Longitudinal string excitation

Hadron-nucleon inelastic collision

Let us consider collision of two hadrons with their c. m. momenta $P_1 = \{E^{+}_1,m^2_1/E^{+}_1,{\bf0}\}$ and $P_2 = \{E^{-}_2,m^2_2/E^{-}_2,{\bf0}\}$, 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 $p_2 = \{0, x^{-}E^{-}_2,{\bf0}\}$, respectively.

The diffractive string excitation

In the diffractive string excitation (the Fritiof approach [9]) only momentum can be transferred:

$$\begin{array}{cc} P^{\prime}_1 = P_1 + q\\ P^{\prime}_2 = P_2 -q, \end{array}$$ (22.18)

where

$$q=\{-q^2_t/(x^{-}E^{-}_2),q^2_t/(x^{+}E^{+}_1),\bf q_t \}$$ (22.19)

is parton momentum transferred and ${\bf q_t}$ is its transverse component. We use the Fritiof approach to simulate the diffractive excitation of particles.

The string excitation by parton exchange

For this case the parton exchange (rearrangement) and the momentum exchange are allowed [10],[11],[7]:

$$\begin{array}{cc} P^{\prime}_1 = P_1 - p_1 + p_2 + q \\ P^{\prime}_2 = P_2 + p_1 - p_2 - q, \end{array}$$ (22.20)

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.

Transverse momentum sampling

The transverse component of the parton momentum transferred is generated according to probability

$$P({\bf q_t})d{\bf q_t} = \sqrt{\frac{a}{\pi}} \exp{(-aq^2_t)}d{\bf q_t},$$ (22.21)

where parameter $a = 0.6$ GeV$^{-2}$.

Sampling x-plus and x-minus

Light cone parton quantities $x^{+}$ and $x^{-}$ are generated independently and according to distribution:

$$u(x) \sim x^{\alpha}(1 - x)^{\beta},$$ (22.22)

where $x=x^{+}$ or $x=x^{-}$. Parameters $\alpha =-1$ and $\beta = 0$ are chosen for the FRITIOF approach [9]. In the case of the QGSM approach [7] $\alpha = -0.5$ and $\beta = 1.5$ or $\beta = 2.5$. Masses of the excited strings should satisfy the kinematical constraints:

$$P^{\prime +}_1 P^{\prime -}_1 \geq m^2_{h1} + q^2_t$$ (22.23)

and

$$P^{\prime +}_2 P^{\prime -}_2 \geq m^2_{h2} + q^2_t,$$ (22.24)

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.

The diffractive string excitation

In the diffractive string excitation (the FRITIOF approach [9]) 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:

$$[E_1^{+} -\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_1^2}{E^{+}_1} + \frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t$$ (22.25)

and

$$[E_2^{-} +\frac{q^2_t}{x^{-}E^{-}_2}][\frac{m_2^2}{E^{-}_2} - \frac{q^2_t}{x^{+}E^{+}_1}]\geq m^2_{h1} + q^2_t.$$ (22.26)

The string excitation by parton rearrangement

In this approach [7] 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:

$$f^{h}(x_1,x_2,...,x_{2n})=f_{0}\prod_{i=1}^{2n}u^h_{q_i}(x_i) \delta{(1-\sum_{i=1}^{2n}x_i)},$$ (22.27)

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:

$$u^h_{q_v}(x_v)=x_v^{\alpha_v},\ u^h_{q_s}(x_s)=x_s^{\alpha_s},\ u^h_{qq}(x_{qq}) =x_{qq}^{\beta_{qq}},$$ (22.28)

where $\alpha_v = -0.5$ and $\alpha_s = -0.5$ [10] 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. $m^2_i \neq 0$.