1 | %\documentclass[12pt,a4paper,oneside]{book} |
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4 | %%% \usepackage[dvips]{epsfig} |
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5 | %\title{Physics Reference Manual} |
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6 | %\pagestyle{plain} |
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7 | %\begin{document} |
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8 | %{ |
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9 | %\maketitle |
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10 | %\pagestyle {empty} |
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11 | %\setcounter{page}{-10} |
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12 | %\tableofcontents |
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14 | %\pagestyle {empty} |
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15 | %} |
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16 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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17 | |
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18 | \chapter{Chiral Invariant Phase Space Decay.} |
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19 | % \textheight 8.75in |
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20 | % \textwidth 6.5in |
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21 | % \parskip 1.45ex |
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22 | |
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23 | \newtheorem{theorem}{Theorem} |
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24 | \newtheorem{acknowledgement}[theorem]{Acknowledgement} |
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25 | \newtheorem{algorithm}[theorem]{Algorithm} |
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26 | \newtheorem{axiom}[theorem]{Axiom} |
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27 | \newtheorem{claim}[theorem]{Claim} |
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28 | \newtheorem{conclusion}[theorem]{Conclusion} |
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29 | \newtheorem{condition}[theorem]{Condition} |
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30 | \newtheorem{conjecture}[theorem]{Conjecture} |
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31 | \newtheorem{corollary}[theorem]{Corollary} |
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32 | \newtheorem{criterion}[theorem]{Criterion} |
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33 | \newtheorem{definition}[theorem]{Definition} |
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34 | \newtheorem{example}[theorem]{Example} |
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35 | \newtheorem{exercise}[theorem]{Exercise} |
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36 | \newtheorem{lemma}[theorem]{Lemma} |
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37 | \newtheorem{notation}[theorem]{Notation} |
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38 | \newtheorem{problem}[theorem]{Problem} |
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39 | \newtheorem{proposition}[theorem]{Proposition} |
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40 | \newtheorem{remark}[theorem]{Remark} |
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41 | \newtheorem{solution}[theorem]{Solution} |
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42 | \newtheorem{summary}[theorem]{Summary} |
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43 | |
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44 | % \title{Manual for the CHIPS event generator in GEANT4} |
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45 | %\author{M.V.Kossov} |
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46 | %\address{Mikhail.Kossov@itep.ru, Mikhail.Kossov@cern.ch, kossov@jlab.org,\\ |
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47 | %kossov@post.kek.jp} |
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48 | % \date{\today} |
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49 | % \maketitle |
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50 | |
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51 | \section{Introduction} |
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52 | |
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53 | \noindent \qquad |
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54 | The CHIPS computer code is a quark-level event generator for the |
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55 | fragmentation of hadronic systems into hadrons. In contrast to other parton |
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56 | models \cite{Parton_Models} CHIPS is nonperturbative and |
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57 | three-dimensional. It is based on the Chiral Invariant Phase Space |
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58 | (ChIPS) model \cite{CHIPS1,CHIPS2,CHIPS3} which employs a |
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59 | 3D quark-level SU(3) approach. Thus Chiral Invariant Phase Space refers |
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60 | to the phase space of massless partons and hence only light (u, d, s) |
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61 | quarks can be considered. The c, b, and t quarks are not implemented |
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62 | in the model directly, while they can be created in the model as a |
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63 | result of the gluon-gluon or photo-gluon fusion. The main parameter of |
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64 | the CHIPS model is the critical temperature $T_c\approx 200~MeV$. The |
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65 | probability of finding a quark with energy $E$ drops with the energy |
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66 | approximately as $e^{-E/T}$, which is why the heavy flavors of quarks |
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67 | are suppressed in the Chiral Invariant Phase Space. The s quarks, |
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68 | which have masses less then the critical temperature, have an |
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69 | effective suppression factor in the model. |
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70 | |
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71 | The critical temperature $T_c$ defines the number of 3D partons in |
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72 | the hadronic system with total energy $W$. If masses of all partons |
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73 | are zero then the number of partons can be found from the equation |
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74 | $W^2=4T_c^2(n-1)n$. The mean squared total energy can be calculated |
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75 | for any ``parton'' mass (partons are usually massless). The |
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76 | corresponding formula can be found in \cite{hadronMasses}. In this |
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77 | treatment the masses of light hadrons are fitted better than by the |
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78 | chiral bag model of hadrons~\cite{Chiral_Bag} with the same number of |
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79 | parameters. In both models any hadron consists of a few quark-partons, |
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80 | but in the CHIPS model the critical temperature defines the mass of |
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81 | the hadron, consisting of $N$ quark-partons, while in the bag |
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82 | model the hadronic mass is defined by the balance between the |
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83 | quark-parton internal pressure (which according to the uncertainty |
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84 | principle increases when the radius of the ``bag'' decreases) and the |
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85 | external pressure ($B$) of the nonperturbative vacuum, which has |
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86 | negative energy density. |
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87 | |
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88 | In CHIPS the interactions between hadrons are defined by the Isgur |
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89 | quark-exchange diagrams, and the decay of excited hadronic |
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90 | systems in vacuum is treated as the fusion of quark-antiquark or |
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91 | quark-diquark partons. An important feature of the model is the |
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92 | homogeneous distribution of asymptotically free quark-partons over the |
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93 | invariant phase space, as applied to the fragmentation of various |
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94 | types of excited hadronic systems. In this sense the CHIPS model may |
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95 | be considered as a generalization of the well-known hadronic phase |
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96 | space distribution \cite{GENBOD} approach, but it generates not only |
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97 | angular and momentum distributions for a given set of hadrons, but |
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98 | also the multiplicity distributions for different kinds of hadrons, |
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99 | which is defined by the multistep energy dissipation (decay) process. |
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100 | |
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101 | The CHIPS event generator may be applied to nucleon excitations, |
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102 | hadronic systems produced in $e^{+}e^{-}$ and $p\bar p$ annihilation, |
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103 | and high energy nuclear excitations, among others. Despite its quark |
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104 | nature, the nonperturbative CHIPS model can also be used successfully |
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105 | at very low energies. It is valid for photon and hadron projectiles |
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106 | and for hadron and nuclear targets. Exclusive event generation models |
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107 | multiple hadron production, conserving energy, momentum, and other |
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108 | quantum numbers. This generally results in a good description of |
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109 | particle multiplicities, inclusive spectra, and kinematic correlations |
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110 | in multihadron fragmentation processes. Thus, it is possible to use |
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111 | the CHIPS event generator in exclusive modeling of hadron cascades in |
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112 | materials. |
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113 | |
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114 | In the CHIPS model, the result of a hadronic or nuclear interaction is |
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115 | the creation of a quasmon which is essentially an intermediate state |
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116 | of excited hadronic matter. When the interaction occurs in vacuum the |
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117 | quasmon can dissipate energy by radiating particles according to the |
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118 | quark fusion mechanism~\cite{CHIPS1} described in section \ref{annil}. |
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119 | When the interaction occurs in nuclear matter, the energy dissipation |
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120 | of a quasmon can be the result of quark exchange with surrounding |
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121 | nucleons or clusters of nucleons \cite{CHIPS2} (section \ref{picap}), |
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122 | in addition to the vacuum quark fusion mechanism. |
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123 | |
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124 | In this sense the CHIPS model can be a successful competitor of the |
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125 | cascade models, because it does not break the projectile, instead it |
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126 | captures it, creating a quasmon, and then decays the quasmon in |
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127 | nuclear matter. The perturbative mechanisms in deep inelastic |
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128 | scattering are in some sense similar to the cascade calculations, |
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129 | while the parton splitting functions are used instead of |
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130 | interactions. The nonperturbative CHIPS approach is making a ``short |
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131 | cut'' for the perturbative calculations too. Similar to the time-like |
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132 | $s=W^2$ evolution of the number of partons in the nonperturbative |
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133 | chiral phase space (mentioned above) the space-like $Q^2$ evolution of |
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134 | the number of partons is given by $N(Q^2)=n_V+\frac{1}{2\alpha_s(Q^2)}$, |
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135 | where $n_V$ is the number of valence quark-partons. The running |
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136 | $\alpha_s(Q^2)$ value is calculated in CHIPS as |
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137 | $\alpha_s(Q^2)=\frac{4\pi}{\beta_0ln(1+Q^2/T_c^2)}$, where |
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138 | $\beta_0^{n_f=3)=9}$. In other words, the critical temperature $T_c$ |
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139 | plays the role of $\Lambda_QCD$ and still cuts out heavy flavors of |
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140 | quark-partons and high orders of the QCD calculation (NLO, NNLO, |
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141 | N$^3$LO, etc.), substituting for them the effective LO ``short cut''. |
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142 | This simple approximation of $\alpha_s$ fits all the present measurements |
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143 | of this value (Fig.~\ref{alphas}). |
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144 | It is very important that |
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145 | $\alpha_s$ is defined in CHIPS for any $Q^2$, and that the number of partons |
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146 | at $Q^2=0$ converges to the number of valence quarks. |
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147 | |
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148 | \begin{figure} |
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149 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps, height=3.5in, width=4.5in}} |
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150 | % \resizebox{1.00\textwidth}{!} |
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151 | %{ |
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152 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/alpha.eps} |
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153 | %\includegraphics[angle=0,scale=0.6]{plots/alpha.eps} |
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154 | %} |
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155 | \caption{The CHIPS fit of the $\alpha_s$ measurements.} |
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156 | \label{alphas} |
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157 | \end{figure} |
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158 | |
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159 | The effective $\alpha_s$ is defined for all $Q^2$, but at $Q^2=0$ it |
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160 | is infinite. In other words at $Q^2=0$ the number of the virtual |
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161 | interacting partons goes to infinity. This means that on the boundary |
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162 | between perturbative and non-perturbative vacuums a virtual |
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163 | ``thermostate'' of gluons with an effective temperature $T_c$ |
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164 | exists. This ``virtual thermostate'' defines the phase space |
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165 | distribution of partons, and the ``thermalization'' can happen very |
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166 | quickly. On the other hand, the CHIPS nonperturbative approach can be used |
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167 | below $Q^2~=~1~GeV^2$. This was done for the neutrino-nuclear |
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168 | interactions (section \ref{numunuc}). |
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169 | |
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170 | \section{Fundamental Concepts} |
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171 | |
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172 | The CHIPS model is an attempt to use a set of simple rules which govern |
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173 | microscopic quark-level behavior to model macroscopic hadronic systems with |
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174 | a large number of degrees of freedom. The invariant phase space distribution |
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175 | as a paradigm of thermalized chaos is applied to quarks, and simple |
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176 | kinematic mechanisms are used to model the hadronization of quarks into |
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177 | hadrons. Along with relativistic kinematics and the conservation of quantum |
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178 | numbers, the following concepts are used: |
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179 | |
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180 | \begin{itemize} |
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181 | \item {\bf Quasmon:} in the CHIPS model, a quasmon is any excited hadronic |
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182 | system; it can be viewed as a continuous spectrum of a generalized |
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183 | hadron. At the constituent level, a quasmon may be thought of as a |
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184 | bubble of quark-parton plasma in which the quarks are massless and the |
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185 | quark-partons in the quasmon are homogeneously distributed over the |
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186 | invariant phase space. It may also be considered as a bubble of the |
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187 | three-dimensional Feynman-Wilson \cite{Feynman-Wilson} parton gas. The |
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188 | traditional hadron is a particle defined by quantum numbers and a |
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189 | fixed mass or a mass with a width. The quark content of the hadron is |
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190 | a secondary concept constrained by the quantum numbers. The quasmon, |
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191 | however, is defined by its quark content and its mass, and the concept |
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192 | of a well defined particle with quantum numbers (a discrete spectrum) |
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193 | is of secondary importance. A given quasmon hadronic state with fixed |
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194 | mass and quark content can be considered as a superposition of |
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195 | traditional hadrons, with the quark content of the superimposed |
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196 | hadrons being the same as the quark content of the quasmon. |
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197 | |
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198 | \item {\bf Quark fusion:} the quark fusion hypothesis determines the rules |
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199 | of final state hadron production, with energy spectra reflecting the |
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200 | momentum distribution of the quarks in the system. Fusion occurs when a |
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201 | quark-parton in a quasmon joins with another quark-parton from the same |
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202 | quasmon and forms a new white hadron, which can be radiated. If a |
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203 | neighboring nucleon (or the nuclear cluster) is present, quark-partons |
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204 | may also be exchanged between the quasmon and the neighboring nucleon |
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205 | (cluster). The kinematic condition applied to these mechanisms is that |
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206 | the resulting hadrons are produced on their mass shells. The model |
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207 | assumes that the u, d and s quarks are effectively massless, which |
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208 | allows the integrals of the hadronization process to be done easily |
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209 | and the modeling decay algorithm to be accelerated. The quark mass is |
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210 | taken into account indirectly in the masses of outgoing hadrons. The |
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211 | type of the outgoing hadron is selected using combinatoric and |
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212 | kinematic factors consistent with conservation laws. In the present |
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213 | version of CHIPS all mesons with three-digit PDG Monte Carlo codes |
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214 | \cite{CH.PDG} up to spin $4$, and all baryons with four-digit PDG |
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215 | codes up to spin $\frac{7}{2}$ are implemented. |
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216 | |
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217 | \item {\bf Critical temperature} the only non-kinematic concept of the model |
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218 | is the hypothesis of the critical temperature of the quasmon. This has a |
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219 | 40-year history, starting with Ref.~\cite{Hagedorn} and is based on the |
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220 | experimental observation of regularities in the inclusive spectra of hadrons |
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221 | produced in different reactions at high energies. Qualitatively, the |
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222 | hypothesis of a critical temperature assumes that the quark-gluon hadronic |
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223 | system (quasmon) cannot be heated above a certain temperature. Adding more |
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224 | energy to the hadronic system increases only the number of constituent |
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225 | quark-partons while the temperature remains constant. The critical |
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226 | temperature is the principal parameter of the model and is used to |
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227 | calculate the number of quark-partons in a quasmon. In an infinite |
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228 | thermalized system, for example, the mean energy of partons is $2T$ |
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229 | per particle, the same as for the dark body radiation. |
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230 | |
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231 | \end{itemize} |
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232 | |
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233 | \section{Code Development} |
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234 | |
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235 | Because the CHIPS event generator was originally developed only for final |
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236 | state hadronic fragmentation, the initial interaction of projectiles with |
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237 | targets requires further development. Hence, the first applications of |
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238 | CHIPS described interactions at rest, for which the interaction cross |
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239 | section is not important \cite{CHIPS1}, \cite{CHIPS2}, and low energy |
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240 | photonuclear reactions \cite{CHIPS3}, for which the interaction cross |
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241 | section can be calculated easily \cite{photNuc}. With modification of |
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242 | the first interaction algorithm the CHIPS event generator can be used |
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243 | for all kinds of hadronic interaction. The Geant4 String Model |
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244 | interface to the CHIPS generator \cite{GEANT4}, \cite{MC2000} also |
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245 | makes it possible to use the CHIPS code for nuclear fragmentation at |
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246 | extremely high energies. |
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247 | |
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248 | In the first published versions of the CHIPS event generator the class |
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249 | {\tt G4Quasmon} was the head of the model and all initial interactions |
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250 | were hidden in its constructor. More complicated applications of the |
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251 | model such as anti-proton nuclear capture at rest and the Geant4 |
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252 | String Model interface to CHIPS led to the multi-quasmon version of |
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253 | the model. This required a change in the structure of the CHIPS event |
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254 | generator classes. In the case of at-rest anti-proton annihilation in |
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255 | a nucleus, for example, the first interaction occurs on the nuclear |
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256 | periphery. After this initial interaction, a fraction (defined by a |
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257 | special parameter of the model) of the secondary mesons independently |
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258 | penetrate the nucleus. Each of these mesons can create a separate |
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259 | quasmon in the interior of the nucleus. In this case the class {\tt |
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260 | G4Quasmon} can no longer be the head of the model. A new head class, |
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261 | {\tt G4QEnvironment}, was developed which can adopt a vector of |
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262 | projectile hadrons ({\tt G4QHadronVector}) and create a vector of |
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263 | quasmons, {\tt G4QuasmonVector}. All newly created quasmons then begin |
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264 | the energy dissipation process in parallel in the same nucleus. The |
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265 | {\tt G4QEnvironment} instance can be used both for vacuum and for nuclear |
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266 | matter. If {\tt G4QEnvironment} is created in vacuum, it is practically |
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267 | identical to the {\tt G4Quasmon} class, because in this case only one |
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268 | instance of {\tt G4Quasmon} is allowed. This leaves the model unchanged |
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269 | for hadronic interactions. |
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270 | |
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271 | The convention adopted for the CHIPS model requires all its class names |
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272 | to use the prefix {\tt G4Q} in order to distinguish them from other Geant4 |
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273 | classes, most of which use the {\tt G4} prefix. The intent is that the |
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274 | {\tt G4Q} prefix will not be used by other Geant4 projects. |
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275 | |
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276 | \section{Nucleon-Antinucleon Annihilation at Rest} \label{annil} |
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277 | |
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278 | In order to generate hadron spectra from the annihilation of a proton |
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279 | with an anti-proton at rest, the number of partons in the system must be |
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280 | found. For a finite system of $N$ partons with a total center-of-mass energy |
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281 | $M$, the invariant phase space integral, $\Phi_N$, is proportional to |
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282 | $M^{2N-4}$. According to the dimensional counting rule, $2N$ comes from |
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283 | $\prod\limits_{i=1}^{N}\frac{d^{3}p_{i}}{E_{i}}$, and $4$ comes from |
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284 | the energy and momentum conservation function, $\delta ^{4}($\b{P}$-\sum |
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285 | $\b{p}$_{i})$. At a temperature $T$ the statistical density of states is |
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286 | proportional to $e^{-\frac{M}{T}}$ so that the probability to find a system |
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287 | of $N$ quark-partons in a state with mass $M$ is $dW \propto |
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288 | M^{2N-4}e^{-\frac{M}{T}}dM$. For this kind of probability distribution the |
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289 | mean value of $M^{2}$ is |
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290 | \begin{equation} |
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291 | <M^{2}>=4N(N-1)\cdot T^{2}. \label{temperature} |
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292 | \end{equation} |
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293 | When $N$ goes to infinity one obtains for massless particles the |
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294 | well-known $<M>\equiv \sqrt{<M^{2}>}=2NT$ result. |
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295 | |
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296 | After a nucleon absorbs an incident quark-parton, such as a real or |
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297 | virtual photon, for example, the newly formed quasmon has a total of $N$ |
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298 | quark-partons, where $N$ is determined by Eq. \ref{temperature}. |
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299 | Choosing one of these quark-partons with energy $k$ in the center of mass |
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300 | system (CMS) of $N$ partons, the spectrum of the remaining $N-1$ |
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301 | quark-partons is given by |
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302 | \begin{equation} |
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303 | \frac{dW}{kdk} \propto (M_{N-1})^{2N-6}, |
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304 | \end{equation} |
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305 | where $M_{N-1}$ is the effective mass of the $N-1$ quark-partons. |
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306 | This result was obtained by applying the above phase-space relation |
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307 | ($\Phi_N \propto M^{2N-4}$) to the residual $N-1$ quarks. The effective |
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308 | mass is a function of the total mass $M$, |
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309 | \begin{equation} |
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310 | M_{N-1}^{2}=M^{2}-2kM , \label{m_n-1} |
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311 | \end{equation} |
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312 | so that the resulting equation for the quark-parton |
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313 | spectrum is: |
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314 | \begin{equation} |
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315 | \frac{dW}{kdk}\propto (1-\frac{2k}{M})^{N-3}. \label{spectrum_1} |
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316 | \end{equation} |
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317 | |
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318 | \subsection{Meson Production} |
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319 | |
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320 | In this section, only the quark fusion mechanism of hadronization is |
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321 | considered. The quark exchange mechanism can take place only in |
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322 | nuclear matter where a quasmon has neighboring nucleons. In order to |
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323 | decompose a quasmon into an outgoing hadron and a residual quasmon, one |
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324 | needs to calculate the probability of two quark-partons combining to |
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325 | produce the effective mass of the outgoing hadron. This requires that |
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326 | the spectrum of the second quark-parton be calculated. This is done by |
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327 | following the same argument used to determine Eq.~\ref{spectrum_1}. |
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328 | One quark-parton is chosen from the residual $N-1$. It has an energy |
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329 | $q$ in the CMS of the $N-1$ quark-partons. The spectrum is obtained by |
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330 | substituting $N-1$ for $N$ and $M_{N-1}$ for $M$ in |
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331 | Eq.~\ref{spectrum_1} and then using Eq.~\ref{m_n-1} to get |
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332 | \begin{equation} |
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333 | \frac{dW}{q dq }\propto \left( 1-\frac{2q }{M\sqrt{1- |
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334 | \frac{2k}{M}}}\right) ^{N-4}. \label{spectrum_2} |
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335 | \end{equation} |
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336 | |
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337 | Next, one of the residual quark-partons must be selected from this spectrum |
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338 | such that its fusion with the primary quark-parton makes a hadron of |
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339 | mass $\mu$. This selection is performed by the mass shell condition for |
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340 | the outgoing hadron, |
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341 | \begin{equation} |
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342 | \mu^2 = 2 \frac{k}{\sqrt{1-\frac{2k}{M}}} |
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343 | \cdot q \cdot (1-\cos \theta ) . \label{hadron} |
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344 | \end{equation} |
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345 | Here $\theta$ is the angle between the momenta, {\bf k} and {\bf q} of |
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346 | the two quark-partons in the CMS of $N-1$ quarks. Now the kinematic quark |
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347 | fusion probability can be calculated for any primary quark-parton with |
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348 | energy $k$: |
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349 | \begin{eqnarray} |
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350 | P(k,M,\mu )=&&\int \left( 1-\frac{2q }{M\sqrt{1-\frac{2k}{M}}}\right) |
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351 | ^{N-4} \nonumber\\ |
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352 | && \times\ \delta \left( \mu ^{2}-\frac{2kq (1-\cos \theta )}{\sqrt{1- |
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353 | \frac{2k}{M}}}\right) q dq d\cos \theta .\ \ \ \ |
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354 | \end{eqnarray} |
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355 | Using the $\delta$-function\footnote{\protect{ |
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356 | If $g(x_0)$=0, $\int f(x)\delta\left[g(x)\right]dx = |
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357 | \int \frac{f(x)\delta\left[g(x)\right]}{g^\prime(x)} dg(x) = |
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358 | \frac{f(x_0)}{g^\prime(x_0)}$ |
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359 | }} |
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360 | to perform the integration over $q$ one gets: |
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361 | \begin{eqnarray} |
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362 | P(k,M,\mu )=&&\int \left( 1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}\right) |
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363 | ^{N-4} \nonumber\\ |
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364 | && \times\ \left( \frac{\mu ^{2}\sqrt{1-\frac{2k}{M}}}{2k(1-\cos \theta )} |
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365 | \right)^{2}d\left(\frac{1-\cos \theta }{\mu ^{2}}\right) ,\ \ |
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366 | \end{eqnarray} |
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367 | or |
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368 | \begin{eqnarray} |
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369 | P(k,M,\mu )=&&\frac{M-2k}{4k}\int \left(1-\frac{\mu ^{2}}{Mk(1 - |
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370 | \cos\theta)}\right) ^{N-4} \nonumber\\ |
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371 | && \times\ d\left(1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}\right). |
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372 | \end{eqnarray} |
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373 | After the substitution |
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374 | $z=1-\frac{2q }{M_{N-1}}=1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}$, this |
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375 | becomes |
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376 | \begin{equation} |
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377 | P(k,M,\mu ) = \frac{M-2k}{4k} \int z^{N-4} dz , |
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378 | \end{equation} |
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379 | where the limits of integration are $0$ when |
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380 | $\cos\theta = 1 - \frac{\mu ^{2}}{M\cdot k}$, and |
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381 | \begin{equation} |
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382 | z_{\max }=1-\frac{\mu^2}{2Mk}, \label{z_max} |
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383 | \end{equation} |
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384 | when $\cos \theta =-1$. The resulting range of $\theta$\ is therefore |
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385 | $-1<\cos \theta < 1-\frac{\mu ^{2}}{M\cdot k}$. Integrating from $0$ to |
---|
386 | $z$ yields |
---|
387 | \begin{equation} |
---|
388 | \frac{M-2k}{4k\cdot (N-3)}\cdot z^{N-3}, \label{z_probab} |
---|
389 | \end{equation} |
---|
390 | and integrating from $0$ to $z_{max}$ yields the total kinematic |
---|
391 | probability for hadronization of a quark-parton with energy $k$ into a |
---|
392 | hadron with mass $\mu$: |
---|
393 | \begin{equation} |
---|
394 | \frac{M-2k}{4k \cdot (N-3)} \cdot z_{\max}^{N-3} . |
---|
395 | \label{tot_kin_probab} |
---|
396 | \end{equation} |
---|
397 | The ratio of expressions \ref{z_probab} and \ref{tot_kin_probab} can be |
---|
398 | treated as a random number, $R$, uniformly distributed on the interval |
---|
399 | [0,1]. Solving for $z$ then gives |
---|
400 | \begin{equation} |
---|
401 | z=\sqrt[N-3]{R}\cdot z_{\max }. \label{z_random} |
---|
402 | \end{equation} |
---|
403 | |
---|
404 | In addition to the kinematic selection of the two quark-partons in the |
---|
405 | fusion process, the quark content of the quasmon and the spin of the |
---|
406 | candidate final hadron are used to determine the probability that a |
---|
407 | given type of hadron is produced. Because only the relative hadron |
---|
408 | formation probabilities are necessary, overall normalization factors can |
---|
409 | be dropped. Hence the relative probability can |
---|
410 | be written as |
---|
411 | \begin{equation} |
---|
412 | P_h(k,M,\mu )=(2s_h+1)\cdot z_{\max }^{N-3}\cdot C_{Q}^{h} . |
---|
413 | \label{rel_prob} |
---|
414 | \end{equation} |
---|
415 | Here, only the factor $z_{\max }^{N-3}$ is used since the other factors |
---|
416 | in equation \ref{tot_kin_probab} are constant for all candidates for the |
---|
417 | outgoing hadron. The factor $2s_h+1$ counts the spin states of a |
---|
418 | candidate hadron of spin $s_h$, and $C_{Q}^{h}$ is the number of ways the |
---|
419 | candidate hadron can be formed from combinations of the quarks within the |
---|
420 | quasmon. In making these combinations, the standard quark wave functions |
---|
421 | for pions and kaons were used. For $\eta$ and $\eta^{\prime }$ mesons the |
---|
422 | quark wave functions |
---|
423 | $\eta=\frac{\bar{u}u+\bar{d}d}{2}-\frac{\bar{s}s}{\sqrt{2}}$ and |
---|
424 | $\eta^{\prime }=\frac{\bar{u}u+\bar{d}d}{2}+\frac{\bar{s}s}{\sqrt{2}}$ |
---|
425 | were used. No mixing was assumed for the $\omega $\ and $\phi $\ meson |
---|
426 | states, hence $\omega =\frac{ \bar{u}u+\bar{d}d}{\sqrt{2}}$ and |
---|
427 | $\varphi=\bar{s}s$. |
---|
428 | |
---|
429 | A final model restriction is applied to the hadronization process: |
---|
430 | after a hadron is emitted, the quark content of the residual quasmon |
---|
431 | must have a quark content corresponding to either one or two real |
---|
432 | hadrons. When the quantum numbers of a quasmon, determined by its quark |
---|
433 | content, cannot be represented by the quantum numbers of a real hadron, |
---|
434 | the quasmon is considered to be a virtual hadronic molecule such as |
---|
435 | $\pi ^{+}\pi ^{+}$ or $K^{+}\pi ^{+}$, in which case it is defined in |
---|
436 | the CHIPS model to be a Chipolino pseudo-particle. |
---|
437 | |
---|
438 | To fuse quark-partons and create the decay of a quasmon into a hadron and |
---|
439 | residual quasmon, one needs to generate randomly the residual quasmon mass |
---|
440 | $m$, which in fact is the mass of the residual $N-2$ quarks. Using an |
---|
441 | equation similar to \ref{m_n-1}) one finds that |
---|
442 | \begin{equation} |
---|
443 | m^{2}=z\cdot (M^{2}-2kM). \label{m(z)} |
---|
444 | \end{equation} |
---|
445 | Using Eqs. \ref{z_random} and \ref{z_max}, the mass of the residual |
---|
446 | quasmon can be expressed in terms of the random number $R$: |
---|
447 | \begin{equation} |
---|
448 | m^{2}=(M-2k)\cdot (M-\frac{\mu ^{2}}{2k})\cdot \sqrt[N-3]{R} . |
---|
449 | \label{res_quasmon} |
---|
450 | \end{equation} |
---|
451 | At this point, the decay of the original quasmon into a final state |
---|
452 | hadron and a residual quasmon of mass $m$ has been simulated. The process |
---|
453 | may now be repeated on the residual quasmon. |
---|
454 | |
---|
455 | This iterative hadronization process continues as long as the residual |
---|
456 | quasmon mass remains greater than $m_{\min }$, whose value depends on the |
---|
457 | type of quasmon. For hadron-type residual quasmons |
---|
458 | \begin{equation} |
---|
459 | m_{\min }=m_{\min }^{QC}+m_{\pi ^{0}}, \label{m_min} |
---|
460 | \end{equation} |
---|
461 | where $m_{\min }^{QC}$ is the minimum hadron mass for the residual |
---|
462 | quark content (QC). For Chipolino-type residual quasmons consisting |
---|
463 | of hadrons $h_1$ and $h_2$, |
---|
464 | \begin{equation} |
---|
465 | m_{\min }=m_{h_1}+m_{h_2}. \label{m_min_chipolino} |
---|
466 | \end{equation} |
---|
467 | These conditions insure that the quasmon always has enough energy to decay |
---|
468 | into at least two final state hadrons, conserving four-momentum and charge. |
---|
469 | |
---|
470 | If the remaining CMS energy of the residual quasmon falls below $m_{\min}$, |
---|
471 | then the hadronization process terminates with a final two-particle decay. |
---|
472 | If the parent quasmon is a Chipolino consisting of hadrons $h_1$ and $h_2$, |
---|
473 | then a binary decay of the parent quasmon into $m_{h_1}$ and $m_{h_2}$ |
---|
474 | takes place. If the parent quasmon is not a Chipolino then a decay into |
---|
475 | $m_{\min}^{QC}$ and $m_h$ takes place. The decay into $m_{\min}^{QC}$ and |
---|
476 | $m_\pi^0$ is always possible in this case because of condition \ref{m_min}. |
---|
477 | |
---|
478 | If the residual quasmon is not Chipolino-type, and $m>m_{\min}$, the |
---|
479 | hadronization loop can still be finished by the resonance production |
---|
480 | mechanism, which is modeled following the concept of parton-hadron |
---|
481 | duality \cite{Duality}. If the residual quasmon has a mass in the vicinity |
---|
482 | of a resonance with the same quark content ($\rho$ or $K^{\ast}$ for |
---|
483 | example), there is a probability for the residual quasmon to convert to |
---|
484 | this resonance.\footnote{When comparing quark contents, the quark content |
---|
485 | of the quasmon is reduced by canceling quark-antiquark pairs of the same |
---|
486 | flavor.} |
---|
487 | In the present version of the CHIPS event generator the probability of |
---|
488 | convert to the resonance is given by |
---|
489 | \begin{equation} |
---|
490 | P_{\rm{res}}=\frac{m_{\min }^{2}}{m^{2}}. \label{res_probab} |
---|
491 | \end{equation} |
---|
492 | Hence the resonance with the mass-squared value $m_{r}^{2}$ closest to |
---|
493 | $m^{2}$ is selected, and the binary decay of the quasmon into $m_{h}$ |
---|
494 | and $m_{r}$ takes place. |
---|
495 | |
---|
496 | With more detailed experimental data, it will be possible to take into |
---|
497 | account angular momentum conservation, as well as $C$-, $P$- and |
---|
498 | $G$-parity conservation. In the present version of the generator, $\eta$ |
---|
499 | and $\eta ^{\prime }$ are suppressed by a factor of $0.3$. This factor |
---|
500 | was tuned using data from experiments on antiproton annihilation at rest |
---|
501 | in liquid hydrogen and can be different for other hadronic reactions. It |
---|
502 | is possible to vary it when describing other reactions. |
---|
503 | |
---|
504 | Another parameter, $s/u$, controls the suppression of heavy quark |
---|
505 | production \cite{JETSET}. For proton-antiproton annihilation at rest the |
---|
506 | strange quark-antiquark sea was found to be suppressed by the factor |
---|
507 | $s/u = 0.1$. In the JETSET \cite{JETSET} event generator, the default |
---|
508 | value for this parameter is $s/u = 0.3$. The lower value may be due to |
---|
509 | quarks and anti-quarks of colliding hadrons initially forming a non-strange |
---|
510 | sea, with the strange sea suppressed by the OZI rule \cite{OZI}. This |
---|
511 | question is still under discussion \cite{OZI_violation} and demands further |
---|
512 | experimental measurements. The $s/u$ parameter may differ for other |
---|
513 | reactions. In particular, for e$^{+}$e$^{-}$ reactions it can be closer to |
---|
514 | 0.3. |
---|
515 | |
---|
516 | Finally, the temperature parameter has been fixed at $T=180$ MeV. In |
---|
517 | earlier versions of the model it was found that this value successfully |
---|
518 | reproduced spectra of outgoing hadrons in different types of medium-energy |
---|
519 | reactions. |
---|
520 | |
---|
521 | \begin{figure} |
---|
522 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps, height=3.5in, width=4.5in}} |
---|
523 | % \resizebox{1.00\textwidth}{!} |
---|
524 | %{ |
---|
525 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/mommul.eps} |
---|
526 | %\includegraphics[angle=0,scale=0.6]{plots/mommul.eps} |
---|
527 | %} |
---|
528 | \caption{(a) (left): momentum distribution of charged pions produced in |
---|
529 | proton-antiproton annihilation at rest. The experimental data are from |
---|
530 | \protect\cite{pispectrum}, and the histogram was produced by the CHIPS |
---|
531 | Monte Carlo. The experimental spectrum is normalized to the measured |
---|
532 | average charged pion multiplicity, 3.0. (b) (right): pion multiplicity |
---|
533 | distribution. Data points were taken from compilations of experimental |
---|
534 | data \protect\cite{pap_exdata}, and the histogram was produced by the |
---|
535 | CHIPS Monte Carlo. The number of events with kaons in the final state is |
---|
536 | shown in pion multiplicity bin 9, where no real 9-pion events are |
---|
537 | generated or observed experimentally. In the model, the percentage of |
---|
538 | annihilation events with kaons is close to the experimental value of |
---|
539 | 6\% \cite{pap_exdata}. |
---|
540 | } |
---|
541 | \label{mommul} |
---|
542 | \end{figure} |
---|
543 | |
---|
544 | \begin{figure} |
---|
545 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/channels.eps, height=3.5in, width=4.5in}} |
---|
546 | % \resizebox{1.00\textwidth}{!} |
---|
547 | %{ |
---|
548 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/channels.eps} |
---|
549 | %\includegraphics[angle=0,scale=0.6]{plots/channels.eps} |
---|
550 | %} |
---|
551 | \caption{Branching probabilities for different channels in |
---|
552 | proton-antiproton annihilation at rest. The experimental data are from |
---|
553 | \protect\cite {pap_exdata}, and the histogram was produced by the CHIPS |
---|
554 | Monte Carlo. } |
---|
555 | \label{channels} |
---|
556 | \end{figure} |
---|
557 | |
---|
558 | The above parameters were used to fit not only the spectrum of pions |
---|
559 | Fig.~\ref{mommul},a and the multiplicity distribution for pions |
---|
560 | Fig.~\ref{mommul},b but also branching ratios of various measured |
---|
561 | \cite{pispectrum,pap_exdata} exclusive channels as shown in Figs. |
---|
562 | ~\ref{channels},~\ref{threechan},~\ref{twochan}. In Fig.~\ref{twochan} |
---|
563 | one can see many decay channels with higher meson resonances. The |
---|
564 | relative contribution of events with meson resonances produced in the |
---|
565 | final state is 30 - 40 percent, roughly in agreement with experiment. The |
---|
566 | agreement between the model and experiment for particular decay modes is |
---|
567 | within a factor of 2-3 except for the branching ratios to higher |
---|
568 | resonances. In these cases it is not completely clear how the resonance |
---|
569 | is defined in a concrete experiment. In particular, for the |
---|
570 | $a_{2}\omega $ channel the mass sum of final hadrons is 2100 MeV with a |
---|
571 | full width of about 110 MeV while the total initial energy of the p\={p} |
---|
572 | annihilation reaction is only 1876.5 MeV. This decay channel can be |
---|
573 | formally simulated by an event generator using the tail of the Breit-Wigner |
---|
574 | distribution for the $a_{2}$ resonance, but it is difficult to imagine how |
---|
575 | the $a_{2}$ resonance can be experimentally identified $2\Gamma $ away |
---|
576 | from its mean mass value. |
---|
577 | |
---|
578 | \subsection{Baryon Production} |
---|
579 | |
---|
580 | To model fragmentation into baryons the POPCORN idea \cite{POPCORN} was |
---|
581 | used, which assumes the existence of diquark-partons. The assumption of |
---|
582 | massless diquarks is somewhat inconsistent at low energies, as is the |
---|
583 | assumption of massless s-quarks, but it is simple and it helps to generate |
---|
584 | baryons in the same way as mesons. |
---|
585 | |
---|
586 | Baryons are heavy, and the baryon production in $p\bar p$ annihilation |
---|
587 | reactions at medium energies is very sensitive to the value of the |
---|
588 | temperature. If the temperature is low, the baryon yield is small, and |
---|
589 | the mean multiplicity of pions increases very noticeably with center-of-mass |
---|
590 | energy as seen in Fig.~\ref{apcmul}. For higher temperature values the baryon |
---|
591 | yield reduces the pion multiplicity at higher energies. The existing |
---|
592 | experimental data \cite{Energy_Dep}, shown in Fig.~\ref{apcmul}, can be |
---|
593 | considered as a kind of ``thermometer'' for the model. This thermometer |
---|
594 | confirms that the critical temperature is about 200 MeV. |
---|
595 | |
---|
596 | \begin{figure} |
---|
597 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/threechn.eps, height=4.5in, width=4.5in}} |
---|
598 | % \resizebox{1.00\textwidth}{!} |
---|
599 | %{ |
---|
600 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/threechn.eps} |
---|
601 | %\includegraphics[angle=0,scale=0.6]{plots/threechn.eps} |
---|
602 | %} |
---|
603 | \caption{Branching probabilities for different channels with |
---|
604 | three-particle final states in proton-antiproton annihilation at |
---|
605 | rest. The points are experimental data \protect\cite{pap_exdata} and the |
---|
606 | histogram is from the CHIPS Monte Carlo. } |
---|
607 | \label{threechan} |
---|
608 | \end{figure} |
---|
609 | \begin{figure} |
---|
610 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/twochn.eps, height=4.5in, width=4.5in}} |
---|
611 | % \resizebox{1.00\textwidth}{!} |
---|
612 | %{ |
---|
613 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/twochn.eps} |
---|
614 | %\includegraphics[angle=0,scale=0.6]{plots/twochn.eps} |
---|
615 | %} |
---|
616 | \caption{Branching probabilities for different channels with |
---|
617 | two-particle final states in proton-antiproton annihilation at |
---|
618 | rest. The points are experimental data \protect\cite{pap_exdata} and the |
---|
619 | histogram is from the CHIPS Monte Carlo. } |
---|
620 | \label{twochan} |
---|
621 | \end{figure} |
---|
622 | \begin{figure} |
---|
623 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/apcmul.eps, height=4.5in, width=4.5in}} |
---|
624 | % \resizebox{1.00\textwidth}{!} |
---|
625 | %{ |
---|
626 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/apcmul.eps} |
---|
627 | %\includegraphics[angle=0,scale=0.6]{plots/apcmul.eps} |
---|
628 | %} |
---|
629 | \caption{Average meson multiplicities in proton-antiproton and in |
---|
630 | electron-positron annihilation, as a function of the center-of-mass energy of |
---|
631 | the interacting hadronic system. The points are experimental data |
---|
632 | \protect\cite {Energy_Dep} and the lines are CHIPS Monte Carlo calculations |
---|
633 | at several values of the critical temperature parameter $T$. } |
---|
634 | \label{apcmul} |
---|
635 | \end{figure} |
---|
636 | |
---|
637 | It can be used as a tool for the Monte Carlo simulation of a wide variety |
---|
638 | of hadronic reactions. The CHIPS event generator can be used not only for |
---|
639 | ``phase-space background'' calculations in place of the standard GENBOD |
---|
640 | routine \cite{GENBOD}, but even for taking into account the reflection of |
---|
641 | resonances in connected final hadron combinations. Thus it can be useful |
---|
642 | for physics analysis too, even though its main range of application is the |
---|
643 | simulation of the evolution of hadronic and electromagnetic showers in |
---|
644 | matter at medium energies. |
---|
645 | |
---|
646 | \section{Nuclear Pion Capture at Rest and Photonuclear Reactions Below the |
---|
647 | Delta(3,3) Resonance} \label{picap} |
---|
648 | |
---|
649 | When compared with the first ``in vacuum'' version of the model, described |
---|
650 | in Section \ref{annil}, modeling hadronic fragmentation in nuclear matter |
---|
651 | is more complicated because of the much greater number of possible |
---|
652 | secondary fragments. However, the hadronization process itself |
---|
653 | is simpler in a way. In vacuum, the quark-fusion mechanism requires a |
---|
654 | quark-parton partner from the external (as in |
---|
655 | JETSET \cite{JETSET}) or internal (the quasmon itself, Section \ref{annil}) |
---|
656 | quark-antiquark sea. In nuclear matter, there is a second possibility: |
---|
657 | quark exchange with a neighboring hadronic system, which could be a nucleon |
---|
658 | or multinucleon cluster. |
---|
659 | |
---|
660 | In nuclear matter the spectra of secondary hadrons and nuclear fragments |
---|
661 | reflect the quark-parton energy spectrum within a quasmon. In the case of |
---|
662 | inclusive spectra that are decreasing steeply with energy, and |
---|
663 | correspondingly steeply decreasing spectra of the quark-partons in a quasmon, |
---|
664 | only those secondary hadrons which get the maximum energy from the primary |
---|
665 | quark-parton of energy $k$ contribute to the inclusive spectra. This |
---|
666 | extreme situation requires the exchanged quark-parton with energy $q$, |
---|
667 | coming toward the quasmon from the cluster, to move in a direction |
---|
668 | opposite to that of the primary quark-parton. As a result the |
---|
669 | hadronization quark exchange process becomes one-dimensional along the |
---|
670 | direction of $k$. If a neighboring nucleon or nucleon cluster with bound |
---|
671 | mass $\tilde{\mu}$ absorbs the primary quark-parton and radiates the |
---|
672 | exchanged quark-parton in the opposite direction, then the energy of the |
---|
673 | outgoing fragment is $E=\tilde{\mu}+k-q$, and the momentum is $p=k+q$. |
---|
674 | Both the energy and the momentum of the outgoing nuclear fragment are known, |
---|
675 | as is the mass $\tilde{\mu}$ of the nuclear fragment in nuclear matter, so |
---|
676 | the momentum of the primary quark-parton can be reconstructed using the |
---|
677 | approximate relation |
---|
678 | \begin{equation} |
---|
679 | k=\frac{p+E-B\cdot m_{N}}{2} . \label{k} |
---|
680 | \end{equation} |
---|
681 | Here $B$ is the baryon number of the outgoing fragment |
---|
682 | ($\tilde{\mu}\approx B\cdot m_{N}$) and $m_N$ is the nucleon mass. In |
---|
683 | Ref.~\cite{K_parameter} it was shown that the invariant inclusive spectra of |
---|
684 | pions, protons, deuterons, and tritons in proton-nucleus reactions at |
---|
685 | 400~GeV \cite{FNAL} not only have the same exponential slope but almost |
---|
686 | coincide when they are plotted as a function of $k=\frac{p+E_{\rm{kin}}}{2}$. |
---|
687 | Using data at 10~GeV \cite{FAS}, it was shown that the parameter $k$, defined |
---|
688 | by Eq.~\ref{k}, is also appropriate for the description of secondary |
---|
689 | anti-protons produced in high energy nuclear reactions. This means that the |
---|
690 | extreme assumption of one-dimensional hadronization is a good approximation. |
---|
691 | |
---|
692 | The same approximation is also valid for the quark fusion mechanism. In |
---|
693 | the one-dimensional case, assuming that $q$ is the momentum of the second |
---|
694 | quark fusing with the primary quark-parton of energy $k$, the total energy |
---|
695 | of the outgoing hadron is $E=q+k$ and the momentum is $p=k-q$. In the |
---|
696 | one-dimensional case the secondary quark-parton must move in the opposite |
---|
697 | direction with respect to the primary quark-parton, otherwise the mass of |
---|
698 | the outgoing hadron would be zero. So, for mesons $k=\frac{p+E}{2}$, in |
---|
699 | accordance with Eq.~\ref{k}. In the case of anti-proton radiation, the |
---|
700 | baryon number of the quasmon is increased by one, and the primary |
---|
701 | antiquark-parton will spend its energy to build up the mass of the |
---|
702 | antiproton by picking up an anti-diquark. Thus, the energy conservation |
---|
703 | law for antiproton radiation looks like $E+m_{N}=q+k$ and |
---|
704 | $k=\frac{p+E+m_{N}}{2}$, which is again in accordance with Eq.~\ref{k}. |
---|
705 | |
---|
706 | The one-dimensional quark exchange mechanism was proposed in 1984 |
---|
707 | \cite{K_parameter}. Even in its approximate form it was useful in the |
---|
708 | analysis of inclusive spectra of hadrons and nuclear fragments in |
---|
709 | hadron-nuclear reactions at high energies. Later the same approach was |
---|
710 | used in the analysis of nuclear fragmentation in electro-nuclear |
---|
711 | reactions \cite{TPC}. Also in 1984 the quark-exchange mechanism developed |
---|
712 | in the framework of the non-relativistic quark model was found to be |
---|
713 | important for the explanation of the short distance features of $NN$ |
---|
714 | interactions \cite{NN QEX}. Later it was successfully applied to |
---|
715 | $K^{-}p$ interactions \cite{Kp QUEX}. The idea of the quark exchange |
---|
716 | mechanism between nucleons was useful even for the explanation of the EMC |
---|
717 | effect \cite{EMC}. For the non-relativistic quark model, the quark |
---|
718 | exchange technique was developed as an alternative to the Feynman diagram |
---|
719 | technique at short distances \cite{QUEX}. |
---|
720 | |
---|
721 | The CHIPS event generator models quark exchange processes, taking into |
---|
722 | account kinematic and combinatorial factors for asymptotically free |
---|
723 | quark-partons. In the naive picture of the quark-exchange mechanism, |
---|
724 | one quark-parton tunnels from the asymptotically free region of one hadron |
---|
725 | to the asymptotically free region of another hadron. To conserve color, |
---|
726 | another quark-parton from the neighboring hadron must replace the first |
---|
727 | quark-parton in the quasmon. This makes the tunneling mutual, and the |
---|
728 | resulting process is quark exchange. |
---|
729 | |
---|
730 | The experimental data available for multihadron production at high energies |
---|
731 | show regularities in the secondary particle spectra that can be related to |
---|
732 | the simple kinematic, combinatorial, and phase space rules of such quark |
---|
733 | exchange and fusion mechanisms. The CHIPS model combines these mechanisms |
---|
734 | consistently. |
---|
735 | |
---|
736 | \begin{figure}[tbp] |
---|
737 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram.eps, height=2.5in, width=2.5in}} |
---|
738 | %\resizebox{1.00\textwidth}{!} |
---|
739 | %{ |
---|
740 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram.eps} |
---|
741 | %\includegraphics[angle=0,scale=0.6]{plots/diagram.eps} |
---|
742 | %} |
---|
743 | \caption{Diagram of the quark exchange mechanism. } |
---|
744 | \label{diagram} |
---|
745 | \end{figure} |
---|
746 | |
---|
747 | Fig.~\ref{diagram} shows a quark exchange diagram which helps to keep track |
---|
748 | of the kinematics of the process. It was shown in Section \ref{annil} that |
---|
749 | a quasmon, $Q$ is kinematically determined by a few asymptotically free |
---|
750 | quark-partons homogeneously distributed over the invariant phase space. The |
---|
751 | quasmon mass $M$ is related to the number of quark-partons $N$ through |
---|
752 | \begin{equation} |
---|
753 | <M^{2}>=4N(N-1)\cdot T^{2}, \label{temperatureII} |
---|
754 | \end{equation} |
---|
755 | where $T$ is the temperature of the system. |
---|
756 | |
---|
757 | The spectrum of quark partons can be calculated as |
---|
758 | \begin{equation} |
---|
759 | \frac{dW}{k^{\ast }dk^{\ast}}\propto |
---|
760 | \left(1-\frac{2k^{\ast}}{M} \right)^{N-3}, \label{spectrum_1II} |
---|
761 | \end{equation} |
---|
762 | where $k^{\ast}$ is the energy of the primary quark-parton in the |
---|
763 | center-of-mass system of the quasmon. After the primary quark-parton |
---|
764 | is randomized and the neighboring cluster, or parent cluster, $PC$, with |
---|
765 | bound mass $\tilde{\mu}$\ is selected, the quark exchange process begins. |
---|
766 | To follow the process kinematically one should imagine a colored, compound |
---|
767 | system consisting of a stationary, bound parent cluster and the primary quark. |
---|
768 | The primary quark has energy $k$ in the lab system, |
---|
769 | \begin{equation} |
---|
770 | k=k^{\ast }\cdot \frac{E_{N}+p_{N}\cdot \cos (\theta _{k})}{M_{N}}, |
---|
771 | \end{equation} |
---|
772 | where $M_N$, $E_N$ and $p_N$ are the mass, energy, and momentum of the |
---|
773 | quasmon in the lab frame. The mass of the compound system, $CF$, is |
---|
774 | $\mu _{c}=\sqrt{(\tilde{\mu}+k)^{2}}$, where $\tilde{\mu}$ and $k$ are the |
---|
775 | corresponding four-vectors. This colored compound system decays into a |
---|
776 | free outgoing nuclear fragment, $F$, with mass $\mu$ and a recoiling quark |
---|
777 | with energy $q$. $q$ is measured in the CMS of $\tilde{\mu}$, which |
---|
778 | coincides with the lab frame in the present version of the model because no |
---|
779 | cluster motion is considered. At this point one should recall that a colored |
---|
780 | residual quasmon, $CRQ$, with mass $M_{N-1}$ remains after the radiation of |
---|
781 | $k$. $CRQ$ is finally fused with the recoil quark $q$ to form the residual |
---|
782 | quasmon $RQ$. The minimum mass of $RQ$ should be greater than $M_{\min}$, |
---|
783 | which is determined by the minimum mass of a hadron (or Chipolino |
---|
784 | double-hadron as defined in Section \ref{annil}) with the same quark content. |
---|
785 | |
---|
786 | All quark-antiquark pairs with the same flavor should be canceled in the |
---|
787 | minimum mass calculations. This imposes a restriction, which in the |
---|
788 | center-of-mass system of $\mu_{c}$, can be written as |
---|
789 | \begin{equation} |
---|
790 | 2q\cdot (E-p\cdot \cos \theta_{qCQ})+M_{N-1}^{2}>M_{\min }^{2}, |
---|
791 | \label{min_mass} |
---|
792 | \end{equation} |
---|
793 | where $E$ is the energy and $p$ is the momentum of the colored residual |
---|
794 | quasmon with mass $M_{N-1}$ in the CMS of $\mu _{c}$. The restriction for |
---|
795 | $\cos\theta_{qCQ}$ then becomes |
---|
796 | \begin{equation} |
---|
797 | \cos \theta _{qCQ}<\frac{2qE-M_{\min }^{2}+M_{N-1}^{2}}{2qp}, |
---|
798 | \label{cost_restriction} |
---|
799 | \end{equation} |
---|
800 | which implies |
---|
801 | \begin{equation} |
---|
802 | q>\frac{M_{N-1}^{2}-M_{\min }^{2}}{2\cdot (E+p)}. \label{resid_rest} |
---|
803 | \end{equation} |
---|
804 | |
---|
805 | A second restriction comes from the nuclear Coulomb barrier for charged |
---|
806 | particles. The Coulomb barrier can be calculated in the simple form: |
---|
807 | \begin{equation} |
---|
808 | E_{CB}=\frac{Z_{F}\cdot |
---|
809 | Z_{R}}{A_{F}^{\frac{1}{3}}+A_{R}^{\frac{1}{3}}}\ (\rm{MeV}), |
---|
810 | \label{CoulBar} |
---|
811 | \end{equation} |
---|
812 | where $Z_F$ and $A_F$ are the charge and atomic weight of the fragment, and |
---|
813 | $Z_R$ and $A_R$ are the charge and atomic weight of the residual nucleus. |
---|
814 | The obvious restriction is |
---|
815 | \begin{equation} |
---|
816 | q<k+\tilde{\mu}-\mu -E_{CB}. \label{cb_rest} |
---|
817 | \end{equation} |
---|
818 | |
---|
819 | In addition to \ref{resid_rest} and \ref{cb_rest}, the quark |
---|
820 | exchange mechanism imposes restrictions which are calculated below. The |
---|
821 | spectrum of recoiling quarks is similar to the $k^{\ast}$ spectrum in the |
---|
822 | quasmon (\ref{spectrum_1II}): |
---|
823 | \begin{equation} |
---|
824 | \frac{dW}{q\ dq\ d\cos \theta }\propto |
---|
825 | \left(1-\frac{2q}{\tilde{\mu}}\right)^{n-3}, \label{spectrum_2II} |
---|
826 | \end{equation} |
---|
827 | where $n$ is the number of quark-partons in the nucleon cluster. It is |
---|
828 | assumed that $n = 3 A_C$, where $A_C$ is the atomic weight of the parent |
---|
829 | cluster. The tunneling of quarks from one nucleon to another provides a |
---|
830 | common phase space for all quark-partons in the cluster. |
---|
831 | |
---|
832 | An additional equation follows from the mass shell condition for the |
---|
833 | outgoing fragment, |
---|
834 | \begin{equation} |
---|
835 | \mu ^{2}=\tilde{\mu}^{2}+2\tilde{\mu}\cdot k-2\tilde{\mu}\cdot q-2k\cdot |
---|
836 | q\cdot (1-\cos \theta _{kq}), \label{hadronII} |
---|
837 | \end{equation} |
---|
838 | where $\theta _{kq}$ is the angle between quark-parton momenta in the lab |
---|
839 | frame. From this equation $q$ can be calculated as |
---|
840 | \begin{equation} |
---|
841 | q=\frac{\tilde{\mu}\cdot (k-\Delta )}{\tilde{\mu}+k\cdot |
---|
842 | (1-\cos \mathit{\theta }_{\mathit{kq}})}, \label{q-cos} |
---|
843 | \end{equation} |
---|
844 | where $\Delta $ is the covariant binding energy of the cluster |
---|
845 | $\Delta =\frac{\mu ^{2}-\tilde{\mu}^{2}}{2\tilde{\mu}}$. |
---|
846 | The quark exchange probability integral can be then written in the form: |
---|
847 | \begin{eqnarray} |
---|
848 | &&P(k,\tilde{\mu},\mu )= \nonumber \\ |
---|
849 | &&\int \delta \left[ \mu ^{2}-\tilde{\mu}^{2}-2\tilde{\mu}\cdot k+2\tilde{\mu |
---|
850 | }\cdot q+2k\cdot {q}\cdot (1-\cos \theta _{kq})\right] \nonumber \\ |
---|
851 | &&\ \ \ \ \ \ \ \ \times \ \left( 1-\frac{2{q}}{\tilde{\mu}}\right) ^{n-3}{q} |
---|
852 | d{q\cdot }d\cos \theta _{kq}. |
---|
853 | \end{eqnarray} |
---|
854 | Using the $\delta$-function to perform the integration over $q$ one obtains |
---|
855 | \begin{eqnarray} |
---|
856 | P(k,\tilde{\mu},\mu ) &=&\int \left( 1-\frac{2(k-\Delta )}{\tilde{\mu} |
---|
857 | +k(1-\cos \theta _{\mathit{kq}})}\right) ^{n-3} \nonumber \\ |
---|
858 | &&\times \ \frac{\tilde{\mu}(k-\Delta )}{2[\tilde{\mu}+k(1-\cos \mathit{ |
---|
859 | \theta }_{\mathit{kq}})]^{2}}d\mathit{\cos \theta }_{\mathit{kq}} |
---|
860 | \end{eqnarray} |
---|
861 | or |
---|
862 | \begin{eqnarray} |
---|
863 | P(k,\tilde{\mu},\mu ) &=&\int \left( 1-\frac{2(k-\Delta )}{\tilde{\mu} |
---|
864 | +k(1-\cos \theta _{\mathit{kq}})}\right) ^{n-3} \nonumber \\ |
---|
865 | &&\times \ \left( \frac{\tilde{\mu}(k-\Delta )}{\tilde{\mu}+k(1-\cos |
---|
866 | \mathit{\theta }_{\mathit{kq}})}\right) ^{2} \nonumber \\ |
---|
867 | &&\times \ d \left( \frac{\tilde{\mu}+k(1-\cos |
---|
868 | \mathit{\theta }_{\mathit{kq}})}{\tilde{\mu}(k-\Delta )}\right). |
---|
869 | \end{eqnarray} |
---|
870 | The result of the integration is |
---|
871 | \begin{eqnarray} |
---|
872 | &&P(k,\tilde{\mu},\mu )=\frac{\tilde{\mu}}{4k(n-2)} \nonumber \\ |
---|
873 | &&\times \ \left[ \left( 1-\frac{2(k-\Delta )}{\tilde{\mu}+2k}\right) |
---|
874 | _{\rm{high}}^{n-2}-\left( 1-\frac{2(k-\Delta )}{\tilde{\mu}}\right) |
---|
875 | _{\rm{low}}^{n-2}% |
---|
876 | \right] . \label{QUEX_Int} |
---|
877 | \end{eqnarray} |
---|
878 | For randomization it is convenient to make $z$ a random parameter |
---|
879 | \begin{equation} |
---|
880 | z=1-\frac{2(k-\Delta )}{\tilde{\mu}+k(1-\cos |
---|
881 | \theta_{\mathit{kq}})}=1-\frac{2{q}}{\tilde{\mu}}. \label{z(q)} |
---|
882 | \end{equation} |
---|
883 | From (\ref{QUEX_Int}) one can find the high and the low limits of the |
---|
884 | randomization. The first limit is for $k$: $k>\Delta$. It is |
---|
885 | similar to the restriction for quasmon fragmentation in vacuum: |
---|
886 | $k^{\ast}>\frac{\mu^{2}}{2M}$. The second limit is |
---|
887 | $k=\frac{\mu^{2}}{2\tilde{\mu}}$, when the low limit of randomization |
---|
888 | becomes equal to zero. If $k<\frac{\mu^{2}}{2\tilde{\mu}}$, then |
---|
889 | $-1<\cos\theta_{kq}<1$\ and |
---|
890 | $z_{\rm{low}}=1-\frac{2(k-\Delta)}{\tilde{\mu}}$. If |
---|
891 | $k>\frac{\mu^{2}}{2\tilde{\mu}}$, then the range of $\cos\theta |
---|
892 | _{kq}$\ is $-1<\cos\theta_{kq}<\frac{\mu^{2}}{k\tilde{\mu}}-1$\ and |
---|
893 | $z_{\rm{low}}=0$. This value of $z_{\rm{low}}$\ should be corrected |
---|
894 | using the Coulomb barrier restriction (\ref{cb_rest}), and the value of |
---|
895 | $z_{\rm{high}}$ should be corrected using the minimum residual quasmon |
---|
896 | restriction (\ref{resid_rest}). In the case of a quasmon with momentum much |
---|
897 | less than $k$ it is possible to impose tighter restrictions than |
---|
898 | (\ref{resid_rest}) because the direction of motion of the CRQ is |
---|
899 | opposite to $k$. So |
---|
900 | $\cos\theta_{qCQ}=-\cos\mathit{\theta}_{\mathit{kq}}$, and from |
---|
901 | (\ref{q-cos}) one can find that |
---|
902 | \begin{equation} |
---|
903 | \cos \theta_{qCQ} =1-\frac{\tilde{\mu}\cdot (k-\Delta -q)}{k\cdot q}. |
---|
904 | \label{cos_q} |
---|
905 | \end{equation} |
---|
906 | So in this case the equation (\ref{resid_rest})\ can be replaced by |
---|
907 | the more stringent one: |
---|
908 | \begin{equation} |
---|
909 | q>\frac{M_{N-1}^{2}-M_{\min }^{2}+2\frac{p\cdot |
---|
910 | \tilde{\mu}}{k}(k-\Delta )}{2\cdot (E+p+\frac{p\cdot \tilde{\mu}}{k})}. |
---|
911 | \end{equation} |
---|
912 | |
---|
913 | The integrated kinematical quark exchange probability (in the range |
---|
914 | from $z_{\rm{low}}$ to $z_{\rm{high}}$) is |
---|
915 | \begin{equation} |
---|
916 | \frac{\tilde{\mu}}{4k(n-2)}\cdot z^{n-2}, \label{z_probabII} |
---|
917 | \end{equation} |
---|
918 | and the total kinematic probability of hadronization of the quark-parton |
---|
919 | with energy $k$ into a nuclear fragment with mass\ $\mu $ is |
---|
920 | \begin{equation} |
---|
921 | \frac{\tilde{\mu}}{4k(n-2)}\cdot |
---|
922 | \left( z_{\rm{high}}^{n-2}-z_{\rm{low}}^{n-2}\right). |
---|
923 | \label{tot_kin_probabII} |
---|
924 | \end{equation} |
---|
925 | This can be compared with the vacuum probability of the quark fusion mechanism |
---|
926 | from Section \ref{annil}: |
---|
927 | \begin{equation} |
---|
928 | \frac{M-2k}{4k(N-3)}z_{\max }^{N-3}. |
---|
929 | \end{equation} |
---|
930 | The similarity is very important, as the absolute probabilities |
---|
931 | define the competition between vacuum and nuclear channels. |
---|
932 | |
---|
933 | Equations (\ref{z_probabII})\ and (\ref{tot_kin_probabII})\ can be used for |
---|
934 | randomization of $z$: |
---|
935 | \begin{equation} |
---|
936 | z=z_{\rm{low}}+\sqrt[n-2]{R}\cdot (z_{\rm{high}}-z_{\rm{low}}), |
---|
937 | \label{z_randomII} |
---|
938 | \end{equation} |
---|
939 | where $R$\ is a random number, uniformly distributed in the interval (0,1). |
---|
940 | |
---|
941 | Eq. (\ref{tot_kin_probabII})\ can be used to control the competition |
---|
942 | between different nuclear fragments and hadrons in the hadronization |
---|
943 | process, but in contrast to the case of ``in vacuum'' hadronization |
---|
944 | it is not enough to take into account only the quark combinatorics of the |
---|
945 | quasmon and the outgoing hadron. In the case of hadronization in nuclear |
---|
946 | matter, different parent bound clusters should be taken into account as well. |
---|
947 | For example, tritium can be radiated as a result of quark exchange with a |
---|
948 | bound tritium cluster or as a result of quark exchange with a bound $^3$He |
---|
949 | cluster. |
---|
950 | |
---|
951 | To calculate the yield of fragments it is necessary to calculate the |
---|
952 | probability to find a cluster with certain proton and neutron content |
---|
953 | in a nucleus. One could consider any particular probability as an |
---|
954 | independent parameter, but in such a case the process of tuning the model |
---|
955 | would be difficult. We proposed the following scenario of |
---|
956 | clusterization. A gas of quasi-free nucleons is close to the phase |
---|
957 | transition to a liquid bound by strong quark exchange forces. Precursors of |
---|
958 | the liquid phase are nuclear clusters, which may |
---|
959 | be considered as ``drops'' of the liquid phase within the nucleus. Any |
---|
960 | cluster can meet another nucleon and absorb it (making it bigger), |
---|
961 | or it can release one of the nucleons (making it smaller). The |
---|
962 | first parameter $\varepsilon_{1}$\ is the percentage of quasi-free |
---|
963 | nucleons not involved in the clusterization process. The rest of the |
---|
964 | nucleons ($1-\varepsilon_{1}$) clusterize. |
---|
965 | We assume that since on the periphery of the nucleus the density |
---|
966 | is lower, one can consider only dibaryon clusters, and neglect |
---|
967 | triple-baryon clusters. Still we denote the number of nucleons |
---|
968 | clusterized in dibaryons on the periphery by the parameter |
---|
969 | $\varepsilon_{2}$. In the dense part of the nucleus, strong quark |
---|
970 | exchange forces make clusters out of quasi-free nucleons with high |
---|
971 | probability. To characterize the distribution of clusters the |
---|
972 | clusterization probability parameter $\omega$ was used. |
---|
973 | |
---|
974 | If the number of nucleons involved in clusterization is |
---|
975 | $a=(1-\varepsilon_{1}-\varepsilon _{2})\cdot A$, then the probability |
---|
976 | to find a cluster consisting of $\nu$\ nucleons is defined by the |
---|
977 | distribution |
---|
978 | \begin{equation} |
---|
979 | P_{\nu }\propto C_{\nu }^{a}\cdot \omega ^{\nu -1}, |
---|
980 | \end{equation} |
---|
981 | where $C_{\nu }^{a}$ is the corresponding binomial coefficient. |
---|
982 | The coefficient of proportionality can be found from the equation |
---|
983 | \begin{equation} |
---|
984 | a=b\cdot \sum\limits_{\nu =1}^{a}\nu \cdot C_{\nu }^{a}\cdot \omega ^{\nu |
---|
985 | -1}=b\cdot a\cdot (1+\omega )^{a-1}. |
---|
986 | \end{equation} |
---|
987 | Thus, the number of clusters consisting of $\nu$\ nucleons is |
---|
988 | \begin{equation} |
---|
989 | P_{\nu }=\frac{C_{\nu }^{a}\cdot \omega ^{\nu -1}}{(1+\omega )^{a-1}}. |
---|
990 | \end{equation} |
---|
991 | For clusters with an even number of nucleons we used only isotopically |
---|
992 | symmetric configurations ($\nu=2n$, $n$\ protons and $n$\ neutrons) and |
---|
993 | for odd clusters ($\nu =2n+1$) we used only two configurations: $n$\ |
---|
994 | neutrons with $n+1$\ protons and $n+1$\ neutrons with $n$\ protons. This |
---|
995 | restriction, which we call ``isotopic focusing'', can be considered an |
---|
996 | empirical rule of the CHIPS model which helps to describe data. It is |
---|
997 | applied in the case of nuclear |
---|
998 | clusterization (isotopically symmetric clusters) and in the case of |
---|
999 | hadronization in nuclear matter. In the hadronization process the |
---|
1000 | quasmon is shifted from the isotopic symmetric state (e.g., by capturing |
---|
1001 | a negative pion) and transfers excess charge to the outgoing nuclear |
---|
1002 | cluster. This tendency is symmetric with respect to the quasmon and |
---|
1003 | the parent cluster. |
---|
1004 | |
---|
1005 | The temperature parameter used to calculate the number of |
---|
1006 | quark-partons in a quasmon (see equation~\ref{temperatureII}) was chosen |
---|
1007 | to be $T=180$ MeV, which is the same as in Section \ref{annil}. |
---|
1008 | |
---|
1009 | CHIPS is mostly a model of fragmentation, conserving energy, momentum, and |
---|
1010 | charge. But to compare it with experimental data one needs to model also the |
---|
1011 | first interaction of the projectile with the |
---|
1012 | nucleus. For proton-antiproton annihilation this was easy, as we |
---|
1013 | assumed that in the interaction at rest, a proton and antiproton always |
---|
1014 | create a quasmon. In the case of pion capture the pion can be captured by |
---|
1015 | different clusters. We assumed that the probability of capture is |
---|
1016 | proportional to the number of nucleons in a cluster. After the |
---|
1017 | capture the quasmon is formed, and the CHIPS generator produces |
---|
1018 | fragments consecutively and recursively, choosing at each step the |
---|
1019 | quark-parton four-momentum $k$, the type of parent and outgoing fragment, |
---|
1020 | and the four-momentum of the exchange quark-parton $q$, to produce |
---|
1021 | a final state hadron and the new quasmon with less energy. |
---|
1022 | |
---|
1023 | In the CHIPS model we consider this process as a chaotic process |
---|
1024 | with large number of degrees of freedom and do not take into account |
---|
1025 | any final state interactions of outgoing hadrons. Nevertheless, when |
---|
1026 | the excitation energy dissipates, and in some step the quasmon mass |
---|
1027 | drops below the mass shell, the quark-parton mechanism of hadronization |
---|
1028 | fails. To model the event exclusively, it becomes necessary to |
---|
1029 | continue fragmentation at the hadron level. Such a fragmentation process |
---|
1030 | is known as nuclear evaporation. It is modeled using the |
---|
1031 | non-relativistic phase space approach. In the non-relativistic case the |
---|
1032 | phase space of nucleons can be integrated as well as in the |
---|
1033 | ultra-relativistic case of quark-partons. |
---|
1034 | |
---|
1035 | The general formula for the non-relativistic phase space can be found starting |
---|
1036 | with the phase space for two particles $\tilde{\Phi}_{2}$. It is |
---|
1037 | proportional to the center-of-mass momentum: |
---|
1038 | \begin{equation} |
---|
1039 | \tilde{\Phi}_2(W_2) \propto \sqrt{W_2}, \label{F2} |
---|
1040 | \end{equation} |
---|
1041 | where $W_2$\ is a total kinetic energy of the two non-relativistic |
---|
1042 | particles. If the phase space integral is known for $n-1$\ hadrons |
---|
1043 | then it is possible to calculate the phase space integral for $n$\ |
---|
1044 | hadrons: |
---|
1045 | \begin{eqnarray} |
---|
1046 | \tilde{\Phi}_{n}(W_n) &=&\int \tilde{\Phi}_{n-1}(W_{n-1}) \cdot |
---|
1047 | \delta (W_{n}-W_{n-1}-E_{\rm{kin}}) \nonumber \\ |
---|
1048 | &&\times \sqrt{E_{\rm{kin}}}dE_{\rm{kin}} dW_{n-1}. \label{Fn} |
---|
1049 | \end{eqnarray} |
---|
1050 | Using (\ref{F2})\ and (\ref{Fn})\ one can find that |
---|
1051 | \begin{equation} |
---|
1052 | \tilde{\Phi}_{n}(W_n)\propto W_{n}^{\frac{3}{2}n-\frac{5}{2}} |
---|
1053 | \end{equation} |
---|
1054 | and the spectrum of hadrons, defined by the phase space of residual |
---|
1055 | $n-1$ nucleons, can be written as |
---|
1056 | \begin{equation} |
---|
1057 | \frac{dN}{\sqrt{E_{\rm{kin}}}dE_{\rm{kin}}} \propto |
---|
1058 | \left(1-\frac{E_{\rm{kin}}}{W_{n}}\right)^{\frac{3}{2}n-4}. |
---|
1059 | \label{evap_spectr} |
---|
1060 | \end{equation} |
---|
1061 | This spectrum can be randomized. The only problem is from which level one |
---|
1062 | should measure the thermal kinetic energy when most nucleons in nuclei |
---|
1063 | are filling nuclear levels with zero temperature. To model the evaporation |
---|
1064 | process we used this unknown level as a parameter $U$\ of the evaporation |
---|
1065 | process. Comparison with experimental data gives $U=1.7$ MeV. Thus, the |
---|
1066 | total kinetic energy of $A$\ nucleons is |
---|
1067 | \begin{equation} |
---|
1068 | W_{A}=U\cdot A+E_{\rm{ex}}, |
---|
1069 | \end{equation} |
---|
1070 | where $E_{\rm{ex}}$ is the excitation energy of the nucleus. |
---|
1071 | |
---|
1072 | To\ be\ radiated,\ \ the nucleon\ \ should\ \ overcome\ \ the threshold |
---|
1073 | \begin{equation} |
---|
1074 | U_{\rm{thresh}}=U+U_{\rm{bind}}+E_{CB}, |
---|
1075 | \end{equation} |
---|
1076 | where $U_{\rm{bind}}$\ is the separation energy of the nucleon, and |
---|
1077 | $E_{CB}$\ is the Coulomb barrier energy which is non-zero only for |
---|
1078 | positive particles and can be calculated using formula |
---|
1079 | (\ref{CoulBar}). |
---|
1080 | |
---|
1081 | From several experimental investigations of nuclear pion capture at |
---|
1082 | rest, four published results have been selected here, which |
---|
1083 | constitute, in our opinion, a representative data set covering a wide |
---|
1084 | range of target nuclei, types of produced hadrons and nuclear |
---|
1085 | fragments, and their energy range. In the first publication |
---|
1086 | \cite{MIPHI}\ the spectra of charged fragments (protons, deuterons, |
---|
1087 | tritons, $^{3}$He, $^{4}$He) in pion capture were measured on |
---|
1088 | 17 nuclei within one experimental setup. To verify the spectra we |
---|
1089 | compared them for a carbon target with detailed measurements of the |
---|
1090 | spectra of charged fragments given in Ref.~\cite{Mechtersheimer}. In |
---|
1091 | addition, we took $^{6}$Li spectra for a carbon |
---|
1092 | target from the same paper. |
---|
1093 | |
---|
1094 | The neutron spectra were added from Ref.~\cite{Cernigoi} and |
---|
1095 | Ref.~\cite{Madey}. We present data and Monte Carlo distributions as |
---|
1096 | the invariant phase space function |
---|
1097 | $f=\frac{d\sigma}{pdE}$\ depending on the variable |
---|
1098 | $k=\frac{p+E_{\rm{kin}}}{2}$\ as defined in equation~(\ref{k}). |
---|
1099 | |
---|
1100 | Spectra on $^{9}$Be, $^{12}$C, $^{28}$Si ($^{27}$Al for secondary |
---|
1101 | neutrons), $^{59}$Co ($^{64}$Cu for secondary neutrons), and |
---|
1102 | $^{181}$Ta\ are shown in Figs.~\ref{be0405}\ through~\ref{ta73108}. |
---|
1103 | The data are well-described, including the total energy spent in the |
---|
1104 | reaction to yield the particular type of fragments. |
---|
1105 | |
---|
1106 | The evaporation process for nucleons is also well-described. It is |
---|
1107 | exponential in $k$, and looks especially impressive for Si/Al and |
---|
1108 | Co/Cu data, where the Coulomb barrier is low, and one can see proton |
---|
1109 | evaporation as a continuation of the evaporation spectra from |
---|
1110 | secondary neutrons. This way the exponential behavior of the |
---|
1111 | evaporation process can be followed over 3 orders of |
---|
1112 | magnitude. Clearly seen is\ the\ transition region at\ \ $k \approx |
---|
1113 | 90$\ MeV\ \ (kinetic energy $15-20$\ MeV)\ \ between the quark-level |
---|
1114 | hadronization process and the hadron-level evaporation process. For |
---|
1115 | light target nuclei the evaporation process becomes much less |
---|
1116 | prominent. |
---|
1117 | |
---|
1118 | The $^{6}$Li spectrum on a carbon target exhibits an interesting regularity |
---|
1119 | when plotted as a function of $k$: it practically coincides with the |
---|
1120 | spectrum of $^{4}$He fragments, and shows exponential behavior in a |
---|
1121 | wide range of $k$, corresponding to a few orders of magnitude in the |
---|
1122 | invariant cross section. To keep the figure readable, the $^{6}$Li |
---|
1123 | spectrum generated by CHIPS was not plotted. It coincides with the |
---|
1124 | $^{4}$He spectrum at $k > 200$\ MeV, and under-estimates lithium |
---|
1125 | emission at lower energies, similarly to the $^{3}$He and tritium data. |
---|
1126 | |
---|
1127 | Between the region where hadron-level processes dominate and the |
---|
1128 | kinematic limit, all hadronic spectrum slopes become similar when plotted |
---|
1129 | as a function of $k$. In addition to this general behavior there is |
---|
1130 | the effect of strong proton-neutron splitting. For protons and neutrons |
---|
1131 | it reaches almost an order of magnitude. To model such splitting in |
---|
1132 | the CHIPS generator, the mechanism of ``isotopic focusing'' was used, |
---|
1133 | which locally transfers the negative charge from the pion to the first |
---|
1134 | radiated nuclear fragment. |
---|
1135 | |
---|
1136 | \begin{table} |
---|
1137 | \caption{Clusterization parameters} |
---|
1138 | \label{tab:1} |
---|
1139 | \begin{tabular}{llllll} |
---|
1140 | \hline\noalign{\smallskip} |
---|
1141 | & $^{9}$Be & $^{12}$C & $^{28}$Si & $^{59}$Co & $^{181}$Ta \\ |
---|
1142 | \noalign{\smallskip}\hline\noalign{\smallskip} |
---|
1143 | $\varepsilon_{1}$ & 0.45 & 0.40 & 0.35 & 0.33 & 0.33 \\ |
---|
1144 | $\varepsilon_{2}$ & 0.15 & 0.15 & 0.05 & 0.03 & 0.02 \\ |
---|
1145 | $\omega $ & 5.00 & 5.00 & 5.00 & 5.00 & 5.00 \\ |
---|
1146 | \noalign{\smallskip}\hline |
---|
1147 | \end{tabular} |
---|
1148 | \end{table} |
---|
1149 | |
---|
1150 | Thus, the model qualitatively describes all typical features of the |
---|
1151 | pion capture process. The question is what can be extracted from the |
---|
1152 | experimental data with this tool. The clusterization parameters are |
---|
1153 | listed in Table~\ref{tab:1}. No formal fitting procedure has been |
---|
1154 | performed. A balanced qualitative agreement with all data was used to |
---|
1155 | tune the parameters. The difference between the $\frac{\varepsilon |
---|
1156 | _{2}}{\varepsilon _{1}}$\ ratio and the parameter $\omega$\ (which is |
---|
1157 | the same for all nuclei) is an indication that there is a |
---|
1158 | phase transition between the gas phase and the liquid phase of the |
---|
1159 | nucleus. The large value of the parameter $\omega$, determining the |
---|
1160 | average size of a nuclear cluster, is critical in describing |
---|
1161 | the model spectra |
---|
1162 | at large $k$, where the fragment spectra approach the kinematic |
---|
1163 | limits. |
---|
1164 | |
---|
1165 | Using the same parameters of clusterization, the $\gamma$\ absorption |
---|
1166 | data \cite{Ryckbosch} on Al and Ca nuclei were compared in |
---|
1167 | Fig.~\ref{gam62}) to the CHIPS results. One can see that the |
---|
1168 | spectra of secondary protons and deuterons are qualitatively described |
---|
1169 | by the CHIPS model. |
---|
1170 | |
---|
1171 | \begin{figure}[tbp] |
---|
1172 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/be0405k.eps, height=4.5in, width=4.5in}} |
---|
1173 | %\resizebox{1.00\textwidth}{!} |
---|
1174 | %{ |
---|
1175 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/be0405k.eps} |
---|
1176 | %\includegraphics[angle=0,scale=0.6]{plots/be0405k.eps} |
---|
1177 | %} |
---|
1178 | \caption{\protect{Comparison of the CHIPS model results with |
---|
1179 | experimental data on proton, neutron, and nuclear fragment production |
---|
1180 | in the capture of negative pions on $^9$Be. |
---|
1181 | Proton~\cite{MIPHI} and neutron~\cite{Cernigoi}\ experimental spectra |
---|
1182 | are shown in the upper left panel by open circles and open squares, |
---|
1183 | respectively. The model calculations are shown by the two |
---|
1184 | corresponding solid lines. The same arrangement |
---|
1185 | is used to present $^{3}$He~\cite{MIPHI} |
---|
1186 | and tritium~\cite{MIPHI} |
---|
1187 | spectra in the lower left panel. Deuterium~\cite{MIPHI} |
---|
1188 | and $^{4}$He~\cite{MIPHI} spectra are |
---|
1189 | shown in the right panels of the figure by open squares |
---|
1190 | and lines (CHIPS model). The average kinetic |
---|
1191 | energy carried away by each nuclear fragment is shown in the panels |
---|
1192 | by the two numbers: first is the average calculated using the |
---|
1193 | experimental data shown; second is the model result.}} |
---|
1194 | \label{be0405} |
---|
1195 | \end{figure} |
---|
1196 | |
---|
1197 | \begin{figure}[tbp] |
---|
1198 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/c0606k.eps, height=4.5in, width=4.5in}} |
---|
1199 | %\resizebox{1.00\textwidth}{!} |
---|
1200 | %{ |
---|
1201 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/c0606k.eps} |
---|
1202 | %\includegraphics[angle=0,scale=0.6]{plots/c0606k.eps} |
---|
1203 | %} |
---|
1204 | \caption{\protect{Same as in Figure~\ref{be0405}, for |
---|
1205 | pion capture on $^{12}$C. The experimental neutron spectrum |
---|
1206 | is taken from \cite{Madey}. In addition, the detailed data on |
---|
1207 | charged particle production, including the $^{6}$Li spectrum, taken from |
---|
1208 | Ref.~\cite{Mechtersheimer}, are superimposed on the plots as a series of |
---|
1209 | dots.}} |
---|
1210 | \label{c0606} |
---|
1211 | \end{figure} |
---|
1212 | |
---|
1213 | \begin{figure}[tbp] |
---|
1214 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/si1414k.eps, height=4.5in, width=4.5in}} |
---|
1215 | %\resizebox{1.00\textwidth}{!} |
---|
1216 | %{ |
---|
1217 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/si1414k.eps} |
---|
1218 | %\includegraphics[angle=0,scale=0.6]{plots/si1414k.eps} |
---|
1219 | %} |
---|
1220 | \caption{\protect{Same as in Figure~\ref{be0405}, for |
---|
1221 | pion capture on $^{28}$Si nucleus. The experimental neutron spectrum |
---|
1222 | is taken from~\cite{Madey}, for the reaction on $^{27}$Al.}} |
---|
1223 | \label{si1414} |
---|
1224 | \end{figure} |
---|
1225 | |
---|
1226 | \begin{figure}[tbp] |
---|
1227 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/co2732k.eps, height=4.5in, width=4.5in}} |
---|
1228 | %\resizebox{1.00\textwidth}{!} |
---|
1229 | %{ |
---|
1230 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/co2732k.eps} |
---|
1231 | %\includegraphics[angle=0,scale=0.6]{plots/co2732k.eps} |
---|
1232 | %} |
---|
1233 | \caption{\protect{Same as in Figure~\ref{be0405}, for |
---|
1234 | pion capture on $^{59}$Co. The experimental neutron spectrum |
---|
1235 | is taken from~\cite{Madey}, for the reaction on $^{64}$Cu.}} |
---|
1236 | \label{co2732} |
---|
1237 | \end{figure} |
---|
1238 | |
---|
1239 | \begin{figure}[tbp] |
---|
1240 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/ta73108k.eps, height=4.5in, width=4.5in}} |
---|
1241 | %\resizebox{1.00\textwidth}{!} |
---|
1242 | %{ |
---|
1243 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/ta73108k.eps} |
---|
1244 | %\includegraphics[angle=0,scale=0.6]{plots/ta73108k.eps} |
---|
1245 | %} |
---|
1246 | \caption{\protect{Same as in Figure~\ref{be0405}, for |
---|
1247 | pion capture on $^{181}$Ta. The experimental neutron |
---|
1248 | spectrum is taken from~\cite{Madey}.}} |
---|
1249 | \label{ta73108} |
---|
1250 | \end{figure} |
---|
1251 | |
---|
1252 | \begin{figure}[tbp] |
---|
1253 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps, height=4.5in, width=4.5in}} |
---|
1254 | %\resizebox{0.70\textwidth}{!} |
---|
1255 | %{ |
---|
1256 | \includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps} |
---|
1257 | %\includegraphics[angle=0,scale=0.75]{plots/gam62.eps} |
---|
1258 | %} |
---|
1259 | \caption{\protect{Comparison of CHIPS model with |
---|
1260 | experimental data~\cite{Ryckbosch} on |
---|
1261 | proton and deuteron production at $90^{\circ}$ |
---|
1262 | in photonuclear reactions on $^{27}$Al |
---|
1263 | and $^{40}$Ca at 59 -- 65 MeV. Open circles and solid squares represent the |
---|
1264 | experimental proton and deuteron spectra, |
---|
1265 | respectively. Solid |
---|
1266 | and dashed lines show the results of the corresponding CHIPS model |
---|
1267 | calculation. Statistical errors in the CHIPS results are not shown and |
---|
1268 | can be judged by the point-to-point variations in the lines. The |
---|
1269 | comparison is absolute, using the values of total |
---|
1270 | photonuclear cross section 3.6 mb for Al and 5.4 mb for Ca, |
---|
1271 | as given in Ref.~\cite{Ahrens}. |
---|
1272 | }} |
---|
1273 | \label{gam62} |
---|
1274 | \end{figure} |
---|
1275 | |
---|
1276 | The CHIPS model covers a wide spectrum of hadronic reactions with a |
---|
1277 | large number of degrees of freedom. In the case of nuclear reactions |
---|
1278 | the CHIPS generator helps to understand phenomena such as the |
---|
1279 | order-of-magnitude splitting of neutron and proton spectra, the high |
---|
1280 | yield of energetic nuclear fragments, and the emission of nucleons |
---|
1281 | which kinematically can be produced only if seven or more nucleons |
---|
1282 | are involved in the reaction. |
---|
1283 | |
---|
1284 | The CHIPS generator allows the extraction of collective parameters of |
---|
1285 | a nucleus such as clusterization. The qualitative conclusion based on |
---|
1286 | the fit to the experimental data is that most of the nucleons are |
---|
1287 | clusterized, at least in heavy nuclei. The nuclear clusters can be |
---|
1288 | considered as drops of a liquid nuclear phase. The quark exchange |
---|
1289 | makes the phase space of quark-partons of each cluster common, |
---|
1290 | stretching the kinematic limits for particle production. |
---|
1291 | |
---|
1292 | The hypothetical quark exchange process is important not only for |
---|
1293 | nuclear clusterization, but also for the nuclear hadronization |
---|
1294 | process. The quark exchange between the excited cluster (quasmon) |
---|
1295 | and a neighboring nuclear cluster, even at low excitation level, |
---|
1296 | operates with quark-partons at energies comparable with the nucleon |
---|
1297 | mass. As a result it easily reaches the kinematic limits of the |
---|
1298 | reaction, revealing the multi-nucleon nature of the process. |
---|
1299 | |
---|
1300 | Up to now the most under-developed part of the model has been the |
---|
1301 | initial interaction between projectile and target. That is why we |
---|
1302 | started with proton-antiproton annihilation and pion capture on |
---|
1303 | nuclei at rest, because the interaction cross section is not involved. |
---|
1304 | The further development of the model will require a better |
---|
1305 | understanding of the mechanism of the first interaction. However, |
---|
1306 | we believe that even the basic model will be useful in the |
---|
1307 | understanding the nature of multi-hadron fragmentation. Because |
---|
1308 | of the model's features, it is a suitable candidate for the hadron |
---|
1309 | production and hadron cascade parts of the newly developed event |
---|
1310 | generation and detector simulation Monte Carlo computer codes. |
---|
1311 | |
---|
1312 | \section{Modeling of real and virtual photon |
---|
1313 | interactions with nuclei below pion production threshold.} |
---|
1314 | |
---|
1315 | In the example of |
---|
1316 | the photonuclear reaction discussed in the Appendix D, namely |
---|
1317 | the description of $90^{\circ}$ proton and deuteron spectra in |
---|
1318 | $A({\gamma},X)$ reactions at $E_{\gamma} = 59-65$ MeV, the assumption |
---|
1319 | on the initial Quasmon excitation mechanism was the same. The |
---|
1320 | description of the $90^{\circ}$ data was satisfactory, but the |
---|
1321 | generated data showed very little angular dependence, because the |
---|
1322 | velocity of the quasmons produced in the initial state was small, |
---|
1323 | and the fragmentation process was almost isotropic. Experimentally, |
---|
1324 | the angular dependence of secondary protons in photo-nuclear reactions |
---|
1325 | is quite strong even at low energies (see, for example, |
---|
1326 | Ref.~\cite{Ryckebusch}). This is a challenging experimental fact which |
---|
1327 | is difficult to explain in any model. It's enough to say that if the |
---|
1328 | angular dependence of secondary protons in the $\gamma ^{40}$Ca |
---|
1329 | interaction at 60 MeV is analyzed in terms of relativistic boost, then |
---|
1330 | the velocity of the source should reach $0.33 c$; hence the mass |
---|
1331 | of the source should be less than pion mass. The main point of this |
---|
1332 | discussion is to show that the quark-exchange mechanism used in the |
---|
1333 | CHIPS model can not only model the clusterization of nucleons in nuclei |
---|
1334 | and hadronization of intranuclear excitations into nuclear fragments, |
---|
1335 | but it can also model complicated mechanisms of the interaction of |
---|
1336 | photons and hadrons in nuclear matter. |
---|
1337 | |
---|
1338 | In Ref. Appendix D a quark-exchange diagram was defined which |
---|
1339 | helps to keep track of the kinematics of the quark-exchange process |
---|
1340 | (see Fig.~1 in Apendix D). To apply the same diagram to |
---|
1341 | the first interaction of a photon with a nucleus, it is necessary to |
---|
1342 | assume that the quark-exchange process takes place in nuclei |
---|
1343 | continuously, even |
---|
1344 | without any external interaction. Nucleons with high momenta do not |
---|
1345 | leave the nucleus because of the lack of excess energy. The |
---|
1346 | hypothesis of the CHIPS model is that the quark-exchange forces |
---|
1347 | between nucleons \cite{NN QEX}\ continuously create clusters in normal |
---|
1348 | nuclei. Since a low-energy photon (below the pion production threshold) |
---|
1349 | cannot be absorbed by a free nucleon, other absorption mechanisms |
---|
1350 | involving more than one nucleon have to be used. |
---|
1351 | |
---|
1352 | The simplest scenario is photon absorption by a quark-parton in |
---|
1353 | the nucleon. At low energies and in vacuum this does not work because |
---|
1354 | there is no corresponding excited baryonic state. But in nuclear matter |
---|
1355 | it is possible to exchange this quark with a neighboring nucleon |
---|
1356 | or a nuclear cluster. The diagram for the process is shown in |
---|
1357 | Fig.~\ref{diagram1}. In this case the photon is absorbed by a |
---|
1358 | quark-parton from the parent cluster $\rm{PC}_1$, and then |
---|
1359 | the secondary nucleon or cluster $\rm{PC}_2$ |
---|
1360 | absorbs the entire momentum of the quark and photon. The exchange |
---|
1361 | quark-parton $q$ restores the balance of color, producing the |
---|
1362 | final-state hadron F and the residual Quasmon RQ. The process looks like a |
---|
1363 | knockout of a quasi-free nucleon or cluster out of the nucleus. It should be |
---|
1364 | emphasized that in this scenario the CHIPS event generator |
---|
1365 | produces not only ``quasi-free'' nucleons but ``quasi-free'' fragments |
---|
1366 | as well. The yield of these quasi-free nucleons or fragments is |
---|
1367 | concentrated in the forward direction. |
---|
1368 | |
---|
1369 | The second scenario which provides for an angular dependence is the absorption |
---|
1370 | of the photon by a colored fragment ($\rm{CF}_2$ |
---|
1371 | in Fig.~\ref{diagram2}). In this |
---|
1372 | scenario, both the primary quark-parton with momentum $k$ and the photon |
---|
1373 | with momentum $q_{\gamma}$ are absorbed by a parent cluster ($\rm{PC}_2$ in |
---|
1374 | Fig.~\ref{diagram2}), and the recoil quark-parton with momentum $q$ |
---|
1375 | cannot fully compensate the momentum $k+q_{\gamma}$. |
---|
1376 | As a result the radiation of the |
---|
1377 | secondary fragment in the forward direction becomes more probable. |
---|
1378 | |
---|
1379 | In both cases the angular dependence is defined by the first act of |
---|
1380 | hadronization. Further fragmentation of the residual quasmon is |
---|
1381 | almost isotropic. |
---|
1382 | |
---|
1383 | \begin{figure}[tbp] |
---|
1384 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram1.eps, height=2.5in, width=2.5in}} |
---|
1385 | %\resizebox{0.70\textwidth}{!} |
---|
1386 | %{ |
---|
1387 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram1.eps} |
---|
1388 | %\includegraphics[angle=0,scale=0.6]{plots/diagram1.eps} |
---|
1389 | %} |
---|
1390 | \caption{\protect{Diagram of photon absorption in the quark |
---|
1391 | exchange mechanism. $\rm{PC}_{1,2}$ stand for parent clusters |
---|
1392 | with bound masses |
---|
1393 | $\tilde{\mu}_{1,2}$, participating in the quark-exchange. $\rm{CF}_{1,2}$ |
---|
1394 | stand for the colored nuclear fragments in the process of quark |
---|
1395 | exchange. F($\mu$) denotes the outgoing hadron with mass $\mu$ in the |
---|
1396 | final state. RQ is the residual Quasmon which carries the rest of the |
---|
1397 | excitation energy and momentum. $M_{\min}$ characterizes |
---|
1398 | its minimum mass defined by its quark content. Dashed lines indicate |
---|
1399 | colored objects. The photon is absorbed by a |
---|
1400 | quark-parton $k$ from the parent cluster $\rm{PC}_1$. |
---|
1401 | }} |
---|
1402 | \label{diagram1} |
---|
1403 | \end{figure} |
---|
1404 | |
---|
1405 | \begin{figure}[tbp] |
---|
1406 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram2.eps, height=2.5in, width=2.5in}} |
---|
1407 | %\resizebox{0.70\textwidth}{!} |
---|
1408 | %{ |
---|
1409 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/diagram2.eps} |
---|
1410 | %\includegraphics[angle=0,scale=0.6]{plots/diagram2.eps} |
---|
1411 | %} |
---|
1412 | \caption{\protect{Diagram of photon absorption in the |
---|
1413 | quark-exchange mechanism. The notation is the same as in |
---|
1414 | Fig.~\ref{diagram1}. The photon is absorbed by the colored fragment |
---|
1415 | $\rm{CF}_2$. |
---|
1416 | }} |
---|
1417 | \label{diagram2} |
---|
1418 | \end{figure} |
---|
1419 | |
---|
1420 | It was shown in Section \ref{annil} that the energy spectrum of quark |
---|
1421 | partons in a quasmon can be calculated as |
---|
1422 | \begin{equation} |
---|
1423 | \frac{dW}{k^{\ast }dk^{\ast }}\propto |
---|
1424 | \left(1-\frac{2k^{\ast }}{M} \right)^{N-3}, \label{spectrum_1III} |
---|
1425 | \end{equation} |
---|
1426 | where $k^{\ast }$ is the energy of the primary quark-parton in the |
---|
1427 | center-of-mass system of the quasmon, $M$\ is the mass of the quasmon. |
---|
1428 | The number $N$ of quark-partons in the quasmon can be calculated |
---|
1429 | from the equation |
---|
1430 | \begin{equation} |
---|
1431 | <M^{2}>=4\cdot N\cdot (N-1)\cdot T^{2}. \label{temperatureIII} |
---|
1432 | \end{equation} |
---|
1433 | Here $T$ is the temperature of the system. |
---|
1434 | |
---|
1435 | In the first scenario of the $\gamma A$ interaction |
---|
1436 | (Fig.~\ref{diagram1}), because both interacting particles are massless, |
---|
1437 | we assumed that the cross section for the interaction of a photon with |
---|
1438 | a particular quark-parton is proportional to the charge of the |
---|
1439 | quark-parton squared, and inversely proportional to the mass of the |
---|
1440 | photon-parton system $s$, which can be calculated as |
---|
1441 | \begin{equation} |
---|
1442 | s=2\omega k(1-\cos (\theta _{k})). \label{s} |
---|
1443 | \end{equation} |
---|
1444 | Here $\omega $\ is the energy of the photon, and $k$ is the energy of |
---|
1445 | the quark-parton in the laboratory system (LS): |
---|
1446 | \begin{equation} |
---|
1447 | k=k^{\ast }\cdot \frac{E_{N}+p_{N}\cdot \cos (\theta _{k})}{M_{N}}. |
---|
1448 | \end{equation} |
---|
1449 | For a virtual photon, equation~(\ref{s}) can be written as |
---|
1450 | \begin{equation} |
---|
1451 | s=2k(\omega -q_{\gamma}\cdot \cos (\theta _{k})), |
---|
1452 | \end{equation} |
---|
1453 | where $q_{\gamma}$ is the momentum of the virtual photon. In both cases |
---|
1454 | equation~(\ref{spectrum_1III}) transforms into |
---|
1455 | \begin{equation} |
---|
1456 | \frac{dW}{dk^{\ast }}\propto \left(1-\frac{2k^{\ast }}{M} \right)^{N-3}, |
---|
1457 | \end{equation} |
---|
1458 | and the angular distribution in $\cos (\theta _{k})$\ converges to a |
---|
1459 | $\delta $-function. In the case of a real photon |
---|
1460 | $\cos (\theta _{k})=1$, and in the case of a virtual photon |
---|
1461 | $\cos (\theta _{k})=\frac{\omega }{q_{\gamma}}$. |
---|
1462 | |
---|
1463 | In the second scenario for the photon interaction |
---|
1464 | (Fig.~\ref{diagram2}) we assumed that both the photon and the primary |
---|
1465 | quark-parton, randomized according to |
---|
1466 | Eq.~(\ref{spectrum_1III}), enter the parent cluster $\rm{PC}_2$, |
---|
1467 | and after that the normal procedure of quark exchange |
---|
1468 | continues, in which the recoiling quark-parton $q$ returns |
---|
1469 | to the first cluster. |
---|
1470 | |
---|
1471 | An additional parameter in the model is the relative contribution of |
---|
1472 | both mechanisms. As a first approximation we assumed equal |
---|
1473 | probability, but in the future, when more detailed data are obtained, |
---|
1474 | this parameter can be adjusted. |
---|
1475 | |
---|
1476 | \begin{figure}[tbp] |
---|
1477 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps, height=4.5in, width=4.5in}} |
---|
1478 | %\resizebox{0.80\textwidth}{!} |
---|
1479 | %{ |
---|
1480 | \includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gam62.eps} |
---|
1481 | %\includegraphics[angle=0,scale=0.75]{plots/gam62.eps} |
---|
1482 | %} |
---|
1483 | \caption{\protect{Comparison of the CHIPS model results (lines) with the |
---|
1484 | experimental data~\cite{Ryckbosch} on proton spectra at $90^{\circ}$ |
---|
1485 | in the photonuclear reactions on $^{40}$Ca at 59--65 MeV (open |
---|
1486 | circles), |
---|
1487 | and proton spectra at $60^{\circ}$ (triangles) and $150^{\circ}$ |
---|
1488 | (diamonds). |
---|
1489 | Statistical errors in the CHIPS results are not shown but |
---|
1490 | can be judged by the point-to-point variations in the lines. The |
---|
1491 | comparison is absolute, using the value of the total |
---|
1492 | photonuclear cross section of 5.4 mb for Ca, as given in Ref.~\cite{Ahrens}. |
---|
1493 | } } |
---|
1494 | \label{gam62III} |
---|
1495 | \end{figure} |
---|
1496 | |
---|
1497 | \begin{figure}[tbp] |
---|
1498 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e123.eps, height=4.5in, width=4.5in}} |
---|
1499 | %\resizebox{0.80\textwidth}{!} |
---|
1500 | %{ |
---|
1501 | \includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e123.eps} |
---|
1502 | %\includegraphics[angle=0,scale=0.75]{plots/gamm_c0606_e123.eps} |
---|
1503 | %} |
---|
1504 | \caption{\protect{Comparison of the CHIPS model results (lines) with the |
---|
1505 | experimental data~\cite{Harty} on |
---|
1506 | proton spectra at $57^{\circ}$, $77^{\circ}$, $97^{\circ}$, |
---|
1507 | $117^{\circ}$, and $127^{\circ}$ |
---|
1508 | in the photonuclear reactions on $^{12}$C at 123 MeV (open |
---|
1509 | circles). The value of the total photonuclear cross section was set to 1.8 mb. |
---|
1510 | } } |
---|
1511 | \label{gam_123} |
---|
1512 | \end{figure} |
---|
1513 | |
---|
1514 | \begin{figure}[tbp] |
---|
1515 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e151.eps, height=4.5in, width=4.5in}} |
---|
1516 | %\resizebox{0.80\textwidth}{!} |
---|
1517 | %{ |
---|
1518 | \includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gamm_c0606_e151.eps} |
---|
1519 | %\includegraphics[angle=0,scale=0.75]{plots/gamm_c0606_e151.eps} |
---|
1520 | %} |
---|
1521 | \caption{\protect{Same as in Fig.~\ref{gam_123}, for the photon energy 151 MeV.} |
---|
1522 | } |
---|
1523 | \label{gam_151} |
---|
1524 | \end{figure} |
---|
1525 | |
---|
1526 | \begin{figure}[tbp] |
---|
1527 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}} |
---|
1528 | %\resizebox{0.80\textwidth}{!} |
---|
1529 | %{ |
---|
1530 | \includegraphics[angle=0,scale=0.75]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps} |
---|
1531 | %\includegraphics[angle=0,scale=0.75]{plots/vgam_c0606k.eps} |
---|
1532 | %} |
---|
1533 | \caption{\protect{Comparison of the CHIPS model results (line) with |
---|
1534 | the experimental data~\cite{Bates} (open circles) on the |
---|
1535 | proton spectrum measured in parallel kinematics in the |
---|
1536 | $^{12}$C(e,e$^{\prime}$p)\ reaction at an energy transfer equal to 210 |
---|
1537 | MeV and momentum transfer equal to 585 MeV/$c$. Statistical errors in |
---|
1538 | the CHIPS result are not shown but can be judged by the point-to-point |
---|
1539 | variations in the line. The relative normalization is arbitrary. |
---|
1540 | } } |
---|
1541 | \label{vgam} |
---|
1542 | \end{figure} |
---|
1543 | |
---|
1544 | We begin the comparison with the data on proton production in the |
---|
1545 | $^{40}$Ca$(\gamma,X)$\ reaction at $90^{\circ}$\ and 59--65 MeV |
---|
1546 | \cite{Ryckbosch}, and at $60^{\circ}$\ and $150^{\circ}$\ and 60 MeV |
---|
1547 | \cite{Abeele}. We analyzed these data together to compare the angular |
---|
1548 | dependence generated by CHIPS with experimental data. The data are |
---|
1549 | presented as a function of the invariant inclusive cross section |
---|
1550 | $f=\frac{d\sigma }{p_{p}dE_{p}}$\ depending on the variable |
---|
1551 | $k=\frac{T_{p}+p_{p}}{2}$, |
---|
1552 | where $T_{p}$\ and $p_{p}$\ are the kinetic energy and momentum of the |
---|
1553 | secondary proton. As one can see from Fig.~\ref{gam62III}, the angular |
---|
1554 | dependence of the proton yield in photoproduction on $^{40}$Ca at |
---|
1555 | $60$ MeV is reproduced quite well by the CHIPS event generator. |
---|
1556 | |
---|
1557 | The second set of measurements that we use for the benchmark |
---|
1558 | comparison deals with the secondary proton yields in |
---|
1559 | $^{12}$C$(\gamma,X)$ reactions at 123 and 151 MeV \cite{Harty}, |
---|
1560 | which is still below the pion production threshold on |
---|
1561 | a free nucleon. Inclusive spectra of protons have been measured in |
---|
1562 | $\gamma ^{12}$C reactions at $57^{\circ}$, $77^{\circ}$, $97^{\circ}$, |
---|
1563 | $117^{\circ}$, and $127^{\circ}$. |
---|
1564 | Originally, these data were presented as a function of |
---|
1565 | the missing energy. We present the data in Figs.~\ref{gam_123} |
---|
1566 | and \ref{gam_151} together with CHIPS calculations in |
---|
1567 | the form of the invariant inclusive cross section dependent on $k$. |
---|
1568 | All parameters of the model such as temperature $T$ and parameters |
---|
1569 | of clusterization for the particular nucleus were the same as in |
---|
1570 | Appendix D, where pion capture spectra were fitted. |
---|
1571 | The agreement between the experimental data and the CHIPS model results |
---|
1572 | is quite remarkable. Both data and calculations show significant strength |
---|
1573 | in the proton yield cross section up to the kinematic limits of the |
---|
1574 | reaction. The angular distribution in the model is not as prominent as |
---|
1575 | in the experimental data, but agrees well qualitatively. |
---|
1576 | |
---|
1577 | Using the same parameters, we applied the CHIPS event generator to the |
---|
1578 | $^{12}$C(e,e$^{\prime }$p) reaction measured in Ref.\cite{Bates}. The |
---|
1579 | proton spectra were measured in parallel kinematics in the interaction |
---|
1580 | of virtual photons with energy $\omega = 210$ MeV and momentum |
---|
1581 | $q_{\gamma} = 585$ MeV/$c$. To account for the experimental conditions |
---|
1582 | in the CHIPS event generator, we have selected protons generated in |
---|
1583 | the forward direction with respect to the direction of the virtual |
---|
1584 | photon, with the relative angle $\Theta_{qp} < 6^{\circ}$. The CHIPS |
---|
1585 | generated distribution and the experimental data are shown in |
---|
1586 | Fig.~\ref{vgam} in the form of the invariant inclusive cross section as a |
---|
1587 | function of $k$. The CHIPS event generator works only with ground |
---|
1588 | states of nuclei so we did not expect any narrow peaks for |
---|
1589 | $^{1}p_{3/2}$-shell knockout or for other shells. Nevertheless we |
---|
1590 | found that the CHIPS event generator fills in the so-called |
---|
1591 | ``$^{1}s_{1/2}$-shell knockout'' region, which is usually artificially |
---|
1592 | smeared by a Lorentzian~\cite{Lorentzian}. In the regular |
---|
1593 | fragmentation scenario the spectrum of protons below $k = 300$ MeV is |
---|
1594 | normal; it falls down to the kinematic limit. The additional yield at |
---|
1595 | $k > 300$ MeV is a reflection of the specific first act of |
---|
1596 | hadronization with the quark exchange kinematics. The slope increase |
---|
1597 | with momentum is approximated well by the model, but it is obvious |
---|
1598 | that the yield close to the kinematic limit of the $2 \rightarrow 2$ |
---|
1599 | reaction can only be described in detail if the excited states of the |
---|
1600 | residual nucleus are taken into account. |
---|
1601 | |
---|
1602 | The angular dependence of the proton yield in low-energy photo-nuclear |
---|
1603 | reactions is described in the CHIPS model and event generator. The |
---|
1604 | most important assumption in the description is the hypothesis of a |
---|
1605 | direct interaction of the photon with an asymptotically free quark in |
---|
1606 | the nucleus, even at low energies. This means that asymptotic freedom of |
---|
1607 | QCD and dispersion sum rules~\cite{sum_rules} can in some way be |
---|
1608 | generalized for low energies. The knockout of a proton from a nuclear |
---|
1609 | shell or the homogeneous distributions of nuclear evaporation cannot |
---|
1610 | explain significant angular dependences at low energies. |
---|
1611 | |
---|
1612 | The same mechanism appears to be capable of modeling proton yields in |
---|
1613 | such reactions as the $^{16}$C(e,e$^{\prime }$p) reaction measured at MIT |
---|
1614 | Bates \cite{Bates}, where it was shown that the region of missing |
---|
1615 | energy above 50 MeV reflects ``two-or-more-particle knockout'' (or the |
---|
1616 | ``continuum'' in terms of the shell model). The CHIPS model may help |
---|
1617 | to understand and model such phenomena. |
---|
1618 | |
---|
1619 | \section{Chiral invariant phase-space decay in high energy hadron nuclear |
---|
1620 | reactions} |
---|
1621 | |
---|
1622 | \noindent \qquad Chiral invariant phase-space decay can be used to |
---|
1623 | de-excite an excited hadronic system. This possibility can be exploited |
---|
1624 | to replace the intra-nuclear cascading after a high energy primary |
---|
1625 | interaction takes place. The basic assumption in this is that the energy |
---|
1626 | loss of the high energy hadron in nuclear matter is approximately |
---|
1627 | constant per unit path length (about 1 GeV/fm). This energy is extracted |
---|
1628 | from the soft part of the particle spectrum of the primary interaction, |
---|
1629 | and from particles with formation times that place them within the |
---|
1630 | nuclear boundaries. |
---|
1631 | |
---|
1632 | Several approaches of transfering this energy into quasmons were studied, |
---|
1633 | and comparisons with energy spectra of particles emitted in the backward |
---|
1634 | hemisphere were made for a range of materials. Best results were achieved |
---|
1635 | with a model that creates one quasmon per particle absorbed in the nucleus. |
---|
1636 | |
---|
1637 | |
---|
1638 | \section{Neutrino-nuclear interactions} |
---|
1639 | \label{numunuc} |
---|
1640 | |
---|
1641 | The simulation of DIS reactions includes reactions with high $Q^2$. The |
---|
1642 | first approximation of the $Q^2$-dependent photonuclear cross-sections |
---|
1643 | at high $Q^2$ was made in \cite{photNuc}, where the modified photonuclear |
---|
1644 | cross sections of virtual photons \cite{Electronuc} were used. The |
---|
1645 | structure functions of protons and deuterons have been approximated in |
---|
1646 | CHIPS by the sum of |
---|
1647 | non-perturbative multiperipheral and non-perturbative direct |
---|
1648 | interactions of virtual photons with hadronic partons: |
---|
1649 | \begin{figure}[tbp] |
---|
1650 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}} |
---|
1651 | %\resizebox{0.80\textwidth}{!} |
---|
1652 | %{ |
---|
1653 | \includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/gabsa.eps} |
---|
1654 | %\includegraphics[angle=0,scale=0.60]{plots/gabsa.eps} |
---|
1655 | %} |
---|
1656 | \caption{ |
---|
1657 | Fit of $\gamma A$ cross sections with different $H$ values. Data are |
---|
1658 | from \cite{photNuc}. |
---|
1659 | } |
---|
1660 | \label{gamC} |
---|
1661 | \end{figure} |
---|
1662 | \begin{equation} |
---|
1663 | F_2(x,Q^2)=[A(Q^2)\cdot x^{-\Delta(Q^2)}+B(Q^2)\cdot |
---|
1664 | x]\cdot(1-x)^{N(Q^2)-2}, |
---|
1665 | \label{DIS} |
---|
1666 | \end{equation} |
---|
1667 | where $A(Q^2)=\bar{e^2_S}\cdot D\cdot U$, $B(Q^2)=\bar{e^2_V}\cdot(1-D)\cdot V$, |
---|
1668 | $\bar{e^2}_{V(p)}=\frac{1}{3}$, $\bar{e^2}_{V(d)}=\frac{5}{18}$, |
---|
1669 | $\bar{e^2_S}=\frac{1}{3}-\frac{\frac{1}{3}-\frac{5}{18}}{1+m^2_\phi/Q^2} |
---|
1670 | +\frac{\frac{1}{3}-\frac{5}{18}}{1+m^2_{J/\psi}/Q^2}- |
---|
1671 | \frac{\frac{1}{3}-\frac{19}{63}}{1+m^2_{\Upsilon}/Q^2}$, |
---|
1672 | $N=3+\frac{0.5}{\alpha_s(Q^2)}$, |
---|
1673 | $\alpha_s(Q^2)=\frac{4\pi}{\beta_0 ln(1+\frac{Q^2}{\Lambda^2})}$, |
---|
1674 | $\beta_0^{(n_f=3)}=9$, $\Lambda=200~MeV$, |
---|
1675 | $U=\frac{(3~C(Q^2)+N-3)\cdot\Gamma(N-\Delta)} |
---|
1676 | {N\cdot\Gamma(N-1)\cdot\Gamma(1-\Delta)}$, $V=3(N-1)$, |
---|
1677 | $D(Q^2)=H\cdot S(Q^2)\left(1-\frac{1}{2}S(Q^2)\frac{\bar{e^2_V}}{\bar{e^2_S}} |
---|
1678 | \right)$, |
---|
1679 | $S={\left(1+\frac{m^2_\rho}{Q^2}\right)^{-\alpha_P(Q^2)}}$, |
---|
1680 | $\alpha_P=1+\Delta(Q^2)$, $\Delta=\frac{1+r}{12.5+2r}$, |
---|
1681 | $r=\left(\frac{Q^2}{1.66}\right)^{1/2}$, $C=\frac{1+f}{g\cdot (1+f/.24)}$, |
---|
1682 | $f=\left(\frac{Q^2}{0.08}\right)^2$, $g=1+\frac{Q^2}{21.6}$. |
---|
1683 | The parton distributions are normalized to the unit total momentum |
---|
1684 | fraction. |
---|
1685 | |
---|
1686 | \begin{figure}[tbp] |
---|
1687 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}} |
---|
1688 | %\resizebox{0.80\textwidth}{!} |
---|
1689 | %{ |
---|
1690 | \includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/f23nud.eps} |
---|
1691 | %\includegraphics[angle=0,scale=0.60]{plots/f23nud.eps} |
---|
1692 | %} |
---|
1693 | \caption{ |
---|
1694 | Fit of $f_{2d}(x,Q^2)$ (filled circles, solid lines) and |
---|
1695 | $f_{3d}(x,Q^2)$ (open circles, dashed lines) structure functions |
---|
1696 | measured by the WA25 experiment \cite{WA25}. |
---|
1697 | } |
---|
1698 | \label{nuD} |
---|
1699 | \end{figure} |
---|
1700 | |
---|
1701 | The photonuclear cross sections are calculated by the eikonal formula: |
---|
1702 | \begin{equation} |
---|
1703 | \sigma_\gamma^{tot}=\left[\frac{4\pi\alpha}{Q^2}F_2\left(\frac{Q^2} |
---|
1704 | {2M\nu},Q^2\right)\right]^{\nu=E}_{Q^2=0}, |
---|
1705 | \label{eikonal} |
---|
1706 | \end{equation} |
---|
1707 | An example of the approximation is shown in Fig.~\ref{gamC}. One can |
---|
1708 | see that the hadronic resonances are ``melted'' in nuclear matter and |
---|
1709 | the multi-peripheral part of the cross section (high energy) is |
---|
1710 | shadowed. |
---|
1711 | |
---|
1712 | The differential cross section of the $(\nu,\mu)$ reaction was |
---|
1713 | approximated as |
---|
1714 | \begin{equation} |
---|
1715 | \frac{yd^2\sigma^{\nu,\bar\nu}}{dydQ^2}=\frac{G^2_F\cdot M^4_W}{4\pi\cdot |
---|
1716 | (Q^2+M^2_W)^2}\left[c_1(y)\cdot f_2(x,Q^2)\pm c_2(y)\cdot xf_3(x,Q^2)\right], |
---|
1717 | \label{difsec} |
---|
1718 | \end{equation} |
---|
1719 | where $c_1(y)=2-2y+\frac{y^2}{1+R}$, $R=\frac{\sigma_L}{\sigma_T}$, |
---|
1720 | $c_2(y)=y(2-y)$. As $\bar{e^2_V}=\bar{e^2_S}=1$ in |
---|
1721 | Eq.\ref{DIS}, hence $f_2(x,Q^2)=\left[D\cdot U\cdot |
---|
1722 | x^{-\Delta}+(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$, |
---|
1723 | $xf_3(x,Q^2)=\left[ D\cdot U_{f3}\cdot x^{-\Delta} |
---|
1724 | +(1-D)\cdot V\cdot x\right]\cdot(1-x)^{N-2}$, with |
---|
1725 | $D=H\cdot S(Q^2)\cdot\left(1-\frac{1}{2}S(Q^2)\right)$ and |
---|
1726 | $U_{f3}=\frac{3\cdot C(Q^2)\cdot\Gamma(N-\Delta)} |
---|
1727 | {N\cdot\Gamma(N-1)\Gamma(1-\Delta)}$. The approximation is compared |
---|
1728 | with data in Fig.\ref{nuD} for deuterium \cite{WA25} and in |
---|
1729 | Fig.\ref{nuFe} for iron \cite{CDHSW,CCFR}. It must be emphasized |
---|
1730 | that the CHIPS parton distributions are the same as for |
---|
1731 | electromagnetic reactions. |
---|
1732 | |
---|
1733 | \begin{figure}[tbp] |
---|
1734 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}} |
---|
1735 | %\resizebox{0.80\textwidth}{!} |
---|
1736 | %{ |
---|
1737 | \includegraphics[angle=0,scale=0.6]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/f23nufe.eps} |
---|
1738 | %\includegraphics[angle=0,scale=0.6]{plots/f23nufe.eps} |
---|
1739 | %} |
---|
1740 | \caption{ |
---|
1741 | Fit of $f_{2Fe}(x,Q^2)$ (filled markers, solid lines) and |
---|
1742 | $f_{3Fe}(x,Q^2)$ (open markers, dashed lines) structure functions |
---|
1743 | measured by the CDHSW \cite{CDHSW} (circles) and CCFR \cite{CCFR} |
---|
1744 | (squares) experiments. |
---|
1745 | } |
---|
1746 | \label{nuFe} |
---|
1747 | \end{figure} |
---|
1748 | |
---|
1749 | For the $(\nu,\mu)$ amplitudes one can not apply the optical theorem, |
---|
1750 | To calculate the total cross sections, it is therefore necessary to |
---|
1751 | integrate the differential cross sections first over $x$ and then over |
---|
1752 | $Q^2$. For the $(\nu,\mu)$ reactions the differential cross section |
---|
1753 | can be integrated with good accuracy even for low energies because it |
---|
1754 | does not have the $\frac{1}{Q^4}$ factor of the boson propagator. The |
---|
1755 | quasi-elastic part of the total cross-section can be calculated for |
---|
1756 | $W<m_N+m_\pi$. The total $(\nu,\mu)$ cross-sections are shown in |
---|
1757 | Fig.\ref{totqe}(a,b). The dashed curve corresponds to the GRV \cite{GRV} |
---|
1758 | approximation of parton distributions and the dash-dotted curves |
---|
1759 | correspond to the KMRS \cite{KMRS} approximation. Neither approximation |
---|
1760 | fits low energies, because the perturbative calculations |
---|
1761 | give parton distributions only for $Q^2 > 1~GeV^2$. In \cite{Comby} an |
---|
1762 | attempt was made to freeze the DIS parton distributions at $Q^2=1$ and |
---|
1763 | to use them at low $Q^2$. The $W<1.4~GeV$ part of DIS was replaced by |
---|
1764 | the quasi-elastic and one pion production contributions, calculated on |
---|
1765 | the basis of the low energy models. The results of \cite{Comby} are |
---|
1766 | shown by the dotted lines. The nonperturbative CHIPS approximation |
---|
1767 | (solid curves) fits both total and quasi-elastic cross sections even at |
---|
1768 | low energies. |
---|
1769 | |
---|
1770 | \begin{figure}[tbp] |
---|
1771 | % \centerline{\epsfig{file=hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/vgam_c0606k.eps, height=4.5in, width=4.5in}} |
---|
1772 | %\resizebox{0.80\textwidth}{!} |
---|
1773 | %{ |
---|
1774 | \includegraphics[angle=0,scale=0.60]{hadronic/theory_driven/ChiralInvariantPhaseSpace/plots/numu_cs.eps} |
---|
1775 | %\includegraphics[angle=0,scale=0.60]{plots/numu_cs.eps} |
---|
1776 | %} |
---|
1777 | \caption{ |
---|
1778 | Fit of total (a,b) and quasi-elastic (c,d) cross-sections of |
---|
1779 | $(\nu,\mu)$ reactions (Geant4 database). The solid line |
---|
1780 | is the CHIPS approximation (for other lines see text). |
---|
1781 | } |
---|
1782 | \label{totqe} |
---|
1783 | \end{figure} |
---|
1784 | |
---|
1785 | The quasi-elastic $(\nu,\mu)$ cross sections are shown in |
---|
1786 | Fig.\ref{totqe}(c,d). The CHIPS approximation (solid line) is compared |
---|
1787 | with calculations made in \cite{Comby} (the dotted line) and the best |
---|
1788 | fit of the $V-A$ theory was made in \cite{VMA} (the dashed lines). One |
---|
1789 | can see that CHIPS gives reasonable agreement. |
---|
1790 | |
---|
1791 | The $Q^2$ spectra for each energy are known as an intermediate result |
---|
1792 | of the calculation of total or quasi-elastic cross sections. For the |
---|
1793 | quasi-elastic interactions ($W<m_N+m_\pi$) one can use $x=1$ and |
---|
1794 | simulate a binary reaction. In the final state the recoil nucleon has |
---|
1795 | some probability of interacting with the nucleus. If $W>m_N+m_\pi$ the |
---|
1796 | $Q^2$ value is randomized and therefore the $Q^2$ dependent |
---|
1797 | coefficients (the number of partons in non-perturbative phase space |
---|
1798 | $N$, the Pomeron intercept $\alpha_P$, the fraction of the direct |
---|
1799 | interactions, etc.) can be calculated. Then for fixed energy and |
---|
1800 | $Q^2$ the neutrino interaction with quark-partons (directly or through |
---|
1801 | the Pomeron ladder) can be randomized and the secondary parton |
---|
1802 | distribution can be calculated. In vacuum or in nuclear matter the |
---|
1803 | secondary partons are creating quasmons \cite{CHIPS1,CHIPS2} which |
---|
1804 | decay to secondary hadrons. |
---|
1805 | |
---|
1806 | \section{Conclusion.} |
---|
1807 | |
---|
1808 | \noindent \qquad For users who would like to improve the |
---|
1809 | interaction part of the CHIPS event generator for their own |
---|
1810 | specific reactions, some advice concerning data presentation |
---|
1811 | is useful. |
---|
1812 | |
---|
1813 | It is a good idea to use a normalized invariant function $\rho (k)$% |
---|
1814 | \[ |
---|
1815 | \rho =\frac{2E\cdot d^{3}\sigma }{\sigma _{tot}\cdot d^{3}p}\propto \frac{% |
---|
1816 | d\sigma }{\sigma _{tot}\cdot pdE}, |
---|
1817 | \] |
---|
1818 | where $\sigma _{tot}$\ is the total cross section of the reaction. |
---|
1819 | The simple rule, then, is to divide the distribution over the hadron |
---|
1820 | energy $E$ by the momentum and by the reaction cross section. The argument |
---|
1821 | $k$ can be calculated for any outgoing hadron or fragment as |
---|
1822 | \[ |
---|
1823 | k=\frac{E+p-B\cdot m_{N}}{2}, |
---|
1824 | \] |
---|
1825 | which is the energy of the primary quark-parton. Because the spectrum |
---|
1826 | of the quark-partons is universal for all the secondary hadrons or |
---|
1827 | fragments, the distributions over this parameter have a similar shape |
---|
1828 | for all the secondaries. They should differ only when the kinematic |
---|
1829 | limits are approached or in the evaporation region. This feature is |
---|
1830 | useful for any analysis of experimental data, independent of the CHIPS |
---|
1831 | model. |
---|
1832 | |
---|
1833 | % The released version of the CHIPS event generator is not perfect yet, |
---|
1834 | % so in case of an error it is necessary to distinguish between the error |
---|
1835 | % of the test program ({\bf CHIPStest.cc}) and the error in the body of |
---|
1836 | % the generator. Usually the error printing contains the address of the |
---|
1837 | % routine, but sometimes the name is abbreviated so that instead of |
---|
1838 | % {\bf G4QEnvironment}, {\bf G4Quasmon}, or {\bf G4QNucleus}, one will |
---|
1839 | % find {\bf G4QE}, {\bf G4Q}, or {\bf G4QN}. The errors in |
---|
1840 | % {\bf CHIPStest.cc} can be easily analyzed. Even if sometimes energy or |
---|
1841 | % charge is not conserved, this check can be excluded in order to keep |
---|
1842 | % going. On the other hand, if the error is in the body it is difficult |
---|
1843 | % to fix. The normal procedure is to uncomment the flags of the debugging |
---|
1844 | % prints in the corresponding part of the source code and try to find out |
---|
1845 | % the reason. Anyway inform authors about the error. Do not forget to attach the |
---|
1846 | % {\bf CHIPStest.cc} and the {\bf chipstest.in} files. |
---|
1847 | |
---|
1848 | Some concluding remarks should be made about the parameters of the model. |
---|
1849 | The main parameter, the critical temperature T$_{c}$, should not be varied. |
---|
1850 | A large set of data confirms the value {\bf 180 MeV} while from the mass |
---|
1851 | spectrum of hadrons it can be found more precisely as 182 MeV. The |
---|
1852 | clusterization parameter is {\bf 4.} which is just about 4$\pi /3.$ |
---|
1853 | If the quark exchange starts at the mean distance between baryons in the |
---|
1854 | dense part of the nucleus, then the radius of the clusterization sphere is |
---|
1855 | twice the ''the radius of the space occupied by the baryon''. |
---|
1856 | It gives 8 for the parameter, but the space occupied by the baryon can not |
---|
1857 | be spherical; only cubic subdivision of space is possible so the factor |
---|
1858 | $\pi/6 $ appears. But this is a rough estimate, so {\bf 4} or even {\bf 5} |
---|
1859 | can be tried. The surface parameter $fD$ varies slightly with $A$, |
---|
1860 | growing from 0 to 0.04. For the present CHIPS version the recommended |
---|
1861 | parameters for low energies are: |
---|
1862 | |
---|
1863 | \begin{tabular}{llllllllll} |
---|
1864 | {\bf A} & {\bf T} & {\bf s/u} & {\bf eta} & {\bf noP} & {\bf fN} & {\bf fD} |
---|
1865 | & {\bf Cp} & {\bf rM} & {\bf sA} \\ |
---|
1866 | {\bf Li} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ |
---|
1867 | {\bf Be} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ |
---|
1868 | {\bf C} & 180. & 0.1 & 0.3 & 223 & .4 & .00 & 4. & 1.0 & 0.4 \\ |
---|
1869 | {\bf O} & 180. & 0.1 & 0.3 & 223 & .4 & .02 & 4. & 1.0 & 0.4 \\ |
---|
1870 | {\bf F} & 180. & 0.1 & 0.3 & 223 & .4 & .03 & 4. & 1.0 & 0.4 \\ |
---|
1871 | {\bf Al} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ |
---|
1872 | {\bf Ca} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ |
---|
1873 | {\bf Cu} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ |
---|
1874 | {\bf Ta} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 \\ |
---|
1875 | {\bf U} & 180. & 0.1 & 0.3 & 223 & .4 & .04 & 4. & 1.0 & 0.4 |
---|
1876 | \end{tabular} |
---|
1877 | |
---|
1878 | The vacuum hadronization weight parameter can be bigger for light |
---|
1879 | nuclei and smaller for heavy nuclei, but $1.0$ is a good guess. The |
---|
1880 | s/u parameter is not yet tuned, as it demands strange particle |
---|
1881 | production data. A guess is that if there are as many $u\bar{u}$ |
---|
1882 | and $d\bar{d}$ pairs in the reaction as in the $p\bar{p}$ |
---|
1883 | interaction, the parameter can be 0.1. In other cases it is closer |
---|
1884 | to 0.3 as in other event generators. But it is bestnot to touch any |
---|
1885 | parameters for the first experience with the CHIPS event generator. |
---|
1886 | Only the incident momentum, the PDG code of the projectile, and the |
---|
1887 | CHIPS style PDG code of the target need be changed. |
---|
1888 | |
---|
1889 | |
---|
1890 | \section{Status of this document} |
---|
1891 | |
---|
1892 | 02.12.05 neutrino interactions section and figures added by M.V. Kossov \\ |
---|
1893 | 26.04.03 first four sections re-written by D.H. Wright \\ |
---|
1894 | 01.01.01 created by M.V. Kossov and H.P. Wellisch \\ |
---|
1895 | |
---|
1896 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%**************************** |
---|
1897 | |
---|
1898 | \begin{latexonly} |
---|
1899 | |
---|
1900 | \begin{thebibliography}{} |
---|
1901 | |
---|
1902 | % \bibitem{STAND_ALONE} \noindent M. V. Kossov, Manual for the CHIPS |
---|
1903 | % event generator,High Energy Accelerator Research Organization (KEK) |
---|
1904 | % Internal 2000-17, February 2001, H/R |
---|
1905 | |
---|
1906 | \bibitem{Parton_Models} B. Andersson, G. Gustafson, G. Ingelman, |
---|
1907 | T. Sj\"{o}strand, Phys. Rep. {\textbf{97}} (1983) 31 |
---|
1908 | |
---|
1909 | \bibitem{CHIPS1} \noindent P. V. Degtyarenko, M. V. Kossov, and H.P. |
---|
1910 | Wellisch, Chiral invariant phase space event generator, I. |
---|
1911 | Nucleon-antinucleon annihilation at rest, Eur. Phys. J. A 8 (2000) 217. |
---|
1912 | |
---|
1913 | \bibitem{CHIPS2} P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
---|
1914 | Chiral invariant phase space event generator, II.Nuclear pion capture at |
---|
1915 | rest, Eur. Phys. J. A 9 (2000) 411. |
---|
1916 | |
---|
1917 | \bibitem{CHIPS3} P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
---|
1918 | Chiral invariant phase space event generator, III Photonuclear reactions |
---|
1919 | below $\Delta $(3,3) excitation, Eur. Phys. J. A 9, (2000) 421. |
---|
1920 | |
---|
1921 | \bibitem{hadronMasses} M. V. Kossov, Chiral invariant phase space |
---|
1922 | model, I Masses of hadrons, Eur. Phys. J. A 14 (2002) 265. |
---|
1923 | |
---|
1924 | \bibitem{Chiral_Bag} C.A.Z. Vasconcellos et al., Eur. Phys. J. C |
---|
1925 | {\textbf{4}} (1998) 115; |
---|
1926 | G.A. Miller, A.W. Thomas, S. Theberge, Phys. Lett. B {\textbf{91}} (1980) |
---|
1927 | 192; |
---|
1928 | C.E. de Tar, Phys. Rev. D {\textbf{24}} (1981) 752; |
---|
1929 | M.A.B. B\'{e}g, G.T. Garvey, Comments Nucl. Part. Phys. {\textbf{18}} |
---|
1930 | (1988) 1 |
---|
1931 | |
---|
1932 | \bibitem{GENBOD} F. James, \textit{Monte Carlo Phase Space}, CERN 68-15 |
---|
1933 | (1968) |
---|
1934 | |
---|
1935 | \bibitem{Feynman-Wilson} K.G. Wilson, Proc. Fourteenth Scottish |
---|
1936 | Universities Summer School in Physics (1973), eds R. L. Crawford, R. |
---|
1937 | Jennings (Academic Press, New York, 1974) |
---|
1938 | |
---|
1939 | \bibitem{CH.PDG} Monte Carlo particle numbering scheme, in: |
---|
1940 | Particle Data Group, \textit{Review of Particle Physics}, |
---|
1941 | Eur. Phys. J. C {\textbf{3}} (1998) 180 |
---|
1942 | |
---|
1943 | \bibitem{Hagedorn} R. Hagedorn, Nuovo Cimento Suppl. {\textbf{3}} |
---|
1944 | (1965) 147 |
---|
1945 | |
---|
1946 | \bibitem{photNuc} M. V. Kossov, Approximation of photonuclear |
---|
1947 | interaction cross-sections, Eur. Phys. J. A 14 (2002) 377. |
---|
1948 | |
---|
1949 | \bibitem{GEANT4} S. Giani et al., Geant4: Object Oriented Toolkit for |
---|
1950 | Simulation in HEP, LCB status report CERN/LHCC/98-44, November 1998. |
---|
1951 | |
---|
1952 | \bibitem{MC2000} J. P. Wellisch, On hadronic models in GEANT4, Program |
---|
1953 | and Book of Abstracts.International Conference on Advanced Monte Carlo for |
---|
1954 | Radiation Physics, Particle Transport Simulation and Applications, 23-26 |
---|
1955 | October 2000, IST,Lisbon, Portugal, p. 330. |
---|
1956 | |
---|
1957 | \bibitem{Duality} Yu.L. Dokshitzer, V.S. Fadin and V.A. Khoze, |
---|
1958 | Phys. Lett. {\textbf{115B}} (1982) 242L |
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1959 | |
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1960 | \bibitem{JETSET} T. Sj\"{o}strand, Comp. Phys. Comm. {\textbf{92}} (1994) |
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1961 | 74 |
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1962 | |
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1963 | \bibitem{OZI} S. Ocubo, Phys. Lett. {\textbf{5}} (1963) 165; |
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1964 | G. Zweig, CERN Preprint 8419/TH-412 (1964); |
---|
1965 | I. Iizuka, Progr. Theor. Phys. Suppl. {\textbf{37}} (1966) 21 |
---|
1966 | |
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1967 | \bibitem{OZI_violation} V.E. Markushin, M.P. Locher, |
---|
1968 | Eur. Phys. J. A {\textbf{1}} (1998) 91 |
---|
1969 | |
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1970 | \bibitem{pispectrum} J. Sedlak and V. Simak, Sov. J. Part. Nucl. |
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1971 | {\textbf{19}} (1988) 191 |
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1972 | |
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1973 | \bibitem{pap_exdata} C. Amsler, Rev.Mod.Phys. {\textbf{70}} (1998) 1293; |
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1974 | C. Amsler and F. Myher, Annu. Rev. Nucl. Part. Sci. {\textbf{41}} (1991) |
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1975 | 219 |
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1976 | |
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1977 | \bibitem{POPCORN} B. Andersson, G. Gustafson, T. Sj\"{o}strand, Nucl. |
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1978 | Phys. B {\textbf{197}}(1982) 45; |
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1979 | B. Andersson, G. Gustafson, T. Sj\"{o}strand, Physica Scripta {\textbf{32}} |
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1980 | (1985) 574 |
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1981 | |
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1982 | \bibitem{Energy_Dep} P. Gregory et al., Nucl. Phys. B {\textbf{102}} (1976) |
---|
1983 | 189 |
---|
1984 | |
---|
1985 | \bibitem{K_parameter} M.V. Kossov and L.M. Voronina, Preprint ITEP |
---|
1986 | 165-84, Moscow (1984) |
---|
1987 | |
---|
1988 | \bibitem{FNAL} V.I.~Efremenko et al., Phys. Rev. C \textbf{22} (1980) 700. |
---|
1989 | |
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1990 | \bibitem{FAS} S.V~Boyarinov et al., Phys. At. Nucl. \textbf{56} |
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1991 | (1993) 72. |
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1992 | |
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1993 | \bibitem{TPC} P.V. Degtyarenko et al., Phys. Rev. C {\textbf{50}} (1994) |
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1994 | R541 |
---|
1995 | |
---|
1996 | \bibitem{NN QEX} K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{29} (1984) 952. |
---|
1997 | |
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1998 | \bibitem{Kp QUEX} K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{34} (1986) |
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1999 | 1372. |
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2000 | |
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2001 | \bibitem{EMC} P.~Hoodbhoy and R.~J.~Jaffe, Phys. Rev. D \textbf{35} |
---|
2002 | (1987) 113. |
---|
2003 | |
---|
2004 | \bibitem{QUEX} N.~Isgur, Nucl. Phys. \textbf{A497} (1989) 91. |
---|
2005 | |
---|
2006 | %%%%%%%%%%%%%%% |
---|
2007 | |
---|
2008 | \bibitem{massSpectr} M. V. Kossov, CHIPS: masses of hadrons. (be |
---|
2009 | published). |
---|
2010 | |
---|
2011 | \bibitem{eqPhotons} L. D. Landau, E. M. Lifshitz, ``Course of |
---|
2012 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
---|
2013 | Pergamon Press, paragraph 96, The method of equivalent photons. |
---|
2014 | |
---|
2015 | \bibitem{Shadowing} J. Eickmeyer et al. Phys. Rev. Letters {\bf 36 }(1976) |
---|
2016 | 289-291. |
---|
2017 | |
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2018 | \bibitem{Guilo} D'Agostini, Hard Scattering Process in High Energy |
---|
2019 | Gamma-Induced Reactions, DESY 94-169, September 1994. |
---|
2020 | |
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2021 | \bibitem{Electronuc} F. W. Brasse et al. Nuclear Physics {\bf B39 }(1972) |
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2026 | |
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2029 | |
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2030 | \bibitem{CCFR} E. Oltman {\textit {et~al}}, Z. Phys C {\textbf{53}}, |
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2032 | |
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2037 | 3645 (1990) |
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2038 | |
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2039 | \bibitem{Comby} P. Lipari {\textit {et~al}}, Phys. Rev. Let. {\textbf{74}}, |
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2041 | |
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2046 | 221-239. |
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2047 | |
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2048 | \bibitem{DINREG} P.V. Degtyarenko and M.V. Kossov, Preprint ITEP |
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2049 | 11-92, Moscow (1992) |
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2050 | |
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2051 | \bibitem{ARGUS} P.V. Degtyarenko et al., Z. Phys. A - Atomic Nuclei, |
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2053 | |
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2054 | \bibitem{GDINR} P.V. Degtyarenko, \textit{Applications of the photonuclear |
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2055 | fragmentation model to radiation protection problems}, in: |
---|
2056 | Proceedings of Second Specialist's Meeting on Shielding Aspects of |
---|
2057 | Accelerators, Targets and Irradiation Facilities (SATIF-2), CERN, |
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2058 | Geneva, Switzerland, 12-13 October 1995, published by Nuclear Energy |
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2059 | Agency, Organization for Economic Co-operation and Development, pages |
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2060 | 67 - 91 (1996) |
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2064 | |
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2065 | E. Poggio, H. Quinn, and S. Weinberg, Phys. Rev. D \textbf{13} (1976) 1958 |
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2066 | |
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2067 | \bibitem{MIPHI} A.~I.~Amelin et al., ``Energy spectra of charged particles |
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2068 | in the reaction of $\pi^-$ absorption at rest by $^{6,7}$Li, $^{9}$Be, $% |
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2069 | ^{10,11}$B, $^{12}$C, $^{28}$Si, $^{40}$Ca, $^{59}$Co, $^{93}$Nb, $% |
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2070 | ^{114,117,120,124}$Sn, $^{169}$Tm, $^{181}$Ta and $^{209}$Bi nuclei'', |
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2071 | Moscow Physics and Engineering Institute Preprint No. 034-90, Moscow, 1990. |
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2072 | |
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2101 | |
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2102 | \end{thebibliography} |
---|
2103 | |
---|
2104 | \end{latexonly} |
---|
2105 | |
---|
2106 | \begin{htmlonly} |
---|
2107 | |
---|
2108 | \section{Bibliography} |
---|
2109 | |
---|
2110 | \begin{enumerate} |
---|
2111 | % \bibitem{STAND_ALONE} \noindent M. V. Kossov, Manual for the CHIPS |
---|
2112 | % event generator,High Energy Accelerator Research Organization (KEK) |
---|
2113 | % Internal 2000-17, February 2001, H/R |
---|
2114 | |
---|
2115 | \item B. Andersson, G. Gustafson, G. Ingelman, |
---|
2116 | T. Sj\"{o}strand, Phys. Rep. {\textbf{97}} (1983) 31 |
---|
2117 | |
---|
2118 | \item \noindent P. V. Degtyarenko, M. V. Kossov, and H.P. |
---|
2119 | Wellisch, Chiral invariant phase space event generator, I. |
---|
2120 | Nucleon-antinucleon annihilation at rest, Eur. Phys. J. A {\bf 8}, 217-222 |
---|
2121 | (2000). |
---|
2122 | |
---|
2123 | \item P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
---|
2124 | Chiral invariant phase space event generator, II.Nuclear pion capture at |
---|
2125 | rest, Eur. Phys. J. A 9, (2001). |
---|
2126 | |
---|
2127 | \item P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
---|
2128 | Chiral invariant phase space event generator, III Photonuclear reactions |
---|
2129 | below $\Delta $(3,3) excitation, Eur. Phys. J. A 9, (2001). |
---|
2130 | |
---|
2131 | \item C.A.Z. Vasconcellos et al., Eur. Phys. J. C |
---|
2132 | {\textbf{4}} (1998) 115; |
---|
2133 | G.A. Miller, A.W. Thomas, S. Theberge, Phys. Lett. B {\textbf{91}} (1980) |
---|
2134 | 192; |
---|
2135 | C.E. de Tar, Phys. Rev. D {\textbf{24}} (1981) 752; |
---|
2136 | M.A.B. B\'{e}g, G.T. Garvey, Comments Nucl. Part. Phys. {\textbf{18}} |
---|
2137 | (1988) 1 |
---|
2138 | |
---|
2139 | \item F. James, \textit{Monte Carlo Phase Space}, CERN 68-15 |
---|
2140 | (1968) |
---|
2141 | |
---|
2142 | \item K.G. Wilson, Proc. Fourteenth Scottish |
---|
2143 | Universities Summer School in Physics (1973), eds R. L. Crawford, R. |
---|
2144 | Jennings (Academic Press, New York, 1974) |
---|
2145 | |
---|
2146 | \item Monte Carlo particle numbering scheme, in: |
---|
2147 | Particle Data Group, \textit{Review of Particle Physics}, |
---|
2148 | Eur. Phys. J. C {\textbf{3}} (1998) 180 |
---|
2149 | |
---|
2150 | \item R. Hagedorn, Nuovo Cimento Suppl. {\textbf{3}} (1965) 147 |
---|
2151 | |
---|
2152 | \item S. Giani et al., Geant4: Object Oriented Toolkit for |
---|
2153 | Simulation in HEP, LCB status report CERN/LHCC/98-44, November 1998. |
---|
2154 | |
---|
2155 | \item J. P. Wellisch, On hadronic models in GEANT4, Program |
---|
2156 | and Book of Abstracts.International Conference on Advanced Monte Carlo for |
---|
2157 | Radiation Physics, Particle Transport Simulation and Applications, 23-26 |
---|
2158 | October 2000, IST,Lisbon, Portugal, p. 330. |
---|
2159 | |
---|
2160 | \item Yu.L. Dokshitzer, V.S. Fadin and V.A. Khoze, |
---|
2161 | Phys. Lett. {\textbf{115B}} (1982) 242L |
---|
2162 | |
---|
2163 | \item T. Sj\"{o}strand, Comp. Phys. Comm. {\textbf{92}} (1994) |
---|
2164 | 74 |
---|
2165 | |
---|
2166 | \item S. Ocubo, Phys. Lett. {\textbf{5}} (1963) 165; |
---|
2167 | G. Zweig, CERN Preprint 8419/TH-412 (1964); |
---|
2168 | I. Iizuka, Progr. Theor. Phys. Suppl. {\textbf{37}} (1966) 21 |
---|
2169 | |
---|
2170 | \item V.E. Markushin, M.P. Locher, |
---|
2171 | Eur. Phys. J. A {\textbf{1}} (1998) 91 |
---|
2172 | |
---|
2173 | \item J. Sedlak and V. Simak, Sov. J. Part. Nucl. |
---|
2174 | {\textbf{19}} (1988) 191 |
---|
2175 | |
---|
2176 | \item C. Amsler, Rev.Mod.Phys. {\textbf{70}} (1998) 1293; |
---|
2177 | C. Amsler and F. Myher, Annu. Rev. Nucl. Part. Sci. {\textbf{41}} (1991) |
---|
2178 | 219 |
---|
2179 | |
---|
2180 | \item B. Andersson, G. Gustafson, T. Sj\"{o}strand, Nucl. |
---|
2181 | Phys. B {\textbf{197}}(1982) 45; |
---|
2182 | B. Andersson, G. Gustafson, T. Sj\"{o}strand, Physica Scripta {\textbf{32}} |
---|
2183 | (1985) 574 |
---|
2184 | |
---|
2185 | \item P. Gregory et al., Nucl. Phys. B {\textbf{102}} (1976) |
---|
2186 | 189 |
---|
2187 | |
---|
2188 | \item M.V. Kossov and L.M. Voronina, Preprint ITEP |
---|
2189 | 165-84, Moscow (1984) |
---|
2190 | |
---|
2191 | \item V.I.~Efremenko et al., Phys. Rev. C \textbf{22} (1980) 700. |
---|
2192 | |
---|
2193 | \item S.V~Boyarinov et al., Phys. At. Nucl. \textbf{56} |
---|
2194 | (1993) 72. |
---|
2195 | |
---|
2196 | \item P.V. Degtyarenko et al., Phys. Rev. C {\textbf{50}} (1994) |
---|
2197 | R541 |
---|
2198 | |
---|
2199 | \item K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{29} (1984) 952. |
---|
2200 | |
---|
2201 | \item K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{34} (1986) |
---|
2202 | 1372. |
---|
2203 | |
---|
2204 | \item P.~Hoodbhoy and R.~J.~Jaffe, Phys. Rev. D \textbf{35} |
---|
2205 | (1987) 113. |
---|
2206 | |
---|
2207 | \item N.~Isgur, Nucl. Phys. \textbf{A497} (1989) 91. |
---|
2208 | |
---|
2209 | %%%%%%%%%%%%%%% |
---|
2210 | |
---|
2211 | \item M. V. Kossov, CHIPS: masses of hadrons. (be |
---|
2212 | published). |
---|
2213 | |
---|
2214 | \item L. D. Landau, E. M. Lifshitz, ``Course of |
---|
2215 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
---|
2216 | Pergamon Press, paragraph 96, The method of equivalent photons. |
---|
2217 | |
---|
2218 | \item J. Eickmeyer et al. Phys. Rev. Letters {\bf 36 }(1976) |
---|
2219 | 289-291. |
---|
2220 | |
---|
2221 | \item D'Agostini, Hard Scattering Process in High Energy |
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2223 | |
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2224 | \item F. W. Brasse et al. Nuclear Physics {\bf B39 }(1972) |
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2226 | |
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2227 | \item A. Lepretre et al. Nuclear Physics {\bf A390 }(1982) |
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2228 | 221-239. |
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2229 | |
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2230 | \item P.V. Degtyarenko and M.V. Kossov, Preprint ITEP |
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2231 | 11-92, Moscow (1992) |
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2232 | |
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2233 | \item P.V. Degtyarenko et al., Z. Phys. A - Atomic Nuclei, |
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2234 | {\textbf{335}} (1990) 231 |
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2235 | |
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2236 | \item P.V. Degtyarenko, \textit{Applications of the photonuclear |
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2237 | fragmentation model to radiation protection problems}, in: |
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2238 | Proceedings of Second Specialist's Meeting on Shielding Aspects of |
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2239 | Accelerators, Targets and Irradiation Facilities (SATIF-2), CERN, |
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2240 | Geneva, Switzerland, 12-13 October 1995, published by Nuclear Energy |
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2241 | Agency, Organization for Economic Co-operation and Development, pages |
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2242 | 67 - 91 (1996) |
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2243 | |
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2244 | \item C. Bernard, A. Duncan, J. LoSecco, and S. Weinberg, |
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2245 | Phys. Rev. D \textbf{12} (1975) 792; |
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2246 | |
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2247 | E. Poggio, H. Quinn, and S. Weinberg, Phys. Rev. D \textbf{13} (1976) 1958 |
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2248 | |
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2249 | \item A.~I.~Amelin et al., ``Energy spectra of charged particles |
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2250 | in the reaction of $\pi^-$ absorption at rest by $^{6,7}$Li, $^{9}$Be, $% |
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2251 | ^{10,11}$B, $^{12}$C, $^{28}$Si, $^{40}$Ca, $^{59}$Co, $^{93}$Nb, $% |
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2252 | ^{114,117,120,124}$Sn, $^{169}$Tm, $^{181}$Ta and $^{209}$Bi nuclei'', |
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2253 | Moscow Physics and Engineering Institute Preprint No. 034-90, Moscow, 1990. |
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2254 | |
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2255 | \item G.~Mechtersheimer et al., Nucl. Phys. |
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2256 | \textbf{A324} (1979) 379. |
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2257 | |
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2258 | \item C.~Cernigoi et al., Nucl. Phys. \textbf{A456} (1986) 599. |
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2259 | |
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2261 | |
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2267 | 2704. |
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2268 | |
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2269 | \item C.~Van~den~Abeele; private communication cited |
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2270 | in the reference: Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994) |
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2271 | 2704. |
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2272 | |
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2273 | \item P.D.~Harty et al. (unpublished); |
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2274 | private communication cited |
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2275 | in the reference: Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994) |
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2277 | |
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2278 | \item L.B.~Weinstein et al., Phys. Rev. Lett. \textbf{64} (1990) |
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2279 | 1646. |
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2280 | |
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2281 | \item J.P.~Jeukenne and C.~Mahaux, Nucl. Phys. A \textbf{394} |
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2282 | (1983) 445. |
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2283 | |
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2284 | \end{enumerate} |
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2285 | |
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2286 | \end{htmlonly} |
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