# Gravity based on internal symmetry of quantum fields

Mikko Partanen  Photonics Group, Department of Electronics and
Nanoengineering, Aalto University, P.O. Box 13500, 00076 Aalto, Finland  Jukka
Tulkki  Engineered Nanosystems Group, School of Science, Aalto University,
P.O. Box 12200, 00076 Aalto, Finland

(November 21, 2023)

###### Abstract

The standard model of particle physics describes electromagnetic, weak, and
strong interactions, which are three of the four known fundamental forces of
nature. The unification of the fourth interaction, gravity, with the standard
model has been challenging due to incompatibilities of the underlying theories
– general relativity and quantum field theory. While quantum field theory
utilizes symmetries associated with internal vector spaces of quantum fields,
general relativity is based on external space-time symmetries. Here we show
how the internal special unitary symmetry of quantum fields can be employed to
extend the standard model to describe gravity. The resulting theory, called
unified gravity, is, in the classical regime for weak gauge coupling,
equivalent to teleparallel gravity and general relativity, while beyond-
general-relativity studies are enabled in the case of finite gauge coupling.

##  I Introduction

Quantum field theory is a theoretical framework, which synthesizes classical
field theory, quantum mechanics, and special relativity Peskin2018 ;
Weinberg1996 . The standard model of particle physics arises on the basis of
this framework through unitary symmetries related to invariances of a physical
system Schwartz2014 . The gauge invariance of electrodynamics, related to the
Abelian phase rotation transformations, is the most trivial example of such a
symmetry. The Yang–Mills theory extends the gauge theory to non-Abelian
special unitary symmetries Hooft2005 ; Peskin2018 ; Weinberg1996 ;
Schwartz2014 , which enable mutually interacting force carriers. It describes
the behavior of the other fundamental interactions of the standard model being
at the core of the unification of electrodynamics to weak and strong
interactions Schwartz2014 . A similar internal special-unitary-symmetry-based
approach to the description of gravity has remained unknown Livio2012 ;
Maldacena2014 ; Anderson2023 ; Misner1973 ; Dirac1975 . Therefore, alternative
approaches, such as string theory Becker2007 ; Green1987  and loop quantum
gravity Ashtekar1986 ; Jacobson1988 ; Rovelli1990 , are being developed. Many
authors have approached the problem by attempting to reformulate space-time
symmetries in a way compatible with the Yang-Mills theory Utiyama1956 ;
Kibble1961 ; MacDowell1977 ; Kaku1977 ; Stelle1980 ; Ivanov1982 ; Fradkin1985
; Witten1988 ; Vasiliev2001 ; Fabi2014 ; Ho2016 . The difference between
external space-time symmetries and internal vector space symmetries of quantum
fields, however, represents a challenge for this gauge gravitation theory
approach Blagojevic2013 ; Clifton2012 ; Karananas2021 ; Westman2013 ;
Sciama1964 ; Neville1980 .

In this work, we employ the internal special unitary symmetry of the eight-
spinor formulation of quantum fields in the standard model Partanen2023b ;
supplemental . We utilize the Lie algebra associated with this symmetry to
couple the quantum fields to a new tensor gauge field in a way that is
analogous to the gauge couplings of the fields in the electromagnetic, weak,
and strong interactions of the standard model. Once the new gauge field is
introduced, the powerful machinery of the Yang-Mills theory leads to dynamical
equations, which enable the quantum field generalization of the equations of
teleparallel gravity and Einstein’s field equations of general relativity
Bahamonde2023 ; Aldrovandi2012 . The ensuing Yang-Mills gauge theory of
gravity, called unified gravity, is subject to quantization using the approach
of the Yang-Mills theory Peskin2018 ; Weinberg1996 . The quanta of the gauge
field, the gravitons, are spin-2 tensor bosons. These quanta are to be added
in the spectrum of the known elementary particles extending the existing
standard model to describe gravity. The detailed study of this quantization is
left to future works.

Gravity couples to all fields and matter. Therefore, one cannot exclude any
field or matter from the complete dynamical description of gravity. However,
to make our theory of unified gravity more transparent and easier to
understand for nonexpert readers, we limit, in the main text, our study to the
coupling between gravity and electrodynamics. The system of the
electromagnetic field, Dirac electron-positron field, and the gravitational
field provides all the insight needed for obtaining a unified description of
gravity on equal footing with the other known fundamental forces of nature.
The extension of unified gravity to cover the full standard model is presented
in the supplemental material supplemental .

We start with the eight-spinor formulation of quantum electrodynamics
Partanen2023b . The eight-spinors reveal a profound connection between the
internal special unitary symmetry of quantum fields and the symmetric stress-
energy tensor. Since the symmetric stress-energy tensor is the source of the
gravitational field in general relativity, it becomes obvious that the gauge
theory based on the special unitary symmetry should describe gravitational
interaction. This is the basis for the development of the Yang-Mills gauge
theory of gravity in the present work.

##  II Generating Lagrangian density of gravity

The electromagnetic field is described by an eight-component spinor
Partanen2023b , given in terms of the conventional three-component real-valued
electric and magnetic fields $\mathbf{E}_{\Re}$ and $\mathbf{B}_{\Re}$
Jackson1999  by
$\Psi_{\Re}=\sqrt{\varepsilon_{0}/2}\,[0,\mathbf{E}_{\Re},0,ic\mathbf{B}_{\Re}]^{T}$.
Here the superscript $T$ denotes the transpose, $\varepsilon_{0}$ is the
vacuum permittivity, and $c$ is the speed of light in vacuum. The four-
component Dirac spinor field $\psi$ and its adjoint
$\bar{\psi}=\psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{F}}^{0}$ are used in
their conventional forms. The $4\times 4$ Dirac gamma matrices
$\boldsymbol{\gamma}_{\mathrm{F}}^{a}$ and the electromagnetic-gauge-covariant
derivatives $\vec{D}_{a}$ form eight-component spinors
$\boldsymbol{\gamma}_{\mathrm{F}}$ and $\vec{D}$, given in the supplemental
material supplemental  and in Ref. Partanen2023b . The latin indices are
associated with the tangent space and the Lie algebra, to be developed below.
They range over the four Lie algebra dimensions, i.e., $a\in\\{0,x,y,z\\}$.
Below, the Greek indices will denote the general space-time indices, which
range over the four general space-time dimensions
$\mu\in\\{x^{0},x^{1},x^{2},x^{3}\\}$.

The eight-spinor theory is formulated in terms of four $8\times 8$ bosonic
gamma matrices $\boldsymbol{\gamma}_{\mathrm{B}}^{a}$ and
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}=i\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{x}\boldsymbol{\gamma}_{\mathrm{B}}^{y}\boldsymbol{\gamma}_{\mathrm{B}}^{z}$.
These matrices are explicitly presented in Ref. Partanen2023b . They satisfy
the Dirac algebra, i.e., the Clifford algebra
$\mathcal{C}\ell_{1,3}(\mathbb{C})$. The defining property of the Dirac
algebra of $\boldsymbol{\gamma}_{\mathrm{B}}^{a}$ is the anticommutation
relation
$\\{\boldsymbol{\gamma}_{\mathrm{B}}^{a},\boldsymbol{\gamma}_{\mathrm{B}}^{b}\\}=2\eta^{ab}\mathbf{I}_{8}$,
where $\mathbf{I}_{8}$ is the $8\times 8$ identity matrix and $\eta^{ab}$ is
the Minkowski metric tensor with $\eta^{00}=1$ and
$\eta^{xx}=\eta^{yy}=\eta^{zz}=-1$. The general space-time gamma matrices
$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$ can be constructed from
$\boldsymbol{\gamma}_{\mathrm{B}}^{a}$ by the tetrad field $g_{a}^{\;\,\mu}$
Blagojevic2013 ; Sciama1964  as
$\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}=g_{a}^{\;\,\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{a}$.
In addition, we use $\mathbf{e}_{8}$ to represent the timelike unit eight-
spinor. Using $\boldsymbol{\gamma}_{\mathrm{B}}^{\mu}$ and
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ and denoting
$\bar{\Psi}=\Psi^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$ and
$\bar{\mathbf{I}}_{8}=\boldsymbol{\gamma}_{\mathrm{B}}^{0}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{0}$,
the generating Lagrangian density of gravity for the electromagnetic and Dirac
fields is given by Partanen2023b

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=\Big{[}\frac{\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}(\reflectbox{$\vec{\reflectbox{$D$}}$}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\mathbf{I}_{8}\vec{D})\psi$
$\displaystyle\hskip
11.38092pt\\!+\\!\frac{im_{\mathrm{e}}c^{2}}{2g_{\mathrm{g}}}\bar{\psi}\bar{\mathbf{e}}_{8}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}\mathbf{e}_{8}\psi\\!+\\!\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\partial}_{\nu}\bar{\mathbf{I}}_{8}^{\dagger}\Psi_{\Re}$
$\displaystyle\hskip
11.38092pt\\!+m_{\mathrm{e}}c^{2}\bar{\psi}\psi+\bar{\Psi}_{\Re}\Psi_{\Re}\Big{]}\sqrt{-g}.$
(1)

Here $\hbar$ is the reduced Planck constant, $m_{\mathrm{e}}$ is the electron
rest mass, and $g=\det(g_{\mu\nu})$ is the determinant of the metric tensor
$g_{\mu\nu}$. See the supplemental material supplemental  for technical
definitions of quantities. The constant $g_{\mathrm{g}}$ is _the coupling
constant of unified gravity_ , given in units of inverse distance. This
coupling constant is associated with the Yang-Mills gauge theory and it
follows from the internal symmetry of quantum fields associated with the
gauge-covariant derivatives to be introduced below. This constant does not
appear in the conventional general relativity and makes the present theory
fundamentally different. The tensor gauge field to be introduced will be a
quantum field in the same way as the fields of the other fundamental
interactions described by the standard model.

The first three terms under the square brackets of Eq. (1) represent the
generating Lagrangian densities of gravity for the kinetic and mass terms of
the Dirac and electromagnetic fields. They have been determined by reverse
engineering so that the variation of $\mathcal{L}_{0}$ in the special unitary
transformation, to be introduced below, becomes proportional to the symmetric
stress-energy tensor Partanen2023b ; supplemental . This leads to a Yang-Mills
gauge theory, which accounts for both the quantum electrodynamics and the
dynamics of the tensor gauge field. Since the symmetric stress-energy tensor
appearing in the Yang-Mills gauge theory is the source term of the
gravitational field, it follows that the emerging gauge field describes the
gravitational interaction of the particle fields of the standard model. One
should not be confused with the fact that the partial derivatives
$\vec{\partial}_{\nu}$ in Eq. (1) act on $\mathbf{I}_{8}$ and do not extend to
the spinors $\Psi_{\Re}$ and $\psi$. Even though these terms are seemingly
zero in $\mathcal{L}_{0}$, they give an essential contribution to the complete
gauge-invariant Lagrangian density after the special unitary symmetry
transformation and the gauge-covariant derivatives are introduced below.

##  III Special unitary symmetry

We follow the conventional Yang-Mills theory to seek for global symmetries
with respect to which the generating Lagrangian density of gravity in Eq. (1)
is invariant. Then, we introduce gauge fields to make these symmetries local.
The generating Lagrangian density trivially satisfies the U(1) symmetry of
quantum electrodynamics. This symmetry is satisfied locally since the
electromagnetic gauge field is included and the electromagnetic-gauge-
covariant derivative $\vec{D}$ is used. We next apply the special unitary
symmetry transformation, given by Partanen2023b

$\mathbf{I}_{8}\rightarrow\mathbf{U}\mathbf{I}_{8},\hskip
14.22636pt\text{where }\mathbf{U}=e^{i\phi_{a}\mathbf{t}^{a}}.$ (2)

Here $\phi_{a}$ is a real-valued four-component vector describing the symmetry
transformation parameters. The symmetry transformation matrix $\mathbf{U}$ has
determinant 1. The transformation generators $\mathbf{t}^{a}$ are constant
traceless Hermitian matrices given in terms of the complex-conjugated gamma
matrices as
$\mathbf{t}^{a}=(\boldsymbol{\gamma}_{\mathrm{B}}^{0}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{a})^{*}$.
This set of matrices generates the Clifford algebra
$\mathcal{C}\ell_{4,0}(\mathbb{C})$ with the anticommutation relation
$\\{\mathbf{t}^{a},\mathbf{t}^{b}\\}=2\delta^{ab}\mathbf{I}_{8}$, where
$\delta^{ab}$ is the Kronecker delta. The commutation relation is given by
$[\mathbf{t}^{a},\mathbf{t}^{b}]=if_{c}^{\;\,ab}\mathbf{t}^{c}$, where
$f_{c}^{\;\,ab}=-\frac{i}{8}\mathrm{Tr}(\mathbf{t}^{c}[\mathbf{t}^{a},\mathbf{t}^{b}])=2\varepsilon^{0cab}$
are real-valued constants, in which $\varepsilon^{abcd}$ is the Levi-Civita
symbol. The traces satisfy
$\mathrm{Tr}(\mathbf{t}^{a}\mathbf{t}^{b})=8\delta^{ab}$. The matrices
$\mathbf{t}^{a}$ are also generators of the Lie algebra, where the commutator
has the role of the Lie bracket and $f_{c}^{\;\,ab}$ are the totally
antisymmetric structure constants. This fact allows us to use the powerful
machinery of the Yang-Mills gauge theory Peskin2018 . All that follows is,
thus, a direct consequence of the gauge invariance with respect to the
symmetry transformation in Eq. (2) following the conventional approach of
quantum field theory.

The diagonal Lie algebra metric for raising and lowering the Lie algebra
indices supplemental  is defined in terms of the Kronecker delta and the
Killing form. For the inverse metric, we have
$\eta^{ab}=\delta^{ab}+\frac{1}{4}f_{c}^{\;\,ad}f_{d}^{\;\,bc}$. Thus, it
follows that the Lie algebra indices are raised and lowered by the constant
Minkowski metric $\eta_{ab}$ and its inverse $\eta^{ab}$. The space-time
indices are raised and lowered by the space-time-dependent metric $g_{\mu\nu}$
and its inverse $g^{\mu\nu}$, to be defined below. Therefore, the present
theory can be termed bimetric.

While the standard model of particles physics is based on the unitary group
U(1) and the special unitary groups SU(2) and SU(3), the symmetry group
corresponding to the transformation in Eq. (2) is a four-dimensional subgroup
of SU(8), which we denote SU(8)4D. The transformation in Eq. (2) commutes with
$\boldsymbol{\gamma}_{\mathrm{B}}^{5}$ and pairs of gamma matrices as
$[\mathbf{U},\boldsymbol{\gamma}_{\mathrm{B}}^{5}]=\mathbf{0}$ and
$[\mathbf{U},\boldsymbol{\gamma}_{\mathrm{B}}^{\rho}\boldsymbol{\gamma}_{\mathrm{B}}^{\sigma}]=\mathbf{0}$.
These relations are needed for the invariance of the Lagrangian density in the
gauge theory that follows. Furthermore, $\mathbf{U}$ commutes with the tensor-
field Lorentz transformation
$\boldsymbol{\Lambda}_{\mathrm{J}}=e^{\frac{1}{8}\Omega_{\rho\sigma}[\boldsymbol{\gamma}_{\mathrm{B}}^{\rho},\boldsymbol{\gamma}_{\mathrm{B}}^{\sigma}]}$
as $[\mathbf{U},\boldsymbol{\Lambda}_{\mathrm{J}}]=\mathbf{0}$, where
$\Omega_{\rho\sigma}$ is a matrix parametrizing the Lorentz transformation
Partanen2023b . A more detailed study of the symmetry properties of
$\mathbf{U}$ is left as a topic of further work.

##  IV Gauge-covariant derivative

The symmetry transformation in Eq. (2) is global for constant values of
$\phi_{a}$. To promote this global symmetry to a local symmetry, we allow
$\phi_{a}(x^{0},x^{1},x^{2},x^{3})$ to be space-time dependent. Since the
symmetry transformation matrices $\mathbf{U}$ are noncommuting for different
values of $\phi_{a}$, the symmetry transformation represents a noncommuting
symmetry, and our field theory is termed a non-Abelian gauge theory. A
prototype example of a non-Abelian gauge theory is the original theory of Yang
and Mills considering the proton-neutron doublet transformed under isotopic
spin Yang1954 ; Hooft2005 ; Peskin2018 ; Weinberg1996 . Following the standard
approach of the Yang-Mills theory Peskin2018 ; Weinberg1996 , it follows that
the generating Lagrangian density of gravity in Eq. (1) can be made locally
invariant in the symmetry transformation in Eq. (2) when we introduce the
gauge-covariant derivatives
$\vec{\boldsymbol{\mathcal{D}}}_{\nu}\mathbf{I}_{8}$ and $\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}\bar{\mathbf{I}}_{8}^{\dagger}$
as

$\displaystyle\vec{\boldsymbol{\mathcal{D}}}_{\nu}=\mathbf{I}_{8}\vec{\partial}_{\nu}-ig_{\mathrm{g}}\mathbf{g}_{\nu},\hskip
14.22636pt\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}=\mathbf{I}_{8}\vec{\partial}_{\nu}-ig_{\mathrm{g}}\bar{\mathbf{g}}_{\nu},$
$\displaystyle\text{where
}\mathbf{g}_{\nu}=g_{a\nu}\mathbf{t}^{a}=\boldsymbol{\eta}_{\nu}+\mathbf{h}_{\nu}=\eta_{ab}\partial_{\nu}x^{b}\mathbf{t}^{a}+h_{a\nu}\mathbf{t}^{a}.$
(3)

Here the Hermitian gauge field $\mathbf{g}_{\nu}$ is given in terms of
$\mathbf{t}^{a}$ and real-valued components $g_{a\nu}$. It is split into the
coordinate-related term
$\boldsymbol{\eta}_{\nu}=\eta_{ab}\partial_{\nu}x^{b}\mathbf{t}^{a}$, which
contains a derivative of the tangent-space coordinates, and the true
gravitational field term $\mathbf{h}_{\nu}=h_{a\nu}\mathbf{t}^{a}$. The gauge
field $\mathbf{g}_{\nu}$ and its parts $\boldsymbol{\eta}_{\nu}$ and
$\mathbf{h}_{\nu}$ are invariant in the Lorentz transformation as
$\boldsymbol{\Lambda}_{\mathrm{J}}\mathbf{g}_{\nu}\boldsymbol{\Lambda}_{\mathrm{J}}^{-1}=\mathbf{g}_{\nu}$.
With definitions above, the general space-time-dependent metric tensor is
given by $g_{\mu\nu}=\eta^{ab}g_{a\mu}g_{b\nu}$ and the inverse tetrad field
by $g_{\;\,\nu}^{a}=\eta^{ab}g_{b\nu}$. The consistency of these relations
becomes obvious from the field equations derived below. The gauge-covariant
derivatives transform by matrices $\mathbf{U}$ and
$\bar{\mathbf{U}}^{\dagger}$ as
$\vec{\boldsymbol{\mathcal{D}}}_{\nu}\mathbf{I}_{8}\rightarrow\mathbf{U}\vec{\boldsymbol{\mathcal{D}}}_{\nu}\mathbf{I}_{8}$
and $\mathstrut\mkern 1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}\bar{\mathbf{I}}_{8}^{\dagger}\rightarrow\bar{\mathbf{U}}^{\dagger}\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}\bar{\mathbf{I}}_{8}^{\dagger}$.
These relations require that the transformation of $\mathbf{g}_{\nu}$ is given
by
$\mathbf{g}_{\nu}\rightarrow(\mathbf{U}\mathbf{g}_{\nu}-\frac{i}{g_{\mathrm{g}}}\partial_{\nu}\mathbf{U})\mathbf{U}^{\dagger}$.
Using the gauge-covariant derivative operators
$\vec{\boldsymbol{\mathcal{D}}}_{\nu}$ and $\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}$ in place of the partial
derivatives $\vec{\partial}_{\nu}$ makes the generating Lagrangian density of
gravity in Eq. (1) invariant with respect to the local form of the symmetry
transformation in Eq. (2).

##  V Gauge-invariant Lagrangian density

To write the complete gauge-invariant Lagrangian density, we must also include
a gauge-invariant term that depends only on the true gravitational gauge field
$\mathbf{h}_{\nu}$. Utilizing the Yang-Mills gauge theory Peskin2018 , this
can be obtained from the commutator of the gauge-covariant derivatives
Peskin2018 ; Weinberg1996 . Excluding any coordinate-related inertial effects
by removing the dependence of $\boldsymbol{\eta}_{\nu}$, the relation
$[\vec{\boldsymbol{\mathcal{D}}}_{\mu}+ig_{\mathrm{g}}\boldsymbol{\eta}_{\mu},\vec{\boldsymbol{\mathcal{D}}}_{\nu}+ig_{\mathrm{g}}\boldsymbol{\eta}_{\nu}]=-ig_{\mathrm{g}}\mathbf{H}_{\mu\nu}$
is used to define an antisymmetric field strength tensor $\mathbf{H}_{\mu\nu}$
as

$\displaystyle\mathbf{H}_{\mu\nu}=\partial_{\mu}\mathbf{h}_{\nu}-\partial_{\nu}\mathbf{h}_{\mu}-ig_{\mathrm{g}}[\mathbf{h}_{\mu},\mathbf{h}_{\nu}]=H_{a\mu\nu}\mathbf{t}^{a},$
$\displaystyle\hskip
5.69046ptH_{a\mu\nu}=\partial_{\mu}h_{a\nu}-\partial_{\nu}h_{a\mu}+g_{\mathrm{g}}f^{\;\,bc}_{a}h_{b\mu}h_{c\nu}.$
(4)

The commutator term of $\mathbf{H}_{\mu\nu}$ is one of the prime novelties of
the Yang-Mills theory since it leads to direct interaction between the gauge
field quanta Peskin2018 . As conventional in the Yang-Mills theory, the field
strength tensor is not a gauge-invariant quantity. In the present case, the
redundant degrees of freedom are associated with the fact that the gauge field
$h_{a\nu}$ is not restricted to be symmetric, but the metric tensor, given in
terms of $h_{a\nu}$, is symmetric.

Using the inverse tetrad $g^{a}_{\;\,\rho}$, we can use $H_{a\mu\nu}$ to form
$H_{\rho\mu\nu}=g^{a}_{\;\,\rho}H_{a\mu\nu}$. Since this quantity has three
space-time indices instead of the two indices of the gauge theories of the
standard model, we have more possibilities to form gauge-invariant quantities.
Accordingly, following teleparallel gravity Bahamonde2023 ; Aldrovandi2012 ;
Krssak2019 ; Jimenez2020  in a form known as the teleparallel equivalent of
general relativity, we write the gauge-invariant Lagrangian density term
$\mathcal{L}_{\mathrm{g}}$ for the gauge field with the auxiliary quantities
$S^{\rho\mu\nu}$ and $K^{\rho\mu\nu}$, called the superpotential and
contortion, as

$\displaystyle\mathcal{L}_{\mathrm{g}}$
$\displaystyle=\frac{\sqrt{-g}}{4\kappa}S^{\rho\mu\nu}H_{\rho\mu\nu},$
$\displaystyle S^{\rho\mu\nu}$
$\displaystyle=-S^{\rho\nu\mu}=K^{\mu\nu\rho}-g^{\rho\nu}H_{\;\;\;\;\,\sigma}^{\sigma\mu}+g^{\rho\mu}H_{\;\;\;\;\,\sigma}^{\sigma\nu},$
$\displaystyle K^{\mu\nu\rho}$
$\displaystyle=\frac{1}{2}(H^{\nu\mu\rho}+H^{\rho\mu\nu}-H^{\mu\nu\rho}).$ (5)

Here $\kappa=8\pi G/c^{4}$ is Einstein’s constant, where $G$ is the
gravitational constant. In the limit where the coupling constant
$g_{\mathrm{g}}$ of unified gravity approaches zero, it is known from the
teleparallel equivalent of general relativity Bahamonde2023 ; Aldrovandi2012 ;
Krssak2019 ; Jimenez2020  that the Lagrangian density term in Eq. (5), agrees
with the Einstein-Hilbert Lagrangian density of general relativity up to a
total divergence. Consequently, in the classical regime, the dynamical
equations of gravity, following from teleparallel gravity and from the present
theory in the limit $g_{\mathrm{g}}\rightarrow 0$, are identical to those of
general relativity as discussed below.

Using the gauge-covariant derivatives and adding the gauge field term in Eq.
(5), the complete gauge-invariant generalization of the generating Lagrangian
density of gravity in Eq. (1) is given by

$\displaystyle\mathcal{L}$ $\displaystyle\\!=\\!\Big{[}\frac{\hbar
c}{4g_{\mathrm{g}}}\bar{\psi}(\reflectbox{$\vec{\reflectbox{$D$}}$}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\boldsymbol{\mathcal{D}}}_{\nu}\mathbf{I}_{8}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\bar{\mathbf{I}}_{8}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\vec{\boldsymbol{\mathcal{D}}}_{\nu}\mathbf{I}_{8}\vec{D})\psi$
$\displaystyle\hskip
11.38092pt\\!\\!+\\!\frac{im_{\mathrm{e}}c^{2}}{2g_{\mathrm{g}}}\bar{\psi}\bar{\mathbf{e}}_{8}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}\bar{\mathbf{I}}_{8}^{\dagger}\mathbf{e}_{8}\psi\\!+\\!\frac{i}{g_{\mathrm{g}}}\bar{\Psi}_{\Re}\mathbf{I}_{8}^{\dagger}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathstrut\mkern
1.5mu\vec{\boldsymbol{\mathcal{D}}}\mkern-13.0mu\raise
9.47217pt\hbox{$\scriptscriptstyle-$}_{\nu}^{\mkern
3.0mu\raise-3.91806pt\hbox{$\scriptstyle{\dagger}$}}\bar{\mathbf{I}}_{8}^{\dagger}\Psi_{\Re}$
$\displaystyle\hskip
11.38092pt\\!\\!+m_{\mathrm{e}}c^{2}\bar{\psi}\psi+\bar{\Psi}_{\Re}\Psi_{\Re}+\frac{1}{4\kappa}S^{\rho\mu\nu}H_{\rho\mu\nu}\Big{]}\sqrt{-g}.$
(6)

As shown in the supplemantal material supplemental , this quantity is equal to
the sum of the Lagrangian density of quantum electrodynamics and the tensor
gauge field term. Then, it is equal to the Lagrangian density of quantum
electrodynamics in the Minkowski metric limit $h_{a\nu}\rightarrow 0$. In the
general case, the gauge field $h_{a\nu}$ is obtained as a solution of the
field equations derived below.

##  VI Dynamical equations

Through the well-known Euler-Lagrange equations, one can derive the dynamical
equations for all fields appearing in the Lagrangian density in Eq. (6). The
resulting generalized Maxwell’s and Dirac’s equations for the pertinent fields
are presented in the supplemental material supplemental . The Euler-Lagrange
equations for $h_{a\nu}$ are given by $\partial\mathcal{L}/\partial
h_{a\nu}-\partial_{\rho}[\partial\mathcal{L}/\partial(\partial_{\rho}h_{a\nu})]=0$.
After some algebra, using the derivative rules of $h_{a\nu}$ supplemental , we
obtain the dynamical equation

$\nabla_{\rho}(S^{a\nu\rho})+g_{\mathrm{g}}f_{c}^{\;\,ab}h_{b\rho}S^{c\nu\rho}=\kappa(T^{a\nu}+\mathfrak{T}^{a\nu}).$
(7)

Here we use the coordinate-covariant derivative $\nabla_{\mu}$, which, due to
the antisymmetry of the last two indices of $S^{a\nu\rho}$, satisfies
$\nabla_{\rho}(S^{a\nu\rho})=\frac{1}{\sqrt{-g}}\partial_{\rho}(\sqrt{-g}\,S^{a\nu\rho})$.
From the equations above, it is obvious that the equations of the teleparallel
equivalent of general relativity are obtained from the present theory in the
limit $g_{\mathrm{g}}\rightarrow 0$.

The source term $T^{a\nu}=g^{a}_{\;\,\mu}T^{\mu\nu}$ in Eq. (7) is the total
symmetric stress-energy tensor of the electromagnetic and Dirac fields
Partanen2023b . In terms of the electromagnetic and Dirac field spinors, we
obtain

$\displaystyle T^{\mu\nu}$ $\displaystyle=\frac{i\hbar
c}{4}\bar{\psi}(\reflectbox{$\vec{\reflectbox{$D$}}$}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathbf{t}^{\mu}\boldsymbol{\gamma}_{\mathrm{F}}-\bar{\boldsymbol{\gamma}}_{\mathrm{F}}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\mathbf{t}^{\mu}\vec{D})\psi$
$\displaystyle\hskip
11.38092pt+\frac{m_{\mathrm{e}}c^{2}}{2}g^{\mu\nu}\bar{\psi}\psi+\bar{\Psi}_{\Re}\mathbf{t}^{\mu}\boldsymbol{\gamma}_{\mathrm{B}}^{\nu}\boldsymbol{\gamma}_{\mathrm{B}}^{5}\Psi_{\Re}.$
(8)

As shown in Ref. Partanen2023b , the terms associated with $\psi$ and
$\Psi_{\Re}$ agree, respectively, with the well-known stress-energy tensors of
the Dirac and electromagnetic fields Peskin2018 .

The second source term
$\mathfrak{T}^{a\nu}=g^{a}_{\;\,\mu}\mathfrak{T}^{\mu\nu}$ in Eq. (7) is the
total symmetric stress-energy tensor of the tensor gauge field. It is given by

$\mathfrak{T}^{\mu\nu}=\frac{1}{\kappa}\Big{(}S^{\sigma\rho\nu}H^{\;\,\;\;\mu}_{\sigma\rho}-\frac{1}{4}g^{\mu\nu}S^{\rho\sigma\lambda}H_{\rho\sigma\lambda}\Big{)}.$
(9)

This tensor is of the same form as the corresponding quantity in the
teleparallel equivalent of general relativity Bahamonde2023 ; Aldrovandi2012 ;
Krssak2019 .

##  VII Discussion

Before comparison of unified gravity with previous gravity theories, we want
to emphasize that a direct quantitative comparison is challenging since the
standard model, which is the starting point of unified gravity, is a quantum
field theory, and therefore, its Lagrangian density and Euler-Lagrange
equations cannot be directly compared with the corresponding particle systems
in classical physics. Such a comparison is possible only through coherent
coupled quantum field states. This quantization and the coherent states of the
coupled quantum fields are a topic of further work.

It is obvious that Eq. (7) is subject to quantization using the methods of
quantizing Yang-Mills theories Hooft1971a ; Hooft1971b ; Peskin2018 ;
Weinberg1996 . The resulting quanta of the field should be elementary
particles. Since the present gauge field is a tensor field, its quanta must be
spin-2 tensor bosons Misner1973 . Thus, the particle associated with the gauge
field $h_{a\nu}$ must be the graviton, the force carrier for gravity. As
characteristic for Yang-Mills theories, the gravitons are mutually interacting
through the commutator term of the field strength tensor. After replacing the
classical fields with pertinent quantum operators, Eq. (7) can be viewed as a
quantum field theoretical generalization of Einstein’s field equations of
general relativity.

With the above limitation in mind, the dynamical equation of unified gravity
in Eq. (7) becomes, for weak coupling, $g_{\mathrm{g}}\rightarrow 0$,
equivalent to the dynamical equation of the teleparallel equivalent of general
relativity Bahamonde2023 . Therefore, in this limit, the predictions of
unified gravity agree with those of general relativity. However, the theories
are in general fundamentally different. Detailed study of unified gravity in
the regime of strong gravitational fields for nonzero $g_{\mathrm{g}}$ is left
as a topic of further work. It is expected that, in the case of strong fields,
the commutator term of the field strength tensor in Eq. (4) becomes
substantial when the coupling constant $g_{\mathrm{g}}$ is nonzero, even if it
is small. Such strong gravitational fields are encountered in black holes and
at the possible beginning of time. Therefore, unified gravity provides a tool
for the investigation of intense gravitational fields beyond the applicability
of general relativity.

##  VIII Conclusion

The internal SU(8)4D symmetry of the eight-spinor formulation of quantum
fields in the standard model Partanen2023b ; supplemental  leads to unified
gravity, the Yang-Mills gauge theory of gravity. Through its relation to the
teleparallel equivalent of general relativity for $g_{\mathrm{g}}\rightarrow
0$, unified gravity agrees with general relativity in the limit of weak
coupling. Unified gravity is based on the generating Lagrangian density of
gravity, which is related to the conservation of the stress-energy tensor, and
which can be made locally invariant in the special unitary transformation by
introducing a tensor gauge field. The entire dynamics of the known particles
and fields, including gravity, can be described by a single master Lagrangian
of the Universe supplemental  through the Euler-Lagrange equations in a
unified way. Therefore, our work opens a new era of quantum science of
gravity. After rigorous quantization of unified gravity using known methods to
quantize Yang-Mills theories, physicists may finally have the long-sought tool
for the investigation of intense gravitational fields in black holes and the
possible beginning of time. Unified gravity includes a single undetermined
parameter, which is the coupling constant $g_{\mathrm{g}}$. The determination
of the value of this parameter is of large interest.

###### Acknowledgements.

This work has been funded by the Academy of Finland under Contract No. 349971.
Wolfram Mathematica has been extensively used to verify the equations of the
present work. We thank Will Barker and Grigory Volovik for useful discussions.
Will Barker pointed out Barker2023  that the theory following from the
Lagrangian density term $\mathcal{L}_{\mathrm{g}}$ of the first version of our
manuscript was not directly comparable to the known theories of gravity, which
led us to the redefinition of this term by using degrees of freedom enabled by
the theory.

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aainstitutetext: University of Zurich, Switzerlandbbinstitutetext: ETH Zurich,
Switzerlandccinstitutetext: University of California, Berkeley,
USAddinstitutetext: Fermi National Accelerator Laboratory, Batavia, IL 60510,
USAeeinstitutetext: European Organization for Nuclear Research (CERN),
Switzerland

# Autoencoders for Semivisible Jet Detection

Florencia Canelli b Annapaola de Cosa c Luc Le Pottier b Jeremi Niedziela d
Kevin Pedro e Maurizio Pierini<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The production of dark matter particles from confining dark sectors may lead
to many novel experimental signatures. Depending on the details of the theory,
dark quark production in proton-proton collisions could result in semivisible
jets of particles: collimated sprays of dark hadrons of which only some are
detectable by particle collider experiments. The experimental signature is
characterised by the presence of reconstructed missing momentum collinear with
the visible components of the jets. This complex topology is sensitive to
detector inefficiencies and mis-reconstruction that generate artificial
missing momentum. With this work, we propose a signal-agnostic strategy to
reject ordinary jets and identify semivisible jets via anomaly detection
techniques. A deep neural autoencoder network with jet substructure variables
as input proves highly useful for analyzing anomalous jets. The study focuses
on the semivisible jet signature; however, the technique can apply to any new
physics model that predicts signatures with jets from non-Standard Model
particles.

##  1 Introduction

Hidden Valley models of dark matter allow for a strong interaction in the dark
sector [1]. Such a strong interaction would cause showering of the dark
particles, producing both invisible dark matter and Standard Model (SM)
hadrons. As a result, so-called semivisible (SV) jets would be formed,
containing some fraction $r_{inv}$ of invisible particles [2]. Depending on
the mass of the mediator $m_{{\text{Z}}^{\prime}}$, SV jets could be produced
at the Large Hadron Collider (LHC) energies and registered in detectors [3].

Events containing such jets would be characterised by missing transverse
energy $\cancel{E}_{T}$ being aligned with a jet, which is a scenario
difficult to distinguish from detector issues and therefore usually neglected
in LHC data analyses. Moreover, the details of the Hidden Valley physics
depend on a number of parameters. There are four parameters that most
influence the kinematic behavior in the final state:

  * •

$\alpha_{\text{dark}}$– the coupling constant of the dark sector strong
interaction equivalent,

  * •

$m_{{\text{Z}}^{\prime}}$– the mass of the mediator,

  * •

$m_{\text{dark}}$– the mass of the dark hadrons,

  * •

$r_{\text{inv}}$– the fraction of stable, invisible dark hadrons.

These parameters could be a function of other parameters in a fully-specified
Hidden Valley model, as well as parameters that impact the modeling of strong
dynamics in the event generator (hadronization, fragmentation). This leads to
an extremely large model space with a huge number of possible scenarios that
can easily evade any constraints from e.g. cosmological measurements. It is
clearly impractical to perform dedicated searches for all possible model
variations.

In order to gain robustness against detector effects, as well as details of
the model implementation, anomaly detection techniques such as unsupervised
deep learning can be used. Those are designed to detect objects significantly
different from the training sample, without prior knowledge of signal
characteristics. The presented approach is suitable to search for a wide class
of new physics models containing anomalous jets, including the SV jets
stemming from Hidden Valley models, which are the main focus of this work.

Here, an autoencoder-based analysis for SV jets detection will be presented.
Unlike previous attempts [4, 5, 6, 7] that use jet images (four-momenta of jet
constituents), we incorporate high-level jet features and substructure
variables computed based on the four-momenta of the jet constituents: Energy
Flow Polynomials (EFPs) [8], Energy Correlation Functions (ECFs) [9] and their
ratios: C2 and D2, as well as the jet $p_{T}$ dispersion $p_{T}D$ [10] and jet
axes [11]. These prove to be highly useful in discriminating between the QCD
background and the SV jet signal [12].

The trained model can then be used to tag anomalous jets, as the autoencoder
has been trained to recognize QCD jets in the signal region, but not SV signal
jets (or any other anomalous jets, such as emerging jets [13]).

It is worth noting that, while the performance is assessed using SV jet models
as a benchmark, the training process is independent of the signal
implementation, unlike in case of the supervised learning. Given that the
landscape of possible Hidden Valley scenarios is so vast, using one classifier
(even parametric [14]) per signal model becomes impractical.

##  2 Data samples

###  2.1 Generation

Both multi-jet QCD and dark sector samples are generated using PYTHIA8 [15]
and reconstructed with DELPHES [16] providing a detector response
approximating that of CMS. Proton-proton collisions at a centre-of-mass energy
of 13 TeV are considered, with approximately 50 collisions per bunch crossing.
The particle flow reconstruction algorithm distributed with DELPHES is used.
In particular, jets are clustered from reconstructed particles and pileup
reduction is applied.

To examine the robustness of autoencoder models to varying signal parameters,
multiple SV jet samples were generated with values of the fraction of stable
jet hadrons $r_{\text{inv}}\in\\{0.3,0.5,0.7\\}$ and the ${\text{Z}}^{\prime}$
boson mass $m_{{\text{Z}}^{\prime}}\in\\{1.5,2.0,2.5,3.0,3.5,4.0\\}$. The
number of events generated varies depending on the sample, accounting for the
different selection efficiency (see Sec. 2.2):

  * •

QCD: $3.7\cdot 10^{7}$ events,

  * •

SV jets, $m_{{\text{Z}}^{\prime}}>1.5$ TeV: 50k events per sample,

  * •

SV jets, $m_{{\text{Z}}^{\prime}}=1.5$ TeV, $r_{\text{inv}}\in\\{0.3,0.5\\}$:
100k events per sample,

  * •

SV jets, $m_{{\text{Z}}^{\prime}}=1.5$ TeV, $r_{\text{inv}}=0.7$: 140k events
per sample.

###  2.2 Preselection

A set of preselection criteria (based on those outlined in Ref. [2]) is
applied to both training and testing samples on an event-by-event basis. These
requirements are aimed at isolating the SV jet signal, as well as reproducing
the realistic triggering capabilities of LHC experiments. The signal region is
defined to include events passing the following criteria:

  * •

at least 2 jets with $|\eta|<2.4$ and $p_{T}>200$ GeV

  * •

then, for the two leading jets:

    * –

$|\Delta\eta|<1.5$,

    * –

$m_{T}>1500$ GeV,

    * –

$\cancel{E}_{T}/m_{T}>0.25$,

where $\eta$ is the pseudorapidity, $p_{T}$ is the transverse momentum for
each jet, $\Delta\eta$ is the absolute dijet $\eta$ difference,
$\cancel{E}_{T}$ is the missing momentum in the transverse plane for the
event, and $m_{T}$ is the transverse mass of the leading dijet system and the
$\cancel{E}_{T}$.

The selection efficiency was found to be at the level of $0.13\%$ for the QCD
events and between $0.2$ and $15.3\%$ for SV jets, depending on the signal
model parameters. The number of QCD jets after applying selections is
$\approx$100k (split between training, validation, and testing) and varies
between 500 and 15k for SV jets, depending on the model parameters.

###  2.3 Feature Selection

Since the goal of this study is to find anomalies on the basis of tagging
individual anomalous jets rather than anomalous events, a set of jet-level and
jet substructure variables is determined for the training.

The coordinates of each jet ($\eta$ and $\phi$) are included in the training
to allow the network to learn about problematic regions of the detector and
avoid tagging noise or other detector failures as anomalous signals.

In order to avoid bias, the transverse momentum of the jet was not included in
the training. Moreover, the jet $p_{T}$ distribution was flattened (by
applying appropriate weights) for the training.

A number of other jet-level and substructure variables have been considered
for the training. A fraction of them were rejected due to poor signal-
background discrimination. The correlations between remaining variables were
tested, as autoencoders tend to perform better with uncorrelated inputs. It
was found that for QCD and SV jets, the EFPs are fully correlated with each
other, as well as with the girth [17], Les-Houches Angularity (LHA) [18], and
normalized ECFs: e2 and e3. Because of that, it was decided to only keep one
of those variables for the training, namely EFP1 [8] which exploits pairwise
relations between jet constituents.

The jet constituent four-momenta have also been considered; however, they
proved not to bring any significant gain in performance, while increasing the
complexity (${\approx}100$ instead of ${\approx}10$ input features) and
training time of the autoencoder. This can be understood since EFPs and ECFs
are included in the training and provide a linear basis for characterizing jet
constituent substructure, making up for the discrimination that is lost when
using high-level jet features rather than jet constituents.

Ultimately, the following set of jet-related input variables is selected:
$\eta$, $\phi$, invariant mass $m_{j}$, jet fragmentation function $p_{T}D$,
jet ellipse minor and major axes, EFP1, and ECF ratios: C2 and D2. The
distributions of those variables for the QCD background and selected signals
is presented in Fig. 1.

The first three of these features are provided directly by the DELPHES
reconstruction algorithm. The remaining ones are calculated using a list of
jet constituents, which consists of all particles in the event which lie
within a cone of radius $R=0.8$ from the jet axis.

All of the selected high level features discussed here provide some meaningful
discrimination power, with the exception of the angular coordinates, $\eta$
and $\phi$. Since those are randomly distributed, they provide no meaningful
information about QCD jet structure. However, given the ability of
autoencoders to learn a compressed representation of input signals, it is
feasible that they might also learn the locations and behaviors of defective
detector components. For instance, they could learn location of energy spikes
in certain detector locations as normal behavior that does not indicate an
anomaly associated with a potential signal. Such detector malfunctions are not
included in the DELPHES reconstruction; however, they may be present in the
collision data, on which the autoencoder can be trained.

Figure 1: Distributions of input variables for QCD background and selected
signal models.

##  3 Models

All unsupervised models presented in the following sections are trained using
the QCD data sample described in Sec. 2 as background, with a 70:15:15 split
between training, validation, and testing, respectively.

###  3.1 Autoencoders

An autoencoder is a neural network which approximates the identity
transformation as the application of a function $f$ from the input space to a
latent space (encoding) and a function $f^{-1}$ from the latent space back to
the input space (decoding), which provides an approximation of the input as
the output. The constraint forces the autoencoder to find hidden relationships
between features in the input data set. In the case of a neural autoencoder,
this constraint takes the form of a bottleneck in the network architecture: an
input vector $\vec{x}\in\mathbb{R}^{n}$ is compressed into a latent space
vector $\vec{l}\in\mathbb{R}^{m}$ with $m<n$, enforced by an $m$-node layer
within the architecture of an autoencoder with input dimension $n$. This
latent space vector is then reconstructed by a decoding network in an attempt
to match the input values.

Figure 2: Neural Network architecture.

The autoencoder is trained to minimize the reconstruction error of each
sample, defined for an $n$-dimensional input feature vector $\vec{x}$ with
components $x_{1},...,x_{n}$. A number of loss functions were considered and
the mean absolute error was selected, as it had the best performance on the
considered benchmark data sets:

$\text{MAE}=\frac{1}{N}\sum_{i}^{N}\left|x^{\prime}_{i}-x_{i}\right|$ (1)

where $x_{i}$ is the original value and $x^{\prime}_{i}$ is the reconstructed
value for the $i^{th}$ feature of a jet.

The training data consists of the simulated QCD background events that fall
within our signal region as per the cuts described in Sec. 2.

Once a model is trained to recognize the background, one can evaluate it on
unlabeled data and use the reconstruction errors as a measure of similarity of
a given sample to the training data set. Typically, a model trained on QCD
will return low reconstruction loss for samples similar to those it was
trained on, and high loss for samples that are more complex [19]. In case of a
search for new physics, an autoencoder would be trained and evaluated, with
the largest 10%, 1%, and 0.1% of its reconstruction losses isolated and
analyzed.

###  3.2 Alternative Models

Along with neural autoencoders - the simplest form of autoencoder - other
types of autoencoders were considered:

  * •

Variational Autoencoders [20] (VAE): provide probabilistic representation of
the input data, which can give higher performance in some applications.

  * •

Principal Component Analysis (PCA) [21]: can be viewed as a linear autoencoder
with no hidden layers, which allows determining the weights analytically.

The results obtained with the VAE and PCA will be reported in this work.
Sparse Autoencoders (SAE) [22] were investigated in the early stages of this
study and found to give worse results than the dense autoencoder; therefore,
they will not be included.

###  3.3 Model Optimization

In order to determine the optimal architecture, normalization, and hyper-
parameters for the SV jet search, a large number of models were trained and
evaluated. The optimization was focused on maximizing the Area Under Receiver
Operating Characteristic (ROC) Curve (AUC). For each set of hyper-parameters
and each choice of architecture, 80% of best models were selected and the
average AUC over all signal samples and all models was used to choose the
highest performing one.

It was found that the best results are obtained by standardizing the data,
i.e., by normalizing them such that the mean of the distribution is at zero
and the standard deviation is unity. An extensive architecture scan was
performed and the architecture with two hidden layers of size 10, followed by
the bottleneck of size 6 was found to be optimal. The activation function of
the hidden layers and the bottleneck is ‘elu’ [23] and the output layer uses
linear activation.

The autoencoder models were implemented in Keras/TensorFlow [24, 25]. Each
model was trained for 200 epochs (with early stopping enabled), with a batch
size of 256. The learning rate was set to $10^{-6}$ and the optimizer found to
yield the best performance was ‘Nadam’ [26].

##  4 Results

In this section, the results from the best autoencoder will be presented. The
best autoencoder was found by training 150 models with the optimal hyper-
parameters and varying random seed and selecting the best performing one in
terms of AUC. No signs of overfitting were observed, and the loss evolution
was smooth and saturated after around 100 epochs.

###  4.1 Reconstructed Distributions

The differences between input and reconstructed jet features are shown for
both SVJ and QCD data in Fig. 3. One can observe that, in almost all cases,
the difference is larger for the signals than for the background. What is
more, the distributions for the signal do not change with varying mass;
however there is a slight difference in shape when moving from
$r_{\text{inv}}$ = 0.3 to 0.7.

Figure 3: Distributions of difference between input and reconstructed jet
features for QCD and a few example SVJ signals.

###  4.2 Reconstruction Loss

An important figure of merit is the distribution of losses for the background
and signal samples. As shown in Fig. 4, QCD tends to have lower loss values
than SVJ, providing good discrimination.

In order to achieve the best possible performance, alternative approaches to
autoencoder training and evaluation were studied. For those, reference
distributions were built for each node of the network from the training data
set, using the reconstructed variables, MAE values, or even values from the
latent space nodes. The performance was then evaluated by comparing values
from the tested jets to those reference distributions.

The comparison was done using the negative log-likelihood, as well as the
Mahalanobis distance [27]. It was found that none of the six combinations
performed better than the MAE loss function.

Figure 4: Reconstruction loss for QCD and different SVJ signals.

###  4.3 Discrimination Power

In order to compare discrimination power of different architectures, one can
examine the ROC curves and AUC values for each generated signal. As a
reference for the best-case performance, we considered a Boosted Decision
Trees (BDT), providing a strong baseline for supervised algorithms. The BDT
was implemented using SKLearn’s AdaBoostClassifier, trained on a mixture of
all signals, and had its hyper-parameters optimized for maximum AUC. The input
features were the same as for the autoencoder. For the final comparison, 10
models were trained and the best one was selected (although the performance
spread between different models was negligible).

A comparison of ROC curves for the autoencoder, BDT, PCA and VAE is shown in
Fig. 5. As can be seen, the BDT classifier provides the best performance
regardless of the signal considered, although it is rather closely followed by
the autoencoder. What is more, as will be shown in the further part of this
section, the AE can become more powerful than the BDT if the latter was
trained on a wrong signal hypothesis. The VAE has been trained predominantly
on the reconstruction loss and evaluated on reconstruction loss exclusively,
neglecting the KL divergence [28]; therefore, it could be further improved
with a dedicated study. The PCA performance is significantly worse than those
of autoencoders and BDT.

Figure 5: Comparison of the ROC curves for the autoencoder, PCA, BDT and VAE
for a few selected signals.

Figure 6 presents a comparison of the AUC values between the autoencoder and
BDT for each of the signals. It is clear from this plot that signals with
lower values of $r_{\text{inv}}$ are much more efficiently identified,
although all signals can be tagged at relatively high rates. Moreover, even
though the autoencoder performs worse than the BDT in all signal cases, it is
still quite good, with the added benefit of requiring no particular signal to
train against.

###  4.4 Robustness

As an additional test, the autoencoder performance was compared to that of a
BDT trained on the wrong signal hypothesis. This allows to study the
generalization power of the autoencoder against that of a supervised model and
assess their ability to discover “by accident” a signal other than the
targeted one. As can be seen in the rightmost panel of Fig.6, a clear decrease
in the performance can be observed in two cases, up to $19\%$ worse when
trained on $m_{{\text{Z}}^{\prime}}=2$ TeV, $r_{\text{inv}}=0.7$, but tested
on $m_{{\text{Z}}^{\prime}}=4$ TeV, $r_{\text{inv}}=0.3$. What is remarkable
is that in this particular case, the BDT performs worse than the autoencoder,
which was trained without any information about the signal characteristics.

Another robustness test was to evaluate both the BDT and the AE on signal
samples with varying $m_{\text{dark}}$ values, with $m_{{\text{Z}}^{\prime}}$
fixed to 3 TeV. The results of this study are presented in Fig.7. Here, one
can see again that in certain cases the BDT trained on a wrong signal
hypothesis can be outperformed by the AE. In this scenario, the AE becomes
more powerful than a classifier for $m_{\text{dark}}=100$ GeV, when the BDT
training was performed on a mixture of 18 signal samples (with varying
$m_{{\text{Z}}^{\prime}}$ and $r_{\text{inv}}$) with $m_{\text{dark}}=20$ GeV.
More details on the signal samples with varying $m_{\text{dark}}$ values can
be found in Appendix A.

We noticed that training the BDT on a specific signal point with low
$r_{\text{inv}}$ leads to higher efficiency for signals with higher
$r_{\text{inv}}$ with respect to training on a mixture of all signal points.
This result comes from the fact that low $r_{\text{inv}}$ jets are more
similar to background jets. Such conclusions suggest that it might be a better
strategy to train a classifier on low $r_{\text{inv}}$ signals rather than on
a mixture of low and high parameter values.

From this test, we conclude that it would be preferable for an LHC signal-
specific search to have a signal-agnostic equivalent exploiting autoencoders
to complement model-dependent searches, as opposed to relying on accidental
generalization properties of the signal-specific selection algorithm. The
synergy of the two approaches would enhance the chances of discovering a
signal.

Figure 6: Left and middle panels: comparison of AUC values of the autoencoder
and BDT. Right panel: AUC values for a BDT trained on a signal with parameters
different from those it was tested on. E.g., the AUC value presented in the
top left corner of the table comes from a model trained on the lower right
corner sample. Figure 7: Comparison of AUC values of the autoencoder and BDT
with varying $m_{\text{dark}}$ values. The BDT was trained on a mixture of all
signals with $m_{\text{dark}}=20$ GeV.

###  4.5 Sensitivity to Semivisible Jet Models

As a last step, a threshold was optimized for the SVJ signal samples to tag
jets with loss > 0.06 as anomalous. Doing so, one can mimic the impact of the
AE algorithm on a real analysis, estimating expected upper limits. Events
passing the preselection were grouped in categories with 0, 1 or 2 SV jets,
and dijet transverse mass distributions for the signal and background in these
categories were used to set approximate exclusion limits on the cross section,
depending on the ${\text{Z}}^{\prime}$ mass. The results, which take into
account basic uncertainties (luminosity and trigger), are presented in Figs.
8a and 8b and compared with the example theoretical cross section. The results
without applying an autoencoder-based tagger are also shown.

As expected, the limits are stronger for lower values of $r_{\text{inv}}$ and
the improvement from using the tagger is more pronounced in this region. The
presented results show that one can expect an approach based on the
autoencoder to be able to cover a large area of the phase space.

(a) $r_{\text{inv}}$ = 0.3

(b) $r_{\text{inv}}$ = 0.7

Figure 8: The cross section limits vs. $m_{{\text{Z}}^{\prime}}$ for different
values of $r_{\text{inv}}$, scaled to the luminosity of the full Run 2 of the
LHC.

##  5 Conclusion

This study shows that the usage of a neural autoencoder based on physics-
motivated high-level features, despite its simplicity, is extremely effective
as an anomaly detection model. This is especially true in cases of a signal
with widely varying parameters, such as semivisible jets, as the model is
trained on the background sample only and is therefore highly model
independent. The projections of the exclusion limits for this particular class
of models show a 40-60% improvement in the search reach compared to a study
without such an anomalous jet tagger. What is more, the autoencoder could be
trained on the real data, accounting for any effects of an imperfect detector
simulation.

This result demonstrates the usefulness of the autoencoder network and jet
substructure variables such as Energy Flow Polynomials, Energy Correlation
Functions, the jet $p_{T}$ dispersion $p_{T}D$, and the jet ellipse axes for
the detection of anomalous jets. It is also worthwhile to acknowledge certain
aspects that should be further investigated in the future.

The assumption that the network should be insensitive to detector effects
could be tested by simulating such issues and contaminating the training
sample with affected events. Next, in order to train autoencoder on the
collision data, one has to verify that a small admixture of signal events in
the training sample does not influence the performance. Finally, one could try
to further exploit the four-momenta of the jet constituents using
convolutional or graph autoencoders, which should preserve information about
the spatial distribution of particles within the jet.

## Acknowledgements

M. Pierini is supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (Grant Agreement
No. 772369). A. de Cosa and J. Niedziela are supported by the Swiss National
Science Fundation (SNFS) under the SNSF Eccellenza program (PCEFP2_186878). L.
Le Pottier is supported by the Swiss State Secretariat for Education, Research
and Innovation (SERI) under the ThinkSwiss program. K. Pedro is supported by
the Fermi National Accelerator Laboratory, managed and operated by Fermi
Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S.
Department of Energy.

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##  Appendix A $m_{\text{dark}}$ Variations

Figure 9 displays the input variables used in training the tagger algorithms
for signal samples with different $m_{\text{dark}}$ values. In particular, the
effect of a large $m_{\text{dark}}$ value can be seen in the $m_{j}$
distribution. Figure 10 displays the reconstruction error for these variables
and signal samples after applying the AE. The signal distributions show only
moderate differences over a large range of $m_{\text{dark}}$ values. These
figures can be compared, respectively, to Figs. 1 and 3, which show the
variations in $r_{\text{inv}}$ and $m_{{\text{Z}}^{\prime}}$.

Figure 9: Distributions of input variables for QCD background and signal
models with $m_{{\text{Z}}^{\prime}}$ = 3 TeV and varying $m_{\text{dark}}$
and $r_{\text{inv}}$ values. Figure 10: Distributions of difference between
input and reconstructed jet features for QCD and signal models at
$m_{{\text{Z}}^{\prime}}$ = 3 TeV and varying $m_{\text{dark}}$ and
$r_{\text{inv}}$ values.