# ivrit.ai: A Comprehensive Dataset of Hebrew Speech for AI Research and
Development

Yanir Marmor
Weizmann Institute of Science
<EMAIL_ADDRESS>Kinneret Misgav
Hadassah Research Fund
<EMAIL_ADDRESS>Yair Lifshitz

<EMAIL_ADDRESS>
###### Abstract

We introduce ivrit.ai, a comprehensive Hebrew speech dataset, addressing the
distinct lack of extensive, high-quality resources for advancing Automated
Speech Recognition (ASR) technology in Hebrew. With over 3,300 speech hours
and a over a thousand diverse speakers, ivrit.ai offers a substantial
compilation of Hebrew speech across various contexts. It is delivered in three
forms to cater to varying research needs: raw unprocessed audio; data post-
Voice Activity Detection, and partially transcribed data. The dataset stands
out for its legal accessibility, permitting use at no cost, thereby serving as
a crucial resource for researchers, developers, and commercial entities.
ivrit.ai opens up numerous applications, offering vast potential to enhance AI
capabilities in Hebrew. Future efforts aim to expand ivrit.ai further, thereby
advancing Hebrew’s standing in AI research and technology.

##  1 Introduction

Automated Speech Recognition (ASR; also known as speech-to-text) technology
holds vast potential for enhancing various processes involving human speech.
Nevertheless, its effectiveness is not uniformly distributed across languages.
While some languages significantly benefit from ASR tools, others, such as
Hebrew, find these technologies to be underwhelming. This often results in the
continued use of human scribes when transcriptions are not available.

The initial phase of creating precise and efficient ASR tools requires an
extensive corpus of high-quality speech data. To our knowledge, such a dataset
for Hebrew has not been publicly available until now. We introduce ivrit.ai
("Ivrit" is how Hebrew speakers pronounce the name of their language), a
comprehensive Hebrew speech dataset designed for AI research and development.
ivrit.ai is designed to empower advanced AI technologies to function smoothly
in Hebrew, enabling AI to read, write, listen to, and articulate the language
with fluency. We view this as a strategic initiative to provide Hebrew
speakers and their communities with access to superior AI technology in a
practical manner under a favorable license, thus enabling interested
commercial parties to use this data at no cost. ivrit.ai contains
approximately 3,300 hours of Hebrew speech, collected from a diverse range of
online platforms including podcasts and other audio content. Given the wide
variety of speech types and topics within ivrit.ai, it serves as a valuable
resource for advancing AI in Hebrew.

###  1.1 Automated Speech Recognition

Oral communication is a crucial component of human interaction. It
communicates words and meaning through intonation, pitch (i.e., how high or
low the voice sounds), pace, and subtle non-verbal cues, together forming the
rich tapestry of human conversation. Humans carry out all these complex
processes intuitively, instantaneously, and responsively.

Although spoken language is a naturally comfortable mode of communication for
humans, ASR technology aims to harness this ease and transform it into a
format beneficial for computers. It accomplishes this by converting speech
signals into written text, thus transcribing oral communication into a
computer-readable input.

Whereas oral communication is natural for humans, it presents significant
challenges for computers due to its complexities. These complexities include
physical aspects such as diverse accents, dialects, pitch, multiple speakers,
and overlapping speech. Specific environmental factors can also add
difficulties, such as channel distortions, background noises, and limitations
in sampling and compression techniques. Additional complexities are associated
with cognitive process aspects. Since spoken language carries inherent
meaning, it’s crucial for machines to understand words in their respective
contexts to yield accurate transcriptions (e.g., distinguishing between knight
and night). Further complexities arise during the transcription phase, which
involves converting spoken words into written text, such as the need for
appropriate punctuation.

Previous studies have addressed these challenges by using algorithmic
approaches that leverage extensive datasets. For instance, Christensen et al.
(2001) introduced statistical prosody models for punctuation annotation, while
Wang and Chen (2018) surveyed models for speaker-noise separation. However,
these solutions frequently encounter issues with generalization across various
settings, including differences in speakers, languages, and environmental
factors. Furthermore, these methodologies often face limitations in their
accuracy and performance, which can hinder their effectiveness in diverse and
complex real-world applications.

###  1.2 ASR in the age of Large Language Models

Large language models (LLMs; e.g., ChatGPT by OpenAI Roumeliotis and Tselikas
(2023), Bard by Google Manyika (2023)) are revolutionizing the tools we use
and also hold promise for the ASR task. Based on transformer-based
architecture Zhao et al. (2023) and vast datasets, LLMs are currently
recognized as state-of-the-art (SOTA) for typical natural language processing
(NLP) tasks (e.g., Hendy et al. (2023), Alarcon et al. (2021), Luo et al.
(2022)).

The remarkable success of LLMs in text-based tasks has sparked interest in the
field of speech processing to develop solutions for the ASR deficits mentioned
earlier, based on similar principles: transformer-based architecture Latif et
al. (2023) and large corpora. Currently, ASR models based on these principles
(e.g., Whisper Radford et al. (2023), SpeechT5 Ao et al. (2021)) are
considered SOTA for most ASR tasks.

To fully realize the significant benefits of ASR and its wide range of
applications, large, high-quality datasets are essential. These datasets
should include a wide vocabulary, diverse speakers, and a variety of topics.
Since languages differ in multiple aspects (e.g., phonetics, syntax, and
semantic structure), one may utilize datasets tailored to each specific
language the ASR should support. At present, only a few languages have access
to datasets of sufficient size and quality. Many languages suffer from a lack
of resources, which prevents their speakers from maximizing the potential of
existing ASR technology. In these languages, a lack of resources poses a major
barrier to the effective adoption and utilization of ASR technology. In light
of these challenges, it’s crucial to support languages in the digital world
for fostering linguistic diversity and inclusivity. Making more speech and
transcribed speech data available for use could significantly benefit less-
prevalent languages. Specifically, in this project, we aim to provide such
support for Hebrew speakers.

###  1.3 Processing Hebrew Speech

The processing of Hebrew speech is challenged by the lack of available data.
Some innovative monolingual datasets do exist in Hebrew (Sharoni et al.
(2023)) and there are some multilingual datasets that include Hebrew Black
(2019). However, their scope and quality do not meet the requirements needed
for training and optimizing robust ASR models. As a result, Hebrew speakers
cannot fully benefit from advancements in ASR technology. Therefore, it is
crucial to collect and develop comprehensive datasets in Hebrew.

###  1.4 The Present Dataset

The ivrit.ai dataset that we present here has the potential to contribute
substantially to various speech and text tasks in Hebrew. This dataset
includes over 3,300 hours of Hebrew speech, collected from multiple online
sources. The dataset includes a wide range of speakers, varying by gender,
origin, and education level, as well as a variety of speech styles (e.g.,
official lectures, informal conversations), and topics (e.g., sports podcasts,
Talmud lessons). Approximately 2.8 million utterances and about 30 million
words are included in the dataset. The audio data and transcripts are
available for research and development in AI modeling, encompassing both non-
commercial and commercial uses.

##  2 Related works

Significant advances in ASR and NLP have been made in recent years, as a
result of numerous datasets and research studies. However, the Hebrew language
has received very little attention in this domain.

###  2.1 General Speech Datasets and Research

Monolingual speech datasets have been developed for specific domains and
communication situations, including the ATIS corpus for air travel information
requests Hemphill et al. (1990), as well as other corpora containing
recordings of meetings Garofolo et al. (2004b), telephone conversations
Canavan et al. (1997), and broadcast media Garofolo et al. (2004a). The
original TIMIT collection Garofolo (1993) and many of the broadcast news
corpora provide relatively clean audio scenarios in which a single speaker
reads from a prepared text, such as TREC Garofolo et al. (2000) and CLEF
Federico and Jones (2004). This characteristic restricts the usefulness of the
data in more dynamic environments and in more spontaneous situations. There
are also collections of more naturally occurring conversational material, such
as the CALLHOME corpus Canavan et al. (1997), the Santa Barbara Corpus of
Spoken American English Du Bois et al. (2000), and the TED talks corpus Hasebe
(2015), as well as podcast corpora, for instance, Spotify Clifton et al.
(2020) and MSP Lotfian and Busso (2017) Martinez-Lucas et al. (2020). These
collections capture unscripted and spontaneously organized discourse in a
conversational setting, including turns, interviews, stretches of monologue,
and argumentation. Due to this aspect, this data is more useful for dealing
with natural language patterns and real-world dialogue dynamics. None of these
datasets include Hebrew speech.

###  2.2 Multilingual Speech Datasets and Research

Current multilingual speech datasets with transcriptions, such as those drawn
from conversational telephone speech (IARPA Babel Program Cui et al. (2013)),
political speech Wang et al. (2021), and audiobooks Panayotov et al. (2015)
Pratap et al. (2020), cover a variety of domains and span approximately 100
languages. However, the representation of Hebrew in these datasets is
relatively low. This limitation also extends to multilingual datasets without
transcriptions, such as VoxLingua107 Valk and Alumäe (2021) and VoxPopuli Wang
et al. (2021), which span multiple languages and contain large amounts of
unlabeled data. Even in datasets like the one utilizing read versions of the
New Testament (MSU Black (2019) and recently MMS Pratap et al. (2023)), which
cover many languages and provide high-quality alignments, the Hebrew content
is not extensive. Unfortunately, the authors don’t detail the Hebrew content,
but generally suggest around 25 hours per language. The data from these
datasets is used to train self-supervised models, build speech recognition
systems, and develop language identification models Pratap et al. (2023) Wang
et al. (2021). Despite the availability of numerous monolingual and
multilingual speech datasets, each with their unique characteristics,
limitations, and areas of focus, the representation of Hebrew remains limited.
The field continues to evolve with ongoing efforts to develop more
comprehensive and diverse datasets that can support a wider range of ASR and
NLP research. Thus, there is a clear need for more extensive Hebrew content in
these resources.

###  2.3 Prior Hebrew Speech Datasets

Corpus | Hours | Speakers | Trans. | Type |  Topics |  License
---|---|---|---|---|---|---
SASPEECH Sharoni et al. (2023) | $30$ | 1 | 4M/26A | Mixed |  Economy, Politics |  Non-commercial
HUJI Corpus Marmorstein and Matalon (2022) | 3.8 | 60 | M | Conversations |  General Lifestyle |  CC BY 4.0
CoSIH Izre’el et al. (2001) | 12.3 | $\pm 140$ | M | Conversations |  General Lifestyle |  Non-commercial
MaTaCOp Azogui et al. (2016) | 5.3 | 16 | M | Conversations |  Map Task framework Anderson et al. (1991) |  Non-commercial
MLS Pratap et al. (2020) | 126 | $13$ | M | Reading |  Classic Books |  CC BY 4.0
CMU Black (2019) | $\pm 25$ | - | M | Reading |  New Testament |  -
MMS Pratap et al. (2023) | $\pm 25$ | - | M | Reading |  New Testament |  -
Whisper Radford et al. (2023) | $688$ | - | - | - |  - |  Not available
ivrit.ai | 3,300 | $+1000$ | A | Mixed |  Wide range (economy, politics, science, bible, philosophy, technology, history, etc.) |  Augmented CC BY 4.0 (see Availability section)
Table 1: Comparison of various speech datasets. The columns show the name of
the corpus, total hours of audio available, the number of speakers included,
whether the dataset is transcribed or not (M for manual transcription, and A
for automatic transcription, the type of speech included (reading,
conversations, mixed), the topics covered in the dataset, and the terms of the
license for using the dataset. Dash (-) indicates that the data is not
available for the corresponding field.

An array of datasets spanning academic to non-academic ventures provide
resources for Hebrew ASR research. These datasets can be monolingual,
containing Hebrew exclusively, or multilingual, encompassing Hebrew along with
other languages. Table 1 furnishes a detailed overview of these available
Hebrew speech datasets, both mono- and multilingual, outlining their distinct
characteristics and inherent limitations.

###  2.4 Natural Language Processing and Speech Recognition Research in Hebrew

Substantial progress has been made in the field of Hebrew NLP, with models
like Hebert Chriqui and Yahav (2022), AlephBERT Seker et al. (2021), and
AlephBERTGimmel Guetta et al. (2022) setting benchmarks in various textual
tasks. These advancements illustrate the potential for an intricate
understanding of written Hebrew texts.

The current landscape of Hebrew Automatic Speech Recognition (ASR) is largely
supported by commercial ASR services like Google Cloud, Microsoft (e.g., via
Word), Samsung, IBM Watson, and WIT.ai. These services offer engines with
accessible APIs that support Hebrew Silber-Varod et al. (2021). Additionally,
openly accessible models like OpenAI’s Whisper Radford et al. (2023) and Meta
AI’s multilingual model Pratap et al. (2023) are available as open sources.
This availability allows developers to create their own models for specific
tasks. Expanding on these resources, we have contributed to Hebrew ASR
development by introducing the ivrit.ai dataset. This open-access dataset,
free for all interested parties, equips researchers and developers with the
means to further improve their Hebrew ASR models.

###  2.5 Unlabeled Speech Datasets and Research

In the field of ASR, significant progress has been achieved through the
application and evolution of unsupervised speech pre-training techniques Wu et
al. (2020), semi-supervised learning (self-training) Liu et al. (2022), and
the combination of these techniques Xu et al. (2021). However, the research
community focuses primarily on ASR tasks utilizing English as the main input.
Through their proficient use of a copious amount of unlabeled English speech
data, these methods have improved English-centric ASR applications. A recently
published unlabeled speech corpus, covering 23 languages, demonstrated the
potential of these techniques for more languages Wang et al. (2021). However,
Hebrew was not included in this corpus. These results underscore the urgent
need for large datasets of Hebrew speech, even if they are not labeled.
Significant progress can be expected in Hebrew ASR if these datasets are
developed and utilized effectively.

##  3 Dataset Creation

DownloadVAD ProcessingTranscription Figure 1: Illustration of the Data
Pipeline: (1) Downloading the data in accordance with data creator agreements,
(2) Processing Voice Activity Detection (VAD) to segment audio based on
silences, and (3) Transcribing the segmented audio, which currently relies
solely on machine-based process but is planned to incorporate human
transcription.

The data and code are openly available at Hugging Face
(https://huggingface.co/ivrit-ai) and the GitHub repository
(https://github.com/yairl/ivrit.ai). Throughout this section, we will describe
the process of creating ivrit.ai. This resource, with its variety of speakers
and audio qualities, offers a comprehensive representation of the Hebrew
language in multiple contexts. Detailed information on how it was collected
and preprocessed will be provided. Figure 1 shows a schematic diagram of the
dataset creation pipeline.

###  3.1 Data Acquisition

We gathered audio clips from a variety of sources, encompassing both
individual and institutional content creators.

ivrit.ai’s license is specifically designed to enable commercial use of this
corpus for training AI models, such as speech-to-text or LLM models, while
preserving the intellectual property right of the content owner. Every item in
the corpus has been provided by a content owner who signed permission to use
this item under ivrit.ai’s license, thereby permitting the use of their work
in this manner.

The advantages of such an agreement are twofold. First, it ensures fairness
towards the content creators by explicitly stating in the agreement that their
work can be used. Second, it provides certainty for research and development
entities by confirming that the data has been collected with permission and is
distributed within a suitable legal framework.

###  3.2 Data Processing

The ivrit.ai data we’ve collected are being released in three distinct
datasets, catering to various research needs:

  * •

Raw Data: This is the original, unprocessed collection of audio clips.

  * •

Data Post-VAD: For this dataset, we have run Voice Activity Detection (VAD) on
the raw audio Team (2021). This operation segregates short units, ranging from
a few seconds to a minute, pinpointing parts where speakers were actively
involved. Figure 2 provides insights into the length distribution of the audio
pieces post-VAD.

  * •

Partially Transcribed Data: This dataset provides a portion of the audio data
along with their corresponding transcriptions.

Figure 2: Distribution of post-VAD audio clips in ivrit.ai corpus. The x-axis
represents the segment length in seconds, while the y-axis indicates the
frequency of each duration range

###  3.3 Transcribed Speech

The ivrit.ai dataset has been transcribed using the Whisper ASR tool (Radford
et al. (2023), using the whisper-small model). The transcription process is
applied to numerous short audio segments, resulting in 2.8 million transcribed
utterances and about 30 million words.

##  4 Dataset Description

As the data was collected from multiple contributors and represents a wide
range of recordings (narrative podcast, conversation, lesson), we are not able
to provide any precise information regarding the speakers. However, we can
provide some general information. The ivrit.ai dataset comprises over 3,300
hours of speech from a thousand diverse speakers. The dataset encompasses a
wide range of audio file lengths, and Figure 3 provides insights into the
distribution of episode lengths within the dataset.

Figure 3: Histogram depicting the distribution of episode durations in
ivrit.ai corpus. The x-axis represents the episode duration in minutes, while
the y-axis indicates the frequency of each duration range

Speakers’ ages ranged between the 20s and 70s. While some speakers are native
Hebrew speakers, for others, Hebrew is the second language (with English,
Russian, Arabic, and other languages being their native languages).

Languages other than Hebrew, predominantly English, can appear within the
corpus at three levels of magnitude. First, as single words borrowed for
specific uses, technical or otherwise; second, as slightly longer phrases that
have gained popularity in Hebrew usage (for instance, the phrase "having said
that" has been adopted into Hebrew conversations as is); and third, there are
entire episodes that are conducted solely in English.

##  5 Availability

The dataset is publicly accessible on the ivrit.ai website and is distributed
under an ivrit.ai license, an augmented CC-BY 4.0 license tailored to allow AI
model training for commercial use. Detailed licensing terms can be found in
Appendix A.

##  6 Discussion

We present here the ivrit.ai dataset, a comprehensive collection of over 3,300
hours of high-quality Hebrew speech, curated to advance AI research in the
Hebrew language. This novel dataset consists of a wide range of speech types
and topics, ready for use in various applications such as emergency response
systems, accessibility tools for the disabled, medical transcription services,
and digital voice assistants in the service industry, among others.

The ivrit.ai dataset stands out among other Hebrew datasets in its size,
diversity, and coverage of different speech styles and domains. Furthermore,
the ivrit.ai dataset offers more legal accessibility than many other datasets,
as it is available for industrial use, making it a valuable resource for
researchers and developers. Researchers can leverage this dataset to train and
evaluate their models, while also utilizing it as a benchmark for performance
comparison.

Among the limitations of the project, the dataset may be biased in aspects
such as gender or age imbalances, which may affect the performance of AI
models trained on the dataset. Additionally, the dataset’s diversity,
considered above as a strength, introduces variability in recording means,
speaker characteristics, and background noises. Moreover, deficits in data
collection and transcription could impact the dataset’s quality or usability.

##  7 Conclusion and Future Work

ASR technology holds vast potential for enhancing various human processes.
Although generating high-quality, efficient ASR tools is well known, model
quality depends on the dataset size. Despite the benefits that can be obtained
from ASR tools for some languages, others, such as Hebrew, are underwhelmed by
the technology.

We introduced the ivrit.ai dataset, a comprehensive collection of over 3,300
hours of Hebrew speech, designed to advance AI research in Hebrew. With a wide
range of speech types and topics, the dataset offers many possibilities. In
our view, the availability of such a diverse and extensive dataset is a
significant step forward in the field of Hebrew ASR and NLP research. This
dataset has the potential to improve multiple ASR-based systems’ accuracy and
performance.

Dataset acquisition tends to be effort-intensive, and fraught with legal
difficulties due to copyright requirements that often conflict with standard
licenses. ivrit.ai aims to create the world’s largest freely-available audio
dataset in Hebrew, fully transcribed, and fully available for the specific
purpose of training ASR and AI models.

Looking forward, we plan to further expand the ivrit.ai dataset, increase the
corpus by another order of magnitude and promote applied developments based on
the dataset, particularly in specific domains. Community involvement and
collaboration will be crucial to these efforts. By making the dataset widely
accessible, the aim is to place Hebrew at the forefront of AI research and
technology.

##  8 Acknowledgments

We would like to express our deepest gratitude to all the content creators who
generously allowed us to use their data for this project. Their contributions
have been invaluable in advancing AI research in Hebrew. The full list of data
contributors is updated and available on the ivrit.ai website.

We also extend our heartfelt thanks to Adv. Eli Greenbaum from Yigal Arnon &
Co., who generously provided his legal expertise pro bono to draft the license
for this open data project. His contribution has been instrumental in ensuring
the accessibility and wide distribution of the ivrit.ai dataset.

Your collective support and contributions have been instrumental in the
success of this project, and we look forward to seeing the advancements in AI
research that the ivrit.ai dataset will facilitate.

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##  Appendix A License Appendix

This material and data is licensed under the terms of the Creative Commons
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Notwithstanding the foregoing, this material and data may only be used,
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subject to the foregoing restriction. In addition, this material and data may
not be used in order to create audiovisual material that simulates the voice
or likeness of the specific individuals appearing or speaking in such
materials and data (a “deep-fake”). To the extent this paragraph is
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govern.

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aainstitutetext: II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149,
D-22761 Hamburg, Germanybbinstitutetext: Departamento de Fisica, Universidad
Tecnica Federico Santa Maria, Avda. Espana 1680, Casilla 110-V, Valparaiso,
Chile ccinstitutetext: INFN Sezione di Bologna and DIFA, via Irnerio 46,
I-40126 Bologna, Italy

# Emergent symmetries at criticality in multi field RFT/DP

Jochen Bartels b Carlos Contreras c Gian Paolo Vacca<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study multifield extensions of Reggeon Field Theory (also equivalent to
Directed Percolation model) at criticality in the perturbative
$\epsilon$-expansion below the upper critical dimension $D_{c}=4$ at one loop,
for the special case when all fields have the same scale (anomalous)
dimensions. Analyzing all the fixed points of the renormalization group flow
for $N=2$ flavors and some for $N=3$, we find scale invariant solutions which
are characterized by specific emergent global symmetries of the interacting
potential. We also study two infinite families as a function of $N$ having
$\mathbb{Z}_{N}$ and $S_{N+1}$ symmetries.

##  1 Introduction

In the formulation and study of phenomena in critical and almost-critical
physical systems quantum and statistical field theories serve as an important
and effective framework. Several aspects of both high energy physics of
particles with fundamental interactions and low energy physics of systems
subject to quantum/statistical fluctuations can be succesfully described, and
even useful descriptions beyond physics towards interdisciplinary domains can
be constructed. The most useful paradigm in this framework is given by the
renormalization group which, focusing on the description of a system at
different scales, permits to link different fundamental properties of a system
and in particular scaling, universality and symmetry. Typical systems which
are investigated undergo a continuous second order phase transition at
criticality, where the correlation lengths become infinite. In many critical
systems studied scale invariance is believed to be lifted to conformal
invariance (proof of that being available only in rare cases) making available
other means of investigations such as the conformal bootstrap in case of
unitary theories. The most famous and successful example of this is the QFT
description of the Ising model in three dimensions. On the other hand there
are systems of interest which are not conformal at criticality, in particular
the ones presenting anisotropic scaling, which are simply scale invariant.

In this work we focus on quantum/statistical models of this kind and in
particular on extensions of the so called Reggeon Field Theory, which was
formulated more that sixty years ago during the quest of understanding high
energy scattering of strongly interacting particles. After the formulation of
the theory Gribov:1967vfb ; Gribov:1968uy  several works were already devoted
to the perturbative investigation of its critical properties Abarbanel:1973pq
; Sugar:1974td . In this theory the role of time is played by the rapidity
while the space is the transverse space with respect to the scattering
direction. Even if nowadays strong interactions are successfully described by
QCD as a fundamental asymptotically free theory, still one expects that high
energy scattering in the Regge limit could be effectively described by some
kind of RFT derived from QCD. This idea is difficult to be investigated
because of the need of a non perturbative analysis, which has led to first
attempts based on the use of an exact functional renormalization group (FRG)
framework Bartels:2015gou ; Bartels:2018pin . This QFT is interesting also
because it was shown Cardy:1980rr  to describe completely different physical
phenomena in the framework of non equilibrium phase transitions in statistical
physics, known as Directed Percolation (DP). For a general introduction to the
subject and this paradigmatic model see HHL . Such equivalence with models of
non equilibrium physics has increased the interest in this theory Canet:2003yu
, with also a recent perturbative analyses up to three loops Adzhemyan:2023wbz
. All the perturbative analyses are typically performed using dimensional
regularization in the $\epsilon$-expansion below the upper critical dimension.

We remind here the form of a single field RFT, which is described by the
action

$S=\int d\tau
d^{D}x\Big{[}\psi^{{\dagger}}\left(i\partial{\tau}+\alpha^{\prime}\nabla^{2}+\alpha^{\prime}\mu\right)\psi-\frac{i}{2}\alpha^{\prime}\lambda\psi^{{\dagger}}\psi\left(\psi+\psi^{\dagger}\right)\Big{]}\,,$

where the fields live in $1+D$ dimensions and imaginary time is related to the
rapidity. Here $\alpha^{\prime}$ is the slope and $\mu$ is the intercept minus
one of the Regge trajectory. In high energy scattering the physical
(transverse) space dimension $D$ is two, while for Directed Percolation also
other dimensions are of interest. The dynamics respects the so called
projectile-target symmetry in the scattering, or in other words the rapidity-
reversal symmetry.

In high energy scattering quantum properties of the particles involved imply
the need to have several reggeon fields in the description. First even in the
QCD Pomeron analysis one may encounter a set of Pomeron states which will
eventually translate in an effective multifield RFT. More in general the
dominant contributions at high energies are given by the so called Pomeron and
the Odderon. These objects have their perturbative counterpart in simplified
analysis in QCD Fadin:1975cb ; Balitsky:1978ic  and Bartels:1999yt ,
respectively, but in full interacting QCD theory is difficult to formulate a
reliable description. Because of that an idea of considering an effective RFT
for these two fields with all the possible specific interactions was proposed
and the behavior at criticality studied both pertubatively Bartels:2016ecw ;
Braun:2023vos  and non perturbatively, with FRG methods Bartels:2016ecw . It
is not yet known if this model can have a physical interesting realization in
non equilibium statistical physics generalizing the Directed Percolation
model. Neverthelss one can find in the literature analyses tauber1 ;
Janssen:2000ds  related to multispecies populations close to criticality,
generalizing the DP universality class. This goes in the direction that we
want to investigate in this work.

We shall consider here a multi-field extension of Reggeon Field Theory (RFT)
and Directed Percolation models with $N$ fields flavors (reggeons or order
parameters/species), focusing on specific families of theories which, allowing
real rotations in the flavor space, maintain a global $O(N)$ symmetry at
criticality in the two point function, i.e. the non interacting part of the
action. Such a choice corresponds in RFT to the picture of having $N$
interacting reggeons with the same Regge trajectories. We shall allow a
completely general interacting critical potential and see how the O(N)
symmetry is broken at criticality to some smaller, typically discrete,
symmetry so that there will exist a orbit in the space of potentials
(couplings) invariant under a subgroup of $O(N)$ of physically equivalent
scale invariant theories. This kind of analysis has been performed for example
in an exaustive way, eventually with the help of numerical techniques, in the
search of fixed point in interacting standard multi-field scalar theories up
to $N=3$ fields Osborn:2017ucf ; Codello:2019isr  for cubic interactions and
up to $N=4$ fields Osborn:2017ucf ; Codello:2020lta  with quartic
interactions, with further searches pushed up to $N=7$ fields Osborn:2020cnf .

We shall restrict our analysis to the one-loop perturbative
$\epsilon$-expansion, which is nevertheless enough to grasp the main
qualitative features of these systems.

Let us stress that the set of theories studied here are covering only a part
of the possible multi-field theories: for example the Pomeron-Odderon system
does not satisfy the constraints adopted here, since the two reggeons at
criticality have been shown to have different slopes, both in the one loop
approximation and in non perturbative settings as well. Indeed the system that
we investigate in this work must satisfy additional constraints on top of the
fixed point conditions of the RG flow.

The symmetries of the scale invariant theories found, can be considered
emergent in the sense that we have not imposed them from the start but they
were found as the ones characterizing the quantum scale invariant solutions.

Outline of the work: In Section 2 we start by defining the set of multi-field
theories we want to investigate, discussing some of the features which we need
to take into account during the study. Section 3 is the core of this work and
contains the results of our investigations. After discussing some general
strategy we present an analysis of the scale invariant theories for $N=2$,
adding some details in the discussion of the polynomial invariants, build from
the couplings of the potential, under $O(2)$ transformations. This is followed
by a partial analysis of the case with $N=3$ fields, which is much more
involved. One of the important features we analyze are the global symmetries
of the scaling invariant solutions. This allows us from what found for $N=2$
and $N=3$ to guess the form of two families of solution for a generic number
$N$ of fields. These two families are then generally investigated in Sections
3.3 and 3.4. After our conclusions two Appendices are given. In the first
Appendix A we collect the derivation of the perturbative RG flow equations and
conditions used in the analysis of Section 3. In the second Appendix B one can
find some definitions and relations used to study in Section 3.3 one of the
infinite families for general $N$.

##  2 Multi-field RFT/DP model

Let us define here the theory which we want to investigate. We shall consider
a set of fields and their conjugate $(\psi_{i},\psi_{i}^{\dagger})$, for
$i=1,\cdots,N$, whose evolution and interactions are defined by an action of
the form

$\displaystyle S=\int d\tau
d^{D}x\Big{[}\sum_{j}(\psi_{j}^{{\dagger}})\left(i\partial{\tau}+\alpha^{\prime}\nabla^{2}+\alpha^{\prime}\mu\right)\psi_{j}-V(\psi,\psi^{\dagger})\Big{]},$
(1)

where

$\displaystyle
V(\psi,\psi^{\dagger})=\sum_{jkl}\frac{i}{2}\alpha^{\prime}\lambda_{j,kl}\left(\psi_{j}^{{\dagger}}\psi_{k}\psi_{l}+\psi_{j}\psi_{k}^{{\dagger}}\psi_{l}^{{\dagger}}\right)\,.$
(2)

Here $\psi_{i}$ is an element of the vector $\psi$. In general
$\alpha^{\prime}$ (the so called regge slope) and $\mu$ (mass parameter
related to the intercept) can be real symmetric matrices (with one of them
always diagonalizable), but in this analysis we shall consider the case for
which all the fields have the same properties, i.e. the kinetic and mass terms
of the action enjoy a full $O(N)$ symmetry in flavor space (requiring this
property to be true even with the interactions turned on) and $\psi$ and
$\psi^{\dagger}$ transform as vectors under rotations. Other cases for which
such a symmetry is not present from the start are of course physically
relevant and an example in RFT is given by an already mentioned system of two
reggeons, i.e. a Pomeron interacting with an Odderon Bartels:2016ecw ;
Braun:2023vos .

In the present analysis we therefore make also a further choice: we enforce
the conditions ${\cal C}$ that radiative quantum/statistical corrections do
not spoil the $O(N)$ invariance of the quadratic part of the effective action
at criticality. In this way the full $O(N)$ transformation on the field
vector, $\psi^{\prime}=R_{r}\psi$ and
$\psi^{\dagger}{}^{\prime}=R_{r}\psi^{\dagger}$ for $r\in O(N)$, induces in
general a corresponding transformation on the interacting potential
$V=\lambda^{T}P(\psi,\psi^{\dagger})$, where $\lambda$ is a vector of
couplings and $P$ is a vector of all possible monomial composite operators of
the fields. Indeed the action of $r\in O(N)$ induces a transformation
$P^{\prime}=\tilde{R}_{r}P$ and therefore
$\lambda^{\prime}=\tilde{R}^{T}_{r}\lambda$ . One can also decompose this
space of couplings $\lambda$ according to the irreducible representations of
$O(N)$ and also construct invariants of different order in the couplings. We
shall briefly discuss this for the specific $N=2$ case in Section 3.1.1.

The goal of this work is to investigate with renormalization group techniques
the possible critical theories, obtained from the fixed points as zeros of the
beta functions (and satisfy also the additional constraints ${\cal C}$) which
are collected in a vector transforming also in the same irreducible
representation of $O(N)$ as the coupling vector.

Each scale invariant solution, with couplings $\lambda_{*}$, is characterized
by a global symmetry defined in terms of a subgroup $G\subset O(N)$ which
leaves the interacting potential invariant, i.e.
$\lambda_{*}=\tilde{R}_{g}\lambda_{*}$ for $g\in G$. It is then natural,
considering the action of an $O(N)$ transformation, to define a coset $O(N)/G$
which characterizes an orbit in the space of the couplings $\lambda$
(potential $V$) with the same physical properties (same universal critical
exponents).

Therefore the framework we consider is also interesting from the theoretically
point of view since we show that any scale invariant solutions may have an
emergent symmetry, typically realized by some finite point group $G\subset
O(N)$.

The renormalization group analysis is based on the study of the beta functions
$\beta_{\lambda}$ which are derived in Appendix A, together with the anomalous
dimensions and the constraints ${\cal C}$ needed to preserve the $O(N)$
symmetry in the part of the action quadratic in the fields. In particular we
have: the anomalous dimensions

$\displaystyle\gamma_{ij}$ $\displaystyle=$
$\displaystyle-\frac{1}{8}\frac{1}{(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}=\gamma\,\delta_{ij}$
(3)

and

$\displaystyle\tilde{\zeta}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\frac{1}{8(4\pi)^{2}}\lambda_{i,lm}\lambda_{i,lm}=-\gamma$
(4)

together with $\kappa_{\perp}=-3\gamma$ and the beta functions

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\beta_{\lambda_{i,jk}}=\beta_{i,jk}$
$\displaystyle=$
$\displaystyle-\frac{\epsilon}{2}\lambda_{i,jk}+\tilde{\zeta}\lambda_{i,jk}+3\gamma\lambda_{i,jk}+\frac{1}{2}\frac{1}{(4\pi)^{2}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)\,.$
(5)

These equations have to be completed adding the relations ${\cal C}$ of Eq.
(48) which imply the special form of the anomalous dimension matrix
$\gamma_{ij}$.

We also introduce the stability matrix $M_{pq}=\frac{\partial\beta_{\lambda
p}}{\partial\lambda_{q}}$ whose eigenvalues give the universal and invariant
critical exponents which characterizes the stability properties of a fixed
point, following the evolution under the RG flow after having added a
perturbation.

Les us count in general the number of all possible couplings associated to
cubic interactions and the number of relations that we have. For general $N$
we have $\frac{N^{2}(N+1)}{2}$ different couplings and the same number of beta
functions, which are cubic polynomials in the couplings. Moreover we have
$(N-1)+\frac{N(N-1)}{2}=\frac{(N+2)(N-1)}{2}$ quadratic contraints ${\cal C}$
coming from the requirement of the existence of a single quadratic invariant
making all the field anomalous dimension equal, which are given in Eq. (48).
Finally the continuous $SO(N)\subset O(N)$ symmetry has $\frac{N(N-1)}{2}$
generators, which can imply that the $O(N)$ symmetry reduces the number of
independent physically couplings, since the orbits generated are corresponding
to the same physics. Let us summarize these numbers for $N$ up to $4$:

$\displaystyle\left(\begin{array}[]{cccc}N&\\#{\rm couplings}&\,\,\\#O(N){\rm
generators}&\\#{\rm constraints}\\\ 1&1&0&0\\\ 2&6&1&2\\\ 3&18&3&5\\\
4&40&6&9\end{array}\right)$ (11)

We finally remind the one loop results for a single reggeon (Pomeron), i.e.
$N=1$, which has a single coupling and just one scaling solution, fixed point
of the beta function:

$\displaystyle\beta_{\lambda}=-\frac{\epsilon}{2}\lambda+\frac{3}{4}\frac{\lambda^{3}}{(4\pi)^{2}}.$
(12)

At the fixed point one has the following values for the coupling, anomalous
dimensions and critical exponents:

$\displaystyle\frac{\lambda_{*}^{2}}{(4\pi)^{2}}=\frac{2}{3}\epsilon\,,\quad\gamma=-\frac{1}{12}\epsilon\,,\quad\tilde{\zeta}=\frac{1}{12}\epsilon\,,\quad\nu_{\perp}=\frac{1}{2}+\frac{1}{16}\epsilon\,,\quad\frac{\partial\beta_{\lambda}}{\partial\lambda}=\epsilon\,.$
(13)

##  3 Search strategy and results

In this section we study the system of $N$ fields at criticality using the one
loop RG flow equations in the $\epsilon$-expansion below the upper critical
dimension $D_{c}=4$. First we consider the case $N=2$ and perform a systematic
search of fixed points of the RG flow giving all possible solutions. Then we
perform a partial analysis for the case $N=3$ and present here all the
solutions which are charaterized by up to $5$ independent monomial composite
operators in the critical potential. Finally, observing the emergent pattern
of symmetries, we then consider and study two infinite families of critical
theories for any number $N$ of field flavors.

In general we show only the interacting solutions which are truly new and not
factorizable in known solutions of a smaller number of fields. Since each
solution is actually an orbit, given the underlining $O(N)$ symmetry which we
require for the quadratic part of the effective action, i.e. for the two point
function, we present them with the simplest representative we can find.

To distinguish among physically inequivalent fixed point solutions, which
belong to different orbits, an important guide is given by the study of the
invariant quantities under $O(N)$ transformations. In particular we have
considered the anomalous dimension and the set of critical exponents, which
are the eigenvalues of the stability matrix (gradient of the beta function
vectors). As a further criterion we have considered only solutions for which
exists a representative with couplings that can be or purely real or purely
imaginary but not fully complex. We stress that since the perturbative
analysis is restricted to one loop, higher loop investigations could
eventually disantangle possible degeracies present in fixed points found at
the lowest one loop order.

###  3.1 Scale invariant solutions for $N=2$

The case $N=2$ which has the six different couplings

$\\!\\!\\!\\!\\!(\lambda_{1},\,\,\lambda_{2},\,\,\lambda_{3},\,\,\lambda_{4},\,\,\lambda_{5},\,\,\lambda_{6})=(\lambda_{1,11},\,\,\lambda_{1,22},\,\,\lambda_{2,11},\,\,\lambda_{2,22},\,\,\lambda_{1,21}=\lambda_{1,12},\,\,\lambda_{2,21}=\lambda_{2,12})\,.$
(14)

Here we are interested in fixed points where possibly only one, two, three,
four, or five couplings are different from zero, since by an $O(2)$ rotation
at least one coupling can be set to zero. Using the couplings in Eq. (14) the
critical potential in general reads

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!V=\frac{i}{2}\Biggl{(}$
$\displaystyle\lambda_{1}\psi_{1}^{{\dagger}}(\psi_{1}^{{\dagger}}+\psi_{1})\psi_{1}+\lambda_{4}\psi_{2}^{{\dagger}}(\psi_{2}^{{\dagger}}+\psi_{2})\psi_{2}+\lambda_{2}(\psi_{2}^{{\dagger}}{}^{2}\psi_{1}+\psi_{1}^{{\dagger}}\psi_{2}{}^{2})+$
(15)
$\displaystyle\lambda_{3}(\psi_{1}^{{\dagger}}{}^{2}\psi_{2}+\psi_{2}^{{\dagger}}\psi_{1}{}^{2})+2\lambda_{5}(\psi_{1}^{{\dagger}}\psi_{1}\psi_{2}+\psi_{1}^{{\dagger}}\psi_{2}^{{\dagger}}\psi_{1})+2\lambda_{6}(\psi_{2}^{{\dagger}}\psi_{1}\psi_{2}+\psi_{1}^{{\dagger}}\psi_{2}^{{\dagger}}\psi_{2})\Biggr{)}$

so that the exchange of the two field flavors $(1,2)$ corresponds to the
exchanges $(\lambda_{1},\lambda_{4})$, $(\lambda_{2},\lambda_{3})$ and
$(\lambda_{5},\lambda_{6})$.

First of all we have imposed the conditions ${\cal C}$ (48) of having
counterterms to the two point function proportional to the unity $2\times 2$
matrix, which amount to two extra equations, to be considered together with
the six fixed point conditions.

We present the solutions found in Table 1 and in particular give for each
critical theory a representative of the fixed point potential, its symmetry,
and the anomalous dimensions $\gamma$ and $\tilde{\zeta}$. Moreover for each
fixed point we have also computed the eigenvalues $e_{n}$ of the stability
matrix (related to the six cubic operators which can be present in the
potential). Positive eigenvalues indicate which are the IR stable eigen-
directions in the theory space charted by the couplings while UV stable
directions correspond to the negative values. Moreover the solutions presented
here (and in Table 2 as well) have
$\nu_{\perp}=\frac{1}{2}+\frac{\kappa_{\perp}}{4}=\frac{1}{2}+\frac{3}{4}\tilde{\zeta}$.

We remind that the form of the potential of the solutions can change without
affecting the physics thanks to the freedom of performing an $SO(2)(\subset
O(2))$ rotation, since the kinetic terms are invariant. Such a rotation has
one generator and can be used to set to zero one coupling to simplify the
searching procedure and picking up possibly a simpler representative of the
orbit associated to a scaling solution.

Therefore we show a convenient representative for each different solution.
Anomalous dimensions and eigenvalues of the stability matrix are clearly
invariant under $O(2)$ field transformations.

We have included in the first row of the table also the ”trivial” decoupled
system of two self interacting single field (pomeron) solutions to show the
stability properties. In the following we shall not show any other solution
which can be factorized in terms of solutions with a smaller number of fields.
Clearly such a solution as a $\mathbb{Z}_{2}$ symmetry for the exchange of the
two fields and a $\mathbb{Z}_{PT}$ for the change of sign of the fields is
combined with complex conjugation.

Then we have found four new non trivial scaling solution. The solution in the
second row has a symmetry associated to the change of sign of $\psi_{2}$
combined with complex conjugation, again named $\mathbb{Z}_{PT}$ with the two
flavors interacting among each other. It has the same anomalous dimension but
different stability properties compared to the first solution. The solution in
the third row has the same anomalous dimensions of the previous two, other
stability properties and the symmetry is a bit enhanced, since, apart a
$\mathbb{Z}_{PT}$ symmetry realized as before, there is also a
$\mathbb{Z}_{2}$ component associated to the change of sign of $\psi_{1}$.

In the fourth and fifth rows of Table 1 we find two solution with different
anomalous dimensions and new kind of symmetries. In the fourth row the
solution has a full diehedral point group symmetry $\mathbb{D}_{3}=S_{3}$
which we shall see that lies at the bottom of a tower of solutions with
permutational symmetry. Finally another solution with a cyclic symmetry has
been found and report in the fifth row, which is also the first element of a
family of solutions with more fields to be discussed later.

Table 1: Scale invariant solutions given by a representative of the cubic potential of each orbit, together with its finite symmetry encoded in the finite subgroup $G\subset O(2)$, the eigenvalues of the stability matrix and the anomalous dimensions $\tilde{\zeta}$ and $\gamma$.  | $V\frac{4\pi}{\sqrt{\epsilon}}$ | $G\subset O(2)$ | $(e_{1},e_{2},e_{3},e_{4},e_{5},e_{6})$ | $\tilde{\zeta}/\epsilon$ | $\gamma/\epsilon$
---|---|---|---|---|---
1 | $\frac{i}{\sqrt{6}}\left[\psi_{1}^{\dagger}\psi_{1}(\psi_{1}+\psi_{1}^{\dagger})+\psi_{2}^{\dagger}\psi_{2}(\psi_{2}+\psi_{2}^{\dagger})\right]$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{PT}$ | $\frac{1}{6}(-2,-2,-1,0,5,6)$ | $\frac{1}{12}$ | $-\frac{1}{12}$
2 |  | $\frac{i}{\sqrt{6}}\psi_{1}^{\dagger}\psi_{1}(\psi_{1}+\psi_{1}^{\dagger})+i\sqrt{\frac{2}{3}}\psi_{1}^{\dagger}\psi_{1}(\psi_{2}+\psi_{2}^{\dagger})$
---
$+\frac{i}{\sqrt{6}}\psi_{2}^{\dagger}\psi_{2}(\psi_{2}+\psi_{2}^{\dagger})$
$\mathbb{Z}_{PT}$ | $\frac{1}{6}(-1,0,0,2,5,6)$ |  |
3 |  | $\frac{i}{2\sqrt{3}}\Bigl{[}\psi_{2}^{\dagger}\psi_{2}(\psi_{2}+\psi_{2}^{\dagger})+2\psi_{1}^{\dagger}\psi_{1}(\psi_{2}+\psi_{2}^{\dagger})$
---
$-(\psi_{2}^{\dagger}\psi_{1}^{2}+\psi_{1}^{\dagger}{}^{2}\psi_{2})\Bigr{]}$
$\mathbb{Z}_{2}\times\mathbb{Z}_{PT}$ | $\frac{1}{6}(-2,-2,-1,-1,0,6)$ |  |
4 |  | $-\frac{1}{2}\psi_{2}^{\dagger}\psi_{2}(\psi_{2}+\psi_{2}^{\dagger})+\psi_{1}^{\dagger}\psi_{1}(\psi_{2}+\psi_{2}^{\dagger})$
---
$+\frac{1}{2}(\psi_{2}^{\dagger}\psi_{1}^{2}+\psi_{1}^{\dagger}{}^{2}\psi_{2})$
$\mathbb{D}_{3}=S_{3}$ | $\frac{1}{2}(-7,-7,0,2,2,2)$ | $-\frac{1}{4}$ | $\frac{1}{4}$
5 | $\frac{1}{\sqrt{2}}\left[(\psi_{2}^{\dagger}\psi_{1}^{2}+\psi_{1}^{\dagger}{}^{2}\psi_{2})+(\psi_{1}^{\dagger}\psi_{2}^{2}+\psi_{2}^{\dagger}{}^{2}\psi_{1})\right]$ | $\mathbb{Z}_{2}$ | $\frac{1}{2}(-7,-2,-1,0,2,2)$ |  |

####  3.1.1 Invariants

In the following, even if we do not really need here, we briefly remind how
one may construct explicit irreducible representations of $SO(2)$ under which
transforms the potential in the basis of the couplings (14). Considering an
infinitesimal rotation of the fields in $\theta$

$\psi^{\prime}=R\psi\,,\quad\psi^{\dagger}{}^{\prime}=R\psi^{\dagger}\,,\quad
R=\left(\begin{array}[]{cc}1&\theta\\\ -\theta&1\\\ \end{array}\right)\,,$
(16)

one finds the induced transformation on the coupling vector (potential)

$\tilde{R}_{\theta}^{T}=\left(\begin{array}[]{cccccc}1&0&\theta&0&\theta&0\\\
0&1&0&\theta&-\theta&0\\\ -\theta&0&1&0&0&\theta\\\ 0&-\theta&0&1&0&-\theta\\\
-2\theta&2\theta&0&0&1&\theta\\\ 0&0&-2\theta&2\theta&-\theta&1\\\
\end{array}\right),$ (17)

which has eigenvalues $\rho_{j}$ and the corresponding eigenvectors $u_{j}$ in
the complex form given by

$\left(\begin{array}[]{ccl}\rho_{1}=1-i\theta&u_{1}^{T}=&(0,-1,i,0,-i,1)\\\
\rho_{2}=1-i\theta&u_{2}^{T}=&(i,i,1,1,0,0)\\\
\rho_{3}=1-3i\theta&u_{3}^{T}=&\left(-\frac{1}{2},\frac{1}{2},\frac{i}{2},-\frac{i}{2},i,1\right)\\\
\rho_{1}^{*}=1+i\theta&u_{1}^{*T}=&(0,-1,-i,0,i,1)\\\
\rho_{2}^{*}=1+i\theta&u_{2}^{*T}=&(-i,-i,1,1,0,0)\\\
\rho_{3}^{*}=1+3i\theta&u_{3}^{*T}=&\left(-\frac{1}{2},\frac{1}{2},-\frac{i}{2},\frac{i}{2},-i,1\right)\\\
\end{array}\right)\,.$ (18)

With these eigenvectors we can construct the combination of couplings which
transforms diagonally under an infinitesimal $SO(2)$ rotations as
$u_{i}^{T}\lambda\to\rho_{i}u_{i}^{T}\lambda$. Note that the eigenvalues for a
finite transformation are essentially phase factors. Therefore monomials
constructed as a product of powers the $u_{i}^{T}\lambda$ and
$u_{i}^{*T}\lambda$, for which the product of the corresponding phase factor
in the transformations becomes unity, are invariants. In this case it is
evident that we can construct easily both quadratic and quartic invariants.
The anomalous dimensions and the eigenvalues of the stability matrix can
therefore be written in terms of an independent set of the invariants.

###  3.2 Scale invariant solutions for $N=3$

We perform here a partial investigation for the three field case ($N=3$). In
this case the most generic potential is characterized by $18$ couplings. The
conditions ${\cal C}$ of having counterterms from the two point function
proportional to the unity $3\times 3$ matrix, amount to five extra equations.
The $O(3)$ symmetry group contains the continuous $SO(3)$ group which has
three generators, and can be used to constrain three couplings. But in the
following we are not attempting to perform an exhaustive search, for example
generalizing what was done in Codello:2019isr ; Codello:2020lta  and we shall
ignore this possibility. Our strategy will consider the search of solutions
characterized by few non zero couplings, most likely the simpler ones which we
expect to be the most symmetric, and will stop at some point, leaving for
future investigations a possible complete analysis.

Table 2: Scale invariant solutions given by a representative of a cubic potential of each orbit, together with its finite symmetry encoded in the finite subgroup $G\subset O(2)$. The corresponding eigenvalues of the stability matrix are given in Table 3. | $V\frac{4\pi}{\sqrt{\epsilon}}$ | $G\subset O(2)$ | $\tilde{\zeta}/\epsilon$ | $\gamma/\epsilon$
---|---|---|---|---
1 | $\frac{1}{\sqrt{2}}\left(\psi_{1}\psi^{\dagger}_{2}{}^{2}+\psi_{2}\psi^{\dagger}_{3}{}^{2}+\psi_{3}\psi^{\dagger}_{1}{}^{2}\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{3}$ |  -$\frac{1}{4}$ | $\frac{1}{4}$
2 | $i\left(\psi_{1}\psi^{\dagger}_{2}\psi^{\dagger}_{3}+\psi_{2}\psi^{\dagger}_{3}\psi^{\dagger}_{1}+\psi_{3}\psi^{\dagger}_{1}\psi^{\dagger}_{2}\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $S_{4}\times\mathbb{Z}_{PT}$ | $\frac{1}{4}$ |  -$\frac{1}{4}$
3 | $i\left(\psi_{1}\psi^{\dagger}_{2}\psi^{\dagger}_{3}-\psi_{2}\psi^{\dagger}_{3}\psi^{\dagger}_{1}+\psi_{3}\psi^{\dagger}_{1}\psi^{\dagger}_{2}\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{2}^{3}\times\mathbb{Z}_{PT}$ |  |
4 | $i\left(\psi_{1}\psi^{\dagger}_{2}\psi^{\dagger}_{3}+\psi_{2}\psi^{\dagger}_{1}\psi^{\dagger}_{3}+\frac{1}{2}(\psi_{1}^{2}+\psi_{2}^{2})\psi_{3}^{\dagger})\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{PT}$ |  |
5 | $\frac{i}{\sqrt{3}}\psi^{\dagger}_{1}\psi_{1}\psi_{3}+\frac{i}{2\sqrt{3}}\psi^{\dagger}_{3}(\psi_{3}^{2}-\psi_{1}^{2})+\frac{i}{\sqrt{2}}\psi^{\dagger}_{2}\psi_{2}^{2}-\sqrt{\frac{2}{3}}\psi_{1}\psi_{2}\psi^{\dagger}_{2}+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{PT}$ | $\frac{1}{12}$ | $-\frac{1}{12}$
6 | $\frac{i}{2}\sqrt{\frac{3}{2}}\psi_{3}\left(\psi^{\dagger}_{1}\psi_{1}+\psi^{\dagger}_{2}\psi_{2}\right)+\frac{i}{\sqrt{6}}\psi_{3}\psi^{\dagger}_{3}{}^{2}+\frac{1}{2\sqrt{6}}\psi_{3}\left(\psi_{1}\psi^{\dagger}_{2}-\psi_{2}\psi^{\dagger}_{1}\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{8}$ |  |
7 | $i\sqrt{\frac{2}{3}}\psi_{3}\left(\psi^{\dagger}_{1}\psi_{1}+\psi^{\dagger}_{2}\psi_{2}\right)+\frac{i}{\sqrt{6}}\psi_{3}\psi^{\dagger}_{3}{}^{2}-\frac{1}{\sqrt{6}}\left(\psi_{1}\psi^{\dagger}_{1}{}^{2}+\psi_{2}\psi^{\dagger}_{2}{}^{2}\right)+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{PT}$ |  |
8 | $\frac{i}{2\sqrt{3}}\psi_{3}\left(\psi^{\dagger}_{3}{}^{2}-\psi^{\dagger}_{2}{}^{2}\right)+i\psi_{3}\left(\frac{\sqrt{3}}{2}\psi_{1}\psi^{\dagger}_{1}+\frac{1}{\sqrt{3}}\psi_{2}\psi^{\dagger}_{2}\right)-\frac{1}{2}\sqrt{\frac{5}{3}}\psi_{1}\psi^{\dagger}_{1}\psi_{2}+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})$ | $\mathbb{Z}_{2}\times\mathbb{Z}_{PT}$ |  |

We present them in Table 2 and give the eigenvalues of the stability matrix in
Table 3. We do not show the solution with three equal decoupled pomerons,
which is clearly present. It has $3$ positive, $3$ null and $12$ negative
eigenvalues of the stability matrix. We do not show also all the solutions
characterized by one reggeon system decoupled from a coupled system of the
other two, since they can be constructed by combining a single pomeron system
solution with the solutions given in the previous subsection. The only piece
of information which could be of interest is the one regarding the stability
properties looking at the eigenvalues of the stability matrix, which can be
easily obtained in case of need.

We consider and present here all the non trivial solutions (independent
orbits) which have up to five non zero couplings.

With three non zero couplings we find three solutions, which in Table 2 are
numbered from $1$ to $3$. The first one has a real interacting potential,
cyclic symmetric in the flavor, corresponding to a $\mathbb{Z}_{3}$ symmetry.
We note that a solution with such symmetry property has been discussed in
Janssen:2000ds . It is the second member of a general family of cyclic
symmetric scaling solutions for any $N$ which will be discussed later. The
second solution has an $S_{4}\times\mathbb{Z}_{PT}$ , where with
$\mathbb{Z}_{PT}$ we means a change of sign and complex conjugation. Also this
is the second instance of a solution belonging to a general family, this time
with $S_{N+1}$ permutation symmetry. They will be discussed in general in the
next subsection. The third solution in Table 2 is the last non trivial one
that we have found allowing only three non zero couplings. It looks similar to
the previous one apart from a sign changed which makes it is less symmetric,
specifically with a $\mathbb{Z}_{2}^{3}\times\mathbb{Z}_{PT}$ symmetry. The
$3$ factors $\mathbb{Z}_{2}$ denote the symmetry under the exchange of the
fields $(1,3)$, the change of sign of both of them and the change of sign of
another pair of fields, e.g. $(1,2)$.

Allowing a fourth non zero coupling we find another non trivial solution
(number $4$ in Table 2) with symmetry
$\mathbb{Z}_{2}^{2}\times\mathbb{Z}_{PT}$, realized by exchange of the fields
$(1,2)$ and changing the sign of both of them. As usual the rest of the
symmetry present is related to a change of sign and complex conjugation.

Finally allowing five non zero couplings we find four more solutions numbered
from $5$ to $8$ in Table 2. The first (number $5$) has just a
$\mathbb{Z}_{PT}$ symmetry realized by changing sign of the pair of fields
$(2,3)$ and performing a complex conjugation. The solution number $6$ has a
cyclic $\mathbb{Z}_{8}$ symmetry describing the invariance under rotation with
an angle $\pi/4$ in the plane of the field flavors $(1,2)$. The solution in
the $7$th row of Table 2 is symmetric under exchange of fields $(1,2)$ and
under change of sign of field $3$ together with complex conjugation. The last
solution (number $8$) has a $\mathbb{Z}_{2}$ factor because it is symmetric
under the change of sign of field $1$ and the same realization of
$\mathbb{Z}_{PT}$ as in the previous solution.

The degree of stability under perturbations according to the RG flow of all
these $8$ fixed point solutions is reported in Table 3.

We have also analyzed the case with $6$ non zero couplings finding other
independent solutions but we do not find it worth to report them here.

Table 3: Eigenvalues of the stability matrix of the solutions presented in Table 2. For the solution 1 and 5 we give the numerical values when the analytic form in constructed from roots of quartic equations. |  $(e_{1},\cdots,e_{18})$
---|---
1 |  | $(-1.76181,-1.76181,-1.76181,-0.720398-0.832342i,-0.720398-0.832342i,-0.720398-0.832342i,-0.720398+0.832342i,$
---
$0.720398+0.832342i,-0.720398+0.832342i,0.,0.,0.,0.\,-0.866025i,0.\,+0.866025i,0.702608,0.702608,0.702608,1.)$
2 | $(-\sqrt{7},-\sqrt{7},-\sqrt{7},-2,-2,-2,-1,-1,-1,-1,-1,0,0,0,1,\sqrt{7},\sqrt{7},\sqrt{7})$
3 | $\left(-\sqrt{7},-\frac{1}{2}(3+\sqrt{5}),-\frac{1}{2}(3+\sqrt{5}),-2,-1,-1,-1,-1,-1,\frac{1}{2}(\sqrt{5}-3),\frac{1}{2}(\sqrt{5}-3),0,0,0,0,0,1,\sqrt{7}\right)$
4 | $\left(-\sqrt{7},-\sqrt{7},-\frac{1}{2}(3+\sqrt{5}),-\frac{1}{2}(3+\sqrt{5}),-2,-1,-1,-1,-1,\frac{1}{2}\left(\sqrt{5}-3\right),\frac{1}{2}\left(\sqrt{5}-3\right),0,0,0,0,1,\sqrt{7},\sqrt{7}\right)$
5 |  | $\bigl{(}\,-\frac{1}{3}(1+\sqrt{2}),-\frac{2}{3},-0.603553-0.441845i,-0.603553+0.441845i,-0.383164,-\frac{1}{3},-\frac{1}{3},$
---
$-\frac{1}{6},-\frac{1}{6},0,0,0,0,\frac{1}{3}(\sqrt{2}-1),0.590271,1,\frac{1}{12}(15-i\sqrt{15}),\frac{1}{12}(15+i\sqrt{15})\,\bigr{)}$
6 | $\left(-\frac{1}{6},0,0,0,0,0,0,0,\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{3},\frac{1}{3},\frac{1}{2},\frac{1}{2},\frac{5}{6},\frac{5}{6},1\right)$
7 | $\left(-\frac{1}{6},-\frac{1}{6},0,0,0,0,0,\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{2}{3},\frac{5}{6},\frac{5}{6},\frac{5}{6},1\right)$
8 | $\left(-\frac{1}{3},-\frac{1}{3},\frac{1}{6}(1-\sqrt{5}),-\frac{1}{6},-\frac{1}{6},-\frac{1}{6},\frac{1}{12}(1-\sqrt{5},0,0,0,0,0,\frac{1}{12}(1+\sqrt{5}),\frac{1}{6}(4-\sqrt{5}),\frac{1}{2},\frac{1}{6}(1+\sqrt{5}),1,\frac{1}{6}(4+\sqrt{5})\right)$

###  3.3 A family of $S_{N+1}$ symmetric fixed point solutions

Having observed that for $N=2$ exists a fixed point solution ($4$th row in
Table 1) with $S_{3}$ symmetry and for $N=3$ one with $S_{4}$ symmetry, it is
natural to expect the existence for $N$ fields of a fixed point invariant
under point group $S_{N+1}$.

For this highly symmetric family of solutions the interacting potential can be
described by a single coupling according to

$\lambda_{i,jk}=\lambda\,q^{(3)}_{ijk}$ (19)

where the fully symmetric $q^{(3)}$ tensor is defined and discussed in
Appendix B. Let us stress that this representation is valid for all $N\geq 2$,
while the case $N=1$ is not included in this description.

As previously discussed we remind that the explicit form of the potential of
these critical theories is not uniquely determined as we know that it is
associated to a full orbit of physically equivalent fixed point potentials
transforming under a representation of the coset $O(N)/S_{N\\!+\\!1}$. For
example let us consider the two solutions with this symmetry previously found
for $N=2$ and $N=3$. The case $N=2$ can be obtained considering the following
set of three vectors of ${\mathbb{R}}^{2}$

$e^{1}=\sqrt{2}(0,-1)^{T}\,,\,e^{2}=\frac{\sqrt{2}}{2}(\sqrt{3},1)^{T}\,,\,e^{3}=\frac{\sqrt{2}}{2}(-\sqrt{3},1)^{T}$
(20)

which correspond to the vertices of an equilateral triangle, while for the
case $N=3$ the explicit solution found correspond to the set of the four
vectors of ${\mathbb{R}}^{3}$

$e^{1}=(1,1,-1)^{T}\,,\,e^{2}=(-1,-1,-1)^{T}\,,e^{3}=(1,-1,1)^{T}\,,\,e^{4}=(-1,1,1)^{T}$
(21)

which correspond to the vertices of a tetrahedron.

Using the relation (73) the counterterms for the two point function becomes
diagonal in the flavor and automatically satisfy the constraints in Eq. (48).
In particular we obtain

$\gamma=-\frac{N-1}{8}\frac{\lambda^{2}}{(4\pi)^{2}}\,,\quad\tilde{\zeta}=\frac{N-1}{8}\frac{\lambda^{2}}{(4\pi)^{2}}\,,\quad\kappa_{\perp}=\frac{3}{8}(N-1)\frac{\lambda^{2}}{(4\pi)^{2}}\,.$
(22)

The counterterm associated to the triple interation is strongly simplified
thanks to the relation (74). Putting all the results together as in Appendix A
one obtains a simple form for the beta function for the coupling $\lambda$. It
is convenient to introduce the rescaled coupling
$\tilde{\lambda}=\lambda/(4\pi)$, for which the beta function reads

$\beta_{\tilde{\lambda}}=-\frac{\epsilon}{2}\tilde{\lambda}-\frac{N\\!-\\!1}{4}\tilde{\lambda}^{3}+(N\\!-\\!2)\tilde{\lambda}^{3}\,.$
(23)

We note that for $N=2$ the vertex correction at one loop is zero and the fixed
point is non trivial because of a non zero anomalous dimension. At the fixed
point one has

$\tilde{\lambda}_{*}^{2}=\frac{2\epsilon}{3N\\!-\\!7}\,,\quad-\gamma=\tilde{\zeta}=\kappa_{\perp}/3=\frac{N\\!-\\!1}{3N\\!-\\!7}\frac{\epsilon}{4}\,,\quad\frac{\partial\beta_{\tilde{\lambda}}}{\partial\tilde{\lambda}}=\epsilon\,.$
(24)

One can easily check that for example for $N=2,3,4$ one has
$\gamma=\frac{\epsilon}{4},-\frac{\epsilon}{4},-\frac{3}{20}\epsilon$ so that
the previously obtained results for $N=2$ and $N=3$ are reproduced. One
obtains also

$\nu_{\perp}=\frac{1}{2}+\frac{3}{16}\frac{N\\!-\\!1}{3N\\!-\\!7}\epsilon\,.$
(25)

For $N\geq 3$ the fixed point coupling $\lambda_{*}$ is real and therefore the
symmetry is enhanced to include a global change of sign of the fields together
with a complex conjugation, which we name with $\mathbb{Z}_{PT}$. Curiously,
one can notice that the properties associated to $N=1$, single pomeron case,
are reproduced in the limit $N\to\infty$.

It is well known that an Euclidean theory of a scalar field multiplet with a
symmetric cubic interaction as above is describing a Potts model and there
exist meaningful cases for the limit $N\to 0$ and $N\to-1$, associated to
percolation and spanning forest statistical models respectively. It is not
clear if in our nonisotropic case such limits can have some kind of physical
interpretation.

###  3.4 A family of $\mathbb{Z}_{N}$ symmetric fixed point solutions

Solutions with a cyclic symmetry were found for $N=2$ ($5$th row in Table 1)
and $N=3$ ($1$st row in Table 2). It is natural to expect in general the
existence of a cyclic symmetric ($\mathbb{Z}_{N}$) scale invariant solution,
and so we test here this idea. A simple realization of a cyclic symmetric
potential is given by

$V_{\rm
cyclic}=c\left[\psi_{1}\psi^{\dagger}_{2}{}^{2}+\psi_{2}\psi^{\dagger}_{3}{}^{2}+\cdots+\psi_{N}\psi^{\dagger}_{1}{}^{2}+(\psi_{i}\leftrightarrow\psi^{\dagger}_{i})\right]\,,$
(26)

a form which can be obtained inserting in (2) the coupling

$\lambda_{i,jk}=\lambda\,\delta_{jk}\delta_{j,1+i\,{\rm mod}\,N}\,.$ (27)

From this definition we find for the coupling dependence in the one loop two-
and three- point functions (and therefore in the counterterms) that

$\lambda_{i,ml}\lambda_{j,ml}=\lambda^{2}\delta_{ij}\,,\quad\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}=0\,.$
(28)

Using these results and having again defined $\tilde{\lambda}=\lambda/(4\pi)$,
we find

$-\gamma=\tilde{\zeta}=\kappa_{\perp}/3=\frac{\tilde{\lambda}^{2}}{8}$ (29)

and, with the vertex correction being zero, the beta function acquires a very
simple form

$\beta_{\tilde{\lambda}}=-\frac{\epsilon}{2}\tilde{\lambda}+\tilde{\zeta}\lambda+3\gamma\lambda=-\frac{\epsilon}{2}\tilde{\lambda}-\frac{\tilde{\lambda}^{3}}{4}\,,$
(30)

giving at the fixed point, independently from $N$ 111This is in agreement with
a statement in Janssen:2000ds .,

$\tilde{\lambda}_{*}^{2}=-2\epsilon\,,\quad\gamma=-\tilde{\zeta}=-\kappa_{\perp}/3=\frac{\epsilon}{4}\,,\quad\nu_{\perp}=\frac{1}{2}-\frac{3}{16}\epsilon\,,\quad\frac{\partial\beta_{\tilde{\lambda}}}{\partial\tilde{\lambda}}=\epsilon\,.$
(31)

This is a phenomenon similar to what has been seen to happen in the first not
trivial order of perturbation theory of cubic higher derivative shift
symmetric theories Safari:2021ocb .

##  4 Conclusions

We have explored specific realizations of multi-field RFT having the Reggeon
fields with the same Regge trajectory, preserved by the interactions, which
are assumed a priori arbitrary. The analysis of the scale invariant critical
solutions has been performed with renormalization group techniques in the
first non trivial order in the $\epsilon$-expansion below the upper critical
dimension.

Our investigations have covered the case of $N=2$ and $N=3$ fields (non
exautive for the latter) and revealed the pattern of critical theories with
their global symmetries. The zoology of solutions grows rapidly with $N$ and
even if it is possible to push the search further, it would reach soon the
possibilities of an analytical search and would require heavy numerical
investigations. We have prolonged solutions found for the two and three field
cases to construct two scale invariant models for an arbitrary $N$.

Given the equivalence among RFT and out-of-equilibrium Directed Percolation
models, this analysis can be of interest in multi-field extensions of these
statistical models. These could be interesting since the role of symmetries
could help in the understanding of non equilibrium dynamics close to
criticality.

We have chosen to impose on the system the conditions of having a single Regge
trajectory to see in theories as simple as possible the pattern of symmetries
compatible with scale inviariance, but certainly it will be interesting and
deserves future investigations the case where such apriori conditions are
removed. In this way all possible scaling solutions can be investigated,
taking into account the possible residual $O(m<N)$ symmetry of any RG fixed
point if $m<N$ Regge trajectories happen to coincide.

A natural step to improve the current understanding is to push the
investigation beyond perturbation theory. Apart from numerical montecarlo
studies the alternative theoretical approach is to perform a Wilsonian RG
analysis using for example functional renormalization group methods along the
line previoulsy employed in the study of the pomeron odderon system
Bartels:2016ecw . We hope to address this problem in a near future but of
course in the multifield case this is a computationally hard task in its
generality, and possibly easier fixing the specific global symmetries to be
investigated. One has also to keep in mind that given that in this theory a
criticality is not a CFT (and is also not unitary) there is no possibility to
use conformal bootstrap Rattazzi:2008pe  methods.

##  5 Acknowledgement

This research was supported by the Fondecyt Fund (Chile) grands N:1231829.
J.B. thanks the support of the International Collaboration Fondecyt
F:1231829/Chile and also the hospitality of the Federico Santa María Technical
University. C.C also appreciates the support of the INFN and the hospitality
of the University of Bologna and the Hamburg University.

##  Appendix A One-loop beta functions and anomalous dimensions

We study here the one loop renormalization properites of multi-field Reggeon
Field Theory (equivalent to the multi-field Directed Percolation) focusing on
a special highly symmetric case with just $O(N)$ invariant quadratic
operators. The theory is defined in terms of $N$ reggeon fields $\psi_{i}$ and
$\psi_{i}^{\dagger}$ (order parameters) on a $D+1$ dimensional domain,
constructed as an euclidean $D$-dimensional transverse space with coordinate
$x$ times a one dimensional time space with coordinate $\tau$. The dual
variables will be momentum $q$ with mass dimension $M$ and energy $E$.

The bare action

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d\tau
d^{D}x\Big{[}\psi_{B}^{{\dagger}}{}^{T}\left(i\partial{\tau}+\alpha^{\prime}_{B}\nabla^{2}+\tilde{\alpha}^{\prime}_{B}\mu_{B}\right)\psi_{B}]$
(32) $\displaystyle-$
$\displaystyle\frac{i}{2}\tilde{\alpha}^{\prime}_{B}\lambda_{B\,i,jk}\left(\psi_{B;i}^{{\dagger}}\psi_{B;j}\psi_{B;k}+\psi_{B,i}\psi_{B,j}^{{\dagger}}\psi_{B;k}^{{\dagger}}\right)\Big{]}$

is written in terms of the bare quantities. Here $\psi_{i}$
($\psi_{i}^{\dagger}$) is an element of the $N$-component field vector $\psi$
($\psi^{\dagger}$), ”$T$” indicates transposition. In the most general case
$\alpha^{\prime}$ and $\mu$ are symmetric matrices and one of them can always
be made diagonal ($\alpha^{\prime}$) employing the $O(N)$ symmetry of the
$\tau$ derivative term in the two-point function. We have introduced the
scalar quantity $\tilde{\alpha}^{\prime}$ in flavor space (number) constructed
from $\alpha^{\prime}$, which is needed for dimensional reasons and defines
the different anomalous scaling in the transverse space with respect to the
time space.

In this work we shall consider the special restricted case with such matrices
proportional to the identity, so that $\tilde{\alpha}^{\prime}$ and
$\alpha^{\prime}$ in practice coincide with a trivial abuse of notation,
making bare slopes and masses independent of the flavor $i$. This translates
into a requirement on the form of the counterterms of the two-point function
which, instead of being generically symmetric, have to be proportional to the
identity matrix (i.e. one has to impose $N(N+1)/2-1$ conditions on the
couplings at the fixed point). We shall also specialize to the case $\mu=0$
which denotes scale invariance in dimensional regularization. The
$\lambda_{i,jk}=\lambda_{i,kj}$ are symmetric in $(j,k)$ and summation over
the flavor indices is understood.

From dimensional analysis we have

$\displaystyle[\tau]=E^{-1}\,,\quad[x]=M^{-1}\,,\quad[\psi_{B}]=M^{D/2}\,,\quad[\alpha^{\prime}]=E/M^{2}\,,\quad[\mu]=M^{2}\,,\quad[\lambda_{i,jk}]=M^{2-D/2}$

from which one can see that the critical dimension for a dimensionless
coupling associated to the cubic interaction is $D_{c}=4$.

The renormalized form of the action, for $D=D_{c}-\epsilon$, is given by

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int d\tau
d^{D}x\Big{[}\psi^{{\dagger}}{}^{T}Z_{\psi}i\partial_{\tau}\psi+{\psi^{{\dagger}}}^{T}Z_{\psi}^{\frac{1}{2}}Z_{\alpha^{\prime}}\alpha^{\prime}\nabla^{2}Z_{\psi}^{\frac{1}{2}}\psi+{\psi^{{\dagger}}}^{T}Z_{\psi}^{\frac{1}{2}}Z_{\mu}\tilde{\alpha}^{\prime}\mu
Z_{\psi}^{\frac{1}{2}}\psi$ (33)
$\displaystyle-\frac{i}{2}\tilde{\alpha}^{\prime}\left(\lambda_{i,jk}+{\delta\lambda}_{i,jk}\right)M^{\frac{\epsilon}{2}}\left(\psi_{i}^{{\dagger}}\psi_{j}\psi_{k}+\psi_{i}\psi_{j}^{{\dagger}}\psi_{k}^{{\dagger}}\right)\Big{]}\,,$

where all the $Z^{\prime}$s, constructed by suitable counterterms, are in
general matrices while the vertex counterterm ${\delta\lambda}_{i,jk}$ is a
three index tensor. The vector field is renormalized as

$\displaystyle\psi_{B}=Z_{\psi}^{\frac{1}{2}}\psi\,,\quad
Z_{\psi}=1+\delta_{\psi}\,$ (34)

with $Z_{\psi}$ a symmetric matrix constructed from the counterterm
$\delta_{\psi}$.

Keeping this relation into account one can obtain the usual relations among
bare and renormalized quantities. We define the matrices

$Z_{\alpha^{\prime}}\alpha^{\prime}=\alpha_{B}^{\prime}\,,\quad
Z_{\mu}\tilde{\alpha}^{\prime}\mu=\tilde{\alpha}^{\prime}_{B}\mu_{B},$ (35)
$Z_{\alpha^{\prime}}=1+\delta_{\alpha^{\prime}}\,,\quad
Z_{\mu}=1+\delta_{\mu}\,.$ (36)

As said we also define

$\tilde{Z}_{\alpha^{\prime}}\tilde{\alpha}^{\prime}=\tilde{\alpha}_{B}^{\prime}\,,\quad\tilde{Z}_{\alpha,}=1+\tilde{\delta}_{\alpha^{\prime}},$
(37)

where we choose $\tilde{\alpha}^{\prime}$ according to

$\displaystyle\tilde{Z}_{\alpha^{\prime}}=\frac{1}{N}tr\left(Z_{\alpha^{\prime}}\right)=1+\frac{1}{N}tr\left(\delta_{\alpha^{\prime}}\right)\,.$
(38)

For the couplings one has

$\displaystyle\lambda_{B;i,jk}=M^{\frac{\epsilon}{2}}\tilde{Z}_{\alpha^{\prime}}^{-1}\left(\lambda_{i^{\prime},j^{\prime}k^{\prime}}+\delta\lambda_{i^{\prime},j^{\prime}k^{\prime}}\right)(Z_{\psi}^{-\frac{1}{2}})_{i^{\prime}i}(Z_{\psi}^{-\frac{1}{2}})_{j^{\prime}j}(Z_{\psi}^{-\frac{1}{2}})_{k^{\prime}k},$
(39)

and to derive the beta function we impose the usual condition
$M\frac{\partial}{\partial M}\lambda_{B;i,jk}=0$ which leads to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Big{[}\frac{\epsilon}{2}(\lambda+\delta\lambda)_{i,jk}-M\frac{\partial}{\partial
M}\log\tilde{Z}_{\alpha^{\prime}}(\lambda+\delta\lambda)_{i,jk}+M\frac{\partial}{\partial
M}(\lambda+\delta\lambda)_{i,jk}$ (40) $\displaystyle\hskip
28.45274pt-\frac{1}{2}\left((M\frac{\partial}{\partial M}\log
Z_{\psi})_{i^{\prime\prime}i}(\lambda+\delta\lambda)_{i^{\prime\prime},jk}+(M\frac{\partial}{\partial
M}\log
Z_{\psi})_{j^{\prime\prime}j}(\lambda+\delta\lambda)_{i,j^{\prime\prime}k}\right.$
$\displaystyle\left.+(M\frac{\partial}{\partial M}\log
Z_{\psi})_{k^{\prime\prime}k}(\lambda+\delta\lambda)_{i,jk^{\prime\prime}}\right)\Big{]}\,.$

With the anomalous dimension matrix

$\displaystyle\gamma=\frac{1}{2}M\frac{\partial}{\partial M}\log Z_{\psi}$
(41)

and with

$\displaystyle\tilde{\zeta}_{\alpha}=M\frac{\partial}{\partial
M}\tilde{Z}_{\alpha}$ (42)

and

$\displaystyle\beta_{i,jk}=M\frac{\partial}{\partial
M}\lambda_{i,jk}=-\frac{\epsilon}{2}\lambda_{i,jk}+\tilde{\beta}_{i,jk}$ (43)

and

$\displaystyle M\frac{\partial}{\partial
M}\delta\lambda_{i,jk}=\beta_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\delta\lambda_{i,jk}\approx-\frac{\epsilon}{2}\lambda_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\delta\lambda_{i,jk}\,,$
(44)

we get

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\tilde{\zeta}_{\alpha}(\lambda+\delta\lambda)_{i,jk}-\gamma_{i^{\prime}i}(\lambda+\delta\lambda)_{i^{\prime},jk}-\gamma_{j^{\prime}j}(\lambda+\delta\lambda)_{i,j^{\prime}k}-\gamma_{k^{\prime}k}(\lambda+\delta\lambda)_{i,jk^{\prime}}$
(45)
$\displaystyle+\tilde{\beta}_{i,jk}+\frac{\epsilon}{2}\delta\lambda_{i,jk}+\beta_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\delta\lambda_{i,jk}$

or

$\displaystyle\tilde{\beta}_{i,jk}$ $\displaystyle\approx$
$\displaystyle\tilde{\zeta}_{\alpha}\lambda_{i,jk}+(\gamma_{i^{\prime}i}\lambda_{i^{\prime},jk}+\gamma_{j^{\prime}j}\lambda_{i,j^{\prime}k}+\gamma_{k^{\prime}k}\lambda_{i,jk^{\prime}})$
(46)
$\displaystyle-\frac{\epsilon}{2}\delta\lambda_{i,jk}+\frac{\epsilon}{2}\lambda_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\delta\lambda_{i,jk}\,.$

Let us now derive explicitely the counterterms computing the one loop
integrals. We begin with the integral associated to the 2-point diagram of
Fig.1:

Figure 1: One loop 2-point function

In leading order $\epsilon$ one obtains

$\displaystyle i(\Gamma^{(1,1)}_{kl;div})_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{4\epsilon}\frac{\lambda_{i,ml}\lambda_{j,ml}}{(4\pi)^{2}}\Big{[}E-\frac{\alpha^{\prime}}{2}q^{2}+2\tilde{\alpha^{\prime}}\mu\Big{]}\,.$
(47)

Clearly all the three counterterms are proportional to the same symmetric
matrix. At this point we impose here the conditions ${\cal C}$ to have all of
them proportional to the identity matrix:

$\displaystyle\lambda_{i,ml}\lambda_{j,ml}=0\,,{\rm for}\,i\neq j$
$\displaystyle\lambda_{1,ml}\lambda_{1,ml}=\lambda_{2,ml}\lambda_{2,ml}\cdots=\lambda_{N,ml}\lambda_{N,ml},$
(48)

which imply that

$\lambda_{i,ml}\lambda_{j,ml}=\frac{1}{N}\lambda_{k,ml}\lambda_{k,ml}\delta_{ij}\,.$
(49)

In general these one loop corrections then lead to:

$\displaystyle Z_{\psi_{ij}}$ $\displaystyle=$ $\displaystyle
1+\delta\psi_{ij}=1+\frac{1}{\epsilon}\frac{1}{4(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}$
(50)

and

$\displaystyle\left(Z_{\psi}^{\frac{1}{2}}Z_{\alpha^{\prime}}Z_{\psi}^{\frac{1}{2}}\right)_{ij}=1+\frac{1}{\epsilon}\frac{1}{8(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}$
(51)

which implies

$\displaystyle
Z_{\alpha^{\prime}}=1-\frac{1}{\epsilon}\frac{1}{8(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}$
(52)

and

$\displaystyle\tilde{Z}_{\alpha^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{N}tr(Z_{\alpha^{\prime}})=1-\frac{1}{\epsilon}\frac{1}{8(4\pi)^{2}}\frac{1}{N}\lambda_{i,lm}\lambda_{i,lm}\,.$
(53)

The third term:

$\displaystyle\left(Z_{\psi}^{\frac{1}{2}}Z_{\mu}Z_{\psi}^{\frac{1}{2}}\right)_{ij}=1+\frac{1}{\epsilon}\frac{1}{2(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm},$
(54)

which implies

$\displaystyle(Z_{\mu})_{ij}=1+\frac{1}{\epsilon}\frac{1}{4(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}$
(55)

and

$\displaystyle\left(\tilde{Z}_{\alpha^{\prime}}^{-1}Z_{\mu}\right)_{ij}=1+\frac{1}{\epsilon}\frac{3}{8}\frac{1}{(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}\,.$
(56)

From this one can compute the anomalous dimensions. In our specific case,
because of the conditions (48) all the fields have the same anomalous
dimensions and there is also just one anomalous dimension for the
$\psi^{\dagger}{}^{T}\psi$ O(N) symmetric composite operator related to the
”mass” $\mu$.

We can therefore obtain in a usual way the anomalous dimensions:

$\displaystyle\gamma_{ij}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(-\frac{\epsilon}{2}\lambda_{r,st}\right)\frac{\partial}{\partial\lambda_{r,st}}\delta\psi_{ij}=-\frac{1}{8}\frac{1}{(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}=\gamma\,\delta_{ij}\,,$
(57)

where the last equality derives from the condition (48), and

$\displaystyle\tilde{\zeta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2N}\lambda_{r,st}\frac{\partial}{\partial\lambda_{r,st}}\frac{-1}{8(4\pi)^{2}}\lambda_{i,lm}\lambda_{i,lm}=\frac{1}{N}\frac{1}{8(4\pi)^{2}}\lambda_{i,lm}\lambda_{i,lm}\,.$
(58)

This model is characterized by a second order phase transition and approaching
criticality there are two different correlation lengths, one for the
transverse space $\xi_{\perp}\sim|\mu|^{-\nu_{\perp}}$ and one for time
(rapidity) $\xi_{\parallel}\sim|\mu|^{-\nu_{\parallel}}$, with the usual
definition $z=\nu_{\parallel}/\nu_{\perp}$. At leading order one has
$z=2-\tilde{\zeta}$. Another useful quantity is

${\kappa_{\perp}}_{ij}=-\left(-\frac{\epsilon}{2}\lambda_{r,st}\right)\frac{\partial}{\partial\lambda_{r,st}}\ln{\left(\tilde{Z}_{\alpha^{\prime}}^{-1}Z_{\mu}\right)}_{ij}=\frac{3}{8}\frac{1}{(4\pi)^{2}}\lambda_{i,lm}\lambda_{j,lm}=\kappa_{\perp}\,\delta_{ij}\,,$
(59)

from which one can compute at one loop Bartels:2015gou  the critical exponent

$\nu_{\perp}=(2-\kappa_{\perp})^{-1}\simeq\frac{1}{2}+\frac{1}{4}\kappa_{\perp}\,.$
(60)

Clearly $\gamma$, $\tilde{\zeta}$ and $\kappa_{\perp}$ are all proportional up
to a numerical factor.

The counterterms for the triple interations can be computed analyzing the one
loop contributions to the three point function in Fig.2:

Figure 2: One loop 3-point function

In leading order $\epsilon$:

$\displaystyle
i(\Gamma^{(1,2)}_{div;abc})_{i,jk}=-i\tilde{\alpha}^{\prime}M^{\frac{\epsilon}{2}}\frac{1}{2(4\pi)^{2}}\frac{1}{\epsilon}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)$
(61)

and

$\displaystyle\delta\lambda_{i,jk}=\frac{1}{2(4\pi)^{2}}\frac{1}{\epsilon}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)\,.$
(62)

Now we return to (46) and consider the last two terms

$\displaystyle-\frac{\epsilon}{2}\delta\lambda_{i,jk}+\frac{\epsilon}{2}\lambda_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\delta\lambda_{i,jk}$
$\displaystyle=-\frac{1}{4}\frac{1}{(4\pi)^{2}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)$
$\displaystyle\hskip
28.45274pt+\frac{1}{4}\frac{1}{(4\pi)^{2}}\lambda_{l,mn}\frac{\partial}{\partial\lambda_{l,mn}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)$
(63)

where the expression in the last line is given by

$\displaystyle\frac{3}{4}\frac{1}{(4\pi)^{2}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)\,.$
(64)

Therefore the $\beta$ function becomes:

$\displaystyle\beta_{i,jk}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{2}\lambda_{i,jk}+\tilde{\zeta}\lambda_{i,jk}+(\gamma_{i^{\prime}i}\lambda_{i^{\prime},jk}+\gamma_{j^{\prime}j}\lambda_{i,j^{\prime}k}+\gamma_{k^{\prime}k}\lambda_{i,jk^{\prime}})$
(65)
$\displaystyle+\frac{1}{2}\frac{1}{(4\pi)^{2}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)$

or more explicitly

$\displaystyle\beta_{i,jk}$ $\displaystyle=$
$\displaystyle-\frac{\epsilon}{2}\lambda_{i,jk}+\frac{1}{N}\frac{1}{8(4\pi)^{2}}\lambda_{l,mn}\lambda_{l,mn}\lambda_{i,jk}$
(66)
$\displaystyle-\frac{1}{8}\frac{1}{(4\pi)^{2}}\left(\lambda_{i^{\prime},lm}\lambda_{i,lm}\lambda_{i^{\prime}jk}+\lambda_{j^{\prime},lm}\lambda_{j,lm}\lambda_{i,j^{\prime}k}+\lambda_{k^{\prime},lm}\lambda_{k,lm}\lambda_{ijk^{\prime}}\right)$
$\displaystyle+\frac{1}{2}\frac{1}{(4\pi)^{2}}\left(\lambda_{i,ab}\lambda_{j,ac}\lambda_{b,ck}+\lambda_{i,ab}\lambda_{k,ac}\lambda_{b,cj}\right)\,,$

to be considered together with the constraints given in (48).

##  Appendix B $S_{N+1}$ symmetric representation

In order to construct QFTs with $N$ fields and $S_{N+1}$-invariant
interactions, let us first describe a useful representation introducing a set
of $N+1$ vectors $e^{\sigma}$ which point in the directions of the vertices of
a $N$-simplex, i.e. a simplex in ${\mathbb{R}}^{N}$. Without specifying the
explicit components with respect to some given basis, the set of vectors
satisfies the following properties

$\displaystyle e^{\sigma}\cdot
e^{\sigma^{\prime}}=\sum_{i=1}^{N}e_{i}^{\sigma}e_{i}^{\sigma^{\prime}}=(N+1)\delta_{\sigma,\sigma^{\prime}}-1$
(67)
$\displaystyle\sum_{\sigma=1}^{N+1}e_{i}^{\sigma}=0\,,\qquad\sum_{\sigma=1}^{N+1}e_{i}^{\sigma}e_{j}^{\sigma}=(N+1)\delta_{ij}\,.$
(68)

Indeed these relations also determine the vectors $e^{\sigma}$ uniquely, up to
rotations and $S_{q}$ transformations.

It is convenient to introduce the following tensors

$q^{(p)}_{i_{1},\dots,i_{p}}=\frac{1}{N\\!+\\!1}Q^{(p)}_{i_{1},\dots,i_{p}}\quad,\quad
Q^{(p)}_{i_{1},\dots,i_{p}}=\sum_{\alpha}e_{i_{1}}^{\alpha}\dots
e_{i_{p}}^{\alpha}\,,$ (69)

noticing that the first two orders are trivially given by

$\displaystyle q^{(1)}_{i_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{N\\!+\\!1}\sum_{\alpha}e_{i_{1}}^{\alpha}=0\,,\quad
q^{(2)}_{i_{1}i_{2}}=\frac{1}{N\\!+\\!1}\sum_{\alpha}e_{i_{1}}^{\alpha}e_{i_{2}}^{\alpha}=\delta_{i_{1}i_{2}}\,.$
(70)

We can compute in general contractions of product of $q^{(p)}$ tensors of
various orders appearing in the diagrammatic expansion with a recursive
algorithm for reduction of the tensors $q^{(p)}$ Codello:2018nbe , without
using a specific realization of the $e^{\sigma}$ vectors and $q$-tensors.

The algorithm uses the consecutive iteration of the fusion rule of two
$q$-tensors

$\displaystyle
q^{(p)}_{i_{1},\dots,i_{p-1},j}\,q^{(r)}_{i_{p},\dots,i_{p+r-2}}{}^{j}$
$\displaystyle=$ $\displaystyle
q^{(p+r-2)}_{i_{1},\dots,i_{p+r-2}}-q^{(p-1)}_{i_{1},\dots,i_{p-1}}\,q^{(r-1)}_{i_{p},\dots,i_{p+r-2}},$
(71)

and the trace rule

$\displaystyle q^{(p)}_{i_{1},\dots,i_{p-2},j}{}^{j}$ $\displaystyle=$
$\displaystyle N\,q^{(p-2)}_{i_{1},\dots,i_{p-2}}.$ (72)

While the fusion rule generally increases the index of the fused $q$-tensor as
$p\times r\to p+r-2$, the traces of the indices of the Feynman diagrams ensure
that the trace rule reduces the order as $p\to p-2$. The procedure terminates
whenever the cases $q^{(1)}_{i}=0$ and $q^{(2)}_{ij}=\delta_{ij}$ are
encountered.

For our needs we have to compute the following two cases. The first is given
by product of two $q^{(3)}$:

$q^{(3)}_{i_{1},j_{1},j_{2}}\,q^{(3)}_{i_{2}}{}^{j_{1},j_{2}}=q^{(4)}_{i_{1},j_{1},i_{2}}{}^{j_{1}}-q^{(2)}_{i_{1},j_{1}}q^{(2)}_{i_{2}}{}^{j_{1}}=(N-1)\,\delta_{i_{1},i_{2}},$
(73)

where in the first equation we have fused $3\times 3\to 4$, and in the second
equation we have traced $4\to 2$ and fused $2\times 2\to 2$.

The second quantity we need is the product of three copies of $q^{(3)}$:

$\displaystyle
q^{(3)}_{i_{1},j_{1}}{}^{j_{2}}\,q^{(3)}_{i_{2},j_{2}}{}^{j_{3}}\,q^{(3)}_{i_{3},j_{3}}{}^{j_{1}}$
$\displaystyle=$ $\displaystyle
q^{(3)}_{i_{1},j_{1}}{}^{j_{2}}\,q^{(4)}_{i_{2},j_{2},i_{3},}{}^{j_{1}}-q^{(3)}_{i_{1},i_{2},i_{3}}$
(74) $\displaystyle=$ $\displaystyle
q^{(5)}_{i_{1},j_{1},i_{2},i_{3}}{}^{j_{1}}-2q^{(3)}_{i_{1},i_{2},i_{3}}=(N-2)\,q^{(3)}_{i_{1},i_{2},i_{3}}.$

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