The more general transformation (137) is a dynamical symmetry, meaning that it
leaves the stochastic dynamics invariant, precisely if the right-hand side of
the stochastic-quantum dictionary (26) remains unchanged:

$\mathrm{tr}(\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\tilde{P}_{i}\Theta\mathopen{}\mathclose{{}\left(t}\right)\tilde{P}_{j})=\mathrm{tr}(\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\Theta\mathopen{}\mathclose{{}\left(t}\right)P_{j}).$
(138)

This condition is equivalent to the statement that

$\mathrm{tr}(\tilde{\Theta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)P_{j})=\mathrm{tr}(\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\Theta\mathopen{}\mathclose{{}\left(t}\right)P_{j}),$
(139)

where

$\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)\equiv
V\Theta\mathopen{}\mathclose{{}\left(t}\right)V^{\dagger}.$ (140)

Re-expressing both sides of the equivalent condition (139) in terms of squared
absolute values, as in (17), one sees that (140) is a dynamical symmetry
precisely if

$|\tilde{\Theta}_{ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}=|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}.$
(141)

It follows immediately that
$\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)$ can differ from
$\Theta\mathopen{}\mathclose{{}\left(t}\right)$ by at most a Schur-Hadamard
gauge transformation (45), so a necessary and sufficient condition for a
unitary matrix $V$ to give a dynamical symmetry is that

$V\Theta\mathopen{}\mathclose{{}\left(t}\right)V^{\dagger}=\Theta\mathopen{}\mathclose{{}\left(t}\right)\odot\begin{pmatrix}e^{i\theta_{11}\mathopen{}\mathclose{{}\left(t}\right)}&e^{i\theta_{12}\mathopen{}\mathclose{{}\left(t}\right)}\\\
e^{i\theta_{21}\mathopen{}\mathclose{{}\left(t}\right)}&\ddots\\\
&&e^{i\theta_{NN}\mathopen{}\mathclose{{}\left(t}\right)}\end{pmatrix}.$ (142)

As special cases, this condition includes _unitary_ dynamical symmetries,

$V\Theta\mathopen{}\mathclose{{}\left(t}\right)V^{\dagger}=\Theta\mathopen{}\mathclose{{}\left(t}\right),$
(143)

as well as _anti-unitary_ dynamical symmetries,

$V\Theta\mathopen{}\mathclose{{}\left(t}\right)V^{\dagger}=\overline{\Theta\mathopen{}\mathclose{{}\left(t}\right)},$
(144)

but (142) may also open up the possible existence of dynamical symmetries
distinct from these two cases.

For the specific case of an anti-unitary dynamical symmetry, note that if one
redefines $V\mapsto\overline{V}$, which is still unitary, then one can re-
express (144) in the somewhat more conventional form

$VK\Theta\mathopen{}\mathclose{{}\left(t}\right)KV^{\dagger}=\Theta\mathopen{}\mathclose{{}\left(t}\right).$
(145)

Here $K$ denotes the _complex-conjugation operator_ , meaning that $K$ is an
_involution_ ,

$K^{2}=1,$ (146)

and, for any $N\times N$ matrix $X$, one has

$KXK=\overline{X}.$ (147)

The composite operator $VK$ as a whole is then said to be an _anti-unitary
operator_. Anti-unitary operators play an important role in describing _time-
reversal symmetries_.222222Intriguingly, because $K$ anticommutes with $i$,
meaning that $Ki=-iK$, the three mathematical objects $i$, $K$, and $iK$
satisfy
$-i^{2}=K^{2}=\mathopen{}\mathclose{{}\left(iK}\right)^{2}=iK\mathopen{}\mathclose{{}\left(iK}\right)=1$,
and therefore generate a Clifford algebra isomorphic to the _pseudo-
quaternions_ [64]. In a sense, then, the Hilbert spaces of quantum systems are
actually defined not over the complex numbers alone, but over the pseudo-
quaternions.

If
$\Theta\mathopen{}\mathclose{{}\left(t}\right)=U\mathopen{}\mathclose{{}\left(t}\right)$
is unitary, then $V\Theta\mathopen{}\mathclose{{}\left(t}\right)V^{\dagger}$
will likewise be unitary. In that case, suppose either that $V$ is
continuously connected to the identity matrix $\mymathbb{1}$ by some smooth
parameter, with a corresponding self-adjoint generator $G=G^{\dagger}$, or,
alternatively, that $V$ is an involution, meaning that $V^{2}=\mymathbb{1}$,
in which case $G\equiv V=V^{\dagger}$ is _itself_ self-adjoint. Either way,
the expectation value $\mathopen{}\mathclose{{}\left\langle
G\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle$ is a physically
meaningful quantity, and _Noether’s theorem_ easily follows as the statement
that this expectation value is constant in time, or _conserved_ :

$\mathopen{}\mathclose{{}\left\langle
G\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle=\mathrm{tr}(GU\mathopen{}\mathclose{{}\left(t}\right)\rho\mathopen{}\mathclose{{}\left(0}\right)U^{\dagger}\mathopen{}\mathclose{{}\left(t}\right))=\mathopen{}\mathclose{{}\left\langle
G\mathopen{}\mathclose{{}\left(0}\right)}\right\rangle.$ (148)

###  VI.2 Dilations

In most textbook treatments of quantum theory, a quantum system is
axiomatically defined as a particular Hilbert space together with a preferred
set of self-adjoint operators designated as observables with predetermined
physical meanings, along with a particular Hamiltonian to define the system’s
time evolution.232323In some circumstances, it may turn out to be more
convenient to define a quantum system by a formal C*-algebra of observables
alone, without picking a specific Hilbert-space representation [65, 6, 66].
From that point of view, modifying a system’s Hilbert-space formulation in any
nontrivial way would necessarily mean fundamentally modifying the system
itself.

From the alternative point of view developed in this paper, by contrast, a
Hilbert-space formulation is merely a collection of mathematical tools for
constructing the dynamics of a given generalized stochastic system. The
generalized stochastic system itself is ultimately defined by a configuration
space and a dynamical law that stand apart from any arbitrary choice of
Hilbert-space formulation. As a consequence, one is free to modify a
generalized stochastic system’s Hilbert-space formulation as needed, much like
changing from one gauge choice to another in a gauge theory, or like adding
physically meaningless variables to the Lagrangian formulation of a
deterministic classical system.

With this motivation in place, recall again the basic stochastic-quantum
dictionary (26):

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}(\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\Theta\mathopen{}\mathclose{{}\left(t}\right)P_{j}).$
(149)

The Hilbert-space formulation expressed by the right-hand side can be
manipulated for convenience, provided that the left-hand side of the
dictionary remains unchanged.

In particular, for any integer $D\geq 2$, one can freely enlarge, or _dilate_
, the Hilbert-space formulation to a larger dimension $ND$ by the following
_dilation transformation_ :

$\mathopen{}\mathclose{{}\left.\begin{aligned}
\Theta\mathopen{}\mathclose{{}\left(t}\right)&\mapsto\Theta\mathopen{}\mathclose{{}\left(t}\right)\otimes\mymathbb{1}^{\mathcal{I}},\\\
P_{i}\mathopen{}\mathclose{{}\left(t}\right)&\mapsto
P_{i}\mathopen{}\mathclose{{}\left(t}\right)\otimes\mymathbb{1}^{\mathcal{I}},\\\
P_{j}\mathopen{}\mathclose{{}\left(t}\right)&\mapsto
P_{j}\mathopen{}\mathclose{{}\left(t}\right)\otimes
P_{\gamma}^{\mathcal{I}}.\end{aligned}\qquad}\right\\}$ (150)

Here $\mymathbb{1}^{\mathcal{I}}$ is the $D\times D$ identity matrix on a new
_internal_ Hilbert space $\mathcal{H}_{\mathcal{I}}$, and
$P_{1}^{\mathcal{I}},\dots,P_{D}^{\mathcal{I}}$ collectively form any PVM on
that internal Hilbert space satisfying the usual conditions of mutual
exclusivity,

$P_{\gamma}^{\mathcal{I}}P_{\gamma^{\prime}}^{\mathcal{I}}=\delta_{\gamma\gamma^{\prime}}P_{\gamma}^{\mathcal{I}},$
(151)

and completeness,

$\sum_{\gamma=1}^{D}P_{\gamma}^{\mathcal{I}}=\mymathbb{1}^{\mathcal{I}}.$
(152)

It is then a mathematical identity that one can rewrite the stochastic-quantum
dictionary (26) as

$\mathopen{}\mathclose{{}\left.\begin{aligned}
\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)&=\mathrm{tr}\mathopen{}\mathclose{{}\left(\mathrm{tr}_{\mathcal{I}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\otimes\mymathbb{1}^{\mathcal{I}}}\right]\mathopen{}\mathclose{{}\left[P_{i}\otimes\mymathbb{1}^{\mathcal{I}}}\right]}\right.}\right.\\\
&\qquad\times\mathopen{}\mathclose{{}\left.\mathopen{}\mathclose{{}\left.\mathopen{}\mathclose{{}\left[\Theta\mathopen{}\mathclose{{}\left(t}\right)\otimes\mymathbb{1}^{\mathcal{I}}}\right]\mathopen{}\mathclose{{}\left[P_{j}\otimes
P_{\gamma}^{\mathcal{I}}}\right]}\right)}\right),\end{aligned}\negthickspace\negthickspace\negthickspace\negthickspace\qquad}\right\\}$
(153)

with a second trace, or _partial trace_ , over the internal Hilbert space
$\mathcal{H}_{\mathcal{I}}$. The choice of value for the label $\gamma$ here
is immaterial, with different choices of $\gamma$ related by gauge
transformations.

One can equivalently write the dilated form (153) of the dictionary in _block-
matrix form_ as

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}_{\mathcal{I}}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}\dagger}\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}P_{\gamma}^{\mathcal{I}}}\right).$
(154)

Here
$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}$
is a diagonal $D\times D$ matrix consisting of repeated copies of the specific
entry $\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ (for fixed $i,j$)
along the diagonal:

$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}\equiv\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)\,\mymathbb{1}^{\mathcal{I}}.$
(155)

Meanwhile, the adjoint operation $\dagger$ in (154) acts on this $D\times D$
block matrix
$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}$,
so it does not transpose the indices $i$ and $j$ on the $N\times N$ matrix
$\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ itself:

$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}\dagger}\equiv\overline{\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]}^{\mathcal{I}}.$
(156)

It follows that

$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}\dagger}\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}P_{\gamma}^{\mathcal{I}}=|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}P_{\gamma}^{\mathcal{I}},$
(157)

so the trace over $\mathcal{H}_{\mathcal{I}}$ indeed yields
$|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}=\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$,
as required.

In this dilated version of the Hilbert-space formulation, Schur-Hadamard gauge
transformations (45) are enhanced to the following local-in-time gauge
transformations, which have not yet been described in the research literature
and therefore constitute another new result:

$\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}\mapsto
V_{\mathopen{}\mathclose{{}\left(ij}\right)}^{\mathcal{I}}\mathopen{}\mathclose{{}\left(t}\right)\mathopen{}\mathclose{{}\left[\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right]^{\mathcal{I}}.$
(158)

Here
$V_{\mathopen{}\mathclose{{}\left(ij}\right)}^{\mathcal{I}}\mathopen{}\mathclose{{}\left(t}\right)$
are a set of $N^{2}$ unitary, $D\times D$ matrices, where each such unitary
matrix as a whole is labeled by a specific pair
$\mathopen{}\mathclose{{}\left(ij}\right)$ of configuration labels:

$V_{\mathopen{}\mathclose{{}\left(ij}\right)}^{\mathcal{I}\dagger}\mathopen{}\mathclose{{}\left(t}\right)=(V_{\mathopen{}\mathclose{{}\left(ij}\right)}^{\mathcal{I}}\mathopen{}\mathclose{{}\left(t}\right))^{-1}.$
(159)

The gauge transformations (158) will not generally preserve the factorization
$\Theta\mathopen{}\mathclose{{}\left(t}\right)\otimes\mymathbb{1}^{\mathcal{I}}$
appearing in (153), so they motivate considering more general $ND\times ND$
time-evolution operators
$\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)$, in terms of which the
dilated dictionary (153) takes the form

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}\mathopen{}\mathclose{{}\left(\mathrm{tr}_{\mathcal{I}}\mathopen{}\mathclose{{}\left(\tilde{\Theta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\mathopen{}\mathclose{{}\left[P_{i}\otimes\mymathbb{1}^{\mathcal{I}}}\right]\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)\mathopen{}\mathclose{{}\left[P_{j}\otimes
P_{\gamma}^{\mathcal{I}}}\right]}\right)}\right).$ (160)

Any $ND\times ND$ matrix
$\tilde{\Theta}\mathopen{}\mathclose{{}\left(t}\right)$ appearing on the
right-hand side of this dictionary and satisfying the natural generalization
of the summation condition (18) is guaranteed to lead to a valid transition
matrix $\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ on the left-hand
side, so working with a dilated Hilbert-space formulation essentially provides
a larger ‘canvas’ for designing transition matrices.

As a simple example of a dilation for the case $D=2$, one can formally
eliminate the complex numbers from a quantum system’s Hilbert space [67].
Specifically, by increasing the system’s Hilbert-space dimension from $N$ to
$2N$, one can replace the imaginary unit $i\equiv\sqrt{-1}$ with the real-
valued $2\times 2$ matrix
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}}\right)$, with the enhanced version (158) of Schur-
Hadamard gauge transformations now consisting of two-dimensional rotations of
the internal Hilbert space $\mathcal{H}_{\mathcal{I}}$.242424Importantly,
notice that the proof of the uncertainty principle presented in Footnote 20
works just as well with the imaginary unit $i$ represented by a $2\times 2$
matrix in this way. One can then represent the complex-conjugation operator
$K$ appearing in (145) as the real-valued $2\times 2$ matrix
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\right)$. The result is that all unitary and anti-unitary
operators become $2N\times 2N$ real orthogonal matrices. One cost of using
this ‘real’ representation, however, is that the Hilbert spaces of composite
systems will not factorize as neatly into Hilbert spaces for their constituent
subsystems.252525Without increasing the dimension $N$ of a system’s Hilbert
space, one could instead attempt to limit the appearance of the complex
numbers in a system’s Hilbert-space formulation by using the original Schur-
Hadamard gauge transformation (45) to make all the entries of the system’s
time-evolution operator $\Theta\mathopen{}\mathclose{{}\left(t}\right)$ real-
valued. In this alternative approach, however, a unistochastic transition
matrix $\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ may not be expressible
in terms of a unitary or orthogonal time-evolution operator, and the complex
numbers will generally still be needed anyway to define various observables.

As a much more significant application of dilations, recall that any
transition matrix $\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ has a
Kraus decomposition (51):

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\sum_{\beta=1}^{N}\mathrm{tr}(K_{\beta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)P_{j}).$
(161)

The _Stinespring dilation theorem_ [68, 69] then guarantees that by an
appropriate dilation to a larger Hilbert space if necessary, one can express
$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ in terms of a unitary
time-evolution operator $\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$:

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}\mathopen{}\mathclose{{}\left(\mathrm{tr}_{\mathcal{I}}\mathopen{}\mathclose{{}\left(\tilde{U}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\mathopen{}\mathclose{{}\left[P_{i}\otimes\mymathbb{1}^{\mathcal{I}}}\right]\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)\mathopen{}\mathclose{{}\left[P_{j}\otimes
P_{\gamma}^{\mathcal{I}}}\right]}\right)}\right).$ (162)

This fact makes clear the inevitability of unitary time evolution in quantum
theory. As mentioned earlier, it also implies at the level of a theorem [50]
that every generalized stochastic system can be regarded as a subsystem of a
unistochastic system.262626From the starting assumptions presented here, one
can sketch the following proof: Given $N\times N$ Kraus operators
$K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)$, with $\beta=1,\dots,N$,
define an $N^{3}\times N^{2}$ matrix
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$ according to
$\tilde{V}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(jl}\right)}\mathopen{}\mathclose{{}\left(t}\right)\equiv
K_{\beta,ij}\mathopen{}\mathclose{{}\left(t}\right)\delta_{lm}$, treating
$\mathopen{}\mathclose{{}\left(i\beta m}\right)$ as the first index of
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$ and treating
$\mathopen{}\mathclose{{}\left(jl}\right)$ as its second index. One can show
that this matrix satisfies
$\tilde{V}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)=\mymathbb{1}_{N^{2}\times
N^{2}}$, so it defines a partial isometry, which can always be extended to a
unitary $N^{3}\times N^{3}$ matrix
$\tilde{U}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(ja}\right)}\mathopen{}\mathclose{{}\left(t}\right)$
by adding $N^{3}-N^{2}$ additional columns that are mutually orthogonal with
each other and with the previous $N^{2}$ columns already in
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$, where the new index $a$
runs through $N^{2}$ possible values. These additional columns can always be
chosen so that at the initial time $0$, where
$V\mathopen{}\mathclose{{}\left(0}\right)=\mymathbb{1}$ is the $N\times N$
identity matrix, they make $\tilde{U}\mathopen{}\mathclose{{}\left(0}\right)$
coincide with the $N^{3}\times N^{3}$ identity matrix. The last step is to
show that $\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$ satisfies (162),
whose right-hand side reduces to
$\sum_{\beta,m}|\tilde{U}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(j\gamma}\right)}\mathopen{}\mathclose{{}\left(t}\right)|^{2}=\sum_{\beta}|K_{\beta,ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}$.
QED

As yet another key application of dilations, a dilated Hilbert-space
formulation can make it possible to describe new kinds of emergeables. Some of
these _dilation-emergeables_ may be observables that can yield definite
results in measurement processes, along the lines of Subsection V.2, despite
not having a direct meaning solely at the level of the system’s underlying
configuration space.

In this way, a generalized stochastic system based on a configuration space
can easily accommodate emergent observables that describe empirically
meaningful patterns in the dynamics and that model all kinds of quantum
phenomena. Indeed, obtaining a unitary time-evolution operator for a given
system may require dilating the Hilbert space in just this way, as in (162).

It is important to keep in mind that whether or not one actually carries out
this formal dilation of the Hilbert-space formulation, the stochastic dynamics
of the underlying generalized stochastic system will still be the same. Any
emergent patterns in the system’s stochastic dynamics that are made manifest
by the dilation, as represented by any new dilation-emergeables that arise,
were always there all along, albeit in a non-manifest way.

An important example of this last application is _intrinsic spin_. To
introduce spin as a dilation-emergeable, one merely dilates the Hilbert space
to $ND$ dimensions, introduces a $D$-dimensional representation of
$SO\mathopen{}\mathclose{{}\left(3}\right)$ for the internal Hilbert space,
and then requires that the dilated time-evolution operator has the appropriate
form of rotation symmetry. This approach to representing spin ensures that
despite picking an arbitrary three-dimensional coordinate axis in the process
of formally carrying out the dilation of the Hilbert space—such as by choosing
the spin-$z$ operator to be diagonal on the dilated Hilbert space—the
underlying generalized stochastic system does not fundamentally involve any
preferred direction or entail any basic violation of rotation invariance.

###  VI.3 Nonlocality

This paper has shown that systems based on trajectories in configuration
spaces and evolving according to generically indivisible stochastic dynamics
have Hilbert-space representations and can replicate the usual mathematical
formalism and empirical predictions of quantum theory.

Technically speaking, the configurations in this new picture for quantum
theory play the role of _hidden variables_ , meaning physical parameters that
exist separately from wave functions and density matrices. Of course, one
could argue that configurations should more properly be called ___physical
variables_ , given that wave functions and density matrices arise from the
stochastic-quantum correspondence merely as secondary, representational
constructs, rather than as fundamental entities in their own right. Either
way, any mention of ‘hidden variables’ immediately raises questions about the
potential invocation of _nonlocality_ , the study of which has motivated
famous papers like that of Einstein, Podolsky, and Rosen [70], and has led to
the development of a number of important _no-go theorems_ [71, 72, 73, 74].

Before assessing the implications of these no-go theorems for the picture
described in this paper, it will be important to note that these theorems do
not rule out the possibility of hidden variables altogether. Nor do these
theorems imply that introducing hidden variables would necessarily make
quantum theory any more dynamically nonlocal than it already is.

Being mindful of these caveats, there is ample reason to probe the question of
nonlocal dynamics in the approach to quantum theory taken in this paper. After
all, looking back at the discussion of entanglement in Subsection IV.8, a pair
of systems that interact at some time will generically exhibit what look like
nonlocal stochastic dynamics after that time, at least until the later
occurrence of a division event due to decoherence by an external system.

In what follows, it will be important to be keep in mind the distinction
between _deterministic_ hidden-variables theories and _stochastic_ hidden-
variables theories.

Bell’s original nonlocality theorem, as formulated and proved in 1964 [71],
only addressed the case of a _deterministic_ hidden-variables theory.
Specifically, Bell showed that if one assumes that a theory’s hidden variables
uniquely determine measurement outcomes, and if one further assumes that the
hidden variables are local in the sense that measurement results should not
depend on the settings of faraway measuring devices, then one arrives at an
inequality that is expressly violated by quantum theory. Bell’s 1964 theorem
therefore establishes that any purported formulation of quantum theory based
on local deterministic hidden variables is ruled out empirically.

At first glance, there might have seemed to be just two available options in
response to Bell’s nonlocality theorem. Either one could accept a theory of
_nonlocal_ deterministic hidden variables, or one could _deny_ the existence
of nonlocal deterministic hidden variables and thereby try to avoid having to
introduce any ostensibly new dynamical nonlocality into quantum theory.

However, for a hidden-variables theory based on _stochastic_ dynamics rather
than on _deterministic_ dynamics, the question of dynamical nonlocality
becomes murkier. The generalization to stochastic dynamics means that one
needs to rely on more abstract, statistical conditions for establishing
whether or not the theory’s hidden variables behave in a dynamically local
manner.

The most frequently cited statistical locality criterion for stochastic
hidden-variables theories was formulated by Bell later on, in 1975 [73, 75,
76]. That statistical locality criterion is a statement about how rich a
theory’s hidden variables should be in order for the theory to qualify as
dynamically local.

To formulate Bell’s statistical locality criterion, one starts by considering
the case of a measurement outcome $x$ based on local measurement settings $a$,
and a far-separated measurement outcome $y$ based on local measurement
settings $b$. Then one supposes that the joint probabilities
$p\mathopen{}\mathclose{{}\left(x,y|a,b}\right)$ for the measurement results
$x$ and $y$, conditioned on the measurement settings $a$ and $b$, show a
statistical correlation. Bell argued that in order for the theory in question
to be considered dynamically local, the theory should contain enough hidden
variables to account for the statistical correlation in the following precise
sense: if one conditions on all the hidden variables $\lambda$ in the past
light cone of the two measurements, then those hidden variables should screen
off the correlation between the measurement results, meaning that the joint
probabilities should factorize according to

$p\mathopen{}\mathclose{{}\left(x,y|a,b,\lambda}\right)=p\mathopen{}\mathclose{{}\left(x|a,\lambda}\right)p\mathopen{}\mathclose{{}\left(y|b,\lambda}\right).$
(163)

Bell’s statistical locality criterion is precisely the condition that the
theory in question should have enough hidden variables to ensure that the
factorization (163) is always possible. Based on this statistical locality
criterion, which should hold even in cases of ‘one-shot’ measurements in which
certain measurement outcomes can be assigned a 100% probability [74], one can
again derive predictions that are violated by quantum theory, just as in the
case of a deterministic hidden-variables theory.

However, Bell’s statistical locality criterion is broader than the conditions
he studied in his 1964 theorem on deterministic hidden-variables theories.
Bell’s statistical locality condition is so broad, in fact, that Bell used it
to argue that textbook quantum theory is _itself_ already dynamically nonlocal
[73, 77].

To understand why, observe that textbook quantum theory is committed to the
existence of measurement settings and definite measurement outcomes that end
up behaving precisely as a (highly incomplete) set of stochastically evolving
hidden variables. In other words, although textbook quantum theory is not a
_deterministic_ hidden-variables theory, it is, in fact, a _stochastic_
hidden-variables theory.

The stochastic-quantum correspondence derived in this paper makes these
commitments by textbook quantum theory manifest. Indeed, one can regard
textbook quantum theory as the insistence that for any measurement set-up
consisting of a subject system $\mathcal{S}$, a measuring device
$\mathcal{D}$, and an environment $\mathcal{E}$, as laid out in Subsection
V.2, the configurations of $\mathcal{D}$ are to be treated as hidden variables
(that is, as beables), whereas the configurations of $\mathcal{S}$ and
$\mathcal{E}$ are to be regarded merely as emergeables.

The stochastic-quantum correspondence therefore brings newfound clarity to the
outstanding foundational problems of textbook quantum theory. Specifically,
one sees from this novel perspective that the seemingly arbitrary division of
the world into measuring devices, which purportedly have underlying
configurations, and all other systems, which purportedly do not, leads
directly to all the usual mysteries about the measurement process according to
textbook quantum theory. After all, what, in the end, determines whether a
given system counts as a measuring device, and therefore merits having
underlying configurations?

More relevant to the present discussion is that because textbook quantum
theory includes stochastic hidden variables for measuring devices, and because
those stochastic hidden variables are insufficient to ensure the factorization
property (163), the nonlocality theorems that employ Bell’s statistical
locality criterion imply that textbook quantum theory is itself dynamically
nonlocal. Hence, there is no real cost to upgrading the configurations of
$\mathcal{S}$ and $\mathcal{E}$ to being hidden variables on an equal footing
with the configurations of $\mathcal{D}$. These additional hidden variables do
not lead to the factorization property (163) either, but they also do not lead
to any trouble for the _no-communication theorem_ [78, 79], which precludes
using quantum theory to send controllable signals faster than light.

The main conclusion of this analysis is that if one takes Bell’s statistical
locality criterion seriously as the proper way to identify dynamical
nonlocality, then textbook quantum theory is already dynamically nonlocal, so
adding some additional hidden variables to the theory will not ultimately make
that dynamical nonlocality any worse. Alternatively, one could instead dispute
that Bell’s statistical locality criterion is an acceptable criterion for
locality in the first place, perhaps by arguing that local correlation-
producing interactions are valid common-causes explanations, but are simply
not the sorts of things that can be conditioned on. However, if one denies the
validity of Bell’s criterion, then it cannot be used to argue that the picture
of quantum theory presented in this paper is dynamically nonlocal. Either way,
the approach taken toward quantum theory in this paper is no more or less
dynamically nonlocal than textbook quantum theory already is.

It may be that a more empirically meaningful notion of dynamical locality is
the condition that new statistical correlations can only arise between systems
that are in local contact, either directly or through the mediation of other
systems. As the no-communication theorem shows, this empirical locality
criterion is satisfied by textbook quantum theory, as well as by any physical
theory or interpretative framework that reproduces the predictions of textbook
quantum theory, and ultimately ensures the impossibility of sending
controllable messages faster than light. One could argue that any stronger
locality requirements, such as Bell’s statistical locality condition, go
beyond what is empirically verifiable, and are therefore matters of
metaphysics.

A number of other important no-go theorems have been proved over the years,
including von Neumann’s early no-go theorem [80, 17], the Kochen-Specker
theorem [81], the Pusey-Barrett-Rudolph theorem [82], and Myrvold’s no-go
theorem [83]. These theorems either assume that _all_ observables are true
random variables (that is, beables) that exist at the level of the given
system’s configuration space, or they assume that measurements are _passive
operations_ that merely reveal pre-existing values of observables without
altering the behavior of measured systems in the process, or they assume that
measurements are fundamentally _irreversible_ , or they assume the existence
of _additional_ probability formulas. Because the picture of quantum theory
introduced in this paper refrains from making any of these assumptions, it is
consistent with these theorems.

##  VII Discussion and Future Work

This paper has shown that one can reconstruct the mathematical formalism and
all the empirical predictions of quantum theory using simpler, more physically
transparent axioms than the standard Dirac-von Neumann axioms. Rather than
postulating Hilbert spaces and their ingredients from the beginning, one
instead posits a physical model, called a _generalized stochastic system_ ,
based on trajectories in configuration spaces following generically
indivisible stochastic dynamics. The _stochastic-quantum correspondence_ then
connects generalized stochastic systems with quantum systems in a fundamental
way, showing that every quantum system can be viewed as the Hilbert-space
representation of an underlying generalized stochastic system.

This perspective deflates some of the most mysterious features of quantum
theory. In particular, one sees that density matrices, wave functions, and all
the other appurtenances of Hilbert spaces, while highly useful, are merely
gauge variables. These appurtenances should therefore not be assigned direct
physical meanings or treated as though they directly represent physical
objects, any more than Lagrangians or Hamilton’s principal functions directly
represent physical objects. Superposition is then not a literal smearing of
physical objects, but is merely a mathematical artifact of catching a system
in the middle of an indivisible stochastic process, as represented using a
Hilbert-space formulation and wave functions.

Moreover, from this standpoint, _canonical quantization_ need not be regarded
as the promotion of classical observables to noncommutative operators by fiat,
but can be implemented (when mathematically feasible) simply by generalizing a
classical system’s dynamics from being deterministic to being stochastic, with
all the exotic features of quantum theory then emerging automatically. As a
consequence, this formulation of canonical quantization potentially offers
more straightforward techniques for coupling classical systems to quantum
systems in real-world applications.

In an important sense, the stochastic-quantum correspondence also legitimizes
many standard practices followed in physics and in other scientific areas,
like astronomy, chemistry, biology, and paleontology. To see why, notice that
according to the thoroughly instrumentalist and operationalist Dirac-von
Neumann axioms, the _only_ predictions provided by textbook quantum theory are
predictions of measurement outcomes, probabilities of measurement outcomes,
and expectation values that are averages of measurement outcomes statistically
weighted by measurement-outcome probabilities [36, 37, 38, 39, 40]. Meanwhile,
scientists in all areas of research talk about other kinds of phenomena—from
the mixing of gases in the primordial universe to the spontaneous appearance
of genetic mutations—that presumably just _happen_ in some way, according to
_happening_ probabilities, in the past, present, or future. Strictly speaking,
however, the _happening_ of phenomena lies outside the axiomatic ambit of
textbook quantum theory, which refers only to connecting the measurement
settings of chemical detectors and telescopes to the probabilities of their
measurement outcomes. The inability of textbook quantum theory to account for
the happening of phenomena either means that scientists are not speaking
honestly or coherently about their research, or that textbook quantum theory
is inadequate as a physical theory.

Decoherence alone cannot bridge the gap between measurement-outcome
probabilities and happening probabilities, because decoherence can only
temporarily change whatever orthonormal basis momentarily diagonalizes a
system’s density matrix (and, after all, every density matrix is always
diagonal in _some_ orthonormal basis). After a system undergoes decoherence,
textbook quantum theory then still requires one to make a direct appeal to the
measurement axioms to translate the final density matrix into a statement
about probabilities, which will then axiomatically end up being measurement-
outcome probabilities, rather than happening probabilities.

Nor can appealing to some sort of _thermodynamic limit_ resolve the
discrepancy either. In order for a limit in a physical context to make sense,
there should be clearly physical ingredients or constituents involved.
Furthermore, the end result of the limit should gradually emerge as a better
and better physical approximation at _finite_ stages of the limiting process,
simply because a rigorous limit consists of inequalities between finite (if
arbitrarily large or small) parameters. For example, in the _hydrodynamic
limit_ of a system of classical interacting particles, the particles are the
physical ingredients, and one sees fluid-like behavior gradually emerge as a
better and better physical approximation as the number of particles
progressively increases. In the case of textbook quantum theory, by contrast,
every finite stage of any purported thermodynamic limit features only
measurement outcomes and measurement-outcome probabilities, so there are no
clearly physical ingredients or constituents, and the gap between measurement
outcomes and the happening of phenomena never closes.

The stochastic-quantum correspondence yields a much richer version of quantum
theory in which physical phenomena really happen, with probabilities that are
really happening probabilities, and therefore vindicates the ways that
scientists talk about the world. Measurement-outcome probabilities are then
merely a special case, arising when what is actually happening is a change to
the configuration of a measuring device.

Because this paper’s approach invokes hidden variables in the form of
underlying physical configurations, this framework for quantum theory shares
some aspects with the _de Broglie-Bohm formulation_ , or _Bohmian mechanics_
[84, 85, 86]. However, in contrast to this paper’s approach, Bohmian mechanics
employs deterministic dynamics, and features a fundamental guiding equation
that explicitly breaks Lorentz invariance by singling out a preferred
foliation of spacetime into spacelike hypersurfaces. This paper instead takes
seriously what experiments strongly suggest—that the dynamics of quantum
theory is indeterministic, and that there is no fundamentally preferred
foliation of spacetime. The formulation of quantum theory in this paper is
also more flexible and model-independent than Bohmian mechanics, and works for
all kinds of quantum systems, from particles to fields and beyond.

In contrast with the _Everett interpretation_ [87, 88], also known as the
_‘many worlds’ interpretation_ , the framework presented in this paper assumes
that quantum systems, like classical systems, have definite configurations in
configuration spaces, and does not attempt to derive probability from non-
probabilistic assumptions or grapple with fundamental aspects of personal
identity in a universe continuously branching into large (and somewhat
undefined) numbers of parallel worlds. The approach in this paper is therefore
more modest, metaphysically speaking, than the Everett interpretation.

Neither this paper’s approach nor the Everett interpretation satisfies the
statistical locality criterion formulated by Bell and described in Subsection
VI.3, but the Everett interpretation arguably exhibits a different notion of
dynamical locality at a level of description that _transcends_ its individual
world-branches [89]. However, because each individual world-branch looks no
more or less nonlocal than the world according to textbook quantum theory, it
is not clear whether the Everett interpretation’s dynamical locality is more
than a metaphysical statement, nor is it easy to discern what concrete
benefits it truly provides.

Unlike _stochastic-collapse theories_ [14, 15], this paper does not invoke any
fundamental violations of unitarity, nor does it require introducing any new
constants of nature to specify dynamical-collapse rates.

Looking forward, it would be interesting to see what implications the
stochastic-quantum correspondence could have for both phenomenological
stochastic processes, like those in biology or finance, as well as for future
work in fundamental physics, like quantum gravity.

More broadly, by recasting the Hilbert-space formulation of quantum theory as
merely a convenient way to represent a large class of stochastic processes,
one opens the door to searching for totally different representations that
might look nothing at all like Hilbert spaces and that could allow for the
construction of more general kinds of stochastic processes. Perhaps one could
even find a way to generalize beyond stochastic processes altogether. work.

## Acknowledgments

The author would especially like to acknowledge Emily Adlam, David Albert,
Howard Georgi, David Kagan, and Logan McCarty for extensive discussions during
the writing of this paper. The author would also like to thank Scott Aaronson,
Guido Bacciagaluppi, David Baker, Craig Callender, Ignacio Cirac, Iris Cong,
Jordan Cotler, Erik Curiel, Louis Deslauriers, Melissa Franklin, Philip Goyal,
David Griffiths, Mina Himwich, Jenann Ismael, Daniel Jafferis, David Kaiser,
Efthimios Kaxiras, Gregory Kestin, Matthew Kleban, Serhii Kryhin, Barry
Loewer, Curtis McMullen, Simon Milz, Arjun Mirani, Wayne Myrvold, Filip
Niewinski, Jill North, John Norton, Matthew Reece, Vivishek Sudhir, Noel
Swanson, Xi Yin, and Nicole Yunger Halpern for helpful conversations.

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Search for Higgs bosons decaying into new spin-0 or spin-1 particles in four-
lepton final states with the ATLAS detector with 139 fb-1 of $pp$ collision
data at $\sqrt{s}=13$ TeV Searches are conducted for new spin-0 or spin-1
bosons using events where a Higgs boson with mass $125$ GeV decays into four
leptons ($\ell=$ $e$, $\mu$). This decay is presumed to occur via an
intermediate state which contains two on-shell, promptly decaying bosons:
$H\rightarrow XX/ZX\rightarrow 4\ell$, where the new boson $X$ has a mass
between 1 and 60 GeV. The search uses $pp$ collision data collected with the
ATLAS detector at the LHC with an integrated luminosity of 139 fb-1 at a
centre-of-mass energy $\sqrt{s}=13$ TeV. The data are found to be consistent
with Standard Model expectations. Limits are set on fiducial cross sections
and on the branching ratio of the Higgs boson to decay into $XX/ZX$, improving
those from previous publications by a factor between two and four. Limits are
also set on mixing parameters relevant in extensions of the Standard Model
containing a dark sector where $X$ is interpreted to be a dark boson.
HDBS-2018-55 JHEP JHEP 03 (2022) 041 10.1007/JHEP03(2022)041 CERN-EP-2021-193

###### Contents

  1. 1 Introduction
  2. 2 Benchmark models
    1. 2.1 Dark bosons
    2. 2.2 Extended Higgs sectors
  3. 3 ATLAS detector
  4. 4 Data and simulated event samples
  5. 5 Event reconstruction and selection
    1. 5.1 Lepton reconstruction
    2. 5.2 Invariant kinematic mass variables
    3. 5.3 Common event selection

##  1 Introduction

Although the Higgs boson was discovered at the Large Hadron Collider (LHC) in
2012 [HIGG-2012-27, CMS-HIG-12-028], there is good reason to believe that the
description of the Higgs sector of the Standard Model (SM) is still
incomplete. Besides the well-known issues of naturalness and baryon asymmetry,
astrophysical observations implying the existence of dark matter motivate
extensions to the Higgs sector of the SM, particularly those that propose the
existence of a ‘dark’ (i.e., hidden) sector, with its own hidden-sector
particles [Martin2014, Cline2014].

An attractive way to search for new physics in the Higgs sector is through
non-standard (‘exotic’) decays of the Higgs boson. Existing precision
measurements of the properties of the Higgs boson still allow a branching
ratio of up to about 30% to non-standard decays (assuming that the couplings
of the Higgs boson to the $W$ and $Z$ bosons are not larger than their SM
values) [HIGG-2014-06, CMS-HIG-14-009, HIGG-2015-07]. Further, since the SM
predicts a very narrow decay width for the Higgs boson, even a small coupling
to a new light state could result in a significant branching ratio to that
state. In addition, new hidden-sector particles may preferentially couple to
the Higgs boson, making it a ‘portal’ to explore this new physics
[Shrock:1982kd, Strassler:2006im, Schabinger:2005ei, Patt:2006fw]. Such exotic
decays of the Higgs boson are predicted by many proposed extensions to the SM,
including models with a first-order electroweak phase transition
[Profumo:2007wc, Blinov:2015sna], models with neutral naturalness
[Burdman:2006tz, Craig:2015pha, Curtin:2015fna], and models with a hidden
sector [Fayet:2004bw, Finkbeiner:2007kk, ArkaniHamed:2008qn, Dudas:2012t1,
Curtin:2014cca, Curtin:2013fra, Davoudiasl:2013aya, Davoudiasl:2012ag,
wells2008find, gopalakrishna2008higgs, Evans:2017kti], as well as by several
models of dark matter [Silveira:1985rk, Pospelov:2007mp, Draper:2010ew,
Ipek:2014gua, Martin:2014sxa, Clowe], including some posited to explain
observed excesses of astrophysical positrons [Adriani:2008zr, ATIC:2008t1,
Accardo:2014lma]. They are also predicted by the Next-to-Minimal
Supersymmetric Standard Model (NMSSM) [Dobrescu:2000yn, Ellwanger:2003jt,
Dermisek:2005ar, Chang:2008cw, Morrissey:2008gm, Dawson:2018dcd].

This paper reports three related searches, each of which looks for a SM Higgs
boson $H$ decaying via a new boson into a final state consisting of four
charged leptons ($\ell\equiv e$, $\mu$). All use the full LHC Run 2 data set
of about $139\text{\,}{\mathrm{fb}}^{-1}$ that the ATLAS detector collected
from proton–proton collisions at a centre-of-mass energy of
$\sqrt{s}=$13\text{\,}\text{T\mathrm{ eV }}$$. Following the models motivating
these analyses, the new boson could be either a dark-sector vector boson or a
scalar boson, denoted by $X$. The three searches considered are:

  * •

High-mass (HM): $H\rightarrow XX\rightarrow 4\ell$
($$15\text{\,}\text{G\mathrm{ eV }}$<m_{X}<$60\text{\,}\text{G\mathrm{ eV
}}$$).

  * •

Low-mass (LM): $H\rightarrow XX\rightarrow 4\mu$ ($$1\text{\,}\text{G\mathrm{
eV }}$<m_{X}<$15\text{\,}\text{G\mathrm{ eV }}$$).

  * •

Single $Z$ boson (ZX): $H\rightarrow ZX\rightarrow 4\ell$
($$15\text{\,}\text{G\mathrm{ eV }}$<m_{X}<$55\text{\,}\text{G\mathrm{ eV
}}$$).

The LM analysis uses only the $4\mu$ final state because the selection
efficiency for isolated muons is significantly larger than that for isolated
electrons in this mass range (see LABEL:sec:HM_Selection). These searches are
sensitive to any intermediate bosons within the considered mass ranges that
are narrow, on-shell, and decay promptly. This paper provides model-
independent fiducial cross-section limits, as well as limits based on the
specific models described in Section 2.

This work extends previous searches performed by ATLAS with
$20\text{\,}{\mathrm{fb}}^{-1}$ of data collected at
$\sqrt{s}=$8\text{\,}\text{T\mathrm{ eV }}$$ [EXOT-2013-15] and with
$36\text{\,}{\mathrm{fb}}^{-1}$ of data collected at
$\sqrt{s}=$13\text{\,}\text{T\mathrm{ eV }}$$ [EXOT-2016-22]. In addition to a
larger data sample and improved lepton identification, the signal region
selection of the HM analysis has been re-optimized. Other similar searches,
including searches for pairs of light bosons decaying into muons,
$\tau$-leptons, photons, and/or jets, as well as searches for a single light
boson decaying into a pair of muons, using both
$\sqrt{s}=$8\text{\,}\text{T\mathrm{ eV }}$$ and data, have been performed by
ATLAS [HDBS-2018-47, HIGG-2017-09, HIGG-2017-05, HIGG-2016-03, HIGG-2014-02],
CMS [CMS-HIG-13-010, CMS-HIG-16-015, CMS-HIG-18-024, CMS-EXO-20-014], and LHCb
[LHCB-PAPER-2017-038]. Further searches for a SM Higgs boson decaying into
undetected particles are reported in Refs. [HIGG-2018-54, CMS-HIG-17-023].

This section is followed by a summary of the theoretical models used in the
interpretation of the results (Section 2). Next, the detector is described
(Section 3), followed by discussions of features that are common to all three
analyses, including the samples of data and simulated events (Section 4), the
reconstruction of lepton candidates and of their combinations (Sections 5.1
and 5.2), the event selections (Section 5.3), and the common systematic
uncertainties (LABEL:sec:systematics). Next, aspects specific to each analysis
are described (LABEL:sec:HM_analysis, LABEL:sec:LM_analysis and
LABEL:sec:ZX_analysis). Finally, the ways the analyses are combined to extract
limits, and the interpretations of the results in terms of the theoretical
models, are presented in LABEL:sec:Limits, and a summary is given in
LABEL:sec:Conclusion.

##  2 Benchmark models

###  2.1 Dark bosons

Many theories of dark matter posit a hidden sector [Fayet:2004bw,
Finkbeiner:2007kk, ArkaniHamed:2008qn, Dudas:2012t1, Curtin:2014cca,
Curtin:2013fra, Davoudiasl:2013aya, Davoudiasl:2012ag, wells2008find,
gopalakrishna2008higgs, Alexander:2016aln], which does not interact with SM
particles except via a mediator or portal interaction (besides gravity). A
concrete realization of such a mediator involves adding a field with a
U(1)${}_{\text{d}}$ dark gauge symmetry [Curtin:2014cca, Curtin:2013fra,
Davoudiasl:2013aya, Davoudiasl:2012ag, wells2008find, gopalakrishna2008higgs]
which mixes kinetically with the SM U(1)${}_{\text{Y}}$ hypercharge gauge
field with some strength $\epsilon$ [Galison:1983pa, Holdom:1985ag,
Dienes:1996zr]. The gauge boson of this symmetry is the $Z_{\text{d}}$ vector
boson, also called a ‘dark photon’.

The coupling strength of the $Z_{\text{d}}$ boson to SM particles, and hence
its lifetime (assuming no significant decays to non-SM particles), is
determined by the mixing parameter $\epsilon$. The decays of the
$Z_{\text{d}}$ boson, on the other hand, are determined by the gauge
couplings, and the decay branching ratios are largely independent of
$\epsilon$ for $\epsilon\ll 1$. Over the $Z_{\text{d}}$ mass range
$$1\text{\,}\text{G\mathrm{ eV
}}$<m_{Z_{\text{d}}}<$60\text{\,}\text{G\mathrm{ eV }}$$, the branching ratio
for decays into electron or muon pairs can be 10%–15% [Curtin:2014cca]. Over
the same mass range, the decay is prompt for $\epsilon\gtrsim 10^{-5}$
[Curtin:2014cca]. For smaller values of $\epsilon$, the decay vertex would be
significantly displaced from the interaction point, while for
$\epsilon\lesssim 10^{-8}$ the lifetime of the $Z_{\text{d}}$ boson becomes
long enough for it to likely escape the detector. Also, the decay width of the
$Z_{\text{d}}$ boson is very small ($\ll$1\text{\,}\mathrm{GeV}$$) for
$\epsilon\ll 1$ and $m_{Z_{\text{d}}}<$60\text{\,}\mathrm{GeV}$$. ATLAS and
CMS have searched for these long-lifetime signatures in collisions at energies
of both [EXOT-2013-22, EXOT-2014-09, SUSY-2014-02, CMS-EXO-12-037] and
[EXOT-2017-03, SUSY-2017-04, EXOT-2017-28, EXOT-2017-32, CMS-HIG-18-003].

If the U(1)${}_{\text{d}}$ symmetry is broken by an additional dark Higgs
boson $s$, then there could be mixing with strength $\kappa$ between the SM
Higgs boson and the dark Higgs boson [Curtin:2014cca, Curtin:2013fra,
Davoudiasl:2013aya, Davoudiasl:2012ag, wells2008find, gopalakrishna2008higgs].
The observed Higgs boson would be one of the mass eigenstates and could also
decay into dark-sector particles, including dark Higgs bosons that
subsequentially decay into SM fermions. The dark Higgs boson would inherit the
Yukawa couplings from the SM Higgs boson and decay preferentially into high-
mass fermion pairs.

A further possibility is mass mixing between the $Z_{\text{d}}$ boson and the
SM $Z$ boson [Davoudiasl:2013aya, Davoudiasl:2012ag]. If the mass term for
this mixing is written as $\epsilon_{Z}m_{Z}^{2}ZZ_{\text{d}}$, with
$\epsilon_{Z}=\delta m_{Z_{\text{d}}}/m_{Z}$, then $\delta$ is the model
parameter describing the mixing.

The processes probed in this paper that involve a SM Higgs boson decaying into
$Z_{\text{d}}$ bosons are depicted in Figures 1(a) and 1(b) and are included
in the Hidden Abelian Higgs Model (HAHM) [Curtin:2014cca]. The decay
$H\rightarrow ZZ_{\text{d}}$ is sensitive to the parameters $\epsilon$ and
$m_{Z_{\text{d}}}$, but does not depend on $\kappa$. However, the presence of
an irreducible background from the SM $H\rightarrow ZZ^{*}$ process means that
this signal can be observed only as a peak in the dilepton mass spectrum over
the background. The process $H\rightarrow Z_{\text{d}}Z_{\text{d}}$, in
contrast, is much more easily separated from SM backgrounds and hence is
potentially sensitive to smaller values of the kinetic mixing $\epsilon$,
where it is only required that the mixing be large enough for the
$Z_{\text{d}}$ boson to decay promptly. However, this process does require
mixing between the SM and dark-sector Higgs bosons and thus depends on
$\kappa$.

Limits on the kinetic mixing of $\epsilon\lesssim 0.03$ have been set from
precision electroweak measurements [Curtin:2014cca, Hook:2010tw,
Pospelov:2008zw] over the range $$1\text{\,}\text{G\mathrm{ eV
}}$<m_{Z_{\text{d}}}<$200\text{\,}\text{G\mathrm{ eV }}$$. Searches for
dilepton resonances, $pp\rightarrow Z_{\text{d}}\rightarrow\ell\ell$, at the
LHC for $m_{Z_{\text{d}}}<m_{Z}$ imply that $\epsilon\lesssim 0.005$–$0.020$
for $$20\text{\,}\text{G\mathrm{ eV
}}$<m_{Z_{\text{d}}}<$80\text{\,}\text{G\mathrm{ eV }}$$ [Hoenig:2014dsa].
Other searches rule out $\epsilon\gtrsim 10^{-3}$ for
$$10\text{\,}\text{M\mathrm{ eV
}}$<m_{Z_{\text{d}}}<$10\text{\,}\text{G\mathrm{ eV }}$$ [Batley:2009aa,
Lees:2012ra, PhysRevLett.59.755, PhysRevD.38.3375, PhysRevLett.67.2942,
Adler:2004hp]. The $H\rightarrow XX\rightarrow 4\ell$ analyses constrain the
Higgs mixing parameter $\kappa$, while the $H\rightarrow
ZZ_{\text{d}}\rightarrow 4\ell$ analysis provides information about the
kinetic mixing parameter $\epsilon$.

((a))

((b))

((c))

Figure 1: Exotic decays of the Higgs boson into four leptons induced by
intermediate dark vector bosons via (a) the hypercharge portal (to which the
ZX analysis is sensitive) and (b) the Higgs portal, where $s$ is a dark Higgs
boson [Curtin:2014cca] (to which the HM and LM analyses are sensitive). The
$Z_{\text{d}}$ gauge boson decays into SM particles through kinetic mixing
with the hypercharge field (with branching ratios that are nearly independent
of $\epsilon$). The $HZZ_{\text{d}}$ vertex factor is proportional to
$\epsilon$ whereas the $HZ_{\text{d}}Z_{\text{d}}$ vertex factor is
proportional to $\kappa$. (c) illustrates the decay of a Higgs boson into dark
Higgs scalars $s$ or pseudoscalars $a$ that couple to SM particles through
mixing with the SM Higgs field in models with an extended Higgs sector
(Section 2.2).

###  2.2 Extended Higgs sectors

Models containing two Higgs doublets and an additional scalar field (2HDM+S)
[Curtin:2013fra, a0] are also relevant for the search for $H\rightarrow
XX\rightarrow 4\mu$. Two-Higgs-doublet models (2HDMs) generically contain two
neutral scalars $H_{1,2}$, two charged scalars $H^{\pm}$, and one neutral
pseudoscalar $A$. The lighter of the neutral scalars $H_{1}$ is identified as
the observed Higgs boson $H$, while the other states are constrained to be
heavy by existing data [Gunion:1997aq, Belanger:2013xza]. Adding a complex
scalar singlet that mixes weakly with $H_{1,2}$ gives two additional states, a
scalar $s$ and a pseudoscalar $a$. If these are lighter than $m_{H}/2$, then
$H\rightarrow aa$ and $H\rightarrow ss$ decays are allowed (Figure 1(c)). This
paper probes the process $H\rightarrow aa\rightarrow 4\mu$, but limits on
$H\rightarrow aa\rightarrow 4\mu$ also apply to $H\rightarrow ss\rightarrow
4\mu$.

The decays of the scalar and pseudoscalar into fermions are determined by
their Yukawa couplings [Curtin:2013fra], implying that the branching ratio to
electrons is very small, and that the branching ratio to muons is smaller than
that of the $Z_{\text{d}}$ vector bosons described previously. Branching
ratios for $H\rightarrow aa$ and $a\rightarrow\mu\mu$ can be significant in
the range $2m_{\mu}<m_{a}<2m_{\tau}$, ranging from $10^{-2}$ to $10^{-1}$ in
some regions of parameter space [Curtin:2013fra]. In 2HDMs, there are several
possible ways for the Higgs sector to couple to fermions. Of these, type-III
models (in which leptons and quarks couple to different Higgs doublets) at
large $\tan\beta$ (where $\tan\beta$ is the ratio of the vacuum expectation
values of the two Higgs doublets) are particularly interesting for these
analyses. A light pseudoscalar can correspond to the $R$-symmetry limit of the
NMSSM [Ellis:1988er, Dermisek:2006wr], which reduces the need for fine-tuning
and addresses the $\mu$-problem [Giudice:1988yz]. Searches for exotic decays
of the Higgs boson into new light scalars or pseudoscalars have been carried
out for a variety of mass ranges and final states with both LHC [HIGG-2016-03,
HIGG-2017-05, HIGG-2017-09, EXOT-2016-22, CMS-HIG-18-006, CMS-HIG-18-011, CMS-
HIG-18-003, CMS-HIG-17-024, CMS-HIG-17-029, HIGG-2014-02, EXOT-2013-15,
EXOT-2013-24, CMS-HIG-13-010, CMS-HIG-16-015, CMS-HIG-14-019, CMS-HIG-16-015]
and Tevatron [Abazov:2009yi] data.

##  3 ATLAS detector

The ATLAS detector [PERF-2007-01] at the LHC covers nearly the entire solid
angle around the collision point.111 ATLAS uses a right-handed coordinate
system with its origin at the nominal interaction point (IP) in the centre of
the detector and the $z$-axis along the beam pipe. The $x$-axis points from
the IP to the centre of the LHC ring, and the $y$-axis points upwards.
Cylindrical coordinates $(r,\phi)$ are used in the transverse plane, $\phi$
being the azimuthal angle around the $z$-axis. The pseudorapidity is defined
in terms of the polar angle $\theta$ as $\eta=-\ln\tan(\theta/2)$, and the
rapidity is defined in terms of energy $E$ and momentum $p$ as
$y=(1/2)[(E+p_{z})/(E-p_{z})]$. Angular distance is measured in units of
$\Delta R\equiv\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}$. It consists of an
inner tracking detector surrounded by a thin superconducting solenoid,
electromagnetic and hadron calorimeters, and a muon spectrometer incorporating
three large superconducting air-core toroidal magnets.

The inner-detector system (ID) is immersed in a $2\text{\,}\mathrm{T}$ axial
magnetic field and provides charged-particle tracking in the range
$|\eta|<2.5$. The high-granularity silicon pixel detector covers the vertex
region and typically provides four measurements per track, the first hit
normally being in the insertable B-layer (IBL) installed before Run 2 [ATLAS-
TDR-19, PIX-2018-001]. It is followed by the silicon microstrip tracker, which
usually provides eight measurements per track. These silicon detectors are
complemented by the transition radiation tracker (TRT), which enables radially
extended track reconstruction up to $|\eta|=2.0$. The TRT also provides
electron identification information based on the fraction of hits (typically
30 in total) above a higher energy-deposit threshold corresponding to
transition radiation.

The calorimeter system covers the pseudorapidity range $|\eta|<4.9$. Within
the region $|\eta|<3.2$, electromagnetic calorimetry is provided by barrel and
endcap high-granularity lead/liquid-argon (LAr) calorimeters, with an
additional thin LAr presampler covering $|\eta|<1.8$ to correct for energy
loss in material upstream of the calorimeters. Hadron calorimetry is provided
by the steel/scintillator-tile calorimeter, segmented into three barrel
structures within $|\eta|<1.7$, and two copper/LAr hadron endcap calorimeters.
The solid angle coverage is completed with forward copper/LAr and tungsten/LAr
calorimeter modules optimized for electromagnetic and hadronic energy
measurements respectively.

The muon spectrometer (MS) comprises separate trigger and high-precision
tracking chambers measuring the deflection of muons in a magnetic field
generated by the superconducting air-core toroidal magnets. The field integral
of the toroids ranges between $2.0$ and
$6.0\text{\,}\mathrm{T}\text{\,}\mathrm{m}$ across most of the detector. A set
of precision chambers covers the region $|\eta|<2.7$ with three layers of
monitored drift tubes, complemented by cathode-strip chambers in the forward
region, where the background is highest. The muon trigger system covers the
range $|\eta|<2.4$ with resistive-plate chambers in the barrel, and thin-gap
chambers in the endcap regions.

Interesting events are selected by the first-level trigger system implemented
in custom hardware, followed by selections made by algorithms implemented in
software in the high-level trigger [TRIG-2016-01]. The first-level trigger
accepts events from the $40\text{\,}\mathrm{MHz}$ bunch crossings at a rate
below $100\text{\,}\mathrm{kHz}$, which the high-level trigger reduces in
order to record events to disk at about $1\text{\,}\mathrm{kHz}$.

An extensive software suite [ATL-SOFT-PUB-2021-001] is used in the
reconstruction and analysis of real and simulated data, in detector
operations, and in the trigger and data acquisition systems of the experiment.

##  4 Data and simulated event samples

The results in this paper are based on $139\text{\,}{\mathrm{fb}}^{-1}$ of
$\sqrt{s}=$13\text{\,}\mathrm{TeV}$$ proton–proton ($pp$) collision data
collected with the ATLAS detector over the period 2015–2018.

Monte Carlo (MC) simulation is used to determine expected contributions from
both the signal processes and most background processes. For most samples,
detector effects were included using a Geant4 [Agostinelli:2002hh] simulation
of the ATLAS detector [SOFT-2010-01]. The $H\to aa$, $H\to ZZ_{\text{d}}$, and
$H\to Za$ signal samples, as well as a portion of the $gg\to ZZ^{*}$ and
triboson background samples, instead used a fast simulation [SOFT-2010-01]
which relies on a parameterization of the calorimeter response [ATL-PHYS-
PUB-2010-013]. The effects of pile-up (additional $pp$ collisions in the same
or a neighbouring bunch crossing) are included in the simulation. Weights are
applied to the simulated events to correct for small differences between data
and simulation in the reconstruction, identification, isolation, and impact
parameter efficiencies for electrons and muons [EGAM-2018-01, PERF-2017-01,
MUON-2018-03-pub]. Further, the lepton momentum scales and resolutions in the
simulation are adjusted to match the data [EGAM-2018-01, PERF-2017-03,
MUON-2018-03-pub]. Table 1 summarizes the samples and generators used, which
include MadGraph5_aMC@NLO version 2.2.2 [Alwall:2014hca], Powheg Box v2
[Hamilton:2013fea, Hamilton:2015nsa, Alioli:2010xd, Nason:2004rx,
Frixione:2007vw], Pythia 8.186 [Sjostrand:2014zea] (along with EvtGen 1.2.0
[Lange:2001uf] to decay heavy-flavour hadrons), and Sherpa 2.2.0, 2.2.1, and
2.2.2 [Bothmann:2019yzt].

Table 1: Overview of the event generators used for the simulated signal and background samples. For each process, the table lists the matrix element (ME) generator used along with the parton distribution function (PDF), the model used to implement parton showering (PS), the underlying event (UE), and the decay of heavy-flavour hadrons (HF), as well as the set of tuned parameters used to model the UE. The text gives the full version numbers of the generators. Process | ME generator | ME PDF | PS/UE/HF model | UE tune
---|---|---|---|---
$H\to Z_{\text{d}}Z_{\text{d}}$ / $ZZ_{\text{d}}$ [t] |  MadGraph5_aMC@NLO |  NNPDF2.3lo [Ball:2012cx] |  Pythia/EvtGen |  A14 [ATL-PHYS-PUB-2014-021]
$H\to aa$ [b] | Powheg Box | PDF4LHC15nnlo |  Pythia/EvtGen |  AZNLO [STDM-2012-23]
|  |  [Butterworth:2015oua] |  |
$H\to Za$ [b] | Powheg Box | PDF4LHC15nnlo |  Pythia/EvtGen | AZNLO
ggF[t] | Powheg Box | PDF4LHC15nnlo |  Pythia/EvtGen | AZNLO
VBF | Powheg Box |  CT10nlo [Lai:2010vv] |  Pythia/EvtGen | AZNLO
$VH$ | Pythia |  NNPDF2.3lo [Ball:2012cx] |  Pythia/EvtGen | A14
$ggZH$ | Powheg Box | NNPDF3.0nlo |  Pythia/EvtGen | AZNLO
$b\bar{b}H$ |  MadGraph5_aMC@NLO | NNPDF2.3lo |  Pythia/EvtGen | A14
$t\bar{t}H$ | Powheg Box | NNPDF2.3lo |  Pythia/EvtGen | A14
$ZZ$ | Sherpa | NNPDF3.0nnlo | Sherpa |  Sherpa default
$VVV$ | Sherpa | NNPDF3.0nnlo | Sherpa |  Sherpa default
$t\bar{t}Z$ | Sherpa | NNPDF3.0nnlo | Sherpa |  Sherpa default
$Z+\textrm{jets}$ | Sherpa | NNPDF3.0nnlo | Sherpa |  Sherpa default
$t\bar{t}$ | Powheg Box | NNPDF3.0nlo |  Pythia/EvtGen | A14
$WZ$ | Powheg Box | CT10nlo |  Pythia/EvtGen | A14

Signal samples involving a $Z_{\text{d}}$ vector boson were generated
according to the HAHM [wells2008find, gopalakrishna2008higgs, Curtin:2013fra,
Curtin:2014cca] implementation in MadGraph5_aMC@NLO, with the Higgs bosons
being produced via gluon–gluon fusion (ggF) and the Higgs boson mass set to
$m_{H}=$125\text{\,}\mathrm{GeV}$$. For the $H\rightarrow
Z_{\text{d}}Z_{\text{d}}$ process, $\epsilon$ and $\kappa$ were both set to
$10^{-4}$ and samples were generated with
$m_{Z_{\text{d}}}=$0.5\text{\,}\mathrm{GeV}$$, $1\text{\,}\mathrm{GeV}$,
$2\text{\,}\mathrm{GeV}$, and every $5\text{\,}\mathrm{GeV}$ in the range
$$5\text{\,}\text{G\mathrm{ eV }}$\leq
m_{Z_{\text{d}}}\leq$60\text{\,}\mathrm{GeV}$$. For the $H\rightarrow
ZZ_{\text{d}}$ process, $\kappa$ was changed to $10^{-10}$, and samples were
generated every $5\text{\,}\mathrm{GeV}$ in the range
$$15\text{\,}\text{G\mathrm{ eV }}$\leq
m_{Z_{\text{d}}}\leq$55\text{\,}\mathrm{GeV}$$. Final states with
$\tau$-leptons were not included; the change in signal region yield due to the
omission of these decays was below 1% and thus neglected. The much smaller
production of signal events by vector-boson fusion (VBF), $VH$, and
$t\bar{t}H$ was also omitted.

Samples for $H\to aa$ were simulated using Powheg Box at next-to-next-to-
leading order (NNLO) for arbitrary inclusive $gg\to H$ observables by
reweighting the Higgs boson rapidity to that of HNNLO [Hamilton:2013fea,
Hamilton:2012np, Campbell:2012am, Hamilton:2012rf, Catani:2007vq-fixed].
Again, only the ggF production process was considered. Higgs boson decays into
two scalars and thence into muons were simulated at leading order (LO) using
Pythia. Samples were generated for $m_{a}=0.5$, 1, 2, 2.5, 4, 6, 8, 10, 15,
30, 45, and $60\text{\,}\mathrm{GeV}$. Samples for $H\to Za$ were generated
similarly, for $m_{a}=1$, 2, 4, 6, 8, 10, 15, 20, 25, and
$30\text{\,}\mathrm{GeV}$.

Prompt-lepton backgrounds are estimated directly from MC simulations. They
arise primarily from the SM $H\to ZZ^{*}\to 4\ell$ process along with the non-
resonant $ZZ^{*}\to 4\ell$ process. Smaller leptonic backgrounds arise from
triboson production as well as $t\bar{t}+Z$ decays. Decays involving
$Z\rightarrow\tau\tau$ were found to contribute negligibly to the background
yields and are thus not included in the simulation. Backgrounds with jets
misidentified as leptons are estimated with data-driven methods, detailed
below in the individual analysis sections.

The $H\to ZZ^{*}\to 4\ell$ background process comprises various Higgs boson
production modes. The ggF process $gg\rightarrow H$ [ATL-PHYS-PUB-2016-006,
Hamilton:2013fea, Hamilton:2015nsa, Alioli:2010xd, Nason:2004rx,
Frixione:2007vw] was simulated in the same way as the $H\rightarrow aa$ signal
sample described above. The prediction was normalized to the next-to-next-to-
next-to-leading-order (N3LO) cross section in QCD with next-to-leading-order
(NLO) electroweak corrections [deFlorian:2016spz, Anastasiou:2016cez,
Anastasiou:2015ema, Dulat:2018rbf, Harlander:2009mq, Harlander:2009bw,
Harlander:2009my, Czakon:2021yub, Pak:2009dg, Actis:2008ug, Actis:2008ts,
Bonetti:2018ukf]. The VBF process [Nason:2009ai, Alioli:2010xd, Nason:2004rx,
Frixione:2007vw] was simulated using Powheg Box at NLO. The prediction was
normalized to an approximate-NNLO cross section in QCD with NLO electroweak
corrections [Ciccolini:2007jr, Ciccolini:2007ec, Bolzoni:2010xr]. Associated
production with a vector boson ($VH$) [Luisoni:2013kna, Ciccolini:2003jy,
Brein:2003wg, Ferrera:2011bk, Brein:2011vx, Ferrera:2013yga, Ferrera:2014lca,
Campbell:2016jau] was simulated at LO, while $t\bar{t}$ and $b\bar{b}$
associated production ($t\bar{t}H$, $b\bar{b}H$) [ATL-PHYS-PUB-2016-005,
Alwall:2014hca], as well as loop-induced Higgs and $Z$ boson production
($ggZH$) [Kniehl:1990iva], were simulated at NLO.

The non-resonant $q\bar{q}\to ZZ^{*}\to 4\ell$ background process [ATL-PHYS-
PUB-2017-005] was simulated using Sherpa 2.2.2 at NLO for up to one additional
parton and at LO for up to three additional partons. Matrix element
calculations were matched and merged with the Sherpa parton shower based on
the Catani–Seymour dipole factorization [Gleisberg:2008fv, Schumann:2007mg]
using the MEPS@NLO prescription [Hoeche:2011fd, Hoeche:2012yf, Catani:2001cc,
Hoeche:2009rj]. The virtual QCD corrections are provided by the OpenLoops
library [Buccioni:2019sur, Cascioli:2011va, Denner:2016kdg]. The gluon-
initiated process ($gg\to ZZ^{*}\to 4\ell$) was simulated in the same manner,
except that it was at LO, and the s-channel $H$ diagrams were omitted to avoid
double-counting. The gluon-initiated process has a large QCD correction at
NLO, so the cross section was scaled by a NLO/LO $K$-factor of $1.70\pm 0.15$
[Caola:2015psa]. Interference between the $gg\to ZZ^{*}\to 4\ell$ and $gg\to
H\to ZZ^{*}\to 4\ell$ processes is neglected.

Higher-order electroweak processes include triboson production ($VVV$) and
vector-boson scattering (VBS). These processes can yield final states with
four leptons along with two additional particles. They were generated with
Sherpa 2.2.2 at NLO for the inclusive processes and at LO for up to two
additional partons, with the same treatment as for $q\bar{q}\to ZZ^{*}\to
4\ell$. Higgs boson production via VBF was subtracted from these samples in
order to avoid double-counting.

The process $t\bar{t}+(Z\to\ell\ell)$ was generated with Sherpa 2.2.0 at LO
with up to one additional parton emission.

Other, reducible, backgrounds have fewer than four prompt leptons in the final
state, but can be accepted by the signal selection if there are additional
leptons from heavy-flavour decay or jets misidentified as leptons. The
$Z+\textrm{jets}$ process was generated with Sherpa 2.2.1 using NLO matrix
elements for up to two partons and LO matrix elements for up to four partons.
The $t\bar{t}$ process was generated with Powheg Box at NLO with the
$h_{\textrm{damp}}$ parameter, which regulates the high-$p_{\text{T}}$
radiation against which the $t\bar{t}$ system recoils, set to
$1.5m_{\textrm{top}}=$258\text{\,}\mathrm{GeV}$$ [ATL-PHYS-PUB-2016-020]. The
$WZ$ process was also generated with Powheg Box at NLO with the CT10nlo PDF.

##  5 Event reconstruction and selection

###  5.1 Lepton reconstruction

For the analyses considered in this paper, the final-state objects of interest
are electrons and muons.

Electrons are reconstructed and identified from charged-particle tracks in the
ID that match energy deposits in the calorimeters [EGAM-2018-01]. The
identification algorithm, based on a likelihood analysis, corresponds to the
‘Loose’ selection described in Ref. [EGAM-2018-01]. The reconstruction and
identification efficiency for electrons from $Z\to ee$ decays is about 90% per
electron [EGAM-2018-01].

Muon reconstruction [MUON-2018-03-pub] begins by independently finding tracks
in both the ID and MS. These track candidates are combined in a second step
along with information from the calorimeters to form muon candidates of
different types. Combined muons have matching tracks in both the MS and ID.
Segment-tagged muons have an ID track but only a single-chamber track segment
in the MS. Calorimeter-tagged (CT) muons have no MS track but have a pattern
of energy deposition in the calorimeters consistent with a muon; this is used
only in regions where the MS is not fully instrumented ($|\eta|<0.1$).
Finally, stand-alone (SA) muons have an MS track but no ID track, and are used
in regions beyond the coverage of the ID, $2.5<|\eta|<2.7$. Due to the reduced
performance of the latter two types, no more than one CT or SA muon may be
used in an event. Muons are then identified by imposing quality requirements,
corresponding to the ‘Loose’ selection in Ref. [MUON-2018-03-pub]. The
reconstruction and identification efficiency for muons from $W\to\mu\nu$
decays is greater than 98% [MUON-2018-03-pub].

To avoid identifying the same detector signature as multiple particles, an
electron candidate that has the same ID track as a muon candidate is ignored,
unless the muon is only calorimeter-tagged, in which case the muon is ignored
instead. Electrons that have the same track or cluster as a
higher-$p_{\text{T}}$ electron are also ignored.

###  5.2 Invariant kinematic mass variables

All three analyses considered in this paper involve looking for mass
resonances in final states consisting of a quadruplet of two same-flavour
opposite-sign (SFOS) lepton pairs: $(e^{+}e^{-}+e^{+}e^{-})$,
($e^{+}e^{-}+\mu^{+}\mu^{-})$, or $(\mu^{+}\mu^{-}+\mu^{+}\mu^{-})$. The
invariant masses of the two pairs are denoted by $m_{12}$ and $m_{34}$, where
$m_{12}$ is taken to be the one closer in mass to the $Z$ boson,
$|m_{12}-m_{Z}|<|m_{34}-m_{Z}|$.

If all four leptons have the same flavour, then for a given $m_{12}$ and
$m_{34}$ labelling, alternative SFOS pairings can also be defined. An
invariant mass $m_{14}$ is constructed from the positively charged lepton of
the $m_{12}$ pair and the negatively charged lepton of the $m_{34}$ pair. The
other alternative pairing $m_{23}$ is constructed analogously.

###  5.3 Common event selection

The analyses all involve a Higgs boson decaying into a pair of new bosons $X$,
or into a new boson $X$ along with a $Z$ boson, which in turn decay into pairs
of leptons. The $X$ bosons are presumed to be on-shell, so the strategy is to
search for resonances in the relevant dilepton mass distributions. Each
analysis defines a signal region (SR) via a series of selections on measured
quantities which maximizes the sensitivity to the signal.

All three analyses share a common preselection, but differ in the subsequent
steps of selecting the candidate final-state leptons, forming them into
quadruplets, selecting one of those quadruplets, and applying further
requirements to the selected quadruplet. LABEL:tab:eventpreselectionSummary
shows the event selections of the different analyses.

Table 2: Summary of event selection requirements for the ZX, HM, and LM
analyses. The quarkonia masses are taken to be
$m_{J/\Psi}=$3.096\text{\,}\mathrm{GeV}$$,
$m_{\Psi(2S)}=$3.686\text{\,}\mathrm{GeV}$$,
$m_{\Upsilon(1S)}=$9.461\text{\,}\mathrm{GeV}$$, and
$m_{\Upsilon(3S)}=$10.355\text{\,}\mathrm{GeV}$$ [pdg20]. The text provides
other definitions.