# LGTM: Local-to-Global Text-Driven Human Motion
Diffusion Model

Haowen Sun  0000-0002-6450-7746 Shenzhen UniversityChina<EMAIL_ADDRESS>,
Ruikun Zheng  Shenzhen UniversityChina<EMAIL_ADDRESS>,  Haibin Huang
0000-0002-7787-6428 Kuaishou TechnologyChina<EMAIL_ADDRESS>,
Chongyang Ma  0000-0002-8243-9513 ByteDance Inc.USA<EMAIL_ADDRESS>,
Hui Huang  0000-0003-3212-0544 Shenzhen UniversityChina<EMAIL_ADDRESS>and
Ruizhen Hu  0000-0002-6798-0336 Shenzhen UniversityChina<EMAIL_ADDRESS>
(2024)

###### Abstract.

In this paper, we introduce LGTM, a novel Local-to-Global pipeline for Text-
to-Motion generation. LGTM utilizes a diffusion-based architecture and aims to
address the challenge of accurately translating textual descriptions into
semantically coherent human motion in computer animation. Specifically,
traditional methods often struggle with semantic discrepancies, particularly
in aligning specific motions to the correct body parts. To address this issue,
we propose a two-stage pipeline to overcome this challenge: it first employs
large language models (LLMs) to decompose global motion descriptions into
part-specific narratives, which are then processed by independent body-part
motion encoders to ensure precise local semantic alignment. Finally, an
attention-based full-body optimizer refines the motion generation results and
guarantees the overall coherence. Our experiments demonstrate that LGTM gains
significant improvements in generating locally accurate, semantically-aligned
human motion, marking a notable advancement in text-to-motion applications.
Code and data for this paper are available at https://github.com/L-Sun/LGTM

Motion Synthesis; Diffusion Model; Text-Driven Generation.

††submissionid: 415††journalyear: 2024††copyright: acmlicensed††conference:
Special Interest Group on Computer Graphics and Interactive Techniques
Conference Conference Papers ’24; July 27-August 1, 2024; Denver, CO,
USA††booktitle: Special Interest Group on Computer Graphics and Interactive
Techniques Conference Conference Papers ’24 (SIGGRAPH Conference Papers ’24),
July 27-August 1, 2024, Denver, CO, USA††doi: 10.1145/3641519.3657422††isbn:
979-8-4007-0525-0/24/07††ccs: Computing methodologies Motion processing Figure
1. LGTM decomposes input text description at the part level using LLMs and
generates output motion from local to global with both part description and
full-body descriptions.

##  1\. Introduction

In this paper, we address the problem of text-to-motion, i.e., given a textual
description of movements for a character, we aim to automatically generate
plausible and realistic 3D human motions. The successful automation of this
process holds significant potential for a variety of downstream applications,
including the creation of content for augmented and virtual reality
environments, advancements in robotics, and enhancements in human-machine
interactions (Scanlon et al., 2023; Lan et al., 2023; Zhao et al., 2022; Chen
et al., 2021).

As a longstanding challenge at the confluence of natural language processing,
machine learning, and computer graphics, text-to-motion generation has
garnered significant attention in recent research (Petrovich et al., 2022;
Tevet et al., 2022a; Jiang et al., 2023). The advent of diffusion models, as
highlighted in various studies (Rombach et al., 2022; Poole et al., 2022;
Alexanderson et al., 2023), has propelled notable advancements in this field
(Tevet et al., 2022b). Despite these strides, the task of generating motions
that are both locally semantic accurate and globally coherent from textual
descriptions remains a formidable hurdle. Current methods often face
difficulties in effectively capturing the nuanced local semantics embedded in
motion descriptions and in producing motions that align accurately with these
semantic cues.

In particular, existing approaches in text-to-motion synthesis often encounter
issues such as local semantic leakage and missing elements (Tevet et al.,
2022b; Chen et al., 2023a). For instance, when prompted with a description
like “a man kicks something with his left leg”, these methods might
erroneously generate a motion that corresponds to a “right kick”. Similarly,
prompts involving complex actions requiring coordination of multiple body
parts frequently result in motions with certain parts omitted. Our
observations reveal two primary shortcomings in these methods. Firstly, most
existing techniques utilize a single global text descriptor for all local body
motions. This approach requires the network to learn the association between
local motion semantics and respective body parts from a unified global text
source. This process proves challenging, especially when the textual content
bears similarity across different body parts, leading to difficulties in
differentiating specific actions for each part. Secondly, the text encoders
used in these methods exhibit limited effectiveness in encoding motion-related
text. This limitation is apparent in the high feature similarity observed
among different motion texts, as detailed in recent studies (Petrovich et al.,
2023). This homogeneity in encoded text features further exacerbates the
network’s struggle to discern and accurately represent subtle variations in
local textual semantics.

Towards this end, we present a novel diffusion-based text-to-motion generation
architecture, LGTM, adept at producing motions that are both in alignment with
textual descriptions and precise in local semantic accuracy. LGTM operates
through a local-to-global approach, structured in two main stages. The first
stage implements an efficient strategy to tackle the issue of local semantic
accuracy. Here, we introduce a partition module that employs large language
models (LLMs) to dissect global motion descriptions into narratives specific
to each body part. Subsequently, dedicated body-part motion encoders
independently process these part-specific narratives. This focused approach
effectively circumvents local semantic inaccuracies by reducing redundant
information and preventing semantic leakage, thus maintaining a sharp focus on
relevant local semantics. However, as each body-part motion encoder functions
independently, without awareness of other parts’ movements, it is imperative
to synchronize these individual motions to avoid full-body coordination
issues. To address this, the second stage of LGTM introduces an attention-
based full-body optimizer. This component is specifically designed to
facilitate the integration of information among different body parts, ensuring
that the overall motion is not only locally precise but also globally coherent
and fluid.

To evaluate the effectiveness of LGTM, we further conduct experiments on text-
driven motion generation and provide both quantitative and qualitative
results. Our experiments show that our proposed LGTM can generate faithful
motions that better align with the input text both locally and globally, and
outperform state-of-the-art methods.

To summarize, our contributions are as follows:

  * •

We present LGTM, a novel diffusion-based architecture that translate textual
descriptions into accurate and coherent human motions, marking a significant
improvement over previous text-to-motion approaches.

  * •

LGTM introduces a unique partition module that utilizes LLMs to decompose
complex motion descriptions into part-specific narratives. This significantly
enhances local semantic accuracy in motion generation.

  * •

Our experiments demonstrate the effective integration of independent body-part
motion encoders with an attention-based full-body optimizer, ensuring both
local precision and global coherence in generated motions, providing a
promising improvement for text-to-motion generation.

##  2\. Related Work

The generation of motion sequences is a longstanding challenge within the
domain of computer graphics, where the objective is to produce a series of
motion frames guided by conditional control signals. Given that our approach
is centered on body-partition-based text-to-motion synthesis, we explore
relevant literature across two primary aspects: body partition modeling and
text-to-motion generation.

#### Part-based motion modeling.

Partitioning the human body into distinct segments facilitates the control of
motion synthesis at a more granular level, allowing for localized adjustments.

Several studies have explored the concept of combining motions of individual
body parts to synthesize novel motions. (Hecker et al., 2008) introduced a
retargeting algorithm that composes motions at the level of individual body
parts to generate diverse character animations. (Jang et al., 2008) divided
motions into upper and lower body segments, merging them through an algorithm
to augment their motion database. (Soga et al., 2016) synthesized dance
motions from existing datasets by focusing on body partitions. (Jang et al.,
2022) performed style transfer at the part level, utilizing a graph
convolutional network to assemble different body part motions into new,
coherent sequences, preserving local styles while transferring them to
specific body parts without compromising the integrity of other parts or the
entire body. However, these methods rely on pre-existing motion data, and
hence are more accurately described as synthesis rather than generation.

For more detailed local control, (Starke et al., 2020) proposed a local phase
model based on body partitions used to generate basketball player movements,
achieving higher local fidelity compared to global phase approaches (Zhang et
al., 2018; Starke et al., 2019). (Starke et al., 2021) introduced a neural
animation layering technique that combines trajectories of different body
parts produced by control modules, providing animators with more granular
control and enabling the creation of high-quality motion. (Lee et al., 2022)
developed an algorithm for reassembling physically-based part motions,
allowing the combination of partial movements from characters with varying
skeletal structures. By operating in a physically simulated virtual
environment, they employed part-wise timewarping and optimization-based
assembly to ensure improved spatial and temporal alignment. (Bae et al., 2023)
utilized part-wise motion discriminators to enhance motion variety and a
global control policy to maintain the physical realism of the movements.

#### Text-to-motion generation.

Text provides a user-friendly interface for directing motion generation due to
its ease of use and editing capabilities. However, a significant challenge
arises from the difficulty in precisely controlling the outcome of the
generated motion through text. In this subsection, we examine text-to-motion
generation techniques and identify their limitations.

Certain text-to-motion approaches are founded on the encoder-decoder
architecture and focus on aligning modalities within a unified latent space.
(Ahuja and Morency, 2019) trained their network by alternating between
encoding motions and texts, then decoding them back into motion, thereby
implicitly aligning the two modalities. (Ghosh et al., 2021; Petrovich et al.,
2022) encoded text and motion concurrently and decoded them into motion,
employing additional loss functions to bring the modalities closer within the
latent space. These methods struggle with generating motions from lengthy
textual descriptions. (Athanasiou et al., 2022) tackled long motion generation
by producing short motion clips in an auto-regressive fashion, but this
requires manual segmentation of long textual descriptions into shorter
segments and specification of action duration. To utilize visual priors,
(Tevet et al., 2022a) employed a frozen CLIP (Radford et al., 2021) text
encoder to encode motion descriptions and aligned the motion latent space with
that of CLIP. Nevertheless, the images used for alignment, rendered from
random motion frames, can confuse the network when the frames are not
representative. Moreover, (Petrovich et al., 2023) observed that motion
descriptions tend to cluster closely in the CLIP latent space, as the
distribution of motion-related text is narrower than that of the broader text
datasets used to train CLIP.

Figure 2. Overview of our LGTM framework, which consists of three major
components.  (1) The partition module utilizes ChatGPT to deconstruct motion
descriptions $T$ into body part level text $T_{\mathrm{part}}$, and decomposes
full-body motion $\mathbf{M}$ to body part motion
$\mathbf{M}_{\mathrm{part}}$;  (2) The part motion encoders encodes part-level
motions with corresponding part-level text independently and a diffusion time
step $n$;  (3) The full-body motion optimizer utilizes an attention module to
optimize fused body part motion with full-body text semantic.

Recent developments in neural diffusion models for image generation have
inspired text-to-motion methods that leverage these models to achieve superior
quality. (Tevet et al., 2022b; Zhang et al., 2022) utilized Transformer to
denoise motion conditioned on text. (Chen et al., 2023b) introduced a U-Net-
based DDIM generative model to denoise motion in latent space, resulting in
expedited generation. However, these methods lack the ability to control
localized motion generation through masking. Additionally, they struggle to
learning correct mapping of the local semantics because all body parts share
the same textual information, which potentially lead to semantically
mismatched part motions.

An alternative approach to motion generation involves processing motion in a
discrete space through token prediction (Guo et al., 2022b; Jiang et al.,
2023; Yao et al., [n. d.]). But the limitations of these works are that the
expressive capacity of the codebook can restrict the diversity of the
generated motions, potentially causing the text input to be mapped to
unintended motions.

The challenges in controlling local motion semantics stem from:  (1) the
sharing of textual information across all body parts, and  (2) the difficulty
networks face in distinguishing text latent codes encoded by CLIP.  These
factors contribute to the difficulty of achieving precise local semantic
control in motion generation, leading to issues such as semantic leakage.

Drawing inspiration from the technological advancements and challenges
identified in prior research, we propose a novel framework that combines body-
part partitioning with independent local motion semantic injection and a
global semantic joint optimization strategy. This framework is designed to
enhance the fidelity and controllability of text-to-motion synthesis,
addressing the need for more nuanced and accurate motion generation.

##  3\. Method

In this section, we delve into the specifics of LGTM, as illustrated in Figure
2. LGTM is structured as a local-to-global generation framework that initially
creates local, part-level motion, followed by a global fusion and optimization
process to produce the final full-body motion. At its core, LGTM operates by
subdividing the full-body text and motion spaces into body-part-specific
subspaces. Such subdivision is adeptly handled by a dedicated Partition
Module.

For each of these subspaces, we have developed specialized part motion
encoders. These encoders are trained to learn independently a series of
mappings between part-level motions and part-level text. This strategy
effectively mitigates the issues of incorrect local semantic mapping seen in
previous methods. Following the localized encoding, LGTM introduces a full-
body motion optimizer to establish correlations among the various subspaces
and ensure the consistency and coherence of the final full-body motion. Below,
we provide a detailed explanation of the functionalities and details of each
module in LGTM.

###  3.1. Preliminary: Human Motion Diffusion Model

#### Input representation.

We define the input pair for our method as $\left(\mathbf{M},T\right)$, where
$\mathbf{M}$ represents full-body motion data and $T$ denotes the raw full-
body text description. Specifically, we use the HumanML3D representation
proposed by (Guo et al., 2022a) as our motion data representation, which is
calculated from the SMPL motion data (Loper et al., 2015) and includes
redundant motion features that are helpful for network training. A full-body
motion data $\mathbf{M}$ contains $F$ frames and $J=22$ joints. Specifically,
we denote
$\mathbf{M}=[\dot{\mathbf{r}}_{\mathrm{root}},v_{\mathrm{root}},h,\mathbf{p},\mathbf{r},\mathbf{v},\mathbf{c}]$,
where $\dot{\mathbf{r}}_{\mathrm{root}}\in\mathbb{R}^{F\times 1}$,
$v_{\mathrm{root}}\in\mathbb{R}^{F\times 2}$ and $h\in\mathbb{R}^{F\times 1}$
are the angular velocity around y-axis, linear velocity on x-z plane, and
height of the root joint, $\mathbf{p}\in\mathbb{R}^{F\times(J-1)\times 3}$ and
$\mathbf{r}\in\mathbb{R}^{F\times(J-1)\times 6}$ are local position and 6D
rotation (Zhang et al., 2018) of all joints except root joint,
$\mathbf{v}\in\mathbb{R}^{F\times J\times 3}$ is the local velocity of all
joints, and $\mathbf{c}\in\mathbb{R}^{F\times 4}$ is the contact signal of
feet.

#### Diffusion model.

Our method is built upon a text-conditional diffusion model. In the training
stage, this model adds noise to a clean motion $\mathbf{{M}}$ following the
Markov process and trains a network to predict the added noise with an L2
loss. In the sampling stage, this model gradually reduces noise from a purely
noised motion $\mathbf{M}^{n}$ with the predicted noise. We use the DDIM (Song
et al., 2022) as our diffusion model to accelerate the sampling process. More
details is provided in the supplementary material.

###  3.2. Partition Module

The Partition Module is designed to inject local semantics into each body part
for Part Motion Encoders. In practice, an input pair
$\left(\mathbf{M},T\right)$ is divided into six parts, including head, left
arm, right arm, torso, left leg, and right leg.

The motion $\mathbf{M}$ is decomposed as follows:

$\displaystyle\mathbf{M}_{\mathrm{head}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{head}},\mathbf{r}_{\mathrm{head}},\mathbf{v}_{\mathrm{head}}\right]\in\mathbb{R}^{F\times
24}$ $\displaystyle\mathbf{M}_{\mathrm{left\\_arm}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{left\\_arm}},\mathbf{r}_{\mathrm{left\\_arm}},\mathbf{v}_{\mathrm{left\\_arm}}\right]\in\mathbb{R}^{F\times
48}$ $\displaystyle\mathbf{M}_{\mathrm{right\\_arm}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{right\\_arm}},\mathbf{r}_{\mathrm{right\\_arm}},\mathbf{v}_{\mathrm{right\\_arm}}\right]\in\mathbb{R}^{F\times
48}$ $\displaystyle\mathbf{M}_{\mathrm{torso}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{torso}},\mathbf{r}_{\mathrm{torso}},\mathbf{v}_{\mathrm{torso}},\dot{\mathbf{r}}_{\mathrm{root}},v_{\mathrm{root}},h\right]\in\mathbb{R}^{F\times
43}$ $\displaystyle\mathbf{M}_{\mathrm{left\\_leg}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{left\\_leg}},\mathbf{r}_{\mathrm{left\\_leg}},\mathbf{v}_{\mathrm{left\\_leg}},\mathbf{c}_{\mathrm{left\\_leg}}\right]\in\mathbb{R}^{F\times
50}$ $\displaystyle\mathbf{M}_{\mathrm{right\\_leg}}$
$\displaystyle=\left[\mathbf{p}_{\mathrm{right\\_leg}},\mathbf{r}_{\mathrm{right\\_leg}},\mathbf{v}_{\mathrm{right\\_leg}},\mathbf{c}_{\mathrm{right\\_leg}}\right]\in\mathbb{R}^{F\times
50},$

where the subscript indicates where the feature from. For example,
$\mathbf{p}_{\mathrm{right\\_leg}}$ includes all local positions of joints
from the right leg.

For the motion description $T$, we leverage the knowledge inference
capabilities of LLMs to decompose it into six parts: $T_{\mathrm{head}}$,
$T_{\mathrm{left\\_arm}}$, $T_{\mathrm{right\\_arm}}$, $T_{\mathrm{torso}}$,
$T_{\mathrm{left\\_leg}}$ and $T_{\mathrm{right\\_leg}}$ using crafted
prompts. The prompt includes three sections: task definition, output
requirements, and some output examples. The task definition instructs LLMs to
extract principal descriptions for each motion part. The output requirements
tell LLMs that we need structured output such as JSON format, body part
naming, etc. Then, we employ a few-shot approach to guide LLMs in generating
the desired output. More details of our prompts can be found in the
supplementary materials. A decomposed description example is shown in Table 1.

Table 1. An example of decomposing full-body motion description: “a person waves the right hand and then slightly bends down to the right and takes a few steps forward.” Part name | Part description
---|---
head | dose nothing
left arm | dose nothing
right arm | waves hand
torso | slightly bends down
left leg | takes a few steps forward
right leg | takes a few steps forward

###  3.3. Part Motion Encoders

The part motion encoders,
$\\{E_{\mathrm{head}},\dots,E_{\mathrm{right\\_leg}}\\}$, aim to learn local
semantic mapping from part-level input pairs
$\left(\mathbf{M}_{\mathrm{part}}^{n},T_{\mathrm{part}}\right)$ independently.
Since each encoder obtains information only from its corresponding part-level
input pair and cannot access information from other body parts, the issue of
semantic leakage is effectively alleviated. We denote the part-level encoding
process as follows:

(1)
$\mathbf{z}^{n}_{\mathrm{part}}=E_{\mathrm{part}}\left(\mathbf{M}^{n}_{\mathrm{part}},T_{\mathrm{part}},n\right),$

where each part motion encoder, $E_{\mathrm{part}}$, consists of three
components: a linear layer, a text encoder, and a Conformer (Gulati et al.,
2020). The linear layer aims to align the size of the latent dimension with
that of the text encoder. We use six different frozen part-level TMR text
encoders (Petrovich et al., 2023), each corresponding to one of the six body
parts, which are pretrained on part-level motion-text pairs
$\left(\mathbf{M}_{\mathrm{part}},T_{\mathrm{part}}\right)$ respectively.
Since the TMR model is trained only on motion description and motion data, and
not on large visual datasets, the motion-related text embedding encoded by TMR
is easier for the network to distinguish than that by CLIP. The projected
motion and text embedding are then fused and processed by a Conformer(Gulati
et al., 2020). The Conformer incorporates convolution blocks into the
Transformer (Vaswani et al., 2017) architecture to better capture temporal
local features. Moreover, previous work (Alexanderson et al., 2023) shows the
success of Conformer on music-to-dance task.

Figure 3. The structure of an attention encoder block.

###  3.4. Full-Body Motion Optimizer

Since each part’s motion and text are independently encoded to
$\left\\{\mathbf{z}_{\mathrm{head}}^{n},\cdots,\mathbf{z}_{\mathrm{left\\_leg}}^{n}\right\\}$
independently, the network will ignore the correlations between the different
body parts, therefore, we propose that the full-body motion optimizer $G$
establishes correlations by adjusting the movements of each body part based on
full-body text information.

Specifically, we first concatenate all body part latent codes into a full-body
latent code $\mathbf{z}^{n}$ whose shape is $\left(F,S\right)=\left(F,6\times
128\right)$, and then fuse it with the global text embedding encoded by
freezing the full-body level TMR text encoder. Next, we use an attention
encoder (Vaswani et al., 2017) to compute a delta that adjusts each part in
the latent code $\mathbf{z}^{n}$. The attention encoder is where the exchange
of spatio-temporal information actually occurs. It consists of several
attention encoder blocks, each containing a multi-head attention block and a
feed-forward layer, as shown in Figure 3. Since the latent code
$\mathbf{z}^{n}$ is processed by a multi-head attention block on the temporal
dimension $F$, and feed-forward layers (FFN) operate on the spatial dimension
$S$, the latent code for each body part can continuously exchange temporal and
spatial information. Next, we use a SmoothNet (Zeng et al., 2022) to reduce
jitter, which contains a stacked MLP with residual connections and operates on
the temporal dimension, acting as a low-pass filter in the latent space.

Finally, we project the latent code to origin feature dimension, and get a
clean motion $\hat{\mathbf{M}}^{0}$. The full-body motion optimizer can be
formulated as

(2) $\displaystyle\hat{\mathbf{M}}^{0}$
$\displaystyle=G\left(\mathbf{z}_{\mathrm{head}}^{n},\cdots,\mathbf{z}_{\mathrm{left\\_leg}}^{n},T\right)$
$\displaystyle=\mathrm{Linear}(\mathrm{SoothNet}(\mathbf{z}^{n}+\mathrm{AttentionEncoder}(\mathbf{z}_{\mathrm{text}}+\mathbf{z}^{n})))$

##  4\. Results

In this section, we present the motions generated by our method and conduct a
comparative analysis with other text-driven motion generation methods.
Additionally, we perform several ablation studies to highlight the
contributions of individual components within our framework.

Figure 4. Example results generated by our method.

###  4.1. Implementation Details

The part-level motion description is generated by ChatGPT.
(gpt-3.5-turbo-1106) model. Our model is trained with AdamW optimizer with
learning rate decaying strategy of fast warm cosine decay. The initial
learning rate is $10^{-4}$ and the batch size is 64. The number of diffusion
steps is 1K. The training time of our model on the HumanML3D dataset is about
8 hours on 3 NVIDIA RTX 4090 GPUs.

###  4.2. Qualitative Results

Figure 4 shows several example results generated by our method. We can see
that our method can generate motion with precise local semantics, such as body
part semantic correspondence and action timing order, as our method injects
local semantic information into corresponding parts independently, and the
whole-body optimizer builds correct relationships between body parts in both
spatial and temporal domains. For example, the result of “a man leans forward
and jumps high” shows that the character does lean and jump in the correct
order. The result of“a man lock his hands to his face, and do a dance move net
with his legs” shows that the character keeps correct spatial relationship
between hand and face while dancing. The result of “a person doing air kicks
with his right feet” shows that the character do kick with correct body part.

We also provide some visual comparisons to two baselines, including MDM (Tevet
et al., 2022b) and MLD (Chen et al., 2023b). Figure 5 shows that our method
can generate more semantic well-matched motion. In the first row, the
character can pick something with both hands in our result, but with just left
hand in MDM. In the second row, the character only jumps on the left foot
correctly in our result, but jumps on both feet in MDM and dose not jump in
MLD. In the third row, the result of MDM contains weird pose and the MLD dose
not contain “claps”, but our result is more correct. The last row shows that,
for more complex text inputs, our method is able to generate more semantic
accurate results than those two baselines.

Figure 5. Qualitative comparison of results generated by our method with those
from MDM (Tevet et al., 2022b) and MLD (Chen et al., 2023b).

###  4.3. Quantitative Evaluation

#### Evaluation metrics.

To quantitatively evaluate our method, we use the metrics suggested by (Guo et
al., 2022a) which includes  (1) Fréchet Inception Distance (FID)that evaluates
the generated motion quality against real motion distribution;  (2) Diversity
(DIV)that calculates the variance of generated motion;  (3) R Precisionthat
calculates the top-n matching accuracy between generated motion and the
corresponding text description;  (4) Multi-Modal Distance (MM Dist)that
calculates the distance between paired motion and text;  (5) Part-level Multi-
Modal Similarity (PMM Sim)that calculates the normalized cosine similarity
between part-level paired motion and text.  These metrics are calculated in
the latent space using the text encoder and motion encoder from T2M (Guo et
al., 2022a) as in previous works. As our method provides detailed control of
generated motions, we also compare our method to baselines in terms of part-
level motion quality using Part-level Multi-Modal Similarity (PMM Sim), by
training both part-level text encoder and motion encoder with contrastive
learning as in TMR (Petrovich et al., 2023), which we believe makes motion
samples in the latent space more dispersed allowing dissimilar motions can be
distinguished more easily. Specifically, we calculate the PMM Sim in the TMR
latent space as follows:

(3)
$s_{\mathrm{part}}=\frac{1}{2}\left(\frac{z^{\mathrm{M}}_{\mathrm{part}}\cdot
z^{T}_{\mathrm{part}}}{\|z^{\mathrm{M}}_{\mathrm{part}}\|\|z^{T}_{\mathrm{part}}\|}+1\right)$

where both $z^{\mathrm{M}}_{\mathrm{part}}$ and $z^{T}_{\mathrm{part}}$ are
obtained by encoding part-level motion and text through TMR encoders. Although
we mainly focus on semantically controllable generation, we also evaluate
common artifacts in text-to-motion synthesis. We assess the generated motions
using three specific metrics: sliding, penetration, and floating, as
introduced by (Yuan et al., 2022).

#### Comparison results.

The comparison results for full-body motion are presented in Tables 2, and the
comparison results for part-level motion are presented in Table 3. The FID and
DIV in Tables 2 indicate that our method generates more realistic and diverse
motion. The R Precision and MM Dist indicate that our method can generate
better globally semantically matching motion. Table 3 also shows that our
method achieves the best local semantic matching, with performance very close
to that of real data. Our local-to-global design injects local semantic
information independently into body parts and refines it with global
semantics, which provides more accurate and structured semantic information to
the network to help generation and thus achieve higher quality. For artifact
evaluation, as shown in Table 4, we can see that each method exhibits
performance very close to the ground truth (the Real row) at the millimeter
scale. The artifacts can be attributed to the dataset’s intrinsic quality
variances.

Table 2. Comparison of the visual quality and degree of semantic matching
between input text and output full-body motion. These metrics are computed in
the latent space of the T2M model (Guo et al., 2022a).

Method | FID $\downarrow$ | DIV$\uparrow$ | R Precision$\uparrow$ | MM Dist $\downarrow$
---|---|---|---|---
Top 1 | Top 2 | Top 3
Real | 0.000 | 9.831 | 0.513 | 0.708 | 0.804 | 2.939
MotionDiffuse(2022) | 0.687 | 8.894 | 0.318 | 0.531 | 0.677 | 3.118
MDM(2022b) | 0.747 | 9.462 | 0.390 | 0.581 | 0.695 | 3.635
MLD(2023b) | 1.753 | 8.970 | 0.383 | 0.573 | 0.687 | 3.682
Ours (LGTM) | 0.218 | 9.638 | 0.490 | 0.689 | 0.788 | 3.013

Table 3. Comparison of text-to-motion generation using PMM Sim. These metrics
are calculated in the latent space of the part-level TRM encoder. Higher
values indicate better performance.

Method | head | left arm | right arm | torso | left leg | right leg
---|---|---|---|---|---|---
Real | 0.803 | 0.716 | 0.723 | 0.759 | 0.755 | 0.760
MotionDiffuse(2022) | 0.789 | 0.687 | 0.712 | 0.735 | 0.728 | 0.739
MDM(2022b) | 0.783 | 0.699 | 0.691 | 0.740 | 0.717 | 0.723
MLD(2023b) | 0.771 | 0.675 | 0.702 | 0.717 | 0.723 | 0.726
Ours (LGTM) | 0.799 | 0.719 | 0.724 | 0.763 | 0.755 | 0.763

Table 4. Comparison of text to motion generation using metrics on artifact.

Method | sliding (cm/s) $\downarrow$ | penetration (cm) $\downarrow$ | floating (cm) $\downarrow$
---|---|---|---
Real | 0.743 | 1.442 | 0.079
MotionDiffuse(2022) | 1.359 | 1.783 | 0.051
MDM(2022b) | 0.721 | 1.622 | 0.102
MLD(2023b) | 0.949 | 2.392 | 0.064
Ours (LGTM) | 0.854 | 1.247 | 0.046

###  4.4. Ablation Studies

We have designed two main experiments to assess the impact of different
components of our approach. The first experiment investigates the influence of
different text encoders on the motion quality. The second experiment evaluates
the effect of the full-body motion optimizer on the the quality of motions
generated by our method.

#### The importance of text encoder.

We test our method by replacing our pre-trained text encoder with CLIP as an
alternative, demonstrating that the TMR text encoder we use can capture more
detailed semantics. Furthermore, we also present the results obtained by MDM
using either CLIP or the TMR text encoder for comparison.

Table 5 and Table 6 evaluate full-body and part-level motion quality,
respectively. In general, we observe that using the TMR text encoder
consistently produces better results than using CLIP, for both our method and
MDM as well as both local and global quality. When comparing our method to MDM
using the same text encoder, our method generally performs better, further
demonstrating the superiority of our local-to-global design.

Table 5. Comparison of the impact of different text encoders on full-body
metrics computed in the latent space of T2M model (Guo et al., 2022a).

Method | FID $\downarrow$ | DIV$\uparrow$ | R Precision$\uparrow$ | MM Dist $\downarrow$
---|---|---|---|---
Top 1 | Top 2 | Top 3
MDM + CLIP | 0.747 | 9.462 | 0.390 | 0.581 | 0.695 | 3.635
MDM + TMR | 0.403 | 9.687 | 0.455 | 0.653 | 0.759 | 3.266
Ours + CLIP | 0.331 | 9.386 | 0.391 | 0.569 | 0.674 | 3.699
Ours + TMR | 0.218 | 9.638 | 0.490 | 0.689 | 0.788 | 3.013

Table 6. Comparison of the impact of different text encoders on PMM Sim
computed using the part-level TRM encoder. The greater the value, the better.

Method | head | left arm | right arm | torso | left leg | right leg
---|---|---|---|---|---|---
MDM + CLIP | 0.783 | 0.699 | 0.691 | 0.740 | 0.717 | 0.723
MDM + TMR | 0.803 | 0.704 | 0.707 | 0.756 | 0.734 | 0.743
Ours + CLIP | 0.795 | 0.693 | 0.694 | 0.752 | 0.725 | 0.732
Ours + TMR | 0.799 | 0.719 | 0.724 | 0.763 | 0.755 | 0.763

#### The impact of Conformer.

The goal of replacing Transformer with Conformer in Part Motion Encoders is to
improve the motion quality. To validate the improvement, we compare both
configurations on global quality metrics. From Table 7 and Table 8, we observe
that LGTM with Conformer can achieves better quality and semantic matching
performance than with Transformer. This improvement can be attributed to the
convolution blocks of Conformer, which capture local features better than
self-attention.

Table 7. Comparison of the impact of using Conformer versus Transformer in
Part Motion Encoders on global quality.

Method | FID $\downarrow$ | DIV$\uparrow$ | R Precision$\uparrow$ | MM Dist $\downarrow$
---|---|---|---|---
Top 1 | Top 2 | Top 3
Transformer | 1.814 | 8.578 | 0.373 | 0.567 | 0.680 | 3.688
Conformer | 0.218 | 9.638 | 0.490 | 0.689 | 0.788 | 3.013

Table 8. Comparison of the impact of using Conformer versus Transformer in
Part Motion Encoders on PMM Sim. Higher values indicate better performance.

Method | head | left arm | right arm | torso | left leg | right leg
---|---|---|---|---|---|---
Transformer | 0.784 | 0.712 | 0.718 | 0.750 | 0.728 | 0.732
Conformer | 0.799 | 0.719 | 0.724 | 0.763 | 0.755 | 0.763

#### The importance of full-body motion optimizer.

The goal of our full-body motion optimizer is to establish correlations among
different body part movements and improve the coordination of full-body
movements. To validate the effect, we compare it to the setting “w/o opt”,
where we remove the key component of our full-body optimizer, namely, the
attention encoder From Table 9 and Table 10, we can see that the local motion
quality drops, and the full-body motion quality is also much worse without the
optimizer; see Figure 6 for one example result. Without the full-body
optimizer, the character’s two feet cannot coordinate well to step alternately
during movement due to the lack of information exchange.

Table 9. Comparison of the impact of attention encoder on global quality.
These metric are calculated using the T2M model (Guo et al., 2022a).

Method | FID $\downarrow$ | DIV$\uparrow$ | R Precision$\uparrow$ | MM Dist $\downarrow$
---|---|---|---|---
Top 1 | Top 2 | Top 3
w/o opt | 7.384 | 11.552 | 0.219 | 0.360 | 0.454 | 5.227
w/ opt | 0.218 | 9.638 | 0.490 | 0.689 | 0.788 | 3.013

Table 10. Comparison of the impact of attention encoder on PMM Sim using the
part-level TMR encoder. The greater the value, the better.

Method | head | left arm | right arm | torso | left leg | right leg
---|---|---|---|---|---|---
w/o opt | 0.783 | 0.715 | 0.700 | 0.735 | 0.699 | 0.709
w/ opt | 0.799 | 0.719 | 0.724 | 0.763 | 0.755 | 0.763

(a) With full-body optimizer.

(b) Without full-body optimizer.

Figure 6. Motions generation by our method with and without the full-body
optimizer for “a person walks upstairs, turns left, and walks back
downstairs.”

##  5\. Conclusion

Figure 7. A failure case. The corresponding input prompt is someone imitating
a golf swing.

In this study, we propose LGTM for text-to-motion generation, which
significantly improves the accuracy and coherence of 3D human motion derived
from textual descriptions. By integrating large language models with a local-
to-global generation framework, our method effectively addresses key
challenges in semantic mapping and motion coherence.

#### Limitation and future work.

As we use ChatGPT for motion description decomposition, the local semantic
mapping depends on the reasoning ability of ChatGPT. Incorrect decomposition
or mapping may lead to unsatisfactory motion generation results. For example,
when generating the “golf swing” motion, which requires high-level and full-
body coordination, LGTM struggles because ChatGPT identifies that the right
hand swings the golf club but fails to decompose this reasoning into a series
of low-level actions for each body part. The result is that the network
generates an implausible motion, as shown in Figure 7. Also, ambiguous texts
in the dataset can confuse the network during training. For example, the
phrase “a person performs action A and action B” could imply that these
actions occur simultaneously or sequentially, leading to output that may not
align with user expectations. This issue could be mitigated by providing more
detailed temporal descriptions. Furthermore, due to the limited length of
samples in the dataset, our current framework cannot consistently generate
long-term motions with high quality. For future work, one promising direction
is to incorporate our local-to-global idea with those VQ-VAE based approaches
such as TM2T (Guo et al., 2022b) and MotionGPT (Jiang et al., 2023) by
onstructing part-level motion clips as motion tokens for more detailed motion
generation with different part-level motion combinations.

###### Acknowledgements.

We thank the anonymous reviewers for their valuable comments. This work was
supported in parts by NSFC (62322207, 62161146005, U2001206), Guangdong
Natural Science Foundation (2021B1515020085), Shenzhen Science and Technology
Program (RCYX20210609103121030), DEGP Innovation Team (2022KCXTD025),
Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ) and
Scientific Development Funds of Shenzhen University.

## References

  * (1)
  * Ahuja and Morency (2019) Chaitanya Ahuja and Louis-Philippe Morency. 2019.  Language2pose: Natural Language Grounded Pose Forecasting. In _2019 International Conference on 3D Vision (3DV)_. IEEE, 719–728.
  * Alexanderson et al. (2023) Simon Alexanderson, Rajmund Nagy, Jonas Beskow, and Gustav Eje Henter. 2023.  Listen, Denoise, Action! Audio-Driven Motion Synthesis with Diffusion Models.  _ACM Transactions on Graphics_ 42, 4 (Aug. 2023), 1–20.  https://doi.org/10.1145/3592458
  * Athanasiou et al. (2022) Nikos Athanasiou, Mathis Petrovich, Michael J. Black, and Gül Varol. 2022.  TEACH: Temporal Action Composition for 3D Humans. In _2022 International Conference on 3D Vision (3DV)_. IEEE Computer Society, 414–423.  https://doi.org/10.1109/3DV57658.2022.00053
  * Bae et al. (2023) Jinseok Bae, Jungdam Won, Donggeun Lim, Cheol-Hui Min, and Young Min Kim. 2023.  PMP: Learning to Physically Interact with Environments Using Part-wise Motion Priors. In _ACM SIGGRAPH 2023 Conference Proceedings_ _(SIGGRAPH ’23)_. Association for Computing Machinery, New York, NY, USA, 1–10.  https://doi.org/10.1145/3588432.3591487
  * Chen et al. (2021) Lu Chen, Sida Peng, and Xiaowei Zhou. 2021.  Towards Efficient and Photorealistic 3D Human Reconstruction: A Brief Survey.  _Visual Informatics_ 5, 4 (Dec. 2021), 11–19.  https://doi.org/10.1016/j.visinf.2021.10.003
  * Chen et al. (2023a) Xin Chen, Biao Jiang, Wen Liu, Zilong Huang, Bin Fu, Tao Chen, and Gang Yu. 2023a.  Executing Your Commands via Motion Diffusion in Latent Space. In _2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_. IEEE, Vancouver, BC, Canada, 18000–18010.  https://doi.org/10.1109/CVPR52729.2023.01726
  * Chen et al. (2023b) Xin Chen, Biao Jiang, Wen Liu, Zilong Huang, Bin Fu, Tao Chen, Jingyi Yu, and Gang Yu. 2023b.  Executing Your Commands via Motion Diffusion in Latent Space.  https://doi.org/10.48550/arXiv.2212.04048 arXiv:2212.04048 [cs]
  * Ghosh et al. (2021) Anindita Ghosh, Noshaba Cheema, Cennet Oguz, Christian Theobalt, and Philipp Slusallek. 2021.  Synthesis of Compositional Animations From Textual Descriptions. In _Proceedings of the IEEE/CVF International Conference on Computer Vision_. 1396–1406.
  * Gulati et al. (2020) Anmol Gulati, James Qin, Chung-Cheng Chiu, Niki Parmar, Yu Zhang, Jiahui Yu, Wei Han, Shibo Wang, Zhengdong Zhang, Yonghui Wu, and Ruoming Pang. 2020.  Conformer: Convolution-augmented Transformer for Speech Recognition.  arXiv:2005.08100 [cs, eess]
  * Guo et al. (2022a) Chuan Guo, Shihao Zou, Xinxin Zuo, Sen Wang, Wei Ji, Xingyu Li, and Li Cheng. 2022a.  Generating Diverse and Natural 3D Human Motions From Text. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_. 5152–5161.
  * Guo et al. (2022b) Chuan Guo, Xinxin Zuo, Sen Wang, and Li Cheng. 2022b.  TM2T: Stochastic and Tokenized Modeling for the Reciprocal Generation of 3D Human Motions and Texts. In _Computer Vision – ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XXXV_. Springer-Verlag, Berlin, Heidelberg, 580–597.  https://doi.org/10.1007/978-3-031-19833-5_34
  * Hecker et al. (2008) Chris Hecker, Bernd Raabe, Ryan W. Enslow, John DeWeese, Jordan Maynard, and Kees van Prooijen. 2008.  Real-Time Motion Retargeting to Highly Varied User-Created Morphologies.  _ACM Transactions on Graphics_ 27, 3 (Aug. 2008), 1–11.  https://doi.org/10.1145/1360612.1360626
  * Jang et al. (2022) Deok-Kyeong Jang, Soomin Park, and Sung-Hee Lee. 2022.  Motion Puzzle: Arbitrary Motion Style Transfer by Body Part.  _ACM Transactions on Graphics_ (Jan. 2022).  https://doi.org/10.1145/3516429
  * Jang et al. (2008) Won-Seob Jang, Won-Kyu Lee, In-Kwon Lee, and Jehee Lee. 2008.  Enriching a Motion Database by Analogous Combination of Partial Human Motions.  _The Visual Computer_ 24, 4 (April 2008), 271–280.  https://doi.org/10.1007/s00371-007-0200-1
  * Jiang et al. (2023) Biao Jiang, Xin Chen, Wen Liu, Jingyi Yu, Gang Yu, and Tao Chen. 2023.  MotionGPT: Human Motion as a Foreign Language.  https://doi.org/10.48550/arXiv.2306.14795 arXiv:2306.14795 [cs]
  * Lan et al. (2023) Chong Lan, Yongsheng Wang, Chengze Wang, Shirong Song, and Zheng Gong. 2023.  Application of ChatGPT-Based Digital Human in Animation Creation.  _Future Internet_ 15, 9 (Sept. 2023), 300.  https://doi.org/10.3390/fi15090300
  * Lee et al. (2022) Seyoung Lee, Jiye Lee, and Jehee Lee. 2022.  Learning Virtual Chimeras by Dynamic Motion Reassembly.  _ACM Transactions on Graphics_ 41, 6 (Nov. 2022), 182:1–182:13.  https://doi.org/10.1145/3550454.3555489
  * Loper et al. (2015) Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. 2015.  SMPL: A Skinned Multi-Person Linear Model.  _ACM Transactions on Graphics_ 34, 6 (Oct. 2015), 248:1–248:16.  https://doi.org/10.1145/2816795.2818013
  * Petrovich et al. (2022) Mathis Petrovich, Michael J. Black, and Gül Varol. 2022.  TEMOS: Generating Diverse Human Motions from Textual Descriptions. In _Computer Vision – ECCV 2022: 17th European Conference, Tel Aviv, Israel, October 23–27, 2022, Proceedings, Part XXII_. Springer-Verlag, Berlin, Heidelberg, 480–497.  https://doi.org/10.1007/978-3-031-20047-2_28
  * Petrovich et al. (2023) Mathis Petrovich, Michael J. Black, and Gül Varol. 2023.  TMR: Text-to-Motion Retrieval Using Contrastive 3D Human Motion Synthesis.  arXiv:2305.00976 [cs]
  * Poole et al. (2022) Ben Poole, Ajay Jain, Jonathan T. Barron, and Ben Mildenhall. 2022.  DreamFusion: Text-to-3D Using 2D Diffusion.  https://doi.org/10.48550/arXiv.2209.14988 arXiv:2209.14988 [cs, stat]
  * Radford et al. (2021) Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, Gretchen Krueger, and Ilya Sutskever. 2021.  Learning Transferable Visual Models From Natural Language Supervision.  https://doi.org/10.48550/arXiv.2103.00020 arXiv:2103.00020 [cs]
  * Rombach et al. (2022) Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bjorn Ommer. 2022.  High-Resolution Image Synthesis with Latent Diffusion Models. In _2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_. IEEE, New Orleans, LA, USA, 10674–10685.  https://doi.org/10.1109/CVPR52688.2022.01042
  * Scanlon et al. (2023) Mark Scanlon, Frank Breitinger, Christopher Hargreaves, Jan-Niclas Hilgert, and John Sheppard. 2023.  ChatGPT for Digital Forensic Investigation: The Good, the Bad, and the Unknown.  _Forensic Science International: Digital Investigation_ 46 (Oct. 2023), 301609.  https://doi.org/10.1016/j.fsidi.2023.301609
  * Soga et al. (2016) Asako Soga, Yuho Yazaki, Bin Umino, and Motoko Hirayama. 2016.  Body-Part Motion Synthesis System for Contemporary Dance Creation. In _ACM SIGGRAPH 2016 Posters_. ACM, Anaheim California, 1–2.  https://doi.org/10.1145/2945078.2945107
  * Song et al. (2022) Jiaming Song, Chenlin Meng, and Stefano Ermon. 2022.  Denoising Diffusion Implicit Models.  https://doi.org/10.48550/arXiv.2010.02502 arXiv:2010.02502 [cs]
  * Starke et al. (2019) Sebastian Starke, He Zhang, Taku Komura, and Jun Saito. 2019.  Neural State Machine for Character-Scene Interactions.  _ACM Transactions on Graphics_ 38, 6 (Nov. 2019), 209:1–209:14.  https://doi.org/10.1145/3355089.3356505
  * Starke et al. (2020) Sebastian Starke, Yiwei Zhao, Taku Komura, and Kazi Zaman. 2020.  Local Motion Phases for Learning Multi-Contact Character Movements.  _ACM Transactions on Graphics_ 39, 4 (July 2020), 54:1–54:13.  https://doi.org/10.1145/3386569.3392450
  * Starke et al. (2021) Sebastian Starke, Yiwei Zhao, Fabio Zinno, and Taku Komura. 2021.  Neural Animation Layering for Synthesizing Martial Arts Movements.  _ACM Transactions on Graphics_ 40, 4 (July 2021), 92:1–92:16.  https://doi.org/10.1145/3450626.3459881
  * Tevet et al. (2022a) Guy Tevet, Brian Gordon, Amir Hertz, Amit H. Bermano, and Daniel Cohen-Or. 2022a.  MotionCLIP: Exposing Human Motion Generation to CLIP Space. In _Computer Vision – ECCV 2022_ _(Lecture Notes in Computer Science)_ , Shai Avidan, Gabriel Brostow, Moustapha Cissé, Giovanni Maria Farinella, and Tal Hassner (Eds.). Springer Nature Switzerland, Cham, 358–374.  https://doi.org/10.1007/978-3-031-20047-2_21
  * Tevet et al. (2022b) Guy Tevet, Sigal Raab, Brian Gordon, Yoni Shafir, Daniel Cohen-or, and Amit Haim Bermano. 2022b.  Human Motion Diffusion Model. In _The Eleventh International Conference on Learning Representations_.
  * Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017.  Attention Is All You Need. In _Advances in Neural Information Processing Systems_ , Vol. 30. Curran Associates, Inc.
  * Yao et al. ([n. d.]) Heyuan Yao, Zhenhua Song, Yuyang Zhou, Tenglong Ao, Baoquan Chen, and Libin Liu. [n. d.].  MoConVQ: Unified Physics-Based Motion Control via Scalable Discrete Representations.  ([n. d.]).
  * Yuan et al. (2022) Ye Yuan, Jiaming Song, Umar Iqbal, Arash Vahdat, and Jan Kautz. 2022.  PhysDiff: Physics-Guided Human Motion Diffusion Model.  https://doi.org/10.48550/arXiv.2212.02500 arXiv:2212.02500 [cs]
  * Zeng et al. (2022) Ailing Zeng, Lei Yang, Xuan Ju, Jiefeng Li, Jianyi Wang, and Qiang Xu. 2022.  SmoothNet: A Plug-and-Play Network for Refining Human Poses in Videos. In _European Conference on Computer Vision_. Springer.
  * Zhang et al. (2018) He Zhang, Sebastian Starke, Taku Komura, and Jun Saito. 2018.  Mode-Adaptive Neural Networks for Quadruped Motion Control.  _ACM Transactions on Graphics_ 37, 4 (July 2018), 145:1–145:11.  https://doi.org/10.1145/3197517.3201366
  * Zhang et al. (2022) Mingyuan Zhang, Zhongang Cai, Liang Pan, Fangzhou Hong, Xinying Guo, Lei Yang, and Ziwei Liu. 2022.  MotionDiffuse: Text-Driven Human Motion Generation with Diffusion Model.  arXiv:2208.15001 [cs]
  * Zhao et al. (2022) Yuheng Zhao, Jinjing Jiang, Yi Chen, Richen Liu, Yalong Yang, Xiangyang Xue, and Siming Chen. 2022.  Metaverse: Perspectives from Graphics, Interactions and Visualization.  _Visual Informatics_ 6, 1 (March 2022), 56–67.  https://doi.org/10.1016/j.visinf.2022.03.002


# Lattice QCD calculation of $D_{s}^{*}$ radiative decay using the 2+1 Wilson
Clover fermion

Yu Meng<EMAIL_ADDRESS>School of Physics and Microelectronics, Zhengzhou
University, Zhengzhou, Henan 450001, China  Jin-Long Dang  School of Physics,
Peking University, Beijing 100871, China  Chuan Liu<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, China  Center for High
Energy Physics, Peking University, Beijing 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing 100871, China  Zhaofeng Liu
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
People’s Republic of China  Center for High Energy Physics, Peking University,
Beijing 100871, China  Tinghong Shen  School of Physics and Technology, Wuhan
University, Wuhan 430072, China  Haobo Yan  School of Physics, Peking
University, Beijing 100871, China  Ke-Long Zhang  Computer Network Information
Center, Chinese Academy of Sciences

(August 31, 2024)

###### Abstract

We perform a lattice calculation on the radiative decay of $D_{s}^{*}$ using
the 2+1 Wilson Clover gauge ensembles generated by CLQCD collaboration. The
radiative transition $D_{s}^{*}\rightarrow D_{s}\gamma$ and Dalitz decay
$D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$ are studied respectively. After a
continuum extrapolation using three lattice spacings, we obtain
$\Gamma(D_{s}^{*}\rightarrow D_{s}\gamma)=0.0549(54)$ keV, which is consistent
with previous lattice calculations but with much improved precision. The
Dalitz decay rate is also calculated for the first time and the ratio with the
radiative transition is found to be $R_{ee}=0.624(3)\%$. A total decay width
of $D_{s}^{*}$ can then be determined as 0.0587(54) keV taking into account
the experimental branching fraction. Combining with the most recent
experimental measurement on the branching fraction of the purely leptonic
decay $D_{s}^{+,*}\rightarrow e^{+}\nu_{e}$, we obtain
$f_{D_{s}^{*}}|V_{cs}|=(190.5^{+55.1}_{-41.7_{\textrm{stat.}}}\pm
12.6_{\textrm{syst.}})$ MeV, with a significantly improved systematic
uncertainty compared to $42.7_{\textrm{syst.}}$ obtained using previous
lattice prediction of total decay width $0.070(28)$ keV as input.

##  I Introduction

Testing the standard model precisely and searching for signals or even hints
for new physics beyond the standard model is one of the major goals of
contemporary particle physics. Various flavor-changing weak decay processes
can be used to extract relevant Cabibbo-Kobayashi-Maskawa(CKM) matrix elements
and then test them under 3 flavor unitarity, which has become a well-known and
extremely important direction in flavor physics [1]. For the electromagnetic
decays without involving the CKM matrix elements, the decay rate can be
measured experimentally and also be calculated theoretically, hence providing
an alternative and even more direct way to test the standard model. In recent
works on $\eta_{c}\rightarrow 2\gamma$ [2, 3], for example, the decay rate has
been calculated precisely and it appears to differ significantly from the
Particle Data Group’s reported value. It therefore leaves some interesting
physics in this channel.

In this work, we focus on the radiative decay of the excited strange charm
meson, the vector meson $D_{s}^{*}$ with quark content $c\bar{s}$. Though the
particle mass and the branching fraction of the known decay channels have been
measured, the total decay width of $D_{s}^{*}$ is not determined
experimentally. A possible way to extract the total decay width is to combine
the calculated partial decay width, such as $D_{s}^{*}\rightarrow D_{s}\gamma$
and its experimental branching fraction. On the theoretical side, an
impressive achievement comes from the lattice calculation [4], in which the
authors obtained the radiative decay width as $\Gamma(D_{s}^{*}\rightarrow
D_{s}\gamma)=0.066(26)$ keV. It thereby determines the total decay width
$\Gamma_{D_{s}^{*}}^{\textrm{total}}=0.070(28)$ keV. A most recent
experimental measurement on the branching fraction of the purely leptonic
decay $D_{s}^{*,+}\rightarrow e^{+}\nu_{e}$ gives
$\operatorname{Br}(D_{s}^{*,+}\rightarrow
e^{+}\nu_{e})=(2.1^{+1.2}_{-0.9_{\textrm{stat.}}}\pm
0.2_{\textrm{syst.}})\times 10^{-5}$ [5]. Combining this branching fraction
and the total decay width, it obtains
$f_{D_{s}^{*}}|V_{cs}|=(207.9^{+59.4}_{-44.6_{\textrm{stat.}}}\pm
42.7_{\textrm{syst.}})$ MeV. The systematic uncertainty comes from the
uncertainties in the experimental measurement and the total decay width. Since
the precision of the theoretical total decay width is about 40%, significantly
larger than the experimental 10%, it is therefore urgent to reduce the
theoretical uncertainty for a more precise extraction of the quantity
$f_{D_{s}^{*}}|V_{cs}|$.

The aim of this work is to further improve upon the previous lattice study of
this radiative decay. Several improvements are made to obtain a more accurate
result. i) We adopt a novel method to extract the on-shell transition factor.
The new method allows a calculation of an off-shell transition factor with
zero transfer momentum. When the continuous momentum extrapolation is
performed, the accuracy of the on-shell factor is well controlled by this
point; ii) We consider a large number of time separations between the initial
and final particles in our calculation. A correlated fit to a constant at
large time separation is performed and the excited-state contamination is well
removed; iii) We utilize three gauge ensembles with different lattice
spacings, the finest of which is 0.052 fm, leading to a well-controlled
continuous limit $a^{2}\rightarrow 0$. These efforts finally allow us to
obtain the decay width with a precision of about 9.8%.

The rest of this paper is organized as follows. In Sec. II, we introduce the
methodology for calculating the radiative decay width in this work. This
section is divided into three parts: in Sec. II.1 the theoretical framework is
given; in Sec. II.2 the hadronic function is extracted from the lattice data;
in Sec. II.3 the decay widths of $D_{s}^{*}\rightarrow D_{s}^{*}\gamma$ and
$D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$ are obtained using the form factors
calculated on the lattice. In Sec. III we give details of the simulations and
show the main results. This section is further divided into three parts: in
Sec. III.1 the numerical values of $D_{s}^{*}$ and $D_{s}$ masses, together
with the dispersion relation of $D_{s}$ particle are presented; in Sec. III.2
the results of $D_{s}^{*}\rightarrow D_{s}\gamma$ are obtained, a continuum
extrapolation under three lattice spacings is performed; in Sec. III.3 the
results of $D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$ are summarized; Finally, we
conclude in Sec. IV.

##  II Methodology

The lattice study on the radiative transition process is quite mature, either
for the traditional momentum extrapolation, or the latter twisted boundary
condition [6, 7]. We won’t go into these details in this paper. Instead, we
would proceed in another way, which is called the scalar function method in
recent years. Such a method has been widely applied to various physical
processes [8, 9, 10, 2, 11, 12, 13, 14, 15] and achieved great successes. For
more detailed derivations in this paper, we refer to the supplementary
materials in our previous study on the charmonium two-photon decay[2], where
the same parameterization of the form factor is utilized.

###  II.1 Scalar function method

We start with a Euclidean hadronic function in the infinite volume

$H_{\mu\nu}(\vec{x},t)=\langle
0|\mathcal{O}_{D_{s}}(\vec{x},t)J_{\nu}^{\textrm{em}}(0)|D_{s,\mu}^{*}(p^{\prime})\rangle,t>0$
(1)

where $J_{\nu}^{em}=\sum_{q}e_{q}\,\bar{q}\gamma_{\nu}q$
($e_{q}=2/3,-1/3,-1/3,2/3$ for $q=u,d,s,c$. The
$|D_{s,\mu}^{*}(p^{\prime})\rangle$ is a $D_{s}^{*}$ state with momentum
$p^{\prime}$ and $\mathcal{O}_{D_{s}}$ is the interpolating operator of
$D_{s}$. At large time $t$, the hadronic function is saturated by the single
$D_{s}$ state

$\displaystyle H_{\mu\nu}(x)\doteq
H_{\mu\nu}^{D_{s}}(x)=\int\frac{d^{3}\vec{p}}{(2\pi)^{3}}\frac{1}{2E_{D_{s}}}e^{-E_{D_{s}}t+i\vec{p}\cdot\vec{x}}$
$\displaystyle\times\langle
0|\mathcal{O}_{D_{s}}(0)|D_{s}(\vec{p})\rangle\langle
D_{s}(\vec{p})|J_{\nu}(0)|D_{s,\mu}^{*}(p^{\prime})\rangle$ (2)

Considering the following parameterizations

$\displaystyle\langle 0|\mathcal{O}_{D_{s}}(0)|D_{s}(\vec{p})\rangle$
$\displaystyle=$ $\displaystyle Z_{D_{s}}$ $\displaystyle\langle
D_{s}(p)|J_{\nu}^{\textrm{em}}(0)|D_{s,\mu}^{*}(p^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\frac{2V_{\textrm{eff}}(q^{2})}{m_{D_{s}}+m_{D_{s}^{*}}}\epsilon_{\mu\nu\alpha\beta}p_{\alpha}p^{\prime}_{\beta}$
(3)

where an effective transition factor $V_{\textrm{eff}}(q^{2})$ is introduced
and the square of transfer momentum $q^{2}$ is determined by
$q^{2}=(m_{D_{s}^{*}}-E_{D_{s}})^{2}-|\vec{p}|^{2}$ as $D_{s}^{*}$ is at rest.
Then, the spatial Fourier transform of $H_{\mu\nu}(\vec{x},t)$ yields

$\displaystyle\tilde{H}_{\mu\nu}(\vec{p},t)$
$\displaystyle\doteq\tilde{H}_{\mu\nu}^{D_{s}}(\vec{p},t)=\frac{1}{m_{D_{s}}+m_{D_{s}^{*}}}\frac{Z_{D_{s}}}{E_{D_{s}}}e^{-E_{D_{s}}t}$
(4)
$\displaystyle\times\epsilon_{\mu\nu\alpha\beta}p_{\alpha}p^{\prime}_{\beta}V_{\textrm{eff}}(q^{2})$

The conventional straightforward way to extract the $V_{\textrm{eff}}(q^{2})$
is to utilize the above equation with a series of nonzero lattice momentum
$\vec{p}=2\pi\vec{n}/L,\vec{n}\neq 0$. The on-shell transition factor
$V_{\textrm{eff}}(0)$ is obtained by the momentum extrapolation with these
discrete $q^{2}$. We remark that the specific momentum point $\vec{p}=0$
closest to the on-shell condition $q^{2}=0$ is missed in such a way.
Therefore, an extrapolation including this point will improve the precision
and, more importantly, the reliability of the momentum extrapolation. To this
end, we construct a scalar function given below

$\displaystyle\mathcal{I}(t,|\vec{p}|)$ $\displaystyle=$
$\displaystyle\frac{1}{m_{D_{s}^{*}}|\vec{p}|^{2}}\epsilon_{\mu\nu\alpha\beta}p_{\alpha}p^{\prime}_{\beta}\tilde{H}_{\mu\nu}(\vec{p},t)$
(5)

On the one hand, it relates to the transition factor $V_{\textrm{eff}}(q^{2})$
by

$\mathcal{I}(t,|\vec{p}|)=\frac{-2Z_{D_{s}}m_{D_{s}^{*}}}{m_{D_{s}}+m_{D_{s}^{*}}}V_{\textrm{eff}}(q^{2})\frac{e^{-E_{D_{s}}t}}{E_{D_{s}}}$
(6)

On the other hand, it can be calculated directly only using the hadronic
function $H_{\mu\nu}(\vec{x},t)$ as an input,

$\displaystyle\mathcal{I}(t,|\vec{p}|)=\int
d^{3}\vec{x}\frac{j_{1}(|\vec{p}||\vec{x}|)}{|\vec{p}||\vec{x}|}\epsilon_{\mu\nu\alpha
0}x_{\alpha}H_{\mu\nu}(\vec{x},t)$ (7)

where $j_{n}(x)$ are the spherical Bessel functions. Finally, it arrives at

$\displaystyle V_{\textrm{eff}}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{-(m_{D_{s}}+m_{D_{s}^{*}})E_{D_{s}}}{2Z_{D_{s}}m_{D_{s}^{*}}}e^{E_{D_{s}}t}$
(8) $\displaystyle\times$ $\displaystyle\int
d^{3}\vec{x}\frac{j_{1}(|\vec{p}||\vec{x}|)}{|\vec{p}||\vec{x}|}\epsilon_{\mu\nu\alpha
0}x_{\alpha}H_{\mu\nu}(\vec{x},t)$

It is easy to verify the momentum $|\vec{p}|=0$ is immediately accessible
since $j_{1}(x)/x$ tends to a finite value as $x\rightarrow 0$. The on-shell
transition factor $V_{\textrm{eff}}(0)$ can be determined by a general
polynomial extrapolation,

$V_{\textrm{eff}}(q^{2})=d_{0}+d_{1}\cdot\frac{q^{2}}{m_{D_{s}^{*}}^{2}}+d_{2}\cdot\frac{q^{4}}{m_{D_{s}^{*}}^{4}}+\mathcal{O}(q^{6}/m_{D_{s}^{*}}^{6})$
(9)

where the coefficients $d_{i}$ are introduced and $V_{\textrm{eff}}(0)\equiv
d_{0}$. Since $q^{2}=(m_{D_{s}^{*}}-m_{D_{s}})^{2}\equiv(\delta m)^{2}$ as
$\vec{p}=0$, the difference of $V_{\textrm{eff}}(0)$ and
$V_{\textrm{eff}}((\delta m)^{2})$ is thereby a very small quantity with the
consideration of $(\delta m)^{2}/m_{D_{s}^{*}}^{2}\sim 0.46\%$. It is
therefore expected that the extrapolation precision with $\vec{p}=0$ included
can be significantly improved.

###  II.2 Hadronic function $H_{\mu\nu}(\vec{x},t)$

The hadronic function $H_{\mu\nu}(\vec{x},t)$ can be extracted from a three-
point function $C_{\mu\nu}^{(3)}(\vec{x},t)$

$\displaystyle
C_{\mu\nu}^{(3)}(\vec{x},t)=\langle\mathcal{O}_{D_{s}^{+}}(\vec{x},t)J_{\nu}^{\textrm{em}}(0)\mathcal{O}_{D_{s,\mu}^{+,*}}^{\dagger}(-t)\rangle$
(10)

where interpolating operators are chosen as
$\mathcal{O}_{D_{s,\mu}^{+,*}}^{\dagger}=-\bar{c}\gamma_{\mu}s$ and
$\mathcal{O}_{D_{s}^{+}}=\bar{s}\gamma_{5}c$. If only the connected Wick
contractions are considered, then it has

$\displaystyle C_{\mu\nu}^{(3)}(\vec{x},t)=$ $\displaystyle-
e_{c}\langle\gamma_{5}\gamma_{\mu}S_{s}^{\dagger}(\vec{x},t;-t)S_{c}(\vec{x},t;0)\gamma_{\nu}S_{c}(0,-t)\rangle$
$\displaystyle+e_{s}\langle\gamma_{5}\gamma_{\mu}S_{s}^{\dagger}(0,-t)\gamma_{\nu}S_{s}^{\dagger}(\vec{x},t;0)S_{c}(\vec{x},t;-t)\rangle$
(11)

Thus, the hadronic function $H_{\mu\nu}(x)$ is given by

$\displaystyle
H_{\mu\nu}(\vec{x},t)=\frac{2m_{D_{s}}}{Z_{D_{s}}}e^{m_{D_{s}}t}C_{\mu\nu}^{(3)}(\vec{x},t)$
(12)

where $m_{D_{s}}$ and $Z_{D_{s}}$ are extracted from the two-point function
$C^{(2)}(\vec{p},t)=\sum\limits_{\vec{x}}\cos(\vec{p}\cdot\vec{x})\langle\mathcal{O}_{h}(\vec{x},t)\mathcal{O}_{h}^{\dagger}(0)\rangle$
by a single-state fit

$\displaystyle
C^{(2)}(\vec{p},t)=\frac{Z_{h}^{2}}{2E_{h}}\left(e^{-E_{h}t}+e^{-E_{h}(T-t)}\right)$
(13)

with $m_{h}=E_{h}(\vec{p}=0)$ the ground-state energy and $Z_{h}=\langle
h|\mathcal{O}^{\dagger}_{h}|0\rangle$ is the overlap amplitude for the ground
state. The symbol $h$ denotes the hadron, for example, $D_{s}$ or $D_{s}^{*}$
in this paper. For the computation of the three-point function
$C_{\mu\nu}^{(3)}(\vec{x},t)$, we place the point source propagator on the
current and wall source propagator on the initial hadron. All the propagators
are produced on a large number of time slices by average to increase the
statistics based on time translation invariance.

###  II.3 Decay width of $D_{s}^{*}\rightarrow D_{s}\gamma$ and
$D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$

The amplitude of $D_{s}^{+,*}\rightarrow\gamma D_{s}^{+}$ can be written as

$i\mathcal{M}(\lambda^{\prime},\lambda)=ie\epsilon^{\mu}_{D_{s}^{*}}(p^{\prime},\lambda^{\prime})\epsilon^{\nu}_{\gamma}(q,\lambda)\langle
D_{s}(p)|J_{\nu}^{\textrm{em}}(0)|D_{s,\mu}^{*}(p^{\prime})\rangle$ (14)

where $\epsilon^{\mu}_{D_{s}^{*}}(p^{\prime},\lambda^{\prime})$ is the
polarization vector of the $D_{s}^{*}$ particle and
$\epsilon^{\nu}_{\gamma}(q,\lambda)$ is the photon polarization with the four-
vector momentum $q=p^{\prime}-p$. These polarizations satisfy the following
identities:

$\displaystyle\sum\limits_{\lambda}\epsilon^{\mu}_{D_{s}^{*}}(p^{\prime},\lambda)\epsilon^{\nu}_{D_{s}^{*}}(p^{\prime},\lambda)$
$\displaystyle=$ $\displaystyle-
g_{\mu\nu}+\frac{p^{\prime}_{\mu}p^{\prime}_{\nu}}{m_{D_{s}^{*}}^{2}}$ (15)
$\displaystyle\sum\limits_{\lambda}\epsilon^{\mu}_{\gamma}(p,\lambda)\epsilon^{\nu}_{\gamma}(p,\lambda)$
$\displaystyle=$ $\displaystyle-g_{\mu\nu}$ (16)

Combining the parameterization in Eq. II.1, it immediately leads to the
following decay width

$\displaystyle\Gamma(D_{s}^{*}\rightarrow\gamma D_{s})$
$\displaystyle=\frac{1}{2m_{D_{s}^{*}}}\int\frac{d^{3}\vec{q}}{(2\pi)^{3}2|\vec{q}|}\int\frac{d^{3}\vec{p}}{(2\pi)^{3}2E_{D_{s}}}$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(p^{\prime}-p-q)\times\frac{1}{3}\sum\limits_{\lambda^{\prime}}\sum\limits_{\lambda}|\mathcal{M}(\lambda^{\prime},\lambda)|^{2}$
$\displaystyle=\frac{4}{3}\frac{\alpha(\delta
m)^{3}}{(m_{D_{s}}+m_{D_{s}^{*}})^{2}}|V_{\textrm{eff}}(0)|^{2}$ (17)

where $\alpha\equiv e^{2}/4\pi$. The factor $1/3$ in the third line denotes
the average over three polarizations of $D^{*}_{s}$ in its rest frame and the
final photon polarization has been summed up.

The amplitude of the Dalitz decay $D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$ has
additionally photon propagator $-ig_{\nu\nu^{\prime}}/q^{2}$ and leptonic
current $\bar{u}_{e}\gamma_{\nu^{\prime}}u_{e}$ compared with the amplitude of
the radiative decay $D_{s}^{*}\rightarrow D_{s}\gamma$ in Eq. (14). A direct
calculation similar to the above gives the Dalitz decay widths normalized to
the corresponding radiative decay as follows [16]

$\displaystyle R_{ee}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(D_{s}^{*}\rightarrow
D_{s}e^{+}e^{-})}{\Gamma(D_{s}^{*}\rightarrow D_{s}\gamma)}$ (18)
$\displaystyle=$
$\displaystyle\frac{\alpha}{3\pi}\int\frac{dq^{2}}{q^{2}}\Big{|}\frac{V_{\textrm{eff}}(q^{2})}{V_{\textrm{eff}}(0)}\Big{|}^{2}\left(1-\frac{4m_{e}^{2}}{q^{2}}\right)^{\frac{1}{2}}\left(1+\frac{2m_{e}^{2}}{q^{2}}\right)$
$\displaystyle\times$
$\displaystyle\left[\left(1+\frac{q^{2}}{m_{D_{s}^{*}}^{2}-m_{D_{s}}^{2}}\right)^{2}-\frac{4m_{D_{s}^{*}}^{2}q^{2}}{(m_{D_{s}^{*}}^{2}-m_{D_{s}}^{2})^{2}}\right]^{\frac{3}{2}}$

##  III Simulations and results

Ensemble | C24P29 | C32P30 | C48P32
---|---|---|---
$a(\textrm{fm})$ | 0.10530(18) | 0.07746(18) | 0.05187(26)
$a\mu_{s}$ | -0.2400 | -0.2050 | -0.1700
$a\mu_{c}$ | 0.4479 | 0.2079 | 0.0581
$L^{3}\times T$ | $24^{3}\times 72$ | $32^{3}\times 96$ | $48^{3}\times 144$
$N_{\textrm{cfg}}\times N_{\textrm{src}}$ | $450\times 72$ | $377\times 96$ | $306\times 72$
$m_{\pi}(\textrm{MeV})$ | 292.7(1.2) | 303.2(1.3) | 317.2(0.9)
$m_{J/\psi}(\textrm{MeV})$ | 3098.6(0.3) | 3094.9(0.4) | 3096.5(0.3)
$t$ | 3-18 | 2-22 | 8-30
$Z_{V}$ | 0.79814(23) | 0.83548(12) | 0.86855(04)
Table 1:  Parameters of gauge ensembles used in this work. From top to bottom,
we list the ensemble name, the lattice spacing $a$, the bare quark mass
including the strange quark $a\mu_{s}$ and charm quark $a{\mu}_{c}$, the
spatial and temporal lattice size $L$ and $T$, the number of the measurements
of the correlation function for each ensemble $N_{\textrm{cfg}}\times
N_{\textrm{src}}$, the pion mass $m_{\pi}$, the $J/\psi$ mass $m_{J/\psi}$,
the range of the time separation $t$ between the initial hadron and the
electromagnetic current, and the vector normalization constant $Z_{V}$. Here,
$L$, $T$ and $t$ are given in lattice units.

Figure 1:  For the ensembles of C24P29, C32P30, and C48P32 from left to right,
the mass spectra of $D_{s}^{*}$ and $D_{s}$ particles extracted from the two-
point functions calculated using wall source propagators (top) and the energy
levels of $D_{s}$ particle with different momentum
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$ extracted from two-point
functions calculated using point source propagators (bottom). The horizontal
gray bands denote the fitting ranges.

We employ three 2+1 flavor Wilson clover gauge ensembles generated by the
CLQCD collaboration with lattice spacings $a\approx 0.1053,0.0775,0.0519$ fm,
the parameters of which are shown in Table. 1. For more details, we refer to
Ref. [17]. All the ensembles have similar volumes and physical pion masses and
are expected to provide a fully well-controlled continuous extrapolation. The
bare valence charm quark mass has not been presented in the original paper, so
we determine its value by demanding the lattice result of $J/\psi$ mass to
reproduce its physical value. This is due to the fact that the annihilation
effect of $J/\psi$ particle is verified to be much smaller than the $\eta_{c}$
meson [18]. The latter is expected to cause a $1-4$ MeV mass shift.

###  III.1 Mass spectrum

Ensemble | C24P29 | C32P30 | C48P32
---|---|---|---
$m_{D_{s}^{*}}$(MeV) | 2086.5(1.2) | 2098.1(1.2) | 2117.1(2.5)
$m_{D_{s}}$(MeV) | 1998.2(0.5) | 1989.4(0.5) | 1983.8(0.6)
$Z_{D_{s}^{*}}$ | 0.1474(9) | 0.0858(5) | 0.0414(7)
$Z_{D_{s}}$ | 0.2175(5) | 0.1388(3) | 0.0722(2)
Table 2: Mass spectra $m_{D_{s}^{*}/D_{s}}$ and overlap function
$Z_{D_{s}^{*}/D_{s}}$ for $D_{s}^{*}$ and $D_{s}$ particles, which are
extracted from the two-point funtion calculated by the wall source
propagators. $\delta m\equiv m_{D_{s}^{*}}-m_{D_{s}}$ is the hyperfine
splitting.

The ground-state energies of the particle $D_{s}$ and $D_{s}^{*}$ are
extracted from the two-point functions which are calculated by the wall source
propagators. It is found in our study that the uncertainty is reduced by
$30\%\sim 60\%$ by using a wall propagator compared to that using the point
source propagator. For the determination of energy levels especially with
nonzero momenta, we calculated them directly by the point source propagators.
A single-state correlated fit with the formula Eq. (13) is utilized and the
numerical fitting results of the spectra are summarized in the Tab. 2 and Tab.
3. The effective levels of the particle $D_{s}$ and $D_{s}^{*}$ are both shown
in Fig. 1 for all the ensembles and the horizontal gray bands therein denote
the fitting center values and statistical errors estimated by the jackknife
method. The ground state masses are shown by the upper panels in Fig. 1 and
the energy levels with a series of momenta are illustrated in the lower
panels.

We also check the dispersion relation of $D_{s}$ particle using the energy
levels summarized in Tab. 3. This verification is crucial since the energy of
$D_{s}$ at nonzero momenta, namely, $E_{D_{s}}$, directly enters our
calculation of the transition factor in Eq. (8). It is found that the discrete
dispersion relation

$4\sinh^{2}\frac{E_{D_{s}}}{2}=4\sinh^{2}\frac{m_{D_{s}}}{2}+\mathcal{Z}_{\textrm{latt}}^{D_{s}}\cdot
4\sum\limits_{i}\sin^{2}\frac{\vec{p}_{i}}{2}$ (19)

describes the energies and momenta well and a nice linear behavior between
$4\sinh^{2}(E_{D_{s}}/2)$ and $4\sum\limits_{i}\sin^{2}(\vec{p}_{i}/2)$ is
obtained as illustrated in Fig. 2. The numerical values of the slope are well
consistent with one, leading to a well-satisfying discrete dispersion relation
in our simulations.

Figure 2:  The dispersion relation of $D_{s}$ meson for the ensembles C24P29, C32P30, and C48P32, respectively.  Ensemble | C24P29 | C32P30 | C48P32
---|---|---|---
$aE_{D_{s}}(|\vec{n}|^{2}=0)$ | 1.0657(4) | 0.7811(3) | 0.5214(2)
$aE_{D_{s}}(|\vec{n}|^{2}=1)$ | 1.0926(5) | 0.8035(3) | 0.5370(3)
$aE_{D_{s}}(|\vec{n}|^{2}=2)$ | 1.1187(6) | 0.8251(5) | 0.5523(5)
$aE_{D_{s}}(|\vec{n}|^{2}=3)$ | 1.1441(9) | 0.8461(8) | 0.5672(9)
$aE_{D_{s}}(|\vec{n}|^{2}=4)$ | 1.1671(14) | 0.8656(12) | 0.5821(18)
$\mathcal{Z}_{\textrm{latt}}^{D_{s}}$ | 1.0305(89) | 1.0230(90) | 1.0182(130)
Table 3: Numerical results $E_{D_{s}}(\vec{p})$ with
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$. The coefficient
$\mathcal{Z}_{\textrm{latt}}^{D_{s}}$ is defined by the discrete dispersion
relation. All the results are extracted from the two-point function calculated
by the point source propagators.

###  III.2 $D_{s}^{*}\rightarrow D_{s}\gamma$

Figure 3:  The transition factors $V_{c/s}(q^{2})$ with different momentum
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$. The horizontal gray bands
denote the fitting results.

There is a total of two contributions for the effective transition factor
$V_{\textrm{eff}}(q^{2})$, one is that the photon is radiated from the charm
quark, and the other is from a strange quark. This can also be read directly
from the Wick contraction in Eq. (II.2). In the following, we divide the
effective transition factor into two parts, $V_{c}(q^{2})$ and $V_{s}(q^{2})$.
The former denotes the photon has escaped from the charm quark, and the latter
from the strange quark. Specifically, it has

$V_{\textrm{eff}}(q^{2})=\frac{1}{3}V_{s}(q^{2})-\frac{2}{3}V_{c}(q^{2})$ (20)

Since these two parts come from different Wick contractions, each of them can
be calculated separately. The lattice results of $V_{c}(q^{2})$ and
$V_{s}(q^{2})$ as a function of the time separation $t$ are shown in Fig. 3,
together with a series of momenta
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$. Here we present all the
results from the three ensembles, i.e. C24P29, C32P30, and C48P32 from top to
bottom, respectively. It shows that $V_{c}(q^{2})$ and $V_{s}(q^{2})$ have
obvious $t$ dependence in the small time region, indicating sizable excited-
state effects associated with the initial and final hadrons. With enough time
intervals utilized in this work, we could observe obvious plateaus in a large
enough time region. Therefore, excited-state effects are well controlled in
our calculations. All results of $V_{c}(q^{2})$ and $V_{s}(q^{2})$ are
obtained by a correlated fit of the lattice data to a constant at a suitable
time region, which is denoted by the gray bands in the figure. The fitting
values are summarized in Table. 4.

Ensemble | C24P29 | C32P30 | C48P32
---|---|---|---
$V_{c}(|\vec{n}|^{2}=0)$ | 0.6406(35) | 0.7675(78) | 0.9898(106)
$V_{c}(|\vec{n}|^{2}=1)$ | 0.6170(37) | 0.7445(78) | 0.9547(106)
$V_{c}(|\vec{n}|^{2}=2)$ | 0.5945(36) | 0.7197(79) | 0.9192(107)
$V_{c}(|\vec{n}|^{2}=3)$ | 0.5738(38) | 0.6945(80) | 0.8825(118)
$V_{c}(|\vec{n}|^{2}=4)$ | 0.5464(43) | 0.6655(82) | 0.8361(149)
$V_{s}(|\vec{n}|^{2}=0)$ | 2.8381(123) | 2.6166(183) | 2.8021(246)
$V_{s}(|\vec{n}|^{2}=1)$ | 2.2767(90) | 2.1297(142) | 2.3074(190)
$V_{s}(|\vec{n}|^{2}=2)$ | 1.8327(72) | 1.7270(115) | 1.8928(156)
$V_{s}(|\vec{n}|^{2}=3)$ | 1.4789(69) | 1.4039(95) | 1.5519(144)
$V_{s}(|\vec{n}|^{2}=4)$ | 1.1724(69) | 1.1411(82) | 1.2703(154)
$V_{\textrm{eff}}(|\vec{n}|^{2}=0)$ | 0.5190(41) | 0.3605(58) | 0.2742(79)
$V_{\textrm{eff}}(|\vec{n}|^{2}=1)$ | 0.3476(32) | 0.2136(50) | 0.1327(68)
$V_{\textrm{eff}}(|\vec{n}|^{2}=2)$ | 0.2146(28) | 0.0959(47) | 0.0182(64)
$V_{\textrm{eff}}(|\vec{n}|^{2}=3)$ | 0.1104(28) | 0.0050(47) | -0.0711(67)
$V_{\textrm{eff}}(|\vec{n}|^{2}=4)$ | 0.0265(31) | -0.0633(48) | -0.1340(86)
Table 4: Numerical results of $V_{c}(q^{2})$, $V_{s}(q^{2})$, and
$V_{\textrm{eff}}(q^{2})$ with
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$.

Combining the transition factors $V_{c}(q^{2})$ and $V_{s}(q^{2})$, the
effective transition factor $V_{\textrm{eff}}(q^{2})$ can be calculated
immediately, the numerical values of which are summarized in Tab. 4. The on-
shell factor $V_{\textrm{eff}}(0)$ is then extracted by a momentum
extrapolation in $q^{2}\rightarrow 0$ with the five momentum modes taken into
account. The numerical results of the coefficients $d_{0},d_{1}$ and $d_{2}$
are summarized in Tab. 5. In this work, it is found that the polynomial
formula in Eq. (9) can describe the lattice data very well, as shown in Fig.
4. It is seen that due to similar masses of the initial and final particles,
i.e. $D_{s}^{*}$ and $D_{s}$, the on-shell transition factor is very close to
that with zero momentum. Therefore, the statistical error of the on-shell
transition factor is almost dominated by the error of the off-shell transition
factor with zero momentum.

Ensemble | C24P29 | C32P30 | C48P32
---|---|---|---
$d_{0}$ | 0.512(4) | 0.353(6) | 0.264(8)
$d_{1}$ | 3.177(30) | 2.682(34) | 2.638(58)
$d_{2}$ | 4.419(103) | 3.693(88) | 3.746(242)
Table 5: Numerical results of $d_{i}$ with $i=0,1,2$ are extracted by the
momentum extrapolation in Eq. (9). Figure 4:  The momentum extrapolation for
transition factors $V_{\textrm{eff}}(q^{2})$ with different momentum
$\vec{p}=2\pi\vec{n}/L,|\vec{n}|^{2}=0,1,2,3,4$. The black squares denote the
on-shell transition factor $V_{\textrm{eff}}(0)$. Figure 5:  The lattice
results of $V_{\textrm{eff}}(0)$ as a function of the lattice spacing. The
errors of lattice spacing have been included in the continuous limit, which
are presented by the horizontal error bars. The symbols of the blue cross and
black triangle indicate the results given by HPQCD and this work,
respectively.

The lattice results for the on-shell effective transition factor at different
lattice spacings are shown in Fig. 5, together with an extrapolation that is
linear in $a^{2}$. This linear behavior is expected since the ensembles used
in the work have adopted the tadpole-improved tree-level Symanzik gauge action
and the tadpole-improved tree-level Clover fermion action. It is also seen
that the fitting curves describe the lattice data well. After the continuous
extrapolation, we obtain

$V_{\textrm{eff}}(0)=0.178(9)$ (21)

Compared to the previous lattice calculation given by HPQCD [4], the charm
quark in our simulation still contains a relatively large discretization
error. However, with several improvements in our calculation, for example, the
scalar function methodology, the especially finer lattice spacings, and many
time separation utilized, we finally obtain $V_{\textrm{eff}}(0)$ with a
significantly improved precision, where the statistical error is more than 4
times smaller than before. Using the physical transition factor
$V_{\textrm{eff}}(0)=0.178(9)$ as input, and considering the physical masses
of $D_{s}^{*}$ and $D_{s}$, i.e. $m_{D_{s}^{*}}=2112.2$ MeV,
$m_{D_{s}}=1968.35$ MeV [19], the decay width of the radiative decay
$D_{s}^{*}\rightarrow\gamma D_{s}$ appears to be

$\Gamma(D_{s}^{*}\rightarrow\gamma D_{s})=0.0549(54)\leavevmode\nobreak\
\textrm{keV}$ (22)

where the error only comes from the statistical error of the transition factor
$V_{\textrm{eff}}(0)$ and the mass uncertainties of $D_{s}^{*}$ and $D_{s}$
are ignored. For the first time, the accuracy of lattice calculation is up to
a percent level. It is seen our current result is well consistent with
previous lattice calculations but with the statistical error of only a fifth
of the former.

With the input of the branching fraction
$\operatorname{Br}(D_{s}^{*}\rightarrow D_{s}\gamma)=93.5(7)\%$, it
immediately obtains the total decay width
$\Gamma^{\textrm{total}}_{D_{s}^{*}}=0.0587(54)$ keV. Recently, the BESIII has
reported the first experimental study of the purely leptonic decay
$D_{s}^{+,*}\rightarrow e^{+}\nu_{e}$, and gives the branching fraction of
this decay as $(2.1^{+1.2}_{-0.9_{\textrm{stat.}}}\pm
0.2_{\textrm{syst.}})\times 10^{-5}$ [5]. Combining this branching fraction
with our lattice calculation on the total decay width, we obtain
$f_{D_{s}^{*}}|V_{cs}|=(190.5^{+55.1}_{-41.7_{\textrm{stat.}}}\pm
12.6_{\textrm{syst.}})$ MeV, with a much smaller systematic error compared to
that extracted $42.7_{\textrm{syst.}}$ by previous lattice QCD prediction of
the total decay width $0.070(28)$ keV. At present, the updated systematic
uncertainty is mainly from the uncertainties in the measured
$\operatorname{Br}(D_{s}^{+,*}\rightarrow e^{+}\nu_{e})(9.5\%)$ and the LQCD
updated $\Gamma^{\textrm{total}}_{D_{s}^{*}}(9.2\%)$.

###  III.3 $D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$

Figure 6:  The lattice results of $R_{ee}$ as a function of the lattice spacing. The black triangle denotes the result in the continuous limit $a^{2}\rightarrow 0$. Ensemble | C24P29 | C32P30 | C48P32 | Cont.Limit
---|---|---|---|---
$R_{ee}\times 10^{3}$ | 5.66(2) | 5.93(2) | 6.10(3) | 6.24(3)
Table 6: Numerical value of $R_{ee}$ from three gauge ensembles, together with
the physical result in a continuous limit.

For the Dalitz decay of $D_{s}^{*}$, a virtual photon is internally converted
to a leptonic pair $l^{+}l^{-}$. Since the $\mu$ mass is larger than the mass
splitting of $D_{s}^{*}$ and $D_{s}$, the only possible decay mode is the
$e^{+}e^{-}$ pair. Taking into account the transition factor
$V_{\textrm{eff}}(q^{2})$ already obtained in the calculation of
$D_{s}^{*}\rightarrow D_{s}\gamma$, the ratio $R_{ee}$ defined in Eq. (18) is
calculated straightforwardly. The results of the ratio for all ensembles are
shown in Fig. 6 and the numerical values are presented in Tab. 6. A linear
behavior in $a^{2}$ can well describe the lattice data as expected. Finally,
we have

$R_{ee}=\frac{\Gamma(D_{s}^{*}\rightarrow
D_{s}e^{+}e^{-})}{\Gamma(D_{s}^{*}\rightarrow D_{s}\gamma)}=0.624(3)\%$ (23)

which is consistent with the PDG value $0.67(16)\%$. The errors of the
subtracting transition factors $V_{\textrm{eff}}(q^{2})$ and
$V_{\textrm{eff}}(0)$ almost completely cancel, eventually leading to a per
mill level calculation.

As far as we know, three main decay channels of $D_{s}^{*}$ are listed by PDG,
and they are $D_{s}^{*}\rightarrow D_{s}\gamma$, $D_{s}^{*}\rightarrow
D_{s}\pi^{0}$, and $D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$. Their respective
branching fractions are determined from only two relative ratios, i.e.
$R_{ee}=\Gamma(D_{s}^{*}\rightarrow
D_{s}e^{+}e^{-})/\Gamma(D_{s}^{*}\rightarrow D_{s}\gamma)$ and
$R_{Ds\pi^{0}}=\Gamma(D_{s}^{*}\rightarrow
D_{s}\pi^{0})/\Gamma(D_{s}^{*}\rightarrow D_{s}\gamma)$. By assuming the sum
of the three branching fractions is equal to 1, one has
$\operatorname{Br}(D_{s}^{*}\rightarrow
D_{s}\gamma)=1/(1+R_{ee}+R_{D_{s}\pi})$,
$\operatorname{Br}(D_{s}^{*}\rightarrow
D_{s}\pi^{0})=R_{D_{s}\pi^{0}}/(1+R_{ee}+R_{D_{s}\pi})$, and
$\operatorname{Br}(D_{s}^{*}\rightarrow
D_{s}e^{+}e^{-})=R_{ee}/(1+R_{ee}+R_{D_{s}\pi})$. It is seen that the quantity
$R_{ee}$ is directly measurable. Therefore, such a physical quantity
especially with ultra-high precision serves as an excellent ground for testing
the standard model. More accurate experimental measurements on $R_{ee}$ are
welcome in the future.

##  IV Conclusion

In this work, we present a lattice QCD calculation on the radiative decay of
$D_{s}^{*}$ particle. The transition process $D_{s}^{*}\rightarrow
D_{s}\gamma$ and Dalitz decay $D_{s}^{*}\rightarrow D_{s}e^{+}e^{-}$ are
studied respectively. Using the 2+1 Wilson Clover gauge ensembles under three
different lattice spacings, we finally obtain $\Gamma(D_{s}^{*}\rightarrow
D_{s}\gamma)=0.0549(54)$ keV, with an error significantly reduced compared to
previous lattice calculation. The Dalitz decay of $D_{s}^{*}$ is also studied
for the first time, and the ratio is obtained as $R_{ee}=0.624(3)\%$, with a
precision better than one percent.

To reach a percent level calculation, further improvements in several aspects
are adopted. Firstly, we utilize a scalar function method to calculate the
effective transition factor of $D_{s}^{*}\rightarrow D_{s}\gamma$, where the
zero transfer momentum can be projected directly without any ambiguity. As the
mass splitting of $D_{s}^{*}$ and $D_{s}$ is relatively small, the on-shell
transition factor is very close to that with zero momentum. Therefore, the
precision of the on-shell transition factor is almost dominated by the off-
shell factor with zero momentum after a momentum extrapolation. Secondly, a
large number of time separations $t$ have been utilized in our calculation,
the excited-state contamination caused by the initial and final states are
therefore removed and the transition factor is obtained by a correlated fit to
a constant at large $t$. Thirdly, we have used three ensembles with different
lattice spacings to perform a continuous limit, especially including a very
fine spacing with only 0.052 fm. Taking into account of the above-mentioned
improvements, we managed to obtain a result for the decay width in Eq. (22)
with an error of about 9.8%.

A precise determination of $D_{s}^{*}$ total decay width plays a vital role in
extracting the CKM matrix element $V_{cs}$. Combining with a recent
experimental measurement on $D_{s}^{*,+}\rightarrow e^{+}\nu_{e}$, the
quantity $f_{D_{s}^{*}}|V_{cs}|$ is estimated and found to be
$(190.5^{+55.1}_{-41.7_{\textrm{stat.}}}\pm 12.6_{\textrm{syst.}})$ MeV. The
systematic error here is significantly reduced compared to 42.7 which is
extracted using the previous lattice result
$\Gamma_{D_{s}^{*}}^{\textrm{total}}=0.070(28)$ keV as input. A further
improvement on the measurement of $D_{s}^{*}$ purely leptonic decay, both
statistically and systematically, is expected for the next generation of the
super-collider, such as Super Tau Charm Facility [20].

###### Acknowledgements.

We gratefully acknowledge the helpful discussions with CLQCD members. Y.M.
thanks Liuming Liu, Wei Sun, and Yi-Bo Yang for providing the helps on the
chroma software [21]. Y.M. and C.L. are supported by NSFC of China under Grant
No.12293060, No.12293063, No. 12305094, No.11935017, and No.12070131001. Z. L.
is supported by NSFC of China under Grant No. 12075253 and No. 12192264. The
calculation is supported by SongShan supercomputer at the National
Supercomputing Center in Zhengzhou.

## References

  * [1] Flavour Lattice Averaging Group (FLAG), Y. Aoki et al.,  Eur. Phys. J. C 82, 869 (2022), 2111.09849.
  * [2] Y. Meng, X. Feng, C. Liu, T. Wang, and Z. Zou,  Sci. Bull. 68, 1880 (2023), 2109.09381.
  * [3] HPQCD Collaboration, B. Colquhoun, L. J. Cooper, C. T. H. Davies, and G. P. Lepage,  Phys. Rev. D 108, 014513 (2023), 2305.06231.
  * [4] G. C. Donald, C. T. H. Davies, J. Koponen, and G. P. Lepage,  Phys. Rev. Lett. 112, 212002 (2014), 1312.5264.
  * [5] BESIII, M. Ablikim et al.,  Phys. Rev. Lett. 131, 141802 (2023), 2304.12159.
  * [6] P. F. Bedaque,  Phys. Lett. B 593, 82 (2004), nucl-th/0402051.
  * [7] G. M. de Divitiis, R. Petronzio, and N. Tantalo,  Phys. Lett. B 595, 408 (2004), hep-lat/0405002.
  * [8] X. Feng, Y. Fu, and L.-C. Jin,  Phys. Rev. D 101, 051502 (2020), 1911.04064.
  * [9] X. Feng, M. Gorchtein, L.-C. Jin, P.-X. Ma, and C.-Y. Seng,  Phys. Rev. Lett. 124, 192002 (2020), 2003.09798.
  * [10] X.-Y. Tuo, X. Feng, L.-C. Jin, and T. Wang,  Phys. Rev. D 105, 054518 (2022), 2103.11331.
  * [11] Z. Zou, Y. Meng, and C. Liu,  Chin. Phys. C 46, 053102 (2022), 2111.00768.
  * [12] Y. Fu, X. Feng, L.-C. Jin, and C.-F. Lu,  Phys. Rev. Lett. 128, 172002 (2022), 2202.01472.
  * [13] X.-Y. Tuo, X. Feng, and L.-C. Jin,  Phys. Rev. D 106, 074510 (2022), 2206.00879.
  * [14] N. Christ, X. Feng, L. Jin, C. Tu, and Y. Zhao,  Phys. Rev. Lett. 130, 191901 (2023), 2208.03834.
  * [15] Y. Meng,  Lattice QCD calculation of the invisible decay $J/\psi\rightarrow\gamma\nu\bar{\nu}$,  in 40th International Symposium on Lattice Field Theory, 2023, 2309.15436.
  * [16] L. Landsberg,  Physics Reports 128, 301 (1985).
  * [17] Z.-C. Hu et al.,  (2023), 2310.00814.
  * [18] L. Levkova and C. DeTar,  Phys. Rev. D 83, 074504 (2011), 1012.1837.
  * [19] Particle Date Group, R. L. Workman et al.,  Prog. Theor. Exp. Phys. 2022, 083C01 (2022).
  * [20] M. Achasov et al.,  Front. Phys. (Beijing) 19, 14701 (2024), 2303.15790.
  * [21] SciDAC, LHPC, UKQCD, R. G. Edwards and B. Joo,  Nucl. Phys. B Proc. Suppl. 140, 832 (2005), hep-lat/0409003.