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\begin{titlepage}

\noindent
\begin{flushleft}
{\tt DESY 12-100 \hfill ISSN 0418-9833} \\
{\tt June 2012}                  \\
\end{flushleft}



\noindent


\vspace{2cm}
\begin{center}
\begin{Large}

{\bf Combined inclusive diffractive cross sections measured with 
forward proton spectrometers in deep inelastic {\boldmath $ep$} scattering 
at HERA\\}

\vspace{2cm}

H1 and ZEUS Collaborations


\end{Large}
\end{center}

\vspace{2cm}

\begin{abstract}
  A combination of the inclusive diffractive cross section measurements made by the H1 and ZEUS Collaborations at HERA is 
 presented. 
 The analysis uses samples of diffractive deep inelastic $ep$ scattering data at a centre-of-mass energy $\sqrt{s}$~=~318 
GeV where leading protons  are 
  detected by dedicated spectrometers. Correlations of systematic uncertainties are taken into account, 
 resulting in an improved precision of the cross section measurement which reaches 6\% for the most precise points.
 The combined data cover the range $2.5 < Q^2 < 200$ GeV$^2$ in photon virtuality, $0.00035 < 
 \xip < 0.09$ in proton fractional 
 momentum loss, $0.09 < |t| < 0.55$ GeV$^2$ in squared four-momentum transfer at the proton vertex and $0.0018 < \beta < 0.816$ 
 in $\beta=x/\xip$, where $x$ is the Bjorken scaling variable.

\end{abstract}


\vspace{1.5cm}

%\begin{center}
%Draft 2.0 (after 1st EB, circulated to EB)
%\end{center}
\begin{center}
  accepted by \EJC
\end{center}


\end{titlepage}


\begin{flushleft}
  \input{DESY-09-100_auth}

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\newpage
\linenumbers

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec-int}

Diffractive collisions in deep inelastic electron-proton 
scattering (DIS), $ep \to eXp$, where the proton in the final state carries most of the beam momentum and $X$ represents all other
final state particles, have been studied extensively at the 
HERA collider.  
%and provide an essential input for the understanding of quantum 
%chromodynamics (QCD) at high parton densities. 
They can be viewed as resulting from processes in which a photon exchanged between the electron and the proton probes a colour-singlet combination of partons 
 with vacuum quantum numbers emitted by the proton. 
The negative four-momentum squared of the virtual photon, $Q^2$,
 supplies a hard scale, which 
allows the application of perturbative quantum chromodynamics (QCD).
Diffractive reactions in DIS are a tool 
to investigate low-momentum partons in the proton, notably through the
study of diffractive parton distribution functions (DPDFs), determined by a QCD analysis of the data.

In diffractive $ep$ scattering the virtual photon dissociates 
at a photon-proton centre-of-mass energy $W$ and squared four-momentum transfer $t$ at 
the proton vertex (figure~\ref{fig-diagram}), producing a hadronic system 
$X$ with mass $M_X$.  
The fractional longitudinal
momentum loss of the proton is
denoted as $\xip$, while the fraction of this momentum 
%carried by the struck quark 
taking part in the interaction with the photon is denoted as $\beta$. These variables are related to Bjorken $x$ by $x=\beta \, \xip$.
%and the mass of the dissociated system $X$ is denoted $M_X$. 
The variable $\beta$ is related to $M_X$, $t$ and $Q^2$ by $\beta=Q^2/(Q^2+M_X^2-t)$. The variable $\xip$ is given by 
$\xip=(Q^2+M_X^2-t)/(Q^2+W^2-m_p^2)$, where $m_p$ is the proton mass. 
The variables $W$, $Q^2$ and the fractional energy loss $y$ of the electron in the proton rest frame
are related by $W^2\simeq sy-Q^2$, where $s$ is the square of the $ep$ centre-of-mass energy.
\unitlength1cm
\begin{figure}[!h]
\vfill
\begin{center}
\hskip 3.0cm
 \includegraphics[scale = 0.3]{figs/fig1ref.eps}
\end{center}
\vspace*{-.3cm}
\caption{
Diagram of the reaction $ep \rightarrow eXp$.
}
\label{fig-diagram}
\vfill
\end{figure}

%In processes of the type $ep \to eXp$, the exchanged virtual 

%The diffractive dissociation of virtual photons, $\gamma^{\star}p \to Xp$ 
%(figure~\ref{fig:diagram}), produces a hadronic system $X$ with mass $M_X$ 

Similarly to inclusive DIS,
%cross section measurements for the reaction $ep \to eXp$
diffractive cross section measurements 
are conventionally expressed in terms of the
reduced diffractive cross section, $\sigma_r^{D(4)}$, which is related
to the measured $ep$ cross section by
\begin{eqnarray}
\frac{{\rm d} \sigma^{ep \rightarrow eXp}}{{\rm d} \beta {\rm d} Q^2 {\rm d} \xip {\rm d} t} =
\frac{4\pi\alpha^2}{\beta Q^4} \ \ \left[1-y+\frac{y^2}{2}\right] \ \
\sigma_r^{D(4)}(\beta,Q^2,\xip,t) \ .
\label{sigma-2}
\end{eqnarray}
The reduced cross section $\sigma_r^{D(3)}(\beta,Q^2,\xip)$ is obtained by integrating $\sigma_r^{D(4)}(\beta,Q^2,\xip,t)$ over $t$.
% in the $t$-range of the measurement.
The diffractive reduced cross section is related to the diffractive structure functions by:
\begin{equation}
\sigma_r^{D(3)}(\xip, \beta, Q^2, y) = F_2^{D(3)}(\xip, \beta, Q^2) - \frac{y^2}{1 + (1-y)^2}F_L^{D(3)}(\xip, \beta, Q^2).
\label{eqn:sigma}
\end{equation}

Experimentally, diffractive $ep$ scattering is 
characterised by the presence of a leading proton in the final state 
and by a depletion of 
hadronic activity in the pseudo-rapidity\footnote{The pseudo-rapidity is defined as $\eta=-\ln\tan\theta/2$ where the 
polar angle $\theta$ is measured with respect to the proton beam direction.} distribution of particles 
(large rapidity gap, LRG) in the forward (proton) direction. 
Both of these signatures have been exploited in various analyses 
by H1 and ZEUS to select diffractive samples either  
by tagging the outgoing proton in dedicated proton 
spectrometers~\cite{h1-fps1,h1-fps2,zeus-lps1,zeus-lrglps}  
or by requiring the presence of a large rapidity 
gap~\cite{zeus-lrglps,h1-lrg1,h1-lrg2}. 
%distribution of hadronic final state particles 
%in the forward region. 
%Alternatively, a method is 
%used to determine statistically the number of diffractive events, based on 
%the expected difference in shape of the distributions of the invariant 
%mass, $M_X$, for diffractive and non-diffractive 
%events~\cite{h1-mx, zeus-mx1,zeus-mx2}. 
The two methods differ partially in the accessible kinematic ranges 
%(higher $\xpom$ for the proton-tagged data)
(lower $\xpom$ reach for the LRG data)
 and substantially in their 
dominant sources of systematic uncertainties. 
In LRG-based measurements, 
 the largest uncertainty arises from proton dissociative events, 
 $ep \rightarrow eXN$, in which 
 the proton dissociates into a low mass state $N$. 
%
%the admixture of low mass
%leading baryon systems other than protons. These include 
%proton excitations to low mass states as well as leading
%neutrons produced via charge exchange reactions. All such 
%contributions are collectively referred to here as `proton dissociation', 
%$ep \rightarrow eXN$, with the baryon state
%$N$ having mass $M_N$. 
%
%Due to the limitations in the instrumentation very near to the 
%beam-pipe,
% Such events cannot always be distinguished from those 
% in which the proton is scattered elastically and provide 
% a background which affects the normalisation of the 
% measured cross section.
%Events with 
%high $M_X$ are not accessible since the 
%non-diffractive background grows with $M_X$ and the rapidity gap becomes 
%more and more forward (and eventually becomes confined to the 
%beam-pipe).
%; a large rapidity
%gap is still present and the $M_X$ measurement is unaffected. 
% Conversely, 
Low $\xip$ samples selected by the proton spectrometers 
have little or no proton dissociation contribution, 
but their precision is limited statistically by the  
small acceptances and systematically by large uncertainties in the proton
tagging efficiency, which strongly depends on the proton-beam optics.
%Proton spectrometers also allow a measurement of $t$, 
%and give access to higher $x_{\spom}$ than the other methods, 
The results from both methods are found to be 
consistent~\cite{h1-fps1,h1-fps2,zeus-lrglps,h1-lrg2,martapaul}.

 Combining  measurements can provide more precise and 
kinematically extended data than the individual measurements. 
% The combined results can 
% eventually be used to extract the DPDFs. A first step,
% presented in this paper, is taken towards such a perspective by 
In this paper, a combination of 
the H1~\cite{h1-fps2, h1-fps1} and the ZEUS~\cite{zeus-lrglps,zeus-lps1} proton spectrometer results is presented.
%\footnote{
%In these studies the main detectors are used to reconstruct $Q^2$ and $x$ 
%whereas $M_X$, $\beta$, $\xip$ and $t$ derive from the proton spectrometers 
%or from the joint information of the proton spectrometers and the main 
%detectors.} 
%(the Forward Proton Spectrometer, FPS, for H1 and 
%the Leading Proton Spectrometer, LPS, for ZEUS).
% At low $\xip$, these combined proton-tagged data, which are free from proton dissociation contribution, can be used 
% to assess the normalisation of the diffractive cross section in DIS. 
%Because of the variety of the selection methods, subject to different 
%systematic uncertainties, 
%the complementary features of the data and the statistical limitations 
%of some samples.
The combination is performed using the weighted averaging method 
introduced in~\cite{glazov} and extended in~\cite{:2009wt,Aaron:2009bp}. 
%This also provides a model independent 
%check of the data consistency. 
The correlated systematic uncertainties and global normalisations are constrained in the fit such that one consistent data set is 
obtained. Since H1 
and ZEUS have employed different experimental techniques, using different detectors and methods of kinematic reconstruction, the combination leads to significantly reduced uncertainties. 
%The kinematic range of the combined data is: photon virtualities $2.5 < Q^2 < 200$ GeV$^2$, values of the fraction of the proton's momentum carried by the diffractive exchange of $0.0009 < \xip < 0.09$, values of the fraction of the momentum of the diffractive exchange carried by the struck parton of $0.0018 < \beta < 0.56$
%and values of the square of the four-momentum exchanged at the proton vertex $0.09 < |t| < 0.55$ GeV$^2$.
 The kinematic range of the combined data is: $2.5 \leq Q^2 \leq 200$~GeV$^2$, $0.0018 \leq \beta \leq 0.816$, $0.00035 \leq 
 x_{\pom} \leq 0.09$ and $0.09 < |t| < 0.55$~GeV$^2$. The latter range restricts the analysis to the $t$ values directly accessible 
by both the H1 and ZEUS proton spectrometers. 
%$2.5 \le Q^2 \le 200$ GeV$^2$, $0.0009 \le \xip \le 0.09$, $0.0018 \le \beta \le 0.56$ and $0.09 \le |t| \le 0.55$ GeV$^2$.

%To select diffractive events, either the final-state proton 
%can be detected (LPS method) or the different characteristics 
%of the system $X$ in diffractive and non-diffractive events (hadronic 
%methods) can be exploited.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Combination of the H1 and ZEUS measurements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Data samples}

The H1~\cite{H1_det} and ZEUS~\cite{ZEUS_det} detectors were general purpose instruments which consisted of 
tracking systems surrounded by electromagnetic and hadronic calorimeters and muon detectors, ensuring close to 4$\pi$ 
coverage about the $ep$ interaction point.
Both detectors were equipped with proton spectrometers; the Leading Proton 
Spectrometer (LPS) for ZEUS, the Forward Proton Spectrometer (FPS) and the 
Very Forward Proton Spectrometer (VFPS) for H1. The LPS and FPS spectrometers were located between $60$ and $90$~m away from the main 
detectors in the forward (proton beam) direction. The VFPS spectrometer was located around $220$~m away from the main H1 detector in the forward direction. 
%A detailed descriptions of the H1 and ZEUS detectors can be found elsewhere~\cite{H1_det,ZEUS_det}. 

The combination is based on the cross sections measured with the H1 FPS~\cite{h1-fps1,h1-fps2} and the ZEUS 
LPS~\cite{zeus-lps1,zeus-lrglps}. 
The bulk of the data~\cite{h1-fps1,h1-fps2,zeus-lrglps} was taken at electron and proton beam energies of 
$E_e\simeq27.5$~GeV and $E_p=920$~GeV, 
respectively, corresponding to an $ep$ centre-of-mass energy of
$\sqrt{s} = 318$~GeV. The earlier ZEUS LPS data~\cite{zeus-lps1} collected at $E_p=820$~GeV are corrected   
to a common $\sqrt{s} = 318$~GeV by using the extrapolation procedure described in section~\ref{sec-grid}. 
The three-fold differential reduced cross sections, $\sigma_r^{D(3)}$($\beta$,~$Q^2$,~$\xip$), are combined.
For the original measurements, the main H1 and ZEUS detectors are used 
to reconstruct $Q^2, W$ and $x$, whereas $M_X$, $\beta$, $\xip$ and $t$ are 
derived from the proton spectrometer measurements  
or from combined information of the proton spectrometers and the main 
detectors.
%A summary of the features of the data is given in table~\ref{tab-data}.
In table~\ref{tab-data} the data sets used for the combination are listed together with their kinematic ranges and  
integrated luminosities.
%The H1 data taken with the VFPS~\cite{h1-vfps} are still under analysis and are 
%not included in the combination; anyhow, they cover
%a different kinematic region and are expected to have little
%influence in the range of the combined FPS+LPS data.

\begin{table}
\begin{center}
\begin{footnotesize}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
 Data Set & $Q^2$ range & $\xip$ range & $y$ range & $\beta$ range & $t$ range & Luminosity & Ref.\\
 & [GeV$^2$] & & & & [GeV$^2$] & [pb$^{-1}$] & \\
\hline     
 H1 FPS HERA II & $4 - 700$ & $< 0.1$ & $0.03 - 0.8$ & $0.001 - 1$ & $0.1 - 0.7$ & $156.6$ & ~\cite{h1-fps2}\\                                  
\hline 
H1 FPS HERA I & $2 - 50$ & $< 0.1$ & $0.02 - 0.6$ & $0.004 - 1$ & $0.08 - 0.5$ & $28.4$ & ~\cite{h1-fps1}\\                                  
\hline\hline
 & & & $W$ range & $M_X$ range & & & \\
 & & & [GeV] & [GeV] & & & \\\hline
ZEUS LPS 2 & $2.5 - 120$ & $0.0002 - 0.1$ & $40 - 240$ & $2 - 40$ & $0.09 - 0.55$ & $32.6$ &~\cite{zeus-lrglps} \\
\hline 
ZEUS LPS 1 & $2 - 100$ & $<0.1$ & $25 - 240$ & $>1.5$ & $0.075 - 0.35$ & $3.6$ &~\cite{zeus-lps1}\\  
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\vspace{-0.3cm}

\caption{H1 and ZEUS data sets used for the combination of the measurements.}
\label{tab-data}
\end{table}

\subsubsection{Restricted {\boldmath $t$} range}
\label{sec-t}

In the individual analyses~\cite{h1-fps1,h1-fps2,zeus-lrglps,zeus-lps1} the 
reduced cross sections are directly measured for ranges of the squared 
four-momentum 
transfer $t$ visible to the proton spectrometers (see table~\ref{tab-data}) 
and extrapolated to the range\footnote{The smallest kinematically accessible value of 
$|t|$ is denoted as $|t_{min}|$.} $|t_{min}| < |t| < 1$ GeV$^2$ (denoted in the
following as `the full $t$ range'), assuming an exponential 
$t$ dependence of the diffractive cross section 
and using the exponential slope measured from the data. 
Due to the uncertainties of the slope parameters measured by H1~\cite{h1-fps2, h1-fps1} and 
ZEUS~\cite{zeus-lrglps,zeus-lps1}, 
this extrapolation 
introduces an additional uncertainty in the normalisation of the cross section. 
%Such extrapolation, which depends on the exponential slope of the $t$ 
%distribution and on its uncertainty, introduces an extra uncertainty in the 
%normalisation of the cross section. 
To reduce this source of
systematic uncertainty, the H1 and ZEUS cross sections are 
combined in the restricted $t$ range $0.09 < |t| < 0.55$ GeV$^2$ 
covered by the proton spectrometer acceptances of both detectors for the bulk of the data. 
%MK
 The correction factors from the visible $t$ range of the `FPS HERA I' and `LPS 1' data samples to the 
restricted $t$ 
range are evaluated by
using the  $t$ dependencies as a function of $x_{\pom}$ measured for each sample.  
The correction factors for the most precise `FPS HERA II' data are applied in bins of $\beta, Q^2$ and $\xpom$.
For the `LPS 2' sample 
the restricted range coincides with the visible range. 
%MK
Because of the uncertainty on the exponential
 slope parameter, such factors introduce uncertainties of
2.2\%, 1.1\% and 5\% on the `FPS HERA II', `FPS HERA I' and `LPS 1' data, 
respectively, which are included in the normalisation uncertainty on each 
sample. 
The total normalisation uncertainties of the data samples are listed in table~\ref{tab-t}.
In the restricted $t$ range, these uncertainties 
are in general smaller 
and the average normalisations are in better agreement than in the full $t$ range; 
the ratio of the `FPS HERA II' to the `LPS 2' data averaged over the measured 
data points, which is 
$0.85 \pm 0.01$(stat) $\pm 0.03 $(sys) $^{+0.09}_{-0.12}$(norm)~ 
in the full $t$ range~\cite{h1-fps2}, becomes 
$0.91 \pm 0.01$ (stat) $\pm 0.03$ (sys) $\pm 0.08$ (norm)~
in the restricted $t$ range. Within the uncertainties, the ratio does not show any 
significant $\beta$, $Q^2$ or $\xip$ dependence. 
%The correction factors from the full $t$ range to the restricted $t$ range are within $\pm5$\% for `FPS HERA I', within 
%(+5\%/-8\%) for `FPS HERA II' and within (+17\%/-3\%) for `LPS 1'. The restricted range coincides with the visible range of the 
%`LPS 2' data. The correction factors are evaluated 
%using the  $t$ dependence measured at each $\beta, Q^2, x_{\pom}$ and introduce 
%an uncertainty which is summed up in the normalisation uncertainty of each sample. 

%\vspace*{1.0cm}

\begin{table}
\begin{center}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{|c|c|c|}
\hline
 Data Set & $|t_{min}| < |t| < 1$ GeV$^2$ & $0.09 < |t| < 0.55$ GeV$^2$ \\
\hline
FPS HERA II & $\pm 6 \%$ & $\pm 5\%$ \\
FPS HERA I & $\pm 10 \%$ & $\pm 10 \%$\\
LPS 2 & $+11\%$, $-7\%$ & $\pm 7 \%$\\
LPS 1    & $+12\%$, $-10\%$& $\pm 11 \%$\\ 
\hline
\end{tabular}
\vspace{-0.3cm}

\end{center}
\caption{Normalisation uncertainties in 
the full range $|t| < 1$ GeV$^2$ and in the restricted $t$ range for the data used for the combination of the measurements.}
\label{tab-t}
\end{table}

\subsubsection{Extrapolation to a common ({\boldmath $Q^2$, $\xip$, $\beta$}) grid}
\label{sec-grid}

%The averaging procedure works under the assumption that there is 
%a single true value of the cross section for a single data point.~Therefore, 
%cross section values from different data sets are needed
%at the same ($\beta$, $Q^2$, $\xip$) bin.
The original binning schemes of the $\sigma_r^{D(3)}$
measurements are very different for H1 and ZEUS.
In the H1 case the measurements are extracted at fixed $\beta$, whereas for ZEUS the cross section is measured 
at fixed $M_X$; also the $Q^2$~and 
$\xip$~central values differ. Therefore, prior to the combination, the H1 and 
ZEUS data are transformed to a common grid of ($\beta, Q^2, x_{\pom}$)
points.
%\footnote{The same procedure could not be applied to the $\sigma_r^{D(4)}$ 
%measurements, for which reaching a common  
%($\beta, Q^2, \xip, t$) grid would imply extrapolating in 
%$\xip$ and $t$ separately: this cannot be done rigorously as the $\xip$ 
%and $t$ dependences are connected through the slope of the Pomeron 
%trajectory~(Section~\ref{sec-kine}).}
The grid points are based on the original binning scheme 
of the `FPS HERA II' data. 
The ($Q^2, \xip$) grid points at the lowest $Q^2$ value 
of $2.5$~GeV$^2$ and at the lowest and highest $ x_{\pom}$ values, which are beyond the `FPS HERA II' data grid, are 
taken from the `LPS 2' measurement.
%The older H1 FPS HERA-1~\cite{H1FPS1} and ZEUS LPS1~\cite{ZEUSLPS1} data are extrapolated to the selected common 
%grid of points.
%MK
% The matching is done between
% the original grid and the common grid of ($\beta, Q^2, x_{\pom}$) using the nearest grid point.
% taken for all the variables linearly. 
%MK

The transformation of a measurement from the original $i^{th}$ point 
($\beta_i, Q^2_i, \xip_i$)  
to the nearest grid point ($\beta_{grid}, Q^2_{grid}, \xip_{grid}$) is performed by 
multiplying the measured cross section by the ratio $\sigma_r^{D(3)}$($\beta_{grid}, Q^2_{grid}, \xip_{grid}$)/ $\sigma_r^{D(3)}$($\beta_{i}, Q^2_{i}, \xip_{i}$) 
calculated with the Next-to-Leading-Order (NLO) DPDF `ZEUS SJ' parameterisation~\cite{zeus-qcdfits}. 
Most of the corrections are smaller than $10$\%, while a few points undergo 
corrections up to $\sim~30$\%. The procedure is checked by using the  
NLO DPDF `H1 Fit B' parameterisation~\cite{h1-lrg1}. 
%The resulting cross section difference is of the order of 
%$1.2$\% on average and up to $8$\% for a few data points. 
The resulting difference is treated as a procedural uncertainty on the
combined cross section, as discussed in Section~\ref{sec-procedurals}.

The cross sections from all the data sets are shown in figure~\ref{fig-q2dep-allinputs} after correcting to $0.09 < |t| < 0.55$ GeV$^2$ and transforming to the 
common 
grid. 

\subsection{Combination method}
\label{sec-comb}
The combination is based on the $\chisq$~minimisation method
described in~\cite{glazov} and used for previous combined HERA 
results~\cite{:2009wt}. The averaging procedure is based on 
the assumption that at a given kinematic
 point the H1 and ZEUS experiments are 
measuring the same cross section.
~The correlated systematic uncertainties are floated coherently.
~The procedure allows a model independent check of the data consistency and 
leads to a significant reduction of the correlated uncertainties. 

 For an individual data set, the $\chisq$~function is defined as:
\begin{equation}
\chi^2_{exp} (\pmb{m},\pmb{b}) = \sum_i 
\frac{\left[ m^i - \sum_j \gamma^i_j m^i b_j - \mu^i \right]^2}
{\delta^2_{i,stat} \mu^i \left( m^i - \sum_j \gamma^i_j m^i b_j \right) 
+ \left( \delta_{i,uncor} m^i \right)^2} + \sum_j b^2_j~.
\label{eq-chi}
\end{equation}

\noindent
Here $\mu^i$ is the measured cross section value at a point $i$ 
($\beta_i$,~$Q^2_i$,~$\xip_i$), and $\gamma^i_j$, 
$\delta_{i,stat}$ and $\delta_{i,uncor}$ are the relative correlated systematic, 
relative statistical and relative uncorrelated systematic uncertainties, 
respectively. The vector $\pmb{m}$ of quantities $m^i$ expresses the values 
of the combined cross section for each point $i$ and the vector 
$\pmb{b}$ of quantities $b_j$ expresses the shifts 
of the correlated systematic uncertainty sources, $j$, in units of the standard deviation. 
% The summation over $j$ extends over all correlated systematic sources. 
The relative uncertainties $\gamma^i_j$ and 
$\delta_{i,uncor}$ are multiplied by the %expectation 
combined cross section $m^i$ in order to take into account the fact 
that the correlated and uncorrelated systematic uncertainties are to a good 
approximation proportional to the central values (multiplicative uncertainties). 
On the other hand, the statistical uncertainties scale with the square root of the 
expected number of events, which is determined by the expected cross 
section, corrected for the biases due to the correlated 
systematic uncertainties. This is taken into account by the 
$\delta^2_{i,stat} \mu^i ( m^i - \sum_j \gamma^i_j m^i b_j )$ 
term. 

%The second term in the expression for $\chi^2_{exp}$ is the sum of the 
%resulting shifts of the systematic uncertainties. 

%The $\chisq$~function defined in Eq.~(\ref{eq-chi}) has by construction a 
%minimum $\chi^2_{exp} = 0$~for $m^i = \mu^i$ and $b_j = 0$. 

If several analyses provide measurements at the same ($\beta$,~$Q^2$, $\xip$) 
values, 
% they can be combined by using the formula above, generalised
% for the case of multiple data sets. Then 
a $\chi^2_{tot}$ is built~\cite{Aaron:2009bp} from the sum of the $\chi^2_{exp}$ of each data set, assuming  the 
individual data sets to be statistically uncorrelated. The $\chi^2_{tot}$ is minimised with respect to the $m^i$ and 
$b_j$ from each data set 
with an iterative procedure.
~The ratio $\chi^2_{min} / n_{dof}$ is a measure of the 
consistency of the data sets.~The number of degrees of freedom, $n_{dof}$, is
calculated as the difference between the total number of measurements and the 
 number of averaged points. The uncertainties of the combined cross sections are evaluated from the 
 $\chi^2_{min}+1$ 
criteria~\cite{glazov,Aaron:2009bp,:2009wt}. 
% Possible tensions between the data sets can be investigated by means of 
% unbiased pull distributions, defined as in~\cite{:2009wt}.
% such that for data with no correlated systematic uncertainties they are expected to follow 
% Gaussian distributions with zero mean and unit width. 
% When correlated systematic uncertainties are present and 
% since the corresponding parameters are floated, this can lead to narrowed pull distributions. 
For some of the ($\beta, Q^2, x_{\pom}$) points there is only one measurement;
%Note that through 
however, because of the systematic 
uncertainty correlations such measurements may %nevertheless 
be shifted with 
respect to the original values, and the uncertainties may be reduced.
% in the averaging procedure. 

%~For the measurement at the point $i$ of the data set $e$, the pull is defined as:
%\begin{equation}
%p^{i,e} = \frac{\mu^{i,e} - \mu^{i,ave} (1 - \sum_j \gamma^{i,e}_j b_{j,ave})}
%{\sqrt{ \Delta^2_{i,e} - \Delta^2_{i,ave}}}~,
%\label{eq-pull}
%\end{equation}
%where $ \Delta^2_{i,e}$ ($ \Delta^2_{i,ave}$) is the quadratic sum of the
%statistical and uncorrelated systematic uncertainty for the measurement $e$ (average).~The
%pull value is $p^{i,e} \neq 0$ under the condition that measurements from more
%than one data set average at the same point $i$.~This means that if only one
%input data set $e$ contributes to the point $i$, the value of the combined
%cross section in that point is $\mu^{i,ave} = \mu^{i,e} / (1 - \sum_j \gamma^{i,e}_j b_{j,ave})$.

%For the correlated systematic source $j$ the pull is given by:
%The definition of pull for the correlated systematic sources in given by:
%\begin{equation}
%p_s = \frac{b_{j,ave}}
%{\sqrt{ 1 - \Delta^2_{b_j,ave}}}~,
%\label{eq:pull:s}
%\end{equation}
%where $\Delta^2_{b_j,ave}$ is the uncertainty on the source $j$ after the averaging procedure.

\subsection{Uncertainties}

\subsubsection{Experimental systematic uncertainties and their correlations}
\label{sec-corrsys}


The input cross sections are published with their statistical and 
 systematic uncertainties. The statistical uncertainties correspond to $\delta_{i,stat}$ in Eq.~(\ref{eq-chi}). The systematic 
 uncertainties are classified as point-to-point correlated or point-to-point uncorrelated, corresponding to $\gamma^i_j$ and 
 $\delta_{i,uncor}$ respectively, according to the information provided in the corresponding publications, as follows: 
\begin{itemize}
\item
 for the two older analyses, `FPS HERA I' and `LPS 1', only the total systematic 
 uncertainties are given~\cite{h1-fps1,zeus-lps1}, with no information on the single contributions and 
 point-to-point 
 correlations. For these two samples only the normalisation uncertainties~(table~\ref{tab-t}) are considered among the correlated 
systematics, while the remaining uncertainties are treated as uncorrelated;
\item
 for the sample `FPS HERA II' all the systematic sources discussed in~\cite{h1-fps2} are treated as point-to-point correlated. 
 The hadronic energy scale uncertainty is taken as correlated separately for 
 $\xip < 0.012$ and $\xip > 0.012$. This is to account for the different sensitivity to this 
 systematic source for the two $\xip$ regions, where different methods are
 used to reconstruct the variable $\beta$, which are typically sensitive to different  
 regions of the H1 central  calorimeter. 
 For $\xpom<0.012$, where 
 the mass  $M_X$ of the hadronic final state is used to reconstruct $\beta$, the 
 effect on the cross section due to the hadronic energy scale uncertainty is 4\% on average and reaches 
 6.7\%. For $\xpom>0.012$, where $\beta$ is reconstructed with the 
 leading proton energy measured by the FPS, the cross section shows almost 
 no sensitivity to the hadronic energy scale;
%where the scaling variable $x$ is obtained by combining the information from
%the scattered electron and from the angles of the electron and the hadronic final state and the variable      
%$\xip$ is evaluated using the leading proton energy.
%: an average effect of 4\% reaching 6-7\% in some bins for $\xip<0.012$, where $\beta$ is reconstructed with the main 
%detector, and almost no sensitivity for $\xip>0.012$, where $\beta$ is reconstructed 
%with the FPS; 
\item
 in the `LPS 2' case, the total systematic uncertainties quoted in~\cite{zeus-lrglps} are %split between 
decomposed in
 correlated and uncorrelated 
%sources 
following the prescriptions in~\cite{zeus-qcdfits}. They are symmetrised
%MK
%\footnote{The result of 
% the average is found to be insensitive to the details of the symmetrisation procedure.} 
 by taking the average of the positive and negative uncertainties. 
\end{itemize}

In the H1 `FPS HERA II' analysis, the systematic effects related to the leading proton measurement are considered 
as correlated and derived from the variation of the acceptance in the $\xpom$ and $t$ bins when 
%applying the positive and negative shifts of 
shifting
the FPS energy scale 
and transverse momentum within the estimated uncertainties~\cite{h1-fps2}.   
% In the ZEUS `LPS 2' case the effect of this uncertainty on the
% cross section is derived from the uncertainty on the alignment procedure of the LPS stations and the beam optics. 
% the key variable measured with the LPS is the 
% longitudinal momentum of the scattered proton, the uncertainty on which 
% derives from the alignment procedure. 
% In the ZEUS `LPS 2' case, the systematic uncertainty of the leading proton measurement 
%%on the cross section is covered by the uncertainty on the alignment procedure of the LPS stations 
%is expressed in terms of the uncertainty on the LPS station alignment and that on the proton beam optics 
% simulation. 
In the ZEUS 'LPS 2' case, the systematic uncertainty
related to the leading proton measurement is dominated by the
incomplete knowledge of the beam optics, of the position of the beamline
aperture limitations and of the intrinsic transverse-momentum spread of the proton beam at the interaction point. 
The beam optics contribution is largely independent of the kinematic variables and therefore is taken as a
normalisation uncertainty~\cite{zeus-lrglps}.
The other contributions are quantified by
varying the cut on the distance of closest approach of the reconstructed proton track to the beampipe, and the value of the intrinsic transverse-momentum spread assumed in the simulation. They are treated as uncorrelated uncertainties.

%The former uncertainty is treated as uncorrelated and 
% the resulting uncertainty on the spectrometer acceptance is estimated
% by varying the distance of closest approach of the LPS 
% stations to the proton beam. 
%The effect of the latter uncertainty is found to be largely independent of 
%the kinematic variables and therefore 
% is taken as a normalisation uncertainty~\cite{zeus-lrglps}. 
%%MK of the LPS stations to the proton beam ??? or of the scattered proton to the beam pipe ???
%largely covered by the uncertainty on the beam optics. 

%All the systematic uncertainties are treated as independent between the H1 and ZEUS measurements.
All the H1 systematic uncertainties are treated as independent of the ZEUS uncertainties, and vice versa. 
Possible effects due to correlations between the two experiments are 
taken into account in the procedural uncertainties, 
discussed in Section~\ref{sec-procedurals}.
In total, 23 independent sources of correlated systematic
uncertainties are considered, including the global normalisation for each sample. The full list is given in 
table~\ref{tab-sys}.


\subsubsection{Procedural uncertainties}
\label{sec-procedurals}

The following uncertainties on the combined cross sections due to the combination procedure 
 are studied:
\begin{itemize}
\item
The $\chi^2$ function given by Eq.~(\ref{eq-chi}) treats all 
systematic uncertainties as multiplicative, i.e. proportional 
to the expected central values. While this generally holds for 
the normalisation uncertainties, it may not be the case for the 
other uncertainties.~To study the sensitivity of the average 
result to this issue, an alternative averaging is performed. 
Only the normalisation uncertainty and those related to the $t$ 
reconstruction 
% ($\delta_{p_x,FPS}$, $\delta_{p_y,FPS}$, $\delta_{t,FPS}$, $\delta_{t,LPS}$), 
(the uncertainties on the `FPS HERA II' proton $p_x$, $p_y$ reconstruction and on the `FPS HERA II' and `LPS 2' $t$ 
reweighting) 
which, for the reasons explained in Section~\ref{sec-t}, 
can affect the normalisation, are taken as multiplicative,  
while all other uncertainties are treated as additive. The difference 
between this average and the nominal result is of the order of 1\% 
on average and 6.4\% at most.
\item
The H1 and ZEUS experiments use similar methods for detector calibration, apply similar 
reweighting to the Monte Carlo models used for the acceptance corrections and employ similar Monte Carlo models for QED 
radiative corrections, 
for the hadronic final state simulation and for background subtraction. 
Such similarities may lead to 
correlations between the measurements of the two experiments. 
%Three systematic sources are identified which are most likely to be correlated between the two experiments. 
Three systematic source are identified as the most likely to be correlated between the two experiments.
These are %related to 
the electromagnetic energy scale 
and the reweighting of the 
simulation in $\xip$ and $t$. Averages are formed for each of the $2^3$ possible
assumptions on 
the presence of correlations of these systematic uncertainties 
%whether these systematics are correlated or uncorrelated
between the experiments and are compared with the nominal average for which all sources are assumed to be 
uncorrelated. 
 The maximum difference
      between the nominal and the alternative averages is
      taken as an uncertainty.
It is 1.4\% on average and 6.6\% at most, 
with no particular dependence on the kinematics.

\item
The bias introduced by transforming the data to the common grid
 (see Section~\ref{sec-grid}) is studied by using 
correction factors obtained from the NLO DPDF `H1 Fit B'~\cite{h1-lrg1} 
parameterisation. 
For a few bins this changes the result by 
up to 8\%, but 
% Though in a few bins the difference increases up to 8\%, 
the average effect is 1.2\%.

 \item The averaging procedure shifts the H1 hadronic energy scale at $\xip < 0.012$ by substantially more than $1\sigma$ 
of the nominal value 
(see Section~\ref{sec-res}). To study the 
sensitivity of the average result to the treatment of the uncertainty on the H1 
hadronic energy scale, an alternative averaging is performed for which this uncertainty is considered as 
point-to-point uncorrelated. The difference between the alternative and nominal results is 0.9\% on average and 
reaches 
8.7\% at low 
$\xpom$. 
\end{itemize}

For each combined data point the difference between the average obtained by considering 
each of the procedural effects and the nominal average is calculated and summed 
in quadrature. The effect of the procedural 
uncertainties is 2.9\% on average and 9.3\% at most. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\label{sec-res}

In the minimisation procedure, 352 data 
points are combined to 191 cross section measurements.~The data show good 
consistency, with 
$\chi^2_{min} / n_{dof} = 133 / 161$.
%~For a standard weighted average the resulting $\chi^2 / n_{dof}$ is 134/161. 
The distributions of pulls~\cite{:2009wt}, shown in figure~\ref{fig-pulls} for each data set, 
 exhibit no significant tensions.
 For data with no correlated systematic uncertainties pulls are expected to follow 
 Gaussian distributions with zero mean and unit width. 
 Correlated systematic uncertainties lead to narrowed pull distributions. 

~The effects of the combination on the correlated systematic 
uncertainties are summarised in table~\ref{tab-sys} in terms of shifts 
in units of the original uncertainty and of values of the final uncertainties 
as percentages of the originals. 
%in $\sigma$ units and in percent of the original uncertainty. 
The combined cross section values are given in table~\ref{tab-results}  
together with statistical, uncorrelated systematic, correlated systematic, experimental, procedural   
and total uncertainties. The experimental uncertainty is obtained as the quadratic sum of the statistical, 
uncorrelated 
systematic and correlated systematic uncertainties. The total 
uncertainty is defined as the  quadratic sum of the experimental and procedural uncertainties. The 
full information about correlations 
can be found elsewhere~\cite{website}. As the global 
normalisations of the input data sets are fitted as correlated systematic uncertainties, the normalisation uncertainty on the 
combined data is included in the correlated systematic uncertainty given in table~\ref{tab-results}. 

Most of the 23 correlated systematic uncertainties shift by less than $0.5~\sigma$ of the 
nominal value in the averaging procedure. None of them shifts by substantially more than 
$1 \sigma$, with the exception of the hadronic energy scale for $\xip<0.012$ 
for the `FPS HERA II' sample. 
Detailed studies show that there is a tension between the H1 `FPS HERA II' and ZEUS `LPS 2' data at low $\xip$; the 
average ratio of the H1 to ZEUS cross sections is above $1.0$ for 
$\beta>0.1$ and below $0.9$ for $\beta<0.1$. The H1
cross section 
uncertainty is positively correlated with the hadronic energy scale for $\beta>0.1$ and anti-correlated for $\beta<0.1$. As
a result, the combination shifts the H1 cross sections for $\xip<0.012$ in the direction opposite to the cross section uncertainty due to the 
H1 hadronic energy scale. Conversely the combined statistical and uncorrelated uncertainty on the ZEUS data is much 
larger than the 
ZEUS hadronic energy scale uncertainty; consequently 
the fit is less sensitive to the ZEUS hadronic energy scale.
%If such error source is taken as correlated 
%across the whole $\xip$ region (i.e. without splitting between $\xip<0.012$ and 
%$\xip > 0.012$) the resulting shift is $1.7 \sigma$. This significant effect, 
%together with the observation that only the lowest $\xip$ combined points 
%show relevant displacements (up to 15\%) with respect to the input measurements, is the reason to split the $\xip$ range in the 
%fit of this error parameter. 

The influence of several correlated systematic uncertainties is reduced 
significantly for the combined result. Specifically, the uncertainty on the 
FPS proton energy measurement and the normalisation uncertainties on the 
`FPS HERA I' and `LPS 1' samples are reduced by more than a factor of 2.
The H1 hadronic energy scale uncertainty for the low $\xip$-range ($\xip<0.012$) and the 
ZEUS hadronic energy scale uncertainty are reduced to around 55\% of those for the individual data sets. 
% Two facts explain these significant reductions. 
Since H1 and ZEUS use different 
reconstruction methods, similar systematic sources influence the measured 
 cross section differently. % as a function of the kinematics.~
Therefore, requiring the cross sections to be consistent at all 
($\beta$, $Q^2$, $\xip$) points constrains the systematic uncertainties efficiently.
%~In addition, the `FPS HERA II' sample has superior precision compared to the others for most regions of the phase space. The less precise measurements are fitted to the more precise one, with a simultaneous reduction of the correlated systematic uncertainty.~This reduction propagates also to the average points based only on the less precise data sets.
Due to this cross calibration effect,
the combined measurement shows an average improvement of the experimental uncertainty of about 27\% with respect 
to the most precise single data set, 
`FPS HERA II', though the latter data set contains five times more events than
the second largest data set, `LPS 2'. 
The correlated  part of the experimental uncertainty is reduced from about 69\% in~\cite{h1-fps2} to 49\% in the 
combined 
measurement. The statistical, experimental and procedural uncertainties on the combined data are  on average 11\%, 13.8\% and 
2.9\%, respectively. The total uncertainty on the cross section is 14.3\% on average and is 6\% for the most precise
 points. The normalisation uncertainty, which contributes to 
the correlated systematic uncertainty on the combined data, 
is on average 4\%.
The combined result extends the kinematic 
coverage with respect to the H1 and ZEUS measurements taken separately 
%such that 
and the resulting
cross section covers the region $2.5 \leq Q^2 \leq 200$~GeV$^2$, $0.0018 \leq \beta \leq 0.816$
and $0.00035 \leq x_{\pom} \leq 0.09$, for $0.09 < |t| < 0.55$~GeV$^2$.
Figure~\ref{fig-q2sel-all} shows the HERA combined cross section as a function of $Q^2$~at
 $x_{\pom} = 0.05$, for different values of $\beta$, compared with the individual measurements used for the 
combination.~The reduction of the 
total uncertainty
of the HERA measurement compared to the input cross sections is visible.
~The derivative of the reduced cross section as a function of $\log(Q^2)$ decreases with 
$\beta$,
%~The rate of the 
%growth of the reduced cross section with $Q^2$ ~depends on $\beta$~
a feature characteristic of the scaling violations in
diffractive DIS, which are now measured precisely from proton-tagged as well
as LRG data.
Figures~\ref{fig-q2dep-comb} and~\ref{fig-xpdep-comb} show the HERA combined diffractive reduced cross sections as a 
function of $Q^2$ 
and $\xip$, respectively. 

At low $\xip \lesssim 0.01$, 
% where in the framework of Regge phenomenology~\cite{regge} 
% the diffractive reactions are ascribed to the exchange of a leading (pomeron) trajectory and 
where the proton spectrometer data are free from proton dissociation contributions, the combined 
data provide the most precise determination of the 
absolute normalisation of the diffractive cross section. 
%This combined data are very valuable in the scenario of inclusive diffraction 
%at HERA. Diffractive interactions are often discussed in the framework of Regge 
%phenomenology~\cite{regge} in terms of  the exchange of a `pomeron' with 
%vacuum quantum numbers. Similar reactions, in which
%sub-leading Reggeon and pion trajectories are exchanged, 
%have a negligible cross section at the smallest $\xip$ values.
%The LPS and FPS data, and hence the combined data, 
%extend to $\xip \sim 0.1$ and are therefore also 
%sensitive to such non-leading contributions. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The reduced diffractive cross sections, $\sigma_r^{D(3)}(ep \to eXp)$, measured by the H1 and 
ZEUS Collaborations by using proton spectrometers to detect the leading protons  
are combined. The input data 
 from the two experiments are consistent with a $\chi^2_{min} / n_{dof} = 133 / 161$. 
% Correlations of systematic uncertainties are taken into account by the combination method,
 The combination of the measurements results in more precise and kinematically extended diffractive DIS data in the 
$t$-range $0.09 < |t| < 0.55$~GeV$^2$. The total uncertainty on the cross section measurement is 6\% for the most 
 precise  points.
The combined data provide the most precise determination of the absolute normalisation of the $ep\rightarrow eXp$ cross section.
%  $2.5 \leq Q^2 \leq 200$~GeV$^2$, $0.0018 \leq \beta \leq 0.816$, $0.00035 \leq x_{\pom} \leq 0.09$ and $0.09 < |t| < 
%  0.55$~GeV$^2$.

%These combined data are very valuable in the scenario of inclusive diffraction at HERA and beyond.~They can provide the absolute normalisation of the diffractive cross section in DIS and can help to quantify background contributions in the samples selected with hadronic methods (large rapidity gap technique and $M_X$ method). 

%The combination of H1 and ZEUS 
%diffractive cross sections measured with proton spectrometers 
%results in a dataset for sigma of improved precision which is 
%compatible with the original datasets. This is the first step 
%towards a combination of diffractive cross sections from other 
%experimental methods and may eventually result in a common fit 
%and the extraction of unique diffractive deep inelastic 
%scattering structure functions."


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Acknowledgements}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We are grateful to the HERA machine group whose outstanding
efforts have made these experiments possible.
We appreciate the contributions to the construction and maintenance of the H1 and ZEUS detectors of many people who are not listed as authors.
We thank our funding agencies for financial 
support, the DESY technical staff for continuous assistance and the 
DESY directorate for their support and for the hospitality they extended to the non-DESY members of the collaborations. 
% WE EXPECIALLY THANK THE HERA MACHINE GROUPS FOR THE SUPPORT 
% WITH THE LPS AND FPS DATA TAKING?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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  H1 Collaboration, I.~Abt {\it et al.},
%   ``The Tracking, calorimeter and muon detectors of the H1 experiment at HERA,''
  Nucl.\ Instrum.\ Meth.\ {\bf A386} (1997) 348; \\
%
   R.~D.~Appuhn {\it et al.}  [H1 SPACAL Group],
   %``The H1 lead/scintillating-fibre calorimeter,''
   Nucl.\ Instrum.\ Meth.\  {\bf A386} (1997) 397. 

\bibitem{ZEUS_det}
ZEUS Collaboration, U.~Holm (ed.), {\it The ZEUS Detector}, Status Report (unpublished), DESY (1993), available on 
http://www-zeus.desy.de/bluebook/bluebook.html.

\bibitem{zeus-qcdfits}
ZEUS Collaboration, S.~Chekanov {\it et al.},
Nucl. Phys.{} {\bf B831}~(2010)~1.

\bibitem{website}
The combined data together with the full correlation information are provided 
at the URL http://www.desy.de/h1zeus.

% \bibitem{regge}
% P.D.B.~Collins,
% {\em An Introduction to {Regge} Theory and High Energy Physics,\rm{
%  Cambridge University Press, Cambridge (1977).}}

\end{thebibliography}

%%%%%%%%%%%%%%%%%%%%%%%%%%% TABLES  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage

% \begin{table}[!ht]
% \begin{center}
% \begin{footnotesize}
% \begin{tabular}{|l|c|c|c|} 
% \hline
% Source & Symbol & Shift ($\sigma$ units) & Reduction \% \\ 
% \hline
% FPS HERA II hadronic energy scale $\xip<0.012$   & $\delta_{had1,H1}$ & -1.61 & 56.9      \\
% FPS HERA II hadronic energy scale $\xip>0.012$   & $\delta_{had12,H1}$ & 0.13 & 99.8      \\
% FPS HERA II electromagnetic energy scale    & $\delta_{ele,H1}$ & 0.49 & 85.9      \\
% FPS HERA II electron angle            & $\delta_{\theta,H1}$ & 0.67 & 66.6   \\
% FPS HERA II $\beta$~reweighting       & $\delta_{\beta,H1}$  & 0.15 & 90.4   \\
% FPS HERA II $\xip$~reweighting        & $\delta_{\xip,H1}$ & 0.05 & 98.3 \\
% FPS HERA II $t$~reweighting           & $\delta_{t,FPS}$ & 0.70 & 79.8        \\
% FPS HERA II $Q^2$~reweighting         & $\delta_{Q^2,H1}$ & 0.09 & 97.6     \\
% FPS HERA II proton energy             & $\delta_{E_p,FPS}$ & 0.05 & 45.6     \\
% FPS HERA II proton $p_x$              & $\delta_{p_x,FPS}$ & 0.62  & 74.5    \\
% FPS HERA II proton $p_y$              & $\delta_{p_y,FPS}$ & 0.27  & 86.5   \\
% FPS HERA II vertex reconstruction     & $\delta_{vtx,FPS}$ & 0.07  & 97.0   \\
% FPS HERA II background subtraction    & $\delta_{bgd,FPS}$ & 0.84  & 89.9   \\
% FPS HERA II bin centre corrections    & $\delta_{bcc,FPS}$ & -1.05  & 87.3   \\
% FPS HERA II global normalisation      & $\delta_{norm,FPSHERAII}$ & -0.39 & 84.4     \\
% FPS HERA I global normalisation      & $\delta_{norm,FPSHERAI}$ &  0.81 & 48.9     \\
% \hline
% LPS 2 hadronic energy scale           & $\delta_{had,ZEUS}$ & -0.02 & 55.0     \\
% LPS 2 electromagnetic energy scale    & $\delta_{ele,ZEUS}$ & -0.14  & 62.4   \\
% LPS 2 $\xip$~reweighting        & $\delta_{x_{\pom},ZEUS}$ & -0.32 & 98.2\\
% LPS 2 $t$~reweighting           & $\delta_{t,LPS}$ & -0.26 & 86.4       \\
% LPS 2 background subtraction & $\delta_{bgd,ZEUS}$ & 0.40 & 94.9     \\
% LPS 2 global normalisation      & $\delta_{norm,LPS2}$  & -0.53 & 67.7  \\
% LPS 1 global normalisation         & $\delta_{norm,LPS1}$  & 0.86 & 44.1   \\
% \hline

\begin{table}[!ht]
\begin{center}
\begin{normalsize}
\begin{tabular}{|l|r|c|} 
\hline
Source & Shift ($\sigma$ units) & Reduction factor \% \\ 
\hline
FPS HERA II hadronic energy scale $\xip<0.012$  & $-1.61~~~~$ & $56.9$      \\
FPS HERA II hadronic energy scale $\xip>0.012$  & $~0.13~~~~$ & $99.8$      \\
FPS HERA II electromagnetic energy scale        & $~0.49~~~~$ & $85.9$      \\
FPS HERA II electron angle                      & $~0.67~~~~$ & $66.6$   \\
FPS HERA II $\beta$~reweighting                 & $~0.15~~~~$ & $90.4$   \\
FPS HERA II $\xip$~reweighting                  & $~0.05~~~~$ & $98.3$ \\
FPS HERA II $t$~reweighting                     & $~0.70~~~~$ & $79.8$        \\
FPS HERA II $Q^2$~reweighting                   & $~0.09~~~~$ & $97.6$     \\
FPS HERA II proton energy                       & $~0.05~~~~$ & $45.6$     \\
FPS HERA II proton $p_x$                        & $~0.62~~~~$ & $74.5$    \\
FPS HERA II proton $p_y$                        & $~0.27~~~~$ & $86.5$   \\
FPS HERA II vertex reconstruction               & $~0.07~~~~$ & $97.0$   \\
FPS HERA II background subtraction              & $~0.84~~~~$ & $89.9$   \\
FPS HERA II bin centre corrections              & $-1.05~~~~$ & $87.3$   \\
FPS HERA II global normalisation                & $-0.39~~~~$ & $84.4$     \\
FPS HERA I global normalisation                 & $~0.81~~~~$ & $48.9$     \\
\hline
LPS 2 hadronic energy scale                     & $-0.02~~~~$ & $55.0$     \\
LPS 2 electromagnetic energy scale              & $-0.14~~~~$ & $62.4$   \\
LPS 2 $\xip$~reweighting                        & $-0.32~~~~$ & $98.2$ \\
LPS 2 $t$~reweighting                           & $-0.26~~~~$ & $86.4$       \\
LPS 2 background subtraction                    & $~0.40~~~~$ & $94.9$     \\
LPS 2 global normalisation                      & $-0.53~~~~$ & $67.7$  \\
LPS 1 global normalisation                      & $~0.86~~~~$ & $44.1$   \\
\hline
\end{tabular}
\end{normalsize}
\caption{Sources of point-to-point correlated systematic uncertainties 
considered in the combination. For each source the shifts resulting 
from the combination in units of the original uncertainty 
and the values of the final uncertainties as percentages of the original 
are given.}
\label{tab-sys}
\end{center}
\end{table}

%\include{tab_latex_03corFR_kk}
\include{tab_latex_03corFR2}

%%%%%%%%%%%%%%%%%%%%%%%%%%% FIGURES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[hb]
\begin{center}
%\includegraphics[scale=0.8]{figs/all_q2.eps}
\includegraphics[scale=0.8]{figs/h1zeus_q2depFR.eps}
\caption{Reduced diffractive  cross section $x_{\pom} \, \sigma_r^{D(3)}(\beta,Q^2,x_{\pom})$
for $0.09 < |t| < 0.55 \ {\rm GeV^2}$
as a function of $Q^2$ for different values of $\beta$ and $\xip$.
 The H1 `FPS~HERA II'~\cite{h1-fps2},  H1 `FPS~HERA I'~\cite{h1-fps1}, ZEUS `LPS~2'~\cite{zeus-lrglps} and ZEUS 
 `LPS~1'~\cite{zeus-lps1} data are presented. The inner error bars indicate the statistical uncertainties, while the outer 
 error bars show the statistical and systematic
 uncertainties  added in quadrature. Normalisation uncertainties are not included in the error bars of the individual 
measurements. }
\label{fig-q2dep-allinputs}
\end{center}
\end{figure}

\begin{figure}[hb]
\begin{center}
\includegraphics[scale=0.8]{figs/pullsFR.eps}
\caption{Pull distributions %of pulls 
for the individual data sets.
% ~There are no entries outside the histogram range.
 The RMS values give the root mean square of the distributions.
% The curves show the result of a Gaussian fit to the distributions.
}
\label{fig-pulls}
\end{center}
\end{figure}


%\begin{figure}[hb]
%\begin{center}
%\includegraphics[width=15cm,height=15cm]{figs/allcomb_q2dep.eps}
%\caption{Reduced diffractive  cross section $x_{\pom} \, \sigma_r^{D(3)}(\beta,Q^2,x_{\pom})$
%for $0.09 < |t| < 0.55 \ {\rm GeV^2}$,
%shown as a function of $Q^2$ for different values of $\beta$ and $\xip$. The combined data are compared to the H1 and ZEUS data input to the averaging procedure: the H1 `FPS~HERA II'~\cite{h1-fps2},  H1 `FPS~HERA I'~\cite{h1-fps1}, ZEUS `LPS~2'~\cite{zeus-lrglps} and ZEUS `LPS~1'~\cite{zeus-lps1} data. The error bars indicate the statistical and systematic errors  added in quadrature.}
%\label{fig-q2dep-all}
%\end{center}
%\end{figure}

\begin{figure}[hb]
\begin{center}
%\includegraphics[scale=0.8]{figs/all_q2sel.eps}
\includegraphics[scale=0.8]{figs/h1zeus_q2selFR.eps}
\caption{Reduced diffractive  cross section $x_{\pom} \, \sigma_r^{D(3)}(\beta,Q^2,x_{\pom})$ for $0.09 < |t| < 0.55 \ {\rm GeV^2}$
as a function of $Q^2$ for different values of $\beta$ at $\xip = 0.05$. 
 The HERA combined data are compared to the H1 and ZEUS data inputs to the averaging procedure.
% the H1 `FPS~HERA II'~\cite{h1-fps2},  
% H1 `FPS~HERA I'~\cite{h1-fps1}, ZEUS `LPS~2'~\cite{zeus-lrglps} and ZEUS `LPS~1'~\cite{zeus-lps1} data. The points for 
% which only one data set contributes are also shown.  
 The error bars indicate 
 the statistical and systematic uncertainties  added in quadrature for the input measurements and the statistical, 
systematic and 
 procedural uncertainties added in quadrature for the combined points. Normalisation uncertainties are not included in 
the error bars of 
 the individual measurements, whereas they are included in the error bars of the combined points.}
\label{fig-q2sel-all}
\end{center}
\end{figure}

\begin{figure}[hb]
\begin{center}
%\includegraphics[scale=0.8]{figs/comb_q2dep.eps}
\includegraphics[scale=0.8]{figs/h1zeus_q2combFR.eps}
\caption{HERA combined reduced diffractive cross section 
 $x_{\pom} \, \sigma_r^{D(3)}(\beta,Q^2,x_{\pom})$ for $0.09 < |t| < 0.55 \ {\rm GeV^2}$ as a function of $Q^2$ for 
 different values of $\beta$ and $\xip$. 
 The error bars indicate the statistical, systematic and procedural uncertainties added in quadrature. The normalisation 
uncertainty is 
 included.}
\label{fig-q2dep-comb}
\end{center}
\end{figure}

\begin{figure}[hb]
\begin{center}
%\includegraphics[scale=0.8]{figs/comb_xpom.eps}
\includegraphics[scale=0.8]{figs/h1zeus_xpcombFR.eps}
\caption{HERA combined reduced diffractive cross section 
 $x_{\pom} \, \sigma_r^{D(3)}(\beta,Q^2,x_{\pom})$ for $0.09 < |t| < 0.55 \ {\rm GeV^2}$ as a function of $\xip$ for 
 different values of $\beta$ and $Q^2$. The error bars indicate the statistical, systematic and procedural uncertainties 
added in 
 quadrature. The normalisation uncertainty is included.}
\label{fig-xpdep-comb}
\end{center}
\end{figure}

\end{document}




