Semileptonic decays of heavy mesons with artificial neural networks

Experimental checks of the second row unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix involve extractions of the matrix element $V_{cd}$, which may be obtained from semileptonic decay rates of $D$ to $\pi$. These decay rates are proportional to hadronic form factors which parametrize how the quark $c\to d$ transition is realized in $D\to \pi$ meson decays. The form factors cannot yet be analytically computed over the whole range of available momentum transfer $q^2$, but can be parametrized with a varying degree of model dependency. We propose analysis of the form factor shapes using a system of artificial neural networks trained from experimental pseudodata and averaged together to predict their shapes with a prescribed uncertainty. We comment on the parameters of several commonly-used model parametrizations of semileptonic form factors. We extract shape parameters and use unitarity to bound the form factor at a given $q^2$, which then allows us to bound the CKM matrix element $|V_{cd}|$.

Figure 1

A sample structure of an artificial neural network with two hidden layers. In this work we used the ANN structure (2,3,4,1), i.e., a network with two nodes in the input layer, three (four) nodes in the first (second) hidden layer, and one node in the output layer. The function ${\xi }^{\ell }$ is defined in Eq. (13).

Figure 2

Our averaged ANN result for the differential decay rate plotted against the experimental measurement. The purple data points are the experimental data from [30]. The black and cyan curves are the average value and one standard deviation, respectively, from the output of our averaged ANN.

Figure 3

ANN fits for $|V_{cd}{F}_{+}(q^2)|$ plotted against the three models described in the text. The black and cyan curves are the average value and one standard deviation, respectively, from the output of our neural network. The dotted red curve is the simple pole model. The dot-dashed green curve is the modified pole model. The dashed magenta curve is the BZ model. The purple data points are calculated from the experimental data in Ref. [30].

Figure 4

Impact of the number of epochs per neural network training. ANN structure (2,3,4,1): (a) ANN fit for 20 epochs, (b) ANN fit for 30 epochs, (c) ANN fit for 40 epochs, and (d) ANN fit for 50 epochs.

Figure 5

Impact of the number of hidden nodes per neural network for 20 epochs: (a) ANN fit for (2,1,2,1), (b) ANN fit for (2,3,4,1), (c) ANN fit for (2,6,8,1), and (d) ANN fit for (2,15,12,1).

Figure 6

Impact of the number of hidden nodes per neural network for 20 epochs: (a) ANN fit for (2,20,23,1), (b) ANN fit for (2,30,37,1).

Figure 7

Impact of the number of hidden nodes per neural network for 50 epochs: (a) ANN fit for (2,1,2,1), (b) ANN fit for (2,3,4,1), (c) ANN fit for (2,6,8,1), and (d) ANN fit for (2,15,12,1).

Figure 8

Impact of the number of hidden nodes per neural network for 50 epochs: (a) ANN fit for (2,20,23,1), (b) ANN fit for (2,30,37,1).

Figure 9

Impact of the 180 pseudodata points per $q^2$ bin at 20 epochs and 50 epochs for ANN structure (2,3,4,1): (a) ANN fit for 20 epochs, and (b) ANN fit for 50 epochs.

Figure 10

Impact of the 1800 pseudo data per $q^2$ bin at 20 epochs and 50 epochs for ANN structure (2,3,4,1): (a) ANN fit for 20 epochs, and (b) ANN fit for 50 epochs.