# Self-supervised Cross-view Representation Reconstruction for
Change Captioning

Yunbin Tu1, Liang Li2,6111Corresponding authors, Li Su1,3111Corresponding
authors, Zheng-Jun Zha4, Chenggang Yan5,6, Qingming Huang1,2,3
1University of Chinese Academy of Sciences, Beijing, China
2Key Lab of Intelligent Information Processing, ICT, CAS, Beijing, China
3Peng Cheng Laboratory, Shenzhen, China
4University of Science and Technology of China, Hefei, China
5Hangzhou Dianzi University, Hangzhou, China
6Lishui Institute of Hangzhou Dianzi University, Hangzhou, China
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Change captioning aims to describe the difference between a pair of similar
images. Its key challenge is how to learn a stable difference representation
under pseudo changes caused by viewpoint change. In this paper, we address
this by proposing a self-supervised cross-view representation reconstruction
(SCORER) network. Concretely, we first design a multi-head token-wise matching
to model relationships between cross-view features from similar/dissimilar
images. Then, by maximizing cross-view contrastive alignment of two similar
images, SCORER learns two view-invariant image representations in a self-
supervised way. Based on these, we reconstruct the representations of
unchanged objects by cross-attention, thus learning a stable difference
representation for caption generation. Further, we devise a cross-modal
backward reasoning to improve the quality of caption. This module reversely
models a “hallucination” representation with the caption and “before”
representation. By pushing it closer to the “after” representation, we enforce
the caption to be informative about the difference in a self-supervised
manner. Extensive experiments show our method achieves the state-of-the-art
results on four datasets. The code is available at
https://github.com/tuyunbin/SCORER.

##  1 Introduction

Change captioning is a new task of vision and language, which requires not
only understanding the contents of two similar images, but also describing
their difference with natural language. In real world, this task brings a
variety of applications, such as generating elaborated reports about monitored
facilities [8, 10] and pathological changes [18, 14].

Figure 1: The examples of change captioning. (a) is from a surveillance scene
with underlying illumination change. (b) is from an image editing scene. (c)
shows that with both object move and moderate viewpoint change. (d) shows that
with both object move and extreme viewpoint change. Changed objects and
referents are shown in red and green boxes, respectively.

While single-image captioning is already regarded as a very challenging task,
change captioning carries additional difficulties. Simply locating
inconspicuous differences is one such challenge (Fig. 1 (a) (b)). Further, in
a dynamic environment, it is common to acquire two images under different
viewpoints, which leads to pseudo changes about objects’ scale and location
(Fig. 1 (c) (d)). As such, change captioning needs to characterize the real
change while resisting pseudo changes. To locate change, the most intuitive
way is to subtract two images [22, 7], but this risks computing difference
features with noise if two images are unaligned [31]. Recently, researchers
[25] find that same objects from different viewpoints would have similar
features, so they match object features between two images to predict
difference features. This paradigm has been followed by some of the recent
works [11, 24, 38, 31, 30].

Despite the progress, current match-based methods suffer from learning stable
difference features under pseudo changes. In detail, the matching is directly
modeled between two image features, usually by cross-attention. However, the
features of corresponding objects might shift under pseudo change. This case
is more severe under drastic viewpoint changes (Fig. 1 (d)). Such feature
shift appearing in most objects would overwhelm the local feature change, thus
making it less effective to directly match two images.

For this challenge, we have two new observations. (1) While the feature
difference might be ignored between a pair of similar images, it is hard to be
overwhelmed between two images from different pairs. As such, contrastive
difference learning between similar/dissimilar images can help the model focus
more on the change of feature and resist feature shift. (2) Pseudo changes are
essentially different distortions of objects, so they just construct cross-
view comparison between two similar images, rather than affecting their
similarity. Motivated by these, we study cross-view feature matching between
similar/dissimilar images, and maximize the alignment of similar ones, so as
to learn two view-invariant image representations. Based on these, we can
reconstruct the representations of unchanged objects and learn a stable
difference representation.

In this paper, we tackle the above challenge with a novel Self-supervised
CrOss-view REpresentation Reconstruction (SCORER) network, which learns a
stable difference representation while resisting pseudo changes for caption
generation. Concretely, given two similar images, we first devise a multi-head
token-wise matching (MTM) to model relationships between cross-view features
from similar/dissimilar images, via fully interacting different feature
subspaces. Then, by maximizing cross-view contrastive alignment of the given
image pair, SCORER learns their representations that are invariant to pseudo
changes in a self-supervised way. Based on these, SCORER mines their reliable
common features by cross-attention, so as to reconstruct the representations
of unchanged objects. Next, we fuse the representations into two images to
highlight the unchanged objects and implicitly infer the difference. Through
this manner, we can obtain the difference representation that not only
captures the change, but also conserves referent information, thus generating
a high-level linguistic sentence with a transformer decoder.

To improve the quality of sentence, we further design a cross-modal backward
reasoning (CBR) module. CBR first reversely produces a “hallucination”
representation with the full representations of sentence and “before” image,
where the “hallucination” is modeled based on the viewpoint of “before”. Then,
we push it closer to the “after” representation by maximizing their cross-view
contrastive alignment. Through this self-supervised manner, we ensure that the
generated sentence is informative about the difference.

Our key contributions are: (1) We propose SCORER to learn two view-invariant
image representations for reconstructing the representations of unchanged
objects, so as to model a stable difference representation under pseudo
changes. (2) We devise MTM to model relationships between cross-view images by
fully interacting their different feature subspaces, which plays a critical
role in view-invariant representation learning. (3) We design CBR to improve
captioning quality by enforcing the generated caption is informative about the
difference. (4) Our method performs favorably against the state-of-the-art
methods on four public datasets with different change scenarios.

Figure 2: The architecture of the proposed method, including a pre-trained CNN
model, the self-supervised cross-view representation reconstruction network, a
transformer decoder, and the cross-modal backward reasoning module.
$\tilde{X}^{\prime}_{bef}$ and $\tilde{X}^{\prime}_{aft}$ denote the “before”
and “after” image features from different pairs in the training batch. $B$ is
the batch size; $N$ indicates the feature number in each image.

##  2 Related Work

Change Captioning  is a new task in vision-language understanding and
generation [13, 19, 17, 29, 5, 35]. The pioneer works [10, 27] describe the
difference between two aligned images (Fig. 1 (a) (b)). Since there usually
exist viewpoint changes in a dynamic environment, recent works [22, 11]
collect two datasets to simulate moderate (Fig. 1 (c)) and extreme viewpoint
changes (Fig. 1 (d)). To describe the difference under viewpoint changes,
previous works [22, 15] compute the difference by direct subtraction, which
could compute difference with noise [25]. Recent methods [11, 24, 31, 30, 28,
39] directly match two images to predict difference features. However, due to
the influence of pseudo changes, these methods are hard to learn stable
difference features. In contrast, our SCORER first learns two view-invariant
image representations by maximizing their cross-view contrastive alignment.
Then, it mines their common features to reconstruct the representations of
unchanged objects, thus learning a stable difference representation for
caption generation. We note that the latest work [38] pre-trains the model
with three self-supervised tasks, in order to improve cross-modal alignment.
Different from it, we enforce the cross-modal alignment by implementing cross-
modal backward reasoning in a self-supervised way. Meanwhile, our overall
architecture is trained in an end-to-end manner, which improves the training
efficiency.

Token-wise Matching has been used in latest image/video retrieval works [37,
36] to compute cross-modal interaction between image/video and text features.
However, since pseudo changes would induce feature shift between object pairs,
it is insufficient to only match cross-view features at token level. Hence, we
further design a multi-head token-wise matching for finer-level interaction
between different feature subspaces of cross-view images. This is key to learn
the view-invariant representations.

Cross-modal Consistency Constraint  is to verify the quality of caption by
using it and “before” image to rebuild “after” image. This idea has been tried
by the latest works [7, 11]. However, both works only enforce the consistency
among the caption, the changed object in “before” and “after” images, while
ignoring the constraint for referents. If the changed object is similar to
other objects (Fig. 1 (c) (d)), describing both the change and its referent is
helpful to convey accurate change information. Considering this, we perform
backward reasoning with the full representations of “before” and “after”
images, which helps generate a high-level sentence about the change and its
referent.

##  3 Methodology

As shown in Fig. 2, our method consists of four parts: (1) A pre-trained CNN
encodes a pair of cross-view images into two representations. (2) The proposed
SCORER learns two view-invariant representations to reconstruct the
representations of unchanged objects and model the difference representation.
(3) A transformer decoder translates the difference representation into a
high-level linguistic sentence. (4) The proposed CBR improves the quality of
sentence via enforcing it to be informative about the difference.

###  3.1 Cross-view Image Pair Encoding

Formally, given a pair of images “before” $I_{bef}$ and “after” $I_{aft}$, we
utilize a pre-trained CNN model to extract their grid features, denoted as
$X_{bef}$ and $X_{aft}$, where $X\in\mathbb{R}^{C\times H\times W}$. $C$, $H$,
$W$ indicate the number of channels, height, and width. We first project both
representations into a low-dimensional embedding space of $\mathbb{R}^{D}$:

$\tilde{X}_{o}=\text{conv}_{2}({X}_{o})+pos({X}_{o}),\\\ $ (1)

where $o\in(bef,aft)$. $\text{conv}_{2}$ denotes a 2D-convolutional layer;
$pos$ is a learnable position embedding layer.

###  3.2 Self-supervised Cross-view Representation Reconstruction

The core module of SCORER is the multi-head token wise matching (MTM). MTM
aims to model relationships between cross-view images by performing fine-
grained interaction between different feature subspaces, which plays a key
role in view-invariant representation learning. In the following, we first
elaborate MTM and then describe how to use it for view-invariant
representation learning. Finally, we introduce how to reconstruct the
representations of unchanged objects for difference representation learning.

####  3.2.1 Multi-head Token-wise Matching.

We first introduce the single-head token-wise matching (TM) and then extend it
into the multi-head version. Formally, given a query $Q\in\mathbb{R}^{N\times
D}$ and a key $K\in\mathbb{R}^{N\times D}$, we first compute the similarity of
$i$-th query token with all key tokens and select the maximum one as its
token-wise maximum similarity with $K$. Then, we perform average pooling over
the token-wise maximum similarity of all query tokens to obtain the similarity
of $Q$ to $K$. By analogy, we compute the average token-wise maximum
similarity of $K$ to $Q$, which ensures capturing correct relationships
between them. The above computation is formulated as follows:

$\begin{gathered}\text{TM}(Q,K)=\left[\frac{1}{N}\sum_{i=1}^{N}\max_{j=1}^{N}\left(e_{i,j}\right)+\frac{1}{N}\sum_{j=1}^{N}\max_{i=1}^{N}\left(e_{i,j}\right)\right]/2,\\\
e_{i,j}=\left(q_{i}\right)^{\top}k_{j}.\end{gathered}$ (2)

Further, we extend TM into a multi-head version to jointly match different
feature subspaces of $Q$ and $K$, so as to perform fine-grained interaction
between them:

$\begin{gathered}\text{MTM}(Q,K)=\text{ Concat }_{i^{\prime}=1\ldots
h}\left(\text{ head }_{i^{\prime}}\right),\\\ \text{ head
}_{i^{\prime}}=\text{TM}\left(QW_{i^{\prime}}^{Q},KW_{i^{\prime}}^{K}\right).\end{gathered}$
(3)

####  3.2.2 View-invariant Representation Learning

In a training batch, we sample $B$ image pairs of “before” and “after”. For
$k$-th “before” image $\tilde{X}_{k}^{b}$, $k$-th “after” image
$\tilde{X}_{k}^{a}$ is its positive, while other “after” images will be the
negatives in this batch. First, we reshape $\tilde{X}\in\mathbb{R}^{D\times
H\times W}$ to $\tilde{X}\in\mathbb{R}^{N\times D}$, where $N=HW$ denotes the
number of features. Then, we use MTM to compute similarity ($B\times B$
matrix) of “before” to “after” and “after” to “before”, respectively. Next, we
maximize cross-view contrastive alignment between $\tilde{X}_{k}^{b}$ and
$\tilde{X}_{k}^{a}$ while minimizing the alignment of non-similar images, by
the InfoNCE loss [20]:

$\begin{gathered}\mathcal{L}_{b2a}=-\frac{1}{B}\sum_{k}^{B}\log\frac{\exp\left(\text{MTM}\left(\tilde{X}_{k}^{b},\tilde{X}_{k}^{a}\right)/\tau\right)}{\sum_{r}^{B}\exp\left(\text{MTM}\left(\tilde{X}_{k}^{b},\tilde{X}_{r}^{a}\right)/\tau\right)},\\\
\mathcal{L}_{a2b}=-\frac{1}{B}\sum_{k}^{B}\log\frac{\exp\left(\text{MTM}\left(\tilde{X}_{k}^{a},\tilde{X}_{k}^{b}\right)/\tau\right)}{\sum_{r}^{B}\exp\left(\text{MTM}\left(\tilde{X}_{k}^{a},\tilde{X}_{r}^{b}\right)/\tau\right)},\\\
\mathcal{L}_{\text{cv
}}=\frac{1}{2}(\mathcal{L}_{b2a}+\mathcal{L}_{a2b}),\end{gathered}$ (4)

where $\tau$ is the temperature hyper-parameter. In this self-supervised way,
we can make the representations of $\tilde{X}_{bef}$ and $\tilde{X}_{aft}$
invariant to pseudo changes, so as to facilitate the following cross-view
representation reconstruction.

####  3.2.3 Cross-view Representation Reconstruction

Based on the two view-invariant representations $\tilde{X}_{bef}$ and
$\tilde{X}_{aft}$, we use a multi-head cross-attention (MHCA) [33] to mine
their common features for reconstructing the representations of unchanged
objects in each image. Here, representation reconstruction indicates that the
unchanged representations of each image are distilled from the other one,
e.g., the unchanged representations of $\tilde{X}_{bef}$ are computed by
transferring similar features on $\tilde{X}_{aft}$ back to the corresponding
positions on $\tilde{X}_{bef}$. In this way, we reconstruct the unchanged
representations for each image, respectively:

$\displaystyle\tilde{X}_{bef}^{u}=\text{MHCA
}(\tilde{X}_{bef},\tilde{X}_{aft},\tilde{X}_{aft}),$ (5)
$\displaystyle\tilde{X}_{aft}^{u}=\text{
MHCA}(\tilde{X}_{aft},\tilde{X}_{bef},\tilde{X}_{bef}).$

Then, instead of subtracting them from image representations [25, 31, 30],
which leads to information (_e.g.,_ referents) loss, we integrate them into
image representations to highlight the unchanged objects and deduce the
difference information, so as to learn the stable difference representation in
each image:

$\tilde{X}_{o}^{c}=\text{LN}(\tilde{X}_{o}+\tilde{X}_{o}^{u}).\\\ $ (6)

Herein, $o\in(bef,aft)$ and LN is short for LayerNorm [2]. Finally, we obtain
the difference representation between two images by fusing
$\tilde{X}_{bef}^{c}$ and $\tilde{X}_{aft}^{c}$, which is implemented by a
fully-connected layer with the ReLU function:

$\tilde{X}_{c}=\operatorname{ReLU}\left(\left[\tilde{X}_{bef}^{c};\tilde{X}_{aft}^{c}\right]W_{h}+b_{h}\right),$
(7)

where [;] is a concatenation operation.

###  3.3 Caption Generation

After leaning $\tilde{X}_{c}\in\mathbb{R}^{N\times D}$, we use a transformer
decoder [33] to translate it into a sentence. First, the multi-head self-
attention takes the word features $E[W]=\\{E[w_{1}],...,E[w_{m}]\\}$ (ground-
truth words during training, predicted words during inference) as inputs and
computes a set of intra-relation embedded word features, denoted as
$\hat{E}[W]$. Then, the decoder utilizes $\hat{E}[W]$ to query the most
related features $\hat{H}$ from $\tilde{X}_{c}$ via the multi-head cross-
attention. Afterward, the $\hat{H}$ is passed to a feed-forward network to
obtain an enhanced representation $\tilde{H}$. Finally, the probability
distributions of target words are calculated by:

$\tilde{W}=\operatorname{Softmax}\left(\tilde{H}W_{c}+{b}_{c}\right),$ (8)

where $W_{c}\in\mathbb{R}^{D\times U}$ and $b_{c}\in\mathbb{R}^{U}$ are the
parameters to be learned; $U$ is the dimension of vocabulary size.

###  3.4 Cross-modal Backward Reasoning

To improve the quality of generated sentence, we devise the CBR to first
reversely model a “hallucination” representation with the sentence and
“before” image. Then, we push it closer to the “after” representation to
enforce the sentence to be informative about the difference. Concretely, we
first fuse $\tilde{H}\in\mathbb{R}^{m\times D}$ by the mean-pooling operation
to obtain a sentence feature $\tilde{T}$. Then, we broadcast
$\tilde{T}\in\mathbb{R}^{D}$ as $\tilde{T}\in\mathbb{R}^{D\times H\times W}$
and concatenate it with $\tilde{X}_{bef}$, so as to obtain the “hallucination”
$\hat{X}_{hal}$:

$\hat{X}_{hal}=\text{conv}_{2}([\tilde{X}_{bef};\tilde{T}]),\hat{X}_{hal}\in\mathbb{R}^{D\times
H\times W}.$ (9)

$\hat{X}_{hal}$ and $\tilde{X}_{bef}$ are kept as the same shape to ensure
that spatial information is not collapsed. Next, we capture the relationships
between different locations in $\hat{X}_{hal}$ based on the multi-head self-
attention (MHSA), which is essential for backward reasoning and computed by:

$\tilde{X}_{hal}=\text{conv}_{2}[\text{MHSA
}(\hat{X}_{hal},\hat{X}_{hal},\hat{X}_{hal})],$ (10)

Since the “hallucination” representation is produced based on the viewpoint of
“before” representation, it is less effective to directly match it with the
“after” representation.

To this end, we sample unrelated representations of “hallucination” and
“after” from different pairs, which are as erroneous candidates for CBR.
Similarly, in each batch, for $k$-th “hallucination” $\tilde{X}_{k}^{h}$,
$k$-th “after” $\tilde{X}_{k}^{a}$ is its positive, while the other “after”
images will be the negatives. Also, we use MTM to capture relationships
between positive/negative pairs. Subsequently, we maximize cross-view
contrastive alignment of positive pairs by the InfoNCE loss [20], which is
similar to Eq. (4):

$\mathcal{L}_{\text{cm}}=\frac{1}{2}(\mathcal{L}_{h2a}+\mathcal{L}_{a2h}).$
(11)

Through this self-supervised manner, we make the sentence sufficiently
describe the difference information.

###  3.5 Joint Training

The proposed overall network is trained in an end-to-end manner by maximizing
the likelihood of the observed word sequence. Given the ground-truth words
$\left(w_{1}^{*},\ldots,w_{m}^{*}\right)$, we minimize the negative log-
likelihood loss:

$\mathcal{L}_{cap}(\theta)=-\sum_{t=1}^{m}\log p_{\theta}\left(w_{t}^{*}\mid
w_{<t}^{*}\right),$ (12)

where $p_{\theta}\left(w_{t}^{*}\mid w_{<t}^{*}\right)$ is computed by Eq.
(8), and $\theta$ are the parameters of the network. Besides, the network is
self-supervised by the losses of two contrastive alignments. Hence, the final
loss function is optimized as follows:

$\mathcal{L}=\mathcal{L}_{cap}+\lambda_{v}\mathcal{L}_{\text{cv}}+\lambda_{m}\mathcal{L}_{\text{cm}},$
(13)

where $\lambda_{v}$ and $\lambda_{m}$ are the trade-off parameters, which are
discussed in the supplementary material.

##  4 Experiments

###  4.1 Datasets

CLEVR-Change is a large-scale dataset [22] with moderate viewpoint change. It
has 79,606 image pairs, including five change types, _i.e._ , “Color”,
“Texture”, “Add”, “Drop”, and “Move”.We use the official split with 67,660 for
training, 3,976 for validation and 7,970 for testing.

CLEVR-DC is a large-scale dataset [11] with extreme viewpoint shift. It
includes 48,000 pairs with same change types as CLEVR-Change. We use the
official split with 85% for training, 5% for validation, and 10% for testing.

Image Editing Request dataset [27] includes 3,939 aligned image pairs with
5,695 editing instructions. We use the official split with 3,061 image pairs
for training, 383 for validation, and 495 for testing.

Spot-the-Diff dataset [10] includes 13,192 aligned image pairs from
surveillance cameras. Following SOTA methods, we mainly evaluate our model in
a single change setting. Based on the official split, the dataset is split
into training, validation, and testing with a ratio of 8:1:1.

###  4.2 Evaluation Metrics

Following the current state-of-the-art methods, five metrics are used to
evaluate the generated sentences, _i.e._ , BLEU-4 (B) [21], METEOR (M) [3],
ROUGE-L (R) [16], CIDEr (C) [34], and SPICE (S) [1]. The results are computed
based on the Microsoft COCO evaluation server [4].

###  4.3 Implementation Details

For a fair comparison, we follow the SOTA methods to use a pre-trained
ResNet-101 [6] to extract grid features of an image pair, with the dimension
of 1024 $\times$ 14 $\times$ 14\. We first project these features into a lower
dimension of 512. The hidden size in the overall model and word embedding size
are set to 512 and 300. The proper head and layer numbers of SCORER are
discussed below. The head and layer numbers in the decoder are set to 8 and 2
on the four datasets. During training, We use Adam optimizer [12] to minimize
the negative log-likelihood loss of Eq. (13). During inference, the greedy
decoding strategy is used to generate captions. Both training and inference
are implemented with PyTorch [23] on an RTX 3090 GPU. More implementation
details are described in the supplementary material.

###  4.4 Performance Comparison

####  4.4.1 Results on the CLEVR-Change Dataset.

We compare with the state-of-the-art methods in: 1) total performance under
both semantic and pseudo changes; 2) semantic change; 3) different change
types. The comparison methods are categorized into 1) end-to-end training:
DUDA [22], DUDA+ [7], R3Net+SSP [31], VACC [11], SRDRL+AVS [32], SGCC [15],
MCCFormers-D [24], IFDC [9], BDLSCR [26], NCT [30], and VARD-Trans [28]; 2)
reinforcement learning: M-VAM+RAF [25]; 3) pre-training: PCL w/ pre-training
[38].

| Total | Semantic Change
---|---|---
Method | B | M | R | C | S | B | M | R | C | S
PCL w/ Pre-training (AAAI 2022) [38] | 51.2 | 36.2 | 71.7 | 128.9 | - | - | - | - | - | -
M-VAM+RAF (ECCV 2020) [25] | 51.3 | 37.8 | 70.4 | 115.8 | 30.7 | - | - | - | - | -
DUDA (ICCV 2019) [22] | 47.3 | 33.9 | - | 112.3 | 24.5 | 42.9 | 29.7 | - | 94.6 | 19.9
DUDA+ (CVPR 2021) [7] | 51.2 | 37.7 | 70.5 | 115.4 | 31.1 | 49.9 | 34.3 | 65.4 | 101.3 | 27.9
R3Net+SSP (EMNLP 2021) [31] | 54.7 | 39.8 | 73.1 | 123.0 | 32.6 | 52.7 | 36.2 | 69.8 | 116.6 | 30.3
VACC (ICCV 2021) [11] | 52.4 | 37.5 | - | 114.2 | 31.0 | - | - | - | - | -
SGCC (ACM MM 2021) [15] | 51.1 | 40.6 | 73.9 | 121.8 | 32.2 | - | - | - | - | -
SRDRL+AVS (ACL 2021) [32] | 54.9 | 40.2 | 73.3 | 122.2 | 32.9 | 52.7 | 36.4 | 69.7 | 114.2 | 30.8
MCCFormers-D (ICCV 2021) [24] | 52.4 | 38.3 | - | 121.6 | 26.8 | - | - | - | - | -
IFDC (TMM 2022) [9] | 49.2 | 32.5 | 69.1 | 118.7 | - | 47.2 | 29.3 | 63.7 | 105.4 | -
NCT (TMM 2023) [30] | 55.1 | 40.2 | 73.8 | 124.1 | 32.9 | 53.1 | 36.5 | 70.7 | 118.4 | 30.9
VARD-Trans (TIP 2023) [28] | 55.4 | 40.1 | 73.8 | 126.4 | 32.6 | - | - | - | - | -
SCORER (Ours) | 55.8 | 40.8 | 74.0 | 126.0 | 33.0 | 54.1 | 37.4 | 71.5 | 122.0 | 31.2
SCORER+CBR (Ours) | 56.3 | 41.2 | 74.5 | 126.8 | 33.3 | 54.4 | 37.6 | 71.7 | 122.4 | 31.6
Table 1: Comparison with the state-of-the-art methods on CLEVR-Change under the settings of total performance and semantic change. | CIDEr
---|---
Method | CL | T | A | D | MV
PCL w/ PT | 131.2 | 101.1 | 133.3 | 116.5 | 81.7
M-VAM+RAF | 122.1 | 98.7 | 126.3 | 115.8 | 82.0
DUDA | 120.4 | 86.7 | 108.2 | 103.4 | 56.4
DUDA+ | 120.8 | 89.9 | 119.8 | 123.4 | 62.1
R3Net+SSP | 139.2 | 123.5 | 122.7 | 121.9 | 88.1
SRDRL+AVS | 136.1 | 122.7 | 121.0 | 126.0 | 78.9
BDLSCR | 136.1 | 122.7 | 121.0 | 126.0 | 78.9
IFDC | 133.2 | 99.1 | 128.2 | 118.5 | 82.1
NCT | 140.2 | 128.8 | 128.4 | 129.0 | 86.0
SCORER | 143.2 | 135.2 | 129.4 | 132.6 | 91.6
SCORER+CBR | 146.2 | 133.7 | 131.1 | 133.9 | 92.2
Table 2: A detailed breakdown of evaluation on CLEVR-Change with different
change types: “(CL) Color”, “(T) Textur”, “(A) Add”, “(D) Drop”, and “(MV)
Move”. PT is short for pre-training.

In Table 1, our method achieves the best results on all metrics against the
end-to-end training methods. Besides, our method performs much better than
these two methods augmented by pre-training and reinforcement learning. We
note that SCORER outperforms MCCFormers-D by a large margin. MCCFormers-D is a
classic match-based method that directly correlates two image representations
to learn a difference representation, which is then fed into a transformer
decoder for caption generation. Different from it, our SCORER first learns two
view-invariant image representations by maximizing their cross-view
contrastive alignment. Then, SCORER reconstructs the representations of
unchanged objects, so as to learn a stable difference representation under
pseudo changes for caption generation.

In Table 2, under the detailed change types, our method surpasses the current
methods by a large margin in almost every category. Under the most difficult
type “Move”, our SCORER+CBR achieves the relative improvement of 4.7% against
R3Net+SSP. This validates the necessary of view-invariant representation
learning. Moreover, under different settings, CBR helps yield an extra
performance boost, which shows it does improve captioning quality.

####  4.4.2 Results on the CLEVR-DC Dataset

On CLEVR-DC with extreme viewpoint changes, we compare SCORER/SCORER+CBR with
several state-of-the-art methods: DUDA/DUDA+CC [22], M-VAM/M-VAM+CC [25],
VA/VACC [11], MCCFormers-D [24], NCT [30], and VARD-Trans [28]. For fair-
comparison, we compare them based on the usage of cross-modal consistency
constraint. We implement MCCFormers-D based on the released code on CLEVR-DC
and Image Editing Request datasets.

The results are shown in Table 3. Our SCORER achieves the best results on most
metrics. This benefits from learning two view-invariant representations to
reconstruct representations of unchanged objects, thus learning a stable
difference representation under extreme viewpoint changes. When we implement
CBR, the performance of SCORER+CBR is further boosted, especially achieving
16.7% improvement against VACC on CIDEr. This shows that our CBR can calibrate
the model to generate a linguistic sentence describing the change and its
referent.

Method | B | M | C | S
---|---|---|---|---
DUDA [22] | 40.3 | 27.1 | 56.7 | 16.1
M-VAM [25] | 40.9 | 27.1 | 60.1 | 15.8
VA [11] | 44.5 | 29.2 | 70.0 | 17.1
MCCFormers-D [24] | 46.9 | 31.7 | 71.6 | 14.6
NCT [30] | 47.5 | 32.5 | 76.9 | 15.6
VARD-Trans [28] | 48.3 | 32.4 | 77.6 | 15.4
SCORER | 49.5 | 33.4 | 82.4 | 15.8
DUDA+CC [22] | 41.7 | 27.5 | 62.0 | 16.4
M-VAM+CC [25] | 41.0 | 27.2 | 62.0 | 15.7
VACC [11] | 45.0 | 29.3 | 71.7 | 17.6
SCORER+CBR | 49.4 | 33.4 | 83.7 | 16.2
Table 3: Comparison with the SOTA methods on CLEVR-DC.

####  4.4.3 Results on the Image Editing Reques Dataset

To validate the generalization of our method, we conduct the experiment on a
challenging dataset of Image Editing Request (IER). We compare with the
following SOTA methods: DUDA [22], Dyn rel-att [27], MCCFormers-D [24], BDLSCR
[26], NCT [30], and VARD-Trans [28].

Table 11 shows SCORER+CBR outperforms the SOTA methods on most metrics.
Especially on BLEU-4, SCORER+CBR obtains the relative improvement of 23.5%
against the latest method NCT (TMM 2023). The edited objects are usually
inconspicuous. This indicates that the proposed method can fully mine the
common features by maximizing cross-view contrastive alignment between two
images, so as to accurately describe which part of the “before” image has been
edited. Further, the generated sentence is refined in the process of cross-
modal backward reasoning.

Method | B | M | R | C
---|---|---|---|---
DUDA [22] | 6.5 | 12.4 | 37.3 | 22.8
Dyn rel-att [27] | 6.7 | 12.8 | 37.5 | 26.4
MCCFormers-D [24] | 8.3 | 14.3 | 39.2 | 30.2
BDLSCR [26] | 6.9 | 14.6 | 38.5 | 27.7
NCT [30] | 8.1 | 15.0 | 38.8 | 34.2
VARD-Trans [28] | 10.0 | 14.8 | 39.0 | 35.7
SCORER | 9.6 | 14.6 | 39.5 | 31.0
SCORER+CBR | 10.0 | 15.0 | 39.6 | 33.4
Table 4: Comparison with the SOTA methods on IER.

####  4.4.4 Results on the Spot-the-Diff Dataset

To further validate the generalization, we conduct the experiment on Spot-the-
Diff that includes aligned image pairs from the surveillance cameras. The
following SOTA methods are compared: DUDA+ [7], M-VAM/M-VAM+RAF [25], VACC
[11], SRDRL+AVS [32], MCCFormers-D [24], IFDC [9], BDLSCR [26], and VARD-Trans
[28].

In Table 5, our method achieves superior results on most metrics, which shows
its generalization on different scenarios. Besides, our method performs lower
on METEOR and SPICE when implementing CBR. Our conjecture is that image pairs
on this dataset actually contain one or more changes. For fair-comparison, we
conduct experiments mainly based on the single-change setup. This makes the
“hallucination” representation, which is reversely modeled by the “before”
representation and single-change caption, not fully matched with the “after”
representation. As such, SCORER+CBR does not gain significant improvement.

In short, compared with the state-of-the-art methods in different change
scenarios, our method achieves the impressive performance. The superiority
mainly results from that 1) SCORER learns two view-invariant image
representations for reconstructing the representations of unchanged objects,
so as to learn a stable difference representation for generating a linguistic
sentence; 2) CBR can further improve the quality of generated sentence.

Method | B | M | C | S
---|---|---|---|---
M-VAM+RAF [25] | 11.1 | 12.9 | 43.5 | 17.1
M-VAM [25] | 10.1 | 12.4 | 38.1 | 14.0
DUDA+ [7] | 8.1 | 12.5 | 34.5 | -
VACC [11] | 9.7 | 12.6 | 41.5 | -
SRDRL+AVS [32] | - | 13.0 | 35.3 | 18.0
MCCFormers-D [24] | 10.0 | 12.4 | 43.1 | 18.3
IFDC [9] | 8.7 | 11.7 | 37.0 | -
BDLSCR [26] | 6.6 | 10.6 | 42.2 | -
VARD-Trans [28] | - | 12.5 | 30.3 | 17.3
SCORER | 9.4 | 13.8 | 38.5 | 19.3
SCORER+CBR | 10.2 | 12.2 | 38.9 | 18.4
Table 5: Comparison with the SOTA methods on Spot-the-Diff. Ablation | B | M | R | C | S
---|---|---|---|---|---
Subtraction | 53.3 | 38.8 | 72.1 | 119.7 | 31.8
RR | 55.1 | 40.5 | 73.6 | 123.8 | 32.5
SCORER | 55.8 | 40.8 | 74.0 | 126.0 | 33.0
RR+CBR | 55.8 | 41.0 | 74.2 | 125.5 | 32.9
SCORER+CBR | 56.3 | 41.2 | 74.5 | 126.8 | 33.3
Table 6: Ablation on CLEVR-Change under Total Performance. | Semantic Change | Only Pseudo Change
---|---|---
Method | B | M | R | C | S | B | M | R | C | S
Subtraction | 50.2 | 34.1 | 67.1 | 108.0 | 28 | 57.3 | 48.4 | 74.7 | 113.8 | 34.0
RR | 53.3 | 37.1 | 70.8 | 119.1 | 30.4 | 61.1 | 50.7 | 76.4 | 114.9 | 34.6
SCORER | 54.3 | 37.5 | 71.5 | 122.0 | 31.2 | 61.4 | 50.6 | 76.5 | 116.4 | 34.7
RR+CBR | 54.1 | 37.4 | 71.5 | 122.4 | 31.2 | 60.7 | 51.2 | 76.9 | 114.9 | 34.6
SCORER+CBR | 54.4 | 37.6 | 71.7 | 122.4 | 31.6 | 62.0 | 51.7 | 77.4 | 117.9 | 35.0
Table 7: Ablation study on CLEVR-Change under the evaluation of semantic
change and only pseudo change.

###  4.5 Ablation Study and Analysis

Ablation Study of Each Module on CLEVR-Change. Table 6 shows ablation study of
each module under total performance. Subtraction indicates directly
subtracting two images; RR means vanilla representation reconstruction. We
find that RR is much better than Subtraction, showing that match-based
strategy is more reliable than direct subtraction under pseudo changes. When
we maximize cross-view contrastive alignment of two images, SCORER yields a
further performance boost. This shows that it is important to learn the
representations invariant under pseudo changes, which is key to learn a stable
difference representation. Besides, when we augment RR and SCORER with CBR,
both RR+CBR and SCORER+CBR achieve better performances. This not only
validates that CBR improves captioning quality, but also proves that CBR is
generalizable.

Table 7 shows the ablation study of each module under semantic change and only
pseudo change, separately. We can obtain observations similar to the total
performance. Besides, we find that SCORER is much better than RR under
semantic change, but under only pseudo change, SCORER brings less gain. This
results from that in this case, the learned difference representation contains
less information, making SCORER difficult to align it with words. By contrast,
SCORER+CBR significantly improves RR on both settings, which shows that SCORER
and CBR supplement each other. More ablation studies on the other datasets are
in the supplementary material.

Figure 3: Ablation studies of MTM on four datasets. Figure 4: Effect of head
number of SCORER on four datasets. Figure 5: Effect of layer number of SCORER
on four datasets. Figure 6: Captioning and change localization of varied
viewpoints. Figure 7: Three cases from different scenarios, where the
generated captions along with the attention weight at each word are
visualized.

Ablation Study of MTM. Instead of using MTM to perform fine-grained matching
between different feature subspaces of cross-view images, we use max/mean-
pooling to obtain the global feature of each image and compute their
similarity. Besides, we implement TM without multi-head operation. The results
in Fig. 3 show that MTM achieves the best results, which demonstrates that it
plays a critical role in view-invariant representation learning. Besides, only
implementing token-wise matching is not better than simple mean-pooling. Our
conjecture is that the changed object commonly appears in a local region with
weak feature, so it is insufficient to reveal this slight difference by only
interacting features at token level. As such, it is necessary to match two
image features at finer level, _i.e.,_ subspace level.

Effect of Head Number of SCORER. We further investigate the effect of head
number for SCORER, i.e., the head number of MTM and MHCA (Eq. (5)). The
results are shown in Fig. 4. We find that the best results are achieved on the
four datasets when setting the head number as 8.

Effect of Layer Number of SCORER. We investigate the effect of layer number
for SCORER in Fig. 5. On four datasets, we find that increasing the layer
number does not bring better performance, because deeper layers could result
in the problem of over-fitting. Besides, the layer number is the deepest on
Spot-the-Diff. Our conjecture is that objects have no good postures and
background information is more complex in a surveillance scenario. As such, we
empirically set proper layer number of 2, 1, 3, and 2 on four datasets.

Figure 8: Visualization of the alignment of unchanged objects computed by
MCCFormers-D [24] and our SCORER.

###  4.6 Captioning and change localization results with varied viewpoints

To intuitively evaluate the efficacy of our method to handle pseudo changes,
we show the captioning (Fig. 6 (a)) and change localization (Fig. 6 (b))
results of SCORER+CBR and SOTA method MCCFormers-D [24] with varied
viewpoints. The amount of viewpoint change is measured by the IoUs of objects’
bounding boxes across an image pair (lower IoU means higher difficulty). For
change localization, the pioneer work DUDA [22] tried Pointing Game to
evaluate attention maps of change localization, where maps are computed by
using the captured difference to directly query related regions on each image.
In contrast, we consider simultaneously evaluating change localization and
cross-modal alignment, _i.e._ , attention maps of cross-modal alignment, to
check whether the model can locate changed regions when generating
corresponding words. This is more challenging but more reasonable. In Fig. 6,
we find that our method outperforms MCCFormers-D and shows better robustness
with varied viewpoint changes on both evaluations, which benefit from view-
invariant representation learning and cross-modal backward reasoning.

###  4.7 Qualitative Analysis

To intuitively evaluate our method, we conduct qualitative analysis on the
four datasets. Fig. 7 illustrates three cases in different change scenarios.
For each case, we visualize the generated caption along with the attention
weight at each word. When the weight is higher, the region is brighter. We
observe that when generating the words about the changed object or its
referents, SCORER+CBR can adaptively locate the corresponding regions. In Fig.
8, we visualize the alignment between unchanged objects under different change
scenes. The compared method is the SOTA method MCCFormers-D [24]. We implement
it based on the released code. We find that when directly correlating two
image features, MCCFormers-D only aligns salient objects between two images.
Instead, our SCORER first learns two view-invariant representations in a self-
supervised way. Based on these, SCORER can better align and reconstruct the
representations of unchanged objects, so as to facilitate subsequent
difference representation learning. More qualitative examples are shown in the
supplementary material.

##  5 Conclusion

This paper proposes a novel SCORER to learn a stable difference representation
while resisting pseudo changes. SCORER first learns two view-invariant image
representations in a self-supervised way, by maximizing the cross-view
contrastive alignment of two images. Based on these, SCORER mines their common
features to reconstruct the representations of unchanged objects. This helps
learn a stable difference representation for caption generation. Further, we
design the CBR to improve captioning quality by enforcing the yielded caption
is informative about the difference in a self-supervised manner. Extensive
experiments show that our method achieves the state-of-the-art results on four
public datasets with different change scenarios.

## Acknowledgements

This work was supported by the National Key Research and Development Program
of China under Grant (2018AAA0102000), National Nature Science Foundation of
China (62322211, U21B2024, 61931008, 62071415, 62236008, U21B2038),
Fundamental Research Funds for the Central Universities, “Pioneer”, Zhejiang
Provincial Natural Science Foundation of China (LDT23F01011F01,
LDT23F01015F01, LDT23F01014F01) and “Leading Goose” R&D Program of Zhejiang
Province (2022C01068), and Youth Innovation Promotion Association of Chinese
Academy of Sciences (2020108).

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##  6 Supplementary Material

In this supplementary material, we will show more experimental results. First,
we show the implementation details on the four datasets. Second, we provide
the discussion of trade-off parameters on the four datasets. Next, we show
more ablation studies on the four datasets. Finally, we show more qualitative
examples on the four datasets.

###  6.1 Implementation Details

We provide more implementation details of our method. During training, the
batch sizes and learning rates of our method on the four datasets are shown in
Table 8. We train the model to convergence with 10K iterations in total. Both
training and inference are implemented with PyTorch on an RTX 3090 GPU. The
used resources on the four datasets are shown in Table 9. We can find that our
method does not need much training time and GPU memory, so it can be easily
reproduced by other researchers.

| Batch Size | Learning Rate
---|---|---
CLEVR-Change | 128 | 2 $\times$ $10^{-4}$
CLEVR-DC | 128 | 2 $\times$ $10^{-4}$
Spot-the-Diff | 32 | 2 $\times$ $10^{-4}$
Image Editing Request | 16 | 1 $\times$ $10^{-4}$
Table 8: The training parameters on the four datasets. | Training Time | GPU Memory
---|---|---
CLEVR-Change | 3 hours | 20G
CLEVR-DC | 1.5 hours | 15.6G
Spot-the-Diff | 20 minutes | 5G
Image Editing Request | 15 minutes | 4.3G
Table 9: Used training time and GPU memory on the four datasets. Figure 9: The effects of $\lambda_{v}$ and $\lambda_{m}$ on CLEVR-Change.  Figure 10: The effects of $\lambda_{v}$ and $\lambda_{m}$ on CLEVR-DC.  Figure 11: The effects of $\lambda_{v}$ and $\lambda_{m}$ on Image Editing Request.  Figure 12: The effects of $\lambda_{v}$ and $\lambda_{m}$ on Spot-the-Diff.  Ablation | B | M | R | C | S
---|---|---|---|---|---
Subtraction | 46.2 | 31.5 | 63.4 | 68.5 | 13.9
RR | 46.9 | 31.7 | 64.2 | 71.6 | 14.6
SCORER | 49.5 | 33.4 | 66.0 | 82.4 | 15.8
RR+CBR | 47.2 | 32.3 | 64.4 | 73.0 | 15.0
SCORER+CBR | 49.4 | 33.4 | 66.1 | 83.7 | 16.2
Table 10: Ablation study on CLEVR-DC. Ablation | B | M | R | C | S
---|---|---|---|---|---
Subtraction | 6.7 | 13.7 | 37.4 | 22.1 | 8.7
RR | 8.3 | 14.3 | 39.2 | 30.2 | 12.4
SCORER | 9.6 | 14.6 | 39.5 | 31.0 | 12.6
RR+CBR | 7.1 | 14.6 | 40.6 | 31.9 | 12.3
SCORER+CBR | 10.0 | 15.0 | 39.6 | 33.4 | 12.6
Table 11: Ablation study on Image Editing Request. Ablation | B | M | R | C | S
---|---|---|---|---|---
Subtraction | 7.2 | 11.9 | 28.9 | 29.4 | 13.0
RR | 7.4 | 12.0 | 27.6 | 26.2 | 14.2
SCORER | 9.4 | 13.8 | 32.0 | 38.5 | 19.3
RR+CBR | 8.1 | 11.7 | 30.6 | 32.4 | 15.6
SCORER+CBR | 10.2 | 12.2 | 31.9 | 38.9 | 18.4
Table 12: Ablation study on Spot-the-Diff.

###  6.2 Study on the Trade-off Parameters

In this section, we discuss the trade-off parameters $\lambda_{v}$ and
$\lambda_{m}$ in Eq. (13) of the main paper on the four datasets. Both
parameters are to balance the contributions from the caption generator,
SCORER, and CBR. In Figure 9, We first analyze the effect of $\lambda_{v}$ on
the CLEVR-Change dataset. We find that the performance of SCORER changes under
different values, because the model will focus much on one part but ignore the
supervision from the other. We empirically set $\lambda_{v}$ to 0.1. Then, we
fix $\lambda_{v}$ to discuss the effect of $\lambda_{m}$ for SCORER+CBR and
set it as $0.001$. In Figure 10, we also first analyze the effect of
$\lambda_{v}$ on the CLEVR-DC dataset. We empirically set $\lambda_{v}$ to
0.01. Then, we fix $\lambda_{v}$ to discuss $\lambda_{m}$. We find that the
CIDEr scores are identical when setting $\lambda_{m}$ as 0.002 and 0.004.
Thus, we further compare their SPICE scores (16.0 vs. 16.2) and set
$\lambda_{m}$ to 0.004. By that analogy, we discuss the effect of
$\lambda_{v}$ and $\lambda_{m}$ on the Image Editing Request and Spot-the-Diff
datasets. We empirically set $\lambda_{v}$ and $\lambda_{m}$ as 0.003 and 0.06
on Image Editing Request; 0.3 and 0.5 on Spot-the-Diff.

###  6.3 Ablation Study

We carry out ablation studies to validate the effectiveness of our method. (1)
Subtraction is a transformer-based baseline model which computes difference
features by direct subtraction. (2) RR refers to vanilla representation
reconstruction without cross-view contrastive alignment. (3) SCORER is the
proposed self-supervised cross-view representation reconstruction network. (4)
CBR means the proposed module of cross-modal backward reasoning.

Results on CLEVR-DC. Table 10 shows the ablation studies of our method on the
CLEVR-DC dataset, which are evaluated in terms of total performance. We can
draw the same conclusion from the ablative variants. Compared with the
baseline model of Subtraction, it is effective to first compute the aligned
properties and then deduce the difference features. The proposed SCORER first
learns the representations that are invariant under extreme viewpoint changes
for a pair of similar images, by maximizing their cross-view contrastive
alignment. Then, SCORER can fully mine their common features to reconstruct
the representations of unchanged objects, thereby learning a stable difference
representation for caption generation. Besides, CBR is helpful to improve the
quality of generated sentences, which shows that it does enforce the yielded
sentence to be informative about the learned difference.

Results on Image Editing Request. Table 11 shows the ablation studies of our
method on the Image Editing Request dataset, where two images in the pair are
aligned and the edited objects on this dataset are usually inconspicuous. We
can obtain the same observations. Match-based strategy (RR) performs better
than the strategy of direct subtraction. The proposed SCORER can fully align
and mine the common features between two images, so as to reconstruct reliable
unchanged representations for learning a stable difference representation.
When we implement CBR, the performance of SCORER+CBR is further boosted, which
shows that CBR is helpful to improve captioning quality.

Results on Spot-the-Diff. Table 12 shows the ablation studies of our method on
the Spot-the-Diff dataset. We can find that compared with the baseline model
of Subtraction, the improvement is not significant when using the model of
representation reconstruction. Our conjecture is that image pairs on this
dataset are well-aligned, so direct subtraction also can capture some salient
changes. When we perform cross-view contrastive alignment, the performance of
SCORER is significantly boosted, which shows that it facilitates correct
alignment between unchanged objects, so as to help locate fine-grained
changes. When introducing CBR, we observe that the results of RR+CBR and
SCORER+CBR are not improved significantly. As we discussed in the main paper,
the image pairs on this dataset actually contain one or more changes. For
fair-comparison, we conduct experiments mainly based on the single-change
setup. This makes the “hallucination” representation, which is reversely
modeled by the “before” representation and single-change caption, not fully
matched with the “after” representation. In this situation, the performances
of RR+CBR and SCORER+CBR do not gain significant improvement.

Effect of Cross-view Contrastive Alignment. We study the effect of cross-view
contrastive alignment, which is key to learn view-invariant image
representations. Besides, we try $L_{2}$ distance metric to achieve this goal
by only maximizing alignment of similar images. Fig. 13 shows comparison
results among RR (without alignment constraint), RR+$L_{2}$, and SCORER. We
find that SCORER achieves the best result, while the performance of RR+$L_{2}$
is the worst. The comparison results validate that it is necessary to build
contrastive alignment between similar/dissimilar images, which helps the model
focus more on the change of feature and resist feature shift. As a result, the
model can capture the stable difference representation between two images for
caption generation.

Figure 13: Effect of cross-view contrastive alignment on four datasets.

Study of Different Fusion Strategies to Model Difference Representation. In
the Eq. (7) of main paper, we obtain the difference representation between two
images by concatenating the changed features of each image. In order to
validate whether concatenation is a good choice in the case of extreme
viewpoint changes, besides concatenation (“cat”), we try to use other fusion
strategies to model the difference representation between two images: sum,
hadamard product, and respective interaction with words and then “cat”. The
experiment is conducted on the CLEVR-DC dataset with extreme viewpoint
changes. Here, we report the comparison results under CIDEr metric: “cat”
(82.4), sum (61.9), hadamard product (61.6), respective interaction with words
and then “cat” (81.3). The results validate the effectiveness of our choice:
using “cat” to construct an omni-representation of change between cross-view
images. With this omni-representation, our model can accurately locate changed
regions on two images during word generation, even under extreme viewpoint
changes. This benefits from view-invariant representation learning and cross-
modal backward reasoning.

###  6.4 Qualitative Analysis

In this section, we provide more visualization results about the alignment of
unchanged objects, and the generated captions along with the attention weight
at each word on the four datasets.

In Figure 14 \- 17, we visualize the alignment between unchanged objects under
different change types on CLEVR-Change, CLEVR-DC, Image Editing Request, and
Spot-the-Diff, respectively. The compared method is MCCFormers-D that is a
state-of-the-art method based on transformer. To fully match cross-view
images, both SCORER and MCCFormers-D respectively use one image to query the
shared objects on the other one, so obtaining two attention maps about cross-
view alignment. We find that when directly matching two image features,
MCCFormers-D mainly attends to some salient objects. Instead, our SCORER first
learns two view-invariant image representations in a self-supervised way, by
maximizing their cross-view contrastive alignment. Based on these, SCORER can
better align and reconstruct the representations of unchanged objects, so as
to facilitate subsequent difference representation learning.

In Figure 18 \- 21, we visualize the generated captions along with the
attention weight at each word under different change types on CLEVR-Change,
CLEVR-DC, Image Editing Request, and Spot-the-Diff, respectively. When
predicting the next word, the decoder uses generated words to compute
attention over the learned difference representation, which yields a single
attention map about cross-modal alignment. We interpolate it on each image to
show the localization of before- and after-changed object during word
generation. When the attention weight is higher, the localized region is
brighter. We observe that when generating the words about the changed object
or its referent, SCORER+CBR can adaptively attend to the corresponding
regions. This superiority mainly results from the facts that 1) SCORER learns
two view-invariant image representations for reconstructing the
representations of unchanged objects, so as to learn a stable difference
representation for caption generation; 2) cross-modal backward reasoning can
improve the quality of generated captions by enforcing the caption to be
informative about the learned difference.

Figure 14: Visualization of the alignment of unchanged objects on CLEVR-
Change, computed by MCCFormers-D and our SCORER.  Figure 15: Visualization of
the alignment of unchanged objects on CLEVR-DC, computed by MCCFormers-D and
our SCORER.  Figure 16: Visualization of the alignment of unchanged objects on
Image Editing Request, computed by MCCFormers-D and our SCORER.  Figure 17:
Visualization of the alignment of unchanged objects on Spot-the-Diff, computed
by MCCFormers-D and our SCORER.  Figure 18: Three cases about “Color Change”,
“Add”, and “Move” from CLEVR-Change, where the generated captions along with
the attention weight at each word are visualized.  Figure 19: Three cases
about “Drop”, “Add”, and “Move” from CLEVR-DC, where the generated captions
along with the attention weight at each word are visualized.  Figure 20: Three
cases about “Drop”, “Color Change”, and “Add” from Image Editing Request,
where the generated captions along with the attention weight at each word are
visualized.  Figure 21: Three cases about “Drop”, “Add”, and “Move” from Spot-
the-Diff, where the generated captions along with the attention weight at each
word are visualized.


# The masses and decay widths of the $S$-wave $\Lambda_{c}\bar{\Lambda}_{c}$
bound states

Shi-Ji Cao  Physics Department, Ningbo University, Zhejiang 315211, China
Jing-Juan Qi<EMAIL_ADDRESS>College of Information and Intelligence
Engineering, Zhejiang Wanli University, Zhejiang 315101, China  Xin-Heng Guo
<EMAIL_ADDRESS>College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China  Zhen-Yang Wang<EMAIL_ADDRESS>Physics Department, Ningbo University, Zhejiang 315211, China

###### Abstract

In this work, we investigate possible bound states of the
$\Lambda_{c}\bar{\Lambda}_{c}$ system in the Bethe-Salpeter formalism in the
ladder and instantaneous approximations. By numerically solving the Bethe-
Salpeter equation, we confirm the existence of $\Lambda_{c}\bar{\Lambda}_{c}$
bound states with quantum numbers $J^{PC}=0^{-+}$ and $J^{PC}=1^{--}$. We
further investigate the partial decay widths of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound states into $N\bar{N}$, $D\bar{D}$,
$D\bar{D}^{\ast}$, $D^{\ast}\bar{D}^{\ast}$, $\pi\bar{\pi}$, and $K\bar{K}$.
Our results indicate that the decay width of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state with $J^{PC}=1^{--}$ is much larger
than that with $J^{PC}=0^{-+}$, and among their decay channels, the
$D\bar{D}^{\ast}$ final state is the main decay mode. We suggest experiments
to search for the $\Lambda_{c}\bar{\Lambda}_{c}$ bound states in the
$D\bar{D}^{\ast}$ final state.

##  I Introduction

The emergence of new techniques and the availability of data with larger
statistics from modern facilities have sparked a resurgence of interest in the
study of spin-1/2 baryon production processes in $e^{+}e^{-}$ annihilations,
particularly around the threshold regions. Notably, efforts have been
intensified to secure more precise experimental measurements of near-threshold
cross sections for reactions such as $e^{+}e^{-}\rightarrow p\bar{p}$ [1, 2,
3, 4], $e^{+}e^{-}\rightarrow n\bar{n}$ [5, 6, 7], and
$e^{+}e^{-}\rightarrow\Lambda\bar{\Lambda}$ [8, 9, 10]. A common feature for
these processes is the enhancement or plateau of the near-threshold scattering
cross section, which has led to various theoretical interpretations, such as
final-state interactions [11, 12, 13], an attractive Coulomb interaction [14,
15], and bound states or mesonlike resonances [16, 17, 18, 19, 20, 21, 22].

The BESIII Collaboration has recently carried out precision measurements of
the cross section for the $e^{+}e^{-}\rightarrow\Lambda_{c}\bar{\Lambda}_{c}$
process at four energy points just above the $\Lambda_{c}\bar{\Lambda}_{c}$
threshold [23]. These measurements, depicted in Fig. 1, reveal an intriguing
pattern: a discernible non-zero cross section close to the
$\Lambda_{c}\bar{\Lambda}_{c}$ threshold and an almost constant profile. This
contrasts sharply with the earlier results obtained by the Belle Collaboration
using the initial-state radiation method [24], which are also illustrated in
Fig. 1 and were plagued by significant uncertainties in both the center-of-
mass energy and the cross section.

Figure 1: Cross section of $e^{+}e^{-}\rightarrow\Lambda_{c}\bar{\Lambda}_{c}$
obtained by BESIII and Belle.

In Ref. [25], it is proposed that the flat cross section behavior near the
threshold may be attributed to the Sommerfeld factor, which ensures a non-zero
distribution even at the threshold, and the presence of a near-threshold pole,
which balances the increasing phase space trend multiplied by the Sommerfeld
factor, resulting in a nearly constant distribution. However, since only the
four data points near the threshold from BESIII are considered, it cannot be
determined on which Riemann sheet the pole is located. By fitting the BESIII
data [23, 26] on the cross section of $\Lambda_{c}\bar{\Lambda}_{c}$ pair
production in $e^{+}e^{-}$ annihilation and corresponding electromagnetic form
factors, the authors predict existence of a bound state in the system
$\Lambda_{c}\bar{\Lambda}_{c}$ at energy $\sim$38 MeV below the threshold.
Ref. [27] utilizes a model incorporating a Breit-Wigner resonance and
$\Lambda_{c}\bar{\Lambda}_{c}$ four-point contact interactions to explain the
enhancement above the $\Lambda_{c}\bar{\Lambda}_{c}$ threshold as a
consequence of a virtual pole. Nonetheless, if the
$\Lambda_{c}\bar{\Lambda}_{c}$ contact coupling were larger, this pole would
become a bound state. Therefore, determining whether the
$\Lambda_{c}\bar{\Lambda}_{c}$ can form a bound state is crucial for
comprehending the flat cross section behavior near the threshold of the
$e^{+}e^{-}\rightarrow\Lambda_{c}\bar{\Lambda}_{c}$ process.

The existence of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound state has been
corroborated by numerous theoretical works. Within the one-boson-exchange
model [28, 29], the $\Lambda_{c}\bar{\Lambda}_{c}$ system can exist as bound
state and the binding energy of this system is very sensitive to the cutoff.
Similarly, the quasipotential Bethe-Salpeter (BS) equation approach also
supports the existence of a $\Lambda_{c}\bar{\Lambda}_{c}$ bound state [30].
The existence of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound state is also
supported by the effective field theory [31]. Additionally, based on the heavy
baryon chiral perturbation theory it is suggested that the
$\Lambda_{c}\bar{\Lambda}_{c}$ system can form a bound state through two-pion
exchange interaction potentials [32]. In Ref. [25], the authors state that the
binding energy of this system may range from a few to several tens of MeV
depending on the cutoff within the BS equation approach.

As a formally exact equation to describe the relativistic bound system, the BS
equation is formulated in Minkowski space based on the relativistic quantum
theory [33, 34]. Over the past decades, this formalism has been successfully
used to investigate heavy mesons, heavy baryons, and exotic states [35, 36,
37, 38, 39, 40]. In this work we will establish the BS equation for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system. The interaction kernel will be derived
from the four-point Green function with the relevant Lagrangians. Since the
strong interaction vertices are determined by the physical particles and the
off-shell exchanged particles, a form factor is introduced to account for the
finite size effects of the interacting hadrons. Subsequently, the BS equations
will be numerically solved under the covariant instantaneous approximation.
Moreover, we will explore some possible partial decay widths of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state.

The remainder of this paper is organized as follows. Section II will establish
the BS equation and the normalization conditions for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system. In Sec. III, the partial decay widths
of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound states to various final states
will be investigated. The numerical results will be presented in Sec. IV. The
final section will offer our summary and discussions.

##  II Bethe-Salpeter equation for the $\Lambda_{c}\bar{\Lambda}_{c}$ system

As we discussed in Introduction, we will study the possible
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state. The $S$-wave
$\Lambda_{c}\bar{\Lambda}_{c}$ system may form two bound states with quantum
numbers $J^{PC}=0^{-+}$ and $1^{--}$. The BS wave function for the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state is defined as:

$\chi_{P}(x_{1},x_{2},P)=\langle
0|T\Lambda_{c}(x_{1})\bar{\Lambda}_{c}(x_{2})|P\rangle,$ (1)

where $P$ ($=Mv$) is the total momentum of the $\Lambda_{c}\bar{\Lambda}_{c}$
bound state and $v$ represents its velocity. The BS wave function in the
momentum space is defined as

$\chi_{P}(x_{1},x_{2},P)=e^{-iPX}\int\frac{d^{4}p}{(2\pi)^{4}}\chi_{P}(p)e^{-ipx},$
(2)

where $X=\lambda_{1}x_{1}-\lambda_{2}x_{2}$ and $x=x_{1}-x_{2}$ are the
center-of-mass coordinate and the relative coordinate of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state with
$\lambda_{1(2)}=\frac{m_{1(2)}}{m_{1}+m_{2}}=\frac{1}{2}$ and $m_{1(2)}$ is
the mass of $\Lambda_{c}$ ($\bar{\Lambda}_{c}$), and $p$ is the relative
momentum of the bound state. The momenta of $\Lambda_{c}$ and
$\bar{\Lambda}_{c}$ can be expressed in terms of the relative momentum $p$ and
the total momentum $P$ as $p_{1}=\lambda_{1}P+p$ and $p_{2}=\lambda_{2}P-p$,
respectively.

The BS equation in the momentum space can be written as follows [41, 42]

$\chi_{P}(p)=S(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}\bar{K}(P,p,q)\chi_{P}(q)S(-p_{2}),$
(3)

where $K(P,p,q)$ is the interaction kernel from the irreducible Feynman
diagrams, $S(p_{1})$ and $S(-p_{2})$ are the propagators of $\Lambda_{c}$ and
$\bar{\Lambda}_{c}$, respectively. For convenience, we define $p_{l}\equiv
v\cdot p$ as the longitudinal projection of $p$ along $v$ and $p_{t}\equiv
p-p_{l}v$ which is transverse to $v$.

In general, the normalization condition of the BS wave function for the
$\Lambda_{c}$ and $\bar{\Lambda}_{c}$ system is

$i\int\frac{d^{4}p}{(2\pi)^{4}}\chi_{P}(p)\frac{d^{4}q}{(2\pi)^{4}}\frac{\partial}{\partial
P^{0}}\left[I(P,p,q)+\bar{K}(P,p,q)\right]\chi_{P}(q)=2E_{\mathbf{P}},\,\,P^{0}=E_{\mathbf{P}},$
(4)

where $I(P,p,q)=(2\pi)^{4}\delta^{4}(p-q)S^{-1}(p_{1})S^{-1}(-p_{2})$.

In the heavy quark limit, the propagators of $\Lambda_{c}$ and
$\bar{\Lambda}_{c}$ can be expressed as the following forms:

$S(p_{1})=i\frac{m_{1}(1+\not{v})}{2w_{1}(\lambda_{1}M+p_{l}-w_{1}+i\epsilon)},$
(5)

and

$S(p_{2})=i\frac{m_{2}(1+\not{v})}{2w_{2}(\lambda_{2}M-p_{l}-w_{2}+i\epsilon)},$
(6)

where the energy $w_{1(2)}=\sqrt{m_{1(2)}-p_{t}^{2}}$, and $\epsilon$ is the
infinitesimal.

Substituting Eqs. (5) and (6) into Eq. (3), we obtain the following two
constraint relations for the BS wave function $\chi_{P}(p)$:

$\not{v}\chi_{P}(p)=\chi_{P}(p),$ (7) $\chi_{P}(p)\not{v}=-\chi_{P}(p).$ (8)

Then taking into account these two constraint relations and other restrictions
from Lorentz covariance and parity transformation on the form of
$\chi_{P}(p)$, the BS wave functions for the $S$-wave pseudoscalar
($J^{PC}=0^{-+}$) and vector ($J^{PC}=1^{--}$) $\Lambda_{c}\bar{\Lambda}_{c}$
bound state can be parametrized in the following forms, respectively:

$\chi_{P}(p)=(1+\not{v})\gamma_{5}f_{1},$ (9)

and

$\chi^{(r)}_{P}(p)=(1+\not{v})\not{\xi}^{(r)}f_{2},$ (10)

where $f_{1}$ and $f_{2}$ are the Lorentz-scalar functions of $p_{t}^{2}$ and
$p_{l}$, and $\xi^{(r)}_{\mu}$ is the polarization vector of the vector
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state.

To simplify the BS equation (3), we impose the so-called covariant
instantaneous approximation in the kernel: $p_{l}=q_{l}=0$. In this
approximation the projection of the momentum of each constituent particle
along the total momentum $P$ is not changed, the energy exchanged between the
constituent particles of the binding system is neglected. This approximation
is appropriate since we consider the binding energy of
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state to be in the range (0-50) MeV,
which is very small compared to the constituent particle masses. Under this
approximation, the kernel in the BS equation is reduced to $K(P,p_{t},q_{t})$,
which will be used in the following calculations.

After some algebra, we find that the BS scalar wave functions for both the
pseudoscalar $\Lambda_{c}\bar{\Lambda}_{c}$ bound state ($f_{1}$) and the
vector $\Lambda_{c}\bar{\Lambda}_{c}$ bound state ($f_{2}$) satisfy the same
integral equation as follows:

$f(p)=\frac{-m_{1}m_{2}}{w_{1}w_{2}(\lambda_{1}M+p_{l}-w_{1}+i\epsilon)(\lambda_{2}M-p_{l}-w_{2}+i\epsilon)}\int\frac{d^{4}q}{(2\pi)^{4}}\bar{K}(P,p_{t},q_{t})f(q).$
(11)

We integrate both sides of above equation with respect to $p_{l}$ to obtain

$\tilde{f}(p_{t})=\frac{-im_{1}m_{2}}{w_{1}w_{2}(M-w_{1}-w_{2})}\int\frac{d^{3}q_{t}}{(2\pi)^{3}}\bar{K}(P,p_{t},q_{t})\tilde{f}(q_{t}),$
(12)

where we define $\tilde{f}(p_{t})=\int dp_{l}f(p)$.

For later convenience we also write out $f(p)$ in term of $\tilde{f}(p_{t})$.
From Eqs. (11) and (12) we have

$f(p)=-i\frac{M-w_{1}-w_{2}}{2\pi(\lambda_{1}M+p_{l}-w_{1}+i\epsilon)(\lambda_{2}M-p_{l}-w_{2}+i\epsilon)}\tilde{f}(p_{t}).$
(13)

As the $\Lambda_{c}$ is an isoscalar state, the total interaction kernel
arises from the exchanges of $\omega$ and $\sigma$. The related Lagrangians,
constructed with heavy quark and chiral symmetries [43, 44], are presented as
follows:

$\begin{split}\mathcal{L}_{\Lambda_{c}\Lambda_{c}\omega}&=-i\frac{g_{V}\beta_{B}}{4m_{\Lambda_{c}}}\omega^{\mu}\bar{\Lambda}_{c}\overleftrightarrow{\partial}\Lambda_{c},\\\
\mathcal{L}_{\Lambda_{c}\Lambda_{c}\sigma}&=i\ell_{B}\sigma\bar{\Lambda}_{c}\Lambda_{c},\end{split}$
(14)

where the adopted coupling constants are $g_{V}$ = 5.8, $\beta_{B}$ = 0.87,
and $\ell_{B}=-3.1$ .

Then the interaction kernel can be derived in the lowest-order form as
follows:

$\begin{split}\bar{K}_{\omega}(P,p,q)&=(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})\left(\frac{g_{V}\beta_{B}}{4m_{\Lambda_{c}}}\right)^{2}(p_{1}+q_{1})^{\mu}(p_{2}+q_{2})^{\nu}\Delta_{\mu\nu}^{\omega}(k),\\\
\bar{K}_{\sigma}(P,p,q)&=(2\pi)^{4}\delta^{4}(q_{1}+q_{2}-p_{1}-p_{2})\ell_{B}^{2}\Delta^{\sigma}(k),\end{split}$
(15)

where $\Delta_{\mu\nu}^{\omega}(k)$ and $\Delta^{\sigma}(k)$ are the
propagators for the exchanged $\omega$ and $\sigma$ mesons, respectively, and
$k$ represents the momentum of the exchanged meson.

To take into account the structure and finite size effect of the interacting
hadrons, it is necessary to introduce a form factor at the vertices. For
$t$-channel vertices, we take a monopole form factors as,

$F(k^{2})=\frac{\Lambda^{2}-m^{2}}{\Lambda^{2}-k^{2}},$ (16)

where $m$ and $\Lambda$ represent the mass and cutoff parameter of the
exchanged meson, respectively. The cutoff $\Lambda$ can be further
reparameterized as $\Lambda=m+\alpha\Lambda_{QCD}$ with $\Lambda_{QCD}$ = 220
MeV, and the parameter $\alpha$ being of order unity. The value of $\alpha$
depends on the exchanged and external particles involved in the strong
interaction vertex and cannot be obtained from the first principle.

##  III The partial decay widths of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound
state

In this section, we will investigate the decay widths of the $S$-wave
$\Lambda_{c}\bar{\Lambda}_{c}$ bound states, thereby providing theoretical
support for the experimental search for $\Lambda_{c}\bar{\Lambda}_{c}$ bound
states. For the $\Lambda_{c}\bar{\Lambda}_{c}$ bound state with quantum
numbers $J^{PC}=1^{--}$, some possible strong decay channels include
$N\bar{N}$, $D\bar{D}$, $D\bar{D}^{\ast}$, $D^{\ast}\bar{D}^{\ast}$,
$\pi\bar{\pi}$, and $K\bar{K}$. Due to parity constraints, final states such
as $D\bar{D}$, $\pi\bar{\pi}$, and $K\bar{K}$ are forbidden for the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state with $J^{PC}=0^{-+}$, while only
$N\bar{N}$, $D\bar{D}^{\ast}$, and $D^{\ast}\bar{D}^{\ast}$ final states are
allowed.

The strong decays of $S$-wave $\Lambda_{c}\bar{\Lambda}_{c}$ bound states into
$N\bar{N}$, $D^{(\ast)}\bar{D}^{(\ast)}$, $\pi\bar{\pi}$, and $K\bar{K}$ can
occur via exchanging $D/D^{\ast}$, $N$ (nucleon), $\Sigma_{c}$, and
$\Xi^{\prime}_{c}$, respectively. The relevant interaction vertices are [45,
46, 47, 48]

$\begin{split}\mathcal{L}_{DN\Lambda_{c}}=&ig_{DN\Lambda_{c}}\bar{\Lambda}_{c}\gamma^{5}DN+\rm{H.c.},\\\
\mathcal{L}_{D^{\ast}N\Lambda_{c}}=&g_{D^{\ast}N\Lambda_{c}}\bar{\Lambda}_{c}\gamma^{\mu}ND^{\ast}_{\mu}+\rm{H.c.},\\\
\mathcal{L}_{\Lambda_{c}\Sigma_{c}\pi}=&g_{\Lambda_{c}\Sigma_{c}\pi}\bar{\Lambda}_{c}\partial^{\mu}\pi\gamma_{5}\gamma_{\mu}\Sigma_{c}+\rm{H.c.},\\\
\mathcal{L}_{\Lambda_{c}\Xi^{\prime}_{c}K}=&g_{\Lambda_{c}\Xi^{\prime}_{c}K}\bar{\Lambda}_{c}\partial^{\mu}K\gamma_{5}\gamma_{\mu}\Sigma_{c}+\rm{H.c.},\\\
\end{split}$ (17)

where $\rm{H.c.}$ represent the Hermitian conjugate of the previous terms. The
coupling constants $g_{\Lambda_{c}ND}=-13.98$ and
$g_{\Lambda_{c}ND^{\ast}}=-5.2$ are determined from flavour-SU(4) symmetry
[49, 50], $g_{\Lambda_{c}\Sigma_{c}\pi}=\frac{g_{2}}{\sqrt{2}f_{\pi}}$ and
$g_{\Lambda_{c}\Xi^{\prime}_{c}K}=\frac{g_{2}}{2f_{\pi}}$ [51], where
$f_{\pi}=93$ MeV is the pion decay constant, and $g_{2}=0.565$ is determined
from the $\Sigma^{++}_{c}\rightarrow\Lambda^{+}_{c}\pi^{+}$ decay [52].

In the rest frame, the two-body decay width of the bound state is expressed as

$d\Gamma=\frac{1}{32\pi^{2}}|\mathcal{M}|^{2}\frac{|\mathbf{p}^{\prime}|}{M^{2}}d\Omega,$
(18)

where $\mathcal{M}$ is the Lorentz invariant decay amplitude of the process.
$|\mathbf{p}^{\prime}|$ is the magnitude of the three-momentum of the final
state particles in the rest frame of the bound state, defined by

$|\mathbf{p}^{\prime}|=\frac{1}{2M}\sqrt{\lambda(M^{2},m_{1}^{2},m_{2}^{2})},$
(19)

with $\lambda(a,b,c)=a^{2}+b^{2}+c^{2}-2ab-2ac-2bc$ being the
K$\mathrm{\ddot{a}}$ll$\mathrm{\acute{e}}$n function.

The lowest order Lorentz-invariant decay amplitude for the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state decaying into $N\bar{N}$ is

$\begin{split}\mathcal{M}_{N\bar{N}}=&\mathcal{M}_{N\bar{N}}^{D}+\mathcal{M}_{N\bar{N}}^{D^{\ast}}\\\
=&\int\frac{d^{4}p}{(2\pi)^{4}}g_{D^{\ast}N\Lambda_{c}}^{2}\bar{u}(p^{\prime}_{1})\gamma^{5}\chi_{P}(p)\gamma^{5}v(p^{\prime}_{2})\Delta_{D}(k,m_{D})F(|\bm{k}|^{2},m_{D})^{2}\\\
&-\int\frac{d^{4}p}{(2\pi)^{4}}g_{D^{\ast}N\Lambda_{c}}^{2}\bar{u}(p^{\prime}_{1})\gamma_{\mu}\chi_{P}(p)\gamma_{\nu}v(p^{\prime}_{2})\Delta_{D^{\ast}}^{\mu\nu}(k,m_{D^{\ast}})F(|\bm{k}|^{2},m_{D^{\ast}})^{2},\end{split}$
(20)

where $p^{\prime}_{1(2)}=\left(E_{1(2)},(-)\mathbf{p}^{\prime}\right)$ denotes
the momentum of $N(\bar{N})$ in the final state, $\bm{k}$ is the three-
momentum transfer in the decay process, $\bar{u}$ and $v$ are the Dirac
spinors of $N$ and $\bar{N}$, respectively, $\Delta_{D}(k,m_{D})$ and
$\Delta_{D^{\ast}}^{\mu\nu}(k,m_{D^{\ast}})$ are the propagators for the
exchanged mesons. For convenience, we define
$p^{\prime}\equiv\lambda_{2}p^{\prime}_{1}-\lambda_{1}p^{\prime}_{2}$ which is
not the relative momentum of the final particles, and is given by
$p^{\prime}=(\lambda_{2}E_{1}-\lambda_{1}E_{2},\mathbf{p}^{\prime})$. The BS
wave function $\chi_{P}(p)$ could be either pseudoscalar or vector.

The lowest-order Lorentz-invariant decay amplitudes are

$\mathcal{M}_{D\bar{D}}=\int\frac{d^{4}p}{(2\pi)^{4}}g_{D^{\ast}N\Lambda_{c}}^{2}\gamma_{5}\Delta_{N}(k,m_{N})\gamma_{5}\chi_{P}(p)F(|\bm{k}|^{2},m_{N})^{2},$
(21)
$\mathcal{M}_{D\bar{D}^{\ast}}=-i\int\frac{d^{4}p}{(2\pi)^{4}}g_{DN\Lambda_{c}}g_{D^{\ast}N\Lambda_{c}}\epsilon^{\ast}_{\mu}(p^{\prime}_{2})\gamma_{5}\chi_{P}(p)(k)\gamma^{\mu}\Delta_{N}(k,m_{N})F(|\bm{k}|^{2},m_{N})^{2},$
(22)

and

$\mathcal{M}_{D^{\ast}\bar{D}^{\ast}}=-\int\frac{d^{4}p}{(2\pi)^{4}}g_{D^{\ast}N\Lambda_{c}}^{2}\epsilon^{\ast}_{\mu}(p^{\prime}_{1})\gamma^{\mu}\chi_{P}(p)\gamma^{\nu}\epsilon_{\nu}^{\ast}(p^{\prime}_{2})\Delta_{N}(k,m_{N})F(|\bm{k}|^{2},m_{N})^{2},$
(23)

for the $D\bar{D}$ (only allowed for the vector $\Lambda_{c}\bar{\Lambda}_{c}$
bound state), $DD^{\ast}$, and $D^{\ast}\bar{D^{\ast}}$ final states,
respectively, $\epsilon$ is the polarization vector of $D^{\ast}$ or
$\bar{D}^{\ast}$.

For the $\pi\bar{\pi}$ and $K\bar{K}$ final states, the lowest-order Lorentz-
invariant decay amplitudes are given by

$\mathcal{M}_{\pi\bar{\pi}}=\int\frac{d^{4}p}{(2\pi)^{4}}g_{\Lambda_{c}\Sigma_{c}\pi}^{2}\gamma_{\mu}\gamma^{5}\chi_{P}(p)\gamma_{5}\gamma_{\nu}p^{{}^{\prime}\mu}_{1}p^{{}^{\prime}\nu}_{2}\Delta_{\Sigma_{c}}(k,m_{\Sigma_{c}})F(|\bm{k}|^{2},m_{\Sigma_{c}})^{2},$
(24)

and

$\mathcal{M}_{K\bar{K}}=\int\frac{d^{4}p}{(2\pi)^{4}}g_{\Lambda_{c}\Xi^{\prime}_{c}K}^{2}\gamma_{\mu}\gamma^{5}\chi_{P}(p)\gamma_{5}\gamma_{\nu}p^{{}^{\prime}\mu}_{1}p^{{}^{\prime}\nu}_{2}\Delta_{\Xi^{\prime}_{c}}(k,m_{\Xi^{\prime}_{c}})F(|\bm{k}|^{2},m_{\Xi^{\prime}_{c}})^{2},$
(25)

respectively. These two decay processes are exclusive to the vector
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state.

##  IV Numerical results

To show the numerical results, we begin by presenting the masses of mesons and
baryons in Table 1 [53], which are essential for studying the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound states and their potential decay
channels. Our model includes a single free parameter, $\alpha$, which is
influenced by the exchanged particle and the external particles at the strong
interaction vertex. It is expected that $\alpha$ is order unity and cannot be
decided from the first principle. Consequently, the binding energy $E_{b}$
(defined as $E_{b}=m_{1}+m_{2}-M$, where we consider
$\Lambda_{c}\bar{\Lambda}_{c}$ as a shallow bound state system with $E_{b}$
ranging from 0 to 50 MeV) of the $\Lambda_{c}\bar{\Lambda}_{c}$ system cannot
be determined, as it depends on the value of the parameter $\alpha$. To
explore possible solutions for the $\Lambda_{c}\bar{\Lambda}_{c}$ bound
states, we vary $\alpha$ over a broader range (0.5-3).

Table 1: Masses (unit: MeV) of the relevant mesons and baryons. $\Lambda_{c}^{+}$ | $\Sigma_{c}^{++}$ | $\Sigma_{c}^{+}$ | $\Sigma_{c}^{0}$ | $\Xi^{{}^{\prime}+}_{c}$ | $\Xi^{{}^{\prime}0}_{c}$ | $p$ | $n$ |  |
---|---|---|---|---|---|---|---|---|---
2286.46 | 2453.97 | 2452.65 | 2453.75 | 2578.2 | 2578.70 | 938.27 | 939.57 |  |
$D^{0}$ | $D^{\pm}$ | $D^{\ast 0}$ | $D^{\ast\pm}$ | $\pi^{0}$ | $\pi^{\pm}$ | $K^{0}$ | $K^{\pm}$ | $\omega$ | $\sigma$
1864.84 | 1869.66 | 2006.85 | 2010.26 | 134.98 | 139.57 | 497.61 | 493.68 | 782.66 | 500

To solve the integral equation (12), we discretize the integration region into
$n$ pieces (with $n$ being sufficiently large), transforming Eq. (12) to an
eigenvalue equation for the $n$ dimensional vector $\tilde{f}$. After solving
the eigenvalue equation, the $\Lambda_{c}\bar{\Lambda}_{c}$ system can exist
as a bound state within a binding energy range of 0-50 MeV, with the parameter
$\alpha$ ranging from 1.03 to 1.78. This indicates that the contributions from
the exchanges of $\omega$ and $\sigma$ are sufficient to allow the
$\Lambda_{c}\bar{\Lambda}_{c}$ system to form bound states. The values of
$\alpha$ and the corresponding binding energy $E_{b}$ are depicted in Fig. 2,
indicating that $\alpha$ increases with the binding energy $E_{b}$. This trend
is attributed to the increase in momentum exchange between the constituent
particles of the bound state as $E_{b}$ increases, necessitating a larger
momentum cutoff $\Lambda$, hence, an increase in $\alpha$ with $E_{b}$. As we
discussed in Sec.II, for both the pseudoscalar and vector
$\Lambda_{c}\bar{\Lambda}_{c}$ systems, they satisfy the same scalar BS
equation (11), and therefore, they yield identical results.

Figure 2: Values of $\alpha$ and $E_{b}$ for the possible bound states for the
$\Lambda_{c}\bar{\Lambda}_{c}$ system.

Taking into account the constraints of quantum numbers such as spin and
parity, the $\Lambda_{c}\bar{\Lambda}_{c}$ bound states with $J^{PC}=0^{-+}$
can decay through strong interactions into $N\bar{N}$, $D\bar{D}^{\ast}$, and
$D^{\ast}\bar{D}^{\ast}$ final states. The estimated partial decay widths for
these final states are $1.54\times 10^{-10}\sim 1.43\times 10^{-4}$ MeV,
$2.43\times 10^{-9}\sim 1.65\times 10^{-3}$ MeV, and $1.04\times 10^{-9}\sim
6.89\times 10^{-4}$ MeV, respectively, with the binding energy ranging from 0
to 50 MeV. The partial decay widths are also illustrated in Fig. 3. It is
observed that the decay widths to $N\bar{N}$, $D\bar{D}^{\ast}$, and
$D^{\ast}\bar{D}^{\ast}$ final states exhibit a similar trend of increasing
with the binding energy $E_{b}$. The decay to the $N\bar{N}$ final state is
suppressed by the OZI rule due to $c\bar{c}$ annihilation, resulting in the
smallest decay width among the considered channels. The decays of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state with $J^{PC}=0^{-+}$ to both
$D\bar{D}^{\ast}$ and $D^{\ast}\bar{D}^{\ast}$ final states are mediated by
$N$ exchange. Despite the decay to $D\bar{D}^{\ast}$ being a $P$-wave process
and to $D^{\ast}\bar{D}^{\ast}$ an $S$-wave process the stronger coupling at
the $\Lambda_{c}DN$ vertex and the larger phase space for the
$D\bar{D}^{\ast}$ final state make this decay width the most significant.

Figure 3: The particle decay widths of the $\Lambda_{c}\bar{\Lambda}_{c}$
bound state with $J^{PC}=0^{-+}$ to $N\bar{N}$ (a), $D\bar{D}^{\ast}$ (b), and
$D^{\ast}\bar{D}^{\ast}$ (c) final states, respectively.

The partial decay widths of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound state
with $J^{PC}=1^{--}$ to $N\bar{N}$, $D\bar{D}$, $D\bar{D}^{\ast}$,
$D^{\ast}\bar{D}^{\ast}$, $\pi\bar{\pi}$, and $K\bar{K}$ final states range
from $9.72\times 10^{-11}\sim 1.90\times 10^{-4}$ MeV, $6.80\times 10^{-8}\sim
6.31\times 10^{-2}$ MeV, $1.69\times 10^{-3}\sim 45.33$ MeV, $1.12\times
10^{-8}\sim 6.55\times 10^{-3}$ MeV, $6.67\times 10^{-15}\sim 1.41\times
10^{-8}$ MeV, and $2.10\times 10^{-13}\sim 4.62\times 10^{-7}$ MeV,
respectively, for a binding energy range of 0 to 50 MeV. The
$\Lambda_{c}\bar{\Lambda}_{c}$ bound state with $J^{PC}=1^{--}$ exhibits
larger decay widths compared to the state with $J^{PC}=0^{-+}$. The decay
processes to $\pi\bar{\pi}$ and $K\bar{K}$ final states, involving the
$c\bar{c}$ annihilation, are suppressed by the OZI rule, and the coupling
strengths at the $\Lambda_{c}\Sigma_{c}\pi$ and $\Lambda_{c}\Xi^{\prime}_{c}K$
vertices are relatively small, leading to minimal decay widths as anticipated.
The decay of the $\Lambda_{c}\bar{\Lambda}_{c}$ bound state with
$J^{PC}=1^{--}$ to $N\bar{N}$ also includes $c\bar{c}$ annihilation, hence
this process is also suppressed. For decays to $D\bar{D}^{\ast}$ and
$D^{\ast}\bar{D}^{\ast}$, both being $S$-wave decays, the significantly
stronger coupling at the $\Lambda_{c}DN$ vertex compared to the
$\Lambda_{c}D^{\ast}N$ vertex results in a much larger decay width to the
$D\bar{D}^{\ast}$ final state than to the $D^{\ast}\bar{D}^{\ast}$ final
state.

Figure 4: The particle decay widths of the $\Lambda_{c}\bar{\Lambda}_{c}$
bound state with $J^{PC}=1^{--}$ to $N\bar{N}$ (a), $D\bar{D}$ (b),
$D\bar{D}^{\ast}$ (c), $D^{\ast}\bar{D}^{\ast}$ (d), $\pi\bar{\pi}$ (e), and
$K\bar{K}$ (f) final states, respectively.

##  V Summary

In this study, we have presented a comprehensive analysis of the possible
$\Lambda_{c}\bar{\Lambda}_{c}$ bound states and some possible strong decays of
these bound states. Our theoretical framework is based on the BS equation,
which provides a relativistically consistent description of the bound state
system. The interaction kernel of the BS equation was constructed from
relevant Lagrangians that describe the strong interaction vertices among the
exchanged ($\omega$ and $\sigma$) and external ($\Lambda_{c}$) particles.

Our numerical results indicate that $\Lambda_{c}\bar{\Lambda}_{c}$ bound
states could exist. However, we cannot determine the mass of the bound state
precisely, as it depends on the value of the parameter $\alpha$, which is not
determined from first principles and reflects the non-perturbative nature of
QCD at low energies. Our results indicate that when the binding energy is
between 0 and 50 MeV, there exist bound states within the range of 1.03 to
1.78 for the parameter $\alpha$.

We have also calculated the partial decay widths of the
$\Lambda_{c}\bar{\Lambda}_{c}$ bound states for various decay channels. For
the $J^{PC}=0^{-+}$ state, the decays to $N\bar{N}$, $D\bar{D}^{\ast}$, and
$D^{\ast}\bar{D}^{\ast}$ final states were considered. The decay widths
increase with the binding energy, with the decay to the $D\bar{D}^{\ast}$
final state being the most significant one due to the larger phase space and
stronger coupling at the $\Lambda_{c}DN$ vertex compared to the
$\Lambda_{c}D^{\ast}N$ vertex. The OZI rule suppresses the decay to the
$N\bar{N}$ final state, resulting in the smallest decay width for this
channel.

For the $J^{PC}=1^{--}$ state, we investigated the decay channels to
$N\bar{N}$, $D\bar{D}$, $D\bar{D}^{\ast}$, $D^{\ast}\bar{D}^{\ast}$,
$\pi\bar{\pi}$, and $K\bar{K}$ final states. The decay widths for the
$\pi\bar{\pi}$ and $K\bar{K}$ channels are the smallest due to the suppression
by the OZI rule and the relatively small coupling constants at the
$\Lambda_{c}\Sigma_{c}\pi$ and $\Lambda_{c}\Xi^{\prime}_{c}K$ vertices.
Similar to the $J^{PC}=0^{-+}$ state, the decay to the $D\bar{D}^{\ast}$ final
state is the most prominent one for the $J^{PC}=1^{--}$ state.

The results presented in this work have important implications for the
experimental search for $\Lambda_{c}\bar{\Lambda}_{c}$ bound states. The
predicted decay widths can serve as a guide for future experiments at
facilities such as BESIII, LHCb, and Belle II. Further theoretical and
experimental investigations are needed to validate the existence of these
bound states and to refine the understanding of their properties.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China
(Project Nos. 12105149 and 12275024).

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