tcb@breakable

Visual Hallucinations of Multi-modal Large Language Models

Wen Huang∗ |  |  | Hongbin Liu∗
---|---|---|---
University of Science & Technology of China |  |  | Duke University
<EMAIL_ADDRESS>|  |  |<EMAIL_ADDRESS>
Minxin Guo |  |  | Neil Zhenqiang Gong
The University of Hong Kong |  |  | Duke University
<EMAIL_ADDRESS>|  |  |<EMAIL_ADDRESS>
**footnotetext: Equal contributions.

###### Abstract

_Visual hallucination (VH)_ means that a multi-modal LLM (MLLM) imagines
incorrect details about an image in visual question answering. Existing
studies find VH instances only in existing image datasets, which results in
biased understanding of MLLMs’ performance under VH due to limited diversity
of such VH instances. In this work, we propose a tool called VHTest to
generate a diverse set of VH instances. Specifically, VHTest finds some
initial VH instances in existing image datasets (e.g., COCO), generates a text
description for each VH mode, and uses a text-to-image generative model (e.g.,
DALL·E-3) to generate VH images based on the text descriptions. We collect a
benchmark dataset with 1,200 VH instances in 8 VH modes using VHTest. We find
that existing MLLMs such as GPT-4V, LLaVA-1.5, and MiniGPT-v2 hallucinate for
a large fraction of the instances in our benchmark. Moreover, we find that
fine-tuning an MLLM using our benchmark dataset reduces its likelihood to
hallucinate without sacrificing its performance on other benchmarks. Our
benchmarks are publicly available: https://github.com/wenhuang2000/VHTest.

##  1 Introduction

A multi-modal LLM (MLLM) [27, 30, 1, 6, 23] generates a _text response_ for a
given _image_ and _question_. An MLLM typically comprises three components: a
vision encoder, a vision-language connector, and an LLM. The vision encoder
(e.g., CLIP [18]) converts an image into an embedding vector. The vision-
language connector projects an image embedding vector into the LLM’s word
embedding space. The projected vector is concatenated with the token
embeddings of the question to form an input to the LLM, which generates a text
response.

However, MLLMs often generate text responses containing factually incorrect
details about an image, known as _visual hallucination (VH)_ [10, 13]. Figure
1 shows an example where the MLLM hallucinates two lamps, contradicting the
three lamps in the image. VHs in MLLMs pose obstacles to developing safe and
trustworthy AI, which is emphasized in a recent U.S. Executive Order calling
for rigorous testing to address potential harms from advanced AI systems [22].

Figure 1: An example of MLLM’s visual hallucination. The text in RED
highlights the hallucinated detail in the image, where there are three lamps.

Prior works have tried to benchmark MLLMs’ VHs related to object existence
[10, 12], optical character recognition (OCR), object counting, object
positions comparing [3], orientation, and viewpoint [25] (concurrent to ours).
However, they collect VH images only from existing image datasets like COCO
[11]. This limits the diversity of VH images since they can only find a
limited number of them. Moreover, existing image datasets may have been used
to pre-train an MLLM, leading to data contamination [7, 21]. As a result, such
VH images lead to a biased understanding of an MLLM’s performance, e.g., an
MLLM is incorrectly concluded to perform well under VH.

#### Our Work

We propose VHTest, a tool that generates VH instances to test MLLMs. A VH
instance is a triple (an image, a question, a reference answer). Our VHTest
has three key steps. Step I finds initial VH instances using existing image
datasets like COCO. Specifically, we first identify image pairs with high CLIP
embedding similarity but low DINO v2 [16] embedding similarity. Such image
pairs have contradictory similarities from two powerful vision encoders,
indicating potential VHs. We note that Step I is also used in a concurrent
work [25]. We then manually design questions and reference answers for these
images to obtain initial VH instances. Step II generates a text description
for a VH mode using the initial VH instances. A text description describes
visual properties of a VH image. Finally, Step III uses a text-to-image
generative model (e.g., DALL-E 3) to generate new images based on the text
descriptions. Moreover, based on some templates, we design questions and
reference answers for the generated images to construct VH instances.

Figure 2: Example VH instances generated by our VHTest and the text responses
of three MLLMs for them. Figure 7 in Appendix shows examples for the other
four VH modes.

Using our VHTest, we construct a new benchmark dataset for evaluating VHs in
MLLMs. Our benchmark contains 1,200 VH instances covering 8 VH modes. The 8 VH
modes are related to _existence_ , _shape_ , _color_ , _orientation_ , _OCR_ ,
_size_ , _position_ , and _counting_ of visual objects in an image. Note that
shape and size VH modes are formulated by us, while the other 6 VH modes were
also considered in prior [27] and concurrent [25] studies. Figure 2 and Figure
7 show some VH instances generated by VHTest.

We comprehensively evaluate state-of-the-art MLLMs, including GPT-4V,
LLaVA-1.5, and MiniGPT-v2 on our benchmark. Our results show that MLLMs
hallucinate for a large fraction of the VH instances in our benchmark. For
example, GPT-4V, LLaVA-1.5, and MiniGPT-v2 only achieve overall accuracy of
0.383, 0.229, and 0.075 on our benchmark, respectively. We also find that
MLLMs have different performance across VH modes. For example, GPT-4V is most
prone to orientation VH with 0.153 accuracy; while LLaVA-1.5 and MiniGPT-v2
are most susceptible to OCR VH with 0.127 and 0.000 accuracy, respectively.

Finally, we show that fine-tuning an MLLM using our benchmark dataset
mitigates VH. Specifically, we divide our benchmark into the training/testing
splits with a ratio 80%/20%. We then fine-tune the LLaVA-1.5 model on the
training split. After fine-tuning, we evaluate the model on the testing split.
Our results show fine-tuning reduces the likelihood for an MLLM to
hallucinate. For example, in position VH mode, the fine-tuned LLaVA-1.5 gains
0.200 accuracy from 0.333 to 0.533. Moreover, fine-tuning maintains model
performance on other benchmark datasets.

##  2 Definitions

#### VH Modes

We can categorize visual properties of objects in an image into _individual_
properties, which can be attributed to individual objects (e.g., existence,
shape, color, orientation, and OCR), and _group_ properties, which emerge from
comparisons across multiple objects (e.g., relative size, relative position,
and counting). Based on such categorization, we have 8 VH modes as follows. In
particular, each VH mode occurs when an MLLM’s text response is factually
incorrect with respect to the corresponding visual property in an image.

  1. 1.

Existence VH: $O$ is the set of objects in image $I$, and $O^{\prime}$ is the
set of objects an MLLM identifies in $I$. Existence VH occurs if: $\exists
o_{i}\in O\,\,\>\>s.t.\>\>o_{i}\notin O^{\prime}\>\>\>or\>\>\>\exists
o^{\prime}_{j}\in O^{\prime}\>\>s.t.\>\>o^{\prime}_{j}\notin O$. In other
words, the MLLM misses at least one object in $I$ or fabricates at least one
nonexistent object.

  2. 2.

Shape VH: Let $S=\\{s(o_{i})\\}_{i=1}^{n}$ denote the list of shapes for
objects $\\{o_{i}\\}_{i=1}^{n}$ in $I$, and
$S^{\prime}=\\{s^{\prime}(o_{i})\\}_{i=1}^{n}$ is the corresponding list of
shapes identified by an MLLM. A shape VH occurs if: $\exists
o_{i}\>\>s.t.\>\>s(o_{i})\neq s^{\prime}(o_{i})$. Intuitively, the MLLM fails
to accurately describe the shape of at least one object in $I$.

  3. 3.

Color VH: Let $C=\\{c(o_{i})\\}_{i=1}^{n}$ denote the list of colors for
objects $\\{o_{i}\\}_{i=1}^{n}$ in $I$, and
$C^{\prime}=\\{c^{\prime}(o_{i})\\}_{i=1}^{n}$ is the corresponding list of
colors identified by an MLLM. A color VH occurs if the MLLM fails to
accurately identify the color of at least one object in $I$. Formally, we
have: $\exists o_{i}\>\>s.t.\>\>c(o_{i})\neq c^{\prime}(o_{i})$.

  4. 4.

Orientation VH: An LLM fails to precisely recognize the facing orientation of
at least one object in an image.

  5. 5.

OCR VH: An MLLM fails to accurately identify at least one character in an
image.

  6. 6.

Size VH: An MLLM fails to accurately compare the relative sizes of multiple
objects in an image.

  7. 7.

Position VH: An MLLM fails to accurately identify spatial relationships
between objects in an image.

  8. 8.

Counting VH: An MLLM exhibits a counting VH mode when it cannot accurately
enumerate the number of objects in an image.

#### VH Instance

An VH instance is a triple $\\{x_{i},x_{t},y_{r}\\}$, where $x_{i}$ is an
image, $x_{t}$ is a question, and $y_{r}$ is a reference answer. We say a VH
instance _succeeds_ for an MLLM if and only if the MLLM’s text response for
$x_{i}$ and $x_{t}$ is factually incorrect compared to the reference answer
$y_{r}$. For instance, in the example shown in Figure 1, the reference answer
is “three lamps”, while the MLLM’s text response indicates two lamps.

##  3 Our VHTest

Figure 3: Pipeline of our VHTest. The human-head symbol means a human worker
manually generates a question-answer pair for an image.

###  3.1 Step I: Finding Initial VH Instances

Since the CLIP vision encoder is the backbone of many popular MLLMs such as
LLaVA [14], LLaMA-Adapter [4], and mPLUG-Owl [28], we leverage CLIP to find
initial VH instances. Our goal is to find images in an existing image dataset
(e.g., COCO) that are incorrectly embedded by the CLIP vision encoder to have
high similarity, despite differences in visual semantics. Such images may lead
to VH for MLLMs due to their incorrect embeddings.

Specifically, given an image pair $(x_{1},x_{2})$, we compute the cosine
similarity between their CLIP embedding vectors, i.e.,
$cos(f_{C}(x_{1}),f_{C}(x_{2}))$, where $f_{C}(\cdot)$ is an embedding vector
output by CLIP. Moreover, we also use DINO v2 [16] as a reference vision
encoder to compute their embedding vectors and compute their cosine
similarity, i.e., $cos(f_{D}(x_{1}),f_{D}(x_{2}))$, where $f_{D}(\cdot)$ is an
embedding vector output by DINO v2. Then, we find image pairs that have large
cosine similarity under CLIP but small cosine similarity under DINO v2. In
particular, we find image pairs that satisfy
$cos(f_{C}(x_{1}),f_{C}(x_{2}))\geq 0.9$ and
$cos(f_{D}(x_{1}),f_{D}(x_{2}))\leq 0.55$. Such image pairs are our
candidates.

Among the candidates, we further select the top 200 pairs with the largest
cosine similarity under CLIP. Moreover, we manually design questions and
reference answers to form 800 initial VH instances, where 100 for each VH mode
and an image may be used in multiple VH instances. Finally, we test an MLLM
(called _testing MLLM_ , e.g., GPT-4V) on them. With manual verification,
GPT-4V hallucinates on 204 of them (called successful initial VH instances).
Table 1 shows the number of successful initial instances in each VH mode.

###  3.2 Step II: Text Description Generation

Given the initial VH instances, we use an MLLM (called _description-generation
MLLM_ , e.g., GPT-4V) to generate a text description for each VH mode. The
text description aims to guide a text-to-image generative model (in Step III)
to generate more images that are likely to trigger VH in MLLMs.

#### Using a Successful Initial VH Instance

We first describe how to leverage prompt engineering to generate a text
description based on one successful initial VH instance. Specifically, we
construct an example comprised of: 1) the VH instance’s image and question, 2)
the testing MLLM’s hallucinated response, and 3) the detected hallucination in
the response. We then add an additional prompt asking the description-
generation MLLM to generate a text description that explains potential causes
underlying the observed VH and describes how to generate more images. We show
a summary version of our prompt in Figure 4 while the full prompt is shown in
Figure 8 in Appendix.

Summary version PROVIDE: {image, question, hallucinated response, detected
hallucination in the response} PROMPT: [Through the example, explain the
potential causes of such hallucination and how to produce more images and
questions.] Figure 4: Summary of our prompt to generate a text description
based on a successful initial VH instance.

#### Using an Unsuccessful Initial VH Instance

We also design a prompt to generate a text description based on an
unsuccessful initial VH instance for which the testing MLLM does not
hallucinate. Such prompt is needed when not enough successful initial VH
instances are available for a VH mode. Our idea is to construct a prompt with
some _hypothetical hallucinated detail_ regarding the VH image. Specifically,
we construct an example comprised of: 1) the VH instance’s image and question,
2) a hypothetical hallucinated response, and 3) the detected hallucination in
the hypothetical response. Given the example, we also add an additional prompt
like we discussed above. We show a summary version of our prompt in Figure 5
while the full prompt is shown in Figure 9 in Appendix.

Summary version PROVIDE: {image, question, hypothetical hallucinated response,
detected hallucination in the hypothetical response} PROMPT: [Through the
example, explain the potential causes of the hallucination and how to generate
more images and questions.] Figure 5: Summary of our prompt to generate a text
description based on an unsuccessful initial VH instance.

#### Integrating Multiple VH Instances

To increase diversity of our text description, for each VH mode, we generate
10 text descriptions based on 10 successful initial VH instances. If a VH mode
has less than 10 successful initial VH instances, we generate the remaining
text descriptions based on unsuccessful ones. Then, for each VH mode, we use a
prompt in Figure 10 in Appendix for a description-generation MLLM to summarize
the 10 text descriptions as the final one. Appendix A shows our generated text
description for each VH mode.

###  3.3 Step III: Generating More VH Instances

We generate more VH instances based on the text descriptions. Recall that a VH
instance consists of an image, a question, and a reference answer. Therefore,
we describe how to generate each component in the following.

#### VH Image

We use a text-to-image generative model (e.g., DALL·E-3 in our experiments) to
generate VH images based on the text descriptions. Specifically, to generate
an image in a VH mode, we append the corresponding text description to the
prompt in Figure 11 in Appendix for the text-to-image generative model to
generate an image.

#### VH Question

Given a VH image, we prepare a question based on it. Specifically, we leverage
object-driven templates to create diverse and relevant VH questions. For
instance, “Describe the shape of the [object] in the picture.” is a template
for the shape VH mode. We curate question templates for each VH mode and they
are shown in Appendix C. Given a template for a VH mode, a human worker
generates a question via manually analyzing the objects in the VH image. For
instance, the human worker may replace the [object] as “pear” in the template
above when the VH image contains a pear. The human worker verifies that the
question should have a non-ambiguous answer based on the VH image. If no
questions with non-ambiguous answers can be constructed, the VH image is
discarded.

#### VH Reference Answer

Given a VH image and a question, a human worker also provides a factually
correct answer as a reference answer via manually analyzing the VH image. The
triple (image, question, reference answer) is a VH instance.

###  3.4 Benchmark Construction

We use VHTest to build two benchmarks. Specifically, we find initial VH
instances in COCO [11] and we generate 150 VH instances for each VH mode,
which results in 1,200 VH instances across 8 VH modes in total. This benchmark
consists of “open-ended question” (OEQ), which requires manually labeling the
responses of MLLMs when testing them on the benchmark. Therefore, to
facilitate automatic evaluation when testing MLLMs, we also construct a
closed-ended “yes/no question” (YNQ) version of the benchmark.

Specifically, for each VH instance, we convert the open-ended question into a
binary “yes/no” question. For instance, the open-ended question “Describe the
shape of the pear in the picture.” is converted into “Is the shape of the pear
in the picture a square?”. Moreover, the reference answer is converted into
yes or no. We construct the binary questions to ensure that the YNQ benchmark
has a 50/50 split of “yes”/“no” reference answers.

Our benchmark construction took approximately 300 human-hours in total.

##  4 Experiments

Table 1: Accuracy of GPT-4V on the initial VH instances from COCO. Each VH mode has 100 initial VH instances. | Existence | Shape | Color | Orientation | OCR | Size | Position | Counting | Average
---|---|---|---|---|---|---|---|---|---
Accuracy | 0.880 | 1.000 | 0.920 | 0.280 | 0.600 | 0.980 | 0.700 | 0.600 | 0.745
Number of successful initial VH instances | 12 | 0 | 8 | 72 | 40 | 2 | 30 | 40 | 204

###  4.1 Experimental Setup

#### MLLMs

We evaluate three state-of-the-art MLLMs on our benchmarks: GPT-4V with its
“gpt-4-vision-preview” version, LLaVA-1.5-13b [14], and MiniGPT-v2 [2].

#### Evaluation Metric

We use _accuracy_ as an evaluation metric. Given an MLLM, we use it to produce
a text response for each VH instance (more precisely, the image and question
in a VH instance) in our OEQ benchmark; we manually analyze the text responses
and compare with the reference answers; and accuracy is the fraction of the VH
instances for which the MLLM’s text responses are factually correct. For the
YNQ benchmark, we can automatically calculate the accuracy of an MLLM since
the reference answers are just yes or no. Note that a smaller accuracy
indicates that an MLLM is more likely to hallucinate on our benchmarks, which
shows that our VHTest is better at generating successful VH instances.

###  4.2 Testing VH in MLLMs

#### Initial VH Instances are Insufficient

Table 1 shows the accuracy of GPT-4V on the 100 initial VH instances from COCO
for each VH mode, and the number of successful initial VH instances in each VH
mode. We observe that a large fraction of the initial VH instances are not
successful. In particular, the average accuracy of GPT-4V across the 8 VH
modes is 0.745, which means that only 204 of the 800 initial VH instances make
GPT-4V hallucinate. These results show that VH instances in existing image
datasets are insufficient at testing MLLMs. For instance, given the results in
Table 1, one may conclude that GPT-4V does not hallucinate for the shape VH
mode since its accuracy is 1.000 for this VH mode. However, as we will discuss
in the following, GPT-4V hallucinates substantially in the shape VH mode on
our benchmarks.

Table 2: Accuracy of GPT-4V, LLaVA-1.5 and MiniGPT-v2 on our OEQ benchmark.  | GPT-4V | LLaVA-1.5 | MiniGPT-v2 | Average
---|---|---|---|---
Existence | 0.427 | 0.240 | 0.013 | 0.227
Shape | 0.487 | 0.167 | 0.093 | 0.249
Color | 0.460 | 0.267 | 0.053 | 0.260
Orientation | 0.153 | 0.140 | 0.127 | 0.140
OCR | 0.367 | 0.127 | 0.000 | 0.164
Size | 0.413 | 0.353 | 0.140 | 0.302
Position | 0.547 | 0.347 | 0.147 | 0.347
Counting | 0.213 | 0.193 | 0.027 | 0.144
Average | 0.383 | 0.229 | 0.075 | 0.229
Table 3: Accuracy of GPT-4V, LLaVA-1.5 and MiniGPT-v2 on our YNQ benchmark.  | GPT-4V | LLaVA-1.5 | MiniGPT-v2 | Average
---|---|---|---|---
Existence | 0.627 | 0.640 | 0.540 | 0.602
Shape | 0.760 | 0.513 | 0.487 | 0.587
Color | 0.587 | 0.593 | 0.487 | 0.556
Orientation | 0.560 | 0.500 | 0.527 | 0.529
OCR | 0.573 | 0.420 | 0.487 | 0.493
Size | 0.687 | 0.587 | 0.540 | 0.604
Position | 0.580 | 0.687 | 0.513 | 0.593
Counting | 0.513 | 0.520 | 0.527 | 0.520
Average | 0.611 | 0.558 | 0.513 | 0.561

#### MLLMs Hallucinate on our VHTest Benchmarks

Table 2 and Table 3 show the accuracy of GPT-4V, LLaVA-1.5, and MiniGPT-v2 on
our OEQ and YNQ benchmarks, respectively. We observe that these MLLMs achieve
a strikingly low average accuracy across the eight VH modes: on average 0.229
and 0.561 for all MLLMs on OEQ and YNQ benchmarks, respectively. For example,
on average, 925 out of the 1,200 VH instances in our OEQ benchmark induce
these MLLMs to hallucinate. It is worth noting that random guessing can
achieve an accuracy of 0.5 on our YNQ benchmark since it is a balanced yes/no
benchmark. Our results indicate that VHTest is highly effective at generating
successful VH instances.

Among the three MLLMs, GPT-4V and MiniGPT-v2 achieve the highest and lowest
accuracy on our benchmarks, respectively. This suggests that GPT-4V is the
most truthful while MiniGPT-v2 is the least truthful on our benchmarks.
Furthermore, we observe that MLLMs perform the poorest on the orientation,
counting, and OCR VH modes based on their average accuracy for each VH mode.
For example, the average accuracy of the three MLLMs on orientation VH mode is
only 0.14. This implies that orientation is the most challenging VH mode,
causing more VHs in MLLMs compared to other VH modes.

###  4.3 Ablation Study

Unless otherwise mentioned, we use the counting VH mode due to the GPT-4V
query limitation.

Table 4: Accuracy of the 3 MLLMs on VH instances generated by 4 different text-to-image generative models. | DALL·E-3 | Midjourney 6 |  Stable Diffusion XL 1.0 |  Stable Diffusion 2.1
---|---|---|---|---
GPT-4V | 0.233 | 0.267 | 0.500 | 0.429
LLaVA-1.5 | 0.167 | 0.133 | 0.233 | 0.233
MiniGPT-v2 | 0.033 | 0.067 | 0.033 | 0.000
Average | 0.144 | 0.156 | 0.256 | 0.221

#### Using Different Text-to-image Generative Models in Step III

We use DALL·E-3 to generate VH images in Step III of our VHTest. We also
evaluate other text-to-image generative models, including Midjourney 6 [15],
Stable Diffusion XL 1.0 [17], and Stable Diffusion 2.1 [20] by using each of
them to generate 30 VH instances, respectively. DALL·E-3 rewrites a prompt
automatically to add more details, whereas the other three models lack this
capability. We find that other text-to-image generative models cannot generate
high-quality images when directly using the text description from Step II as a
prompt. Therefore, we use GPT-4 to generate prompts given the text description
for other text-to-image generative models. Specifically, we use the prompt in
Figure 12 for GPT-4 to generate 30 prompts. Then, we use each text-to-image
generative model to generate 30 VH images with the generated prompts. Finally,
we manually craft the questions and reference answers to form 30 VH instances.
Table 4 shows the accuracy of the three MLLMs on the 30 VH instances generated
by different text-to-image generative models. We observe that these MLLMs
achieve the lowest average accuracy of 0.144 when using DALL·E-3 in Step III.
This indicates that DALL·E-3 is the most effective tool in generating VH
instances that are likely to trigger VHs in MLLMs.

Table 5: Accuracy of the 3 MLLMs on the VH instances generated by our VHTest using successful initial VH instances and unsuccessful initial VH instances. |  | Successful initial
---
VH instances
| Unsuccessful initial
---
VH instances
GPT-4V | 0.300 | 0.700
LLaVA-1.5 | 0.200 | 0.500
MiniGPT-v2 | 0.100 | 0.300
Average | 0.200 | 0.500

#### Do Successful Initial VH Instances Help?

We analyze the impact of successful initial VH instances on generating VH
instances. Specifically, we use Step II of our VHTest to generate a text
description based on three successful initial VH instances and a text
description based on three unsuccessful initial VH instances in the counting
VH mode. Then, we use Step III of our VHTest to generate 10 VH instances based
on each text description. Table 5 shows the accuracy of the 3 MLLMs on the 10
VH instances in the two scenarios. Our results show that the VH instances
generated using successful initial VH instances substantially reduce accuracy
across GPT-4V, LLaVA-1.5, and MiniGPT-v2. A lower accuracy indicates a higher
degree of VH in MLLMs. Our results show that using successful initial VH
instances, our VHTest is more likely to generate successful VH instances.

###  4.4 Mitigating VH in MLLMs

Table 6: Results of LLaVA-1.5 before and after fine-tuning on our benchmarks
and existing ones.

|  Before Fine-tuning |  After Fine-tuning
---|---|---
Existence | 0.233 | 0.267
Shape | 0.167 | 0.333
Color | 0.267 | 0.267
Orientation | 0.133 | 0.167
OCR | 0.133 | 0.167
Size | 0.367 | 0.367
Position | 0.333 | 0.533
Counting | 0.200 | 0.267
Average | 0.229 | 0.296
(a)

|  Before Fine-tuning |  After Fine-tuning
---|---|---
Existence | 0.633 | 0.600
Shape | 0.423 | 0.538
Color | 0.733 | 0.700
Orientation | 0.500 | 0.567
OCR | 0.433 | 0.467
Size | 0.567 | 0.700
Position | 0.700 | 0.700
Counting | 0.467 | 0.433
Average | 0.557 | 0.588
(b)

|  Before Fine-tuning |  After Fine-tuning
---|---|---
MME Perception | 1531.3 | 1556.6
MME Cognition | 295.4 | 288.2
POPE | 85.9 | 84.8
(c)

(a)

(b)

(c)

Figure 6: Ablation study on fine-tuning LLaVA-1.5.

#### Fine-tuning LLaVA-1.5 on our Benchmark

We also study whether fine-tuning an MLLM on our benchmark makes it less
likely to hallucinate. Towards this goal, we use the open-source LLaVA-1.5.
Specifically, we split our OEQ benchmark into a training set comprising 80%
(120 VH instances for each VH mode) and a testing set comprising 20% (30 VH
instances for each VH mode) of the data. To make the testing set performance
as close as possible to the full OEQ benchmark dataset performance before
fine-tuning, for every VH mode, we randomly divide the VH instances into
80%/20% split 100 times and select the split whose testing set accuracy is the
closest to the accuracy on the full OEQ benchmark.

We follow LLaVA-1.5’s limited task-specific fine-tuning setting. However, we
_unfreeze_ the vision encoder since VH may result from its incorrect embedding
vectors for images. We set the learning rate to 4e-6 during fine-tuning; and
we fine-tune on the 960 training VH instances for one epoch (more
hyperparameters and fine-tuning details are shown in Appendix E). Our fine-
tuning took only 18 minutes on a single A6000.

#### Fine-tuning Results

Table 6 shows the accuracy of LLaVA-1.5 and fine-tuned LLaVA-1.5 on the
testing VH instances of our OEQ and YNQ benchmarks. Table 6 also shows the
results on MME [3] and POPE [10], two popular existing benchmarks (not
necessarily VH) to evaluate the performance of an MLLM. We find that fine-
tuned LLaVA-1.5 achieves higher average accuracy than LLaVA-1.5 in both our
benchmarks, while they achieve comparable results on MME and POPE benchmarks.
Our results indicate that fine-tuning an MLLM on our benchmark makes it less
likely to hallucinate.

#### Ablation Study on Fine-tuning

Figure 6 shows ablation study results on the MME benchmark and the testing set
of our YNQ benchmark when fine-tuning LLaVA-1.5. We use our YNQ benchmark
instead of OEQ because it supports automatic evaluation. Since an MLLM has
three key components: vision encoder, vision-language connector, and LLM, we
explore fine-tuning different components. Our results show that fine-tuning
all components achieves the best overall results. As for fine-tuning epochs,
the results show that fine-tuning for one epoch minimizes overfitting to our
benchmark, retaining the best performance on MME. For learning rate, both
excessively large and small learning rates lead to a decline in performance on
our benchmark and MME.

##  5 Related Work

#### Hallucinations

Hallucinations are well-known issues for generative AI, including LLMs [8, 5],
MLLMs [13, 19, 25], and text-to-image generative models [24]. In general,
hallucination refers to a generative model imagines factually incorrect
details in its response for a given input. VH occurs when an MLLM imagines
incorrect details about an image in visual question answering.

#### VH Benchmarks in MLLMs

Prior works have tried to benchmark MLLMs’ VHs [10, 12, 3, 25]. However, they
collect VH images only from existing image datasets. This limits the diversity
of VH images. Moreover, existing image datasets may have been used to pre-
train an MLLM, leading to data contamination [7, 21]. Our VHTest can generate
a diverse set of new VH images that do not appear in existing benchmarks.
Moreover, the shape and size VH modes are formulated by us for the first time.

#### Mitigating VH in MLLMs

Existing works on mitigating VHs in MLLMs can be categorized into _fine-
tuning-phase_ and _testing-phase_ mitigation. Fine-tuning-phase mitigation
focuses on improving the fine-tuning data quality [26, 12] and/or model
structure [25]. These works typically freeze the vision encoder during fine-
tuning, following the standard fine-tuning setting of LLaVA-1.5. We find that
fine-tuning the vision encoder together reduces VHs in MLLMs. Testing-phase
mitigation leverages prompt engineering with more visual evidence [9] or
correction tools for hallucinated responses [29]. Testing-phase mitigation is
complementary to fine-tuning-phase mitigation.

##  6 Conclusion and Future Work

We propose VHTest to generate VH instances to test MLLMs. We collect VH
benchmarks using VHTest and we find that state-of-the-art MLLMs exhibit high
hallucination rates on our benchmarks. Moreover, fine-tuning MLLMs on our
benchmark reduces hallucination without sacrificing their other
performance/capability. We acknowledge that our VHTest still requires human
workers to manually generate a question-answer pair for an automatically
generated VH image. An interesting future work is to make VHTest fully
automatic so it can generate as many VH instances as needed.

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Figure 7: VH instances generated by our VHTest for the other four VH modes and
the text responses of three MLLMs for them.

##  Appendix A Generated Text Descriptions of VH Modes

The following shows the text description for each VH mode generated by our
VHTest.

  1. 1.

Existence VH: In existence hallucination, a multi-modal large language model
(MLLM) may encounter two types of hallucinations. The first type is when an
object exists in an image, but the MLLM asserts that it does not exist in the
picture, called negation existence hallucination. The other type is when a
certain object does not exist in the image, but the MLLM creates or infers
details, objects, or its attributes in an image, known as extrinsic
hallucination.

As for the negation existence hallucination, an MLLM may exhibit challenges in
recognizing the existence of certain objects within an image, particularly
when these objects are not prominently featured within the picture or are
partially obscured. The issue becomes apparent when the model disregards
objects that are present but do not constitute the primary focus of the image.
Factors contributing to negation existence hallucination include objects being
small, distant, having low contrast with the background, or being located at
the periphery of the frame. Such conditions can cause the MLLM to miss or
ignore these elements, leading to an incomplete or inaccurate understanding of
the scene. Negation existence hallucination is particularly evident in complex
environments where multiple objects coexist but some are less dominant or
visually prominent.

Extrinsic hallucination occurs when an MLLM creates or infers details,
objects, or attributes in an image that are not actually present. This is
often due to the model’s reliance on learned patterns and associations from
its training data rather than the specific content of the image it’s
analyzing. MLLMs may "hallucinate" details or objects based on what they have
learned from other contexts, leading to extrinsic hallucinations. Also, the
model readily associates certain scenes with typical objects even when those
objects are not present, especially for scenes containing other complex
patterns.

  2. 2.

Shape VH: A multi-modal large language model (MLLM) misconstrues shapes,
particularly when typical simple shape and undulating strange shape are
crowded. For example, a plate contains a banana and other fruit, some of which
is undulating and coiled. When faced with non-standard shapes, it often
simplifies them to more common, recognizable forms due to biases in training
data. In instances where multiple shapes are present, especially with varying
levels of detail or color intensity, the MLLM might get diverted towards the
more attention-grabbing elements, overlooking or misreading other shapes in
the process. Additionally, the MLLM faces difficulty in discerning subtle
differences in shapes, often generalizing them into broad categories based on
prominent features. These biases indicate a challenge in the MLLM’s ability to
accurately interpret and distinguish between shapes, which could lead to
incorrect generalizations and potential misinterpretations of image data.

  3. 3.

Color VH: A multi-modal large language model (MLLM) struggles with recognizing
or accurately identifying colors or color variations present in an image. It
can misinterpret the sequence of color transition, omitting or inaccurately
describing colors. This can be due to factors like subtle variations,
intertwined colors, lighting, shadows, or adjacent color interference.
Moreover, the recognition of color in little items may be more challenging,
especially when the MLLM’s attention is attracted by a multitude of various
elements inside the image.

  4. 4.

Orientation VH: Orientation hallucination in a multi-modal large language
model (MLLM) often results from a combination of factors that lead to
misinterpretation of the direction in which objects, like vehicles, are
facing. The primary issues include perspective ambiguities where 3D spatial
orientation is not easily conveyed in a 2D image, dense object overlap, and
environments with complex, busy backgrounds that distract from or mimic the
shape and color of the target object. The MLLM might also falter when the
object of interest is situated among others facing various directions,
confusing the model’s directionality cues. Such errors could stem from the
MLLM either overlooking dominant visual cues that indicate orientation or
mistakenly assigning equal importance to all elements in the scene. Underlying
these issues may be a lack of focused training on discerning object’s
orientation especially vehicle’s orientation, which causes the MLLM to
underperform in scenarios where human observers would rely on subtle
contextual clues to determine directionality. This gap between the model’s
interpretation and human perspective underscores the challenge in encoding and
analyzing orientation within complex visual contexts.

  5. 5.

OCR VH: OCR hallucination often stems from a complex interplay of factors. A
multi-modal large language model (MLLM) falters when faced with unconventional
visual scenarios, such as irregular character spacing, similar-looking
letters, vertical arrangement of characters, or interference from nearby
visual elements. These issues are compounded when the text contains
intentional misspellings, typos, or uncommon character combinations that the
system attempts to auto-correct based on standard language patterns. The
ability to accurately recognize characters is further challenged by the
influence of adjacent characters and the overall visual context.

  6. 6.

Size VH: Size comparison in images requires the multi-modal large language
model (MLLM) to recognize and accurately compare objects based on visual
scale. Challenges in this task arise from factors like perspective,
distortions, overlapping of objects, intricate patterns, complex backgrounds,
discrepancies in expected real-world proportions, and excessive focus on
foreground objects that causes the size of enormous background objects to be
overlooked. Especially, when conducting a comparison among multiple objects of
similar types and sizes, with other different types of objects in the scene
that disturb the MLLM’s attention, the task becomes even more challenging. The
MLLM misjudges which object is larger or might rank objects incorrectly in
terms of size.

  7. 7.

Position VH: A multi-modal large language model (MLLM) encounters difficulties
in accurately assessing the spatial positioning relationship between objects
within an image. This is exacerbated in scenarios where objects are placed in
non-linear configurations, such as circular layouts, which can disrupt the
model’s ability to apply standard left-to-right reading patterns. Moreover,
when the objects are set against complex or similar backgrounds that lack
clear demarcation lines, the MLLM’s spatial parsing capabilities can be
further compromised. Contributing factors such as overlapping objects,
inconsistent scaling, and deceptive perspective or angled viewpoints also
intensify this challenge. In addition, the presence of shadows or uneven
lighting may cast ambiguity on the object’s precise location, thereby leading
to misinterpretation. Such spatial hallucination can be attributed not just to
the inherent complexity of object arrangement but also to the MLLM’s
processing of visual cues that inform depth, orientation, and the relationship
of elements within the image space.

  8. 8.

Counting VH: A multi-modal large language model (MLLM) has difficulty in
accurately counting the quantity of specific elements or attributes in an
image, especially when the objects are closely packed, overlap, are of
different sizes, are partially visible, or have varying orientations. The task
becomes increasingly complex when attempting to count specific subtype objects
within a larger type category. Such conditions can lead the model to overlook
certain objects, mistakenly merge similar items, or misinterpret the image
data, thereby providing an inaccurate counting of numbers, amounts, or values.

##  Appendix B Prompts Used in VHTest

We use the following prompt in Figure 8 for text description generation in
Step II, based on a successful initial VH instance.

Prompt for text description generation using a successful initial VH instance:
[image] Question: [question] Multi-modal LLM (MLLM)’s response: [testing
MLLM’s hallucinated response] Detected hallucination in the response:
[detected hallucination in the response] Focus on the following elements:
image, question, MLLM’s response, and the detected hallucination in the
response. I’m trying to identify visual hallucinations in the MLLM associated
with its visual process. Through the specific example, are there any general
types of hallucination modes you notice the MLLM makes, or any visual features
that MLLM fails to encode, ultimately leading to the errors in MLLM’s
response? Try to give hallucination modes that are specific enough that
someone could enable consistent reproduction of images and corresponding
questions. Please try to include as many general hallucination modes as
possible. These hallucination modes will be used later to generate images or
videos. In your hallucination modes, please clearly explain why the this
hallucination mode would cause difficulties for the vision encoder of MLLM to
understand images related to this hallucination mode. I will further use this
reason to precisely generate images that match the hallucination mode and are
able to mislead the MLLM. Please encapsulate the essence of the examples
provided, summarize as many as possible and stick to the examples. Figure 8:
Full prompt to generate a text description based on a successful initial VH
instance.

We utilize the following prompt in Figure 9 for text description generation in
Step II, given an unsuccessful initial VH instance.

Prompt for text description generation using an unsuccessful initial VH
instance: [image] Question: [question] A hypothetical response from the multi-
modal LLM (MLLM): [a hypothetical hallucinated response] Detected
hallucination in the hypothetical response: [detected hallucination in the
hypothetical response] Focus on the following elements: image, question,
MLLM’s hypothetical response, and detected hallucination in the hypothetical
response. I’m trying to identify visual hallucinations in the MLLM associated
with its visual process. Through the specific example, are there any general
types of hallucination modes you notice the MLLM makes, or any visual features
that MLLM fails to encode, ultimately leading to the errors in MLLM’s
response? Try to give hallucination modes that are specific enough that
someone could enable consistent reproduction of images and corresponding
questions. Please try to include as many general hallucination modes as
possible. These hallucination modes will be used later to generate images or
videos. In your hallucination modes, please clearly explain why the this
hallucination mode would cause difficulties for the vision encoder of MLLM to
understand images related to this hallucination mode. I will further use this
reason to precisely generate images that match the hallucination mode and are
able to mislead the MLLM. Please encapsulate the essence of the examples
provided, summarize as many as possible and stick to the examples. Figure 9:
Full prompt to generate a text description based on an unsuccessful initial VH
instance.

We use the following prompt in Figure 10 for text description integration in
Step II.

Prompt for text description integration: I want you to focus on the [VH mode]:
[the definition of VH mode]. Try to summarize the [VH mode] text description
in ONE paragraph less than 200 words to explain the causes of the
hallucination based on the information below. [N text descriptions] Figure 10:
Prompt to integrate N text descriptions into a final text description for a VH
mode.

We use the following prompt in Figure 11 for a text-to-image generative model
to generate a VH image in Step III.

Prompt for VH image generation: Generate an image that reflects the given
hallucination mode. After your image generation, we will manually generate
questions based on the image. Your MAIN GOAL is to ensure that the image
aligns with the hallucination mode well so that querying the multi-modal LLM
with the combination of manual questions and the image can effectively induce
hallucinations. You will be evaluated on how well you actually perform. The
generated image should ideally align with the hallucination mode, but there’s
room for creativity. Be both creative and cautious. You can try to create
image with different scenes and objects that align with the hallucination
mode. Moreover, when you generate images, remember you are a very clever
expert in exploiting the hallucination mode. For future debugging purposes,
ensure that the generated images with our manual questions MUST cause multi-
modal LLMs to provide incorrect responses. Hallucination Mode: [Text
description of a VH mode] Figure 11: Prompt for a text-to-image generative
model to generate a VH image under a VH mode.

We use the following prompt in Figure 12 for an LLM to generate prompts for
text-to-image generative models. We only append extra text at the end compared
with the prompt in Figure 11.

Prompt for an LLM to generate prompts for text-to-image generative models:
Generate an image that reflects the given hallucination mode. After your image
generation, we will manually generate questions based on the image. Your MAIN
GOAL is to ensure that the image aligns with the hallucination mode well so
that querying the multi-modal LLM with the combination of manual questions and
the image can effectively induce hallucinations. You will be evaluated on how
well you actually perform. The generated image should ideally align with the
hallucination mode, but there’s room for creativity. Be both creative and
cautious. You can try to create image with different scenes and objects that
align with the hallucination mode. Moreover, when you generate images,
remember you are a very clever expert in exploiting the hallucination mode.
For future debugging purposes, ensure that the generated images with our
manual questions MUST cause multi-modal LLMs to provide incorrect responses.
Hallucination Mode: [Text description of a VH mode] As an LLM, DO NOT generate
images, but generate prompts for text-to-image generative models aligning with
the hallucination mode. I will use your generated prompts for text-to-image
generative models later. Figure 12: Prompt for an LLM to generate prompts for
text-to-image generative models under a VH mode.

##  Appendix C Question Templates

We provide question templates for all eight VH modes below.

Question templates for existence VH mode 1\. Could you describe [object] in
the picture? 2\. Describe the [property] of [object] in the picture. Question
templates for shape VH mode 1\. Describe the shape(s) of [object] in the
picture. 2\. In the photo, is [object] depicted with a conventional, typical
shape? Question templates for color VH mode 1\. Describe the color(s) of
[object] in the picture. 2\. Describe [n] main color(s) of [object] in the
picture. 3\. Describe all the colors of [object] in the picture. Do not miss
any color. 4\. Describe the original color of [object] in the picture.
Question templates for orientation VH mode 1\. From the camera/viewer’s
perspective, please choose the direction that is most proximate to the
orientation in which [object] is facing in the picture. [Left, 45 degrees
leftwards away from the camera/viewer, Away from the camera/viewer, 45 degrees
rightwards away from the camera/viewer, Right, 45 degrees rightwards towards
the camera/viewer, Towards the camera/viewer, 45 degrees leftwards towards the
camera/viewer] 2\. From the camera/viewer’s perspective, please choose the
direction that is most proximate to the orientation in which [object] is
facing in the picture. [Left, Left and upward within the pictorial plane,
Upward within the pictorial plane, Right and upward within the pictorial
plane, Right, Right and downward within the pictorial plane, Downward within
the pictorial plane, Left and downward within the pictorial plane] 3\. From
the camera/viewer’s perspective, please choose the direction that is most
proximate to the orientation in which [object] is facing in the picture. [Away
from the camera/viewer, Towards the camera/viewer] Question templates for OCR
VH mode 1\. Read the (English) character(s)/word(s)/ caption [at WHERE] in the
picture. Question templates for size VH mode 1\. Could you identify and locate
[object] with the largest/smallest size in the picture? 2\. Could you identify
and locate [object] with the tallest/shortest height in the picture? 3\. Could
you identify and locate [object] with the second largest/smallest size in the
picture? 4\. Could you identify and locate [object] with the largest/smallest
and the second largest/smallest size in the picture? 5\. Could you identify
and locate [object] with the tallest/shortest and the second tallest/shortest
height in the picture? 6\. Could you identify and locate the 2/3/4
largest/smallest [object] in the picture? 7\. Which one is
larger/smaller/taller/shorter, [object A] or [object B]? 8\. What is the
largest [object] in the picture? Question templates for position VH mode 1\.
Is [object A] to the left/right of/on the top of/being placed
on/above/under/inside/outside [object B]? 2\. Which one is on the
left/right/top, [object A] or [object B]? 3\. Which one is closer to/further
from the camera or viewer perspective, [object A] or [object B]? 4\. Which one
is positioned under/positioned above/closer to [reference object], [object A]
or [object B]? 5\. Regardless of the positional relationship of the actual
scene in reality, from the camera’s perspective in the photo, is [object A]
positioned above/below [object B]? 6\. Is [object A] on and touching [object
B]? Question templates for counting VH mode 1\. How many [object] are
depicted/there/visible/can be seen [at WHERE] in the picture?

##  Appendix D Special Cases in Step III

In existence VH mode, a human worker uses the prompt in Figure 13 to find non-
existent objects in a VH image with the aid of an MLLM, such as GPT-4V. The
non-existent objects found in this way are more likely to trigger VHs in
MLLMs.

Prompt to find names of non-existent objects in a VH image in the existence VH
mode [image] First, list all the names of objects in the picture. And then
return some names of objects according to Requirement 1 or Requirement 2:
Requirement 1: Objects you associate with the scene that should be there but
are not actually there in the picture. Requirement 2: Objects that look
similar to an object in the picture but do not actually exist in the picture.
Figure 13: Prompt to find names of non-existent objects in a VH image in the
existence VH mode.

##  Appendix E Details of Fine-tuning Experiments

###  E.1 Hyperparamters

When fine-tuning LLaVA-1.5 on the training set of VHTest dataset, we follow
LLaVA-1.5’s limited task-specific fine-tuning setting. Additionally, we
unfreeze the vision encoder. The hyperparameters are shown in Table 7.

Table 7: Hyperparamters for fine-tuning LLaVA-1.5. Hyperparameter | Setting
---|---
Batch size | 16
Vision encoder lr | 4e-6
Projection lr | 4e-6
LLM lr | 4e-6
Lr schedule | cosine decay
Lr warmup ratio | 0.03
Weight decay | 0
Epoch | 1
Optimizer | AdamW
DeepSpeed stage | 3

###  E.2 Pre-processing of the Training Set

As mentioned in [14], the short-form data overfit an MLLM behaviorally to
short-form answers. We follow this work to do pre-processing on both counting
and position VH modes. To be more specific, for counting VH mode instances, we
use the prompt below in Figure 14 for an LLM (e.g., GPT-4) to modify every
number reference answer into a reference sentence. As for position VH mode
instances, we add the sentence “Answer the question using a single word or
phrase.” after all the questions in every position VH instance.

Prompt for reference answer modification in counting VH mode Don’t talk
nonsense. Turn the ground-truth numbers into extremely correct, extremely
accurate, and only a little of diverse sentences based on the corresponding
question. Sentences should be independent of each other, with each sentence
occupying one line. Figure 14: Prompt for reference answer modification in
counting VH modeduring pre-processing of training set.


# The decay $\eta^{\prime}\to\pi\rho$ in the chiral NJL model

M. K. Volkov1<EMAIL_ADDRESS>A. A. Pivovarov1<EMAIL_ADDRESS>K.
Nurlan1,2,3<EMAIL_ADDRESS>1 BLTP, Joint Institute for Nuclear Research,
Dubna, 141980 Russia
2 Institute of Nuclear Physics, Almaty, 050032, Kazakhstan
3 L.N. Gumilyov Eurasian National University, Nur-Sultan, 01008, Kazakhstan

###### Abstract

The branching fraction of the decay $\eta^{\prime}\to\pi\rho$ is calculated in
the chiral quark NJL model. The decay exists due to the mass difference
between $u$\- and $d$-quarks leading to $\pi^{0}-\eta^{{}^{\prime}}$ mixing.
Two different approaches are applied: the approximate calculation of this
process taking into account the transition $\pi^{0}-\eta^{{}^{\prime}}$
explicitly without dioganalisation and a more precise approach consisting in
the diagonalisation of the singlet and octet states leading to the physical
fields $\pi^{0}$, $\eta$ and $\eta^{{}^{\prime}}$. The obtained results are in
satisfactory agreement with current experimental data.

Recently, the collaboration BESIII reported new experimental data for the
decays $\eta^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$ and
$\eta^{\prime}\to\pi^{0}\pi^{0}\pi^{0}$ [1]. These experimental measurements
allowed one to estimate the partial width of the decay
$\eta^{\prime}\to\pi^{\pm}\rho^{\mp}$ with the result
$Br(\eta^{\prime}\to\pi^{\pm}\rho^{\mp})=(7.44\pm 0.60)\times 10^{-4}$. This
process is a dominant intermediate channel of the decay
$\eta^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$[1]. The theoretical description of the
decay $\eta^{\prime}\to\pi^{+}\pi^{-}\pi^{0}$ was presented in the work [2].

The decay $\eta^{\prime}\to\pi\rho$ is interesting for the following reasons.
It exists due to the mass difference between the $u$\- and $d$-quarks. As a
result, a possibility of transition between the fields $\pi$ and
$\eta^{\prime}$ appears. In present time, there is a significant difference in
the estimations of the value of this transition (see Table 1). The calculation
of the decay $\eta^{\prime}\to\pi\rho$ is a good possibility to verify these
estimations. Indeed, the width of this decay depends on two parameters: the
constant $g_{\rho}=6$ of the decay $\rho\to\pi\pi$ [3, 4] and the value of the
$\pi^{0}-\eta^{\prime}$ transition. For this reason, in the present work, the
main attention is paid to the consideration of this decay from different
points of view.

In the present paper, the calculation of the decay $\eta^{\prime}\to\pi\rho$
is carried out in the framework of the Nambu–Jona-Lasinio (NJL) model [5, 6,
7, 8, 9, 4]. The difference between the $u$\- and $d$-quark masses determining
the value of this decay can be found by using the experimental width of the
non-standard decay $\omega\to\pi\pi$. The corresponding calculation was
described in the review [5] and then with the new corrected parameters in the
recent review [4]. As a result, the value 4.2 MeV has been obtained for the
quark mass difference.

The found quark mass difference allows one to calculate the values of the
transitions $\pi^{0}-\eta$ and $\pi^{0}-\eta^{{}^{\prime}}$. These transitions
also make such a decay as $\rho\to\pi\eta$ possible. This process was
described in the paper [10] and also in the paper [11]. More precise
estimations for this process will be given in this work. Unfortunately, at the
present time, the experimental estimation of this process gives only the upper
limit. That is why, the main purpose of the present paper is to describe the
decay $\eta^{\prime}\to\pi\rho$.

For a description of the transitions $\pi^{0}-\eta$ and
$\pi^{0}-\eta^{\prime}$ we use the quark one-loop approximation with the
parameters taken from the NJL model [4] (Fig. 1). The description of these
transitions is carried out by using the parts of the Lagrangian presented
bellow containing the interactions of the $\pi$, $\eta$, and $\eta^{\prime}$
mesons with quarks. Also, we present the Lagrangian of the interaction of the
vector $\rho$-meson with quarks which is necessary for the description of the
decay $\eta^{\prime}\to\pi\rho$.

$\displaystyle\Delta{\mathcal{L}}=\bar{q}\biggl{[}ig_{\pi}\gamma^{5}\left(\lambda_{0}^{\pi}\pi+\lambda_{+}^{\pi}\pi+\lambda_{-}^{\pi}\pi^{-}\right)+\frac{g_{\rho}}{2}\gamma^{\mu}\left(\lambda_{-}^{\rho}{\rho}^{-}_{\mu}+\lambda_{+}^{\rho}{\rho}^{+}_{\mu}\right)+i\sin(\bar{\theta})g_{\eta_{u}}\gamma^{5}\lambda_{u}\eta$
$\displaystyle\qquad+i\cos(\bar{\theta})g_{\eta_{s}}\gamma^{5}\lambda_{s}\eta+i\cos(\bar{\theta})g_{\eta_{u}}\gamma^{5}\lambda_{u}\eta^{\prime}-i\sin(\bar{\theta})g_{\eta_{s}}\gamma^{5}\lambda_{s}\eta^{\prime}\biggr{]}q,$
(1)

where $q$ and $\bar{q}$ are fields of u, d and s quarks with constituent
masses $m_{u}=270$ MeV, $m_{d}=274.2$ MeV and $m_{s}=420$ MeV; $\lambda$
matrices are linear combinations of the Gell-Mann matrices [4].
$\bar{\theta}=\theta_{0}-\theta$, where $\theta=-19^{\circ}$ is the deviation
from the ideal mixing angle $\theta_{0}=35.3^{\circ}$ for the $\eta$ and
$\eta^{\prime}$ mesons [4].

The renormalization constants for the meson fields take the form

$\displaystyle g_{\pi}=\sqrt{\frac{Z_{\pi}}{4I_{2}(u)}},\quad
g_{\rho}=\sqrt{\frac{3}{2I_{2}(u)}},\quad
g_{\eta_{u}}=\sqrt{\frac{Z_{\eta_{u}}}{4I_{2}(u)}},\quad
g_{\eta_{s}}=\sqrt{\frac{Z_{\eta_{s}}}{4I_{2}(s)}}.$ (2)

where $Z_{\pi}$, $Z_{\eta_{u}}$ and $Z_{\eta_{s}}$ are additional
renormalization constants arising in the transitions $\pi-a_{1}$ and
$\eta(\eta^{\prime})-f_{1}(1285)(f_{1}(1420))$ [5, 4].

The integrals appearing in the coupling constants have the form

$\displaystyle
I_{n}(q)=-i\frac{N_{c}}{(2\pi)^{4}}\int\frac{\Theta(\Lambda_{4}^{2}+k^{2})}{(m_{q}^{2}-k^{2})^{n}}\mathrm{d}^{4}k,$
(3)

where $\Lambda_{4}=1.65$ GeV is the ultraviolet cutoff parameter [4].

$\pi^{0}$$\eta(\eta^{\prime})$ Figure 1: Diagram describing
$\pi^{0}\to\eta(\eta^{\prime})$ transitions in the NJL model.

Using the Lagrangian in (The decay $\eta^{\prime}\to\pi\rho$ in the chiral NJL
model) for the constant describing the $\pi-\eta$ transition, we obtain the
following expression:

$\displaystyle\epsilon_{\pi\eta}=\frac{3g_{\pi}^{2}\sin{(\bar{\theta})}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[\int\frac{Tr\left\\{\gamma^{5}\left(\hat{k}+\frac{\hat{p}}{2}+m_{u}\right)\gamma^{5}\left(\hat{k}-\frac{\hat{p}}{2}+m_{u}\right)\right\\}}{\left[\left(k+\frac{p}{2}\right)^{2}-m_{u}^{2}\right]\left[\left(k-\frac{p}{2}\right)^{2}-m_{u}^{2}\right]}\frac{d^{4}k}{\left(2\pi\right)^{4}}-\int\frac{Tr\left\\{\gamma^{5}\left(\hat{k}+\frac{\hat{p}}{2}+m_{d}\right)\gamma^{5}\left(\hat{k}-\frac{\hat{p}}{2}+m_{d}\right)\right\\}}{\left[\left(k+\frac{p}{2}\right)^{2}-m_{d}^{2}\right]\left[\left(k-\frac{p}{2}\right)^{2}-m_{d}^{2}\right]}\frac{d^{4}k}{\left(2\pi\right)^{4}}\right].$

We use the standard procedure for the NJL model and expand these integrals in
a series in powers of external momentum and keep only the divergent parts.
Then the transition takes the form:

$\displaystyle\epsilon_{\pi\eta}=\frac{g_{\pi}^{2}\sin{(\bar{\theta})}}{M_{\eta}^{2}-M_{\pi}^{2}}\left[4I_{1}(u)+2M_{\eta}^{2}I_{2}(u)-\left(4I_{1}(d)+2M_{\eta}^{2}I_{2}(d)\right)\right].$
(5)

Having carried out minor transformations of this expression and excluding
converging terms, we can obtain for this transition:

$\displaystyle\epsilon_{\pi\eta}=4g_{\pi}^{2}\sin{(\bar{\theta})}\left(m_{d}^{2}-m_{u}^{2}\right)I_{2}(u)\frac{1}{M_{\eta}^{2}-M_{\pi}^{2}}\approx(0.93\pm
0.15)\cdot 10^{-2}.$ (6)

For the $\pi-\eta^{\prime}$ transition, we similarly obtain

$\displaystyle\epsilon_{\pi\eta^{\prime}}=4g_{\pi}^{2}\cos{(\bar{\theta})}\left(m_{d}^{2}-m_{u}^{2}\right)I_{2}(u)\frac{1}{M_{\eta^{\prime}}^{2}-M_{\pi}^{2}}\approx(2.07\pm
0.35)\cdot 10^{-3}.$ (7)

The inaccuracy of the version of the NJL model used in this paper is estimated
at $\sim$ 17% [4].

The values $\epsilon_{\pi\eta}$ and $\epsilon_{\pi\eta^{\prime}}$ can be
considered as rotation angles of mixing of the states $\pi^{0}-\eta$ and
$\pi^{0}-\eta^{{}^{\prime}}$.

$\eta^{\prime}$$\pi^{+}$$\rho^{-}$$\pi^{0}$ Figure 2: Diagram for
$\eta^{\prime}\to\pi^{+}\rho^{-}$ decay in the NJL model.

Using the obtained transition value $\pi^{0}-\eta$, for the decay
$\rho\to\pi\eta$ we obtain the amplitude of the following form:

$\displaystyle\mathcal{M}(\rho^{\pm}\to\pi^{\pm}\eta)=g_{\rho}\epsilon_{\pi\eta}e_{\mu}(p_{\rho})(p_{\eta}-p_{\pi})^{\mu}.$
(8)

Then the branching fraction of this process is:

$\displaystyle Br(\rho^{\pm}\to\pi^{\pm}\eta)\approx(6.29\pm 1.07)\times
10^{-6}.$ (9)

The experiment gives only the upper limit [12]:

$\displaystyle Br(\rho^{\pm}\to\pi^{\pm}\eta)_{exp}<6\cdot 10^{-3}.$ (10)

The diagram describing the decay $\eta^{\prime}\to\pi^{+}\rho^{-}$ is
presented in Figure 2. For this decay, we obtain the following expression:

$\displaystyle\mathcal{M}(\eta^{\prime}\to\pi^{+}\rho^{-})=2g_{\rho}\epsilon_{\pi\eta^{{}^{\prime}}}e^{*}_{\mu}(p_{\rho})p_{\pi}^{\mu}.$
(11)

For the $\eta^{\prime}\to\pi^{-}\rho^{+}$ process, the amplitude looks
similar. The resulting branching fraction will be the sum of the partial
widths of these processes:

$\displaystyle Br(\eta^{\prime}\to\pi^{\pm}\rho^{\mp})\approx(5.3\pm
0.9)\times 10^{-4}.$ (12)

The experimental data [12]:

$\displaystyle Br(\eta^{\prime}\to\pi^{\pm}\rho^{\mp})_{exp}=(7.4\pm
2.3)\times 10^{-4}.$ (13)

As we can see, our result is in satisfactory agreement with the experiment
taking into account the accuracies of measurement and model.

It is interesting to note that in the case of the $\eta^{\prime}\to\pi\rho$
decay, there is an experimental value of the branching fractions, while for
the $\rho\to\pi\eta$ decay, there is currently only the upper bound.

The above approach took into account the mixing between $\eta-\pi^{0}$ and
$\eta^{{}^{\prime}}-\pi^{0}$ mesons. The states $\eta$ and
$\eta^{{}^{\prime}}$ were already considered to be diagonalized after mixing
due to the ’t Hooft interaction [9, 7]. A more precise approach would be to
consider mixing of all three states at the same time. For these purposes, the
singlet state $\phi_{0}$ and the octet states $\phi_{3}$ and $\phi_{8}$ can be
used. The quark-meson Lagrangian containing these states takes the following
form:

$\displaystyle\Delta{\mathcal{L}}=i\bar{q}\gamma^{5}\left[\lambda_{0}\phi_{0}+\lambda_{3}\phi_{3}+\lambda_{8}\phi_{8}\right]q,$
(14)

where

$\displaystyle\lambda_{0}=\sqrt{\frac{2}{3}}\left(\begin{tabular}[]{ccc}1&0&0\\\
0&1&0\\\
0&0&1\end{tabular}\right),\quad\lambda_{3}=\left(\begin{tabular}[]{ccc}1&0&0\\\
0&-1&0\\\
0&0&0\end{tabular}\right),\quad\lambda_{8}=\frac{1}{\sqrt{3}}\left(\begin{tabular}[]{ccc}1&0&0\\\
0&1&0\\\ 0&0&-2\end{tabular}\right).$ (24)

The fields $\phi_{0}$, $\phi_{3}$ and $\phi_{8}$ can be mixed. In this case,
for a correct description of the mixing of the $\phi_{0}$ and $\phi_{8}$
states, it is necessary to take into account the ’t Hooft interaction in the
initial four-quark Lagrangian. This results in a deviation from the ideal
mixing angle by $\theta=-19^{\circ}$. Taking into account the mass difference
between the $u$\- and $d$-quarks leads to the mixing of the
$\phi_{0}-\phi_{3}$ and $\phi_{8}-\phi_{3}$ fields. Since this difference is
small, the angle of rotation in the $\phi_{0}-\phi_{3}$ and
$\phi_{8}-\phi_{3}$ planes will also be small, and therefore, the field
transformation can be presented in the following form (as it was done, for
example, in [13, 14]):

$\displaystyle\phi_{3}$ $\displaystyle=$
$\displaystyle\pi^{0}-\epsilon_{\pi\eta}\eta-\epsilon_{\pi\eta^{\prime}}\eta^{\prime},$
$\displaystyle\phi_{8}$ $\displaystyle=$
$\displaystyle\left(\epsilon_{\pi\eta}\cos{\theta}+\epsilon_{\pi\eta^{\prime}}\sin{\theta}\right)\pi^{0}+\cos{\theta}\eta+\sin{\theta}\eta^{\prime},$
$\displaystyle\phi_{0}$ $\displaystyle=$
$\displaystyle\left(\epsilon_{\pi\eta^{\prime}}\cos{\theta}-\epsilon_{\pi\eta}\sin{\theta}\right)\pi^{0}-\sin{\theta}\eta+\cos{\theta}\eta^{\prime},$
(25)

where $\epsilon_{\pi\eta}$ and $\epsilon_{\pi\eta^{\prime}}$ are the rotation
angles in the planes $\phi_{0}-\phi_{3}$ and $\phi_{8}-\phi_{3}$. Then for
these angles one can obtain the expressions

$\displaystyle\epsilon_{\pi\eta}=\frac{M_{03}\sin{\theta}-M_{38}\cos{\theta}}{M_{\eta}^{2}-M_{\pi}^{2}},\quad\epsilon_{\pi\eta^{\prime}}=-\frac{M_{03}\cos{\theta}+M_{38}\sin{\theta}}{M_{\eta^{\prime}}^{2}-M_{\pi}^{2}},$
(26)

where $M_{03}$ and $M_{38}$ are the transition amplitudes between the
corresponding states. Taking into account the diagram shown in Figure 1, the
following expressions are obtained for these amplitudes:

$\displaystyle
M_{03}=M_{38}=4g_{\pi}^{2}\left(m_{d}^{2}-m_{u}^{2}\right)I_{2}(u).$ (27)

Then the desired angles are $\epsilon_{\pi\eta}\approx(1.45\pm 0.25)\times
10^{-2}$, $\epsilon_{\pi\eta^{\prime}}\approx(2.21\pm 0.38)\times 10^{-3}$.
Substituting these values into the amplitudes (8) and (11), one can obtain for
the decays $\rho^{\pm}\to\pi^{\pm}\eta$ and
$\eta^{\prime}\to\pi^{\pm}\rho^{\mp}$ the following results:

$\displaystyle Br(\rho^{\pm}\to\pi^{\pm}\eta)\approx(1.54\pm 0.26)\times
10^{-5},$ (28) $\displaystyle
Br(\eta^{\prime}\to\pi^{\pm}\rho^{\mp})\approx(6.00\pm 1.01)\times 10^{-4}.$
(29)

As we can see, the obtained results still do not contradict the experimental
data.

| NJL | Paver and Raizuddin [11, 15] | Osipov [13] | Kroll [14]
---|---|---|---|---
$\epsilon_{\pi\eta}\times 10^{2}$ | $0.93\pm 0.15(1.45\pm 0.25)$ | $1.34$ | $1.3\pm 0.1$ | $1.7\pm 0.2$
$\epsilon_{\pi\eta^{\prime}}\times 10^{3}$ | $2.07\pm 0.35(2.21\pm 0.38)$ | $3\pm 1$ | $3.9\pm 0.3$ | $4.0\pm 1.0$
Table 1: Comparison of the values of the rotation angles $\epsilon_{\pi\eta}$
and $\epsilon_{\pi\eta^{\prime}}$. For the case of the NJL model, two results
obtained by different methods are presented.

The obtained values for the rotation angles $\epsilon_{\pi\eta}$ and
$\epsilon_{\pi\eta^{\prime}}$ can be compared with the results of other
authors [11, 13, 15, 14] (see Table 1).

Our results have been obtained without introducing additional arbitrary
parameters and are in satisfactory agreement with the experimental data for
the $\eta^{\prime}\to\pi^{\pm}\rho^{\mp}$ process. The existing discrepancies
with other theoretical works for the values $\epsilon_{\pi\eta}$ and
$\epsilon_{\pi\eta^{\prime}}$ can be explained by the use of different models.

## Acknowledgments

The authors are grateful to A.B. Arbuzov and A.A. Osipov for useful
discussions.

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