Ziyang Zhang, Edgar Simo-Serra C¿p0.20  Ziyang Zhang† Edgar Simo-Serra‡ Waseda
University E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Neural Scene Baking for Permutation Invariant Transparency Rendering with
Real-time Global Illumination

###### Abstract

Neural rendering provides a fundamentally new way to render photorealistic
images. Similar to traditional light-baking methods, neural rendering utilizes
neural networks to bake representations of scenes, materials, and lights into
latent vectors learned from path-tracing ground truths. However, existing
neural rendering algorithms typically use G-buffers to provide position,
normal, and texture information of scenes, which are prone to occlusion by
transparent surfaces, leading to distortions and loss of detail in the
rendered images. To address this limitation, we propose a novel neural
rendering pipeline that accurately renders the scene behind transparent
surfaces with global illumination and variable scenes. Our method separates
the G-buffers of opaque and transparent objects, retaining G-buffer
information behind transparent objects. Additionally, to render the
transparent objects with permutation invariance, we designed a new
permutation-invariant neural blending function. We integrate our algorithm
into an efficient custom renderer to achieve real-time performance. Our
results show that our method is capable of rendering photorealistic images
with variable scenes and viewpoints, accurately capturing complex transparent
structures along with global illumination. Our renderer can achieve real-time
performance ($256\times 256$ at 63 FPS and $512\times 512$ at 32 FPS) on
scenes with multiple variable transparent objects.

Our primary results. Our custom neural rendering framework addresses
transparency issues in existing neural rendering approaches. The loss of
details of our method is significant lower than the Active Exploration[1].

##  1 Introduction

Figure 1: Overview of our neural scene rendering framework. First, our
rasterization-based renderer will first render the G-buffer, direct lighting,
and transparency buffers. Then, a neural network, which we denote as GlassNet,
will use those buffers and rendering results as inputs to synthesize high
quality images with global illumination and accurate transparency. Details of
GlassNet can be found in section 3.4 and figure 2.  Figure 2: Architecture of
GlassNet. Our proposed method, GlassNet, contains four building blocks
including the scene encoder, $\mathcal{F}$; the permutation invariant
transparency buffer blending function, $\mathcal{T}$; the final blending
network, $\mathcal{B}$; and the rendering network, $\mathcal{R}$. All parts
are trained jointly.

Global illumination rendering has been a fundamental challenge in computer
graphics due to the complexity of the physical phenomena involved.
Traditionally, path tracing algorithms [2, 3] have been the main method to
render photorealistic images with high-quality global illumination effects.
However, such methods are known to be expensive and time-consuming; rendering
one image may take minutes. Baked global illumination techniques, on the other
hand, provide efficient solutions for interactive rendering by pre-computing
lighting information. Methods such as lightmap baking and light probes [4]
have been commonly used in game engines to generate real-time global
illumination, but they have limitations on dynamic scenes.

Recently, neural rendering approaches have emerged as promising alternative
for realistic image synthesis. Using neural networks to represent scenes,
materials, and lighting can be seen as one type of pre-computation method. In
the early stage, neural rendering methods were only capable of rendering fixed
scenes under limited types of lighting with complex spatial inputs [5]. More
recently, by using G-buffer as inputs, newer methods avoid using complex
spatial data such as point clouds, and are able to render scenes with variable
geometries, materials, and lighting [6, 7, 8, 1, 9]. However, the usages of
G-buffer re-introduces the classic problem of transparency rendering. When
generating G-buffers, the shading information will be overwritten by
transparent surfaces in the front, as shown in figure 3. Without such crucial
shading information, rendering models can only hard code all the information
into the neural network based on ground truths, which is unrealistic and
ineffective, leading to poor quality.

To address this limitation, we propose a novel neural rendering framework that
separately renders opaque objects and semi-transparent surfaces into two
groups of G-buffers. By doing this, all necessary shading information of both
opaque and transparent surfaces are preserved and can be used for further
rendering. Furthermore, we design a neural blending function to encode the
transparency buffer into a latent representation with permutation invariance
to achieve Order-Independent-Transparency (OIT). Finally, we use a deep CNN
(Convolutional Neural Network) to encode scenes and blend them with
transparency representations, rendering the resulting image with global
illumination.

Our results show that we significantly improve the transparency quality,
exceeding the most recent neural approaches, Active Exploration [1] and Neural
Global Illumination [9]. Our method can accurately render the geometry and
texture details under transparent surfaces, whereas methods without
transparency buffers converge to low-quality results.

In summary, our main contributions are:

  1. 1.

A novel neural rendering framework preserves both opaque and transparent
information simultaneously on buffers, providing accurate shading information
on rendering transparency.

  2. 2.

A neural blending function that encodes transparent surfaces with permutation
invariant, achieving OIT.

  3. 3.

A real-time renderer capable of rendering images with high performance enabled
by the above algorithms.

##  2 Related Work

###  2.1 Traditional Transparency Rendering

In the traditional forward rendering pipeline, various techniques have been
used to handle transparency. One widely used algorithm is alpha blending [10],
which utilizes a separate alpha channel to control the color composition of
semi-transparent objects with opaque objects on each fragment. However, alpha
blending is order-dependent and requires sorting of objects by depth, which
can be impractical. The high computational complexity is not the only problem
of those sorting-based algorithms. When transparent objects intersect with
each other, it is impossible to sort the objects globally unless the sorting
is done by fragments, which again greatly increases the computational
complexity. Another algorithm called the A-Buffer algorithm [11] stores a list
of relevant fragments in each pixel, but sorting of all fragments is required
with unbounded memory usage. The $Z^{3}$ algorithm [12] is a similar algorithm
with bounded memory usage but comes with a trade-off in image quality. In
contrast, our proposed method does not require any sorting, and does not
generate a large amount of frame buffers during rendering procedure.

To avoid sorting, several techniques have been introduced. Depth peeling [13]
renders objects in multiple passes, peeling transparent objects based on the
z-buffer. However, the number of passes required by depth peeling increases
the computational cost.

These algorithms often encounter challenges in the modern deferred rendering
pipeline. Deferred rendering [14] is a technique that separates the geometric
stage and lighting stage into two passes. In the geometry pass, all the
geometry information is rendered into G-buffers to be used by the lighting
pass for shading. While deferred rendering significantly improves rendering
performance with multiple light sources, it introduces complexity to
transparency rendering. Rendering geometry information into the G-buffer
causes the objects behind transparent surfaces to be occluded, making it
impossible to retain all geometry information without a prohibitively large
number of buffers. Several algorithms solve this problem with worsened image
quality. [15, 16]. By applying a permutation-invariant blending algorithm, our
method can combine generated transparency information during inference, while
only requiring two frame buffers regardless of the number of transparent
objects, significantly lowering the memory usage.

Several neural rendering approaches utilize the G-buffers as input for their
neural networks [7, 1, 9, 17], resulting in these methods also encountering
transparency occlusion issues with the G-buffers, causing bad image quality.
In contrast, our method is not affected by such issues and supports OIT
rendering.

###  2.2 Baked Global Illumination

Different from expensive path tracing methods, which are the main approach to
render photorealistic images in the modern film industry [18], baked or
precomputed global illumination methods are more often used in interactive or
real-time settings [19].

Lightmap baking is a popular technique used widely in modern game engines
[20]. It precomputes the diffuse global illumination and uses interpolation to
generate global illumination during rendering. Several methods have been
proposed to generate the lightmap offline, such as by radiosity [21] or path-
tracing. Naively, baked lightmap only supports static scenes with diffuse
objects. Light probes technique was introduced to enable global illumination
for dynamic objects [4]. McGuire et al. [22] extended light probes to support
glossy surfaces. However, such methods are limited to either static lighting
or a limited amount of light transportation types.

Several neural approaches use neural networks to encode the scene
representation into neural vectors and synthesize global illumination such as
Compositional Neural Scene Representations (CNSR) [7], Active Exploration and
Neural Global Illumination. Similar to such neural approaches, our method can
render global illumination with dynamic scenes with various light
transportation types, with support for transparent objects.

###  2.3 Neural Rendering

Using neural networks for rendering is a rapidly developing research topic
that provides fundamentally new approaches to photorealistic rendering. We
will introduce the closest related paper to this work. We refer to [23] for a
more detailed neural rendering survey.

Initially, Ren et al. [5] used neural networks to learn indirect lights from
dynamic lighting and roughness with fixed geometry, complex spatial data, and
only point lights. Later, Eslami et al. [6] proposed a method that trains an
encoder-decoder neural network to represent scenes as latent vectors. CNSR by
Granskog et al. further improved the interpretability of the neural scene
representation by partitioning the scene representation vectors into sections
of lighting, materials, geometry, etc. Similar to our method, most recent
solutions [7, 8, 1, 9] use G-buffers as the input to the neural networks to
produce indirect lighting and global illumination. Using G-buffers instead of
point clouds or voxel data simplifies the input data. Such algorithms can
usually be seen as precomputed light transportation and a form of scene
baking. Using neural networks to precompute wavelet function has also shown
strong improvement on glossy global illumination [24]. Attempts on high
performance real-time neural rendering are also made by Xin et al. [17], but
only limited on single-bounce diffuse indirect illumination

However, using G-buffers as inputs brings back the classic issue caused by
transparent objects. The G-buffers cannot provide information on objects
behind transparent surfaces, ending with poor rendering quality by such
methods. In our framework, we use independent G-buffers for the transparent
objects and a permutation invariant neural network to achieve OIT. Note that
the recent studies in neural rendering are mainly orthogonal to our proposed
method with different goals. Our method can be directly combined with those
methods to improve the rendering quality further.

###  2.4 Permutation Invariant Machine Learning

PointNet [25] uses a multi-layer perceptron network to transform the input
into latent vectors and uses a symmetric function to compose such vectors,
which is proved to be a simple but strong method. PointNet inspires how our
rendering neural network can process each transparency input to cancel the
permutation order dependency. We refer a more detailed review of permutation-
invariant neural network to Section 6 of [26].

##  3 Proposed Approach

###  3.1 Overview

Given a scene, the goal of our approach is to bake a neural scene
representation into a neural network that can correctly approximate global
illumination with variable parameters of material, geometry, and lighting
without being affected by transparent surfaces. For any shading position, $x$,
on surfaces, the outgoing radiance, $L_{o}$, can be calculated by the
Rendering Equation[2]:

$\displaystyle L_{o}(x,\omega_{o})=$
$\displaystyle\int_{\Omega}L_{i}(x,\omega_{i})f_{x}(x,\omega_{o},\omega_{i})|n_{x}\cdot\omega_{i}|\,\mathrm{d}\omega_{i}$
(1) $\displaystyle+L_{e}(x,\omega_{o})$

where $f_{x}$ and $n_{x}$ represent the BSDF and normal; $\omega_{o}$
indicates outgoing or viewing direction, $\omega_{i}$ indicates the incident
direction; $L_{i}$ and $L_{e}$ denote the incident radiance and emission
radiance. $\Omega$ denotes a hemisphere.

The Rendering Equation can also be written into the following form such as the
direct lighting and the indirect lighting are separated:

$L_{o}(x,\omega_{o})=L_{e}(x,\omega_{o})+L_{d}(x,\omega_{o})+L_{g}(x,\omega_{i})$
(2)

where $L_{d}$ is the direct lighting, and $L_{g}$ is the indirect lighting
which contributes mainly to the global illumination. Current methods use high-
dimension complex non-linear functions, _i.e_., neural networks, to predict
$L_{o}(x,\omega_{o})$ given all the necessary geometry, lighting, and material
parameters as inputs. In contrast, our approach consists of a neural network,
which we denote as GlassNet, to predict the indirect lighting, $L_{g}$, as
shown in figure 1. We use Linearly Transformed Cosines (LTCs) [27] as a
sufficient approximation to the direct illumination term $L_{d}$ as an input
to our neural network to achieve efficiency. Notice that as an input to the
neural network, $L_{d}$ is not necessarily to be analytically accurate because
the neural network will learn the complexity of the lighting transportation
from the path tracing ground truths.

Most recent methods provide necessary shading information such as
$\omega_{o}$, position, texture, and material parameters through G-buffers.
However, the naive approach using G-buffers as input brings a significant
problem: as shown in figure 3, important shading information is hidden by
transparent surfaces. We propose a new neural rendering framework that
separately draws opaque G-buffers and transparency buffers, preserving all the
information, including the transparent surfaces and the objects behind those.
Then, we introduce a symmetric neural function to cancel the permutation
dependency in the transparency buffers and generate a neural transparency
representations as shown in section 3.3.

###  3.2 Transparency Buffers

Figure 3: Our G-buffer scene representation approach. Instead of naively
rendering the entire scene to a single set of G-buffers, we use separate
buffers for the transparent objects, allowing us to represent complex
transparency visibility in scenes accurately.

For a given scene, we rasterize each transparent surface into a list of
G-buffers containing all the information required by a traditional path tracer
to render $L_{o}$, including position, surface normal, albedo, $\omega_{o}$,
and material parameters as shown in figure 4. We generate a set of buffers
with a size of $(w,h,c*(t+1))$, where $w$, $h$ and $c$ denote the width,
height and channel of images. $t$ is the number of transparent objects in the
scene. Finally, our rasterization-based renderer creates the opaque G-buffers,
and use LTCs to compute the direct lighting $L_{d}$. By doing so, we can
preserve all the necessary information on both normal surfaces and transparent
surfaces.

However, further processing is necessary for the naive transparency buffer to
be a good representation as a neural network input. Generally, neural networks
are known to depend on the order of inputs[25]. Inputting the naive
transparency buffer will cause our rendering network to be order-dependent on
transparency. With modern rendering engine optimization features such as batch
rendering[28] and occlusion culling[29], the order of rendered objects cannot
be guaranteed. The surfaces can even be culled during rendering. It is also
infeasible to sort the rendered buffers after rendering. In order to solve
this problem, we designed a symmetric neural network invariant to permutation
to blend the transparency buffer.

###  3.3 Permutation Invariant Neural Representation

We design a blending invariant algorithm to cancel any sensitivity to the
order of the inputs while significantly lowering the memory usage during
inference as shown in section 4.4.1, inspired by traditional transparency
algorithms such as alpha blending. In alpha blending, the function, $T$, of
compositing transparent objects can be described as[10, 30, 31]:

$T\Bigl{(}\bigl{\\{}(C_{1},\alpha_{1}),(C_{2},\alpha_{2}),\ldots,(C_{t},\alpha_{t})\bigr{\\}}\Bigr{)}=\sum_{i=1}^{t}\alpha_{i}\,C_{i}\,z_{i}$
(3)

where $C_{i}$ and $\alpha_{i}$ respectively are the color and the alpha value
of each transparent object. $z_{i}$ is the product of all alpha values in
front of the object:

$z_{i}=\begin{cases}1&\text{if }i=t\\\
\prod_{j=1}^{t-i}(1-\alpha_{j})&\text{otherwise}\end{cases}$ (4)

This function is clearly input-order-dependent, but it can be revised into a
function with permutation invariant to inputs.

Based on the alpha blending function with the inspiration from the PointNet,
we design our permutation invariant transparency buffer blending function,
$\mathcal{T}$, to generate the neural representation, $\tau$:

$\displaystyle\mathcal{T}\Bigl{(}\\{b_{1},b_{2},\ldots,b_{t}\\},\sigma\Bigr{)}$
$\displaystyle\approx
g\Bigl{(}h(b_{1},\sigma),h(b_{2},\sigma),\ldots,h(b_{t},\sigma)\Bigr{)}\noindent$
$\displaystyle=\sum_{i=1}^{t}h(b_{i},\sigma)=\tau$ (5)

where $b$ is the transparency buffer, $g$ is a symmetric function, $\sigma$ is
a scene representation, and $h$ is a neural network shared by all the
transparency buffers generating neural representations. We use addition as $g$
to match the alpha blending function, which clearly makes this function
mathematically permutation invariant to inputs. This allows the neural network
to blend all the transparency buffers into a single latent vector that is
independent of the input order.

| Normal | Position | Albedo | Roughness | $\omega_{o}$ | Depth | $L_{d}$ |
---|---|---|---|---|---|---|---
|  |  |  |  |  |  |
G-Buffer |
Normal | Position | Albedo | Roughness | $\omega_{o}$ | Depth | Relative Depth |
|  |  |  |  |  |  |
Transparency Buffer |
| Ground Truth
---
Figure 4: G-buffer, transparency buffer, and ground truth. Our method can
overcome the noise in the ground truth.

###  3.4 Rendering Model

We introduce our rendering model, GlassNet as shown in figure 1. GlassNet has
four components: the scene encoder, $\mathcal{F}$; the permutation invariant
transparency buffer blending function, $\mathcal{T}$; the final blending
network, $\mathcal{B}$; and the rendering network, $\mathcal{R}$. Both
$\mathcal{F}$ and $\mathcal{B}$ use U-Net [32] as the network backend, whereas
other networks use multi-layer CNN without down-sampling. All four components
are trained jointly as one neural network. Full details of the model are shown
in figure 2.

The scene encoding network, $\mathcal{F}$, accepts tensors containing the
position, normal, texture, material parameters, $\omega_{o}$, depth buffer,
and direct lighting. It then generates a neural scene representation,
$\sigma$. Similarly, each transparency buffer is processed by $\mathcal{T}$,
generating the neural representation of transparent surfaces $\rho$. All the
inputs use positional encoding by Mildenhall et al. [33] to capture high-
frequency information. Afterward, the blending network $\mathcal{B}$ blends
the scene representation $\sigma$ and transparent object representations
$\rho$ and processes visibility. By doing so, $\mathcal{B}$ generates the
final neural buffer, $\phi$.

Finally, the rendering network $\mathcal{R}$ takes the neural buffer $\phi$
with additional direct lighting as input to render the final image. Although
previous works, such as Neural Global Illumination, report using convectional
neural network structure on the rendering network can lead to distortion of
shapes, we find that this problem can usually be solved by using kernels with
the size of $1\times 1$ at the end of the network, and adding larger kernel
can help with global effects such as blurring. For balancing the render
quality and performance, the GlassNet is trained with lower resolution
($256\times 256$) textures, G-buffers, and ground truths. A multi-layer CNN is
used as a super-sampling network that super-samples the result at $\times 2$
the original resolution. Given that we can cheaply get high-resolution texture
and G-buffer from the rasterizer, the super-sampling network can be simple to
achieve real-time performance.

###  3.5 Training

We train the model in an end-to-end fashion to bake a specific scene by
randomly sampling camera angles and positions within certain ranges of each
scene following uniform distributions to render ground truth images. In
particular, for each scene, we generate output image and G-buffer pairs for
training. The ground truth is rendered by the modern path tracer, Mitsuba 3
[34]. For the G-buffer and direct lighting generation, we implemented our
custom rasterization renderer using OpenGL in C++, with CUDA support to
support neural network inference. As a loss function, we choose is the
combination of Structural Similarity Index Measure (SSIM) [35] and $L_{1}$
loss.

##  4 Experiments

Figure 5: Randomly selected images from datasets. The transparent areas are highlighted by the blue masks in the second row. | Ground Truth | Ours | Active Exploration
---|---|---|---
BATHROOM |  |  |
DESK |  |  |
CORNELLBOX |  |  |
Figure 6: Qualitative comparison. We compare with Active Exploration in
several challenging scenes. Note that our method can preserve details of
objects behind transparent surfaces with the synthesis of global illumination.
All scenes are rendered on test sets with different camera views from the
training set. The transparent areas are indicated by yellow dashed lines in
the column of ground truths.

###  4.1 Experimental Settings

Scene Method Metrics MAE $\downarrow$ LIPIPS $\downarrow$ DSSIM $\downarrow$
PSNR $\uparrow$ T.MAE $\downarrow$ T.PSNR $\uparrow$ BATHROOM Active Exp.
0.01475 0.02995 0.04840 30.85680 0.06170 23.72751 Neural G.I. 0.00934 0.00961
0.02280 35.03145 0.03888 28.31231 Naive 0.02131 0.04409 0.05622 29.17893
0.08208 22.41524 Ours 0.00808 0.00686 0.01887 36.81351 0.02738 32.46141
CORNELLBOX Active Exp. 0.03237 0.18584 0.20583 25.21868 0.10491 19.78125
Neural G.I. 0.04093 0.12171 0.19213 24.13943 0.13141 17.84513 Naive 0.04988
0.20916 0.23296 22.90424 0.15471 17.80871 Ours 0.02142 0.01595 0.05436
29.75453 0.06087 25.44725 DESK Active Exp. 0.01739 0.05523 0.06117 27.87549
0.15792 16.31566 Neural G.I. 0.01733 0.04327 0.04985 29.34936 0.12360 17.96721
Naive 0.03274 0.07845 0.07987 26.16205 0.19205 15.75951 Ours 0.01658 0.01150
0.01782 34.82217 0.05487 28.24681 CORNELLBOX8 Active Exp. 0.02911 0.17719
0.20583 26.16428 0.08924 21.14504 Neural G.I. 0.04307 0.19579 0.27657 22.83434
0.13061 17.90722 Naive 0.05093 0.28167 0.31939 21.84069 0.15147 17.09835 Ours
0.02064 0.04152 0.07311 28.56370 0.05721 24.25974

Table 1: Quantitative comparison. We compare our approach with the naive
baseline, Active Exploration and Neural Global Illumination on several scenes
with diverse metrics. T.MAE and T.PSNR denote the MAE loss and PSNR on the
areas of the image with transparent objects. Best result is denoted in bold.

The input and output resolution used for training are both $256\times 256$ px.
The super-sampling network uses $512\times 512$ textures to super-sample the
$256\times 256$ GlassNet rendered results to $512\times 512$ px, improving the
visual quality. In addition, all inputs are positional encoded using the
method proposed by Mildenhall et al.. All types of inputs and ground truth
images are shown in Figure 4. We used a relatively low number of path tracing
samples and did not pre-process the data with a denoiser.

We used Adam optimizer [36] to perform gradient descent with different
hyperparameters on each scene. The dataset details and hyperparameters such as
train/validation/test splits, learning rate and regularization for each scene
can be found in Appendix A. The training time varies from 5 to 10 hours using
8 RTX 2080 Ti GPUs, depending on the scene.

As evaluation metrics, we use Mean Absolute Error (MAE), LPIPS [37], DSSIM
[38], and Peak Signal-to-Noise Ratio (PSNR). For transparency quality
evaluation, we use PSNR and MAE computed on pixels rendered with transparency
(T.MAE and T.PSNR).

###  4.2 Comparison with the State-of-the-Art

We compared with Active Exploration and Neural Global Illumination, as they
are the closest recent work. Comparisons were carried out in 4 challenging
scenes: BATHROOM, CORNELLBOX, DESK, and CORNELLBOX8 as shown in figure 5. The
BATHROOM was modified from the Bitterli dataset [39], while DESK and
CORNELLBOX were creating using existing models [40], to focus on transparency
effects. In CORNELLBOX8, We added 4 more transparent objects to CORNELLBOX to
explore the performance under the condition of high number of transparent
objects. Qualitative results are shown in Neural Scene Baking for Permutation
Invariant Transparency Rendering with Real-time Global Illumination and figure
6, while quantitative results are shown in table 1. A comparison with real-
time path tracing method is also demonstrated in LABEL:fig:RTRT

From the qualitative comparison in figure 6 we can see that although Active
Exploration performed well in many areas, without transparency buffers, the
neural network could not synthesize the area behind the semi-transparent
surfaces without losing fundamental details. In the DESK and CORNELLBOX
scenes, the text texture was almost impossible to render without provided
texture information. Instead, our method could synthesize all the texture
accurately. Furthermore, in the BATHROOM scene, all the detailed shapes behind
the glass such as taps were missing in the Active Exploration, where our
method preserved them well. The quantitative comparison in table 1
corroborates the visual improvement of our method, which achieves better
values on most metrics, with more significant improvements on parts of the
image that need transparency. In addition, the rendering results and
comparison video are available in Video S1 in the Electronic Supplementary
Material.

Overall, our method preserves the details of texture and geometry of shape
well. The global illumination effects such as mirror reflection are also well
synthesized. Our approach shows strong performance on preserve all the high
frequency texture, which is also shown on the wood texture of the BATHROOM
scene. Such details are extremely hard to be synthesized behind nearly
transparent surfaces without transparency buffers.

Our experiments show that the transparency buffer is essential for neural
network to render objects behind transparent surfaces accurately. Although the
global illumination effects are not the main focus of our method, we also
demonstrate good quality on indirect diffuse lighting, mirror reflection, and
soft shadows on par with existing approaches.

Scene Method Metrics DSSIM $\downarrow$ T.MAE $\downarrow$ T.PSNR $\uparrow$
CORNELLBOX CNSR 0.1502 0.0973 20.4854 Ours 0.0451 0.0457 26.2100 DESK CNSR
0.0539 0.1115 18.5453 Ours 0.0174 0.0533 28.6498

Table 2: Quantitative comparison with CNSR. Best result is highlighted in bold. | Ground Truth | Ours | CNSR
---|---|---|---
CORNELLBOX |  |  |
Figure 9: Rendering results of ours and Granskog et al. Our method preserves
the transparency quality better. The transparent areas are indicated by yellow
dashed lines.

We also compared our method to CNSR to further verify the effectiveness of our
approach to transparency. We generated the dataset in the format CNSR requires
and used the identical dataset to train the CNSR model and our model. The
results are shown in figure 9 and table 2. The transparency quality rendered
by our approach is significantly better. Note that this comparison experiment
is only supplementary since the main focus of the two methods is different.

In addition, we compared our approach to the real-time path tracing method in
LABEL:fig:RTRT. We used 32 samples per pixel to generate the path tracing
image with a standard Optix denoiser [41]. The Optix denoiser failed to
reconstruct the wood texture and the tape behind the glass, while our
algorithm rendered such objects accurately. Due to the glass, the denoiser can
no longer rely on albedo and normal AOV to gain extra information, causing the
loss of details.

Size Type # of Trans. Obj. 1 2 4 8 $256\times 256$ Framerate (FPS) 100 77 63
45 Frametime (ms) 10 13 16 22 $512\times 512$ Framerate (FPS) 50 37 32 23
Frametime (ms) 20 27 31 43 $1024\times 1024_{\text{CARN}}^{\dagger}$ Framerate
(FPS) 24 21 20 18 Frametime (ms) 41 47 50 55 $1024\times
1024_{\text{NSRR}}^{\dagger}$ Framerate (FPS) 17 16 15 12 Frametime (ms) 56 62
66 78

Table 3: Performance evaluation. We can interactively render all test cases.
$\dagger$ $1024\times 1024$ achieves by an external supersampling network,
CARN or NSRR with temporal stability, as a post-processing procedure.

###  4.3 Ablation Study

| MAE$\downarrow$ | LIPIPS$\downarrow$ | DSSIM$\downarrow$ | PSNR$\uparrow$
---|---|---|---|---
w/o P.E. | 0.023711 | 0.026580 | 0.059718 | 28.937529
L1 Loss | 0.023923 | 0.033820 | 0.072169 | 25.218679
Ours | 0.021424 | 0.015953 | 0.054358 | 29.754528
Table 4: Impact of positional encoding and loss functions. Our experiments show the combination of positional encoding and L1SSIM loss function achieves the best performance of the network. Best result is highlighted in bold. | MAE | MAE (inv. order) | Difference
---|---|---|---
BATHROOM | 0.00805 | 0.00805 | 0.00000
DESK | 0.01658 | 0.01658 | 0.00000
CORNELLBOX | 0.02125 | 0.02125 | 0.00000
Table 5: MAE on different transparency buffer order. We invert the order of
transparency buffer, and achieve identical rendering results.

Although various previous works such as Active Exploration and CNSR reported
that the PixelGenerator has better up-sampling performance than CNN, we found
that PixelGenerator is not able to effectively synthesize the transparent
structure given the same amount of training time as CNN. We implemented the
same rendering network structure with PixelGenerator as the backbone network.
LABEL:fig:ablation_pg shows that PixelGenerator failed to reconstruct the
glossy effects on transparent objects given $400$ training iterations.

To overcome the bad up-sampling performance of CNN, an up-sampling network is
solely trained to super-sample the rendered images at two times the
resolution. Our experiment shows fine-tuning the model with a smaller high-
resolution dataset cannot achieve good visual effects.
LABEL:fig:albation_retrain shows that the image generated by our up-sampling
network has better visual quality, especially on high-roughness transparent
objects, than retraining the CNN.

We studied the influence of positional encoding on our method. Mildenhall et
al.show that positional encoding can help neural networks learn high-frequency
information; therefore, all the inputs to our neural network are positional
encoded. We also demonstrate that using the combination of MAE loss and DSSIM
loss can achieve the best quality, with quantitative results are shown in
table 4. Furthermore, we verified the effectiveness of our permutation-
invariant algorithm. Table 5 shows that shuffled transparency buffer causes on
effect on our model performance.

###  4.4 Performance Evaluation

Figure 11: Result in combining with other supersampling models.  We
demonstrate that using external supersampling models such as NSRR can further
supersample our results to $1024\times 1024$.

####  4.4.1 Real-time Performance

We tested our renderer on one RTX 4090, and the performance is shown in table
3. The computation time of our algorithm does not significantly grow w.r.t.
the complexity of meshes because the transparent buffers only separately
record each transparent object instead of each primitive. The CORNELLBOX and
DESK (30,344 and 38,400 triangles of transparent objects) show that our method
can efficiently render complex meshes.

Furthermore, the neural blending algorithm shown in section 3.3 also grows
linearly w.r.t. the number of transparent objects given a specific resolution
because $h$ is n $\mathcal{O}(1)$ function under specific input size. With the
resolution of $512\times 512$, one extra transparent object costs $3.3$ ms
more time on average. We believe this cost can be further optimized by pruning
the model.

Our approach can be further supersampled by other existing fast supersampling
networks. As examples, we demonstrate that by using CARN[42] and NSRR[43] as
post-processing models, our method can reach $1024\times 1024$ resolution with
anti-aliasing effects. The results can be found in table 3, figure 11, and
Video S1 in the Electronic Supplementary Material.

$1$$2$$4$$8$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$22$$24$$26$$28$$30$$32$$34$$36$Number
of transparent objectsRequired memory (Megabytes)OursTraditional Figure 12:
Buffer memory usage comparison. Under $256\times 256$, one transparent object
buffer causes around 4.5 megabytes of memory. Our blending algorithm is
summation based and thus do not require space to store each transparency
buffer.

####  4.4.2 Memory Efficiency

The permutation invariant blending algorithm also has a memory efficiency
advantage over existing approaches. With traditional methods, the model has to
receive all transparency buffers in order to make inference, so that the
memory costs grow linearly w.r.t. the number of transparent objects. However,
with our proposed algorithm, there is no additional framebuffer costs because
the model can operate on a single buffer of each transparent objects, and then
add it to the summed latent representation. The memory usage comparison is
plotted in figure 12 approximated by framebuffer usage. We can see that our
permutation invariant blending algorithm has constant space growth w.r.t. the
number of transparent objects.

| RAM (GB) | VRAM (GB)
---|---|---
Active Exploration | 2.50 | 11
Ours | 0.68 | 4
Table 6: Runtime memory usage comparison. We compare our method with Active
Exploration on $512\times 512$ with three transparent objects. Least memory
usage is highlighted in bold.

We also compare the actual runtime RAM and VRAM usage of our method with that
of Active Exploration. The experiment uses the DESK scene with the resolution
of $512\times 512$ to test the memory performance. Table 6 shows our
implementation significantly reduces the required runtime memory.

##  5 Limitations and Discussion

Although our method shows significant improvements in neural rendering of
scenes with transparent objects with full OIT, our method relies on two
fundamental assumptions about scenes: the presence of transparent surfaces
with a low refraction index and the absence of participating media. However,
when refraction occurs, the shading information provided by the G-buffer is no
longer accurate, which becomes an even more significant issue on spherical
surfaces. This limitation is illustrated in LABEL:fig:limitation. Our
rendering network currently struggles to produce correct results when strong
refraction effects are present. Additionally, due to the use of G-buffers, our
framework is unable to represent participating media, leading to the
ineffective synthesis of volumetric effects.

As a problem of neural rendering in general, current neural rendering
approaches perform relatively poorly on high-resolution images. Although our
method uses CARN or NSRR as a post-processing supersampling procedure to boost
the resolution further to $1024\times 1024$ as described in section 4.4.1,
further attempts to raise resolution will significantly affect the
performance, causing low framerates. Further research may focus on the
performance of neural rendering in high-resolution settings.

Another potential direction for future research is incorporating path
prediction techniques, such as the method proposed by Li et al. [44], into
scene representation. Furthermore, exploring efficient approaches for
representing participating media on both the input and neural network sides
presents an intriguing research topic.

Several orthogonal methods can be combined with our work. For example, the
Monte Carlo sampling technique proposed in Active Exploration can be utilized
to improve the effects such as perfect mirror reflection and caustic effects.
Neural Global Illumination uses radiance cues as inputs to the model, which
improves the scene representation learning.

### Appendix

Table 7 records the key scene settings and hyperparameters during the dataset
generation, training, and evaluation.

Scene | Train | Validation | Test | S.P.P. | L.R. | W.D. | # of Trans. Obj. | Variable
---|---|---|---|---|---|---|---|---
BATHROOM | 1800 | 200 | 128 | 8192 | 0.0001 | 0.00075 | 1 | 1 (Roughness)
CORNELLBOX | 1800 | 200 | 128 | 4096 | 0.0001 | 0.0001 | 4 | 4 (Position and Lighting)
DESK | 1800 | 200 | 128 | 4096 | 0.0001 | 0.0001 | 3 | 3 (Color)
CORNELLBOX8 | 900 | 100 | 128 | 4096 | 0.0001 | 0.0001 | 8 | 7 (Position and Lighting)
Table 7: Scene settings and hyperparameter choices. S.P.P. indicates the
number of samples per pixel using during the ground truth generation. L.R.
refers to learning rate and W.D. refers to weight decay.

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# Hubble selection of the weak scale from QCD quantum phase transition

Sunghoon Jung<EMAIL_ADDRESS>Center for Theoretical Physics, Department
of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Astronomy Research Center, Seoul National University, Seoul 08826, Korea
TaeHun Kim<EMAIL_ADDRESS>Center for Theoretical Physics, Department of
Physics and Astronomy, Seoul National University, Seoul 08826, Korea

###### Abstract

If the strange quark were lighter, QCD phase transition could have been first
order. This implies that QCD may have quantum critical points as the Higgs vev
$v_{h}$ is varied from its Standard Model value. We show that inflationary
quantum evolution of $v_{h}$ with the relaxion can drive our universe toward
those critical points, realizing the weak scale close to the observed value
while explaining its closeness to $\Lambda_{\rm QCD}$. We first explore
quantum critical points of $N_{f}=3$ QCD, parameterized by $v_{h}$ at $T=0$,
and present a basic model for the weak scale. It results in a sharply
localized probability distribution of the weak scale, which is critical not to
the crossover at zero but to the quantum transition at ${\sim}\Lambda_{\rm
QCD}$.

##  I Introduction

The Planck-weak scale hierarchy may be addressed by the near-criticality of
the Higgs mass parameter [1, 2]. As it crosses zero, the universe transitions
between broken and unbroken phases of the electroweak symmetry. The crossing
could generate some SM backreactions [3, 4, 5, 7, 6] that allow dynamical
selection of the critical weak scale. However, this transition itself is
crossover at zero temperature $T$, providing only a smooth selection rule.
Meanwhile, the Higgs mass near-criticality could also be a result of anthropic
[7, 8, 9, 10, 11] or entropic selections [12, 13, 14, 15].

Recently, a powerful mechanism for dynamical selection of criticality was
developed in [16], where inflationary quantum-dominated evolution of the
relaxion inevitably drives a theory close to a quantum critical point. It is
crucial to be the critical point between discrete phases, because large energy
contrast there is what allows exponential localization by large Hubble rate
difference – Hubble selection.

Then we ask: to what quantum critical points does the Higgs mass have
relevance? [16] discussed its renormalization scale dependence through the
Higgs self coupling, as the electroweak vacuum is barrier separated from the
global one near the Planck scale [17, 18]; and [5] studied a prototype model
with multiple axions.

We note that if the strange quark were lighter, QCD chiral phase transition
(finite-$T$) could have been first order. Although it is not yet firmly
established by lattice calculation [19, 20, 21, 26, 22, 23, 24, 25], it has
been argued based on the (non-)existence of IR fixed points in the 3D linear
sigma model [27, 28]. In other words, QCD at $T=0$ too may have rich vacuum
structure, as a function of quark masses (hence, the Higgs vev $v_{h}$). But
it is not well known currently.

In this work, we present a cosmological account of the weak scale criticality
from possible quantum critical points of QCD. We first explore such critical
points as a function of $v_{h}$ for the first time, and calculate their roles
for the weak scale criticality that is realized through cosmological Hubble
selection. As a result, the weak scale is near-critical, not to the
electroweak crossover, but to the quantum phase transition of the chiral
symmetry, whose generic scale $\Lambda_{\rm QCD}$ happens to be close to the
observed weak scale.

The paper discusses the following in order: basic model ingredients, QCD
quantum critical points, Hubble selection mechanism, and realization of the
weak scale criticality. Then we conclude with potential improvements of the
scenario.

##  II Model

The model consists of the relaxion $\phi$ [3], the Higgs $h$, and the meson
field $\Sigma$: $V_{\rm tot}\,=\,V_{\phi}+V_{h}+V_{\Sigma}$. The relaxion
couples only to the Higgs sector, but the change of $v_{h}$ (scanned by
$\phi$) induces a change in the $\Sigma$ sector. The desired quantum critical
point $v_{h}^{*}$ is developed in the $\Sigma$ sector. Then Hubble selection
(acting on $\phi$) self-organizes the universe to the critical point.

The relaxion potential is axion-like:

$V_{\phi}\,=\,\Lambda_{\phi}^{4}\cos\frac{\phi}{f_{\phi}}.$ (1)

For Hubble selection, its field range $f_{\phi}$ shall exceed the Planck scale
(see later), which is possible with multiple axions [29, 30, 31].

The Higgs potential takes the SM form ($\lambda_{h}\simeq 0.13$) plus the
coupling to the relaxion

$V_{h}\,=\,\frac{1}{2}(M^{2}-g\widetilde{\phi})h^{2}+\frac{\lambda_{h}}{4}h^{4}\,\,\to\,\,-\frac{1}{2}(g\phi)h^{2}+\frac{\lambda_{h}}{4}h^{4},$
(2)

where $h$ is the real Higgs field in unitary gauge. We shift $\phi$ such that
the quadratic term $\mu_{h}^{2}=-g\phi$ vanishes at $\phi{=}0$.
$v_{h}^{2}\equiv-\mu_{h}^{2}/\lambda_{h}$ is used to label the relaxion
scanning ($v_{\rm EW}\,{=}\,246$ GeV).111The QCD backreaction $V_{h}\ni
y_{q}h\langle\bar{q}q\rangle/\sqrt{2}$ is ignored, as it is sizable only from
the $SU(3)_{V}$ vacuum for $v_{h}\lesssim\Lambda_{\rm QCD}$ (which is not
Hubble selected), while only deepening the $V_{h}$ minimum (increasing the
critical energy drop). The required field range of $\phi$ to scan
$\mu_{h}^{2}$ up to the cutoff $M^{2}$ is $\delta\phi\sim M^{2}/g$, thus we
set $f_{\phi}=M^{2}/g.$ The dimensionful coupling $g$ is a spurion of the
relaxion shift symmetry, thus can be small naturally.

Below $\Lambda_{\rm QCD}=200\,{\rm MeV}$, scalar meson fields $\Sigma_{ij}(x)$
are relevant degrees of freedom, and their condensate is an order parameter
for the chiral symmetry breaking. The linear sigma model (LSM) is given by
[33, 32, 34]

$\displaystyle V_{\Sigma}$ $\displaystyle\,=\,$ $\displaystyle\mu^{2}{\rm
Tr}[\Sigma\Sigma^{\dagger}]+\lambda_{1}({\rm
Tr}[\Sigma\Sigma^{\dagger}])^{2}+\lambda_{2}{\rm
Tr}[(\Sigma\Sigma^{\dagger})^{2}]$ (3) $\displaystyle-c({\rm det}\Sigma+{\rm
det}\Sigma^{\dagger})-{\rm Tr}[{\cal H}(\Sigma+\Sigma^{\dagger})],$

where fields and parameters are decomposed as
$\Sigma=(\sigma_{a}+i\pi_{a})T^{a},\,{\cal H}=h_{a}T^{a}$ with generators
$T^{a}$ satisfying ${\rm Tr}[T^{a}T^{b}]=\delta^{ab}/2$ for
$a=0,...,N_{f}^{2}-1$. Without losing generality, $\lambda_{1,2},h_{a}$ are
real, $c>0$, and $\mu^{2}$ can take either sign.

LSM describes the $U(N_{f})\times U(N_{f})$ flavor symmetry of QCD, under
which $\Sigma$ is bi-fundamental. The first line conserves
$SU(N_{f})_{L}\times SU(N_{f})_{R}$; $\lambda_{2}$ is non-zero, otherwise
symmetry is enhanced to $O(2N_{f}^{2})$. $U(1)_{V}$ is the conserved baryon
number; we omit to write it. The instanton contribution $c$ breaks the
anomalous $U(1)_{A}$ down to $Z_{A}(N_{f})$ [35, 36]. ${\cal H}$ is the
leading chiral-symmetry breaking mass term. We fix $N_{f}=3$ with the isospin
symmetry $m_{u}=m_{d}$; only $h_{0},h_{8}\neq 0$.

Remarkably, QCD LSM presents necessary features for quantum critical points at
$T=0$. $c$ is a cubic term for $N_{f}=3$, creating local vacua even with
$\mu^{2}>0$; this would not have been possible for $N_{f}=2$. In addition, the
linear term ${\cal H}$ plays the role of external magnetic fields, as a
function of the quark mass (or $v_{h}$), destabilizing the local vacuum at a
critical point $v_{h}^{*}$.

##  III QCD quantum critical points

How does QCD vacuum structure respond to the relaxion scanning of $v_{h}$? The
partially conserved axial-vector currents yield

$\partial_{\mu}j^{\mu
5}_{a}\,=\,m_{\pi_{a}}^{2}f_{\pi_{a}}\pi_{a}\,=\,\pi_{b}h_{c}d_{abc},$ (4)

where the last equality is from the variation under chiral transformations.
For the pion $\pi^{0}=\pi_{3}$ with
$d_{3b0}=\sqrt{2/3}\delta_{b3},d_{3b8}=1/\sqrt{3}\delta_{b3}$, and for
$K^{0}=(\pi_{6}+i\pi_{7})/\sqrt{2}$ with
$d_{Kb0}=\sqrt{2/3}\delta_{bK},d_{Kb8}=-1/\sqrt{12}\delta_{bK}$, we have

$m_{\pi}^{2}f_{\pi}=\sqrt{\frac{2}{3}}h_{0}+\frac{h_{8}}{\sqrt{3}},\quad
m_{K}^{2}f_{K}=\sqrt{\frac{2}{3}}h_{0}-\frac{h_{8}}{2\sqrt{3}}.$ (5)

Using these most precisely measured properties (Table. 1), we fix the “SM
point” as [37]

$h_{0}(v_{\rm EW})=(287\,{\rm MeV})^{3},\,\,\,\,h_{8}(v_{\rm EW})=-(312\,{\rm
MeV})^{3}.$ (6)

From the chiral Lagrangian, $m_{\pi}^{2}f_{\pi}\propto m_{q}$. The comparison
with current mass terms, ${\cal L}\ni-m_{q}(\bar{u}u+\bar{d}d)-m_{s}\bar{s}s$,
also yields the same ratio
$h_{8}/h_{0}\simeq\left(\frac{m_{q}-m_{s}}{\sqrt{3}}\right)/\left(\frac{2m_{q}+m_{s}}{\sqrt{6}}\right)\simeq-1.3,$
using $m_{q}=(m_{u}+m_{d})/2=3.45\,{\rm MeV}$, $m_{s}=93\,{\rm MeV}$ [38].
Thus, we assume that ${\cal H}$ is linear to $v_{h}$ as

$h_{0,8}(v_{h})\,=\,h_{0,8}(v_{\rm EW})\frac{v_{h}}{v_{\rm EW}}.$ (7)

This is the main response of QCD vacua to the relaxion.

Figure 1: Vacuum energies of benchmark coexisting QCD vacua at $T=0$, as functions of $v_{h}$. The quantum critical point is at $v_{h}^{*}\simeq 20\,{\rm MeV}$, with $\Lambda_{\rm QCD}$-scale energy contrast.  parameter | $f_{\pi}$ | $f_{K}$ | $m_{\pi}$ | $m_{K}$ | $m_{\eta}$ | $m_{\eta^{\prime}}$ | $m_{a_{0}}$ | $m_{f_{0}(500)}$ | $m_{f_{0}(1370)}$ | $m_{K^{*}(1430)}$
---|---|---|---|---|---|---|---|---|---|---
measured | 92.4 | 113 | $139.57^{\pm 0.005}$ | $497.61^{\pm 0.013}$ | $547.86^{\pm 0.017}$ | $957.78^{\pm 0.06}$ | $980^{\pm 20}$ | $500^{\pm 150}$ | $1370^{\pm 150}$ | $1425^{\pm 50}$
benchmark | 92.4 | 113 | 137 | 491 | 534 | 973 | 1050 | 731 | 1260 | 1140
Table 1: Tree-level description of the benchmark SM point in Eq. (8) and (6).
In units of $\,{\rm MeV}$. Data from PDG [38].

QCD vacua at $T=0$ are found by minimizing $V_{\Sigma}$ and considering
stability along all 18 field directions. It is known that LSM can have three
types of vacua at ${\cal H}=0$ [39, 40]: $SU(3)_{L}\times SU(3)_{R}$
($s_{1}=s_{3}=0$), $SU(3)_{V}$ ($s_{1}=s_{3}\neq 0$), and $SU(2)_{L}\times
SU(2)_{R}\times U(1)_{V}$ ($s_{1}=0,s_{3}\neq 0$), where
$\langle\Sigma\rangle\,=\,\sigma_{0}T^{0}+\sigma_{8}T^{8}\,=\,{\rm
diag}(s_{1},s_{1},s_{3})$. In particular, the $SU(3)$ global vacuum (that we
live today) and the $SU(3)\times SU(3)$ local vacuum _coexist_ [40] if
$\mu^{2}>0$ and
$K\,\equiv\,\frac{c^{2}}{2\mu^{2}(3\lambda_{1}+\lambda_{2})}\,>\,4.5$ with
$3\lambda_{1}+\lambda_{2}>0.$ This parameter space is our focus. Currently, it
is not known whether this can be realized near the SM.

By scanning LSM parameters with $K>4.5$, we have successfully found a range of
desired possibilities (App. A). The following benchmark SM point (with Eq.
(6))

$\mu^{2}=(60\,{\rm MeV})^{2},\,\,\,c=4800\,{\rm
MeV},\,\,\,\lambda_{1}=7,\,\,\,\lambda_{2}=46$ (8)

has $K=47.8$, and the tree-level description of meson spectrum (Table. 1)
yields the goodness-of-fit $\chi^{2}$/dof = 0.44 with the first 7 observables
and 3.11 including all. The first 7 are most reliable, while the last 3 are
under dispute; their measurements are not as precise and identities are
unclear [38]. 5% theoretical uncertainties are assumed, as typical size of
perturbative corrections. It is as good as example benchmarks’ in the
literature: [39] (with $\mu^{2}>0$) yields 0.20 and 2.84, respectively, and
[41] ($\mu^{2}<0$) yields 1.22 and 4.64.

As ${\cal H}$ or $v_{h}$ increases from zero with Eq. (7), the metastable
$SU(3)\times SU(3)$ vacuum becomes shallower and unstable at the critical
point, which is found at $v_{h}^{*}\simeq 20\,{\rm MeV}$ for the benchmark. In
fact, a wider range of $v_{h}^{*}$, at least ${\cal O}(1\sim 100)\,{\rm MeV}$,
is found (see App. A). The vacuum energies of coexisting QCD vacua are shown
in Fig. 1; parameterized by $v_{h}$ as Eq. (14) and (15). The critical energy
contrast between those vacua is $93\,{\rm MeV}$, comparable to $\Lambda_{\rm
QCD}$.

Actually, LSM is over-constrained at tree-level, so it does not yield perfect,
single best-fit. As LSM (and essentially all models) is an effective model for
non-perturbative effects, lattice calculation is eventually needed to verify
QCD quantum critical points.

##  IV Hubble selection

Inflationary quantum fluctuations on the scalar field value allow to access
higher potential regime. Although the field in each Hubble patch always rolls
down in average, larger Hubble rates at higher potentials can make a
difference in the global field-value distribution among patches.

The Hubble (volume) weighted field distribution $\rho(\phi,t)$ obeys the
modified Fokker-Planck equation (FPV)[47, 48, 49, 50]

$\frac{\partial\rho(\phi,t)}{\partial
t}\,=\,\frac{\partial}{\partial\phi}\left(\frac{V^{\prime}}{3H}\rho\right)\,+\,\frac{1}{8\pi^{2}}\frac{\partial^{2}(H^{3}\rho)}{\partial\phi^{2}}\,+\,3\Delta
H\rho.$ (9)

The $\phi$ dependence of the Hubble constant $H=H_{0}+\Delta H(\phi)$ in the
last term with $\Delta H(\phi)=\frac{V(\phi)}{6M_{\rm Pl}^{2}H_{0}}\ll H_{0}$
accounts for the volume weights within the distribution. $M_{\rm Pl}\simeq
2.4\times 10^{18}\,{\rm GeV}$. It is solved by (for a linear potential without
boundary conditions)

$\rho(\phi,t)\propto\exp\left\\{-\frac{1}{2\sigma_{\phi}}\Big{[}\phi-\big{(}\phi_{0}+\dot{\phi}_{c}t+\tfrac{3}{2}(\Delta
H)^{\prime}\sigma_{\phi}^{2}t\big{)}\Big{]}^{2}\right\\},$ (10)

where $\dot{\phi}_{c}=-V^{\prime}/3H$ is classical rolling while quantum
effects are captured by diffusion
$\sigma_{\phi}(t)^{2}=\frac{H^{2}}{4\pi^{2}}Ht$ due to the de-Sitter
temperature $H/2\pi$ [52, 51] and additional velocity $\dot{\phi}_{H}=3(\Delta
H)^{\prime}\sigma_{\phi}^{2}$ due to the Hubble volume effect. The terms
inside the parenthesis describe the motion of a peak.

“Hubble selection” starts to operate as the peak of the distribution starts to
_climb up_ the potential: $\sigma_{\phi}^{2}\simeq\frac{2}{3}M_{\rm Pl}^{2}.$
The width at this moment is always Planckian, reflecting quantum nature. The
field excursion by this moment is non-negligible
$\Delta\phi\sim\frac{4\pi^{2}M_{\rm
Pl}}{9}\frac{\Lambda_{\phi}^{4}}{H^{4}}\frac{M_{\rm Pl}}{M^{2}/g}$. Thus, for
a peak to climb, the field range $\delta\phi\sim M^{2}/g$ must accommodate
both the field excursion (stronger condition) and the width, yielding
respectively

$g\,\lesssim\,H\frac{H}{M_{\rm
Pl}}\frac{M^{2}}{\Lambda_{\phi}^{2}}\,\lesssim\,\frac{M^{2}}{M_{\rm Pl}}.$
(11)

We call this condition _global_ Quantum Beats Classical (QBC). It is stronger
than the usual local QBC, $V^{\prime}\lesssim H^{3}$ requiring $g\lesssim
H\frac{H^{2}}{\Lambda_{\phi}^{2}}\frac{M^{2}}{\Lambda_{\phi}^{2}}$, since
$\Lambda_{\phi}^{2}\lesssim HM_{\rm Pl}$ from Condition 1 later. It also has
different meanings as it involves the field range while the local one depends
only on the potential slope. It turns out to be equivalent to the QV condition
in [16] (App. B) that accounts for Hubble volume effects. If it is not
satisfied, $\rho$ makes an equilibrium at the bottom of a potential [53], with
the width $\sigma_{\phi}\sim H^{2}/m_{\phi}\sim
H^{2}M^{2}/\Lambda_{\phi}^{2}g\lesssim M_{\rm Pl}$ being consistently sub-
Planckian. Thus, we require the global QBC on a whole $\phi$ field range,
which reads Eq. (11).

The e-folding until this moment $\Delta N\simeq\frac{8\pi^{2}M_{\rm
Pl}^{2}}{3H^{2}}$ already saturates the upper bound for finite inflation,
$\frac{2\pi^{2}}{3}\frac{M_{\rm Pl}^{2}}{H^{2}}$, given by the de-Sitter
entropy [54, 55, 56]. Thus, Hubble selection needs eternal inflation, and the
universe eventually reaches a stationary state [57, 58, 59, 60]. Probability
distributions are to be defined among Hubble patches that have reached
reheating [61, 62, 63]. Then since the latest patches dominate the ensemble
with an exponentially larger number, only the stationary or equilibrium
distribution can be relevant to us.

$\rho(\phi,t)$ makes an equilibrium somewhere near the top of a potential,
which is the critical point $\phi_{*}$ in this work. The distribution can be
especially narrow, of the Planckian order (Eq. (19)), if the top is a critical
point where energy sharply drops after it. This is how Hubble selection self-
organizes the universe toward critical points [16].

The flatter the potential is (with stronger quantum effects), the closer to
$\phi_{*}$ the equilibrium is. The closest possible field distance is
Planckian, again reflecting quantum uncertainty principle. For even flatter
potentials, the equilibrium distribution rather spreads away from $\phi_{*}$.
It is expected as distributions will be flat in the limit $V^{\prime}\to 0$.
Discarding Hubble patches with $\phi>\phi_{*}$, i.e. $\rho(\phi=\phi_{*})=0$,
induces repulsive motion $\dot{\phi}_{b}\sim-H^{3}/(8\pi^{2}\sigma_{\phi})$
(App. B) so that the balance now reads
$\dot{\phi}_{c}+\dot{\phi}_{H}+\dot{\phi}_{b}\sim 0$. When the boundary
repulsion is stronger than classical rolling, an equilibrium is reached with

$\sigma_{\phi}\,\simeq\,\left(\frac{M_{\rm
Pl}^{2}H^{4}}{4\pi^{2}V^{\prime}}\right)^{1/3}\,\simeq\,\left(\frac{M_{\rm
Pl}^{2}M^{2}H^{4}}{4\pi^{2}g\Lambda_{\phi}^{4}}\right)^{1/3},$ (12)

which is the width in the Q2V regime [16]. $\sigma_{\phi}$ indeed increases as
$V$ flattens; nevertheless, the $v_{h}$ distribution can be arbitrarily
narrowed, as will be discussed. In this regime, one also expects $|\phi_{\rm
peak}-\phi_{*}|\sim\sigma_{\phi}$ from dimensional consideration. These
heuristic discussions on Q2V will be more carefully demonstrated with the
method of images in App. B.

A theory enters the Q2V regime when the balance width becomes larger than
$M_{\rm Pl}$:

$g\,\lesssim\,H\frac{H}{M_{\rm Pl}}\frac{H^{2}M^{2}}{\Lambda_{\phi}^{4}}.$
(13)

This is equivalent to $V^{\prime}\lesssim H^{3}H/M_{\rm Pl}$ [16], thus field-
range independent local balance near $\phi_{*}$. Q2V is typically stronger
than the global QBC in Eq. (11) and is not absolutely needed for Hubble
selection, but will be useful for efficient localization of $v_{h}$.

##  V The weak scale criticality

What is the final equilibrium distribution of $v_{h}$?

For $\mu_{h}^{2}>0$ ($\phi<0$), $V_{h}=V_{\Sigma}=0$ remain unchanged with
$\phi$, so the $\phi$ dynamics is quantum dominated. But it must not dominate
the inflation dynamics (Condition 1):
$\Lambda_{\phi}^{4}\,\lesssim\,H^{2}M_{\rm Pl}^{2}.$

As soon as $\mu_{h}^{2}\leq 0$ ($\phi>0$), the Higgs gets vev $v_{h}\geq 0$,
and $V_{h},V_{\Sigma}$ minima evolve with $\phi$:
$V_{h}(h=v_{h})=-\frac{\lambda_{h}}{4}v_{h}^{4}$, and, at leading orders in
$h$ with Eq. (7),

$\displaystyle V_{\Sigma(L\times R)}$ $\displaystyle\,\simeq\,$
$\displaystyle-a_{1}\Lambda_{\rm QCD}^{4}\frac{v_{h}^{2}}{v_{\rm EW}^{2}},$
(14) $\displaystyle V_{\Sigma(V)}$ $\displaystyle\,\simeq\,$
$\displaystyle-\Lambda_{\rm QCD}^{4}(a_{2}+a_{3}\frac{v_{h}}{v_{\rm EW}}),$
(15)

where $a_{1}\simeq 114,a_{2}\simeq 0.059,a_{3}\simeq 1.38$ for the benchmark
(Fig. 1). $\langle\sigma_{0}\rangle_{(L\times
R)}\simeq-\langle\sigma_{8}\rangle\simeq a_{4}v_{h}$ with $a_{4}\simeq 0.025$.
Note that these are decreasing with $\phi$, as opposed to
$V_{\phi}\,\simeq\,\frac{g\Lambda_{\phi}^{4}}{M^{2}}\phi$.

Above all, as $\phi$ slowly evolves, $h$ and $\Sigma$ must always follow their
highest minima for given $\phi$. Since potential shapes change slowly with
$\phi$, it only requires equilibrium widths to be small enough:
$H^{2}/m_{h,\Sigma}\lesssim\Lambda_{\rm QCD}$ with $m_{h}\sim v_{h}$ and
$m_{\Sigma}\sim\Lambda_{\rm QCD}$, where the Hubble-selected higher-energy
$SU(3)\times SU(3)$ vacuum is used. For $v_{h}^{*}\lesssim\Lambda_{\rm QCD}$
from Condition 3, we obtain (Condition 2): $H\,\lesssim\,v_{h}^{*}.$

Figure 2: (Top): total potential energy near the critical point as a function
of $\phi$, for the benchmark Eq. (18); dashed line for comparison. (Inset):
zoom-in near $\phi_{*}$. (Bottom): individual contribution from $V_{\phi}$,
$|V_{\Sigma}|$, and $|V_{h}|$.

For Hubble selection, the increasing $V_{\phi}$ needs to dominate the $\phi$
dynamics for $v_{h}\leq v_{h}^{*}$. Consider potential variations with respect
to $\phi$

$\displaystyle\frac{\delta V_{\phi}}{\delta\phi}$ $\displaystyle\,\sim\,$
$\displaystyle g\frac{\Lambda_{\phi}^{4}}{M^{2}},\quad\frac{\delta
V_{\Sigma(L\times
R)}}{\delta\phi}\,\sim\,-a_{1}\frac{g}{\lambda_{h}}\frac{\Lambda_{\rm
QCD}^{4}}{v_{\rm EW}^{2}},$ (16) $\displaystyle\frac{\delta
V_{h}}{\delta\phi}$ $\displaystyle\,\sim\,$
$\displaystyle-\frac{g}{2}v_{h}^{2},\quad\frac{\delta
V_{\Sigma(V)}}{\delta\phi}\,\sim\,-a_{3}\frac{g}{\lambda_{h}}\frac{\Lambda_{\rm
QCD}^{4}}{v_{h}v_{\rm EW}}.$

Unless $v_{h}$ is too small, the $\delta V_{\phi}/\delta\phi$ dominance
requires $\Lambda_{\phi}^{2}/M\gtrsim v_{h}^{*}$. Furthermore, the increase
$\Delta V_{\phi}$ after $v_{h}\geq v_{h}^{*}$ should never compensate the
$\Lambda_{\rm QCD}$-scale critical energy drop. As the decrease of
$V_{h}\propto-\phi^{2}$ dominates for $v_{h}\gtrsim\Lambda_{\rm QCD}$,
suppressing $\Delta V_{\phi}$ for $v_{h}^{*}\lesssim v_{h}\lesssim\Lambda_{\rm
QCD}$ yields $\Delta
V_{\phi}\,\simeq\,\frac{g\Lambda_{\phi}^{4}}{M^{2}}\cdot\frac{\lambda_{h}\Lambda_{\rm
QCD}^{2}}{g}\,\lesssim\,\Lambda_{\rm QCD}^{4}.$ Thus, $V_{\phi}$ cannot be too
flat nor steep relative to $V_{h,\Sigma}$ (Condition 3):

$v_{h}^{*}\lesssim\Lambda_{\phi}^{2}/M\lesssim\Lambda_{\rm QCD}.$ (17)

Notably, it requires a mild separation of $\Lambda_{\phi}$ and $M$ (see
later); if $M\sim\Lambda_{\phi}$ strictly, $M\lesssim\Lambda_{\rm QCD}$. As
$\Lambda_{\phi}$ and $g$ are only upper bounded, a large range of parameter
space is consistent.

We choose a benchmark ($v_{h}^{*}$ from LSM)

$\displaystyle v_{h}^{*}$ $\displaystyle\,\simeq\,$ $\displaystyle 20\,{\rm
MeV},\quad H\,=\,v_{h}^{*},\quad M\,=\,3\times 10^{-3}M_{\rm Pl},$
$\displaystyle\Lambda_{\phi}^{2}$ $\displaystyle\,=\,$ $\displaystyle
10^{-2}HM_{\rm Pl},\quad g\,=\,10^{-3}H^{2}/M_{\rm Pl},$ (18)

satisfying the global QBC (Eq. (11)), Q2V (Eq. (13)) marginally, and
Conditions 1 $\sim$ 3\. The total potential energy near $v_{h}^{*}$ is shown
in Fig. 2. As desired, the energy peaks sharply at the critical point, drops
significantly after that, and never comes back up; for much smaller or larger
$V_{\phi}^{\prime}$, it would not sharply peak at the critical point.

Figure 3: Volume-weighted equilibrium distribution of $v_{h}$ near
$v_{h}^{*}\simeq 20\,{\rm MeV}$. The width is $\sigma_{v_{h}}\simeq 0.5\,{\rm
MeV}$ for the benchmark Eq. (18); dashed line for comparison.

In equilibrium, most Hubble patches need to have $v_{h}$ near $v_{h}^{*}$ (or,
$\phi$ near $\phi_{*}=\lambda_{h}v_{h}^{*2}/g$). Following [16, 53], we solve
the FPV at large time, by approximating $V_{\phi}$ linearly and ignoring
Hubble patches with $v_{h}\,{>}\,v_{h}^{*}$. Results are shown in Fig. 3. The
width is estimated by

$\sigma_{\phi}\simeq\phi_{*}\left(\frac{3\phi_{*}^{2}}{2M_{\rm
Pl}^{2}}\right)^{-1/2}\sim M_{\rm Pl},$ (19)

for global QBC, which is indeed Planckian, but is broader for Q2V , as
discussed. In any case, the width of $v_{h}$ is suppressed by $g$ as

$\sigma_{v_{h}}\,\simeq\,\frac{g\sigma_{\phi}}{2\lambda_{h}v_{h}^{*}},$ (20)

thus can be arbitrarily narrow at the price of arbitrarily large $\phi$ range.
For the benchmark satisfying Q2V marginally, $\sigma_{v_{h}}\simeq 0.5\,{\rm
MeV}\ll v_{h}^{*}$ is narrow enough; but for 10-100 times larger $g$ with only
global QBC, $\sigma_{v_{h}}\,{\sim}\,v_{h}^{*}$.

Lastly, a small difference between $v_{h}^{*}$ and $v_{\rm EW}$ remains. After
inflation, the global QCD vacuum is reached, and $\phi$ slow-rolls to the
today’s $v_{h}=246$ GeV from the decreasing $V_{\rm tot}(\phi)$. $\phi$ could
be safely slow-rolling today or trapped by SM backreactions. Signals from
phase transition or time-dependent $v_{h}$ may be produced. In any case,
$v_{\rm EW}$ is generically close to $\Lambda_{\rm QCD}$.

##  VI Discussion

The minimal model discussed does not solve the hierarchy problem even though
$v_{h}$ can be finely Hubble selected. The separation $\Lambda_{\phi}\lesssim
M$ needed in Eq. (17) is not stable against quantum corrections, from Higgs
loop diagrams with the Higgs-relaxion interaction. Thus, it is another
manifestation of the (little) hierarchy problem. Assuming a fine-tuning of
order $\epsilon\equiv\Lambda_{\phi}/M<1$, the cutoff can be as high as
$M\lesssim\Lambda_{\rm QCD}/\epsilon^{2}$. Once with this fine-tuning, small
$g$ is especially versatile for Hubble selection. As it appears all and only
in $\phi$-dependent dynamics and is only upper bounded, small $g$ makes it all
possible that $\phi$ is quantum dominated while $h$ and $\Sigma$ instantly
follow their local minima. Only resulting hierarchy $f_{\phi}\gg M_{\rm Pl}$
needs to be generated consistently [29, 30, 31].

We have explored possible quantum critical points of the QCD $\Sigma$ sector,
resting on several assumptions. Most relevantly, $N_{f}>3$ and other responses
to the $v_{h}$ scanning need to be studied. The vacuum structure with
$N_{f}>3$ is likely to be more rich and complicated; in this sense, our work
is a conservative exploration. But these are not easy to improve just with
perturbative calculations within LSM, whose parameters are not dictated by
symmetries and inherently non-perturbative [42, 43, 44, 46, 45]. Eventually,
lattice calculation is needed as a first-principle method [19, 20, 21, 26, 22,
23, 24, 25] to verify and develop the scenario.

The near-criticality of the SM may have quantum cosmological origins. Using a
toy model utilizing QCD critical points, we have shown how such can be
realized in nature with Hubble selection. Having identified both potentials
and problems, we await for further developments.

###### Acknowledgements.

We thank Sang Hui Im, Hyung Do Kim, Choonkyu Lee, Ke-Pan Xie for valuable
conversations. We are supported by Grant Korea NRF-2019R1C1C1010050, and SJ
also by POSCO Science Fellowship.

##  Appendix A Quantum critical points in LSM

We present our initial study of the $N_{f}=3$ LSM parameter space at tree-
level. Quantum critical points are found in a wide range of parameter space,
at a range of $v_{h}^{*}$ values relevant to this work. More dedicated studies
shall be done with lattice calculation.

We fix LSM parameters to the benchmark in Eq. (8) and (6), while varying
$\lambda_{1}$ and $\mu^{2}>0$. It turns out that large uncertainties of the
last 3 observables in Table 1 leave some freedom in choosing $\lambda_{1}$ and
$\mu^{2}$. And surprisingly, these two free parameters are already good enough
to find reasonable SM points with critical points. We should focus on the
parameter space with $K>4.5$ so that coexisting vacua are present at ${\cal
H}=0$; this roughly requires $\mu^{2}\lesssim$ 10 times the benchmark value.

Fig. 4 shows the numerical results of the critical point $v_{h}^{*}$ (upper
panel) and $\chi^{2}$/dof (lower) as a function of $\lambda_{1}$ and
$\mu^{2}$; the scale factors are relative to the benchmark values. We found
that $v_{h}^{*}$ is mostly sensitive only to $\mu^{2}$, while $\chi^{2}$ only
to $\lambda_{1}$. As alluded, $v_{h}^{*}={\cal O}(1-100)\,{\rm MeV}$ can be
obtained with the $\mu^{2}$ scale factor = 0.3 $\sim$ 3, and $\chi^{2}$/dof
can be minimized to $3\sim 4$ with the $\lambda_{1}$ scale factor = $0.5\sim
1.5$. The energy contrast at the critical point does not vary much,
70$\sim$120 MeV close to $\Lambda_{\rm QCD}$, in the parameter space shown in
the figure.

Figure 4: (Top): LSM quantum critical point $v_{h}^{*}$ with $\mu^{2}$. Other
parameters are fixed to benchmark values. Scale factors are relative to the
benchmark. (Bottom): $\chi^{2}$/dof with $\lambda_{1}$ using the same data as
in Table 1 of the main text.

##  Appendix B Quantum regimes

Quantum regimes for Hubble selection are categorized into two [16]: the global
QBC (or QV) and Q2V . What distinguishes them? We discuss this by first
describing the scaling behaviors of QBC and Q2V ; second, showing the
equivalence between the global QBC and QV; and finally demonstrating the
boundary effects in Q2V . Refer to [16, 53, 7] and references therein for
other details on FPV solutions.

###  B.1 Scaling

Consider a linear potential

$V\,=\,V^{\prime}\phi\qquad\textrm{for}\,|\phi|\leq\phi_{0},$ (21)

where the field range $\phi_{0}$ does not have to be the full range, and the
slope is constant. QV and Q2V conditions are expressed as $\alpha\beta\gtrsim
1$ and $\alpha^{2}\beta\gtrsim 1$, respectively, in terms of [16]

$\alpha\,=\,\frac{3}{4\pi^{2}}\frac{H^{4}}{V^{\prime}\phi_{0}},\qquad\beta\,=\,\frac{3}{2}\frac{\phi_{0}^{2}}{M_{\rm
Pl}^{2}}.$ (22)

First, scale the field range by $a$: $\phi_{0}\to a\phi_{0}$. Under this,

$\alpha\,\propto\,a^{-1},\qquad\beta\,\propto\,a^{2}.$ (23)

Thus, the QV condition is not scale invariant, $\alpha\beta\propto a$,
reflecting that it is equivalent to the global QBC (see below) which was
derived by considering the necessary field range for Hubble selection. On the
other hand, the Q2V condition is invariant $\alpha^{2}\beta\propto a^{0}$, as
it measures only the local balance near a critical point (the upper boundary),
as derived in the main text.

But physical peak properties at equilibrium are also expressed in terms of
$\alpha$ and $\beta$. The physical width and the peak location from the
boundary must be scale invariant, which are indeed satisfied as

$\sigma_{\phi}\,\sim\,\phi_{0}\beta^{-1/2}\,\propto\,a^{0},\qquad\sigma_{\phi}\,\sim\,\phi_{0}(\alpha/\beta)^{1/3}\,\propto\,a^{0}$
(24)

for QV and Q2V [16], and

$\frac{\phi_{0}}{2\alpha\beta}\,\sim\,\frac{4\pi^{2}}{9}\frac{V^{\prime}M_{\rm
Pl}^{2}}{H^{4}}\,\propto\,a^{0}$ (25)

for QV. Note that the last expression is the minimum field excursion
$\Delta\phi$ for a peak to start climbing up. This led to the global QBC in
Eq. (11).

The scaling also dictates how the peak location and width change with, e.g.
the potential slope (the $V^{\prime}$ factor), in the global QBC regime. It
can be convenient when numerically solving for equilibrium distributions since
they can be localized in a very narrow range, away from the boundary.

###  B.2 Consistency

Restoring the factors of our model, the full field range $\phi_{0}=M^{2}/g$
and $V^{\prime}=\Lambda_{\phi}^{4}/f_{\phi}$ with $f_{\phi}=M^{2}/g$, we have

$\alpha\,=\,\frac{3}{4\pi^{2}}\frac{H^{4}}{\Lambda_{\phi}^{4}},\qquad\beta\,=\,\frac{3}{2}\frac{M^{4}}{g^{2}M_{\rm
Pl}^{2}}.$ (26)

Thus, QV and Q2V read, respectively,

$\displaystyle g\lesssim H\frac{H}{M_{\rm
Pl}}\frac{M^{2}}{\Lambda_{\phi}^{2}},\qquad g\lesssim H\frac{H}{M_{\rm
Pl}}\frac{H^{2}M^{2}}{\Lambda_{\phi}^{4}}.$ (27)

They agree with our global QBC in Eq. (11) (not exactly with the local one)
and the derivation in Eq. (13), respectively.

Figure 5: Effects of the boundary at $\phi_{0}=0$ on the volume-weighted
distribution $\rho(\phi)$. For several choices of $\sigma_{\phi}$ with fixed
$\Delta\phi=100$ in the solution Eq. (28) obtained by the method of images. As
the boundary begins to matter (Q2V ), the distribution becomes pushed away.

###  B.3 Boundary effects

Lastly, we demonstrate the boundary effects on Q2V regime discussed
heuristically in the main text. The absorbing boundary condition at $\phi_{0}$
is relevant to critical points. It means that any Hubble patches with
$\phi>\phi_{0}$ are out of Hubble selection, hence discarded, as the potential
energy sharply peaks at $\phi_{0}$.

The absorbing boundary condition can be imposed by the method of images.
Putting two charges at $\phi_{0}\mp\Delta\phi$, and ignoring Hubble expansion
for simplicity, the solution is of the form [63]

$\rho(\phi)\,\propto\,e^{-\frac{[\phi-(\phi_{0}-\Delta\phi)]^{2}}{2\sigma_{\phi}^{2}}}-e^{-\frac{[\phi-(\phi_{0}+\Delta\phi)]^{2}}{2\sigma_{\phi}^{2}}},$
(28)

where $\phi<\phi_{0}$ is the physical region. It describes that, as
$\Delta\phi$ and $\sigma_{\phi}$ become comparable, or equivalently as the
significant portion of the would-be distribution passes the boundary, the
boundary becomes relevant, and the final distribution is displaced away from
it. Fig. 5 demonstrates this behavior as a function of $\sigma_{\phi}$ for
fixed $\Delta\phi$.

Physically, why/how does a boundary affect a peak, located away from it? The
boundary condition generates the asymmetric absorbing probability, skewing the
distribution, which otherwise would be symmetrically quantum diffused. The
peak location $\phi_{\rm peak}$ is at the extremum satisfying

$e^{\frac{2(\phi_{0}-\phi_{\rm
peak})\Delta\phi}{\sigma_{\phi}^{2}}}\,=\,\frac{(\phi_{0}-\phi_{\rm
peak})+\Delta\phi}{(\phi_{0}-\phi_{\rm peak})-\Delta\phi}.$ (29)

When $\sigma_{\phi}\gtrsim\Delta\phi$, the boundary is most important, and

$\phi_{0}-\phi_{\rm peak}\,\simeq\,\sigma_{\phi}.$ (30)

This must be true as $\sigma_{\phi}$ is the only dimensionful parameter in
this limit. This effectively means repulsive motion

$\dot{\phi}_{b}\,\sim\,-\frac{d\sigma_{\phi}}{dt}\,=\,-\frac{H^{3}}{8\pi^{2}\sigma_{\phi}},$
(31)

justifying the formula used in the main text and the peak location $|\phi_{\rm
peak}-\phi_{0}|\sim\sigma_{\phi}$ in the Q2V regime.

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