source: trunk/documents/UserDoc/DocBookUsersGuides/PhysicsReferenceManual/latex/hadronic/theory_driven/PartonString/LongitudinalStringExcitation.tex

\section{Longitudinal string excitation}

\subsection{Hadron--nucleon inelastic collision}

\hspace{1.0em}Let us consider collision of two hadrons with their c. m. momenta 
$P_1 = \{E^{+}_1,m^2_1/E^{+}_1,{\bf 0}\}$ and $P_2 =
\{E^{-}_2,m^2_2/E^{-}_2,{\bf 0}\}$, where the light-cone variables
$E^{\pm}_{1,2} = E_{1,2} \pm P_{z1,2}$ are defined through hadron
energies $E_{1,2}=\sqrt{m^2_{1,2} + P^2_{z1,2}}$, hadron longitudinal
momenta $P_{z1,2}$ and hadron masses $m_{1,2}$, respectively. Two
hadrons collide by two partons with momenta $p_1 = \{x^{+}E^{+}_1,0,{\bf
0}\}$ and $p_2 = \{0, x^{-}E^{-}_2,{\bf 0}\}$, respectively.

\subsection{The diffractive string excitation} 

In the diffractive string excitation (the Fritiof approach \cite{FRITIOF87})
 only momentum can be transferred:
\begin{equation}
\label{LSE1}
\begin{array}{cc}
P^{\prime}_1 = P_1 + q\\
P^{\prime}_2 = P_2 -q,
\end{array}
\end{equation}
where 
\begin{equation}
\label{LSE2}q=\{-q^2_t/(x^{-}E^{-}_2),q^2_t/(x^{+}E^{+}_1),\bf q_t \}
\end{equation}
 is parton 
momentum transferred and ${\bf q_t}$ is its transverse component.
We use the Fritiof approach to simulate the diffractive excitation of 
particles.
 
\subsection{The string excitation by parton exchange}

\hspace{1.0em}For this case the parton exchange (rearrangement) and
the momentum exchange are allowed \cite{QGSM82},\cite{DPM94},\cite{Am86}:
\begin{equation}
\label{LSE3}
\begin{array}{cc}
P^{\prime}_1 = P_1 - p_1 + p_2 + q \\
P^{\prime}_2 = P_2 + p_1 - p_2 - q,
\end{array}
\end{equation}
where $q= \{0,0, {\bf q_t}\}$ is parton momentum transferred, i. e. only
its transverse components ${\bf q_t} = 0$ is taken into account.

\subsection{Transverse momentum sampling}

\hspace{1.0em}The transverse component of the parton momentum
transferred is generated according to probability
\begin{equation}
\label{LSE4}P({\bf q_t})d{\bf q_t} = 
\sqrt{\frac{a}{\pi}} \exp{(-aq^2_t)}d{\bf q_t},
\end{equation}
where  parameter $a = 0.6$ GeV$^{-2}$.

\subsection{Sampling x-plus and x-minus}

Light cone parton quantities $x^{+}$ and $x^{-}$ are generated
independently and according to distribution:
\begin{equation}
\label{LSE5} u(x) \sim x^{\alpha}(1 - x)^{\beta},
\end{equation}
where $x=x^{+}$ or $x=x^{-}$.
Parameters $\alpha =-1$ and $\beta = 0$ are chosen for the FRITIOF approach 
\cite{FRITIOF87}. In the case of the QGSM approach \cite{Am86} $\alpha = -0.5$ 
and $\beta = 1.5$ or $\beta = 2.5$.  Masses of the excited strings
should satisfy the kinematical constraints:
\begin{equation}
\label{LSE6} P^{\prime +}_1 P^{\prime -}_1 \geq m^2_{h1} + q^2_t
\end{equation}
and
\begin{equation}
\label{LSE7} P^{\prime +}_2 P^{\prime -}_2 \geq m^2_{h2} + q^2_t,
\end{equation}
where hadronic masses $m_{h1}$ and $m_{h2}$ (model parameters) are
defined by string quark contents.  Thus, the random selection of the
values $x^{+}$ and $x^{-}$ is limited by above constraints.

\subsection{The diffractive string excitation} 

\hspace{1.0em}In the diffractive string excitation (the 
FRITIOF approach \cite{FRITIOF87}) for each
inelastic hadron--nucleon collision we have to select
randomly the transverse momentum transferred ${\bf q_t}$ (in accordance
with the probability given by Eq. ($\ref{LSE4}$)) and select randomly
the values of $x^{\pm}$ (in accordance with distribution defined by
Eq. ($\ref{LSE5}$)). Then we have to calculate the parton momentum
transferred $q$ using Eq. ($\ref{LSE2}$) and update scattered hadron
and nucleon or scatterred nucleon and nucleon momenta using
Eq. ($\ref{LSE3}$). For each collision we have to check the constraints
($\ref{LSE6}$) and ($\ref{LSE7}$), which can be written more
explicitly:
\begin{equation}
\label{LSE8} [E_1^{+} -\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_1^2}{E^{+}_1} + 
\frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t
\end{equation}
and
\begin{equation}
\label{LSE9} [E_2^{-} +\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_2^2}{E^{-}_2} - 
\frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t.
\end{equation}

\subsection{The string excitation by parton rearrangement} 

\hspace{1.0em}In this approach \cite{Am86} strings (as result of parton
rearrangement) should be spanned not only between valence quarks of
colliding hadrons, but also between valence and sea quarks and between
sea quarks.  The each participant hadron or nucleon should be splitted
into set of partons: valence quark and antiquark for meson or valence
quark (antiquark) and diquark (antidiquark) for baryon (antibaryon) and
additionaly the $(n-1)$ sea quark-antiquark pairs (their flavours are
selected according to probability ratios $ u:d:s = 1:1:0.35$), if hadron
or nucleon is participating in the $n$ inelastic collisions.  Thus for
each participant hadron or nucleon we have to generate a set of light
cone variables $x_{2n}$, where $x_{2n}=x^{+}_{2n}$ or
$x_{2n}=x^{-}_{2n}$ according to distribution:
\begin{equation}
\label{LS10} f^{h}(x_1,x_2,...,x_{2n})=f_{0}\prod_{i=1}^{2n}u^h_{q_i}(x_i)
\delta{(1-\sum_{i=1}^{2n}x_i)},
\end{equation}
where $f_0$ is the normalization constant.
Here, the quark structure functions $u_{q_i}^h(x_i)$ for valence quark 
(antiquark) $q_v$, 
sea quark and antiquark $q_s$ and valence diquark (antidiquark) $qq$ are:
\begin{equation}
\label{LS11}
u^h_{q_v}(x_v)=x_v^{\alpha_v},\ u^h_{q_s}(x_s)=x_s^{\alpha_s},\ u^h_{qq}(x_{qq})
=x_{qq}^{\beta_{qq}},
\end{equation}
where $\alpha_v = -0.5$ and $\alpha_s = -0.5$ \cite{QGSM82} 
 for the non-strange quarks (antiquarks) and $\alpha_v =
0$ and $\alpha_s = 0$ for strange quarks (antiquarks), $\beta_{uu} =
1.5$ and $\beta_{ud} = 2.5$ for proton (antiproton) and $\beta_{dd} =
1.5$ and $\beta_{ud} = 2.5$ for neutron (antineutron).  Usualy $x_i$ are
selected between $x^{min}_i \leq x_i \leq 1$, where model parameter
$x^{min}$ is a function of initial energy, to prevent from production of
strings with low masses (less than hadron masses), when whole selection
procedure should be repeated.  Then the transverse momenta of partons
${\bf q_{it}}$ are generated according to the Gaussian probability
Eq. ($\ref{LSE4}$) with $a = 1/4\Lambda(s)$ and under the constraint: $\sum_{i=1}^{2n}{\bf
q_{it}}=0$. The partons are considered as the off-shell partons,
i. e. $m^2_i \neq 0$.