# Toward identifiability of total effects in summary causal graphs with latent
confounders: an extension of the front-door criterion

Charles K. Assaad
Sorbonne Université, INSERM,
Institut Pierre Louis d’Epidémiologie et de Santé Publique
F75012, Paris, France

###### Abstract

Conducting experiments to estimate total effects can be challenging due to
cost, ethical concerns, or practical limitations. As an alternative,
researchers often rely on causal graphs to determine if it is possible to
identify these effects from observational data. Identifying total effects in
fully specified non-temporal causal graphs has garnered considerable
attention, with Pearl’s front-door criterion enabling the identification of
total effects in the presence of latent confounding even when no variable set
is sufficient for adjustment. However, specifying a complete causal graph is
challenging in many domains. Extending these identifiability results to
partially specified graphs is crucial, particularly in dynamic systems where
causal relationships evolve over time. This paper addresses the challenge of
identifying total effects using a specific and well-known partially specified
graph in dynamic systems called a summary causal graph, which does not specify
the temporal lag between causal relations and can contain cycles. In
particular, this paper presents sufficient graphical conditions for
identifying total effects from observational data, even in the presence of
hidden confounding and when no variable set is sufficient for adjustment,
contributing to the ongoing effort to understand and estimate causal effects
from observational data using summary causal graphs.

##  1 Introduction

Causal questions arise when we seek to understand the effects of
interventions, such as asking, "If we administer today a hypertension
treatment to a patient with kidney insufficiency, will the kidney function
(represented by the creatinine level) improve tomorrow?". These questions,
often referred to as total causal effects or simply total effects (Pearl et
al., 2000), are denoted as $\Pr(creatinine_{tomorrow}\mid
do(hypertension_{today}))$. They differ from associational relationships,
$\Pr(creatinine_{tomorrow}\mid hypertension_{today})$, as they isolate the
direct effect of an intervention, disregarding other influencing factors such
as sodium intake or protein intake or stress level. Experimentation is known
as the traditional approach across various fields to estimate the total effect
of interventions free from bias (Neyman et al., 1990). However, conducting
experiments is not always feasible due to cost, ethical considerations, or
practical limitations. Consequently, scientists often resort to estimating
effects of interventions from observational data. This process relies on
specific assumptions and typically involves two sequential steps:
identifiability and estimation (Pearl, 2019).

Graphical models, provide a framework for identifying total effects from
causal graphs which encode variables as vertices and causal relationships as
arrows, allowing researchers to visualize and analyze complex causal
structures (Pearl et al., 2000). Identifying total effects in fully specified
non-temporal causal graphs has been a subject of considerable attention
(Pearl, 1993, 1995; Spirtes et al., 2000; Pearl et al., 2000; Shpitser and
Pearl, 2008; Shpitser et al., 2010). Adjusting for covariates is one method
among several that allow us to estimate causal effects from observational
data. Pearl (1995) has provided examples where no variable set is sufficient
for adjustment, yet the causal effect can still be consistently estimated. In
such cases, estimation involves multi-stage adjustments. Pearl’s introduction
of front-door criteria enables the identification of total effects even in the
presence of latent confounding, assuming the causal graph is a directed
acyclic graph (DAG).

These results have been extended to fully specified temporal graphs (Blondel
et al., 2016) which represent causal relations in dynamic systems where causal
relationships evolve over time. However, constructing a fully scpecified
temporal graph requires knowledge of all causal relationships among observed
variables, which is often unavailable, especially in many real-world
applications. However, experts may know that one variable causes another
without knowing the exact temporal lag. For example, understanding the
transmission of SARS-CoV-2 from younger to older individuals, and vice versa,
can help define interventions most likely to reduce the number of deaths.
Indeed, it has been shown that younger adults tended to be highly infected
during the first wave of the pandemic, while older individuals faced a higher
risk of death if infected (Carrat et al., 2021; Lapidus et al., 2021; Glemain
et al., 2024). Considering sufficiently large time intervals (several weeks)
as in repeated serosurveys, like in Wiegand et al. (2023), it is not clear if
the number of new infections in one age group during a time interval
(incidence) may be influenced by incidence in the other age group during the
same interval. Incidence in an age group can also be influenced by incidence
during the previous time interval in any age group. Therefore, constructing a
fully specified causal graph is difficult.

In such cases, partially specified causal graphs can be useful. A very well
known and useful partially specified causal graph is the summary causal graph
(SCG) which represents causal relations without including temporal
information, i.e., each vertex represents a time series. Both medical and
epidemiological examples given above can be represented by an SCG with a
cyclic relationship representing the interplay between creatinine and
hypertension in the first example and between the two age groups in the second
example. Recently, there has been new interest in extending identifiability
results to partially specified graphs Maathuis and Colombo (2013); Perkovic
(2020); Anand et al. (2023). Specifically for SCGs, assuming no instantaneous
relations, Eichler and Didelez (2007) demonstrated that the total effect is
identifiable, while Assaad et al. (2023) established identifiability in the
presence of instantaneous relations for acyclic SCGs. Assaad et al. (2024)
addressed the identifiability of total effects under more general conditions
in SCGs that includes cycles and instantaneous relations. However, non of
these works considered the case where the total effect is not identifiable by
adjustment due to hidden confounding.

In this paper, we address the challenge of identifying total effects when a
fully specified temporal causal graph is not available, and instead, we have
access to an SCG while allowing instantaneous relations, cycles, and hidden
confounding. By leveraging the front-door criterion (Pearl et al., 2000), we
develop sufficient conditions for identifying total effects from observational
data in the case where it is not identifiable by adjustment due to hidden
confounding.

The remainder of the paper is organized as follows: Section 2 introduces
necessary terminology and tools and it formalizes the problem. Section 3
presents the main result of this paper and Section 4 discusses several
examples of non-identifiability. Finally, Section 5 concludes the paper.

##  2 Preliminaries and problem setup

In this section, we first introduce some terminology and tools which are
standard for the major part and then, formalize the problem we are going to
solve. In the remainder, the set of all integers is represented by
$\mathbb{Z}$. We start by defining the causal model that we consider.

$L_{t}$$L_{t-1}$$L_{t-2}$$W_{t}$$W_{t-1}$$W_{t-2}$$Y_{t}$$Y_{t-1}$$Y_{t-2}$$X_{t}$$X_{t-1}$$X_{t-2}$
(a) FTCG.

$W_{t}$$W_{t-1}$$W_{t-2}$$Y_{t}$$Y_{t-1}$$Y_{t-2}$$X_{t}$$X_{t-1}$$X_{t-2}$
(b) FT-ADMG

$X$$Y$$W$ (c) SCG

Figure 1: A full time directed acyclic graph (a) where black vertices
represent latent variables, a full time acyclic directed mixed graph (b)
compatible with (a) and where $L_{t-2}$, $L_{t-1}$ and $L_{t}$ are eliminated
and replaced by dashed edges, and a summary causal graph (c) compatible with
(b).

######  Definition 1 (Dynamic structural causal model (DSCM)).

A dynamic structural causal model is a tuple
$\mathcal{M}=(\mathbb{L},\mathbb{V},\mathbb{F},P(\mathbb{l}))$, where
$\mathbb{L}=\mathbb{L}^{v^{1}}\cup\cdots\cup\mathbb{L}^{v^{d}}$ such that
$\forall i\in\\{1,\cdots,d\\}$,
$\mathbb{L}^{v^{i}}=\\{(L^{v^{i}})_{t\in\mathbb{Z}}\\}$, is a set of sets of
exogenous variables, which cannot be observed, but which affect the rest of
the model . $\mathbb{V}=\mathbb{V}^{1}\cup\cdots\cup\mathbb{V}^{d}$ such that
$\forall i\in\\{1,\cdots,d\\}$,
$\mathbb{V}^{i}=\\{(V^{i}_{t})_{t\in\mathbb{Z}}\\}$, is a set of sets of
endogenous variables, which are observed and which are functionally depednent
on some subset $\mathbb{L}^{v_{i}}\cup\mathbb{V}$. $\mathbb{F}$ is a set of
functions such that each $f^{v_{i}}_{t}$ is a mapping from subset
$\mathbb{L}^{v^{i}}\cup\mathbb{V}\backslash\\{V^{i}_{t}\\}$ to $V^{i}_{t}$.
$P(\mathbb{l})$ is a joint probability distribution over $\mathbb{L}$.

We assume that each time series representing hidden confounders does not
directly cause itself, as stated below.

######  Assumption 1.

Consider a DSCM
$\mathcal{M}=(\mathbb{L},\mathbb{V},\mathbb{F},P(\mathbb{l}))$, where
$\mathbb{L}=\mathbb{L}^{v^{1}}\cup\cdots\cup\mathbb{L}^{v^{d}}$. $\forall
i\in\\{1,\cdots,d\\}$. $\forall
L^{v^{i}}_{t},L^{v^{i}}_{t^{\prime}}\in\mathbb{L}^{v^{i}}$, such that $t\neq
t^{\prime},L^{v^{i}}_{t}\rotatebox[origin={c}]{90.0}{$\models$}L^{v^{i}}_{t^{\prime}}$.

Furthermore, we assume stationarity.

######  Assumption 2.

Consider a DSCM
$\mathcal{M}=(\mathbb{L},\mathbb{V},\mathbb{F},P(\mathbb{l}))$. $\forall
f^{v^{i}}_{t},f^{v^{i}}_{t^{\prime}}\in\mathbb{F}$,
$f^{v^{i}}_{t}=f^{v^{i}}_{t^{\prime}}$.

Assumption 2 entails that if $Y_{t}=f^{y}_{t}(X_{t-1},W_{t-1})$, then $\forall
i\in\mathbb{Z}$, $Y_{t-i}=f^{y}_{t}(X_{t-1-i},W_{t-1-i})$. Note that in case
Assumption 2 is violated then finding a unique total effect would be ill-
posed, as we assume a dynamic system with only one multivariate observational
time series. Violating this assumption would imply that the total effect
changes over time. However, this assumption can be relaxed if multiple
observations of each variable in $\mathbb{V}$ are available.

The full time causal graph which qualitativly represents the direct causal
relations in a DSCM can be represented by a full-time acyclic directed mixed
graph.

######  Definition 2 (Full-Time Acyclic Directed Mixed Graph).

Consider a DSCM $\mathcal{M}$. The _full-time acyclic directed mixed graph
(FT-ADMG)_ $\mathcal{G}=(\mathbb{V},\mathbb{E})$ induced by $\mathcal{M}$ is
defined in the following way:

$\displaystyle\mathbb{E}^{1}$ $\displaystyle:=$
$\displaystyle\\{X_{t-\gamma}\rightarrow Y_{t}$ $\displaystyle|$
$\displaystyle\forall Y_{t}\in\mathbb{V},~{}X_{t}\in\mathbb{X}\text{ such that
}Y_{t}:=f^{y}_{t}(\mathbb{X},\mathbb{L}^{y})\text{ in }\mathcal{M}\text{ and
}\mathbb{X}\subset\mathbb{V}\backslash\\{Y_{t}\\}\\},$
$\displaystyle\mathbb{E}^{2}$ $\displaystyle:=$
$\displaystyle\\{X_{t-\gamma}\mathrel{\leavevmode\hbox to18.9pt{\vbox
to4.3pt{\pgfpicture\makeatletter\hbox{\hskip 1.19998pt\lower-1.59998pt\hbox
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$\displaystyle|$ $\displaystyle\forall
X_{t-\gamma},Y_{t}\in\mathbb{V}~{}\text{ such that
}\mathbb{L}^{x}\cap\mathbb{L}^{y}\neq\emptyset\\}.$

where $\mathbb{E}=\mathbb{E}^{1}\cup\mathbb{E}^{2}$.

### FT-ADMG notions

For an FT-ADMG $\mathcal{G}$, a _path_ from $X$ to $Y$ in $\mathcal{G}$ is a
sequence of distinct vertices $\langle X,\ldots,Y\rangle$ in which every pair
of successive vertices is adjacent. A _directed path_ from $X$ to $Y$ is a
path from $X$ to $Y$ in which all edges are directed towards $Y$ in
$\mathcal{G}$, that is $X\rightarrow\ldots\rightarrow Y$. A _backdoor path_
between $X$ and $Y$ is a path between $X$ and $Y$ with an arrowhead into $X$
in $\mathcal{G}$. If $X\rightarrow Y$, then $X$ is a _parent_ of $Y$. If there
is a directed path from $X$ to $Y$, then $X$ is an _ancestor_ of $Y$, and $Y$
is a _descendant_ of $X$. A vertex counts as its own descendant and as its own
ancestor. The sets of parents, ancestors and descendants of $X$ in
$\mathcal{G}$ are denoted by $\text{Par}(X,\mathcal{G})$,
$\text{Anc}(X,\mathcal{G})$ and $\text{Desc}(X,\mathcal{G})$ respectively. If
a path $\pi$ contains $X\rightarrow W\leftarrow Y$ as a subpath, then $W$ is a
_collider_ on $\pi$. A path $\pi$ from $X$ to $Y$ is _active_ given a vertex
set $\mathbb{W}$, with $X,Y\notin\mathbb{W}$ if every non-collider on $\pi$ is
not in $\mathbb{W}$, and every collider on $\pi$ has a descendant in
$\mathbb{W}$. Otherwise, $\mathbb{W}$ _blocks_ $\pi$. Lastly, we call each
vertex in an FT-ADMG as a micro vertex.

The FT-ADMG is supposed to be a DAG with bidirected edges representing hidden
confounding. If all variables representing hidden confounding become observed
then the FT-ADMG becomes a full time DAG, commonly known as a full time causal
graph (FTCG). Figure 1(a) presents an FTCG and Figure 1(b) presents its
corresponding FT-ADMG. Furthermore, Assumption 2 allows us to fix the maximum
temporal lag between a cause and an effect, denoted as $\gamma_{\max}$.

$X$$Y$$W$ (a) SCG.

$W_{t}$$W_{t-1}$$W_{t-2}$$Y_{t}$$Y_{t-1}$$Y_{t-2}$$X_{t}$$X_{t-1}$$X_{t-2}$
(b) Candidate FT-ADMG 1.

$W_{t}$$W_{t-1}$$W_{t-2}$$Y_{t}$$Y_{t-1}$$Y_{t-2}$$X_{t}$$X_{t-1}$$X_{t-2}$
(c) Candidate FT-ADMG 2.

Figure 2: An SCG in (a) where $\\{W\\}$ does not satisfy condition 4.a of
Definition 6 for the total effect $\Pr(y_{t}|do(x_{t-1}))$ since
$Cycles(X,\mathcal{G}^{s})\neq\emptyset$ and $\gamma\neq 0$ with two candidate
FT-ADMG in (b) and (c). Each pair of red and blue vertices in the FT-ADMGs
represents the total effect we are interested in.

The _total effect_ (Pearl et al., 2000) between two micro variables is written
as $P(Y_{t}=y_{t}|do(X_{t-\gamma}=x_{t-\gamma}))$ (as well as
$P(y_{t}|do(x_{t-\gamma}))$ by a slight abuse of notation). $Y_{t}$
corresponds to the response and $do(X_{t-\gamma}=x_{t-\gamma})$ represents an
intervention (as defined in Pearl et al. (2000) and Eichler and Didelez (2007,
Assumption 2.3)) on the variable $X$ at time $t-\gamma$, with $\gamma\geq 0$.
In the remainder of the paper, we consider $\gamma$ to be in
$\\{0,\gamma_{\max}\\}$. The identifiability of the total effect in FT-ADMGs
is defined as follows.

######  Definition 3 (Identifiability of total effects in FT-ADMGs).

Let $X_{t-\gamma}$ and $Y_{t}$ be disjoint vertices in an FT-ADMG
$\mathcal{G}=(\mathbb{V},\mathbb{E})$. The total effect of $X_{t-\gamma}$ on
$Y_{t}$ is identifiable in $\mathcal{G}$ if $\Pr(y_{t}|do(x_{t-\gamma}))$ is
uniquely computable from any observational distribution consistent with
$\mathcal{G}$.

A total effect is uniquely computable if $\Pr(y_{t}\mid do(x_{t-\gamma}))$ can
be expressed using a _do-free formula_. Given a fully specified FT-ADMG, there
exists many tools to identify the total effect. For example the standard
backoor criterion (Pearl, 1993, 1995) can be used to find a set of covariates
$\mathbb{B}$ that is sufficient for adjustment; in such case the total effect
is written as $\sum_{\mathbb{b}}\Pr(y_{t}\mid
x_{t-\gamma},\mathbb{b})\Pr(\mathbb{b})$. When such a set does not exist due
to hidden confounding, the standard frontdoor criterion, as introduced by
Pearl (1995), can sometimes enable the derivation of an alternative do-free
formula.

However, in many real-world applications such as medicine or epidemiology,
experts often cannot provide the FT-ADMG. Furthermore, discovering (Spirtes et
al., 2000) the true FT-ADMG from real observational data is often not
satisfactory (Aït-Bachir et al., 2023) due to the additional non-testable
strong assumptions (Assaad et al., 2022). Therefore, we typically rely on a
partially specified representation of the FT-ADMG, known as a summary causal
graph.

######  Definition 4 (Summary Causal Graph with possible latent confounding).

Consider an FT-ADMG $\mathcal{G}=(\mathbb{V},\mathbb{E})$. The _summary causal
graph (SCG)_ $\mathcal{G}^{s}=(\mathbb{S},\mathbb{E}^{{s}})$ compatible with
$\mathcal{G}$ is defined in the following way:

$\displaystyle\mathbb{S}$ $\displaystyle:=$ $\displaystyle\\{Y$
$\displaystyle|$ $\displaystyle\forall Y_{t}\in\mathbb{V}\\},$
$\displaystyle\mathbb{E}^{s1}$ $\displaystyle:=$ $\displaystyle\\{X\rightarrow
Y$ $\displaystyle|$ $\displaystyle\forall X,Y\in\mathbb{S},~{}\exists
t^{\prime}\leq t\in\mathbb{Z}\text{ such that }X_{t^{\prime}}\rightarrow
Y_{t}\in\mathbb{E}\\},$ $\displaystyle\mathbb{E}^{s2}$ $\displaystyle:=$
$\displaystyle\\{X\mathrel{\leavevmode\hbox to17.9pt{\vbox
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}Y$ $\displaystyle|$
$\displaystyle\forall X,Y\in\mathbb{S},~{}X\neq Y\text{ and }\exists
t^{\prime}\leq t\in\mathbb{Z}\text{ such that
}X_{t^{\prime}}\mathrel{\leavevmode\hbox to17.9pt{\vbox
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}Y_{t}\in\mathbb{E}\\},$

where $\mathbb{E}^{s}=\mathbb{E}^{s1}\cup\mathbb{E}^{s2}$.

SCG notations For an SCG $\mathcal{G}^{s}$, a directed path from $X$ to $Y$
and the edge $Y\rightarrow X$ form a _directed cycle_ in $\mathcal{G}$. We
denote by $Cycles(X,\mathcal{G})$ the set of all directed cycles containing
$X$ in $\mathcal{G}$. A _directed path_ between $X$ and $Y$ is a path between
$X$ and $Y$ which starts by $X\rightarrow$ and does not contain any arrow on
the path pointing strictly towards $X$. A _backdoor path_ between $X$ and $Y$
is a path between $X$ and $Y$ which starts by either $X\leftarrow$ or
$X\leftrightarrows$. If $X\rightarrow Y$ or $X\leftrightarrows Y$, then $X$ is
a _parent_ of $Y$. The notions of ancestors and descendants are defined
similarly as in the case of FT-ADMGs. Since SCGs can contain cycles we will
use the notion of $\sigma$-blocked path (Forré and Mooij, 2017). A path in an
SCG is $\sigma$-blocked given a set $\mathbb{W}$ if it contains a strict
collider at $W$ (i.e., $\rightarrow W\leftarrow$, and not $\leftrightarrows
W\leftarrow$ or $\rightarrow W\leftrightarrows$) such that $\mathbb{W}\cup
Desc(W,\mathcal{G}^{s})=\emptyset$ or if it contains strict non-collider at
$W$ (i.e., $\rightarrow W\rightarrow$ or $\leftarrow W\rightarrow$ or
$\leftarrow W\leftrightarrows$ , and not $\leftrightarrows W\leftarrow$) such
that $W\in\mathbb{W}$ and there exists a directed edge pointing from $W$ to a
vertex on the path that does not form a cycle with $W$. A path in an SCG is
activated if it is not $\sigma$-blocked. In particular, a path is activated by
an empty set if it does not contain any strict collider. Lastly, we call each
vertex in an SCG as a macro vertex.

Figure 1(c) presents the SCG compatible with the FT-ADMG in Figure 1(b). For a
given SCG $\mathcal{G}^{s}$, we call any FT-ADMG from which $\mathcal{G}^{s}$
can be derived as a candidate FT-ADMG for $\mathcal{G}^{s}$. For example, in
Figure 2, we present an SCG with two of its candidate FT-ADMGs. The set of all
candidate FT-ADMGs for $\mathcal{G}^{s}$ is denoted by
$\mathcal{C}(\mathcal{G}^{s})$.

We focus in this paper on identifying the total effect _when the only
knowledge one has of the underlying DSCM consists in the SCG derived from the
unknown, true FT-ADMG_. In this setting, the identifiability of the total
effect in SCGs is defined as follows:

######  Definition 5 (Identifiability of total effects in SCGs).

Consider an SCG $\mathcal{G}^{s}$. Let $X_{t-\gamma}$ and $Y_{t}$ be disjoint
vertices in every candidate FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$. The
total effect of $X_{t-\gamma}$ on $Y_{t}$ is identifiable in $\mathcal{G}^{s}$
if $\Pr(y_{t}|do(x_{t-\gamma}))$ is uniquely computable from any observational
distribution consistent with any FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$.

Obviously, when the true FT-ADMG is unknown but we have access to the
compatible SCG, we can enumurate all candidates FT-ADMGs and then search for a
do-free formula applicable for each of those FT-ADMGs. Within this approch, it
is possible to identify the total effect if we can find a set of micro
vertices that satisfies the front-door criterion over all FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$. However, enumerating all candidate FT-ADMGs is
computationally expensive (Robinson, 1977), even when considering the
constraints given by an SCG. Therefore, we address the following technical
problem:

######  Problem 1.

Consider an SCG $\mathcal{G}^{s}$ and the total effect
$\Pr(y_{t}|do(x_{t-\gamma}))$. We aim to find sufficient conditions for
identifying $\Pr(y_{t}|do(x_{t-\gamma}))$ using an SCG with latent confounding
without enumerating all candidate FT-ADMGs in $\mathcal{C}(\mathcal{G}^{s})$.

##  3 The identifiability of total effects in SCGs with latent confounders

In this section, we present the main result of the paper. We start by giving
our extension of the frontdoor criterion for SCGs.

######  Definition 6 (Front-door criterion for SCGs).

Consider an SCG $\mathcal{G}^{s}$. A set of macro vertices $\mathbb{W}$ in
$\mathcal{G}^{s}$ satisfy the front-door criterion for SCGs relative to a pair
of micro vertices $(X_{t-\gamma},Y_{t})$ compatible with a pair of macro
vertices $(X,Y)$ in $\mathcal{G}^{s}$ if:

  1. 1.

$\mathbb{W}$ intercepts all activated directed paths from $X$ to $Y$;

  2. 2.

there is no activated backdoor path from $X$ to $\mathbb{W}$;

  3. 3.

all backdoor paths from $\mathbb{W}$ to $Y$ are $\sigma$-blocked by $X$;

  4. 4.

one of the following holds:

    1. (a)

$Cycles(X,\mathcal{G}^{s})=\emptyset$ ; or

    2. (b)

$\gamma=0$.

Conditions 1-3 in Definition 6 correspond to the three conditions in the
standard frontdoor citerion (Pearl, 1995). For an illusrtation of Definition
6, we provide in Figure 3 several examples of SCGs where Definition 6 is
satisfied for $W$ relative to $(X_{t-\gamma},Y_{t})$. Notice that Definition 6
remains satisfied for $W$ relative to $(X_{t},Y_{t})$ if we add a cycle on $X$
that does not involve any other vertex in the presented SCGs. Additionally, we
provide in Figure 4 several examples of SCGs where Definition 6 is not
satisfied for any vertex.

$X$$Y$$W$ (a)

$X$$Y$$W$$Z$ (b)

$X$$U$$Z$$W$$Y$ (c)

$X$$W$$Z$$U$$Y$ (d)

Figure 3: Four SCGs satisfying Definition 6 for $W$ relative to the micro of
vertices $(X_{t-\gamma},Y_{t})$. Each pair of red and blue vertices in the
FTCGs represents the total effect we are interested in.

In the following, we present lemmas that form the building blocks for the
theorem introduced at the end of the section. The first lemma asserts that if
a set of macro vertices intercepts all directed paths between $X$ and $Y$ in
an SCG, then there exists a finite set of micro vertices that intercepts all
directed paths between $X_{t-\gamma}$ and $Y_{t}$ in any candidate FT-ADMG.
This mirrors Condition 1 for micro vertices.

######  Lemma 3.1.

Consider an SCG $\mathcal{G}^{s}$. If a set of macro vertices $\mathbb{W}$
intercepts all directed paths from $X$ to $Y$ in $\mathcal{G}^{s}$ then
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$ intercepts all
directed paths from $X_{t-\gamma}$ to $Y_{t}$ in any candidate FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$.

###### Proof.

Let $\mathcal{G}$ be a candidate FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$.
Consider a directed path $\pi$ from $X_{t-\gamma}$ to $Y_{t}$ in
$\mathcal{G}$. Suppose $\pi$ does not contain any vertex from
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$. Then, $\pi$ must
include at least one vertex from
$\\{(\mathbb{W}_{t-\gamma+\ell})_{\ell\in\mathbb{Z}\backslash{\\{0,\cdots,\gamma\\}}}\\}$,
as all paths from $X$ to $Y$ are intercepted by $\mathbb{W}$ in
$\mathcal{G}^{s}$. Consider the case where there exists $W_{t-\gamma+\ell}$ in
$\pi$ such that $\ell<0$. In this case, $\pi$ cannot be a directed path
because by temporal priority, there exists no directed path from
$X_{t-\gamma}$ to $W_{t-\gamma+\ell}$. Consider the case where there exists
$W_{t-\gamma+\ell}$ in $\pi$ such that $\ell>\gamma$. In this case, $\pi$
cannot be a directed path because by temporal priority, there exists no
directed path from $W_{t-\gamma+\ell}$ to $Y_{t}$. Therefore, it must be the
case that $\pi$ includes at least one vertex from
${(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}}$. Since $\mathcal{G}$
and $\pi$ are arbitrary, this conclusion applies to all directed paths between
from $X_{t-\gamma}$ to $Y_{t}$ in any candidate FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$. ∎

The second lemma asserts that given Conditions 1-3, along with Condition 4a,
it is guaranteed that there exists a finite set that blocks all backdoor paths
from $X_{t-\gamma}$ to some micro vertices that intercept all directed paths
from $X_{t-\gamma}$ to $Y_{t}$ in any candidate FT-ADMG. Furthermore, this
finite set does not contain any descendants of $X_{t-\gamma}$. This mirrors
Condition 2 for micro vertices.

######  Lemma 3.2.

Consider an SCG $\mathcal{G}^{s}=(\mathbb{S},\mathbb{E}^{s})$ and the pair of
micro vertices $(X_{t-\gamma},Y_{t})$ compatible with the macro vertices
$(X,Y)$. Suppose $\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$ is
a set of micro vertices that intercepts all directed paths from $X_{t-\gamma}$
to $Y_{t}$ in any candidate FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$ and for
which their set of compatible macro vertices $\mathbb{W}\subset\mathbb{S}$
statisfies Conditions 2 and 3 of Definition 6 in $\mathcal{G}^{s}$. If
$Cycles(X)=\emptyset$ then
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}\cup\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}\cup\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Par(X,\mathcal{G}^{s})\backslash\\{X\\}\\}$ blocks all backdoor paths from
$X_{t-\gamma}$ to $\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$
and does not contain any descendant of $X_{t-\gamma}$ in any candidate FT-ADMG
in $\mathcal{C}(\mathcal{G}^{s})$.

###### Proof.

Consider an arbitrary micro vertex
$W_{t-\lambda}\in\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$. By
definition of $\gamma_{\max}$, the set
$\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Par(X,\mathcal{G}^{s})\\}$ contains all observed parents of $X_{t-\gamma}$ in
any FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$. Given that
$Cycles(X,\mathcal{G}^{s})=\emptyset$, if $B\in Par(X,\mathcal{G}^{s})$, then
$B\not\in Desc(X,\mathcal{G}^{s})$, implying that
$\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Par(X,\mathcal{G}^{s})\\}$ cannot include any descendant of $X_{t-\gamma}$.
Therefore, this set blocks all backdoor paths from $X_{t-\gamma}$ to
$W_{t-\lambda}$ that begin with an undashed left arrow, i.e.,
"$X_{t-\gamma}\leftarrow$" without activating a new path between
$X_{t-\gamma}$ and $W_{t-\lambda}$ . Consequently, the only backdoor paths
from $X_{t-\gamma}$ to $W_{t-\lambda}$ that are not blocked by
$\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma{\max}}|B\in
Par(X,\mathcal{G}^{s})\\}$ are those starting with
"$X_{t-\gamma}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$". Let us consider
two cases: Case 1 where a backdoor path from $X_{t-\gamma}$ to $W_{t-\lambda}$
starts with $X_{t-\gamma}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and passes by
$X_{t-\lambda^{\prime}}$ such that $\lambda^{\prime}\neq\gamma$; and Case 2
where a backdoor path from $X_{t-\gamma}$ to $W_{t-\lambda}$ starts with
$X_{t-\lambda}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ but does not pass
by $X_{t-\lambda^{\prime}}$ for all $\lambda^{\prime}\neq\gamma$.

Case 1: the case where $\lambda^{\prime}>\gamma+\gamma_{\max}$, cannot exists,
because otherwise Assumption 1 is violated. In case
$\gamma+1\leq\lambda^{\prime}\leq\gamma+\gamma_{\max}$, then by Assumption 1,
$\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}$ blocks the back-door
path. In case $0\leq\lambda^{\prime}\leq\gamma$, then by Assumption 1,
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}$ blocks the back-door path.
Furthermore, by temporal priority,
$\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}$ cannot contain
descendants of $X_{t-\gamma}$ and since $Cycles(X,\mathcal{G}^{s})=0$,
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}$ cannot contain descendants
of $X_{t-\gamma}$.

Case 2: If the backdoor path does not pass by any $X_{t-\lambda^{\prime}}$
with $\lambda^{\prime}\neq\gamma$ then by Condition 2, the backdoor path
should be blocked by the empty set. ∎

Note that this lemma does not hold if Assumption 1 is violated. In such a
scenario, a backdoor path from $X_{t-\gamma}$ to $W_{t-\gamma}$ could start
with $\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and pass through a
hidden confounder $L_{t-\gamma}$. This path might eventually involve another
hidden confounder $L_{t-\gamma-i}$ (due to the violation of this assumption).
If $i$ is sufficiently large, we can envision an FT-ADMG where a directed path
exists from $L_{t-\gamma-i}$ to $W_{t-\gamma}$ that does not pass through
$\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma}\\}$.

Additionally, note that if $Cycle(X,\mathcal{G}^{s})\neq\emptyset$, then
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}$ can include a descendant of
$X_{t-\gamma}$. Furthermore, there would exist an FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$ where
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}$ is required to block all
backdoor paths between $X_{t-\gamma}$ and
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$. For instance, in
Figure 2, since there is a cycle on $X$ in the SCG, it is possible to have an
FT-ADMG with the backdoor path $\langle X_{t-1},X_{t-2},W_{t-1}\rangle$.
$X_{t-2}$ is the only micro vertex that can block this path, but conditioning
on $X_{t-2}$ would also activate a path between $X_{t-3}$ and $X_{t-1}$,
creating a new backdoor path $\langle X_{t-1},X_{t-3},W_{t-2},W_{t-1}\rangle$.
However, it turns out this is not the case when $\gamma=0$, as stated in the
following lemma.

######  Lemma 3.3.

Consider an SCG $\mathcal{G}^{s}=(\mathbb{S},\mathbb{E}^{s})$ and the pair of
micro vertices $(X_{t},Y_{t})$ compatible with the macro vertices $(X,Y)$.
Suppose $\mathbb{W}_{t}$ is a set of micro vertices that intercepts all
directed paths from $X_{t}$ to $Y_{t}$ in any candidate FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$ and for which their set of compatible macro
vertices $\mathbb{W}\subset\mathbb{S}$ statisfies Conditions 2 and 3 of
Definition 6 in $\mathcal{G}^{s}$. In any candidate FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$,
$\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash
Desc(X,\mathcal{G})\\}\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$ blocks all backdoor paths
from $X_{t}$ to $\mathbb{W}_{t}$ and does not contain any descendant of
$X_{t}$.

###### Proof.

Consider an arbitrary micro vertex $W_{t}\in\mathbb{W}_{t}$. Condition 2
implies that there exists no active backdoor path $\pi^{s}=\langle
V^{1}=X,\dots,V^{n}=W\rangle$ from $X$ to $W$ in $\mathcal{G}^{s}$ such that
$\langle V^{2},\dots,V^{n-1}\rangle\subseteq Desc(X,\mathcal{G}^{s})$ which
means that there are no cycles that contain $X$ and another vertex on a path
from $X$ to $W$. Thus all backdoor paths from $X_{t}$ to $W_{t}$ in any FT-
ADMG $\mathcal{G}$ in $\mathcal{C}(\mathcal{G}^{s})$ that pass by a vertex
$Z_{t-\lambda}$ such that $Z\in Cycles(X,\mathcal{G}^{s})$ has to pass by
another vertex $X_{t-\lambda^{\prime}}$ with $\lambda^{\prime}\neq 0$. If
$\lambda^{\prime}<0$, then obviously the backdoor path is necesserly blocked
by the empty set since $Z_{t-\lambda^{\prime}}$ cannot be an ancestor of
$W_{t}$ nor an ancestor of $X_{t}$. If $\lambda^{\prime}>0$, the backdoor path
can be blocked by
$X_{t-\lambda^{\prime}}\in\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$. In addition, by
construction, $\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash
Desc(X,\mathcal{G})\\}\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$ cannot contain any
descendant of $X_{t}$. Which means that
$\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash
Desc(X,\mathcal{G})\\}\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$ blocks all backdoor paths
from $X_{t}$ to $W_{t}$ starting with an undashed right arrow, i.e,
"$X_{t}\leftarrow$". This means that the only backdoor paths from $X_{t}$ to
$W_{t}$ that are not blocked are the ones statring with
"$X_{t}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$". As before, let us
consider two cases: Case 1 where a backdoor path from $X_{t}$ to $W_{t}$
starts with $X_{t}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ and passes by
$X_{t-\lambda^{\prime}}$ such that $\lambda^{\prime}\neq 0$; and Case 2 where
a backdoor path from $X_{t}$ to $W_{t}$ starts with
$X_{t}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
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}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}$ but does not pass
by $X_{t-\lambda^{\prime}}$ for all $\lambda^{\prime}\neq\gamma$.

Case 1: the case where $\lambda^{\prime}>\gamma_{\max}$, cannot exists,
because otherwise Assumption 1 would be violated. In case
$\gamma+1\leq\lambda^{\prime}\leq\gamma+\gamma_{\max}$, then by Assumption 1,
$\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}$ blocks the back-door
path. Notice that $\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}$ is
necessarly in $\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash
Desc(X,\mathcal{G})\\}\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$.

Case 2: If the backdoor path does not pass by any $X_{t-\lambda^{\prime}}$
with $\lambda^{\prime}\neq\gamma$ then by Condition 2, the backdoor path
should be blocked by the empty set. ∎

The last lemma asserts that given Conditions 1-3, it is guaranteed that there
exists a finite set that blocks all backdoor paths from some micro vertices
(that intercept all directed paths from $X_{t-\gamma}$ to $Y_{t}$ in any
candidate FT-ADMG) to $Y_{t}$. Furthermore, this finite set does not contain
any descendants of the micro vertices that intercept all directed paths. This
mirrors Condition 3 for micro vertices.

######  Lemma 3.4.

Consider an SCG $\mathcal{G}^{s}=(\mathbb{S},\mathbb{E}^{s})$ and the pair of
micro vertices $(X_{t-\gamma},Y_{t})$ compatible with the macro vertices
$(X,Y)$. Suppose $\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$ is
a set of micro vertices that intercepts all directed paths from $X_{t-\gamma}$
to $Y_{t}$ in any candidate FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$ and for
which their set of compatible macro vertices $\mathbb{W}\subset\mathbb{S}$
statisfies Conditions 2 and 3 of Definition 6 in $\mathcal{G}^{s}$. Then
$\\{(X_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}\\}\cup\\{(B_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\cup
Desc(W,\mathcal{G}^{s})\\}\\}\cup\\{(B_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\cap
Desc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\\}\\}$ blocks all
backdoor paths from $\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$
to $Y_{t}$ and does not contain any descendent of
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$.

###### Proof.

Consider an arbtrary macro vertex $W\in\mathbb{W}$. All backdoor path in any
FT-ADMG in $\mathcal{C}(\mathcal{G}^{s})$ from $W_{t-\lambda}$ to $Y_{t}$ that
pass by at least one vertex in
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$ is blocked by the
intervention. Now we consider the remaining backdoor paths. If there is a
cycle that includes $W$ and any other vertex on a directed path from $X$ to
$Y$ that is intercepted by $W$, then there exists a backdoor path from $W$ to
$X$ or a backdoor path from $W$ to $Y$ that does not pass through $W$. By
Conditions 2 and 3, this cannot be true. Therefore, it must be the case that
all cycles containing $W$ do not include any other vertex on a directed path
from $X$ to $Y$. Thus, all remaining backdoor paths are by construction
blocked by the set of ancestors of
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$. Notice that the
first vertex after $X_{t-\gamma}$ on any directed path from $X_{t-\gamma}$ to
$W_{t-\lambda}$ cannot be a descendant of $X_{t-\gamma}$ because otherwise
Condition 2 will be violated. Which means that the set of ancestors can be
reduced to
$\\{(X_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}\\}\cup\\{(B_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\cup
Desc(W,\mathcal{G}^{s})\\}\\}\cup\\{(B_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\cap
Desc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\\}\\}$.
Additionally, by construction
$\\{(X_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}\\}\cup\\{(B_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\cup
Desc(W,\mathcal{G}^{s})\\}\\}\cup\\{(B_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\cap
Desc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\\}\\}$ does not
contain any decendent of
$\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$. ∎

$X$$Y$$W$ (a)

$X$$Y$$W$ (b)

$X$$Y$$W$ (c)

$X$$Y$$W$$Z$ (d) SCG.

$X$$Y$$W$$Z$ (e) SCG.

Figure 4: Five SCGs not satisfying Definition 6 for any vertex relative to the
micro of vertices $(X_{t-\gamma},Y_{t})$. Each pair of red and blue vertices
in the FTCGs represents the total effect we are interested in.

In the following we present a sound theorem for identifying total effects even
in the presence of hidden confounding.

######  Theorem 1.

If a set of vertices $\mathbb{W}$ satisfies Definition 6 for SCGs relative to
$(X_{t-\gamma},Y_{t})$ and if $\Pr(x_{t-\gamma},y_{t})>0$ then the total
effect of $X$ on $Y$ is identifiable and is given by the formula

$\displaystyle\Pr(y_{t}\mid do(x_{t-\gamma}))=$
$\displaystyle\sum_{\mathbb{f}}\sum_{\mathbb{b}^{xf},\mathbb{b}^{x}}\Pr(\mathbb{f}\mid
x_{t-\gamma},\mathbb{b}^{xf},\mathbb{b}^{x})\Pr(\mathbb{b}^{xf},\mathbb{b}^{x})$
(1)
$\displaystyle\sum_{x_{t-\gamma}^{\prime},\mathbb{b}^{f}}\Pr(y_{t}\mid\mathbb{f},x_{t-\gamma}^{\prime},\mathbb{b}^{xf},\mathbb{b}^{f})\Pr(x_{t-\gamma}^{\prime},\mathbb{b}^{xf},\mathbb{b}^{f})$

where $\mathbb{F}=\\{(\mathbb{W}_{t-\gamma+\ell})_{0\leq\ell\leq\gamma}\\}$,
$\mathbb{B}^{xf}=\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}\cup\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}$,

$\displaystyle\mathbb{B}^{x}=$
$\displaystyle\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash Desc(X,\mathcal{G})\\}$
$\displaystyle\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\},$

and

$\displaystyle\mathbb{B}^{f}=$
$\displaystyle\\{(B_{t-\gamma-\ell})_{-\gamma\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\cup
Desc(W,\mathcal{G}^{s})\\}\\}$
$\displaystyle\cup\\{(B_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(W,\mathcal{G}^{s})\cap
Desc(W,\mathcal{G}^{s})\backslash\\{Anc(X,\mathcal{G}^{s})\\}\\}$

###### Proof.

By Lemma 3.1, since $\mathbb{W}$ intercepts all directed paths from $X$ to $Y$
in the SCG, then $\mathbb{F}$ intercepts all directed paths from
$X_{t-\gamma}$ to $Y_{t}$ in any candidate FT-ADMG in
$\mathcal{C}(\mathcal{G}^{s})$. Given this result, by the law of total
probability, the total effect $\Pr(y_{t}\mid do(x_{t-\gamma}))$ can be
computed in two steps: first, by computing $\Pr(w_{t-\gamma},\cdots,w_{t}\mid
do(x_{t-\gamma}))$, and then by computing $\Pr(y_{t}\mid
do(w_{t-\gamma}),\cdots,do(w_{t}))$, and subsequently multiplying the two
quantities together while summing over $\mathbb{f}$.

Notice that the set $\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Par(X,\mathcal{G}^{s})\\}$ used in Lemma 3.2 is a subset of $\mathbb{B}^{x}$
and at the same time when $\gamma>0$ $\mathbb{B}^{x}$ cannot contain any
descendant of $X_{t-\gamma}$. Therefore, by Lemma 3.2 and 3.3,
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}\cup\\{(X_{t-\gamma-\ell})_{1\leq\ell\leq\gamma_{\max}}\\}\cup\mathbb{B}^{x}$
statisfies standard the backdoor criterion (Pearl, 1995) relative to
$(X_{t-\gamma},\\{\mathbb{W}_{t-\gamma},\cdots,\mathbb{W}_{t}\\})$ which means
that by Pearl (1995, Theorem 1)
$\Pr(\mathbb{w}_{t-\gamma},\cdots,\mathbb{w}_{t}\mid
do(x_{t-\gamma}))=\sum_{\mathbb{b}^{xf},\mathbb{b}^{x}}\Pr(\mathbb{w}_{t-\gamma},\cdots,\mathbb{w}_{t}\mid
x_{t-\gamma},\mathbb{b}^{xf},\mathbb{b}^{x})\Pr(\mathbb{b}^{xf},\mathbb{b}^{x}).$

By Lemma 3.4,
$\\{(X_{t-\gamma+\ell})_{1\leq\ell\leq\gamma}\\}\cup\\{(B_{t-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\backslash
Desc(X,\mathcal{G})\\}\cup\\{(B_{t-\ell})_{1\leq\ell\leq\gamma_{\max}}|B\in
Anc(X,\mathcal{G}^{s})\cap Desc(X,\mathcal{G})\\}$ statisfies the standard
backdoor criterion (Pearl, 1995) relative to
$(\\{\mathbb{W}_{t-\gamma},\cdots,\mathbb{W}_{t}\\},Y_{t})$ which means that
by Pearl (1995, Theorem 1) $\Pr(y_{t}\mid
do(\mathbb{w}_{t-\gamma}),\cdots,do(\mathbb{w}_{t}))=\sum_{\mathbb{b}^{xf},x_{t-\gamma}^{\prime},\mathbb{b}^{f}}\Pr(y_{t}\mid\mathbb{w}_{t-\gamma},\cdots,\mathbb{w}_{t},x_{t-\gamma}^{\prime},\mathbb{b}^{xf},\mathbb{b}^{f})\Pr(x_{t-\gamma}^{\prime},\mathbb{b}^{xf},\mathbb{b}^{f}).$
∎

The sets used in Equation1 of Theorem 1 are applicable whether Condition 4a or
Condition 4b of Definition 6 is statisfied. Moreover, by Lemma 3.2 the set
$\mathbb{B}^{x}$ in Theorem 1 can be reduced to
$\\{(B_{t-\gamma-\ell})_{0\leq\ell\leq\gamma_{\max}}|B\in
Par(X,\mathcal{G}^{s})\\}$ when Condition 4a is satisfied.

##  4 Towards non identifiability results

In this section, we explain why the total effect $\Pr(y_{t}\mid
do(x_{t-\gamma}))$ is not identifiable in any of the SCGs shown in Figure 4.

Consider any SCG in Figure 4. $\Pr(y_{t}\mid do(x_{t-\gamma}))$ is not
identifiable by the backdoor criterion (Pearl, 1995) because the backdoor path
$X_{t-\gamma}\mathrel{\leavevmode\hbox to17.9pt{\vbox
to0.4pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower 2.29721pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{{}{}}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{2.5pt,1.0pt}{0.0pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.45999pt}{2.49721pt}\pgfsys@lineto{17.04004pt}{2.49721pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.45999pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.04004pt}{2.49721pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}Y_{t}$ cannot be
blocked by any observed micro vertex. In the following, we demonstrate that
for each SCG, the total effect cannot be identified by decomposing it into
other total effects and then multiplying them together.

Consider the SCG $\mathcal{G}^{s}$ in Figure 4(a). We need to show that if
$X\leftrightarrows W$ in $\mathcal{G}^{s}$, then there exists at least one
$\lambda$ where $0\leq\lambda<\gamma_{\max}$ such that $\Pr(w_{t-\lambda}\mid
do(x_{t-\gamma}))$ is not identifiable. For $\gamma=\lambda$, there exists an
FT-ADMG $\mathcal{G}_{1}\in\mathcal{C}(\mathcal{G}^{s})$ such that $X_{t}\in
Par(W_{t},\mathcal{G}_{1})$ and another FT-ADMG
$\mathcal{G}_{2}\in\mathcal{C}(\mathcal{G}^{s})$ where $W_{t}\in
Par(X_{t},\mathcal{G}_{2})$. Thus, the total effect is not identifiable. We
can apply the same logic to the SCGs in Figure 4(c) and a similar logic for
Figure 4(b) to show that $\Pr(y_{t}\mid do(w_{t}))$ is not identifiable.

Consider the SCG $\mathcal{G}^{s}$ in Figure 4(d). We need to demonstrate that
if $X\rightarrow W\rightarrow Z\rightarrow X$ exists in $\mathcal{G}^{s}$,
then there exists at least one $\lambda$ where $0\leq\lambda<\gamma_{\max}$
such that $\Pr(w_{t-\lambda}\mid do(x_{t-\gamma}))$ is not identifiable. Let’s
take $\gamma=\lambda$. Since there is a backdoor path $\pi^{s}=\langle
X,Z,W\rangle$, there exists an FT-ADMG $\mathcal{G}_{1}$ with the backdoor
path $\pi_{1}=X_{t}\leftarrow Z_{t}\leftarrow W_{t}$, which would need to be
blocked by conditioning on $Z_{t}$. However, there is also a directed path
$\pi^{s}=\langle X,W,Z\rangle$, which corresponds to an FT-ADMG
$\mathcal{G}_{2}$ with the directed path $\pi_{2}=X_{t}\rightarrow
W_{t}\rightarrow Z_{t}$, where conditioning on $Z_{t}$ would bias the
estimation of the total effect. This ambiguity around $Z_{t}$ implies that the
total effect is not identifiable. A similar argument can be made for the SCG
in Figure 4(e), to show that $\Pr(y_{t}\mid do(w_{t}))$ is not identifiable.

##  5 Conclusion

In this paper, we focus on identifying total effects from summary causal
graphs with latent confounding. Through Definition 6, we establish graphical
conditions that are sufficient, under any underlying probability distribution,
for the identifiability of the total effect. Additionally, in cases where
identifiability is established, through Theorem 1, we provide a do-free
formula for estimating the total effect. These results contribute to the
ongoing effort to understand and estimate total effects from observational
data using summary causal graphs. The main limitaions of our Theorem 1 is that
it is sound but not complete for identifying total effects using summary
causal graphs. Therefore, for future work, it would be interesting to provide
a _complete_ identifiability result, taking into account latent confounding
and cycles.

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#  Charge gap in SU(3) Yang-Mills-plus-nonlinear-spinor-field theory

Vladimir Dzhunushaliev<EMAIL_ADDRESS>Department of Theoretical
and Nuclear Physics, Al-Farabi Kazakh National University, Almaty 050040,
Kazakhstan  Academician J. Jeenbaev Institute of Physics of the NAS of the
Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan  Vladimir
Folomeev<EMAIL_ADDRESS>Academician J. Jeenbaev Institute of Physics of
the NAS of the Kyrgyz Republic, 265 a, Chui Street, Bishkek 720071, Kyrgyzstan
Daulet Berkimbayev  Department of Theoretical and Nuclear Physics, Al-Farabi
Kazakh National University, Almaty 050040, Kazakhstan

###### Abstract

We obtain particlelike solutions in SU(3) gauge Yang-Mills theory with color
magnetic and electric fields sourced by a nonlinear spinor field. The
asymptotic behavior of these fields is studied. It is shown that the electric
field exhibits the Coulomb asymptotic behavior, and one of color components of
the magnetic field behaves asymptotically as the field of a magnetic dipole;
this enables one to determine the corresponding charge and magnetic moment. It
is shown that profiles of the color charge and magnetic moment have global
minima, which may be called charge and magnetic moment gaps. The relationship
between the total energy of the system and the color charge is obtained. We
discuss the physical reason for the appearance of the mass, charge, and
magnetic moment gaps in the theory under consideration.

SU(3) Yang-Mills theory, nonlinear spinor field, particlelike solutions,
charge and magnetic moment gaps

###### pacs:

11.15.Tk, 12.38.Lg, 11.15.-q

##  I Introduction

In quantum chromodynamics, there is a well-known problem which states that in
non-Abelian quantum Yang-Mills theory there should exist a minimum value of
mass, called a mass gap. In this connection, one might wonder whether minimum
values of other physical quantities in quantum field theories do exist. For
instance, one may consider the question of the existence of a charge gap. It
would be logical to assume that the presence of a gap for any physical
quantity should be related to the presence of a mass gap. If such a
relationship does exist, one might expect that if in some theory there is a
global minimum in the energy spectrum of particlelike solutions sourced by a
charged field (a spinor field or a complex scalar field), for such a spectrum,
there would exist a global minimum for a charge of such particlelike
solutions.

In Ref. [1], it is shown how weak coupling Feynman diagrams can contain the
information of a nonperturbative mass gap in an asymptotically free theory. In
the paper [2], it is demonstrated that if one takes a QCD glueball as a
massive spin-2 state to be mixed with a massless graviton, then strong gravity
will predict a mass gap of any object composed of particles that couple to the
glueball. In Ref. [3], it is shown that holographic duals of gauge theories
with a mass gap also exhibit confinement. In Ref. [4], the behavior of a mass
gap in the presence of a magnetic field is studied, and it is shown that the
quark stays around $\Lambda_{\text{QCD}}$ or less, provided that it is mainly
generated by the nonperturbative part of the gluon exchange. However, that
still leaves open the question of whether there exist minimum values of other
physical quantities.

Consistent with this, we will demonstrate here the existence of a global
minimum for a charge, i.e., the presence of a charge gap. A crucial role in
our investigations will be played by nonlinear spinor fields. In the 1950’s,
the nonlinear Dirac equation was employed by W. Heisenberg in constructing a
model of an electron [5]. In turn, various modifications of this equation were
used within the Nambu-Jona-Lasinio model (for a review, see Refs. [6, 7]).

In the present study we generalize the equations obtained in Refs. [8, 9]
[where a non-Abelian magnetic field belongs to the group SU(2), which can be
regarded as a subgroup $\text{SU(2)}\subset\text{SU(3)}$] by introducing a
$t$-component of a non-Abelian gauge potential belonging to the subgroup
$\text{U(1)}\subset\text{SU(3)}$. The presence of such a component results in
the appearance of a color electric field sourced by a nonlinear spinor field
$\psi$. As was demonstrated in Refs. [8, 9], in this theory, the energy
spectrum of particlelike solutions has a global minimum – a mass gap. It was
shown in Ref. [9] that the dominant contribution to the energy density is
given by the term $\psi^{\dagger}\psi$ that is obviously related to the
density of color charge as well. This fact enables us to assume that for a
family of regular solutions belonging to the subgroup
$\text{SU(2)}\times\text{U(1)}\subset\text{SU(3)}$ (with a color electric
field included) obtained in the present paper, there should exist a global
minimum of a color charge – a charge gap.

##  II General equations and field $\mathfrak{Ans\ddot{a}tze}$

In this section we closely follow Ref. [8]. The Lagrangian describing a system
consisting of a non-Abelian SU(3) field $A^{a}_{\mu}$ interacting with
nonlinear spinor field $\psi$ can be taken in the form

$\begin{split}\mathcal{L}=&-\frac{1}{4}F^{a}_{\mu\nu}F^{a\mu\nu}+i\hbar
c\bar{\psi}\gamma^{\mu}D_{\mu}\psi-
m_{f}c^{2}\bar{\psi}\psi+\frac{l_{0}^{2}}{2}\hbar
c\left(\bar{\psi}\psi\right)^{2}.\end{split}$ (1)

Here $m_{f}$ is the mass of the spinor field;
$D_{\mu}=\partial_{\mu}-i\frac{g}{2}\lambda^{a}A^{a}_{\mu}$ is the gauge-
covariant derivative, where $g$ is the coupling constant and $\lambda^{a}$ are
the SU(3) generators (the Gell-Mann matrices);
$F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf_{abc}A^{b}_{\mu}A^{c}_{\nu}$
is the field strength tensor for the SU(3) field, where $f_{abc}$ are the
SU(3) structure constants; $l_{0}$ is a constant; $\gamma^{\mu}$ are the Dirac
matrices in the standard representation; $a,b,c=1,\cdots,8$ are color indices
and $\mu,\nu=0,1,2,3$ are spacetime indices.

Using the Lagrangian (1), one can find the corresponding field equations,

$\displaystyle D_{\nu}F^{a\mu\nu}$ $\displaystyle=$ $\displaystyle
j^{a\mu}=\frac{g\hbar c}{2}\bar{\psi}\gamma^{\mu}\lambda^{a}\psi,$ (2)
$\displaystyle i\hbar\gamma^{\mu}D_{\mu}\psi-
m_{f}c\psi+l_{0}^{2}\hbar\psi\left(\bar{\psi}\psi\right)$ $\displaystyle=$
$\displaystyle 0.$ (3)

Here $j^{a\mu}$ is a color current created by the spinor field $\psi$.
Particlelike solutions are sought in the form of

$\displaystyle A^{a}_{i}=$
$\displaystyle\frac{1}{g}\left[1-f(r)\right]\begin{pmatrix}0&\phantom{-}\sin\varphi&\sin\theta\cos\theta\cos\varphi\\\
0&-\cos\varphi&\sin\theta\cos\theta\sin\varphi\\\
0&0&-\sin^{2}\theta\end{pmatrix},\quad i=r,\theta,\varphi\text{ (in polar
coordinates)},\quad a=1,2,3,$ (4) $\displaystyle A^{8}_{t}=$
$\displaystyle\frac{\chi(r)}{gr},$ (5)

where (4) is the standard $\mathfrak{Ansatz}$ used in describing the ’t Hooft-
Polyakov monopole, and the $\mathfrak{Ansatz}$ for the spinor field is taken
in the form

$\psi^{T}=\frac{e^{-i\frac{Et}{\hbar}}}{gr\sqrt{2}}\begin{Bmatrix}\begin{pmatrix}0\\\
-u\\\ 0\\\ \end{pmatrix},\begin{pmatrix}u\\\ 0\\\ 0\\\
\end{pmatrix},\begin{pmatrix}iv\sin\theta e^{-i\varphi}\\\ -iv\cos\theta\\\
0\\\ \end{pmatrix},\begin{pmatrix}-iv\cos\theta\\\ -iv\sin\theta
e^{i\varphi}\\\ 0\\\ \end{pmatrix}\end{Bmatrix}.$ (6)

This is the SU(3) generalization of the SU(2) $\mathfrak{Ansatz}$ given in
Refs. [10, 11]. Here $E/\hbar$ is the spinor frequency and the functions $u$
and $v$ depend on the radial coordinate $r$ only.

For the $\mathfrak{Ans\ddot{a}tze}$ (4) and (5), we have the following
nonvanishing components of electric and magnetic fields:

$\displaystyle E^{8}_{r}=$ $\displaystyle\frac{\chi-r\chi^{\prime}}{gr^{2}},$
(7) $\displaystyle H^{1}_{i}=$
$\displaystyle\frac{1}{g}\left\\{\frac{\sin\theta\cos\varphi}{r^{2}}\left(1-f^{2}\right),-\cos\theta\cos\varphi\,f^{\prime},\sin\theta\sin\varphi\,f^{\prime}\right\\},$
(8) $\displaystyle H^{2}_{i}=$
$\displaystyle\frac{1}{g}\left\\{\frac{\sin\theta\sin\varphi}{r^{2}}\left(1-f^{2}\right),-\cos\theta\sin\varphi\,f^{\prime},-\sin\theta\cos\varphi\,f^{\prime}\right\\},$
(9) $\displaystyle H^{3}_{i}=$
$\displaystyle\frac{1}{g}\left\\{\frac{\cos\theta}{r^{2}}\left(1-f^{2}\right),\sin\theta\,f^{\prime},0\right\\},$
(10)

where the spatial index $i=r,\theta,\varphi$.

Substituting the expressions (4)-(6) in the field equations (2) and (3), one
can obtain equations for the unknown functions $f,u$, $v$, and $\chi$:

$\displaystyle-f^{\prime\prime}+\frac{f\left(f^{2}-1\right)}{x^{2}}=$
$\displaystyle-\tilde{g}^{2}\frac{\tilde{u}\tilde{v}}{x},$ (11)
$\displaystyle-\frac{\chi^{\prime\prime}}{x}=$
$\displaystyle-\tilde{g}^{2}\frac{\tilde{u}^{2}+\tilde{v}^{2}}{2\sqrt{3}\,x^{2}},$
(12) $\displaystyle\tilde{v}^{\prime}+\frac{f\tilde{v}}{x}=$
$\displaystyle\tilde{u}\left(-1+\tilde{E}+\frac{\chi}{2\sqrt{3}\,x}+\frac{\tilde{u}^{2}-\tilde{v}^{2}}{x^{2}}\right),$
(13) $\displaystyle\tilde{u}^{\prime}-\frac{f\tilde{u}}{x}=$
$\displaystyle\tilde{v}\left(-1-\tilde{E}-\frac{\chi}{2\sqrt{3}\,x}+\frac{\tilde{u}^{2}-\tilde{v}^{2}}{x^{2}}\right),$
(14)

written in terms of the following dimensionless variables: $x=r/\lambda_{c}$,
$\tilde{u}=u\sqrt{l_{0}^{2}/\lambda_{c}g^{2}},\tilde{v}=v\sqrt{l_{0}^{2}/\lambda_{c}g^{2}},\tilde{E}=\lambda_{c}E/(\hbar
c),\tilde{g}^{2}=\left(\bar{g}\lambda_{c}/l_{0}\right)^{2}$, where
$\lambda_{c}=\hbar/(m_{f}c)$ is the Compton wavelength and
${\bar{g}}^{2}=g^{2}\hbar c$ is a dimensionless coupling constant. The prime
denotes differentiation with respect to $x$. Notice that Eq. (12) is in fact
the Maxwell equation $\divergence\vec{E}=\rho$ with the density of electric
charge
$\rho=-\tilde{g}^{2}\frac{\tilde{u}^{2}+\tilde{v}^{2}}{2\sqrt{3}\,x^{2}}$.

The total energy density of the system is

$\tilde{\epsilon}=\frac{1}{\tilde{g}^{2}}\left[\frac{{f^{\prime}}^{2}}{x^{2}}+\frac{\left(f^{2}-1\right)^{2}}{2x^{4}}+\frac{\left(\chi-x\chi^{\prime}\right)^{2}}{2x^{4}}\right]+\left[\tilde{E}\frac{\tilde{u}^{2}+\tilde{v}^{2}}{x^{2}}+\frac{\left(\tilde{u}^{2}-\tilde{v}^{2}\right)^{2}}{2x^{4}}\right],$
(15)

where the expressions in the first square brackets correspond to the energy
density of color magnetic and electric fields and the expressions in the
second square brackets describe the energy density of the nonlinear spinor
field. Then the total energy of the system under consideration is calculated
as

$\tilde{W}_{t}\equiv\frac{\lambda_{c}g^{2}}{\tilde{g}^{2}}W_{t}=4\pi\int\limits_{0}^{\infty}x^{2}\tilde{\epsilon}dx.$
(16)

##  III Numerical solutions, electric charge and magnetic moment

A solution of Eqs. (11)-(14) in the vicinity of the center $x=0$ can be
obtained from the expansion of the functions in a Taylor series:

$f=1+\frac{f_{2}}{2}x^{2}+\ldots,\quad\tilde{u}=\tilde{u}_{1}x+\frac{\tilde{u}_{3}}{3!}x^{3}+\ldots,\quad\tilde{v}=\frac{\tilde{v}_{2}}{2}x^{2}+\frac{\tilde{v}_{4}}{4!}x^{4}+\ldots,\quad\chi=\frac{\chi_{3}}{3!}x^{3}+\ldots$

Substitution of these expansions in the field equations yields the following
values of the expansion coefficients:

$\tilde{v}_{2}=\frac{2}{3}\tilde{u}_{1}\left(\tilde{E}-1+\tilde{u}_{1}^{2}\right),\quad\chi_{3}=\frac{\tilde{g}^{2}}{2\sqrt{3}}\tilde{u}_{1}^{2},$

whereas the parameters $\tilde{u}_{1}$ and $f_{2}$ are free. Solutions
possessing a finite total energy do exist not for all values of
$\tilde{u}_{1}$ and $f_{2}$, which are eigenvalues of the problem. To find
them, we solved Eqs. (11)-(14) numerically using the shooting method to obtain
regular solutions with finite energy. The typical solutions are depicted in
Fig. 1.

Figure 1:  Left panel: the gauge potentials $f,\chi$ and the spinor functions
$u,v$. Middle panel: the radial component of the color electric field
$\tilde{E}^{8}_{r}$, the radial part $(1-f^{2})/x^{2}$ of the radial
components of the color magnetic field $H^{1,2,3}_{r}$, and the radial part
$f^{\prime}$ of the tangential components of the color magnetic field
$H^{1,2,3}_{\theta,\varphi}$. Right panel: the profile of the energy density
(15). For all panels, $\tilde{g}=0.7$, $\tilde{E}=0.8$, $f_{2}=-0.206395$,
$\tilde{u}_{1}=1.1971179$.

Using the results of numerical calculations, one can show that asymptotically
(as $x\to\infty$) the functions $f(x)$ and $\chi(x)$ behave as

$\displaystyle f\approx$ $\displaystyle 1-\frac{\tilde{\mathfrak{m}}_{c}}{x},$
(17) $\displaystyle\tilde{A}^{8}_{t}\equiv
g\lambda_{c}A^{8}_{t}=\frac{\chi}{x}\approx$
$\displaystyle\chi_{\infty}+\frac{\tilde{Q}_{c}}{x},$ (18)

where $\tilde{\mathfrak{m}}_{c}$ and $\tilde{Q}_{c}$ are constants whose
physical meaning is defined below.

As it follows from Eq. (18), the color electric field has a Coulomb-law fall
off at large distances, as expected due to the fact that Eq. (12) describes in
practice an Abelian electric field. Then the electric field strength has the
following asymptotic behavior:

$\tilde{E}^{8}_{r}\approx\frac{\tilde{Q}_{c}}{x^{2}}.$ (19)

Thus the parameter $\tilde{Q}_{c}$ appearing in the asymptotic expressions
(18) and (19) is a color electric charge for the U(1) field. To find its
numerical values, one can construct profiles of the distribution of
$\tilde{E}^{8}_{r}x^{2}$ as a function of the radial coordinate $x$, and then
determine the asymptotic value of this distribution. Alternatively, one can
calculate the integral of the density of color charge

$\tilde{Q}_{c}=-\frac{\tilde{g}^{2}}{2\sqrt{3}}\int\limits_{0}^{\infty}\left(\tilde{u}^{2}+\tilde{v}^{2}\right)dx.$
(20)

It is seen that the color charge is negative; in order to obtain a positive
value of the charge, it is necessary to take an opposite sign of the
$\mathfrak{Ansatz}$ (5) for the potential $A^{8}_{t}$.

We have checked that the values of the charge $\tilde{Q}_{c}$ determined
according to Eqs. (19) and (20) are the same for $\tilde{E}=0.7$. Next, we
have studied the dependence of the modulus of the color charge
$\left|\tilde{Q}_{c}\right|$ on the parameter $\tilde{E}$ (shown in Fig. 3).
It is seen from this figure that the color charge $\left|\tilde{Q}_{c}\right|$
has a global minimum, which may be called a charge gap. This gap is present in
SU(3) Yang-Mills theory under consideration with a source of a color electric
field taken in the form of a nonlinear spinor field.

Figure 2: The dependence of the modulus of the color charge
$\left|\tilde{Q}_{c}\right|$ and of the color magnetic dipole moment
$\tilde{\mathfrak{m}}_{c}$ on the parameter $\tilde{E}$.

Figure 3: The energy spectrum of the solutions $\tilde{W}_{t}$ and the
normalization integral
$N\left(\tilde{E}\right)=4\pi\tilde{E}\int\limits_{0}^{\infty}\left(\tilde{u}^{2}+\tilde{v}^{2}\right)dx$
which is proportional to the absolute value of the color charge from Eq. (20).

Fig. 3 shows the distributions of the energy [given by Eq. (16)] and of the
normalization integral as functions of the spinor frequency $\tilde{E}$. In
turn, using the analytic expression for the charge (20), one can obtain the
following relationship between the total energy (16) and the color charge
(20):

$\tilde{W}_{t}=\tilde{W}_{em}+8\pi\sqrt{3}\frac{\left|\tilde{Q}_{c}\right|}{\tilde{g}^{2}}\tilde{E}+\tilde{W}_{nl}.$

Here $\tilde{W}_{em}$ is the energy of the Yang-Mills field [the integral over
the whole volume of the terms in the first square brackets in Eq. (15)];
$\tilde{W}_{nl}$ is the energy of the spinor field associated with the
nonlinearity of this field [the integral over the whole volume of the second
term appearing in the second square brackets in Eq. (15)]. Note that for
$\tilde{E}\rightarrow 1$ the total energy of the particlelike object under
consideration and its color charge coincide, apart from a constant:

$\tilde{W}_{t}\approx
8\pi\sqrt{3}\frac{\left|\tilde{Q}_{c}\right|}{\tilde{g}^{2}}\tilde{E}\quad\text{
for }\quad\tilde{E}\rightarrow 1.$

To ascertain the physical meaning of the parameter $\tilde{\mathfrak{m}}_{c}$,
consider the asymptotic behavior of the color magnetic field. Substituting the
expression for the potential (17) in the expressions (8)-(10), we have

$\displaystyle\tilde{H}^{1}_{i}\equiv$ $\displaystyle
g\lambda_{c}^{2}H^{1}_{i}\approx\frac{\tilde{\mathfrak{m}}_{c}}{x^{3}}\left\\{2\sin\theta\cos\varphi,-\cos\theta\cos\varphi,\sin\varphi\right\\},$
(21) $\displaystyle\tilde{H}^{2}_{i}\equiv$ $\displaystyle
g\lambda_{c}^{2}H^{2}_{i}\approx\frac{\tilde{\mathfrak{m}}_{c}}{x^{3}}\left\\{2\sin\theta\sin\varphi,-\cos\theta\sin\varphi,-\cos\varphi\right\\},$
(22) $\displaystyle\tilde{H}^{3}_{i}\equiv$ $\displaystyle
g\lambda_{c}^{2}H^{3}_{i}\approx\frac{\tilde{\mathfrak{m}}_{c}}{x^{3}}\left\\{2\cos\theta,\sin\theta,0\right\\},$
(23)

where we have introduced the physical components of the magnetic field vector
$H^{a}_{i}=\sqrt{H^{a}_{i}H^{ai}}$ (no summing over $a$ and $i$). It is seen
from Eq. (23) that the asymptotic behavior of the color magnetic field
$\vec{H}^{3}$ coincides exactly with the field of a magnetic dipole in
Maxwell’s magnetostatic. This enables us to refer to the parameter
$\tilde{\mathfrak{m}}_{c}$ as the color magnetic “moment”. We have enclosed
the word “moment” in quotation marks since the asymptotic behavior of the
color fields $\vec{H}^{1,2}$ does not coincide with the magnetic field of
Maxwell’s magnetic dipole. As one sees from Fig. 3, the color magnetic
“moment”, as well as the color charge, has a global minimum, which may be
called a magnetic moment gap.

Notice also that, in a color space, the asymptotic behavior of the scalar
product of the color vector of the magnetic field $H^{a}_{i}$ and the color
vector $l^{a}=\left\\{\cos\varphi,\sin\varphi,0\right\\}$ coincides with the
magnetic field of a magnetic dipole as well:

$g\lambda_{c}^{2}H^{a}_{i}l^{a}\approx\frac{\tilde{\mathfrak{m}}_{c}}{x^{3}}\left\\{2\sin\theta,-\cos\theta,0\right\\},$

but with the change in the direction of the magnetic moment as
$\theta=\theta^{\prime}+\pi/2$.

##  IV Discussion and conclusions

In the present study, we have demonstrated the existence of charge and
magnetic moment gaps in SU(3) Yang-Mills theory with a source of a gauge field
in the form of a nonlinear spinor field. We have found particlelike solutions
with color electric and magnetic fields belonging to the subgroup
$\text{SU(2)}\times\text{U(1)}\subset\text{SU(3)}$. By the investigation of
the asymptotic behavior of such fields, we have shown that the electric field
has a Coulomb asymptotic form, and the $\vec{H}^{3}$ color component of the
magnetic field behaves asymptotically as the field of a magnetic dipole.

In the dimensionless form, the regular solutions obtained form a family
labeled by the dimensionless spinor frequency $\tilde{E}$. For this family,
one can construct profiles of the dependence of the color electric charge
$\tilde{Q}_{c}$ and of the color magnetic moment $\tilde{\mathfrak{m}}_{c}$ on
the parameter $\tilde{E}$, which represents the solutions in this family. As
in the case of the energy spectrum, these profiles have global minima, which
may be called charge and magnetic moment gaps.

Summarizing the results obtained,

  * •

The equations of non-Abelian SU(3) Yang-Mills theory with a source in the form
of a nonlinear spinor field have regular particlelike solutions with non-
Abelian $\text{ SU(2)}\subset\text{SU(3)}$ magnetic and Abelian
$\text{U(1)}\subset\text{SU(3)}$ electric fields.

  * •

Asymptotically, the color electric field exhibits the Coulomb behavior; this
enables one to introduce the corresponding charge.

  * •

The asymptotic behavior of the color magnetic field is the same as that of a
magnetic dipole in Maxwell’s electrodynamics; this enables one to determine
the corresponding color magnetic moment.

  * •

The profiles of the color charge and magnetic moment, being functions of the
system parameter $\tilde{E}$ (which represents the solution), have global
minima (see Fig. 3); by analogy with the energy spectrum of these solutions,
this enables one to speak about the presence in the system of charge and
magnetic moment gaps.

The reason for the appearance of the mass, charge, and magnetic moment gaps in
the theory under consideration is of great importance. In our opinion, the
main and the only reason is the presence of a nonlinear spinor field. A mass
gap for a system supported by a nonlinear spinor field uncoupled to other
fields was first obtained in Refs. [12, 13]. According to our results obtained
in the present paper and earlier (see, e.g., Refs. [8, 9]), the gaps are also
present when other fields are involved; that is, this effect is inherent to a
nonlinear spinor field, in contrast to, for example, a system involving a
nonlinear scalar field where a mass gap is absent.

## Acknowledgments

This research has been funded by the Science Committee of the Ministry of
Education and Science of the Republic of Kazakhstan. We are also grateful to
the Research Group Linkage Programme of the Alexander von Humboldt Foundation
for the support of this research and would like to thank the Carl von
Ossietzky University of Oldenburg for hospitality while this work was carried
out.

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