# Generating 3D structures from a 2D slice with GAN-based dimensionality
expansion

Steve Kench<EMAIL_ADDRESS>Dyson School of Design Engineering,
Imperial College London, London SW7 2DB Samuel J. Cooper
<EMAIL_ADDRESS>Dyson School of Design Engineering, Imperial
College London, London SW7 2DB

###### Abstract

Generative adversarial networks (GANs) can be trained to generate 3D image
data, which is useful for design optimisation. However, this conventionally
requires 3D training data, which is challenging to obtain. 2D imaging
techniques tend to be faster, higher resolution, better at phase
identification and more widely available. Here, we introduce a generative
adversarial network architecture, SliceGAN, which is able to synthesise high
fidelity 3D datasets using a single representative 2D image. This is
especially relevant for the task of material microstructure generation, as a
cross-sectional micrograph can contain sufficient information to statistically
reconstruct 3D samples. Our architecture implements the concept of uniform
information density, which both ensures that generated volumes are equally
high quality at all points in space, and that arbitrarily large volumes can be
generated. SliceGAN has been successfully trained on a diverse set of
materials, demonstrating the widespread applicability of this tool. The
quality of generated micrographs is shown through a statistical comparison of
synthetic and real datasets of a battery electrode in terms of key
microstructural metrics. Finally, we find that the generation time for a
$10^{8}$ voxel volume is on the order of a few seconds, yielding a path for
future studies into high-throughput microstructural optimisation.

##  1 Introduction

The properties and behaviour of a material depend strongly on its
microstructure, which in turn is controlled by conditions of manufacture.
Physical simulations play an important role in offering insight into these
relationships to help inform the design of next generation materials.
Importantly, many material behaviours, such as deformation under stress or
fluid flow through a porous medium, are inherently volumetric and cannot be
accurately modelled using 2D data alone. The fidelity of simulation techniques
used to extract these properties is thus partly determined by the quality of
available 3D microstructural datasets. However, as compared to their 3D
equivalents, 2D micrographs are often easier to obtain, higher resolution and
can contain a more representative distribution of features based on a larger
field of view. Accurate methods for the statistical reconstruction of 3D
volumes using 2D micrographs are thus highly desirable.

Generative adversarial networks (GANs) are a promising candidate model for
this task. A GAN consists of two neural nets: a generator, $G$, to synthesise
fake samples, f, and a discriminator, $D$, to distinguish between f and real
samples, , r, from the dataset. During training, $G$ and $D$ are updated
iteratively, enabling the generator to capture features of the real dataset
and ultimately, if successful, produce realistic images 9, 5, 19. Since their
introduction, extensive work has led to improved image quality, feature
diversity and training stability 2. The motivation behind this paper is to
apply this powerful technique to the task of dimensionality expansion.
Crucially, in order to achieve this, the incompatibility between a 3D
generator versus 2D training data must be solved. Furthermore, important
considerations regarding the effect of information density on image quality
are raised. The key contributions of this work are summarised as follows:

  1. 1.

In section 3, we introduce a novel GAN architecture, _SliceGAN_ , which is
designed to address the challenge of generating 3D volumetric data from 2D
slices of an isotropic material. The ability to statistically reconstruct
anisotropic microstructures with a simple extension is also demonstrated.

  2. 2.

In section 4, the low quality regions found at the outer edges of GAN
generated images is discussed. This issue is found to be associated with non-
uniform information density in the generator. The authors define a set of
requirements for the parameters of transpose convolutional operations to avoid
this issue.

  3. 3.

In section 5, _SliceGAN_ is applied to a range of 2D microstructures, and
shown to produce 3D volumes with visually indistinguishable cross sections
from training data. The seven example training sets include a synthetic
crystalline microstructure, an anisotropic polymer membranes and a three phase
battery electrode material. This final example is further validated through a
comparison of three key metrics to demonstrate the statistical similarity
between synthetic volumes and the original dataset.

##  2 Background

###  2.1 GANs and their application in material science

Deep convolutional generative adversarial networks use a layered architecture
to identify and reproduce the hierarchical features of an image dataset. When
trained successfully, they offer high resolution, diverse image generation.
Since the introduction of GANs, numerous adaptions have been proposed, falling
broadly into two categories. First, there are those dedicated to improved
training of adversarial nets, such as Wasserstein GANs (WGANs), which increase
training stability and generated image diversity by addressing the challenge
of vanishing gradients 2, 12. Second, there are approaches that aim to bring
new capabilities such as conditional GANs (cGANs), which are used to condition
a generator based on image labels, thus allowing the synthesis of multiple
image categories using a single generator 17. A further relevant example of
this latter category developed a method for spatial model generation from a
set of 3D training samples 23. The diverse and rapid evolution of these
different methodologies indicates the flexibility and wide applicability of
GANs.

It is thus unsurprising that GANs have already proven useful in the field of
material science 18. A recent study by Gayon-Lombardo et al. used GANs to
generate the microstructure of energy storage materials based on 3D training
data 7. The synthetic volumes were not only much larger than the training set,
but also had periodic boundaries, which makes them particularly well suited to
generating representative domains for multi-physics simulation. GANs have also
been applied to the task of material design. Li $et.$ $al$ trained a generator
to synthesise realistic micrographs of a semi-conductor surface 26. A batch of
instances can then be produced by the generator, and the property of interest,
in this case absorption co-efficient, calculated for each sample. A Gaussian
processes algorithm was then used to optimise for this property through an
informed exploration of the microstructural space. This offers an efficient
alternative compared to a conventional experimental materials design approach.
These studies are an indication of the broad potential applications of GANs in
the field of material science.

###  2.2 Current microstructural reconstruction approaches

Due to the widespread potential applications of 2D to 3D reconstruction
algorithms, a range of techniques have been developed in previous studies. For
example, given a particle size distribution measured from a 2D image,
simulated annealing algorithms can be used alongside ellipsoid packing to
emulate powder processing 24. Equally, tessellation algorithms are
incorporated in software tools such as Dream3D, which uses extracted 2D
properties such as grain size and orientation to simulate 3D grain growth 11.
Unfortunately, these techniques are designed for, and thus limited to, powder
and polycrystalline microstructures respectively. Furthermore, a loss of
geometric information is inevitable, as irregular particles are approximated
to ellipsoids, and complex grain shapes are characterised by mean diameters
and aspect ratios.

Figure 1: _SliceGAN_ training procedure. First, real images are sampled from a
2D training sample. Second, a fake volume f is generated and sliced along $x$,
$y$ and $z$. This yields a compatible pair of datasets, which can both be fed
to a 2D discriminator.

Other approaches have been developed based on correlation statistics. Given an
image containing $i$ phases, the two-point correlation function $S_{2}$
describes the probability that two randomly selected pixels, separated by
distance $r$, are of the same phase 22. Importantly, it has been shown
analytically that for a given isotropic material, the same two-point
correlation function is recovered whether calculated for a 2D or 3D dataset21.
This led to the development of stochastic algorithms which take initially
random 3D volumes and iteratively attempt to minimise the difference between
the correlation functions of the real and generated data 13, 15. Whilst
successful in some cases, the degeneracy of two-point correlations can lead to
unrealistic results 8. Furthermore, even state-of-the-art methods are
computationally expensive with simulation times to synthesise a modest
$10^{6}$ voxel sample on the order of hours using a 2.8 GHz, 16GB RAM
computer, making this approach unsuitable for materials optimisation
approaches 27.

The GANs described in this work take a similar time span to train (c. 4 hours
on an NVIDIA Titan Xp GPU); however, once trained, the generator is able to
synthesis $10^{8}$ voxel instances in seconds. This $10^{5}$ acceleration in
generation speed compared to 27 enables both high-throughput optimisation and
large scale modelling. Furthermore, GANs are highly flexible compared to the
tessellation algorithms described above. Indeed, a key strength of deep
convolutional neural nets is their ability to efficiently capture the diverse
and complex features of an image without any user defined statistical metrics
as inputs. Another important advantage of using GANs is the breadth of
available variants that could be incorporated. Adaptations might aim to
decrease training time, for example through transfer learning, or to
facilitate new capabilities, for example using conditional GANs to allow the
generation of different microstructures with specific features indicated by a
label. _SliceGAN_ is thus a building block for a powerful tool set of machine
learning methods for microstructural characterisation, modelling and
optimisation.

##  3 SliceGAN

The fundamental role of _SliceGAN_ is to resolve the dimensionality
incompatibility between a 2D training image and 3D generated volumes. This is
achieved by incorporating a slicing step before fake instances from the 3D
generator are sent to the 2D discriminator. For a generated cubic volume of
edge length $l$ voxels, $3l$ 2D images are obtained by taking slices along the
$x,y$ and $z$ directions at 1 voxel increments. During the training of $D$,
for each fake 2D slice, a real 2D image is also sampled and fed to the
discriminator, such that it learns equally from real and fake instances. To
train $G$, the same slicing procedure is followed; however, a larger batch
size ($m_{G}$) compared to the discriminator ($m_{D}$) is used. This
rebalances the effect of training $D$ on a large number of slices per
generated sample. We find that $m_{G}=2m_{D}$ typically results in the best
efficiency. Importantly, the particular architecture of $G$ used in this study
means that each slice in any given set of 32 adjacent planes is synthesised
through a unique combination of kernel elements (see section 4). As such, a
minimum of 32 slices in each direction must be shown to $D$ in order to ensure
each path through the generator is trained. In practice, we find training to
be both more reliable and efficient when $D$ is applied to all 64 slices in
each direction, despite the doubling of discriminator operations that this
entails. After slicing, a standard Wasserstein loss function is used as
proposed in 2 to encourage stable training. The complete training process for
_SliceGAN_ is described in Algorithm 1, with the data sampling, generation and
slicing stages depicted in Figure 1.

1:$w$ and $\theta$, the trainable parameters for $D$ and $G$ respectively; the
number of discriminator iterations per generator iteration $n_{D}$; the batch
sizes $m_{D}$ and $m_{G}$ for $D$ and $G$ respectively; the gradient penalty
coefficient, $\lambda$; volume edge length, $l$; and Adam optimizer
hyperparameters 16, $\alpha,\beta 1,\beta 2$.

2:while $\theta$ has not converged do

3: Discriminator training:

4: for $t=0,...,n_{{D}}$ do

5: for $i=1,...,m_{D}$ do

6: Sample a latent vector from a normal distribution $\textbf{z}\sim
p(\textbf{z})$

7: $\textbf{f}\leftarrow G_{\theta}(\textbf{z})$ generate a 3D volume

8: for $a=1,2,3$ do

9: for $d=1,...,l$ do

10: $\textbf{f}_{\textbf{s}}\leftarrow$ 2D slice of f at depth $d$ along axis
$a$

11: Sample an $l\times l$ image from the real dataset
$\textbf{r}\sim\mathbf{P}_{\textbf{r}}$

12: Sample a random number $\epsilon\sim\mathbb{U}[0,1]$

13:
$\textbf{k}\leftarrow\epsilon\textbf{f}_{\textbf{s}}+(1-\epsilon)\textbf{r}$

14: $L_{D}^{(a,d)}\leftarrow
D_{w}(\textbf{f}_{\textbf{s}})-D_{w}(\textbf{r})+\lambda(\|\nabla_{\textbf{k}}D_{w}(\textbf{k})\|_{2}-1)^{2}$

15:
$w\leftarrow\text{Adam}(\nabla_{w}\frac{1}{m_{D}}\sum_{a=1}^{3}\sum_{d=1}^{l}L_{D}^{(a,d)})$

16: Generator Training:

17: for $j=1,...,m_{G}$ do

18: Sample a latent vector from a normal distribution $\textbf{z}\sim
p(\textbf{z})$

19: $\textbf{f}\leftarrow G_{\theta}(\textbf{z})$ generate a 3D volume

20: for $a=1,2,3$ do

21: for $d=1,...,l$ do

22: $\textbf{f}_{\textbf{s}}\leftarrow$ 2D slice of f at depth $d$ along axis
$a$

23: $L_{G}^{(a,d)}\leftarrow-D_{w}(\textbf{f}_{\textbf{s}})$

24:
$\theta\leftarrow\text{Adam}(\nabla_{\theta}\frac{1}{m_{G}}\sum_{a=1}^{3}\sum_{d=1}^{l}L_{G}^{(a,d)})$

Algorithm 1 SliceGAN algorithm for isotropic materials

Importantly, the above approach is only suitable for isotropic materials,
where a single 2D image can be used to train slices from the $x,y$ and $z$
orientations of f. An extension of the architecture is required for
anisotropic materials. For example, given a microstructure containing a matrix
with rods oriented parallel to the $z$ axis, two training images are needed:
one perpendicular to the $z$ axis containing circular features and another
parallel to the $z$ axis containing long rectangular features (whilst these
are the simplest orientations for this example, any set of perpendicular
planes could be chosen in principal). During training, images $\textbf{r}_{z}$
are sampled from the former and images $\textbf{r}_{x,y}$ from the latter,
whilst slices from the fake samples are also divided into those taken along
the $z$ axis, $\textbf{f}_{z}$, and those taken from $x$ or $y$,
$\textbf{f}_{x,y}$. Separate discriminators are then taught to capture the
distribution of features along the different orientations, and thus teach $G$
to synthesise volumes that look like $\textbf{r}_{z}$ along the $z$ axis, and
$\textbf{r}_{x,y}$ along the $x$ and $y$ axis. The adjusted _SliceGAN_
algorithm for anisotropic materials is given in Supplementary Information S1.
Successful examples of this approach are presented in section 5.

##  4 Generator information density

Figure 2: a) An example of how information density is calculated for a single
transpose convolution with input size = $3\times 3$, $k$ = 4, $s$ = 2 and $p$
= 0. The overlap from the outputs of individual pixels is depicted, showing a
information density gradient at the edges of the final image. b) Illustration
of how information density gradients propagate and are exacerbated in a full
neural net as an input is passed through multiple transpose convolutional
layers.

During the development stages of _SliceGAN_ , significantly lower visual
quality was observed near sample edges compared to the bulk. This may not be
problematic for GANs applications such as face generation, where key objects
often lie in the centre of an image and the edges are of less importance.
However, for microstructural synthesis this is not the case, as although the
relative position of features is important, their absolute location is often
arbitrary. Edges are thus of equal importance to the center, and can strongly
influence the measured physical properties of a microstructure. To understand
the cause of edge artifacts, a simple test case is proposed to explore the
nature of transpose convolutions, which are the key operator within the
convolutional generator neural nets used in this work. We first define a
$3\times 3$ input with a single channel, and pixel values set to 1. The
transpose convolution parameters are selected with kernel size of $k=4$;
stride, $s=2$; padding, $p=0$ (NB. In a generator, $p$ refers to the removal
of near edge layers, rather than their addition) and output channels
$C_{\text{out}}=1$. The kernel weights are set to 1. Figure 2a depicts this
simple scenario, and shows how the transpose convolution operates on the input
pixel by pixel. We first observe the output from the top left pixel, which is
distributed over a region equal to the kernel size. As the contribution from
subsequent pixels is added, an overlap of information is seen. The final
output shows a central region with flat information density of 4, and lower
information content towards the edges and corners. Figure 2b shows the
additive effect of this phenomenon in a typical generator with multiple
convolutional operations. Indeed, the number of parameters contributing to
central pixels is 3 orders of magnitude greater than for edge pixels, which
explains the poor image quality in this region.

In order to avoid artifacts of this nature, three rules can be used to define
acceptable combinations of $k,s$ and $p$ such that a uniform information
density is ensured (Although an important consideration for image quality,
$C_{\text{out}}$ does not effect information density gradients):

  1. 1.

$s<k$ (Although $s\leq k$ is a sufficient requirement for uniformity, $s<k$
ensures that there is kernel overlap which is crucial for avoiding feature
discontinuities in the output.)

  2. 2.

$k\bmod s=0$ (If $k$ is not divisible by $s$, the kernel output regions from
nearby input pixels never lie flush to each other, resulting in a checkerboard
information density 20. An example of this behaviour with $k=4$, $s=3$ is
shown in Supplementary Information S2.)

  3. 3.

$p\geq k-s$ (The region of edge voxels with less than the central uniform
information density is of width $k-s$, and must thus be removed by applying a
padding of this value or greater.)

These constraints leave a small set of practical transpose convolutions. We
first discard any prime value of $k$, as in this scenario rule 1 and 2
together only allow for $s=1$. Such configurations result in no expansion of
the spatial dimensions and are thus not useful. In general, $k>6$ is also
impractical, especially in 3D convolutional nets, as the number of parameters
scales with $k^{3}$. Assuming $p$ is chosen to be as small as possible to
minimise the amount of cropping at each layer (which equates to wasted
computational expense), this leaves three practical sets for $\\{k,s,p\\}$:
$\\{4,2,2\\},\\{6,3,3\\},\\{6,2,4\\}$. In the work presented here, the
$\\{4,2,2\\}$ set of parameters are used for most transpose convolutions to
avoid the computational expense of $k=6$. An alternative approach to avoid
edge artifacts could be the use of resize-convolution generator architectures
with a different set of parameter rules (most importantly, the use of any
zero-padding results in undesirable information gradients). Unfortunately,
under these constraints resize-convolution has significantly larger memory
requirements for 3D volume generation than transpose convolutions. A sacrifice
in either training time (by reducing batch size) or image quality (by reducing
the number of filters) thus becomes necessary, as shown in Supplementary
Information S3.

Table 1: Table showing the architecture of the 3D generator (memory size $\approx$ 50 MB) and 2D discriminator (memory size $\approx$ 11 MB) used in this work.  Generator | Discriminator
---|---
Layer | $k,s,p$ | Output shape | Layer | $k,s,p$ | Output shape
Input $\mathbf{z}$ | - | $64\times 4\times 4\times 4$ | Input | - | $3\times 64\times 64$
1 | 4,2,2 | $512\times 6\times 6\times 6$ | 1 | 4,2,1 | $64\times 32\times 32$
2 | 4,2,2 | $256\times 10\times 10\times 10$ | 2 | 4,2,1 | $128\times 16\times 16$
3 | 4,2,2 | $128\times 18\times 18\times 18$ | 3 | 4,2,1 | $256\times 8\times 8$
4 | 4,2,2 | $64\times 34\times 34\times 34$ | 4 | 4,2,1 | $512\times 4\times 4$
5 | 4,2,3 | $3\times 64\times 64\times 64$ | 5 | 4,2,0 | $1\times 1\times 1$
softmax | - | $3\times 64\times 64\times 64$ |  |  |
Table 2: Synthesis details for the examples presented in Figure 3. Label | Material | Imaging technique | Image type | Isotropy | Ref
---|---|---|---|---|---
A | Polycrystalline grains | Synthetic | Two-phase | Isotropic | -
B | Ceramic (perovskite) | Kelvin probe force topography | Two-phase | Isotropic | 10
C | Carbon fibre rods | Secondary electron microscopy | Two-phase | Anisotropic | 3
D | Battery seperator | X-ray tomography reconstruction | Three-phase | Anisotropic | 6
E | Steel | Electron back-scatter microscopy | Colour | Isotropic | 1
F | Grain boundary | Synthetic | Three-phase | Anisotropic | -
G | NMC battery cathode | X-ray tomography | Three-phase | Isotropic | 14

Another crucial attribute when designing a GAN architecture for material
science applications is the ability to generate different size volumes during
inference. This is important as often the standard training volume (normally
64 voxels cubed to allow reasonably large batch sizes under memory
constraints) is not large enough for use in multiphysics simulations. A
previous study suggests that after training, bespoke volumes can be
synthesised by varying the spatial size of the latent vector $\mathbf{z}$ 7.
However, this approach results in lower quality microstructures with
distortions. This observation can be explained by the fact that during
training, $\mathbf{z}$ has a spatial dimension of 1, making the first
transpose convolution of $G$ unique in that it operates on a single voxel per
channel. In this scenario, no kernel overlap occurs in the first layer output.
Increasing the spatial dimension of $\mathbf{z}$ during inference changes this
behaviour and introduces overlap. To avoid this problem, we choose to train
into the first generator layer an understanding of overlap; thus an input
vector with spatial size 4 is used. The resulting information spread is
depicted in Supplementary Information S4, with the final network architectures
given in Table 1. Interestingly, this alteration leads to periodicity within
the generator, whereby sets of every 32${}^{\text{nd}}$ plane are generated
from the same combination of kernel elements, as shown in Supplementary
Information S5. This does not make the synthetic volume itself periodic, as
each plane is associated with a different region of the input seed. However,
it does mean that it is not possible to train $G$ to capture anisotropy within
a given plane (e.g a graded density).

##  5 Empirical results

###  5.1 Data pre-processing

Whilst it is possible to train _SliceGAN_ using RGB or greyscale micrographs
(for which no pre-processing is necessary), the approach has been most
successful with one-hot encoded representations of segmented $n$-phase
microstructural data. For a training dataset, the encoding process entails the
creation of $n$ separate channels, each containing 1s where the material phase
they correspond to is present and 0s otherwise, as shown in Supplementary
Information S6. In generated instances, $n$ channels are again required,
though they now represent the probability of finding a given phase at a
location in an image. This is implemented using a softmax function as the
final layer of the generator. In either case, an $n$-phase microstructure can
always be recovered from a one-hot array through a max function over the
channel axis.

###  5.2 Application to microstructures

Figure 3: Application of _SliceGAN_ to a variety of microstructures, with
details specified in Table 2. From left to right, the original training
dataset, a 3D statistical reconstruction and 2 examples of slices through an
example volume taken at different angles.

To demonstrate the capabilities of _SliceGAN_ , a range of micrographs were
chosen to be statistically reconstructed, as shown in Figure 3. The first
column shows a region of the 2D training data used. The second column shows an
example of a generated 3D instance, with the volume edge lengths selected to
best display the microstructural features. Next, slices of a synthetic volume
taken i) parallel to a discriminated plane to show the dataset quality and ii)
at a 45∘ angle to the discriminators to demonstrate the consistency of the
microstructure in directions not involved in training. For each case, the
material, imaging technique and training details are given in Table 2. The
first example presented in row A of Figure 3 is a synthetic grain structure,
chosen for its well defined features: perfectly straight grain boundaries and
exclusively convex grains. _SliceGAN_ performs reasonably well on this data,
generating realistic looking volumes, though some disconnected and curved
grain boundaries are observed. These shortcomings are not specific to the
_SliceGAN_ approach, as similar features are also observed in synthetic 2D
images from a conventional GAN. The next training image (row B) is a real
micrograph from surface analysis of a ceramic, where the black phase
correspond to areas with high Caesium content 10. Again, a visually realistic
sample is generated through _SliceGAN_.

Figure 4: Statistical analysis of three electrochemical and general materials
properties, showing that the real properties of the dataset are captured well
by both a 3D to 3D GAN and _SliceGAN_. Boxes depict the interquartile range
(IQR) from the lower quartile, Q1 to the upper quartile, Q3). Each box
contains a horizontal line and a small square to indicate the median and mean
respectively. The upper and lower whiskers show the last datum greater than
$\text{Q3}+(1.5\times\text{IQR})$ and less than
$\text{Q1}-(1.5\times\text{IQR})$ respectively. Outlier data points beyond
this range are represented by solid black diamonds.

Two images of fibre reinforced composite taken from perpendicular views
demonstrate the ability to synthesise anisotropic images 3. A similar approach
is taken for a polymer battery separator material 6, 25. This demonstrates the
flexibility of _SliceGAN_ , and will enable its application to many common
microstructures (rows C and D). A micrograph of polycrystalline single phase
steel is considered next (row E) 1. Previous work was focused on segmented 2
or 3 phase materials; here the extension to colour images (where color encodes
grain orientation) is presented. Image quality is somewhat reduced compared to
previous examples. This is likely because our 12 GB GPU memory constraint
means that the networks are too small to capture continuous label spaces
(instead of the segmented materials in other examples). These volumes can be
made to look more like the training data by applying filters (e.g. mode), but
this kind of post-processing is beyond the scope of this study and would need
to be selected on a case by case based. In row F, a test on a synthetic
material interface is presented where a large 128 $\times 10^{4}$ pixel image
was generated using a bounded random walk algorithm to generate an interface,
followed by seeding of randomly size boundary particles. In this case, the 2D
example images shown are small crops of the full training dataset. This was an
opportunity to test whether 2 discriminators, both perpendicular to the
material interface, were sufficient for training. Interestingly, whilst the
general surface topology of the synthesised material reflects the real dataset
well, the interface particles form long ridges on the surface at a 45∘ angle.
This is due to insufficient constraint without the top view of the material,
which allows the generator to effectively trick the discriminator. This could
potentially be resolved by implementing discriminated slices at 45∘ angles.

Finally, open access tomographic data collected from a Li-ion NMC cathode
sample is used as shown in row G 14. Although the dataset is 3D, only a random
subsample of 2D sections are used for _SliceGAN_ training. This experiment is
particularly useful as 3D properties of the original dataset can be compared
to those of the synthesised microstructure. Furthermore, instances from a 3D
to 3D GAN can also be evaluated. Thus, to quantify the performance of
_SliceGAN_ , 100 sample batches were taken from each of i) the real dataset
ii) the 3D to 3D dataset generated by Gayon-Lombardo et al 7 and iii)
_SliceGAN_. Three relevant metrics were chosen and calculated using
_TauFactor_ 4 (an open-source microstructural analysis package) for each set
of microstructures, as shown in Figure 4. Two-point correlation functions and
triple phase boundary densities are also compared in Supplementary Information
S7.

For all of these conventional microstructural metrics, both GAN architectures
reproduce the training set well. Although the medians correspond very closely
in each case, the distributions, in particular in the case of the relative
diffusivity, are not exactly the same as the training set. Crucially, unlike
the other metrics, the effective diffusivity is an emergent property of the
generated 3D volume as a whole, and is not measurable from any individual 2D
slice. As such, it’s particularly impressive that _SliceGAN_ can reproduce
this metric.

##  6 Conclusions

The results presented in this study show the ability of _SliceGAN_ to
faithfully synthesise 3D $n$-phase media from 2D micrographs. This tool has
three key uses: First, in the field of material characterisation as a means to
visualise a statistically realistic 3D instance of a 2D micrograph. This is
especially important considering the limitations of direct 3D imaging, namely
in terms of resolution and field of view. Second, for electrochemical
simulation; 2D micrographs are insufficient for a range of desirable
electrochemical and mechanical simulation techniques. Finally, for materials
optimisation purposes, as the speed of the generator allows rapid exploration
of the space of possible 3D microstructures. Future work will aim to develop
this tool set in combination with other common GANs approaches such as
conditional GANs and transfer learning. The former will enable interpolation
between microstructures with differing properties, whilst the latter aims to
speed the training process. Thus, there is significant potential for expansion
of this approach as a tool for material characterisation and optimisation.

## Data Availability

The study used open-access training data available from the following sources:
ceramic 10, carbon fibre rods 3, battery separator 6, 25, steel 1, and NMC
battery cathode 14.

## Code Availability

The codes used in this manuscript, are available at:
https://github.com/stke9/SliceGAN. All other generated data used is available
from the authors on request.

Correspondence and requests for materials should be directed to Steven Kench
at<EMAIL_ADDRESS>
## Acknowledgements

This work was supported by funding from the EPSRC Faraday Institution Multi-
Scale Modelling project (https://faraday.ac.uk/; EP/S003053/1, grant number
FIRG003 received by SK). The Titan Xp GPU used for this research was kindly
donated by the NVIDIA Corporation through their GPU Grant program to SC. The
authors would also like to thank Aron Walsh, Andrea Gayon Lombardo and Daniel
Figueroa for their valuable comments, as well at the rest of the tldr group.

## Authorship

SK designed and developed the code for $SliceGAN$, trained the models,
performed the statistical analysis and drafted the manuscript. SJC contributed
to the development of the concepts presented in all sections of this work,
helped with data interpretation in section 5 and made substantial revisions
and edits to all sections of the draft manuscript.

## Competing interests

The authors declare no competing interests.

## References

##

  * 1 Oxford instruments grain size and grain boundary characterisation in sem, 2020.
  * 2 M. Arjovsky, S. Chintala, and L. Bottou.  Wasserstein gan.  arXiv preprint arXiv:1701.07875, 2017.
  * 3 D. Armentrout, M. Kumosa, and T. Mcquarrie.  Boron-free fibers for prevention of acid induced brittle fracture of composite insulator grp rods.  Power Delivery, IEEE Transactions on, 18:684 – 693, 08 2003.
  * 4 S. Cooper, A. Bertei, P. Shearing, J. Kilner, and N. Brandon.  Taufactor: An open-source application for calculating tortuosity factors from tomographic data.  SoftwareX, 5:203 – 210, 2016.
  * 5 A. Creswell, T. White, V. Dumoulin, K. Arulkumaran, B. Sengupta, and A. A. Bharath.  Generative adversarial networks: An overview.  IEEE Signal Processing Magazine, 35(1):53–65, 2018.
  * 6 D. Finegan, S. Cooper, B. Tjaden, O. Taiwo, J. Gelb, G. Hinds, D. Brett, and P. Shearing.  Microstructure reconstruction of battery polymer separators by fusing 2d and 3d image data for transport property analysis.  Journal of Power Sources, 333:184–192, 2016.
  * 7 A. Gayon-Lombardo, L. Mosser, N. P. Brandon, and S. J. Cooper.  Pores for thought: The use of generative adversarial networks for the stochastic reconstruction of 3d multi-phase electrode microstructures with periodic boundaries, 2020.
  * 8 C. J. Gommes, Y. Jiao, and S. Torquato.  Microstructural degeneracy associated with a two-point correlation function and its information content.  Physical Review E, 85(5):051140, 2012.
  * 9 I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.  Generative adversarial nets.  In Advances in neural information processing systems, pages 2672–2680, 2014.
  * 10 P. Gratia, I. Zimmermann, P. Schouwink, J.-H. Yum, J.-N. Audinot, K. Sivula, T. Wirtz, and M. K. Nazeeruddin.  The many faces of mixed ion perovskites: unraveling and understanding the crystallization process.  ACS Energy Letters, 2(12):2686–2693, 2017.
  * 11 M. A. Groeber and M. A. Jackson.  Dream. 3d: a digital representation environment for the analysis of microstructure in 3d.  Integrating materials and manufacturing innovation, 3(1):5, 2014\.
  * 12 I. Gulrajani, F. Ahmed, M. Arjovsky, V. Dumoulin, and A. C. Courville.  Improved training of wasserstein gans.  CoRR, abs/1704.00028, 2017.
  * 13 A. Hasanabadi, M. Baniassadi, K. Abrinia, M. Safdari, and H. Garmestani.  3d microstructural reconstruction of heterogeneous materials from 2d cross sections: a modified phase-recovery algorithm.  Computational Materials Science, 111:107–115, 2016.
  * 14 T. Hsu, W. K. Epting, R. Mahbub, N. T. Nuhfer, S. Bhattacharya, Y. Lei, H. M. Miller, P. R. Ohodnicki, K. R. Gerdes, H. W. Abernathy, et al.  Mesoscale characterization of local property distributions in heterogeneous electrodes.  Journal of Power Sources, 386:1–9, 2018.
  * 15 H. Izadi, M. Baniassadi, A. Hasanabadi, B. Mehrgini, H. Memarian, H. Soltanian-Zadeh, and K. Abrinia.  Application of full set of two point correlation functions from a pair of 2d cut sections for 3d porous media reconstruction.  Journal of Petroleum Science and Engineering, 149:789–800, 2017\.
  * 16 D. P. Kingma and J. Ba.  Adam: A method for stochastic optimization, 2017.
  * 17 M. Mirza and S. Osindero.  Conditional generative adversarial nets.  arXiv preprint arXiv:1411.1784, 2014.
  * 18 L. Mosser, O. Dubrule, and M. J. Blunt.  Reconstruction of three-dimensional porous media using generative adversarial neural networks.  Phys. Rev. E, 96:043309, Oct 2017.
  * 19 A. Odena.  Semi-supervised learning with generative adversarial networks.  arXiv preprint arXiv:1606.01583, 2016.
  * 20 A. Odena, V. Dumoulin, and C. Olah.  Deconvolution and checkerboard artifacts.  Distill, 2016.
  * 21 S. Torquato and H. Haslach Jr.  Random heterogeneous materials: microstructure and macroscopic properties.  Appl. Mech. Rev., 55(4):B62–B63, 2002.
  * 22 S. Torquato and G. Stell.  Microstructure of two-phase random media. i. the n-point probability functions.  The Journal of Chemical Physics, 77(4):2071–2077, 1982.
  * 23 J. Wu, C. Zhang, T. Xue, W. T. Freeman, and J. B. Tenenbaum.  Learning a probabilistic latent space of object shapes via 3d generative-adversarial modeling.  CoRR, abs/1610.07584, 2016.
  * 24 H. Xu, D. A. Dikin, C. Burkhart, and W. Chen.  Descriptor-based methodology for statistical characterization and 3d reconstruction of microstructural materials.  Computational Materials Science, 85:206–216, 2014.
  * 25 H. Xu, F. Usseglio-Viretta, S. Kench, S. J. Cooper, and D. P. Finegan.  Microstructure reconstruction of battery polymer separators by fusing 2d and 3d image data for transport property analysis.  Journal of Power Sources, Accepted 2020.
  * 26 Z. Yang, X. Li, L. Catherine Brinson, A. N. Choudhary, W. Chen, and A. Agrawal.  Microstructural materials design via deep adversarial learning methodology.  Journal of Mechanical Design, 140(11), 2018.
  * 27 Y. Zhang, M. Yan, Y. Wan, Z. Jiao, Y. Chen, F. Chen, C. Xia, and M. Ni.  High-throughput 3d reconstruction of stochastic heterogeneous microstructures in energy storage materials.  npj Computational Materials, 5(1):1–8, 2019.


11institutetext: National Institute of Science Education and Research (NISER),
Bhubaneswar 752050, Odisha, India22institutetext: Institute of Physics,
Sachivalaya Marg, Bhubaneshwar, India-75100533institutetext: Homi Bhabha
National Institute, Training School Complex, Anushaktinagar, Mumbai 400094,
India44institutetext: Centre for High Energy Physics, Indian Institute of
Science, C.V. Raman Avenue, Bangalore 560012, India

# MHV Gluon Scattering in the Massive Scalar Background and Celestial OPE

Shamik Banerjee 1,3 Raju Mandal 2,3 Akavoor Manu 4 and Partha Paul
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we study the tree-level OPE between two positive helicity
outgoing gluons in the celestial CFT for the Yang-Mills theory chirally
coupled to a massive scalar background. This theory breaks the translation as
well as scale invariance. We compute the subleading terms in the OPE expansion
and show that they are same as the subleading terms of the OPE expansions in
the MHV sector. As a result the amplitudes of this theory also satisfy the set
of differential equations obtained previously for MHV amplitudes in pure YM
theory. This is not surprising because the symmetries coming from the leading
and subleading soft gluon theorems do not change in the presence of a massive
scalar background.

##  1 Introduction

Scattering amplitudes are an important set of observables in any quantum field
theory in four dimensional asymptotically flat spacetime. Usually these
amplitudes are written in momentum eigen-basis in which translation invariance
is manifest. However, instead of the momentum eigenstates if one uses the
conformal primary or the boost eigen-basis, scattering amplitudes transform as
a correlation function of a 2D conformal field theory under the (Lorentz)
$SL(2,C)$ transformations Pasterski:2016qvg ; Pasterski:2017kqt ;
Banerjee:2018gce ; Freidel:2022skz ; Cotler:2023qwh 111The Lorentz group
$SL(2,\mathbb{C})$ act on the celestial sphere as the group of global
conformal transformations.. The amplitudes written in the conformal primary
basis are known as celestial amplitudes. For massless particles they are
obtained by Mellin transformations with respect to the energies of the
external particles Pasterski:2017kqt ; Banerjee:2018gce .

Now the fact that they transform like conformal correlators in 2D allows us to
use powerful techniques familiar from the study of the 2D CFT. In particular,
momentum space soft theorems can be reinterpreted as Ward identities for
various current algebra symmetries on the celestial sphere Strominger:2014as ;
Strominger:2014ass ; Banerjee:2020zlg ; Donnay:2018neh ; Pate:2019mfs ;
Fan:2019emx ; Nandan:2019jas ; Adamo:2019ipt ; Banerjee:2020vnt ; He:2014cra ;
Guevara:2021abz ; Strominger:2021lvk ; Strominger:2021mtt ; Kapec:2015ena ;
Kapec:2014zla ; Campiglia:2015qka ; Nande:2017dba ; He:2020ifr ; Kapec:2014opa
; He:2017fsb ; Donnay:2020guq ; Stieberger:2018onx ; Banerjee:2022wht ;
Himwich:2021dau . For example, the leading soft theorem for a gluon can be
interpreted as the Ward identity for the $SU(N)$ Kac-Moody algebra
Strominger:2014ass ; Donnay:2018neh ; Pate:2019mfs ; Fan:2019emx ;
Nandan:2019jas ; Adamo:2019ipt ; He:2014cra ; Kapec:2015ena ; Kapec:2014zla ;
Campiglia:2015qka ; Nande:2017dba ; He:2020ifr . Similarly, the subleading
soft gluon theorem can also be interpreted as the Ward identity of an infinite
dimensional current algebra Banerjee:2020vnt ; Guevara:2021abz ;
Strominger:2021lvk ; Strominger:2021mtt . The symmetry algebras coming from
leading and subleading soft gluon theorems contain null states
Banerjee:2020vnt . Decoupling of these null states gives rise to a set of
differential equations for the celestial amplitude Banerjee:2020vnt 222Please
see Hu:2021lrx  for the derivation of the differential equations in the pure
YM case using BCFW shift. 333For a similar story in the case of gravitons
please see Banerjee:2020zlg ; Banerjee:2021cly ; Banerjee:2023zip .. Hence the
amplitudes are heavily constrained due to the presence of infinite dimensional
symmetries coming from soft gluon theorems. One gets these null states in case
of gravitons as well as gluons by demanding consistency of the celestial OPE
with the soft theorems Banerjee:2020zlg ; Banerjee:2021cly ; Banerjee:2020vnt
; Banerjee:2023zip . For more discussions on celestial OPE please see
Fan:2019emx ; Pate:2019lpp ; Banerjee:2020kaa ; Banerjee:2021dlm ;
Bhardwaj:2022anh ; Raclariu:2021zjz ; Pasterski:2021rjz ; McLoughlin:2022ljp .

Now for gluons, the Mellin transform of tree level three and four point
scattering amplitudes are distributional in nature due to the momentum
conserving delta function and the conformal invariance of the tree level Yang-
Mills theory. The momentum conservation constraint can be removed by supplying
background momentum to the YM theory which breaks translational invariance
explicitly Fan:2022vbz . In Fan:2022vbz , this was achieved by coupling the YM
theory to background massless dilaton field. But this does not break the
conformal invariance of the YM theory. Translation invariance breaking
solution was also considered in Costello:2022wso .

A somewhat different set up was considered in Casali:2022fro . They chirally
coupled the Yang-Mills theory to a massive dilaton background. As a result the
space-time translation as well as the scale invariance of the tree-level Yang-
Mills theory was explicitly broken but the (Lorentz) 2D conformal invariance
was preserved. The interesting fact about this coupled theory is that, when
written in the celestial basis, the 3-point amplitudes take the usual 2D CFT
form. It was shown Casali:2022fro  that the leading soft gluon theorem and the
leading OPE structure remain unchanged for this chirally coupled theory. In
this paper we show that the subleading soft gluon theorem and subleading OPE
structrures also remain unchanged, giving rise to the same null state relation
as obtained in Banerjee:2020vnt . We expect this to be true as the leading and
subleading soft gluon theorems don’t require space-time translation or scale
invariance.

This paper is organized as follows. In section 3, following Casali:2022fro  we
briefly review the Yang-Mills amplitude chirally coupled to a massive dilaton
background and Mellin transform it to get the celestial amplitude. OPE
factorisations of the 4-point celestial amplitude is given in section 4. In
section 5 we show that the subleading soft gluon theorem remains the same for
this coupled theory. By demanding the consistency of the OPE at order 1 with
the soft theorem we get the same null state relation under the soft current
algebra as obtained in Banerjee:2020vnt . We also show that the 3point
amplitude satisfies the BG equation in section B. Finally we end with
discussion and future directions in section 6.

##  2 Notations and Conventions

We will work in (-,+,-,+) signature in four spacetime dimensions. This is also
called Klein space. The geometry of this space has been discussed in great
detail in Atanasov:2021oyu . The scattering amplitudes, written in the boost
eigenstates, behave as a correlation functions in a Lorentzian CFT on the
celestial torus Atanasov:2021oyu ; Atanasov:2021cje . We review some of the
elementary equations of the celestial amplitude for this paper to be self-
contained.

In Klein space the null momentum ($p_{i}$) of $i$-th hard massless particle is
parametrized as:

$p_{i}=\epsilon_{i}\omega_{i}q({z_{i},\bar{z}_{i}})$ (1)

where $\omega_{i}$ is a real positive number, $\epsilon_{i}=+1(-1)$
corresponds to an outgoing (incoming) particle and

$q({z_{i},\bar{z}_{i}})=\\{1+z_{i}\bar{z}_{i},z_{i}+\bar{z}_{i},z_{i}-\bar{z}_{i},1-z_{i}\bar{z}_{i}\\}$
(2)

$z_{i},\bar{z}_{i}$ are two real independent variables. The map from a
creation/annihilation operator of a massless particle in the bulk to an
operator on a celestial torus is given by the Mellin transformation

$\begin{gathered}\phi^{a,\epsilon}_{h,\bar{h}}(z,\bar{z})=\int_{0}^{\infty}d\omega\,\omega^{\Delta-1}A^{a}(\epsilon\omega,\sigma,z,\bar{z})\end{gathered}$
(3)

where $A^{a}(\epsilon\omega,\sigma,z,\bar{z})$ is an annihilaton
($\epsilon=+1$)/creation ($\epsilon=-1$) operator in the adjoint
representation of an $SU(N)$ gauge theory, $\sigma$ is the helicity of the
corresponding massless particle and

$h=\frac{\Delta+\sigma}{2},\,\bar{h}=\frac{\Delta-\sigma}{2}$ (4)

The momentum space amplitudes for $n$ number of external massless states
written in the celestial conformal primary basis (3) then takes the form of a
2D conformal correlator

$\displaystyle\mathcal{M}_{n}(\\{a_{i},\epsilon_{i},z_{i},\bar{z}_{i},h_{i},\bar{h}_{i}\\})$
$\displaystyle=$
$\displaystyle\left<\prod_{j=1}^{n}\phi^{a_{j},\epsilon_{j}}_{h_{j},\bar{h}_{j}}(z_{j},\bar{z}_{j})\right>$
(5) $\displaystyle=$ $\displaystyle\left(\prod_{j=1}^{n}\int
d\omega_{j}\,\omega_{j}^{\Delta_{j}-1}\right)\mathcal{A}_{n}(a_{i},\epsilon_{i}\omega_{i},z_{i},\bar{z}_{i},\sigma_{i})$

For an on-shell massive scalar particle, the conformal primary wavefunction is
obtained through the bulk-to-boundary propagator Pasterski:2016qvg ;
Pasterski:2017kqt . For concreteness we take the outgoing particle with unit
mass. In $(2,2)$ signature Klein space, the momentum $Q^{\mu}$ of a massive
scalar of unit mass satisfying the onshell condition $Q^{2}=-1$ can be
parametrized using the coordinates of $\textnormal{AdS}_{3}/\mathbb{Z}$ as
444The metric on the $\textnormal{AdS}_{3}/\mathbb{Z}$ is given by
$ds^{2}_{H_{3}}=\frac{dy^{2}+dzd\bar{z}}{y^{2}}$ (6)

$Q^{\mu}=\frac{1}{2y}\\{1+y^{2}+z\bar{z},z+\bar{z},(z-\bar{z}),1-y^{2}-z\bar{z}\\}$
(7)

The mapping of a massive scalar particle to a conformal primary wavefunction
is given through the bulk-to-boundary propagator

$\begin{gathered}\Phi^{\epsilon}_{\Delta}(X,z,\bar{z})=\int_{0}^{\infty}\frac{dy}{y^{3}}\int
dzd\bar{z}\,G_{\Delta}(y,z,\bar{z};w,\bar{w})\,e^{i\epsilon Q^{\mu}\cdot
X_{\mu}}\end{gathered}$ (8)

where the bulk-to-boundary propagator is given by

$G_{\Delta}(y,z,\bar{z};w,\bar{w})=\left(\frac{y}{y^{2}+(z+\bar{w})(z-\bar{w})}\right)^{\Delta}$
(9)

##  3 Celestial MHV Amplitudes in the Massive Scalar Background

An $n$-point MHV gluon amplitude in a massive complex scalar background was
computed in Casali:2022fro . The theory considered there was a Yang-Mills
theory chirally coupled to a massive complex scalar. The massive scalar was
coupled to the anti-self dual curvature tensor. An $(n+1)$-point scalar-gluon
(one scalar and $n$ gluons) single trace, color-ordered tree-level amplitude
in this set up is given by

$\begin{split}\mathcal{A}_{n+1}(\phi,1^{\epsilon_{1}}_{-},2^{\epsilon_{2}}_{-},3^{\epsilon_{3}}_{+},\cdots,n^{\epsilon_{n}}_{+})=\frac{\left<12\right>^{4}}{\left<12\right>\left<23\right>\cdots\left<n1\right>}\delta^{4}\left(\sum_{i=1}^{n}p_{i}+Q\right)\end{split}$
(10)

The above amplitude is exactly the same as $n$-point MHV amplitude except that
the scalar momentum now appears in the momentum conserving delta function. The
scalar field can be treated as a background and the amplitude (10) is coupled
to this background by integrating over the scalar phase space Casali:2022fro

$\mathcal{A}_{n}^{\phi}(1^{\epsilon_{1}}_{-},2^{\epsilon_{2}}_{-},3^{\epsilon_{3}}_{+},\cdots,n^{\epsilon_{n}}_{+})=\int\widetilde{d^{3}Q}\,g(Q)\mathcal{A}_{n+1}(\phi,1^{\epsilon_{1}}_{-},2^{\epsilon_{2}}_{-},3^{\epsilon_{3}}_{+},\cdots,n^{\epsilon_{n}}_{+})$
(11)

where $g(Q)$ is the Fourier coefficient of the scalar field $\phi(X)$ and
$\widetilde{d^{3}Q}$ is the invariant measure. This amplitude is called the
$n$-point MHV amplitude in the massive scalar background and hence is denoted
by
$\mathcal{A}_{n}^{\phi}(1^{a_{1},\epsilon_{1}}_{-},2^{a_{2},\epsilon_{2}}_{-},3^{a_{3},\epsilon_{3}}_{+},\cdots,n^{a_{n},\epsilon_{n}}_{+})$.
We are interested in the OPE factorization of this amplitude on the celestial
torus. Hence we Mellin transform the amplitude (11) and get $n$-point
correlators on the celestial torus. To simplify calculations we will work with
the 5-point scalar gluon amplitude, i.e., the 4-point MHV amplitude in the
massive scalar background.

##  4 OPE Factorisation from the 4-point Celestial Amplitude

In this section we factorize the $4$-point amplitude into $3$-point amplitude
and determine the leading and subleading terms in the OPE between two positive
helicity outgoing ($\epsilon_{3}=\epsilon_{4}=+1$) gluons. We show that the
OPE remains the same as the MHV case Banerjee:2020vnt  . Let us start with the
full 5-point scalar-gluon amplitude, given by

$\displaystyle\mathcal{A}_{5}(\phi,1^{a_{1},\epsilon_{1}}_{-},2^{a_{2},\epsilon_{2}}_{-},3^{a_{3},\epsilon_{3}}_{+},4^{a_{4},\epsilon_{4}}_{+})=\left\\{\mathcal{A}_{4}[1^{\epsilon_{1}}_{-}2^{\epsilon_{2}}_{-}3^{\epsilon_{3}}_{+}4^{\epsilon_{4}}_{+}]\
tr(T^{a_{1}}T^{a_{2}}T^{a_{3}}T^{a_{4}})+\textnormal{perm (234)}\right\\}$
$\displaystyle\times\delta^{(4)}\left(\sum_{i=1}^{4}p_{i}+Q\right)\hskip
28.45274pt$ (12)

where
$\mathcal{A}_{4}[i^{\epsilon_{i}}_{\sigma_{i}}j^{\epsilon_{j}}_{\sigma_{j}}k^{\epsilon_{k}}_{\sigma_{k}}l^{\epsilon_{l}}_{\sigma_{l}}]$
are color ordered partial MHV amplitudes given by

$\mathcal{A}_{4}[i^{\epsilon_{i}}_{-}j^{\epsilon_{j}}_{+}k^{\epsilon_{k}}_{-}l^{\epsilon_{l}}_{+}]=\frac{\left<ik\right>^{4}}{\left<ij\right>\left<jk\right>\left<kl\right>\left<li\right>}$
(13)

After substituting the explicit form of the color ordered amplitude (13) in
(12) and using $[T^{a},T^{b}]=if^{abc}T^{c},\ tr(T^{a}T^{b})=\delta^{ab}$ we
get

$\displaystyle\mathcal{A}_{5}(\phi,1^{a_{1},\epsilon_{1}}_{-},2^{a_{2},\epsilon_{2}}_{-},3^{a_{3},\epsilon_{3}}_{+},4^{a_{4},\epsilon_{4}}_{+})=\frac{\left<12\right>^{3}}{\left<23\right>\left<34\right>\left<41\right>}\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-\frac{z_{14}z_{23}}{z_{13}z_{24}}f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\right)\delta^{(4)}\left(\sum_{i=1}^{4}p_{i}+Q\right)$
(14)

Substituting this expression in (11) for the $n=4$ and Mellin integrating over
the energies we finally get the 4-point celestial MHV amplitude in the massive
scalar background, given by Casali:2022fro

$\displaystyle\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)=\frac{\mathcal{N}_{4}}{(2\pi)^{4}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-\frac{z_{12}z_{34}}{z_{13}z_{24}}f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\right)\Gamma(\Delta_{1}+1)$
$\displaystyle\times\,\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\Gamma(\Delta_{4}-1)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}$
$\displaystyle\times\,(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{4}+1}\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{4}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{4}}-1\right)$ (15)

where
$\beta_{4}=\sum_{j=1}^{4}\left({\Delta_{j}}-1\right),\,\mathcal{N}_{4}=\prod_{j=1}^{4}(-i\epsilon_{j})^{\Delta_{j}-\sigma_{j}}$.
To clarify the notations let us note that a bulk point $X^{\mu}$ is
parameterized as $\tau\hat{x}^{\mu}$ with $\hat{x}^{2}=-1$.
$\widetilde{d^{3}\hat{x}}$ is a measure on the $\hat{x}^{2}=-1$ slice 555For
more details please see Casali:2022fro .. One can also compute the 3-point
function in the same way and it is given by

$\begin{split}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)=\frac{\widetilde{\mathcal{N}}_{3}}{(2\pi)^{4}}\frac{2z_{12}^{3}}{z_{24}z_{41}}\left(if^{a_{1}a_{2}a_{4}}\right)\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{4}-1)\\\
\times\
\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{4}+1}\\\
\times\
\int_{0}^{i\infty}d\tau\,\tau^{-1-\tilde{\beta}_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\tilde{\beta}_{3}}-1\right)\end{split}$ (16)

where

$\tilde{\beta}_{3}=\sum_{j=1,j\neq
3}^{4}\left({\Delta_{j}}-1\right),\,\widetilde{\mathcal{N}}_{3}=\prod_{j=1,j\neq
3}^{4}(-i\epsilon_{j})^{\Delta_{j}-\sigma_{j}}$ (17)

.

We now take the OPE limit $z_{3}\to z_{4},\,\bar{z}_{3}\to\bar{z}_{4}$ in
(15). To do that let us first note that in the OPE limit we have

$(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}=(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{3}+1}\left[1+\frac{\Delta_{3}-1}{y^{2}+|z-z_{4}|^{2}}((\bar{z}-\bar{z}_{4})z_{34}+(z-z_{4})\bar{z}_{34}+z_{34}\bar{z}_{34})\right]+\cdots$
(18)

This will be useful in the next two subsections to extract the OPE from (15).

###  4.1 Leading Order

The leading order OPE between two positive helicity gluons was computed in
Casali:2022fro  and it was shown that the leading term doesn’t get any
correction in the massive scalar background. Here for the sake of completeness
we reproduce their results and then we move to the subleading terms in the
next subsection. The leading order term of (15) in the OPE expansion is

$\displaystyle\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)=\frac{\mathcal{N}_{4}}{(2\pi)^{4}}\frac{z_{12}^{3}}{z_{24}z_{41}}\frac{1}{z_{34}}f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\Gamma(\Delta_{4}-1)$
$\displaystyle\times\,\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{3}-\Delta_{4}+2}$
$\displaystyle\times\,\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{4}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{4}}-1\right)\hskip 28.45274pt$ (19)

Replacing $\Delta_{4}\to\Delta_{3}+\Delta_{4}-1$ in the 3-point amplitude
(16), we get

$\begin{split}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},4^{a_{4},\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1,+}\right)=\frac{\mathcal{N}_{4}}{(2\pi)^{4}}\frac{2z_{12}^{3}}{z_{24}z_{41}}\left(if^{a_{1}a_{2}a_{4}}\right)\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\\\
\times\
\Gamma(\Delta_{3}+\Delta_{4}-2)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{3}+\Delta_{4}+2}\\\
\times\
\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{4}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{4}}-1\right)\end{split}$ (20)

Hence at leading order we can write

$\begin{gathered}\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)=-\frac{1}{2z_{34}}B(\Delta_{3}-1,\Delta_{4}-1)if^{xa_{3}a_{4}}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1}},2^{a_{2},\epsilon_{2}}_{\Delta_{2}},4^{x,\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1}\right)\\\
\Rightarrow\left<\mathcal{O}^{a_{1},\epsilon_{1}}_{\Delta_{1},-}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2},\epsilon_{2}}_{\Delta_{2},-}(z_{2},\bar{z}_{2})\mathcal{O}^{a_{3},\epsilon_{3}}_{\Delta_{3},+}(z_{3},\bar{z}_{3})\mathcal{O}^{a_{4},\epsilon_{4}}_{\Delta_{4},+}(z_{4},\bar{z}_{4})\right>\\\
=-\frac{1}{2z_{34}}B(\Delta_{3}-1,\Delta_{4}-1)if^{xa_{3}a_{4}}\left<\mathcal{O}^{a_{1},\epsilon_{1}}_{\Delta_{1},-}(z_{1},\bar{z}_{1})\mathcal{O}^{a_{2},\epsilon_{2}}_{\Delta_{2},-}(z_{2},\bar{z}_{2})\mathcal{O}^{a_{4},\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1,+}(z_{4},\bar{z}_{4})\right>\\\
\end{gathered}$

At the level of OPE the above equation reads

$\mathcal{O}^{a_{3},+1}_{\Delta_{3},+}(z_{3},\bar{z}_{3})\mathcal{O}^{a_{4},+1}_{\Delta_{4},+}(z_{4},\bar{z}_{4})\sim-\frac{1}{2z_{34}}B(\Delta_{3}-1,\Delta_{4}-1)if^{a_{3}a_{4}x}\mathcal{O}^{x,+1}_{\Delta_{3}+\Delta_{4}-1,+}(z_{4},\bar{z}_{4})$
(21)

###  4.2 Subleading Terms: $\mathcal{O}(1)$

In this section we show that the subleading ($\mathcal{O}(1)$) term in the OPE
expansion remains same as the MHV case. As we know from the study of MHV gluon
amplitudes Banerjee:2020vnt , the descendants of the leading soft gluon
symmetry algebra appears at $\mathcal{O}(1)$ in the OPE expansion. Here we
only write down the action of the relevant operators on the celestial gluon
amplitude. The leading conformally soft gluon operator Donnay:2018neh ;
Pate:2019mfs ; Fan:2019emx ; Nandan:2019jas ; Adamo:2019ipt ; Guevara:2021abz
; Strominger:2021lvk ; Strominger:2021mtt  for positive helicity is defined as

$R^{1,a}_{0}(z)=\lim_{\Delta\to
1}(\Delta-1)\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ (22)

where $\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ is a positive helicity primary
gluon operator with scaling dimension $\Delta$. The soft current
$R^{1,a}_{0}(z)$ is a Kac-Moody current Strominger:2014ass ; He:2014cra ;
Kapec:2015ena ; Kapec:2014zla ; Campiglia:2015qka ; Nande:2017dba ; He:2020ifr
. The modes of the current $R^{1,a}_{0}(z)$ are denoted by $R^{1,a}_{p,0}$.
For our purpose we only mention the correlation function of the descendants
$R^{1,a}_{-p,0}\,\mathcal{O}^{b}_{\Delta,\sigma}(z,\bar{z}),\ p\geq 1$ with a
collection of gluon primaries. These are given by666Here we are using the
notation of Strominger:2021lvk . Banerjee:2020vnt

$\begin{split}\left<R^{1,a}_{-p,0}\,\mathcal{O}^{b}_{\Delta,\sigma}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})\right>=\mathcal{R}^{1,a}_{-p,0}(z)\left<\mathcal{O}^{b}_{\Delta,\sigma}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})\right>\end{split}$
(23)

where the operator $\mathcal{R}^{1,a}_{-p,0}(z)$ is defined as

$\begin{split}\mathcal{R}^{1,a}_{-p,0}(z)\left<\mathcal{O}^{b}_{\Delta,\sigma}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})\right>=\sum_{k=1}^{n}\frac{T^{a}_{k}}{(z_{k}-z)^{p}}\left<\mathcal{O}^{b}_{\Delta,\sigma}(z,\bar{z})\prod_{i=1}^{n}\mathcal{O}^{a_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})\right>\end{split}$
(24)

Let us now consider the $\mathcal{O}(1)$ term in the OPE expansion of (15),
given by

$\displaystyle\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)\big{|}_{\mathcal{O}(1)}=\frac{\mathcal{N}_{4}}{(2\pi)^{4}}\frac{z_{12}^{3}}{z_{24}z_{41}}\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\Gamma(\Delta_{4}-1)$
$\displaystyle\times\int\widetilde{d^{3}\hat{x}}\left[\frac{f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}}{z_{14}}+\frac{f^{a_{2}a_{3}x}f^{xa_{4}a_{1}}}{z_{24}}+(\Delta_{3}-1)\frac{(\bar{z}-\bar{z}_{4})}{y^{2}+|z-z_{4}|^{2}}f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}\right](-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}$
$\displaystyle\times\,(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}\,(-q(z_{4},\bar{z}_{4})\cdot\hat{x})^{-\Delta_{3}-\Delta_{4}+2}\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{4}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{4}}-1\right)\hskip 28.45274pt$

With the help of the operator (24), RHS of the above equation can be written
as

$\begin{split}\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)\big{|}_{\mathcal{O}(1)}=\frac{1}{2}B(\Delta_{3}-1,\Delta_{4}-1)\left[-\frac{(\Delta_{3}-1)}{(\Delta_{3}+\Delta_{4}-2)}if^{xa_{3}a_{4}}\mathcal{L}_{-1}(4)\right.\\\
\left.\times\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1}},2^{a_{2},\epsilon_{2}}_{\Delta_{2}},4^{x,\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1}\right)+\frac{(\Delta_{4}-1)}{(\Delta_{3}+\Delta_{4}-2)}\mathcal{R}^{1,a_{3}}_{-1,0}(4)\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1}},2^{a_{2},\epsilon_{2}}_{\Delta_{2}},4^{a_{4},\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1}\right)\right.\\\
\left.+\frac{(\Delta_{3}-1)}{(\Delta_{3}+\Delta_{4}-2)}\mathcal{R}^{1,a_{4}}_{-1,0}(4)\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1}},2^{a_{2},\epsilon_{2}}_{\Delta_{2}},4^{a_{3},\epsilon_{4}}_{\Delta_{3}+\Delta_{4}-1}\right)\right]\end{split}$
(25)

where the argument $(4)$ in the operators
$\mathcal{L}_{-1},\mathcal{R}^{1,a}_{-1,0}$ implies that these modes are
acting on the last particle of the 3-point amplitude. At the level of the OPE
we have 777We would like to emphasize that (26) does not hold beyond MHV
sector. In the $N^{k}MHV$ sector the soft symmetry algebra changes because of
the existence of the negative helicity soft gluons and as a result the
$O(z^{0}\bar{z}^{0})$ term in the OPE has to change. This is also the case for
pure YM theory and has nothing to do with the existence of the massive scalar
background.

$\begin{split}\mathcal{O}^{a_{3},+1}_{\Delta_{3},+}(z_{3},\bar{z}_{3})\mathcal{O}^{a_{4},+1}_{\Delta_{4},+}(z_{4},\bar{z}_{4})\big{|}_{\mathcal{O}(1)}\sim\frac{1}{2}B(\Delta_{3}-1,\Delta_{4}-1)\left[-\frac{(\Delta_{3}-1)}{(\Delta_{3}+\Delta_{4}-2)}if^{xa_{3}a_{4}}L_{-1}\right.\\\
\left.+\left(\frac{(\Delta_{4}-1)}{(\Delta_{3}+\Delta_{4}-2)}\delta^{a_{3}y}\delta^{a_{4}x}+\frac{(\Delta_{3}-1)}{(\Delta_{3}+\Delta_{4}-2)}\delta^{a_{4}y}\delta^{a_{3}x}\right)R^{1,y}_{-1,0}\right]\mathcal{O}^{x,+1}_{\Delta_{3}+\Delta_{4}-1,+}(z_{4},\bar{z}_{4})\end{split}$
(26)

One can recognize that this is the $\mathcal{O}(1)$ term in the OPE obtained
by Banerjee:2020vnt ; Ebert:2020nqf  in the MHV case. Thus we can see that
$\mathcal{O}(1)$ OPE also doesn’t change in the presence of the massive scalar
background.

##  5 Subleading Soft Gluon Theorem in Massive Scalar Background

In this section we show that the subleading conformal soft gluon theorem
remains same in a massive scalar background. More precisely, we show that the
subleading conformal soft limit, $\Delta_{4}\rightarrow 0$, of (15) is
equivalent to the action of the subleading soft operator Donnay:2018neh ;
Pate:2019mfs ; Fan:2019emx ; Nandan:2019jas ; Adamo:2019ipt  of the $4$-th
particle, on the $3$-pt correlation function (16).

Hence we start with the 4-point amplitude (15) and take the conformal soft
limit $\Delta_{4}\rightarrow 0$ to get

$\displaystyle\lim_{\Delta_{4}\rightarrow 0}\
\Delta_{4}\widetilde{\mathcal{M}}_{4}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+}\right)=-i\frac{\mathcal{N}_{3}}{(2\pi)^{4}}\frac{z_{12}^{3}}{z_{23}z_{34}z_{41}}\left(f^{a_{1}a_{2}x}f^{xa_{3}a_{4}}-\frac{z_{12}z_{34}}{z_{13}z_{24}}f^{a_{1}a_{3}x}f^{xa_{2}a_{4}}\right)$
$\displaystyle\times\,\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}$
$\displaystyle\times\,(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})\int_{0}^{i\infty}d\tau\,\tau^{-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)$ (27)

The above result can be written as

$\begin{split}\lim_{\Delta_{4}\rightarrow 0}\
\Delta_{4}\widetilde{\mathcal{M}}^{\Phi}_{4}(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+})=\bigg{(}\frac{c_{1}}{z_{14}}+\frac{c_{2}}{z_{24}}+\frac{c_{3}}{z_{34}}\bigg{)}\
i\frac{{\mathcal{N}_{3}}}{(2\pi)^{4}}\ \frac{z_{12}^{3}}{z_{23}z_{31}}\\\
\times\,\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}\\\
\times\,(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})\int_{0}^{i\infty}d\tau\,\tau^{-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)\end{split}$ (28)

where $c_{1}=f^{xa_{2}a_{3}}f^{a_{4}a_{1}x}$,
$c_{2}=f^{a_{1}xa_{3}}f^{a_{4}a_{2}x}$ and
$c_{3}=f^{a_{1}a_{2}x}f^{a_{4}a_{3}x}$.

We shall now show that the above expression for the conformal soft limit of
the celestial correlation function is the same as the action of the subleading
soft operator on the $3$-pt correlation function. The subleading soft gluon
theorem in Mellin space is given by Pate:2019lpp ,

$\begin{split}\left<R^{0,a_{4}}(z_{4},\bar{z}_{4})\prod_{i=1}^{3}\mathcal{O}_{\Delta_{i},\sigma_{i}}^{a_{i},\epsilon_{i}}(z_{i},\bar{z}_{i})\right>=-\sum_{k=1}^{3}\
\frac{\epsilon_{k}}{z_{4k}}(-2\bar{h}_{k}+1+\bar{z}_{4k}\bar{\partial}_{k})\
T^{a_{4}}_{k}\mathcal{P}^{-1}_{k}\left<\prod_{i=1}^{3}\mathcal{O}_{\Delta_{i},\sigma_{i}}^{a_{i},\epsilon_{i}}(z_{i},\bar{z}_{i})\right>\end{split}$
(29)

where $R^{0,a}(z,\bar{z})$ is the subleading conformally soft gluon operator
defined by

$R^{0,a}(z,\bar{z})=\lim_{\Delta\to
0}\Delta\mathcal{O}^{a}_{\Delta,+}(z,\bar{z})$ (30)

$T^{a}_{k}$ is the lie algebra generator in the adjoint representation of the
gauge group and $\mathcal{P}^{-1}_{k}$ is a dimension lowering operator acting
on the $k$-th primary field. The action of both is

$T^{a}_{k}\mathcal{O}^{a_{i},\epsilon_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})=if^{aa_{i}x}\,\mathcal{O}^{x,\epsilon_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})\,\delta_{k,i},\
\mathcal{P}^{-1}_{k}\mathcal{O}^{a_{i},\epsilon_{i}}_{\Delta_{i},\sigma_{i}}(z_{i},\bar{z}_{i})=\mathcal{O}^{a_{i},\epsilon_{i}}_{\Delta_{i}-1,\sigma_{i}}(z_{i},\bar{z}_{i})\delta_{ki}$
(31)

By repeated use of the following equation,

$(-q_{i}\cdot\hat{x})=\frac{y^{2}+(z-z_{i})(\bar{z}-\bar{z}_{i})}{y}$ (32)

and the explicit expression for 3-point function given in (16) on the RHS of
(29), one can show that (appendix A)

$\begin{split}-\sum_{k=1}^{3}\
\frac{\epsilon_{k}}{z_{4k}}(-2\bar{h}_{k}+1+\bar{z}_{4k}\bar{\partial}_{k})\
T^{a_{4}}_{k}\mathcal{P}^{-1}_{k}\left<\prod_{i=1}^{3}\mathcal{O}_{\Delta_{i},\sigma_{i}}^{a_{i},\epsilon_{i}}(z_{i},\bar{z}_{i})\right>=\bigg{(}\frac{c_{1}}{z_{14}}+\frac{c_{2}}{z_{24}}+\frac{c_{3}}{z_{34}}\bigg{)}\
i\frac{{\mathcal{N}_{3}}}{(2\pi)^{4}}\ \frac{2z_{12}^{3}}{z_{23}z_{31}}\\\
\times\,\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}\\\
\times\,(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})\int_{0}^{i\infty}d\tau\,\tau^{-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)\end{split}$ (33)

Comparing (28), (29) and (33) we get

$\begin{split}\lim_{\Delta_{4}\rightarrow 0}\
\Delta_{4}\widetilde{\mathcal{M}}^{\Phi}_{4}(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+},4^{a_{4},\epsilon_{4}}_{\Delta_{4},+})=\left<R^{0,a_{4}}(z_{4},\bar{z}_{4})\prod_{i=1}^{3}\mathcal{O}_{\Delta_{i},\sigma_{i}}^{a_{i},\epsilon_{i}}(z_{i},\bar{z}_{i})\right>\\\
=-\frac{1}{2}\sum_{k=1}^{3}\
\frac{\epsilon_{k}}{z_{4k}}(-2\bar{h}_{k}+1+\bar{z}_{4k}\bar{\partial}_{k})\
T^{a_{4}}_{k}\mathcal{P}^{-1}_{k}\left<\prod_{i=1}^{3}\mathcal{O}_{\Delta_{i},\sigma_{i}}^{a_{i},\epsilon_{i}}(z_{i},\bar{z}_{i})\right>\end{split}$
(34)

Thus we see that subleading soft gluon theorem for a positive helicity soft
gluon does not change if we couple the Yang-Mills with the massive complex
scalar in a chiral way mentioned in section 3.

###  5.1 BG Equations in Massive Scalar Background

A set of differential equations for the gluon MHV amplitudes were obtained in
Banerjee:2020vnt  by demanding the consistency between the subleading soft
gluon theorem and OPE factorisation at $\mathcal{O}(1)$ 888Although these
equations look very similar to the KZ equation which appears in WZW models,
they are qualitatively different. In particular, they cannot be derived by any
Sugawara construction on the celestial sphere. One way to see this is the
following. Sugawara construction leads to the quantization of dimensions of
primary operators which we know is not the case in celestial holography.. Here
we have shown that, even in the case of MHV amplitudes in a massive scalar
background the subleading soft gluon theorem and the OPE factorisation at
$\mathcal{O}(1)$ do not change. Hence we conclude that the celestial MHV
amplitudes coupled to a massive scalar background also satisfy BG equations.
This is not surprising because the existence of the null state, which gives
rise to the differential equations, is guaranteed by the leading and the
subleading soft gluon theorems and $SL(2,C)$ invariance. The presence of a
massive dilaton background breaks the scaling as well as translational
invariance but the soft gluon theorems remain unchanged. So the BG equations
should also not change.

##  6 Discussions

In this paper we extracted the OPE between two positive helicity outgoing
gluons from the Mellin amplitude of the Yang-Mills theory chirally coupled to
massive scalar background. The leading order term was already computed in
Casali:2022fro . Here we have computed a subleading term. One of our
motivation behind this work was to check if the scattering amplitudes in this
theory is also a solution of the BG equations. We have shown in this paper
that this is indeed the case999This seems to raise a puzzle because we are
saying that the MHV amplitudes in pure YM and the YM coupled to massive
dilaton both satisfy the BG equation. The resolution of this is the following:
The MHV amplitudes in pure YM are not only the solutions of BG equations but
they also have to satisfy the Ward identities coming from the space time
translation invariance and the scale invariance of the YM theory. But when
coupled to the massive dilaton background both these Ward identities are no
longer valid and so we get a different kind of MHV amplitude by solving the BG
equation.. Though the scaling and translation symmetry were explicitly broken
for the theory we considered, the leading and subleading soft theorems remain
unchanged. The OPE factorization at $\mathcal{O}(1)$ is completely determined
in terms of the descendants of the $SL(2,C)$ and the leading soft symmetry
algebra in the same way as the MHV amplitudes. On the other hand we have also
shown that the subleading soft gluon theorem does not change in the massive
background. Thus by comparing the $\mathcal{O}(1)$ OPE with the subleading
soft gluon theorem we get the same BG equations as the MHV amplitudes. More
generally, we can say that the scattering amplitudes of all the theories which
respect the symmetries coming from the leading and subleading soft gluon
theorems should satisfy the BG equations.

However, if one considers the graviton scattering amplitude and breaks some of
the symmetries considered here, the situation will change completely. This is
because of the fact that the leading soft graviton theorem is a consequence of
supertranslation invariance. So if in a gravitational theory the translation
symmetry is broken, the leading soft graviton theorem would no longer holds.
Then one can ask the question that what happens to the decoupling equations
obtained for the MHV graviton scattering amplitudes in Banerjee:2020zlg .

It would also be interesting to check how the analysis of our work would
change in the context of deformed soft algebras for gauge theories recently
considered in Melton:2022fsf . We hope to answer some of these questions in
the near future.

##  7 Acknowledgement

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 and by SERB grant MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-
Space Holography). PP would like to thank the organizers and the participants
of the CHEP in-house symposium, 2022 held in IISc, Bangalore where part of
this work was presented. The work of PP is supported by an IOE endowed
postdoctoral position at IISc, Bangalore, India.

##  Appendix A Subleading Soft Gluon Theorem

In this appendix we derive (33). Let us start with the $3$-point correlation
function given by (16),

$\begin{split}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+}\right)=\frac{{\mathcal{N}}_{3}}{(2\pi)^{4}}\frac{2z_{12}^{3}}{z_{23}z_{31}}\left(if^{a_{1}a_{2}a_{3}}\right)\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\\\
\times\
\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}\\\
\times\
\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)\end{split}$ (35)

where

$\beta_{3}=\sum_{j=1}^{3}\left({\Delta_{j}}-1\right),\,{\mathcal{N}}_{3}=\prod_{j=1}^{3}(-i\epsilon_{j})^{\Delta_{j}-\sigma_{j}}$
(36)

.

So, the action of the subleading soft operator on the $3$-point correlation
function is

$\begin{split}&-\sum_{k=1}^{3}\
\frac{\epsilon_{k}}{z_{4k}}(-2\bar{h}_{k}+1+\bar{z}_{4k}\bar{\partial}_{k})\
T^{a_{4}}_{k}\mathcal{P}^{-1}_{k}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+}\right)\\\
&=\epsilon_{1}\frac{c_{1}}{z_{41}}(-\Delta_{1}+\bar{z}_{41}\bar{\partial}_{1})\
\widetilde{\mathcal{M}}_{3}^{\Phi}(1_{\Delta_{1}-1,-},2_{\Delta_{2},-},3_{\Delta_{3},+})+\epsilon_{2}\frac{c_{2}}{z_{42}}(-\Delta_{2}+\bar{z}_{42}\bar{\partial}_{2})\
\widetilde{\mathcal{M}}_{3}^{\Phi}(1_{\Delta_{1},-},2_{\Delta_{2}-1,-},3_{\Delta_{3},+})\\\
&\hskip
85.35826pt+\epsilon_{3}\frac{c_{3}}{z_{43}}(-\Delta_{3}+2+\bar{z}_{43}\bar{\partial}_{3})\
\widetilde{\mathcal{M}}_{3}^{\Phi}(1_{\Delta_{1},-},2_{\Delta_{2},-},3_{\Delta_{3}-1,+})\end{split}$
(37)

We have pulled out the colour factors ($if$) out of the amplitude and
$\widetilde{\mathcal{M}}^{\Phi}_{3}(1_{\Delta_{1}-1,-},2_{\Delta_{2},-},3_{\Delta_{3},+})$
etc. are the colour stripped amplitude in eq.(LABEL:eq:313). The right hand
side of eq.(LABEL:eq:313) can be written as

$\begin{split}\bigg{(}\frac{c_{1}}{z_{14}}+\frac{c_{2}}{z_{24}}+\frac{c_{3}}{z_{34}}+(c_{1}+c_{2}+c_{3})\partial_{4}\bigg{)}\
i\frac{{\mathcal{N}}_{3}}{(2\pi)^{4}}\frac{2z_{12}^{3}}{z_{23}z_{31}}\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\\\
\times\
\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}(-q(z_{4},\bar{z}_{4})\cdot\hat{x})\\\
\times\
\int_{0}^{i\infty}d\tau\,\tau^{-1-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)\end{split}$ (38)

where we have used the following equation

$\bar{z}_{4k}\bar{\partial}_{k}(-q_{k}\cdot\hat{x})=-(-q_{k}\cdot\hat{x})+(1+z_{k4}\partial_{4})(-q_{4}\cdot\hat{x})$
(39)

Now using the Jacobi identity

$c_{1}+c_{2}+c_{3}=0$ (40)

we finally get

$\begin{split}-\sum_{k=1}^{3}\
\frac{\epsilon_{k}}{z_{4k}}(-2\bar{h}_{k}+1+\bar{z}_{4k}\bar{\partial}_{k})\
T^{a_{4}}_{k}\mathcal{P}^{-1}_{k}\widetilde{\mathcal{M}}_{3}^{\Phi}\left(1^{a_{1},\epsilon_{1}}_{\Delta_{1},-},2^{a_{2},\epsilon_{2}}_{\Delta_{2},-},3^{a_{3},\epsilon_{3}}_{\Delta_{3},+}\right)=\bigg{(}\frac{c_{1}}{z_{14}}+\frac{c_{2}}{z_{24}}+\frac{c_{3}}{z_{34}}\bigg{)}\
i\frac{{\mathcal{N}}_{3}}{(2\pi)^{4}}\frac{2z_{12}^{3}}{z_{23}z_{31}}\\\
\times\Gamma(\Delta_{1}+1)\Gamma(\Delta_{2}+1)\Gamma(\Delta_{3}-1)\int\widetilde{d^{3}\hat{x}}(-q(z_{1},\bar{z}_{1})\cdot\hat{x})^{-\Delta_{1}-1}(-q(z_{2},\bar{z}_{2})\cdot\hat{x})^{-\Delta_{2}-1}\
(-q(z_{3},\bar{z}_{3})\cdot\hat{x})^{-\Delta_{3}+1}\\\
\times(-q(z_{4},\bar{z}_{4})\cdot\hat{x})\
\int_{0}^{i\infty}d\tau\,\tau^{-\beta_{3}}\phi_{B}(\tau)\left(e^{2\pi
i\beta_{3}}-1\right)\end{split}$ (41)

which is same as (33).

##  Appendix B Solution of the BG Equations

In subsection 5.1, we have argued that the BG equations remain same if we put
the MHV amplitudes in a massive scalar background. Here we show explicitly
that the three point MHV amplitude in the massive background satisfies the BG
equations. We first write down the most general form of the 3-point amplitude
using the $SL(2,C)$ symmetry and then derive the constraints for the 3-point
coefficient imposed by the BG equations.

Let us start with the color ordered $SL(2,C)$-covariant 3-point amplitude
given by,

$\displaystyle\widetilde{\mathcal{M}}_{3}(1^{-}_{\Delta_{1}}2^{-}_{\Delta_{2}}3^{+}_{\Delta_{3}})=C(\Delta_{1},\Delta_{2},\Delta_{3})z_{12}^{h_{3}-h_{1}-h_{2}}z_{13}^{h_{2}-h_{1}-h_{3}}z_{23}^{h_{1}-h_{2}-h_{3}}\bar{z}_{12}^{\bar{h}_{3}-\bar{h}_{1}-\bar{h}_{2}}\bar{z}_{13}^{\bar{h}_{2}-\bar{h}_{1}-\bar{h}_{3}}\bar{z}_{23}^{\bar{h}_{1}-\bar{h}_{2}-\bar{h}_{3}}$
(42)

There are two sets of decoupling equations for the color ordered amplitudes
Banerjee:2020vnt ; Hu:2021lrx . They are

$\begin{split}\left(\partial_{3}-\frac{\Delta_{3}}{z_{13}}-\frac{1}{z_{23}}\right)\widetilde{\mathcal{M}}_{3}(1^{-}_{\Delta_{1}}2^{-}_{\Delta_{2}}3^{+}_{\Delta_{3}})+\epsilon_{1}\epsilon_{3}\frac{\Delta_{1}-\sigma_{1}-1+\bar{z}_{13}\bar{\partial}_{1}}{z_{13}}\widetilde{\mathcal{M}}_{3}(1^{-}_{\Delta_{1}-1}2^{-}_{\Delta_{2}}3^{+}_{\Delta_{3}+1})=0\\\
\left(\partial_{3}-\frac{\Delta_{3}}{z_{23}}-\frac{1}{z_{13}}\right)\widetilde{\mathcal{M}}_{3}(1^{-}_{\Delta_{1}}2^{-}_{\Delta_{2}}3^{+}_{\Delta_{3}})+\epsilon_{2}\epsilon_{3}\frac{\Delta_{2}-\sigma_{2}-1+\bar{z}_{23}\bar{\partial}_{2}}{z_{23}}\widetilde{\mathcal{M}}_{3}(1^{-}_{\Delta_{1}}2^{-}_{\Delta_{2}-1}3^{+}_{\Delta_{3}+1})=0\end{split}$
(43)

Using (42) in (43) we get the following constraints on the 3-point coefficient

$\displaystyle
C(\Delta_{1}-1,\Delta_{2},\Delta_{3}+1)=\epsilon_{1}\epsilon_{3}\frac{(\Delta_{1}-\Delta_{2}-\Delta_{3}+1)}{(\Delta_{3}-\Delta_{1}-\Delta_{2}-1)}C(\Delta_{1},\Delta_{2},\Delta_{3})$
(44) $\displaystyle
C(\Delta_{1},\Delta_{2}-1,\Delta_{3}+1)=\epsilon_{2}\epsilon_{3}\frac{(\Delta_{2}-\Delta_{1}-\Delta_{3}+1)}{(\Delta_{3}-\Delta_{1}-\Delta_{2}-1)}C(\Delta_{1},\Delta_{2},\Delta_{3})$
(45)

Now, one can check that the 3-point coefficient given by Casali:2022fro

$C(\Delta_{1},\Delta_{2},\Delta_{3})=\mathcal{N}_{3}\,\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}+3}{2}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}-1}{2}\right)\Gamma\left(\frac{-\Delta_{1}+\Delta_{2}+\Delta_{3}-1}{2}\right)f(\beta)$
(46)

satisfies (44), (45), where $f(\beta)$ is any function with
$\beta=\sum_{i=1}^{3}\Delta_{i}$ and $\mathcal{N}_{3}$ is given by (36).

## References

  * (1) S. Pasterski, S. H. Shao and A. Strominger, “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere,” Phys. Rev. D 96 (2017) no.6, 065026 doi:10.1103/PhysRevD.96.065026 [arXiv:1701.00049 [hep-th]].
  * (2) S. Pasterski and S. H. Shao, “Conformal basis for flat space amplitudes,” Phys. Rev. D 96 (2017) no.6, 065022 doi:10.1103/PhysRevD.96.065022 [arXiv:1705.01027 [hep-th]].
  * (3) S. Banerjee, “Null Infinity and Unitary Representation of The Poincare Group,” JHEP 01 (2019), 205 doi:10.1007/JHEP01(2019)205 [arXiv:1801.10171 [hep-th]].
  * (4) L. Freidel, D. Pranzetti and A. M. Raclariu, “A discrete basis for celestial holography,” [arXiv:2212.12469 [hep-th]].
  * (5) J. Cotler, N. Miller and A. Strominger, “An Integer Basis for Celestial Amplitudes,” [arXiv:2302.04905 [hep-th]].
  * (6) 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]].
  * (7) 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]].
  * (8) S. Ebert, A. Sharma and D. Wang, “Descendants in celestial CFT and emergent multi-collinear factorization,” JHEP 03 (2021), 030 doi:10.1007/JHEP03(2021)030 [arXiv:2009.07881 [hep-th]].
  * (9) 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]].
  * (10) A. Strominger, “w(1+infinity) and the Celestial Sphere,” [arXiv:2105.14346 [hep-th]].
  * (11) A. Strominger, “$w_{1+\infty}$ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries,” Phys. Rev. Lett. 127 (2021) no.22, 221601 doi:10.1103/PhysRevLett.127.221601
  * (12) 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]].
  * (13) 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]].
  * (14) 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]].
  * (15) L. Donnay, S. Pasterski and A. Puhm, “Asymptotic Symmetries and Celestial CFT,” [arXiv:2005.08990 [hep-th]].
  * (16) 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]].
  * (17) S. Banerjee and S. Pasterski, “Revisiting the Shadow Stress Tensor in Celestial CFT,” [arXiv:2212.00257 [hep-th]].
  * (18) A. Strominger, “On BMS Invariance of Gravitational Scattering,” JHEP 1407, 152 (2014) doi:10.1007/JHEP07(2014)152 [arXiv:1312.2229 [hep-th]].
  * (19) A. Strominger, “Asymptotic Symmetries of Yang-Mills Theory,” JHEP 1407, 151 (2014) doi:10.1007/JHEP07(2014)151 [arXiv:1308.0589 [hep-th]].
  * (20) W. Fan, A. Fotopoulos and T. R. Taylor, “Soft Limits of Yang-Mills Amplitudes and Conformal Correlators,” JHEP 05 (2019), 121 doi:10.1007/JHEP05(2019)121 [arXiv:1903.01676 [hep-th]].
  * (21) L. Donnay, A. Puhm and A. Strominger, “Conformally Soft Photons and Gravitons,” JHEP 01 (2019), 184 doi:10.1007/JHEP01(2019)184 [arXiv:1810.05219 [hep-th]].
  * (22) 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]].
  * (23) 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]].
  * (24) 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]].
  * (25) T. He, P. Mitra, A. P. Porfyriadis and A. Strominger, “New Symmetries of Massless QED,” JHEP 10 (2014), 112 doi:10.1007/JHEP10(2014)112 [arXiv:1407.3789 [hep-th]].
  * (26) D. Kapec, M. Pate and A. Strominger, “New Symmetries of QED,” Adv. Theor. Math. Phys. 21 (2017), 1769-1785 doi:10.4310/ATMP.2017.v21.n7.a7 [arXiv:1506.02906 [hep-th]].
  * (27) D. Kapec, V. Lysov and A. Strominger, “Asymptotic Symmetries of Massless QED in Even Dimensions,” Adv. Theor. Math. Phys. 21 (2017), 1747-1767 doi:10.4310/ATMP.2017.v21.n7.a6 [arXiv:1412.2763 [hep-th]].
  * (28) M. Campiglia and A. Laddha, “Asymptotic symmetries of QED and Weinberg’s soft photon theorem,” JHEP 07 (2015), 115 doi:10.1007/JHEP07(2015)115 [arXiv:1505.05346 [hep-th]].
  * (29) A. Nande, M. Pate and A. Strominger, “Soft Factorization in QED from 2D Kac-Moody Symmetry,” JHEP 02 (2018), 079 doi:10.1007/JHEP02(2018)079 [arXiv:1705.00608 [hep-th]].
  * (30) T. He and P. Mitra, “Covariant Phase Space and Soft Factorization in Non-Abelian Gauge Theories,” JHEP 03 (2021), 015 doi:10.1007/JHEP03(2021)015 [arXiv:2009.14334 [hep-th]].
  * (31) W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, “Elements of celestial conformal field theory,” JHEP 08 (2022), 213 doi:10.1007/JHEP08(2022)213 [arXiv:2202.08288 [hep-th]]. 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]].
  * (32) K. Costello and N. M. Paquette, “Celestial holography meets twisted holography: 4d amplitudes from chiral correlators,” JHEP 10 (2022), 193 doi:10.1007/JHEP10(2022)193 [arXiv:2201.02595 [hep-th]].
  * (33) E. Casali, W. Melton and A. Strominger, “Celestial Amplitudes as AdS-Witten Diagrams,” [arXiv:2204.10249 [hep-th]].
  * (34) A. Atanasov, A. Ball, W. Melton, A. M. Raclariu and A. Strominger, “(2, 2) Scattering and the celestial torus,” JHEP 07 (2021), 083 doi:10.1007/JHEP07(2021)083 [arXiv:2101.09591 [hep-th]].
  * (35) 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]].
  * (36) 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]].
  * (37) S. Banerjee, S. Ghosh and R. Gonzo, “BMS symmetry of celestial OPE,” JHEP 04 (2020), 130 doi:10.1007/JHEP04(2020)130 [arXiv:2002.00975 [hep-th]].
  * (38) S. Banerjee, S. Ghosh and P. Paul, “(Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes,” JHEP 09 (2022), 236 doi:10.1007/JHEP09(2022)236 [arXiv:2108.04262 [hep-th]].
  * (39) R. Bhardwaj, L. Lippstreu, L. Ren, M. Spradlin, A. Yelleshpur Srikant and A. Volovich, “Loop-level gluon OPEs in celestial holography,” JHEP 11 (2022), 171 doi:10.1007/JHEP11(2022)171 [arXiv:2208.14416 [hep-th]].
  * (40) A. M. Raclariu, “Lectures on Celestial Holography,” [arXiv:2107.02075 [hep-th]].
  * (41) S. Pasterski, “Lectures on celestial amplitudes,” Eur. Phys. J. C 81 (2021) no.12, 1062 doi:10.1140/epjc/s10052-021-09846-7 [arXiv:2108.04801 [hep-th]]. S. Pasterski, M. Pate and A. M. Raclariu, “Celestial Holography,” [arXiv:2111.11392 [hep-th]].
  * (42) T. McLoughlin, A. Puhm and A. M. Raclariu, “The SAGEX review on scattering amplitudes chapter 11: soft theorems and celestial amplitudes,” J. Phys. A 55 (2022) no.44, 443012 doi:10.1088/1751-8121/ac9a40 [arXiv:2203.13022 [hep-th]].
  * (43) 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]].
  * (44) S. Banerjee, H. Kulkarni and P. Paul, “An infinite family of $w_{1+\infty}$ invariant theories on the celestial sphere,” [arXiv:2301.13225 [hep-th]].
  * (45) Y. Hu, L. Ren, A. Y. Srikant and A. Volovich, “Celestial dual superconformal symmetry, MHV amplitudes and differential equations,” JHEP 12 (2021), 171 doi:10.1007/JHEP12(2021)171 [arXiv:2106.16111 [hep-th]]. Y. Hu and S. Pasterski, “Celestial recursion,” JHEP 01, 151 (2023) doi:10.1007/JHEP01(2023)151 [arXiv:2208.11635 [hep-th]]
  * (46) W. Melton, S. A. Narayanan and A. Strominger, “Deforming Soft Algebras for Gauge Theory,” [arXiv:2212.08643 [hep-th]].