[1208] | 1 | %\documentclass[12pt,a4paper,oneside]{book} |
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| 2 | %\usepackage[dvips]{graphicx} |
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| 3 | %\usepackage{html} |
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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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| 13 | %\setcounter{page}{-0} |
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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 - |
---|
| 370 | \cos\theta)}\right) ^{N-4} \nonumber\\ |
---|
| 371 | && \times\ d\left(1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}\right). |
---|
| 372 | \end{eqnarray} |
---|
| 373 | After the substitution |
---|
| 374 | $z=1-\frac{2q }{M_{N-1}}=1-\frac{\mu ^{2}}{Mk(1-\cos \theta )}$, this |
---|
| 375 | becomes |
---|
| 376 | \begin{equation} |
---|
| 377 | P(k,M,\mu ) = \frac{M-2k}{4k} \int z^{N-4} dz , |
---|
| 378 | \end{equation} |
---|
| 379 | where the limits of integration are $0$ when |
---|
| 380 | $\cos\theta = 1 - \frac{\mu ^{2}}{M\cdot k}$, and |
---|
| 381 | \begin{equation} |
---|
| 382 | z_{\max }=1-\frac{\mu^2}{2Mk}, \label{z_max} |
---|
| 383 | \end{equation} |
---|
| 384 | when $\cos \theta =-1$. The resulting range of $\theta$\ is therefore |
---|
| 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) |
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| 1927 | 192; |
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| 1928 | C.E. de Tar, Phys. Rev. D {\textbf{24}} (1981) 752; |
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| 1929 | M.A.B. B\'{e}g, G.T. Garvey, Comments Nucl. Part. Phys. {\textbf{18}} |
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| 1930 | (1988) 1 |
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| 1931 | |
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| 1932 | \bibitem{GENBOD} F. James, \textit{Monte Carlo Phase Space}, CERN 68-15 |
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| 1933 | (1968) |
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| 1934 | |
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| 1935 | \bibitem{Feynman-Wilson} K.G. Wilson, Proc. Fourteenth Scottish |
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| 1936 | Universities Summer School in Physics (1973), eds R. L. Crawford, R. |
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| 1937 | Jennings (Academic Press, New York, 1974) |
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| 1938 | |
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| 1939 | \bibitem{CH.PDG} Monte Carlo particle numbering scheme, in: |
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| 1940 | Particle Data Group, \textit{Review of Particle Physics}, |
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| 1941 | Eur. Phys. J. C {\textbf{3}} (1998) 180 |
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| 1942 | |
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| 1946 | \bibitem{photNuc} M. V. Kossov, Approximation of photonuclear |
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| 1948 | |
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| 1949 | \bibitem{GEANT4} S. Giani et al., Geant4: Object Oriented Toolkit for |
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| 1951 | |
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| 1952 | \bibitem{MC2000} J. P. Wellisch, On hadronic models in GEANT4, Program |
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| 1953 | and Book of Abstracts.International Conference on Advanced Monte Carlo for |
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| 1954 | Radiation Physics, Particle Transport Simulation and Applications, 23-26 |
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| 1955 | October 2000, IST,Lisbon, Portugal, p. 330. |
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| 1956 | |
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| 1983 | 189 |
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| 1984 | |
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| 1985 | \bibitem{K_parameter} M.V. Kossov and L.M. Voronina, Preprint ITEP |
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| 1986 | 165-84, Moscow (1984) |
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| 1987 | |
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| 1988 | \bibitem{FNAL} V.I.~Efremenko et al., Phys. Rev. C \textbf{22} (1980) 700. |
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| 1989 | |
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| 1991 | (1993) 72. |
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| 1992 | |
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| 2003 | |
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| 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 |
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| 2012 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
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| 2013 | Pergamon Press, paragraph 96, The method of equivalent photons. |
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| 2014 | |
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| 2016 | 289-291. |
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| 2017 | |
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| 2018 | \bibitem{Guilo} D'Agostini, Hard Scattering Process in High Energy |
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| 2019 | Gamma-Induced Reactions, DESY 94-169, September 1994. |
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---|
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| 2101 | |
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| 2102 | \end{thebibliography} |
---|
| 2103 | |
---|
| 2104 | \end{latexonly} |
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| 2105 | |
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| 2106 | \begin{htmlonly} |
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| 2107 | |
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| 2108 | \section{Bibliography} |
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| 2109 | |
---|
| 2110 | \begin{enumerate} |
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| 2111 | % \bibitem{STAND_ALONE} \noindent M. V. Kossov, Manual for the CHIPS |
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| 2112 | % event generator,High Energy Accelerator Research Organization (KEK) |
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| 2113 | % Internal 2000-17, February 2001, H/R |
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| 2114 | |
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| 2115 | \item B. Andersson, G. Gustafson, G. Ingelman, |
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| 2116 | T. Sj\"{o}strand, Phys. Rep. {\textbf{97}} (1983) 31 |
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| 2117 | |
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| 2118 | \item \noindent P. V. Degtyarenko, M. V. Kossov, and H.P. |
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| 2119 | Wellisch, Chiral invariant phase space event generator, I. |
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| 2120 | Nucleon-antinucleon annihilation at rest, Eur. Phys. J. A {\bf 8}, 217-222 |
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| 2121 | (2000). |
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| 2122 | |
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| 2123 | \item P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
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| 2124 | Chiral invariant phase space event generator, II.Nuclear pion capture at |
---|
| 2125 | rest, Eur. Phys. J. A 9, (2001). |
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| 2126 | |
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| 2127 | \item P. V. Degtyarenko, M. V. Kossov, and H. P. Wellisch, |
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| 2128 | Chiral invariant phase space event generator, III Photonuclear reactions |
---|
| 2129 | below $\Delta $(3,3) excitation, Eur. Phys. J. A 9, (2001). |
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| 2130 | |
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| 2131 | \item C.A.Z. Vasconcellos et al., Eur. Phys. J. C |
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| 2137 | (1988) 1 |
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| 2138 | |
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| 2139 | \item F. James, \textit{Monte Carlo Phase Space}, CERN 68-15 |
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| 2140 | (1968) |
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| 2141 | |
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| 2142 | \item K.G. Wilson, Proc. Fourteenth Scottish |
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| 2143 | Universities Summer School in Physics (1973), eds R. L. Crawford, R. |
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| 2144 | Jennings (Academic Press, New York, 1974) |
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| 2145 | |
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| 2146 | \item Monte Carlo particle numbering scheme, in: |
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| 2147 | Particle Data Group, \textit{Review of Particle Physics}, |
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| 2148 | Eur. Phys. J. C {\textbf{3}} (1998) 180 |
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| 2149 | |
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| 2150 | \item R. Hagedorn, Nuovo Cimento Suppl. {\textbf{3}} (1965) 147 |
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| 2151 | |
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| 2152 | \item S. Giani et al., Geant4: Object Oriented Toolkit for |
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| 2153 | Simulation in HEP, LCB status report CERN/LHCC/98-44, November 1998. |
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| 2154 | |
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| 2155 | \item J. P. Wellisch, On hadronic models in GEANT4, Program |
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| 2156 | and Book of Abstracts.International Conference on Advanced Monte Carlo for |
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| 2157 | Radiation Physics, Particle Transport Simulation and Applications, 23-26 |
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| 2158 | October 2000, IST,Lisbon, Portugal, p. 330. |
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| 2159 | |
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| 2160 | \item Yu.L. Dokshitzer, V.S. Fadin and V.A. Khoze, |
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| 2161 | Phys. Lett. {\textbf{115B}} (1982) 242L |
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| 2170 | \item V.E. Markushin, M.P. Locher, |
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| 2171 | Eur. Phys. J. A {\textbf{1}} (1998) 91 |
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| 2172 | |
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| 2177 | C. Amsler and F. Myher, Annu. Rev. Nucl. Part. Sci. {\textbf{41}} (1991) |
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| 2180 | \item B. Andersson, G. Gustafson, T. Sj\"{o}strand, Nucl. |
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| 2181 | Phys. B {\textbf{197}}(1982) 45; |
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| 2182 | B. Andersson, G. Gustafson, T. Sj\"{o}strand, Physica Scripta {\textbf{32}} |
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| 2184 | |
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| 2185 | \item P. Gregory et al., Nucl. Phys. B {\textbf{102}} (1976) |
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| 2186 | 189 |
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| 2187 | |
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| 2188 | \item M.V. Kossov and L.M. Voronina, Preprint ITEP |
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| 2189 | 165-84, Moscow (1984) |
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| 2190 | |
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| 2191 | \item V.I.~Efremenko et al., Phys. Rev. C \textbf{22} (1980) 700. |
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| 2192 | |
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| 2194 | (1993) 72. |
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| 2195 | |
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| 2196 | \item P.V. Degtyarenko et al., Phys. Rev. C {\textbf{50}} (1994) |
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| 2197 | R541 |
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| 2198 | |
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| 2199 | \item K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{29} (1984) 952. |
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| 2200 | |
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| 2201 | \item K.~Maltman and N.~Isgur, Phys. Rev. D \textbf{34} (1986) |
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| 2202 | 1372. |
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| 2203 | |
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| 2204 | \item P.~Hoodbhoy and R.~J.~Jaffe, Phys. Rev. D \textbf{35} |
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| 2205 | (1987) 113. |
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| 2206 | |
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| 2207 | \item N.~Isgur, Nucl. Phys. \textbf{A497} (1989) 91. |
---|
| 2208 | |
---|
| 2209 | %%%%%%%%%%%%%%% |
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| 2210 | |
---|
| 2211 | \item M. V. Kossov, CHIPS: masses of hadrons. (be |
---|
| 2212 | published). |
---|
| 2213 | |
---|
| 2214 | \item L. D. Landau, E. M. Lifshitz, ``Course of |
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| 2215 | Theoretical Physics'' v.4, part 1, ``Relativistic Quantum Theory'', |
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| 2216 | Pergamon Press, paragraph 96, The method of equivalent photons. |
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| 2217 | |
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| 2218 | \item J. Eickmeyer et al. Phys. Rev. Letters {\bf 36 }(1976) |
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| 2219 | 289-291. |
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| 2220 | |
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| 2221 | \item D'Agostini, Hard Scattering Process in High Energy |
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| 2222 | Gamma-Induced Reactions, DESY 94-169, September 1994. |
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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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| 2225 | 421-431. |
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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 | |
---|
| 2230 | \item P.V. Degtyarenko and M.V. Kossov, Preprint ITEP |
---|
| 2231 | 11-92, Moscow (1992) |
---|
| 2232 | |
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| 2233 | \item P.V. Degtyarenko et al., Z. Phys. A - Atomic Nuclei, |
---|
| 2234 | {\textbf{335}} (1990) 231 |
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| 2235 | |
---|
| 2236 | \item P.V. Degtyarenko, \textit{Applications of the photonuclear |
---|
| 2237 | fragmentation model to radiation protection problems}, in: |
---|
| 2238 | Proceedings of Second Specialist's Meeting on Shielding Aspects of |
---|
| 2239 | Accelerators, Targets and Irradiation Facilities (SATIF-2), CERN, |
---|
| 2240 | Geneva, Switzerland, 12-13 October 1995, published by Nuclear Energy |
---|
| 2241 | Agency, Organization for Economic Co-operation and Development, pages |
---|
| 2242 | 67 - 91 (1996) |
---|
| 2243 | |
---|
| 2244 | \item C. Bernard, A. Duncan, J. LoSecco, and S. Weinberg, |
---|
| 2245 | Phys. Rev. D \textbf{12} (1975) 792; |
---|
| 2246 | |
---|
| 2247 | E. Poggio, H. Quinn, and S. Weinberg, Phys. Rev. D \textbf{13} (1976) 1958 |
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| 2248 | |
---|
| 2249 | \item A.~I.~Amelin et al., ``Energy spectra of charged particles |
---|
| 2250 | in the reaction of $\pi^-$ absorption at rest by $^{6,7}$Li, $^{9}$Be, $% |
---|
| 2251 | ^{10,11}$B, $^{12}$C, $^{28}$Si, $^{40}$Ca, $^{59}$Co, $^{93}$Nb, $% |
---|
| 2252 | ^{114,117,120,124}$Sn, $^{169}$Tm, $^{181}$Ta and $^{209}$Bi nuclei'', |
---|
| 2253 | Moscow Physics and Engineering Institute Preprint No. 034-90, Moscow, 1990. |
---|
| 2254 | |
---|
| 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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| 2260 | \item R.~Madey et al., Phys. Rev. C \textbf{25} (1982) 3050. |
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| 2261 | |
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| 2262 | \item D.~Ryckbosch et al., Phys. Rev. C \textbf{42} (1990) 444. |
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| 2263 | |
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| 2264 | \item J.~Ahrens et al., Nucl. Phys. \textbf{A446} (1985) 229c. |
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| 2265 | |
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| 2266 | \item Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994) |
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| 2267 | 2704. |
---|
| 2268 | |
---|
| 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) |
---|
| 2271 | 2704. |
---|
| 2272 | |
---|
| 2273 | \item P.D.~Harty et al. (unpublished); |
---|
| 2274 | private communication cited |
---|
| 2275 | in the reference: Jan~Ryckebusch et al., Phys. Rev. C \textbf{49} (1994) |
---|
| 2276 | 2704. |
---|
| 2277 | |
---|
| 2278 | \item L.B.~Weinstein et al., Phys. Rev. Lett. \textbf{64} (1990) |
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| 2279 | 1646. |
---|
| 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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