Deep-Learned Event Variables for Collider Phenomenology

In this letter, we shall focus on the last stage of this dimensionality reduction chain, namely, the optimal construction of kinematic variables, which is essential to expedite the discovery of new physics and/or to improve the precision of parameter measurements. By now, the experimentalist’s toolbox contains a large number of kinematic variables, which have been thoroughly tested in analyses with real data (see [2, 3, 4, 5] for reviews). The latest important addition to this set are the so-called “singularity variables” [6, 7, 8, 9, 10], which are applicable to missing energy events — the harbingers of dark matter production at colliders. In the machine learning era, a myriad of algorithms have been invented or adopted to tackle various tasks that arise in the analysis of collider data, e.g., signal–background discrimination (see [11] for a continuously updated complete review of the literature). Under the hood, the machines trained in these techniques could learn to construct useful features from the low-level event description, because they are relevant to the task at hand. But it is difficult to interpret what exactly the machines have learned in the process [12, 13]. Furthermore, it is rarely studied whether the human-engineered features are indeed the best event variables for certain purposes, and whether machines can outperform theorists at constructing event variables.

These two issues, explainability and optimality, are precisely the two questions which we shall address in this letter. We shall introduce a new technique for training neural networks to directly output useful features or event variables (which offer sensitivity over a range of unknown parameter values). This allows for explainability of the machine’s output by comparison against known features in the data. At the same time, it is important to verify that the variables obtained using our technique are indeed the optimal choice, and we will test this by directly comparing them against the human-engineered variables that are known to be optimal for their respective event topologies. Once we have validated our training procedure in this way, we could extend it to more complex event topologies and derive novel kinematic variables in interesting and difficult scenarios.

Understanding how and what a neural network (NN) learns is a difficult task. Here we shall consider relatively simple physics examples that are nevertheless highly non-trivial from a machine learning point of view: (1) visible two-body decay (to two visible daughter particles); (2) semi-invisible two-body decay (to one visible and one invisible daughter particle); (3) semi-invisible two-body decays of pair-produced particles. It is known that the relevant variables in those situations are the invariant mass $m$, the transverse mass $m_{T}$ [14, 15] and the stransverse mass $m_{T2}$ [16], respectively. We will demonstrate that in each case, the NN can be trained to learn the corresponding physics variable in the reduced latent space. The method can be readily generalized to more complex cases to derive deep-learned, next-generation event variables.

The mutual information of $V$ and $\Theta$ is given by

where $p_{V}$ and $p_{\Theta}$ are the probability distribution functions of $V$ and $\Theta$, respectively, and $p_{(V,\Theta)}$ is their joint distribution. $\mathcal{V}$ and $\Omega$ are the domains of $V$ and $\Theta$, respectively. One can think of $p_{\Theta}$ as the prior distribution of $\Theta$.¹¹1For convenience, we will adopt the Bayesian interpretation of probability in the presentation of this work. The distributions $p_{(V,\Theta)}$ and $p_{V}$ can then be derived from $p_{\Theta}$ and the conditional distribution $p_{V|\Theta}(v|\theta)$.

The mutual information $I(V;\Theta)$ quantifies the amount of information contained in $V$ about $\Theta$. Therefore, a good event variable $V$ should have relatively high values of $I(V;\Theta)$. From Eq. (1), one can see that $I(V;\Theta)$ is nothing but the Kullback–Leibler (KL) divergence from (a) the factorized distribution $p_{V}\otimes p_{\Theta}$ to (b) the joint distribution $p_{(V,\Theta)}$. The KL divergence, in turn, is a measure of how distinguishable the two distributions (a) and (b) are.

These observations lead to the following strategy schematically outlined in FIG. 1: train the event variable network so that the distributions $p_{V}\otimes p_{\Theta}$ and $p_{(V,\Theta)}$ are highly distinguishable. An auxiliary classifier network can then be used for evaluating the distinguishability of the two distributions. The basic blueprint of our training technique will be described next.

$p_{X|\Theta}$ is the distribution of the event $X$ conditional on $\Theta$. General purpose event generators can be used to sample from this distribution. The overall distribution of $X$, namely $p_{X}$, is given by

Our training data consists of two classes, whose generation is illustrated in the left (green) block of FIG. 1. Each training data point is given by a 2-tuple $(X,\Theta)$ along with the class label $y_{\mathrm{target}}\in\{0,1\}$ of the data point. Under class 0, $X$ and $\Theta$ are independent of each other and their joint distribution is given by $p_{X}\otimes p_{\Theta}$. This is accomplished by simply replacing the true value of $\Theta$ used to generate $X$ with a fake one for the datapoints in class 0. Under class 1, the joint distribution of $(X,\Theta)$ is given by

The information bottleneck [18] $V(X)$ created by the EVN module is optimized by simply training the composite network as a classifier for the input data $(X,\Theta)$, using the class labels $y_{\mathrm{target}}$ as the supervisory signal.

The prior $p_{\Theta}$ is chosen to sample $m_{A}$ uniformly in the range $[100,500]$ GeV. For each sampled value of $m_{A}$, we generate an event as follows. A generic boost for the parent particle $A$ is obtained by isotropically picking the direction for its momentum and uniformly sampling its lab frame energy in the range $[m_{A},1500\text{ GeV}]$. Subsequently, $A$ is decayed on-shell into two massless particles (isotropically in its own rest frame), so that the input data consists of the lab-frame final state 4-momenta $\{p_{b},p_{c}\}\equiv\{E_{b},\vec{p}_{b},E_{c},\vec{p}_{c}\}$.

All the neural networks used in this work were implemented in TensorFlow [21]. For the event variable network, we used a sequential fully connected architecture with 5 hidden layers. The hidden layers, in order, have 128, 64, 64, 64, and 32 nodes, all of them using ReLU as the activation function. The output layer has one node with no activation function. The classifier network is a fully connected network with 3 hidden layers (16 nodes each, ReLU activation). The output layer has one node with sigmoid activation. These two networks were combined as shown in the right (blue) block of FIG. 1 and trained with 2.5 million events total (50–50 split between classes 0 and 1), out of which 20% was set aside for validation. The network was trained for 20 epochs with a mini-batch size of 50, using the Adam optimizer and the binary crossentropy loss function.

For the event topology considered in this example, it is known that the event variable most sensitive to the value of $m_{A}$ is the invariant mass of the daughter particles

as well as any variable that is in one-to-one correspondence with it. In order to test whether our artificial event variable $V$ learned by the NN correlates with $m_{bc}$, we show a heatmap of the joint distribution of $(V,m_{bc})$ in the upper-left panel of FIG. 2. Here, and in what follows, the heatmap is generated using a separate test dataset with $10^{5}$ events. In the plot we also show two nonparametric correlation coefficients, namely Kendall’s $\tau$ coefficient [22] and Spearman’s rank correlation coefficients $r_{s}$ [23]. A value of $\pm 1$ for them would indicate one-to-one correspondence. Our results depict an almost perfect correspondence between $V$ and $m_{bc}$. Here, and in what follows, we append an overall minus sign to $V$ if needed, to make the correlations positive and the plots in FIG. 2 intuitive.

In practice, the artificial variable can be used to compare the data against templates simulated for different values of $\Theta$. To illustrate this usage, in the lower-left panel of FIG. 2 we show unit-normalized distributions of the deep-learned variable $V$ for several different values of $m_{A}=\{200,280,320,400\}$ GeV. It is seen that the distributions are highly sensitive to the parameter choice $m_{A}$ and, if needed, $V$ can be calibrated so that the peak location directly corresponds to $m_{A}$. The observed spread around the peak values in the histogram, as well as the less than perfect correspondence between $V$ and $m_{bc}$, are due to limitations in the NN architecture and training.

We generate $(m_{A},m_{C})$ by uniformly sampling $(m_{A},\delta_{m})$ in the region defined by $100\text{ GeV}\leq m_{A}\leq 500\text{ GeV}$ and $0\leq\delta_{m}\leq m_{A}$, where $\delta_{m}\equiv(m_{A}-m_{C})^{2}/m_{A}$. This choice of prior ensures that the relevant mass difference parameter in this event topology $\mu\equiv(m_{A}^{2}-m_{C}^{2})/m_{A}$ is adequately sampled in the range $[0,500]$ GeV. For the given value of $m_{A}$, we generate an event as follows. The parent particle $A$ is boosted along the beam axis $\pm z$ (with equal probability) to an energy chosen uniformly in the range $[m_{A},1500\text{ GeV}]$. The particle $A$ is decayed on-shell into $b$ and $C$, isotropically in its own rest frame. The details of network architectures and training are the same as in Example 1.

The relevant variable for this event topology is the transverse mass $m_{T}$, which in our setup is given by

where the choice of mass ansatz $\tilde{m}_{C}$ for the mass of the invisible particle $C$ does not affect the rank ordering of the events. For concreteness in what follows we shall use $\tilde{m}_{C}=0$. The corresponding heatmap of the joint distribution $(V,m_{T})$ and unit-normalized distributions of the variable $V$ for several choices of $m_{A}=\{200,280,320,400\}$ GeV and $m_{C}=100$ GeV are shown in the middle panels of FIG. 2. Once again, we observe an almost perfect correlation between $V$ and $m_{T}$, and a high sensitivity of the $V$ distributions to the input masses.

The masses $(m_{A},m_{C})$ are generated as in Example 2. In order to avoid fine turning the network to the details of a particular collider, we uniformly sampled the invariant mass $m_{A_{1}A_{2}}$ of the $A_{1}A_{2}$ system in the range $[2m_{A},1500\text{ GeV}]$ and the lab-frame energy of the $A_{1}A_{2}$ system in the range $[m_{A_{1}A_{2}},2500\text{ GeV}]$. The direction of the system was chosen to be along $\pm z$ with equal probability. The direction of $A_{1}$ is chosen isotropically in the rest frame of the $A_{1}A_{2}$ system. $A_{1}$ and $A_{2}$ are both decayed on-shell, isotropically in their respective rest frames. The details of network architectures and training are the same as in Example 1.

The straightforward generalization of the idea of the transverse mass to the considered event topology leads to the stransverse mass variable $m_{T2}(\tilde{m}_{C})$ [16]. In the upper-right panel of FIG. 2 we show a heatmap of the joint distribution of $(V,m_{T2}(0))$, which reveals reasonably good, but not perfect correlation, implying that the artificial event variable encapsulates information beyond $m_{T2}$. This could have been expected for the following two reasons: 1) unlike the previous two examples of singular variables with sharp features in their distributions, $m_{T2}$ does not belong to the class of singular variables [10]; 2) $m_{T2}$ only uses a subset of the available kinematic information, namely the transverse momentum components. In contrast, the artificial kinematic variable can use all of the available information, and in a more optimal way. The lower-right panel of FIG. 2 displays unit-normalized distributions of the artificial variable for several choices of $m_{A}$ and fixed $m_{C}=100$ GeV, again demonstrating the sensitivity of $V$ to the mass spectrum.