[1208] | 1 | |
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| 2 | \section{Muon Photonuclear Interaction} |
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| 3 | |
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| 4 | %muon photonuclear interaction |
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| 5 | %(not deep inelastic scattering) |
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| 6 | |
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| 7 | The inelastic interaction of muons with nuclei is important at high muon |
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| 8 | energies ($E \geq 10$~ GeV), and at relatively high energy transfers $\nu$ |
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| 9 | ($\nu / E \geq 10^{-2}$). It is especially important for light materials |
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| 10 | and for the study of detector response to high energy muons, muon propagation |
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| 11 | and muon-induced hadronic background. The average energy loss for this |
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| 12 | process increases almost lineary with energy, and at TeV muon energies |
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| 13 | constitutes about 10\% of the energy loss rate.\\ |
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| 14 | |
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| 15 | \noindent |
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| 16 | The main contribution to the cross section $\sigma ( E, \nu $) and energy |
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| 17 | loss comes from the low $Q^{2}$--region ( $Q^{2} \ll 1~{\rm GeV}^{2}$). |
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| 18 | In this domain, many simplifications can be made in the theoretical |
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| 19 | consideration of the process in order to obtain convenient and simple |
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| 20 | formulae for the cross section. Most widely used are the expressions given |
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| 21 | by Borog and Petrukhin \cite{munu.bor75}, and Bezrukov and Bugaev |
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| 22 | \cite{munu.bez81}. Results from these authors agree within 10\% for the |
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| 23 | differential cross section and within about 5\% for the average energy loss, |
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| 24 | provided the same photonuclear cross section, $\sigma_{\gamma N}$, is used |
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| 25 | in the calculations. |
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| 26 | |
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| 27 | \subsection{Differential Cross Section} |
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| 28 | |
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| 29 | The Borog and Petrukhin formula for the cross section is based on: |
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| 30 | \begin{itemize} |
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| 31 | \item Hand's formalism \cite{munu.hand63} for inelastic muon scattering, |
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| 32 | \item a semi-phenomenological inelastic form factor, which is a Vector |
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| 33 | Dominance Model with parameters estimated from experimental data, and |
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| 34 | \item nuclear shadowing effects with a reasonable theoretical |
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| 35 | parameterization \cite{munu.brod72}. |
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| 36 | \end{itemize} |
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| 37 | |
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| 38 | \noindent |
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| 39 | For $E\geq10$~GeV, the Borog and Petrukhin cross section |
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| 40 | \mbox{(${\rm cm}^{2}$/g GeV)}, differential in transferred energy, is |
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| 41 | \begin{equation} |
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| 42 | \label{munu.1} |
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| 43 | \sigma(E,\nu ) = \Psi(\nu ) \Phi( E,v ) , |
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| 44 | \end{equation} |
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| 45 | |
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| 46 | \begin{equation} |
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| 47 | \label{munu.2} |
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| 48 | \Psi(\nu) = \frac{\alpha}{\pi} \frac{A_{\rm eff} N_{AV}}{A} |
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| 49 | \sigma_{\gamma N}(\nu) \frac{1}{\nu} , |
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| 50 | \end{equation} |
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| 51 | |
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| 52 | \begin{equation} |
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| 53 | \label{munu.3} |
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| 54 | \Phi(E,v) = v-1 + \left[1-v+\frac{v^{2}}{2} |
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| 55 | \left(1+\frac{2\mu^{2}}{\Lambda^{2}}\right)\right] |
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| 56 | \ln\frac{\displaystyle\frac{E^{2}(1-v)}{\mu^{2}}\left(1+\displaystyle\frac{\mu^{2} |
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| 57 | v^{2}}{\Lambda^{2}(1-v)}\right)} |
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| 58 | {\displaystyle |
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| 59 | 1+\frac{Ev}{\Lambda}\left(1+\frac{\Lambda}{2M}+\frac{Ev}{\Lambda}\right)}, |
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| 60 | \end{equation} |
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| 61 | % |
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| 62 | where $\nu$ is the energy lost by the muon, $v = \nu /$E, and $\mu$ and $M$ |
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| 63 | are the muon and nucleon (proton) masses, respectively. $\Lambda$ is a |
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| 64 | Vector Dominance Model parameter in the inelastic form factor which is |
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| 65 | estimated to be $\Lambda^{2}=0.4 \;{\rm GeV}^{2}$. \\ |
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| 66 | |
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| 67 | \noindent |
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| 68 | For $A_{\rm eff}$, which includes the effect of nuclear shadowing, the |
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| 69 | parameterization \cite{munu.brod72} |
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| 70 | \begin{equation} |
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| 71 | \label{munu.4} |
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| 72 | A_{\rm eff} = 0.22 A + 0.78 A^{0.89} |
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| 73 | \end{equation} |
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| 74 | is chosen. \\ |
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| 75 | |
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| 76 | \noindent |
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| 77 | A reasonable choice for the photonuclear cross section, $\sigma_{\gamma N}$, |
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| 78 | is the parameterization obtained by Caldwell et al. \cite{munu.cald79} |
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| 79 | based on the experimental data on photoproduction by real photons: |
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| 80 | \begin{equation} |
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| 81 | \label{munu.5} |
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| 82 | \sigma_{\gamma N} = ( 49.2 + 11.1 \ln K + 151.8/ \sqrt{K} ) \cdot 10^{-30} {\rm cm}^{2} \quad K~~\mbox{in GeV} . |
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| 83 | \end{equation} |
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| 84 | The upper limit of the transferred energy is taken to be |
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| 85 | $\nu_{\rm max}$ = $E - M/2$. The choice of the lower limit $\nu_{\rm min}$ |
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| 86 | is less certain since the formula \ref{munu.1}, \ref{munu.2}, \ref{munu.3} is |
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| 87 | not valid in this domain. Fortunately, $\nu_{\rm min}$ influences the total |
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| 88 | cross section only logarithmically and has no practical effect on the average |
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| 89 | energy loss for high energy muons. Hence, a reasonable choice for |
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| 90 | $\nu_{\rm min}$ is 0.2~GeV. \\ |
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| 91 | |
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| 92 | \noindent |
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| 93 | In Eq.~\ref{munu.2}, $A_{\rm eff}$ and $\sigma_{\gamma N}$ appear as |
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| 94 | factors. A more rigorous theoretical approach may lead to some dependence |
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| 95 | of the shadowing effect on $\nu$ and $E$; therefore in the differential |
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| 96 | cross section and in the sampling procedure, this possibility is forseen |
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| 97 | and the atomic weight $A$ of the element is kept as an explicit parameter. \\ |
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| 98 | |
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| 99 | \noindent |
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| 100 | The total cross section is obtained by integration of Eq.~\ref{munu.1} |
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| 101 | between $\nu_{\min}$ and $\nu_{\max}$; to facilitate the computation, |
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| 102 | a $\ln (\nu )$--substitution is used. |
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| 103 | |
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| 104 | \subsection{Sampling} |
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| 105 | |
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| 106 | \subsubsection{Sampling the Transferred Energy} |
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| 107 | |
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| 108 | The muon photonuclear interaction is always treated as a discrete process |
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| 109 | with its mean free path determined by the total cross section. The total |
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| 110 | cross section is obtained by the numerical integration of Eq.~\ref{munu.1} |
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| 111 | within the limits $\nu_{\min}$ and $\nu_{\max}$. The process is considered |
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| 112 | for muon energies $1 \rm{GeV} \leq T \leq 1000 \rm{PeV}$, though it should |
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| 113 | be noted that above 100~TeV the extrapolation (Eq.~\ref{munu.5}) of |
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| 114 | $\sigma_{\gamma N}$ may be too crude. \\ |
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| 115 | |
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| 116 | \noindent |
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| 117 | The random transferred energy, $\nu_{p}$, is found from the numerical |
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| 118 | solution of the equation : |
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| 119 | \begin{equation} |
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| 120 | \label{munu.6} |
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| 121 | P = \int_{\nu_{p}}^{\nu_{\rm max}} \sigma(E,\nu ) d \nu |
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| 122 | \left/ \int_{\nu_{\rm min}}^{\nu_{\rm max}} \sigma(E,\nu) d\nu \right. . |
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| 123 | \end{equation} |
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| 124 | % |
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| 125 | Here $P$ is the random uniform probability, with $\nu_{\rm max}= E-M/2$ |
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| 126 | and $\nu_{\rm min}=0.2$~GeV. \\ |
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| 127 | |
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| 128 | \noindent |
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| 129 | For fast sampling, the solution of Eq.~\ref{munu.6} is tabulated at |
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| 130 | initialization time. During simulation, the sampling method returns a value |
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| 131 | of $\nu_{p}$ corresponding to the probability $P$, by interpolating the |
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| 132 | table. The tabulation routine uses Eq.~\ref{munu.1} for the differential |
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| 133 | cross section. The table contains values of |
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| 134 | \begin{equation} |
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| 135 | \label{munu.7} |
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| 136 | x_p = \ln (\nu_p / \nu_{\rm max})/\ln (\nu_{\rm max}/\nu_{\rm min}), |
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| 137 | \end{equation} |
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| 138 | calculated at each point on a three-dimensional grid with constant spacings |
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| 139 | in $\ln (T)$, $\ln(A)$ and $\ln(P)$ . The sampling uses linear |
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| 140 | interpolations in $\ln(T)$ and $\ln(A)$, and a cubic interpolation in |
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| 141 | $\ln(P)$. Then the transferred energy is calculated from the inverse |
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| 142 | transformation of Eq.~\ref{munu.7}, |
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| 143 | $\nu_{p}=\nu_{\rm max}(\nu_{\rm max}/\nu_{\rm min})^{x_{p}}$. |
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| 144 | Tabulated parameters reproduce the theoretical dependence to better than |
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| 145 | 2\% for $T > 1$~GeV and better than 1\% for $ T >10$~GeV. |
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| 146 | |
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| 147 | \subsubsection{Sampling the Muon Scattering Angle} |
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| 148 | |
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| 149 | According to Refs.~\cite{munu.bor75, munu.bor77}, in the region where the |
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| 150 | four-momentum transfer is not very large ($Q^{2} \leq 3 {\rm GeV}^{2}$), |
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| 151 | the $t$~--~dependence of the cross section may be described as: |
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| 152 | \begin{equation} |
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| 153 | \label{munu.8} |
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| 154 | \frac{d \sigma }{dt} \sim \frac{(1- t / t_{\rm max}) } |
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| 155 | {t (1+t/ \nu^{2})(1+t/m^{2}_{0})} [(1-y)(1-t_{\rm min}/t)+y^{2}/2] , |
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| 156 | \end{equation} |
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| 157 | where $t$ is the square of the four-momentum transfer, |
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| 158 | $Q^{2} = 2 (EE' - PP'\cos\theta - \mu^{2})$. Also, |
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| 159 | $t_{\rm min} = (\mu y)^{2} / (1-y)$, $y=\nu /E$ and |
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| 160 | $t_{\rm max} = 2M\nu $. $\nu = E-E'$ is the energy lost by the muon and |
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| 161 | $E$ is the total initial muon energy. $M$ is the nucleon (proton) mass and |
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| 162 | $m_{0}^{2} \equiv \Lambda^{2} \simeq 0.4\;{\rm GeV}^{2}$ is |
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| 163 | a phenomenological parameter determing the behavior of the inelastic form |
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| 164 | factor. Factors which depend weakly, or not at all, on $t$ are omitted. \\ |
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| 165 | |
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| 166 | \noindent |
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| 167 | To simulate random $t$ and hence the random muon deflection angle, it is |
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| 168 | convenient to represent Eq.~\ref{munu.8} in the form : |
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| 169 | % |
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| 170 | \begin{equation} |
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| 171 | \label{munu.9} |
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| 172 | \sigma( t ) \sim f(t) g(t), |
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| 173 | \end{equation} |
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| 174 | where |
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| 175 | \begin{eqnarray} |
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| 176 | \label{munu.10} |
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| 177 | f(t) = \frac{1}{t(1+ t/t_{1} )} , \\ |
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| 178 | g(t) = \frac{1-t/t_{\max}}{1+t/t_{2}} \cdot |
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| 179 | \frac{(1-y)(1-t_{\min}/t)+y^{2}/2} |
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| 180 | {(1-y)+y^{2}/2}, \nonumber |
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| 181 | \end{eqnarray} |
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| 182 | and |
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| 183 | \begin{equation} |
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| 184 | \label{munu.11} |
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| 185 | t_{1} = {\min} (\nu^{2}, m_{0}^{2}) \quad |
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| 186 | t_{2} ={\max} (\nu^{2}, m_{0}^{2}) . |
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| 187 | \end{equation} |
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| 188 | |
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| 189 | \noindent |
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| 190 | $t_{P}$ is found analytically from Eq.~\ref{munu.10} : |
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| 191 | $$t_{P} = \frac{t_{\max}t_{1}} |
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| 192 | {\displaystyle(t_{\max}+t_{1})\left[\frac{t_{\max}(t_{\min}+t_{1})} |
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| 193 | {t_{\min}(t_{\max}+t_{1})}\right]^{P}-t_{\max}} , |
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| 194 | $$ |
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| 195 | % |
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| 196 | where $P$ is a random uniform number between 0 and 1, which is accepted with |
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| 197 | probability $g(t)$. The conditions of Eq.~\ref{munu.11} make use of the |
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| 198 | symmetry between $\nu^{2}$ and $m_{0}^{2}$ in Eq.~\ref{munu.8} and allow |
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| 199 | increased selection efficiency, which is typically $\geq 0.7$. The polar |
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| 200 | muon deflection angle $\theta$ can easily be found from |
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| 201 | \footnote{This convenient formula has been shown to the authors by |
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| 202 | D.A. Timashkov.} |
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| 203 | |
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| 204 | $$\sin^{2}(\theta /2) = \frac {t_{P} - t_{\rm min}} |
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| 205 | {4\,(EE'- \mu^{2}) - 2\,t_{\rm min}}. $$ |
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| 206 | The hadronic vertex is generated by the hadronic processes taking |
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| 207 | into account the four-momentum transfer. |
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| 208 | |
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| 209 | \subsection{Status of this document} |
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| 210 | |
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| 211 | 12.10.98 created by R.Kokoulin, A.Rybin.\\ |
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| 212 | 18.05.00 edited by S.Kelner, R.Kokoulin, and A.Rybin. \\ |
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| 213 | 07.12.02 re-worded by D.H. Wright \\ |
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| 214 | 30.08.04 correction of eq. 8.24 (to 1/sqrt) from H. Araujo \\ |
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| 215 | % |
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| 216 | |
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| 217 | \begin{latexonly} |
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| 218 | |
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| 219 | \begin{thebibliography}{99} |
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| 220 | |
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| 221 | \bibitem{munu.bor75} V.V.Borog and A.A.Petrukhin, |
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| 222 | Proc. 14th Int.Conf. on Cosmic Rays, Munich,1975, |
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| 223 | {\bf vol.6}, p.1949. |
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| 224 | \bibitem{munu.bez81} L.B.Bezrukov and E.V.Bugaev, |
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| 225 | Sov. J. Nucl. Phys., {\bf 33}, 1981, p.635. |
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| 226 | \bibitem{munu.hand63} L.N.Hand. Phys. Rev., {\bf 129}, 1834 (1963). |
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| 227 | \bibitem{munu.brod72} S.J.Brodsky, F.E.Close and J.F.Gunion, |
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| 228 | Phys. Rev. {\bf D6}, 177 (1972). |
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| 229 | \bibitem{munu.cald79} D.O. Caldwell et al., |
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| 230 | Phys. Rev. Lett., {\bf 42}, 553 (1979). |
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| 231 | \bibitem{munu.bor77} V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin, |
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| 232 | Sov. J. Nucl. Phys., {\bf 25}, 1977, p.46. |
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| 233 | |
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| 234 | |
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| 235 | \end{thebibliography} |
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| 236 | |
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| 237 | \end{latexonly} |
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| 238 | |
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| 239 | \begin{htmlonly} |
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| 240 | |
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| 241 | \subsection{Bibliography} |
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| 242 | |
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| 243 | \begin{enumerate} |
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| 244 | \item V.V.Borog and A.A.Petrukhin, |
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| 245 | Proc. 14th Int.Conf. on Cosmic Rays, Munich,1975, |
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| 246 | {\bf vol.6}, p.1949. |
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| 247 | \item L.B.Bezrukov and E.V.Bugaev, |
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| 248 | Sov. J. Nucl. Phys., {\bf 33}, 1981, p.635. |
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| 249 | \item L.N.Hand. Phys. Rev., {\bf 129}, 1834 (1963). |
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| 250 | \item S.J.Brodsky, F.E.Close and J.F.Gunion, |
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| 251 | Phys. Rev. {\bf D6}, 177 (1972). |
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| 252 | \item D.O. Caldwell et al., |
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| 253 | Phys. Rev. Lett., {\bf 42}, 553 (1979). |
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| 254 | \item V.V.Borog, V.G.Kirillov-Ugryumov, A.A.Petrukhin, |
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| 255 | Sov. J. Nucl. Phys., {\bf 25}, 1977, p.46. |
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| 256 | \end{enumerate} |
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| 257 | |
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| 258 | \end{htmlonly} |
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| 259 | |
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