Reweighting Monte Carlo

For any Monte Carlo event a probability for it occurring, i.e. it's normalised cross-section, can be calculated from the square of the amplitude for this event. The probability can be calculated of it being a Standard Model event or an event in the presence of an anomalous coupling. The ratio of these two probabilities can be used to reweight a set of Standard Model events into a sample corresponding to the non-Standard Model anomalous coupling.

The WVCXME [100] program was developed from the EXCALIBUR four-fermion Monte Carlo generator. It uses the matrix elements from the generator program to calculate the amplitude squared. All it needs to know is the four-vectors of the four-fermions in the event and then it can calculate the amplitude squared for the Standard Model or with an anomalous coupling present. The generator includes all the features discussed in section 5.2.1. EXCALIBUR is a four-fermion generator, so WVCXME also takes into account the non-CC03 events that are sensitive to TGCs, and also the interference between the four-fermion final states.

An example of the relevant angles for the SDM analysis calculated from a fully detector simulated Standard Model four-fermion EXCALIBUR sample of Monte Carlo events, that have been reweighted to a sample with $\Delta\kappa_{\gamma}$$=-2.0$, is shown in figure 6.2. Also shown on the same figure are the angles calculated from a fully detector simulated sample of EXCALIBUR Monte Carlo events that were generated with a coupling of $\Delta\kappa_{\gamma}$$=-2.0$. The angles from the reweighted sample agree well with those from the sample generated with the anomalous coupling.

Figure 6.2: Comparison of angular distributions from EXCALIBUR Monte Carlo generated with anomalous coupling $\Delta\kappa_{\gamma}$$=-2.0$ (green line) and a Standard Model sample reweighted to a coupling of $\Delta\kappa_{\gamma}$$=-2.0$ using WVCXME (red line) and BILGOU (blue points). Also shown is the distribution of angles from Standard Model Monte Carlo. This has been normalised to the other samples so the event shapes can be compared.

EXCALIBUR, and thus WVCXME, only contains the CP-conserving couplings, so this technique cannot be employed to produce samples of Monte Carlo with CP-violating couplings. However a similar reweighting technique can be used to make samples of Monte Carlo with CP-violating couplings using a different method to calculate the weights.

The analytical expression of the 5-fold differential cross-section for the CC03 events, given by equation 3.27, can be used to calculate weights for the W-pair events within a Monte Carlo sample. It does not contain any of the add-ons that WVCXME has, such as accounting for the finite W width, and it can only calculate weights for the CC03 events in the four-fermion sample. However, within these limitations, as it contains all 14 anomalous couplings, it can be used to calculate weights for CP-violating couplings.

ISR can be accounted for by using the generated four-vectors of the four-fermions to boost the input angles, used in calculating the weight, back into the true centre-of-mass frame. Using equation 3.27, the Standard Model cross-section for a Standard Model Monte Carlo event ($SM$) can be calculated and also the cross-section for an anomalous coupling ($\alpha$) being present. The ratio of these two is then the equivalent weight, shown in equation 6.9. In equation 6.9 $\Omega$ represents the set of five angles; $\cos\theta_{\rm W}$, $\cos\theta_{f_{1}}$, $\cos\theta_{\bar{f_{4}}}$, $\phi_{f_{1}}$ and $\phi_{\bar{f}_{4}}$.

$${\rm wgt} = \left(\frac{{\rm d}\sigma}{{\rm d}\Omega}\right)^{\alpha}\left/\left(\frac{{\rm d}\sigma}{{\rm d}\Omega}\right)^{SM}\right.$$ (6.9)

As this method is based on analytical expressions from papers by Bilenkii and Gounaris, the reweighting scheme is known as the BILGOU reweighting scheme. The angular distributions for a Standard Model EXCALIBUR Monte Carlo sample, reweighted to an anomalous coupling of $\Delta\kappa_{\gamma}$$=-2.0$, using the BILGOU reweighting scheme are shown in figure 6.2. The distributions agree well with both those generated with a coupling of $\Delta\kappa_{\gamma}$$=-2.0$ and those reweighted using WVCXME.

The single W SDM elements extracted from a sample of fully detector simulated EXCALIBUR Monte Carlo generated with an anomalous coupling of $\Delta\kappa_{\gamma}$$=-2.0$ are shown in figure 6.3. Also shown are the SDM elements extracted from a Standard Model sample which has been reweighted using both methods of reweighting. Good agreement is seen in most cases.

Figure 6.3: Comparison of the SDM elements extracted from EXCALIBUR Monte Carlo, generated with anomalous coupling $\Delta\kappa_{\gamma}$$=-2.0$ (black line) and a Standard Model sample, reweighted to a coupling of $\Delta\kappa_{\gamma}$$=-2.0$, using WVCXME (red points) and BILGOU (green points).