# FedDMF: Privacy-Preserving User Attribute Prediction using Deep Matrix
Factorization

Ming Cheung  Beta Labs, The Lane Crawford Joyce Group
Hong Kong, China
<EMAIL_ADDRESS>
###### Abstract

User attribute prediction is a crucial task in various industries. However,
sharing user data across different organizations faces challenges due to
privacy concerns and legal requirements regarding personally identifiable
information. Regulations such as the General Data Protection Regulation (GDPR)
in the European Union and the Personal Information Protection Law of the
People’s Republic of China impose restrictions on data sharing. To address the
need for utilizing features from multiple clients while adhering to legal
requirements, federated learning algorithms have been proposed. These
algorithms aim to predict user attributes without directly sharing the data.
However, existing approaches typically rely on matching users across
companies, which can result in dishonest partners discovering user lists or
the inability to utilize all available features. In this paper, we propose a
novel algorithm for predicting user attributes without requiring user
matching. Our approach involves training deep matrix factorization models on
different clients and sharing only the item vectors. This allows us to predict
user attributes without sharing the user vectors themselves. The algorithm is
evaluated using the publicly available MovieLens dataset and demonstrate that
it achieves similar performance to the FedAvg algorithm, reaching 96% of a
single model’s accuracy. The proposed algorithm is particularly well-suited
for improving customer targeting and enhancing the overall customer
experience. This paper presents a valuable contribution to the field of user
attribute prediction by offering a novel algorithm that addresses some of the
most pressing privacy concerns in this area.

##  I Introduction

User attribute prediction plays a crucial role in various industries,
including retail, as it enables improved customer targeting and enhances the
overall customer experience. However, sharing user data across different
organizations poses challenges due to privacy concerns and legal requirements
regarding Personal Identifiable Information (PII). The General Data Protection
Regulation (GDPR) is a European Union regulation that safeguards personal data
of individuals within the EU and imposes requirements on organizations
worldwide that process personal data. Similarly, the Personal Information
Protection Law (PIPL) is a Chinese law focused on protecting personal
information within China’s territory, applicable to both domestic and foreign
organizations processing personal information of Chinese residents.
Consequently, sharing data among companies and organizations is becoming
increasingly difficult.
To address these challenges, federated learning algorithms have been proposed
to train models without directly sharing data. However, previous algorithms
often require matching users across companies or fail to utilize all available
features. Fig. 1 illustrates an example of federated learning for two
companies. In Fig. 1 (a), two companies have distinct sets of users and
features, with some common users and features existing in both clients.
Although Company 2 possesses labels that Client 1 requires, they cannot be
shared due to legal requirements, and vice versa. These labels represent user
attributes such as gender and transaction records indicating whether a user
has made a purchase. Horizontal federated learning focuses on extracting
common features with all users for training a model [1]. As a result, all
users have similar feature vectors but it does not utilize all features for
effective learning. On the other hand, vertical federated learning extracts
common users with all features for model training [2]. These users possess a
complete set of features across both companies. However, the two clients must
match users before training, which can pose privacy risks.

Figure 1: Conventional federated learning and the proposed.

To overcome these limitations, this paper proposes a novel algorithm that
utilizes all features without requiring user matching. Instead of training a
model with a subset of users or features, a deep matrix factorization (DMF)
model is trained on different clients, and the feature vectors are shared to
predict user attributes [3]. Such common features can be easily found from
e-commerce websites without privacy concerns. By employing this algorithm,
accurate predictions can be achieved without the need to share user vectors,
thereby preserving user privacy. To demonstrate the effectiveness of the
proposed algorithm, experiments were conducted on the publicly available
MovieLens dataset. The results indicate that the proposed algorithm performs
similarly to the FedAvg approach and achieves 96% of a single model’s
performance. This algorithm is particularly well-suited for the retail
industry, where it can enhance customer targeting and improve the overall
customer experience. This paper contributes in the following ways:

  * •

Proposes a new algorithm called FedDMF, based on DMF, that eliminates the need
for sharing and matching users.

  * •

Conducts experiments to demonstrate the effectiveness of the proposed
algorithm.

The rest of the paper is structured as follows: Section II provides a brief
overview of related work, Section III describes the proposed algorithm in
detail, Section IV presents the experimental setup and results, and Section V
concludes the paper and discusses future work.

##  II Related Works

Federated learning (FL) has emerged as a promising algorithm for learning from
decentralized data sources while preserving user privacy [4]. There are
applications in various domains, including healthcare [5, 6], finance [7], and
the retail industry [8, 9], for tasks such as recommendations, attribution,
and user segmentation. FL encompasses two commonly used settings: vertical FL
and horizontal FL. In horizontal FL [1], the data is partitioned horizontally,
with each party holding a subset of the features for all users. However, in
this setting, the common features are limited to a small set, as depicted in
Fig. 1 (a). If there are more than two clients involved, the number of
features may be insufficient to train meaningful models. On the other hand,
vertical FL [2, 10, 11] involves partitioning the data vertically, whereby
each party holds a subset of the samples. While vertical FL can utilize all
features, it requires matching users across different clients, and there may
be a limited subset of common users.
While FL aims to preserve user privacy by keeping data decentralized, using FL
still poses privacy risks. During the training process, information leakage
can occur, enabling adversaries to reconstruct sensitive data or infer private
information about individuals [12]. Adversaries with access to the model
updates or aggregated data may attempt to extract sensitive information [13,
14]. To address these concerns, algorithms have been developed [15]. However,
previous federated learning algorithms for predicting user attributes often
required matching users across different companies, leading to time-consuming
processes and potential violations of user privacy. Even using private set
intersection may introduce potential information leakage and computational
overhead [16], as the user identity can be recovered during the matching
process.
Recently, there has been growing interest in using deep matrix factorization
(DMF) models for user attribute prediction [17]. These models have proven
effective in capturing latent factors that influence user preferences and
modeling user-feature interactions. In this paper, a novel algorithm is
proposed to predict user attributes across different clients without the need
for user matching, while utilizing all features across different clients. This
approach involves training a deep MF model on different clients and sharing
only the feature vectors.

Figure 2: Proposed algorithm: FedDMF.

##  III Proposed Algorithm

This section presents a novel algorithm for model training that eliminates the
need to match users across datasets of different clients, thereby avoiding
data leakage. The proposed algorithm is introduced after providing an overview
of deep matrix factorization (DMF) and federated learning (FL).

###  III-A Deep matrix factorization

Fig. 2 illustrates the ideas behind the proposed algorithm. DMF, as shown in
Fig. 2(a), is a technique that combines the power of deep learning and matrix
factorization to learn low-dimensional representations of high-dimensional
data [3]. It employs neural networks to model complex relationships and
factorize the data into latent factors, facilitating efficient representation
learning and recommendation systems. In a recommendation scenario, the items
are considered as features. The model outputs depend on whether a user has
interacted with an item, with an interaction represented by a value of 1 and
no interaction represented by a value of 0. The objective function to be
minimized can be defined as follows:

$\hat{y}_{ui}=f(\mathbf{p}_{u},\mathbf{q}_{i},\mathbf{W})$ (1)

where $\hat{y}_{ui}$ represents the predicted output for user $u$ and item
$i$. The symbols $\mathbf{p}_{u}$ and $\mathbf{q}_{i}$ denote the learnable
embedding vectors for user $u$ and item $i$, respectively. The function
$f(\cdot)$ represents the deep learning model that takes the user embedding
and item embedding as inputs and predicts the output, with $\mathbf{W}$ is the
learnable weight of the deep learning model. The model can be trained using
the mean squared error (MSE), which measures the average squared difference
between the predicted ratings and the actual ratings in the training data:

$L_{MSE}=\frac{1}{N}\sum_{(u,i)\in\text{Training
Data}}(\hat{y}{u,i}-y{u,i})^{2}$ (2)

where $N$ is the total number of training samples, and $y_{u,i}$ is the actual
value for user $u$ and item $i$ in the training data. The model is updated
using the computed gradient $\theta$ through approaches such as stochastic
gradient descent [18].

###  III-B Federated Learning

In the federated learning (FL) setting, each client trains a separate model
using its local dataset, as data cannot be shared among different clients. The
loss is calculated separately on each client using its own data. Once the loss
is computed, the losses from all clients are combined, and the models on each
client are updated accordingly. The local models are then aggregated to obtain
a global model that captures knowledge from all clients while keeping
sensitive data within each party, addressing privacy concerns. The aggregation
of local models can be performed using various techniques, such as FedAvg.
Mathematically, the federated averaging algorithm can be expressed as follows:

$\theta_{l+1}=\frac{1}{N_{p}}\sum_{p=1}^{N_{p}}\theta_{p}$ (3)

where, $\theta_{l+1}$ represents the gradient at iteration $l+1$, $\theta_{p}$
denotes the local model of party $p$, and $N_{p}$ is the total number of
clients participating in the federated learning process. The local models can
be updated using gradients. However, as discussed earlier, both horizontal and
vertical FL suffer from limitations such as a limited number of features or
users, as well as privacy risks. In the next section, the proposed FedDMF
algorithm will be introduced to address these challenges.

Figure 3: Experimental Settings. Figure 4: Model structure, (a)
initialization, (b) model structure.

###  III-C Proposed FedDMF

This section introduces the proposed FedDMF algorithm, which is based on DMF.
Unlike conventional algorithms that require clients to match the identity of
users across different clients, the proposed algorithm allows each client to
have a separate model with non-shared parameters, minimizing the risk of data
sharing. While the user vectors may have different inputs and shapes across
clients, the feature vectors must have the same shape across all clients. It
is assumed that the features among different clients can be matched, and the
feature vectors can be shared. This assumption is valid in scenarios such as
e-commerce websites, where all websites share the same features and can be
easily matched across platforms.
Before training, the features are matched, and all clients have a copy of the
feature matching. Each client then initializes a separate model with the
agreed feature vector size. During training, each model is trained using the
respective client’s data. In addition to the mean squared error (MSE) loss in
Eq. 2, there is an additional loss term that measures the cosine distance
between vectors of the same feature across different clients. It is the
embedding similarity loss:

${L_{emb}}=\frac{\beta}{|\textbf{P}|}.\sum_{(i_{p_{1}},i_{p_{2}})\in\textbf{P}}max(1-cos(\mathbf{q}_{i_{p_{1}}},\mathbf{q}_{i_{p_{2}}}),m)$
(4)

The modified loss function is as follows:

${L_{a}}=L_{MSE}+{L_{emb}}$ (5)

where P represents the set of feature pairs that exist in both clients. The
new loss term is the sum of the cosine distances between all pairs
($i_{p_{1}}$, $i_{p_{2}}$) that belong to the set P. For each pair
($i_{p_{1}}$, $i_{p_{2}}$) in P, the loss is the maximum value between the
cosine distance and $m$. The parameter $\beta$ is the weight of the cosine
distance and prevents the pairs from being overfit, thus affecting the
performance. Finally, the sum is divided by the size of the set P using the
fraction.
This additional loss term helps to project the feature vectors among different
clients onto the same hyperspace, enabling the prediction of a vector from
another model even if the model has not seen the vector before. The proposed
algorithm allows the training of a joint model without the need to match a set
of users from different clients. This is an improvement over previous
algorithms, as the proposed FedDMF does not require matching users, utilizes
all features, and does not require sharing model parameters. In the retail
industry, the features could be user-feature transaction records. Although the
transaction records are kept private, the feature matching can be easily
obtained from platforms such as e-commerce websites. The feature vectors can
then be shared between clients during training. The next section introduces
the experimental setup to evaluate the performance of the proposed FedDMF
algorithm.

##  IV Experimental Results

This section presents the experimental results of the proposed FedDMF
algorithm. The dataset used in the experiments is first introduced, followed
by the experimental settings.

###  IV-A Dataset

The movielens dataset, specifically the ml-latest-small version, was used in
our experiments. This public dataset contains a subset of movie ratings by
users111http://files.grouplens.org/datasets/movielens/ml-latest-small.zip. The
dataset was created by randomly selecting a subset of ratings from the
original movielens dataset. It includes ratings for over 9,000 movies by more
than 600 users, where each movie serves as a feature in the experiments.

###  IV-B Settings

To simulate the Federated Learning (FL) training scenario, the dataset is
split into two parts, representing two clients. The experimental settings are
illustrated in Fig. 3. The features are divided based on the number of
features each client has, and a certain percentage of features are selected as
common features to be used in training. The data is further split into
different parts: client 1 and client 2. While there are some common features
between the two clients, there are no common users. For example, in Fig. 3,
$A_{1,1}$, $A_{1,C}$, and $A_{1,2}$ represent the data parts for user 1.
$A_{1,1}$ and $A_{1,C}$ are used by client 1 for training, while $A_{1,2}$
serves as the testing set for client 1. The difference between $A_{1,1}$ and
$A_{1,C}$ is that $A_{1,C}$ consists of the shared features with client 2,
while $A_{1,1}$ includes the features not present in the training set of
client 2. The features of $A_{1,2}$ belong to client 2. During the training
process, the prediction does not include the common features, but rather the
features from the other client. For example, client 1 will predict the user-
feature pair $A_{1,2}$, while client 2 will predict the user-feature pair
$A_{2,1}$. The proposed algorithm trains the model using a classification
approach, where a user who has rated a movie is classified as 1 for that user-
movie pair and 0 otherwise.
The DeepMF model structure, which is a neural network-based model, is depicted
in Fig. 4. The model takes as input the number of users, number of features,
embedding size, and hidden size. The inputs to the model are the user
embedding, $\mathbf{p}_{u}$, and the feature embedding, $\mathbf{q}_{i}$.
These inputs are passed through the user and feature fully connected (FC)
layers, followed by the ReLU activation function. Dropout is applied to the
output of the FC1 layer. The result is then passed through the FC2 layer,
followed by the sigmoid activation function, to obtain the final output, which
represents the predicted value, $\hat{y}_{u,i}$. The embedding size and hidden
size are set to 32 and 64, respectively. The batch size is 32 with a learning
rate of 0.001. Each training process consists of 30 epochs, with the value of
$m$ set to 0.2 and $\beta$ set to 100. Once the model is trained, it is used
to predict the features of other clients. If the predicted value is greater
than 1, it is classified as 1; otherwise, it is classified as 0. The
experiments are repeated three times with different training and testing sets,
and the mean result is recorded.

Figure 5: Experimental Results with baselines: (a) training set, (b) testing
set. Figure 6: Experimental Results with different percentages of features
that client 1 has: (a) training set, (b) testing set.

###  IV-C Results

This section presents the experimental results of the proposed algorithm and
compares its effectiveness with two baselines. The first baseline is DMF,
which represents the conventional algorithm when all data is accessible and
serves as the upper bound. The second baseline is FedAvg, where the gradient
is averaged in each epoch to train the same model for both clients.
Additionally, a random approach (rand) is included as another baseline. The
results are illustrated in Fig. 5. In the experiments, 50% of the features
were used as common features, and each client had 50% of the remaining
features and 50% of the users. Fig. 5(a) and (b) show the training and testing
$F1$ scores, respectively. The results indicate that DMF achieves the best
performance, followed by FedAvg as the second best. The proposed algorithm
demonstrates similar performance to the FedAvg algorithm, achieving 96% of the
performance achieved by a single model. These results confirm that the
proposed algorithm is capable of training a high-quality model. In the
subsequent sections, The performance of the proposed algorithm with different
percentages of common features will be investigated. This analysis aims to
provide insights into the algorithm’s performance under varying conditions.

###  IV-D Results of proportion of users

This experiment investigates how the proportion of users affects the training
results. Similar to the previous experiments, the results are measured in
terms of F1 score, with 50% of common features and 50% of users from client 1.
The results are shown in Fig. 6. In Fig. 6(a) and (b), the predictions of the
features in the training and testing sets are presented, respectively. The
x-axis represents the percentage of features that client 1 has, ranging from
0.1 to 0.5. The blue curve represents the results for the first client, while
the green curve represents the results for the second client. From Fig. 6(a),
it is observed that when the proportion of a client is small, the F1 score is
high. This can be attributed to the fact that with fewer user-feature pairs to
predict, the model can allocate more resources to accurately predict those
pairs. Moving to Fig. 6(b), it can be seen that the algorithm achieves the
best results when both clients have an equal number of users. This suggests
that having an equal distribution of users provides a balanced and
representative training set, leading to improved prediction performance.

Figure 7: Experimental Results with different percentage of common features:
(a) training set, (b) testing set. Figure 8: Experimental Results with
different percentage of client 1 users: (a) training set, (b) testing set.

###  IV-E Common Features

The percentage of common features plays a significant role in model training.
An experiment was conducted to investigate how the percentage of common
features affects the model’s performance, and the results are presented in
Fig. 7. It is important to note that in the prediction of both the training
and testing sets, the common features are not considered. The focus is on
evaluating the impact of the percentage of common features on the performance
of predicting features from each client and features from the other client.
The results show that the $F1$ score initially increases with the percentage
of common features for both types of features. This can be attributed to the
fact that having more common features implies a higher likelihood of better
training. As more features are available for training, the model can learn
more effectively and improve its prediction performance. However, beyond a
certain threshold, the $F1$ score starts to drop. One possible reason for this
decline is that, as the prediction does not consider common features, there
are only a few features available in the ground truth for evaluation. This
scarcity of features can lead to inaccurate predictions and lower $F1$ scores.
Another notable result is that the Rand index increases with the percentage of
common features. This outcome can be explained by the fact that the prediction
does not consider common features. As a result, there are fewer features to be
predicted, as well as a reduced number of user-feature pairs in the ground
truth. Consequently, the prediction has a higher recall and the same
precision, resulting in a higher F1 score.

###  IV-F Results of % of client 1 features

This section investigates how the proportion of features in client 1 affects
the model’s performance. It is expected that when a client contains more
users, the training results for that client should be better. To validate this
hypothesis, an experiment was conducted, and the results are presented in Fig.
8. In Fig. 8(a), the training $F1$ score reaches a high value at and after
0.5, indicating that when a client has 50% of the users, it achieves a higher
$F1$ score. However, when a client has less than 50% of the users, its $F1$
score is lower. The performance for the training features remains consistent
across different proportions. The testing $F1$ score is shown in Fig. 8(b). It
is observed that it reaches its maximum value at a proportion of 0.5. This
suggests that having an equal proportion of users in both clients leads to the
best performance. One possible reason for the increasing $F1$ score in testing
features is that the algorithms require enough features to train a general
model. Therefore, a higher proportion of features helps improve the
performance. However, if the proportion becomes too large, the $F1$ score
drops, indicating that having too many features to predict can hinder the
model’s performance. The results demonstrate that if both clients have a
similar number of users, the algorithm performs the best. This finding
emphasizes the importance of achieving a balanced distribution of users across
clients to optimize the model performance in federated learning scenarios.

###  IV-G Discussion and Future Directions

The primary objective of this research was to address privacy concerns
associated with sharing user data across organizations. The proposed algorithm
successfully eliminates the need for user matching while still utilizing all
features, thereby significantly enhancing user privacy. However, there are
several aspects that should be further investigated and improved to advance
the research in this area. Firstly, it is important to conduct additional
experiments to assess whether the sharing of feature vectors still indirectly
discloses certain information about users. While the proposed algorithm
eliminates the need for explicit user matching, future research should be
conducted on developing advanced privacy-preserving techniques to minimize the
disclosure of user information during the sharing of feature vectors. Another
direction is to evaluate and generalize the proposed algorithm beyond the
MovieLens dataset. It is important to assess the algorithm’s performance on
real-world datasets, for evaluating the algorithm’s effectiveness and
robustness across various domains and datasets. Furthermore, expanding the
experiments to involve three or more clients would be beneficial. This would
provide insights into the algorithm’s performance as the number of
participating clients increases, as well as its scalability and effectiveness
in real-world scenarios.

##  V Conclusion

This paper presents FedDMF, an algorithm that tackles user attribute
prediction while considering privacy concerns and legal requirements. FedDMF
eliminates the need for user matching, safeguarding user privacy and
preventing unauthorized access to sensitive information. By training deep
matrix factorization models on individual clients and sharing only feature
vectors, accurate user attribute prediction is achieved without compromising
privacy. Experimental results on the MovieLens dataset show that FedDMF
performs comparably to FedAvg, achieving 96% of the accuracy of a single
model. The proposed algorithm enhances customer targeting and improves the
overall customer experience. FedDMF combines federated learning and deep
matrix factorization, providing a promising solution for privacy-preserving
user attribute prediction. It enables organizations to collaborate and
leverage collective data while maintaining privacy and compliance. Future
research should evaluate the algorithm on diverse datasets and explore
optimization techniques for scalability and efficiency. FedDMF offers a
practical and effective approach for leveraging user data while respecting
privacy concerns.

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# Nuclear transverse momentum imbalance in the color dipole approach at the
LHC regime

F. G. Ben<EMAIL_ADDRESS>M. V. T. Machado  Universidade Federal
do Rio Grande do Sul, Institute of Physics, Porto Alegre, RS, Brazil

###### Abstract

Transverse momentum broadening of a parton propagating through a large nucleus
is evaluated in the color dipole approach using different models for the
dipole cross section or unintegrated gluon distribution, which lead to
different values of the coefficient $C_{\mathcal{F}}(0,s)$. Numerical
calculations are compared to data extracted from LHCb and ALICE experiments
for nuclear broadening of $J/\psi$. We find that different models which
describe the small-$x$ data predict values of $\Delta{p_{T}^{2}}$ that agree
reasonably well with experiment, specially for forward rapidity. The
centrality dependence was also analysed and the models are consistent with the
ALICE measurements.

Dipole cross section; nuclear broadening

##  I Introduction

High energy partons propagating through nuclear matter experience an increase
in transverse momentum due to multiple interactions with the medium. This
broadening in transverse momentum is higher in proton-nucleus ($pA$)
collisions than in proton-proton ($pp$) collisions, and the increase of the
mean transverse momentum squared of the produced particle in $pA$ collisions,
compared to $pp$ collisions, is defined as nuclear broadening:

$\Delta{p_{T}^{2}}=\langle p_{T}^{2}\rangle_{pA}-\langle
p_{T}^{2}\rangle_{pp}.$ (1)

Different theoretical approaches have been used to describe this broadening.
Among them, one has QCD color dipole approach, Glauber multiple scattering
Gyulassy  _et al._ (2002), the Color Glass Condensate (CGC) framework Dumitru
and Jalilian-Marian (2002); Rezaeian (2013) and the high-twist expansion of
matrix elementsXing  _et al._ (2012); Kang  _et al._ (2016, 2015) as well. In
particular, in the color dipole approach Dolejší  _et al._ (1993); Johnson
_et al._ (2001), as well as in the BDMPS approach Baier  _et al._ (1997), the
parton acquires transverse momentum through a random walk through nuclear
matter, undergoing multiple rescatterings. In fact, as shown in Ref. Raufeisen
(2003), both descriptions are equivalent and may be also related to the higher
twist factorization formalism, in which broadening arises from the exchange of
a single soft gluon. Although all of these descriptions rely on
nonperturbative inputs, the color dipole approach has the advantage of relying
on a well developed phenomenology from deep-inelastic scattering Johnson  _et
al._ (2001). The purpose of this Letter is to evaluate $\Delta{p_{T}^{2}}$ in
the color dipole approach using different phenomenological models as an input
and compare its predictions. We also compare the numerical results to data
extracted from the LHC for nuclear broadening of $J/\psi$.

Let us begin by briefly summarizing the main points from the dipole approach.
Transverse momentum broadening of a high energy parton propagating through
large nuclear matter is given by

$\Delta{p_{T}^{2}}^{\mathcal{R}}=\langle T_{A}\rangle C_{\mathcal{R}}(0,s),$
(2)

where $\langle T_{A}\rangle=\int d^{2}b\,T_{A}^{2}(b)/A$ is the nuclear
thickness function averaged over impact parameter $b$. Assuming uniform
nuclear density, $\langle T_{A}\rangle=2\rho_{A}L$, where $\rho_{A}=0.16$ fm-3
is the nuclear density, $L=3R_{A}/4$, and $R_{A}$ is the nuclear radius. We
follow the notation of Ref. Raufeisen (2003), in which the index $\mathcal{R}$
represents the projectile parton, i.e., $\mathcal{R}=\mathcal{F}$ for a quark
and $\mathcal{R}=\mathcal{A}$ for a gluon. The coefficient $C_{\mathcal{R}}$
contains the nonperturbative physics and it arises from the expression for the
total cross section $\sigma_{q\bar{q}}(r_{T},s)$ for the interaction between a
nucleon and a colorless dipole $q\bar{q}$ having transverse separation $r_{T}$
and c.m. energy squared $s$:

$\displaystyle C_{\mathcal{F}}(0,s)$ $\displaystyle=$
$\displaystyle\frac{d}{dr_{T}^{2}}\sigma_{q\bar{q}}(r_{T},s)\Bigr{|}_{r_{T}\rightarrow
0}$ (3) $\displaystyle C_{\mathcal{A}}(0,s)$ $\displaystyle=$ $\displaystyle
9C_{\mathcal{F}}(0,s)/4.$ (4)

As shown in Ref. Johnson  _et al._ (2001), the effect of broadening increases
with energy, and it will depend on the phenomenological model for
$\sigma_{q\bar{q}}(r_{T},s)$.

One can now look for different models for the dipole cross section and compare
its predictions. We start with the parametrization of Kopeliovich, Schäfer and
Tarasov (KST) Kopeliovich  _et al._ (2000), the one used in Ref. Raufeisen
(2003), which has a saturated form of the dipole cross section adjusted to
low-$Q^{2}$ deep inelastic scattering (DIS) data:

$\sigma_{q\bar{q}}(r_{T},s)=\sigma_{0}(s)\Bigr{[}1-\exp\Bigr(-\frac{r_{T}^{2}}{R_{0}^{2}(s)}\Bigr{missing})\Bigr{]},$
(5)

in which an explicit energy dependence is introduced through

$\sigma_{0}(s)=\sigma^{\pi
p}_{\mathrm{tot}}(s)\Bigr{(}1+\frac{3R_{0}^{2}(s)}{8\langle
r_{ch}^{2}\rangle_{\pi}}\Bigr{)},$ (6)

where $\sigma^{\pi p}_{\mathrm{tot}}(s)=23.6\times(s/s_{0})^{0.08}$, $\langle
r_{ch}^{2}\rangle_{\pi}=0.44\pm 0.01$ fm2 is the mean-squared pion charge
radius, and $R_{0}(s)=0.88\,\mathrm{fm}\times(s/s_{0})^{-\lambda/2}$ with
$\lambda=0.28$ and $s_{0}=1000$ GeV2 is the energy-dependent radius. Using Eq.
(3), this leads to

$C_{\mathcal{F}}^{\mathrm{KST}}(0,s)=\frac{\sigma_{0}(s)}{R_{0}^{2}(s)}.$ (7)

In a different model, by Schildknecht, Surrow and Tentyukov (SST) Schildknecht
_et al._ (2001); Schildknecht (2021), we have

$\sigma_{q\bar{q}}(r_{T},s)=\sigma^{(\infty)}\frac{1}{24}r_{T}^{2}\Lambda^{2}(s),$
(8)

where $\Lambda^{2}(s)=C_{1}(s+W_{0}^{2})^{C_{2}}$,
$\sigma^{(\infty)}=48\,\mathrm{GeV}^{-2}=18.7\,\mathrm{mb}$, $C_{1}=0.34\pm
0.05$ and $C_{2}=0.27\pm 0.01$. The SST model provides simple analytic
expressions for proton structure function $F_{2}$, $F_{L}$, as well as for the
photoabsorption cross section. It includes color transparency and saturation
properties that are dependent on the relative magnitude of $Q^{2}$ and $W$ in
the small-$x$ domain. The results are compared to a global analysis of all
experimental data available and agreement is quite good. Applying Eq. (8) to
Eq. (3), we find that

$C_{\mathcal{F}}^{\mathrm{SST}}(0,s)=\frac{\sigma^{(\infty)}}{24}\Lambda^{2}(s).$
(9)

In the model by Donnachie and Dosch (DD) Donnachie and Dosch (2001, 2002), the
dipole cross section is given by

$\sigma_{d}(r_{T})=0.098(\langle g^{2}FF\rangle
a^{4})^{2}r_{T}\Bigr{[}1-\exp(-\frac{r_{T}}{3.1a})\Bigr{]},$ (10)

with
$\sigma_{q\bar{q}}(r_{T},s)=\sigma_{d}\times(\dfrac{s}{s_{o}})^{\epsilon}$.
This model is based on the expectation value of two light-like Wilson loops
and the dipole cross section is obtained from dipole-dipole interaction. The
quantity $\langle g^{2}FF\rangle$ describes the gluon condensate and the
parameter $a$ is the correlation length of the two-gluon correlator. At small
$r_{T}$, the hard Pomeron contribution dominates and then
$\epsilon=\epsilon_{hard}=0.42$. The dimensionless constant $\langle
g^{2}FF\rangle a^{4}$ has the numerical value of $23.8$, $s_{o}=20^{2}$ GeV2
and $a=0.346$ fm. These parameters were taken from lattice results and
calibrated to high energy proton-proton scattering. If $a$ and $r_{T}$ are
measured in fm, the result is in milibarn. At small $r_{T}$, it leads to

$C_{\mathcal{F}}^{\mathrm{DD}}(0,s)=\frac{0.0098(\langle g^{2}FF\rangle
a^{4})^{2}[\mathrm{fm}]}{3.1a}\Bigr{(}\frac{s}{s_{o}}\Bigr{)}^{0.42}.$ (11)

The DD model is the basis for the posterior FKS model Forshaw  _et al._ (1999,
2000); McDermott  _et al._ (2002), a two-component model in which
$\sigma(s,r)=\sigma_{\mathrm{soft}}(s,r)+\sigma_{\mathrm{hard}}(s,r)$. Physics
at small $r$ is dominated by the hard term

$\sigma_{\mathrm{hard}}=(a_{2}^{H}r^{2}+a_{6}^{H}r^{6})\exp(-\nu_{H}r)(r^{2}s)^{\lambda_{H}}.$
(12)

Due to more complex shape on $r$ for both hard and soft Pomeron contribution,
FKS is not directly suitable for our analysis.

As presented in Ref. Johnson  _et al._ (2001), the coefficient
$C_{\mathcal{F}}(0,s)$ may also be obtained directly from the unintegrated
gluon density

$C_{\mathcal{F}}(0,s)=\frac{\pi}{3}\int
d^{2}k\frac{\alpha_{s}(k^{2})k^{2}}{k^{4}}\mathcal{F}(x,k^{2}),$ (13)

where $\mathcal{F}(x,k^{2})=\partial G(x,k^{2})/\partial(\ln k^{2})$. We shall
consider the recent model by Moriggi, Peccini and Machado (MPM) Moriggi  _et
al._ (2020), in which $\mathcal{F}(x,k^{2})$ is modeled as

$\mathcal{F}^{\mathrm{MPM}}(x,k^{2})=\frac{3\sigma_{o}}{4\pi^{2}\alpha_{s}}\frac{1+\delta
n}{Q_{s}^{2}}\frac{k^{4}}{(1+\tau)^{2+\delta n}},$ (14)

with $Q_{s}^{2}=1\,[\mathrm{GeV}^{2}]\times(x/xo)^{0.33}$ and $\delta
n=a\tau^{b}$, where $\tau=k^{2}/Q_{s}^{2}$. The parameters $\sigma_{0}$,
$x_{o}$, $a$ and $b$ were fitted against DIS data for $x<0.01$ (see Ref.
Moriggi  _et al._ (2020) for details). This model is based on the geometric
scaling scaling property and it is constructed in order to describe DIS data
and $p_{T}$ spectra of produced hadrons at high energy proton-proton
simultaneously. It has also been extended to proton-nucleus and nucleus-
nucleus collisions in Ref. Moriggi  _et al._ (2021). In order to calculate the
energy dependence of $C_{\mathcal{F}}(0,s)$ in this model we need to know the
value of $x$. We shall follow the approach of Ref. Johnson  _et al._ (2001)
and use the minimal value of $x$ permitted by kinematics, $x=4k^{2}/s$, in the
integral in Eq. (13). The corresponding value of
$C_{\mathcal{F}}^{\mathrm{MPM}}(0,s)$ should be an upper bound in this model.

Figure 1: Coefficient $C_{\mathcal{F}}(0,s)$ using KST, SST, MPM and DD
models, as a function of $\sqrt{s}$.

##  II Comparing different models

Experiment |  $y$ range |  $\Delta{p_{T}^{2}}$ Exp. |  $\Delta{p_{T}^{2}}$ KST |  $\Delta{p_{T}^{2}}$ SST |  $\Delta{p_{T}^{2}}$ MPM |  $\Delta{p_{T}^{2}}$ DD
---|---|---|---|---|---|---
LHCb | $-4.5<y<-2.5$ | $0.79\pm 0.12$ | $0.84$ | $0.81$ | $1.01$ | $1.23$
| $2<y<4$ | $2.05\pm 0.12$ | $3.90$ | $2.81$ | $3.54$ | $13.23$
ALICE | $-4.46<y<-2.96$ | $0.68\pm 0.33$ | $0.70$ | $0.72$ | $0.82$ | $0.86$
| $2.03<y<3.53$ | $1.91\pm 0.42$ | $2.91$ | $2.22$ | $2.85$ | $9.14$
PHENIX | $-2.2<y<-1.2$ | $0.43\pm 0.08$ | $0.57$ | $0.67$ | $0.59$ | $0.48$
| $-0.35<y<0.35$ | $0.71\pm 0.2$ | $0.74$ | $0.75$ | $0.88$ | $0.97$
| $1.2<y<2.2$ | $0.43\pm 0.08$ | $0.95$ | $0.86$ | $1.16$ | $1.71$
Table 1: Values of $\Delta{p_{T}^{2}}$ from LHCb, ALICE and PHENIX, from Ref.
Arleo and Naïm (2020), for nuclear broadening of $J/\psi$ at
$\sqrt{s_{h}}=8.16$ TeV, $\sqrt{s_{h}}=5.02$ TeV amd $\sqrt{s_{h}}=200$ GeV
respectively (labeled ‘$\Delta{p_{T}^{2}}$ Exp.’). The remaining columns
present the prediction for $\Delta{p_{T}^{2}}$ using KST, SST, MPM and DD
models. All $\Delta{p_{T}^{2}}$ values are in GeV2.

We now proceed to compare the predictions of $C_{\mathcal{F}}(0,s)$ using the
models described in the previous section. Fig. 1 shows the energy dependence
of $C_{\mathcal{F}}(0,s)$ as a function of $\sqrt{s}$. Notice that the
relevant energy scale is not the hadronic c.m. energy $s_{h}$, but rather the
energy of parton-target system, i.e., $s=x_{1}s_{h}$ (see, for instance, Ref.
Raufeisen (2004)).

Our main goal is now to evaluate $\Delta{p_{T}^{2}}$ using these models and
compare the results to actual data. Numerical values of $\Delta{p_{T}^{2}}$
for $J/\psi$ production in pp and pPb collisions were extracted from PHENIX,
ALICE and LHCb in Ref. Arleo and Naïm (2020) and are presented in the third
column of Table 1. As noted in Ref. Raufeisen (2003), broadening in $J/\psi$
production is equal to broadening for gluons (assuming final states effects
are negligible). We therefore evaluate $\Delta{p_{T}^{2}}$ using Eq. (2), with
$\mathcal{R}=\mathcal{A}$ and $A=208$. The value of $s=x_{1}s_{h}$ is obtained
using $x_{1}=e^{y}M/\sqrt{s_{h}}$, with the average value of $y$ in each bin.
We also take into account suppression due to gluon shadowing, as presented in
Ref. Kopeliovich  _et al._ (2000); Raufeisen (2004). While negligible for
$x_{2}\sim 10^{-2}$, for $x_{2}\sim 10^{-5}$ it leads to a reduction of
$C_{\mathcal{F}}(0,s)$ by about 1/3. The predictions for $\Delta{p_{T}^{2}}$
using KST, SST, MPM and DD models are presented in Table 1.

In the LHC regime, KST, SST and MPM models lead to reasonable and similar
predictions for $\Delta{p_{T}^{2}}$, while the DD model largely overestimates
broadening in the positive $y$ range. We trace this to the large exponential
growth $(s/s_{o})^{0.42}$ in Eq. (11), meaning that $C_{\mathcal{F}}^{DD}$
grows faster with $\sqrt{s}=\sqrt{x_{1}s_{h}}$ than in the other models, as
shown in Fig. 1. Although KST, SST and MPM models perform significantly well
for backward rapidity (leading to $x_{2}\sim 10^{-2}$), we find that SST
performs better in the forward rapidity intervals (corresponding to $x_{2}\sim
10^{-5}$).

Data for $\Delta{p_{T}^{2}}$ in $J/\psi$ production in pPb collisions at
$\sqrt{s_{h}}=5.02$ TeV is also available from ALICE as a function of
centrality in Ref. Adam  _et al._ (2015). In each centrality class, we used
Eq. (2) with $\langle T_{A}\rangle$ equal to the average value of the nuclear
overlap function $\langle T_{pPb}^{\mathrm{mult}}\rangle$ (see Table 1 in Ref.
Adam  _et al._ (2015)). Table 2 presents $\Delta{p_{T}^{2}}$ using data from
ALICE, in the range $2.03<y<3.53$, and the predictions using KST, SST and MPM
models, in each centrality class. We restrict the analysis just for forward
rapidities, where all models are within their range of validity. We checked
that the description for backward rapidities are still reasonable. As expected
the SST model remains with good adherence to experimental data within the
errors. The other models are not excluded given the experimental
uncertainties.

##  III Discussions and Conclusions

ZN class |  $\Delta{p_{T}^{2}}$ ALICE |  $\Delta{p_{T}^{2}}$ KST |  $\Delta{p_{T}^{2}}$ SST |  $\Delta{p_{T}^{2}}$ MPM
---|---|---|---|---
2-10% | $2.15\pm 0.51$ | $2.82$ | $2.14$ | $2.75$
10-20% | $2.38\pm 0.51$ | $2.65$ | $2.01$ | $2.59$
20-40% | $2.08\pm 0.49$ | $2.30$ | $1.74$ | $2.25$
40-60% | $1.70\pm 0.49$ | $1.71$ | $1.30$ | $1.67$
60-80% | $0.94\pm 0.49$ | $1.03$ | $0.78$ | $1.01$
80-100% | $0.72\pm 0.52$ | $0.51$ | $0.39$ | $0.50$
Table 2: Values of $\Delta{p_{T}^{2}}$ in $J/\psi$ production in pPb
collisions at $\sqrt{s_{h}}=5.02$ TeV from ALICE, with $2.03<y<3.53$, from
Ref. Adam  _et al._ (2015), as a function of centrality. We also present the
prediction for $\Delta{p_{T}^{2}}$ using KST, SST and MPM models. All
$\Delta{p_{T}^{2}}$ values are in GeV2.

Before conclusions, we would like to compare the present calculation to other
approaches where multiple scattering of the projectile partons is the
underlying dynamics. One of them is the Color Glass Condensate formalism.
Specifically, the production of quarkonium at forward rapidities for proton
and nuclear targets was addressed in Ref. Ducloué  _et al._ (2015). By using a
proper treatment of the nuclear geometry, one obtains a nuclear suppression
that is in very good agreement to the experimental data. An extension of these
investigations has been done in Ref. Ducloué  _et al._ (2016), where the mean
transverse momentum, $\langle p_{T}\rangle$, was analyzed in terms of meson
rapidity and collision centrality. It was found an intense variation of
$\Delta p_{T}^{2}$ as a function of $N_{\mathrm{coll}}$. The main ingredient
of these approaches is the nuclear saturation scale, in which
$p_{T}$-broadening in the saturation regime is roughly expressed as $\langle
p_{T}^{2}\rangle\approx Q_{s,A}^{2}/4$ Dumitru and Jalilian-Marian (2002);
Schäfer and Zhou (2013). Here, the nuclear saturation scale squared is
$Q_{s,A}^{2}\sim A^{1/3}s^{\lambda}$ with $\lambda\simeq 0.3$.

The LHC data for broadening can also be described through approaches that take
into account initial/final-state parton’s multiple scattering in the nuclear
medium Kang and Qiu (2008, 2013) or those that include coherent energy loss
effects from the incoming and outgoing partons Arleo  _et al._ (2013, 2019).
Both predictions are consistent with the ALICE data, with the multiple
scattering approach doing a better job for forward rapidities compared to the
energy loss model. The collider data for prompt quarkonium production have
allowed accurate extraction of the $x$-dependence of the transport
coefficient, $\hat{q}(x)$ Naïm (2020); Arleo and Naïm (2020). In this case,
the nuclear broadening is related to $\hat{q}_{0}=\hat{q}(x=x_{0})$ (with
$x_{0}=10^{-2}$) in the following form:

$\displaystyle\Delta
p_{T}^{2}=\frac{\hat{q}(x)}{N_{c}}C_{\mathrm{oc,sg}}\Delta
L,\quad\hat{q}(x)=\hat{q}_{0}\left(\frac{x_{0}}{x}\right)^{\lambda_{g}},$ (15)

where one assumes a power-like behavior for the gluon distribution,
$xG(x,\mu^{2})\propto x^{-\lambda_{g}}$ with $\lambda_{g}\simeq 0.25$. The
color factors for quarkonium are $C_{\mathrm{oc,sg}}=N_{c},N_{c}/2$ for octet
and singlet color states, respectively. The nuclear medium length $L_{A}$
enters into the quantity $\Delta L=L_{A}-L_{p}^{\prime}$. The value found for
the transport coefficient is $\hat{q}_{0}\simeq 0.050$ GeV2/fm Naïm (2020);
Arleo and Naïm (2020). This value is somewhat higher than that obtained by a
global analysis of the transport coefficient for cold nuclear matter within
the generalized QCD factorization formalism Ru  _et al._ (2021), with the best
fit giving $\hat{q}_{0}\simeq 0.02$ GeV2/fm. However, a different functional
form for the $\hat{q}(x,\mu^{2})$ is employed in Ref. Ru  _et al._ (2021).

Concerning the present work, the results obtained by using the KST model are
consistent with those predicted in Ref. Kopeliovich  _et al._ (2017), namely
$\Delta p_{T}^{2}(-4.46\leq y\leq-2.96)=0.35$, $\Delta p_{T}^{2}(-1.37\leq
y\leq-0.43)=0.73$ and $\Delta p_{T}^{2}(2.03\leq y\leq 3.53)=2.27$ GeV2 at
$\sqrt{s}=5.02$ TeV. We find that, although the models KST, SST and MPM
predict values of $\Delta{p_{T}^{2}}$ that agree reasonably well with
experiment, the SST model leads to better results, specially for forward
rapidity. The DD model predicts stronger broadening than the others. The
reason is the high value for the hard Pomeron intercept,
$\alpha_{I\\!\\!P}(0)=1.42$. The dependence on centrality measured by ALICE is
correctly described by the different models. Although some of those models
include gluon saturation corrections, the experimental results are not able to
clearly disentangle the different QCD dynamics embedded in the considered
models. In fact this is a current challenge, and intense work is ongoing in
order to find different ways of identifying the saturated gluon state Morreale
and Salazar (2021).

## Acknowledgements

This work was financed by the Brazilian funding agency CNPq and in part by
CAPES - Finance Code 001.

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