HERA data on inclusive $e^{\pm}p$ Deep Inelastic Scattering %(DIS) cross sections are used for a next-to-leading order (NLO) %Quantum Chromodynamic (QCD) analysis %to obtain a set of parton density functions (PDFs), %HERAPDF1.0~\cite{h1zeus:2009wt}. In this set, the combination of the Neutral %and Charged Current (NC and CC) data allows for the following PDF separation: % gluon $xg(x)$; valence $xu_v(x)$, $xd_v(x)$ as well as $u-$ and $d$ type sea %which below the b~quark mass threshold are %%$x\bar{U}(x)= x\bar{u}(x)$, $x\bar{D}(x) = x \bar{d}(x) + x\bar{s}(x)$ densities. %$x\bar{U}(x)= x\bar{u}(x) + x\bar{c}(x)$, $x\bar{D}(x) = x \bar{d}(x) + x\bar{s}(x)$ densities. %The strange quark density is assumed to be equal to a %fraction $f_S$ of the total $x\bar{D}(x)$. %The heavy quark densities are calculated following various implementations of the %variable flavour number schemes (VFNS). At NLO, VFNS have a significant ambiguity in describing the onset of the heavy quark densities at the scales $Q$ comparable with the heavy quark pole masses for the charm and bottom quarks, $m_c$ and $m_b$, respectively. Different approaches for the interpolation function and counting of orders in $\alpha_S$ lead to a number of VFNSs, four of which, S-ACOT-$\chi$~\cite{ref:SACOTchi}, ACOT-full~\cite{ref:ACOTfull}, RT~\cite{ref:RT,ThornePrivComm}, RT-optimised~\cite{ThornePrivComm}, are considered in this note. The zero mass variable flavour number scheme (ZMVFNS) was also considered in this analysis. In all schemes, the onset of the heavy quarks is controlled by the parameter \mbct\ . %In the massive VFN schemes this parameter is related to the pole %mass of the heavy quark (see e.g.~\cite{Martin:2010}), %in the ZMVFN scheme it corresponds to a threshold parameter. In DIS heavy quarks are produced dominantly via boson-gluon fusion. At HERA, the charm contribution to the total DIS cross section is significant and reaches about 30\% at large values of $Q^2$. The beauty contribution is an order of magnitude smaller. In the HERAPDF1.0 analysis, the uncertainty due to the heavy flavour treatment is estimated by varying \mbct\ within the ranges of $1.35<$ \mct\ $<1.65$ and $4.3<$ \mbt\ $<5.0$ GeV. The variation of \mbt\ shows a small influence on the resulting PDFs (apart from $x\bar{b}(x)$ itself). In contrary, the variation of \mct\ results in significant change of the gluon distribution which leads consequently to the change of the sea quark densities. In particular, the suppression of $x\bar{c}(x)$ in the sea distribution is compensated by the increase of $x\bar{u}(x)$. %the variation of \mct\ gives rise to a significant changes in both $x\bar{u}(x)$ and %$x\bar{c}(x)$ which propagate to significant uncertainty ($\sim 3\%$) for the $W^{\pm}$ %and $Z$ boson production cross section at the LHC~\cite{Mandy:DIS10}. %The reason for this dependence is well understood: values of %Bjorken-$x$ corresponding to central production of $W,Z$ at the LHC %are measured at HERA around the charm threshold where the contribution %of \ftcc\ to the total inclusive $F_2$ is significant (up to $30\%$ at large $Q^2$). % %At the same time charm quarks close to \mct\ are produced via boson-gluon fusion %which leads to the increased $g(x)$ density. % %A recent preliminary combination of the H1 and ZEUS \ftcc\ data~\cite{ref:f2charmdata} reach %precision of $5-10\%$. These data can be used to compare with VFNSs %predictions. The kinematic range of the data, $2\le Q^2 \le~1000$~GeV$^2$, covers the region sensitive %to the heavy quark thresholds. %The accuracy of the data should allow to reduce the ambiguity in the separation %of $x\bar{U}(x)$ into $x\bar{u}(x)$ and $x\bar{c}(x)$ contributions. % %This note is dedicated to an evaluation of the uncertainties due to the %charm treatment in the PDF fits using the combined \ftcc\ data. %A recent preliminary combination of the H1 and ZEUS \ftcc\ data~\cite{ref:f2charmdata} %covers the kinematic range $2\le Q^2 \le~1000$~GeV$^2$ and reaches %precision of $5-10\%$. %The VFNS predictions for these data are highly sensitive to \mct\ . %A QCD analysis using the combined \ftcc\ data together with the published NC and CC data from %HERA I is preformed to extact PDFs and determine the optimal \mct\ value for e%ach of the VFNS %mentioned above. The resulting PDFs are propagated to calculate predictions %for the $W^{\pm}$ and $Z$ cross sections at the LHC. %Finally it is checked that with %optimal \mct\ the spread of $W^{\pm}$ and $Z$ predictions can be reduced compared %to the predictions based on HERAPDF1.0 with the charm mass variation as given above. In a recent preliminary combination of the H1 and ZEUS \ftcc\ measurements~\cite{ref:f2charmdata} a precision of $5-10\%$ has been achieved. The kinematic range of the data, $2\le Q^2 \le 1000$~GeV$^2$, include the region sensitive to the heavy quark thresholds. In this note, these charm data are used together with different implementations of VFNSs. The accuracy of the data should allow to reduce the ambiguity in the separation of $x\bar{U}(x)$ into $x\bar{u}(x)$ and $x\bar{c}(x)$ contributions. A QCD analysis is therefore performed including the charm data together with inclusive DIS cross sections in order to extract PDFs and to determine the optimal \mct\ value for each of the VFNS investigated. With the resulting PDFs predictions for the $W^\pm$ and $Z$ cross sections at the LHC are calculated. %The resulting PDFs are propagated to calculate predictions for the $W^{\pm}$ and $Z$ cross sections at the LHC. The note is organised as follows. In \Sec~\ref{sec:fit}, the data sets and the QCD fit are introduced. In \SSec~\ref{sec:mcfit} the determination of \mct\ for each VFNS and extraction of the corresponding PDF sets are described. %\SSec~\ref{sec:mcfm} gives the settings %of the {\tt MCFM} programme which is used to calculate predictions at the LHC. The predictions for $W^{\pm}$ and $Z$ boson production cross sections are presented in \Sec~\ref{sec:lhc}. %The results of this investigation are summarised in \Sec~\ref{sec:conclusions}. \section{Data and QCD Fit Settings} %\input{qcdfit} \label{sec:fit} The preliminary combined charm data are used together with the published combined measurement of inclusive DIS cross sections at HERA~\cite{h1zeus:2009wt} as input to a QCD fit based on the DGLAP NLO evolution scheme. For this purpose, the program QCDNUM~\cite{qcdnum} is used. The minimum invariant mass $W$ of the hadronic system is $15$~GeV and the value of the Bjorken scaling variable is restricted to $x \le 0.65$. Therefore target mass corrections and higher twist contributions are expected to be small. Furthermore, the analysis is restricted to $Q^2_{min}=3.5$~GeV$^2$ to assure the applicability of pQCD. The consistency of the input data set and the control on the systematic uncertainties enable the calculation of the experimental uncertainties on the PDFs using the $\chi^2$ tolerance of $\Delta\chi^2=1$. In the fit procedure, the following independent PDFs are chosen: $xu_v(x)$, $xd_v(x)$, $xg(x)$ and $x\bar{U}(x)$, $x\bar{D}(x)$, where $x\bar{U}(x) = x\bar{u}(x)$, and $x\bar{D}(x) = x\bar{d}(x) +x\bar{s}(x)$ at the starting scale $Q_0$ below the charm mass. The default PDFs at the starting scale $Q_0^2=1.9$~GeV$^2$, is given by \begin{equation} xf(x) = A x^{B} (1-x)^{C} (1 + E x^2). \label{eqn:pdf} \end{equation} The parameterisation and the constraints on the parameters are the same as for the central PDF of HERAPDF1.0~\cite{h1zeus:2009wt}. The PDFs are evolved using DGLAP evolution equations~\cite{Gribov:1972ri,Gribov:1972rt,Lipatov:1974qm,Dokshitzer:1977sg,Altarelli:1977zs} at NLO~\cite{Curci:1980uw,Furmanski:1980cm} in the $\overline{\rm MS}$ scheme with the renormalisation and factorisation scales set to $Q^2$ and the strong coupling set to $\alpha_s(M_Z) = 0.1176$~\cite{pdg}. The QCD predictions for the structure functions are obtained by convolution of the PDFs with the NLO coefficient functions calculated using different implementations of the general mass variable favour number scheme: ACOT full~\cite{ref:ACOTfull} as used for the CTEQHQ releases of PDFs, S-ACOT-$\chi$~\cite{ref:SACOTchi} as used for the latest CTEQ releases of PDFs, RT scheme~\cite{ref:RT,ThornePrivComm} as used for the MRST and MSTW releases of PDFs, as well as an optimised RT scheme providing a smoother behaviour across thresholds~\cite{ThornePrivComm}. The ZMVFNS as implemented by the NNPDF group~\cite{NNPDF} is also used. Since the study involves variations of the charm mass parameter down to $\mct=1.2$~GeV and the evolution starting scale must be chosen below $(\mct)^2$, fits are performed with a starting scale of $Q^2_0=1.4$~GeV$^2$. As it was shown in~\cite{h1zeus:2009wt}, the gluon density at this starting scale could not be well described by a parameterisation form of \Eq~\ref{eqn:pdf} and it is therefore modified to \begin{equation} x g(x) = A_g x^{B_g}(1-x)^{C_g} - A'_g x^{B'_g}(1-x)^{25}\,, \label{eq:flex} \end{equation} which allows for more flexible shapes of the gluon PDF and the low starting scale. The choice of the exponent in the $(1-x)$-term is motivated by the approach of the MSTW group. Other variants of the PDF parameterisations which proved to have an effect on HERAPDF1.0 have been considered and included in the evaluation of the systematic uncertainties for \mct. \section{Determination of Optimal \mct} %\input{mcfit} \label{sec:mcfit} In each heavy flavour scheme PDF fits were performed by varying $\mct$ from 1.2 GeV to 1.8 GeV. For each fit the $\chi^2$ value is calculated and the optimal value $\mcto$ is subsequently determined from a parabolic fit to the $\chi^2$ data of a form \begin{equation} \chi^2(\mct) = \chi^2_{\rm min} + \left(\frac{\mct - \mcto}{\Delta_{\mcto}}\right)^2\,, \end{equation} where $\chi^2_{\rm min}$ is the $\chi^2$ value at the minimum and $\Delta_{\mcto}$ is the experimental uncertainty on $\mcto$. %Standard and flexible gluon parameterisations were tried in this analysis as described in the previous section. %Both standard and flexible gluon parameterisation are tried in this analysis for $\mct\ge 1.4$~GeV. %For the standard parameterisation, the starting scale of the evolution is chosen to be %$Q^2_0=1.9$~GeV$^2$ while the fits with the flexible gluon parameterisation use $Q^2_0=1.4$~GeV$^2$. %Low value of the starting scale of the evolution $Q^2_0$ for the flexible parameterisation permits scans of $\mct$ down to $\mct=1.2$~GeV. The scanning procedure is illustrated in \Figs~\ref{fig:chnocharmrt}-\ref{fig:chcharmrt} for the standard RT scheme. First the \mbox{HERAPDF1.0} datasets, i.e. the H1 and ZEUS combined inclusive NC and CC cross sections from HERA I~\cite{h1zeus:2009wt} are used (\Fig~\ref{fig:chnocharmrt}), not including the charm $\ftcc$ data. In this case, $\chi^2$ varies only little with $\mct$ in the range $1.2$~GeV to $1.8$~GeV. When the charm data are included, $\chi^2$ is much more sensitive to $\mct$ (\Fig~\ref{fig:chcharmrt}). Fits using the standard and flexible gluon parameterisation show very similar behaviour. \FFigs~\ref{fig:chcharmrto}-\ref{fig:chcharmz} illustrate the $\chi^2$ scans for the optimised RT, ACOT full, S-ACOT-$\chi$ and ZMVFN schemes. \FFig~\ref{fig:chall} summarises the study by showing the $\mct$ scanning results for all schemes together. It is interesting to observe that $\chi^2_{\rm min}$ values are comparable for all schemes despite different values of $\mcto$. Values of $\chi^2_{\rm min}$ are almost identical for standard RT, optimised RT and ACOT full schemes, and are worse by $\sim 20$ units for the S-ACOT~$\chi$ scheme and by $\sim 50$ units for the ZMVFN scheme. %\FFigs~\ref{fig:data5}-\ref{fig:dataz} compare the $\ftcc$ data with fits using different VFN schemes The experimental uncertainty on $\mcto$ includes the model, parameterisation and $\alpha_S$ uncertainty, following the prescription of~\cite{h1zeus:2009wt}. In this procedure, for each VFN scheme the fitting assumptions are varied one by one and $\mct$ scans are repeated around the $\mcto$ value obtained from the central fits. Typically, uncertainties in $\mcto$ are increased by factor two compared to experimental errors only with exception of the ZMVFN scheme for which the uncertainty is increased by a factor of $\sim 3$. In \FFig~\ref{fig:data5} the $\ftcc$ data are compared with fits using different VFN schemes which are performed for $\mct=\mcto$. The comparison of the predictions of the individual schemes to the $\ftcc$ data are shown in \FFigs~\ref{fig:datart}-\ref{fig:dataz}. These predictions describe the data well with the exception of the ZMVFN scheme at $Q^2\le 12$~GeV$^2$.The uncertainties on $\mcto$ are given in \Tab~\ref{sigma_tab}. %\section{MCFM Settings} %\input{mcfm} \section{$W^{\pm},Z$ Production Cross Sections at the LHC} %\input{lhc} \label{sec:lhc} The PDFs obtained from fits to the HERA data by the $\mct$ scanning procedure are used to calculate predictions for $W^{\pm}$ and $Z$ production cross sections at the LHC. These predictions are calculated for $1.2\le \mct \le 1.8$~GeV in $0.1$~GeV steps for each of the VFN schemes using the MCFM program, version 5.7, with the same conditions as for the PDF4LHC benchmarking~\cite{pdf4lhc}. The $W^\pm$ and $Z^0$ cross sections as a function of $\mct$ for the different schemes are shown in \FFigs~\ref{fig:wp}-\ref{fig:z} and the values for the optimal choice $\mcto$ are summarised in \Tab~\ref{sigma_tab}. %The results of this study are summarised in \FFigs~\ref{fig:wp}-\ref{fig:z} and \Tab~\ref{sigma_tab}. %For all VFN schemes, approximately similar dependence of the production cross sections is observed. For all VFN schemes a similar monotonic dependence of the $W^{\pm}$ and $Z^0$ boson production cross sections is observed. There is, however, a sizable offset between the predictions if they are considered for a fixed value of $\mct$: if the ZMVFN scheme is included (excluded) the difference reaches $7\%$ ($4.5\%$). Similarly, for each scheme the change of the prediction varies by about $7\%$ for $\mct$ raising from $1.2$ to $1.8$~GeV. However, when using the $\mcto$ the spread of predictions is reduced to $0.7\%$ ($2.3\%$) when excluding (including) the ZMVFNS calculations. %The spread of the predictions is much reduced if for each VFN scheme model the corresponding value of $\mcto$ %is used to calculate them. In this case the spread does not exceed $0.7\%$ ($2.3\%$) if ZMVFN scheme %excluded (included) in the comparison. %\FFig~\ref{fig:chivssigma} shows $\chi^2$ of the $\ftcc$ data as a function of ratios of the $Z,W^{\pm}$ %production cross section for each model divided by the prediction based on the standard RT fit. %Models which have smaller $\chi^2$ show less spread of the predictions. The ZMVFN scheme describes the data worst and differs significantly from the other schemes in $W^\pm$ and $Z^0$ predictions. Furthermore, the ZMVFNS does not describe the charm production in DIS at HERA even at high $Q^2$~\cite{dstar}. \\ \begin{table}[!ht] \centering \linespread{1.3} \normalsize \begin{tabular}[h]{|c|c|c|c||c|c|c|} \hline $scheme$ & \mcto\,& $\chi^2/dof$ & $\chi^2/ndp$ & $\sigma_{Z}$(nb) & $\sigma_{W^{+}}$(nb) & $\sigma_{W^{-}}$(nb) \\ & GeV & & $F_2^{c\bar{c}}$ & & & \\ \hline \hline RT standard & 1.58$^{+0.02}_{-0.03}$& 620.3/621& 42.0/41& 29.27$^{+0.07}_{-0.11}$& 57.82$^{+0.14}_{-0.22}$ & 40.22$^{+0.10}_{-0.15}$\\ RT optimised & 1.46$^{+0.02}_{-0.04}$& 621.6/621& 46.5/41& 29.17$^{+0.07}_{-0.13}$& 57.75$^{+0.14}_{-0.26}$ & 40.15$^{+0.10}_{-0.18}$\\ ACOT-full & 1.58$^{+0.03}_{-0.04}$& 621.2/621& 59.9/41& 29.28$^{+0.10}_{-0.13}$& 57.93$^{+0.18}_{-0.24}$ & 40.16$^{+0.12}_{-0.16}$\\ S-ACOT-$\chi$ & 1.26$^{+0.02}_{-0.04}$& 639.7/621& 68.5/41& 29.37$^{+0.08}_{-0.15}$& 58.06$^{+0.16}_{-0.30}$ & 40.23$^{+0.11}_{-0.21}$\\ ZMVFNS & 1.68$^{+0.06}_{-0.07}$& 667.4/621& 88.1/41& 28.71$^{+0.19}_{-0.20}$& 56.77$^{+0.33}_{-0.34}$ & 39.46$^{+0.24}_{-0.25}$\\ [2pt] \hline \end{tabular} \linespread{1.0} \caption{ The \mcto\ as determined from the \mct\ scans in different heavy flavour schemes. The corresponding $\chi^2$ per degrees of freedom $dof$ (per number of data points $ndp$) values for the complete data set using inclusive and charm data (for the charm data only) are presented, as obtained using the flexible gluon parametrisation. The predictions of the $Z/W$ cross sections at LHC are given.} \label{sigma_tab} \end{table} \section{Conclusions} %\input{conclusions} \label{sec:conclusions} Using recent preliminary $\ftcc$ data together with the published HERA I combined data, a NLO QCD analysis was performed based on different implementations of the variable flavour number scheme. For each implementation, an optimal value of the charm mass parameter $\mct$ was determined. The values of optimal $\mct$ show sizable spread, ranging between $1.26$~GeV and $1.68$~GeV. Apart from the ZMVFN scheme, all schemes were found to describe the data well, with comparable $\chi^2/\dof$, as long as $\mct$ was taken at corresponding optimal values. PDFs obtained from fits with different $\mct$ were used to predict $W^{\pm}$ and $Z$ production cross sections at the LHC. A sizable spread in the predictions was observed for each model when $\mct$ was varied between $1.2$ and $1.8$~GeV, as well as when considering different schemes at a fixed value of $\mct$. This spread is significantly reduced when the optimal value of $\mct$ is used in each model. This analysis has shown that the inclusion of the preliminary $\ftcc$ measurements helps to reduce the uncertainties due to the heavy flavour treatment in the PDF fit on the $W^{\pm}$ and $Z$ production cross section at LHC to below $1.0\%$ in a model independent way. \bibliography{charmfit.bib} \clearpage %\input{figures} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig1.eps} \caption{\it $\chi^2$ of the HERA I data fit (HERAPDF1.0) in the standard RT scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chnocharmrt} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig2.eps} \caption{\it $\chi^2$ of the HERA I+\ftcc\ fit in the standard RT scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chcharmrt} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig3.eps} \caption{\it $\chi^2$ of the HERA I + \ftcc\ fit in the optimised RT scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chcharmrto} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig4.eps} \caption{\it $\chi^2$ of the HERA I + \ftcc\ fit in the ACOT-full scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chcharmacotf} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig5.eps} \caption{\it $\chi^2$ of the HERA I + \ftcc\ fit in the S-ACOT-$\chi$ scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chcharmacotc} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=9cm]{H1prelim-10-143.fig6.eps} \caption{\it $\chi^2$ of the HERA I + \ftcc\ fit in the ZMVFN scheme as a function of \mct\ . Open and closed symbols represent flexible and standard parametrisation repsectively (see text for the explanation).} \label{fig:chcharmz} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=12cm]{H1prelim-10-143.fig7.eps} \caption{\it Comparison of the $\chi^2$ of HERA I + \ftcc\ fits using different heavy flavour schemes represented as lines of different styles. The flexible parmetrisation was used for the fits shown in the figure. } \label{fig:chall} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig8.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fits using different heavy flavour schemes obtained at \mcto\ of each scheme. The data are shown with the uncorrelated uncertainties.} \label{fig:data5} \end{center} \end{figure} %\begin{figure}[!ht] % \begin{center} % \includegraphics[width=12cm]{figures/f2c_4schemes.eps} % \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fits using different % HF schemes obtained at \mcto\ of each scheme. % The data are shown with the uncorrelated uncertainties.} % \label{fig:data4} % \end{center} %\end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig9.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fit using the RT standard scheme obtained at \mcto\ . The data are shown with the uncorrelated uncertainties.} \label{fig:datart} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig10.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fit using the RT optimised scheme obtained at \mcto\ . The data are shown with the uncorrelated uncertainties.} \label{fig:datarto} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig11.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fit using the ACOT-full scheme obtained at \mcto\ . The data are shown with the uncorrelated uncertainties.} \label{fig:dataacotf} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig12.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fit using the S-ACOT-$\chi$ scheme obtained at \mcto\ . The data are shown with the uncorrelated uncertainties.} \label{fig:dataacotc} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=14cm]{H1prelim-10-143.fig13.eps} \caption{\it \ftcc\ as a function of $x$ in $Q^2$ bins compared to QCD fit using the ZMVFN scheme obtained at \mcto\ . The data are shown with the uncorrelated uncertainties.} \label{fig:dataz} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=\linewidth]{H1prelim-10-143.fig14.eps} \caption{\it $W^{+}$ production cross section $\sigma_{W^+}$ at the LHC for $\sqrt{s}=7$~TeV as a function of \mct. The lines show predictions for different VFN schemes as inidcated by the legend. The stars show position of the corresponding $\mcto$ values. The thick dashed horizontal lines indicate the range of $\sigma_{W^+}$, determined for $\mct=\mcto$, if massive VFN schemes are considered. The thin dashed horizontal line corresponds to the prediction using ZMVFN scheme for $\mct=\mcto$. } \label{fig:wp} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=\linewidth]{H1prelim-10-143.fig15.eps} \caption{\it $W^{-}$ production cross section $\sigma_{W^-}$ at the LHC for $\sqrt{s}=7$~TeV as a function of \mct. The lines show predictions for different VFN schemes as inidcated by the legend. The stars show position of the corresponding $\mcto$ values. The thick dashed horizontal lines indicate the range of $\sigma_{W^-}$, determined for $\mct=\mcto$, if massive VFN schemes are considered. The thin dashed horizontal line corresponds to the prediction using ZMVFN scheme for $\mct=\mcto$. } \label{fig:wm} \end{center} \end{figure} \begin{figure}[!ht] \begin{center} \includegraphics[width=\linewidth]{H1prelim-10-143.fig16.eps} \caption{\it $Z$ production cross section $\sigma_{Z}$ at the LHC for $\sqrt{s}=7$~TeV as a function of \mct. The lines show predictions for different VFN schemes as inidcated by the legend. The stars show position of the corresponding $\mcto$ values. The thick dashed horizontal lines indicate the range of $\sigma_{Z}$, determined for $\mct=\mcto$, if massive VFN schemes are considered. The thin dashed horizontal line corresponds to the prediction using ZMVFN scheme for $\mct=\mcto$. } \label{fig:z} \end{center} \end{figure} %\begin{figure}[!ht] % \begin{center} % \includegraphics[width=\linewidth]{figures/chi2_vs_rat.eps} % \caption{\it Dependence of partial $\chi^2$ from the $\ftcc$ data on %the ratio of the $Z,W^{\pm}$ production cross sections for a given % model to %the prediction based on the standard RT fit. Vertical lines indicate %$0.5\%$ deviation for the cross-section ratio. %} %\label{fig:chivssigma} %\end{center} %\end{figure} %\clearpage %\input{tables} \end{document}