# Cascading Waves of Fluctuation in Time-delay Multi-agent Rendezvous

Guangyi Liu, Vivek Pandey, Christoforos Somarakis, and Nader Motee  This work
was supported by the ONR N00014-19-1-2478. G. Liu, V. Pandey, and N. Motee are
with the Department of Mechanical Engineering and Mechanics, Lehigh
University, Bethlehem, PA, 18015, USA<EMAIL_ADDRESS>Somarakis is with the Intelligent Systems Lab, Palo Alto Research Center - a
Xerox Company, Palo Alto, CA, 94304<EMAIL_ADDRESS>
###### Abstract

We develop a framework to assess the risk of cascading failures when a team of
agents aims to rendezvous in time in the presence of exogenous noise and
communication time-delay. The notion of value-at-risk (VaR) measure is used to
evaluate the risk of cascading failures (i.e., waves of large fluctuations)
when agents have failed to rendezvous. Furthermore, an efficient explicit
formula is obtained to calculate the risk of higher-order cascading failures
recursively. Finally, from a risk-aware design perspective, we report an
evaluation of the most vulnerable sequence of agents in various communication
graphs.

##  I Introduction

In the past decades, consensus networks have shown spectacular advantages in
coordinating the networked control systems, e.g., [6, 4]. However, like “every
coin has two sides”, consensus networks or cyber-physical systems also show
their vulnerability to large-scale failures [3, 18, 7]. Given that most
consensus networks suffer from communication or other physically induced
limitations (e.g., time-delay), as well as external disturbances (e.g., biased
opinion or noisy sensor measurements), individuals within the network will be
inevitably driven away from the consensus and enter failure states. These
failures are prone to cascade in the system and cause new failures to emerge
[8, 9]; then, one needs to investigate how the existing failures are going to
impact the rest of the system, i.e., the cascading (domino) effect, as well as
developing systemic techniques to quantify such effects.

The domino effects in the systemic failure navigate our interests towards the
cascading failures [14, 25, 26], and how their emergence will impact the
system’s safety. In other words, we aim to investigate how resilient the
system is while multiple failures have already occurred in the system, as well
as how likely the failure will cascade. Probing into the cascading failures
and their emergence have many practical meanings from the system design
perspective: It is essential to evaluate the performance of the networked
control system under the impact of the existing failures, or, in other words,
the ability to isolate the propagation of existing failures and keep the rest
of the system safe.

In this paper, we will develop a theoretical framework based on systemic risk
analysis [16, 17] to evaluate the possibility of cascading failures in the
consensus network. Our objective is to utilize the risk quantification regime,
e.g., the value-at-risk measure, to assess how likely the existing failures in
the consensus network can trigger future failures throughout the system.
Providing the quantification of such effects will give valuable insights into
the networked control systems design and highlight the potential of a
networked system to suffer from cascading systemic failures.

We consider the motivational scenario as a network of autonomous agents that
aim to achieve rendezvous in time [21]. It is assumed that agents communicate
over a time-invariant network with lagged transmission and processing of
information from other agents. The network is also perturbed by statistical
noise applied to each agent independently. The noise models the effect of
external perturbations and turns the system into a stochastic dynamical
network. We will focus on studying the event of a large fluctuation, i.e.,
failing to reach the rendezvous in time, given that some agents have already
deviated from the consensus.

Our Contributions:  As a natural outgrowth of our recent works on first-order
linear consensus networks [22, 19, 21, 9], this paper generalizes the concept
of the cascading failures of two agents [8, 9] into the situation when there
exists an arbitrary number of failures [10]. First, we examine the value-at-
risk measure of cascading failures in the steady-state distribution of
observables for a time-delayed linear consensus network. In particular,
explicit formulas are obtained for the risk of large fluctuation, given that
other agents have failed to reach the consensus. Then, with extensive
simulations, we evaluate the most vulnerable sequence of agents in a network,
which is exclusively obtained with the generalized cascading risk for multiple
existing failures.

##  II Mathematical Notation

We denote the non-negative orthant of the Euclidean space $\mathbb{R}^{n}$ by
$\mathbb{R}_{+}^{n}$, its set of standard Euclidean basis by
$\\{\bm{e}_{1},\dots,\bm{e}_{n}\\}$, and the vector of all ones by
$\bm{1}_{n}=[1,\dots,1]^{T}$. The absolute value operation for vectors is
defined element-wise, i.e., $|{\bm{z}}|=[|z_{1}|,...,|z_{n}|]^{T}$.

Algebraic Graph Theory: A weighted graph is defined by
$\mathcal{G}=(\mathcal{V},\mathcal{E},\omega)$, where $\mathcal{V}$ is the set
of nodes, $\mathcal{E}$ is the set of edges (feedback links), and
$\omega:\mathcal{V}\times\mathcal{V}\rightarrow\mathbb{R}_{+}$ is the weight
function that assigns a non-negative number (feedback gain) to every link. Two
nodes are directly connected if and only if $(i,j)\in\mathcal{E}$.

######  Assumption 1.

Every graph in this paper is connected. In addition, for every
$i,j\in\mathcal{V}$, the following properties hold:

  * •

$\omega(i,j)>0$ if and only if $(i,j)\in\mathcal{E}$.

  * •

$\omega(i,j)=\omega(j,i)$, i.e., links are undirected.

  * •

$\omega(i,i)=0$, i.e., links are simple.

The Laplacian matrix of $\mathcal{G}$ is a $n\times n$ matrix $L=[l_{ij}]$
with elements

$l_{ij}:=\begin{cases}\;-k_{i,j}&\text{if }\;i\neq j\\\
\;k_{i,1}+\ldots+k_{i,n}&\text{if }\;i=j\end{cases},$

where $k_{i,j}:=\omega(i,j)$. The Laplacian matrix of a graph is symmetric and
positive semi-definite [24]. Assumption 1 implies the smallest Laplacian
eigenvalue is zero with algebraic multiplicity one. The spectrum of $L$ can be
ordered as $0=\lambda_{1}<\lambda_{2}\leq\dots\leq\lambda_{n}.$ The
eigenvector of $L$ corresponding to $\lambda_{k}$ is denoted by $\bm{q}_{k}$.
By letting $Q=[\bm{q}_{1}|\dots|\bm{q}_{n}]$, it follows that $L=Q\Lambda
Q^{T}$ with $\Lambda=\text{diag}[0,\lambda_{2},\dots,\lambda_{n}]$. We
normalize the Laplacian eigenvectors such that $Q$ becomes an orthogonal
matrix, i.e., $Q^{T}Q=QQ^{T}=I_{n}$ with
$\bm{q}_{1}=\frac{1}{\sqrt{n}}\bm{1}_{n}$.

Probability Theory: Let $\mathcal{L}^{2}(\mathbb{R}^{q})$ be the set of all
$\mathbb{R}^{q}-$valued random vectors $\bm{z}=[z^{(1)},\dots,z^{(q)}]^{T}$ of
a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with finite second
moments. A normal random variable $\bm{y}\in\mathbb{R}^{q}$ with mean
$\bm{\mu}\in\mathbb{R}^{q}$ and $q\times q$ covariance matrix $\Sigma$ is
represented by $\bm{y}\sim\mathcal{N}(\bm{\mu},\Sigma)$. The error function
$\text{erf}:\mathbb{R}\rightarrow(-1,1)$ is
$\text{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}\text{d}t,$ which is
invertible on its range as $\text{erf}^{-1}(x)$. We employ standard notation
$\text{d}\bm{\xi}_{t}$ for the formulation of stochastic differential
equations.

##  III Problem Statement

We consider the class of time-delay linear consensus networks that have found
wide applications in engineering, such as clock synchronization in sensor
networks, rendezvous in space or time, and heading alignment in swarm
robotics; we refer to [15, 11] for more details. As a motivational
application, we consider the rendezvous problem in time where the group
objective is to meet simultaneously at a prespecified location known to all
agents.111Rendezvous in space is very similar to rendezvous in time by
switching the role of time and location. Agents do not have a priori knowledge
of the meeting time as it may have to be adjusted in response to unexpected
emergencies or exogenous uncertainties [15]. Thus, all agents should agree on
a rendezvous time by achieving consensus. The consensus can be accomplished by
each agent $i=1,\dots,n$ creating a state variable, say $x_{i}\in\mathbb{R}$,
representing its belief of the rendezvous time. Each agent’s initial belief is
set to its preferred time that it can rendezvous with others. Then, the
rendezvous dynamics for each agent evolve in time according to the following
stochastic differential equations:

$\text{d}x_{t}^{(i)}=u_{t}^{(i)}\,\text{d}t+b\,\text{d}\xi_{t}^{(i)},$ (1)

for all $i=1,\dots,n$. Each agent’s control input is $u_{i}(t)\in\mathbb{R}$.
The source of uncertainty is diffused in the network as additive stochastic
noise, and its magnitude is uniformly scaled by the diffusion coefficient
$b\in\mathbb{R}$. The impact of uncertain environments on the dynamics of
agents is modeled by independent Brownian motions $\xi_{1},\dots,\xi_{n}$. In
most real-world situations, agents experience a time-delay accessing,
computing, or sharing their own and neighboring agents’ state information
[15]. Hence, all agents are assumed to experience an identical time-delay
$\tau\in\mathbb{R}_{+}$. The control inputs are determined via a negotiation
process by forming a linear consensus network over a communication graph using
feedback law

$u_{t}^{(i)}=\sum_{j=1}^{n}k_{ij}\left(x_{t-\tau}^{(j)}-x_{t-\tau}^{(i)}\right),$
(2)

where $k_{ij}\in\mathbb{R}_{+}$ are non-negative feedback gains. Let us denote
the state vector by $\bm{x}_{t}=[x_{t}^{(1)},\dots,x_{t}^{(n)}]^{T}$, and the
vector of exogenous disturbance by
$\bm{\xi}_{t}=[\xi_{t}^{(1)},\dots,\xi_{t}^{(n)}]^{T}$. The dynamics of the
resulting closed-loop network can be cast as a linear consensus network that
is governed by the time-delayed stochastic differential equation

$\text{d}\bm{x}_{t}=-L\,\bm{x}_{t-\tau}\,\text{d}t+B\,\text{d}\bm{\xi}_{t}$
(3)

for all $t\geq 0$, where the initial function $\bm{x}_{t}=\phi(t)$ is
deterministically given for $t\in[-\tau,0]$ and $B=bI_{n}$. The underlying
coupling structure of the consensus network (3) is a graph $\mathcal{G}$ that
satisfies Assumption 1 with Laplacian matrix $L$. It is considered that the
communication graph $\mathcal{G}$ is time-invariant such that the network of
agents aim to reach a consensus on a rendezvous time before they perform
motion planning to get to the meeting location. Upon reaching a consensus, a
properly designed internal feedback control mechanism steers each agent toward
the rendezvous time.

######  Assumption 2.

The time-delay satisfies $\tau<\frac{\pi}{2\lambda_{n}}$.

When there is no noise, i.e., $b=0$, it is known [12] that under Assumption 1
and 2, states of all agents converge to the average of all initial states
$\frac{1}{n}\bm{1}_{n}^{T}\bm{x}_{0}$; whereas in the presence of input noise,
state variables fluctuate around the average value
$\frac{1}{n}\bm{1}_{n}^{T}\bm{x}_{t}$. To quantify the quality of rendezvous
and its fragility features, we consider the vector of observables

$\bm{y}_{t}=M_{n}\,\bm{x}_{t},$ (4)

in which $M_{n}=I_{n}-\frac{1}{n}\bm{1}_{n}\bm{1}_{n}^{T}$ is the centering
matrix and the observable $\bm{y}_{t}=(y_{t}^{(1)},...,y_{t}^{(n)})^{T}$
measures the agents’ deviations from consensus. Assumption 1 implies that one
of the modes of network (3) is marginally stable. The marginally stable mode,
which corresponds to the zero eigenvalue of $L$, is unobservable from the
output (4), which keeps $\bm{y}_{t}$ bounded in the steady-state. When the
noise is absent, we have $\bm{y}_{t}\rightarrow 0$ as $t\rightarrow\infty$.
Consequently, the exogenous noise excites the observable modes of the network,
and the output fluctuates around zero. This implies that agents will not agree
upon an exact rendezvous time. A practical resolution is to allow a tolerance
interval for agents to concur.

######  Definition 1.

For a given tolerance level $c\in\mathbb{R}_{+}$, the consensus network (3)
reaches $c$-consensus if it can tolerate some degrees of disagreement such
that

$\lim_{t\rightarrow\infty}|\bm{y}_{t}|\leq c\,\bm{1}_{n}$ (5)

holds with high probability222The high probability means a probability larger
than a predefined cut-off number close to one..

The notion of $c$-consensus means that all agents have agreement on all points
in $\\{\bm{x}\in\mathbb{R}^{n}\,\big{|}\,|M_{n}\bm{x}|\leq c\bm{1}_{n}\\}$.
Suppose that event (5) holds, then the network of agents will achieve a
$c$-consensus over the rendezvous time in the following sense. In steady-
state, the $i$’th agent is assured that by $x_{t}^{(i)}\pm\,c$ units of time,
all other agents will arrive and meet each other in that time interval with
high probability. At the same time, some undesirable situations that we refer
to as systemic failures may also happen.

######  Definition 2.

For a given tolerance level $c\in\mathbb{R}_{+}$, an agent with observable
$y_{t}^{(i)}$ is prone to systemic failure, i.e., the large fluctuation, if
the probability of the stochastic event

$|y_{t}^{(i)}|>c$ (6)

is nonzero as $t\rightarrow\infty$.

The problem is to quantify the risk of large fluctuations conditioned on the
knowledge that a subset of agents has already deviated from the consensus. We
refer to this quantity as the cascading risk of large fluctuations, and our
goal is to characterize its value as a function of the underlying
communication graph topology, time-delay, and statistics of noise. To this
end, we will develop a general framework to assess the cascading risk of large
fluctuations using the steady-state statistics of the closed-loop system (3).

The rest of the paper is organized as follows. First, in §IV, we present a few
key preliminary results on the steady-state behavior and statistics of the
noisy consensus network. These results help us shape the closed-form risk
formula of cascading failures in §V, which constitutes the main contribution
of this work. Finally, in §VI, we also present an update law that reveals the
evolution of the cascading risk when new failures are discovered in the
system. All proofs of theoretical results are shown in the appendix of this
paper.

##  IV Preliminary Results

In this section, we present the steady-state statistics of network observables
under external disturbances and time-delay. Such statistics form the
foundation for the latter results on the value-at-risk measure.

###  IV-A Steady-State Statistics of Observables

Let us denote the steady-state value of the observable $y_{i}(t)$ by
$\bar{y}_{i}:=\lim_{t\rightarrow+\infty}y_{t}^{(i)}$ whenever it exists. It is
known from [21] that if Assumption 1 and 2 are satisfied, as
$t\rightarrow\infty$, the statistics of steady-state observables (4) follows a
normal distribution $\bm{\bar{y}}\sim\mathcal{N}(0,\Sigma)$, where $\Sigma$ is
a $n\times n$ covariance matrix .

######  Lemma 1.

The elements of $\Sigma=[\sigma_{ij}]$ are given by

$\displaystyle\sigma_{ij}=\frac{1}{2}b^{2}\sum_{k=2}^{n}\frac{\cos(\lambda_{k}\tau)}{\lambda_{k}(1-\sin(\lambda_{k}\tau))}(\bm{m}_{i}^{T}\bm{q}_{k})(\bm{m}_{j}^{T}\bm{q}_{k}),$
(7)

where $\bm{m}_{i}$ denotes the $i$’th column of $M_{n}$ for all
$i,j=1,\dots,n$.

For simplicity, we write the diagonal elements $\sigma_{ii}$ as
$\sigma_{i}^{2}$. One can also find out that the cross-correlation
$\rho_{ij}=\frac{\sigma_{ij}}{\sigma_{i}\sigma_{j}}$ is independent of $b$,
the magnitude of external noises. This observation implies that the
probabilistic interrelation between two agents originates from the time-delay
and the communication graph but not the magnitude of the stochastic
perturbation.

###  IV-B Value-at-Risk Measure

To quantify the uncertainty level encapsulated in agents’ observables, we
employ the notion of Value-at-Risk (VaR) measure [5, 16, 22, 20]. The VaR
indicates the chance of a random variable landing inside an undesirable set of
values that characterizes a systemic failure. The set of such undesirable
values is referred to as a systemic set and denoted by $U\subset\mathbb{R}$.
In probability space $(\Omega,\mathcal{F},\mathbb{P})$, the set of systemic
events of random variable $y:\Omega\rightarrow\mathbb{R}$ is defined as
$\\{\omega\in\Omega~{}|~{}y(\omega)\in U\\}$. We define a collections of
supersets $\\{U_{\delta}~{}|~{}\delta\in[0,\infty]\\}$ of $U$ that satisfy the
following conditions for any sequence $\\{\delta_{n}\\}^{\infty}_{n=1}$ with
property $\lim_{n\rightarrow\infty}\delta_{n}=\infty$

  * •

$U_{\delta_{2}}\subset U_{\delta_{1}}$ when $\delta_{1}<\delta_{2}$

  * •

$\lim_{n\rightarrow\infty}U_{\delta_{n}}=\bigcap_{n=1}^{\infty}U_{\delta_{n}}=U$.

In practice, one can tailor the super-sets to cover a suitable neighborhood of
$U$ to characterize alarm zones as a random variable approaches $U$. For a
given $\delta>0$, the chance of $\\{y\in U_{\delta}\\}$ indicates how close
$y$ can get to $U$ in probability. For a given design parameter
$\varepsilon\in(0,1)$, the VaR measure
$\mathcal{R}_{\varepsilon}:\mathcal{F}\rightarrow\mathbb{R}_{+}$ is defined by

$\mathcal{R}_{\varepsilon}:=\inf\left\\{\delta>0~{}\big{|}~{}\mathbb{P}\left\\{y\in
U_{\delta}\right\\}<\varepsilon\right\\},$

where a smaller $\varepsilon$ indicates a higher level of confidence on random
variable $y$ to stay away from $U_{\delta}$. Let us elaborate and interpret
what typical values of $\mathcal{R}_{\varepsilon}$ imply. The case
$\mathcal{R}_{\varepsilon}=0$ signifies that the probability of observing $y$
dangerously close to $U$ is less than $\varepsilon$. We have
$\mathcal{R}_{\varepsilon}>0$ if and only if $y\in U_{\delta}$ for
$\delta>\mathcal{R}_{\varepsilon}$ with probability less than $\varepsilon$.
In addition to several interesting properties (see for instance [1, 2, 20]),
the risk measure is non-increasing with $\varepsilon$. We refer to Fig. 1 for
an example.

##  V Risk of Cascading Failures

Figure 1: This figure depicts the concept of the risk set $U_{\delta}$ and how
the distribution of $\bar{y}_{j}$ changes when
$\bar{\bm{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}$ is introduced.

As a natural outgrowth of our previous work [9], we generalize the definition
of the “cascading failures” from assuming only one existing failure into an
arbitrary number of existing failures [10]. To this end, let us consider the
risk of cascading large fluctuation for the $j$’th agent, assuming that agents
with ordered indices $\mathcal{I}_{m}=\\{i_{1},\cdots,i_{m}\\}$ for some $m<n$
have failed to reach the $c-$consensus, where $j\notin\mathcal{I}_{m}$ and
$i_{1}<i_{2}<...<i_{m}$. To evaluate the effect of the prior large
fluctuations on the $j$’th agent, let us form a $(m+1)\times(m+1)$ block
covariance matrix

$\tilde{\Sigma}=\begin{bmatrix}\,\tilde{\Sigma}_{11}&\tilde{\Sigma}_{12}\\\
\tilde{\Sigma}_{21}&\tilde{\Sigma}_{22}\,\end{bmatrix},$ (8)

where
$\tilde{\Sigma}_{11}=\sigma_{j}^{2},\,\tilde{\Sigma}_{12}=\tilde{\Sigma}_{21}^{T}=[\sigma_{ji_{1}},...,\sigma_{ji_{m}}]$,
and
$\tilde{\Sigma}_{22}=[\sigma_{k_{1}k_{2}}]_{k_{1},k_{2}\in\mathcal{I}_{m}}\in\mathbb{R}^{m\times
m}$. The values of $\sigma_{ij}$s are computed using the result from (7).

We also denote that observables of the failed agent are represented by vector
$\bm{y}_{f}=[y_{f_{1}},...,y_{f_{m}}]^{T}$. For instance, when the $i$’th
agent failed to each the $c-$consensus, one may set the value of $y_{f_{i}}$
to satisfy $|y_{f_{i}}|>c$. To obtain the statistics of the observable when
multiple failures occur, one should calculate the steady-state conditional
probability distribution of $\bar{y}_{j}$ given
$\bm{\bar{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}$ within a multivariate normal
distribution, where
$\bm{\bar{y}}_{\mathcal{I}_{m}}=[\bar{y}_{i_{1}},...,\bar{y}_{i_{m}}]^{T}$.

######  Lemma 2.

Suppose that the system (3) reaches the steady-state, the conditional
distribution of $\bar{y}_{j}$ given
$\bm{\bar{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}$ follows a normal distribution
$\mathcal{N}(\tilde{\mu},\tilde{\sigma}^{2})$ with

$\tilde{\mu}=\tilde{\Sigma}_{12}\tilde{\Sigma}_{22}^{-1}\bm{y}_{f},\text{ and
}~{}\tilde{\sigma}^{2}=\tilde{\Sigma}_{11}-\tilde{\Sigma}_{12}\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21},$
(9)

where the sub-blocks $\tilde{\Sigma}_{11},...,\tilde{\Sigma}_{22}$ are defined
in (8).

The above lemma reveals the statistics of the observables when there exist
multiple large fluctuations (failures) in the consensus network. In this
scenario, we aim to evaluate the risk of cascading failure on the $j$’th
agent. The corresponding event is shown by

$\left\\{\bar{y}_{j}\in
U_{\delta}~{}|~{}\bar{\bm{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}\right\\},$

where the family of risk set333It should be noted that in our definition of
$U_{\delta}$ for $c$-consensus, the family of supersets obviously depends on
$c$ but its dependency to this parameter is omitted from $U_{\delta}$ for
notational simplicity. of cascading failure for the $j$’th agent is given by

$U_{\delta}=\left(-\infty,-\delta-c\right)\bigcup\left(\delta+c,\infty\right),$

with $c\in\mathbb{R}_{+}$ and $\delta\in[0,\infty)$. Then, the value-at-risk
measure for the cascading failure on the $j$’th agent is defined as

$\displaystyle\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}=\inf\big{\\{}\delta>0\;\big{|}\;\mathbb{P}\\{\bar{y}_{j}\in
U_{\delta}\;\big{|}\;\bar{\bm{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}\\}<\varepsilon\big{\\}},$
(10)

where $\varepsilon$ takes value in $(0,1)$. The cascading risk
$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}$ takes value in
$[0,\infty)$, and it measures the chance of an agent failing to reach the
$c-$consensus given that $m$ other agents have failed to rendezvous.

For the exposition of the next result, we introduce the following notation:

$\displaystyle\kappa_{\delta,\pm}^{\mathcal{I}_{m},j}=\frac{(\delta+c)\pm\tilde{\mu}}{\sqrt{2}\tilde{\sigma}},$
(11)

with $\tilde{\mu}$ and $\tilde{\sigma}$ defined in Lemma 2. The notation below
is also introduced for simplicity,

$\displaystyle
S(\delta)=\inf\left\\{\delta>0\,\bigg{|}\,\text{erf}(\kappa_{\delta,+}^{\mathcal{I}_{m},j})+\text{erf}(\kappa_{\delta,-}^{\mathcal{I}_{m},j})>2(1-\varepsilon)\right\\}.$
(12)

######  Theorem 1.

Suppose that the consensus network (3) reaches the steady-state and $m$ agents
with labels $\mathcal{I}_{m}$ have failed to reach the $c$-consensus with the
vector of observable $\bar{\bm{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}$. The risk of
cascading large fluctuation of agent $j$ is

$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}:=\begin{cases}0,&\text{if}~{}1-\frac{1}{2}\left(\text{erf}(\kappa_{0,+}^{\mathcal{I}_{m},j})+\text{erf}(\kappa_{0,-}^{\mathcal{I}_{m},j})\right)\leq\varepsilon\\\
S(\delta),&\text{otherwise}\end{cases}$

where $j=1,\dots,n$, $j\notin\mathcal{I}_{m}$, $|\bm{y}_{f}|>c\,\bm{1}_{m}$,
and $S(\delta)$ is defined by (12).

The two cases of the cascading risk value are self-explanatory. The case
$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}=0$ indicates the scenario
when the probability of the $j$’th agent failing to reach $c-$consensus is
always less than $\varepsilon$, which commonly corresponds to a low confidence
level or the conditional distribution that lands away from the undesired set
of the observable values. In the other case, the risk obtains a positive value
based on the solution of (12). A higher value of
$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}$ usually indicates a
higher chance that the cascading failure will occur in the system. The value
of $c$ will also play the role as a design parameter when
$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}=0$ since the system will
reach the $c-$consensus with the confidence level $1-\varepsilon$.

When the existing failures are not correlated to the $j$’th agent, i.e.,
$\tilde{\Sigma}_{12}=\tilde{\Sigma}_{21}^{T}=\bm{0}$, the cascading risk
expression $\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}$ boils down to
the risk of large fluctuation for a single agent to rendezvous,

$\mathcal{R}^{j}_{\varepsilon}=\sqrt{2}\,\sigma_{j}\iota_{\varepsilon}-c,\text{
if }\sigma_{j}>\dfrac{c}{\sqrt{2}\iota_{\varepsilon}},$

where $\iota_{\varepsilon}=\text{erf}^{-1}(1-\varepsilon)$. This formula
expands the risk of large fluctuations expression derived in [21], which
considers deviations on $c$-consensus state for $c=0$.

Given that $m$ agents have failed to reach the $c$-consensus, one can
formulate the risk profile of the network by stacking up all the cascading
risk $\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}$ among remaining
agents with a systemic risk profile vector
$\bm{\mathcal{R}}^{\mathcal{I}_{m}}_{\varepsilon}\in\mathbb{R}^{n}$ as follows

$\bm{\mathcal{R}}^{\mathcal{I}_{m}}_{\varepsilon}=\big{[}\mathcal{R}^{\mathcal{I}_{m},1}_{\varepsilon},\dots,\mathcal{R}^{\mathcal{I}_{m},n}_{\varepsilon}\big{]}^{T},$

where $\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}=0$ if
$j\in\mathcal{I}_{m}$.

##  VI The Update Law for the Cascading Risk Computation

(a) The path graph.

(b) The $2-$cycle graph.

(c) The $5-$cycle graph.

(d) The $7-$cycle graph.

(e) The complete graph.

Figure 2: The risk profile of cascading failure
$\mathcal{R}^{\mathcal{I}_{m}}_{\varepsilon}$ when the existing large
fluctuations have occurred at agents $\\{9,10,11,12\\}$.

In this section, let us consider the scenario where agents with labels
$\mathcal{I}_{m}$ are found in failure states, and we aim to update the
statistics of the agent of interest, i.e.,
$\bar{y}_{j}|\bar{\bm{y}}_{\mathcal{I}_{m}}=\bm{y}_{f}$, when a new failure at
agent $k|_{k\notin\mathcal{I}_{m}}$ is discovered. For the exposition of the
next result, let us introduce the following notations

$\displaystyle\tilde{\mu}_{j}=\tilde{\Sigma}_{12}(j)\tilde{\Sigma}_{22}^{-1}\bm{y}_{f},~{}~{}\tilde{\sigma}^{2}_{j}=\sigma_{j}^{2}-\tilde{\Sigma}_{12}(j)\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(j),$
(13)
$\displaystyle\tilde{\mu}_{k}=\tilde{\Sigma}_{12}(k)\tilde{\Sigma}_{22}^{-1}\bm{y}_{f},~{}~{}\tilde{\sigma}^{2}_{k}=\sigma_{k}^{2}-\tilde{\Sigma}_{12}(k)\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(k),$

where the value of $\sigma_{j},\sigma_{k}$ are computed using (7), the values
of $\tilde{\Sigma}_{12}(k)=\tilde{\Sigma}_{21}^{T}(k)$ are given by (8) in the
view of the agent $k$ 444The values of
$\tilde{\Sigma}_{12}(j)=\tilde{\Sigma}_{21}^{T}(j)$ are obtained with agent
$j$, respectively., and the matrix $\tilde{\Sigma}_{22}^{-1}$ is computed
using (8).

######  Theorem 2.

Suppose that $\bar{y}_{j}$ follows
$\mathcal{N}(\tilde{\mu}_{j},\tilde{\sigma}_{j}^{2})$ when $m$ agents have
already failed with labels $\mathcal{I}_{m}$. The updated conditional
distribution of $\bar{y}_{j}$ when a new agent fails, i.e., agent
$k\notin\mathcal{I}_{m}$ with observable $|y_{f_{k}}|>c$, is given by
$\mathcal{N}(\tilde{\mu}^{\prime},\tilde{\sigma}^{\prime 2})$, such that

$\displaystyle\tilde{\mu}^{\prime}=\tilde{\mu}_{j}-\frac{\tilde{\sigma}_{jk}}{\tilde{\sigma}_{k}^{2}}(\tilde{\mu}_{k}-y_{f_{k}}),\text{
and }~{}\tilde{\sigma}^{\prime
2}=\tilde{\sigma}_{j}^{2}-\frac{\tilde{\sigma}_{jk}}{\tilde{\sigma}_{k}^{2}},$
(14)

where

$\tilde{\sigma}_{jk}=\sigma_{jk}-\tilde{\Sigma}_{12}(j)\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(k),$

and the value of
$\tilde{\mu}_{j},\tilde{\mu}_{k},\tilde{\sigma}_{j},\tilde{\sigma_{k}}$ are
computed using (13).

The above result provides a good insight into the change in the conditional
mean and covariance of an agent when the number of failed agents increase from
$m$ to $m+1$. Interestingly, the conditional covariance of the agent $j$ is
independent of the value of the deviation of the failed agent $k$. However,
the conditional mean of the agent $j$ depends on the difference between the
conditional mean of the failed agent $k$, when $m$ failures have occurred, and
its current value of the failed observable, $y_{f_{k}}$. In the unlikely case
of $y_{f_{k}}=\tilde{\mu}_{k}$, the conditional mean of the agent $j$ remains
unchanged.

In addition, the above result can be applied to find the sequence of agents
with maximum risk after each additional failure. This information can be very
helpful in designing an optimal communication network to protect a sequence of
agents that is most vulnerable to cascading systemic failures.

##  VII Case Studies

We discuss the case studies for the rendezvous problem with the consensus
dynamics governed by (3) among the complete, path, and the p-cycle
communication graphs [24]. In all case studies, we consider agents with labels
$\mathcal{I}_{m}$ cannot achieve the $c$-consensus and have been found with
large fluctuations $\bm{y}_{f}=y_{f}\bm{1}_{m}$. Let us consider
$n=20,c=0.1,y_{f}=2,b=4,\tau=0.05\text{ and }\varepsilon=0.1$ for all
simulations.

###  VII-A Risk of Cascading Large Fluctuation

The risk profile of the cascading failure to reach $c$-consensus
$\bm{\mathcal{R}}^{\mathcal{I}_{m}}_{\varepsilon}$ is computed using the
closed-form expression derived in Theorem 1 for various unweighted
communication graph topologies, which is shown in Fig. 2. There are a few
interesting results worth reporting: In path graph [24], it is shown that the
riskiest agent is found at both ends of the graph when there exist multiple
failures in the system; In $p-$cycle graphs [24], one can observe that the
risk profile pattern converges to the complete graph as $p$ increases.
Finally, the complete graph shows that the risk value remains unchanged
regardless of the agent of the interest.

###  VII-B Characteristics of Existing Failures

In the cascading risk, the characteristics of the existing failures will
profoundly affect the cascading risk profile. For instance, the number of
existing failures and their spatial distribution in the network reshapes the
cascading risk profile in various networks. This phenomenon is examined in
Fig. 3, in which a different number of failures and various failures
distribution are considered. Some of these results can be validated with our
theoretical findings, e.g., in a complete graph, the risk profile obtains a
constant value with a fixed number of existing failures, regardless of their
spatial distribution in the communication graph.

(a) The path graph.

(b) The $2-$cycle graph.

(c) The $5-$cycle graph.

(d) The complete graph.

(e) The path graph.

(f) The $2-$cycle graph.

(g) The $5-$cycle graph.

(h) The complete graph.

Figure 3: The cascading risk profile with (a)-(d) a different number of
failures; (e) - (h) various spatial distributions. The agents that obtain the
risk value as $0$ represent existing failures; the red color denotes when the
risk value reaches $+\infty$.

###  VII-C Most Vulnerable Sequence

(a) The path graph.

(b) The $1-$cycle graph.

(c) The $5-$cycle graph.

(d) The $8-$cycle graph.

(e) The $9-$cycle graph.

(f) The complete graph.

Figure 4: The most vulnerable sequence among various graphs.

We also explore the emergence of the most vulnerable sequence of agents prone
to cascading failures (Fig. 4). The most vulnerable sequence can be assessed
by first finding the agent that obtains the highest risk and then finding the
next most vulnerable agent by evaluating the cascading risk with the
assumption that the previously found agents have failed to reach the
$c-$consensus. The result in the complete graph indicates that any arbitrary
sequence of agents is identically vulnerable since the agents will always
obtain the same value of cascading risk in a complete graph. The investigation
of the most vulnerable sequence provides information from the network design
perspective such that one can adjust the systemic design based on the priority
of the agents in the network.

##  VIII Conclusion

To investigate the emergence of the cascading waves of fluctuations in the
time-delay multi-agent systems, we develop a methodology to evaluate the risk
of cascading failures using the notion of value-at-risk measure. The proposed
cascading risk framework is a natural outgrowth of single event risk measure
[9], and it shows a significant advantage compared to the previous works as it
can explore the scenarios when multiple system failures exist. In addition, to
provide insights from the design perspective, we explore the time-delay
induced hard limits on the best achievable levels of cascading risk as well as
the update law for risk computation. Furthermore, the cascading risk framework
reveals its advantages in the design of networks as it enables the designer to
assess the sequence of the most vulnerable agents in a given network.

####  -1 Proof of Lemma 1

The result is an immediate extension of the steady-state statistics of the
observables in [21] by considering a centering matrix $M_{n}$. $\square$

####  -2 Proof of Lemma 2

The result follows directly after Lemma 1 and the conditional distribution of
a bi-variate normal random variable as in [23]. $\square$

####  -3 Proof of Theorem 1

Considering Lemma 2, one can rewrite the RHS of (10) as

$\inf\big{\\{}\delta>0\big{|}\int_{-\infty}^{-\kappa_{\delta,+}^{\mathcal{I}_{m},j}}e^{-t^{2}}dt+\int_{-\infty}^{-\kappa_{\delta,-}^{\mathcal{I}_{m},j}}e^{-t^{2}}dt<\sqrt{\pi}\varepsilon\big{\\}}.$

The result follows by taking cases on the risk value. More specifically, the
case when $\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}=0$ is equivalent
to

$\displaystyle\int_{-\infty}^{-\kappa_{0,+}^{\mathcal{I}_{m},j}}e^{-t^{2}}\,dt+\int_{-\infty}^{-\kappa_{0,-}^{\mathcal{I}_{m},j}}e^{-t^{2}}\,dt\leq\sqrt{\pi}\varepsilon,$

and the first branch condition follows from the definition of the error
function.

For the second branch, when
$\mathcal{R}^{\mathcal{\bm{I}}_{m},j}_{\varepsilon}\in(0,\infty]$, the
monotonicity condition on $\kappa_{\delta,+}^{\mathcal{I}_{m},j}$ and
$\kappa_{\delta,-}^{\mathcal{I}_{m},j}$ implies there is a unique
$\delta^{*}>0$ such that

$\int_{-\infty}^{-\kappa_{\delta^{*},+}^{\mathcal{I}_{m},j}}e^{-t^{2}}\,dt+\int_{-\infty}^{-\kappa_{\delta^{*},-}^{\mathcal{I}_{m},j}}e^{-t^{2}}\,dt=\sqrt{\pi}\varepsilon,$

the result follows by solving the above equation for $\delta^{*}$. $\square$

####  -4 Proof of Theorem 2

To characterize the effect of the failures of $m+1$ agents, let us consider
the block covariance matrix

$\tilde{\Sigma}_{22}^{\prime}=\begin{bmatrix}\tilde{\Sigma}_{22}&\tilde{\Sigma}_{21}(k)\\\
\tilde{\Sigma}_{12}(k)&\tilde{\Sigma}_{kk}\end{bmatrix},$

where
$\tilde{\Sigma}_{kk}=\sigma_{k}^{2},\tilde{\Sigma}_{21}(k)=\tilde{\Sigma}_{12}^{T}(k)=\begin{bmatrix}\sigma_{ki_{1}}&\dots\sigma_{ki_{m}}\end{bmatrix}$,
and $\tilde{\Sigma}_{22}$ is obtained from (8). Since $\tilde{\Sigma}_{22}$ is
invertible, we have

$\displaystyle\tilde{\Sigma}_{22}^{\prime-1}=$ $\displaystyle\hskip
2.84526pt\begin{bmatrix}\tilde{\Sigma}_{22}^{-1}(I_{m}+\tilde{\Sigma}_{21}(k)S^{-1}\tilde{\Sigma}_{12}(k)\tilde{\Sigma}_{22}^{-1})&-\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(k)S^{-1}\\\
S^{-1}\tilde{\Sigma}_{12}(k)\tilde{\Sigma}_{22}^{-1}&S^{-1}\end{bmatrix},$

where
$S=\tilde{\Sigma}_{22}^{\prime}/\tilde{\Sigma}_{22}=\tilde{\Sigma}_{kk}-\tilde{\Sigma}_{12}(k)\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(k)=\tilde{\sigma}_{k}^{2}$
is the Schur complement [13] of block $\tilde{\Sigma}_{22}$ of the matrix
$\tilde{\Sigma}_{22}^{\prime}$. Let us consider the vector of failed
observables of $(m+1)$ agents as $[\bar{\bm{y}}_{f}~{}\bar{{y}}_{f_{k}}]^{T},$
where $\bm{\bar{y}}_{f}=[\bar{y}_{f_{1}},...,\bar{y}_{f_{m}}]^{T}$ is the
vector of failed observables of $m$ agents and $\bar{{y}}_{f_{k}}$ is the
failed observable of agent k, i.e., $(m+1)^{th}$ agent. Consider the following
vectors,
$\tilde{\Sigma}_{12}^{\prime}=[\tilde{\Sigma}_{12}~{}\tilde{\Sigma}_{12}(k)]=\tilde{\Sigma}_{12}^{\prime
T}$ and the conditional cross-covariance of agents $j$ and $k$ after $m$
agents have failed
$\tilde{\sigma}_{jk}=\sigma_{jk}-\tilde{\Sigma}_{12}(j)\tilde{\Sigma}_{22}^{-1}\tilde{\Sigma}_{21}(k)$,
the result then follows directly by applying Lemma 2. $\square$

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# Twin Pati-Salam theory of flavour for a new picture of $B$-anomalies

M. Fernández Navarro  School of Physics & Astronomy, University of
Southampton,
Southampton SO17 1BJ, UK

###### Abstract

We discuss a theory of flavour which can simultaneously explain and connect
the origin of the Standard Model flavour hierarchies with the
$U_{1}(\mathbf{3},\mathrm{\mathbf{1},2/3})$ leptoquark explanation of the
$B$-decay flavour anomalies. First, we motivate the vector leptoquark $U_{1}$
as an excellent mediator to address the updated picture of $B$-anomalies.
Secondly, we introduce the Twin Pati-Salam theory of flavour, which naturally
provides a low scale 4321 gauge group with a TeV scale $U_{1}$. We show that
both the couplings of $U_{1}$ to SM fermions and the effective SM Yukawa
couplings arise from the same physics, naturally providing a pattern of
dominantly left-handed $U_{1}$ couplings as most preferred by current data.
Finally, we discuss the most promising signals at low energy processes that
can test the model in the near future.

##  1 Introduction

The high number of flavour parameters in the Standard Model (SM) and their
suspicious hierarchical pattern might be understood in the form of physics
beyond the SM. A well-motivated assumption is that the SM Yukawa couplings
arise as effective remnants of a more fundamental theory of flavour,
explaining their hierarchical pattern. However, the energy scale $\Lambda$ of
such a theory is commonly unspecified, since theories of flavour usually
accommodate the flavour parameters of the SM by fixing the ratios
$\left\langle\phi\right\rangle/\Lambda$, where $\phi$ is associated to heavy
Higgs bosons spontaneously breaking the gauge structures of the new theory.
Nevertheless, such a theory is expected to predict non-generic fermion mixing
in each charged sector. In absence of a very particular GIM-like suppression,
it is possible that the new massive degrees of freedom from the theory of
flavour could mediate sizable flavour-changing currents. Given the sensitivity
of flavour observables to very high energy scales, we might be able to
disentangle the high scales $\left\langle\phi\right\rangle$ or $\Lambda$ from
the amount of data obtained in flavour physics experiments.

In this direction, a conspicuous series of anomalies in flavour observables
has emerged in the last decade. On the one hand, significant discrepancies
between the SM and the experiments in several $b\rightarrow s\mu\mu$
observables point to a possible new physics (NP) effect. Nevertheless, such NP
must couple similarly to muon and electron pairs in order to pass the lepton
flavour universality (LFU) test of the recently updated $R_{K^{(*)}}$ ratios
[1]. On the other hand, more discrepancies between the experiment and the SM
appear in $b\rightarrow c\ell\nu$ charged current transitions, where the
remarkable effect is the possible breaking of LFU suggested by the
measurements of the $R_{D^{(*)}}$ ratios, which hint to a preference for the
decay $b\rightarrow c\tau\nu$ with respect to the transitions involving light
charged leptons. Although the measurement of $R_{D^{(*)}}$ is more challenging
from the experimental point of view, the world average of current data
suggests a departure from the SM at the $3.2\sigma$ significance level [1].
All things considered, both sets of anomalies together seem to prefer a
sizable new physics effect affecting third family fermions, competing with a
tree-level SM charged current, along with a suppressed and flavour universal
effect for the light families that explains the neutral current anomalies.

In the following, we will introduce the vector leptoquark
$U_{1}(\mathbf{3},\mathrm{\mathbf{1},2/3})$ as an excellent mediator to
address the current picture of anomalies, which moreover is naturally
connected to the origin of flavour hierarchies in the SM.

##  2 The $U_{1}$ vector leptoquark and the 4321 embedding

The vector leptoquark $U_{1}(\mathbf{3},\mathrm{\mathbf{1},2/3})$ contributes
at tree-level to the $R_{D^{(*)}}$ ratios via the contact interaction
$\left(\bar{c}_{L}\gamma^{\mu}b_{L}\right)\left(\bar{\tau}_{L}\gamma^{\mu}\nu_{L}\right)$.
This contribution is also connected to the contact interaction
$\left(\bar{s}_{L}\gamma^{\mu}b_{L}\right)\left(\bar{\tau}_{L}\gamma^{\mu}\tau_{L}\right)$
via $SU(2)_{L}$ invariance. Remarkably, at the 1-loop level the latter
provides a contribution to the LFU operator
$\left(\bar{s}_{L}\gamma^{\mu}b_{L}\right)\left(\bar{\ell}\gamma^{\mu}\ell\right)$
[2]. Therefore, a vector leptoquark
$U_{1}(\mathbf{3},\mathrm{\mathbf{1},2/3})$ explaining the $R_{D^{(*)}}$
anomalies naturally delivers a relevant LFU contribution to $b\rightarrow
s\ell\ell$ that can reduce the tension with the experiment.

Nevertheless, the explanation of the $R_{D^{(*)}}$ anomalies requires a TeV
scale $U_{1}$. Such a light vector leptoquark arising from traditional Pati-
Salam (PS) embeddings would be at odds with data from flavour-violating kaon
decays. However, model building efforts in recent years led to the so-called
“4321” gauge group [3] (for alternative constructions see e.g. [2]):

$SU(4)\times SU(3)^{\prime}\times SU(2)_{L}\times U(1)_{Y^{\prime}}\,,$ (1)

which can provide a TeV scale $U_{1}$ as long as the first and second fermion
families are singlets under $SU(4)$. The theory further predicts a heavy
$Z^{\prime}$ and coloron $g^{\prime}$ with masses in the TeV ballpark as well,
plus heavy vector-like (VL) fermions that induce $U_{1}$ couplings to the
light families through mixing.

In recent years, several theories of flavour have been proposed assuming that
the third SM family transforms non-trivially under $SU(4)$, see e.g. [4].
However, all of them predict a sizable coupling of third family right-handed
(RH) fermions to $U_{1}$, which is tightly constrained by direct searches at
LHC, see e.g. the recent review [5]. Instead, if the third family is a singlet
under $SU(4)$, then the $U_{1}$ currents can be dominantly left-handed (LH), a
scenario compatible with current high energy data. In this so-called
“fermiophobic” scenario, the couplings of $U_{1}$ to SM fermions are all
induced through mixing with vector-like fermions that transform non-trivially
under $SU(4)$. Nevertheless, the flavour structure of such a model is ad-hoc,
worsening the flavour puzzle with a bunch of extra parameters that give no
information about the origin of flavour in the SM. We will show in the
following a theory of flavour featuring a low scale fermiophobic 4321 model,
which addresses the origin of flavour hierarchies in the SM and predicts an
even richer phenomenology.

##  3 The Twin Pati-Salam theory of flavour

Given the structure of the fermiophobic 4321 group described in the previous
section, where a set of vector-like fermions transforms under $SU(4)$ while
SM-like fermions transform under $SU(3)^{\prime}$, a natural UV completion
would be that of two commuting Pati-Salam groups, one for vector-like fermions
and another one for SM-like fermions. This is the basic structure of the Twin
Pati-Salam gauge group [6, 7] introduced in Fig. 1, which includes a scalar
sector linking both PS sites.

---
(a)

Field | $SU(4)_{PS}^{I}$ | $SU(2)_{L}^{I}$ | $SU(2)_{R}^{I}$ | $SU(4)_{PS}^{II}$ | $SU(2)_{L}^{II}$ | $SU(2)_{R}^{II}$
---|---|---|---|---|---|---
$\Psi_{L(1,2,3)}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$
$\psi_{R(1,2,3)}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$
$\Psi_{L(4,5,6)}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$\tilde{\Psi}_{R(4,5,6)}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$\tilde{\psi}_{L(4,5,6}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$\psi_{R(4,5,6)}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$
$\phi$ | $\mathbf{\overline{4}}$ | $\mathbf{\overline{2}}$ | $\mathbf{1}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$
$\overline{\phi}$ | $\mathbf{4}$ | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$
$H$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$
$\overline{H}$ | $\mathbf{\overline{4}}$ | $\mathbf{1}$ | $\mathbf{\overline{2}}$ | $\mathbf{4}$ | $\mathbf{2}$ | $\mathbf{1}$
$\Omega_{15}$ | $\mathbf{15}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$ | $\mathbf{1}$

(b)

Figure 1: The model is based on two copies of the PS gauge group
$SU(4)_{PS}\times SU(2)_{L}\times SU(2)_{R}$. The circles represent the gauge
groups with the indicated symmetry breaking. Three families of SM-like
(chiral) fermions, arranged in Pati-Salam multiplets, transform under
$\mathrm{PS}^{II}$ (in blue), while three families of vector-like fermions
(denoted 4, 5 and 6) transform under $\mathrm{PS}^{I}$ (in red). The Twin
Pati-Salam symmetry is spontaneously broken down to the 4321 group at high
energies $M_{\mathrm{High}}\apprge 1\,\mathrm{PeV}$ (via scalar content not
shown here, we refer the reader to [6, 7]), then the 4321 group is broken to
the SM at the scale $M_{\mathrm{low}}\sim\mathcal{O}(\mathrm{TeV})$ (via the
scalars $\phi$ and $\bar{\phi}$ connecting both PS groups). The Higgs $H$ and
$\bar{H}$ decompose in a multi-Higgs doublet model at low energies and perform
electroweak symmetry breaking, see [6, 7].

###  3.1 Effective Yukawa and $U_{1}$ couplings

The scalar sector of the model is chosen such that renormalisable Yukawa
couplings for SM-like fermions are forbidden. Nevertheless, the scalar sector
acts as a link between the two PS groups, generating mixing between vector-
like and SM-like fermions which provides effective Yukawa couplings and
effective $U_{1}$ couplings for SM-like fermions 111In the simplified model
discussed here, we work in the first family massless approximation (since only
second and thid family fermions are relevant for our description of the
$B$-anomalies) and we do not discuss the extended Higgs sector of the model.
We refer the interested reader to [6, 7] where the complete model is presented
and discussed.. In Fig. 2 we display this mechanism at work. The third family
effective Yukawa couplings are connected to the ratio of the TeV scale VEVs of
the scalars $\phi_{3}$ and $\phi_{1}$ over the mass of $SU(2)_{L}$ doublet
fourth family vector-like fermions $M_{4}^{Q,L}$. The effective $U_{1}$
couplings to third family fermions arise from this mixing as well, so they are
connected to the same ratios of NP scales. As an example, we include below the
effective top Yukawa and LH third family $U_{1}$ coupling as

$y_{t}\sim\frac{\left\langle\phi_{3}\right\rangle}{M_{4}^{Q}}\sim\mathcal{O}(1)\Longrightarrow\beta_{Q_{3}L_{3}}^{L}\sim\frac{\left\langle\phi_{3}\right\rangle}{M_{4}^{Q}}\frac{\left\langle\phi_{1}\right\rangle}{M_{4}^{L}}\sim\mathcal{O}(1)\,,$
(2)

where the ratio $\left\langle\phi_{1}\right\rangle/M_{4}^{L}$ is also expected
to be $\mathcal{O}(1)$ since the twin PS symmetry enforces $M_{4}^{Q}\sim
M_{4}^{L}$. Therefore, the prediction of an $\mathcal{O}(1)$ top Yukawa
coupling leads to an $\mathcal{O}(1)$ third family LH coupling to $U_{1}$ as
well. In contrast, the effective Yukawa couplings for second family fermions
arise from mixing with $SU(2)_{L}$ singlet fourth family vector-like fermions.
As a consequence, they are connected to ratios of the VEVs of other scalars
$\bar{\phi}_{3}$ and $\bar{\phi}_{1}$ (whose VEVs are also TeV scale since
they also participate in 4321 spontaneous breaking) over the mass of the
$SU(2)_{L}$ singlet vector-like fermions $M_{4}^{u,d,e}$. This mixing leads to
suppressed $U_{1}$ couplings to RH SM-like fermions, since they are connected
to the origin of second family fermion masses,

$y_{c}\sim\frac{\left\langle\bar{\phi}_{3}\right\rangle}{M_{4}^{u}}\ll
1\Rightarrow\beta_{b\tau}^{R}\sim\frac{\left\langle\bar{\phi}_{3}\right\rangle}{M_{4}^{d}}\frac{\left\langle\bar{\phi}_{1}\right\rangle}{M_{4}^{e}}\ll
1\,,$ (3)

where again the Twin PS symmetry enforces $M_{4}^{u}\sim M_{4}^{d}\sim
M_{4}^{e}$. This way, just by predicting flavour hierarchies we obtain the
desired pattern of dominantly left-handed $U_{1}$ couplings. Remarkably, the
$SU(2)_{L}$ doublet fermions in the fourth family have to live at the TeV
scale in order to explain the fermion masses and $U_{1}$ couplings, within the
reach of LHC. Instead, the $SU(2)_{L}$ singlet fourth family vector-like
fermions are expected to be much heavier in order to explain the lighter
second family fermions.

$Q_{L3}$$M^{Q}_{4}$$u_{R3}$$\phi_{3}$$H_{t}$$\tilde{Q}_{R4}$$Q_{L4}$

(a)

$Q_{Li}$$M^{u}_{4}$$u_{Rj}$$H_{c}$$\overline{\phi}_{3}$$\tilde{u}_{L4}$$u_{R4}$

(b)

$Q_{L3}$$M^{Q}_{4}$$M^{L}_{4}$$L_{L3}$$\phi_{3}$$U_{1}$$\phi_{1}$$\tilde{Q}_{R4}$$Q_{L4}$$L_{L4}$$\tilde{Q}_{R4}$

(c)

$d_{R3}$$M^{d}_{4}$$M^{e}_{4}$$e_{R3}$$\overline{\phi}_{3}$$U_{1}$$\overline{\phi}_{1}$$\tilde{d}_{L4}$$d_{R4}$$e_{R4}$$\tilde{e}_{L4}$

(d)

Figure 2: (Top) Diagrams leading to effective Yukawa couplings for the top
quark (left), charm quark and their mixing (right), with $i,j=2,3$ (Bottom)
Diagrams leading to effective $U_{1}$ couplings to third family LH fermions
(left) and RH fermions (right).

###  3.2 GIM-like mechanism at work

In the previous section we have introduced how large and dominantly LH
couplings for third family fermions arise in the model, while couplings to RH
fermions are naturally suppressed. So far, only the fourth family vector-like
fermions have been introduced. From now on, we shall assume that another
(fifth) family of vector-like fermions mixes with the fourth family, while a
sixth family remains unmixed. This mixing is different between the quark and
lepton sector (thanks to the $\Omega_{15}$ scalar), the particular mechanism
explained in detail in Ref. [7]. Such a pattern of VL fermion mixing predicts
a CKM-like matrix for the $U_{1}$ interactions with VL fermions,

$\mathcal{L}_{U_{1}}=\frac{g_{4}}{\sqrt{2}}\left(\begin{array}[]{ccc}\overline{Q}_{L4}&\overline{Q}_{L5}&\overline{Q}_{L6}\end{array}\right)\gamma_{\mu}\left(\begin{array}[]{ccc}c_{\theta_{LQ}}&-s_{\theta_{LQ}}&0\\\
s_{\theta_{LQ}}&c_{\theta_{LQ}}&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{c}L_{L4}\\\ L_{L5}\\\
L_{L6}\end{array}\right)U_{1}^{\mu}+\mathrm{h.c.}$ (4)

The CKM-like matrix above, containing the $s_{\theta_{LQ}}$ mixing, cancels
for the neutral currents mediated by the $Z^{\prime}$ and $g^{\prime}$, which
at this point remain flavour universal in vector-like fermion flavour space.
Now we introduce the 3-4 mixing discussed in the previous section, plus 2-5
and 1-6 fermion mixing in a similar manner, obtaining

$\mathcal{L}_{U_{1}}=\frac{g_{4}}{\sqrt{2}}\left(\begin{array}[]{ccc}\overline{Q}_{L1}&\overline{Q}_{L2}&\overline{Q}_{L3}\end{array}\right)\gamma_{\mu}\left(\begin{array}[]{ccc}s_{16}^{Q}s_{26}^{Q}&0&0\\\
0&c_{\theta_{LQ}}s_{25}^{Q}s_{25}^{Q}&s_{\theta_{LQ}}s_{25}^{Q}s_{34}^{L}\\\
0&-s_{\theta_{LQ}}s_{34}^{Q}s_{25}^{L}&c_{\theta_{LQ}}s_{34}^{Q}s_{34}^{Q}\end{array}\right)\left(\begin{array}[]{c}L_{L1}\\\
L_{L2}\\\ L_{L3}\end{array}\right)U_{1}^{\mu}+\mathrm{h.c.}$ (5)

where one can now read the relevant couplings participating in the
$B$-anomalies as

$\left.\begin{array}[]{c}\beta_{b\tau}^{L}=c_{\theta_{LQ}}s_{34}^{Q}s_{34}^{L}\\\
\\\
\beta_{s\tau}^{L}=s_{\theta_{LQ}}s_{25}^{Q}s_{34}^{L}\approx\beta_{c\nu_{\tau}}\end{array}\right\\}\Rightarrow\delta
R_{D^{(*)}}=\frac{R_{D^{(*)}}}{R_{D^{(*)}}^{\mathrm{SM}}}\propto\beta_{b\tau}^{L}\beta_{s\tau}^{L}\,,$
(6)

where a value $s_{\theta_{LQ}}\approx 1/\sqrt{2}$ allows for the largest
contribution to $\delta R_{D^{(*)}}$, and provides $\beta^{L}_{b\tau}\approx
1/\sqrt{2}$. The fact that $\beta^{L}_{b\tau}$ is smaller than 1 ameliorates
constraints from direct searches at LHC and LFU tests of $\tau$ decays.
Remarkably, the model predicts $\delta R_{D}=\delta R_{D^{*}}$.

In the basis of Eq. 5, the neutral currents mediated by the $Z^{\prime}$ and
$g^{\prime}$ become flavour diagonal (up to CKM and lepton mixing). Enforcing
$s_{34}^{Q}=s_{25}^{Q}=s_{16}^{Q}$ would suppress all FCNCs, but given the
large mixing angles $s_{34}^{Q,L}\approx 1$ required to explain the heaviness
of the top, this condition would lead to unsuppressed coloron production at
the LHC. Instead, we can enforce $s_{34}^{Q}=s_{25}^{Q}$, which protects from
the most dangerous FCNCs but allows sizable 2-3 transitions.

##  4 Low energy phenomenology

(a)

(b)

Figure 3: (Left) Parameter space in the plane ($x_{34}^{\psi}$,
$x_{25}^{\psi}$) compatible with $R_{D^{(*)}}$ and $R_{K^{(*)}}$ at 1$\sigma$,
with the remaining parameters are fixed as in the benchmark of [7]. The blue
region is excluded at 95% CL by $\Delta M_{s}$. (Right) $C_{9}^{U}$ as a
function of $x_{25}^{\psi}$. Regions preferred by the $B$-anomalies are
depicted. Data for the global fit to $b\rightarrow s\ell\ell$ is taken from
[8].

Having introduced the model and fixed all the different NP scales in the
previous section, now we explore the parameter space of $s_{34}^{Q,L}\propto
x_{34}^{\psi}$ and $s_{25}^{Q,L}\propto x_{25}^{\psi}$, where $x_{34}^{\psi}$
and $x_{25}^{\psi}$ are the dimensionless couplings that control VL-SM fermion
mixing. As depicted in Fig. 3a, the model can accommodate the $R_{D^{(*)}}$
anomalies within the $1\sigma$ region preferred by the experiment. We
highlight the regions preferred by $R_{K^{(*)}}$ according to the 2021 and
2022 measurements, respectively. A large contribution to the mass difference
$\Delta M_{s}$ from $B_{s}-\bar{B}_{s}$ meson mixing arises in our model at
1-loop, mediated by $U_{1}$. Nevertheless, this constraint is relaxed if
vector-like fermions running in the loop are lighter than 1 TeV, giving a
clear prediction for direct searches at LHC.

In Fig. 3b, we depict the values of the Wilson coefficient $C_{9}^{U}$
associated to the LFU operator
$\left(\bar{s}_{L}\gamma^{\mu}b_{L}\right)\left(\bar{\ell}\gamma^{\mu}\ell\right)$
that are preferred by $R_{D^{(*)}}$ and $R_{K^{(*)}}$. Due to the correlations
with $R_{K^{(*)}}$, the model can deliver $C_{9}^{U}\approx-0.4$, which is
close to the values preferred at $2\sigma$ by the global fit [8].

(a)

(b)

Figure 4: (Left) $\mathcal{B}\left(\tau\rightarrow 3\mu\right)$ as a function
of the $\mu-\tau$ mixing sine $s_{23}^{e}$. The purple region denotes the
$Z^{\prime}$ contribution while the blue region denotes the $U_{1}$
contribution. (Right) $\delta(B\rightarrow
K\bar{\nu}\nu)=\mathcal{B}(B\rightarrow
K\bar{\nu}\nu)/\mathcal{B}(B\rightarrow K\bar{\nu}\nu)_{\mathrm{SM}}-1$ as a
function of the 5th family vector-like mass term.

Furthermore, the model predicts a sizable enhancement of several observables
above their SM prediction. In Fig. 4a, we highlight the enhancement of
$\tau\rightarrow 3\mu$ mediated by the heavy $Z^{\prime}$ boson. This signal
is connected to non-generic $\mu-\tau$ mixing predicted by the Twin PS theory
of flavour, being a consequence of the mechanism explaining flavour
hierarchies. In Fig. 4b, we highlight the enhancement of $B\rightarrow
K^{(*)}\bar{\nu}\nu$ over the SM prediction, which is within the future reach
of the Belle II collaboration. Remarkably, given the dominantly LH nature of
$U_{1}$ couplings in the Twin PS model, this signal is much more enhanced with
respect to alternative constructions [4].

##  5 Conclusions

The Twin Pati-Salam theory of flavour naturally predicts a TeV scale
$U_{1}(\mathbf{3},\mathrm{\mathbf{1},2/3})$ leptoquark with the proper
features to address the current picture of $B$-anomalies, and connects the
origin of the $U_{1}$ couplings to SM fermions with the origin of effective
Yukawa couplings in the SM. A relevant enhancement of key low energy
observables will allow to test the model in the near future, along with a rich
spectrum of gauge bosons and vector-like fermions at the TeV scale.

## Acknowledgments

I thank my collaborator Steve King and I thank Xavier Ponce Díaz for support
during the preparation of my talk and valuable comments about my slides. I am
very grateful to the organisers of the Moriond conference, and in particular
to Jean-Marie Frère, for the inspiring atmosphere during the conference, for
the kind invitation and for trusting me for a plenary talk at such an early
stage of my career. This work has received funding from the European Union’s
Horizon 2020 Research and Innovation Programme under Marie Skłodowska-Curie
grant agreement HIDDeN European ITN project (H2020-MSCA-
ITN-2019//860881-HIDDeN).

## References

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