The new method of interference contributions accounting for inelastic scattering diagrams.
I.V.~Sharph
O.S.~Potiienko
M.A.~Deliyergiyev
K.K.~Merkotan
N.O.~Chudak
D.A.~Ptashynskyy
G.O.~Sokhrannyi
A.V.~Tykhonov
Yu.V.~Volkotrub
U.V.~Shabatura
V.D.~Rusov
A new method of interference contributions accounting is proposed. This method makes it possible to solve the problem of accounting the particles identity in the final state of the scattering process. This problem requires the calculation of a huge number of multidimensional integrals that express the scattering cross-section through the square modulus of the scattering amplitude. Our method is based on the physical features of the scattering amplitude. We show that the set of all interference contributions may be divided into subsets with the similar contributions. Therefore one does not need to calculate each of them separately. Within this approach it is possible to describe inelastic scattering processes with large number of secondary particles. The method was also verified within the $\phi^3$ model using the Laplace method for the calculation of multidimensional integrals in order to compute the inclusive rapidity cross-sections. A qualitative description of experimental data was obtained. An important result of this work is the explanation of peaks behavior in the inclusive rapidity distributions and their possible interference origin.
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\title{The new method of interference contributions accounting for inelastic scattering diagrams.}

\author{I.V.~Sharph, O.S.~Potiienko, M.A.~Deliyergiyev, K.K.~Merkotan,\\
        N.O.~Chudak, D.A.~Ptashynskyy, G.O.~Sokhrannyi, A.V.~Tykhonov,\\
        Yu.V.~Volkotrub, U.V.~Shabatura, V.D.~Rusov}%1
%\affil{Odessa National Polytechnic University, Ukraine}%институт 1
%\addres{Schevchenko av. 1, Odessa, 65044, Ukraine}




\begin{document}


\maketitle

\begin{center}
\textit{Department of Theoretical and Experimental Nuclear Physics,\\
        Odessa National Polytechnic University, Odessa, Ukraine}
\end{center}

\begin{abstract}

A new method of interference contributions accounting is proposed. This method makes it possible to solve the problem of accounting the particles identity in the final state of the scattering process. This problem requires the calculation of a huge number of multidimensional integrals that express the scattering cross-section through the square modulus of the scattering amplitude. Our method is based on the physical features of the scattering amplitude. We show that the set of all interference contributions may be divided into subsets with the similar contributions. Therefore one does not need to calculate each of them separately. Within this approach it is possible to describe inelastic scattering processes with large number of secondary particles. 
 The method was also verified within the $\phi^3$ model using the Laplace method for the calculation of multidimensional integrals in order to compute the inclusive rapidity cross-sections. A qualitative description of experimental data was obtained.

	An important result of this work is the explanation of peaks behavior in the inclusive rapidity distributions and their possible interference origin.
	
\end{abstract}

\section{Introduction}

There are two problems when considering an inelastic scattering processes: the first problem is  the calculation of a multidimensional integral (\ref{crosssection}); and the second problem  is the large number of such integrals. 

\begin{eqnarray}
\sigma_n=\frac{\left(2\pi\right)^4}{4n!I}\int\frac{d\vec{P}_3}{(2\pi)^32P_{30}}\frac{d\vec{P}_4}{(2\pi)^32P_{40}} \prod_{a = 1}^{n}\frac{d\vec{p}_a}{(2\pi)^32p_{a0}}\times \nonumber\\
\times\delta\left(P_{initial}-P_{final}\right)\vert A\left(P_1,P_2,P_3,P_4,p_1,p_2,\dots,p_n\right)\vert^2
\label{crosssection}
\end{eqnarray}

 The first problem may be solved using the Laplace method.
 
 Now let us consider the so called disconnected diagram (Fig. \ref{fig01}). In the language of diagrams: we need to connect the lines of secondary particles $p_1,p_2,...,n$ to the diagram vertices $1,2,...,n$ in all possible ways. For each way of connection we need to calculate the interference contribution i.e. the integral (\ref{crosssection}). There are $n!$ possible ways of such connections. It means that we need to calculate $n!$ multidimentional integrals. It is not a problem for a small number of secondary particles (e.g. for five or seven). However, for the process with a large number (20,50,60...) of secondary particles, it is impossible to calculate all interference contributions directly.
 
 
 \begin{figure}[htb]        
   \begin{center}
        \includegraphics[width=0.55\linewidth]{diagram.pdf}
    \end{center}
    \caption{Disconnected Feynman diagram.}
    \label{fig01}
  \end{figure} 
 
\begin{table}[htb]
\begin{center}
\begin{tabular}{| l | l |}
\hline
Secondary particles number & Number of interference contributions \\
\hline 
n = 5 & n! = 120 \\ 
\hline
n = 7 & n! = 5040 \\
\hline
n = 15 & n! = $13 \cdot 10^{11} $ \\
\hline
n = 60 & n! = $8.32 \cdot 10^{81}$ \\
\hline
\end{tabular}
\end{center}

\caption{The number of the interference contributions for the different secondary particles number.}
\end{table}		

Obviously, it is impossible to build all the permutations even for 30 particles nowadays, and thus to account all 30! possible interference contributions. Our method makes it possible to calculate the interference contributions for up to hundred secondary particles. 

  
 
  
 \section{Laplace method}
 
 The scattering amplitude $A\left(P_1,P_2,P_3,P_4,p_1,p_2,\dots,p_n\right)$ depends on the four-momenta of the secondary paritcles. It is convenient to introduce the $(3n+2)$-dimentional vector $X$, which includes all the necessary variables $y_1,y_2,\dots,y_n$, $p_{1x},p_{2x},\dots,p_{nx}$, $p_{1y},p_{2y},\dots,p_{ny}$ and $P_{ax}=(P_{3x}-P_{4x})/2,P_{ay}=(P_{3y}-P_{4y})/2$. And so let us write $A(\sqrt{s},X)$.


The scattering amplitude may be given in the following form:

 \begin{equation}
   A(\sqrt{s},X) = e^{lnA(\sqrt{s},X)}
   \label{explnform}
 \end{equation}
 
 As it was shown in \cite{ref1}, the scattering amplitude $A(\sqrt{s},X)$ has a maximum point $X_0$ at fixed energy $\sqrt{s}$, and by applying Taylor expansion near the maximum point $X_0$ up to second derivatives:

 \begin{equation}
   A(\sqrt{s},X) = A(\sqrt{s},X_0)e^{\frac{1}{2}D_{ab}(X_a-X_{0a})(X_b-X_{0b})}
 \end{equation}

\noindent where

 \begin{equation}
   D_{ab} = \left.\frac{\partial^2 lnA(\sqrt{s},X)}{\partial X_a \partial X_b}\right|_{X=X_0}
 \end{equation}
 
 The diagonalization of matrix $D_{ab}$ yields several one-dimentional integrals instead of a single multidimentional one.

 
\section{Definitions and important details}
First of all we need to introduce the definition of the "normal connection" or "normal order". Let us connect first line to the first vertex, second line to the second vertex and so on. From now on let us call it a \textit{"normal order"} or a \textit{"normal connection"}. Now we can consider all the possible connections as the permutations of the seconrady particles in normal order.

  \begin{figure}[htb]
    \begin{center}            
       \includegraphics[width=0.29\linewidth]{normal_connection.pdf}
       \hspace{40pt}
       \includegraphics[width=0.30\linewidth]{arithmetic.pdf}
    \end{center}
    %\caption{Different connections, but same maximum point $X_0$.}
    \label{normal}
    \caption{Normal connection of secondary particles (left), and secondary particles rapidities at the maxtimum point (right).}
  \end{figure}



It is important that the secondary particle rapidities make up the arithmetic progression at the maximum point of scattering amplitude (Fig. \ref{normal} right), while $y_{max} = - y_{min}$ and the central particle has a zero rapidity. 

 
  
\section{The essence of the new method}

The permutations of the secondary particles (different connections) at low energies lead to the same maximum point  of scattering amplitude (Fig. \ref{samepoint}), thus there are many similar contributions which are easy to calculate. 

  \begin{figure}[htb]
    \begin{center}            
       \includegraphics[width=0.23\linewidth]{connection1.pdf}
       \hspace{40pt}
       \includegraphics[width=0.23\linewidth]{connection2.pdf}
    \end{center}
    \caption{Low energies: different connections but the same maximum point.}
    \label{samepoint}
  \end{figure}

This is not true for the high energies. However, at high energies it is possible to combine the secondary particles into the groups in such a way that the permutations of the particles inside the groups do not change the maximum point. It means, that the rapidities of the particles from the same group are equal.

\vspace{10pt}

The problem of accounting of all $n!$ interference contributions may be solved in two steps:

 \begin{enumerate}
   \item{ The first step is the consideration of all possible permutations inside the groups - \textit{inner permutations}}
   \item{ The second step is the consideration of all possible ways (methods) of groups filling - \textit{external permutations}.}
 \end{enumerate}

 \begin{equation}
   A(\sqrt{s},P,p)= \sum_{external\;perm.}\sum_{inner\;perm.}a(\sqrt{s},P_3,P_4,p_1,p_2,...,p_n)
 \end{equation}

%Now we use Laplace method not to each contribution like (1), but to the partial summ of %interference contributions
% From now the Laplace method may be used not to the each interference contribution, but to the partial summ of one:
From now on the Laplace method may be applied to the partial sum of the interference contributions:

\begin{equation}
\sum_{inner\;perm.}a(\sqrt{s},P_3,P_4,p_1,p_2,...,p_n)
\label{partial}
\end{equation}

%At first we are trying to find the maximum point of partial summ (\ref{partial}), when the %rapidities of secondary particles from the same group are equal. 


All inner permutations may be accounted by the combinatorial coefficients. Each external permutation may be represented by $N \times N$-dimentional matrix.

\begin{equation}
  M_{ik}
\end{equation}

The matrix element $M_{ik}$ shows how many patricles are taken from the $i^{th}$ group of the normal order and then pushed to the $k^{th}$ group of the new permutation. Each of these contributions has a weighting factor $P_M$ :
\begin{equation}
P_M = \prod_{i=1}^{N}\prod_{j=1}^{k}C_{n_i-\sum_{l}^{j-1}M_{il}}^{M_{ij}}
\label{weightingfactor}
\end{equation} 
 
In this way the number of necessary calculations of the multidimentional integrals is reduced by finding the similar interference contributions and accounting all of them by combinatorial coefficients. So it becomes possible to calculate all the interference contributions for the processes with a large number (up to hundred) of the secondary particles.


\section{Results and Conclusions}

As a result of this method application, a qualitative description of inclusive rapidity distribution was achieved. 


%\begin{figure}[h]
%\begin{minipage}[h]{0.45\linewidth}
%\center{\includegraphics[width=1.0\linewidth]{inclExper.jpg} \\ a) Experimental data}
%\end{minipage}
%\hfill
%\begin{minipage}[h]{0.52\linewidth}
%\center{\includegraphics[width=1.0\linewidth]{rapidities.pdf} \\ b) Theory prediction }
%\end{minipage}
%\caption{Inclusive rapidity distribution.}
%\label{inclusive}
%\end{figure}

%\begin{figure}[h]
%\begin{minipage}[h]{0.43\linewidth}
%\center{\includegraphics[width=1.0\linewidth]{eta_both_2013.pdf} \\ a) Experimental data}
%\end{minipage}
%\hfill
%\begin{minipage}[h]{0.57\linewidth}
%\center{\includegraphics[width=1.0\linewidth]{rapidities.pdf} \\ b) Theory prediction }
%\end{minipage}
%\caption{Inclusive rapidity distribution. The experimental data are taken from \cite{ref2}}
%\label{inclusive}
%\end{figure}

\begin{figure}[h]
\begin{minipage}[h]{0.48\linewidth}
\center{\includegraphics[width=1.0\linewidth]{experimental.pdf} \\ a) Experimental data}
\end{minipage}
\hfill
\begin{minipage}[h]{0.52\linewidth}
\center{\includegraphics[width=1.0\linewidth]{rapidities2.pdf} \\ b) Theory prediction }
\end{minipage}
\caption{Inclusive rapidity distribution. The experimental data are taken from \cite{ref2}.}
\label{inclusive}
\end{figure}

It is important to note that there is a single peak for low energies (Fig.~\ref{inclusive}), which splits into two peaks more and more as energy grows.

\subsection{An explanation of peaks behaviour}

 Let us consider the so called cut diagram (Fig.~\ref{cut}). There are two types of lines connecting the secondary particle lines from the left side to the right side:
 
   \begin{itemize}
   \item{\textcolor{green}{\textit{Green (dotted) lines}} connect the particle lines from the central group on left side to the particles from the central group on the right side (Fig.~\ref{cut}). It is these lines that produce the maximum at zero point, since the particles from the central groups have zero rapidities.}
   \item{\textcolor{red}{\textit{Red lines}} connect the particle lines from/to the non-central groups (Fig.~\ref{cut}). The lines of this type shift the maximum from zero, because the particles in non-central groups have nonzero rapidities.}
  \end{itemize}
  
  

 \begin{figure}[htb]
    \begin{center}            
       \includegraphics[width=0.7\linewidth]{cut.pdf}
    \end{center}
    \caption{Cut diagram}
    \label{cut}
  \end{figure}
  
%But what happens with the energy $\sqrt{s}$  growth?
%\begin{itemize}
%\item{The possible number of secondary particles ($n$).}
%\item{Increases a difference of rapidities arithmetic progression.}
%\end{itemize}

There are two factors impact on the peaks behaviour with the energy $\sqrt{s}$ growth:
\begin{itemize}
\item{The possible number of the secondary particles ($n$) increases, and thus the number of the red and green lines increases.}
\item{A difference of the rapidities arithmetic progression increases.}
\end{itemize}



However, the number of red lines grows faster than the number of green lines as the secondary particles number increases. This fact provides the peak splitting. 



\begin{thebibliography}{9}

\bibitem{ref1}
I. Sharf, A. Tykhonov, G. Sokhrannyi, M. Deliyergiyev, N. Podolyan and V. Rusov, "On the Role of Longitudinal Momenta in High Energy Hadron-Hadron Scattering", \textit{Central Eur.J.Phys.}, Vol. 10, 2012, pp.858-887. \href{http://arxiv.org/abs/1110.4945v2}{arXiv:1110.4945 [hep-ph]}.

%\bibitem{ref2}
%K.A. Olive \textit{et al.} (Particle Data Group), \textit{Chin.~Phys.~C}, Vol.38, 2014, p.531.

\bibitem{ref2}
  Ajay Kumar Dash and Bedangadas Mohanty, Extrapolation of multiplicity distribution in {$p+p(\bar{p})$} collisions to LHC energies, \textit{Journal of Physics G: Nuclear and Particle Physics}, Vol.~37, 2010, pp.~025102.

%@article{0954-3899-37-2-025102,
%  author={Ajay Kumar Dash and Bedangadas Mohanty},
%  title={Extrapolation of multiplicity distribution in ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/toc_jpg327139ieqn1.gif] {p+p(\bar{p})} collisions to LHC energies},
%  journal={Journal of Physics G: Nuclear and Particle Physics},
%  volume={37},
%  number={2},
%  pages={025102},
%  url={http://stacks.iop.org/0954-3899/37/i=2/a=025102},
%  year={2010},
%  abstract={The multiplicity ( N ch ) and pseudorapidity distribution (d N ch /dη) of primary charged particles in p + p collisions at Large Hadron Collider (LHC) energies of ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn3.gif] {\sqrt{s} =} 10 and 14 TeV are obtained from extrapolation of existing measurements at lower ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn4.gif] {\sqrt{s}} . These distributions are then compared to calculations from PYTHIA and PHOJET models. The existing ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn5.gif] {\sqrt{s}} measurements are unable to distinguish between a logarithmic and power law dependence of the average charged particle multiplicity ( ##IMG## [http://ej.iop.org/icons/Entities/lang.gif] {lang} N ch ##IMG## [http://ej.iop.org/icons/Entities/rang.gif] {rang} ) on ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn6.gif] {\sqrt{s}} , and their extrapolation to energies accessible at LHC give very different values. Assuming a reasonably good description of inclusive charged particle multiplicity distributions by negative binomial distribution (NBD) at lower ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn7.gif] {\sqrt{s}} to hold for LHC energies, we observe that the logarithmic ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn8.gif] {\sqrt{s}} dependences of ##IMG## [http://ej.iop.org/icons/Entities/lang.gif] {lang} N ch ##IMG## [http://ej.iop.org/icons/Entities/rang.gif] {rang} are favored by the models at midrapidity. The d N ch /dη versus η distributions for the existing measurements are found to be reasonably well described by a function with three parameters which accounts for the basic features of the distribution, height at midrapidity, central rapidity plateau and the higher rapidity fall-off. Extrapolation of these parameters as a function of ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn9.gif] {\sqrt{s}} is used to predict the pseudorapidity distributions of charged particles at LHC energies. d N ch /dη calculations from PYTHIA and PHOJET models are found to be lower compared to those obtained from the extrapolated d N ch /dη versus η distributions for a broad η range.}
%}

\end{thebibliography}



\end{document}


1509.04329.extracted_bib
{ref1 I. Sharf, A. Tykhonov, G. Sokhrannyi, M. Deliyergiyev, N. Podolyan and V. Rusov, "On the Role of Longitudinal Momenta in High Energy Hadron-Hadron Scattering", Central Eur.J.Phys., Vol. 10, 2012, pp.858-887. \href{http://arxiv.org/abs/1110.4945v2{arXiv:1110.4945 [hep-ph]. %
1509.04329.extracted_bib
{ref2 %K.A. Olive et al. (Particle Data Group), Chin.~Phys.~C, Vol.38, 2014, p.531.
1509.04329.extracted_bib
{ref2 Ajay Kumar Dash and Bedangadas Mohanty, Extrapolation of multiplicity distribution in {$p+p(\bar{p)$ collisions to LHC energies, Journal of Physics G: Nuclear and Particle Physics, Vol.~37, 2010, pp.~025102. %@article{0954-3899-37-2-025102, % author={Ajay Kumar Dash and Bedangadas Mohanty, % title={Extrapolation of multiplicity distribution in ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/toc_jpg327139ieqn1.gif] {p+p(\bar{p) collisions to LHC energies, % journal={Journal of Physics G: Nuclear and Particle Physics, % volume={37, % number={2, % pages={025102, % url={http://stacks.iop.org/0954-3899/37/i=2/a=025102, % year={2010, % abstract={The multiplicity ( N ch ) and pseudorapidity distribution (d N ch /dη) of primary charged particles in p + p collisions at Large Hadron Collider (LHC) energies of ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn3.gif] {\sqrt{s = 10 and 14 TeV are obtained from extrapolation of existing measurements at lower ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn4.gif] {\sqrt{s . These distributions are then compared to calculations from PYTHIA and PHOJET models. The existing ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn5.gif] {\sqrt{s measurements are unable to distinguish between a logarithmic and power law dependence of the average charged particle multiplicity ( ##IMG## [http://ej.iop.org/icons/Entities/lang.gif] {lang N ch ##IMG## [http://ej.iop.org/icons/Entities/rang.gif] {rang ) on ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn6.gif] {\sqrt{s , and their extrapolation to energies accessible at LHC give very different values. Assuming a reasonably good description of inclusive charged particle multiplicity distributions by negative binomial distribution (NBD) at lower ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn7.gif] {\sqrt{s to hold for LHC energies, we observe that the logarithmic ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn8.gif] {\sqrt{s dependences of ##IMG## [http://ej.iop.org/icons/Entities/lang.gif] {lang N ch ##IMG## [http://ej.iop.org/icons/Entities/rang.gif] {rang are favored by the models at midrapidity. The d N ch /dη versus η distributions for the existing measurements are found to be reasonably well described by a function with three parameters which accounts for the basic features of the distribution, height at midrapidity, central rapidity plateau and the higher rapidity fall-off. Extrapolation of these parameters as a function of ##IMG## [http://ej.iop.org/images/0954-3899/37/2/025102/jpg327139ieqn9.gif] {\sqrt{s is used to predict the pseudorapidity distributions of charged particles at LHC energies. d N ch /dη calculations from PYTHIA and PHOJET models are found to be lower compared to those obtained from the extrapolated d N ch /dη versus η distributions for a broad η range. %