Gflash Shower Parameterizations

The computing time needed for the simulation of high energy electromagnetic showers can become very large, since it increases approximately linearly with the energy absorbed in the detector. Using parameterizations instead of individual particle tracking for electromagnetic (sub)showers can speed up the simulations considerably without sacrificing much precision. The Gflash package allows the parameterization of electron and positron showers in homogeneous (for the time being) calorimeters and is based on the parameterization described in Ref. [GP93] .

Parameterization Ansatz

The spatial energy distribution of electromagnetic showers is given by three probability density functions (pdf),

\[dE (\vec{r}) \, = \, E\, f(t)dt\, f(r)dr\, f(\phi) d\phi ,\]

describing the longitudinal, radial, and azimuthal energy distributions. Here \(t\) denotes the longitudinal shower depth in units of radiation length, \(r\) measures the radial distance from the shower axis in Molière units, and \(\phi\) is the azimuthal angle. The start of the shower is defined by the space point where the electron or positron enters the calorimeter, which is different from the original Gflash. A gamma distribution is used for the parameterization of the longitudinal shower profile, \(f(t)\). The radial distribution \(f(r)\), is described by a two-component ansatz. In \(\phi\), it is assumed that the energy is distributed uniformly: \(f(\phi) = 1/2\pi\).

Longitudinal Shower Profiles

The average longitudinal shower profiles can be described by a gamma distribution [LS75]:

\[\left\langle \frac{1}{E} \frac{dE(t)}{dt} \right\rangle \, = \, f(t) \, = \, \frac{ (\beta t)^{\alpha -1} \beta \exp(-\beta t) } { \Gamma(\alpha) }.\]

The center of gravity, \(\langle t \rangle\), and the depth of the maximum, \(T\), are calculated from the shape parameter \(\alpha\) and the scaling parameter \(\beta\) according to:

(217)\[\begin{split}\langle t \rangle & = \frac{\alpha}{\beta}\\ T & = \frac{\alpha-1}{\beta}.\end{split}\]

In the parameterization all lengths are measured in units of radiation length \((X_0)\), and energies in units of the critical energy (\(E_c= 2.66 \left( X_0 \frac{Z}{A} \right)^{1.1}\)). This allows material independence, since the longitudinal shower moments are equal in different materials, according to Ref. [Ros52]. The following equations are used for the energy dependence of \(T_{hom}\) and \((\alpha_{hom})\), with \(y = E/E_c\) and \(t=x/X_0\), \(x\) being the longitudinal shower depth:

\[\begin{split}T_{hom} & = \ln y + t_1 \\ \alpha_{hom} & = a_1 + (a_2 + a_3/Z) \ln y.\end{split}\]

The \(y\)-dependence of the fluctuations can be described by:

(218)\[\sigma \, = \, ( s_1 + s_2 \ln y )^{-1} .\]

The correlation between \(\ln T_{hom}\) and \(\ln \alpha_{hom}\) is given by:

(219)\[\rho(\ln T_{hom}, \ln \alpha_{hom}) \, \equiv \, \rho \, = \, r_1 + r_2 \ln y .\]

From these formulae, correlated and varying parameters \(\alpha_i\) and \(\beta_i\) are generated according to

\[\begin{split}\left( \begin{array}{c} \ln T_i \\ \ln \alpha_i \end{array} \right) \, = \, \left( \begin{array}{c} \langle \ln T \rangle \\ \langle \ln \alpha \rangle \end{array} \right) + C \left( \begin{array}{c} z_1 \\ z_2 \end{array} \right)\end{split}\]

with

\[\begin{split}C \, = \, \left( \begin{array}{cc} \sigma (\ln T) & 0 \\ 0 & \sigma (\ln \alpha) \end{array} \right) \left( \begin{array}{cc} \sqrt{\frac{1+\rho}{2}} & \sqrt{\frac{1-\rho}{2}} \\ \sqrt{\frac{1+\rho}{2}} & - \sqrt{\frac{1-\rho}{2}} \end{array} \right) \,\end{split}\]

\(\sigma (\ln \alpha)\) and \(\sigma (\ln T)\) are the fluctuations of \(T_{hom}\) and \((\alpha_{hom}\). The values of the coefficients can be found in Ref.[GP93].

Radial Shower Profiles

For the description of average radial energy profiles,

\[f(r) \, = \, \frac{1}{dE(t)} \frac{dE(t,r)}{dr},\]

a variety of different functions can be found in the literature. In Gflash the following two-component ansatz, an extension of that in Ref. [GRP90], was used:

\[\begin{split}f(r) & = p f_C(r) + (1-p) f_T(r) \\ & = p \frac{2 r R_C^2}{(r^2 + R_C^2)^2} + (1-p) \frac{2 r R_T^2}{(r^2 + R_T^2)^2}\end{split}\]

with

\[0 \leq p \leq 1 .\]

Here \(R_C\) (\(R_T\)) is the median of the core (tail) component and \(p\) is a probability giving the relative weight of the core component. The variable \(\tau = t/T\), which measures the shower depth in units of the depth of the shower maximum, is used in order to generalize the radial profiles. This makes the parameterization more convenient and separates the energy and material dependence of various parameters. The median of the core distribution, \(R_C\), increases linearly with \(\tau\). The weight of the core, \(p\), is maximal around the shower maximum, and the width of the tail, \(R_T\), is minimal at \(\tau \approx 1\).

The following formulae are used to parameterize the radial energy density distribution for a given energy and material:

\[\begin{split}R_{C,hom}(\tau) & = z_1 + z_2 \tau \\ R_{T,hom}(\tau) & = k_1 \{ \exp (k_3(\tau -k_2)) + \exp (k_4(\tau -k_2)) \} \\ p_{hom}(\tau) & = p_1 \exp \left\{ \frac{p_2-\tau}{p_3} - \exp \left( \frac{p_2-\tau}{p_3} \right) \right\}\end{split}\]

The parameters \(z_1 \cdots p_3\) are either constant or simple functions of \(\ln E\) or \(Z\).

Radial shape fluctuations are also taken into account. A detailed explanation of this procedure, as well as a list of all the parameters used in Gflash, can be found in Ref. [GP93].

Gflash Performance

The parameters used in this Gflash implementation were extracted from full simulation studies with Geant 3. They also give good results inside the Geant4 fast shower framework when compared with the full electromagnetic shower simulation. However, if more precision or higher particle energies are required, retuning may be necessary. For the longitudinal profiles the difference between full simulation and Gflash parameterization is at the level of a few percent. Because the radial profiles are slightly broader in Geant3 than in Geant4, the differences may reach >10%. The gain in speed, on the other hand, is impressive. The simulation of a 1 TeV electron in a PbWO₄ cube is 160 times faster with Gflash. Gflash can also be used to parameterize electromagnetic showers in sampling calorimeters. So far, however, only homogeneous materials are supported.

Bibliography

GRP90

G. Grindhammer, M. Rudowicz, and S. Peters. The fast simulation of electromagnetic and hadronic showers. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 290(2-3):469–488, may 1990. URL: https://doi.org/10.1016/0168-9002(90)90566-O, doi:10.1016/0168-9002(90)90566-o.

GP93(1,2,3)

Guenter Grindhammer and S. Peters. The Parameterized simulation of electromagnetic showers in homogeneous and sampling calorimeters. In International Conference on Monte Carlo Simulation in High-Energy and Nuclear Physics - MC 93 Tallahassee, Florida, February 22-26, 1993. 1993. arXiv:hep-ex/0001020.

LS75

Egidio Longo and Ignazio Sestili. Monte carlo calculation of photon-initiated electromagnetic showers in lead glass. Nuclear Instruments and Methods, 128(2):283–307, oct 1975. URL: https://doi.org/10.1016/0029-554X(75)90679-5, doi:10.1016/0029-554x(75)90679-5.

Ros52

B. Rossi. High Energy Particles. Prentice-Hall, edition, 1952.