Deep neural network application: Higgs boson $CP$ state mixing angle in $H\to \tau \tau$ decay and at the LHC

The consecutive steps of cascade decay initiated by $H\to \tau \tau$ can be useful for the measurement of Higgs couplings and in particular of the Higgs boson parity. In the previous papers we have found that multidimensional signatures of the ${\tau }^{\pm }\to {\pi }^{\pm }{\pi }^0\nu$ and ${\tau }^{\pm }\to 3{\pi }^{\pm }\nu$ decays can be used to distinguish between scalar and pseudoscalar Higgs state. The machine learning techniques (ML) of binary classification, offered break-through opportunities to manage such complex multidimensional signatures. The classification between two possible $CP$ states: scalar and pseudoscalar, is now extended to the measurement of the hypothetical mixing angle of Higgs boson parity states. The functional dependence of $H\to \tau \tau$ matrix element on the mixing angle is predicted by theory. The potential to determine preferred mixing angle of the Higgs boson events sample including $\tau$-decays is studied using deep neural network. The problem is addressed as classification or regression with the aim to determine the per-event: (a) probability distribution (spin weight) of the mixing angle; (b) parameters of the functional form of the spin weight; (c) the most preferred mixing angle. Performance of proposed methods is evaluated and compared.

Figure 1

Distributions of the formula (5) $C_0$, $C_1$, $C_2$ coefficients, for the $H\to \tau \tau$ events sample.

Figure 2

The spin weight $wt$ (top plot) and only its ${\alpha }^{CP}$ dependent component (bottom plot) for five $H\to \tau \tau$ events of Table 1. Note the vertical scale change between top and bottom plots.

Figure 3

Cross-check distributions of the spin weight $wt$ calculated at generation (blue points) and from functional form of Eq. (5) (orange line), as a function of $CP$ mixing parameter ${\alpha }^{CP}$. For top and bottom plots two different example events were used. Coefficients $C_i$ are reconstructed from Eq. (5) and $wt$ is taken at three different ${\alpha }^{CP}$.

Figure 4

The AUC score for binary classification between ${\mathcal{H}}_0$ and ${\mathcal{H}}_{{\alpha }^{CP}}$ hypotheses and corresponding oracle predictions.

Figure 5

Normalized to probability spin weight $w{t}^{\mathrm{norm}}$, predicted (orange steps) and true (blue line), as a function of ${\alpha }_i^{CP}$ for two example events (top and bottom plots). $DNN$ was trained with $N_{\text{class}}=21$ spanning range $(0,2\pi )$.

Figure 6

The $l_2$ norm, quantifying difference between true and predicted spin weight $w{t}^{\mathrm{norm}}$, as a function of class multiplicity $N_{\text{class}}$.

Figure 7

Distribution of ${\mathrm{\Delta }}_{\text{class}}^{\max }$ between predicted most probable class and true most probable class for $N_{\text{class}}=21$ (top) and 51 (bottom). The mean and std are calculated in units of class index [idx] or units of radians [rad].

Figure 8

Difference between true and predicted coefficients $C_0$, $C_1$, $C_2$ of formula (5). For $DNN$ training the granularity of $N_{\text{class}}=21$ was used.

Figure 9

The difference between true and predicted most probable mixing angle ${\alpha }_{\max }^{CP}$, calculated using formula (5) and coefficients $C_0$, $C_1$, $C_2$ learned with classification method. The granularity of ${\alpha }_{\max }^{CP}$, $N_{\text{class}}=21$ and 51 was used respectively for top and bottom plot.

Figure 10

Distributions of true and predicted coefficients $C_0$, $C_1$, $C_2$ of formula (5). For $DNN$ training the granularity of $N_{\text{class}}=21$ was used.

Figure 11

Distributions (top plot) of true and predicted most preferred mixing angle ${\alpha }^{CP}$. The distribution of per-event difference of the two is shown on the bottom plot. The granularity of $N_{\text{class}}=21$ was used for training $DNN$.

Figure 12

Example plots with $DNN$ regression results: the spin weight $wt$, predicted (orange steps) and true (blue line), as a function of ${\alpha }_i^{CP}$ for two example events (top and bottom plots). $DNN$ was trained with $N_{\text{class}}=51$ spanning range $(0,2\pi )$.

Figure 13

The $l_2$ norm for predicted spin weight $wt$ (top) and $w{t}^{\mathrm{norm}}$ (bottom) as a function of $N_{\text{class}}$.

Figure 14

Distribution of ${\mathrm{\Delta }}_{\text{class}}$ between most probable predicted class and true most probable class. The $N_{\text{class}}=21$ and 51 are used for respectively top and bottom plot. The mean and std standard deviation are calculated in units of class index [idx] and units of radians [rad].

Figure 15

Difference between true and predicted coefficients $C_0$, $C_1$, $C_2$ of formula (5).

Figure 16

Distributions (top plot) of true (black dashed line) and predicted (orange line) most preferred mixing angle ${\alpha }^{CP}$. The prediction was based on coefficients $C_0$, $C_1$, $C_2$. The distribution of per-event difference of the two is shown on the bottom plot.

Figure 17

Distributions of true and predicted coefficients $C_0$, $C_1$, $C_2$ of formula (5).

Figure 18

Distributions (top plot) of true (black dashed line) and predicted (orange line) most preferred mixing angle ${\alpha }^{CP}$. The distribution of per-event difference of the two is shown on the bottom plot.

Figure 19

Performance of $wt$ fitting ($l_2$) for different number of layers and nodes, assuming $N_{\text{class}}=21$ (top) and $N_{\text{class}}=51$ (bottom).

Figure 20

The $DNN$ loss for classification (left-side) and regression (right-side), as function of number of epochs used for training. It is shown for learning spin weight (top plots), $C_i$ coefficients (middle plots) and most likely mixing angle ${\alpha }_{\max }^{CP}$ (bottom plots). For the classification, $N_{\text{class}}=21$ was used.