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