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\begin{document}
\title{Data-driven Low-energy Generic Generator for CMD-3}

\author{A~A~Korobov$^{1,2\dag}$, S~I~Eidelman$^{1,2,3}$}
\address{1. Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, 630090, Russia\\
2. Novosibirsk State University, Novosibirsk, 630090, Russia\\
3. Lebedev Physical Institute, RAS, Moscow, 119333, Russia\\}


\ead{korobovfech@gmail.com$^{\dag}$}

\begin{abstract}
A Monte Carlo generator to simulate events of single-photon
annihilation to hadrons at center-of-mass energies below 2 GeV
is described. The generator is based on existing data on cross sections
of various exclusive channels of $e^+e^-$ annihilation obtained in scan
and ISR experiments. It is extensively used in the software packages for 
analysis of experiments at the Novosibirsk $e^+e^-$ colliders VEPP-2000 
and VEPP-4M aimed at high-precision measurements of hadronic cross sections 
with different applications, e.g. to calculations of the hadronic vacuum 
polarization for the muon anomalous magnetic moment.
\end{abstract}
\section{Introduction}
\label{section1}


Since 2010 the VEPP-2000 electron-positron collider in Budker Institute 
of Nuclear Physics has been operated in the center-of-mass (c.m.) energy 
range $\sqrt{s}=320-2000$ MeV~\cite{vepp}. The VEPP-2000 has two interaction 
points in which the Cryogenic Magnetic Detector (CMD-3)~\cite{cmd} and 
Spherical Neutral Detector (SND)~\cite{snd} are located. The physics program 
of experiments includes high-precision measurements of the total and 
exclusive cross sections of $e^{+}e^{-} \rightarrow$ hadrons ($R$ measurement),
studies of excitations of $\rho,~\omega,~\phi$ vector mesons, tests of
conservation of vector current in $e^{+}e^{-}$ annihilation and $\tau$ decays, 
production of nucleon-antinucleon pairs near threshold, two-photon physics. 
Cross sections of $e^+e^- \rightarrow$ hadrons have numerous applications,
e.g. to evaluation of the hadronic contribution to the muon 
anomalous magnetic moment $a_{\mu}$~\cite{amu}.
 
Monte Carlo (MC) simulation of signal and background is an important part of 
data analysis allowing determination of the detection efficiency and 
comparison of data with expectations to be performed. In this work a new 
generic MC generator of multihadron production in $e^{+}e^{-}$ annihilation, 
MHG2000, is discussed, in which all hadronic final states kinematically 
allowed at $\sqrt{s} <$ 2 GeV can be generated. Since perturbative Quantum 
Chromodynamic fails at these energies, standard generators of multihadron 
production like LUND~\cite{lund} or PYTHIA~\cite{pyth} cannot be used. For 
this reason, simulation of background, the main task of MHG2000, is performed
by constructing a generator based on experimental data.

The structure of the work is the following. In Section 2, we highlight
some details of MHG2000 related to experimental data. In Section 3, all the 
necessary information for a user is listed on how to run the MHG2000 code. 
Section 4 describes the generator output and gives some examples of its
application. In Section 5, some concluding remarks are made.

\section{Use of experimental data}
\label{section2}
In the MHG2000 generator all experimentally measured processes of $e^+e^-$ 
annihilation into hadrons up to 2 GeV are used to calculate a total cross 
section at a given c.m. energy and events of each final state are
sampled with a probability proportional to the  fraction of its measured 
cross section in the total one.
To simplify the calculations, all the data are approximated by the functions,
which are motivated by underlying physics.
Information about the final state and corresponding references for the data 
used in the generator are listed in the table \ref{tab1} below:\\
\\
\begin{table}[h]
\caption{\label{tab1}List of the final states in MHG2000}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l} 
\hline
Process & Experimental data & Process & Experimental data \\
\hline
$\pi^+\pi^-$ & BaBar\cite{babar2pi} & $\pi^+\pi^-\pi^0\pi^0$ & DM2\cite{DM2pipi2pi0}, CMD2\cite{cmd4pi} \\
$\pi^0\gamma$ & SND\cite{sndpig} & $\pi^+\pi^-\pi^0\eta$ & CMD3\cite{cmd3pieta} \\
$\eta\gamma$ & SND\cite{sndetagamma}, CMD2\cite{cmdetagamma} & $K^+K^-\pi^+\pi^-$ & BaBar\cite{kkpipic},CMD3\cite{Shemyakin} \\
$p\overline{p}$ & BaBar\cite{babarpp},CMD3\cite{cmdpantip} & $K^+K^-\pi^0\pi^0$ & BaBar\cite{kkpipic}\\
$n\overline{n}$ & SND\cite{sndnn} & $K^{-}K^{0}\pi^+\pi^0$ & BaBar\cite{kck0pipi0} \\
$K^+K^-$ & BaBar\cite{babarkpkn} CMD2\cite{cmdkpkn} & $K^0_SK^0_L\pi^{+}\pi^{-}$ & BaBar\cite{solodov}\\
$K^0_SK^0_L$ & BaBar\cite{solodov}, CMD2\cite{cmdKLKS}, SND\cite{sndKLKS} & $K^0_SK^0_S\pi^{+}\pi^{-}$ & BaBar\cite{solodov}\\
$\pi^+\pi^-\pi^0$ & SND\cite{snd3pi}, CMD2\cite{cmd3pi}, BaBar\cite{babar3pi} & $K^{0}_L\overline{K^{0}_S}\pi^{0}\pi^{0}$ & BaBar\cite{klks2pi0}\\
$\pi^+\pi^-\eta$ & SND\cite{sndetapipi} & $K^{+}\overline{K^{0}}\pi^{-}\pi^{0}$ & BaBar\cite{kck0pipi0}\\
$K^+K^-\pi^0$ & BaBar\cite{babarkkpi} & $2\pi^{+}2\pi^{-}\pi^{0}$ & BaBar\cite{Babar5pi} \\
$K^+K^-\eta$ & BaBar\cite{babarkkpi} & $\pi^{+}\pi^{-}3\pi^{0}$ & BaBar\cite{pipi3pi0}\\
$K^\pm K^0_S\pi^\mp$ & BaBar\cite{babarkkpi} & $3\pi^{+}3\pi^{-}$ & BaBar\cite{Babar6pi},CMD3\cite{cmd6pi}\\
$K^\pm K^0_L\pi^\mp$ & BaBar\cite{babarkkpi} & $2\pi^{+}2\pi^{-}2\pi^{0}$ & BaBar\cite{Babar6pi}\\
$\pi^+\pi^-\pi^+\pi^-$ & BaBar\cite{Babar4pic},CMD3\cite{2pi2pi} & $\pi^{+}\pi^{-}4\pi^{0}$ & IR\\
\hline
\end{tabular}
\end{center}
\end{table}
\\
\\
IR in the Table means that at the present time data for that final state are
not yet available, so that we use the isospin relations to evaluate the 
corresponding cross section.

The procedure of selecting a final state to generate consists of three main 
parts: approximation of all the data at a given energy, calculation of the 
probabilities of various processes and sampling of the number of the process
that will be generated.  

An example of the approximation is shown in Fig.~\ref{fig1}.

\begin{figure}[h]
  \includegraphics[width=0.7\textwidth]{2pibabarfitlast.pdf}\\
\caption{Energy dependence of the cross section of the process  
$e^{+}e^{-}\rightarrow \pi^{+}\pi^{-}$ of the $\pi^{+}\pi^{-}$ invariant mass}\label{fig1}
\end{figure}

The data shown in Fig.~\ref{fig1} are taken from the BaBar 
work~\cite{babar2pi}. These data are approximated by the following function:\\
\begin{eqnarray}
A(s)*\Bigl|\frac{BW^{GS}_{\rho}(s,m_{\rho},\Gamma_{\rho})\frac{1+c_{\omega}BW^{KS}_{\omega}(s,m_{\omega},\Gamma_{\omega})}{1+c_{\omega}}+c_{\rho'}BW^{GS}_{\rho'}(s,m_{\rho'},\Gamma_{\rho'})}{1+c_{\rho'}+c_{\rho''}+c_{\rho'''}}+\nonumber\\
+\frac{c_{\rho''}BW^{GS}_{\rho''}(s,m_{\rho''},\Gamma_{\rho''})+c_{\rho'''}BW^{GS}_{\rho'''}(s,m_{\rho'''},\Gamma_{\rho'''})}{1+c_{\rho'}+c_{\rho''}+c_{\rho'''}}\Bigr|^{2}
\end{eqnarray}
where\\
$A(s)=scale/(3s)*(1-4m_{\pi}/s)^{3/2}$,\\
$BW^{KS}(s,m,\Gamma)=\frac{m^2}{s-m^2-im\Gamma}$ where
$c_{\omega}=|c_{\omega}|e^{i\phi_w}$ and $s=x^{2}$
Parameters of this function obtained from the approximation are shown in 
Fig.~\ref{fig1}.
The Gounaris-Sakurai parametrization $BW^{GS}(s,m,\Gamma)$ is described in detail in the 
paper~\cite{babar2pi}.

For each final state such functions are required to evaluate the total hadron 
cross section at any c.m. energy up to 2 GeV.
The probability of any process at a given c.m. energy is determined as a ratio 
of its cross section to the total hadron cross section.  The number of the 
final state (process) $n$ is a discrete random variable with a probability
density given by the calculated probabilities of all the processes. 
After selecting a process by sampling this discrete variable an event of
this process is generated.
 
MHG2000 can also generate the final state with an additional ISR photon. In 
this case the selection of process is changed.
For each process its cross section is corrected for radiation by convolving
the hadron cross section with the radiator function. The radiator function 
$D(x,s)$ defined in Ref.~\cite{fadin} is given by the following formula:\\
$D(x,s)=\frac{1}{2}\beta(1-x)^{\beta/2-1}(1+\frac{3}{8}*\beta-\frac{1}{48}\beta^{2}(\frac{1}{3}L+\pi^{2}-\frac{47}{8}))-\frac{1}{4}\beta(1+x)+\frac{1}{32}\beta^{2}(4(1+x)ln\frac{1}{1-x}+\frac{1+3x^{2}}{1-x}ln\frac{1}{x}-5-x)$ with $\beta=\frac{2\alpha}{\pi}(L-1); L=ln\frac{s}{m^{2}}$\\
Here $x$ is a variable and $s$ is the square of c. m.\  energy.

\section{Input of MHG2000}
\label{section3}
The following input parameters should be set to use MHG2000.
First of all, it is necessary to set the collision c.m. energy in GeV. Then 
the generation mode is selected: event generation without an ISR photon (the
corresponding flag equals zero), event generation with an ISR photon
(flag equals one). 
Then we can create the list of final states that will be generated. For 
example, only one process $e^{+}e^{-}\rightarrow\pi^{+}\pi^{-}$ is generated or 
all the rest 30 processes are generated. The default value corresponds to 
generation of all the processes currently included. Such a possibility is 
useful when it is necessary to simulate all background events without the 
signal. 
\\
\section{Output of MHG2000}
\label{section4}
The output of the MHG2000 generator is an event, which is a random set of the 
four-momenta of all final particles (values of energy and momentum in GeV) and
the process number. In the programming language an event from MHG2000 is a
structure, which is an element of GenEvent class from HepMC lib. This structure
consists of the vertices. The vertex is the element of GenVertex subclass. The
vertex is added to the event by GenEvent class method add\_vertex. Several 
vertices are used to simulate events produced at hadron accelerators but only 
one vertex is used to simulate events produced at $e^+e^-$ machines. 
The vertex structure consists of the particles. The particle is the element 
of the GenParticle subclass. The particle is added to the vertex by GenVertex 
subclass method add\_particle\_out. The particle includes its code 
(Particle\_ID) and its four-momentum.
The correspondence between the particle code (Particle ID) and its PDG name is 
described in more detail in Ref.~\cite{PartID}. 
The classes and methods of HepMC library are described in Ref.~\cite{HepMC1}.

\section{First applications of MHG2000}

\begin{figure}
\includegraphics[width=0.8\linewidth]{mgg1_eta_multihadrons.png}\\
		\caption{\label{fig6}Two-photon invariant mass in the 
$\pi^+\pi^-\pi^0\eta~(\eta \to2\gamma)$ study~\cite{cmd3pieta}: MHG2000 
simulation without a signal, sum of signal MC and background, data are shown 
by the full and empty histograms and dots, respectively.}
\end{figure}
\begin{figure}
\includegraphics[width=0.8\linewidth]{expvsmhgvsmc1ddeltaen4tr.pdf}\\
		\caption{\label{fig7}Difference between the collision energy 
and the sum of four particle energies, assuming a pion mass in the 
$\pi^+\pi^-\pi^+\pi^-$ study: MHG2000 simulation without a signal, signal MC, data are shown by the vertical, horizontal shaded histograms and dots, 
respectively.}
\end{figure}
\begin{figure}
  \includegraphics[width=0.6\textwidth]{picNtrDistPoint1.pdf}
  \caption{\label{fig5}Properties of hadronic events produced in the $uds$ 
continuum at 1.84 GeV.  Here $N$ is the number of events, $N_{trk}$ is the 
number of tracks in the event. The experimental distribution and 
MC simulation based on LUARLW, JETSET and MHG2000 are plotted.  
All distributions are normalized to unity.}
\end{figure}

The main application of MHG2000 is estimation of the hadronic background 
during data analysis. 
The generator was applied for estimation of the background in various 
CMD-3 studies, e.g. those described in Refs.~\cite{Shemyakin,cmd3pieta}. 
Fig.~\ref{fig6} shows the invariant mass of two photons for the final state 
$\pi^+\pi^-\pi^0\eta~(\eta \to2\gamma)$~\cite{cmd3pieta}. It can be seen that
energy deposition due to neutral final states is currently somewhat 
underestimated. Another example is the difference between the collision energy 
and the sum of four particle energies in the study of the 
$\pi^+\pi^-\pi^+\pi^-$ final state, see Fig.~\ref{fig7}. 

The MHG2000 generator was also applied to the measurement of $R$ between 1.84 
and 3.05 GeV at the KEDR detector~\cite{kedrnew}, where it was used to 
estimate systematic uncertainties. The multiplicity distribution produced by 
various MC generators is presented in Fig.~\ref{fig5}. It is clear that
the MC distribution produced by MHG2000  agrees reasonably well with the 
experimental one.

\section{Conclusion}
\label{section5}
The Monte Carlo code MHG2000 provides generation of hadron production in 
electron positron annihilation at the c.m. energy below 2 GeV.
This generator has been successfully applied in several CMD-3 data analyses as 
an instrument for estimation of the background contribution. 
Further applications of the code include its usage at other $e^{+}e^{-}$ 
colliders. Work is in process to extend the list of available hadronic states
and expand the c.m. energy range up to 2.5 GeV.

\ack
The authors acknowledge support of A. L. Kataev who initiated this talk. We 
are grateful to our CMD-3 colleagues for useful discussions and tests 
of MHG2000 in their analysis. This work is partially supported by the Russian 
Foundation for Basic Research grant 17-52-50064.
\\
\section*{References}
\begin{thebibliography}{99}
\bibitem{vepp} D. E. Berkaev et al., JETP {\bf 113}, 213 (2011).
\bibitem{cmd} B. I. Khazin et al. [CMD-3 Collaboration], Nucl. Phys. B (Proc. 
Suppl.) {\bf 181-182}, 376 (2008).
\bibitem{snd} M. N. Achasov et al. [SND Collaboration], Nucl. Instrum. Methods 
Phys. Res., Sect. A {\bf 449}, 125 (2000).
\bibitem{amu} F. Jegerlehner and A. Nyffeler, Phys. Rep. {\bf 477}, 1 (2009).
\bibitem{lund} H. U. Bengtsson, G. Ingelman, Comput. Phys. Commun. {\bf 34}, 
521 (1985).
\bibitem{pyth} http://home.thep.lu.se/~torbjorn/Pythia.html
\bibitem{babar2pi} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D 
{\bf 86}, 032013 (2012).
\bibitem{sndpig} M. N.  Achasov et al. [SND Collaboration], Phys. Lett. B
{\bf 559}, 171 (2003).
\bibitem{sndetagamma} M. N. Achasov et al. [SND Collaboration], Phys. Rev. D
{\bf 74}, 014016 (2006).
\bibitem{cmdetagamma} R. R. Akhmetshin et al. [CMD-2 Collaboration], 
Phys. Lett. B {\bf 605}, 26 (2005).
\bibitem{babarpp} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D
{\bf 87}, 092005 (2013).
\bibitem{cmdpantip} R. R. Akhmetshin et al. [CMD-3 Collaboration], 
Phys. Lett. B {\bf 759}, 634 (2016).
\bibitem{sndnn} M. N. Achasov et al. [SND Collaboration], Phys. Rev. D 
{\bf 90}, 112007 (2014).
\bibitem{babarkpkn} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D
{\bf 88}, 032013 (2013).
\bibitem{cmdkpkn} R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B
{\bf 466}, 385 (1999).
\bibitem{solodov} B. Aubert et al. [BaBar Collaboration],  Phys. Rev. D
{\bf 89}, 092002 (2014).
\bibitem{cmdKLKS} R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B
{\bf 669}, 217 (2008).
\bibitem{sndKLKS} M. N. Achasov et al. [SND Collaboration], 
Zh. Eksp. Teor. Fiz. {\bf 103}, 831 (2006).
\bibitem{snd3pi} M. N. Achasov et al. [SND Collaboration], Phys.Rev. D
{\bf 68}, 052006 (2003).
\bibitem{cmd3pi} R. R. Akhmetshin et al. [CMD-2 Collaboration], Phys. Lett. B
{\bf 642}, 203 (2006).
\bibitem{babar3pi} B. Aubert et al. [BaBar Collaboration], Phys. Rev. D 
{\bf 70}, 072004 (2004).
\bibitem{sndetapipi} V. M. Aulchenko et al. [SND Collaboration], Phys. Rev. D
{\bf 91}, 052013 (2015).
\bibitem{babarkkpi} B. Aubert et al. [BaBar Collaboration],  Phys. Rev. D
{\bf 77}, 092002 (2008).
\bibitem{Babar4pic} B. Aubert et al. [BaBar Collaboration], Phys. Rev. D
{\bf 71}, 052001 (2005).
\bibitem{2pi2pi} R. R. Akhmetshin et al. [CMD-3 Collaboration], Phys. Lett. B
{\bf 768}, 345 (2017).
\bibitem{DM2pipi2pi0} D. Bisello et al. [DM2 Collaboration],  Nucl. Phys. Proc.
Suppl. {\bf 21}, 111 (1991).
\bibitem{cmd4pi} R. R. Akhmetshin et al. [CMD-2 Collaboration],  Phys. Lett. B
{\bf 466}, 392 (1999).
\bibitem{cmd3pieta} R. R. Akhmetshin et al. [CMD-3 Collaboration], 
Phys. Lett. B {\bf 773}, 150 (2017).
\bibitem{kkpipic} B. Aubert et al. [BaBar Collaboration], Phys. Rev. D
{\bf 76}, 012008 (2007).
\bibitem{Shemyakin} D. N. Shemyakin et al. [CMD-3 Collaboration], Phys. Lett. B
{\bf 756}, 153 (2016).
\bibitem{kck0pipi0} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D
{\bf 95}, 092005 (2017).
\bibitem{klks2pi0} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D
{\bf 95}, 052001 (2017).
\bibitem{Babar5pi} B. Aubert et al. [BaBar Collaboration], Phys. Rev. D
{\bf 76}, 092005 (2007).
\bibitem{pipi3pi0} J. P. Lees et al. [BaBar Collaboration], Phys. Rev. D
{\bf 98}, 112015 (2018).
\bibitem{Babar6pi} B. Aubert et al. [BaBar Collaboration], Phys. Rev. D
{\bf 73}, 052003 (2006).
\bibitem{cmd6pi} R. R. Akhmetshin et al. [CMD-3 Collaboration], 
Phys. Lett. B {\bf 723}, 82-89 (2013).
\bibitem{fadin} E.A. Kuraev and V,S. Fadin, Sov. J. Nucl. Phys. 
{\bf 41}, 466 (1985).
\bibitem{PartID} http://pdg.lbl.gov
\bibitem{HepMC1} http://hepmc.web.cern.ch/hepmc/
\bibitem{kedrnew} V. V. Anashin et al. [KEDR Collaboration], Phys. Lett. B
{\bf 770}, 174 (2017).
%\bibitem{rev} V. P. Druzhinin, S. I. Eidelman, S. I. Serednyakov, 
%E. P. Solodov, Rev. Mod. Phys. {\bf 83}, 1545 (2011).
\end{thebibliography}
\end{document}
