After consultation with the Geant4 authors, it was suggested the most likely reason for the observed resolution changes is alterations to the multiple scattering models. Therefore, a dedicated test of multiple scattering was needed. It has been seen that the modelling of the multiple scattering process has a strong influence on the accuracy of the Impact Parameter (IP) resolution. As the IP is used extensively to isolate secondary vertices in the LHCb detector, it is crucial the modelling of the IP is not degraded.
When a charged particle traverses material there is a non-zero probability that it will undergo elastic Coulomb scattering from a nucleus within the material. The differential cross section for this process is given by
dσ dΩ = 1 4π0 2 z2e4 M2c4β4 1 sin4(θ/2), (3.3)
Figure 3.4: The lateral displacement and angular dispersion when a charged particle traverses a medium [1].
of the charged particle and θ is the angle through which the charged particle is scattered [99].
Except for cases where the scattering material is a very thin film, the charged particle will scatter multiple times before exiting the material. Hence, multiple coulomb scattering, which is more commonly known as just multiple scattering, occurs. The net effect is a lateral displacement as well as a scattering angle, as depicted in Figure 3.4. In this case a statistical treatment has to be used to obtain a distribution for the scattering angle, which is defined as θ in Figure 3.4. One such statistical treatment is Moli`ere theory, which has been shown to give very good agreement with data over a wide range of particles, materials and energies [100,101]. Several other theories have been shown to produce consistent results; Lewis theory also provides moments for the spatial displacement distribution [102]. Both the Moliere and Lewis theories give a scattering angle distribution that is Gaussian for the central 98% of scattering angle values, but the tails of the distribution fall off more slowly than a Gaussian function due to the 1/ sin4(θ/2) term in Equation 3.3. The width of the central Gaussian is defined as θ0 which can be approximated by the Highland
formula θ0 = 14.1 MeV pv z r L LR 1 + 1 9log10 L LR , (3.4)
where p is the incident particle’s momentum, v is the incident particle’s velocity, L is the length of the material and Lr is the radiation length of the material. This
formula is an empirical formula that arises from fits to Moli`ere theory [103].
When one simulates multiple scattering, in a similar way to the theoretical models, it is rarely possible to simulate every individual collision. It is only possible if the number of scatters is small and a large amount of CPU time is available. For the latter reason, this type of simulation based on simulating every single scatter was not implemented until 2005. Unfortunately, there is still a limited number of applications where this is a viable option. The UrbanMsc models get around this by using a “condensed” simulation of multiple scattering, which involves simulating one step of the particle’s path at a time and applying net effects at the end of each step [95]. More specifically, the angle through which the particle has been scattered and the lateral displacement are applied at the end of each step as part of the multiple scattering simulation. The scattering angle is sampled from distributions calculated using Lewis theory but no theory of a full displacement distribution exists. Therefore, Geant4 uses its own, approximate, algorithms to calculate the lateral displacement after each step [104].
Another approach to simulating multiple scattering, which has been implemented more recently, is to use a “mixed” approach. This involves sampling scatters, where the scattering angle θ is below a threshold θmax, in a similar way to the “con-
densed” approach discussed previously. However, if the scattering angle is above θmax then a single scattering approach is used. This is implemented in Geant4 as
the WentzelVI model.
To carry out a direct investigation of multiple scattering, a test based on an example provided by the authors of Geant4 was setup to fire particles into a square sheet
θ
Figure 3.5: The setup of the multiple scattering test. θ is the angle under investi- gation by this test.
of material at normal incidence and study the angle of the scattered particle to the normal as it exits the material on the opposite side. A diagram of this is shown in Figure 3.5. The type of material used, the width of the material and the thickness of the material can all be specified. In this case the setup used was a 300µm thick sheet of silicon designed to model the LHCb VELO as closely as possible. This is the area of the detector where precise tracking measurements sensitive to multiple scattering take place; the IP is largely dictated by measurements in the VELO. The aim of this test is to measure the parameter θ0 of the scattered particles’ angular
distribution for electrons at a range of energies and use it as a metric with which to
compare Geant4 versions. The θ0 parameter is then estimated at each energy by
calculating the standard deviation of the central 98% of scattering angles, effectively measuring the width of the central Gaussian component of the scattering angle distribution. In order to estimate an uncertainty on the θ0 parameter the test is
re-run 1000 times at each energy, the mean of the resulting θ0 values is used for
Energy/GeV
1
0
0.2
0.4
0.6
0.8
1
/mrad
0θ
0
0.1
0.2
0.3
0.4
0.5
0.6
Geant4 v9.5.2 Geant4 v9.6.4 Geant4 v10.3.3Figure 3.6: θ0 as a function of the inverse of the incident electron energy.
3.4.1
Multiple Scattering Test Results
This test has been performed at 14 energies in the range 1 – 100 GeV for the three versions of Geant4 introduced in Section 3.3 (with the EMLHCb PL): v9.5.2, v9.6.4 and v10.3.3. The θ0 parameter as a function of the inverse of the incident
electron energy can be seen in Figure 3.6, and the numerical results can be found in Table 3.2. These results show, firstly, that there is no change in the scattering angle between Geant4 v9.6.4 and v10.3.3. However, it does show that there was a difference in scattering angle between Geant4 v9.5.2 and v9.6.4. This change is prevalent at low energies. The most likely reason for this change is the fact that the multiple scattering model used for electrons and positrons above 100 MeV changed from the UrbanMsc95 model in v9.5.2 to the WentzelVI model in v9.6.4. This observation meant that particularly close attention was paid to reconstructed physics quantities which are sensitive to multiple scattering, such as the IP resolution, when the validation of the full simulation package was carried out.
Energy/ GeV θ0/mrad v 9.5.2 v9.6.4 v10.3.3 1 0.5809± 0.0031 0.6065± 0.0031 0.6051± 0.0029 2 0.2901± 0.0015 0.3033± 0.0015 0.3025± 0.0015 3 0.1931± 0.0009 0.2023± 0.0010 0.2017± 0.0010 4 0.1447± 0.0008 0.1517± 0.0008 0.1512± 0.0007 5 0.1156± 0.0006 0.1214± 0.0006 0.1210± 0.0006 7 0.0824± 0.0004 0.0867± 0.0004 0.0865± 0.0004 9 0.0641± 0.0003 0.0674± 0.0004 0.0672± 0.0003 12 0.0480± 0.0003 0.0506± 0.00025 0.0504± 0.0003 15 0.0383± 0.0002 0.0405± 0.00020 0.0404± 0.0002 20 0.0287± 0.0002 0.0304± 0.00015 0.0303± 0.0002 25 0.0229± 0.0001 0.0243± 0.00012 0.0242± 0.0001 30 0.0191± 0.0001 0.0202± 0.00010 0.0202± 0.0001 40 0.01431± 0.00008 0.01524 ± 0.00008 0.0151± 0.0001 100 0.00577± 0.00003 0.00607 ± 0.00002 0.00605 ± 0.00003
Table 3.2: Results from the multiple scattering test, showing θ0 at various energies
for the three Geant4 versions tested.