# Localization vs. Semantics:
How Can Language Benefit Visual Representation Learning?

Zhuowan Li$1$ Cihang Xie$2$ Benjamin Van Durme$1$ Alan Yuille$1$
$1$ Johns Hopkins University $2$ University of California, Santa Cruz

###### Abstract

Despite the superior performance brought by vision-and-language pretraining,
it remains unclear whether learning with multi-modal data can help understand
each individual modality. In this work, we investigate how language can help
with visual representation learning from a probing perspective. Specifically,
we compare vision-and-language and vision-only models by probing their visual
representations on a broad range of tasks, in order to assess the quality of
the learned representations in a fine-grained manner. Interestingly, our
probing results suggest that vision-and-language models are better at label
prediction tasks like object and attribute prediction, while vision-only
models are stronger at dense prediction tasks that require more localized
information. With further analysis using detailed metrics, our study suggests
that language helps vision models learn better semantics, but not
localization. Code is released at https://github.com/Lizw14/visual_probing.

##  1 Introduction

Humans learn about the world through both vision and language, and the
understanding of both modalities offers mutual benefits. For example, pointing
to images helps children learn the meaning of words [7], and specific
descriptions of pictures (_e.g_., the yellow is to the left of the black) help
children memorize visual features better [12]. For machines, the joint
understanding of vision and language is greatly promoted by recent
developments in vision-and-language pretraining (VLP).

Recent VLP methods achieve impressive performance, not only on multi-modal
tasks like visual question answering, but also on uni-modal vision tasks or
language tasks (_e.g_., ImageNet classification [11], GLUE [53] language
understanding). For examples, contrastive methods like CLIP [39] and ALIGN
[24] learn strong visual models for transfer learning; unified foundation
models like OFA [54] and FLAVA [43] target tasks in vision, language, and
vision-and-language all at once with a single model. Unlike earlier VLP
methods (_e.g_., LXMERT [45], OSCAR [29]) that rely on image features
extracted by separately trained features extractors, these recent developments
in VLP learn feature representations directly from raw image pixels in an end-
to-end manner. This learning paradigm enables textual knowledge to propagate
into the visual encoder, potentially allowing visual understanding to benefit
from learning together with language.

Figure 1: We compare the visual representations of vision-and-language models
with vision-only models from a probing perspective. Probing results on five
tasks suggest that language helps vision models learn better semantics, but
not localization.

Despite the superior performance, there is little understanding of how multi-
modal training can benefit visual learning. Meanwhile, interestingly, we note
the opposite question of how vision can help language has already been
extensively studied in NLP—existing studies suggest that vision helps language
by improving grounding [46, 23], reducing reporting bias [16], and enhancing
commonsense knowledge [59], _etc_. Inspired by these works, we take a similar
but further step—we aim to understand _how language can benefit visual
learning_ by comparing multi-modal and uni-modal visual representations in
terms of a broad spectrum of abilities, _e.g_., predicting labels and dense
maps, understanding single/multi-object images.

To this end, we compare vision-and-language (VL) models and vision-only (V)
models from a probing perspective. Probing, which is to train a low-capacity
model to predict linguistic properties from frozen representations, is a
commonly used technique in NLP for interpreting the representations of trained
models [42, 6]. Using probing, language model representations have been found
to encode a broad range of linguistic properties like part-of-speech [5] or
sentence length [1]. Following this paradigm, we first extract image features
using different pretrained models, and then train a simple prediction head to
align the model’s representation space with the label space of interest. We
make the head as simple as possible based on the intuition that less
expressive heads can more selectively reflect the quality of the
representations [21]. The probing is done on various tasks and datasets:
object name classification on the Visual Genome dataset [26], attribute
prediction on the VAW dataset [37], object detection and instance segmentation
on the MSCOCO dataset [31], and semantic object part segmentation on the
PartImageNet dataset [17]. With these probing tasks, we compare advanced
vision-only models including MAE [18] and MOCOv3 [8], with vision-and-language
pretrained models including OFA [54], FLAVA [43] and CLIP [39].

Our experiments draw several interesting conclusions. Firstly, we find that VL
models are much better at label prediction tasks (_e.g_., object/attribute
prediction), while vision-only models are stronger at dense prediction tasks
like detection and segmentation. In other words, language improves the
semantic information in visual representations to better predict fine-grained
labels, but does not enrich the localization information that is required by
spatial-aware tasks. We further verify our findings with a more detailed
analysis. Secondly, our results on attribute prediction reveal distinct
expertise of vision and VL models: vision models are better for predicting
visually grounded attributes like textures, while VL models are better at more
abstract attributes like actions. We also study the influence of finetuning on
downstream tasks, and find that finetuning degrades the probing results to
varying degrees, depending on the downstream task. More analysis is described
in Sec. 4.

In summary, this work aims to understand the role of language in visual
learning from a holistic perspective, which goes beyond the high performance
of VLP. To achieve this goal, we borrow the idea of probing from the NLP
field, and probe the visual representation in pretrained models on a broad
spectrum of tasks that measure various properties of the representations. As
shown in Fig. 1, our probing results suggest that training with language helps
vision models learn better semantics, but not localization. We hope our
findings provide insights for improving vision models with multi-modal
knowledge.

##  2 Related work

#### Pretrained models.

Inspired by the success of pretrained transformers [51] in NLP like BERT [25,
34], the multi-modal and computer vision fields have shifted to explore the
transformer-based pretraining strategy. Earlier multi-modal pretraining
methods take image features extracted by separately trained vision models like
Faster-RCNN [19] or Resnet [20] as input, then train transformers to fuse the
object visual features and language. Representative works, including LXMERT
[45], UNITER [9], OSCAR [29], VinVL [60], etc., perform well on multi-modal
downstream tasks like visual question answering [3] and image captioning [52].
Recently, thanks to the invention of vision transformer ViT [13] and the
follow-up improvements [49, 4, 35, 18], end-to-end pretraining from raw images
pixels is made possible. End-to-end trained VLP models not only perform well
on multi-modal tasks, but also uni-modal tasks like image classification and
language understanding. For example, dual encoders trained with a contrastive
loss like CLIP [39] and ALIGN [24] achieve superior visual learning
performance. The unified transformer architecture for language and images
greatly boosts the development of unified foundation models that solve
language, vision, and multi-modal tasks at one time. Representative works
include OFA [54], Florence [58], FLAVA [43], Unified-IO [36], CoCa [57], and
SimVLM [55] etc. We refer readers to [15] for more details.

#### Vision and language benefit each other.

Several recent works in NLP suggest that vision can help language
understanding. Vokenization [46], Z-LaVI [56] and VIDLANKD [47] show language
understanding performance can be improved by better grounding, visual
imagination, or knowledge distillation from videos. Recent work [59] analyzes
language and multi-modal models and shows that vision can help language models
learn better commonsense knowledge and mitigate reporting bias. However, there
is little understanding of how language can help vision, which goes beyond the
high performance achieved by CLIP, ALIGN, etc. This paper aims to analyze
these advanced VLP methods and concretely understand their advantages in
learning.

#### Probing.

Probing is a widely used strategy in NLP for interpreting representations [42,
6]. Given a fixed source of representations (_e.g_., word embeddings from
language models) and a linguistic property of interest (_e.g_., part-of-
speech), a probing head with restricted capacity is trained to predict the
property from the representations. Note that only the probing head is trained
while the representations from pretrained models are frozen during the process
[2]. Various works use probing to show that language representations encode a
broad range of properties like part-of-speech [5], syntax [22], semantics
[27], sentence length [1], etc., and to compare different language models in
those properties [48]. Simple probing heads are most commonly used based on
the intuition that we want to assess implicit and easily accessible
information in a representation [2, 33]. Recently, more complex probing heads
(_e.g_., multi-layer perceptron) are also suggested [38, 10], and the effects
of probing complexity have been studied [21].

Probing has also been adopted to understand multimodal representations in
terms of the capacity for instance retrieval [23], inter-modality knowledge
[41], understanding of verbs [32], entity and syntactic grounding [28], and
visual commonsense knowledge [59], etc. With probing, multi-modal VL models
are compared with uni-modal language models to assess the advantage of multi-
modal learning. In computer vision, linear probing performance is used as a
fast on-the-fly metric for model evaluation [13, 18, 8], which is
complementary to fine-tuning considering the computational cost. While there
is little understanding of visual representation using probing, we hereby
design various visual prediction tasks to assess the encoded visual knowledge
in the representations. To our knowledge, we are the first to compare VL
models and vision-only models using probing.

##  3 Method

To analyze the capacity of the learned representations of different models, we
choose a set of tasks to probe the models. For each task, we first extract
features using the pretrained models, then we train a simple standard head to
predict the results. We try to keep the head as simple as possible, because by
using a simple head with limited capacity, the probing performance can be
restricted by the amount of information that are explicitly learned and
encoded in the representation, instead of being learned by the head (as
suggested in [21]).

Mathematically, for every image $I\in\mathbb{R}^{3\times w\times h}$ and its
features $f$, where $f\in\mathbb{R}^{C\times W\times H}$, a prediction head
$H$ is trained to predict the task-specific results. Here $(w,h)$ is the size
of the input image and $(C,W,H)$ is the size of the feature. In the whole
process, only the head is trained while the pretrained model (_i.e_., feature
extractor) is frozen.

In this section, we will first describe the probing tasks, datasets and the
design of the prediction head for each task (Sec. 3.1), then we describe the
models we evaluated (Sec. 3.2), and finally how to make the comparison
settings fair for every model (Sec. 3.3).

###  3.1 Probing tasks and datasets

Task | Dataset | # of classes | Metric | Prediction head
---|---|---|---|---
object name prediction | Visual Genome [26] | 151 | accuracy | linear classifier on ROI features
attribute prediction | VAW [37] | 620 | mAP | linear classifier on ROI features
part semantic segmentation | PartImageNet [17] | 40 | mIOU | head from Segmenter [44]
object detection | MSCOCO [31] | 80 | mAP | head from VitDet [30]
instance segmentation | MSCOCO [31] | 80 | mAP | head from VitDet [30]

Table 1: The details of {dataset, number of classes, metric, prediction head}
for the five probing tasks.

We choose five probing tasks: object name prediction, attribute prediction,
object detection, instance segmentation and semantic segmentation for object
parts. Among the five tasks, object name and attribute prediction focus more
on predicting the semantic labels, while the others are dense prediction tasks
that highly rely on spatial information.

#### Object name prediction.

Information about object names is a key component in various multi-modal
downstream tasks like VQA and image captioning, in which text descriptions
refer to objects by their names. Given an image and a bounding box, object
name prediction requires predicting the name of the object in the box. We use
the Visual Genome dataset [26] for training and evaluation in this task.
Images in Visual Genome mostly come from MSCOCO [31] and contain multiple
objects. For each object, the annotations provide its bounding box, name and
attributes (color, material, etc.). The annotations cover 151 object classes
for 1.3 million objects in 108k images.

A simple linear classifier is used to predict object names. More specifically,
for each object, we first use ROI-Pooling [40] to average pool the features
according to its box, then use a linear layer on top of the pooled features to
predict the name class of the object. Cross entropy loss is used to train the
head. Note that the ground-truth bounding box coordinates are provided to the
head for both training and testing.

#### Object attribute prediction.

Similar to object name prediction, attribute prediction requires predicting
attributes for the object in the given bounding box. As shown in [60], visual
features with better-encoded attribute information can substantially improve
the performance of multi-modal tasks. This motivates us to treat the attribute
as an important axis for evaluating visual representation. The VAW dataset
[37] is used for object attribute prediction. VAW improves the noisy attribute
annotations in Visual Genome. VAW annotates 620 attributes belonging to 8
categories, including color, shape, size, material, texture, action, state,
and others. Every attribute is annotated as positive, negative, or unknown for
each instance. The annotation covers 260k instances from 72k images, which is
a subset of Visual Genome images. Mean average precision (mAP) is used to
evaluate the prediction results following [37].

Since attribute prediction is formulated as a multi-label classification
problem, the prediction head is similar to object name prediction, but has
several differences. First, binary cross entropy loss is used for training
instead of cross entropy. Second, since the attributes naturally come with a
long-tailed distribution, to prevent the rare attributes (_e.g_., playing)
from being overriden by the frequent ones (_e.g_., black), we assign higher
weights to rare attributes and lower weights to frequent ones. Third, for the
attributes labeled as unknown, we treat them as negative labels with a small
(0.001) weight. Those strategies are borrowed from [37].

#### Object detection and instance segmentation.

While object name/attribute prediction tests the ability to predict class
labels when the object bounding box is given, we are also interested in tasks
that focus more on locating the objects. We choose object detection and
instance segmentation on MSCOCO [31] for this purpose. MSCOCO contains 330K
images with 1.5 million object instances in 80 categories. The bounding box
and segmentation mask are annotated for each instance. mAP, _i.e_., mean of
average precision for each category, is adopted as the evaluation metric.

Because detection and segmentation cannot be completed using a simple head
like a linear layer, we adopt the prediction head in VitDet [30] as our
probing head. While the widely used Mask-RCNN is based on convolutional neural
network (CNN) features, [30] propose a variant that is more suitable for non-
hierarchical transformer features. Considering the fact that most of our
evaluated models are transformer-based, we adopt this VitDet head for probing
in our work. Unless specified, all the experiment settings are kept the same
as [30].

#### Part semantic segmentation.

While image classification accuracy on ImageNet dataset [11] is the most
commonly used metric for evaluating visual representations, the recent
PartImageNet dataset [17] provides additional annotations for the ImageNet
images, thus enables finer-grained evaluation. PartImageNet annotates
segmentation masks of 40 object parts (_e.g_., head, body, tail) for 11
categories of objects on 24k images. Using this dataset, we perform semantic
segmentation of object parts as an additional probing task that requires
localization information.

For the segmentation head, we use the mask transformer decoder in Segmenter
[44] due to its simplicity and impressive performance on standard datasets.
[44] adapts transformers for semantic segmentation with the proposed “mask
transformer decoder” on top of the embeddings produced by the transformer
encoder (standard ViT). In our probing, we replace their transformer encoder
with the pretrained models to be evaluated and train the mask transformer
decoder to output the semantic segmentation map. Because our goal is to fairly
compare different models instead of achieving high performance, we reduce the
input image size (from $1024\times 1024$ to $224\times 224$). A linear layer
is used to match the feature’s dimensions and bilinear upsampling is used to
match feature’s spatial sizes. All the other training settings are kept the
same.

###  3.2 Evaluated models

We evaluate five models: three representative VL models including CLIP, OFA
and FLAVA, and two vision-only models including MAE and MOCOv3. Among the five
models, CLIP and MOCOv3 are trained using contrastive loss, while the others
are trained with sequence modeling losses. We choose these models because they
are representative and highly popular, and their pretrained weights and code
are publicly available. In the following, we describe the models, especially
their visual components, and how we extract features from them.

#### CLIP [39].

CLIP is a dual encoder model trained with contrastive loss using 400M image-
text pairs. The image embeddings produced by the image encoder, which can be
either a ResNet or a transformer, and the text embeddings produced by the text
encoder are trained to be closer with each other in the embedding space when
the image and text pair matches. The learned image embeddings are shown to
have superior transferability on various downstream tasks. In our study, image
features are extracted using the pretrained image encoder.

#### OFA [54].

OFA is a unified model that targets both uni-modal and multi-modal tasks. The
vision tasks (image classification and object detection), language tasks, and
multi-modal tasks (VQA, region/image captioning, visual grounding) are all
formulated into a sequence-to-sequence generation problem. In particular,
special visual tokens from discrete-VAE [50, 14] are used for image infilling
and the object bounding box coordinates are also discretized into special
tokens. The OFA model first uses a ResNet (Res101 for OFAbase) to encode
images, then use the transformer encoder and decoder to generate the target
sequence from image and text features. Cross entropy loss is used as
supervision. OFA is pretrained using 20M image-text pairs with additional uni-
modal data. To obtain visual representations of images, we feed the model with
only the image (_i.e_., empty text input), send it through the ResNet, and
take the output of the transformer encoder.

#### FLAVA [43].

FLAVA is a fully transformer-based unified model. Similar to OFA, the model
can solve both uni-modal and multi-modal tasks. However, the differences lie
in (a) tasks, (b) model architecture, and (c) training loss. (a) FLAVA does
not have bounding boxes in the vocabulary, and thus does not support box-
related tasks like object detection, visual grounding or region captioning.
(b) FLAVA is fully based on transformers; it first uses two separate
transformer encoders to encode images and texts, then have several more
transformer layers for multi-modal fusion. (c) FLAVA uses multiple losses
including CLIP-like contrastive loss, masked image/text/multi-modal modeling
losses, and image-text matching loss. FLAVA is pretrained on 70M image and
text pairs. We take the output of the visual transformer encoder as image
representations.

#### MAE [18].

Masked Auto-Encoder (MAE) is a self-supervised vision model trained with a
masked image modeling task. MAE encodes masked image patches with a
transformer encoder and reconstructs the missing pixels with a lightweight
decoder trained with MSE loss. Unlike OFA and FLAVA, the reconstruction for
MAE happens in the continuous pixel space, which does not require dVAE to
generate discretized image tokens. MAE is trained only with ImageNet-1k data
and shows promising transfer performance to downstream tasks.

#### MOCOv3 [8].

We choose MOCOv3 to represent self-supervised vision transformers trained with
contrastive loss. During training, two crops for each image under random data
augmentation are encoded by two encoders, a key encoder and a query encoder,
into two vectors named “key” and “query” respectively. During training, the
goal is to retrieve the corresponding “key” by the “query”. Similar to MAE,
MOCOv3 is trained using ImageNet-1k.

###  3.3 Comparison settings

To make the comparison fair, we carefully choose the model size and input
size, and ensure different methods are comparable. As probing tasks are highly
sensitive to image size and feature’s spatial size, for all the models on all
the tasks, we fix the input image resolution to be 224*224. We choose this
size because 224*224 is the input size for pretraining for all the models
except OFA (OFA is pretrained with size 384 for base version and 480 for
large). For dense tasks, although the original detection and segmentation
models (_i.e_., VitDet and Segmenter) use larger input image sizes for better
performance, we unify the input size because our goal is to fairly compare
models, rather than achieving the best performance.

The probing results are also sensitive to the models’ input patch size,
because different patch sizes produces features with different spatial sizes.
For example, for input images of 224*224, ViT-B/16 produces visual
representations with size 768*14*14, while ViT-B/14 gives feature size
768*16*16, which will affect probing. Therefore, considering the availability
of pretrained checkpoints with different model sizes and input patch sizes, we
evaluate with the ViT-B/16 backbone by default. Because OFA is not purely
transformer-based, we evaluate on the base size, which has a ResNet +
transformer encoder with 120M parameters (comparable to ViT-B/16 with 86M
parameters).

##  4 Experiments

In this section, we first compare the overall results of the five evaluated
methods on each probing task. Next, we conduct a more detailed analysis of the
segmentation and attribute prediction results. Lastly, we study the effects of
model size and finetuning on downstream tasks.

###  4.1 Implementation details

For object name classification and attribute prediction, the model is trained
with a learning rate of 0.001 and batch size of 64 for 200 epochs. We adopt
early stopping based on validation performance, then report performance on the
test split using the best model. For object detection and segmentation on the
COCO dataset, the model is trained for 120k iterations with batch size 20. The
learning rate is first set to 8e-5, then decay twice at step 100k and 115k
with a factor of 0.1. For part segmentation, we train the model with a
learning rate of 0.01 and batch size of 128 for 200 epochs. The validation
performance for the final checkpoint is reported.

###  4.2 Probing results

We probe the five models on each of the five probing tasks. We first save the
features extracted by each pretrained model for both train/val and test
splits, then train the head to predict the probing results based on the saved
features. We make sure that the experiment settings, including model size,
input size, training protocol and data splits, are well aligned for every
model in order to make fair comparisons. The probing results are shown in Tab.
2. We also include the ImageNet finetuning accuracy and linear probing
accuracy of each model for reference, because they are widely-used metrics for
model evaluation.

On each task, we compare the VL models and vision-only models. Note that the
evaluation metric for each task is different (as in Tab. 1), performance on
different tasks cannot be compared and we only compare numbers in each column
separately.

| Task | VG Obj. | VAW Attr. | COCO Det. | COCO Seg. | Part Seg. | IN1k ft. | IN1k probe
---|---|---|---|---|---|---|---|---
V+L | OFA | 57.13 | 61.67 | 25.04 | 19.38 | 33.11 | 82.2 | -
FLAVA | 54.29 | 61.51 | 21.06 | 17.20 | 34.77 | - | 75.5
CLIP | 51.54 | 61.15 | 19.55 | 15.56 | 40.61 | - | 80.2
V | MAE | 49.52 | 52.59 | 25.29 | 22.05 | 42.30 | 83.6 | 68.0
MOCOv3 | 47.81 | 54.44 | 20.31 | 16.96 | 40.11 | 83.2 | 76.7

Table 2: Probing results on the five probing tasks. VL models perform better
on label prediction tasks, while vision-only models perform better on dense
prediction tasks. Finetuning and linear probing results on ImageNet for each
model (cited from original papers) are also shown for reference.

| MSCOCO | PartImageNet
---|---|---
| mAP | Semantic | Localization | mIOU | Semantic | Localization
OFA | 19.38 | 60.02 | 17.41 | 33.11 | 71.71 | 84.15
FLAVA | 17.20 | 61.48 | 14.67 | 34.77 | 75.28 | 83.76
CLIP | 15.56 | 68.24 | 13.25 | 40.61 | 80.21 | 86.80
MAE | 22.05 | 46.85 | 20.69 | 42.30 | 75.03 | 89.50
MOCOv3 | 16.96 | 49.80 | 15.08 | 40.11 | 76.18 | 86.08

Table 3: Detailed analysis of instance segmentation and part segmentation
results. We evaluate the segmentation results (standard metric mAP, mIOU) from
two additional perspectives: semantics (F1 score for semantic class
prediction) and localization (mAP/mIOU for foreground/background
segmentation). While vision-only models are better on the standard metrics, VL
models are better when evaluated with semantics metrics.

For object name prediction and attribute prediction, VL models consistently
perform better than vision-only models. For object name prediction on Visual
Genome, VL models all achieve more than 51% accuracy while vision-only models
get accuracy less than 50%; for attribute prediction on VAW, mAP for VL models
are higher than 61% while lower than 55% for vision-only models. This suggests
that representations from VL models capture richer semantic information about
the objects in each image, which can be decoded using a simple linear layer.
In contrast, in vision-only models the name and attribute information are not
explicit enough.

For the dense prediction tasks, MAE performs the best on all three tasks. For
part semantic segmentation on PartImageNet, MOCOv3 and CLIP also get decent
performance ($>40\%$ mIOU) that is close to MAE (42%), while the other two VL
models are lower by a large margin ($<35\%$). For object detection on MSCOCO,
OFA gets close mAP (25.0) to MAE (25.3) while the performance of the other
three models are much lower; however, when it comes to instance segmentation,
the advantage of MAE is more clear, surpassing all the other models with a
margin larger than 2.7%.

Interestingly, comparing the object detection and instance segmentation
results on COCO, we find that the performance drops of vision-only models are
consistently smaller than VL models, which indicates that vision-only models
learn better localized representations. For example, for OFA, the mIOU for
segmentation is 5.7% (25.04-19.38) lower than that for detection; while the
drop MAE and MOCOv3 are smaller (3.2%, 3.3%). Because segmentation requires
more localized features than detection to find the boundary of objects, the
performance gap between detection and segmentation can be an indicator of the
localized information in the representations, considering those two tasks are
based on the same dataset. With the more-localized representations, the model
can better predict the mask boundary. Therefore, the smaller gap of vision-
only models suggests they learn more localized representations.

Figure 2: Comparison of segmentation results using different models. Compared
to vision-and-language models, vision-only models more accurately predict the
boundary of segmentation masks, but make mistakes in labeling the regions.

To further verify this finding, we next take a closer look into segmentation
results, which more clearly compare the semantics and localization information
in different models.

#### A closer look at the segmentation results.

We evaluate the instance segmentation results on COCO and semantic
segmentation results on PartImageNet using two more metrics: (a) the label
prediction metric, and (b) the foreground-background segmentation metric,
where (a) is an indicator for semantics and (b) for localization. The
motivation is that the segmentation metrics (mAP for instance segmentation,
mIOU for semantic segmentation) require correctly predicting both the class
label and the boundary, so the quality of both determines the score.
Therefore, we propose two additional metrics to measure the two factors
separately. For (a), for each image, we transform its predicted segmentation
map into label predictions, and evaluate the quality using the multi-label
prediction metric. In particular, we treat the appeared classes in the
predicted segmentation map as positive labels and the others as negative; then
the label predictions are evaluated using the F1 score. F1 score is defined as
$\frac{2*\text{precision}*\text{recall}}{(\text{precision}+\text{recall})}$,
where precision and recall are averaged over label classes. For (b), we merge
all the different object categories and process the segmentation map into
binary labels, _i.e_., foreground and background, then report the mIOU (for
instance segmentation) or mAP (for semantic segmentation) of the binary
segmentation maps.

Tab. 3 shows the segmentation results on COCO and PartImageNet evaluated using
the above two metrics. Although MAE achieves the best performance on both
datasets, when looking at the semantic and localization results, we find that
its advantage mainly comes from better localization, rather than semantics. In
terms of semantics, VL models perform much better than MAE. For example, on
the MSCOCO dataset, VL models achieve F1 scores higher than 60, while MAE and
MOCOv3 are lower than 50. The results suggest that while MAE is better at
finding the object boundaries when predicting segmentation masks, VL models
are better at predicting labels for the objects.

In Fig. 2, we show several examples of the part segmentation results on
PartImageNet. In the examples, MAE captures the object’s shape more
accurately, like the curly snake body, the shark’s small fin, and the
quadruped contour. However, MAE and MOCOv3 make more mistakes in labeling the
regions compared to VL models. For example, MAE wrongly predicts the shark fin
as a reptile foot, and the quadruped as a reptile; MOCOv3 confuses the
quadruped head and foot as the fish head and fins. Those examples more
explicitly compare the semantics and localization knowledge learned by VL and
vision-only models.

#### Analysis on different attribute groups.

We further decompose the attribute prediction results into different attribute
groups. In the VAW dataset, attributes are categorized into 8 groups: action,
texture, shape, size, color, material, state, and others. The results are
shown in Fig. 3. Interestingly, despite the overall better results of VL
models, we find that their advantages differ in different groups. For example,
the gap between VL and vision-only models in the “action” category is more
significant than in the “texture” category. Intuitively, “action” is less
visually grounded then “texture” requires more context and semantic
information, on which VL models is better at, suggesting that while vision-
only ones are better at predicting highly visually grounded local attributes
(_e.g_., texture), VL models are better at more abstract ones.

Figure 3: A closer look at the attribute prediction results by separately
evaluating different types of attributes. The advantage of VL models is more
significant in the more abstract categories (_e.g_., action) than visually
grounded categories (_e.g_., texture).

#### Semantics vs. localization.

To summarize, by probing the models on the five tasks, a detailed look at the
segmentation results, and a finer evaluation of different attribute
categories, we find that the VL model is better at predicting semantic labels,
while vision-only models are better at localized dense prediction.

###  4.3 More analysis

#### Findings of contrastive training.

The results also show that contrastive models perform relatively better on
localization for single-object images than multi-object images. Among the five
tasks, part segmentation on PartImageNet dataset are based on single-object
images from ImageNet, while the other four tasks are based on COCO-style
multi-object images. In Tab. 3, comparing the contrastively trained models
(CLIP, MOCOv3) and the models trained with sequence modeling objectives (OFA,
FLAVE, MAE), we find that contrastive models perform relatively better on
PartImageNet than MSCOCO. For example, on PartImageNet, CLIP outperforms the
other two VL models (_i.e_., OFA and FLAVA) by a large margin (more than 6%
mIOU); on MSCOCO, it under-performs them. The semantic and localization
evaluation suggests that this difference is mainly caused by localization,
_e.g_., the localization results of CLIP is much better than OFA and FLAVA on
PartImageNet. A similar observation can be obtained by comparing MOCOv3 and
MAE: although MOCOv3 underperforms MAE on both datasets, the gap is much
smaller on PartImageNet than MSCOCO (2.2 vs. 5.1). Therefore, we suggest that
the localization ability of contrastive models is relatively stronger on
single-object images.

#### The Effect of model size.

To study the effect of model size, in Tab. 4, we show the probing results with
size base and large for MAE and OFA.

For MAE, a larger model size improves performance on all the probing tasks in
parallel for 1% to 2%. However, note that this improvement is less significant
compared to the big gaps between different model types. For OFA, except for
the marginal improvement in attribute prediction, the larger model size hurts
probing results on the other four tasks. The reason for the decrease is that
the OFAlarge is pretrained with a larger input image size (480*480) compared
with OFAbase model (384*384). Because we probe all models with the same image
size (224*224) for a fair comparison, the gap in image size between
pretraining and probing is more significant for OFAlarge. In summary, the
effect of model size is less considerable than other factors like model type
or input image size.

| obj. | attr. | det. | seg. | p-seg.
---|---|---|---|---|---
MAEbase | 49.52 | 52.59 | 25.29 | 22.05 | 42.30
MAElarge | 51.91 | 53.38 | 29.67 | 25.63 | 44.85
OFAbase | 57.13 | 61.67 | 25.04 | 19.38 | 33.11
OFAlarge | 52.33 | 62.01 | 21.23 | 16.51 | 32.04

Table 4: Probing results of model sizes. The influence of model size is less
considerable than other factors like model type.

#### The effect of downstream finetuning.

Tab. 5 compares probing results of models with and without finetuning on
downstream tasks. For MAE, the results are based on the base size; for OFA,
the results are on large size, due to the availability of publicly released
model checkpoints.

For both models, finetuning on image classification on ImageNet-1k and VQA on
VQAv2 hurts the probing performance to varying degrees (except for attribute
prediction). This indicates that while in pretraining, the model learns
features that capture various fine-grained information about the image, during
finetuning towards a specific task, only information useful for the task is
kept and other information is dropped. Moreover, compared with ImageNet
finetuning, finetuning on VQA leads to a much smaller performance decrease in
probing results, suggesting that the change in probing results depends on the
nature of downstream tasks. In this case, VQA requires more fine-grained
information about objects, attributes, etc., resulting in a smaller drop than
ImageNet finetuning.

| obj. | attr. | det. | seg. | p-seg.
---|---|---|---|---|---
MAE | 49.52 | 52.59 | 25.29 | 22.05 | 42.30
MAEIN1k | 45.16 | 53.82 | 21.41 | 17.74 | 35.62
OFA | 52.33 | 62.01 | 21.23 | 16.51 | 32.04
OFAIN1k | 50.54 | 60.74 | 18.91 | 14.67 | 27.56
OFAVQA | 51.42 | 63.40 | 19.01 | 14.22 | 28.34

Table 5: Probing results of models finetuned on downstream tasks including
ImageNet-1k and VQA. The MAE results are based on base size and OFA results
are on large size. Finetuning hurts the probing performance in most cases.

#### Limitations.

This study is limited by the coverage of pretrained models. We only evaluate
models which have publicly accessible checkpoints, and which can be aligned in
terms of model sizes, patch sizes, etc. Because we do not have enough
computational resources to retrain the models, our comparisons are restricted
by the released ones.

##  5 Conclusion

This work studies how language can help visual learning with feature probing
under well-aligned settings. By comparing three representative VL models and
two vision-only models on five probing tasks, we find that VL models are
stronger in label prediction tasks, while vision-only models are better in
dense prediction tasks. With detailed analysis, we conclude that language
helps vision models learn better semantics, but not localization. We hope our
diagnostic findings inspire future works in improving visual learning with the
help of language.

## Acknowledgements

This work is partially supported by a gift from the JHU+Amazon Initiative for
Interactive AI. This work is also supported with Cloud TPUs from Google’s TPU
Research Cloud (TRC) program. We would like to thank Elias Stengel-Eskin, Kate
Sanders, David Etter, Reno Kriz, and Chen Wei for their helpful suggestions.

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# Momentum dependence of $\phi$ meson’s spin alignment

Xin-Li Sheng  INFN-Firenze, Via Giovanni Sansone, 1, 50019 Sesto Fiorentino
FI, Italy  Shi Pu  Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Qun Wang  Department of
Modern Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  School of Mechanics and Physics, Anhui University of Science
and Technology, Huainan,Anhui 232001, China

###### Abstract

We study the rapidity and azimuthal angle dependences of the global spin
alignment $\rho_{00}$ for $\phi$ mesons with respect to the reaction plane in
Au+Au collisions at RHIC by the relativistic coalescence model in the spin
transport theory. The global spin alignment of $\phi$ mesons arises from local
fluctuations of strong force fields whose values are extracted from the STAR’s
data. The calculated results show that $\rho_{00}<1/3$ at the rapidity $Y=0$,
and then it increases with rapidity and becomes $\rho_{00}>1/3$ at $Y=1$. Such
a rapidity dependence is dominated by the relative motion of the $\phi$ meson
in the bulk matter. We also give prediction for the azimuthal angle dependence
of $\rho_{00}$ at different rapidities.

##  I Introduction

In noncentral heavy-ion collisions, the colliding nuclei carry a global
orbital angluar momentum (OAM). A small portion of the global OAM is
transferred into the quark-gluon plasma (QGP) in the form of vorticity fields.
Quarks are then polarized by vorticity fields through spin-orbit couplings and
form polarized hadrons by coalescence or recombination (Liang and Wang, 2005a;
Voloshin, 2004; Betz et al., 2007; Becattini et al., 2008; Gao et al., 2008),
see Refs. (Wang, 2017; Florkowski et al., 2019; Gao et al., 2021; Huang et
al., 2021; Becattini and Lisa, 2020; Becattini, 2022) for recent reviews. The
effect of the vorticity field is supported by the global spin polarization of
$\Lambda$ and $\overline{\Lambda}$ hyperons observed by the STAR collaboration
in Au+Au collisions (Adamczyk et al., 2017; Adam et al., 2018). According to
the spin-flavor wave functions of $\Lambda$ and $\overline{\Lambda}$ in the
quark model (Liang and Wang, 2005a; Yang et al., 2018), the spins of $\Lambda$
and $\overline{\Lambda}$ are carried by $s$ and $\overline{s}$ quarks
respectively, indicating that strange quarks in the QGP are also globally
polarized.

It was proposed by Liang and Wang in 2005 that the polarized quarks can also
form vector mesons by recombination with nonvanishing spin alignment (Liang
and Wang, 2005b). The spin of a vector meson is described by a $3\times 3$
spin density matrix $\rho$ with elements $\rho_{\lambda_{1}\lambda_{2}}$,
where subscripts $\lambda_{1},\lambda_{2}=0,\pm 1$ label the spin quantum
number along a specific spin quantization direction. The spin alignment refers
to the element $\rho_{00}$, which denotes the probablity for the spin state
with $\lambda=0$. The polarization vector of the vector meson is preferrably
aligned in the spin quantization direction when $\rho_{00}>1/3$, while it is
preferrably aligned in the perpendicular direction when $\rho_{00}<1/3$. In
experiments, the spin alignment can be measured through polar angle
distributions of daughter particles in p-wave strong decays, such as
$\phi\rightarrow K^{+}+K^{-}$ (Schilling et al., 1970; Liang and Wang, 2005b;
Yang et al., 2018; Acharya et al., 2020; Abdallah et al., 2023), or dilepton
decays, such as $J/\psi\rightarrow\mu^{+}+\mu^{-}$ (Faccioli et al., 2010; De
Moura et al., 2023; Acharya et al., 2021, 2023).

Rencently, the STAR collaboration has measured the global spin alignment of
$\phi$ mesons with respect to the reaction plane in Au+Au collisions (Abdallah
et al., 2023), and the results show a significant positive deviation from 1/3.
Such a deviation is much larger than contributions from conventional
mechanisms such as vorticity fields and magnetic fields (Liang and Wang,
2005b; Yang et al., 2018; Xia et al., 2021; Sheng et al., 2022a; Wei and
Huang, 2023). Other possible contributions are proposed in (Gao, 2021; Müller
and Yang, 2022; Li and Liu, 2022; Wagner et al., 2023; Kumar et al., 2023) but
without quantitative results that can be compared with experimental data. Upto
now, the effect of local correlation or fluctuation of a kind of the strong
force field called the $\phi$ field (Sheng et al., 2020, 2022b, 2023a) is the
only mechanism that can quantitatively explain the SATR data. According to the
chiral quark model (Manohar and Georgi, 1984; Fernandez et al., 1993; Li et
al., 1997; Zhao et al., 1998; Zacchi et al., 2015, 2017), the SU(3) octet
vector fields in the form of a $3\times 3$ matrix can be induced by vector
currents of pseudo-Goldstone bosons that surround $s$ and $\overline{s}$
quarks in the hadronization stage. The $\phi$ field is just the “33” component
of the SU(3) octet vector fields that is coupled to $s$ and $\overline{s}$.

In this paper, we study the momentum dependence of $\rho_{00}^{y}$ in the
relativistic coalescence model based on the spin Boltzmann equation in non-
equilibrium transport theory (Sheng et al., 2023a, 2022b). Here the
superscript $i=x,y,z$ in $\rho_{00}^{i}$ denotes the spin quantization
direction. In this paper, we choose the beam direction along the $z$-axis, the
normal direction of the reaction plane along the $y$-axis, and the impact
parameter along the $x$-axis. The parameters for the $\phi$ field’s local
fluctuations are extracted by fitting the STAR data for $\rho_{00}^{y}$ and
$\rho_{00}^{x}$ as functions of collision energy (Abdallah et al., 2023), and
the values of these parameters are the same as in Ref. (Sheng et al., 2023a).
Some recent reviews on the STAR experiment and theoretical models can be found
in Refs. (Chen et al., 2023; Wang, 2023; Sheng et al., 2023b).

The paper is organized as follows. In Sec. II we give the formula for
$\rho_{00}$ as functions of the $\phi$ field’s fluctuations. Assuming that the
fluctuations are isotropic in the lab frame, we derive a compact formula for
the momentum dependence of $\rho_{00}$. In Sec. III we compute the rapidity
and azimuthal angle dependence of the global spin alignment $\rho_{00}^{y}$
for $\phi$ mesons in Au+Au collisions. Finally we make a summary and a
discussion on the our result in Sec. IV. As notational convention, we use
boldface symbols for three-dimensional vectors, such as ${\bf
p}=(p_{x},p_{y},p_{z})$.

##  II Theoretical model

In non-relativistic quark coalescence model, the spin alignment of the $\phi$
meson depends on the spin polarizations of its constituent quark and antiquark
(Liang and Wang, 2005b; Yang et al., 2018; Sheng et al., 2020). In Refs.
(Sheng et al., 2023a, 2022b), some of us constructed a relativistic quark
coalescence model based on spin Boltzmann equation in transport theory to
describe the spin phenomena of vector mesons. Considering the quark
polarization induced by $F_{\phi}^{\mu\nu}$, the field strength tensor of the
$\phi$ field, the spin alignment for the $\phi$ meson is given by (Sheng et
al., 2023a, 2022b)

$\displaystyle\rho_{00}(x,{\bf p})$ $\displaystyle=$
$\displaystyle\frac{1}{3}-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\text{h}}^{2}}C_{1}\left[\frac{1}{3}{\bf
B}_{\phi}^{\prime}\cdot{\bf
B}_{\phi}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot{\bf
B}_{\phi}^{\prime})^{2}\right]$ (1)
$\displaystyle-\frac{4g_{\phi}^{2}}{m_{\phi}^{2}T_{\text{h}}^{2}}C_{1}\left[\frac{1}{3}{\bf
E}_{\phi}^{\prime}\cdot{\bf
E}_{\phi}^{\prime}-(\boldsymbol{\epsilon}_{0}\cdot{\bf
E}_{\phi}^{\prime})^{2}\right]\,,$

where $C_{1}$ and $C_{2}$ are two coefficients depending on $m_{\phi}$ (the
$\phi$ meson’s mass) and $m_{s}$ (mass of constituent strange quark),
$T_{\text{h}}$ is the local temperature at the hadronization time, and
$\boldsymbol{\epsilon}_{0}$ denotes the unit vector along the measuring (spin
quantization) direction which is also the $\phi$ meson’s polarization vector
for the spin state $\lambda=0$. Here ${\bf E}_{\phi}^{\prime}$ and ${\bf
B}_{\phi}^{\prime}$ are electric and magnetic parts of the $\phi$ field in the
meson’s rest frame, which are functions of spacetime. When boosted to the lab
frame, ${\bf E}_{\phi}^{\prime}$ and ${\bf B}_{\phi}^{\prime}$ also depend on
the meson’s momentum ${\bf p}$, i.e., they can be expressed in terms of fields
in the lab frame ${\bf B}_{\phi}$ and ${\bf E}_{\phi}$ as

$\displaystyle{\bf B}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\bf B}_{\phi}-\gamma{\bf v}\times{\bf
E}_{\phi}+(1-\gamma)\frac{{\bf v}\cdot{\bf B}_{\phi}}{v^{2}}{\bf v}\,,$
$\displaystyle{\bf E}_{\phi}^{\prime}$ $\displaystyle=$
$\displaystyle\gamma{\bf E}_{\phi}+\gamma{\bf v}\times{\bf
B}_{\phi}+(1-\gamma)\frac{{\bf v}\cdot{\bf E}_{\phi}}{v^{2}}{\bf v}\,,$ (2)

where $\gamma=E_{{\bf p}}^{\phi}/m_{\phi}$ is the Lorentz contraction factor
and ${\bf v}={\bf p}/m_{\phi}$ is the $\phi$ meson’s velocity. Substituting
Eq. (2) into Eq. (1), we are able to express $\rho_{00}$ in terms of ${\bf
B}_{\phi}$ and ${\bf E}_{\phi}$ which depend only on spacetime but not on
momentum.

We learn from Eq. (1) that the deviation from $1/3$ for $\rho_{00}$ is caused
by the anisotropy of local field fluctuations in the meson’s rest frame. The
deviation is positive (negative) when the fluctuation of ${\bf
B}_{\phi}^{\prime}$ or ${\bf E}_{\phi}^{\prime}$ in the measuring direction
$\boldsymbol{\epsilon}_{0}$ is larger (smaller) than the average fluctuation
in directions perpendicular to $\boldsymbol{\epsilon}_{0}$. We assume that the
fluctuations in the lab frame are parameterized in an anisotropic form,

$\left\langle g_{\phi}^{2}{\bf B}_{\phi}^{i}{\bf
B}_{\phi}^{j}/T_{\text{h}}^{2}\right\rangle=\left\langle g_{\phi}^{2}{\bf
E}_{\phi}^{i}{\bf
E}_{\phi}^{j}/T_{\text{h}}^{2}\right\rangle=F^{2}\delta^{ij}+\Delta\hat{{\bf
a}}^{i}\hat{{\bf a}}^{j}\,,$ (3)

while the correlation between ${\bf B}_{\phi}$ and ${\bf E}_{\phi}$ is
neglected, $\left\langle g_{\phi}^{2}{\bf B}_{\phi}^{i}{\bf
E}_{\phi}^{j}/T_{\text{h}}^{2}\right\rangle=0$. Here
$\left\langle\cdots\right\rangle$ denotes the spacetime average, $\hat{{\bf
a}}$ denotes the direction of anisotropy, $F^{2}$ is the isotropic part of the
fluctuation, and $\Delta$ denotes the difference between the fluctuation in
$\hat{{\bf a}}$ and the average fluctuation in directions perpendicular to
$\hat{{\bf a}}$. The field configuration in Eq. (3) becomes isotropic when
$\Delta=0$. The field fluctuations in the meson’s rest frame can be obtained
using Eqs. (2) and (3).

By substituting Eq. (2) into Eq. (1), averaging over spacetime, and then
applying Eq. (3) with $\Delta=0$, we obtain the spin alignment in the
$y$-direction, the direction of the global OAM with
$\boldsymbol{\epsilon}_{0}=(0,1,0)$,

$\left\langle\delta\rho_{00}^{y}\right\rangle({\bf
p})=\frac{8}{3m_{\phi}^{4}}(C_{1}+C_{2})F^{2}\left(\frac{p_{x}^{2}+p_{z}^{2}}{2}-p_{y}^{2}\right)\,,$
(4)

where $\delta\rho_{00}^{y}\equiv\rho_{00}^{y}-1/3$ and $(C_{1}+C_{2})F^{2}$ is
a positive number. Obviously, for static $\phi$ mesons with ${\bf p}=0$ we
should have $\rho_{00}^{y}=1/3$ because field fluctuations are isotropic in
the lab frame. However, the motion of the $\phi$ meson will break the
symmetry. According to the Lorentz transformation of fields in Eq. (2), field
components perpendicular to the motion direction are enhanced by the $\gamma$
factor, while the component in the motion direction is not. Therefore when
observing in the meson’s rest frame, the fluctuation in the direction of
motion will be smaller than fluctuations in perpendicular directions. For a
meson with $p_{x}=p_{z}=0$ and $p_{y}\neq 0$, we obtain the relation
$\left\langle({\bf B}_{\phi,y}^{\prime})^{2}\right\rangle<\left\langle({\bf
B}_{\phi,x}^{\prime})^{2}\right\rangle=\left\langle({\bf
B}_{\phi,z}^{\prime})^{2}\right\rangle$, leading to $\rho_{00}^{y}<1/3$
according to Eq. (1) or (4). Similarly, motions along $x$\- and $z$-directions
lead to $\rho_{00}^{y}>1/3$.

We can also rewrite the result in Eq. (4) in terms of the transverse momentum
$p_{T}$, the azimuthal angle $\varphi$, and the rapidity $Y$,

$\left\langle\delta\rho_{00}^{y}\right\rangle({\bf
p})\propto\frac{1}{2}p_{T}^{2}\left[3\cos(2\varphi)-1\right]+\sqrt{m_{\phi}^{2}+p_{T}^{2}}\sinh^{2}Y\,.$
(5)

The azimuthal angle dependence shows a $\cos(2\varphi)$ structure, which was
first derived in Ref. (Sheng et al., 2023a). We also find that the spin
alignment increases with rapidity. This is because the anisotropy of field
fluctuations in the meson’s rest frame becomes more significant in the
direction of the meson’s larger momentum.

We note that Eqs. (4) and (5) are based on the assumption (3) with vanishing
anisotropy $\Delta=0$ in the lab frame. A nonzero $\Delta$ will contribute to
$\rho_{00}^{y}$, but the analytical formula is too complicated to be given
here. Our numerical results for nonzero $\Delta$ in the next section will show
that its contribution is small compared with the results for $\Delta=0$. So
the spin alignment is dominated by the isotropic part of the field
fluctuations.

##  III Numerical results

Similar to the previous work (Sheng et al., 2023a) by some of us, we consider
the anisotropy with respect to the $z$-direction or the beam direction in
heavy-ion collisions. So we assume that transverse and longitudinal
fluctuations are different $F_{T}^{2}=F^{2}$ and $F_{z}^{2}=F^{2}+\Delta$,
which are regarded as parameters that can be extracted from the STAR’s data on
momentum integrated $\rho_{00}^{y}$ (Abdallah et al., 2023). At collision
energies $\sqrt{s_{\text{NN}}}=$ 11.5, 19.6, 27, 39, 62.4, and 200 GeV, the
extracted values of $F^{2}$ and $\Delta$ are $F^{2}/m_{\pi}^{2}=$ 16.5, 3.74,
3.42, 1.02, 2.85, 0.359 and $\Delta/m_{\pi}^{2}=$ 3.33, $-0.468$, $-0.9$,
0.218, 0.336, 0.128, respectively.

Figure 1: The global spin alignment $\rho_{00}^{y}$ (solid lines) as functions
of rapidity at collision energies $\sqrt{s_{\text{NN}}}=$11.5, 19.6, 27, 39,
62.4, and 200 GeV. The shaded areas are error bands from fitting parameters
for local fluctuations in strong force fields. The dashed lines indicate the
value of 1/3 without spin alignment. Figure 2: Contributions from the
anisotropy parameter $\Delta$ in the lab frame to the $\rho_{00}^{y}$ versus
rapidity at different collision energies.

In Fig. 1, we present $\rho_{00}^{y}$ as functions of the $\phi$ meson’s
rapidity $Y$ at collision energies $\sqrt{s_{\text{NN}}}=$11.5 - 200 GeV. At
each rapidity and energy, we assume that the spacetime averages of
fluctuations follow Eq. (3). We take averages over the meson’s transverse
momentum in the range 1.2 GeV$<p_{T}<$5.4 GeV and the azimuthal angle in the
range $0<\varphi<2\pi$, weighted by the $\phi$ meson’s momentum spectra

$E_{{\bf p}}\frac{d^{3}N}{d^{3}{\bf p}}=\frac{d^{2}N}{2\pi
p_{T}dp_{T}dY}\left[1+2v_{2}(p_{T})\cos(2\varphi)\right],$ (6)

where the transverse momentum spectra and $v_{2}(p_{T})$ are taken from STAR’s
data (Abelev et al., 2007, 2009; Adamczyk et al., 2013; Adam et al., 2020). At
all energies, the derivation from 1/3 is negative at the mid-rapidity $Y=0$,
while it increases at larger $Y$. To see the influence of the anisotropy in
the field fluctuation, we isolate the contribution from $\Delta$ to
$\rho_{00}^{y}$ by taking the difference between results in Fig. 1 with non-
vanishing $\Delta$ and those obtained with the same $F^{2}$ but $\Delta=0$. As
shown in Fig. 2, the effects of non-vanishing $\Delta$ are one order of
magnitude smaller than $\rho_{00}^{y}-1/3$ in Fig. 1. This indicates that the
rapidity dependence in Fig. 1 is dominated by the isotropic part $F^{2}$ in
Eq. (3).

The theoretical model in Sec. II provides a clear picture for the rapidity
dependence of the spin alignment. The bulk matter in which $\phi$ mesons are
produced can be treated as a nearly-isotropic medium with a small anisotropy
along the $z$-direction in fluctuations of strong force fields described by
Eq. (3), i.e. $F^{2}\gg\Delta$. The $\phi$ meson’s motion relative to the bulk
matter breaks the rotational symmetry in meson’s rest frame, leading to a
larger probability for spin $\pm 1$ states than the spin 0 state with respect
to the motion direction. For example, if mesons move in the $z$-direction, we
have $\rho_{00}^{z}<1/3$, or equivalently $\rho_{00}^{x}=\rho_{00}^{y}>1/3$
because of the normalization condition
$\rho_{00}^{x}+\rho_{00}^{y}+\rho_{00}^{z}=1$.

Figure 3: The contour plots for $\delta\rho_{00}^{y}$ in the transverse
momentum plane $(p_{x},p_{y})$ for $\phi$ mesons at $Y=0$ (left panel) and
$Y=1$ (right panel) in Au+Au collisions at 200 GeV.  Figure 4: Azimuthal angle
$\varphi$ dependence of $\delta\rho_{00}^{y}$ at rapidity $Y=0$ (blue lines)
and $Y=1$ (orange lines) at 200 GeV. Solid lines are calculated using
fluctuations parameters $F^{2}$ and $\Delta$ given in the first paragraph of
Sec. III and dashed lines are calculated using the same $F^{2}$ but
$\Delta=0$.

In order to study the transverse momentum dependence of $\rho_{00}^{y}$ in
different rapidity regions, we present in Fig. 3 the contour plot for the
deviation $\delta\rho_{00}^{y}$ in $(p_{x},p_{y})$ plane at $Y=0$ (left panel)
and $Y=1$ (right panel) at 200 GeV. We observe a significant quadrupole
structure: the mesons with $|p_{x}|\gg|p_{y}|$ have $\delta\rho_{00}^{y}>0$,
while those with $|p_{x}|\ll|p_{y}|$ have $\delta\rho_{00}^{y}<0$. Such a
structure is the result of Eq. (4) or (5).

We also calculated the azimuthal angle dependence of $\delta\rho_{00}^{y}$ at
a fixed transverse momentum $p_{T}=2$ GeV, see Fig. 4. For mesons at $Y=0$,
$\delta\rho_{00}^{y}$ is positive at $\varphi=0$ and decreases to a negative
minimum value at $\varphi=\pi/2$. The curve shows a $\cos(2\varphi)$ behavior,
as expected from Eq. (5). At a more forward rapidity $Y=1$,
$\delta\rho_{00}^{y}$ is shifted by a positive value relative to the $Y=0$
curve, which is also described by Eq. (5). The effects of the anisotropy in
strong force field fluctuations in the lab frame, which are quantified by
$\Delta$ in Eq. (3), are also shown in Fig. 4. We see that the effects of non-
vanishing $\Delta$ are small, implying that the azimuthal angle dependence is
dominated by the isotropic part $F^{2}$ in Eq. (3).

##  IV Summary

The momentum dependence of the $\phi$ meson’s global spin alignment in heavy-
ion collisions is studied. According to the relativistic quark coalescence
model in the spin transport theory for vector mesons (Sheng et al., 2022b,
2023a), the derivation from $1/3$ for $\rho_{00}$ for the $\phi$ meson is
driven by the anisotropy of local fluctuations of strong force fields in the
vector meson’s rest frame, which can be related to fluctuations in the lab
frame through Lorentz transformation. From the geometry of the quark-gluon
plasma produced in heavy-ion collisions, it is natural to assume that
fluctuations are nearly isotropic in the lab frame, with tiny anisotropy in
the $z$-direction or beam direction. By neglecting the anisotropy part, we
derive an analytical expression for $\rho_{00}^{y}-1/3$ which is proportional
to $(p_{x}^{2}+p_{z}^{2})/2-p_{y}^{2}$, indicating that the meson’s motion
along $x$\- and $z$-directions will enhance $\rho_{00}^{y}$, while the motion
along $y$-direction will decrease $\rho_{00}^{y}$. We then predict the
rapidity dependence of $\rho_{00}^{y}$ using fluctuation parameters that are
extracted from the STAR experiment data on momentum-integrated $\rho_{00}^{y}$
(Abdallah et al., 2023). Our results show that $\rho_{00}^{y}$ has a negative
derivation from $1/3$ at mid-rapidity $Y=0$ and a positive derivation at
slightly forward rapidity $Y=1$. Predictions for the azimuthal angle
dependence of $\rho_{00}^{y}$ at these two rapidities have also been made.

Although we assume that fluctuations in the lab frame contain an isotropic
part $F^{2}$ and an anisotropic part $\Delta$, we find in our calculation that
the contribution from the anisotropic part is negligible compared with the
isotropic part, as manifested in Fig. 4. This means that we may safely set
$\Delta=0$ and still obtain nearly the same results as Fig. 1. The momentum
dependence is mainly caused by the broken rotational symmetry due to the
motion of the $\phi$ meson relative to the bulk matter. Such a mechanism can
be tested in future experiments.

## Acknowledgement

We thank Jinhui Chen, Aihong Tang, Gavin Wilks and Xu Sun for helpful
discussions. This work is supported in part by the National Natural Science
Foundation of China (NSFC) under Grant No. 12135011 and 12075235, by the
Strategic Priority Research Program of the Chinese Academy of Sciences (CAS)
under Grant No. XDB34030102.

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