(100) $\displaystyle=$ $\displaystyle
m_{1}^{2}+m_{2}^{2}+2(E_{1T}E_{2T}+\vec{p}_{1T}\cdot\vec{p}_{2T})\,,$

where $E_{i}$ and $\vec{p}_{i}$ are the visible energy and three-momentum in
the $i$th branch, and $E_{iT}$ and $\vec{p}_{iT}$ are the corresponding
transverse energy and transverse momentum. They satisfy $M_{C}\geq M_{CT}$,
just like $M\geq M_{T}$. An interesting property of the $M_{C}$ variable is
that it is invariant under the back-to-back boost of the two visible systems.

Similar to the stransverse mass $M_{T2}$, the constransverse mass variable
$M_{CT2}$ is defined as [254, 255, 256]

$\displaystyle
M_{CT2}=\min_{\vec{q}_{1T}+\vec{q}_{2T}=\not{p}_{T}}\left[\max\left\\{M_{CT}^{(1)},M_{CT}^{(2)}\right\\}\right]\,,$
(101)

where each $M_{CT}^{(i)}$ ($i=1,2$) is applied to the semi-invisible decay of
parent particle $P_{i}$.

Another mass variable, $M_{2C}$, is defined as the minimum four-dimensional
mass

$\displaystyle M_{2C}^{2}=\min_{q_{1},q_{2}}\Big{(}p_{1}+q_{1}\Big{)}^{2}\,,$
(102)

under the following constraints

$\displaystyle\big{(}p_{1}+q_{1}\big{)}^{2}$ $\displaystyle=$
$\displaystyle\big{(}p_{2}+q_{2}\big{)}^{2}$ (103) $\displaystyle q_{1}^{2}$
$\displaystyle=$ $\displaystyle q_{2}^{2}$ (104)
$\displaystyle\vec{q}_{1T}+\vec{q}_{2T}$ $\displaystyle=$
$\displaystyle\not{\vec{p}}_{T}$ (105)
$\displaystyle\sqrt{\big{(}p_{1}+q_{1}\big{)}^{2}}-\sqrt{\big{(}q_{1}\big{)}^{2}}$
$\displaystyle=$ $\displaystyle m_{P}-m_{0}\,,$ (106)

where the parent-daughter mass difference in the last constraint is assumed to
be known from a preliminary measurement of an invariant mass endpoint (52)
[244]. The $M_{2C}$ mass variable is bounded by the parent mass, $M_{2C}\leq
m_{P}$.

Another set of kinematic variables extensively used by the CMS Collaboration
in its searches for supersymmetry is the razor kinematic variables [257, 258,
259]. They are known to be sensitive to large mass differences between the
parent particle and the invisible particles at the end of a decay chain. The
razor variables are defined as [177]

$\displaystyle M_{R}^{2}$ $\displaystyle=$
$\displaystyle(E_{1}+E_{2})^{2}-(q_{1z}+q_{2z})^{2}\,,$ (107)
$\displaystyle(M_{T}^{R})^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\not{E}_{T}(q_{1T}+q_{2T})-\not{\vec{p}}_{T}\cdot(\vec{q}_{1T}+\vec{q}_{2T})\right]\,,$
(108) $\displaystyle R^{2}$ $\displaystyle=$
$\displaystyle\left(\frac{M_{T}^{R}}{M_{R}}\right)^{2}\,.$ (109)

For QCD multijet background events, the distributions in both $M_{R}$ and
$R^{2}$ fall exponentially, while for signal events they peak at finite
values.

###  VII.3 Singularity Variables

The geometrical features of the high-dimensional phase space available to a
given event topology are largely washed out when projecting to a single one-
dimensional event observable. The so-called singularity variables [125, 260,
261, 262], however, provide an intuitive way to retain high-dimensional
features as singularities in the corresponding one-dimensional kinematic
distributions. The origin of such singularities is very well
understood—similar to the phenomenon of caustics in optics, they are formed at
points where the projection of the allowed phase space onto the observable
space gets folded; see Figure 16 for a cartoon illustration.

Figure 16: A 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. Taken
from Ref. [125].

As previously discussed, such projections are inevitable in the presence of
invisible momenta in the event, $\\{q\\}$, since event observables must be
constructed out of visible momenta $\\{p\\}$ only. The geometrical structure
in the allowed higher-dimensional phase space is imposed by a certain set of
kinematic conditions (constraints)

$f_{\alpha}(\\{p\\},\\{q\\})=0,~{}\qquad\alpha=1,2,\dots$ (110)

which arise in the assumed event topology. Mathematically the singularity
condition is then expressed as the reduction in the rank of the Jacobian
matrix of the coordinate transformation from the relevant set of kinematic
constraints $f_{\alpha}$ to $\\{q\\}$, which is why such singularities are
sometimes known as Jacobian peaks. Explicitly,

$\det\left(\frac{\partial f_{\alpha}}{\partial q^{\mu}}\right)=0.$ (111)

Given an event topology, the general procedure for deriving the singularity
coordinate from Eq. (111) was discussed and illustrated in Ref. [262] for the
case of square Jacobian matrices and in Ref. [125] for the general case. Among
the set of singularity variables one finds well-known examples like the
transverse invariant mass $M_{T}$ discussed in Section VII.2 [260, 261, 262],
the invariant mass $M_{a_{2}a_{1}}$ of the visible decay products in the two-
step two-body cascade decay of Figure 7($a$) in three [125] or two [262]
spatial dimensions, and the Cambridge $M_{T2}$ variable from Section VII.2
[263]. However, there are also more recently discovered singularity variables
like the $\Delta_{4}$ variable [264, 265], which is applicable to the three-
step two-body decay chain in Figure 7(c), the $\Delta_{\rm antler}$ variable
[262, 266] relevant to the event topology of Figure 7(b), and the
$\Delta_{t\bar{t}}$ variable [267, 262] relevant for the $t\bar{t}$ dilepton
topology of Figure 9(b). The singularity variables are excellent analysis
tools and can serve a dual purpose: in the age of discovery, they can be used
to target signal-rich regions of phase space, and post-discovery, they form
the basis for the focus point method for mass measurements discussed below in
Section X.2.

##  VIII Exclusive event variables: energy, time, distance

In this section, we discuss a few basic quantities that are directly available
in experiments and their non-trivial utilization. We begin with the energy
variable followed by the timing and distance variables.

###  VIII.1 Energy peak

While energy is not a Lorentz-invariant quantity, the peak position in the
energy distribution of a visible particle coming from a two-body decay of the
heavier resonance/parent particle carries a boost-distribution-invariant
property. Suppose that particle $A_{1}$ is a scalar or produced in an
unpolarized way and decays into a massless visible particle $a_{1}$ and
another paricle $a_{0}$ which may be visible or invisible. In the rest frame
of $A_{1}$, the energy of $a_{1}$ $E_{a_{1}}^{*}$ is simply given by

$E_{a_{1}}^{*}=\frac{m_{A_{1}}^{2}-m_{a_{0}}^{2}}{2m_{A_{1}}}\,.$ (112)

In the laboratory frame, one should perform a Lorentz transformation to find
the laboratory-frame energy of $a_{1}$ $E_{a_{1}}$:

$E_{a_{1}}=\gamma_{A_{1}}E_{a_{1}}^{*}(1+\beta_{A_{1}}\cos\theta_{a_{1}}^{*})\,,$
(113)

where $\gamma_{A_{1}}=(1-\beta_{A_{1}}^{2})^{-1/2}$ is the boost factor of
$A_{1}$ in the laboratory frame and $\theta_{a_{1}}$ is the emission angle of
$a_{1}$ in the $A_{1}$ rest frame with respect to $\vec{\beta}_{A_{1}}$. One
can see that for any $\gamma_{A_{1}}$, $E_{a_{1}}^{*}$ is the (only) commonly
included value in the distribution of $E_{a_{1}}$. Since $A_{1}$ is assumed
unpolarized or scalar, $\cos\theta_{a_{1}}^{*}$ is a flat variable and so is
$E_{a_{1}}$. Therefore, whatever distribution of $\gamma_{A_{1}}$ is given,
the final $E_{a_{1}}$ distribution shows a peak at $E_{a_{1}}^{*}$ [64]. This
observation was made in the context of the cosmic $\pi^{0}$ decay [268], and
then extended and generalized to the two-body decay of an unpolarized
resonance at colliders [64, 269].

This energy-peak feature can be viewed in the logarithmic energy space. In Eq.
(113), the maximum and minimum laboratory-frame energy values
$E_{a_{1}}^{\pm}$ arise at $\cos\theta^{\ast}_{a_{1}}=\pm 1$, resulting in

$E_{a_{1}}^{\pm}=\gamma_{A_{1}}E_{a_{1}}^{\ast}(1\pm\beta_{A_{1}})\,,$ (114)

from which one can see that the rest-frame energy $E_{a_{1}}^{\ast}$ is the
geometric mean of the maximum and minimum laboratory-frame energoies
$E_{a_{1}}^{\pm}$ for any boost factor $\gamma_{A_{1}}$:

$\left(E_{a_{1}}^{\ast}\right)^{2}=E_{a_{1}}^{+}E_{a_{1}}^{-}\,.$ (115)

This further implies that $\ln E_{a_{1}}^{\ast}$ is the mean of $\ln
E_{a_{1}}^{+}$ and $\ln E_{a_{1}}^{-}$ and the rectangular distribution in
$E_{a}$ for a given $\gamma_{A_{1}}$ is log-symmetric with respect to $\ln
E_{a_{1}}^{\ast}$. Once such log-symmetric rectangular distributions, which
are weighted by the $\gamma_{A_{1}}$ distribution, are stacked up, the final
$E_{a_{1}}$ distribution is automatically log-symmetric with respect to $\ln
E_{a_{1}}^{\ast}$.

As suggested by Eq. (112), the extraction of $E_{a_{1}}^{*}$ implies the
measurement of a mass relation between $A_{1}$ and $a_{0}$. This kinematic
feature is particularly useful in a hadron collider environment where the
longitudinal boosts of individual events are a priori unknown, i.e., the
$\gamma_{A_{1}}$ profile is unknown. In addition, since the method involves no
combinatorial ambiguity, its applicability is nearly unaffected by high
particle multiplicity. If the $a_{0}$ mass is known through independent
measurements and if the peak in the $a_{1}$ energy distribution is extracted,
the mass of $A_{1}$ can be readily determined using Eq. (112). A well-
motivated and practical physics application is the top quark mass measurement
in the top quark decay, $t\to bW$, through the $b$-jet energy-peak method [64]
and the weight function method [81]. The CMS Collaboration has measured the
top quark mass by extracting the peak in the $b$-jet energy distribution in
the $e\mu$ channel [270] as shown in Figure 17 where the aforementioned log-
symmetric feature is evident. Another SM example is the $W$ mass
determination, using the lepton energy spectrum in the case of associate
production of leptonic $W$ along with other particles, i.e., $pp\to
WX,W\to\ell\nu_{\ell}$ [126]. The method is not just restricted to SM
processes but straightforwardly applicable to new particles mass measurements;
examples include mass measurements of new resonances in models of
supersymmetry [271, 272, 22, 273, 126] and in the context of potential cosmic
$\gamma$-ray excesses [274, 275, 276, 277, 278].

The crucial assumptions to retain the boost-invariant feature of the energy
peak are that the visible decay product is massless and it comes from a two-
body decay of an unpolarized (or scalar) heavy resonance. As some of them
loosen, the validity of the method would be gradually degraded and a certain
extent of prescriptions would be needed.

First of all, if the visible decay product has a non-zero mass, the relation
in Eq. (113) is modified to

$E_{a_{1}}=\gamma_{A_{1}}(E_{a_{1}}^{*}+p_{a_{1}}^{*}\beta_{A_{1}}\cos\theta_{a_{1}}^{*})\,,$
(116)

where $p_{a_{1}}^{*}$ is the magnitude of the $a_{1}$ momentum measured in the
$A_{1}$ rest frame. Unlike the massless case, $E_{a_{1}}^{*}$ is no more
commonly contained in the $E_{a_{1}}$ distribution for any boost
$\gamma_{A_{1}}$. One can find that if $\gamma_{A_{1}}>\gamma_{A_{1}}^{\rm
cr}\equiv 2\gamma_{a_{1}}^{*}-1$ with $\gamma_{a_{1}}^{*}$ being the boost
factor of $a_{1}$ in the $A_{1}$ rest frame, the minimum $E_{a_{1}}$ occurring
with $\cos\theta_{a_{1}}^{*}=-1$ becomes larger than $E_{a_{1}}^{*}$ [64].
Therefore, for the $\gamma_{A_{1}}$ profile extending beyond
$\gamma_{A_{1}}^{\rm cr}$, the peak in the overall $E_{a_{1}}$ distribution
may be larger than $E_{a_{1}}^{*}$. In the example of top quark decay,
$\gamma_{b}^{*}\approx 15$ and hence $\gamma_{t}^{\rm cr}\approx 450$. At the
LHC, such a large boost factor of top quark is kinematically inaccessible, so
the energy-peak method can safely go through for the top quark decay at the
LHC. However, if $m_{a_{1}}$ is too large with respect to
$m_{A_{1}}-m_{a_{0}}$, the shift of the energy peak is unavoidable.
Nevertheless, one can still extract $E_{a_{1}}^{*}$, modeling the energy
distribution by appropriately accommodating the shift. We refer to Ref. [273]
for a more detailed discussion.

Figure 17: The 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 $\ln E_{b}^{\ast}$, and the fit
was performed with a Gaussian template. Taken from Ref. [270].

Second, once the decay of interest accompanies additional decay products,
$\cos\theta_{a_{1}}^{*}$ is no more a flat variable and hence the argument
breaks down. It was analytically demonstrated that the resulting peak position
is always less than $E_{a_{1}}^{*}$ predicted in the associated multi-body
decay process of $A_{1}$ [279]. In the top quark decay, the $b$ quark often
comes with a hard gluon emission in the final state which is not captured as
part of the $b$ jet, and thus the contamination from such $t\to bWg$
inevitably induces a systematic error in the $E_{b}^{*}$ extraction.555By
contrast, initial state radiation from either incoming partons or top quark
itself simply reshuffles the $\gamma_{t}$ profile, and hence it does not ruin
the boost-invariant feature [64]. The systematics was carefully assessed in
the top quark decay at the next-to-leading order in Ref. [280] and it claimed
that a $\lesssim 0.5\%$ level of the associated systematic error would be
achievable for a 1% jet energy scaling uncertainty. Another approach to treat
the multi-body decays involving an invisible decay product is to interpret the
$A_{1}$ decay as an effective two-body decay to $a_{0}$ and a composite
visible system. Assuming that the visible particles are all massless, their
invariant mass $M_{\rm vis}$ spans from 0 to $m_{A_{1}}-m_{a_{0}}$. One can
then divide $M_{\rm vis}$ space into pieces, and for each such phase-space
slice, the method for the massive visible particle that is briefly discussed
earlier can be applied [22].

Finally, polarized production of (non-scalar) $A_{1}$ induces a non-trivial
angular dependence of $a_{1}$, i.e., $\cos\theta_{a_{1}}^{*}$ is non-flat.
Therefore, the peak position in the overall $E_{a_{1}}$ distribution can be
larger or smaller than $E_{a_{1}}^{*}$ depending on the underlying decay
dynamics [64]. At the LHC, the top quarks are predominantly produced via QCD,
hence they are unpolarized. However, if certain new-physics dynamics produces
polarized top quarks, an appreciable deviation from the $b$-jet energy peak
can be interpreted as a sign of new physics. This further implies that the
energy variable can be utilized as a cut to isolate the signal from SM
backgrounds. This aspect of the energy peak was extensively investigated in
Ref. [271] in the context of supersymmetric top quark decays.

Energy peaks can also be applied in lepton colliders, where the predominance
of electroweak interactions can potentially bring in more effects related to
polarization. As one of the main top quark sources in $e^{+}e^{-}$ programs is
the production close to the threshold, where particular care is needed to
account for the production of bound state and slowly-moving top quarks.
Reference [105] has studied threshold effects on the energy spectrum of a $b$
quark at a 380 GeV $e^{+}e^{-}$ collider. As the production of top quarks at
threshold strengthens the validity of the arguments behind the invariance of
the energy peak, the correction to the energy distribution are found to be
very localized compared to the corrections to the transverse momentum.

The utility of the energy peak is not limited to particle mass measurements
and cutting for signal versus background discrimination. A representative
example is to distinguish the two-body decay topology from the $\geq 3$-body
ones. Imagine two scenarios where each of the pair-produced $A_{1}$’s follows
either $A_{1}\to a_{1}a_{0}$ (i.e., two-body topology) or $A_{1}\to
a_{1}a_{0}a_{0}^{\prime}$ (i.e., $\geq 3$-body topology) with $A_{0}^{\prime}$
representing additional invisible particle(s). It was demonstrated that half
the $\mu$ parameter extracted from the $M_{T2}$ distribution is the same as
(greater than) the peak position $E_{a_{1}}^{\rm peak}$ for the two-body
(three-body) decay scenario [279]:

$\frac{\mu}{2}=\frac{m_{A_{1}}^{2}-m_{a_{0}}^{2}}{2m_{A_{1}}}\left\\{\begin{array}[]{l
c}=E_{a_{1}}^{\rm peak}&\hbox{for the two-body decay,}\\\\[10.00002pt]
>E_{a_{1}}^{\rm peak}&\hbox{for the three-body decay.}\end{array}\right.$
(117)

###  VIII.2 Timing

The variables in the preceding sections involve quantities in energy-momentum
space. Likewise, one may utilize the information in time-position space. These
next two subsections are devoted to discussing variables designed with time
and position information.

In principle, the timing information is useful for the following situations
including: i) the case where the signals of interest differ from the unwanted
signals (or backgrounds) by the timing at which they hit the detector system;
ii) the case where the subprocesses of the signal process come along in a
time-ordered manner (e.g., sequential decays of heavy resonances). In
practice, the timing information becomes useful when the resolution in the
timing measurement is sufficiently good and the uncertainty stemming from the
particle-beam pulse spread is small enough. In many of the collider
experiments (more generally accelerator-based experiments) including the LHC,
beam parameters are well under control and sufficiently narrow beam pulses can
be generated. When it comes to the timing resolution, it is a few hundred
pico-seconds as of Run II of the LHC and thus its utilization is somewhat
limited especially for the (new) physics processes where all relevant hard
interactions take place instantly.

The use of timing information is receiving increasing attention, however, as
higher-resolution timing information allows for improved pile-up-origin
background suppression [281, 282] and it provides a unique handle in the
search for long-lived particles (LLPs) [283, 284, 285] which were targeted
marginally in the earlier LHC operation as its detectors (e.g., ATLAS, CMS,
and LHCb) were designed to be optimal to prompt processes. Therefore, the
ATLAS [281], CMS [282], and LHCb [286] Collaborations are planning to install
dedicated timing modules and develop appropriate trigger algorithms, expecting
them to operate from Run IV.

Indeed, it has been pointed out that the timing variable allows for a powerful
separation between delayed new physics signal events and SM background events,
given projected upgrades and implementations of high-capability timing modules
and dedicated triggers at the LHC detectors [283, 284, 285]. For example,
massive enough LLPs at the LHC can travel for a finite amount of time such
that their decay products arrive at detectors with time delays around
nanosecond scale, unlike the light SM particles. The strategies for utilizing
this time delay feature can be applied in the search for LLPs, using the
initial state radiation as a way of setting a reference timing and requiring
at least one LLP to decay within the detector [283]. It has been demonstrated
that the strategies can improve the sensitivity to the lifetime of the LLPs by
two orders of magnitude or more [283], in comparison to conventional search
strategies for LLPs [287, 288, 289], e.g., displaced vertex searches. Example
sensitivity reaches for the scenario where the SM Higgs decays to a pair of
LLPs $X$ are shown in Figure 18 in terms of limits on the branching fraction
of $h\to XX$ as a function of the proper decay length of $X$ [283].

Figure 18: The 95% CL limits on BR$(h\to XX)$ for signal process $pp\to jh$
followed by subsequent decays $h\to XX$ and $X\to jj$ with $X$ being a new
particle. Taken from Ref. [283].

The timing information is useful not only for the discovery opportunities of
LLPs but for the resonance mass reconstruction of the new particles involved
in the associated decay process. The authors in Ref. [284] investigated the
case where LLPs are pair-produced and each of them decays to an invisible
particle and visible particle(s). They showed that the particle mass spectrum
can be completely determined event by event, using the missing transverse
momentum condition and the timing information, even in the case where the
pair-produced LLPs are not identical. Another possibility that allows for the
event-by-event mass measurement is the so-called “tumbler” scenario where a
LLP decays to lighter LLPs sequentially. The authors in Ref. [285] studied the
simplest tumbler process, $A_{2}\to a_{2}A_{1}$ followed by $A_{1}\to
a_{1}a_{0}$ with $A_{2}$ and $A_{1}$ being LLPs, $a_{2}$ and $a_{1}$ being
visible particle systems, and $a_{0}$ being a collider-stable particle or
dark-matter candidate. They demonstrated that the masses of $A_{2}$, $A_{1}$,
and $a_{0}$ can be determined with the aid of timing information but without
the recourse to the missing transverse momentum measurement.

Timing is also considered as a discriminating variable to suppress the beam-
induced background activities due to muon decays in muon beam machines [290,
291, 292, 293]. Due to the unstable nature of the particles in the beam, the
muon beams are a source of particles that go outside the control of the
machine optics. The primary particles come from muon decay products: electron
and neutrinos. They can interact with the accelerator apparatus and every
material surrounding the beam, generating secondary radiation which can
eventually reach the detectors. The tracking of these secondary particles is
usually quite difficult, as they are a very large number of low-momentum
particles. They can still be discriminated based on the fact that the
production of secondary particles happens during the whole time of beam
circulation, while interesting particles from the hard collisions appear only
at beam crossing. The discrimination is further helped from the fact that only
a limited length before the detector, hence a specific time window, actually
behaves as a source of secondaries that are potentially harmful for the
physics analyses.

###  VIII.3 Distance variables

Mass measurements are inherently energy scale measurements. However, excellent
measurements of masses can, in principle, be conducted by accurately measuring
the distances travelled by particles. The key relation is

$\left\langle
d\right\rangle=c\beta\cdot\gamma\tau_{0}=\frac{\sqrt{E^{2}-m^{2}}}{m}\cdot
c\tau_{0}\,,$ (118)

where $\left\langle d\right\rangle$ is the average distance travelled before
decaying, $\tau_{0}$ is the proper lifetime of the particle at hand, and
$\gamma=E/m=\left(1-\beta^{2}\right)^{-1/2}$ is the usual Lorentz boost factor
that governs the time dilation and length contraction. In essence a mass
measurement can be carried out from a sample of particles all of which have
identical boost factor – or equivalently identical energy – by measuring the
average decay length of the sample and measuring elsewhere (or using theory
predictions for) the proper lifetime $\tau_{0}$.

A practical difficulty of pursuing this method is that in general it is very
hard to come across a sample of particles perfectly monochromatic in energy. A
non-monochromatic particle sample can be used to carry out a mass measurement
if the energy distribution of these particles is sufficiently well known. The
case of particles produced by parton collisions in $pp$ or $p\bar{p}$
colliders clearly shows how measurements of this sort need very accurate
knowledge of the source (i.e., PDFs in this case).

This idea can be exploited even for particles whose lifetime is short enough
to make it impossible to measure $\left\langle d\right\rangle$. If the prompt-
decaying particle produces an unstable long-lived particle as a decay product,
it can be viewed as the source of a new sample of particles as it inherits a
certain extent of relevant information of the decaying particle. The average
decay length of the unstable decay product can now be measured and turned into
a mass measurement, provided that the proper lifetime and the energy
distribution of the measurable decay product are sufficiently well known.

Measurements of this sort have been proposed for SM particles such as the top
quark [55]. Concrete results for this strategy appeared from the CDF [294] and
CMS [56] experiments. A great deal of work in these measurements is devoted in
understanding the source of the unstable particles whose decay length is
measured and a number of issues having something to do with the formation of
hadrons and other aspects of QCD that impact the kinematics of top quark decay
products.

In fact, in the case of the top quark mass measurement the accessible unstable
states are B-flavored hadrons (e.g., $B^{+}$, $B_{s}^{0}$, $B^{0}$,
$\Lambda_{b}^{0}$, …) whose $c\tau_{0}$’s are in the range of 100 $\mu$m hence
can be measured in modern detectors, especially when they enjoy typical
Lorentz boost factor of $\mathcal{O}(10)$ as expected from top quark decays.
Relating the top quark mass to the observed decay lengths of the $B$-hadrons
has significant complications with respect to the simple one-step
monochromatic case sketched at the beginning of this section. Indeed, one has
to deal with multiple species of $B$ hadrons, each of which has a different
mass and a different proper lifetime. As a consequence, the actual yield of
each type of $B$-hadrons does affect a measurement that is blind to the
identification of each species. For this reason, the analyses of CDF [294] and
CMS [56] are filled with details on the treatment of hadronization effects,
which become of primary interest when a sub-percent measurement is attempted
to be competitive with other top quark mass determinations. We refer to the
above studies for a detailed discussion of these effects.

Concerning the use of kinematic variables, we remark that so far the CDF [294]
and CMS [56] Collaborations have managed to relate the top quark mass and the
length distribution of the $B$-hadrons only by producing templates of the
length distribution with a full chain of MC simulations. However, the insights
on the peak of the energy distribution can be translated into properties of
the length distribution, as entailed by the relation among energy, mass, and
decay length as in Eq. (118). Work is underway to formulate new mass
measurement strategies exploiting this insight [295, 296].

In addition to mass measurements, distance variables are useful discriminators
for the identification of heavy flavor quarks and leptons. Particles with a
measurably long proper lifetime, e.g., $B$-hadrons, result in a measurable
impact parameter, defined as the transverse distance of closest approach of a
track to the primary interaction vertex. This quantity can be shown [297] to
be largely unaffected by the boost of the decaying particle, as time dilation
contributes to displace further the decay point in the laboratory frame, but
at the same time length contraction makes the direction of the decay product
tend to align with the decaying particle momentum. As a consequence, the
characteristic decay times of particles are translated to the characteristic
impact parameters, which are very useful for particle identification.

Modern experiments use impact parameter information, among many other inputs,
to give a likelihood for particle identification. For a detailed explanation
of the role played by the impact parameter in heavy flavor tagging, we refer
to e.g., Ref. [298]. A modern incarnation that leverages the impact parameter
in a neural network classifier is described in Ref. [299].

A major difficulty in analyzing collider events is the necessity to
disentangle the useful particles potentially bearing information on
interesting phenomena from the particles stemming out of ordinary collisions
not carrying any useful information. Especially at hadron colliders the
collision rate is so high that a number collisions can happen for each bunch
crossing. A basic tool to discriminate particles from the collisions recorded
at once is the position from which these particles momenta originate if
extrapolated to the beam axis [300, 301]. This simple observable keeps being a
basic ingredient for current and future experiments and it is used in
conjunction with the most theoretically sophisticated tools [302] to remove
pile-up effects.

New physics models have provided many examples of signatures involving LLPs
and other exotic states (e.g., Refs. [303, 304, 305, 306, 307, 308, 309, 310,
311, 312, 285]) that can be analyzed with observables referring to length
measurements. The most basic measurements involve the euclidian distance
between the primary interaction point and the displaced vertex where the
exotic particle decays. A summary of the power of this approach to search for
exotic states is given, for example, in Ref. [313] together with a comparison
of the coverage of new physics models parameter space of the equivalent
“prompt” searches not exploiting length measurements. The typical prompt
search gets quickly ineffective when one considers distances greater than 100
$\mu$m $\div$ 1 cm for the lifetime of the exotic state, or the equivalent
parameter that controls the appearance of displaced vertices in more
complicated models. The threshold for the beginning of degradation of the
prompt searches is process-dependent, but the general message that the
displaced vertex searches can fully fill in this gap is very robust. As a
matter of fact, when the experiments have looked for these exotic signals
(e.g., Refs. [314, 315, 316, 317, 318]), the bounds from displaced vertices
are stronger than the prompt counterparts.

##  IX Other exclusive event variables

###  IX.1 Dimensionless variables

The variables in the preceding sections are dimensionful hence they allow to
infer the scale information of the underlying physics processes. By contrast,
dimensionless variables make it possible to extract scale-independent
information. Here we briefly review a few dimensionless exclusive event
variables most of which are developed for particular processes and/or event
topologies.

The first example is the $\alpha_{T}$ variable [319] which is introduced to
efficiently reduce multijet events without a significant missing transverse
momentum $\not{\vec{p}}_{T}$. For dijet events, it can be defined as [320]

$\alpha_{T}=\frac{E_{T}^{j_{2}}}{m_{T}}\,,$ (119)

where $E_{T}^{j_{2}}$ is the transverse energy of the second hardest jet and
$m_{T}=\sqrt{(E_{T}^{j_{1}}+E_{T}^{j_{2}})^{2}-(\vec{p}_{T}^{\,j_{1}}+\vec{p}_{T}^{\,j_{2}})^{2}}$
is the transverse mass of the dijet system. For a perfect dijet system where
the two jets are back-to-back, $E_{T}^{j_{1}}=E_{T}^{j_{2}}$ and
$\vec{p}_{T}^{\,j_{1}}=-\vec{p}_{T}^{\,j_{2}}$ which leads to
$m_{T}=2E_{T}^{j_{2}}$ and $\alpha_{T}=0.5$. $\alpha_{T}$ is significantly
larger than 0.5, when the two jets are not back-to-back, recoiling against
$\not{\vec{p}}_{T}$.

In the case of events with three or more jets, one can form an equivalent
dijet system by combining the jets in the event into two pseudo-jets. One
chooses the combination such that the $E_{T}$ difference ($\Delta H_{T}$)
between the two pseudo-jets is minimized. This simple clustering criterion
provides a good separation between QCD multijet events and events with true
$\not{\vec{p}}_{T}$. In this case, the $\alpha_{T}$ is generalized as

$\displaystyle\alpha_{T}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{H_{T}-\Delta
H_{T}}{\sqrt{H_{T}^{2}-\not{H}_{T}^{2}}}$ (120) $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{1-\Delta
H_{T}/H_{T}}{\sqrt{1-(\not{H}_{T}/H_{T})^{2}}}\,,$ (121)

where $H_{T}=\sum_{j=1}^{N_{\rm jets}}E_{T}^{j}$ and
$\not{H}_{T}=|\sum_{j=1}^{N_{\rm jets}}\vec{p}_{T}^{j}|$. Here $N_{\rm jets}$
is the number of jets with $E_{T}$ greater than a certain threshold, typically
chosen to be 50 GeV [321].

The second example is topness and higgsness for which the main idea is to use
the value of $\chi^{2}$ as a cut. This method becomes more powerful,
especially when one can define two (or more) independent $\chi^{2}$ values.
Topness was originally introduced to reduce the $t\bar{t}$ background in the
search for supersymmetric top quarks [322] and later further fine-tuned in the
search for double Higgs production [323, 324, 325]. Topness basically aims to
check the consistency of a given event with $t\bar{t}$ production. It is a
minimized chi-square value constructed by using four on-shell constraints,
$m_{t}$, $m_{\bar{t}}$, $m_{W^{+}}$ and $m_{W^{-}}$, and 6 unknowns (the
three-momenta of the two neutrinos, $\vec{p}_{\nu}$ and $\vec{p}_{\bar{\nu}}$)

$\displaystyle\chi^{2}_{ij}$ $\displaystyle\equiv$
$\displaystyle\min_{\tiny\not{\vec{p}}_{T}=\vec{p}_{\nu
T}+\vec{p}_{\bar{\nu}T}}\left[\frac{\left(m^{2}_{b_{i}\ell^{+}\nu}-m^{2}_{t}\right)^{2}}{\sigma_{t}^{4}}\,+\frac{\left(m^{2}_{\ell^{+}\nu}-m^{2}_{W}\right)^{2}}{\sigma_{W}^{4}}\,\right.$
(122) $\displaystyle\hskip
14.22636pt\left.+\frac{\left(m^{2}_{b_{j}\ell^{-}\bar{\nu}}-m^{2}_{t}\right)^{2}}{\sigma_{t}^{4}}\,+\frac{\left(m^{2}_{\ell^{-}\bar{\nu}}-m^{2}_{W}\right)^{2}}{\sigma_{W}^{4}}\right],$

subject to the constraint, $\not{\vec{p}}_{T}=\vec{p}_{\nu
T}+\vec{p}_{\bar{\nu}T}$. Here $\sigma_{t}$ and $\sigma_{W}$ determine the
relative weight of the on-shell conditions, and should not be less than
typical resolutions. Due to the twofold ambiguity in paring a $b$-jet (out of
$b_{1}$ and $b_{2}$) and a lepton (out of $\ell^{+}$ and $\ell^{-}$), we
define Topness as the smaller of the two possible chi-square values,
$\chi_{12}^{2}$ and $\chi_{21}^{2}$:

$\displaystyle T$ $\displaystyle\equiv$
$\displaystyle{\min}\left(\chi^{2}_{12}\,,\,\chi^{2}_{21}\right)\,.$ (123)

Similarly, Higgsness aims to probe the consistency of a given event with
double Higgs production. The challenge here is to find the sufficient number
of constraints, as there are four unknowns, while there are only two
intermediate on-shell particles.666See Ref. [326] for the heavy Higgs decaying
to two on-shell $W$ bosons. The Higgsness is defined by

$\displaystyle H$ $\displaystyle\equiv$ $\displaystyle{\rm
min}\left[\frac{\left(m^{2}_{\ell^{+}\ell^{-}\nu\bar{\nu}}-m^{2}_{h}\right)^{2}}{\sigma_{h_{\ell}}^{4}}\right.+\frac{\left(m_{\nu\bar{\nu}}^{2}-m_{\nu\bar{\nu},peak}^{2}\right)^{2}}{\sigma^{4}_{\nu}}$
$\displaystyle+{\min}\left(\frac{\left(m^{2}_{\ell^{+}\nu}-m^{2}_{W}\right)^{2}}{\sigma_{W}^{4}}+\frac{\left(m^{2}_{\ell^{-}\bar{\nu}}-m^{2}_{W^{*},peak}\right)^{2}}{\sigma_{W_{*}}^{4}}\,,\right.$
$\displaystyle\left.\left.\frac{\left(m^{2}_{\ell^{-}\bar{\nu}}-m^{2}_{W}\right)^{2}}{\sigma_{W}^{4}}+\frac{\left(m^{2}_{\ell^{+}\nu}-m^{2}_{W^{*},{\rm
peak}}\right)^{2}}{\sigma_{W_{*}}^{4}}\right)\right]\,,$

where $m_{W^{*}}$ is bounded from above, $m_{W^{*}}\leq m_{h}-m_{W}$, and its
location of the peak can be estimated by

$\displaystyle m_{W^{*}}^{\rm peak}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\sqrt{2\left(m_{h}^{2}+m_{W}^{2}\right)-\sqrt{m_{h}^{4}+14m_{h}^{2}m_{W}^{2}+m_{W}^{4}}}$
(125) $\displaystyle\approx$ $\displaystyle 40\text{ GeV}\,.$

The $m_{\nu\bar{\nu}}^{\rm peak}=m_{\ell\ell}^{\rm peak}\approx 30$ GeV is the
location of the peak in the invariant mass distribution of two neutrinos
$\frac{d\sigma}{dm_{\nu\bar{\nu}}}$ (or $\frac{d\sigma}{dm_{\ell\ell}}$),
which is bounded from above by
$m_{\nu\bar{\nu}}^{\max}=m_{\ell\ell}^{\max}=\sqrt{m_{h}^{2}-m_{W}^{2}}$. The
phase space distribution of $\frac{d\sigma}{dm_{\nu\bar{\nu}}}$ is given by

$\frac{d\sigma}{dm_{\nu\bar{\nu}}}\propto\int
dm_{W^{*}}^{2}\lambda^{1/2}(m_{h}^{2},m_{W}^{2},m_{W^{*}}^{2})f(m_{\nu\bar{\nu}}),$
(126)

where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2yz-2zx$ is the kinematic
triangular function and $f(m_{\nu\bar{\nu}})$ is the invariant mass
distribution of the antler topology (see also Section VII.1) with $h\to
WW^{*}\to\ell^{+}\ell^{-}\nu\bar{\nu}$

$\displaystyle f(m_{\nu\bar{\nu}})\sim\left\\{\begin{array}[]{l
l}\eta\,m_{\nu\bar{\nu}}\,,&0\leq m_{\nu\bar{\nu}}\leq
e^{-\eta}E,\\\\[2.84526pt]
m_{\nu\bar{\nu}}\ln(E/m_{\nu\bar{\nu}})\,,&e^{-\eta}E\leq m_{\nu\bar{\nu}}\leq
E,\end{array}\right.$ (129) (130)

where the endpoint $E$ and the parameter $\eta$ are defined in terms of the
particle masses as [199]

$\displaystyle E$ $\displaystyle=$
$\displaystyle\sqrt{m_{W}m_{W^{*}}\,e^{\eta}},$ (131) $\displaystyle\cosh\eta$
$\displaystyle=$
$\displaystyle\left(\frac{m_{h}^{2}-m_{W}^{2}-m_{W^{*}}^{2}}{2m_{W}m_{W^{*}}}\right).$
(132)

The actual peak of 30 GeV is slightly less than the result for pure phase
space due to a helicity suppression in the $W$-$\ell$-$\nu$ vertex.

The definitions of Topness and Higgsness involve $\sigma$ hyperparameters
which represent experimental uncertainties and particle widths. However, in
principle, they can be taken as free parameters. The precise values of these
parameters are not crucial, as results are not sensitive to their numerical
values.

As mentioned previously, the $\chi^{2}$ method becomes more useful when
applied in more than one dimension. For example, Figure 19 shows a scatter
plot distribution of ($\ln H$, $\ln T$) for double Higgs production and
backgrounds. The black-solid curves in both panels are the same and represent
the optimized cut.

Figure 19:  Scatter plot distribution of ($\ln H$, $\ln T$) for double Higgs
production ($hh$) and backgrounds ($t\bar{t}$, $t\bar{t}h$, $t\bar{t}V$,
$\ell\ell bj$ and $\tau\tau bb$) after loose baseline selection cuts. The
black-solid curves in both panels are the same and represent the optimized
cut. Taken from Ref. [323]

The third example is ratios of energy and transverse momenta which have been
recognized as useful kinematic variables very early on, especially for
problems in which part of the information on the kinematics is not accessible
due to production of invisible particles. The ratio of energies and $p_{T}$
from cascade decays $A_{2}\to A_{1}a_{2}\to a_{1}a_{0}$ can provide extra
information [327] on top of the “classic” invariant mass $M_{a_{1}a_{2}}$ of
the visible decay products. Indeed, by properly combining information from the
transverse energy ratio $\ln E_{1,T}/E_{2,T}$ and the invariant mass
$M_{a_{1}a_{2}}$, it is possible to reconstruct the full information on the
three masses involved in the cascade decay [328].

Another interesting application is for the case where the numbers of final
state invisible particles are different in the decays of pair-produced heavy
resonances. For example, dark-matter “partners” charged under a $Z_{2}$
symmetry decay to a single dark-matter candidate, e.g., $A_{1}\to a_{1}a_{0}$
with $A_{1}$, $a_{1}$, and $a_{0}$ being dark-matter partner, visible
particle(s), and dark-matter candidate, respectively. By contrast, those
charged under a $Z_{3}$ symmetry can decay with one or two dark-matter
candidates, namely, $A_{1}\to a_{1}a_{0}a_{0}$ or $A_{1}\to a_{1}a_{0}$. This
implies that if $Z_{3}$-charged $A_{1}$’s are pair-produced, each of their
decays terminates with different numbers of $a_{0}$, the ratios of energy or
$p_{T}$ of $a_{1}$ in both decay sides are likely to be unbalanced, whereas it
is more likely to be balanced in the $Z_{2}$ case. The authors of Ref. [232]
defined the ratio $R_{p_{T}}$ as

$R_{p_{T}}=\frac{\max\left(p_{T,a_{1}}^{(1)},p_{T,a_{1}}^{(2)}\right)}{\min\left(p_{T,a_{1}}^{(1)},p_{T,a_{1}}^{(2)}\right)},$
(133)

where $p_{T,a_{1}}^{(i)}$ denotes the $p_{T}$ of the visible particle in the
$i$-th decay side ($i=1,2$). Therefore, if the underlying physics is $Z_{3}$
($Z_{2}$), the ratios are typically larger than (close to) 1, and thus these
models can be distinguished [232].

Finally, likelihood methods such as the MELA and other matrix-element based
techniques can be categorized as dimensionless variables. Although fitting in
the definition of dimensionless variables suitable for this section, they will
be discussed separately in Section X.4.

###  IX.2 ISR methods

In this subsection, we discuss kinematic effects due to the presence of
initial state radiation (ISR). We begin with a method which attempts to
identify ISR. At hadron colliders, the production of heavy new particles is
often accompanied by additional jets with a significant transverse momentum.
These extra jets make the combinatorial problem worse and complicate the
reconstruction of new particle masses. In Ref. [19], a novel technique was
discussed to reduce these effects and to reconstruct a clear kinematical
endpoint, taking gluino pair production and decay at the LHC as an example.
That analysis considered the three-body decay of the gluino, which leads to 4
jets plus missing transverse momentum, in which case the mass reconstruction
is done via $M_{T2}$. To isolate the ISR jet, the authors introduced
$M_{T2}^{\min}=\min\limits_{i=1,\cdots,5}M_{T2}(i)$, where $M_{T2}(i)$ is
calculated from the five highest $p_{T}$ jets, excluding the $i$-th highest
$p_{T}$ jet. Then the $i_{\min}$-th jet which satisfies
$M_{T2}(i_{\min})=M_{T2}^{\min}$ is tagged as the ISR jet. After all this, a
strong correlation was found between the reconstructed ISR jet and true ISR
jet.

Occasionally, the ISR helps measurement of particle masses. The author of Ref.
[223] considered the two-body decay of a particle at a hadron collider into a
visible and an invisible particle, generalizing $W\to\ell\nu_{\ell}$, where
the masses of the decaying particle and the invisible daughter particle are
unknown. It was proved analytically that the transverse mass, when maximized
over all possible kinematic configurations, can be used to determine both of
the unknown masses. The authors of Ref. [224] generalized the idea for more
complex decays of a singly-produced mother particle and for pair-produced
particles. On the other hand, in the absence of ISR, one can in principle
consider the upstream transverse momentum (UTM) playing the role of ISR,
placing the system of interest under different momentum configurations [219].
In all cases, the mass variables ($M_{T}$, $M_{T2}$, or total invariant mass)
optimized over all possible momentum configurations, which are given by either
ISR or UTM, exhibit a kink structure at the true values of the mother and
daughter particle masses, as illustrated in Figure 20. An application of these
ISR and kink methods is illustrated in Ref. [329] to determine the masses of
the supersymmetric chargino and sneutrino in an inclusive manner, i.e. using
the two well measured lepton momenta, while treating all other upstream
objects in the event as a single entity of total transverse momentum
$\not{\vec{p}}_{T}$. This method takes full advantage of the large production
rates of colored superpartners, but does not rely on the poorly measured
hadronic jets, and avoids any jet combinatorics problems.

Figure 20:  The transverse mass of the mother particle decaying semi-invisibly
as a function of the daughter particle mass for the phase space Monte Carlo in
which the mother has been constrained to be at rest in the laboratory frame
with vanishing $p_{T}$ (left) and for the phase space Monte Carlo in which the
mother can have large transverse momentum in the laboratory frame (right).
Taken from Ref. [224].

ISR plays a crucial role in studying dark-matter production at colliders as
well. Under the hypothesis of classical weakly interacting massive particles
(WIMPs), one can consider pair annihilation of dark matter ($\chi$) into a
pair of SM particles, $\chi+\chi\to X_{i}+\bar{X}_{i}$, with
$X_{i}=\ell,q,g,\cdots$. The detailed balancing equation [330, 331] relates
the pair-annihilation cross section and its inverse:

$\frac{\sigma(\chi+\chi\to
X_{i}+\bar{X}_{i})}{\sigma(X_{i}+\bar{X}_{i}\to\chi+\chi)}\,=\,2\,\frac{v_{X}^{2}(2S_{X}+1)^{2}}{v_{\chi}^{2}(2S_{\chi}+1)^{2}}\,,$
(134)

where $v_{i}$ and $S_{i}$ denote the velocity of initial-state species $i$ and
the spin number of species $i$, respectively, and where the cross sections are
averaged over spins but not other quantum numbers such as color. Then the WIMP
production rate can be obtained as

$\displaystyle\sigma(X_{i}\bar{X}_{i}\to 2\chi)=$ $\displaystyle
2^{2(J_{0}-1)}\,\kappa_{i}\sigma_{\rm ann}\,\frac{(2S_{\chi}+1)^{2}}{(2S_{\rm
X}+1)^{2}}\left(1-\frac{4m_{\chi}^{2}}{s}\right)^{1/2+J_{0}},$ (135)

where the initial state particles are assumed to be relativistic ($m_{X}\ll
m_{\chi}$). Eq. (135) is written in terms of a small number of parameters with
a clear physical meaning: the mass $m_{\chi}$ and the spin $S_{\chi}$ of the
WIMP, the value of $J_{0}$ (either 0 or 1, depending on whether dark-matter
annihilation is $s$-wave or $p$-wave annihilation), and the annihilation
fraction $\kappa_{i}$ for the given initial state. Importantly, the overall
scale for this prediction, the total annihilation cross section quantity
$\sigma_{\rm ann}$ is provided by cosmology. This formula is only valid at
center of mass energies slightly above the $2\chi$ threshold,
$v=2v_{\chi}=2\sqrt{1-4m_{\chi}^{2}/s}\ll 1$, and receives corrections of
order $v^{2}$. Taking $X_{i}=q$ or $g$ (or even $W$, $Z$) for a hadron
collider or $X_{i}=e$ for an electron-positron machine, Eq. (135) provides a
prediction of the WIMP production rate.

Unfortunately, this dark-matter production process is not directly observalbe
at colliders. At least one detectable particle is required for the event to
pass the triggers and be recorded on tape. Therefore it is desirable to
consider the production of two WIMPs in association with a photon or a gluon
radiated from the known initial state. We consider a simple example given in
Ref. [331], $e^{+}e^{-}\to 2\chi+\gamma$. If the emitted photon is either soft
or collinear with the incoming electron or positron, soft/collinear
factorization theorems provide a model-independent relation. The emission of
collinear photons is given by

$\frac{d\sigma(e^{+}e^{-}\to 2\chi+\gamma)}{dx\,d\cos\theta}\approx{\cal
F}(x,\cos\theta)\,\hat{\sigma}(e^{+}e^{-}\to 2\chi),$ (136)

where $x=2E_{\gamma}/\sqrt{s}$ ($E_{\gamma}$ is the photon energy), $\theta$
is the angle between the photon direction and the direction of the incoming
electron beam, ${\cal F}$ denotes the collinear factor:

${\cal
F}(x,\cos\theta)=\frac{\alpha}{\pi}\frac{1+(1-x)^{2}}{x}\frac{1}{\sin^{2}\theta}\,,$
(137)

and $\hat{\sigma}$ is the WIMP pair-production cross section evaluated at the
reduced center of mass energy, $\hat{s}=(1-x)s$. Note that upon integration
over $\theta$, the above equation reproduces the familiar Weizsacker-Williams
distribution function. The factor ${\cal F}$ is universal: it does not depend
on the nature of the (electrically neutral) particles produced in association
with the photon. Combining Eq. (135) and Eq. (136), one can easily obtain the
expression for $\dfrac{d\sigma(e^{+}e^{-}\to 2\chi+\gamma)}{dx\,d\cos\theta}$.
The left panel in Figure 21 shows the comparison between the photon spectra
from the process $e^{+}e^{-}\to\not{E}_{T}+\gamma$ in the explicit
supersymmetric models (red) and the spectra predicted by above procedure
[331]. At hadron colliders, the corresponding distributions show different
shapes, as illustrated in the right panel, which shows the jet $p_{T}$
distribution for an EFT leading to mono-jet plus $\not{\vec{p}}_{T}$ at the
Tevatron [332].

Figure 21: (left) Comparison between the photon spectra from the process
$e^{+}e^{-}\to\not{E}_{T}+\gamma$ in the explicit supersymmetric models (red)
and the spectra predicted by Eq. (136) (Eq. (9) in Ref. [331]) for a
p-annihilator of the corresponding mass. Taken from Ref. [331]. (right)
Comparisons of simulated signal events from two different Monte-Carlo tools
for the parton and the particle-level at Tevatron. Taken from Ref. [332].

Similar to $\gamma+\not{E}_{T}$ at LC, one can consider dark-matter production
at hadron colliders. Dark-matter particles could lead to events with the large
missing transverse momentum, if another visible object (e.g., an energetic
jet) is produced at the same time. Such mono-jet process has been widely
studied at the Tevatron, LHC and future colliders [332, 333, 334, 335]. The
same idea has been extended to other standard model particles (such as $W/Z$,
$b/t$, $h$, etc and a new particle such as $Z^{\prime}$) being produced
together with dark-matter candidates. Such searches are often called mono-X
searches [336] and are one of primary methods looking for dark-matter
particles at the LHC. Figure 22 (taken from Ref. [337].) shows the missing
transverse momentum comparison between data and the background prediction in
the monojet (left) and monoV (right) signal region before and after the
simultaneous fit. A similar result can be found from Ref. [338]. (see Ref.
[339] for summary of recent results.)

Figure 22:  Comparison between data and the background prediction in the
monojet (left) and monoV (right) signal region before and after the
simultaneous fit. Taken from Ref. [337].

ISR plays an important role in the search for new physics models, where the
mass spectrum is generate (i.e., see Refs. [340, 341] for compressed
supersymmetry and Refs. [186, 342, 343] for universal extra dimensions). For
example, when the squark ($\tilde{q}$) and the lightest neutralino
($\tilde{\chi}_{1}^{0}$) are degenerate, the pair-produced squark leads to
very soft decay products with little missing transverse momentum, which will
be completely hidden under the QCD backgrounds. By requiring a hard ISR, one
can boost the squark system with substantial $p_{T}$, and improve the signal
sensitivity, especially in the region of $m_{\tilde{\chi}_{1}^{0}}\approx
m_{\tilde{q}}$.

Another situation where the ISR is crucial is the low mass dijet resonance
searches. The low mass dijet resonance is completely hidden under QCD
backgrounds. Similarly to SUSY searches with degenerate mass spectrum, one
requires a hard radiation ($\gamma$, $Z$, $j$ etc) from the initial state,
which boosts the resonance to a high transverse momentum. Such boosted
resonance will appear as unresolved fat-jet. The appropriate tagging algorithm
and the hard radiation can overcome the huge QCD background [344, 345]. Mono-
jet searches can therefore be interpreted in several different ways — as dark-
matter production in association with a jet, squark production in SUSY with
degenerate spectrum, or $Z^{\prime}$ production with invisible decay.

Finally, initial state radiation is also useful in the measurement of particle
properties such as top quark mass measurement at the LHC [62, 346] and ILC
[107].

##  X Variables and methods using ensembles of events

So far we have considered kinematic variables which can be calculated on an
event-by-event basis. In this section, we introduce a variety of different
types of kinematic methods, which use ensembles of events. For concreteness,
let us consider the generic event topology in a collider analysis displayed in
Figure 23. The particles $X_{i},1\leq i\leq n$, are BSM particles which appear
as promptly decaying, on-shell intermediate resonances. The particles $x_{i}$
are the corresponding SM decay products, which are all visible in the
detector. We begin with the so-called polynomial method [347, 348, 241].

Figure 23: The generic event topology under consideration in Section X. The
particles $X_{i},1\leq i\leq n$, are BSM particles which appear as 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 $\vec{p}_{T}$. $X_{0}$
is a BSM particle which is invisible in the detector. The integer $n$ counts
the total number of intermediate BSM particles in each chain, so that the
total number of BSM particles in each chain is $n+1$. For simplicity, in this
review we shall only consider symmetric events, in which the two decay chains
are identical. The generalization of the methods discussed here to asymmetric
decay chains is straightforward [216, 217]. Taken from Ref. [219].

###  X.1 Polynomial method

We use the experimentally measured four-momenta $p^{\mu}_{i}$ (of all SM
particles) as well as the missing transverse momentum $\not{\vec{p}}_{T}$ in
the event. We then impose the mass shell constraints for the intermediate BSM
particles $X_{i}$ and attempt to solve the resulting system of equations for
the 8 unknown components of the four-momenta $q_{0}^{\mu}$ of the two missing
particles $X_{0}$. With the $(n+1)$ unknown BSM masses $m_{i}$ for $X_{i}$,
the number of unknown parameters $N_{p}$ is given by

$N_{p}=8+(n+1)=n+9\,.$ (138)

The number of measurements (constraints) $N_{m}$ includes the two components
of the missing transverse momentum condition and $2(n+1)$ mass-shell
conditions (for each BSM particle $X_{i}$ belonging to one of the two decay
chains shown in Figure 23):

$N_{m}=2(n+1)+2=2n+4\,.$ (139)

Then the number of undetermined parameters for any given event is readily
obtained from Eqs. (138) and (139)

$N_{p}-N_{m}=5-n.$ (140)

Therefore if $n\geq 5$, one can in principle solve for the momenta of the
invisible particles and reconstruct the entire final state (up to the
combinatorial issue mentioned in Section II.2).

However, one might do better than this, by combining the information from two
or more events [349, 218, 348, 350]. For example, consider another event of
the same type. Since the $(n+1)$ unknown masses were already counted in Eq.
(138), the second event introduces only 8 new parameters (the four-momenta of
the two $X_{0}$ particles in the second event), bringing up the total number
of unknowns in the two events to

$N_{p}=8+8+(n+1)=n+17\ .$ (141)

At the same time, all the constraints are still valid for the second event,
which results in $(2n+4)$ additional constraints. This brings the total number
of constraints to

$N_{m}=(2n+4)+(2n+4)=4n+8\ .$ (142)

Subtracting (141) and (142), we get

$N_{p}-N_{m}=9-3n.$ (143)

Comparing the previous result (140) with (143), we see that the latter
decreases much faster with $n$. Therefore, when using the polynomial method,
combining information from two different events is beneficial for large $n$
(in this example, for $n\geq 3$).

Following the same logic, one can generalize this parameter counting to the
case where the polynomial method is applied for a group of $N_{\rm eve}$
different events of the same type at a time. The number of unknown parameters
is

$N_{p}=n+1+8N_{\rm eve},$ (144)

the number of constraints is

$N_{m}=(2n+4)N_{\rm eve},$ (145)

and therefore, the number of undetermined parameters is given by

$N_{p}-N_{m}=n+1-2(n-2)N_{\rm eve}.$ (146)

For $N_{\rm eve}=1$ and $N_{\rm eve}=2$ this equation reduces to Eqs. (140)
and (143), respectively. What is the optimal number of events $N_{\rm eve}$
for the polynomial method? The answer can be readily obtained from Eq. (146),
where $N_{\rm eve}$ enters the last term on the right-hand side. If this term
is negative, increasing $N_{\rm eve}$ would decrease the number of
undetermined parameters, and therefore it would be beneficial to combine
information from more and more different events. From Eq. (146) we see that
this would be the case if the decay chain is sufficiently long, i.e., $n\geq
3$. On the other hand, when $n=1$, considering more than one event at a time
is actually detrimental – we are adding more unknowns than constraints. In the
case of $n=2$, the number of undetermined parameters $N_{p}-N_{m}$ is actually
independent of $N_{\rm eve}$ and one might as well consider the simplest case
of $N_{\rm eve}=1$.

Figure 24: The 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 Figure 23, for various mass determination methods:
$M_{T2}$ method (green, open squares), endpoint method (red, open circles),
polynomial method for $N_{\rm eve}=2$ (blue, $\times$ symbols), or a hybrid
method which is a combination of the latter two methods (magenta, $\otimes$
symbols). 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. Taken from Ref. [219].

Figure 24 summarizes the 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 Figure 23, for various mass determination methods:
$M_{T2}$ method (green, open squares), endpoint method (red, open circles),
polynomial method for $N_{\rm eve}=2$ (blue, $\times$ symbols), or a hybrid
method which is a combination of the latter two methods (magenta, $\otimes$
symbols) [219]. 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. For readers who
are interested in counting the number of undetermined parameters of various
methods, we refer to Ref. [219] for more details including a hybrid method
combining the techniques of the polynomial and endpoint methods [351]. A
similar idea on the mass determination in sequential particle decay chains is
discussed in Ref. [352].

The polynomial method described in this section relies on the presence of a
sufficient number of kinematic constraints, so that the event kinematics
becomes exactly solvable for the components of the invisible momenta. This
typically requires complex event topologies, with several successive decays in
each decay chain. Therefore, the method cannot be applied to simpler event
topologies with fewer kinematics constraints, and new ideas are needed. Some
of those alternative techniques are described in the following subsections.

###  X.2 Focus point method

Figure 25: The left panel shows a plot of 20 extreme curves in the
$(\tilde{m}_{A_{1}},\tilde{m}_{A_{2}})$ plane for fixed
$\tilde{m}_{a_{0}}=m_{a_{0}}=700$ GeV, for the the $t\bar{t}$-like topology of
Figure 9. The right panel shows the fractional density of extreme curves,
i.e., the fraction of events whose extreme curves pass through a given
$10\times 10$ GeV pixel. (Taken from Ref. [267].)

The focus point method for mass measurement proposed in [267, 262] can be
applied to certain event topologies with underconstrained kinematics. The
method relies on the fact that the projection onto the visible space will
result in a relatively large number of events in the vicinity of a
singularity, as illustrated in Figure 16. Turning the argument around, one
could ask, for any given event, what choice of the unknown mass parameters
would place a singularity at that point. This condition delineates a
hypersurface in mass parameter space, and we have one such “extreme” surface
for each event in the data. As shown in [267], the extreme surfaces for many
of the events in the dataset pass close to the true values of the unknown
masses. This leads to a technique for estimating the unknown masses simply as
the “focus-point” of the extreme surfaces in the mass-parameter space. This is
illustrated in Figure 25 for the $t\bar{t}$-like topology of Figure 9($b$),
for $m_{A_{2}}=1000$ GeV, $m_{A_{1}}=800$ GeV and $m_{a_{0}}=700$ GeV. The
left panel in the figure shows the kinematic boundaries of 100 events in the
trial mass parameter space $(\tilde{m}_{A_{1}},\tilde{m}_{A_{2}})$, with
$\tilde{m}_{a_{0}}$ fixed to its true value $m_{a_{0}}=700$ GeV. Notice that
the kinematic boundaries tend to focus at the true values of the masses of the
parent particles $A_{2}$ and $A_{1}$ in this example. The right panel in
Figure 25 shows a heatmap of the density of extreme surfaces (curves in this
case) per $10\times 10$ GeV bin. The bright spot in the figure clearly marks
the true values of the masses. This technique can be readily generalized from
$t\bar{t}$ events to more general event topologies in SUSY and beyond [267].

###  X.3 Kinematic endpoint methods

As we have discussed in preceding sections, the distributions of kinematic
variables often allow us to infer the mass values of the (new) particles
involved in the physics process of interest. For example, the endpoints of
invariant mass, transverse mass, $M_{T2}$, and $M_{2}$ variables and energy
peaks are determined purely by kinematics, regardless of detailed dynamics
(see also Section VII). Therefore, those observables have received great
attention and a host of ideas have been proposed especially in the context of
mass measurements of new particles because they do not require any prior
knowledge about the exact details of the associated new physics.

The classic supersymmetric “$q\ell\ell$-chain” introduced in Section VII.1 is
one of the most extensively studied benchmark processes in this regard [27,
202, 203, 204, 205, 206, 207, 208, 23, 209]. There are up to four unknown mass
parameters, namely, $m_{\tilde{q}}$, $m_{\tilde{\chi}_{2}^{0}}$,
$m_{\tilde{\ell}}$, and $m_{\tilde{\chi}_{1}^{0}}$, and four kinematic
endpoints are readily available, e.g., $M_{q\ell_{n}}^{\max}$,
$M_{q\ell_{f}}^{\max}$, $M_{\ell\ell}^{\max}$, and $M_{q\ell\ell}^{\max}$. To
avoid the combinatorial ambiguity between $\ell_{n}$ and $\ell_{f}$, one uses
instead the set $M_{q\ell}^{<,\max}$, $M_{q\ell}^{>,\max}$,
$M_{\ell\ell}^{\max}$, and $M_{q\ell\ell}^{\max}$ (see Section VII.1).
Therefore, in principle, all the unknown mass parameters can be completely
determined by “inverting” the four endpoint measurements. However, this mass
determination can be sometimes ambiguous or underconstrained, and several
degenerate solutions for the masses may arise [203, 353, 354, 355, 206].

There are several reasons for these challenges in the mass determination.
First of all, depending on the underlying mass spectrum, the four observables
are not completely independent; in certain regions of parameter space, the
following relation holds [203]:

$\left(M_{q\ell\ell}^{\max}\right)^{2}=\left(M_{q\ell}^{>,\max}\right)^{2}+\left(M_{\ell\ell}^{\max}\right)^{2}\,.$
(147)

In this case, another independent measurement is needed, and one such example
is the lower kinematic endpoint $M_{q\ell\ell,{\rm const.}}^{\min}$ [27]. Here
the constrained variable $M_{q\ell\ell,{\rm const.}}$ is the usual
$M_{q\ell\ell}$ subject to the condition
$M_{\ell\ell}^{\max}/\sqrt{2}<M_{\ell\ell}<M_{\ell\ell}^{\max}$ which forces
to choose events where the opening angle between the two leptons is greater
than $\pi/2$ in the rest frame of $\tilde{\ell}$. Second of all, the finite
detector resolution smears each of the measured endpoint values away from the
theoretically predicted ones, resulting in multiple and/or unphysical
solutions [203, 204, 354]. Similarly, if an endpoint is identified as the
(long) tail of the associated distribution, its measurement is highly
sensitive to the data statistics and “false” solutions can emerge. However,
even in the ideal case of a perfect experiment, it is possible that different
mass spectra can result in the same set of endpoints so that an application of
relevant inversion formulas (see Ref. [207] for the full sets of formulas)
would yield “fake” solutions [207]. A possible way of resolving this ambiguity
is to study the shapes of the boundaries of the bivariate distributions in
$\left\\{M_{q\ell}^{<},M_{q\ell}^{>}\right\\}$ and
$\left\\{M_{\ell\ell},M_{q\ell\ell}\right\\}$ [207].

The above discussion requires one sufficiently long decay chain to determine
the full mass spectrum. However, for the event topologies involving pair-
produced heavy resonances, the mass determination can often be done in
combination with transverse variables such as $M_{T2}$. A well-studied
prototypical example is the fully-leptonic SM $t\bar{t}$ production. The
$b\ell$ invariant mass endpoint encodes the mass relation

$M_{b\ell}^{\max}=\frac{1}{m_{W}}\sqrt{(m_{t}^{2}-m_{W}^{2})(m_{W}^{2}-m_{\nu}^{2})}\,,$
(148)

and the two combinatorics-free subsystems, leptonic and bottom respectively,
when applied to $M_{T2}$ or $M_{T2,\perp}$, allow one to extract two
independent mass relations $\mu^{(\ell\ell)}$ and $\mu^{(bb)}$, respectively
(see also Section VII.2):

$\displaystyle\mu^{(\ell\ell)}$ $\displaystyle=$
$\displaystyle\frac{m_{W}}{2}\left(1-\frac{m_{\nu}^{2}}{m_{W}^{2}}\right)\,,$
(149) $\displaystyle\mu^{(bb)}$ $\displaystyle=$
$\displaystyle\frac{m_{t}}{2}\left(1-\frac{m_{W}^{2}}{m_{t}^{2}}\right)\,.$
(150)

The CMS Collaboration performed the top quark mass measurement using this idea
at $\sqrt{s}=7$ TeV [236, 356] and $\sqrt{s}=8$ TeV [237], and achieved a
$\sim 2$ GeV-level and a $\lesssim 1$ GeV-level systematics in the respective
measurements by constraining the $W$ and $\nu$ masses from other independent
measurements.

In a similar fashion, the three mass parameters appearing in the two-step two-
body cascade decay topology shown in Figure 7($a$) can be determined in
combination with the peak values in the $a_{2}$ and $a_{1}$ energy
distributions, if both of $A_{2}$ and $A_{1}$ are either scalar or
unpolarized. Again, the kinematic endpoint in the $M_{a_{2}a_{1}}$
distribution is

$M_{a_{2}a_{1}}^{\max}=\frac{1}{m_{A_{1}}}\sqrt{(m_{A_{2}}^{2}-m_{A_{1}}^{2})(m_{A_{1}}^{2}-m_{a_{0}}^{2})}\,,$
(151)

and the two energy-peak values are given by

$\displaystyle E_{a_{2}}^{\rm peak}$ $\displaystyle=$
$\displaystyle\frac{m_{A_{2}}^{2}-m_{A_{1}}^{2}}{2m_{A_{2}}}\,,$ (152)
$\displaystyle E_{a_{1}}^{\rm peak}$ $\displaystyle=$
$\displaystyle\frac{m_{A_{1}}^{2}-m_{a_{0}}^{2}}{2m_{A_{1}}}\,.$ (153)

The last three mass relations are completely independent, allowing one to
determine the three mass parameters. The generic idea was first proposed in
Ref. [272], and applied to a supersymmetric gluino decay process,
$\tilde{g}\to b\tilde{b},\tilde{b}\to b\tilde{\chi}_{1}^{0}$, with
combinatorial ambiguity in the $b$-jet energy distribution appropriately
prescribed.

###  X.4 Matrix element and likelihood methods

The Matrix Element Method (MEM) is one of the likelihood methods which
utilizes the quantum amplitude of a process. The probability to observe
visible particles $\\{P_{i}^{\rm vis}\\}$, $i=1,\cdots,N_{\rm vis}$ under the
assumed process and parameters $\\{\alpha\\}$ for a single event is given by

$\displaystyle{\cal P}(\\{P_{i}^{\rm
vis}\\}|\alpha)=\frac{1}{\sigma_{\alpha}}\biggr{[}\prod_{i=1}^{N_{\rm
vis}}\int\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}\biggr{]}W(\\{P_{i}^{\rm
vis}\\},\\{p_{i}\\})$ $\displaystyle\times\biggr{[}\prod_{j=1}^{N_{\rm
inv}}\int\frac{d^{3}q_{j}}{(2\pi)^{3}2E_{j}}\biggr{]}\sum_{a,b}\frac{f_{a}(x_{1})f_{b}(x_{2})}{2sx_{1}x_{2}}|{\cal
M}_{\alpha}(\\{p_{i}\\},\\{q_{j}\\})|^{2}$
$\displaystyle\times(2\pi)^{4}\delta^{4}\left[p_{a}+p_{b}-\left(\sum_{i}p_{i}+\sum_{j}q_{j}^{\rm
inv}\right)\right],$ (154)

where $\\{p_{a},p_{b}\\}$ are the four-momenta of the initial-state partons
$a$ and $b$ and $\\{f_{a},f_{b}\\}$ are their corresponding PDFs. Here one
integrates out the unknown momenta $\\{q_{j}\\}$ of invisible particles
$j=1,\cdots,N_{\rm inv}$ and considers various non-partonic effects including
detector response and QCD activity with a transfer function $W(\\{P_{i}^{\rm
vis}\\},\\{p_{i}\\})$ between the parton-level momentum $\\{p_{i}\\}$ and the
reconstructed momentum $\\{P_{i}^{\rm vis}\\}$. Integrating the transfer
functions often takes up most of the computing resources for a MEM analys. In
the case where the visible particles consist of only light leptons (electron
and muon), we can neglect the transfer functions, which leads to an important
simplification for purely leptonic channels like Higgs to four leptons [357,
358, 359]. We can construct the likelihood specific to $\\{\alpha\\}$,
$\mathcal{L}_{\alpha}$, for a set of $N$ events with individual likelihoods
for each event $n$ as

$\mathcal{L}_{\alpha}\equiv\prod_{n}^{N}{\cal P}(\\{P_{i}^{\rm
vis}\\}_{n}|\alpha).$ (155)

Thus one can expect to find the model parameters $\\{\alpha\\}$ by maximizing
$\mathcal{L}_{\alpha}$. By constructing a likelihood function, one can measure
not only particle properties such as the mass and width of new particles but
also coupling structures in the interaction vertices as demonstrated in [360].

###  X.5 Edge detection

The identification of kinematic endpoints in a certain one-dimensional
distribution is often indicative of boundaries in the high-dimensional phase
space that is being projected onto the one-dimensional subspace (see Figure
16). At the same time, such kinematic endpoints can also be formed by signal
events piling up on top of a smooth background distribution, and thus can be
used for discovery [361].

The traditional endpoint techniques (usually in the context of supersymmetry,
see Ref. [209] for a recent review) were typically applied to one-dimensional
histograms, but that does not necessarily have to be the case — while
dimensional reduction increases the statistics near the kinematic endpoint, it
may also result in the loss of useful information. This is why recent work
advocated for edge detection in a two-dimensional space of observables [362,
363, 206, 207] and even for a (surface) boundary detection in three or more
dimensions [264, 364, 365].

The traditional approach to edge detection is to bin the data in a lower-
dimensional observable space and identify a kinematic edge by comparing the
counts in adjacent bins, looking for a significant variation [232, 366].
Conventional edge-detection algorithms for machine vision have been developed
mostly for two-dimensional image data and are not necessarily aligned with the
goals of particle physics analyses, which need to account for smearing of the
edges due to detector resolution, particle widths, etc.

When it comes to detecting kinematic features in the data, an alternative
approach to binning is offered by the tessellation of the data, where we can
treat the full set of collider events as a point pattern in the observable
multi-dimensional kinematic space. There exist different tessellation methods.
For example, Ref. [361] proposed a phase-space edge detection method based on
the Voronoi tessellation, which divides the original space into non-
overlapping regions (Voronoi cells) so that the points within each region are
closest to one of the original data points [367]. It was shown that the value
of the scaled standard deviation,

$\frac{\sigma_{a}}{\bar{a}}=\frac{1}{\bar{a}}\sqrt{\sum_{n\in
N_{i}}\frac{(a_{n}-\bar{a})^{2}}{|N_{i}|-1}},$ (156)

where $N_{i}$ is the set of neighbors of the $i$-th Voronoi cell and
$\bar{a}(N_{i})$ is their mean area, is indicative of whether the $i$-th cell
is close to a boundary. This result can be easily understood intuitively by
noting that for boundary cells, the neighbors on the dense side have small
areas while the neighbors on the sparse side have large areas. Therefore, edge
cells are expected to show relatively large scaled standard deviation. The
method was subsequently tested on the classic supersymmetric
“$q\ell\ell"chain$, $\tilde{q}\to\tilde{\chi}_{2}^{0}j$,
$\tilde{\chi}_{2}^{0}\to\tilde{\ell}^{\pm}\ell^{\mp}$,
$\tilde{\ell}^{\pm}\to\tilde{\chi}_{1}^{0}\ell^{\pm}$: Reference [361]
considered edge detection in the two-dimensional space
$\left\\{m_{\ell\ell}^{2},(m_{j\ell\ell}^{2}-m_{\ell\ell}^{2})/6\right\\}$,
while Ref. [368] demonstrated a surface boundary detection in the three-
dimensional space
$\left\\{m_{j\ell_{n}}^{2},m_{\ell\ell}^{2},m_{j\ell_{f}}^{2}\right\\}$. The
method can also be adapted to mass measurements of new particles [364] and for
enhancing the discovery opportunities in combination with the $\Delta_{4}$
variable [265].

The Delaunay triangulation is a tessellation that is the dual graph of a
Voronoi tessellation. Therefore, Ref. [369] proposed an alternative edge
detection method which utilizes the Delaunay tessellation of the data instead.
Since edge detection necessarily involves computing the gradient of the phase
space density, the Delaunay cells, being formed by several neighboring data
points, are the natural objects for computing local gradients.

###  X.6 Interference effects

When high precision is required in measurements of masses or other kinematic
properties, subtle quantum effects can modify the theoretical predictions in
ways that can be hard to predict or to interpret without a full theoretical
understanding of kinematic variables.

Even relatively simple observables like the invariant mass can be affected by
subtle effects which may give apparently inconsistent results between
measurements of the mass of a particle in two different channels. This is the
case for instance for the Higgs boson whose most precise mass measurements are
in the $\gamma\gamma$ and $4\ell$ channels. The key difference between these
two channels is that the $4\ell$ channel is essentially free from background,
whereas the $\gamma\gamma$ channel has substantial background from QED+QCD
production of two photons. The presence of large background opens the
possibility to have non-resonant features to redefine the expected signal
shape even for perfect detectors. The studies in Refs. [370, 371, 372, 373,
374, 375] have pointed out that the $\gamma\gamma$ peak in the
$M_{\gamma\gamma}$ distribution is expected to be shifted and broadened by the
interference between $gg\to\gamma\gamma$ and $gg\to h\to\gamma\gamma$.
Therefore, the mass measured as the peak of the $M_{\gamma\gamma}$
distribution will differ from that measured in background-free channels such
as $4\ell$. The subtraction of the peak from $4\ell$ events into the
extraction of the peak in the $M_{\gamma\gamma}$ distribution may help to
highlight this effects in a model-independent way. As the effect has to do
with the Higgs boson width, it has also been pointed out that the mass shift
can be used to constrain the Higgs boson width [376, 377].

The shift also depends on experimental conditions such as the the diphoton
mass resolution. Depending on how the measurement in the $\gamma\gamma$
channel is performed, the shift may range from a fraction of 100 MeV to a
fraction of 1 GeV, which would be significantly large to be observed [371].
The ATLAS Collaboration has evaluated the impact of this theoretical effect on
a realistic mass measurement and has found a non-vanishing effect around 35
MeV for the specific procedure used to extract the Higgs boson mass in the
$\gamma\gamma$ channel [378].

The present Higgs boson mass appearing in the up-to-date Particle Data Group
report is $125.25\pm 0.17$ [379], stemming from slightly disagreeing
measurements from CMS [380] $m_{h}^{\gamma\gamma}=125.78\pm 0.26$ GeV,
$m_{h}^{4\ell}=125.46\pm 0.16$ GeV and ATLAS [381] $m_{h}^{4\ell}=124.79\pm
0.37$ GeV, $m_{h}^{\gamma\gamma}=124.93\pm 0.40$ GeV. Future HL-LHC
measurements, exploiting a dataset about two orders of magnitude larger, could
reach a measurement with a much smaller uncertainty than the present one.
Systematic uncertainties will be relevant for such a large dataset, but a
purely statistical rescaling of the present measurement would hint at the
necessity to take into account these subtle effects from interference. For
example, in Ref. [382] a precision on $m_{h}$ in the $\gamma\gamma$ channel
around 10-20 MeV is foreseen [383, 384], thus calling for a careful evaluation
of the interference effects.

##  XI Kinematic variables in the machine learning era

Figure 26:  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.

Recently, there has been an explosion of studies employing ML for various
tasks in the analysis of high energy data. A collection of references is
maintained at HEPML-LivingReview [385]. So far we have discussed traditional
(i.e., non-ML) analysis techniques using kinematic variables [Figure 26($a$)].
Here we will discuss the multifaceted synergy between ML and kinematic
variables in particle physics.

###  XI.1 Feature engineering

Feature Engineering for ML. Kinematic variables are often used as input event-
features in ML approaches. While in principle, one can train machines using
only raw or low-level data as inputs [Figure 26($b$)], the dimensionality of
such feature vectors will typically be large. When we feed low-level data, the
machine could spend a lot of resources trying to extract useful information
from it. Furthermore, inaccuracies in the simulation models could lead to
unknown and unquantified errors in the high-dimensional low-level simulated
data, which could lead to unquantified errors in the subsequent ML-based
analysis. Both these issues can be ameliorated by using reconstructed objects
and/or human-engineered high-level variables as inputs in ML applications
[Figure 26($c$)]. Optionally, one can also pass the low-level information to
the machine, in addition to the high-level features. Carefully chosen high-
level input features can efficiently retain the information from low-level
data that is relevant to the task at hand, and facilitate efficient training
of ML approaches. Furthermore, reducing the dimensionality of the input allows
for easier and more meaningful validation of the simulation models (in the
low-dimensional input space), for the purposes of the analysis at hand.

Feature Engineering with ML. An interesting development in the last few years
is the construction of sensitive analysis variables using machine learning
techniques [Figure 26($d$)]. Any ML-approach, in which a machine (e.g., neural
network, boosted decision tree) takes individual events as input and returns
an output, can be thought of as constructing an analysis variable or
observable. However discussing all such applications of machine learning,
including classifiers [386], likelihood ratio estimators [387], etc. is beyond
the scope of this paper. Instead, we will focus on ML-approaches for
constructing event observables, which are functionally similar to more
traditional event observables.

At an abstract level, many ML-approaches for constructing collider observables
share the following basic procedure. 1) Construct a trainable, machine-
learning-based function which maps a high-dimensional event description to a
low-dimensional observable. 2) Construct an evaluation metric to quantify the
performance of the ML-based observable, for some task at hand. 3) Train the
ML-based observable by optimizing the evaluation metric. Several novel ML-
based observables have been proposed in recent years, each differing in the
implementation details of the steps listed above [388, 31].

For example, in Ref. [388], jet observables are constructed as products of
powers of $N$-subjettiness variables with unknown (trainable) exponents. The
performance of the observable for distinguishing between different signatures
($H\rightarrow bb$ vs. $g\rightarrow bb$; $Z^{\prime}$ vs. quarks and gluons)
is used as the performance metric to be minimized in order to choose the
exponents in the observable.

In Ref. [31], kinematic variables for different event topologies are
constructed as neural-network-based functions of reconstructed parton-level
data. The variables are trained by maximizing the sensitivity of their
distributions to the value of underlying parameters in the corresponding event
topologies, as captured by the mutual information between the parameters and
the variable. Variables trained using such an approach are sensitive over a
range of unknown theory parameter values, and can subsequently be used for
signal discovery or parameter measurement analyses involving the concerned
topology.

###  XI.2 Domain-inspired machine learning

Several HEP-inspired neural network architectures have been invented for use
in ML for HEP. For example, Energy Flow Networks and Particle Flow Networks
[389], based on Deep Sets [390] and Energy Flow Polynomials [391], are neural
network architectures designed for learning from collider events represented
as unordered, variable-length sets of particles. As another example, Lorentz
Boost Networks (LBN) [392, 393, 394] allow for the construction of composite
particles and rest frames (both represented by combinations of particles)
within the trainable layers of the network, using four-momenta of final state
particles as input. The LBNs also Lorentz boost the composite particles into
the constructed rest frames. The features thus constructed within the layers
of the network can then be used to perform relevant physics tasks like
classification or regression. Such domain-specific network architectures have
been observed to outperform other domain-unspecific neural network
architectures for collider event classification tasks [392]. Furthermore, they
also allow for interpretation of the intermediate layers of the trained neural
network.

###  XI.3 Interpretability and explainability

Most machine learning approaches, including neural networks and boosted
decision trees, act as blackbox systems with varying degrees of
“blackboxness”, depending on their architecture. This poses a challenge to the
trustworthiness of ML-based analyses. This problem can be approached from
three directions. The first approach is to interpret and explain the ML
blackbox. The second approach is to try and make the machine less of a black
box. The third approach is to design ML-based analyses techniques that are
robust despite the blackbox nature of the machine. Kinematic variables can
play a role in each of these approaches. For example, one can use kinematic
variables to interpret and explain the decisions made by the machine learning
algorithms [395, 396, 397, 398]. Using kinematic-variable-inspired neural
architectures like energy flow and particle flow networks [389] and lorentz
boost networks [392, 393, 394] can reduce the black-box nature of neural
networks. Finally, using machine learning to construct _low-dimensional event_
observables that are not tuned to specific study points (i.e., are sensitive
to the underlying physics over a range of unknown model parameters) [31], akin
to kinematic variables, allow for meaningful control-region validation of the
simulations which in turn leads to robust analysis techniques.

###  XI.4 Advantage of quantum computation for identifying event-topologies

So far, most kinematic analyses are based on an assumed event topology which
makes it possible to optimize a kinematic variable in each case. Due to the
complicated structure of phase space and the limited information from
invisible particles, various machine learning algorithms become useful, but
their training data is also generated only for particular event topologies.

Table 4:  The number of inequivalent event-topologies as a function of $1\leq N_{v}\leq 4$ and $1\leq N_{\chi}\leq 5$. Taken from Ref. [199]. | $N_{\chi}$
---|---
$N_{v}$ | 1 | 2 | 3 | 4 | 5
1 | 1 | 2 | 4 | 8 | 16
2 | 2 | 7 | 20 | 55 | 142
3 | 4 | 20 | 78 | 270 | 860
4 | 8 | 55 | 270 | 1138 | 4294

Given the lack of any clear signal of new physics so far at the LHC, we need
to ask how one can perform an optimized analysis without any assumptions on
the new physics model. This question is related to identifying the event-
topology of the signal from data alone. Checking all possible event-topologies
can be a very time-consuming task. For example, the number of possible event-
topologies with $N_{v}=5$ visible particles and $1\leq N_{\chi}\leq 4$
“assumed” invisible particles is $\mathcal{O}(5000)$ as shown in Table 4
[199]. As the number of visible and invisible particles increases, the number
of possible event-topologies grows exponentially, making the problem of
identifying the event-topology an NP-hard problem. Since it is in the category
of combinatorial optimization problems, where various quantum algorithms have
been introduced [399, 400] and shown to be successful, this suggests the use
of quantum computers for this task.

When the produced particles are boosted, their decay products are organized
into groups exhibiting characteristic structures and substructures. One can
then utilize a shape variable used in clustering a jet with either a gate-type
quantum computer or a quantum annealer [401, 402, 403, 404]. Unlike QCD jet
activity, the phase space of a hard process can exhibit a more complicated
structure, in which case one can try to minimize some basic kinematic
quantity, for example, the total invariant mass or sum/difference of invariant
masses of the clusters. If one restricts to a $2\to 2$ process, one can
directly use a Quadratic Unconstrained Binary Optimization (QUBO) with an
Ising model to cluster visible particles as in Figure 27 [405]. Once we
identify the event-topology, we can proceed to optimize the analysis to
measure the masses and spins of the new particles as usual.

Figure 27: (a) $n$-observed particles, (b) dividing $n$ particles into two
groups for a $2\to 2$ process, and (c) identified event-topology with $A$ and
$B$. Taken from Ref. [405].

##  XII Kinematic variables in different experiments

While most of the recent developments in kinematic variables have been
motivated by the phenomenology in collider experiments, especially hadron
colliders, they are readily applicable to other experiments including
accelerator-based experiments [e.g, fixed target or beam-dump type (neutrino)
experiments], reactor-based (neutrino) experiments, dark-matter (in)direct
detection experiments, and cosmic-particle telescopes. Here we briefly review
existing usage of kinematic variables in non-collider experiments, and discuss
future prospects of applications of kinematic variables to upcoming
experiments.

One of the crucial differences of non-collider experiments from collider-based
experiments is that the transverse plane for a given event is usually ill-
defined. In typical collider experiments, the initial-state particles have
their momenta aligned with the beam axis, so that the transverse plane is
literally transverse to the beam direction and most of the transverse
variables are defined with respect to this plane. By contrast, this is not
always the case for non-collider experiments. For example, in the beam-focused
neutrino experiments, source particles of neutrinos (e.g., charged pions,
kaons, and muons) are focused and aligned to the particle beam axis by the
magnets in the horn system and then decay to neutrinos. Although the source
particles are highly boosted in the forward direction, the neutrinos come with
a non-zero angular spread with respect to the beam axis and as a consequence,
we do not know the “neutrino beam” direction event-by-event. Likewise, in
dark-matter or cosmic-ray detection experiments, the incoming direction of
dark matter or cosmic particles is not known a priori. In these experiments,
therefore, any variables defined on the beam-transverse plane do not allow for
robust physical interpretations unless they are re-defined with appropriate
prescriptions. Instead, basic quantities (e.g., energy, timing, etc),
invariant quantities (e.g., invariant mass), or their combinations are more
straightforwardly applicable.

Energy: Energy is one of the most widely used variables in particle physics
experiments, as it is a basic physics quantity to measure at detectors. A few
example uses follow.

  1. (a)

In conventional dark-matter direct detection experiments targeting WIMP dark-
matter candidates, the shape of the nuclear recoil energy spectrum carries
information about the dark-matter properties. One can estimate the mass scale
of dark matter [406] or test whether the observed events are caused by
inelastic dark matter [407].

  2. (b)

In stopped-pion neutrino experiments (e.g., COHERENT [408, 409, 410] and CCM
[411]), the energy is used to eliminate the pion-induced muon neutrino events
in the search for low-mass dark matter. Since the beam energy of these
experiments is small, $\pi^{+}$’s produced in the beam target material lose
their kinetic energy and stop before decaying to a $\mu^{+}$ and a
$\nu_{\mu}$. The energy of $\nu_{\mu}$ is single-valued at $\sim 30$ MeV so
that the recoil energy of $\nu_{\mu}$ scattering events is bounded from above.
By contrast, vector-portal dark matter coming from the $\pi^{0}$ decay through
a dark photon is typically more energetic hence deposits more energy in the
detector. An energy cut rejects $\nu_{\mu}$-induced events, leading to a dark-
matter signal-rich region [412, 413, 414, 411, 415].

  3. (c)

The energy peak is useful in the energy distribution of cosmic photons from
neutral pion decays. In a cosmic shower, $\pi^{0}$’s are produced with various
boost factors. Since $\pi^{0}$ is a scalar and its decay is a two-body
process, the peak position is identified as half the $\pi^{0}$ mass [268] as
also discussed in Section VIII.1 in a more general context.

Timing: Timing is also a readily accessible quantity in many of the
aforementioned experiments. As discussed in Section VIII.2, it is important to
set the reference time (i.e., $t_{0}$) in order to render the timing values
meaningful. This is deeply connected to the event triggering. In fixed target
type experiments, the beam-on time is often set to be $t_{0}$. Three example
applications are given below.

  1. (a)

In the stopped pion neutrino experiments mentioned earlier whose proton beam
energy is $\sim 1$ GeV, beam-related neutrons would give rise to an enormous
amount of background. However, due to the scale of the beam energy, most of
the produced neutrons are slowly moving so that their arrival timing at the
detector is rather delayed, compared to the pion-induced neutrino events.
Therefore, restricting to the prompt region, one can reject beam-related
neutrons very efficiently (see, e.g., Refs. [411, 415]).

  2. (b)

In a similar manner, neutrinos from muon decays can be vetoed using a timing
cut in the stopped pion neutrino experiments. Since a muon is much longer-
lived than a charged pion, the muon-induced neutrinos typically arrive at the
detector much later than a pion-induced neutrino. In other words, the muon-
induced neutrino events usually fall in delayed timing bins. The low-mass dark
matter mentioned before is also prompt as it comes from the rare $\pi^{0}$
decay, and therefore, a timing cut can reduce significantly not only beam-
related neutron backgrounds but also muon-induced “delayed” neutrino events,
while keeping as many dark-matter events as possible [413, 414].

  3. (c)

An inverted timing cut can be utilized in the beam-focused neutrino
experiments. The MiniBooNE Collaboration sets the limit for WIMP dark-matter
candidates, based on the fact that no significant number of events are
observed in their delayed timing bins [416, 417]. If a WIMP candidate with a
sub-GeV or greater mass were produced in the MiniBooNE target, it would travel
rather slowly because the beam energy is as small as $\sim 8$ GeV.

Angle or directionality: The angle variable is useful if the source point is
well known, the momenta of visible particles can be measured, and the angular
resolution is good enough. In typical accelerator-based experiments, the angle
is usually defined with respect to the beam axis, visible particles are
energetic enough to measure their three-momentum, and detectors (e.g.,
calorimeters, liquid argon time projection chamber detectors) are capable of
measuring angles precisely.

An angle can also be defined in the search for cosmogenic signals. For
example, a class of boosted dark matter models built upon a non-minimal dark-
sector framework [418] predict that a certain dark-matter component can be
produced with a significant boost factor in the present universe by the pair-
annihilation of the halo dark-matter component [419]. Therefore, visible
particles (e.g., recoil electrons) induced by the scattering of boosted dark
matter can be not only energetic enough to allow for the measurement of their
three-momenta but forward-directed enough to be aligned with the momentum of
the incoming boosted dark matter. The dominant fraction of the flux of this
so-called boosted dark matter comes from the regions where halo dark matter is
densely populated, e.g., galactic center [419, 420, 234], dwarf galaxies
[421], and the sun [422, 423, 420] in the solar-captured boosted dark matter
scenarios. Therefore, the line extended between these source points and the
detector location can be taken as a reference axis. If a detector features a
good angular resolution, one may define a signal-rich region by selecting the
events in which the angles of visible particles with respect to the reference
axis are within a certain range. The Super-Kamiokande Collaboration used angle
cuts to perform a search for galactic boosted dark matter and solar-captured
boosted dark matter that is interacting with electrons, and set the first
limits for models of two-component boosted dark matter [424].

Complex variables: Beyond the basic quantities that have been discussed thus
far, complex kinematic variables constructed with basic quantities are being
used in a wide range of non-collider experiments as they are equipped with
high-capability detectors with good angular, spatial, and/or energy-momentum
resolutions. A couple of examples are given below.

  1. (a)

A wide range of dark-sector scenarios predict the production of low-mass dark
matter (say, $\chi$) in various beam-induced neutrino experiments. An
extensively investigated detection channel is the elastic scattering of dark
matter off an electron inside the detector material:
$\chi+e^{-}\to\chi+e^{-}$. One of the major backgrounds to this signal is the
charged-current quasi-elastic (CCQE) scattering of electron neutrinos, i.e.,
$\nu_{e}+n\to e^{-}+p$ or $\bar{\nu}_{e}+p\to e^{+}+n$, where the final-state
nucleon is not energetic enough to be detected. It was demonstrated that this
type of signal-faking events can be significantly rejected by an application
of the $E_{e}\theta_{e}^{2}$ cut with $E_{e}$ and $\theta_{e}$ being the
energy and the angle of the final-state electron. For example, a
$E_{e}\theta_{e}^{2}<5$ MeV$\cdot$rad2 cut can reduce about 99% of CCQE events
at the NO$\nu$A near detector [425] and a $E_{e}\theta_{e}^{2}<2$
MeV$\cdot$rad2 cut can suppress the CCQE events by $\sim 99.9\%$ at the DUNE
near detector [426].

  2. (b)

A class of well-motivated new physics models that can be tested at neutrino
experiments predict upscattering of the incoming particles: for example,
upscattering of a SM neutrino to a heavier sterile neutrino (say, $N_{R}$)
through mixing [427] and upscattering of dark matter to a heavier dark-sector
state (say, $\chi^{\prime}$) [407, 428, 429]. These upscattered states may
decay into a set of visible particles in addition to the recoiling particle
emerging from the primary scattering of the incoming neutrino or dark matter.
Thus there can be multiple visible particles in the final state, allowing for
constructing complex variables such as invariant masses [429].

##  XIII Conclusions and outlook

In general, the outcome of any particle physics experiment (whether studying
scattering or decay processes) is a measured probability distribution in the
relevant phase space of the final state. Unfortunately, in typical situations,
the phase space is high-dimensional, and the observed features are difficult
to visualize. Furthermore, for many interesting signals, some information,
e.g., related to the kinematics of invisible particles like neutrinos or dark-
matter candidates, may be missing. In that case it makes sense to perform
dimensional reduction to the lower-dimensional observable slice of the phase
space. In doing so, a major goal is to use the proper kinematic variables
which retain as much information as possible about the underlying physics,
features, etc.

The higher-level variables which are derived from the measured particle
kinematic information are generically referred to as “kinematic variables”.
Depending on the type of experimental signature and/or the goal of the
analysis, many different kinematic variables have been introduced and
discussed over the years in the particle phenomenology literature. The main
purpose of this review was to collect and summarize all those recent
developments in one place. We also provided the motivation for introducing
each variable, its applicability and limitations, together with a guide to the
relevant references.

Many of the traditional questions and approaches in particle kinematics are
now being reevaluated using machine learning. The ability of ML to better
capture the high-dimensional correlations in the data may lead to superior
performance, at the expense of introducing unphysical hyperparameters and
perhaps less transparency and interpretability. At the same time, kinematic
variables can be incorporated into the ML approaches, thus boosting their
performance and interpretability. The general methods which have guided
particle phenomenologists in deriving these kinematic variables can be used in
other fields of science and are therefore of interest outside the domain of
particle physics.

##  Appendix A Tools and codes for kinematic variables

For the benefit of the users of kinematic variables, in Table 5 we list a few
popular public codes for numerically computing some of the kinematic variables
described in the main text. Note that the name of each code in the table is
hyperlinked to the respective web-page or repository. We also provide the
corresponding reference, language, and system requirements.

Code | Language/Requirements | Kinematic variables
---|---|---
$M_{T2}$ [242] | Python3 | $M_{T2}$
Oxbridge Kinetics Library [430] | Python, C++, Root |  $\alpha_{T}$, $M_{T2}$, $M_{TGen}$ $M_{2C}$, $M_{C}$, $M_{CT}$, $M_{CT2}$
OPTIMASS [239] | C++, Python2.7, Root |  $M_{2}$ and friends
YAM2 [240] | C++ | $M_{2}$
KLFitter [431] | C++, Root | Top quark reconstruction
RestFrames [24] | C++, Root | Recursive Jigsaw Reconstruction
MoMEMta [432] | C++, Root | Modular Matrix Element Method implementation.
Root [433, 434] | Root | Basic kinematic variables and $M_{T}$
WIMPMASS [350, 348, 241, 347] | C++, Root |  $M_{T2}$ using bisection method
CLHEP [435] | C++ | Basic kinematic variables and $M_{T}$
FASTJET [436] | C++ | Basic kinematic variables and $E_{T}$
Table 5:  A summary of public codes for numerically computing kinematic
variables.

## Acknowledgements

This review was written at the request of the Theory Frontier Collider
Phenomenology group (TF07) of the Snowmass 2021-22 Particle Physics Community
Planning Exercise [437]. We would like to thank Roberto Di Nardo, Lance Dixon,
Roy Forestano, Stephen Martin, Federico Meloni and Eyup Unlu for discussions
and comments on the manuscript. KK, KM, and PS would like to thank the Aspen
Center for Physics for hospitality during the completion of this work,
supported in part by National Science Foundation grant PHY-1607611. DK is
supported by the DOE Grant No. DE-SC0010813. KK is supported in part by DOE
DE-SC0021447. KM is supported in part by DOE DE-SC0022148. MP is supported by
NRF-2021R1A2C4002551. PS is partially supported by the U.S. Department of
Energy, Office of Science, Office of High Energy Physics QuantISED program
under the grants “HEP Machine Learning and Optimization Go Quantum”, Award
Number 0000240323, and “DOE QuantiSED Consortium QCCFP-QMLQCF”, Award Number
DE-SC0019219. This manuscript has been authored by Fermi Research Alliance,
LLC under Contract No. DEAC02-07CH11359 with the U.S. Department of Energy,
Office of Science, Office of High Energy Physics.

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