Muon Photonuclear Interaction

The inelastic interaction of muons with nuclei is important at high muon energies (\(E \geq 10\) GeV), and at relatively high energy transfers \(\nu\) (\(\nu / E \geq 10^{-2}\)). It is especially important for light materials and for the study of detector response to high energy muons, muon propagation and muon-induced hadronic background. The average energy loss for this process increases almost linearly with energy, and at TeV muon energies constitutes about 10% of the energy loss rate.

The main contribution to the cross section \(\sigma ( E, \nu\)) and energy loss comes from the low \(Q^{2}\)–region ( \(Q^{2} \ll 1~{\rm GeV}^{2}\)). In this domain, many simplifications can be made in the theoretical consideration of the process in order to obtain convenient and simple formulae for the cross section. Most widely used are the expressions given by Borog and Petrukhin [BP75], and Bezrukov and Bugaev [BB81]. Results from these authors agree within 10% for the differential cross section and within about 5% for the average energy loss, provided the same photonuclear cross section, \(\sigma_{\gamma N}\), is used in the calculations.

Differential Cross Section

The Borog and Petrukhin formula for the cross section is based on:

  • Hand’s formalism [Han63] for inelastic muon scattering,

  • a semi-phenomenological inelastic form factor, which is a Vector Dominance Model with parameters estimated from experimental data, and

  • nuclear shadowing effects with a reasonable theoretical parameterization [BCG72].

For \(E\geq10\) GeV, the Borog and Petrukhin cross section, differential in transferred energy, is

(161)\[\sigma(E,\nu ) = \Psi(\nu ) \Phi( E,v ) ,\]
(162)\[\Psi(\nu) = \frac{\alpha}{\pi} \frac{A_{\rm eff} N_{AV}}{A} \sigma_{\gamma N}(\nu) \frac{1}{\nu} ,\]
(163)\[ \Phi(E,v) = v-1 + \left[1-v+\frac{v^{2}}{2} \left(1+\frac{2\mu^{2}}{\Lambda^{2}}\right)\right] \ln\frac{\frac{E^{2}(1-v)}{\mu^{2}}\left(1+\frac{\mu^{2} v^{2}}{\Lambda^{2}(1-v)}\right)} { 1+\frac{Ev}{\Lambda}\left(1+\frac{\Lambda}{2M}+\frac{Ev}{\Lambda}\right)},\]

where \(\nu\) is the energy lost by the muon, \(v = \nu /\)E, and \(\mu\) and \(M\) are the muon and nucleon (proton) masses, respectively. \(\Lambda\) is a Vector Dominance Model parameter in the inelastic form factor which is estimated to be \(\Lambda^{2}=0.4 \;{\rm GeV}^{2}\).

For \(A_{\rm eff}\), which includes the effect of nuclear shadowing, the parameterization [BCG72]

\[A_{\rm eff} = 0.22 A + 0.78 A^{0.89}\]

is chosen.

A reasonable choice for the photonuclear cross section, \(\sigma_{\gamma N}\), is the parameterization obtained by Caldwell et al. [eal79] based on the experimental data on photoproduction by real photons:

(164)\[\sigma_{\gamma N} = ( 49.2 + 11.1 \ln K + 151.8/ \sqrt{K} ) \cdot 10^{-30} {\rm cm}^{2}, \quad K~~\mbox{in GeV} .\]

The upper limit of the transferred energy is taken to be \(\nu_{\rm max}\) = \(E - M/2\). The choice of the lower limit \(\nu_{\rm min}\) is less certain since the formula (161), (162), (163) is not valid in this domain. Fortunately, \(\nu_{\rm min}\) influences the total cross section only logarithmically and has no practical effect on the average energy loss for high energy muons. Hence, a reasonable choice for \(\nu_{\rm min}\) is 0.2 GeV.

In Eq.(162), \(A_{\rm eff}\) and \(\sigma_{\gamma N}\) appear as factors. A more rigorous theoretical approach may lead to some dependence of the shadowing effect on \(\nu\) and \(E\); therefore in the differential cross section and in the sampling procedure, this possibility is foreseen and the atomic weight \(A\) of the element is kept as an explicit parameter.

The total cross section is obtained by integration of Eq.(161) between \(\nu_{\rm min}\) and \(\nu_{\rm max}\); to facilitate the computation, a \(\ln (\nu)\) substitution is used.

Sampling

Sampling the Transferred Energy

The muon photonuclear interaction is always treated as a discrete process with its mean free path determined by the total cross section. The total cross section is obtained by the numerical integration of Eq.(161) within the limits \(\nu_{\min}\) and \(\nu_{\max}\). The process is considered for muon energies \(1 \rm{ GeV} \leq T \leq 1000\) PeV, though it should be noted that above 100 TeV the extrapolation (Eq.(164)) of \(\sigma_{\gamma N}\) may be too crude. The random transferred energy, \(\nu_{p}\), is found from the numerical solution of the equation :

(165)\[P = \int_{\nu_{p}}^{\nu_{\rm max}} \sigma(E,\nu ) d \nu \left/ \int_{\nu_{\rm min}}^{\nu_{\rm max}} \sigma(E,\nu) d\nu \right. .\]

Here \(P\) is the random uniform probability, with \(\nu_{\rm max}= E-M/2\) and \(\nu_{\rm min}=0.2\) GeV. For fast sampling, the solution of Eq.(165) is tabulated at initialization time. During simulation, the sampling method returns a value of \(\nu_{p}\) corresponding to the probability \(P\), by interpolating the table. The tabulation routine uses Eq.(161) for the differential cross section. The table contains values of

(166)\[ x_p = \ln (\nu_p / \nu_{\rm max})/\ln (\nu_{\rm max}/\nu_{\rm min}),\]

calculated at each point on a three-dimensional grid with constant spacings in \(\ln (T)\), \(\ln(A)\) and \(\ln(P)\) . The sampling uses linear interpolations in \(\ln(T)\) and \(\ln(A)\), and a cubic interpolation in \(\ln(P)\). Then the transferred energy is calculated from the inverse transformation of Eq.(166), \(\nu_{p}=\nu_{\rm max}(\nu_{\rm max}/\nu_{\rm min})^{x_{p}}\). Tabulated parameters reproduce the theoretical dependence to better than 2% for \(T > 1\) GeV and better than 1% for \(T >10\) GeV.

Sampling the Muon Scattering Angle

According to Refs. [BKUP77, BP75], in the region where the four-momentum transfer is not very large (\(Q^{2} \leq 3 {\rm GeV}^{2}\)), the \(t\)-dependence of the cross section may be described as:

(167)\[ \frac{d \sigma }{dt} \sim \frac{(1- t / t_{\rm max}) } {t (1+t/ \nu^{2})(1+t/m^{2}_{0})} [(1-y)(1-t_{\rm min}/t)+y^{2}/2] ,\]

where \(t\) is the square of the four-momentum transfer, \(Q^{2} = 2 (EE' - PP'\cos\theta - \mu^{2})\). Also, \(t_{\rm min} = (\mu y)^{2} / (1-y)\), \(y=\nu /E\) and \(t_{\rm max} = 2M\nu\). \(\nu = E-E'\) is the energy lost by the muon and \(E\) is the total initial muon energy. \(M\) is the nucleon (proton) mass and \(m_{0}^{2} \equiv \Lambda^{2} \simeq 0.4\;{\rm GeV}^{2}\) is a phenomenological parameter determining the behavior of the inelastic form factor. Factors which depend weakly, or not at all, on \(t\) are omitted.

To simulate random \(t\) and hence the random muon deflection angle, it is convenient to represent Eq.(167) in the form :

\[\sigma( t ) \sim f(t) g(t),\]

where

(168)\[\begin{split}f(t) &= \frac{1}{t(1+ t/t_{1} )} , \\ g(t) &= \frac{1-t/t_{\max}}{1+t/t_{2}} \cdot \frac{(1-y)(1-t_{\min}/t)+y^{2}/2} {(1-y)+y^{2}/2},\end{split}\]

and

(169)\[ t_{1} = {\min} (\nu^{2}, m_{0}^{2}) \quad t_{2} ={\max} (\nu^{2}, m_{0}^{2}) .\]

\(t_{P}\) is found analytically from Eq.(168) :

\[t_{P} = \frac{t_{\max}t_{1}} {(t_{\max}+t_{1})\left[\frac{t_{\max}(t_{\min}+t_{1})} {t_{\min}(t_{\max}+t_{1})}\right]^{P}-t_{\max}} ,\]

where \(P\) is a random uniform number between 0 and 1, which is accepted with probability \(g(t)\). The conditions of Eq.(169) make use of the symmetry between \(\nu^{2}\) and \(m_{0}^{2}\) in Eq.(167) and allow increased selection efficiency, which is typically \(\geq 0.7\). The polar muon deflection angle \(\theta\) can easily be found from 1.

\[\sin^{2}(\theta /2) = \frac {t_{P} - t_{\rm min}} {4\,(EE'- \mu^{2}) - 2\,t_{\rm min}}.\]

The hadronic vertex is generated by the hadronic processes taking into account the four-momentum transfer.

Bibliography

BB81

L.B. Bezrukov and E.V. Bugaev. Nucleon shadowing effects in photon nucleus interaction. (in russian). Yad. Fiz., 33:1195–1207, 1981. Sov. J. Nucl. Phys. 33, 635 (1981) - english translation.

BKUP77

V. V Borog, V. G. Kirillov-Ugryumov, and A. A Petrukhin. Inelastic Interactions of Muons with Fe Nuclei at Small q**2 in the Energy Region Above 200-GeV. Yad. Fiz., 25:85–93, 1977. Sov. J. Nucl. Phys.25, 46-51 (1977) - english translation.

BP75(1,2)

V.V. Borog and A.A. Petrukhin. The cross-section of the nuclear interaction of high energy muons. International Cosmic Ray Conference, 6:1949–1954, August 1975.

BCG72(1,2)

Stanley J. Brodsky, Francis E. Close, and J. F. Gunion. Phenomenology of photon processes, vector dominance, and crucial tests for parton models. Physical Review D, 6(1):177–189, jul 1972. URL: https://doi.org/10.1103/PhysRevD.6.177, doi:10.1103/physrevd.6.177.

eal79

D. O. Caldwell et al. Measurement of shadowing in photon-nucleus total cross sections from 20 to 185 GeV. Physical Review Letters, 42(9):553–556, feb 1979. URL: https://doi.org/10.1103/PhysRevLett.42.553, doi:10.1103/physrevlett.42.553.

Han63

L. N. Hand. Experimental investigation of pion electroproduction. Physical Review, 129(4):1834–1846, feb 1963. URL: https://doi.org/10.1103/PhysRev.129.1834, doi:10.1103/physrev.129.1834.

1

This convenient formula has been shown to the authors by D.A. Timashkov.