# Reversible Graph Neural Network-based Reaction Distribution Learning for
Multiple Appropriate Facial Reactions Generation

Tong Xu, Micol Spitale, Hao Tang, Lu Liu, Hatice Gune, and Siyang Song∗ Tong
Xu, Lu Liu and Siyang Song are with the School of Computing and Mathematical
Sciences, University of Leicester, Leicester, LE2 7RH, United Kingdom. E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
(∗ Corresponding Author: Siyang Song, E-mail<EMAIL_ADDRESS>
Micol Spitale, Hatice Gunes and Siyang Song are with the AFAR Lab, Department
of Computer Science and Technology, University of Cambridge, Cambridge, CB3
0FT, United Kingdom. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Hao Tang is with the Department of Information Technology and Electrical
Engineering, ETH Zurich, Zurich 8092, Switzerland. E-mail:
<EMAIL_ADDRESS>
###### Abstract

Generating facial reactions in a human-human dyadic interaction is complex and
highly dependent on the context since more than one facial reactions can be
appropriate for the speaker’s behaviour. This has challenged existing machine
learning (ML) methods, whose training strategies enforce models to reproduce a
specific (not multiple) facial reaction from each input speaker behaviour.
This paper proposes the first multiple appropriate facial reaction generation
framework that re-formulates the one-to-many mapping facial reaction
generation problem as a one-to-one mapping problem. This means that we
approach this problem by considering the generation of a distribution of the
listener’s appropriate facial reactions instead of multiple different
appropriate facial reactions, i.e., ’many’ appropriate facial reaction labels
are summarised as ’one’ distribution label during training. Our model consists
of a perceptual processor, a cognitive processor, and a motor processor. The
motor processor is implemented with a novel Reversible Multi-dimensional Edge
Graph Neural Network (REGNN). This allows us to obtain a distribution of
appropriate real facial reactions during the training process, enabling the
cognitive processor to be trained to predict the appropriate facial reaction
distribution. At the inference stage, the REGNN decodes an appropriate facial
reaction by using this distribution as input. Experimental results demonstrate
that our approach outperforms existing models in generating more appropriate,
realistic, and synchronized facial reactions. The improved performance is
largely attributed to the proposed appropriate facial reaction distribution
learning strategy and the use of a REGNN. The code is available at
https://github.com/TongXu-05/REGNN-Multiple-Appropriate-Facial-Reaction-
Generation.

###### Index Terms:

Multiple appropriate facial reaction generation, Reversible Graph Neural
Network, Facial reaction distribution learning

##  I Introduction

Figure 1: Our approach predicts an distribution representing multiple
different but appropriate facial reactions from each input speaker behaviour,
based on which multiple different but appropriate, realistic, and synchronized
human listener facial reactions could be generated.

Non-verbal behaviour interaction plays a key role in human-human communication
[1], with facial reactions providing important cues for understanding each
other’s emotional states. In dyadic interactions, a facial reaction refers to
the listener’s non-verbal facial behaviours in response to the speaker’s
verbal and non-verbal behaviours (e.g., facial muscle movements) [2, 3].
Previous studies [4, 5] have shown that the generation of listener’s facial
reactions to a speaker’s behaviour in dyadic interaction consists of three
main stages: Firstly, the listener’s perceptual system (e.g., ears and eyes)
receives external signals expressed by the speaker, which are pre-processed
before being transmitted to the brain for further analysis. Then, the
cognitive processor processes the pre-processed signals by taking personalized
perception bias into account, resulting in the generation of personalized
reaction signals. Finally, the motor processor decodes these personalized
signals to the facial muscles, producing corresponding facial reactions.

In contrast to most machine learning tasks, the generation of listener’s
facial reactions to a specific speaker behaviour are characterized by
variability and uncertainty [6, 7], meaning that different facial reactions
can be expressed by listeners in response to the same speaker behaviour.
Existing machine learning (ML)-based Facial Reaction Generation (FRG) models
aim to reproduce the real facial reaction expressed under a specific context
(called “GT reaction” in this paper) in response to each given speaker
behaviour. These models – including Generative Adversarial Networks (GAN) [8,
9], VQ-VAE [7], and person-specific FRG networks [10, 11] – are trained by
minimizing L1 or L2 loss between the generated and GT facial reactions.
However, this training strategy creates an ill-posed problem for existing FRG
models where similar inputs (speaker behaviours) are paired with different
labels (listener facial reactions), resulting in a “one-to-many mapping”
problem in the training phase. This limitation makes it theoretically very
challenging for existing approaches to learn good FRG models that can generate
diverse, appropriate, and photo-realistic facial reactions in response to
speaker behaviours.

In this paper, we propose the first deep learning framework to generate
multiple appropriate facial reactions in response to each speaker behaviour.
Rather than simply reproducing the GT facial reaction expressed by the
corresponding listener, our approach generates multiple different but
appropriate, realistic, and synchronized facial reactions from the given
context. Inspired by the theoretical framework of the human model processor
[12], our approach is designed to consist of three modules: (i) a perceptual
processor that encodes the input speaker audio and facial signals; (ii) a
cognitive processor that predicts an appropriate facial reaction distribution
from the encoded speaker audio-facial representation, which represents
multiple different but appropriate facial reactions; and (iii) a reversible
Graph Neural Network (GNN)-based motor processor that decodes an appropriate
facial reaction from the learned distribution.

To address the “one-to-many mapping” problem during the training phase, we
propose a novel Reversible Multi-dimensional Edge Graph Neural Network (REGNN)
as the motor processor. The REGNN is designed to summarize a distribution that
represents all appropriate real facial reactions displayed by listeners in
training set in response to each input speaker behaviour. The summarised
distribution is then used to supervise the training of the cognitive processor
by enforcing it to learn an appropriate facial reaction distribution from the
input speaker behaviour. This distribution learning strategy transforms the
ill-posed training problem, where one input speaker behaviour corresponds to
multiple appropriate listener facial reactions, into a well-posed problem,
where one input speaker behaviour corresponds to one distribution representing
all appropriate facial reactions. We illustrate our approach in Fig. 1 and
Fig. 2. The main contributions and novelties of this paper are summarised as
follows:

  * •

To the best of our knowledge, we present the first deep learning framework
capable of generating multiple appropriate, realistic, and synchronized facial
reactions in response to a speaker behaviour. Our framework introduces a novel
appropriate facial reaction distribution learning (AFRDL) strategy that
addresses the ill-posed “one-to-many mapping ” problem.This strategy
reformulates the problem as a “one-to-one mapping” problem, thus providing a
well-defined learning objective.

  * •

We propose a novel Reversible Multi-dimensional Edge Graph Neural Network
(REGNN) that can forwardly summarise a distribution from multiple real
appropriate facial reactions at the training stage, and reversely decode a
facial reaction from the predicted facial reaction distribution at the
inference stage.

  * •

We generated – using the proposed approach – appropriate, realistic, and
synchronized facial reactions achieving better performances compared to other
existing related solutions, and we provide the first open-source code for the
multiple appropriate facial reaction generation task.

##  II Related Work

###  II-A Facial reaction theory

During dyadic interactions, the facial reactions of a listener are shaped by a
combination of facial muscle movements. These movements are controlled by
person-specific cognitive processes that are primarily influenced by the
behaviours expressed by the corresponding speaker [13]. Research conducted by
Hess et al. [14] also found that the generation of facial reactions is
predominantly influenced by individual-specific cognitive processes, which are
not only influenced by the speaker’s behaviour but also by the listener’s
personality [15] and emotional states [16]. For instance, individuals who
frequently experience fear possess more sensitive and easily stimulated
amygdalae, rendering them more prone to displaying facial reactions indicative
of fear. Similarly, experiencing pleasant emotions triggers the contraction of
the zygomatic major muscle, resulting in a smiling facial reaction, while
confusion enhances the activity of the corrugator muscle, leading to a
furrowed brow expression. Therefore, as summarised in [6], in dyadic
interactions, a broad spectrum of different facial reactions might be
appropriate in response to a speaker behaviour according to the internal
states of the listener. This is because human behavioural responses are
stimulated by the context the listener experiences [3], which lead to
different but appropriate facial reactions expressed by not only different
listeners but also the same listener under different contexts (e.g., external
environments or internal states) [6, 17, 18]. A similar hypothesis has been
mentioned in a recent facial reaction generation study [7].

###  II-B Automatic facial reaction generation

To the best of our knowledge, there have been few studies [8, 9, 19, 10, 11,
20, 7, 21] on automatic facial reaction generation. An early approach [9]
proposed a two-stage conditional GAN to generate facial reaction sketches
based on the speaker’s facial action units (AUs). Their later works [8, 19]
exploited more speaker emotion-related features (e.g., facial expression
features) to reproduce better facial reactions expressed by listeners. To
consider personalized factors in expressing facial reactions, Song et al. [10,
11] used Neural Architecture Search (NAS) to explore a person-specific network
for each listener adaptively. Ng et al. [22] extended and combined the cross-
attention transformer with VQ-variational auto-encoder (VQ-VAE) [23] model to
generate a wide range of diverse facial reactions for each listener by
leveraging multi-modal speaker behaviours. Besides generating facial
reactions, some studies aiming to generate non-verbal head [21] or gesture
reactions [24]. However, none of these have attempted to generate multiple
appropriate facial reactions from a speaker behaviour [6].

Note that the approach proposed in this paper is different from previous
facial expression/display generation methods [25, 26, 27, 28, 29, 30, 31, 32],
where the facial images are generated based on manually defined conditions
such pre-defined AUs, landmarks and audio behaviours without considering
interaction scenarios (i.e., they do not predict reactions from speaker
behaviours).

##  III Generation Task definition

Given a speaker behaviour
$B_{S}^{t_{1},t_{2}}=\\{A_{S}^{t_{1},t_{2}},F_{S}^{t_{1},t_{2}}\\}$ at the
time $[t_{1},t_{2}]$, the goal is to learn a ML model $\mathcal{H}$ that can
generate multiple different spatio-temporal human facial reactions
$P(F_{L}|B_{S}^{t_{1},t_{2}})=\\{p(F_{L}|B_{S}^{t_{1},t_{2}})_{1},\cdots,p(F_{L}|B_{S}^{t_{1},t_{2}})_{N}\\}$
that are appropriate for responding to $B_{S}^{t_{1},t_{2}}$, which is
formulated as:

$P(F_{L}|B_{S}^{t_{1},t_{2}})=\mathcal{H}(B_{S}^{t_{1},t_{2}}),$ (1)

where
$P(F_{L}|B_{S}^{t_{1},t_{2}})=\\{p(F_{L}|B_{S}^{t_{1},t_{2}})_{1}\neq\cdots\neq
p(F_{L}|B_{S}^{t_{1},t_{2}})_{N}\\}$ and each generated facial reaction
$p(F_{L}|B_{S}^{t_{1},t_{2}})_{n}$ should be similar to at least one real
facial reaction $f_{L}(B_{S}^{t_{1},t_{2}})_{m}$ that is appropriate in
response to $B_{S}^{t_{1},t_{2}}$ in the training set:

$p(F_{L}|B_{S}^{t_{1},t_{2}})_{n}\approx f_{L}(B_{S}^{t_{1},t_{2}})_{m}\in
F_{L}(B_{S}^{t_{1},t_{2}}),$ (2)

where
$F_{L}(B_{S}^{t_{1},t_{2}})=\\{f_{L}(B_{S}^{t_{1},t_{2}})_{1},\cdots,f_{L}(B_{S}^{t_{1},t_{2}})_{M}\\}$
denotes a set of real facial reactions expressed by listeners in the training
set, which are appropriate in response to the speaker behaviour
$B_{S}^{t_{1},t_{2}}$. The above definition corresponds to the offline facial
reaction generation task defined by [6] (See[6] for details).

Figure 2: Overview of the proposed multiple appropriate facial reaction
generation framework. Step 1: the Perceptual Processor first encodes facial
and audio representations from the perceived audio-visual speaker behaviours.
Step 2: the Cognitive Processor then predicts a distribution from the combined
audio-visual representation, which represents all appropriate facial reactions
in response to the input speaker behaviour. Step 3: the REGNN-based Motor
Processor finally samples and reversely decodes multiple appropriate facial
reactions from the learned distribution.

##  IV The proposed approach

This section presents our novel FRG approach. We first introduce the pipeline
in Sec. IV-A, and then detail the proposed appropriate facial reaction
distribution learning (AFRDL) strategy in Sec. IV-B. Finally, we provide the
details of the novel REGNN in Sec. IV-C, which represents one of the main
components of the AFRDL strategy.

###  IV-A Facial reaction generation framework

This section develops an appropriate FRG model,
$\mathcal{H}=\\{\textbf{Enc},\textbf{Cog},\textbf{Mot}\\}$, which can generate
multiple diverse and appropriate human facial reactions in response to each
speaker audio-facial behaviour. As shown in Fig. 2, our model consists of
three main modules inspired by the Human Model Processor (HMP) [12]: (i)
Perceptual Processor
($\textbf{Enc}=\\{\textbf{Enc}_{\text{A}},\textbf{Enc}_{\text{F}}\\}$) that
encodes each raw speaker audio and facial behaviours into a pair of latent
audio and facial representations; (ii) Cognitive Processor (Cog) that predicts
a distribution representing all appropriate listener facial reactions based on
the produced speaker audio and facial representations; and (iii) the REGNN-
based Motor Processor (Mot) that samples and generates an appropriate facial
reaction from the predicted distribution. We also illustrate the model
pipeline in Fig. 2.

Specifically, the Perceptual Processor is a two branch encoder consisting of a
facial encoder $\textbf{Enc}_{\text{F}}$ (Swin-Transformer [33]) and an audio
encoder $\textbf{Enc}_{\text{A}}$ (VGGish [34]). It takes a speaker’s audio
and facial signals $A_{S}^{t_{1},t_{2}}$ and $F_{S}^{t_{1},t_{2}}$ expressed
at the time interval $[t_{1},t_{2}]$ as the input, and generates a pair of
latent audio and facial representations $\bar{A}_{S}^{t_{1},t_{2}}$ and
$\bar{F}_{S}^{t_{1},t_{2}}$ as:

$\begin{split}\bar{A}_{S}^{t_{1},t_{2}}=\textbf{Enc}_{\text{A}}(A_{S}^{t_{1},t_{2}}),\\\
\bar{F}_{S}^{t_{1},t_{2}}=\textbf{Enc}_{\text{F}}(F_{S}^{t_{1},t_{2}}).\end{split}$
(3)

Figure 3: Illustration of the proposed AFRDL strategy (Pseudo-code is provided
in the supplementary material). Given a speaker behaviour, REGNN first encodes
all appropriate real facial reactions as a set of latent graph
representations. These representations are then summarised as an appropriate
real facial reaction distribution to supervise the Cognitive Processor’s
training, where the summarised distribution is a graph representation
consisting of multiple nodes, and each node is represented by a Gaussian
Mixture Model (GMM) summarising multiple facial attribute time-series
corresponding to multiple appropriate real facial reactions. Here, the MSE
loss function is employed to enforce the distribution predicted by the
Cognitive Processor to be similar to the summarised real facial reaction
distribution.

Based on $\bar{A}_{S}^{t_{1},t_{2}}$ and $\bar{F}_{S}^{t_{1},t_{2}}$, the
Cognitive Processor first aligns and combines them as a latent speaker audio-
facial behaviour representation $\bar{B}_{S}^{t_{1},t_{2}}$ based on the same
attention-based strategy introduced in [35]. Rather than predicting a specific
facial reaction from $\bar{B}_{S}^{t_{1},t_{2}}$, we propose to predict an
appropriate facial reaction distribution graph representation
$\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$ from $\bar{B}_{S}^{t_{1},t_{2}}$, as
multiple facial reactions may be appropriate for responding to each input
speaker behaviour. This means that $Z_{p}(B_{S}^{t_{1},t_{2}})$ represents the
distribution of all appropriate facial reactions. This is achieved by $I$
projection heads (i.e., $I$ fully connected (FC) layers), where each head
learns a $D$-dimensional vector that is specifically treated as a node feature
for $\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$. These processes can be formulated as:

$\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})=\textbf{COG}(\bar{B}_{S}^{t_{1},t_{2}}),$
(4)

where $\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})\in\mathbb{R}^{I\times P}$ ($I$ nodes).
This way, the “one-to-many mapping” problem occurring in the FRG task is
addressed by re-formulating it into a ’one-to-one mapping’ problem at the
training stage (one speaker behaviour-to-one distribution representing
multiple appropriate facial reactions). Then, we feed all node features to our
multi-dimensional edge feature learning (MEFL) block that consists of $D$
attention operations, where each attention operation generates an $I\times I$
attention map describing a specific type of mutual relationship between a pair
of nodes. Consequently, $D$ attention maps describing $D$ types of
relationship cues would be produced. Thus, a pair of multi-dimensional
directed edge features can be obtained to describe the relationship between
each pair of nodes, i.e., each multi-dimensional edge feature
$e^{\bar{Z}_{p}}_{i,j}$ is a $D$ dimensional vector produced by concatenating
the values at the $i_{\text{th}}$ row and $j_{\text{th}}$ column of $D$
attention maps. Here, we only keep $\mathcal{K}$ directed edges starting from
each node, which has largest norms.

Since human facial behaviours can be interpreted by various medium and high
level primitives (e.g., facial Action Units (AUs) and affects) which are
usually mutually correlated [36]), we propose to describe each spatio-temporal
facial reaction by $I$ facial attribute time-series primitives, which are
represented as a graph consisting of $I$ nodes, where the presence of each
directed edge and its multi-dimensional edge feature is also defined by a MEFL
block. Consequently, the relationship between each pair of facial attribute
time-series (nodes) can be explicitly described by a pair of multi-dimensional
edge features. To allow such facial reaction graph representation to be
generated from the distribution $\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$, we propose
a novel Reversible Multi-dimensional Edge Graph Neural Network (REGNN) as the
Motor Processor. It first samples an appropriate facial reaction latent graph
representation $\bar{g}_{p}(B_{S}^{t_{1},t_{2}})_{n}\in\mathbb{R}^{I\times D}$
from the predicted distribution $\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$, and then
decodes it as a facial reaction graph representation
$\hat{g}_{p}(B_{S}^{t_{1},t_{2}})_{n}\in\mathbb{R}^{I\times D}$. Here,
different facial reaction latent graph representations can be sampled from
$\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$, and thus multiple different facial
reactions
$\hat{G}_{p}(B_{S}^{t_{1},t_{2}})=\\{\hat{g}_{p}(B_{S}^{t_{1},t_{2}})_{1},\cdots,\hat{g}_{p}(B_{S}^{t_{1},t_{2}})_{N}\\}$
can be generated. This can be formulated as:

$\hat{g}_{p}(B_{S}^{t_{1},t_{2}})_{n}=\textbf{Mot}^{-1}(Z_{p}(B_{S}^{t_{1},t_{2}})),$
(5)

where $\text{Mot}^{-1}$ denotes that motor processors that reversely infers
facial reactions from the predicted distribution $Z_{p}(B_{S}^{t_{1},t_{2}})$.

Subsequently, a 2D facial reaction sequence $p(F_{L}|B_{S}^{t_{1},t_{2}})_{n}$
can be produced from $\hat{g}_{p}(B_{S}^{t_{1},t_{2}})_{n}$. To train the
Cognitive Processor that can accurately predict the distribution of all
appropriate facial reactions in response to each input speaker behaviour, we
propose a novel Reversible Multi-dimensional Edge Graph Neural Network (REGNN)
as the motor processor, which learns an appropriate real facial reaction
distribution graph representation $Z_{L}(B_{S}^{t_{1},t_{2}})$ for each
speaker behaviour $B_{S}^{t_{1},t_{2}}$, representing all listeners’ real
facial reactions that are appropriate as the corresponding facial reaction to
$B_{S}^{t_{1},t_{2}}$.

###  IV-B Appropriate facial reaction distribution learning

Our facial reaction distribution generation strategy aims to address the “one-
to-many mapping” problem occurring in FRG models’ training (i.e., one input
speaker behaviour corresponds to multiple appropriate facial reaction labels)
by re-formulating it as a “one-to-one mapping” problem (i.e., one input
speaker behaviour corresponds to one distribution representing multiple
appropriate facial reactions).

As shown in Fig. 3, given an audio-visual speaker behaviour
$B_{S}^{t_{1},t_{2}}=\\{A_{S}^{t_{1},t_{2}},F_{S}^{t_{1},t_{2}}\\}$, and its
corresponding multiple appropriate real facial reactions
$F_{L}(B_{S}^{t_{1},t_{2}})$ expressed by human listeners of the training set,
we first construct a set of real facial reaction graph representations
$G_{L}(B_{S}^{t_{1},t_{2}})=\\{g_{L}(B_{S}^{t_{1},t_{2}})_{1},\cdots,g_{L}(B_{S}^{t_{1},t_{2}})_{M}\\}$
to represent all appropriate real facial reactions defined by
$F_{L}(B_{S}^{t_{1},t_{2}})$, where each node in a graph representation
describes a facial attribute time-series and each edge explicitly describes
the relationship between a pair of nodes. At the same time, the presence of
each directed edge and its multi-dimensional edge feature is also defined by a
MEFL block. Since these representations have the same property, i.e., each
describes an appropriate facial reaction in response to $B_{S}^{t_{1},t_{2}}$,
we hypothesize that they are drawn from the same distribution. Subsequently,
we train the REGNN by enforcing it to map all appropriate real facial reaction
graph representations in response to the same speaker behaviour onto a
’ground-truth’ (GT) real appropriate facial reaction distribution
$Z_{L}(B_{S}^{t_{1},t_{2}})$ as:

$\begin{split}&\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{m}=\text{Mot}(g_{L}(B_{S}^{t_{1},t_{2}})_{m}),\\\
&\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{m}\sim Z_{L}(B_{S}^{t_{1},t_{2}}),\quad
m=1,2,\cdots M\\\ &\textbf{subject to}~{}~{}f_{L}(B_{S}^{t_{1},t_{2}})_{m}\in
F_{L}(B_{S}^{t_{1},t_{2}}),\end{split}$ (6)

where $\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{m}$ denotes a latent graph
representation produced from $g_{L}(B_{S}^{t_{1},t_{2}})_{m}$, and all latent
graph representations are expected to follow the same distribution
$Z_{L}(B_{S}^{t_{1},t_{2}})$. This is achieved by minimizing the sum of L1
distances obtained from all the corresponding latent graph representation
pairs in an unsupervised manner:

$\mathcal{L}_{1}=\sum_{m_{1}=1}^{M-1}\sum_{m_{2}=m_{1}+1}^{M}L1(\hat{g}_{L}(B_{S}^{t_{1},t_{2}})_{m_{2}},\hat{g}_{L}(B_{S}^{t_{1},t_{2}})_{m_{1}}).$
(7)

Inspired by the fact that Guassian Mixture Model (GMM) is powerful to describe
distributed subpopulations (e.g., individual appropriate facial reactions)
within an overall population (e.g., all appropriate facial reactions), we
propose a novel Gaussian Mixture Graph Distribution (GMGD) to represent
$Z_{L}(B_{S}^{t_{1},t_{2}})=\\{v_{1}^{Z},v_{2}^{Z},\cdots,v_{I}^{Z}\\}$, where
each node $v_{i}^{Z}$ in $Z_{L}(B_{S}^{t_{1},t_{2}})$ is represented by a
Gaussian Mixture Model (GMM) consisting of $M$ Gaussian distributions (defined
as
$\mathcal{N}(\\{\mu^{1}_{i},\cdots,\mu^{M}_{i}\\},\\{\sigma^{1}_{i},\cdots,\sigma^{M}_{i}\\})$).
For each node, the mean values $\mu^{1}_{i},\cdots,\mu^{M}_{i}$ of $M$
Gaussian distributions are defined by the $i_{\text{th}}$ node features
$\bar{v}(L)_{i}^{1},\cdots,\bar{v}(L)_{i}^{M}$ of the corresponding $M$ latent
graph representations
$\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{1},\cdots,\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{M}$
produced by the motor processor, while standard deviations
$\\{\sigma(L)^{1}_{i},\cdots,\sigma(L)^{M}_{i}\\})$ are empirically defined.
These can be formulated as:

$\begin{split}&\mu^{m}_{i}=v_{i}^{m}\in\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{m},~{}~{}m=1,2,\cdots,M\\\
\end{split}$ (8)

where $v_{i}^{m}$ is the $i_{\text{th}}$ node in the $m_{\text{th}}$ latent
graph representation $\bar{g}_{L}(B_{S}^{t_{1},t_{2}})_{m}$.

As a result, the Cognitive Processor is trained under the supervision of the
GT real facial reaction distribution $Z_{L}(B_{S}^{t_{1},t_{2}})$, i.e., it is
trained to predict an appropriate facial reaction distribution graph
representation $\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$ from $B_{S}^{t_{1},t_{2}}$
(formulated in Eq. (4)). This training process is achieved by minimizing the
L2 distance between $Z_{L}(B_{S}^{t_{1},t_{2}})$ and
$\bar{Z}_{L}(B_{S}^{t_{1},t_{2}})$ as:

$\begin{split}\mathcal{L}_{2}=\text{MSE}(\bar{Z}_{p}(B_{S}^{t_{1},t_{2}}),Z_{L}(B_{S}^{t_{1},t_{2}}))\end{split}$
(9)

where MSE denotes the Mean Square Error. At the inference stage, the well-
trained motor processor first samples a facial reaction graph representation
$\bar{g}_{p}(B_{S}^{t_{1},t_{2}})_{n}$ from the distribution
$\bar{Z}_{p}(B_{S}^{t_{1},t_{2}})$ predicted by the Cognitive Processor, and
then reversely decodes it as facial reaction.

###  IV-C Reversible Multi-dimensional Edge Graph Neural Network

The REGNN can forwardly encode a GT facial reaction distribution to describe
all appropriate real facial reactions in response to the input speaker
behaviour at the training stage, which plays a key role in supervising the
cognitive processor’s training. As shown in Fig. 4, the REGNN network consists
of $N$ REGNN layers, which forwardly takes a graph
$\mathcal{G}^{0}(\mathcal{V}^{0},\mathcal{E}^{0})$ as the input and generates
a graph $\mathcal{G}^{N}(\mathcal{V}^{N},\mathcal{E}^{N})$ as the output,
where $\mathcal{V}^{0}=\\{v_{1}^{0},v_{2}^{0},\cdots,v_{I}^{0}\\}$ and
$\mathcal{E}=\\{e_{i,j}^{0}|v_{i}^{0},v_{j}^{0}\in\mathcal{V}\quad\&\quad\mathcal{A}_{i,j}=1\\}$
denote a set of node and edge features contained in the input graph
$\mathcal{G}^{0}(\mathcal{V}^{0},\mathcal{E}^{0})$, respectively, and
$\mathcal{A}$ is the adjacency matrix that defines the connectivity between
nodes. Importantly, the REGNN can also reversely output
$\mathcal{G}^{0}(\mathcal{V}^{0},\mathcal{E}^{0})$ from
$\mathcal{G}^{N}(\mathcal{V}^{N},\mathcal{E}^{N})$. In this paper, since only
node features represent the target facial attributes/distributions, the REGNN
is designed for node feature reasoning. In the following, we present the
details of both the forward propagation and reverse propagation mechanisms of
the REGNN.

Figure 4: The proposed Reversible Multi-dimensional Edge Graph Neural Network
(REGNN) is made up of $N$ REGNN layers, where each layer can forwardly and
reversely propagate node representations. Pseudo-code of its propagation is
provided in the supplementary material.

Forward propagation: The $n_{\text{th}}$ REGNN layer takes: (i) the node
feature set $\mathcal{V}^{n-1}$ generated from the ${n-1}_{th}$ layer as its
input node features, which is pre-processed by a normalization layer and a
Sigmoid activation;; and (ii) the initial edge feature set $\mathcal{E}^{0}$
as its input edge features, and then outputs a graph
$\mathcal{G}^{n}(\mathcal{V}^{n},\mathcal{E}^{n})$. This setting ensures the
reversibility of the REGNN (explained and derived in the Supplementary
Material). Specifically, the $n_{\text{th}}$ REGNN layer first learns a set of
edge features $\mathcal{E}^{n}$ based on not only $\mathcal{E}^{0}$ but also
$\mathcal{V}^{n-1}$, allowing $\mathcal{V}^{n-1}$-related edge features to be
used for updating $\mathcal{V}^{n-1}$ to $\mathcal{V}^{n}$, i.e., it computes
each directed edge feature $e^{n}_{j,i}\in\mathcal{E}^{n}$ starting from the
node $v_{j}^{n-1}$ to $v_{i}^{n-1}$ as:

$e^{n}_{j,i}=\frac{a^{n}_{j,i}e^{0}_{j,i}}{\sum_{v^{n-1}_{k}\in\mathcal{N}_{v^{n-1}_{i}}}a^{n}_{k,i}e^{0}_{k,i}},$
(10)

where $e^{0}_{j,i}\in\mathcal{E}_{0}$ denotes an initial directed edge
feature. The employed $a^{n}_{j,i}$ is a learnable relationship coefficient
that captures correlations between the node $v^{n-1}_{i}$ and its high-order
neighboring nodes to define $\mathcal{V}^{n-1}$-related and context-aware
(i.e., aware of the high-order neighbors) edge feature $e^{n}_{j,i}$, which
contributes to updating $v^{n-1}_{j}$ to $v^{n}_{j}$. The term
$\sum_{v^{n-1}_{k}\in\mathcal{N}_{v^{n-1}_{i}}}a^{n}_{k,i}e^{0}_{k,i}$
regularizes the obtained edge feature. Here, the $a_{j,i}^{n}$ can be computed
as:

$a_{j,i}^{n}=\frac{\mathrm{exp}(v_{i}^{n-1}\mathbf{W}_{q}^{n})(v_{j}^{n-1}\mathbf{W}_{m})^{\top}}{\sum_{v^{n-1}_{k}\in\mathcal{N}_{v^{n-1}_{i}}}\mathrm{exp}\left((v_{i}^{n-1}\mathbf{W}_{q}^{n})(v^{n-1}_{k}\mathbf{W}_{m}^{n})^{\top}\right)},$
(11)

where $\mathbf{W}_{q}$ and $\mathbf{W}_{m}$ are learnable weight vectors and
$\mathcal{N}_{i}$ denotes the adjacent node set of $v_{i}^{n-1}$. This way,
the learned $a_{j,i}^{n}$ captures information from not only the relationship
cues between the corresponding node features $v^{n-1}_{j}$ and $v^{n-1}_{i}$
but also the context of the node $v^{n-1}_{i}$ (i.e., its high-order
neighboring nodes $\mathcal{N}_{v^{n-1}_{i}}$).

Building upon these learned edge features, the REGNN layer then update each
node feature $v_{i}^{n}$ based on: (i) node feature
$v_{i}^{n-1}\in\mathcal{G}^{n-1}$ itself; and (ii) the message
$\hat{v}_{i}^{n-1}$ aggregated from all its adjacent node, which is decided by
its adjacent node feature set ($\mathcal{N}_{v_{i}^{n-1}}$) and the
corresponding updated directed edge features $e^{n}_{j,i}\in\mathcal{E}^{n}$
that point to $v_{i}^{n-1}$ as:

$v_{i}^{n}=v_{i}^{n-1}+{\hat{v}}_{i}^{n-1},\\\ $ (12)

where the message $\hat{v}_{i}^{n-1}$ is computed as:

$\hat{v}_{i}^{n-1}=\mathbf{W}_{e}^{n}\sum_{v^{n-1}_{j}\in\mathcal{N}_{v_{i}^{n-1}}}\left(e_{j,i}^{n}\circ
v^{n-1}_{j}\right),$ (13)

where $\mathbf{W}_{e}^{n}\in\mathbb{R}^{1\times D}$ is a learnable weight
vector that combines messages passed by all dimensions ($D$ dimensions) of the
multi-dimensional edge $e_{j,i}^{n}$.

Reverse propagation: The proposed REGNN layer is able to reversely infer each
input node feature $v_{i}^{n-1}$ from its output node features $v_{i}^{n}$,
which can be formulated as:

$\displaystyle x_{k}=v^{n}_{i}-\varphi(x_{k-1}),$ (14) $\displaystyle
v^{n-1}_{i}=x_{k},\quad\quad\textbf{subject to}\quad\quad x_{k}=x_{k-1},$

where the function $\varphi$ is defined as the combination of the Sigmoid
activation, edge updating (Eqa. 10) and message passing functions (Eqa. 13);
$x_{k}$ is computed iteratively until it is converged to $x_{k}=x_{k-1}$.
Here, the $x_{0}$ is set as a non-zero random value. To achieve the
aforementioned reversibility (i.e., converged at the $x_{k}=x_{k-1}$), which
is the key for decoding an appropriate facial reaction from the predicted
distribution, the $\mathbf{W}_{e}^{n}$ defined in Eq. (13), as well as
$W_{q}^{n}$ and $W_{k}^{n}$ are optimized to enable the function $varphi$ used
for both forward and reverse propagation to satisfy the following condition:

$\varphi(v_{i}^{n-1})=\frac{\varphi(\text{Sig}(v_{i}^{n-1}))}{1+2\|\
\mathbf{W}_{q}^{n}\mathbf{W_{k}^{n}}^{\top}\|_{2}},$ (15)

where Sig denotes the Sigmoid activation function. This setting allows the
function $\varphi$ to be a contraction mapping function that satisfies the
Lipschitz continuous and Lipschitz constant less than $1$ [37, 38], and thus
adhering to the fixed point theorem, i.e., the function $\varphi$ is learned
to ensure a fixed point $x$ to satisfy the equation $x=v_{i}^{n}-\varphi(x)$.
The proof and derivation for defining Eq. (15) for ensuring the reversibility
of the proposed REGNN are provided in the supplementary material.

##  V Experiments

This section first provides the details of experimental settings in Sec. V-A,
then, Sec. V-B compares our approach with previous facial reaction generation
solutions. Finally, we conduct a series of ablation studies to investigate the
contributions of different modules.

###  V-A Experimental settings

Dataset. This paper evaluates the proposed approach based on the video
conference clips recorded under various dyadic interaction settings, which are
provided by two publicly available datasets: NoXI [39] and RECOLA [40]. As
there are only three valid video pairs in RECOLA dataset, we follow a public
facial reaction challenge 111https://sites.google.com/cam.ac.uk/react2023/home
to combine these two datasets, resulting in 2962 pairs of audio-visual
speaker-listener dyadic interaction clips, including 1594 pairs of training
clips, 562 pairs of validation clips, and 806 pairs of test clips, where each
clip is 30 seconds long. All appropriateness labels are obtained by the same
strategy provided in [6]. Note that the employed dataset is different from the
challenge, as the UDIVA dataset [41] is not included because it is recorded
under in-person dyadic interactions (with a lateral camera view which captures
both participants), where the profile of participants’ faces are frequently
recorded.

Implementation details. In our experiments, the Perceptual Processor takes the
facial image sequence of an entire video clip and log-mel spectrogram features
of an entire audio clip as input. At the training stage, we employ the Swin-
Transformer [42] pre-trained on FER2013 [43] and pre-trained Vggish model [34]
as the initial facial and audio feature extraction models. Then, the Adam
optimizer [44] with a stochastic optimization strategy was employed to train
the entire framework in an end-to-end manner using an initial learning rate of
$10^{-4}$ and weight decay of $5\times 10^{-4}$. The maximum number of epochs
was set to $100$, with learning rate decay performed at the $20^{th}$ and
$50^{th}$ epochs, each with a decay rate of $0.1$. During the inference stage,
we empirically set the $\sigma$ of all GMM in the GMGD as $0.6$, based on
which the REGNN reversely samples and decodes the facial reactions. In this
paper, we use the model proposed in [25] to generate facial images from all
predicted AUs.

Evaluation metrics. We employed the four sets of metrics defined in [6] to
evaluate the generated facial reactions, namely in terms of: (i)
Appropriateness: the distance and correlation between generated facial
reactions and their most similar appropriate facial reaction using FRDist and
FRCorr. In addition, we also report the Pearson Correlation Coefficient (PCC)
between predictions and their most similar appropriate real facial reaction;
(ii) Diversity: the variation among 1) frames in each generated facial
reaction (FRVar), 2) multiple facial reactions generated from the same speaker
behaviour (FRDiv), and 3) facial reactions generated for different speaker
behaviours (FRDvs); (iii) Realism: employing Fréchet Inception Distance (FID)
used in [6] (FRRea); and (iv) Synchrony is measured by Time Lagged Cross
Correlation (TLCC) between the speaker facial behaviour and the corresponding
generated facial reaction.

###  V-B Comparison to related works

Figure 5: Visualisation of the facial reactions generated from different
approaches, where early approaches [8, 9, 45] generated some very low-quality
facial images, while the predictions of a recent approach [7] is quite
different from the ground-truth (i.e., low appropriateness and synchrony). Our
approach generated multiple diverse but appropriate, realistic, and
synchronized facial reactions from the input speaker behaviour. Figure 6:
Examples of the low-quality facial images generated from the facial reaction
attributes predicted by competitors.

As this represents the first study aiming to generate multiple appropriate
facial reactions, we compare our approach with different reproduced baselines
that have been previously used for generating facial reactions:

  * •

Huang et al. (S) [8]: This method employs a two-stage conditional GAN. In the
first stage, feature vectors composed of $17$ different facial action units of
the speaker’s facial behaviour are used as the condition to predict the
listener’s facial landmarks. Then, in the second stage, the obtained facial
landmarks are employed as the condition to generate the corresponding facial
reaction sequence. Here, the speaker’s facial behaviour from time $t-9$ to $t$
is used to predict the $t_{\text{th}}$ facial reaction frame. Consequently,
this method cannot predict the first ten facial reaction frames. Therefore, in
our experiments, its input size is set to $(750,25)$ and the output size is
set to $(740,25)$.

  * •

Huang et al. (C) [9]: Building upon [8], this method utilizes a feature vector
composed of $8$ distinct facial expressions of the speaker behaviour as a
conditioning input. The final facial reaction is generated using a two-stage
GAN similar to [9]. As both [8] and [9] require the inputs from time $t-9$ to
$t$ to predict the outputs at time $t$, they cannot make predictions for the
first ten frames of the listener’s facial reaction. Therefore, in our
experiments, their input size is set to $(750,25)$, and the output is set to
$(740,25)$.

  * •

UNet [45]: UNet is a widely-used model in generative tasks, consisting of the
symmetric encoder and decoder components. In this experiment, both the encoder
and decoder are set to four layers. The input and output of the network are
both time series data with a shape of $(750,25)$.

  * •

Ng et al. [7]: This approach trains a VQ-VAE model to predict a discrete
encoding from the input speaker behaviour, representing the corresponding
listener’s facial reaction. At the inference stage, a transformer-based
predictor module is utilized to perform a lookup on the discrete encoding to
predict the listener’s facial reaction. In our experiments, we customized the
size of hidden layers in the model to accommodate the input sequence size of
$(750,25)$ and output size of $(750,25)$.

The implementation details of these reproduced approaches are provided in the
supplementary material.

TABLE I: Comparison between the proposed approach and several reproduced existing related works. Methods | FRDist $\downarrow$ | FRCorr $\uparrow$ | PCC $\uparrow$ | FRRea $\downarrow$ | FRVar $\uparrow$ | FRDvs $\uparrow$ | FRDiv $\uparrow$ | Synchrony $\downarrow$
---|---|---|---|---|---|---|---|---
Huang et al. (S) [8] | 14.91 | 0.023 | 0.045 | 42.44 | 0.092 | 0.186 | 0.181 | 42.83
Huang et al. (C) [9] | 11.99 | 0.035 | 0.058 | 32.53 | 0.054 | 0.114 | 0.102 | 43.83
UNet [45] | 8.31 | 0.067 | 0.141 | 23.81 | 0.015 | 0.049 | 0.0 | 40.19
Ng et al. [7] | 10.36 | 0.082 | 0.146 | 27.33 | 0.075 | 0.064 | 0.061 | 39.22
Ours | 7.62 | 0.106 | 0.153 | 21.58 | 0.077 | 0.121 | 0.048 | 38.76

Table I shares the results achieved in our experiments demonstrates that our
approach outperforms all competitors in generating appropriate, realistic, and
synchronized facial reactions, as indicated by lower FRDist distances,
Realism, Synchrony values, and higher correlations with the most similar real
facial reactions. Specifically, our approach achieves improvements of $4.37$
and $2.74$ in FRDist, $0.071$ and $0.024$ in FRCorr, and $0.095$ and $0.007$
in PCC over the condition GAN-based approach [9] and the recently proposed VQ-
VAE based approach [7], respectively. We also compare our predictions with
predictions achieved by [7] in Fig. 7. Furthermore, our approach generates
diverse facial reactions in response to different speaker behaviours, as well
as decent diversity among frames of each generated facial reaction. Our
approach can also generate different facial reactions in response to each
speaker behaviour (visualized in Fig. 5). These results demonstrate the
effectiveness of our AFRDL strategy in generating multiple different but
appropriate, realistic, and synchronized facial reactions. It should be noted
that while our approach did not generate facial reactions with as much
diversity as C-GAN based [8, 9] and VQ-VAE based approaches [7], such high
diversity results are partially associated to the generation of abnormal
facial reactions (i.e., facial behaviours that are not appropriate or cannot
be properly expressed by humans (illustrated in Fig. 6)), which is reflected
by their much worse appropriateness, realism, and synchrony performances.

Figure 7: Visualisation of the facial attribute predictions. Although our
approach is trained to generate appropriate facial reactions, the facial
attributes predicted by our approach are still highly correlated with the GT
real facial reaction.

###  V-C Ablation studies

In this section, we first conduct a series of ablation studies to evaluate the
effectiveness/importance of (i) each modality of the speaker behaviour; (ii)
the proposed appropriate facial reaction distribution learning (AFRDL)
strategy; (iii) the proposed reversible graph model (REGNN); and (iv) the
multi-dimensional edge feature learning (MEFL) module. We also provide the
sensitivity analysis for two main variables: (i) $\sigma$ used in defining the
Gaussian Mixture Graph Distribution (GMGD) $Z_{L}(B_{S}^{t_{1},t_{2}})$; and
(ii) the dimension $D$ of the multi-dimensional edge feature output from the
MEFL module, in the Supplementary Material.

Contributions of different modalities. The experimental results presented in
Table II reveal that both audio and facial modalities of the speaker behaviour
offer valuable cues for generating appropriate facial reactions. In
particular, the facial behaviours of the speaker exhibit a greater impact on
the performance, with better performances in terms of appropriateness,
diversity, and realism. Since feeding both audio and visual speaker behaviours
results in the best performance, it suggests that audio and visual cues from
the speaker behaviour are complementary and relevant for the generation of
appropriate facial reactions. Importantly, the proposed approach outperforms
several existing methods [8, 9, 7] even when either audio or visual modality
of the speaker behaviour alone is utilized, further validating our proposed
framework’s effectiveness.

TABLE II: Results achieved for four ablation studies. | Methods | FRDist $\downarrow$ | FRCorr $\uparrow$ | PCC $\uparrow$ | FRRea $\downarrow$ | FRVar $\uparrow$ | FRDvs $\uparrow$ | FRDiv $\uparrow$ | Synchrony $\downarrow$
---|---|---|---|---|---|---|---|---|---
Modalities | Video | 8.33 | 0.101 | 0.129 | 24.97 | 0.013 | 0.026 | 0.052 | 38.62
Audio | 8.81 | 0.031 | 0.062 | 30.54 | 0.023 | 0.047 | 0.081 | 40.33
Audio+Video | 7.62 | 0.106 | 0.153 | 21.58 | 0.077 | 0.121 | 0.048 | 38.76
AFRDL | Without Graph | 11.6 | 0.048 | 0.070 | 29.71 | 0.075 | 0.128 | 0.076 | 42.43
With AFRDL | 7.62 | 0.106 | 0.153 | 21.58 | 0.077 | 0.121 | 0.048 | 38.76
Motor processor | i-ResNet [46] | 15.52 | 0.051 | 0.077 | 35.20 | 0.090 | 0.193 | 0.169 | 42.12
REGNN | 7.62 | 0.106 | 0.153 | 21.58 | 0.077 | 0.121 | 0.048 | 38.76
Edge feature | Without MEFL | 8.97 | 0.045 | 0.096 | 20.33 | 0.031 | 0.110 | 0.062 | 38.91
With MEFL | 7.62 | 0.106 | 0.153 | 21.58 | 0.077 | 0.121 | 0.048 | 38.76

Appropriate facial reaction distribution learning (AFRDL) strategy. Table II
reports a comparative analysis of the effectiveness of the proposed AFRDL
strategy, where we developed a variant of our framework with the same
architecture but different training strategy. The variant was trained using
MSE loss, which minimizes the difference between the generated facial reaction
and the GT real facial reaction of the input speaker behaviour. The results
indicate that the proposed AFRDL strategy is crucial in generating high-
quality facial reactions, as the variant achieved significantly worse
performance in terms of appropriateness, realism, and synchrony.

Figure 8: Visualisation of the learned distributions. It is clear that the
distribution learned by the proposed REGNN (depicted in (b)) are more
discriminative than i-ResNet (depicted in (a)).

REGNN vs. reversible CNN. Table II and Fig. 8 also compare the performance
achieved by the proposed reversible GNN (REGNN) with a widely-used reversible
CNN (i-ResNet [46]) to investigate the effectiveness of our REGNN, while
keeping the rest of the framework and training strategy unchanged. The results
demonstrate that the REGNN-based systems (i.e., single-value edge graph-based
system and multi-dimensional edge graph-based system) outperform the i-ResNet-
based system with substantial improvements across all metrics of
appropriateness, realism, and synchrony. As discussed in Sec. V-B, the
i-ResNet-based system sometimes generates abnormal facial reactions, which may
account for its better diversity performance (illustrated in Fig. 6). In other
words, the proposed REGNN predicts more appropriate facial reactions and
allows the generated facial image sequences to be more realistic. We
hypothesis that this is because the REGNN-based system can explicitly
represents the task-specific relationship between each pair of facial
attributes in the form graph representation, and thereby they can be better
modeled.

MEFL module. Finally, we found that the proposed MEFL module also provides
clear improvement for facial reaction generation. As seen in Table II, the
additional usage of the multi-dimensional edge features generated by our MEFL
module provides large improvements in terms of all appropriateness metrics
(i.e., $15\%,135\%$, and $59\%$ relative improvements in FRDist, FRCorr, and
PCC, respectively) as well as two diversity metrics (i.e., FRVar and FRDvs),
showing that the task-specific relationship between facial attributes is
complex, and thus could be better modeled by multi-dimensional edge features
rather than single-value edge features. In contrast, the multi-dimensional
edge features only have small impacts on the generated facial reactions’
realism and synchrony performances.

##  VI Conclusion

In this paper, we propose the first deep learning-based framework that opens
up a new avenue of research for predicting multiple appropriate human facial
reactions to a speaker behaviour. It is the first work that reformulates the
“one-to-many mapping” problem occurring when training FRG models as a ‘one-to-
one mapping’ problem (one speaker behaviour-to-one distribution representing
multiple appropriate facial reactions), where a novel reversible GNN (REGNN)
and a multiple appropriate facial reaction distribution learning (AFRDL)
strategy are proposed. As the first specifically designed multiple appropriate
FRG model, experimental results show that: (i) our approach can generate
multiple diverse but appropriate, realistic, and synchronized facial reactions
in response to each speaker behaviour, and achieve greater performance in
appropriateness, realism, and synchrony metrics as compared to all the
reproduced existing works; (ii) the proposed REGNN-based facial reaction
distribution learning contributes substantially to the promising
appropriateness, realism, and synchrony performances achieved by our approach;
(iii) both audio and facial speaker behaviours provide relevant and
complementary information; (iv) the proposed REGNN is crucial for the success
of the AFRDL strategy; and (v) the MEFL module is crucial in generating
appropriate facial reactions, as multi-dimensional edge features can
comprehensively model task-specific relationships among facial attributes.

Limitations and future work: As the first multiple appropriate FRG framework,
this paper only predicted reactions based on speakers’ non-verbal behaviours,
which did not achieved very decent performance. Another limitation is that the
facial reaction distribution of different speaker behaviours sometimes are
similar and thus not discriminative. Our future work will focus on (i)
developing more advanced generative algorithms; (ii) considering both verbal
and non-verbal behaviours of speakers; and (iii) investigating better ways to
represent appropriate facial reaction distributions.

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# First-order formalism for non-Abelian string

E. Acalapati<EMAIL_ADDRESS>H. S. Ramadhan111Corresponding author.
<EMAIL_ADDRESS>Departemen Fisika, FMIPA, Universitas Indonesia, Depok,
16424, Indonesia.

###### Abstract

We construct a method of obtaining the Bogomolny equations for general non-
Abelian multi-field vortices. This is done by generalizing the well-known
first-order formalism to the case of non-Abelian strings. We do not assume any
specific gauge group nor the shape of the kinetic term function, but require
only that the fields are axially-symmetric and static. With this formalism we
reproduce the BPS equations of $SU(2)\times U(1)$ Alice strings
Chatterjee:2017jsi , and present their corresponding numerical solutions.

##  I Introduction

In field theory, the Bogomolny-Prasad-Sommerfield (BPS)222Also equivalently
called the Bogomolny equations. equations are a very convenient way of solving
field equations for certain configurations of topological solitons, since they
reduce the second-order Euler-Lagrange equations into first-order
Bogomolny:1975de ; Prasad:1975kr . Physically, the the solutions represent the
lowest global energy configuration and thus stable. The original Bogomolny
mechanism Bogomolny:1975de  relies so much on “trial-and-error” method.
Although successful for the standard canonical defects, it is of little help
in the face of various non-canonical or exotic non-standard defects. To
overcome this difficulty, several formalism have been proposed to build
systematic algorithm for obtaining the BPS equations. Bazeia, Losano, and
Menezes observed that stability in static field theory implies the stressless
condition, $T_{ij}=0$ Bazeia:2008tj ; Bazeia:2007df . In Adam:2013hza  the
authors generalize the notion of self-duality to construct the BPS equations
and their corresponding topological charge. One of us (HSR) and collaborators
showed that the BPS condition can be derived directly from the Euler-Lagrange
equations via separation of variable without the need to appeal to the
Hamiltonian; we dubbed the on-shell method Atmaja:2014fha ; Atmaja:2015lia .
In Atmaja:2015umo , Atmaja suggests a method for obtaining BPS equations by
relying on the so-called BPS energy function $Q$. This method is suitable for
BPS defects with non-canonical kinetic terms. In Bazeia:2017nas  the authors
developed the so-called first-order formalism, which was based on the concept
of strong necessary condition Sokalski:2001wk , to obtain BPS equations for
generalized vortices. Interestingly, they did not start from the stressless
condition as in Bazeia:2008tj ; Bazeia:2007df . Rather, by assuming axial
symmetry of the energy-momentum tensor, they show that the stressless
condition is the result of stability under energy rescaling.

The first-order formalism Bazeia:2017nas  was constructed for Abelian
vortices. It is not surprising since topologically stable vortices exist when
the first homotopy group of the corresponding vacuum manifold is nontrivial,
$\pi_{1}\left(\mathcal{M}\right)\neq 0$. The simplest group that satisfies
this condition is when a $U(1)$ symmetry spontaneously broken by a complex
scalar field, thus Abelian. On the other hand, vortices (or strings) can also
be shown to exist in certain non-Abelian theories. Perhaps the simplest non-
Abelian string is the $Z_{N}$-strings, discussed by Nielsen and Olesen in the
very same paper that pointed out the existence of Abelian vortex
Nielsen:1973cs . This string can be formed through a maximal breaking of
$SU(N)\rightarrow Z_{N}$. Within the the context of Super QCD, the BPS
equations for $Z_{N}$ strings were obtained in Kneipp:2001tp ; Kneipp:2002yv ;
Kneipp:2002gk ; Kneipp:2008dc . Note that non-Abelian strings can exist
despite topologically trivial. Vachaspati and Achuccaro show that string
coming from the breaking of $SU(2)_{g}\times U(1)_{l}\rightarrow U(1)_{g}$ can
be stable even though the vacuum manifold is $S^{3}$ Vachaspati:1991dz ,
dubbed the semilocal string. This is due fact that there is gradient energy
cost to consider when deforming it to vacuum. When the $SU(2)$ is gauged the
defects become electroweak string Vachaspati:1992jk . The readers are advised
to consult Vilenkin and Shellard Vilenkin:2000jqa  for a more complete
discussion on non-Abelian strings.

Four decades ago, Schwarz found a class of non-Abelian vortices, dubbed the
Alice strings, where the unbroken symmetries become multi-valued when
parallel-transported around it Schwarz:1982ec ; Benson:2004ue . This happens
because the unbroken group $H$ contains both the usual $U(1)$ electrodynamics
(with charge operator $Q$) and an element $X$ of the disconnected component of
$H$ such that $XQX^{-1}=-Q$ Bucher:1991qhl . This property leads to a peculiar
behavior: a test particle shall flip its charge’s sign after circumnavigating
the string once. Another exotic feature of this defects is that when the
string forms a closed loop it can be endowed with charge that produces long-
range field as in the ordinary electrodynamics, but without well-localized
source. This strangely-behaved charge is dubbed the Cheshire charge
Alford:1990mk ; Bucher:1991bc . Though still hypothetical, the Alice string
concept found interesting applications in various areas of theoretical
physics. In condensed matter, the Bose-Einstein Condensates (BEC) is known to
contain the global Alice vortices Leonhardt:2000km ; Ruostekoski:2003qx ;
Kawaguchi:2012ii . In cosmology, the possible role of Alice string as ‘mirror’
particle was suggested in Blinnikov:1982eh ; Ignatiev:2003sy . More recently,
axionic Alice string is thought to solve the domain walls problem Sato:2018nqy
; Chatterjee:2019rch . It is also worth to mention that Okada and Sakatani
proposed an interesting argument that defect branes can be perceived as a
realization of Alice string in string theory Okada:2014wma .

The simplest model that contains Alice string in its spectrum is an $SO(3)$
gauge theory with a Higgs field $\Phi$ that belongs to its $5$-dimensional
irreducible representation, $\Phi\rightarrow S\Phi S^{-1}$, where $S\in SO(3)$
Bucher:1991qhl ; Bucher:1990fq . With appropriate choice of ground state, the
symmetry is spontaneously broken to $SO(3)\rightarrow
U(1)\times\mathbb{Z}_{2}$. The first homotopy group of the vacuum manifold is
nontrivial, $\pi_{1}\left(\mathcal{M}\right)\cong\mathbb{Z}_{2}$ Postma:1997 .
This $\mathbb{Z}_{2}$-charge is topologically stable, but continuously
deformable when any two of them collide. Consequently, two Alice strings
annihilate each other. Thus, in this simplest model the Alice cannot be BPS
where multi-vortices can stably exist. However, the $SO(3)$ gauge is not the
only model where Alice string can be embedded. The BEC vortices discussed
previously is one immediate counter-example Leonhardt:2000km ;
Ruostekoski:2003qx ; Kawaguchi:2012ii , where Alice string can be found in the
$U(1)\times SO(3)\rightarrow O(2)$ symmetry-breaking scenario. The homotopy
group is non-trivial, $\pi_{1}(\mathcal{M})\cong\mathbb{Z}$. This
$\mathbb{Z}$-string can exist in multi-configuration, as opposed to its
$\mathbb{Z}_{2}$-counterparts, and thus is BPS-compatible.

Recently, Chatterjee and Nitta found the BPS states of Alice strings in the
$U(1)\times SO(3)$ gauge theory coupled with charged complex scalar fields in
the vector representation Chatterjee:2017jsi . This model is a gauged version
of the BEC global Alice vortices Leonhardt:2000km . The Bogomolny equations
are achieved by the usual Bogomolny completion of the corresponding energy
integral Bogomolny:1975de . The BPS solutions are also numerically. Not long
after that, the BPS zero modes and its effective action are discussed in
Chatterjee:2017hya . With this BPS string’s discovery, the possibility of
having multi Alice vortices as well as their exotic dynamics are wide open,
both in field theory as well as in SUSY embedding.

In this work, we shall extend the first-order formalism Bazeia:2017nas  to the
case of non-Abelian strings. It is then applied to obtaining BPS Alice strings
for several models. In Sec. II we construct the first-order formalism for non-
Abelian multi-field vortices. We apply the formalism to the case of Alice
string in the $U(1)\times SO(3)$ gauge theory coupled with charged complex
scalar fields in the vector representation in Sec. III. Finally we summarize
our findings in Sec. IV.

##  II First-order Formalism for Non-Abelian Multi-field Vortices

Consider the most general Lagrangian containing multi Higgs and gauge fields
of co-dimension-2. The gauge fields can either be: Maxwell and $(m-1)$ Yang-
Mills, or pure $m$-Yang-Mills. The Lagrangian is as follows:

$\mathcal{L}=\mathcal{L}(|\Phi|,A^{\mu}_{j},X,Y^{(j)}).$ (1)

Some explanations are in order. Here,

$\Phi\equiv\left(\phi_{1}\ \phi_{2}....\phi_{n}\right)^{T},\ \
A^{\mu}_{j}\equiv\left(A^{\mu}_{1},A^{\mu}_{2},...\right)\equiv\left(a^{\mu},A^{a\mu},...\right),$
(2)

with $\phi_{i}$ complex scalar fields, $i=1,2,...,n$, $j=1,2,...,m$, and where
$a^{\mu}$, $A^{a\mu},...$ denote the Maxwell and Yang-Mills fields,
respectively. We also define

$X\equiv\operatorname{Tr}D_{\mu}\Phi^{\dagger}D^{\mu}\Phi,\ \ \
Y^{(j)}\propto\operatorname{Tr}G_{\mu\nu(j)}G^{\mu\nu}_{~{}(j)},$ (3)

where no summation is implied on $j$, and $G^{\mu\nu}_{j}$ denotes the field
tensor for either Maxwell
$G^{\mu\nu}_{(j)}\equiv\partial^{\mu}a^{\nu}-\partial^{\nu}a^{\mu}$ or Yang-
Mills
$G^{\mu\nu}_{(j)}\equiv\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}-ig_{j}[A^{\mu},A^{\nu}]$
with $g_{j}$ the coupling constant of the corresponding symmetry group.
Details on how many fields are involved depend on the specific models
considered. Here,
$\mathcal{L_{M}}\equiv\partial\mathcal{L}/\partial\mathcal{M}$. The
corresponding Euler-Lagrange equations and energy-momentum tensors are

$\displaystyle D_{\mu}\left(\mathcal{L}_{X}D^{\mu}\Phi\right)$
$\displaystyle=$ $\displaystyle\frac{\Phi}{2|\Phi|}\mathcal{L}_{|\Phi|},$
$\displaystyle\partial_{\mu}\left(\mathcal{L}_{Y^{(j)}}G^{\mu\nu}_{(j)}\right)$
$\displaystyle=$ $\displaystyle-\mathcal{L}_{A_{\nu j}},$ (4)

and

$T_{\mu\nu}=\mathcal{L}_{X}\operatorname{Tr}\left(D_{\mu}\Phi^{{\dagger}}D_{\nu}\Phi+D_{\nu}\Phi^{{\dagger}}D_{\mu}\Phi\right)+\sum\limits^{m}_{j}\mathcal{L}_{Y^{(j)}}G_{\mu\lambda(j)}G^{\lambda}_{\
\nu(j)}-g_{\mu\nu}\mathcal{L},$ (5)

respectively.

The ansatz chosen shall depend on the corresponding symmetry group employed.
However, as long as axial symmetry and staticity are employed, they can be
written as

$\displaystyle\Phi_{i}\propto f_{i}(r),\ \ A^{\mu}_{j}\propto A_{j}(r).$ (6)

In these forms,the kinetic terms will be:

$\displaystyle X$ $\displaystyle=$
$\displaystyle-\sum\limits^{n}_{i}\left(f_{i}^{\prime
2}+\frac{\left(A_{1}+\sum\limits^{m}_{j}(-1)^{i+j}A_{j+1}\right)^{2}}{r^{2}}f_{i}^{2}\right),$
$\displaystyle Y^{(j)}$ $\displaystyle=$ $\displaystyle-\frac{A_{j}^{\prime
2}}{2g_{j}^{2}r^{2}},$ (7)

where ′ denotes the derivative wrt $r$. The alternating sign
$\left(-1\right)^{j+1}$ will depend on the specific model and the number of
non-Abelian gauge fields $A_{j}$ involved. The off-diagonal components of the
energy-momentum tensor will be of the form:

$T_{12}=\mathcal{L}_{X}\sum\limits^{n}_{i}\left(f_{i}^{\prime
2}-\frac{\left(A_{1}+\sum\limits^{m}_{j}(-1)^{i+j}A_{j+1}\right)^{2}}{r^{2}}f_{i}^{2}\right)\sin
2\theta.$ (8)

The first-order formalism relies on two fundamental assumptions: axial
symmetry and energy stability under spatial rescaling Bazeia:2017nas . The
axial symmetry implies

$\sum\limits^{n}_{i}\left(f_{i}^{\prime
2}-\frac{\left(A_{1}+\sum\limits^{m}_{j}(-1)^{i+j}A_{j+1}\right)^{2}}{r^{2}}f_{i}^{2}\right)=0.$
(9)

For ordinary (single-field) vortex, the condition (9) simply reduces to

$f^{\prime 2}=\frac{a^{2}f^{2}}{r^{2}}.$ (10)

For multi-field system, however, the situation becomes tricky. In this work,
we assume that each Higgs field is independent, i.e.,

$f_{i}^{\prime
2}=\frac{\left(A_{1}+\sum\limits^{m}_{j}(-1)^{i+j}A_{j+1}\right)^{2}}{r^{2}}f_{i}^{2}.$
(11)

Note that this may not be the most general equations derived from (9).
However, in the next section we show that this yields the correct BPS
conditions for the known Alice strings. Eq. (11) is the first of the two
Bogomolny equations. With this condition,

$X=-2\sum\limits^{n}_{i}f_{i}^{\prime
2}=-2\sum\limits^{n}_{i}\frac{\left(A_{1}+\sum\limits^{m}_{j}(-1)^{i+j}A_{j+1}\right)^{2}}{r^{2}}f_{i}^{2}.$
(12)

The second fundamental assumption, i.e., the stability under rescaling, is
essentially the Derrick’s theorem for static defects Derrick:1964ww . The
argument is as follows. For static axially-symmetric vortices, the total
energy is

$E=-2\pi\int r\ dr\ \mathcal{L}\left[\Phi_{i},A_{j},X,Y^{(j)}\right].$ (13)

Since, under rescaling $r\rightarrow\lambda r$, the kinetic terms rescale as
$X\rightarrow\lambda^{2}X$ and $Y^{(j)}\rightarrow\lambda^{4}Y^{(j)}$, the
energy does also rescale as

$E^{(\lambda)}=-\frac{2\pi}{\lambda^{2}}\int r\ dr\
\mathcal{L}\left[\Phi_{i},A_{j},\lambda^{2}X,\lambda^{4}Y^{(j)}\right].$ (14)

Derrick’s theorem states that the condition for stability implies

$\frac{dE^{(\lambda)}}{d\lambda}\bigg{|}_{\lambda=1}=0.$ (15)

This results in

$\mathcal{L}-\mathcal{L}_{X}X-2\sum\limits^{m}_{j}\mathcal{L}_{Y^{(j)}}Y^{(j)}=0.$
(16)

Taking its derivative wrt to $r$, we obtain

$\displaystyle 2\sum\limits^{n}_{i}\frac{f_{i}^{\prime
2}}{r^{2}}\left[r\left(r\mathcal{L}_{X}f_{i}^{\prime}\right)^{\prime}-\mathcal{L}_{X}\left(A_{1}+\sum\limits^{m}_{j}\left(-1\right)^{i+j}A_{j+1}\right)^{2}f_{i}+\frac{r^{2}f_{i}\mathcal{L}_{|\Phi|}}{2\sqrt{\sum\limits^{n}_{\ell=1}f_{\ell}^{2}}}\right]$
$\displaystyle+\sum\limits^{m}_{j}\frac{A_{j}^{\prime}}{g_{j}^{2}r^{2}}\left[r\left(\mathcal{L}_{Y^{(j)}}\frac{A_{j}^{\prime}}{g_{j}r}\right)^{\prime}-2g_{j}\mathcal{L}_{X}\sum\limits^{n}_{i}\left(A_{1}+\left(-1\right)^{i+j}A_{j+1}\right)^{2}f_{i}^{2}\right]=0.$
(17)

Here we express $|\Phi|=\sqrt{\sum\limits^{n}_{\ell=1}f_{\ell}^{2}}$. It can
be shown that when setting each square bracket to zero it gives us the field
equations, as derived from the Euler-Lagrange (II). By inserting Eq. (11) into
(II), the first-order equation for $A_{j}$ can be extracted. Generally, it
will be in the form

$\frac{A_{j}^{\prime}}{r}=M\left(f_{i},r\right),$ (18)

where $M\left(f_{i},r\right)$ in general is an algebraic function of $f_{i}$
(and $r$) but not their derivative. Eq. (18) constitutes the second Bogomolny
eqiations.

The BPS energy can be also be calculated directly. Using the condition (16),
the energy density can be re-written as

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}X+2\sum\limits^{m}_{j}\mathcal{L}_{Y^{(j)}}Y^{(j)}$
(19) $\displaystyle=$ $\displaystyle
2\mathcal{L}_{X}\sum\limits^{n}_{i}f_{i}^{\prime
2}+\sum\limits^{m}_{j}\mathcal{L}_{Y^{(j)}}\frac{A_{j}^{\prime
2}}{g_{j}^{2}r^{2}},$

where in the last line we have made use of Eq. (11). We can define an
auxiliary function $W=W(f_{i},A_{j})$ such that

$W_{f_{i}}\equiv\frac{dW}{df_{i}}\equiv 2r\mathcal{L}_{X}f_{i}^{\prime},\ \ \
W_{A_{j}}\equiv\frac{dW}{dA_{j}}\equiv\mathcal{L}_{Y}\frac{A_{j}^{\prime}}{g_{j}^{2}r}.$
(20)

With these definitions, the energy density (19) can be writen as

$\displaystyle\rho=\frac{1}{r}\frac{dW}{dr},$ (21)

which implies

$\displaystyle E=2\pi\int rdr\ \rho=2\pi\ W(f_{i},A_{j})\bigg{|}_{boundary}.$
(22)

One generic feature of various Bogomolnyi alternative formalisms Bazeia:2008tj
; Bazeia:2007df ; Adam:2013hza ; Atmaja:2014fha ; Atmaja:2015lia ;
Atmaja:2015umo ; Bazeia:2017nas  is that, unlike the original Bogomolny’s
trick Bogomolny:1975de , they do not start from a given potential. Instead,
the form of the potential is deduced such that it satisfies the Bogomolny
equations and the corresponding non-trivial topology suitable for the
existence of the defects. In the next section we shall apply these mechanisms
to several models of Alice strings and obtaining their corresponding BPS
configurations.

##  III BPS Alice Strings

This Alice Lagrangian considered by Chatterjee and Nitta Chatterjee:2017jsi
is given by

$\mathcal{L}=|\operatorname{Tr}D_{\mu}\Phi|^{2}-\frac{1}{2}\operatorname{Tr}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}+V,$
(23)

where $D_{\mu}\Phi\equiv\partial_{\mu}\Phi-iea_{\mu}\Phi-
ig\left[A_{\mu},\Phi\right]$,
$F_{\mu\nu}\equiv\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig\left[A_{\mu},A_{\nu}\right]$,
and $f_{\mu\nu}\equiv\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$. The gauge
fields $a_{\mu}$ and $A_{\mu}$ refers to the Maxwell and Yang-Mills fields,
respectively. Here $V=V\left(\Phi,\Phi^{{\dagger}}\right)$, and note that we
do not specify the potential beforehand.

The appropriate static and axially-symmetric ansatze are:

$\displaystyle\Phi=\Phi(r,\theta)$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&f_{1}(r)e^{in\theta}\\\
f_{2}(r)&0\end{pmatrix},$ (24) $\displaystyle a_{\mu}=a_{i}(r,\theta)$
$\displaystyle=$
$\displaystyle-\epsilon_{ij}\frac{x_{j}}{er^{2}}\biggl{(}a(r)-\frac{n}{2}\biggr{)},$
(25) $\displaystyle A_{\mu}=A_{i}(r,\theta)$ $\displaystyle=$
$\displaystyle-\epsilon_{ij}\frac{x_{j}}{gr^{2}}\biggl{(}A(r)-\frac{n}{2}\biggr{)}\frac{\sigma^{3}}{2}.$
(26)

In our language, $f_{i}=\left(f_{1},f_{2}\right)$ and
$A_{j}=\left(a,A\right)$, thus $Y^{(j)}\equiv\left(y,Y\right)$. Notice that
the ansatz for gauge fields are slightly different from Chatterjee:2017jsi .
The following boundary conditions are imposed to ensure regularity at the core
and appropriate behavior in the vacuum:

$\displaystyle f_{1}(0)$ $\displaystyle=$ $\displaystyle f_{2}^{\prime}(0)=0,\
\ \ \ \ \ \ \ f_{1}(\infty)=f_{2}(\infty)=const.\equiv\xi,$ $\displaystyle
a(0)$ $\displaystyle=$ $\displaystyle A(0)=\frac{n}{2},\ \ \ \ \ \ \ \
a(\infty)=A(\infty)=0.$ (27)

In these ansatze, (12) can be written as

$\displaystyle X$ $\displaystyle=$ $\displaystyle-\Biggl{[}f_{1}^{\prime
2}+\frac{\bigl{(}a+A\bigr{)}^{2}}{r^{2}}f_{1}^{2}+f_{2}^{\prime
2}+\frac{\bigl{(}a-A\bigr{)}^{2}}{r^{2}}f_{2}^{2}\Biggr{]},$ (28)
$\displaystyle y$ $\displaystyle=$ $\displaystyle-\frac{a^{\prime
2}}{2e^{2}r^{2}},$ (29) $\displaystyle Y$ $\displaystyle=$
$\displaystyle-\frac{A^{\prime 2}}{2g^{2}r^{2}}.$ (30)

The Euler-Lagrange’s equations (II) becomes

$\displaystyle\frac{1}{r}\bigl{(}r\mathcal{L}_{X}f_{1}^{\prime}\bigr{)}^{\prime}-\frac{\mathcal{L}_{X}\bigl{(}a+A\bigr{)}^{2}f_{1}}{r^{2}}+\frac{f_{1}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{1}{r}\bigl{(}r\mathcal{L}_{X}f_{2}^{\prime}\bigr{)}^{\prime}-\frac{\mathcal{L}_{X}\bigl{(}a-A\bigr{)}^{2}f_{2}}{r^{2}}+\frac{f_{2}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
r\left(\mathcal{L}_{y}\frac{a^{\prime}}{er^{2}}\right)^{\prime}-2e\mathcal{L}_{X}\left[\left(a+A\right)f_{1}^{2}+\left(a-A\right)f_{2}^{2}\right]$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
r\left(\mathcal{L}_{Y}\frac{A^{\prime}}{r}\right)^{\prime}-2g^{2}\mathcal{L}_{X}\Bigl{[}\bigl{(}a+A\bigr{)}f_{1}^{2}-\bigl{(}a-A\bigr{)}f_{2}^{2}\Bigr{]}$
$\displaystyle=$ $\displaystyle 0.$ (31)

The condition (II) in these ansatze can also be written as

$\displaystyle
2f_{1}^{\prime}\biggl{[}\frac{1}{r}\bigl{(}r\mathcal{L}_{X}f_{1}^{\prime}\bigr{)}^{\prime}-\frac{\mathcal{L}_{X}\bigl{(}a+A\bigr{)}^{2}f_{1}}{r^{2}}+\frac{f_{1}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}\biggr{]}$
$\displaystyle+2f_{2}^{\prime}\biggl{[}\frac{1}{r}\bigl{(}r\mathcal{L}_{X}f_{2}^{\prime}\bigr{)}^{\prime}-\frac{\mathcal{L}_{X}\bigl{(}a-A\bigr{)}^{2}f_{2}}{r^{2}}+\frac{f_{2}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}\biggr{]}$
$\displaystyle+\frac{a^{\prime}}{e^{2}r^{2}}\biggl{[}r\bigl{(}\mathcal{L}_{y}\frac{a^{\prime}}{er^{2}}\bigr{)}^{\prime}-2e\mathcal{L}_{X}\Bigl{\\{}\bigl{(}a+A\bigr{)}f_{1}^{2}+\bigl{(}a-A\bigr{)}f_{2}^{2}\Bigr{\\}}\biggr{]}$
$\displaystyle+\frac{A^{\prime}}{g^{2}r^{2}}\biggl{[}r\bigl{(}\mathcal{L}_{y}\frac{A^{\prime}}{gr^{2}}\bigr{)}^{\prime}-2g\mathcal{L}_{X}\Bigl{\\{}\bigl{(}a+A\bigr{)}f_{1}^{2}-\bigl{(}a-A\bigr{)}f_{2}^{2}\Bigr{\\}}\biggr{]}=0.$
(32)

Notice that each term in the square bracket is equivalent to the lhs of each
Euler-Lagrange equations above, Eq. (III). For completeness, we also show the
spatial components of (5) as:

$\displaystyle T_{11}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{y}\frac{a^{\prime
2}}{e^{2}r^{2}}+\mathcal{L}_{Y}\frac{A^{\prime
2}}{g^{2}r^{2}}+2\mathcal{L}_{X}\Bigl{\\{}\bigl{(}f_{1}^{\prime
2}+f_{2}^{\prime 2}\bigr{)}\cos^{2}\theta$
$\displaystyle+\Bigl{[}\bigl{(}a+A\bigr{)}^{2}f_{1}^{2}+\bigl{(}a-A\bigr{)}^{2}f_{2}^{2}\Bigr{]}\sin^{2}\theta\Bigr{\\}}+\mathcal{L},$
$\displaystyle T_{12}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}\Biggl{[}f_{1}^{\prime
2}-\frac{\bigl{(}a+A\bigr{)}^{2}f_{1}^{2}}{r^{2}}+f_{2}^{\prime
2}-\frac{\bigl{(}a-A\bigr{)}^{2}f_{2}^{2}}{r^{2}}\Biggr{]}\sin 2\theta,$
$\displaystyle T_{22}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{y}\frac{a^{\prime
2}}{e^{2}r^{2}}+\mathcal{L}_{Y}\frac{A^{\prime
2}}{g^{2}r^{2}}+2\mathcal{L}_{X}\Bigl{\\{}\bigl{(}f_{1}^{\prime
2}+f_{2}^{\prime 2}\bigr{)}\sin^{2}\theta$ (33)
$\displaystyle+\Bigr{[}\bigl{(}a+A\bigr{)}^{2}f_{1}^{2}+\bigl{(}a-A\bigr{)}^{2}f_{2}^{2}\Bigr{]}\cos^{2}\theta\Bigl{\\}}+\mathcal{L}.$

The first Bogomolny equation, (11), becomes

$\displaystyle f_{1}^{\prime 2}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}a+A\bigr{)}^{2}f_{1}^{2}}{r^{2}},$ (34)
$\displaystyle f_{2}^{\prime 2}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}a-A\bigr{)}^{2}f_{2}^{2}}{r^{2}}.$ (35)

Deriving Eqs. (34)-(35) wrt to r, and then inserting them into Eq. (III)
results in

$\displaystyle\frac{\bigl{(}a+A\bigr{)}f_{1}}{r}\Biggl{[}\mathcal{L}_{XX}X^{\prime}+\mathcal{L}_{Xy}y^{\prime}+\mathcal{L}_{XY}Y^{\prime}+\mathcal{L}_{X\Phi}\frac{\bigl{(}a+A\bigr{)}f_{1}^{2}/r+\bigl{(}a-A\bigr{)}f_{2}^{2}/r}{\sqrt{f_{1}^{2}+f_{2}^{2}}}\Biggr{]}$
$\displaystyle+\frac{a^{\prime}+A^{\prime}}{r}f_{1}\mathcal{L}_{X}+\frac{f_{1}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}=0.$
(36)
$\displaystyle\frac{\bigl{(}a-A\bigr{)}f_{1}}{r}\Biggl{[}\mathcal{L}_{XX}X^{\prime}+\mathcal{L}_{Xy}y^{\prime}+\mathcal{L}_{XY}Y^{\prime}+\mathcal{L}_{X\Phi}\frac{\bigl{(}a+A\bigr{)}f_{1}^{2}/r+\bigl{(}a-A\bigr{)}f_{2}^{2}/r}{\sqrt{f_{1}^{2}+f_{2}^{2}}}\Biggr{]}$
$\displaystyle+\frac{a^{\prime}-A^{\prime}}{r}f_{1}\mathcal{L}_{X}+\frac{f_{2}}{2\sqrt{f_{1}^{2}+f_{2}^{2}}}\mathcal{L}_{|\Phi|}=0.$
(37)

Judging from the form of the Lagrangian (23), $\mathcal{L}=X+Y+y-V$, it is not
difficult to show that Eqs. (III)-(III) reduce to the following:

$\displaystyle\frac{a^{\prime}+A^{\prime}}{r}f_{1}+\frac{1}{2}\mathcal{L}_{f_{1}}$
$\displaystyle=$ $\displaystyle 0,$ (38)
$\displaystyle\frac{a^{\prime}-A^{\prime}}{r}f_{2}+\frac{1}{2}\mathcal{L}_{f_{2}}$
$\displaystyle=$ $\displaystyle 0.$ (39)

Eqs. (38) are the second Bogomolny equations in this model. With these
conditions, (16) becomes

$-y-Y-V=0.$ (40)

Now here comes the tricky parts. First, we assume that the potential can be
separated into two terms $V=V_{1}+V_{2}$. Second, each can be identified as
follows:

$\displaystyle V_{1}$ $\displaystyle=$ $\displaystyle-y\ \ \ \ \ \
\rightarrow\ \ \ \frac{a^{\prime}}{r}=e\sqrt{2V_{1}},$ (41) $\displaystyle
V_{2}$ $\displaystyle=$ $\displaystyle-Y\ \ \ \ \ \rightarrow\ \ \
\frac{A^{\prime}}{r}=g\sqrt{2V_{2}}.$ (42)

Combining all the previous equations, we have the constraint conditions

$\displaystyle\Bigl{(}e\sqrt{2V_{1}}+g\sqrt{2V_{2}}\Bigr{)}f_{1}-\frac{1}{2}V_{1,f_{1}}-\frac{1}{2}V_{2,f_{1}}$
$\displaystyle=$ $\displaystyle 0,$ (43)
$\displaystyle\Bigl{(}e\sqrt{2V_{1}}-g\sqrt{2V_{2}}\Bigr{)}f_{2}-\frac{1}{2}V_{1,f_{2}}-\frac{1}{2}V_{2,f_{2}}$
$\displaystyle=$ $\displaystyle 0.$ (44)

Third, the constraints above can be satisfied by taking

$\displaystyle\frac{\partial V_{1}}{\partial f_{1}}$ $\displaystyle=$
$\displaystyle 2e\sqrt{2V_{1}}f_{1},\ \ \ \ \ \ \ \frac{\partial
V_{1}}{\partial f_{2}}=2e\sqrt{2V_{1}}f_{2},$ (45)
$\displaystyle\frac{\partial V_{2}}{\partial f_{1}}$ $\displaystyle=$
$\displaystyle 2g\sqrt{2V_{2}}f_{1},\ \ \ \ \ \ \ \frac{\partial
V_{2}}{\partial f_{2}}=2g\sqrt{2V_{2}}f_{2}.$ (46)

These configurations are chosen to guarantee the Schwarz’s integrability
condition. The solution for Eq. (45) is

$\displaystyle V_{1}(f_{1},f_{2})$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{2}\left(f_{1}^{2}+f_{2}^{2}-2\xi^{2}\right)^{2},$
(47)

where the integration constant $2\xi^{2}$ can be identified as the VEV
$\langle\Phi\rangle$. In terms of the $\Phi$ field it can be written as

$V_{1}\left(\Phi\right)=\frac{e^{2}}{2}\left(\operatorname{Tr}\Phi^{{\dagger}}\Phi-2\xi^{2}\right)^{2}.$
(48)

Similarly, the solution for Eq. (46) is

$V_{2}(f_{1},f_{2})=\frac{g^{2}}{2}\left(f_{1}^{2}-f_{2}^{2}+C\right)^{2},$
(49)

or

$V_{2}(\Phi)=\frac{g^{2}}{4}\operatorname{Tr}\left([\Phi^{{\dagger}},\Phi]+C\right)^{2},$
(50)

where $C$ is another integration constant. this time, without loss of
generality we can set $C=0$. Thus, we will have the following result for the
general potential that can generate BPS equations.

$\displaystyle V(\Phi)$ $\displaystyle=$ $\displaystyle
V_{1}(\Phi)+V_{2}(\Phi),$ (51) $\displaystyle=$
$\displaystyle\frac{e^{2}}{2}\Bigl{(}\operatorname{Tr}\Phi^{\dagger}\Phi-2\xi^{2}\Bigl{)}^{2}+\frac{g^{2}}{4}\operatorname{Tr}\Bigl{(}[\Phi^{\dagger},\Phi]\Bigr{)}^{2}.$
(52)

Note that we reproduce the same BPS potential as in Chatterjee:2017jsi .

To summarize, by means of Eqs. (34), (35), (41), (42), (47), and (49), the BPS
equations are

$\displaystyle f_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}a+A\bigr{)}f_{1}}{r},$ $\displaystyle
f_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\bigl{(}a-A\bigr{)}f_{2}}{r},$
$\displaystyle\frac{a^{\prime}}{r}$ $\displaystyle=$ $\displaystyle
e^{2}\bigl{(}f_{1}^{2}+f_{2}^{2}-2\xi^{2}\bigr{)},$
$\displaystyle\frac{A^{\prime}}{r}$ $\displaystyle=$ $\displaystyle
g^{2}\bigl{(}f_{1}^{2}-f_{2}^{2}\bigr{)}.$ (53)

Imposing the boundary conditions (III) we solve Eqs. (III) numerically. To get
the profiles of Alice strings, we set $n=1$. The typical results are shown in
Figs. 1-2. As in Chatterjee:2017jsi  for the parameter value $\ell\equiv
e/g=1$, the boundary conditions force the $f_{2}(r)$ profile to be constant
and the $a(r)$ coincides with $A(r)$, as shown in Fig. 1. For other value of
$\ell$ we can see, for example in Fig. 2, that all profile functions
interpolate distinctly between the boundaries.

Figure 1: Profiles of scalar and gauge fields for $n=1$ and $\ell=1$. Figure
2: Profiles of scalar and gauge fields for $n=1$ and $\ell=2$.

##  IV Summary

In this work we enlarge the first-order formalism of obtaining Bogomolnyi
equations to non-Abelian vortices. Here we neither assume any specific group
nor the number of gauge fields involved. The kinetic terms are not restricted
to canonical form and may take any arbitrary non-canonical function. We only
require axial symmetry and staticity. As in the Abelian case developed by
Bazeia et al. Bazeia:2017nas , this formalism relies on axial symmetry and
Derrick’s theorem. The former implies BPS equations for the Higgs fields,
while the latter yields the equations for the gauge fields. Both must be
consistent in a sense that they should satisfy the constraint equation. In
this formalism, the potential is not put by hand. Rather, we start from
unspecified potential and move forward by applying the algorithm to obtain the
suitable BPS equations. The potential is obtained from the equation(s) of
constraint. The “tricky” part in this generalization is that it assumes that
each Higgs field is independent on each other. Thus the first Bogomolny
equation can be written as in Eq. (11). We do not claim that this postulate is
the most general we can have, though we do not have the proof to rule it our
either, but certainly for the cases considered here it does work.

We apply the formalism to the $U(1)\times SO(3)$ gauge theory coupled with
charged complex scalar field. Chatterjee and Nitta already derived the BPS
equations for such a model Chatterjee:2017jsi  using the usual Bogomolny’s
trick Bogomolny:1975de . Here we derive the same BPS equations using the
first-order formalism, and obtain the same potential up to some constant.
Theoretically, the scope of this work’s formalism is somehow limited, since
not many non-Abelian strings are topologically nontrivial. Alice strings are
one of the few non-Abelian vortices with genuine topological stability, thus
our choice of model studied in this work. Phenomenologically, however, this
formalism might shed light on the field of condensed matter and, more
especially, in the application of Alice strings in superfluids
Fujimoto:2020dsa -Masaki:2021hmk . We hope to come to this topics in the near
future.

###### Acknowledgements.

We thank Ardian Atmaja and Ilham Prasetyo for the fruitful discussion. HSR is
funded by the Hibah Riset FMIPA UI No. PKS-026/UN2.F3.D/PPM.00.02/2023.

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