Kinematic variables and feature engineering for particle phenomenology

Kinematic variables play an important role in collider phenomenology, as they expedite discoveries of new particles by separating signal events from unwanted background events and allow for measurements of particle properties such as masses, couplings, and spins. For the past ten years, an enormous number of kinematic variables have been designed and proposed, primarily for the experiments at the CERN Large Hadron Collider, allowing for a drastic reduction of high-dimensional experimental data to lower-dimensional observables, from which one can readily extract underlying features of phase space and develop better-optimized data-analysis strategies. Recent developments in the area of phase-space kinematics are reviewd, and new kinematic variables with important phenomenological implications and physics applications are summarized. Recently proposed analysis methods and techniques specifically designed to leverage new kinematic variables are also reviewed. As machine learning is currently percolating through many fields of particle physics, including collider phenomenology, the interconnection and mutual complementarity of kinematic variables and machine-learning techniques are discussed. Finally, the manner in which utilization of kinematic variables originally developed for colliders can be extended to other high-energy physics experiments, including neutrino experiments, is discussed.

Figure 1

Illustration of the dimensionality reduction in collider experiments and phenomenological studies. The common goal is to find the optimal low-dimensional kinematic observables. On the experimental side (the red boxes on the left) this is accomplished by reconstructing the low-level detector data into progressively more physically motivated quantities. On the theory side (the blue boxes on the right) the kinematic variables (or features in their distributions) are meant to reflect fundamental parameters in the theory Lagrangian. The solid (dotted) arrows indicate the typical flow (simplifying shortcuts) in the high-energy physics simulation chain.

Figure 2

The standard geometry of a collider experiment. The $z$ axis (in blue) is oriented along the beam, while the $x$ and $y$ axes (in red) define the transverse plane. Any three-dimensional vector $\overrightarrow{p}$ can be uniquely decomposed into a longitudinal component $p_z$ and a transverse component ${\overrightarrow{p}}_T$. From [109].

Figure 3

The generic event topology of a collider event. The result of the initial collision (for definitiveness, we show the case of hadron colliders where the beams consist of protons $p$ or antiprotons $\overline{p}$) is a set of final-state particles that can be grouped into collections $A,B,C,\dots$, each containing sets of visible and invisible daughters. Adapted from [109].

Figure 4

Generic event topology illustrating the use of inclusive event variables from Sec. 4. From [342].

Figure 5

Unit-normalized distributions of various inclusive event variables (${\cancel{E}}_T$, $M_{\mathrm{eff}}$, $H_T$, $M_1$, and ${\sqrt{\widehat{s}}}_{\min }$) for top quark pair production ($pp\to t\overline{t}\to b\overline{b}{\ell }^{+}{\ell }^{-}\nu \overline{\nu }$). Unlike the variables plotted in the right panel, those in the left panel require an ansatz for the invisible mass parameter $M_{\mathrm{inv}}$, which was taken to be zero. The yellow-shaded histogram shows the true $\sqrt{\widehat{s}}$ distribution in the sample. From [109].

Figure 6

Left panel: diphoton invariant mass distribution of $h\to \gamma \gamma$ events. From [16]. Right panel: $M_{b\ell }$ invariant mass distribution of the leptonic top quark decay events. From [167].

Figure 7

(a) Two-step, two-body cascade decay topology. (b) Antler decay topology. (c) Three-step, two-body cascade decay topology.

Figure 8

Unit-normalized distribution of the rescaled invariant mass $\widehat{m}=M_{a_2{a}_1}/M_{a_2{a}_1}^{\max }$ for the two-step, two-body decay topology in Fig. 7. The solid line shows the pure phase-space distribution from Eq. (42), while the dotted (dashed) line corresponds to the left-handed (right-handed) chiral coupling. From [107].

Figure 9

Left panel: the nine event topologies considered for the decay of a heavy resonance $A$ into two visible particles and up to two invisible particles. Right panel: the disambiguation of the nine decay topologies in the $\mathbf{(}P/E,(2/\pi ){\tan }^{-1}{R}_2\mathbf{)}$ space. From [187].

Figure 10

Distribution of $M_T$ for $W$ boson to $\mu {\nu }_{\mu }$ (left panel) and $e{\nu }_e$ (right panel) final states for $m_{\mathrm{inv}}=0$. Simulation and data points are shown as histogram and data points, respectively. From [30].

Figure 11

Event topologies of pair-produced heavy resonances, each of which undergoes (a) a one-step, two-body decay, (b) a two-step, two-body decay, and (c) a three-step, two-body decay.

Figure 12

Left panel: $M_{T2}$ distribution for slepton production $pp\to \tilde{\ell }{\tilde{\ell }}^{*}\to {\ell }^{+}{\ell }^{-}+{\cancel{\overrightarrow{p}}}_T$ at the LHC, assuming the actual value for the test mass $\tilde{m}$. The slepton and neutralino masses are set at $m_{\tilde{\ell }}=157.1\mathrm{GeV}$ and $m_{{\tilde{\chi }}_1^0}=121.5\mathrm{GeV}$, respectively. Right panel: values of $m_{\tilde{\ell }}$ as a function of test mass. From [355].

Figure 13

$M_{T2}^{\max }$ as a function of the test mass for gluino pair production at the LHC $pp\to \tilde{g}\tilde{g}\to jjjj+{\cancel{\overrightarrow{p}}}_T$. In this example, the squarks are assumed to be heavy, so the gluino decay is three body: $\tilde{g}\to q\overline{q}{\tilde{\chi }}_1^0$. From [180].

Figure 14

The distributions of ${\tilde{q}}^{\pm }-q^{\mathrm{true}}$ for the full event set (left panel), and for the top 10% events closest to the $M_{T2}$ end point [Eq. (61)] (right panel). Here the MAOS momenta were constructed with $\tilde{m}=0$. From [182].

Figure 15

Distributions of various doubly projected variables in the case of dilepton top quark production: $M_{2\top \perp }$ (black curve), $M_{\top 2\perp }$ (cyan curve), $M_{\top \perp 2}$ (magenta curve), $M_{2°\perp }$ (red curve), $M_{\circ 2\perp }$ (green curve), and $M_{\circ \perp 2}$ (blue curve). The yellow-shaded distribution gives the average top quark mass in the event. The different types of projections (mass-preserving $\perp$, speed-preserving $\vee$, and massless $\circ$) were defined by [109]. From [109].

Figure 16

Comparison of the MAOS and $M_2$-assisted methods for top mass reconstruction. Left panel: distributions of the reconstructed top mass ${\tilde{M}}_t$ with methods that use two mass inputs (the $W$-boson mass and the neutrino mass): the three MAOS methods $\mathrm{MAOS}4(ab)$ (blue solid line), $\mathrm{MAOS}1(b)$ (green dot-dashed line), and $\mathrm{MAOS}4(a)$ (cyan dotted line) and the two $M_2$-based methods, $M_{2CR}(ab)$ (red solid line) and $M_{2CR}(a)$ (orange dashed line). Right panel: distributions of the reconstructed top mass ${\tilde{M}}_t$ with methods that use a single mass input (the neutrino mass): $\mathrm{MAOS}2(b)$ (blue dotted line) and $\mathrm{MAOS}3(b)$ (green dot-dashed line), as well as $M_{2CX}(b)$ (orange dashed line) and $M_{2CC}(b)$ (red solid line). From [326].

Figure 17

Correlations between $\mathrm{\Delta }{q}_z$ and $\mathrm{\Delta }{q}_x$ for $\mathrm{MAOS}1(b\ell ;m_t)$ (left panel) and $M_{2Ct}^{(\ell )}$ (right panel). From [226].

Figure 18

Schematic depiction of the projection of the full phase space onto the space of observable momenta illustrating how folds in the allowed full phase space result in wall singularities in the observable space. From [332].

Figure 19

CMS top quark mass measurement using the energy-peak method in the energy distribution of $b$ jets from the top quark decay $t\to bW$. The distribution is log symmetric with respect to $\ln E_b^{*}$, and the fit was performed with a Gaussian template. From [200].

Figure 20

95% C.L. limits on $\mathrm{BR}(h\to XX)$ for signal process $pp\to jh$ followed by subsequent decays $h\to XX$ and $X\to jj$, with $X$ a new particle. From [362].

Figure 21

Scatterplot distribution of $(\ln H,\ln T)$ for (left panel) double Higgs production ($hh$) and (right panel) the SM backgrounds ($t\overline{t}$, $t\overline{t}h$, $t\overline{t}V$, $\ell \ell bj$, and $\tau \tau bb$) after loose baseline selection cuts. The solid black curves in both panels are the same and represent the optimized cuts. From [334].

Figure 22

The transverse mass of the mother particle decaying semi-invisibly as a function of the daughter particle mass for the phase-space Monte Carlo calculation in which the mother has been constrained to be at rest in the laboratory frame with vanishing $p_T$ (left panel) and for the phase-space Monte Carlo calculation in which the mother can have large transverse momentum in the laboratory frame (right panel). From [111].

Figure 23

Left panel: comparison between the photon spectra from the process $e^{+}{e}^{-}\to {\cancel{E}}_T+\gamma$ in the explicit supersymmetric models [lower (red) line] and the spectra predicted by Eq. (112) [Eq. (9) of [135]; i.e., the upper (green) line] for a $p$ annihilator of the corresponding mass. From [135]. Right panel: comparisons of simulated signal events for the $p\overline{p}\to {\cancel{E}}_T+j$ process at the Tevatron from two different Monte Carlo tools at the parton level (solid lines) and after detector simulation (dashed line). From [99].

Figure 24

Comparison between the data and the background prediction in the monojet (left panel) and mono-$V$ (right panel) signal regions before and after the simultaneous fit. From [442].

Figure 25

The generic event topology under consideration in Sec. 8. The particles $X_i,1\le i\le n$, are BSM particles that appear to be promptly decaying, on-shell intermediate resonances. The particles $x_i$ are the corresponding SM decay products, which are all visible in the detector; i.e., we assume that there are no neutrinos among them. ISR stands for generic initial-state radiation with total transverse momentum ${\overrightarrow{p}}_T$. $X_0$ is a BSM particle that is invisible in the detector. The integer $n$ counts the total number of intermediate BSM particles in each chain, so the total number of BSM particles in each chain is $n+1$. For simplicity, in this review we consider only symmetric events, in which the two decay chains are identical. The generalization of the methods discussed here to asymmetric decay chains is straightforward ([112]; [343]). From [150].

Figure 26

Dependence of the number of undetermined parameters $N_p-N_m$ as a function of the number $n$ of intermediate heavy resonances in the decay chains of Fig. 25 for various mass determination approaches: the $M_{T2}$ method (green open squares), the end-point method (red open circles), the polynomial method for $N_{\mathrm{eve}}=2$ (blue multiplication signs), and a hybrid method that combines the last two methods (magenta circled multiplication signs). Within the yellow-shaded region the number of unknowns $N_p$ does not exceed the number of measurements $N_m$ for the corresponding method and the mass spectrum can be completely determined. From [150].

Figure 27

Left panel: plot of 20 extreme curves in the $({\tilde{m}}_{A_1},{\tilde{m}}_{A_2})$ plane for a fixed ${\tilde{m}}_{a_0}=m_{a_0}=700\mathrm{GeV}$ for the $t\overline{t}$-like topology of Fig. 11. Right panel: the fractional density of extreme curves, i.e., the fraction of events whose extreme curves pass through a given $10\times 10\mathrm{GeV}$ pixel. From [328].

Figure 28

Examples of extreme momentum configurations of the visible decay products of a resonance: (a) pencil-like, (b) equilateral trianglelike, and (c) pyramidlike.

Figure 29

Various possible analysis chains for some physics-motivated task. (a) The traditional (non-ML) analysis technique using kinematic variables. (b) ML-based analysis using only raw or low-level data as inputs. (c) ML-based analysis using in addition reconstructed objects and/or human-engineered high-level variables as inputs. (d) Construction of sensitive analysis variables using machine-learning techniques.

Figure 30

(a) $n$-observed particles. (b) Divsion of $n$ particles into two groups for a $2\to 2$ process. (c) Identified event topology with $A$ and $B$. From [335].

Figure 31

Top quark mass from a high precision rate calculation at $e^{+}{e}^{-}$ collider. From [34].