# Hi-Fi Multi-Speaker English TTS Dataset

###### Abstract

This paper introduces a new multi-speaker English dataset for training text-
to-speech models. The dataset is based on LibriVox audiobooks and Project
Gutenberg texts, both in the public domain. The new dataset contains about 292
hours of speech from 10 speakers with at least 17 hours per speaker sampled at
44.1 kHz. To select speech samples with high quality, we considered audio
recordings with a signal bandwidth of at least 13 kHz and a signal-to-noise
ratio (SNR) of at least 32 dB. The dataset is publicly released at
“http://www.openslr.org/109/”.

##  1 Introduction

Table 1: English text-to-speech datasets Dataset | Num of | Avg num of | Sampling | SNR analysis | License | Purpose
---|---|---|---|---|---|---
speakers | hours/speaker | rate, kHz
LJSpeech | 1 | 24 | 22.05 | - | Public Domain | single-speaker TTS
M-AILABS | 3 | 34 | 16 | - | Permissive | single- and multi-speaker TTS
VCTK | 109 | 0.4 | 48 | - | CC BY 4.0 | multi-speaker / adaptive TTS
LibriTTS | 2456 | 4.2 | 24 | Y | CC BY 4.0 | multi-speaker TTS
Blizzard-2013 | 1 | 319 | 44.1 | professional speaker | Non-commercial | single-speaker TTS
Hi-Fi TTS | 10 | 29.2 | 44.1 | Y | CC BY 4.0 | high-quality multi-speaker TTS

Neural network (NN) based text-to-speech (TTS) systems are capable of
synthesizing speech that sounds very close to natural speech [1, 2, 3]. To
produce high-quality speech, TTS models require speech samples recorded in
professional studio with usually a few dozens of hours per speaker. Public
datasets for training TTS systems are mostly constructed from LibriVox audio
books [4] and text from Project Gutenberg111http://gutenberg.org/. Among the
most popular datasets are the LJSpeech [5] and the M-AILABS [6]. The LJSpeech
is a single-speaker TTS dataset derived from LibriVox books. The corpus
contains about 24 hours of speech sampled at 22.05 kHz. The M-AILABS is a
multi-language corpus also based on LibriVox. The English subset of the
M-AILABS contains about 102 hours of audio data for 3 speakers (2 female and 1
male). These two datasets are frequently used for single-speaker TTS research
since they provide a relatively large number of hours per speaker [7, 8, 9,
10, 11]. However, low sampling rates and the lack of audio quality analysis
limit the usability of the datasets for producing high-quality speech. The
largest high-quality English dataset for speech synthesis was provided by The
Voice Factory as part of the Blizzard Challenge 2013 [12]. The dataset has
over 300 hours of speech from a professional speaker, but this corpus is
distributed under a non-commercial license.

Two highly popular datasets for multi-speaker TTS are the LibriTTS corpus [13]
and voice cloning toolkit (VCTK) from CSTR [14]. The LibriTTS is a revised
version of the LibriSpeech [15] that excludes examples with low SNR. The
LibriTTS is often used for research in low resource multi-speaker speech
synthesis [16, 17] and voice adaptation [11]. The English multi-speaker part
of VCTK contains short studio quality audio clips recorded by 109 native
English speakers. Each speaker reads out about 400 sentences, which were
selected from a newspaper, the rainbow passage and an elicitation paragraph
used for the speech accent archive. The total duration of the dataset is about
44 hours. All recordings were done with a sampling frequency of 96 kHz, and
then down-sampled to 48 kHz. The dataset was intended for speaker-adaptive and
multi-speaker TTS [18, 19, 20, 11]. Both datasets gave a significant boost to
the speech research thanks to large amount of data, wide speaker diversity,
and a non-restrictive license. However, building high-quality TTS systems from
such small amount of data per speaker is challenging. Table 1 summarizes the
existing datasets.

To accelerate the TTS research, we built a new dataset with the following
goals in mind:

  * •

speech signal’s bandwidth of at least 13 kHz

  * •

SNR of at least 32 dB in the 0.3 - 4 kHz band (where most speech energy is
concentrated)

  * •

high-quality reference texts, i.e. texts are normalized and text-audio match
verified

  * •

minimum sampling frequency of 44.1 kHz

  * •

minimum of 10 native English speakers

  * •

gender diversity

  * •

at least 15 hours of audio per speaker

We fulfilled our goals by constructing the Hi-Fi Multi-Speaker English TTS
(Hi-Fi TTS) dataset. The dataset includes speech data for 10 speakers (6
female and 4 male) with at least 17 hours per speaker (17.7-58.0 hours). The
quality of the reference texts was checked by running inference with ASR
models and including only samples with zero Word Error Rate (WER).

We expect the Hi-Fi TTS dataset to facilitate training of TTS models that 1)
generalize better, i.e. have a broader range of pitch and fewer
mispronunciations 2) have higher overall quality, i.e. less noisy and contain
fewer artifacts, and 3) are more expressive and provide a wider range of voice
variations to choose from.

Although the source material for the Hi-Fi TTS dataset is in the public
domain, when we selected a voice sample for a text-to-speech (TTS) application
we sought the written consent of the speaker. We believe it is ethically
responsible to get the speaker’s consent before using their voice for
synthetic replication and we strongly encourage other developers to seek such
consent.

The rest of the paper is organized as follows. Section 2 summarizes the
process of speaker selection based on the audio quality, defines the data pre-
processing steps and the text-audio alignment approach. Section 3 shows the
final dataset statistics and available subsets.

##  2 The dataset construction

###  2.1 Speaker selections

This section describes the process of selecting LibriVox readers with high-
quality audio and sufficient amount of speech.

####  2.1.1 Downloading metadata and audio samples

First we download book and speaker metadata for all available English audio
books from LibriVox.org. We use LibriLight [21], LibriVox
API222https://librivox.org/api/info and internal scripts for parsing LibriVox
book pages to get the required information about books and speakers and verify
that the book links to the reference texts are valid. We compute the total
number of hours available per speaker and keep only readers with at least 50
hours of audio. Next, we download the most recent audio samples for the
selected readers. Note, each book page on Librivox.org contains links to both
individual chapters and the entire book in ZIP format. The ZIP archive
contains audio files in MP3 format at a sampling rate of 22.05 kHz, while the
individual chapter links redirect to 128 kbps MP3 audio files sampled at 44.1
kHz. We use the latter option to download 44.1 kHz audio files. At the end of
this stage we obtain the most recent audio recordings for 250 readers with at
least 50 hours of audio data and valid reference book links.

Figure 1: Spectrograms of LibriVox recordings: (a) good audio quality
(wideband signal, low noise); (b) not acceptable audio quality (narrowband
signal); (c) not acceptable audio quality (low SNR); (d) not acceptable audio
quality (both narrowband and noisy).

####  2.1.2 Audio quality analysis

To select speakers with high audio quality, we estimate bandwidth and SNR to
guide the selection process. We use the first 30 seconds of the audio samples
for analysis.

To estimate the speech signal’s bandwidth, we take the mean of the power
spectrogram and find the highest frequency that has at least -50 dB level
relative to the peak value of the spectrum.

Figure 2: Energy-based VAD applied to a LibriVox recording.

To estimate SNR we 1) pass the signal through a simple energy-based voice
activity detector (VAD), 2) estimate the noise power in non-speech segments,
3) gauge the signal-plus-noise power in speech segments, and finally 4)
compute SNR. This algorithm assumes that the noise is stationary. Energy-based
VAD is demonstrated in Figure 2. This approach works fairly well in scenarios
where SNR is not very low.

We consider four frequency bands for our SNR estimation. The frequency band
from 100 Hz to 1 kHz helps to detect low-frequency tones and noises. The 300
Hz to 4 kHz band contains most of the speech energy, and we require at least
32 dB SNR in this band. Finally, the bands from 4 kHz to 10 kHz and from 10
kHz to 15 kHz are useful to spot high-frequency noises and to check if the
signal’s bandwidth is indeed wide.

The LibriVox readers were sorted based on SNR calculated for the most recently
added book, and only the readers with the highest SNR values are considered.
We assume that the recordings that were added last have the highest quality
for the particular speaker.

We observe that many samples have a narrow signal bandwidth despite having a
sampling rate of 44.1 kHz (see Figure 1.d). Further, in some cases non-speech
segments of audio samples were denoised so that the assumption that noise is
stationary was violated (see Figure 1.b). To verify that the selected speakers
are a good match for the Hi-Fi TTS dataset, we listened to audio samples from
each reader.

After selecting an initial set of readers with acceptable audio quality, we
calculate SNR for all the books of the selected readers since we observed that
the reader’s audio quality may differ across the books. Based on SNR we select
books that have low noise and high bandwidth into the clean subset of the
dataset. The books with noticeable noise or narrower bandwidth are included in
the other subset (see Section 3). We anticipate that some research project
could benefit from additional data for readers even though the audio quality
is not as high. The SNR-based filtering along with the careful analysis of
spectrogram reduces the number of the readers from 250 to 10 with 3 readers in
the clean set, and 7 readers in the other set, see Table 2.

Table 2: Hi-Fi Multi-Speaker TTS Dataset statistics.

Reader ID | Reader Name | Gender | Clean, hrs | Other, hrs | Total, hrs
---|---|---|---|---|---
92 | Cori Samuel | F | 27.3 | - | 27.3
6097 | Phil Benson | M | 30.1 | 3.4 | 33.5
9017 | John Van Stan | M | 58.0 | - | 58.0
6670 | Mike Pelton | M | - | 17.7 | 17.7
6671 | Tony Oliva | M | - | 24.1 | 24.1
8051 | Maria Kasper | F | - | 30.4 | 30.4
9136 | Helen Taylor | F | - | 24.3 | 24.3
11614 | Sylviamb | F | - | 22.2 | 22.2
11697 | Celine Major | F | - | 26.8 | 26.8
12787 | LikeManyWaters | F | - | 27.3 | 27.3
|  |  | 115.4 | 176.2 | 291.6

###  2.2 Text pre-processing

The text pre-processing pipeline can be summarized as follows:

  1. 1.

Download reference text files. In most cases, a LibriVox book page contains a
link to the corresponding reference book in TXT format hosted at the Gutenberg
Project. Rarely, LibriVox redirects to the reference book in PDF format, in
such cases, we use the output of an Optical Character Recognition (OCR) from
the PDF of the book.

  2. 2.

Split reference text files into chapters based on the LibriVox audio files.
The original book text is a single file while the audio book is usually
divided into chapters. To segment the original text into chapters that
correspond to the chapter level audio recordings, we transcribe audio files
with the QuartzNet ASR model [22]. Then, we match the chapter level audio with
the corresponding text snippet using the transcripts along with the chapter
titles information from LibriVox pages.

  3. 3.

Remove footnotes or inaudible sections in square or curly brackets from the
reference texts.

  4. 4.

Normalize text. All abbreviations and non-standard words [23] are expanded to
their full spoken equivalent with the publicly available NeMo normalization
tool [24]. For example, ”on the 18th” is converted to ”on the eighteenth”,
”Hon.” to ”Honorable”, etc.

  5. 5.

Separate long chapter-level reference texts into chunks based on the end of
sentence (EOS) punctuation marks to ensure that sufficient amount of samples
represent complete sentences. As discussed in Zen et al. [13], audio samples
split on sentence breaks should facilitate learning long-term characteristics
of speech such as sentence-level prosody features. The threshold of 60
characters was chosen empirically to ensure the average duration of the final
segments is no longer than 20 seconds. The text chunks longer than 60
characters are further divided with semi-colon, dash or colon punctuation
marks.

###  2.3 Audio segmentation and text-audio alignment

Next, we split long LibriVox audio files of the selected high-quality readers
into smaller fragments suitable for TTS model training and align them with the
corresponding reference texts. The length of LibriVox audio files varies from
a few minutes to an hour. At the same time, TTS models are trained on short
audio clips that are usually up to 20 seconds long. To segment long audio
files into short audio segments and the corresponding text fragments, we
follow the work of Kürzinger et al. on Connectionist Temporal Classification
(CTC) [25] based segmentation [26].333https://pypi.org/project/ctc-
segmentation/

LibriVox audio files contain preamble segments where readers briefly talk
about the LibriVox project and state their name and book information. The
preamble text is missing from the reference texts. To handle such cases, the
CTC-Segmentation [26] algorithm allows the alignment to start at any point of
the audio file by setting the transition cost for the beginning of the segment
to zero. However, when an utterance is mentioned in the untranscribed preamble
and then repeated again as part of the actual recording the alignment results
could be negatively affected. We observe that cutting off 3 seconds from the
beginning of the audio helps to resolve this issue.

The CTC-Segmentation [26] consists of two stages:

  1. 1.

Forward pass. Character log probabilities for each time step are obtained from
an ASR model pre-trained with CTC loss. For this step we downsample the
original 44.1 kHz MP3 audio files to 16 kHz and convert them to WAV format.
Then we run the audio files through the
QuartzNet444https://ngc.nvidia.com/catalog/models/nvidia:nemospeechmodels ASR
model [22] to extract the log probabilities (greedy decoding strategy without
language model). The QuartzNet model was trained on six datasets: LibriSpeech
[15], Mozilla Common Voice [27], the Wall Street Journal dataset [28], The
Fisher corpus [29], Switchboard [30], and the Singapore English subset of the
National Speech Corpus [31].

  2. 2.

Backward pass. At this stage, the most probable time step that corresponds to
the last character of the text segment is determined and then the full path is
recovered.

Along with the split intervals the CTC-Segmentation technique emits a
confidence score for each aligned segment. This score represents the
probability of finding the correct alignment of the short text segment within
the audio clip. The alignment confidence score goes down when the words in the
reference text and audio clips do not match, for example when a reader flips,
misses, or repeats words. We apply a threshold score value of $-2$ to remove
poorly aligned utterances (the score value lies in log space).

###  2.4 Post-processing and sample selection

After aligning the text-audio pairs, we observe that a mismatch between audio
transcript and reference text is still present, i.e. merely applying filtering
based on the confidence alignment score is not sufficient. This is usually the
case when the reference book version is changed or when a reader modifies the
reference text while reading. To filter out such cases we run inference with
both the QuartzNet and Citrinet [32] ASR models (greedy decoding, no language
model) to calculate WER. Only samples with zero WER are included in the
dataset.

##  3 Dataset splits and statistics

Figure 3: Violin plots of the audio lengths per speaker:
a) clean set, b) other set.

The Hi-Fi TTS dataset contains audio clips sampled at 44.1 kHz in WAV format
for 10 speakers (6 female and 4 male speakers), see(Table 2). Each speaker has
at least 17 hours of speech. All samples were verified and have zero WER (see
Section 2.4). The Hi-Fi TTS corpus contains two parts:

  * •

The clean subset contains high-quality audio books with relatively good voice
attributes based on the SNR analysis (at least 40 dB), see Section 2.1.2.

  * •

The other set includes books with lower SNR (at least 32 dB). Some readers are
present in both subsets as the audio quality varies from book to book.

Figure 3 demonstrates violin plots [33] of the duration of the audio clips in
the Hi-Fi TTS for each speaker.

We also created dev and test splits to facilitate model comparison. We provide
original text and the corresponding normalized version of the text (with
letter capitalization preserved). At the time of writing, the Hi-Fi TTS does
not overlap with the M-AILABS and the LJSpeech datasets. The LibriTTS most
likely contains some audio samples from speakers included in the Hi-Fi TTS.

##  4 Conclusions

This paper introduces the Hi-Fi Multi-Speaker TTS dataset created from the
LibriVox audiobooks and the Gutenberg project texts. The readers included in
the dataset were carefully selected using SNR and spectrogram analysis. The
quality of the text-audio alignments and the accuracy of the reference texts
were validated with ASR model predictions (only samples with zero WER are
accepted). To our knowledge, this is the first multi-speaker dataset of such
quality.

Despite the careful selection of the speakers and their books, the audio
quality of LibriVox recordings is still lagging far behind professional audio
recordings. The next step of the project could be to apply voice processing
techniques to improve the quality of the sound, for example by removing
hissing and metallic sound in samples with boosted high frequencies.

The code for normalization, segmentation and interactive data analysis is
publicly available via NeMo toolkit [34].555https://github.com/NVIDIA/NeMo

##  5 Acknowledgements

The authors would like to thank Rafael Valle, Adrian Lancucki, Jocelyn Huang
and the rest of the NVIDIA TTS team for review, helpful comments and feedback.
We also thank the LibriVox volunteers without whom this project would not have
been possible.

## References

  * [1] R. Prenger, R. Valle, and B. Catanzaro, “WaveGlow: A flow-based generative network for speech synthesis,” _arXiv:1811.00002_ , 2018.
  * [2] G. Yang, S. Yang, K. Liu, P. Fang, W. Chen, and L. Xie, “Multi-band MelGAN: Faster waveform generation for high-quality text-to-speech,” _arXiv:2005.05106_ , 2020.
  * [3] J. Kong, J. Kim, and J. Bae, “HiFi-GAN: Generative adversarial networks for efficient and high fidelity speech synthesis,” _arXiv:2010.05646_ , 2020\.
  * [4] “LibriVox - Free public domain audiobooks,” https://librivox.org/.
  * [5] K. Ito and L. Johnson, “The LJ Speech Dataset,” https://keithito.com/LJ-Speech-Dataset/, 2017.
  * [6] Munich Artificial Intelligence Laboratories GmbH, “The m-ailabs speech dataset,” https://www.caito.de/2019/01/the-m-ailabs-speech-dataset/, 2017\.
  * [7] S. Beliaev, Y. Rebryk, and B. Ginsburg, “Talknet: Fully-convolutional non-autoregressive speech synthesis model,” _arXiv:2005.05514_ , 2020.
  * [8] Z. Zeng, J. Wang, N. Cheng, and J. Xiao, “Melglow: Efficient waveform generative network based on location-variable convolution,” _arXiv:2012.01684_ , 2020.
  * [9] J. Kong, J. Kim, and J. Bae, “Hifi-gan: Generative adversarial networks for efficient and high fidelity speech synthesis,” _arXiv:2010.05646_ , 2020\.
  * [10] A. Łańcucki, “FastPitch: Parallel text-to-speech with pitch prediction,” _arXiv:2006.06873_ , 2021.
  * [11] M. Chen, X. Tan, B. Li, Y. Liu, T. Qin, S. Zhao, and T.-Y. Liu, “AdaSpeech: Adaptive text to speech for custom voice,” _arXiv:2103.00993_ , 2021.
  * [12] S. King and V. Karaiskos, “The blizzard challenge 2013,” 2013.
  * [13] H. Zen, V. Dang, R. Clark, Y. Zhang, R. J. Weiss, Y. Jia, Z. Chen, and Y. Wu, “LibriTTS: A corpus derived from LibriSpeech for text-to-speech,” _arXiv:1904.02882_ , 2019.
  * [14] J. Yamagishi, C. Veaux, and K. MacDonald, “CSTR VCTK Corpus: English Multi-speaker Corpus for CSTR Voice Cloning Toolkit (version 0.92),” 2019.
  * [15] V. Panayotov, G. Chen, D. Povey, and S. Khudanpur, “Librispeech: an ASR corpus based on public domain audio books,” in _ICASSP_ , 2015.
  * [16] J. Park, K. Zhao, K. Peng, and W. Ping, “Multi-speaker end-to-end speech synthesis,” _arXiv:1907.04462_ , 2019.
  * [17] R. Valle, J. Li, R. Prenger, and B. Catanzaro, “Mellotron: Multispeaker expressive voice synthesis by conditioning on rhythm, pitch and global style tokens,” in _ICASSP_ , 2020.
  * [18] W. Ping, K. Peng, A. Gibiansky, S. O. Arik, A. Kannan, S. Narang, J. Raiman, and J. Miller, “Deep voice 3: Scaling text-to-speech with convolutional sequence learning,” _arXiv:1710.07654_ , 2017.
  * [19] Y. Jia, Y. Zhang, R. J. Weiss, Q. Wang, J. Shen, F. Ren, Z. Chen, P. Nguyen, R. Pang, I. L. Moreno _et al._ , “Transfer learning from speaker verification to multispeaker text-to-speech synthesis,” _arXiv:1806.04558_ , 2018.
  * [20] M. Chen, X. Tan, Y. Ren, J. Xu, H. Sun, S. Zhao, and T. Qin, “Multispeech: Multi-speaker text to speech with transformer,” _arXiv:2006.04664_ , 2020\.
  * [21] J. Kahn, M. Rivière, W. Zheng, E. Kharitonov, Q. Xu, P. E. Mazaré, J. Karadayi, V. Liptchinsky, R. Collobert, C. Fuegen, T. Likhomanenko, G. Synnaeve, A. Joulin, A. Mohamed, and E. Dupoux, “Libri-Light: A benchmark for ASR with limited or no supervision,” in _ICASSP_ , 2020.
  * [22] S. Kriman, S. Beliaev, B. Ginsburg, J. Huang, O. Kuchaiev, V. Lavrukhin, R. Leary, J. Li, and Y. Zhang, “Quartznet: Deep automatic speech recognition with 1d time-channel separable convolutions,” in _ICASSP_ , 2020.
  * [23] P. Taylor, “Text-to-Speech Synthesis,” in _Cambridge university press_ , 2009.
  * [24] Y. Zhang, E. Bakhturina, K. Gorman, and B. Ginsburg, “Nemo inverse text normalization: From development to production,” _arXiv preprint arXiv:2104.05055_ , 2021.
  * [25] A. Graves, S. Fernández, F. Gomez, and J. Schmidhuber, “Connectionist temporal classification: labelling unsegmented sequence data with recurrent neural networks,” in _ICML_ , 2006.
  * [26] L. Kürzinger, D. Winkelbauer, L. Li, T. Watzel, and G. Rigoll, “CTC-Segmentation of Large Corpora for German End-to-End Speech Recognition,” in _Speech and Computer_ , A. Karpov and R. Potapova, Eds. Springer International Publishing, 2020.
  * [27] R. Ardila, M. Branson, K. Davis, M. Henretty, M. Kohler, J. Meyer, R. Morais, L. Saunders, F. M. Tyers, and G. Weber, “Common voice: A massively-multilingual speech corpus,” _arXiv:1912.06670_ , 2019.
  * [28] D. B. Paul and J. Baker, “The design for the Wall Street Journal-based CSR corpus,” in _Speech and Natural Language: Proceedings of a Workshop Held at Harriman, New York, February 23-26, 1992_ , 1992.
  * [29] C. Cieri, D. Miller, and K. Walker, “The Fisher corpus: a resource for the next generations of speech-to-text.” in _LREC_ , 2004.
  * [30] J. J. Godfrey, E. C. Holliman, and J. McDaniel, “SWITCHBOARD: Telephone speech corpus for research and development,” in _ICASSP_ , 1992.
  * [31] J. X. Koh, A. Mislan, K. Khoo, B. Ang, W. Ang, C. Ng, and Y. Tan, “Building the singapore english national speech corpus,” _Malay_ , 2019.
  * [32] S. Majumdar, J. Balam, O. Hrinchuk, V. Lavrukhin, V. Noroozi, and B. Ginsburg, “Citrinet: Closing the Gap between Non-Autoregressive and Autoregressive End-to-End Models for Automatic Speech Recognition,” 2021.
  * [33] J. L. Hintze and R. D. Nelson, “Violin plots: a box plot-density trace synergism,” _The American Statistician_ , vol. 52, no. 2, pp. 181–184, 1998\.
  * [34] O. Kuchaiev, J. Li, H. Nguyen, O. Hrinchuk, R. Leary, B. Ginsburg, S. Kriman, S. Beliaev, V. Lavrukhin, J. Cook _et al._ , “Nemo: a toolkit for building ai applications using neural modules,” _arXiv:1909.09577_ , 2019\.


aainstitutetext: National Institute of Science Education and Research (NISER),
Bhubaneswar 752050, Odisha, Indiabbinstitutetext: Homi Bhabha National
Institute, Training School Complex, Anushaktinagar, Mumbai 400094,
Indiaccinstitutetext: Centre For Cosmology and Science Popularization (CCSP),
SGT University, Gurugram, Delhi- NCR, Haryana- 122505, Indiaddinstitutetext:
Centre for High Energy Physics, Indian Institute of Science, C.V. Raman
Avenue, Bangalore 560012, India

# All $S$ invariant gluon OPEs on the celestial sphere

Shamik Banerjee${}^{\,a,b}$, Raju Mandal${}^{\,a,b}$, Sagnik
Misra${}^{\,a,b}$, Sudhakar Panda${}^{\,a,c}$, Partha Paul${}^{\,d}$
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

$S$ algebra is an infinite dimensional Lie algebra which is known to be the
symmetry algebra of some gauge theories. It is a "coloured version" of the
$w_{1+\infty}$. In this paper we write down all possible $S$ invariant
(celestial) OPEs between two positive helicity outgoing gluons and also find
the Knizhnik-Zamolodchikov type null states for these theories. Our analysis
hints at the existence of an infinite number of $S$ invariant gauge theories
which include the (tree-level) MHV-sector and the self-dual Yang-Mills theory.

##  1 Introduction

S-matrix is an important observable in any quantum field theory in
asymptotically flat spacetime. In fact in quantum theory of gravity S-matrix
is argued to be the only observable. Therefore any holographic dual theory has
to compute the S-matrix elements or scattering amplitudes in the bulk.
Celestial holography is an attempt in this direction Strominger:2017zoo ;
Pasterski:2021rjz ; Donnay:2023mrd . The realization that the soft theorems of
gauge theories and gravity are the Ward identities for different asymptotic
symmetries Strominger:2013lka ; Strominger:2013jfa ; He ; Strominger:2014pwa ;
Campiglia:2016jdj ; Kapec:2016jld ; Kapec:2014opa ; Banerjee:2020vnt ;
He:2017fsb ; Banerjee:2022wht ; Donnay:2021wrk ; Donnay:2020guq ;
Stieberger:2018onx ; Banerjee:2020kaa ; Banerjee:2020zlg ; Banerjee:2021cly ;
Gupta:2021cwo ; Guevara:2021abz ; Strominger:2021mtt ; Himwich:2021dau ;
Melton:2022fsf ; Adamo:2021lrv ; Costello:2022wso ; Costello:2022upu ;
Costello:2023vyy ; Garner:2023izn ; Stieberger:2022zyk ; Stieberger:2023fju
has led to important insights into the study of scattering amplitudes
rewritten as correlators of a conformal field theory on the 2D celstial
sphere. This CFT is commonly known as the celestial conformal field theory
(CCFT). The map from scattering amplitudes to the correlators of CCFT is done
via the Mellin transformation. Usually scattering amplitudes are written in
momentum basis. The job of Mellin transformation is to change the momentum
eigenbasis into boost eigenbasis Pasterski:2016qvg ; Pasterski:2017kqt ;
Banerjee:2018gce ; Banerjee:2019prz . The isomorphism between the global
conformal group in 2D and the Lorentz group in 4D is at the heart of this
basis change.

Operator Product Expansion (OPE) in CCFT correspond to the collinear limit in
the bulk and it plays a very important role in the study of the dual theory
Fan:2019emx ; Banerjee:2020zlg ; Banerjee:2021cly ; Guevara:2021abz ;
Himwich:2021dau ; Costello:2022upu ; Banerjee:2023zip ; Ebert:2020nqf ;
Adamo:2022wjo ; Ren:2023trv ; Hu:2021lrx ; Fan:2022vbz ; Fan:2022kpp ;
Bhardwaj:2022anh ; Krishna:2023ukw ; Hu:2022bpa ; Pate:2019lpp ;
Atanasov:2021cje ; Banerjee:2023jne ; Banerjee:2023rni . In a previous paper
Banerjee:2023zip , we have studied the $w_{1+\infty}$ invariant OPEs in
theories of gravity. Our analysis showed that there are an infinite number of
theories on the celestial sphere which are $w_{1+\infty}$ invariant. By
deriving the OPE from graviton scattering amplitudes we have explicitly shown
in Banerjee:2023jne  that the self dual gravity is one example of this
infinite family.

In this paper we perform a similar analysis for gluons. In the case of gluons
the infinite symmetry algebra is know as the $S$ algebra Guevara:2021abz ;
Strominger:2021mtt . We write down all possible $S$ invariant OPE structures
between two positive helicity outgoing gluons. We find that there is a
(discrete) infinite number of such structures and presumably, each one of them
corresponds to a $S$ invariant theory of gluons in the bulk. However, a more
explicit Lagrangian description of these theories are not known to us.

There is an important difference between the analyses of $w_{1+\infty}$ and
$S$-invariant theories, which we want to point out. $S$ algebra does not
contain the Poincare generators. Therefore the consistent OPEs need not be
Poincare invariant. However in this paper we make sure that all the OPEs are
(conformal) Lorentz invariant and this plays an important role. This is along
the line of Fan:2022vbz ; Fan:2022kpp ; Banerjee:2023rni ; Casali:2022fro .

We start with a brief review of the soft gluon symmetry algebra known as the
$S$ algebra in section 2. In section 3, the general structure of the OPE
between two positive helicity outgoing gluons on the celestial sphere has been
discussed. We have argued, how the null states of the MHV-sector can be used
to write down the general OPE. In section 4, we have written down the null
states that appear at $\mathcal{O}(z^{0}\bar{z}^{0})$ of the gluon-gluon OPE
in the MHV-sector. These are not the complete set of null states that the MHV
sector has at $\mathcal{O}(z^{0}\bar{z}^{0})$. There are more of them. We talk
about them later in section 8, where we have discussed the Knizhnik-
Zamolodchikov (KZ)-type null states. Section 5 explicitly shows how to
organise the OPE at every order. For simplicity we focus on the
$\mathcal{O}(z^{0}\bar{z}^{0})$ terms in the OPE. We have also discussed the
transformation properties of MHV-null states under the $S$ algebra in this
section, which are required to organise the OPE. Section 6 shows the
invariance of the $\mathcal{O}(z^{0}\bar{z}^{0})$ OPE under $S$ algebra. In
section 7, we have argued, how an infinite number theories can exist on the
celestial sphere. We conclude the paper with the discussion of the results
found in this paper and some future directions in section 8.1.

##  2 The S algebra

We start by describing the soft symmetry algebra which follows from the
universal singular terms in the OPE between two positive helicity outgoing
gluons Guevara:2021abz ; Strominger:2021mtt . Let
$O^{a,+}_{\Delta}(z,\bar{z})$ denote a positive helicity outgoing gluon
conformal primary operator of dimension $\Delta$ at the point $(z,\bar{z})$ on
the celestial sphere. The universal singular terms in the OPE are given by

$O^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})O^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})\sim\frac{-if^{abc}}{z_{12}}\sum_{n=0}^{\infty}B(\Delta_{1}-1+n,\Delta_{2}-1)\frac{\bar{z}_{12}^{n}}{n!}\bar{\partial}^{n}O^{c,+}_{\Delta_{1}+\Delta_{2}-1}(z_{2},\bar{z}_{2})$
(1)

The interesting fact about these singular OPE coefficients is that it allows
us to define an infinite tower of conformally soft Donnay:2018neh ;
Pate:2019mfs ; Fan:2019emx ; Nandan:2019jas ; Adamo:2019ipt ; Puhm:2019zbl ;
Guevara:2019ypd  gluons Guevara:2021abz ; Strominger:2021mtt  by

$R^{k,a}(z,\bar{z}):=\lim_{\Delta\to k}(\Delta-k)O^{a,+}_{\Delta}(z,\bar{z}),\
k=1,0,-1,\cdots$ (2)

Now it follows from the structure of the OPE (1) that one can introduce the
following truncated mode expansion

$R^{k,a}(z,\bar{z})=\sum_{n=\frac{k-1}{2}}^{\frac{1-k}{2}}\frac{R^{k,a}_{n}(z)}{\bar{z}^{n+\frac{k-1}{2}}}$
(3)

where the expansion coefficients $R^{k,a}_{n}(z)$ are the conserved
holomorphic currents. For fixed values of $k$ and $a$ there are $(2-k)$ such
currents and they transform in the $(2-k)$ dimensional representation of the
$\overline{SL_{2}(R)}$.

The holomorphic currents $R^{k,a}_{n}(z)$ can be further mode expanded as

$R^{k,a}_{n}(z)=\sum_{\alpha\in\mathbb{Z}-\frac{k+1}{2}}\frac{R^{k,a}_{\alpha,n}}{z^{\alpha+\frac{k+1}{2}}}$
(4)

The algebra of these modes can be obtained from the singular terms (1) of the
$RR$ OPE and is given by Guevara:2021abz ,

$\begin{gathered}\left[R^{k,a}_{\alpha,n},R^{l,b}_{\alpha^{\prime},n^{\prime}}\right]=-if^{abc}\frac{\left(\frac{1-k}{2}-n+\frac{1-l}{2}-n^{\prime}\right)!}{\left(\frac{1-k}{2}-n\right)!\left(\frac{1-l}{2}-n^{\prime}\right)!}\frac{\left(\frac{1-k}{2}+n+\frac{1-l}{2}+n^{\prime}\right)!}{\left(\frac{1-k}{2}+n\right)!\left(\frac{1-l}{2}+n^{\prime}\right)!}R^{k+l-1,c}_{\alpha+\alpha^{\prime},n+n^{\prime}}\end{gathered}$
(5)

Now one can define the following generators Strominger:2021mtt

$S^{q,a}_{\alpha,m}=\left(q-m-1\right)!\left(q+m-1\right)!R^{3-2q,a}_{\alpha,m}$
(6)

in terms of which the algebra (5) simplifies to

$[S^{p,a}_{\alpha,m},S^{q,b}_{\beta,n}]=-if^{abc}S^{p+q-1,c}_{\alpha+\beta,m+n}$
(7)

This infinite dimensional algebra of the conformally soft gluons, known as the
$S$ algebra, plays a central role in this paper.111In this paper we write the
OPE in terms of of the descendants of the $R$ algebra (5). However, we will
continue to refer to (5) as the $S$ algebra.

In this paper we want to classify all possible OPEs between two positive
helicity outgoing gluons which are invariant under the $S$ algebra. The
strategy we adopt here is similar to that in the gravity case Banerjee:2023zip
but the details are very different. For example, in the gravity case the soft
symmetry algebra which is isomorphic to the $w_{1+\infty}$ contains all four
global space-time translations. But, this is not the case for the $S$ algebra
and so there are $S$ invariant theories which are not space-time
translationally invariant Fan:2022vbz ; Fan:2022kpp ; Banerjee:2023rni ;
Casali:2022fro . However, we preserve Lorentz invariance because it translates
into conformal invariance on the celestial sphere and the structure of the $S$
algebra depends on that.

##  3 General structure of the OPE Between two positive helicity outgoing
gluons

We can write the general structure of the OPE between two positive helicity
gluons invariant under the $S$ algebra as

$\begin{gathered}O^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})O^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})=\frac{-if^{abc}}{z_{12}}\sum_{n=0}^{\infty}B(\Delta_{1}-1+n,\Delta_{2}-1)\frac{\bar{z}_{12}^{n}}{n!}\bar{\partial}^{n}O^{c,+}_{\Delta_{1}+\Delta_{2}-1}(z_{2},\bar{z}_{2})\\\
+\sum_{p,q=0}^{\infty}z_{12}^{p}\bar{z}_{12}^{q}\sum_{k=1}^{\tilde{n}_{p,q}}\tilde{C}_{p,q}^{k}(\Delta_{1},\Delta_{2})\tilde{O}^{ab}_{k,p,q}(\Delta_{1},\Delta_{2},z_{2},\bar{z}_{2})\end{gathered}$
(8)

where in the second line we have now added the $S$ algebra descendants of a
positive helicity gluon. The sum over $k$ could be finite or infinite
depending on the theory. Our goal is to determine the descendants
$\tilde{O}^{ab}_{k,p,q}$ and the OPE coefficients $\tilde{C}_{p,q}^{k}$ in a
general $S$-invariant theory.

In the gravity case Banerjee:2023zip  we found that any $w$-invariant OPE can
be written in terms of the MHV OPE and the MHV null states. We have also
checked by detailed calculation that this structure holds in the self-dual
gravity theory Banerjee:2023jne  which is $w$ invariant. The same reasoning
also goes through for the $S$ algebra and gluons. We summarize the argument
below.

Since the $S$ algebra is universal, i.e the same 222For example, this is not
true in the conventional $2-D$ CFTs because different CFTs have different
Virasoro central charges and so different conformal symmetry algebras. algebra
holds in any $S$ invariant theory, it is reasonable to assume that there is a
Master OPE which holds in all $S$ invariant theories. Let us now consider the
gluon-gluon OPE in the (tree-level) MHV sector of the pure YM theory. Since
the MHV sector is $S$ invariant the Master OPE, when inserted in a MHV gluon
scattering amplitude, should reproduce the known MHV sector OPE. Therefore one
can write

$\text{Master OPE}=\text{MHV-sector OPE}+R$ (9)

where $R$ should vanish inside an MHV scattering amplitude. This is possible
only if $R$ is a linear combination of MHV null states. Now since the MHV-
sector OPE already contains the universal singular terms (1) of the gluon-
gluon OPE, $R$ consists only of non-singular terms. So we can write,

$\begin{gathered}O^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})O^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})\big{|}_{\textnormal{Any
Theory}}=O^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})O^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})\big{|}_{\textnormal{MHV}}\\\
+\sum_{p,q=0}^{\infty}z_{12}^{p}\bar{z}_{12}^{q}\sum_{k=1}^{\tilde{n}_{p,q}}\tilde{C}_{p,q}^{k}(\Delta_{1},\Delta_{2})M^{ab}_{k,p,q}(\Delta_{1},\Delta_{2},z_{2},\bar{z}_{2})\end{gathered}$
(10)

where $M^{ab}_{k,p,q}$ are the MHV null states. So when "Any Theory" is taken
to be the MHV sector, $M^{ab}_{k,p,q}$ vanishes and we get back the MHV sector
OPE by construction.

We now describe the MHV null states which are of interest to us. In this paper
we apply this general procedure to write down the OPE at
$\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$.

##  4 Null states in the MHV sector

The general null state at order $z_{12}^{0}\bar{z}_{12}^{0}$ is given by
333These null states can be obtained by taking soft limits of the gluon-gluon
MHV OPE Banerjee:2020vnt . The relevant terms in the gluon-gluon OPE in the
MHV sector which gives rise to these null states are given in (23).

$\begin{gathered}\Psi^{ab}_{j}(\Delta)=R^{-j,a}_{\frac{j-1}{2},\frac{j+1}{2}}\mathcal{O}^{b,+}_{\Delta+j}-\frac{(-1)^{j}j}{\Gamma(j+2)}\frac{\Gamma(\Delta+j-1)}{\Gamma(\Delta-2)}R^{1,a}_{-1,0}\mathcal{O}^{b,+}_{\Delta-1}\\\
-\frac{(-1)^{j}}{\Gamma(j+1)}\frac{\Gamma(\Delta+j-1)}{\Gamma(\Delta-1)}R^{0,a}_{-\frac{1}{2},\frac{1}{2}}\mathcal{O}^{b,+}_{\Delta}\end{gathered}$
(11)

Here we have ignored the $(p,q)$ index and have simply written $M^{ab}_{k}$
instead of $M^{ab}_{k,0,0}$ for the order $z_{12}^{0}\bar{z}_{12}^{0}$ MHV
null states.

Now it turns out that the following basis of null states

$\begin{gathered}M_{k}^{ab}(\Delta)=\sum_{i=1}^{k}\frac{1}{\Gamma(k-i+1)}\frac{\Gamma(\Delta+k-1)}{\Gamma(\Delta+i-1)}\Psi_{i}^{ab}(\Delta)\end{gathered}$
(12)

is more convenient because they transform nicely under the $S$-algebra. We
will discuss their transformation law in the next section.

We conclude this section by defining the antisymmetric part of the null states
$M_{k}^{ab}(\Delta)$ as

$M^{a}_{k}(\Delta)=f^{abc}M_{k}^{bc},$ (13)

##  5 Organizing the OPE at every order

Since the MHV sector is $S$ invariant the MHV null states must form a
representation of the $S$ algebra. In other words every generator of the $S$
algebra must map any MHV null state to another MHV null state. Our analysis
shows that this representation is reducible and different $S$ invariant
theories correspond to different irreducible components of this
representation. So our first job is to study the action of the $S$ algebra
generators on the MHV null states. This is facilitated by the following
observation.

We have discussed in Section 2 that the $S$ algebra is generated by an
infinite number of holomorphic soft currents $\\{R^{k,a}_{p}(z)\\}$ 444Here
$-p$ is the antiholomorphic weight of the current. where $k=1,0,-1,-2,\cdots.$
is the dimension $\Delta$ of the soft operator and $\frac{k-1}{2}\leq
p\leq\frac{1-k}{2}$. For a fixed $k$, the soft currents
$R^{k,a}_{\frac{1-k}{2}},\cdots,R^{k,a}_{\frac{k-1}{2}}$ transform in a
$(2-k)$ dimensional representation of the $\overline{sl_{2}(R)}$ 555Note that
we are assuming the theory to be (conformal) Lorentz invariant.. This can be
seen from the following commutation relations:

$\begin{gathered}\left[H^{0}_{0,-1},R^{k,a}_{m,p}\right]=\frac{1}{2}(2p+k-3)R^{k,a}_{m,p-1}\
\textnormal{for}\
p>\frac{k-1}{2},\qquad\left[H^{0}_{0,-1},R^{k,a}_{m,\frac{k-1}{2}}\right]=0\\\
\left[H^{0}_{0,0},R^{k,a}_{m,p}\right]=-2p\,R^{k,a}_{m,p}\\\
\left[H^{0}_{0,1},R^{k,a}_{m,p}\right]=\frac{1}{2}(2p-k+3)R^{k,a}_{m,p+1}\
\textnormal{for}\
p<-\frac{k-1}{2},\qquad\left[H^{0}_{0,1},R^{k,a}_{m,-\frac{k-1}{2}}\right]=0\end{gathered}$
(14)

Now let us consider the currents
$R^{1,a}_{0},R^{0,a}_{\frac{1}{2}},R^{-1,a}_{1},\cdots$ with the lowest
$\overline{sl_{2}(R)}$ weights. Starting from $R^{1,a}_{0}$ all the currents
in this family can be obtained by applying the global subleading soft gluon
operator $R^{0,b}_{\frac{1}{2},\frac{1}{2}}$. This can be seen from the the
following commutation relations

$\left[R^{0,a}_{\frac{1}{2},\frac{1}{2}},R^{k,b}_{m,\frac{1-k}{2}}\right]=-if^{abc}(2-k)R^{k-1,c}_{m+\frac{1}{2},\frac{2-k}{2}}$
(15)

Equations (14) and (15) show that we can write any generator of the $S$
algebra as a sum of products of the generators
$\left(R^{1,a}_{n,0},R^{0,a}_{\frac{1}{2},\frac{1}{2}},H^{0}_{0,0},H^{0}_{0,\pm
1}\right)$. Therefore in order to study the action of the $S$ algebra
generators on the MHV null states we just need to focus on these finite number
of generators.

Figure 1: The figure shows the soft gluon currents arranged in representations
of $\overline{sl_{2}(R)}$. The $\overline{sl_{2}(R)}$ generators move the
currents horizontally in both directions whereas the global subleading soft
gluon symmetry generator $R^{0,a}_{\frac{1}{2},\frac{1}{2}}$ moves the
currents vertically downward.

###  5.1 Transformation properties of the null states under
$\overline{sl_{2}(R)}$ algebra

Using the action of different generators of $\overline{sl_{2}(R)}$ algebra on
the gluon primary operators and the commutation relations (14) it is easy to
show that

$\begin{gathered}H^{0}_{0,1}\Psi^{ab}_{k}(\Delta)=0.\end{gathered}$ (16)

Thus (12) implies that

$\begin{gathered}H^{0}_{0,1}M^{ab}_{k}(\Delta)=0.\end{gathered}$ (17)

Therefore the null states $M^{ab}_{k}$ are $\overline{sl_{2}(R)}$ primaries.

###  5.2 Transformation properties of the null states under the leading soft
gluon current algebra

One can easily check that under the leading soft gluon current algebra, the
null states (12) transform as

$\begin{gathered}R^{1,a}_{0,0}M_{k}^{bc}(\Delta)=-if^{abd}M_{k}^{dc}(\Delta)-if^{acd}M_{k}^{bd}(\Delta)\\\
R^{1,a}_{n,0}M_{k}^{bc}(\Delta)=0,\ n>0\end{gathered}$ (18)

Therefore the null states $M^{ab}_{k}$ are the leading soft gluon current
algebra primaries.

###  5.3 Transformation properties of the null states under subleading soft
gluon operator $R^{0,a}_{\frac{1}{2},\frac{1}{2}}$

This is perhaps the most important transformation property because it mixes
the null states $M^{ab}_{k}$ with different $k$ values. The action of
$R^{0,a}_{\frac{1}{2},\frac{1}{2}}$ on $M_{k}^{bc}(\Delta)$ is given by,

$\begin{gathered}R^{0,a}_{\frac{1}{2},\frac{1}{2}}M_{k}^{bc}(\Delta)=-(k+2)if^{abx}M_{k+1}^{xc}(\Delta-1)+(\Delta+k-2)\left[if^{acx}M_{k}^{bx}(\Delta-1)+if^{abx}M_{k}^{xc}(\Delta-1)\right]\end{gathered}$
(19)

We have used (37) to derive the above equation.

Now let us consider the set of null states:

$M_{k}^{bc}(\Delta),\ k=1,2,\cdots,n.$ (20)

From (19) we can see that if we set

$\begin{gathered}M^{ab}_{k+1}(\Delta)=0,k\geq n\geq 0.\end{gathered}$ (21)

then the set (20) is closed under the action of
$R^{0,a}_{\frac{1}{2},\frac{1}{2}}$. Moreover it follows from (19) that the
infinite set of equations (21) is also invariant under the action of
$R^{0,a}_{\frac{1}{2},\frac{1}{2}}$ because the index $k$ mixes only with
$k^{\prime}\geq k$. Therefore the truncation (21) is $S$ algebra invariant and
we can get an $S$ invariant OPE if we keep only the finite set (20). Let us
emphasize that the integer $n$ is in no way restricted by the $S$ invariance.

##  6 $\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$ OPE and its invariance under
the $S$ algebra

Let us now consider the $\mathcal{O}(1)$ terms in the OPE when we keep only
the finite set of MHV null states (20). In particular, we show that the
$\mathcal{O}(1)$ terms in the OPE with the following coefficients are
$S$-invariant:

$\boxed{\begin{gathered}\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\mathcal{O}(1)}=\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\textnormal{MHV
OPE at }\mathcal{O}(1)}\\\
+\sum_{k=1}^{n}B(\Delta_{1}+k,\Delta_{2}-1)M_{k}^{ab}(\Delta_{1}+\Delta_{2})\end{gathered}}$
(22)

where
$\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\textnormal{MHV
OPE at}\ \mathcal{O}(1)}$ is given by Banerjee:2020vnt ; Ebert:2020nqf

$\begin{gathered}\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\textnormal{MHV
OPE at}\
\mathcal{O}(1)}=B(\Delta_{1}-1,\Delta_{2}-1)\left[\Delta_{1}\,R^{1,a}_{-1,0}\mathcal{O}^{b,+}_{\Delta_{1}+\Delta_{2}-1}(0,0)\right.\\\
\left.+\frac{\Delta_{1}-1}{\Delta_{1}+\Delta_{2}-2}R^{0,a}_{-\frac{1}{2},\frac{1}{2}}\mathcal{O}^{b,+}_{\Delta_{1}+\Delta_{2}}(0,0)\right]\end{gathered}$
(23)

Let us first apply $R^{0,a}_{\frac{1}{2},\frac{1}{2}}$ on the OPE (22). After
some straightforward algebra we get,

$\begin{gathered}R^{0,x}_{\frac{1}{2},\frac{1}{2}}\left(\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\mathcal{O}(1)}\right)-R^{0,x}_{\frac{1}{2},\frac{1}{2}}\left[\mathcal{O}^{a,+}_{\Delta_{1}}(z,\bar{z})\mathcal{O}^{b,+}_{\Delta_{2}}(0,0)\big{|}_{\textnormal{MHV
OPE at }\mathcal{O}(1)}\right.\\\
\left.+\sum_{k=1}^{n}B(\Delta_{1}+k,\Delta_{2}-1)M_{k}^{ab}(\Delta_{1}+\Delta_{2})\right]=if^{xay}(n+2)B(\Delta_{1}+n,\Delta_{2}-1)M^{yb}_{n+1}(\Delta_{1}+\Delta_{2}-1)\end{gathered}$
(24)

Now, we have argued in the previous section that if the $\mathcal{O}(1)$ OPE
of a $S$ invariant theory truncates at $k=n$, then $M^{ab}_{n+1}(\Delta)$ will
be a null state of that theory. Thus, we can set the RHS of (24) to 0 and
hence (22) is invariant under the action of
$R^{0,a}_{\frac{1}{2},\frac{1}{2}}$. Using (18), one can also verify that,
(22) is invariant under the actions of $R^{1,a}_{n,0}$.

In Banerjee:2020vnt , it was shown that the OPE in the MHV-sector is invariant
under the action of $H^{0}_{0,1}$. We can also see from (16) that the null
states $M_{k}^{ab}(\Delta)$ are annihilated by $H^{0}_{0,1}$. Thus, the OPE
(22) is also invariant under $H^{0}_{0,1}$. Hence we conclude that the
truncated OPE (22) is invariant under the $S$ algebra.

##  7 Infinite family of $S$ invariant theories

In section (5.3) we have shown that the following set of equations

$\begin{gathered}M^{ab}_{k+1}=0,k\geq n\geq 0.\end{gathered}$ (25)

are $S$ invariant. Thus at $\mathcal{O}(z^{0}\bar{z}^{0})$ we can truncate the
OPE (10) at an arbitrary $n$ in an $S$-invariant way. That is to say
$S$-invariance does not fix the value of the integer $n$. Hence we can get a
discrete infinite family of $S$-invariant OPEs for different choices of the
integer $n$. Each of these consistent OPEs correspond to a $S$ invariant
theory of gluons. But, at present we do not know the Lagrangian description of
these theories except perhaps the self-dual Yang-Mills theory.

##  8 Knizhnik-Zamolodchikov type null states

Knizhnik-Zamolodchikov (KZ) type null states contain descendants of the
holomorphic translation generator $L_{-1}$ on the celestial sphere. They can
be obtained algebraically by determining the relevant primary descendant but
in our case we can bypass this tedious procedure if we use the OPE
commutativity

$\begin{gathered}\mathcal{O}^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})\mathcal{O}^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})=\mathcal{O}^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})\mathcal{O}^{a,+}_{\Delta_{1}}(z_{1},\bar{z}_{1})\end{gathered}$
(26)

The reason behind this is that the $\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$
terms of the OPE, as written in (22), are not manifestly symmetric under the
exchange (26). Therefore OPE commutativity imposes non-trivial constraints on
the OPE coefficients and one such constraint is essentially the KZ equation.
The process can be further simplified if make the operator
$\mathcal{O}^{b,+}_{\Delta_{2}}(z_{2},\bar{z}_{2})$ leading soft by taking the
limit $\Delta_{2}\to 1$. Now a straightforward calculation gives the KZ type
null state

$\boxed{\begin{split}K^{a}(\Delta)=\xi^{a}(\Delta)-i\sum_{k=1}^{n}M^{a}_{k}(\Delta+1)=0.\end{split}}$
(27)

where

$\xi^{a}(\Delta)=C_{A}L_{-1}\mathcal{O}^{a,+}_{\Delta}-(\Delta+1)R^{1,b}_{-1,0}R^{1,b}_{0,0}\mathcal{O}^{a,+}_{\Delta}-R^{0,b}_{-\frac{1}{2},\frac{1}{2}}R^{1,b}_{0,0}\mathcal{O}^{a,+}_{\Delta+1}$
(28)

is the KZ type null state in the MHV-sector Banerjee:2020vnt  and
$M^{a}_{k}(\Delta)$ is the antisymmetric part of the null state
$M^{ab}_{k}(\Delta)$ defined in (13). We have also used the identity
$f^{abx}f^{aby}=C_{A}\delta^{xy}$ in deriving the KZ type null state equation
(27).

Another null state equation involving the descendant
$L_{-1}\mathcal{O}^{a,+}_{\Delta}$ can be obtained from (26) in a similar way
by taking the subleading conformal soft limit $\Delta_{2}\to 0$. It is given
by

$\begin{split}(\Delta-1)\xi^{a}(\Delta)-\sum_{k=1}^{n}(\Delta+k)M^{a}_{k}(\Delta+1)=0.\end{split}$
(29)

Now multiplying equation (27) by $(\Delta-1)$ and then subtracting it from
(29) we get the following (current algebra) null state

$\chi^{1,a}_{n}(\Delta)=\sum_{k=1}^{n}(k+1)M_{k}^{a}(\Delta)=0$ (30)

One can continue this procedure and get other current algebra null states by
taking conformal soft limits $\Delta_{2}\rightarrow k,\ k\leq-1$. We can
denote them by $\\{\chi^{1}_{n}(\Delta),\chi^{2}_{n}(\Delta),\cdots\\}$.
However, it can be shown that after a finite iteration this procedure stops
due to the truncation (21).

###  8.1 S invariance of the KZ type null state

In this section we show that the KZ type null state (27) is $S$-invariant.

First of all, the states $M_{k}^{ab}(\Delta)$, and as a result
$M^{a}_{k}(\Delta)=f^{abc}M^{bc}_{k}(\Delta)$, are annihilated by
$H^{0}_{0,1}$. Therefore the state $K^{a}(\Delta)$ is a primary of
$\overline{sl_{2}(R)}$ because the KZ type null state (28) in the MHV-sector
is annihilated Banerjee:2020vnt  by $H^{0}_{0,1}$.

Similarly, one can show after some algebra that the following relation holds

$\begin{gathered}R^{0,c}_{\frac{1}{2},\frac{1}{2}}\
K^{a}(\Delta)=(\Delta-2)if^{cax}K^{x}(\Delta-1)-(n+2)f^{cbx}f^{aby}M^{xy}_{n+1}(\Delta)+f^{cax}\chi^{1,x}_{n}(\Delta)\\\
+f^{cby}f^{bax}\left[\sum_{k=1}^{n}(k+1)M_{k}^{yx}(\Delta)+2E^{yx}(\Delta)\right]+f^{cab}f^{byx}E^{yx}(\Delta)\end{gathered}$
(31)

where

$E^{yx}(\Delta)=(\Delta-2)R^{1,y}_{-1,0}\mathcal{O}^{x,+}_{\Delta-1}+R^{0,y}_{-\frac{1}{2},\frac{1}{2}}\mathcal{O}^{x,+}_{\Delta}$
(32)

Now we know that, in a theory in which the
$\mathcal{O}(z_{12}^{0}\bar{z}_{12}^{0})$ OPE truncates at $k=n$, i.e, (21)
holds, both $M_{n+1}^{bc}(\Delta)$ and $\chi^{1,a}_{n}(\Delta)$ are null
states. Thus we can set them to 0 and get,

$\begin{gathered}R^{0,c}_{\frac{1}{2},\frac{1}{2}}\
K^{a}(\Delta)=(\Delta-2)if^{cax}K^{x}(\Delta-1)+f^{cby}f^{bax}\left[\sum_{k=1}^{n}(k+1)M_{k}^{yx}(\Delta)+2E^{yx}(\Delta)\right]+f^{cab}f^{byx}E^{yx}(\Delta)\end{gathered}$
(33)

We show in Appendix (C) that the second and the third terms on the RHS of (33)
are actually zero. Taking this into account we get

$\begin{gathered}R^{0,c}_{\frac{1}{2},\frac{1}{2}}\
K^{a}(\Delta)=(\Delta-2)if^{cax}K^{x}(\Delta-1)\end{gathered}$ (34)

Thus we see that $R^{0,c}_{\frac{1}{2},\frac{1}{2}}$ maps the KZ type null
state $K^{a}(\Delta)$ to linear combination of other null states in the
theory. Hence, the null state equation

$K^{a}(\Delta)=0$ (35)

is $S$ invariant.

##  9 Acknowledgement

SB would like to thank the participants of the Kickoff Workshop for the Simons
Collaboration on Celestial Holography for helpful comments. The work of SB is
partially supported by the Swarnajayanti Fellowship (File No-
SB/SJF/2021-22/14) of the Department of Science and Technology and SERB,
India. The work of PP is supported by an IOE endowed Postdoctoral position at
IISc, Bengaluru, India. SP is supported by the INSA Senior scientist position
at NISER, Bhubaneswar through the Grant number INSA/SP/SS/2023.

##  Appendix A $S$ algebra primaries

In this Appendix, we write down the conditions on the primary operators that
follow from the OPE between two positive helicity outgoing gluon primaries
(1). They are obtained by taking different soft limits in (1) and comparing
both the sides of the OPE:

$\begin{split}R^{k,a}_{p-\frac{k+1}{2},-q-\frac{k-1}{2}}\mathcal{O}^{b,+}_{\Delta}(0,0)&=0,p\geq
2\\\
R^{k,a}_{\frac{1-k}{2},-q-\frac{k-1}{2}}\mathcal{O}^{b,+}_{\Delta}(0,0)&=-if^{abc}\frac{(-1)^{k+q+1}}{\Gamma(-k-q+2)}\frac{\Gamma(\Delta-1)}{\Gamma(\Delta+q+k-2)}\frac{\bar{\partial}^{q}}{q!}\mathcal{O}^{c,+}_{\Delta+k-1}(0,0)\end{split}$
(36)

where $0\leq q\leq 1-k,\ k=1,0,-1,\cdots$. These conditions have been used in
writing down the transformation properties of the MHV null states. For more
details of how to obtain these conditions one can check Appendix F of
Banerjee:2023jne . The analyses there was done for $w_{1+\infty}$ primaries,
but the methodology is same for $S$ algebra also.

##  Appendix B Transformation properties of the $\Psi_{j}^{bc}$-null states
under the leading soft gluon operator $R^{1,a}_{0,0}$ and the subleading soft
gluon operator $R^{0,a}_{\frac{1}{2},\frac{1}{2}}$

Using (11), (5) and (36), one can show that,

$\begin{split}R^{1,a}_{0,0}\Psi^{bc}_{j}(\Delta)&=-if^{abx}\Psi^{xc}_{j}(\Delta)-if^{acx}\Psi^{bx}_{j}(\Delta)\\\
R^{0,a}_{\frac{1}{2},\frac{1}{2}}\Psi^{bc}_{j}(\Delta)&=-(j+2)\,if^{abx}\Psi^{xc}_{j+1}(\Delta-1)+(\Delta+j-2)\,if^{acx}\Psi^{bx}_{j}(\Delta-1)\\\
&+2\frac{(-1)^{j}}{\Gamma(j+1)}\frac{\Gamma(\Delta+j-1)}{\Gamma(\Delta-1)}if^{abx}\Psi^{xc}_{1}(\Delta-1)\end{split}$
(37)

These equations have been used in sections 5.2 and 5.3.

##  Appendix C Proof that the KZ-type null states are closed under the action
of $R^{0,a}_{\frac{1}{2},\frac{1}{2}}$

We write the second and third term in (33) as,

$\begin{gathered}\Sigma^{ca}(\Delta)=f^{cby}f^{bax}\left[\sum_{k=1}^{n}(k+1)M_{k}^{yx}(\Delta)+2E^{yx}(\Delta)\right]+f^{cab}f^{byx}E^{yx}\left(\Delta\right)\end{gathered}$
(38)

The above equation can be decomposed into symmetric and antisymmetric part in
the following way,

$\Sigma^{ca}(\Delta)=\Sigma_{A}^{ca}(\Delta)+\Sigma_{S}^{ca}(\Delta)$ (39)

where

$\begin{split}\Sigma_{A}^{ca}(\Delta)&=\frac{1}{2}\left[\Sigma^{ca}(\Delta)-\Sigma^{ac}(\Delta)\right]\\\
\Sigma_{S}^{ca}(\Delta)&=\frac{1}{2}\left[\Sigma^{ca}(\Delta)+\Sigma^{ac}(\Delta)\right]\end{split}$
(40)

Now, using the Jacobi identity

$f^{acb}f^{bxy}+f^{xab}f^{bcy}+f^{xcb}f^{aby}=0$ (41)

one can show that

$\Sigma_{A}^{ca}(\Delta)=-\frac{1}{2}f^{cab}\chi^{1,b}_{n}(\Delta)=0$ (42)

We now simplify the symmetric part (40) and get,

$\Sigma_{S}^{ca}(\Delta)=\frac{1}{2}f^{cby}f^{bax}\left[\sum_{k=1}^{n}(k+1)\left(M_{k}^{xy}(\Delta)+M_{k}^{yx}(\Delta)\right)+2\left(E^{xy}(\Delta)+E^{yx}(\Delta)\right)\right]$
(43)

The leading and subleading soft limits of (26) and some straightforward
algebra then gives,

$\sum_{k=1}^{n}(k+1)\left(M_{k}^{xy}(\Delta)+M_{k}^{yx}(\Delta)\right)+2\left(E^{xy}(\Delta)+E^{yx}(\Delta)\right)=0.$
(44)

Hence we conclude that,

$\Sigma^{ca}(\Delta)=0.$ (45)

## References

  * (1) A. Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,” [arXiv:1703.05448 [hep-th]].
  * (2) S. Pasterski, “Lectures on celestial amplitudes,” Eur. Phys. J. C 81, no.12, 1062 (2021) doi:10.1140/epjc/s10052-021-09846-7 [arXiv:2108.04801 [hep-th]]
  * (3) L. Donnay, “Celestial holography: An asymptotic symmetry perspective,” [arXiv:2310.12922 [hep-th]].
  * (4) S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D 96, no. 6, 065026 (2017) doi:10.1103/PhysRevD.96.065026 [arXiv:1701.00049 [hep-th]].
  * (5) S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96, no. 6, 065022 (2017) doi:10.1103/PhysRevD.96.065022 [arXiv:1705.01027 [hep-th]].
  * (6) S. Banerjee, “Null Infinity and Unitary Representation of The Poincare Group,” JHEP 1901, 205 (2019) doi:10.1007/JHEP01(2019)205 [arXiv:1801.10171 [hep-th]].
  * (7) S. Banerjee, S. Ghosh, P. Pandey and A. P. Saha, “Modified celestial amplitude in Einstein gravity,” JHEP 03 (2020), 125 doi:10.1007/JHEP03(2020)125 [arXiv:1909.03075 [hep-th]].
  * (8) W. Fan, A. Fotopoulos and T. R. Taylor, “Soft Limits of Yang-Mills Amplitudes and Conformal Correlators,” JHEP 1905, 121 (2019) doi:10.1007/JHEP05(2019)121 [arXiv:1903.01676 [hep-th]].
  * (9) M. Pate, A. M. Raclariu, A. Strominger and E. Y. Yuan, “Celestial operator products of gluons and gravitons,” Rev. Math. Phys. 33 (2021) no.09, 2140003 doi:10.1142/S0129055X21400031 [arXiv:1910.07424 [hep-th]].
  * (10) A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 1407, 151 (2014) doi:10.1007/JHEP07(2014)151 [arXiv:1308.0589 [hep-th]].
  * (11) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407, 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]].
  * (12) T. He, V. Lysov, P. Mitra and A. Strominger, “BMS supertranslations and Weinberg’s soft graviton theorem,” JHEP 1505, 151 (2015) doi:10.1007/JHEP05(2015)151 [arXiv:1401.7026 [hep-th]].
  * (13) A. Strominger and A. Zhiboedov, “Gravitational Memory, BMS Supertranslations and Soft Theorems,” JHEP 1601, 086 (2016) doi:10.1007/JHEP01(2016)086 [arXiv:1411.5745 [hep-th]].
  * (14) M. Campiglia and A. Laddha, “Sub-subleading soft gravitons: New symmetries of quantum gravity?,” Phys. Lett. B 764 (2017), 218-221 doi:10.1016/j.physletb.2016.11.046 [arXiv:1605.09094 [gr-qc]].
  * (15) D. Kapec, P. Mitra, A. M. Raclariu and A. Strominger, “2D Stress Tensor for 4D Gravity,” Phys. Rev. Lett. 119, no. 12, 121601 (2017) doi:10.1103/PhysRevLett.119.121601 [arXiv:1609.00282 [hep-th]].
  * (16) D. Kapec, V. Lysov, S. Pasterski and A. Strominger, “Semiclassical Virasoro symmetry of the quantum gravity $\mathcal{S}$-matrix,” JHEP 1408, 058 (2014) doi:10.1007/JHEP08(2014)058 [arXiv:1406.3312 [hep-th]].
  * (17) T. He, D. Kapec, A. M. Raclariu and A. Strominger, “Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity,” JHEP 1708, 050 (2017) doi:10.1007/JHEP08(2017)050 [arXiv:1701.00496 [hep-th]].
  * (18) S. Banerjee and S. Pasterski, “Revisiting the shadow stress tensor in celestial CFT,” JHEP 04 (2023), 118 doi:10.1007/JHEP04(2023)118 [arXiv:2212.00257 [hep-th]].
  * (19) S. Banerjee, S. Ghosh and R. Gonzo, “BMS symmetry of celestial OPE,” JHEP 04, 130 (2020) doi:10.1007/JHEP04(2020)130 [arXiv:2002.00975 [hep-th]].
  * (20) S. Banerjee, S. Ghosh and P. Paul, “MHV graviton scattering amplitudes and current algebra on the celestial sphere,” JHEP 02 (2021), 176 doi:10.1007/JHEP02(2021)176 [arXiv:2008.04330 [hep-th]].
  * (21) S. Banerjee and S. Ghosh, “MHV gluon scattering amplitudes from celestial current algebras,” JHEP 10 (2021), 111 doi:10.1007/JHEP10(2021)111 [arXiv:2011.00017 [hep-th]].
  * (22) N. Gupta, P. Paul and N. V. Suryanarayana, “ $\widehat{sl_{2}}$ Symmetry of ${\mathbb{R}}^{1,3}$ Gravity,” Phys. Rev. D 108 (2023) no.8, 086029 doi:10.1103/PhysRevD.108.086029 [arXiv:2109.06857 [hep-th]].
  * (23) A. Guevara, E. Himwich, M. Pate and A. Strominger, “Holographic symmetry algebras for gauge theory and gravity,” JHEP 11 (2021), 152 doi:10.1007/JHEP11(2021)152 [arXiv:2103.03961 [hep-th]].
  * (24) A. Strominger, “$w_{1+\infty}$ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries,” Phys. Rev. Lett. 127, no.22, 221601 (2021) doi:10.1103/PhysRevLett.127.221601
  * (25) E. Himwich, M. Pate and K. Singh, “Celestial operator product expansions and w1+∞ symmetry for all spins,” JHEP 01, 080 (2022) doi:10.1007/JHEP01(2022)080 [arXiv:2108.07763 [hep-th]].
  * (26) W. Melton, S. A. Narayanan and A. Strominger, “Deforming soft algebras for gauge theory,” JHEP 03, 233 (2023) doi:10.1007/JHEP03(2023)233 [arXiv:2212.08643 [hep-th]].
  * (27) T. Adamo, L. Mason and A. Sharma, “Celestial $w_{1+\infty}$ Symmetries from Twistor Space,” SIGMA 18, 016 (2022) doi:10.3842/SIGMA.2022.016 [arXiv:2110.06066 [hep-th]].
  * (28) K. Costello and N. M. Paquette, “Celestial holography meets twisted holography: 4d amplitudes from chiral correlators,” JHEP 10, 193 (2022) doi:10.1007/JHEP10(2022)193 [arXiv:2201.02595 [hep-th]].
  * (29) K. Costello and N. M. Paquette, “Associativity of One-Loop Corrections to the Celestial Operator Product Expansion,” Phys. Rev. Lett. 129, no.23, 231604 (2022) doi:10.1103/PhysRevLett.129.231604 [arXiv:2204.05301 [hep-th]].
  * (30) S. Banerjee, S. Ghosh and S. S. Samal, “Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes,” JHEP 08 (2021), 067 doi:10.1007/JHEP08(2021)067 [arXiv:2104.02546 [hep-th]].
  * (31) L. Donnay and R. Ruzziconi, “BMS flux algebra in celestial holography,” JHEP 11, 040 (2021) doi:10.1007/JHEP11(2021)040 [arXiv:2108.11969 [hep-th]].
  * (32) L. Donnay, S. Pasterski and A. Puhm, “Asymptotic Symmetries and Celestial CFT,” JHEP 09 (2020), 176 doi:10.1007/JHEP09(2020)176 [arXiv:2005.08990 [hep-th]].
  * (33) S. Stieberger and T. R. Taylor, “Symmetries of Celestial Amplitudes,” Phys. Lett. B 793, 141 (2019) doi:10.1016/j.physletb.2019.03.063 [arXiv:1812.01080 [hep-th]].
  * (34) K. J. Costello, “Bootstrapping two-loop QCD amplitudes,” [arXiv:2302.00770 [hep-th]].
  * (35) N. Garner and N. M. Paquette, “Twistorial monopoles & chiral algebras,” JHEP 08, 088 (2023) doi:10.1007/JHEP08(2023)088 [arXiv:2305.00049 [hep-th]].
  * (36) S. Stieberger, T. R. Taylor and B. Zhu, “Celestial Liouville theory for Yang-Mills amplitudes,” Phys. Lett. B 836, 137588 (2023) doi:10.1016/j.physletb.2022.137588 [arXiv:2209.02724 [hep-th]].
  * (37) S. Stieberger, T. R. Taylor and B. Zhu, “Yang-Mills as a Liouville theory,” Phys. Lett. B 846, 138229 (2023) doi:10.1016/j.physletb.2023.138229 [arXiv:2308.09741 [hep-th]].
  * (38) A. Ball, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, “Supersymmetry and the Celestial Jacobi Identity,” [arXiv:2311.01364 [hep-th]].
  * (39) S. Banerjee, H. Kulkarni and P. Paul, “An infinite family of w1+∞ invariant theories on the celestial sphere,” JHEP 05 (2023), 063 doi:10.1007/JHEP05(2023)063 [arXiv:2301.13225 [hep-th]].
  * (40) S. Ebert, A. Sharma and D. Wang, “Descendants in celestial CFT and emergent multi-collinear factorization,” JHEP 03, 030 (2021) doi:10.1007/JHEP03(2021)030 [arXiv:2009.07881 [hep-th]].
  * (41) T. Adamo, W. Bu, E. Casali and A. Sharma, “All-order celestial OPE in the MHV sector,” JHEP 03, 252 (2023) doi:10.1007/JHEP03(2023)252 [arXiv:2211.17124 [hep-th]]
  * (42) L. Ren, A. Schreiber, A. Sharma and D. Wang, “All-order celestial OPE from on-shell recursion,” JHEP 10, 080 (2023) doi:10.1007/JHEP10(2023)080 [arXiv:2305.11851 [hep-th]].
  * (43) Y. Hu, L. Ren, A. Y. Srikant and A. Volovich, “Celestial dual superconformal symmetry, MHV amplitudes and differential equations,” JHEP 12, 171 (2021) doi:10.1007/JHEP12(2021)171 [arXiv:2106.16111 [hep-th]].
  * (44) W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, “Elements of celestial conformal field theory,” JHEP 08, 213 (2022) doi:10.1007/JHEP08(2022)213 [arXiv:2202.08288 [hep-th]].
  * (45) W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, “Celestial Yang-Mills amplitudes and D = 4 conformal blocks,” JHEP 09, 182 (2022) doi:10.1007/JHEP09(2022)182 [arXiv:2206.08979 [hep-th]].
  * (46) E. Casali, W. Melton and A. Strominger, “Celestial amplitudes as AdS-Witten diagrams,” JHEP 11 (2022), 140 doi:10.1007/JHEP11(2022)140 [arXiv:2204.10249 [hep-th]].
  * (47) S. Banerjee, R. Mandal, A. Manu and P. Paul, “MHV gluon scattering in the massive scalar background and celestial OPE,” JHEP 10 (2023), 007 doi:10.1007/JHEP10(2023)007 [arXiv:2302.10245 [hep-th]].
  * (48) R. Bhardwaj, L. Lippstreu, L. Ren, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, “Loop-level gluon OPEs in celestial holography,” JHEP 11, 171 (2022) doi:10.1007/JHEP11(2022)171 [arXiv:2208.14416 [hep-th]].
  * (49) H. Krishna, “Celestial gluon and graviton OPE at loop level,” [arXiv:2310.16687 [hep-th]].
  * (50) Y. Hu and S. Pasterski, “Celestial recursion,” JHEP 01 (2023), 151 doi:10.1007/JHEP01(2023)151 [arXiv:2208.11635 [hep-th]].
  * (51) A. Atanasov, W. Melton, A. M. Raclariu and A. Strominger, “Conformal block expansion in celestial CFT,” Phys. Rev. D 104 (2021) no.12, 126033 doi:10.1103/PhysRevD.104.126033 [arXiv:2104.13432 [hep-th]].
  * (52) S. Banerjee, H. Kulkarni and P. Paul, “Celestial OPE in Self Dual Gravity,” [arXiv:2311.06485 [hep-th]].
  * (53) S. Ghosh, P. Raman and A. Sinha, “Celestial insights into the S-matrix bootstrap,” JHEP 08, 216 (2022) doi:10.1007/JHEP08(2022)216 [arXiv:2204.07617 [hep-th]]
  * (54) L. Donnay, A. Puhm and A. Strominger, “Conformally Soft Photons and Gravitons,” JHEP 1901, 184 (2019) doi:10.1007/JHEP01(2019)184 [arXiv:1810.05219 [hep-th]].
  * (55) M. Pate, A. M. Raclariu and A. Strominger, “Conformally Soft Theorem in Gauge Theory,” Phys. Rev. D 100 (2019) no.8, 085017 doi:10.1103/PhysRevD.100.085017 [arXiv:1904.10831 [hep-th]].
  * (56) D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, “Celestial Amplitudes: Conformal Partial Waves and Soft Limits,” JHEP 10 (2019), 018 doi:10.1007/JHEP10(2019)018 [arXiv:1904.10940 [hep-th]].
  * (57) T. Adamo, L. Mason and A. Sharma, “Celestial amplitudes and conformal soft theorems,” Class. Quant. Grav. 36 (2019) no.20, 205018 doi:10.1088/1361-6382/ab42ce [arXiv:1905.09224 [hep-th]].
  * (58) A. Puhm, “Conformally Soft Theorem in Gravity,” JHEP 09 (2020), 130 doi:10.1007/JHEP09(2020)130 [arXiv:1905.09799 [hep-th]].
  * (59) A. Guevara, “Notes on Conformal Soft Theorems and Recursion Relations in Gravity,” arXiv:1906.07810 [hep-th].