Bremsstrahlung

Bremsstrahlung dominates other muon interaction processes in the region of catastrophic collisions (\(v \geq 0.1\) ), that is at “moderate” muon energies above the kinematic limit for knock–on electron production. At high energies (\(E \geq 1\) TeV) this process contributes about 40% of the average muon energy loss.

Differential Cross Section

The differential cross section for muon bremsstrahlung (in units of cm²/(g GeV) can be written as

\[\begin{split}\frac{d \sigma (E,\epsilon,Z,A)}{d \epsilon} &= \frac{16}{3} \alpha N_A (\frac{m}{\mu} r_{e})^2 \frac{1}{\epsilon A} Z(Z \Phi_n + \Phi_e)(1-v+\frac{3}{4} v^2) \\ &= 0 \quad \mbox{if } \quad \epsilon \geq \epsilon_{\rm max} = E-\mu ,\end{split}\]

where \(\mu\) and \(m\) are the muon and electron masses, \(Z\) and \(A\) are the atomic number and atomic weight of the material, and \(N_{A}\) is Avogadro’s number. If \(E\) and \(T\) are the initial total and kinetic energy of the muon, and \(\epsilon\) is the emitted photon energy, then \(\epsilon = E - E'\) and the relative energy transfer \(v = \epsilon /E\).

\(\Phi_{n}\) represents the contribution of the nucleus and can be expressed as

\[\begin{split}\Phi_{n} &= \ln \frac {BZ^{-1/3}(\mu + \delta (D_{n}' \sqrt{e} -2))} {D_{n}'(m+ \delta \sqrt{e}BZ^{-1/3})} ; \\ &= 0 \quad \mbox{if negative}.\end{split}\]

\(\Phi_{e}\) represents the contribution of the electrons and can be expressed as

\[\begin{split}\Phi_{e} &= \ln \frac {B'Z^{-2/3} \mu } {\left(1+ \displaystyle\frac{\delta \mu}{m^{2} \sqrt{e}}\right)(m+ \delta \sqrt{e} B'Z^{-2/3}) }; \\ &= 0 \quad \mbox{if} \epsilon \geq \epsilon'_{\rm max} = E/(1+ \mu^{2}/2mE); \\ &= 0 \quad \mbox{if negative}.\end{split}\]

In \(\Phi_n\) and \(\Phi_e\), for all nuclei except hydrogen,

\[\begin{split}\delta &= \mu^{2} \epsilon /2EE' = \mu^{2} v/2(E- \epsilon);\\ D'_{n} &= D_{n}^{(1-1/Z)}, \quad D_{n}= 1.54A^{0.27}; \\ B &= 183, \\ B' &= 1429, \\ \sqrt{e} &= 1.648(721271).\end{split}\]

For hydrogen (\(Z\)=1) \(B = 202.4,\: B' = 446, \: D_{n}' = D_{n}\).

These formulae are taken mostly from Refs. [KKP95] and [SRKP97]. They include improved nuclear size corrections in comparison with Ref. [PS68] in the region \(v \sim 1\) and low \(Z\). Bremsstrahlung on atomic electrons (taking into account target recoil and atomic binding) is introduced instead of a rough substitution \(Z(Z+1)\). A correction for processes with nucleus excitation is also included [ABB94].

Applicability and Restrictions of the Method

The above formulae assume that:

\(E \gg \mu\), hence the ultrarelativistic approximation is used;
\(E \leq 10^{20}\) eV; above this energy, LPM suppression can be expected;
\(v \geq 10^{-6}\) ; below \(10^{-6}\) Ter-Mikaelyan suppression takes place. However, in the latter region the cross section of muon bremsstrahlung is several orders of magnitude less than that of other processes.

The Coulomb correction (for high \(Z\)) is not included. However, existing calculations [AB97] show that for muon bremsstrahlung this correction is small.

Continuous Energy Loss

The restricted energy loss for muon bremsstrahlung \((d E/ dx)_{\rm rest}\) with relative transfers \(v = \epsilon / (T+ \mu) \leq v_{\rm cut}\) can be calculated as follows :

\[\left(\frac{d E}{d x}\right)_{\rm rest} = \int_{0}^{\epsilon_{\rm cut}} \epsilon\,\sigma (E,\epsilon )\,d\epsilon = (T+\mu ) \int_{0}^{v_{\rm cut}}\epsilon\,\sigma (E,\epsilon)\,dv\,.\]

If the user cut \(v_{\rm cut} \geq v_{\rm max}=T/(T+ \mu)\), the total average energy loss is calculated. Integration is done using Gaussian quadratures, and binning provides an accuracy better than about 0.03% for \(T = 1\) GeV, \(Z=1\). This rapidly improves with increasing \(T\) and \(Z\).

Total Cross Section

The integration of the differential cross section over \(d\epsilon\) gives the total cross section for muon bremsstrahlung:

\[\sigma_{\rm tot} (E,\epsilon_{\rm cut}) = \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}}\sigma (E,\epsilon ) d \epsilon = \int_{\ln v_{\rm cut}}^{\ln v_{\rm max}}\epsilon \sigma (E,\epsilon) d(\ln v) ,\]

where \(v_{\rm max}=T/(T+ \mu)\). If \(v_{\rm cut} \geq v_{\rm max}\) , \(\sigma_{\rm tot}=0\).

Sampling

The photon energy \(\epsilon_{p}\) is found by numerically solving the equation :

\[P \:= \int_{\epsilon_{p}}^{\epsilon_{\rm max}} \sigma (E,\epsilon,Z,A) \, d \epsilon \left/ \int_{\epsilon_{\rm cut}}^{\epsilon_{\rm max}} \sigma (E,\epsilon,Z,A ) \, d \epsilon\right. .\]

Here \(P\) is the random uniform probability, \(\epsilon_{\rm max}=T\), and \(\epsilon_{\rm cut}=(T+\mu) \cdot v_{\rm cut}\). \(v_{min.cut}=10^{-5}\) is the minimal relative energy transfer adopted in the algorithm.

For fast sampling, the solution of the above equation is tabulated at initialization time for selected \(Z\), \(T\) and \(P\). During simulation, this table is interpolated in order to find the value of \(\epsilon_{p}\) corresponding to the probability \(P\).

The tabulation routine uses accurate functions for the differential cross section. The table contains values of

(156)\[ x_p = \ln (v_p / v_{\rm max})/\ln (v_{\rm max}/v_{\rm cut}) ,\]

where \(v_{p} = \epsilon_{p}/(T+ \mu)\) and \(v_{\rm max} = T/(T+ \mu)\). Tabulation is performed in the range \(1 \leq Z \leq 128\), \(1 \leq T \leq 1000\) PeV, \(10^{-5} \leq P \leq 1\) with constant logarithmic steps. Atomic weight (which is a required parameter in the cross section) is estimated here with an iterative solution of the approximate relation:

\[A = Z\,(2+0.015\,A^{2/3}).\]

For \(Z=1\), \(A=1\) is used.

To find \(x_{p}\) (and thus \(\epsilon_{p}\)) corresponding to a given probability \(P\), the sampling method performs a linear interpolation in \(\ln Z\) and \(\ln T\), and a cubic, 4 point Lagrangian interpolation in \(\ln P\). For \(P \leq P_{\rm min}\), a linear interpolation in \((P,x)\) coordinates is used, with \(x = 0\) at \(P = 0\). Then the energy \(\epsilon_p\) is obtained from the inverse transformation of (156) :

\[\epsilon_{p} = (T+ \mu ) v_{\rm max} (v_{\rm max}/v_{\rm cut})^{x_{p}}\]

The algorithm with the parameters described above has been tested for various \(Z\) and \(T\). It reproduces the differential cross section to within 0.2 – 0.7 % for \(T \geq 10\) GeV. The average total energy loss is accurate to within 0.5%. While accuracy improves with increasing \(T\), satisfactory results are also obtained for \(1 \leq T \leq 10\) GeV.

It is important to note that this sampling scheme allows the generation of \(\epsilon_{p}\) for different user cuts on \(v\) which are above \(v_{\rm min.cut}\). To perform such a simulation, it is sufficient to define a new probability variable

\[P' = P \: \sigma_{\rm tot} \: (v_{\rm user.cut}) / \sigma_{\rm tot} (v_{\rm min.cut})\]

and use it in the sampling method. Time consuming re-calculation of the 3-dimensional table is therefore not required because only the tabulation of \(\sigma_{\rm tot}(v_{\rm user.cut})\) is needed.

The small-angle, ultrarelativistic approximation is used for the simulation (with about 20% accuracy at \(\theta\le\theta^*\approx1\)) of the angular distribution of the final state muon and photon. Since the target recoil is small, the muon and photon are directed symmetrically (with equal transverse momenta and coplanar with the initial muon):

\[p_{\perp \mu} = p_{\perp \gamma}, \quad {\rm where} \quad p_{\perp \mu}= E' \theta_{\mu}, \quad p_{\perp \gamma} = \epsilon \theta_{\gamma} .\]

\(\theta_{\mu}\) and \(\theta_{\gamma}\) are muon and photon emission angles. The distribution in the variable \(r=E\theta_{\gamma}/\mu\) is given by

\[f(r) dr \sim r dr/(1+r^2)^2 .\]

Random angles are sampled as follows:

\[\theta_{\gamma} = \frac{\mu}{E} r \quad \theta_{\mu} = \frac{\epsilon}{E'} \theta_{\gamma} ,\]

where

\[r=\sqrt{\frac{a}{1-a}}\,,\quad a=\xi\,\frac{r_{\rm \max}^2}{1+r_{\rm max}^2}\,,\quad r_{\max}=\min(1,E'/\epsilon)\cdot E\,\theta^*/\mu\,,\]

and \(\xi\) is a random number uniformly distributed between 0 and 1.

Bibliography

ABB94: Y. M. Andreev, L. B. Bezrukov, and E. V. Bugaev. Excitation of a target in muon bremsstrahlung. Physics of Atomic Nuclei, 57:2066–2074, December 1994.
AB97: Yu. M. Andreev and E. V. Bugaev. Muon bremsstrahlung on heavy atoms. Physical Review D, 55(3):1233–1243, feb 1997. URL: https://doi.org/10.1103/PhysRevD.55.1233, doi:10.1103/physrevd.55.1233.
KKP95: SR Kelner, RP Kokoulin, and AA Petrukhin. About cross section for high-energy muon bremsstrahlung. Technical Report, MEphI, 1995. Preprint MEPhI 024-95, Moscow, 1995, CERN SCAN-9510048.
PS68: A. A. Petrukhin and V. V. Shestakov. The influence of the nuclear and atomic form factors on the muon bremsstrahlung cross section. Canadian Journal of Physics, 46(10):S377–S380, may 1968. URL: https://doi.org/10.1139/p68-251, doi:10.1139/p68-251.
SRKP97: R.P. Kokoulin S.R. Kelner and A.A. Petrukhin. Bremsstrahlung from muons scattered by atomic electrons. Physics of Atomic Nuclei, 60:576–583, April 1997.