# Fine-Grained Multi-View Hand Reconstruction Using Inverse Rendering

Qijun Gan, Wentong Li, Jinwei Ren, Jianke Zhu Corresponding author.

###### Abstract

Reconstructing high-fidelity hand models with intricate textures plays a
crucial role in enhancing human-object interaction and advancing real-world
applications. Despite the state-of-the-art methods excelling in texture
generation and image rendering, they often face challenges in accurately
capturing geometric details. Learning-based approaches usually offer better
robustness and faster inference, which tend to produce smoother results and
require substantial amounts of training data. To address these issues, we
present a novel fine-grained multi-view hand mesh reconstruction method that
leverages inverse rendering to restore hand poses and intricate details.
Firstly, our approach predicts a parametric hand mesh model through Graph
Convolutional Networks (GCN) based method from multi-view images. We further
introduce a novel Hand Albedo and Mesh (HAM) optimization module to refine
both the hand mesh and textures, which is capable of preserving the mesh
topology. In addition, we suggest an effective mesh-based neural rendering
scheme to simultaneously generate photo-realistic image and optimize mesh
geometry by fusing the pre-trained rendering network with vertex features. We
conduct the comprehensive experiments on InterHand2.6M, DeepHandMesh and
dataset collected by ourself, whose promising results show that our proposed
approach outperforms the state-of-the-art methods on both reconstruction
accuracy and rendering quality. Code and dataset are publicly available at
https://github.com/agnJason/FMHR.

## Introduction

3D human reconstruction has attracted considerable research attentions (Saito
et al. 2019; Peng et al. 2021a; Weng et al. 2022; Chen et al. 2021; Noguchi et
al. 2021; Bhatnagar et al. 2020) in recent years. While there have been
promising advancements in reconstructing the human body and face (Lei et al.
2023; Grassal et al. 2022; Peng et al. 2021b; Xiu et al. 2022), it still
remains a formidable challenge to achieve highly accurate hand reconstruction
due to the inherent complexity of joint variations. By taking consideration of
the distinctive nature for hands, it is essential to investigate the hand
geometry and rendering for obtaining the realistic and fine-grained
representations.

Conventional model-based methods, such as MANO (Romero, Tzionas, and Black
2017) and Nimble (Li et al. 2022b), often rely on smoothing meshes and texture
maps for hand representation. Nevertheless, it typically requires costly
scanning data and artistic expertise in order to achieve intricate and
personalized hand meshes with texture maps. Moreover, the complex nature of
hand movements and challenges posed by occlusions hinder the faithful
restoration of hand mesh. Model-free approaches like LISA (Corona et al. 2022)
aim to address these challenges by reconstructing coherent hands from image
sequences, while HandAvatar (Chen, Wang, and Shum 2023) focuses on
reconstructing and rendering hands in arbitrary poses by disentangling
reflectance and lighting. Nonetheless, the resulting mesh often exhibits
smoothness due to large pose variations across different frames.

Figure 1: Our proposed approach focuses on reconstructing hands from multi-
view images, allowing for the generation of precise poses, geometry, and
photo-realistic rendering.

Neural rendering-based methods, such as NeRF (Mildenhall et al. 2021) and NeuS
(Wang et al. 2021; Fu et al. 2022), have been widely used in synthesizing
static objects from multi-view images. As for the dynamic objects like hands,
these methods have the difficulties in obtaining a fixed topological structure
through implicit surface representation. Despite their impressive rendering
results, NeuS (Wang et al. 2021) requires a large amount of training time due
to sampling along the ray. Meanwhile, NeuralBody (Peng et al. 2021b) attaches
latent codes to SMPL (Loper et al. 2015) vertices, which enables to diffuse
into space through sparse convolution. However, this may lead to some
artifacts on the predicted mesh. To address these limitations, HandNeRF (Guo
et al. 2023) proposes a pose-driven deformation field to render photo-
realistic hands from various views and poses, while it does not explicitly
provide a hand mesh for animation and real-time rendering.

To overcome the above challenges, we propose an effective coarse-to-fine
approach to hand mesh reconstruction from multi-view images. By synergizing
the benefits of parametric models and mesh-based rendering, our method
achieves high-fidelity reconstruction results while maintaining fast training
time with only 1.5 minutes. By incorporating multi-layer perceptrons into
Graph Convolutional Networks (GCN) (Kipf and Welling 2017), we can
simultaneously recover hand mesh and MANO parameters. Unlike learning-based
methods that demand extensive training data, we introduce a Hand Albedo and
Mesh (HAM) optimization module by leveraging inverse rendering to enhance
level of details. The resulting fine-grained mesh preserves the flexibility
provided by the parametric model MANO (Romero, Tzionas, and Black 2017), which
enables to repose it into novel configurations. To further enhance rendering
quality and refine the mesh, we suggest a mesh-based neural rendering scheme
by fusing a pre-trained rendering network with vertex features. Our proposed
method is evaluated on the InterHand2.6M, DeepHandMesh and dataset collected
by ourself. The promising experimental results demonstrate its efficacy in
reconstructing high-fidelity hands from multi-view images, as illustrated in
Fig. 1.

Our main contributions are summarized as below.

  * •

We propose a coarse-to-fine approach to accurately recover the fine-grained
hand mesh model from multi-view images by taking advantage of inverse
rendering.

  * •

A novel HAM optimization module is presented to refine the over-smoothing
results of parametric hand models.

  * •

We devise an effective mesh-based neural rendering scheme to simultaneously
generate photo-realistic image and optimize mesh geometry by fusing the pre-
trained rendering network with vertex features.

## Related Work

Figure 2: Overview of our coarse-to-fine framework. Given a set of calibrated
images, we initialize MANO parameters and refine the mesh using our proposed
HAM module and inverse rendering to achieve geometric details. By jointly
optimizing the mesh using a model-based neural rendering, a fine-grained mesh
can be obtained along with its hyper-realistic rendered images.

### Model-Based Hand Reconstruction

Parametric models have been widely used in representing objects with the fixed
typologies, such as the human body (Loper et al. 2015; Pavlakos et al. 2019;
Osman, Bolkart, and Black 2020; Chen et al. 2022), face (Li et al. 2017; Hong
et al. 2022), hands (Romero, Tzionas, and Black 2017; Li et al. 2022b), and
animals (Zuffi et al. 2017). These models enable the transformation of mesh
geometry by adjusting model parameters corresponding to pose and shape
variations. Moreover, hand pose can be effectively estimated by images (Li,
Gao, and Sang 2021) or point clouds (Cheng et al. 2022; Ren et al. 2023). In
hand reconstruction, the parametric models like MANO (Romero, Tzionas, and
Black 2017), NIMBLE (Li et al. 2022b), have been used to recover the hand in
the input image (Boukhayma, de Bem, and Torr 2019; Hasson et al. 2019; Kong et
al. 2022; Cao et al. 2021; Doosti et al. 2020; Hasson et al. 2020). In (Fan et
al. 2021; Ren, Zhu, and Zhang 2023; Kim, Kim, and Baek 2021; Zhang et al.
2021; Li et al. 2022a), the parametric models are employed to reconstruct two
hands, where hand interactions and gestures could be simulated. Recently,
(Chen et al. 2023) employ the HandTrackNet to track the variations of MANO
parameters, which is utilized for hand-object interactions. While parametric
methods are able to recover hand poses and shapes, the resulting meshes lack
the capability to represent geometric textures.

### Model-Free Hand Reconstruction

Parametric models (Romero, Tzionas, and Black 2017) are valuable for
incorporating the prior knowledge of pose and shape, while their
representation power is constrained by imposing the template shape and
details. To overcome this limitation, various approaches have been explored.
Instead of directly regressing the MANO parameters, I2L-MeshNet (Moon and Lee
2020) predicts a 1D heatmap for each vertex coordinate, and (Ge et al. 2019;
Kulon et al. 2020) employ GCN-based methods to recover hand meshes. On the
other hand, DeepHandMesh (Moon, Shiratori, and Lee 2020) make use of an
encoder-decoder framework to generate highly detailed hand meshes. To achieve
photo-realistic hand rendering, HARP (Karunratanakul et al. 2023) suggest an
optimization-based approach to recover both normal and albedo maps. Recently,
(Luan et al. 2023) introduce a frequency decomposition loss to capture
personalized hand from a single image, which address the problem of data
scarcity through multi-view reshaping. HandAvatar (Chen, Wang, and Shum 2023)
yields occupancy and illumination field to generate free-pose photo-realistic
hand avatar.

### Neural Rendering-Based Reconstruction

In past few years, rapid progress in 3D modeling and image synthesis has been
obtained through neural implicit representations (Mescheder et al. 2019;
Mildenhall et al. 2021). In contrast to the classical discrete representations
like meshes, point clouds, and voxels, neural implicit representations
leverage neural networks to model scenes, which offer continuous results with
higher fidelity and flexibility. Among the various methods, Neural Radiance
Fields (NeRF) (Mildenhall et al. 2021) gains great popularity and demonstrates
impressive performance across various tasks. Nevertheless, the traditional
NeRF-based methods face challenges in dealing with objects having temporal
changes (Fu et al. 2022; Wang et al. 2021).

Neural Body (Peng et al. 2021b) combines NeRF with the parametric SMPL body
model (Loper et al. 2015), which is able to recover dynamic objects. Moreover,
(Liu et al. 2021) leverage NeRF to learn pose-related geometric deformations
and textures in canonical space from multi-view videos. Recently, LISA (Corona
et al. 2022) fuses volumetric rendering with hand geometric priors to capture
animatable hand appearances. HandNeRF (Guo et al. 2023) represents the
interactive hands by deformable neural radiance fields to generate photo-
realistic images.

## Method

Our objective is to achieve fine-grained 3D hand mesh reconstruction and
synthesize photo-realistic novel views. To tackle this challenge, our method
consists of three key steps. Firstly, we estimate an initial coarse hand mesh
and the parameters of MANO model (Romero, Tzionas, and Black 2017), which are
recovered from multi-view images through incorporating multi-layer perceptrons
(MLPs) into GCN. Secondly, an HAM optimization module is introduced to restore
a fine-grained mesh with the albedo map and surface details by leveraging the
power of inverse rendering. Finally, we suggest a mesh-based neural rendering
scheme to efficiently generate photo-realistic images and refine the mesh
through joint optimization. Fig. 2 illustrates the entire pipeline of our
presented method.

### Coarse Initialization

In hand reconstruction, it is crucial to accurately predict the 3D pose and
shape due to the high flexibility of the hand. Since generating 3D poses is
typically challenging in the wild cases, some approaches like HandAvatar
(Chen, Wang, and Shum 2023) employ the annotated hand poses and shapes as
initialization. Instead, we aim to estimate the consistent 3D hand poses from
multi-view images.

Given a set of images $\mathcal{I}=\\{I^{1},\cdots,I^{n}\\}$ captured by the
calibrated cameras and their corresponding 2D joints
$\mathcal{J}_{2D}=\\{J_{2D}^{1},\cdots,J_{2D}^{n}\\}$, 3D hand joints
$J_{3D}\in\mathbb{R}^{B\times 3}$ in world space with $B$ per-bone parts can
be estimated. To address the limitations of representing joint Euler angles
directly with 3D joints information, we introduce a GCN-based network
$\mathcal{G}$ to recover the MANO model as in (Choi, Moon, and Lee 2020).

$\mathcal{M}(\hat{\theta},\hat{\beta})=\mathcal{G}(J_{3D}),$ (1)

where $\hat{\theta}\in\mathbb{R}^{B\times 3}$ and
$\hat{\beta}\in\mathbb{R}^{10}$ represent pose and shape parameters of MANO
model, respectively. The network $\mathcal{G}$ is designed as a four-layer GCN
with a MANO head. The MANO head consists of MLPs to obtain the corresponding
MANO parameters via the features of the first three GCN layers, as shown in
Fig. 3.

Given $n$-view images $I^{n}$ along with their corresponding camera parameters
$\pi^{n}$ and the positions of the hand joints $J_{2D}^{n}$ in the images,
$J_{3D}$ can be obtained by minimizing the reprojection error across different
views. The camera parameters include the intrinsic matrix $K$ and extrinsic
matrix $T$. In the $i$-th view, $J_{2D}^{i}$ can be calculated by
$J_{2D}^{i}=\pi^{i}(J_{3D})$. Since it is difficult to circumvent the issue of
imperfect estimation of $J_{2D}$, we recover $J_{3D}$ by minimizing the
following objective

$\mathcal{L}_{joints}=\sum_{i=1}^{n}\frac{1}{n}||J_{2D}^{i}-\pi^{i}(\hat{J}_{3D})||^{2}.$
(2)

Once the 3D joints $\hat{J}_{3D}$ with multi-view consistency is obtained, the
GCN-based network $\mathcal{G}$ can be used to recover the MANO parameters.
$\mathcal{G}$ is trained using the annotated datasets by minimizing the
following energy function

$\mathcal{L}_{\mathcal{G}}=\mathcal{L}_{v}+\mathcal{L}_{n}+\mathcal{L}_{j}+\mathcal{L}_{MANO}.$
(3)

Specifically, $\mathcal{L}_{v}$ measures the $L_{1}$ distance between the
predicted vertices $\hat{V}$ and annotated MANO vertices $V$ as below

$\mathcal{L}_{v}=\sum|\hat{V}-V|.$ (4)

$\mathcal{L}_{n}$ ensures the alignment of mesh normals $\mathbf{n}$ with
MANO, which is defined as

$\mathcal{L}_{n}=\sum|\hat{\mathbf{n}}-\mathbf{n}|.$ (5)

$\mathcal{L}_{j}$ constrains the discrepancy between the generated MANO joints
$\hat{J_{3D}}$ and the input joints $J_{3D}$ as follows

$\mathcal{L}_{j}=\sum|\hat{J_{3D}}-J_{3D}|.$ (6)

Additionally, $\mathcal{L}_{MANO}$ is used to supervise the generation of MANO
parameters, which is formulated as below

$\mathcal{L}_{MANO}=\sum|(\hat{\theta},\hat{\beta})-(\theta,\beta))|.$ (7)

While the GCN-based network $\mathcal{G}$ excels at accurately recovering the
pose by utilizing 3D joints as input, it lacks information about hand fatness
or leanness. To overcome this limitation, we incorporate segmentation map
obtained by promptable SAM (Kirillov et al. 2023) in order to optimize the
shape parameters $\hat{\beta}$. This extra optimization step further enhances
the accuracy of the MANO parameters.

Figure 3: Our GCN-based network. The four-layer GCN progressively doubles the
number of vertices and MANO head outputs the corresponding MANO parameters.

### Mesh Refinement

The MANO model generated by $\mathcal{G}$ represents a smoothing mesh without
geometric textures. Inspired by the Shape from Shading (SFS) algorithm (Horn
1970) and PatchShading (Lin et al. 2024; Lin, Zhu, and Zhang 2022), we
accurately capture the folds and textures on the coarse hand mesh. Assuming
that the diffuse reflection of human skin follows Lambertian reflectance, we
propose a Hand Albedo and Mesh (HAM) optimization module to refine the coarse
mesh, which takes advantage of both SFS and inverse rendering. To ensure the
coherence in the mesh optimization process, we incorporate four effective
regularization terms described in the following.

We utilize $\mathcal{M}_{r}$ as the initial model that is obtained from the
MANO subdivision with 49,281 vertices and 98,432 faces. The albedo values
$\rho$ are assigned to each vertex and an inverse renderer $\zeta$ is employed
to generate albedo map $\mathbf{C}_{\rho}$ and normal map $\mathbf{C}_{N}$.
The inverse renderer $\zeta$ is represented as follows

$\mathbf{C}=\zeta(c,V,F,\pi),$ (8)

where $V$ and $F$ denote the vertices and faces of mesh $\mathcal{M}_{r}$,
respectively. The feature at each vertex, such as normal and albedo, is
represented by $c$. The rendering map $\mathbf{C}$ is obtained by the inverse
renderer $\zeta$. With the calibrated images $I^{i}$, the HAM optimizes the
vertex positions $V$ and vertex albedo $\rho$ by minimizing the texture loss
$\mathcal{L}_{t}$ in the following equation

$\mathcal{L}_{t}=\sum\limits_{n}|B(\pi^{i})-I^{i}|,$ (9)

where $B(\pi^{i})$ represents the rendered image under camera parameters
$\pi^{i}$. It is obtained by computing the illumination matrix $\mathbf{G}$,
the normal map $\mathbf{C}_{N}$ and the albedo map $\mathbf{C}_{\rho}$ as
below

$B(\pi^{i})=\mathbf{C}_{\rho}\cdot SH(\mathbf{G},\mathbf{C}_{N}),$ (10)

where $SH(\cdot)$ represents sphere harmonic (SH) function with the third
order. Considering the variations in lighting across different views, we
optimize the lighting matrix $\mathbf{G}$ during the process. We compute the
texture loss between $B(\pi^{i})$ and the original image $I^{i}$ as the loss
$\mathcal{L}_{t}$.

To enhance the efficacy of extracting geometric information from shadows, the
regularization term $\mathcal{L}_{\rho}$ is introduced. This term is
especially designed to align with the observation that human skin tends to
exhibit the consistent color. $\mathcal{L}_{\rho}$ is defined as follows

$\mathcal{L}_{\rho}=\lambda_{1}L*\rho,$ (11)

where $L$ denotes the Laplacian matrix. $\lambda$ is the balanced weight. To
ensure the optimized vertices to be smoothing, we introduce the following term
for conforming to the geometric characteristics of the hand

$\displaystyle\mathcal{L}_{r}$
$\displaystyle=\lambda_{2}L*V+\lambda_{3}\mathcal{L}_{mask}+\lambda_{4}\mathcal{L}_{e}+\lambda_{5}\mathcal{L}_{d}$
(12) $\displaystyle=\lambda_{2}L*V+\lambda_{3}\sum|\hat{M}-M_{MANO}|$
$\displaystyle+\lambda_{4}\sum\limits_{i,j}||E_{ij}||^{2}+\lambda_{5}\sum\limits_{i}||\triangle{V_{i}}||^{2}.$

Specifically, $\mathcal{L}_{mask}$ represents the $L_{1}$ loss between the
rendered mask and the original MANO mask. $E_{ij}$ is obtained by calculating
the Euclidean distance $||\cdot||^{2}$ between adjacent vertices $V_{i}$ and
$V_{j}$ on the mesh edges. $\mathcal{L}_{e}$ is employed to restrict the
length of $E_{ij}$. Let $\triangle{V_{i}}$ represent the displacement distance
of vertices $V_{i}$. $\mathcal{L}_{d}$ is utilized to ensure that the
optimized hand remains close to the MANO model. Each term is assigned with a
constant coefficient denoted by $\lambda$. The overall loss function
$\mathcal{L}_{total}$ is defined as below

$\mathcal{L}_{total}=\mathcal{L}_{t}+\mathcal{L}_{\rho}+\mathcal{L}_{r}.$ (13)

The HAM module facilitates the refinement of the hand model by jointly
optimizing both the mesh vertices $\hat{V}$ and albedo $\hat{\rho}$. This
results in a high-quality output $\mathcal{M}_{f}$ that retains the consistent
topology.

### Texture Optimization and Joint Refinement

While having achieved geometric alignment of the fine mesh, there are still
some limitations in image rendering and mesh refinement. To alleviate this
issue, we adopt a mesh-based neural rendering method. Leveraging the preserved
topology of our refined hands, the neural rendering model can be pre-trained
with diverse hand data, thereby reducing the training time for each individual
hand. We propose an efficient strategy that involves the pre-training a neural
rendering network using a large amount of hand data, followed by fine-tuning
on individual data, and ultimately conducting joint optimization with the mesh
to achieve hyper-realistic rendering and accurate geometry.

Having acquired a refined mesh denoted as $\mathcal{M}_{f}(\hat{V},F)$ along
with its corresponding vertex albedo $\hat{\rho}$, we aim to enhance the
accuracy of the rendered texture. To this end, we design a neural renderer
$\mathcal{T}$ defined as below

$\mathbf{t}(\mathbf{r})=\mathcal{T}(\mathbf{x},\mathbf{f},\rho,\mathbf{n}),$
(14)

where the output pixel value $\mathbf{t}$ is determined by the direction of
the light ray $\mathbf{r}$. At a given pixel position, the texture field $t$
is adjusted with respect to the position $\mathbf{x}$, feature vector
$\mathbf{f}$, albedo $\rho$, and normal $\mathbf{n}$, which are obtained by
inverse rendering $\zeta$. For the neural renderer, we employ the MLPs with
four layers, each of which consists of 256 dimensions. The length of vertex
features $\mathbf{f}$ is set to 20. The training loss is defined as follows

$\mathcal{L}_{tex}=\sum|\hat{\mathbf{t}}_{i}-I_{i}|.$ (15)

Pre-training. During the pre-training, we conduct sampling from the subdivided
MANO mesh $\mathcal{M}_{r}$ instead of utilizing $\mathcal{M}_{f}$. Through
the pre-training, we obtain a well-trained neural renderer that unifies the
vertices features $\mathbf{f}$ independent from position $\mathbf{x}$, albedo
$\rho$ and normal $\mathbf{n}$.

Fine-tuning. Given the diversity of data and potential overfitting during
training, the pre-trained neural rendering model often yields average results,
which is unable to capture intricate texture details. To synthesize high-
fidelity hand images and achieve promising rendering quality, it is necessary
to fine-tune the model for each specific dataset. The advantage of the pre-
trained model lies in its capacity to accelerate neural rendering, allowing
for efficient completion of fine-tuning within minutes. During the fine-tuning
process, we employ the vertices of mesh $\mathcal{M}_{f}$ obtained from Mesh
Refinement along with its corresponding albedo for training. Both the vertex
features $\mathbf{f}$ and the neural renderer $\mathcal{T}$ are set to be
learnable.

Joint Optimization. After achieving fine geometric structures and realistic
image rendering, it becomes necessary to perform joint optimization on both
mesh and texture to further enhance the overall quality. Drawing inspiration
from (Walker et al. 2023), we adopt a geometry-based shader
$\tilde{\mathbf{t}}$ with the detached output $\hat{\mathbf{t}}$ of
$\mathbf{t}(\mathbf{r})$, which is illustrated as follows

$\tilde{\mathbf{t}}(\mathbf{r})=\tilde{\mathcal{T}}(\hat{\mathbf{t}},f,\rho,\mathbf{n}).$
(16)

To fine-tune the geometry and train the geometry-based shader, the loss
functions $\mathcal{L}_{e}$ and $\mathcal{L}_{d}$ depicted in Eq. 12 are
employed to ensure surface smoothness. Moreover, the vertices are designated
to be learnable. The shader loss is formulated as follows

$\mathcal{L}_{geo}=\sum|\tilde{\mathbf{t}}_{i}-I_{i}|+\gamma_{1}\mathcal{L}_{e}+\gamma_{2}\mathcal{L}_{d}.$
(17)

where $\gamma_{1}$ and $\gamma_{2}$ are weights for ${L}_{e}$ and ${L}_{d}$,
respectively.

## Experiments

### Datasets

InterHand2.6M. InterHand2.6M (Moon et al. 2020) is a large-scale dataset,
comprising images of size $512\times 334$ pixels and associated MANO
annotations. The dataset contains multi-view temporal sequences of single hand
as well as interacting hands. Our experiments primarily focus on the 5 FPS
version of the InterHand2.6M dataset. In all experiments, both the GCN-based
network and pre-trained neural rendering network are trained using the data
from the training set.

DeepHandMesh. The DeepHandMesh dataset (Moon, Shiratori, and Lee 2020)
consists of images captured from five different views with the same size as
the images in InterHand2.6M dataset. Additionally, this dataset provides the
corresponding 3D hand scans, which enables the validation of the mesh
reconstruction quality against 3D ground truth.

Our Dataset. Due to the restricted resolution of the above datasets, the
attainment of higher geometric precision and color fidelity requires high-
resolution hand images. To address this issue, we collect a dataset using 16
calibrated cameras, which captures the synchronized images at a resolution of
$1280\times 1024$ pixels at 15 FPS. The cameras are distributed mainly in a
semi-circle and placed at various heights to ensure a comprehensive visual
coverage.

Figure 4: Qualitative performance comparison. We show the rendering results of single hand (first two rows) and dual hands (last four rows), which are optimized and trained from 10-view images. The hands rendered with pure white color represent the shading in order to highlight the level of mesh detail. Method | test/Capture0-Single | test/Capture1-Single | test/Capture0-Interacting
---|---|---|---
LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$ | LPIPS $\downarrow$ | PSNR $\uparrow$ | SSIM $\uparrow$
Ani-NeRF | 0.0621 | 31.78 | 0.968 | - | - | - | 0.0798 | 29.35 | 0.949
HandAvatar | 0.0504 | 33.01 | 0.933 | 0.0425 | 32.12 | 0.938 | - | - | -
HandNerf | 0.0375 | 32.70 | 0.974 | - | - | - | 0.0367 | 30.62 | 0.958
NeuS | 0.0197 | 35.34 | 0.986 | 0.0211 | 34.98 | 0.984 | 0.0743 | 31.46 | 0.927
Ours | 0.0069 | 37.26 | 0.992 | 0.0098 | 37.91 | 0.991 | 0.0126 | 37.08 | 0.986
Table 1: Rendering quality comparisons among our method and prior arts on the
InterHand2.6M dataset.

### Implementation Details and Metrics

Optimization. To achieve the fine-grained mesh, we adopt a subdivision
technique inspired by (Chen, Wang, and Shum 2023), which expands the original
778 vertices in the MANO model to a total of 49,281 vertices. During the
optimization process, we utilize the Adam optimizer (Kingma and Ba 2014) with
the balanced weights of $\lambda_{1}=20$, $\lambda_{2}=40$, $\lambda_{3}=20$,
$\lambda_{4}=100$, and $\lambda_{5}=2$ to jointly optimize the vertices,
vertex albedo, and lighting coefficients over 100 iterations. This
optimization process takes approximately 20 seconds. Additionally, the neural
renderer is pre-trained on InterHand2.6M dataset for 20 epochs. Subsequently,
for fine-tuning and joint optimization, each process requires 100 epochs of
training with $\gamma_{1}=100$ and $\gamma_{2}=2$, respectively. Notably, the
entire optimization pipeline is computationally efficient, which takes
approximately 90 seconds on a single NVIDIA 3090Ti GPU.

Evaluation Metrics. We evaluate the accuracy of the reconstructed 3D surface
by computing the average point-to-surface Euclidean distance (P2S) between the
vertices of the recovered surface and their corresponding ground truth, which
are measured in millimetre. Due to the disparate size ranging between the
generated hand mesh and the 3D scans in the DeepHandMesh data, the Chamfer
distance metric is considered unsuitable. In line with prior research in
neural rendering, peak signal-to-noise ratio (PSNR), structural similarity
index (SSIM), and learned perceptual image patch similarity (LPIPS) are
adopted as evaluation metrics to gauge the fidelity of the synthesized image
results.

### Results on Hand Reconstruction

P2S $\downarrow$ |  Two occlusions |  Thumb tuck |  Shake- speare |  Total Average
---|---|---|---|---
DHM | 3.467 | 2.624 | 1.773 | 2.937
w/o.HAM | 1.637 | 2.317 | 1.668 | 1.873
Ours | 1.532 | 1.281 | 1.271 | 1.456
Table 2: Mesh reconstruction quality comparison among ours and DHM (Moon,
Shiratori, and Lee 2020) on DeepHandMesh dataset with 5 views.

Results of InterHand2.6M. To evaluate the quality of novel view synthesis, we
conduct experiments on the InterHand2.6M dataset using 10 views and evaluate
on the rest views. Table 1 summarizes the evaluation results on rendering
quality. Fig. 4 illustrates the visual comparisons against HandNeRF (Guo et
al. 2023), HandAvatar (Chen, Wang, and Shum 2023) and NeuS (Wang et al. 2021).
The metrics of Ani-NeRF are extracted from the data presented in (Guo et al.
2023). It is important to note that HandAvatar lacks support for interactive
hands, while HandNeRF is not able to directly predict geometry. Both HandNeRF
and HandAvatar rely on learning from video sequence for voxel rendering with
large pose variations, which may result in smoothing texture. By taking
advantage of the design of our topology-consistent hand mesh and the mesh-
based neural rendering network, our presented method achieves PSNR of 37dB
with just about one minute of fine-tuning. Comparing to NeuS, some semi-
transparent mist-like artifacts are observed around the rendered hand, as
shown in Fig. 4.

Results of DeepHandMesh. In the DeepHandMesh dataset, five views are utilized
to estimate MANO parameters and optimize the mesh. 3D scans of DeepHandMesh is
employed to assess the geometric quality. Table 2 reports the mesh geometry
quality, and Fig. 5 shows the generated mesh. Unlike DeepHandMesh relying on
weakly supervised learning from depth maps, our method leverages the diffuse
reflection assumption and incorporates the HAM module to achieve fine-grained
hand reconstruction with sparse views. It is challenging for learning-based
methods to capture the personalized details on DeepHandMesh, which leads to
smoothing results due to their generalization capability. In contrast, the
multi-view reconstruction method, NeuS (Wang et al. 2021), fails due to the
insufficient number of views.

Init | HAM | Joint refine | P2S $\downarrow$ | PSNR $\uparrow$
---|---|---|---|---
✓ |  |  | 1.873 | -
| ✓ | ✓ | 5.506 | 31.96
✓ | ✓ |  | 1.457 | 35.68
✓ | ✓ | ✓ | 1.456 | 37.91
Table 3: The impact of coarse initialization, HAM and joint refinement on
rendering quality and mesh geometry.

Results of Our Dataset. Fig. 6 shows the results of our dataset. Our dataset
exhibits a more authentic color quality. The experimental results indicate
that our proposed approach is effective for various datasets with different
capture devices. Due to the absence of hand priors, the mesh generated by NeuS
(Wang et al. 2021) may not conform to the characteristics typically associated
with hands.

### Ablation Study

Coarse Initialization. We directly compare our results with the annotations in
the InterHand2.6M dataset. As illustrated in Fig. 7, the MANO model generated
by our GCN-based network exhibits a closer alignment with the hands in the
images. Moreover, Table 3 demonstrates that directly using the MANO parameters
provided by the dataset may introduce errors, which will subsequently affect
the rendering quality and reconstruction results.

Figure 5: Comparison on mesh quality. The generated meshes are compared in
terms of geometric quality using 5 different views on the DeepHandMesh (Moon,
Shiratori, and Lee 2020) dataset. JR represents the Joint Refinement. Figure
6: Results of our dataset. To evaluate both geometric accuracy and rendering
quality, we compare our framework with NeuS (Wang et al. 2021) on our dataset.

Effects of HAM. The results reported in Fig. 8 provide clear evidence of the
substantial impact of mesh subdivision and the HAM module in achieving fine-
detailed hand reconstructions. The mesh subdivision process effectively
increases the number of vertices in MANO model, while the HAM module plays a
crucial role in capturing the surface wrinkles and intricate details of the
reconstructed hand. As indicated in Table 3, the inclusion of HAM module
greatly contributes to generating the surface reconstructions that closely
resemble the real 3D scans.

Texture Optimization and Joint Refinement. Due to variations in viewpoints,
the mesh generated by HAM may exhibit artifacts resulting from shading
inconsistencies and ambiguous reflectance. Comparison results in Table 3
clearly demonstrate the significant improvement in image quality achieved
through joint refinement. The visualizations presented in Fig. 4 further
illustrate that neural rendering excels in reconstructing accurate lighting
conditions compared to the results of HAM module. Although the geometric
optimization achieves the subtle metrics changes in terms of rendering
results, it effectively eliminates the non-smoothing singularities, as shown
in Fig. 5.

Figure 7: Comparison on MANO meshes. We compare our proposed GCN-based network
with the annotations from the InterHand2.6M dataset. Figure 8: The visual
results of different methods on mesh geometry quality.

## Conclusion

In this paper, we introduced a novel fine-grained multi-view hand mesh
reconstruction method by leveraging effective inverse rendering to restore
hand poses and intricate details. Our approach predicted a parametric hand
mesh model by a GCN-based network while refining both the hand mesh and
textures through the Hand Albedo and Mesh (HAM) optimization module. To
generate photo-realistic image, we suggested an effective mesh-based neural
rendering scheme by fusing the pre-trained rendering network with vertex
features. Through extensive experiments on diverse datasets, the promising
results demonstrated the efficacy of our proposed approach.

## Acknowledgments

This work is supported by National Natural Science Foundation of China under
Grants (62376244, 61831015). It is also supported by the Information
Technology Center and State Key Lab of CAD&CG, Zhejiang University.

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# Composite Hybrid Inflation : Dilaton and Waterfall Pions

Giacomo Cacciapaglia  Dhong Yeon Cheong  Aldo Deandrea  Wanda Isnard  Seong
Chan Park

###### Abstract

We investigate the possibility that inflation originates from a composite
field theory, in terms of an effective chiral Lagrangian involving a dilaton
and pions. The walking dynamics of the theory constrain the potential in a
specific way, where the anomalous dimensions of operators involving pions play
a crucial role. For realistic values of the anomalous dimensions, we find a
successful hybrid inflation occurring via the dilaton-inflaton, with the pions
acting as waterfall fields. Compositeness consistency strongly constrain the
model, predicting a dilaton scale $f_{\chi}\sim\mathcal{O}(1)$ in unit of the
Planck scale, an inflation scale $H_{\text{inf}}\sim 10^{10}$ GeV, and the
pion scale around $10^{14}$ GeV. We further discuss possible phenomenological
consequences of this theory.

##  1 Introduction

Cosmic inflation, responsible for the accelerated expansion in our early
universe [1, 2, 3, 4, 5, 6, 7], has been pursued for decades (see for example
[8, 9, 10, 11] for reviews and detailed bibliography). Pushing the frontier of
inflation both theoretically and observationally lead to great progress in
understanding the physics involved in this process. However, pinpointing the
microscopic origin of the inflation mechanism still remains an open question
despite numerous attempts to describe its nature.

In this work, we consider the idea of a composite origin for inflation based
on a fundamental gauge theory. Such theories consist of an underlying gauge
symmetry that confines at low scales (while being asymptotically free) and
that couples to some fundamental fermions. Generally, in the confined phase,
such theories contain two classes of “light” composite scalars, originating
from the spontaneous breaking of global symmetries. The breaking of the chiral
symmetry of the fundamental fermions leads to the presence of Nambu-Goldstone
bosons (pions). Furthermore, if the theory has a near-conformal dynamics, a
light scalar resonance emerges in the form of a dilaton [12, 13]. A
constrained potential for the latter is generated by the scale anomaly [14,
15], and it naturally offers a flat direction useful for generating inflation
in the early Universe. Evidence for walking dynamics has been collected both
phenomenologically and from lattice data ( see [16] and references therein).
Note that here “light” refers to the condensation scale, where other
resonances also appear.

Composite inflation has been considered before in the literature, and we refer
the reader to these reviews for more complete details [17, 18]. One early tool
that was used to describe a dilaton-like inflaton is holography [19], in the
form of a D3/D7 brane system [20]. In more general composite theories, both
mesons [21] and glueballs with dilaton-like dynamics [22] have been considered
in association to non-minimal coupling to gravity. Another approach involves
Nambu–Jona-Lasinio descriptions of the strong dynamics [23, 24]. The
possibility of a Nambu-Goldstone inflaton has been explored in [25, 26].
Finally, the dilaton potential from Refs [14, 15] has been considered in Ref.
[27]. In this paper, the inflaton is associated with the dilaton, while pions
play a role in determining the effective potential for the inflaton, hence
providing an example of two-field inflation. In this work, we reanalyze and
extend the inflaton model of Ref. [27], focusing on a consistent origin of the
inflaton potential from composite dynamics. In particular, we consider a case
where the couplings of the dilaton-inflaton to pions are determined in terms
of the scaling dimensions of some composite operators. Hence, realistic values
for such scaling dimensions show that new pion couplings are necessary to
ensure a timely end of inflation. This finally leads to a waterfall mechanism,
triggered by the pions, while inflation occurs along the dilaton direction. We
remark that the couplings of the dilaton to pions may receive other sizable
corrections, making them independent from the anomalous dimensions and hence
justifying the values chosen, for instance, in Ref. [27].

This paper is organized as follows: We first introduce in Sec. 2 our benchmark
theory originating from a composite model, identifying the parameters and
their corresponding role. In Sec. 3 we review the single-field case, extending
previous work to understand the impact of the composite scaling dimensions.
Finally we consider the complete model in Sec. 4, talking into account the
waterfall end of inflation due to the pions and imposing consistency
conditions to make sure of the composite origin of the potential. This leads
to a well-determined and constrained favourable parameter space. We finally
offer our conclusions and discuss the outlook in Sec. 5.

##  2 The Model

For concreteness, we consider a $SU(N_{c})$ gauge theory coupled to $N_{f}$
Dirac fermions in the fundamental representation of the gauge group, even
though the effective model can be associated to any underlying gauge dynamics
[16, 28]. For certain values of $N_{f}$ and $N_{c}$, the theory will flow to
an Infra-Red (IR) fixed point [29, 30], hence generating a phase of near-
scale-invariant dynamics (walking regime) [31, 12]. Further to the IR, fermion
and/or gluon condensates can form, leading to the spontaneous breaking of the
chiral and scale symmetry. Note that the two symmetries can be broken by the
same condensate, or not. The Nambu-Goldstone bosons associated to the breaking
of these symmetries are the pions $\phi(x)=\phi^{a}(x)T^{a}$ and the dilaton
$\chi(x)$, respectively. The pions are usually parameterized in the effective
chiral Lagrangian in the non-linear form:

$U(x)=\exp\left(\frac{i\phi}{f_{\phi}}\right)\,,$ (2.1)

where $f_{\phi}$ denotes the decay constant of the $\phi$ fields. The dynamics
of the pions is described by the usual chiral Lagrangian in a derivative
expansion, with leading term ${\rm
Tr}\left[\partial_{\mu}U^{\dagger}\partial^{\mu}U\right]$. The dilaton
couplings, instead, are determined by scale invariance and controlled by a
decay constant $f_{\chi}$, which could be very different from that of the
pions $f_{\phi}$. The dilaton also obtains a potential from the scale anomaly.
At low energies, where inflation is expected to take place, the resulting
Lagrangian for these composite fields is given by:

$\begin{split}\frac{\mathcal{L}}{\sqrt{-g}}\supset&\frac{1}{2}R-\frac{1}{2}\partial_{\mu}\chi\partial^{\mu}\chi-\frac{f_{\phi}^{2}}{2}\left(\frac{\chi}{f_{\chi}}\right)^{2}{\rm
Tr}\left[\partial_{\mu}U^{\dagger}\partial^{\mu}U\right]-\frac{\lambda_{\chi}}{4}\chi^{4}\left(\log\frac{\chi}{f_{\chi}}-A\right)\\\
&+\frac{\lambda_{\chi}\delta_{1}f_{\chi}^{4}}{2}\left(\frac{\chi}{f_{\chi}}\right)^{3-\gamma_{m}}{\rm
Tr}\left[U+U^{\dagger}\right]+\frac{\lambda_{\chi}\delta_{2}f_{\chi}^{4}}{4}\left(\frac{\chi}{f_{\chi}}\right)^{2(3-\gamma_{4f})}{\rm
Tr}\left[(U-U^{\dagger})^{2}\right]\\\ &-V_{0}\,,\end{split}$ (2.2)

where we set $M_{P}=1/\sqrt{8\pi G}=1$ and the metric at the flat limit
$g_{\mu\nu}=(-,+,+,+)$. $V_{0}$ is a numerical value that sets the vacuum
energy to zero.

The first line of Eq. (2.2) stems from the composite dynamics of the dilaton
and pions. In particular, the last term proportional to $\lambda_{\chi}$
represents the dilaton potential [14, 15], which is fully determined by the
scale anomaly in the theory. The constant $A\sim 1/4$ is chosen so that the
dilaton potential has a minimum at
$\langle\chi\rangle\equiv\chi_{0}=f_{\chi}$, the dilaton decay constant, which
is the order parameter breaking scale invariance. Finally, $\lambda_{\chi}$ is
a dimensionless coupling constant measuring the explicit breaking of scale
invariance.

The second line in Eq. (2.2) contains the effective potential for the pions,
generated by explicit breaking terms of the chiral symmetry of the underlying
model. For later convenience, we defined the dimensionless coupling constants
$\delta_{1}$ and $\delta_{2}$ relative to $\lambda_{\chi}$, even though they
are of completely different origin. The first term, controlled by
$\delta_{1}$, is typically generated by a bare mass term for the confining
fermions. The second, controlled by $\delta_{2}$, is generated by four-fermion
interactions and, depending on the sign, it could destabilize the minimum of
the pion potential away from $\langle\phi\rangle=0$. The interplay between
these two terms has been used in composite Goldstone Higgs models [32], while
here this mechanism can determine the end of inflation via a waterfall
potential, as we will see later. In this work, we assume that the couplings of
the dilaton $\chi$ to these terms are determined by the scaling dimension of
the originating fermionic operators, parameterized in terms of the anomalous
dimensions $\gamma_{m}$ and $\gamma_{4f}$. Their values are fixed by the
walking dynamics of the theory, and they can be computed non-perturbatively
via lattice results. For instance, computations of the mass anomalous
dimension $\gamma_{m}$ in various walking theories seem to suggest that
$0<\gamma_{m}\lesssim 1$ [33]. Similar considerations should apply to
$\gamma_{4f}$, even though no lattice data is available for this operator. In
the following, therefore, we will consider the range

$0<\gamma_{m}\,,\;\gamma_{4f}\lesssim 1,$ (2.3)

as compatible with the composite nature of inflation. However, as it was
stressed in Ref. [27], it is important to note that other scalar mesons in the
confined theory or additional contributions from the U$(1)$ axial anomaly
could modify the power of the dilaton field $\chi$ in the second line of Eq.
(2.2). This may destroy the relation to the anomalous dimensions, hence
allowing arbitrary exponents in the power-law terms in $\chi$, as considered
in Refs [34, 35, 27].

Another consistency relation with compositeness stems from the pion potential
in the second line of Eq. (2.2). At the minimum for the dilaton potential,
$\chi=f_{\chi}$, the coefficient of these two terms should be small
perturbations on the chiral Lagrangian and be determined in terms of the pion
decay constant alone, $f_{\phi}$. Hence, a composite origin of the Lagrangian
(2.2) requires

$\lambda_{\chi}\delta_{1,2}f_{\chi}^{4}\lesssim\mathcal{O}(1)\,f_{\phi}^{4}\,.$
(2.4)

The relation above can be expressed as a constraint on the parameter space

${\rm
max}(\delta_{1},\delta_{2})\lesssim\frac{1}{\lambda_{\chi}}\left(\frac{f_{\phi}}{f_{\chi}}\right)^{4}\,,$
(2.5)

which we will apply below. Note that this is based on an order of magnitude
estimate, hence it should not be considered as a strict bound but rather as a
consistency criterion on the values of the model parameters.

##  3 Single Field Inflation and anomalous mass dimension

We first revisit the single-field case, where the pion minimum is always
$\phi=0$ during inflation: this is achieved for $\delta_{2}\leq 0$ (we will
fix $\delta_{2}=0$ for simplicity). Hence, inflation is triggered by the slow-
roll of the dilaton $\chi$. This case has been considered in Ref. [27] but the
reference value for the dilaton power was chosen to be $1$, corresponding to
$\gamma_{m}=2$ in the ideal case. As we discussed in the previous section,
this is unrealistic from the point of view of composite dynamics, henceforth
we will check if a successful inflation can be obtained within the range
$0<\gamma_{m}\lesssim 1$.

Under the simplifying assumption $\delta_{2}=0$, the inflationary potential
becomes

$\displaystyle
V(\phi=0,\chi)=-\lambda_{\chi}\delta_{1}f_{\chi}^{4}\left(\frac{\chi}{f_{\chi}}\right)^{3-\gamma_{m}}+\frac{\lambda_{\chi}}{4}\chi^{4}\left(\log\frac{\chi}{f_{\chi}}-A^{\rm
single}\right)+V_{0}^{\rm single}\,,$ (3.1)

with $A^{\rm single}=\frac{1}{4}+{\delta_{1}(3-\gamma_{m})}$ and $V_{0}^{\rm
single}=\frac{1}{16}\lambda_{\chi}f_{\chi}^{4}\left(1+4\delta_{1}(1+\gamma_{m})\right)$.
We recall that the values of the two constants stem from the conditions

$V_{\chi}=\frac{\partial V}{\partial\chi}=0\,,\quad V=0\,,$ (3.2)

at the minimum $\langle\phi\rangle=0$ and $\langle\chi\rangle=f_{\chi}$. The
former defines $f_{\chi}$ as the minimum value of the dilaton field, while the
latter sets to zero the vacuum energy. Note that $\delta_{1}=0$ leads to the
pure Coleman-Weinberg (CW) inflation, where the correlation between the e-fold
number and the spectral index is given in Ref. [36].

Slow-roll inflation occurs in the small $\chi$-field regime. Using the
potential given in Eq. (3.1), we compute the usual slow-roll parameters as a
function of the field value:

$\epsilon_{\chi}=\left.\frac{1}{2}\left(\frac{V_{\chi}}{V}\right)^{2}\right|_{\phi=0}\,,~{}~{}\eta_{\chi}=\left.\frac{V_{\chi\chi}}{{V}}\right|_{\phi=0}\,,~{}~{}\xi_{\chi}=\left.\frac{V_{\chi}V_{\chi\chi}}{{V^{2}}}\right|_{\phi=0}.$
(3.3)

The inflationary observables can be expressed in terms of these slow-roll
parameters: the spectral index $n_{s}=1-6\epsilon_{\chi}+2\eta_{\chi}$, the
tensor-to-scalar ratio $r=16\epsilon_{\chi}$, the spectral index running
$\frac{{\rm d}n_{s}}{{\rm d}\ln
k}=-2\xi_{\chi}+16\epsilon_{\chi}\eta_{\chi}-24\epsilon_{\chi}^{2}$, the
curvature power spectrum
$\mathcal{P}_{\zeta}(k_{*})=\frac{1}{24\pi^{2}}\frac{V_{0}}{\epsilon_{\chi}}$,
and the number of e-folds

$N_{e}=\int_{\chi_{*}}^{\chi_{\text{end}}}{\rm
d}\chi\frac{1}{\sqrt{2\epsilon_{\chi}}}\,,$ (3.4)

where $\chi_{\ast}$ is the field value at the pivot scale
$k_{\ast}=0.05~{}\text{Mpc}^{-1}$ and $\chi_{\text{end}}$ at the end of
inflation, defined as the field value where the slow-roll parameters approach
unity. The corresponding CMB observations from Planck18+BK18+BAO [37] give the
values

$\begin{split}n_{s}&=0.9649\pm 0.0042,\\\ r&<0.036\,,\\\
\ln(10^{10}\mathcal{P}_{\zeta}(k_{*}))&=3.040\pm 0.0016\,,\\\ \frac{{\rm
d}n_{s}}{{\rm d}\ln k}&=-0.0045\pm 0.0067\,.\end{split}$ (3.5)

The CMB pivot scale e-folds is expressed in terms of the potential value
during inflation and the reheating temperature as

$\displaystyle N_{e}\simeq
61+\frac{2}{3}\ln\frac{V_{0}^{1/4}}{10^{16}~{}\text{GeV}}+\frac{1}{3}\ln\frac{T_{\rm
reh}}{10^{16}~{}\text{GeV}}\,.$ (3.6)

The above equation can be used to constrain the number of e-folds in
inflationary scenarios.

Figure 1: Number of e-folds versus spectral index for a single field slow-roll
inflation with the pure Coleman-Weinberg potential (in red, $\delta_{1}=0$)
and varying the coefficient $\delta_{1}\in[0,1]\times 10^{-6}$ (in blue) for
four different choices of the anomalous mass dimension $\gamma_{m}=(2,\ 1.5,\
1,\ 0.5)$. The black line, dark gray and light gray shaded areas correspond to
the Planck18+BK18+BAO spectral index central value, $1\sigma$ and $2\sigma$
contours respectively. $\gamma_{m}=2$ allows optimal compatibility with CMB
measurements [27], whereas other values allow marginal $2\sigma$
compatibility. However, lattice studies suggest $\gamma_{m}\lesssim 1$, for
which inflation is disfavored.

By fixing the anomalous dimension $\gamma_{m}$ and $f_{\chi}$, we convert the
remaining parameters to the observables
$(\mathcal{P}_{\zeta}(k_{*}),n_{s}(k_{*}),N_{e}(k_{*}))$. We extract the
parameter sets that satisfy correct CMB power spectrum normalization as in Eq.
(3.5), then obtain a relation between $n_{s}(k_{*})$ and $N_{e}(k_{*})$. In
Fig. 1, we depict the spectral index $n_{s}$ as a function of the number of
e-folds, with fixed CMB normalization, all evaluated at the CMB pivot scale.
The four panels correspond to different values of the mass anomalous dimension
$\gamma_{m}$. We note that for $\gamma_{m}=2$, good values of the e-fold
number can be obtained [27]. As the top-left panel shows, increasing
$\delta_{1}$ reduces the number of e-folding, which is typically too large in
the pure CW case. However, for smaller values of $\gamma_{m}$, this behavior
changes and it is not possible to go below a minimal number of e-folds. In
particular, the bottom panels show that for values of the anomalous dimension
compatible with the compositeness criterion in Eq. (2.3), i.e.
$0<\gamma_{m}\lesssim 1$, a marginal 2$\sigma$ compatibility is allowed for
$\gamma_{m}\sim 1$. Smaller values of $\gamma_{m}$ tend to request larger
values of e-folds, $N_{e}\geq 70$, which often leads to unacceptably high
reheating temperatures.

From this analysis we can conclude, therefore, that a single field slow-roll
inflation scenario in the case of a composite dynamical origin for the
inflaton field is not favored when considering realistic values for the
anomalous mass dimension $\gamma_{m}$.

##  4 Hybrid Inflation and Observables

As the values of the mass anomalous dimension compatible with compositeness
are not favored for a single field inflation, we now include the dynamical
effect of the pions. Henceforth, we allow both $\delta_{1}\neq 0$ and
$\delta_{2}\neq 0$ in this section to study the more general case suggested by
composite dynamics. From Eq. (2.2), the potential that we will use for
inflation is explicitly given by:

$\begin{split}V(\phi,\chi)=&-\lambda_{\chi}\delta_{1}f_{\chi}^{4}\left(\frac{\chi}{f_{\chi}}\right)^{3-\gamma_{m}}\cos\frac{\phi}{f_{\phi}}-\lambda_{\chi}\delta_{2}f_{\chi}^{4}\left(\frac{\chi}{f_{\chi}}\right)^{2(3-\gamma_{4f})}\sin^{2}\frac{\phi}{f_{\phi}}\\\
&+\frac{\lambda_{\chi}}{4}\chi^{4}\left(\log\frac{\chi}{f_{\chi}}-A\right)+V_{0}\,.\end{split}$
(4.1)

From the requirement that the minimum remains at
$\langle\chi\rangle\equiv\chi_{0}=f_{\chi}$ and at zero vacuum energy,

$V(\phi_{0},\chi_{0})=0\,,\text{~{}~{}~{}~{}~{}}V_{\chi}(\phi_{0},\chi_{0})=0\text{~{}~{}~{}~{}~{}and~{}~{}~{}~{}~{}}V_{\phi}(\phi_{0},\chi_{0})=0\,,$
(4.2)

one can obtain the expression for $V_{0}$, $A$ and the vacuum expectation
value of the pion fields $\langle\phi\rangle\equiv\phi_{0}$:

$\begin{split}V_{0}&=\frac{\lambda_{\chi}f_{\chi}^{4}}{16}\left[1+2\frac{\delta_{1}^{2}}{\delta_{2}}(2+\gamma_{m}-\gamma_{4f})-8\delta_{2}(1-\gamma_{4f})\right]\,,\\\
A&=\frac{1}{4}\left[1+2\frac{\delta_{1}^{2}}{\delta_{2}}(\gamma_{m}-\gamma_{4f})-8\delta_{2}(3-\gamma_{4f})\right]\,,\\\
\phi_{0}&=f_{\phi}\arccos\frac{\delta_{1}}{2\delta_{2}}\,.\end{split}$ (4.3)

We request $V_{0}>0$ to ensure a de-Sitter vacuum. Note also that the non-
trivial $\phi_{0}$ vacuum exists for $0<\delta_{1}<2\delta_{2}$ 111For
$\delta_{1}\geq 2\delta_{2}$, the vacuum is stuck at
$\phi_{0}/f_{\phi}=\pi/2$. Also, for $\delta_{2}<0$, the minimum remains at
$\phi_{0}=0$..

Figure 2: Schematic representation of the inflationary potential, with the
black line indicating the trajectory of the fields starting from
$\chi_{\text{in}}\ll f_{\chi}$ and $\phi_{\text{in}}=0$. The inflaton field
slowly rolls towards $\chi_{0}=f_{\chi}$, however at the turning point (green
dot) the pion direction is destabilized.

Figure 2 offers a schematic representation of the inflaton potential. At small
$\chi$ values, the $\delta_{1}$ term dominates the pion potential, hence
forcing $\phi=0$. The potential, therefore, consists in a Coleman-Weinberg-
like potential in the dilaton $\chi$ direction, where the potential
asymptotically goes to a constant value in the small field regime $V(0,\chi\ll
f_{\chi})\simeq V_{0}$, supporting small-field inflation. Starting from small
dilaton field values, the inflaton rolls down the pion valley, inducing
inflation, until a critical value $\chi_{c}$ where the pions develop a
tachyonic mass. The pions, due to the tachyonic instability, act as waterfall
fields terminating inflation and rolling down the waterfall potential,
settling into the true vacuum. The trajectory along the pion direction is
given by:

$\phi(\chi)=\left\\{\begin{array}[]{cc}0&\text{when $\chi<\chi_{c}$}\\\
f_{\phi}\arccos\frac{\delta_{1}}{2\delta_{2}}\left(\frac{f_{\chi}}{\chi}\right)^{3+\gamma_{m}-2\gamma_{4f}}&\text{when
$\chi>\chi_{c}$}\end{array}\right.,$ (4.4)

where a transition ($V_{\phi\phi}$ changes sign) takes place at

$\chi=\chi_{c}\equiv\left(\frac{\delta_{1}}{2\delta_{2}}\right)^{\frac{1}{3+\gamma_{m}-2\gamma_{4f}}}f_{\chi}\,,$
(4.5)

which is depicted as the sudden turn in the trajectory in Fig 2. The end of
inflation is controlled by $\chi_{c}\equiv\chi_{\text{end}}$. As shown in Fig.
2, the potential in Eq. (4.1) has two nonequivalent degenerate vacua, hence
the formation of domain walls may occur at the end of inflation [38]. This can
be easily avoided by lifting the degeneracy by additional pion potential
terms, like for instance a term linear in $\sin\phi/f_{\phi}$:

$\Delta
V=-\lambda_{\chi}\delta_{3}f_{\chi}^{4}\left(\frac{\chi}{f_{\chi}}\right)^{p}\sin\frac{\phi}{f_{\phi}}\,.$
(4.6)

As long as $\delta_{3}\ll\delta_{1,2}$, this term will not significantly
affect the properties of inflation.

In this model there are 5 parameters that determine the inflationary
specifics, namely

$(\delta_{1},~{}\delta_{2},~{}\lambda_{\chi},~{}f_{\chi},~{}\chi_{*}),$ (4.7)

with $\chi_{*}$ denoting the inflaton field value at the pivot scale,
$k=0.05~{}\text{Mpc}^{-1}$. As $\phi=0$ during inflation, $\delta_{2}$ only
enters in the determination of the number of e-folds by fixing the end of
inflation at $\chi=\chi_{\text{end}}$. To explore numerically the parameter
space of the model, we choose to impose 3 constraints on the following
inflationary observables:

$\mathcal{P}_{\zeta}(k_{*})=2.1\times
10^{-9}\,,\;\;\;n_{s}=0.965\,,\;\;\;N_{e}=50\,,$ (4.8)

chosen to satisfy the experimental observations in Eq. (3.5) and a consistent
duration of inflation. This allows us to determine the values of $\delta_{1}$,
$\delta_{2}$ and $\chi^{*}$, while keeping $\lambda_{\chi}$ and $f_{\chi}$ as
free parameters.

Figure 3:  Parameter space dependence on the anomalous mass dimension
$\gamma_{m}$ setting $\gamma_{4f}=1$ (left) and on the anomalous 4-Fermi
dimension $\gamma_{4f}$ setting $\gamma_{m}=1$ (right). The allowed region
corresponds to the regions sandwiched between the solid $(\delta_{2}=10^{-4})$
and dashed lines $(\delta_{2}=1)$. The region gets restricted by $\delta_{2}$
values that satisfy $V_{0}>0$ .

Firstly, we investigate the dependence of the inflation dynamics on the
anomalous dimensions $\gamma_{m}$ and $\gamma_{4f}$. In Fig. 3 we show the
allowed values of $f_{\chi}$ and $\lambda_{\chi}$, comprised between the solid
and dashed lines, for various values of $\gamma_{m}$ (left) and $\gamma_{4f}$
(right), while the other anomalous dimension is fixed to 1. By varying the
ratio between $\delta_{1}$ and $\delta_{2}$, we obtain a parameter set
$(\delta_{2},\lambda_{\chi},f_{\chi})$ that satisfies Eq. (4.8). We focus on
the $\delta_{2}$ region $0<\delta_{2}\leq 1$ in order to guarantee that the
potential is still dominated by the Coleman-Weinberg like potential, hence the
parameter space being confined between a band. We note that for small values
of $\delta_{2}\sim 10^{-4}$, the prediction between $\lambda_{\chi}$ and
$f_{\chi}$ become independent from the anomalous dimensions. On the other
limit, as $\delta_{2}$ increases, the potential shape differs from the
Coleman-Weinberg potential resulting in anomalous dimension dependent
predictions. We remark that reducing the values of the anomalous dimensions
from 1 to zero also reduces the allowed parameter space where a successful
inflation can occur.

###  4.1 Tachyonic Preheating and Waterfall Termination

In this setup, the mass of the pions takes the following form as a function of
the inflaton field $\chi$:

$m_{\phi}^{2}=\frac{\lambda_{\chi}\delta_{1}f_{\chi}^{1+\gamma_{m}}\chi^{3-\gamma_{m}}}{f_{\phi}^{2}}\cos\left({\frac{\phi}{f_{\phi}}}\right)-\frac{2\lambda_{\chi}\delta_{2}f_{\chi}^{2(\gamma_{4f}-1)}\chi^{2(3-\gamma_{4f})}}{f_{\phi}^{2}}\cos\left({\frac{2\phi}{f_{\phi}}}\right)\,.$
(4.9)

A hybrid inflation model is realized where the dilaton $\chi$ is the (main)
inflaton and the pions $\phi$ are waterfall fields to assist the end of
inflation. In the inflationary valley and at the minimum after inflation, we
obtain the simplified relations:

$m_{\phi}^{2}=\lambda_{\chi}\frac{f_{\chi}^{4}}{f_{\phi}^{2}}\times\left\\{\begin{array}[]{cc}\delta_{1}f_{\chi}^{\gamma_{m}-3}\chi^{3-\gamma_{m}}-2\delta_{2}f_{\chi}^{2(\gamma_{4f}-3)}\chi^{2(3-\gamma_{4f})}&\text{if
$\phi=0\,,~{}\chi<\chi_{c}$}\,,\\\
\left(2\delta_{2}-\frac{\delta_{1}^{2}}{2\delta_{2}}\right)&\text{if
$\phi=\phi_{0}\,,~{}\chi=f_{\chi}$}\,.\end{array}\right.$ (4.10)

During inflation, the waterfall fields are kept at their local minimum
$\phi=\phi_{0}=0$, while they become tachyonic when $\chi>\chi_{c}$. At this
point, the perturbations $\delta\phi$ grow exponentially due to the tachyonic
instability, leading to a rapid production of pions 222Tachyonic instability
can cause significant density perturbation leading to primordial black hole
production and stochastic gravitational wave [39]. Recently, a hint is
suggested from pulsar timing array observations [40].. The inflaton rapidly
falls down in the $\phi$ direction leading to the end of inflation in the form
of tachyonic preheating [41, 42, 43]. By controlling $\chi_{c}$ via
$\delta_{2}$, we realize a proper number of e-folds, $N_{e}\approx 50$.
However, in order to terminate inflation rapidly, the growth of this
exponential mode must be compensated by the Hubble expansion. This criterion
can be expressed as [44]

$\displaystyle|m_{\phi}^{2}|_{\phi=0,\chi=\chi_{c}+\Delta\chi}\gg H^{2}\,,$
(4.11)

where $\Delta\chi$ corresponds to the field excursion of the dilaton field and
$H$ is the Hubble constant at $\chi_{c}$.

The field excursion is the distance reached in approximately one Hubble time.
Recalling that $H^{2}\simeq\frac{V}{3}\simeq\frac{V_{0}}{3}$, the Hubble time
is:

$\displaystyle\Delta t\sim
H^{-1}\simeq\frac{4\sqrt{3}}{f_{\chi}^{2}\sqrt{\left(\frac{2\delta_{1}^{2}}{\delta_{2}}(2-\gamma_{4f}+\gamma_{m})+8\delta_{2}(\gamma_{4f}-1)+1\right)\lambda_{\chi}}}.$
(4.12)

Using the slow-roll approximation $3H\dot{\chi}\simeq-V_{\chi}$ we can
calculate the field excursion:

$\displaystyle\Delta\chi$
$\displaystyle\sim-\frac{V_{\chi}}{3H^{2}}\simeq-\left.\frac{16~{}\delta_{2}~{}V_{\chi}}{\left(2(2-\gamma_{4f}+\gamma_{m})\delta_{1}^{2}+\delta_{2}(1+8(\gamma_{4f}-1)\delta_{2})\right)\lambda_{\chi}f_{\chi}^{4}}\right|_{\phi=0,\chi=\chi_{c}}.$
(4.13)

As the pion mass squared in Eq. (4.9) is inversely proportional to
$f_{\phi}^{2}$, the criterion in Eq. (4.11) will always provide an upper bound
on the pion decay constant $f_{\phi}<f_{\phi}^{\text{waterfall}}$. Combined
with the compositeness criterion in Eq. (2.5), which provides a lower bound
$f_{\phi}>f_{\phi}^{\text{composite}}$, we can constrain the values of the
pion decay constant compatible with the composite nature of inflation. Note
that for some parameters points

$f_{\phi}^{\text{composite}}>f_{\phi}^{\text{waterfall}}\,,$ (4.14)

hence we will deem such points as incompatible with compositeness.

###  4.2 Benchmark $\gamma_{m}=\gamma_{4f}=1$

To be more concrete, we consider here a benchmark with
$\gamma_{m}=\gamma_{4f}=1$, which are the maximal values of the anomalous
dimensions allowed by compositeness. As we showed, this also corresponds to
the widest parameter space favorable for inflation. Following the same
prescription as in the previous subsection, the favorable parameter space in
terms of $\lambda_{\chi}$ and $f_{\chi}$ is provided in Fig. 4.

For $\delta_{1}<2\delta_{2}$, as required by the existence of a non-trivial
pion vacuum, the mass of the pions reduces to:

$|m_{\phi}^{2}|_{\phi=0,\chi=\chi_{c}+\Delta\chi}=\frac{4\lambda_{\chi}f_{\chi}^{2}\delta_{1}^{3}}{f_{\phi}^{2}\delta_{2}^{2}}\log\left(\frac{2\delta_{2}}{\delta_{1}}\right)\,.$
(4.15)

The criterion given in the Eq. (4.11) allows us to set the following upper
bound on the pion decay constant $f_{\phi}$:

$f_{\phi}\ll\frac{8\sqrt{3}\delta_{1}^{3/2}}{\delta_{2}f_{\chi}}\log^{1/2}\left(\frac{2\delta_{2}}{\delta_{1}}\right)\,.$
(4.16)

Together with the compositeness criterion in Eq. (2.5), we obtain a constraint
on the two free parameter $f_{\chi}$ and $\lambda_{\chi}$:

$f_{\chi}\lambda_{\chi}^{1/4}\lesssim\frac{8\sqrt{3}\delta_{1}^{3/2}}{\delta_{2}^{5/4}}\log^{1/2}\left(\frac{2\delta_{2}}{\delta_{1}}\right)\,.$
(4.17)

Only points roughly consistent with this bound are then compatible with a
composite origin of the inflationary potential in Eq. (4.1). This region is
indicated in Fig. 4 by the gray area, which mainly requires the dilaton scale
$f_{\chi}$ to remain of the order of the Planck mass.

Figure 4:  Parameter space allowed considering the CMB observations and
$\gamma_{m}=\gamma_{4f}=1$. The horizontal axis is in units of
$\lambda_{\chi}\times 10^{14}$. The gray region corresponds to the preferred
parameter space suggested by the compositeness condition in Eq. (4.17).

We further consider a specific benchmark point within the area consistent with
compositeness and providing a successful inflation, in order to quantify
reasonable values for the pion decay constant $f_{\phi}$. As such, we consider
the following parameter values:

$\begin{split}&f_{\chi}=0.5~{}M_{P}\,,\;\;\;\lambda_{\chi}=1.42\times
10^{-14}\,,\;\;\;\chi_{*}=0.004~{}M_{P}\,,\\\
&\delta_{1}=10^{-5}\,,\;\;\;\delta_{2}=10^{-1}\,;\end{split}$ (4.18)

for which the pion decay constant must be within the following range

$\displaystyle 9.7\times 10^{-5}~{}M_{P}\lesssim f_{\phi}\lesssim 2.7\times
10^{-4}~{}M_{P}\,.$ (4.19)

For this particular benchmark point, we can also calculate the Hubble scale at
the end of inflation as

$\displaystyle H_{\text{inf}}=4.34\times 10^{-9}~{}M_{P}\simeq
10^{10}~{}\text{ GeV}\,.$ (4.20)

Once choosing $f_{\phi}=10^{-4}~{}M_{P}$, we can also numerically evaluate the
masses for the dilaton and the the pions once they relaxed at the true
minimum:

$\displaystyle\left.m_{\chi}^{2}\right|_{\chi=f_{\chi},\phi=\phi_{0}}$
$\displaystyle=$
$\displaystyle\lambda_{\chi}f_{\chi}^{2}+2\lambda_{\chi}\frac{\delta_{1}^{2}f_{\chi}^{2}}{\delta_{2}}\sim
3.57\times 10^{-15}~{}M_{P}^{2}\,,$ (4.21)
$\displaystyle\left.m_{\phi}^{2}\right|_{\chi=f_{\chi},\phi=\phi_{0}}$
$\displaystyle\sim$ $\displaystyle 1.8\times 10^{-8}~{}M_{P}^{2}\,.$ (4.22)

Expressing them in GeV we obtain

$m_{\chi}\sim 10^{11}~{}\text{GeV,~{}~{}~{}~{}~{}}m_{\phi}\sim
10^{14}~{}\text{GeV,~{}~{}for~{}~{}}f_{\phi}=2.44\times
10^{14}~{}\text{GeV}\,.$ (4.23)

Note that, as the parameter space in Fig 4 is very constrained, the specific
numerical value for these parameters cannot vary a lot around the values
obtained in this specific benchmark. As we have seen, reducing the anomalous
dimensions can further limit the favorable value of the parameters. In this
sense, the composite origin for the inflation model in Eq. (4.1) _predicts_ a
composite scale $f_{\chi}\sim\mathcal{O}(1)~{}M_{P}$ and an inflation scale
$H_{\text{inf}}\sim\mathcal{O}(10^{10})~{}\text{GeV}$, while the pion mass and
decay constant are typically a few orders of magnitude below the Planck scale.

##  5 Conclusions and Outlook

We examined whether a composite theory can trigger cosmological inflation via
the dynamical interplay of a dilaton and pions. Inspired by a $SU(N_{c})$
model with fermions in the fundamental representation exhibiting a walking
dynamics, we impose consistency criteria on the dilaton-pion potential to
ensure its composite origin. In particular, the values of the anomalous
dimensions, controlling the coupling of the dilaton to pions, are constrained
to be smaller than unity as suggested by recent lattice studies. We show that
the composite inflation is compatible with the latest Planck18+BK18+BAO
observations, only if the pions play the role of waterfall field terminating
inflation. The inflaton is played mainly by the dilaton, which slowly rolls
from small-field values. The consistency with the composite origin of the
potential constrains the parameter space to ensure inflation. The inflation
scale is around $H_{\text{inf}}\sim 10^{10}$ GeV, while the pion masses and
decay constant are around $f_{\phi}\sim m_{\phi}\sim 10^{14}$ GeV. Henceforth,
the inflationary dynamics occurs below the Planck scale, in a controllable
field-theory regime. The pion potential is also constrained by the
inflationary observables, leading to constraints on the interactions that
generate such terms by breaking explicitly the chiral symmetry of the
composite theory. Hence, the couplings of the Standard Model to the composite
sector, once specified, need to respect such constraints.

These stark predictions can lead to interesting consequences for the dynamics
of the Universe right after inflation. Indeed, the waterfall end of inflation
will inject energy in the pion/dilaton system below the Planck scale. The
subsequent reheating of the Standard Model particles will crucially depend on
their couplings to the composite sector. For example, some pions may remain
stable and contribute to the Dark Matter density, behaving like wimpzillas due
to their mass [45, 46, 47]. Pion and dilaton decays into the Standard Model
may then guarantee the reheating of the visible sector, whose dynamics
crucially depends on the allowed strength of their coupling to composite
operators, contributing to the inflationary potential. We leave this
investigation to future studies. We also note that the turbulent end of
inflation could lead to sizable production of primordial black holes and
stochastic gravitational waves. Furthermore, the symmetry breaking structure
of a specific theory may lead to the production of unstable topological
defects, specifically domain walls, which may involve copious production of
gravitational waves. Finally, we want to point out that non-minimal couplings
between the Ricci scalar and the composite states may lead to possible
observational consequences [48, 49, 50, 51].

## Acknowledgments

We thank Maria Mylova for helpful discussions. This work was supported by
National Research Foundation grants funded by the Korean government
(NRF-2021R1A4A2001897) and (NRF-2019R1A2C1089334) (SCP). GC and AD were
partially supported by a Campus France PHC STAR grant. AD acknowledges partial
support from the National Research Foundation in South Africa.

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