aainstitutetext: SLAC National Accelerator Laboratory, Stanford University,
Stanford, CA 94309, USAbbinstitutetext: Department of Physics, Harvard
University, Cambridge, MA 02138, USAccinstitutetext: Department of Physics,
University of California, Santa Barbara, CA 93106, USAddinstitutetext: Kavli
Institute for Theoretical Physics, Santa Barbara, CA 93106,
USAeeinstitutetext: Department of Physics and Astronomy, University of
California, Los Angeles, CA 90095, USA
Mani L. Bhaumik Institute for Theoretical Physics, University of California,
Los Angeles, CA 90095, USA

#  The Phase Space Distance Between Collider Events

Tianji Cai b Junyi Cheng c,d Nathaniel Craig c Giacomo Koszegi e and Andrew J.
Larkoski<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

How can one fully harness the power of physics encoded in relativistic
$N$-body phase space? Topologically, phase space is isomorphic to the product
space of a simplex and a hypersphere and can be equipped with explicit
coordinates and a Riemannian metric. This natural structure that scaffolds the
space on which all collider physics events live opens up new directions for
machine learning applications and implementation. Here we present a detailed
construction of the phase space manifold and its differential line element,
identifying particle ordering prescriptions that ensure that the metric
satisfies necessary properties. We apply the phase space metric to several
binary classification tasks, including discrimination of high-multiplicity
resonance decays or boosted hadronic decays of electroweak bosons from QCD
processes, and demonstrate powerful performance on simulated data. Our work
demonstrates the many benefits of promoting phase space from merely a
background on which calculations take place to being geometrically entwined
with a theory’s dynamics.

##  1 Introduction

Novel representations of collider data play an increasingly important role in
the modern machine learning (ML) era Larkoski:2017jix ; Guest:2018yhq ;
Albertsson:2018maf ; Radovic:2018dip ; Carleo:2019ptp ; Bourilkov:2019yoi ;
Schwartz:2021ftp ; Karagiorgi:2021ngt ; Shanahan:2022ifi . In particular, a
physically-motivated notion of the distance between collider events is a
powerful input to ML-based methods for event classification and anomaly
detection. Recently, optimal transport distances between energy flow
distributions Komiske:2019fks ; Komiske_2020 ; PhysRevD.105.076003 ;
Cai:2020vzx  or spectral functions Larkoski:2023qnv  have given rise to a new
geometry of collider events, with widespread applications Komiske:2019jim ;
Cesarotti:2020hwb ; Cesarotti:2020ngq ; Stein:2020rou ; Kasieczka:2021xcg ;
Howard:2021pos ; DiGuglielmo:2021ide ; Kansal:2021cqp ; Collins:2021pld ;
Cheng:2020dal ; CrispimRomao:2020ejk ; ATLAS:2023mny ; Craig:2024rlv .
However, there is another natural distance between collider events that has
yet to be fully explored: the one furnished by the phase space manifold on
which scattering amplitudes are defined.

Traditionally, little attention has been paid to the intrinsic geometry of the
phase space manifold itself, except within the context of the structure of
S-matrix elements (especially at the boundaries of the phase space). The
deeper potential of the phase space manifold was recently highlighted in Ref.
Larkoski:2020thc  (as well as Refs. Cox:2018wce ; Henning:2019mcv ;
Henning:2019enq ), which presented a covariant description of the massless
$N$-body phase space manifold $\Pi_{N}$ and demonstrated that it is isomorphic
to the product space of an $N-1$-dimensional simplex $\Delta_{N-1}$ and a
$2N-3$-dimensional hypersphere $S^{2N-3}$, $\Pi_{N}\cong\Delta_{N-1}\times
S^{2N-3}$. Explicit global coordinates can then be constructed on the phase
space, promoting it to a Riemannian metric space. This offers a new way to
organize collider data that is firmly grounded in the underlying physics and
furnishes a novel representation of collider data for any downstream machine
learning application.

In this paper we develop and demonstrate a practical metric for collider
events using the global phase space coordinates presented in Ref.
Larkoski:2020thc , tailored in part to the structure of events at hadron
colliders. Constructing a true metric from these phase space coordinates poses
a number of challenges, which we address in turn. Given that the dimension of
the phase space manifold depends on the particle multiplicity, exclusive
clustering of events into a fixed number of $N$ “particles” enables the
calculation of distances between different events on a phase space manifold of
uniform dimension $3N-4$.111A common practice in analyzing variable particle
number events in collider physics is to fix the multiplicity to be some very
large number, expected to be larger than that in any given event. Then, events
are zero-padded with momentum four-vectors to ensure that every event has the
same number of particles. Here, we explicitly study the resolved multiplicity
dependence of the phase space metric as a tool for discrimination, but this
more inclusive approach is simpler in practice as it does not require
knowledge of the preferred or natural dimensionality of phase space. Ensuring
that the phase space metric satisfies the identity of indiscernibles requires
minimizing distances over $N!$ permutations of particles for each pair of
events, which is an exponentially computationally intensive task. Instead, we
efficiently approximate this exact minimization by first minimizing distances
on the simplex part of the phase space manifold and then azimuthally rotating
events to minimize distances on the hypersphere part. Together, these steps
ensure that the metric on phase space satisfies the identity of indiscernibles
and defines a physically principled prescription for event pre-processing. We
further tailor the metric to global phase space coordinates relevant to hadron
colliders, including both the center-of-mass frame for full events and the
collinear or infinite momentum frame for individual boosted jets.

Different types of collider events exhibit distinctive structures in phase
space, making the phase space distance between events a potentially powerful
tool for event classification and anomaly detection. We demonstrate this
potential with a number of examples in which the phase space distances among
ensembles of signal and background events are inputs to simple ML algorithms
for classification. Distances on the phase space manifold lead to competitive
classification both at the level of entire collider events and within the
substructure of boosted jets. Taken together, our results suggest that
distances on a suitably-defined phase space manifold provide a practical and
effective representation of collider data with numerous applications.

This paper is organized as follows: In Sec. 2 we review the prescription for
constructing the phase space manifold and explicit global coordinates in the
center-of-mass frame presented in Ref. Larkoski:2020thc  and develop an
analogous prescription in the collinear limit for boosted objects. We then
address the practical aspects of constructing a phase space metric for
collider events in Sec. 3, including, among others, the representation of
different events in terms of a fixed number of particles and the efficient
minimization of distances over particle permutations. We demonstrate the
classification power of the phase space metric coupled to simple ML algorithms
in Sec. 4 with an event-level Standard Model $t\bar{t}$ vs. QCD dijet study,
an event-level beyond-the-Standard Model Hidden Valley isotropic resonance
decay vs. QCD dijet study, and a boosted, hadronically-decaying $W$ boson vs.
QCD jet study. We summarize our conclusions and sketch future directions in
Sec. 5, reserving technical details about the construction of coordinates and
the minimization over particle permutations, as well as the full result
tables, for the three appendices.

##  2 Intrinsic Geometry of the $N$-body Phase Space

In this section, we review the formalism of Ref. Larkoski:2020thc  for
establishing the phase space manifold in the center-of-mass (CM) frame. We
restrict our study to assume that experimentally detected particles are all
massless, which is a good approximation in the high energy collider
environment of the LHC. For our purposes in this paper, we will need more
explicit results for application on realistic collider event data. We will
thus provide all necessary mappings from momentum space coordinates to the
natural coordinates on the simplex-hypersphere formulation of the phase space
and the corresponding Riemannian metric. Additionally, we will construct the
manifold on the phase space of a boosted jet, effectively in the infinite-
momentum frame. Here the coordinate of the jet axis is special, breaking the
event’s O(3) rotational invariance in the CM frame. Further, the jet mass is
not necessarily measured and constrained, which introduces a new coordinate.

###  2.1 Phase Space in the CM Frame: A Product Manifold

We start with the volume form of the $N$-body phase space in terms of the
four-momenta $p_{k}$ of the $N$ massless particles constituting an event in
Cartesian coordinates, where

$\displaystyle
d\Pi_{N}=(2\pi)^{4-3N}\left[\prod_{k=1}^{N}d^{4}p_{k}\,\delta^{+}(p_{k}^{2})\right]\delta^{(4)}\left(Q-\sum_{k=1}^{N}p_{k}\right)\,.$
(1)

Here $Q=(Q,0,0,0)$ is the total energy of the event in the CM frame and
$\delta^{+}(p_{k}^{2})=\delta(p_{k}^{2})\,\Theta(p_{0k})$ enforcing that all
particles are on-shell with positive energy. The on-shell constraints can be
satisfied by re-expressing the four-momenta in terms of the two spinors for
each particle, where

$\displaystyle\lambda_{k}^{\alpha}=\frac{1}{\sqrt{p_{0k}-p_{3k}}}\begin{pmatrix}p_{0k}-p_{3k}\\\
p_{1k}+ip_{2k}\end{pmatrix}\,,\quad\quad\tilde{\lambda}_{k}^{\dot{\alpha}}=\frac{1}{\sqrt{p_{0k}-p_{3k}}}\begin{pmatrix}p_{0k}-p_{3k}\\\
p_{1k}-ip_{2k}\end{pmatrix}\,,$ (2)

with $k=1,\dots,N$ labeling the particle and $\alpha,\dot{\alpha}=1,2$
labeling the spinor components. Note that we have
$\tilde{\lambda}_{k}^{\dot{\alpha}}=\lambda_{k}^{\alpha*}$ because the
momentum matrix $p_{k}\cdot\sigma=\lambda_{k}\tilde{\lambda}_{k}$ is
Hermitian.

We now define two $N$-component vectors

$\displaystyle\vec{u}\equiv\frac{1}{\sqrt{\sum_{j=1}^{N}(p_{0j}-p_{3j})}}(\lambda_{1}^{1}~{}\lambda_{2}^{1}~{}\dots~{}\lambda_{N}^{1})\text{
, with components }u_{k}=\frac{1}{\sqrt{Q}}\sqrt{p_{0k}-p_{3k}}\,,$ (3)
$\displaystyle\vec{v}\equiv\frac{1}{\sqrt{\sum_{j=1}^{N}(p_{0j}+p_{3j})}}(\lambda_{1}^{2}~{}\lambda_{2}^{2}~{}\dots~{}\lambda_{N}^{2})\text{
, with components
}v_{k}=\frac{1}{\sqrt{Q}}\frac{p_{1k}+ip_{2k}}{\sqrt{p_{0k}-p_{3k}}}\,,$ (4)

in the CM frame. Notice that $\vec{u}$ is a real vector while $\vec{v}$ is
complex. Written in terms of these $\vec{u},\vec{v}$ vectors, Eq. (1) now
becomes

$\displaystyle\begin{split}d\Pi_{N}=&(2\pi)^{4-3N}Q^{2N-4}\left[\prod_{k=1}^{N}\frac{d^{2}\lambda_{k}^{1}d^{2}\lambda_{k}^{2}}{\text{U}(1)}\right]\\\
=&(2\pi)^{4-3N}Q^{2N-4}\,\frac{d^{N}u\,d^{N}v}{\text{U}(1)^{N}}\,\delta(1-|\vec{u}|^{2})\,\delta(1-|\vec{v}|^{2})\,\delta^{(2)}(\vec{u}^{\dagger}\vec{v})\,,\end{split}$
(5)

where the division by U(1) represents implicit restriction to one element of
the little group action on the spinors.

By introducing the new coordinate

$\displaystyle\rho_{k}\equiv u_{k}^{2}\,,$ (6)

the $\vec{u}$ part of the volume form reduces to

$\displaystyle\frac{d^{N}u}{\text{U}(1)^{N}}\,\delta(1-|\vec{u}|^{2})=\frac{1}{2^{N}}\prod_{k=1}^{N}[d\rho_{k}]\,\delta\left(1-\sum_{k=1}^{N}\rho_{k}\right)=\frac{1}{2^{N}}\,d\Delta_{N-1}\,,$
(7)

where $d\Delta_{N-1}$ is the flat measure on a $N-1$-dimensional simplex. We
therefore obtain the simplex part of the phase space manifold.

For the terms in Eq. (5) involving $\vec{v}$, we first enforce the
orthogonality condition $\delta^{(2)}(\vec{u}^{\dagger}\vec{v})$ to reduce the
dimension of $\vec{v}$ by 1 via the removal of the last component of
$\vec{v}$, i.e., $v_{N}$. Thus, we have

$\displaystyle
d^{N}v\,\delta(1-|\vec{v}|^{2})\,\delta^{(2)}(\vec{u}^{\dagger}\vec{v})=\frac{d^{N-1}v}{\rho_{N}}\,\delta(1-|\vec{v}|^{2}-|v_{N}|^{2})\,.$
(8)

Note that the $\vec{v}$ on the right hand side of Eq. (8) now has only $N-1$
components, and $v_{N}$ can be expressed in terms of the components of
$\vec{u}$ and $\vec{v}$. We then perform a change of variables from $\vec{v}$
to $\vec{v}^{\prime}$, where $\vec{v}^{\prime}$ is an $N-1$-component complex
unit vector,

$\displaystyle\frac{d^{N-1}v}{\rho_{N}}\,\delta(1-|\vec{v}|^{2}-|v_{N}|^{2})=d^{N-1}v^{\prime}\,\delta(1-|\vec{v}^{\prime}|^{2})=dS^{2N-3}\,,$
(9)

which is manifestly the measure for the sphere $S^{2N-3}$.

Putting it all together, we have shown that the phase space is a product
manifold of the simplex $\Delta_{N-1}$ and the sphere $S^{2N-3}$:

$\displaystyle
d\Pi_{N}=\frac{(2\pi)^{4}}{Q^{4}}\left(\frac{Q^{2}}{16\pi^{3}}\right)^{N}\,d\Delta_{N-1}\,dS^{2N-3}\,.$
(10)

One can easily check that dimensionality works since the sum of the simplex
and sphere dimensions gives us $(N-1)+(2N-3)=3N-4$, which is the correct phase
space dimension for a system of $N$ massless particles. Also, because we will
always compare events with the same total energy $Q$, we will often rescale
energies so that $Q=1$, which eliminates the overall dimensionality of the
phase space metric.

###  2.2 Phase Space in the CM Frame: Metric and Explicit Coordination

From this factorization of the phase space manifold into a simplex and a
sphere, we can then introduce explicit, global coordinates. First, let’s
explicitly work out the full transformation from $\vec{v}$ to
$\vec{v}^{\prime}$. This can be expressed by

$\begin{gathered}\begin{pmatrix}v_{1}^{\prime}\\\ v_{2}^{\prime}\\\ \vdots\\\
v_{N-1}^{\prime}\end{pmatrix}=\frac{1}{u_{N}(1-u_{N}^{2})}\,\hat{R}\begin{pmatrix}v_{1}\\\
v_{2}\\\ \vdots\\\ v_{N-1},\end{pmatrix}\end{gathered}$ (11)

where the matrix $\hat{R}$ is

$\displaystyle\hat{R}$ (12) $\displaystyle\hskip
7.11317pt=\begin{pmatrix}u_{1}^{2}+u_{N}(1-u_{1}^{2}-u_{N}^{2})&u_{1}u_{2}(1-u_{N})&\cdots&u_{1}u_{N-1}(1-u_{N})\\\
u_{1}u_{2}(1-u_{N})&u_{2}^{2}+u_{N}(1-u_{2}^{2}-u_{N}^{2})&\cdots&u_{2}u_{N-1}(1-u_{N})\\\
\vdots&\vdots&\ddots&\vdots\\\
u_{1}u_{N-1}(1-u_{N})&u_{2}u_{N-1}(1-u_{N})&\cdots&u_{N-1}^{2}+u_{N}(1-u_{N-1}^{2}-u_{N}^{2})\end{pmatrix}\,.$

Note that the above matrix $\hat{R}$ is a real, symmetric and positive-
definite, and the reality of $\hat{R}$ ensures that the real and imaginary
components of $\vec{v}$ are not mixed with each other. Furthermore, the
determinant of the matrix is $u_{N}^{-1}$, and inverting $\hat{R}$ gives a
Jacobian $J=u_{N}^{2}$, which ensures that
$d^{N-1}v=u_{N}^{2}\,d^{N-1}v^{\prime}=\rho_{N}\,d^{N-1}v^{\prime}$. The
matrix of Eq. (12) was not explicitly constructed in Ref. Larkoski:2020thc ,
so we present its derivation in App. A.

The vector $\vec{v}^{\prime}$ can now be conveniently expressed by a
generalized spherical coordinate system,

$\begin{gathered}v^{\prime}_{1}=e^{-i\xi_{1}}\cos\eta_{1}\\\
v^{\prime}_{2}=e^{-i\xi_{2}}\sin\eta_{1}\cos\eta_{2}\\\ \vdots\\\
v^{\prime}_{N-2}=e^{-i\xi_{N-2}}\sin\eta_{1}\cdots\sin\eta_{N-3}\cos\eta_{N-2}\\\
v^{\prime}_{N-1}=e^{-i\xi_{N-1}}\sin\eta_{1}\cdots\sin\eta_{N-3}\sin\eta_{N-2},\end{gathered}$
(13)

where $\xi_{k}\in[0,2\pi]$ and $\eta_{k}\in[0,\pi/2]$. These coordinates
explicitly enforce the normalization $|\vec{v}^{\prime}|^{2}=1$. In the case
of $N=2$, the sphere part of the manifold is $S^{2\cdot 2-3}=S^{1}$, and so no
$\eta$ coordinate remains, and we’re left with only one parameter for the
sphere part, i.e., $\xi_{1}$.

To summarize, we have shown that the $(3N-4)$ degrees of freedom of the phase
space manifold are composed of two groups of coordinates, with $(N-1)$ $\rho$
coordinates on the simplex $\Delta^{N-1}$, and $(N-1)$ $\xi$ together with
$(N-2)$ $\eta$ coordinates on the hypersphere $S^{2N-3}$.

Armed with a system of explicit coordinates, we can now easily define the
corresponding metric on the phase space manifold. The line element of the
simplex is

$\displaystyle ds^{2}_{\Delta_{N-1}}=\sum_{k=1}^{N}d\rho_{k}^{2}.$ (14)

The summation above is over $N$ coordinates despite the fact that the simplex
has dimension $N-1$. Indeed, these coordinates are not all independent but are
instead constrained via the $\delta$-function in Eq. (7). With a suitable
change of variables, the simplex line element could be written using only
$N-1$ coordinates, but this is not necessary for our purposes. The line
element of the hypersphere follows a recursive description

$\displaystyle
ds^{2}_{S^{2N-3}}=d\eta_{N-2}^{2}+\cos^{2}\eta_{N-2}\;d\xi_{N-1}^{2}+\sin^{2}\eta_{N-2}\;ds^{2}_{S^{2N-5}}\,,$
(15)

starting from $ds^{2}_{S^{1}}=d\xi_{1}^{2}$.

In this paper, we are more interested in the distances between pairs of events
assigned according to their positions on the phase space manifold, so these
line elements can be promoted to honest Riemannian metrics that satisfy
symmetry, identity of indiscernibles, and the triangle inequality. For two
events that we label $A$ and $B$, their metric distance on the simplex is

$\displaystyle
d_{\Delta}(\vec{\rho}_{A},\vec{\rho}_{B})\equiv\sqrt{(\vec{\rho}_{A}-\vec{\rho}_{B})^{2}}=\sqrt{\sum_{k=1}^{N}(\rho_{kA}-\rho_{kB})^{2}}\,.$
(16)

Distances on the hypersphere are determined by the angle between the two
events, and therefore the metric on the hypersphere is

$\displaystyle
d_{S}(\vec{v}_{A}^{\prime},\vec{v}_{B}^{\prime})\equiv\cos^{-1}(\Re{\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}})=\cos^{-1}\left(\Re{\sum_{k=1}^{N-1}v_{kA}^{\prime*}v_{kB}^{\prime}}\right)\,,$
(17)

where $\Re{\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}}$ is the real part
of the inner product of the vectors $\vec{v}^{\prime}_{A}$ and
$\vec{v}^{\prime}_{B}$. Any convex linear combination of these two metrics is
itself a metric, and could therefore be a metric on the phase space manifold.
However, the phase space volume element is fixed, which constrains the
determinant of the metric.

By demanding that the metric on the phase space reproduces the dependence on
multiplicity $N$ in the volume element of Eq. (10), we can express the phase
space metric between two events $A$ and $B$ as dependent on a single parameter
$c>0$, where

$\displaystyle
d_{\Pi}(\vec{\rho}_{A},\vec{v}_{A}^{\prime};\vec{\rho}_{B},\vec{v}_{B}^{\prime})=\sqrt{\frac{1}{16\pi^{2}}\left(\frac{c}{4}\right)^{\frac{3-2N}{3N-4}}d^{2}_{\Delta}(\vec{\rho}_{A},\vec{\rho}_{B})+\frac{1}{4\pi^{2}}\left(\frac{c}{4}\right)^{\frac{N-1}{3N-4}}d^{2}_{S}(\vec{v}_{A}^{\prime},\vec{v}_{B}^{\prime})}\,.$
(18)

Varying the value of $c$ then reweights the simplex and sphere contributions
to the distance between events. With only the volume of the phase space as a
constraint, there is no natural value for $c$ and all choices give rise to
equally valid phase space distances.

Indeed, to ensure that $d_{\Pi}$ is a metric on the phase space, we must
further enforce permutation invariance of the particles in the events.
Specifically, if the events were identical, $A=B$, then for two distinct
orderings of the particles, the metric as defined now would in general be non-
zero. This would immediately fail the identity of indiscernibles requirement,
and so we must additionally enforce that the metric is minimized over all
permutations of particles in the events. In practice, global minimization over
$N!$ permutations is exponentially computationally intensive and unfeasible
even for reasonably small $N$. In Sec. 3, we will discuss simple preprocessing
steps based on physical considerations that can be implemented to approximate
the global permutation minimum, and, at the very least, ensure that the
distance from an event to itself is 0.

###  2.3 Phase Space in the Collinear Limit

The phase space for an isolated jet—as a collimated, high-energy collection of
particles—has its many distinct features. Our starting approximation for a jet
is in the infinite momentum limit, in which momentum transverse to its axis is
parametrically smaller than the momentum along the axis Fubini:1964boa ;
Weinberg:1966jm ; Susskind:1967rg ; Bardakci:1968zqb ; Chang:1968bh ;
Kogut:1969xa . In this limit, there is no Lorentz transformation that can be
performed to boost to the rest frame of the jet, and so it requires a new
analysis of the corresponding manifold on which particle momenta live.

Without loss of generality, we assume that the net momentum of the jet lies
along the $+\hat{z}$ axis and introduce the lightcone momentum components
$p_{0}+p_{3}$ and $p_{0}-p_{3}$. For our purposes here, we define a jet as a
collection of particles for which their net $p_{0}+p_{3}$ component is
parametrically larger than the $p_{0}-p_{3}$ component, i.e., $p_{0}+p_{3}\gg
p_{0}-p_{3}$. To find a jet, we require some net $p_{0}+p_{3}$ but are
inclusive to the value of $p_{0}-p_{3}$, without further restrictions on the
structure of the jet. Additionally, as the axis of the jet is defined to be
the direction of net momentum, the net momentum transverse to the axis is 0.
We denote these components of momentum transverse to the jet axis as
$\vec{p}_{\perp}$.

With this set-up, the differential phase space volume of a jet that consists
of $N$ massless particles is then:

$\displaystyle d\Pi_{N}^{(J)}=$
$\displaystyle(2\pi)^{4-3N}\left[\prod_{k=1}^{N}d^{4}p_{k}\,\delta^{+}(p_{k}^{2})\right]d(p_{0}-p_{3})\,\delta\left(p_{0}+p_{3}-\sum_{k=1}^{N}(p_{0k}+p_{3k})\right)$
(19) $\displaystyle\hskip
28.45274pt\times\delta\left(p_{0}-p_{3}-\sum_{k=1}^{N}(p_{0k}-p_{3k})\right)\delta^{(2)}\left(\sum_{k=1}^{N}\vec{p}_{\perp
k}\right).$

As with phase space in the CM frame, we introduce vectors $\vec{u}$ and
$\vec{v}$ to automatically incorporate the on-shell constraints, where now the
individual components are:

$\displaystyle u_{k}=\sqrt{\frac{p_{0k}-p_{3k}}{p_{0}-p_{3}}}\,,$
$\displaystyle
v_{k}=\frac{p_{1k}+i\,p_{2k}}{\sqrt{(p_{0}+p_{3})(p_{0k}-p_{3k})}}\,.$ (20)

Note that unlike their definition for phase space in the CM frame, these
vectors are dimensionless, which is preferred because to good approximation
jets are scale invariant. Phase space volume in these coordinates then becomes

$\displaystyle\begin{split}d\Pi_{N}^{(J)}=&(2\pi)^{4-3N}(p_{0}+p_{3})^{2N-4}\left(\frac{p_{0}-p_{3}}{p_{0}+p_{3}}\right)^{N-2}\frac{d^{N}u\,d^{N}v}{\text{U}(1)^{N}}\,d(p_{0}-p_{3})\\\
&\times\delta(1-|\vec{u}|^{2})\,\delta(1-|\vec{v}|^{2})\,\delta^{(2)}\left(\vec{u}^{\dagger}\vec{v}\right).\end{split}$
(21)

Here we can go a bit further and simplify the phase space volume by
incorporating some additional physics. In the soft and/or collinear limit that
dominates the description of a jet, perturbative QCD is approximately scale-
invariant, which imposes structure on the squared matrix element
$|\mathcal{M}|^{2}$ as a function of the present scales. By dimensional
analysis, the squared matrix element for $N$ final state particles must have
overall dependence on $p_{0}+p_{3}$ as

$\displaystyle|\mathcal{M}|^{2}\propto(p_{0}+p_{3})^{4-2N}\,,$ (22)

in order to ensure that probabilities are dimensionless. Further, assuming
scale invariance, probabilities are unchanged if the ratio

$\displaystyle\chi\equiv\frac{p_{0}-p_{3}}{p_{0}+p_{3}}\,,$ (23)

is rescaled as $\chi\to\lambda\chi$, for any $\lambda>0$, and so the matrix
element must have dependence on $\chi$ as

$\displaystyle|\mathcal{M}|^{2}\sim\chi^{1-N}(p_{0}+p_{3})^{4-2N}\,.$ (24)

Including this knowledge of the matrix element, the phase space now becomes

$\displaystyle d\Pi_{N}^{(J)}\,|{\cal
M}|^{2}\sim(2\pi)^{4-3N}\,d\ln\chi\,\frac{d^{N}u\,d^{N}v}{\text{U}(1)^{N}}\,\delta\left(1-|\vec{u}|^{2}\right)\delta\left(1-|\vec{v}|^{2}\right)\delta^{(2)}\left(\vec{u}^{\dagger}\vec{v}\right)\,.$
(25)

If one is inclusive over particle number $N$, then this exact scale invariance
in QCD is modified by an anomalous dimension (and the running coupling),
proportional to $\alpha_{s}$. In general, the matrix element will exhibit
singularities on the phase space when pairs of particles become collinear, but
those details depend precisely on the specific dynamics of the jet ensemble of
interest.

One can now proceed to construct explicit global coordinates on the jet’s
phase space manifold. As in the case of events in the CM frame, we will have
coordinates $\rho,v^{\prime},\xi,\eta$ expressed with the same formulae as in
Eqs. (6), (11), and (13). Therefore, we again end up with $(N-1)$ $\rho$
coordinates for the simplex and $(2N-3)$ $\xi$ and $\eta$ coordinates for the
hypersphere. For a jet, we will also have an additional $\chi$ coordinate
whose form is dictated by the specific structure of the squared matrix
element. As a result, the phase space for collinear jets is a product of three
parts: the simplex, the hypersphere, and a scale factor.

For two jets $A$ and $B$, the metric of the scale factor $\chi$ is simply the
Euclidean metric on the real line:

$\displaystyle
d_{\chi}(\chi_{A},\chi_{B})=\left|\log(\chi_{A})-\log(\chi_{B})\right|\,.$
(26)

Now, with three components to the jet phase space manifold and one constraint
on its volume form of Eq. (25), we can express the metric as a convex sum in
terms of two parameters $c_{1},c_{2}>0$, where

$\displaystyle
d^{2}_{\Pi^{(J)}}(\vec{\rho}_{A},\vec{v}_{A}^{\prime},\chi_{A};\vec{\rho}_{B},\vec{v}_{B}^{\prime},\chi_{B})$
(27) $\displaystyle\hskip
14.22636pt=\frac{1}{16\pi^{2}}\left(\frac{c_{1}}{4}\right)^{\frac{3-2N}{3N-3}}\left(\frac{c_{2}}{16\pi^{2}}\right)^{-\frac{1}{3N-3}}d^{2}_{\Delta}(\vec{\rho}_{A},\vec{\rho}_{B})+\frac{1}{4\pi^{2}}\left(\frac{c_{1}}{4}\right)^{\frac{N}{3N-3}}\left(\frac{c_{2}}{16\pi^{2}}\right)^{-\frac{1}{3N-3}}d^{2}_{S}(\vec{v}_{A}^{\prime},\vec{v}_{B}^{\prime})$
$\displaystyle\hskip
28.45274pt+\left(\frac{c_{1}}{4}\right)^{\frac{3-2N}{3N-3}}\left(\frac{c_{2}}{16\pi^{2}}\right)^{\frac{3N-4}{3N-3}}d_{\chi}^{2}(\chi_{A},\chi_{B}),$

where $d_{\Delta}$ is again given by Eq. (16) and $d_{S}$ by Eq. (17). In some
jet studies, there is an expected natural mass scale, for example, when
searching for resonances that decay hadronically, and in those cases, one
would impose a narrow cut on the jet mass. As such, the value of $\chi$ would
vary very little on a jet-by-jet basis in the relevant ensemble, and so the
contribution to the metric from $d_{\chi}$ can typically be ignored, as is
done in our later analysis. Additionally, same as events in the CM frame,
permutation invariance of the particles must be imposed externally to ensure
that the distance between a jet and itself is 0.

##  3 Implementation of the Phase Space Metric for Classification

In this section, we describe the practical implementation of the phase space
metric that we will explore in specific case studies later. It involves
significant pre-processing steps that go a long way toward realizing the
permutation invariance and exact minimization required for the phase space
distance to satisfy the requirements of a true metric. We also describe the
general techniques employed for post-processing the phase space representation
of events and jets for classification, where machine learning is introduced to
establish boundaries between signal- and background-rich regions.

Pre- and post-processing of particle physics collider event data for any
analysis, with machine learning or not, has long been employed to simplify the
representation of the data, see, e.g., Refs. Cogan:2014oua ; Almeida:2015jua ;
deOliveira:2015xxd ; Guest:2016iqz ; Louppe:2017ipp ; Cheng:2017rdo ;
Datta:2017rhs ; Komiske:2017aww ; Butter:2017cot . Typically, pre-processing
is done in an ad hoc manner, modding out by symmetries that are expected to
hold to good approximation, which is sometimes challenging to justify
quantitatively. On the phase space manifold with explicit global coordinates
and a local metric, these pre-processing steps are motivated by the necessary
minimization over particle ordering and orientation to enforce the properties
of the metric. The steps discussed here can therefore inform a principled pre-
processing prescription in other contexts.

There are necessarily many moving pieces in simplifying the event
representation, evaluating the metric, and eventually classifying events. In
the interest of clarity, we dedicate a subsection to each main processing
step. Note that here we refer to both full collision events and individual
jets excised from events as “events” and their clustered constituents as
“jets”, with the understanding that the appropriate description of the phase
space is used depending on the particular application.

###  3.1 Pre-processing of Collider Events

We first discuss the pre-processing to perform on events. Such pre-processing
steps ensure that the phase space distance between an event and itself
vanishes.

####  3.1.1 Event Representation

Our formulation of a metric on the phase space explicitly fixes the particle
multiplicity $N$. The first step is therefore to map any given event onto an
$N$-body phase space. We accomplish this by clustering the event (jet) into
$N$ jets (subjets) with the exclusive $k_{T}$ clustering algorithm
Catani:1991hj ; Catani:1993hr ; Ellis:1993tq , where $N$ is a pre-specified
fixed number. Further, the formalism requires each of the $N$ constituents to
be massless. We therefore manually substitute the energy of each jet by the
new value $E=|\vec{p}|$, and verify that the ratio $m/E$ for each resulting
object is indeed almost exactly zero (allowing for machine precision).

For small $N$, we expect that this masslessness restriction is not a good
approximation, as each of the $N$ constituents may consist of relatively hard,
relatively wide-angle particles that generate a relatively large mass for the
constituent as a whole. However, as multiplicity $N$ increases, it is expected
that this masslessness restriction weakens as small clusters of hadrons are
resolved that have high energy, but are collinear with one another. This is
justified a posteriori by plotting the distributions of the ratio $m/E$ for
the jets as a function of $N$ (before manually setting $m$ to be 0), which we
show later in Sec. 4 for the specific classification tasks. In principle, mass
can be incorporated in the phase space manifold and calculations in
perturbative QCD always assume massless partons. We leave a detailed analysis
of the limitations of this massless approximation to future work.

####  3.1.2 Particle Ordering

Events on which only energies and momenta are measured have no natural
ordering of their constituent particles; so any function of them must be
permutation invariant. For the phase space metric, this permutation invariance
should in principle be implemented as a minimization of the distance between
two events over all possible permutations of particles, which ensures that the
metric then satisfies both the triangle inequality and the identity of
indiscernibles.

However, for every pair of events with $N$ particles, we would have to scan
over all possible $N!$ distinct orderings to identify the minimum. Such an
exhaustive search is clearly computationally impossible even for moderate $N$.
While minimization over an exponentially-large number of permuations is ripe
for machine learning, here we propose instead a much more pedestrian approach
that is provably correct for the simplex submanifold and thus motivates the
prescription on the entire phase space.

Recall that the metric on the simplex is the Euclidean metric in $N$
dimensions:

$\displaystyle d^{2}_{\Delta}(\vec{\rho}_{A},\vec{\rho}_{B})=\min_{\sigma\in
S_{N}}\sum_{i=1}^{N}(\rho_{iA}-\rho_{\sigma(i)B})^{2}\,.$ (28)

Of course, the simplex is $N-1$ dimensional by energy conservation, but we
will work with this manifestly permutation invariant form. Also, here we have
explicitly included the minimization over the permutations of particles in
event $B$, where $\sigma$ is an element of the symmetric group $S_{N}$.
Without loss of generality, we can assume that the particles in event $A$ are,
say, ordered monotonically decreasing in $\rho$ value. Then, the permutation
of the particles of event $B$ that minimizes this distance is also when the
particles in event $B$ are ordered in decreasing $\rho$ value. That is,

$\displaystyle
d^{2}_{\Delta}(\vec{\rho}_{A},\vec{\rho}_{B})=\sum_{\begin{subarray}{c}i=1\\\
\rho_{iA}>\rho_{(i+1)A}\\\
\rho_{iB}>\rho_{(i+1)B}\end{subarray}}^{N}(\rho_{iA}-\rho_{iB})^{2}\,.$ (29)

We provide an elementary proof that this ordering does indeed render the
distance a metric on the simplex in App. B.

The story is more complicated for the hypersphere, since there is a rotation
from $v$ to $v^{\prime}$ coordinates. Of course, ultimately $d^{2}_{\Pi}$ is
the correct distance to be minimized for each pair of events. Since
$d^{2}_{\Pi}$ is a convex linear combination of the squared metrics on the
simplex and the hypersphere, ordering particles according to their $\rho$
value does not guarantee minimization over permutations of the complete metric
on the phase space.

Nevertheless, this pre-processing step is very simple and computationally
cheap, and at the very least approximates a local minimum over permutations.
To practically implement this ordering, recall that the definition of $\rho$
is

$\displaystyle\rho=p_{0}+p_{3}\,,$ (30)

i.e., the sum of the energy and $z$-component of the particle momentum. For
events in the CM frame, there is possibly an O(3) rotation symmetry of
particle momenta. For events at a hadron collider, this O(3) symmetry is
broken to O(2) about the beams and reflection of the beams, but still, the
direction of the $+z$-axis is not unambiguously defined. In our analyses,
then, we will consider the following three different particle orderings to
probe the sensitivity to the choice of axis:

  * •

Descending order in $p_{0}+p_{3}$ ,

  * •

Descending order in $p_{0}-p_{3}$ ,

  * •

Descending order in $p_{T}\equiv\sqrt{p_{1}^{2}+p_{2}^{2}}$ .

Note that there is little a priori reason why one particle ordering is
preferred for the full phase space distance. Our proposal here is inspired by
the structure of the simplex alone, and we hope that the above physics-
grounded orderings already give satisfying performance and further that the
performance does not depend critically on the orderings. Future study of a
better and more specialized way to search over the particle ordering space
will hopefully improve the performance even more.

####  3.1.3 Azimuthal Rotation

Aside from the particle ordering, there are still minimization ambiguities on
the hypersphere that can be addressed with pre-processing. Recall that the
metric on the hypersphere is

$\displaystyle
d_{S}(\vec{v}_{A}^{\prime},\vec{v}_{B}^{\prime})=\min_{\arg(\vec{v}_{B}^{\prime})}\cos^{-1}\left(\Re\left[\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}\right]\right)\,.$
(31)

Here the minimization is a bit subtle. Event-by-event, there is an O(2)
azimuthal symmetry about the beam, for which the momentum transverse to the
beam can be rotated by a common angle or reflected. In the metric, this SO(2)
rotation invariance is manifest as minimization over the argument of the
vector $\vec{v}_{B}^{\prime}$ (we fix the vector $\vec{v}_{A}^{\prime}$
because only their relative phase is important). We will address reflection
invariance shortly.

Because $\cos^{-1}$ monotonically decreases as its argument increases from 0,
the minimization over the phase of $\vec{v}_{B}^{\prime}$ is implemented by
maximizing

$\displaystyle\max_{\arg(\vec{v}_{B}^{\prime})}\Re\left[\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}\right]\leq\left|\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}\right|\,.$
(32)

Because the real part of this vector dot product is bounded from above by its
absolute value, even when maximizing over the argument of
$\vec{v}_{B}^{\prime}$, we should just use the absolute value in calculating
the metric on the hypersphere. This prescription will minimize the distance on
the hypersphere for any given particle ordering, but does not ensure that the
ordering itself renders a global minimum. Nevertheless, with the ordering
prescription on the simplex, this procedure will produce at least a local
minimum of the metric on the entire phase space manifold.

To account for reflections in the transverse space, note that a reflection of
$p_{y}\to-p_{y}$ corresponds to complex conjugation in the vector $\vec{v}$.
Therefore, to minimize the hypersphere metric over reflections with a given
particle ordering, we simply choose the larger of
$\left|\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}\right|$ and
$\left|\vec{v}_{A}^{\prime\intercal}\vec{v}_{B}^{\prime}\right|$. Again, we
just need to reflect (i.e., complex conjugate) vector $\vec{v}_{A}^{\prime}$
in the product because the metric is only sensitive to the relative
orientation between events.

For a given particle ordering, the final form of the metric on the hypersphere
is thus

$\displaystyle
d_{S}(\vec{v}_{A}^{\prime},\vec{v}_{B}^{\prime})=\cos^{-1}\left(\max\left[\left|\vec{v}_{A}^{\prime\dagger}\vec{v}_{B}^{\prime}\right|,\left|\vec{v}_{A}^{\prime\intercal}\vec{v}_{B}^{\prime}\right|\right]\right)\,.$
(33)

Crucially, the particle ordering on the simplex ensures that the distance
between an event and itself vanishes, and the above prescription for
minimization over azimuthal symmetries ensures that the distance on the
hypersphere also vanishes. Taken together, these pre-processing prescriptions
explicitly ensure that the metric on the phase space satisfies the quality of
identity of indiscernibles.

###  3.2 Processing the Phase Space Manifold

With the pre-processing steps defined above, we now turn to discuss practical
evaluation of the coordinates on the phase space manifold and the
corresponding distances between realistic events. Here we encounter the
additional freedom in the definition of the simplex and sphere coordinates.

If events enjoy invariance to O(3) transformations of the celestial sphere and
all particles are detected, then there is a continuous infinity of possible
choices for the $\vec{u}$ and $\vec{v}$ coordinates, all of which are equally
valid. This is because, in the center-of-mass frame, the net momentum is 0 and
so the net momentum transverse to any fixed axis is also 0; therefore any axis
can legitimately serve as the $z$-axis, leading to a valid metric.

However, here we either consider events produced at a hadron collider, for
which numerous particles are lost down the beam region and undetected, or
individual jets for which a center-of-mass frame does not exist. These systems
explicitly break the infinite degeneracy of coordinate systems and reduce the
$\vec{u}$ and $\vec{v}$ coordinates to just a couple of choices that respect
the symmetries of the events in the lab frame.

These possible choices are exclusively set by the definition of the $\vec{u}$
coordinates on the simplex, or equivalently, $\rho$. Given fixed coordinates
for which the $\hat{z}$-axis lies along the beams and one beam is fixed to be
in the $+\hat{z}$ direction, there are two possibilities:

$\displaystyle\rho=p_{0}+p_{3}\,,\qquad\text{or}\qquad\rho=p_{0}-p_{3}\,.$
(34)

Note that either choice produces a valid metric, because there is no absolute
frame for the coordinates on the phase space. In principle, any valid
definition of the phase space metric contains the same information content and
therefore should be equally powerful for the downstream classification. Later
we will study the response of a few classification tasks with both of the
above choices of the $\vec{u}$ coordinates, denoted as “\+ on simplex” and “\-
on simplex” accordingly.

Note that this choice of coordinates ensures that the momentum transverse to
the $z$-axis, which defines the coordinates $\vec{v}$, is (very nearly) 0, up
to the small residual transverse momentum lost down the beams. However, the
visible particles in general have a large net momentum along the $z$-axis,
which will need to be boosted away to render the events in the center-of-mass
frame. Such a boost does not affect the direction of the $z$-axis, and further
the $\rho$ coordinate transforms homogeneously under a $z$-axis boost, i.e.,

$\displaystyle\rho\to e^{\eta}\rho\,,$ (35)

where the (pseudo)rapidity of the visible particles in the lab frame is
$\eta$.

Given a choice for the $\vec{u}$ coordinates, the $\vec{v}$ coordinates are
then unique. Conservation of transverse momentum requires that $\vec{v}$ is
transverse to $\vec{u}$, i.e., $\vec{u}^{\dagger}\vec{v}=0$; so $\vec{v}$ must
be formed from a complex linear combination of $p_{1}$ and $p_{2}$. We can in
principle rotate the $x$ and $y$ axes about the $z$ axis however we desire,
naively producing another continuous infinity of axis choices. However,
focused on ultimately evaluating the metric distance between events, we have
already optimized this azimuthal rotation in the implementation of the metric
on the hypersphere. Global event rotations about the $z$ axis, as would be
used to define new $(x,y)$ axes, cannot affect the metric distance. Therefore
we can happily take whatever axes are provided to us in the representation of
the data.

For studies on jets, we simply note that all of the above considerations apply
there as well. The only modification is that we set the jet axis to be the
$+\hat{z}$ direction, and the two choices of $\rho$ become the only legitimate
ones, because the covariant phase space derivation relies on there being zero
total momentum transverse to whatever momentum direction appears in the
simplex coordinate; otherwise, we would have $\vec{u}^{\dagger}\vec{v}\neq 0$.
We can then proceed to compute the phase space coordinates according to the
prescriptions in Sec 2.

###  3.3 Post-processing for Classification

After obtaining the simplex and the sphere distances, further post-processing
steps are necessary in order to accomplish the task of supervised event
classification. First, one needs to choose the relative weighting between the
simplex and sphere parts (and $\chi$ if included) to define the total phase
space metric. One then needs to pick a classifier which takes the phase space
distance as the input. The two steps are indeed coupled together in the sense
that a good choice of the relative weighting is determined solely by the
performance of the classifier. In principle, different classifiers may prefer
different values for the relative weighting, although one would expect a good
classifier to deliver relatively stable performance across a range of
reasonable values for the relative weighting.

####  3.3.1 Relative Weighting between the Simplex and the Sphere

To define the phase space distance between any pair of events, we need to
specify the free coefficient $c$ in Eq. (18) (or $c_{1}$ and $c_{2}$ in Eq.
(27)). To simplify our analysis, here we do not consider variation of the
weighting in front of $\chi$. Therefore, the only free parameter to tune is
$c$ in Eq. (18).

Obviously, there are infinitely many choices and any finite value for $c$
gives rise to a valid metric. In a perfect world with infinite computational
power, we would want to continuously vary the weighting to maximize tagging
performance, quantified in a single number called Area under the ROC Curve
(AUC), where an AUC of 1 means a perfect classifier and a value close to 0.5
signifies a random guess.

Fortunately, varying the weighting does not entail redoing the actual distance
matrix computation. Rather, we can just take the existing two distance
matrices (one for the simplex and the other for the hypersphere) and create
different linear combinations of them. In practice, we need to ensure
relatively balanced contributions from each part to the total phase space
distance. Therefore, in our study we only consider three reasonable values for
the relative weighting, i.e., sphere : simplex = $1:4$, $1:1$, and $4:1$.

####  3.3.2 Machine Learning Classifier Choice

In this study, we choose Support Vector Machine (SVM) cortes1995support  as
the classifier for event and jet tagging, thanks to its simplicity and
robustness.222We have also explored the performance of the $k$-nearest
neighbors ($k$NN) algorithm, another simple and robust supervised learning
classifier. We find that SVM consistently outperforms $k$NN in all
classification tasks, which presumably reflects the natural clustering of
events of different types on the phase space manifold. In general, we expect
any algorithm (including neural networks) that can take distance matrices as
inputs to work well with the phase space formalism, and encourage further
exploration of more advanced methods.

SVM separates data points into two classes by finding a hyperplane whose
distances to the nearest data points are maximized on both sides. The
generalized notion of distance is known as kernel. We choose the commonly-used
Radial Basis Function (RBF) kernel powell1977restart ;
broomhead1988multivariable  $k(\vec{x}_{A};\vec{x}_{B})=\exp\\{-\gamma
d(\vec{x}_{A};\vec{x}_{B})^{2}\\}$, where in our case
$d(\vec{x}_{A};\vec{x}_{B})=d_{\Pi}(\vec{\rho}_{A},\vec{v}^{\prime}_{A};\vec{\rho}_{B},\vec{v}^{\prime}_{B})$
or
$d(\vec{x}_{A};\vec{x}_{B})=d_{\Pi}(\vec{\rho}_{A},\vec{v}^{\prime}_{A},\chi_{A};\vec{\rho}_{B},\vec{v}^{\prime}_{B},\chi_{B})$
is the phase space distance.

There are two hyperparameters $C$ and $\gamma$ in the SVM model. The kernel
parameter $\gamma$ can be interpreted as the inverse of the radius of
influence of training data points. On the other hand, the soft margin
parameter $C$ controls the relative importance between correct classification
and large separation, with a small $C$ valuing a large separation between the
optimal hyperplane and nearest data points over correctly classifying all data
points and vice versa.

For the event and jet classification tasks described in Sec. 4, we generate
10k events or jets in each case, and further separate them into a training
dataset of size 5000, a validation dataset of size 2500 for choosing the best
model hyperparameters (i.e., $\gamma$ and $C$), and a test dataset of size
2500 to evaluate the model performance. We restrict the ranges of
hyperparameters to be $C\in[10^{-2},10^{3}]$ and $\gamma\in[10^{-2},10^{5}]$
with a multiplication of 10 in each step. These values are chosen based on
empirical observations with the goal to include the best combination of
hyperparameters.

##  4 Phase Space Manifold for Event Classification and Jet Tagging

The separation of signal events from backgrounds is a key task for the search
of new physics beyond the Standard Model (BSM) at the LHC. Oftentimes, the
majority of background processes are made up of QCD events, whose dynamics in
the perturbative regime is dominated by the emission of soft and collinear
quarks and gluons due to the small ’t Hooft coupling $\lambda=g_{s}^{2}N_{C}$
of QCD at high collider scales. Much of the kinematic information from this
initial perturbative showering is retained through subsequent hadronization,
and the final state particles of QCD events, mostly populated by light mesons,
display jet-like structure that occupy some certain marked region in the phase
space manifold.

In contrast, various kinds of signals (including both SM and BSM events) may
show distinctively different phase space structures due to the unique nature
of their respective underlying physics. For example, top-quark pair
production, a typical SM process, gives rise to an event configuration of six
prongs on average, inducing a more uniform distribution than the QCD
background. This will in turn differentiate the phase space region occupied by
$t\bar{t}$ events from that of QCD events.

One can further consider exotic BSM scenarios where high-multiplicity events
are produced whose final state particles have uniform distribution on the
$N$-body phase space. Indeed, many models can produce such high-multiplicity
events. One general class is Hidden Valley models Strassler_2007 , which
emerges from solutions to string theory constructions and the hierarchy
problem. With large ’t Hooft couplings in contrast to QCD, Hidden Valleys can
yield uniform radiation patterns and is a good signature to compare against SM
backgrounds such as QCD events.

Here we demonstrate the utility of the phase space distance in three
classification tasks: event-level classification of top pair production vs.
QCD dijet events in Sec. 4.1; event-level classification of a Hidden Valley
scenario with high-multiplicity uniformly-distributed events vs. QCD dijet
background events in Sec. 4.2; and jet-level classification of two-pronged
boosted $W$ boson jets vs. QCD background jets in Sec. 4.3. The event-level
classification tasks are tailored in part for comparison with benchmarks for
the event isotropy variable explored in Ref. Cesarotti:2020hwb .

For each classification task, we examine different definitions of the phase
space distances, as detailed in Sec. 3. Specifically, we try both definitions
of the $\vec{u}$ coordinates, i.e., “\- on simplex” and “\+ on simplex”, as
well as three relative weightings between the simplex and the sphere parts,
denoted as “sphere:simplex = 1:4”, “sphere:simplex = 1:1”, and “sphere:simplex
= 4:1”. This gives us 6 sub-tasks, all giving rise to equally valid phase
space distances. One would expect the classification performance to vary
minimally no matter which distance definition is used. This is because the
exact same information, though combined in different ways, is encoded in any
valid phase space distance and a good classifier should be able to utilize
them equally well. Our intuition is largely supported by the three case
studies.

In contrast, within each sub-task, the choice of particle ordering and the
number of jets (subjets) representing an event (jet), i.e., $N$, do make a
material difference, with the former giving different approximations to the
exact phase space distance and the latter defining distinct phase space
manifolds themselves. The particle ordering and (sub)jet number that maximize
tagging performance vary from case to case depending on the underlying
collisions. In general, the masslessness assumption for the individual
components in an event holds to better accuracy as $N$ grows higher. One
therefore expects the current phase space formulation to give a more precise
description of the actual events consisting of massive particles under the
large $N$-representations, potentially leading to better downstream
classification performance with increasing $N$.

We perform all six sub-tasks for each of the three classification tasks,
amounting to $\textbf{2}\text{ ($\vec{u}$ coordinates
definitions)}\times\textbf{3}\text{ (particle
orderings)}\times\textbf{8}\text{ (event representations)}=\textbf{48}$ rounds
of phase space distance calculations. The relative weighting does not require
re-calculation of the simplex and the sphere distances, as they can be
combined according to the specific ratio to directly generate the total phase
space distance. The calculation of the simplex and sphere distances for the
entire dataset of 10k events takes a couple of hours on a single desktop
(Intel Core i7-8700 CPU) for $N\leq 5$, with computing time increasing sharply
for larger $N>10$, e.g. taking about 50 hours for $N=40$. However, we expect
optimization of the codes to significantly improve efficiency, and the use of
cluster will further decrease computational time given that our method is
highly amenable to parallel computing.

Performance is always quoted as the best AUC on the test set with the optimal
hyperparameters chosen by the validation set. All results are shown as plots
of AUC vs. $N$-representation, with the actual AUC scores given in tables in
App. C.

###  4.1 Top Pair Production vs. QCD Dijet Events

Proton-proton collision events at $\sqrt{s}=14$ TeV are generated in madgraph5
2.9.6 Alwall_2014 , with top quarks being pair produced via $pp\to t\bar{t}$,
and QCD dijet events generated through $pp\to jj$ where $j$ represents a quark
($u,d,c,s$) or gluon. The top (or anti-top) quarks subsequently decay via
$t\to W^{+}q$ (or $\bar{t}\to W^{-}\bar{q}$) where $q=b,s,d$ and $\bar{q}$
represents the corresponding anti-quarks. The $W$ bosons further decay
hadronically, i.e., $W\rightarrow jj$, so that the top events in principle
have six jets if all $j$’s are resolved. Particles are then hadronized and
further decayed in Pythia 8.303 Sj_strand_2015 , where default tuning and
showering parameters are used. Only final state particles with pseudo-rapidity
$|y|<2$ in the lab frame are kept. We select events whose scalar transverse
momentum is $\sum p_{T}>400$ GeV and cluster them into jets using FastJet
3.4.0 Cacciari_2012 . In total, we have 10k simulated events, with about half
being signal.

To construct the phase space manifold, we first cluster each event into a
certain number of jets using the $k_{T}$ exclusive algorithm. For each event,
only the $N=3,4,5,10,15,20,25,30$ hardest jets are retained, where a jet is
treated as a pseudo-particle with its kinematic information given by the jet
axis and its mass manually set to zero. We then center each event by boosting
it to its CM frame, and normalize the $(E,p_{x},p_{y},p_{z})$ of each
constituent jet by the total energy of the event when computing the phase
space coordinates.

Fig. 1 visualizes a top pair production event and a QCD dijet event randomly
selected from our 10k dataset. The events with all the constituent particles
are shown first, followed by the various $N$-representations on the $y-\phi$
plane with $N=3,4,5,10,15,20,25,30$. Next, to demonstrate the appropriateness
of the masslessness approximation, Fig. 2 shows the distributions of each
jet’s $m/p_{T}$ on a log-log scale, before jet mass is manually set to zero.
Clearly, most jets have negligible mass comparing to their transverse momenta,
validating the massless phase space approximation especially at large $N$.

Figure 1: A $t\bar{t}$ event (red) and QCD dijet event (green), represented by
all the final state particles (leftmost plot) or $N$ exclusive jets
($N=3,4,5,10,15,20,25,30$). The size of each dot is proportional to the
$p_{T}$ of each constituent, plotted on the $y-\phi$ plane.

Figure 2: Histograms for each jet’s $m/p_{T}$ (before manually setting $m$ to
zero) of the QCD (left) and top (right) events. Note that both axes are on a
log basis, and the colors denote different numbers $N$ of jets per event.

####  4.1.1 Results

We first show some sample histograms of the total phase space distances for
our top and QCD events under a particular set of metric definitions (\- on
simplex, sphere:simplex = 1:1, $(p_{0}-p_{3})$-ordering); see Fig. 3. Here we
also separately plot the pairwise distances between two QCD events, two top
events, and a QCD event vs. a top event. Since the histograms aggregate all
pairwise distances, the plots for the distances among different types of
events look rather similar, even though the discriminative power of the total
phase space distance for top vs. QCD events is indeed very high. The
difference of histograms for different types of events is, however, more
noticeable when $N=4,5$, indicating better classification performance around
these $N$ values to be confirmed below.

Figure 3: From left to right: Histograms of the total phase space distances
between two arbitrary events in the dataset, between two QCD events, between
two top events, and between a QCD event and a top event. The particular set of
metric definitions is chosen to be \- on simplex, sphere:simplex = 1:1, and
$(p_{0}-p_{3})$-ordering. The colors denote different numbers $N$ of jets per
event.

Fig. 4 shows the classification performance of the total phase space distances
coupled with SVM. The highest AUC obtained among all definitions is $\sim
0.86$, corresponding to the simplex definition of $\rho=p_{0}-p_{3}$; a
relative weighting of sphere:simplex = 1:1; the $(p_{0}+p_{3})$-ordering; and
an event representation of $N=4$ jets per event. This is quite remarkable,
given that all we have utilized is the “locations” on the phase space manifold
of different types of events. No explicit information about the underlying
collision process is provided to the model.

Here we make three overall observations. First, as noted above, both the
definitions of the simplex coordinates and the relative weighting between the
simplex and the sphere make little difference on the final AUC score,
suggesting that any valid phase space distance contains comparable information
recoverable by SVM. Of course, this is only true within reasonable variation
of the sphere-to-simplex weighting, as extreme weights asymptote to the
sphere-only and simplex-only distances.

Second, there is no obvious best particle ordering for approximating the exact
phase space distance. All three orderings render roughly the same
classification performance, especially in the low $N$ region ($N\leq 10$)
where the best AUC is achieved. As more jets are used to represent an event
($N>10$), the overall performance degrades slightly and in most cases the
descending $p_{T}$-ordering becomes the best choice among the three, giving
stable performance across the entire range of $N$. But again, the difference
is relatively small to draw statistically significant conclusions about the
best particle ordering scheme. One would expect that an exact minimization of
the phase space distance over all particle permutations would further improve
its tagging power, although a more accurate (in this case smaller) distance
does not necessarily entail a higher AUC, as will be shown momentarily.

Instead, the largest impact on classification arises from event representation
by $N$ (massless) jets, which defines the phase space manifold itself with a
dimension of $3N-4$. The best performance is achieved when $N\sim 4,5$ and the
performance consistently deteriorates with increasing $N$. As exemplified in
Fig. 1, around 5 jets can capture most of the hard components inside a QCD
dijet event or a $t\bar{t}$ event, which matches our theoretical
understanding. Adding more objects means including soft particles which mainly
originate from QCD for both signal and background, the inclusion of which may
confound the downstream classifier. Another explanation could be that higher
$N$ entails a phase space with higher dimensionality, potentially making it
harder for SVM to classify the data. However, the other two classification
tasks will soon show that this is not the case, as there the classification
performance indeed increases with larger $N$. The $N$-dependence of
classification appears to be more a feature of the underlying events being
compared than the dimensionality of the phase space manifold itself.

Figure 4: AUC values for QCD dijet events vs. top pair production events,
using SVM as the classifier. The number of jets $N$ to represent each event
(i.e., the $x$-axis) is varied to be $N=3,4,5,10,15,20,25,30$. Each subplot
corresponds to a distinct pair of $\vec{u}$ coordinates definition coupled
with the sphere-to-simplex weighting ratio. The blue, red, and green curves
represent results with constituents ordered by decreasing $(p_{0}-p_{3})$,
$(p_{0}+p_{3})$, and $p_{T}$, respectively.

We now zoom in to examine the simplex and sphere submanifolds separately under
the two $\vec{u}$ coordinates definitions. Fig. 5 shows the AUC scores if the
input distance matrix only consists of either the simplex or the sphere part
of the phase space, which is equivalent to setting sphere:simplex as 0:1 or
1:0, respectively. The goal is to disentangle the information content
contained in the two submanifolds and see which part contributes most to the
classification task.

For both simplex and sphere parts, the general trend of the AUC scores follows
that of the total phase space, with performance again peaking around $N\sim
4,5$ and gradually declining with larger $N$ by about $10\sim 15\%$. Here the
sphere part of the phase space contains more discrimination power than the
simplex, where the best achievable AUC by the sphere distance is more than
$20\%$ higher than that of the simplex distance under the same definitions.
Indeed, the sphere distance alone gives almost comparable AUC as the total
phase space distance, with a mere $\sim 3\%$ difference between their
respective optimal scores. The major difference, however, is that the total
phase space distance depends less on the specific particle ordering, whereas
the difference among the three orderings is much more pronounced for the two
submanifolds, especially on the simplex.

Interestingly, given a particular definition of the $\vec{u}$ coordinates, the
best particle ordering corresponding to the exact minimum distance on the
simplex nonetheless performs the worst when only the simplex distance matrix
is inputted to the model. For example, with $\rho$ defined to be
$(p_{0}+p_{3})$ (i.e., \+ on simplex), the particle ordering that minimizes
the distance on the simplex is the $(p_{0}+p_{3})$-ordering, which
consistently gives the lowest AUC scores across all values of $N$ for the
simplex alone. In this case, it is always the $(p_{0}-p_{3})$-ordering that
delivers the best performing simplex distance, quite contrary to our
intuition.

On the other hand, the particle ordering that minimizes the simplex distance
gives rise to a sphere distance that performs the best for lower $N$ ($N\leq
15$), where there is no obvious reason why one ordering is favored more than
another on the sphere. The intermediate $p_{T}$-ordering turns out to generate
the most stable performance over all $N$, both for the simplex alone and for
the sphere alone. Such behavior is common to all three classification tasks,
and does not appear to be an artifact of the particular events under
consideration.

Of course, analyzing the two submanifolds of the phase space separately may
not be physically meaningful. Rather one should always use the total phase
space distance, in which case all three particle orderings give rise to
similar performance with a slight preference for the $p_{T}$-ordering, thanks
to its stability across $N$. Yet one general lesson we can learn from the
simplex-only analysis is that an exact minimum distance does not guarantee
best performance for classification.

Figure 5: AUC values of QCD dijet events vs. top pair production events, where
the input distance matrix is the simplex (left) or the sphere (right) part
alone under both $\vec{u}$ coordinates definitions (upper for \- on simplex
and lower for \+ on simplex).

###  4.2 A New Physics Toy Model with Uniform $N$-body Phase Space Events

We now consider as our signal a Hidden Valley scenario with a heavy mediator
$Z^{\prime}$ that generates high-multiplicity events uniformly distributed in
$N$-body phase space. The backgrounds are still QCD dijet events, generated in
the same way as before, except that the scalar transverse momentum cut is now
raised to $\sum p_{T}>650$ GeV.

The Hidden Valley model is generated via HiddenValley:ffbar2Zv in Pythia 8.303
with default tuning and showering parameters. The nominal mass of the
$Z^{\prime}$ is set to be 1 TeV with a width of 20 GeV, and a mass cut of
$m_{Z^{\prime}}\in[750,1250]$ GeV. Only final states with $|y|<2$ are kept.
The particle $Z^{\prime}$ is decayed using the Rambo algorithm KLEISS1986359 ,
where its $N_{Z^{\prime}}$ decay products are distributed uniformly in the
rest frame of the $Z^{\prime}$. Here we pick $N_{Z^{\prime}}=10$ to
demonstrate the high-multiplicity of the new physics events. We then boost the
decay products of $Z^{\prime}$ back to the lab frame. After applying again a
$|y|<2$ cut for these particles, we combine the remaining decay products with
the rest of the Pythia-generated final-state particles in the same event.
Finally, we apply a cut $\sum p_{T}>650$ GeV on the scalar transverse momentum
of each event.

Both signal and background events are clustered into $N$ jets in FastJet using
the $k_{T}$ exclusive algorithm, with $N=5,10,15,20,25,30,35,40$ hardest jets
in each event to form various phase space representations. Fig. 6 visualizes
on the $y-\phi$ plane a new physics event represented as $N$ jets, and Fig. 7
again displays the distributions of $m/p_{T}$ for the jets on a log-log scale.
The massless phase space approximation is clearly appropriate for the majority
of the events, even at low $N$.

Figure 6: A new physics event on the $y-\phi$ plane clustered into
$N=5,10,15,20,25,30,35,40$ jets (i.e., the number of dots in each plot), where
the size of every dot is proportional to the $p_{T}$ of the constituent
particle.

Figure 7: Histograms for each jet’s $m/p_{T}$ (before manually setting $m$ to
zero) of the QCD (left) and new physics (right) events. Note that both axes
are on a log basis, and the colors denote different numbers $N$ of jets per
event.

####  4.2.1 Results

Again, we first show in Fig. 8 a few sample histograms of the total phase
space distances for the QCD dijet events and the new physics scenario under
the set of metric definitions (\- on simplex, sphere:simplex = 1:1,
$(p_{0}-p_{3})$-ordering). The difference in the histograms for the distances
among various types of events is now more pronounced, hinting at potentially
better classification performance as confirmed below.

Figure 8: From left to right: Histograms of the total phase space distances
between two arbitrary events in the dataset, between two QCD events, between
two new physics events (indicated by “Zv”), and between a QCD event and a new
physics event. The particular set of metric definitions is chosen to be \- on
simplex, sphere:simplex = 1:1, and $(p_{0}-p_{3})$-ordering. The colors denote
different numbers $N$ of jets per event.

Fig. 9 shows the AUC scores of the total phase space distances. The
performance is excellent across a wide range of $N$ values, with higher $N$
giving an AUC score close to 1. This indicates that the current signals and
backgrounds are extremely well separable on the phase space manifold, and that
the higher the dimension of the phase space, the better the separability. Such
observations correspond well with our physics intuition, since the new physics
events have a much higher multiplicity than the QCD events.

We again note that both definitions of the $\vec{u}$ coordinates give
comparable performance and the sphere-to-simplex relative weighting has little
impact on classification, corroborating our hypothesis that any valid phase
space distance definition contains equivalent information for a reasonable
classifier to utilize. Furthermore, there is very little variation in the
final performance for the three particle orderings, suggesting that the local
minima found by the approximation schemes are likely very close to the global
minimum.

Unlike the previous top vs. QCD event classification task, here the
performance is relatively stable with respect to the number of jets used to
represent an event. Apart from the slight drop at $N\sim 10-15$ when the
simplex gets a higher weight, the performance shows a modest yet steady
increase when more objects are included, contrary to the trend observed in the
top vs. QCD task. This suggests that the $N$-dependence of classification is a
function of the specific underlying physics, rather than the overall
properties of the $N$-dependent phase space manifold.

Figure 9: AUC values for QCD dijet events vs. new physics events, using SVM as
the classifier. The number of jets to represent each event (i.e., $x$-axis) is
varied to be $N=5,10,15,20,25,30,35,40$. Each subplot corresponds to a
distinct pair of the $\vec{u}$ coordinates definition coupled with a specific
sphere-to-simplex weighting ratio. The blue, red, and green curves represent
results with constituents ordered by decreasing $(p_{0}-p_{3})$,
$(p_{0}+p_{3})$, and $p_{T}$, respectively.

Fig. 10 studies the simplex and sphere submanifolds separately for both
$\vec{u}$ coordinates definitions. In general, results are qualitatively the
same as in the previous task. The sphere part once again proves to be more
useful than the simplex, where its highest attainable AUC is less than $2\%$
lower than the optimal AUC obtained by the total phase space distance. The
three particle orderings give roughly the same performance on the sphere for
all $N$, with the $p_{T}$-ordering being slightly better for larger $N$.

On the simplex, we again notice the phenomenon that the particle ordering that
minimizes the simplex distance performs the worst and a larger performance gap
($\sim 20\%$) now exists for different orderings. For both $(p_{0}+p_{3})$ and
$(p_{0}-p_{3})$ orderings, the performance has a noticeable drop when $N$
approaches 15, which is around the number of hard components in the new
physics events (recall $N_{Z^{\prime}}=10$). On the other hand, the
$p_{T}$-ordering does not have this effect and its performance grows steadily
with increasing $N$.

Figure 10: AUC values of QCD dijet events vs. new physics events, where the
input distance matrix is the simplex (left) or the sphere (right) part alone
under both $\vec{u}$ coordinates definitions (upper for \- on simplex and
lower for \+ on simplex).

###  4.3 High Energy Collinear QCD vs. $W$ Boson Jets

We now focus on single jet tagging and apply the phase space formalism in the
collinear limit, as described in Sec. 2.3. Proton-proton collision events are
first simulated at $\sqrt{s}=14$ TeV in MadGraph, with $W$ bosons being pair
produced via $pp\to W^{+}W^{-}$, gluons being generated via $q\overline{q}\to
Z\to\nu\overline{\nu}g$, and quarks generated via $qg\to
Z\to\nu\overline{\nu}q$. The particles are then hadronized and decayed in
Pythia with default tuning and showering parameters. Afterwards, events are
clustered into jets using FastJet 3.4.0 with the anti-$k_{T}$ inclusive
algorithm and a jet radius of $R=1$. The jet transverse momentum is selected
to be $p_{T}\in[500,550]$ GeV and jet mass is set within the range of
$m\in[60,100]$ GeV. At most two jets are kept with $|y|\leq 1.7$ and
$|\phi|\leq\frac{\pi}{2}$.

To fix the number of constituents, we re-cluster each jet into $N$ subjets in
FastJet using the exclusive $k_{T}$ algorithm. Only the hardest
$N=3,4,5,10,15,20,25,30$ subjets are kept and are further manually set to be
massless. Here the lower limit $N=3$ is chosen to reflect the two-pronged
structure of a $W$ boson jet, whereas the upper limit is constrained by the
inability of FastJet to find more than 30 subjets. We then rotate the
3-momentum of each jet so that its axis is aligned with the $+z$ direction.
The jet constituents are represented in the $(E,p_{x},p_{y},p_{z})$
coordinates, with their $E$ normalized by the total energy of the jet.

Fig. 11 displays a random $W$ boson jet and a QCD jet with different $N$
subjets on the $y-\phi$ plane. Fig. 12 shows the distributions of each
subjet’s $m/p_{T}$ on a log-log scale. Clearly with increasing $N$, more and
more subjets have their mass becoming negligible comparing to $p_{T}$, again
validating the masslessness assumption especially for large $N$.

Figure 11: A $W$ (red; row 1) or QCD (green; row 2) jet as represented by
$N=3,4,5,10,15,20,25,30$ exclusive subjets in the $y-\phi$ plane. The size of
each dot is proportional to the $p_{T}$ of each subjet.

Figure 12: Histograms for each subjet’s $m/p_{T}$ (before manually setting $m$
to zero) of the QCD (left) and $W$ (right) jets. Note that both axes are on a
log basis, and the colors denote different numbers $N$ of subjets per jet.

####  4.3.1 Results

Fig. 13 displays the histograms of the total phase space distances for the $W$
and QCD jets under the metric definitions (\- on simplex, sphere:simplex =
1:1, $(p_{0}-p_{3})$-ordering). The tagging results are displayed in Fig. 14.
Overall, the performance is excellent, with the highest AUC score approaching
0.9. As before, the $\vec{u}$ coordinates definitions and relative simplex-to-
sphere weighting have little impact on the classification, and all three
particle orderings result in similar performance. The AUC score enjoys a minor
increase at higher values of $N$, where the masslessness assumption holds
better. In general, performance stabilizes at its peak after $N>10$.

Figure 13: From left to right: Histograms of the total phase space distances
between two arbitrary jets in the dataset, between two QCD jets, between two
$W$ jets, and between a QCD jet and a $W$ jet. The particular set of metric
definitions is chosen to be \- on simplex, sphere:simplex = 1:1, and
$(p_{0}-p_{3})$-ordering. The colors denote different numbers $N$ of jets per
event. Figure 14: AUC values for $W$ vs. QCD jets using SVM as the classifier.
The number of subjets to represent each jet (i.e., the $x$-axis) is varied to
be $N=3,4,5,10,15,20,25,30$. Each subplot corresponds to a distinct pair of
$\vec{u}$ coordinates definition coupled with the sphere-to-simplex weighting
ratio. The blue, red, and green curves represent results with constituents
ordered by decreasing $(p_{0}-p_{3})$, $(p_{0}+p_{3})$, and $p_{T}$,
respectively.

When zooming in to the performance of the simplex and the sphere submanifolds
(see Fig. 15), we observe qualitatively similar phenomena as in the previous
two event classification tasks, except that here the simplex part contains as
much as, or even more, discrimination information than the sphere part,
especially for $N>10$ with the best performing particle ordering. This
difference highlights the importance of the simplex coordinates in the
collinear limit, as now the $z$-axis is defined to be aligned with the jet
axis with the value of $p_{3}$ carrying physical information about the jet
structure.

We also notice here that compared to the previous event classification tasks,
different particle orderings now result in larger gaps in the AUC score of
more than $20\%$. On the other hand, the ordering that exactly minimizes the
simplex distance continues to give the worst performance for the simplex-only
analysis, but is still the best performer for the sphere-only analysis. In
this case, combining simplex and sphere into a total phase space distance has
the distinctive advantage of stabilizing the performance for different
particle orderings, as well as for a wide range of $N$ values with significant
improvement at lower $N$.

Figure 15: AUC values of $W$ vs. QCD jets, where the input distance matrix is
the simplex (left) or the sphere (right) part alone under both $\vec{u}$
coordinates definitions (upper for \- on simplex and lower for \+ on simplex).

##  5 Conclusions and Future Directions

The phase space manifold furnishes a natural and powerful notion of the
distance between collider events. In this paper, we have developed a practical
prescription for defining and computing phase space distances at hadron
colliders, both for boosted objects and entire events. These distances
illuminate the distinctive phase space structures of different scattering
amplitudes and lead to competitive event classification when coupled to simple
machine learning algorithms.

Specifically, we have found that any valid definition of the phase space
distance gives rise to comparably good classification performance, reflecting
the fact that the distance captures the same underlying physics. Furthermore,
the particle ordering scheme to approximate the exact minimization required in
the definition of the total phase space distance also bears relatively little
impact on the downstream performance. This is an encouraging sign, suggesting
that a future exact minimization would most likely give similar results with
lower risk of qualitatively changing the conclusions drawn in the current
study.

Instead, the key factor that determines the classification performance is the
number of objects used to represent an event. This is in part due to the
validity of the masslessness assumption, as higher $N$ results in a lower mass
of the individual clustered objects. More importantly, $N$ defines the
dimensionality of the phase space manifold itself. Naturally, the performance
of the distance varies with $N$ and is dependent on the specific tagging task
under consideration.

There are a number of promising directions for future study. In converting
explicit global coordinates on the phase space manifold to a practical metric,
we have made certain choices that leave room for exploration. For example, as
mentioned above, we have fixed the dimensionality of the phase space manifold
for a given ensemble of events by exclusively clustering into $N$ objects,
mapping events onto an $N$-object representation with crucial influence over
the effectiveness of ML-based classification. Alternative approaches to
treating events of different intrinsic dimensionality may prove fruitful.

Furthermore, the phase space distances computed between these $N$-object
representations assume a massless phase space, which is not guaranteed to be a
good approximation in all regimes. This could be circumvented by generalizing
the phase space metric to massive particles. Finally, we forego an exact (yet
computationally prohibitive) minimization of the distance over particle
permutations in favor of an approximate procedure that is only guaranteed to
find local minima. Alternative approaches to the minimization problem
(potentially using machine learning) may prove effective.

Further opportunities also abound within our prescription for computing phase
space distances. Analytic and numerical study of the phase space structure of
different scattering amplitudes is likely to be productive. And while we have
demonstrated the value of phase space distances in classification tasks, they
are equally likely to be useful in other collider physics applications such as
unsupervised anomaly detection and simulation-based inference.

Long a backdrop to the calculation of scattering amplitudes, the geometry of
phase space is beyond ready to step into the limelight.

###### Acknowledgements.

We would like to thank Liam Brennan, Brian Henning, Joe Incandela, Tamas Vami,
and Danyi Zhang for useful discussions. The work of TC, NC, and GK was
supported in part by the U.S. Department of Energy under the grant DE-
SC0011702 and performed in part at the Kavli Institute for Theoretical
Physics, supported by the National Science Foundation under Grant No. NSF
PHY-1748958. TC is additionally supported by the U.S. Department of Energy
Awards No. DE-FOA-0002705, KA/OR55/22 (AIHEP) and No. DE-AC02-76SF00515.

##  Appendix A Coordinates on the Phase Space Hypersphere Submanifold

In this appendix, we derive the change of variables from $\vec{v}$ to
$\vec{v}^{\prime}$. For compactness, we will make use of index notation, where
repeated indices are implicitly summed over $1,\ldots,N-1$. Consider the
arguments of the delta functions in Eq. 9. In order to have equality, we need

$\displaystyle
v^{*}_{i}v_{i}+|v_{N}|^{2}=v^{\prime*}_{k}v^{\prime}_{k}=\left(v_{i}^{*}A_{ki}^{*}\right)\left(A_{kj}v_{j}\right)$
(36)

where the matrix $A$ represents the change of variables we wish to find.
Recall that, at this point, all components of $\vec{u}$ are real and strictly
positive, while $v_{N}$ is defined via

$\displaystyle v_{N}=-\frac{1}{u_{N}}\left(u_{i}v_{i}\right).$ (37)

Plugging Eq. 37 into the left hand side of Eq. 36, we have

$\displaystyle v^{*}_{i}v_{i}+|v_{N}|^{2}$
$\displaystyle=v^{*}_{i}v_{i}+\frac{1}{u_{N}^{2}}\left(v_{i}^{*}u_{i}\right)\left(u_{j}v_{j}\right)$
(38)
$\displaystyle=v_{i}^{*}\delta_{ij}v_{j}+\frac{1}{u_{N}^{2}}\left(v_{i}^{*}u_{i}\right)\left(u_{j}v_{j}\right)$
(39)
$\displaystyle=v_{i}^{*}\left[\frac{1}{u_{N}^{2}}\left(u_{i}u_{j}+u_{N}^{2}\delta_{ij}\right)\right]v_{j}.$
(40)

The quantity in brackets is essentially the square of the matrix we seek. With
malice aforethought, we proceed to “complete the square,” making use of the
$\vec{u}$ normalization condition, $u_{i}u_{i}=1-u_{N}^{2}$. We then have

$\displaystyle=v_{i}^{*}\left\\{\frac{1}{u_{N}^{2}}\left[u_{i}u_{j}\left(\frac{1-u_{N}^{2}}{(1+u_{N})^{2}}+2\frac{u_{N}}{1+u_{N}}\right)+u_{N}^{2}\delta_{ij}\right]\right\\}v_{j}$
(41)
$\displaystyle=v_{i}^{*}\left[\frac{1}{u_{N}^{2}}\left(u_{i}u_{j}\frac{u_{k}u_{k}}{(1+u_{N})^{2}}+u_{k}u_{j}\delta_{ik}\frac{u_{N}}{1+u_{N}}+u_{i}u_{k}\delta_{kj}\frac{u_{N}}{1+u_{N}}+u_{N}^{2}\delta_{ik}\delta_{kj}\right)\right]v_{j}$
(42)
$\displaystyle=v_{i}^{*}\left[\frac{1}{u_{N}^{2}}\left(\frac{u_{i}u_{k}}{1+u_{N}}+u_{N}\delta_{ik}\right)\left(\frac{u_{k}u_{j}}{1+u_{N}}+u_{N}\delta_{kj}\right)\right]v_{j},$
(43)

from which we identify the real, symmetric transformation matrix

$\displaystyle A_{ij}$
$\displaystyle=\frac{1}{u_{N}}\left(\frac{u_{i}u_{j}}{1+u_{N}}+u_{N}\delta_{ij}\right)$
(44) $\displaystyle=\frac{1}{u_{N}(1-u_{N}^{2})}\,\hat{R}_{ij},$ (45)

with $\hat{R}$ given by Eq. 12, as desired.

##  Appendix B Proof of Minimal Ordering for Euclidean Metric

In this appendix, we present a simple proof regarding the permutation-
invariant Euclidean metric.

######  Lemma 1.

The permutation-invariant Euclidean metric distance between two sets of points
$\\{x_{i}\\}_{i=1}^{N},\\{y_{i}\\}_{i=1}^{N}$ can be expressed as

$\displaystyle
d^{2}\left(\\{x_{i}\\}_{i=1}^{N},\\{y_{i}\\}_{i=1}^{N}\right)=\min_{\sigma\in
S_{N}}\sum_{i=1}^{N}\left(x_{i}-y_{\sigma(i)}\right)^{2}=\sum_{\begin{subarray}{c}i=1\\\
x_{i}>x_{i+1}\\\ y_{i}>y_{i+1}\end{subarray}}^{N}(x_{i}-y_{i})^{2}\,.$ (46)

That is, one ordering of points that minimizes their distance is when they are
both in descending order.

###### Proof.

We will assume that all points are distinct, i.e., $x_{i}=x_{j}$ implies
$i=j$. With probability 1, all points are distinct in our application of this
theorem to the study of phase space because the number of points is finite and
drawn from a continuous probability distribution. The case with degeneracies
can be considered as well, but the main result does not change.

Without loss of generality, we can assume that the $x_{i}$ points are ordered,
$x_{1}>x_{2}>\cdots>x_{N}$, while the $y_{i}$ points are in random order.
Expanding out the square, the metric can be expressed as

$\displaystyle\min_{\sigma\in
S_{N}}\sum_{i=1}^{N}\left(x_{i}-y_{\sigma(i)}\right)^{2}=\sum_{i=1}^{N}\left(x_{i}^{2}+y_{i}^{2}\right)-2\max_{\sigma\in
S_{N}}\sum_{i=1}^{N}x_{i}y_{\sigma(i)}\,.$ (47)

We only need to consider the final term on the right for the minimization of
the metric. Let’s consider two terms in that sum and determine if permuting
them increases its value. That is, we will consider $x_{i}>x_{j}$ for $j>i$
and focus on the terms

$\displaystyle\sum_{i=1}^{N}x_{i}y_{\sigma(i)}\supset
x_{i}y_{\sigma(i)}+x_{j}y_{\sigma(j)}\,.$ (48)

Now, there are obviously two cases: either $y_{\sigma(i)}>y_{\sigma(j)}$ or
$y_{\sigma(j)}>y_{\sigma(i)}$. Further, we can assume, without loss of
generality, that $\sigma(j)>\sigma(i)$. First, if
$y_{\sigma(i)}>y_{\sigma(j)}$, then note that

$\displaystyle(x_{i}-x_{j})(y_{\sigma(i)}-y_{\sigma(j)})>0\,,$ (49)

or that

$\displaystyle
x_{i}y_{\sigma(i)}+x_{j}y_{\sigma(j)}>x_{i}y_{\sigma(j)}+x_{j}y_{\sigma(i)}\,.$
(50)

That is, the points are already ordered to minimize the metric, with $i<j$ and
$\sigma(i)<\sigma(j)$. In the other case, where $y_{\sigma(j)}>y_{\sigma(i)}$,
we have

$\displaystyle(x_{i}-x_{j})(y_{\sigma(j)}-y_{\sigma(i)})>0\,,$ (51)

or that

$\displaystyle
x_{i}y_{\sigma(j)}+x_{j}y_{\sigma(i)}>x_{i}y_{\sigma(i)}+x_{j}y_{\sigma(j)}\,.$
(52)

In this case, we can swap the indices on the $y$ points,
$\sigma(i)\leftrightarrow\sigma(j)$, and after this swap, the metric is
minimized with the index ordering $i<j$ and $\sigma(i)<\sigma(j)$.

We can perform this pairwise comparison of points and their contribution to
the metric, swapping indices on the $y$ points as necessary, until no swap
increases the value of the sum in Eq. 48. After this procedure, we have
therefore reordered all $y$ points such that $y_{i}>y_{j}$ with $i<j$, for all
$1\leq i<j\leq N$. That is, all $y$ points have been sorted in descending
order, just like the $x$ points. Further, note that this sorting is the global
minimum because any other ordering necessarily increases the metric distance.
This then proves the equality of Eq. 46. ∎

##  Appendix C AUC Score Tables for the Classification Tasks

In this appendix, we provide all the AUC scores for the three classification
tasks discussed in the main text from Fig. 4 to Fig. 15.

Distance |  $\vec{u}$ Definition | Sphere : Simplex | Particle Ordering | $N=3$ | $N=4$ | $N=5$ | $N=10$ | $N=15$ | $N=20$ | $N=25$ | $N=30$
---|---|---|---|---|---|---|---|---|---|---|---
Total | \+ on simplex | 1:4 | $p_{0}+p_{3}$ | 0.828 | 0.857 | 0.858 | 0.808 | 0.785 | 0.767 | 0.697 | 0.701
$p_{T}$ | 0.825 | 0.857 | 0.849 | 0.813 | 0.805 | 0.778 | 0.795 | 0.790
$p_{0}-p_{3}$ | 0.824 | 0.852 | 0.845 | 0.808 | 0.785 | 0.779 | 0.770 | 0.755
1:1 | $p_{0}+p_{3}$ | 0.809 | 0.851 | 0.855 | 0.823 | 0.814 | 0.801 | 0.785 | 0.781
$p_{T}$ | 0.828 | 0.849 | 0.843 | 0.818 | 0.814 | 0.804 | 0.801 | 0.809
$p_{0}-p_{3}$ | 0.823 | 0.848 | 0.844 | 0.809 | 0.767 | 0.761 | 0.764 | 0.759
4:1 | $p_{0}+p_{3}$ | 0.813 | 0.846 | 0.845 | 0.825 | 0.824 | 0.812 | 0.800 | 0.797
$p_{T}$ | 0.823 | 0.844 | 0.832 | 0.809 | 0.813 | 0.807 | 0.811 | 0.814
$p_{0}-p_{3}$ | 0.810 | 0.838 | 0.829 | 0.786 | 0.775 | 0.756 | 0.756 | 0.744
\- on simplex | 1:4 | $p_{0}+p_{3}$ | 0.827 | 0.850 | 0.851 | 0.797 | 0.793 | 0.779 | 0.770 | 0.759
$p_{T}$ | 0.821 | 0.852 | 0.853 | 0.807 | 0.798 | 0.794 | 0.782 | 0.781
$p_{0}-p_{3}$ | 0.831 | 0.850 | 0.850 | 0.792 | 0.772 | 0.757 | 0.730 | 0.710
1:1 | $p_{0}+p_{3}$ | 0.820 | 0.860 | 0.845 | 0.803 | 0.772 | 0.767 | 0.770 | 0.763
$p_{T}$ | 0.821 | 0.853 | 0.848 | 0.816 | 0.809 | 0.807 | 0.805 | 0.798
$p_{0}-p_{3}$ | 0.826 | 0.838 | 0.849 | 0.811 | 0.808 | 0.797 | 0.799 | 0.783
4:1 | $p_{0}+p_{3}$ | 0.818 | 0.847 | 0.834 | 0.769 | 0.781 | 0.761 | 0.753 | 0.755
$p_{T}$ | 0.832 | 0.848 | 0.834 | 0.797 | 0.805 | 0.812 | 0.808 | 0.801
$p_{0}-p_{3}$ | 0.824 | 0.846 | 0.835 | 0.825 | 0.814 | 0.810 | 0.808 | 0.801
Simplex | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.600 | 0.652 | 0.641 | 0.620 | 0.603 | 0.606 | 0.602 | 0.595
$p_{T}$ | 0.618 | 0.663 | 0.666 | 0.655 | 0.642 | 0.642 | 0.645 | 0.637
$p_{0}-p_{3}$ | 0.691 | 0.758 | 0.763 | 0.727 | 0.726 | 0.715 | 0.687 | 0.692
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.689 | 0.728 | 0.737 | 0.719 | 0.702 | 0.705 | 0.700 | 0.682
$p_{T}$ | 0.614 | 0.649 | 0.659 | 0.647 | 0.634 | 0.624 | 0.636 | 0.639
$p_{0}-p_{3}$ | 0.612 | 0.636 | 0.645 | 0.609 | 0.598 | 0.586 | 0.578 | 0.583
Sphere | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.774 | 0.822 | 0.829 | 0.800 | 0.757 | 0.751 | 0.726 | 0.717
$p_{T}$ | 0.729 | 0.759 | 0.789 | 0.787 | 0.786 | 0.777 | 0.778 | 0.783
$p_{0}-p_{3}$ | 0.725 | 0.758 | 0.768 | 0.756 | 0.732 | 0.726 | 0.730 | 0.715
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.729 | 0.767 | 0.778 | 0.738 | 0.731 | 0.726 | 0.713 | 0.709
$p_{T}$ | 0.731 | 0.774 | 0.805 | 0.781 | 0.772 | 0.775 | 0.782 | 0.777
$p_{0}-p_{3}$ | 0.759 | 0.832 | 0.829 | 0.803 | 0.758 | 0.747 | 0.738 | 0.718
Table 1: AUC scores for top vs. QCD events Distance |  $\vec{u}$ Definition | Sphere : Simplex | Particle Ordering | $N=5$ | $N=10$ | $N=15$ | $N=20$ | $N=25$ | $N=30$ | $N=35$ | $N=40$
---|---|---|---|---|---|---|---|---|---|---|---
Total | \+ on simplex | 1:4 | $p_{0}+p_{3}$ | 0.945 | 0.915 | 0.899 | 0.924 | 0.933 | 0.951 | 0.961 | 0.963
$p_{T}$ | 0.941 | 0.923 | 0.917 | 0.947 | 0.963 | 0.963 | 0.978 | 0.982
$p_{0}-p_{3}$ | 0.938 | 0.919 | 0.907 | 0.914 | 0.933 | 0.946 | 0.947 | 0.958
1:1 | $p_{0}+p_{3}$ | 0.944 | 0.931 | 0.937 | 0.952 | 0.956 | 0.970 | 0.972 | 0.979
$p_{T}$ | 0.935 | 0.925 | 0.922 | 0.942 | 0.955 | 0.962 | 0.977 | 0.981
$p_{0}-p_{3}$ | 0.941 | 0.920 | 0.919 | 0.916 | 0.931 | 0.946 | 0.955 | 0.962
4:1 | $p_{0}+p_{3}$ | 0.928 | 0.938 | 0.941 | 0.949 | 0.960 | 0.966 | 0.972 | 0.975
$p_{T}$ | 0.921 | 0.919 | 0.923 | 0.939 | 0.948 | 0.959 | 0.968 | 0.972
$p_{0}-p_{3}$ | 0.923 | 0.914 | 0.920 | 0.916 | 0.923 | 0.940 | 0.952 | 0.953
\- on simplex | 1:4 | $p_{0}+p_{3}$ | 0.944 | 0.914 | 0.906 | 0.923 | 0.931 | 0.936 | 0.954 | 0.970
$p_{T}$ | 0.937 | 0.910 | 0.927 | 0.949 | 0.966 | 0.972 | 0.974 | 0.979
$p_{0}-p_{3}$ | 0.941 | 0.911 | 0.902 | 0.925 | 0.929 | 0.942 | 0.959 | 0.965
1:1 | $p_{0}+p_{3}$ | 0.943 | 0.924 | 0.917 | 0.932 | 0.933 | 0.944 | 0.958 | 0.968
$p_{T}$ | 0.929 | 0.930 | 0.923 | 0.945 | 0.955 | 0.966 | 0.974 | 0.981
$p_{0}-p_{3}$ | 0.937 | 0.936 | 0.935 | 0.949 | 0.956 | 0.959 | 0.964 | 0.971
4:1 | $p_{0}+p_{3}$ | 0.935 | 0.918 | 0.913 | 0.927 | 0.932 | 0.940 | 0.948 | 0.957
$p_{T}$ | 0.918 | 0.919 | 0.924 | 0.941 | 0.950 | 0.963 | 0.970 | 0.970
$p_{0}-p_{3}$ | 0.928 | 0.939 | 0.939 | 0.953 | 0.963 | 0.965 | 0.971 | 0.976
Simplex | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.767 | 0.732 | 0.690 | 0.684 | 0.701 | 0.707 | 0.721 | 0.755
$p_{T}$ | 0.785 | 0.778 | 0.778 | 0.788 | 0.820 | 0.843 | 0.842 | 0.858
$p_{0}-p_{3}$ | 0.852 | 0.847 | 0.827 | 0.840 | 0.851 | 0.861 | 0.884 | 0.888
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.866 | 0.839 | 0.813 | 0.827 | 0.850 | 0.871 | 0.882 | 0.887
$p_{T}$ | 0.775 | 0.779 | 0.781 | 0.810 | 0.813 | 0.838 | 0.846 | 0.853
$p_{0}-p_{3}$ | 0.763 | 0.737 | 0.685 | 0.694 | 0.700 | 0.705 | 0.729 | 0.739
Sphere | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.916 | 0.897 | 0.888 | 0.913 | 0.916 | 0.935 | 0.947 | 0.948
$p_{T}$ | 0.891 | 0.891 | 0.907 | 0.927 | 0.945 | 0.947 | 0.959 | 0.960
$p_{0}-p_{3}$ | 0.891 | 0.876 | 0.888 | 0.897 | 0.915 | 0.927 | 0.941 | 0.946
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.879 | 0.877 | 0.882 | 0.907 | 0.916 | 0.928 | 0.943 | 0.952
$p_{T}$ | 0.893 | 0.900 | 0.908 | 0.925 | 0.937 | 0.944 | 0.958 | 0.963
$p_{0}-p_{3}$ | 0.918 | 0.897 | 0.890 | 0.912 | 0.923 | 0.926 | 0.938 | 0.943
Table 2: AUC scores for QCD vs. New Physics events Distance |  $\vec{u}$ Definition | Sphere : Simplex | Particle Ordering | $N=3$ | $N=4$ | $N=5$ | $N=10$ | $N=15$ | $N=20$ | $N=25$ | $N=30$
---|---|---|---|---|---|---|---|---|---|---|---
Total | \+ on simplex | 1:4 | $p_{0}+p_{3}$ | 0.821 | 0.831 | 0.849 | 0.878 | 0.878 | 0.888 | 0.885 | 0.879
$p_{T}$ | 0.823 | 0.823 | 0.834 | 0.864 | 0.869 | 0.870 | 0.871 | 0.860
$p_{0}-p_{3}$ | 0.825 | 0.831 | 0.838 | 0.870 | 0.876 | 0.878 | 0.872 | 0.867
1:1 | $p_{0}+p_{3}$ | 0.825 | 0.830 | 0.841 | 0.879 | 0.881 | 0.887 | 0.888 | 0.879
$p_{T}$ | 0.821 | 0.823 | 0.832 | 0.868 | 0.864 | 0.862 | 0.873 | 0.862
$p_{0}-p_{3}$ | 0.825 | 0.827 | 0.822 | 0.870 | 0.873 | 0.881 | 0.876 | 0.864
4:1 | $p_{0}+p_{3}$ | 0.819 | 0.831 | 0.853 | 0.879 | 0.882 | 0.884 | 0.889 | 0.880
$p_{T}$ | 0.820 | 0.818 | 0.835 | 0.861 | 0.857 | 0.858 | 0.858 | 0.845
$p_{0}-p_{3}$ | 0.821 | 0.827 | 0.814 | 0.868 | 0.859 | 0.868 | 0.851 | 0.853
\- on simplex | 1:4 | $p_{0}+p_{3}$ | 0.823 | 0.839 | 0.851 | 0.878 | 0.889 | 0.889 | 0.883 | 0.886
$p_{T}$ | 0.821 | 0.829 | 0.845 | 0.874 | 0.862 | 0.856 | 0.839 | 0.832
$p_{0}-p_{3}$ | 0.817 | 0.831 | 0.838 | 0.862 | 0.863 | 0.856 | 0.863 | 0.858
1:1 | $p_{0}+p_{3}$ | 0.818 | 0.841 | 0.827 | 0.870 | 0.885 | 0.886 | 0.886 | 0.885
$p_{T}$ | 0.819 | 0.837 | 0.847 | 0.869 | 0.865 | 0.857 | 0.867 | 0.851
$p_{0}-p_{3}$ | 0.824 | 0.841 | 0.836 | 0.869 | 0.876 | 0.876 | 0.881 | 0.873
4:1 | $p_{0}+p_{3}$ | 0.825 | 0.840 | 0.827 | 0.868 | 0.879 | 0.874 | 0.871 | 0.859
$p_{T}$ | 0.819 | 0.834 | 0.828 | 0.861 | 0.845 | 0.830 | 0.825 | 0.820
$p_{0}-p_{3}$ | 0.826 | 0.828 | 0.848 | 0.868 | 0.878 | 0.879 | 0.877 | 0.869
Simplex | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.646 | 0.665 | 0.660 | 0.687 | 0.701 | 0.709 | 0.719 | 0.725
$p_{T}$ | 0.698 | 0.751 | 0.767 | 0.805 | 0.808 | 0.817 | 0.816 | 0.808
$p_{0}-p_{3}$ | 0.737 | 0.805 | 0.831 | 0.863 | 0.865 | 0.869 | 0.856 | 0.848
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.725 | 0.792 | 0.824 | 0.876 | 0.885 | 0.888 | 0.887 | 0.875
$p_{T}$ | 0.666 | 0.738 | 0.779 | 0.839 | 0.838 | 0.848 | 0.848 | 0.833
$p_{0}-p_{3}$ | 0.579 | 0.594 | 0.634 | 0.697 | 0.706 | 0.698 | 0.697 | 0.691
Sphere | \+ on simplex | N/A | $p_{0}+p_{3}$ | 0.750 | 0.806 | 0.834 | 0.854 | 0.878 | 0.874 | 0.880 | 0.867
$p_{T}$ | 0.715 | 0.764 | 0.789 | 0.833 | 0.812 | 0.808 | 0.804 | 0.800
$p_{0}-p_{3}$ | 0.695 | 0.740 | 0.770 | 0.824 | 0.816 | 0.808 | 0.799 | 0.808
\- on simplex | N/A | $p_{0}+p_{3}$ | 0.697 | 0.729 | 0.742 | 0.802 | 0.797 | 0.800 | 0.804 | 0.795
$p_{T}$ | 0.730 | 0.738 | 0.774 | 0.805 | 0.783 | 0.762 | 0.755 | 0.744
$p_{0}-p_{3}$ | 0.774 | 0.817 | 0.840 | 0.859 | 0.852 | 0.854 | 0.863 | 0.843
Table 3: AUC scores for QCD vs. W jets

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