Reconstruction of decays to merged photons using end-to-end deep learning with domain continuation in the CMS detector

A novel technique based on machine learning is introduced to reconstruct the decays of highly Lorentz-boosted particles. Using an end-to-end deep learning strategy, the technique bypasses existing rule-based particle reconstruction methods typically used in high energy physics analyses. It uses minimally processed detector data as input and directly outputs particle properties of interest. The new technique is demonstrated for the reconstruction of the invariant mass of particles decaying in the CMS detector. The decay of a hypothetical scalar particle $\mathcal{A}$ into two photons, $\mathcal{A}\to \gamma \gamma$, is chosen as a benchmark decay. Lorentz boosts ${\gamma }_{\mathrm{L}}=60–600$ are considered, ranging from regimes where both photons are resolved to those where the photons are closely merged as one object. A training method using domain continuation is introduced, enabling the invariant mass reconstruction of unresolved photon pairs in a novel way. The new technique is validated using ${\pi }^0\to \gamma \gamma$ decays in LHC collision data.

Figure 1

Simulation results for the decay chain $H\to \mathcal{A}\mathcal{A}$, $\mathcal{A}\to \gamma \gamma$ at various boosts: (upper plots) barely resolved, $m_{\mathcal{A}}=1.0\mathrm{GeV}$, ${\gamma }_{\mathrm{L}}=50$; (middle plots) shower merged, $m_{\mathcal{A}}=0.4\mathrm{GeV}$, ${\gamma }_{\mathrm{L}}=150$; and (lower plots) instrumentally merged, $m_{\mathcal{A}}=0.1\mathrm{GeV}$, ${\gamma }_{\mathrm{L}}=625$. The left column shows the normalized distribution of opening angles between the leading (${\gamma }_1$) and subleading (${\gamma }_2$) photons from the particle $\mathcal{A}$ decay, expressed by the number of crystals in the $\eta$ direction, $\mathrm{\Delta }\eta ({\gamma }_1,{\gamma }_2{)}^{\mathrm{gen}}$, versus the $\varphi$ direction, $\mathrm{\Delta }\varphi ({\gamma }_1,{\gamma }_2{)}^{\mathrm{gen}}$. Note that the distributions include contributions outside of the plotted ranges and thus may not sum to unity within the displayed ranges. The right column displays the ECAL energy shower pattern for a single $\mathcal{A}\to \gamma \gamma$ decay, plotted in relative ECAL crystal index coordinates and color-coded by energy. In all cases, only decays reconstructed as a single PF photon candidate passing selection criteria are used.

Figure 2

Left: the regressed mass $m_{\mathrm{\Gamma }}$ vs the generated $m_{\mathcal{A}}$ value for simulated $\mathcal{A}\to \gamma \gamma$ decays generated uniformly in $(p_{\mathrm{T}},m_{\mathcal{A}})$ before domain continuation is implemented. The regressed $m_{\mathrm{\Gamma }}$ is normalized in 0.025 GeV vertical slices of the generated $m_{\mathcal{A}}$. The color scale to the right of the plot gives the normalized number of events per vertical slice in 0.025 GeV bins of $m_{\mathrm{\Gamma }}$. Right: the regressed $m_{\mathrm{\Gamma }}$ distribution for simulated single-photon samples only, before domain continuation, resulting in a distinct peak in the low-$m_{\mathrm{\Gamma }}$ region. The distribution is normalized to unity with the vertical bars on the points indicating the statistical uncertainty.

Figure 3

Pictorial representation of the $m_{\mathcal{A}}\to 0$ boundary problem occurring when attempting to regress below the mass resolution. Left: the distribution of physically observable $\mathcal{A}\to \gamma \gamma$ invariant masses (${\mathrm{f}}_{\mathrm{obs}}$) vs the generated $m_{\mathcal{A}}$. When $m_{\mathcal{A}}\approx \sigma (m_{\mathcal{A}})$, the left tail of the mass distribution becomes underrepresented in the training set. Middle: as $m_{\mathcal{A}}\to 0$, only half of the mass distribution is represented. The regressor subsequently defaults to the last full mass distribution at $m_{\mathcal{A}}\approx \sigma (m_{\mathcal{A}})$. Right: with domain continuation, the generated mass distribution of the original training samples ($\mathcal{A}\to \gamma \gamma$, red region) is augmented with topologically similar samples that are randomly assigned nonphysical masses ($\gamma$, blue region). This allows the regressor to see a full mass distribution over the entire region of interest (unhatched region). Predictions in the black hatched regions are discarded.

Figure 4

Mass regression performance for simulated $\mathcal{A}\to \gamma \gamma$ samples generated uniformly in $(p_{\mathrm{T}},m_{\mathcal{A}})$, corresponding to mean boosts in the range $⟨{\gamma }_{\mathrm{L}}⟩=600–50$ for $m_{\mathcal{A}}=0.1–1.2\mathrm{GeV}$. Upper: regressed $m_{\mathrm{\Gamma }}$ vs generated $m_{\mathcal{A}}$. The regressed $m_{\mathrm{\Gamma }}$ is normalized in 0.025 GeV vertical slices of the generated $m_{\mathcal{A}}$. The color scale to the right of the plot gives the normalized number of events per vertical slice in 0.025 GeV bins of $m_{\mathrm{\Gamma }}$. Lower left: the MAE (blue circles, use left scale) and MRE (red squares, use right scale) vs the generated $m_{\mathcal{A}}$. For clarity, the MRE for $m_{\mathcal{A}}<0.1\mathrm{GeV}$ is not shown since its value diverges as $m_{\mathcal{A}}\to 0$. Lower right: the $m_{\mathcal{A}}$ regression efficiency as a function of the generated $m_{\mathcal{A}}$. The hatched region shows the efficiency for single photons. The vertical bars on the points show the statistical uncertainty in the simulated sample.

Figure 5

Reconstructed mass spectra for end-to-end (left column), photon NN (middle column), and $3\times 3$ algorithms (right column) for $\mathcal{A}\to \gamma \gamma$ decays with $m_{\mathcal{A}}=1.0\mathrm{GeV}$ (upper row), $m_{\mathcal{A}}=0.4\mathrm{GeV}$ (second row), $m_{\mathcal{A}}=0.1\mathrm{GeV}$ (third row), and for isolated single photons (lower row). For each panel, the mass spectra are separated by reconstructed $p_{\mathrm{T},\mathrm{\Gamma }}$ value into ranges of 30–55 GeV (red circles, low-$p_{\mathrm{T},\mathrm{\Gamma }}$), 55–70 GeV (gray triangles, mid-$p_{\mathrm{T},\mathrm{\Gamma }}$), 70–100 GeV (blue squares, high-$p_{\mathrm{T},\mathrm{\Gamma }}$), and $>100\mathrm{GeV}$ (green inverted triangles, ultra-$p_{\mathrm{T},\mathrm{\Gamma }}$). The vertical bars on the points give the statistical uncertainties. All the mass spectra are normalized to unity, including samples outside $m_{\mathcal{A}}\text{-ROI}$. The vertical dotted line shows the input $m_{\mathcal{A}}$ value.

Figure 6

Reconstructed mass $m_{\mathrm{\Gamma }}$ for end-to-end (red circles), photon NN (blue squares), and $3\times 3$ (gray triangles) algorithms for hadronic jets from data enriched with ${\pi }^0\to \gamma \gamma$ decays. All distributions are normalized to the same number of events, including those outside $m_{\mathcal{A}}\text{-ROI}$. The statistical uncertainties in the distributions are negligible.

Figure 7

Regressed mass from ${\pi }^0\to \gamma \gamma$ data events vs $p_{\mathrm{T},\mathrm{\Gamma }}$ (left) and ${\eta }_{\mathrm{\Gamma }}$ (right). For both plots, the regressed mass distributions are normalized in vertical slices of the accompanying kinematic quantity to highlight the intrinsic dependence on the quantity. The relative contribution over each vertical slice is given by the color scale to the right of each plot.

Figure 8

Upper: two-dimensional plot of the regressed mass for ${\pi }^0\to \gamma \gamma$ data events vs the amount of pileup (PU). The mass distribution is normalized in vertical slices of the amount of pileup. The relative contribution over each vertical slice is given by the color scale to the right of the plot. Lower left: the regressed mass distributions for the start (gray circles), middle (blue squares), and end (red triangles) of data taking during the year 2017. Lower right: the regressed mass distributions for the 2017 (gray circles) and 2018 (blue squares) data-taking periods. The lower two plots are normalized to unity and the vertical bars on the points show the statistical uncertainties. The lower panel for the lower left plot gives the ratio of distributions for the middle to the start (blue squares) and the end to the start (red triangles) of the 2017 data-taking period. The lower panel for the lower right plot gives the ratio (blue squares) for the 2018 and 2017 data-taking periods. The vertical bars on the points in both lower panels show the statistical uncertainties in the numerator quantity, and the gray bands give the similar uncertainty in the denominator quantity.

Figure 9

The agreement in the regressed $m_{\mathrm{\Gamma }}$ spectrum between electrons in data versus simulation. Left: contours of 68% (solid line) and 95% (dotted line) confidence level (CL) in the ${\chi }^2$ test statistic as a function of the ($s_{\text{scale}}$, $s_{\text{smear}}$) hypothesis. The best fit point ($s_{\text{scale}}=1.040$, $s_{\text{smear}}=0\mathrm{MeV}$) is indicated by the red diamond. Right: the regressed mass distributions in data (points) and the best fit Monte Carlo (MC) simulation (blue region) for electrons from $Z\to e^{+}{e}^{-}$ events. The difference between the simulated distribution under the null scale and smearing hypothesis versus the best fit hypothesis (Syst) is plotted as a green band. Each of the distributions are normalized to unity, including samples outside $m_{\mathcal{A}}\text{-ROI}$. Statistical uncertainties in the data distribution are negligible. The lower panel shows the ratio of the data to the simulation under the best fit hypothesis (points). The statistical uncertainties in the latter are plotted as a blue band. The ratio of the simulated distribution for the null to the best fit hypothesis is displayed as a green band.

Figure 10

A typical $\mathcal{A}\to \gamma \gamma$ decay using minimally processed (left) and clustered (right) data. The energy distributions are plotted in relative ECAL crystal index coordinates of the pseudorapidity $\eta$ versus the azimuthal angle $\varphi$ and color coded by energy.

Figure 11

Regressed mass spectra for the mass regressor trained on minimally processed data (upper) versus clustered data (lower) at shower-merged boosts (left) and barely resolved boosts (right). For each scenario, the mass regressor is run on the same set of $\mathcal{A}\to \gamma \gamma$ decays, composed either of minimally processed (blue circles, all) or clustered (red squares, clustered) data. The vertical bars on the points give the statistical uncertainties and the vertical dotted line shows the input $m_{\mathcal{A}}$ value.

Figure 12

A typical hadronic jet sample reconstructed by the $3\times 3$ algorithm with a mass near the $\eta$ meson peak, using minimally processed (all, left) and $3\times 3$ clustered ($3\times 3$, right) data. The energy distributions are plotted in relative ECAL crystal index coordinates of the azimuthal angle $\varphi$ versus the pseudorapidity $\eta$ and color coded by energy.

Figure 13

Regressed mass spectra vs generated $p_{\mathrm{T},\mathcal{A}}$ (left), generated ${\eta }_{\mathcal{A}}$ (center), and amount of pileup (right), for $\mathcal{A}\to \gamma \gamma$ decays that are barely resolved (upper), shower merged (middle), or instrumentally merged (lower). In all plots, the regressed mass distribution is normalized in vertical slices of the quantity of interest. The relative contribution over each vertical slice is given by the color scale to the right of each plot.