WZ polarisation

This thesis presents the first measurement of the longitudinal polarisation states of one or both bosons in $\mathrm{WZjj}\text{-EW}$ processes, using the data recorded by the ATLAS experiment during the full Run-2 of the LHC, corresponding to an integrated luminosity of $140\,\mathrm{fb}^{-1}$ at a centre-of-mass energy of $13\,\mathrm{TeV}$, combined with the first part of Run-3, corresponding to $164\,\mathrm{fb}^{-1}$ at $13.6\,\mathrm{TeV}$.

The polarisation being only a statistical property of the bosons, it is not possible to measure it directly via an observable, but it can be inferred from the angular distribution of the decay products of the bosons. Three distinct measurements are considered: both bosons longitudinally polarised ($f_{\mathrm{00}}$), the most challenging, with only $1\,\%$ of the total event yield; only the $\mathrm{W}$ boson longitudinally polarised ($f_{\mathrm{0X}}$); and only the $\mathrm{Z}$ boson longitudinally polarised ($f_{\mathrm{X0}}$).

Deep neural networks are trained to separate $\mathrm{WZjj}\text{-EW}$ events from the background and then to disentangle the polarisation states. All DNNs are feed-forward fully connected NNs with a ReLU activation function for the hidden layers, and a dropout layer between the hidden layers to prevent overfitting. The hyperparameters of each DNN are optimised using a Bayesian optimisation, and the networks are trained in five folds, so that each event entering the final template is evaluated by a DNN that was not trained on that event.

More: constraints on the DNN architecture

The DNN architecture was subject to constraints due to portability to the analysis framework in C++ but also to Fortran, in which MoCaNLO is written. MoCaNLO is a fixed-order integrator, not an event generator, so the non-negligible NLO EW correction to the measured observable can only be obtained by evaluating the DNNs and the Voronoi binning inside the integration, event by event. More elaborate architectures could therefore not be used. However, a study comparing the present setup to an adaptation of the transformer using the GN2 network developed for jet tagging was carried out. No improvement was observed with respect to the present methods.

The final discriminant planes are built using the output of the S. vs B. DNN with the $\mathrm{WZjj}\text{-EW}$ output (giving us the probability of having the signal) on the $x$-axis and the output of one of the polarisation DNNs on the $y$-axis (giving us the probability of having the desired polarisation state). The signal is expected to be concentrated in the top right corner of the plane, while the background is expected to be concentrated in the bottom left corner. The bottom right corner is expected to be the signal $\mathrm{WZjj}\text{-EW}$ but with the non-targeted polarisation states.

The retained method works in two steps. The first part is to form a 2D histogram showing the ratio of signal over the square root of the background, $\frac{S}{\sqrt{B}}$, using a fine binning (here $75\times75$ bins). Then we compute the $k$-means clusters on the signal only, using the $\frac{S}{\sqrt{B}}$ value in each bin as a per-event weight, so that signal-rich bins with a large $\frac{S}{\sqrt{B}}$ exert a stronger pull on the cluster centres. In this way the algorithm concentrates the centres in the regions of the discriminant plane with the highest signal significance.

The second step then consists of applying a Voronoi binning using the cluster centres as seeds: each Voronoi region is the set of points that are closer to a particular cluster centre than to any other cluster centre. The number of clusters is set to $10$, presenting a good compromise between having enough bins to exploit the DNN discrimination power and keeping a reasonable number of bins to avoid large statistical uncertainty in each bin. The bins are ordered by increasing $\frac{S}{S+B}$.

The unrolled Voronoi templates of the signal region are fitted together with two control regions, with a binned maximum-likelihood fit. The parameters of interest are the polarisation fraction and the fiducial signal strength $\mu_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}}$, which gives the fiducial cross-section $$\sigma_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}} = \mu_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}}\cdot \frac{N_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}}{4 \mathcal{L}},$$ with $N_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}$ the total number of events in the fiducial phase-space and $\mathcal{L}$ the integrated luminosity; the factor $4$ accounts for the four decay channels $\mathrm{e}\mathrm{e}\mathrm{e}$, $\mathrm{e}\mathrm{\mu}\mathrm{\mu}$, $\mathrm{\mu}\mathrm{e}\mathrm{e}$, and $\mathrm{\mu}\mathrm{\mu}\mathrm{\mu}$. The polarised and non-polarised components are then normalised by $$\mu_{\mathrm{XX}} = \frac{\mu_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}}\cdot 4\mathcal{L} \cdot \sigma_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}}{N_{\mathrm{W_XZ_Xjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}} \times f_{\mathrm{XX}}^{\mathrm{fid.}} , \qquad \mu_{\overline{\mathrm{XX}}} = \frac{\mu_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}}\cdot 4\mathcal{L} \cdot \sigma_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}}{N_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}} - N_{\mathrm{W_XZ_Xjj}\text{-}\mathrm{EW}}^{\mathrm{fid.},\mathrm{MC}}} \times (1 - f_{\mathrm{XX}}^{\mathrm{fid.}}),$$ so that only $\mu_{\mathrm{WZjj}\text{-}\mathrm{EW}}^{\mathrm{fid.}}$ and $f_{\mathrm{XX}}^{\mathrm{fid.}}$ float freely.